Derivative function：
Enter an original function (that is, the function to be derived), then set the variable to be derived and the order of the derivative, and click the "Next" button to obtain the derivative function of the corresponding order of the function.
Note that the input function supports mathematical functions and other constants.
Enter an original function (that is, the function to be derived), then set the variable to be derived and the order of the derivative, and click the "Next" button to obtain the derivative function of the corresponding order of the function.
Note that the input function supports mathematical functions and other constants.
Current location：Derivative function > Derivative function calculation history > Answer
There are 1 questions in this calculation: for each question, the 3 derivative of s is calculated.
Note that variables are case sensitive.
\[ \begin{equation}\begin{split}[1/1]Find\ the\ third\ derivative\ of\ function\ \frac{({e}^{(st)}(1 + Cs))}{((1 + As)(1 + Bs))}\ with\ respect\ to\ s：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &Primitive\ function\ = \frac{Cs{e}^{(ts)}}{(Bs + As + ABs^{2} + 1)} + \frac{{e}^{(ts)}}{(Bs + As + ABs^{2} + 1)}\\&\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( \frac{Cs{e}^{(ts)}}{(Bs + As + ABs^{2} + 1)} + \frac{{e}^{(ts)}}{(Bs + As + ABs^{2} + 1)}\right)}{ds}\\=&(\frac{-(B + A + AB*2s + 0)}{(Bs + As + ABs^{2} + 1)^{2}})Cs{e}^{(ts)} + \frac{C{e}^{(ts)}}{(Bs + As + ABs^{2} + 1)} + \frac{Cs({e}^{(ts)}((t)ln(e) + \frac{(ts)(0)}{(e)}))}{(Bs + As + ABs^{2} + 1)} + (\frac{-(B + A + AB*2s + 0)}{(Bs + As + ABs^{2} + 1)^{2}}){e}^{(ts)} + \frac{({e}^{(ts)}((t)ln(e) + \frac{(ts)(0)}{(e)}))}{(Bs + As + ABs^{2} + 1)}\\=& - \frac{CBs{e}^{(ts)}}{(Bs + As + ABs^{2} + 1)^{2}} - \frac{CAs{e}^{(ts)}}{(Bs + As + ABs^{2} + 1)^{2}} - \frac{2CABs^{2}{e}^{(ts)}}{(Bs + As + ABs^{2} + 1)^{2}} + \frac{C{e}^{(ts)}}{(Bs + As + ABs^{2} + 1)} + \frac{tCs{e}^{(ts)}}{(Bs + As + ABs^{2} + 1)} - \frac{B{e}^{(ts)}}{(Bs + As + ABs^{2} + 1)^{2}} - \frac{A{e}^{(ts)}}{(Bs + As + ABs^{2} + 1)^{2}} - \frac{2ABs{e}^{(ts)}}{(Bs + As + ABs^{2} + 1)^{2}} + \frac{t{e}^{(ts)}}{(Bs + As + ABs^{2} + 1)}\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( - \frac{CBs{e}^{(ts)}}{(Bs + As + ABs^{2} + 1)^{2}} - \frac{CAs{e}^{(ts)}}{(Bs + As + ABs^{2} + 1)^{2}} - \frac{2CABs^{2}{e}^{(ts)}}{(Bs + As + ABs^{2} + 1)^{2}} + \frac{C{e}^{(ts)}}{(Bs + As + ABs^{2} + 1)} + \frac{tCs{e}^{(ts)}}{(Bs + As + ABs^{2} + 1)} - \frac{B{e}^{(ts)}}{(Bs + As + ABs^{2} + 1)^{2}} - \frac{A{e}^{(ts)}}{(Bs + As + ABs^{2} + 1)^{2}} - \frac{2ABs{e}^{(ts)}}{(Bs + As + ABs^{2} + 1)^{2}} + \frac{t{e}^{(ts)}}{(Bs + As + ABs^{2} + 1)}\right)}{ds}\\=& - (\frac{-2(B + A + AB*2s + 0)}{(Bs + As + ABs^{2} + 1)^{3}})CBs{e}^{(ts)} - \frac{CB{e}^{(ts)}}{(Bs + As + ABs^{2} + 1)^{2}} - \frac{CBs({e}^{(ts)}((t)ln(e) + \frac{(ts)(0)}{(e)}))}{(Bs + As + ABs^{2} + 1)^{2}} - (\frac{-2(B + A + AB*2s + 0)}{(Bs + As + ABs^{2} + 1)^{3}})CAs{e}^{(ts)} - \frac{CA{e}^{(ts)}}{(Bs + As + ABs^{2} + 1)^{2}} - \frac{CAs({e}^{(ts)}((t)ln(e) + \frac{(ts)(0)}{(e)}))}{(Bs + As + ABs^{2} + 1)^{2}} - 2(\frac{-2(B + A + AB*2s + 0)}{(Bs + As + ABs^{2} + 1)^{3}})CABs^{2}{e}^{(ts)} - \frac{2CAB*2s{e}^{(ts)}}{(Bs + As + ABs^{2} + 1)^{2}} - \frac{2CABs^{2}({e}^{(ts)}((t)ln(e) + \frac{(ts)(0)}{(e)}))}{(Bs + As + ABs^{2} + 1)^{2}} + (\frac{-(B + A + AB*2s + 0)}{(Bs + As + ABs^{2} + 1)^{2}})C{e}^{(ts)} + \frac{C({e}^{(ts)}((t)ln(e) + \frac{(ts)(0)}{(e)}))}{(Bs + As + ABs^{2} + 1)} + (\frac{-(B + A + AB*2s + 0)}{(Bs + As + ABs^{2} + 1)^{2}})tCs{e}^{(ts)} + \frac{tC{e}^{(ts)}}{(Bs + As + ABs^{2} + 1)} + \frac{tCs({e}^{(ts)}((t)ln(e) + \frac{(ts)(0)}{(e)}))}{(Bs + As + ABs^{2} + 1)} - (\frac{-2(B + A + AB*2s + 0)}{(Bs + As + ABs^{2} + 1)^{3}})B{e}^{(ts)} - \frac{B({e}^{(ts)}((t)ln(e) + \frac{(ts)(0)}{(e)}))}{(Bs + As + ABs^{2} + 1)^{2}} - (\frac{-2(B + A + AB*2s + 0)}{(Bs + As + ABs^{2} + 1)^{3}})A{e}^{(ts)} - \frac{A({e}^{(ts)}((t)ln(e) + \frac{(ts)(0)}{(e)}))}{(Bs + As + ABs^{2} + 1)^{2}} - 2(\frac{-2(B + A + AB*2s + 0)}{(Bs + As + ABs^{2} + 1)^{3}})ABs{e}^{(ts)} - \frac{2AB{e}^{(ts)}}{(Bs + As + ABs^{2} + 1)^{2}} - \frac{2ABs({e}^{(ts)}((t)ln(e) + \frac{(ts)(0)}{(e)}))}{(Bs + As + ABs^{2} + 1)^{2}} + (\frac{-(B + A + AB*2s + 0)}{(Bs + As + ABs^{2} + 1)^{2}})t{e}^{(ts)} + \frac{t({e}^{(ts)}((t)ln(e) + \frac{(ts)(0)}{(e)}))}{(Bs + As + ABs^{2} + 1)}\\=&\frac{2CB^{2}s{e}^{(ts)}}{(Bs + As + ABs^{2} + 1)^{3}} + \frac{4CABs{e}^{(ts)}}{(Bs + As + ABs^{2} + 1)^{3}} + \frac{8CAB^{2}s^{2}{e}^{(ts)}}{(Bs + As + ABs^{2} + 1)^{3}} - \frac{2CB{e}^{(ts)}}{(Bs + As + ABs^{2} + 1)^{2}} - \frac{2tCBs{e}^{(ts)}}{(Bs + As + ABs^{2} + 1)^{2}} + \frac{2CA^{2}s{e}^{(ts)}}{(Bs + As + ABs^{2} + 1)^{3}} + \frac{8CA^{2}Bs^{2}{e}^{(ts)}}{(Bs + As + ABs^{2} + 1)^{3}} - \frac{2CA{e}^{(ts)}}{(Bs + As + ABs^{2} + 1)^{2}} - \frac{2tCAs{e}^{(ts)}}{(Bs + As + ABs^{2} + 1)^{2}} + \frac{8CA^{2}B^{2}s^{3}{e}^{(ts)}}{(Bs + As + ABs^{2} + 1)^{3}} - \frac{6CABs{e}^{(ts)}}{(Bs + As + ABs^{2} + 1)^{2}} - \frac{4tCABs^{2}{e}^{(ts)}}{(Bs + As + ABs^{2} + 1)^{2}} + \frac{2tC{e}^{(ts)}}{(Bs + As + ABs^{2} + 1)} + \frac{t^{2}Cs{e}^{(ts)}}{(Bs + As + ABs^{2} + 1)} + \frac{2B^{2}{e}^{(ts)}}{(Bs + As + ABs^{2} + 1)^{3}} + \frac{4AB{e}^{(ts)}}{(Bs + As + ABs^{2} + 1)^{3}} + \frac{8AB^{2}s{e}^{(ts)}}{(Bs + As + ABs^{2} + 1)^{3}} - \frac{2tB{e}^{(ts)}}{(Bs + As + ABs^{2} + 1)^{2}} + \frac{2A^{2}{e}^{(ts)}}{(Bs + As + ABs^{2} + 1)^{3}} + \frac{8A^{2}Bs{e}^{(ts)}}{(Bs + As + ABs^{2} + 1)^{3}} - \frac{2tA{e}^{(ts)}}{(Bs + As + ABs^{2} + 1)^{2}} + \frac{8A^{2}B^{2}s^{2}{e}^{(ts)}}{(Bs + As + ABs^{2} + 1)^{3}} - \frac{2AB{e}^{(ts)}}{(Bs + As + ABs^{2} + 1)^{2}} - \frac{4tABs{e}^{(ts)}}{(Bs + As + ABs^{2} + 1)^{2}} + \frac{t^{2}{e}^{(ts)}}{(Bs + As + ABs^{2} + 1)}\\\\ &\color{blue}{The\ third\ derivative\ of\ function:} \\&\frac{d\left( \frac{2CB^{2}s{e}^{(ts)}}{(Bs + As + ABs^{2} + 1)^{3}} + \frac{4CABs{e}^{(ts)}}{(Bs + As + ABs^{2} + 1)^{3}} + \frac{8CAB^{2}s^{2}{e}^{(ts)}}{(Bs + As + ABs^{2} + 1)^{3}} - \frac{2CB{e}^{(ts)}}{(Bs + As + ABs^{2} + 1)^{2}} - \frac{2tCBs{e}^{(ts)}}{(Bs + As + ABs^{2} + 1)^{2}} + \frac{2CA^{2}s{e}^{(ts)}}{(Bs + As + ABs^{2} + 1)^{3}} + \frac{8CA^{2}Bs^{2}{e}^{(ts)}}{(Bs + As + ABs^{2} + 1)^{3}} - \frac{2CA{e}^{(ts)}}{(Bs + As + ABs^{2} + 1)^{2}} - \frac{2tCAs{e}^{(ts)}}{(Bs + As + ABs^{2} + 1)^{2}} + \frac{8CA^{2}B^{2}s^{3}{e}^{(ts)}}{(Bs + As + ABs^{2} + 1)^{3}} - \frac{6CABs{e}^{(ts)}}{(Bs + As + ABs^{2} + 1)^{2}} - \frac{4tCABs^{2}{e}^{(ts)}}{(Bs + As + ABs^{2} + 1)^{2}} + \frac{2tC{e}^{(ts)}}{(Bs + As + ABs^{2} + 1)} + \frac{t^{2}Cs{e}^{(ts)}}{(Bs + As + ABs^{2} + 1)} + \frac{2B^{2}{e}^{(ts)}}{(Bs + As + ABs^{2} + 1)^{3}} + \frac{4AB{e}^{(ts)}}{(Bs + As + ABs^{2} + 1)^{3}} + \frac{8AB^{2}s{e}^{(ts)}}{(Bs + As + ABs^{2} + 1)^{3}} - \frac{2tB{e}^{(ts)}}{(Bs + As + ABs^{2} + 1)^{2}} + \frac{2A^{2}{e}^{(ts)}}{(Bs + As + ABs^{2} + 1)^{3}} + \frac{8A^{2}Bs{e}^{(ts)}}{(Bs + As + ABs^{2} + 1)^{3}} - \frac{2tA{e}^{(ts)}}{(Bs + As + ABs^{2} + 1)^{2}} + \frac{8A^{2}B^{2}s^{2}{e}^{(ts)}}{(Bs + As + ABs^{2} + 1)^{3}} - \frac{2AB{e}^{(ts)}}{(Bs + As + ABs^{2} + 1)^{2}} - \frac{4tABs{e}^{(ts)}}{(Bs + As + ABs^{2} + 1)^{2}} + \frac{t^{2}{e}^{(ts)}}{(Bs + As + ABs^{2} + 1)}\right)}{ds}\\=&2(\frac{-3(B + A + AB*2s + 0)}{(Bs + As + ABs^{2} + 1)^{4}})CB^{2}s{e}^{(ts)} + \frac{2CB^{2}{e}^{(ts)}}{(Bs + As + ABs^{2} + 1)^{3}} + \frac{2CB^{2}s({e}^{(ts)}((t)ln(e) + \frac{(ts)(0)}{(e)}))}{(Bs + As + ABs^{2} + 1)^{3}} + 4(\frac{-3(B + A + AB*2s + 0)}{(Bs + As + ABs^{2} + 1)^{4}})CABs{e}^{(ts)} + \frac{4CAB{e}^{(ts)}}{(Bs + As + ABs^{2} + 1)^{3}} + \frac{4CABs({e}^{(ts)}((t)ln(e) + \frac{(ts)(0)}{(e)}))}{(Bs + As + ABs^{2} + 1)^{3}} + 8(\frac{-3(B + A + AB*2s + 0)}{(Bs + As + ABs^{2} + 1)^{4}})CAB^{2}s^{2}{e}^{(ts)} + \frac{8CAB^{2}*2s{e}^{(ts)}}{(Bs + As + ABs^{2} + 1)^{3}} + \frac{8CAB^{2}s^{2}({e}^{(ts)}((t)ln(e) + \frac{(ts)(0)}{(e)}))}{(Bs + As + ABs^{2} + 1)^{3}} - 2(\frac{-2(B + A + AB*2s + 0)}{(Bs + As + ABs^{2} + 1)^{3}})CB{e}^{(ts)} - \frac{2CB({e}^{(ts)}((t)ln(e) + \frac{(ts)(0)}{(e)}))}{(Bs + As + ABs^{2} + 1)^{2}} - 2(\frac{-2(B + A + AB*2s + 0)}{(Bs + As + ABs^{2} + 1)^{3}})tCBs{e}^{(ts)} - \frac{2tCB{e}^{(ts)}}{(Bs + As + ABs^{2} + 1)^{2}} - \frac{2tCBs({e}^{(ts)}((t)ln(e) + \frac{(ts)(0)}{(e)}))}{(Bs + As + ABs^{2} + 1)^{2}} + 2(\frac{-3(B + A + AB*2s + 0)}{(Bs + As + ABs^{2} + 1)^{4}})CA^{2}s{e}^{(ts)} + \frac{2CA^{2}{e}^{(ts)}}{(Bs + As + ABs^{2} + 1)^{3}} + \frac{2CA^{2}s({e}^{(ts)}((t)ln(e) + \frac{(ts)(0)}{(e)}))}{(Bs + As + ABs^{2} + 1)^{3}} + 8(\frac{-3(B + A + AB*2s + 0)}{(Bs + As + ABs^{2} + 1)^{4}})CA^{2}Bs^{2}{e}^{(ts)} + \frac{8CA^{2}B*2s{e}^{(ts)}}{(Bs + As + ABs^{2} + 1)^{3}} + \frac{8CA^{2}Bs^{2}({e}^{(ts)}((t)ln(e) + \frac{(ts)(0)}{(e)}))}{(Bs + As + ABs^{2} + 1)^{3}} - 2(\frac{-2(B + A + AB*2s + 0)}{(Bs + As + ABs^{2} + 1)^{3}})CA{e}^{(ts)} - \frac{2CA({e}^{(ts)}((t)ln(e) + \frac{(ts)(0)}{(e)}))}{(Bs + As + ABs^{2} + 1)^{2}} - 2(\frac{-2(B + A + AB*2s + 0)}{(Bs + As + ABs^{2} + 1)^{3}})tCAs{e}^{(ts)} - \frac{2tCA{e}^{(ts)}}{(Bs + As + ABs^{2} + 1)^{2}} - \frac{2tCAs({e}^{(ts)}((t)ln(e) + \frac{(ts)(0)}{(e)}))}{(Bs + As + ABs^{2} + 1)^{2}} + 8(\frac{-3(B + A + AB*2s + 0)}{(Bs + As + ABs^{2} + 1)^{4}})CA^{2}B^{2}s^{3}{e}^{(ts)} + \frac{8CA^{2}B^{2}*3s^{2}{e}^{(ts)}}{(Bs + As + ABs^{2} + 1)^{3}} + \frac{8CA^{2}B^{2}s^{3}({e}^{(ts)}((t)ln(e) + \frac{(ts)(0)}{(e)}))}{(Bs + As + ABs^{2} + 1)^{3}} - 6(\frac{-2(B + A + AB*2s + 0)}{(Bs + As + ABs^{2} + 1)^{3}})CABs{e}^{(ts)} - \frac{6CAB{e}^{(ts)}}{(Bs + As + ABs^{2} + 1)^{2}} - \frac{6CABs({e}^{(ts)}((t)ln(e) + \frac{(ts)(0)}{(e)}))}{(Bs + As + ABs^{2} + 1)^{2}} - 4(\frac{-2(B + A + AB*2s + 0)}{(Bs + As + ABs^{2} + 1)^{3}})tCABs^{2}{e}^{(ts)} - \frac{4tCAB*2s{e}^{(ts)}}{(Bs + As + ABs^{2} + 1)^{2}} - \frac{4tCABs^{2}({e}^{(ts)}((t)ln(e) + \frac{(ts)(0)}{(e)}))}{(Bs + As + ABs^{2} + 1)^{2}} + 2(\frac{-(B + A + AB*2s + 0)}{(Bs + As + ABs^{2} + 1)^{2}})tC{e}^{(ts)} + \frac{2tC({e}^{(ts)}((t)ln(e) + \frac{(ts)(0)}{(e)}))}{(Bs + As + ABs^{2} + 1)} + (\frac{-(B + A + AB*2s + 0)}{(Bs + As + ABs^{2} + 1)^{2}})t^{2}Cs{e}^{(ts)} + \frac{t^{2}C{e}^{(ts)}}{(Bs + As + ABs^{2} + 1)} + \frac{t^{2}Cs({e}^{(ts)}((t)ln(e) + \frac{(ts)(0)}{(e)}))}{(Bs + As + ABs^{2} + 1)} + 2(\frac{-3(B + A + AB*2s + 0)}{(Bs + As + ABs^{2} + 1)^{4}})B^{2}{e}^{(ts)} + \frac{2B^{2}({e}^{(ts)}((t)ln(e) + \frac{(ts)(0)}{(e)}))}{(Bs + As + ABs^{2} + 1)^{3}} + 4(\frac{-3(B + A + AB*2s + 0)}{(Bs + As + ABs^{2} + 1)^{4}})AB{e}^{(ts)} + \frac{4AB({e}^{(ts)}((t)ln(e) + \frac{(ts)(0)}{(e)}))}{(Bs + As + ABs^{2} + 1)^{3}} + 8(\frac{-3(B + A + AB*2s + 0)}{(Bs + As + ABs^{2} + 1)^{4}})AB^{2}s{e}^{(ts)} + \frac{8AB^{2}{e}^{(ts)}}{(Bs + As + ABs^{2} + 1)^{3}} + \frac{8AB^{2}s({e}^{(ts)}((t)ln(e) + \frac{(ts)(0)}{(e)}))}{(Bs + As + ABs^{2} + 1)^{3}} - 2(\frac{-2(B + A + AB*2s + 0)}{(Bs + As + ABs^{2} + 1)^{3}})tB{e}^{(ts)} - \frac{2tB({e}^{(ts)}((t)ln(e) + \frac{(ts)(0)}{(e)}))}{(Bs + As + ABs^{2} + 1)^{2}} + 2(\frac{-3(B + A + AB*2s + 0)}{(Bs + As + ABs^{2} + 1)^{4}})A^{2}{e}^{(ts)} + \frac{2A^{2}({e}^{(ts)}((t)ln(e) + \frac{(ts)(0)}{(e)}))}{(Bs + As + ABs^{2} + 1)^{3}} + 8(\frac{-3(B + A + AB*2s + 0)}{(Bs + As + ABs^{2} + 1)^{4}})A^{2}Bs{e}^{(ts)} + \frac{8A^{2}B{e}^{(ts)}}{(Bs + As + ABs^{2} + 1)^{3}} + \frac{8A^{2}Bs({e}^{(ts)}((t)ln(e) + \frac{(ts)(0)}{(e)}))}{(Bs + As + ABs^{2} + 1)^{3}} - 2(\frac{-2(B + A + AB*2s + 0)}{(Bs + As + ABs^{2} + 1)^{3}})tA{e}^{(ts)} - \frac{2tA({e}^{(ts)}((t)ln(e) + \frac{(ts)(0)}{(e)}))}{(Bs + As + ABs^{2} + 1)^{2}} + 8(\frac{-3(B + A + AB*2s + 0)}{(Bs + As + ABs^{2} + 1)^{4}})A^{2}B^{2}s^{2}{e}^{(ts)} + \frac{8A^{2}B^{2}*2s{e}^{(ts)}}{(Bs + As + ABs^{2} + 1)^{3}} + \frac{8A^{2}B^{2}s^{2}({e}^{(ts)}((t)ln(e) + \frac{(ts)(0)}{(e)}))}{(Bs + As + ABs^{2} + 1)^{3}} - 2(\frac{-2(B + A + AB*2s + 0)}{(Bs + As + ABs^{2} + 1)^{3}})AB{e}^{(ts)} - \frac{2AB({e}^{(ts)}((t)ln(e) + \frac{(ts)(0)}{(e)}))}{(Bs + As + ABs^{2} + 1)^{2}} - 4(\frac{-2(B + A + AB*2s + 0)}{(Bs + As + ABs^{2} + 1)^{3}})tABs{e}^{(ts)} - \frac{4tAB{e}^{(ts)}}{(Bs + As + ABs^{2} + 1)^{2}} - \frac{4tABs({e}^{(ts)}((t)ln(e) + \frac{(ts)(0)}{(e)}))}{(Bs + As + ABs^{2} + 1)^{2}} + (\frac{-(B + A + AB*2s + 0)}{(Bs + As + ABs^{2} + 1)^{2}})t^{2}{e}^{(ts)} + \frac{t^{2}({e}^{(ts)}((t)ln(e) + \frac{(ts)(0)}{(e)}))}{(Bs + As + ABs^{2} + 1)}\\=&\frac{-6CB^{3}s{e}^{(ts)}}{(Bs + As + ABs^{2} + 1)^{4}} - \frac{18CAB^{2}s{e}^{(ts)}}{(Bs + As + ABs^{2} + 1)^{4}} - \frac{36CAB^{3}s^{2}{e}^{(ts)}}{(Bs + As + ABs^{2} + 1)^{4}} + \frac{6CB^{2}{e}^{(ts)}}{(Bs + As + ABs^{2} + 1)^{3}} + \frac{6tCB^{2}s{e}^{(ts)}}{(Bs + As + ABs^{2} + 1)^{3}} - \frac{18CA^{2}Bs{e}^{(ts)}}{(Bs + As + ABs^{2} + 1)^{4}} - \frac{72CA^{2}B^{2}s^{2}{e}^{(ts)}}{(Bs + As + ABs^{2} + 1)^{4}} + \frac{12CAB{e}^{(ts)}}{(Bs + As + ABs^{2} + 1)^{3}} + \frac{12tCABs{e}^{(ts)}}{(Bs + As + ABs^{2} + 1)^{3}} - \frac{72CA^{2}B^{3}s^{3}{e}^{(ts)}}{(Bs + As + ABs^{2} + 1)^{4}} + \frac{36CAB^{2}s{e}^{(ts)}}{(Bs + As + ABs^{2} + 1)^{3}} + \frac{24tCAB^{2}s^{2}{e}^{(ts)}}{(Bs + As + ABs^{2} + 1)^{3}} - \frac{6tCB{e}^{(ts)}}{(Bs + As + ABs^{2} + 1)^{2}} - \frac{3t^{2}CBs{e}^{(ts)}}{(Bs + As + ABs^{2} + 1)^{2}} - \frac{6CA^{3}s{e}^{(ts)}}{(Bs + As + ABs^{2} + 1)^{4}} - \frac{36CA^{3}Bs^{2}{e}^{(ts)}}{(Bs + As + ABs^{2} + 1)^{4}} + \frac{6CA^{2}{e}^{(ts)}}{(Bs + As + ABs^{2} + 1)^{3}} + \frac{6tCA^{2}s{e}^{(ts)}}{(Bs + As + ABs^{2} + 1)^{3}} - \frac{72CA^{3}B^{2}s^{3}{e}^{(ts)}}{(Bs + As + ABs^{2} + 1)^{4}} + \frac{36CA^{2}Bs{e}^{(ts)}}{(Bs + As + ABs^{2} + 1)^{3}} + \frac{24tCA^{2}Bs^{2}{e}^{(ts)}}{(Bs + As + ABs^{2} + 1)^{3}} - \frac{6tCA{e}^{(ts)}}{(Bs + As + ABs^{2} + 1)^{2}} - \frac{3t^{2}CAs{e}^{(ts)}}{(Bs + As + ABs^{2} + 1)^{2}} - \frac{48CA^{3}B^{3}s^{4}{e}^{(ts)}}{(Bs + As + ABs^{2} + 1)^{4}} + \frac{48CA^{2}B^{2}s^{2}{e}^{(ts)}}{(Bs + As + ABs^{2} + 1)^{3}} + \frac{24tCA^{2}B^{2}s^{3}{e}^{(ts)}}{(Bs + As + ABs^{2} + 1)^{3}} - \frac{6CAB{e}^{(ts)}}{(Bs + As + ABs^{2} + 1)^{2}} - \frac{18tCABs{e}^{(ts)}}{(Bs + As + ABs^{2} + 1)^{2}} - \frac{6t^{2}CABs^{2}{e}^{(ts)}}{(Bs + As + ABs^{2} + 1)^{2}} + \frac{3t^{2}C{e}^{(ts)}}{(Bs + As + ABs^{2} + 1)} + \frac{t^{3}Cs{e}^{(ts)}}{(Bs + As + ABs^{2} + 1)} - \frac{6B^{3}{e}^{(ts)}}{(Bs + As + ABs^{2} + 1)^{4}} - \frac{18AB^{2}{e}^{(ts)}}{(Bs + As + ABs^{2} + 1)^{4}} - \frac{36AB^{3}s{e}^{(ts)}}{(Bs + As + ABs^{2} + 1)^{4}} + \frac{6tB^{2}{e}^{(ts)}}{(Bs + As + ABs^{2} + 1)^{3}} - \frac{18A^{2}B{e}^{(ts)}}{(Bs + As + ABs^{2} + 1)^{4}} - \frac{72A^{2}B^{2}s{e}^{(ts)}}{(Bs + As + ABs^{2} + 1)^{4}} + \frac{12tAB{e}^{(ts)}}{(Bs + As + ABs^{2} + 1)^{3}} - \frac{72A^{2}B^{3}s^{2}{e}^{(ts)}}{(Bs + As + ABs^{2} + 1)^{4}} + \frac{12AB^{2}{e}^{(ts)}}{(Bs + As + ABs^{2} + 1)^{3}} + \frac{24tAB^{2}s{e}^{(ts)}}{(Bs + As + ABs^{2} + 1)^{3}} - \frac{3t^{2}B{e}^{(ts)}}{(Bs + As + ABs^{2} + 1)^{2}} - \frac{6A^{3}{e}^{(ts)}}{(Bs + As + ABs^{2} + 1)^{4}} - \frac{36A^{3}Bs{e}^{(ts)}}{(Bs + As + ABs^{2} + 1)^{4}} + \frac{6tA^{2}{e}^{(ts)}}{(Bs + As + ABs^{2} + 1)^{3}} - \frac{72A^{3}B^{2}s^{2}{e}^{(ts)}}{(Bs + As + ABs^{2} + 1)^{4}} + \frac{12A^{2}B{e}^{(ts)}}{(Bs + As + ABs^{2} + 1)^{3}} + \frac{24tA^{2}Bs{e}^{(ts)}}{(Bs + As + ABs^{2} + 1)^{3}} - \frac{3t^{2}A{e}^{(ts)}}{(Bs + As + ABs^{2} + 1)^{2}} - \frac{48A^{3}B^{3}s^{3}{e}^{(ts)}}{(Bs + As + ABs^{2} + 1)^{4}} + \frac{24A^{2}B^{2}s{e}^{(ts)}}{(Bs + As + ABs^{2} + 1)^{3}} + \frac{24tA^{2}B^{2}s^{2}{e}^{(ts)}}{(Bs + As + ABs^{2} + 1)^{3}} - \frac{6tAB{e}^{(ts)}}{(Bs + As + ABs^{2} + 1)^{2}} - \frac{6t^{2}ABs{e}^{(ts)}}{(Bs + As + ABs^{2} + 1)^{2}} + \frac{t^{3}{e}^{(ts)}}{(Bs + As + ABs^{2} + 1)}\\ \end{split}\end{equation} \]
Note that variables are case sensitive.
\[ \begin{equation}\begin{split}[1/1]Find\ the\ third\ derivative\ of\ function\ \frac{({e}^{(st)}(1 + Cs))}{((1 + As)(1 + Bs))}\ with\ respect\ to\ s：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &Primitive\ function\ = \frac{Cs{e}^{(ts)}}{(Bs + As + ABs^{2} + 1)} + \frac{{e}^{(ts)}}{(Bs + As + ABs^{2} + 1)}\\&\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( \frac{Cs{e}^{(ts)}}{(Bs + As + ABs^{2} + 1)} + \frac{{e}^{(ts)}}{(Bs + As + ABs^{2} + 1)}\right)}{ds}\\=&(\frac{-(B + A + AB*2s + 0)}{(Bs + As + ABs^{2} + 1)^{2}})Cs{e}^{(ts)} + \frac{C{e}^{(ts)}}{(Bs + As + ABs^{2} + 1)} + \frac{Cs({e}^{(ts)}((t)ln(e) + \frac{(ts)(0)}{(e)}))}{(Bs + As + ABs^{2} + 1)} + (\frac{-(B + A + AB*2s + 0)}{(Bs + As + ABs^{2} + 1)^{2}}){e}^{(ts)} + \frac{({e}^{(ts)}((t)ln(e) + \frac{(ts)(0)}{(e)}))}{(Bs + As + ABs^{2} + 1)}\\=& - \frac{CBs{e}^{(ts)}}{(Bs + As + ABs^{2} + 1)^{2}} - \frac{CAs{e}^{(ts)}}{(Bs + As + ABs^{2} + 1)^{2}} - \frac{2CABs^{2}{e}^{(ts)}}{(Bs + As + ABs^{2} + 1)^{2}} + \frac{C{e}^{(ts)}}{(Bs + As + ABs^{2} + 1)} + \frac{tCs{e}^{(ts)}}{(Bs + As + ABs^{2} + 1)} - \frac{B{e}^{(ts)}}{(Bs + As + ABs^{2} + 1)^{2}} - \frac{A{e}^{(ts)}}{(Bs + As + ABs^{2} + 1)^{2}} - \frac{2ABs{e}^{(ts)}}{(Bs + As + ABs^{2} + 1)^{2}} + \frac{t{e}^{(ts)}}{(Bs + As + ABs^{2} + 1)}\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( - \frac{CBs{e}^{(ts)}}{(Bs + As + ABs^{2} + 1)^{2}} - \frac{CAs{e}^{(ts)}}{(Bs + As + ABs^{2} + 1)^{2}} - \frac{2CABs^{2}{e}^{(ts)}}{(Bs + As + ABs^{2} + 1)^{2}} + \frac{C{e}^{(ts)}}{(Bs + As + ABs^{2} + 1)} + \frac{tCs{e}^{(ts)}}{(Bs + As + ABs^{2} + 1)} - \frac{B{e}^{(ts)}}{(Bs + As + ABs^{2} + 1)^{2}} - \frac{A{e}^{(ts)}}{(Bs + As + ABs^{2} + 1)^{2}} - \frac{2ABs{e}^{(ts)}}{(Bs + As + ABs^{2} + 1)^{2}} + \frac{t{e}^{(ts)}}{(Bs + As + ABs^{2} + 1)}\right)}{ds}\\=& - (\frac{-2(B + A + AB*2s + 0)}{(Bs + As + ABs^{2} + 1)^{3}})CBs{e}^{(ts)} - \frac{CB{e}^{(ts)}}{(Bs + As + ABs^{2} + 1)^{2}} - \frac{CBs({e}^{(ts)}((t)ln(e) + \frac{(ts)(0)}{(e)}))}{(Bs + As + ABs^{2} + 1)^{2}} - (\frac{-2(B + A + AB*2s + 0)}{(Bs + As + ABs^{2} + 1)^{3}})CAs{e}^{(ts)} - \frac{CA{e}^{(ts)}}{(Bs + As + ABs^{2} + 1)^{2}} - \frac{CAs({e}^{(ts)}((t)ln(e) + \frac{(ts)(0)}{(e)}))}{(Bs + As + ABs^{2} + 1)^{2}} - 2(\frac{-2(B + A + AB*2s + 0)}{(Bs + As + ABs^{2} + 1)^{3}})CABs^{2}{e}^{(ts)} - \frac{2CAB*2s{e}^{(ts)}}{(Bs + As + ABs^{2} + 1)^{2}} - \frac{2CABs^{2}({e}^{(ts)}((t)ln(e) + \frac{(ts)(0)}{(e)}))}{(Bs + As + ABs^{2} + 1)^{2}} + (\frac{-(B + A + AB*2s + 0)}{(Bs + As + ABs^{2} + 1)^{2}})C{e}^{(ts)} + \frac{C({e}^{(ts)}((t)ln(e) + \frac{(ts)(0)}{(e)}))}{(Bs + As + ABs^{2} + 1)} + (\frac{-(B + A + AB*2s + 0)}{(Bs + As + ABs^{2} + 1)^{2}})tCs{e}^{(ts)} + \frac{tC{e}^{(ts)}}{(Bs + As + ABs^{2} + 1)} + \frac{tCs({e}^{(ts)}((t)ln(e) + \frac{(ts)(0)}{(e)}))}{(Bs + As + ABs^{2} + 1)} - (\frac{-2(B + A + AB*2s + 0)}{(Bs + As + ABs^{2} + 1)^{3}})B{e}^{(ts)} - \frac{B({e}^{(ts)}((t)ln(e) + \frac{(ts)(0)}{(e)}))}{(Bs + As + ABs^{2} + 1)^{2}} - (\frac{-2(B + A + AB*2s + 0)}{(Bs + As + ABs^{2} + 1)^{3}})A{e}^{(ts)} - \frac{A({e}^{(ts)}((t)ln(e) + \frac{(ts)(0)}{(e)}))}{(Bs + As + ABs^{2} + 1)^{2}} - 2(\frac{-2(B + A + AB*2s + 0)}{(Bs + As + ABs^{2} + 1)^{3}})ABs{e}^{(ts)} - \frac{2AB{e}^{(ts)}}{(Bs + As + ABs^{2} + 1)^{2}} - \frac{2ABs({e}^{(ts)}((t)ln(e) + \frac{(ts)(0)}{(e)}))}{(Bs + As + ABs^{2} + 1)^{2}} + (\frac{-(B + A + AB*2s + 0)}{(Bs + As + ABs^{2} + 1)^{2}})t{e}^{(ts)} + \frac{t({e}^{(ts)}((t)ln(e) + \frac{(ts)(0)}{(e)}))}{(Bs + As + ABs^{2} + 1)}\\=&\frac{2CB^{2}s{e}^{(ts)}}{(Bs + As + ABs^{2} + 1)^{3}} + \frac{4CABs{e}^{(ts)}}{(Bs + As + ABs^{2} + 1)^{3}} + \frac{8CAB^{2}s^{2}{e}^{(ts)}}{(Bs + As + ABs^{2} + 1)^{3}} - \frac{2CB{e}^{(ts)}}{(Bs + As + ABs^{2} + 1)^{2}} - \frac{2tCBs{e}^{(ts)}}{(Bs + As + ABs^{2} + 1)^{2}} + \frac{2CA^{2}s{e}^{(ts)}}{(Bs + As + ABs^{2} + 1)^{3}} + \frac{8CA^{2}Bs^{2}{e}^{(ts)}}{(Bs + As + ABs^{2} + 1)^{3}} - \frac{2CA{e}^{(ts)}}{(Bs + As + ABs^{2} + 1)^{2}} - \frac{2tCAs{e}^{(ts)}}{(Bs + As + ABs^{2} + 1)^{2}} + \frac{8CA^{2}B^{2}s^{3}{e}^{(ts)}}{(Bs + As + ABs^{2} + 1)^{3}} - \frac{6CABs{e}^{(ts)}}{(Bs + As + ABs^{2} + 1)^{2}} - \frac{4tCABs^{2}{e}^{(ts)}}{(Bs + As + ABs^{2} + 1)^{2}} + \frac{2tC{e}^{(ts)}}{(Bs + As + ABs^{2} + 1)} + \frac{t^{2}Cs{e}^{(ts)}}{(Bs + As + ABs^{2} + 1)} + \frac{2B^{2}{e}^{(ts)}}{(Bs + As + ABs^{2} + 1)^{3}} + \frac{4AB{e}^{(ts)}}{(Bs + As + ABs^{2} + 1)^{3}} + \frac{8AB^{2}s{e}^{(ts)}}{(Bs + As + ABs^{2} + 1)^{3}} - \frac{2tB{e}^{(ts)}}{(Bs + As + ABs^{2} + 1)^{2}} + \frac{2A^{2}{e}^{(ts)}}{(Bs + As + ABs^{2} + 1)^{3}} + \frac{8A^{2}Bs{e}^{(ts)}}{(Bs + As + ABs^{2} + 1)^{3}} - \frac{2tA{e}^{(ts)}}{(Bs + As + ABs^{2} + 1)^{2}} + \frac{8A^{2}B^{2}s^{2}{e}^{(ts)}}{(Bs + As + ABs^{2} + 1)^{3}} - \frac{2AB{e}^{(ts)}}{(Bs + As + ABs^{2} + 1)^{2}} - \frac{4tABs{e}^{(ts)}}{(Bs + As + ABs^{2} + 1)^{2}} + \frac{t^{2}{e}^{(ts)}}{(Bs + As + ABs^{2} + 1)}\\\\ &\color{blue}{The\ third\ derivative\ of\ function:} \\&\frac{d\left( \frac{2CB^{2}s{e}^{(ts)}}{(Bs + As + ABs^{2} + 1)^{3}} + \frac{4CABs{e}^{(ts)}}{(Bs + As + ABs^{2} + 1)^{3}} + \frac{8CAB^{2}s^{2}{e}^{(ts)}}{(Bs + As + ABs^{2} + 1)^{3}} - \frac{2CB{e}^{(ts)}}{(Bs + As + ABs^{2} + 1)^{2}} - \frac{2tCBs{e}^{(ts)}}{(Bs + As + ABs^{2} + 1)^{2}} + \frac{2CA^{2}s{e}^{(ts)}}{(Bs + As + ABs^{2} + 1)^{3}} + \frac{8CA^{2}Bs^{2}{e}^{(ts)}}{(Bs + As + ABs^{2} + 1)^{3}} - \frac{2CA{e}^{(ts)}}{(Bs + As + ABs^{2} + 1)^{2}} - \frac{2tCAs{e}^{(ts)}}{(Bs + As + ABs^{2} + 1)^{2}} + \frac{8CA^{2}B^{2}s^{3}{e}^{(ts)}}{(Bs + As + ABs^{2} + 1)^{3}} - \frac{6CABs{e}^{(ts)}}{(Bs + As + ABs^{2} + 1)^{2}} - \frac{4tCABs^{2}{e}^{(ts)}}{(Bs + As + ABs^{2} + 1)^{2}} + \frac{2tC{e}^{(ts)}}{(Bs + As + ABs^{2} + 1)} + \frac{t^{2}Cs{e}^{(ts)}}{(Bs + As + ABs^{2} + 1)} + \frac{2B^{2}{e}^{(ts)}}{(Bs + As + ABs^{2} + 1)^{3}} + \frac{4AB{e}^{(ts)}}{(Bs + As + ABs^{2} + 1)^{3}} + \frac{8AB^{2}s{e}^{(ts)}}{(Bs + As + ABs^{2} + 1)^{3}} - \frac{2tB{e}^{(ts)}}{(Bs + As + ABs^{2} + 1)^{2}} + \frac{2A^{2}{e}^{(ts)}}{(Bs + As + ABs^{2} + 1)^{3}} + \frac{8A^{2}Bs{e}^{(ts)}}{(Bs + As + ABs^{2} + 1)^{3}} - \frac{2tA{e}^{(ts)}}{(Bs + As + ABs^{2} + 1)^{2}} + \frac{8A^{2}B^{2}s^{2}{e}^{(ts)}}{(Bs + As + ABs^{2} + 1)^{3}} - \frac{2AB{e}^{(ts)}}{(Bs + As + ABs^{2} + 1)^{2}} - \frac{4tABs{e}^{(ts)}}{(Bs + As + ABs^{2} + 1)^{2}} + \frac{t^{2}{e}^{(ts)}}{(Bs + As + ABs^{2} + 1)}\right)}{ds}\\=&2(\frac{-3(B + A + AB*2s + 0)}{(Bs + As + ABs^{2} + 1)^{4}})CB^{2}s{e}^{(ts)} + \frac{2CB^{2}{e}^{(ts)}}{(Bs + As + ABs^{2} + 1)^{3}} + \frac{2CB^{2}s({e}^{(ts)}((t)ln(e) + \frac{(ts)(0)}{(e)}))}{(Bs + As + ABs^{2} + 1)^{3}} + 4(\frac{-3(B + A + AB*2s + 0)}{(Bs + As + ABs^{2} + 1)^{4}})CABs{e}^{(ts)} + \frac{4CAB{e}^{(ts)}}{(Bs + As + ABs^{2} + 1)^{3}} + \frac{4CABs({e}^{(ts)}((t)ln(e) + \frac{(ts)(0)}{(e)}))}{(Bs + As + ABs^{2} + 1)^{3}} + 8(\frac{-3(B + A + AB*2s + 0)}{(Bs + As + ABs^{2} + 1)^{4}})CAB^{2}s^{2}{e}^{(ts)} + \frac{8CAB^{2}*2s{e}^{(ts)}}{(Bs + As + ABs^{2} + 1)^{3}} + \frac{8CAB^{2}s^{2}({e}^{(ts)}((t)ln(e) + \frac{(ts)(0)}{(e)}))}{(Bs + As + ABs^{2} + 1)^{3}} - 2(\frac{-2(B + A + AB*2s + 0)}{(Bs + As + ABs^{2} + 1)^{3}})CB{e}^{(ts)} - \frac{2CB({e}^{(ts)}((t)ln(e) + \frac{(ts)(0)}{(e)}))}{(Bs + As + ABs^{2} + 1)^{2}} - 2(\frac{-2(B + A + AB*2s + 0)}{(Bs + As + ABs^{2} + 1)^{3}})tCBs{e}^{(ts)} - \frac{2tCB{e}^{(ts)}}{(Bs + As + ABs^{2} + 1)^{2}} - \frac{2tCBs({e}^{(ts)}((t)ln(e) + \frac{(ts)(0)}{(e)}))}{(Bs + As + ABs^{2} + 1)^{2}} + 2(\frac{-3(B + A + AB*2s + 0)}{(Bs + As + ABs^{2} + 1)^{4}})CA^{2}s{e}^{(ts)} + \frac{2CA^{2}{e}^{(ts)}}{(Bs + As + ABs^{2} + 1)^{3}} + \frac{2CA^{2}s({e}^{(ts)}((t)ln(e) + \frac{(ts)(0)}{(e)}))}{(Bs + As + ABs^{2} + 1)^{3}} + 8(\frac{-3(B + A + AB*2s + 0)}{(Bs + As + ABs^{2} + 1)^{4}})CA^{2}Bs^{2}{e}^{(ts)} + \frac{8CA^{2}B*2s{e}^{(ts)}}{(Bs + As + ABs^{2} + 1)^{3}} + \frac{8CA^{2}Bs^{2}({e}^{(ts)}((t)ln(e) + \frac{(ts)(0)}{(e)}))}{(Bs + As + ABs^{2} + 1)^{3}} - 2(\frac{-2(B + A + AB*2s + 0)}{(Bs + As + ABs^{2} + 1)^{3}})CA{e}^{(ts)} - \frac{2CA({e}^{(ts)}((t)ln(e) + \frac{(ts)(0)}{(e)}))}{(Bs + As + ABs^{2} + 1)^{2}} - 2(\frac{-2(B + A + AB*2s + 0)}{(Bs + As + ABs^{2} + 1)^{3}})tCAs{e}^{(ts)} - \frac{2tCA{e}^{(ts)}}{(Bs + As + ABs^{2} + 1)^{2}} - \frac{2tCAs({e}^{(ts)}((t)ln(e) + \frac{(ts)(0)}{(e)}))}{(Bs + As + ABs^{2} + 1)^{2}} + 8(\frac{-3(B + A + AB*2s + 0)}{(Bs + As + ABs^{2} + 1)^{4}})CA^{2}B^{2}s^{3}{e}^{(ts)} + \frac{8CA^{2}B^{2}*3s^{2}{e}^{(ts)}}{(Bs + As + ABs^{2} + 1)^{3}} + \frac{8CA^{2}B^{2}s^{3}({e}^{(ts)}((t)ln(e) + \frac{(ts)(0)}{(e)}))}{(Bs + As + ABs^{2} + 1)^{3}} - 6(\frac{-2(B + A + AB*2s + 0)}{(Bs + As + ABs^{2} + 1)^{3}})CABs{e}^{(ts)} - \frac{6CAB{e}^{(ts)}}{(Bs + As + ABs^{2} + 1)^{2}} - \frac{6CABs({e}^{(ts)}((t)ln(e) + \frac{(ts)(0)}{(e)}))}{(Bs + As + ABs^{2} + 1)^{2}} - 4(\frac{-2(B + A + AB*2s + 0)}{(Bs + As + ABs^{2} + 1)^{3}})tCABs^{2}{e}^{(ts)} - \frac{4tCAB*2s{e}^{(ts)}}{(Bs + As + ABs^{2} + 1)^{2}} - \frac{4tCABs^{2}({e}^{(ts)}((t)ln(e) + \frac{(ts)(0)}{(e)}))}{(Bs + As + ABs^{2} + 1)^{2}} + 2(\frac{-(B + A + AB*2s + 0)}{(Bs + As + ABs^{2} + 1)^{2}})tC{e}^{(ts)} + \frac{2tC({e}^{(ts)}((t)ln(e) + \frac{(ts)(0)}{(e)}))}{(Bs + As + ABs^{2} + 1)} + (\frac{-(B + A + AB*2s + 0)}{(Bs + As + ABs^{2} + 1)^{2}})t^{2}Cs{e}^{(ts)} + \frac{t^{2}C{e}^{(ts)}}{(Bs + As + ABs^{2} + 1)} + \frac{t^{2}Cs({e}^{(ts)}((t)ln(e) + \frac{(ts)(0)}{(e)}))}{(Bs + As + ABs^{2} + 1)} + 2(\frac{-3(B + A + AB*2s + 0)}{(Bs + As + ABs^{2} + 1)^{4}})B^{2}{e}^{(ts)} + \frac{2B^{2}({e}^{(ts)}((t)ln(e) + \frac{(ts)(0)}{(e)}))}{(Bs + As + ABs^{2} + 1)^{3}} + 4(\frac{-3(B + A + AB*2s + 0)}{(Bs + As + ABs^{2} + 1)^{4}})AB{e}^{(ts)} + \frac{4AB({e}^{(ts)}((t)ln(e) + \frac{(ts)(0)}{(e)}))}{(Bs + As + ABs^{2} + 1)^{3}} + 8(\frac{-3(B + A + AB*2s + 0)}{(Bs + As + ABs^{2} + 1)^{4}})AB^{2}s{e}^{(ts)} + \frac{8AB^{2}{e}^{(ts)}}{(Bs + As + ABs^{2} + 1)^{3}} + \frac{8AB^{2}s({e}^{(ts)}((t)ln(e) + \frac{(ts)(0)}{(e)}))}{(Bs + As + ABs^{2} + 1)^{3}} - 2(\frac{-2(B + A + AB*2s + 0)}{(Bs + As + ABs^{2} + 1)^{3}})tB{e}^{(ts)} - \frac{2tB({e}^{(ts)}((t)ln(e) + \frac{(ts)(0)}{(e)}))}{(Bs + As + ABs^{2} + 1)^{2}} + 2(\frac{-3(B + A + AB*2s + 0)}{(Bs + As + ABs^{2} + 1)^{4}})A^{2}{e}^{(ts)} + \frac{2A^{2}({e}^{(ts)}((t)ln(e) + \frac{(ts)(0)}{(e)}))}{(Bs + As + ABs^{2} + 1)^{3}} + 8(\frac{-3(B + A + AB*2s + 0)}{(Bs + As + ABs^{2} + 1)^{4}})A^{2}Bs{e}^{(ts)} + \frac{8A^{2}B{e}^{(ts)}}{(Bs + As + ABs^{2} + 1)^{3}} + \frac{8A^{2}Bs({e}^{(ts)}((t)ln(e) + \frac{(ts)(0)}{(e)}))}{(Bs + As + ABs^{2} + 1)^{3}} - 2(\frac{-2(B + A + AB*2s + 0)}{(Bs + As + ABs^{2} + 1)^{3}})tA{e}^{(ts)} - \frac{2tA({e}^{(ts)}((t)ln(e) + \frac{(ts)(0)}{(e)}))}{(Bs + As + ABs^{2} + 1)^{2}} + 8(\frac{-3(B + A + AB*2s + 0)}{(Bs + As + ABs^{2} + 1)^{4}})A^{2}B^{2}s^{2}{e}^{(ts)} + \frac{8A^{2}B^{2}*2s{e}^{(ts)}}{(Bs + As + ABs^{2} + 1)^{3}} + \frac{8A^{2}B^{2}s^{2}({e}^{(ts)}((t)ln(e) + \frac{(ts)(0)}{(e)}))}{(Bs + As + ABs^{2} + 1)^{3}} - 2(\frac{-2(B + A + AB*2s + 0)}{(Bs + As + ABs^{2} + 1)^{3}})AB{e}^{(ts)} - \frac{2AB({e}^{(ts)}((t)ln(e) + \frac{(ts)(0)}{(e)}))}{(Bs + As + ABs^{2} + 1)^{2}} - 4(\frac{-2(B + A + AB*2s + 0)}{(Bs + As + ABs^{2} + 1)^{3}})tABs{e}^{(ts)} - \frac{4tAB{e}^{(ts)}}{(Bs + As + ABs^{2} + 1)^{2}} - \frac{4tABs({e}^{(ts)}((t)ln(e) + \frac{(ts)(0)}{(e)}))}{(Bs + As + ABs^{2} + 1)^{2}} + (\frac{-(B + A + AB*2s + 0)}{(Bs + As + ABs^{2} + 1)^{2}})t^{2}{e}^{(ts)} + \frac{t^{2}({e}^{(ts)}((t)ln(e) + \frac{(ts)(0)}{(e)}))}{(Bs + As + ABs^{2} + 1)}\\=&\frac{-6CB^{3}s{e}^{(ts)}}{(Bs + As + ABs^{2} + 1)^{4}} - \frac{18CAB^{2}s{e}^{(ts)}}{(Bs + As + ABs^{2} + 1)^{4}} - \frac{36CAB^{3}s^{2}{e}^{(ts)}}{(Bs + As + ABs^{2} + 1)^{4}} + \frac{6CB^{2}{e}^{(ts)}}{(Bs + As + ABs^{2} + 1)^{3}} + \frac{6tCB^{2}s{e}^{(ts)}}{(Bs + As + ABs^{2} + 1)^{3}} - \frac{18CA^{2}Bs{e}^{(ts)}}{(Bs + As + ABs^{2} + 1)^{4}} - \frac{72CA^{2}B^{2}s^{2}{e}^{(ts)}}{(Bs + As + ABs^{2} + 1)^{4}} + \frac{12CAB{e}^{(ts)}}{(Bs + As + ABs^{2} + 1)^{3}} + \frac{12tCABs{e}^{(ts)}}{(Bs + As + ABs^{2} + 1)^{3}} - \frac{72CA^{2}B^{3}s^{3}{e}^{(ts)}}{(Bs + As + ABs^{2} + 1)^{4}} + \frac{36CAB^{2}s{e}^{(ts)}}{(Bs + As + ABs^{2} + 1)^{3}} + \frac{24tCAB^{2}s^{2}{e}^{(ts)}}{(Bs + As + ABs^{2} + 1)^{3}} - \frac{6tCB{e}^{(ts)}}{(Bs + As + ABs^{2} + 1)^{2}} - \frac{3t^{2}CBs{e}^{(ts)}}{(Bs + As + ABs^{2} + 1)^{2}} - \frac{6CA^{3}s{e}^{(ts)}}{(Bs + As + ABs^{2} + 1)^{4}} - \frac{36CA^{3}Bs^{2}{e}^{(ts)}}{(Bs + As + ABs^{2} + 1)^{4}} + \frac{6CA^{2}{e}^{(ts)}}{(Bs + As + ABs^{2} + 1)^{3}} + \frac{6tCA^{2}s{e}^{(ts)}}{(Bs + As + ABs^{2} + 1)^{3}} - \frac{72CA^{3}B^{2}s^{3}{e}^{(ts)}}{(Bs + As + ABs^{2} + 1)^{4}} + \frac{36CA^{2}Bs{e}^{(ts)}}{(Bs + As + ABs^{2} + 1)^{3}} + \frac{24tCA^{2}Bs^{2}{e}^{(ts)}}{(Bs + As + ABs^{2} + 1)^{3}} - \frac{6tCA{e}^{(ts)}}{(Bs + As + ABs^{2} + 1)^{2}} - \frac{3t^{2}CAs{e}^{(ts)}}{(Bs + As + ABs^{2} + 1)^{2}} - \frac{48CA^{3}B^{3}s^{4}{e}^{(ts)}}{(Bs + As + ABs^{2} + 1)^{4}} + \frac{48CA^{2}B^{2}s^{2}{e}^{(ts)}}{(Bs + As + ABs^{2} + 1)^{3}} + \frac{24tCA^{2}B^{2}s^{3}{e}^{(ts)}}{(Bs + As + ABs^{2} + 1)^{3}} - \frac{6CAB{e}^{(ts)}}{(Bs + As + ABs^{2} + 1)^{2}} - \frac{18tCABs{e}^{(ts)}}{(Bs + As + ABs^{2} + 1)^{2}} - \frac{6t^{2}CABs^{2}{e}^{(ts)}}{(Bs + As + ABs^{2} + 1)^{2}} + \frac{3t^{2}C{e}^{(ts)}}{(Bs + As + ABs^{2} + 1)} + \frac{t^{3}Cs{e}^{(ts)}}{(Bs + As + ABs^{2} + 1)} - \frac{6B^{3}{e}^{(ts)}}{(Bs + As + ABs^{2} + 1)^{4}} - \frac{18AB^{2}{e}^{(ts)}}{(Bs + As + ABs^{2} + 1)^{4}} - \frac{36AB^{3}s{e}^{(ts)}}{(Bs + As + ABs^{2} + 1)^{4}} + \frac{6tB^{2}{e}^{(ts)}}{(Bs + As + ABs^{2} + 1)^{3}} - \frac{18A^{2}B{e}^{(ts)}}{(Bs + As + ABs^{2} + 1)^{4}} - \frac{72A^{2}B^{2}s{e}^{(ts)}}{(Bs + As + ABs^{2} + 1)^{4}} + \frac{12tAB{e}^{(ts)}}{(Bs + As + ABs^{2} + 1)^{3}} - \frac{72A^{2}B^{3}s^{2}{e}^{(ts)}}{(Bs + As + ABs^{2} + 1)^{4}} + \frac{12AB^{2}{e}^{(ts)}}{(Bs + As + ABs^{2} + 1)^{3}} + \frac{24tAB^{2}s{e}^{(ts)}}{(Bs + As + ABs^{2} + 1)^{3}} - \frac{3t^{2}B{e}^{(ts)}}{(Bs + As + ABs^{2} + 1)^{2}} - \frac{6A^{3}{e}^{(ts)}}{(Bs + As + ABs^{2} + 1)^{4}} - \frac{36A^{3}Bs{e}^{(ts)}}{(Bs + As + ABs^{2} + 1)^{4}} + \frac{6tA^{2}{e}^{(ts)}}{(Bs + As + ABs^{2} + 1)^{3}} - \frac{72A^{3}B^{2}s^{2}{e}^{(ts)}}{(Bs + As + ABs^{2} + 1)^{4}} + \frac{12A^{2}B{e}^{(ts)}}{(Bs + As + ABs^{2} + 1)^{3}} + \frac{24tA^{2}Bs{e}^{(ts)}}{(Bs + As + ABs^{2} + 1)^{3}} - \frac{3t^{2}A{e}^{(ts)}}{(Bs + As + ABs^{2} + 1)^{2}} - \frac{48A^{3}B^{3}s^{3}{e}^{(ts)}}{(Bs + As + ABs^{2} + 1)^{4}} + \frac{24A^{2}B^{2}s{e}^{(ts)}}{(Bs + As + ABs^{2} + 1)^{3}} + \frac{24tA^{2}B^{2}s^{2}{e}^{(ts)}}{(Bs + As + ABs^{2} + 1)^{3}} - \frac{6tAB{e}^{(ts)}}{(Bs + As + ABs^{2} + 1)^{2}} - \frac{6t^{2}ABs{e}^{(ts)}}{(Bs + As + ABs^{2} + 1)^{2}} + \frac{t^{3}{e}^{(ts)}}{(Bs + As + ABs^{2} + 1)}\\ \end{split}\end{equation} \]