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 4 derivative of x is calculated.
Note that variables are case sensitive.
\[ \begin{equation}\begin{split}[1/1]Find\ the\ 4th\ derivative\ of\ function\ {arctan(x)}^{x}\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( {arctan(x)}^{x}\right)}{dx}\\=&({arctan(x)}^{x}((1)ln(arctan(x)) + \frac{(x)((\frac{(1)}{(1 + (x)^{2})}))}{(arctan(x))}))\\=&{arctan(x)}^{x}ln(arctan(x)) + \frac{x{arctan(x)}^{x}}{(x^{2} + 1)arctan(x)}\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( {arctan(x)}^{x}ln(arctan(x)) + \frac{x{arctan(x)}^{x}}{(x^{2} + 1)arctan(x)}\right)}{dx}\\=&({arctan(x)}^{x}((1)ln(arctan(x)) + \frac{(x)((\frac{(1)}{(1 + (x)^{2})}))}{(arctan(x))}))ln(arctan(x)) + \frac{{arctan(x)}^{x}(\frac{(1)}{(1 + (x)^{2})})}{(arctan(x))} + \frac{(\frac{-(2x + 0)}{(x^{2} + 1)^{2}})x{arctan(x)}^{x}}{arctan(x)} + \frac{{arctan(x)}^{x}}{(x^{2} + 1)arctan(x)} + \frac{x({arctan(x)}^{x}((1)ln(arctan(x)) + \frac{(x)((\frac{(1)}{(1 + (x)^{2})}))}{(arctan(x))}))}{(x^{2} + 1)arctan(x)} + \frac{x{arctan(x)}^{x}(\frac{-(1)}{arctan^{2}(x)(1 + (x)^{2})})}{(x^{2} + 1)}\\=&{arctan(x)}^{x}ln^{2}(arctan(x)) + \frac{2x{arctan(x)}^{x}ln(arctan(x))}{(x^{2} + 1)arctan(x)} + \frac{2{arctan(x)}^{x}}{(x^{2} + 1)arctan(x)} - \frac{2x^{2}{arctan(x)}^{x}}{(x^{2} + 1)^{2}arctan(x)} + \frac{x^{2}{arctan(x)}^{x}}{(x^{2} + 1)^{2}arctan^{2}(x)} - \frac{x{arctan(x)}^{x}}{(x^{2} + 1)^{2}arctan^{2}(x)}\\\\ &\color{blue}{The\ third\ derivative\ of\ function:} \\&\frac{d\left( {arctan(x)}^{x}ln^{2}(arctan(x)) + \frac{2x{arctan(x)}^{x}ln(arctan(x))}{(x^{2} + 1)arctan(x)} + \frac{2{arctan(x)}^{x}}{(x^{2} + 1)arctan(x)} - \frac{2x^{2}{arctan(x)}^{x}}{(x^{2} + 1)^{2}arctan(x)} + \frac{x^{2}{arctan(x)}^{x}}{(x^{2} + 1)^{2}arctan^{2}(x)} - \frac{x{arctan(x)}^{x}}{(x^{2} + 1)^{2}arctan^{2}(x)}\right)}{dx}\\=&({arctan(x)}^{x}((1)ln(arctan(x)) + \frac{(x)((\frac{(1)}{(1 + (x)^{2})}))}{(arctan(x))}))ln^{2}(arctan(x)) + \frac{{arctan(x)}^{x}*2ln(arctan(x))(\frac{(1)}{(1 + (x)^{2})})}{(arctan(x))} + \frac{2(\frac{-(2x + 0)}{(x^{2} + 1)^{2}})x{arctan(x)}^{x}ln(arctan(x))}{arctan(x)} + \frac{2{arctan(x)}^{x}ln(arctan(x))}{(x^{2} + 1)arctan(x)} + \frac{2x({arctan(x)}^{x}((1)ln(arctan(x)) + \frac{(x)((\frac{(1)}{(1 + (x)^{2})}))}{(arctan(x))}))ln(arctan(x))}{(x^{2} + 1)arctan(x)} + \frac{2x{arctan(x)}^{x}(\frac{(1)}{(1 + (x)^{2})})}{(x^{2} + 1)(arctan(x))arctan(x)} + \frac{2x{arctan(x)}^{x}ln(arctan(x))(\frac{-(1)}{arctan^{2}(x)(1 + (x)^{2})})}{(x^{2} + 1)} + \frac{2(\frac{-(2x + 0)}{(x^{2} + 1)^{2}}){arctan(x)}^{x}}{arctan(x)} + \frac{2({arctan(x)}^{x}((1)ln(arctan(x)) + \frac{(x)((\frac{(1)}{(1 + (x)^{2})}))}{(arctan(x))}))}{(x^{2} + 1)arctan(x)} + \frac{2{arctan(x)}^{x}(\frac{-(1)}{arctan^{2}(x)(1 + (x)^{2})})}{(x^{2} + 1)} - \frac{2(\frac{-2(2x + 0)}{(x^{2} + 1)^{3}})x^{2}{arctan(x)}^{x}}{arctan(x)} - \frac{2*2x{arctan(x)}^{x}}{(x^{2} + 1)^{2}arctan(x)} - \frac{2x^{2}({arctan(x)}^{x}((1)ln(arctan(x)) + \frac{(x)((\frac{(1)}{(1 + (x)^{2})}))}{(arctan(x))}))}{(x^{2} + 1)^{2}arctan(x)} - \frac{2x^{2}{arctan(x)}^{x}(\frac{-(1)}{arctan^{2}(x)(1 + (x)^{2})})}{(x^{2} + 1)^{2}} + \frac{(\frac{-2(2x + 0)}{(x^{2} + 1)^{3}})x^{2}{arctan(x)}^{x}}{arctan^{2}(x)} + \frac{2x{arctan(x)}^{x}}{(x^{2} + 1)^{2}arctan^{2}(x)} + \frac{x^{2}({arctan(x)}^{x}((1)ln(arctan(x)) + \frac{(x)((\frac{(1)}{(1 + (x)^{2})}))}{(arctan(x))}))}{(x^{2} + 1)^{2}arctan^{2}(x)} + \frac{x^{2}{arctan(x)}^{x}(\frac{-2(1)}{arctan^{3}(x)(1 + (x)^{2})})}{(x^{2} + 1)^{2}} - \frac{(\frac{-2(2x + 0)}{(x^{2} + 1)^{3}})x{arctan(x)}^{x}}{arctan^{2}(x)} - \frac{{arctan(x)}^{x}}{(x^{2} + 1)^{2}arctan^{2}(x)} - \frac{x({arctan(x)}^{x}((1)ln(arctan(x)) + \frac{(x)((\frac{(1)}{(1 + (x)^{2})}))}{(arctan(x))}))}{(x^{2} + 1)^{2}arctan^{2}(x)} - \frac{x{arctan(x)}^{x}(\frac{-2(1)}{arctan^{3}(x)(1 + (x)^{2})})}{(x^{2} + 1)^{2}}\\=&\frac{6{arctan(x)}^{x}ln(arctan(x))}{(x^{2} + 1)arctan(x)} - \frac{6x^{2}{arctan(x)}^{x}ln(arctan(x))}{(x^{2} + 1)^{2}arctan(x)} + {arctan(x)}^{x}ln^{3}(arctan(x)) + \frac{3x{arctan(x)}^{x}ln^{2}(arctan(x))}{(x^{2} + 1)arctan(x)} + \frac{3x^{2}{arctan(x)}^{x}ln(arctan(x))}{(x^{2} + 1)^{2}arctan^{2}(x)} + \frac{6x{arctan(x)}^{x}}{(x^{2} + 1)^{2}arctan^{2}(x)} - \frac{3x{arctan(x)}^{x}ln(arctan(x))}{(x^{2} + 1)^{2}arctan^{2}(x)} - \frac{8x{arctan(x)}^{x}}{(x^{2} + 1)^{2}arctan(x)} - \frac{3{arctan(x)}^{x}}{(x^{2} + 1)^{2}arctan^{2}(x)} + \frac{8x^{3}{arctan(x)}^{x}}{(x^{2} + 1)^{3}arctan(x)} - \frac{6x^{3}{arctan(x)}^{x}}{(x^{2} + 1)^{3}arctan^{2}(x)} + \frac{6x^{2}{arctan(x)}^{x}}{(x^{2} + 1)^{3}arctan^{2}(x)} + \frac{x^{3}{arctan(x)}^{x}}{(x^{2} + 1)^{3}arctan^{3}(x)} - \frac{3x^{2}{arctan(x)}^{x}}{(x^{2} + 1)^{3}arctan^{3}(x)} + \frac{2x{arctan(x)}^{x}}{(x^{2} + 1)^{3}arctan^{3}(x)}\\\\ &\color{blue}{The\ 4th\ derivative\ of\ function:} \\&\frac{d\left( \frac{6{arctan(x)}^{x}ln(arctan(x))}{(x^{2} + 1)arctan(x)} - \frac{6x^{2}{arctan(x)}^{x}ln(arctan(x))}{(x^{2} + 1)^{2}arctan(x)} + {arctan(x)}^{x}ln^{3}(arctan(x)) + \frac{3x{arctan(x)}^{x}ln^{2}(arctan(x))}{(x^{2} + 1)arctan(x)} + \frac{3x^{2}{arctan(x)}^{x}ln(arctan(x))}{(x^{2} + 1)^{2}arctan^{2}(x)} + \frac{6x{arctan(x)}^{x}}{(x^{2} + 1)^{2}arctan^{2}(x)} - \frac{3x{arctan(x)}^{x}ln(arctan(x))}{(x^{2} + 1)^{2}arctan^{2}(x)} - \frac{8x{arctan(x)}^{x}}{(x^{2} + 1)^{2}arctan(x)} - \frac{3{arctan(x)}^{x}}{(x^{2} + 1)^{2}arctan^{2}(x)} + \frac{8x^{3}{arctan(x)}^{x}}{(x^{2} + 1)^{3}arctan(x)} - \frac{6x^{3}{arctan(x)}^{x}}{(x^{2} + 1)^{3}arctan^{2}(x)} + \frac{6x^{2}{arctan(x)}^{x}}{(x^{2} + 1)^{3}arctan^{2}(x)} + \frac{x^{3}{arctan(x)}^{x}}{(x^{2} + 1)^{3}arctan^{3}(x)} - \frac{3x^{2}{arctan(x)}^{x}}{(x^{2} + 1)^{3}arctan^{3}(x)} + \frac{2x{arctan(x)}^{x}}{(x^{2} + 1)^{3}arctan^{3}(x)}\right)}{dx}\\=&\frac{6(\frac{-(2x + 0)}{(x^{2} + 1)^{2}}){arctan(x)}^{x}ln(arctan(x))}{arctan(x)} + \frac{6({arctan(x)}^{x}((1)ln(arctan(x)) + \frac{(x)((\frac{(1)}{(1 + (x)^{2})}))}{(arctan(x))}))ln(arctan(x))}{(x^{2} + 1)arctan(x)} + \frac{6{arctan(x)}^{x}(\frac{(1)}{(1 + (x)^{2})})}{(x^{2} + 1)(arctan(x))arctan(x)} + \frac{6{arctan(x)}^{x}ln(arctan(x))(\frac{-(1)}{arctan^{2}(x)(1 + (x)^{2})})}{(x^{2} + 1)} - \frac{6(\frac{-2(2x + 0)}{(x^{2} + 1)^{3}})x^{2}{arctan(x)}^{x}ln(arctan(x))}{arctan(x)} - \frac{6*2x{arctan(x)}^{x}ln(arctan(x))}{(x^{2} + 1)^{2}arctan(x)} - \frac{6x^{2}({arctan(x)}^{x}((1)ln(arctan(x)) + \frac{(x)((\frac{(1)}{(1 + (x)^{2})}))}{(arctan(x))}))ln(arctan(x))}{(x^{2} + 1)^{2}arctan(x)} - \frac{6x^{2}{arctan(x)}^{x}(\frac{(1)}{(1 + (x)^{2})})}{(x^{2} + 1)^{2}(arctan(x))arctan(x)} - \frac{6x^{2}{arctan(x)}^{x}ln(arctan(x))(\frac{-(1)}{arctan^{2}(x)(1 + (x)^{2})})}{(x^{2} + 1)^{2}} + ({arctan(x)}^{x}((1)ln(arctan(x)) + \frac{(x)((\frac{(1)}{(1 + (x)^{2})}))}{(arctan(x))}))ln^{3}(arctan(x)) + \frac{{arctan(x)}^{x}*3ln^{2}(arctan(x))(\frac{(1)}{(1 + (x)^{2})})}{(arctan(x))} + \frac{3(\frac{-(2x + 0)}{(x^{2} + 1)^{2}})x{arctan(x)}^{x}ln^{2}(arctan(x))}{arctan(x)} + \frac{3{arctan(x)}^{x}ln^{2}(arctan(x))}{(x^{2} + 1)arctan(x)} + \frac{3x({arctan(x)}^{x}((1)ln(arctan(x)) + \frac{(x)((\frac{(1)}{(1 + (x)^{2})}))}{(arctan(x))}))ln^{2}(arctan(x))}{(x^{2} + 1)arctan(x)} + \frac{3x{arctan(x)}^{x}*2ln(arctan(x))(\frac{(1)}{(1 + (x)^{2})})}{(x^{2} + 1)(arctan(x))arctan(x)} + \frac{3x{arctan(x)}^{x}ln^{2}(arctan(x))(\frac{-(1)}{arctan^{2}(x)(1 + (x)^{2})})}{(x^{2} + 1)} + \frac{3(\frac{-2(2x + 0)}{(x^{2} + 1)^{3}})x^{2}{arctan(x)}^{x}ln(arctan(x))}{arctan^{2}(x)} + \frac{3*2x{arctan(x)}^{x}ln(arctan(x))}{(x^{2} + 1)^{2}arctan^{2}(x)} + \frac{3x^{2}({arctan(x)}^{x}((1)ln(arctan(x)) + \frac{(x)((\frac{(1)}{(1 + (x)^{2})}))}{(arctan(x))}))ln(arctan(x))}{(x^{2} + 1)^{2}arctan^{2}(x)} + \frac{3x^{2}{arctan(x)}^{x}(\frac{(1)}{(1 + (x)^{2})})}{(x^{2} + 1)^{2}(arctan(x))arctan^{2}(x)} + \frac{3x^{2}{arctan(x)}^{x}ln(arctan(x))(\frac{-2(1)}{arctan^{3}(x)(1 + (x)^{2})})}{(x^{2} + 1)^{2}} + \frac{6(\frac{-2(2x + 0)}{(x^{2} + 1)^{3}})x{arctan(x)}^{x}}{arctan^{2}(x)} + \frac{6{arctan(x)}^{x}}{(x^{2} + 1)^{2}arctan^{2}(x)} + \frac{6x({arctan(x)}^{x}((1)ln(arctan(x)) + \frac{(x)((\frac{(1)}{(1 + (x)^{2})}))}{(arctan(x))}))}{(x^{2} + 1)^{2}arctan^{2}(x)} + \frac{6x{arctan(x)}^{x}(\frac{-2(1)}{arctan^{3}(x)(1 + (x)^{2})})}{(x^{2} + 1)^{2}} - \frac{3(\frac{-2(2x + 0)}{(x^{2} + 1)^{3}})x{arctan(x)}^{x}ln(arctan(x))}{arctan^{2}(x)} - \frac{3{arctan(x)}^{x}ln(arctan(x))}{(x^{2} + 1)^{2}arctan^{2}(x)} - \frac{3x({arctan(x)}^{x}((1)ln(arctan(x)) + \frac{(x)((\frac{(1)}{(1 + (x)^{2})}))}{(arctan(x))}))ln(arctan(x))}{(x^{2} + 1)^{2}arctan^{2}(x)} - \frac{3x{arctan(x)}^{x}(\frac{(1)}{(1 + (x)^{2})})}{(x^{2} + 1)^{2}(arctan(x))arctan^{2}(x)} - \frac{3x{arctan(x)}^{x}ln(arctan(x))(\frac{-2(1)}{arctan^{3}(x)(1 + (x)^{2})})}{(x^{2} + 1)^{2}} - \frac{8(\frac{-2(2x + 0)}{(x^{2} + 1)^{3}})x{arctan(x)}^{x}}{arctan(x)} - \frac{8{arctan(x)}^{x}}{(x^{2} + 1)^{2}arctan(x)} - \frac{8x({arctan(x)}^{x}((1)ln(arctan(x)) + \frac{(x)((\frac{(1)}{(1 + (x)^{2})}))}{(arctan(x))}))}{(x^{2} + 1)^{2}arctan(x)} - \frac{8x{arctan(x)}^{x}(\frac{-(1)}{arctan^{2}(x)(1 + (x)^{2})})}{(x^{2} + 1)^{2}} - \frac{3(\frac{-2(2x + 0)}{(x^{2} + 1)^{3}}){arctan(x)}^{x}}{arctan^{2}(x)} - \frac{3({arctan(x)}^{x}((1)ln(arctan(x)) + \frac{(x)((\frac{(1)}{(1 + (x)^{2})}))}{(arctan(x))}))}{(x^{2} + 1)^{2}arctan^{2}(x)} - \frac{3{arctan(x)}^{x}(\frac{-2(1)}{arctan^{3}(x)(1 + (x)^{2})})}{(x^{2} + 1)^{2}} + \frac{8(\frac{-3(2x + 0)}{(x^{2} + 1)^{4}})x^{3}{arctan(x)}^{x}}{arctan(x)} + \frac{8*3x^{2}{arctan(x)}^{x}}{(x^{2} + 1)^{3}arctan(x)} + \frac{8x^{3}({arctan(x)}^{x}((1)ln(arctan(x)) + \frac{(x)((\frac{(1)}{(1 + (x)^{2})}))}{(arctan(x))}))}{(x^{2} + 1)^{3}arctan(x)} + \frac{8x^{3}{arctan(x)}^{x}(\frac{-(1)}{arctan^{2}(x)(1 + (x)^{2})})}{(x^{2} + 1)^{3}} - \frac{6(\frac{-3(2x + 0)}{(x^{2} + 1)^{4}})x^{3}{arctan(x)}^{x}}{arctan^{2}(x)} - \frac{6*3x^{2}{arctan(x)}^{x}}{(x^{2} + 1)^{3}arctan^{2}(x)} - \frac{6x^{3}({arctan(x)}^{x}((1)ln(arctan(x)) + \frac{(x)((\frac{(1)}{(1 + (x)^{2})}))}{(arctan(x))}))}{(x^{2} + 1)^{3}arctan^{2}(x)} - \frac{6x^{3}{arctan(x)}^{x}(\frac{-2(1)}{arctan^{3}(x)(1 + (x)^{2})})}{(x^{2} + 1)^{3}} + \frac{6(\frac{-3(2x + 0)}{(x^{2} + 1)^{4}})x^{2}{arctan(x)}^{x}}{arctan^{2}(x)} + \frac{6*2x{arctan(x)}^{x}}{(x^{2} + 1)^{3}arctan^{2}(x)} + \frac{6x^{2}({arctan(x)}^{x}((1)ln(arctan(x)) + \frac{(x)((\frac{(1)}{(1 + (x)^{2})}))}{(arctan(x))}))}{(x^{2} + 1)^{3}arctan^{2}(x)} + \frac{6x^{2}{arctan(x)}^{x}(\frac{-2(1)}{arctan^{3}(x)(1 + (x)^{2})})}{(x^{2} + 1)^{3}} + \frac{(\frac{-3(2x + 0)}{(x^{2} + 1)^{4}})x^{3}{arctan(x)}^{x}}{arctan^{3}(x)} + \frac{3x^{2}{arctan(x)}^{x}}{(x^{2} + 1)^{3}arctan^{3}(x)} + \frac{x^{3}({arctan(x)}^{x}((1)ln(arctan(x)) + \frac{(x)((\frac{(1)}{(1 + (x)^{2})}))}{(arctan(x))}))}{(x^{2} + 1)^{3}arctan^{3}(x)} + \frac{x^{3}{arctan(x)}^{x}(\frac{-3(1)}{arctan^{4}(x)(1 + (x)^{2})})}{(x^{2} + 1)^{3}} - \frac{3(\frac{-3(2x + 0)}{(x^{2} + 1)^{4}})x^{2}{arctan(x)}^{x}}{arctan^{3}(x)} - \frac{3*2x{arctan(x)}^{x}}{(x^{2} + 1)^{3}arctan^{3}(x)} - \frac{3x^{2}({arctan(x)}^{x}((1)ln(arctan(x)) + \frac{(x)((\frac{(1)}{(1 + (x)^{2})}))}{(arctan(x))}))}{(x^{2} + 1)^{3}arctan^{3}(x)} - \frac{3x^{2}{arctan(x)}^{x}(\frac{-3(1)}{arctan^{4}(x)(1 + (x)^{2})})}{(x^{2} + 1)^{3}} + \frac{2(\frac{-3(2x + 0)}{(x^{2} + 1)^{4}})x{arctan(x)}^{x}}{arctan^{3}(x)} + \frac{2{arctan(x)}^{x}}{(x^{2} + 1)^{3}arctan^{3}(x)} + \frac{2x({arctan(x)}^{x}((1)ln(arctan(x)) + \frac{(x)((\frac{(1)}{(1 + (x)^{2})}))}{(arctan(x))}))}{(x^{2} + 1)^{3}arctan^{3}(x)} + \frac{2x{arctan(x)}^{x}(\frac{-3(1)}{arctan^{4}(x)(1 + (x)^{2})})}{(x^{2} + 1)^{3}}\\=&\frac{-32x{arctan(x)}^{x}ln(arctan(x))}{(x^{2} + 1)^{2}arctan(x)} + \frac{12{arctan(x)}^{x}ln^{2}(arctan(x))}{(x^{2} + 1)arctan(x)} + \frac{24x{arctan(x)}^{x}ln(arctan(x))}{(x^{2} + 1)^{2}arctan^{2}(x)} + \frac{12{arctan(x)}^{x}}{(x^{2} + 1)^{2}arctan^{2}(x)} - \frac{12{arctan(x)}^{x}ln(arctan(x))}{(x^{2} + 1)^{2}arctan^{2}(x)} + \frac{32x^{3}{arctan(x)}^{x}ln(arctan(x))}{(x^{2} + 1)^{3}arctan(x)} - \frac{12x^{2}{arctan(x)}^{x}ln^{2}(arctan(x))}{(x^{2} + 1)^{2}arctan(x)} - \frac{24x^{3}{arctan(x)}^{x}ln(arctan(x))}{(x^{2} + 1)^{3}arctan^{2}(x)} - \frac{56x^{2}{arctan(x)}^{x}}{(x^{2} + 1)^{3}arctan^{2}(x)} + \frac{24x^{2}{arctan(x)}^{x}ln(arctan(x))}{(x^{2} + 1)^{3}arctan^{2}(x)} + \frac{4x{arctan(x)}^{x}ln^{3}(arctan(x))}{(x^{2} + 1)arctan(x)} + \frac{6x^{2}{arctan(x)}^{x}ln^{2}(arctan(x))}{(x^{2} + 1)^{2}arctan^{2}(x)} - \frac{6x{arctan(x)}^{x}ln^{2}(arctan(x))}{(x^{2} + 1)^{2}arctan^{2}(x)} + \frac{4x^{3}{arctan(x)}^{x}ln(arctan(x))}{(x^{2} + 1)^{3}arctan^{3}(x)} + \frac{12x^{2}{arctan(x)}^{x}}{(x^{2} + 1)^{3}arctan^{3}(x)} - \frac{12x^{2}{arctan(x)}^{x}ln(arctan(x))}{(x^{2} + 1)^{3}arctan^{3}(x)} - \frac{24x{arctan(x)}^{x}}{(x^{2} + 1)^{3}arctan^{3}(x)} + {arctan(x)}^{x}ln^{4}(arctan(x)) + \frac{8x{arctan(x)}^{x}ln(arctan(x))}{(x^{2} + 1)^{3}arctan^{3}(x)} + \frac{56x^{2}{arctan(x)}^{x}}{(x^{2} + 1)^{3}arctan(x)} - \frac{8{arctan(x)}^{x}}{(x^{2} + 1)^{2}arctan(x)} + \frac{32x{arctan(x)}^{x}}{(x^{2} + 1)^{3}arctan^{2}(x)} + \frac{8{arctan(x)}^{x}}{(x^{2} + 1)^{3}arctan^{3}(x)} - \frac{48x^{4}{arctan(x)}^{x}}{(x^{2} + 1)^{4}arctan(x)} + \frac{44x^{4}{arctan(x)}^{x}}{(x^{2} + 1)^{4}arctan^{2}(x)} - \frac{44x^{3}{arctan(x)}^{x}}{(x^{2} + 1)^{4}arctan^{2}(x)} - \frac{12x^{4}{arctan(x)}^{x}}{(x^{2} + 1)^{4}arctan^{3}(x)} + \frac{36x^{3}{arctan(x)}^{x}}{(x^{2} + 1)^{4}arctan^{3}(x)} - \frac{24x^{2}{arctan(x)}^{x}}{(x^{2} + 1)^{4}arctan^{3}(x)} + \frac{x^{4}{arctan(x)}^{x}}{(x^{2} + 1)^{4}arctan^{4}(x)} - \frac{6x^{3}{arctan(x)}^{x}}{(x^{2} + 1)^{4}arctan^{4}(x)} + \frac{11x^{2}{arctan(x)}^{x}}{(x^{2} + 1)^{4}arctan^{4}(x)} - \frac{6x{arctan(x)}^{x}}{(x^{2} + 1)^{4}arctan^{4}(x)}\\ \end{split}\end{equation} \]
Note that variables are case sensitive.
\[ \begin{equation}\begin{split}[1/1]Find\ the\ 4th\ derivative\ of\ function\ {arctan(x)}^{x}\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( {arctan(x)}^{x}\right)}{dx}\\=&({arctan(x)}^{x}((1)ln(arctan(x)) + \frac{(x)((\frac{(1)}{(1 + (x)^{2})}))}{(arctan(x))}))\\=&{arctan(x)}^{x}ln(arctan(x)) + \frac{x{arctan(x)}^{x}}{(x^{2} + 1)arctan(x)}\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( {arctan(x)}^{x}ln(arctan(x)) + \frac{x{arctan(x)}^{x}}{(x^{2} + 1)arctan(x)}\right)}{dx}\\=&({arctan(x)}^{x}((1)ln(arctan(x)) + \frac{(x)((\frac{(1)}{(1 + (x)^{2})}))}{(arctan(x))}))ln(arctan(x)) + \frac{{arctan(x)}^{x}(\frac{(1)}{(1 + (x)^{2})})}{(arctan(x))} + \frac{(\frac{-(2x + 0)}{(x^{2} + 1)^{2}})x{arctan(x)}^{x}}{arctan(x)} + \frac{{arctan(x)}^{x}}{(x^{2} + 1)arctan(x)} + \frac{x({arctan(x)}^{x}((1)ln(arctan(x)) + \frac{(x)((\frac{(1)}{(1 + (x)^{2})}))}{(arctan(x))}))}{(x^{2} + 1)arctan(x)} + \frac{x{arctan(x)}^{x}(\frac{-(1)}{arctan^{2}(x)(1 + (x)^{2})})}{(x^{2} + 1)}\\=&{arctan(x)}^{x}ln^{2}(arctan(x)) + \frac{2x{arctan(x)}^{x}ln(arctan(x))}{(x^{2} + 1)arctan(x)} + \frac{2{arctan(x)}^{x}}{(x^{2} + 1)arctan(x)} - \frac{2x^{2}{arctan(x)}^{x}}{(x^{2} + 1)^{2}arctan(x)} + \frac{x^{2}{arctan(x)}^{x}}{(x^{2} + 1)^{2}arctan^{2}(x)} - \frac{x{arctan(x)}^{x}}{(x^{2} + 1)^{2}arctan^{2}(x)}\\\\ &\color{blue}{The\ third\ derivative\ of\ function:} \\&\frac{d\left( {arctan(x)}^{x}ln^{2}(arctan(x)) + \frac{2x{arctan(x)}^{x}ln(arctan(x))}{(x^{2} + 1)arctan(x)} + \frac{2{arctan(x)}^{x}}{(x^{2} + 1)arctan(x)} - \frac{2x^{2}{arctan(x)}^{x}}{(x^{2} + 1)^{2}arctan(x)} + \frac{x^{2}{arctan(x)}^{x}}{(x^{2} + 1)^{2}arctan^{2}(x)} - \frac{x{arctan(x)}^{x}}{(x^{2} + 1)^{2}arctan^{2}(x)}\right)}{dx}\\=&({arctan(x)}^{x}((1)ln(arctan(x)) + \frac{(x)((\frac{(1)}{(1 + (x)^{2})}))}{(arctan(x))}))ln^{2}(arctan(x)) + \frac{{arctan(x)}^{x}*2ln(arctan(x))(\frac{(1)}{(1 + (x)^{2})})}{(arctan(x))} + \frac{2(\frac{-(2x + 0)}{(x^{2} + 1)^{2}})x{arctan(x)}^{x}ln(arctan(x))}{arctan(x)} + \frac{2{arctan(x)}^{x}ln(arctan(x))}{(x^{2} + 1)arctan(x)} + \frac{2x({arctan(x)}^{x}((1)ln(arctan(x)) + \frac{(x)((\frac{(1)}{(1 + (x)^{2})}))}{(arctan(x))}))ln(arctan(x))}{(x^{2} + 1)arctan(x)} + \frac{2x{arctan(x)}^{x}(\frac{(1)}{(1 + (x)^{2})})}{(x^{2} + 1)(arctan(x))arctan(x)} + \frac{2x{arctan(x)}^{x}ln(arctan(x))(\frac{-(1)}{arctan^{2}(x)(1 + (x)^{2})})}{(x^{2} + 1)} + \frac{2(\frac{-(2x + 0)}{(x^{2} + 1)^{2}}){arctan(x)}^{x}}{arctan(x)} + \frac{2({arctan(x)}^{x}((1)ln(arctan(x)) + \frac{(x)((\frac{(1)}{(1 + (x)^{2})}))}{(arctan(x))}))}{(x^{2} + 1)arctan(x)} + \frac{2{arctan(x)}^{x}(\frac{-(1)}{arctan^{2}(x)(1 + (x)^{2})})}{(x^{2} + 1)} - \frac{2(\frac{-2(2x + 0)}{(x^{2} + 1)^{3}})x^{2}{arctan(x)}^{x}}{arctan(x)} - \frac{2*2x{arctan(x)}^{x}}{(x^{2} + 1)^{2}arctan(x)} - \frac{2x^{2}({arctan(x)}^{x}((1)ln(arctan(x)) + \frac{(x)((\frac{(1)}{(1 + (x)^{2})}))}{(arctan(x))}))}{(x^{2} + 1)^{2}arctan(x)} - \frac{2x^{2}{arctan(x)}^{x}(\frac{-(1)}{arctan^{2}(x)(1 + (x)^{2})})}{(x^{2} + 1)^{2}} + \frac{(\frac{-2(2x + 0)}{(x^{2} + 1)^{3}})x^{2}{arctan(x)}^{x}}{arctan^{2}(x)} + \frac{2x{arctan(x)}^{x}}{(x^{2} + 1)^{2}arctan^{2}(x)} + \frac{x^{2}({arctan(x)}^{x}((1)ln(arctan(x)) + \frac{(x)((\frac{(1)}{(1 + (x)^{2})}))}{(arctan(x))}))}{(x^{2} + 1)^{2}arctan^{2}(x)} + \frac{x^{2}{arctan(x)}^{x}(\frac{-2(1)}{arctan^{3}(x)(1 + (x)^{2})})}{(x^{2} + 1)^{2}} - \frac{(\frac{-2(2x + 0)}{(x^{2} + 1)^{3}})x{arctan(x)}^{x}}{arctan^{2}(x)} - \frac{{arctan(x)}^{x}}{(x^{2} + 1)^{2}arctan^{2}(x)} - \frac{x({arctan(x)}^{x}((1)ln(arctan(x)) + \frac{(x)((\frac{(1)}{(1 + (x)^{2})}))}{(arctan(x))}))}{(x^{2} + 1)^{2}arctan^{2}(x)} - \frac{x{arctan(x)}^{x}(\frac{-2(1)}{arctan^{3}(x)(1 + (x)^{2})})}{(x^{2} + 1)^{2}}\\=&\frac{6{arctan(x)}^{x}ln(arctan(x))}{(x^{2} + 1)arctan(x)} - \frac{6x^{2}{arctan(x)}^{x}ln(arctan(x))}{(x^{2} + 1)^{2}arctan(x)} + {arctan(x)}^{x}ln^{3}(arctan(x)) + \frac{3x{arctan(x)}^{x}ln^{2}(arctan(x))}{(x^{2} + 1)arctan(x)} + \frac{3x^{2}{arctan(x)}^{x}ln(arctan(x))}{(x^{2} + 1)^{2}arctan^{2}(x)} + \frac{6x{arctan(x)}^{x}}{(x^{2} + 1)^{2}arctan^{2}(x)} - \frac{3x{arctan(x)}^{x}ln(arctan(x))}{(x^{2} + 1)^{2}arctan^{2}(x)} - \frac{8x{arctan(x)}^{x}}{(x^{2} + 1)^{2}arctan(x)} - \frac{3{arctan(x)}^{x}}{(x^{2} + 1)^{2}arctan^{2}(x)} + \frac{8x^{3}{arctan(x)}^{x}}{(x^{2} + 1)^{3}arctan(x)} - \frac{6x^{3}{arctan(x)}^{x}}{(x^{2} + 1)^{3}arctan^{2}(x)} + \frac{6x^{2}{arctan(x)}^{x}}{(x^{2} + 1)^{3}arctan^{2}(x)} + \frac{x^{3}{arctan(x)}^{x}}{(x^{2} + 1)^{3}arctan^{3}(x)} - \frac{3x^{2}{arctan(x)}^{x}}{(x^{2} + 1)^{3}arctan^{3}(x)} + \frac{2x{arctan(x)}^{x}}{(x^{2} + 1)^{3}arctan^{3}(x)}\\\\ &\color{blue}{The\ 4th\ derivative\ of\ function:} \\&\frac{d\left( \frac{6{arctan(x)}^{x}ln(arctan(x))}{(x^{2} + 1)arctan(x)} - \frac{6x^{2}{arctan(x)}^{x}ln(arctan(x))}{(x^{2} + 1)^{2}arctan(x)} + {arctan(x)}^{x}ln^{3}(arctan(x)) + \frac{3x{arctan(x)}^{x}ln^{2}(arctan(x))}{(x^{2} + 1)arctan(x)} + \frac{3x^{2}{arctan(x)}^{x}ln(arctan(x))}{(x^{2} + 1)^{2}arctan^{2}(x)} + \frac{6x{arctan(x)}^{x}}{(x^{2} + 1)^{2}arctan^{2}(x)} - \frac{3x{arctan(x)}^{x}ln(arctan(x))}{(x^{2} + 1)^{2}arctan^{2}(x)} - \frac{8x{arctan(x)}^{x}}{(x^{2} + 1)^{2}arctan(x)} - \frac{3{arctan(x)}^{x}}{(x^{2} + 1)^{2}arctan^{2}(x)} + \frac{8x^{3}{arctan(x)}^{x}}{(x^{2} + 1)^{3}arctan(x)} - \frac{6x^{3}{arctan(x)}^{x}}{(x^{2} + 1)^{3}arctan^{2}(x)} + \frac{6x^{2}{arctan(x)}^{x}}{(x^{2} + 1)^{3}arctan^{2}(x)} + \frac{x^{3}{arctan(x)}^{x}}{(x^{2} + 1)^{3}arctan^{3}(x)} - \frac{3x^{2}{arctan(x)}^{x}}{(x^{2} + 1)^{3}arctan^{3}(x)} + \frac{2x{arctan(x)}^{x}}{(x^{2} + 1)^{3}arctan^{3}(x)}\right)}{dx}\\=&\frac{6(\frac{-(2x + 0)}{(x^{2} + 1)^{2}}){arctan(x)}^{x}ln(arctan(x))}{arctan(x)} + \frac{6({arctan(x)}^{x}((1)ln(arctan(x)) + \frac{(x)((\frac{(1)}{(1 + (x)^{2})}))}{(arctan(x))}))ln(arctan(x))}{(x^{2} + 1)arctan(x)} + \frac{6{arctan(x)}^{x}(\frac{(1)}{(1 + (x)^{2})})}{(x^{2} + 1)(arctan(x))arctan(x)} + \frac{6{arctan(x)}^{x}ln(arctan(x))(\frac{-(1)}{arctan^{2}(x)(1 + (x)^{2})})}{(x^{2} + 1)} - \frac{6(\frac{-2(2x + 0)}{(x^{2} + 1)^{3}})x^{2}{arctan(x)}^{x}ln(arctan(x))}{arctan(x)} - \frac{6*2x{arctan(x)}^{x}ln(arctan(x))}{(x^{2} + 1)^{2}arctan(x)} - \frac{6x^{2}({arctan(x)}^{x}((1)ln(arctan(x)) + \frac{(x)((\frac{(1)}{(1 + (x)^{2})}))}{(arctan(x))}))ln(arctan(x))}{(x^{2} + 1)^{2}arctan(x)} - \frac{6x^{2}{arctan(x)}^{x}(\frac{(1)}{(1 + (x)^{2})})}{(x^{2} + 1)^{2}(arctan(x))arctan(x)} - \frac{6x^{2}{arctan(x)}^{x}ln(arctan(x))(\frac{-(1)}{arctan^{2}(x)(1 + (x)^{2})})}{(x^{2} + 1)^{2}} + ({arctan(x)}^{x}((1)ln(arctan(x)) + \frac{(x)((\frac{(1)}{(1 + (x)^{2})}))}{(arctan(x))}))ln^{3}(arctan(x)) + \frac{{arctan(x)}^{x}*3ln^{2}(arctan(x))(\frac{(1)}{(1 + (x)^{2})})}{(arctan(x))} + \frac{3(\frac{-(2x + 0)}{(x^{2} + 1)^{2}})x{arctan(x)}^{x}ln^{2}(arctan(x))}{arctan(x)} + \frac{3{arctan(x)}^{x}ln^{2}(arctan(x))}{(x^{2} + 1)arctan(x)} + \frac{3x({arctan(x)}^{x}((1)ln(arctan(x)) + \frac{(x)((\frac{(1)}{(1 + (x)^{2})}))}{(arctan(x))}))ln^{2}(arctan(x))}{(x^{2} + 1)arctan(x)} + \frac{3x{arctan(x)}^{x}*2ln(arctan(x))(\frac{(1)}{(1 + (x)^{2})})}{(x^{2} + 1)(arctan(x))arctan(x)} + \frac{3x{arctan(x)}^{x}ln^{2}(arctan(x))(\frac{-(1)}{arctan^{2}(x)(1 + (x)^{2})})}{(x^{2} + 1)} + \frac{3(\frac{-2(2x + 0)}{(x^{2} + 1)^{3}})x^{2}{arctan(x)}^{x}ln(arctan(x))}{arctan^{2}(x)} + \frac{3*2x{arctan(x)}^{x}ln(arctan(x))}{(x^{2} + 1)^{2}arctan^{2}(x)} + \frac{3x^{2}({arctan(x)}^{x}((1)ln(arctan(x)) + \frac{(x)((\frac{(1)}{(1 + (x)^{2})}))}{(arctan(x))}))ln(arctan(x))}{(x^{2} + 1)^{2}arctan^{2}(x)} + \frac{3x^{2}{arctan(x)}^{x}(\frac{(1)}{(1 + (x)^{2})})}{(x^{2} + 1)^{2}(arctan(x))arctan^{2}(x)} + \frac{3x^{2}{arctan(x)}^{x}ln(arctan(x))(\frac{-2(1)}{arctan^{3}(x)(1 + (x)^{2})})}{(x^{2} + 1)^{2}} + \frac{6(\frac{-2(2x + 0)}{(x^{2} + 1)^{3}})x{arctan(x)}^{x}}{arctan^{2}(x)} + \frac{6{arctan(x)}^{x}}{(x^{2} + 1)^{2}arctan^{2}(x)} + \frac{6x({arctan(x)}^{x}((1)ln(arctan(x)) + \frac{(x)((\frac{(1)}{(1 + (x)^{2})}))}{(arctan(x))}))}{(x^{2} + 1)^{2}arctan^{2}(x)} + \frac{6x{arctan(x)}^{x}(\frac{-2(1)}{arctan^{3}(x)(1 + (x)^{2})})}{(x^{2} + 1)^{2}} - \frac{3(\frac{-2(2x + 0)}{(x^{2} + 1)^{3}})x{arctan(x)}^{x}ln(arctan(x))}{arctan^{2}(x)} - \frac{3{arctan(x)}^{x}ln(arctan(x))}{(x^{2} + 1)^{2}arctan^{2}(x)} - \frac{3x({arctan(x)}^{x}((1)ln(arctan(x)) + \frac{(x)((\frac{(1)}{(1 + (x)^{2})}))}{(arctan(x))}))ln(arctan(x))}{(x^{2} + 1)^{2}arctan^{2}(x)} - \frac{3x{arctan(x)}^{x}(\frac{(1)}{(1 + (x)^{2})})}{(x^{2} + 1)^{2}(arctan(x))arctan^{2}(x)} - \frac{3x{arctan(x)}^{x}ln(arctan(x))(\frac{-2(1)}{arctan^{3}(x)(1 + (x)^{2})})}{(x^{2} + 1)^{2}} - \frac{8(\frac{-2(2x + 0)}{(x^{2} + 1)^{3}})x{arctan(x)}^{x}}{arctan(x)} - \frac{8{arctan(x)}^{x}}{(x^{2} + 1)^{2}arctan(x)} - \frac{8x({arctan(x)}^{x}((1)ln(arctan(x)) + \frac{(x)((\frac{(1)}{(1 + (x)^{2})}))}{(arctan(x))}))}{(x^{2} + 1)^{2}arctan(x)} - \frac{8x{arctan(x)}^{x}(\frac{-(1)}{arctan^{2}(x)(1 + (x)^{2})})}{(x^{2} + 1)^{2}} - \frac{3(\frac{-2(2x + 0)}{(x^{2} + 1)^{3}}){arctan(x)}^{x}}{arctan^{2}(x)} - \frac{3({arctan(x)}^{x}((1)ln(arctan(x)) + \frac{(x)((\frac{(1)}{(1 + (x)^{2})}))}{(arctan(x))}))}{(x^{2} + 1)^{2}arctan^{2}(x)} - \frac{3{arctan(x)}^{x}(\frac{-2(1)}{arctan^{3}(x)(1 + (x)^{2})})}{(x^{2} + 1)^{2}} + \frac{8(\frac{-3(2x + 0)}{(x^{2} + 1)^{4}})x^{3}{arctan(x)}^{x}}{arctan(x)} + \frac{8*3x^{2}{arctan(x)}^{x}}{(x^{2} + 1)^{3}arctan(x)} + \frac{8x^{3}({arctan(x)}^{x}((1)ln(arctan(x)) + \frac{(x)((\frac{(1)}{(1 + (x)^{2})}))}{(arctan(x))}))}{(x^{2} + 1)^{3}arctan(x)} + \frac{8x^{3}{arctan(x)}^{x}(\frac{-(1)}{arctan^{2}(x)(1 + (x)^{2})})}{(x^{2} + 1)^{3}} - \frac{6(\frac{-3(2x + 0)}{(x^{2} + 1)^{4}})x^{3}{arctan(x)}^{x}}{arctan^{2}(x)} - \frac{6*3x^{2}{arctan(x)}^{x}}{(x^{2} + 1)^{3}arctan^{2}(x)} - \frac{6x^{3}({arctan(x)}^{x}((1)ln(arctan(x)) + \frac{(x)((\frac{(1)}{(1 + (x)^{2})}))}{(arctan(x))}))}{(x^{2} + 1)^{3}arctan^{2}(x)} - \frac{6x^{3}{arctan(x)}^{x}(\frac{-2(1)}{arctan^{3}(x)(1 + (x)^{2})})}{(x^{2} + 1)^{3}} + \frac{6(\frac{-3(2x + 0)}{(x^{2} + 1)^{4}})x^{2}{arctan(x)}^{x}}{arctan^{2}(x)} + \frac{6*2x{arctan(x)}^{x}}{(x^{2} + 1)^{3}arctan^{2}(x)} + \frac{6x^{2}({arctan(x)}^{x}((1)ln(arctan(x)) + \frac{(x)((\frac{(1)}{(1 + (x)^{2})}))}{(arctan(x))}))}{(x^{2} + 1)^{3}arctan^{2}(x)} + \frac{6x^{2}{arctan(x)}^{x}(\frac{-2(1)}{arctan^{3}(x)(1 + (x)^{2})})}{(x^{2} + 1)^{3}} + \frac{(\frac{-3(2x + 0)}{(x^{2} + 1)^{4}})x^{3}{arctan(x)}^{x}}{arctan^{3}(x)} + \frac{3x^{2}{arctan(x)}^{x}}{(x^{2} + 1)^{3}arctan^{3}(x)} + \frac{x^{3}({arctan(x)}^{x}((1)ln(arctan(x)) + \frac{(x)((\frac{(1)}{(1 + (x)^{2})}))}{(arctan(x))}))}{(x^{2} + 1)^{3}arctan^{3}(x)} + \frac{x^{3}{arctan(x)}^{x}(\frac{-3(1)}{arctan^{4}(x)(1 + (x)^{2})})}{(x^{2} + 1)^{3}} - \frac{3(\frac{-3(2x + 0)}{(x^{2} + 1)^{4}})x^{2}{arctan(x)}^{x}}{arctan^{3}(x)} - \frac{3*2x{arctan(x)}^{x}}{(x^{2} + 1)^{3}arctan^{3}(x)} - \frac{3x^{2}({arctan(x)}^{x}((1)ln(arctan(x)) + \frac{(x)((\frac{(1)}{(1 + (x)^{2})}))}{(arctan(x))}))}{(x^{2} + 1)^{3}arctan^{3}(x)} - \frac{3x^{2}{arctan(x)}^{x}(\frac{-3(1)}{arctan^{4}(x)(1 + (x)^{2})})}{(x^{2} + 1)^{3}} + \frac{2(\frac{-3(2x + 0)}{(x^{2} + 1)^{4}})x{arctan(x)}^{x}}{arctan^{3}(x)} + \frac{2{arctan(x)}^{x}}{(x^{2} + 1)^{3}arctan^{3}(x)} + \frac{2x({arctan(x)}^{x}((1)ln(arctan(x)) + \frac{(x)((\frac{(1)}{(1 + (x)^{2})}))}{(arctan(x))}))}{(x^{2} + 1)^{3}arctan^{3}(x)} + \frac{2x{arctan(x)}^{x}(\frac{-3(1)}{arctan^{4}(x)(1 + (x)^{2})})}{(x^{2} + 1)^{3}}\\=&\frac{-32x{arctan(x)}^{x}ln(arctan(x))}{(x^{2} + 1)^{2}arctan(x)} + \frac{12{arctan(x)}^{x}ln^{2}(arctan(x))}{(x^{2} + 1)arctan(x)} + \frac{24x{arctan(x)}^{x}ln(arctan(x))}{(x^{2} + 1)^{2}arctan^{2}(x)} + \frac{12{arctan(x)}^{x}}{(x^{2} + 1)^{2}arctan^{2}(x)} - \frac{12{arctan(x)}^{x}ln(arctan(x))}{(x^{2} + 1)^{2}arctan^{2}(x)} + \frac{32x^{3}{arctan(x)}^{x}ln(arctan(x))}{(x^{2} + 1)^{3}arctan(x)} - \frac{12x^{2}{arctan(x)}^{x}ln^{2}(arctan(x))}{(x^{2} + 1)^{2}arctan(x)} - \frac{24x^{3}{arctan(x)}^{x}ln(arctan(x))}{(x^{2} + 1)^{3}arctan^{2}(x)} - \frac{56x^{2}{arctan(x)}^{x}}{(x^{2} + 1)^{3}arctan^{2}(x)} + \frac{24x^{2}{arctan(x)}^{x}ln(arctan(x))}{(x^{2} + 1)^{3}arctan^{2}(x)} + \frac{4x{arctan(x)}^{x}ln^{3}(arctan(x))}{(x^{2} + 1)arctan(x)} + \frac{6x^{2}{arctan(x)}^{x}ln^{2}(arctan(x))}{(x^{2} + 1)^{2}arctan^{2}(x)} - \frac{6x{arctan(x)}^{x}ln^{2}(arctan(x))}{(x^{2} + 1)^{2}arctan^{2}(x)} + \frac{4x^{3}{arctan(x)}^{x}ln(arctan(x))}{(x^{2} + 1)^{3}arctan^{3}(x)} + \frac{12x^{2}{arctan(x)}^{x}}{(x^{2} + 1)^{3}arctan^{3}(x)} - \frac{12x^{2}{arctan(x)}^{x}ln(arctan(x))}{(x^{2} + 1)^{3}arctan^{3}(x)} - \frac{24x{arctan(x)}^{x}}{(x^{2} + 1)^{3}arctan^{3}(x)} + {arctan(x)}^{x}ln^{4}(arctan(x)) + \frac{8x{arctan(x)}^{x}ln(arctan(x))}{(x^{2} + 1)^{3}arctan^{3}(x)} + \frac{56x^{2}{arctan(x)}^{x}}{(x^{2} + 1)^{3}arctan(x)} - \frac{8{arctan(x)}^{x}}{(x^{2} + 1)^{2}arctan(x)} + \frac{32x{arctan(x)}^{x}}{(x^{2} + 1)^{3}arctan^{2}(x)} + \frac{8{arctan(x)}^{x}}{(x^{2} + 1)^{3}arctan^{3}(x)} - \frac{48x^{4}{arctan(x)}^{x}}{(x^{2} + 1)^{4}arctan(x)} + \frac{44x^{4}{arctan(x)}^{x}}{(x^{2} + 1)^{4}arctan^{2}(x)} - \frac{44x^{3}{arctan(x)}^{x}}{(x^{2} + 1)^{4}arctan^{2}(x)} - \frac{12x^{4}{arctan(x)}^{x}}{(x^{2} + 1)^{4}arctan^{3}(x)} + \frac{36x^{3}{arctan(x)}^{x}}{(x^{2} + 1)^{4}arctan^{3}(x)} - \frac{24x^{2}{arctan(x)}^{x}}{(x^{2} + 1)^{4}arctan^{3}(x)} + \frac{x^{4}{arctan(x)}^{x}}{(x^{2} + 1)^{4}arctan^{4}(x)} - \frac{6x^{3}{arctan(x)}^{x}}{(x^{2} + 1)^{4}arctan^{4}(x)} + \frac{11x^{2}{arctan(x)}^{x}}{(x^{2} + 1)^{4}arctan^{4}(x)} - \frac{6x{arctan(x)}^{x}}{(x^{2} + 1)^{4}arctan^{4}(x)}\\ \end{split}\end{equation} \]