Optimal. Leaf size=79 \[ -\frac{1}{2} i \text{PolyLog}\left (2,1-\frac{2}{1+i x}\right )-\frac{x}{4 \left (x^2+1\right )}+\frac{\tan ^{-1}(x)}{2 \left (x^2+1\right )}-\frac{1}{2} i \tan ^{-1}(x)^2-\frac{1}{4} \tan ^{-1}(x)-\log \left (\frac{2}{1+i x}\right ) \tan ^{-1}(x) \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.11268, antiderivative size = 79, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 8, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.615, Rules used = {4964, 4920, 4854, 2402, 2315, 4930, 199, 203} \[ -\frac{1}{2} i \text{PolyLog}\left (2,1-\frac{2}{1+i x}\right )-\frac{x}{4 \left (x^2+1\right )}+\frac{\tan ^{-1}(x)}{2 \left (x^2+1\right )}-\frac{1}{2} i \tan ^{-1}(x)^2-\frac{1}{4} \tan ^{-1}(x)-\log \left (\frac{2}{1+i x}\right ) \tan ^{-1}(x) \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 4964
Rule 4920
Rule 4854
Rule 2402
Rule 2315
Rule 4930
Rule 199
Rule 203
Rubi steps
\begin{align*} \int \frac{x^3 \tan ^{-1}(x)}{\left (1+x^2\right )^2} \, dx &=-\int \frac{x \tan ^{-1}(x)}{\left (1+x^2\right )^2} \, dx+\int \frac{x \tan ^{-1}(x)}{1+x^2} \, dx\\ &=\frac{\tan ^{-1}(x)}{2 \left (1+x^2\right )}-\frac{1}{2} i \tan ^{-1}(x)^2-\frac{1}{2} \int \frac{1}{\left (1+x^2\right )^2} \, dx-\int \frac{\tan ^{-1}(x)}{i-x} \, dx\\ &=-\frac{x}{4 \left (1+x^2\right )}+\frac{\tan ^{-1}(x)}{2 \left (1+x^2\right )}-\frac{1}{2} i \tan ^{-1}(x)^2-\tan ^{-1}(x) \log \left (\frac{2}{1+i x}\right )-\frac{1}{4} \int \frac{1}{1+x^2} \, dx+\int \frac{\log \left (\frac{2}{1+i x}\right )}{1+x^2} \, dx\\ &=-\frac{x}{4 \left (1+x^2\right )}-\frac{1}{4} \tan ^{-1}(x)+\frac{\tan ^{-1}(x)}{2 \left (1+x^2\right )}-\frac{1}{2} i \tan ^{-1}(x)^2-\tan ^{-1}(x) \log \left (\frac{2}{1+i x}\right )-i \operatorname{Subst}\left (\int \frac{\log (2 x)}{1-2 x} \, dx,x,\frac{1}{1+i x}\right )\\ &=-\frac{x}{4 \left (1+x^2\right )}-\frac{1}{4} \tan ^{-1}(x)+\frac{\tan ^{-1}(x)}{2 \left (1+x^2\right )}-\frac{1}{2} i \tan ^{-1}(x)^2-\tan ^{-1}(x) \log \left (\frac{2}{1+i x}\right )-\frac{1}{2} i \text{Li}_2\left (1-\frac{2}{1+i x}\right )\\ \end{align*}
Mathematica [A] time = 0.0673323, size = 64, normalized size = 0.81 \[ \frac{1}{2} i \text{PolyLog}\left (2,-e^{2 i \tan ^{-1}(x)}\right )+\frac{1}{2} i \tan ^{-1}(x)^2-\tan ^{-1}(x) \log \left (1+e^{2 i \tan ^{-1}(x)}\right )-\frac{1}{8} \sin \left (2 \tan ^{-1}(x)\right )+\frac{1}{4} \tan ^{-1}(x) \cos \left (2 \tan ^{-1}(x)\right ) \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
Maple [B] time = 0.013, size = 139, normalized size = 1.8 \begin{align*}{\frac{\arctan \left ( x \right ) \ln \left ({x}^{2}+1 \right ) }{2}}+{\frac{\arctan \left ( x \right ) }{2\,{x}^{2}+2}}-{\frac{x}{4\,{x}^{2}+4}}-{\frac{\arctan \left ( x \right ) }{4}}-{\frac{i}{4}}\ln \left ( -{\frac{i}{2}} \left ( x+i \right ) \right ) \ln \left ( x-i \right ) -{\frac{i}{8}} \left ( \ln \left ( x-i \right ) \right ) ^{2}+{\frac{i}{4}}\ln \left ( x-i \right ) \ln \left ({x}^{2}+1 \right ) -{\frac{i}{4}}{\it dilog} \left ( -{\frac{i}{2}} \left ( x+i \right ) \right ) +{\frac{i}{4}}\ln \left ({\frac{i}{2}} \left ( x-i \right ) \right ) \ln \left ( x+i \right ) +{\frac{i}{8}} \left ( \ln \left ( x+i \right ) \right ) ^{2}-{\frac{i}{4}}\ln \left ( x+i \right ) \ln \left ({x}^{2}+1 \right ) +{\frac{i}{4}}{\it dilog} \left ({\frac{i}{2}} \left ( x-i \right ) \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{3} \arctan \left (x\right )}{{\left (x^{2} + 1\right )}^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{x^{3} \arctan \left (x\right )}{x^{4} + 2 \, x^{2} + 1}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: RecursionError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{3} \arctan \left (x\right )}{{\left (x^{2} + 1\right )}^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]