Optimal. Leaf size=67 \[ \frac{1}{2} i \text{PolyLog}\left (2,1-\frac{2}{1+i x}\right )+\frac{1}{2} x^2 \tan ^{-1}(x)-\frac{x}{2}+\frac{1}{2} i \tan ^{-1}(x)^2+\frac{1}{2} \tan ^{-1}(x)+\log \left (\frac{2}{1+i x}\right ) \tan ^{-1}(x) \]
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Rubi [A] time = 0.0937433, antiderivative size = 67, 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 = {4916, 4852, 321, 203, 4920, 4854, 2402, 2315} \[ \frac{1}{2} i \text{PolyLog}\left (2,1-\frac{2}{1+i x}\right )+\frac{1}{2} x^2 \tan ^{-1}(x)-\frac{x}{2}+\frac{1}{2} i \tan ^{-1}(x)^2+\frac{1}{2} \tan ^{-1}(x)+\log \left (\frac{2}{1+i x}\right ) \tan ^{-1}(x) \]
Antiderivative was successfully verified.
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Rule 4916
Rule 4852
Rule 321
Rule 203
Rule 4920
Rule 4854
Rule 2402
Rule 2315
Rubi steps
\begin{align*} \int \frac{x^3 \tan ^{-1}(x)}{1+x^2} \, dx &=\int x \tan ^{-1}(x) \, dx-\int \frac{x \tan ^{-1}(x)}{1+x^2} \, dx\\ &=\frac{1}{2} x^2 \tan ^{-1}(x)+\frac{1}{2} i \tan ^{-1}(x)^2-\frac{1}{2} \int \frac{x^2}{1+x^2} \, dx+\int \frac{\tan ^{-1}(x)}{i-x} \, dx\\ &=-\frac{x}{2}+\frac{1}{2} x^2 \tan ^{-1}(x)+\frac{1}{2} i \tan ^{-1}(x)^2+\tan ^{-1}(x) \log \left (\frac{2}{1+i x}\right )+\frac{1}{2} \int \frac{1}{1+x^2} \, dx-\int \frac{\log \left (\frac{2}{1+i x}\right )}{1+x^2} \, dx\\ &=-\frac{x}{2}+\frac{1}{2} \tan ^{-1}(x)+\frac{1}{2} x^2 \tan ^{-1}(x)+\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}{2}+\frac{1}{2} \tan ^{-1}(x)+\frac{1}{2} x^2 \tan ^{-1}(x)+\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 = 0.0323023, size = 57, normalized size = 0.85 \[ \frac{1}{2} \left (i \text{PolyLog}\left (2,\frac{x+i}{x-i}\right )+\left (x^2+2 \log \left (-\frac{2 i}{x-i}\right )+1\right ) \tan ^{-1}(x)-x+i \tan ^{-1}(x)^2\right ) \]
Antiderivative was successfully verified.
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Maple [B] time = 0.031, size = 128, normalized size = 1.9 \begin{align*}{\frac{{x}^{2}\arctan \left ( x \right ) }{2}}-{\frac{\arctan \left ( x \right ) \ln \left ({x}^{2}+1 \right ) }{2}}-{\frac{x}{2}}+{\frac{\arctan \left ( x \right ) }{2}}+{\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.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{3} \arctan \left (x\right )}{x^{2} + 1}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{x^{3} \arctan \left (x\right )}{x^{2} + 1}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{3} \operatorname{atan}{\left (x \right )}}{x^{2} + 1}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{3} \arctan \left (x\right )}{x^{2} + 1}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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