Optimal. Leaf size=67 \[ -\frac {x}{2}+\frac {1}{2} \tan ^{-1}(x)+\frac {1}{2} x^2 \tan ^{-1}(x)+\frac {1}{2} i \tan ^{-1}(x)^2+\tan ^{-1}(x) \log \left (\frac {2}{1+i x}\right )+\frac {1}{2} i \text {Li}_2\left (1-\frac {2}{1+i x}\right ) \]
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Rubi [A]
time = 0.07, antiderivative size = 67, normalized size of antiderivative = 1.00, 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 = {5036, 4946,
327, 209, 5040, 4964, 2449, 2352} \begin {gather*} \frac {1}{2} i \text {PolyLog}\left (2,1-\frac {2}{1+i x}\right )+\frac {1}{2} x^2 \text {ArcTan}(x)+\frac {1}{2} i \text {ArcTan}(x)^2+\frac {\text {ArcTan}(x)}{2}+\text {ArcTan}(x) \log \left (\frac {2}{1+i x}\right )-\frac {x}{2} \end {gather*}
Antiderivative was successfully verified.
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Rule 209
Rule 327
Rule 2352
Rule 2449
Rule 4946
Rule 4964
Rule 5036
Rule 5040
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 \text {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*}
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Mathematica [A]
time = 0.02, size = 57, normalized size = 0.85 \begin {gather*} \frac {1}{2} \left (-x+i \tan ^{-1}(x)^2+\tan ^{-1}(x) \left (1+x^2+2 \log \left (-\frac {2 i}{-i+x}\right )\right )+i \text {Li}_2\left (\frac {i+x}{-i+x}\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Both result and optimal contain complex but leaf count of result is larger than twice
the leaf count of optimal. 127 vs. \(2 (53 ) = 106\).
time = 0.08, size = 128, normalized size = 1.91
method | result | size |
risch | \(\frac {\arctan \left (x \right )}{2}-\frac {x}{2}+\frac {i x^{2} \ln \left (-i x +1\right )}{4}-\frac {i \ln \left (\frac {1}{2}+\frac {i x}{2}\right ) \ln \left (-i x +1\right )}{4}+\frac {i \dilog \left (\frac {1}{2}-\frac {i x}{2}\right )}{4}-\frac {i \ln \left (-i x +1\right )^{2}}{8}-\frac {i x^{2} \ln \left (i x +1\right )}{4}+\frac {i \ln \left (\frac {1}{2}-\frac {i x}{2}\right ) \ln \left (i x +1\right )}{4}-\frac {i \dilog \left (\frac {1}{2}+\frac {i x}{2}\right )}{4}+\frac {i \ln \left (i x +1\right )^{2}}{8}\) | \(113\) |
default | \(\frac {x^{2} \arctan \left (x \right )}{2}-\frac {\arctan \left (x \right ) \ln \left (x^{2}+1\right )}{2}-\frac {x}{2}+\frac {\arctan \left (x \right )}{2}-\frac {i \ln \left (x -i\right ) \ln \left (x^{2}+1\right )}{4}+\frac {i \dilog \left (-\frac {i \left (x +i\right )}{2}\right )}{4}+\frac {i \ln \left (x -i\right ) \ln \left (-\frac {i \left (x +i\right )}{2}\right )}{4}+\frac {i \ln \left (x -i\right )^{2}}{8}+\frac {i \ln \left (x +i\right ) \ln \left (x^{2}+1\right )}{4}-\frac {i \dilog \left (\frac {i \left (x -i\right )}{2}\right )}{4}-\frac {i \ln \left (x +i\right ) \ln \left (\frac {i \left (x -i\right )}{2}\right )}{4}-\frac {i \ln \left (x +i\right )^{2}}{8}\) | \(128\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{3} \operatorname {atan}{\left (x \right )}}{x^{2} + 1}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {x^3\,\mathrm {atan}\left (x\right )}{x^2+1} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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