Optimal. Leaf size=67 \[ \frac {x}{2}+\frac {1}{2} x^2 \cot ^{-1}(x)-\frac {1}{2} i \cot ^{-1}(x)^2-\frac {\text {ArcTan}(x)}{2}+\cot ^{-1}(x) \log \left (\frac {2}{1+i x}\right )-\frac {1}{2} i \text {PolyLog}\left (2,1-\frac {2}{1+i x}\right ) \]
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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 = {5037, 4947,
327, 209, 5041, 4965, 2449, 2352} \begin {gather*} -\frac {\text {ArcTan}(x)}{2}-\frac {1}{2} i \text {Li}_2\left (1-\frac {2}{i x+1}\right )+\frac {1}{2} x^2 \cot ^{-1}(x)+\frac {x}{2}-\frac {1}{2} i \cot ^{-1}(x)^2+\log \left (\frac {2}{1+i x}\right ) \cot ^{-1}(x) \end {gather*}
Antiderivative was successfully verified.
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Rule 209
Rule 327
Rule 2352
Rule 2449
Rule 4947
Rule 4965
Rule 5037
Rule 5041
Rubi steps
\begin {align*} \int \frac {x^3 \cot ^{-1}(x)}{1+x^2} \, dx &=\int x \cot ^{-1}(x) \, dx-\int \frac {x \cot ^{-1}(x)}{1+x^2} \, dx\\ &=\frac {1}{2} x^2 \cot ^{-1}(x)-\frac {1}{2} i \cot ^{-1}(x)^2+\frac {1}{2} \int \frac {x^2}{1+x^2} \, dx+\int \frac {\cot ^{-1}(x)}{i-x} \, dx\\ &=\frac {x}{2}+\frac {1}{2} x^2 \cot ^{-1}(x)-\frac {1}{2} i \cot ^{-1}(x)^2+\cot ^{-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} x^2 \cot ^{-1}(x)-\frac {1}{2} i \cot ^{-1}(x)^2-\frac {1}{2} \tan ^{-1}(x)+\cot ^{-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} x^2 \cot ^{-1}(x)-\frac {1}{2} i \cot ^{-1}(x)^2-\frac {1}{2} \tan ^{-1}(x)+\cot ^{-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 [B] Both result and optimal contain complex but leaf count is larger than twice
the leaf count of optimal. \(241\) vs. \(2(67)=134\).
time = 0.05, size = 241, normalized size = 3.60 \begin {gather*} \frac {x}{2}+\frac {1}{2} x^2 \cot ^{-1}(x)-\frac {\text {ArcTan}(x)}{2}+\frac {1}{8} i \log ^2(i-x)-\frac {1}{4} i \log (i-x) \log \left (-\frac {i-x}{x}\right )-\frac {1}{4} i \log (i-x) \log \left (-\frac {1}{2} i (i+x)\right )+\frac {1}{4} i \log \left (-\frac {1}{2} i (i-x)\right ) \log (i+x)-\frac {1}{4} i \log \left (-\frac {i-x}{x}\right ) \log (i+x)-\frac {1}{8} i \log ^2(i+x)+\frac {1}{4} i \log (i-x) \log \left (\frac {i+x}{x}\right )+\frac {1}{4} i \log (i+x) \log \left (\frac {i+x}{x}\right )-\frac {1}{4} i \text {PolyLog}\left (2,-\frac {1}{2} i (i-x)\right )+\frac {1}{4} i \text {PolyLog}\left (2,-\frac {1}{2} i (i+x)\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 = 0.12, size = 128, normalized size = 1.91
method | result | size |
default | \(\frac {x^{2} \mathrm {arccot}\left (x \right )}{2}-\frac {\mathrm {arccot}\left (x \right ) \ln \left (x^{2}+1\right )}{2}+\frac {x}{2}-\frac {\arctan \left (x \right )}{2}+\frac {i \ln \left (x -i\right ) \ln \left (x^{2}+1\right )}{4}-\frac {i \dilog \left (-\frac {i \left (i+x \right )}{2}\right )}{4}-\frac {i \ln \left (x -i\right ) \ln \left (-\frac {i \left (i+x \right )}{2}\right )}{4}-\frac {i \ln \left (x -i\right )^{2}}{8}-\frac {i \ln \left (i+x \right ) \ln \left (x^{2}+1\right )}{4}+\frac {i \dilog \left (\frac {i \left (x -i\right )}{2}\right )}{4}+\frac {i \ln \left (i+x \right ) \ln \left (\frac {i \left (x -i\right )}{2}\right )}{4}+\frac {i \ln \left (i+x \right )^{2}}{8}\) | \(128\) |
risch | \(\frac {\pi \,x^{2}}{4}+\frac {\pi }{4}-\frac {\pi \ln \left (x^{2}+1\right )}{4}-\frac {i \ln \left (-i x +1\right ) x^{2}}{4}-\frac {i \dilog \left (\frac {1}{2}-\frac {i x}{2}\right )}{4}-\frac {i \ln \left (\frac {1}{2}-\frac {i x}{2}\right ) \ln \left (i x +1\right )}{4}-\frac {i \ln \left (i x +1\right )^{2}}{8}+\frac {i \ln \left (-i x +1\right )^{2}}{8}+\frac {x}{2}+\frac {i \ln \left (i x +1\right ) x^{2}}{4}-\frac {\arctan \left (x \right )}{2}+\frac {i \ln \left (\frac {1}{2}+\frac {i x}{2}\right ) \ln \left (-i x +1\right )}{4}+\frac {i \dilog \left (\frac {1}{2}+\frac {i x}{2}\right )}{4}\) | \(131\) |
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 {acot}{\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 {acot}\left (x\right )}{x^2+1} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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