Optimal. Leaf size=67 \[ -\frac {1}{2} i \text {Li}_2\left (1-\frac {2}{i x+1}\right )+\frac {1}{2} x^2 \cot ^{-1}(x)+\frac {x}{2}-\frac {1}{2} \tan ^{-1}(x)-\frac {1}{2} i \cot ^{-1}(x)^2+\log \left (\frac {2}{1+i x}\right ) \cot ^{-1}(x) \]
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Rubi [A] time = 0.09, 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 = {4917, 4853, 321, 203, 4921, 4855, 2402, 2315} \[ -\frac {1}{2} i \text {PolyLog}\left (2,1-\frac {2}{1+i x}\right )+\frac {1}{2} x^2 \cot ^{-1}(x)+\frac {x}{2}-\frac {1}{2} \tan ^{-1}(x)-\frac {1}{2} i \cot ^{-1}(x)^2+\log \left (\frac {2}{1+i x}\right ) \cot ^{-1}(x) \]
Antiderivative was successfully verified.
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Rule 203
Rule 321
Rule 2315
Rule 2402
Rule 4853
Rule 4855
Rule 4917
Rule 4921
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 \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} 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] time = 0.07, size = 241, normalized size = 3.60 \[ -\frac {1}{4} i \text {Li}_2\left (-\frac {1}{2} i (i-x)\right )+\frac {1}{4} i \text {Li}_2\left (-\frac {1}{2} i (x+i)\right )+\frac {1}{2} x^2 \cot ^{-1}(x)+\frac {x}{2}+\frac {1}{8} i \log ^2(-x+i)-\frac {1}{8} i \log ^2(x+i)-\frac {1}{4} i \log (-x+i) \log \left (-\frac {-x+i}{x}\right )-\frac {1}{4} i \log (-x+i) \log \left (-\frac {1}{2} i (x+i)\right )+\frac {1}{4} i \log \left (-\frac {1}{2} i (-x+i)\right ) \log (x+i)-\frac {1}{4} i \log \left (-\frac {-x+i}{x}\right ) \log (x+i)+\frac {1}{4} i \log (-x+i) \log \left (\frac {x+i}{x}\right )+\frac {1}{4} i \log (x+i) \log \left (\frac {x+i}{x}\right )-\frac {1}{2} \tan ^{-1}(x) \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.39, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {x^{3} \operatorname {arccot}\relax (x)}{x^{2} + 1}, x\right ) \]
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 \[ \int \frac {x^{3} \operatorname {arccot}\relax (x)}{x^{2} + 1}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.18, size = 128, normalized size = 1.91 \[ \frac {x^{2} \mathrm {arccot}\relax (x )}{2}-\frac {\mathrm {arccot}\relax (x ) \ln \left (x^{2}+1\right )}{2}+\frac {x}{2}-\frac {\arctan \relax (x )}{2}+\frac {i \ln \left (x -i\right ) \ln \left (x^{2}+1\right )}{4}-\frac {i \ln \left (x -i\right )^{2}}{8}-\frac {i \dilog \left (-\frac {i \left (x +i\right )}{2}\right )}{4}-\frac {i \ln \left (x -i\right ) \ln \left (-\frac {i \left (x +i\right )}{2}\right )}{4}-\frac {i \ln \left (x +i\right ) \ln \left (x^{2}+1\right )}{4}+\frac {i \ln \left (x +i\right )^{2}}{8}+\frac {i \dilog \left (\frac {i \left (x -i\right )}{2}\right )}{4}+\frac {i \ln \left (x +i\right ) \ln \left (\frac {i \left (x -i\right )}{2}\right )}{4} \]
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 \[ \int \frac {x^{3} \operatorname {arccot}\relax (x)}{x^{2} + 1}\,{d x} \]
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 \[ \int \frac {x^3\,\mathrm {acot}\relax (x)}{x^2+1} \,d x \]
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 \[ \int \frac {x^{3} \operatorname {acot}{\relax (x )}}{x^{2} + 1}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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