Optimal. Leaf size=89 \[ i \operatorname {PolyLog}\left (2,1-\frac {2}{1+i x}\right )+\frac {x}{4 \left (x^2+1\right )}+\frac {1}{2} x^2 \tan ^{-1}(x)-\frac {\tan ^{-1}(x)}{2 \left (x^2+1\right )}-\frac {x}{2}+i \tan ^{-1}(x)^2+\frac {3}{4} \tan ^{-1}(x)+2 \log \left (\frac {2}{1+i x}\right ) \tan ^{-1}(x) \]
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Rubi [A] time = 0.23, antiderivative size = 89, normalized size of antiderivative = 1.00, number of steps used = 17, number of rules used = 11, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.846, Rules used = {4964, 4916, 4852, 321, 203, 4920, 4854, 2402, 2315, 4930, 199} \[ i \text {PolyLog}\left (2,1-\frac {2}{1+i x}\right )+\frac {x}{4 \left (x^2+1\right )}+\frac {1}{2} x^2 \tan ^{-1}(x)-\frac {\tan ^{-1}(x)}{2 \left (x^2+1\right )}-\frac {x}{2}+i \tan ^{-1}(x)^2+\frac {3}{4} \tan ^{-1}(x)+2 \log \left (\frac {2}{1+i x}\right ) \tan ^{-1}(x) \]
Antiderivative was successfully verified.
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Rule 199
Rule 203
Rule 321
Rule 2315
Rule 2402
Rule 4852
Rule 4854
Rule 4916
Rule 4920
Rule 4930
Rule 4964
Rubi steps
\begin {align*} \int \frac {x^5 \tan ^{-1}(x)}{\left (1+x^2\right )^2} \, dx &=-\int \frac {x^3 \tan ^{-1}(x)}{\left (1+x^2\right )^2} \, dx+\int \frac {x^3 \tan ^{-1}(x)}{1+x^2} \, dx\\ &=\int x \tan ^{-1}(x) \, dx+\int \frac {x \tan ^{-1}(x)}{\left (1+x^2\right )^2} \, dx-2 \int \frac {x \tan ^{-1}(x)}{1+x^2} \, dx\\ &=\frac {1}{2} x^2 \tan ^{-1}(x)-\frac {\tan ^{-1}(x)}{2 \left (1+x^2\right )}+\frac {1}{2} \int \frac {1}{\left (1+x^2\right )^2} \, dx-\frac {1}{2} \int \frac {x^2}{1+x^2} \, dx-2 \left (-\frac {1}{2} i \tan ^{-1}(x)^2-\int \frac {\tan ^{-1}(x)}{i-x} \, dx\right )\\ &=-\frac {x}{2}+\frac {x}{4 \left (1+x^2\right )}+\frac {1}{2} x^2 \tan ^{-1}(x)-\frac {\tan ^{-1}(x)}{2 \left (1+x^2\right )}+\frac {1}{4} \int \frac {1}{1+x^2} \, dx+\frac {1}{2} \int \frac {1}{1+x^2} \, dx-2 \left (-\frac {1}{2} i \tan ^{-1}(x)^2-\tan ^{-1}(x) \log \left (\frac {2}{1+i x}\right )+\int \frac {\log \left (\frac {2}{1+i x}\right )}{1+x^2} \, dx\right )\\ &=-\frac {x}{2}+\frac {x}{4 \left (1+x^2\right )}+\frac {3}{4} \tan ^{-1}(x)+\frac {1}{2} x^2 \tan ^{-1}(x)-\frac {\tan ^{-1}(x)}{2 \left (1+x^2\right )}-2 \left (-\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 )\right )\\ &=-\frac {x}{2}+\frac {x}{4 \left (1+x^2\right )}+\frac {3}{4} \tan ^{-1}(x)+\frac {1}{2} x^2 \tan ^{-1}(x)-\frac {\tan ^{-1}(x)}{2 \left (1+x^2\right )}-2 \left (-\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 )\right )\\ \end {align*}
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Mathematica [A] time = 0.23, size = 70, normalized size = 0.79 \[ \frac {1}{8} \left (-8 i \operatorname {PolyLog}\left (2,-e^{2 i \tan ^{-1}(x)}\right )+4 \left (x^2+1\right ) \tan ^{-1}(x)-4 x-8 i \tan ^{-1}(x)^2+16 \tan ^{-1}(x) \log \left (1+e^{2 i \tan ^{-1}(x)}\right )+\sin \left (2 \tan ^{-1}(x)\right )-2 \tan ^{-1}(x) \cos \left (2 \tan ^{-1}(x)\right )\right ) \]
Warning: Unable to verify antiderivative.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^5 \tan ^{-1}(x)}{\left (1+x^2\right )^2} \, dx \]
Verification is Not applicable to the result.
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fricas [F] time = 1.01, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {x^{5} \arctan \relax (x)}{x^{4} + 2 \, x^{2} + 1}, x\right ) \]
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 \[ \int \frac {x^{5} \arctan \relax (x)}{{\left (x^{2} + 1\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.34, size = 149, normalized size = 1.67
method | result | size |
default | \(\frac {x^{2} \arctan \relax (x )}{2}-\arctan \relax (x ) \ln \left (x^{2}+1\right )-\frac {\arctan \relax (x )}{2 \left (x^{2}+1\right )}-\frac {x}{2}+\frac {x}{4 x^{2}+4}+\frac {3 \arctan \relax (x )}{4}-\frac {i \ln \left (x -i\right ) \ln \left (x^{2}+1\right )}{2}+\frac {i \dilog \left (-\frac {i \left (x +i\right )}{2}\right )}{2}+\frac {i \ln \left (x -i\right ) \ln \left (-\frac {i \left (x +i\right )}{2}\right )}{2}+\frac {i \ln \left (x -i\right )^{2}}{4}+\frac {i \ln \left (x +i\right ) \ln \left (x^{2}+1\right )}{2}-\frac {i \dilog \left (\frac {i \left (x -i\right )}{2}\right )}{2}-\frac {i \ln \left (x +i\right ) \ln \left (\frac {i \left (x -i\right )}{2}\right )}{2}-\frac {i \ln \left (x +i\right )^{2}}{4}\) | \(149\) |
risch | \(-\frac {x}{2}+\frac {5 \arctan \relax (x )}{8}-\frac {i \dilog \left (\frac {1}{2}+\frac {i x}{2}\right )}{2}-\frac {i \ln \left (i x +1\right ) x^{2}}{4}+\frac {i}{8 i x +8}-\frac {i \ln \left (-i x +1\right )}{8 \left (-i x +1\right )}+\frac {i \dilog \left (\frac {1}{2}-\frac {i x}{2}\right )}{2}+\frac {i \ln \left (-i x +1\right ) x^{2}}{4}-\frac {i \ln \left (-i x +1\right )^{2}}{4}+\frac {i \ln \left (i x +1\right )^{2}}{4}+\frac {\ln \left (i x +1\right ) x}{16 i x -16}+\frac {\ln \left (-i x +1\right ) x}{-16 i x -16}-\frac {i \ln \left (\frac {1}{2}+\frac {i x}{2}\right ) \ln \left (-i x +1\right )}{2}+\frac {i \ln \left (-i x +1\right )}{-16 i x -16}-\frac {i \ln \left (i x +1\right )}{16 \left (i x -1\right )}+\frac {i \ln \left (\frac {1}{2}-\frac {i x}{2}\right ) \ln \left (i x +1\right )}{2}-\frac {i}{8 \left (-i x +1\right )}+\frac {i \ln \left (i x +1\right )}{8 i x +8}\) | \(243\) |
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 \[ \int \frac {x^{5} \arctan \relax (x)}{{\left (x^{2} + 1\right )}^{2}}\,{d x} \]
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 \[ \int \frac {x^5\,\mathrm {atan}\relax (x)}{{\left (x^2+1\right )}^2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: RecursionError} \]
Verification of antiderivative is not currently implemented for this CAS.
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