Optimal. Leaf size=53 \[ \frac {1}{4} x^4 \tan ^{-1}(x)^2-\frac {1}{6} x^3 \tan ^{-1}(x)+\frac {x^2}{12}-\frac {1}{3} \log \left (x^2+1\right )+\frac {1}{2} x \tan ^{-1}(x)-\frac {1}{4} \tan ^{-1}(x)^2 \]
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Rubi [A] time = 0.12, antiderivative size = 53, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 7, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.875, Rules used = {4852, 4916, 266, 43, 4846, 260, 4884} \[ \frac {x^2}{12}-\frac {1}{3} \log \left (x^2+1\right )+\frac {1}{4} x^4 \tan ^{-1}(x)^2-\frac {1}{6} x^3 \tan ^{-1}(x)+\frac {1}{2} x \tan ^{-1}(x)-\frac {1}{4} \tan ^{-1}(x)^2 \]
Antiderivative was successfully verified.
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Rule 43
Rule 260
Rule 266
Rule 4846
Rule 4852
Rule 4884
Rule 4916
Rubi steps
\begin {align*} \int x^3 \tan ^{-1}(x)^2 \, dx &=\frac {1}{4} x^4 \tan ^{-1}(x)^2-\frac {1}{2} \int \frac {x^4 \tan ^{-1}(x)}{1+x^2} \, dx\\ &=\frac {1}{4} x^4 \tan ^{-1}(x)^2-\frac {1}{2} \int x^2 \tan ^{-1}(x) \, dx+\frac {1}{2} \int \frac {x^2 \tan ^{-1}(x)}{1+x^2} \, dx\\ &=-\frac {1}{6} x^3 \tan ^{-1}(x)+\frac {1}{4} x^4 \tan ^{-1}(x)^2+\frac {1}{6} \int \frac {x^3}{1+x^2} \, dx+\frac {1}{2} \int \tan ^{-1}(x) \, dx-\frac {1}{2} \int \frac {\tan ^{-1}(x)}{1+x^2} \, dx\\ &=\frac {1}{2} x \tan ^{-1}(x)-\frac {1}{6} x^3 \tan ^{-1}(x)-\frac {1}{4} \tan ^{-1}(x)^2+\frac {1}{4} x^4 \tan ^{-1}(x)^2+\frac {1}{12} \operatorname {Subst}\left (\int \frac {x}{1+x} \, dx,x,x^2\right )-\frac {1}{2} \int \frac {x}{1+x^2} \, dx\\ &=\frac {1}{2} x \tan ^{-1}(x)-\frac {1}{6} x^3 \tan ^{-1}(x)-\frac {1}{4} \tan ^{-1}(x)^2+\frac {1}{4} x^4 \tan ^{-1}(x)^2-\frac {1}{4} \log \left (1+x^2\right )+\frac {1}{12} \operatorname {Subst}\left (\int \left (1+\frac {1}{-1-x}\right ) \, dx,x,x^2\right )\\ &=\frac {x^2}{12}+\frac {1}{2} x \tan ^{-1}(x)-\frac {1}{6} x^3 \tan ^{-1}(x)-\frac {1}{4} \tan ^{-1}(x)^2+\frac {1}{4} x^4 \tan ^{-1}(x)^2-\frac {1}{3} \log \left (1+x^2\right )\\ \end {align*}
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Mathematica [A] time = 0.02, size = 37, normalized size = 0.70 \[ \frac {1}{12} \left (3 \left (x^4-1\right ) \tan ^{-1}(x)^2+x^2-4 \log \left (x^2+1\right )-2 \left (x^2-3\right ) x \tan ^{-1}(x)\right ) \]
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int x^3 \tan ^{-1}(x)^2 \, dx \]
Verification is Not applicable to the result.
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fricas [A] time = 1.01, size = 36, normalized size = 0.68 \[ \frac {1}{4} \, {\left (x^{4} - 1\right )} \arctan \relax (x)^{2} + \frac {1}{12} \, x^{2} - \frac {1}{6} \, {\left (x^{3} - 3 \, x\right )} \arctan \relax (x) - \frac {1}{3} \, \log \left (x^{2} + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.09, size = 41, normalized size = 0.77 \[ \frac {1}{4} \, x^{4} \arctan \relax (x)^{2} - \frac {1}{6} \, x^{3} \arctan \relax (x) + \frac {1}{12} \, x^{2} + \frac {1}{2} \, x \arctan \relax (x) - \frac {1}{4} \, \arctan \relax (x)^{2} - \frac {1}{3} \, \log \left (x^{2} + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.10, size = 42, normalized size = 0.79
method | result | size |
default | \(\frac {x^{2}}{12}+\frac {x \arctan \relax (x )}{2}-\frac {x^{3} \arctan \relax (x )}{6}-\frac {\arctan \relax (x )^{2}}{4}+\frac {x^{4} \arctan \relax (x )^{2}}{4}-\frac {\ln \left (x^{2}+1\right )}{3}\) | \(42\) |
risch | \(-\frac {\left (\frac {x^{4}}{4}-\frac {1}{4}\right ) \ln \left (i x +1\right )^{2}}{4}-\frac {\left (-\frac {x^{4} \ln \left (-i x +1\right )}{2}-\frac {i x^{3}}{3}+i x +\frac {\ln \left (-i x +1\right )}{2}\right ) \ln \left (i x +1\right )}{4}-\frac {x^{4} \ln \left (-i x +1\right )^{2}}{16}+\frac {\ln \left (-i x +1\right )^{2}}{16}-\frac {i x^{3} \ln \left (-i x +1\right )}{12}+\frac {i \ln \left (-i x +1\right ) x}{4}+\frac {x^{2}}{12}-\frac {\ln \left (x^{2}+1\right )}{3}\) | \(123\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.00, size = 44, normalized size = 0.83 \[ \frac {1}{4} \, x^{4} \arctan \relax (x)^{2} + \frac {1}{12} \, x^{2} - \frac {1}{6} \, {\left (x^{3} - 3 \, x + 3 \, \arctan \relax (x)\right )} \arctan \relax (x) + \frac {1}{4} \, \arctan \relax (x)^{2} - \frac {1}{3} \, \log \left (x^{2} + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.34, size = 41, normalized size = 0.77 \[ \frac {x^4\,{\mathrm {atan}\relax (x)}^2}{4}-\frac {x^3\,\mathrm {atan}\relax (x)}{6}-\frac {{\mathrm {atan}\relax (x)}^2}{4}-\frac {\ln \left (x^2+1\right )}{3}+\frac {x\,\mathrm {atan}\relax (x)}{2}+\frac {x^2}{12} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.62, size = 44, normalized size = 0.83 \[ \frac {x^{4} \operatorname {atan}^{2}{\relax (x )}}{4} - \frac {x^{3} \operatorname {atan}{\relax (x )}}{6} + \frac {x^{2}}{12} + \frac {x \operatorname {atan}{\relax (x )}}{2} - \frac {\log {\left (x^{2} + 1 \right )}}{3} - \frac {\operatorname {atan}^{2}{\relax (x )}}{4} \]
Verification of antiderivative is not currently implemented for this CAS.
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