Optimal. Leaf size=35 \[ \frac {1}{2} \log \left (x^2+1\right )+\frac {1}{2} x^2 \tan ^{-1}(x)^2+\frac {1}{2} \tan ^{-1}(x)^2-x \tan ^{-1}(x) \]
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Rubi [A] time = 0.06, antiderivative size = 35, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.833, Rules used = {4852, 4916, 4846, 260, 4884} \[ \frac {1}{2} \log \left (x^2+1\right )+\frac {1}{2} x^2 \tan ^{-1}(x)^2+\frac {1}{2} \tan ^{-1}(x)^2-x \tan ^{-1}(x) \]
Antiderivative was successfully verified.
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Rule 260
Rule 4846
Rule 4852
Rule 4884
Rule 4916
Rubi steps
\begin {align*} \int x \tan ^{-1}(x)^2 \, dx &=\frac {1}{2} x^2 \tan ^{-1}(x)^2-\int \frac {x^2 \tan ^{-1}(x)}{1+x^2} \, dx\\ &=\frac {1}{2} x^2 \tan ^{-1}(x)^2-\int \tan ^{-1}(x) \, dx+\int \frac {\tan ^{-1}(x)}{1+x^2} \, dx\\ &=-x \tan ^{-1}(x)+\frac {1}{2} \tan ^{-1}(x)^2+\frac {1}{2} x^2 \tan ^{-1}(x)^2+\int \frac {x}{1+x^2} \, dx\\ &=-x \tan ^{-1}(x)+\frac {1}{2} \tan ^{-1}(x)^2+\frac {1}{2} x^2 \tan ^{-1}(x)^2+\frac {1}{2} \log \left (1+x^2\right )\\ \end {align*}
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Mathematica [A] time = 0.01, size = 26, normalized size = 0.74 \[ \frac {1}{2} \left (\log \left (x^2+1\right )+\left (x^2+1\right ) \tan ^{-1}(x)^2-2 x \tan ^{-1}(x)\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.43, size = 25, normalized size = 0.71 \[ \frac {1}{2} \, {\left (x^{2} + 1\right )} \arctan \relax (x)^{2} - x \arctan \relax (x) + \frac {1}{2} \, \log \left (x^{2} + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.02, size = 29, normalized size = 0.83 \[ \frac {1}{2} \, x^{2} \arctan \relax (x)^{2} - x \arctan \relax (x) + \frac {1}{2} \, \arctan \relax (x)^{2} + \frac {1}{2} \, \log \left (x^{2} + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 30, normalized size = 0.86 \[ \frac {x^{2} \arctan \relax (x )^{2}}{2}-x \arctan \relax (x )+\frac {\arctan \relax (x )^{2}}{2}+\frac {\ln \left (x^{2}+1\right )}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.26, size = 34, normalized size = 0.97 \[ \frac {1}{2} \, x^{2} \arctan \relax (x)^{2} - {\left (x - \arctan \relax (x)\right )} \arctan \relax (x) - \frac {1}{2} \, \arctan \relax (x)^{2} + \frac {1}{2} \, \log \left (x^{2} + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.12, size = 29, normalized size = 0.83 \[ \frac {\ln \left (x^2+1\right )}{2}+\frac {{\mathrm {atan}\relax (x)}^2}{2}+\frac {x^2\,{\mathrm {atan}\relax (x)}^2}{2}-x\,\mathrm {atan}\relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.36, size = 29, normalized size = 0.83 \[ \frac {x^{2} \operatorname {atan}^{2}{\relax (x )}}{2} - x \operatorname {atan}{\relax (x )} + \frac {\log {\left (x^{2} + 1 \right )}}{2} + \frac {\operatorname {atan}^{2}{\relax (x )}}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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