Optimal. Leaf size=23 \[ -\frac {1}{2} \log \left (x^2+1\right )-\frac {1}{2} \tan ^{-1}(x)^2+x \tan ^{-1}(x) \]
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Rubi [A] time = 0.05, antiderivative size = 23, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {4916, 4846, 260, 4884} \[ -\frac {1}{2} \log \left (x^2+1\right )-\frac {1}{2} \tan ^{-1}(x)^2+x \tan ^{-1}(x) \]
Antiderivative was successfully verified.
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Rule 260
Rule 4846
Rule 4884
Rule 4916
Rubi steps
\begin {align*} \int \frac {x^2 \tan ^{-1}(x)}{1+x^2} \, dx &=\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-\int \frac {x}{1+x^2} \, dx\\ &=x \tan ^{-1}(x)-\frac {1}{2} \tan ^{-1}(x)^2-\frac {1}{2} \log \left (1+x^2\right )\\ \end {align*}
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Mathematica [A] time = 0.02, size = 23, normalized size = 1.00 \[ -\frac {1}{2} \log \left (x^2+1\right )-\frac {1}{2} \tan ^{-1}(x)^2+x \tan ^{-1}(x) \]
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^2 \tan ^{-1}(x)}{1+x^2} \, dx \]
Verification is Not applicable to the result.
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fricas [A] time = 0.87, size = 19, normalized size = 0.83 \[ x \arctan \relax (x) - \frac {1}{2} \, \arctan \relax (x)^{2} - \frac {1}{2} \, \log \left (x^{2} + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.99, size = 19, normalized size = 0.83 \[ x \arctan \relax (x) - \frac {1}{2} \, \arctan \relax (x)^{2} - \frac {1}{2} \, \log \left (x^{2} + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.30, size = 20, normalized size = 0.87
method | result | size |
default | \(x \arctan \relax (x )-\frac {\arctan \relax (x )^{2}}{2}-\frac {\ln \left (x^{2}+1\right )}{2}\) | \(20\) |
risch | \(\frac {\ln \left (i x +1\right )^{2}}{8}+\frac {i \left (-x +\frac {i \ln \left (-i x +1\right )}{2}\right ) \ln \left (i x +1\right )}{2}+\frac {\ln \left (-i x +1\right )^{2}}{8}+\frac {i \ln \left (-i x +1\right ) x}{2}-\frac {\ln \left (x^{2}+1\right )}{2}\) | \(67\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.99, size = 24, normalized size = 1.04 \[ {\left (x - \arctan \relax (x)\right )} \arctan \relax (x) + \frac {1}{2} \, \arctan \relax (x)^{2} - \frac {1}{2} \, \log \left (x^{2} + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.22, size = 19, normalized size = 0.83 \[ -\frac {{\mathrm {atan}\relax (x)}^2}{2}+x\,\mathrm {atan}\relax (x)-\frac {\ln \left (x^2+1\right )}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.39, size = 19, normalized size = 0.83 \[ x \operatorname {atan}{\relax (x )} - \frac {\log {\left (x^{2} + 1 \right )}}{2} - \frac {\operatorname {atan}^{2}{\relax (x )}}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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