Optimal. Leaf size=23 \[ x \tan ^{-1}(x)-\frac {1}{2} \tan ^{-1}(x)^2-\frac {1}{2} \log \left (1+x^2\right ) \]
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Rubi [A]
time = 0.03, 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 = {5036, 4930,
266, 5004} \begin {gather*} -\frac {1}{2} \log \left (x^2+1\right )-\frac {1}{2} \tan ^{-1}(x)^2+x \tan ^{-1}(x) \end {gather*}
Antiderivative was successfully verified.
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Rule 266
Rule 4930
Rule 5004
Rule 5036
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 = 0.01, size = 23, normalized size = 1.00 \begin {gather*} x \tan ^{-1}(x)-\frac {1}{2} \tan ^{-1}(x)^2-\frac {1}{2} \log \left (1+x^2\right ) \end {gather*}
Antiderivative was successfully verified.
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Mathics [A]
time = 2.07, size = 19, normalized size = 0.83 \begin {gather*} x \text {ArcTan}\left [x\right ]-\frac {\text {Log}\left [1+x^2\right ]}{2}-\frac {\text {ArcTan}\left [x\right ]^2}{2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.08, size = 20, normalized size = 0.87
method | result | size |
default | \(x \arctan \left (x \right )-\frac {\arctan \left (x \right )^{2}}{2}-\frac {\ln \left (x^{2}+1\right )}{2}\) | \(20\) |
risch | \(\frac {\ln \left (i x +1\right )^{2}}{8}+\frac {i \left (-x +\frac {i \ln \left (-i x +1\right )}{2}\right ) \ln \left (i x +1\right )}{2}+\frac {\ln \left (-i x +1\right )^{2}}{8}+\frac {i x \ln \left (-i x +1\right )}{2}-\frac {\ln \left (x^{2}+1\right )}{2}\) | \(67\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.37, size = 24, normalized size = 1.04 \begin {gather*} {\left (x - \arctan \left (x\right )\right )} \arctan \left (x\right ) + \frac {1}{2} \, \arctan \left (x\right )^{2} - \frac {1}{2} \, \log \left (x^{2} + 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.32, size = 19, normalized size = 0.83 \begin {gather*} x \arctan \left (x\right ) - \frac {1}{2} \, \arctan \left (x\right )^{2} - \frac {1}{2} \, \log \left (x^{2} + 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.13, size = 19, normalized size = 0.83 \begin {gather*} x \operatorname {atan}{\left (x \right )} - \frac {\log {\left (x^{2} + 1 \right )}}{2} - \frac {\operatorname {atan}^{2}{\left (x \right )}}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.00, size = 21, normalized size = 0.91 \begin {gather*} \frac {2 x \arctan x-\arctan ^{2}x-\ln \left (x^{2}+1\right )}{2} \end {gather*}
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 \begin {gather*} -\frac {{\mathrm {atan}\left (x\right )}^2}{2}+x\,\mathrm {atan}\left (x\right )-\frac {\ln \left (x^2+1\right )}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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