Optimal. Leaf size=21 \[ -\frac {x}{2}+\frac {1}{2} \tan ^{-1}(x)+\frac {1}{2} x^2 \tan ^{-1}(x) \]
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Rubi [A]
time = 0.01, antiderivative size = 21, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.750, Rules used = {4946, 327, 209}
\begin {gather*} \frac {1}{2} x^2 \text {ArcTan}(x)+\frac {\text {ArcTan}(x)}{2}-\frac {x}{2} \end {gather*}
Antiderivative was successfully verified.
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Rule 209
Rule 327
Rule 4946
Rubi steps
\begin {align*} \int x \tan ^{-1}(x) \, dx &=\frac {1}{2} x^2 \tan ^{-1}(x)-\frac {1}{2} \int \frac {x^2}{1+x^2} \, dx\\ &=-\frac {x}{2}+\frac {1}{2} x^2 \tan ^{-1}(x)+\frac {1}{2} \int \frac {1}{1+x^2} \, dx\\ &=-\frac {x}{2}+\frac {1}{2} \tan ^{-1}(x)+\frac {1}{2} x^2 \tan ^{-1}(x)\\ \end {align*}
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Mathematica [A]
time = 0.00, size = 21, normalized size = 1.00 \begin {gather*} -\frac {x}{2}+\frac {1}{2} \tan ^{-1}(x)+\frac {1}{2} x^2 \tan ^{-1}(x) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.00, size = 16, normalized size = 0.76
method | result | size |
default | \(-\frac {x}{2}+\frac {\arctan \left (x \right )}{2}+\frac {x^{2} \arctan \left (x \right )}{2}\) | \(16\) |
meijerg | \(-\frac {x}{2}+\frac {\left (3 x^{2}+3\right ) \arctan \left (x \right )}{6}\) | \(16\) |
risch | \(-\frac {i x^{2} \ln \left (i x +1\right )}{4}+\frac {i x^{2} \ln \left (-i x +1\right )}{4}-\frac {x}{2}+\frac {\arctan \left (x \right )}{2}\) | \(35\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 2.02, size = 15, normalized size = 0.71 \begin {gather*} \frac {1}{2} \, x^{2} \arctan \left (x\right ) - \frac {1}{2} \, x + \frac {1}{2} \, \arctan \left (x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.81, size = 13, normalized size = 0.62 \begin {gather*} \frac {1}{2} \, {\left (x^{2} + 1\right )} \arctan \left (x\right ) - \frac {1}{2} \, x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.07, size = 15, normalized size = 0.71 \begin {gather*} \frac {x^{2} \operatorname {atan}{\left (x \right )}}{2} - \frac {x}{2} + \frac {\operatorname {atan}{\left (x \right )}}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.68, size = 15, normalized size = 0.71 \begin {gather*} \frac {1}{2} \, x^{2} \arctan \left (x\right ) - \frac {1}{2} \, x + \frac {1}{2} \, \arctan \left (x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.02, size = 14, normalized size = 0.67 \begin {gather*} \mathrm {atan}\left (x\right )\,\left (\frac {x^2}{2}+\frac {1}{2}\right )-\frac {x}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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