Optimal. Leaf size=32 \[ \frac {x}{4 \left (x^2+1\right )}-\frac {\tan ^{-1}(x)}{2 \left (x^2+1\right )}+\frac {1}{4} \tan ^{-1}(x) \]
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Rubi [A] time = 0.03, antiderivative size = 32, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {4930, 199, 203} \[ \frac {x}{4 \left (x^2+1\right )}-\frac {\tan ^{-1}(x)}{2 \left (x^2+1\right )}+\frac {1}{4} \tan ^{-1}(x) \]
Antiderivative was successfully verified.
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Rule 199
Rule 203
Rule 4930
Rubi steps
\begin {align*} \int \frac {x \tan ^{-1}(x)}{\left (1+x^2\right )^2} \, dx &=-\frac {\tan ^{-1}(x)}{2 \left (1+x^2\right )}+\frac {1}{2} \int \frac {1}{\left (1+x^2\right )^2} \, dx\\ &=\frac {x}{4 \left (1+x^2\right )}-\frac {\tan ^{-1}(x)}{2 \left (1+x^2\right )}+\frac {1}{4} \int \frac {1}{1+x^2} \, dx\\ &=\frac {x}{4 \left (1+x^2\right )}+\frac {1}{4} \tan ^{-1}(x)-\frac {\tan ^{-1}(x)}{2 \left (1+x^2\right )}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 21, normalized size = 0.66 \[ \frac {\left (x^2-1\right ) \tan ^{-1}(x)+x}{4 \left (x^2+1\right )} \]
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x \tan ^{-1}(x)}{\left (1+x^2\right )^2} \, dx \]
Verification is Not applicable to the result.
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fricas [A] time = 1.10, size = 19, normalized size = 0.59 \[ \frac {{\left (x^{2} - 1\right )} \arctan \relax (x) + x}{4 \, {\left (x^{2} + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.06, size = 26, normalized size = 0.81 \[ \frac {x}{4 \, {\left (x^{2} + 1\right )}} - \frac {\arctan \relax (x)}{2 \, {\left (x^{2} + 1\right )}} + \frac {1}{4} \, \arctan \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.36, size = 27, normalized size = 0.84
method | result | size |
default | \(\frac {x}{4 x^{2}+4}+\frac {\arctan \relax (x )}{4}-\frac {\arctan \relax (x )}{2 \left (x^{2}+1\right )}\) | \(27\) |
risch | \(\frac {i \ln \left (i x +1\right )}{4 x^{2}+4}-\frac {i \left (2 \ln \left (-i x +1\right )+\ln \left (x -i\right ) x^{2}+\ln \left (x -i\right )-\ln \left (x +i\right ) x^{2}-\ln \left (x +i\right )+2 i x \right )}{8 \left (x +i\right ) \left (x -i\right )}\) | \(79\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.96, size = 26, normalized size = 0.81 \[ \frac {x}{4 \, {\left (x^{2} + 1\right )}} - \frac {\arctan \relax (x)}{2 \, {\left (x^{2} + 1\right )}} + \frac {1}{4} \, \arctan \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.08, size = 21, normalized size = 0.66 \[ \frac {\mathrm {atan}\relax (x)}{4}+\frac {\frac {x}{4}-\frac {\mathrm {atan}\relax (x)}{2}}{x^2+1} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.64, size = 31, normalized size = 0.97 \[ \frac {x^{2} \operatorname {atan}{\relax (x )}}{4 x^{2} + 4} + \frac {x}{4 x^{2} + 4} - \frac {\operatorname {atan}{\relax (x )}}{4 x^{2} + 4} \]
Verification of antiderivative is not currently implemented for this CAS.
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