Optimal. Leaf size=34 \[ -\frac {1}{4 \left (x^2+1\right )}-\frac {x \tan ^{-1}(x)}{2 \left (x^2+1\right )}+\frac {1}{4} \tan ^{-1}(x)^2 \]
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Rubi [A] time = 0.05, antiderivative size = 34, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {4934, 4884} \[ -\frac {1}{4 \left (x^2+1\right )}-\frac {x \tan ^{-1}(x)}{2 \left (x^2+1\right )}+\frac {1}{4} \tan ^{-1}(x)^2 \]
Antiderivative was successfully verified.
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Rule 4884
Rule 4934
Rubi steps
\begin {align*} \int \frac {x^2 \tan ^{-1}(x)}{\left (1+x^2\right )^2} \, dx &=-\frac {1}{4 \left (1+x^2\right )}-\frac {x \tan ^{-1}(x)}{2 \left (1+x^2\right )}+\frac {1}{2} \int \frac {\tan ^{-1}(x)}{1+x^2} \, dx\\ &=-\frac {1}{4 \left (1+x^2\right )}-\frac {x \tan ^{-1}(x)}{2 \left (1+x^2\right )}+\frac {1}{4} \tan ^{-1}(x)^2\\ \end {align*}
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Mathematica [A] time = 0.03, size = 28, normalized size = 0.82 \[ \frac {\left (x^2+1\right ) \tan ^{-1}(x)^2-2 x \tan ^{-1}(x)-1}{4 \left (x^2+1\right )} \]
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^2 \tan ^{-1}(x)}{\left (1+x^2\right )^2} \, dx \]
Verification is Not applicable to the result.
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fricas [A] time = 0.73, size = 26, normalized size = 0.76 \[ \frac {{\left (x^{2} + 1\right )} \arctan \relax (x)^{2} - 2 \, x \arctan \relax (x) - 1}{4 \, {\left (x^{2} + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{2} \arctan \relax (x)}{{\left (x^{2} + 1\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.39, size = 29, normalized size = 0.85
method | result | size |
default | \(-\frac {1}{4 \left (x^{2}+1\right )}-\frac {x \arctan \relax (x )}{2 \left (x^{2}+1\right )}+\frac {\arctan \relax (x )^{2}}{4}\) | \(29\) |
risch | \(-\frac {\ln \left (i x +1\right )^{2}}{16}+\frac {\left (x^{2} \ln \left (-i x +1\right )+\ln \left (-i x +1\right )+2 i x \right ) \ln \left (i x +1\right )}{8 x^{2}+8}-\frac {x^{2} \ln \left (-i x +1\right )^{2}+\ln \left (-i x +1\right )^{2}+4 i \ln \left (-i x +1\right ) x +4}{16 \left (x +i\right ) \left (x -i\right )}\) | \(101\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.98, size = 40, normalized size = 1.18 \[ -\frac {1}{2} \, {\left (\frac {x}{x^{2} + 1} - \arctan \relax (x)\right )} \arctan \relax (x) - \frac {{\left (x^{2} + 1\right )} \arctan \relax (x)^{2} + 1}{4 \, {\left (x^{2} + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.06, size = 23, normalized size = 0.68 \[ \frac {{\mathrm {atan}\relax (x)}^2}{4}-\frac {\frac {x\,\mathrm {atan}\relax (x)}{2}+\frac {1}{4}}{x^2+1} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: RecursionError} \]
Verification of antiderivative is not currently implemented for this CAS.
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