Optimal. Leaf size=19 \[ \frac {1}{4} \tanh ^{-1}(x)-\frac {x}{4 \left (x^2+1\right )} \]
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Rubi [A] time = 0.01, antiderivative size = 19, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {471, 206} \begin {gather*} \frac {1}{4} \tanh ^{-1}(x)-\frac {x}{4 \left (x^2+1\right )} \end {gather*}
Antiderivative was successfully verified.
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Rule 206
Rule 471
Rubi steps
\begin {align*} \int \frac {x^2}{\left (1-x^2\right ) \left (1+x^2\right )^2} \, dx &=-\frac {x}{4 \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} \tanh ^{-1}(x)\\ \end {align*}
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Mathematica [A] time = 0.01, size = 27, normalized size = 1.42 \begin {gather*} \frac {1}{8} \left (-\frac {2 x}{x^2+1}-\log (1-x)+\log (x+1)\right ) \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^2}{\left (1-x^2\right ) \left (1+x^2\right )^2} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [B] time = 0.65, size = 34, normalized size = 1.79 \begin {gather*} \frac {{\left (x^{2} + 1\right )} \log \left (x + 1\right ) - {\left (x^{2} + 1\right )} \log \left (x - 1\right ) - 2 \, x}{8 \, {\left (x^{2} + 1\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.30, size = 30, normalized size = 1.58 \begin {gather*} -\frac {1}{4 \, {\left (x + \frac {1}{x}\right )}} + \frac {1}{16} \, \log \left ({\left | x + \frac {1}{x} + 2 \right |}\right ) - \frac {1}{16} \, \log \left ({\left | x + \frac {1}{x} - 2 \right |}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 24, normalized size = 1.26 \begin {gather*} -\frac {x}{4 \left (x^{2}+1\right )}-\frac {\ln \left (x -1\right )}{8}+\frac {\ln \left (x +1\right )}{8} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.65, size = 23, normalized size = 1.21 \begin {gather*} -\frac {x}{4 \, {\left (x^{2} + 1\right )}} + \frac {1}{8} \, \log \left (x + 1\right ) - \frac {1}{8} \, \log \left (x - 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.14, size = 17, normalized size = 0.89 \begin {gather*} \frac {\mathrm {atanh}\relax (x)}{4}-\frac {x}{4\,\left (x^2+1\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.12, size = 20, normalized size = 1.05 \begin {gather*} - \frac {x}{4 x^{2} + 4} - \frac {\log {\left (x - 1 \right )}}{8} + \frac {\log {\left (x + 1 \right )}}{8} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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