Optimal. Leaf size=21 \[ \frac {x}{2 \left (1-x^2\right )}+\frac {1}{2} \tanh ^{-1}(x) \]
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Rubi [A]
time = 0.00, antiderivative size = 21, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {205, 213}
\begin {gather*} \frac {x}{2 \left (1-x^2\right )}+\frac {1}{2} \tanh ^{-1}(x) \end {gather*}
Antiderivative was successfully verified.
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Rule 205
Rule 213
Rubi steps
\begin {align*} \int \frac {1}{\left (-1+x^2\right )^2} \, dx &=\frac {x}{2 \left (1-x^2\right )}-\frac {1}{2} \int \frac {1}{-1+x^2} \, dx\\ &=\frac {x}{2 \left (1-x^2\right )}+\frac {1}{2} \tanh ^{-1}(x)\\ \end {align*}
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Mathematica [A]
time = 0.00, size = 27, normalized size = 1.29 \begin {gather*} \frac {1}{4} \left (-\frac {2 x}{-1+x^2}-\log (1-x)+\log (1+x)\right ) \end {gather*}
Antiderivative was successfully verified.
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Mathics [A]
time = 1.76, size = 30, normalized size = 1.43 \begin {gather*} \frac {-2 x+\left (-1+x^2\right ) \left (\text {Log}\left [1+x\right ]-\text {Log}\left [-1+x\right ]\right )}{-4+4 x^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.07, size = 28, normalized size = 1.33
method | result | size |
meijerg | \(-\frac {i \left (\frac {2 i x}{-2 x^{2}+2}+i \arctanh \left (x \right )\right )}{2}\) | \(23\) |
norman | \(-\frac {x}{2 \left (x^{2}-1\right )}-\frac {\ln \left (-1+x \right )}{4}+\frac {\ln \left (1+x \right )}{4}\) | \(24\) |
risch | \(-\frac {x}{2 \left (x^{2}-1\right )}-\frac {\ln \left (-1+x \right )}{4}+\frac {\ln \left (1+x \right )}{4}\) | \(24\) |
default | \(-\frac {1}{4 \left (-1+x \right )}-\frac {\ln \left (-1+x \right )}{4}-\frac {1}{4 \left (1+x \right )}+\frac {\ln \left (1+x \right )}{4}\) | \(28\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 23, normalized size = 1.10 \begin {gather*} -\frac {x}{2 \, {\left (x^{2} - 1\right )}} + \frac {1}{4} \, \log \left (x + 1\right ) - \frac {1}{4} \, \log \left (x - 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 34 vs.
\(2 (15) = 30\).
time = 0.32, size = 34, normalized size = 1.62 \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}{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 = 0.05, size = 20, normalized size = 0.95 \begin {gather*} - \frac {x}{2 x^{2} - 2} - \frac {\log {\left (x - 1 \right )}}{4} + \frac {\log {\left (x + 1 \right )}}{4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.00, size = 29, normalized size = 1.38 \begin {gather*} -\frac {\ln \left |x-1\right |}{4}+\frac {\ln \left |x+1\right |}{4}-\frac {x}{2 \left (x^{2}-1\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.09, size = 17, normalized size = 0.81 \begin {gather*} \frac {\mathrm {atanh}\left (x\right )}{2}-\frac {x}{2\,\left (x^2-1\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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