Optimal. Leaf size=8 \[ \frac {1}{x}-\tanh ^{-1}(x) \]
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Rubi [A]
time = 0.00, antiderivative size = 8, 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 = {1607, 331, 213}
\begin {gather*} \frac {1}{x}-\tanh ^{-1}(x) \end {gather*}
Antiderivative was successfully verified.
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Rule 213
Rule 331
Rule 1607
Rubi steps
\begin {align*} \int \frac {1}{-x^2+x^4} \, dx &=\int \frac {1}{x^2 \left (-1+x^2\right )} \, dx\\ &=\frac {1}{x}+\int \frac {1}{-1+x^2} \, dx\\ &=\frac {1}{x}-\tanh ^{-1}(x)\\ \end {align*}
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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(22\) vs. \(2(8)=16\).
time = 0.00, size = 22, normalized size = 2.75 \begin {gather*} \frac {1}{x}+\frac {1}{2} \log (1-x)-\frac {1}{2} \log (1+x) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.04, size = 17, normalized size = 2.12
method | result | size |
meijerg | \(-\frac {i \left (\frac {2 i}{x}-2 i \arctanh \left (x \right )\right )}{2}\) | \(16\) |
default | \(\frac {1}{x}+\frac {\ln \left (-1+x \right )}{2}-\frac {\ln \left (1+x \right )}{2}\) | \(17\) |
norman | \(\frac {1}{x}+\frac {\ln \left (-1+x \right )}{2}-\frac {\ln \left (1+x \right )}{2}\) | \(17\) |
risch | \(\frac {1}{x}+\frac {\ln \left (-1+x \right )}{2}-\frac {\ln \left (1+x \right )}{2}\) | \(17\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 1.44, size = 16, normalized size = 2.00 \begin {gather*} \frac {1}{x} - \frac {1}{2} \, \log \left (x + 1\right ) + \frac {1}{2} \, \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. 20 vs.
\(2 (8) = 16\).
time = 1.31, size = 20, normalized size = 2.50 \begin {gather*} -\frac {x \log \left (x + 1\right ) - x \log \left (x - 1\right ) - 2}{2 \, x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 15 vs.
\(2 (5) = 10\).
time = 0.03, size = 15, normalized size = 1.88 \begin {gather*} \frac {\log {\left (x - 1 \right )}}{2} - \frac {\log {\left (x + 1 \right )}}{2} + \frac {1}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 18 vs.
\(2 (8) = 16\).
time = 0.44, size = 18, normalized size = 2.25 \begin {gather*} \frac {1}{x} - \frac {1}{2} \, \log \left ({\left | x + 1 \right |}\right ) + \frac {1}{2} \, \log \left ({\left | x - 1 \right |}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.17, size = 8, normalized size = 1.00 \begin {gather*} \frac {1}{x}-\mathrm {atanh}\left (x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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