Optimal. Leaf size=13 \[ -\frac {1}{2} \tan ^{-1}(x)-\frac {1}{2} \tanh ^{-1}(x) \]
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Rubi [A] time = 0.00, antiderivative size = 13, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {212, 206, 203} \[ -\frac {1}{2} \tan ^{-1}(x)-\frac {1}{2} \tanh ^{-1}(x) \]
Antiderivative was successfully verified.
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Rule 203
Rule 206
Rule 212
Rubi steps
\begin {align*} \int \frac {1}{-1+x^4} \, dx &=-\left (\frac {1}{2} \int \frac {1}{1-x^2} \, dx\right )-\frac {1}{2} \int \frac {1}{1+x^2} \, dx\\ &=-\frac {1}{2} \tan ^{-1}(x)-\frac {1}{2} \tanh ^{-1}(x)\\ \end {align*}
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Mathematica [A] time = 0.00, size = 25, normalized size = 1.92 \[ \frac {1}{4} \log (1-x)-\frac {1}{4} \log (x+1)-\frac {1}{2} \tan ^{-1}(x) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.40, size = 17, normalized size = 1.31 \[ -\frac {1}{2} \, \arctan \relax (x) - \frac {1}{4} \, \log \left (x + 1\right ) + \frac {1}{4} \, \log \left (x - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 1.03, size = 19, normalized size = 1.46 \[ -\frac {1}{2} \, \arctan \relax (x) - \frac {1}{4} \, \log \left ({\left | x + 1 \right |}\right ) + \frac {1}{4} \, \log \left ({\left | x - 1 \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 10, normalized size = 0.77 \[ -\frac {\arctanh \relax (x )}{2}-\frac {\arctan \relax (x )}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.41, size = 17, normalized size = 1.31 \[ -\frac {1}{2} \, \arctan \relax (x) - \frac {1}{4} \, \log \left (x + 1\right ) + \frac {1}{4} \, \log \left (x - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.16, size = 9, normalized size = 0.69 \[ -\frac {\mathrm {atan}\relax (x)}{2}-\frac {\mathrm {atanh}\relax (x)}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.12, size = 17, normalized size = 1.31 \[ \frac {\log {\left (x - 1 \right )}}{4} - \frac {\log {\left (x + 1 \right )}}{4} - \frac {\operatorname {atan}{\relax (x )}}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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