Optimal. Leaf size=13 \[ -\frac{1}{2} \tan ^{-1}(x)-\frac{1}{2} \tanh ^{-1}(x) \]
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Rubi [A] time = 0.0031654, antiderivative size = 13, normalized size of antiderivative = 1., 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 212
Rule 206
Rule 203
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*}
Mathematica [A] time = 0.0035192, 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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Maple [A] time = 0., size = 10, normalized size = 0.8 \begin{align*} -{\frac{\arctan \left ( x \right ) }{2}}-{\frac{{\it Artanh} \left ( x \right ) }{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.41753, size = 23, normalized size = 1.77 \begin{align*} -\frac{1}{2} \, \arctan \left (x\right ) - \frac{1}{4} \, \log \left (x + 1\right ) + \frac{1}{4} \, \log \left (x - 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.88316, size = 68, normalized size = 5.23 \begin{align*} -\frac{1}{2} \, \arctan \left (x\right ) - \frac{1}{4} \, \log \left (x + 1\right ) + \frac{1}{4} \, \log \left (x - 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.114985, size = 17, normalized size = 1.31 \begin{align*} \frac{\log{\left (x - 1 \right )}}{4} - \frac{\log{\left (x + 1 \right )}}{4} - \frac{\operatorname{atan}{\left (x \right )}}{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.08715, size = 26, normalized size = 2. \begin{align*} -\frac{1}{2} \, \arctan \left (x\right ) - \frac{1}{4} \, \log \left ({\left | x + 1 \right |}\right ) + \frac{1}{4} \, \log \left ({\left | x - 1 \right |}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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