Optimal. Leaf size=14 \[ x-\frac{1}{2} \tan ^{-1}(x)-\frac{1}{2} \tanh ^{-1}(x) \]
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Rubi [A] time = 0.0063513, antiderivative size = 14, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.364, Rules used = {321, 212, 206, 203} \[ x-\frac{1}{2} \tan ^{-1}(x)-\frac{1}{2} \tanh ^{-1}(x) \]
Antiderivative was successfully verified.
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Rule 321
Rule 212
Rule 206
Rule 203
Rubi steps
\begin{align*} \int \frac{x^4}{-1+x^4} \, dx &=x+\int \frac{1}{-1+x^4} \, dx\\ &=x-\frac{1}{2} \int \frac{1}{1-x^2} \, dx-\frac{1}{2} \int \frac{1}{1+x^2} \, dx\\ &=x-\frac{1}{2} \tan ^{-1}(x)-\frac{1}{2} \tanh ^{-1}(x)\\ \end{align*}
Mathematica [A] time = 0.0041189, size = 26, normalized size = 1.86 \[ x+\frac{1}{4} \log (1-x)-\frac{1}{4} \log (x+1)-\frac{1}{2} \tan ^{-1}(x) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 19, normalized size = 1.4 \begin{align*} x+{\frac{\ln \left ( -1+x \right ) }{4}}-{\frac{\ln \left ( 1+x \right ) }{4}}-{\frac{\arctan \left ( x \right ) }{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.40145, size = 24, normalized size = 1.71 \begin{align*} x - \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 = 1.90322, size = 72, normalized size = 5.14 \begin{align*} x - \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 = 0.113575, size = 19, normalized size = 1.36 \begin{align*} x + \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 [A] time = 1.05932, size = 27, normalized size = 1.93 \begin{align*} x - \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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