Optimal. Leaf size=8 \[ \frac{1}{4} \tanh ^{-1}\left (x^4\right ) \]
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Rubi [A] time = 0.0041538, antiderivative size = 8, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {275, 206} \[ \frac{1}{4} \tanh ^{-1}\left (x^4\right ) \]
Antiderivative was successfully verified.
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Rule 275
Rule 206
Rubi steps
\begin{align*} \int \frac{x^3}{1-x^8} \, dx &=\frac{1}{4} \operatorname{Subst}\left (\int \frac{1}{1-x^2} \, dx,x,x^4\right )\\ &=\frac{1}{4} \tanh ^{-1}\left (x^4\right )\\ \end{align*}
Mathematica [B] time = 0.0034813, size = 23, normalized size = 2.88 \[ \frac{1}{8} \log \left (x^4+1\right )-\frac{1}{8} \log \left (1-x^4\right ) \]
Antiderivative was successfully verified.
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Maple [B] time = 0.005, size = 30, normalized size = 3.8 \begin{align*} -{\frac{\ln \left ( -1+x \right ) }{8}}-{\frac{\ln \left ( 1+x \right ) }{8}}-{\frac{\ln \left ({x}^{2}+1 \right ) }{8}}+{\frac{\ln \left ({x}^{4}+1 \right ) }{8}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 0.973019, size = 23, normalized size = 2.88 \begin{align*} \frac{1}{8} \, \log \left (x^{4} + 1\right ) - \frac{1}{8} \, \log \left (x^{4} - 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.2715, size = 50, normalized size = 6.25 \begin{align*} \frac{1}{8} \, \log \left (x^{4} + 1\right ) - \frac{1}{8} \, \log \left (x^{4} - 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 0.105728, size = 15, normalized size = 1.88 \begin{align*} - \frac{\log{\left (x^{4} - 1 \right )}}{8} + \frac{\log{\left (x^{4} + 1 \right )}}{8} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.13302, size = 24, normalized size = 3. \begin{align*} \frac{1}{8} \, \log \left (x^{4} + 1\right ) - \frac{1}{8} \, \log \left ({\left | x^{4} - 1 \right |}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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