Optimal. Leaf size=32 \[ \frac{x^6}{4 \left (1-x^4\right )}+\frac{3 x^2}{4}-\frac{3}{4} \tanh ^{-1}\left (x^2\right ) \]
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Rubi [A] time = 0.0130546, antiderivative size = 32, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.312, Rules used = {28, 275, 288, 321, 207} \[ \frac{x^6}{4 \left (1-x^4\right )}+\frac{3 x^2}{4}-\frac{3}{4} \tanh ^{-1}\left (x^2\right ) \]
Antiderivative was successfully verified.
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Rule 28
Rule 275
Rule 288
Rule 321
Rule 207
Rubi steps
\begin{align*} \int \frac{x^9}{1-2 x^4+x^8} \, dx &=\int \frac{x^9}{\left (-1+x^4\right )^2} \, dx\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{x^4}{\left (-1+x^2\right )^2} \, dx,x,x^2\right )\\ &=\frac{x^6}{4 \left (1-x^4\right )}+\frac{3}{4} \operatorname{Subst}\left (\int \frac{x^2}{-1+x^2} \, dx,x,x^2\right )\\ &=\frac{3 x^2}{4}+\frac{x^6}{4 \left (1-x^4\right )}+\frac{3}{4} \operatorname{Subst}\left (\int \frac{1}{-1+x^2} \, dx,x,x^2\right )\\ &=\frac{3 x^2}{4}+\frac{x^6}{4 \left (1-x^4\right )}-\frac{3}{4} \tanh ^{-1}\left (x^2\right )\\ \end{align*}
Mathematica [A] time = 0.0236068, size = 39, normalized size = 1.22 \[ \frac{1}{8} \left (2 \left (\frac{1}{1-x^4}+2\right ) x^2+3 \log \left (1-x^2\right )-3 \log \left (x^2+1\right )\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.009, size = 41, normalized size = 1.3 \begin{align*}{\frac{{x}^{2}}{2}}-{\frac{1}{8\,{x}^{2}+8}}-{\frac{3\,\ln \left ({x}^{2}+1 \right ) }{8}}-{\frac{1}{8\,{x}^{2}-8}}+{\frac{3\,\ln \left ({x}^{2}-1 \right ) }{8}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.01735, size = 46, normalized size = 1.44 \begin{align*} \frac{1}{2} \, x^{2} - \frac{x^{2}}{4 \,{\left (x^{4} - 1\right )}} - \frac{3}{8} \, \log \left (x^{2} + 1\right ) + \frac{3}{8} \, \log \left (x^{2} - 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.45559, size = 115, normalized size = 3.59 \begin{align*} \frac{4 \, x^{6} - 6 \, x^{2} - 3 \,{\left (x^{4} - 1\right )} \log \left (x^{2} + 1\right ) + 3 \,{\left (x^{4} - 1\right )} \log \left (x^{2} - 1\right )}{8 \,{\left (x^{4} - 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.126685, size = 34, normalized size = 1.06 \begin{align*} \frac{x^{2}}{2} - \frac{x^{2}}{4 x^{4} - 4} + \frac{3 \log{\left (x^{2} - 1 \right )}}{8} - \frac{3 \log{\left (x^{2} + 1 \right )}}{8} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.1067, size = 47, normalized size = 1.47 \begin{align*} \frac{1}{2} \, x^{2} - \frac{x^{2}}{4 \,{\left (x^{4} - 1\right )}} - \frac{3}{8} \, \log \left (x^{2} + 1\right ) + \frac{3}{8} \, \log \left ({\left | x^{2} - 1 \right |}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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