Optimal. Leaf size=24 \[ \frac{2}{4-x^2}+\frac{1}{2} \log \left (4-x^2\right ) \]
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Rubi [A] time = 0.0147868, antiderivative size = 24, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.188, Rules used = {28, 266, 43} \[ \frac{2}{4-x^2}+\frac{1}{2} \log \left (4-x^2\right ) \]
Antiderivative was successfully verified.
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Rule 28
Rule 266
Rule 43
Rubi steps
\begin{align*} \int \frac{x^3}{16-8 x^2+x^4} \, dx &=\int \frac{x^3}{\left (-4+x^2\right )^2} \, dx\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{x}{(-4+x)^2} \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (\frac{4}{(-4+x)^2}+\frac{1}{-4+x}\right ) \, dx,x,x^2\right )\\ &=\frac{2}{4-x^2}+\frac{1}{2} \log \left (4-x^2\right )\\ \end{align*}
Mathematica [A] time = 0.0055415, size = 20, normalized size = 0.83 \[ \frac{1}{2} \log \left (x^2-4\right )-\frac{2}{x^2-4} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.049, size = 19, normalized size = 0.8 \begin{align*} -2\, \left ({x}^{2}-4 \right ) ^{-1}+{\frac{\ln \left ({x}^{2}-4 \right ) }{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.982657, size = 24, normalized size = 1. \begin{align*} -\frac{2}{x^{2} - 4} + \frac{1}{2} \, \log \left (x^{2} - 4\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.66334, size = 59, normalized size = 2.46 \begin{align*} \frac{{\left (x^{2} - 4\right )} \log \left (x^{2} - 4\right ) - 4}{2 \,{\left (x^{2} - 4\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.089023, size = 14, normalized size = 0.58 \begin{align*} \frac{\log{\left (x^{2} - 4 \right )}}{2} - \frac{2}{x^{2} - 4} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.15778, size = 26, normalized size = 1.08 \begin{align*} -\frac{2}{x^{2} - 4} + \frac{1}{2} \, \log \left ({\left | x^{2} - 4 \right |}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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