Optimal. Leaf size=25 \[ \frac{1}{2} \log \left (1-x^2\right )+\frac{3}{2} \log \left (4-x^2\right ) \]
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Rubi [A] time = 0.0293958, antiderivative size = 25, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {1593, 1247, 632, 31} \[ \frac{1}{2} \log \left (1-x^2\right )+\frac{3}{2} \log \left (4-x^2\right ) \]
Antiderivative was successfully verified.
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Rule 1593
Rule 1247
Rule 632
Rule 31
Rubi steps
\begin{align*} \int \frac{-7 x+4 x^3}{4-5 x^2+x^4} \, dx &=\int \frac{x \left (-7+4 x^2\right )}{4-5 x^2+x^4} \, dx\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{-7+4 x}{4-5 x+x^2} \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{-1+x} \, dx,x,x^2\right )+\frac{3}{2} \operatorname{Subst}\left (\int \frac{1}{-4+x} \, dx,x,x^2\right )\\ &=\frac{1}{2} \log \left (1-x^2\right )+\frac{3}{2} \log \left (4-x^2\right )\\ \end{align*}
Mathematica [A] time = 0.0054111, size = 25, normalized size = 1. \[ \frac{1}{2} \log \left (1-x^2\right )+\frac{3}{2} \log \left (4-x^2\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.005, size = 18, normalized size = 0.7 \begin{align*}{\frac{\ln \left ({x}^{2}-1 \right ) }{2}}+{\frac{3\,\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 = 0.945906, size = 34, normalized size = 1.36 \begin{align*} \frac{3}{2} \, \log \left (x + 2\right ) + \frac{1}{2} \, \log \left (x + 1\right ) + \frac{1}{2} \, \log \left (x - 1\right ) + \frac{3}{2} \, \log \left (x - 2\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.47928, size = 50, normalized size = 2. \begin{align*} \frac{1}{2} \, \log \left (x^{2} - 1\right ) + \frac{3}{2} \, \log \left (x^{2} - 4\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.110134, size = 17, normalized size = 0.68 \begin{align*} \frac{3 \log{\left (x^{2} - 4 \right )}}{2} + \frac{\log{\left (x^{2} - 1 \right )}}{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.08159, size = 26, normalized size = 1.04 \begin{align*} \frac{1}{2} \, \log \left ({\left | x^{2} - 1 \right |}\right ) + \frac{3}{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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