Optimal. Leaf size=22 \[ -\frac{1}{2 a \log (f) \left (a f^{2 x}+b\right )} \]
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Rubi [A] time = 0.0199484, antiderivative size = 22, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {2282, 261} \[ -\frac{1}{2 a \log (f) \left (a f^{2 x}+b\right )} \]
Antiderivative was successfully verified.
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Rule 2282
Rule 261
Rubi steps
\begin{align*} \int \frac{1}{\left (b f^{-x}+a f^x\right )^2} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{x}{\left (b+a x^2\right )^2} \, dx,x,f^x\right )}{\log (f)}\\ &=-\frac{1}{2 a \left (b+a f^{2 x}\right ) \log (f)}\\ \end{align*}
Mathematica [A] time = 0.020285, size = 23, normalized size = 1.05 \[ -\frac{1}{2 a^2 f^{2 x} \log (f)+2 a b \log (f)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.001, size = 21, normalized size = 1. \begin{align*} -{\frac{1}{2\,\ln \left ( f \right ) a \left ( a \left ({f}^{x} \right ) ^{2}+b \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.09949, size = 31, normalized size = 1.41 \begin{align*} \frac{1}{2 \,{\left (a b + \frac{b^{2}}{f^{2 \, x}}\right )} \log \left (f\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.54727, size = 54, normalized size = 2.45 \begin{align*} -\frac{1}{2 \,{\left (a^{2} f^{2 \, x} \log \left (f\right ) + a b \log \left (f\right )\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.246905, size = 22, normalized size = 1. \begin{align*} \frac{1}{2 a b \log{\left (f \right )} + 2 b^{2} f^{- 2 x} \log{\left (f \right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.23908, size = 27, normalized size = 1.23 \begin{align*} -\frac{1}{2 \,{\left (a f^{2 \, x} + b\right )} a \log \left (f\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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