Optimal. Leaf size=22 \[ -\frac {1}{2 a \log (f) \left (a f^{2 x}+b\right )} \]
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Rubi [A] time = 0.02, antiderivative size = 22, normalized size of antiderivative = 1.00, 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 261
Rule 2282
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*}
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Mathematica [A] time = 0.02, 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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fricas [A] time = 0.41, size = 21, normalized size = 0.95 \[ -\frac {1}{2 \, {\left (a^{2} f^{2 \, x} \log \relax (f) + a b \log \relax (f)\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.34, size = 20, normalized size = 0.91 \[ -\frac {1}{2 \, {\left (a f^{2 \, x} + b\right )} a \log \relax (f)} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 21, normalized size = 0.95 \[ -\frac {1}{2 \left (a \,f^{2 x}+b \right ) a \ln \relax (f )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.43, size = 23, normalized size = 1.05 \[ \frac {1}{2 \, {\left (a b + \frac {b^{2}}{f^{2 \, x}}\right )} \log \relax (f)} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.61, size = 20, normalized size = 0.91 \[ -\frac {1}{2\,a\,\ln \relax (f)\,\left (b+a\,f^{2\,x}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.12, size = 24, normalized size = 1.09 \[ - \frac {1}{2 a^{2} f^{2 x} \log {\relax (f )} + 2 a b \log {\relax (f )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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