Optimal. Leaf size=63 \[ -\frac{\log \left (a f^{2 x}+b\right )}{4 a b \log ^2(f)}-\frac{x}{2 a \log (f) \left (a f^{2 x}+b\right )}+\frac{x}{2 a b \log (f)} \]
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Rubi [A] time = 0.0783578, antiderivative size = 63, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.353, Rules used = {2283, 2191, 2282, 36, 29, 31} \[ -\frac{\log \left (a f^{2 x}+b\right )}{4 a b \log ^2(f)}-\frac{x}{2 a \log (f) \left (a f^{2 x}+b\right )}+\frac{x}{2 a b \log (f)} \]
Antiderivative was successfully verified.
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Rule 2283
Rule 2191
Rule 2282
Rule 36
Rule 29
Rule 31
Rubi steps
\begin{align*} \int \frac{x}{\left (b f^{-x}+a f^x\right )^2} \, dx &=\int \frac{f^{2 x} x}{\left (b+a f^{2 x}\right )^2} \, dx\\ &=-\frac{x}{2 a \left (b+a f^{2 x}\right ) \log (f)}+\frac{\int \frac{1}{b+a f^{2 x}} \, dx}{2 a \log (f)}\\ &=-\frac{x}{2 a \left (b+a f^{2 x}\right ) \log (f)}+\frac{\operatorname{Subst}\left (\int \frac{1}{x (b+a x)} \, dx,x,f^{2 x}\right )}{4 a \log ^2(f)}\\ &=-\frac{x}{2 a \left (b+a f^{2 x}\right ) \log (f)}-\frac{\operatorname{Subst}\left (\int \frac{1}{b+a x} \, dx,x,f^{2 x}\right )}{4 b \log ^2(f)}+\frac{\operatorname{Subst}\left (\int \frac{1}{x} \, dx,x,f^{2 x}\right )}{4 a b \log ^2(f)}\\ &=\frac{x}{2 a b \log (f)}-\frac{x}{2 a \left (b+a f^{2 x}\right ) \log (f)}-\frac{\log \left (b+a f^{2 x}\right )}{4 a b \log ^2(f)}\\ \end{align*}
Mathematica [A] time = 0.0572333, size = 48, normalized size = 0.76 \[ \frac{\frac{2 x f^{2 x} \log (f)}{a f^{2 x}+b}-\frac{\log \left (a f^{2 x}+b\right )}{a}}{4 b \log ^2(f)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.016, size = 56, normalized size = 0.9 \begin{align*}{\frac{x \left ({{\rm e}^{x\ln \left ( f \right ) }} \right ) ^{2}}{2\,b\ln \left ( f \right ) \left ( \left ({{\rm e}^{x\ln \left ( f \right ) }} \right ) ^{2}a+b \right ) }}-{\frac{\ln \left ( \left ({{\rm e}^{x\ln \left ( f \right ) }} \right ) ^{2}a+b \right ) }{4\, \left ( \ln \left ( f \right ) \right ) ^{2}ab}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.06419, size = 73, normalized size = 1.16 \begin{align*} \frac{f^{2 \, x} x}{2 \,{\left (a b f^{2 \, x} \log \left (f\right ) + b^{2} \log \left (f\right )\right )}} - \frac{\log \left (\frac{a f^{2 \, x} + b}{a}\right )}{4 \, a b \log \left (f\right )^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.50443, size = 144, normalized size = 2.29 \begin{align*} \frac{2 \, a f^{2 \, x} x \log \left (f\right ) -{\left (a f^{2 \, x} + b\right )} \log \left (a f^{2 \, x} + b\right )}{4 \,{\left (a^{2} b f^{2 \, x} \log \left (f\right )^{2} + a b^{2} \log \left (f\right )^{2}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.296525, size = 54, normalized size = 0.86 \begin{align*} \frac{x}{2 a b \log{\left (f \right )} + 2 b^{2} f^{- 2 x} \log{\left (f \right )}} - \frac{x}{2 a b \log{\left (f \right )}} - \frac{\log{\left (\frac{a}{b} + f^{- 2 x} \right )}}{4 a b \log{\left (f \right )}^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x}{{\left (a f^{x} + \frac{b}{f^{x}}\right )}^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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