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 = 0.08, antiderivative size = 63, normalized size of antiderivative = 1.00, 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 29
Rule 31
Rule 36
Rule 2191
Rule 2282
Rule 2283
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*}
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Mathematica [A] time = 0.06, 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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fricas [A] time = 0.42, size = 61, normalized size = 0.97 \[ \frac {2 \, a f^{2 \, x} x \log \relax (f) - {\left (a f^{2 \, x} + b\right )} \log \left (a f^{2 \, x} + b\right )}{4 \, {\left (a^{2} b f^{2 \, x} \log \relax (f)^{2} + a b^{2} \log \relax (f)^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x}{{\left (a f^{x} + \frac {b}{f^{x}}\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 56, normalized size = 0.89 \[ \frac {x \,{\mathrm e}^{2 x \ln \relax (f )}}{2 \left (a \,{\mathrm e}^{2 x \ln \relax (f )}+b \right ) b \ln \relax (f )}-\frac {\ln \left (a \,{\mathrm e}^{2 x \ln \relax (f )}+b \right )}{4 a b \ln \relax (f )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.50, size = 54, normalized size = 0.86 \[ \frac {f^{2 \, x} x}{2 \, {\left (a b f^{2 \, x} \log \relax (f) + b^{2} \log \relax (f)\right )}} - \frac {\log \left (\frac {a f^{2 \, x} + b}{a}\right )}{4 \, a b \log \relax (f)^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.61, size = 51, normalized size = 0.81 \[ \frac {f^{2\,x}\,x}{2\,\left (b^2\,\ln \relax (f)+a\,b\,f^{2\,x}\,\ln \relax (f)\right )}-\frac {\ln \left (b+a\,f^{2\,x}\right )}{4\,a\,b\,{\ln \relax (f)}^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.19, size = 53, normalized size = 0.84 \[ - \frac {x}{2 a^{2} f^{2 x} \log {\relax (f )} + 2 a b \log {\relax (f )}} + \frac {x}{2 a b \log {\relax (f )}} - \frac {\log {\left (f^{2 x} + \frac {b}{a} \right )}}{4 a b \log {\relax (f )}^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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