Optimal. Leaf size=110 \[ -\frac{i \text{PolyLog}\left (2,-\frac{i \sqrt{a} f^x}{\sqrt{b}}\right )}{2 \sqrt{a} \sqrt{b} \log ^2(f)}+\frac{i \text{PolyLog}\left (2,\frac{i \sqrt{a} f^x}{\sqrt{b}}\right )}{2 \sqrt{a} \sqrt{b} \log ^2(f)}+\frac{x \tan ^{-1}\left (\frac{\sqrt{a} f^x}{\sqrt{b}}\right )}{\sqrt{a} \sqrt{b} \log (f)} \]
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Rubi [A] time = 0.0952606, antiderivative size = 110, 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 = {2282, 205, 2266, 12, 4848, 2391} \[ -\frac{i \text{PolyLog}\left (2,-\frac{i \sqrt{a} f^x}{\sqrt{b}}\right )}{2 \sqrt{a} \sqrt{b} \log ^2(f)}+\frac{i \text{PolyLog}\left (2,\frac{i \sqrt{a} f^x}{\sqrt{b}}\right )}{2 \sqrt{a} \sqrt{b} \log ^2(f)}+\frac{x \tan ^{-1}\left (\frac{\sqrt{a} f^x}{\sqrt{b}}\right )}{\sqrt{a} \sqrt{b} \log (f)} \]
Antiderivative was successfully verified.
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Rule 2282
Rule 205
Rule 2266
Rule 12
Rule 4848
Rule 2391
Rubi steps
\begin{align*} \int \frac{x}{b f^{-x}+a f^x} \, dx &=\frac{x \tan ^{-1}\left (\frac{\sqrt{a} f^x}{\sqrt{b}}\right )}{\sqrt{a} \sqrt{b} \log (f)}-\int \frac{\tan ^{-1}\left (\frac{\sqrt{a} f^x}{\sqrt{b}}\right )}{\sqrt{a} \sqrt{b} \log (f)} \, dx\\ &=\frac{x \tan ^{-1}\left (\frac{\sqrt{a} f^x}{\sqrt{b}}\right )}{\sqrt{a} \sqrt{b} \log (f)}-\frac{\int \tan ^{-1}\left (\frac{\sqrt{a} f^x}{\sqrt{b}}\right ) \, dx}{\sqrt{a} \sqrt{b} \log (f)}\\ &=\frac{x \tan ^{-1}\left (\frac{\sqrt{a} f^x}{\sqrt{b}}\right )}{\sqrt{a} \sqrt{b} \log (f)}-\frac{\operatorname{Subst}\left (\int \frac{\tan ^{-1}\left (\frac{\sqrt{a} x}{\sqrt{b}}\right )}{x} \, dx,x,f^x\right )}{\sqrt{a} \sqrt{b} \log ^2(f)}\\ &=\frac{x \tan ^{-1}\left (\frac{\sqrt{a} f^x}{\sqrt{b}}\right )}{\sqrt{a} \sqrt{b} \log (f)}-\frac{i \operatorname{Subst}\left (\int \frac{\log \left (1-\frac{i \sqrt{a} x}{\sqrt{b}}\right )}{x} \, dx,x,f^x\right )}{2 \sqrt{a} \sqrt{b} \log ^2(f)}+\frac{i \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{i \sqrt{a} x}{\sqrt{b}}\right )}{x} \, dx,x,f^x\right )}{2 \sqrt{a} \sqrt{b} \log ^2(f)}\\ &=\frac{x \tan ^{-1}\left (\frac{\sqrt{a} f^x}{\sqrt{b}}\right )}{\sqrt{a} \sqrt{b} \log (f)}-\frac{i \text{Li}_2\left (-\frac{i \sqrt{a} f^x}{\sqrt{b}}\right )}{2 \sqrt{a} \sqrt{b} \log ^2(f)}+\frac{i \text{Li}_2\left (\frac{i \sqrt{a} f^x}{\sqrt{b}}\right )}{2 \sqrt{a} \sqrt{b} \log ^2(f)}\\ \end{align*}
Mathematica [A] time = 0.062469, size = 108, normalized size = 0.98 \[ \frac{i \left (-\text{PolyLog}\left (2,-\frac{i \sqrt{a} f^x}{\sqrt{b}}\right )+\text{PolyLog}\left (2,\frac{i \sqrt{a} f^x}{\sqrt{b}}\right )+x \log (f) \left (\log \left (1-\frac{i \sqrt{a} f^x}{\sqrt{b}}\right )-\log \left (1+\frac{i \sqrt{a} f^x}{\sqrt{b}}\right )\right )\right )}{2 \sqrt{a} \sqrt{b} \log ^2(f)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.041, size = 134, normalized size = 1.2 \begin{align*}{\frac{x}{2\,\ln \left ( f \right ) }\ln \left ({ \left ( -a{f}^{x}+\sqrt{-ab} \right ){\frac{1}{\sqrt{-ab}}}} \right ){\frac{1}{\sqrt{-ab}}}}-{\frac{x}{2\,\ln \left ( f \right ) }\ln \left ({ \left ( a{f}^{x}+\sqrt{-ab} \right ){\frac{1}{\sqrt{-ab}}}} \right ){\frac{1}{\sqrt{-ab}}}}+{\frac{1}{2\, \left ( \ln \left ( f \right ) \right ) ^{2}}{\it dilog} \left ({ \left ( -a{f}^{x}+\sqrt{-ab} \right ){\frac{1}{\sqrt{-ab}}}} \right ){\frac{1}{\sqrt{-ab}}}}-{\frac{1}{2\, \left ( \ln \left ( f \right ) \right ) ^{2}}{\it dilog} \left ({ \left ( a{f}^{x}+\sqrt{-ab} \right ){\frac{1}{\sqrt{-ab}}}} \right ){\frac{1}{\sqrt{-ab}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.72942, size = 247, normalized size = 2.25 \begin{align*} -\frac{x \sqrt{-\frac{a}{b}} \log \left (f^{x} \sqrt{-\frac{a}{b}} + 1\right ) \log \left (f\right ) - x \sqrt{-\frac{a}{b}} \log \left (-f^{x} \sqrt{-\frac{a}{b}} + 1\right ) \log \left (f\right ) - \sqrt{-\frac{a}{b}}{\rm Li}_2\left (f^{x} \sqrt{-\frac{a}{b}}\right ) + \sqrt{-\frac{a}{b}}{\rm Li}_2\left (-f^{x} \sqrt{-\frac{a}{b}}\right )}{2 \, a \log \left (f\right )^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{f^{x} x}{a f^{2 x} + b}\, dx \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}{a f^{x} + \frac{b}{f^{x}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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