Integrand size = 17, antiderivative size = 110 \[ \int \frac {x}{b f^{-x}+a f^x} \, dx=\frac {x \arctan \left (\frac {\sqrt {a} f^x}{\sqrt {b}}\right )}{\sqrt {a} \sqrt {b} \log (f)}-\frac {i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {a} f^x}{\sqrt {b}}\right )}{2 \sqrt {a} \sqrt {b} \log ^2(f)}+\frac {i \operatorname {PolyLog}\left (2,\frac {i \sqrt {a} f^x}{\sqrt {b}}\right )}{2 \sqrt {a} \sqrt {b} \log ^2(f)} \]
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Time = 0.06 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.353, Rules used = {2320, 211, 2298, 12, 4940, 2438} \[ \int \frac {x}{b f^{-x}+a f^x} \, dx=\frac {x \arctan \left (\frac {\sqrt {a} f^x}{\sqrt {b}}\right )}{\sqrt {a} \sqrt {b} \log (f)}-\frac {i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {a} f^x}{\sqrt {b}}\right )}{2 \sqrt {a} \sqrt {b} \log ^2(f)}+\frac {i \operatorname {PolyLog}\left (2,\frac {i \sqrt {a} f^x}{\sqrt {b}}\right )}{2 \sqrt {a} \sqrt {b} \log ^2(f)} \]
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Rule 12
Rule 211
Rule 2298
Rule 2320
Rule 2438
Rule 4940
Rubi steps \begin{align*} \text {integral}& = \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 {\text {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 \text {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 \text {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*}
Time = 0.08 (sec) , antiderivative size = 108, normalized size of antiderivative = 0.98 \[ \int \frac {x}{b f^{-x}+a f^x} \, dx=\frac {i \left (x \log (f) \left (\log \left (1-\frac {i \sqrt {a} f^x}{\sqrt {b}}\right )-\log \left (1+\frac {i \sqrt {a} f^x}{\sqrt {b}}\right )\right )-\operatorname {PolyLog}\left (2,-\frac {i \sqrt {a} f^x}{\sqrt {b}}\right )+\operatorname {PolyLog}\left (2,\frac {i \sqrt {a} f^x}{\sqrt {b}}\right )\right )}{2 \sqrt {a} \sqrt {b} \log ^2(f)} \]
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Time = 0.04 (sec) , antiderivative size = 134, normalized size of antiderivative = 1.22
method | result | size |
risch | \(\frac {x \ln \left (\frac {-a \,f^{x}+\sqrt {-a b}}{\sqrt {-a b}}\right )}{2 \ln \left (f \right ) \sqrt {-a b}}-\frac {x \ln \left (\frac {a \,f^{x}+\sqrt {-a b}}{\sqrt {-a b}}\right )}{2 \ln \left (f \right ) \sqrt {-a b}}+\frac {\operatorname {dilog}\left (\frac {-a \,f^{x}+\sqrt {-a b}}{\sqrt {-a b}}\right )}{2 \ln \left (f \right )^{2} \sqrt {-a b}}-\frac {\operatorname {dilog}\left (\frac {a \,f^{x}+\sqrt {-a b}}{\sqrt {-a b}}\right )}{2 \ln \left (f \right )^{2} \sqrt {-a b}}\) | \(134\) |
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Time = 0.26 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.02 \[ \int \frac {x}{b f^{-x}+a f^x} \, dx=-\frac {x \sqrt {-\frac {a}{b}} \log \left (f^{x} \sqrt {-\frac {a}{b}} + 1\right ) \log \left (f\right ) - x \sqrt {-\frac {a}{b}} \log \left (-f^{x} \sqrt {-\frac {a}{b}} + 1\right ) \log \left (f\right ) - \sqrt {-\frac {a}{b}} {\rm Li}_2\left (f^{x} \sqrt {-\frac {a}{b}}\right ) + \sqrt {-\frac {a}{b}} {\rm Li}_2\left (-f^{x} \sqrt {-\frac {a}{b}}\right )}{2 \, a \log \left (f\right )^{2}} \]
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\[ \int \frac {x}{b f^{-x}+a f^x} \, dx=\int \frac {f^{x} x}{a f^{2 x} + b}\, dx \]
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\[ \int \frac {x}{b f^{-x}+a f^x} \, dx=\int { \frac {x}{a f^{x} + \frac {b}{f^{x}}} \,d x } \]
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\[ \int \frac {x}{b f^{-x}+a f^x} \, dx=\int { \frac {x}{a f^{x} + \frac {b}{f^{x}}} \,d x } \]
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Timed out. \[ \int \frac {x}{b f^{-x}+a f^x} \, dx=\int \frac {x}{\frac {b}{f^x}+a\,f^x} \,d x \]
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