Integrand size = 27, antiderivative size = 203 \[ \int \frac {x}{a+b f^{-c-d x}+c f^{c+d x}} \, dx=\frac {x \log \left (1+\frac {2 c f^{c+d x}}{a-\sqrt {a^2-4 b c}}\right )}{\sqrt {a^2-4 b c} d \log (f)}-\frac {x \log \left (1+\frac {2 c f^{c+d x}}{a+\sqrt {a^2-4 b c}}\right )}{\sqrt {a^2-4 b c} d \log (f)}+\frac {\operatorname {PolyLog}\left (2,-\frac {2 c f^{c+d x}}{a-\sqrt {a^2-4 b c}}\right )}{\sqrt {a^2-4 b c} d^2 \log ^2(f)}-\frac {\operatorname {PolyLog}\left (2,-\frac {2 c f^{c+d x}}{a+\sqrt {a^2-4 b c}}\right )}{\sqrt {a^2-4 b c} d^2 \log ^2(f)} \]
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Time = 0.23 (sec) , antiderivative size = 203, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, Rules used = {2299, 2296, 2221, 2317, 2438} \[ \int \frac {x}{a+b f^{-c-d x}+c f^{c+d x}} \, dx=\frac {\operatorname {PolyLog}\left (2,-\frac {2 c f^{c+d x}}{a-\sqrt {a^2-4 b c}}\right )}{d^2 \log ^2(f) \sqrt {a^2-4 b c}}-\frac {\operatorname {PolyLog}\left (2,-\frac {2 c f^{c+d x}}{a+\sqrt {a^2-4 b c}}\right )}{d^2 \log ^2(f) \sqrt {a^2-4 b c}}+\frac {x \log \left (\frac {2 c f^{c+d x}}{a-\sqrt {a^2-4 b c}}+1\right )}{d \log (f) \sqrt {a^2-4 b c}}-\frac {x \log \left (\frac {2 c f^{c+d x}}{\sqrt {a^2-4 b c}+a}+1\right )}{d \log (f) \sqrt {a^2-4 b c}} \]
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Rule 2221
Rule 2296
Rule 2299
Rule 2317
Rule 2438
Rubi steps \begin{align*} \text {integral}& = \int \frac {f^{c+d x} x}{b+a f^{c+d x}+c f^{2 (c+d x)}} \, dx \\ & = \frac {(2 c) \int \frac {f^{c+d x} x}{a-\sqrt {a^2-4 b c}+2 c f^{c+d x}} \, dx}{\sqrt {a^2-4 b c}}-\frac {(2 c) \int \frac {f^{c+d x} x}{a+\sqrt {a^2-4 b c}+2 c f^{c+d x}} \, dx}{\sqrt {a^2-4 b c}} \\ & = \frac {x \log \left (1+\frac {2 c f^{c+d x}}{a-\sqrt {a^2-4 b c}}\right )}{\sqrt {a^2-4 b c} d \log (f)}-\frac {x \log \left (1+\frac {2 c f^{c+d x}}{a+\sqrt {a^2-4 b c}}\right )}{\sqrt {a^2-4 b c} d \log (f)}-\frac {\int \log \left (1+\frac {2 c f^{c+d x}}{a-\sqrt {a^2-4 b c}}\right ) \, dx}{\sqrt {a^2-4 b c} d \log (f)}+\frac {\int \log \left (1+\frac {2 c f^{c+d x}}{a+\sqrt {a^2-4 b c}}\right ) \, dx}{\sqrt {a^2-4 b c} d \log (f)} \\ & = \frac {x \log \left (1+\frac {2 c f^{c+d x}}{a-\sqrt {a^2-4 b c}}\right )}{\sqrt {a^2-4 b c} d \log (f)}-\frac {x \log \left (1+\frac {2 c f^{c+d x}}{a+\sqrt {a^2-4 b c}}\right )}{\sqrt {a^2-4 b c} d \log (f)}-\frac {\text {Subst}\left (\int \frac {\log \left (1+\frac {2 c x}{a-\sqrt {a^2-4 b c}}\right )}{x} \, dx,x,f^{c+d x}\right )}{\sqrt {a^2-4 b c} d^2 \log ^2(f)}+\frac {\text {Subst}\left (\int \frac {\log \left (1+\frac {2 c x}{a+\sqrt {a^2-4 b c}}\right )}{x} \, dx,x,f^{c+d x}\right )}{\sqrt {a^2-4 b c} d^2 \log ^2(f)} \\ & = \frac {x \log \left (1+\frac {2 c f^{c+d x}}{a-\sqrt {a^2-4 b c}}\right )}{\sqrt {a^2-4 b c} d \log (f)}-\frac {x \log \left (1+\frac {2 c f^{c+d x}}{a+\sqrt {a^2-4 b c}}\right )}{\sqrt {a^2-4 b c} d \log (f)}+\frac {\text {Li}_2\left (-\frac {2 c f^{c+d x}}{a-\sqrt {a^2-4 b c}}\right )}{\sqrt {a^2-4 b c} d^2 \log ^2(f)}-\frac {\text {Li}_2\left (-\frac {2 c f^{c+d x}}{a+\sqrt {a^2-4 b c}}\right )}{\sqrt {a^2-4 b c} d^2 \log ^2(f)} \\ \end{align*}
Time = 0.26 (sec) , antiderivative size = 149, normalized size of antiderivative = 0.73 \[ \int \frac {x}{a+b f^{-c-d x}+c f^{c+d x}} \, dx=\frac {d x \log (f) \left (\log \left (1+\frac {2 c f^{c+d x}}{a-\sqrt {a^2-4 b c}}\right )-\log \left (1+\frac {2 c f^{c+d x}}{a+\sqrt {a^2-4 b c}}\right )\right )+\operatorname {PolyLog}\left (2,\frac {2 c f^{c+d x}}{-a+\sqrt {a^2-4 b c}}\right )-\operatorname {PolyLog}\left (2,-\frac {2 c f^{c+d x}}{a+\sqrt {a^2-4 b c}}\right )}{\sqrt {a^2-4 b c} d^2 \log ^2(f)} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(432\) vs. \(2(187)=374\).
Time = 0.09 (sec) , antiderivative size = 433, normalized size of antiderivative = 2.13
method | result | size |
risch | \(-\frac {\ln \left (\frac {-2 b \,f^{-d x} f^{-c}+\sqrt {a^{2}-4 c b}-a}{-a +\sqrt {a^{2}-4 c b}}\right ) x}{d \ln \left (f \right ) \sqrt {a^{2}-4 c b}}+\frac {\ln \left (\frac {2 b \,f^{-d x} f^{-c}+\sqrt {a^{2}-4 c b}+a}{a +\sqrt {a^{2}-4 c b}}\right ) x}{d \ln \left (f \right ) \sqrt {a^{2}-4 c b}}-\frac {\ln \left (\frac {-2 b \,f^{-d x} f^{-c}+\sqrt {a^{2}-4 c b}-a}{-a +\sqrt {a^{2}-4 c b}}\right ) c}{d^{2} \ln \left (f \right ) \sqrt {a^{2}-4 c b}}+\frac {\ln \left (\frac {2 b \,f^{-d x} f^{-c}+\sqrt {a^{2}-4 c b}+a}{a +\sqrt {a^{2}-4 c b}}\right ) c}{d^{2} \ln \left (f \right ) \sqrt {a^{2}-4 c b}}+\frac {\operatorname {dilog}\left (\frac {-2 b \,f^{-d x} f^{-c}+\sqrt {a^{2}-4 c b}-a}{-a +\sqrt {a^{2}-4 c b}}\right )}{d^{2} \ln \left (f \right )^{2} \sqrt {a^{2}-4 c b}}-\frac {\operatorname {dilog}\left (\frac {2 b \,f^{-d x} f^{-c}+\sqrt {a^{2}-4 c b}+a}{a +\sqrt {a^{2}-4 c b}}\right )}{d^{2} \ln \left (f \right )^{2} \sqrt {a^{2}-4 c b}}+\frac {2 c \arctan \left (\frac {2 b \,f^{-d x} f^{-c}+a}{\sqrt {-a^{2}+4 c b}}\right )}{d^{2} \ln \left (f \right ) \sqrt {-a^{2}+4 c b}}\) | \(433\) |
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Time = 0.31 (sec) , antiderivative size = 353, normalized size of antiderivative = 1.74 \[ \int \frac {x}{a+b f^{-c-d x}+c f^{c+d x}} \, dx=\frac {b c \sqrt {\frac {a^{2} - 4 \, b c}{b^{2}}} \log \left (2 \, c f^{d x + c} + b \sqrt {\frac {a^{2} - 4 \, b c}{b^{2}}} + a\right ) \log \left (f\right ) - b c \sqrt {\frac {a^{2} - 4 \, b c}{b^{2}}} \log \left (2 \, c f^{d x + c} - b \sqrt {\frac {a^{2} - 4 \, b c}{b^{2}}} + a\right ) \log \left (f\right ) + {\left (b d x + b c\right )} \sqrt {\frac {a^{2} - 4 \, b c}{b^{2}}} \log \left (f\right ) \log \left (\frac {{\left (b \sqrt {\frac {a^{2} - 4 \, b c}{b^{2}}} + a\right )} f^{d x + c} + 2 \, b}{2 \, b}\right ) - {\left (b d x + b c\right )} \sqrt {\frac {a^{2} - 4 \, b c}{b^{2}}} \log \left (f\right ) \log \left (-\frac {{\left (b \sqrt {\frac {a^{2} - 4 \, b c}{b^{2}}} - a\right )} f^{d x + c} - 2 \, b}{2 \, b}\right ) + b \sqrt {\frac {a^{2} - 4 \, b c}{b^{2}}} {\rm Li}_2\left (-\frac {{\left (b \sqrt {\frac {a^{2} - 4 \, b c}{b^{2}}} + a\right )} f^{d x + c} + 2 \, b}{2 \, b} + 1\right ) - b \sqrt {\frac {a^{2} - 4 \, b c}{b^{2}}} {\rm Li}_2\left (\frac {{\left (b \sqrt {\frac {a^{2} - 4 \, b c}{b^{2}}} - a\right )} f^{d x + c} - 2 \, b}{2 \, b} + 1\right )}{{\left (a^{2} - 4 \, b c\right )} d^{2} \log \left (f\right )^{2}} \]
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\[ \int \frac {x}{a+b f^{-c-d x}+c f^{c+d x}} \, dx=\int \frac {x}{a + b f^{- c - d x} + c f^{c + d x}}\, dx \]
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Exception generated. \[ \int \frac {x}{a+b f^{-c-d x}+c f^{c+d x}} \, dx=\text {Exception raised: ValueError} \]
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\[ \int \frac {x}{a+b f^{-c-d x}+c f^{c+d x}} \, dx=\int { \frac {x}{c f^{d x + c} + b f^{-d x - c} + a} \,d x } \]
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Timed out. \[ \int \frac {x}{a+b f^{-c-d x}+c f^{c+d x}} \, dx=\int \frac {x}{a+c\,f^{c+d\,x}+\frac {b}{f^{c+d\,x}}} \,d x \]
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