Integrand size = 18, antiderivative size = 63 \[ \int x \log \left (1+e \left (f^{c (a+b x)}\right )^n\right ) \, dx=-\frac {x \operatorname {PolyLog}\left (2,-e \left (f^{c (a+b x)}\right )^n\right )}{b c n \log (f)}+\frac {\operatorname {PolyLog}\left (3,-e \left (f^{c (a+b x)}\right )^n\right )}{b^2 c^2 n^2 \log ^2(f)} \]
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Time = 0.03 (sec) , antiderivative size = 63, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {2611, 2320, 6724} \[ \int x \log \left (1+e \left (f^{c (a+b x)}\right )^n\right ) \, dx=\frac {\operatorname {PolyLog}\left (3,-e \left (f^{c (a+b x)}\right )^n\right )}{b^2 c^2 n^2 \log ^2(f)}-\frac {x \operatorname {PolyLog}\left (2,-e \left (f^{c (a+b x)}\right )^n\right )}{b c n \log (f)} \]
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Rule 2320
Rule 2611
Rule 6724
Rubi steps \begin{align*} \text {integral}& = -\frac {x \text {Li}_2\left (-e \left (f^{c (a+b x)}\right )^n\right )}{b c n \log (f)}+\frac {\int \text {Li}_2\left (-e \left (f^{c (a+b x)}\right )^n\right ) \, dx}{b c n \log (f)} \\ & = -\frac {x \text {Li}_2\left (-e \left (f^{c (a+b x)}\right )^n\right )}{b c n \log (f)}+\frac {\text {Subst}\left (\int \frac {\text {Li}_2\left (-e x^n\right )}{x} \, dx,x,f^{c (a+b x)}\right )}{b^2 c^2 n \log ^2(f)} \\ & = -\frac {x \text {Li}_2\left (-e \left (f^{c (a+b x)}\right )^n\right )}{b c n \log (f)}+\frac {\text {Li}_3\left (-e \left (f^{c (a+b x)}\right )^n\right )}{b^2 c^2 n^2 \log ^2(f)} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 63, normalized size of antiderivative = 1.00 \[ \int x \log \left (1+e \left (f^{c (a+b x)}\right )^n\right ) \, dx=-\frac {x \operatorname {PolyLog}\left (2,-e \left (f^{c (a+b x)}\right )^n\right )}{b c n \log (f)}+\frac {\operatorname {PolyLog}\left (3,-e \left (f^{c (a+b x)}\right )^n\right )}{b^2 c^2 n^2 \log ^2(f)} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(261\) vs. \(2(63)=126\).
Time = 0.63 (sec) , antiderivative size = 262, normalized size of antiderivative = 4.16
method | result | size |
risch | \(\frac {x^{2} \ln \left (1+e \left (f^{c \left (b x +a \right )}\right )^{n}\right )}{2}-\frac {\ln \left (1+f^{x b c n} f^{-x b c n} \left (f^{c \left (b x +a \right )}\right )^{n} e \right ) x^{2}}{2}-\frac {\operatorname {Li}_{2}\left (-f^{x b c n} f^{-x b c n} \left (f^{c \left (b x +a \right )}\right )^{n} e \right ) \ln \left (f^{c \left (b x +a \right )}\right )}{c^{2} b^{2} \ln \left (f \right )^{2} n}+\frac {\operatorname {Li}_{3}\left (-f^{x b c n} f^{-x b c n} \left (f^{c \left (b x +a \right )}\right )^{n} e \right )}{c^{2} b^{2} \ln \left (f \right )^{2} n^{2}}-\frac {\operatorname {dilog}\left (1+f^{x b c n} f^{-x b c n} \left (f^{c \left (b x +a \right )}\right )^{n} e \right ) x}{c b \ln \left (f \right ) n}+\frac {\operatorname {dilog}\left (1+f^{x b c n} f^{-x b c n} \left (f^{c \left (b x +a \right )}\right )^{n} e \right ) \ln \left (f^{c \left (b x +a \right )}\right )}{c^{2} b^{2} \ln \left (f \right )^{2} n}\) | \(262\) |
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Time = 0.33 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.92 \[ \int x \log \left (1+e \left (f^{c (a+b x)}\right )^n\right ) \, dx=-\frac {b c n x {\rm Li}_2\left (-e f^{b c n x + a c n}\right ) \log \left (f\right ) - {\rm polylog}\left (3, -e f^{b c n x + a c n}\right )}{b^{2} c^{2} n^{2} \log \left (f\right )^{2}} \]
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\[ \int x \log \left (1+e \left (f^{c (a+b x)}\right )^n\right ) \, dx=\int x \log {\left (e \left (f^{a c + b c x}\right )^{n} + 1 \right )}\, dx \]
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Time = 0.23 (sec) , antiderivative size = 117, normalized size of antiderivative = 1.86 \[ \int x \log \left (1+e \left (f^{c (a+b x)}\right )^n\right ) \, dx=\frac {1}{2} \, x^{2} \log \left (e f^{{\left (b x + a\right )} c n} + 1\right ) - \frac {b^{2} c^{2} n^{2} x^{2} \log \left (e f^{b c n x} f^{a c n} + 1\right ) \log \left (f\right )^{2} + 2 \, b c n x {\rm Li}_2\left (-e f^{b c n x} f^{a c n}\right ) \log \left (f\right ) - 2 \, {\rm Li}_{3}(-e f^{b c n x} f^{a c n})}{2 \, b^{2} c^{2} n^{2} \log \left (f\right )^{2}} \]
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\[ \int x \log \left (1+e \left (f^{c (a+b x)}\right )^n\right ) \, dx=\int { x \log \left (e {\left (f^{{\left (b x + a\right )} c}\right )}^{n} + 1\right ) \,d x } \]
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Timed out. \[ \int x \log \left (1+e \left (f^{c (a+b x)}\right )^n\right ) \, dx=\int x\,\ln \left (e\,{\left (f^{c\,\left (a+b\,x\right )}\right )}^n+1\right ) \,d x \]
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