Optimal. Leaf size=35 \[ \frac{\text{PolyLog}\left (n+1,e \left (\frac{a+b x}{c+d x}\right )^n\right )}{n (b c-a d)} \]
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Rubi [A] time = 0.071321, antiderivative size = 35, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 34, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.029, Rules used = {6610} \[ \frac{\text{PolyLog}\left (n+1,e \left (\frac{a+b x}{c+d x}\right )^n\right )}{n (b c-a d)} \]
Antiderivative was successfully verified.
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Rule 6610
Rubi steps
\begin{align*} \int \frac{\text{Li}_n\left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{(a+b x) (c+d x)} \, dx &=\frac{\text{Li}_{1+n}\left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{(b c-a d) n}\\ \end{align*}
Mathematica [A] time = 0.0198299, size = 34, normalized size = 0.97 \[ \frac{\text{PolyLog}\left (n+1,e \left (\frac{a+b x}{c+d x}\right )^n\right )}{b c n-a d n} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.891, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{ \left ( bx+a \right ) \left ( dx+c \right ) }{\it polylog} \left ( n,e \left ({\frac{bx+a}{dx+c}} \right ) ^{n} \right ) }\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\rm Li}_{n}(e \left (\frac{b x + a}{d x + c}\right )^{n})}{{\left (b x + a\right )}{\left (d x + c\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\rm polylog}\left (n, e \left (\frac{b x + a}{d x + c}\right )^{n}\right )}{b d x^{2} + a c +{\left (b c + a d\right )} x}, x\right ) \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{\operatorname{Li}_{n}\left (e \left (\frac{a}{c + d x} + \frac{b x}{c + d x}\right )^{n}\right )}{\left (a + b x\right ) \left (c + d x\right )}\, 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{{\rm Li}_{n}(e \left (\frac{b x + a}{d x + c}\right )^{n})}{{\left (b x + a\right )}{\left (d x + c\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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