Optimal. Leaf size=35 \[ -\frac{\text{PolyLog}\left (3,\frac{b c-a d}{d (a+b x)}+1\right )}{b c-a d} \]
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Rubi [A] time = 0.0638609, antiderivative size = 35, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 38, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.026, Rules used = {6610} \[ -\frac{\text{PolyLog}\left (3,\frac{b c-a d}{d (a+b x)}+1\right )}{b c-a d} \]
Antiderivative was successfully verified.
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Rule 6610
Rubi steps
\begin{align*} \int \frac{\text{Li}_2\left (1+\frac{b c-a d}{d (a+b x)}\right )}{(a+b x) (c+d x)} \, dx &=-\frac{\text{Li}_3\left (1+\frac{b c-a d}{d (a+b x)}\right )}{b c-a d}\\ \end{align*}
Mathematica [A] time = 0.0115641, size = 30, normalized size = 0.86 \[ \frac{\text{PolyLog}\left (3,\frac{b (c+d x)}{d (a+b x)}\right )}{a d-b c} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.062, size = 36, normalized size = 1. \begin{align*}{\frac{1}{ad-bc}{\it polylog} \left ( 3,1-{\frac{ad-bc}{d \left ( bx+a \right ) }} \right ) } \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}_2\left (\frac{b c - a d}{{\left (b x + a\right )} d} + 1\right )}{{\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 Li}_2\left (\frac{b c - a d}{b d x + a d} + 1\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(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \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}_2\left (\frac{b c - a d}{{\left (b x + a\right )} d} + 1\right )}{{\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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