Optimal. Leaf size=84 \[ -\frac{b \text{PolyLog}(2,c (a+b x))}{a}-\frac{\text{PolyLog}(2,c (a+b x))}{x}-\frac{b \text{PolyLog}\left (2,1-\frac{b c x}{1-a c}\right )}{a}-\frac{b \log \left (\frac{b c x}{1-a c}\right ) \log (-a c-b c x+1)}{a} \]
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Rubi [A] time = 0.116564, antiderivative size = 84, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 9, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.692, Rules used = {6598, 36, 29, 31, 2416, 2394, 2315, 2393, 2391} \[ -\frac{b \text{PolyLog}(2,c (a+b x))}{a}-\frac{\text{PolyLog}(2,c (a+b x))}{x}-\frac{b \text{PolyLog}\left (2,1-\frac{b c x}{1-a c}\right )}{a}-\frac{b \log \left (\frac{b c x}{1-a c}\right ) \log (-a c-b c x+1)}{a} \]
Antiderivative was successfully verified.
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Rule 6598
Rule 36
Rule 29
Rule 31
Rule 2416
Rule 2394
Rule 2315
Rule 2393
Rule 2391
Rubi steps
\begin{align*} \int \frac{\text{Li}_2(c (a+b x))}{x^2} \, dx &=-\frac{\text{Li}_2(c (a+b x))}{x}-b \int \frac{\log (1-a c-b c x)}{x (a+b x)} \, dx\\ &=-\frac{\text{Li}_2(c (a+b x))}{x}-b \int \left (\frac{\log (1-a c-b c x)}{a x}-\frac{b \log (1-a c-b c x)}{a (a+b x)}\right ) \, dx\\ &=-\frac{\text{Li}_2(c (a+b x))}{x}-\frac{b \int \frac{\log (1-a c-b c x)}{x} \, dx}{a}+\frac{b^2 \int \frac{\log (1-a c-b c x)}{a+b x} \, dx}{a}\\ &=-\frac{b \log \left (\frac{b c x}{1-a c}\right ) \log (1-a c-b c x)}{a}-\frac{\text{Li}_2(c (a+b x))}{x}+\frac{b \operatorname{Subst}\left (\int \frac{\log (1-c x)}{x} \, dx,x,a+b x\right )}{a}-\frac{\left (b^2 c\right ) \int \frac{\log \left (-\frac{b c x}{-1+a c}\right )}{1-a c-b c x} \, dx}{a}\\ &=-\frac{b \log \left (\frac{b c x}{1-a c}\right ) \log (1-a c-b c x)}{a}-\frac{b \text{Li}_2(c (a+b x))}{a}-\frac{\text{Li}_2(c (a+b x))}{x}-\frac{b \text{Li}_2\left (1-\frac{b c x}{1-a c}\right )}{a}\\ \end{align*}
Mathematica [A] time = 0.0519712, size = 73, normalized size = 0.87 \[ -\frac{(a+b x) \text{PolyLog}(2,c (a+b x))+b x \left (\text{PolyLog}\left (2,\frac{a c+b c x-1}{a c-1}\right )+\log \left (\frac{b c x}{1-a c}\right ) \log (-a c-b c x+1)\right )}{a x} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.13, size = 85, normalized size = 1. \begin{align*} -{\frac{{\it polylog} \left ( 2,xbc+ac \right ) }{x}}-{\frac{b\ln \left ( -xbc-ac+1 \right ) }{a}\ln \left ( -{\frac{xbc}{ac-1}} \right ) }-{\frac{b}{a}{\it dilog} \left ( -{\frac{xbc}{ac-1}} \right ) }-{\frac{b{\it dilog} \left ( -xbc-ac+1 \right ) }{a}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.981607, size = 154, normalized size = 1.83 \begin{align*} \frac{{\left (\log \left (b c x + a c\right ) \log \left (-b c x - a c + 1\right ) +{\rm Li}_2\left (-b c x - a c + 1\right )\right )} b}{a} - \frac{{\left (\log \left (-b c x - a c + 1\right ) \log \left (-\frac{b c x + a c - 1}{a c - 1} + 1\right ) +{\rm Li}_2\left (\frac{b c x + a c - 1}{a c - 1}\right )\right )} b}{a} - \frac{{\rm Li}_2\left (b c x + a c\right )}{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 (b c x + a c\right )}{x^{2}}, 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 ({\left (b x + a\right )} c\right )}{x^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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