Optimal. Leaf size=84 \[ -\frac {b \text {Li}_2(c (a+b x))}{a}-\frac {\text {Li}_2(c (a+b x))}{x}-\frac {b \text {Li}_2\left (1-\frac {b c x}{1-a c}\right )}{a}-\frac {b \log \left (\frac {b c x}{1-a c}\right ) \log (-a c-b c x+1)}{a} \]
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Rubi [A] time = 0.12, antiderivative size = 84, normalized size of antiderivative = 1.00, 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 29
Rule 31
Rule 36
Rule 2315
Rule 2391
Rule 2393
Rule 2394
Rule 2416
Rule 6598
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*}
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Mathematica [A] time = 0.05, size = 73, normalized size = 0.87 \[ -\frac {(a+b x) \text {Li}_2(c (a+b x))+b x \left (\text {Li}_2\left (\frac {a c+b x c-1}{a c-1}\right )+\log \left (\frac {b c x}{1-a c}\right ) \log (-a c-b c x+1)\right )}{a x} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.90, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\rm Li}_2\left (b c x + a c\right )}{x^{2}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\rm Li}_2\left ({\left (b x + a\right )} c\right )}{x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 85, normalized size = 1.01 \[ -\frac {\polylog \left (2, b c x +a c \right )}{x}-\frac {b \dilog \left (-b c x -a c +1\right )}{a}-\frac {b \ln \left (-b c x -a c +1\right ) \ln \left (-\frac {x b c}{a c -1}\right )}{a}-\frac {b \dilog \left (-\frac {x b c}{a c -1}\right )}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.32, size = 114, normalized size = 1.36 \[ \frac {{\left (\log \left (b c x + a c\right ) \log \left (-b c x - a c + 1\right ) + {\rm Li}_2\left (-b c x - a c + 1\right )\right )} b}{a} - \frac {{\left (\log \left (-b c x - a c + 1\right ) \log \left (-\frac {b c x + a c - 1}{a c - 1} + 1\right ) + {\rm Li}_2\left (\frac {b c x + a c - 1}{a c - 1}\right )\right )} b}{a} - \frac {{\rm Li}_2\left (b c x + a c\right )}{x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {\mathrm {polylog}\left (2,c\,\left (a+b\,x\right )\right )}{x^2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\operatorname {Li}_{2}\left (a c + b c x\right )}{x^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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