Optimal. Leaf size=84 \[ -\frac {b \log \left (\frac {b c x}{1-a c}\right ) \log (1-a c-b c x)}{a}-\frac {b \text {PolyLog}(2,c (a+b x))}{a}-\frac {\text {PolyLog}(2,c (a+b x))}{x}-\frac {b \text {PolyLog}\left (2,1-\frac {b c x}{1-a c}\right )}{a} \]
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Rubi [A]
time = 0.09, 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 = {6733, 36, 29,
31, 2463, 2441, 2352, 2440, 2438} \begin {gather*} -\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} \end {gather*}
Antiderivative was successfully verified.
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Rule 29
Rule 31
Rule 36
Rule 2352
Rule 2438
Rule 2440
Rule 2441
Rule 2463
Rule 6733
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 \text {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 = 0.04, size = 73, normalized size = 0.87 \begin {gather*} -\frac {(a+b x) \text {PolyLog}(2,c (a+b x))+b x \left (\log \left (\frac {b c x}{1-a c}\right ) \log (1-a c-b c x)+\text {PolyLog}\left (2,\frac {-1+a c+b c x}{-1+a c}\right )\right )}{a x} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 1.31, size = 94, normalized size = 1.12
method | result | size |
derivativedivides | \(c b \left (-\frac {\polylog \left (2, x b c +a c \right )}{x b c}-\frac {\dilog \left (-\frac {x b c}{a c -1}\right )+\ln \left (-x b c -a c +1\right ) \ln \left (-\frac {x b c}{a c -1}\right )}{a c}-\frac {\dilog \left (-x b c -a c +1\right )}{a c}\right )\) | \(94\) |
default | \(c b \left (-\frac {\polylog \left (2, x b c +a c \right )}{x b c}-\frac {\dilog \left (-\frac {x b c}{a c -1}\right )+\ln \left (-x b c -a c +1\right ) \ln \left (-\frac {x b c}{a c -1}\right )}{a c}-\frac {\dilog \left (-x b c -a c +1\right )}{a c}\right )\) | \(94\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 114, normalized size = 1.36 \begin {gather*} \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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \frac {\operatorname {Li}_{2}\left (a c + b c x\right )}{x^{2}}\, dx \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \frac {\mathrm {polylog}\left (2,c\,\left (a+b\,x\right )\right )}{x^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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