Optimal. Leaf size=173 \[ \frac {b^2 \text {Li}_2(c (a+b x))}{2 a^2}+\frac {b^2 \text {Li}_2\left (1-\frac {b c x}{1-a c}\right )}{2 a^2}+\frac {b^2 \log \left (\frac {b c x}{1-a c}\right ) \log (-a c-b c x+1)}{2 a^2}+\frac {b^2 c \log (x)}{2 a (1-a c)}-\frac {b^2 c \log (-a c-b c x+1)}{2 a (1-a c)}-\frac {\text {Li}_2(c (a+b x))}{2 x^2}+\frac {b \log (-a c-b c x+1)}{2 a x} \]
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Rubi [A] time = 0.18, antiderivative size = 173, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 11, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.846, Rules used = {6598, 44, 2416, 2395, 36, 29, 31, 2394, 2315, 2393, 2391} \[ \frac {b^2 \text {PolyLog}(2,c (a+b x))}{2 a^2}+\frac {b^2 \text {PolyLog}\left (2,1-\frac {b c x}{1-a c}\right )}{2 a^2}-\frac {\text {PolyLog}(2,c (a+b x))}{2 x^2}+\frac {b^2 \log \left (\frac {b c x}{1-a c}\right ) \log (-a c-b c x+1)}{2 a^2}+\frac {b^2 c \log (x)}{2 a (1-a c)}-\frac {b^2 c \log (-a c-b c x+1)}{2 a (1-a c)}+\frac {b \log (-a c-b c x+1)}{2 a x} \]
Antiderivative was successfully verified.
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Rule 29
Rule 31
Rule 36
Rule 44
Rule 2315
Rule 2391
Rule 2393
Rule 2394
Rule 2395
Rule 2416
Rule 6598
Rubi steps
\begin {align*} \int \frac {\text {Li}_2(c (a+b x))}{x^3} \, dx &=-\frac {\text {Li}_2(c (a+b x))}{2 x^2}-\frac {1}{2} b \int \frac {\log (1-a c-b c x)}{x^2 (a+b x)} \, dx\\ &=-\frac {\text {Li}_2(c (a+b x))}{2 x^2}-\frac {1}{2} b \int \left (\frac {\log (1-a c-b c x)}{a x^2}-\frac {b \log (1-a c-b c x)}{a^2 x}+\frac {b^2 \log (1-a c-b c x)}{a^2 (a+b x)}\right ) \, dx\\ &=-\frac {\text {Li}_2(c (a+b x))}{2 x^2}-\frac {b \int \frac {\log (1-a c-b c x)}{x^2} \, dx}{2 a}+\frac {b^2 \int \frac {\log (1-a c-b c x)}{x} \, dx}{2 a^2}-\frac {b^3 \int \frac {\log (1-a c-b c x)}{a+b x} \, dx}{2 a^2}\\ &=\frac {b \log (1-a c-b c x)}{2 a x}+\frac {b^2 \log \left (\frac {b c x}{1-a c}\right ) \log (1-a c-b c x)}{2 a^2}-\frac {\text {Li}_2(c (a+b x))}{2 x^2}-\frac {b^2 \operatorname {Subst}\left (\int \frac {\log (1-c x)}{x} \, dx,x,a+b x\right )}{2 a^2}+\frac {\left (b^2 c\right ) \int \frac {1}{x (1-a c-b c x)} \, dx}{2 a}+\frac {\left (b^3 c\right ) \int \frac {\log \left (-\frac {b c x}{-1+a c}\right )}{1-a c-b c x} \, dx}{2 a^2}\\ &=\frac {b \log (1-a c-b c x)}{2 a x}+\frac {b^2 \log \left (\frac {b c x}{1-a c}\right ) \log (1-a c-b c x)}{2 a^2}+\frac {b^2 \text {Li}_2(c (a+b x))}{2 a^2}-\frac {\text {Li}_2(c (a+b x))}{2 x^2}+\frac {b^2 \text {Li}_2\left (1-\frac {b c x}{1-a c}\right )}{2 a^2}+\frac {\left (b^2 c\right ) \int \frac {1}{x} \, dx}{2 a (1-a c)}+\frac {\left (b^3 c^2\right ) \int \frac {1}{1-a c-b c x} \, dx}{2 a (1-a c)}\\ &=\frac {b^2 c \log (x)}{2 a (1-a c)}-\frac {b^2 c \log (1-a c-b c x)}{2 a (1-a c)}+\frac {b \log (1-a c-b c x)}{2 a x}+\frac {b^2 \log \left (\frac {b c x}{1-a c}\right ) \log (1-a c-b c x)}{2 a^2}+\frac {b^2 \text {Li}_2(c (a+b x))}{2 a^2}-\frac {\text {Li}_2(c (a+b x))}{2 x^2}+\frac {b^2 \text {Li}_2\left (1-\frac {b c x}{1-a c}\right )}{2 a^2}\\ \end {align*}
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Mathematica [A] time = 0.16, size = 131, normalized size = 0.76 \[ \frac {b x \left (b x (a c-1) \text {Li}_2\left (\frac {a c+b x c-1}{a c-1}\right )-a b c x \log (x)+\left (a (a c+b c x-1)+b x (a c-1) \log \left (\frac {b c x}{1-a c}\right )\right ) \log (-a c-b c x+1)\right )-(a c-1) \left (a^2-b^2 x^2\right ) \text {Li}_2(c (a+b x))}{2 a^2 x^2 (a c-1)} \]
Antiderivative was successfully verified.
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fricas [F] time = 1.28, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\rm Li}_2\left (b c x + a c\right )}{x^{3}}, x\right ) \]
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 \[ \int \frac {{\rm Li}_2\left ({\left (b x + a\right )} c\right )}{x^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 195, normalized size = 1.13 \[ -\frac {\polylog \left (2, b c x +a c \right )}{2 x^{2}}+\frac {b^{2} \dilog \left (-b c x -a c +1\right )}{2 a^{2}}-\frac {b^{2} c \ln \left (-b c x \right )}{2 a \left (a c -1\right )}+\frac {b^{2} c \ln \left (-b c x -a c +1\right )}{2 a \left (a c -1\right )}+\frac {b c \ln \left (-b c x -a c +1\right )}{2 \left (a c -1\right ) x}-\frac {b \ln \left (-b c x -a c +1\right )}{2 a \left (a c -1\right ) x}+\frac {b^{2} \ln \left (-b c x -a c +1\right ) \ln \left (-\frac {x b c}{a c -1}\right )}{2 a^{2}}+\frac {b^{2} \dilog \left (-\frac {x b c}{a c -1}\right )}{2 a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.31, size = 193, normalized size = 1.12 \[ -\frac {b^{2} c \log \relax (x)}{2 \, {\left (a^{2} c - a\right )}} - \frac {{\left (\log \left (b c x + a c\right ) \log \left (-b c x - a c + 1\right ) + {\rm Li}_2\left (-b c x - a c + 1\right )\right )} b^{2}}{2 \, a^{2}} + \frac {{\left (\log \left (-b c x - a c + 1\right ) \log \left (-\frac {b c x + a c - 1}{a c - 1} + 1\right ) + {\rm Li}_2\left (\frac {b c x + a c - 1}{a c - 1}\right )\right )} b^{2}}{2 \, a^{2}} - \frac {{\left (a^{2} c - a\right )} {\rm Li}_2\left (b c x + a c\right ) - {\left (b^{2} c x^{2} + {\left (a b c - b\right )} x\right )} \log \left (-b c x - a c + 1\right )}{2 \, {\left (a^{2} c - a\right )} x^{2}} \]
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 \[ \int \frac {\mathrm {polylog}\left (2,c\,\left (a+b\,x\right )\right )}{x^3} \,d x \]
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 \[ \int \frac {\operatorname {Li}_{2}\left (a c + b c x\right )}{x^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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