Optimal. Leaf size=276 \[ -\frac {b^3 \text {Li}_2(c (a+b x))}{3 a^3}-\frac {b^3 \text {Li}_2\left (1-\frac {b c x}{1-a c}\right )}{3 a^3}-\frac {b^3 \log \left (\frac {b c x}{1-a c}\right ) \log (-a c-b c x+1)}{3 a^3}-\frac {b^3 c \log (x)}{3 a^2 (1-a c)}+\frac {b^3 c \log (-a c-b c x+1)}{3 a^2 (1-a c)}-\frac {b^2 \log (-a c-b c x+1)}{3 a^2 x}+\frac {b^3 c^2 \log (x)}{6 a (1-a c)^2}-\frac {b^3 c^2 \log (-a c-b c x+1)}{6 a (1-a c)^2}-\frac {b^2 c}{6 a x (1-a c)}-\frac {\text {Li}_2(c (a+b x))}{3 x^3}+\frac {b \log (-a c-b c x+1)}{6 a x^2} \]
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Rubi [A] time = 0.27, antiderivative size = 276, normalized size of antiderivative = 1.00, number of steps used = 14, 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} \[ -\frac {b^3 \text {PolyLog}(2,c (a+b x))}{3 a^3}-\frac {b^3 \text {PolyLog}\left (2,1-\frac {b c x}{1-a c}\right )}{3 a^3}-\frac {\text {PolyLog}(2,c (a+b x))}{3 x^3}-\frac {b^3 c \log (x)}{3 a^2 (1-a c)}-\frac {b^3 \log \left (\frac {b c x}{1-a c}\right ) \log (-a c-b c x+1)}{3 a^3}+\frac {b^3 c \log (-a c-b c x+1)}{3 a^2 (1-a c)}-\frac {b^2 \log (-a c-b c x+1)}{3 a^2 x}+\frac {b^3 c^2 \log (x)}{6 a (1-a c)^2}-\frac {b^3 c^2 \log (-a c-b c x+1)}{6 a (1-a c)^2}-\frac {b^2 c}{6 a x (1-a c)}+\frac {b \log (-a c-b c x+1)}{6 a x^2} \]
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^4} \, dx &=-\frac {\text {Li}_2(c (a+b x))}{3 x^3}-\frac {1}{3} b \int \frac {\log (1-a c-b c x)}{x^3 (a+b x)} \, dx\\ &=-\frac {\text {Li}_2(c (a+b x))}{3 x^3}-\frac {1}{3} b \int \left (\frac {\log (1-a c-b c x)}{a x^3}-\frac {b \log (1-a c-b c x)}{a^2 x^2}+\frac {b^2 \log (1-a c-b c x)}{a^3 x}-\frac {b^3 \log (1-a c-b c x)}{a^3 (a+b x)}\right ) \, dx\\ &=-\frac {\text {Li}_2(c (a+b x))}{3 x^3}-\frac {b \int \frac {\log (1-a c-b c x)}{x^3} \, dx}{3 a}+\frac {b^2 \int \frac {\log (1-a c-b c x)}{x^2} \, dx}{3 a^2}-\frac {b^3 \int \frac {\log (1-a c-b c x)}{x} \, dx}{3 a^3}+\frac {b^4 \int \frac {\log (1-a c-b c x)}{a+b x} \, dx}{3 a^3}\\ &=\frac {b \log (1-a c-b c x)}{6 a x^2}-\frac {b^2 \log (1-a c-b c x)}{3 a^2 x}-\frac {b^3 \log \left (\frac {b c x}{1-a c}\right ) \log (1-a c-b c x)}{3 a^3}-\frac {\text {Li}_2(c (a+b x))}{3 x^3}+\frac {b^3 \operatorname {Subst}\left (\int \frac {\log (1-c x)}{x} \, dx,x,a+b x\right )}{3 a^3}+\frac {\left (b^2 c\right ) \int \frac {1}{x^2 (1-a c-b c x)} \, dx}{6 a}-\frac {\left (b^3 c\right ) \int \frac {1}{x (1-a c-b c x)} \, dx}{3 a^2}-\frac {\left (b^4 c\right ) \int \frac {\log \left (-\frac {b c x}{-1+a c}\right )}{1-a c-b c x} \, dx}{3 a^3}\\ &=\frac {b \log (1-a c-b c x)}{6 a x^2}-\frac {b^2 \log (1-a c-b c x)}{3 a^2 x}-\frac {b^3 \log \left (\frac {b c x}{1-a c}\right ) \log (1-a c-b c x)}{3 a^3}-\frac {b^3 \text {Li}_2(c (a+b x))}{3 a^3}-\frac {\text {Li}_2(c (a+b x))}{3 x^3}-\frac {b^3 \text {Li}_2\left (1-\frac {b c x}{1-a c}\right )}{3 a^3}+\frac {\left (b^2 c\right ) \int \left (-\frac {1}{(-1+a c) x^2}+\frac {b c}{(-1+a c)^2 x}-\frac {b^2 c^2}{(-1+a c)^2 (-1+a c+b c x)}\right ) \, dx}{6 a}-\frac {\left (b^3 c\right ) \int \frac {1}{x} \, dx}{3 a^2 (1-a c)}-\frac {\left (b^4 c^2\right ) \int \frac {1}{1-a c-b c x} \, dx}{3 a^2 (1-a c)}\\ &=-\frac {b^2 c}{6 a (1-a c) x}+\frac {b^3 c^2 \log (x)}{6 a (1-a c)^2}-\frac {b^3 c \log (x)}{3 a^2 (1-a c)}-\frac {b^3 c^2 \log (1-a c-b c x)}{6 a (1-a c)^2}+\frac {b^3 c \log (1-a c-b c x)}{3 a^2 (1-a c)}+\frac {b \log (1-a c-b c x)}{6 a x^2}-\frac {b^2 \log (1-a c-b c x)}{3 a^2 x}-\frac {b^3 \log \left (\frac {b c x}{1-a c}\right ) \log (1-a c-b c x)}{3 a^3}-\frac {b^3 \text {Li}_2(c (a+b x))}{3 a^3}-\frac {\text {Li}_2(c (a+b x))}{3 x^3}-\frac {b^3 \text {Li}_2\left (1-\frac {b c x}{1-a c}\right )}{3 a^3}\\ \end {align*}
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Mathematica [A] time = 0.29, size = 210, normalized size = 0.76 \[ -\frac {b \left (-\frac {a^2 \log (-a c-b c x+1)}{x^2}-\frac {a^2 b c (-b c x \log (-a c-b c x+1)+a c+b c x \log (x)-1)}{x (a c-1)^2}+2 b^2 \text {Li}_2(c (a+b x))+2 b^2 \text {Li}_2\left (\frac {a c+b x c-1}{a c-1}\right )-\frac {2 a b^2 c (\log (x)-\log (-a c-b c x+1))}{a c-1}+2 b^2 \log \left (\frac {b c x}{1-a c}\right ) \log (-a c-b c x+1)+\frac {2 a b \log (-a c-b c x+1)}{x}\right )}{6 a^3}-\frac {\text {Li}_2(a c+b x c)}{3 x^3} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.81, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\rm Li}_2\left (b c x + a c\right )}{x^{4}}, x\right ) \]
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 \[ \int \frac {{\rm Li}_2\left ({\left (b x + a\right )} c\right )}{x^{4}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 376, normalized size = 1.36 \[ -\frac {\polylog \left (2, b c x +a c \right )}{3 x^{3}}-\frac {b^{3} \dilog \left (-b c x -a c +1\right )}{3 a^{3}}+\frac {b^{2} c^{2}}{6 \left (a c -1\right )^{2} x}-\frac {b^{2} c}{6 a \left (a c -1\right )^{2} x}+\frac {b^{3} c^{2} \ln \left (-b c x \right )}{6 a \left (a c -1\right )^{2}}-\frac {b^{3} c^{2} \ln \left (-b c x -a c +1\right )}{6 a \left (a c -1\right )^{2}}+\frac {b \,c^{2} \ln \left (-b c x -a c +1\right ) a}{6 x^{2} \left (a c -1\right )^{2}}-\frac {b c \ln \left (-b c x -a c +1\right )}{3 x^{2} \left (a c -1\right )^{2}}+\frac {b \ln \left (-b c x -a c +1\right )}{6 a \,x^{2} \left (a c -1\right )^{2}}+\frac {b^{3} c \ln \left (-b c x \right )}{3 a^{2} \left (a c -1\right )}-\frac {b^{3} c \ln \left (-b c x -a c +1\right )}{3 a^{2} \left (a c -1\right )}-\frac {b^{2} c \ln \left (-b c x -a c +1\right )}{3 a \left (a c -1\right ) x}+\frac {b^{2} \ln \left (-b c x -a c +1\right )}{3 a^{2} \left (a c -1\right ) x}-\frac {b^{3} \ln \left (-b c x -a c +1\right ) \ln \left (-\frac {x b c}{a c -1}\right )}{3 a^{3}}-\frac {b^{3} \dilog \left (-\frac {x b c}{a c -1}\right )}{3 a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.32, size = 302, normalized size = 1.09 \[ \frac {{\left (\log \left (b c x + a c\right ) \log \left (-b c x - a c + 1\right ) + {\rm Li}_2\left (-b c x - a c + 1\right )\right )} b^{3}}{3 \, a^{3}} - \frac {{\left (\log \left (-b c x - a c + 1\right ) \log \left (-\frac {b c x + a c - 1}{a c - 1} + 1\right ) + {\rm Li}_2\left (\frac {b c x + a c - 1}{a c - 1}\right )\right )} b^{3}}{3 \, a^{3}} + \frac {{\left (3 \, a b^{3} c^{2} - 2 \, b^{3} c\right )} \log \relax (x)}{6 \, {\left (a^{4} c^{2} - 2 \, a^{3} c + a^{2}\right )}} + \frac {{\left (a^{2} b^{2} c^{2} - a b^{2} c\right )} x^{2} - 2 \, {\left (a^{4} c^{2} - 2 \, a^{3} c + a^{2}\right )} {\rm Li}_2\left (b c x + a c\right ) - {\left ({\left (3 \, a b^{3} c^{2} - 2 \, b^{3} c\right )} x^{3} + 2 \, {\left (a^{2} b^{2} c^{2} - 2 \, a b^{2} c + b^{2}\right )} x^{2} - {\left (a^{3} b c^{2} - 2 \, a^{2} b c + a b\right )} x\right )} \log \left (-b c x - a c + 1\right )}{6 \, {\left (a^{4} c^{2} - 2 \, a^{3} c + a^{2}\right )} x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {\mathrm {polylog}\left (2,c\,\left (a+b\,x\right )\right )}{x^4} \,d x \]
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 \[ \int \frac {\operatorname {Li}_{2}\left (a c + b c x\right )}{x^{4}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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