Optimal. Leaf size=276 \[ -\frac{b^3 \text{PolyLog}(2,c (a+b x))}{3 a^3}-\frac{b^3 \text{PolyLog}\left (2,1-\frac{b c x}{1-a c}\right )}{3 a^3}-\frac{\text{PolyLog}(2,c (a+b x))}{3 x^3}-\frac{b^3 c \log (x)}{3 a^2 (1-a c)}-\frac{b^3 \log \left (\frac{b c x}{1-a c}\right ) \log (-a c-b c x+1)}{3 a^3}+\frac{b^3 c \log (-a c-b c x+1)}{3 a^2 (1-a c)}-\frac{b^2 \log (-a c-b c x+1)}{3 a^2 x}+\frac{b^3 c^2 \log (x)}{6 a (1-a c)^2}-\frac{b^3 c^2 \log (-a c-b c x+1)}{6 a (1-a c)^2}-\frac{b^2 c}{6 a x (1-a c)}+\frac{b \log (-a c-b c x+1)}{6 a x^2} \]
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Rubi [A] time = 0.272902, antiderivative size = 276, normalized size of antiderivative = 1., 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 6598
Rule 44
Rule 2416
Rule 2395
Rule 36
Rule 29
Rule 31
Rule 2394
Rule 2315
Rule 2393
Rule 2391
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*}
Mathematica [A] time = 0.303672, size = 210, normalized size = 0.76 \[ -\frac{b \left (2 b^2 \text{PolyLog}(2,c (a+b x))+2 b^2 \text{PolyLog}\left (2,\frac{a c+b c x-1}{a c-1}\right )-\frac{a^2 \log (-a c-b c x+1)}{x^2}-\frac{a^2 b c (-b c x \log (-a c-b c x+1)+a c+b c x \log (x)-1)}{x (a c-1)^2}-\frac{2 a b^2 c (\log (x)-\log (-a c-b c x+1))}{a c-1}+2 b^2 \log \left (\frac{b c x}{1-a c}\right ) \log (-a c-b c x+1)+\frac{2 a b \log (-a c-b c x+1)}{x}\right )}{6 a^3}-\frac{\text{PolyLog}(2,a c+b c x)}{3 x^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.214, size = 376, normalized size = 1.4 \begin{align*} -{\frac{{\it polylog} \left ( 2,xbc+ac \right ) }{3\,{x}^{3}}}-{\frac{{b}^{3}\ln \left ( -xbc-ac+1 \right ) }{3\,{a}^{3}}\ln \left ( -{\frac{xbc}{ac-1}} \right ) }-{\frac{{b}^{3}}{3\,{a}^{3}}{\it dilog} \left ( -{\frac{xbc}{ac-1}} \right ) }+{\frac{{b}^{3}{c}^{2}\ln \left ( -xbc \right ) }{6\,a \left ( ac-1 \right ) ^{2}}}+{\frac{{b}^{2}{c}^{2}}{6\, \left ( ac-1 \right ) ^{2}x}}-{\frac{{b}^{2}c}{6\,a \left ( ac-1 \right ) ^{2}x}}-{\frac{{b}^{3}{c}^{2}\ln \left ( -xbc-ac+1 \right ) }{6\,a \left ( ac-1 \right ) ^{2}}}+{\frac{b{c}^{2}\ln \left ( -xbc-ac+1 \right ) a}{6\,{x}^{2} \left ( ac-1 \right ) ^{2}}}-{\frac{bc\ln \left ( -xbc-ac+1 \right ) }{3\,{x}^{2} \left ( ac-1 \right ) ^{2}}}+{\frac{b\ln \left ( -xbc-ac+1 \right ) }{6\,a{x}^{2} \left ( ac-1 \right ) ^{2}}}-{\frac{{b}^{3}{\it dilog} \left ( -xbc-ac+1 \right ) }{3\,{a}^{3}}}+{\frac{c{b}^{3}\ln \left ( -xbc \right ) }{3\,{a}^{2} \left ( ac-1 \right ) }}-{\frac{c{b}^{3}\ln \left ( -xbc-ac+1 \right ) }{3\,{a}^{2} \left ( ac-1 \right ) }}-{\frac{{b}^{2}c\ln \left ( -xbc-ac+1 \right ) }{3\,a \left ( ac-1 \right ) x}}+{\frac{{b}^{2}\ln \left ( -xbc-ac+1 \right ) }{3\,{a}^{2} \left ( ac-1 \right ) x}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.02696, size = 408, normalized size = 1.48 \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^{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 \left (x\right )}{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}} \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^{4}}, 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^{4}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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