Optimal. Leaf size=364 \[ \text{PolyLog}\left (3,\frac{c (a+b x)}{a (c+d x)}\right )-\text{PolyLog}\left (3,\frac{d (a+b x)}{b (c+d x)}\right )+\log \left (\frac{a (c+d x)}{c (a+b x)}\right ) \text{PolyLog}\left (2,\frac{c (a+b x)}{a (c+d x)}\right )-\log \left (\frac{a (c+d x)}{c (a+b x)}\right ) \text{PolyLog}\left (2,\frac{d (a+b x)}{b (c+d x)}\right )+\text{PolyLog}\left (2,\frac{b x}{a}+1\right ) \left (\log (c+d x)-\log \left (\frac{a (c+d x)}{c (a+b x)}\right )\right )+\text{PolyLog}\left (2,\frac{d x}{c}+1\right ) \left (\log \left (\frac{a (c+d x)}{c (a+b x)}\right )+\log (a+b x)\right )-\text{PolyLog}\left (3,\frac{b x}{a}+1\right )-\text{PolyLog}\left (3,\frac{d x}{c}+1\right )+\frac{1}{2} \left (\log \left (\frac{b c-a d}{b (c+d x)}\right )-\log \left (-\frac{x (b c-a d)}{a (c+d x)}\right )+\log \left (-\frac{b x}{a}\right )\right ) \log ^2\left (\frac{a (c+d x)}{c (a+b x)}\right )-\frac{1}{2} \left (\log \left (-\frac{b x}{a}\right )-\log \left (-\frac{d x}{c}\right )\right ) \left (\log \left (\frac{a (c+d x)}{c (a+b x)}\right )+\log (a+b x)\right )^2+\log \left (-\frac{b x}{a}\right ) \log (a+b x) \log (c+d x) \]
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Rubi [A] time = 0.0545108, antiderivative size = 364, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.062, Rules used = {2435} \[ \text{PolyLog}\left (3,\frac{c (a+b x)}{a (c+d x)}\right )-\text{PolyLog}\left (3,\frac{d (a+b x)}{b (c+d x)}\right )+\log \left (\frac{a (c+d x)}{c (a+b x)}\right ) \text{PolyLog}\left (2,\frac{c (a+b x)}{a (c+d x)}\right )-\log \left (\frac{a (c+d x)}{c (a+b x)}\right ) \text{PolyLog}\left (2,\frac{d (a+b x)}{b (c+d x)}\right )+\text{PolyLog}\left (2,\frac{b x}{a}+1\right ) \left (\log (c+d x)-\log \left (\frac{a (c+d x)}{c (a+b x)}\right )\right )+\text{PolyLog}\left (2,\frac{d x}{c}+1\right ) \left (\log \left (\frac{a (c+d x)}{c (a+b x)}\right )+\log (a+b x)\right )-\text{PolyLog}\left (3,\frac{b x}{a}+1\right )-\text{PolyLog}\left (3,\frac{d x}{c}+1\right )+\frac{1}{2} \left (\log \left (\frac{b c-a d}{b (c+d x)}\right )-\log \left (-\frac{x (b c-a d)}{a (c+d x)}\right )+\log \left (-\frac{b x}{a}\right )\right ) \log ^2\left (\frac{a (c+d x)}{c (a+b x)}\right )-\frac{1}{2} \left (\log \left (-\frac{b x}{a}\right )-\log \left (-\frac{d x}{c}\right )\right ) \left (\log \left (\frac{a (c+d x)}{c (a+b x)}\right )+\log (a+b x)\right )^2+\log \left (-\frac{b x}{a}\right ) \log (a+b x) \log (c+d x) \]
Antiderivative was successfully verified.
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Rule 2435
Rubi steps
\begin{align*} \int \frac{\log (a+b x) \log (c+d x)}{x} \, dx &=\log \left (-\frac{b x}{a}\right ) \log (a+b x) \log (c+d x)+\frac{1}{2} \left (\log \left (-\frac{b x}{a}\right )+\log \left (\frac{b c-a d}{b (c+d x)}\right )-\log \left (-\frac{(b c-a d) x}{a (c+d x)}\right )\right ) \log ^2\left (\frac{a (c+d x)}{c (a+b x)}\right )-\frac{1}{2} \left (\log \left (-\frac{b x}{a}\right )-\log \left (-\frac{d x}{c}\right )\right ) \left (\log (a+b x)+\log \left (\frac{a (c+d x)}{c (a+b x)}\right )\right )^2+\left (\log (c+d x)-\log \left (\frac{a (c+d x)}{c (a+b x)}\right )\right ) \text{Li}_2\left (1+\frac{b x}{a}\right )+\log \left (\frac{a (c+d x)}{c (a+b x)}\right ) \text{Li}_2\left (\frac{c (a+b x)}{a (c+d x)}\right )-\log \left (\frac{a (c+d x)}{c (a+b x)}\right ) \text{Li}_2\left (\frac{d (a+b x)}{b (c+d x)}\right )+\left (\log (a+b x)+\log \left (\frac{a (c+d x)}{c (a+b x)}\right )\right ) \text{Li}_2\left (1+\frac{d x}{c}\right )-\text{Li}_3\left (1+\frac{b x}{a}\right )+\text{Li}_3\left (\frac{c (a+b x)}{a (c+d x)}\right )-\text{Li}_3\left (\frac{d (a+b x)}{b (c+d x)}\right )-\text{Li}_3\left (1+\frac{d x}{c}\right )\\ \end{align*}
Mathematica [A] time = 0.0943344, size = 394, normalized size = 1.08 \[ \text{PolyLog}\left (3,\frac{a (c+d x)}{c (a+b x)}\right )-\text{PolyLog}\left (3,\frac{b (c+d x)}{d (a+b x)}\right )+\log \left (\frac{a (c+d x)}{c (a+b x)}\right ) \left (\text{PolyLog}\left (2,\frac{b (c+d x)}{d (a+b x)}\right )-\text{PolyLog}\left (2,\frac{a (c+d x)}{c (a+b x)}\right )\right )+\text{PolyLog}\left (2,\frac{b x}{a}+1\right ) \left (\log (c+d x)-\log \left (\frac{a (c+d x)}{c (a+b x)}\right )\right )+\text{PolyLog}\left (2,\frac{d x}{c}+1\right ) \left (\log \left (\frac{a (c+d x)}{c (a+b x)}\right )+\log (a+b x)\right )-\text{PolyLog}\left (3,\frac{b x}{a}+1\right )-\text{PolyLog}\left (3,\frac{d x}{c}+1\right )+\frac{1}{2} \left (\log \left (\frac{a d-b c}{d (a+b x)}\right )-\log \left (\frac{b c x-a d x}{a c+b c x}\right )+\log \left (-\frac{b x}{a}\right )\right ) \log ^2\left (\frac{a (c+d x)}{c (a+b x)}\right )+\log \left (\frac{d x}{c}+1\right ) \left (\log \left (-\frac{d x}{c}\right )-\log \left (-\frac{b x}{a}\right )\right ) \log \left (\frac{a (c+d x)}{c (a+b x)}\right )+\log \left (-\frac{b x}{a}\right ) \log (a+b x) \log (c+d x)+\frac{1}{2} \log \left (\frac{d x}{c}+1\right ) \left (\log \left (-\frac{b x}{a}\right )-\log \left (-\frac{d x}{c}\right )\right ) \left (\log \left (\frac{d x}{c}+1\right )-2 \log (a+b x)\right ) \]
Antiderivative was successfully verified.
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Maple [F] time = 0.075, size = 0, normalized size = 0. \begin{align*} \int{\frac{\ln \left ( bx+a \right ) \ln \left ( dx+c \right ) }{x}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\log \left (b x + a\right ) \log \left (d x + c\right )}{x}\,{d x} \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{\log \left (b x + a\right ) \log \left (d x + c\right )}{x}, 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{\log \left (b x + a\right ) \log \left (d x + c\right )}{x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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