Optimal. Leaf size=82 \[ \text{PolyLog}\left (3,-\frac{a}{b x}\right )+\log (x) \text{PolyLog}\left (2,-\frac{a}{b x}\right )+\frac{1}{2} \log ^2(x) \log \left (\frac{a}{x (b c-a d)}+\frac{b}{b c-a d}\right )-\frac{1}{2} \log ^2(x) \log \left (\frac{a}{b x}+1\right ) \]
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Rubi [A] time = 0.176511, antiderivative size = 82, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.192, Rules used = {2380, 2375, 2337, 2374, 6589} \[ \text{PolyLog}\left (3,-\frac{a}{b x}\right )+\log (x) \text{PolyLog}\left (2,-\frac{a}{b x}\right )+\frac{1}{2} \log ^2(x) \log \left (\frac{a}{x (b c-a d)}+\frac{b}{b c-a d}\right )-\frac{1}{2} \log ^2(x) \log \left (\frac{a}{b x}+1\right ) \]
Antiderivative was successfully verified.
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Rule 2380
Rule 2375
Rule 2337
Rule 2374
Rule 6589
Rubi steps
\begin{align*} \int \frac{\log (x) \log \left (\frac{a+b x}{(b c-a d) x}\right )}{x} \, dx &=\int \frac{\log \left (\frac{b}{b c-a d}+\frac{a}{(b c-a d) x}\right ) \log (x)}{x} \, dx\\ &=\frac{1}{2} \log \left (\frac{b}{b c-a d}+\frac{a}{(b c-a d) x}\right ) \log ^2(x)+\frac{a \int \frac{\log ^2(x)}{\left (\frac{b}{b c-a d}+\frac{a}{(b c-a d) x}\right ) x^2} \, dx}{2 (b c-a d)}\\ &=-\frac{1}{2} \log \left (1+\frac{a}{b x}\right ) \log ^2(x)+\frac{1}{2} \log \left (\frac{b}{b c-a d}+\frac{a}{(b c-a d) x}\right ) \log ^2(x)+\int \frac{\log \left (1+\frac{a}{b x}\right ) \log (x)}{x} \, dx\\ &=-\frac{1}{2} \log \left (1+\frac{a}{b x}\right ) \log ^2(x)+\frac{1}{2} \log \left (\frac{b}{b c-a d}+\frac{a}{(b c-a d) x}\right ) \log ^2(x)+\log (x) \text{Li}_2\left (-\frac{a}{b x}\right )-\int \frac{\text{Li}_2\left (-\frac{a}{b x}\right )}{x} \, dx\\ &=-\frac{1}{2} \log \left (1+\frac{a}{b x}\right ) \log ^2(x)+\frac{1}{2} \log \left (\frac{b}{b c-a d}+\frac{a}{(b c-a d) x}\right ) \log ^2(x)+\log (x) \text{Li}_2\left (-\frac{a}{b x}\right )+\text{Li}_3\left (-\frac{a}{b x}\right )\\ \end{align*}
Mathematica [A] time = 5.02716, size = 66, normalized size = 0.8 \[ \text{PolyLog}\left (3,-\frac{b x}{a}\right )-\log (x) \text{PolyLog}\left (2,-\frac{b x}{a}\right )+\frac{1}{6} \log ^2(x) \left (3 \log \left (\frac{a+b x}{b c x-a d x}\right )-3 \log \left (\frac{b x}{a}+1\right )+\log (x)\right ) \]
Antiderivative was successfully verified.
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Maple [C] time = 0.121, size = 450, normalized size = 5.5 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \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 (x\right ) \log \left (\frac{b x + a}{{\left (b c - a d\right )} x}\right )}{x}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \frac{a \int \frac{\log{\left (x \right )}^{2}}{a x + b x^{2}}\, dx}{2} + \frac{\log{\left (x \right )}^{2} \log{\left (\frac{a + b x}{x \left (- a d + b c\right )} \right )}}{2} \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 (x\right ) \log \left (\frac{b x + a}{{\left (b c - a d\right )} x}\right )}{x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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