Optimal. Leaf size=82 \[ \frac {1}{2} \log ^2(x) \log \left (\frac {a}{x (b c-a d)}+\frac {b}{b c-a d}\right )+\text {Li}_3\left (-\frac {a}{b x}\right )+\log (x) \text {Li}_2\left (-\frac {a}{b x}\right )-\frac {1}{2} \log ^2(x) \log \left (\frac {a}{b x}+1\right ) \]
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Rubi [A] time = 0.18, antiderivative size = 82, normalized size of antiderivative = 1.00, 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 2337
Rule 2374
Rule 2375
Rule 2380
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*}
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Mathematica [A] time = 5.03, size = 66, normalized size = 0.80 \[ \frac {1}{6} \log ^2(x) \left (3 \log \left (\frac {a+b x}{b c x-a d x}\right )-3 \log \left (\frac {b x}{a}+1\right )+\log (x)\right )+\text {Li}_3\left (-\frac {b x}{a}\right )-\log (x) \text {Li}_2\left (-\frac {b x}{a}\right ) \]
Antiderivative was successfully verified.
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fricas [F] time = 0.63, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\log \relax (x) \log \left (\frac {b x + a}{{\left (b c - a d\right )} x}\right )}{x}, x\right ) \]
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 \[ \int \frac {\log \relax (x) \log \left (\frac {b x + a}{{\left (b c - a d\right )} x}\right )}{x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.18, size = 450, normalized size = 5.49 \[ -\frac {i \pi \,\mathrm {csgn}\left (\frac {i}{x}\right ) \mathrm {csgn}\left (\frac {i \left (b x +a \right )}{a d -b c}\right ) \mathrm {csgn}\left (\frac {i \left (b x +a \right )}{\left (a d -b c \right ) x}\right ) \ln \relax (x )^{2}}{4}+\frac {i \pi \,\mathrm {csgn}\left (\frac {i}{x}\right ) \mathrm {csgn}\left (\frac {i \left (b x +a \right )}{\left (a d -b c \right ) x}\right )^{2} \ln \relax (x )^{2}}{4}-\frac {i \pi \,\mathrm {csgn}\left (i \left (b x +a \right )\right ) \mathrm {csgn}\left (\frac {i}{a d -b c}\right ) \mathrm {csgn}\left (\frac {i \left (b x +a \right )}{a d -b c}\right ) \ln \relax (x )^{2}}{4}+\frac {i \pi \,\mathrm {csgn}\left (i \left (b x +a \right )\right ) \mathrm {csgn}\left (\frac {i \left (b x +a \right )}{a d -b c}\right )^{2} \ln \relax (x )^{2}}{4}+\frac {i \pi \,\mathrm {csgn}\left (\frac {i}{a d -b c}\right ) \mathrm {csgn}\left (\frac {i \left (b x +a \right )}{a d -b c}\right )^{2} \ln \relax (x )^{2}}{4}-\frac {i \pi \mathrm {csgn}\left (\frac {i \left (b x +a \right )}{a d -b c}\right )^{3} \ln \relax (x )^{2}}{4}+\frac {i \pi \,\mathrm {csgn}\left (\frac {i \left (b x +a \right )}{a d -b c}\right ) \mathrm {csgn}\left (\frac {i \left (b x +a \right )}{\left (a d -b c \right ) x}\right )^{2} \ln \relax (x )^{2}}{4}+\frac {i \pi \mathrm {csgn}\left (\frac {i \left (b x +a \right )}{\left (a d -b c \right ) x}\right )^{3} \ln \relax (x )^{2}}{4}-\frac {i \pi \mathrm {csgn}\left (\frac {i \left (b x +a \right )}{\left (a d -b c \right ) x}\right )^{2} \ln \relax (x )^{2}}{2}-\frac {\ln \relax (x )^{3}}{3}-\frac {\ln \relax (x )^{2} \ln \left (\frac {b x}{a}+1\right )}{2}+\frac {\ln \relax (x )^{2} \ln \left (b x +a \right )}{2}-\frac {\ln \relax (x )^{2} \ln \left (a d -b c \right )}{2}-\polylog \left (2, -\frac {b x}{a}\right ) \ln \relax (x )+\frac {i \pi \ln \relax (x )^{2}}{2}+\polylog \left (3, -\frac {b x}{a}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
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 \[ \int \frac {\ln \left (-\frac {a+b\,x}{x\,\left (a\,d-b\,c\right )}\right )\,\ln \relax (x)}{x} \,d x \]
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 \[ \frac {a \int \frac {\log {\relax (x )}^{2}}{a x + b x^{2}}\, dx}{2} + \frac {\log {\relax (x )}^{2} \log {\left (\frac {a + b x}{x \left (- a d + b c\right )} \right )}}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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