Optimal. Leaf size=67 \[ \frac {(a x+b) \log ^2\left (a c+\frac {b c}{x}\right )}{a}-\frac {2 b \log \left (-\frac {b}{a x}\right ) \log \left (c \left (a+\frac {b}{x}\right )\right )}{a}-\frac {2 b \text {Li}_2\left (\frac {b}{a x}+1\right )}{a} \]
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Rubi [A] time = 0.07, antiderivative size = 67, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.385, Rules used = {2453, 2449, 2454, 2394, 2315} \[ -\frac {2 b \text {PolyLog}\left (2,\frac {b}{a x}+1\right )}{a}+\frac {(a x+b) \log ^2\left (a c+\frac {b c}{x}\right )}{a}-\frac {2 b \log \left (-\frac {b}{a x}\right ) \log \left (c \left (a+\frac {b}{x}\right )\right )}{a} \]
Antiderivative was successfully verified.
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Rule 2315
Rule 2394
Rule 2449
Rule 2453
Rule 2454
Rubi steps
\begin {align*} \int \log ^2\left (\frac {c (b+a x)}{x}\right ) \, dx &=\int \log ^2\left (a c+\frac {b c}{x}\right ) \, dx\\ &=\frac {(b+a x) \log ^2\left (a c+\frac {b c}{x}\right )}{a}+\frac {(2 b) \int \frac {\log \left (a c+\frac {b c}{x}\right )}{x} \, dx}{a}\\ &=\frac {(b+a x) \log ^2\left (a c+\frac {b c}{x}\right )}{a}-\frac {(2 b) \operatorname {Subst}\left (\int \frac {\log (a c+b c x)}{x} \, dx,x,\frac {1}{x}\right )}{a}\\ &=\frac {(b+a x) \log ^2\left (a c+\frac {b c}{x}\right )}{a}-\frac {2 b \log \left (c \left (a+\frac {b}{x}\right )\right ) \log \left (-\frac {b}{a x}\right )}{a}+\frac {\left (2 b^2 c\right ) \operatorname {Subst}\left (\int \frac {\log \left (-\frac {b x}{a}\right )}{a c+b c x} \, dx,x,\frac {1}{x}\right )}{a}\\ &=\frac {(b+a x) \log ^2\left (a c+\frac {b c}{x}\right )}{a}-\frac {2 b \log \left (c \left (a+\frac {b}{x}\right )\right ) \log \left (-\frac {b}{a x}\right )}{a}-\frac {2 b \text {Li}_2\left (1+\frac {b}{a x}\right )}{a}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 63, normalized size = 0.94 \[ \frac {\log \left (\frac {c (a x+b)}{x}\right ) \left ((a x+b) \log \left (\frac {c (a x+b)}{x}\right )-2 b \log \left (-\frac {b}{a x}\right )\right )-2 b \text {Li}_2\left (\frac {b}{a x}+1\right )}{a} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.73, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\log \left (\frac {a c x + b c}{x}\right )^{2}, x\right ) \]
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 \[ \int \log \left (\frac {{\left (a x + b\right )} c}{x}\right )^{2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.62, size = 0, normalized size = 0.00 \[ \int \ln \left (\frac {\left (a x +b \right ) c}{x}\right )^{2}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.66, size = 113, normalized size = 1.69 \[ x \log \left (\frac {{\left (a x + b\right )} c}{x}\right )^{2} + \frac {2 \, b \log \left (a x + b\right ) \log \left (\frac {{\left (a x + b\right )} c}{x}\right )}{a} + \frac {{\left (\frac {c \log \left (a x + b\right )^{2}}{a} - \frac {2 \, {\left (\log \left (\frac {a x}{b} + 1\right ) \log \relax (x) + {\rm Li}_2\left (-\frac {a x}{b}\right )\right )} c}{a}\right )} b - \frac {2 \, {\left (c \log \left (a x + b\right ) - c \log \relax (x)\right )} b \log \left (a x + b\right )}{a}}{c} \]
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 \[ \int {\ln \left (\frac {c\,\left (b+a\,x\right )}{x}\right )}^2 \,d x \]
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 \[ 2 b \int \frac {\log {\left (a c + \frac {b c}{x} \right )}}{a x + b}\, dx + x \log {\left (\frac {c \left (a x + b\right )}{x} \right )}^{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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