Optimal. Leaf size=67 \[ \frac{8 b \text{PolyLog}\left (2,1-\frac{b}{a x+b}\right )}{a}+x \log ^2\left (\frac{c (a x+b)^2}{x^2}\right )-\frac{4 b \log \left (\frac{b}{a x+b}\right ) \log \left (\frac{c (a x+b)^2}{x^2}\right )}{a} \]
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Rubi [A] time = 0.156801, antiderivative size = 67, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4, Rules used = {2486, 2488, 2411, 2343, 2333, 2315} \[ \frac{8 b \text{PolyLog}\left (2,1-\frac{b}{a x+b}\right )}{a}+x \log ^2\left (\frac{c (a x+b)^2}{x^2}\right )-\frac{4 b \log \left (\frac{b}{a x+b}\right ) \log \left (\frac{c (a x+b)^2}{x^2}\right )}{a} \]
Antiderivative was successfully verified.
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Rule 2486
Rule 2488
Rule 2411
Rule 2343
Rule 2333
Rule 2315
Rubi steps
\begin{align*} \int \log ^2\left (\frac{c (b+a x)^2}{x^2}\right ) \, dx &=x \log ^2\left (\frac{c (b+a x)^2}{x^2}\right )+(4 b) \int \frac{\log \left (\frac{c (b+a x)^2}{x^2}\right )}{b+a x} \, dx\\ &=-\frac{4 b \log \left (\frac{b}{b+a x}\right ) \log \left (\frac{c (b+a x)^2}{x^2}\right )}{a}+x \log ^2\left (\frac{c (b+a x)^2}{x^2}\right )-\frac{\left (8 b^2\right ) \int \frac{\log \left (\frac{b}{b+a x}\right )}{x (b+a x)} \, dx}{a}\\ &=-\frac{4 b \log \left (\frac{b}{b+a x}\right ) \log \left (\frac{c (b+a x)^2}{x^2}\right )}{a}+x \log ^2\left (\frac{c (b+a x)^2}{x^2}\right )-\frac{\left (8 b^2\right ) \operatorname{Subst}\left (\int \frac{\log \left (\frac{b}{x}\right )}{x \left (-\frac{b}{a}+\frac{x}{a}\right )} \, dx,x,b+a x\right )}{a^2}\\ &=-\frac{4 b \log \left (\frac{b}{b+a x}\right ) \log \left (\frac{c (b+a x)^2}{x^2}\right )}{a}+x \log ^2\left (\frac{c (b+a x)^2}{x^2}\right )+\frac{\left (8 b^2\right ) \operatorname{Subst}\left (\int \frac{\log (b x)}{\left (-\frac{b}{a}+\frac{1}{a x}\right ) x} \, dx,x,\frac{1}{b+a x}\right )}{a^2}\\ &=-\frac{4 b \log \left (\frac{b}{b+a x}\right ) \log \left (\frac{c (b+a x)^2}{x^2}\right )}{a}+x \log ^2\left (\frac{c (b+a x)^2}{x^2}\right )+\frac{\left (8 b^2\right ) \operatorname{Subst}\left (\int \frac{\log (b x)}{\frac{1}{a}-\frac{b x}{a}} \, dx,x,\frac{1}{b+a x}\right )}{a^2}\\ &=-\frac{4 b \log \left (\frac{b}{b+a x}\right ) \log \left (\frac{c (b+a x)^2}{x^2}\right )}{a}+x \log ^2\left (\frac{c (b+a x)^2}{x^2}\right )+\frac{8 b \text{Li}_2\left (\frac{a x}{b+a x}\right )}{a}\\ \end{align*}
Mathematica [A] time = 0.0156204, size = 106, normalized size = 1.58 \[ \frac{8 b \text{PolyLog}\left (2,\frac{a x+b}{b}\right )}{a}+x \log ^2\left (\frac{c (a x+b)^2}{x^2}\right )-\frac{4 b \log \left (\frac{b}{a x+b}\right ) \log \left (\frac{c (a x+b)^2}{x^2}\right )}{a}-\frac{4 b \log ^2\left (\frac{b}{a x+b}\right )}{a}-\frac{8 b \log \left (-\frac{a x}{b}\right ) \log \left (\frac{b}{a x+b}\right )}{a} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.526, size = 0, normalized size = 0. \begin{align*} \int \left ( \ln \left ({\frac{c \left ( ax+b \right ) ^{2}}{{x}^{2}}} \right ) \right ) ^{2}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.22382, size = 159, normalized size = 2.37 \begin{align*} x \log \left (\frac{{\left (a x + b\right )}^{2} c}{x^{2}}\right )^{2} + \frac{4 \, b \log \left (a x + b\right ) \log \left (\frac{{\left (a x + b\right )}^{2} c}{x^{2}}\right )}{a} + \frac{4 \,{\left ({\left (\frac{c \log \left (a x + b\right )^{2}}{a} - \frac{2 \,{\left (\log \left (\frac{a x}{b} + 1\right ) \log \left (x\right ) +{\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 \left (x\right )\right )} b \log \left (a x + b\right )}{a}\right )}}{c} \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 (\log \left (\frac{a^{2} c x^{2} + 2 \, a b c x + b^{2} c}{x^{2}}\right )^{2}, 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*} 4 b \int \frac{\log{\left (a^{2} c + \frac{2 a b c}{x} + \frac{b^{2} c}{x^{2}} \right )}}{a x + b}\, dx + x \log{\left (\frac{c \left (a x + b\right )^{2}}{x^{2}} \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 \log \left (\frac{{\left (a x + b\right )}^{2} c}{x^{2}}\right )^{2}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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