Optimal. Leaf size=30 \[ \text{PolyLog}\left (2,-\frac{c x}{b}\right )+\log (x) \log \left (\frac{c x}{b}+1\right )+\frac{\log ^2(x)}{2} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0988166, antiderivative size = 30, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.21, Rules used = {2357, 2301, 2317, 2391} \[ \text{PolyLog}\left (2,-\frac{c x}{b}\right )+\log (x) \log \left (\frac{c x}{b}+1\right )+\frac{\log ^2(x)}{2} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2357
Rule 2301
Rule 2317
Rule 2391
Rubi steps
\begin{align*} \int \frac{(b+2 c x) \log (x)}{x (b+c x)} \, dx &=\int \left (\frac{\log (x)}{x}+\frac{c \log (x)}{b+c x}\right ) \, dx\\ &=c \int \frac{\log (x)}{b+c x} \, dx+\int \frac{\log (x)}{x} \, dx\\ &=\frac{\log ^2(x)}{2}+\log (x) \log \left (1+\frac{c x}{b}\right )-\int \frac{\log \left (1+\frac{c x}{b}\right )}{x} \, dx\\ &=\frac{\log ^2(x)}{2}+\log (x) \log \left (1+\frac{c x}{b}\right )+\text{Li}_2\left (-\frac{c x}{b}\right )\\ \end{align*}
Mathematica [A] time = 0.0102847, size = 31, normalized size = 1.03 \[ \text{PolyLog}\left (2,-\frac{c x}{b}\right )+\log (x) \log \left (\frac{b+c x}{b}\right )+\frac{\log ^2(x)}{2} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.013, size = 31, normalized size = 1. \begin{align*}{\frac{ \left ( \ln \left ( x \right ) \right ) ^{2}}{2}}+\ln \left ( x \right ) \ln \left ({\frac{cx+b}{b}} \right ) +{\it dilog} \left ({\frac{cx+b}{b}} \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.02865, size = 66, normalized size = 2.2 \begin{align*}{\left (\log \left (c x + b\right ) + \log \left (x\right )\right )} \log \left (x\right ) - \log \left (c x + b\right ) \log \left (x\right ) + \log \left (\frac{c x}{b} + 1\right ) \log \left (x\right ) - \frac{1}{2} \, \log \left (x\right )^{2} +{\rm Li}_2\left (-\frac{c x}{b}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\left (2 \, c x + b\right )} \log \left (x\right )}{c x^{2} + b x}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [C] time = 83.6595, size = 192, normalized size = 6.4 \begin{align*} b \left (\begin{cases} - \frac{1}{c x} & \text{for}\: b = 0 \\\frac{\begin{cases} \log{\left (c \right )} \log{\left (x \right )} + \operatorname{Li}_{2}\left (\frac{b e^{i \pi }}{c x}\right ) & \text{for}\: \left |{x}\right | < 1 \\- \log{\left (c \right )} \log{\left (\frac{1}{x} \right )} + \operatorname{Li}_{2}\left (\frac{b e^{i \pi }}{c x}\right ) & \text{for}\: \frac{1}{\left |{x}\right |} < 1 \\-{G_{2, 2}^{2, 0}\left (\begin{matrix} & 1, 1 \\0, 0 & \end{matrix} \middle |{x} \right )} \log{\left (c \right )} +{G_{2, 2}^{0, 2}\left (\begin{matrix} 1, 1 & \\ & 0, 0 \end{matrix} \middle |{x} \right )} \log{\left (c \right )} + \operatorname{Li}_{2}\left (\frac{b e^{i \pi }}{c x}\right ) & \text{otherwise} \end{cases}}{b} & \text{otherwise} \end{cases}\right ) - b \left (\begin{cases} \frac{1}{c x} & \text{for}\: b = 0 \\\frac{\log{\left (\frac{b}{x} + c \right )}}{b} & \text{otherwise} \end{cases}\right ) \log{\left (x \right )} - 2 c \left (\begin{cases} \frac{x}{b} & \text{for}\: c = 0 \\\frac{\begin{cases} \log{\left (b \right )} \log{\left (x \right )} - \operatorname{Li}_{2}\left (\frac{c x e^{i \pi }}{b}\right ) & \text{for}\: \left |{x}\right | < 1 \\- \log{\left (b \right )} \log{\left (\frac{1}{x} \right )} - \operatorname{Li}_{2}\left (\frac{c x e^{i \pi }}{b}\right ) & \text{for}\: \frac{1}{\left |{x}\right |} < 1 \\-{G_{2, 2}^{2, 0}\left (\begin{matrix} & 1, 1 \\0, 0 & \end{matrix} \middle |{x} \right )} \log{\left (b \right )} +{G_{2, 2}^{0, 2}\left (\begin{matrix} 1, 1 & \\ & 0, 0 \end{matrix} \middle |{x} \right )} \log{\left (b \right )} - \operatorname{Li}_{2}\left (\frac{c x e^{i \pi }}{b}\right ) & \text{otherwise} \end{cases}}{c} & \text{otherwise} \end{cases}\right ) + 2 c \left (\begin{cases} \frac{x}{b} & \text{for}\: c = 0 \\\frac{\log{\left (b + c x \right )}}{c} & \text{otherwise} \end{cases}\right ) \log{\left (x \right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (2 \, c x + b\right )} \log \left (x\right )}{{\left (c x + b\right )} x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]