Optimal. Leaf size=28 \[ \frac{\text{PolyLog}\left (2,\frac{b c-a d}{b (c+d x)}\right )}{d f} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0224578, antiderivative size = 29, normalized size of antiderivative = 1.04, number of steps used = 1, number of rules used = 1, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.034, Rules used = {2447} \[ \frac{\text{PolyLog}\left (2,1-\frac{d (a+b x)}{b (c+d x)}\right )}{d f} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2447
Rubi steps
\begin{align*} \int \frac{\log \left (\frac{d (a+b x)}{b (c+d x)}\right )}{c f+d f x} \, dx &=\frac{\text{Li}_2\left (1-\frac{d (a+b x)}{b (c+d x)}\right )}{d f}\\ \end{align*}
Mathematica [B] time = 0.050042, size = 114, normalized size = 4.07 \[ \frac{\log \left (\frac{b c-a d}{b c+b d x}\right ) \left (2 \log \left (\frac{d (a+b x)}{a d-b c}\right )-2 \log \left (\frac{d (a+b x)}{b (c+d x)}\right )+\log \left (\frac{b c-a d}{b c+b d x}\right )\right )-2 \text{PolyLog}\left (2,\frac{b (c+d x)}{b c-a d}\right )}{2 d f} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.048, size = 30, normalized size = 1.1 \begin{align*}{\frac{1}{df}{\it dilog} \left ( 1+{\frac{ad-bc}{b \left ( dx+c \right ) }} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [B] time = 1.23762, size = 213, normalized size = 7.61 \begin{align*} -\frac{b{\left (\frac{\log \left (d x + c\right )^{2}}{b f} - \frac{2 \,{\left (\log \left (b x + a\right ) \log \left (\frac{b d x + a d}{b c - a d} + 1\right ) +{\rm Li}_2\left (-\frac{b d x + a d}{b c - a d}\right )\right )}}{b f}\right )}}{2 \, d} - \frac{b{\left (\frac{d \log \left (b x + a\right )}{b} - \frac{d \log \left (d x + c\right )}{b}\right )} \log \left (d f x + c f\right )}{d^{2} f} + \frac{\log \left (d f x + c f\right ) \log \left (\frac{{\left (b x + a\right )} d}{{\left (d x + c\right )} b}\right )}{d f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 0.974562, size = 63, normalized size = 2.25 \begin{align*} \frac{{\rm Li}_2\left (-\frac{b d x + a d}{b d x + b c} + 1\right )}{d f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \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{\log \left (\frac{{\left (b x + a\right )} d}{{\left (d x + c\right )} b}\right )}{d f x + c f}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]