Optimal. Leaf size=32 \[ -\frac {2}{a c (1-a x)}-\frac {\log (1-a x)}{a c} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.06, antiderivative size = 32, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {6167, 6129, 43} \[ -\frac {2}{a c (1-a x)}-\frac {\log (1-a x)}{a c} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 43
Rule 6129
Rule 6167
Rubi steps
\begin {align*} \int \frac {e^{2 \coth ^{-1}(a x)}}{c-a c x} \, dx &=-\int \frac {e^{2 \tanh ^{-1}(a x)}}{c-a c x} \, dx\\ &=-\frac {\int \frac {1+a x}{(1-a x)^2} \, dx}{c}\\ &=-\frac {\int \left (\frac {2}{(-1+a x)^2}+\frac {1}{-1+a x}\right ) \, dx}{c}\\ &=-\frac {2}{a c (1-a x)}-\frac {\log (1-a x)}{a c}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.02, size = 30, normalized size = 0.94 \[ -\frac {\frac {2}{a (1-a x)}+\frac {\log (1-a x)}{a}}{c} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.47, size = 29, normalized size = 0.91 \[ -\frac {{\left (a x - 1\right )} \log \left (a x - 1\right ) - 2}{a^{2} c x - a c} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.13, size = 31, normalized size = 0.97 \[ -\frac {\log \left ({\left | a x - 1 \right |}\right )}{a c} + \frac {2}{{\left (a x - 1\right )} a c} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.04, size = 31, normalized size = 0.97 \[ -\frac {\ln \left (a x -1\right )}{c a}+\frac {2}{c a \left (a x -1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.31, size = 30, normalized size = 0.94 \[ \frac {2}{a^{2} c x - a c} - \frac {\log \left (a x - 1\right )}{a c} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 1.20, size = 29, normalized size = 0.91 \[ -\frac {2}{a\,\left (c-a\,c\,x\right )}-\frac {\ln \left (a\,x-1\right )}{a\,c} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 0.13, size = 20, normalized size = 0.62 \[ \frac {2}{a^{2} c x - a c} - \frac {\log {\left (a x - 1 \right )}}{a c} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________