Optimal. Leaf size=20 \[ \frac {x}{c}-\frac {\log (a x+1)}{a c} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.12, antiderivative size = 20, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {6167, 6131, 6129, 43} \[ \frac {x}{c}-\frac {\log (a x+1)}{a c} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 43
Rule 6129
Rule 6131
Rule 6167
Rubi steps
\begin {align*} \int \frac {e^{-2 \coth ^{-1}(a x)}}{c-\frac {c}{a x}} \, dx &=-\int \frac {e^{-2 \tanh ^{-1}(a x)}}{c-\frac {c}{a x}} \, dx\\ &=\frac {a \int \frac {e^{-2 \tanh ^{-1}(a x)} x}{1-a x} \, dx}{c}\\ &=\frac {a \int \frac {x}{1+a x} \, dx}{c}\\ &=\frac {a \int \left (\frac {1}{a}-\frac {1}{a (1+a x)}\right ) \, dx}{c}\\ &=\frac {x}{c}-\frac {\log (1+a x)}{a c}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.02, size = 22, normalized size = 1.10 \[ \frac {a \left (\frac {x}{a}-\frac {\log (a x+1)}{a^2}\right )}{c} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.93, size = 19, normalized size = 0.95 \[ \frac {a x - \log \left (a x + 1\right )}{a c} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.13, size = 21, normalized size = 1.05 \[ \frac {x}{c} - \frac {\log \left ({\left | a x + 1 \right |}\right )}{a c} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.03, size = 21, normalized size = 1.05 \[ \frac {x}{c}-\frac {\ln \left (a x +1\right )}{a c} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.31, size = 20, normalized size = 1.00 \[ \frac {x}{c} - \frac {\log \left (a x + 1\right )}{a c} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 1.20, size = 19, normalized size = 0.95 \[ -\frac {\ln \left (a\,x+1\right )-a\,x}{a\,c} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 0.10, size = 17, normalized size = 0.85 \[ a \left (\frac {x}{a c} - \frac {\log {\left (a x + 1 \right )}}{a^{2} c}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________