Optimal. Leaf size=25 \[ -\frac {c \log (x)}{a}+\frac {4 c \log (1-a x)}{a}+c x \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.08, antiderivative size = 25, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {6167, 6131, 6129, 72} \[ -\frac {c \log (x)}{a}+\frac {4 c \log (1-a x)}{a}+c x \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 72
Rule 6129
Rule 6131
Rule 6167
Rubi steps
\begin {align*} \int e^{4 \coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right ) \, dx &=\int e^{4 \tanh ^{-1}(a x)} \left (c-\frac {c}{a x}\right ) \, dx\\ &=-\frac {c \int \frac {e^{4 \tanh ^{-1}(a x)} (1-a x)}{x} \, dx}{a}\\ &=-\frac {c \int \frac {(1+a x)^2}{x (1-a x)} \, dx}{a}\\ &=-\frac {c \int \left (-a+\frac {1}{x}-\frac {4 a}{-1+a x}\right ) \, dx}{a}\\ &=c x-\frac {c \log (x)}{a}+\frac {4 c \log (1-a x)}{a}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.04, size = 25, normalized size = 1.00 \[ -\frac {c \log (x)}{a}+\frac {4 c \log (1-a x)}{a}+c x \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.66, size = 23, normalized size = 0.92 \[ \frac {a c x + 4 \, c \log \left (a x - 1\right ) - c \log \relax (x)}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [B] time = 0.12, size = 55, normalized size = 2.20 \[ \frac {{\left (a x - 1\right )} c}{a} - \frac {3 \, c \log \left (\frac {{\left | a x - 1 \right |}}{{\left (a x - 1\right )}^{2} {\left | a \right |}}\right )}{a} - \frac {c \log \left ({\left | -\frac {1}{a x - 1} - 1 \right |}\right )}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.04, size = 25, normalized size = 1.00 \[ c x +\frac {4 c \ln \left (a x -1\right )}{a}-\frac {c \ln \relax (x )}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.31, size = 24, normalized size = 0.96 \[ c x + \frac {4 \, c \log \left (a x - 1\right )}{a} - \frac {c \log \relax (x)}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.07, size = 24, normalized size = 0.96 \[ c\,x-\frac {c\,\ln \relax (x)}{a}+\frac {4\,c\,\ln \left (a\,x-1\right )}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 0.21, size = 17, normalized size = 0.68 \[ c x + \frac {c \left (- \log {\relax (x )} + 4 \log {\left (x - \frac {1}{a} \right )}\right )}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________