Optimal. Leaf size=36 \[ \frac {x}{c}+\frac {1}{a c (1-a x)}+\frac {2 \log (1-a x)}{a c} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.10, antiderivative size = 36, 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 = {6302, 6292,
6285, 45} \begin {gather*} \frac {1}{a c (1-a x)}+\frac {2 \log (1-a x)}{a c}+\frac {x}{c} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 45
Rule 6285
Rule 6292
Rule 6302
Rubi steps
\begin {align*} \int \frac {e^{2 \coth ^{-1}(a x)}}{c-\frac {c}{a^2 x^2}} \, dx &=-\int \frac {e^{2 \tanh ^{-1}(a x)}}{c-\frac {c}{a^2 x^2}} \, dx\\ &=\frac {a^2 \int \frac {e^{2 \tanh ^{-1}(a x)} x^2}{1-a^2 x^2} \, dx}{c}\\ &=\frac {a^2 \int \frac {x^2}{(1-a x)^2} \, dx}{c}\\ &=\frac {a^2 \int \left (\frac {1}{a^2}+\frac {1}{a^2 (-1+a x)^2}+\frac {2}{a^2 (-1+a x)}\right ) \, dx}{c}\\ &=\frac {x}{c}+\frac {1}{a c (1-a x)}+\frac {2 \log (1-a x)}{a c}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.02, size = 28, normalized size = 0.78 \begin {gather*} \frac {x+\frac {1}{a-a^2 x}+\frac {2 \log (1-a x)}{a}}{c} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.10, size = 37, normalized size = 1.03
method | result | size |
risch | \(\frac {x}{c}-\frac {1}{a c \left (a x -1\right )}+\frac {2 \ln \left (a x -1\right )}{a c}\) | \(36\) |
default | \(\frac {a^{2} \left (\frac {x}{a^{2}}-\frac {1}{a^{3} \left (a x -1\right )}+\frac {2 \ln \left (a x -1\right )}{a^{3}}\right )}{c}\) | \(37\) |
norman | \(\frac {\frac {a \,x^{2}}{c}-\frac {2 x}{c}}{a x -1}+\frac {2 \ln \left (a x -1\right )}{a c}\) | \(39\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.26, size = 35, normalized size = 0.97 \begin {gather*} \frac {x}{c} - \frac {1}{a^{2} c x - a c} + \frac {2 \, \log \left (a x - 1\right )}{a c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 0.34, size = 40, normalized size = 1.11 \begin {gather*} \frac {a^{2} x^{2} - a x + 2 \, {\left (a x - 1\right )} \log \left (a x - 1\right ) - 1}{a^{2} c x - a c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A]
time = 0.07, size = 36, normalized size = 1.00 \begin {gather*} a^{2} \left (- \frac {1}{a^{4} c x - a^{3} c} + \frac {x}{a^{2} c} + \frac {2 \log {\left (a x - 1 \right )}}{a^{3} c}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
time = 0.39, size = 36, normalized size = 1.00 \begin {gather*} \frac {x}{c} + \frac {2 \, \log \left ({\left | a x - 1 \right |}\right )}{a c} - \frac {1}{{\left (a x - 1\right )} a c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 0.05, size = 33, normalized size = 0.92 \begin {gather*} \frac {x}{c}+\frac {1}{a\,\left (c-a\,c\,x\right )}+\frac {2\,\ln \left (a\,x-1\right )}{a\,c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________