Optimal. Leaf size=43 \[ -\frac {1}{c x}+\frac {a}{c (1-a x)}+\frac {2 a \log (x)}{c}-\frac {2 a \log (1-a x)}{c} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.07, antiderivative size = 43, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, Rules used = {6285, 46}
\begin {gather*} \frac {a}{c (1-a x)}+\frac {2 a \log (x)}{c}-\frac {2 a \log (1-a x)}{c}-\frac {1}{c x} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 46
Rule 6285
Rubi steps
\begin {align*} \int \frac {e^{2 \tanh ^{-1}(a x)}}{x^2 \left (c-a^2 c x^2\right )} \, dx &=\frac {\int \frac {1}{x^2 (1-a x)^2} \, dx}{c}\\ &=\frac {\int \left (\frac {1}{x^2}+\frac {2 a}{x}+\frac {a^2}{(-1+a x)^2}-\frac {2 a^2}{-1+a x}\right ) \, dx}{c}\\ &=-\frac {1}{c x}+\frac {a}{c (1-a x)}+\frac {2 a \log (x)}{c}-\frac {2 a \log (1-a x)}{c}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.02, size = 35, normalized size = 0.81 \begin {gather*} \frac {-\frac {1}{x}+\frac {a}{1-a x}+2 a \log (x)-2 a \log (1-a x)}{c} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.06, size = 35, normalized size = 0.81
method | result | size |
default | \(\frac {-\frac {1}{x}+2 a \ln \left (x \right )-\frac {a}{a x -1}-2 a \ln \left (a x -1\right )}{c}\) | \(35\) |
risch | \(\frac {-2 a x +1}{c x \left (a x -1\right )}-\frac {2 a \ln \left (a x -1\right )}{c}+\frac {2 a \ln \left (-x \right )}{c}\) | \(44\) |
norman | \(\frac {\frac {1}{c}-\frac {2 a^{2} x^{2}}{c}-\frac {a^{3} x^{3}}{c}}{x \left (a^{2} x^{2}-1\right )}+\frac {2 a \ln \left (x \right )}{c}-\frac {2 a \ln \left (a x -1\right )}{c}\) | \(63\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.26, size = 42, normalized size = 0.98 \begin {gather*} -\frac {2 \, a \log \left (a x - 1\right )}{c} + \frac {2 \, a \log \left (x\right )}{c} - \frac {2 \, a x - 1}{a c x^{2} - c x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 0.35, size = 57, normalized size = 1.33 \begin {gather*} -\frac {2 \, a x + 2 \, {\left (a^{2} x^{2} - a x\right )} \log \left (a x - 1\right ) - 2 \, {\left (a^{2} x^{2} - a x\right )} \log \left (x\right ) - 1}{a c x^{2} - c x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A]
time = 0.12, size = 31, normalized size = 0.72 \begin {gather*} \frac {2 a \left (\log {\left (x \right )} - \log {\left (x - \frac {1}{a} \right )}\right )}{c} + \frac {- 2 a x + 1}{a c x^{2} - c x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
time = 0.42, size = 45, normalized size = 1.05 \begin {gather*} -\frac {2 \, a \log \left ({\left | a x - 1 \right |}\right )}{c} + \frac {2 \, a \log \left ({\left | x \right |}\right )}{c} - \frac {2 \, a x - 1}{{\left (a x^{2} - x\right )} c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 0.07, size = 34, normalized size = 0.79 \begin {gather*} \frac {2\,a\,x-1}{c\,x-a\,c\,x^2}+\frac {4\,a\,\mathrm {atanh}\left (2\,a\,x-1\right )}{c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________