Optimal. Leaf size=20 \[ a \log (x)-a \log (1-a x)-\frac {1}{x} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.09, antiderivative size = 20, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {6150, 44} \[ a \log (x)-a \log (1-a x)-\frac {1}{x} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 44
Rule 6150
Rubi steps
\begin {align*} \int \frac {e^{\tanh ^{-1}(a x)}}{x^2 \sqrt {1-a^2 x^2}} \, dx &=\int \frac {1}{x^2 (1-a x)} \, dx\\ &=\int \left (\frac {1}{x^2}+\frac {a}{x}-\frac {a^2}{-1+a x}\right ) \, dx\\ &=-\frac {1}{x}+a \log (x)-a \log (1-a x)\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.01, size = 20, normalized size = 1.00 \[ a \log (x)-a \log (1-a x)-\frac {1}{x} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.58, size = 22, normalized size = 1.10 \[ -\frac {a x \log \left (a x - 1\right ) - a x \log \relax (x) + 1}{x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.19, size = 21, normalized size = 1.05 \[ -a \log \left ({\left | a x - 1 \right |}\right ) + a \log \left ({\left | x \right |}\right ) - \frac {1}{x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.03, size = 20, normalized size = 1.00 \[ -\frac {1}{x}+a \ln \relax (x )-a \ln \left (a x -1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.31, size = 19, normalized size = 0.95 \[ -a \log \left (a x - 1\right ) + a \log \relax (x) - \frac {1}{x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.87, size = 16, normalized size = 0.80 \[ 2\,a\,\mathrm {atanh}\left (2\,a\,x-1\right )-\frac {1}{x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 0.14, size = 15, normalized size = 0.75 \[ - a \left (- \log {\relax (x )} + \log {\left (x - \frac {1}{a} \right )}\right ) - \frac {1}{x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________