Optimal. Leaf size=19 \[ -2 a \log (x)+2 a \log (1-a x)+\frac{1}{x} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0445641, antiderivative size = 19, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {6167, 6126, 77} \[ -2 a \log (x)+2 a \log (1-a x)+\frac{1}{x} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 6167
Rule 6126
Rule 77
Rubi steps
\begin{align*} \int \frac{e^{2 \coth ^{-1}(a x)}}{x^2} \, dx &=-\int \frac{e^{2 \tanh ^{-1}(a x)}}{x^2} \, dx\\ &=-\int \frac{1+a x}{x^2 (1-a x)} \, dx\\ &=-\int \left (\frac{1}{x^2}+\frac{2 a}{x}-\frac{2 a^2}{-1+a x}\right ) \, dx\\ &=\frac{1}{x}-2 a \log (x)+2 a \log (1-a x)\\ \end{align*}
Mathematica [A] time = 0.0110726, size = 19, normalized size = 1. \[ -2 a \log (x)+2 a \log (1-a x)+\frac{1}{x} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.044, size = 19, normalized size = 1. \begin{align*}{x}^{-1}-2\,a\ln \left ( x \right ) +2\,a\ln \left ( ax-1 \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.13019, size = 24, normalized size = 1.26 \begin{align*} 2 \, a \log \left (a x - 1\right ) - 2 \, a \log \left (x\right ) + \frac{1}{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.9424, size = 58, normalized size = 3.05 \begin{align*} \frac{2 \, a x \log \left (a x - 1\right ) - 2 \, a x \log \left (x\right ) + 1}{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 2.05294, size = 15, normalized size = 0.79 \begin{align*} 2 a \left (- \log{\left (x \right )} + \log{\left (x - \frac{1}{a} \right )}\right ) + \frac{1}{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.16738, size = 27, normalized size = 1.42 \begin{align*} 2 \, a \log \left ({\left | a x - 1 \right |}\right ) - 2 \, a \log \left ({\left | x \right |}\right ) + \frac{1}{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]