Optimal. Leaf size=35 \[ -\frac{1}{a c (a x+1)}-\frac{2 \log (a x+1)}{a c}+\frac{x}{c} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.155279, antiderivative size = 35, normalized size of antiderivative = 1., 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 = {6167, 6157, 6150, 43} \[ -\frac{1}{a c (a x+1)}-\frac{2 \log (a x+1)}{a c}+\frac{x}{c} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 6167
Rule 6157
Rule 6150
Rule 43
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 = 0.0309475, size = 28, normalized size = 0.8 \[ \frac{-\frac{1}{a^2 x+a}-\frac{2 \log (a x+1)}{a}+x}{c} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.045, size = 36, normalized size = 1. \begin{align*}{\frac{x}{c}}-{\frac{1}{ac \left ( ax+1 \right ) }}-2\,{\frac{\ln \left ( ax+1 \right ) }{ac}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.02637, size = 46, normalized size = 1.31 \begin{align*} \frac{x}{c} - \frac{1}{a^{2} c x + a c} - \frac{2 \, \log \left (a x + 1\right )}{a c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.19498, size = 86, normalized size = 2.46 \begin{align*} \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{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 0.320006, size = 36, normalized size = 1.03 \begin{align*} 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{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.15606, size = 49, normalized size = 1.4 \begin{align*} \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{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]