Optimal. Leaf size=28 \[ -\frac{\sqrt{1-\frac{1}{a^2 x^2}}}{c^2 \left (a-\frac{1}{x}\right )} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.101127, antiderivative size = 28, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {6175, 6178, 651} \[ -\frac{\sqrt{1-\frac{1}{a^2 x^2}}}{c^2 \left (a-\frac{1}{x}\right )} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 6175
Rule 6178
Rule 651
Rubi steps
\begin{align*} \int \frac{e^{-\coth ^{-1}(a x)}}{(c-a c x)^2} \, dx &=\frac{\int \frac{e^{-\coth ^{-1}(a x)}}{\left (1-\frac{1}{a x}\right )^2 x^2} \, dx}{a^2 c^2}\\ &=-\frac{\operatorname{Subst}\left (\int \frac{1}{\left (1-\frac{x}{a}\right ) \sqrt{1-\frac{x^2}{a^2}}} \, dx,x,\frac{1}{x}\right )}{a^2 c^2}\\ &=-\frac{\sqrt{1-\frac{1}{a^2 x^2}}}{c^2 \left (a-\frac{1}{x}\right )}\\ \end{align*}
Mathematica [A] time = 0.050955, size = 27, normalized size = 0.96 \[ -\frac{x \sqrt{1-\frac{1}{a^2 x^2}}}{c^2 (a x-1)} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.043, size = 36, normalized size = 1.3 \begin{align*} -{\frac{ax+1}{ \left ( ax-1 \right ) a{c}^{2}}\sqrt{{\frac{ax-1}{ax+1}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.03071, size = 31, normalized size = 1.11 \begin{align*} -\frac{1}{a c^{2} \sqrt{\frac{a x - 1}{a x + 1}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.58156, size = 78, normalized size = 2.79 \begin{align*} -\frac{{\left (a x + 1\right )} \sqrt{\frac{a x - 1}{a x + 1}}}{a^{2} c^{2} x - a c^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \frac{\int \frac{\sqrt{\frac{a x}{a x + 1} - \frac{1}{a x + 1}}}{a^{2} x^{2} - 2 a x + 1}\, dx}{c^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \mathit{undef} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]