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 = 0.10, antiderivative size = 28, normalized size of antiderivative = 1.00, 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 651
Rule 6175
Rule 6178
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 = 0.06, 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]
________________________________________________________________________________________
fricas [A] time = 0.57, size = 39, normalized size = 1.39 \[ -\frac {{\left (a x + 1\right )} \sqrt {\frac {a x - 1}{a x + 1}}}{a^{2} c^{2} x - a c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \mathit {undef} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.04, size = 36, normalized size = 1.29 \[ -\frac {\sqrt {\frac {a x -1}{a x +1}}\, \left (a x +1\right )}{\left (a x -1\right ) a \,c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.30, size = 23, normalized size = 0.82 \[ -\frac {1}{a c^{2} \sqrt {\frac {a x - 1}{a x + 1}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.03, size = 23, normalized size = 0.82 \[ -\frac {1}{a\,c^2\,\sqrt {\frac {a\,x-1}{a\,x+1}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {\int \frac {\sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a^{2} x^{2} - 2 a x + 1}\, dx}{c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________