Optimal. Leaf size=19 \[ \frac {x \sqrt {1-\frac {1}{a^2 x^2}}}{c} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.04, antiderivative size = 19, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {6177, 264} \[ \frac {x \sqrt {1-\frac {1}{a^2 x^2}}}{c} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 264
Rule 6177
Rubi steps
\begin {align*} \int \frac {e^{-\coth ^{-1}(a x)}}{c-\frac {c}{a x}} \, dx &=-\frac {\operatorname {Subst}\left (\int \frac {1}{x^2 \sqrt {1-\frac {x^2}{a^2}}} \, dx,x,\frac {1}{x}\right )}{c}\\ &=\frac {\sqrt {1-\frac {1}{a^2 x^2}} x}{c}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.07, size = 19, normalized size = 1.00 \[ \frac {x \sqrt {1-\frac {1}{a^2 x^2}}}{c} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 1.50, size = 27, normalized size = 1.42 \[ \frac {{\left (a x + 1\right )} \sqrt {\frac {a x - 1}{a x + 1}}}{a c} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.14, size = 24, normalized size = 1.26 \[ \frac {\sqrt {a^{2} x^{2} - 1} \mathrm {sgn}\left (a x + 1\right )}{a c} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.04, size = 28, normalized size = 1.47 \[ \frac {\sqrt {\frac {a x -1}{a x +1}}\, \left (a x +1\right )}{a c} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [B] time = 0.31, size = 44, normalized size = 2.32 \[ -\frac {2 \, a \sqrt {\frac {a x - 1}{a x + 1}}}{\frac {{\left (a x - 1\right )} a^{2} c}{a x + 1} - a^{2} c} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.05, size = 39, normalized size = 2.05 \[ \frac {2\,\sqrt {\frac {a\,x-1}{a\,x+1}}}{a\,c-\frac {a\,c\,\left (a\,x-1\right )}{a\,x+1}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {a \int \frac {x \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x - 1}\, dx}{c} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________