Optimal. Leaf size=72 \[ \frac {x \sqrt {1-\frac {1}{a^2 x^2}}}{\sqrt {c-\frac {c}{a^2 x^2}}}-\frac {\sqrt {1-\frac {1}{a^2 x^2}} \log (a x+1)}{a \sqrt {c-\frac {c}{a^2 x^2}}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.11, antiderivative size = 72, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {6197, 6193, 43} \[ \frac {x \sqrt {1-\frac {1}{a^2 x^2}}}{\sqrt {c-\frac {c}{a^2 x^2}}}-\frac {\sqrt {1-\frac {1}{a^2 x^2}} \log (a x+1)}{a \sqrt {c-\frac {c}{a^2 x^2}}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 43
Rule 6193
Rule 6197
Rubi steps
\begin {align*} \int \frac {e^{-\coth ^{-1}(a x)}}{\sqrt {c-\frac {c}{a^2 x^2}}} \, dx &=\frac {\sqrt {1-\frac {1}{a^2 x^2}} \int \frac {e^{-\coth ^{-1}(a x)}}{\sqrt {1-\frac {1}{a^2 x^2}}} \, dx}{\sqrt {c-\frac {c}{a^2 x^2}}}\\ &=\frac {\left (a \sqrt {1-\frac {1}{a^2 x^2}}\right ) \int \frac {x}{1+a x} \, dx}{\sqrt {c-\frac {c}{a^2 x^2}}}\\ &=\frac {\left (a \sqrt {1-\frac {1}{a^2 x^2}}\right ) \int \left (\frac {1}{a}-\frac {1}{a (1+a x)}\right ) \, dx}{\sqrt {c-\frac {c}{a^2 x^2}}}\\ &=\frac {\sqrt {1-\frac {1}{a^2 x^2}} x}{\sqrt {c-\frac {c}{a^2 x^2}}}-\frac {\sqrt {1-\frac {1}{a^2 x^2}} \log (1+a x)}{a \sqrt {c-\frac {c}{a^2 x^2}}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.03, size = 45, normalized size = 0.62 \[ \frac {\sqrt {1-\frac {1}{a^2 x^2}} (a x-\log (a x+1))}{a \sqrt {c-\frac {c}{a^2 x^2}}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.50, size = 26, normalized size = 0.36 \[ \frac {\sqrt {a^{2} c} {\left (a x - \log \left (a x + 1\right )\right )}}{a^{2} c} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \mathit {sage}_{2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.05, size = 59, normalized size = 0.82 \[ -\frac {\sqrt {\frac {a x -1}{a x +1}}\, \left (a x +1\right ) \left (-a x +\ln \left (a x +1\right )\right )}{\sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2} x^{2}}}\, x \,a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {\frac {a x - 1}{a x + 1}}}{\sqrt {c - \frac {c}{a^{2} x^{2}}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {\sqrt {\frac {a\,x-1}{a\,x+1}}}{\sqrt {c-\frac {c}{a^2\,x^2}}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {\frac {a x - 1}{a x + 1}}}{\sqrt {- c \left (-1 + \frac {1}{a x}\right ) \left (1 + \frac {1}{a x}\right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________