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 = 0.113005, antiderivative size = 72, normalized size of antiderivative = 1., 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 6197
Rule 6193
Rule 43
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 = 0.0301584, 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]
________________________________________________________________________________________
Maple [A] time = 0.219, size = 59, normalized size = 0.8 \begin{align*} -{\frac{ \left ( ax+1 \right ) \left ( -ax+\ln \left ( ax+1 \right ) \right ) }{{a}^{2}x}\sqrt{{\frac{ax-1}{ax+1}}}{\frac{1}{\sqrt{{\frac{c \left ({a}^{2}{x}^{2}-1 \right ) }{{a}^{2}{x}^{2}}}}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{\frac{a x - 1}{a x + 1}}}{\sqrt{c - \frac{c}{a^{2} x^{2}}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.88431, size = 57, normalized size = 0.79 \begin{align*} \frac{\sqrt{a^{2} c}{\left (a x - \log \left (a x + 1\right )\right )}}{a^{2} c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \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{sage}_{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]