Optimal. Leaf size=118 \[ \frac{2 x \sqrt{1-\frac{1}{a x}} \sqrt{\frac{1}{a x}+1}}{\sqrt{c-a c x}}-\frac{2 \sqrt{2} \sqrt{1-\frac{1}{a x}} \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{\frac{1}{x}}}{\sqrt{a} \sqrt{\frac{1}{a x}+1}}\right )}{\sqrt{a} \sqrt{\frac{1}{x}} \sqrt{c-a c x}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.166492, antiderivative size = 118, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.278, Rules used = {6176, 6181, 94, 93, 206} \[ \frac{2 x \sqrt{1-\frac{1}{a x}} \sqrt{\frac{1}{a x}+1}}{\sqrt{c-a c x}}-\frac{2 \sqrt{2} \sqrt{1-\frac{1}{a x}} \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{\frac{1}{x}}}{\sqrt{a} \sqrt{\frac{1}{a x}+1}}\right )}{\sqrt{a} \sqrt{\frac{1}{x}} \sqrt{c-a c x}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 6176
Rule 6181
Rule 94
Rule 93
Rule 206
Rubi steps
\begin{align*} \int \frac{e^{\coth ^{-1}(a x)}}{\sqrt{c-a c x}} \, dx &=\frac{\left (\sqrt{1-\frac{1}{a x}} \sqrt{x}\right ) \int \frac{e^{\coth ^{-1}(a x)}}{\sqrt{1-\frac{1}{a x}} \sqrt{x}} \, dx}{\sqrt{c-a c x}}\\ &=-\frac{\sqrt{1-\frac{1}{a x}} \operatorname{Subst}\left (\int \frac{\sqrt{1+\frac{x}{a}}}{x^{3/2} \left (1-\frac{x}{a}\right )} \, dx,x,\frac{1}{x}\right )}{\sqrt{\frac{1}{x}} \sqrt{c-a c x}}\\ &=\frac{2 \sqrt{1-\frac{1}{a x}} \sqrt{1+\frac{1}{a x}} x}{\sqrt{c-a c x}}-\frac{\left (2 \sqrt{1-\frac{1}{a x}}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{x} \left (1-\frac{x}{a}\right ) \sqrt{1+\frac{x}{a}}} \, dx,x,\frac{1}{x}\right )}{a \sqrt{\frac{1}{x}} \sqrt{c-a c x}}\\ &=\frac{2 \sqrt{1-\frac{1}{a x}} \sqrt{1+\frac{1}{a x}} x}{\sqrt{c-a c x}}-\frac{\left (4 \sqrt{1-\frac{1}{a x}}\right ) \operatorname{Subst}\left (\int \frac{1}{1-\frac{2 x^2}{a}} \, dx,x,\frac{\sqrt{\frac{1}{x}}}{\sqrt{1+\frac{1}{a x}}}\right )}{a \sqrt{\frac{1}{x}} \sqrt{c-a c x}}\\ &=\frac{2 \sqrt{1-\frac{1}{a x}} \sqrt{1+\frac{1}{a x}} x}{\sqrt{c-a c x}}-\frac{2 \sqrt{2} \sqrt{1-\frac{1}{a x}} \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{\frac{1}{x}}}{\sqrt{a} \sqrt{1+\frac{1}{a x}}}\right )}{\sqrt{a} \sqrt{\frac{1}{x}} \sqrt{c-a c x}}\\ \end{align*}
Mathematica [A] time = 0.0616523, size = 99, normalized size = 0.84 \[ \frac{2 x \sqrt{1-\frac{1}{a x}} \left (\sqrt{a} \sqrt{\frac{1}{a x}+1}-\sqrt{2} \sqrt{\frac{1}{x}} \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{\frac{1}{x}}}{\sqrt{a} \sqrt{\frac{1}{a x}+1}}\right )\right )}{\sqrt{a} \sqrt{c-a c x}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.154, size = 83, normalized size = 0.7 \begin{align*} 2\,{\frac{\sqrt{-c \left ( ax-1 \right ) }}{\sqrt{-c \left ( ax+1 \right ) }ca} \left ( \sqrt{c}\sqrt{2}\arctan \left ( 1/2\,{\frac{\sqrt{-c \left ( ax+1 \right ) }\sqrt{2}}{\sqrt{c}}} \right ) -\sqrt{-c \left ( ax+1 \right ) } \right ){\frac{1}{\sqrt{{\frac{ax-1}{ax+1}}}}}} \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{1}{\sqrt{-a c x + c} \sqrt{\frac{a x - 1}{a x + 1}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.57809, size = 568, normalized size = 4.81 \begin{align*} \left [\frac{\sqrt{2}{\left (a c x - c\right )} \sqrt{-\frac{1}{c}} \log \left (-\frac{a^{2} x^{2} - 2 \, \sqrt{2} \sqrt{-a c x + c}{\left (a x + 1\right )} \sqrt{\frac{a x - 1}{a x + 1}} \sqrt{-\frac{1}{c}} + 2 \, a x - 3}{a^{2} x^{2} - 2 \, a x + 1}\right ) - 2 \, \sqrt{-a c x + c}{\left (a x + 1\right )} \sqrt{\frac{a x - 1}{a x + 1}}}{a^{2} c x - a c}, -\frac{2 \,{\left (\sqrt{-a c x + c}{\left (a x + 1\right )} \sqrt{\frac{a x - 1}{a x + 1}} - \frac{\sqrt{2}{\left (a c x - c\right )} \arctan \left (\frac{\sqrt{2} \sqrt{-a c x + c} \sqrt{\frac{a x - 1}{a x + 1}}}{{\left (a x - 1\right )} \sqrt{c}}\right )}{\sqrt{c}}\right )}}{a^{2} c x - a c}\right ] \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 [C] time = 1.20736, size = 123, normalized size = 1.04 \begin{align*} \frac{2 \,{\left (\frac{\sqrt{2} \sqrt{c} \arctan \left (\frac{\sqrt{2} \sqrt{-a c x - c}}{2 \, \sqrt{c}}\right ) - \sqrt{-a c x - c}}{\mathrm{sgn}\left (-a c x - c\right )} - \frac{-i \, \sqrt{2} \sqrt{-c} \arctan \left (-i\right ) + \sqrt{2} \sqrt{-c}}{\mathrm{sgn}\left (c\right )}\right )}}{a c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]