Optimal. Leaf size=152 \[ \frac{x \sqrt{1-\frac{1}{a x}} \sqrt{\frac{1}{a x}+1}}{\sqrt{c-\frac{c}{a x}}}+\frac{3 \sqrt{1-\frac{1}{a x}} \tanh ^{-1}\left (\sqrt{\frac{1}{a x}+1}\right )}{a \sqrt{c-\frac{c}{a x}}}-\frac{2 \sqrt{2} \sqrt{1-\frac{1}{a x}} \tanh ^{-1}\left (\frac{\sqrt{\frac{1}{a x}+1}}{\sqrt{2}}\right )}{a \sqrt{c-\frac{c}{a x}}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.134815, antiderivative size = 152, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.318, Rules used = {6182, 6179, 99, 156, 63, 208, 206} \[ \frac{x \sqrt{1-\frac{1}{a x}} \sqrt{\frac{1}{a x}+1}}{\sqrt{c-\frac{c}{a x}}}+\frac{3 \sqrt{1-\frac{1}{a x}} \tanh ^{-1}\left (\sqrt{\frac{1}{a x}+1}\right )}{a \sqrt{c-\frac{c}{a x}}}-\frac{2 \sqrt{2} \sqrt{1-\frac{1}{a x}} \tanh ^{-1}\left (\frac{\sqrt{\frac{1}{a x}+1}}{\sqrt{2}}\right )}{a \sqrt{c-\frac{c}{a x}}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 6182
Rule 6179
Rule 99
Rule 156
Rule 63
Rule 208
Rule 206
Rubi steps
\begin{align*} \int \frac{e^{\coth ^{-1}(a x)}}{\sqrt{c-\frac{c}{a x}}} \, dx &=\frac{\sqrt{1-\frac{1}{a x}} \int \frac{e^{\coth ^{-1}(a x)}}{\sqrt{1-\frac{1}{a x}}} \, dx}{\sqrt{c-\frac{c}{a x}}}\\ &=-\frac{\sqrt{1-\frac{1}{a x}} \operatorname{Subst}\left (\int \frac{\sqrt{1+\frac{x}{a}}}{x^2 \left (1-\frac{x}{a}\right )} \, dx,x,\frac{1}{x}\right )}{\sqrt{c-\frac{c}{a x}}}\\ &=\frac{\sqrt{1-\frac{1}{a x}} \sqrt{1+\frac{1}{a x}} x}{\sqrt{c-\frac{c}{a x}}}-\frac{\sqrt{1-\frac{1}{a x}} \operatorname{Subst}\left (\int \frac{\frac{3}{2 a}+\frac{x}{2 a^2}}{x \left (1-\frac{x}{a}\right ) \sqrt{1+\frac{x}{a}}} \, dx,x,\frac{1}{x}\right )}{\sqrt{c-\frac{c}{a x}}}\\ &=\frac{\sqrt{1-\frac{1}{a x}} \sqrt{1+\frac{1}{a x}} x}{\sqrt{c-\frac{c}{a x}}}-\frac{\left (2 \sqrt{1-\frac{1}{a x}}\right ) \operatorname{Subst}\left (\int \frac{1}{\left (1-\frac{x}{a}\right ) \sqrt{1+\frac{x}{a}}} \, dx,x,\frac{1}{x}\right )}{a^2 \sqrt{c-\frac{c}{a x}}}-\frac{\left (3 \sqrt{1-\frac{1}{a x}}\right ) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{1+\frac{x}{a}}} \, dx,x,\frac{1}{x}\right )}{2 a \sqrt{c-\frac{c}{a x}}}\\ &=\frac{\sqrt{1-\frac{1}{a x}} \sqrt{1+\frac{1}{a x}} x}{\sqrt{c-\frac{c}{a x}}}-\frac{\left (3 \sqrt{1-\frac{1}{a x}}\right ) \operatorname{Subst}\left (\int \frac{1}{-a+a x^2} \, dx,x,\sqrt{1+\frac{1}{a x}}\right )}{\sqrt{c-\frac{c}{a x}}}-\frac{\left (4 \sqrt{1-\frac{1}{a x}}\right ) \operatorname{Subst}\left (\int \frac{1}{2-x^2} \, dx,x,\sqrt{1+\frac{1}{a x}}\right )}{a \sqrt{c-\frac{c}{a x}}}\\ &=\frac{\sqrt{1-\frac{1}{a x}} \sqrt{1+\frac{1}{a x}} x}{\sqrt{c-\frac{c}{a x}}}+\frac{3 \sqrt{1-\frac{1}{a x}} \tanh ^{-1}\left (\sqrt{1+\frac{1}{a x}}\right )}{a \sqrt{c-\frac{c}{a x}}}-\frac{2 \sqrt{2} \sqrt{1-\frac{1}{a x}} \tanh ^{-1}\left (\frac{\sqrt{1+\frac{1}{a x}}}{\sqrt{2}}\right )}{a \sqrt{c-\frac{c}{a x}}}\\ \end{align*}
Mathematica [A] time = 0.0647422, size = 93, normalized size = 0.61 \[ \frac{\sqrt{1-\frac{1}{a x}} \left (a x \sqrt{\frac{1}{a x}+1}+3 \tanh ^{-1}\left (\sqrt{\frac{1}{a x}+1}\right )-2 \sqrt{2} \tanh ^{-1}\left (\frac{\sqrt{\frac{1}{a x}+1}}{\sqrt{2}}\right )\right )}{a \sqrt{c-\frac{c}{a x}}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.171, size = 151, normalized size = 1. \begin{align*}{\frac{x}{2\,c}\sqrt{{\frac{c \left ( ax-1 \right ) }{ax}}} \left ( 2\,\sqrt{ \left ( ax+1 \right ) x}{a}^{3/2}\sqrt{{a}^{-1}}-2\,\sqrt{2}\ln \left ({\frac{2\,\sqrt{2}\sqrt{{a}^{-1}}\sqrt{ \left ( ax+1 \right ) x}a+3\,ax+1}{ax-1}} \right ) \sqrt{a}+3\,\ln \left ( 1/2\,{\frac{2\,\sqrt{ \left ( ax+1 \right ) x}\sqrt{a}+2\,ax+1}{\sqrt{a}}} \right ) a\sqrt{{a}^{-1}} \right ){\frac{1}{\sqrt{{\frac{ax-1}{ax+1}}}}}{a}^{-{\frac{3}{2}}}{\frac{1}{\sqrt{ \left ( ax+1 \right ) x}}}{\frac{1}{\sqrt{{a}^{-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{c - \frac{c}{a x}} \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 = 2.70694, size = 1137, normalized size = 7.48 \begin{align*} \left [\frac{3 \,{\left (a x - 1\right )} \sqrt{c} \log \left (-\frac{8 \, a^{3} c x^{3} - 7 \, a c x + 4 \,{\left (2 \, a^{3} x^{3} + 3 \, a^{2} x^{2} + a x\right )} \sqrt{c} \sqrt{\frac{a x - 1}{a x + 1}} \sqrt{\frac{a c x - c}{a x}} - c}{a x - 1}\right ) + 4 \,{\left (a^{2} x^{2} + a x\right )} \sqrt{\frac{a x - 1}{a x + 1}} \sqrt{\frac{a c x - c}{a x}} + \frac{2 \, \sqrt{2}{\left (a c x - c\right )} \log \left (-\frac{17 \, a^{3} x^{3} - 3 \, a^{2} x^{2} - 13 \, a x - \frac{4 \, \sqrt{2}{\left (3 \, a^{3} x^{3} + 4 \, a^{2} x^{2} + a x\right )} \sqrt{\frac{a x - 1}{a x + 1}} \sqrt{\frac{a c x - c}{a x}}}{\sqrt{c}} - 1}{a^{3} x^{3} - 3 \, a^{2} x^{2} + 3 \, a x - 1}\right )}{\sqrt{c}}}{4 \,{\left (a^{2} c x - a c\right )}}, \frac{2 \, \sqrt{2}{\left (a c x - c\right )} \sqrt{-\frac{1}{c}} \arctan \left (\frac{2 \, \sqrt{2}{\left (a^{2} x^{2} + a x\right )} \sqrt{\frac{a x - 1}{a x + 1}} \sqrt{-\frac{1}{c}} \sqrt{\frac{a c x - c}{a x}}}{3 \, a^{2} x^{2} - 2 \, a x - 1}\right ) - 3 \,{\left (a x - 1\right )} \sqrt{-c} \arctan \left (\frac{2 \,{\left (a^{2} x^{2} + a x\right )} \sqrt{-c} \sqrt{\frac{a x - 1}{a x + 1}} \sqrt{\frac{a c x - c}{a x}}}{2 \, a^{2} c x^{2} - a c x - c}\right ) + 2 \,{\left (a^{2} x^{2} + a x\right )} \sqrt{\frac{a x - 1}{a x + 1}} \sqrt{\frac{a c x - c}{a x}}}{2 \,{\left (a^{2} c x - a c\right )}}\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 [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{c - \frac{c}{a x}} \sqrt{\frac{a x - 1}{a x + 1}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]