Optimal. Leaf size=140 \[ \frac{x \sqrt{c-\frac{c}{a x}}}{\sqrt{1-\frac{1}{a x}} \sqrt{\frac{1}{a x}+1}}+\frac{9 \sqrt{c-\frac{c}{a x}}}{a \sqrt{1-\frac{1}{a x}} \sqrt{\frac{1}{a x}+1}}-\frac{7 \sqrt{c-\frac{c}{a x}} \tanh ^{-1}\left (\sqrt{\frac{1}{a x}+1}\right )}{a \sqrt{1-\frac{1}{a x}}} \]
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Rubi [A] time = 0.117883, antiderivative size = 140, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {6182, 6179, 89, 78, 63, 208} \[ \frac{x \sqrt{c-\frac{c}{a x}}}{\sqrt{1-\frac{1}{a x}} \sqrt{\frac{1}{a x}+1}}+\frac{9 \sqrt{c-\frac{c}{a x}}}{a \sqrt{1-\frac{1}{a x}} \sqrt{\frac{1}{a x}+1}}-\frac{7 \sqrt{c-\frac{c}{a x}} \tanh ^{-1}\left (\sqrt{\frac{1}{a x}+1}\right )}{a \sqrt{1-\frac{1}{a x}}} \]
Antiderivative was successfully verified.
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Rule 6182
Rule 6179
Rule 89
Rule 78
Rule 63
Rule 208
Rubi steps
\begin{align*} \int e^{-3 \coth ^{-1}(a x)} \sqrt{c-\frac{c}{a x}} \, dx &=\frac{\sqrt{c-\frac{c}{a x}} \int e^{-3 \coth ^{-1}(a x)} \sqrt{1-\frac{1}{a x}} \, dx}{\sqrt{1-\frac{1}{a x}}}\\ &=-\frac{\sqrt{c-\frac{c}{a x}} \operatorname{Subst}\left (\int \frac{\left (1-\frac{x}{a}\right )^2}{x^2 \left (1+\frac{x}{a}\right )^{3/2}} \, dx,x,\frac{1}{x}\right )}{\sqrt{1-\frac{1}{a x}}}\\ &=\frac{\sqrt{c-\frac{c}{a x}} x}{\sqrt{1-\frac{1}{a x}} \sqrt{1+\frac{1}{a x}}}-\frac{\sqrt{c-\frac{c}{a x}} \operatorname{Subst}\left (\int \frac{-\frac{7}{2 a}+\frac{x}{a^2}}{x \left (1+\frac{x}{a}\right )^{3/2}} \, dx,x,\frac{1}{x}\right )}{\sqrt{1-\frac{1}{a x}}}\\ &=\frac{9 \sqrt{c-\frac{c}{a x}}}{a \sqrt{1-\frac{1}{a x}} \sqrt{1+\frac{1}{a x}}}+\frac{\sqrt{c-\frac{c}{a x}} x}{\sqrt{1-\frac{1}{a x}} \sqrt{1+\frac{1}{a x}}}+\frac{\left (7 \sqrt{c-\frac{c}{a x}}\right ) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{1+\frac{x}{a}}} \, dx,x,\frac{1}{x}\right )}{2 a \sqrt{1-\frac{1}{a x}}}\\ &=\frac{9 \sqrt{c-\frac{c}{a x}}}{a \sqrt{1-\frac{1}{a x}} \sqrt{1+\frac{1}{a x}}}+\frac{\sqrt{c-\frac{c}{a x}} x}{\sqrt{1-\frac{1}{a x}} \sqrt{1+\frac{1}{a x}}}+\frac{\left (7 \sqrt{c-\frac{c}{a x}}\right ) \operatorname{Subst}\left (\int \frac{1}{-a+a x^2} \, dx,x,\sqrt{1+\frac{1}{a x}}\right )}{\sqrt{1-\frac{1}{a x}}}\\ &=\frac{9 \sqrt{c-\frac{c}{a x}}}{a \sqrt{1-\frac{1}{a x}} \sqrt{1+\frac{1}{a x}}}+\frac{\sqrt{c-\frac{c}{a x}} x}{\sqrt{1-\frac{1}{a x}} \sqrt{1+\frac{1}{a x}}}-\frac{7 \sqrt{c-\frac{c}{a x}} \tanh ^{-1}\left (\sqrt{1+\frac{1}{a x}}\right )}{a \sqrt{1-\frac{1}{a x}}}\\ \end{align*}
Mathematica [A] time = 0.0462443, size = 67, normalized size = 0.48 \[ \frac{\sqrt{c-\frac{c}{a x}} \left (a x-7 \sqrt{\frac{1}{a x}+1} \tanh ^{-1}\left (\sqrt{\frac{1}{a x}+1}\right )+9\right )}{a \sqrt{1-\frac{1}{a^2 x^2}}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.184, size = 146, normalized size = 1. \begin{align*} -{\frac{ \left ( ax+1 \right ) x}{2\, \left ( ax-1 \right ) ^{2}}\sqrt{{\frac{c \left ( ax-1 \right ) }{ax}}} \left ({\frac{ax-1}{ax+1}} \right ) ^{{\frac{3}{2}}} \left ( -2\,{a}^{3/2}x\sqrt{ \left ( ax+1 \right ) x}+7\,\ln \left ( 1/2\,{\frac{2\,\sqrt{ \left ( ax+1 \right ) x}\sqrt{a}+2\,ax+1}{\sqrt{a}}} \right ) xa-18\,\sqrt{ \left ( ax+1 \right ) x}\sqrt{a}+7\,\ln \left ( 1/2\,{\frac{2\,\sqrt{ \left ( ax+1 \right ) x}\sqrt{a}+2\,ax+1}{\sqrt{a}}} \right ) \right ){\frac{1}{\sqrt{ \left ( ax+1 \right ) x}}}{\frac{1}{\sqrt{a}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{c - \frac{c}{a x}} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{3}{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.10432, size = 640, normalized size = 4.57 \begin{align*} \left [\frac{7 \,{\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} + 9 \, a x\right )} \sqrt{\frac{a x - 1}{a x + 1}} \sqrt{\frac{a c x - c}{a x}}}{4 \,{\left (a^{2} x - a\right )}}, \frac{7 \,{\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} + 9 \, a x\right )} \sqrt{\frac{a x - 1}{a x + 1}} \sqrt{\frac{a c x - c}{a x}}}{2 \,{\left (a^{2} x - a\right )}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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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.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{c - \frac{c}{a x}} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{3}{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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