Optimal. Leaf size=117 \[ \frac{c^3 x \left (1-\frac{1}{a^2 x^2}\right )^{3/2}}{\left (c-\frac{c}{a x}\right )^{3/2}}+\frac{c^2 \sqrt{1-\frac{1}{a^2 x^2}}}{a \sqrt{c-\frac{c}{a x}}}-\frac{c^{3/2} \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{1-\frac{1}{a^2 x^2}}}{\sqrt{c-\frac{c}{a x}}}\right )}{a} \]
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Rubi [A] time = 0.223204, antiderivative size = 117, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.227, Rules used = {6177, 879, 865, 875, 208} \[ \frac{c^3 x \left (1-\frac{1}{a^2 x^2}\right )^{3/2}}{\left (c-\frac{c}{a x}\right )^{3/2}}+\frac{c^2 \sqrt{1-\frac{1}{a^2 x^2}}}{a \sqrt{c-\frac{c}{a x}}}-\frac{c^{3/2} \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{1-\frac{1}{a^2 x^2}}}{\sqrt{c-\frac{c}{a x}}}\right )}{a} \]
Antiderivative was successfully verified.
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Rule 6177
Rule 879
Rule 865
Rule 875
Rule 208
Rubi steps
\begin{align*} \int e^{\coth ^{-1}(a x)} \left (c-\frac{c}{a x}\right )^{3/2} \, dx &=-\left (c \operatorname{Subst}\left (\int \frac{\sqrt{c-\frac{c x}{a}} \sqrt{1-\frac{x^2}{a^2}}}{x^2} \, dx,x,\frac{1}{x}\right )\right )\\ &=\frac{c^3 \left (1-\frac{1}{a^2 x^2}\right )^{3/2} x}{\left (c-\frac{c}{a x}\right )^{3/2}}+\frac{c^2 \operatorname{Subst}\left (\int \frac{\sqrt{1-\frac{x^2}{a^2}}}{x \sqrt{c-\frac{c x}{a}}} \, dx,x,\frac{1}{x}\right )}{2 a}\\ &=\frac{c^2 \sqrt{1-\frac{1}{a^2 x^2}}}{a \sqrt{c-\frac{c}{a x}}}+\frac{c^3 \left (1-\frac{1}{a^2 x^2}\right )^{3/2} x}{\left (c-\frac{c}{a x}\right )^{3/2}}+\frac{c \operatorname{Subst}\left (\int \frac{\sqrt{c-\frac{c x}{a}}}{x \sqrt{1-\frac{x^2}{a^2}}} \, dx,x,\frac{1}{x}\right )}{2 a}\\ &=\frac{c^2 \sqrt{1-\frac{1}{a^2 x^2}}}{a \sqrt{c-\frac{c}{a x}}}+\frac{c^3 \left (1-\frac{1}{a^2 x^2}\right )^{3/2} x}{\left (c-\frac{c}{a x}\right )^{3/2}}+\frac{c^3 \operatorname{Subst}\left (\int \frac{1}{-\frac{c}{a^2}+\frac{c^2 x^2}{a^2}} \, dx,x,\frac{\sqrt{1-\frac{1}{a^2 x^2}}}{\sqrt{c-\frac{c}{a x}}}\right )}{a^3}\\ &=\frac{c^2 \sqrt{1-\frac{1}{a^2 x^2}}}{a \sqrt{c-\frac{c}{a x}}}+\frac{c^3 \left (1-\frac{1}{a^2 x^2}\right )^{3/2} x}{\left (c-\frac{c}{a x}\right )^{3/2}}-\frac{c^{3/2} \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{1-\frac{1}{a^2 x^2}}}{\sqrt{c-\frac{c}{a x}}}\right )}{a}\\ \end{align*}
Mathematica [A] time = 0.0518382, size = 70, normalized size = 0.6 \[ \frac{c \sqrt{c-\frac{c}{a x}} \left (\sqrt{\frac{1}{a x}+1} (a x+2)-\tanh ^{-1}\left (\sqrt{\frac{1}{a x}+1}\right )\right )}{a \sqrt{1-\frac{1}{a x}}} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.178, size = 105, normalized size = 0.9 \begin{align*} -{\frac{c}{2}\sqrt{{\frac{c \left ( ax-1 \right ) }{ax}}} \left ( -2\,{a}^{3/2}x\sqrt{ \left ( ax+1 \right ) x}+\ln \left ({\frac{1}{2} \left ( 2\,\sqrt{ \left ( ax+1 \right ) x}\sqrt{a}+2\,ax+1 \right ){\frac{1}{\sqrt{a}}}} \right ) xa-4\,\sqrt{ \left ( ax+1 \right ) x}\sqrt{a} \right ){\frac{1}{\sqrt{{\frac{ax-1}{ax+1}}}}}{a}^{-{\frac{3}{2}}}{\frac{1}{\sqrt{ \left ( ax+1 \right ) x}}}} \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 \frac{{\left (c - \frac{c}{a x}\right )}^{\frac{3}{2}}}{\sqrt{\frac{a x - 1}{a x + 1}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.21834, size = 667, normalized size = 5.7 \begin{align*} \left [\frac{{\left (a c x - c\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} c x^{2} + 3 \, a c x + 2 \, c\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{{\left (a c x - c\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} c x^{2} + 3 \, a c x + 2 \, c\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 \frac{{\left (c - \frac{c}{a x}\right )}^{\frac{3}{2}}}{\sqrt{\frac{a x - 1}{a x + 1}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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