Optimal. Leaf size=158 \[ \frac{x \left (a-\frac{1}{x}\right )^2 \left (c-\frac{c}{a x}\right )^{3/2}}{a^2 \left (1-\frac{1}{a x}\right )^{3/2} \sqrt{\frac{1}{a x}+1}}+\frac{\left (21 a+\frac{1}{x}\right ) \left (c-\frac{c}{a x}\right )^{3/2}}{a^2 \left (1-\frac{1}{a x}\right )^{3/2} \sqrt{\frac{1}{a x}+1}}-\frac{9 \left (c-\frac{c}{a x}\right )^{3/2} \tanh ^{-1}\left (\sqrt{\frac{1}{a x}+1}\right )}{a \left (1-\frac{1}{a x}\right )^{3/2}} \]
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Rubi [A] time = 0.133294, antiderivative size = 158, 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, 98, 146, 63, 208} \[ \frac{x \left (a-\frac{1}{x}\right )^2 \left (c-\frac{c}{a x}\right )^{3/2}}{a^2 \left (1-\frac{1}{a x}\right )^{3/2} \sqrt{\frac{1}{a x}+1}}+\frac{\left (21 a+\frac{1}{x}\right ) \left (c-\frac{c}{a x}\right )^{3/2}}{a^2 \left (1-\frac{1}{a x}\right )^{3/2} \sqrt{\frac{1}{a x}+1}}-\frac{9 \left (c-\frac{c}{a x}\right )^{3/2} \tanh ^{-1}\left (\sqrt{\frac{1}{a x}+1}\right )}{a \left (1-\frac{1}{a x}\right )^{3/2}} \]
Antiderivative was successfully verified.
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Rule 6182
Rule 6179
Rule 98
Rule 146
Rule 63
Rule 208
Rubi steps
\begin{align*} \int e^{-3 \coth ^{-1}(a x)} \left (c-\frac{c}{a x}\right )^{3/2} \, dx &=\frac{\left (c-\frac{c}{a x}\right )^{3/2} \int e^{-3 \coth ^{-1}(a x)} \left (1-\frac{1}{a x}\right )^{3/2} \, dx}{\left (1-\frac{1}{a x}\right )^{3/2}}\\ &=-\frac{\left (c-\frac{c}{a x}\right )^{3/2} \operatorname{Subst}\left (\int \frac{\left (1-\frac{x}{a}\right )^3}{x^2 \left (1+\frac{x}{a}\right )^{3/2}} \, dx,x,\frac{1}{x}\right )}{\left (1-\frac{1}{a x}\right )^{3/2}}\\ &=\frac{\left (a-\frac{1}{x}\right )^2 \left (c-\frac{c}{a x}\right )^{3/2} x}{a^2 \left (1-\frac{1}{a x}\right )^{3/2} \sqrt{1+\frac{1}{a x}}}+\frac{\left (c-\frac{c}{a x}\right )^{3/2} \operatorname{Subst}\left (\int \frac{\left (\frac{9}{2 a}-\frac{x}{2 a^2}\right ) \left (1-\frac{x}{a}\right )}{x \left (1+\frac{x}{a}\right )^{3/2}} \, dx,x,\frac{1}{x}\right )}{\left (1-\frac{1}{a x}\right )^{3/2}}\\ &=\frac{\left (21 a+\frac{1}{x}\right ) \left (c-\frac{c}{a x}\right )^{3/2}}{a^2 \left (1-\frac{1}{a x}\right )^{3/2} \sqrt{1+\frac{1}{a x}}}+\frac{\left (a-\frac{1}{x}\right )^2 \left (c-\frac{c}{a x}\right )^{3/2} x}{a^2 \left (1-\frac{1}{a x}\right )^{3/2} \sqrt{1+\frac{1}{a x}}}+\frac{\left (9 \left (c-\frac{c}{a x}\right )^{3/2}\right ) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{1+\frac{x}{a}}} \, dx,x,\frac{1}{x}\right )}{2 a \left (1-\frac{1}{a x}\right )^{3/2}}\\ &=\frac{\left (21 a+\frac{1}{x}\right ) \left (c-\frac{c}{a x}\right )^{3/2}}{a^2 \left (1-\frac{1}{a x}\right )^{3/2} \sqrt{1+\frac{1}{a x}}}+\frac{\left (a-\frac{1}{x}\right )^2 \left (c-\frac{c}{a x}\right )^{3/2} x}{a^2 \left (1-\frac{1}{a x}\right )^{3/2} \sqrt{1+\frac{1}{a x}}}+\frac{\left (9 \left (c-\frac{c}{a x}\right )^{3/2}\right ) \operatorname{Subst}\left (\int \frac{1}{-a+a x^2} \, dx,x,\sqrt{1+\frac{1}{a x}}\right )}{\left (1-\frac{1}{a x}\right )^{3/2}}\\ &=\frac{\left (21 a+\frac{1}{x}\right ) \left (c-\frac{c}{a x}\right )^{3/2}}{a^2 \left (1-\frac{1}{a x}\right )^{3/2} \sqrt{1+\frac{1}{a x}}}+\frac{\left (a-\frac{1}{x}\right )^2 \left (c-\frac{c}{a x}\right )^{3/2} x}{a^2 \left (1-\frac{1}{a x}\right )^{3/2} \sqrt{1+\frac{1}{a x}}}-\frac{9 \left (c-\frac{c}{a x}\right )^{3/2} \tanh ^{-1}\left (\sqrt{1+\frac{1}{a x}}\right )}{a \left (1-\frac{1}{a x}\right )^{3/2}}\\ \end{align*}
Mathematica [C] time = 0.0411834, size = 71, normalized size = 0.45 \[ \frac{c \sqrt{c-\frac{c}{a x}} \left (9 a x \text{Hypergeometric2F1}\left (-\frac{1}{2},1,\frac{1}{2},\frac{1}{a x}+1\right )+a^2 x^2+10 a x+2\right )}{a^2 x \sqrt{1-\frac{1}{a^2 x^2}}} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.189, size = 169, normalized size = 1.1 \begin{align*}{\frac{c \left ( ax+1 \right ) }{2\, \left ( ax-1 \right ) ^{2}} \left ({\frac{ax-1}{ax+1}} \right ) ^{{\frac{3}{2}}}\sqrt{{\frac{c \left ( ax-1 \right ) }{ax}}} \left ( 2\,{a}^{5/2}{x}^{2}\sqrt{ \left ( ax+1 \right ) x}+38\,{a}^{3/2}x\sqrt{ \left ( ax+1 \right ) x}-9\,\ln \left ( 1/2\,{\frac{2\,\sqrt{ \left ( ax+1 \right ) x}\sqrt{a}+2\,ax+1}{\sqrt{a}}} \right ){x}^{2}{a}^{2}-9\,\ln \left ( 1/2\,{\frac{2\,\sqrt{ \left ( ax+1 \right ) x}\sqrt{a}+2\,ax+1}{\sqrt{a}}} \right ) xa+4\,\sqrt{ \left ( ax+1 \right ) x}\sqrt{a} \right ){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{\left (c - \frac{c}{a x}\right )}^{\frac{3}{2}} \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 = 2.12392, size = 675, normalized size = 4.27 \begin{align*} \left [\frac{9 \,{\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} + 19 \, 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{9 \,{\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} + 19 \, 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{\left (c - \frac{c}{a x}\right )}^{\frac{3}{2}} \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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