Optimal. Leaf size=117 \[ -\frac{2 x \sqrt{c-\frac{c}{a x}}}{c^2 \sqrt{1-\frac{1}{a^2 x^2}}}-\frac{3 \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{1-\frac{1}{a^2 x^2}}}{\sqrt{c-\frac{c}{a x}}}\right )}{a c^{3/2}}+\frac{3 x \sqrt{1-\frac{1}{a^2 x^2}}}{c \sqrt{c-\frac{c}{a x}}} \]
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Rubi [A] time = 0.222819, antiderivative size = 117, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208, Rules used = {6177, 869, 873, 875, 208} \[ -\frac{2 x \sqrt{c-\frac{c}{a x}}}{c^2 \sqrt{1-\frac{1}{a^2 x^2}}}-\frac{3 \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{1-\frac{1}{a^2 x^2}}}{\sqrt{c-\frac{c}{a x}}}\right )}{a c^{3/2}}+\frac{3 x \sqrt{1-\frac{1}{a^2 x^2}}}{c \sqrt{c-\frac{c}{a x}}} \]
Antiderivative was successfully verified.
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Rule 6177
Rule 869
Rule 873
Rule 875
Rule 208
Rubi steps
\begin{align*} \int \frac{e^{-3 \coth ^{-1}(a x)}}{\left (c-\frac{c}{a x}\right )^{3/2}} \, dx &=-\frac{\operatorname{Subst}\left (\int \frac{\left (c-\frac{c x}{a}\right )^{3/2}}{x^2 \left (1-\frac{x^2}{a^2}\right )^{3/2}} \, dx,x,\frac{1}{x}\right )}{c^3}\\ &=-\frac{2 \sqrt{c-\frac{c}{a x}} x}{c^2 \sqrt{1-\frac{1}{a^2 x^2}}}-\frac{3 \operatorname{Subst}\left (\int \frac{\sqrt{c-\frac{c x}{a}}}{x^2 \sqrt{1-\frac{x^2}{a^2}}} \, dx,x,\frac{1}{x}\right )}{c^2}\\ &=\frac{3 \sqrt{1-\frac{1}{a^2 x^2}} x}{c \sqrt{c-\frac{c}{a x}}}-\frac{2 \sqrt{c-\frac{c}{a x}} x}{c^2 \sqrt{1-\frac{1}{a^2 x^2}}}+\frac{3 \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 c^2}\\ &=\frac{3 \sqrt{1-\frac{1}{a^2 x^2}} x}{c \sqrt{c-\frac{c}{a x}}}-\frac{2 \sqrt{c-\frac{c}{a x}} x}{c^2 \sqrt{1-\frac{1}{a^2 x^2}}}+\frac{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{3 \sqrt{1-\frac{1}{a^2 x^2}} x}{c \sqrt{c-\frac{c}{a x}}}-\frac{2 \sqrt{c-\frac{c}{a x}} x}{c^2 \sqrt{1-\frac{1}{a^2 x^2}}}-\frac{3 \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{1-\frac{1}{a^2 x^2}}}{\sqrt{c-\frac{c}{a x}}}\right )}{a c^{3/2}}\\ \end{align*}
Mathematica [C] time = 0.0446935, size = 64, normalized size = 0.55 \[ \frac{2 \left (1-\frac{1}{a x}\right )^{3/2} \text{Hypergeometric2F1}\left (-\frac{1}{2},2,\frac{1}{2},\frac{1}{a x}+1\right )}{a \sqrt{\frac{1}{a x}+1} \left (c-\frac{c}{a x}\right )^{3/2}} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.182, size = 149, normalized size = 1.3 \begin{align*} -{\frac{ \left ( ax+1 \right ) x}{2\, \left ( ax-1 \right ) ^{2}{c}^{2}} \left ({\frac{ax-1}{ax+1}} \right ) ^{{\frac{3}{2}}}\sqrt{{\frac{c \left ( ax-1 \right ) }{ax}}} \left ( -2\,{a}^{3/2}x\sqrt{ \left ( ax+1 \right ) x}+3\,\ln \left ( 1/2\,{\frac{2\,\sqrt{ \left ( ax+1 \right ) x}\sqrt{a}+2\,ax+1}{\sqrt{a}}} \right ) xa-6\,\sqrt{ \left ( ax+1 \right ) x}\sqrt{a}+3\,\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 \frac{\left (\frac{a x - 1}{a x + 1}\right )^{\frac{3}{2}}}{{\left (c - \frac{c}{a x}\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 = 2.23823, size = 662, normalized size = 5.66 \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} + 3 \, a x\right )} \sqrt{\frac{a x - 1}{a x + 1}} \sqrt{\frac{a c x - c}{a x}}}{4 \,{\left (a^{2} c^{2} x - a c^{2}\right )}}, \frac{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} + 3 \, a x\right )} \sqrt{\frac{a x - 1}{a x + 1}} \sqrt{\frac{a c x - c}{a x}}}{2 \,{\left (a^{2} c^{2} x - a c^{2}\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(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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