Optimal. Leaf size=37 \[ -\frac{2 a c^2 \left (1-\frac{1}{a^2 x^2}\right )^{3/2}}{3 \left (c-\frac{c}{a x}\right )^{3/2}} \]
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Rubi [A] time = 0.147761, antiderivative size = 37, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.08, Rules used = {6178, 649} \[ -\frac{2 a c^2 \left (1-\frac{1}{a^2 x^2}\right )^{3/2}}{3 \left (c-\frac{c}{a x}\right )^{3/2}} \]
Antiderivative was successfully verified.
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Rule 6178
Rule 649
Rubi steps
\begin{align*} \int \frac{e^{\coth ^{-1}(a x)} \sqrt{c-\frac{c}{a x}}}{x^2} \, dx &=-\left (c \operatorname{Subst}\left (\int \frac{\sqrt{1-\frac{x^2}{a^2}}}{\sqrt{c-\frac{c x}{a}}} \, dx,x,\frac{1}{x}\right )\right )\\ &=-\frac{2 a c^2 \left (1-\frac{1}{a^2 x^2}\right )^{3/2}}{3 \left (c-\frac{c}{a x}\right )^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.0872397, size = 45, normalized size = 1.22 \[ -\frac{2 a \sqrt{1-\frac{1}{a^2 x^2}} (a x+1) \sqrt{c-\frac{c}{a x}}}{3 a x-3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.117, size = 41, normalized size = 1.1 \begin{align*} -{\frac{2\,ax+2}{3\,x}\sqrt{{\frac{c \left ( ax-1 \right ) }{ax}}}{\frac{1}{\sqrt{{\frac{ax-1}{ax+1}}}}}} \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{\sqrt{c - \frac{c}{a x}}}{x^{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 = 1.63512, size = 122, normalized size = 3.3 \begin{align*} -\frac{2 \,{\left (a^{2} x^{2} + 2 \, a x + 1\right )} \sqrt{\frac{a x - 1}{a x + 1}} \sqrt{\frac{a c x - c}{a x}}}{3 \,{\left (a x^{2} - x\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{\sqrt{c - \frac{c}{a x}}}{x^{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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