Optimal. Leaf size=77 \[ \frac{2 a^2 c \left (1-\frac{1}{a^2 x^2}\right )^{3/2}}{5 \sqrt{c-\frac{c}{a x}}}-\frac{2 a^2 c^2 \left (1-\frac{1}{a^2 x^2}\right )^{3/2}}{15 \left (c-\frac{c}{a x}\right )^{3/2}} \]
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Rubi [A] time = 0.188534, antiderivative size = 77, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.12, Rules used = {6178, 795, 649} \[ \frac{2 a^2 c \left (1-\frac{1}{a^2 x^2}\right )^{3/2}}{5 \sqrt{c-\frac{c}{a x}}}-\frac{2 a^2 c^2 \left (1-\frac{1}{a^2 x^2}\right )^{3/2}}{15 \left (c-\frac{c}{a x}\right )^{3/2}} \]
Antiderivative was successfully verified.
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Rule 6178
Rule 795
Rule 649
Rubi steps
\begin{align*} \int \frac{e^{\coth ^{-1}(a x)} \sqrt{c-\frac{c}{a x}}}{x^3} \, dx &=-\left (c \operatorname{Subst}\left (\int \frac{x \sqrt{1-\frac{x^2}{a^2}}}{\sqrt{c-\frac{c x}{a}}} \, dx,x,\frac{1}{x}\right )\right )\\ &=\frac{2 a^2 c \left (1-\frac{1}{a^2 x^2}\right )^{3/2}}{5 \sqrt{c-\frac{c}{a x}}}-\frac{1}{5} (a 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 )\\ &=-\frac{2 a^2 c^2 \left (1-\frac{1}{a^2 x^2}\right )^{3/2}}{15 \left (c-\frac{c}{a x}\right )^{3/2}}+\frac{2 a^2 c \left (1-\frac{1}{a^2 x^2}\right )^{3/2}}{5 \sqrt{c-\frac{c}{a x}}}\\ \end{align*}
Mathematica [A] time = 0.102677, size = 58, normalized size = 0.75 \[ \frac{2 a \sqrt{1-\frac{1}{a^2 x^2}} \left (2 a^2 x^2-a x-3\right ) \sqrt{c-\frac{c}{a x}}}{15 x (a x-1)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.118, size = 47, normalized size = 0.6 \begin{align*}{\frac{ \left ( 2\,ax+2 \right ) \left ( 2\,ax-3 \right ) }{15\,{x}^{2}}\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^{3} \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 = 1.62791, size = 140, normalized size = 1.82 \begin{align*} \frac{2 \,{\left (2 \, a^{3} x^{3} + a^{2} x^{2} - 4 \, a x - 3\right )} \sqrt{\frac{a x - 1}{a x + 1}} \sqrt{\frac{a c x - c}{a x}}}{15 \,{\left (a x^{3} - x^{2}\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^{3} \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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