Optimal. Leaf size=150 \[ -\frac{a^2 \left (c-\frac{c}{a x}\right )^{7/2}}{c^3 \sqrt{1-\frac{1}{a^2 x^2}}}-\frac{7 a^2 \sqrt{1-\frac{1}{a^2 x^2}} \left (c-\frac{c}{a x}\right )^{3/2}}{5 c}-\frac{56}{15} a^2 \sqrt{1-\frac{1}{a^2 x^2}} \sqrt{c-\frac{c}{a x}}-\frac{224 a^2 c \sqrt{1-\frac{1}{a^2 x^2}}}{15 \sqrt{c-\frac{c}{a x}}} \]
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Rubi [A] time = 0.266011, antiderivative size = 150, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.148, Rules used = {6178, 789, 657, 649} \[ -\frac{a^2 \left (c-\frac{c}{a x}\right )^{7/2}}{c^3 \sqrt{1-\frac{1}{a^2 x^2}}}-\frac{7 a^2 \sqrt{1-\frac{1}{a^2 x^2}} \left (c-\frac{c}{a x}\right )^{3/2}}{5 c}-\frac{56}{15} a^2 \sqrt{1-\frac{1}{a^2 x^2}} \sqrt{c-\frac{c}{a x}}-\frac{224 a^2 c \sqrt{1-\frac{1}{a^2 x^2}}}{15 \sqrt{c-\frac{c}{a x}}} \]
Antiderivative was successfully verified.
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Rule 6178
Rule 789
Rule 657
Rule 649
Rubi steps
\begin{align*} \int \frac{e^{-3 \coth ^{-1}(a x)} \sqrt{c-\frac{c}{a x}}}{x^3} \, dx &=-\frac{\operatorname{Subst}\left (\int \frac{x \left (c-\frac{c x}{a}\right )^{7/2}}{\left (1-\frac{x^2}{a^2}\right )^{3/2}} \, dx,x,\frac{1}{x}\right )}{c^3}\\ &=-\frac{a^2 \left (c-\frac{c}{a x}\right )^{7/2}}{c^3 \sqrt{1-\frac{1}{a^2 x^2}}}-\frac{(7 a) \operatorname{Subst}\left (\int \frac{\left (c-\frac{c x}{a}\right )^{5/2}}{\sqrt{1-\frac{x^2}{a^2}}} \, dx,x,\frac{1}{x}\right )}{2 c^2}\\ &=-\frac{7 a^2 \sqrt{1-\frac{1}{a^2 x^2}} \left (c-\frac{c}{a x}\right )^{3/2}}{5 c}-\frac{a^2 \left (c-\frac{c}{a x}\right )^{7/2}}{c^3 \sqrt{1-\frac{1}{a^2 x^2}}}-\frac{(28 a) \operatorname{Subst}\left (\int \frac{\left (c-\frac{c x}{a}\right )^{3/2}}{\sqrt{1-\frac{x^2}{a^2}}} \, dx,x,\frac{1}{x}\right )}{5 c}\\ &=-\frac{56}{15} a^2 \sqrt{1-\frac{1}{a^2 x^2}} \sqrt{c-\frac{c}{a x}}-\frac{7 a^2 \sqrt{1-\frac{1}{a^2 x^2}} \left (c-\frac{c}{a x}\right )^{3/2}}{5 c}-\frac{a^2 \left (c-\frac{c}{a x}\right )^{7/2}}{c^3 \sqrt{1-\frac{1}{a^2 x^2}}}-\frac{1}{15} (112 a) \operatorname{Subst}\left (\int \frac{\sqrt{c-\frac{c x}{a}}}{\sqrt{1-\frac{x^2}{a^2}}} \, dx,x,\frac{1}{x}\right )\\ &=-\frac{224 a^2 c \sqrt{1-\frac{1}{a^2 x^2}}}{15 \sqrt{c-\frac{c}{a x}}}-\frac{56}{15} a^2 \sqrt{1-\frac{1}{a^2 x^2}} \sqrt{c-\frac{c}{a x}}-\frac{7 a^2 \sqrt{1-\frac{1}{a^2 x^2}} \left (c-\frac{c}{a x}\right )^{3/2}}{5 c}-\frac{a^2 \left (c-\frac{c}{a x}\right )^{7/2}}{c^3 \sqrt{1-\frac{1}{a^2 x^2}}}\\ \end{align*}
Mathematica [A] time = 0.113466, size = 70, normalized size = 0.47 \[ -\frac{2 a \sqrt{1-\frac{1}{a^2 x^2}} \left (158 a^3 x^3+79 a^2 x^2-16 a x+3\right ) \sqrt{c-\frac{c}{a x}}}{15 x \left (a^2 x^2-1\right )} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.128, size = 70, normalized size = 0.5 \begin{align*} -{\frac{ \left ( 2\,ax+2 \right ) \left ( 158\,{x}^{3}{a}^{3}+79\,{a}^{2}{x}^{2}-16\,ax+3 \right ) }{15\,{x}^{2} \left ( ax-1 \right ) ^{2}}\sqrt{{\frac{c \left ( ax-1 \right ) }{ax}}} \left ({\frac{ax-1}{ax+1}} \right ) ^{{\frac{3}{2}}}} \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}} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{3}{2}}}{x^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.65063, size = 150, normalized size = 1. \begin{align*} -\frac{2 \,{\left (158 \, a^{3} x^{3} + 79 \, a^{2} x^{2} - 16 \, 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}} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{3}{2}}}{x^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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