Optimal. Leaf size=70 \[ -\frac{2}{3} a \sqrt{1-\frac{1}{a^2 x^2}} \sqrt{c-\frac{c}{a x}}-\frac{8 a c \sqrt{1-\frac{1}{a^2 x^2}}}{3 \sqrt{c-\frac{c}{a x}}} \]
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Rubi [A] time = 0.184962, antiderivative size = 70, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {6178, 657, 649} \[ -\frac{2}{3} a \sqrt{1-\frac{1}{a^2 x^2}} \sqrt{c-\frac{c}{a x}}-\frac{8 a c \sqrt{1-\frac{1}{a^2 x^2}}}{3 \sqrt{c-\frac{c}{a x}}} \]
Antiderivative was successfully verified.
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Rule 6178
Rule 657
Rule 649
Rubi steps
\begin{align*} \int \frac{e^{-\coth ^{-1}(a x)} \sqrt{c-\frac{c}{a x}}}{x^2} \, dx &=-\frac{\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 )}{c}\\ &=-\frac{2}{3} a \sqrt{1-\frac{1}{a^2 x^2}} \sqrt{c-\frac{c}{a x}}-\frac{4}{3} \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{8 a c \sqrt{1-\frac{1}{a^2 x^2}}}{3 \sqrt{c-\frac{c}{a x}}}-\frac{2}{3} a \sqrt{1-\frac{1}{a^2 x^2}} \sqrt{c-\frac{c}{a x}}\\ \end{align*}
Mathematica [A] time = 0.0815907, size = 46, normalized size = 0.66 \[ -\frac{2 a \sqrt{1-\frac{1}{a^2 x^2}} (5 a x-1) \sqrt{c-\frac{c}{a x}}}{3 a x-3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.125, size = 54, normalized size = 0.8 \begin{align*} -{\frac{ \left ( 2\,ax+2 \right ) \left ( 5\,ax-1 \right ) }{ \left ( 3\,ax-3 \right ) x}\sqrt{{\frac{c \left ( ax-1 \right ) }{ax}}}\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}} \sqrt{\frac{a x - 1}{a x + 1}}}{x^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.55286, size = 124, normalized size = 1.77 \begin{align*} -\frac{2 \,{\left (5 \, a^{2} x^{2} + 4 \, 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}} \sqrt{\frac{a x - 1}{a x + 1}}}{x^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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