Optimal. Leaf size=82 \[ -\frac{2 a \left (c-\frac{c}{a x}\right )^{3/2}}{3 c}-4 a \sqrt{c-\frac{c}{a x}}+4 \sqrt{2} a \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c-\frac{c}{a x}}}{\sqrt{2} \sqrt{c}}\right ) \]
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Rubi [A] time = 0.37594, antiderivative size = 82, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 8, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.296, Rules used = {6167, 6133, 25, 514, 444, 50, 63, 208} \[ -\frac{2 a \left (c-\frac{c}{a x}\right )^{3/2}}{3 c}-4 a \sqrt{c-\frac{c}{a x}}+4 \sqrt{2} a \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c-\frac{c}{a x}}}{\sqrt{2} \sqrt{c}}\right ) \]
Antiderivative was successfully verified.
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Rule 6167
Rule 6133
Rule 25
Rule 514
Rule 444
Rule 50
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{e^{-2 \coth ^{-1}(a x)} \sqrt{c-\frac{c}{a x}}}{x^2} \, dx &=-\int \frac{e^{-2 \tanh ^{-1}(a x)} \sqrt{c-\frac{c}{a x}}}{x^2} \, dx\\ &=-\int \frac{\sqrt{c-\frac{c}{a x}} (1-a x)}{x^2 (1+a x)} \, dx\\ &=\frac{a \int \frac{\left (c-\frac{c}{a x}\right )^{3/2}}{x (1+a x)} \, dx}{c}\\ &=\frac{a \int \frac{\left (c-\frac{c}{a x}\right )^{3/2}}{\left (a+\frac{1}{x}\right ) x^2} \, dx}{c}\\ &=-\frac{a \operatorname{Subst}\left (\int \frac{\left (c-\frac{c x}{a}\right )^{3/2}}{a+x} \, dx,x,\frac{1}{x}\right )}{c}\\ &=-\frac{2 a \left (c-\frac{c}{a x}\right )^{3/2}}{3 c}-(2 a) \operatorname{Subst}\left (\int \frac{\sqrt{c-\frac{c x}{a}}}{a+x} \, dx,x,\frac{1}{x}\right )\\ &=-4 a \sqrt{c-\frac{c}{a x}}-\frac{2 a \left (c-\frac{c}{a x}\right )^{3/2}}{3 c}-(4 a c) \operatorname{Subst}\left (\int \frac{1}{(a+x) \sqrt{c-\frac{c x}{a}}} \, dx,x,\frac{1}{x}\right )\\ &=-4 a \sqrt{c-\frac{c}{a x}}-\frac{2 a \left (c-\frac{c}{a x}\right )^{3/2}}{3 c}+\left (8 a^2\right ) \operatorname{Subst}\left (\int \frac{1}{2 a-\frac{a x^2}{c}} \, dx,x,\sqrt{c-\frac{c}{a x}}\right )\\ &=-4 a \sqrt{c-\frac{c}{a x}}-\frac{2 a \left (c-\frac{c}{a x}\right )^{3/2}}{3 c}+4 \sqrt{2} a \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c-\frac{c}{a x}}}{\sqrt{2} \sqrt{c}}\right )\\ \end{align*}
Mathematica [A] time = 0.0702351, size = 69, normalized size = 0.84 \[ \frac{2 (1-7 a x) \sqrt{c-\frac{c}{a x}}}{3 x}+4 \sqrt{2} a \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c-\frac{c}{a x}}}{\sqrt{2} \sqrt{c}}\right ) \]
Antiderivative was successfully verified.
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Maple [B] time = 0.176, size = 254, normalized size = 3.1 \begin{align*}{\frac{1}{3\,{x}^{2}}\sqrt{{\frac{c \left ( ax-1 \right ) }{ax}}} \left ( 6\,\sqrt{ \left ( ax-1 \right ) x}{a}^{5/2}\sqrt{{a}^{-1}}{x}^{3}-18\,\sqrt{a{x}^{2}-x}{a}^{5/2}\sqrt{{a}^{-1}}{x}^{3}+12\,{a}^{3/2} \left ( a{x}^{2}-x \right ) ^{3/2}x\sqrt{{a}^{-1}}+9\,\ln \left ( 1/2\,{\frac{2\,\sqrt{a{x}^{2}-x}\sqrt{a}+2\,ax-1}{\sqrt{a}}} \right ) \sqrt{{a}^{-1}}{x}^{3}{a}^{2}-6\,{a}^{3/2}\sqrt{2}\ln \left ({\frac{2\,\sqrt{2}\sqrt{{a}^{-1}}\sqrt{ \left ( ax-1 \right ) x}a-3\,ax+1}{ax+1}} \right ){x}^{3}-9\,\ln \left ( 1/2\,{\frac{2\,\sqrt{ \left ( ax-1 \right ) x}\sqrt{a}+2\,ax-1}{\sqrt{a}}} \right ) \sqrt{{a}^{-1}}{x}^{3}{a}^{2}-2\, \left ( a{x}^{2}-x \right ) ^{3/2}\sqrt{a}\sqrt{{a}^{-1}} \right ){\frac{1}{\sqrt{ \left ( ax-1 \right ) x}}}{\frac{1}{\sqrt{a}}}{\frac{1}{\sqrt{{a}^{-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{{\left (a x - 1\right )} \sqrt{c - \frac{c}{a x}}}{{\left (a x + 1\right )} x^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.59768, size = 374, normalized size = 4.56 \begin{align*} \left [\frac{2 \,{\left (3 \, \sqrt{2} a \sqrt{c} x \log \left (-\frac{2 \, \sqrt{2} a \sqrt{c} x \sqrt{\frac{a c x - c}{a x}} + 3 \, a c x - c}{a x + 1}\right ) -{\left (7 \, a x - 1\right )} \sqrt{\frac{a c x - c}{a x}}\right )}}{3 \, x}, -\frac{2 \,{\left (6 \, \sqrt{2} a \sqrt{-c} x \arctan \left (\frac{\sqrt{2} \sqrt{-c} \sqrt{\frac{a c x - c}{a x}}}{2 \, c}\right ) +{\left (7 \, a x - 1\right )} \sqrt{\frac{a c x - c}{a x}}\right )}}{3 \, x}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{- c \left (-1 + \frac{1}{a x}\right )} \left (a x - 1\right )}{x^{2} \left (a x + 1\right )}\, dx \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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