Optimal. Leaf size=125 \[ \frac{2 a c^2 \left (1-\frac{1}{a^2 x^2}\right )^{3/2}}{3 \left (c-\frac{c}{a x}\right )^{3/2}}+\frac{4 a c \sqrt{1-\frac{1}{a^2 x^2}}}{\sqrt{c-\frac{c}{a x}}}-4 \sqrt{2} a \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{1-\frac{1}{a^2 x^2}}}{\sqrt{2} \sqrt{c-\frac{c}{a x}}}\right ) \]
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Rubi [A] time = 0.239222, antiderivative size = 125, 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, 665, 661, 208} \[ \frac{2 a c^2 \left (1-\frac{1}{a^2 x^2}\right )^{3/2}}{3 \left (c-\frac{c}{a x}\right )^{3/2}}+\frac{4 a c \sqrt{1-\frac{1}{a^2 x^2}}}{\sqrt{c-\frac{c}{a x}}}-4 \sqrt{2} a \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{1-\frac{1}{a^2 x^2}}}{\sqrt{2} \sqrt{c-\frac{c}{a x}}}\right ) \]
Antiderivative was successfully verified.
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Rule 6178
Rule 665
Rule 661
Rule 208
Rubi steps
\begin{align*} \int \frac{e^{3 \coth ^{-1}(a x)} \sqrt{c-\frac{c}{a x}}}{x^2} \, dx &=-\left (c^3 \operatorname{Subst}\left (\int \frac{\left (1-\frac{x^2}{a^2}\right )^{3/2}}{\left (c-\frac{c x}{a}\right )^{5/2}} \, 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}}-\left (2 c^2\right ) \operatorname{Subst}\left (\int \frac{\sqrt{1-\frac{x^2}{a^2}}}{\left (c-\frac{c x}{a}\right )^{3/2}} \, dx,x,\frac{1}{x}\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}}+\frac{4 a c \sqrt{1-\frac{1}{a^2 x^2}}}{\sqrt{c-\frac{c}{a x}}}-(4 c) \operatorname{Subst}\left (\int \frac{1}{\sqrt{c-\frac{c x}{a}} \sqrt{1-\frac{x^2}{a^2}}} \, dx,x,\frac{1}{x}\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}}+\frac{4 a c \sqrt{1-\frac{1}{a^2 x^2}}}{\sqrt{c-\frac{c}{a x}}}+\frac{\left (8 c^2\right ) \operatorname{Subst}\left (\int \frac{1}{-\frac{2 c}{a^2}+\frac{c^2 x^2}{a^2}} \, dx,x,\frac{\sqrt{1-\frac{1}{a^2 x^2}}}{\sqrt{c-\frac{c}{a x}}}\right )}{a}\\ &=\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}}+\frac{4 a c \sqrt{1-\frac{1}{a^2 x^2}}}{\sqrt{c-\frac{c}{a x}}}-4 \sqrt{2} a \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{1-\frac{1}{a^2 x^2}}}{\sqrt{2} \sqrt{c-\frac{c}{a x}}}\right )\\ \end{align*}
Mathematica [A] time = 0.247701, size = 155, normalized size = 1.24 \[ \frac{2 a \left (\sqrt{1-\frac{1}{a^2 x^2}} (7 a x+1) \sqrt{c-\frac{c}{a x}}-3 \sqrt{2} \sqrt{c} (a x-1) \log \left (2 \sqrt{2} a^2 \sqrt{c} x^2 \sqrt{1-\frac{1}{a^2 x^2}} \sqrt{c-\frac{c}{a x}}+c \left (3 a^2 x^2-2 a x-1\right )\right )+3 \sqrt{2} \sqrt{c} (a x-1) \log \left ((a x-1)^2\right )\right )}{3 a x-3} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.181, size = 141, normalized size = 1.1 \begin{align*} -{\frac{2\,ax-2}{ \left ( 3\,ax+3 \right ) x}\sqrt{{\frac{c \left ( ax-1 \right ) }{ax}}} \left ( 3\,a\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}^{2}-7\,a\sqrt{{a}^{-1}}x\sqrt{ \left ( ax+1 \right ) x}-\sqrt{{a}^{-1}}\sqrt{ \left ( ax+1 \right ) x} \right ) \left ({\frac{ax-1}{ax+1}} \right ) ^{-{\frac{3}{2}}}{\frac{1}{\sqrt{ \left ( ax+1 \right ) x}}}{\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{\sqrt{c - \frac{c}{a x}}}{x^{2} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.95061, size = 770, normalized size = 6.16 \begin{align*} \left [\frac{3 \, \sqrt{2}{\left (a^{2} x^{2} - a x\right )} \sqrt{c} \log \left (-\frac{17 \, a^{3} c x^{3} - 3 \, a^{2} c x^{2} - 13 \, a c x - 4 \, \sqrt{2}{\left (3 \, a^{3} x^{3} + 4 \, a^{2} x^{2} + a x\right )} \sqrt{c} \sqrt{\frac{a x - 1}{a x + 1}} \sqrt{\frac{a c x - c}{a x}} - c}{a^{3} x^{3} - 3 \, a^{2} x^{2} + 3 \, a x - 1}\right ) + 2 \,{\left (7 \, a^{2} x^{2} + 8 \, 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 )}}, \frac{2 \,{\left (3 \, \sqrt{2}{\left (a^{2} x^{2} - a x\right )} \sqrt{-c} \arctan \left (\frac{2 \, \sqrt{2}{\left (a^{2} x^{2} + a x\right )} \sqrt{-c} \sqrt{\frac{a x - 1}{a x + 1}} \sqrt{\frac{a c x - c}{a x}}}{3 \, a^{2} c x^{2} - 2 \, a c x - c}\right ) +{\left (7 \, a^{2} x^{2} + 8 \, a x + 1\right )} \sqrt{\frac{a x - 1}{a x + 1}} \sqrt{\frac{a c x - c}{a x}}\right )}}{3 \,{\left (a x^{2} - x\right )}}\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} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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