Optimal. Leaf size=94 \[ \frac{x \sqrt{c-\frac{c}{a x}}}{c^2}-\frac{\tanh ^{-1}\left (\frac{\sqrt{c-\frac{c}{a x}}}{\sqrt{c}}\right )}{a c^{3/2}}+\frac{\sqrt{2} \tanh ^{-1}\left (\frac{\sqrt{c-\frac{c}{a x}}}{\sqrt{2} \sqrt{c}}\right )}{a c^{3/2}} \]
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Rubi [A] time = 0.21546, antiderivative size = 94, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 10, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.417, Rules used = {6167, 6133, 25, 514, 375, 103, 21, 83, 63, 208} \[ \frac{x \sqrt{c-\frac{c}{a x}}}{c^2}-\frac{\tanh ^{-1}\left (\frac{\sqrt{c-\frac{c}{a x}}}{\sqrt{c}}\right )}{a c^{3/2}}+\frac{\sqrt{2} \tanh ^{-1}\left (\frac{\sqrt{c-\frac{c}{a x}}}{\sqrt{2} \sqrt{c}}\right )}{a c^{3/2}} \]
Antiderivative was successfully verified.
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Rule 6167
Rule 6133
Rule 25
Rule 514
Rule 375
Rule 103
Rule 21
Rule 83
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{e^{-2 \coth ^{-1}(a x)}}{\left (c-\frac{c}{a x}\right )^{3/2}} \, dx &=-\int \frac{e^{-2 \tanh ^{-1}(a x)}}{\left (c-\frac{c}{a x}\right )^{3/2}} \, dx\\ &=-\int \frac{1-a x}{\left (c-\frac{c}{a x}\right )^{3/2} (1+a x)} \, dx\\ &=\frac{a \int \frac{x}{\sqrt{c-\frac{c}{a x}} (1+a x)} \, dx}{c}\\ &=\frac{a \int \frac{1}{\left (a+\frac{1}{x}\right ) \sqrt{c-\frac{c}{a x}}} \, dx}{c}\\ &=-\frac{a \operatorname{Subst}\left (\int \frac{1}{x^2 (a+x) \sqrt{c-\frac{c x}{a}}} \, dx,x,\frac{1}{x}\right )}{c}\\ &=\frac{\sqrt{c-\frac{c}{a x}} x}{c^2}+\frac{\operatorname{Subst}\left (\int \frac{\frac{c}{2}-\frac{c x}{2 a}}{x (a+x) \sqrt{c-\frac{c x}{a}}} \, dx,x,\frac{1}{x}\right )}{c^2}\\ &=\frac{\sqrt{c-\frac{c}{a x}} x}{c^2}+\frac{\operatorname{Subst}\left (\int \frac{\sqrt{c-\frac{c x}{a}}}{x (a+x)} \, dx,x,\frac{1}{x}\right )}{2 c^2}\\ &=\frac{\sqrt{c-\frac{c}{a x}} x}{c^2}+\frac{\operatorname{Subst}\left (\int \frac{1}{x \sqrt{c-\frac{c x}{a}}} \, dx,x,\frac{1}{x}\right )}{2 a c}-\frac{\operatorname{Subst}\left (\int \frac{1}{(a+x) \sqrt{c-\frac{c x}{a}}} \, dx,x,\frac{1}{x}\right )}{a c}\\ &=\frac{\sqrt{c-\frac{c}{a x}} x}{c^2}-\frac{\operatorname{Subst}\left (\int \frac{1}{a-\frac{a x^2}{c}} \, dx,x,\sqrt{c-\frac{c}{a x}}\right )}{c^2}+\frac{2 \operatorname{Subst}\left (\int \frac{1}{2 a-\frac{a x^2}{c}} \, dx,x,\sqrt{c-\frac{c}{a x}}\right )}{c^2}\\ &=\frac{\sqrt{c-\frac{c}{a x}} x}{c^2}-\frac{\tanh ^{-1}\left (\frac{\sqrt{c-\frac{c}{a x}}}{\sqrt{c}}\right )}{a c^{3/2}}+\frac{\sqrt{2} \tanh ^{-1}\left (\frac{\sqrt{c-\frac{c}{a x}}}{\sqrt{2} \sqrt{c}}\right )}{a c^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.0517848, size = 94, normalized size = 1. \[ \frac{x \sqrt{c-\frac{c}{a x}}}{c^2}-\frac{\tanh ^{-1}\left (\frac{\sqrt{c-\frac{c}{a x}}}{\sqrt{c}}\right )}{a c^{3/2}}+\frac{\sqrt{2} \tanh ^{-1}\left (\frac{\sqrt{c-\frac{c}{a x}}}{\sqrt{2} \sqrt{c}}\right )}{a c^{3/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.17, size = 134, normalized size = 1.4 \begin{align*} -{\frac{x}{2\,{c}^{2}}\sqrt{{\frac{c \left ( ax-1 \right ) }{ax}}} \left ( -2\,\sqrt{ \left ( ax-1 \right ) x}{a}^{3/2}\sqrt{{a}^{-1}}+\ln \left ({\frac{1}{2} \left ( 2\,\sqrt{ \left ( ax-1 \right ) x}\sqrt{a}+2\,ax-1 \right ){\frac{1}{\sqrt{a}}}} \right ) a\sqrt{{a}^{-1}}+\sqrt{2}\ln \left ({\frac{1}{ax+1} \left ( 2\,\sqrt{2}\sqrt{{a}^{-1}}\sqrt{ \left ( ax-1 \right ) x}a-3\,ax+1 \right ) } \right ) \sqrt{a} \right ){\frac{1}{\sqrt{ \left ( ax-1 \right ) x}}}{a}^{-{\frac{3}{2}}}{\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{a x - 1}{{\left (a x + 1\right )}{\left (c - \frac{c}{a x}\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 = 1.98137, size = 531, normalized size = 5.65 \begin{align*} \left [\frac{2 \, a x \sqrt{\frac{a c x - c}{a x}} + \sqrt{2} \sqrt{c} \log \left (-\frac{\frac{2 \, \sqrt{2} a x \sqrt{\frac{a c x - c}{a x}}}{\sqrt{c}} + 3 \, a x - 1}{a x + 1}\right ) + \sqrt{c} \log \left (-2 \, a c x + 2 \, a \sqrt{c} x \sqrt{\frac{a c x - c}{a x}} + c\right )}{2 \, a c^{2}}, -\frac{\sqrt{2} c \sqrt{-\frac{1}{c}} \arctan \left (\frac{\sqrt{2} a x \sqrt{-\frac{1}{c}} \sqrt{\frac{a c x - c}{a x}}}{a x - 1}\right ) - a x \sqrt{\frac{a c x - c}{a x}} - \sqrt{-c} \arctan \left (\frac{\sqrt{-c} \sqrt{\frac{a c x - c}{a x}}}{c}\right )}{a c^{2}}\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{a x - 1}{\left (- c \left (-1 + \frac{1}{a x}\right )\right )^{\frac{3}{2}} \left (a x + 1\right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.25617, size = 176, normalized size = 1.87 \begin{align*} -a c{\left (\frac{\sqrt{2} \arctan \left (\frac{\sqrt{2} \sqrt{\frac{a c x - c}{a x}}}{2 \, \sqrt{-c}}\right )}{a^{2} \sqrt{-c} c^{2}} - \frac{\arctan \left (\frac{\sqrt{\frac{a c x - c}{a x}}}{\sqrt{-c}}\right )}{a^{2} \sqrt{-c} c^{2}} - \frac{\sqrt{\frac{a c x - c}{a x}}}{a^{2}{\left (c - \frac{a c x - c}{a x}\right )} c^{2}}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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