Optimal. Leaf size=92 \[ x \sqrt{c-\frac{c}{a x}}-\frac{5 \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c-\frac{c}{a x}}}{\sqrt{c}}\right )}{a}+\frac{4 \sqrt{2} \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c-\frac{c}{a x}}}{\sqrt{2} \sqrt{c}}\right )}{a} \]
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Rubi [A] time = 0.198132, antiderivative size = 92, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 9, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.375, Rules used = {6167, 6133, 25, 514, 375, 98, 156, 63, 208} \[ x \sqrt{c-\frac{c}{a x}}-\frac{5 \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c-\frac{c}{a x}}}{\sqrt{c}}\right )}{a}+\frac{4 \sqrt{2} \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c-\frac{c}{a x}}}{\sqrt{2} \sqrt{c}}\right )}{a} \]
Antiderivative was successfully verified.
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Rule 6167
Rule 6133
Rule 25
Rule 514
Rule 375
Rule 98
Rule 156
Rule 63
Rule 208
Rubi steps
\begin{align*} \int e^{-2 \coth ^{-1}(a x)} \sqrt{c-\frac{c}{a x}} \, dx &=-\int e^{-2 \tanh ^{-1}(a x)} \sqrt{c-\frac{c}{a x}} \, dx\\ &=-\int \frac{\sqrt{c-\frac{c}{a x}} (1-a x)}{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}}{a+\frac{1}{x}} \, dx}{c}\\ &=-\frac{a \operatorname{Subst}\left (\int \frac{\left (c-\frac{c x}{a}\right )^{3/2}}{x^2 (a+x)} \, dx,x,\frac{1}{x}\right )}{c}\\ &=\sqrt{c-\frac{c}{a x}} x+\frac{\operatorname{Subst}\left (\int \frac{\frac{5 c^2}{2}-\frac{3 c^2 x}{2 a}}{x (a+x) \sqrt{c-\frac{c x}{a}}} \, dx,x,\frac{1}{x}\right )}{c}\\ &=\sqrt{c-\frac{c}{a x}} x+\frac{(5 c) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{c-\frac{c x}{a}}} \, dx,x,\frac{1}{x}\right )}{2 a}-\frac{(4 c) \operatorname{Subst}\left (\int \frac{1}{(a+x) \sqrt{c-\frac{c x}{a}}} \, dx,x,\frac{1}{x}\right )}{a}\\ &=\sqrt{c-\frac{c}{a x}} x-5 \operatorname{Subst}\left (\int \frac{1}{a-\frac{a x^2}{c}} \, dx,x,\sqrt{c-\frac{c}{a x}}\right )+8 \operatorname{Subst}\left (\int \frac{1}{2 a-\frac{a x^2}{c}} \, dx,x,\sqrt{c-\frac{c}{a x}}\right )\\ &=\sqrt{c-\frac{c}{a x}} x-\frac{5 \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c-\frac{c}{a x}}}{\sqrt{c}}\right )}{a}+\frac{4 \sqrt{2} \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c-\frac{c}{a x}}}{\sqrt{2} \sqrt{c}}\right )}{a}\\ \end{align*}
Mathematica [A] time = 0.0480419, size = 92, normalized size = 1. \[ x \sqrt{c-\frac{c}{a x}}-\frac{5 \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c-\frac{c}{a x}}}{\sqrt{c}}\right )}{a}+\frac{4 \sqrt{2} \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c-\frac{c}{a x}}}{\sqrt{2} \sqrt{c}}\right )}{a} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.165, size = 190, normalized size = 2.1 \begin{align*} -{\frac{x}{2}\sqrt{{\frac{c \left ( ax-1 \right ) }{ax}}} \left ( 2\,\sqrt{a{x}^{2}-x}{a}^{3/2}\sqrt{{a}^{-1}}-4\,\sqrt{ \left ( ax-1 \right ) x}{a}^{3/2}\sqrt{{a}^{-1}}-\ln \left ({\frac{1}{2} \left ( 2\,\sqrt{a{x}^{2}-x}\sqrt{a}+2\,ax-1 \right ){\frac{1}{\sqrt{a}}}} \right ) a\sqrt{{a}^{-1}}+4\,\sqrt{2}\ln \left ({\frac{2\,\sqrt{2}\sqrt{{a}^{-1}}\sqrt{ \left ( ax-1 \right ) x}a-3\,ax+1}{ax+1}} \right ) \sqrt{a}+6\,\ln \left ( 1/2\,{\frac{2\,\sqrt{ \left ( ax-1 \right ) x}\sqrt{a}+2\,ax-1}{\sqrt{a}}} \right ) a\sqrt{{a}^{-1}} \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{{\left (a x - 1\right )} \sqrt{c - \frac{c}{a x}}}{a x + 1}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.62031, size = 508, normalized size = 5.52 \begin{align*} \left [\frac{2 \, a x \sqrt{\frac{a c x - c}{a x}} + 4 \, \sqrt{2} \sqrt{c} \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 ) + 5 \, \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}, \frac{a x \sqrt{\frac{a c x - c}{a x}} - 4 \, \sqrt{2} \sqrt{-c} \arctan \left (\frac{\sqrt{2} \sqrt{-c} \sqrt{\frac{a c x - c}{a x}}}{2 \, c}\right ) + 5 \, \sqrt{-c} \arctan \left (\frac{\sqrt{-c} \sqrt{\frac{a c x - c}{a x}}}{c}\right )}{a}\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 )}{a x + 1}\, 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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