Optimal. Leaf size=74 \[ 2 \sqrt{c-a c x}+2 \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c-a c x}}{\sqrt{c}}\right )-4 \sqrt{2} \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c-a c x}}{\sqrt{2} \sqrt{c}}\right ) \]
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Rubi [A] time = 0.232492, antiderivative size = 74, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 8, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.348, Rules used = {6167, 6130, 21, 84, 156, 63, 208, 206} \[ 2 \sqrt{c-a c x}+2 \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c-a c x}}{\sqrt{c}}\right )-4 \sqrt{2} \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c-a c x}}{\sqrt{2} \sqrt{c}}\right ) \]
Antiderivative was successfully verified.
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Rule 6167
Rule 6130
Rule 21
Rule 84
Rule 156
Rule 63
Rule 208
Rule 206
Rubi steps
\begin{align*} \int \frac{e^{-2 \coth ^{-1}(a x)} \sqrt{c-a c x}}{x} \, dx &=-\int \frac{e^{-2 \tanh ^{-1}(a x)} \sqrt{c-a c x}}{x} \, dx\\ &=-\int \frac{(1-a x) \sqrt{c-a c x}}{x (1+a x)} \, dx\\ &=-\frac{\int \frac{(c-a c x)^{3/2}}{x (1+a x)} \, dx}{c}\\ &=2 \sqrt{c-a c x}-\frac{\int \frac{a c^2-3 a^2 c^2 x}{x (1+a x) \sqrt{c-a c x}} \, dx}{a c}\\ &=2 \sqrt{c-a c x}-c \int \frac{1}{x \sqrt{c-a c x}} \, dx+(4 a c) \int \frac{1}{(1+a x) \sqrt{c-a c x}} \, dx\\ &=2 \sqrt{c-a c x}-8 \operatorname{Subst}\left (\int \frac{1}{2-\frac{x^2}{c}} \, dx,x,\sqrt{c-a c x}\right )+\frac{2 \operatorname{Subst}\left (\int \frac{1}{\frac{1}{a}-\frac{x^2}{a c}} \, dx,x,\sqrt{c-a c x}\right )}{a}\\ &=2 \sqrt{c-a c x}+2 \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c-a c x}}{\sqrt{c}}\right )-4 \sqrt{2} \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c-a c x}}{\sqrt{2} \sqrt{c}}\right )\\ \end{align*}
Mathematica [A] time = 0.0325196, size = 74, normalized size = 1. \[ 2 \sqrt{c-a c x}+2 \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c-a c x}}{\sqrt{c}}\right )-4 \sqrt{2} \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c-a c x}}{\sqrt{2} \sqrt{c}}\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.051, size = 58, normalized size = 0.8 \begin{align*} 2\,{\it Artanh} \left ({\frac{\sqrt{-acx+c}}{\sqrt{c}}} \right ) \sqrt{c}-4\,{\it Artanh} \left ( 1/2\,{\frac{\sqrt{-acx+c}\sqrt{2}}{\sqrt{c}}} \right ) \sqrt{2}\sqrt{c}+2\,\sqrt{-acx+c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.34585, size = 416, normalized size = 5.62 \begin{align*} \left [2 \, \sqrt{2} \sqrt{c} \log \left (\frac{a c x + 2 \, \sqrt{2} \sqrt{-a c x + c} \sqrt{c} - 3 \, c}{a x + 1}\right ) + \sqrt{c} \log \left (\frac{a c x - 2 \, \sqrt{-a c x + c} \sqrt{c} - 2 \, c}{x}\right ) + 2 \, \sqrt{-a c x + c}, 4 \, \sqrt{2} \sqrt{-c} \arctan \left (\frac{\sqrt{2} \sqrt{-a c x + c} \sqrt{-c}}{2 \, c}\right ) - 2 \, \sqrt{-c} \arctan \left (\frac{\sqrt{-a c x + c} \sqrt{-c}}{c}\right ) + 2 \, \sqrt{-a c x + c}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 5.38679, size = 80, normalized size = 1.08 \begin{align*} - \frac{2 c \operatorname{atan}{\left (\frac{\sqrt{- a c x + c}}{\sqrt{- c}} \right )}}{\sqrt{- c}} + \frac{4 \sqrt{2} c \operatorname{atan}{\left (\frac{\sqrt{2} \sqrt{- a c x + c}}{2 \sqrt{- c}} \right )}}{\sqrt{- c}} + 2 \sqrt{- a c x + c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.15313, size = 90, normalized size = 1.22 \begin{align*} \frac{4 \, \sqrt{2} c \arctan \left (\frac{\sqrt{2} \sqrt{-a c x + c}}{2 \, \sqrt{-c}}\right )}{\sqrt{-c}} - \frac{2 \, c \arctan \left (\frac{\sqrt{-a c x + c}}{\sqrt{-c}}\right )}{\sqrt{-c}} + 2 \, \sqrt{-a c x + c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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