Optimal. Leaf size=58 \[ \frac{2 \sqrt{2} \tanh ^{-1}\left (\frac{\sqrt{c-a c x}}{\sqrt{2} \sqrt{c}}\right )}{a \sqrt{c}}-\frac{2 \sqrt{c-a c x}}{a c} \]
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Rubi [A] time = 0.0985863, antiderivative size = 58, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3, Rules used = {6167, 6130, 21, 50, 63, 206} \[ \frac{2 \sqrt{2} \tanh ^{-1}\left (\frac{\sqrt{c-a c x}}{\sqrt{2} \sqrt{c}}\right )}{a \sqrt{c}}-\frac{2 \sqrt{c-a c x}}{a c} \]
Antiderivative was successfully verified.
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Rule 6167
Rule 6130
Rule 21
Rule 50
Rule 63
Rule 206
Rubi steps
\begin{align*} \int \frac{e^{-2 \coth ^{-1}(a x)}}{\sqrt{c-a c x}} \, dx &=-\int \frac{e^{-2 \tanh ^{-1}(a x)}}{\sqrt{c-a c x}} \, dx\\ &=-\int \frac{1-a x}{(1+a x) \sqrt{c-a c x}} \, dx\\ &=-\frac{\int \frac{\sqrt{c-a c x}}{1+a x} \, dx}{c}\\ &=-\frac{2 \sqrt{c-a c x}}{a c}-2 \int \frac{1}{(1+a x) \sqrt{c-a c x}} \, dx\\ &=-\frac{2 \sqrt{c-a c x}}{a c}+\frac{4 \operatorname{Subst}\left (\int \frac{1}{2-\frac{x^2}{c}} \, dx,x,\sqrt{c-a c x}\right )}{a c}\\ &=-\frac{2 \sqrt{c-a c x}}{a c}+\frac{2 \sqrt{2} \tanh ^{-1}\left (\frac{\sqrt{c-a c x}}{\sqrt{2} \sqrt{c}}\right )}{a \sqrt{c}}\\ \end{align*}
Mathematica [A] time = 0.0263095, size = 58, normalized size = 1. \[ \frac{2 \sqrt{2} \tanh ^{-1}\left (\frac{\sqrt{c-a c x}}{\sqrt{2} \sqrt{c}}\right )}{a \sqrt{c}}-\frac{2 \sqrt{c-a c x}}{a c} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.043, size = 45, normalized size = 0.8 \begin{align*} -2\,{\frac{1}{ac} \left ( \sqrt{-acx+c}-{\it Artanh} \left ( 1/2\,{\frac{\sqrt{-acx+c}\sqrt{2}}{\sqrt{c}}} \right ) \sqrt{2}\sqrt{c} \right ) } \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 = 1.95542, size = 293, normalized size = 5.05 \begin{align*} \left [\frac{\sqrt{2} \sqrt{c} \log \left (\frac{a x - \frac{2 \, \sqrt{2} \sqrt{-a c x + c}}{\sqrt{c}} - 3}{a x + 1}\right ) - 2 \, \sqrt{-a c x + c}}{a c}, \frac{2 \,{\left (\sqrt{2} c \sqrt{-\frac{1}{c}} \arctan \left (\frac{\sqrt{2} \sqrt{-a c x + c} \sqrt{-\frac{1}{c}}}{a x - 1}\right ) - \sqrt{-a c x + c}\right )}}{a c}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 20.1008, size = 60, normalized size = 1.03 \begin{align*} - \frac{2 \sqrt{- a c x + c}}{a c} - \frac{2 \sqrt{2} \operatorname{atan}{\left (\frac{\sqrt{2}}{\sqrt{- \frac{1}{c}} \sqrt{- a c x + c}} \right )}}{a c \sqrt{- \frac{1}{c}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.1126, size = 69, normalized size = 1.19 \begin{align*} -\frac{2 \, \sqrt{2} \arctan \left (\frac{\sqrt{2} \sqrt{-a c x + c}}{2 \, \sqrt{-c}}\right )}{a \sqrt{-c}} - \frac{2 \, \sqrt{-a c x + c}}{a c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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