Optimal. Leaf size=37 \[ \frac{\sqrt{2} \tanh ^{-1}\left (\frac{\sqrt{c-a c x}}{\sqrt{2} \sqrt{c}}\right )}{a c^{3/2}} \]
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Rubi [A] time = 0.0927469, antiderivative size = 37, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {6167, 6130, 21, 63, 206} \[ \frac{\sqrt{2} \tanh ^{-1}\left (\frac{\sqrt{c-a c x}}{\sqrt{2} \sqrt{c}}\right )}{a c^{3/2}} \]
Antiderivative was successfully verified.
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Rule 6167
Rule 6130
Rule 21
Rule 63
Rule 206
Rubi steps
\begin{align*} \int \frac{e^{-2 \coth ^{-1}(a x)}}{(c-a c x)^{3/2}} \, dx &=-\int \frac{e^{-2 \tanh ^{-1}(a x)}}{(c-a c x)^{3/2}} \, dx\\ &=-\int \frac{1-a x}{(1+a x) (c-a c x)^{3/2}} \, dx\\ &=-\frac{\int \frac{1}{(1+a x) \sqrt{c-a c x}} \, dx}{c}\\ &=\frac{2 \operatorname{Subst}\left (\int \frac{1}{2-\frac{x^2}{c}} \, dx,x,\sqrt{c-a c x}\right )}{a c^2}\\ &=\frac{\sqrt{2} \tanh ^{-1}\left (\frac{\sqrt{c-a c x}}{\sqrt{2} \sqrt{c}}\right )}{a c^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.0184084, size = 37, normalized size = 1. \[ \frac{\sqrt{2} \tanh ^{-1}\left (\frac{\sqrt{c-a c x}}{\sqrt{2} \sqrt{c}}\right )}{a c^{3/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.052, size = 29, normalized size = 0.8 \begin{align*}{\frac{\sqrt{2}}{a}{\it Artanh} \left ({\frac{\sqrt{2}}{2}\sqrt{-acx+c}{\frac{1}{\sqrt{c}}}} \right ){c}^{-{\frac{3}{2}}}} \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 = 1.89117, size = 231, normalized size = 6.24 \begin{align*} \left [\frac{\sqrt{2} \log \left (\frac{a x - \frac{2 \, \sqrt{2} \sqrt{-a c x + c}}{\sqrt{c}} - 3}{a x + 1}\right )}{2 \, a c^{\frac{3}{2}}}, \frac{\sqrt{2} \sqrt{-\frac{1}{c}} \arctan \left (\frac{\sqrt{2} \sqrt{-a c x + c} \sqrt{-\frac{1}{c}}}{a x - 1}\right )}{a c}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 19.8034, size = 41, normalized size = 1.11 \begin{align*} - \frac{\sqrt{2} \operatorname{atan}{\left (\frac{\sqrt{2} \sqrt{- a c x + c}}{2 \sqrt{- c}} \right )}}{a c \sqrt{- c}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.15827, size = 49, normalized size = 1.32 \begin{align*} -\frac{\sqrt{2} \arctan \left (\frac{\sqrt{2} \sqrt{-a c x + c}}{2 \, \sqrt{-c}}\right )}{a \sqrt{-c} c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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