Optimal. Leaf size=57 \[ \frac{\tanh ^{-1}\left (\frac{\sqrt{c-a c x}}{\sqrt{2} \sqrt{c}}\right )}{\sqrt{2} a c^{5/2}}-\frac{1}{a c^2 \sqrt{c-a c x}} \]
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Rubi [A] time = 0.103479, antiderivative size = 57, 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, 51, 63, 206} \[ \frac{\tanh ^{-1}\left (\frac{\sqrt{c-a c x}}{\sqrt{2} \sqrt{c}}\right )}{\sqrt{2} a c^{5/2}}-\frac{1}{a c^2 \sqrt{c-a c x}} \]
Antiderivative was successfully verified.
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Rule 6167
Rule 6130
Rule 21
Rule 51
Rule 63
Rule 206
Rubi steps
\begin{align*} \int \frac{e^{-2 \coth ^{-1}(a x)}}{(c-a c x)^{5/2}} \, dx &=-\int \frac{e^{-2 \tanh ^{-1}(a x)}}{(c-a c x)^{5/2}} \, dx\\ &=-\int \frac{1-a x}{(1+a x) (c-a c x)^{5/2}} \, dx\\ &=-\frac{\int \frac{1}{(1+a x) (c-a c x)^{3/2}} \, dx}{c}\\ &=-\frac{1}{a c^2 \sqrt{c-a c x}}-\frac{\int \frac{1}{(1+a x) \sqrt{c-a c x}} \, dx}{2 c^2}\\ &=-\frac{1}{a c^2 \sqrt{c-a c x}}+\frac{\operatorname{Subst}\left (\int \frac{1}{2-\frac{x^2}{c}} \, dx,x,\sqrt{c-a c x}\right )}{a c^3}\\ &=-\frac{1}{a c^2 \sqrt{c-a c x}}+\frac{\tanh ^{-1}\left (\frac{\sqrt{c-a c x}}{\sqrt{2} \sqrt{c}}\right )}{\sqrt{2} a c^{5/2}}\\ \end{align*}
Mathematica [C] time = 0.022137, size = 37, normalized size = 0.65 \[ -\frac{\text{Hypergeometric2F1}\left (-\frac{1}{2},1,\frac{1}{2},\frac{1}{2} (1-a x)\right )}{a c^2 \sqrt{c-a c x}} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.049, size = 50, normalized size = 0.9 \begin{align*} -2\,{\frac{1}{ac} \left ( 1/2\,{\frac{1}{c\sqrt{-acx+c}}}-1/4\,{\frac{\sqrt{2}}{{c}^{3/2}}{\it Artanh} \left ( 1/2\,{\frac{\sqrt{-acx+c}\sqrt{2}}{\sqrt{c}}} \right ) } \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 = 1.82303, size = 362, normalized size = 6.35 \begin{align*} \left [\frac{\sqrt{2}{\left (a x - 1\right )} \sqrt{c} \log \left (\frac{a c x - 2 \, \sqrt{2} \sqrt{-a c x + c} \sqrt{c} - 3 \, c}{a x + 1}\right ) + 4 \, \sqrt{-a c x + c}}{4 \,{\left (a^{2} c^{3} x - a c^{3}\right )}}, -\frac{\sqrt{2}{\left (a x - 1\right )} \sqrt{-c} \arctan \left (\frac{\sqrt{2} \sqrt{-a c x + c} \sqrt{-c}}{2 \, c}\right ) - 2 \, \sqrt{-a c x + c}}{2 \,{\left (a^{2} c^{3} x - a c^{3}\right )}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 15.727, size = 61, normalized size = 1.07 \begin{align*} - \frac{1}{a c^{2} \sqrt{- a c x + c}} - \frac{\sqrt{2} \operatorname{atan}{\left (\frac{\sqrt{2} \sqrt{- a c x + c}}{2 \sqrt{- c}} \right )}}{2 a c^{2} \sqrt{- c}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.16371, size = 73, normalized size = 1.28 \begin{align*} -\frac{\sqrt{2} \arctan \left (\frac{\sqrt{2} \sqrt{-a c x + c}}{2 \, \sqrt{-c}}\right )}{2 \, a \sqrt{-c} c^{2}} - \frac{1}{\sqrt{-a c x + c} a c^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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