Optimal. Leaf size=78 \[ \frac{\sqrt{c-a c x}}{x}-5 a \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c-a c x}}{\sqrt{c}}\right )+4 \sqrt{2} a \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c-a c x}}{\sqrt{2} \sqrt{c}}\right ) \]
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Rubi [A] time = 0.226114, antiderivative size = 78, 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, 98, 156, 63, 208, 206} \[ \frac{\sqrt{c-a c x}}{x}-5 a \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c-a c x}}{\sqrt{c}}\right )+4 \sqrt{2} a \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c-a c x}}{\sqrt{2} \sqrt{c}}\right ) \]
Antiderivative was successfully verified.
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Rule 6167
Rule 6130
Rule 21
Rule 98
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^2} \, dx &=-\int \frac{e^{-2 \tanh ^{-1}(a x)} \sqrt{c-a c x}}{x^2} \, dx\\ &=-\int \frac{(1-a x) \sqrt{c-a c x}}{x^2 (1+a x)} \, dx\\ &=-\frac{\int \frac{(c-a c x)^{3/2}}{x^2 (1+a x)} \, dx}{c}\\ &=\frac{\sqrt{c-a c x}}{x}+\frac{\int \frac{\frac{5 a c^2}{2}-\frac{3}{2} a^2 c^2 x}{x (1+a x) \sqrt{c-a c x}} \, dx}{c}\\ &=\frac{\sqrt{c-a c x}}{x}+\frac{1}{2} (5 a c) \int \frac{1}{x \sqrt{c-a c x}} \, dx-\left (4 a^2 c\right ) \int \frac{1}{(1+a x) \sqrt{c-a c x}} \, dx\\ &=\frac{\sqrt{c-a c x}}{x}-5 \operatorname{Subst}\left (\int \frac{1}{\frac{1}{a}-\frac{x^2}{a c}} \, dx,x,\sqrt{c-a c x}\right )+(8 a) \operatorname{Subst}\left (\int \frac{1}{2-\frac{x^2}{c}} \, dx,x,\sqrt{c-a c x}\right )\\ &=\frac{\sqrt{c-a c x}}{x}-5 a \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c-a c x}}{\sqrt{c}}\right )+4 \sqrt{2} a \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c-a c x}}{\sqrt{2} \sqrt{c}}\right )\\ \end{align*}
Mathematica [A] time = 0.0411239, size = 78, normalized size = 1. \[ \frac{\sqrt{c-a c x}}{x}-5 a \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c-a c x}}{\sqrt{c}}\right )+4 \sqrt{2} a \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c-a c x}}{\sqrt{2} \sqrt{c}}\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.051, size = 71, normalized size = 0.9 \begin{align*} -2\,ac \left ( -1/2\,{\frac{\sqrt{-acx+c}}{acx}}+5/2\,{\frac{1}{\sqrt{c}}{\it Artanh} \left ({\frac{\sqrt{-acx+c}}{\sqrt{c}}} \right ) }-2\,{\frac{\sqrt{2}}{\sqrt{c}}{\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 = 1.32769, size = 455, normalized size = 5.83 \begin{align*} \left [\frac{4 \, \sqrt{2} a \sqrt{c} x \log \left (\frac{a c x - 2 \, \sqrt{2} \sqrt{-a c x + c} \sqrt{c} - 3 \, c}{a x + 1}\right ) + 5 \, a \sqrt{c} x \log \left (\frac{a c x + 2 \, \sqrt{-a c x + c} \sqrt{c} - 2 \, c}{x}\right ) + 2 \, \sqrt{-a c x + c}}{2 \, x}, -\frac{4 \, \sqrt{2} a \sqrt{-c} x \arctan \left (\frac{\sqrt{2} \sqrt{-a c x + c} \sqrt{-c}}{2 \, c}\right ) - 5 \, a \sqrt{-c} x \arctan \left (\frac{\sqrt{-a c x + c} \sqrt{-c}}{c}\right ) - \sqrt{-a c x + c}}{x}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 6.87575, size = 162, normalized size = 2.08 \begin{align*} - \frac{a c^{2} \sqrt{\frac{1}{c^{3}}} \log{\left (- c^{2} \sqrt{\frac{1}{c^{3}}} + \sqrt{- a c x + c} \right )}}{2} + \frac{a c^{2} \sqrt{\frac{1}{c^{3}}} \log{\left (c^{2} \sqrt{\frac{1}{c^{3}}} + \sqrt{- a c x + c} \right )}}{2} + \frac{6 a c \operatorname{atan}{\left (\frac{\sqrt{- a c x + c}}{\sqrt{- c}} \right )}}{\sqrt{- c}} - \frac{4 \sqrt{2} a c \operatorname{atan}{\left (\frac{\sqrt{2} \sqrt{- a c x + c}}{2 \sqrt{- c}} \right )}}{\sqrt{- c}} + \frac{\sqrt{- a c x + c}}{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.15494, size = 96, normalized size = 1.23 \begin{align*} -\frac{4 \, \sqrt{2} a c \arctan \left (\frac{\sqrt{2} \sqrt{-a c x + c}}{2 \, \sqrt{-c}}\right )}{\sqrt{-c}} + \frac{5 \, a c \arctan \left (\frac{\sqrt{-a c x + c}}{\sqrt{-c}}\right )}{\sqrt{-c}} + \frac{\sqrt{-a c x + c}}{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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