Optimal. Leaf size=68 \[ -\frac{2 c \sqrt{1-a^2 x^2}}{\sqrt{c-a c x}}-2 \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{1-a^2 x^2}}{\sqrt{c-a c x}}\right ) \]
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Rubi [A] time = 0.158214, antiderivative size = 68, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.174, Rules used = {6128, 881, 875, 208} \[ -\frac{2 c \sqrt{1-a^2 x^2}}{\sqrt{c-a c x}}-2 \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{1-a^2 x^2}}{\sqrt{c-a c x}}\right ) \]
Antiderivative was successfully verified.
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Rule 6128
Rule 881
Rule 875
Rule 208
Rubi steps
\begin{align*} \int \frac{e^{-\tanh ^{-1}(a x)} \sqrt{c-a c x}}{x} \, dx &=\frac{\int \frac{(c-a c x)^{3/2}}{x \sqrt{1-a^2 x^2}} \, dx}{c}\\ &=-\frac{2 c \sqrt{1-a^2 x^2}}{\sqrt{c-a c x}}+\int \frac{\sqrt{c-a c x}}{x \sqrt{1-a^2 x^2}} \, dx\\ &=-\frac{2 c \sqrt{1-a^2 x^2}}{\sqrt{c-a c x}}+\left (2 a^2 c^2\right ) \operatorname{Subst}\left (\int \frac{1}{-a^2 c+a^2 c^2 x^2} \, dx,x,\frac{\sqrt{1-a^2 x^2}}{\sqrt{c-a c x}}\right )\\ &=-\frac{2 c \sqrt{1-a^2 x^2}}{\sqrt{c-a c x}}-2 \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{1-a^2 x^2}}{\sqrt{c-a c x}}\right )\\ \end{align*}
Mathematica [A] time = 0.0198962, size = 44, normalized size = 0.65 \[ -\frac{2 c \sqrt{1-a x} \left (\sqrt{a x+1}+\tanh ^{-1}\left (\sqrt{a x+1}\right )\right )}{\sqrt{c-a c x}} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.097, size = 69, normalized size = 1. \begin{align*} 2\,{\frac{\sqrt{-c \left ( ax-1 \right ) }\sqrt{-{a}^{2}{x}^{2}+1}}{ \left ( ax-1 \right ) \sqrt{c \left ( ax+1 \right ) }} \left ( \sqrt{c}{\it Artanh} \left ({\frac{\sqrt{c \left ( ax+1 \right ) }}{\sqrt{c}}} \right ) +\sqrt{c \left ( ax+1 \right ) } \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{-a^{2} x^{2} + 1} \sqrt{-a c x + c}}{{\left (a x + 1\right )} x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.90054, size = 417, normalized size = 6.13 \begin{align*} \left [\frac{{\left (a x - 1\right )} \sqrt{c} \log \left (-\frac{a^{2} c x^{2} + a c x + 2 \, \sqrt{-a^{2} x^{2} + 1} \sqrt{-a c x + c} \sqrt{c} - 2 \, c}{a x^{2} - x}\right ) + 2 \, \sqrt{-a^{2} x^{2} + 1} \sqrt{-a c x + c}}{a x - 1}, -\frac{2 \,{\left ({\left (a x - 1\right )} \sqrt{-c} \arctan \left (\frac{\sqrt{-a^{2} x^{2} + 1} \sqrt{-a c x + c} \sqrt{-c}}{a^{2} c x^{2} - c}\right ) - \sqrt{-a^{2} x^{2} + 1} \sqrt{-a c x + c}\right )}}{a x - 1}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{- c \left (a x - 1\right )} \sqrt{- \left (a x - 1\right ) \left (a x + 1\right )}}{x \left (a x + 1\right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.21521, size = 109, normalized size = 1.6 \begin{align*} \frac{2 \,{\left ({\left (\frac{c \arctan \left (\frac{\sqrt{a c x + c}}{\sqrt{-c}}\right )}{\sqrt{-c}} - \sqrt{a c x + c}\right )} c - \frac{c^{2} \arctan \left (\frac{\sqrt{2} \sqrt{c}}{\sqrt{-c}}\right ) - \sqrt{2} \sqrt{-c} c^{\frac{3}{2}}}{\sqrt{-c}}\right )}{\left | c \right |}}{c^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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