Optimal. Leaf size=97 \[ -\frac{\sqrt{\frac{1}{a x}+1} \sqrt{c-a c x}}{x \sqrt{1-\frac{1}{a x}}}-\frac{\sqrt{a} \sqrt{\frac{1}{x}} \sqrt{c-a c x} \sinh ^{-1}\left (\frac{\sqrt{\frac{1}{x}}}{\sqrt{a}}\right )}{\sqrt{1-\frac{1}{a x}}} \]
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Rubi [A] time = 0.194801, antiderivative size = 97, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.238, Rules used = {6176, 6181, 50, 54, 215} \[ -\frac{\sqrt{\frac{1}{a x}+1} \sqrt{c-a c x}}{x \sqrt{1-\frac{1}{a x}}}-\frac{\sqrt{a} \sqrt{\frac{1}{x}} \sqrt{c-a c x} \sinh ^{-1}\left (\frac{\sqrt{\frac{1}{x}}}{\sqrt{a}}\right )}{\sqrt{1-\frac{1}{a x}}} \]
Antiderivative was successfully verified.
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Rule 6176
Rule 6181
Rule 50
Rule 54
Rule 215
Rubi steps
\begin{align*} \int \frac{e^{\coth ^{-1}(a x)} \sqrt{c-a c x}}{x^2} \, dx &=\frac{\sqrt{c-a c x} \int \frac{e^{\coth ^{-1}(a x)} \sqrt{1-\frac{1}{a x}}}{x^{3/2}} \, dx}{\sqrt{1-\frac{1}{a x}} \sqrt{x}}\\ &=-\frac{\left (\sqrt{\frac{1}{x}} \sqrt{c-a c x}\right ) \operatorname{Subst}\left (\int \frac{\sqrt{1+\frac{x}{a}}}{\sqrt{x}} \, dx,x,\frac{1}{x}\right )}{\sqrt{1-\frac{1}{a x}}}\\ &=-\frac{\sqrt{1+\frac{1}{a x}} \sqrt{c-a c x}}{\sqrt{1-\frac{1}{a x}} x}-\frac{\left (\sqrt{\frac{1}{x}} \sqrt{c-a c x}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{x} \sqrt{1+\frac{x}{a}}} \, dx,x,\frac{1}{x}\right )}{2 \sqrt{1-\frac{1}{a x}}}\\ &=-\frac{\sqrt{1+\frac{1}{a x}} \sqrt{c-a c x}}{\sqrt{1-\frac{1}{a x}} x}-\frac{\left (\sqrt{\frac{1}{x}} \sqrt{c-a c x}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{1+\frac{x^2}{a}}} \, dx,x,\sqrt{\frac{1}{x}}\right )}{\sqrt{1-\frac{1}{a x}}}\\ &=-\frac{\sqrt{1+\frac{1}{a x}} \sqrt{c-a c x}}{\sqrt{1-\frac{1}{a x}} x}-\frac{\sqrt{a} \sqrt{\frac{1}{x}} \sqrt{c-a c x} \sinh ^{-1}\left (\frac{\sqrt{\frac{1}{x}}}{\sqrt{a}}\right )}{\sqrt{1-\frac{1}{a x}}}\\ \end{align*}
Mathematica [A] time = 0.0387791, size = 76, normalized size = 0.78 \[ -\frac{\sqrt{\frac{1}{x}} \sqrt{c-a c x} \left (\sqrt{\frac{1}{x}} \sqrt{\frac{1}{a x}+1}+\sqrt{a} \sinh ^{-1}\left (\frac{\sqrt{\frac{1}{x}}}{\sqrt{a}}\right )\right )}{\sqrt{1-\frac{1}{a x}}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.144, size = 78, normalized size = 0.8 \begin{align*} -{\frac{1}{x} \left ( \arctan \left ({\sqrt{-c \left ( ax+1 \right ) }{\frac{1}{\sqrt{c}}}} \right ) xac+\sqrt{-c \left ( ax+1 \right ) }\sqrt{c} \right ) \sqrt{-c \left ( ax-1 \right ) }{\frac{1}{\sqrt{{\frac{ax-1}{ax+1}}}}}{\frac{1}{\sqrt{-c \left ( ax+1 \right ) }}}{\frac{1}{\sqrt{c}}}} \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 c x + c}}{x^{2} \sqrt{\frac{a x - 1}{a x + 1}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.57579, size = 514, normalized size = 5.3 \begin{align*} \left [\frac{{\left (a^{2} x^{2} - a x\right )} \sqrt{-c} \log \left (-\frac{a^{2} c x^{2} + a c x + 2 \, \sqrt{-a c x + c}{\left (a x + 1\right )} \sqrt{-c} \sqrt{\frac{a x - 1}{a x + 1}} - 2 \, c}{a x^{2} - x}\right ) - 2 \, \sqrt{-a c x + c}{\left (a x + 1\right )} \sqrt{\frac{a x - 1}{a x + 1}}}{2 \,{\left (a x^{2} - x\right )}}, -\frac{{\left (a^{2} x^{2} - a x\right )} \sqrt{c} \arctan \left (\frac{\sqrt{-a c x + c} \sqrt{c} \sqrt{\frac{a x - 1}{a x + 1}}}{a c x - c}\right ) + \sqrt{-a c x + c}{\left (a x + 1\right )} \sqrt{\frac{a x - 1}{a x + 1}}}{a x^{2} - x}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [C] time = 1.19978, size = 162, normalized size = 1.67 \begin{align*} -\frac{a^{2} c^{2}{\left (\frac{\arctan \left (\frac{\sqrt{-a c x - c}}{\sqrt{c}}\right )}{\sqrt{c} \mathrm{sgn}\left (-a c x - c\right )} + \frac{\sqrt{-a c x - c}}{a c x \mathrm{sgn}\left (-a c x - c\right )}\right )} - \frac{-i \, a^{2} \sqrt{-c} c \arctan \left (-i \, \sqrt{2}\right ) - \sqrt{2} a^{2} \sqrt{-c} c}{\mathrm{sgn}\left (c\right )}}{a c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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