Optimal. Leaf size=51 \[ x \left (-\sqrt{c-\frac{c}{a x}}\right )-\frac{3 \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c-\frac{c}{a x}}}{\sqrt{c}}\right )}{a} \]
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Rubi [A] time = 0.101261, antiderivative size = 51, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.292, Rules used = {6133, 25, 514, 375, 78, 63, 208} \[ x \left (-\sqrt{c-\frac{c}{a x}}\right )-\frac{3 \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c-\frac{c}{a x}}}{\sqrt{c}}\right )}{a} \]
Antiderivative was successfully verified.
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Rule 6133
Rule 25
Rule 514
Rule 375
Rule 78
Rule 63
Rule 208
Rubi steps
\begin{align*} \int e^{2 \tanh ^{-1}(a x)} \sqrt{c-\frac{c}{a x}} \, dx &=\int \frac{\sqrt{c-\frac{c}{a x}} (1+a x)}{1-a x} \, dx\\ &=-\frac{c \int \frac{1+a x}{\sqrt{c-\frac{c}{a x}} x} \, dx}{a}\\ &=-\frac{c \int \frac{a+\frac{1}{x}}{\sqrt{c-\frac{c}{a x}}} \, dx}{a}\\ &=\frac{c \operatorname{Subst}\left (\int \frac{a+x}{x^2 \sqrt{c-\frac{c x}{a}}} \, dx,x,\frac{1}{x}\right )}{a}\\ &=-\sqrt{c-\frac{c}{a x}} x+\frac{(3 c) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{c-\frac{c x}{a}}} \, dx,x,\frac{1}{x}\right )}{2 a}\\ &=-\sqrt{c-\frac{c}{a x}} x-3 \operatorname{Subst}\left (\int \frac{1}{a-\frac{a x^2}{c}} \, dx,x,\sqrt{c-\frac{c}{a x}}\right )\\ &=-\sqrt{c-\frac{c}{a x}} x-\frac{3 \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c-\frac{c}{a x}}}{\sqrt{c}}\right )}{a}\\ \end{align*}
Mathematica [A] time = 0.0301668, size = 51, normalized size = 1. \[ x \left (-\sqrt{c-\frac{c}{a x}}\right )-\frac{3 \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c-\frac{c}{a x}}}{\sqrt{c}}\right )}{a} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.131, size = 120, normalized size = 2.4 \begin{align*}{\frac{x}{2}\sqrt{{\frac{c \left ( ax-1 \right ) }{ax}}} \left ( 2\,\sqrt{a{x}^{2}-x}\sqrt{a}-4\,\sqrt{ \left ( ax-1 \right ) x}\sqrt{a}-\ln \left ({\frac{1}{2} \left ( 2\,\sqrt{a{x}^{2}-x}\sqrt{a}+2\,ax-1 \right ){\frac{1}{\sqrt{a}}}} \right ) -2\,\ln \left ( 1/2\,{\frac{2\,\sqrt{ \left ( ax-1 \right ) x}\sqrt{a}+2\,ax-1}{\sqrt{a}}} \right ) \right ){\frac{1}{\sqrt{ \left ( ax-1 \right ) x}}}{\frac{1}{\sqrt{a}}}} \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{{\left (a x + 1\right )}^{2} \sqrt{c - \frac{c}{a x}}}{a^{2} x^{2} - 1}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.19437, size = 275, normalized size = 5.39 \begin{align*} \left [-\frac{2 \, a x \sqrt{\frac{a c x - c}{a x}} - 3 \, \sqrt{c} \log \left (-2 \, a c x + 2 \, a \sqrt{c} x \sqrt{\frac{a c x - c}{a x}} + c\right )}{2 \, a}, -\frac{a x \sqrt{\frac{a c x - c}{a x}} - 3 \, \sqrt{-c} \arctan \left (\frac{\sqrt{-c} \sqrt{\frac{a c x - c}{a x}}}{c}\right )}{a}\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 - \frac{c}{a x}}}{a x - 1}\, dx - \int \frac{a x \sqrt{c - \frac{c}{a x}}}{a x - 1}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.20874, size = 132, normalized size = 2.59 \begin{align*} -\frac{3 \, \sqrt{c} \log \left ({\left | a \right |} \sqrt{{\left | c \right |}}\right ) \mathrm{sgn}\left (x\right )}{2 \, a} + \frac{3 \, \sqrt{c} \log \left ({\left | -2 \,{\left (\sqrt{a^{2} c} x - \sqrt{a^{2} c x^{2} - a c x}\right )}{\left | a \right |} + a \sqrt{c} \right |}\right )}{2 \, a \mathrm{sgn}\left (x\right )} - \frac{\sqrt{a^{2} c x^{2} - a c x}{\left | a \right |}}{a^{2} \mathrm{sgn}\left (x\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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