Optimal. Leaf size=96 \[ -\frac{x \sqrt{c-\frac{c}{a x}}}{c}+\frac{3 \tanh ^{-1}\left (\frac{\sqrt{c-\frac{c}{a x}}}{\sqrt{c}}\right )}{a \sqrt{c}}-\frac{2 \sqrt{2} \tanh ^{-1}\left (\frac{\sqrt{c-\frac{c}{a x}}}{\sqrt{2} \sqrt{c}}\right )}{a \sqrt{c}} \]
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Rubi [A] time = 0.161138, antiderivative size = 96, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 8, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {6133, 25, 514, 375, 99, 156, 63, 208} \[ -\frac{x \sqrt{c-\frac{c}{a x}}}{c}+\frac{3 \tanh ^{-1}\left (\frac{\sqrt{c-\frac{c}{a x}}}{\sqrt{c}}\right )}{a \sqrt{c}}-\frac{2 \sqrt{2} \tanh ^{-1}\left (\frac{\sqrt{c-\frac{c}{a x}}}{\sqrt{2} \sqrt{c}}\right )}{a \sqrt{c}} \]
Antiderivative was successfully verified.
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Rule 6133
Rule 25
Rule 514
Rule 375
Rule 99
Rule 156
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{e^{-2 \tanh ^{-1}(a x)}}{\sqrt{c-\frac{c}{a x}}} \, dx &=\int \frac{1-a x}{\sqrt{c-\frac{c}{a x}} (1+a x)} \, dx\\ &=-\frac{a \int \frac{\sqrt{c-\frac{c}{a x}} x}{1+a x} \, dx}{c}\\ &=-\frac{a \int \frac{\sqrt{c-\frac{c}{a x}}}{a+\frac{1}{x}} \, dx}{c}\\ &=\frac{a \operatorname{Subst}\left (\int \frac{\sqrt{c-\frac{c x}{a}}}{x^2 (a+x)} \, dx,x,\frac{1}{x}\right )}{c}\\ &=-\frac{\sqrt{c-\frac{c}{a x}} x}{c}+\frac{\operatorname{Subst}\left (\int \frac{-\frac{3 c}{2}+\frac{c x}{2 a}}{x (a+x) \sqrt{c-\frac{c x}{a}}} \, dx,x,\frac{1}{x}\right )}{c}\\ &=-\frac{\sqrt{c-\frac{c}{a x}} x}{c}-\frac{3 \operatorname{Subst}\left (\int \frac{1}{x \sqrt{c-\frac{c x}{a}}} \, dx,x,\frac{1}{x}\right )}{2 a}+\frac{2 \operatorname{Subst}\left (\int \frac{1}{(a+x) \sqrt{c-\frac{c x}{a}}} \, dx,x,\frac{1}{x}\right )}{a}\\ &=-\frac{\sqrt{c-\frac{c}{a x}} x}{c}+\frac{3 \operatorname{Subst}\left (\int \frac{1}{a-\frac{a x^2}{c}} \, dx,x,\sqrt{c-\frac{c}{a x}}\right )}{c}-\frac{4 \operatorname{Subst}\left (\int \frac{1}{2 a-\frac{a x^2}{c}} \, dx,x,\sqrt{c-\frac{c}{a x}}\right )}{c}\\ &=-\frac{\sqrt{c-\frac{c}{a x}} x}{c}+\frac{3 \tanh ^{-1}\left (\frac{\sqrt{c-\frac{c}{a x}}}{\sqrt{c}}\right )}{a \sqrt{c}}-\frac{2 \sqrt{2} \tanh ^{-1}\left (\frac{\sqrt{c-\frac{c}{a x}}}{\sqrt{2} \sqrt{c}}\right )}{a \sqrt{c}}\\ \end{align*}
Mathematica [A] time = 0.0474643, size = 96, normalized size = 1. \[ -\frac{x \sqrt{c-\frac{c}{a x}}}{c}+\frac{3 \tanh ^{-1}\left (\frac{\sqrt{c-\frac{c}{a x}}}{\sqrt{c}}\right )}{a \sqrt{c}}-\frac{2 \sqrt{2} \tanh ^{-1}\left (\frac{\sqrt{c-\frac{c}{a x}}}{\sqrt{2} \sqrt{c}}\right )}{a \sqrt{c}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.126, size = 136, normalized size = 1.4 \begin{align*} -{\frac{x}{2\,c}\sqrt{{\frac{c \left ( ax-1 \right ) }{ax}}} \left ( 2\,\sqrt{ \left ( ax-1 \right ) x}{a}^{3/2}\sqrt{{a}^{-1}}-3\,\ln \left ( 1/2\,{\frac{2\,\sqrt{ \left ( ax-1 \right ) x}\sqrt{a}+2\,ax-1}{\sqrt{a}}} \right ) a\sqrt{{a}^{-1}}-2\,\sqrt{2}\ln \left ({\frac{2\,\sqrt{2}\sqrt{{a}^{-1}}\sqrt{ \left ( ax-1 \right ) x}a-3\,ax+1}{ax+1}} \right ) \sqrt{a} \right ){\frac{1}{\sqrt{ \left ( ax-1 \right ) x}}}{a}^{-{\frac{3}{2}}}{\frac{1}{\sqrt{{a}^{-1}}}}} \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{a^{2} x^{2} - 1}{{\left (a x + 1\right )}^{2} \sqrt{c - \frac{c}{a x}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.78072, size = 535, normalized size = 5.57 \begin{align*} \left [-\frac{2 \, a x \sqrt{\frac{a c x - c}{a x}} - 2 \, \sqrt{2} \sqrt{c} \log \left (\frac{\frac{2 \, \sqrt{2} a x \sqrt{\frac{a c x - c}{a x}}}{\sqrt{c}} - 3 \, a x + 1}{a x + 1}\right ) - 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 c}, \frac{2 \, \sqrt{2} c \sqrt{-\frac{1}{c}} \arctan \left (\frac{\sqrt{2} a x \sqrt{-\frac{1}{c}} \sqrt{\frac{a c x - c}{a x}}}{a x - 1}\right ) - 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 c}\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{a x}{a x \sqrt{c - \frac{c}{a x}} + \sqrt{c - \frac{c}{a x}}}\, dx - \int - \frac{1}{a x \sqrt{c - \frac{c}{a x}} + \sqrt{c - \frac{c}{a x}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.32843, size = 227, normalized size = 2.36 \begin{align*} a c{\left (\frac{2 \, \sqrt{2} \arctan \left (\frac{\sqrt{2} \sqrt{-\frac{c - \frac{2 \, c}{a x + 1}}{\frac{1}{a x + 1} - 1}}}{2 \, \sqrt{-c}}\right )}{a^{2} \sqrt{-c} c} - \frac{3 \, \arctan \left (\frac{\sqrt{-\frac{c - \frac{2 \, c}{a x + 1}}{\frac{1}{a x + 1} - 1}}}{\sqrt{-c}}\right )}{a^{2} \sqrt{-c} c} - \frac{\sqrt{-\frac{c - \frac{2 \, c}{a x + 1}}{\frac{1}{a x + 1} - 1}}}{a^{2}{\left (c + \frac{c - \frac{2 \, c}{a x + 1}}{\frac{1}{a x + 1} - 1}\right )} c}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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