Optimal. Leaf size=134 \[ -\frac{2 \sqrt{\frac{1}{a x}+1} \sqrt{c-\frac{c}{a x}}}{\sqrt{1-\frac{1}{a x}}}-\frac{8 \sqrt{c-\frac{c}{a x}}}{\sqrt{1-\frac{1}{a x}} \sqrt{\frac{1}{a x}+1}}+\frac{2 \sqrt{c-\frac{c}{a x}} \tanh ^{-1}\left (\sqrt{\frac{1}{a x}+1}\right )}{\sqrt{1-\frac{1}{a x}}} \]
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Rubi [A] time = 0.261511, antiderivative size = 134, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.185, Rules used = {6182, 6180, 87, 63, 208} \[ -\frac{2 \sqrt{\frac{1}{a x}+1} \sqrt{c-\frac{c}{a x}}}{\sqrt{1-\frac{1}{a x}}}-\frac{8 \sqrt{c-\frac{c}{a x}}}{\sqrt{1-\frac{1}{a x}} \sqrt{\frac{1}{a x}+1}}+\frac{2 \sqrt{c-\frac{c}{a x}} \tanh ^{-1}\left (\sqrt{\frac{1}{a x}+1}\right )}{\sqrt{1-\frac{1}{a x}}} \]
Antiderivative was successfully verified.
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Rule 6182
Rule 6180
Rule 87
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{e^{-3 \coth ^{-1}(a x)} \sqrt{c-\frac{c}{a x}}}{x} \, dx &=\frac{\sqrt{c-\frac{c}{a x}} \int \frac{e^{-3 \coth ^{-1}(a x)} \sqrt{1-\frac{1}{a x}}}{x} \, dx}{\sqrt{1-\frac{1}{a x}}}\\ &=-\frac{\sqrt{c-\frac{c}{a x}} \operatorname{Subst}\left (\int \frac{\left (1-\frac{x}{a}\right )^2}{x \left (1+\frac{x}{a}\right )^{3/2}} \, dx,x,\frac{1}{x}\right )}{\sqrt{1-\frac{1}{a x}}}\\ &=-\frac{\sqrt{c-\frac{c}{a x}} \operatorname{Subst}\left (\int \left (-\frac{4}{a \left (1+\frac{x}{a}\right )^{3/2}}+\frac{1}{a \sqrt{1+\frac{x}{a}}}+\frac{1}{x \sqrt{1+\frac{x}{a}}}\right ) \, dx,x,\frac{1}{x}\right )}{\sqrt{1-\frac{1}{a x}}}\\ &=-\frac{8 \sqrt{c-\frac{c}{a x}}}{\sqrt{1-\frac{1}{a x}} \sqrt{1+\frac{1}{a x}}}-\frac{2 \sqrt{1+\frac{1}{a x}} \sqrt{c-\frac{c}{a x}}}{\sqrt{1-\frac{1}{a x}}}-\frac{\sqrt{c-\frac{c}{a x}} \operatorname{Subst}\left (\int \frac{1}{x \sqrt{1+\frac{x}{a}}} \, dx,x,\frac{1}{x}\right )}{\sqrt{1-\frac{1}{a x}}}\\ &=-\frac{8 \sqrt{c-\frac{c}{a x}}}{\sqrt{1-\frac{1}{a x}} \sqrt{1+\frac{1}{a x}}}-\frac{2 \sqrt{1+\frac{1}{a x}} \sqrt{c-\frac{c}{a x}}}{\sqrt{1-\frac{1}{a x}}}-\frac{\left (2 a \sqrt{c-\frac{c}{a x}}\right ) \operatorname{Subst}\left (\int \frac{1}{-a+a x^2} \, dx,x,\sqrt{1+\frac{1}{a x}}\right )}{\sqrt{1-\frac{1}{a x}}}\\ &=-\frac{8 \sqrt{c-\frac{c}{a x}}}{\sqrt{1-\frac{1}{a x}} \sqrt{1+\frac{1}{a x}}}-\frac{2 \sqrt{1+\frac{1}{a x}} \sqrt{c-\frac{c}{a x}}}{\sqrt{1-\frac{1}{a x}}}+\frac{2 \sqrt{c-\frac{c}{a x}} \tanh ^{-1}\left (\sqrt{1+\frac{1}{a x}}\right )}{\sqrt{1-\frac{1}{a x}}}\\ \end{align*}
Mathematica [A] time = 0.373846, size = 131, normalized size = 0.98 \[ -\frac{2 a x \sqrt{1-\frac{1}{a^2 x^2}} (5 a x+1) \sqrt{c-\frac{c}{a x}}}{a^2 x^2-1}+\sqrt{c} \log \left (2 a^2 \sqrt{c} x^2 \sqrt{1-\frac{1}{a^2 x^2}} \sqrt{c-\frac{c}{a x}}+c \left (2 a^2 x^2-a x-1\right )\right )-\sqrt{c} \log (1-a x) \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.182, size = 151, normalized size = 1.1 \begin{align*} -{\frac{ax+1}{ \left ( ax-1 \right ) ^{2}} \left ({\frac{ax-1}{ax+1}} \right ) ^{{\frac{3}{2}}}\sqrt{{\frac{c \left ( ax-1 \right ) }{ax}}} \left ( 10\,{a}^{3/2}x\sqrt{ \left ( ax+1 \right ) x}-\ln \left ({\frac{1}{2} \left ( 2\,\sqrt{ \left ( ax+1 \right ) x}\sqrt{a}+2\,ax+1 \right ){\frac{1}{\sqrt{a}}}} \right ){x}^{2}{a}^{2}-\ln \left ({\frac{1}{2} \left ( 2\,\sqrt{ \left ( ax+1 \right ) x}\sqrt{a}+2\,ax+1 \right ){\frac{1}{\sqrt{a}}}} \right ) xa+2\,\sqrt{ \left ( ax+1 \right ) x}\sqrt{a} \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{\sqrt{c - \frac{c}{a x}} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{3}{2}}}{x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.81085, size = 609, normalized size = 4.54 \begin{align*} \left [\frac{{\left (a x - 1\right )} \sqrt{c} \log \left (-\frac{8 \, a^{3} c x^{3} - 7 \, a c x + 4 \,{\left (2 \, a^{3} x^{3} + 3 \, a^{2} x^{2} + a x\right )} \sqrt{c} \sqrt{\frac{a x - 1}{a x + 1}} \sqrt{\frac{a c x - c}{a x}} - c}{a x - 1}\right ) - 4 \,{\left (5 \, a x + 1\right )} \sqrt{\frac{a x - 1}{a x + 1}} \sqrt{\frac{a c x - c}{a x}}}{2 \,{\left (a x - 1\right )}}, -\frac{{\left (a x - 1\right )} \sqrt{-c} \arctan \left (\frac{2 \,{\left (a^{2} x^{2} + a x\right )} \sqrt{-c} \sqrt{\frac{a x - 1}{a x + 1}} \sqrt{\frac{a c x - c}{a x}}}{2 \, a^{2} c x^{2} - a c x - c}\right ) + 2 \,{\left (5 \, a x + 1\right )} \sqrt{\frac{a x - 1}{a x + 1}} \sqrt{\frac{a c x - c}{a x}}}{a x - 1}\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 [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{c - \frac{c}{a x}} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{3}{2}}}{x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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