Optimal. Leaf size=128 \[ -\frac{a x \sqrt{\frac{1}{a x}+1} \left (1-\frac{1}{a x}\right )^{3/2}}{\left (a-\frac{1}{x}\right ) (c-a c x)^{3/2}}-\frac{\sqrt{a} \left (1-\frac{1}{a x}\right )^{3/2} \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{\frac{1}{x}}}{\sqrt{a} \sqrt{\frac{1}{a x}+1}}\right )}{\sqrt{2} \left (\frac{1}{x}\right )^{3/2} (c-a c x)^{3/2}} \]
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Rubi [A] time = 0.184957, antiderivative size = 128, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.278, Rules used = {6176, 6181, 94, 93, 206} \[ -\frac{a x \sqrt{\frac{1}{a x}+1} \left (1-\frac{1}{a x}\right )^{3/2}}{\left (a-\frac{1}{x}\right ) (c-a c x)^{3/2}}-\frac{\sqrt{a} \left (1-\frac{1}{a x}\right )^{3/2} \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{\frac{1}{x}}}{\sqrt{a} \sqrt{\frac{1}{a x}+1}}\right )}{\sqrt{2} \left (\frac{1}{x}\right )^{3/2} (c-a c x)^{3/2}} \]
Antiderivative was successfully verified.
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Rule 6176
Rule 6181
Rule 94
Rule 93
Rule 206
Rubi steps
\begin{align*} \int \frac{e^{\coth ^{-1}(a x)}}{(c-a c x)^{3/2}} \, dx &=\frac{\left (\left (1-\frac{1}{a x}\right )^{3/2} x^{3/2}\right ) \int \frac{e^{\coth ^{-1}(a x)}}{\left (1-\frac{1}{a x}\right )^{3/2} x^{3/2}} \, dx}{(c-a c x)^{3/2}}\\ &=-\frac{\left (1-\frac{1}{a x}\right )^{3/2} \operatorname{Subst}\left (\int \frac{\sqrt{1+\frac{x}{a}}}{\sqrt{x} \left (1-\frac{x}{a}\right )^2} \, dx,x,\frac{1}{x}\right )}{\left (\frac{1}{x}\right )^{3/2} (c-a c x)^{3/2}}\\ &=-\frac{a \left (1-\frac{1}{a x}\right )^{3/2} \sqrt{1+\frac{1}{a x}} x}{\left (a-\frac{1}{x}\right ) (c-a c x)^{3/2}}-\frac{\left (1-\frac{1}{a x}\right )^{3/2} \operatorname{Subst}\left (\int \frac{1}{\sqrt{x} \left (1-\frac{x}{a}\right ) \sqrt{1+\frac{x}{a}}} \, dx,x,\frac{1}{x}\right )}{2 \left (\frac{1}{x}\right )^{3/2} (c-a c x)^{3/2}}\\ &=-\frac{a \left (1-\frac{1}{a x}\right )^{3/2} \sqrt{1+\frac{1}{a x}} x}{\left (a-\frac{1}{x}\right ) (c-a c x)^{3/2}}-\frac{\left (1-\frac{1}{a x}\right )^{3/2} \operatorname{Subst}\left (\int \frac{1}{1-\frac{2 x^2}{a}} \, dx,x,\frac{\sqrt{\frac{1}{x}}}{\sqrt{1+\frac{1}{a x}}}\right )}{\left (\frac{1}{x}\right )^{3/2} (c-a c x)^{3/2}}\\ &=-\frac{a \left (1-\frac{1}{a x}\right )^{3/2} \sqrt{1+\frac{1}{a x}} x}{\left (a-\frac{1}{x}\right ) (c-a c x)^{3/2}}-\frac{\sqrt{a} \left (1-\frac{1}{a x}\right )^{3/2} \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{\frac{1}{x}}}{\sqrt{a} \sqrt{1+\frac{1}{a x}}}\right )}{\sqrt{2} \left (\frac{1}{x}\right )^{3/2} (c-a c x)^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.107615, size = 116, normalized size = 0.91 \[ \frac{x \sqrt{1-\frac{1}{a x}} \left (2 \sqrt{a} \sqrt{\frac{1}{a x}+1}+\sqrt{2} \sqrt{\frac{1}{x}} (a x-1) \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{\frac{1}{x}}}{\sqrt{a} \sqrt{\frac{1}{a x}+1}}\right )\right )}{2 \sqrt{a} c (a x-1) \sqrt{c-a c x}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.141, size = 118, normalized size = 0.9 \begin{align*} -{\frac{1}{ \left ( 2\,ax-2 \right ) a}\sqrt{-c \left ( ax-1 \right ) } \left ( \sqrt{2}\arctan \left ({\frac{\sqrt{2}}{2}\sqrt{-c \left ( ax+1 \right ) }{\frac{1}{\sqrt{c}}}} \right ) xac-\sqrt{2}\arctan \left ({\frac{\sqrt{2}}{2}\sqrt{-c \left ( ax+1 \right ) }{\frac{1}{\sqrt{c}}}} \right ) c+2\,\sqrt{-c \left ( ax+1 \right ) }\sqrt{c} \right ){\frac{1}{\sqrt{{\frac{ax-1}{ax+1}}}}}{\frac{1}{\sqrt{-c \left ( ax+1 \right ) }}}{c}^{-{\frac{5}{2}}}} \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{1}{{\left (-a c x + c\right )}^{\frac{3}{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 = 1.6742, size = 664, normalized size = 5.19 \begin{align*} \left [-\frac{\sqrt{2}{\left (a^{2} x^{2} - 2 \, a x + 1\right )} \sqrt{-c} \log \left (-\frac{a^{2} c x^{2} + 2 \, a c x - 2 \, \sqrt{2} \sqrt{-a c x + c}{\left (a x + 1\right )} \sqrt{-c} \sqrt{\frac{a x - 1}{a x + 1}} - 3 \, c}{a^{2} x^{2} - 2 \, a x + 1}\right ) + 4 \, \sqrt{-a c x + c}{\left (a x + 1\right )} \sqrt{\frac{a x - 1}{a x + 1}}}{4 \,{\left (a^{3} c^{2} x^{2} - 2 \, a^{2} c^{2} x + a c^{2}\right )}}, -\frac{\sqrt{2}{\left (a^{2} x^{2} - 2 \, a x + 1\right )} \sqrt{c} \arctan \left (\frac{\sqrt{2} \sqrt{-a c x + c} \sqrt{c} \sqrt{\frac{a x - 1}{a x + 1}}}{a c x - c}\right ) + 2 \, \sqrt{-a c x + c}{\left (a x + 1\right )} \sqrt{\frac{a x - 1}{a x + 1}}}{2 \,{\left (a^{3} c^{2} x^{2} - 2 \, a^{2} c^{2} x + a c^{2}\right )}}\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 [A] time = 1.16659, size = 96, normalized size = 0.75 \begin{align*} -\frac{\frac{\sqrt{2} \arctan \left (\frac{\sqrt{2} \sqrt{-a c x - c}}{2 \, \sqrt{c}}\right )}{\sqrt{c}} + \frac{2 \, \sqrt{-a c x - c}}{a c x - c}}{2 \, a c \mathrm{sgn}\left (-a c x - c\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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