Optimal. Leaf size=85 \[ \frac{2 x \sqrt{1-\frac{1}{a x}}}{\sqrt{\frac{1}{a x}+1} \sqrt{c-a c x}}+\frac{6 \sqrt{1-\frac{1}{a x}}}{a \sqrt{\frac{1}{a x}+1} \sqrt{c-a c x}} \]
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Rubi [A] time = 0.143442, antiderivative size = 85, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {6176, 6181, 78, 37} \[ \frac{2 x \sqrt{1-\frac{1}{a x}}}{\sqrt{\frac{1}{a x}+1} \sqrt{c-a c x}}+\frac{6 \sqrt{1-\frac{1}{a x}}}{a \sqrt{\frac{1}{a x}+1} \sqrt{c-a c x}} \]
Antiderivative was successfully verified.
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Rule 6176
Rule 6181
Rule 78
Rule 37
Rubi steps
\begin{align*} \int \frac{e^{-3 \coth ^{-1}(a x)}}{\sqrt{c-a c x}} \, dx &=\frac{\left (\sqrt{1-\frac{1}{a x}} \sqrt{x}\right ) \int \frac{e^{-3 \coth ^{-1}(a x)}}{\sqrt{1-\frac{1}{a x}} \sqrt{x}} \, dx}{\sqrt{c-a c x}}\\ &=-\frac{\sqrt{1-\frac{1}{a x}} \operatorname{Subst}\left (\int \frac{1-\frac{x}{a}}{x^{3/2} \left (1+\frac{x}{a}\right )^{3/2}} \, dx,x,\frac{1}{x}\right )}{\sqrt{\frac{1}{x}} \sqrt{c-a c x}}\\ &=\frac{2 \sqrt{1-\frac{1}{a x}} x}{\sqrt{1+\frac{1}{a x}} \sqrt{c-a c x}}+\frac{\left (3 \sqrt{1-\frac{1}{a x}}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{x} \left (1+\frac{x}{a}\right )^{3/2}} \, dx,x,\frac{1}{x}\right )}{a \sqrt{\frac{1}{x}} \sqrt{c-a c x}}\\ &=\frac{6 \sqrt{1-\frac{1}{a x}}}{a \sqrt{1+\frac{1}{a x}} \sqrt{c-a c x}}+\frac{2 \sqrt{1-\frac{1}{a x}} x}{\sqrt{1+\frac{1}{a x}} \sqrt{c-a c x}}\\ \end{align*}
Mathematica [A] time = 0.0312703, size = 48, normalized size = 0.56 \[ \frac{2 \sqrt{1-\frac{1}{a x}} (a x+3)}{a \sqrt{\frac{1}{a x}+1} \sqrt{c-a c x}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.041, size = 47, normalized size = 0.6 \begin{align*} 2\,{\frac{ \left ( ax+3 \right ) \left ( ax+1 \right ) }{ \left ( ax-1 \right ) a\sqrt{-acx+c}} \left ({\frac{ax-1}{ax+1}} \right ) ^{3/2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.08734, size = 65, normalized size = 0.76 \begin{align*} \frac{2 \,{\left (a^{2} x^{2} + 4 \, a x + 3\right )}{\left (a x - 1\right )}}{{\left (a^{2} \sqrt{-c} x - a \sqrt{-c}\right )}{\left (a x + 1\right )}^{\frac{3}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.53825, size = 99, normalized size = 1.16 \begin{align*} -\frac{2 \, \sqrt{-a c x + c}{\left (a x + 3\right )} \sqrt{\frac{a x - 1}{a x + 1}}}{a^{2} c x - a c} \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 = 1.18049, size = 49, normalized size = 0.58 \begin{align*} \frac{2 \,{\left (\sqrt{-a c x - c} - \frac{2 \, c}{\sqrt{-a c x - c}}\right )}{\left | c \right |}}{a c^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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