Optimal. Leaf size=29 \[ -\frac{2 (a x+1) e^{-3 \coth ^{-1}(a x)}}{a (c-a c x)^{3/2}} \]
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Rubi [A] time = 0.0380544, antiderivative size = 29, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.05, Rules used = {6174} \[ -\frac{2 (a x+1) e^{-3 \coth ^{-1}(a x)}}{a (c-a c x)^{3/2}} \]
Antiderivative was successfully verified.
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Rule 6174
Rubi steps
\begin{align*} \int \frac{e^{-3 \coth ^{-1}(a x)}}{(c-a c x)^{3/2}} \, dx &=-\frac{2 e^{-3 \coth ^{-1}(a x)} (1+a x)}{a (c-a c x)^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.0286373, size = 41, normalized size = 1.41 \[ -\frac{2 x \left (1-\frac{1}{a x}\right )^{3/2}}{\sqrt{\frac{1}{a x}+1} (c-a c x)^{3/2}} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.046, size = 35, normalized size = 1.2 \begin{align*} -2\,{\frac{ax+1}{a \left ( -acx+c \right ) ^{3/2}} \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 = 1.09125, size = 61, normalized size = 2.1 \begin{align*} -\frac{2 \,{\left (a \sqrt{-c} x + \sqrt{-c}\right )}{\left (a x - 1\right )}}{{\left (a^{2} c^{2} x - a c^{2}\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 = 1.54254, size = 90, normalized size = 3.1 \begin{align*} -\frac{2 \, \sqrt{-a c x + c} \sqrt{\frac{a x - 1}{a x + 1}}}{a^{2} c^{2} x - a c^{2}} \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 = 1.15873, size = 55, normalized size = 1.9 \begin{align*} \frac{{\left (\frac{\sqrt{2}}{a \sqrt{-c}} - \frac{2}{\sqrt{-a c x - c} a}\right )}{\left | c \right |} \mathrm{sgn}\left (a x + 1\right )}{c^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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