Optimal. Leaf size=33 \[ -\frac{2 \sqrt{c-a c x}}{a c^2 \sqrt{1-a^2 x^2}} \]
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Rubi [A] time = 0.0504776, antiderivative size = 33, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, Rules used = {6127, 649} \[ -\frac{2 \sqrt{c-a c x}}{a c^2 \sqrt{1-a^2 x^2}} \]
Antiderivative was successfully verified.
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Rule 6127
Rule 649
Rubi steps
\begin{align*} \int \frac{e^{-3 \tanh ^{-1}(a x)}}{(c-a c x)^{3/2}} \, dx &=\frac{\int \frac{(c-a c x)^{3/2}}{\left (1-a^2 x^2\right )^{3/2}} \, dx}{c^3}\\ &=-\frac{2 \sqrt{c-a c x}}{a c^2 \sqrt{1-a^2 x^2}}\\ \end{align*}
Mathematica [A] time = 0.0235171, size = 35, normalized size = 1.06 \[ -\frac{2 (1-a x)^{3/2}}{a \sqrt{a x+1} (c-a c x)^{3/2}} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.031, size = 34, normalized size = 1. \begin{align*} -2\,{\frac{ \left ( -{a}^{2}{x}^{2}+1 \right ) ^{3/2}}{ \left ( -acx+c \right ) ^{3/2} \left ( ax+1 \right ) ^{2}a}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.984977, size = 38, normalized size = 1.15 \begin{align*} -\frac{2 \, \sqrt{a x + 1} \sqrt{c}}{a^{2} c^{2} x + a c^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.57041, size = 82, normalized size = 2.48 \begin{align*} \frac{2 \, \sqrt{-a^{2} x^{2} + 1} \sqrt{-a c x + c}}{a^{3} c^{2} x^{2} - a c^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.19356, size = 41, normalized size = 1.24 \begin{align*} \frac{{\left (\frac{\sqrt{2}}{a \sqrt{c}} - \frac{2}{\sqrt{a c x + c} a}\right )}{\left | c \right |}}{c^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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