Optimal. Leaf size=67 \[ \frac{2 (c-a c x)^{3/2}}{a c^2 \sqrt{1-a^2 x^2}}-\frac{8 \sqrt{c-a c x}}{a c \sqrt{1-a^2 x^2}} \]
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Rubi [A] time = 0.0639084, antiderivative size = 67, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15, Rules used = {6127, 657, 649} \[ \frac{2 (c-a c x)^{3/2}}{a c^2 \sqrt{1-a^2 x^2}}-\frac{8 \sqrt{c-a c x}}{a c \sqrt{1-a^2 x^2}} \]
Antiderivative was successfully verified.
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Rule 6127
Rule 657
Rule 649
Rubi steps
\begin{align*} \int \frac{e^{-3 \tanh ^{-1}(a x)}}{\sqrt{c-a c x}} \, dx &=\frac{\int \frac{(c-a c x)^{5/2}}{\left (1-a^2 x^2\right )^{3/2}} \, dx}{c^3}\\ &=\frac{2 (c-a c x)^{3/2}}{a c^2 \sqrt{1-a^2 x^2}}+\frac{4 \int \frac{(c-a c x)^{3/2}}{\left (1-a^2 x^2\right )^{3/2}} \, dx}{c^2}\\ &=-\frac{8 \sqrt{c-a c x}}{a c \sqrt{1-a^2 x^2}}+\frac{2 (c-a c x)^{3/2}}{a c^2 \sqrt{1-a^2 x^2}}\\ \end{align*}
Mathematica [A] time = 0.0237497, size = 40, normalized size = 0.6 \[ -\frac{2 \sqrt{1-a x} (a x+3)}{a \sqrt{a x+1} \sqrt{c-a c x}} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.029, size = 46, normalized size = 0.7 \begin{align*} 2\,{\frac{ \left ( -{a}^{2}{x}^{2}+1 \right ) ^{3/2} \left ( ax+3 \right ) }{\sqrt{-acx+c} \left ( ax+1 \right ) ^{2} \left ( ax-1 \right ) a}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.987793, size = 41, normalized size = 0.61 \begin{align*} -\frac{2 \,{\left (a x + 3\right )} \sqrt{a x + 1}}{a^{2} \sqrt{c} x + a \sqrt{c}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.57468, size = 90, normalized size = 1.34 \begin{align*} \frac{2 \, \sqrt{-a^{2} x^{2} + 1} \sqrt{-a c x + c}{\left (a x + 3\right )}}{a^{3} c x^{2} - a c} \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 = 1.21528, size = 59, normalized size = 0.88 \begin{align*} \frac{4 \, \sqrt{2}{\left | c \right |}}{a c^{\frac{3}{2}}} - \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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