Optimal. Leaf size=52 \[ \frac{2 (1-a x)}{3 a \left (c-a^2 c x^2\right )^{3/2}}-\frac{x}{3 c \sqrt{c-a^2 c x^2}} \]
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Rubi [A] time = 0.104115, antiderivative size = 52, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {6167, 6142, 653, 191} \[ \frac{2 (1-a x)}{3 a \left (c-a^2 c x^2\right )^{3/2}}-\frac{x}{3 c \sqrt{c-a^2 c x^2}} \]
Antiderivative was successfully verified.
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Rule 6167
Rule 6142
Rule 653
Rule 191
Rubi steps
\begin{align*} \int \frac{e^{-2 \coth ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^{3/2}} \, dx &=-\int \frac{e^{-2 \tanh ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^{3/2}} \, dx\\ &=-\left (c \int \frac{(1-a x)^2}{\left (c-a^2 c x^2\right )^{5/2}} \, dx\right )\\ &=\frac{2 (1-a x)}{3 a \left (c-a^2 c x^2\right )^{3/2}}-\frac{1}{3} \int \frac{1}{\left (c-a^2 c x^2\right )^{3/2}} \, dx\\ &=\frac{2 (1-a x)}{3 a \left (c-a^2 c x^2\right )^{3/2}}-\frac{x}{3 c \sqrt{c-a^2 c x^2}}\\ \end{align*}
Mathematica [A] time = 0.0351886, size = 63, normalized size = 1.21 \[ \frac{\sqrt{1-a x} (a x+2) \sqrt{1-a^2 x^2}}{3 a c (a x+1)^{3/2} \sqrt{c-a^2 c x^2}} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.044, size = 31, normalized size = 0.6 \begin{align*}{\frac{ \left ( ax-1 \right ) ^{2} \left ( ax+2 \right ) }{3\,a} \left ( -{a}^{2}c{x}^{2}+c \right ) ^{-{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [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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Fricas [A] time = 1.70498, size = 97, normalized size = 1.87 \begin{align*} \frac{\sqrt{-a^{2} c x^{2} + c}{\left (a x + 2\right )}}{3 \,{\left (a^{3} c^{2} x^{2} + 2 \, a^{2} c^{2} x + a c^{2}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{a x - 1}{\left (- c \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac{3}{2}} \left (a x + 1\right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.2031, size = 200, normalized size = 3.85 \begin{align*} -\frac{{\left (a c + 3 \, \sqrt{-a^{2} c} \sqrt{c}\right )} \mathrm{sgn}\left (x\right )}{3 \,{\left (a^{2} c^{\frac{5}{2}} + \sqrt{-a^{2} c} a c^{2}\right )}} + \frac{2 \,{\left (2 \, a^{2} c - 3 \, a \sqrt{c}{\left (\sqrt{-a^{2} c + \frac{c}{x^{2}}} - \frac{\sqrt{c}}{x}\right )} + 3 \,{\left (\sqrt{-a^{2} c + \frac{c}{x^{2}}} - \frac{\sqrt{c}}{x}\right )}^{2}\right )}}{3 \,{\left (a \sqrt{c} - \sqrt{-a^{2} c + \frac{c}{x^{2}}} + \frac{\sqrt{c}}{x}\right )}^{3} c \mathrm{sgn}\left (x\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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