Optimal. Leaf size=46 \[ -\frac{\sqrt{1-a^2 x^2}}{2 a c (a x+1)^2 \sqrt{c-a^2 c x^2}} \]
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Rubi [A] time = 0.0812225, antiderivative size = 46, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {6143, 6140, 32} \[ -\frac{\sqrt{1-a^2 x^2}}{2 a c (a x+1)^2 \sqrt{c-a^2 c x^2}} \]
Antiderivative was successfully verified.
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Rule 6143
Rule 6140
Rule 32
Rubi steps
\begin{align*} \int \frac{e^{-3 \tanh ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^{3/2}} \, dx &=\frac{\sqrt{1-a^2 x^2} \int \frac{e^{-3 \tanh ^{-1}(a x)}}{\left (1-a^2 x^2\right )^{3/2}} \, dx}{c \sqrt{c-a^2 c x^2}}\\ &=\frac{\sqrt{1-a^2 x^2} \int \frac{1}{(1+a x)^3} \, dx}{c \sqrt{c-a^2 c x^2}}\\ &=-\frac{\sqrt{1-a^2 x^2}}{2 a c (1+a x)^2 \sqrt{c-a^2 c x^2}}\\ \end{align*}
Mathematica [A] time = 0.0434178, size = 53, normalized size = 1.15 \[ \frac{\sqrt{1-a^2 x^2} \sqrt{c-a^2 c x^2}}{2 a c^2 (a x-1) (a x+1)^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.027, size = 38, normalized size = 0.8 \begin{align*} -{\frac{1}{2\, \left ( ax+1 \right ) ^{2}a} \left ( -{a}^{2}{x}^{2}+1 \right ) ^{{\frac{3}{2}}} \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 [A] time = 0.996838, size = 39, normalized size = 0.85 \begin{align*} -\frac{1}{2 \,{\left (a^{3} c^{\frac{3}{2}} x^{2} + 2 \, a^{2} c^{\frac{3}{2}} x + a c^{\frac{3}{2}}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.45908, size = 146, normalized size = 3.17 \begin{align*} -\frac{\sqrt{-a^{2} c x^{2} + c} \sqrt{-a^{2} x^{2} + 1}{\left (a x^{2} + 2 \, x\right )}}{2 \,{\left (a^{4} c^{2} x^{4} + 2 \, a^{3} c^{2} x^{3} - 2 \, a c^{2} x - 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{\left (- \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac{3}{2}}}{\left (- c \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac{3}{2}} \left (a x + 1\right )^{3}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (-a^{2} x^{2} + 1\right )}^{\frac{3}{2}}}{{\left (-a^{2} c x^{2} + c\right )}^{\frac{3}{2}}{\left (a x + 1\right )}^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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