Optimal. Leaf size=38 \[ -\frac{(2-a x) e^{-2 \tan ^{-1}(a x)}}{5 a c \sqrt{a^2 c x^2+c}} \]
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Rubi [A] time = 0.0392175, antiderivative size = 38, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.043, Rules used = {5069} \[ -\frac{(2-a x) e^{-2 \tan ^{-1}(a x)}}{5 a c \sqrt{a^2 c x^2+c}} \]
Antiderivative was successfully verified.
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Rule 5069
Rubi steps
\begin{align*} \int \frac{e^{-2 \tan ^{-1}(a x)}}{\left (c+a^2 c x^2\right )^{3/2}} \, dx &=-\frac{e^{-2 \tan ^{-1}(a x)} (2-a x)}{5 a c \sqrt{c+a^2 c x^2}}\\ \end{align*}
Mathematica [A] time = 0.0213199, size = 37, normalized size = 0.97 \[ \frac{(a x-2) e^{-2 \tan ^{-1}(a x)}}{5 a c \sqrt{a^2 c x^2+c}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.036, size = 41, normalized size = 1.1 \begin{align*}{\frac{ \left ({a}^{2}{x}^{2}+1 \right ) \left ( ax-2 \right ) }{5\,a{{\rm e}^{2\,\arctan \left ( ax \right ) }}} \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] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{e^{\left (-2 \, \arctan \left (a x\right )\right )}}{{\left (a^{2} c x^{2} + c\right )}^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.94746, size = 103, normalized size = 2.71 \begin{align*} \frac{\sqrt{a^{2} c x^{2} + c}{\left (a x - 2\right )} e^{\left (-2 \, \arctan \left (a x\right )\right )}}{5 \,{\left (a^{3} c^{2} x^{2} + a c^{2}\right )}} \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 [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{e^{\left (-2 \, \arctan \left (a x\right )\right )}}{{\left (a^{2} c x^{2} + c\right )}^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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