Optimal. Leaf size=37 \[ \frac{(a x+2) e^{2 \tan ^{-1}(a x)}}{5 a c \sqrt{a^2 c x^2+c}} \]
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Rubi [A] time = 0.0397945, antiderivative size = 37, 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} \[ \frac{(a x+2) e^{2 \tan ^{-1}(a x)}}{5 a c \sqrt{a^2 c x^2+c}} \]
Antiderivative was successfully verified.
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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 = 0.0160972, size = 37, normalized size = 1. \[ \frac{(a x+2) e^{2 \tan ^{-1}(a x)}}{5 a c \sqrt{a^2 c x^2+c}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.039, size = 39, normalized size = 1.1 \begin{align*}{\frac{ \left ({a}^{2}{x}^{2}+1 \right ) \left ( ax+2 \right ){{\rm e}^{2\,\arctan \left ( ax \right ) }}}{5\,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] 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 = 2.02885, size = 101, normalized size = 2.73 \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] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{e^{2 \operatorname{atan}{\left (a x \right )}}}{\left (c \left (a^{2} x^{2} + 1\right )\right )^{\frac{3}{2}}}\, 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{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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