Optimal. Leaf size=35 \[ \frac{(a x+1) e^{\tan ^{-1}(a x)}}{2 a c \sqrt{a^2 c x^2+c}} \]
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Rubi [A] time = 0.0361105, antiderivative size = 35, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.048, Rules used = {5069} \[ \frac{(a x+1) e^{\tan ^{-1}(a x)}}{2 a c \sqrt{a^2 c x^2+c}} \]
Antiderivative was successfully verified.
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Rule 5069
Rubi steps
\begin{align*} \int \frac{e^{\tan ^{-1}(a x)}}{\left (c+a^2 c x^2\right )^{3/2}} \, dx &=\frac{e^{\tan ^{-1}(a x)} (1+a x)}{2 a c \sqrt{c+a^2 c x^2}}\\ \end{align*}
Mathematica [A] time = 0.0148304, size = 35, normalized size = 1. \[ \frac{(a x+1) e^{\tan ^{-1}(a x)}}{2 a c \sqrt{a^2 c x^2+c}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.038, size = 37, normalized size = 1.1 \begin{align*}{\frac{ \left ({a}^{2}{x}^{2}+1 \right ) \left ( ax+1 \right ){{\rm e}^{\arctan \left ( ax \right ) }}}{2\,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 (\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 = 1.87468, size = 99, normalized size = 2.83 \begin{align*} \frac{\sqrt{a^{2} c x^{2} + c}{\left (a x + 1\right )} e^{\left (\arctan \left (a x\right )\right )}}{2 \,{\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^{\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 (\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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