Optimal. Leaf size=46 \[ -\frac{(1-i) 2^{-1+i} (1-i a x)^{1-i} \, _2F_1\left (-i,1-i;2-i;\frac{1}{2} (1-i a x)\right )}{a} \]
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Rubi [A] time = 0.0108576, antiderivative size = 46, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {5061, 69} \[ -\frac{(1-i) 2^{-1+i} (1-i a x)^{1-i} \, _2F_1\left (-i,1-i;2-i;\frac{1}{2} (1-i a x)\right )}{a} \]
Antiderivative was successfully verified.
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Rule 5061
Rule 69
Rubi steps
\begin{align*} \int e^{-2 \tan ^{-1}(a x)} \, dx &=\int (1-i a x)^{-i} (1+i a x)^i \, dx\\ &=-\frac{(1-i) 2^{-1+i} (1-i a x)^{1-i} \, _2F_1\left (-i,1-i;2-i;\frac{1}{2} (1-i a x)\right )}{a}\\ \end{align*}
Mathematica [A] time = 0.0209654, size = 37, normalized size = 0.8 \[ -\frac{(1+i) e^{(-2+2 i) \tan ^{-1}(a x)} \, _2F_1\left (1+i,2;2+i;-e^{2 i \tan ^{-1}(a x)}\right )}{a} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.051, size = 0, normalized size = 0. \begin{align*} \int{{\rm e}^{-2\,\arctan \left ( ax \right ) }}\, dx \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 e^{\left (-2 \, \arctan \left (a x\right )\right )}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (e^{\left (-2 \, \arctan \left (a x\right )\right )}, x\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 e^{- 2 \operatorname{atan}{\left (a x \right )}}\, 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 e^{\left (-2 \, \arctan \left (a x\right )\right )}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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