Optimal. Leaf size=60 \[ \frac{\left (\frac{1}{5}+\frac{2 i}{5}\right ) 2^{1-\frac{i}{2}} (1-i a x)^{1+\frac{i}{2}} \, _2F_1\left (\frac{i}{2},1+\frac{i}{2};2+\frac{i}{2};\frac{1}{2} (1-i a x)\right )}{a} \]
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Rubi [A] time = 0.0113063, antiderivative size = 60, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 6, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {5061, 69} \[ \frac{\left (\frac{1}{5}+\frac{2 i}{5}\right ) 2^{1-\frac{i}{2}} (1-i a x)^{1+\frac{i}{2}} \, _2F_1\left (\frac{i}{2},1+\frac{i}{2};2+\frac{i}{2};\frac{1}{2} (1-i a x)\right )}{a} \]
Antiderivative was successfully verified.
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Rule 5061
Rule 69
Rubi steps
\begin{align*} \int e^{\tan ^{-1}(a x)} \, dx &=\int (1-i a x)^{\frac{i}{2}} (1+i a x)^{-\frac{i}{2}} \, dx\\ &=\frac{\left (\frac{1}{5}+\frac{2 i}{5}\right ) 2^{1-\frac{i}{2}} (1-i a x)^{1+\frac{i}{2}} \, _2F_1\left (\frac{i}{2},1+\frac{i}{2};2+\frac{i}{2};\frac{1}{2} (1-i a x)\right )}{a}\\ \end{align*}
Mathematica [A] time = 0.0154955, size = 45, normalized size = 0.75 \[ \frac{\left (\frac{4}{5}-\frac{8 i}{5}\right ) e^{(1+2 i) \tan ^{-1}(a x)} \, _2F_1\left (1-\frac{i}{2},2;2-\frac{i}{2};-e^{2 i \tan ^{-1}(a x)}\right )}{a} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.051, size = 0, normalized size = 0. \begin{align*} \int{{\rm e}^{\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 (\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 (\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^{\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 (\arctan \left (a x\right )\right )}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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