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 = 0.01, antiderivative size = 46, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, 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 69
Rule 5061
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*}
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Mathematica [A] time = 0.02, size = 37, normalized size = 0.80 \[ \frac {(1-i) e^{(2+2 i) \tan ^{-1}(a x)} \, _2F_1\left (1-i,2;2-i;-e^{2 i \tan ^{-1}(a x)}\right )}{a} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.55, size = 0, normalized size = 0.00 \[ {\rm integral}\left (e^{\left (2 \, \arctan \left (a x\right )\right )}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \mathit {sage}_{0} x \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.06, size = 0, normalized size = 0.00 \[ \int {\mathrm e}^{2 \arctan \left (a x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int e^{\left (2 \, \arctan \left (a x\right )\right )}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int {\mathrm {e}}^{2\,\mathrm {atan}\left (a\,x\right )} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int e^{2 \operatorname {atan}{\left (a x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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