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 = {5169, 71}
\begin {gather*} \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 {gather*}
Antiderivative was successfully verified.
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Rule 71
Rule 5169
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 \begin {gather*} \frac {(1-i) e^{(2+2 i) \text {ArcTan}(a x)} \, _2F_1\left (1-i,2;2-i;-e^{2 i \text {ArcTan}(a x)}\right )}{a} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.00, 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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int e^{2 \operatorname {atan}{\left (a x \right )}}\, dx \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int {\mathrm {e}}^{2\,\mathrm {atan}\left (a\,x\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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