Optimal. Leaf size=28 \[ (1-i) e^{(1+i) x} \, _2F_1\left (1-i,2;2-i;-e^{i x}\right ) \]
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Rubi [A]
time = 0.02, antiderivative size = 28, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {4542, 4536}
\begin {gather*} (1-i) e^{(1+i) x} \text {Hypergeometric2F1}\left (1-i,2,2-i,-e^{i x}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 4536
Rule 4542
Rubi steps
\begin {align*} \int \frac {e^x}{1+\cos (x)} \, dx &=\frac {1}{2} \int e^x \sec ^2\left (\frac {x}{2}\right ) \, dx\\ &=(1-i) e^{(1+i) x} \, _2F_1\left (1-i,2;2-i;-e^{i x}\right )\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 28, normalized size = 1.00 \begin {gather*} (1-i) e^{(1+i) x} \, _2F_1\left (1-i,2;2-i;-e^{i x}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.01, size = 0, normalized size = 0.00 \[\int \frac {{\mathrm e}^{x}}{1+\cos \left (x \right )}\, dx\]
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 \frac {e^{x}}{\cos {\left (x \right )} + 1}\, 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 = -0.04 \begin {gather*} \int \frac {{\mathrm {e}}^x}{\cos \left (x\right )+1} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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