Optimal. Leaf size=26 \[ (-1+i) e^{(1+i) x} \, _2F_1\left (1-i,2;2-i;e^{i x}\right ) \]
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Rubi [A]
time = 0.02, antiderivative size = 26, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {4543, 4538}
\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 4538
Rule 4543
Rubi steps
\begin {align*} \int \frac {e^x}{1-\cos (x)} \, dx &=\frac {1}{2} \int e^x \csc ^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 [B] Both result and optimal contain complex but leaf count is larger than twice
the leaf count of optimal. \(84\) vs. \(2(26)=52\).
time = 0.06, size = 84, normalized size = 3.23 \begin {gather*} \frac {(1+i) e^x \sin \left (\frac {x}{2}\right ) \left ((1-i) \cos \left (\frac {x}{2}\right )+(1+i) \, _2F_1\left (-i,1;1-i;e^{i x}\right ) \sin \left (\frac {x}{2}\right )+e^{i x} \, _2F_1\left (1,1-i;2-i;e^{i x}\right ) \sin \left (\frac {x}{2}\right )\right )}{-1+\cos (x)} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.02, 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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