Optimal. Leaf size=41 \[ (-2+2 i) e^{(1+i) x} \, _2F_1\left (1-i,2;2-i;e^{i x}\right )+\frac {e^x \sin (x)}{1-\cos (x)} \]
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Rubi [A]
time = 0.08, antiderivative size = 45, normalized size of antiderivative = 1.10, number of steps
used = 7, number of rules used = 6, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {4551, 4549,
4528, 2225, 2283, 2326} \begin {gather*} -4 i e^x \text {Hypergeometric2F1}\left (-i,1,1-i,e^{i x}\right )+2 i e^x-\frac {e^x \sin (x)}{1-\cos (x)} \end {gather*}
Antiderivative was successfully verified.
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Rule 2225
Rule 2283
Rule 2326
Rule 4528
Rule 4549
Rule 4551
Rubi steps
\begin {align*} \int \frac {e^x (1+\sin (x))}{1-\cos (x)} \, dx &=2 \int \frac {e^x \sin (x)}{1-\cos (x)} \, dx+\int \frac {e^x (1-\sin (x))}{1-\cos (x)} \, dx\\ &=-\frac {e^x \sin (x)}{1-\cos (x)}+2 \int e^x \cot \left (\frac {x}{2}\right ) \, dx\\ &=-\frac {e^x \sin (x)}{1-\cos (x)}-2 i \int \left (-e^x-\frac {2 e^x}{-1+e^{i x}}\right ) \, dx\\ &=-\frac {e^x \sin (x)}{1-\cos (x)}+2 i \int e^x \, dx+4 i \int \frac {e^x}{-1+e^{i x}} \, dx\\ &=2 i e^x-4 i e^x \, _2F_1\left (-i,1;1-i;e^{i x}\right )-\frac {e^x \sin (x)}{1-\cos (x)}\\ \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. \(100\) vs. \(2(41)=82\).
time = 0.15, size = 100, normalized size = 2.44 \begin {gather*} \frac {2 e^x \sin \left (\frac {x}{2}\right ) \left (\cos \left (\frac {x}{2}\right )+2 i \, _2F_1\left (-i,1;1-i;e^{i x}\right ) \sin \left (\frac {x}{2}\right )+(1+i) e^{i x} \, _2F_1\left (1,1-i;2-i;e^{i x}\right ) \sin \left (\frac {x}{2}\right )\right ) (1+\sin (x))}{(-1+\cos (x)) \left (\cos \left (\frac {x}{2}\right )+\sin \left (\frac {x}{2}\right )\right )^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.05, size = 0, normalized size = 0.00 \[\int \frac {{\mathrm e}^{x} \left (\sin \left (x \right )+1\right )}{1-\cos \left (x \right )}\, dx\]
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 - \int \frac {e^{x} \sin {\left (x \right )}}{\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 = -0.02 \begin {gather*} \int -\frac {{\mathrm {e}}^x\,\left (\sin \left (x\right )+1\right )}{\cos \left (x\right )-1} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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