Optimal. Leaf size=41 \[ \frac {e^x \sin (x)}{1-\cos (x)}-(2-2 i) e^{(1+i) x} \, _2F_1\left (1-i,2;2-i;e^{i x}\right ) \]
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Rubi [A] time = 0.11, 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 = {4463, 4461, 4443, 2194, 2251, 2288} \[ -4 i e^x \text {Hypergeometric2F1}\left (-i,1,1-i,e^{i x}\right )+2 i e^x-\frac {e^x \sin (x)}{1-\cos (x)} \]
Antiderivative was successfully verified.
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Rule 2194
Rule 2251
Rule 2288
Rule 4443
Rule 4461
Rule 4463
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] time = 0.22, size = 100, normalized size = 2.44 \[ \frac {2 e^x \sin \left (\frac {x}{2}\right ) (\sin (x)+1) \left (2 i \, _2F_1\left (-i,1;1-i;e^{i x}\right ) \sin \left (\frac {x}{2}\right )+(1+i) e^{i x} \, _2F_1\left (1,1-i;2-i;e^{i x}\right ) \sin \left (\frac {x}{2}\right )+\cos \left (\frac {x}{2}\right )\right )}{(\cos (x)-1) \left (\sin \left (\frac {x}{2}\right )+\cos \left (\frac {x}{2}\right )\right )^2} \]
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {e^x (1+\sin (x))}{1-\cos (x)} \, dx \]
Verification is Not applicable to the result.
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fricas [F] time = 1.42, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {e^{x} \sin \relax (x) + e^{x}}{\cos \relax (x) - 1}, x\right ) \]
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 \[ \int -\frac {{\left (\sin \relax (x) + 1\right )} e^{x}}{\cos \relax (x) - 1}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.07, size = 0, normalized size = 0.00 \[\int \frac {{\mathrm e}^{x} \left (1+\sin \relax (x )\right )}{1-\cos \relax (x )}\, dx\]
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 \[ \frac {2 \, {\left (2 \, {\left (\cos \relax (x)^{2} + \sin \relax (x)^{2} - 2 \, \cos \relax (x) + 1\right )} \int \frac {e^{x} \sin \relax (x)}{\cos \relax (x)^{2} + \sin \relax (x)^{2} - 2 \, \cos \relax (x) + 1}\,{d x} - e^{x} \sin \relax (x)\right )}}{\cos \relax (x)^{2} + \sin \relax (x)^{2} - 2 \, \cos \relax (x) + 1} \]
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 \[ \int -\frac {{\mathrm {e}}^x\,\left (\sin \relax (x)+1\right )}{\cos \relax (x)-1} \,d x \]
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 \[ - \int \frac {e^{x}}{\cos {\relax (x )} - 1}\, dx - \int \frac {e^{x} \sin {\relax (x )}}{\cos {\relax (x )} - 1}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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