Optimal. Leaf size=30 \[ (1+i) e^{(1+i) x} \, _2F_1\left (1-i,2;2-i;-i e^{i x}\right ) \]
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Rubi [A] time = 0.03, antiderivative size = 30, 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 = {4456, 4450} \[ (1+i) e^{(1+i) x} \text {Hypergeometric2F1}\left (1-i,2,2-i,-i e^{i x}\right ) \]
Antiderivative was successfully verified.
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Rule 4450
Rule 4456
Rubi steps
\begin {align*} \int \frac {e^x}{1-\sin (x)} \, dx &=\frac {1}{2} \int e^x \sec ^2\left (\frac {\pi }{4}+\frac {x}{2}\right ) \, dx\\ &=(1+i) e^{(1+i) x} \, _2F_1\left (1-i,2;2-i;-i e^{i x}\right )\\ \end {align*}
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Mathematica [B] time = 0.68, size = 61, normalized size = 2.03 \[ \frac {2 e^x \sin \left (\frac {x}{2}\right )}{\cos \left (\frac {x}{2}\right )-\sin \left (\frac {x}{2}\right )}+(1+i) (\sinh (x)+\cosh (x)) (1-(1+i) \, _2F_1(-i,1;1-i;\sin (x)-i \cos (x))) \]
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {e^x}{1-\sin (x)} \, dx \]
Verification is Not applicable to the result.
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fricas [F] time = 1.06, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {e^{x}}{\sin \relax (x) - 1}, x\right ) \]
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 \[ \int -\frac {e^{x}}{\sin \relax (x) - 1}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.04, size = 0, normalized size = 0.00 \[\int \frac {{\mathrm e}^{x}}{1-\sin \relax (x )}\, dx\]
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 \[ \frac {2 \, {\left (\cos \relax (x) e^{x} - {\left (\cos \relax (x)^{2} + \sin \relax (x)^{2} - 2 \, \sin \relax (x) + 1\right )} \int \frac {\cos \relax (x) e^{x}}{\cos \relax (x)^{2} + \sin \relax (x)^{2} - 2 \, \sin \relax (x) + 1}\,{d x}\right )}}{\cos \relax (x)^{2} + \sin \relax (x)^{2} - 2 \, \sin \relax (x) + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.03 \[ -\int \frac {{\mathrm {e}}^x}{\sin \relax (x)-1} \,d x \]
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 \[ - \int \frac {e^{x}}{\sin {\relax (x )} - 1}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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