Optimal. Leaf size=43 \[ \frac {e^x \cos (x)}{\sin (x)+1}-(2+2 i) e^{(1+i) x} \, _2F_1\left (1-i,2;2-i;i e^{i x}\right ) \]
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Rubi [A] time = 0.13, antiderivative size = 47, normalized size of antiderivative = 1.09, number of steps used = 7, number of rules used = 6, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {4462, 4459, 4442, 2194, 2251, 2288} \[ 4 i e^x \text {Hypergeometric2F1}\left (i,1,1+i,-i e^{-i x}\right )-2 i e^x-\frac {e^x \cos (x)}{\sin (x)+1} \]
Antiderivative was successfully verified.
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Rule 2194
Rule 2251
Rule 2288
Rule 4442
Rule 4459
Rule 4462
Rubi steps
\begin {align*} \int \frac {e^x (1+\cos (x))}{1+\sin (x)} \, dx &=2 \int \frac {e^x \cos (x)}{1+\sin (x)} \, dx+\int \frac {e^x (1-\cos (x))}{1+\sin (x)} \, dx\\ &=-\frac {e^x \cos (x)}{1+\sin (x)}+2 \int e^x \tan \left (\frac {\pi }{4}-\frac {x}{2}\right ) \, dx\\ &=-\frac {e^x \cos (x)}{1+\sin (x)}+2 i \int \left (-e^x+\frac {2 e^x}{1+e^{2 i \left (\frac {\pi }{4}-\frac {x}{2}\right )}}\right ) \, dx\\ &=-\frac {e^x \cos (x)}{1+\sin (x)}-2 i \int e^x \, dx+4 i \int \frac {e^x}{1+e^{2 i \left (\frac {\pi }{4}-\frac {x}{2}\right )}} \, dx\\ &=-2 i e^x+4 i e^x \, _2F_1\left (i,1;1+i;-i e^{-i x}\right )-\frac {e^x \cos (x)}{1+\sin (x)}\\ \end {align*}
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Mathematica [A] time = 0.20, size = 73, normalized size = 1.70 \[ \frac {1}{2} (\cos (x)+1) \sec ^2\left (\frac {x}{2}\right ) \left (\frac {e^x \left ((1+2 i) \tan \left (\frac {x}{2}\right )-(1-2 i)\right )}{\tan \left (\frac {x}{2}\right )+1}-4 i (\sinh (x)+\cosh (x)) \, _2F_1(-i,1;1-i;i \cos (x)-\sin (x))\right ) \]
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {e^x (1+\cos (x))}{1+\sin (x)} \, dx \]
Verification is Not applicable to the result.
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fricas [F] time = 1.12, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (\cos \relax (x) + 1\right )} e^{x}}{\sin \relax (x) + 1}, x\right ) \]
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 \[ \int \frac {{\left (\cos \relax (x) + 1\right )} e^{x}}{\sin \relax (x) + 1}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.07, size = 0, normalized size = 0.00 \[\int \frac {{\mathrm e}^{x} \left (1+\cos \relax (x )\right )}{1+\sin \relax (x )}\, dx\]
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 \[ -\frac {2 \, {\left (\cos \relax (x) e^{x} - 2 \, {\left (\cos \relax (x)^{2} + \sin \relax (x)^{2} + 2 \, \sin \relax (x) + 1\right )} \int \frac {\cos \relax (x) e^{x}}{\cos \relax (x)^{2} + \sin \relax (x)^{2} + 2 \, \sin \relax (x) + 1}\,{d x}\right )}}{\cos \relax (x)^{2} + \sin \relax (x)^{2} + 2 \, \sin \relax (x) + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {{\mathrm {e}}^x\,\left (\cos \relax (x)+1\right )}{\sin \relax (x)+1} \,d x \]
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 \[ \int \frac {\left (\cos {\relax (x )} + 1\right ) e^{x}}{\sin {\relax (x )} + 1}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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