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 = 0.0340965, antiderivative size = 30, normalized size of antiderivative = 1., 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 4456
Rule 4450
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*}
Mathematica [B] time = 0.62303, 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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Maple [F] time = 0.054, size = 0, normalized size = 0. \begin{align*} \int{\frac{{{\rm e}^{x}}}{1-\sin \left ( x \right ) }}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \frac{2 \,{\left (\cos \left (x\right ) e^{x} -{\left (\cos \left (x\right )^{2} + \sin \left (x\right )^{2} - 2 \, \sin \left (x\right ) + 1\right )} \int \frac{\cos \left (x\right ) e^{x}}{\cos \left (x\right )^{2} + \sin \left (x\right )^{2} - 2 \, \sin \left (x\right ) + 1}\,{d x}\right )}}{\cos \left (x\right )^{2} + \sin \left (x\right )^{2} - 2 \, \sin \left (x\right ) + 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-\frac{e^{x}}{\sin \left (x\right ) - 1}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} - \int \frac{e^{x}}{\sin{\left (x \right )} - 1}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int -\frac{e^{x}}{\sin \left (x\right ) - 1}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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