Optimal. Leaf size=12 \[ \frac{e^x \sin (x)}{\cos (x)+1} \]
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Rubi [A] time = 0.0277275, antiderivative size = 12, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.071, Rules used = {2288} \[ \frac{e^x \sin (x)}{\cos (x)+1} \]
Antiderivative was successfully verified.
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Rule 2288
Rubi steps
\begin{align*} \int \frac{e^x (1+\sin (x))}{1+\cos (x)} \, dx &=\frac{e^x \sin (x)}{1+\cos (x)}\\ \end{align*}
Mathematica [A] time = 0.180259, size = 10, normalized size = 0.83 \[ e^x \tan \left (\frac{x}{2}\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.034, size = 8, normalized size = 0.7 \begin{align*}{{\rm e}^{x}}\tan \left ({\frac{x}{2}} \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.20976, size = 30, normalized size = 2.5 \begin{align*} \frac{2 \, e^{x} \sin \left (x\right )}{\cos \left (x\right )^{2} + \sin \left (x\right )^{2} + 2 \, \cos \left (x\right ) + 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.95081, size = 34, normalized size = 2.83 \begin{align*} \frac{e^{x} \sin \left (x\right )}{\cos \left (x\right ) + 1} \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{\left (\sin{\left (x \right )} + 1\right ) e^{x}}{\cos{\left (x \right )} + 1}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.15052, size = 9, normalized size = 0.75 \begin{align*} e^{x} \tan \left (\frac{1}{2} \, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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