Optimal. Leaf size=14 \[ \frac{e^x \cos (x)}{1-\sin (x)} \]
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Rubi [A] time = 0.0240413, antiderivative size = 14, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.062, Rules used = {2288} \[ \frac{e^x \cos (x)}{1-\sin (x)} \]
Antiderivative was successfully verified.
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Rule 2288
Rubi steps
\begin{align*} \int \frac{e^x (1+\cos (x))}{1-\sin (x)} \, dx &=\frac{e^x \cos (x)}{1-\sin (x)}\\ \end{align*}
Mathematica [A] time = 0.0748232, size = 23, normalized size = 1.64 \[ -\frac{e^x \left (\tan \left (\frac{x}{2}\right )+1\right )}{\tan \left (\frac{x}{2}\right )-1} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.069, size = 53, normalized size = 3.8 \begin{align*}{ \left ( -{{\rm e}^{x}}\tan \left ({\frac{x}{2}} \right ) -{{\rm e}^{x}} \left ( \tan \left ({\frac{x}{2}} \right ) \right ) ^{2}-{{\rm e}^{x}} \left ( \tan \left ({\frac{x}{2}} \right ) \right ) ^{3}-{{\rm e}^{x}} \right ) \left ( \left ( \tan \left ({\frac{x}{2}} \right ) \right ) ^{2}+1 \right ) ^{-1} \left ( \tan \left ({\frac{x}{2}} \right ) -1 \right ) ^{-1}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.21475, size = 30, normalized size = 2.14 \begin{align*} \frac{2 \, \cos \left (x\right ) e^{x}}{\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 [A] time = 1.81892, size = 74, normalized size = 5.29 \begin{align*} \frac{{\left (\cos \left (x\right ) + 1\right )} e^{x} + e^{x} \sin \left (x\right )}{\cos \left (x\right ) - \sin \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{e^{x}}{\sin{\left (x \right )} - 1}\, dx - \int \frac{e^{x} \cos{\left (x \right )}}{\sin{\left (x \right )} - 1}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.14211, size = 27, normalized size = 1.93 \begin{align*} -\frac{e^{x} \tan \left (\frac{1}{2} \, x\right ) + e^{x}}{\tan \left (\frac{1}{2} \, x\right ) - 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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