Optimal. Leaf size=10 \[ -\frac{\cos (x)}{\sin (x)+1} \]
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Rubi [A] time = 0.005755, antiderivative size = 10, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 6, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {2648} \[ -\frac{\cos (x)}{\sin (x)+1} \]
Antiderivative was successfully verified.
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Rule 2648
Rubi steps
\begin{align*} \int \frac{1}{1+\sin (x)} \, dx &=-\frac{\cos (x)}{1+\sin (x)}\\ \end{align*}
Mathematica [B] time = 0.009546, size = 23, normalized size = 2.3 \[ \frac{2 \sin \left (\frac{x}{2}\right )}{\sin \left (\frac{x}{2}\right )+\cos \left (\frac{x}{2}\right )} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.006, size = 11, normalized size = 1.1 \begin{align*} -2\, \left ( 1+\tan \left ( x/2 \right ) \right ) ^{-1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.938928, size = 20, normalized size = 2. \begin{align*} -\frac{2}{\frac{\sin \left (x\right )}{\cos \left (x\right ) + 1} + 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.66611, size = 62, normalized size = 6.2 \begin{align*} -\frac{\cos \left (x\right ) - \sin \left (x\right ) + 1}{\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 [A] time = 0.376199, size = 8, normalized size = 0.8 \begin{align*} - \frac{2}{\tan{\left (\frac{x}{2} \right )} + 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.07258, size = 14, normalized size = 1.4 \begin{align*} -\frac{2}{\tan \left (\frac{1}{2} \, x\right ) + 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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