Optimal. Leaf size=10 \[ -\frac {\cos (x)}{\sin (x)+1} \]
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Rubi [A] time = 0.01, antiderivative size = 10, normalized size of antiderivative = 1.00, 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*}
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Mathematica [B] time = 0.01, size = 23, normalized size = 2.30 \[ \frac {2 \sin \left (\frac {x}{2}\right )}{\sin \left (\frac {x}{2}\right )+\cos \left (\frac {x}{2}\right )} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.41, size = 18, normalized size = 1.80 \[ -\frac {\cos \relax (x) - \sin \relax (x) + 1}{\cos \relax (x) + \sin \relax (x) + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.04, size = 10, normalized size = 1.00 \[ -\frac {2}{\tan \left (\frac {1}{2} \, x\right ) + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 11, normalized size = 1.10 \[ -\frac {2}{\tan \left (\frac {x}{2}\right )+1} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.42, size = 15, normalized size = 1.50 \[ -\frac {2}{\frac {\sin \relax (x)}{\cos \relax (x) + 1} + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.02, size = 10, normalized size = 1.00 \[ -\frac {2}{\mathrm {tan}\left (\frac {x}{2}\right )+1} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.36, size = 8, normalized size = 0.80 \[ - \frac {2}{\tan {\left (\frac {x}{2} \right )} + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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