Optimal. Leaf size=11 \[ x+\frac {\cos (x)}{\sin (x)+1} \]
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Rubi [A] time = 0.02, antiderivative size = 11, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {2735, 2648} \[ x+\frac {\cos (x)}{\sin (x)+1} \]
Antiderivative was successfully verified.
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Rule 2648
Rule 2735
Rubi steps
\begin {align*} \int \frac {\sin (x)}{1+\sin (x)} \, dx &=x-\int \frac {1}{1+\sin (x)} \, dx\\ &=x+\frac {\cos (x)}{1+\sin (x)}\\ \end {align*}
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Mathematica [B] time = 0.04, size = 25, normalized size = 2.27 \[ x-\frac {2 \sin \left (\frac {x}{2}\right )}{\sin \left (\frac {x}{2}\right )+\cos \left (\frac {x}{2}\right )} \]
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sin (x)}{1+\sin (x)} \, dx \]
Verification is Not applicable to the result.
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fricas [B] time = 1.03, size = 24, normalized size = 2.18 \[ \frac {{\left (x + 1\right )} \cos \relax (x) + {\left (x - 1\right )} \sin \relax (x) + x + 1}{\cos \relax (x) + \sin \relax (x) + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.08, size = 12, normalized size = 1.09 \[ x + \frac {2}{\tan \left (\frac {1}{2} \, x\right ) + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.05, size = 15, normalized size = 1.36
method | result | size |
risch | \(x +\frac {2}{{\mathrm e}^{i x}+i}\) | \(15\) |
default | \(2 \arctan \left (\tan \left (\frac {x}{2}\right )\right )+\frac {2}{\tan \left (\frac {x}{2}\right )+1}\) | \(19\) |
norman | \(\frac {x +x \tan \left (\frac {x}{2}\right )+x \left (\tan ^{2}\left (\frac {x}{2}\right )\right )+x \left (\tan ^{3}\left (\frac {x}{2}\right )\right )+2 \left (\tan ^{2}\left (\frac {x}{2}\right )\right )+2}{\left (1+\tan ^{2}\left (\frac {x}{2}\right )\right ) \left (\tan \left (\frac {x}{2}\right )+1\right )}\) | \(53\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.96, size = 28, normalized size = 2.55 \[ \frac {2}{\frac {\sin \relax (x)}{\cos \relax (x) + 1} + 1} + 2 \, \arctan \left (\frac {\sin \relax (x)}{\cos \relax (x) + 1}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.24, size = 12, normalized size = 1.09 \[ x+\frac {2}{\mathrm {tan}\left (\frac {x}{2}\right )+1} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.54, size = 29, normalized size = 2.64 \[ \frac {x \tan {\left (\frac {x}{2} \right )}}{\tan {\left (\frac {x}{2} \right )} + 1} + \frac {x}{\tan {\left (\frac {x}{2} \right )} + 1} + \frac {2}{\tan {\left (\frac {x}{2} \right )} + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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