Optimal. Leaf size=11 \[ x+\frac {\cos (x)}{1+\sin (x)} \]
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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 = {2814, 2727}
\begin {gather*} x+\frac {\cos (x)}{\sin (x)+1} \end {gather*}
Antiderivative was successfully verified.
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Rule 2727
Rule 2814
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] Leaf count is larger than twice the leaf count of optimal. \(25\) vs. \(2(11)=22\).
time = 0.03, size = 25, normalized size = 2.27 \begin {gather*} x-\frac {2 \sin \left (\frac {x}{2}\right )}{\cos \left (\frac {x}{2}\right )+\sin \left (\frac {x}{2}\right )} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.03, size = 19, normalized size = 1.73
method | result | size |
risch | \(x +\frac {2}{{\mathrm e}^{i x}+i}\) | \(15\) |
default | \(2 \arctan \left (\tan \left (\frac {x}{2}\right )\right )+\frac {2}{1+\tan \left (\frac {x}{2}\right )}\) | \(19\) |
norman | \(\frac {x +x \tan \left (\frac {x}{2}\right )+x \left (\tan ^{2}\left (\frac {x}{2}\right )\right )+x \left (\tan ^{3}\left (\frac {x}{2}\right )\right )+2 \left (\tan ^{2}\left (\frac {x}{2}\right )\right )+2}{\left (1+\tan ^{2}\left (\frac {x}{2}\right )\right ) \left (1+\tan \left (\frac {x}{2}\right )\right )}\) | \(53\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 28 vs.
\(2 (11) = 22\).
time = 2.97, size = 28, normalized size = 2.55 \begin {gather*} \frac {2}{\frac {\sin \left (x\right )}{\cos \left (x\right ) + 1} + 1} + 2 \, \arctan \left (\frac {\sin \left (x\right )}{\cos \left (x\right ) + 1}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 24 vs.
\(2 (11) = 22\).
time = 0.49, size = 24, normalized size = 2.18 \begin {gather*} \frac {{\left (x + 1\right )} \cos \left (x\right ) + {\left (x - 1\right )} \sin \left (x\right ) + x + 1}{\cos \left (x\right ) + \sin \left (x\right ) + 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 29 vs.
\(2 (8) = 16\).
time = 0.23, size = 29, normalized size = 2.64 \begin {gather*} \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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.30, size = 12, normalized size = 1.09 \begin {gather*} x + \frac {2}{\tan \left (\frac {1}{2} \, x\right ) + 1} \end {gather*}
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 \begin {gather*} x+\frac {2}{\mathrm {tan}\left (\frac {x}{2}\right )+1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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