Optimal. Leaf size=17 \[ \frac {x}{a}+\frac {\cos (x)}{a \sin (x)+a} \]
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Rubi [A] time = 0.03, antiderivative size = 17, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {2735, 2648} \[ \frac {x}{a}+\frac {\cos (x)}{a \sin (x)+a} \]
Antiderivative was successfully verified.
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Rule 2648
Rule 2735
Rubi steps
\begin {align*} \int \frac {\sin (x)}{a+a \sin (x)} \, dx &=\frac {x}{a}-\int \frac {1}{a+a \sin (x)} \, dx\\ &=\frac {x}{a}+\frac {\cos (x)}{a+a \sin (x)}\\ \end {align*}
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Mathematica [B] time = 0.04, size = 42, normalized size = 2.47 \[ \frac {\left (\sin \left (\frac {x}{2}\right )+\cos \left (\frac {x}{2}\right )\right ) \left ((x-2) \sin \left (\frac {x}{2}\right )+x \cos \left (\frac {x}{2}\right )\right )}{a (\sin (x)+1)} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.46, size = 28, normalized size = 1.65 \[ \frac {{\left (x + 1\right )} \cos \relax (x) + {\left (x - 1\right )} \sin \relax (x) + x + 1}{a \cos \relax (x) + a \sin \relax (x) + a} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.92, size = 19, normalized size = 1.12 \[ \frac {x}{a} + \frac {2}{a {\left (\tan \left (\frac {1}{2} \, x\right ) + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.07, size = 25, normalized size = 1.47 \[ \frac {2 \arctan \left (\tan \left (\frac {x}{2}\right )\right )}{a}+\frac {2}{a \left (\tan \left (\frac {x}{2}\right )+1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.82, size = 32, normalized size = 1.88 \[ \frac {2 \, \arctan \left (\frac {\sin \relax (x)}{\cos \relax (x) + 1}\right )}{a} + \frac {2}{a + \frac {a \sin \relax (x)}{\cos \relax (x) + 1}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 6.54, size = 19, normalized size = 1.12 \[ \frac {2}{a\,\left (\mathrm {tan}\left (\frac {x}{2}\right )+1\right )}+\frac {x}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.84, size = 34, normalized size = 2.00 \[ \frac {x \tan {\left (\frac {x}{2} \right )}}{a \tan {\left (\frac {x}{2} \right )} + a} + \frac {x}{a \tan {\left (\frac {x}{2} \right )} + a} + \frac {2}{a \tan {\left (\frac {x}{2} \right )} + a} \]
Verification of antiderivative is not currently implemented for this CAS.
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