Optimal. Leaf size=13 \[ \frac {x}{a}-\frac {\sin (x)}{a} \]
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Rubi [A] time = 0.04, antiderivative size = 13, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {2682, 8} \[ \frac {x}{a}-\frac {\sin (x)}{a} \]
Antiderivative was successfully verified.
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Rule 8
Rule 2682
Rubi steps
\begin {align*} \int \frac {\sin ^2(x)}{a+a \cos (x)} \, dx &=-\frac {\sin (x)}{a}+\frac {\int 1 \, dx}{a}\\ &=\frac {x}{a}-\frac {\sin (x)}{a}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 17, normalized size = 1.31 \[ \frac {2 \left (\frac {x}{2}-\frac {\sin (x)}{2}\right )}{a} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.85, size = 10, normalized size = 0.77 \[ \frac {x - \sin \relax (x)}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.51, size = 25, normalized size = 1.92 \[ \frac {x}{a} - \frac {2 \, \tan \left (\frac {1}{2} \, x\right )}{{\left (\tan \left (\frac {1}{2} \, x\right )^{2} + 1\right )} a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.04, size = 31, normalized size = 2.38 \[ -\frac {2 \tan \left (\frac {x}{2}\right )}{a \left (\tan ^{2}\left (\frac {x}{2}\right )+1\right )}+\frac {2 \arctan \left (\tan \left (\frac {x}{2}\right )\right )}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.84, size = 42, normalized size = 3.23 \[ \frac {2 \, \arctan \left (\frac {\sin \relax (x)}{\cos \relax (x) + 1}\right )}{a} - \frac {2 \, \sin \relax (x)}{{\left (a + \frac {a \sin \relax (x)^{2}}{{\left (\cos \relax (x) + 1\right )}^{2}}\right )} {\left (\cos \relax (x) + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.29, size = 10, normalized size = 0.77 \[ \frac {x-\sin \relax (x)}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.37, size = 46, normalized size = 3.54 \[ \frac {x \tan ^{2}{\left (\frac {x}{2} \right )}}{a \tan ^{2}{\left (\frac {x}{2} \right )} + a} + \frac {x}{a \tan ^{2}{\left (\frac {x}{2} \right )} + a} - \frac {2 \tan {\left (\frac {x}{2} \right )}}{a \tan ^{2}{\left (\frac {x}{2} \right )} + a} \]
Verification of antiderivative is not currently implemented for this CAS.
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