Optimal. Leaf size=10 \[ -\frac {\log (\cos (x)+1)}{a} \]
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Rubi [A] time = 0.02, antiderivative size = 10, 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 = {2667, 31} \[ -\frac {\log (\cos (x)+1)}{a} \]
Antiderivative was successfully verified.
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Rule 31
Rule 2667
Rubi steps
\begin {align*} \int \frac {\sin (x)}{a+a \cos (x)} \, dx &=-\frac {\operatorname {Subst}\left (\int \frac {1}{a+x} \, dx,x,a \cos (x)\right )}{a}\\ &=-\frac {\log (1+\cos (x))}{a}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 12, normalized size = 1.20 \[ -\frac {2 \log \left (\cos \left (\frac {x}{2}\right )\right )}{a} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.76, size = 12, normalized size = 1.20 \[ -\frac {\log \left (\frac {1}{2} \, \cos \relax (x) + \frac {1}{2}\right )}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.37, size = 10, normalized size = 1.00 \[ -\frac {\log \left (\cos \relax (x) + 1\right )}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 13, normalized size = 1.30 \[ -\frac {\ln \left (a +a \cos \relax (x )\right )}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.63, size = 12, normalized size = 1.20 \[ -\frac {\log \left (a \cos \relax (x) + a\right )}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.06, size = 10, normalized size = 1.00 \[ -\frac {\ln \left (\cos \relax (x)+1\right )}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.12, size = 8, normalized size = 0.80 \[ - \frac {\log {\left (\cos {\relax (x )} + 1 \right )}}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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