Optimal. Leaf size=12 \[ -\frac {\log (a+b \cos (x))}{b} \]
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Rubi [A] time = 0.02, antiderivative size = 12, 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 = {2668, 31} \[ -\frac {\log (a+b \cos (x))}{b} \]
Antiderivative was successfully verified.
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Rule 31
Rule 2668
Rubi steps
\begin {align*} \int \frac {\sin (x)}{a+b \cos (x)} \, dx &=-\frac {\operatorname {Subst}\left (\int \frac {1}{a+x} \, dx,x,b \cos (x)\right )}{b}\\ &=-\frac {\log (a+b \cos (x))}{b}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 12, normalized size = 1.00 \[ -\frac {\log (a+b \cos (x))}{b} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.73, size = 15, normalized size = 1.25 \[ -\frac {\log \left (-b \cos \relax (x) - a\right )}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.13, size = 13, normalized size = 1.08 \[ -\frac {\log \left ({\left | b \cos \relax (x) + a \right |}\right )}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 13, normalized size = 1.08 \[ -\frac {\ln \left (a +b \cos \relax (x )\right )}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.31, size = 12, normalized size = 1.00 \[ -\frac {\log \left (b \cos \relax (x) + a\right )}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.06, size = 12, normalized size = 1.00 \[ -\frac {\ln \left (a+b\,\cos \relax (x)\right )}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.33, size = 17, normalized size = 1.42 \[ \begin {cases} - \frac {\log {\left (\frac {a}{b} + \cos {\relax (x )} \right )}}{b} & \text {for}\: b \neq 0 \\- \frac {\cos {\relax (x )}}{a} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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