Optimal. Leaf size=12 \[ -\frac{\log (a+b \cos (x))}{b} \]
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Rubi [A] time = 0.0223243, antiderivative size = 12, normalized size of antiderivative = 1., 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 2668
Rule 31
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*}
Mathematica [A] time = 0.0182928, size = 12, normalized size = 1. \[ -\frac{\log (a+b \cos (x))}{b} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.007, size = 13, normalized size = 1.1 \begin{align*} -{\frac{\ln \left ( a+b\cos \left ( x \right ) \right ) }{b}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.968939, size = 16, normalized size = 1.33 \begin{align*} -\frac{\log \left (b \cos \left (x\right ) + a\right )}{b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.06818, size = 31, normalized size = 2.58 \begin{align*} -\frac{\log \left (-b \cos \left (x\right ) - a\right )}{b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.386041, size = 17, normalized size = 1.42 \begin{align*} \begin{cases} - \frac{\log{\left (\frac{a}{b} + \cos{\left (x \right )} \right )}}{b} & \text{for}\: b \neq 0 \\- \frac{\cos{\left (x \right )}}{a} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.07803, size = 18, normalized size = 1.5 \begin{align*} -\frac{\log \left ({\left | b \cos \left (x\right ) + a \right |}\right )}{b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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