Optimal. Leaf size=11 \[ \frac{\log (a+b \sin (x))}{b} \]
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Rubi [A] time = 0.0218965, antiderivative size = 11, 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} \[ \frac{\log (a+b \sin (x))}{b} \]
Antiderivative was successfully verified.
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Rule 2668
Rule 31
Rubi steps
\begin{align*} \int \frac{\cos (x)}{a+b \sin (x)} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{1}{a+x} \, dx,x,b \sin (x)\right )}{b}\\ &=\frac{\log (a+b \sin (x))}{b}\\ \end{align*}
Mathematica [A] time = 0.0062776, size = 11, normalized size = 1. \[ \frac{\log (a+b \sin (x))}{b} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.014, size = 12, normalized size = 1.1 \begin{align*}{\frac{\ln \left ( a+b\sin \left ( x \right ) \right ) }{b}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.944746, size = 15, normalized size = 1.36 \begin{align*} \frac{\log \left (b \sin \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 = 2.07247, size = 28, normalized size = 2.55 \begin{align*} \frac{\log \left (b \sin \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 = 0.359504, size = 14, normalized size = 1.27 \begin{align*} \begin{cases} \frac{\log{\left (\frac{a}{b} + \sin{\left (x \right )} \right )}}{b} & \text{for}\: b \neq 0 \\\frac{\sin{\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 = 1.09398, size = 16, normalized size = 1.45 \begin{align*} \frac{\log \left ({\left | b \sin \left (x\right ) + a \right |}\right )}{b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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