Optimal. Leaf size=18 \[ \frac{\log (a+b \sin (c+d x))}{b d} \]
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Rubi [A] time = 0.0267588, antiderivative size = 18, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105, Rules used = {2668, 31} \[ \frac{\log (a+b \sin (c+d x))}{b d} \]
Antiderivative was successfully verified.
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Rule 2668
Rule 31
Rubi steps
\begin{align*} \int \frac{\cos (c+d x)}{a+b \sin (c+d x)} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{1}{a+x} \, dx,x,b \sin (c+d x)\right )}{b d}\\ &=\frac{\log (a+b \sin (c+d x))}{b d}\\ \end{align*}
Mathematica [A] time = 0.0067112, size = 18, normalized size = 1. \[ \frac{\log (a+b \sin (c+d x))}{b d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.013, size = 19, normalized size = 1.1 \begin{align*}{\frac{\ln \left ( a+b\sin \left ( dx+c \right ) \right ) }{bd}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.932619, size = 24, normalized size = 1.33 \begin{align*} \frac{\log \left (b \sin \left (d x + c\right ) + a\right )}{b d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.37353, size = 42, normalized size = 2.33 \begin{align*} \frac{\log \left (b \sin \left (d x + c\right ) + a\right )}{b d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.618084, size = 41, normalized size = 2.28 \begin{align*} \begin{cases} \frac{x \cos{\left (c \right )}}{a} & \text{for}\: b = 0 \wedge d = 0 \\\frac{x \cos{\left (c \right )}}{a + b \sin{\left (c \right )}} & \text{for}\: d = 0 \\\frac{\sin{\left (c + d x \right )}}{a d} & \text{for}\: b = 0 \\\frac{\log{\left (\frac{a}{b} + \sin{\left (c + d x \right )} \right )}}{b d} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.10082, size = 26, normalized size = 1.44 \begin{align*} \frac{\log \left ({\left | b \sin \left (d x + c\right ) + a \right |}\right )}{b d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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