Optimal. Leaf size=16 \[ \frac{\log (\sin (c+d x)+1)}{a d} \]
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Rubi [A] time = 0.0255872, antiderivative size = 16, 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 = {2667, 31} \[ \frac{\log (\sin (c+d x)+1)}{a d} \]
Antiderivative was successfully verified.
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Rule 2667
Rule 31
Rubi steps
\begin{align*} \int \frac{\cos (c+d x)}{a+a \sin (c+d x)} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{1}{a+x} \, dx,x,a \sin (c+d x)\right )}{a d}\\ &=\frac{\log (1+\sin (c+d x))}{a d}\\ \end{align*}
Mathematica [A] time = 0.0107324, size = 16, normalized size = 1. \[ \frac{\log (\sin (c+d x)+1)}{a d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.012, size = 19, normalized size = 1.2 \begin{align*}{\frac{\ln \left ( a+a\sin \left ( dx+c \right ) \right ) }{da}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.932045, size = 24, normalized size = 1.5 \begin{align*} \frac{\log \left (a \sin \left (d x + c\right ) + a\right )}{a d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.69334, size = 39, normalized size = 2.44 \begin{align*} \frac{\log \left (\sin \left (d x + c\right ) + 1\right )}{a d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.545384, size = 24, normalized size = 1.5 \begin{align*} \begin{cases} \frac{\log{\left (\sin{\left (c + d x \right )} + 1 \right )}}{a d} & \text{for}\: d \neq 0 \\\frac{x \cos{\left (c \right )}}{a \sin{\left (c \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 = 1.33604, size = 26, normalized size = 1.62 \begin{align*} \frac{\log \left ({\left | a \sin \left (d x + c\right ) + a \right |}\right )}{a d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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