Optimal. Leaf size=31 \[ \frac{\sin (c+d x)}{a d}-\frac{\log (\sin (c+d x)+1)}{a d} \]
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Rubi [A] time = 0.0460874, antiderivative size = 31, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.12, Rules used = {2833, 12, 43} \[ \frac{\sin (c+d x)}{a d}-\frac{\log (\sin (c+d x)+1)}{a d} \]
Antiderivative was successfully verified.
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Rule 2833
Rule 12
Rule 43
Rubi steps
\begin{align*} \int \frac{\cos (c+d x) \sin (c+d x)}{a+a \sin (c+d x)} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{x}{a (a+x)} \, dx,x,a \sin (c+d x)\right )}{a d}\\ &=\frac{\operatorname{Subst}\left (\int \frac{x}{a+x} \, dx,x,a \sin (c+d x)\right )}{a^2 d}\\ &=\frac{\operatorname{Subst}\left (\int \left (1-\frac{a}{a+x}\right ) \, dx,x,a \sin (c+d x)\right )}{a^2 d}\\ &=-\frac{\log (1+\sin (c+d x))}{a d}+\frac{\sin (c+d x)}{a d}\\ \end{align*}
Mathematica [A] time = 0.0202694, size = 25, normalized size = 0.81 \[ \frac{\sin (c+d x)-\log (\sin (c+d x)+1)}{a d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.02, size = 32, normalized size = 1. \begin{align*} -{\frac{\ln \left ( 1+\sin \left ( dx+c \right ) \right ) }{da}}+{\frac{\sin \left ( dx+c \right ) }{da}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.07792, size = 41, normalized size = 1.32 \begin{align*} -\frac{\frac{\log \left (\sin \left (d x + c\right ) + 1\right )}{a} - \frac{\sin \left (d x + c\right )}{a}}{d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.36042, size = 63, normalized size = 2.03 \begin{align*} -\frac{\log \left (\sin \left (d x + c\right ) + 1\right ) - \sin \left (d x + c\right )}{a d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.967933, size = 37, normalized size = 1.19 \begin{align*} \begin{cases} - \frac{\log{\left (\sin{\left (c + d x \right )} + 1 \right )}}{a d} + \frac{\sin{\left (c + d x \right )}}{a d} & \text{for}\: d \neq 0 \\\frac{x \sin{\left (c \right )} \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 = 1.32545, size = 42, normalized size = 1.35 \begin{align*} -\frac{\frac{\log \left ({\left | \sin \left (d x + c\right ) + 1 \right |}\right )}{a} - \frac{\sin \left (d x + c\right )}{a}}{d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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