Optimal. Leaf size=44 \[ \frac{B \log (\sin (c+d x)+1)}{a^2 d}-\frac{A-B}{d \left (a^2 \sin (c+d x)+a^2\right )} \]
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Rubi [A] time = 0.0678509, antiderivative size = 44, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.069, Rules used = {2833, 43} \[ \frac{B \log (\sin (c+d x)+1)}{a^2 d}-\frac{A-B}{d \left (a^2 \sin (c+d x)+a^2\right )} \]
Antiderivative was successfully verified.
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Rule 2833
Rule 43
Rubi steps
\begin{align*} \int \frac{\cos (c+d x) (A+B \sin (c+d x))}{(a+a \sin (c+d x))^2} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{A+\frac{B x}{a}}{(a+x)^2} \, dx,x,a \sin (c+d x)\right )}{a d}\\ &=\frac{\operatorname{Subst}\left (\int \left (\frac{A-B}{(a+x)^2}+\frac{B}{a (a+x)}\right ) \, dx,x,a \sin (c+d x)\right )}{a d}\\ &=\frac{B \log (1+\sin (c+d x))}{a^2 d}-\frac{A-B}{d \left (a^2+a^2 \sin (c+d x)\right )}\\ \end{align*}
Mathematica [A] time = 0.0733169, size = 41, normalized size = 0.93 \[ \frac{\frac{B \log (\sin (c+d x)+1)}{a}-\frac{A-B}{a \sin (c+d x)+a}}{a d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.048, size = 56, normalized size = 1.3 \begin{align*} -{\frac{A}{d{a}^{2} \left ( 1+\sin \left ( dx+c \right ) \right ) }}+{\frac{B}{d{a}^{2} \left ( 1+\sin \left ( dx+c \right ) \right ) }}+{\frac{B\ln \left ( 1+\sin \left ( dx+c \right ) \right ) }{d{a}^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.02342, size = 58, normalized size = 1.32 \begin{align*} -\frac{\frac{A - B}{a^{2} \sin \left (d x + c\right ) + a^{2}} - \frac{B \log \left (\sin \left (d x + c\right ) + 1\right )}{a^{2}}}{d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.4452, size = 112, normalized size = 2.55 \begin{align*} \frac{{\left (B \sin \left (d x + c\right ) + B\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) - A + B}{a^{2} d \sin \left (d x + c\right ) + a^{2} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.0167, size = 121, normalized size = 2.75 \begin{align*} \begin{cases} - \frac{A}{a^{2} d \sin{\left (c + d x \right )} + a^{2} d} + \frac{B \log{\left (\sin{\left (c + d x \right )} + 1 \right )} \sin{\left (c + d x \right )}}{a^{2} d \sin{\left (c + d x \right )} + a^{2} d} + \frac{B \log{\left (\sin{\left (c + d x \right )} + 1 \right )}}{a^{2} d \sin{\left (c + d x \right )} + a^{2} d} + \frac{B}{a^{2} d \sin{\left (c + d x \right )} + a^{2} d} & \text{for}\: d \neq 0 \\\frac{x \left (A + B \sin{\left (c \right )}\right ) \cos{\left (c \right )}}{\left (a \sin{\left (c \right )} + a\right )^{2}} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.33329, size = 103, normalized size = 2.34 \begin{align*} -\frac{\frac{B{\left (\frac{\log \left (\frac{{\left | a \sin \left (d x + c\right ) + a \right |}}{{\left (a \sin \left (d x + c\right ) + a\right )}^{2}{\left | a \right |}}\right )}{a} - \frac{1}{a \sin \left (d x + c\right ) + a}\right )}}{a} + \frac{A}{{\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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