Optimal. Leaf size=37 \[ \frac{1}{d \left (a^2 \sin (c+d x)+a^2\right )}+\frac{\log (\sin (c+d x)+1)}{a^2 d} \]
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Rubi [A] time = 0.048023, antiderivative size = 37, 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} \[ \frac{1}{d \left (a^2 \sin (c+d x)+a^2\right )}+\frac{\log (\sin (c+d x)+1)}{a^2 d} \]
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))^2} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{x}{a (a+x)^2} \, dx,x,a \sin (c+d x)\right )}{a d}\\ &=\frac{\operatorname{Subst}\left (\int \frac{x}{(a+x)^2} \, dx,x,a \sin (c+d x)\right )}{a^2 d}\\ &=\frac{\operatorname{Subst}\left (\int \left (-\frac{a}{(a+x)^2}+\frac{1}{a+x}\right ) \, dx,x,a \sin (c+d x)\right )}{a^2 d}\\ &=\frac{\log (1+\sin (c+d x))}{a^2 d}+\frac{1}{d \left (a^2+a^2 \sin (c+d x)\right )}\\ \end{align*}
Mathematica [A] time = 0.0278844, size = 27, normalized size = 0.73 \[ \frac{\frac{1}{\sin (c+d x)+1}+\log (\sin (c+d x)+1)}{a^2 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.03, size = 35, normalized size = 1. \begin{align*}{\frac{1}{d{a}^{2} \left ( 1+\sin \left ( dx+c \right ) \right ) }}+{\frac{\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 = 1.40492, size = 46, normalized size = 1.24 \begin{align*} \frac{\frac{1}{a^{2} \sin \left (d x + c\right ) + a^{2}} + \frac{\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 = 1.43138, size = 104, normalized size = 2.81 \begin{align*} \frac{{\left (\sin \left (d x + c\right ) + 1\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) + 1}{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 = 1.02931, size = 95, normalized size = 2.57 \begin{align*} \begin{cases} \frac{\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{\log{\left (\sin{\left (c + d x \right )} + 1 \right )}}{a^{2} d \sin{\left (c + d x \right )} + a^{2} d} + \frac{1}{a^{2} d \sin{\left (c + d x \right )} + a^{2} d} & \text{for}\: d \neq 0 \\\frac{x \sin{\left (c \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 = 1.28639, size = 76, normalized size = 2.05 \begin{align*} -\frac{\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}}{a d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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