Optimal. Leaf size=34 \[ \frac{\sin (c+d x)}{b d}-\frac{a \log (a+b \sin (c+d x))}{b^2 d} \]
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Rubi [A] time = 0.050181, antiderivative size = 34, 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} \[ \frac{\sin (c+d x)}{b d}-\frac{a \log (a+b \sin (c+d x))}{b^2 d} \]
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+b \sin (c+d x)} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{x}{b (a+x)} \, dx,x,b \sin (c+d x)\right )}{b d}\\ &=\frac{\operatorname{Subst}\left (\int \frac{x}{a+x} \, dx,x,b \sin (c+d x)\right )}{b^2 d}\\ &=\frac{\operatorname{Subst}\left (\int \left (1-\frac{a}{a+x}\right ) \, dx,x,b \sin (c+d x)\right )}{b^2 d}\\ &=-\frac{a \log (a+b \sin (c+d x))}{b^2 d}+\frac{\sin (c+d x)}{b d}\\ \end{align*}
Mathematica [A] time = 0.0229036, size = 33, normalized size = 0.97 \[ -\frac{\frac{a \log (a+b \sin (c+d x))}{b^2}-\frac{\sin (c+d x)}{b}}{d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.026, size = 35, normalized size = 1. \begin{align*} -{\frac{a\ln \left ( a+b\sin \left ( dx+c \right ) \right ) }{d{b}^{2}}}+{\frac{\sin \left ( dx+c \right ) }{bd}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.988223, size = 45, normalized size = 1.32 \begin{align*} -\frac{\frac{a \log \left (b \sin \left (d x + c\right ) + a\right )}{b^{2}} - \frac{\sin \left (d x + c\right )}{b}}{d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.52505, size = 74, normalized size = 2.18 \begin{align*} -\frac{a \log \left (b \sin \left (d x + c\right ) + a\right ) - b \sin \left (d x + c\right )}{b^{2} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.869154, size = 66, normalized size = 1.94 \begin{align*} \begin{cases} \frac{x \sin{\left (c \right )} \cos{\left (c \right )}}{a} & \text{for}\: b = 0 \wedge d = 0 \\- \frac{\cos ^{2}{\left (c + d x \right )}}{2 a d} & \text{for}\: b = 0 \\\frac{x \sin{\left (c \right )} \cos{\left (c \right )}}{a + b \sin{\left (c \right )}} & \text{for}\: d = 0 \\- \frac{a \log{\left (\frac{a}{b} + \sin{\left (c + d x \right )} \right )}}{b^{2} d} + \frac{\sin{\left (c + d x \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 = 1.19328, size = 46, normalized size = 1.35 \begin{align*} -\frac{\frac{a \log \left ({\left | b \sin \left (d x + c\right ) + a \right |}\right )}{b^{2}} - \frac{\sin \left (d x + c\right )}{b}}{d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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