Optimal. Leaf size=41 \[ \frac{(A b-a B) \log (a+b \sin (c+d x))}{b^2 d}+\frac{B \sin (c+d x)}{b d} \]
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Rubi [A] time = 0.0701739, antiderivative size = 41, 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} \[ \frac{(A b-a B) \log (a+b \sin (c+d x))}{b^2 d}+\frac{B \sin (c+d x)}{b d} \]
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+b \sin (c+d x)} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{A+\frac{B x}{b}}{a+x} \, dx,x,b \sin (c+d x)\right )}{b d}\\ &=\frac{\operatorname{Subst}\left (\int \left (\frac{B}{b}+\frac{A b-a B}{b (a+x)}\right ) \, dx,x,b \sin (c+d x)\right )}{b d}\\ &=\frac{(A b-a B) \log (a+b \sin (c+d x))}{b^2 d}+\frac{B \sin (c+d x)}{b d}\\ \end{align*}
Mathematica [A] time = 0.0450554, size = 39, normalized size = 0.95 \[ \frac{\frac{(A b-a B) \log (a+b \sin (c+d x))}{b}+B \sin (c+d x)}{b d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.034, size = 56, normalized size = 1.4 \begin{align*}{\frac{B\sin \left ( dx+c \right ) }{bd}}+{\frac{\ln \left ( a+b\sin \left ( dx+c \right ) \right ) A}{bd}}-{\frac{\ln \left ( a+b\sin \left ( dx+c \right ) \right ) Ba}{{b}^{2}d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.971681, size = 54, normalized size = 1.32 \begin{align*} \frac{\frac{B \sin \left (d x + c\right )}{b} - \frac{{\left (B a - A b\right )} \log \left (b \sin \left (d x + c\right ) + a\right )}{b^{2}}}{d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.48445, size = 89, normalized size = 2.17 \begin{align*} \frac{B b \sin \left (d x + c\right ) -{\left (B a - A b\right )} \log \left (b \sin \left (d x + c\right ) + a\right )}{b^{2} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.818668, size = 104, normalized size = 2.54 \begin{align*} \begin{cases} \frac{x \left (A + B \sin{\left (c \right )}\right ) \cos{\left (c \right )}}{a} & \text{for}\: b = 0 \wedge d = 0 \\\frac{\frac{A \sin{\left (c + d x \right )}}{d} - \frac{B \cos ^{2}{\left (c + d x \right )}}{2 d}}{a} & \text{for}\: b = 0 \\\frac{x \left (A + B \sin{\left (c \right )}\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 d} - \frac{B a \log{\left (\frac{a}{b} + \sin{\left (c + d x \right )} \right )}}{b^{2} d} + \frac{B \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 = 1.22548, size = 55, normalized size = 1.34 \begin{align*} \frac{\frac{B \sin \left (d x + c\right )}{b} - \frac{{\left (B a - A b\right )} \log \left ({\left | b \sin \left (d x + c\right ) + a \right |}\right )}{b^{2}}}{d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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