Optimal. Leaf size=48 \[ \frac{B \log (a+b \sin (c+d x))}{b^2 d}-\frac{A b-a B}{b^2 d (a+b \sin (c+d x))} \]
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Rubi [A] time = 0.0767375, antiderivative size = 48, 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} \[ \frac{B \log (a+b \sin (c+d x))}{b^2 d}-\frac{A b-a B}{b^2 d (a+b \sin (c+d x))} \]
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))^2} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{A+\frac{B x}{b}}{(a+x)^2} \, dx,x,b \sin (c+d x)\right )}{b d}\\ &=\frac{\operatorname{Subst}\left (\int \left (\frac{A b-a B}{b (a+x)^2}+\frac{B}{b (a+x)}\right ) \, dx,x,b \sin (c+d x)\right )}{b d}\\ &=\frac{B \log (a+b \sin (c+d x))}{b^2 d}-\frac{A b-a B}{b^2 d (a+b \sin (c+d x))}\\ \end{align*}
Mathematica [A] time = 0.0868795, size = 42, normalized size = 0.88 \[ \frac{\frac{a B-A b}{a+b \sin (c+d x)}+B \log (a+b \sin (c+d x))}{b^2 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.056, size = 63, normalized size = 1.3 \begin{align*}{\frac{B\ln \left ( a+b\sin \left ( dx+c \right ) \right ) }{{b}^{2}d}}-{\frac{A}{bd \left ( a+b\sin \left ( dx+c \right ) \right ) }}+{\frac{aB}{{b}^{2}d \left ( a+b\sin \left ( dx+c \right ) \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.992207, size = 65, normalized size = 1.35 \begin{align*} \frac{\frac{B a - A b}{b^{3} \sin \left (d x + c\right ) + a b^{2}} + \frac{B \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 = 1.42007, size = 128, normalized size = 2.67 \begin{align*} \frac{B a - A b +{\left (B b \sin \left (d x + c\right ) + B a\right )} \log \left (b \sin \left (d x + c\right ) + a\right )}{b^{3} d \sin \left (d x + c\right ) + a b^{2} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.27253, size = 178, normalized size = 3.71 \begin{align*} \begin{cases} \frac{x \left (A + B \sin{\left (c \right )}\right ) \cos{\left (c \right )}}{a^{2}} & \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^{2}} & \text{for}\: b = 0 \\\frac{x \left (A + B \sin{\left (c \right )}\right ) \cos{\left (c \right )}}{\left (a + b \sin{\left (c \right )}\right )^{2}} & \text{for}\: d = 0 \\- \frac{A b}{a b^{2} d + b^{3} d \sin{\left (c + d x \right )}} + \frac{B a \log{\left (\frac{a}{b} + \sin{\left (c + d x \right )} \right )}}{a b^{2} d + b^{3} d \sin{\left (c + d x \right )}} + \frac{B a}{a b^{2} d + b^{3} d \sin{\left (c + d x \right )}} + \frac{B b \log{\left (\frac{a}{b} + \sin{\left (c + d x \right )} \right )} \sin{\left (c + d x \right )}}{a b^{2} d + b^{3} d \sin{\left (c + d x \right )}} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.35983, size = 108, normalized size = 2.25 \begin{align*} -\frac{\frac{B{\left (\frac{\log \left (\frac{{\left | b \sin \left (d x + c\right ) + a \right |}}{{\left (b \sin \left (d x + c\right ) + a\right )}^{2}{\left | b \right |}}\right )}{b} - \frac{a}{{\left (b \sin \left (d x + c\right ) + a\right )} b}\right )}}{b} + \frac{A}{{\left (b \sin \left (d x + c\right ) + a\right )} b}}{d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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