Optimal. Leaf size=20 \[ -\frac{1}{b d (a+b \sin (c+d x))} \]
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Rubi [A] time = 0.0266825, antiderivative size = 20, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105, Rules used = {2668, 32} \[ -\frac{1}{b d (a+b \sin (c+d x))} \]
Antiderivative was successfully verified.
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Rule 2668
Rule 32
Rubi steps
\begin{align*} \int \frac{\cos (c+d x)}{(a+b \sin (c+d x))^2} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{1}{(a+x)^2} \, dx,x,b \sin (c+d x)\right )}{b d}\\ &=-\frac{1}{b d (a+b \sin (c+d x))}\\ \end{align*}
Mathematica [A] time = 0.023615, size = 20, normalized size = 1. \[ -\frac{1}{b d (a+b \sin (c+d x))} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.023, size = 21, normalized size = 1.1 \begin{align*} -{\frac{1}{bd \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 = 0.932654, size = 27, normalized size = 1.35 \begin{align*} -\frac{1}{{\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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Fricas [A] time = 2.37971, size = 45, normalized size = 2.25 \begin{align*} -\frac{1}{b^{2} d \sin \left (d x + c\right ) + a b d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.14095, size = 51, normalized size = 2.55 \begin{align*} \begin{cases} \frac{x \cos{\left (c \right )}}{a^{2}} & \text{for}\: b = 0 \wedge d = 0 \\\frac{\sin{\left (c + d x \right )}}{a^{2} d} & \text{for}\: b = 0 \\\frac{x \cos{\left (c \right )}}{\left (a + b \sin{\left (c \right )}\right )^{2}} & \text{for}\: d = 0 \\- \frac{1}{a b d + b^{2} 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 = 1.10155, size = 27, normalized size = 1.35 \begin{align*} -\frac{1}{{\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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