Optimal. Leaf size=21 \[ -\frac{1}{d \left (a^2 \sin (c+d x)+a^2\right )} \]
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Rubi [A] time = 0.0261668, antiderivative size = 21, 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 = {2667, 32} \[ -\frac{1}{d \left (a^2 \sin (c+d x)+a^2\right )} \]
Antiderivative was successfully verified.
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Rule 2667
Rule 32
Rubi steps
\begin{align*} \int \frac{\cos (c+d x)}{(a+a \sin (c+d x))^2} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{1}{(a+x)^2} \, dx,x,a \sin (c+d x)\right )}{a d}\\ &=-\frac{1}{d \left (a^2+a^2 \sin (c+d x)\right )}\\ \end{align*}
Mathematica [A] time = 0.0704079, size = 31, normalized size = 1.48 \[ -\frac{1}{a^2 d \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.015, size = 21, normalized size = 1. \begin{align*} -{\frac{1}{d \left ( a+a\sin \left ( dx+c \right ) \right ) a}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.960756, size = 27, normalized size = 1.29 \begin{align*} -\frac{1}{{\left (a \sin \left (d x + c\right ) + a\right )} a d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.85078, size = 45, normalized size = 2.14 \begin{align*} -\frac{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 = 0.990904, size = 32, normalized size = 1.52 \begin{align*} \begin{cases} - \frac{1}{a^{2} d \sin{\left (c + d x \right )} + a^{2} d} & \text{for}\: d \neq 0 \\\frac{x \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 = 1.15113, size = 27, normalized size = 1.29 \begin{align*} -\frac{1}{{\left (a \sin \left (d x + c\right ) + a\right )} a d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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