Optimal. Leaf size=22 \[ -\frac{1}{2 a d (a \sin (c+d x)+a)^2} \]
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Rubi [A] time = 0.0246114, antiderivative size = 22, 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} \[ -\frac{1}{2 a d (a \sin (c+d x)+a)^2} \]
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))^3} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{1}{(a+x)^3} \, dx,x,a \sin (c+d x)\right )}{a d}\\ &=-\frac{1}{2 a d (a+a \sin (c+d x))^2}\\ \end{align*}
Mathematica [A] time = 0.0569452, size = 33, normalized size = 1.5 \[ -\frac{1}{2 a^3 d \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )^4} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.014, size = 21, normalized size = 1. \begin{align*} -{\frac{1}{2\,da \left ( a+a\sin \left ( dx+c \right ) \right ) ^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.94394, size = 27, normalized size = 1.23 \begin{align*} -\frac{1}{2 \,{\left (a \sin \left (d x + c\right ) + a\right )}^{2} a d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.83076, size = 82, normalized size = 3.73 \begin{align*} \frac{1}{2 \,{\left (a^{3} d \cos \left (d x + c\right )^{2} - 2 \, a^{3} d \sin \left (d x + c\right ) - 2 \, a^{3} d\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.66904, size = 51, normalized size = 2.32 \begin{align*} \begin{cases} - \frac{1}{2 a^{3} d \sin ^{2}{\left (c + d x \right )} + 4 a^{3} d \sin{\left (c + d x \right )} + 2 a^{3} d} & \text{for}\: d \neq 0 \\\frac{x \cos{\left (c \right )}}{\left (a \sin{\left (c \right )} + a\right )^{3}} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.19321, size = 27, normalized size = 1.23 \begin{align*} -\frac{1}{2 \,{\left (a \sin \left (d x + c\right ) + a\right )}^{2} a d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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