Optimal. Leaf size=22 \[ -\frac{2}{a d \sqrt{a \sin (c+d x)+a}} \]
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Rubi [A] time = 0.0335983, antiderivative size = 22, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095, Rules used = {2667, 32} \[ -\frac{2}{a d \sqrt{a \sin (c+d x)+a}} \]
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))^{3/2}} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{1}{(a+x)^{3/2}} \, dx,x,a \sin (c+d x)\right )}{a d}\\ &=-\frac{2}{a d \sqrt{a+a \sin (c+d x)}}\\ \end{align*}
Mathematica [A] time = 0.0299117, size = 22, normalized size = 1. \[ -\frac{2}{a d \sqrt{a \sin (c+d x)+a}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 21, normalized size = 1. \begin{align*} -2\,{\frac{1}{da\sqrt{a+a\sin \left ( dx+c \right ) }}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.948987, size = 27, normalized size = 1.23 \begin{align*} -\frac{2}{\sqrt{a \sin \left (d x + c\right ) + a} a d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.23214, size = 78, normalized size = 3.55 \begin{align*} -\frac{2 \, \sqrt{a \sin \left (d x + c\right ) + a}}{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 = 2.86371, size = 46, normalized size = 2.09 \begin{align*} \begin{cases} \text{NaN} & \text{for}\: c = \frac{3 \pi }{2} \wedge d = 0 \\\frac{x \cos{\left (c \right )}}{\left (a \sin{\left (c \right )} + a\right )^{\frac{3}{2}}} & \text{for}\: d = 0 \\\text{NaN} & \text{for}\: c = - d x + \frac{3 \pi }{2} \\- \frac{2}{a d \sqrt{a \sin{\left (c + d x \right )} + a}} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.13427, size = 27, normalized size = 1.23 \begin{align*} -\frac{2}{\sqrt{a \sin \left (d x + c\right ) + a} a d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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