Optimal. Leaf size=22 \[ -\frac{2}{b d \sqrt{a+b \sin (c+d x)}} \]
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Rubi [A] time = 0.0377284, 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 = {2668, 32} \[ -\frac{2}{b d \sqrt{a+b \sin (c+d x)}} \]
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))^{3/2}} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{1}{(a+x)^{3/2}} \, dx,x,b \sin (c+d x)\right )}{b d}\\ &=-\frac{2}{b d \sqrt{a+b \sin (c+d x)}}\\ \end{align*}
Mathematica [A] time = 0.0140742, size = 22, normalized size = 1. \[ -\frac{2}{b d \sqrt{a+b \sin (c+d x)}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 21, normalized size = 1. \begin{align*} -2\,{\frac{1}{bd\sqrt{a+b\sin \left ( dx+c \right ) }}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.945956, size = 27, normalized size = 1.23 \begin{align*} -\frac{2}{\sqrt{b \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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Fricas [A] time = 1.93253, size = 78, normalized size = 3.55 \begin{align*} -\frac{2 \, \sqrt{b \sin \left (d x + c\right ) + a}}{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 = 3.5954, size = 56, normalized size = 2.55 \begin{align*} \begin{cases} \frac{x \cos{\left (c \right )}}{a^{\frac{3}{2}}} & \text{for}\: b = 0 \wedge d = 0 \\\frac{x \cos{\left (c \right )}}{\left (a + b \sin{\left (c \right )}\right )^{\frac{3}{2}}} & \text{for}\: d = 0 \\\frac{\sin{\left (c + d x \right )}}{a^{\frac{3}{2}} d} & \text{for}\: b = 0 \\- \frac{2}{b d \sqrt{a + b \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 = 1.08248, size = 27, normalized size = 1.23 \begin{align*} -\frac{2}{\sqrt{b \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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