Optimal. Leaf size=24 \[ -\frac{2}{3 b d (a+b \sin (c+d x))^{3/2}} \]
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Rubi [A] time = 0.0422203, antiderivative size = 24, 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} \[ -\frac{2}{3 b d (a+b \sin (c+d x))^{3/2}} \]
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))^{5/2}} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{1}{(a+x)^{5/2}} \, dx,x,b \sin (c+d x)\right )}{b d}\\ &=-\frac{2}{3 b d (a+b \sin (c+d x))^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.0187543, size = 24, normalized size = 1. \[ -\frac{2}{3 b d (a+b \sin (c+d x))^{3/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 21, normalized size = 0.9 \begin{align*} -{\frac{2}{3\,bd} \left ( a+b\sin \left ( dx+c \right ) \right ) ^{-{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.943273, size = 27, normalized size = 1.12 \begin{align*} -\frac{2}{3 \,{\left (b \sin \left (d x + c\right ) + a\right )}^{\frac{3}{2}} b d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.26043, size = 130, normalized size = 5.42 \begin{align*} \frac{2 \, \sqrt{b \sin \left (d x + c\right ) + a}}{3 \,{\left (b^{3} d \cos \left (d x + c\right )^{2} - 2 \, a b^{2} d \sin \left (d x + c\right ) -{\left (a^{2} b + b^{3}\right )} d\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 29.4815, size = 87, normalized size = 3.62 \begin{align*} \begin{cases} \frac{x \cos{\left (c \right )}}{a^{\frac{5}{2}}} & \text{for}\: b = 0 \wedge d = 0 \\\frac{\sin{\left (c + d x \right )}}{a^{\frac{5}{2}} d} & \text{for}\: b = 0 \\\frac{x \cos{\left (c \right )}}{\left (a + b \sin{\left (c \right )}\right )^{\frac{5}{2}}} & \text{for}\: d = 0 \\- \frac{2}{3 a b d \sqrt{a + b \sin{\left (c + d x \right )}} + 3 b^{2} d \sqrt{a + b \sin{\left (c + d x \right )}} \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 = 1.09457, size = 27, normalized size = 1.12 \begin{align*} -\frac{2}{3 \,{\left (b \sin \left (d x + c\right ) + a\right )}^{\frac{3}{2}} b d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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