Optimal. Leaf size=24 \[ \frac{2 (a+b \sin (c+d x))^{3/2}}{3 b d} \]
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Rubi [A] time = 0.0357776, 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} \[ \frac{2 (a+b \sin (c+d x))^{3/2}}{3 b d} \]
Antiderivative was successfully verified.
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Rule 2668
Rule 32
Rubi steps
\begin{align*} \int \cos (c+d x) \sqrt{a+b \sin (c+d x)} \, dx &=\frac{\operatorname{Subst}\left (\int \sqrt{a+x} \, dx,x,b \sin (c+d x)\right )}{b d}\\ &=\frac{2 (a+b \sin (c+d x))^{3/2}}{3 b d}\\ \end{align*}
Mathematica [A] time = 0.0158773, size = 24, normalized size = 1. \[ \frac{2 (a+b \sin (c+d x))^{3/2}}{3 b d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.007, 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 = 0.942176, size = 27, normalized size = 1.12 \begin{align*} \frac{2 \,{\left (b \sin \left (d x + c\right ) + a\right )}^{\frac{3}{2}}}{3 \, b d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.75918, size = 51, normalized size = 2.12 \begin{align*} \frac{2 \,{\left (b \sin \left (d x + c\right ) + a\right )}^{\frac{3}{2}}}{3 \, b d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.517719, size = 83, normalized size = 3.46 \begin{align*} \begin{cases} \sqrt{a} x \cos{\left (c \right )} & \text{for}\: b = 0 \wedge d = 0 \\x \sqrt{a + b \sin{\left (c \right )}} \cos{\left (c \right )} & \text{for}\: d = 0 \\\frac{\sqrt{a} \sin{\left (c + d x \right )}}{d} & \text{for}\: b = 0 \\\frac{2 a \sqrt{a + b \sin{\left (c + d x \right )}}}{3 b d} + \frac{2 \sqrt{a + b \sin{\left (c + d x \right )}} \sin{\left (c + d x \right )}}{3 d} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.07427, size = 27, normalized size = 1.12 \begin{align*} \frac{2 \,{\left (b \sin \left (d x + c\right ) + a\right )}^{\frac{3}{2}}}{3 \, b d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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