Integrand size = 21, antiderivative size = 24 \[ \int \cos (c+d x) \sqrt {a+b \sin (c+d x)} \, dx=\frac {2 (a+b \sin (c+d x))^{3/2}}{3 b d} \]
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Time = 0.02 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {2747, 32} \[ \int \cos (c+d x) \sqrt {a+b \sin (c+d x)} \, dx=\frac {2 (a+b \sin (c+d x))^{3/2}}{3 b d} \]
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Rule 32
Rule 2747
Rubi steps \begin{align*} \text {integral}& = \frac {\text {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*}
Time = 0.01 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00 \[ \int \cos (c+d x) \sqrt {a+b \sin (c+d x)} \, dx=\frac {2 (a+b \sin (c+d x))^{3/2}}{3 b d} \]
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Time = 0.30 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.88
method | result | size |
derivativedivides | \(\frac {2 \left (a +b \sin \left (d x +c \right )\right )^{\frac {3}{2}}}{3 b d}\) | \(21\) |
default | \(\frac {2 \left (a +b \sin \left (d x +c \right )\right )^{\frac {3}{2}}}{3 b d}\) | \(21\) |
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Time = 0.28 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.83 \[ \int \cos (c+d x) \sqrt {a+b \sin (c+d x)} \, dx=\frac {2 \, {\left (b \sin \left (d x + c\right ) + a\right )}^{\frac {3}{2}}}{3 \, b d} \]
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Leaf count of result is larger than twice the leaf count of optimal. 83 vs. \(2 (19) = 38\).
Time = 0.16 (sec) , antiderivative size = 83, normalized size of antiderivative = 3.46 \[ \int \cos (c+d x) \sqrt {a+b \sin (c+d x)} \, dx=\begin {cases} \sqrt {a} x \cos {\left (c \right )} & \text {for}\: b = 0 \wedge d = 0 \\\frac {\sqrt {a} \sin {\left (c + d x \right )}}{d} & \text {for}\: b = 0 \\x \sqrt {a + b \sin {\left (c \right )}} \cos {\left (c \right )} & \text {for}\: d = 0 \\\frac {2 a \sqrt {a + b \sin {\left (c + d x \right )}}}{3 b d} + \frac {2 \sqrt {a + b \sin {\left (c + d x \right )}} \sin {\left (c + d x \right )}}{3 d} & \text {otherwise} \end {cases} \]
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Time = 0.19 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.83 \[ \int \cos (c+d x) \sqrt {a+b \sin (c+d x)} \, dx=\frac {2 \, {\left (b \sin \left (d x + c\right ) + a\right )}^{\frac {3}{2}}}{3 \, b d} \]
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Time = 0.29 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.83 \[ \int \cos (c+d x) \sqrt {a+b \sin (c+d x)} \, dx=\frac {2 \, {\left (b \sin \left (d x + c\right ) + a\right )}^{\frac {3}{2}}}{3 \, b d} \]
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Time = 4.66 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.83 \[ \int \cos (c+d x) \sqrt {a+b \sin (c+d x)} \, dx=\frac {2\,{\left (a+b\,\sin \left (c+d\,x\right )\right )}^{3/2}}{3\,b\,d} \]
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