Optimal. Leaf size=24 \[ \frac {2 (a+b \sin (c+d x))^{3/2}}{3 b d} \]
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Rubi [A] time = 0.04, antiderivative size = 24, normalized size of antiderivative = 1.00, 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 32
Rule 2668
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*}
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Mathematica [A] time = 0.02, size = 24, normalized size = 1.00 \[ \frac {2 (a+b \sin (c+d x))^{3/2}}{3 b d} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.02, size = 20, normalized size = 0.83 \[ \frac {2 \, {\left (b \sin \left (d x + c\right ) + a\right )}^{\frac {3}{2}}}{3 \, b d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.45, size = 20, normalized size = 0.83 \[ \frac {2 \, {\left (b \sin \left (d x + c\right ) + a\right )}^{\frac {3}{2}}}{3 \, b d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 21, normalized size = 0.88 \[ \frac {2 \left (a +b \sin \left (d x +c \right )\right )^{\frac {3}{2}}}{3 b d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.33, size = 20, normalized size = 0.83 \[ \frac {2 \, {\left (b \sin \left (d x + c\right ) + a\right )}^{\frac {3}{2}}}{3 \, b d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 5.20, size = 20, normalized size = 0.83 \[ \frac {2\,{\left (a+b\,\sin \left (c+d\,x\right )\right )}^{3/2}}{3\,b\,d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.55, size = 83, normalized size = 3.46 \[ \begin {cases} \sqrt {a} x \cos {\relax (c )} & \text {for}\: b = 0 \wedge d = 0 \\\frac {\sqrt {a} \sin {\left (c + d x \right )}}{d} & \text {for}\: b = 0 \\x \sqrt {a + b \sin {\relax (c )}} \cos {\relax (c )} & \text {for}\: d = 0 \\\frac {2 a \sqrt {a + b \sin {\left (c + d x \right )}}}{3 b d} + \frac {2 \sqrt {a + b \sin {\left (c + d x \right )}} \sin {\left (c + d x \right )}}{3 d} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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