Optimal. Leaf size=24 \[ \frac {2 (a+a \sin (c+d x))^{3/2}}{3 a d} \]
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Rubi [A]
time = 0.02, 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 = {2746, 32}
\begin {gather*} \frac {2 (a \sin (c+d x)+a)^{3/2}}{3 a d} \end {gather*}
Antiderivative was successfully verified.
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Rule 32
Rule 2746
Rubi steps
\begin {align*} \int \cos (c+d x) \sqrt {a+a \sin (c+d x)} \, dx &=\frac {\text {Subst}\left (\int \sqrt {a+x} \, dx,x,a \sin (c+d x)\right )}{a d}\\ &=\frac {2 (a+a \sin (c+d x))^{3/2}}{3 a d}\\ \end {align*}
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Mathematica [A]
time = 0.06, size = 44, normalized size = 1.83 \begin {gather*} \frac {2 \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^2 \sqrt {a (1+\sin (c+d x))}}{3 d} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.04, size = 21, normalized size = 0.88
method | result | size |
derivativedivides | \(\frac {2 \left (a +a \sin \left (d x +c \right )\right )^{\frac {3}{2}}}{3 d a}\) | \(21\) |
default | \(\frac {2 \left (a +a \sin \left (d x +c \right )\right )^{\frac {3}{2}}}{3 d a}\) | \(21\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 20, normalized size = 0.83 \begin {gather*} \frac {2 \, {\left (a \sin \left (d x + c\right ) + a\right )}^{\frac {3}{2}}}{3 \, a d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.35, size = 25, normalized size = 1.04 \begin {gather*} \frac {2 \, \sqrt {a \sin \left (d x + c\right ) + a} {\left (\sin \left (d x + c\right ) + 1\right )}}{3 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 58 vs.
\(2 (19) = 38\).
time = 0.12, size = 58, normalized size = 2.42 \begin {gather*} \begin {cases} \frac {2 \sqrt {a \sin {\left (c + d x \right )} + a} \sin {\left (c + d x \right )}}{3 d} + \frac {2 \sqrt {a \sin {\left (c + d x \right )} + a}}{3 d} & \text {for}\: d \neq 0 \\x \sqrt {a \sin {\left (c \right )} + a} \cos {\left (c \right )} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 5.80, size = 38, normalized size = 1.58 \begin {gather*} \frac {4 \, \sqrt {2} \sqrt {a} \cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}{3 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 4.57, size = 20, normalized size = 0.83 \begin {gather*} \frac {2\,{\left (a\,\left (\sin \left (c+d\,x\right )+1\right )\right )}^{3/2}}{3\,a\,d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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