Optimal. Leaf size=26 \[ -\frac{2 a \cos (c+d x)}{d \sqrt{a \sin (c+d x)+a}} \]
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Rubi [A] time = 0.0131947, antiderivative size = 26, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.071, Rules used = {2646} \[ -\frac{2 a \cos (c+d x)}{d \sqrt{a \sin (c+d x)+a}} \]
Antiderivative was successfully verified.
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Rule 2646
Rubi steps
\begin{align*} \int \sqrt{a+a \sin (c+d x)} \, dx &=-\frac{2 a \cos (c+d x)}{d \sqrt{a+a \sin (c+d x)}}\\ \end{align*}
Mathematica [B] time = 0.0325276, size = 65, normalized size = 2.5 \[ \frac{2 \sqrt{a (\sin (c+d x)+1)} \left (\sin \left (\frac{1}{2} (c+d x)\right )-\cos \left (\frac{1}{2} (c+d x)\right )\right )}{d \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.332, size = 43, normalized size = 1.7 \begin{align*} 2\,{\frac{a \left ( 1+\sin \left ( dx+c \right ) \right ) \left ( \sin \left ( dx+c \right ) -1 \right ) }{\cos \left ( dx+c \right ) \sqrt{a+a\sin \left ( dx+c \right ) }d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{a \sin \left (d x + c\right ) + a}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.44243, size = 136, normalized size = 5.23 \begin{align*} -\frac{2 \, \sqrt{a \sin \left (d x + c\right ) + a}{\left (\cos \left (d x + c\right ) - \sin \left (d x + c\right ) + 1\right )}}{d \cos \left (d x + c\right ) + d \sin \left (d x + c\right ) + d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{a \sin{\left (c + d x \right )} + a}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{a \sin \left (d x + c\right ) + a}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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