Optimal. Leaf size=26 \[ -\frac{2 a \cos (e+f x)}{f \sqrt{a \sin (e+f x)+a}} \]
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Rubi [A] time = 0.0138016, 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} \[ -\frac{2 a \cos (e+f x)}{f \sqrt{a \sin (e+f x)+a}} \]
Antiderivative was successfully verified.
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Rule 2646
Rubi steps
\begin{align*} \int \sqrt{a+a \sin (e+f x)} \, dx &=-\frac{2 a \cos (e+f x)}{f \sqrt{a+a \sin (e+f x)}}\\ \end{align*}
Mathematica [B] time = 0.0334138, size = 65, normalized size = 2.5 \[ \frac{2 \sqrt{a (\sin (e+f x)+1)} \left (\sin \left (\frac{1}{2} (e+f x)\right )-\cos \left (\frac{1}{2} (e+f x)\right )\right )}{f \left (\sin \left (\frac{1}{2} (e+f x)\right )+\cos \left (\frac{1}{2} (e+f x)\right )\right )} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.467, size = 43, normalized size = 1.7 \begin{align*} 2\,{\frac{a \left ( 1+\sin \left ( fx+e \right ) \right ) \left ( -1+\sin \left ( fx+e \right ) \right ) }{\cos \left ( fx+e \right ) \sqrt{a+a\sin \left ( fx+e \right ) }f}} \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 (f x + e\right ) + a}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.77519, size = 136, normalized size = 5.23 \begin{align*} -\frac{2 \, \sqrt{a \sin \left (f x + e\right ) + a}{\left (\cos \left (f x + e\right ) - \sin \left (f x + e\right ) + 1\right )}}{f \cos \left (f x + e\right ) + f \sin \left (f x + e\right ) + f} \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 (e + f 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 (f x + e\right ) + a}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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