Optimal. Leaf size=62 \[ -\frac{2 a (3 c+d) \cos (e+f x)}{3 f \sqrt{a \sin (e+f x)+a}}-\frac{2 d \cos (e+f x) \sqrt{a \sin (e+f x)+a}}{3 f} \]
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Rubi [A] time = 0.0554464, antiderivative size = 62, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.08, Rules used = {2751, 2646} \[ -\frac{2 a (3 c+d) \cos (e+f x)}{3 f \sqrt{a \sin (e+f x)+a}}-\frac{2 d \cos (e+f x) \sqrt{a \sin (e+f x)+a}}{3 f} \]
Antiderivative was successfully verified.
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Rule 2751
Rule 2646
Rubi steps
\begin{align*} \int \sqrt{a+a \sin (e+f x)} (c+d \sin (e+f x)) \, dx &=-\frac{2 d \cos (e+f x) \sqrt{a+a \sin (e+f x)}}{3 f}+\frac{1}{3} (3 c+d) \int \sqrt{a+a \sin (e+f x)} \, dx\\ &=-\frac{2 a (3 c+d) \cos (e+f x)}{3 f \sqrt{a+a \sin (e+f x)}}-\frac{2 d \cos (e+f x) \sqrt{a+a \sin (e+f x)}}{3 f}\\ \end{align*}
Mathematica [A] time = 0.123096, size = 82, normalized size = 1.32 \[ -\frac{2 \sqrt{a (\sin (e+f x)+1)} \left (\cos \left (\frac{1}{2} (e+f x)\right )-\sin \left (\frac{1}{2} (e+f x)\right )\right ) (3 c+d \sin (e+f x)+2 d)}{3 f \left (\sin \left (\frac{1}{2} (e+f x)\right )+\cos \left (\frac{1}{2} (e+f x)\right )\right )} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.614, size = 58, normalized size = 0.9 \begin{align*}{\frac{ \left ( 2+2\,\sin \left ( fx+e \right ) \right ) a \left ( -1+\sin \left ( fx+e \right ) \right ) \left ( d\sin \left ( fx+e \right ) +3\,c+2\,d \right ) }{3\,f\cos \left ( fx+e \right ) }{\frac{1}{\sqrt{a+a\sin \left ( fx+e \right ) }}}} \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}{\left (d \sin \left (f x + e\right ) + c\right )}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.83614, size = 225, normalized size = 3.63 \begin{align*} -\frac{2 \,{\left (d \cos \left (f x + e\right )^{2} +{\left (3 \, c + 2 \, d\right )} \cos \left (f x + e\right ) +{\left (d \cos \left (f x + e\right ) - 3 \, c - d\right )} \sin \left (f x + e\right ) + 3 \, c + d\right )} \sqrt{a \sin \left (f x + e\right ) + a}}{3 \,{\left (f \cos \left (f x + e\right ) + f \sin \left (f x + e\right ) + f\right )}} \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 \left (\sin{\left (e + f x \right )} + 1\right )} \left (c + d \sin{\left (e + f x \right )}\right )\, 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}{\left (d \sin \left (f x + e\right ) + c\right )}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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