Optimal. Leaf size=138 \[ \frac{2 a \sqrt{c+d \sin (e+f x)} E\left (\frac{1}{2} \left (e+f x-\frac{\pi }{2}\right )|\frac{2 d}{c+d}\right )}{d f \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}-\frac{2 a (c-d) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left (\frac{1}{2} \left (e+f x-\frac{\pi }{2}\right )|\frac{2 d}{c+d}\right )}{d f \sqrt{c+d \sin (e+f x)}} \]
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Rubi [A] time = 0.122268, antiderivative size = 138, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {2752, 2663, 2661, 2655, 2653} \[ \frac{2 a \sqrt{c+d \sin (e+f x)} E\left (\frac{1}{2} \left (e+f x-\frac{\pi }{2}\right )|\frac{2 d}{c+d}\right )}{d f \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}-\frac{2 a (c-d) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left (\frac{1}{2} \left (e+f x-\frac{\pi }{2}\right )|\frac{2 d}{c+d}\right )}{d f \sqrt{c+d \sin (e+f x)}} \]
Antiderivative was successfully verified.
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Rule 2752
Rule 2663
Rule 2661
Rule 2655
Rule 2653
Rubi steps
\begin{align*} \int \frac{a+a \sin (e+f x)}{\sqrt{c+d \sin (e+f x)}} \, dx &=\frac{a \int \sqrt{c+d \sin (e+f x)} \, dx}{d}+\frac{(-a c+a d) \int \frac{1}{\sqrt{c+d \sin (e+f x)}} \, dx}{d}\\ &=\frac{\left (a \sqrt{c+d \sin (e+f x)}\right ) \int \sqrt{\frac{c}{c+d}+\frac{d \sin (e+f x)}{c+d}} \, dx}{d \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}+\frac{\left ((-a c+a d) \sqrt{\frac{c+d \sin (e+f x)}{c+d}}\right ) \int \frac{1}{\sqrt{\frac{c}{c+d}+\frac{d \sin (e+f x)}{c+d}}} \, dx}{d \sqrt{c+d \sin (e+f x)}}\\ &=\frac{2 a E\left (\frac{1}{2} \left (e-\frac{\pi }{2}+f x\right )|\frac{2 d}{c+d}\right ) \sqrt{c+d \sin (e+f x)}}{d f \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}-\frac{2 a (c-d) F\left (\frac{1}{2} \left (e-\frac{\pi }{2}+f x\right )|\frac{2 d}{c+d}\right ) \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}{d f \sqrt{c+d \sin (e+f x)}}\\ \end{align*}
Mathematica [C] time = 6.24661, size = 880, normalized size = 6.38 \[ a \left (\frac{\sec (e) \left (-\frac{F_1\left (-\frac{1}{2};-\frac{1}{2},-\frac{1}{2};\frac{1}{2};-\frac{\csc (e) \left (c+d \cos \left (f x-\tan ^{-1}(\cot (e))\right ) \sqrt{\cot ^2(e)+1} \sin (e)\right )}{d \sqrt{\cot ^2(e)+1} \left (1-\frac{c \csc (e)}{d \sqrt{\cot ^2(e)+1}}\right )},-\frac{\csc (e) \left (c+d \cos \left (f x-\tan ^{-1}(\cot (e))\right ) \sqrt{\cot ^2(e)+1} \sin (e)\right )}{d \sqrt{\cot ^2(e)+1} \left (-\frac{c \csc (e)}{d \sqrt{\cot ^2(e)+1}}-1\right )}\right ) \cot (e) \sin \left (f x-\tan ^{-1}(\cot (e))\right )}{\sqrt{\cot ^2(e)+1} \sqrt{\frac{\cos \left (f x-\tan ^{-1}(\cot (e))\right ) \sqrt{\cot ^2(e)+1} d+\sqrt{\cot ^2(e)+1} d}{d \sqrt{\cot ^2(e)+1}-c \csc (e)}} \sqrt{\frac{d \sqrt{\cot ^2(e)+1}-d \cos \left (f x-\tan ^{-1}(\cot (e))\right ) \sqrt{\cot ^2(e)+1}}{\sqrt{\cot ^2(e)+1} d+c \csc (e)}} \sqrt{c+d \cos \left (f x-\tan ^{-1}(\cot (e))\right ) \sqrt{\cot ^2(e)+1} \sin (e)}}-\frac{\frac{2 d \sin (e) \left (c+d \cos \left (f x-\tan ^{-1}(\cot (e))\right ) \sqrt{\cot ^2(e)+1} \sin (e)\right )}{d^2 \cos ^2(e)+d^2 \sin ^2(e)}-\frac{\cot (e) \sin \left (f x-\tan ^{-1}(\cot (e))\right )}{\sqrt{\cot ^2(e)+1}}}{\sqrt{c+d \cos \left (f x-\tan ^{-1}(\cot (e))\right ) \sqrt{\cot ^2(e)+1} \sin (e)}}\right ) (\sin (e+f x)+1)}{f \left (\cos \left (\frac{e}{2}+\frac{f x}{2}\right )+\sin \left (\frac{e}{2}+\frac{f x}{2}\right )\right )^2}+\frac{2 \sqrt{c+d \sin (e+f x)} \tan (e) (\sin (e+f x)+1)}{d f \left (\cos \left (\frac{e}{2}+\frac{f x}{2}\right )+\sin \left (\frac{e}{2}+\frac{f x}{2}\right )\right )^2}+\frac{2 F_1\left (\frac{1}{2};\frac{1}{2},\frac{1}{2};\frac{3}{2};-\frac{\sec (e) \left (c+d \cos (e) \sin \left (f x+\tan ^{-1}(\tan (e))\right ) \sqrt{\tan ^2(e)+1}\right )}{d \sqrt{\tan ^2(e)+1} \left (1-\frac{c \sec (e)}{d \sqrt{\tan ^2(e)+1}}\right )},-\frac{\sec (e) \left (c+d \cos (e) \sin \left (f x+\tan ^{-1}(\tan (e))\right ) \sqrt{\tan ^2(e)+1}\right )}{d \sqrt{\tan ^2(e)+1} \left (-\frac{c \sec (e)}{d \sqrt{\tan ^2(e)+1}}-1\right )}\right ) \sec (e) \sec \left (f x+\tan ^{-1}(\tan (e))\right ) \sqrt{\frac{d \sqrt{\tan ^2(e)+1}-d \sin \left (f x+\tan ^{-1}(\tan (e))\right ) \sqrt{\tan ^2(e)+1}}{\sqrt{\tan ^2(e)+1} d+c \sec (e)}} \sqrt{\frac{\sin \left (f x+\tan ^{-1}(\tan (e))\right ) \sqrt{\tan ^2(e)+1} d+\sqrt{\tan ^2(e)+1} d}{d \sqrt{\tan ^2(e)+1}-c \sec (e)}} \sqrt{c+d \cos (e) \sin \left (f x+\tan ^{-1}(\tan (e))\right ) \sqrt{\tan ^2(e)+1}} (\sin (e+f x)+1)}{d f \left (\cos \left (\frac{e}{2}+\frac{f x}{2}\right )+\sin \left (\frac{e}{2}+\frac{f x}{2}\right )\right )^2 \sqrt{\tan ^2(e)+1}}\right ) \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.8, size = 203, normalized size = 1.5 \begin{align*} -2\,{\frac{a \left ( c-d \right ) }{{d}^{2}\cos \left ( fx+e \right ) \sqrt{c+d\sin \left ( fx+e \right ) }f}\sqrt{{\frac{c+d\sin \left ( fx+e \right ) }{c-d}}}\sqrt{-{\frac{ \left ( -1+\sin \left ( fx+e \right ) \right ) d}{c+d}}}\sqrt{-{\frac{d \left ( 1+\sin \left ( fx+e \right ) \right ) }{c-d}}} \left ({\it EllipticE} \left ( \sqrt{{\frac{c+d\sin \left ( fx+e \right ) }{c-d}}},\sqrt{{\frac{c-d}{c+d}}} \right ) c+{\it EllipticE} \left ( \sqrt{{\frac{c+d\sin \left ( fx+e \right ) }{c-d}}},\sqrt{{\frac{c-d}{c+d}}} \right ) d-2\,{\it EllipticF} \left ( \sqrt{{\frac{c+d\sin \left ( fx+e \right ) }{c-d}}},\sqrt{{\frac{c-d}{c+d}}} \right ) d \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 \frac{a \sin \left (f x + e\right ) + a}{\sqrt{d \sin \left (f x + e\right ) + c}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{a \sin \left (f x + e\right ) + a}{\sqrt{d \sin \left (f x + e\right ) + c}}, x\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*} a \left (\int \frac{\sin{\left (e + f x \right )}}{\sqrt{c + d \sin{\left (e + f x \right )}}}\, dx + \int \frac{1}{\sqrt{c + d \sin{\left (e + f x \right )}}}\, dx\right ) \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 \frac{a \sin \left (f x + e\right ) + a}{\sqrt{d \sin \left (f x + e\right ) + c}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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