Optimal. Leaf size=66 \[ -\frac{2 a \cos (e+f x) \sin ^{-n}(e+f x) (d \sin (e+f x))^n \, _2F_1\left (\frac{1}{2},-n;\frac{3}{2};1-\sin (e+f x)\right )}{f \sqrt{a \sin (e+f x)+a}} \]
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Rubi [A] time = 0.0745947, antiderivative size = 66, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.12, Rules used = {2776, 67, 65} \[ -\frac{2 a \cos (e+f x) \sin ^{-n}(e+f x) (d \sin (e+f x))^n \, _2F_1\left (\frac{1}{2},-n;\frac{3}{2};1-\sin (e+f x)\right )}{f \sqrt{a \sin (e+f x)+a}} \]
Antiderivative was successfully verified.
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Rule 2776
Rule 67
Rule 65
Rubi steps
\begin{align*} \int (d \sin (e+f x))^n \sqrt{a+a \sin (e+f x)} \, dx &=\frac{\left (a^2 \cos (e+f x)\right ) \operatorname{Subst}\left (\int \frac{(d x)^n}{\sqrt{a-a x}} \, dx,x,\sin (e+f x)\right )}{f \sqrt{a-a \sin (e+f x)} \sqrt{a+a \sin (e+f x)}}\\ &=\frac{\left (a^2 \cos (e+f x) \sin ^{-n}(e+f x) (d \sin (e+f x))^n\right ) \operatorname{Subst}\left (\int \frac{x^n}{\sqrt{a-a x}} \, dx,x,\sin (e+f x)\right )}{f \sqrt{a-a \sin (e+f x)} \sqrt{a+a \sin (e+f x)}}\\ &=-\frac{2 a \cos (e+f x) \, _2F_1\left (\frac{1}{2},-n;\frac{3}{2};1-\sin (e+f x)\right ) \sin ^{-n}(e+f x) (d \sin (e+f x))^n}{f \sqrt{a+a \sin (e+f x)}}\\ \end{align*}
Mathematica [C] time = 0.408685, size = 266, normalized size = 4.03 \[ \frac{(1+i) e^{-\frac{1}{2} i f x} \sqrt{a (\sin (e+f x)+1)} (d \sin (e+f x))^n \left (\sin ^2(e) e^{2 i f x}-i \sin (2 e) e^{2 i f x}+\cos ^2(e) \left (-e^{2 i f x}\right )+1\right )^{-n} \left ((2 n+1) e^{i f x} \left (\cos \left (\frac{e}{2}\right )+i \sin \left (\frac{e}{2}\right )\right ) \, _2F_1\left (\frac{1}{4} (1-2 n),-n;\frac{1}{4} (5-2 n);e^{2 i f x} (\cos (e)+i \sin (e))^2\right )+(2 n-1) \left (\sin \left (\frac{e}{2}\right )+i \cos \left (\frac{e}{2}\right )\right ) \, _2F_1\left (\frac{1}{4} (-2 n-1),-n;\frac{1}{4} (3-2 n);e^{2 i f x} (\cos (e)+i \sin (e))^2\right )\right )}{f (2 n-1) (2 n+1) \left (\sin \left (\frac{1}{2} (e+f x)\right )+\cos \left (\frac{1}{2} (e+f x)\right )\right )} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.161, size = 0, normalized size = 0. \begin{align*} \int \left ( d\sin \left ( fx+e \right ) \right ) ^{n}\sqrt{a+a\sin \left ( fx+e \right ) }\, dx \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 )\right )^{n}\,{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 (\sqrt{a \sin \left (f x + e\right ) + a} \left (d \sin \left (f x + e\right )\right )^{n}, 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*} \int \sqrt{a \left (\sin{\left (e + f x \right )} + 1\right )} \left (d \sin{\left (e + f x \right )}\right )^{n}\, 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 )\right )^{n}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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