Optimal. Leaf size=46 \[ -\frac {2 a \cos (e+f x) \, _2F_1\left (\frac {1}{2},-n;\frac {3}{2};1-\sin (e+f x)\right )}{f \sqrt {a \sin (e+f x)+a}} \]
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Rubi [A] time = 0.06, antiderivative size = 46, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {2776, 65} \[ -\frac {2 a \cos (e+f x) \, _2F_1\left (\frac {1}{2},-n;\frac {3}{2};1-\sin (e+f x)\right )}{f \sqrt {a \sin (e+f x)+a}} \]
Antiderivative was successfully verified.
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Rule 65
Rule 2776
Rubi steps
\begin {align*} \int \sin ^n(e+f x) \sqrt {a+a \sin (e+f x)} \, dx &=\frac {\left (a^2 \cos (e+f x)\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 )}{f \sqrt {a+a \sin (e+f x)}}\\ \end {align*}
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Mathematica [C] time = 4.29, size = 264, normalized size = 5.74 \[ \frac {(1+i) e^{-\frac {1}{2} i f x} \sqrt {a (\sin (e+f x)+1)} \sin ^n(e+f x) \left (\sin ^2(e) e^{2 i f x}-i \sin (2 e) e^{2 i f x}+\cos ^2(e) \left (-e^{2 i f x}\right )+1\right )^{-n} \left ((2 n+1) e^{i f x} \left (\cos \left (\frac {e}{2}\right )+i \sin \left (\frac {e}{2}\right )\right ) \, _2F_1\left (\frac {1}{4} (1-2 n),-n;\frac {1}{4} (5-2 n);e^{2 i f x} (\cos (e)+i \sin (e))^2\right )+(2 n-1) \left (\sin \left (\frac {e}{2}\right )+i \cos \left (\frac {e}{2}\right )\right ) \, _2F_1\left (\frac {1}{4} (-2 n-1),-n;\frac {1}{4} (3-2 n);e^{2 i f x} (\cos (e)+i \sin (e))^2\right )\right )}{f (2 n-1) (2 n+1) \left (\sin \left (\frac {1}{2} (e+f x)\right )+\cos \left (\frac {1}{2} (e+f x)\right )\right )} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.48, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\sqrt {a \sin \left (f x + e\right ) + a} \sin \left (f x + e\right )^{n}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {a \sin \left (f x + e\right ) + a} \sin \left (f x + e\right )^{n}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.22, size = 0, normalized size = 0.00 \[ \int \left (\sin ^{n}\left (f x +e \right )\right ) \sqrt {a +a \sin \left (f x +e \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {a \sin \left (f x + e\right ) + a} \sin \left (f x + e\right )^{n}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int {\sin \left (e+f\,x\right )}^n\,\sqrt {a+a\,\sin \left (e+f\,x\right )} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {a \left (\sin {\left (e + f x \right )} + 1\right )} \sin ^{n}{\left (e + f x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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