Optimal. Leaf size=46 \[ -\frac {2 a \cos (e+f x) \, _2F_1\left (\frac {1}{2},-n;\frac {3}{2};1-\sin (e+f x)\right )}{f \sqrt {a+a \sin (e+f x)}} \]
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Rubi [A]
time = 0.05, 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 = {2855, 67}
\begin {gather*} -\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}} \end {gather*}
Antiderivative was successfully verified.
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Rule 67
Rule 2855
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 ) \text {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] Result contains complex when optimal does not.
time = 2.85, size = 264, normalized size = 5.74 \begin {gather*} \frac {(1+i) e^{-\frac {1}{2} i f x} \left (e^{i f x} (1+2 n) \, _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 ) \left (\cos \left (\frac {e}{2}\right )+i \sin \left (\frac {e}{2}\right )\right )+(-1+2 n) \, _2F_1\left (\frac {1}{4} (-1-2 n),-n;\frac {1}{4} (3-2 n);e^{2 i f x} (\cos (e)+i \sin (e))^2\right ) \left (i \cos \left (\frac {e}{2}\right )+\sin \left (\frac {e}{2}\right )\right )\right ) \left (1-e^{2 i f x} \cos ^2(e)+e^{2 i f x} \sin ^2(e)-i e^{2 i f x} \sin (2 e)\right )^{-n} \sin ^n(e+f x) \sqrt {a (1+\sin (e+f x))}}{f (-1+2 n) (1+2 n) \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.13, 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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \sqrt {a \left (\sin {\left (e + f x \right )} + 1\right )} \sin ^{n}{\left (e + f x \right )}\, dx \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int {\sin \left (e+f\,x\right )}^n\,\sqrt {a+a\,\sin \left (e+f\,x\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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