Integrand size = 25, antiderivative size = 80 \[ \int \frac {(d \sin (e+f x))^n}{\sqrt {a+a \sin (e+f x)}} \, dx=-\frac {\operatorname {AppellF1}\left (\frac {1}{2},-n,1,\frac {3}{2},1-\sin (e+f x),\frac {1}{2} (1-\sin (e+f x))\right ) \cos (e+f x) \sin ^{-n}(e+f x) (d \sin (e+f x))^n}{f \sqrt {a+a \sin (e+f x)}} \]
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Time = 0.13 (sec) , antiderivative size = 80, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {2866, 2865, 2864, 129, 440} \[ \int \frac {(d \sin (e+f x))^n}{\sqrt {a+a \sin (e+f x)}} \, dx=-\frac {\cos (e+f x) \sin ^{-n}(e+f x) \operatorname {AppellF1}\left (\frac {1}{2},-n,1,\frac {3}{2},1-\sin (e+f x),\frac {1}{2} (1-\sin (e+f x))\right ) (d \sin (e+f x))^n}{f \sqrt {a \sin (e+f x)+a}} \]
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Rule 129
Rule 440
Rule 2864
Rule 2865
Rule 2866
Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {1+\sin (e+f x)} \int \frac {(d \sin (e+f x))^n}{\sqrt {1+\sin (e+f x)}} \, dx}{\sqrt {a+a \sin (e+f x)}} \\ & = \frac {\left (\sin ^{-n}(e+f x) (d \sin (e+f x))^n \sqrt {1+\sin (e+f x)}\right ) \int \frac {\sin ^n(e+f x)}{\sqrt {1+\sin (e+f x)}} \, dx}{\sqrt {a+a \sin (e+f x)}} \\ & = -\frac {\left (\cos (e+f x) \sin ^{-n}(e+f x) (d \sin (e+f x))^n\right ) \text {Subst}\left (\int \frac {(1-x)^n}{(2-x) \sqrt {x}} \, dx,x,1-\sin (e+f x)\right )}{f \sqrt {1-\sin (e+f x)} \sqrt {a+a \sin (e+f x)}} \\ & = -\frac {\left (2 \cos (e+f x) \sin ^{-n}(e+f x) (d \sin (e+f x))^n\right ) \text {Subst}\left (\int \frac {\left (1-x^2\right )^n}{2-x^2} \, dx,x,\sqrt {1-\sin (e+f x)}\right )}{f \sqrt {1-\sin (e+f x)} \sqrt {a+a \sin (e+f x)}} \\ & = -\frac {\operatorname {AppellF1}\left (\frac {1}{2},-n,1,\frac {3}{2},1-\sin (e+f x),\frac {1}{2} (1-\sin (e+f x))\right ) \cos (e+f x) \sin ^{-n}(e+f x) (d \sin (e+f x))^n}{f \sqrt {a+a \sin (e+f x)}} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(242\) vs. \(2(80)=160\).
Time = 0.72 (sec) , antiderivative size = 242, normalized size of antiderivative = 3.02 \[ \int \frac {(d \sin (e+f x))^n}{\sqrt {a+a \sin (e+f x)}} \, dx=\frac {\cos (e+f x) \sin ^n(e+f x) (d \sin (e+f x))^n \left (-\sin ^2(e+f x)\right )^{-n} \sqrt {a (1+\sin (e+f x))} \left (1-\frac {1}{1+\sin (e+f x)}\right )^{-n} \left (4 \operatorname {AppellF1}\left (-\frac {1}{2}-n,-\frac {1}{2},-n,\frac {1}{2}-n,\frac {2}{1+\sin (e+f x)},\frac {1}{1+\sin (e+f x)}\right ) (-\sin (e+f x))^n \sqrt {\frac {-1+\sin (e+f x)}{1+\sin (e+f x)}}-(1+2 n) \operatorname {AppellF1}\left (1,\frac {1}{2},-n,2,\frac {1}{2} (1+\sin (e+f x)),1+\sin (e+f x)\right ) \sqrt {2-2 \sin (e+f x)} \left (1-\frac {1}{1+\sin (e+f x)}\right )^n\right )}{4 a f (1+2 n) (-1+\sin (e+f x))} \]
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\[\int \frac {\left (d \sin \left (f x +e \right )\right )^{n}}{\sqrt {a +a \sin \left (f x +e \right )}}d x\]
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\[ \int \frac {(d \sin (e+f x))^n}{\sqrt {a+a \sin (e+f x)}} \, dx=\int { \frac {\left (d \sin \left (f x + e\right )\right )^{n}}{\sqrt {a \sin \left (f x + e\right ) + a}} \,d x } \]
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\[ \int \frac {(d \sin (e+f x))^n}{\sqrt {a+a \sin (e+f x)}} \, dx=\int \frac {\left (d \sin {\left (e + f x \right )}\right )^{n}}{\sqrt {a \left (\sin {\left (e + f x \right )} + 1\right )}}\, dx \]
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\[ \int \frac {(d \sin (e+f x))^n}{\sqrt {a+a \sin (e+f x)}} \, dx=\int { \frac {\left (d \sin \left (f x + e\right )\right )^{n}}{\sqrt {a \sin \left (f x + e\right ) + a}} \,d x } \]
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\[ \int \frac {(d \sin (e+f x))^n}{\sqrt {a+a \sin (e+f x)}} \, dx=\int { \frac {\left (d \sin \left (f x + e\right )\right )^{n}}{\sqrt {a \sin \left (f x + e\right ) + a}} \,d x } \]
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Timed out. \[ \int \frac {(d \sin (e+f x))^n}{\sqrt {a+a \sin (e+f x)}} \, dx=\int \frac {{\left (d\,\sin \left (e+f\,x\right )\right )}^n}{\sqrt {a+a\,\sin \left (e+f\,x\right )}} \,d x \]
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