Integrand size = 21, antiderivative size = 91 \[ \int (d \sin (e+f x))^n (1+\sin (e+f x))^m \, dx=-\frac {2^{\frac {1}{2}+m} \operatorname {AppellF1}\left (\frac {1}{2},-n,\frac {1}{2}-m,\frac {3}{2},1-\sin (e+f x),\frac {1}{2} (1-\sin (e+f x))\right ) \cos (e+f x) \sin ^{-n}(e+f x) (d \sin (e+f x))^n}{f \sqrt {1+\sin (e+f x)}} \]
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Time = 0.07 (sec) , antiderivative size = 91, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {2865, 2864, 138} \[ \int (d \sin (e+f x))^n (1+\sin (e+f x))^m \, dx=-\frac {2^{m+\frac {1}{2}} \cos (e+f x) \sin ^{-n}(e+f x) (d \sin (e+f x))^n \operatorname {AppellF1}\left (\frac {1}{2},-n,\frac {1}{2}-m,\frac {3}{2},1-\sin (e+f x),\frac {1}{2} (1-\sin (e+f x))\right )}{f \sqrt {\sin (e+f x)+1}} \]
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Rule 138
Rule 2864
Rule 2865
Rubi steps \begin{align*} \text {integral}& = \left (\sin ^{-n}(e+f x) (d \sin (e+f x))^n\right ) \int \sin ^n(e+f x) (1+\sin (e+f x))^m \, dx \\ & = -\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)^{-\frac {1}{2}+m}}{\sqrt {x}} \, dx,x,1-\sin (e+f x)\right )}{f \sqrt {1-\sin (e+f x)} \sqrt {1+\sin (e+f x)}} \\ & = -\frac {2^{\frac {1}{2}+m} \operatorname {AppellF1}\left (\frac {1}{2},-n,\frac {1}{2}-m,\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 {1+\sin (e+f x)}} \\ \end{align*}
\[ \int (d \sin (e+f x))^n (1+\sin (e+f x))^m \, dx=\int (d \sin (e+f x))^n (1+\sin (e+f x))^m \, dx \]
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\[\int \left (d \sin \left (f x +e \right )\right )^{n} \left (\sin \left (f x +e \right )+1\right )^{m}d x\]
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\[ \int (d \sin (e+f x))^n (1+\sin (e+f x))^m \, dx=\int { \left (d \sin \left (f x + e\right )\right )^{n} {\left (\sin \left (f x + e\right ) + 1\right )}^{m} \,d x } \]
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\[ \int (d \sin (e+f x))^n (1+\sin (e+f x))^m \, dx=\int \left (d \sin {\left (e + f x \right )}\right )^{n} \left (\sin {\left (e + f x \right )} + 1\right )^{m}\, dx \]
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\[ \int (d \sin (e+f x))^n (1+\sin (e+f x))^m \, dx=\int { \left (d \sin \left (f x + e\right )\right )^{n} {\left (\sin \left (f x + e\right ) + 1\right )}^{m} \,d x } \]
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\[ \int (d \sin (e+f x))^n (1+\sin (e+f x))^m \, dx=\int { \left (d \sin \left (f x + e\right )\right )^{n} {\left (\sin \left (f x + e\right ) + 1\right )}^{m} \,d x } \]
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Timed out. \[ \int (d \sin (e+f x))^n (1+\sin (e+f x))^m \, dx=\int {\left (d\,\sin \left (e+f\,x\right )\right )}^n\,{\left (\sin \left (e+f\,x\right )+1\right )}^m \,d x \]
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