Integrand size = 27, antiderivative size = 131 \[ \int \frac {(3+3 \sin (e+f x))^m}{\sqrt {c+d \sin (e+f x)}} \, dx=\frac {\sqrt {2} \operatorname {AppellF1}\left (\frac {1}{2}+m,\frac {1}{2},\frac {1}{2},\frac {3}{2}+m,\frac {1}{2} (1+\sin (e+f x)),-\frac {d (1+\sin (e+f x))}{c-d}\right ) \cos (e+f x) (3+3 \sin (e+f x))^m \sqrt {\frac {c+d \sin (e+f x)}{c-d}}}{f (1+2 m) \sqrt {1-\sin (e+f x)} \sqrt {c+d \sin (e+f x)}} \]
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Time = 0.11 (sec) , antiderivative size = 131, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.148, Rules used = {2867, 145, 144, 143} \[ \int \frac {(3+3 \sin (e+f x))^m}{\sqrt {c+d \sin (e+f x)}} \, dx=\frac {\sqrt {2} \cos (e+f x) (a \sin (e+f x)+a)^m \sqrt {\frac {c+d \sin (e+f x)}{c-d}} \operatorname {AppellF1}\left (m+\frac {1}{2},\frac {1}{2},\frac {1}{2},m+\frac {3}{2},\frac {1}{2} (\sin (e+f x)+1),-\frac {d (\sin (e+f x)+1)}{c-d}\right )}{f (2 m+1) \sqrt {1-\sin (e+f x)} \sqrt {c+d \sin (e+f x)}} \]
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Rule 143
Rule 144
Rule 145
Rule 2867
Rubi steps \begin{align*} \text {integral}& = \frac {\left (a^2 \cos (e+f x)\right ) \text {Subst}\left (\int \frac {(a+a x)^{-\frac {1}{2}+m}}{\sqrt {a-a x} \sqrt {c+d 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) \sqrt {\frac {a-a \sin (e+f x)}{a}}\right ) \text {Subst}\left (\int \frac {(a+a x)^{-\frac {1}{2}+m}}{\sqrt {\frac {1}{2}-\frac {x}{2}} \sqrt {c+d x}} \, dx,x,\sin (e+f x)\right )}{\sqrt {2} f (a-a \sin (e+f x)) \sqrt {a+a \sin (e+f x)}} \\ & = \frac {\left (a^2 \cos (e+f x) \sqrt {\frac {a-a \sin (e+f x)}{a}} \sqrt {\frac {a (c+d \sin (e+f x))}{a c-a d}}\right ) \text {Subst}\left (\int \frac {(a+a x)^{-\frac {1}{2}+m}}{\sqrt {\frac {1}{2}-\frac {x}{2}} \sqrt {\frac {a c}{a c-a d}+\frac {a d x}{a c-a d}}} \, dx,x,\sin (e+f x)\right )}{\sqrt {2} f (a-a \sin (e+f x)) \sqrt {a+a \sin (e+f x)} \sqrt {c+d \sin (e+f x)}} \\ & = \frac {\sqrt {2} \operatorname {AppellF1}\left (\frac {1}{2}+m,\frac {1}{2},\frac {1}{2},\frac {3}{2}+m,\frac {1}{2} (1+\sin (e+f x)),-\frac {d (1+\sin (e+f x))}{c-d}\right ) \cos (e+f x) (a+a \sin (e+f x))^m \sqrt {\frac {c+d \sin (e+f x)}{c-d}}}{f (1+2 m) \sqrt {1-\sin (e+f x)} \sqrt {c+d \sin (e+f x)}} \\ \end{align*}
\[ \int \frac {(3+3 \sin (e+f x))^m}{\sqrt {c+d \sin (e+f x)}} \, dx=\int \frac {(3+3 \sin (e+f x))^m}{\sqrt {c+d \sin (e+f x)}} \, dx \]
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\[\int \frac {\left (a +a \sin \left (f x +e \right )\right )^{m}}{\sqrt {c +d \sin \left (f x +e \right )}}d x\]
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\[ \int \frac {(3+3 \sin (e+f x))^m}{\sqrt {c+d \sin (e+f x)}} \, dx=\int { \frac {{\left (a \sin \left (f x + e\right ) + a\right )}^{m}}{\sqrt {d \sin \left (f x + e\right ) + c}} \,d x } \]
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\[ \int \frac {(3+3 \sin (e+f x))^m}{\sqrt {c+d \sin (e+f x)}} \, dx=\int \frac {\left (a \left (\sin {\left (e + f x \right )} + 1\right )\right )^{m}}{\sqrt {c + d \sin {\left (e + f x \right )}}}\, dx \]
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\[ \int \frac {(3+3 \sin (e+f x))^m}{\sqrt {c+d \sin (e+f x)}} \, dx=\int { \frac {{\left (a \sin \left (f x + e\right ) + a\right )}^{m}}{\sqrt {d \sin \left (f x + e\right ) + c}} \,d x } \]
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\[ \int \frac {(3+3 \sin (e+f x))^m}{\sqrt {c+d \sin (e+f x)}} \, dx=\int { \frac {{\left (a \sin \left (f x + e\right ) + a\right )}^{m}}{\sqrt {d \sin \left (f x + e\right ) + c}} \,d x } \]
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Timed out. \[ \int \frac {(3+3 \sin (e+f x))^m}{\sqrt {c+d \sin (e+f x)}} \, dx=\int \frac {{\left (a+a\,\sin \left (e+f\,x\right )\right )}^m}{\sqrt {c+d\,\sin \left (e+f\,x\right )}} \,d x \]
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