Integrand size = 28, antiderivative size = 74 \[ \int \frac {(3+3 \sin (e+f x))^m}{(c-c \sin (e+f x))^{5/2}} \, dx=\frac {\cos (e+f x) \operatorname {Hypergeometric2F1}\left (3,\frac {1}{2}+m,\frac {3}{2}+m,\frac {1}{2} (1+\sin (e+f x))\right ) (3+3 \sin (e+f x))^m}{4 c^2 f (1+2 m) \sqrt {c-c \sin (e+f x)}} \]
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Time = 0.12 (sec) , antiderivative size = 74, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.107, Rules used = {2824, 2746, 70} \[ \int \frac {(3+3 \sin (e+f x))^m}{(c-c \sin (e+f x))^{5/2}} \, dx=\frac {\cos (e+f x) (a \sin (e+f x)+a)^m \operatorname {Hypergeometric2F1}\left (3,m+\frac {1}{2},m+\frac {3}{2},\frac {1}{2} (\sin (e+f x)+1)\right )}{4 c^2 f (2 m+1) \sqrt {c-c \sin (e+f x)}} \]
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Rule 70
Rule 2746
Rule 2824
Rubi steps \begin{align*} \text {integral}& = \frac {\cos (e+f x) \int \sec ^5(e+f x) (a+a \sin (e+f x))^{\frac {5}{2}+m} \, dx}{a^2 c^2 \sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}} \\ & = \frac {\left (a^3 \cos (e+f x)\right ) \text {Subst}\left (\int \frac {(a+x)^{-\frac {1}{2}+m}}{(a-x)^3} \, dx,x,a \sin (e+f x)\right )}{c^2 f \sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}} \\ & = \frac {\cos (e+f x) \operatorname {Hypergeometric2F1}\left (3,\frac {1}{2}+m,\frac {3}{2}+m,\frac {1}{2} (1+\sin (e+f x))\right ) (a+a \sin (e+f x))^m}{4 c^2 f (1+2 m) \sqrt {c-c \sin (e+f x)}} \\ \end{align*}
Time = 5.67 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.99 \[ \int \frac {(3+3 \sin (e+f x))^m}{(c-c \sin (e+f x))^{5/2}} \, dx=\frac {3^m \cos (e+f x) \operatorname {Hypergeometric2F1}\left (3,\frac {1}{2}+m,\frac {3}{2}+m,\frac {1}{2} (1+\sin (e+f x))\right ) (1+\sin (e+f x))^m}{4 c^2 (f+2 f m) \sqrt {c-c \sin (e+f x)}} \]
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\[\int \frac {\left (a +a \sin \left (f x +e \right )\right )^{m}}{\left (c -c \sin \left (f x +e \right )\right )^{\frac {5}{2}}}d x\]
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\[ \int \frac {(3+3 \sin (e+f x))^m}{(c-c \sin (e+f x))^{5/2}} \, dx=\int { \frac {{\left (a \sin \left (f x + e\right ) + a\right )}^{m}}{{\left (-c \sin \left (f x + e\right ) + c\right )}^{\frac {5}{2}}} \,d x } \]
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\[ \int \frac {(3+3 \sin (e+f x))^m}{(c-c \sin (e+f x))^{5/2}} \, dx=\int \frac {\left (a \left (\sin {\left (e + f x \right )} + 1\right )\right )^{m}}{\left (- c \left (\sin {\left (e + f x \right )} - 1\right )\right )^{\frac {5}{2}}}\, dx \]
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\[ \int \frac {(3+3 \sin (e+f x))^m}{(c-c \sin (e+f x))^{5/2}} \, dx=\int { \frac {{\left (a \sin \left (f x + e\right ) + a\right )}^{m}}{{\left (-c \sin \left (f x + e\right ) + c\right )}^{\frac {5}{2}}} \,d x } \]
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Exception generated. \[ \int \frac {(3+3 \sin (e+f x))^m}{(c-c \sin (e+f x))^{5/2}} \, dx=\text {Exception raised: TypeError} \]
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Timed out. \[ \int \frac {(3+3 \sin (e+f x))^m}{(c-c \sin (e+f x))^{5/2}} \, dx=\int \frac {{\left (a+a\,\sin \left (e+f\,x\right )\right )}^m}{{\left (c-c\,\sin \left (e+f\,x\right )\right )}^{5/2}} \,d x \]
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