Integrand size = 26, antiderivative size = 76 \[ \int \frac {(3+3 \sin (e+f x))^m}{c-c \sin (e+f x)} \, dx=\frac {2^{\frac {1}{2}+m} \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},\frac {1}{2}-m,\frac {1}{2},\frac {1}{2} (1-\sin (e+f x))\right ) \sec (e+f x) (1+\sin (e+f x))^{\frac {1}{2}-m} (3+3 \sin (e+f x))^m}{c f} \]
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Time = 0.10 (sec) , antiderivative size = 76, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {2815, 2768, 72, 71} \[ \int \frac {(3+3 \sin (e+f x))^m}{c-c \sin (e+f x)} \, dx=\frac {2^{m+\frac {1}{2}} \sec (e+f x) (\sin (e+f x)+1)^{\frac {1}{2}-m} (a \sin (e+f x)+a)^m \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},\frac {1}{2}-m,\frac {1}{2},\frac {1}{2} (1-\sin (e+f x))\right )}{c f} \]
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Rule 71
Rule 72
Rule 2768
Rule 2815
Rubi steps \begin{align*} \text {integral}& = \frac {\int \sec ^2(e+f x) (a+a \sin (e+f x))^{1+m} \, dx}{a c} \\ & = \frac {\left (a \sec (e+f x) \sqrt {a-a \sin (e+f x)} \sqrt {a+a \sin (e+f x)}\right ) \text {Subst}\left (\int \frac {(a+a x)^{-\frac {1}{2}+m}}{(a-a x)^{3/2}} \, dx,x,\sin (e+f x)\right )}{c f} \\ & = \frac {\left (2^{-\frac {1}{2}+m} a \sec (e+f x) \sqrt {a-a \sin (e+f x)} (a+a \sin (e+f x))^m \left (\frac {a+a \sin (e+f x)}{a}\right )^{\frac {1}{2}-m}\right ) \text {Subst}\left (\int \frac {\left (\frac {1}{2}+\frac {x}{2}\right )^{-\frac {1}{2}+m}}{(a-a x)^{3/2}} \, dx,x,\sin (e+f x)\right )}{c f} \\ & = \frac {2^{\frac {1}{2}+m} \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},\frac {1}{2}-m,\frac {1}{2},\frac {1}{2} (1-\sin (e+f x))\right ) \sec (e+f x) (1+\sin (e+f x))^{\frac {1}{2}-m} (a+a \sin (e+f x))^m}{c f} \\ \end{align*}
\[ \int \frac {(3+3 \sin (e+f x))^m}{c-c \sin (e+f x)} \, dx=\int \frac {(3+3 \sin (e+f x))^m}{c-c \sin (e+f x)} \, dx \]
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\[\int \frac {\left (a +a \sin \left (f x +e \right )\right )^{m}}{c -c \sin \left (f x +e \right )}d x\]
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\[ \int \frac {(3+3 \sin (e+f x))^m}{c-c \sin (e+f x)} \, dx=\int { -\frac {{\left (a \sin \left (f x + e\right ) + a\right )}^{m}}{c \sin \left (f x + e\right ) - c} \,d x } \]
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\[ \int \frac {(3+3 \sin (e+f x))^m}{c-c \sin (e+f x)} \, dx=- \frac {\int \frac {\left (a \sin {\left (e + f x \right )} + a\right )^{m}}{\sin {\left (e + f x \right )} - 1}\, dx}{c} \]
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\[ \int \frac {(3+3 \sin (e+f x))^m}{c-c \sin (e+f x)} \, dx=\int { -\frac {{\left (a \sin \left (f x + e\right ) + a\right )}^{m}}{c \sin \left (f x + e\right ) - c} \,d x } \]
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\[ \int \frac {(3+3 \sin (e+f x))^m}{c-c \sin (e+f x)} \, dx=\int { -\frac {{\left (a \sin \left (f x + e\right ) + a\right )}^{m}}{c \sin \left (f x + e\right ) - c} \,d x } \]
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Timed out. \[ \int \frac {(3+3 \sin (e+f x))^m}{c-c \sin (e+f x)} \, dx=\int \frac {{\left (a+a\,\sin \left (e+f\,x\right )\right )}^m}{c-c\,\sin \left (e+f\,x\right )} \,d x \]
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