Integrand size = 25, antiderivative size = 100 \[ \int \frac {(3+3 \sin (e+f x))^m}{(c+d \sin (e+f x))^3} \, dx=\frac {\sqrt {2} \operatorname {AppellF1}\left (\frac {1}{2}+m,\frac {1}{2},3,\frac {3}{2}+m,\frac {1}{2} (1+\sin (e+f x)),-\frac {d (1+\sin (e+f x))}{c-d}\right ) \cos (e+f x) (3+3 \sin (e+f x))^m}{(c-d)^3 f (1+2 m) \sqrt {1-\sin (e+f x)}} \]
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Time = 0.09 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.120, Rules used = {2867, 142, 141} \[ \int \frac {(3+3 \sin (e+f x))^m}{(c+d \sin (e+f x))^3} \, dx=\frac {\sqrt {2} \cos (e+f x) (a \sin (e+f x)+a)^m \operatorname {AppellF1}\left (m+\frac {1}{2},\frac {1}{2},3,m+\frac {3}{2},\frac {1}{2} (\sin (e+f x)+1),-\frac {d (\sin (e+f x)+1)}{c-d}\right )}{f (2 m+1) (c-d)^3 \sqrt {1-\sin (e+f x)}} \]
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Rule 141
Rule 142
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} (c+d x)^3} \, 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}} (c+d x)^3} \, dx,x,\sin (e+f x)\right )}{\sqrt {2} f (a-a \sin (e+f x)) \sqrt {a+a \sin (e+f x)}} \\ & = \frac {\sqrt {2} \operatorname {AppellF1}\left (\frac {1}{2}+m,\frac {1}{2},3,\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}{(c-d)^3 f (1+2 m) \sqrt {1-\sin (e+f x)}} \\ \end{align*}
\[ \int \frac {(3+3 \sin (e+f x))^m}{(c+d \sin (e+f x))^3} \, dx=\int \frac {(3+3 \sin (e+f x))^m}{(c+d \sin (e+f x))^3} \, dx \]
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\[\int \frac {\left (a +a \sin \left (f x +e \right )\right )^{m}}{\left (c +d \sin \left (f x +e \right )\right )^{3}}d x\]
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\[ \int \frac {(3+3 \sin (e+f x))^m}{(c+d \sin (e+f x))^3} \, dx=\int { \frac {{\left (a \sin \left (f x + e\right ) + a\right )}^{m}}{{\left (d \sin \left (f x + e\right ) + c\right )}^{3}} \,d x } \]
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Timed out. \[ \int \frac {(3+3 \sin (e+f x))^m}{(c+d \sin (e+f x))^3} \, dx=\text {Timed out} \]
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\[ \int \frac {(3+3 \sin (e+f x))^m}{(c+d \sin (e+f x))^3} \, dx=\int { \frac {{\left (a \sin \left (f x + e\right ) + a\right )}^{m}}{{\left (d \sin \left (f x + e\right ) + c\right )}^{3}} \,d x } \]
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\[ \int \frac {(3+3 \sin (e+f x))^m}{(c+d \sin (e+f x))^3} \, dx=\int { \frac {{\left (a \sin \left (f x + e\right ) + a\right )}^{m}}{{\left (d \sin \left (f x + e\right ) + c\right )}^{3}} \,d x } \]
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Timed out. \[ \int \frac {(3+3 \sin (e+f x))^m}{(c+d \sin (e+f x))^3} \, dx=\int \frac {{\left (a+a\,\sin \left (e+f\,x\right )\right )}^m}{{\left (c+d\,\sin \left (e+f\,x\right )\right )}^3} \,d x \]
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