Integrand size = 29, antiderivative size = 80 \[ \int (3+3 \sin (e+f x))^m (3+5 \sin (e+f x))^{-1-m} \, dx=-\frac {4^{-1-m} \cos (e+f x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},1+m,\frac {3}{2},\frac {3-3 \sin (e+f x)}{4 (3+3 \sin (e+f x))}\right ) (1+\sin (e+f x))^{-1-m} (3+3 \sin (e+f x))^m}{f} \]
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Time = 0.08 (sec) , antiderivative size = 115, normalized size of antiderivative = 1.44, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.069, Rules used = {2867, 134} \[ \int (3+3 \sin (e+f x))^m (3+5 \sin (e+f x))^{-1-m} \, dx=-\frac {\sqrt {\frac {1-\sin (e+f x)}{\sin (e+f x)+1}} \cos (e+f x) (5 \sin (e+f x)+3)^{-m} (a \sin (e+f x)+a)^m \operatorname {Hypergeometric2F1}\left (\frac {1}{2},-m,1-m,\frac {5 \sin (e+f x)+3}{4 (\sin (e+f x)+1)}\right )}{4 f m (1-\sin (e+f x))} \]
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Rule 134
Rule 2867
Rubi steps \begin{align*} \text {integral}& = \frac {\left (a^2 \cos (e+f x)\right ) \text {Subst}\left (\int \frac {(3+5 x)^{-1-m} (a+a x)^{-\frac {1}{2}+m}}{\sqrt {a-a x}} \, dx,x,\sin (e+f x)\right )}{f \sqrt {a-a \sin (e+f x)} \sqrt {a+a \sin (e+f x)}} \\ & = -\frac {\cos (e+f x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},-m,1-m,\frac {3+5 \sin (e+f x)}{4 (1+\sin (e+f x))}\right ) \sqrt {\frac {1-\sin (e+f x)}{1+\sin (e+f x)}} (3+5 \sin (e+f x))^{-m} (a+a \sin (e+f x))^m}{4 f m (1-\sin (e+f x))} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(199\) vs. \(2(80)=160\).
Time = 7.76 (sec) , antiderivative size = 199, normalized size of antiderivative = 2.49 \[ \int (3+3 \sin (e+f x))^m (3+5 \sin (e+f x))^{-1-m} \, dx=-\frac {3^m \operatorname {Hypergeometric2F1}\left (1+m,1+2 m,2 (1+m),\frac {4 \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )}{3 \cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )}\right ) \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right ) \left (-\frac {\cos \left (\frac {1}{2} (e+f x)\right )+3 \sin \left (\frac {1}{2} (e+f x)\right )}{3 \cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )}\right )^m (1+\sin (e+f x))^m (3+5 \sin (e+f x))^{-m}}{f (1+2 m) \left (3 \cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )} \]
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\[\int \left (3+5 \sin \left (f x +e \right )\right )^{-1-m} \left (a +a \sin \left (f x +e \right )\right )^{m}d x\]
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\[ \int (3+3 \sin (e+f x))^m (3+5 \sin (e+f x))^{-1-m} \, dx=\int { {\left (a \sin \left (f x + e\right ) + a\right )}^{m} {\left (5 \, \sin \left (f x + e\right ) + 3\right )}^{-m - 1} \,d x } \]
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\[ \int (3+3 \sin (e+f x))^m (3+5 \sin (e+f x))^{-1-m} \, dx=\int \left (a \left (\sin {\left (e + f x \right )} + 1\right )\right )^{m} \left (5 \sin {\left (e + f x \right )} + 3\right )^{- m - 1}\, dx \]
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\[ \int (3+3 \sin (e+f x))^m (3+5 \sin (e+f x))^{-1-m} \, dx=\int { {\left (a \sin \left (f x + e\right ) + a\right )}^{m} {\left (5 \, \sin \left (f x + e\right ) + 3\right )}^{-m - 1} \,d x } \]
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\[ \int (3+3 \sin (e+f x))^m (3+5 \sin (e+f x))^{-1-m} \, dx=\int { {\left (a \sin \left (f x + e\right ) + a\right )}^{m} {\left (5 \, \sin \left (f x + e\right ) + 3\right )}^{-m - 1} \,d x } \]
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Timed out. \[ \int (3+3 \sin (e+f x))^m (3+5 \sin (e+f x))^{-1-m} \, dx=\int \frac {{\left (a+a\,\sin \left (e+f\,x\right )\right )}^m}{{\left (5\,\sin \left (e+f\,x\right )+3\right )}^{m+1}} \,d x \]
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