Integrand size = 29, antiderivative size = 128 \[ \int (3+3 \sin (e+f x))^m (c+d \sin (e+f x))^{-1-m} \, dx=-\frac {3\ 2^{\frac {1}{2}+m} \cos (e+f x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{2}-m,\frac {3}{2},\frac {(c-d) (1-\sin (e+f x))}{2 (c+d \sin (e+f x))}\right ) (3+3 \sin (e+f x))^{-1+m} \left (\frac {(c+d) (1+\sin (e+f x))}{c+d \sin (e+f x)}\right )^{\frac {1}{2}-m} (c+d \sin (e+f x))^{-m}}{(c+d) f} \]
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Time = 0.11 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.01, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.069, Rules used = {2867, 134} \[ \int (3+3 \sin (e+f x))^m (c+d \sin (e+f x))^{-1-m} \, dx=-\frac {a 2^{m+\frac {1}{2}} \cos (e+f x) (a \sin (e+f x)+a)^{m-1} \left (\frac {(c+d) (\sin (e+f x)+1)}{c+d \sin (e+f x)}\right )^{\frac {1}{2}-m} (c+d \sin (e+f x))^{-m} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{2}-m,\frac {3}{2},\frac {(c-d) (1-\sin (e+f x))}{2 (c+d \sin (e+f x))}\right )}{f (c+d)} \]
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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 {(a+a x)^{-\frac {1}{2}+m} (c+d x)^{-1-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 {2^{\frac {1}{2}+m} a \cos (e+f x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{2}-m,\frac {3}{2},\frac {(c-d) (1-\sin (e+f x))}{2 (c+d \sin (e+f x))}\right ) (a+a \sin (e+f x))^{-1+m} \left (\frac {(c+d) (1+\sin (e+f x))}{c+d \sin (e+f x)}\right )^{\frac {1}{2}-m} (c+d \sin (e+f x))^{-m}}{(c+d) f} \\ \end{align*}
Result contains complex when optimal does not.
Time = 11.36 (sec) , antiderivative size = 353, normalized size of antiderivative = 2.76 \[ \int (3+3 \sin (e+f x))^m (c+d \sin (e+f x))^{-1-m} \, dx=\frac {3^m \operatorname {Hypergeometric2F1}\left (1+m,1+2 m,2 (1+m),\frac {2 i d \sqrt {c^2-d^2} (\cos (e+f x)+i (1+\sin (e+f x)))}{\left (c-d+\sqrt {c^2-d^2}\right ) \left (-c+\sqrt {c^2-d^2}+i d \cos (e+f x)-d \sin (e+f x)\right )}\right ) (\cos (e+f x)-i \sin (e+f x)) (1+\sin (e+f x))^m (1-i \cos (e+f x)+\sin (e+f x)) (c+d \sin (e+f x))^{-1-m} \left (c+\sqrt {c^2-d^2}-i d \cos (e+f x)+d \sin (e+f x)\right ) \left (\frac {\left (-c+d+\sqrt {c^2-d^2}\right ) \left (c+\sqrt {c^2-d^2}-i d \cos (e+f x)+d \sin (e+f x)\right )}{\left (c-d+\sqrt {c^2-d^2}\right ) \left (-c+\sqrt {c^2-d^2}+i d \cos (e+f x)-d \sin (e+f x)\right )}\right )^m}{\left (c-d+\sqrt {c^2-d^2}\right ) f (1+2 m)} \]
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\[\int \left (a +a \sin \left (f x +e \right )\right )^{m} \left (c +d \sin \left (f x +e \right )\right )^{-1-m}d x\]
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\[ \int (3+3 \sin (e+f x))^m (c+d \sin (e+f x))^{-1-m} \, dx=\int { {\left (a \sin \left (f x + e\right ) + a\right )}^{m} {\left (d \sin \left (f x + e\right ) + c\right )}^{-m - 1} \,d x } \]
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\[ \int (3+3 \sin (e+f x))^m (c+d \sin (e+f x))^{-1-m} \, dx=\int \left (a \left (\sin {\left (e + f x \right )} + 1\right )\right )^{m} \left (c + d \sin {\left (e + f x \right )}\right )^{- m - 1}\, dx \]
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\[ \int (3+3 \sin (e+f x))^m (c+d \sin (e+f x))^{-1-m} \, dx=\int { {\left (a \sin \left (f x + e\right ) + a\right )}^{m} {\left (d \sin \left (f x + e\right ) + c\right )}^{-m - 1} \,d x } \]
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\[ \int (3+3 \sin (e+f x))^m (c+d \sin (e+f x))^{-1-m} \, dx=\int { {\left (a \sin \left (f x + e\right ) + a\right )}^{m} {\left (d \sin \left (f x + e\right ) + c\right )}^{-m - 1} \,d x } \]
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Timed out. \[ \int (3+3 \sin (e+f x))^m (c+d \sin (e+f x))^{-1-m} \, dx=\int \frac {{\left (a+a\,\sin \left (e+f\,x\right )\right )}^m}{{\left (c+d\,\sin \left (e+f\,x\right )\right )}^{m+1}} \,d x \]
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