Integrand size = 36, antiderivative size = 76 \[ \int \frac {\cos ^2(e+f x) (a+a \sin (e+f x))^m}{(c-c \sin (e+f x))^{3/2}} \, dx=\frac {\cos (e+f x) \operatorname {Hypergeometric2F1}\left (1,\frac {3}{2}+m,\frac {5}{2}+m,\frac {1}{2} (1+\sin (e+f x))\right ) (a+a \sin (e+f x))^{1+m}}{a c f (3+2 m) \sqrt {c-c \sin (e+f x)}} \]
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Time = 0.25 (sec) , antiderivative size = 76, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {2920, 2824, 2746, 70} \[ \int \frac {\cos ^2(e+f x) (a+a \sin (e+f x))^m}{(c-c \sin (e+f x))^{3/2}} \, dx=\frac {\cos (e+f x) (a \sin (e+f x)+a)^{m+1} \operatorname {Hypergeometric2F1}\left (1,m+\frac {3}{2},m+\frac {5}{2},\frac {1}{2} (\sin (e+f x)+1)\right )}{a c f (2 m+3) \sqrt {c-c \sin (e+f x)}} \]
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Rule 70
Rule 2746
Rule 2824
Rule 2920
Rubi steps \begin{align*} \text {integral}& = \frac {\int \frac {(a+a \sin (e+f x))^{1+m}}{\sqrt {c-c \sin (e+f x)}} \, dx}{a c} \\ & = \frac {\cos (e+f x) \int \sec (e+f x) (a+a \sin (e+f x))^{\frac {3}{2}+m} \, dx}{a c \sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}} \\ & = \frac {\cos (e+f x) \text {Subst}\left (\int \frac {(a+x)^{\frac {1}{2}+m}}{a-x} \, dx,x,a \sin (e+f x)\right )}{c f \sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}} \\ & = \frac {\cos (e+f x) \operatorname {Hypergeometric2F1}\left (1,\frac {3}{2}+m,\frac {5}{2}+m,\frac {1}{2} (1+\sin (e+f x))\right ) (a+a \sin (e+f x))^{1+m}}{a c f (3+2 m) \sqrt {c-c \sin (e+f x)}} \\ \end{align*}
Time = 50.65 (sec) , antiderivative size = 130, normalized size of antiderivative = 1.71 \[ \int \frac {\cos ^2(e+f x) (a+a \sin (e+f x))^m}{(c-c \sin (e+f x))^{3/2}} \, dx=\frac {\cos (e+f x) (a (1+\sin (e+f x)))^m \left (-6-4 m+(6+4 m) \operatorname {Hypergeometric2F1}\left (1,\frac {1}{2}+m,\frac {3}{2}+m,\frac {1}{2} (1+\sin (e+f x))\right )+(1+2 m) \operatorname {Hypergeometric2F1}\left (1,\frac {3}{2}+m,\frac {5}{2}+m,\frac {1}{2} (1+\sin (e+f x))\right ) (1+\sin (e+f x))\right )}{2 c f (1+2 m) (3+2 m) \sqrt {c-c \sin (e+f x)}} \]
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\[\int \frac {\left (\cos ^{2}\left (f x +e \right )\right ) \left (a +a \sin \left (f x +e \right )\right )^{m}}{\left (c -c \sin \left (f x +e \right )\right )^{\frac {3}{2}}}d x\]
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\[ \int \frac {\cos ^2(e+f x) (a+a \sin (e+f x))^m}{(c-c \sin (e+f x))^{3/2}} \, dx=\int { \frac {{\left (a \sin \left (f x + e\right ) + a\right )}^{m} \cos \left (f x + e\right )^{2}}{{\left (-c \sin \left (f x + e\right ) + c\right )}^{\frac {3}{2}}} \,d x } \]
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\[ \int \frac {\cos ^2(e+f x) (a+a \sin (e+f x))^m}{(c-c \sin (e+f x))^{3/2}} \, dx=\int \frac {\left (a \left (\sin {\left (e + f x \right )} + 1\right )\right )^{m} \cos ^{2}{\left (e + f x \right )}}{\left (- c \left (\sin {\left (e + f x \right )} - 1\right )\right )^{\frac {3}{2}}}\, dx \]
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\[ \int \frac {\cos ^2(e+f x) (a+a \sin (e+f x))^m}{(c-c \sin (e+f x))^{3/2}} \, dx=\int { \frac {{\left (a \sin \left (f x + e\right ) + a\right )}^{m} \cos \left (f x + e\right )^{2}}{{\left (-c \sin \left (f x + e\right ) + c\right )}^{\frac {3}{2}}} \,d x } \]
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Timed out. \[ \int \frac {\cos ^2(e+f x) (a+a \sin (e+f x))^m}{(c-c \sin (e+f x))^{3/2}} \, dx=\text {Timed out} \]
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Timed out. \[ \int \frac {\cos ^2(e+f x) (a+a \sin (e+f x))^m}{(c-c \sin (e+f x))^{3/2}} \, dx=\int \frac {{\cos \left (e+f\,x\right )}^2\,{\left (a+a\,\sin \left (e+f\,x\right )\right )}^m}{{\left (c-c\,\sin \left (e+f\,x\right )\right )}^{3/2}} \,d x \]
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