Integrand size = 36, antiderivative size = 50 \[ \int \frac {\cos ^2(e+f x) (a+a \sin (e+f x))^m}{\sqrt {c-c \sin (e+f x)}} \, dx=\frac {2 \cos (e+f x) (a+a \sin (e+f x))^{1+m}}{a f (3+2 m) \sqrt {c-c \sin (e+f x)}} \]
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Time = 0.18 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, Rules used = {2920, 2817} \[ \int \frac {\cos ^2(e+f x) (a+a \sin (e+f x))^m}{\sqrt {c-c \sin (e+f x)}} \, dx=\frac {2 \cos (e+f x) (a \sin (e+f x)+a)^{m+1}}{a f (2 m+3) \sqrt {c-c \sin (e+f x)}} \]
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Rule 2817
Rule 2920
Rubi steps \begin{align*} \text {integral}& = \frac {\int (a+a \sin (e+f x))^{1+m} \sqrt {c-c \sin (e+f x)} \, dx}{a c} \\ & = \frac {2 \cos (e+f x) (a+a \sin (e+f x))^{1+m}}{a f (3+2 m) \sqrt {c-c \sin (e+f x)}} \\ \end{align*}
Time = 2.05 (sec) , antiderivative size = 85, normalized size of antiderivative = 1.70 \[ \int \frac {\cos ^2(e+f x) (a+a \sin (e+f x))^m}{\sqrt {c-c \sin (e+f x)}} \, dx=\frac {2 \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right ) \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^3 (a (1+\sin (e+f x)))^m}{f (3+2 m) \sqrt {c-c \sin (e+f x)}} \]
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\[\int \frac {\left (\cos ^{2}\left (f x +e \right )\right ) \left (a +a \sin \left (f x +e \right )\right )^{m}}{\sqrt {c -c \sin \left (f x +e \right )}}d x\]
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Leaf count of result is larger than twice the leaf count of optimal. 108 vs. \(2 (48) = 96\).
Time = 0.29 (sec) , antiderivative size = 108, normalized size of antiderivative = 2.16 \[ \int \frac {\cos ^2(e+f x) (a+a \sin (e+f x))^m}{\sqrt {c-c \sin (e+f x)}} \, dx=-\frac {2 \, {\left (\cos \left (f x + e\right )^{2} - {\left (\cos \left (f x + e\right ) + 2\right )} \sin \left (f x + e\right ) - \cos \left (f x + e\right ) - 2\right )} \sqrt {-c \sin \left (f x + e\right ) + c} {\left (a \sin \left (f x + e\right ) + a\right )}^{m}}{2 \, c f m + 3 \, c f + {\left (2 \, c f m + 3 \, c f\right )} \cos \left (f x + e\right ) - {\left (2 \, c f m + 3 \, c f\right )} \sin \left (f x + e\right )} \]
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\[ \int \frac {\cos ^2(e+f x) (a+a \sin (e+f x))^m}{\sqrt {c-c \sin (e+f x)}} \, dx=\int \frac {\left (a \left (\sin {\left (e + f x \right )} + 1\right )\right )^{m} \cos ^{2}{\left (e + f x \right )}}{\sqrt {- c \left (\sin {\left (e + f x \right )} - 1\right )}}\, dx \]
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Timed out. \[ \int \frac {\cos ^2(e+f x) (a+a \sin (e+f x))^m}{\sqrt {c-c \sin (e+f x)}} \, dx=\text {Timed out} \]
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Timed out. \[ \int \frac {\cos ^2(e+f x) (a+a \sin (e+f x))^m}{\sqrt {c-c \sin (e+f x)}} \, dx=\text {Timed out} \]
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Time = 1.01 (sec) , antiderivative size = 68, normalized size of antiderivative = 1.36 \[ \int \frac {\cos ^2(e+f x) (a+a \sin (e+f x))^m}{\sqrt {c-c \sin (e+f x)}} \, dx=-\frac {{\left (a\,\left (\sin \left (e+f\,x\right )+1\right )\right )}^m\,\sqrt {-c\,\left (\sin \left (e+f\,x\right )-1\right )}\,\left (2\,\cos \left (e+f\,x\right )+\sin \left (2\,e+2\,f\,x\right )\right )}{c\,f\,\left (2\,m+3\right )\,\left (\sin \left (e+f\,x\right )-1\right )} \]
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