Integrand size = 38, antiderivative size = 241 \[ \int \frac {\cos ^2(e+f x) (a+a \sin (e+f x))^{7/2}}{(c-c \sin (e+f x))^{9/2}} \, dx=\frac {\cos (e+f x) (a+a \sin (e+f x))^{7/2}}{3 c f (c-c \sin (e+f x))^{7/2}}-\frac {2 a \cos (e+f x) (a+a \sin (e+f x))^{5/2}}{3 c^2 f (c-c \sin (e+f x))^{5/2}}+\frac {2 a^2 \cos (e+f x) (a+a \sin (e+f x))^{3/2}}{c^3 f (c-c \sin (e+f x))^{3/2}}+\frac {8 a^4 \cos (e+f x) \log (1-\sin (e+f x))}{c^4 f \sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}}+\frac {4 a^3 \cos (e+f x) \sqrt {a+a \sin (e+f x)}}{c^4 f \sqrt {c-c \sin (e+f x)}} \]
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Time = 0.56 (sec) , antiderivative size = 241, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {2920, 2818, 2819, 2816, 2746, 31} \[ \int \frac {\cos ^2(e+f x) (a+a \sin (e+f x))^{7/2}}{(c-c \sin (e+f x))^{9/2}} \, dx=\frac {8 a^4 \cos (e+f x) \log (1-\sin (e+f x))}{c^4 f \sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}+\frac {4 a^3 \cos (e+f x) \sqrt {a \sin (e+f x)+a}}{c^4 f \sqrt {c-c \sin (e+f x)}}+\frac {2 a^2 \cos (e+f x) (a \sin (e+f x)+a)^{3/2}}{c^3 f (c-c \sin (e+f x))^{3/2}}-\frac {2 a \cos (e+f x) (a \sin (e+f x)+a)^{5/2}}{3 c^2 f (c-c \sin (e+f x))^{5/2}}+\frac {\cos (e+f x) (a \sin (e+f x)+a)^{7/2}}{3 c f (c-c \sin (e+f x))^{7/2}} \]
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Rule 31
Rule 2746
Rule 2816
Rule 2818
Rule 2819
Rule 2920
Rubi steps \begin{align*} \text {integral}& = \frac {\int \frac {(a+a \sin (e+f x))^{9/2}}{(c-c \sin (e+f x))^{7/2}} \, dx}{a c} \\ & = \frac {\cos (e+f x) (a+a \sin (e+f x))^{7/2}}{3 c f (c-c \sin (e+f x))^{7/2}}-\frac {4 \int \frac {(a+a \sin (e+f x))^{7/2}}{(c-c \sin (e+f x))^{5/2}} \, dx}{3 c^2} \\ & = \frac {\cos (e+f x) (a+a \sin (e+f x))^{7/2}}{3 c f (c-c \sin (e+f x))^{7/2}}-\frac {2 a \cos (e+f x) (a+a \sin (e+f x))^{5/2}}{3 c^2 f (c-c \sin (e+f x))^{5/2}}+\frac {(2 a) \int \frac {(a+a \sin (e+f x))^{5/2}}{(c-c \sin (e+f x))^{3/2}} \, dx}{c^3} \\ & = \frac {\cos (e+f x) (a+a \sin (e+f x))^{7/2}}{3 c f (c-c \sin (e+f x))^{7/2}}-\frac {2 a \cos (e+f x) (a+a \sin (e+f x))^{5/2}}{3 c^2 f (c-c \sin (e+f x))^{5/2}}+\frac {2 a^2 \cos (e+f x) (a+a \sin (e+f x))^{3/2}}{c^3 f (c-c \sin (e+f x))^{3/2}}-\frac {\left (4 a^2\right ) \int \frac {(a+a \sin (e+f x))^{3/2}}{\sqrt {c-c \sin (e+f x)}} \, dx}{c^4} \\ & = \frac {\cos (e+f x) (a+a \sin (e+f x))^{7/2}}{3 c f (c-c \sin (e+f x))^{7/2}}-\frac {2 a \cos (e+f x) (a+a \sin (e+f x))^{5/2}}{3 c^2 f (c-c \sin (e+f x))^{5/2}}+\frac {2 a^2 \cos (e+f x) (a+a \sin (e+f x))^{3/2}}{c^3 f (c-c \sin (e+f x))^{3/2}}+\frac {4 a^3 \cos (e+f x) \sqrt {a+a \sin (e+f x)}}{c^4 f \sqrt {c-c \sin (e+f x)}}-\frac {\left (8 a^3\right ) \int \frac {\sqrt {a+a \sin (e+f x)}}{\sqrt {c-c \sin (e+f x)}} \, dx}{c^4} \\ & = \frac {\cos (e+f x) (a+a \sin (e+f x))^{7/2}}{3 c f (c-c \sin (e+f x))^{7/2}}-\frac {2 a \cos (e+f x) (a+a \sin (e+f x))^{5/2}}{3 c^2 f (c-c \sin (e+f x))^{5/2}}+\frac {2 a^2 \cos (e+f x) (a+a \sin (e+f x))^{3/2}}{c^3 f (c-c \sin (e+f x))^{3/2}}+\frac {4 a^3 \cos (e+f x) \sqrt {a+a \sin (e+f x)}}{c^4 f \sqrt {c-c \sin (e+f x)}}-\frac {\left (8 a^4 \cos (e+f x)\right ) \int \frac {\cos (e+f x)}{c-c \sin (e+f x)} \, dx}{c^3 \sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}} \\ & = \frac {\cos (e+f x) (a+a \sin (e+f x))^{7/2}}{3 c f (c-c \sin (e+f x))^{7/2}}-\frac {2 a \cos (e+f x) (a+a \sin (e+f x))^{5/2}}{3 c^2 f (c-c \sin (e+f x))^{5/2}}+\frac {2 a^2 \cos (e+f x) (a+a \sin (e+f x))^{3/2}}{c^3 f (c-c \sin (e+f x))^{3/2}}+\frac {4 a^3 \cos (e+f x) \sqrt {a+a \sin (e+f x)}}{c^4 f \sqrt {c-c \sin (e+f x)}}+\frac {\left (8 a^4 \cos (e+f x)\right ) \text {Subst}\left (\int \frac {1}{c+x} \, dx,x,-c \sin (e+f x)\right )}{c^4 f \sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}} \\ & = \frac {\cos (e+f x) (a+a \sin (e+f x))^{7/2}}{3 c f (c-c \sin (e+f x))^{7/2}}-\frac {2 a \cos (e+f x) (a+a \sin (e+f x))^{5/2}}{3 c^2 f (c-c \sin (e+f x))^{5/2}}+\frac {2 a^2 \cos (e+f x) (a+a \sin (e+f x))^{3/2}}{c^3 f (c-c \sin (e+f x))^{3/2}}+\frac {8 a^4 \cos (e+f x) \log (1-\sin (e+f x))}{c^4 f \sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}}+\frac {4 a^3 \cos (e+f x) \sqrt {a+a \sin (e+f x)}}{c^4 f \sqrt {c-c \sin (e+f x)}} \\ \end{align*}
Time = 13.80 (sec) , antiderivative size = 260, normalized size of antiderivative = 1.08 \[ \int \frac {\cos ^2(e+f x) (a+a \sin (e+f x))^{7/2}}{(c-c \sin (e+f x))^{9/2}} \, dx=-\frac {a^3 \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^3 \sqrt {a (1+\sin (e+f x))} \left (-563+3 \cos (4 (e+f x))-960 \log \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )+48 \cos (2 (e+f x)) \left (5+12 \log \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )\right )+690 \sin (e+f x)+1440 \log \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right ) \sin (e+f x)+18 \sin (3 (e+f x))-96 \log \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right ) \sin (3 (e+f x))\right )}{24 c^4 f \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right ) (-1+\sin (e+f x))^4 \sqrt {c-c \sin (e+f x)}} \]
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Time = 0.34 (sec) , antiderivative size = 301, normalized size of antiderivative = 1.25
method | result | size |
default | \(-\frac {\left (48 \left (\cos ^{2}\left (f x +e \right )\right ) \sin \left (f x +e \right ) \ln \left (\csc \left (f x +e \right )-\cot \left (f x +e \right )-1\right )-24 \left (\cos ^{2}\left (f x +e \right )\right ) \sin \left (f x +e \right ) \ln \left (\frac {2}{1+\cos \left (f x +e \right )}\right )-3 \left (\cos ^{4}\left (f x +e \right )\right )-144 \left (\cos ^{2}\left (f x +e \right )\right ) \ln \left (\csc \left (f x +e \right )-\cot \left (f x +e \right )-1\right )+72 \left (\cos ^{2}\left (f x +e \right )\right ) \ln \left (\frac {2}{1+\cos \left (f x +e \right )}\right )-49 \left (\cos ^{2}\left (f x +e \right )\right ) \sin \left (f x +e \right )-192 \ln \left (\csc \left (f x +e \right )-\cot \left (f x +e \right )-1\right ) \sin \left (f x +e \right )+96 \ln \left (\frac {2}{1+\cos \left (f x +e \right )}\right ) \sin \left (f x +e \right )+63 \left (\cos ^{2}\left (f x +e \right )\right )+192 \ln \left (\csc \left (f x +e \right )-\cot \left (f x +e \right )-1\right )-96 \ln \left (\frac {2}{1+\cos \left (f x +e \right )}\right )+76 \sin \left (f x +e \right )-60\right ) \sqrt {a \left (1+\sin \left (f x +e \right )\right )}\, a^{3} \sec \left (f x +e \right )}{3 f \left (\cos ^{2}\left (f x +e \right )+2 \sin \left (f x +e \right )-2\right ) \sqrt {-c \left (\sin \left (f x +e \right )-1\right )}\, c^{4}}\) | \(301\) |
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\[ \int \frac {\cos ^2(e+f x) (a+a \sin (e+f x))^{7/2}}{(c-c \sin (e+f x))^{9/2}} \, dx=\int { \frac {{\left (a \sin \left (f x + e\right ) + a\right )}^{\frac {7}{2}} \cos \left (f x + e\right )^{2}}{{\left (-c \sin \left (f x + e\right ) + c\right )}^{\frac {9}{2}}} \,d x } \]
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Timed out. \[ \int \frac {\cos ^2(e+f x) (a+a \sin (e+f x))^{7/2}}{(c-c \sin (e+f x))^{9/2}} \, dx=\text {Timed out} \]
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\[ \int \frac {\cos ^2(e+f x) (a+a \sin (e+f x))^{7/2}}{(c-c \sin (e+f x))^{9/2}} \, dx=\int { \frac {{\left (a \sin \left (f x + e\right ) + a\right )}^{\frac {7}{2}} \cos \left (f x + e\right )^{2}}{{\left (-c \sin \left (f x + e\right ) + c\right )}^{\frac {9}{2}}} \,d x } \]
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Time = 0.33 (sec) , antiderivative size = 170, normalized size of antiderivative = 0.71 \[ \int \frac {\cos ^2(e+f x) (a+a \sin (e+f x))^{7/2}}{(c-c \sin (e+f x))^{9/2}} \, dx=-\frac {2 \, a^{\frac {7}{2}} \sqrt {c} {\left (\frac {3 \, \cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2}}{c^{5} \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )} + \frac {12 \, \log \left (-\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 1\right )}{c^{5} \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )} - \frac {18 \, \cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} - 30 \, \cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 13}{{\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 1\right )}^{3} c^{5} \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )}\right )} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )}{3 \, f} \]
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Timed out. \[ \int \frac {\cos ^2(e+f x) (a+a \sin (e+f x))^{7/2}}{(c-c \sin (e+f x))^{9/2}} \, dx=\int \frac {{\cos \left (e+f\,x\right )}^2\,{\left (a+a\,\sin \left (e+f\,x\right )\right )}^{7/2}}{{\left (c-c\,\sin \left (e+f\,x\right )\right )}^{9/2}} \,d x \]
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