Integrand size = 42, antiderivative size = 357 \[ \int \frac {(g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^{3/2}}{(c-c \sin (e+f x))^{11/2}} \, dx=\frac {4 a (g \cos (e+f x))^{5/2} \sqrt {a+a \sin (e+f x)}}{17 f g (c-c \sin (e+f x))^{11/2}}-\frac {28 a^2 (g \cos (e+f x))^{5/2}}{221 c f g \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{9/2}}+\frac {14 a^2 (g \cos (e+f x))^{5/2}}{663 c^2 f g \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{7/2}}+\frac {14 a^2 (g \cos (e+f x))^{5/2}}{1105 c^3 f g \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{5/2}}+\frac {14 a^2 (g \cos (e+f x))^{5/2}}{1105 c^4 f g \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{3/2}}-\frac {14 a^2 g \sqrt {\cos (e+f x)} \sqrt {g \cos (e+f x)} E\left (\left .\frac {1}{2} (e+f x)\right |2\right )}{1105 c^5 f \sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}} \]
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Time = 1.24 (sec) , antiderivative size = 357, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.119, Rules used = {2929, 2931, 2921, 2721, 2719} \[ \int \frac {(g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^{3/2}}{(c-c \sin (e+f x))^{11/2}} \, dx=-\frac {14 a^2 g \sqrt {\cos (e+f x)} E\left (\left .\frac {1}{2} (e+f x)\right |2\right ) \sqrt {g \cos (e+f x)}}{1105 c^5 f \sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}+\frac {14 a^2 (g \cos (e+f x))^{5/2}}{1105 c^4 f g \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{3/2}}+\frac {14 a^2 (g \cos (e+f x))^{5/2}}{1105 c^3 f g \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{5/2}}+\frac {14 a^2 (g \cos (e+f x))^{5/2}}{663 c^2 f g \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{7/2}}-\frac {28 a^2 (g \cos (e+f x))^{5/2}}{221 c f g \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{9/2}}+\frac {4 a \sqrt {a \sin (e+f x)+a} (g \cos (e+f x))^{5/2}}{17 f g (c-c \sin (e+f x))^{11/2}} \]
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Rule 2719
Rule 2721
Rule 2921
Rule 2929
Rule 2931
Rubi steps \begin{align*} \text {integral}& = \frac {4 a (g \cos (e+f x))^{5/2} \sqrt {a+a \sin (e+f x)}}{17 f g (c-c \sin (e+f x))^{11/2}}-\frac {(7 a) \int \frac {(g \cos (e+f x))^{3/2} \sqrt {a+a \sin (e+f x)}}{(c-c \sin (e+f x))^{9/2}} \, dx}{17 c} \\ & = \frac {4 a (g \cos (e+f x))^{5/2} \sqrt {a+a \sin (e+f x)}}{17 f g (c-c \sin (e+f x))^{11/2}}-\frac {28 a^2 (g \cos (e+f x))^{5/2}}{221 c f g \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{9/2}}+\frac {\left (21 a^2\right ) \int \frac {(g \cos (e+f x))^{3/2}}{\sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{7/2}} \, dx}{221 c^2} \\ & = \frac {4 a (g \cos (e+f x))^{5/2} \sqrt {a+a \sin (e+f x)}}{17 f g (c-c \sin (e+f x))^{11/2}}-\frac {28 a^2 (g \cos (e+f x))^{5/2}}{221 c f g \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{9/2}}+\frac {14 a^2 (g \cos (e+f x))^{5/2}}{663 c^2 f g \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{7/2}}+\frac {\left (7 a^2\right ) \int \frac {(g \cos (e+f x))^{3/2}}{\sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{5/2}} \, dx}{221 c^3} \\ & = \frac {4 a (g \cos (e+f x))^{5/2} \sqrt {a+a \sin (e+f x)}}{17 f g (c-c \sin (e+f x))^{11/2}}-\frac {28 a^2 (g \cos (e+f x))^{5/2}}{221 c f g \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{9/2}}+\frac {14 a^2 (g \cos (e+f x))^{5/2}}{663 c^2 f g \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{7/2}}+\frac {14 a^2 (g \cos (e+f x))^{5/2}}{1105 c^3 f g \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{5/2}}+\frac {\left (7 a^2\right ) \int \frac {(g \cos (e+f x))^{3/2}}{\sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{3/2}} \, dx}{1105 c^4} \\ & = \frac {4 a (g \cos (e+f x))^{5/2} \sqrt {a+a \sin (e+f x)}}{17 f g (c-c \sin (e+f x))^{11/2}}-\frac {28 a^2 (g \cos (e+f x))^{5/2}}{221 c f g \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{9/2}}+\frac {14 a^2 (g \cos (e+f x))^{5/2}}{663 c^2 f g \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{7/2}}+\frac {14 a^2 (g \cos (e+f x))^{5/2}}{1105 c^3 f g \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{5/2}}+\frac {14 a^2 (g \cos (e+f x))^{5/2}}{1105 c^4 f g \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{3/2}}-\frac {\left (7 a^2\right ) \int \frac {(g \cos (e+f x))^{3/2}}{\sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}} \, dx}{1105 c^5} \\ & = \frac {4 a (g \cos (e+f x))^{5/2} \sqrt {a+a \sin (e+f x)}}{17 f g (c-c \sin (e+f x))^{11/2}}-\frac {28 a^2 (g \cos (e+f x))^{5/2}}{221 c f g \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{9/2}}+\frac {14 a^2 (g \cos (e+f x))^{5/2}}{663 c^2 f g \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{7/2}}+\frac {14 a^2 (g \cos (e+f x))^{5/2}}{1105 c^3 f g \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{5/2}}+\frac {14 a^2 (g \cos (e+f x))^{5/2}}{1105 c^4 f g \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{3/2}}-\frac {\left (7 a^2 g \cos (e+f x)\right ) \int \sqrt {g \cos (e+f x)} \, dx}{1105 c^5 \sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}} \\ & = \frac {4 a (g \cos (e+f x))^{5/2} \sqrt {a+a \sin (e+f x)}}{17 f g (c-c \sin (e+f x))^{11/2}}-\frac {28 a^2 (g \cos (e+f x))^{5/2}}{221 c f g \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{9/2}}+\frac {14 a^2 (g \cos (e+f x))^{5/2}}{663 c^2 f g \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{7/2}}+\frac {14 a^2 (g \cos (e+f x))^{5/2}}{1105 c^3 f g \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{5/2}}+\frac {14 a^2 (g \cos (e+f x))^{5/2}}{1105 c^4 f g \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{3/2}}-\frac {\left (7 a^2 g \sqrt {\cos (e+f x)} \sqrt {g \cos (e+f x)}\right ) \int \sqrt {\cos (e+f x)} \, dx}{1105 c^5 \sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}} \\ & = \frac {4 a (g \cos (e+f x))^{5/2} \sqrt {a+a \sin (e+f x)}}{17 f g (c-c \sin (e+f x))^{11/2}}-\frac {28 a^2 (g \cos (e+f x))^{5/2}}{221 c f g \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{9/2}}+\frac {14 a^2 (g \cos (e+f x))^{5/2}}{663 c^2 f g \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{7/2}}+\frac {14 a^2 (g \cos (e+f x))^{5/2}}{1105 c^3 f g \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{5/2}}+\frac {14 a^2 (g \cos (e+f x))^{5/2}}{1105 c^4 f g \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{3/2}}-\frac {14 a^2 g \sqrt {\cos (e+f x)} \sqrt {g \cos (e+f x)} E\left (\left .\frac {1}{2} (e+f x)\right |2\right )}{1105 c^5 f \sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}} \\ \end{align*}
Time = 10.55 (sec) , antiderivative size = 532, normalized size of antiderivative = 1.49 \[ \int \frac {(g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^{3/2}}{(c-c \sin (e+f x))^{11/2}} \, dx=-\frac {14 (g \cos (e+f x))^{3/2} E\left (\left .\frac {1}{2} (e+f x)\right |2\right ) \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^{11} (a (1+\sin (e+f x)))^{3/2}}{1105 f \cos ^{\frac {3}{2}}(e+f x) \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^3 (c-c \sin (e+f x))^{11/2}}+\frac {(g \cos (e+f x))^{3/2} \sec (e+f x) \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^{11} \left (\frac {14}{1105}+\frac {8}{17 \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^8}-\frac {80}{221 \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^6}+\frac {14}{663 \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^4}+\frac {14}{1105 \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^2}+\frac {16 \sin \left (\frac {1}{2} (e+f x)\right )}{17 \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^9}-\frac {160 \sin \left (\frac {1}{2} (e+f x)\right )}{221 \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^7}+\frac {28 \sin \left (\frac {1}{2} (e+f x)\right )}{663 \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^5}+\frac {28 \sin \left (\frac {1}{2} (e+f x)\right )}{1105 \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^3}+\frac {28 \sin \left (\frac {1}{2} (e+f x)\right )}{1105 \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )}\right ) (a (1+\sin (e+f x)))^{3/2}}{f \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^3 (c-c \sin (e+f x))^{11/2}} \]
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Result contains complex when optimal does not.
Time = 3.76 (sec) , antiderivative size = 1578, normalized size of antiderivative = 4.42
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.16 (sec) , antiderivative size = 396, normalized size of antiderivative = 1.11 \[ \int \frac {(g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^{3/2}}{(c-c \sin (e+f x))^{11/2}} \, dx=\frac {2 \, {\left (21 \, a g \cos \left (f x + e\right )^{4} - 266 \, a g \cos \left (f x + e\right )^{2} + 502 \, a g + {\left (105 \, a g \cos \left (f x + e\right )^{2} + 278 \, a g\right )} \sin \left (f x + e\right )\right )} \sqrt {g \cos \left (f x + e\right )} \sqrt {a \sin \left (f x + e\right ) + a} \sqrt {-c \sin \left (f x + e\right ) + c} + 21 \, {\left (5 i \, \sqrt {2} a g \cos \left (f x + e\right )^{4} - 20 i \, \sqrt {2} a g \cos \left (f x + e\right )^{2} + 16 i \, \sqrt {2} a g + {\left (-i \, \sqrt {2} a g \cos \left (f x + e\right )^{4} + 12 i \, \sqrt {2} a g \cos \left (f x + e\right )^{2} - 16 i \, \sqrt {2} a g\right )} \sin \left (f x + e\right )\right )} \sqrt {a c g} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (f x + e\right ) + i \, \sin \left (f x + e\right )\right )\right ) + 21 \, {\left (-5 i \, \sqrt {2} a g \cos \left (f x + e\right )^{4} + 20 i \, \sqrt {2} a g \cos \left (f x + e\right )^{2} - 16 i \, \sqrt {2} a g + {\left (i \, \sqrt {2} a g \cos \left (f x + e\right )^{4} - 12 i \, \sqrt {2} a g \cos \left (f x + e\right )^{2} + 16 i \, \sqrt {2} a g\right )} \sin \left (f x + e\right )\right )} \sqrt {a c g} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (f x + e\right ) - i \, \sin \left (f x + e\right )\right )\right )}{3315 \, {\left (5 \, c^{6} f \cos \left (f x + e\right )^{4} - 20 \, c^{6} f \cos \left (f x + e\right )^{2} + 16 \, c^{6} f - {\left (c^{6} f \cos \left (f x + e\right )^{4} - 12 \, c^{6} f \cos \left (f x + e\right )^{2} + 16 \, c^{6} f\right )} \sin \left (f x + e\right )\right )}} \]
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Timed out. \[ \int \frac {(g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^{3/2}}{(c-c \sin (e+f x))^{11/2}} \, dx=\text {Timed out} \]
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\[ \int \frac {(g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^{3/2}}{(c-c \sin (e+f x))^{11/2}} \, dx=\int { \frac {\left (g \cos \left (f x + e\right )\right )^{\frac {3}{2}} {\left (a \sin \left (f x + e\right ) + a\right )}^{\frac {3}{2}}}{{\left (-c \sin \left (f x + e\right ) + c\right )}^{\frac {11}{2}}} \,d x } \]
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Timed out. \[ \int \frac {(g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^{3/2}}{(c-c \sin (e+f x))^{11/2}} \, dx=\text {Timed out} \]
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Timed out. \[ \int \frac {(g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^{3/2}}{(c-c \sin (e+f x))^{11/2}} \, dx=\int \frac {{\left (g\,\cos \left (e+f\,x\right )\right )}^{3/2}\,{\left (a+a\,\sin \left (e+f\,x\right )\right )}^{3/2}}{{\left (c-c\,\sin \left (e+f\,x\right )\right )}^{11/2}} \,d x \]
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