Integrand size = 42, antiderivative size = 237 \[ \int \frac {(g \cos (e+f x))^{3/2} \sqrt {a+a \sin (e+f x)}}{(c-c \sin (e+f x))^{7/2}} \, dx=\frac {4 a (g \cos (e+f x))^{5/2}}{9 f g \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{7/2}}-\frac {2 a (g \cos (e+f x))^{5/2}}{15 c f g \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{5/2}}-\frac {2 a (g \cos (e+f x))^{5/2}}{15 c^2 f g \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{3/2}}+\frac {2 a g \sqrt {\cos (e+f x)} \sqrt {g \cos (e+f x)} E\left (\left .\frac {1}{2} (e+f x)\right |2\right )}{15 c^3 f \sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}} \]
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Time = 0.81 (sec) , antiderivative size = 237, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.119, Rules used = {2929, 2931, 2921, 2721, 2719} \[ \int \frac {(g \cos (e+f x))^{3/2} \sqrt {a+a \sin (e+f x)}}{(c-c \sin (e+f x))^{7/2}} \, dx=\frac {2 a g \sqrt {\cos (e+f x)} E\left (\left .\frac {1}{2} (e+f x)\right |2\right ) \sqrt {g \cos (e+f x)}}{15 c^3 f \sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}-\frac {2 a (g \cos (e+f x))^{5/2}}{15 c^2 f g \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{3/2}}-\frac {2 a (g \cos (e+f x))^{5/2}}{15 c f g \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{5/2}}+\frac {4 a (g \cos (e+f x))^{5/2}}{9 f g \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{7/2}} \]
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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}}{9 f g \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{7/2}}-\frac {a \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}{3 c} \\ & = \frac {4 a (g \cos (e+f x))^{5/2}}{9 f g \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{7/2}}-\frac {2 a (g \cos (e+f x))^{5/2}}{15 c f g \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{5/2}}-\frac {a \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}{15 c^2} \\ & = \frac {4 a (g \cos (e+f x))^{5/2}}{9 f g \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{7/2}}-\frac {2 a (g \cos (e+f x))^{5/2}}{15 c f g \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{5/2}}-\frac {2 a (g \cos (e+f x))^{5/2}}{15 c^2 f g \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{3/2}}+\frac {a \int \frac {(g \cos (e+f x))^{3/2}}{\sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}} \, dx}{15 c^3} \\ & = \frac {4 a (g \cos (e+f x))^{5/2}}{9 f g \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{7/2}}-\frac {2 a (g \cos (e+f x))^{5/2}}{15 c f g \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{5/2}}-\frac {2 a (g \cos (e+f x))^{5/2}}{15 c^2 f g \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{3/2}}+\frac {(a g \cos (e+f x)) \int \sqrt {g \cos (e+f x)} \, dx}{15 c^3 \sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}} \\ & = \frac {4 a (g \cos (e+f x))^{5/2}}{9 f g \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{7/2}}-\frac {2 a (g \cos (e+f x))^{5/2}}{15 c f g \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{5/2}}-\frac {2 a (g \cos (e+f x))^{5/2}}{15 c^2 f g \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{3/2}}+\frac {\left (a g \sqrt {\cos (e+f x)} \sqrt {g \cos (e+f x)}\right ) \int \sqrt {\cos (e+f x)} \, dx}{15 c^3 \sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}} \\ & = \frac {4 a (g \cos (e+f x))^{5/2}}{9 f g \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{7/2}}-\frac {2 a (g \cos (e+f x))^{5/2}}{15 c f g \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{5/2}}-\frac {2 a (g \cos (e+f x))^{5/2}}{15 c^2 f g \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{3/2}}+\frac {2 a g \sqrt {\cos (e+f x)} \sqrt {g \cos (e+f x)} E\left (\left .\frac {1}{2} (e+f x)\right |2\right )}{15 c^3 f \sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}} \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 4.76 (sec) , antiderivative size = 256, normalized size of antiderivative = 1.08 \[ \int \frac {(g \cos (e+f x))^{3/2} \sqrt {a+a \sin (e+f x)}}{(c-c \sin (e+f x))^{7/2}} \, dx=\frac {4 e^{3 i (e+f x)} \left (e^{-i (e+f x)} \left (1+e^{2 i (e+f x)}\right ) g\right )^{3/2} \left (\sqrt {1+e^{2 i (e+f x)}} \left (-29 i+e^{i (e+f x)}+15 i e^{2 i (e+f x)}-3 e^{3 i (e+f x)}\right )+\left (-i+e^{i (e+f x)}\right )^5 \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {3}{4},\frac {7}{4},-e^{2 i (e+f x)}\right )\right ) \sqrt {a (1+\sin (e+f x))}}{45 c^3 \left (-i+e^{i (e+f x)}\right )^4 \sqrt {i c e^{-i (e+f x)} \left (-i+e^{i (e+f x)}\right )^2} \left (i+e^{i (e+f x)}\right ) \left (1+e^{2 i (e+f x)}\right )^{3/2} f} \]
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Result contains complex when optimal does not.
Time = 1.62 (sec) , antiderivative size = 1090, normalized size of antiderivative = 4.60
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.14 (sec) , antiderivative size = 269, normalized size of antiderivative = 1.14 \[ \int \frac {(g \cos (e+f x))^{3/2} \sqrt {a+a \sin (e+f x)}}{(c-c \sin (e+f x))^{7/2}} \, dx=-\frac {2 \, {\left (3 \, g \cos \left (f x + e\right )^{2} + 9 \, g \sin \left (f x + e\right ) + g\right )} \sqrt {g \cos \left (f x + e\right )} \sqrt {a \sin \left (f x + e\right ) + a} \sqrt {-c \sin \left (f x + e\right ) + c} - 3 \, {\left (-3 i \, \sqrt {2} g \cos \left (f x + e\right )^{2} + {\left (i \, \sqrt {2} g \cos \left (f x + e\right )^{2} - 4 i \, \sqrt {2} g\right )} \sin \left (f x + e\right ) + 4 i \, \sqrt {2} g\right )} \sqrt {a c g} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (f x + e\right ) + i \, \sin \left (f x + e\right )\right )\right ) - 3 \, {\left (3 i \, \sqrt {2} g \cos \left (f x + e\right )^{2} + {\left (-i \, \sqrt {2} g \cos \left (f x + e\right )^{2} + 4 i \, \sqrt {2} g\right )} \sin \left (f x + e\right ) - 4 i \, \sqrt {2} g\right )} \sqrt {a c g} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (f x + e\right ) - i \, \sin \left (f x + e\right )\right )\right )}{45 \, {\left (3 \, c^{4} f \cos \left (f x + e\right )^{2} - 4 \, c^{4} f - {\left (c^{4} f \cos \left (f x + e\right )^{2} - 4 \, c^{4} f\right )} \sin \left (f x + e\right )\right )}} \]
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Timed out. \[ \int \frac {(g \cos (e+f x))^{3/2} \sqrt {a+a \sin (e+f x)}}{(c-c \sin (e+f x))^{7/2}} \, dx=\text {Timed out} \]
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\[ \int \frac {(g \cos (e+f x))^{3/2} \sqrt {a+a \sin (e+f x)}}{(c-c \sin (e+f x))^{7/2}} \, dx=\int { \frac {\left (g \cos \left (f x + e\right )\right )^{\frac {3}{2}} \sqrt {a \sin \left (f x + e\right ) + a}}{{\left (-c \sin \left (f x + e\right ) + c\right )}^{\frac {7}{2}}} \,d x } \]
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Timed out. \[ \int \frac {(g \cos (e+f x))^{3/2} \sqrt {a+a \sin (e+f x)}}{(c-c \sin (e+f x))^{7/2}} \, dx=\text {Timed out} \]
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Timed out. \[ \int \frac {(g \cos (e+f x))^{3/2} \sqrt {a+a \sin (e+f x)}}{(c-c \sin (e+f x))^{7/2}} \, dx=\int \frac {{\left (g\,\cos \left (e+f\,x\right )\right )}^{3/2}\,\sqrt {a+a\,\sin \left (e+f\,x\right )}}{{\left (c-c\,\sin \left (e+f\,x\right )\right )}^{7/2}} \,d x \]
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