Integrand size = 42, antiderivative size = 406 \[ \int (g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^{5/2} (c-c \sin (e+f x))^{5/2} \, dx=\frac {154 a^3 c^3 (g \cos (e+f x))^{5/2}}{585 f g \sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}}+\frac {154 a^3 c^3 g \sqrt {\cos (e+f x)} \sqrt {g \cos (e+f x)} E\left (\left .\frac {1}{2} (e+f x)\right |2\right )}{195 f \sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}}+\frac {22 a^3 c^2 (g \cos (e+f x))^{5/2} \sqrt {c-c \sin (e+f x)}}{195 f g \sqrt {a+a \sin (e+f x)}}+\frac {2 a^3 c (g \cos (e+f x))^{5/2} (c-c \sin (e+f x))^{3/2}}{39 f g \sqrt {a+a \sin (e+f x)}}-\frac {14 a^3 (g \cos (e+f x))^{5/2} (c-c \sin (e+f x))^{5/2}}{117 f g \sqrt {a+a \sin (e+f x)}}-\frac {2 a^2 (g \cos (e+f x))^{5/2} \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{5/2}}{13 f g}-\frac {2 a (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{3/2} (c-c \sin (e+f x))^{5/2}}{13 f g} \]
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Time = 1.46 (sec) , antiderivative size = 406, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {2930, 2921, 2721, 2719} \[ \int (g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^{5/2} (c-c \sin (e+f x))^{5/2} \, dx=\frac {154 a^3 c^3 (g \cos (e+f x))^{5/2}}{585 f g \sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}+\frac {154 a^3 c^3 g \sqrt {\cos (e+f x)} E\left (\left .\frac {1}{2} (e+f x)\right |2\right ) \sqrt {g \cos (e+f x)}}{195 f \sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}+\frac {22 a^3 c^2 \sqrt {c-c \sin (e+f x)} (g \cos (e+f x))^{5/2}}{195 f g \sqrt {a \sin (e+f x)+a}}-\frac {14 a^3 (c-c \sin (e+f x))^{5/2} (g \cos (e+f x))^{5/2}}{117 f g \sqrt {a \sin (e+f x)+a}}+\frac {2 a^3 c (c-c \sin (e+f x))^{3/2} (g \cos (e+f x))^{5/2}}{39 f g \sqrt {a \sin (e+f x)+a}}-\frac {2 a^2 \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{5/2} (g \cos (e+f x))^{5/2}}{13 f g}-\frac {2 a (a \sin (e+f x)+a)^{3/2} (c-c \sin (e+f x))^{5/2} (g \cos (e+f x))^{5/2}}{13 f g} \]
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Rule 2719
Rule 2721
Rule 2921
Rule 2930
Rubi steps \begin{align*} \text {integral}& = -\frac {2 a (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{3/2} (c-c \sin (e+f x))^{5/2}}{13 f g}+\frac {1}{13} (11 a) \int (g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^{3/2} (c-c \sin (e+f x))^{5/2} \, dx \\ & = -\frac {2 a^2 (g \cos (e+f x))^{5/2} \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{5/2}}{13 f g}-\frac {2 a (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{3/2} (c-c \sin (e+f x))^{5/2}}{13 f g}+\frac {1}{13} \left (7 a^2\right ) \int (g \cos (e+f x))^{3/2} \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{5/2} \, dx \\ & = -\frac {14 a^3 (g \cos (e+f x))^{5/2} (c-c \sin (e+f x))^{5/2}}{117 f g \sqrt {a+a \sin (e+f x)}}-\frac {2 a^2 (g \cos (e+f x))^{5/2} \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{5/2}}{13 f g}-\frac {2 a (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{3/2} (c-c \sin (e+f x))^{5/2}}{13 f g}+\frac {1}{39} \left (7 a^3\right ) \int \frac {(g \cos (e+f x))^{3/2} (c-c \sin (e+f x))^{5/2}}{\sqrt {a+a \sin (e+f x)}} \, dx \\ & = \frac {2 a^3 c (g \cos (e+f x))^{5/2} (c-c \sin (e+f x))^{3/2}}{39 f g \sqrt {a+a \sin (e+f x)}}-\frac {14 a^3 (g \cos (e+f x))^{5/2} (c-c \sin (e+f x))^{5/2}}{117 f g \sqrt {a+a \sin (e+f x)}}-\frac {2 a^2 (g \cos (e+f x))^{5/2} \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{5/2}}{13 f g}-\frac {2 a (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{3/2} (c-c \sin (e+f x))^{5/2}}{13 f g}+\frac {1}{39} \left (11 a^3 c\right ) \int \frac {(g \cos (e+f x))^{3/2} (c-c \sin (e+f x))^{3/2}}{\sqrt {a+a \sin (e+f x)}} \, dx \\ & = \frac {22 a^3 c^2 (g \cos (e+f x))^{5/2} \sqrt {c-c \sin (e+f x)}}{195 f g \sqrt {a+a \sin (e+f x)}}+\frac {2 a^3 c (g \cos (e+f x))^{5/2} (c-c \sin (e+f x))^{3/2}}{39 f g \sqrt {a+a \sin (e+f x)}}-\frac {14 a^3 (g \cos (e+f x))^{5/2} (c-c \sin (e+f x))^{5/2}}{117 f g \sqrt {a+a \sin (e+f x)}}-\frac {2 a^2 (g \cos (e+f x))^{5/2} \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{5/2}}{13 f g}-\frac {2 a (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{3/2} (c-c \sin (e+f x))^{5/2}}{13 f g}+\frac {1}{195} \left (77 a^3 c^2\right ) \int \frac {(g \cos (e+f x))^{3/2} \sqrt {c-c \sin (e+f x)}}{\sqrt {a+a \sin (e+f x)}} \, dx \\ & = \frac {154 a^3 c^3 (g \cos (e+f x))^{5/2}}{585 f g \sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}}+\frac {22 a^3 c^2 (g \cos (e+f x))^{5/2} \sqrt {c-c \sin (e+f x)}}{195 f g \sqrt {a+a \sin (e+f x)}}+\frac {2 a^3 c (g \cos (e+f x))^{5/2} (c-c \sin (e+f x))^{3/2}}{39 f g \sqrt {a+a \sin (e+f x)}}-\frac {14 a^3 (g \cos (e+f x))^{5/2} (c-c \sin (e+f x))^{5/2}}{117 f g \sqrt {a+a \sin (e+f x)}}-\frac {2 a^2 (g \cos (e+f x))^{5/2} \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{5/2}}{13 f g}-\frac {2 a (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{3/2} (c-c \sin (e+f x))^{5/2}}{13 f g}+\frac {1}{195} \left (77 a^3 c^3\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 \\ & = \frac {154 a^3 c^3 (g \cos (e+f x))^{5/2}}{585 f g \sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}}+\frac {22 a^3 c^2 (g \cos (e+f x))^{5/2} \sqrt {c-c \sin (e+f x)}}{195 f g \sqrt {a+a \sin (e+f x)}}+\frac {2 a^3 c (g \cos (e+f x))^{5/2} (c-c \sin (e+f x))^{3/2}}{39 f g \sqrt {a+a \sin (e+f x)}}-\frac {14 a^3 (g \cos (e+f x))^{5/2} (c-c \sin (e+f x))^{5/2}}{117 f g \sqrt {a+a \sin (e+f x)}}-\frac {2 a^2 (g \cos (e+f x))^{5/2} \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{5/2}}{13 f g}-\frac {2 a (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{3/2} (c-c \sin (e+f x))^{5/2}}{13 f g}+\frac {\left (77 a^3 c^3 g \cos (e+f x)\right ) \int \sqrt {g \cos (e+f x)} \, dx}{195 \sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}} \\ & = \frac {154 a^3 c^3 (g \cos (e+f x))^{5/2}}{585 f g \sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}}+\frac {22 a^3 c^2 (g \cos (e+f x))^{5/2} \sqrt {c-c \sin (e+f x)}}{195 f g \sqrt {a+a \sin (e+f x)}}+\frac {2 a^3 c (g \cos (e+f x))^{5/2} (c-c \sin (e+f x))^{3/2}}{39 f g \sqrt {a+a \sin (e+f x)}}-\frac {14 a^3 (g \cos (e+f x))^{5/2} (c-c \sin (e+f x))^{5/2}}{117 f g \sqrt {a+a \sin (e+f x)}}-\frac {2 a^2 (g \cos (e+f x))^{5/2} \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{5/2}}{13 f g}-\frac {2 a (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{3/2} (c-c \sin (e+f x))^{5/2}}{13 f g}+\frac {\left (77 a^3 c^3 g \sqrt {\cos (e+f x)} \sqrt {g \cos (e+f x)}\right ) \int \sqrt {\cos (e+f x)} \, dx}{195 \sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}} \\ & = \frac {154 a^3 c^3 (g \cos (e+f x))^{5/2}}{585 f g \sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}}+\frac {154 a^3 c^3 g \sqrt {\cos (e+f x)} \sqrt {g \cos (e+f x)} E\left (\left .\frac {1}{2} (e+f x)\right |2\right )}{195 f \sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}}+\frac {22 a^3 c^2 (g \cos (e+f x))^{5/2} \sqrt {c-c \sin (e+f x)}}{195 f g \sqrt {a+a \sin (e+f x)}}+\frac {2 a^3 c (g \cos (e+f x))^{5/2} (c-c \sin (e+f x))^{3/2}}{39 f g \sqrt {a+a \sin (e+f x)}}-\frac {14 a^3 (g \cos (e+f x))^{5/2} (c-c \sin (e+f x))^{5/2}}{117 f g \sqrt {a+a \sin (e+f x)}}-\frac {2 a^2 (g \cos (e+f x))^{5/2} \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{5/2}}{13 f g}-\frac {2 a (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{3/2} (c-c \sin (e+f x))^{5/2}}{13 f g} \\ \end{align*}
Time = 4.52 (sec) , antiderivative size = 120, normalized size of antiderivative = 0.30 \[ \int (g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^{5/2} (c-c \sin (e+f x))^{5/2} \, dx=\frac {a^2 c^2 (g \cos (e+f x))^{3/2} \sqrt {a (1+\sin (e+f x))} \sqrt {c-c \sin (e+f x)} \left (7392 E\left (\left .\frac {1}{2} (e+f x)\right |2\right )+\sqrt {\cos (e+f x)} (1897 \sin (2 (e+f x))+400 \sin (4 (e+f x))+45 \sin (6 (e+f x)))\right )}{9360 f \cos ^{\frac {5}{2}}(e+f x)} \]
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Result contains complex when optimal does not.
Time = 4.92 (sec) , antiderivative size = 511, normalized size of antiderivative = 1.26
method | result | size |
default | \(\frac {2 \sqrt {-c \left (\sin \left (f x +e \right )-1\right )}\, \sqrt {a \left (1+\sin \left (f x +e \right )\right )}\, \sqrt {g \cos \left (f x +e \right )}\, c^{2} a^{2} g \left (45 \left (\cos ^{5}\left (f x +e \right )\right ) \sin \left (f x +e \right )+45 \left (\cos ^{4}\left (f x +e \right )\right ) \sin \left (f x +e \right )+231 i \sqrt {\frac {\cos \left (f x +e \right )}{1+\cos \left (f x +e \right )}}\, \sqrt {\frac {1}{1+\cos \left (f x +e \right )}}\, E\left (i \left (\csc \left (f x +e \right )-\cot \left (f x +e \right )\right ), i\right )-231 i \sqrt {\frac {\cos \left (f x +e \right )}{1+\cos \left (f x +e \right )}}\, \sqrt {\frac {1}{1+\cos \left (f x +e \right )}}\, F\left (i \left (\csc \left (f x +e \right )-\cot \left (f x +e \right )\right ), i\right )+55 \left (\cos ^{3}\left (f x +e \right )\right ) \sin \left (f x +e \right )+462 i \sqrt {\frac {\cos \left (f x +e \right )}{1+\cos \left (f x +e \right )}}\, \sqrt {\frac {1}{1+\cos \left (f x +e \right )}}\, E\left (i \left (\csc \left (f x +e \right )-\cot \left (f x +e \right )\right ), i\right ) \sec \left (f x +e \right )-462 i \sqrt {\frac {\cos \left (f x +e \right )}{1+\cos \left (f x +e \right )}}\, \sqrt {\frac {1}{1+\cos \left (f x +e \right )}}\, F\left (i \left (\csc \left (f x +e \right )-\cot \left (f x +e \right )\right ), i\right ) \sec \left (f x +e \right )+55 \left (\cos ^{2}\left (f x +e \right )\right ) \sin \left (f x +e \right )+231 i \sqrt {\frac {\cos \left (f x +e \right )}{1+\cos \left (f x +e \right )}}\, \sqrt {\frac {1}{1+\cos \left (f x +e \right )}}\, E\left (i \left (\csc \left (f x +e \right )-\cot \left (f x +e \right )\right ), i\right ) \left (\sec ^{2}\left (f x +e \right )\right )-231 i \sqrt {\frac {\cos \left (f x +e \right )}{1+\cos \left (f x +e \right )}}\, \sqrt {\frac {1}{1+\cos \left (f x +e \right )}}\, F\left (i \left (\csc \left (f x +e \right )-\cot \left (f x +e \right )\right ), i\right ) \left (\sec ^{2}\left (f x +e \right )\right )+77 \cos \left (f x +e \right ) \sin \left (f x +e \right )+77 \sin \left (f x +e \right )+231 \tan \left (f x +e \right )\right )}{585 f \left (1+\cos \left (f x +e \right )\right )}\) | \(511\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.14 (sec) , antiderivative size = 171, normalized size of antiderivative = 0.42 \[ \int (g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^{5/2} (c-c \sin (e+f x))^{5/2} \, dx=\frac {-231 i \, \sqrt {2} \sqrt {a c g} a^{2} c^{2} g {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (f x + e\right ) + i \, \sin \left (f x + e\right )\right )\right ) + 231 i \, \sqrt {2} \sqrt {a c g} a^{2} c^{2} g {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (f x + e\right ) - i \, \sin \left (f x + e\right )\right )\right ) + 2 \, {\left (45 \, a^{2} c^{2} g \cos \left (f x + e\right )^{4} + 55 \, a^{2} c^{2} g \cos \left (f x + e\right )^{2} + 77 \, a^{2} c^{2} g\right )} \sqrt {g \cos \left (f x + e\right )} \sqrt {a \sin \left (f x + e\right ) + a} \sqrt {-c \sin \left (f x + e\right ) + c} \sin \left (f x + e\right )}{585 \, f} \]
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Timed out. \[ \int (g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^{5/2} (c-c \sin (e+f x))^{5/2} \, dx=\text {Timed out} \]
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\[ \int (g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^{5/2} (c-c \sin (e+f x))^{5/2} \, dx=\int { \left (g \cos \left (f x + e\right )\right )^{\frac {3}{2}} {\left (a \sin \left (f x + e\right ) + a\right )}^{\frac {5}{2}} {\left (-c \sin \left (f x + e\right ) + c\right )}^{\frac {5}{2}} \,d x } \]
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\[ \int (g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^{5/2} (c-c \sin (e+f x))^{5/2} \, dx=\int { \left (g \cos \left (f x + e\right )\right )^{\frac {3}{2}} {\left (a \sin \left (f x + e\right ) + a\right )}^{\frac {5}{2}} {\left (-c \sin \left (f x + e\right ) + c\right )}^{\frac {5}{2}} \,d x } \]
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Timed out. \[ \int (g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^{5/2} (c-c \sin (e+f x))^{5/2} \, dx=\int {\left (g\,\cos \left (e+f\,x\right )\right )}^{3/2}\,{\left (a+a\,\sin \left (e+f\,x\right )\right )}^{5/2}\,{\left (c-c\,\sin \left (e+f\,x\right )\right )}^{5/2} \,d x \]
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