Integrand size = 42, antiderivative size = 352 \[ \int (g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^{3/2} (c-c \sin (e+f x))^{5/2} \, dx=\frac {14 a^2 c^3 (g \cos (e+f x))^{5/2}}{45 f g \sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}}+\frac {14 a^2 c^3 g \sqrt {\cos (e+f x)} \sqrt {g \cos (e+f x)} E\left (\left .\frac {1}{2} (e+f x)\right |2\right )}{15 f \sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}}+\frac {2 a^2 c^2 (g \cos (e+f x))^{5/2} \sqrt {c-c \sin (e+f x)}}{15 f g \sqrt {a+a \sin (e+f x)}}+\frac {2 a^2 c (g \cos (e+f x))^{5/2} (c-c \sin (e+f x))^{3/2}}{33 f g \sqrt {a+a \sin (e+f x)}}-\frac {14 a^2 (g \cos (e+f x))^{5/2} (c-c \sin (e+f x))^{5/2}}{99 f g \sqrt {a+a \sin (e+f x)}}-\frac {2 a (g \cos (e+f x))^{5/2} \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{5/2}}{11 f g} \]
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Time = 1.19 (sec) , antiderivative size = 352, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {2930, 2921, 2721, 2719} \[ \int (g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^{3/2} (c-c \sin (e+f x))^{5/2} \, dx=\frac {14 a^2 c^3 (g \cos (e+f x))^{5/2}}{45 f g \sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}+\frac {14 a^2 c^3 g \sqrt {\cos (e+f x)} E\left (\left .\frac {1}{2} (e+f x)\right |2\right ) \sqrt {g \cos (e+f x)}}{15 f \sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}+\frac {2 a^2 c^2 \sqrt {c-c \sin (e+f x)} (g \cos (e+f x))^{5/2}}{15 f g \sqrt {a \sin (e+f x)+a}}+\frac {2 a^2 c (c-c \sin (e+f x))^{3/2} (g \cos (e+f x))^{5/2}}{33 f g \sqrt {a \sin (e+f x)+a}}-\frac {14 a^2 (c-c \sin (e+f x))^{5/2} (g \cos (e+f x))^{5/2}}{99 f g \sqrt {a \sin (e+f x)+a}}-\frac {2 a \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{5/2} (g \cos (e+f x))^{5/2}}{11 f g} \]
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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} \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{5/2}}{11 f g}+\frac {1}{11} (7 a) \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^2 (g \cos (e+f x))^{5/2} (c-c \sin (e+f x))^{5/2}}{99 f g \sqrt {a+a \sin (e+f x)}}-\frac {2 a (g \cos (e+f x))^{5/2} \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{5/2}}{11 f g}+\frac {1}{33} \left (7 a^2\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^2 c (g \cos (e+f x))^{5/2} (c-c \sin (e+f x))^{3/2}}{33 f g \sqrt {a+a \sin (e+f x)}}-\frac {14 a^2 (g \cos (e+f x))^{5/2} (c-c \sin (e+f x))^{5/2}}{99 f g \sqrt {a+a \sin (e+f x)}}-\frac {2 a (g \cos (e+f x))^{5/2} \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{5/2}}{11 f g}+\frac {1}{3} \left (a^2 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 {2 a^2 c^2 (g \cos (e+f x))^{5/2} \sqrt {c-c \sin (e+f x)}}{15 f g \sqrt {a+a \sin (e+f x)}}+\frac {2 a^2 c (g \cos (e+f x))^{5/2} (c-c \sin (e+f x))^{3/2}}{33 f g \sqrt {a+a \sin (e+f x)}}-\frac {14 a^2 (g \cos (e+f x))^{5/2} (c-c \sin (e+f x))^{5/2}}{99 f g \sqrt {a+a \sin (e+f x)}}-\frac {2 a (g \cos (e+f x))^{5/2} \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{5/2}}{11 f g}+\frac {1}{15} \left (7 a^2 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 {14 a^2 c^3 (g \cos (e+f x))^{5/2}}{45 f g \sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}}+\frac {2 a^2 c^2 (g \cos (e+f x))^{5/2} \sqrt {c-c \sin (e+f x)}}{15 f g \sqrt {a+a \sin (e+f x)}}+\frac {2 a^2 c (g \cos (e+f x))^{5/2} (c-c \sin (e+f x))^{3/2}}{33 f g \sqrt {a+a \sin (e+f x)}}-\frac {14 a^2 (g \cos (e+f x))^{5/2} (c-c \sin (e+f x))^{5/2}}{99 f g \sqrt {a+a \sin (e+f x)}}-\frac {2 a (g \cos (e+f x))^{5/2} \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{5/2}}{11 f g}+\frac {1}{15} \left (7 a^2 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 {14 a^2 c^3 (g \cos (e+f x))^{5/2}}{45 f g \sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}}+\frac {2 a^2 c^2 (g \cos (e+f x))^{5/2} \sqrt {c-c \sin (e+f x)}}{15 f g \sqrt {a+a \sin (e+f x)}}+\frac {2 a^2 c (g \cos (e+f x))^{5/2} (c-c \sin (e+f x))^{3/2}}{33 f g \sqrt {a+a \sin (e+f x)}}-\frac {14 a^2 (g \cos (e+f x))^{5/2} (c-c \sin (e+f x))^{5/2}}{99 f g \sqrt {a+a \sin (e+f x)}}-\frac {2 a (g \cos (e+f x))^{5/2} \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{5/2}}{11 f g}+\frac {\left (7 a^2 c^3 g \cos (e+f x)\right ) \int \sqrt {g \cos (e+f x)} \, dx}{15 \sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}} \\ & = \frac {14 a^2 c^3 (g \cos (e+f x))^{5/2}}{45 f g \sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}}+\frac {2 a^2 c^2 (g \cos (e+f x))^{5/2} \sqrt {c-c \sin (e+f x)}}{15 f g \sqrt {a+a \sin (e+f x)}}+\frac {2 a^2 c (g \cos (e+f x))^{5/2} (c-c \sin (e+f x))^{3/2}}{33 f g \sqrt {a+a \sin (e+f x)}}-\frac {14 a^2 (g \cos (e+f x))^{5/2} (c-c \sin (e+f x))^{5/2}}{99 f g \sqrt {a+a \sin (e+f x)}}-\frac {2 a (g \cos (e+f x))^{5/2} \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{5/2}}{11 f g}+\frac {\left (7 a^2 c^3 g \sqrt {\cos (e+f x)} \sqrt {g \cos (e+f x)}\right ) \int \sqrt {\cos (e+f x)} \, dx}{15 \sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}} \\ & = \frac {14 a^2 c^3 (g \cos (e+f x))^{5/2}}{45 f g \sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}}+\frac {14 a^2 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 )}{15 f \sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}}+\frac {2 a^2 c^2 (g \cos (e+f x))^{5/2} \sqrt {c-c \sin (e+f x)}}{15 f g \sqrt {a+a \sin (e+f x)}}+\frac {2 a^2 c (g \cos (e+f x))^{5/2} (c-c \sin (e+f x))^{3/2}}{33 f g \sqrt {a+a \sin (e+f x)}}-\frac {14 a^2 (g \cos (e+f x))^{5/2} (c-c \sin (e+f x))^{5/2}}{99 f g \sqrt {a+a \sin (e+f x)}}-\frac {2 a (g \cos (e+f x))^{5/2} \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{5/2}}{11 f g} \\ \end{align*}
Time = 7.98 (sec) , antiderivative size = 193, normalized size of antiderivative = 0.55 \[ \int (g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^{3/2} (c-c \sin (e+f x))^{5/2} \, dx=\frac {c^2 (g \cos (e+f x))^{3/2} (-1+\sin (e+f x))^2 (a (1+\sin (e+f x)))^{3/2} \sqrt {c-c \sin (e+f x)} \left (3696 E\left (\left .\frac {1}{2} (e+f x)\right |2\right )+\sqrt {\cos (e+f x)} (450 \cos (e+f x)+225 \cos (3 (e+f x))+45 \cos (5 (e+f x))+836 \sin (2 (e+f x))+110 \sin (4 (e+f x)))\right )}{3960 f \cos ^{\frac {3}{2}}(e+f x) \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^5 \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^3} \]
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Result contains complex when optimal does not.
Time = 6.67 (sec) , antiderivative size = 497, normalized size of antiderivative = 1.41
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 g \left (45 \left (\cos ^{5}\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 )}}\, 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 )+45 \left (\cos ^{4}\left (f x +e \right )\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 )}{495 f \left (1+\cos \left (f x +e \right )\right )}\) | \(497\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.14 (sec) , antiderivative size = 164, normalized size of antiderivative = 0.47 \[ \int (g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^{3/2} (c-c \sin (e+f x))^{5/2} \, dx=\frac {-231 i \, \sqrt {2} \sqrt {a c g} a c^{2} g {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (f x + e\right ) + i \, \sin \left (f x + e\right )\right )\right ) + 231 i \, \sqrt {2} \sqrt {a c g} a c^{2} g {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (f x + e\right ) - i \, \sin \left (f x + e\right )\right )\right ) + 2 \, {\left (45 \, a c^{2} g \cos \left (f x + e\right )^{4} + 11 \, {\left (5 \, a c^{2} g \cos \left (f x + e\right )^{2} + 7 \, a c^{2} g\right )} \sin \left (f x + e\right )\right )} \sqrt {g \cos \left (f x + e\right )} \sqrt {a \sin \left (f x + e\right ) + a} \sqrt {-c \sin \left (f x + e\right ) + c}}{495 \, f} \]
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Timed out. \[ \int (g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^{3/2} (c-c \sin (e+f x))^{5/2} \, dx=\text {Timed out} \]
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\[ \int (g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^{3/2} (c-c \sin (e+f x))^{5/2} \, dx=\int { \left (g \cos \left (f x + e\right )\right )^{\frac {3}{2}} {\left (a \sin \left (f x + e\right ) + a\right )}^{\frac {3}{2}} {\left (-c \sin \left (f x + e\right ) + c\right )}^{\frac {5}{2}} \,d x } \]
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\[ \int (g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^{3/2} (c-c \sin (e+f x))^{5/2} \, dx=\int { \left (g \cos \left (f x + e\right )\right )^{\frac {3}{2}} {\left (a \sin \left (f x + e\right ) + a\right )}^{\frac {3}{2}} {\left (-c \sin \left (f x + e\right ) + c\right )}^{\frac {5}{2}} \,d x } \]
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Timed out. \[ \int (g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^{3/2} (c-c \sin (e+f x))^{5/2} \, dx=\int {\left (g\,\cos \left (e+f\,x\right )\right )}^{3/2}\,{\left (a+a\,\sin \left (e+f\,x\right )\right )}^{3/2}\,{\left (c-c\,\sin \left (e+f\,x\right )\right )}^{5/2} \,d x \]
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