Integrand size = 42, antiderivative size = 290 \[ \int (g \cos (e+f x))^{3/2} \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{5/2} \, dx=\frac {22 a c^3 (g \cos (e+f x))^{5/2}}{45 f g \sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}}+\frac {22 a c^3 g \sqrt {\cos (e+f x)} \sqrt {g \cos (e+f x)} E\left (\left .\frac {1}{2} (e+f x)\right |2\right )}{15 f \sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}}+\frac {22 a c^2 (g \cos (e+f x))^{5/2} \sqrt {c-c \sin (e+f x)}}{105 f g \sqrt {a+a \sin (e+f x)}}+\frac {2 a c (g \cos (e+f x))^{5/2} (c-c \sin (e+f x))^{3/2}}{21 f g \sqrt {a+a \sin (e+f x)}}-\frac {2 a (g \cos (e+f x))^{5/2} (c-c \sin (e+f x))^{5/2}}{9 f g \sqrt {a+a \sin (e+f x)}} \]
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Time = 1.03 (sec) , antiderivative size = 290, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {2930, 2921, 2721, 2719} \[ \int (g \cos (e+f x))^{3/2} \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{5/2} \, dx=\frac {22 a c^3 (g \cos (e+f x))^{5/2}}{45 f g \sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}+\frac {22 a c^3 g \sqrt {\cos (e+f x)} E\left (\left .\frac {1}{2} (e+f x)\right |2\right ) \sqrt {g \cos (e+f x)}}{15 f \sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}+\frac {22 a c^2 \sqrt {c-c \sin (e+f x)} (g \cos (e+f x))^{5/2}}{105 f g \sqrt {a \sin (e+f x)+a}}+\frac {2 a c (c-c \sin (e+f x))^{3/2} (g \cos (e+f x))^{5/2}}{21 f g \sqrt {a \sin (e+f x)+a}}-\frac {2 a (c-c \sin (e+f x))^{5/2} (g \cos (e+f x))^{5/2}}{9 f g \sqrt {a \sin (e+f x)+a}} \]
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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} (c-c \sin (e+f x))^{5/2}}{9 f g \sqrt {a+a \sin (e+f x)}}+\frac {1}{3} a \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 c (g \cos (e+f x))^{5/2} (c-c \sin (e+f x))^{3/2}}{21 f g \sqrt {a+a \sin (e+f x)}}-\frac {2 a (g \cos (e+f x))^{5/2} (c-c \sin (e+f x))^{5/2}}{9 f g \sqrt {a+a \sin (e+f x)}}+\frac {1}{21} (11 a c) \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 c^2 (g \cos (e+f x))^{5/2} \sqrt {c-c \sin (e+f x)}}{105 f g \sqrt {a+a \sin (e+f x)}}+\frac {2 a c (g \cos (e+f x))^{5/2} (c-c \sin (e+f x))^{3/2}}{21 f g \sqrt {a+a \sin (e+f x)}}-\frac {2 a (g \cos (e+f x))^{5/2} (c-c \sin (e+f x))^{5/2}}{9 f g \sqrt {a+a \sin (e+f x)}}+\frac {1}{15} \left (11 a 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 {22 a 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 {22 a c^2 (g \cos (e+f x))^{5/2} \sqrt {c-c \sin (e+f x)}}{105 f g \sqrt {a+a \sin (e+f x)}}+\frac {2 a c (g \cos (e+f x))^{5/2} (c-c \sin (e+f x))^{3/2}}{21 f g \sqrt {a+a \sin (e+f x)}}-\frac {2 a (g \cos (e+f x))^{5/2} (c-c \sin (e+f x))^{5/2}}{9 f g \sqrt {a+a \sin (e+f x)}}+\frac {1}{15} \left (11 a 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 {22 a 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 {22 a c^2 (g \cos (e+f x))^{5/2} \sqrt {c-c \sin (e+f x)}}{105 f g \sqrt {a+a \sin (e+f x)}}+\frac {2 a c (g \cos (e+f x))^{5/2} (c-c \sin (e+f x))^{3/2}}{21 f g \sqrt {a+a \sin (e+f x)}}-\frac {2 a (g \cos (e+f x))^{5/2} (c-c \sin (e+f x))^{5/2}}{9 f g \sqrt {a+a \sin (e+f x)}}+\frac {\left (11 a 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 {22 a 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 {22 a c^2 (g \cos (e+f x))^{5/2} \sqrt {c-c \sin (e+f x)}}{105 f g \sqrt {a+a \sin (e+f x)}}+\frac {2 a c (g \cos (e+f x))^{5/2} (c-c \sin (e+f x))^{3/2}}{21 f g \sqrt {a+a \sin (e+f x)}}-\frac {2 a (g \cos (e+f x))^{5/2} (c-c \sin (e+f x))^{5/2}}{9 f g \sqrt {a+a \sin (e+f x)}}+\frac {\left (11 a 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 {22 a 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 {22 a 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 {22 a c^2 (g \cos (e+f x))^{5/2} \sqrt {c-c \sin (e+f x)}}{105 f g \sqrt {a+a \sin (e+f x)}}+\frac {2 a c (g \cos (e+f x))^{5/2} (c-c \sin (e+f x))^{3/2}}{21 f g \sqrt {a+a \sin (e+f x)}}-\frac {2 a (g \cos (e+f x))^{5/2} (c-c \sin (e+f x))^{5/2}}{9 f g \sqrt {a+a \sin (e+f x)}} \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 3.41 (sec) , antiderivative size = 281, normalized size of antiderivative = 0.97 \[ \int (g \cos (e+f x))^{3/2} \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{5/2} \, dx=-\frac {c^3 e^{-4 i (e+f x)} \left (-i+e^{i (e+f x)}\right ) g \sqrt {g \cos (e+f x)} \left (\sqrt {1+e^{2 i (e+f x)}} \left (-35-180 i e^{i (e+f x)}+238 e^{2 i (e+f x)}-540 i e^{3 i (e+f x)}+3696 e^{4 i (e+f x)}-540 i e^{5 i (e+f x)}-238 e^{6 i (e+f x)}-180 i e^{7 i (e+f x)}+35 e^{8 i (e+f x)}\right )-2464 e^{6 i (e+f x)} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {3}{4},\frac {7}{4},-e^{2 i (e+f x)}\right )\right ) \sqrt {a (1+\sin (e+f x))}}{2520 \left (i+e^{i (e+f x)}\right ) \sqrt {1+e^{2 i (e+f x)}} f \sqrt {c-c \sin (e+f x)}} \]
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Result contains complex when optimal does not.
Time = 5.24 (sec) , antiderivative size = 496, normalized size of antiderivative = 1.71
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} g \left (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 )-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 )+35 \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 )}}\, 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 )-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 )-90 \left (\cos ^{3}\left (f x +e \right )\right )+35 \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 )}}\, 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 )-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 )-90 \left (\cos ^{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 )}{315 f \left (1+\cos \left (f x +e \right )\right )}\) | \(496\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.12 (sec) , antiderivative size = 159, normalized size of antiderivative = 0.55 \[ \int (g \cos (e+f x))^{3/2} \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{5/2} \, dx=\frac {-231 i \, \sqrt {2} \sqrt {a c g} c^{2} g {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (f x + e\right ) + i \, \sin \left (f x + e\right )\right )\right ) + 231 i \, \sqrt {2} \sqrt {a c g} c^{2} g {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (f x + e\right ) - i \, \sin \left (f x + e\right )\right )\right ) + 2 \, {\left (90 \, c^{2} g \cos \left (f x + e\right )^{2} - 7 \, {\left (5 \, c^{2} g \cos \left (f x + e\right )^{2} - 11 \, c^{2} g\right )} \sin \left (f x + e\right )\right )} \sqrt {g \cos \left (f x + e\right )} \sqrt {a \sin \left (f x + e\right ) + a} \sqrt {-c \sin \left (f x + e\right ) + c}}{315 \, f} \]
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Timed out. \[ \int (g \cos (e+f x))^{3/2} \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{5/2} \, dx=\text {Timed out} \]
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\[ \int (g \cos (e+f x))^{3/2} \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{5/2} \, dx=\int { \left (g \cos \left (f x + e\right )\right )^{\frac {3}{2}} \sqrt {a \sin \left (f x + e\right ) + a} {\left (-c \sin \left (f x + e\right ) + c\right )}^{\frac {5}{2}} \,d x } \]
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\[ \int (g \cos (e+f x))^{3/2} \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{5/2} \, dx=\int { \left (g \cos \left (f x + e\right )\right )^{\frac {3}{2}} \sqrt {a \sin \left (f x + e\right ) + a} {\left (-c \sin \left (f x + e\right ) + c\right )}^{\frac {5}{2}} \,d x } \]
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Timed out. \[ \int (g \cos (e+f x))^{3/2} \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{5/2} \, dx=\int {\left (g\,\cos \left (e+f\,x\right )\right )}^{3/2}\,\sqrt {a+a\,\sin \left (e+f\,x\right )}\,{\left (c-c\,\sin \left (e+f\,x\right )\right )}^{5/2} \,d x \]
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