Integrand size = 42, antiderivative size = 234 \[ \int \frac {(g \cos (e+f x))^{3/2} (c-c \sin (e+f x))^{5/2}}{\sqrt {a+a \sin (e+f x)}} \, dx=\frac {22 c^3 (g \cos (e+f x))^{5/2}}{15 f g \sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}}+\frac {22 c^3 g \sqrt {\cos (e+f x)} \sqrt {g \cos (e+f x)} E\left (\left .\frac {1}{2} (e+f x)\right |2\right )}{5 f \sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}}+\frac {22 c^2 (g \cos (e+f x))^{5/2} \sqrt {c-c \sin (e+f x)}}{35 f g \sqrt {a+a \sin (e+f x)}}+\frac {2 c (g \cos (e+f x))^{5/2} (c-c \sin (e+f x))^{3/2}}{7 f g \sqrt {a+a \sin (e+f x)}} \]
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Time = 0.75 (sec) , antiderivative size = 234, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {2930, 2921, 2721, 2719} \[ \int \frac {(g \cos (e+f x))^{3/2} (c-c \sin (e+f x))^{5/2}}{\sqrt {a+a \sin (e+f x)}} \, dx=\frac {22 c^3 (g \cos (e+f x))^{5/2}}{15 f g \sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}+\frac {22 c^3 g \sqrt {\cos (e+f x)} E\left (\left .\frac {1}{2} (e+f x)\right |2\right ) \sqrt {g \cos (e+f x)}}{5 f \sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}+\frac {22 c^2 \sqrt {c-c \sin (e+f x)} (g \cos (e+f x))^{5/2}}{35 f g \sqrt {a \sin (e+f x)+a}}+\frac {2 c (c-c \sin (e+f x))^{3/2} (g \cos (e+f x))^{5/2}}{7 f g \sqrt {a \sin (e+f x)+a}} \]
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Rule 2719
Rule 2721
Rule 2921
Rule 2930
Rubi steps \begin{align*} \text {integral}& = \frac {2 c (g \cos (e+f x))^{5/2} (c-c \sin (e+f x))^{3/2}}{7 f g \sqrt {a+a \sin (e+f x)}}+\frac {1}{7} (11 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 c^2 (g \cos (e+f x))^{5/2} \sqrt {c-c \sin (e+f x)}}{35 f g \sqrt {a+a \sin (e+f x)}}+\frac {2 c (g \cos (e+f x))^{5/2} (c-c \sin (e+f x))^{3/2}}{7 f g \sqrt {a+a \sin (e+f x)}}+\frac {1}{5} \left (11 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 c^3 (g \cos (e+f x))^{5/2}}{15 f g \sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}}+\frac {22 c^2 (g \cos (e+f x))^{5/2} \sqrt {c-c \sin (e+f x)}}{35 f g \sqrt {a+a \sin (e+f x)}}+\frac {2 c (g \cos (e+f x))^{5/2} (c-c \sin (e+f x))^{3/2}}{7 f g \sqrt {a+a \sin (e+f x)}}+\frac {1}{5} \left (11 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 c^3 (g \cos (e+f x))^{5/2}}{15 f g \sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}}+\frac {22 c^2 (g \cos (e+f x))^{5/2} \sqrt {c-c \sin (e+f x)}}{35 f g \sqrt {a+a \sin (e+f x)}}+\frac {2 c (g \cos (e+f x))^{5/2} (c-c \sin (e+f x))^{3/2}}{7 f g \sqrt {a+a \sin (e+f x)}}+\frac {\left (11 c^3 g \cos (e+f x)\right ) \int \sqrt {g \cos (e+f x)} \, dx}{5 \sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}} \\ & = \frac {22 c^3 (g \cos (e+f x))^{5/2}}{15 f g \sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}}+\frac {22 c^2 (g \cos (e+f x))^{5/2} \sqrt {c-c \sin (e+f x)}}{35 f g \sqrt {a+a \sin (e+f x)}}+\frac {2 c (g \cos (e+f x))^{5/2} (c-c \sin (e+f x))^{3/2}}{7 f g \sqrt {a+a \sin (e+f x)}}+\frac {\left (11 c^3 g \sqrt {\cos (e+f x)} \sqrt {g \cos (e+f x)}\right ) \int \sqrt {\cos (e+f x)} \, dx}{5 \sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}} \\ & = \frac {22 c^3 (g \cos (e+f x))^{5/2}}{15 f g \sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}}+\frac {22 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 )}{5 f \sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}}+\frac {22 c^2 (g \cos (e+f x))^{5/2} \sqrt {c-c \sin (e+f x)}}{35 f g \sqrt {a+a \sin (e+f x)}}+\frac {2 c (g \cos (e+f x))^{5/2} (c-c \sin (e+f x))^{3/2}}{7 f g \sqrt {a+a \sin (e+f x)}} \\ \end{align*}
Time = 8.87 (sec) , antiderivative size = 174, normalized size of antiderivative = 0.74 \[ \int \frac {(g \cos (e+f x))^{3/2} (c-c \sin (e+f x))^{5/2}}{\sqrt {a+a \sin (e+f x)}} \, dx=\frac {c^2 (g \cos (e+f x))^{3/2} \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right ) (-1+\sin (e+f x))^2 \sqrt {c-c \sin (e+f x)} \left (924 E\left (\left .\frac {1}{2} (e+f x)\right |2\right )+\sqrt {\cos (e+f x)} (515 \cos (e+f x)-3 (5 \cos (3 (e+f x))+42 \sin (2 (e+f x))))\right )}{210 f \cos ^{\frac {3}{2}}(e+f x) \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^5 \sqrt {a (1+\sin (e+f x))}} \]
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Result contains complex when optimal does not.
Time = 2.42 (sec) , antiderivative size = 510, normalized size of antiderivative = 2.18
method | result | size |
default | \(\frac {2 \left (-231 i \sqrt {\frac {1}{1+\cos \left (f x +e \right )}}\, \sqrt {\frac {\cos \left (f x +e \right )}{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 (\cos ^{2}\left (f x +e \right )\right )+231 i \sqrt {\frac {1}{1+\cos \left (f x +e \right )}}\, \sqrt {\frac {\cos \left (f x +e \right )}{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 (\cos ^{2}\left (f x +e \right )\right )+15 \left (\cos ^{5}\left (f x +e \right )\right )-462 i \sqrt {\frac {1}{1+\cos \left (f x +e \right )}}\, \sqrt {\frac {\cos \left (f x +e \right )}{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 ) \cos \left (f x +e \right )+462 i \sqrt {\frac {1}{1+\cos \left (f x +e \right )}}\, \sqrt {\frac {\cos \left (f x +e \right )}{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 ) \cos \left (f x +e \right )+15 \left (\cos ^{4}\left (f x +e \right )\right )+63 \left (\cos ^{3}\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 )-140 \left (\cos ^{3}\left (f x +e \right )\right )+63 \left (\cos ^{2}\left (f x +e \right )\right ) \sin \left (f x +e \right )-140 \left (\cos ^{2}\left (f x +e \right )\right )-231 \cos \left (f x +e \right ) \sin \left (f x +e \right )\right ) \sqrt {-c \left (\sin \left (f x +e \right )-1\right )}\, \sqrt {g \cos \left (f x +e \right )}\, g \,c^{2}}{105 f \left (\sin \left (f x +e \right )-1\right ) \left (1+\cos \left (f x +e \right )\right ) \sqrt {a \left (1+\sin \left (f x +e \right )\right )}}\) | \(510\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.13 (sec) , antiderivative size = 151, normalized size of antiderivative = 0.65 \[ \int \frac {(g \cos (e+f x))^{3/2} (c-c \sin (e+f x))^{5/2}}{\sqrt {a+a \sin (e+f x)}} \, dx=\frac {-231 i \, \sqrt {2} \sqrt {a c g} c^{2} g {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (f x + e\right ) + i \, \sin \left (f x + e\right )\right )\right ) + 231 i \, \sqrt {2} \sqrt {a c g} c^{2} g {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (f x + e\right ) - i \, \sin \left (f x + e\right )\right )\right ) - 2 \, {\left (15 \, c^{2} g \cos \left (f x + e\right )^{2} + 63 \, c^{2} g \sin \left (f x + e\right ) - 140 \, c^{2} g\right )} \sqrt {g \cos \left (f x + e\right )} \sqrt {a \sin \left (f x + e\right ) + a} \sqrt {-c \sin \left (f x + e\right ) + c}}{105 \, a f} \]
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Timed out. \[ \int \frac {(g \cos (e+f x))^{3/2} (c-c \sin (e+f x))^{5/2}}{\sqrt {a+a \sin (e+f x)}} \, dx=\text {Timed out} \]
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\[ \int \frac {(g \cos (e+f x))^{3/2} (c-c \sin (e+f x))^{5/2}}{\sqrt {a+a \sin (e+f x)}} \, dx=\int { \frac {\left (g \cos \left (f x + e\right )\right )^{\frac {3}{2}} {\left (-c \sin \left (f x + e\right ) + c\right )}^{\frac {5}{2}}}{\sqrt {a \sin \left (f x + e\right ) + a}} \,d x } \]
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Timed out. \[ \int \frac {(g \cos (e+f x))^{3/2} (c-c \sin (e+f x))^{5/2}}{\sqrt {a+a \sin (e+f x)}} \, dx=\text {Timed out} \]
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Timed out. \[ \int \frac {(g \cos (e+f x))^{3/2} (c-c \sin (e+f x))^{5/2}}{\sqrt {a+a \sin (e+f x)}} \, dx=\int \frac {{\left (g\,\cos \left (e+f\,x\right )\right )}^{3/2}\,{\left (c-c\,\sin \left (e+f\,x\right )\right )}^{5/2}}{\sqrt {a+a\,\sin \left (e+f\,x\right )}} \,d x \]
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