Integrand size = 42, antiderivative size = 343 \[ \int (g \cos (e+f x))^{3/2} \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{7/2} \, dx=\frac {2 a c^4 (g \cos (e+f x))^{5/2}}{3 f g \sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}}+\frac {2 a c^4 g \sqrt {\cos (e+f x)} \sqrt {g \cos (e+f x)} E\left (\left .\frac {1}{2} (e+f x)\right |2\right )}{f \sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}}+\frac {2 a c^3 (g \cos (e+f x))^{5/2} \sqrt {c-c \sin (e+f x)}}{7 f g \sqrt {a+a \sin (e+f x)}}+\frac {10 a c^2 (g \cos (e+f x))^{5/2} (c-c \sin (e+f x))^{3/2}}{77 f g \sqrt {a+a \sin (e+f x)}}+\frac {2 a c (g \cos (e+f x))^{5/2} (c-c \sin (e+f x))^{5/2}}{33 f g \sqrt {a+a \sin (e+f x)}}-\frac {2 a (g \cos (e+f x))^{5/2} (c-c \sin (e+f x))^{7/2}}{11 f g \sqrt {a+a \sin (e+f x)}} \]
[Out]
Time = 1.25 (sec) , antiderivative size = 343, 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} \[ \int (g \cos (e+f x))^{3/2} \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{7/2} \, dx=\frac {2 a c^4 (g \cos (e+f x))^{5/2}}{3 f g \sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}+\frac {2 a c^4 g \sqrt {\cos (e+f x)} E\left (\left .\frac {1}{2} (e+f x)\right |2\right ) \sqrt {g \cos (e+f x)}}{f \sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}+\frac {2 a c^3 \sqrt {c-c \sin (e+f x)} (g \cos (e+f x))^{5/2}}{7 f g \sqrt {a \sin (e+f x)+a}}+\frac {10 a c^2 (c-c \sin (e+f x))^{3/2} (g \cos (e+f x))^{5/2}}{77 f g \sqrt {a \sin (e+f x)+a}}+\frac {2 a c (c-c \sin (e+f x))^{5/2} (g \cos (e+f x))^{5/2}}{33 f g \sqrt {a \sin (e+f x)+a}}-\frac {2 a (c-c \sin (e+f x))^{7/2} (g \cos (e+f x))^{5/2}}{11 f g \sqrt {a \sin (e+f x)+a}} \]
[In]
[Out]
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))^{7/2}}{11 f g \sqrt {a+a \sin (e+f x)}}+\frac {1}{11} (3 a) \int \frac {(g \cos (e+f x))^{3/2} (c-c \sin (e+f x))^{7/2}}{\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))^{5/2}}{33 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))^{7/2}}{11 f g \sqrt {a+a \sin (e+f x)}}+\frac {1}{11} (5 a c) \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 {10 a c^2 (g \cos (e+f x))^{5/2} (c-c \sin (e+f x))^{3/2}}{77 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))^{5/2}}{33 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))^{7/2}}{11 f g \sqrt {a+a \sin (e+f x)}}+\frac {1}{7} \left (5 a c^2\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 c^3 (g \cos (e+f x))^{5/2} \sqrt {c-c \sin (e+f x)}}{7 f g \sqrt {a+a \sin (e+f x)}}+\frac {10 a c^2 (g \cos (e+f x))^{5/2} (c-c \sin (e+f x))^{3/2}}{77 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))^{5/2}}{33 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))^{7/2}}{11 f g \sqrt {a+a \sin (e+f x)}}+\left (a c^3\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 {2 a c^4 (g \cos (e+f x))^{5/2}}{3 f g \sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}}+\frac {2 a c^3 (g \cos (e+f x))^{5/2} \sqrt {c-c \sin (e+f x)}}{7 f g \sqrt {a+a \sin (e+f x)}}+\frac {10 a c^2 (g \cos (e+f x))^{5/2} (c-c \sin (e+f x))^{3/2}}{77 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))^{5/2}}{33 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))^{7/2}}{11 f g \sqrt {a+a \sin (e+f x)}}+\left (a c^4\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 {2 a c^4 (g \cos (e+f x))^{5/2}}{3 f g \sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}}+\frac {2 a c^3 (g \cos (e+f x))^{5/2} \sqrt {c-c \sin (e+f x)}}{7 f g \sqrt {a+a \sin (e+f x)}}+\frac {10 a c^2 (g \cos (e+f x))^{5/2} (c-c \sin (e+f x))^{3/2}}{77 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))^{5/2}}{33 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))^{7/2}}{11 f g \sqrt {a+a \sin (e+f x)}}+\frac {\left (a c^4 g \cos (e+f x)\right ) \int \sqrt {g \cos (e+f x)} \, dx}{\sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}} \\ & = \frac {2 a c^4 (g \cos (e+f x))^{5/2}}{3 f g \sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}}+\frac {2 a c^3 (g \cos (e+f x))^{5/2} \sqrt {c-c \sin (e+f x)}}{7 f g \sqrt {a+a \sin (e+f x)}}+\frac {10 a c^2 (g \cos (e+f x))^{5/2} (c-c \sin (e+f x))^{3/2}}{77 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))^{5/2}}{33 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))^{7/2}}{11 f g \sqrt {a+a \sin (e+f x)}}+\frac {\left (a c^4 g \sqrt {\cos (e+f x)} \sqrt {g \cos (e+f x)}\right ) \int \sqrt {\cos (e+f x)} \, dx}{\sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}} \\ & = \frac {2 a c^4 (g \cos (e+f x))^{5/2}}{3 f g \sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}}+\frac {2 a c^4 g \sqrt {\cos (e+f x)} \sqrt {g \cos (e+f x)} E\left (\left .\frac {1}{2} (e+f x)\right |2\right )}{f \sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}}+\frac {2 a c^3 (g \cos (e+f x))^{5/2} \sqrt {c-c \sin (e+f x)}}{7 f g \sqrt {a+a \sin (e+f x)}}+\frac {10 a c^2 (g \cos (e+f x))^{5/2} (c-c \sin (e+f x))^{3/2}}{77 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))^{5/2}}{33 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))^{7/2}}{11 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 = 3.96 (sec) , antiderivative size = 311, normalized size of antiderivative = 0.91 \[ \int (g \cos (e+f x))^{3/2} \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{7/2} \, dx=\frac {c^4 e^{-5 i (e+f x)} \left (-i+e^{i (e+f x)}\right ) g \sqrt {g \cos (e+f x)} \left (\sqrt {1+e^{2 i (e+f x)}} \left (-21 i+154 e^{i (e+f x)}+423 i e^{2 i (e+f x)}-308 e^{3 i (e+f x)}+1374 i e^{4 i (e+f x)}-7392 e^{5 i (e+f x)}+1374 i e^{6 i (e+f x)}+308 e^{7 i (e+f x)}+423 i e^{8 i (e+f x)}-154 e^{9 i (e+f x)}-21 i e^{10 i (e+f x)}\right )+4928 e^{7 i (e+f x)} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {3}{4},\frac {7}{4},-e^{2 i (e+f x)}\right )\right ) \sqrt {a (1+\sin (e+f x))}}{3696 \left (i+e^{i (e+f x)}\right ) \sqrt {1+e^{2 i (e+f x)}} f \sqrt {c-c \sin (e+f x)}} \]
[In]
[Out]
Result contains complex when optimal does not.
Time = 7.69 (sec) , antiderivative size = 516, normalized size of antiderivative = 1.50
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^{3} g \left (-21 \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 )-21 \left (\cos ^{4}\left (f x +e \right )\right )-77 \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 )+132 \left (\cos ^{3}\left (f x +e \right )\right )-77 \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 )+132 \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 )}{231 f \left (1+\cos \left (f x +e \right )\right )}\) | \(516\) |
[In]
[Out]
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.13 (sec) , antiderivative size = 172, normalized size of antiderivative = 0.50 \[ \int (g \cos (e+f x))^{3/2} \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{7/2} \, dx=\frac {-231 i \, \sqrt {2} \sqrt {a c g} c^{3} g {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (f x + e\right ) + i \, \sin \left (f x + e\right )\right )\right ) + 231 i \, \sqrt {2} \sqrt {a c g} c^{3} g {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (f x + e\right ) - i \, \sin \left (f x + e\right )\right )\right ) - 2 \, {\left (21 \, c^{3} g \cos \left (f x + e\right )^{4} - 132 \, c^{3} g \cos \left (f x + e\right )^{2} + 77 \, {\left (c^{3} g \cos \left (f x + e\right )^{2} - c^{3} g\right )} \sin \left (f x + e\right )\right )} \sqrt {g \cos \left (f x + e\right )} \sqrt {a \sin \left (f x + e\right ) + a} \sqrt {-c \sin \left (f x + e\right ) + c}}{231 \, f} \]
[In]
[Out]
Timed out. \[ \int (g \cos (e+f x))^{3/2} \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{7/2} \, dx=\text {Timed out} \]
[In]
[Out]
\[ \int (g \cos (e+f x))^{3/2} \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{7/2} \, dx=\int { \left (g \cos \left (f x + e\right )\right )^{\frac {3}{2}} \sqrt {a \sin \left (f x + e\right ) + a} {\left (-c \sin \left (f x + e\right ) + c\right )}^{\frac {7}{2}} \,d x } \]
[In]
[Out]
\[ \int (g \cos (e+f x))^{3/2} \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{7/2} \, dx=\int { \left (g \cos \left (f x + e\right )\right )^{\frac {3}{2}} \sqrt {a \sin \left (f x + e\right ) + a} {\left (-c \sin \left (f x + e\right ) + c\right )}^{\frac {7}{2}} \,d x } \]
[In]
[Out]
Timed out. \[ \int (g \cos (e+f x))^{3/2} \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{7/2} \, dx=\int {\left (g\,\cos \left (e+f\,x\right )\right )}^{3/2}\,\sqrt {a+a\,\sin \left (e+f\,x\right )}\,{\left (c-c\,\sin \left (e+f\,x\right )\right )}^{7/2} \,d x \]
[In]
[Out]