Integrand size = 42, antiderivative size = 343 \[ \int (g \cos (e+f x))^{3/2} \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{7/2} \, dx=\frac {2 a c^4 (g \cos (e+f x))^{5/2}}{3 f g \sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}}+\frac {2 a c^4 g \sqrt {\cos (e+f x)} \sqrt {g \cos (e+f x)} E\left (\left .\frac {1}{2} (e+f x)\right |2\right )}{f \sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}}+\frac {2 a c^3 (g \cos (e+f x))^{5/2} \sqrt {c-c \sin (e+f x)}}{7 f g \sqrt {a+a \sin (e+f x)}}+\frac {10 a c^2 (g \cos (e+f x))^{5/2} (c-c \sin (e+f x))^{3/2}}{77 f g \sqrt {a+a \sin (e+f x)}}+\frac {2 a c (g \cos (e+f x))^{5/2} (c-c \sin (e+f x))^{5/2}}{33 f g \sqrt {a+a \sin (e+f x)}}-\frac {2 a (g \cos (e+f x))^{5/2} (c-c \sin (e+f x))^{7/2}}{11 f g \sqrt {a+a \sin (e+f x)}} \]
10/77*a*c^2*(g*cos(f*x+e))^(5/2)*(c-c*sin(f*x+e))^(3/2)/f/g/(a+a*sin(f*x+e ))^(1/2)+2/33*a*c*(g*cos(f*x+e))^(5/2)*(c-c*sin(f*x+e))^(5/2)/f/g/(a+a*sin (f*x+e))^(1/2)-2/11*a*(g*cos(f*x+e))^(5/2)*(c-c*sin(f*x+e))^(7/2)/f/g/(a+a *sin(f*x+e))^(1/2)+2/3*a*c^4*(g*cos(f*x+e))^(5/2)/f/g/(a+a*sin(f*x+e))^(1/ 2)/(c-c*sin(f*x+e))^(1/2)+2*a*c^4*g*(cos(1/2*f*x+1/2*e)^2)^(1/2)/cos(1/2*f *x+1/2*e)*EllipticE(sin(1/2*f*x+1/2*e),2^(1/2))*cos(f*x+e)^(1/2)*(g*cos(f* x+e))^(1/2)/f/(a+a*sin(f*x+e))^(1/2)/(c-c*sin(f*x+e))^(1/2)+2/7*a*c^3*(g*c os(f*x+e))^(5/2)*(c-c*sin(f*x+e))^(1/2)/f/g/(a+a*sin(f*x+e))^(1/2)
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 \[ \int (g \cos (e+f x))^{3/2} \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{7/2} \, dx=\frac {c^4 e^{-5 i (e+f x)} \left (-i+e^{i (e+f x)}\right ) g \sqrt {g \cos (e+f x)} \left (\sqrt {1+e^{2 i (e+f x)}} \left (-21 i+154 e^{i (e+f x)}+423 i e^{2 i (e+f x)}-308 e^{3 i (e+f x)}+1374 i e^{4 i (e+f x)}-7392 e^{5 i (e+f x)}+1374 i e^{6 i (e+f x)}+308 e^{7 i (e+f x)}+423 i e^{8 i (e+f x)}-154 e^{9 i (e+f x)}-21 i e^{10 i (e+f x)}\right )+4928 e^{7 i (e+f x)} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {3}{4},\frac {7}{4},-e^{2 i (e+f x)}\right )\right ) \sqrt {a (1+\sin (e+f x))}}{3696 \left (i+e^{i (e+f x)}\right ) \sqrt {1+e^{2 i (e+f x)}} f \sqrt {c-c \sin (e+f x)}} \]
(c^4*(-I + E^(I*(e + f*x)))*g*Sqrt[g*Cos[e + f*x]]*(Sqrt[1 + E^((2*I)*(e + f*x))]*(-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))) + 4928*E^((7*I)*(e + f*x))*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(e + f*x))])*Sqrt[a*(1 + Sin[e + f*x])])/(3696*E^((5*I)*(e + f*x))*(I + E^(I*(e + f*x)))*Sqrt[1 + E^((2*I)*(e + f*x))]*f*Sqrt[c - c*Sin[e + f*x]])
Time = 2.72 (sec) , antiderivative size = 354, normalized size of antiderivative = 1.03, number of steps used = 16, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.381, Rules used = {3042, 3330, 3042, 3330, 3042, 3330, 3042, 3330, 3042, 3330, 3042, 3321, 3042, 3121, 3042, 3119}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{7/2} (g \cos (e+f x))^{3/2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{7/2} (g \cos (e+f x))^{3/2}dx\) |
| \(\Big \downarrow \) 3330 |
| \(\displaystyle \frac {3}{11} a \int \frac {(g \cos (e+f x))^{3/2} (c-c \sin (e+f x))^{7/2}}{\sqrt {\sin (e+f x) a+a}}dx-\frac {2 a (c-c \sin (e+f x))^{7/2} (g \cos (e+f x))^{5/2}}{11 f g \sqrt {a \sin (e+f x)+a}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {3}{11} a \int \frac {(g \cos (e+f x))^{3/2} (c-c \sin (e+f x))^{7/2}}{\sqrt {\sin (e+f x) a+a}}dx-\frac {2 a (c-c \sin (e+f x))^{7/2} (g \cos (e+f x))^{5/2}}{11 f g \sqrt {a \sin (e+f x)+a}}\) |
| \(\Big \downarrow \) 3330 |
| \(\displaystyle \frac {3}{11} a \left (\frac {5}{3} c \int \frac {(g \cos (e+f x))^{3/2} (c-c \sin (e+f x))^{5/2}}{\sqrt {\sin (e+f x) a+a}}dx+\frac {2 c (c-c \sin (e+f x))^{5/2} (g \cos (e+f x))^{5/2}}{9 f g \sqrt {a \sin (e+f x)+a}}\right )-\frac {2 a (c-c \sin (e+f x))^{7/2} (g \cos (e+f x))^{5/2}}{11 f g \sqrt {a \sin (e+f x)+a}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {3}{11} a \left (\frac {5}{3} c \int \frac {(g \cos (e+f x))^{3/2} (c-c \sin (e+f x))^{5/2}}{\sqrt {\sin (e+f x) a+a}}dx+\frac {2 c (c-c \sin (e+f x))^{5/2} (g \cos (e+f x))^{5/2}}{9 f g \sqrt {a \sin (e+f x)+a}}\right )-\frac {2 a (c-c \sin (e+f x))^{7/2} (g \cos (e+f x))^{5/2}}{11 f g \sqrt {a \sin (e+f x)+a}}\) |
| \(\Big \downarrow \) 3330 |
| \(\displaystyle \frac {3}{11} a \left (\frac {5}{3} c \left (\frac {11}{7} c \int \frac {(g \cos (e+f x))^{3/2} (c-c \sin (e+f x))^{3/2}}{\sqrt {\sin (e+f x) a+a}}dx+\frac {2 c (c-c \sin (e+f x))^{3/2} (g \cos (e+f x))^{5/2}}{7 f g \sqrt {a \sin (e+f x)+a}}\right )+\frac {2 c (c-c \sin (e+f x))^{5/2} (g \cos (e+f x))^{5/2}}{9 f g \sqrt {a \sin (e+f x)+a}}\right )-\frac {2 a (c-c \sin (e+f x))^{7/2} (g \cos (e+f x))^{5/2}}{11 f g \sqrt {a \sin (e+f x)+a}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {3}{11} a \left (\frac {5}{3} c \left (\frac {11}{7} c \int \frac {(g \cos (e+f x))^{3/2} (c-c \sin (e+f x))^{3/2}}{\sqrt {\sin (e+f x) a+a}}dx+\frac {2 c (c-c \sin (e+f x))^{3/2} (g \cos (e+f x))^{5/2}}{7 f g \sqrt {a \sin (e+f x)+a}}\right )+\frac {2 c (c-c \sin (e+f x))^{5/2} (g \cos (e+f x))^{5/2}}{9 f g \sqrt {a \sin (e+f x)+a}}\right )-\frac {2 a (c-c \sin (e+f x))^{7/2} (g \cos (e+f x))^{5/2}}{11 f g \sqrt {a \sin (e+f x)+a}}\) |
| \(\Big \downarrow \) 3330 |
| \(\displaystyle \frac {3}{11} a \left (\frac {5}{3} c \left (\frac {11}{7} c \left (\frac {7}{5} c \int \frac {(g \cos (e+f x))^{3/2} \sqrt {c-c \sin (e+f x)}}{\sqrt {\sin (e+f x) a+a}}dx+\frac {2 c \sqrt {c-c \sin (e+f x)} (g \cos (e+f x))^{5/2}}{5 f g \sqrt {a \sin (e+f x)+a}}\right )+\frac {2 c (c-c \sin (e+f x))^{3/2} (g \cos (e+f x))^{5/2}}{7 f g \sqrt {a \sin (e+f x)+a}}\right )+\frac {2 c (c-c \sin (e+f x))^{5/2} (g \cos (e+f x))^{5/2}}{9 f g \sqrt {a \sin (e+f x)+a}}\right )-\frac {2 a (c-c \sin (e+f x))^{7/2} (g \cos (e+f x))^{5/2}}{11 f g \sqrt {a \sin (e+f x)+a}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {3}{11} a \left (\frac {5}{3} c \left (\frac {11}{7} c \left (\frac {7}{5} c \int \frac {(g \cos (e+f x))^{3/2} \sqrt {c-c \sin (e+f x)}}{\sqrt {\sin (e+f x) a+a}}dx+\frac {2 c \sqrt {c-c \sin (e+f x)} (g \cos (e+f x))^{5/2}}{5 f g \sqrt {a \sin (e+f x)+a}}\right )+\frac {2 c (c-c \sin (e+f x))^{3/2} (g \cos (e+f x))^{5/2}}{7 f g \sqrt {a \sin (e+f x)+a}}\right )+\frac {2 c (c-c \sin (e+f x))^{5/2} (g \cos (e+f x))^{5/2}}{9 f g \sqrt {a \sin (e+f x)+a}}\right )-\frac {2 a (c-c \sin (e+f x))^{7/2} (g \cos (e+f x))^{5/2}}{11 f g \sqrt {a \sin (e+f x)+a}}\) |
| \(\Big \downarrow \) 3330 |
| \(\displaystyle \frac {3}{11} a \left (\frac {5}{3} c \left (\frac {11}{7} c \left (\frac {7}{5} c \left (c \int \frac {(g \cos (e+f x))^{3/2}}{\sqrt {\sin (e+f x) a+a} \sqrt {c-c \sin (e+f x)}}dx+\frac {2 c (g \cos (e+f x))^{5/2}}{3 f g \sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}\right )+\frac {2 c \sqrt {c-c \sin (e+f x)} (g \cos (e+f x))^{5/2}}{5 f g \sqrt {a \sin (e+f x)+a}}\right )+\frac {2 c (c-c \sin (e+f x))^{3/2} (g \cos (e+f x))^{5/2}}{7 f g \sqrt {a \sin (e+f x)+a}}\right )+\frac {2 c (c-c \sin (e+f x))^{5/2} (g \cos (e+f x))^{5/2}}{9 f g \sqrt {a \sin (e+f x)+a}}\right )-\frac {2 a (c-c \sin (e+f x))^{7/2} (g \cos (e+f x))^{5/2}}{11 f g \sqrt {a \sin (e+f x)+a}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {3}{11} a \left (\frac {5}{3} c \left (\frac {11}{7} c \left (\frac {7}{5} c \left (c \int \frac {(g \cos (e+f x))^{3/2}}{\sqrt {\sin (e+f x) a+a} \sqrt {c-c \sin (e+f x)}}dx+\frac {2 c (g \cos (e+f x))^{5/2}}{3 f g \sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}\right )+\frac {2 c \sqrt {c-c \sin (e+f x)} (g \cos (e+f x))^{5/2}}{5 f g \sqrt {a \sin (e+f x)+a}}\right )+\frac {2 c (c-c \sin (e+f x))^{3/2} (g \cos (e+f x))^{5/2}}{7 f g \sqrt {a \sin (e+f x)+a}}\right )+\frac {2 c (c-c \sin (e+f x))^{5/2} (g \cos (e+f x))^{5/2}}{9 f g \sqrt {a \sin (e+f x)+a}}\right )-\frac {2 a (c-c \sin (e+f x))^{7/2} (g \cos (e+f x))^{5/2}}{11 f g \sqrt {a \sin (e+f x)+a}}\) |
| \(\Big \downarrow \) 3321 |
| \(\displaystyle \frac {3}{11} a \left (\frac {5}{3} c \left (\frac {11}{7} c \left (\frac {7}{5} c \left (\frac {c g \cos (e+f x) \int \sqrt {g \cos (e+f x)}dx}{\sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}+\frac {2 c (g \cos (e+f x))^{5/2}}{3 f g \sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}\right )+\frac {2 c \sqrt {c-c \sin (e+f x)} (g \cos (e+f x))^{5/2}}{5 f g \sqrt {a \sin (e+f x)+a}}\right )+\frac {2 c (c-c \sin (e+f x))^{3/2} (g \cos (e+f x))^{5/2}}{7 f g \sqrt {a \sin (e+f x)+a}}\right )+\frac {2 c (c-c \sin (e+f x))^{5/2} (g \cos (e+f x))^{5/2}}{9 f g \sqrt {a \sin (e+f x)+a}}\right )-\frac {2 a (c-c \sin (e+f x))^{7/2} (g \cos (e+f x))^{5/2}}{11 f g \sqrt {a \sin (e+f x)+a}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {3}{11} a \left (\frac {5}{3} c \left (\frac {11}{7} c \left (\frac {7}{5} c \left (\frac {c g \cos (e+f x) \int \sqrt {g \sin \left (e+f x+\frac {\pi }{2}\right )}dx}{\sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}+\frac {2 c (g \cos (e+f x))^{5/2}}{3 f g \sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}\right )+\frac {2 c \sqrt {c-c \sin (e+f x)} (g \cos (e+f x))^{5/2}}{5 f g \sqrt {a \sin (e+f x)+a}}\right )+\frac {2 c (c-c \sin (e+f x))^{3/2} (g \cos (e+f x))^{5/2}}{7 f g \sqrt {a \sin (e+f x)+a}}\right )+\frac {2 c (c-c \sin (e+f x))^{5/2} (g \cos (e+f x))^{5/2}}{9 f g \sqrt {a \sin (e+f x)+a}}\right )-\frac {2 a (c-c \sin (e+f x))^{7/2} (g \cos (e+f x))^{5/2}}{11 f g \sqrt {a \sin (e+f x)+a}}\) |
| \(\Big \downarrow \) 3121 |
| \(\displaystyle \frac {3}{11} a \left (\frac {5}{3} c \left (\frac {11}{7} c \left (\frac {7}{5} c \left (\frac {c g \sqrt {\cos (e+f x)} \sqrt {g \cos (e+f x)} \int \sqrt {\cos (e+f x)}dx}{\sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}+\frac {2 c (g \cos (e+f x))^{5/2}}{3 f g \sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}\right )+\frac {2 c \sqrt {c-c \sin (e+f x)} (g \cos (e+f x))^{5/2}}{5 f g \sqrt {a \sin (e+f x)+a}}\right )+\frac {2 c (c-c \sin (e+f x))^{3/2} (g \cos (e+f x))^{5/2}}{7 f g \sqrt {a \sin (e+f x)+a}}\right )+\frac {2 c (c-c \sin (e+f x))^{5/2} (g \cos (e+f x))^{5/2}}{9 f g \sqrt {a \sin (e+f x)+a}}\right )-\frac {2 a (c-c \sin (e+f x))^{7/2} (g \cos (e+f x))^{5/2}}{11 f g \sqrt {a \sin (e+f x)+a}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {3}{11} a \left (\frac {5}{3} c \left (\frac {11}{7} c \left (\frac {7}{5} c \left (\frac {c g \sqrt {\cos (e+f x)} \sqrt {g \cos (e+f x)} \int \sqrt {\sin \left (e+f x+\frac {\pi }{2}\right )}dx}{\sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}+\frac {2 c (g \cos (e+f x))^{5/2}}{3 f g \sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}\right )+\frac {2 c \sqrt {c-c \sin (e+f x)} (g \cos (e+f x))^{5/2}}{5 f g \sqrt {a \sin (e+f x)+a}}\right )+\frac {2 c (c-c \sin (e+f x))^{3/2} (g \cos (e+f x))^{5/2}}{7 f g \sqrt {a \sin (e+f x)+a}}\right )+\frac {2 c (c-c \sin (e+f x))^{5/2} (g \cos (e+f x))^{5/2}}{9 f g \sqrt {a \sin (e+f x)+a}}\right )-\frac {2 a (c-c \sin (e+f x))^{7/2} (g \cos (e+f x))^{5/2}}{11 f g \sqrt {a \sin (e+f x)+a}}\) |
| \(\Big \downarrow \) 3119 |
| \(\displaystyle \frac {3}{11} a \left (\frac {2 c (c-c \sin (e+f x))^{5/2} (g \cos (e+f x))^{5/2}}{9 f g \sqrt {a \sin (e+f x)+a}}+\frac {5}{3} c \left (\frac {2 c (c-c \sin (e+f x))^{3/2} (g \cos (e+f x))^{5/2}}{7 f g \sqrt {a \sin (e+f x)+a}}+\frac {11}{7} c \left (\frac {2 c \sqrt {c-c \sin (e+f x)} (g \cos (e+f x))^{5/2}}{5 f g \sqrt {a \sin (e+f x)+a}}+\frac {7}{5} c \left (\frac {2 c (g \cos (e+f x))^{5/2}}{3 f g \sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}+\frac {2 c g \sqrt {\cos (e+f x)} E\left (\left .\frac {1}{2} (e+f x)\right |2\right ) \sqrt {g \cos (e+f x)}}{f \sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}\right )\right )\right )\right )-\frac {2 a (c-c \sin (e+f x))^{7/2} (g \cos (e+f x))^{5/2}}{11 f g \sqrt {a \sin (e+f x)+a}}\) |
(-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]]) + (3*a*((2*c*(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]]) + (5*c*((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]]) + (11*c*((2 *c*(g*Cos[e + f*x])^(5/2)*Sqrt[c - c*Sin[e + f*x]])/(5*f*g*Sqrt[a + a*Sin[ e + f*x]]) + (7*c*((2*c*(g*Cos[e + f*x])^(5/2))/(3*f*g*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]]) + (2*c*g*Sqrt[Cos[e + f*x]]*Sqrt[g*Cos[e + f*x]]*EllipticE[(e + f*x)/2, 2])/(f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*S in[e + f*x]])))/5))/7))/3))/11
3.1.88.3.1 Defintions of rubi rules used
Int[Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/d)*EllipticE[(1/2)* (c - Pi/2 + d*x), 2], x] /; FreeQ[{c, d}, x]
Int[((b_)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(b*Sin[c + d*x]) ^n/Sin[c + d*x]^n Int[Sin[c + d*x]^n, x], x] /; FreeQ[{b, c, d}, x] && Lt Q[-1, n, 1] && IntegerQ[2*n]
Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)/(Sqrt[(a_) + (b_.)*sin[(e_.) + (f_ .)*(x_)]]*Sqrt[(c_) + (d_.)*sin[(e_.) + (f_.)*(x_)]]), x_Symbol] :> Simp[g* (Cos[e + f*x]/(Sqrt[a + b*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]])) Int[(g *Cos[e + f*x])^(p - 1), x], x] /; FreeQ[{a, b, c, d, e, f, g, p}, x] && EqQ [b*c + a*d, 0] && EqQ[a^2 - b^2, 0]
Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x _)])^(m_)*((c_) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[(- b)*(g*Cos[e + f*x])^(p + 1)*(a + b*Sin[e + f*x])^(m - 1)*((c + d*Sin[e + f* x])^n/(f*g*(m + n + p))), x] + Simp[a*((2*m + p - 1)/(m + n + p)) Int[(g* Cos[e + f*x])^p*(a + b*Sin[e + f*x])^(m - 1)*(c + d*Sin[e + f*x])^n, x], x] /; FreeQ[{a, b, c, d, e, f, g, n, p}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0] && GtQ[m, 0] && NeQ[m + n + p, 0] && !LtQ[0, n, m] && IntegersQ[2 *m, 2*n, 2*p]
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 )}\) | \(516\) |
int((g*cos(f*x+e))^(3/2)*(c-c*sin(f*x+e))^(7/2)*(a+a*sin(f*x+e))^(1/2),x,m ethod=_RETURNVERBOSE)
2/231/f*(-c*(sin(f*x+e)-1))^(1/2)*(a*(1+sin(f*x+e)))^(1/2)*(g*cos(f*x+e))^ (1/2)*c^3*g/(1+cos(f*x+e))*(-21*cos(f*x+e)^5+231*I*(cos(f*x+e)/(1+cos(f*x+ e)))^(1/2)*(1/(1+cos(f*x+e)))^(1/2)*EllipticE(I*(csc(f*x+e)-cot(f*x+e)),I) -231*I*(cos(f*x+e)/(1+cos(f*x+e)))^(1/2)*(1/(1+cos(f*x+e)))^(1/2)*Elliptic F(I*(csc(f*x+e)-cot(f*x+e)),I)-21*cos(f*x+e)^4-77*cos(f*x+e)^3*sin(f*x+e)+ 462*I*(cos(f*x+e)/(1+cos(f*x+e)))^(1/2)*(1/(1+cos(f*x+e)))^(1/2)*EllipticE (I*(csc(f*x+e)-cot(f*x+e)),I)*sec(f*x+e)-462*I*(cos(f*x+e)/(1+cos(f*x+e))) ^(1/2)*(1/(1+cos(f*x+e)))^(1/2)*EllipticF(I*(csc(f*x+e)-cot(f*x+e)),I)*sec (f*x+e)+132*cos(f*x+e)^3-77*cos(f*x+e)^2*sin(f*x+e)+231*I*(cos(f*x+e)/(1+c os(f*x+e)))^(1/2)*(1/(1+cos(f*x+e)))^(1/2)*EllipticE(I*(csc(f*x+e)-cot(f*x +e)),I)*sec(f*x+e)^2-231*I*(cos(f*x+e)/(1+cos(f*x+e)))^(1/2)*(1/(1+cos(f*x +e)))^(1/2)*EllipticF(I*(csc(f*x+e)-cot(f*x+e)),I)*sec(f*x+e)^2+132*cos(f* x+e)^2+77*cos(f*x+e)*sin(f*x+e)+77*sin(f*x+e)+231*tan(f*x+e))
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 \[ \int (g \cos (e+f x))^{3/2} \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{7/2} \, dx=\frac {-231 i \, \sqrt {2} \sqrt {a c g} c^{3} g {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (f x + e\right ) + i \, \sin \left (f x + e\right )\right )\right ) + 231 i \, \sqrt {2} \sqrt {a c g} c^{3} g {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (f x + e\right ) - i \, \sin \left (f x + e\right )\right )\right ) - 2 \, {\left (21 \, c^{3} g \cos \left (f x + e\right )^{4} - 132 \, c^{3} g \cos \left (f x + e\right )^{2} + 77 \, {\left (c^{3} g \cos \left (f x + e\right )^{2} - c^{3} g\right )} \sin \left (f x + e\right )\right )} \sqrt {g \cos \left (f x + e\right )} \sqrt {a \sin \left (f x + e\right ) + a} \sqrt {-c \sin \left (f x + e\right ) + c}}{231 \, f} \]
integrate((g*cos(f*x+e))^(3/2)*(c-c*sin(f*x+e))^(7/2)*(a+a*sin(f*x+e))^(1/ 2),x, algorithm="fricas")
1/231*(-231*I*sqrt(2)*sqrt(a*c*g)*c^3*g*weierstrassZeta(-4, 0, weierstrass PInverse(-4, 0, cos(f*x + e) + I*sin(f*x + e))) + 231*I*sqrt(2)*sqrt(a*c*g )*c^3*g*weierstrassZeta(-4, 0, weierstrassPInverse(-4, 0, cos(f*x + e) - I *sin(f*x + e))) - 2*(21*c^3*g*cos(f*x + e)^4 - 132*c^3*g*cos(f*x + e)^2 + 77*(c^3*g*cos(f*x + e)^2 - c^3*g)*sin(f*x + e))*sqrt(g*cos(f*x + e))*sqrt( a*sin(f*x + e) + a)*sqrt(-c*sin(f*x + e) + c))/f
Timed out. \[ \int (g \cos (e+f x))^{3/2} \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{7/2} \, dx=\text {Timed out} \]
\[ \int (g \cos (e+f x))^{3/2} \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{7/2} \, dx=\int { \left (g \cos \left (f x + e\right )\right )^{\frac {3}{2}} \sqrt {a \sin \left (f x + e\right ) + a} {\left (-c \sin \left (f x + e\right ) + c\right )}^{\frac {7}{2}} \,d x } \]
integrate((g*cos(f*x+e))^(3/2)*(c-c*sin(f*x+e))^(7/2)*(a+a*sin(f*x+e))^(1/ 2),x, algorithm="maxima")
\[ \int (g \cos (e+f x))^{3/2} \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{7/2} \, dx=\int { \left (g \cos \left (f x + e\right )\right )^{\frac {3}{2}} \sqrt {a \sin \left (f x + e\right ) + a} {\left (-c \sin \left (f x + e\right ) + c\right )}^{\frac {7}{2}} \,d x } \]
integrate((g*cos(f*x+e))^(3/2)*(c-c*sin(f*x+e))^(7/2)*(a+a*sin(f*x+e))^(1/ 2),x, algorithm="giac")
Timed out. \[ \int (g \cos (e+f x))^{3/2} \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{7/2} \, dx=\int {\left (g\,\cos \left (e+f\,x\right )\right )}^{3/2}\,\sqrt {a+a\,\sin \left (e+f\,x\right )}\,{\left (c-c\,\sin \left (e+f\,x\right )\right )}^{7/2} \,d x \]