Integrand size = 42, antiderivative size = 292 \[ \int \frac {(g \cos (e+f x))^{3/2} \sqrt {a+a \sin (e+f x)}}{(c-c \sin (e+f x))^{9/2}} \, dx=\frac {4 a (g \cos (e+f x))^{5/2}}{13 f g \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{9/2}}-\frac {2 a (g \cos (e+f x))^{5/2}}{39 c f g \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{7/2}}-\frac {2 a (g \cos (e+f x))^{5/2}}{65 c^2 f g \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{5/2}}-\frac {2 a (g \cos (e+f x))^{5/2}}{65 c^3 f g \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{3/2}}+\frac {2 a g \sqrt {\cos (e+f x)} \sqrt {g \cos (e+f x)} E\left (\left .\frac {1}{2} (e+f x)\right |2\right )}{65 c^4 f \sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}} \] Output:
4/13*a*(g*cos(f*x+e))^(5/2)/f/g/(a+a*sin(f*x+e))^(1/2)/(c-c*sin(f*x+e))^(9 /2)-2/39*a*(g*cos(f*x+e))^(5/2)/c/f/g/(a+a*sin(f*x+e))^(1/2)/(c-c*sin(f*x+ e))^(7/2)-2/65*a*(g*cos(f*x+e))^(5/2)/c^2/f/g/(a+a*sin(f*x+e))^(1/2)/(c-c* sin(f*x+e))^(5/2)-2/65*a*(g*cos(f*x+e))^(5/2)/c^3/f/g/(a+a*sin(f*x+e))^(1/ 2)/(c-c*sin(f*x+e))^(3/2)+2/65*a*g*cos(f*x+e)^(1/2)*(g*cos(f*x+e))^(1/2)*E llipticE(sin(1/2*f*x+1/2*e),2^(1/2))/c^4/f/(a+a*sin(f*x+e))^(1/2)/(c-c*sin (f*x+e))^(1/2)
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 6.34 (sec) , antiderivative size = 291, normalized size of antiderivative = 1.00 \[ \int \frac {(g \cos (e+f x))^{3/2} \sqrt {a+a \sin (e+f x)}}{(c-c \sin (e+f x))^{9/2}} \, dx=\frac {4 e^{3 i (e+f x)} \left (e^{-i (e+f x)} \left (1+e^{2 i (e+f x)}\right ) g\right )^{3/2} \left (\sqrt {1+e^{2 i (e+f x)}} \left (-1+149 i e^{i (e+f x)}+44 e^{2 i (e+f x)}-64 i e^{3 i (e+f x)}+21 e^{4 i (e+f x)}+3 i e^{5 i (e+f x)}\right )-i \left (-i+e^{i (e+f x)}\right )^7 \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {3}{4},\frac {7}{4},-e^{2 i (e+f x)}\right )\right ) \sqrt {a (1+\sin (e+f x))}}{195 c^4 \left (1-i e^{i (e+f x)}\right ) \left (-i+e^{i (e+f x)}\right )^6 \sqrt {i c e^{-i (e+f x)} \left (-i+e^{i (e+f x)}\right )^2} \left (1+e^{2 i (e+f x)}\right )^{3/2} f} \] Input:
Integrate[((g*Cos[e + f*x])^(3/2)*Sqrt[a + a*Sin[e + f*x]])/(c - c*Sin[e + f*x])^(9/2),x]
Output:
(4*E^((3*I)*(e + f*x))*(((1 + E^((2*I)*(e + f*x)))*g)/E^(I*(e + f*x)))^(3/ 2)*(Sqrt[1 + E^((2*I)*(e + f*x))]*(-1 + (149*I)*E^(I*(e + f*x)) + 44*E^((2 *I)*(e + f*x)) - (64*I)*E^((3*I)*(e + f*x)) + 21*E^((4*I)*(e + f*x)) + (3* I)*E^((5*I)*(e + f*x))) - I*(-I + E^(I*(e + f*x)))^7*Hypergeometric2F1[1/2 , 3/4, 7/4, -E^((2*I)*(e + f*x))])*Sqrt[a*(1 + Sin[e + f*x])])/(195*c^4*(1 - I*E^(I*(e + f*x)))*(-I + E^(I*(e + f*x)))^6*Sqrt[(I*c*(-I + E^(I*(e + f *x)))^2)/E^(I*(e + f*x))]*(1 + E^((2*I)*(e + f*x)))^(3/2)*f)
Time = 2.34 (sec) , antiderivative size = 300, normalized size of antiderivative = 1.03, number of steps used = 14, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {3042, 3329, 3042, 3331, 3042, 3331, 3042, 3331, 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.
| \(\displaystyle \int \frac {\sqrt {a \sin (e+f x)+a} (g \cos (e+f x))^{3/2}}{(c-c \sin (e+f x))^{9/2}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\sqrt {a \sin (e+f x)+a} (g \cos (e+f x))^{3/2}}{(c-c \sin (e+f x))^{9/2}}dx\) |
| \(\Big \downarrow \) 3329 |
| \(\displaystyle \frac {4 a (g \cos (e+f x))^{5/2}}{13 f g \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{9/2}}-\frac {3 a \int \frac {(g \cos (e+f x))^{3/2}}{\sqrt {\sin (e+f x) a+a} (c-c \sin (e+f x))^{7/2}}dx}{13 c}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {4 a (g \cos (e+f x))^{5/2}}{13 f g \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{9/2}}-\frac {3 a \int \frac {(g \cos (e+f x))^{3/2}}{\sqrt {\sin (e+f x) a+a} (c-c \sin (e+f x))^{7/2}}dx}{13 c}\) |
| \(\Big \downarrow \) 3331 |
| \(\displaystyle \frac {4 a (g \cos (e+f x))^{5/2}}{13 f g \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{9/2}}-\frac {3 a \left (\frac {\int \frac {(g \cos (e+f x))^{3/2}}{\sqrt {\sin (e+f x) a+a} (c-c \sin (e+f x))^{5/2}}dx}{3 c}+\frac {2 (g \cos (e+f x))^{5/2}}{9 f g \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{7/2}}\right )}{13 c}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {4 a (g \cos (e+f x))^{5/2}}{13 f g \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{9/2}}-\frac {3 a \left (\frac {\int \frac {(g \cos (e+f x))^{3/2}}{\sqrt {\sin (e+f x) a+a} (c-c \sin (e+f x))^{5/2}}dx}{3 c}+\frac {2 (g \cos (e+f x))^{5/2}}{9 f g \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{7/2}}\right )}{13 c}\) |
| \(\Big \downarrow \) 3331 |
| \(\displaystyle \frac {4 a (g \cos (e+f x))^{5/2}}{13 f g \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{9/2}}-\frac {3 a \left (\frac {\frac {\int \frac {(g \cos (e+f x))^{3/2}}{\sqrt {\sin (e+f x) a+a} (c-c \sin (e+f x))^{3/2}}dx}{5 c}+\frac {2 (g \cos (e+f x))^{5/2}}{5 f g \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{5/2}}}{3 c}+\frac {2 (g \cos (e+f x))^{5/2}}{9 f g \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{7/2}}\right )}{13 c}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {4 a (g \cos (e+f x))^{5/2}}{13 f g \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{9/2}}-\frac {3 a \left (\frac {\frac {\int \frac {(g \cos (e+f x))^{3/2}}{\sqrt {\sin (e+f x) a+a} (c-c \sin (e+f x))^{3/2}}dx}{5 c}+\frac {2 (g \cos (e+f x))^{5/2}}{5 f g \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{5/2}}}{3 c}+\frac {2 (g \cos (e+f x))^{5/2}}{9 f g \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{7/2}}\right )}{13 c}\) |
| \(\Big \downarrow \) 3331 |
| \(\displaystyle \frac {4 a (g \cos (e+f x))^{5/2}}{13 f g \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{9/2}}-\frac {3 a \left (\frac {\frac {\frac {2 (g \cos (e+f x))^{5/2}}{f g \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{3/2}}-\frac {\int \frac {(g \cos (e+f x))^{3/2}}{\sqrt {\sin (e+f x) a+a} \sqrt {c-c \sin (e+f x)}}dx}{c}}{5 c}+\frac {2 (g \cos (e+f x))^{5/2}}{5 f g \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{5/2}}}{3 c}+\frac {2 (g \cos (e+f x))^{5/2}}{9 f g \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{7/2}}\right )}{13 c}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {4 a (g \cos (e+f x))^{5/2}}{13 f g \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{9/2}}-\frac {3 a \left (\frac {\frac {\frac {2 (g \cos (e+f x))^{5/2}}{f g \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{3/2}}-\frac {\int \frac {(g \cos (e+f x))^{3/2}}{\sqrt {\sin (e+f x) a+a} \sqrt {c-c \sin (e+f x)}}dx}{c}}{5 c}+\frac {2 (g \cos (e+f x))^{5/2}}{5 f g \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{5/2}}}{3 c}+\frac {2 (g \cos (e+f x))^{5/2}}{9 f g \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{7/2}}\right )}{13 c}\) |
| \(\Big \downarrow \) 3321 |
| \(\displaystyle \frac {4 a (g \cos (e+f x))^{5/2}}{13 f g \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{9/2}}-\frac {3 a \left (\frac {\frac {\frac {2 (g \cos (e+f x))^{5/2}}{f g \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{3/2}}-\frac {g \cos (e+f x) \int \sqrt {g \cos (e+f x)}dx}{c \sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}}{5 c}+\frac {2 (g \cos (e+f x))^{5/2}}{5 f g \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{5/2}}}{3 c}+\frac {2 (g \cos (e+f x))^{5/2}}{9 f g \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{7/2}}\right )}{13 c}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {4 a (g \cos (e+f x))^{5/2}}{13 f g \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{9/2}}-\frac {3 a \left (\frac {\frac {\frac {2 (g \cos (e+f x))^{5/2}}{f g \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{3/2}}-\frac {g \cos (e+f x) \int \sqrt {g \sin \left (e+f x+\frac {\pi }{2}\right )}dx}{c \sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}}{5 c}+\frac {2 (g \cos (e+f x))^{5/2}}{5 f g \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{5/2}}}{3 c}+\frac {2 (g \cos (e+f x))^{5/2}}{9 f g \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{7/2}}\right )}{13 c}\) |
| \(\Big \downarrow \) 3121 |
| \(\displaystyle \frac {4 a (g \cos (e+f x))^{5/2}}{13 f g \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{9/2}}-\frac {3 a \left (\frac {\frac {\frac {2 (g \cos (e+f x))^{5/2}}{f g \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{3/2}}-\frac {g \sqrt {\cos (e+f x)} \sqrt {g \cos (e+f x)} \int \sqrt {\cos (e+f x)}dx}{c \sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}}{5 c}+\frac {2 (g \cos (e+f x))^{5/2}}{5 f g \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{5/2}}}{3 c}+\frac {2 (g \cos (e+f x))^{5/2}}{9 f g \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{7/2}}\right )}{13 c}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {4 a (g \cos (e+f x))^{5/2}}{13 f g \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{9/2}}-\frac {3 a \left (\frac {\frac {\frac {2 (g \cos (e+f x))^{5/2}}{f g \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{3/2}}-\frac {g \sqrt {\cos (e+f x)} \sqrt {g \cos (e+f x)} \int \sqrt {\sin \left (e+f x+\frac {\pi }{2}\right )}dx}{c \sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}}{5 c}+\frac {2 (g \cos (e+f x))^{5/2}}{5 f g \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{5/2}}}{3 c}+\frac {2 (g \cos (e+f x))^{5/2}}{9 f g \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{7/2}}\right )}{13 c}\) |
| \(\Big \downarrow \) 3119 |
| \(\displaystyle \frac {4 a (g \cos (e+f x))^{5/2}}{13 f g \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{9/2}}-\frac {3 a \left (\frac {2 (g \cos (e+f x))^{5/2}}{9 f g \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{7/2}}+\frac {\frac {2 (g \cos (e+f x))^{5/2}}{5 f g \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{5/2}}+\frac {\frac {2 (g \cos (e+f x))^{5/2}}{f g \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{3/2}}-\frac {2 g \sqrt {\cos (e+f x)} E\left (\left .\frac {1}{2} (e+f x)\right |2\right ) \sqrt {g \cos (e+f x)}}{c f \sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}}{5 c}}{3 c}\right )}{13 c}\) |
Input:
Int[((g*Cos[e + f*x])^(3/2)*Sqrt[a + a*Sin[e + f*x]])/(c - c*Sin[e + f*x]) ^(9/2),x]
Output:
(4*a*(g*Cos[e + f*x])^(5/2))/(13*f*g*Sqrt[a + a*Sin[e + f*x]]*(c - c*Sin[e + f*x])^(9/2)) - (3*a*((2*(g*Cos[e + f*x])^(5/2))/(9*f*g*Sqrt[a + a*Sin[e + f*x]]*(c - c*Sin[e + f*x])^(7/2)) + ((2*(g*Cos[e + f*x])^(5/2))/(5*f*g* Sqrt[a + a*Sin[e + f*x]]*(c - c*Sin[e + f*x])^(5/2)) + ((2*(g*Cos[e + f*x] )^(5/2))/(f*g*Sqrt[a + a*Sin[e + f*x]]*(c - c*Sin[e + f*x])^(3/2)) - (2*g* Sqrt[Cos[e + f*x]]*Sqrt[g*Cos[e + f*x]]*EllipticE[(e + f*x)/2, 2])/(c*f*Sq rt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]]))/(5*c))/(3*c)))/(13*c)
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[-2 *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*(2*n + p + 1))), x] - Simp[b*((2*m + p - 1)/(d*(2*n + p + 1))) Int[(g*Cos[e + f*x])^p*(a + b*Sin[e + f*x])^(m - 1)*(c + d*Sin[e + f*x])^( n + 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] && GtQ[m, 0] && LtQ[n, -1] && NeQ[2*n + p + 1, 0] && In tegersQ[2*m, 2*n, 2*p]
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*((c + d*Sin[e + f*x])^n/(a* f*g*(2*m + p + 1))), x] + Simp[(m + n + p + 1)/(a*(2*m + p + 1)) Int[(g*C os[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] && LtQ[m, -1] && NeQ[2*m + p + 1, 0] && !LtQ[m, n, -1] && Integers Q[2*m, 2*n, 2*p]
Leaf count of result is larger than twice the leaf count of optimal. \(2025\) vs. \(2(252)=504\).
Time = 49.98 (sec) , antiderivative size = 2026, normalized size of antiderivative = 6.94
Input:
int((g*cos(f*x+e))^(3/2)*(a+a*sin(f*x+e))^(1/2)/(c-c*sin(f*x+e))^(9/2),x,m ethod=_RETURNVERBOSE)
Output:
g/c^4*(2/1155/f/(1+2^(1/2))*(5*(cos(1/2*f*x+1/2*e)+1)*((8*cos(1/2*f*x+1/2* e)^6+4*cos(1/2*f*x+1/2*e)^4-18*cos(1/2*f*x+1/2*e)^2-1)*sin(1/2*f*x+1/2*e)+ cos(1/2*f*x+1/2*e)*(8*cos(1/2*f*x+1/2*e)^6-28*cos(1/2*f*x+1/2*e)^4+14*cos( 1/2*f*x+1/2*e)^2+7))*2^(1/2)*((2^(1/2)*cos(1/2*f*x+1/2*e)-2^(1/2)+2*cos(1/ 2*f*x+1/2*e)-1)/(cos(1/2*f*x+1/2*e)+1))^(1/2)*(-2*(2^(1/2)*cos(1/2*f*x+1/2 *e)-2^(1/2)-2*cos(1/2*f*x+1/2*e)+1)/(cos(1/2*f*x+1/2*e)+1))^(1/2)*Elliptic F((1+2^(1/2))*(csc(1/2*f*x+1/2*e)-cot(1/2*f*x+1/2*e)),-2*2^(1/2)+3)+((40*c os(1/2*f*x+1/2*e)^6-120*cos(1/2*f*x+1/2*e)^4+16*cos(1/2*f*x+1/2*e)^2-137)* sin(1/2*f*x+1/2*e)-40*cos(1/2*f*x+1/2*e)^7+104*cos(1/2*f*x+1/2*e)^3-201*co s(1/2*f*x+1/2*e))*2^(1/2)+(40*cos(1/2*f*x+1/2*e)^6-120*cos(1/2*f*x+1/2*e)^ 4+16*cos(1/2*f*x+1/2*e)^2-137)*sin(1/2*f*x+1/2*e)-40*cos(1/2*f*x+1/2*e)^7+ 104*cos(1/2*f*x+1/2*e)^3-201*cos(1/2*f*x+1/2*e))*(g*(-1+2*cos(1/2*f*x+1/2* e)^2))^(1/2)*((2*cos(1/2*f*x+1/2*e)*sin(1/2*f*x+1/2*e)+1)*a)^(1/2)/(-(2*co s(1/2*f*x+1/2*e)*sin(1/2*f*x+1/2*e)-1)*c)^(1/2)/(16*cos(1/2*f*x+1/2*e)^8*s in(1/2*f*x+1/2*e)-16*cos(1/2*f*x+1/2*e)^9-48*cos(1/2*f*x+1/2*e)^6*sin(1/2* f*x+1/2*e)+16*cos(1/2*f*x+1/2*e)^7+32*cos(1/2*f*x+1/2*e)^4*sin(1/2*f*x+1/2 *e)+16*cos(1/2*f*x+1/2*e)^5-4*cos(1/2*f*x+1/2*e)^2*sin(1/2*f*x+1/2*e)-20*c os(1/2*f*x+1/2*e)^3-sin(1/2*f*x+1/2*e)+5*cos(1/2*f*x+1/2*e))-1/15015/f/(1+ 2^(1/2))/(2*2^(1/2)-3)*(-462+231*(cos(1/2*f*x+1/2*e)^2+2*cos(1/2*f*x+1/2*e )+1)*((8*cos(1/2*f*x+1/2*e)^6+4*cos(1/2*f*x+1/2*e)^4-18*cos(1/2*f*x+1/2...
Result contains complex when optimal does not.
Time = 0.11 (sec) , antiderivative size = 304, normalized size of antiderivative = 1.04 \[ \int \frac {(g \cos (e+f x))^{3/2} \sqrt {a+a \sin (e+f x)}}{(c-c \sin (e+f x))^{9/2}} \, dx=-\frac {2 \, {\left (3 \, \sqrt {\frac {1}{2}} {\left (i \, g \cos \left (f x + e\right )^{4} - 8 i \, g \cos \left (f x + e\right )^{2} + 4 \, {\left (i \, g \cos \left (f x + e\right )^{2} - 2 i \, g\right )} \sin \left (f x + e\right ) + 8 i \, g\right )} \sqrt {a c g} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (f x + e\right ) + i \, \sin \left (f x + e\right )\right )\right ) + 3 \, \sqrt {\frac {1}{2}} {\left (-i \, g \cos \left (f x + e\right )^{4} + 8 i \, g \cos \left (f x + e\right )^{2} + 4 \, {\left (-i \, g \cos \left (f x + e\right )^{2} + 2 i \, g\right )} \sin \left (f x + e\right ) - 8 i \, g\right )} \sqrt {a c g} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (f x + e\right ) - i \, \sin \left (f x + e\right )\right )\right ) - {\left (12 \, g \cos \left (f x + e\right )^{2} - {\left (3 \, g \cos \left (f x + e\right )^{2} - 23 \, g\right )} \sin \left (f x + e\right ) + 7 \, g\right )} \sqrt {g \cos \left (f x + e\right )} \sqrt {a \sin \left (f x + e\right ) + a} \sqrt {-c \sin \left (f x + e\right ) + c}\right )}}{195 \, {\left (c^{5} f \cos \left (f x + e\right )^{4} - 8 \, c^{5} f \cos \left (f x + e\right )^{2} + 8 \, c^{5} f + 4 \, {\left (c^{5} f \cos \left (f x + e\right )^{2} - 2 \, c^{5} f\right )} \sin \left (f x + e\right )\right )}} \] Input:
integrate((g*cos(f*x+e))^(3/2)*(a+a*sin(f*x+e))^(1/2)/(c-c*sin(f*x+e))^(9/ 2),x, algorithm="fricas")
Output:
-2/195*(3*sqrt(1/2)*(I*g*cos(f*x + e)^4 - 8*I*g*cos(f*x + e)^2 + 4*(I*g*co s(f*x + e)^2 - 2*I*g)*sin(f*x + e) + 8*I*g)*sqrt(a*c*g)*weierstrassZeta(-4 , 0, weierstrassPInverse(-4, 0, cos(f*x + e) + I*sin(f*x + e))) + 3*sqrt(1 /2)*(-I*g*cos(f*x + e)^4 + 8*I*g*cos(f*x + e)^2 + 4*(-I*g*cos(f*x + e)^2 + 2*I*g)*sin(f*x + e) - 8*I*g)*sqrt(a*c*g)*weierstrassZeta(-4, 0, weierstra ssPInverse(-4, 0, cos(f*x + e) - I*sin(f*x + e))) - (12*g*cos(f*x + e)^2 - (3*g*cos(f*x + e)^2 - 23*g)*sin(f*x + e) + 7*g)*sqrt(g*cos(f*x + e))*sqrt (a*sin(f*x + e) + a)*sqrt(-c*sin(f*x + e) + c))/(c^5*f*cos(f*x + e)^4 - 8* c^5*f*cos(f*x + e)^2 + 8*c^5*f + 4*(c^5*f*cos(f*x + e)^2 - 2*c^5*f)*sin(f* x + e))
Timed out. \[ \int \frac {(g \cos (e+f x))^{3/2} \sqrt {a+a \sin (e+f x)}}{(c-c \sin (e+f x))^{9/2}} \, dx=\text {Timed out} \] Input:
integrate((g*cos(f*x+e))**(3/2)*(a+a*sin(f*x+e))**(1/2)/(c-c*sin(f*x+e))** (9/2),x)
Output:
Timed out
\[ \int \frac {(g \cos (e+f x))^{3/2} \sqrt {a+a \sin (e+f x)}}{(c-c \sin (e+f x))^{9/2}} \, dx=\int { \frac {\left (g \cos \left (f x + e\right )\right )^{\frac {3}{2}} \sqrt {a \sin \left (f x + e\right ) + a}}{{\left (-c \sin \left (f x + e\right ) + c\right )}^{\frac {9}{2}}} \,d x } \] Input:
integrate((g*cos(f*x+e))^(3/2)*(a+a*sin(f*x+e))^(1/2)/(c-c*sin(f*x+e))^(9/ 2),x, algorithm="maxima")
Output:
integrate((g*cos(f*x + e))^(3/2)*sqrt(a*sin(f*x + e) + a)/(-c*sin(f*x + e) + c)^(9/2), x)
Timed out. \[ \int \frac {(g \cos (e+f x))^{3/2} \sqrt {a+a \sin (e+f x)}}{(c-c \sin (e+f x))^{9/2}} \, dx=\text {Timed out} \] Input:
integrate((g*cos(f*x+e))^(3/2)*(a+a*sin(f*x+e))^(1/2)/(c-c*sin(f*x+e))^(9/ 2),x, algorithm="giac")
Output:
Timed out
Timed out. \[ \int \frac {(g \cos (e+f x))^{3/2} \sqrt {a+a \sin (e+f x)}}{(c-c \sin (e+f x))^{9/2}} \, dx=\int \frac {{\left (g\,\cos \left (e+f\,x\right )\right )}^{3/2}\,\sqrt {a+a\,\sin \left (e+f\,x\right )}}{{\left (c-c\,\sin \left (e+f\,x\right )\right )}^{9/2}} \,d x \] Input:
int(((g*cos(e + f*x))^(3/2)*(a + a*sin(e + f*x))^(1/2))/(c - c*sin(e + f*x ))^(9/2),x)
Output:
int(((g*cos(e + f*x))^(3/2)*(a + a*sin(e + f*x))^(1/2))/(c - c*sin(e + f*x ))^(9/2), x)
\[ \int \frac {(g \cos (e+f x))^{3/2} \sqrt {a+a \sin (e+f x)}}{(c-c \sin (e+f x))^{9/2}} \, dx=\text {too large to display} \] Input:
int((g*cos(f*x+e))^(3/2)*(a+a*sin(f*x+e))^(1/2)/(c-c*sin(f*x+e))^(9/2),x)
Output:
(sqrt(g)*sqrt(c)*sqrt(a)*g*(2*sqrt(sin(e + f*x) + 1)*sqrt( - sin(e + f*x) + 1)*sqrt(cos(e + f*x))*sin(e + f*x) + 18*sqrt(sin(e + f*x) + 1)*sqrt( - s in(e + f*x) + 1)*sqrt(cos(e + f*x)) + 4*int((sqrt(sin(e + f*x) + 1)*sqrt( - sin(e + f*x) + 1)*sqrt(cos(e + f*x))*cos(e + f*x))/(sin(e + f*x)**4 - 2* sin(e + f*x)**3 + 2*sin(e + f*x) - 1),x)*sin(e + f*x)**4*f - 16*int((sqrt( sin(e + f*x) + 1)*sqrt( - sin(e + f*x) + 1)*sqrt(cos(e + f*x))*cos(e + f*x ))/(sin(e + f*x)**4 - 2*sin(e + f*x)**3 + 2*sin(e + f*x) - 1),x)*sin(e + f *x)**3*f + 24*int((sqrt(sin(e + f*x) + 1)*sqrt( - sin(e + f*x) + 1)*sqrt(c os(e + f*x))*cos(e + f*x))/(sin(e + f*x)**4 - 2*sin(e + f*x)**3 + 2*sin(e + f*x) - 1),x)*sin(e + f*x)**2*f - 16*int((sqrt(sin(e + f*x) + 1)*sqrt( - sin(e + f*x) + 1)*sqrt(cos(e + f*x))*cos(e + f*x))/(sin(e + f*x)**4 - 2*si n(e + f*x)**3 + 2*sin(e + f*x) - 1),x)*sin(e + f*x)*f + 4*int((sqrt(sin(e + f*x) + 1)*sqrt( - sin(e + f*x) + 1)*sqrt(cos(e + f*x))*cos(e + f*x))/(si n(e + f*x)**4 - 2*sin(e + f*x)**3 + 2*sin(e + f*x) - 1),x)*f + 10*int((sqr t(sin(e + f*x) + 1)*sqrt( - sin(e + f*x) + 1)*sqrt(cos(e + f*x))*sin(e + f *x)**2)/(cos(e + f*x)*sin(e + f*x)**5 - 3*cos(e + f*x)*sin(e + f*x)**4 + 2 *cos(e + f*x)*sin(e + f*x)**3 + 2*cos(e + f*x)*sin(e + f*x)**2 - 3*cos(e + f*x)*sin(e + f*x) + cos(e + f*x)),x)*sin(e + f*x)**4*f - 40*int((sqrt(sin (e + f*x) + 1)*sqrt( - sin(e + f*x) + 1)*sqrt(cos(e + f*x))*sin(e + f*x)** 2)/(cos(e + f*x)*sin(e + f*x)**5 - 3*cos(e + f*x)*sin(e + f*x)**4 + 2*c...