Integrand size = 42, antiderivative size = 290 \[ \int (g \cos (e+f x))^{3/2} \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{5/2} \, dx=\frac {22 a c^3 (g \cos (e+f x))^{5/2}}{45 f g \sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}}+\frac {22 a c^3 g \sqrt {\cos (e+f x)} \sqrt {g \cos (e+f x)} E\left (\left .\frac {1}{2} (e+f x)\right |2\right )}{15 f \sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}}+\frac {22 a c^2 (g \cos (e+f x))^{5/2} \sqrt {c-c \sin (e+f x)}}{105 f g \sqrt {a+a \sin (e+f x)}}+\frac {2 a c (g \cos (e+f x))^{5/2} (c-c \sin (e+f x))^{3/2}}{21 f g \sqrt {a+a \sin (e+f x)}}-\frac {2 a (g \cos (e+f x))^{5/2} (c-c \sin (e+f x))^{5/2}}{9 f g \sqrt {a+a \sin (e+f x)}} \] Output:
22/45*a*c^3*(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)+22/15*a*c^3*g*cos(f*x+e)^(1/2)*(g*cos(f*x+e))^(1/2)*EllipticE(sin (1/2*f*x+1/2*e),2^(1/2))/f/(a+a*sin(f*x+e))^(1/2)/(c-c*sin(f*x+e))^(1/2)+2 2/105*a*c^2*(g*cos(f*x+e))^(5/2)*(c-c*sin(f*x+e))^(1/2)/f/g/(a+a*sin(f*x+e ))^(1/2)+2/21*a*c*(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)-2/9*a*(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)
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 3.61 (sec) , antiderivative size = 281, normalized size of antiderivative = 0.97 \[ \int (g \cos (e+f x))^{3/2} \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{5/2} \, dx=-\frac {c^3 e^{-4 i (e+f x)} \left (-i+e^{i (e+f x)}\right ) g \sqrt {g \cos (e+f x)} \left (\sqrt {1+e^{2 i (e+f x)}} \left (-35-180 i e^{i (e+f x)}+238 e^{2 i (e+f x)}-540 i e^{3 i (e+f x)}+3696 e^{4 i (e+f x)}-540 i e^{5 i (e+f x)}-238 e^{6 i (e+f x)}-180 i e^{7 i (e+f x)}+35 e^{8 i (e+f x)}\right )-2464 e^{6 i (e+f x)} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {3}{4},\frac {7}{4},-e^{2 i (e+f x)}\right )\right ) \sqrt {a (1+\sin (e+f x))}}{2520 \left (i+e^{i (e+f x)}\right ) \sqrt {1+e^{2 i (e+f x)}} f \sqrt {c-c \sin (e+f x)}} \] Input:
Integrate[(g*Cos[e + f*x])^(3/2)*Sqrt[a + a*Sin[e + f*x]]*(c - c*Sin[e + f *x])^(5/2),x]
Output:
-1/2520*(c^3*(-I + E^(I*(e + f*x)))*g*Sqrt[g*Cos[e + f*x]]*(Sqrt[1 + E^((2 *I)*(e + f*x))]*(-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))) - 2464*E^((6*I)*(e + f*x))*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(e + f*x))])*Sqrt[a*(1 + Sin[e + f*x])])/(E^((4*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 = 2.40 (sec) , antiderivative size = 296, normalized size of antiderivative = 1.02, number of steps used = 14, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {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.
| \(\displaystyle \int \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{5/2} (g \cos (e+f x))^{3/2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{5/2} (g \cos (e+f x))^{3/2}dx\) |
| \(\Big \downarrow \) 3330 |
| \(\displaystyle \frac {1}{3} a \int \frac {(g \cos (e+f x))^{3/2} (c-c \sin (e+f x))^{5/2}}{\sqrt {\sin (e+f x) a+a}}dx-\frac {2 a (c-c \sin (e+f x))^{5/2} (g \cos (e+f x))^{5/2}}{9 f g \sqrt {a \sin (e+f x)+a}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{3} a \int \frac {(g \cos (e+f x))^{3/2} (c-c \sin (e+f x))^{5/2}}{\sqrt {\sin (e+f x) a+a}}dx-\frac {2 a (c-c \sin (e+f x))^{5/2} (g \cos (e+f x))^{5/2}}{9 f g \sqrt {a \sin (e+f x)+a}}\) |
| \(\Big \downarrow \) 3330 |
| \(\displaystyle \frac {1}{3} a \left (\frac {11}{7} c \int \frac {(g \cos (e+f x))^{3/2} (c-c \sin (e+f x))^{3/2}}{\sqrt {\sin (e+f x) a+a}}dx+\frac {2 c (c-c \sin (e+f x))^{3/2} (g \cos (e+f x))^{5/2}}{7 f g \sqrt {a \sin (e+f x)+a}}\right )-\frac {2 a (c-c \sin (e+f x))^{5/2} (g \cos (e+f x))^{5/2}}{9 f g \sqrt {a \sin (e+f x)+a}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{3} a \left (\frac {11}{7} c \int \frac {(g \cos (e+f x))^{3/2} (c-c \sin (e+f x))^{3/2}}{\sqrt {\sin (e+f x) a+a}}dx+\frac {2 c (c-c \sin (e+f x))^{3/2} (g \cos (e+f x))^{5/2}}{7 f g \sqrt {a \sin (e+f x)+a}}\right )-\frac {2 a (c-c \sin (e+f x))^{5/2} (g \cos (e+f x))^{5/2}}{9 f g \sqrt {a \sin (e+f x)+a}}\) |
| \(\Big \downarrow \) 3330 |
| \(\displaystyle \frac {1}{3} a \left (\frac {11}{7} c \left (\frac {7}{5} c \int \frac {(g \cos (e+f x))^{3/2} \sqrt {c-c \sin (e+f x)}}{\sqrt {\sin (e+f x) a+a}}dx+\frac {2 c \sqrt {c-c \sin (e+f x)} (g \cos (e+f x))^{5/2}}{5 f g \sqrt {a \sin (e+f x)+a}}\right )+\frac {2 c (c-c \sin (e+f x))^{3/2} (g \cos (e+f x))^{5/2}}{7 f g \sqrt {a \sin (e+f x)+a}}\right )-\frac {2 a (c-c \sin (e+f x))^{5/2} (g \cos (e+f x))^{5/2}}{9 f g \sqrt {a \sin (e+f x)+a}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{3} a \left (\frac {11}{7} c \left (\frac {7}{5} c \int \frac {(g \cos (e+f x))^{3/2} \sqrt {c-c \sin (e+f x)}}{\sqrt {\sin (e+f x) a+a}}dx+\frac {2 c \sqrt {c-c \sin (e+f x)} (g \cos (e+f x))^{5/2}}{5 f g \sqrt {a \sin (e+f x)+a}}\right )+\frac {2 c (c-c \sin (e+f x))^{3/2} (g \cos (e+f x))^{5/2}}{7 f g \sqrt {a \sin (e+f x)+a}}\right )-\frac {2 a (c-c \sin (e+f x))^{5/2} (g \cos (e+f x))^{5/2}}{9 f g \sqrt {a \sin (e+f x)+a}}\) |
| \(\Big \downarrow \) 3330 |
| \(\displaystyle \frac {1}{3} a \left (\frac {11}{7} c \left (\frac {7}{5} c \left (c \int \frac {(g \cos (e+f x))^{3/2}}{\sqrt {\sin (e+f x) a+a} \sqrt {c-c \sin (e+f x)}}dx+\frac {2 c (g \cos (e+f x))^{5/2}}{3 f g \sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}\right )+\frac {2 c \sqrt {c-c \sin (e+f x)} (g \cos (e+f x))^{5/2}}{5 f g \sqrt {a \sin (e+f x)+a}}\right )+\frac {2 c (c-c \sin (e+f x))^{3/2} (g \cos (e+f x))^{5/2}}{7 f g \sqrt {a \sin (e+f x)+a}}\right )-\frac {2 a (c-c \sin (e+f x))^{5/2} (g \cos (e+f x))^{5/2}}{9 f g \sqrt {a \sin (e+f x)+a}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{3} a \left (\frac {11}{7} c \left (\frac {7}{5} c \left (c \int \frac {(g \cos (e+f x))^{3/2}}{\sqrt {\sin (e+f x) a+a} \sqrt {c-c \sin (e+f x)}}dx+\frac {2 c (g \cos (e+f x))^{5/2}}{3 f g \sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}\right )+\frac {2 c \sqrt {c-c \sin (e+f x)} (g \cos (e+f x))^{5/2}}{5 f g \sqrt {a \sin (e+f x)+a}}\right )+\frac {2 c (c-c \sin (e+f x))^{3/2} (g \cos (e+f x))^{5/2}}{7 f g \sqrt {a \sin (e+f x)+a}}\right )-\frac {2 a (c-c \sin (e+f x))^{5/2} (g \cos (e+f x))^{5/2}}{9 f g \sqrt {a \sin (e+f x)+a}}\) |
| \(\Big \downarrow \) 3321 |
| \(\displaystyle \frac {1}{3} a \left (\frac {11}{7} c \left (\frac {7}{5} c \left (\frac {c g \cos (e+f x) \int \sqrt {g \cos (e+f x)}dx}{\sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}+\frac {2 c (g \cos (e+f x))^{5/2}}{3 f g \sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}\right )+\frac {2 c \sqrt {c-c \sin (e+f x)} (g \cos (e+f x))^{5/2}}{5 f g \sqrt {a \sin (e+f x)+a}}\right )+\frac {2 c (c-c \sin (e+f x))^{3/2} (g \cos (e+f x))^{5/2}}{7 f g \sqrt {a \sin (e+f x)+a}}\right )-\frac {2 a (c-c \sin (e+f x))^{5/2} (g \cos (e+f x))^{5/2}}{9 f g \sqrt {a \sin (e+f x)+a}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{3} a \left (\frac {11}{7} c \left (\frac {7}{5} c \left (\frac {c g \cos (e+f x) \int \sqrt {g \sin \left (e+f x+\frac {\pi }{2}\right )}dx}{\sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}+\frac {2 c (g \cos (e+f x))^{5/2}}{3 f g \sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}\right )+\frac {2 c \sqrt {c-c \sin (e+f x)} (g \cos (e+f x))^{5/2}}{5 f g \sqrt {a \sin (e+f x)+a}}\right )+\frac {2 c (c-c \sin (e+f x))^{3/2} (g \cos (e+f x))^{5/2}}{7 f g \sqrt {a \sin (e+f x)+a}}\right )-\frac {2 a (c-c \sin (e+f x))^{5/2} (g \cos (e+f x))^{5/2}}{9 f g \sqrt {a \sin (e+f x)+a}}\) |
| \(\Big \downarrow \) 3121 |
| \(\displaystyle \frac {1}{3} a \left (\frac {11}{7} c \left (\frac {7}{5} c \left (\frac {c g \sqrt {\cos (e+f x)} \sqrt {g \cos (e+f x)} \int \sqrt {\cos (e+f x)}dx}{\sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}+\frac {2 c (g \cos (e+f x))^{5/2}}{3 f g \sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}\right )+\frac {2 c \sqrt {c-c \sin (e+f x)} (g \cos (e+f x))^{5/2}}{5 f g \sqrt {a \sin (e+f x)+a}}\right )+\frac {2 c (c-c \sin (e+f x))^{3/2} (g \cos (e+f x))^{5/2}}{7 f g \sqrt {a \sin (e+f x)+a}}\right )-\frac {2 a (c-c \sin (e+f x))^{5/2} (g \cos (e+f x))^{5/2}}{9 f g \sqrt {a \sin (e+f x)+a}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{3} a \left (\frac {11}{7} c \left (\frac {7}{5} c \left (\frac {c g \sqrt {\cos (e+f x)} \sqrt {g \cos (e+f x)} \int \sqrt {\sin \left (e+f x+\frac {\pi }{2}\right )}dx}{\sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}+\frac {2 c (g \cos (e+f x))^{5/2}}{3 f g \sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}\right )+\frac {2 c \sqrt {c-c \sin (e+f x)} (g \cos (e+f x))^{5/2}}{5 f g \sqrt {a \sin (e+f x)+a}}\right )+\frac {2 c (c-c \sin (e+f x))^{3/2} (g \cos (e+f x))^{5/2}}{7 f g \sqrt {a \sin (e+f x)+a}}\right )-\frac {2 a (c-c \sin (e+f x))^{5/2} (g \cos (e+f x))^{5/2}}{9 f g \sqrt {a \sin (e+f x)+a}}\) |
| \(\Big \downarrow \) 3119 |
| \(\displaystyle \frac {1}{3} a \left (\frac {2 c (c-c \sin (e+f x))^{3/2} (g \cos (e+f x))^{5/2}}{7 f g \sqrt {a \sin (e+f x)+a}}+\frac {11}{7} c \left (\frac {2 c \sqrt {c-c \sin (e+f x)} (g \cos (e+f x))^{5/2}}{5 f g \sqrt {a \sin (e+f x)+a}}+\frac {7}{5} c \left (\frac {2 c (g \cos (e+f x))^{5/2}}{3 f g \sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}+\frac {2 c g \sqrt {\cos (e+f x)} E\left (\left .\frac {1}{2} (e+f x)\right |2\right ) \sqrt {g \cos (e+f x)}}{f \sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}\right )\right )\right )-\frac {2 a (c-c \sin (e+f x))^{5/2} (g \cos (e+f x))^{5/2}}{9 f g \sqrt {a \sin (e+f x)+a}}\) |
Input:
Int[(g*Cos[e + f*x])^(3/2)*Sqrt[a + a*Sin[e + f*x]]*(c - c*Sin[e + f*x])^( 5/2),x]
Output:
(-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]]) + (a*((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*Sin[e + f*x]])))/5))/7))/3
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]
Leaf count of result is larger than twice the leaf count of optimal. \(976\) vs. \(2(250)=500\).
Time = 20.20 (sec) , antiderivative size = 977, normalized size of antiderivative = 3.37
Input:
int((g*cos(f*x+e))^(3/2)*(a+a*sin(f*x+e))^(1/2)*(c-c*sin(f*x+e))^(5/2),x,m ethod=_RETURNVERBOSE)
Output:
1/315/f*g/(2*2^(1/2)-3)/(1+2^(1/2))*c^2*(2*(-1+2*cos(1/2*f*x+1/2*e)^2)*(7* (80*cos(1/2*f*x+1/2*e)^8+80*cos(1/2*f*x+1/2*e)^7-120*cos(1/2*f*x+1/2*e)^6- 120*cos(1/2*f*x+1/2*e)^5+16*cos(1/2*f*x+1/2*e)^4+16*cos(1/2*f*x+1/2*e)^3+1 2*cos(1/2*f*x+1/2*e)^2+12*cos(1/2*f*x+1/2*e)-33)*sin(1/2*f*x+1/2*e)-720*co s(1/2*f*x+1/2*e)^7-720*cos(1/2*f*x+1/2*e)^6+1080*cos(1/2*f*x+1/2*e)^5+1080 *cos(1/2*f*x+1/2*e)^4-540*cos(1/2*f*x+1/2*e)^3-540*cos(1/2*f*x+1/2*e)^2+90 *cos(1/2*f*x+1/2*e)+90)*2^(1/2)-14*(80*cos(1/2*f*x+1/2*e)^8+80*cos(1/2*f*x +1/2*e)^7-120*cos(1/2*f*x+1/2*e)^6-120*cos(1/2*f*x+1/2*e)^5+16*cos(1/2*f*x +1/2*e)^4+16*cos(1/2*f*x+1/2*e)^3+12*cos(1/2*f*x+1/2*e)^2+12*cos(1/2*f*x+1 /2*e)-33)*(-1+2*cos(1/2*f*x+1/2*e)^2)*sin(1/2*f*x+1/2*e)+180*(8*cos(1/2*f* x+1/2*e)^7+8*cos(1/2*f*x+1/2*e)^6-12*cos(1/2*f*x+1/2*e)^5-12*cos(1/2*f*x+1 /2*e)^4+6*cos(1/2*f*x+1/2*e)^3+6*cos(1/2*f*x+1/2*e)^2-cos(1/2*f*x+1/2*e)-1 )*(-1+2*cos(1/2*f*x+1/2*e)^2)+462*(2+(-cos(1/2*f*x+1/2*e)^2-2*cos(1/2*f*x+ 1/2*e)-1)*2^(1/2)+2*cos(1/2*f*x+1/2*e)^2+4*cos(1/2*f*x+1/2*e))*((2^(1/2)*c os(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)*EllipticF((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)+231*(-cos(1/2*f*x+1/2*e)^2-2*cos(1/2*f*x+1/2*e )-1)*2^(1/2)*EllipticE((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)*(-2*(2^(1/2)*cos(1/2*f*x+1/2*e)-2^(1/2)-2*cos(1/2*f*x+1/...
Result contains complex when optimal does not.
Time = 0.11 (sec) , antiderivative size = 159, normalized size of antiderivative = 0.55 \[ \int (g \cos (e+f x))^{3/2} \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{5/2} \, dx=-\frac {2 \, {\left (231 i \, \sqrt {\frac {1}{2}} \sqrt {a c g} c^{2} g {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (f x + e\right ) + i \, \sin \left (f x + e\right )\right )\right ) - 231 i \, \sqrt {\frac {1}{2}} \sqrt {a c g} c^{2} g {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (f x + e\right ) - i \, \sin \left (f x + e\right )\right )\right ) - {\left (90 \, c^{2} g \cos \left (f x + e\right )^{2} - 7 \, {\left (5 \, c^{2} g \cos \left (f x + e\right )^{2} - 11 \, c^{2} g\right )} \sin \left (f x + e\right )\right )} \sqrt {g \cos \left (f x + e\right )} \sqrt {a \sin \left (f x + e\right ) + a} \sqrt {-c \sin \left (f x + e\right ) + c}\right )}}{315 \, f} \] Input:
integrate((g*cos(f*x+e))^(3/2)*(a+a*sin(f*x+e))^(1/2)*(c-c*sin(f*x+e))^(5/ 2),x, algorithm="fricas")
Output:
-2/315*(231*I*sqrt(1/2)*sqrt(a*c*g)*c^2*g*weierstrassZeta(-4, 0, weierstra ssPInverse(-4, 0, cos(f*x + e) + I*sin(f*x + e))) - 231*I*sqrt(1/2)*sqrt(a *c*g)*c^2*g*weierstrassZeta(-4, 0, weierstrassPInverse(-4, 0, cos(f*x + e) - I*sin(f*x + e))) - (90*c^2*g*cos(f*x + e)^2 - 7*(5*c^2*g*cos(f*x + e)^2 - 11*c^2*g)*sin(f*x + e))*sqrt(g*cos(f*x + e))*sqrt(a*sin(f*x + e) + a)*s qrt(-c*sin(f*x + e) + c))/f
Timed out. \[ \int (g \cos (e+f x))^{3/2} \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{5/2} \, dx=\text {Timed out} \] Input:
integrate((g*cos(f*x+e))**(3/2)*(a+a*sin(f*x+e))**(1/2)*(c-c*sin(f*x+e))** (5/2),x)
Output:
Timed out
\[ \int (g \cos (e+f x))^{3/2} \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{5/2} \, dx=\int { \left (g \cos \left (f x + e\right )\right )^{\frac {3}{2}} \sqrt {a \sin \left (f x + e\right ) + a} {\left (-c \sin \left (f x + e\right ) + c\right )}^{\frac {5}{2}} \,d x } \] Input:
integrate((g*cos(f*x+e))^(3/2)*(a+a*sin(f*x+e))^(1/2)*(c-c*sin(f*x+e))^(5/ 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)^(5/2), x)
\[ \int (g \cos (e+f x))^{3/2} \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{5/2} \, dx=\int { \left (g \cos \left (f x + e\right )\right )^{\frac {3}{2}} \sqrt {a \sin \left (f x + e\right ) + a} {\left (-c \sin \left (f x + e\right ) + c\right )}^{\frac {5}{2}} \,d x } \] Input:
integrate((g*cos(f*x+e))^(3/2)*(a+a*sin(f*x+e))^(1/2)*(c-c*sin(f*x+e))^(5/ 2),x, algorithm="giac")
Output:
integrate((g*cos(f*x + e))^(3/2)*sqrt(a*sin(f*x + e) + a)*(-c*sin(f*x + e) + c)^(5/2), x)
Timed out. \[ \int (g \cos (e+f x))^{3/2} \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{5/2} \, dx=\int {\left (g\,\cos \left (e+f\,x\right )\right )}^{3/2}\,\sqrt {a+a\,\sin \left (e+f\,x\right )}\,{\left (c-c\,\sin \left (e+f\,x\right )\right )}^{5/2} \,d x \] Input:
int((g*cos(e + f*x))^(3/2)*(a + a*sin(e + f*x))^(1/2)*(c - c*sin(e + f*x)) ^(5/2),x)
Output:
int((g*cos(e + f*x))^(3/2)*(a + a*sin(e + f*x))^(1/2)*(c - c*sin(e + f*x)) ^(5/2), x)
\[ \int (g \cos (e+f x))^{3/2} \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{5/2} \, dx=\sqrt {g}\, \sqrt {c}\, \sqrt {a}\, c^{2} g \left (\int \sqrt {\sin \left (f x +e \right )+1}\, \sqrt {-\sin \left (f x +e \right )+1}\, \sqrt {\cos \left (f x +e \right )}\, \cos \left (f x +e \right ) \sin \left (f x +e \right )^{2}d x -2 \left (\int \sqrt {\sin \left (f x +e \right )+1}\, \sqrt {-\sin \left (f x +e \right )+1}\, \sqrt {\cos \left (f x +e \right )}\, \cos \left (f x +e \right ) \sin \left (f x +e \right )d x \right )+\int \sqrt {\sin \left (f x +e \right )+1}\, \sqrt {-\sin \left (f x +e \right )+1}\, \sqrt {\cos \left (f x +e \right )}\, \cos \left (f x +e \right )d x \right ) \] Input:
int((g*cos(f*x+e))^(3/2)*(a+a*sin(f*x+e))^(1/2)*(c-c*sin(f*x+e))^(5/2),x)
Output:
sqrt(g)*sqrt(c)*sqrt(a)*c**2*g*(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)**2,x) - 2*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),x) + int(sqrt(sin(e + f*x) + 1)*sqrt( - sin(e + f*x) + 1)* sqrt(cos(e + f*x))*cos(e + f*x),x))