Integrand size = 27, antiderivative size = 224 \[ \int \frac {\sqrt {c+d \sin (e+f x)}}{(3+3 \sin (e+f x))^2} \, dx=-\frac {c \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{27 (c-d) f (1+\sin (e+f x))}-\frac {\cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 f (3+3 \sin (e+f x))^2}-\frac {c E\left (\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right )|\frac {2 d}{c+d}\right ) \sqrt {c+d \sin (e+f x)}}{27 (c-d) f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}+\frac {(c+d) \operatorname {EllipticF}\left (\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right ),\frac {2 d}{c+d}\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}{27 f \sqrt {c+d \sin (e+f x)}} \]
-1/3*c*cos(f*x+e)*(c+d*sin(f*x+e))^(1/2)/a^2/(c-d)/f/(1+sin(f*x+e))-1/3*co s(f*x+e)*(c+d*sin(f*x+e))^(1/2)/f/(a+a*sin(f*x+e))^2+1/3*c*(sin(1/2*e+1/4* Pi+1/2*f*x)^2)^(1/2)/sin(1/2*e+1/4*Pi+1/2*f*x)*EllipticE(cos(1/2*e+1/4*Pi+ 1/2*f*x),2^(1/2)*(d/(c+d))^(1/2))*(c+d*sin(f*x+e))^(1/2)/a^2/(c-d)/f/((c+d *sin(f*x+e))/(c+d))^(1/2)-1/3*(c+d)*(sin(1/2*e+1/4*Pi+1/2*f*x)^2)^(1/2)/si n(1/2*e+1/4*Pi+1/2*f*x)*EllipticF(cos(1/2*e+1/4*Pi+1/2*f*x),2^(1/2)*(d/(c+ d))^(1/2))*((c+d*sin(f*x+e))/(c+d))^(1/2)/a^2/f/(c+d*sin(f*x+e))^(1/2)
Time = 1.81 (sec) , antiderivative size = 253, normalized size of antiderivative = 1.13 \[ \int \frac {\sqrt {c+d \sin (e+f x)}}{(3+3 \sin (e+f x))^2} \, dx=\frac {\left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^4 \left (-c (c+d \sin (e+f x))+\frac {\left (d \cos \left (\frac {1}{2} (e+f x)\right )-c \cos \left (\frac {3}{2} (e+f x)\right )+(3 c-d) \sin \left (\frac {1}{2} (e+f x)\right )\right ) (c+d \sin (e+f x))}{\left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^3}+c (c+d) E\left (\frac {1}{4} (-2 e+\pi -2 f x)|\frac {2 d}{c+d}\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}-\left (c^2-d^2\right ) \operatorname {EllipticF}\left (\frac {1}{4} (-2 e+\pi -2 f x),\frac {2 d}{c+d}\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}\right )}{27 (c-d) f (1+\sin (e+f x))^2 \sqrt {c+d \sin (e+f x)}} \]
((Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^4*(-(c*(c + d*Sin[e + f*x])) + ((d* Cos[(e + f*x)/2] - c*Cos[(3*(e + f*x))/2] + (3*c - d)*Sin[(e + f*x)/2])*(c + d*Sin[e + f*x]))/(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^3 + c*(c + d)*El lipticE[(-2*e + Pi - 2*f*x)/4, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)] - (c^2 - d^2)*EllipticF[(-2*e + Pi - 2*f*x)/4, (2*d)/(c + d)]*Sqrt[ (c + d*Sin[e + f*x])/(c + d)]))/(27*(c - d)*f*(1 + Sin[e + f*x])^2*Sqrt[c + d*Sin[e + f*x]])
Time = 1.17 (sec) , antiderivative size = 244, normalized size of antiderivative = 1.09, number of steps used = 14, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.519, Rules used = {3042, 3243, 27, 3042, 3457, 3042, 3231, 3042, 3134, 3042, 3132, 3142, 3042, 3140}
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.
| \(\displaystyle \int \frac {\sqrt {c+d \sin (e+f x)}}{(a \sin (e+f x)+a)^2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\sqrt {c+d \sin (e+f x)}}{(a \sin (e+f x)+a)^2}dx\) |
| \(\Big \downarrow \) 3243 |
| \(\displaystyle \frac {\int \frac {a (2 c+d)+a d \sin (e+f x)}{2 (\sin (e+f x) a+a) \sqrt {c+d \sin (e+f x)}}dx}{3 a^2}-\frac {\cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 f (a \sin (e+f x)+a)^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {a (2 c+d)+a d \sin (e+f x)}{(\sin (e+f x) a+a) \sqrt {c+d \sin (e+f x)}}dx}{6 a^2}-\frac {\cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 f (a \sin (e+f x)+a)^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {a (2 c+d)+a d \sin (e+f x)}{(\sin (e+f x) a+a) \sqrt {c+d \sin (e+f x)}}dx}{6 a^2}-\frac {\cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 f (a \sin (e+f x)+a)^2}\) |
| \(\Big \downarrow \) 3457 |
| \(\displaystyle \frac {-\frac {\int \frac {d^2 a^2+c d \sin (e+f x) a^2}{\sqrt {c+d \sin (e+f x)}}dx}{a^2 (c-d)}-\frac {2 c \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{f (c-d) (\sin (e+f x)+1)}}{6 a^2}-\frac {\cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 f (a \sin (e+f x)+a)^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-\frac {\int \frac {d^2 a^2+c d \sin (e+f x) a^2}{\sqrt {c+d \sin (e+f x)}}dx}{a^2 (c-d)}-\frac {2 c \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{f (c-d) (\sin (e+f x)+1)}}{6 a^2}-\frac {\cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 f (a \sin (e+f x)+a)^2}\) |
| \(\Big \downarrow \) 3231 |
| \(\displaystyle \frac {-\frac {a^2 c \int \sqrt {c+d \sin (e+f x)}dx-a^2 \left (c^2-d^2\right ) \int \frac {1}{\sqrt {c+d \sin (e+f x)}}dx}{a^2 (c-d)}-\frac {2 c \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{f (c-d) (\sin (e+f x)+1)}}{6 a^2}-\frac {\cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 f (a \sin (e+f x)+a)^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-\frac {a^2 c \int \sqrt {c+d \sin (e+f x)}dx-a^2 \left (c^2-d^2\right ) \int \frac {1}{\sqrt {c+d \sin (e+f x)}}dx}{a^2 (c-d)}-\frac {2 c \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{f (c-d) (\sin (e+f x)+1)}}{6 a^2}-\frac {\cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 f (a \sin (e+f x)+a)^2}\) |
| \(\Big \downarrow \) 3134 |
| \(\displaystyle \frac {-\frac {\frac {a^2 c \sqrt {c+d \sin (e+f x)} \int \sqrt {\frac {c}{c+d}+\frac {d \sin (e+f x)}{c+d}}dx}{\sqrt {\frac {c+d \sin (e+f x)}{c+d}}}-a^2 \left (c^2-d^2\right ) \int \frac {1}{\sqrt {c+d \sin (e+f x)}}dx}{a^2 (c-d)}-\frac {2 c \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{f (c-d) (\sin (e+f x)+1)}}{6 a^2}-\frac {\cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 f (a \sin (e+f x)+a)^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-\frac {\frac {a^2 c \sqrt {c+d \sin (e+f x)} \int \sqrt {\frac {c}{c+d}+\frac {d \sin (e+f x)}{c+d}}dx}{\sqrt {\frac {c+d \sin (e+f x)}{c+d}}}-a^2 \left (c^2-d^2\right ) \int \frac {1}{\sqrt {c+d \sin (e+f x)}}dx}{a^2 (c-d)}-\frac {2 c \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{f (c-d) (\sin (e+f x)+1)}}{6 a^2}-\frac {\cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 f (a \sin (e+f x)+a)^2}\) |
| \(\Big \downarrow \) 3132 |
| \(\displaystyle \frac {-\frac {\frac {2 a^2 c \sqrt {c+d \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}-a^2 \left (c^2-d^2\right ) \int \frac {1}{\sqrt {c+d \sin (e+f x)}}dx}{a^2 (c-d)}-\frac {2 c \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{f (c-d) (\sin (e+f x)+1)}}{6 a^2}-\frac {\cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 f (a \sin (e+f x)+a)^2}\) |
| \(\Big \downarrow \) 3142 |
| \(\displaystyle \frac {-\frac {\frac {2 a^2 c \sqrt {c+d \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}-\frac {a^2 \left (c^2-d^2\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}} \int \frac {1}{\sqrt {\frac {c}{c+d}+\frac {d \sin (e+f x)}{c+d}}}dx}{\sqrt {c+d \sin (e+f x)}}}{a^2 (c-d)}-\frac {2 c \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{f (c-d) (\sin (e+f x)+1)}}{6 a^2}-\frac {\cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 f (a \sin (e+f x)+a)^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-\frac {\frac {2 a^2 c \sqrt {c+d \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}-\frac {a^2 \left (c^2-d^2\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}} \int \frac {1}{\sqrt {\frac {c}{c+d}+\frac {d \sin (e+f x)}{c+d}}}dx}{\sqrt {c+d \sin (e+f x)}}}{a^2 (c-d)}-\frac {2 c \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{f (c-d) (\sin (e+f x)+1)}}{6 a^2}-\frac {\cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 f (a \sin (e+f x)+a)^2}\) |
| \(\Big \downarrow \) 3140 |
| \(\displaystyle \frac {-\frac {\frac {2 a^2 c \sqrt {c+d \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}-\frac {2 a^2 \left (c^2-d^2\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}} \operatorname {EllipticF}\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right ),\frac {2 d}{c+d}\right )}{f \sqrt {c+d \sin (e+f x)}}}{a^2 (c-d)}-\frac {2 c \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{f (c-d) (\sin (e+f x)+1)}}{6 a^2}-\frac {\cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 f (a \sin (e+f x)+a)^2}\) |
-1/3*(Cos[e + f*x]*Sqrt[c + d*Sin[e + f*x]])/(f*(a + a*Sin[e + f*x])^2) + ((-2*c*Cos[e + f*x]*Sqrt[c + d*Sin[e + f*x]])/((c - d)*f*(1 + Sin[e + f*x] )) - ((2*a^2*c*EllipticE[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[c + d*Sin [e + f*x]])/(f*Sqrt[(c + d*Sin[e + f*x])/(c + d)]) - (2*a^2*(c^2 - d^2)*El lipticF[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)])/(f*Sqrt[c + d*Sin[e + f*x]]))/(a^2*(c - d)))/(6*a^2)
3.6.12.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[2*(Sqrt[a + b]/d)*EllipticE[(1/2)*(c - Pi/2 + d*x), 2*(b/(a + b))], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && GtQ[a + b, 0]
Int[Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[Sqrt[a + b*Sin[c + d*x]]/Sqrt[(a + b*Sin[c + d*x])/(a + b)] Int[Sqrt[a/(a + b) + ( b/(a + b))*Sin[c + d*x]], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2 , 0] && !GtQ[a + b, 0]
Int[1/Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/(d*S qrt[a + b]))*EllipticF[(1/2)*(c - Pi/2 + d*x), 2*(b/(a + b))], x] /; FreeQ[ {a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && GtQ[a + b, 0]
Int[1/Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[Sqrt[(a + b*Sin[c + d*x])/(a + b)]/Sqrt[a + b*Sin[c + d*x]] Int[1/Sqrt[a/(a + b) + (b/(a + b))*Sin[c + d*x]], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && !GtQ[a + b, 0]
Int[((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])/Sqrt[(a_) + (b_.)*sin[(e_.) + ( f_.)*(x_)]], x_Symbol] :> Simp[(b*c - a*d)/b Int[1/Sqrt[a + b*Sin[e + f*x ]], x], x] + Simp[d/b Int[Sqrt[a + b*Sin[e + f*x]], x], x] /; FreeQ[{a, b , c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0]
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[b*Cos[e + f*x]*(a + b*Sin[e + f*x])^m* ((c + d*Sin[e + f*x])^n/(a*f*(2*m + 1))), x] - Simp[1/(a*b*(2*m + 1)) Int [(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^(n - 1)*Simp[a*d*n - b*c *(m + 1) - b*d*(m + n + 1)*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e , f}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && LtQ[m, -1] && LtQ[0, n, 1] && (IntegersQ[2*m, 2*n] || (IntegerQ[m] && EqQ[c , 0]))
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Sim p[b*(A*b - a*B)*Cos[e + f*x]*(a + b*Sin[e + f*x])^m*((c + d*Sin[e + f*x])^( n + 1)/(a*f*(2*m + 1)*(b*c - a*d))), x] + Simp[1/(a*(2*m + 1)*(b*c - a*d)) Int[(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^n*Simp[B*(a*c*m + b *d*(n + 1)) + A*(b*c*(m + 1) - a*d*(2*m + n + 2)) + d*(A*b - a*B)*(m + n + 2)*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, n}, x] && NeQ [b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && LtQ[m, -2^(-1)] && !GtQ[n, 0] && IntegerQ[2*m] && (IntegerQ[2*n] || EqQ[c, 0])
Leaf count of result is larger than twice the leaf count of optimal. \(905\) vs. \(2(279)=558\).
Time = 2.36 (sec) , antiderivative size = 906, normalized size of antiderivative = 4.04
(-(-d*sin(f*x+e)-c)*cos(f*x+e)^2)^(1/2)/a^2*(d*(-(-d*sin(f*x+e)^2-c*sin(f* x+e)+d*sin(f*x+e)+c)/(c-d)/((sin(f*x+e)+1)*(sin(f*x+e)-1)*(-d*sin(f*x+e)-c ))^(1/2)-2*d/(2*c-2*d)*(c/d-1)*((c+d*sin(f*x+e))/(c-d))^(1/2)*(d*(1-sin(f* x+e))/(c+d))^(1/2)*(1/(c-d)*(-sin(f*x+e)-1)*d)^(1/2)/(-(-d*sin(f*x+e)-c)*c os(f*x+e)^2)^(1/2)*EllipticF(((c+d*sin(f*x+e))/(c-d))^(1/2),((c-d)/(c+d))^ (1/2))-d/(c-d)*(c/d-1)*((c+d*sin(f*x+e))/(c-d))^(1/2)*(d*(1-sin(f*x+e))/(c +d))^(1/2)*(1/(c-d)*(-sin(f*x+e)-1)*d)^(1/2)/(-(-d*sin(f*x+e)-c)*cos(f*x+e )^2)^(1/2)*((-c/d-1)*EllipticE(((c+d*sin(f*x+e))/(c-d))^(1/2),((c-d)/(c+d) )^(1/2))+EllipticF(((c+d*sin(f*x+e))/(c-d))^(1/2),((c-d)/(c+d))^(1/2))))+( c-d)*(-1/3/(c-d)*(-(-d*sin(f*x+e)-c)*cos(f*x+e)^2)^(1/2)/(sin(f*x+e)+1)^2- 1/3*(-d*sin(f*x+e)^2-c*sin(f*x+e)+d*sin(f*x+e)+c)/(c-d)^2*(c-3*d)/((sin(f* x+e)+1)*(sin(f*x+e)-1)*(-d*sin(f*x+e)-c))^(1/2)+2*d^2/(3*c^2-6*c*d+3*d^2)* (c/d-1)*((c+d*sin(f*x+e))/(c-d))^(1/2)*(d*(1-sin(f*x+e))/(c+d))^(1/2)*(1/( c-d)*(-sin(f*x+e)-1)*d)^(1/2)/(-(-d*sin(f*x+e)-c)*cos(f*x+e)^2)^(1/2)*Elli pticF(((c+d*sin(f*x+e))/(c-d))^(1/2),((c-d)/(c+d))^(1/2))-1/3*d*(c-3*d)/(c -d)^2*(c/d-1)*((c+d*sin(f*x+e))/(c-d))^(1/2)*(d*(1-sin(f*x+e))/(c+d))^(1/2 )*(1/(c-d)*(-sin(f*x+e)-1)*d)^(1/2)/(-(-d*sin(f*x+e)-c)*cos(f*x+e)^2)^(1/2 )*((-c/d-1)*EllipticE(((c+d*sin(f*x+e))/(c-d))^(1/2),((c-d)/(c+d))^(1/2))+ EllipticF(((c+d*sin(f*x+e))/(c-d))^(1/2),((c-d)/(c+d))^(1/2)))))/cos(f*x+e )/(c+d*sin(f*x+e))^(1/2)/f
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.12 (sec) , antiderivative size = 880, normalized size of antiderivative = 3.93 \[ \int \frac {\sqrt {c+d \sin (e+f x)}}{(3+3 \sin (e+f x))^2} \, dx=\text {Too large to display} \]
1/18*((sqrt(2)*(2*c^2 - 3*d^2)*cos(f*x + e)^2 - sqrt(2)*(2*c^2 - 3*d^2)*co s(f*x + e) - (sqrt(2)*(2*c^2 - 3*d^2)*cos(f*x + e) + 2*sqrt(2)*(2*c^2 - 3* d^2))*sin(f*x + e) - 2*sqrt(2)*(2*c^2 - 3*d^2))*sqrt(I*d)*weierstrassPInve rse(-4/3*(4*c^2 - 3*d^2)/d^2, -8/27*(8*I*c^3 - 9*I*c*d^2)/d^3, 1/3*(3*d*co s(f*x + e) - 3*I*d*sin(f*x + e) - 2*I*c)/d) + (sqrt(2)*(2*c^2 - 3*d^2)*cos (f*x + e)^2 - sqrt(2)*(2*c^2 - 3*d^2)*cos(f*x + e) - (sqrt(2)*(2*c^2 - 3*d ^2)*cos(f*x + e) + 2*sqrt(2)*(2*c^2 - 3*d^2))*sin(f*x + e) - 2*sqrt(2)*(2* c^2 - 3*d^2))*sqrt(-I*d)*weierstrassPInverse(-4/3*(4*c^2 - 3*d^2)/d^2, -8/ 27*(-8*I*c^3 + 9*I*c*d^2)/d^3, 1/3*(3*d*cos(f*x + e) + 3*I*d*sin(f*x + e) + 2*I*c)/d) - 3*(-I*sqrt(2)*c*d*cos(f*x + e)^2 + I*sqrt(2)*c*d*cos(f*x + e ) + 2*I*sqrt(2)*c*d + (I*sqrt(2)*c*d*cos(f*x + e) + 2*I*sqrt(2)*c*d)*sin(f *x + e))*sqrt(I*d)*weierstrassZeta(-4/3*(4*c^2 - 3*d^2)/d^2, -8/27*(8*I*c^ 3 - 9*I*c*d^2)/d^3, weierstrassPInverse(-4/3*(4*c^2 - 3*d^2)/d^2, -8/27*(8 *I*c^3 - 9*I*c*d^2)/d^3, 1/3*(3*d*cos(f*x + e) - 3*I*d*sin(f*x + e) - 2*I* c)/d)) - 3*(I*sqrt(2)*c*d*cos(f*x + e)^2 - I*sqrt(2)*c*d*cos(f*x + e) - 2* I*sqrt(2)*c*d + (-I*sqrt(2)*c*d*cos(f*x + e) - 2*I*sqrt(2)*c*d)*sin(f*x + e))*sqrt(-I*d)*weierstrassZeta(-4/3*(4*c^2 - 3*d^2)/d^2, -8/27*(-8*I*c^3 + 9*I*c*d^2)/d^3, weierstrassPInverse(-4/3*(4*c^2 - 3*d^2)/d^2, -8/27*(-8*I *c^3 + 9*I*c*d^2)/d^3, 1/3*(3*d*cos(f*x + e) + 3*I*d*sin(f*x + e) + 2*I*c) /d)) + 6*(c*d*cos(f*x + e)^2 + c*d - d^2 + (2*c*d - d^2)*cos(f*x + e) +...
\[ \int \frac {\sqrt {c+d \sin (e+f x)}}{(3+3 \sin (e+f x))^2} \, dx=\frac {\int \frac {\sqrt {c + d \sin {\left (e + f x \right )}}}{\sin ^{2}{\left (e + f x \right )} + 2 \sin {\left (e + f x \right )} + 1}\, dx}{a^{2}} \]
\[ \int \frac {\sqrt {c+d \sin (e+f x)}}{(3+3 \sin (e+f x))^2} \, dx=\int { \frac {\sqrt {d \sin \left (f x + e\right ) + c}}{{\left (a \sin \left (f x + e\right ) + a\right )}^{2}} \,d x } \]
\[ \int \frac {\sqrt {c+d \sin (e+f x)}}{(3+3 \sin (e+f x))^2} \, dx=\int { \frac {\sqrt {d \sin \left (f x + e\right ) + c}}{{\left (a \sin \left (f x + e\right ) + a\right )}^{2}} \,d x } \]
Timed out. \[ \int \frac {\sqrt {c+d \sin (e+f x)}}{(3+3 \sin (e+f x))^2} \, dx=\int \frac {\sqrt {c+d\,\sin \left (e+f\,x\right )}}{{\left (a+a\,\sin \left (e+f\,x\right )\right )}^2} \,d x \]