Integrand size = 29, antiderivative size = 251 \[ \int \frac {(c+d \sin (e+f x))^{5/2}}{(3+3 \sin (e+f x))^{5/2}} \, dx=-\frac {2 d^{5/2} \arctan \left (\frac {\sqrt {3} \sqrt {d} \cos (e+f x)}{\sqrt {3+3 \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}\right )}{9 \sqrt {3} f}-\frac {\sqrt {c-d} \left (3 c^2+14 c d+43 d^2\right ) \text {arctanh}\left (\frac {\sqrt {\frac {3}{2}} \sqrt {c-d} \cos (e+f x)}{\sqrt {3+3 \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}\right )}{144 \sqrt {6} f}-\frac {(c-d) (3 c+11 d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{48 f (3+3 \sin (e+f x))^{3/2}}-\frac {(c-d) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{4 f (3+3 \sin (e+f x))^{5/2}} \]
-2*d^(5/2)*arctan(cos(f*x+e)*a^(1/2)*d^(1/2)/(a+a*sin(f*x+e))^(1/2)/(c+d*s in(f*x+e))^(1/2))/a^(5/2)/f-1/4*(c-d)*cos(f*x+e)*(c+d*sin(f*x+e))^(3/2)/f/ (a+a*sin(f*x+e))^(5/2)-1/32*(3*c^2+14*c*d+43*d^2)*arctanh(1/2*cos(f*x+e)*a ^(1/2)*(c-d)^(1/2)*2^(1/2)/(a+a*sin(f*x+e))^(1/2)/(c+d*sin(f*x+e))^(1/2))* (c-d)^(1/2)/a^(5/2)/f*2^(1/2)-1/16*(c-d)*(3*c+11*d)*cos(f*x+e)*(c+d*sin(f* x+e))^(1/2)/a/f/(a+a*sin(f*x+e))^(3/2)
Result contains complex when optimal does not.
Time = 15.40 (sec) , antiderivative size = 1845, normalized size of antiderivative = 7.35 \[ \int \frac {(c+d \sin (e+f x))^{5/2}}{(3+3 \sin (e+f x))^{5/2}} \, dx =\text {Too large to display} \]
((Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^5*(-1/4*(c - d)^2/(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^3 - (3*(c - d)*(c + 5*d))/(16*(Cos[(e + f*x)/2] + Sin [(e + f*x)/2])) + (3*(c^2*Sin[(e + f*x)/2] + 4*c*d*Sin[(e + f*x)/2] - 5*d^ 2*Sin[(e + f*x)/2]))/(8*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^2) + (c^2*Si n[(e + f*x)/2] - 2*c*d*Sin[(e + f*x)/2] + d^2*Sin[(e + f*x)/2])/(2*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^4))*Sqrt[c + d*Sin[e + f*x]])/(f*(3 + 3*Sin [e + f*x])^(5/2)) + ((Sqrt[2]*(3*c^3 + 11*c^2*d + 29*c*d^2 - 43*d^3)*Log[1 + Tan[(e + f*x)/2]] - Sqrt[2]*(3*c^3 + 11*c^2*d + 29*c*d^2 - 43*d^3)*Log[ c - d + 2*Sqrt[c - d]*Sqrt[(1 + Cos[e + f*x])^(-1)]*Sqrt[c + d*Sin[e + f*x ]] + (-c + d)*Tan[(e + f*x)/2]] - (32*I)*Sqrt[c - d]*d^(5/2)*(Log[(c - I*( d + (1 + I)*Sqrt[2]*Sqrt[d]*Sqrt[(1 + Cos[e + f*x])^(-1)]*Sqrt[c + d*Sin[e + f*x]]) + ((-I)*c + d)*Tan[(e + f*x)/2])/(16*d^(7/2)*(I + Tan[(e + f*x)/ 2]))] - Log[(c + I*d + (1 + I)*Sqrt[2]*Sqrt[d]*Sqrt[(1 + Cos[e + f*x])^(-1 )]*Sqrt[c + d*Sin[e + f*x]] + (I*c + d)*Tan[(e + f*x)/2])/(16*d^(7/2)*(-I + Tan[(e + f*x)/2]))]))*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^5*((3*c^3)/( 32*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*Sqrt[c + d*Sin[e + f*x]]) + (11*c ^2*d)/(32*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*Sqrt[c + d*Sin[e + f*x]]) + (29*c*d^2)/(32*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*Sqrt[c + d*Sin[e + f*x]]) - (11*d^3)/(32*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*Sqrt[c + d*Sin [e + f*x]]) + (d^3*Sin[e + f*x])/((Cos[(e + f*x)/2] + Sin[(e + f*x)/2])...
Time = 1.40 (sec) , antiderivative size = 272, normalized size of antiderivative = 1.08, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.448, Rules used = {3042, 3244, 27, 3042, 3456, 27, 3042, 3461, 3042, 3254, 218, 3261, 221}
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.
| \(\displaystyle \int \frac {(c+d \sin (e+f x))^{5/2}}{(a \sin (e+f x)+a)^{5/2}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {(c+d \sin (e+f x))^{5/2}}{(a \sin (e+f x)+a)^{5/2}}dx\) |
| \(\Big \downarrow \) 3244 |
| \(\displaystyle -\frac {\int -\frac {\sqrt {c+d \sin (e+f x)} \left (8 a \sin (e+f x) d^2+a (3 c-d) (c+3 d)\right )}{2 (\sin (e+f x) a+a)^{3/2}}dx}{4 a^2}-\frac {(c-d) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{4 f (a \sin (e+f x)+a)^{5/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {\sqrt {c+d \sin (e+f x)} \left (8 a \sin (e+f x) d^2+a (3 c-d) (c+3 d)\right )}{(\sin (e+f x) a+a)^{3/2}}dx}{8 a^2}-\frac {(c-d) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{4 f (a \sin (e+f x)+a)^{5/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {\sqrt {c+d \sin (e+f x)} \left (8 a \sin (e+f x) d^2+a (3 c-d) (c+3 d)\right )}{(\sin (e+f x) a+a)^{3/2}}dx}{8 a^2}-\frac {(c-d) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{4 f (a \sin (e+f x)+a)^{5/2}}\) |
| \(\Big \downarrow \) 3456 |
| \(\displaystyle \frac {\frac {\int \frac {32 a^2 \sin (e+f x) d^3+a^2 (3 c-d) \left (c^2+4 d c+11 d^2\right )}{2 \sqrt {\sin (e+f x) a+a} \sqrt {c+d \sin (e+f x)}}dx}{2 a^2}-\frac {a (c-d) (3 c+11 d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{2 f (a \sin (e+f x)+a)^{3/2}}}{8 a^2}-\frac {(c-d) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{4 f (a \sin (e+f x)+a)^{5/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\int \frac {32 a^2 \sin (e+f x) d^3+a^2 (3 c-d) \left (c^2+4 d c+11 d^2\right )}{\sqrt {\sin (e+f x) a+a} \sqrt {c+d \sin (e+f x)}}dx}{4 a^2}-\frac {a (c-d) (3 c+11 d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{2 f (a \sin (e+f x)+a)^{3/2}}}{8 a^2}-\frac {(c-d) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{4 f (a \sin (e+f x)+a)^{5/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\int \frac {32 a^2 \sin (e+f x) d^3+a^2 (3 c-d) \left (c^2+4 d c+11 d^2\right )}{\sqrt {\sin (e+f x) a+a} \sqrt {c+d \sin (e+f x)}}dx}{4 a^2}-\frac {a (c-d) (3 c+11 d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{2 f (a \sin (e+f x)+a)^{3/2}}}{8 a^2}-\frac {(c-d) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{4 f (a \sin (e+f x)+a)^{5/2}}\) |
| \(\Big \downarrow \) 3461 |
| \(\displaystyle \frac {\frac {a^2 (c-d) \left (3 c^2+14 c d+43 d^2\right ) \int \frac {1}{\sqrt {\sin (e+f x) a+a} \sqrt {c+d \sin (e+f x)}}dx+32 a d^3 \int \frac {\sqrt {\sin (e+f x) a+a}}{\sqrt {c+d \sin (e+f x)}}dx}{4 a^2}-\frac {a (c-d) (3 c+11 d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{2 f (a \sin (e+f x)+a)^{3/2}}}{8 a^2}-\frac {(c-d) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{4 f (a \sin (e+f x)+a)^{5/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {a^2 (c-d) \left (3 c^2+14 c d+43 d^2\right ) \int \frac {1}{\sqrt {\sin (e+f x) a+a} \sqrt {c+d \sin (e+f x)}}dx+32 a d^3 \int \frac {\sqrt {\sin (e+f x) a+a}}{\sqrt {c+d \sin (e+f x)}}dx}{4 a^2}-\frac {a (c-d) (3 c+11 d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{2 f (a \sin (e+f x)+a)^{3/2}}}{8 a^2}-\frac {(c-d) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{4 f (a \sin (e+f x)+a)^{5/2}}\) |
| \(\Big \downarrow \) 3254 |
| \(\displaystyle \frac {\frac {a^2 (c-d) \left (3 c^2+14 c d+43 d^2\right ) \int \frac {1}{\sqrt {\sin (e+f x) a+a} \sqrt {c+d \sin (e+f x)}}dx-\frac {64 a^2 d^3 \int \frac {1}{\frac {a^2 d \cos ^2(e+f x)}{(\sin (e+f x) a+a) (c+d \sin (e+f x))}+a}d\frac {a \cos (e+f x)}{\sqrt {\sin (e+f x) a+a} \sqrt {c+d \sin (e+f x)}}}{f}}{4 a^2}-\frac {a (c-d) (3 c+11 d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{2 f (a \sin (e+f x)+a)^{3/2}}}{8 a^2}-\frac {(c-d) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{4 f (a \sin (e+f x)+a)^{5/2}}\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle \frac {\frac {a^2 (c-d) \left (3 c^2+14 c d+43 d^2\right ) \int \frac {1}{\sqrt {\sin (e+f x) a+a} \sqrt {c+d \sin (e+f x)}}dx-\frac {64 a^{3/2} d^{5/2} \arctan \left (\frac {\sqrt {a} \sqrt {d} \cos (e+f x)}{\sqrt {a \sin (e+f x)+a} \sqrt {c+d \sin (e+f x)}}\right )}{f}}{4 a^2}-\frac {a (c-d) (3 c+11 d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{2 f (a \sin (e+f x)+a)^{3/2}}}{8 a^2}-\frac {(c-d) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{4 f (a \sin (e+f x)+a)^{5/2}}\) |
| \(\Big \downarrow \) 3261 |
| \(\displaystyle \frac {\frac {-\frac {2 a^3 (c-d) \left (3 c^2+14 c d+43 d^2\right ) \int \frac {1}{2 a^2-\frac {a^3 (c-d) \cos ^2(e+f x)}{(\sin (e+f x) a+a) (c+d \sin (e+f x))}}d\frac {a \cos (e+f x)}{\sqrt {\sin (e+f x) a+a} \sqrt {c+d \sin (e+f x)}}}{f}-\frac {64 a^{3/2} d^{5/2} \arctan \left (\frac {\sqrt {a} \sqrt {d} \cos (e+f x)}{\sqrt {a \sin (e+f x)+a} \sqrt {c+d \sin (e+f x)}}\right )}{f}}{4 a^2}-\frac {a (c-d) (3 c+11 d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{2 f (a \sin (e+f x)+a)^{3/2}}}{8 a^2}-\frac {(c-d) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{4 f (a \sin (e+f x)+a)^{5/2}}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {\frac {-\frac {64 a^{3/2} d^{5/2} \arctan \left (\frac {\sqrt {a} \sqrt {d} \cos (e+f x)}{\sqrt {a \sin (e+f x)+a} \sqrt {c+d \sin (e+f x)}}\right )}{f}-\frac {\sqrt {2} a^{3/2} \sqrt {c-d} \left (3 c^2+14 c d+43 d^2\right ) \text {arctanh}\left (\frac {\sqrt {a} \sqrt {c-d} \cos (e+f x)}{\sqrt {2} \sqrt {a \sin (e+f x)+a} \sqrt {c+d \sin (e+f x)}}\right )}{f}}{4 a^2}-\frac {a (c-d) (3 c+11 d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{2 f (a \sin (e+f x)+a)^{3/2}}}{8 a^2}-\frac {(c-d) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{4 f (a \sin (e+f x)+a)^{5/2}}\) |
-1/4*((c - d)*Cos[e + f*x]*(c + d*Sin[e + f*x])^(3/2))/(f*(a + a*Sin[e + f *x])^(5/2)) + (((-64*a^(3/2)*d^(5/2)*ArcTan[(Sqrt[a]*Sqrt[d]*Cos[e + f*x]) /(Sqrt[a + a*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]])])/f - (Sqrt[2]*a^(3/2 )*Sqrt[c - d]*(3*c^2 + 14*c*d + 43*d^2)*ArcTanh[(Sqrt[a]*Sqrt[c - d]*Cos[e + f*x])/(Sqrt[2]*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]])])/f)/ (4*a^2) - (a*(c - d)*(3*c + 11*d)*Cos[e + f*x]*Sqrt[c + d*Sin[e + f*x]])/( 2*f*(a + a*Sin[e + f*x])^(3/2)))/(8*a^2)
3.6.100.3.1 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[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/R t[a/b, 2]], x] /; FreeQ[{a, b}, x] && PosQ[a/b]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x /Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[(b*c - a*d)*Cos[e + f*x]*(a + b*Sin[e + f*x])^m*((c + d*Sin[e + f*x])^(n - 1)/(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 - 2)* Simp[b*(c^2*(m + 1) + d^2*(n - 1)) + a*c*d*(m - n + 1) + d*(a*d*(m - n + 1) + b*c*(m + n))*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] && GtQ[n, 1] && (IntegersQ[2*m, 2*n] || (IntegerQ[m] && EqQ[c, 0]))
Int[Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]]/Sqrt[(c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]], x_Symbol] :> Simp[-2*(b/f) Subst[Int[1/(b + d*x^2), x], x, b*(Cos[e + f*x]/(Sqrt[a + b*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*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]
Int[1/(Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]]*Sqrt[(c_.) + (d_.)*sin[(e _.) + (f_.)*(x_)]]), x_Symbol] :> Simp[-2*(a/f) Subst[Int[1/(2*b^2 - (a*c - b*d)*x^2), x], x, b*(Cos[e + f*x]/(Sqrt[a + b*Sin[e + f*x]]*Sqrt[c + d*S in[e + f*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]
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[(A*b - a*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*(a*d*n - b*c*(m + 1)) - B*(a*c*m + b*d*n) - d*(a*B*(m - n) + A*b*(m + n + 1))*Sin[e + f*x], x], x], x] /; Fre eQ[{a, b, c, d, e, f, A, B}, 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] && (In tegerQ[2*n] || EqQ[c, 0])
Int[((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])/(Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]]*Sqrt[(c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]]), x_Symbol] :> Sim p[(A*b - a*B)/b Int[1/(Sqrt[a + b*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]]) , x], x] + Simp[B/b Int[Sqrt[a + b*Sin[e + f*x]]/Sqrt[c + d*Sin[e + f*x]] , x], x] /; FreeQ[{a, b, c, d, e, f, A, B}, x] && NeQ[b*c - a*d, 0] && EqQ[ a^2 - b^2, 0] && NeQ[c^2 - d^2, 0]
Leaf count of result is larger than twice the leaf count of optimal. \(13852\) vs. \(2(219)=438\).
Time = 1.82 (sec) , antiderivative size = 13853, normalized size of antiderivative = 55.19
\[\text {output too large to display}\]
Leaf count of result is larger than twice the leaf count of optimal. 712 vs. \(2 (219) = 438\).
Time = 0.85 (sec) , antiderivative size = 3855, normalized size of antiderivative = 15.36 \[ \int \frac {(c+d \sin (e+f x))^{5/2}}{(3+3 \sin (e+f x))^{5/2}} \, dx=\text {Too large to display} \]
[1/32*(sqrt(1/2)*((3*a*c^2 + 14*a*c*d + 43*a*d^2)*cos(f*x + e)^3 - 12*a*c^ 2 - 56*a*c*d - 172*a*d^2 + 3*(3*a*c^2 + 14*a*c*d + 43*a*d^2)*cos(f*x + e)^ 2 - 2*(3*a*c^2 + 14*a*c*d + 43*a*d^2)*cos(f*x + e) - (12*a*c^2 + 56*a*c*d + 172*a*d^2 - (3*a*c^2 + 14*a*c*d + 43*a*d^2)*cos(f*x + e)^2 + 2*(3*a*c^2 + 14*a*c*d + 43*a*d^2)*cos(f*x + e))*sin(f*x + e))*sqrt((c - d)/a)*log((4* sqrt(1/2)*sqrt(a*sin(f*x + e) + a)*sqrt(d*sin(f*x + e) + c)*sqrt((c - d)/a )*(cos(f*x + e) - sin(f*x + e) + 1) - (c - 3*d)*cos(f*x + e)^2 - (3*c - d) *cos(f*x + e) + ((c - 3*d)*cos(f*x + e) - 2*c - 2*d)*sin(f*x + e) - 2*c - 2*d)/(cos(f*x + e)^2 - (cos(f*x + e) + 2)*sin(f*x + e) - cos(f*x + e) - 2) ) + 8*(a*d^2*cos(f*x + e)^3 + 3*a*d^2*cos(f*x + e)^2 - 2*a*d^2*cos(f*x + e ) - 4*a*d^2 + (a*d^2*cos(f*x + e)^2 - 2*a*d^2*cos(f*x + e) - 4*a*d^2)*sin( f*x + e))*sqrt(-d/a)*log((128*d^4*cos(f*x + e)^5 + 128*(2*c*d^3 - d^4)*cos (f*x + e)^4 + c^4 + 4*c^3*d + 6*c^2*d^2 + 4*c*d^3 + d^4 - 32*(5*c^2*d^2 - 14*c*d^3 + 13*d^4)*cos(f*x + e)^3 - 32*(c^3*d - 2*c^2*d^2 + 9*c*d^3 - 4*d^ 4)*cos(f*x + e)^2 - 8*(16*d^3*cos(f*x + e)^4 + 24*(c*d^2 - d^3)*cos(f*x + e)^3 - c^3 + 17*c^2*d - 59*c*d^2 + 51*d^3 - 2*(5*c^2*d - 26*c*d^2 + 33*d^3 )*cos(f*x + e)^2 - (c^3 - 7*c^2*d + 31*c*d^2 - 25*d^3)*cos(f*x + e) + (16* d^3*cos(f*x + e)^3 + c^3 - 17*c^2*d + 59*c*d^2 - 51*d^3 - 8*(3*c*d^2 - 5*d ^3)*cos(f*x + e)^2 - 2*(5*c^2*d - 14*c*d^2 + 13*d^3)*cos(f*x + e))*sin(f*x + e))*sqrt(a*sin(f*x + e) + a)*sqrt(d*sin(f*x + e) + c)*sqrt(-d/a) + (...
Timed out. \[ \int \frac {(c+d \sin (e+f x))^{5/2}}{(3+3 \sin (e+f x))^{5/2}} \, dx=\text {Timed out} \]
\[ \int \frac {(c+d \sin (e+f x))^{5/2}}{(3+3 \sin (e+f x))^{5/2}} \, dx=\int { \frac {{\left (d \sin \left (f x + e\right ) + c\right )}^{\frac {5}{2}}}{{\left (a \sin \left (f x + e\right ) + a\right )}^{\frac {5}{2}}} \,d x } \]
Timed out. \[ \int \frac {(c+d \sin (e+f x))^{5/2}}{(3+3 \sin (e+f x))^{5/2}} \, dx=\text {Timed out} \]
Timed out. \[ \int \frac {(c+d \sin (e+f x))^{5/2}}{(3+3 \sin (e+f x))^{5/2}} \, dx=\int \frac {{\left (c+d\,\sin \left (e+f\,x\right )\right )}^{5/2}}{{\left (a+a\,\sin \left (e+f\,x\right )\right )}^{5/2}} \,d x \]