Integrand size = 29, antiderivative size = 362 \[ \int (3+3 \sin (e+f x))^{5/2} (c+d \sin (e+f x))^{5/2} \, dx=-\frac {9 \sqrt {3} (c+d)^3 \left (3 c^2-34 c d+283 d^2\right ) \arctan \left (\frac {\sqrt {3} \sqrt {d} \cos (e+f x)}{\sqrt {3+3 \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}\right )}{128 d^{5/2} f}-\frac {27 (c+d)^2 \left (3 c^2-34 c d+283 d^2\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{128 d^2 f \sqrt {3+3 \sin (e+f x)}}-\frac {9 (c+d) \left (3 c^2-34 c d+283 d^2\right ) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{64 d^2 f \sqrt {3+3 \sin (e+f x)}}-\frac {9 \left (3 c^2-34 c d+283 d^2\right ) \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{80 d^2 f \sqrt {3+3 \sin (e+f x)}}+\frac {81 (c-7 d) \cos (e+f x) (c+d \sin (e+f x))^{7/2}}{40 d^2 f \sqrt {3+3 \sin (e+f x)}}-\frac {9 \cos (e+f x) \sqrt {3+3 \sin (e+f x)} (c+d \sin (e+f x))^{7/2}}{5 d f} \]
-1/128*a^(5/2)*(c+d)^3*(3*c^2-34*c*d+283*d^2)*arctan(cos(f*x+e)*a^(1/2)*d^ (1/2)/(a+a*sin(f*x+e))^(1/2)/(c+d*sin(f*x+e))^(1/2))/d^(5/2)/f-1/192*a^3*( c+d)*(3*c^2-34*c*d+283*d^2)*cos(f*x+e)*(c+d*sin(f*x+e))^(3/2)/d^2/f/(a+a*s in(f*x+e))^(1/2)-1/240*a^3*(3*c^2-34*c*d+283*d^2)*cos(f*x+e)*(c+d*sin(f*x+ e))^(5/2)/d^2/f/(a+a*sin(f*x+e))^(1/2)+3/40*a^3*(c-7*d)*cos(f*x+e)*(c+d*si n(f*x+e))^(7/2)/d^2/f/(a+a*sin(f*x+e))^(1/2)-1/5*a^2*cos(f*x+e)*(c+d*sin(f *x+e))^(7/2)*(a+a*sin(f*x+e))^(1/2)/d/f-1/128*a^3*(c+d)^2*(3*c^2-34*c*d+28 3*d^2)*cos(f*x+e)*(c+d*sin(f*x+e))^(1/2)/d^2/f/(a+a*sin(f*x+e))^(1/2)
Time = 7.50 (sec) , antiderivative size = 398, normalized size of antiderivative = 1.10 \[ \int (3+3 \sin (e+f x))^{5/2} (c+d \sin (e+f x))^{5/2} \, dx=\frac {9 \sqrt {3} (1+\sin (e+f x))^{5/2} \left (\frac {(c+d)^3 \left (3 c^2-34 c d+283 d^2\right ) \left (2 \arctan \left (\frac {\sqrt {2} \sqrt {d} \sin \left (\frac {1}{4} (2 e-\pi +2 f x)\right )}{\sqrt {c+d \sin (e+f x)}}\right )+\text {arctanh}\left (\frac {\sqrt {2} \sqrt {d} \cos \left (\frac {1}{4} (2 e-\pi +2 f x)\right )}{\sqrt {c+d \sin (e+f x)}}\right )-\log \left (\sqrt {2} \sqrt {d} \cos \left (\frac {1}{4} (2 e-\pi +2 f x)\right )+\sqrt {c+d \sin (e+f x)}\right )\right )}{d^{5/2}}+\frac {2 \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right ) \sqrt {c+d \sin (e+f x)} \left (45 c^4-390 c^3 d-8396 c^2 d^2-12762 c d^3-5521 d^4+4 d^2 \left (93 c^2+488 c d+331 d^2\right ) \cos (2 (e+f x))-48 d^4 \cos (4 (e+f x))-30 c^3 d \sin (e+f x)-3322 c^2 d^2 \sin (e+f x)-7774 c d^3 \sin (e+f x)-3874 d^4 \sin (e+f x)+252 c d^3 \sin (3 (e+f x))+348 d^4 \sin (3 (e+f x))\right )}{15 d^2}\right )}{256 f \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^5} \]
(9*Sqrt[3]*(1 + Sin[e + f*x])^(5/2)*(((c + d)^3*(3*c^2 - 34*c*d + 283*d^2) *(2*ArcTan[(Sqrt[2]*Sqrt[d]*Sin[(2*e - Pi + 2*f*x)/4])/Sqrt[c + d*Sin[e + f*x]]] + ArcTanh[(Sqrt[2]*Sqrt[d]*Cos[(2*e - Pi + 2*f*x)/4])/Sqrt[c + d*Si n[e + f*x]]] - Log[Sqrt[2]*Sqrt[d]*Cos[(2*e - Pi + 2*f*x)/4] + Sqrt[c + d* Sin[e + f*x]]]))/d^(5/2) + (2*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])*Sqrt[c + d*Sin[e + f*x]]*(45*c^4 - 390*c^3*d - 8396*c^2*d^2 - 12762*c*d^3 - 5521 *d^4 + 4*d^2*(93*c^2 + 488*c*d + 331*d^2)*Cos[2*(e + f*x)] - 48*d^4*Cos[4* (e + f*x)] - 30*c^3*d*Sin[e + f*x] - 3322*c^2*d^2*Sin[e + f*x] - 7774*c*d^ 3*Sin[e + f*x] - 3874*d^4*Sin[e + f*x] + 252*c*d^3*Sin[3*(e + f*x)] + 348* d^4*Sin[3*(e + f*x)]))/(15*d^2)))/(256*f*(Cos[(e + f*x)/2] + Sin[(e + f*x) /2])^5)
Time = 1.51 (sec) , antiderivative size = 338, normalized size of antiderivative = 0.93, number of steps used = 15, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.483, Rules used = {3042, 3242, 27, 3042, 3460, 3042, 3249, 3042, 3249, 3042, 3249, 3042, 3254, 218}
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.
| \(\displaystyle \int (a \sin (e+f x)+a)^{5/2} (c+d \sin (e+f x))^{5/2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int (a \sin (e+f x)+a)^{5/2} (c+d \sin (e+f x))^{5/2}dx\) |
| \(\Big \downarrow \) 3242 |
| \(\displaystyle \frac {\int \frac {1}{2} \sqrt {\sin (e+f x) a+a} \left (a^2 (c+17 d)-3 a^2 (c-7 d) \sin (e+f x)\right ) (c+d \sin (e+f x))^{5/2}dx}{5 d}-\frac {a^2 \cos (e+f x) \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))^{7/2}}{5 d f}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \sqrt {\sin (e+f x) a+a} \left (a^2 (c+17 d)-3 a^2 (c-7 d) \sin (e+f x)\right ) (c+d \sin (e+f x))^{5/2}dx}{10 d}-\frac {a^2 \cos (e+f x) \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))^{7/2}}{5 d f}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \sqrt {\sin (e+f x) a+a} \left (a^2 (c+17 d)-3 a^2 (c-7 d) \sin (e+f x)\right ) (c+d \sin (e+f x))^{5/2}dx}{10 d}-\frac {a^2 \cos (e+f x) \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))^{7/2}}{5 d f}\) |
| \(\Big \downarrow \) 3460 |
| \(\displaystyle \frac {\frac {a^2 \left (3 c^2-34 c d+283 d^2\right ) \int \sqrt {\sin (e+f x) a+a} (c+d \sin (e+f x))^{5/2}dx}{8 d}+\frac {3 a^3 (c-7 d) \cos (e+f x) (c+d \sin (e+f x))^{7/2}}{4 d f \sqrt {a \sin (e+f x)+a}}}{10 d}-\frac {a^2 \cos (e+f x) \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))^{7/2}}{5 d f}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {a^2 \left (3 c^2-34 c d+283 d^2\right ) \int \sqrt {\sin (e+f x) a+a} (c+d \sin (e+f x))^{5/2}dx}{8 d}+\frac {3 a^3 (c-7 d) \cos (e+f x) (c+d \sin (e+f x))^{7/2}}{4 d f \sqrt {a \sin (e+f x)+a}}}{10 d}-\frac {a^2 \cos (e+f x) \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))^{7/2}}{5 d f}\) |
| \(\Big \downarrow \) 3249 |
| \(\displaystyle \frac {\frac {a^2 \left (3 c^2-34 c d+283 d^2\right ) \left (\frac {5}{6} (c+d) \int \sqrt {\sin (e+f x) a+a} (c+d \sin (e+f x))^{3/2}dx-\frac {a \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{3 f \sqrt {a \sin (e+f x)+a}}\right )}{8 d}+\frac {3 a^3 (c-7 d) \cos (e+f x) (c+d \sin (e+f x))^{7/2}}{4 d f \sqrt {a \sin (e+f x)+a}}}{10 d}-\frac {a^2 \cos (e+f x) \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))^{7/2}}{5 d f}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {a^2 \left (3 c^2-34 c d+283 d^2\right ) \left (\frac {5}{6} (c+d) \int \sqrt {\sin (e+f x) a+a} (c+d \sin (e+f x))^{3/2}dx-\frac {a \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{3 f \sqrt {a \sin (e+f x)+a}}\right )}{8 d}+\frac {3 a^3 (c-7 d) \cos (e+f x) (c+d \sin (e+f x))^{7/2}}{4 d f \sqrt {a \sin (e+f x)+a}}}{10 d}-\frac {a^2 \cos (e+f x) \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))^{7/2}}{5 d f}\) |
| \(\Big \downarrow \) 3249 |
| \(\displaystyle \frac {\frac {a^2 \left (3 c^2-34 c d+283 d^2\right ) \left (\frac {5}{6} (c+d) \left (\frac {3}{4} (c+d) \int \sqrt {\sin (e+f x) a+a} \sqrt {c+d \sin (e+f x)}dx-\frac {a \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{2 f \sqrt {a \sin (e+f x)+a}}\right )-\frac {a \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{3 f \sqrt {a \sin (e+f x)+a}}\right )}{8 d}+\frac {3 a^3 (c-7 d) \cos (e+f x) (c+d \sin (e+f x))^{7/2}}{4 d f \sqrt {a \sin (e+f x)+a}}}{10 d}-\frac {a^2 \cos (e+f x) \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))^{7/2}}{5 d f}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {a^2 \left (3 c^2-34 c d+283 d^2\right ) \left (\frac {5}{6} (c+d) \left (\frac {3}{4} (c+d) \int \sqrt {\sin (e+f x) a+a} \sqrt {c+d \sin (e+f x)}dx-\frac {a \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{2 f \sqrt {a \sin (e+f x)+a}}\right )-\frac {a \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{3 f \sqrt {a \sin (e+f x)+a}}\right )}{8 d}+\frac {3 a^3 (c-7 d) \cos (e+f x) (c+d \sin (e+f x))^{7/2}}{4 d f \sqrt {a \sin (e+f x)+a}}}{10 d}-\frac {a^2 \cos (e+f x) \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))^{7/2}}{5 d f}\) |
| \(\Big \downarrow \) 3249 |
| \(\displaystyle \frac {\frac {a^2 \left (3 c^2-34 c d+283 d^2\right ) \left (\frac {5}{6} (c+d) \left (\frac {3}{4} (c+d) \left (\frac {1}{2} (c+d) \int \frac {\sqrt {\sin (e+f x) a+a}}{\sqrt {c+d \sin (e+f x)}}dx-\frac {a \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{f \sqrt {a \sin (e+f x)+a}}\right )-\frac {a \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{2 f \sqrt {a \sin (e+f x)+a}}\right )-\frac {a \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{3 f \sqrt {a \sin (e+f x)+a}}\right )}{8 d}+\frac {3 a^3 (c-7 d) \cos (e+f x) (c+d \sin (e+f x))^{7/2}}{4 d f \sqrt {a \sin (e+f x)+a}}}{10 d}-\frac {a^2 \cos (e+f x) \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))^{7/2}}{5 d f}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {a^2 \left (3 c^2-34 c d+283 d^2\right ) \left (\frac {5}{6} (c+d) \left (\frac {3}{4} (c+d) \left (\frac {1}{2} (c+d) \int \frac {\sqrt {\sin (e+f x) a+a}}{\sqrt {c+d \sin (e+f x)}}dx-\frac {a \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{f \sqrt {a \sin (e+f x)+a}}\right )-\frac {a \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{2 f \sqrt {a \sin (e+f x)+a}}\right )-\frac {a \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{3 f \sqrt {a \sin (e+f x)+a}}\right )}{8 d}+\frac {3 a^3 (c-7 d) \cos (e+f x) (c+d \sin (e+f x))^{7/2}}{4 d f \sqrt {a \sin (e+f x)+a}}}{10 d}-\frac {a^2 \cos (e+f x) \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))^{7/2}}{5 d f}\) |
| \(\Big \downarrow \) 3254 |
| \(\displaystyle \frac {\frac {a^2 \left (3 c^2-34 c d+283 d^2\right ) \left (\frac {5}{6} (c+d) \left (\frac {3}{4} (c+d) \left (-\frac {a (c+d) \int \frac {1}{\frac {a^2 d \cos ^2(e+f x)}{(\sin (e+f x) a+a) (c+d \sin (e+f x))}+a}d\frac {a \cos (e+f x)}{\sqrt {\sin (e+f x) a+a} \sqrt {c+d \sin (e+f x)}}}{f}-\frac {a \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{f \sqrt {a \sin (e+f x)+a}}\right )-\frac {a \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{2 f \sqrt {a \sin (e+f x)+a}}\right )-\frac {a \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{3 f \sqrt {a \sin (e+f x)+a}}\right )}{8 d}+\frac {3 a^3 (c-7 d) \cos (e+f x) (c+d \sin (e+f x))^{7/2}}{4 d f \sqrt {a \sin (e+f x)+a}}}{10 d}-\frac {a^2 \cos (e+f x) \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))^{7/2}}{5 d f}\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle \frac {\frac {3 a^3 (c-7 d) \cos (e+f x) (c+d \sin (e+f x))^{7/2}}{4 d f \sqrt {a \sin (e+f x)+a}}+\frac {a^2 \left (3 c^2-34 c d+283 d^2\right ) \left (\frac {5}{6} (c+d) \left (\frac {3}{4} (c+d) \left (-\frac {\sqrt {a} (c+d) \arctan \left (\frac {\sqrt {a} \sqrt {d} \cos (e+f x)}{\sqrt {a \sin (e+f x)+a} \sqrt {c+d \sin (e+f x)}}\right )}{\sqrt {d} f}-\frac {a \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{f \sqrt {a \sin (e+f x)+a}}\right )-\frac {a \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{2 f \sqrt {a \sin (e+f x)+a}}\right )-\frac {a \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{3 f \sqrt {a \sin (e+f x)+a}}\right )}{8 d}}{10 d}-\frac {a^2 \cos (e+f x) \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))^{7/2}}{5 d f}\) |
-1/5*(a^2*Cos[e + f*x]*Sqrt[a + a*Sin[e + f*x]]*(c + d*Sin[e + f*x])^(7/2) )/(d*f) + ((3*a^3*(c - 7*d)*Cos[e + f*x]*(c + d*Sin[e + f*x])^(7/2))/(4*d* f*Sqrt[a + a*Sin[e + f*x]]) + (a^2*(3*c^2 - 34*c*d + 283*d^2)*(-1/3*(a*Cos [e + f*x]*(c + d*Sin[e + f*x])^(5/2))/(f*Sqrt[a + a*Sin[e + f*x]]) + (5*(c + d)*(-1/2*(a*Cos[e + f*x]*(c + d*Sin[e + f*x])^(3/2))/(f*Sqrt[a + a*Sin[ e + f*x]]) + (3*(c + d)*(-((Sqrt[a]*(c + d)*ArcTan[(Sqrt[a]*Sqrt[d]*Cos[e + f*x])/(Sqrt[a + a*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]])])/(Sqrt[d]*f)) - (a*Cos[e + f*x]*Sqrt[c + d*Sin[e + f*x]])/(f*Sqrt[a + a*Sin[e + f*x]])) )/4))/6))/(8*d))/(10*d)
3.6.79.3.1 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_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[(-b^2)*Cos[e + f*x]*(a + b*Sin[e + f*x ])^(m - 2)*((c + d*Sin[e + f*x])^(n + 1)/(d*f*(m + n))), x] + Simp[1/(d*(m + n)) Int[(a + b*Sin[e + f*x])^(m - 2)*(c + d*Sin[e + f*x])^n*Simp[a*b*c* (m - 2) + b^2*d*(n + 1) + a^2*d*(m + n) - b*(b*c*(m - 1) - a*d*(3*m + 2*n - 2))*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[m, 1] && !LtQ[ n, -1] && (IntegersQ[2*m, 2*n] || IntegerQ[m + 1/2] || (IntegerQ[m] && EqQ[ c, 0]))
Int[Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]]*((c_.) + (d_.)*sin[(e_.) + ( f_.)*(x_)])^(n_), x_Symbol] :> Simp[-2*b*Cos[e + f*x]*((c + d*Sin[e + f*x]) ^n/(f*(2*n + 1)*Sqrt[a + b*Sin[e + f*x]])), x] + Simp[2*n*((b*c + a*d)/(b*( 2*n + 1))) Int[Sqrt[a + b*Sin[e + f*x]]*(c + d*Sin[e + f*x])^(n - 1), 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] && GtQ[n, 0] && IntegerQ[2*n]
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[Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]]*((A_.) + (B_.)*sin[(e_.) + ( f_.)*(x_)])*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp [-2*b*B*Cos[e + f*x]*((c + d*Sin[e + f*x])^(n + 1)/(d*f*(2*n + 3)*Sqrt[a + b*Sin[e + f*x]])), x] + Simp[(A*b*d*(2*n + 3) - B*(b*c - 2*a*d*(n + 1)))/(b *d*(2*n + 3)) Int[Sqrt[a + b*Sin[e + f*x]]*(c + d*Sin[e + f*x])^n, 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[n, -1]
Timed out.
\[\int \left (a +a \sin \left (f x +e \right )\right )^{\frac {5}{2}} \left (c +d \sin \left (f x +e \right )\right )^{\frac {5}{2}}d x\]
Leaf count of result is larger than twice the leaf count of optimal. 812 vs. \(2 (333) = 666\).
Time = 1.24 (sec) , antiderivative size = 2083, normalized size of antiderivative = 5.75 \[ \int (3+3 \sin (e+f x))^{5/2} (c+d \sin (e+f x))^{5/2} \, dx=\text {Too large to display} \]
[1/15360*(15*(3*a^2*c^5 - 25*a^2*c^4*d + 190*a^2*c^3*d^2 + 750*a^2*c^2*d^3 + 815*a^2*c*d^4 + 283*a^2*d^5 + (3*a^2*c^5 - 25*a^2*c^4*d + 190*a^2*c^3*d ^2 + 750*a^2*c^2*d^3 + 815*a^2*c*d^4 + 283*a^2*d^5)*cos(f*x + e) + (3*a^2* c^5 - 25*a^2*c^4*d + 190*a^2*c^3*d^2 + 750*a^2*c^2*d^3 + 815*a^2*c*d^4 + 2 83*a^2*d^5)*sin(f*x + e))*sqrt(-a/d)*log((128*a*d^4*cos(f*x + e)^5 + a*c^4 + 4*a*c^3*d + 6*a*c^2*d^2 + 4*a*c*d^3 + a*d^4 + 128*(2*a*c*d^3 - a*d^4)*c os(f*x + e)^4 - 32*(5*a*c^2*d^2 - 14*a*c*d^3 + 13*a*d^4)*cos(f*x + e)^3 - 32*(a*c^3*d - 2*a*c^2*d^2 + 9*a*c*d^3 - 4*a*d^4)*cos(f*x + e)^2 - 8*(16*d^ 4*cos(f*x + e)^4 - c^3*d + 17*c^2*d^2 - 59*c*d^3 + 51*d^4 + 24*(c*d^3 - d^ 4)*cos(f*x + e)^3 - 2*(5*c^2*d^2 - 26*c*d^3 + 33*d^4)*cos(f*x + e)^2 - (c^ 3*d - 7*c^2*d^2 + 31*c*d^3 - 25*d^4)*cos(f*x + e) + (16*d^4*cos(f*x + e)^3 + c^3*d - 17*c^2*d^2 + 59*c*d^3 - 51*d^4 - 8*(3*c*d^3 - 5*d^4)*cos(f*x + e)^2 - 2*(5*c^2*d^2 - 14*c*d^3 + 13*d^4)*cos(f*x + e))*sin(f*x + e))*sqrt( a*sin(f*x + e) + a)*sqrt(d*sin(f*x + e) + c)*sqrt(-a/d) + (a*c^4 - 28*a*c^ 3*d + 230*a*c^2*d^2 - 476*a*c*d^3 + 289*a*d^4)*cos(f*x + e) + (128*a*d^4*c os(f*x + e)^4 + a*c^4 + 4*a*c^3*d + 6*a*c^2*d^2 + 4*a*c*d^3 + a*d^4 - 256* (a*c*d^3 - a*d^4)*cos(f*x + e)^3 - 32*(5*a*c^2*d^2 - 6*a*c*d^3 + 5*a*d^4)* cos(f*x + e)^2 + 32*(a*c^3*d - 7*a*c^2*d^2 + 15*a*c*d^3 - 9*a*d^4)*cos(f*x + e))*sin(f*x + e))/(cos(f*x + e) + sin(f*x + e) + 1)) - 8*(384*a^2*d^4*c os(f*x + e)^5 - 45*a^2*c^4 + 360*a^2*c^3*d + 5446*a^2*c^2*d^2 + 6688*a^...
Timed out. \[ \int (3+3 \sin (e+f x))^{5/2} (c+d \sin (e+f x))^{5/2} \, dx=\text {Timed out} \]
\[ \int (3+3 \sin (e+f x))^{5/2} (c+d \sin (e+f x))^{5/2} \, dx=\int { {\left (a \sin \left (f x + e\right ) + a\right )}^{\frac {5}{2}} {\left (d \sin \left (f x + e\right ) + c\right )}^{\frac {5}{2}} \,d x } \]
\[ \int (3+3 \sin (e+f x))^{5/2} (c+d \sin (e+f x))^{5/2} \, dx=\int { {\left (a \sin \left (f x + e\right ) + a\right )}^{\frac {5}{2}} {\left (d \sin \left (f x + e\right ) + c\right )}^{\frac {5}{2}} \,d x } \]
Timed out. \[ \int (3+3 \sin (e+f x))^{5/2} (c+d \sin (e+f x))^{5/2} \, dx=\int {\left (a+a\,\sin \left (e+f\,x\right )\right )}^{5/2}\,{\left (c+d\,\sin \left (e+f\,x\right )\right )}^{5/2} \,d x \]