Integrand size = 25, antiderivative size = 169 \[ \int \frac {(a+a \sin (c+d x))^4}{(e \cos (c+d x))^{11/2}} \, dx=\frac {2 a^4 \sqrt {e \cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{15 d e^6 \sqrt {\cos (c+d x)}}+\frac {4 a^7 (e \cos (c+d x))^{3/2}}{9 d e^7 (a-a \sin (c+d x))^3}-\frac {2 a^8 (e \cos (c+d x))^{3/2}}{15 d e^7 \left (a^2-a^2 \sin (c+d x)\right )^2}-\frac {2 a^8 (e \cos (c+d x))^{3/2}}{15 d e^7 \left (a^4-a^4 \sin (c+d x)\right )} \]
4/9*a^7*(e*cos(d*x+c))^(3/2)/d/e^7/(a-a*sin(d*x+c))^3-2/15*a^8*(e*cos(d*x+ c))^(3/2)/d/e^7/(a^2-a^2*sin(d*x+c))^2-2/15*a^8*(e*cos(d*x+c))^(3/2)/d/e^7 /(a^4-a^4*sin(d*x+c))+2/15*a^4*(cos(1/2*d*x+1/2*c)^2)^(1/2)/cos(1/2*d*x+1/ 2*c)*EllipticE(sin(1/2*d*x+1/2*c),2^(1/2))*(e*cos(d*x+c))^(1/2)/d/e^6/cos( d*x+c)^(1/2)
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 0.10 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.39 \[ \int \frac {(a+a \sin (c+d x))^4}{(e \cos (c+d x))^{11/2}} \, dx=\frac {4\ 2^{3/4} a^4 \operatorname {Hypergeometric2F1}\left (-\frac {9}{4},-\frac {3}{4},-\frac {5}{4},\frac {1}{2} (1-\sin (c+d x))\right ) (1+\sin (c+d x))^{9/4}}{9 d e (e \cos (c+d x))^{9/2}} \]
(4*2^(3/4)*a^4*Hypergeometric2F1[-9/4, -3/4, -5/4, (1 - Sin[c + d*x])/2]*( 1 + Sin[c + d*x])^(9/4))/(9*d*e*(e*Cos[c + d*x])^(9/2))
Time = 0.80 (sec) , antiderivative size = 172, normalized size of antiderivative = 1.02, number of steps used = 12, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.480, Rules used = {3042, 3149, 3042, 3159, 3042, 3160, 3042, 3162, 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.
| \(\displaystyle \int \frac {(a \sin (c+d x)+a)^4}{(e \cos (c+d x))^{11/2}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {(a \sin (c+d x)+a)^4}{(e \cos (c+d x))^{11/2}}dx\) |
| \(\Big \downarrow \) 3149 |
| \(\displaystyle \frac {a^8 \int \frac {(e \cos (c+d x))^{5/2}}{(a-a \sin (c+d x))^4}dx}{e^8}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {a^8 \int \frac {(e \cos (c+d x))^{5/2}}{(a-a \sin (c+d x))^4}dx}{e^8}\) |
| \(\Big \downarrow \) 3159 |
| \(\displaystyle \frac {a^8 \left (\frac {4 e (e \cos (c+d x))^{3/2}}{9 a d (a-a \sin (c+d x))^3}-\frac {e^2 \int \frac {\sqrt {e \cos (c+d x)}}{(a-a \sin (c+d x))^2}dx}{3 a^2}\right )}{e^8}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {a^8 \left (\frac {4 e (e \cos (c+d x))^{3/2}}{9 a d (a-a \sin (c+d x))^3}-\frac {e^2 \int \frac {\sqrt {e \cos (c+d x)}}{(a-a \sin (c+d x))^2}dx}{3 a^2}\right )}{e^8}\) |
| \(\Big \downarrow \) 3160 |
| \(\displaystyle \frac {a^8 \left (\frac {4 e (e \cos (c+d x))^{3/2}}{9 a d (a-a \sin (c+d x))^3}-\frac {e^2 \left (\frac {\int \frac {\sqrt {e \cos (c+d x)}}{a-a \sin (c+d x)}dx}{5 a}+\frac {2 (e \cos (c+d x))^{3/2}}{5 d e (a-a \sin (c+d x))^2}\right )}{3 a^2}\right )}{e^8}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {a^8 \left (\frac {4 e (e \cos (c+d x))^{3/2}}{9 a d (a-a \sin (c+d x))^3}-\frac {e^2 \left (\frac {\int \frac {\sqrt {e \cos (c+d x)}}{a-a \sin (c+d x)}dx}{5 a}+\frac {2 (e \cos (c+d x))^{3/2}}{5 d e (a-a \sin (c+d x))^2}\right )}{3 a^2}\right )}{e^8}\) |
| \(\Big \downarrow \) 3162 |
| \(\displaystyle \frac {a^8 \left (\frac {4 e (e \cos (c+d x))^{3/2}}{9 a d (a-a \sin (c+d x))^3}-\frac {e^2 \left (\frac {\frac {2 (e \cos (c+d x))^{3/2}}{d e (a-a \sin (c+d x))}-\frac {\int \sqrt {e \cos (c+d x)}dx}{a}}{5 a}+\frac {2 (e \cos (c+d x))^{3/2}}{5 d e (a-a \sin (c+d x))^2}\right )}{3 a^2}\right )}{e^8}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {a^8 \left (\frac {4 e (e \cos (c+d x))^{3/2}}{9 a d (a-a \sin (c+d x))^3}-\frac {e^2 \left (\frac {\frac {2 (e \cos (c+d x))^{3/2}}{d e (a-a \sin (c+d x))}-\frac {\int \sqrt {e \sin \left (c+d x+\frac {\pi }{2}\right )}dx}{a}}{5 a}+\frac {2 (e \cos (c+d x))^{3/2}}{5 d e (a-a \sin (c+d x))^2}\right )}{3 a^2}\right )}{e^8}\) |
| \(\Big \downarrow \) 3121 |
| \(\displaystyle \frac {a^8 \left (\frac {4 e (e \cos (c+d x))^{3/2}}{9 a d (a-a \sin (c+d x))^3}-\frac {e^2 \left (\frac {\frac {2 (e \cos (c+d x))^{3/2}}{d e (a-a \sin (c+d x))}-\frac {\sqrt {e \cos (c+d x)} \int \sqrt {\cos (c+d x)}dx}{a \sqrt {\cos (c+d x)}}}{5 a}+\frac {2 (e \cos (c+d x))^{3/2}}{5 d e (a-a \sin (c+d x))^2}\right )}{3 a^2}\right )}{e^8}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {a^8 \left (\frac {4 e (e \cos (c+d x))^{3/2}}{9 a d (a-a \sin (c+d x))^3}-\frac {e^2 \left (\frac {\frac {2 (e \cos (c+d x))^{3/2}}{d e (a-a \sin (c+d x))}-\frac {\sqrt {e \cos (c+d x)} \int \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}dx}{a \sqrt {\cos (c+d x)}}}{5 a}+\frac {2 (e \cos (c+d x))^{3/2}}{5 d e (a-a \sin (c+d x))^2}\right )}{3 a^2}\right )}{e^8}\) |
| \(\Big \downarrow \) 3119 |
| \(\displaystyle \frac {a^8 \left (\frac {4 e (e \cos (c+d x))^{3/2}}{9 a d (a-a \sin (c+d x))^3}-\frac {e^2 \left (\frac {2 (e \cos (c+d x))^{3/2}}{5 d e (a-a \sin (c+d x))^2}+\frac {\frac {2 (e \cos (c+d x))^{3/2}}{d e (a-a \sin (c+d x))}-\frac {2 E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {e \cos (c+d x)}}{a d \sqrt {\cos (c+d x)}}}{5 a}\right )}{3 a^2}\right )}{e^8}\) |
(a^8*((4*e*(e*Cos[c + d*x])^(3/2))/(9*a*d*(a - a*Sin[c + d*x])^3) - (e^2*( (2*(e*Cos[c + d*x])^(3/2))/(5*d*e*(a - a*Sin[c + d*x])^2) + ((-2*Sqrt[e*Co s[c + d*x]]*EllipticE[(c + d*x)/2, 2])/(a*d*Sqrt[Cos[c + d*x]]) + (2*(e*Co s[c + d*x])^(3/2))/(d*e*(a - a*Sin[c + d*x])))/(5*a)))/(3*a^2)))/e^8
3.3.31.3.1 Defintions of rubi rules used
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_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x _)])^(m_), x_Symbol] :> Simp[(a/g)^(2*m) Int[(g*Cos[e + f*x])^(2*m + p)/( a - b*Sin[e + f*x])^m, x], x] /; FreeQ[{a, b, e, f, g}, x] && EqQ[a^2 - b^2 , 0] && IntegerQ[m] && LtQ[p, -1] && GeQ[2*m + p, 0]
Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x _)])^(m_), x_Symbol] :> Simp[2*g*(g*Cos[e + f*x])^(p - 1)*((a + b*Sin[e + f *x])^(m + 1)/(b*f*(2*m + p + 1))), x] + Simp[g^2*((p - 1)/(b^2*(2*m + p + 1 ))) Int[(g*Cos[e + f*x])^(p - 2)*(a + b*Sin[e + f*x])^(m + 2), x], x] /; FreeQ[{a, b, e, f, g}, x] && EqQ[a^2 - b^2, 0] && LeQ[m, -2] && GtQ[p, 1] & & NeQ[2*m + p + 1, 0] && !ILtQ[m + p + 1, 0] && IntegersQ[2*m, 2*p]
Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x _)])^(m_), x_Symbol] :> Simp[b*(g*Cos[e + f*x])^(p + 1)*((a + b*Sin[e + f*x ])^m/(a*f*g*(2*m + p + 1))), x] + Simp[(m + p + 1)/(a*(2*m + p + 1)) Int[ (g*Cos[e + f*x])^p*(a + b*Sin[e + f*x])^(m + 1), x], x] /; FreeQ[{a, b, e, f, g, m, p}, x] && EqQ[a^2 - b^2, 0] && LtQ[m, -1] && NeQ[2*m + p + 1, 0] & & IntegersQ[2*m, 2*p]
Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)/((a_) + (b_.)*sin[(e_.) + (f_.)*(x _)]), x_Symbol] :> Simp[b*((g*Cos[e + f*x])^(p + 1)/(a*f*g*(p - 1)*(a + b*S in[e + f*x]))), x] + Simp[p/(a*(p - 1)) Int[(g*Cos[e + f*x])^p, x], x] /; FreeQ[{a, b, e, f, g, p}, x] && EqQ[a^2 - b^2, 0] && !GeQ[p, 1] && Intege rQ[2*p]
Leaf count of result is larger than twice the leaf count of optimal. \(513\) vs. \(2(177)=354\).
Time = 13.96 (sec) , antiderivative size = 514, normalized size of antiderivative = 3.04
| method | result | size |
| default | \(-\frac {2 \left (96 \left (\sin ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )-48 E\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \left (\sin ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-192 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+96 E\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \left (\sin ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+272 \left (\sin ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )-72 \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, E\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-176 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+24 \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, E\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+144 \left (\sin ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-42 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )-3 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, E\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )-144 \left (\sin ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-4 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{4}}{45 \left (16 \left (\sin ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-32 \left (\sin ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+24 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-8 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+1\right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {-2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) e +e}\, e^{5} d}\) | \(514\) |
| parts | \(\text {Expression too large to display}\) | \(1310\) |
-2/45/(16*sin(1/2*d*x+1/2*c)^8-32*sin(1/2*d*x+1/2*c)^6+24*sin(1/2*d*x+1/2* c)^4-8*sin(1/2*d*x+1/2*c)^2+1)/sin(1/2*d*x+1/2*c)/(-2*sin(1/2*d*x+1/2*c)^2 *e+e)^(1/2)/e^5*(96*sin(1/2*d*x+1/2*c)^10*cos(1/2*d*x+1/2*c)-48*EllipticE( cos(1/2*d*x+1/2*c),2^(1/2))*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*(sin(1/2*d*x+ 1/2*c)^2)^(1/2)*sin(1/2*d*x+1/2*c)^8-192*cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/ 2*c)^8+96*EllipticE(cos(1/2*d*x+1/2*c),2^(1/2))*(2*sin(1/2*d*x+1/2*c)^2-1) ^(1/2)*(sin(1/2*d*x+1/2*c)^2)^(1/2)*sin(1/2*d*x+1/2*c)^6+272*sin(1/2*d*x+1 /2*c)^6*cos(1/2*d*x+1/2*c)-72*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*EllipticE(c os(1/2*d*x+1/2*c),2^(1/2))*(sin(1/2*d*x+1/2*c)^2)^(1/2)*sin(1/2*d*x+1/2*c) ^4-176*sin(1/2*d*x+1/2*c)^4*cos(1/2*d*x+1/2*c)+24*(2*sin(1/2*d*x+1/2*c)^2- 1)^(1/2)*EllipticE(cos(1/2*d*x+1/2*c),2^(1/2))*(sin(1/2*d*x+1/2*c)^2)^(1/2 )*sin(1/2*d*x+1/2*c)^2+144*sin(1/2*d*x+1/2*c)^5-42*sin(1/2*d*x+1/2*c)^2*co s(1/2*d*x+1/2*c)-3*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(2*sin(1/2*d*x+1/2*c)^2-1) ^(1/2)*EllipticE(cos(1/2*d*x+1/2*c),2^(1/2))-144*sin(1/2*d*x+1/2*c)^3-4*si n(1/2*d*x+1/2*c))*a^4/d
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.12 (sec) , antiderivative size = 436, normalized size of antiderivative = 2.58 \[ \int \frac {(a+a \sin (c+d x))^4}{(e \cos (c+d x))^{11/2}} \, dx=\frac {3 \, {\left (i \, \sqrt {2} a^{4} \cos \left (d x + c\right )^{3} + 3 i \, \sqrt {2} a^{4} \cos \left (d x + c\right )^{2} - 2 i \, \sqrt {2} a^{4} \cos \left (d x + c\right ) - 4 i \, \sqrt {2} a^{4} + {\left (-i \, \sqrt {2} a^{4} \cos \left (d x + c\right )^{2} + 2 i \, \sqrt {2} a^{4} \cos \left (d x + c\right ) + 4 i \, \sqrt {2} a^{4}\right )} \sin \left (d x + c\right )\right )} \sqrt {e} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right )\right ) + 3 \, {\left (-i \, \sqrt {2} a^{4} \cos \left (d x + c\right )^{3} - 3 i \, \sqrt {2} a^{4} \cos \left (d x + c\right )^{2} + 2 i \, \sqrt {2} a^{4} \cos \left (d x + c\right ) + 4 i \, \sqrt {2} a^{4} + {\left (i \, \sqrt {2} a^{4} \cos \left (d x + c\right )^{2} - 2 i \, \sqrt {2} a^{4} \cos \left (d x + c\right ) - 4 i \, \sqrt {2} a^{4}\right )} \sin \left (d x + c\right )\right )} \sqrt {e} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right )\right ) - 2 \, {\left (3 \, a^{4} \cos \left (d x + c\right )^{3} - 6 \, a^{4} \cos \left (d x + c\right )^{2} + a^{4} \cos \left (d x + c\right ) + 10 \, a^{4} + {\left (3 \, a^{4} \cos \left (d x + c\right )^{2} + 9 \, a^{4} \cos \left (d x + c\right ) + 10 \, a^{4}\right )} \sin \left (d x + c\right )\right )} \sqrt {e \cos \left (d x + c\right )}}{45 \, {\left (d e^{6} \cos \left (d x + c\right )^{3} + 3 \, d e^{6} \cos \left (d x + c\right )^{2} - 2 \, d e^{6} \cos \left (d x + c\right ) - 4 \, d e^{6} - {\left (d e^{6} \cos \left (d x + c\right )^{2} - 2 \, d e^{6} \cos \left (d x + c\right ) - 4 \, d e^{6}\right )} \sin \left (d x + c\right )\right )}} \]
1/45*(3*(I*sqrt(2)*a^4*cos(d*x + c)^3 + 3*I*sqrt(2)*a^4*cos(d*x + c)^2 - 2 *I*sqrt(2)*a^4*cos(d*x + c) - 4*I*sqrt(2)*a^4 + (-I*sqrt(2)*a^4*cos(d*x + c)^2 + 2*I*sqrt(2)*a^4*cos(d*x + c) + 4*I*sqrt(2)*a^4)*sin(d*x + c))*sqrt( e)*weierstrassZeta(-4, 0, weierstrassPInverse(-4, 0, cos(d*x + c) + I*sin( d*x + c))) + 3*(-I*sqrt(2)*a^4*cos(d*x + c)^3 - 3*I*sqrt(2)*a^4*cos(d*x + c)^2 + 2*I*sqrt(2)*a^4*cos(d*x + c) + 4*I*sqrt(2)*a^4 + (I*sqrt(2)*a^4*cos (d*x + c)^2 - 2*I*sqrt(2)*a^4*cos(d*x + c) - 4*I*sqrt(2)*a^4)*sin(d*x + c) )*sqrt(e)*weierstrassZeta(-4, 0, weierstrassPInverse(-4, 0, cos(d*x + c) - I*sin(d*x + c))) - 2*(3*a^4*cos(d*x + c)^3 - 6*a^4*cos(d*x + c)^2 + a^4*c os(d*x + c) + 10*a^4 + (3*a^4*cos(d*x + c)^2 + 9*a^4*cos(d*x + c) + 10*a^4 )*sin(d*x + c))*sqrt(e*cos(d*x + c)))/(d*e^6*cos(d*x + c)^3 + 3*d*e^6*cos( d*x + c)^2 - 2*d*e^6*cos(d*x + c) - 4*d*e^6 - (d*e^6*cos(d*x + c)^2 - 2*d* e^6*cos(d*x + c) - 4*d*e^6)*sin(d*x + c))
Timed out. \[ \int \frac {(a+a \sin (c+d x))^4}{(e \cos (c+d x))^{11/2}} \, dx=\text {Timed out} \]
\[ \int \frac {(a+a \sin (c+d x))^4}{(e \cos (c+d x))^{11/2}} \, dx=\int { \frac {{\left (a \sin \left (d x + c\right ) + a\right )}^{4}}{\left (e \cos \left (d x + c\right )\right )^{\frac {11}{2}}} \,d x } \]
\[ \int \frac {(a+a \sin (c+d x))^4}{(e \cos (c+d x))^{11/2}} \, dx=\int { \frac {{\left (a \sin \left (d x + c\right ) + a\right )}^{4}}{\left (e \cos \left (d x + c\right )\right )^{\frac {11}{2}}} \,d x } \]
Timed out. \[ \int \frac {(a+a \sin (c+d x))^4}{(e \cos (c+d x))^{11/2}} \, dx=\int \frac {{\left (a+a\,\sin \left (c+d\,x\right )\right )}^4}{{\left (e\,\cos \left (c+d\,x\right )\right )}^{11/2}} \,d x \]