\(\int \frac {\cos ^6(e+f x)}{\sqrt {d \sin (e+f x)} (a+b \sin (e+f x))^{13/2}} \, dx\) [1276]

Optimal result

Integrand size = 35, antiderivative size = 712 \[ \int \frac {\cos ^6(e+f x)}{\sqrt {d \sin (e+f x)} (a+b \sin (e+f x))^{13/2}} \, dx=\frac {2 \cos ^5(e+f x) \sqrt {d \sin (e+f x)}}{11 a d f (a+b \sin (e+f x))^{11/2}}-\frac {20 \left (a^2-b^2\right ) \cos (e+f x) \sqrt {d \sin (e+f x)}}{99 a^2 b^2 d f (a+b \sin (e+f x))^{9/2}}+\frac {80 \left (3 a^2+2 b^2\right ) \cos (e+f x) \sqrt {d \sin (e+f x)}}{693 a^3 b^2 d f (a+b \sin (e+f x))^{7/2}}-\frac {4 \left (5 a^4-17 a^2 b^2+16 b^4\right ) \cos (e+f x) \sqrt {d \sin (e+f x)}}{231 a^4 b^2 \left (a^2-b^2\right ) d f (a+b \sin (e+f x))^{5/2}}-\frac {8 \left (5 a^6-22 a^4 b^2+65 a^2 b^4-32 b^6\right ) \cos (e+f x) \sqrt {d \sin (e+f x)}}{693 a^5 b^2 \left (a^2-b^2\right )^2 d f (a+b \sin (e+f x))^{3/2}}+\frac {16 b \left (93 a^4-93 a^2 b^2+32 b^4\right ) \cos (e+f x)}{693 a^5 \left (a^2-b^2\right )^3 f \sqrt {d \sin (e+f x)} \sqrt {a+b \sin (e+f x)}}-\frac {16 b \left (93 a^4-93 a^2 b^2+32 b^4\right ) \sqrt {\frac {a (1-\csc (e+f x))}{a+b}} \sqrt {\frac {a (1+\csc (e+f x))}{a-b}} E\left (\arcsin \left (\frac {\sqrt {d} \sqrt {a+b \sin (e+f x)}}{\sqrt {a+b} \sqrt {d \sin (e+f x)}}\right )|-\frac {a+b}{a-b}\right ) \tan (e+f x)}{693 a^7 (a-b)^2 (a+b)^{5/2} \sqrt {d} f}-\frac {16 \left (45 a^4-48 a^3 b-69 a^2 b^2+24 a b^3+32 b^4\right ) \sqrt {\frac {a (1-\csc (e+f x))}{a+b}} \sqrt {\frac {a (1+\csc (e+f x))}{a-b}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {d} \sqrt {a+b \sin (e+f x)}}{\sqrt {a+b} \sqrt {d \sin (e+f x)}}\right ),-\frac {a+b}{a-b}\right ) \tan (e+f x)}{693 a^6 (a-b)^2 (a+b)^{5/2} \sqrt {d} f} \]

[Out]

2/11*cos(f*x+e)^5*(d*sin(f*x+e))^(1/2)/a/d/f/(a+b*sin(f*x+e))^(11/2)-20/99*(a^2-b^2)*cos(f*x+e)*(d*sin(f*x+e))
^(1/2)/a^2/b^2/d/f/(a+b*sin(f*x+e))^(9/2)+80/693*(3*a^2+2*b^2)*cos(f*x+e)*(d*sin(f*x+e))^(1/2)/a^3/b^2/d/f/(a+
b*sin(f*x+e))^(7/2)-4/231*(5*a^4-17*a^2*b^2+16*b^4)*cos(f*x+e)*(d*sin(f*x+e))^(1/2)/a^4/b^2/(a^2-b^2)/d/f/(a+b
*sin(f*x+e))^(5/2)-8/693*(5*a^6-22*a^4*b^2+65*a^2*b^4-32*b^6)*cos(f*x+e)*(d*sin(f*x+e))^(1/2)/a^5/b^2/(a^2-b^2
)^2/d/f/(a+b*sin(f*x+e))^(3/2)+16/693*b*(93*a^4-93*a^2*b^2+32*b^4)*cos(f*x+e)/a^5/(a^2-b^2)^3/f/(d*sin(f*x+e))
^(1/2)/(a+b*sin(f*x+e))^(1/2)-16/693*b*(93*a^4-93*a^2*b^2+32*b^4)*EllipticE(d^(1/2)*(a+b*sin(f*x+e))^(1/2)/(a+
b)^(1/2)/(d*sin(f*x+e))^(1/2),((-a-b)/(a-b))^(1/2))*(a*(1-csc(f*x+e))/(a+b))^(1/2)*(a*(1+csc(f*x+e))/(a-b))^(1
/2)*tan(f*x+e)/a^7/(a-b)^2/(a+b)^(5/2)/f/d^(1/2)-16/693*(45*a^4-48*a^3*b-69*a^2*b^2+24*a*b^3+32*b^4)*EllipticF
(d^(1/2)*(a+b*sin(f*x+e))^(1/2)/(a+b)^(1/2)/(d*sin(f*x+e))^(1/2),((-a-b)/(a-b))^(1/2))*(a*(1-csc(f*x+e))/(a+b)
)^(1/2)*(a*(1+csc(f*x+e))/(a-b))^(1/2)*tan(f*x+e)/a^6/(a-b)^2/(a+b)^(5/2)/f/d^(1/2)

Rubi [A] (verified)

Time = 1.82 (sec) , antiderivative size = 712, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {2966, 2970, 3134, 3072, 3077, 2895, 3073} \[ \int \frac {\cos ^6(e+f x)}{\sqrt {d \sin (e+f x)} (a+b \sin (e+f x))^{13/2}} \, dx=-\frac {20 \left (a^2-b^2\right ) \cos (e+f x) \sqrt {d \sin (e+f x)}}{99 a^2 b^2 d f (a+b \sin (e+f x))^{9/2}}-\frac {4 \left (5 a^4-17 a^2 b^2+16 b^4\right ) \cos (e+f x) \sqrt {d \sin (e+f x)}}{231 a^4 b^2 d f \left (a^2-b^2\right ) (a+b \sin (e+f x))^{5/2}}+\frac {80 \left (3 a^2+2 b^2\right ) \cos (e+f x) \sqrt {d \sin (e+f x)}}{693 a^3 b^2 d f (a+b \sin (e+f x))^{7/2}}-\frac {16 b \left (93 a^4-93 a^2 b^2+32 b^4\right ) \tan (e+f x) \sqrt {\frac {a (1-\csc (e+f x))}{a+b}} \sqrt {\frac {a (\csc (e+f x)+1)}{a-b}} E\left (\arcsin \left (\frac {\sqrt {d} \sqrt {a+b \sin (e+f x)}}{\sqrt {a+b} \sqrt {d \sin (e+f x)}}\right )|-\frac {a+b}{a-b}\right )}{693 a^7 \sqrt {d} f (a-b)^2 (a+b)^{5/2}}+\frac {16 b \left (93 a^4-93 a^2 b^2+32 b^4\right ) \cos (e+f x)}{693 a^5 f \left (a^2-b^2\right )^3 \sqrt {d \sin (e+f x)} \sqrt {a+b \sin (e+f x)}}-\frac {8 \left (5 a^6-22 a^4 b^2+65 a^2 b^4-32 b^6\right ) \cos (e+f x) \sqrt {d \sin (e+f x)}}{693 a^5 b^2 d f \left (a^2-b^2\right )^2 (a+b \sin (e+f x))^{3/2}}-\frac {16 \left (45 a^4-48 a^3 b-69 a^2 b^2+24 a b^3+32 b^4\right ) \tan (e+f x) \sqrt {\frac {a (1-\csc (e+f x))}{a+b}} \sqrt {\frac {a (\csc (e+f x)+1)}{a-b}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {d} \sqrt {a+b \sin (e+f x)}}{\sqrt {a+b} \sqrt {d \sin (e+f x)}}\right ),-\frac {a+b}{a-b}\right )}{693 a^6 \sqrt {d} f (a-b)^2 (a+b)^{5/2}}+\frac {2 \cos ^5(e+f x) \sqrt {d \sin (e+f x)}}{11 a d f (a+b \sin (e+f x))^{11/2}} \]

[In]

Int[Cos[e + f*x]^6/(Sqrt[d*Sin[e + f*x]]*(a + b*Sin[e + f*x])^(13/2)),x]

[Out]

(2*Cos[e + f*x]^5*Sqrt[d*Sin[e + f*x]])/(11*a*d*f*(a + b*Sin[e + f*x])^(11/2)) - (20*(a^2 - b^2)*Cos[e + f*x]*
Sqrt[d*Sin[e + f*x]])/(99*a^2*b^2*d*f*(a + b*Sin[e + f*x])^(9/2)) + (80*(3*a^2 + 2*b^2)*Cos[e + f*x]*Sqrt[d*Si
n[e + f*x]])/(693*a^3*b^2*d*f*(a + b*Sin[e + f*x])^(7/2)) - (4*(5*a^4 - 17*a^2*b^2 + 16*b^4)*Cos[e + f*x]*Sqrt
[d*Sin[e + f*x]])/(231*a^4*b^2*(a^2 - b^2)*d*f*(a + b*Sin[e + f*x])^(5/2)) - (8*(5*a^6 - 22*a^4*b^2 + 65*a^2*b
^4 - 32*b^6)*Cos[e + f*x]*Sqrt[d*Sin[e + f*x]])/(693*a^5*b^2*(a^2 - b^2)^2*d*f*(a + b*Sin[e + f*x])^(3/2)) + (
16*b*(93*a^4 - 93*a^2*b^2 + 32*b^4)*Cos[e + f*x])/(693*a^5*(a^2 - b^2)^3*f*Sqrt[d*Sin[e + f*x]]*Sqrt[a + b*Sin
[e + f*x]]) - (16*b*(93*a^4 - 93*a^2*b^2 + 32*b^4)*Sqrt[(a*(1 - Csc[e + f*x]))/(a + b)]*Sqrt[(a*(1 + Csc[e + f
*x]))/(a - b)]*EllipticE[ArcSin[(Sqrt[d]*Sqrt[a + b*Sin[e + f*x]])/(Sqrt[a + b]*Sqrt[d*Sin[e + f*x]])], -((a +
b)/(a - b))]*Tan[e + f*x])/(693*a^7*(a - b)^2*(a + b)^(5/2)*Sqrt[d]*f) - (16*(45*a^4 - 48*a^3*b - 69*a^2*b^2
+ 24*a*b^3 + 32*b^4)*Sqrt[(a*(1 - Csc[e + f*x]))/(a + b)]*Sqrt[(a*(1 + Csc[e + f*x]))/(a - b)]*EllipticF[ArcSi
n[(Sqrt[d]*Sqrt[a + b*Sin[e + f*x]])/(Sqrt[a + b]*Sqrt[d*Sin[e + f*x]])], -((a + b)/(a - b))]*Tan[e + f*x])/(6
93*a^6*(a - b)^2*(a + b)^(5/2)*Sqrt[d]*f)

Rule 2895

Int[1/(Sqrt[(d_.)*sin[(e_.) + (f_.)*(x_)]]*Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]]), x_Symbol] :> Simp[-2*(
Tan[e + f*x]/(a*f))*Rt[(a + b)/d, 2]*Sqrt[a*((1 - Csc[e + f*x])/(a + b))]*Sqrt[a*((1 + Csc[e + f*x])/(a - b))]
*EllipticF[ArcSin[Sqrt[a + b*Sin[e + f*x]]/Sqrt[d*Sin[e + f*x]]/Rt[(a + b)/d, 2]], -(a + b)/(a - b)], x] /; Fr
eeQ[{a, b, d, e, f}, x] && NeQ[a^2 - b^2, 0] && PosQ[(a + b)/d]

Rule 2966

Int[((cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_))/Sqrt[(d_.)*sin[(e_.) +
(f_.)*(x_)]], x_Symbol] :> Simp[(-g)*(g*Cos[e + f*x])^(p - 1)*Sqrt[d*Sin[e + f*x]]*((a + b*Sin[e + f*x])^(m +
1)/(a*d*f*(m + 1))), x] + Dist[g^2*((2*m + 3)/(2*a*(m + 1))), Int[(g*Cos[e + f*x])^(p - 2)*((a + b*Sin[e + f*x
])^(m + 1)/Sqrt[d*Sin[e + f*x]]), x], x] /; FreeQ[{a, b, d, e, f, g}, x] && NeQ[a^2 - b^2, 0] && LtQ[m, -1] &&
EqQ[m + p + 1/2, 0]

Rule 2970

Int[cos[(e_.) + (f_.)*(x_)]^4*((d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)
, x_Symbol] :> Simp[(a^2 - b^2)*Cos[e + f*x]*(a + b*Sin[e + f*x])^(m + 1)*((d*Sin[e + f*x])^(n + 1)/(a*b^2*d*f
*(m + 1))), x] + (-Dist[1/(a^2*b^2*(m + 1)*(m + 2)), Int[(a + b*Sin[e + f*x])^(m + 2)*(d*Sin[e + f*x])^n*Simp[
a^2*(n + 1)*(n + 3) - b^2*(m + n + 2)*(m + n + 3) + a*b*(m + 2)*Sin[e + f*x] - (a^2*(n + 2)*(n + 3) - b^2*(m +
n + 2)*(m + n + 4))*Sin[e + f*x]^2, x], x], x] + Simp[(a^2*(n - m + 1) - b^2*(m + n + 2))*Cos[e + f*x]*(a + b
*Sin[e + f*x])^(m + 2)*((d*Sin[e + f*x])^(n + 1)/(a^2*b^2*d*f*(m + 1)*(m + 2))), x]) /; FreeQ[{a, b, d, e, f,
n}, x] && NeQ[a^2 - b^2, 0] && IntegersQ[2*m, 2*n] && LtQ[m, -1] &&  !LtQ[n, -1] && (LtQ[m, -2] || EqQ[m + n +
4, 0])

Rule 3072

Int[((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])/(Sqrt[(d_.)*sin[(e_.) + (f_.)*(x_)]]*((a_) + (b_.)*sin[(e_.) + (f_
.)*(x_)])^(3/2)), x_Symbol] :> Simp[2*(A*b - a*B)*(Cos[e + f*x]/(f*(a^2 - b^2)*Sqrt[a + b*Sin[e + f*x]]*Sqrt[d
*Sin[e + f*x]])), x] + Dist[d/(a^2 - b^2), Int[(A*b - a*B + (a*A - b*B)*Sin[e + f*x])/(Sqrt[a + b*Sin[e + f*x]
]*(d*Sin[e + f*x])^(3/2)), x], x] /; FreeQ[{a, b, d, e, f, A, B}, x] && NeQ[a^2 - b^2, 0]

Rule 3073

Int[((A_) + (B_.)*sin[(e_.) + (f_.)*(x_)])/(((b_.)*sin[(e_.) + (f_.)*(x_)])^(3/2)*Sqrt[(c_) + (d_.)*sin[(e_.)
+ (f_.)*(x_)]]), x_Symbol] :> Simp[-2*A*(c - d)*(Tan[e + f*x]/(f*b*c^2))*Rt[(c + d)/b, 2]*Sqrt[c*((1 + Csc[e +
f*x])/(c - d))]*Sqrt[c*((1 - Csc[e + f*x])/(c + d))]*EllipticE[ArcSin[Sqrt[c + d*Sin[e + f*x]]/Sqrt[b*Sin[e +
f*x]]/Rt[(c + d)/b, 2]], -(c + d)/(c - d)], x] /; FreeQ[{b, c, d, e, f, A, B}, x] && NeQ[c^2 - d^2, 0] && EqQ
[A, B] && PosQ[(c + d)/b]

Rule 3077

Int[((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])/(((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(3/2)*Sqrt[(c_) + (d_.)*s
in[(e_.) + (f_.)*(x_)]]), x_Symbol] :> Dist[(A - B)/(a - b), Int[1/(Sqrt[a + b*Sin[e + f*x]]*Sqrt[c + d*Sin[e
+ f*x]]), x], x] - Dist[(A*b - a*B)/(a - b), Int[(1 + Sin[e + f*x])/((a + b*Sin[e + f*x])^(3/2)*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] && NeQ[a^2 - b^2, 0] && NeQ[c^2
- d^2, 0] && NeQ[A, B]

Rule 3134

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((A_.) + (B_.)*s
in[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(-(A*b^2 - a*b*B + a^2*C))*Cos[e
+ f*x]*(a + b*Sin[e + f*x])^(m + 1)*((c + d*Sin[e + f*x])^(n + 1)/(f*(m + 1)*(b*c - a*d)*(a^2 - b^2))), x] + D
ist[1/((m + 1)*(b*c - a*d)*(a^2 - b^2)), Int[(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^n*Simp[(m + 1)*
(b*c - a*d)*(a*A - b*B + a*C) + d*(A*b^2 - a*b*B + a^2*C)*(m + n + 2) - (c*(A*b^2 - a*b*B + a^2*C) + (m + 1)*(
b*c - a*d)*(A*b - a*B + b*C))*Sin[e + f*x] - d*(A*b^2 - a*b*B + a^2*C)*(m + n + 3)*Sin[e + f*x]^2, x], x], x]
/; FreeQ[{a, b, c, d, e, f, A, B, C, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] &&
LtQ[m, -1] && ((EqQ[a, 0] && IntegerQ[m] &&  !IntegerQ[n]) ||  !(IntegerQ[2*n] && LtQ[n, -1] && ((IntegerQ[n]
&&  !IntegerQ[m]) || EqQ[a, 0])))

Rubi steps \begin{align*} \text {integral}& = \frac {2 \cos ^5(e+f x) \sqrt {d \sin (e+f x)}}{11 a d f (a+b \sin (e+f x))^{11/2}}+\frac {10 \int \frac {\cos ^4(e+f x)}{\sqrt {d \sin (e+f x)} (a+b \sin (e+f x))^{11/2}} \, dx}{11 a} \\ & = \frac {2 \cos ^5(e+f x) \sqrt {d \sin (e+f x)}}{11 a d f (a+b \sin (e+f x))^{11/2}}-\frac {20 \left (a^2-b^2\right ) \cos (e+f x) \sqrt {d \sin (e+f x)}}{99 a^2 b^2 d f (a+b \sin (e+f x))^{9/2}}+\frac {80 \left (3 a^2+2 b^2\right ) \cos (e+f x) \sqrt {d \sin (e+f x)}}{693 a^3 b^2 d f (a+b \sin (e+f x))^{7/2}}-\frac {40 \int \frac {\frac {1}{4} \left (5 a^2-48 b^2\right )-\frac {7}{2} a b \sin (e+f x)-\frac {1}{4} \left (15 a^2-32 b^2\right ) \sin ^2(e+f x)}{\sqrt {d \sin (e+f x)} (a+b \sin (e+f x))^{7/2}} \, dx}{693 a^3 b^2} \\ & = \frac {2 \cos ^5(e+f x) \sqrt {d \sin (e+f x)}}{11 a d f (a+b \sin (e+f x))^{11/2}}-\frac {20 \left (a^2-b^2\right ) \cos (e+f x) \sqrt {d \sin (e+f x)}}{99 a^2 b^2 d f (a+b \sin (e+f x))^{9/2}}+\frac {80 \left (3 a^2+2 b^2\right ) \cos (e+f x) \sqrt {d \sin (e+f x)}}{693 a^3 b^2 d f (a+b \sin (e+f x))^{7/2}}-\frac {4 \left (5 a^4-17 a^2 b^2+16 b^4\right ) \cos (e+f x) \sqrt {d \sin (e+f x)}}{231 a^4 b^2 \left (a^2-b^2\right ) d f (a+b \sin (e+f x))^{5/2}}-\frac {16 \int \frac {\frac {1}{4} \left (5 a^4-107 a^2 b^2+96 b^4\right ) d-\frac {5}{2} a b \left (a^2-4 b^2\right ) d \sin (e+f x)-\frac {3}{4} \left (5 a^4-17 a^2 b^2+16 b^4\right ) d \sin ^2(e+f x)}{\sqrt {d \sin (e+f x)} (a+b \sin (e+f x))^{5/2}} \, dx}{693 a^4 b^2 \left (a^2-b^2\right ) d} \\ & = \frac {2 \cos ^5(e+f x) \sqrt {d \sin (e+f x)}}{11 a d f (a+b \sin (e+f x))^{11/2}}-\frac {20 \left (a^2-b^2\right ) \cos (e+f x) \sqrt {d \sin (e+f x)}}{99 a^2 b^2 d f (a+b \sin (e+f x))^{9/2}}+\frac {80 \left (3 a^2+2 b^2\right ) \cos (e+f x) \sqrt {d \sin (e+f x)}}{693 a^3 b^2 d f (a+b \sin (e+f x))^{7/2}}-\frac {4 \left (5 a^4-17 a^2 b^2+16 b^4\right ) \cos (e+f x) \sqrt {d \sin (e+f x)}}{231 a^4 b^2 \left (a^2-b^2\right ) d f (a+b \sin (e+f x))^{5/2}}-\frac {8 \left (5 a^6-22 a^4 b^2+65 a^2 b^4-32 b^6\right ) \cos (e+f x) \sqrt {d \sin (e+f x)}}{693 a^5 b^2 \left (a^2-b^2\right )^2 d f (a+b \sin (e+f x))^{3/2}}-\frac {32 \int \frac {-\frac {3}{4} b^2 \left (45 a^4-69 a^2 b^2+32 b^4\right ) d^2+18 a b^3 \left (2 a^2-b^2\right ) d^2 \sin (e+f x)}{\sqrt {d \sin (e+f x)} (a+b \sin (e+f x))^{3/2}} \, dx}{2079 a^5 b^2 \left (a^2-b^2\right )^2 d^2} \\ & = \frac {2 \cos ^5(e+f x) \sqrt {d \sin (e+f x)}}{11 a d f (a+b \sin (e+f x))^{11/2}}-\frac {20 \left (a^2-b^2\right ) \cos (e+f x) \sqrt {d \sin (e+f x)}}{99 a^2 b^2 d f (a+b \sin (e+f x))^{9/2}}+\frac {80 \left (3 a^2+2 b^2\right ) \cos (e+f x) \sqrt {d \sin (e+f x)}}{693 a^3 b^2 d f (a+b \sin (e+f x))^{7/2}}-\frac {4 \left (5 a^4-17 a^2 b^2+16 b^4\right ) \cos (e+f x) \sqrt {d \sin (e+f x)}}{231 a^4 b^2 \left (a^2-b^2\right ) d f (a+b \sin (e+f x))^{5/2}}-\frac {8 \left (5 a^6-22 a^4 b^2+65 a^2 b^4-32 b^6\right ) \cos (e+f x) \sqrt {d \sin (e+f x)}}{693 a^5 b^2 \left (a^2-b^2\right )^2 d f (a+b \sin (e+f x))^{3/2}}+\frac {16 b \left (93 a^4-93 a^2 b^2+32 b^4\right ) \cos (e+f x)}{693 a^5 \left (a^2-b^2\right )^3 f \sqrt {d \sin (e+f x)} \sqrt {a+b \sin (e+f x)}}-\frac {32 \int \frac {-18 a^2 b^3 \left (2 a^2-b^2\right ) d^2-\frac {3}{4} b^3 \left (45 a^4-69 a^2 b^2+32 b^4\right ) d^2+\left (-18 a b^4 \left (2 a^2-b^2\right ) d^2-\frac {3}{4} a b^2 \left (45 a^4-69 a^2 b^2+32 b^4\right ) d^2\right ) \sin (e+f x)}{(d \sin (e+f x))^{3/2} \sqrt {a+b \sin (e+f x)}} \, dx}{2079 a^5 b^2 \left (a^2-b^2\right )^3 d} \\ & = \frac {2 \cos ^5(e+f x) \sqrt {d \sin (e+f x)}}{11 a d f (a+b \sin (e+f x))^{11/2}}-\frac {20 \left (a^2-b^2\right ) \cos (e+f x) \sqrt {d \sin (e+f x)}}{99 a^2 b^2 d f (a+b \sin (e+f x))^{9/2}}+\frac {80 \left (3 a^2+2 b^2\right ) \cos (e+f x) \sqrt {d \sin (e+f x)}}{693 a^3 b^2 d f (a+b \sin (e+f x))^{7/2}}-\frac {4 \left (5 a^4-17 a^2 b^2+16 b^4\right ) \cos (e+f x) \sqrt {d \sin (e+f x)}}{231 a^4 b^2 \left (a^2-b^2\right ) d f (a+b \sin (e+f x))^{5/2}}-\frac {8 \left (5 a^6-22 a^4 b^2+65 a^2 b^4-32 b^6\right ) \cos (e+f x) \sqrt {d \sin (e+f x)}}{693 a^5 b^2 \left (a^2-b^2\right )^2 d f (a+b \sin (e+f x))^{3/2}}+\frac {16 b \left (93 a^4-93 a^2 b^2+32 b^4\right ) \cos (e+f x)}{693 a^5 \left (a^2-b^2\right )^3 f \sqrt {d \sin (e+f x)} \sqrt {a+b \sin (e+f x)}}+\frac {\left (8 \left (45 a^4-48 a^3 b-69 a^2 b^2+24 a b^3+32 b^4\right )\right ) \int \frac {1}{\sqrt {d \sin (e+f x)} \sqrt {a+b \sin (e+f x)}} \, dx}{693 a^5 (a-b)^2 (a+b)^3}+\frac {\left (8 b \left (93 a^4-93 a^2 b^2+32 b^4\right ) d\right ) \int \frac {1+\sin (e+f x)}{(d \sin (e+f x))^{3/2} \sqrt {a+b \sin (e+f x)}} \, dx}{693 a^5 \left (a^2-b^2\right )^3} \\ & = \frac {2 \cos ^5(e+f x) \sqrt {d \sin (e+f x)}}{11 a d f (a+b \sin (e+f x))^{11/2}}-\frac {20 \left (a^2-b^2\right ) \cos (e+f x) \sqrt {d \sin (e+f x)}}{99 a^2 b^2 d f (a+b \sin (e+f x))^{9/2}}+\frac {80 \left (3 a^2+2 b^2\right ) \cos (e+f x) \sqrt {d \sin (e+f x)}}{693 a^3 b^2 d f (a+b \sin (e+f x))^{7/2}}-\frac {4 \left (5 a^4-17 a^2 b^2+16 b^4\right ) \cos (e+f x) \sqrt {d \sin (e+f x)}}{231 a^4 b^2 \left (a^2-b^2\right ) d f (a+b \sin (e+f x))^{5/2}}-\frac {8 \left (5 a^6-22 a^4 b^2+65 a^2 b^4-32 b^6\right ) \cos (e+f x) \sqrt {d \sin (e+f x)}}{693 a^5 b^2 \left (a^2-b^2\right )^2 d f (a+b \sin (e+f x))^{3/2}}+\frac {16 b \left (93 a^4-93 a^2 b^2+32 b^4\right ) \cos (e+f x)}{693 a^5 \left (a^2-b^2\right )^3 f \sqrt {d \sin (e+f x)} \sqrt {a+b \sin (e+f x)}}-\frac {16 b \left (93 a^4-93 a^2 b^2+32 b^4\right ) \sqrt {\frac {a (1-\csc (e+f x))}{a+b}} \sqrt {\frac {a (1+\csc (e+f x))}{a-b}} E\left (\arcsin \left (\frac {\sqrt {d} \sqrt {a+b \sin (e+f x)}}{\sqrt {a+b} \sqrt {d \sin (e+f x)}}\right )|-\frac {a+b}{a-b}\right ) \tan (e+f x)}{693 a^7 (a-b)^2 (a+b)^{5/2} \sqrt {d} f}-\frac {16 \left (45 a^4-48 a^3 b-69 a^2 b^2+24 a b^3+32 b^4\right ) \sqrt {\frac {a (1-\csc (e+f x))}{a+b}} \sqrt {\frac {a (1+\csc (e+f x))}{a-b}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {d} \sqrt {a+b \sin (e+f x)}}{\sqrt {a+b} \sqrt {d \sin (e+f x)}}\right ),-\frac {a+b}{a-b}\right ) \tan (e+f x)}{693 a^6 (a-b)^2 (a+b)^{5/2} \sqrt {d} f} \\ \end{align*}

Mathematica [C] (warning: unable to verify)

Result contains complex when optimal does not.

Time = 8.57 (sec) , antiderivative size = 1906, normalized size of antiderivative = 2.68 \[ \int \frac {\cos ^6(e+f x)}{\sqrt {d \sin (e+f x)} (a+b \sin (e+f x))^{13/2}} \, dx=\frac {\sin (e+f x) \sqrt {a+b \sin (e+f x)} \left (\frac {2 \left (a^4 \cos (e+f x)-2 a^2 b^2 \cos (e+f x)+b^4 \cos (e+f x)\right )}{11 a b^4 (a+b \sin (e+f x))^6}-\frac {4 \left (18 a^4 \cos (e+f x)-13 a^2 b^2 \cos (e+f x)-5 b^4 \cos (e+f x)\right )}{99 a^2 b^4 (a+b \sin (e+f x))^5}+\frac {4 \left (189 a^4 \cos (e+f x)-3 a^2 b^2 \cos (e+f x)+40 b^4 \cos (e+f x)\right )}{693 a^3 b^4 (a+b \sin (e+f x))^4}-\frac {4 \left (42 a^6 \cos (e+f x)-37 a^4 b^2 \cos (e+f x)-17 a^2 b^4 \cos (e+f x)+16 b^6 \cos (e+f x)\right )}{231 a^4 b^4 \left (a^2-b^2\right ) (a+b \sin (e+f x))^3}+\frac {2 \left (63 a^8 \cos (e+f x)-146 a^6 b^2 \cos (e+f x)+151 a^4 b^4 \cos (e+f x)-260 a^2 b^6 \cos (e+f x)+128 b^8 \cos (e+f x)\right )}{693 a^5 b^4 \left (a^2-b^2\right )^2 (a+b \sin (e+f x))^2}-\frac {16 \left (93 a^4 b^2 \cos (e+f x)-93 a^2 b^4 \cos (e+f x)+32 b^6 \cos (e+f x)\right )}{693 a^6 \left (a^2-b^2\right )^3 (a+b \sin (e+f x))}\right )}{f \sqrt {d \sin (e+f x)}}+\frac {8 \sqrt {\sin (e+f x)} \left (\frac {4 a \left (45 a^6-114 a^4 b^2+101 a^2 b^4-32 b^6\right ) \sqrt {\frac {(a+b) \cot ^2\left (\frac {1}{2} \left (-e+\frac {\pi }{2}-f x\right )\right )}{-a+b}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {\csc ^2\left (\frac {1}{2} \left (-e+\frac {\pi }{2}-f x\right )\right ) (a+b \sin (e+f x))}{a}}}{\sqrt {2}}\right ),-\frac {2 a}{-a+b}\right ) \sec (e+f x) \sin ^4\left (\frac {1}{2} \left (-e+\frac {\pi }{2}-f x\right )\right ) \sqrt {-\frac {(a+b) \csc ^2\left (\frac {1}{2} \left (-e+\frac {\pi }{2}-f x\right )\right ) \sin (e+f x)}{a}} \sqrt {\frac {\csc ^2\left (\frac {1}{2} \left (-e+\frac {\pi }{2}-f x\right )\right ) (a+b \sin (e+f x))}{a}}}{(a+b) \sqrt {\sin (e+f x)} \sqrt {a+b \sin (e+f x)}}+4 a \left (-93 a^5 b+93 a^3 b^3-32 a b^5\right ) \left (\frac {\sqrt {\frac {(a+b) \cot ^2\left (\frac {1}{2} \left (-e+\frac {\pi }{2}-f x\right )\right )}{-a+b}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {\csc ^2\left (\frac {1}{2} \left (-e+\frac {\pi }{2}-f x\right )\right ) (a+b \sin (e+f x))}{a}}}{\sqrt {2}}\right ),-\frac {2 a}{-a+b}\right ) \sec (e+f x) \sin ^4\left (\frac {1}{2} \left (-e+\frac {\pi }{2}-f x\right )\right ) \sqrt {-\frac {(a+b) \csc ^2\left (\frac {1}{2} \left (-e+\frac {\pi }{2}-f x\right )\right ) \sin (e+f x)}{a}} \sqrt {\frac {\csc ^2\left (\frac {1}{2} \left (-e+\frac {\pi }{2}-f x\right )\right ) (a+b \sin (e+f x))}{a}}}{(a+b) \sqrt {\sin (e+f x)} \sqrt {a+b \sin (e+f x)}}-\frac {\sqrt {\frac {(a+b) \cot ^2\left (\frac {1}{2} \left (-e+\frac {\pi }{2}-f x\right )\right )}{-a+b}} \operatorname {EllipticPi}\left (-\frac {a}{b},\arcsin \left (\frac {\sqrt {\frac {\csc ^2\left (\frac {1}{2} \left (-e+\frac {\pi }{2}-f x\right )\right ) (a+b \sin (e+f x))}{a}}}{\sqrt {2}}\right ),-\frac {2 a}{-a+b}\right ) \sec (e+f x) \sin ^4\left (\frac {1}{2} \left (-e+\frac {\pi }{2}-f x\right )\right ) \sqrt {-\frac {(a+b) \csc ^2\left (\frac {1}{2} \left (-e+\frac {\pi }{2}-f x\right )\right ) \sin (e+f x)}{a}} \sqrt {\frac {\csc ^2\left (\frac {1}{2} \left (-e+\frac {\pi }{2}-f x\right )\right ) (a+b \sin (e+f x))}{a}}}{b \sqrt {\sin (e+f x)} \sqrt {a+b \sin (e+f x)}}\right )+2 \left (93 a^4 b^2-93 a^2 b^4+32 b^6\right ) \left (\frac {\cos (e+f x) \sqrt {a+b \sin (e+f x)}}{b \sqrt {\sin (e+f x)}}+\frac {i \cos \left (\frac {1}{2} \left (-e+\frac {\pi }{2}-f x\right )\right ) \csc (e+f x) E\left (i \text {arcsinh}\left (\frac {\sin \left (\frac {1}{2} \left (-e+\frac {\pi }{2}-f x\right )\right )}{\sqrt {\sin (e+f x)}}\right )|-\frac {2 a}{-a-b}\right ) \sqrt {a+b \sin (e+f x)}}{b \sqrt {\cos ^2\left (\frac {1}{2} \left (-e+\frac {\pi }{2}-f x\right )\right ) \csc (e+f x)} \sqrt {\frac {\csc (e+f x) (a+b \sin (e+f x))}{a+b}}}+\frac {2 a \left (\frac {a \sqrt {\frac {(a+b) \cot ^2\left (\frac {1}{2} \left (-e+\frac {\pi }{2}-f x\right )\right )}{-a+b}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {\csc ^2\left (\frac {1}{2} \left (-e+\frac {\pi }{2}-f x\right )\right ) (a+b \sin (e+f x))}{a}}}{\sqrt {2}}\right ),-\frac {2 a}{-a+b}\right ) \sec (e+f x) \sin ^4\left (\frac {1}{2} \left (-e+\frac {\pi }{2}-f x\right )\right ) \sqrt {-\frac {(a+b) \csc ^2\left (\frac {1}{2} \left (-e+\frac {\pi }{2}-f x\right )\right ) \sin (e+f x)}{a}} \sqrt {\frac {\csc ^2\left (\frac {1}{2} \left (-e+\frac {\pi }{2}-f x\right )\right ) (a+b \sin (e+f x))}{a}}}{(a+b) \sqrt {\sin (e+f x)} \sqrt {a+b \sin (e+f x)}}-\frac {a \sqrt {\frac {(a+b) \cot ^2\left (\frac {1}{2} \left (-e+\frac {\pi }{2}-f x\right )\right )}{-a+b}} \operatorname {EllipticPi}\left (-\frac {a}{b},\arcsin \left (\frac {\sqrt {\frac {\csc ^2\left (\frac {1}{2} \left (-e+\frac {\pi }{2}-f x\right )\right ) (a+b \sin (e+f x))}{a}}}{\sqrt {2}}\right ),-\frac {2 a}{-a+b}\right ) \sec (e+f x) \sin ^4\left (\frac {1}{2} \left (-e+\frac {\pi }{2}-f x\right )\right ) \sqrt {-\frac {(a+b) \csc ^2\left (\frac {1}{2} \left (-e+\frac {\pi }{2}-f x\right )\right ) \sin (e+f x)}{a}} \sqrt {\frac {\csc ^2\left (\frac {1}{2} \left (-e+\frac {\pi }{2}-f x\right )\right ) (a+b \sin (e+f x))}{a}}}{b \sqrt {\sin (e+f x)} \sqrt {a+b \sin (e+f x)}}\right )}{b}\right )\right )}{693 a^6 (a-b)^3 (a+b)^3 f \sqrt {d \sin (e+f x)}} \]

[In]

Integrate[Cos[e + f*x]^6/(Sqrt[d*Sin[e + f*x]]*(a + b*Sin[e + f*x])^(13/2)),x]

[Out]

(Sin[e + f*x]*Sqrt[a + b*Sin[e + f*x]]*((2*(a^4*Cos[e + f*x] - 2*a^2*b^2*Cos[e + f*x] + b^4*Cos[e + f*x]))/(11
*a*b^4*(a + b*Sin[e + f*x])^6) - (4*(18*a^4*Cos[e + f*x] - 13*a^2*b^2*Cos[e + f*x] - 5*b^4*Cos[e + f*x]))/(99*
a^2*b^4*(a + b*Sin[e + f*x])^5) + (4*(189*a^4*Cos[e + f*x] - 3*a^2*b^2*Cos[e + f*x] + 40*b^4*Cos[e + f*x]))/(6
93*a^3*b^4*(a + b*Sin[e + f*x])^4) - (4*(42*a^6*Cos[e + f*x] - 37*a^4*b^2*Cos[e + f*x] - 17*a^2*b^4*Cos[e + f*
x] + 16*b^6*Cos[e + f*x]))/(231*a^4*b^4*(a^2 - b^2)*(a + b*Sin[e + f*x])^3) + (2*(63*a^8*Cos[e + f*x] - 146*a^
6*b^2*Cos[e + f*x] + 151*a^4*b^4*Cos[e + f*x] - 260*a^2*b^6*Cos[e + f*x] + 128*b^8*Cos[e + f*x]))/(693*a^5*b^4
*(a^2 - b^2)^2*(a + b*Sin[e + f*x])^2) - (16*(93*a^4*b^2*Cos[e + f*x] - 93*a^2*b^4*Cos[e + f*x] + 32*b^6*Cos[e
+ f*x]))/(693*a^6*(a^2 - b^2)^3*(a + b*Sin[e + f*x]))))/(f*Sqrt[d*Sin[e + f*x]]) + (8*Sqrt[Sin[e + f*x]]*((4*
a*(45*a^6 - 114*a^4*b^2 + 101*a^2*b^4 - 32*b^6)*Sqrt[((a + b)*Cot[(-e + Pi/2 - f*x)/2]^2)/(-a + b)]*EllipticF[
ArcSin[Sqrt[(Csc[(-e + Pi/2 - f*x)/2]^2*(a + b*Sin[e + f*x]))/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sec[e + f*x]*Sin[(
-e + Pi/2 - f*x)/2]^4*Sqrt[-(((a + b)*Csc[(-e + Pi/2 - f*x)/2]^2*Sin[e + f*x])/a)]*Sqrt[(Csc[(-e + Pi/2 - f*x)
/2]^2*(a + b*Sin[e + f*x]))/a])/((a + b)*Sqrt[Sin[e + f*x]]*Sqrt[a + b*Sin[e + f*x]]) + 4*a*(-93*a^5*b + 93*a^
3*b^3 - 32*a*b^5)*((Sqrt[((a + b)*Cot[(-e + Pi/2 - f*x)/2]^2)/(-a + b)]*EllipticF[ArcSin[Sqrt[(Csc[(-e + Pi/2
- f*x)/2]^2*(a + b*Sin[e + f*x]))/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sec[e + f*x]*Sin[(-e + Pi/2 - f*x)/2]^4*Sqrt[-
(((a + b)*Csc[(-e + Pi/2 - f*x)/2]^2*Sin[e + f*x])/a)]*Sqrt[(Csc[(-e + Pi/2 - f*x)/2]^2*(a + b*Sin[e + f*x]))/
a])/((a + b)*Sqrt[Sin[e + f*x]]*Sqrt[a + b*Sin[e + f*x]]) - (Sqrt[((a + b)*Cot[(-e + Pi/2 - f*x)/2]^2)/(-a + b
)]*EllipticPi[-(a/b), ArcSin[Sqrt[(Csc[(-e + Pi/2 - f*x)/2]^2*(a + b*Sin[e + f*x]))/a]/Sqrt[2]], (-2*a)/(-a +
b)]*Sec[e + f*x]*Sin[(-e + Pi/2 - f*x)/2]^4*Sqrt[-(((a + b)*Csc[(-e + Pi/2 - f*x)/2]^2*Sin[e + f*x])/a)]*Sqrt[
(Csc[(-e + Pi/2 - f*x)/2]^2*(a + b*Sin[e + f*x]))/a])/(b*Sqrt[Sin[e + f*x]]*Sqrt[a + b*Sin[e + f*x]])) + 2*(93
*a^4*b^2 - 93*a^2*b^4 + 32*b^6)*((Cos[e + f*x]*Sqrt[a + b*Sin[e + f*x]])/(b*Sqrt[Sin[e + f*x]]) + (I*Cos[(-e +
Pi/2 - f*x)/2]*Csc[e + f*x]*EllipticE[I*ArcSinh[Sin[(-e + Pi/2 - f*x)/2]/Sqrt[Sin[e + f*x]]], (-2*a)/(-a - b)
]*Sqrt[a + b*Sin[e + f*x]])/(b*Sqrt[Cos[(-e + Pi/2 - f*x)/2]^2*Csc[e + f*x]]*Sqrt[(Csc[e + f*x]*(a + b*Sin[e +
f*x]))/(a + b)]) + (2*a*((a*Sqrt[((a + b)*Cot[(-e + Pi/2 - f*x)/2]^2)/(-a + b)]*EllipticF[ArcSin[Sqrt[(Csc[(-
e + Pi/2 - f*x)/2]^2*(a + b*Sin[e + f*x]))/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sec[e + f*x]*Sin[(-e + Pi/2 - f*x)/2]
^4*Sqrt[-(((a + b)*Csc[(-e + Pi/2 - f*x)/2]^2*Sin[e + f*x])/a)]*Sqrt[(Csc[(-e + Pi/2 - f*x)/2]^2*(a + b*Sin[e
+ f*x]))/a])/((a + b)*Sqrt[Sin[e + f*x]]*Sqrt[a + b*Sin[e + f*x]]) - (a*Sqrt[((a + b)*Cot[(-e + Pi/2 - f*x)/2]
^2)/(-a + b)]*EllipticPi[-(a/b), ArcSin[Sqrt[(Csc[(-e + Pi/2 - f*x)/2]^2*(a + b*Sin[e + f*x]))/a]/Sqrt[2]], (-
2*a)/(-a + b)]*Sec[e + f*x]*Sin[(-e + Pi/2 - f*x)/2]^4*Sqrt[-(((a + b)*Csc[(-e + Pi/2 - f*x)/2]^2*Sin[e + f*x]
)/a)]*Sqrt[(Csc[(-e + Pi/2 - f*x)/2]^2*(a + b*Sin[e + f*x]))/a])/(b*Sqrt[Sin[e + f*x]]*Sqrt[a + b*Sin[e + f*x]
])))/b)))/(693*a^6*(a - b)^3*(a + b)^3*f*Sqrt[d*Sin[e + f*x]])

Maple [B] (warning: unable to verify)

Leaf count of result is larger than twice the leaf count of optimal. \(37794\) vs. \(2(648)=1296\).

Time = 15.30 (sec) , antiderivative size = 37795, normalized size of antiderivative = 53.08

method	result	size
default	\(\text {Expression too large to display}\)	\(37795\)

[In]

int(cos(f*x+e)^6/(a+b*sin(f*x+e))^(13/2)/(d*sin(f*x+e))^(1/2),x,method=_RETURNVERBOSE)

[Out]

result too large to display

Fricas [F]

\[ \int \frac {\cos ^6(e+f x)}{\sqrt {d \sin (e+f x)} (a+b \sin (e+f x))^{13/2}} \, dx=\int { \frac {\cos \left (f x + e\right )^{6}}{{\left (b \sin \left (f x + e\right ) + a\right )}^{\frac {13}{2}} \sqrt {d \sin \left (f x + e\right )}} \,d x } \]

[In]

integrate(cos(f*x+e)^6/(a+b*sin(f*x+e))^(13/2)/(d*sin(f*x+e))^(1/2),x, algorithm="fricas")

[Out]

integral(sqrt(b*sin(f*x + e) + a)*sqrt(d*sin(f*x + e))*cos(f*x + e)^6/(b^7*d*cos(f*x + e)^8 - (21*a^2*b^5 + 4*
b^7)*d*cos(f*x + e)^6 + (35*a^4*b^3 + 63*a^2*b^5 + 6*b^7)*d*cos(f*x + e)^4 - (7*a^6*b + 70*a^4*b^3 + 63*a^2*b^
5 + 4*b^7)*d*cos(f*x + e)^2 + (7*a^6*b + 35*a^4*b^3 + 21*a^2*b^5 + b^7)*d - (7*a*b^6*d*cos(f*x + e)^6 - 7*(5*a
^3*b^4 + 3*a*b^6)*d*cos(f*x + e)^4 + 7*(3*a^5*b^2 + 10*a^3*b^4 + 3*a*b^6)*d*cos(f*x + e)^2 - (a^7 + 21*a^5*b^2
+ 35*a^3*b^4 + 7*a*b^6)*d)*sin(f*x + e)), x)

Sympy [F(-1)]

Timed out. \[ \int \frac {\cos ^6(e+f x)}{\sqrt {d \sin (e+f x)} (a+b \sin (e+f x))^{13/2}} \, dx=\text {Timed out} \]

[In]

integrate(cos(f*x+e)**6/(a+b*sin(f*x+e))**(13/2)/(d*sin(f*x+e))**(1/2),x)

[Out]

Timed out

Maxima [F]

\[ \int \frac {\cos ^6(e+f x)}{\sqrt {d \sin (e+f x)} (a+b \sin (e+f x))^{13/2}} \, dx=\int { \frac {\cos \left (f x + e\right )^{6}}{{\left (b \sin \left (f x + e\right ) + a\right )}^{\frac {13}{2}} \sqrt {d \sin \left (f x + e\right )}} \,d x } \]

[In]

integrate(cos(f*x+e)^6/(a+b*sin(f*x+e))^(13/2)/(d*sin(f*x+e))^(1/2),x, algorithm="maxima")

[Out]

integrate(cos(f*x + e)^6/((b*sin(f*x + e) + a)^(13/2)*sqrt(d*sin(f*x + e))), x)

Giac [F(-1)]

Timed out. \[ \int \frac {\cos ^6(e+f x)}{\sqrt {d \sin (e+f x)} (a+b \sin (e+f x))^{13/2}} \, dx=\text {Timed out} \]

[In]

integrate(cos(f*x+e)^6/(a+b*sin(f*x+e))^(13/2)/(d*sin(f*x+e))^(1/2),x, algorithm="giac")

[Out]

Timed out

Mupad [F(-1)]

Timed out. \[ \int \frac {\cos ^6(e+f x)}{\sqrt {d \sin (e+f x)} (a+b \sin (e+f x))^{13/2}} \, dx=\int \frac {{\cos \left (e+f\,x\right )}^6}{\sqrt {d\,\sin \left (e+f\,x\right )}\,{\left (a+b\,\sin \left (e+f\,x\right )\right )}^{13/2}} \,d x \]

[In]

int(cos(e + f*x)^6/((d*sin(e + f*x))^(1/2)*(a + b*sin(e + f*x))^(13/2)),x)

[Out]

int(cos(e + f*x)^6/((d*sin(e + f*x))^(1/2)*(a + b*sin(e + f*x))^(13/2)), x)