3.1276 \(\int \frac{\cos ^6(e+f x)}{\sqrt{d \sin (e+f x)} (a+b \sin (e+f x))^{13/2}} \, dx\)

Optimal. Leaf size=712 \[ \frac{16 b \left (-93 a^2 b^2+93 a^4+32 b^4\right ) \cos (e+f x)}{693 a^5 f \left (a^2-b^2\right )^3 \sqrt{d \sin (e+f x)} \sqrt{a+b \sin (e+f x)}}-\frac{8 \left (-22 a^4 b^2+65 a^2 b^4+5 a^6-32 b^6\right ) \cos (e+f x) \sqrt{d \sin (e+f x)}}{693 a^5 b^2 d f \left (a^2-b^2\right )^2 (a+b \sin (e+f x))^{3/2}}-\frac{4 \left (-17 a^2 b^2+5 a^4+16 b^4\right ) \cos (e+f x) \sqrt{d \sin (e+f x)}}{231 a^4 b^2 d f \left (a^2-b^2\right ) (a+b \sin (e+f x))^{5/2}}+\frac{80 \left (3 a^2+2 b^2\right ) \cos (e+f x) \sqrt{d \sin (e+f x)}}{693 a^3 b^2 d f (a+b \sin (e+f x))^{7/2}}-\frac{20 \left (a^2-b^2\right ) \cos (e+f x) \sqrt{d \sin (e+f x)}}{99 a^2 b^2 d f (a+b \sin (e+f x))^{9/2}}-\frac{16 \left (-69 a^2 b^2-48 a^3 b+45 a^4+24 a b^3+32 b^4\right ) \tan (e+f x) \sqrt{\frac{a (1-\csc (e+f x))}{a+b}} \sqrt{\frac{a (\csc (e+f x)+1)}{a-b}} F\left (\sin ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b \sin (e+f x)}}{\sqrt{a+b} \sqrt{d \sin (e+f x)}}\right )|-\frac{a+b}{a-b}\right )}{693 a^6 \sqrt{d} f (a-b)^2 (a+b)^{5/2}}-\frac{16 b \left (-93 a^2 b^2+93 a^4+32 b^4\right ) \tan (e+f x) \sqrt{\frac{a (1-\csc (e+f x))}{a+b}} \sqrt{\frac{a (\csc (e+f x)+1)}{a-b}} E\left (\sin ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b \sin (e+f x)}}{\sqrt{a+b} \sqrt{d \sin (e+f x)}}\right )|-\frac{a+b}{a-b}\right )}{693 a^7 \sqrt{d} f (a-b)^2 (a+b)^{5/2}}+\frac{2 \cos ^5(e+f x) \sqrt{d \sin (e+f x)}}{11 a d f (a+b \sin (e+f x))^{11/2}} \]

[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)

________________________________________________________________________________________

Rubi [A]  time = 2.64462, antiderivative size = 712, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {2887, 2891, 3055, 2993, 2998, 2816, 2994} \[ \frac{16 b \left (-93 a^2 b^2+93 a^4+32 b^4\right ) \cos (e+f x)}{693 a^5 f \left (a^2-b^2\right )^3 \sqrt{d \sin (e+f x)} \sqrt{a+b \sin (e+f x)}}-\frac{8 \left (-22 a^4 b^2+65 a^2 b^4+5 a^6-32 b^6\right ) \cos (e+f x) \sqrt{d \sin (e+f x)}}{693 a^5 b^2 d f \left (a^2-b^2\right )^2 (a+b \sin (e+f x))^{3/2}}-\frac{4 \left (-17 a^2 b^2+5 a^4+16 b^4\right ) \cos (e+f x) \sqrt{d \sin (e+f x)}}{231 a^4 b^2 d f \left (a^2-b^2\right ) (a+b \sin (e+f x))^{5/2}}+\frac{80 \left (3 a^2+2 b^2\right ) \cos (e+f x) \sqrt{d \sin (e+f x)}}{693 a^3 b^2 d f (a+b \sin (e+f x))^{7/2}}-\frac{20 \left (a^2-b^2\right ) \cos (e+f x) \sqrt{d \sin (e+f x)}}{99 a^2 b^2 d f (a+b \sin (e+f x))^{9/2}}-\frac{16 \left (-69 a^2 b^2-48 a^3 b+45 a^4+24 a b^3+32 b^4\right ) \tan (e+f x) \sqrt{\frac{a (1-\csc (e+f x))}{a+b}} \sqrt{\frac{a (\csc (e+f x)+1)}{a-b}} F\left (\sin ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b \sin (e+f x)}}{\sqrt{a+b} \sqrt{d \sin (e+f x)}}\right )|-\frac{a+b}{a-b}\right )}{693 a^6 \sqrt{d} f (a-b)^2 (a+b)^{5/2}}-\frac{16 b \left (-93 a^2 b^2+93 a^4+32 b^4\right ) \tan (e+f x) \sqrt{\frac{a (1-\csc (e+f x))}{a+b}} \sqrt{\frac{a (\csc (e+f x)+1)}{a-b}} E\left (\sin ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b \sin (e+f x)}}{\sqrt{a+b} \sqrt{d \sin (e+f x)}}\right )|-\frac{a+b}{a-b}\right )}{693 a^7 \sqrt{d} f (a-b)^2 (a+b)^{5/2}}+\frac{2 \cos ^5(e+f x) \sqrt{d \sin (e+f x)}}{11 a d f (a+b \sin (e+f x))^{11/2}} \]

Antiderivative was successfully verified.

[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 2887

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] && E
qQ[m + p + 1/2, 0]

Rule 2891

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 3055

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] + Dis
t[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] && Lt
Q[m, -1] && ((EqQ[a, 0] && IntegerQ[m] &&  !IntegerQ[n]) ||  !(IntegerQ[2*n] && LtQ[n, -1] && ((IntegerQ[n] &&
!IntegerQ[m]) || EqQ[a, 0])))

Rule 2993

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 2998

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 2816

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

Rule 2994

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]*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))])/(f*b*c^2), x] /; FreeQ[{b, c, d, e, f, A, B}, x] && NeQ[c^2 - d^2, 0] &&
EqQ[A, B] && PosQ[(c + d)/b]

Rubi steps

\begin{align*} \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{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 (\sin ^{-1}\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}} F\left (\sin ^{-1}\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]  time = 6.93415, size = 1906, normalized size = 2.68 \[ \text{result too large to display} \]

Warning: Unable to verify antiderivative.

[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]])

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Maple [B]  time = 2.328, size = 58449, normalized size = 82.1 \begin{align*} \text{output too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

result too large to display

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\sqrt{b \sin \left (f x + e\right ) + a} \sqrt{d \sin \left (f x + e\right )} \cos \left (f x + e\right )^{6}}{b^{7} d \cos \left (f x + e\right )^{8} -{\left (21 \, a^{2} b^{5} + 4 \, b^{7}\right )} d \cos \left (f x + e\right )^{6} +{\left (35 \, a^{4} b^{3} + 63 \, a^{2} b^{5} + 6 \, b^{7}\right )} d \cos \left (f x + e\right )^{4} -{\left (7 \, a^{6} b + 70 \, a^{4} b^{3} + 63 \, a^{2} b^{5} + 4 \, b^{7}\right )} d \cos \left (f x + e\right )^{2} +{\left (7 \, a^{6} b + 35 \, a^{4} b^{3} + 21 \, a^{2} b^{5} + b^{7}\right )} d -{\left (7 \, a b^{6} d \cos \left (f x + e\right )^{6} - 7 \,{\left (5 \, a^{3} b^{4} + 3 \, a b^{6}\right )} d \cos \left (f x + e\right )^{4} + 7 \,{\left (3 \, a^{5} b^{2} + 10 \, a^{3} b^{4} + 3 \, a b^{6}\right )} d \cos \left (f x + e\right )^{2} -{\left (a^{7} + 21 \, a^{5} b^{2} + 35 \, a^{3} b^{4} + 7 \, a b^{6}\right )} d\right )} \sin \left (f x + e\right )}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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]

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