3.607 \(\int \cos ^{\frac{11}{2}}(c+d x) (a+b \sec (c+d x))^{5/2} (A+B \sec (c+d x)) \, dx\)

Optimal. Leaf size=519 \[ \frac{2 \left (a^2-b^2\right ) \left (285 a^2 A b^2+675 a^4 A+1254 a^3 b B-110 a b^3 B+40 A b^4\right ) \sqrt{\frac{a \cos (c+d x)+b}{a+b}} \text{EllipticF}\left (\frac{1}{2} (c+d x),\frac{2 a}{a+b}\right )}{3465 a^3 d \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}+\frac{2 \left (81 a^2 A+209 a b B+113 A b^2\right ) \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x) \sqrt{a+b \sec (c+d x)}}{693 d}+\frac{2 \left (1145 a^2 A b+539 a^3 B+825 a b^2 B+15 A b^3\right ) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) \sqrt{a+b \sec (c+d x)}}{3465 a d}+\frac{2 \left (1025 a^2 A b^2+675 a^4 A+1793 a^3 b B+55 a b^3 B-20 A b^4\right ) \sin (c+d x) \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}{3465 a^2 d}+\frac{2 \left (255 a^2 A b^3+3705 a^4 A b+3069 a^3 b^2 B+1617 a^5 B-110 a b^4 B+40 A b^5\right ) \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)} E\left (\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right )}{3465 a^3 d \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}+\frac{2 a (11 a B+14 A b) \sin (c+d x) \cos ^{\frac{7}{2}}(c+d x) \sqrt{a+b \sec (c+d x)}}{99 d}+\frac{2 a A \sin (c+d x) \cos ^{\frac{9}{2}}(c+d x) (a+b \sec (c+d x))^{3/2}}{11 d} \]

[Out]

(2*(a^2 - b^2)*(675*a^4*A + 285*a^2*A*b^2 + 40*A*b^4 + 1254*a^3*b*B - 110*a*b^3*B)*Sqrt[(b + a*Cos[c + d*x])/(
a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)])/(3465*a^3*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) + (2*(
3705*a^4*A*b + 255*a^2*A*b^3 + 40*A*b^5 + 1617*a^5*B + 3069*a^3*b^2*B - 110*a*b^4*B)*Sqrt[Cos[c + d*x]]*Ellipt
icE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(3465*a^3*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]) + (2
*(675*a^4*A + 1025*a^2*A*b^2 - 20*A*b^4 + 1793*a^3*b*B + 55*a*b^3*B)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x
]]*Sin[c + d*x])/(3465*a^2*d) + (2*(1145*a^2*A*b + 15*A*b^3 + 539*a^3*B + 825*a*b^2*B)*Cos[c + d*x]^(3/2)*Sqrt
[a + b*Sec[c + d*x]]*Sin[c + d*x])/(3465*a*d) + (2*(81*a^2*A + 113*A*b^2 + 209*a*b*B)*Cos[c + d*x]^(5/2)*Sqrt[
a + b*Sec[c + d*x]]*Sin[c + d*x])/(693*d) + (2*a*(14*A*b + 11*a*B)*Cos[c + d*x]^(7/2)*Sqrt[a + b*Sec[c + d*x]]
*Sin[c + d*x])/(99*d) + (2*a*A*Cos[c + d*x]^(9/2)*(a + b*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(11*d)

________________________________________________________________________________________

Rubi [A]  time = 2.17391, antiderivative size = 519, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 11, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.314, Rules used = {2955, 4025, 4094, 4104, 4035, 3856, 2655, 2653, 3858, 2663, 2661} \[ \frac{2 \left (81 a^2 A+209 a b B+113 A b^2\right ) \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x) \sqrt{a+b \sec (c+d x)}}{693 d}+\frac{2 \left (1145 a^2 A b+539 a^3 B+825 a b^2 B+15 A b^3\right ) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) \sqrt{a+b \sec (c+d x)}}{3465 a d}+\frac{2 \left (1025 a^2 A b^2+675 a^4 A+1793 a^3 b B+55 a b^3 B-20 A b^4\right ) \sin (c+d x) \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}{3465 a^2 d}+\frac{2 \left (a^2-b^2\right ) \left (285 a^2 A b^2+675 a^4 A+1254 a^3 b B-110 a b^3 B+40 A b^4\right ) \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left (\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right )}{3465 a^3 d \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}+\frac{2 \left (255 a^2 A b^3+3705 a^4 A b+3069 a^3 b^2 B+1617 a^5 B-110 a b^4 B+40 A b^5\right ) \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)} E\left (\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right )}{3465 a^3 d \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}+\frac{2 a (11 a B+14 A b) \sin (c+d x) \cos ^{\frac{7}{2}}(c+d x) \sqrt{a+b \sec (c+d x)}}{99 d}+\frac{2 a A \sin (c+d x) \cos ^{\frac{9}{2}}(c+d x) (a+b \sec (c+d x))^{3/2}}{11 d} \]

Antiderivative was successfully verified.

[In]

Int[Cos[c + d*x]^(11/2)*(a + b*Sec[c + d*x])^(5/2)*(A + B*Sec[c + d*x]),x]

[Out]

(2*(a^2 - b^2)*(675*a^4*A + 285*a^2*A*b^2 + 40*A*b^4 + 1254*a^3*b*B - 110*a*b^3*B)*Sqrt[(b + a*Cos[c + d*x])/(
a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)])/(3465*a^3*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) + (2*(
3705*a^4*A*b + 255*a^2*A*b^3 + 40*A*b^5 + 1617*a^5*B + 3069*a^3*b^2*B - 110*a*b^4*B)*Sqrt[Cos[c + d*x]]*Ellipt
icE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(3465*a^3*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]) + (2
*(675*a^4*A + 1025*a^2*A*b^2 - 20*A*b^4 + 1793*a^3*b*B + 55*a*b^3*B)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x
]]*Sin[c + d*x])/(3465*a^2*d) + (2*(1145*a^2*A*b + 15*A*b^3 + 539*a^3*B + 825*a*b^2*B)*Cos[c + d*x]^(3/2)*Sqrt
[a + b*Sec[c + d*x]]*Sin[c + d*x])/(3465*a*d) + (2*(81*a^2*A + 113*A*b^2 + 209*a*b*B)*Cos[c + d*x]^(5/2)*Sqrt[
a + b*Sec[c + d*x]]*Sin[c + d*x])/(693*d) + (2*a*(14*A*b + 11*a*B)*Cos[c + d*x]^(7/2)*Sqrt[a + b*Sec[c + d*x]]
*Sin[c + d*x])/(99*d) + (2*a*A*Cos[c + d*x]^(9/2)*(a + b*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(11*d)

Rule 2955

Int[((a_.) + csc[(e_.) + (f_.)*(x_)]*(b_.))^(m_.)*(csc[(e_.) + (f_.)*(x_)]*(d_.) + (c_))^(n_.)*((g_.)*sin[(e_.
) + (f_.)*(x_)])^(p_.), x_Symbol] :> Dist[(g*Csc[e + f*x])^p*(g*Sin[e + f*x])^p, Int[((a + b*Csc[e + f*x])^m*(
c + d*Csc[e + f*x])^n)/(g*Csc[e + f*x])^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, n, p}, x] && NeQ[b*c - a*d
, 0] &&  !IntegerQ[p] &&  !(IntegerQ[m] && IntegerQ[n])

Rule 4025

Int[(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_)*(csc[(e_.) + (f_.)*(x_)]*
(B_.) + (A_)), x_Symbol] :> Simp[(a*A*Cot[e + f*x]*(a + b*Csc[e + f*x])^(m - 1)*(d*Csc[e + f*x])^n)/(f*n), x]
+ Dist[1/(d*n), Int[(a + b*Csc[e + f*x])^(m - 2)*(d*Csc[e + f*x])^(n + 1)*Simp[a*(a*B*n - A*b*(m - n - 1)) + (
2*a*b*B*n + A*(b^2*n + a^2*(1 + n)))*Csc[e + f*x] + b*(b*B*n + a*A*(m + n))*Csc[e + f*x]^2, x], x], x] /; Free
Q[{a, b, d, e, f, A, B}, x] && NeQ[A*b - a*B, 0] && NeQ[a^2 - b^2, 0] && GtQ[m, 1] && LeQ[n, -1]

Rule 4094

Int[((A_.) + csc[(e_.) + (f_.)*(x_)]*(B_.) + csc[(e_.) + (f_.)*(x_)]^2*(C_.))*(csc[(e_.) + (f_.)*(x_)]*(d_.))^
(n_)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_), x_Symbol] :> Simp[(A*Cot[e + f*x]*(a + b*Csc[e + f*x])^m*(d*
Csc[e + f*x])^n)/(f*n), x] - Dist[1/(d*n), Int[(a + b*Csc[e + f*x])^(m - 1)*(d*Csc[e + f*x])^(n + 1)*Simp[A*b*
m - a*B*n - (b*B*n + a*(C*n + A*(n + 1)))*Csc[e + f*x] - b*(C*n + A*(m + n + 1))*Csc[e + f*x]^2, x], x], x] /;
FreeQ[{a, b, d, e, f, A, B, C}, x] && NeQ[a^2 - b^2, 0] && GtQ[m, 0] && LeQ[n, -1]

Rule 4104

Int[((A_.) + csc[(e_.) + (f_.)*(x_)]*(B_.) + csc[(e_.) + (f_.)*(x_)]^2*(C_.))*(csc[(e_.) + (f_.)*(x_)]*(d_.))^
(n_)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_), x_Symbol] :> Simp[(A*Cot[e + f*x]*(a + b*Csc[e + f*x])^(m +
1)*(d*Csc[e + f*x])^n)/(a*f*n), x] + Dist[1/(a*d*n), Int[(a + b*Csc[e + f*x])^m*(d*Csc[e + f*x])^(n + 1)*Simp[
a*B*n - A*b*(m + n + 1) + a*(A + A*n + C*n)*Csc[e + f*x] + A*b*(m + n + 2)*Csc[e + f*x]^2, x], x], x] /; FreeQ
[{a, b, d, e, f, A, B, C, m}, x] && NeQ[a^2 - b^2, 0] && LeQ[n, -1]

Rule 4035

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

Rule 3856

Int[Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)]/Sqrt[csc[(e_.) + (f_.)*(x_)]*(d_.)], x_Symbol] :> Dist[Sqrt[a +
b*Csc[e + f*x]]/(Sqrt[d*Csc[e + f*x]]*Sqrt[b + a*Sin[e + f*x]]), Int[Sqrt[b + a*Sin[e + f*x]], x], x] /; Free
Q[{a, b, d, e, f}, x] && NeQ[a^2 - b^2, 0]

Rule 2655

Int[Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Dist[Sqrt[a + b*Sin[c + d*x]]/Sqrt[(a + b*Sin[c +
d*x])/(a + b)], Int[Sqrt[a/(a + b) + (b*Sin[c + d*x])/(a + b)], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 -
b^2, 0] &&  !GtQ[a + b, 0]

Rule 2653

Int[Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2*Sqrt[a + b]*EllipticE[(1*(c - Pi/2 + d*x)
)/2, (2*b)/(a + b)])/d, x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && GtQ[a + b, 0]

Rule 3858

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

Rule 2663

Int[1/Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Dist[Sqrt[(a + b*Sin[c + d*x])/(a + b)]/Sqrt[a
+ b*Sin[c + d*x]], Int[1/Sqrt[a/(a + b) + (b*Sin[c + d*x])/(a + b)], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a
^2 - b^2, 0] &&  !GtQ[a + b, 0]

Rule 2661

Int[1/Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2*EllipticF[(1*(c - Pi/2 + d*x))/2, (2*b)
/(a + b)])/(d*Sqrt[a + b]), x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && GtQ[a + b, 0]

Rubi steps

\begin{align*} \int \cos ^{\frac{11}{2}}(c+d x) (a+b \sec (c+d x))^{5/2} (A+B \sec (c+d x)) \, dx &=\left (\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{(a+b \sec (c+d x))^{5/2} (A+B \sec (c+d x))}{\sec ^{\frac{11}{2}}(c+d x)} \, dx\\ &=\frac{2 a A \cos ^{\frac{9}{2}}(c+d x) (a+b \sec (c+d x))^{3/2} \sin (c+d x)}{11 d}-\frac{1}{11} \left (2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{\sqrt{a+b \sec (c+d x)} \left (-\frac{1}{2} a (14 A b+11 a B)-\frac{1}{2} \left (9 a^2 A+11 A b^2+22 a b B\right ) \sec (c+d x)-\frac{1}{2} b (6 a A+11 b B) \sec ^2(c+d x)\right )}{\sec ^{\frac{9}{2}}(c+d x)} \, dx\\ &=\frac{2 a (14 A b+11 a B) \cos ^{\frac{7}{2}}(c+d x) \sqrt{a+b \sec (c+d x)} \sin (c+d x)}{99 d}+\frac{2 a A \cos ^{\frac{9}{2}}(c+d x) (a+b \sec (c+d x))^{3/2} \sin (c+d x)}{11 d}-\frac{1}{99} \left (4 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{-\frac{1}{4} a \left (81 a^2 A+113 A b^2+209 a b B\right )-\frac{1}{4} \left (233 a^2 A b+99 A b^3+77 a^3 B+297 a b^2 B\right ) \sec (c+d x)-\frac{3}{4} b \left (46 a A b+22 a^2 B+33 b^2 B\right ) \sec ^2(c+d x)}{\sec ^{\frac{7}{2}}(c+d x) \sqrt{a+b \sec (c+d x)}} \, dx\\ &=\frac{2 \left (81 a^2 A+113 A b^2+209 a b B\right ) \cos ^{\frac{5}{2}}(c+d x) \sqrt{a+b \sec (c+d x)} \sin (c+d x)}{693 d}+\frac{2 a (14 A b+11 a B) \cos ^{\frac{7}{2}}(c+d x) \sqrt{a+b \sec (c+d x)} \sin (c+d x)}{99 d}+\frac{2 a A \cos ^{\frac{9}{2}}(c+d x) (a+b \sec (c+d x))^{3/2} \sin (c+d x)}{11 d}+\frac{\left (8 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{\frac{1}{8} a \left (1145 a^2 A b+15 A b^3+539 a^3 B+825 a b^2 B\right )+\frac{1}{8} a \left (405 a^3 A+1531 a A b^2+1507 a^2 b B+693 b^3 B\right ) \sec (c+d x)+\frac{1}{2} a b \left (81 a^2 A+113 A b^2+209 a b B\right ) \sec ^2(c+d x)}{\sec ^{\frac{5}{2}}(c+d x) \sqrt{a+b \sec (c+d x)}} \, dx}{693 a}\\ &=\frac{2 \left (1145 a^2 A b+15 A b^3+539 a^3 B+825 a b^2 B\right ) \cos ^{\frac{3}{2}}(c+d x) \sqrt{a+b \sec (c+d x)} \sin (c+d x)}{3465 a d}+\frac{2 \left (81 a^2 A+113 A b^2+209 a b B\right ) \cos ^{\frac{5}{2}}(c+d x) \sqrt{a+b \sec (c+d x)} \sin (c+d x)}{693 d}+\frac{2 a (14 A b+11 a B) \cos ^{\frac{7}{2}}(c+d x) \sqrt{a+b \sec (c+d x)} \sin (c+d x)}{99 d}+\frac{2 a A \cos ^{\frac{9}{2}}(c+d x) (a+b \sec (c+d x))^{3/2} \sin (c+d x)}{11 d}-\frac{\left (16 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{-\frac{3}{16} a \left (675 a^4 A+1025 a^2 A b^2-20 A b^4+1793 a^3 b B+55 a b^3 B\right )-\frac{1}{16} a^2 \left (5055 a^2 A b+2305 A b^3+1617 a^3 B+6655 a b^2 B\right ) \sec (c+d x)-\frac{1}{8} a b \left (1145 a^2 A b+15 A b^3+539 a^3 B+825 a b^2 B\right ) \sec ^2(c+d x)}{\sec ^{\frac{3}{2}}(c+d x) \sqrt{a+b \sec (c+d x)}} \, dx}{3465 a^2}\\ &=\frac{2 \left (675 a^4 A+1025 a^2 A b^2-20 A b^4+1793 a^3 b B+55 a b^3 B\right ) \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)} \sin (c+d x)}{3465 a^2 d}+\frac{2 \left (1145 a^2 A b+15 A b^3+539 a^3 B+825 a b^2 B\right ) \cos ^{\frac{3}{2}}(c+d x) \sqrt{a+b \sec (c+d x)} \sin (c+d x)}{3465 a d}+\frac{2 \left (81 a^2 A+113 A b^2+209 a b B\right ) \cos ^{\frac{5}{2}}(c+d x) \sqrt{a+b \sec (c+d x)} \sin (c+d x)}{693 d}+\frac{2 a (14 A b+11 a B) \cos ^{\frac{7}{2}}(c+d x) \sqrt{a+b \sec (c+d x)} \sin (c+d x)}{99 d}+\frac{2 a A \cos ^{\frac{9}{2}}(c+d x) (a+b \sec (c+d x))^{3/2} \sin (c+d x)}{11 d}+\frac{\left (32 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{\frac{3}{32} a \left (3705 a^4 A b+255 a^2 A b^3+40 A b^5+1617 a^5 B+3069 a^3 b^2 B-110 a b^4 B\right )+\frac{3}{32} a^2 \left (675 a^4 A+3315 a^2 A b^2+10 A b^4+2871 a^3 b B+1705 a b^3 B\right ) \sec (c+d x)}{\sqrt{\sec (c+d x)} \sqrt{a+b \sec (c+d x)}} \, dx}{10395 a^3}\\ &=\frac{2 \left (675 a^4 A+1025 a^2 A b^2-20 A b^4+1793 a^3 b B+55 a b^3 B\right ) \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)} \sin (c+d x)}{3465 a^2 d}+\frac{2 \left (1145 a^2 A b+15 A b^3+539 a^3 B+825 a b^2 B\right ) \cos ^{\frac{3}{2}}(c+d x) \sqrt{a+b \sec (c+d x)} \sin (c+d x)}{3465 a d}+\frac{2 \left (81 a^2 A+113 A b^2+209 a b B\right ) \cos ^{\frac{5}{2}}(c+d x) \sqrt{a+b \sec (c+d x)} \sin (c+d x)}{693 d}+\frac{2 a (14 A b+11 a B) \cos ^{\frac{7}{2}}(c+d x) \sqrt{a+b \sec (c+d x)} \sin (c+d x)}{99 d}+\frac{2 a A \cos ^{\frac{9}{2}}(c+d x) (a+b \sec (c+d x))^{3/2} \sin (c+d x)}{11 d}+\frac{\left (\left (a^2-b^2\right ) \left (675 a^4 A+285 a^2 A b^2+40 A b^4+1254 a^3 b B-110 a b^3 B\right ) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{\sqrt{\sec (c+d x)}}{\sqrt{a+b \sec (c+d x)}} \, dx}{3465 a^3}+\frac{\left (\left (3705 a^4 A b+255 a^2 A b^3+40 A b^5+1617 a^5 B+3069 a^3 b^2 B-110 a b^4 B\right ) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{\sec (c+d x)}} \, dx}{3465 a^3}\\ &=\frac{2 \left (675 a^4 A+1025 a^2 A b^2-20 A b^4+1793 a^3 b B+55 a b^3 B\right ) \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)} \sin (c+d x)}{3465 a^2 d}+\frac{2 \left (1145 a^2 A b+15 A b^3+539 a^3 B+825 a b^2 B\right ) \cos ^{\frac{3}{2}}(c+d x) \sqrt{a+b \sec (c+d x)} \sin (c+d x)}{3465 a d}+\frac{2 \left (81 a^2 A+113 A b^2+209 a b B\right ) \cos ^{\frac{5}{2}}(c+d x) \sqrt{a+b \sec (c+d x)} \sin (c+d x)}{693 d}+\frac{2 a (14 A b+11 a B) \cos ^{\frac{7}{2}}(c+d x) \sqrt{a+b \sec (c+d x)} \sin (c+d x)}{99 d}+\frac{2 a A \cos ^{\frac{9}{2}}(c+d x) (a+b \sec (c+d x))^{3/2} \sin (c+d x)}{11 d}+\frac{\left (\left (a^2-b^2\right ) \left (675 a^4 A+285 a^2 A b^2+40 A b^4+1254 a^3 b B-110 a b^3 B\right ) \sqrt{b+a \cos (c+d x)}\right ) \int \frac{1}{\sqrt{b+a \cos (c+d x)}} \, dx}{3465 a^3 \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}+\frac{\left (\left (3705 a^4 A b+255 a^2 A b^3+40 A b^5+1617 a^5 B+3069 a^3 b^2 B-110 a b^4 B\right ) \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}\right ) \int \sqrt{b+a \cos (c+d x)} \, dx}{3465 a^3 \sqrt{b+a \cos (c+d x)}}\\ &=\frac{2 \left (675 a^4 A+1025 a^2 A b^2-20 A b^4+1793 a^3 b B+55 a b^3 B\right ) \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)} \sin (c+d x)}{3465 a^2 d}+\frac{2 \left (1145 a^2 A b+15 A b^3+539 a^3 B+825 a b^2 B\right ) \cos ^{\frac{3}{2}}(c+d x) \sqrt{a+b \sec (c+d x)} \sin (c+d x)}{3465 a d}+\frac{2 \left (81 a^2 A+113 A b^2+209 a b B\right ) \cos ^{\frac{5}{2}}(c+d x) \sqrt{a+b \sec (c+d x)} \sin (c+d x)}{693 d}+\frac{2 a (14 A b+11 a B) \cos ^{\frac{7}{2}}(c+d x) \sqrt{a+b \sec (c+d x)} \sin (c+d x)}{99 d}+\frac{2 a A \cos ^{\frac{9}{2}}(c+d x) (a+b \sec (c+d x))^{3/2} \sin (c+d x)}{11 d}+\frac{\left (\left (a^2-b^2\right ) \left (675 a^4 A+285 a^2 A b^2+40 A b^4+1254 a^3 b B-110 a b^3 B\right ) \sqrt{\frac{b+a \cos (c+d x)}{a+b}}\right ) \int \frac{1}{\sqrt{\frac{b}{a+b}+\frac{a \cos (c+d x)}{a+b}}} \, dx}{3465 a^3 \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}+\frac{\left (\left (3705 a^4 A b+255 a^2 A b^3+40 A b^5+1617 a^5 B+3069 a^3 b^2 B-110 a b^4 B\right ) \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}\right ) \int \sqrt{\frac{b}{a+b}+\frac{a \cos (c+d x)}{a+b}} \, dx}{3465 a^3 \sqrt{\frac{b+a \cos (c+d x)}{a+b}}}\\ &=\frac{2 \left (a^2-b^2\right ) \left (675 a^4 A+285 a^2 A b^2+40 A b^4+1254 a^3 b B-110 a b^3 B\right ) \sqrt{\frac{b+a \cos (c+d x)}{a+b}} F\left (\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right )}{3465 a^3 d \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}+\frac{2 \left (3705 a^4 A b+255 a^2 A b^3+40 A b^5+1617 a^5 B+3069 a^3 b^2 B-110 a b^4 B\right ) \sqrt{\cos (c+d x)} E\left (\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right ) \sqrt{a+b \sec (c+d x)}}{3465 a^3 d \sqrt{\frac{b+a \cos (c+d x)}{a+b}}}+\frac{2 \left (675 a^4 A+1025 a^2 A b^2-20 A b^4+1793 a^3 b B+55 a b^3 B\right ) \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)} \sin (c+d x)}{3465 a^2 d}+\frac{2 \left (1145 a^2 A b+15 A b^3+539 a^3 B+825 a b^2 B\right ) \cos ^{\frac{3}{2}}(c+d x) \sqrt{a+b \sec (c+d x)} \sin (c+d x)}{3465 a d}+\frac{2 \left (81 a^2 A+113 A b^2+209 a b B\right ) \cos ^{\frac{5}{2}}(c+d x) \sqrt{a+b \sec (c+d x)} \sin (c+d x)}{693 d}+\frac{2 a (14 A b+11 a B) \cos ^{\frac{7}{2}}(c+d x) \sqrt{a+b \sec (c+d x)} \sin (c+d x)}{99 d}+\frac{2 a A \cos ^{\frac{9}{2}}(c+d x) (a+b \sec (c+d x))^{3/2} \sin (c+d x)}{11 d}\\ \end{align*}

Mathematica [C]  time = 19.966, size = 626, normalized size = 1.21 \[ \frac{\cos ^{\frac{5}{2}}(c+d x) (a+b \sec (c+d x))^{5/2} \left (\frac{\left (513 a^2 A+836 a b B+452 A b^2\right ) \sin (3 (c+d x))}{5544}+\frac{\left (3095 a^2 A b+1463 a^3 B+1650 a b^2 B+30 A b^3\right ) \sin (2 (c+d x))}{6930 a}+\frac{\left (9330 a^2 A b^2+6525 a^4 A+16434 a^3 b B+440 a b^3 B-160 A b^4\right ) \sin (c+d x)}{13860 a^2}+\frac{1}{88} a^2 A \sin (5 (c+d x))+\frac{1}{396} a (11 a B+23 A b) \sin (4 (c+d x))\right )}{d (a \cos (c+d x)+b)^2}-\frac{2 \cos ^{\frac{3}{2}}(c+d x) \left (\cos ^2\left (\frac{1}{2} (c+d x)\right ) \sec (c+d x)\right )^{3/2} (a+b \sec (c+d x))^{5/2} \left (i a (a+b) \left (15 a^2 b^2 (19 A+121 B)+6 a^3 b (505 A+209 B)+3 a^4 (225 A+539 B)-10 a b^3 (3 A+11 B)+40 A b^4\right ) \sec ^2\left (\frac{1}{2} (c+d x)\right ) \sqrt{\frac{\sec ^2\left (\frac{1}{2} (c+d x)\right ) (a \cos (c+d x)+b)}{a+b}} \text{EllipticF}\left (i \sinh ^{-1}\left (\tan \left (\frac{1}{2} (c+d x)\right )\right ),\frac{b-a}{a+b}\right )-\left (255 a^2 A b^3+3705 a^4 A b+3069 a^3 b^2 B+1617 a^5 B-110 a b^4 B+40 A b^5\right ) \tan \left (\frac{1}{2} (c+d x)\right ) \sec ^2\left (\frac{1}{2} (c+d x)\right )^{3/2} (a \cos (c+d x)+b)-i (a+b) \left (255 a^2 A b^3+3705 a^4 A b+3069 a^3 b^2 B+1617 a^5 B-110 a b^4 B+40 A b^5\right ) \sec ^2\left (\frac{1}{2} (c+d x)\right ) \sqrt{\frac{\sec ^2\left (\frac{1}{2} (c+d x)\right ) (a \cos (c+d x)+b)}{a+b}} E\left (i \sinh ^{-1}\left (\tan \left (\frac{1}{2} (c+d x)\right )\right )|\frac{b-a}{a+b}\right )\right )}{3465 a^3 d \sec ^{\frac{5}{2}}(c+d x) (a \cos (c+d x)+b)^3} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[Cos[c + d*x]^(11/2)*(a + b*Sec[c + d*x])^(5/2)*(A + B*Sec[c + d*x]),x]

[Out]

(Cos[c + d*x]^(5/2)*(a + b*Sec[c + d*x])^(5/2)*(((6525*a^4*A + 9330*a^2*A*b^2 - 160*A*b^4 + 16434*a^3*b*B + 44
0*a*b^3*B)*Sin[c + d*x])/(13860*a^2) + ((3095*a^2*A*b + 30*A*b^3 + 1463*a^3*B + 1650*a*b^2*B)*Sin[2*(c + d*x)]
)/(6930*a) + ((513*a^2*A + 452*A*b^2 + 836*a*b*B)*Sin[3*(c + d*x)])/5544 + (a*(23*A*b + 11*a*B)*Sin[4*(c + d*x
)])/396 + (a^2*A*Sin[5*(c + d*x)])/88))/(d*(b + a*Cos[c + d*x])^2) - (2*Cos[c + d*x]^(3/2)*(Cos[(c + d*x)/2]^2
*Sec[c + d*x])^(3/2)*(a + b*Sec[c + d*x])^(5/2)*((-I)*(a + b)*(3705*a^4*A*b + 255*a^2*A*b^3 + 40*A*b^5 + 1617*
a^5*B + 3069*a^3*b^2*B - 110*a*b^4*B)*EllipticE[I*ArcSinh[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sec[(c + d*x)/2
]^2*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)] + I*a*(a + b)*(40*A*b^4 - 10*a*b^3*(3*A + 11*B) +
15*a^2*b^2*(19*A + 121*B) + 6*a^3*b*(505*A + 209*B) + 3*a^4*(225*A + 539*B))*EllipticF[I*ArcSinh[Tan[(c + d*x)
/2]], (-a + b)/(a + b)]*Sec[(c + d*x)/2]^2*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)] - (3705*a^4
*A*b + 255*a^2*A*b^3 + 40*A*b^5 + 1617*a^5*B + 3069*a^3*b^2*B - 110*a*b^4*B)*(b + a*Cos[c + d*x])*(Sec[(c + d*
x)/2]^2)^(3/2)*Tan[(c + d*x)/2]))/(3465*a^3*d*(b + a*Cos[c + d*x])^3*Sec[c + d*x]^(5/2))

________________________________________________________________________________________

Maple [B]  time = 1.01, size = 3816, normalized size = 7.4 \begin{align*} \text{output too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(d*x+c)^(11/2)*(a+b*sec(d*x+c))^(5/2)*(A+B*sec(d*x+c)),x)

[Out]

2/3465/d*((b+a*cos(d*x+c))/cos(d*x+c))^(1/2)*cos(d*x+c)^(1/2)*(-1+cos(d*x+c))*(cos(d*x+c)+1)*(-55*B*cos(d*x+c)
^2*((a-b)/(a+b))^(1/2)*a^2*b^4*(1/(cos(d*x+c)+1))^(1/2)-40*A*((a-b)/(a+b))^(1/2)*b^6*(1/(cos(d*x+c)+1))^(1/2)-
3705*A*cos(d*x+c)*((a-b)/(a+b))^(1/2)*a^5*b*(1/(cos(d*x+c)+1))^(1/2)+385*B*cos(d*x+c)^6*((a-b)/(a+b))^(1/2)*a^
6*(1/(cos(d*x+c)+1))^(1/2)+154*B*cos(d*x+c)^4*((a-b)/(a+b))^(1/2)*a^6*(1/(cos(d*x+c)+1))^(1/2)+1078*B*cos(d*x+
c)^2*((a-b)/(a+b))^(1/2)*a^6*(1/(cos(d*x+c)+1))^(1/2)+40*A*cos(d*x+c)*((a-b)/(a+b))^(1/2)*b^6*(1/(cos(d*x+c)+1
))^(1/2)-1617*B*cos(d*x+c)*((a-b)/(a+b))^(1/2)*a^6*(1/(cos(d*x+c)+1))^(1/2)-675*A*((a-b)/(a+b))^(1/2)*a^5*b*(1
/(cos(d*x+c)+1))^(1/2)-3705*A*((a-b)/(a+b))^(1/2)*a^4*b^2*(1/(cos(d*x+c)+1))^(1/2)-1025*A*((a-b)/(a+b))^(1/2)*
a^3*b^3*(1/(cos(d*x+c)+1))^(1/2)-255*A*((a-b)/(a+b))^(1/2)*a^2*b^4*(1/(cos(d*x+c)+1))^(1/2)+20*A*((a-b)/(a+b))
^(1/2)*a*b^5*(1/(cos(d*x+c)+1))^(1/2)-1617*B*((a-b)/(a+b))^(1/2)*a^5*b*(1/(cos(d*x+c)+1))^(1/2)-1793*B*((a-b)/
(a+b))^(1/2)*a^4*b^2*(1/(cos(d*x+c)+1))^(1/2)-3069*B*((a-b)/(a+b))^(1/2)*a^3*b^3*(1/(cos(d*x+c)+1))^(1/2)-55*B
*((a-b)/(a+b))^(1/2)*a^2*b^4*(1/(cos(d*x+c)+1))^(1/2)+110*B*((a-b)/(a+b))^(1/2)*a*b^5*(1/(cos(d*x+c)+1))^(1/2)
+315*A*cos(d*x+c)^7*((a-b)/(a+b))^(1/2)*a^6*(1/(cos(d*x+c)+1))^(1/2)+90*A*cos(d*x+c)^5*((a-b)/(a+b))^(1/2)*a^6
*(1/(cos(d*x+c)+1))^(1/2)+270*A*cos(d*x+c)^3*((a-b)/(a+b))^(1/2)*a^6*(1/(cos(d*x+c)+1))^(1/2)-675*A*cos(d*x+c)
*((a-b)/(a+b))^(1/2)*a^6*(1/(cos(d*x+c)+1))^(1/2)-1617*B*sin(d*x+c)*EllipticF((-1+cos(d*x+c))*((a-b)/(a+b))^(1
/2)/sin(d*x+c),(-(a+b)/(a-b))^(1/2))*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*a^6+1617*B*sin(d*x+c)*(1/
(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*EllipticE((-1+cos(d*x+c))*((a-b)/(a+b))^(1/2)/sin(d*x+c),(-(a+b)/
(a-b))^(1/2))*a^6+675*A*sin(d*x+c)*EllipticF((-1+cos(d*x+c))*((a-b)/(a+b))^(1/2)/sin(d*x+c),(-(a+b)/(a-b))^(1/
2))*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*a^6-40*A*sin(d*x+c)*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+
1))^(1/2)*EllipticE((-1+cos(d*x+c))*((a-b)/(a+b))^(1/2)/sin(d*x+c),(-(a+b)/(a-b))^(1/2))*b^6-3705*A*sin(d*x+c)
*EllipticF((-1+cos(d*x+c))*((a-b)/(a+b))^(1/2)/sin(d*x+c),(-(a+b)/(a-b))^(1/2))*(1/(a+b)*(b+a*cos(d*x+c))/(cos
(d*x+c)+1))^(1/2)*a^5*b+3315*A*sin(d*x+c)*EllipticF((-1+cos(d*x+c))*((a-b)/(a+b))^(1/2)/sin(d*x+c),(-(a+b)/(a-
b))^(1/2))*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*a^4*b^2+430*A*cos(d*x+c)^4*((a-b)/(a+b))^(1/2)*a^5*
b*(1/(cos(d*x+c)+1))^(1/2)+580*A*cos(d*x+c)^4*((a-b)/(a+b))^(1/2)*a^3*b^3*(1/(cos(d*x+c)+1))^(1/2)+1870*B*cos(
d*x+c)^4*((a-b)/(a+b))^(1/2)*a^4*b^2*(1/(cos(d*x+c)+1))^(1/2)+800*A*cos(d*x+c)^3*((a-b)/(a+b))^(1/2)*a^4*b^2*(
1/(cos(d*x+c)+1))^(1/2)-5*A*cos(d*x+c)^3*((a-b)/(a+b))^(1/2)*a^2*b^4*(1/(cos(d*x+c)+1))^(1/2)-255*A*sin(d*x+c)
*EllipticF((-1+cos(d*x+c))*((a-b)/(a+b))^(1/2)/sin(d*x+c),(-(a+b)/(a-b))^(1/2))*(1/(a+b)*(b+a*cos(d*x+c))/(cos
(d*x+c)+1))^(1/2)*a^3*b^3+10*A*sin(d*x+c)*EllipticF((-1+cos(d*x+c))*((a-b)/(a+b))^(1/2)/sin(d*x+c),(-(a+b)/(a-
b))^(1/2))*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*a^2*b^4-40*A*sin(d*x+c)*EllipticF((-1+cos(d*x+c))*(
(a-b)/(a+b))^(1/2)/sin(d*x+c),(-(a+b)/(a-b))^(1/2))*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*a*b^5+3705
*A*sin(d*x+c)*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*EllipticE((-1+cos(d*x+c))*((a-b)/(a+b))^(1/2)/si
n(d*x+c),(-(a+b)/(a-b))^(1/2))*a^5*b-3705*A*sin(d*x+c)*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*Ellipti
cE((-1+cos(d*x+c))*((a-b)/(a+b))^(1/2)/sin(d*x+c),(-(a+b)/(a-b))^(1/2))*a^4*b^2+255*A*sin(d*x+c)*(1/(a+b)*(b+a
*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*EllipticE((-1+cos(d*x+c))*((a-b)/(a+b))^(1/2)/sin(d*x+c),(-(a+b)/(a-b))^(1/
2))*a^3*b^3-255*A*sin(d*x+c)*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*EllipticE((-1+cos(d*x+c))*((a-b)/
(a+b))^(1/2)/sin(d*x+c),(-(a+b)/(a-b))^(1/2))*a^2*b^4+40*A*sin(d*x+c)*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1)
)^(1/2)*EllipticE((-1+cos(d*x+c))*((a-b)/(a+b))^(1/2)/sin(d*x+c),(-(a+b)/(a-b))^(1/2))*a*b^5+2871*B*sin(d*x+c)
*EllipticF((-1+cos(d*x+c))*((a-b)/(a+b))^(1/2)/sin(d*x+c),(-(a+b)/(a-b))^(1/2))*(1/(a+b)*(b+a*cos(d*x+c))/(cos
(d*x+c)+1))^(1/2)*a^5*b-3069*B*sin(d*x+c)*EllipticF((-1+cos(d*x+c))*((a-b)/(a+b))^(1/2)/sin(d*x+c),(-(a+b)/(a-
b))^(1/2))*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*a^4*b^2+1705*B*sin(d*x+c)*EllipticF((-1+cos(d*x+c))
*((a-b)/(a+b))^(1/2)/sin(d*x+c),(-(a+b)/(a-b))^(1/2))*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*a^3*b^3+
110*B*sin(d*x+c)*EllipticF((-1+cos(d*x+c))*((a-b)/(a+b))^(1/2)/sin(d*x+c),(-(a+b)/(a-b))^(1/2))*(1/(a+b)*(b+a*
cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*a^2*b^4-1617*B*sin(d*x+c)*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*El
lipticE((-1+cos(d*x+c))*((a-b)/(a+b))^(1/2)/sin(d*x+c),(-(a+b)/(a-b))^(1/2))*a^5*b+3069*B*sin(d*x+c)*(1/(a+b)*
(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*EllipticE((-1+cos(d*x+c))*((a-b)/(a+b))^(1/2)/sin(d*x+c),(-(a+b)/(a-b))
^(1/2))*a^4*b^2-3069*B*sin(d*x+c)*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*EllipticE((-1+cos(d*x+c))*((
a-b)/(a+b))^(1/2)/sin(d*x+c),(-(a+b)/(a-b))^(1/2))*a^3*b^3-110*B*sin(d*x+c)*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x
+c)+1))^(1/2)*EllipticE((-1+cos(d*x+c))*((a-b)/(a+b))^(1/2)/sin(d*x+c),(-(a+b)/(a-b))^(1/2))*a^2*b^4+110*B*sin
(d*x+c)*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*EllipticE((-1+cos(d*x+c))*((a-b)/(a+b))^(1/2)/sin(d*x+
c),(-(a+b)/(a-b))^(1/2))*a*b^5+1370*A*cos(d*x+c)^5*((a-b)/(a+b))^(1/2)*a^4*b^2*(1/(cos(d*x+c)+1))^(1/2)+1430*B
*cos(d*x+c)^5*((a-b)/(a+b))^(1/2)*a^5*b*(1/(cos(d*x+c)+1))^(1/2)+1535*A*cos(d*x+c)*((a-b)/(a+b))^(1/2)*a^4*b^2
*(1/(cos(d*x+c)+1))^(1/2)-255*A*cos(d*x+c)*((a-b)/(a+b))^(1/2)*a^3*b^3*(1/(cos(d*x+c)+1))^(1/2)+260*A*cos(d*x+
c)*((a-b)/(a+b))^(1/2)*a^2*b^4*(1/(cos(d*x+c)+1))^(1/2)-40*A*cos(d*x+c)*((a-b)/(a+b))^(1/2)*a*b^5*(1/(cos(d*x+
c)+1))^(1/2)-715*B*cos(d*x+c)*((a-b)/(a+b))^(1/2)*a^5*b*(1/(cos(d*x+c)+1))^(1/2)-3069*B*cos(d*x+c)*((a-b)/(a+b
))^(1/2)*a^4*b^2*(1/(cos(d*x+c)+1))^(1/2)+2189*B*cos(d*x+c)*((a-b)/(a+b))^(1/2)*a^3*b^3*(1/(cos(d*x+c)+1))^(1/
2)+110*B*cos(d*x+c)*((a-b)/(a+b))^(1/2)*a^2*b^4*(1/(cos(d*x+c)+1))^(1/2)-110*B*cos(d*x+c)*((a-b)/(a+b))^(1/2)*
a*b^5*(1/(cos(d*x+c)+1))^(1/2)+1120*A*cos(d*x+c)^6*((a-b)/(a+b))^(1/2)*a^5*b*(1/(cos(d*x+c)+1))^(1/2)+902*B*co
s(d*x+c)^3*((a-b)/(a+b))^(1/2)*a^5*b*(1/(cos(d*x+c)+1))^(1/2)+880*B*cos(d*x+c)^3*((a-b)/(a+b))^(1/2)*a^3*b^3*(
1/(cos(d*x+c)+1))^(1/2)+2830*A*cos(d*x+c)^2*((a-b)/(a+b))^(1/2)*a^5*b*(1/(cos(d*x+c)+1))^(1/2)+700*A*cos(d*x+c
)^2*((a-b)/(a+b))^(1/2)*a^3*b^3*(1/(cos(d*x+c)+1))^(1/2)+20*A*cos(d*x+c)^2*((a-b)/(a+b))^(1/2)*a*b^5*(1/(cos(d
*x+c)+1))^(1/2)+2992*B*cos(d*x+c)^2*((a-b)/(a+b))^(1/2)*a^4*b^2*(1/(cos(d*x+c)+1))^(1/2))/a^3/((a-b)/(a+b))^(1
/2)/(b+a*cos(d*x+c))/(1/(cos(d*x+c)+1))^(1/2)/sin(d*x+c)^3

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Maxima [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(d*x+c)^(11/2)*(a+b*sec(d*x+c))^(5/2)*(A+B*sec(d*x+c)),x, algorithm="maxima")

[Out]

Timed out

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (B b^{2} \cos \left (d x + c\right )^{5} \sec \left (d x + c\right )^{3} + A a^{2} \cos \left (d x + c\right )^{5} +{\left (2 \, B a b + A b^{2}\right )} \cos \left (d x + c\right )^{5} \sec \left (d x + c\right )^{2} +{\left (B a^{2} + 2 \, A a b\right )} \cos \left (d x + c\right )^{5} \sec \left (d x + c\right )\right )} \sqrt{b \sec \left (d x + c\right ) + a} \sqrt{\cos \left (d x + c\right )}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^(11/2)*(a+b*sec(d*x+c))^(5/2)*(A+B*sec(d*x+c)),x, algorithm="fricas")

[Out]

integral((B*b^2*cos(d*x + c)^5*sec(d*x + c)^3 + A*a^2*cos(d*x + c)^5 + (2*B*a*b + A*b^2)*cos(d*x + c)^5*sec(d*
x + c)^2 + (B*a^2 + 2*A*a*b)*cos(d*x + c)^5*sec(d*x + c))*sqrt(b*sec(d*x + c) + a)*sqrt(cos(d*x + c)), 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(d*x+c)**(11/2)*(a+b*sec(d*x+c))**(5/2)*(A+B*sec(d*x+c)),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (B \sec \left (d x + c\right ) + A\right )}{\left (b \sec \left (d x + c\right ) + a\right )}^{\frac{5}{2}} \cos \left (d x + c\right )^{\frac{11}{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^(11/2)*(a+b*sec(d*x+c))^(5/2)*(A+B*sec(d*x+c)),x, algorithm="giac")

[Out]

integrate((B*sec(d*x + c) + A)*(b*sec(d*x + c) + a)^(5/2)*cos(d*x + c)^(11/2), x)