\(\int \frac {(a+b \sec (c+d x))^{5/2} (A+B \sec (c+d x))}{\sec ^{\frac {11}{2}}(c+d x)} \, dx\) [455]

Optimal result

Integrand size = 35, antiderivative size = 519 \[ \int \frac {(a+b \sec (c+d x))^{5/2} (A+B \sec (c+d x))}{\sec ^{\frac {11}{2}}(c+d x)} \, dx=\frac {2 \left (a^2-b^2\right ) \left (675 a^4 A+285 a^2 A b^2+40 A b^4+1254 a^3 b B-110 a b^3 B\right ) \sqrt {\frac {b+a \cos (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 a}{a+b}\right ) \sqrt {\sec (c+d x)}}{3465 a^3 d \sqrt {a+b \sec (c+d x)}}+\frac {2 \left (3705 a^4 A b+255 a^2 A b^3+40 A b^5+1617 a^5 B+3069 a^3 b^2 B-110 a b^4 B\right ) E\left (\frac {1}{2} (c+d x)|\frac {2 a}{a+b}\right ) \sqrt {a+b \sec (c+d x)}}{3465 a^3 d \sqrt {\frac {b+a \cos (c+d x)}{a+b}} \sqrt {\sec (c+d x)}}+\frac {2 a (14 A b+11 a B) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{99 d \sec ^{\frac {7}{2}}(c+d x)}+\frac {2 \left (81 a^2 A+113 A b^2+209 a b B\right ) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{693 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {2 \left (1145 a^2 A b+15 A b^3+539 a^3 B+825 a b^2 B\right ) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{3465 a d \sec ^{\frac {3}{2}}(c+d x)}+\frac {2 \left (675 a^4 A+1025 a^2 A b^2-20 A b^4+1793 a^3 b B+55 a b^3 B\right ) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{3465 a^2 d \sqrt {\sec (c+d x)}}+\frac {2 a A (a+b \sec (c+d x))^{3/2} \sin (c+d x)}{11 d \sec ^{\frac {9}{2}}(c+d x)} \]

[Out]

2/11*a*A*(a+b*sec(d*x+c))^(3/2)*sin(d*x+c)/d/sec(d*x+c)^(9/2)+2/3465*(a^2-b^2)*(675*A*a^4+285*A*a^2*b^2+40*A*b
^4+1254*B*a^3*b-110*B*a*b^3)*(cos(1/2*d*x+1/2*c)^2)^(1/2)/cos(1/2*d*x+1/2*c)*EllipticF(sin(1/2*d*x+1/2*c),2^(1
/2)*(a/(a+b))^(1/2))*((b+a*cos(d*x+c))/(a+b))^(1/2)*sec(d*x+c)^(1/2)/a^3/d/(a+b*sec(d*x+c))^(1/2)+2/99*a*(14*A
*b+11*B*a)*sin(d*x+c)*(a+b*sec(d*x+c))^(1/2)/d/sec(d*x+c)^(7/2)+2/693*(81*A*a^2+113*A*b^2+209*B*a*b)*sin(d*x+c
)*(a+b*sec(d*x+c))^(1/2)/d/sec(d*x+c)^(5/2)+2/3465*(1145*A*a^2*b+15*A*b^3+539*B*a^3+825*B*a*b^2)*sin(d*x+c)*(a
+b*sec(d*x+c))^(1/2)/a/d/sec(d*x+c)^(3/2)+2/3465*(675*A*a^4+1025*A*a^2*b^2-20*A*b^4+1793*B*a^3*b+55*B*a*b^3)*s
in(d*x+c)*(a+b*sec(d*x+c))^(1/2)/a^2/d/sec(d*x+c)^(1/2)+2/3465*(3705*A*a^4*b+255*A*a^2*b^3+40*A*b^5+1617*B*a^5
+3069*B*a^3*b^2-110*B*a*b^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)*(a/(a+b))^(1/2))*(a+b*sec(d*x+c))^(1/2)/a^3/d/((b+a*cos(d*x+c))/(a+b))^(1/2)/sec(d*x+c)^(1/2)

Rubi [A] (verified)

Time = 2.88 (sec) , antiderivative size = 519, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {4110, 4179, 4189, 4120, 3941, 2734, 2732, 3943, 2742, 2740} \[ \int \frac {(a+b \sec (c+d x))^{5/2} (A+B \sec (c+d x))}{\sec ^{\frac {11}{2}}(c+d x)} \, dx=\frac {2 \left (81 a^2 A+209 a b B+113 A b^2\right ) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{693 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {2 \left (539 a^3 B+1145 a^2 A b+825 a b^2 B+15 A b^3\right ) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{3465 a d \sec ^{\frac {3}{2}}(c+d x)}+\frac {2 \left (675 a^4 A+1793 a^3 b B+1025 a^2 A b^2+55 a b^3 B-20 A b^4\right ) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{3465 a^2 d \sqrt {\sec (c+d x)}}+\frac {2 \left (a^2-b^2\right ) \left (675 a^4 A+1254 a^3 b B+285 a^2 A b^2-110 a b^3 B+40 A b^4\right ) \sqrt {\sec (c+d x)} \sqrt {\frac {a \cos (c+d x)+b}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 a}{a+b}\right )}{3465 a^3 d \sqrt {a+b \sec (c+d x)}}+\frac {2 \left (1617 a^5 B+3705 a^4 A b+3069 a^3 b^2 B+255 a^2 A b^3-110 a b^4 B+40 A b^5\right ) \sqrt {a+b \sec (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 a}{a+b}\right )}{3465 a^3 d \sqrt {\sec (c+d x)} \sqrt {\frac {a \cos (c+d x)+b}{a+b}}}+\frac {2 a (11 a B+14 A b) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{99 d \sec ^{\frac {7}{2}}(c+d x)}+\frac {2 a A \sin (c+d x) (a+b \sec (c+d x))^{3/2}}{11 d \sec ^{\frac {9}{2}}(c+d x)} \]

[In]

Int[((a + b*Sec[c + d*x])^(5/2)*(A + B*Sec[c + d*x]))/Sec[c + d*x]^(11/2),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)]*Sqrt[Sec[c + d*x]])/(3465*a^3*d*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)*EllipticE[(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)]*Sqrt[Sec[c + d*x]]) + (2
*a*(14*A*b + 11*a*B)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(99*d*Sec[c + d*x]^(7/2)) + (2*(81*a^2*A + 113*A*b
^2 + 209*a*b*B)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(693*d*Sec[c + d*x]^(5/2)) + (2*(1145*a^2*A*b + 15*A*b^
3 + 539*a^3*B + 825*a*b^2*B)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(3465*a*d*Sec[c + d*x]^(3/2)) + (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[a + b*Sec[c + d*x]]*Sin[c + d*x])/(3465*a^2*
d*Sqrt[Sec[c + d*x]]) + (2*a*A*(a + b*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(11*d*Sec[c + d*x]^(9/2))

Rule 2732

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

Rule 2734

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/(a + b))*Sin[c + d*x]], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 -
b^2, 0] &&  !GtQ[a + b, 0]

Rule 2740

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

Rule 2742

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/(a + b))*Sin[c + d*x]], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a
^2 - b^2, 0] &&  !GtQ[a + b, 0]

Rule 3941

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 3943

Int[Sqrt[csc[(e_.) + (f_.)*(x_)]*(d_.)]/Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)], x_Symbol] :> Dist[Sqrt[d*C
sc[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 4110

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 4120

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 4179

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 4189

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]

Rubi steps \begin{align*} \text {integral}& = \frac {2 a A (a+b \sec (c+d x))^{3/2} \sin (c+d x)}{11 d \sec ^{\frac {9}{2}}(c+d x)}-\frac {2}{11} \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) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{99 d \sec ^{\frac {7}{2}}(c+d x)}+\frac {2 a A (a+b \sec (c+d x))^{3/2} \sin (c+d x)}{11 d \sec ^{\frac {9}{2}}(c+d x)}-\frac {4}{99} \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 a (14 A b+11 a B) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{99 d \sec ^{\frac {7}{2}}(c+d x)}+\frac {2 \left (81 a^2 A+113 A b^2+209 a b B\right ) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{693 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {2 a A (a+b \sec (c+d x))^{3/2} \sin (c+d x)}{11 d \sec ^{\frac {9}{2}}(c+d x)}+\frac {8 \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 a (14 A b+11 a B) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{99 d \sec ^{\frac {7}{2}}(c+d x)}+\frac {2 \left (81 a^2 A+113 A b^2+209 a b B\right ) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{693 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {2 \left (1145 a^2 A b+15 A b^3+539 a^3 B+825 a b^2 B\right ) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{3465 a d \sec ^{\frac {3}{2}}(c+d x)}+\frac {2 a A (a+b \sec (c+d x))^{3/2} \sin (c+d x)}{11 d \sec ^{\frac {9}{2}}(c+d x)}-\frac {16 \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 a (14 A b+11 a B) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{99 d \sec ^{\frac {7}{2}}(c+d x)}+\frac {2 \left (81 a^2 A+113 A b^2+209 a b B\right ) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{693 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {2 \left (1145 a^2 A b+15 A b^3+539 a^3 B+825 a b^2 B\right ) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{3465 a d \sec ^{\frac {3}{2}}(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 {a+b \sec (c+d x)} \sin (c+d x)}{3465 a^2 d \sqrt {\sec (c+d x)}}+\frac {2 a A (a+b \sec (c+d x))^{3/2} \sin (c+d x)}{11 d \sec ^{\frac {9}{2}}(c+d x)}+\frac {32 \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 a (14 A b+11 a B) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{99 d \sec ^{\frac {7}{2}}(c+d x)}+\frac {2 \left (81 a^2 A+113 A b^2+209 a b B\right ) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{693 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {2 \left (1145 a^2 A b+15 A b^3+539 a^3 B+825 a b^2 B\right ) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{3465 a d \sec ^{\frac {3}{2}}(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 {a+b \sec (c+d x)} \sin (c+d x)}{3465 a^2 d \sqrt {\sec (c+d x)}}+\frac {2 a A (a+b \sec (c+d x))^{3/2} \sin (c+d x)}{11 d \sec ^{\frac {9}{2}}(c+d x)}+\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 )\right ) \int \frac {\sqrt {\sec (c+d x)}}{\sqrt {a+b \sec (c+d x)}} \, dx}{3465 a^3}+\frac {\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 ) \int \frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {\sec (c+d x)}} \, dx}{3465 a^3} \\ & = \frac {2 a (14 A b+11 a B) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{99 d \sec ^{\frac {7}{2}}(c+d x)}+\frac {2 \left (81 a^2 A+113 A b^2+209 a b B\right ) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{693 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {2 \left (1145 a^2 A b+15 A b^3+539 a^3 B+825 a b^2 B\right ) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{3465 a d \sec ^{\frac {3}{2}}(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 {a+b \sec (c+d x)} \sin (c+d x)}{3465 a^2 d \sqrt {\sec (c+d x)}}+\frac {2 a A (a+b \sec (c+d x))^{3/2} \sin (c+d x)}{11 d \sec ^{\frac {9}{2}}(c+d x)}+\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)} \sqrt {\sec (c+d x)}\right ) \int \frac {1}{\sqrt {b+a \cos (c+d x)}} \, dx}{3465 a^3 \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 {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)} \sqrt {\sec (c+d x)}} \\ & = \frac {2 a (14 A b+11 a B) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{99 d \sec ^{\frac {7}{2}}(c+d x)}+\frac {2 \left (81 a^2 A+113 A b^2+209 a b B\right ) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{693 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {2 \left (1145 a^2 A b+15 A b^3+539 a^3 B+825 a b^2 B\right ) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{3465 a d \sec ^{\frac {3}{2}}(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 {a+b \sec (c+d x)} \sin (c+d x)}{3465 a^2 d \sqrt {\sec (c+d x)}}+\frac {2 a A (a+b \sec (c+d x))^{3/2} \sin (c+d x)}{11 d \sec ^{\frac {9}{2}}(c+d x)}+\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}} \sqrt {\sec (c+d x)}\right ) \int \frac {1}{\sqrt {\frac {b}{a+b}+\frac {a \cos (c+d x)}{a+b}}} \, dx}{3465 a^3 \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 {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}} \sqrt {\sec (c+d x)}} \\ & = \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}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 a}{a+b}\right ) \sqrt {\sec (c+d x)}}{3465 a^3 d \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 ) 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}} \sqrt {\sec (c+d x)}}+\frac {2 a (14 A b+11 a B) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{99 d \sec ^{\frac {7}{2}}(c+d x)}+\frac {2 \left (81 a^2 A+113 A b^2+209 a b B\right ) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{693 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {2 \left (1145 a^2 A b+15 A b^3+539 a^3 B+825 a b^2 B\right ) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{3465 a d \sec ^{\frac {3}{2}}(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 {a+b \sec (c+d x)} \sin (c+d x)}{3465 a^2 d \sqrt {\sec (c+d x)}}+\frac {2 a A (a+b \sec (c+d x))^{3/2} \sin (c+d x)}{11 d \sec ^{\frac {9}{2}}(c+d x)} \\ \end{align*}

Mathematica [A] (verified)

Time = 6.03 (sec) , antiderivative size = 380, normalized size of antiderivative = 0.73 \[ \int \frac {(a+b \sec (c+d x))^{5/2} (A+B \sec (c+d x))}{\sec ^{\frac {11}{2}}(c+d x)} \, dx=\frac {(a+b \sec (c+d x))^{5/2} \left (16 \sqrt {\frac {b+a \cos (c+d x)}{a+b}} \left (a^2 \left (675 a^4 A+3315 a^2 A b^2+10 A b^4+2871 a^3 b B+1705 a b^3 B\right ) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 a}{a+b}\right )+\left (3705 a^4 A b+255 a^2 A b^3+40 A b^5+1617 a^5 B+3069 a^3 b^2 B-110 a b^4 B\right ) \left ((a+b) E\left (\frac {1}{2} (c+d x)|\frac {2 a}{a+b}\right )-b \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 a}{a+b}\right )\right )\right )+a (b+a \cos (c+d x)) \left (2 \left (6525 a^4 A+9330 a^2 A b^2-160 A b^4+16434 a^3 b B+440 a b^3 B\right ) \sin (c+d x)+a \left (4 \left (3095 a^2 A b+30 A b^3+1463 a^3 B+1650 a b^2 B\right ) \sin (2 (c+d x))+5 a \left (\left (513 a^2 A+452 A b^2+836 a b B\right ) \sin (3 (c+d x))+7 a ((46 A b+22 a B) \sin (4 (c+d x))+9 a A \sin (5 (c+d x)))\right )\right )\right )\right )}{27720 a^3 d (b+a \cos (c+d x))^3 \sec ^{\frac {5}{2}}(c+d x)} \]

[In]

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

[Out]

((a + b*Sec[c + d*x])^(5/2)*(16*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*(a^2*(675*a^4*A + 3315*a^2*A*b^2 + 10*A*b^4
+ 2871*a^3*b*B + 1705*a*b^3*B)*EllipticF[(c + d*x)/2, (2*a)/(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)*((a + b)*EllipticE[(c + d*x)/2, (2*a)/(a + b)] - b*EllipticF[(
c + d*x)/2, (2*a)/(a + b)])) + a*(b + a*Cos[c + d*x])*(2*(6525*a^4*A + 9330*a^2*A*b^2 - 160*A*b^4 + 16434*a^3*
b*B + 440*a*b^3*B)*Sin[c + d*x] + a*(4*(3095*a^2*A*b + 30*A*b^3 + 1463*a^3*B + 1650*a*b^2*B)*Sin[2*(c + d*x)]
+ 5*a*((513*a^2*A + 452*A*b^2 + 836*a*b*B)*Sin[3*(c + d*x)] + 7*a*((46*A*b + 22*a*B)*Sin[4*(c + d*x)] + 9*a*A*
Sin[5*(c + d*x)]))))))/(27720*a^3*d*(b + a*Cos[c + d*x])^3*Sec[c + d*x]^(5/2))

Maple [B] (warning: unable to verify)

Leaf count of result is larger than twice the leaf count of optimal. \(8174\) vs. \(2(531)=1062\).

Time = 23.68 (sec) , antiderivative size = 8175, normalized size of antiderivative = 15.75

method	result	size
parts	\(\text {Expression too large to display}\)	\(8175\)
default	\(\text {Expression too large to display}\)	\(8244\)

[In]

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

[Out]

result too large to display

Fricas [C] (verification not implemented)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 0.17 (sec) , antiderivative size = 753, normalized size of antiderivative = 1.45 \[ \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 {\sqrt {2} {\left (-2025 i \, A a^{6} - 5379 i \, B a^{5} b - 2535 i \, A a^{4} b^{2} + 1023 i \, B a^{3} b^{3} + 480 i \, A a^{2} b^{4} - 220 i \, B a b^{5} + 80 i \, A b^{6}\right )} \sqrt {a} {\rm weierstrassPInverse}\left (-\frac {4 \, {\left (3 \, a^{2} - 4 \, b^{2}\right )}}{3 \, a^{2}}, \frac {8 \, {\left (9 \, a^{2} b - 8 \, b^{3}\right )}}{27 \, a^{3}}, \frac {3 \, a \cos \left (d x + c\right ) + 3 i \, a \sin \left (d x + c\right ) + 2 \, b}{3 \, a}\right ) + \sqrt {2} {\left (2025 i \, A a^{6} + 5379 i \, B a^{5} b + 2535 i \, A a^{4} b^{2} - 1023 i \, B a^{3} b^{3} - 480 i \, A a^{2} b^{4} + 220 i \, B a b^{5} - 80 i \, A b^{6}\right )} \sqrt {a} {\rm weierstrassPInverse}\left (-\frac {4 \, {\left (3 \, a^{2} - 4 \, b^{2}\right )}}{3 \, a^{2}}, \frac {8 \, {\left (9 \, a^{2} b - 8 \, b^{3}\right )}}{27 \, a^{3}}, \frac {3 \, a \cos \left (d x + c\right ) - 3 i \, a \sin \left (d x + c\right ) + 2 \, b}{3 \, a}\right ) - 3 \, \sqrt {2} {\left (-1617 i \, B a^{6} - 3705 i \, A a^{5} b - 3069 i \, B a^{4} b^{2} - 255 i \, A a^{3} b^{3} + 110 i \, B a^{2} b^{4} - 40 i \, A a b^{5}\right )} \sqrt {a} {\rm weierstrassZeta}\left (-\frac {4 \, {\left (3 \, a^{2} - 4 \, b^{2}\right )}}{3 \, a^{2}}, \frac {8 \, {\left (9 \, a^{2} b - 8 \, b^{3}\right )}}{27 \, a^{3}}, {\rm weierstrassPInverse}\left (-\frac {4 \, {\left (3 \, a^{2} - 4 \, b^{2}\right )}}{3 \, a^{2}}, \frac {8 \, {\left (9 \, a^{2} b - 8 \, b^{3}\right )}}{27 \, a^{3}}, \frac {3 \, a \cos \left (d x + c\right ) + 3 i \, a \sin \left (d x + c\right ) + 2 \, b}{3 \, a}\right )\right ) - 3 \, \sqrt {2} {\left (1617 i \, B a^{6} + 3705 i \, A a^{5} b + 3069 i \, B a^{4} b^{2} + 255 i \, A a^{3} b^{3} - 110 i \, B a^{2} b^{4} + 40 i \, A a b^{5}\right )} \sqrt {a} {\rm weierstrassZeta}\left (-\frac {4 \, {\left (3 \, a^{2} - 4 \, b^{2}\right )}}{3 \, a^{2}}, \frac {8 \, {\left (9 \, a^{2} b - 8 \, b^{3}\right )}}{27 \, a^{3}}, {\rm weierstrassPInverse}\left (-\frac {4 \, {\left (3 \, a^{2} - 4 \, b^{2}\right )}}{3 \, a^{2}}, \frac {8 \, {\left (9 \, a^{2} b - 8 \, b^{3}\right )}}{27 \, a^{3}}, \frac {3 \, a \cos \left (d x + c\right ) - 3 i \, a \sin \left (d x + c\right ) + 2 \, b}{3 \, a}\right )\right ) + \frac {6 \, {\left (315 \, A a^{6} \cos \left (d x + c\right )^{5} + 35 \, {\left (11 \, B a^{6} + 23 \, A a^{5} b\right )} \cos \left (d x + c\right )^{4} + 5 \, {\left (81 \, A a^{6} + 209 \, B a^{5} b + 113 \, A a^{4} b^{2}\right )} \cos \left (d x + c\right )^{3} + {\left (539 \, B a^{6} + 1145 \, A a^{5} b + 825 \, B a^{4} b^{2} + 15 \, A a^{3} b^{3}\right )} \cos \left (d x + c\right )^{2} + {\left (675 \, A a^{6} + 1793 \, B a^{5} b + 1025 \, A a^{4} b^{2} + 55 \, B a^{3} b^{3} - 20 \, A a^{2} b^{4}\right )} \cos \left (d x + c\right )\right )} \sqrt {\frac {a \cos \left (d x + c\right ) + b}{\cos \left (d x + c\right )}} \sin \left (d x + c\right )}{\sqrt {\cos \left (d x + c\right )}}}{10395 \, a^{4} d} \]

[In]

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

[Out]

1/10395*(sqrt(2)*(-2025*I*A*a^6 - 5379*I*B*a^5*b - 2535*I*A*a^4*b^2 + 1023*I*B*a^3*b^3 + 480*I*A*a^2*b^4 - 220
*I*B*a*b^5 + 80*I*A*b^6)*sqrt(a)*weierstrassPInverse(-4/3*(3*a^2 - 4*b^2)/a^2, 8/27*(9*a^2*b - 8*b^3)/a^3, 1/3
*(3*a*cos(d*x + c) + 3*I*a*sin(d*x + c) + 2*b)/a) + sqrt(2)*(2025*I*A*a^6 + 5379*I*B*a^5*b + 2535*I*A*a^4*b^2
- 1023*I*B*a^3*b^3 - 480*I*A*a^2*b^4 + 220*I*B*a*b^5 - 80*I*A*b^6)*sqrt(a)*weierstrassPInverse(-4/3*(3*a^2 - 4
*b^2)/a^2, 8/27*(9*a^2*b - 8*b^3)/a^3, 1/3*(3*a*cos(d*x + c) - 3*I*a*sin(d*x + c) + 2*b)/a) - 3*sqrt(2)*(-1617
*I*B*a^6 - 3705*I*A*a^5*b - 3069*I*B*a^4*b^2 - 255*I*A*a^3*b^3 + 110*I*B*a^2*b^4 - 40*I*A*a*b^5)*sqrt(a)*weier
strassZeta(-4/3*(3*a^2 - 4*b^2)/a^2, 8/27*(9*a^2*b - 8*b^3)/a^3, weierstrassPInverse(-4/3*(3*a^2 - 4*b^2)/a^2,
8/27*(9*a^2*b - 8*b^3)/a^3, 1/3*(3*a*cos(d*x + c) + 3*I*a*sin(d*x + c) + 2*b)/a)) - 3*sqrt(2)*(1617*I*B*a^6 +
3705*I*A*a^5*b + 3069*I*B*a^4*b^2 + 255*I*A*a^3*b^3 - 110*I*B*a^2*b^4 + 40*I*A*a*b^5)*sqrt(a)*weierstrassZeta
(-4/3*(3*a^2 - 4*b^2)/a^2, 8/27*(9*a^2*b - 8*b^3)/a^3, weierstrassPInverse(-4/3*(3*a^2 - 4*b^2)/a^2, 8/27*(9*a
^2*b - 8*b^3)/a^3, 1/3*(3*a*cos(d*x + c) - 3*I*a*sin(d*x + c) + 2*b)/a)) + 6*(315*A*a^6*cos(d*x + c)^5 + 35*(1
1*B*a^6 + 23*A*a^5*b)*cos(d*x + c)^4 + 5*(81*A*a^6 + 209*B*a^5*b + 113*A*a^4*b^2)*cos(d*x + c)^3 + (539*B*a^6
+ 1145*A*a^5*b + 825*B*a^4*b^2 + 15*A*a^3*b^3)*cos(d*x + c)^2 + (675*A*a^6 + 1793*B*a^5*b + 1025*A*a^4*b^2 + 5
5*B*a^3*b^3 - 20*A*a^2*b^4)*cos(d*x + c))*sqrt((a*cos(d*x + c) + b)/cos(d*x + c))*sin(d*x + c)/sqrt(cos(d*x +
c)))/(a^4*d)

Sympy [F(-1)]

Timed out. \[ \int \frac {(a+b \sec (c+d x))^{5/2} (A+B \sec (c+d x))}{\sec ^{\frac {11}{2}}(c+d x)} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Maxima [F]

\[ \int \frac {(a+b \sec (c+d x))^{5/2} (A+B \sec (c+d x))}{\sec ^{\frac {11}{2}}(c+d x)} \, dx=\int { \frac {{\left (B \sec \left (d x + c\right ) + A\right )} {\left (b \sec \left (d x + c\right ) + a\right )}^{\frac {5}{2}}}{\sec \left (d x + c\right )^{\frac {11}{2}}} \,d x } \]

[In]

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

[Out]

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

Giac [F]

\[ \int \frac {(a+b \sec (c+d x))^{5/2} (A+B \sec (c+d x))}{\sec ^{\frac {11}{2}}(c+d x)} \, dx=\int { \frac {{\left (B \sec \left (d x + c\right ) + A\right )} {\left (b \sec \left (d x + c\right ) + a\right )}^{\frac {5}{2}}}{\sec \left (d x + c\right )^{\frac {11}{2}}} \,d x } \]

[In]

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

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int \frac {(a+b \sec (c+d x))^{5/2} (A+B \sec (c+d x))}{\sec ^{\frac {11}{2}}(c+d x)} \, dx=\int \frac {\left (A+\frac {B}{\cos \left (c+d\,x\right )}\right )\,{\left (a+\frac {b}{\cos \left (c+d\,x\right )}\right )}^{5/2}}{{\left (\frac {1}{\cos \left (c+d\,x\right )}\right )}^{11/2}} \,d x \]

[In]

int(((A + B/cos(c + d*x))*(a + b/cos(c + d*x))^(5/2))/(1/cos(c + d*x))^(11/2),x)

[Out]

int(((A + B/cos(c + d*x))*(a + b/cos(c + d*x))^(5/2))/(1/cos(c + d*x))^(11/2), x)