\(\int \frac {(a+b \sec (c+d x))^{5/2} (A+B \sec (c+d x)+C \sec ^2(c+d x))}{\sec ^{\frac {11}{2}}(c+d x)} \, dx\) [1051]

Optimal result

Integrand size = 45, antiderivative size = 565 \[ \int \frac {(a+b \sec (c+d x))^{5/2} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{\sec ^{\frac {11}{2}}(c+d x)} \, dx=\frac {2 \left (a^2-b^2\right ) \left (40 A b^4+1254 a^3 b B-110 a b^3 B+75 a^4 (9 A+11 C)+15 a^2 b^2 (19 A+33 C)\right ) \sqrt {\frac {b+a \cos (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 a}{a+b}\right ) \sqrt {\sec (c+d x)}}{3465 a^3 d \sqrt {a+b \sec (c+d x)}}+\frac {2 \left (40 A b^5+1617 a^5 B+3069 a^3 b^2 B-110 a b^4 B+15 a^2 b^3 (17 A+33 C)+15 a^4 b (247 A+319 C)\right ) E\left (\frac {1}{2} (c+d x)|\frac {2 a}{a+b}\right ) \sqrt {a+b \sec (c+d x)}}{3465 a^3 d \sqrt {\frac {b+a \cos (c+d x)}{a+b}} \sqrt {\sec (c+d x)}}+\frac {2 \left (5 A b^2+44 a b B+3 a^2 (9 A+11 C)\right ) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{231 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {2 \left (15 A b^3+539 a^3 B+825 a b^2 B+5 a^2 b (229 A+297 C)\right ) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{3465 a d \sec ^{\frac {3}{2}}(c+d x)}-\frac {2 \left (20 A b^4-1793 a^3 b B-55 a b^3 B-75 a^4 (9 A+11 C)-5 a^2 b^2 (205 A+297 C)\right ) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{3465 a^2 d \sqrt {\sec (c+d x)}}+\frac {2 (5 A b+11 a B) (a+b \sec (c+d x))^{3/2} \sin (c+d x)}{99 d \sec ^{\frac {7}{2}}(c+d x)}+\frac {2 A (a+b \sec (c+d x))^{5/2} \sin (c+d x)}{11 d \sec ^{\frac {9}{2}}(c+d x)} \]

[Out]

2/99*(5*A*b+11*B*a)*(a+b*sec(d*x+c))^(3/2)*sin(d*x+c)/d/sec(d*x+c)^(7/2)+2/11*A*(a+b*sec(d*x+c))^(5/2)*sin(d*x
+c)/d/sec(d*x+c)^(9/2)+2/3465*(a^2-b^2)*(40*A*b^4+1254*B*a^3*b-110*B*a*b^3+75*a^4*(9*A+11*C)+15*a^2*b^2*(19*A+
33*C))*(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)+2/231*(5*A*b^2+44*B*a*b+3*a^2*(9*A
+11*C))*sin(d*x+c)*(a+b*sec(d*x+c))^(1/2)/d/sec(d*x+c)^(5/2)+2/3465*(15*A*b^3+539*B*a^3+825*B*a*b^2+5*a^2*b*(2
29*A+297*C))*sin(d*x+c)*(a+b*sec(d*x+c))^(1/2)/a/d/sec(d*x+c)^(3/2)-2/3465*(20*A*b^4-1793*B*a^3*b-55*B*a*b^3-7
5*a^4*(9*A+11*C)-5*a^2*b^2*(205*A+297*C))*sin(d*x+c)*(a+b*sec(d*x+c))^(1/2)/a^2/d/sec(d*x+c)^(1/2)+2/3465*(40*
A*b^5+1617*a^5*B+3069*a^3*b^2*B-110*a*b^4*B+15*a^2*b^3*(17*A+33*C)+15*a^4*b*(247*A+319*C))*(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 = 2.51 (sec) , antiderivative size = 565, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {4179, 4189, 4120, 3941, 2734, 2732, 3943, 2742, 2740} \[ \int \frac {(a+b \sec (c+d x))^{5/2} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{\sec ^{\frac {11}{2}}(c+d x)} \, dx=\frac {2 \sin (c+d x) \left (3 a^2 (9 A+11 C)+44 a b B+5 A b^2\right ) \sqrt {a+b \sec (c+d x)}}{231 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {2 \sin (c+d x) \left (539 a^3 B+5 a^2 b (229 A+297 C)+825 a b^2 B+15 A b^3\right ) \sqrt {a+b \sec (c+d x)}}{3465 a d \sec ^{\frac {3}{2}}(c+d x)}-\frac {2 \sin (c+d x) \left (-75 a^4 (9 A+11 C)-1793 a^3 b B-5 a^2 b^2 (205 A+297 C)-55 a b^3 B+20 A b^4\right ) \sqrt {a+b \sec (c+d x)}}{3465 a^2 d \sqrt {\sec (c+d x)}}+\frac {2 \left (a^2-b^2\right ) \sqrt {\sec (c+d x)} \left (75 a^4 (9 A+11 C)+1254 a^3 b B+15 a^2 b^2 (19 A+33 C)-110 a b^3 B+40 A b^4\right ) \sqrt {\frac {a \cos (c+d x)+b}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 a}{a+b}\right )}{3465 a^3 d \sqrt {a+b \sec (c+d x)}}+\frac {2 \left (1617 a^5 B+15 a^4 b (247 A+319 C)+3069 a^3 b^2 B+15 a^2 b^3 (17 A+33 C)-110 a b^4 B+40 A b^5\right ) \sqrt {a+b \sec (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 a}{a+b}\right )}{3465 a^3 d \sqrt {\sec (c+d x)} \sqrt {\frac {a \cos (c+d x)+b}{a+b}}}+\frac {2 (11 a B+5 A b) \sin (c+d x) (a+b \sec (c+d x))^{3/2}}{99 d \sec ^{\frac {7}{2}}(c+d x)}+\frac {2 A \sin (c+d x) (a+b \sec (c+d x))^{5/2}}{11 d \sec ^{\frac {9}{2}}(c+d x)} \]

[In]

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

[Out]

(2*(a^2 - b^2)*(40*A*b^4 + 1254*a^3*b*B - 110*a*b^3*B + 75*a^4*(9*A + 11*C) + 15*a^2*b^2*(19*A + 33*C))*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*(40*A*b^5 + 1617*a^5*B + 3069*a^3*b^2*B - 110*a*b^4*B + 15*a^2*b^3*(17*A + 33*C) + 15*a^4*
b*(247*A + 319*C))*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*(5*A*b^2 + 44*a*b*B + 3*a^2*(9*A + 11*C))*Sqrt[a + b*Sec[c + d*x]
]*Sin[c + d*x])/(231*d*Sec[c + d*x]^(5/2)) + (2*(15*A*b^3 + 539*a^3*B + 825*a*b^2*B + 5*a^2*b*(229*A + 297*C))
*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(3465*a*d*Sec[c + d*x]^(3/2)) - (2*(20*A*b^4 - 1793*a^3*b*B - 55*a*b^3
*B - 75*a^4*(9*A + 11*C) - 5*a^2*b^2*(205*A + 297*C))*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(3465*a^2*d*Sqrt[
Sec[c + d*x]]) + (2*(5*A*b + 11*a*B)*(a + b*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(99*d*Sec[c + d*x]^(7/2)) + (2*A
*(a + b*Sec[c + d*x])^(5/2)*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 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+b \sec (c+d x))^{5/2} \sin (c+d x)}{11 d \sec ^{\frac {9}{2}}(c+d x)}+\frac {2}{11} \int \frac {(a+b \sec (c+d x))^{3/2} \left (\frac {1}{2} (5 A b+11 a B)+\frac {1}{2} (9 a A+11 b B+11 a C) \sec (c+d x)+\frac {1}{2} b (4 A+11 C) \sec ^2(c+d x)\right )}{\sec ^{\frac {9}{2}}(c+d x)} \, dx \\ & = \frac {2 (5 A b+11 a B) (a+b \sec (c+d x))^{3/2} \sin (c+d x)}{99 d \sec ^{\frac {7}{2}}(c+d x)}+\frac {2 A (a+b \sec (c+d x))^{5/2} \sin (c+d x)}{11 d \sec ^{\frac {9}{2}}(c+d x)}+\frac {4}{99} \int \frac {\sqrt {a+b \sec (c+d x)} \left (\frac {3}{4} \left (5 A b^2+44 a b B+3 a^2 (9 A+11 C)\right )+\frac {1}{4} \left (152 a A b+77 a^2 B+99 b^2 B+198 a b C\right ) \sec (c+d x)+\frac {1}{4} b (56 A b+44 a B+99 b C) \sec ^2(c+d x)\right )}{\sec ^{\frac {7}{2}}(c+d x)} \, dx \\ & = \frac {2 \left (5 A b^2+44 a b B+3 a^2 (9 A+11 C)\right ) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{231 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {2 (5 A b+11 a B) (a+b \sec (c+d x))^{3/2} \sin (c+d x)}{99 d \sec ^{\frac {7}{2}}(c+d x)}+\frac {2 A (a+b \sec (c+d x))^{5/2} \sin (c+d x)}{11 d \sec ^{\frac {9}{2}}(c+d x)}+\frac {8}{693} \int \frac {\frac {1}{8} \left (15 A b^3+539 a^3 B+825 a b^2 B+5 a^2 b (229 A+297 C)\right )+\frac {1}{8} \left (1507 a^2 b B+693 b^3 B+45 a^3 (9 A+11 C)+a b^2 (1531 A+2079 C)\right ) \sec (c+d x)+\frac {1}{8} b \left (836 a b B+36 a^2 (9 A+11 C)+b^2 (452 A+693 C)\right ) \sec ^2(c+d x)}{\sec ^{\frac {5}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}} \, dx \\ & = \frac {2 \left (5 A b^2+44 a b B+3 a^2 (9 A+11 C)\right ) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{231 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {2 \left (15 A b^3+539 a^3 B+825 a b^2 B+5 a^2 b (229 A+297 C)\right ) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{3465 a d \sec ^{\frac {3}{2}}(c+d x)}+\frac {2 (5 A b+11 a B) (a+b \sec (c+d x))^{3/2} \sin (c+d x)}{99 d \sec ^{\frac {7}{2}}(c+d x)}+\frac {2 A (a+b \sec (c+d x))^{5/2} \sin (c+d x)}{11 d \sec ^{\frac {9}{2}}(c+d x)}-\frac {16 \int \frac {\frac {3}{16} \left (20 A b^4-1793 a^3 b B-55 a b^3 B-75 a^4 (9 A+11 C)-5 a^2 b^2 (205 A+297 C)\right )-\frac {1}{16} a \left (1617 a^3 B+6655 a b^2 B+15 a^2 b (337 A+429 C)+5 b^3 (461 A+693 C)\right ) \sec (c+d x)-\frac {1}{8} b \left (15 A b^3+539 a^3 B+825 a b^2 B+5 a^2 b (229 A+297 C)\right ) \sec ^2(c+d x)}{\sec ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}} \, dx}{3465 a} \\ & = \frac {2 \left (5 A b^2+44 a b B+3 a^2 (9 A+11 C)\right ) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{231 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {2 \left (15 A b^3+539 a^3 B+825 a b^2 B+5 a^2 b (229 A+297 C)\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 (20 A b^4-1793 a^3 b B-55 a b^3 B-75 a^4 (9 A+11 C)-5 a^2 b^2 (205 A+297 C)\right ) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{3465 a^2 d \sqrt {\sec (c+d x)}}+\frac {2 (5 A b+11 a B) (a+b \sec (c+d x))^{3/2} \sin (c+d x)}{99 d \sec ^{\frac {7}{2}}(c+d x)}+\frac {2 A (a+b \sec (c+d x))^{5/2} \sin (c+d x)}{11 d \sec ^{\frac {9}{2}}(c+d x)}+\frac {32 \int \frac {\frac {3}{32} \left (40 A b^5+1617 a^5 B+3069 a^3 b^2 B-110 a b^4 B+15 a^2 b^3 (17 A+33 C)+15 a^4 b (247 A+319 C)\right )+\frac {3}{32} a \left (10 A b^4+2871 a^3 b B+1705 a b^3 B+75 a^4 (9 A+11 C)+15 a^2 b^2 (221 A+297 C)\right ) \sec (c+d x)}{\sqrt {\sec (c+d x)} \sqrt {a+b \sec (c+d x)}} \, dx}{10395 a^2} \\ & = \frac {2 \left (5 A b^2+44 a b B+3 a^2 (9 A+11 C)\right ) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{231 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {2 \left (15 A b^3+539 a^3 B+825 a b^2 B+5 a^2 b (229 A+297 C)\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 (20 A b^4-1793 a^3 b B-55 a b^3 B-75 a^4 (9 A+11 C)-5 a^2 b^2 (205 A+297 C)\right ) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{3465 a^2 d \sqrt {\sec (c+d x)}}+\frac {2 (5 A b+11 a B) (a+b \sec (c+d x))^{3/2} \sin (c+d x)}{99 d \sec ^{\frac {7}{2}}(c+d x)}+\frac {2 A (a+b \sec (c+d x))^{5/2} \sin (c+d x)}{11 d \sec ^{\frac {9}{2}}(c+d x)}+\frac {\left (\left (a^2-b^2\right ) \left (40 A b^4+1254 a^3 b B-110 a b^3 B+75 a^4 (9 A+11 C)+15 a^2 b^2 (19 A+33 C)\right )\right ) \int \frac {\sqrt {\sec (c+d x)}}{\sqrt {a+b \sec (c+d x)}} \, dx}{3465 a^3}+\frac {\left (40 A b^5+1617 a^5 B+3069 a^3 b^2 B-110 a b^4 B+15 a^2 b^3 (17 A+33 C)+15 a^4 b (247 A+319 C)\right ) \int \frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {\sec (c+d x)}} \, dx}{3465 a^3} \\ & = \frac {2 \left (5 A b^2+44 a b B+3 a^2 (9 A+11 C)\right ) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{231 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {2 \left (15 A b^3+539 a^3 B+825 a b^2 B+5 a^2 b (229 A+297 C)\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 (20 A b^4-1793 a^3 b B-55 a b^3 B-75 a^4 (9 A+11 C)-5 a^2 b^2 (205 A+297 C)\right ) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{3465 a^2 d \sqrt {\sec (c+d x)}}+\frac {2 (5 A b+11 a B) (a+b \sec (c+d x))^{3/2} \sin (c+d x)}{99 d \sec ^{\frac {7}{2}}(c+d x)}+\frac {2 A (a+b \sec (c+d x))^{5/2} \sin (c+d x)}{11 d \sec ^{\frac {9}{2}}(c+d x)}+\frac {\left (\left (a^2-b^2\right ) \left (40 A b^4+1254 a^3 b B-110 a b^3 B+75 a^4 (9 A+11 C)+15 a^2 b^2 (19 A+33 C)\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 (40 A b^5+1617 a^5 B+3069 a^3 b^2 B-110 a b^4 B+15 a^2 b^3 (17 A+33 C)+15 a^4 b (247 A+319 C)\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 \left (5 A b^2+44 a b B+3 a^2 (9 A+11 C)\right ) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{231 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {2 \left (15 A b^3+539 a^3 B+825 a b^2 B+5 a^2 b (229 A+297 C)\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 (20 A b^4-1793 a^3 b B-55 a b^3 B-75 a^4 (9 A+11 C)-5 a^2 b^2 (205 A+297 C)\right ) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{3465 a^2 d \sqrt {\sec (c+d x)}}+\frac {2 (5 A b+11 a B) (a+b \sec (c+d x))^{3/2} \sin (c+d x)}{99 d \sec ^{\frac {7}{2}}(c+d x)}+\frac {2 A (a+b \sec (c+d x))^{5/2} \sin (c+d x)}{11 d \sec ^{\frac {9}{2}}(c+d x)}+\frac {\left (\left (a^2-b^2\right ) \left (40 A b^4+1254 a^3 b B-110 a b^3 B+75 a^4 (9 A+11 C)+15 a^2 b^2 (19 A+33 C)\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 (40 A b^5+1617 a^5 B+3069 a^3 b^2 B-110 a b^4 B+15 a^2 b^3 (17 A+33 C)+15 a^4 b (247 A+319 C)\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 (40 A b^4+1254 a^3 b B-110 a b^3 B+75 a^4 (9 A+11 C)+15 a^2 b^2 (19 A+33 C)\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 (40 A b^5+1617 a^5 B+3069 a^3 b^2 B-110 a b^4 B+15 a^2 b^3 (17 A+33 C)+15 a^4 b (247 A+319 C)\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 \left (5 A b^2+44 a b B+3 a^2 (9 A+11 C)\right ) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{231 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {2 \left (15 A b^3+539 a^3 B+825 a b^2 B+5 a^2 b (229 A+297 C)\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 (20 A b^4-1793 a^3 b B-55 a b^3 B-75 a^4 (9 A+11 C)-5 a^2 b^2 (205 A+297 C)\right ) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{3465 a^2 d \sqrt {\sec (c+d x)}}+\frac {2 (5 A b+11 a B) (a+b \sec (c+d x))^{3/2} \sin (c+d x)}{99 d \sec ^{\frac {7}{2}}(c+d x)}+\frac {2 A (a+b \sec (c+d x))^{5/2} \sin (c+d x)}{11 d \sec ^{\frac {9}{2}}(c+d x)} \\ \end{align*}

Mathematica [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 6 vs. order 4 in optimal.

Time = 13.58 (sec) , antiderivative size = 7479, normalized size of antiderivative = 13.24 \[ \int \frac {(a+b \sec (c+d x))^{5/2} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{\sec ^{\frac {11}{2}}(c+d x)} \, dx=\text {Result too large to show} \]

[In]

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

[Out]

Result too large to show

Maple [B] (warning: unable to verify)

Leaf count of result is larger than twice the leaf count of optimal. \(10940\) vs. \(2(577)=1154\).

Time = 19.22 (sec) , antiderivative size = 10941, normalized size of antiderivative = 19.36

method	result	size
parts	\(\text {Expression too large to display}\)	\(10941\)
default	\(\text {Expression too large to display}\)	\(11011\)

[In]

int((a+b*sec(d*x+c))^(5/2)*(A+B*sec(d*x+c)+C*sec(d*x+c)^2)/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 = 0.19 (sec) , antiderivative size = 837, normalized size of antiderivative = 1.48 \[ \int \frac {(a+b \sec (c+d x))^{5/2} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{\sec ^{\frac {11}{2}}(c+d x)} \, dx=\text {Too large to display} \]

[In]

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

[Out]

1/10395*(sqrt(2)*(-225*I*(9*A + 11*C)*a^6 - 5379*I*B*a^5*b - 15*I*(169*A + 253*C)*a^4*b^2 + 1023*I*B*a^3*b^3 +
30*I*(16*A + 33*C)*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)*(225*I*(9*A + 11*
C)*a^6 + 5379*I*B*a^5*b + 15*I*(169*A + 253*C)*a^4*b^2 - 1023*I*B*a^3*b^3 - 30*I*(16*A + 33*C)*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 - 15*I*(247*A + 319*C)*a^5*b - 3069
*I*B*a^4*b^2 - 15*I*(17*A + 33*C)*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)) - 3*sqrt(2)*(1617*I*B*a^6 + 15*I*(247*A + 319*C)
*a^5*b + 3069*I*B*a^4*b^2 + 15*I*(17*A + 33*C)*a^3*b^3 - 110*I*B*a^2*b^4 + 40*I*A*a*b^5)*sqrt(a)*weierstrassZe
ta(-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*
(11*B*a^6 + 23*A*a^5*b)*cos(d*x + c)^4 + 5*(9*(9*A + 11*C)*a^6 + 209*B*a^5*b + 113*A*a^4*b^2)*cos(d*x + c)^3 +
(539*B*a^6 + 5*(229*A + 297*C)*a^5*b + 825*B*a^4*b^2 + 15*A*a^3*b^3)*cos(d*x + c)^2 + (75*(9*A + 11*C)*a^6 +
1793*B*a^5*b + 5*(205*A + 297*C)*a^4*b^2 + 55*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. \[ \int \frac {(a+b \sec (c+d x))^{5/2} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{\sec ^{\frac {11}{2}}(c+d x)} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

[In]

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

[Out]

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