3.1011 \(\int \frac{(a+b \sec (c+d x))^4 (A+B \sec (c+d x)+C \sec ^2(c+d x))}{\sec ^{\frac{13}{2}}(c+d x)} \, dx\)

Optimal. Leaf size=516 \[ \frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \text{EllipticF}\left (\frac{1}{2} (c+d x),2\right ) \left (20 a^3 b (9 A+11 C)+330 a^2 b^2 B+45 a^4 B+44 a b^3 (5 A+7 C)+77 b^4 B\right )}{231 d}+\frac{2 \sin (c+d x) \left (11 a^2 (11 A+13 C)+221 a b B+48 A b^2\right ) (a+b \sec (c+d x))^2}{1287 d \sec ^{\frac{7}{2}}(c+d x)}+\frac{2 \sin (c+d x) \left (11 a^2 b^2 (491 A+637 C)+77 a^4 (11 A+13 C)+4004 a^3 b B+3458 a b^3 B+192 A b^4\right )}{6435 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 a \sin (c+d x) \left (a^2 (2518 A b+3146 b C)+1053 a^3 B+2171 a b^2 B+192 A b^3\right )}{9009 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{2 \sin (c+d x) \left (20 a^3 b (9 A+11 C)+330 a^2 b^2 B+45 a^4 B+44 a b^3 (5 A+7 C)+77 b^4 B\right )}{231 d \sqrt{\sec (c+d x)}}+\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \left (78 a^2 b^2 (7 A+9 C)+a^4 (77 A+91 C)+364 a^3 b B+468 a b^3 B+39 b^4 (3 A+5 C)\right )}{195 d}+\frac{2 (13 a B+8 A b) \sin (c+d x) (a+b \sec (c+d x))^3}{143 d \sec ^{\frac{9}{2}}(c+d x)}+\frac{2 A \sin (c+d x) (a+b \sec (c+d x))^4}{13 d \sec ^{\frac{11}{2}}(c+d x)} \]

[Out]

(2*(364*a^3*b*B + 468*a*b^3*B + 39*b^4*(3*A + 5*C) + 78*a^2*b^2*(7*A + 9*C) + a^4*(77*A + 91*C))*Sqrt[Cos[c +
d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(195*d) + (2*(45*a^4*B + 330*a^2*b^2*B + 77*b^4*B + 44*a*b
^3*(5*A + 7*C) + 20*a^3*b*(9*A + 11*C))*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(231*
d) + (2*a*(192*A*b^3 + 1053*a^3*B + 2171*a*b^2*B + a^2*(2518*A*b + 3146*b*C))*Sin[c + d*x])/(9009*d*Sec[c + d*
x]^(5/2)) + (2*(192*A*b^4 + 4004*a^3*b*B + 3458*a*b^3*B + 77*a^4*(11*A + 13*C) + 11*a^2*b^2*(491*A + 637*C))*S
in[c + d*x])/(6435*d*Sec[c + d*x]^(3/2)) + (2*(45*a^4*B + 330*a^2*b^2*B + 77*b^4*B + 44*a*b^3*(5*A + 7*C) + 20
*a^3*b*(9*A + 11*C))*Sin[c + d*x])/(231*d*Sqrt[Sec[c + d*x]]) + (2*(48*A*b^2 + 221*a*b*B + 11*a^2*(11*A + 13*C
))*(a + b*Sec[c + d*x])^2*Sin[c + d*x])/(1287*d*Sec[c + d*x]^(7/2)) + (2*(8*A*b + 13*a*B)*(a + b*Sec[c + d*x])
^3*Sin[c + d*x])/(143*d*Sec[c + d*x]^(9/2)) + (2*A*(a + b*Sec[c + d*x])^4*Sin[c + d*x])/(13*d*Sec[c + d*x]^(11
/2))

________________________________________________________________________________________

Rubi [A]  time = 1.40058, antiderivative size = 516, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 8, integrand size = 43, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.186, Rules used = {4094, 4074, 4047, 3769, 3771, 2641, 4045, 2639} \[ \frac{2 \sin (c+d x) \left (11 a^2 (11 A+13 C)+221 a b B+48 A b^2\right ) (a+b \sec (c+d x))^2}{1287 d \sec ^{\frac{7}{2}}(c+d x)}+\frac{2 \sin (c+d x) \left (11 a^2 b^2 (491 A+637 C)+77 a^4 (11 A+13 C)+4004 a^3 b B+3458 a b^3 B+192 A b^4\right )}{6435 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 a \sin (c+d x) \left (a^2 (2518 A b+3146 b C)+1053 a^3 B+2171 a b^2 B+192 A b^3\right )}{9009 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{2 \sin (c+d x) \left (20 a^3 b (9 A+11 C)+330 a^2 b^2 B+45 a^4 B+44 a b^3 (5 A+7 C)+77 b^4 B\right )}{231 d \sqrt{\sec (c+d x)}}+\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \left (20 a^3 b (9 A+11 C)+330 a^2 b^2 B+45 a^4 B+44 a b^3 (5 A+7 C)+77 b^4 B\right )}{231 d}+\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \left (78 a^2 b^2 (7 A+9 C)+a^4 (77 A+91 C)+364 a^3 b B+468 a b^3 B+39 b^4 (3 A+5 C)\right )}{195 d}+\frac{2 (13 a B+8 A b) \sin (c+d x) (a+b \sec (c+d x))^3}{143 d \sec ^{\frac{9}{2}}(c+d x)}+\frac{2 A \sin (c+d x) (a+b \sec (c+d x))^4}{13 d \sec ^{\frac{11}{2}}(c+d x)} \]

Antiderivative was successfully verified.

[In]

Int[((a + b*Sec[c + d*x])^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/Sec[c + d*x]^(13/2),x]

[Out]

(2*(364*a^3*b*B + 468*a*b^3*B + 39*b^4*(3*A + 5*C) + 78*a^2*b^2*(7*A + 9*C) + a^4*(77*A + 91*C))*Sqrt[Cos[c +
d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(195*d) + (2*(45*a^4*B + 330*a^2*b^2*B + 77*b^4*B + 44*a*b
^3*(5*A + 7*C) + 20*a^3*b*(9*A + 11*C))*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(231*
d) + (2*a*(192*A*b^3 + 1053*a^3*B + 2171*a*b^2*B + a^2*(2518*A*b + 3146*b*C))*Sin[c + d*x])/(9009*d*Sec[c + d*
x]^(5/2)) + (2*(192*A*b^4 + 4004*a^3*b*B + 3458*a*b^3*B + 77*a^4*(11*A + 13*C) + 11*a^2*b^2*(491*A + 637*C))*S
in[c + d*x])/(6435*d*Sec[c + d*x]^(3/2)) + (2*(45*a^4*B + 330*a^2*b^2*B + 77*b^4*B + 44*a*b^3*(5*A + 7*C) + 20
*a^3*b*(9*A + 11*C))*Sin[c + d*x])/(231*d*Sqrt[Sec[c + d*x]]) + (2*(48*A*b^2 + 221*a*b*B + 11*a^2*(11*A + 13*C
))*(a + b*Sec[c + d*x])^2*Sin[c + d*x])/(1287*d*Sec[c + d*x]^(7/2)) + (2*(8*A*b + 13*a*B)*(a + b*Sec[c + d*x])
^3*Sin[c + d*x])/(143*d*Sec[c + d*x]^(9/2)) + (2*A*(a + b*Sec[c + d*x])^4*Sin[c + d*x])/(13*d*Sec[c + d*x]^(11
/2))

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 4074

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_)), x_Symbol] :> Simp[(A*a*Cot[e + f*x]*(d*Csc[e + f*x])^n)/(f*n), x]
+ Dist[1/(d*n), Int[(d*Csc[e + f*x])^(n + 1)*Simp[n*(B*a + A*b) + (n*(a*C + B*b) + A*a*(n + 1))*Csc[e + f*x]
+ b*C*n*Csc[e + f*x]^2, x], x], x] /; FreeQ[{a, b, d, e, f, A, B, C}, x] && LtQ[n, -1]

Rule 4047

Int[(csc[(e_.) + (f_.)*(x_)]*(b_.))^(m_.)*((A_.) + csc[(e_.) + (f_.)*(x_)]*(B_.) + csc[(e_.) + (f_.)*(x_)]^2*(
C_.)), x_Symbol] :> Dist[B/b, Int[(b*Csc[e + f*x])^(m + 1), x], x] + Int[(b*Csc[e + f*x])^m*(A + C*Csc[e + f*x
]^2), x] /; FreeQ[{b, e, f, A, B, C, m}, x]

Rule 3769

Int[(csc[(c_.) + (d_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Simp[(Cos[c + d*x]*(b*Csc[c + d*x])^(n + 1))/(b*d*n), x
] + Dist[(n + 1)/(b^2*n), Int[(b*Csc[c + d*x])^(n + 2), x], x] /; FreeQ[{b, c, d}, x] && LtQ[n, -1] && Integer
Q[2*n]

Rule 3771

Int[(csc[(c_.) + (d_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Dist[(b*Csc[c + d*x])^n*Sin[c + d*x]^n, Int[1/Sin[c + d
*x]^n, x], x] /; FreeQ[{b, c, d}, x] && EqQ[n^2, 1/4]

Rule 2641

Int[1/Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2*EllipticF[(1*(c - Pi/2 + d*x))/2, 2])/d, x] /; FreeQ
[{c, d}, x]

Rule 4045

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

Rule 2639

Int[Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2*EllipticE[(1*(c - Pi/2 + d*x))/2, 2])/d, x] /; FreeQ[{
c, d}, x]

Rubi steps

\begin{align*} \int \frac{(a+b \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{\sec ^{\frac{13}{2}}(c+d x)} \, dx &=\frac{2 A (a+b \sec (c+d x))^4 \sin (c+d x)}{13 d \sec ^{\frac{11}{2}}(c+d x)}+\frac{2}{13} \int \frac{(a+b \sec (c+d x))^3 \left (\frac{1}{2} (8 A b+13 a B)+\frac{1}{2} (11 a A+13 b B+13 a C) \sec (c+d x)+\frac{1}{2} b (3 A+13 C) \sec ^2(c+d x)\right )}{\sec ^{\frac{11}{2}}(c+d x)} \, dx\\ &=\frac{2 (8 A b+13 a B) (a+b \sec (c+d x))^3 \sin (c+d x)}{143 d \sec ^{\frac{9}{2}}(c+d x)}+\frac{2 A (a+b \sec (c+d x))^4 \sin (c+d x)}{13 d \sec ^{\frac{11}{2}}(c+d x)}+\frac{4}{143} \int \frac{(a+b \sec (c+d x))^2 \left (\frac{1}{4} \left (48 A b^2+221 a b B+11 a^2 (11 A+13 C)\right )+\frac{1}{4} \left (226 a A b+117 a^2 B+143 b^2 B+286 a b C\right ) \sec (c+d x)+\frac{1}{4} b (57 A b+39 a B+143 b C) \sec ^2(c+d x)\right )}{\sec ^{\frac{9}{2}}(c+d x)} \, dx\\ &=\frac{2 \left (48 A b^2+221 a b B+11 a^2 (11 A+13 C)\right ) (a+b \sec (c+d x))^2 \sin (c+d x)}{1287 d \sec ^{\frac{7}{2}}(c+d x)}+\frac{2 (8 A b+13 a B) (a+b \sec (c+d x))^3 \sin (c+d x)}{143 d \sec ^{\frac{9}{2}}(c+d x)}+\frac{2 A (a+b \sec (c+d x))^4 \sin (c+d x)}{13 d \sec ^{\frac{11}{2}}(c+d x)}+\frac{8 \int \frac{(a+b \sec (c+d x)) \left (\frac{1}{8} \left (192 A b^3+1053 a^3 B+2171 a b^2 B+a^2 (2518 A b+3146 b C)\right )+\frac{1}{8} \left (2951 a^2 b B+1287 b^3 B+77 a^3 (11 A+13 C)+3 a b^2 (961 A+1287 C)\right ) \sec (c+d x)+\frac{3}{8} b \left (338 a b B+11 a^2 (11 A+13 C)+3 b^2 (73 A+143 C)\right ) \sec ^2(c+d x)\right )}{\sec ^{\frac{7}{2}}(c+d x)} \, dx}{1287}\\ &=\frac{2 a \left (192 A b^3+1053 a^3 B+2171 a b^2 B+a^2 (2518 A b+3146 b C)\right ) \sin (c+d x)}{9009 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{2 \left (48 A b^2+221 a b B+11 a^2 (11 A+13 C)\right ) (a+b \sec (c+d x))^2 \sin (c+d x)}{1287 d \sec ^{\frac{7}{2}}(c+d x)}+\frac{2 (8 A b+13 a B) (a+b \sec (c+d x))^3 \sin (c+d x)}{143 d \sec ^{\frac{9}{2}}(c+d x)}+\frac{2 A (a+b \sec (c+d x))^4 \sin (c+d x)}{13 d \sec ^{\frac{11}{2}}(c+d x)}-\frac{16 \int \frac{-\frac{7}{16} \left (192 A b^4+4004 a^3 b B+3458 a b^3 B+77 a^4 (11 A+13 C)+11 a^2 b^2 (491 A+637 C)\right )-\frac{117}{16} \left (45 a^4 B+330 a^2 b^2 B+77 b^4 B+44 a b^3 (5 A+7 C)+20 a^3 b (9 A+11 C)\right ) \sec (c+d x)-\frac{21}{16} b^2 \left (338 a b B+11 a^2 (11 A+13 C)+3 b^2 (73 A+143 C)\right ) \sec ^2(c+d x)}{\sec ^{\frac{5}{2}}(c+d x)} \, dx}{9009}\\ &=\frac{2 a \left (192 A b^3+1053 a^3 B+2171 a b^2 B+a^2 (2518 A b+3146 b C)\right ) \sin (c+d x)}{9009 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{2 \left (48 A b^2+221 a b B+11 a^2 (11 A+13 C)\right ) (a+b \sec (c+d x))^2 \sin (c+d x)}{1287 d \sec ^{\frac{7}{2}}(c+d x)}+\frac{2 (8 A b+13 a B) (a+b \sec (c+d x))^3 \sin (c+d x)}{143 d \sec ^{\frac{9}{2}}(c+d x)}+\frac{2 A (a+b \sec (c+d x))^4 \sin (c+d x)}{13 d \sec ^{\frac{11}{2}}(c+d x)}-\frac{16 \int \frac{-\frac{7}{16} \left (192 A b^4+4004 a^3 b B+3458 a b^3 B+77 a^4 (11 A+13 C)+11 a^2 b^2 (491 A+637 C)\right )-\frac{21}{16} b^2 \left (338 a b B+11 a^2 (11 A+13 C)+3 b^2 (73 A+143 C)\right ) \sec ^2(c+d x)}{\sec ^{\frac{5}{2}}(c+d x)} \, dx}{9009}-\frac{1}{77} \left (-45 a^4 B-330 a^2 b^2 B-77 b^4 B-44 a b^3 (5 A+7 C)-20 a^3 b (9 A+11 C)\right ) \int \frac{1}{\sec ^{\frac{3}{2}}(c+d x)} \, dx\\ &=\frac{2 a \left (192 A b^3+1053 a^3 B+2171 a b^2 B+a^2 (2518 A b+3146 b C)\right ) \sin (c+d x)}{9009 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{2 \left (192 A b^4+4004 a^3 b B+3458 a b^3 B+77 a^4 (11 A+13 C)+11 a^2 b^2 (491 A+637 C)\right ) \sin (c+d x)}{6435 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 \left (45 a^4 B+330 a^2 b^2 B+77 b^4 B+44 a b^3 (5 A+7 C)+20 a^3 b (9 A+11 C)\right ) \sin (c+d x)}{231 d \sqrt{\sec (c+d x)}}+\frac{2 \left (48 A b^2+221 a b B+11 a^2 (11 A+13 C)\right ) (a+b \sec (c+d x))^2 \sin (c+d x)}{1287 d \sec ^{\frac{7}{2}}(c+d x)}+\frac{2 (8 A b+13 a B) (a+b \sec (c+d x))^3 \sin (c+d x)}{143 d \sec ^{\frac{9}{2}}(c+d x)}+\frac{2 A (a+b \sec (c+d x))^4 \sin (c+d x)}{13 d \sec ^{\frac{11}{2}}(c+d x)}-\frac{1}{231} \left (-45 a^4 B-330 a^2 b^2 B-77 b^4 B-44 a b^3 (5 A+7 C)-20 a^3 b (9 A+11 C)\right ) \int \sqrt{\sec (c+d x)} \, dx-\frac{1}{195} \left (-364 a^3 b B-468 a b^3 B-39 b^4 (3 A+5 C)-78 a^2 b^2 (7 A+9 C)-7 a^4 (11 A+13 C)\right ) \int \frac{1}{\sqrt{\sec (c+d x)}} \, dx\\ &=\frac{2 a \left (192 A b^3+1053 a^3 B+2171 a b^2 B+a^2 (2518 A b+3146 b C)\right ) \sin (c+d x)}{9009 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{2 \left (192 A b^4+4004 a^3 b B+3458 a b^3 B+77 a^4 (11 A+13 C)+11 a^2 b^2 (491 A+637 C)\right ) \sin (c+d x)}{6435 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 \left (45 a^4 B+330 a^2 b^2 B+77 b^4 B+44 a b^3 (5 A+7 C)+20 a^3 b (9 A+11 C)\right ) \sin (c+d x)}{231 d \sqrt{\sec (c+d x)}}+\frac{2 \left (48 A b^2+221 a b B+11 a^2 (11 A+13 C)\right ) (a+b \sec (c+d x))^2 \sin (c+d x)}{1287 d \sec ^{\frac{7}{2}}(c+d x)}+\frac{2 (8 A b+13 a B) (a+b \sec (c+d x))^3 \sin (c+d x)}{143 d \sec ^{\frac{9}{2}}(c+d x)}+\frac{2 A (a+b \sec (c+d x))^4 \sin (c+d x)}{13 d \sec ^{\frac{11}{2}}(c+d x)}-\frac{1}{231} \left (\left (-45 a^4 B-330 a^2 b^2 B-77 b^4 B-44 a b^3 (5 A+7 C)-20 a^3 b (9 A+11 C)\right ) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{1}{\sqrt{\cos (c+d x)}} \, dx-\frac{1}{195} \left (\left (-364 a^3 b B-468 a b^3 B-39 b^4 (3 A+5 C)-78 a^2 b^2 (7 A+9 C)-7 a^4 (11 A+13 C)\right ) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \sqrt{\cos (c+d x)} \, dx\\ &=\frac{2 \left (364 a^3 b B+468 a b^3 B+39 b^4 (3 A+5 C)+78 a^2 b^2 (7 A+9 C)+a^4 (77 A+91 C)\right ) \sqrt{\cos (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \sqrt{\sec (c+d x)}}{195 d}+\frac{2 \left (45 a^4 B+330 a^2 b^2 B+77 b^4 B+44 a b^3 (5 A+7 C)+20 a^3 b (9 A+11 C)\right ) \sqrt{\cos (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \sqrt{\sec (c+d x)}}{231 d}+\frac{2 a \left (192 A b^3+1053 a^3 B+2171 a b^2 B+a^2 (2518 A b+3146 b C)\right ) \sin (c+d x)}{9009 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{2 \left (192 A b^4+4004 a^3 b B+3458 a b^3 B+77 a^4 (11 A+13 C)+11 a^2 b^2 (491 A+637 C)\right ) \sin (c+d x)}{6435 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 \left (45 a^4 B+330 a^2 b^2 B+77 b^4 B+44 a b^3 (5 A+7 C)+20 a^3 b (9 A+11 C)\right ) \sin (c+d x)}{231 d \sqrt{\sec (c+d x)}}+\frac{2 \left (48 A b^2+221 a b B+11 a^2 (11 A+13 C)\right ) (a+b \sec (c+d x))^2 \sin (c+d x)}{1287 d \sec ^{\frac{7}{2}}(c+d x)}+\frac{2 (8 A b+13 a B) (a+b \sec (c+d x))^3 \sin (c+d x)}{143 d \sec ^{\frac{9}{2}}(c+d x)}+\frac{2 A (a+b \sec (c+d x))^4 \sin (c+d x)}{13 d \sec ^{\frac{11}{2}}(c+d x)}\\ \end{align*}

Mathematica [A]  time = 7.14017, size = 658, normalized size = 1.28 \[ \frac{2 \cos ^6(c+d x) (a+b \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \left (2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \text{EllipticF}\left (\frac{1}{2} (c+d x),2\right ) \left (11700 a^3 A b+21450 a^2 b^2 B+14300 a^3 b C+2925 a^4 B+14300 a A b^3+20020 a b^3 C+5005 b^4 B\right )+\frac{2 E\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \left (42042 a^2 A b^2+5929 a^4 A+54054 a^2 b^2 C+28028 a^3 b B+7007 a^4 C+36036 a b^3 B+9009 A b^4+15015 b^4 C\right )}{\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}}\right )}{15015 d (a \cos (c+d x)+b)^4 (A \cos (2 c+2 d x)+A+2 B \cos (c+d x)+2 C)}+\frac{(a+b \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \left (\frac{a^2 \sin (5 (c+d x)) \left (89 a^2 A+52 a^2 C+208 a b B+312 A b^2\right )}{1872}+\frac{1}{77} a \sin (4 (c+d x)) \left (32 a^2 A b+22 a^2 b C+8 a^3 B+33 a b^2 B+22 A b^3\right )+\frac{\sin (c+d x) \left (11856 a^2 A b^2+1897 a^4 A+11232 a^2 b^2 C+7904 a^3 b B+1976 a^4 C+7488 a b^3 B+1872 A b^4\right )}{9360}+\frac{\sin (2 (c+d x)) \left (4164 a^3 A b+6864 a^2 b^2 B+4576 a^3 b C+1041 a^4 B+4576 a A b^3+4928 a b^3 C+1232 b^4 B\right )}{1848}+\frac{\sin (3 (c+d x)) \left (13416 a^2 A b^2+2297 a^4 A+11232 a^2 b^2 C+8944 a^3 b B+2236 a^4 C+7488 a b^3 B+1872 A b^4\right )}{9360}+\frac{1}{88} a^3 (a B+4 A b) \sin (6 (c+d x))+\frac{1}{208} a^4 A \sin (7 (c+d x))\right )}{d \sec ^{\frac{11}{2}}(c+d x) (a \cos (c+d x)+b)^4 (A \cos (2 c+2 d x)+A+2 B \cos (c+d x)+2 C)} \]

Antiderivative was successfully verified.

[In]

Integrate[((a + b*Sec[c + d*x])^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/Sec[c + d*x]^(13/2),x]

[Out]

(2*Cos[c + d*x]^6*((2*(5929*a^4*A + 42042*a^2*A*b^2 + 9009*A*b^4 + 28028*a^3*b*B + 36036*a*b^3*B + 7007*a^4*C
+ 54054*a^2*b^2*C + 15015*b^4*C)*EllipticE[(c + d*x)/2, 2])/(Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) + 2*(11700
*a^3*A*b + 14300*a*A*b^3 + 2925*a^4*B + 21450*a^2*b^2*B + 5005*b^4*B + 14300*a^3*b*C + 20020*a*b^3*C)*Sqrt[Cos
[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])*(a + b*Sec[c + d*x])^4*(A + B*Sec[c + d*x] + C*Sec[c
+ d*x]^2))/(15015*d*(b + a*Cos[c + d*x])^4*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])) + ((a + b*Sec[c
+ d*x])^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*(((1897*a^4*A + 11856*a^2*A*b^2 + 1872*A*b^4 + 7904*a^3*b*B
+ 7488*a*b^3*B + 1976*a^4*C + 11232*a^2*b^2*C)*Sin[c + d*x])/9360 + ((4164*a^3*A*b + 4576*a*A*b^3 + 1041*a^4*B
+ 6864*a^2*b^2*B + 1232*b^4*B + 4576*a^3*b*C + 4928*a*b^3*C)*Sin[2*(c + d*x)])/1848 + ((2297*a^4*A + 13416*a^
2*A*b^2 + 1872*A*b^4 + 8944*a^3*b*B + 7488*a*b^3*B + 2236*a^4*C + 11232*a^2*b^2*C)*Sin[3*(c + d*x)])/9360 + (a
*(32*a^2*A*b + 22*A*b^3 + 8*a^3*B + 33*a*b^2*B + 22*a^2*b*C)*Sin[4*(c + d*x)])/77 + (a^2*(89*a^2*A + 312*A*b^2
+ 208*a*b*B + 52*a^2*C)*Sin[5*(c + d*x)])/1872 + (a^3*(4*A*b + a*B)*Sin[6*(c + d*x)])/88 + (a^4*A*Sin[7*(c +
d*x)])/208))/(d*(b + a*Cos[c + d*x])^4*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sec[c + d*x]^(11/2))

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Maple [B]  time = 2.689, size = 1407, normalized size = 2.7 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*sec(d*x+c))^4*(A+B*sec(d*x+c)+C*sec(d*x+c)^2)/sec(d*x+c)^(13/2),x)

[Out]

-2/45045*((2*cos(1/2*d*x+1/2*c)^2-1)*sin(1/2*d*x+1/2*c)^2)^(1/2)*(-443520*A*a^4*cos(1/2*d*x+1/2*c)*sin(1/2*d*x
+1/2*c)^14+(1330560*A*a^4+1048320*A*a^3*b+262080*B*a^4)*sin(1/2*d*x+1/2*c)^12*cos(1/2*d*x+1/2*c)+(-1798720*A*a
^4-2620800*A*a^3*b-960960*A*a^2*b^2-655200*B*a^4-640640*B*a^3*b-160160*C*a^4)*sin(1/2*d*x+1/2*c)^10*cos(1/2*d*
x+1/2*c)+(1379840*A*a^4+2957760*A*a^3*b+1921920*A*a^2*b^2+411840*A*a*b^3+739440*B*a^4+1281280*B*a^3*b+617760*B
*a^2*b^2+320320*C*a^4+411840*C*a^3*b)*sin(1/2*d*x+1/2*c)^8*cos(1/2*d*x+1/2*c)+(-666512*A*a^4-1815840*A*a^3*b-1
777776*A*a^2*b^2-617760*A*a*b^3-72072*A*b^4-453960*B*a^4-1185184*B*a^3*b-926640*B*a^2*b^2-288288*B*a*b^3-29629
6*C*a^4-617760*C*a^3*b-432432*C*a^2*b^2)*sin(1/2*d*x+1/2*c)^6*cos(1/2*d*x+1/2*c)+(198352*A*a^4+720720*A*a^3*b+
816816*A*a^2*b^2+480480*A*a*b^3+72072*A*b^4+180180*B*a^4+544544*B*a^3*b+720720*B*a^2*b^2+288288*B*a*b^3+60060*
B*b^4+136136*C*a^4+480480*C*a^3*b+432432*C*a^2*b^2+240240*C*a*b^3)*sin(1/2*d*x+1/2*c)^4*cos(1/2*d*x+1/2*c)+(-2
7258*A*a^4-145080*A*a^3*b-144144*A*a^2*b^2-137280*A*a*b^3-18018*A*b^4-36270*B*a^4-96096*B*a^3*b-205920*B*a^2*b
^2-72072*B*a*b^3-30030*B*b^4-24024*C*a^4-137280*C*a^3*b-108108*C*a^2*b^2-120120*C*a*b^3)*sin(1/2*d*x+1/2*c)^2*
cos(1/2*d*x+1/2*c)+35100*A*a^3*b*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*EllipticF(cos(1
/2*d*x+1/2*c),2^(1/2))+42900*A*a*b^3*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*EllipticF(c
os(1/2*d*x+1/2*c),2^(1/2))-17787*A*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*EllipticE(cos
(1/2*d*x+1/2*c),2^(1/2))*a^4-126126*A*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*EllipticE(
cos(1/2*d*x+1/2*c),2^(1/2))*a^2*b^2-27027*A*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*Elli
pticE(cos(1/2*d*x+1/2*c),2^(1/2))*b^4+8775*B*a^4*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)
*EllipticF(cos(1/2*d*x+1/2*c),2^(1/2))+64350*a^2*b^2*B*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(2*sin(1/2*d*x+1/2*c)^2-1)
^(1/2)*EllipticF(cos(1/2*d*x+1/2*c),2^(1/2))+15015*B*b^4*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(2*sin(1/2*d*x+1/2*c)^2-
1)^(1/2)*EllipticF(cos(1/2*d*x+1/2*c),2^(1/2))-84084*B*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(2*sin(1/2*d*x+1/2*c)^2-1)
^(1/2)*EllipticE(cos(1/2*d*x+1/2*c),2^(1/2))*a^3*b-108108*B*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(2*sin(1/2*d*x+1/2*c)
^2-1)^(1/2)*EllipticE(cos(1/2*d*x+1/2*c),2^(1/2))*a*b^3+42900*a^3*b*C*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(2*sin(1/2*
d*x+1/2*c)^2-1)^(1/2)*EllipticF(cos(1/2*d*x+1/2*c),2^(1/2))+60060*C*a*b^3*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(2*sin(
1/2*d*x+1/2*c)^2-1)^(1/2)*EllipticF(cos(1/2*d*x+1/2*c),2^(1/2))-21021*C*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(2*sin(1/
2*d*x+1/2*c)^2-1)^(1/2)*EllipticE(cos(1/2*d*x+1/2*c),2^(1/2))*a^4-162162*C*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(2*sin
(1/2*d*x+1/2*c)^2-1)^(1/2)*EllipticE(cos(1/2*d*x+1/2*c),2^(1/2))*a^2*b^2-45045*C*(sin(1/2*d*x+1/2*c)^2)^(1/2)*
(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*EllipticE(cos(1/2*d*x+1/2*c),2^(1/2))*b^4)/(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d
*x+1/2*c)^2)^(1/2)/sin(1/2*d*x+1/2*c)/(2*cos(1/2*d*x+1/2*c)^2-1)^(1/2)/d

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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((a+b*sec(d*x+c))^4*(A+B*sec(d*x+c)+C*sec(d*x+c)^2)/sec(d*x+c)^(13/2),x, algorithm="maxima")

[Out]

Timed out

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{C b^{4} \sec \left (d x + c\right )^{6} +{\left (4 \, C a b^{3} + B b^{4}\right )} \sec \left (d x + c\right )^{5} + A a^{4} +{\left (6 \, C a^{2} b^{2} + 4 \, B a b^{3} + A b^{4}\right )} \sec \left (d x + c\right )^{4} + 2 \,{\left (2 \, C a^{3} b + 3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} \sec \left (d x + c\right )^{3} +{\left (C a^{4} + 4 \, B a^{3} b + 6 \, A a^{2} b^{2}\right )} \sec \left (d x + c\right )^{2} +{\left (B a^{4} + 4 \, A a^{3} b\right )} \sec \left (d x + c\right )}{\sec \left (d x + c\right )^{\frac{13}{2}}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*sec(d*x+c))^4*(A+B*sec(d*x+c)+C*sec(d*x+c)^2)/sec(d*x+c)^(13/2),x, algorithm="fricas")

[Out]

integral((C*b^4*sec(d*x + c)^6 + (4*C*a*b^3 + B*b^4)*sec(d*x + c)^5 + A*a^4 + (6*C*a^2*b^2 + 4*B*a*b^3 + A*b^4
)*sec(d*x + c)^4 + 2*(2*C*a^3*b + 3*B*a^2*b^2 + 2*A*a*b^3)*sec(d*x + c)^3 + (C*a^4 + 4*B*a^3*b + 6*A*a^2*b^2)*
sec(d*x + c)^2 + (B*a^4 + 4*A*a^3*b)*sec(d*x + c))/sec(d*x + c)^(13/2), 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((a+b*sec(d*x+c))**4*(A+B*sec(d*x+c)+C*sec(d*x+c)**2)/sec(d*x+c)**(13/2),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \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 )}^{4}}{\sec \left (d x + c\right )^{\frac{13}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*sec(d*x+c))^4*(A+B*sec(d*x+c)+C*sec(d*x+c)^2)/sec(d*x+c)^(13/2),x, algorithm="giac")

[Out]

integrate((C*sec(d*x + c)^2 + B*sec(d*x + c) + A)*(b*sec(d*x + c) + a)^4/sec(d*x + c)^(13/2), x)