3.1004 \(\int \sqrt{\sec (c+d x)} (a+b \sec (c+d x))^4 (A+B \sec (c+d x)+C \sec ^2(c+d x)) \, dx\)

Optimal. Leaf size=515 \[ \frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \text{EllipticF}\left (\frac{1}{2} (c+d x),2\right ) \left (66 a^2 b^2 (7 A+5 C)+77 a^4 (3 A+C)+308 a^3 b B+220 a b^3 B+5 b^4 (11 A+9 C)\right )}{231 d}+\frac{2 \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) \left (16 a^2 C+55 a b B+33 A b^2+27 b^2 C\right ) (a+b \sec (c+d x))^2}{231 d}+\frac{2 b \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x) \left (1353 a^2 b B+192 a^3 C+2 a b^2 (891 A+673 C)+539 b^3 B\right )}{3465 d}+\frac{2 \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) \left (9 a^2 b^2 (143 A+101 C)+682 a^3 b B+64 a^4 C+660 a b^3 B+15 b^4 (11 A+9 C)\right )}{693 d}+\frac{2 \sin (c+d x) \sqrt{\sec (c+d x)} \left (12 a^3 b (5 A+3 C)+54 a^2 b^2 B+15 a^4 B+4 a b^3 (9 A+7 C)+7 b^4 B\right )}{15 d}-\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \left (12 a^3 b (5 A+3 C)+54 a^2 b^2 B+15 a^4 B+4 a b^3 (9 A+7 C)+7 b^4 B\right )}{15 d}+\frac{2 (8 a C+11 b B) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) (a+b \sec (c+d x))^3}{99 d}+\frac{2 C \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) (a+b \sec (c+d x))^4}{11 d} \]

[Out]

(-2*(15*a^4*B + 54*a^2*b^2*B + 7*b^4*B + 12*a^3*b*(5*A + 3*C) + 4*a*b^3*(9*A + 7*C))*Sqrt[Cos[c + d*x]]*Ellipt
icE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(15*d) + (2*(308*a^3*b*B + 220*a*b^3*B + 77*a^4*(3*A + C) + 66*a^2*b^2
*(7*A + 5*C) + 5*b^4*(11*A + 9*C))*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(231*d) +
(2*(15*a^4*B + 54*a^2*b^2*B + 7*b^4*B + 12*a^3*b*(5*A + 3*C) + 4*a*b^3*(9*A + 7*C))*Sqrt[Sec[c + d*x]]*Sin[c +
d*x])/(15*d) + (2*(682*a^3*b*B + 660*a*b^3*B + 64*a^4*C + 15*b^4*(11*A + 9*C) + 9*a^2*b^2*(143*A + 101*C))*Se
c[c + d*x]^(3/2)*Sin[c + d*x])/(693*d) + (2*b*(1353*a^2*b*B + 539*b^3*B + 192*a^3*C + 2*a*b^2*(891*A + 673*C))
*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(3465*d) + (2*(33*A*b^2 + 55*a*b*B + 16*a^2*C + 27*b^2*C)*Sec[c + d*x]^(3/2)
*(a + b*Sec[c + d*x])^2*Sin[c + d*x])/(231*d) + (2*(11*b*B + 8*a*C)*Sec[c + d*x]^(3/2)*(a + b*Sec[c + d*x])^3*
Sin[c + d*x])/(99*d) + (2*C*Sec[c + d*x]^(3/2)*(a + b*Sec[c + d*x])^4*Sin[c + d*x])/(11*d)

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Rubi [A]  time = 1.3055, antiderivative size = 515, 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 = {4096, 4076, 4047, 3768, 3771, 2639, 4046, 2641} \[ \frac{2 \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) \left (16 a^2 C+55 a b B+33 A b^2+27 b^2 C\right ) (a+b \sec (c+d x))^2}{231 d}+\frac{2 b \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x) \left (1353 a^2 b B+192 a^3 C+2 a b^2 (891 A+673 C)+539 b^3 B\right )}{3465 d}+\frac{2 \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) \left (9 a^2 b^2 (143 A+101 C)+682 a^3 b B+64 a^4 C+660 a b^3 B+15 b^4 (11 A+9 C)\right )}{693 d}+\frac{2 \sin (c+d x) \sqrt{\sec (c+d x)} \left (12 a^3 b (5 A+3 C)+54 a^2 b^2 B+15 a^4 B+4 a b^3 (9 A+7 C)+7 b^4 B\right )}{15 d}+\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \left (66 a^2 b^2 (7 A+5 C)+77 a^4 (3 A+C)+308 a^3 b B+220 a b^3 B+5 b^4 (11 A+9 C)\right )}{231 d}-\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \left (12 a^3 b (5 A+3 C)+54 a^2 b^2 B+15 a^4 B+4 a b^3 (9 A+7 C)+7 b^4 B\right )}{15 d}+\frac{2 (8 a C+11 b B) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) (a+b \sec (c+d x))^3}{99 d}+\frac{2 C \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) (a+b \sec (c+d x))^4}{11 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*(15*a^4*B + 54*a^2*b^2*B + 7*b^4*B + 12*a^3*b*(5*A + 3*C) + 4*a*b^3*(9*A + 7*C))*Sqrt[Cos[c + d*x]]*Ellipt
icE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(15*d) + (2*(308*a^3*b*B + 220*a*b^3*B + 77*a^4*(3*A + C) + 66*a^2*b^2
*(7*A + 5*C) + 5*b^4*(11*A + 9*C))*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(231*d) +
(2*(15*a^4*B + 54*a^2*b^2*B + 7*b^4*B + 12*a^3*b*(5*A + 3*C) + 4*a*b^3*(9*A + 7*C))*Sqrt[Sec[c + d*x]]*Sin[c +
d*x])/(15*d) + (2*(682*a^3*b*B + 660*a*b^3*B + 64*a^4*C + 15*b^4*(11*A + 9*C) + 9*a^2*b^2*(143*A + 101*C))*Se
c[c + d*x]^(3/2)*Sin[c + d*x])/(693*d) + (2*b*(1353*a^2*b*B + 539*b^3*B + 192*a^3*C + 2*a*b^2*(891*A + 673*C))
*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(3465*d) + (2*(33*A*b^2 + 55*a*b*B + 16*a^2*C + 27*b^2*C)*Sec[c + d*x]^(3/2)
*(a + b*Sec[c + d*x])^2*Sin[c + d*x])/(231*d) + (2*(11*b*B + 8*a*C)*Sec[c + d*x]^(3/2)*(a + b*Sec[c + d*x])^3*
Sin[c + d*x])/(99*d) + (2*C*Sec[c + d*x]^(3/2)*(a + b*Sec[c + d*x])^4*Sin[c + d*x])/(11*d)

Rule 4096

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[(C*Cot[e + f*x]*(a + b*Csc[e + f*x])^m*(d
*Csc[e + f*x])^n)/(f*(m + n + 1)), x] + Dist[1/(m + n + 1), Int[(a + b*Csc[e + f*x])^(m - 1)*(d*Csc[e + f*x])^
n*Simp[a*A*(m + n + 1) + a*C*n + ((A*b + a*B)*(m + n + 1) + b*C*(m + n))*Csc[e + f*x] + (b*B*(m + n + 1) + a*C
*m)*Csc[e + f*x]^2, x], x], x] /; FreeQ[{a, b, d, e, f, A, B, C, n}, x] && NeQ[a^2 - b^2, 0] && GtQ[m, 0] &&
!LeQ[n, -1]

Rule 4076

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[(b*C*Csc[e + f*x]*Cot[e + f*x]*(d*Csc[e + f*x
])^n)/(f*(n + 2)), x] + Dist[1/(n + 2), Int[(d*Csc[e + f*x])^n*Simp[A*a*(n + 2) + (B*a*(n + 2) + b*(C*(n + 1)
+ A*(n + 2)))*Csc[e + f*x] + (a*C + B*b)*(n + 2)*Csc[e + f*x]^2, x], x], x] /; FreeQ[{a, b, d, e, f, A, B, C,
n}, 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 3768

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

Rule 4046

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

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]

Rubi steps

\begin{align*} \int \sqrt{\sec (c+d x)} (a+b \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx &=\frac{2 C \sec ^{\frac{3}{2}}(c+d x) (a+b \sec (c+d x))^4 \sin (c+d x)}{11 d}+\frac{2}{11} \int \sqrt{\sec (c+d x)} (a+b \sec (c+d x))^3 \left (\frac{1}{2} a (11 A+C)+\frac{1}{2} (11 A b+11 a B+9 b C) \sec (c+d x)+\frac{1}{2} (11 b B+8 a C) \sec ^2(c+d x)\right ) \, dx\\ &=\frac{2 (11 b B+8 a C) \sec ^{\frac{3}{2}}(c+d x) (a+b \sec (c+d x))^3 \sin (c+d x)}{99 d}+\frac{2 C \sec ^{\frac{3}{2}}(c+d x) (a+b \sec (c+d x))^4 \sin (c+d x)}{11 d}+\frac{4}{99} \int \sqrt{\sec (c+d x)} (a+b \sec (c+d x))^2 \left (\frac{1}{4} a (99 a A+11 b B+17 a C)+\frac{1}{4} \left (198 a A b+99 a^2 B+77 b^2 B+146 a b C\right ) \sec (c+d x)+\frac{3}{4} \left (33 A b^2+55 a b B+16 a^2 C+27 b^2 C\right ) \sec ^2(c+d x)\right ) \, dx\\ &=\frac{2 \left (33 A b^2+55 a b B+16 a^2 C+27 b^2 C\right ) \sec ^{\frac{3}{2}}(c+d x) (a+b \sec (c+d x))^2 \sin (c+d x)}{231 d}+\frac{2 (11 b B+8 a C) \sec ^{\frac{3}{2}}(c+d x) (a+b \sec (c+d x))^3 \sin (c+d x)}{99 d}+\frac{2 C \sec ^{\frac{3}{2}}(c+d x) (a+b \sec (c+d x))^4 \sin (c+d x)}{11 d}+\frac{8}{693} \int \sqrt{\sec (c+d x)} (a+b \sec (c+d x)) \left (\frac{1}{8} a \left (242 a b B+9 b^2 (11 A+9 C)+a^2 (693 A+167 C)\right )+\frac{1}{8} \left (693 a^3 B+1441 a b^2 B+45 b^3 (11 A+9 C)+a^2 b (2079 A+1381 C)\right ) \sec (c+d x)+\frac{1}{8} \left (1353 a^2 b B+539 b^3 B+192 a^3 C+2 a b^2 (891 A+673 C)\right ) \sec ^2(c+d x)\right ) \, dx\\ &=\frac{2 b \left (1353 a^2 b B+539 b^3 B+192 a^3 C+2 a b^2 (891 A+673 C)\right ) \sec ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{3465 d}+\frac{2 \left (33 A b^2+55 a b B+16 a^2 C+27 b^2 C\right ) \sec ^{\frac{3}{2}}(c+d x) (a+b \sec (c+d x))^2 \sin (c+d x)}{231 d}+\frac{2 (11 b B+8 a C) \sec ^{\frac{3}{2}}(c+d x) (a+b \sec (c+d x))^3 \sin (c+d x)}{99 d}+\frac{2 C \sec ^{\frac{3}{2}}(c+d x) (a+b \sec (c+d x))^4 \sin (c+d x)}{11 d}+\frac{16 \int \sqrt{\sec (c+d x)} \left (\frac{5}{16} a^2 \left (242 a b B+9 b^2 (11 A+9 C)+a^2 (693 A+167 C)\right )+\frac{231}{16} \left (15 a^4 B+54 a^2 b^2 B+7 b^4 B+12 a^3 b (5 A+3 C)+4 a b^3 (9 A+7 C)\right ) \sec (c+d x)+\frac{15}{16} \left (682 a^3 b B+660 a b^3 B+64 a^4 C+15 b^4 (11 A+9 C)+9 a^2 b^2 (143 A+101 C)\right ) \sec ^2(c+d x)\right ) \, dx}{3465}\\ &=\frac{2 b \left (1353 a^2 b B+539 b^3 B+192 a^3 C+2 a b^2 (891 A+673 C)\right ) \sec ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{3465 d}+\frac{2 \left (33 A b^2+55 a b B+16 a^2 C+27 b^2 C\right ) \sec ^{\frac{3}{2}}(c+d x) (a+b \sec (c+d x))^2 \sin (c+d x)}{231 d}+\frac{2 (11 b B+8 a C) \sec ^{\frac{3}{2}}(c+d x) (a+b \sec (c+d x))^3 \sin (c+d x)}{99 d}+\frac{2 C \sec ^{\frac{3}{2}}(c+d x) (a+b \sec (c+d x))^4 \sin (c+d x)}{11 d}+\frac{16 \int \sqrt{\sec (c+d x)} \left (\frac{5}{16} a^2 \left (242 a b B+9 b^2 (11 A+9 C)+a^2 (693 A+167 C)\right )+\frac{15}{16} \left (682 a^3 b B+660 a b^3 B+64 a^4 C+15 b^4 (11 A+9 C)+9 a^2 b^2 (143 A+101 C)\right ) \sec ^2(c+d x)\right ) \, dx}{3465}+\frac{1}{15} \left (15 a^4 B+54 a^2 b^2 B+7 b^4 B+12 a^3 b (5 A+3 C)+4 a b^3 (9 A+7 C)\right ) \int \sec ^{\frac{3}{2}}(c+d x) \, dx\\ &=\frac{2 \left (15 a^4 B+54 a^2 b^2 B+7 b^4 B+12 a^3 b (5 A+3 C)+4 a b^3 (9 A+7 C)\right ) \sqrt{\sec (c+d x)} \sin (c+d x)}{15 d}+\frac{2 \left (682 a^3 b B+660 a b^3 B+64 a^4 C+15 b^4 (11 A+9 C)+9 a^2 b^2 (143 A+101 C)\right ) \sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{693 d}+\frac{2 b \left (1353 a^2 b B+539 b^3 B+192 a^3 C+2 a b^2 (891 A+673 C)\right ) \sec ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{3465 d}+\frac{2 \left (33 A b^2+55 a b B+16 a^2 C+27 b^2 C\right ) \sec ^{\frac{3}{2}}(c+d x) (a+b \sec (c+d x))^2 \sin (c+d x)}{231 d}+\frac{2 (11 b B+8 a C) \sec ^{\frac{3}{2}}(c+d x) (a+b \sec (c+d x))^3 \sin (c+d x)}{99 d}+\frac{2 C \sec ^{\frac{3}{2}}(c+d x) (a+b \sec (c+d x))^4 \sin (c+d x)}{11 d}+\frac{1}{15} \left (-15 a^4 B-54 a^2 b^2 B-7 b^4 B-12 a^3 b (5 A+3 C)-4 a b^3 (9 A+7 C)\right ) \int \frac{1}{\sqrt{\sec (c+d x)}} \, dx+\frac{1}{231} \left (308 a^3 b B+220 a b^3 B+77 a^4 (3 A+C)+66 a^2 b^2 (7 A+5 C)+5 b^4 (11 A+9 C)\right ) \int \sqrt{\sec (c+d x)} \, dx\\ &=\frac{2 \left (15 a^4 B+54 a^2 b^2 B+7 b^4 B+12 a^3 b (5 A+3 C)+4 a b^3 (9 A+7 C)\right ) \sqrt{\sec (c+d x)} \sin (c+d x)}{15 d}+\frac{2 \left (682 a^3 b B+660 a b^3 B+64 a^4 C+15 b^4 (11 A+9 C)+9 a^2 b^2 (143 A+101 C)\right ) \sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{693 d}+\frac{2 b \left (1353 a^2 b B+539 b^3 B+192 a^3 C+2 a b^2 (891 A+673 C)\right ) \sec ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{3465 d}+\frac{2 \left (33 A b^2+55 a b B+16 a^2 C+27 b^2 C\right ) \sec ^{\frac{3}{2}}(c+d x) (a+b \sec (c+d x))^2 \sin (c+d x)}{231 d}+\frac{2 (11 b B+8 a C) \sec ^{\frac{3}{2}}(c+d x) (a+b \sec (c+d x))^3 \sin (c+d x)}{99 d}+\frac{2 C \sec ^{\frac{3}{2}}(c+d x) (a+b \sec (c+d x))^4 \sin (c+d x)}{11 d}+\frac{1}{15} \left (\left (-15 a^4 B-54 a^2 b^2 B-7 b^4 B-12 a^3 b (5 A+3 C)-4 a b^3 (9 A+7 C)\right ) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \sqrt{\cos (c+d x)} \, dx+\frac{1}{231} \left (\left (308 a^3 b B+220 a b^3 B+77 a^4 (3 A+C)+66 a^2 b^2 (7 A+5 C)+5 b^4 (11 A+9 C)\right ) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{1}{\sqrt{\cos (c+d x)}} \, dx\\ &=-\frac{2 \left (15 a^4 B+54 a^2 b^2 B+7 b^4 B+12 a^3 b (5 A+3 C)+4 a b^3 (9 A+7 C)\right ) \sqrt{\cos (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \sqrt{\sec (c+d x)}}{15 d}+\frac{2 \left (308 a^3 b B+220 a b^3 B+77 a^4 (3 A+C)+66 a^2 b^2 (7 A+5 C)+5 b^4 (11 A+9 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 \left (15 a^4 B+54 a^2 b^2 B+7 b^4 B+12 a^3 b (5 A+3 C)+4 a b^3 (9 A+7 C)\right ) \sqrt{\sec (c+d x)} \sin (c+d x)}{15 d}+\frac{2 \left (682 a^3 b B+660 a b^3 B+64 a^4 C+15 b^4 (11 A+9 C)+9 a^2 b^2 (143 A+101 C)\right ) \sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{693 d}+\frac{2 b \left (1353 a^2 b B+539 b^3 B+192 a^3 C+2 a b^2 (891 A+673 C)\right ) \sec ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{3465 d}+\frac{2 \left (33 A b^2+55 a b B+16 a^2 C+27 b^2 C\right ) \sec ^{\frac{3}{2}}(c+d x) (a+b \sec (c+d x))^2 \sin (c+d x)}{231 d}+\frac{2 (11 b B+8 a C) \sec ^{\frac{3}{2}}(c+d x) (a+b \sec (c+d x))^3 \sin (c+d x)}{99 d}+\frac{2 C \sec ^{\frac{3}{2}}(c+d x) (a+b \sec (c+d x))^4 \sin (c+d x)}{11 d}\\ \end{align*}

Mathematica [A]  time = 7.50476, size = 713, normalized size = 1.38 \[ \frac{2 \cos ^6(c+d x) (a+b \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \left (2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \text{EllipticF}\left (\frac{1}{2} (c+d x),2\right ) \left (2310 a^2 A b^2+1155 a^4 A+1650 a^2 b^2 C+1540 a^3 b B+385 a^4 C+1100 a b^3 B+275 A b^4+225 b^4 C\right )+\frac{2 E\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \left (-4620 a^3 A b-4158 a^2 b^2 B-2772 a^3 b C-1155 a^4 B-2772 a A b^3-2156 a b^3 C-539 b^4 B\right )}{\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}}\right )}{1155 d (a \cos (c+d x)+b)^4 (A \cos (2 c+2 d x)+A+2 B \cos (c+d x)+2 C)}+\frac{(a+b \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \left (\frac{4}{15} \sin (c+d x) \left (60 a^3 A b+54 a^2 b^2 B+36 a^3 b C+15 a^4 B+36 a A b^3+28 a b^3 C+7 b^4 B\right )+\frac{4}{77} \sec ^3(c+d x) \left (66 a^2 b^2 C \sin (c+d x)+44 a b^3 B \sin (c+d x)+11 A b^4 \sin (c+d x)+9 b^4 C \sin (c+d x)\right )+\frac{4}{45} \sec ^2(c+d x) \left (54 a^2 b^2 B \sin (c+d x)+36 a^3 b C \sin (c+d x)+36 a A b^3 \sin (c+d x)+28 a b^3 C \sin (c+d x)+7 b^4 B \sin (c+d x)\right )+\frac{4}{231} \sec (c+d x) \left (462 a^2 A b^2 \sin (c+d x)+330 a^2 b^2 C \sin (c+d x)+308 a^3 b B \sin (c+d x)+77 a^4 C \sin (c+d x)+220 a b^3 B \sin (c+d x)+55 A b^4 \sin (c+d x)+45 b^4 C \sin (c+d x)\right )+\frac{4}{9} \sec ^4(c+d x) \left (4 a b^3 C \sin (c+d x)+b^4 B \sin (c+d x)\right )+\frac{4}{11} b^4 C \tan (c+d x) \sec ^4(c+d x)\right )}{d \sec ^{\frac{11}{2}}(c+d x) (a \cos (c+d x)+b)^4 (A \cos (2 c+2 d x)+A+2 B \cos (c+d x)+2 C)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*Cos[c + d*x]^6*((2*(-4620*a^3*A*b - 2772*a*A*b^3 - 1155*a^4*B - 4158*a^2*b^2*B - 539*b^4*B - 2772*a^3*b*C -
2156*a*b^3*C)*EllipticE[(c + d*x)/2, 2])/(Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) + 2*(1155*a^4*A + 2310*a^2*A
*b^2 + 275*A*b^4 + 1540*a^3*b*B + 1100*a*b^3*B + 385*a^4*C + 1650*a^2*b^2*C + 225*b^4*C)*Sqrt[Cos[c + d*x]]*El
lipticF[(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))/(1
155*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)*((4*(60*a^3*A*b + 36*a*A*b^3 + 15*a^4*B + 54*a^2*b^2*B + 7*b^4*B + 36*a^3*
b*C + 28*a*b^3*C)*Sin[c + d*x])/15 + (4*Sec[c + d*x]^4*(b^4*B*Sin[c + d*x] + 4*a*b^3*C*Sin[c + d*x]))/9 + (4*S
ec[c + d*x]^2*(36*a*A*b^3*Sin[c + d*x] + 54*a^2*b^2*B*Sin[c + d*x] + 7*b^4*B*Sin[c + d*x] + 36*a^3*b*C*Sin[c +
d*x] + 28*a*b^3*C*Sin[c + d*x]))/45 + (4*Sec[c + d*x]^3*(11*A*b^4*Sin[c + d*x] + 44*a*b^3*B*Sin[c + d*x] + 66
*a^2*b^2*C*Sin[c + d*x] + 9*b^4*C*Sin[c + d*x]))/77 + (4*Sec[c + d*x]*(462*a^2*A*b^2*Sin[c + d*x] + 55*A*b^4*S
in[c + d*x] + 308*a^3*b*B*Sin[c + d*x] + 220*a*b^3*B*Sin[c + d*x] + 77*a^4*C*Sin[c + d*x] + 330*a^2*b^2*C*Sin[
c + d*x] + 45*b^4*C*Sin[c + d*x]))/231 + (4*b^4*C*Sec[c + d*x]^4*Tan[c + d*x])/11))/(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 = 14.859, size = 1550, normalized size = 3. \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sec(d*x+c)^(1/2)*(a+b*sec(d*x+c))^4*(A+B*sec(d*x+c)+C*sec(d*x+c)^2),x)

[Out]

-(-(-2*cos(1/2*d*x+1/2*c)^2+1)*sin(1/2*d*x+1/2*c)^2)^(1/2)*(2*A*a^4*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*cos(1/2*d
*x+1/2*c)^2+1)^(1/2)/(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)*EllipticF(cos(1/2*d*x+1/2*c),2^(1/2)
)-4/5*a*b*(2*A*b^2+3*B*a*b+2*C*a^2)/(8*sin(1/2*d*x+1/2*c)^6-12*sin(1/2*d*x+1/2*c)^4+6*sin(1/2*d*x+1/2*c)^2-1)/
sin(1/2*d*x+1/2*c)^2*(12*(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))*(sin(1/2*d*x+1
/2*c)^2)^(1/2)*sin(1/2*d*x+1/2*c)^4-24*sin(1/2*d*x+1/2*c)^6*cos(1/2*d*x+1/2*c)-12*(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))*(sin(1/2*d*x+1/2*c)^2)^(1/2)*sin(1/2*d*x+1/2*c)^2+24*sin(1/2*d*x+1/
2*c)^4*cos(1/2*d*x+1/2*c)+3*(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))*(sin(1/2*d*
x+1/2*c)^2)^(1/2)-8*sin(1/2*d*x+1/2*c)^2*cos(1/2*d*x+1/2*c))*(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1
/2)+2*b^2*(A*b^2+4*B*a*b+6*C*a^2)*(-1/56*cos(1/2*d*x+1/2*c)*(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/
2)/(cos(1/2*d*x+1/2*c)^2-1/2)^4-5/42*cos(1/2*d*x+1/2*c)*(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)/(
cos(1/2*d*x+1/2*c)^2-1/2)^2+5/21*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*cos(1/2*d*x+1/2*c)^2+1)^(1/2)/(-2*sin(1/2*d*
x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)*EllipticF(cos(1/2*d*x+1/2*c),2^(1/2)))+2*C*b^4*(-1/352*cos(1/2*d*x+1/2*
c)*(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)/(cos(1/2*d*x+1/2*c)^2-1/2)^6-9/616*cos(1/2*d*x+1/2*c)*
(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)/(cos(1/2*d*x+1/2*c)^2-1/2)^4-15/154*cos(1/2*d*x+1/2*c)*(-
2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)/(cos(1/2*d*x+1/2*c)^2-1/2)^2+15/77*(sin(1/2*d*x+1/2*c)^2)^(
1/2)*(-2*cos(1/2*d*x+1/2*c)^2+1)^(1/2)/(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)*EllipticF(cos(1/2*
d*x+1/2*c),2^(1/2)))+2*b^3*(B*b+4*C*a)*(-1/144*cos(1/2*d*x+1/2*c)*(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^
2)^(1/2)/(cos(1/2*d*x+1/2*c)^2-1/2)^5-7/180*cos(1/2*d*x+1/2*c)*(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^
(1/2)/(cos(1/2*d*x+1/2*c)^2-1/2)^3-14/15*sin(1/2*d*x+1/2*c)^2*cos(1/2*d*x+1/2*c)/(-(-2*cos(1/2*d*x+1/2*c)^2+1)
*sin(1/2*d*x+1/2*c)^2)^(1/2)+7/15*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*cos(1/2*d*x+1/2*c)^2+1)^(1/2)/(-2*sin(1/2*d
*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)*EllipticF(cos(1/2*d*x+1/2*c),2^(1/2))-7/15*(sin(1/2*d*x+1/2*c)^2)^(1/2
)*(-2*cos(1/2*d*x+1/2*c)^2+1)^(1/2)/(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)*(EllipticF(cos(1/2*d*
x+1/2*c),2^(1/2))-EllipticE(cos(1/2*d*x+1/2*c),2^(1/2))))+2*a^2*(6*A*b^2+4*B*a*b+C*a^2)*(-1/6*cos(1/2*d*x+1/2*
c)*(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)/(cos(1/2*d*x+1/2*c)^2-1/2)^2+1/3*(sin(1/2*d*x+1/2*c)^2
)^(1/2)*(-2*cos(1/2*d*x+1/2*c)^2+1)^(1/2)/(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)*EllipticF(cos(1
/2*d*x+1/2*c),2^(1/2)))+2*a^3*(4*A*b+B*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)*(-2*
sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)*EllipticE(cos(1/2*d*x+1/2*c),2^(1/2))+2*(-2*sin(1/2*d*x+1/2*c
)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)*cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2*c)^2)/sin(1/2*d*x+1/2*c)^2/(2*sin(1/2*d*x+1
/2*c)^2-1))/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(sec(d*x+c)^(1/2)*(a+b*sec(d*x+c))^4*(A+B*sec(d*x+c)+C*sec(d*x+c)^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 ({\left (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 )\right )} \sqrt{\sec \left (d x + c\right )}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(d*x+c)^(1/2)*(a+b*sec(d*x+c))^4*(A+B*sec(d*x+c)+C*sec(d*x+c)^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))*sqrt(sec(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(sec(d*x+c)**(1/2)*(a+b*sec(d*x+c))**4*(A+B*sec(d*x+c)+C*sec(d*x+c)**2),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\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} \sqrt{\sec \left (d x + c\right )}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(d*x+c)^(1/2)*(a+b*sec(d*x+c))^4*(A+B*sec(d*x+c)+C*sec(d*x+c)^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*sqrt(sec(d*x + c)), x)