3.725 \(\int \sec ^3(c+d x) (a+b \sec (c+d x))^{5/2} (A+C \sec ^2(c+d x)) \, dx\)

Optimal. Leaf size=650 \[ \frac{2 (a-b) \sqrt{a+b} \left (10 a^3 b^2 (143 A+94 C)+15 a^2 b^3 (1573 A+1175 C)+180 a^4 b C+240 a^5 C-6 a b^4 (2717 A+2174 C)+1617 b^5 (13 A+11 C)\right ) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right ),\frac{a+b}{a-b}\right )}{45045 b^4 d}+\frac{2 \left (15 a^2 C+11 b^2 (13 A+11 C)\right ) \tan (c+d x) \sec ^3(c+d x) \sqrt{a+b \sec (c+d x)}}{1287 d}+\frac{2 a \left (15 a^2 C+2717 A b^2+2209 b^2 C\right ) \tan (c+d x) \sec ^2(c+d x) \sqrt{a+b \sec (c+d x)}}{9009 b d}-\frac{2 \left (-15 a^2 b^2 (715 A+543 C)+90 a^4 C-539 b^4 (13 A+11 C)\right ) \tan (c+d x) \sec (c+d x) \sqrt{a+b \sec (c+d x)}}{45045 b^2 d}+\frac{2 a \left (5 a^2 b^2 (143 A+79 C)+120 a^4 C+b^4 (23309 A+18973 C)\right ) \tan (c+d x) \sqrt{a+b \sec (c+d x)}}{45045 b^3 d}+\frac{2 (a-b) \sqrt{a+b} \left (10 a^4 b^2 (143 A+76 C)-3 a^2 b^4 (13299 A+10223 C)+240 a^6 C-1617 b^6 (13 A+11 C)\right ) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} E\left (\sin ^{-1}\left (\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right )|\frac{a+b}{a-b}\right )}{45045 b^5 d}+\frac{2 C \tan (c+d x) \sec ^3(c+d x) (a+b \sec (c+d x))^{5/2}}{13 d}+\frac{10 a C \tan (c+d x) \sec ^3(c+d x) (a+b \sec (c+d x))^{3/2}}{143 d} \]

[Out]

(2*(a - b)*Sqrt[a + b]*(240*a^6*C - 1617*b^6*(13*A + 11*C) + 10*a^4*b^2*(143*A + 76*C) - 3*a^2*b^4*(13299*A +
10223*C))*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - S
ec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(45045*b^5*d) + (2*(a - b)*Sqrt[a + b]*(240*a^
5*C + 180*a^4*b*C + 1617*b^5*(13*A + 11*C) + 10*a^3*b^2*(143*A + 94*C) + 15*a^2*b^3*(1573*A + 1175*C) - 6*a*b^
4*(2717*A + 2174*C))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqr
t[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(45045*b^4*d) + (2*a*(120*a^4*C + 5
*a^2*b^2*(143*A + 79*C) + b^4*(23309*A + 18973*C))*Sqrt[a + b*Sec[c + d*x]]*Tan[c + d*x])/(45045*b^3*d) - (2*(
90*a^4*C - 539*b^4*(13*A + 11*C) - 15*a^2*b^2*(715*A + 543*C))*Sec[c + d*x]*Sqrt[a + b*Sec[c + d*x]]*Tan[c + d
*x])/(45045*b^2*d) + (2*a*(2717*A*b^2 + 15*a^2*C + 2209*b^2*C)*Sec[c + d*x]^2*Sqrt[a + b*Sec[c + d*x]]*Tan[c +
d*x])/(9009*b*d) + (2*(15*a^2*C + 11*b^2*(13*A + 11*C))*Sec[c + d*x]^3*Sqrt[a + b*Sec[c + d*x]]*Tan[c + d*x])
/(1287*d) + (10*a*C*Sec[c + d*x]^3*(a + b*Sec[c + d*x])^(3/2)*Tan[c + d*x])/(143*d) + (2*C*Sec[c + d*x]^3*(a +
b*Sec[c + d*x])^(5/2)*Tan[c + d*x])/(13*d)

________________________________________________________________________________________

Rubi [A]  time = 2.77039, antiderivative size = 650, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 8, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.229, Rules used = {4097, 4096, 4102, 4092, 4082, 4005, 3832, 4004} \[ \frac{2 \left (15 a^2 C+11 b^2 (13 A+11 C)\right ) \tan (c+d x) \sec ^3(c+d x) \sqrt{a+b \sec (c+d x)}}{1287 d}+\frac{2 a \left (15 a^2 C+2717 A b^2+2209 b^2 C\right ) \tan (c+d x) \sec ^2(c+d x) \sqrt{a+b \sec (c+d x)}}{9009 b d}-\frac{2 \left (-15 a^2 b^2 (715 A+543 C)+90 a^4 C-539 b^4 (13 A+11 C)\right ) \tan (c+d x) \sec (c+d x) \sqrt{a+b \sec (c+d x)}}{45045 b^2 d}+\frac{2 a \left (5 a^2 b^2 (143 A+79 C)+120 a^4 C+b^4 (23309 A+18973 C)\right ) \tan (c+d x) \sqrt{a+b \sec (c+d x)}}{45045 b^3 d}+\frac{2 (a-b) \sqrt{a+b} \left (10 a^3 b^2 (143 A+94 C)+15 a^2 b^3 (1573 A+1175 C)+180 a^4 b C+240 a^5 C-6 a b^4 (2717 A+2174 C)+1617 b^5 (13 A+11 C)\right ) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} F\left (\sin ^{-1}\left (\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right )|\frac{a+b}{a-b}\right )}{45045 b^4 d}+\frac{2 (a-b) \sqrt{a+b} \left (10 a^4 b^2 (143 A+76 C)-3 a^2 b^4 (13299 A+10223 C)+240 a^6 C-1617 b^6 (13 A+11 C)\right ) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} E\left (\sin ^{-1}\left (\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right )|\frac{a+b}{a-b}\right )}{45045 b^5 d}+\frac{2 C \tan (c+d x) \sec ^3(c+d x) (a+b \sec (c+d x))^{5/2}}{13 d}+\frac{10 a C \tan (c+d x) \sec ^3(c+d x) (a+b \sec (c+d x))^{3/2}}{143 d} \]

Antiderivative was successfully verified.

[In]

Int[Sec[c + d*x]^3*(a + b*Sec[c + d*x])^(5/2)*(A + C*Sec[c + d*x]^2),x]

[Out]

(2*(a - b)*Sqrt[a + b]*(240*a^6*C - 1617*b^6*(13*A + 11*C) + 10*a^4*b^2*(143*A + 76*C) - 3*a^2*b^4*(13299*A +
10223*C))*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - S
ec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(45045*b^5*d) + (2*(a - b)*Sqrt[a + b]*(240*a^
5*C + 180*a^4*b*C + 1617*b^5*(13*A + 11*C) + 10*a^3*b^2*(143*A + 94*C) + 15*a^2*b^3*(1573*A + 1175*C) - 6*a*b^
4*(2717*A + 2174*C))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqr
t[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(45045*b^4*d) + (2*a*(120*a^4*C + 5
*a^2*b^2*(143*A + 79*C) + b^4*(23309*A + 18973*C))*Sqrt[a + b*Sec[c + d*x]]*Tan[c + d*x])/(45045*b^3*d) - (2*(
90*a^4*C - 539*b^4*(13*A + 11*C) - 15*a^2*b^2*(715*A + 543*C))*Sec[c + d*x]*Sqrt[a + b*Sec[c + d*x]]*Tan[c + d
*x])/(45045*b^2*d) + (2*a*(2717*A*b^2 + 15*a^2*C + 2209*b^2*C)*Sec[c + d*x]^2*Sqrt[a + b*Sec[c + d*x]]*Tan[c +
d*x])/(9009*b*d) + (2*(15*a^2*C + 11*b^2*(13*A + 11*C))*Sec[c + d*x]^3*Sqrt[a + b*Sec[c + d*x]]*Tan[c + d*x])
/(1287*d) + (10*a*C*Sec[c + d*x]^3*(a + b*Sec[c + d*x])^(3/2)*Tan[c + d*x])/(143*d) + (2*C*Sec[c + d*x]^3*(a +
b*Sec[c + d*x])^(5/2)*Tan[c + d*x])/(13*d)

Rule 4097

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

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 4102

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*d*Cot[e + f*x]*(a + b*Csc[e + f*x])^(m
+ 1)*(d*Csc[e + f*x])^(n - 1))/(b*f*(m + n + 1)), x] + Dist[d/(b*(m + n + 1)), Int[(a + b*Csc[e + f*x])^m*(d*
Csc[e + f*x])^(n - 1)*Simp[a*C*(n - 1) + (A*b*(m + n + 1) + b*C*(m + n))*Csc[e + f*x] + (b*B*(m + n + 1) - a*C
*n)*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] && GtQ[n, 0]

Rule 4092

Int[csc[(e_.) + (f_.)*(x_)]^2*((A_.) + csc[(e_.) + (f_.)*(x_)]*(B_.) + csc[(e_.) + (f_.)*(x_)]^2*(C_.))*(csc[(
e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_), x_Symbol] :> -Simp[(C*Csc[e + f*x]*Cot[e + f*x]*(a + b*Csc[e + f*x])^(m
+ 1))/(b*f*(m + 3)), x] + Dist[1/(b*(m + 3)), Int[Csc[e + f*x]*(a + b*Csc[e + f*x])^m*Simp[a*C + b*(C*(m + 2)
+ A*(m + 3))*Csc[e + f*x] - (2*a*C - b*B*(m + 3))*Csc[e + f*x]^2, x], x], x] /; FreeQ[{a, b, e, f, A, B, C, m
}, x] && NeQ[a^2 - b^2, 0] &&  !LtQ[m, -1]

Rule 4082

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

Rule 4005

Int[(csc[(e_.) + (f_.)*(x_)]*(csc[(e_.) + (f_.)*(x_)]*(B_.) + (A_)))/Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)
], x_Symbol] :> Dist[A - B, Int[Csc[e + f*x]/Sqrt[a + b*Csc[e + f*x]], x], x] + Dist[B, Int[(Csc[e + f*x]*(1 +
Csc[e + f*x]))/Sqrt[a + b*Csc[e + f*x]], x], x] /; FreeQ[{a, b, e, f, A, B}, x] && NeQ[a^2 - b^2, 0] && NeQ[A
^2 - B^2, 0]

Rule 3832

Int[csc[(e_.) + (f_.)*(x_)]/Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)], x_Symbol] :> Simp[(-2*Rt[a + b, 2]*Sqr
t[(b*(1 - Csc[e + f*x]))/(a + b)]*Sqrt[-((b*(1 + Csc[e + f*x]))/(a - b))]*EllipticF[ArcSin[Sqrt[a + b*Csc[e +
f*x]]/Rt[a + b, 2]], (a + b)/(a - b)])/(b*f*Cot[e + f*x]), x] /; FreeQ[{a, b, e, f}, x] && NeQ[a^2 - b^2, 0]

Rule 4004

Int[(csc[(e_.) + (f_.)*(x_)]*(csc[(e_.) + (f_.)*(x_)]*(B_.) + (A_)))/Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)
], x_Symbol] :> Simp[(-2*(A*b - a*B)*Rt[a + (b*B)/A, 2]*Sqrt[(b*(1 - Csc[e + f*x]))/(a + b)]*Sqrt[-((b*(1 + Cs
c[e + f*x]))/(a - b))]*EllipticE[ArcSin[Sqrt[a + b*Csc[e + f*x]]/Rt[a + (b*B)/A, 2]], (a*A + b*B)/(a*A - b*B)]
)/(b^2*f*Cot[e + f*x]), x] /; FreeQ[{a, b, e, f, A, B}, x] && NeQ[a^2 - b^2, 0] && EqQ[A^2 - B^2, 0]

Rubi steps

\begin{align*} \int \sec ^3(c+d x) (a+b \sec (c+d x))^{5/2} \left (A+C \sec ^2(c+d x)\right ) \, dx &=\frac{2 C \sec ^3(c+d x) (a+b \sec (c+d x))^{5/2} \tan (c+d x)}{13 d}+\frac{2}{13} \int \sec ^3(c+d x) (a+b \sec (c+d x))^{3/2} \left (\frac{1}{2} a (13 A+6 C)+\frac{1}{2} b (13 A+11 C) \sec (c+d x)+\frac{5}{2} a C \sec ^2(c+d x)\right ) \, dx\\ &=\frac{10 a C \sec ^3(c+d x) (a+b \sec (c+d x))^{3/2} \tan (c+d x)}{143 d}+\frac{2 C \sec ^3(c+d x) (a+b \sec (c+d x))^{5/2} \tan (c+d x)}{13 d}+\frac{4}{143} \int \sec ^3(c+d x) \sqrt{a+b \sec (c+d x)} \left (\frac{1}{4} a^2 (143 A+96 C)+\frac{1}{2} a b (143 A+116 C) \sec (c+d x)+\frac{1}{4} \left (15 a^2 C+11 b^2 (13 A+11 C)\right ) \sec ^2(c+d x)\right ) \, dx\\ &=\frac{2 \left (15 a^2 C+11 b^2 (13 A+11 C)\right ) \sec ^3(c+d x) \sqrt{a+b \sec (c+d x)} \tan (c+d x)}{1287 d}+\frac{10 a C \sec ^3(c+d x) (a+b \sec (c+d x))^{3/2} \tan (c+d x)}{143 d}+\frac{2 C \sec ^3(c+d x) (a+b \sec (c+d x))^{5/2} \tan (c+d x)}{13 d}+\frac{8 \int \frac{\sec ^3(c+d x) \left (\frac{3}{8} a \left (22 b^2 (13 A+11 C)+a^2 (429 A+318 C)\right )+\frac{1}{8} b \left (77 b^2 (13 A+11 C)+a^2 (3861 A+3057 C)\right ) \sec (c+d x)+\frac{1}{8} a \left (2717 A b^2+15 a^2 C+2209 b^2 C\right ) \sec ^2(c+d x)\right )}{\sqrt{a+b \sec (c+d x)}} \, dx}{1287}\\ &=\frac{2 a \left (2717 A b^2+15 a^2 C+2209 b^2 C\right ) \sec ^2(c+d x) \sqrt{a+b \sec (c+d x)} \tan (c+d x)}{9009 b d}+\frac{2 \left (15 a^2 C+11 b^2 (13 A+11 C)\right ) \sec ^3(c+d x) \sqrt{a+b \sec (c+d x)} \tan (c+d x)}{1287 d}+\frac{10 a C \sec ^3(c+d x) (a+b \sec (c+d x))^{3/2} \tan (c+d x)}{143 d}+\frac{2 C \sec ^3(c+d x) (a+b \sec (c+d x))^{5/2} \tan (c+d x)}{13 d}+\frac{16 \int \frac{\sec ^2(c+d x) \left (\frac{1}{4} a^2 \left (2717 A b^2+15 a^2 C+2209 b^2 C\right )+\frac{1}{16} a b \left (a^2 (9009 A+6753 C)+b^2 (19591 A+16127 C)\right ) \sec (c+d x)-\frac{1}{16} \left (90 a^4 C-539 b^4 (13 A+11 C)-15 a^2 b^2 (715 A+543 C)\right ) \sec ^2(c+d x)\right )}{\sqrt{a+b \sec (c+d x)}} \, dx}{9009 b}\\ &=-\frac{2 \left (90 a^4 C-539 b^4 (13 A+11 C)-15 a^2 b^2 (715 A+543 C)\right ) \sec (c+d x) \sqrt{a+b \sec (c+d x)} \tan (c+d x)}{45045 b^2 d}+\frac{2 a \left (2717 A b^2+15 a^2 C+2209 b^2 C\right ) \sec ^2(c+d x) \sqrt{a+b \sec (c+d x)} \tan (c+d x)}{9009 b d}+\frac{2 \left (15 a^2 C+11 b^2 (13 A+11 C)\right ) \sec ^3(c+d x) \sqrt{a+b \sec (c+d x)} \tan (c+d x)}{1287 d}+\frac{10 a C \sec ^3(c+d x) (a+b \sec (c+d x))^{3/2} \tan (c+d x)}{143 d}+\frac{2 C \sec ^3(c+d x) (a+b \sec (c+d x))^{5/2} \tan (c+d x)}{13 d}+\frac{32 \int \frac{\sec (c+d x) \left (-\frac{1}{16} a \left (90 a^4 C-539 b^4 (13 A+11 C)-15 a^2 b^2 (715 A+543 C)\right )+\frac{1}{32} b \left (30 a^4 C+1617 b^4 (13 A+11 C)+5 a^2 b^2 (17303 A+13723 C)\right ) \sec (c+d x)+\frac{3}{32} a \left (120 a^4 C+5 a^2 b^2 (143 A+79 C)+b^4 (23309 A+18973 C)\right ) \sec ^2(c+d x)\right )}{\sqrt{a+b \sec (c+d x)}} \, dx}{45045 b^2}\\ &=\frac{2 a \left (120 a^4 C+5 a^2 b^2 (143 A+79 C)+b^4 (23309 A+18973 C)\right ) \sqrt{a+b \sec (c+d x)} \tan (c+d x)}{45045 b^3 d}-\frac{2 \left (90 a^4 C-539 b^4 (13 A+11 C)-15 a^2 b^2 (715 A+543 C)\right ) \sec (c+d x) \sqrt{a+b \sec (c+d x)} \tan (c+d x)}{45045 b^2 d}+\frac{2 a \left (2717 A b^2+15 a^2 C+2209 b^2 C\right ) \sec ^2(c+d x) \sqrt{a+b \sec (c+d x)} \tan (c+d x)}{9009 b d}+\frac{2 \left (15 a^2 C+11 b^2 (13 A+11 C)\right ) \sec ^3(c+d x) \sqrt{a+b \sec (c+d x)} \tan (c+d x)}{1287 d}+\frac{10 a C \sec ^3(c+d x) (a+b \sec (c+d x))^{3/2} \tan (c+d x)}{143 d}+\frac{2 C \sec ^3(c+d x) (a+b \sec (c+d x))^{5/2} \tan (c+d x)}{13 d}+\frac{64 \int \frac{\sec (c+d x) \left (-\frac{3}{64} a b \left (60 a^4 C-5 a^2 b^2 (4433 A+3337 C)-3 b^4 (12441 A+10277 C)\right )-\frac{3}{64} \left (240 a^6 C-1617 b^6 (13 A+11 C)+10 a^4 b^2 (143 A+76 C)-3 a^2 b^4 (13299 A+10223 C)\right ) \sec (c+d x)\right )}{\sqrt{a+b \sec (c+d x)}} \, dx}{135135 b^3}\\ &=\frac{2 a \left (120 a^4 C+5 a^2 b^2 (143 A+79 C)+b^4 (23309 A+18973 C)\right ) \sqrt{a+b \sec (c+d x)} \tan (c+d x)}{45045 b^3 d}-\frac{2 \left (90 a^4 C-539 b^4 (13 A+11 C)-15 a^2 b^2 (715 A+543 C)\right ) \sec (c+d x) \sqrt{a+b \sec (c+d x)} \tan (c+d x)}{45045 b^2 d}+\frac{2 a \left (2717 A b^2+15 a^2 C+2209 b^2 C\right ) \sec ^2(c+d x) \sqrt{a+b \sec (c+d x)} \tan (c+d x)}{9009 b d}+\frac{2 \left (15 a^2 C+11 b^2 (13 A+11 C)\right ) \sec ^3(c+d x) \sqrt{a+b \sec (c+d x)} \tan (c+d x)}{1287 d}+\frac{10 a C \sec ^3(c+d x) (a+b \sec (c+d x))^{3/2} \tan (c+d x)}{143 d}+\frac{2 C \sec ^3(c+d x) (a+b \sec (c+d x))^{5/2} \tan (c+d x)}{13 d}+\frac{\left ((a-b) \left (240 a^5 C+180 a^4 b C+1617 b^5 (13 A+11 C)+10 a^3 b^2 (143 A+94 C)+15 a^2 b^3 (1573 A+1175 C)-6 a b^4 (2717 A+2174 C)\right )\right ) \int \frac{\sec (c+d x)}{\sqrt{a+b \sec (c+d x)}} \, dx}{45045 b^3}-\frac{\left (240 a^6 C-1617 b^6 (13 A+11 C)+10 a^4 b^2 (143 A+76 C)-3 a^2 b^4 (13299 A+10223 C)\right ) \int \frac{\sec (c+d x) (1+\sec (c+d x))}{\sqrt{a+b \sec (c+d x)}} \, dx}{45045 b^3}\\ &=\frac{2 (a-b) \sqrt{a+b} \left (240 a^6 C-1617 b^6 (13 A+11 C)+10 a^4 b^2 (143 A+76 C)-3 a^2 b^4 (13299 A+10223 C)\right ) \cot (c+d x) E\left (\sin ^{-1}\left (\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right )|\frac{a+b}{a-b}\right ) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (1+\sec (c+d x))}{a-b}}}{45045 b^5 d}+\frac{2 (a-b) \sqrt{a+b} \left (240 a^5 C+180 a^4 b C+1617 b^5 (13 A+11 C)+10 a^3 b^2 (143 A+94 C)+15 a^2 b^3 (1573 A+1175 C)-6 a b^4 (2717 A+2174 C)\right ) \cot (c+d x) F\left (\sin ^{-1}\left (\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right )|\frac{a+b}{a-b}\right ) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (1+\sec (c+d x))}{a-b}}}{45045 b^4 d}+\frac{2 a \left (120 a^4 C+5 a^2 b^2 (143 A+79 C)+b^4 (23309 A+18973 C)\right ) \sqrt{a+b \sec (c+d x)} \tan (c+d x)}{45045 b^3 d}-\frac{2 \left (90 a^4 C-539 b^4 (13 A+11 C)-15 a^2 b^2 (715 A+543 C)\right ) \sec (c+d x) \sqrt{a+b \sec (c+d x)} \tan (c+d x)}{45045 b^2 d}+\frac{2 a \left (2717 A b^2+15 a^2 C+2209 b^2 C\right ) \sec ^2(c+d x) \sqrt{a+b \sec (c+d x)} \tan (c+d x)}{9009 b d}+\frac{2 \left (15 a^2 C+11 b^2 (13 A+11 C)\right ) \sec ^3(c+d x) \sqrt{a+b \sec (c+d x)} \tan (c+d x)}{1287 d}+\frac{10 a C \sec ^3(c+d x) (a+b \sec (c+d x))^{3/2} \tan (c+d x)}{143 d}+\frac{2 C \sec ^3(c+d x) (a+b \sec (c+d x))^{5/2} \tan (c+d x)}{13 d}\\ \end{align*}

Mathematica [B]  time = 26.4095, size = 4418, normalized size = 6.8 \[ \text{Result too large to show} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[Sec[c + d*x]^3*(a + b*Sec[c + d*x])^(5/2)*(A + C*Sec[c + d*x]^2),x]

[Out]

(Cos[c + d*x]^4*(a + b*Sec[c + d*x])^(5/2)*(A + C*Sec[c + d*x]^2)*((4*(-1430*a^4*A*b^2 + 39897*a^2*A*b^4 + 210
21*A*b^6 - 240*a^6*C - 760*a^4*b^2*C + 30669*a^2*b^4*C + 17787*b^6*C)*Sin[c + d*x])/(45045*b^4) + (4*Sec[c + d
*x]^4*(143*A*b^2*Sin[c + d*x] + 159*a^2*C*Sin[c + d*x] + 121*b^2*C*Sin[c + d*x]))/1287 + (4*Sec[c + d*x]^3*(27
17*a*A*b^2*Sin[c + d*x] + 15*a^3*C*Sin[c + d*x] + 2209*a*b^2*C*Sin[c + d*x]))/(9009*b) + (4*Sec[c + d*x]^2*(10
725*a^2*A*b^2*Sin[c + d*x] + 7007*A*b^4*Sin[c + d*x] - 90*a^4*C*Sin[c + d*x] + 8145*a^2*b^2*C*Sin[c + d*x] + 5
929*b^4*C*Sin[c + d*x]))/(45045*b^2) + (4*Sec[c + d*x]*(715*a^3*A*b^2*Sin[c + d*x] + 23309*a*A*b^4*Sin[c + d*x
] + 120*a^5*C*Sin[c + d*x] + 395*a^3*b^2*C*Sin[c + d*x] + 18973*a*b^4*C*Sin[c + d*x]))/(45045*b^3) + (108*a*b*
C*Sec[c + d*x]^4*Tan[c + d*x])/143 + (4*b^2*C*Sec[c + d*x]^5*Tan[c + d*x])/13))/(d*(b + a*Cos[c + d*x])^2*(A +
2*C + A*Cos[2*c + 2*d*x])) + (4*((4*a^4*A)/(63*b*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - (62*a^2*A*b)/
(35*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - (14*A*b^3)/(15*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]])
+ (32*a^6*C)/(3003*b^3*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) + (304*a^4*C)/(9009*b*Sqrt[b + a*Cos[c +
d*x]]*Sqrt[Sec[c + d*x]]) - (20446*a^2*b*C)/(15015*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - (154*b^3*C)/
(195*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - (248*a^3*A*Sqrt[Sec[c + d*x]])/(315*Sqrt[b + a*Cos[c + d*x
]]) + (4*a^5*A*Sqrt[Sec[c + d*x]])/(63*b^2*Sqrt[b + a*Cos[c + d*x]]) + (76*a*A*b^2*Sqrt[Sec[c + d*x]])/(105*Sq
rt[b + a*Cos[c + d*x]]) - (27968*a^3*C*Sqrt[Sec[c + d*x]])/(45045*Sqrt[b + a*Cos[c + d*x]]) + (32*a^7*C*Sqrt[S
ec[c + d*x]])/(3003*b^4*Sqrt[b + a*Cos[c + d*x]]) + (40*a^5*C*Sqrt[Sec[c + d*x]])/(1287*b^2*Sqrt[b + a*Cos[c +
d*x]]) + (8696*a*b^2*C*Sqrt[Sec[c + d*x]])/(15015*Sqrt[b + a*Cos[c + d*x]]) - (62*a^3*A*Cos[2*(c + d*x)]*Sqrt
[Sec[c + d*x]])/(35*Sqrt[b + a*Cos[c + d*x]]) + (4*a^5*A*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(63*b^2*Sqrt[b +
a*Cos[c + d*x]]) - (14*a*A*b^2*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(15*Sqrt[b + a*Cos[c + d*x]]) - (20446*a^
3*C*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(15015*Sqrt[b + a*Cos[c + d*x]]) + (32*a^7*C*Cos[2*(c + d*x)]*Sqrt[Se
c[c + d*x]])/(3003*b^4*Sqrt[b + a*Cos[c + d*x]]) + (304*a^5*C*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(9009*b^2*S
qrt[b + a*Cos[c + d*x]]) - (154*a*b^2*C*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(195*Sqrt[b + a*Cos[c + d*x]]))*S
qrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*(a + b*Sec[c + d*x])^(5/2)*(A + C*Sec[c + d*x]^2)*((a + b)*((240*a^6*C -
1617*b^6*(13*A + 11*C) + 10*a^4*b^2*(143*A + 76*C) - 3*a^2*b^4*(13299*A + 10223*C))*EllipticE[ArcSin[Tan[(c +
d*x)/2]], (a - b)/(a + b)] + b*(-240*a^5*C + 180*a^4*b*C + 1617*b^5*(13*A + 11*C) - 10*a^3*b^2*(143*A + 94*C)
+ 15*a^2*b^3*(1573*A + 1175*C) + 6*a*b^4*(2717*A + 2174*C))*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b
)])*(Cos[c + d*x]*Sec[(c + d*x)/2]^2)^(3/2)*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)]*Sec[c + d*
x] + (240*a^6*C - 1617*b^6*(13*A + 11*C) + 10*a^4*b^2*(143*A + 76*C) - 3*a^2*b^4*(13299*A + 10223*C))*Cos[c +
d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^4*Tan[(c + d*x)/2]))/(45045*b^4*d*(b + a*Cos[c + d*x])^3*(A + 2*C +
A*Cos[2*c + 2*d*x])*(Sec[(c + d*x)/2]^2)^(3/2)*Sec[c + d*x]^(9/2)*((2*a*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]
*Sin[c + d*x]*((a + b)*((240*a^6*C - 1617*b^6*(13*A + 11*C) + 10*a^4*b^2*(143*A + 76*C) - 3*a^2*b^4*(13299*A +
10223*C))*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + b*(-240*a^5*C + 180*a^4*b*C + 1617*b^5*(13*A
+ 11*C) - 10*a^3*b^2*(143*A + 94*C) + 15*a^2*b^3*(1573*A + 1175*C) + 6*a*b^4*(2717*A + 2174*C))*EllipticF[Arc
Sin[Tan[(c + d*x)/2]], (a - b)/(a + b)])*(Cos[c + d*x]*Sec[(c + d*x)/2]^2)^(3/2)*Sqrt[((b + a*Cos[c + d*x])*Se
c[(c + d*x)/2]^2)/(a + b)]*Sec[c + d*x] + (240*a^6*C - 1617*b^6*(13*A + 11*C) + 10*a^4*b^2*(143*A + 76*C) - 3*
a^2*b^4*(13299*A + 10223*C))*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^4*Tan[(c + d*x)/2]))/(45045*b^
4*(b + a*Cos[c + d*x])^(3/2)*(Sec[(c + d*x)/2]^2)^(3/2)) - (2*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*Tan[(c + d
*x)/2]*((a + b)*((240*a^6*C - 1617*b^6*(13*A + 11*C) + 10*a^4*b^2*(143*A + 76*C) - 3*a^2*b^4*(13299*A + 10223*
C))*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + b*(-240*a^5*C + 180*a^4*b*C + 1617*b^5*(13*A + 11*C
) - 10*a^3*b^2*(143*A + 94*C) + 15*a^2*b^3*(1573*A + 1175*C) + 6*a*b^4*(2717*A + 2174*C))*EllipticF[ArcSin[Tan
[(c + d*x)/2]], (a - b)/(a + b)])*(Cos[c + d*x]*Sec[(c + d*x)/2]^2)^(3/2)*Sqrt[((b + a*Cos[c + d*x])*Sec[(c +
d*x)/2]^2)/(a + b)]*Sec[c + d*x] + (240*a^6*C - 1617*b^6*(13*A + 11*C) + 10*a^4*b^2*(143*A + 76*C) - 3*a^2*b^4
*(13299*A + 10223*C))*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^4*Tan[(c + d*x)/2]))/(15015*b^4*Sqrt[
b + a*Cos[c + d*x]]*(Sec[(c + d*x)/2]^2)^(3/2)) + (2*((a + b)*((240*a^6*C - 1617*b^6*(13*A + 11*C) + 10*a^4*b^
2*(143*A + 76*C) - 3*a^2*b^4*(13299*A + 10223*C))*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + b*(-2
40*a^5*C + 180*a^4*b*C + 1617*b^5*(13*A + 11*C) - 10*a^3*b^2*(143*A + 94*C) + 15*a^2*b^3*(1573*A + 1175*C) + 6
*a*b^4*(2717*A + 2174*C))*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)])*(Cos[c + d*x]*Sec[(c + d*x)/2]
^2)^(3/2)*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)]*Sec[c + d*x] + (240*a^6*C - 1617*b^6*(13*A +
11*C) + 10*a^4*b^2*(143*A + 76*C) - 3*a^2*b^4*(13299*A + 10223*C))*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c +
d*x)/2]^4*Tan[(c + d*x)/2])*(-(Cos[(c + d*x)/2]*Sec[c + d*x]*Sin[(c + d*x)/2]) + Cos[(c + d*x)/2]^2*Sec[c + d
*x]*Tan[c + d*x]))/(45045*b^4*Sqrt[b + a*Cos[c + d*x]]*(Sec[(c + d*x)/2]^2)^(3/2)*Sqrt[Cos[(c + d*x)/2]^2*Sec[
c + d*x]]) + (4*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*(((240*a^6*C - 1617*b^6*(13*A + 11*C) + 10*a^4*b^2*(143*
A + 76*C) - 3*a^2*b^4*(13299*A + 10223*C))*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^6)/2 - a*(240*a^
6*C - 1617*b^6*(13*A + 11*C) + 10*a^4*b^2*(143*A + 76*C) - 3*a^2*b^4*(13299*A + 10223*C))*Cos[c + d*x]*Sec[(c
+ d*x)/2]^4*Sin[c + d*x]*Tan[(c + d*x)/2] - (240*a^6*C - 1617*b^6*(13*A + 11*C) + 10*a^4*b^2*(143*A + 76*C) -
3*a^2*b^4*(13299*A + 10223*C))*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^4*Sin[c + d*x]*Tan[(c + d*x)/2] + 2*(240*
a^6*C - 1617*b^6*(13*A + 11*C) + 10*a^4*b^2*(143*A + 76*C) - 3*a^2*b^4*(13299*A + 10223*C))*Cos[c + d*x]*(b +
a*Cos[c + d*x])*Sec[(c + d*x)/2]^4*Tan[(c + d*x)/2]^2 + (3*(a + b)*((240*a^6*C - 1617*b^6*(13*A + 11*C) + 10*a
^4*b^2*(143*A + 76*C) - 3*a^2*b^4*(13299*A + 10223*C))*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] +
b*(-240*a^5*C + 180*a^4*b*C + 1617*b^5*(13*A + 11*C) - 10*a^3*b^2*(143*A + 94*C) + 15*a^2*b^3*(1573*A + 1175*C
) + 6*a*b^4*(2717*A + 2174*C))*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)])*Sqrt[Cos[c + d*x]*Sec[(c
+ d*x)/2]^2]*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)]*Sec[c + d*x]*(-(Sec[(c + d*x)/2]^2*Sin[c
+ d*x]) + Cos[c + d*x]*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]))/2 + ((a + b)*((240*a^6*C - 1617*b^6*(13*A + 11*C)
+ 10*a^4*b^2*(143*A + 76*C) - 3*a^2*b^4*(13299*A + 10223*C))*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a +
b)] + b*(-240*a^5*C + 180*a^4*b*C + 1617*b^5*(13*A + 11*C) - 10*a^3*b^2*(143*A + 94*C) + 15*a^2*b^3*(1573*A +
1175*C) + 6*a*b^4*(2717*A + 2174*C))*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)])*(Cos[c + d*x]*Sec[
(c + d*x)/2]^2)^(3/2)*Sec[c + d*x]*(-((a*Sec[(c + d*x)/2]^2*Sin[c + d*x])/(a + b)) + ((b + a*Cos[c + d*x])*Sec
[(c + d*x)/2]^2*Tan[(c + d*x)/2])/(a + b)))/(2*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)]) + (a +
b)*(Cos[c + d*x]*Sec[(c + d*x)/2]^2)^(3/2)*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)]*Sec[c + d*
x]*((b*(-240*a^5*C + 180*a^4*b*C + 1617*b^5*(13*A + 11*C) - 10*a^3*b^2*(143*A + 94*C) + 15*a^2*b^3*(1573*A + 1
175*C) + 6*a*b^4*(2717*A + 2174*C))*Sec[(c + d*x)/2]^2)/(2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[1 - ((a - b)*Tan[
(c + d*x)/2]^2)/(a + b)]) + ((240*a^6*C - 1617*b^6*(13*A + 11*C) + 10*a^4*b^2*(143*A + 76*C) - 3*a^2*b^4*(1329
9*A + 10223*C))*Sec[(c + d*x)/2]^2*Sqrt[1 - ((a - b)*Tan[(c + d*x)/2]^2)/(a + b)])/(2*Sqrt[1 - Tan[(c + d*x)/2
]^2])) + (a + b)*((240*a^6*C - 1617*b^6*(13*A + 11*C) + 10*a^4*b^2*(143*A + 76*C) - 3*a^2*b^4*(13299*A + 10223
*C))*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + b*(-240*a^5*C + 180*a^4*b*C + 1617*b^5*(13*A + 11*
C) - 10*a^3*b^2*(143*A + 94*C) + 15*a^2*b^3*(1573*A + 1175*C) + 6*a*b^4*(2717*A + 2174*C))*EllipticF[ArcSin[Ta
n[(c + d*x)/2]], (a - b)/(a + b)])*(Cos[c + d*x]*Sec[(c + d*x)/2]^2)^(3/2)*Sqrt[((b + a*Cos[c + d*x])*Sec[(c +
d*x)/2]^2)/(a + b)]*Sec[c + d*x]*Tan[c + d*x]))/(45045*b^4*Sqrt[b + a*Cos[c + d*x]]*(Sec[(c + d*x)/2]^2)^(3/2
))))

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Maple [B]  time = 3.911, size = 6077, normalized size = 9.4 \begin{align*} \text{output too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sec(d*x+c)^3*(a+b*sec(d*x+c))^(5/2)*(A+C*sec(d*x+c)^2),x)

[Out]

result too large to display

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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)^3*(a+b*sec(d*x+c))^(5/2)*(A+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^{2} \sec \left (d x + c\right )^{7} + 2 \, C a b \sec \left (d x + c\right )^{6} + 2 \, A a b \sec \left (d x + c\right )^{4} + A a^{2} \sec \left (d x + c\right )^{3} +{\left (C a^{2} + A b^{2}\right )} \sec \left (d x + c\right )^{5}\right )} \sqrt{b \sec \left (d x + c\right ) + a}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(d*x+c)^3*(a+b*sec(d*x+c))^(5/2)*(A+C*sec(d*x+c)^2),x, algorithm="fricas")

[Out]

integral((C*b^2*sec(d*x + c)^7 + 2*C*a*b*sec(d*x + c)^6 + 2*A*a*b*sec(d*x + c)^4 + A*a^2*sec(d*x + c)^3 + (C*a
^2 + A*b^2)*sec(d*x + c)^5)*sqrt(b*sec(d*x + c) + a), 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)**3*(a+b*sec(d*x+c))**(5/2)*(A+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} + A\right )}{\left (b \sec \left (d x + c\right ) + a\right )}^{\frac{5}{2}} \sec \left (d x + c\right )^{3}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(d*x+c)^3*(a+b*sec(d*x+c))^(5/2)*(A+C*sec(d*x+c)^2),x, algorithm="giac")

[Out]

integrate((C*sec(d*x + c)^2 + A)*(b*sec(d*x + c) + a)^(5/2)*sec(d*x + c)^3, x)