\(\int \sec ^3(c+d x) (a+b \sec (c+d x))^{5/2} (A+C \sec ^2(c+d x)) \, dx\) [725]

Optimal result

Integrand size = 35, antiderivative size = 650 \[ \int \sec ^3(c+d x) (a+b \sec (c+d x))^{5/2} \left (A+C \sec ^2(c+d x)\right ) \, dx=\frac {2 (a-b) \sqrt {a+b} \left (240 a^6 C-1617 b^6 (13 A+11 C)+10 a^4 b^2 (143 A+76 C)-3 a^2 b^4 (13299 A+10223 C)\right ) \cot (c+d x) E\left (\arcsin \left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right )|\frac {a+b}{a-b}\right ) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (1+\sec (c+d x))}{a-b}}}{45045 b^5 d}+\frac {2 (a-b) \sqrt {a+b} \left (240 a^5 C+180 a^4 b C+1617 b^5 (13 A+11 C)+10 a^3 b^2 (143 A+94 C)+15 a^2 b^3 (1573 A+1175 C)-6 a b^4 (2717 A+2174 C)\right ) \cot (c+d x) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right ) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (1+\sec (c+d x))}{a-b}}}{45045 b^4 d}+\frac {2 a \left (120 a^4 C+5 a^2 b^2 (143 A+79 C)+b^4 (23309 A+18973 C)\right ) \sqrt {a+b \sec (c+d x)} \tan (c+d x)}{45045 b^3 d}-\frac {2 \left (90 a^4 C-539 b^4 (13 A+11 C)-15 a^2 b^2 (715 A+543 C)\right ) \sec (c+d x) \sqrt {a+b \sec (c+d x)} \tan (c+d x)}{45045 b^2 d}+\frac {2 a \left (2717 A b^2+15 a^2 C+2209 b^2 C\right ) \sec ^2(c+d x) \sqrt {a+b \sec (c+d x)} \tan (c+d x)}{9009 b d}+\frac {2 \left (15 a^2 C+11 b^2 (13 A+11 C)\right ) \sec ^3(c+d x) \sqrt {a+b \sec (c+d x)} \tan (c+d x)}{1287 d}+\frac {10 a C \sec ^3(c+d x) (a+b \sec (c+d x))^{3/2} \tan (c+d x)}{143 d}+\frac {2 C \sec ^3(c+d x) (a+b \sec (c+d x))^{5/2} \tan (c+d x)}{13 d} \]

[Out]

2/45045*(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(d*x+c)*
EllipticE((a+b*sec(d*x+c))^(1/2)/(a+b)^(1/2),((a+b)/(a-b))^(1/2))*(a+b)^(1/2)*(b*(1-sec(d*x+c))/(a+b))^(1/2)*(
-b*(1+sec(d*x+c))/(a-b))^(1/2)/b^5/d+2/45045*(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(d*x+c)*EllipticF((a+b*sec(d*x+c))^(1/2)/(a+b)
^(1/2),((a+b)/(a-b))^(1/2))*(a+b)^(1/2)*(b*(1-sec(d*x+c))/(a+b))^(1/2)*(-b*(1+sec(d*x+c))/(a-b))^(1/2)/b^4/d+1
0/143*a*C*sec(d*x+c)^3*(a+b*sec(d*x+c))^(3/2)*tan(d*x+c)/d+2/13*C*sec(d*x+c)^3*(a+b*sec(d*x+c))^(5/2)*tan(d*x+
c)/d+2/45045*a*(120*a^4*C+5*a^2*b^2*(143*A+79*C)+b^4*(23309*A+18973*C))*(a+b*sec(d*x+c))^(1/2)*tan(d*x+c)/b^3/
d-2/45045*(90*a^4*C-539*b^4*(13*A+11*C)-15*a^2*b^2*(715*A+543*C))*sec(d*x+c)*(a+b*sec(d*x+c))^(1/2)*tan(d*x+c)
/b^2/d+2/9009*a*(2717*A*b^2+15*C*a^2+2209*C*b^2)*sec(d*x+c)^2*(a+b*sec(d*x+c))^(1/2)*tan(d*x+c)/b/d+2/1287*(15
*C*a^2+11*b^2*(13*A+11*C))*sec(d*x+c)^3*(a+b*sec(d*x+c))^(1/2)*tan(d*x+c)/d

Rubi [A] (verified)

Time = 3.04 (sec) , antiderivative size = 650, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.229, Rules used = {4182, 4181, 4187, 4177, 4167, 4090, 3917, 4089} \[ \int \sec ^3(c+d x) (a+b \sec (c+d x))^{5/2} \left (A+C \sec ^2(c+d x)\right ) \, dx=\frac {2 \left (15 a^2 C+11 b^2 (13 A+11 C)\right ) \tan (c+d x) \sec ^3(c+d x) \sqrt {a+b \sec (c+d x)}}{1287 d}+\frac {2 a \left (15 a^2 C+2717 A b^2+2209 b^2 C\right ) \tan (c+d x) \sec ^2(c+d x) \sqrt {a+b \sec (c+d x)}}{9009 b d}-\frac {2 \left (90 a^4 C-15 a^2 b^2 (715 A+543 C)-539 b^4 (13 A+11 C)\right ) \tan (c+d x) \sec (c+d x) \sqrt {a+b \sec (c+d x)}}{45045 b^2 d}+\frac {2 a \left (120 a^4 C+5 a^2 b^2 (143 A+79 C)+b^4 (23309 A+18973 C)\right ) \tan (c+d x) \sqrt {a+b \sec (c+d x)}}{45045 b^3 d}+\frac {2 (a-b) \sqrt {a+b} \left (240 a^6 C+10 a^4 b^2 (143 A+76 C)-3 a^2 b^4 (13299 A+10223 C)-1617 b^6 (13 A+11 C)\right ) \cot (c+d x) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (\sec (c+d x)+1)}{a-b}} E\left (\arcsin \left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right )|\frac {a+b}{a-b}\right )}{45045 b^5 d}+\frac {2 (a-b) \sqrt {a+b} \left (240 a^5 C+180 a^4 b C+10 a^3 b^2 (143 A+94 C)+15 a^2 b^3 (1573 A+1175 C)-6 a b^4 (2717 A+2174 C)+1617 b^5 (13 A+11 C)\right ) \cot (c+d x) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (\sec (c+d x)+1)}{a-b}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right )}{45045 b^4 d}+\frac {2 C \tan (c+d x) \sec ^3(c+d x) (a+b \sec (c+d x))^{5/2}}{13 d}+\frac {10 a C \tan (c+d x) \sec ^3(c+d x) (a+b \sec (c+d x))^{3/2}}{143 d} \]

[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 3917

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

Rule 4089

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 + C
sc[e + f*x])/(a - b))]/(b^2*f*Cot[e + f*x]))*EllipticE[ArcSin[Sqrt[a + b*Csc[e + f*x]]/Rt[a + b*(B/A), 2]], (a
*A + b*B)/(a*A - b*B)], x] /; FreeQ[{a, b, e, f, A, B}, x] && NeQ[a^2 - b^2, 0] && EqQ[A^2 - B^2, 0]

Rule 4090

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 4167

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 4177

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 4181

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 4182

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 4187

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]

Rubi steps \begin{align*} \text {integral}& = \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 (\arcsin \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) \operatorname {EllipticF}\left (\arcsin \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] (warning: unable to verify)

Leaf count is larger than twice the leaf count of optimal. \(4551\) vs. \(2(650)=1300\).

Time = 30.99 (sec) , antiderivative size = 4551, normalized size of antiderivative = 7.00 \[ \int \sec ^3(c+d x) (a+b \sec (c+d x))^{5/2} \left (A+C \sec ^2(c+d x)\right ) \, dx=\text {Result too large to show} \]

[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)*(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))*Sqrt[Cos[c + d*x]/(1 + Co
s[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a -
b)/(a + b)] + 2*b*(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))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c
+ d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + (240*a^6*C - 161
7*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]^2*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])*
Sqrt[Sec[(c + d*x)/2]^2]*Sec[c + d*x]^(9/2)*((2*a*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*Sin[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))*Sqrt[Cos[c
+ d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d
*x)/2]], (a - b)/(a + b)] + 2*b*(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))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqr
t[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + (2
40*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]^2*Tan[(c + d*x)/2]))/(45045*b^4*(b + a*Cos[c + d*x])^(3/2)*Sqrt[Sec[(c + d
*x)/2]^2]) - (2*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*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))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqr
t[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + 2*
b*(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))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b
)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(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))*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*
x)/2]^2*Tan[(c + d*x)/2]))/(45045*b^4*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[(c + d*x)/2]^2]) + (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]^4)/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))*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c
+ d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*((Cos[c + d*x]*Sin[c + d*x])/(1 + Cos[c + d*x]
)^2 - Sin[c + d*x]/(1 + Cos[c + d*x])))/Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])] + (b*(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))*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/
(a + b)]*((Cos[c + d*x]*Sin[c + d*x])/(1 + Cos[c + d*x])^2 - Sin[c + d*x]/(1 + Cos[c + d*x])))/Sqrt[Cos[c + d*
x]/(1 + Cos[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))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)
]*(-((a*Sin[c + d*x])/((a + b)*(1 + Cos[c + d*x]))) + ((b + a*Cos[c + d*x])*Sin[c + d*x])/((a + b)*(1 + Cos[c
+ d*x])^2)))/Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))] + (b*(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))
*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*(-((a*Sin[c + d*x]
)/((a + b)*(1 + Cos[c + d*x]))) + ((b + a*Cos[c + d*x])*Sin[c + d*x])/((a + b)*(1 + Cos[c + d*x])^2)))/Sqrt[(b
+ a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))] - 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]^2*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]^2*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))*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]^
2 + (b*(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*(15
73*A + 1175*C) + 6*a*b^4*(2717*A + 2174*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((
a + b)*(1 + Cos[c + d*x]))]*Sec[(c + d*x)/2]^2)/(Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[1 - ((a - b)*Tan[(c + d*x)/
2]^2)/(a + b)]) + ((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))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*Se
c[(c + d*x)/2]^2*Sqrt[1 - ((a - b)*Tan[(c + d*x)/2]^2)/(a + b)])/Sqrt[1 - Tan[(c + d*x)/2]^2]))/(45045*b^4*Sqr
t[b + a*Cos[c + d*x]]*Sqrt[Sec[(c + d*x)/2]^2]) + (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))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d
*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + 2*b*(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))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x
]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(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))*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*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]))/(4
5045*b^4*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[(c + d*x)/2]^2]*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]])))

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(7823\) vs. \(2(604)=1208\).

Time = 214.39 (sec) , antiderivative size = 7824, normalized size of antiderivative = 12.04

method	result	size
parts	\(\text {Expression too large to display}\)	\(7824\)
default	\(\text {Expression too large to display}\)	\(7939\)

[In]

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

[Out]

result too large to display

Fricas [F]

\[ \int \sec ^3(c+d x) (a+b \sec (c+d x))^{5/2} \left (A+C \sec ^2(c+d x)\right ) \, dx=\int { {\left (C \sec \left (d x + c\right )^{2} + A\right )} {\left (b \sec \left (d x + c\right ) + a\right )}^{\frac {5}{2}} \sec \left (d x + c\right )^{3} \,d x } \]

[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)

Sympy [F(-1)]

Timed out. \[ \int \sec ^3(c+d x) (a+b \sec (c+d x))^{5/2} \left (A+C \sec ^2(c+d x)\right ) \, dx=\text {Timed out} \]

[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

Maxima [F(-1)]

Timed out. \[ \int \sec ^3(c+d x) (a+b \sec (c+d x))^{5/2} \left (A+C \sec ^2(c+d x)\right ) \, dx=\text {Timed out} \]

[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

Giac [F]

\[ \int \sec ^3(c+d x) (a+b \sec (c+d x))^{5/2} \left (A+C \sec ^2(c+d x)\right ) \, dx=\int { {\left (C \sec \left (d x + c\right )^{2} + A\right )} {\left (b \sec \left (d x + c\right ) + a\right )}^{\frac {5}{2}} \sec \left (d x + c\right )^{3} \,d x } \]

[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)

Mupad [F(-1)]

Timed out. \[ \int \sec ^3(c+d x) (a+b \sec (c+d x))^{5/2} \left (A+C \sec ^2(c+d x)\right ) \, dx=\int \frac {\left (A+\frac {C}{{\cos \left (c+d\,x\right )}^2}\right )\,{\left (a+\frac {b}{\cos \left (c+d\,x\right )}\right )}^{5/2}}{{\cos \left (c+d\,x\right )}^3} \,d x \]

[In]

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

[Out]

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