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

Optimal. Leaf size=628 \[ -\frac{2 (a-b) \sqrt{a+b} \cot (c+d x) \left (-6 a^2 b^2 (33 A-11 B+24 C)+4 a^3 b (22 B-9 C)-48 a^4 C-3 a b^3 (627 A-143 B+471 C)+3 b^4 (275 A-539 B+225 C)\right ) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right ),\frac{a+b}{a-b}\right )}{3465 b^4 d}+\frac{2 \tan (c+d x) \sec ^2(c+d x) \left (3 a^2 C+110 a b B+99 A b^2+81 b^2 C\right ) \sqrt{a+b \sec (c+d x)}}{693 b d}+\frac{2 \tan (c+d x) \sec (c+d x) \left (33 a^2 b B-18 a^3 C+6 a b^2 (132 A+101 C)+539 b^3 B\right ) \sqrt{a+b \sec (c+d x)}}{3465 b^2 d}-\frac{2 \tan (c+d x) \left (-3 a^2 b^2 (33 A+19 C)+44 a^3 b B-24 a^4 C-968 a b^3 B-75 b^4 (11 A+9 C)\right ) \sqrt{a+b \sec (c+d x)}}{3465 b^3 d}-\frac{2 (a-b) \sqrt{a+b} \cot (c+d x) \left (-18 a^3 b^2 (11 A+6 C)+363 a^2 b^3 B+88 a^4 b B-48 a^5 C+6 a b^4 (451 A+348 C)+1617 b^5 B\right ) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} E\left (\sin ^{-1}\left (\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right )|\frac{a+b}{a-b}\right )}{3465 b^5 d}+\frac{2 (3 a C+11 b B) \tan (c+d x) \sec ^3(c+d x) \sqrt{a+b \sec (c+d x)}}{99 d}+\frac{2 C \tan (c+d x) \sec ^3(c+d x) (a+b \sec (c+d x))^{3/2}}{11 d} \]

[Out]

(-2*(a - b)*Sqrt[a + b]*(88*a^4*b*B + 363*a^2*b^3*B + 1617*b^5*B - 48*a^5*C - 18*a^3*b^2*(11*A + 6*C) + 6*a*b^
4*(451*A + 348*C))*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[
(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(3465*b^5*d) - (2*(a - b)*Sqrt[a + b]
*(4*a^3*b*(22*B - 9*C) - 48*a^4*C - 6*a^2*b^2*(33*A - 11*B + 24*C) + 3*b^4*(275*A - 539*B + 225*C) - 3*a*b^3*(
627*A - 143*B + 471*C))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*
Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(3465*b^4*d) - (2*(44*a^3*b*B -
968*a*b^3*B - 24*a^4*C - 75*b^4*(11*A + 9*C) - 3*a^2*b^2*(33*A + 19*C))*Sqrt[a + b*Sec[c + d*x]]*Tan[c + d*x])
/(3465*b^3*d) + (2*(33*a^2*b*B + 539*b^3*B - 18*a^3*C + 6*a*b^2*(132*A + 101*C))*Sec[c + d*x]*Sqrt[a + b*Sec[c
+ d*x]]*Tan[c + d*x])/(3465*b^2*d) + (2*(99*A*b^2 + 110*a*b*B + 3*a^2*C + 81*b^2*C)*Sec[c + d*x]^2*Sqrt[a + b
*Sec[c + d*x]]*Tan[c + d*x])/(693*b*d) + (2*(11*b*B + 3*a*C)*Sec[c + d*x]^3*Sqrt[a + b*Sec[c + d*x]]*Tan[c + d
*x])/(99*d) + (2*C*Sec[c + d*x]^3*(a + b*Sec[c + d*x])^(3/2)*Tan[c + d*x])/(11*d)

________________________________________________________________________________________

Rubi [A]  time = 2.62743, antiderivative size = 628, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 43, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.163, Rules used = {4096, 4102, 4092, 4082, 4005, 3832, 4004} \[ \frac{2 \tan (c+d x) \sec ^2(c+d x) \left (3 a^2 C+110 a b B+99 A b^2+81 b^2 C\right ) \sqrt{a+b \sec (c+d x)}}{693 b d}+\frac{2 \tan (c+d x) \sec (c+d x) \left (33 a^2 b B-18 a^3 C+6 a b^2 (132 A+101 C)+539 b^3 B\right ) \sqrt{a+b \sec (c+d x)}}{3465 b^2 d}-\frac{2 \tan (c+d x) \left (-3 a^2 b^2 (33 A+19 C)+44 a^3 b B-24 a^4 C-968 a b^3 B-75 b^4 (11 A+9 C)\right ) \sqrt{a+b \sec (c+d x)}}{3465 b^3 d}-\frac{2 (a-b) \sqrt{a+b} \cot (c+d x) \left (-6 a^2 b^2 (33 A-11 B+24 C)+4 a^3 b (22 B-9 C)-48 a^4 C-3 a b^3 (627 A-143 B+471 C)+3 b^4 (275 A-539 B+225 C)\right ) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} F\left (\sin ^{-1}\left (\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right )|\frac{a+b}{a-b}\right )}{3465 b^4 d}-\frac{2 (a-b) \sqrt{a+b} \cot (c+d x) \left (-18 a^3 b^2 (11 A+6 C)+363 a^2 b^3 B+88 a^4 b B-48 a^5 C+6 a b^4 (451 A+348 C)+1617 b^5 B\right ) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} E\left (\sin ^{-1}\left (\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right )|\frac{a+b}{a-b}\right )}{3465 b^5 d}+\frac{2 (3 a C+11 b B) \tan (c+d x) \sec ^3(c+d x) \sqrt{a+b \sec (c+d x)}}{99 d}+\frac{2 C \tan (c+d x) \sec ^3(c+d x) (a+b \sec (c+d x))^{3/2}}{11 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*(a - b)*Sqrt[a + b]*(88*a^4*b*B + 363*a^2*b^3*B + 1617*b^5*B - 48*a^5*C - 18*a^3*b^2*(11*A + 6*C) + 6*a*b^
4*(451*A + 348*C))*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[
(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(3465*b^5*d) - (2*(a - b)*Sqrt[a + b]
*(4*a^3*b*(22*B - 9*C) - 48*a^4*C - 6*a^2*b^2*(33*A - 11*B + 24*C) + 3*b^4*(275*A - 539*B + 225*C) - 3*a*b^3*(
627*A - 143*B + 471*C))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*
Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(3465*b^4*d) - (2*(44*a^3*b*B -
968*a*b^3*B - 24*a^4*C - 75*b^4*(11*A + 9*C) - 3*a^2*b^2*(33*A + 19*C))*Sqrt[a + b*Sec[c + d*x]]*Tan[c + d*x])
/(3465*b^3*d) + (2*(33*a^2*b*B + 539*b^3*B - 18*a^3*C + 6*a*b^2*(132*A + 101*C))*Sec[c + d*x]*Sqrt[a + b*Sec[c
+ d*x]]*Tan[c + d*x])/(3465*b^2*d) + (2*(99*A*b^2 + 110*a*b*B + 3*a^2*C + 81*b^2*C)*Sec[c + d*x]^2*Sqrt[a + b
*Sec[c + d*x]]*Tan[c + d*x])/(693*b*d) + (2*(11*b*B + 3*a*C)*Sec[c + d*x]^3*Sqrt[a + b*Sec[c + d*x]]*Tan[c + d
*x])/(99*d) + (2*C*Sec[c + d*x]^3*(a + b*Sec[c + d*x])^(3/2)*Tan[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 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))^{3/2} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx &=\frac{2 C \sec ^3(c+d x) (a+b \sec (c+d x))^{3/2} \tan (c+d x)}{11 d}+\frac{2}{11} \int \sec ^3(c+d x) \sqrt{a+b \sec (c+d x)} \left (\frac{1}{2} a (11 A+6 C)+\frac{1}{2} (11 A b+11 a B+9 b C) \sec (c+d x)+\frac{1}{2} (11 b B+3 a C) \sec ^2(c+d x)\right ) \, dx\\ &=\frac{2 (11 b B+3 a C) \sec ^3(c+d x) \sqrt{a+b \sec (c+d x)} \tan (c+d x)}{99 d}+\frac{2 C \sec ^3(c+d x) (a+b \sec (c+d x))^{3/2} \tan (c+d x)}{11 d}+\frac{4}{99} \int \frac{\sec ^3(c+d x) \left (\frac{3}{4} a (33 a A+22 b B+24 a C)+\frac{1}{4} \left (198 a A b+99 a^2 B+77 b^2 B+156 a b C\right ) \sec (c+d x)+\frac{1}{4} \left (99 A b^2+110 a b B+3 a^2 C+81 b^2 C\right ) \sec ^2(c+d x)\right )}{\sqrt{a+b \sec (c+d x)}} \, dx\\ &=\frac{2 \left (99 A b^2+110 a b B+3 a^2 C+81 b^2 C\right ) \sec ^2(c+d x) \sqrt{a+b \sec (c+d x)} \tan (c+d x)}{693 b d}+\frac{2 (11 b B+3 a C) \sec ^3(c+d x) \sqrt{a+b \sec (c+d x)} \tan (c+d x)}{99 d}+\frac{2 C \sec ^3(c+d x) (a+b \sec (c+d x))^{3/2} \tan (c+d x)}{11 d}+\frac{8 \int \frac{\sec ^2(c+d x) \left (\frac{1}{2} a \left (99 A b^2+110 a b B+3 a^2 C+81 b^2 C\right )+\frac{1}{8} b \left (1012 a b B+45 b^2 (11 A+9 C)+a^2 (693 A+519 C)\right ) \sec (c+d x)+\frac{1}{8} \left (33 a^2 b B+539 b^3 B-18 a^3 C+6 a b^2 (132 A+101 C)\right ) \sec ^2(c+d x)\right )}{\sqrt{a+b \sec (c+d x)}} \, dx}{693 b}\\ &=\frac{2 \left (33 a^2 b B+539 b^3 B-18 a^3 C+6 a b^2 (132 A+101 C)\right ) \sec (c+d x) \sqrt{a+b \sec (c+d x)} \tan (c+d x)}{3465 b^2 d}+\frac{2 \left (99 A b^2+110 a b B+3 a^2 C+81 b^2 C\right ) \sec ^2(c+d x) \sqrt{a+b \sec (c+d x)} \tan (c+d x)}{693 b d}+\frac{2 (11 b B+3 a C) \sec ^3(c+d x) \sqrt{a+b \sec (c+d x)} \tan (c+d x)}{99 d}+\frac{2 C \sec ^3(c+d x) (a+b \sec (c+d x))^{3/2} \tan (c+d x)}{11 d}+\frac{16 \int \frac{\sec (c+d x) \left (\frac{1}{8} a \left (33 a^2 b B+539 b^3 B-18 a^3 C+6 a b^2 (132 A+101 C)\right )+\frac{1}{16} b \left (2299 a^2 b B+1617 b^3 B+6 a^3 C+18 a b^2 (242 A+191 C)\right ) \sec (c+d x)-\frac{3}{16} \left (44 a^3 b B-968 a b^3 B-24 a^4 C-75 b^4 (11 A+9 C)-3 a^2 b^2 (33 A+19 C)\right ) \sec ^2(c+d x)\right )}{\sqrt{a+b \sec (c+d x)}} \, dx}{3465 b^2}\\ &=-\frac{2 \left (44 a^3 b B-968 a b^3 B-24 a^4 C-75 b^4 (11 A+9 C)-3 a^2 b^2 (33 A+19 C)\right ) \sqrt{a+b \sec (c+d x)} \tan (c+d x)}{3465 b^3 d}+\frac{2 \left (33 a^2 b B+539 b^3 B-18 a^3 C+6 a b^2 (132 A+101 C)\right ) \sec (c+d x) \sqrt{a+b \sec (c+d x)} \tan (c+d x)}{3465 b^2 d}+\frac{2 \left (99 A b^2+110 a b B+3 a^2 C+81 b^2 C\right ) \sec ^2(c+d x) \sqrt{a+b \sec (c+d x)} \tan (c+d x)}{693 b d}+\frac{2 (11 b B+3 a C) \sec ^3(c+d x) \sqrt{a+b \sec (c+d x)} \tan (c+d x)}{99 d}+\frac{2 C \sec ^3(c+d x) (a+b \sec (c+d x))^{3/2} \tan (c+d x)}{11 d}+\frac{32 \int \frac{\sec (c+d x) \left (\frac{3}{32} b \left (22 a^3 b B+2046 a b^3 B-12 a^4 C+75 b^4 (11 A+9 C)+9 a^2 b^2 (187 A+141 C)\right )+\frac{3}{32} \left (88 a^4 b B+363 a^2 b^3 B+1617 b^5 B-48 a^5 C-18 a^3 b^2 (11 A+6 C)+6 a b^4 (451 A+348 C)\right ) \sec (c+d x)\right )}{\sqrt{a+b \sec (c+d x)}} \, dx}{10395 b^3}\\ &=-\frac{2 \left (44 a^3 b B-968 a b^3 B-24 a^4 C-75 b^4 (11 A+9 C)-3 a^2 b^2 (33 A+19 C)\right ) \sqrt{a+b \sec (c+d x)} \tan (c+d x)}{3465 b^3 d}+\frac{2 \left (33 a^2 b B+539 b^3 B-18 a^3 C+6 a b^2 (132 A+101 C)\right ) \sec (c+d x) \sqrt{a+b \sec (c+d x)} \tan (c+d x)}{3465 b^2 d}+\frac{2 \left (99 A b^2+110 a b B+3 a^2 C+81 b^2 C\right ) \sec ^2(c+d x) \sqrt{a+b \sec (c+d x)} \tan (c+d x)}{693 b d}+\frac{2 (11 b B+3 a C) \sec ^3(c+d x) \sqrt{a+b \sec (c+d x)} \tan (c+d x)}{99 d}+\frac{2 C \sec ^3(c+d x) (a+b \sec (c+d x))^{3/2} \tan (c+d x)}{11 d}+\frac{\left (88 a^4 b B+363 a^2 b^3 B+1617 b^5 B-48 a^5 C-18 a^3 b^2 (11 A+6 C)+6 a b^4 (451 A+348 C)\right ) \int \frac{\sec (c+d x) (1+\sec (c+d x))}{\sqrt{a+b \sec (c+d x)}} \, dx}{3465 b^3}+\frac{\left (32 \left (\frac{3}{32} b \left (22 a^3 b B+2046 a b^3 B-12 a^4 C+75 b^4 (11 A+9 C)+9 a^2 b^2 (187 A+141 C)\right )-\frac{3}{32} \left (88 a^4 b B+363 a^2 b^3 B+1617 b^5 B-48 a^5 C-18 a^3 b^2 (11 A+6 C)+6 a b^4 (451 A+348 C)\right )\right )\right ) \int \frac{\sec (c+d x)}{\sqrt{a+b \sec (c+d x)}} \, dx}{10395 b^3}\\ &=-\frac{2 (a-b) \sqrt{a+b} \left (88 a^4 b B+363 a^2 b^3 B+1617 b^5 B-48 a^5 C-18 a^3 b^2 (11 A+6 C)+6 a b^4 (451 A+348 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}}}{3465 b^5 d}-\frac{2 (a-b) \sqrt{a+b} \left (a^3 b (88 B-36 C)-48 a^4 C-6 a^2 b^2 (33 A-11 B+24 C)+3 b^4 (275 A-539 B+225 C)-3 a b^3 (627 A-143 B+471 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}}}{3465 b^4 d}-\frac{2 \left (44 a^3 b B-968 a b^3 B-24 a^4 C-75 b^4 (11 A+9 C)-3 a^2 b^2 (33 A+19 C)\right ) \sqrt{a+b \sec (c+d x)} \tan (c+d x)}{3465 b^3 d}+\frac{2 \left (33 a^2 b B+539 b^3 B-18 a^3 C+6 a b^2 (132 A+101 C)\right ) \sec (c+d x) \sqrt{a+b \sec (c+d x)} \tan (c+d x)}{3465 b^2 d}+\frac{2 \left (99 A b^2+110 a b B+3 a^2 C+81 b^2 C\right ) \sec ^2(c+d x) \sqrt{a+b \sec (c+d x)} \tan (c+d x)}{693 b d}+\frac{2 (11 b B+3 a C) \sec ^3(c+d x) \sqrt{a+b \sec (c+d x)} \tan (c+d x)}{99 d}+\frac{2 C \sec ^3(c+d x) (a+b \sec (c+d x))^{3/2} \tan (c+d x)}{11 d}\\ \end{align*}

Mathematica [A]  time = 21.6226, size = 1087, normalized size = 1.73 \[ \frac{(a+b \sec (c+d x))^{3/2} \left (C \sec ^2(c+d x)+B \sec (c+d x)+A\right ) \left (\frac{4}{99} (11 b B \sin (c+d x)+12 a C \sin (c+d x)) \sec ^4(c+d x)+\frac{4}{11} b C \tan (c+d x) \sec ^4(c+d x)+\frac{4 \left (3 C \sin (c+d x) a^2+110 b B \sin (c+d x) a+99 A b^2 \sin (c+d x)+81 b^2 C \sin (c+d x)\right ) \sec ^3(c+d x)}{693 b}+\frac{4 \left (-18 C \sin (c+d x) a^3+33 b B \sin (c+d x) a^2+792 A b^2 \sin (c+d x) a+606 b^2 C \sin (c+d x) a+539 b^3 B \sin (c+d x)\right ) \sec ^2(c+d x)}{3465 b^2}+\frac{4 \left (24 C \sin (c+d x) a^4-44 b B \sin (c+d x) a^3+99 A b^2 \sin (c+d x) a^2+57 b^2 C \sin (c+d x) a^2+968 b^3 B \sin (c+d x) a+825 A b^4 \sin (c+d x)+675 b^4 C \sin (c+d x)\right ) \sec (c+d x)}{3465 b^3}-\frac{4 \left (48 C a^5-88 b B a^4+198 A b^2 a^3+108 b^2 C a^3-363 b^3 B a^2-2706 A b^4 a-2088 b^4 C a-1617 b^5 B\right ) \sin (c+d x)}{3465 b^4}\right ) \cos ^3(c+d x)}{d (b+a \cos (c+d x)) (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x))}+\frac{4 (a+b \sec (c+d x))^{3/2} \left (C \sec ^2(c+d x)+B \sec (c+d x)+A\right ) \sqrt{\frac{1}{1-\tan ^2\left (\frac{1}{2} (c+d x)\right )}} \left ((a+b) \left (48 C a^5-88 b B a^4+18 b^2 (11 A+6 C) a^3-363 b^3 B a^2-6 b^4 (451 A+348 C) a-1617 b^5 B\right ) E\left (\sin ^{-1}\left (\tan \left (\frac{1}{2} (c+d x)\right )\right )|\frac{a-b}{a+b}\right ) \sqrt{1-\tan ^2\left (\frac{1}{2} (c+d x)\right )} \sqrt{\frac{-a \tan ^2\left (\frac{1}{2} (c+d x)\right )+b \tan ^2\left (\frac{1}{2} (c+d x)\right )+a+b}{a+b}} \left (\tan ^2\left (\frac{1}{2} (c+d x)\right )+1\right )+b (a+b) \left (-48 C a^4+4 b (22 B+9 C) a^3-6 b^2 (33 A+11 B+24 C) a^2+3 b^3 (627 A+143 B+471 C) a+3 b^4 (275 A+539 B+225 C)\right ) \text{EllipticF}\left (\sin ^{-1}\left (\tan \left (\frac{1}{2} (c+d x)\right )\right ),\frac{a-b}{a+b}\right ) \sqrt{1-\tan ^2\left (\frac{1}{2} (c+d x)\right )} \sqrt{\frac{-a \tan ^2\left (\frac{1}{2} (c+d x)\right )+b \tan ^2\left (\frac{1}{2} (c+d x)\right )+a+b}{a+b}} \left (\tan ^2\left (\frac{1}{2} (c+d x)\right )+1\right )+\left (48 C a^5-88 b B a^4+18 b^2 (11 A+6 C) a^3-363 b^3 B a^2-6 b^4 (451 A+348 C) a-1617 b^5 B\right ) \tan \left (\frac{1}{2} (c+d x)\right ) \left (-b \tan ^4\left (\frac{1}{2} (c+d x)\right )+a \left (\tan ^2\left (\frac{1}{2} (c+d x)\right )-1\right )^2+b\right )\right )}{3465 b^4 d (b+a \cos (c+d x))^{3/2} (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) \sec ^{\frac{7}{2}}(c+d x) \left (\tan ^2\left (\frac{1}{2} (c+d x)\right )+1\right )^{3/2} \sqrt{\frac{-a \tan ^2\left (\frac{1}{2} (c+d x)\right )+b \tan ^2\left (\frac{1}{2} (c+d x)\right )+a+b}{\tan ^2\left (\frac{1}{2} (c+d x)\right )+1}}} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(4*(a + b*Sec[c + d*x])^(3/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sqrt[(1 - Tan[(c + d*x)/2]^2)^(-1)]*((a
+ b)*(-88*a^4*b*B - 363*a^2*b^3*B - 1617*b^5*B + 48*a^5*C + 18*a^3*b^2*(11*A + 6*C) - 6*a*b^4*(451*A + 348*C))
*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*(1 + Tan[(c + d*x)/2]^2)*Sq
rt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] + b*(a + b)*(-48*a^4*C + 4*a^3*b*(22*B + 9*C
) - 6*a^2*b^2*(33*A + 11*B + 24*C) + 3*b^4*(275*A + 539*B + 225*C) + 3*a*b^3*(627*A + 143*B + 471*C))*Elliptic
F[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*(1 + Tan[(c + d*x)/2]^2)*Sqrt[(a + b
- a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] + (-88*a^4*b*B - 363*a^2*b^3*B - 1617*b^5*B + 48*a^5*
C + 18*a^3*b^2*(11*A + 6*C) - 6*a*b^4*(451*A + 348*C))*Tan[(c + d*x)/2]*(b - b*Tan[(c + d*x)/2]^4 + a*(-1 + Ta
n[(c + d*x)/2]^2)^2)))/(3465*b^4*d*(b + a*Cos[c + d*x])^(3/2)*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x]
)*Sec[c + d*x]^(7/2)*(1 + Tan[(c + d*x)/2]^2)^(3/2)*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)
/(1 + Tan[(c + d*x)/2]^2)]) + (Cos[c + d*x]^3*(a + b*Sec[c + d*x])^(3/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^
2)*((-4*(198*a^3*A*b^2 - 2706*a*A*b^4 - 88*a^4*b*B - 363*a^2*b^3*B - 1617*b^5*B + 48*a^5*C + 108*a^3*b^2*C - 2
088*a*b^4*C)*Sin[c + d*x])/(3465*b^4) + (4*Sec[c + d*x]^4*(11*b*B*Sin[c + d*x] + 12*a*C*Sin[c + d*x]))/99 + (4
*Sec[c + d*x]^3*(99*A*b^2*Sin[c + d*x] + 110*a*b*B*Sin[c + d*x] + 3*a^2*C*Sin[c + d*x] + 81*b^2*C*Sin[c + d*x]
))/(693*b) + (4*Sec[c + d*x]^2*(792*a*A*b^2*Sin[c + d*x] + 33*a^2*b*B*Sin[c + d*x] + 539*b^3*B*Sin[c + d*x] -
18*a^3*C*Sin[c + d*x] + 606*a*b^2*C*Sin[c + d*x]))/(3465*b^2) + (4*Sec[c + d*x]*(99*a^2*A*b^2*Sin[c + d*x] + 8
25*A*b^4*Sin[c + d*x] - 44*a^3*b*B*Sin[c + d*x] + 968*a*b^3*B*Sin[c + d*x] + 24*a^4*C*Sin[c + d*x] + 57*a^2*b^
2*C*Sin[c + d*x] + 675*b^4*C*Sin[c + d*x]))/(3465*b^3) + (4*b*C*Sec[c + d*x]^4*Tan[c + d*x])/11))/(d*(b + a*Co
s[c + d*x])*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x]))

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Maple [B]  time = 3.062, size = 7208, normalized size = 11.5 \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))^(3/2)*(A+B*sec(d*x+c)+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))^(3/2)*(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 \sec \left (d x + c\right )^{6} +{\left (C a + B b\right )} \sec \left (d x + c\right )^{5} + A a \sec \left (d x + c\right )^{3} +{\left (B a + A b\right )} \sec \left (d x + c\right )^{4}\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))^(3/2)*(A+B*sec(d*x+c)+C*sec(d*x+c)^2),x, algorithm="fricas")

[Out]

integral((C*b*sec(d*x + c)^6 + (C*a + B*b)*sec(d*x + c)^5 + A*a*sec(d*x + c)^3 + (B*a + A*b)*sec(d*x + c)^4)*s
qrt(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))**(3/2)*(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 )}^{\frac{3}{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))^(3/2)*(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)^(3/2)*sec(d*x + c)^3, x)