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

[Out]

-2/3465*(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(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-2/3465*(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(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+2/11*
C*sec(d*x+c)^3*(a+b*sec(d*x+c))^(3/2)*tan(d*x+c)/d-2/3465*(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))*(a+b*sec(d*x+c))^(1/2)*tan(d*x+c)/b^3/d+2/3465*(33*a^2*b*B+539*b^3*B-18*a^3*C+6*a*b^2*(1
32*A+101*C))*sec(d*x+c)*(a+b*sec(d*x+c))^(1/2)*tan(d*x+c)/b^2/d+2/693*(99*A*b^2+110*B*a*b+3*C*a^2+81*C*b^2)*se
c(d*x+c)^2*(a+b*sec(d*x+c))^(1/2)*tan(d*x+c)/b/d+2/99*(11*B*b+3*C*a)*sec(d*x+c)^3*(a+b*sec(d*x+c))^(1/2)*tan(d
*x+c)/d

________________________________________________________________________________________

Rubi [A]  time = 2.63, antiderivative size = 628, normalized size of antiderivative = 1.00, 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 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]

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 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 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 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]

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*}

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Mathematica [A]  time = 21.29, 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 ) F\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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fricas [F]  time = 1.91, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (C b \sec \left (d x + c\right )^{6} + {\left (C a + B b\right )} \sec \left (d x + c\right )^{5} + A a \sec \left (d x + c\right )^{3} + {\left (B a + A b\right )} \sec \left (d x + c\right )^{4}\right )} \sqrt {b \sec \left (d x + c\right ) + a}, x\right ) \]

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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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (C \sec \left (d x + c\right )^{2} + B \sec \left (d x + c\right ) + A\right )} {\left (b \sec \left (d x + c\right ) + a\right )}^{\frac {3}{2}} \sec \left (d x + c\right )^{3}\,{d x} \]

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)

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maple [B]  time = 4.64, size = 7208, normalized size = 11.48 \[ \text {output too large to display} \]

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 = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

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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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\left (a+\frac {b}{\cos \left (c+d\,x\right )}\right )}^{3/2}\,\left (A+\frac {B}{\cos \left (c+d\,x\right )}+\frac {C}{{\cos \left (c+d\,x\right )}^2}\right )}{{\cos \left (c+d\,x\right )}^3} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (a + b \sec {\left (c + d x \right )}\right )^{\frac {3}{2}} \left (A + B \sec {\left (c + d x \right )} + C \sec ^{2}{\left (c + d x \right )}\right ) \sec ^{3}{\left (c + d x \right )}\, dx \]

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]

Integral((a + b*sec(c + d*x))**(3/2)*(A + B*sec(c + d*x) + C*sec(c + d*x)**2)*sec(c + d*x)**3, x)

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