∫ cos2 (c+dx)(A+B sec(c+dx)+C sec2(c+dx))______________________________

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

Optimal. Leaf size=648 \[ \frac {\sin (c+d x) \cos (c+d x) \left (A b^2-a (b B-a C)\right )}{3 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^3}+\frac {x \left (a^2 (A+2 C)-8 a b B+20 A b^2\right )}{2 a^6}-\frac {\sin (c+d x) \cos (c+d x) \left (-4 a^4 C+7 a^3 b B-a^2 b^2 (10 A+C)-2 a b^3 B+5 A b^4\right )}{6 a^2 d \left (a^2-b^2\right )^2 (a+b \sec (c+d x))^2}-\frac {\sin (c+d x) \cos (c+d x) \left (-\left (a^6 (A-6 C)\right )-12 a^5 b B+a^4 b^2 (23 A-2 C)+11 a^3 b^3 B-a^2 b^4 (27 A-C)-4 a b^5 B+10 A b^6\right )}{2 a^4 d \left (a^2-b^2\right )^3}+\frac {\sin (c+d x) \cos (c+d x) \left (12 a^6 C-27 a^5 b B+a^4 b^2 (48 A+C)+20 a^3 b^3 B-a^2 b^4 (53 A-2 C)-8 a b^5 B+20 A b^6\right )}{6 a^3 d \left (a^2-b^2\right )^3 (a+b \sec (c+d x))}+\frac {\sin (c+d x) \left (6 a^7 B-a^6 (24 A b-26 b C)-65 a^5 b^2 B+a^4 b^3 (146 A-17 C)+68 a^3 b^4 B-a^2 b^5 (167 A-6 C)-24 a b^6 B+60 A b^7\right )}{6 a^5 d \left (a^2-b^2\right )^3}+\frac {b \left (-8 a^8 C+20 a^7 b B-8 a^6 b^2 (5 A-C)-35 a^5 b^3 B+7 a^4 b^4 (12 A-C)+28 a^3 b^5 B-a^2 b^6 (69 A-2 C)-8 a b^7 B+20 A b^8\right ) \tanh ^{-1}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{a^6 d \sqrt {a-b} \sqrt {a+b} \left (a^2-b^2\right )^3} \]

[Out]

1/2*(20*A*b^2-8*a*b*B+a^2*(A+2*C))*x/a^6+1/6*(60*A*b^7+6*a^7*B-65*a^5*b^2*B+68*a^3*b^4*B-24*a*b^6*B+a^4*b^3*(1

46*A-17*C)-a^2*b^5*(167*A-6*C)-a^6*(24*A*b-26*C*b))*sin(d*x+c)/a^5/(a^2-b^2)^3/d-1/2*(10*A*b^6-12*a^5*b*B+11*a

^3*b^3*B-4*a*b^5*B-a^6*(A-6*C)+a^4*b^2*(23*A-2*C)-a^2*b^4*(27*A-C))*cos(d*x+c)*sin(d*x+c)/a^4/(a^2-b^2)^3/d+1/

3*(A*b^2-a*(B*b-C*a))*cos(d*x+c)*sin(d*x+c)/a/(a^2-b^2)/d/(a+b*sec(d*x+c))^3-1/6*(5*A*b^4+7*a^3*b*B-2*a*b^3*B-

4*a^4*C-a^2*b^2*(10*A+C))*cos(d*x+c)*sin(d*x+c)/a^2/(a^2-b^2)^2/d/(a+b*sec(d*x+c))^2+1/6*(20*A*b^6-27*a^5*b*B+

20*a^3*b^3*B-8*a*b^5*B-a^2*b^4*(53*A-2*C)+12*a^6*C+a^4*b^2*(48*A+C))*cos(d*x+c)*sin(d*x+c)/a^3/(a^2-b^2)^3/d/(

a+b*sec(d*x+c))+b*(20*A*b^8+20*a^7*b*B-35*a^5*b^3*B+28*a^3*b^5*B-8*a*b^7*B-a^2*b^6*(69*A-2*C)-8*a^6*b^2*(5*A-C

)+7*a^4*b^4*(12*A-C)-8*a^8*C)*arctanh((a-b)^(1/2)*tan(1/2*d*x+1/2*c)/(a+b)^(1/2))/a^6/(a^2-b^2)^3/d/(a-b)^(1/2

)/(a+b)^(1/2)

________________________________________________________________________________________

Rubi [A] time = 12.51, antiderivative size = 648, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 6, integrand size = 41, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.146, Rules used = {4100, 4104, 3919, 3831, 2659, 208} \[ \frac {\sin (c+d x) \left (a^4 b^3 (146 A-17 C)-a^2 b^5 (167 A-6 C)-a^6 (24 A b-26 b C)-65 a^5 b^2 B+68 a^3 b^4 B+6 a^7 B-24 a b^6 B+60 A b^7\right )}{6 a^5 d \left (a^2-b^2\right )^3}-\frac {\sin (c+d x) \cos (c+d x) \left (a^4 b^2 (23 A-2 C)-a^2 b^4 (27 A-C)+a^6 (-(A-6 C))+11 a^3 b^3 B-12 a^5 b B-4 a b^5 B+10 A b^6\right )}{2 a^4 d \left (a^2-b^2\right )^3}+\frac {b \left (-8 a^6 b^2 (5 A-C)+7 a^4 b^4 (12 A-C)-a^2 b^6 (69 A-2 C)-35 a^5 b^3 B+28 a^3 b^5 B+20 a^7 b B-8 a^8 C-8 a b^7 B+20 A b^8\right ) \tanh ^{-1}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{a^6 d \sqrt {a-b} \sqrt {a+b} \left (a^2-b^2\right )^3}+\frac {\sin (c+d x) \cos (c+d x) \left (a^4 b^2 (48 A+C)-a^2 b^4 (53 A-2 C)+20 a^3 b^3 B-27 a^5 b B+12 a^6 C-8 a b^5 B+20 A b^6\right )}{6 a^3 d \left (a^2-b^2\right )^3 (a+b \sec (c+d x))}-\frac {\sin (c+d x) \cos (c+d x) \left (-a^2 b^2 (10 A+C)+7 a^3 b B-4 a^4 C-2 a b^3 B+5 A b^4\right )}{6 a^2 d \left (a^2-b^2\right )^2 (a+b \sec (c+d x))^2}+\frac {\sin (c+d x) \cos (c+d x) \left (A b^2-a (b B-a C)\right )}{3 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^3}+\frac {x \left (a^2 (A+2 C)-8 a b B+20 A b^2\right )}{2 a^6} \]

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

((20*A*b^2 - 8*a*b*B + a^2*(A + 2*C))*x)/(2*a^6) + (b*(20*A*b^8 + 20*a^7*b*B - 35*a^5*b^3*B + 28*a^3*b^5*B - 8

*a*b^7*B - a^2*b^6*(69*A - 2*C) - 8*a^6*b^2*(5*A - C) + 7*a^4*b^4*(12*A - C) - 8*a^8*C)*ArcTanh[(Sqrt[a - b]*T

an[(c + d*x)/2])/Sqrt[a + b]])/(a^6*Sqrt[a - b]*Sqrt[a + b]*(a^2 - b^2)^3*d) + ((60*A*b^7 + 6*a^7*B - 65*a^5*b

^2*B + 68*a^3*b^4*B - 24*a*b^6*B + a^4*b^3*(146*A - 17*C) - a^2*b^5*(167*A - 6*C) - a^6*(24*A*b - 26*b*C))*Sin

[c + d*x])/(6*a^5*(a^2 - b^2)^3*d) - ((10*A*b^6 - 12*a^5*b*B + 11*a^3*b^3*B - 4*a*b^5*B - a^6*(A - 6*C) + a^4*

b^2*(23*A - 2*C) - a^2*b^4*(27*A - C))*Cos[c + d*x]*Sin[c + d*x])/(2*a^4*(a^2 - b^2)^3*d) + ((A*b^2 - a*(b*B -

a*C))*Cos[c + d*x]*Sin[c + d*x])/(3*a*(a^2 - b^2)*d*(a + b*Sec[c + d*x])^3) - ((5*A*b^4 + 7*a^3*b*B - 2*a*b^3

*B - 4*a^4*C - a^2*b^2*(10*A + C))*Cos[c + d*x]*Sin[c + d*x])/(6*a^2*(a^2 - b^2)^2*d*(a + b*Sec[c + d*x])^2) +

((20*A*b^6 - 27*a^5*b*B + 20*a^3*b^3*B - 8*a*b^5*B - a^2*b^4*(53*A - 2*C) + 12*a^6*C + a^4*b^2*(48*A + C))*Co

s[c + d*x]*Sin[c + d*x])/(6*a^3*(a^2 - b^2)^3*d*(a + b*Sec[c + d*x]))

Rule 208

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-(a/b), 2]*ArcTanh[x/Rt[-(a/b), 2]])/a, x] /; FreeQ[{a,

b}, x] && NegQ[a/b]

Rule 2659

Int[((a_) + (b_.)*sin[Pi/2 + (c_.) + (d_.)*(x_)])^(-1), x_Symbol] :> With[{e = FreeFactors[Tan[(c + d*x)/2], x

]}, Dist[(2*e)/d, Subst[Int[1/(a + b + (a - b)*e^2*x^2), x], x, Tan[(c + d*x)/2]/e], x]] /; FreeQ[{a, b, c, d}

, x] && NeQ[a^2 - b^2, 0]

Rule 3831

Int[csc[(e_.) + (f_.)*(x_)]/(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)), x_Symbol] :> Dist[1/b, Int[1/(1 + (a*Sin[e

+ f*x])/b), x], x] /; FreeQ[{a, b, e, f}, x] && NeQ[a^2 - b^2, 0]

Rule 3919

Int[(csc[(e_.) + (f_.)*(x_)]*(d_.) + (c_))/(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)), x_Symbol] :> Simp[(c*x)/a,

x] - Dist[(b*c - a*d)/a, Int[Csc[e + f*x]/(a + b*Csc[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[

b*c - a*d, 0]

Rule 4100

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[((A*b^2 - a*b*B + a^2*C)*Cot[e + f*x]*(a +

b*Csc[e + f*x])^(m + 1)*(d*Csc[e + f*x])^n)/(a*f*(m + 1)*(a^2 - b^2)), x] + Dist[1/(a*(m + 1)*(a^2 - b^2)), I

nt[(a + b*Csc[e + f*x])^(m + 1)*(d*Csc[e + f*x])^n*Simp[a*(a*A - b*B + a*C)*(m + 1) - (A*b^2 - a*b*B + a^2*C)*

(m + n + 1) - a*(A*b - a*B + b*C)*(m + 1)*Csc[e + f*x] + (A*b^2 - a*b*B + a^2*C)*(m + n + 2)*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] && LtQ[m, -1] && !(ILtQ[m + 1/2, 0] &

& ILtQ[n, 0])

Rule 4104

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[(A*Cot[e + f*x]*(a + b*Csc[e + f*x])^(m +

1)*(d*Csc[e + f*x])^n)/(a*f*n), x] + Dist[1/(a*d*n), Int[(a + b*Csc[e + f*x])^m*(d*Csc[e + f*x])^(n + 1)*Simp[

a*B*n - A*b*(m + n + 1) + a*(A + A*n + C*n)*Csc[e + f*x] + A*b*(m + n + 2)*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] && LeQ[n, -1]

Rubi steps

\begin {align*} \int \frac {\cos ^2(c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{(a+b \sec (c+d x))^4} \, dx &=\frac {\left (A b^2-a (b B-a C)\right ) \cos (c+d x) \sin (c+d x)}{3 a \left (a^2-b^2\right ) d (a+b \sec (c+d x))^3}-\frac {\int \frac {\cos ^2(c+d x) \left (5 A b^2-2 a b B-a^2 (3 A-2 C)+3 a (A b-a B+b C) \sec (c+d x)-4 \left (A b^2-a (b B-a C)\right ) \sec ^2(c+d x)\right )}{(a+b \sec (c+d x))^3} \, dx}{3 a \left (a^2-b^2\right )}\\ &=\frac {\left (A b^2-a (b B-a C)\right ) \cos (c+d x) \sin (c+d x)}{3 a \left (a^2-b^2\right ) d (a+b \sec (c+d x))^3}-\frac {\left (5 A b^4+7 a^3 b B-2 a b^3 B-4 a^4 C-a^2 b^2 (10 A+C)\right ) \cos (c+d x) \sin (c+d x)}{6 a^2 \left (a^2-b^2\right )^2 d (a+b \sec (c+d x))^2}+\frac {\int \frac {\cos ^2(c+d x) \left (2 \left (10 A b^4+9 a^3 b B-4 a b^3 B+3 a^4 (A-2 C)-a^2 b^2 (18 A-C)\right )+2 a \left (A b^3+3 a^3 B+2 a b^2 B-a^2 b (6 A+5 C)\right ) \sec (c+d x)-3 \left (5 A b^4+7 a^3 b B-2 a b^3 B-4 a^4 C-a^2 b^2 (10 A+C)\right ) \sec ^2(c+d x)\right )}{(a+b \sec (c+d x))^2} \, dx}{6 a^2 \left (a^2-b^2\right )^2}\\ &=\frac {\left (A b^2-a (b B-a C)\right ) \cos (c+d x) \sin (c+d x)}{3 a \left (a^2-b^2\right ) d (a+b \sec (c+d x))^3}-\frac {\left (5 A b^4+7 a^3 b B-2 a b^3 B-4 a^4 C-a^2 b^2 (10 A+C)\right ) \cos (c+d x) \sin (c+d x)}{6 a^2 \left (a^2-b^2\right )^2 d (a+b \sec (c+d x))^2}+\frac {\left (20 A b^6-27 a^5 b B+20 a^3 b^3 B-8 a b^5 B-a^2 b^4 (53 A-2 C)+12 a^6 C+a^4 b^2 (48 A+C)\right ) \cos (c+d x) \sin (c+d x)}{6 a^3 \left (a^2-b^2\right )^3 d (a+b \sec (c+d x))}-\frac {\int \frac {\cos ^2(c+d x) \left (6 \left (10 A b^6-12 a^5 b B+11 a^3 b^3 B-4 a b^5 B-a^6 (A-6 C)+a^4 b^2 (23 A-2 C)-a^2 b^4 (27 A-C)\right )+a \left (5 A b^5-6 a^5 B-7 a^3 b^2 B-2 a b^4 B-a^2 b^3 (8 A-5 C)+2 a^4 b (9 A+5 C)\right ) \sec (c+d x)-2 \left (20 A b^6-27 a^5 b B+20 a^3 b^3 B-8 a b^5 B-a^2 b^4 (53 A-2 C)+12 a^6 C+a^4 b^2 (48 A+C)\right ) \sec ^2(c+d x)\right )}{a+b \sec (c+d x)} \, dx}{6 a^3 \left (a^2-b^2\right )^3}\\ &=-\frac {\left (10 A b^6-12 a^5 b B+11 a^3 b^3 B-4 a b^5 B-a^6 (A-6 C)+a^4 b^2 (23 A-2 C)-a^2 b^4 (27 A-C)\right ) \cos (c+d x) \sin (c+d x)}{2 a^4 \left (a^2-b^2\right )^3 d}+\frac {\left (A b^2-a (b B-a C)\right ) \cos (c+d x) \sin (c+d x)}{3 a \left (a^2-b^2\right ) d (a+b \sec (c+d x))^3}-\frac {\left (5 A b^4+7 a^3 b B-2 a b^3 B-4 a^4 C-a^2 b^2 (10 A+C)\right ) \cos (c+d x) \sin (c+d x)}{6 a^2 \left (a^2-b^2\right )^2 d (a+b \sec (c+d x))^2}+\frac {\left (20 A b^6-27 a^5 b B+20 a^3 b^3 B-8 a b^5 B-a^2 b^4 (53 A-2 C)+12 a^6 C+a^4 b^2 (48 A+C)\right ) \cos (c+d x) \sin (c+d x)}{6 a^3 \left (a^2-b^2\right )^3 d (a+b \sec (c+d x))}+\frac {\int \frac {\cos (c+d x) \left (2 \left (60 A b^7+6 a^7 B-65 a^5 b^2 B+68 a^3 b^4 B-24 a b^6 B+a^4 b^3 (146 A-17 C)-a^2 b^5 (167 A-6 C)-a^6 (24 A b-26 b C)\right )+2 a \left (10 A b^6-18 a^5 b B+7 a^3 b^3 B-4 a b^5 B-a^2 b^4 (25 A-C)+3 a^6 (A+2 C)+a^4 b^2 (27 A+8 C)\right ) \sec (c+d x)-6 b \left (10 A b^6-12 a^5 b B+11 a^3 b^3 B-4 a b^5 B-a^6 (A-6 C)+a^4 b^2 (23 A-2 C)-a^2 b^4 (27 A-C)\right ) \sec ^2(c+d x)\right )}{a+b \sec (c+d x)} \, dx}{12 a^4 \left (a^2-b^2\right )^3}\\ &=\frac {\left (60 A b^7+6 a^7 B-65 a^5 b^2 B+68 a^3 b^4 B-24 a b^6 B+a^4 b^3 (146 A-17 C)-a^2 b^5 (167 A-6 C)-a^6 (24 A b-26 b C)\right ) \sin (c+d x)}{6 a^5 \left (a^2-b^2\right )^3 d}-\frac {\left (10 A b^6-12 a^5 b B+11 a^3 b^3 B-4 a b^5 B-a^6 (A-6 C)+a^4 b^2 (23 A-2 C)-a^2 b^4 (27 A-C)\right ) \cos (c+d x) \sin (c+d x)}{2 a^4 \left (a^2-b^2\right )^3 d}+\frac {\left (A b^2-a (b B-a C)\right ) \cos (c+d x) \sin (c+d x)}{3 a \left (a^2-b^2\right ) d (a+b \sec (c+d x))^3}-\frac {\left (5 A b^4+7 a^3 b B-2 a b^3 B-4 a^4 C-a^2 b^2 (10 A+C)\right ) \cos (c+d x) \sin (c+d x)}{6 a^2 \left (a^2-b^2\right )^2 d (a+b \sec (c+d x))^2}+\frac {\left (20 A b^6-27 a^5 b B+20 a^3 b^3 B-8 a b^5 B-a^2 b^4 (53 A-2 C)+12 a^6 C+a^4 b^2 (48 A+C)\right ) \cos (c+d x) \sin (c+d x)}{6 a^3 \left (a^2-b^2\right )^3 d (a+b \sec (c+d x))}-\frac {\int \frac {-6 \left (a^2-b^2\right )^3 \left (20 A b^2-8 a b B+a^2 (A+2 C)\right )+6 a b \left (10 A b^6-12 a^5 b B+11 a^3 b^3 B-4 a b^5 B-a^6 (A-6 C)+a^4 b^2 (23 A-2 C)-a^2 b^4 (27 A-C)\right ) \sec (c+d x)}{a+b \sec (c+d x)} \, dx}{12 a^5 \left (a^2-b^2\right )^3}\\ &=\frac {\left (20 A b^2-8 a b B+a^2 (A+2 C)\right ) x}{2 a^6}+\frac {\left (60 A b^7+6 a^7 B-65 a^5 b^2 B+68 a^3 b^4 B-24 a b^6 B+a^4 b^3 (146 A-17 C)-a^2 b^5 (167 A-6 C)-a^6 (24 A b-26 b C)\right ) \sin (c+d x)}{6 a^5 \left (a^2-b^2\right )^3 d}-\frac {\left (10 A b^6-12 a^5 b B+11 a^3 b^3 B-4 a b^5 B-a^6 (A-6 C)+a^4 b^2 (23 A-2 C)-a^2 b^4 (27 A-C)\right ) \cos (c+d x) \sin (c+d x)}{2 a^4 \left (a^2-b^2\right )^3 d}+\frac {\left (A b^2-a (b B-a C)\right ) \cos (c+d x) \sin (c+d x)}{3 a \left (a^2-b^2\right ) d (a+b \sec (c+d x))^3}-\frac {\left (5 A b^4+7 a^3 b B-2 a b^3 B-4 a^4 C-a^2 b^2 (10 A+C)\right ) \cos (c+d x) \sin (c+d x)}{6 a^2 \left (a^2-b^2\right )^2 d (a+b \sec (c+d x))^2}+\frac {\left (20 A b^6-27 a^5 b B+20 a^3 b^3 B-8 a b^5 B-a^2 b^4 (53 A-2 C)+12 a^6 C+a^4 b^2 (48 A+C)\right ) \cos (c+d x) \sin (c+d x)}{6 a^3 \left (a^2-b^2\right )^3 d (a+b \sec (c+d x))}+\frac {\left (b \left (20 A b^8+20 a^7 b B-35 a^5 b^3 B+28 a^3 b^5 B-8 a b^7 B-a^2 b^6 (69 A-2 C)-8 a^6 b^2 (5 A-C)+7 a^4 b^4 (12 A-C)-8 a^8 C\right )\right ) \int \frac {\sec (c+d x)}{a+b \sec (c+d x)} \, dx}{2 a^6 \left (a^2-b^2\right )^3}\\ &=\frac {\left (20 A b^2-8 a b B+a^2 (A+2 C)\right ) x}{2 a^6}+\frac {\left (60 A b^7+6 a^7 B-65 a^5 b^2 B+68 a^3 b^4 B-24 a b^6 B+a^4 b^3 (146 A-17 C)-a^2 b^5 (167 A-6 C)-a^6 (24 A b-26 b C)\right ) \sin (c+d x)}{6 a^5 \left (a^2-b^2\right )^3 d}-\frac {\left (10 A b^6-12 a^5 b B+11 a^3 b^3 B-4 a b^5 B-a^6 (A-6 C)+a^4 b^2 (23 A-2 C)-a^2 b^4 (27 A-C)\right ) \cos (c+d x) \sin (c+d x)}{2 a^4 \left (a^2-b^2\right )^3 d}+\frac {\left (A b^2-a (b B-a C)\right ) \cos (c+d x) \sin (c+d x)}{3 a \left (a^2-b^2\right ) d (a+b \sec (c+d x))^3}-\frac {\left (5 A b^4+7 a^3 b B-2 a b^3 B-4 a^4 C-a^2 b^2 (10 A+C)\right ) \cos (c+d x) \sin (c+d x)}{6 a^2 \left (a^2-b^2\right )^2 d (a+b \sec (c+d x))^2}+\frac {\left (20 A b^6-27 a^5 b B+20 a^3 b^3 B-8 a b^5 B-a^2 b^4 (53 A-2 C)+12 a^6 C+a^4 b^2 (48 A+C)\right ) \cos (c+d x) \sin (c+d x)}{6 a^3 \left (a^2-b^2\right )^3 d (a+b \sec (c+d x))}+\frac {\left (20 A b^8+20 a^7 b B-35 a^5 b^3 B+28 a^3 b^5 B-8 a b^7 B-a^2 b^6 (69 A-2 C)-8 a^6 b^2 (5 A-C)+7 a^4 b^4 (12 A-C)-8 a^8 C\right ) \int \frac {1}{1+\frac {a \cos (c+d x)}{b}} \, dx}{2 a^6 \left (a^2-b^2\right )^3}\\ &=\frac {\left (20 A b^2-8 a b B+a^2 (A+2 C)\right ) x}{2 a^6}+\frac {\left (60 A b^7+6 a^7 B-65 a^5 b^2 B+68 a^3 b^4 B-24 a b^6 B+a^4 b^3 (146 A-17 C)-a^2 b^5 (167 A-6 C)-a^6 (24 A b-26 b C)\right ) \sin (c+d x)}{6 a^5 \left (a^2-b^2\right )^3 d}-\frac {\left (10 A b^6-12 a^5 b B+11 a^3 b^3 B-4 a b^5 B-a^6 (A-6 C)+a^4 b^2 (23 A-2 C)-a^2 b^4 (27 A-C)\right ) \cos (c+d x) \sin (c+d x)}{2 a^4 \left (a^2-b^2\right )^3 d}+\frac {\left (A b^2-a (b B-a C)\right ) \cos (c+d x) \sin (c+d x)}{3 a \left (a^2-b^2\right ) d (a+b \sec (c+d x))^3}-\frac {\left (5 A b^4+7 a^3 b B-2 a b^3 B-4 a^4 C-a^2 b^2 (10 A+C)\right ) \cos (c+d x) \sin (c+d x)}{6 a^2 \left (a^2-b^2\right )^2 d (a+b \sec (c+d x))^2}+\frac {\left (20 A b^6-27 a^5 b B+20 a^3 b^3 B-8 a b^5 B-a^2 b^4 (53 A-2 C)+12 a^6 C+a^4 b^2 (48 A+C)\right ) \cos (c+d x) \sin (c+d x)}{6 a^3 \left (a^2-b^2\right )^3 d (a+b \sec (c+d x))}+\frac {\left (20 A b^8+20 a^7 b B-35 a^5 b^3 B+28 a^3 b^5 B-8 a b^7 B-a^2 b^6 (69 A-2 C)-8 a^6 b^2 (5 A-C)+7 a^4 b^4 (12 A-C)-8 a^8 C\right ) \operatorname {Subst}\left (\int \frac {1}{1+\frac {a}{b}+\left (1-\frac {a}{b}\right ) x^2} \, dx,x,\tan \left (\frac {1}{2} (c+d x)\right )\right )}{a^6 \left (a^2-b^2\right )^3 d}\\ &=\frac {\left (20 A b^2-8 a b B+a^2 (A+2 C)\right ) x}{2 a^6}+\frac {b \left (20 A b^8+20 a^7 b B-35 a^5 b^3 B+28 a^3 b^5 B-8 a b^7 B-a^2 b^6 (69 A-2 C)-8 a^6 b^2 (5 A-C)+7 a^4 b^4 (12 A-C)-8 a^8 C\right ) \tanh ^{-1}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{a^6 \sqrt {a-b} \sqrt {a+b} \left (a^2-b^2\right )^3 d}+\frac {\left (60 A b^7+6 a^7 B-65 a^5 b^2 B+68 a^3 b^4 B-24 a b^6 B+a^4 b^3 (146 A-17 C)-a^2 b^5 (167 A-6 C)-a^6 (24 A b-26 b C)\right ) \sin (c+d x)}{6 a^5 \left (a^2-b^2\right )^3 d}-\frac {\left (10 A b^6-12 a^5 b B+11 a^3 b^3 B-4 a b^5 B-a^6 (A-6 C)+a^4 b^2 (23 A-2 C)-a^2 b^4 (27 A-C)\right ) \cos (c+d x) \sin (c+d x)}{2 a^4 \left (a^2-b^2\right )^3 d}+\frac {\left (A b^2-a (b B-a C)\right ) \cos (c+d x) \sin (c+d x)}{3 a \left (a^2-b^2\right ) d (a+b \sec (c+d x))^3}-\frac {\left (5 A b^4+7 a^3 b B-2 a b^3 B-4 a^4 C-a^2 b^2 (10 A+C)\right ) \cos (c+d x) \sin (c+d x)}{6 a^2 \left (a^2-b^2\right )^2 d (a+b \sec (c+d x))^2}+\frac {\left (20 A b^6-27 a^5 b B+20 a^3 b^3 B-8 a b^5 B-a^2 b^4 (53 A-2 C)+12 a^6 C+a^4 b^2 (48 A+C)\right ) \cos (c+d x) \sin (c+d x)}{6 a^3 \left (a^2-b^2\right )^3 d (a+b \sec (c+d x))}\\ \end {align*}

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Mathematica [C] time = 7.01, size = 658, normalized size = 1.02 \[ \frac {(4 A b-a B) \left (-\frac {\sin (c+d x)}{2 a^5}-\frac {i \cos (c+d x)}{2 a^5}\right )}{d}+\frac {(4 A b-a B) \left (-\frac {\sin (c+d x)}{2 a^5}+\frac {i \cos (c+d x)}{2 a^5}\right )}{d}+\frac {A \sin (2 (c+d x))}{4 a^4 d}+\frac {(c+d x) \left (a^2 A+2 a^2 C-8 a b B+20 A b^2\right )}{2 a^6 d}+\frac {a^2 b^4 C \sin (c+d x)-a b^5 B \sin (c+d x)+A b^6 \sin (c+d x)}{3 a^5 d \left (a^2-b^2\right ) (a \cos (c+d x)+b)^3}+\frac {-12 a^4 b^3 C \sin (c+d x)+15 a^3 b^4 B \sin (c+d x)-18 a^2 A b^5 \sin (c+d x)+7 a^2 b^5 C \sin (c+d x)-10 a b^6 B \sin (c+d x)+13 A b^7 \sin (c+d x)}{6 a^5 d \left (a^2-b^2\right )^2 (a \cos (c+d x)+b)^2}+\frac {36 a^6 b^2 C \sin (c+d x)-60 a^5 b^3 B \sin (c+d x)+90 a^4 A b^4 \sin (c+d x)-32 a^4 b^4 C \sin (c+d x)+71 a^3 b^5 B \sin (c+d x)-122 a^2 A b^6 \sin (c+d x)+11 a^2 b^6 C \sin (c+d x)-26 a b^7 B \sin (c+d x)+47 A b^8 \sin (c+d x)}{6 a^5 d \left (a^2-b^2\right )^3 (a \cos (c+d x)+b)}+\frac {b \left (-8 a^8 C+20 a^7 b B-40 a^6 A b^2+8 a^6 b^2 C-35 a^5 b^3 B+84 a^4 A b^4-7 a^4 b^4 C+28 a^3 b^5 B-69 a^2 A b^6+2 a^2 b^6 C-8 a b^7 B+20 A b^8\right ) \tanh ^{-1}\left (\frac {(b-a) \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )}{a^6 d \sqrt {a^2-b^2} \left (b^2-a^2\right )^3} \]

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

((a^2*A + 20*A*b^2 - 8*a*b*B + 2*a^2*C)*(c + d*x))/(2*a^6*d) + (b*(-40*a^6*A*b^2 + 84*a^4*A*b^4 - 69*a^2*A*b^6

+ 20*A*b^8 + 20*a^7*b*B - 35*a^5*b^3*B + 28*a^3*b^5*B - 8*a*b^7*B - 8*a^8*C + 8*a^6*b^2*C - 7*a^4*b^4*C + 2*a

^2*b^6*C)*ArcTanh[((-a + b)*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(a^6*Sqrt[a^2 - b^2]*(-a^2 + b^2)^3*d) + ((4*A

*b - a*B)*(((-1/2*I)*Cos[c + d*x])/a^5 - Sin[c + d*x]/(2*a^5)))/d + ((4*A*b - a*B)*(((I/2)*Cos[c + d*x])/a^5 -

Sin[c + d*x]/(2*a^5)))/d + (A*b^6*Sin[c + d*x] - a*b^5*B*Sin[c + d*x] + a^2*b^4*C*Sin[c + d*x])/(3*a^5*(a^2 -

b^2)*d*(b + a*Cos[c + d*x])^3) + (-18*a^2*A*b^5*Sin[c + d*x] + 13*A*b^7*Sin[c + d*x] + 15*a^3*b^4*B*Sin[c + d

*x] - 10*a*b^6*B*Sin[c + d*x] - 12*a^4*b^3*C*Sin[c + d*x] + 7*a^2*b^5*C*Sin[c + d*x])/(6*a^5*(a^2 - b^2)^2*d*(

b + a*Cos[c + d*x])^2) + (90*a^4*A*b^4*Sin[c + d*x] - 122*a^2*A*b^6*Sin[c + d*x] + 47*A*b^8*Sin[c + d*x] - 60*

a^5*b^3*B*Sin[c + d*x] + 71*a^3*b^5*B*Sin[c + d*x] - 26*a*b^7*B*Sin[c + d*x] + 36*a^6*b^2*C*Sin[c + d*x] - 32*

a^4*b^4*C*Sin[c + d*x] + 11*a^2*b^6*C*Sin[c + d*x])/(6*a^5*(a^2 - b^2)^3*d*(b + a*Cos[c + d*x])) + (A*Sin[2*(c

+ d*x)])/(4*a^4*d)

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fricas [B] time = 1.07, size = 3454, normalized size = 5.33 \[ \text {result too large to display} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^2*(A+B*sec(d*x+c)+C*sec(d*x+c)^2)/(a+b*sec(d*x+c))^4,x, algorithm="fricas")

[Out]

[1/12*(6*((A + 2*C)*a^13 - 8*B*a^12*b + 8*(2*A - C)*a^11*b^2 + 32*B*a^10*b^3 - 2*(37*A - 6*C)*a^9*b^4 - 48*B*a

^8*b^5 + 4*(29*A - 2*C)*a^7*b^6 + 32*B*a^6*b^7 - (79*A - 2*C)*a^5*b^8 - 8*B*a^4*b^9 + 20*A*a^3*b^10)*d*x*cos(d

*x + c)^3 + 18*((A + 2*C)*a^12*b - 8*B*a^11*b^2 + 8*(2*A - C)*a^10*b^3 + 32*B*a^9*b^4 - 2*(37*A - 6*C)*a^8*b^5

- 48*B*a^7*b^6 + 4*(29*A - 2*C)*a^6*b^7 + 32*B*a^5*b^8 - (79*A - 2*C)*a^4*b^9 - 8*B*a^3*b^10 + 20*A*a^2*b^11)

*d*x*cos(d*x + c)^2 + 18*((A + 2*C)*a^11*b^2 - 8*B*a^10*b^3 + 8*(2*A - C)*a^9*b^4 + 32*B*a^8*b^5 - 2*(37*A - 6

*C)*a^7*b^6 - 48*B*a^6*b^7 + 4*(29*A - 2*C)*a^5*b^8 + 32*B*a^4*b^9 - (79*A - 2*C)*a^3*b^10 - 8*B*a^2*b^11 + 20

*A*a*b^12)*d*x*cos(d*x + c) + 6*((A + 2*C)*a^10*b^3 - 8*B*a^9*b^4 + 8*(2*A - C)*a^8*b^5 + 32*B*a^7*b^6 - 2*(37

*A - 6*C)*a^6*b^7 - 48*B*a^5*b^8 + 4*(29*A - 2*C)*a^4*b^9 + 32*B*a^3*b^10 - (79*A - 2*C)*a^2*b^11 - 8*B*a*b^12

+ 20*A*b^13)*d*x + 3*(8*C*a^8*b^4 - 20*B*a^7*b^5 + 8*(5*A - C)*a^6*b^6 + 35*B*a^5*b^7 - 7*(12*A - C)*a^4*b^8

- 28*B*a^3*b^9 + (69*A - 2*C)*a^2*b^10 + 8*B*a*b^11 - 20*A*b^12 + (8*C*a^11*b - 20*B*a^10*b^2 + 8*(5*A - C)*a^

9*b^3 + 35*B*a^8*b^4 - 7*(12*A - C)*a^7*b^5 - 28*B*a^6*b^6 + (69*A - 2*C)*a^5*b^7 + 8*B*a^4*b^8 - 20*A*a^3*b^9

)*cos(d*x + c)^3 + 3*(8*C*a^10*b^2 - 20*B*a^9*b^3 + 8*(5*A - C)*a^8*b^4 + 35*B*a^7*b^5 - 7*(12*A - C)*a^6*b^6

- 28*B*a^5*b^7 + (69*A - 2*C)*a^4*b^8 + 8*B*a^3*b^9 - 20*A*a^2*b^10)*cos(d*x + c)^2 + 3*(8*C*a^9*b^3 - 20*B*a^

8*b^4 + 8*(5*A - C)*a^7*b^5 + 35*B*a^6*b^6 - 7*(12*A - C)*a^5*b^7 - 28*B*a^4*b^8 + (69*A - 2*C)*a^3*b^9 + 8*B*

a^2*b^10 - 20*A*a*b^11)*cos(d*x + c))*sqrt(a^2 - b^2)*log((2*a*b*cos(d*x + c) - (a^2 - 2*b^2)*cos(d*x + c)^2 -

2*sqrt(a^2 - b^2)*(b*cos(d*x + c) + a)*sin(d*x + c) + 2*a^2 - b^2)/(a^2*cos(d*x + c)^2 + 2*a*b*cos(d*x + c) +

b^2)) + 2*(6*B*a^10*b^3 - 2*(12*A - 13*C)*a^9*b^4 - 71*B*a^8*b^5 + (170*A - 43*C)*a^7*b^6 + 133*B*a^6*b^7 - (

313*A - 23*C)*a^5*b^8 - 92*B*a^4*b^9 + (227*A - 6*C)*a^3*b^10 + 24*B*a^2*b^11 - 60*A*a*b^12 + 3*(A*a^13 - 4*A*

a^11*b^2 + 6*A*a^9*b^4 - 4*A*a^7*b^6 + A*a^5*b^8)*cos(d*x + c)^4 + 3*(2*B*a^13 - 5*A*a^12*b - 8*B*a^11*b^2 + 2

0*A*a^10*b^3 + 12*B*a^9*b^4 - 30*A*a^8*b^5 - 8*B*a^7*b^6 + 20*A*a^6*b^7 + 2*B*a^5*b^8 - 5*A*a^4*b^9)*cos(d*x +

c)^3 + (18*B*a^12*b - 9*(7*A - 4*C)*a^11*b^2 - 132*B*a^10*b^3 + 2*(171*A - 34*C)*a^9*b^4 + 239*B*a^8*b^5 - (5

90*A - 43*C)*a^7*b^6 - 169*B*a^6*b^7 + (421*A - 11*C)*a^5*b^8 + 44*B*a^4*b^9 - 110*A*a^3*b^10)*cos(d*x + c)^2

+ 3*(6*B*a^11*b^2 - (23*A - 20*C)*a^10*b^3 - 59*B*a^9*b^4 + (146*A - 35*C)*a^8*b^5 + 110*B*a^7*b^6 - (263*A -

20*C)*a^6*b^7 - 77*B*a^5*b^8 + 5*(38*A - C)*a^4*b^9 + 20*B*a^3*b^10 - 50*A*a^2*b^11)*cos(d*x + c))*sin(d*x + c

))/((a^17 - 4*a^15*b^2 + 6*a^13*b^4 - 4*a^11*b^6 + a^9*b^8)*d*cos(d*x + c)^3 + 3*(a^16*b - 4*a^14*b^3 + 6*a^12

*b^5 - 4*a^10*b^7 + a^8*b^9)*d*cos(d*x + c)^2 + 3*(a^15*b^2 - 4*a^13*b^4 + 6*a^11*b^6 - 4*a^9*b^8 + a^7*b^10)*

d*cos(d*x + c) + (a^14*b^3 - 4*a^12*b^5 + 6*a^10*b^7 - 4*a^8*b^9 + a^6*b^11)*d), 1/6*(3*((A + 2*C)*a^13 - 8*B*

a^12*b + 8*(2*A - C)*a^11*b^2 + 32*B*a^10*b^3 - 2*(37*A - 6*C)*a^9*b^4 - 48*B*a^8*b^5 + 4*(29*A - 2*C)*a^7*b^6

+ 32*B*a^6*b^7 - (79*A - 2*C)*a^5*b^8 - 8*B*a^4*b^9 + 20*A*a^3*b^10)*d*x*cos(d*x + c)^3 + 9*((A + 2*C)*a^12*b

- 8*B*a^11*b^2 + 8*(2*A - C)*a^10*b^3 + 32*B*a^9*b^4 - 2*(37*A - 6*C)*a^8*b^5 - 48*B*a^7*b^6 + 4*(29*A - 2*C)

*a^6*b^7 + 32*B*a^5*b^8 - (79*A - 2*C)*a^4*b^9 - 8*B*a^3*b^10 + 20*A*a^2*b^11)*d*x*cos(d*x + c)^2 + 9*((A + 2*

C)*a^11*b^2 - 8*B*a^10*b^3 + 8*(2*A - C)*a^9*b^4 + 32*B*a^8*b^5 - 2*(37*A - 6*C)*a^7*b^6 - 48*B*a^6*b^7 + 4*(2

9*A - 2*C)*a^5*b^8 + 32*B*a^4*b^9 - (79*A - 2*C)*a^3*b^10 - 8*B*a^2*b^11 + 20*A*a*b^12)*d*x*cos(d*x + c) + 3*(

(A + 2*C)*a^10*b^3 - 8*B*a^9*b^4 + 8*(2*A - C)*a^8*b^5 + 32*B*a^7*b^6 - 2*(37*A - 6*C)*a^6*b^7 - 48*B*a^5*b^8

+ 4*(29*A - 2*C)*a^4*b^9 + 32*B*a^3*b^10 - (79*A - 2*C)*a^2*b^11 - 8*B*a*b^12 + 20*A*b^13)*d*x - 3*(8*C*a^8*b^

4 - 20*B*a^7*b^5 + 8*(5*A - C)*a^6*b^6 + 35*B*a^5*b^7 - 7*(12*A - C)*a^4*b^8 - 28*B*a^3*b^9 + (69*A - 2*C)*a^2

*b^10 + 8*B*a*b^11 - 20*A*b^12 + (8*C*a^11*b - 20*B*a^10*b^2 + 8*(5*A - C)*a^9*b^3 + 35*B*a^8*b^4 - 7*(12*A -

C)*a^7*b^5 - 28*B*a^6*b^6 + (69*A - 2*C)*a^5*b^7 + 8*B*a^4*b^8 - 20*A*a^3*b^9)*cos(d*x + c)^3 + 3*(8*C*a^10*b^

2 - 20*B*a^9*b^3 + 8*(5*A - C)*a^8*b^4 + 35*B*a^7*b^5 - 7*(12*A - C)*a^6*b^6 - 28*B*a^5*b^7 + (69*A - 2*C)*a^4

*b^8 + 8*B*a^3*b^9 - 20*A*a^2*b^10)*cos(d*x + c)^2 + 3*(8*C*a^9*b^3 - 20*B*a^8*b^4 + 8*(5*A - C)*a^7*b^5 + 35*

B*a^6*b^6 - 7*(12*A - C)*a^5*b^7 - 28*B*a^4*b^8 + (69*A - 2*C)*a^3*b^9 + 8*B*a^2*b^10 - 20*A*a*b^11)*cos(d*x +

c))*sqrt(-a^2 + b^2)*arctan(-sqrt(-a^2 + b^2)*(b*cos(d*x + c) + a)/((a^2 - b^2)*sin(d*x + c))) + (6*B*a^10*b^

3 - 2*(12*A - 13*C)*a^9*b^4 - 71*B*a^8*b^5 + (170*A - 43*C)*a^7*b^6 + 133*B*a^6*b^7 - (313*A - 23*C)*a^5*b^8 -

92*B*a^4*b^9 + (227*A - 6*C)*a^3*b^10 + 24*B*a^2*b^11 - 60*A*a*b^12 + 3*(A*a^13 - 4*A*a^11*b^2 + 6*A*a^9*b^4

- 4*A*a^7*b^6 + A*a^5*b^8)*cos(d*x + c)^4 + 3*(2*B*a^13 - 5*A*a^12*b - 8*B*a^11*b^2 + 20*A*a^10*b^3 + 12*B*a^9

*b^4 - 30*A*a^8*b^5 - 8*B*a^7*b^6 + 20*A*a^6*b^7 + 2*B*a^5*b^8 - 5*A*a^4*b^9)*cos(d*x + c)^3 + (18*B*a^12*b -

9*(7*A - 4*C)*a^11*b^2 - 132*B*a^10*b^3 + 2*(171*A - 34*C)*a^9*b^4 + 239*B*a^8*b^5 - (590*A - 43*C)*a^7*b^6 -

169*B*a^6*b^7 + (421*A - 11*C)*a^5*b^8 + 44*B*a^4*b^9 - 110*A*a^3*b^10)*cos(d*x + c)^2 + 3*(6*B*a^11*b^2 - (23

*A - 20*C)*a^10*b^3 - 59*B*a^9*b^4 + (146*A - 35*C)*a^8*b^5 + 110*B*a^7*b^6 - (263*A - 20*C)*a^6*b^7 - 77*B*a^

5*b^8 + 5*(38*A - C)*a^4*b^9 + 20*B*a^3*b^10 - 50*A*a^2*b^11)*cos(d*x + c))*sin(d*x + c))/((a^17 - 4*a^15*b^2

+ 6*a^13*b^4 - 4*a^11*b^6 + a^9*b^8)*d*cos(d*x + c)^3 + 3*(a^16*b - 4*a^14*b^3 + 6*a^12*b^5 - 4*a^10*b^7 + a^8

*b^9)*d*cos(d*x + c)^2 + 3*(a^15*b^2 - 4*a^13*b^4 + 6*a^11*b^6 - 4*a^9*b^8 + a^7*b^10)*d*cos(d*x + c) + (a^14*

b^3 - 4*a^12*b^5 + 6*a^10*b^7 - 4*a^8*b^9 + a^6*b^11)*d)]

________________________________________________________________________________________

giac [B] time = 0.48, size = 1438, normalized size = 2.22 \[ \text {result too large to display} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^2*(A+B*sec(d*x+c)+C*sec(d*x+c)^2)/(a+b*sec(d*x+c))^4,x, algorithm="giac")

[Out]

-1/6*(6*(8*C*a^8*b - 20*B*a^7*b^2 + 40*A*a^6*b^3 - 8*C*a^6*b^3 + 35*B*a^5*b^4 - 84*A*a^4*b^5 + 7*C*a^4*b^5 - 2

8*B*a^3*b^6 + 69*A*a^2*b^7 - 2*C*a^2*b^7 + 8*B*a*b^8 - 20*A*b^9)*(pi*floor(1/2*(d*x + c)/pi + 1/2)*sgn(-2*a +

2*b) + arctan(-(a*tan(1/2*d*x + 1/2*c) - b*tan(1/2*d*x + 1/2*c))/sqrt(-a^2 + b^2)))/((a^12 - 3*a^10*b^2 + 3*a^

8*b^4 - a^6*b^6)*sqrt(-a^2 + b^2)) + 2*(36*C*a^8*b^2*tan(1/2*d*x + 1/2*c)^5 - 60*B*a^7*b^3*tan(1/2*d*x + 1/2*c

)^5 - 60*C*a^7*b^3*tan(1/2*d*x + 1/2*c)^5 + 90*A*a^6*b^4*tan(1/2*d*x + 1/2*c)^5 + 105*B*a^6*b^4*tan(1/2*d*x +

1/2*c)^5 - 6*C*a^6*b^4*tan(1/2*d*x + 1/2*c)^5 - 162*A*a^5*b^5*tan(1/2*d*x + 1/2*c)^5 + 24*B*a^5*b^5*tan(1/2*d*

x + 1/2*c)^5 + 45*C*a^5*b^5*tan(1/2*d*x + 1/2*c)^5 - 48*A*a^4*b^6*tan(1/2*d*x + 1/2*c)^5 - 117*B*a^4*b^6*tan(1

/2*d*x + 1/2*c)^5 - 6*C*a^4*b^6*tan(1/2*d*x + 1/2*c)^5 + 213*A*a^3*b^7*tan(1/2*d*x + 1/2*c)^5 + 24*B*a^3*b^7*t

an(1/2*d*x + 1/2*c)^5 - 15*C*a^3*b^7*tan(1/2*d*x + 1/2*c)^5 - 48*A*a^2*b^8*tan(1/2*d*x + 1/2*c)^5 + 42*B*a^2*b

^8*tan(1/2*d*x + 1/2*c)^5 + 6*C*a^2*b^8*tan(1/2*d*x + 1/2*c)^5 - 81*A*a*b^9*tan(1/2*d*x + 1/2*c)^5 - 18*B*a*b^

9*tan(1/2*d*x + 1/2*c)^5 + 36*A*b^10*tan(1/2*d*x + 1/2*c)^5 - 72*C*a^8*b^2*tan(1/2*d*x + 1/2*c)^3 + 120*B*a^7*

b^3*tan(1/2*d*x + 1/2*c)^3 - 180*A*a^6*b^4*tan(1/2*d*x + 1/2*c)^3 + 116*C*a^6*b^4*tan(1/2*d*x + 1/2*c)^3 - 236

*B*a^5*b^5*tan(1/2*d*x + 1/2*c)^3 + 392*A*a^4*b^6*tan(1/2*d*x + 1/2*c)^3 - 56*C*a^4*b^6*tan(1/2*d*x + 1/2*c)^3

+ 152*B*a^3*b^7*tan(1/2*d*x + 1/2*c)^3 - 284*A*a^2*b^8*tan(1/2*d*x + 1/2*c)^3 + 12*C*a^2*b^8*tan(1/2*d*x + 1/

2*c)^3 - 36*B*a*b^9*tan(1/2*d*x + 1/2*c)^3 + 72*A*b^10*tan(1/2*d*x + 1/2*c)^3 + 36*C*a^8*b^2*tan(1/2*d*x + 1/2

*c) - 60*B*a^7*b^3*tan(1/2*d*x + 1/2*c) + 60*C*a^7*b^3*tan(1/2*d*x + 1/2*c) + 90*A*a^6*b^4*tan(1/2*d*x + 1/2*c

) - 105*B*a^6*b^4*tan(1/2*d*x + 1/2*c) - 6*C*a^6*b^4*tan(1/2*d*x + 1/2*c) + 162*A*a^5*b^5*tan(1/2*d*x + 1/2*c)

+ 24*B*a^5*b^5*tan(1/2*d*x + 1/2*c) - 45*C*a^5*b^5*tan(1/2*d*x + 1/2*c) - 48*A*a^4*b^6*tan(1/2*d*x + 1/2*c) +

117*B*a^4*b^6*tan(1/2*d*x + 1/2*c) - 6*C*a^4*b^6*tan(1/2*d*x + 1/2*c) - 213*A*a^3*b^7*tan(1/2*d*x + 1/2*c) +

24*B*a^3*b^7*tan(1/2*d*x + 1/2*c) + 15*C*a^3*b^7*tan(1/2*d*x + 1/2*c) - 48*A*a^2*b^8*tan(1/2*d*x + 1/2*c) - 42

*B*a^2*b^8*tan(1/2*d*x + 1/2*c) + 6*C*a^2*b^8*tan(1/2*d*x + 1/2*c) + 81*A*a*b^9*tan(1/2*d*x + 1/2*c) - 18*B*a*

b^9*tan(1/2*d*x + 1/2*c) + 36*A*b^10*tan(1/2*d*x + 1/2*c))/((a^11 - 3*a^9*b^2 + 3*a^7*b^4 - a^5*b^6)*(a*tan(1/

2*d*x + 1/2*c)^2 - b*tan(1/2*d*x + 1/2*c)^2 - a - b)^3) - 3*(A*a^2 + 2*C*a^2 - 8*B*a*b + 20*A*b^2)*(d*x + c)/a

^6 + 6*(A*a*tan(1/2*d*x + 1/2*c)^3 - 2*B*a*tan(1/2*d*x + 1/2*c)^3 + 8*A*b*tan(1/2*d*x + 1/2*c)^3 - A*a*tan(1/2

*d*x + 1/2*c) - 2*B*a*tan(1/2*d*x + 1/2*c) + 8*A*b*tan(1/2*d*x + 1/2*c))/((tan(1/2*d*x + 1/2*c)^2 + 1)^2*a^5))

________________________________________________________________________________________

maple [B] time = 1.17, size = 4523, normalized size = 6.98 \[ \text {output too large to display} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

20/d*b^2/(a^6-3*a^4*b^2+3*a^2*b^4-b^6)/((a-b)*(a+b))^(1/2)*arctanh(tan(1/2*d*x+1/2*c)*(a-b)/((a-b)*(a+b))^(1/2

))*a*B-8/d*b^8/a^5/(a^6-3*a^4*b^2+3*a^2*b^4-b^6)/((a-b)*(a+b))^(1/2)*arctanh(tan(1/2*d*x+1/2*c)*(a-b)/((a-b)*(

a+b))^(1/2))*B-7/d*b^5/a^2/(a^6-3*a^4*b^2+3*a^2*b^4-b^6)/((a-b)*(a+b))^(1/2)*arctanh(tan(1/2*d*x+1/2*c)*(a-b)/

((a-b)*(a+b))^(1/2))*C+20/d*b^9/a^6/(a^6-3*a^4*b^2+3*a^2*b^4-b^6)/((a-b)*(a+b))^(1/2)*arctanh(tan(1/2*d*x+1/2*

c)*(a-b)/((a-b)*(a+b))^(1/2))*A+2/d*b^7/a^4/(a^6-3*a^4*b^2+3*a^2*b^4-b^6)/((a-b)*(a+b))^(1/2)*arctanh(tan(1/2*

d*x+1/2*c)*(a-b)/((a-b)*(a+b))^(1/2))*C+4/d/(a*tan(1/2*d*x+1/2*c)^2-tan(1/2*d*x+1/2*c)^2*b-a-b)^3/(a+b)/(a^3-3

*a^2*b+3*a*b^2-b^3)*tan(1/2*d*x+1/2*c)*C*b^3-4/d/(a*tan(1/2*d*x+1/2*c)^2-tan(1/2*d*x+1/2*c)^2*b-a-b)^3/(a-b)/(

a^3+3*a^2*b+3*a*b^2+b^3)*tan(1/2*d*x+1/2*c)^5*C*b^3-35/d*b^4/a/(a^6-3*a^4*b^2+3*a^2*b^4-b^6)/((a-b)*(a+b))^(1/

2)*arctanh(tan(1/2*d*x+1/2*c)*(a-b)/((a-b)*(a+b))^(1/2))*B+28/d*b^6/a^3/(a^6-3*a^4*b^2+3*a^2*b^4-b^6)/((a-b)*(

a+b))^(1/2)*arctanh(tan(1/2*d*x+1/2*c)*(a-b)/((a-b)*(a+b))^(1/2))*B+5/d/a/(a*tan(1/2*d*x+1/2*c)^2-tan(1/2*d*x+

1/2*c)^2*b-a-b)^3*b^4/(a-b)/(a^3+3*a^2*b+3*a*b^2+b^3)*tan(1/2*d*x+1/2*c)^5*B-5/d/a/(a*tan(1/2*d*x+1/2*c)^2-tan

(1/2*d*x+1/2*c)^2*b-a-b)^3*b^4/(a+b)/(a^3-3*a^2*b+3*a*b^2-b^3)*tan(1/2*d*x+1/2*c)*B-18/d/a^2/(a*tan(1/2*d*x+1/

2*c)^2-tan(1/2*d*x+1/2*c)^2*b-a-b)^3*b^5/(a-b)/(a^3+3*a^2*b+3*a*b^2+b^3)*tan(1/2*d*x+1/2*c)^5*B+2/d/a^3/(a*tan

(1/2*d*x+1/2*c)^2-tan(1/2*d*x+1/2*c)^2*b-a-b)^3*b^6/(a+b)/(a^3-3*a^2*b+3*a*b^2-b^3)*tan(1/2*d*x+1/2*c)*B-2/d/a

^3/(a*tan(1/2*d*x+1/2*c)^2-tan(1/2*d*x+1/2*c)^2*b-a-b)^3*b^6/(a-b)/(a^3+3*a^2*b+3*a*b^2+b^3)*tan(1/2*d*x+1/2*c

)^5*B-18/d/a^2/(a*tan(1/2*d*x+1/2*c)^2-tan(1/2*d*x+1/2*c)^2*b-a-b)^3*b^5/(a+b)/(a^3-3*a^2*b+3*a*b^2-b^3)*tan(1

/2*d*x+1/2*c)*B-2/d*b^6/a^3/(a*tan(1/2*d*x+1/2*c)^2-tan(1/2*d*x+1/2*c)^2*b-a-b)^3/(a+b)/(a^3-3*a^2*b+3*a*b^2-b

^3)*tan(1/2*d*x+1/2*c)*C-12/d*b^7/a^4/(a*tan(1/2*d*x+1/2*c)^2-tan(1/2*d*x+1/2*c)^2*b-a-b)^3/(a^2-2*a*b+b^2)/(a

^2+2*a*b+b^2)*tan(1/2*d*x+1/2*c)^3*B+1/d*b^5/a^2/(a*tan(1/2*d*x+1/2*c)^2-tan(1/2*d*x+1/2*c)^2*b-a-b)^3/(a-b)/(

a^3+3*a^2*b+3*a*b^2+b^3)*tan(1/2*d*x+1/2*c)^5*C+6/d*b^4/a/(a*tan(1/2*d*x+1/2*c)^2-tan(1/2*d*x+1/2*c)^2*b-a-b)^

3/(a+b)/(a^3-3*a^2*b+3*a*b^2-b^3)*tan(1/2*d*x+1/2*c)*C-1/d*b^5/a^2/(a*tan(1/2*d*x+1/2*c)^2-tan(1/2*d*x+1/2*c)^

2*b-a-b)^3/(a+b)/(a^3-3*a^2*b+3*a*b^2-b^3)*tan(1/2*d*x+1/2*c)*C-8/d*b/(a^6-3*a^4*b^2+3*a^2*b^4-b^6)/((a-b)*(a+

b))^(1/2)*arctanh(tan(1/2*d*x+1/2*c)*(a-b)/((a-b)*(a+b))^(1/2))*C*a^2+1/d/a^4*A*arctan(tan(1/2*d*x+1/2*c))+60/

d/a/(a*tan(1/2*d*x+1/2*c)^2-tan(1/2*d*x+1/2*c)^2*b-a-b)^3/(a^2-2*a*b+b^2)/(a^2+2*a*b+b^2)*tan(1/2*d*x+1/2*c)^3

*A*b^4-3/d/a^4/(a*tan(1/2*d*x+1/2*c)^2-tan(1/2*d*x+1/2*c)^2*b-a-b)^3*b^7/(a+b)/(a^3-3*a^2*b+3*a*b^2-b^3)*tan(1

/2*d*x+1/2*c)*A+3/d/a^4/(a*tan(1/2*d*x+1/2*c)^2-tan(1/2*d*x+1/2*c)^2*b-a-b)^3*b^7/(a-b)/(a^3+3*a^2*b+3*a*b^2+b

^3)*tan(1/2*d*x+1/2*c)^5*A-212/3/d/a^3/(a*tan(1/2*d*x+1/2*c)^2-tan(1/2*d*x+1/2*c)^2*b-a-b)^3/(a^2-2*a*b+b^2)/(

a^2+2*a*b+b^2)*tan(1/2*d*x+1/2*c)^3*A*b^6+24/d*a/(a*tan(1/2*d*x+1/2*c)^2-tan(1/2*d*x+1/2*c)^2*b-a-b)^3/(a^2-2*

a*b+b^2)/(a^2+2*a*b+b^2)*tan(1/2*d*x+1/2*c)^3*b^2*C-69/d/a^4/(a^6-3*a^4*b^2+3*a^2*b^4-b^6)/((a-b)*(a+b))^(1/2)

*arctanh(tan(1/2*d*x+1/2*c)*(a-b)/((a-b)*(a+b))^(1/2))*A*b^7+84/d/a^2/(a^6-3*a^4*b^2+3*a^2*b^4-b^6)/((a-b)*(a+

b))^(1/2)*arctanh(tan(1/2*d*x+1/2*c)*(a-b)/((a-b)*(a+b))^(1/2))*A*b^5-40/d/(a*tan(1/2*d*x+1/2*c)^2-tan(1/2*d*x

+1/2*c)^2*b-a-b)^3/(a^2-2*a*b+b^2)/(a^2+2*a*b+b^2)*tan(1/2*d*x+1/2*c)^3*b^3*B+20/d/(a*tan(1/2*d*x+1/2*c)^2-tan

(1/2*d*x+1/2*c)^2*b-a-b)^3/(a+b)/(a^3-3*a^2*b+3*a*b^2-b^3)*tan(1/2*d*x+1/2*c)*b^3*B+20/d/(a*tan(1/2*d*x+1/2*c)

^2-tan(1/2*d*x+1/2*c)^2*b-a-b)^3/(a-b)/(a^3+3*a^2*b+3*a*b^2+b^3)*tan(1/2*d*x+1/2*c)^5*b^3*B-30/d/a/(a*tan(1/2*

d*x+1/2*c)^2-tan(1/2*d*x+1/2*c)^2*b-a-b)^3/(a-b)/(a^3+3*a^2*b+3*a*b^2+b^3)*tan(1/2*d*x+1/2*c)^5*A*b^4+6/d/a^2/

(a*tan(1/2*d*x+1/2*c)^2-tan(1/2*d*x+1/2*c)^2*b-a-b)^3/(a+b)/(a^3-3*a^2*b+3*a*b^2-b^3)*tan(1/2*d*x+1/2*c)*A*b^5

-30/d/a/(a*tan(1/2*d*x+1/2*c)^2-tan(1/2*d*x+1/2*c)^2*b-a-b)^3/(a+b)/(a^3-3*a^2*b+3*a*b^2-b^3)*tan(1/2*d*x+1/2*

c)*A*b^4+34/d/a^3/(a*tan(1/2*d*x+1/2*c)^2-tan(1/2*d*x+1/2*c)^2*b-a-b)^3/(a+b)/(a^3-3*a^2*b+3*a*b^2-b^3)*tan(1/

2*d*x+1/2*c)*A*b^6-12/d*a/(a*tan(1/2*d*x+1/2*c)^2-tan(1/2*d*x+1/2*c)^2*b-a-b)^3/(a-b)/(a^3+3*a^2*b+3*a*b^2+b^3

)*tan(1/2*d*x+1/2*c)^5*b^2*C-6/d/a^2/(a*tan(1/2*d*x+1/2*c)^2-tan(1/2*d*x+1/2*c)^2*b-a-b)^3/(a-b)/(a^3+3*a^2*b+

3*a*b^2+b^3)*tan(1/2*d*x+1/2*c)^5*A*b^5+34/d/a^3/(a*tan(1/2*d*x+1/2*c)^2-tan(1/2*d*x+1/2*c)^2*b-a-b)^3/(a-b)/(

a^3+3*a^2*b+3*a*b^2+b^3)*tan(1/2*d*x+1/2*c)^5*A*b^6-12/d*a/(a*tan(1/2*d*x+1/2*c)^2-tan(1/2*d*x+1/2*c)^2*b-a-b)

^3/(a+b)/(a^3-3*a^2*b+3*a*b^2-b^3)*tan(1/2*d*x+1/2*c)*b^2*C+6/d*b^7/a^4/(a*tan(1/2*d*x+1/2*c)^2-tan(1/2*d*x+1/

2*c)^2*b-a-b)^3/(a-b)/(a^3+3*a^2*b+3*a*b^2+b^3)*tan(1/2*d*x+1/2*c)^5*B+6/d*b^4/a/(a*tan(1/2*d*x+1/2*c)^2-tan(1

/2*d*x+1/2*c)^2*b-a-b)^3/(a-b)/(a^3+3*a^2*b+3*a*b^2+b^3)*tan(1/2*d*x+1/2*c)^5*C+116/3/d*b^5/a^2/(a*tan(1/2*d*x

+1/2*c)^2-tan(1/2*d*x+1/2*c)^2*b-a-b)^3/(a^2-2*a*b+b^2)/(a^2+2*a*b+b^2)*tan(1/2*d*x+1/2*c)^3*B-12/d*b^8/a^5/(a

*tan(1/2*d*x+1/2*c)^2-tan(1/2*d*x+1/2*c)^2*b-a-b)^3/(a+b)/(a^3-3*a^2*b+3*a*b^2-b^3)*tan(1/2*d*x+1/2*c)*A-12/d*

b^8/a^5/(a*tan(1/2*d*x+1/2*c)^2-tan(1/2*d*x+1/2*c)^2*b-a-b)^3/(a-b)/(a^3+3*a^2*b+3*a*b^2+b^3)*tan(1/2*d*x+1/2*

c)^5*A+6/d*b^7/a^4/(a*tan(1/2*d*x+1/2*c)^2-tan(1/2*d*x+1/2*c)^2*b-a-b)^3/(a+b)/(a^3-3*a^2*b+3*a*b^2-b^3)*tan(1

/2*d*x+1/2*c)*B-44/3/d*b^4/a/(a*tan(1/2*d*x+1/2*c)^2-tan(1/2*d*x+1/2*c)^2*b-a-b)^3/(a^2-2*a*b+b^2)/(a^2+2*a*b+

b^2)*tan(1/2*d*x+1/2*c)^3*C+4/d*b^6/a^3/(a*tan(1/2*d*x+1/2*c)^2-tan(1/2*d*x+1/2*c)^2*b-a-b)^3/(a^2-2*a*b+b^2)/

(a^2+2*a*b+b^2)*tan(1/2*d*x+1/2*c)^3*C-2/d*b^6/a^3/(a*tan(1/2*d*x+1/2*c)^2-tan(1/2*d*x+1/2*c)^2*b-a-b)^3/(a-b)

/(a^3+3*a^2*b+3*a*b^2+b^3)*tan(1/2*d*x+1/2*c)^5*C+24/d*b^8/a^5/(a*tan(1/2*d*x+1/2*c)^2-tan(1/2*d*x+1/2*c)^2*b-

a-b)^3/(a^2-2*a*b+b^2)/(a^2+2*a*b+b^2)*tan(1/2*d*x+1/2*c)^3*A+2/d/a^4*arctan(tan(1/2*d*x+1/2*c))*C-1/d/a^4/(1+

tan(1/2*d*x+1/2*c)^2)^2*tan(1/2*d*x+1/2*c)^3*A-40/d*b^3/(a^6-3*a^4*b^2+3*a^2*b^4-b^6)/((a-b)*(a+b))^(1/2)*arct

anh(tan(1/2*d*x+1/2*c)*(a-b)/((a-b)*(a+b))^(1/2))*A+2/d/a^4/(1+tan(1/2*d*x+1/2*c)^2)^2*tan(1/2*d*x+1/2*c)^3*B+

1/d/a^4/(1+tan(1/2*d*x+1/2*c)^2)^2*tan(1/2*d*x+1/2*c)*A+2/d/a^4/(1+tan(1/2*d*x+1/2*c)^2)^2*tan(1/2*d*x+1/2*c)*

B+20/d/a^6*arctan(tan(1/2*d*x+1/2*c))*A*b^2-8/d/a^5*arctan(tan(1/2*d*x+1/2*c))*B*b-8/d/a^5/(1+tan(1/2*d*x+1/2*

c)^2)^2*tan(1/2*d*x+1/2*c)^3*A*b-8/d/a^5/(1+tan(1/2*d*x+1/2*c)^2)^2*tan(1/2*d*x+1/2*c)*A*b+8/d/(a^6-3*a^4*b^2+

3*a^2*b^4-b^6)/((a-b)*(a+b))^(1/2)*arctanh(tan(1/2*d*x+1/2*c)*(a-b)/((a-b)*(a+b))^(1/2))*C*b^3

________________________________________________________________________________________

maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: ValueError} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^2*(A+B*sec(d*x+c)+C*sec(d*x+c)^2)/(a+b*sec(d*x+c))^4,x, algorithm="maxima")

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a

ssume' command before evaluation *may* help (example of legal syntax is 'assume(4*a^2-4*b^2>0)', see `assume?`

for more details)Is 4*a^2-4*b^2 positive or negative?

________________________________________________________________________________________

mupad [B] time = 27.99, size = 21910, normalized size = 33.81 \[ \text {result too large to display} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((tan(c/2 + (d*x)/2)^9*(A*a^8 + 20*A*b^8 - 2*B*a^8 - 59*A*a^2*b^6 + 27*A*a^3*b^5 + 57*A*a^4*b^4 - 21*A*a^5*b^3

- 11*A*a^6*b^2 + 4*B*a^2*b^6 + 24*B*a^3*b^5 - 11*B*a^4*b^4 - 26*B*a^5*b^3 + 6*B*a^6*b^2 + 2*C*a^2*b^6 - C*a^3

*b^5 - 6*C*a^4*b^4 + 4*C*a^5*b^3 + 12*C*a^6*b^2 - 10*A*a*b^7 + 7*A*a^7*b - 8*B*a*b^7 + 2*B*a^7*b))/(a^5*(a + b

)^3*(a - b)) - (2*tan(c/2 + (d*x)/2)^3*(6*A*a^9 - 120*A*b^9 + 6*B*a^9 + 364*A*a^2*b^7 + 71*A*a^3*b^6 - 369*A*a

^4*b^5 - 45*A*a^5*b^4 + 111*A*a^6*b^3 + 3*A*a^7*b^2 + 12*B*a^2*b^7 - 148*B*a^3*b^6 - 29*B*a^4*b^5 + 159*B*a^5*

b^4 + 18*B*a^6*b^3 - 30*B*a^7*b^2 - 12*C*a^2*b^7 - 3*C*a^3*b^6 + 37*C*a^4*b^5 + 8*C*a^5*b^4 - 60*C*a^6*b^3 - 3

0*A*a*b^8 - 21*A*a^8*b + 48*B*a*b^8 - 6*B*a^8*b))/(3*a^5*(a + b)^2*(a - b)^3) - (2*tan(c/2 + (d*x)/2)^7*(6*A*a

^9 + 120*A*b^9 - 6*B*a^9 - 364*A*a^2*b^7 + 71*A*a^3*b^6 + 369*A*a^4*b^5 - 45*A*a^5*b^4 - 111*A*a^6*b^3 + 3*A*a

^7*b^2 + 12*B*a^2*b^7 + 148*B*a^3*b^6 - 29*B*a^4*b^5 - 159*B*a^5*b^4 + 18*B*a^6*b^3 + 30*B*a^7*b^2 + 12*C*a^2*

b^7 - 3*C*a^3*b^6 - 37*C*a^4*b^5 + 8*C*a^5*b^4 + 60*C*a^6*b^3 - 30*A*a*b^8 + 21*A*a^8*b - 48*B*a*b^8 - 6*B*a^8

*b))/(3*a^5*(a + b)^3*(a - b)^2) + (2*tan(c/2 + (d*x)/2)^5*(9*A*a^10 + 180*A*b^10 - 611*A*a^2*b^8 + 740*A*a^4*

b^6 - 324*A*a^6*b^4 + 36*A*a^8*b^2 + 248*B*a^3*b^7 - 320*B*a^5*b^5 + 132*B*a^7*b^3 + 18*C*a^2*b^8 - 62*C*a^4*b

^6 + 110*C*a^6*b^4 - 36*C*a^8*b^2 - 72*B*a*b^9 - 18*B*a^9*b))/(3*a^5*(a + b)^3*(a - b)^3) + (tan(c/2 + (d*x)/2

)*(A*a^8 + 20*A*b^8 + 2*B*a^8 - 59*A*a^2*b^6 - 27*A*a^3*b^5 + 57*A*a^4*b^4 + 21*A*a^5*b^3 - 11*A*a^6*b^2 - 4*B

*a^2*b^6 + 24*B*a^3*b^5 + 11*B*a^4*b^4 - 26*B*a^5*b^3 - 6*B*a^6*b^2 + 2*C*a^2*b^6 + C*a^3*b^5 - 6*C*a^4*b^4 -

4*C*a^5*b^3 + 12*C*a^6*b^2 + 10*A*a*b^7 - 7*A*a^7*b - 8*B*a*b^7 + 2*B*a^7*b))/(a^5*(a + b)*(a - b)^3))/(d*(tan

(c/2 + (d*x)/2)^2*(9*a*b^2 + 3*a^2*b - a^3 + 5*b^3) + tan(c/2 + (d*x)/2)^4*(6*a*b^2 - 6*a^2*b - 2*a^3 + 10*b^3

) - tan(c/2 + (d*x)/2)^6*(6*a*b^2 + 6*a^2*b - 2*a^3 - 10*b^3) + 3*a*b^2 + 3*a^2*b + a^3 + b^3 - tan(c/2 + (d*x

)/2)^10*(3*a*b^2 - 3*a^2*b + a^3 - b^3) + tan(c/2 + (d*x)/2)^8*(3*a^2*b - 9*a*b^2 + a^3 + 5*b^3))) + (atan((((

(8*tan(c/2 + (d*x)/2)*(A^2*a^18 + 800*A^2*b^18 + 4*C^2*a^18 - 800*A^2*a*b^17 - 2*A^2*a^17*b - 8*C^2*a^17*b - 4

720*A^2*a^2*b^16 + 4720*A^2*a^3*b^15 + 11522*A^2*a^4*b^14 - 11522*A^2*a^5*b^13 - 14837*A^2*a^6*b^12 + 14812*A^

2*a^7*b^11 + 10385*A^2*a^8*b^10 - 10430*A^2*a^9*b^9 - 3325*A^2*a^10*b^8 + 3640*A^2*a^11*b^7 - 45*A^2*a^12*b^6

- 350*A^2*a^13*b^5 + 209*A^2*a^14*b^4 - 68*A^2*a^15*b^3 + 35*A^2*a^16*b^2 + 128*B^2*a^2*b^16 - 128*B^2*a^3*b^1

5 - 768*B^2*a^4*b^14 + 768*B^2*a^5*b^13 + 1920*B^2*a^6*b^12 - 1920*B^2*a^7*b^11 - 2600*B^2*a^8*b^10 + 2560*B^2

*a^9*b^9 + 2025*B^2*a^10*b^8 - 1920*B^2*a^11*b^7 - 824*B^2*a^12*b^6 + 768*B^2*a^13*b^5 + 80*B^2*a^14*b^4 - 128

*B^2*a^15*b^3 + 64*B^2*a^16*b^2 + 8*C^2*a^4*b^14 - 8*C^2*a^5*b^13 - 48*C^2*a^6*b^12 + 48*C^2*a^7*b^11 + 117*C^

2*a^8*b^10 - 120*C^2*a^9*b^9 - 164*C^2*a^10*b^8 + 160*C^2*a^11*b^7 + 156*C^2*a^12*b^6 - 120*C^2*a^13*b^5 - 92*

C^2*a^14*b^4 + 48*C^2*a^15*b^3 + 44*C^2*a^16*b^2 + 4*A*C*a^18 - 640*A*B*a*b^17 - 16*A*B*a^17*b - 8*A*C*a^17*b

- 32*B*C*a^17*b + 640*A*B*a^2*b^16 + 3808*A*B*a^3*b^15 - 3808*A*B*a^4*b^14 - 9408*A*B*a^5*b^13 + 9408*A*B*a^6*

b^12 + 12430*A*B*a^7*b^11 - 12320*A*B*a^8*b^10 - 9200*A*B*a^9*b^9 + 8960*A*B*a^10*b^8 + 3360*A*B*a^11*b^7 - 33

60*A*B*a^12*b^6 - 144*A*B*a^13*b^5 + 448*A*B*a^14*b^4 - 240*A*B*a^15*b^3 + 32*A*B*a^16*b^2 + 160*A*C*a^2*b^16

- 160*A*C*a^3*b^15 - 952*A*C*a^4*b^14 + 952*A*C*a^5*b^13 + 2322*A*C*a^6*b^12 - 2352*A*C*a^7*b^11 - 3124*A*C*a^

8*b^10 + 3080*A*C*a^9*b^9 + 2588*A*C*a^10*b^8 - 2240*A*C*a^11*b^7 - 1284*A*C*a^12*b^6 + 840*A*C*a^13*b^5 + 276

*A*C*a^14*b^4 - 112*A*C*a^15*b^3 + 60*A*C*a^16*b^2 - 64*B*C*a^3*b^15 + 64*B*C*a^4*b^14 + 384*B*C*a^5*b^13 - 38

4*B*C*a^6*b^12 - 948*B*C*a^7*b^11 + 960*B*C*a^8*b^10 + 1306*B*C*a^9*b^9 - 1280*B*C*a^10*b^8 - 1128*B*C*a^11*b^

7 + 960*B*C*a^12*b^6 + 592*B*C*a^13*b^5 - 384*B*C*a^14*b^4 - 160*B*C*a^15*b^3 + 64*B*C*a^16*b^2))/(a^20*b + a^

21 - a^10*b^11 - a^11*b^10 + 5*a^12*b^9 + 5*a^13*b^8 - 10*a^14*b^7 - 10*a^15*b^6 + 10*a^16*b^5 + 10*a^17*b^4 -

5*a^18*b^3 - 5*a^19*b^2) + (((4*(4*A*a^27 + 8*C*a^27 - 80*A*a^12*b^15 + 40*A*a^13*b^14 + 516*A*a^14*b^13 - 24

8*A*a^15*b^12 - 1404*A*a^16*b^11 + 640*A*a^17*b^10 + 2076*A*a^18*b^9 - 896*A*a^19*b^8 - 1764*A*a^20*b^7 + 724*

A*a^21*b^6 + 816*A*a^22*b^5 - 316*A*a^23*b^4 - 160*A*a^24*b^3 + 52*A*a^25*b^2 + 32*B*a^13*b^14 - 16*B*a^14*b^1

3 - 208*B*a^15*b^12 + 100*B*a^16*b^11 + 572*B*a^17*b^10 - 252*B*a^18*b^9 - 868*B*a^19*b^8 + 348*B*a^20*b^7 + 7

72*B*a^21*b^6 - 292*B*a^22*b^5 - 380*B*a^23*b^4 + 144*B*a^24*b^3 + 80*B*a^25*b^2 - 8*C*a^14*b^13 + 4*C*a^15*b^

12 + 52*C*a^16*b^11 - 28*C*a^17*b^10 - 140*C*a^18*b^9 + 60*C*a^19*b^8 + 220*C*a^20*b^7 - 60*C*a^21*b^6 - 220*C

*a^22*b^5 + 40*C*a^23*b^4 + 128*C*a^24*b^3 - 24*C*a^25*b^2 - 32*B*a^26*b - 32*C*a^26*b))/(a^25*b + a^26 - a^15

*b^11 - a^16*b^10 + 5*a^17*b^9 + 5*a^18*b^8 - 10*a^19*b^7 - 10*a^20*b^6 + 10*a^21*b^5 + 10*a^22*b^4 - 5*a^23*b

^3 - 5*a^24*b^2) - (8*tan(c/2 + (d*x)/2)*(A*b^2*10i + a^2*((A*1i)/2 + C*1i) - B*a*b*4i)*(8*a^25*b - 8*a^12*b^1

4 + 8*a^13*b^13 + 48*a^14*b^12 - 48*a^15*b^11 - 120*a^16*b^10 + 120*a^17*b^9 + 160*a^18*b^8 - 160*a^19*b^7 - 1

20*a^20*b^6 + 120*a^21*b^5 + 48*a^22*b^4 - 48*a^23*b^3 - 8*a^24*b^2))/(a^6*(a^20*b + a^21 - a^10*b^11 - a^11*b

^10 + 5*a^12*b^9 + 5*a^13*b^8 - 10*a^14*b^7 - 10*a^15*b^6 + 10*a^16*b^5 + 10*a^17*b^4 - 5*a^18*b^3 - 5*a^19*b^

2)))*(A*b^2*10i + a^2*((A*1i)/2 + C*1i) - B*a*b*4i))/a^6)*(A*b^2*10i + a^2*((A*1i)/2 + C*1i) - B*a*b*4i)*1i)/a

^6 + (((8*tan(c/2 + (d*x)/2)*(A^2*a^18 + 800*A^2*b^18 + 4*C^2*a^18 - 800*A^2*a*b^17 - 2*A^2*a^17*b - 8*C^2*a^1

7*b - 4720*A^2*a^2*b^16 + 4720*A^2*a^3*b^15 + 11522*A^2*a^4*b^14 - 11522*A^2*a^5*b^13 - 14837*A^2*a^6*b^12 + 1

4812*A^2*a^7*b^11 + 10385*A^2*a^8*b^10 - 10430*A^2*a^9*b^9 - 3325*A^2*a^10*b^8 + 3640*A^2*a^11*b^7 - 45*A^2*a^

12*b^6 - 350*A^2*a^13*b^5 + 209*A^2*a^14*b^4 - 68*A^2*a^15*b^3 + 35*A^2*a^16*b^2 + 128*B^2*a^2*b^16 - 128*B^2*

a^3*b^15 - 768*B^2*a^4*b^14 + 768*B^2*a^5*b^13 + 1920*B^2*a^6*b^12 - 1920*B^2*a^7*b^11 - 2600*B^2*a^8*b^10 + 2

560*B^2*a^9*b^9 + 2025*B^2*a^10*b^8 - 1920*B^2*a^11*b^7 - 824*B^2*a^12*b^6 + 768*B^2*a^13*b^5 + 80*B^2*a^14*b^

4 - 128*B^2*a^15*b^3 + 64*B^2*a^16*b^2 + 8*C^2*a^4*b^14 - 8*C^2*a^5*b^13 - 48*C^2*a^6*b^12 + 48*C^2*a^7*b^11 +

117*C^2*a^8*b^10 - 120*C^2*a^9*b^9 - 164*C^2*a^10*b^8 + 160*C^2*a^11*b^7 + 156*C^2*a^12*b^6 - 120*C^2*a^13*b^

5 - 92*C^2*a^14*b^4 + 48*C^2*a^15*b^3 + 44*C^2*a^16*b^2 + 4*A*C*a^18 - 640*A*B*a*b^17 - 16*A*B*a^17*b - 8*A*C*

a^17*b - 32*B*C*a^17*b + 640*A*B*a^2*b^16 + 3808*A*B*a^3*b^15 - 3808*A*B*a^4*b^14 - 9408*A*B*a^5*b^13 + 9408*A

*B*a^6*b^12 + 12430*A*B*a^7*b^11 - 12320*A*B*a^8*b^10 - 9200*A*B*a^9*b^9 + 8960*A*B*a^10*b^8 + 3360*A*B*a^11*b

^7 - 3360*A*B*a^12*b^6 - 144*A*B*a^13*b^5 + 448*A*B*a^14*b^4 - 240*A*B*a^15*b^3 + 32*A*B*a^16*b^2 + 160*A*C*a^

2*b^16 - 160*A*C*a^3*b^15 - 952*A*C*a^4*b^14 + 952*A*C*a^5*b^13 + 2322*A*C*a^6*b^12 - 2352*A*C*a^7*b^11 - 3124

*A*C*a^8*b^10 + 3080*A*C*a^9*b^9 + 2588*A*C*a^10*b^8 - 2240*A*C*a^11*b^7 - 1284*A*C*a^12*b^6 + 840*A*C*a^13*b^

5 + 276*A*C*a^14*b^4 - 112*A*C*a^15*b^3 + 60*A*C*a^16*b^2 - 64*B*C*a^3*b^15 + 64*B*C*a^4*b^14 + 384*B*C*a^5*b^

13 - 384*B*C*a^6*b^12 - 948*B*C*a^7*b^11 + 960*B*C*a^8*b^10 + 1306*B*C*a^9*b^9 - 1280*B*C*a^10*b^8 - 1128*B*C*

a^11*b^7 + 960*B*C*a^12*b^6 + 592*B*C*a^13*b^5 - 384*B*C*a^14*b^4 - 160*B*C*a^15*b^3 + 64*B*C*a^16*b^2))/(a^20

*b + a^21 - a^10*b^11 - a^11*b^10 + 5*a^12*b^9 + 5*a^13*b^8 - 10*a^14*b^7 - 10*a^15*b^6 + 10*a^16*b^5 + 10*a^1

7*b^4 - 5*a^18*b^3 - 5*a^19*b^2) - (((4*(4*A*a^27 + 8*C*a^27 - 80*A*a^12*b^15 + 40*A*a^13*b^14 + 516*A*a^14*b^

13 - 248*A*a^15*b^12 - 1404*A*a^16*b^11 + 640*A*a^17*b^10 + 2076*A*a^18*b^9 - 896*A*a^19*b^8 - 1764*A*a^20*b^7

+ 724*A*a^21*b^6 + 816*A*a^22*b^5 - 316*A*a^23*b^4 - 160*A*a^24*b^3 + 52*A*a^25*b^2 + 32*B*a^13*b^14 - 16*B*a

^14*b^13 - 208*B*a^15*b^12 + 100*B*a^16*b^11 + 572*B*a^17*b^10 - 252*B*a^18*b^9 - 868*B*a^19*b^8 + 348*B*a^20*

b^7 + 772*B*a^21*b^6 - 292*B*a^22*b^5 - 380*B*a^23*b^4 + 144*B*a^24*b^3 + 80*B*a^25*b^2 - 8*C*a^14*b^13 + 4*C*

a^15*b^12 + 52*C*a^16*b^11 - 28*C*a^17*b^10 - 140*C*a^18*b^9 + 60*C*a^19*b^8 + 220*C*a^20*b^7 - 60*C*a^21*b^6

- 220*C*a^22*b^5 + 40*C*a^23*b^4 + 128*C*a^24*b^3 - 24*C*a^25*b^2 - 32*B*a^26*b - 32*C*a^26*b))/(a^25*b + a^26

- a^15*b^11 - a^16*b^10 + 5*a^17*b^9 + 5*a^18*b^8 - 10*a^19*b^7 - 10*a^20*b^6 + 10*a^21*b^5 + 10*a^22*b^4 - 5

*a^23*b^3 - 5*a^24*b^2) + (8*tan(c/2 + (d*x)/2)*(A*b^2*10i + a^2*((A*1i)/2 + C*1i) - B*a*b*4i)*(8*a^25*b - 8*a

^12*b^14 + 8*a^13*b^13 + 48*a^14*b^12 - 48*a^15*b^11 - 120*a^16*b^10 + 120*a^17*b^9 + 160*a^18*b^8 - 160*a^19*

b^7 - 120*a^20*b^6 + 120*a^21*b^5 + 48*a^22*b^4 - 48*a^23*b^3 - 8*a^24*b^2))/(a^6*(a^20*b + a^21 - a^10*b^11 -

a^11*b^10 + 5*a^12*b^9 + 5*a^13*b^8 - 10*a^14*b^7 - 10*a^15*b^6 + 10*a^16*b^5 + 10*a^17*b^4 - 5*a^18*b^3 - 5*

a^19*b^2)))*(A*b^2*10i + a^2*((A*1i)/2 + C*1i) - B*a*b*4i))/a^6)*(A*b^2*10i + a^2*((A*1i)/2 + C*1i) - B*a*b*4i

)*1i)/a^6)/((8*(8000*A^3*b^19 - 4000*A^3*a*b^18 + 32*C^3*a^18*b - 50800*A^3*a^2*b^17 + 24400*A^3*a^3*b^16 + 13

5260*A^3*a^4*b^15 - 62030*A^3*a^5*b^14 - 193689*A^3*a^6*b^13 + 82337*A^3*a^7*b^12 + 155991*A^3*a^8*b^11 - 5734

5*A^3*a^9*b^10 - 64479*A^3*a^10*b^9 + 16999*A^3*a^11*b^8 + 8281*A^3*a^12*b^7 + 204*A^3*a^13*b^6 + 1396*A^3*a^1

4*b^5 - 40*A^3*a^15*b^4 + 40*A^3*a^16*b^3 - 512*B^3*a^3*b^16 + 256*B^3*a^4*b^15 + 3328*B^3*a^5*b^14 - 1600*B^3

*a^6*b^13 - 9152*B^3*a^7*b^12 + 4352*B^3*a^8*b^11 + 13888*B^3*a^9*b^10 - 6408*B^3*a^10*b^9 - 12352*B^3*a^11*b^

8 + 5120*B^3*a^12*b^7 + 6080*B^3*a^13*b^6 - 1920*B^3*a^14*b^5 - 1280*B^3*a^15*b^4 + 8*C^3*a^6*b^13 - 4*C^3*a^7

*b^12 - 52*C^3*a^8*b^11 + 22*C^3*a^9*b^10 + 140*C^3*a^10*b^9 - 68*C^3*a^11*b^8 - 220*C^3*a^12*b^7 + 132*C^3*a^

13*b^6 + 220*C^3*a^14*b^5 - 128*C^3*a^15*b^4 - 128*C^3*a^16*b^3 + 96*C^3*a^17*b^2 - 9600*A^2*B*a*b^18 + 32*A*C

^2*a^18*b + 8*A^2*C*a^18*b + 3840*A*B^2*a^2*b^17 - 1920*A*B^2*a^3*b^16 - 24768*A*B^2*a^4*b^15 + 11904*A*B^2*a^

5*b^14 + 67392*A*B^2*a^6*b^13 - 31680*A*B^2*a^7*b^12 - 100368*A*B^2*a^8*b^11 + 45148*A*B^2*a^9*b^10 + 86512*A*

B^2*a^10*b^9 - 34567*A*B^2*a^11*b^8 - 40368*A*B^2*a^12*b^7 + 11960*A*B^2*a^13*b^6 + 7440*A*B^2*a^14*b^5 + 80*A

*B^2*a^15*b^4 + 320*A*B^2*a^16*b^3 + 4800*A^2*B*a^2*b^17 + 61440*A^2*B*a^3*b^16 - 29520*A^2*B*a^4*b^15 - 16538

4*A^2*B*a^5*b^14 + 76812*A^2*B*a^6*b^13 + 241596*A^2*B*a^7*b^12 - 105755*A^2*B*a^8*b^11 - 201479*A^2*B*a^9*b^1

0 + 77359*A^2*B*a^10*b^9 + 88721*A^2*B*a^11*b^8 - 24711*A^2*B*a^12*b^7 - 13929*A^2*B*a^13*b^6 - 255*A^2*B*a^14

*b^5 - 1345*A^2*B*a^15*b^4 + 20*A^2*B*a^16*b^3 - 20*A^2*B*a^17*b^2 + 240*A*C^2*a^4*b^15 - 120*A*C^2*a^5*b^14 -

1548*A*C^2*a^6*b^13 + 684*A*C^2*a^7*b^12 + 4152*A*C^2*a^8*b^11 - 1983*A*C^2*a^9*b^10 - 6336*A*C^2*a^10*b^9 +

3448*A*C^2*a^11*b^8 + 5944*A*C^2*a^12*b^7 - 3196*A*C^2*a^13*b^6 - 3156*A*C^2*a^14*b^5 + 1760*A*C^2*a^15*b^4 +

672*A*C^2*a^16*b^3 + 32*A*C^2*a^17*b^2 + 2400*A^2*C*a^2*b^17 - 1200*A^2*C*a^3*b^16 - 15360*A^2*C*a^4*b^15 + 70

80*A^2*C*a^5*b^14 + 41046*A^2*C*a^6*b^13 - 19233*A^2*C*a^7*b^12 - 60729*A^2*C*a^8*b^11 + 29513*A^2*C*a^9*b^10

+ 53039*A^2*C*a^10*b^9 - 24901*A^2*C*a^11*b^8 - 25211*A^2*C*a^12*b^7 + 9657*A^2*C*a^13*b^6 + 4359*A^2*C*a^14*b

^5 + 192*A^2*C*a^15*b^4 + 448*A^2*C*a^16*b^3 - 8*A^2*C*a^17*b^2 - 96*B*C^2*a^5*b^14 + 48*B*C^2*a^6*b^13 + 624*

B*C^2*a^7*b^12 - 276*B*C^2*a^8*b^11 - 1692*B*C^2*a^9*b^10 + 816*B*C^2*a^10*b^9 + 2628*B*C^2*a^11*b^8 - 1452*B*

C^2*a^12*b^7 - 2532*B*C^2*a^13*b^6 + 1380*B*C^2*a^14*b^5 + 1404*B*C^2*a^15*b^4 - 816*B*C^2*a^16*b^3 - 336*B*C^

2*a^17*b^2 + 384*B^2*C*a^4*b^15 - 192*B^2*C*a^5*b^14 - 2496*B^2*C*a^6*b^13 + 1152*B^2*C*a^7*b^12 + 6816*B^2*C*

a^8*b^11 - 3264*B^2*C*a^9*b^10 - 10464*B^2*C*a^10*b^9 + 5298*B^2*C*a^11*b^8 + 9696*B^2*C*a^12*b^7 - 4752*B^2*C

*a^13*b^6 - 5088*B^2*C*a^14*b^5 + 2208*B^2*C*a^15*b^4 + 1152*B^2*C*a^16*b^3 - 1920*A*B*C*a^3*b^16 + 960*A*B*C*

a^4*b^15 + 12384*A*B*C*a^5*b^14 - 5712*A*B*C*a^6*b^13 - 33456*A*B*C*a^7*b^12 + 15852*A*B*C*a^8*b^11 + 50436*A*

B*C*a^9*b^10 - 25034*A*B*C*a^10*b^9 - 45404*A*B*C*a^11*b^8 + 21788*A*B*C*a^12*b^7 + 22716*A*B*C*a^13*b^6 - 929

2*A*B*C*a^14*b^5 - 4548*A*B*C*a^15*b^4 - 112*A*B*C*a^16*b^3 - 208*A*B*C*a^17*b^2))/(a^25*b + a^26 - a^15*b^11

- a^16*b^10 + 5*a^17*b^9 + 5*a^18*b^8 - 10*a^19*b^7 - 10*a^20*b^6 + 10*a^21*b^5 + 10*a^22*b^4 - 5*a^23*b^3 - 5

*a^24*b^2) - (((8*tan(c/2 + (d*x)/2)*(A^2*a^18 + 800*A^2*b^18 + 4*C^2*a^18 - 800*A^2*a*b^17 - 2*A^2*a^17*b - 8

*C^2*a^17*b - 4720*A^2*a^2*b^16 + 4720*A^2*a^3*b^15 + 11522*A^2*a^4*b^14 - 11522*A^2*a^5*b^13 - 14837*A^2*a^6*

b^12 + 14812*A^2*a^7*b^11 + 10385*A^2*a^8*b^10 - 10430*A^2*a^9*b^9 - 3325*A^2*a^10*b^8 + 3640*A^2*a^11*b^7 - 4

5*A^2*a^12*b^6 - 350*A^2*a^13*b^5 + 209*A^2*a^14*b^4 - 68*A^2*a^15*b^3 + 35*A^2*a^16*b^2 + 128*B^2*a^2*b^16 -

128*B^2*a^3*b^15 - 768*B^2*a^4*b^14 + 768*B^2*a^5*b^13 + 1920*B^2*a^6*b^12 - 1920*B^2*a^7*b^11 - 2600*B^2*a^8*

b^10 + 2560*B^2*a^9*b^9 + 2025*B^2*a^10*b^8 - 1920*B^2*a^11*b^7 - 824*B^2*a^12*b^6 + 768*B^2*a^13*b^5 + 80*B^2

*a^14*b^4 - 128*B^2*a^15*b^3 + 64*B^2*a^16*b^2 + 8*C^2*a^4*b^14 - 8*C^2*a^5*b^13 - 48*C^2*a^6*b^12 + 48*C^2*a^

7*b^11 + 117*C^2*a^8*b^10 - 120*C^2*a^9*b^9 - 164*C^2*a^10*b^8 + 160*C^2*a^11*b^7 + 156*C^2*a^12*b^6 - 120*C^2

*a^13*b^5 - 92*C^2*a^14*b^4 + 48*C^2*a^15*b^3 + 44*C^2*a^16*b^2 + 4*A*C*a^18 - 640*A*B*a*b^17 - 16*A*B*a^17*b

- 8*A*C*a^17*b - 32*B*C*a^17*b + 640*A*B*a^2*b^16 + 3808*A*B*a^3*b^15 - 3808*A*B*a^4*b^14 - 9408*A*B*a^5*b^13

+ 9408*A*B*a^6*b^12 + 12430*A*B*a^7*b^11 - 12320*A*B*a^8*b^10 - 9200*A*B*a^9*b^9 + 8960*A*B*a^10*b^8 + 3360*A*

B*a^11*b^7 - 3360*A*B*a^12*b^6 - 144*A*B*a^13*b^5 + 448*A*B*a^14*b^4 - 240*A*B*a^15*b^3 + 32*A*B*a^16*b^2 + 16

0*A*C*a^2*b^16 - 160*A*C*a^3*b^15 - 952*A*C*a^4*b^14 + 952*A*C*a^5*b^13 + 2322*A*C*a^6*b^12 - 2352*A*C*a^7*b^1

1 - 3124*A*C*a^8*b^10 + 3080*A*C*a^9*b^9 + 2588*A*C*a^10*b^8 - 2240*A*C*a^11*b^7 - 1284*A*C*a^12*b^6 + 840*A*C

*a^13*b^5 + 276*A*C*a^14*b^4 - 112*A*C*a^15*b^3 + 60*A*C*a^16*b^2 - 64*B*C*a^3*b^15 + 64*B*C*a^4*b^14 + 384*B*

C*a^5*b^13 - 384*B*C*a^6*b^12 - 948*B*C*a^7*b^11 + 960*B*C*a^8*b^10 + 1306*B*C*a^9*b^9 - 1280*B*C*a^10*b^8 - 1

128*B*C*a^11*b^7 + 960*B*C*a^12*b^6 + 592*B*C*a^13*b^5 - 384*B*C*a^14*b^4 - 160*B*C*a^15*b^3 + 64*B*C*a^16*b^2

))/(a^20*b + a^21 - a^10*b^11 - a^11*b^10 + 5*a^12*b^9 + 5*a^13*b^8 - 10*a^14*b^7 - 10*a^15*b^6 + 10*a^16*b^5

+ 10*a^17*b^4 - 5*a^18*b^3 - 5*a^19*b^2) + (((4*(4*A*a^27 + 8*C*a^27 - 80*A*a^12*b^15 + 40*A*a^13*b^14 + 516*A

*a^14*b^13 - 248*A*a^15*b^12 - 1404*A*a^16*b^11 + 640*A*a^17*b^10 + 2076*A*a^18*b^9 - 896*A*a^19*b^8 - 1764*A*

a^20*b^7 + 724*A*a^21*b^6 + 816*A*a^22*b^5 - 316*A*a^23*b^4 - 160*A*a^24*b^3 + 52*A*a^25*b^2 + 32*B*a^13*b^14

- 16*B*a^14*b^13 - 208*B*a^15*b^12 + 100*B*a^16*b^11 + 572*B*a^17*b^10 - 252*B*a^18*b^9 - 868*B*a^19*b^8 + 348

*B*a^20*b^7 + 772*B*a^21*b^6 - 292*B*a^22*b^5 - 380*B*a^23*b^4 + 144*B*a^24*b^3 + 80*B*a^25*b^2 - 8*C*a^14*b^1

3 + 4*C*a^15*b^12 + 52*C*a^16*b^11 - 28*C*a^17*b^10 - 140*C*a^18*b^9 + 60*C*a^19*b^8 + 220*C*a^20*b^7 - 60*C*a

^21*b^6 - 220*C*a^22*b^5 + 40*C*a^23*b^4 + 128*C*a^24*b^3 - 24*C*a^25*b^2 - 32*B*a^26*b - 32*C*a^26*b))/(a^25*

b + a^26 - a^15*b^11 - a^16*b^10 + 5*a^17*b^9 + 5*a^18*b^8 - 10*a^19*b^7 - 10*a^20*b^6 + 10*a^21*b^5 + 10*a^22

*b^4 - 5*a^23*b^3 - 5*a^24*b^2) - (8*tan(c/2 + (d*x)/2)*(A*b^2*10i + a^2*((A*1i)/2 + C*1i) - B*a*b*4i)*(8*a^25

*b - 8*a^12*b^14 + 8*a^13*b^13 + 48*a^14*b^12 - 48*a^15*b^11 - 120*a^16*b^10 + 120*a^17*b^9 + 160*a^18*b^8 - 1

60*a^19*b^7 - 120*a^20*b^6 + 120*a^21*b^5 + 48*a^22*b^4 - 48*a^23*b^3 - 8*a^24*b^2))/(a^6*(a^20*b + a^21 - a^1

0*b^11 - a^11*b^10 + 5*a^12*b^9 + 5*a^13*b^8 - 10*a^14*b^7 - 10*a^15*b^6 + 10*a^16*b^5 + 10*a^17*b^4 - 5*a^18*

b^3 - 5*a^19*b^2)))*(A*b^2*10i + a^2*((A*1i)/2 + C*1i) - B*a*b*4i))/a^6)*(A*b^2*10i + a^2*((A*1i)/2 + C*1i) -

B*a*b*4i))/a^6 + (((8*tan(c/2 + (d*x)/2)*(A^2*a^18 + 800*A^2*b^18 + 4*C^2*a^18 - 800*A^2*a*b^17 - 2*A^2*a^17*b

- 8*C^2*a^17*b - 4720*A^2*a^2*b^16 + 4720*A^2*a^3*b^15 + 11522*A^2*a^4*b^14 - 11522*A^2*a^5*b^13 - 14837*A^2*

a^6*b^12 + 14812*A^2*a^7*b^11 + 10385*A^2*a^8*b^10 - 10430*A^2*a^9*b^9 - 3325*A^2*a^10*b^8 + 3640*A^2*a^11*b^7

- 45*A^2*a^12*b^6 - 350*A^2*a^13*b^5 + 209*A^2*a^14*b^4 - 68*A^2*a^15*b^3 + 35*A^2*a^16*b^2 + 128*B^2*a^2*b^1

6 - 128*B^2*a^3*b^15 - 768*B^2*a^4*b^14 + 768*B^2*a^5*b^13 + 1920*B^2*a^6*b^12 - 1920*B^2*a^7*b^11 - 2600*B^2*

a^8*b^10 + 2560*B^2*a^9*b^9 + 2025*B^2*a^10*b^8 - 1920*B^2*a^11*b^7 - 824*B^2*a^12*b^6 + 768*B^2*a^13*b^5 + 80

*B^2*a^14*b^4 - 128*B^2*a^15*b^3 + 64*B^2*a^16*b^2 + 8*C^2*a^4*b^14 - 8*C^2*a^5*b^13 - 48*C^2*a^6*b^12 + 48*C^

2*a^7*b^11 + 117*C^2*a^8*b^10 - 120*C^2*a^9*b^9 - 164*C^2*a^10*b^8 + 160*C^2*a^11*b^7 + 156*C^2*a^12*b^6 - 120

*C^2*a^13*b^5 - 92*C^2*a^14*b^4 + 48*C^2*a^15*b^3 + 44*C^2*a^16*b^2 + 4*A*C*a^18 - 640*A*B*a*b^17 - 16*A*B*a^1

7*b - 8*A*C*a^17*b - 32*B*C*a^17*b + 640*A*B*a^2*b^16 + 3808*A*B*a^3*b^15 - 3808*A*B*a^4*b^14 - 9408*A*B*a^5*b

^13 + 9408*A*B*a^6*b^12 + 12430*A*B*a^7*b^11 - 12320*A*B*a^8*b^10 - 9200*A*B*a^9*b^9 + 8960*A*B*a^10*b^8 + 336

0*A*B*a^11*b^7 - 3360*A*B*a^12*b^6 - 144*A*B*a^13*b^5 + 448*A*B*a^14*b^4 - 240*A*B*a^15*b^3 + 32*A*B*a^16*b^2

+ 160*A*C*a^2*b^16 - 160*A*C*a^3*b^15 - 952*A*C*a^4*b^14 + 952*A*C*a^5*b^13 + 2322*A*C*a^6*b^12 - 2352*A*C*a^7

*b^11 - 3124*A*C*a^8*b^10 + 3080*A*C*a^9*b^9 + 2588*A*C*a^10*b^8 - 2240*A*C*a^11*b^7 - 1284*A*C*a^12*b^6 + 840

*A*C*a^13*b^5 + 276*A*C*a^14*b^4 - 112*A*C*a^15*b^3 + 60*A*C*a^16*b^2 - 64*B*C*a^3*b^15 + 64*B*C*a^4*b^14 + 38

4*B*C*a^5*b^13 - 384*B*C*a^6*b^12 - 948*B*C*a^7*b^11 + 960*B*C*a^8*b^10 + 1306*B*C*a^9*b^9 - 1280*B*C*a^10*b^8

- 1128*B*C*a^11*b^7 + 960*B*C*a^12*b^6 + 592*B*C*a^13*b^5 - 384*B*C*a^14*b^4 - 160*B*C*a^15*b^3 + 64*B*C*a^16

*b^2))/(a^20*b + a^21 - a^10*b^11 - a^11*b^10 + 5*a^12*b^9 + 5*a^13*b^8 - 10*a^14*b^7 - 10*a^15*b^6 + 10*a^16*

b^5 + 10*a^17*b^4 - 5*a^18*b^3 - 5*a^19*b^2) - (((4*(4*A*a^27 + 8*C*a^27 - 80*A*a^12*b^15 + 40*A*a^13*b^14 + 5

16*A*a^14*b^13 - 248*A*a^15*b^12 - 1404*A*a^16*b^11 + 640*A*a^17*b^10 + 2076*A*a^18*b^9 - 896*A*a^19*b^8 - 176

4*A*a^20*b^7 + 724*A*a^21*b^6 + 816*A*a^22*b^5 - 316*A*a^23*b^4 - 160*A*a^24*b^3 + 52*A*a^25*b^2 + 32*B*a^13*b

^14 - 16*B*a^14*b^13 - 208*B*a^15*b^12 + 100*B*a^16*b^11 + 572*B*a^17*b^10 - 252*B*a^18*b^9 - 868*B*a^19*b^8 +

348*B*a^20*b^7 + 772*B*a^21*b^6 - 292*B*a^22*b^5 - 380*B*a^23*b^4 + 144*B*a^24*b^3 + 80*B*a^25*b^2 - 8*C*a^14

*b^13 + 4*C*a^15*b^12 + 52*C*a^16*b^11 - 28*C*a^17*b^10 - 140*C*a^18*b^9 + 60*C*a^19*b^8 + 220*C*a^20*b^7 - 60

*C*a^21*b^6 - 220*C*a^22*b^5 + 40*C*a^23*b^4 + 128*C*a^24*b^3 - 24*C*a^25*b^2 - 32*B*a^26*b - 32*C*a^26*b))/(a

^25*b + a^26 - a^15*b^11 - a^16*b^10 + 5*a^17*b^9 + 5*a^18*b^8 - 10*a^19*b^7 - 10*a^20*b^6 + 10*a^21*b^5 + 10*

a^22*b^4 - 5*a^23*b^3 - 5*a^24*b^2) + (8*tan(c/2 + (d*x)/2)*(A*b^2*10i + a^2*((A*1i)/2 + C*1i) - B*a*b*4i)*(8*

a^25*b - 8*a^12*b^14 + 8*a^13*b^13 + 48*a^14*b^12 - 48*a^15*b^11 - 120*a^16*b^10 + 120*a^17*b^9 + 160*a^18*b^8

- 160*a^19*b^7 - 120*a^20*b^6 + 120*a^21*b^5 + 48*a^22*b^4 - 48*a^23*b^3 - 8*a^24*b^2))/(a^6*(a^20*b + a^21 -

a^10*b^11 - a^11*b^10 + 5*a^12*b^9 + 5*a^13*b^8 - 10*a^14*b^7 - 10*a^15*b^6 + 10*a^16*b^5 + 10*a^17*b^4 - 5*a

^18*b^3 - 5*a^19*b^2)))*(A*b^2*10i + a^2*((A*1i)/2 + C*1i) - B*a*b*4i))/a^6)*(A*b^2*10i + a^2*((A*1i)/2 + C*1i

) - B*a*b*4i))/a^6))*(A*b^2*10i + a^2*((A*1i)/2 + C*1i) - B*a*b*4i)*2i)/(a^6*d) + (b*atan(((b*((8*tan(c/2 + (d

*x)/2)*(A^2*a^18 + 800*A^2*b^18 + 4*C^2*a^18 - 800*A^2*a*b^17 - 2*A^2*a^17*b - 8*C^2*a^17*b - 4720*A^2*a^2*b^1

6 + 4720*A^2*a^3*b^15 + 11522*A^2*a^4*b^14 - 11522*A^2*a^5*b^13 - 14837*A^2*a^6*b^12 + 14812*A^2*a^7*b^11 + 10

385*A^2*a^8*b^10 - 10430*A^2*a^9*b^9 - 3325*A^2*a^10*b^8 + 3640*A^2*a^11*b^7 - 45*A^2*a^12*b^6 - 350*A^2*a^13*

b^5 + 209*A^2*a^14*b^4 - 68*A^2*a^15*b^3 + 35*A^2*a^16*b^2 + 128*B^2*a^2*b^16 - 128*B^2*a^3*b^15 - 768*B^2*a^4

*b^14 + 768*B^2*a^5*b^13 + 1920*B^2*a^6*b^12 - 1920*B^2*a^7*b^11 - 2600*B^2*a^8*b^10 + 2560*B^2*a^9*b^9 + 2025

*B^2*a^10*b^8 - 1920*B^2*a^11*b^7 - 824*B^2*a^12*b^6 + 768*B^2*a^13*b^5 + 80*B^2*a^14*b^4 - 128*B^2*a^15*b^3 +

64*B^2*a^16*b^2 + 8*C^2*a^4*b^14 - 8*C^2*a^5*b^13 - 48*C^2*a^6*b^12 + 48*C^2*a^7*b^11 + 117*C^2*a^8*b^10 - 12

0*C^2*a^9*b^9 - 164*C^2*a^10*b^8 + 160*C^2*a^11*b^7 + 156*C^2*a^12*b^6 - 120*C^2*a^13*b^5 - 92*C^2*a^14*b^4 +

48*C^2*a^15*b^3 + 44*C^2*a^16*b^2 + 4*A*C*a^18 - 640*A*B*a*b^17 - 16*A*B*a^17*b - 8*A*C*a^17*b - 32*B*C*a^17*b

+ 640*A*B*a^2*b^16 + 3808*A*B*a^3*b^15 - 3808*A*B*a^4*b^14 - 9408*A*B*a^5*b^13 + 9408*A*B*a^6*b^12 + 12430*A*

B*a^7*b^11 - 12320*A*B*a^8*b^10 - 9200*A*B*a^9*b^9 + 8960*A*B*a^10*b^8 + 3360*A*B*a^11*b^7 - 3360*A*B*a^12*b^6

- 144*A*B*a^13*b^5 + 448*A*B*a^14*b^4 - 240*A*B*a^15*b^3 + 32*A*B*a^16*b^2 + 160*A*C*a^2*b^16 - 160*A*C*a^3*b

^15 - 952*A*C*a^4*b^14 + 952*A*C*a^5*b^13 + 2322*A*C*a^6*b^12 - 2352*A*C*a^7*b^11 - 3124*A*C*a^8*b^10 + 3080*A

*C*a^9*b^9 + 2588*A*C*a^10*b^8 - 2240*A*C*a^11*b^7 - 1284*A*C*a^12*b^6 + 840*A*C*a^13*b^5 + 276*A*C*a^14*b^4 -

112*A*C*a^15*b^3 + 60*A*C*a^16*b^2 - 64*B*C*a^3*b^15 + 64*B*C*a^4*b^14 + 384*B*C*a^5*b^13 - 384*B*C*a^6*b^12

- 948*B*C*a^7*b^11 + 960*B*C*a^8*b^10 + 1306*B*C*a^9*b^9 - 1280*B*C*a^10*b^8 - 1128*B*C*a^11*b^7 + 960*B*C*a^1

2*b^6 + 592*B*C*a^13*b^5 - 384*B*C*a^14*b^4 - 160*B*C*a^15*b^3 + 64*B*C*a^16*b^2))/(a^20*b + a^21 - a^10*b^11

- a^11*b^10 + 5*a^12*b^9 + 5*a^13*b^8 - 10*a^14*b^7 - 10*a^15*b^6 + 10*a^16*b^5 + 10*a^17*b^4 - 5*a^18*b^3 - 5

*a^19*b^2) + (b*((a + b)^7*(a - b)^7)^(1/2)*((4*(4*A*a^27 + 8*C*a^27 - 80*A*a^12*b^15 + 40*A*a^13*b^14 + 516*A

*a^14*b^13 - 248*A*a^15*b^12 - 1404*A*a^16*b^11 + 640*A*a^17*b^10 + 2076*A*a^18*b^9 - 896*A*a^19*b^8 - 1764*A*

a^20*b^7 + 724*A*a^21*b^6 + 816*A*a^22*b^5 - 316*A*a^23*b^4 - 160*A*a^24*b^3 + 52*A*a^25*b^2 + 32*B*a^13*b^14

- 16*B*a^14*b^13 - 208*B*a^15*b^12 + 100*B*a^16*b^11 + 572*B*a^17*b^10 - 252*B*a^18*b^9 - 868*B*a^19*b^8 + 348

*B*a^20*b^7 + 772*B*a^21*b^6 - 292*B*a^22*b^5 - 380*B*a^23*b^4 + 144*B*a^24*b^3 + 80*B*a^25*b^2 - 8*C*a^14*b^1

3 + 4*C*a^15*b^12 + 52*C*a^16*b^11 - 28*C*a^17*b^10 - 140*C*a^18*b^9 + 60*C*a^19*b^8 + 220*C*a^20*b^7 - 60*C*a

^21*b^6 - 220*C*a^22*b^5 + 40*C*a^23*b^4 + 128*C*a^24*b^3 - 24*C*a^25*b^2 - 32*B*a^26*b - 32*C*a^26*b))/(a^25*

b + a^26 - a^15*b^11 - a^16*b^10 + 5*a^17*b^9 + 5*a^18*b^8 - 10*a^19*b^7 - 10*a^20*b^6 + 10*a^21*b^5 + 10*a^22

*b^4 - 5*a^23*b^3 - 5*a^24*b^2) - (4*b*tan(c/2 + (d*x)/2)*((a + b)^7*(a - b)^7)^(1/2)*(20*A*b^8 - 8*C*a^8 - 69

*A*a^2*b^6 + 84*A*a^4*b^4 - 40*A*a^6*b^2 + 28*B*a^3*b^5 - 35*B*a^5*b^3 + 2*C*a^2*b^6 - 7*C*a^4*b^4 + 8*C*a^6*b

^2 - 8*B*a*b^7 + 20*B*a^7*b)*(8*a^25*b - 8*a^12*b^14 + 8*a^13*b^13 + 48*a^14*b^12 - 48*a^15*b^11 - 120*a^16*b^

10 + 120*a^17*b^9 + 160*a^18*b^8 - 160*a^19*b^7 - 120*a^20*b^6 + 120*a^21*b^5 + 48*a^22*b^4 - 48*a^23*b^3 - 8*

a^24*b^2))/((a^20 - a^6*b^14 + 7*a^8*b^12 - 21*a^10*b^10 + 35*a^12*b^8 - 35*a^14*b^6 + 21*a^16*b^4 - 7*a^18*b^

2)*(a^20*b + a^21 - a^10*b^11 - a^11*b^10 + 5*a^12*b^9 + 5*a^13*b^8 - 10*a^14*b^7 - 10*a^15*b^6 + 10*a^16*b^5

+ 10*a^17*b^4 - 5*a^18*b^3 - 5*a^19*b^2)))*(20*A*b^8 - 8*C*a^8 - 69*A*a^2*b^6 + 84*A*a^4*b^4 - 40*A*a^6*b^2 +

28*B*a^3*b^5 - 35*B*a^5*b^3 + 2*C*a^2*b^6 - 7*C*a^4*b^4 + 8*C*a^6*b^2 - 8*B*a*b^7 + 20*B*a^7*b))/(2*(a^20 - a^

6*b^14 + 7*a^8*b^12 - 21*a^10*b^10 + 35*a^12*b^8 - 35*a^14*b^6 + 21*a^16*b^4 - 7*a^18*b^2)))*((a + b)^7*(a - b

)^7)^(1/2)*(20*A*b^8 - 8*C*a^8 - 69*A*a^2*b^6 + 84*A*a^4*b^4 - 40*A*a^6*b^2 + 28*B*a^3*b^5 - 35*B*a^5*b^3 + 2*

C*a^2*b^6 - 7*C*a^4*b^4 + 8*C*a^6*b^2 - 8*B*a*b^7 + 20*B*a^7*b)*1i)/(2*(a^20 - a^6*b^14 + 7*a^8*b^12 - 21*a^10

*b^10 + 35*a^12*b^8 - 35*a^14*b^6 + 21*a^16*b^4 - 7*a^18*b^2)) + (b*((8*tan(c/2 + (d*x)/2)*(A^2*a^18 + 800*A^2

*b^18 + 4*C^2*a^18 - 800*A^2*a*b^17 - 2*A^2*a^17*b - 8*C^2*a^17*b - 4720*A^2*a^2*b^16 + 4720*A^2*a^3*b^15 + 11

522*A^2*a^4*b^14 - 11522*A^2*a^5*b^13 - 14837*A^2*a^6*b^12 + 14812*A^2*a^7*b^11 + 10385*A^2*a^8*b^10 - 10430*A

^2*a^9*b^9 - 3325*A^2*a^10*b^8 + 3640*A^2*a^11*b^7 - 45*A^2*a^12*b^6 - 350*A^2*a^13*b^5 + 209*A^2*a^14*b^4 - 6

8*A^2*a^15*b^3 + 35*A^2*a^16*b^2 + 128*B^2*a^2*b^16 - 128*B^2*a^3*b^15 - 768*B^2*a^4*b^14 + 768*B^2*a^5*b^13 +

1920*B^2*a^6*b^12 - 1920*B^2*a^7*b^11 - 2600*B^2*a^8*b^10 + 2560*B^2*a^9*b^9 + 2025*B^2*a^10*b^8 - 1920*B^2*a

^11*b^7 - 824*B^2*a^12*b^6 + 768*B^2*a^13*b^5 + 80*B^2*a^14*b^4 - 128*B^2*a^15*b^3 + 64*B^2*a^16*b^2 + 8*C^2*a

^4*b^14 - 8*C^2*a^5*b^13 - 48*C^2*a^6*b^12 + 48*C^2*a^7*b^11 + 117*C^2*a^8*b^10 - 120*C^2*a^9*b^9 - 164*C^2*a^

10*b^8 + 160*C^2*a^11*b^7 + 156*C^2*a^12*b^6 - 120*C^2*a^13*b^5 - 92*C^2*a^14*b^4 + 48*C^2*a^15*b^3 + 44*C^2*a

^16*b^2 + 4*A*C*a^18 - 640*A*B*a*b^17 - 16*A*B*a^17*b - 8*A*C*a^17*b - 32*B*C*a^17*b + 640*A*B*a^2*b^16 + 3808

*A*B*a^3*b^15 - 3808*A*B*a^4*b^14 - 9408*A*B*a^5*b^13 + 9408*A*B*a^6*b^12 + 12430*A*B*a^7*b^11 - 12320*A*B*a^8

*b^10 - 9200*A*B*a^9*b^9 + 8960*A*B*a^10*b^8 + 3360*A*B*a^11*b^7 - 3360*A*B*a^12*b^6 - 144*A*B*a^13*b^5 + 448*

A*B*a^14*b^4 - 240*A*B*a^15*b^3 + 32*A*B*a^16*b^2 + 160*A*C*a^2*b^16 - 160*A*C*a^3*b^15 - 952*A*C*a^4*b^14 + 9

52*A*C*a^5*b^13 + 2322*A*C*a^6*b^12 - 2352*A*C*a^7*b^11 - 3124*A*C*a^8*b^10 + 3080*A*C*a^9*b^9 + 2588*A*C*a^10

*b^8 - 2240*A*C*a^11*b^7 - 1284*A*C*a^12*b^6 + 840*A*C*a^13*b^5 + 276*A*C*a^14*b^4 - 112*A*C*a^15*b^3 + 60*A*C

*a^16*b^2 - 64*B*C*a^3*b^15 + 64*B*C*a^4*b^14 + 384*B*C*a^5*b^13 - 384*B*C*a^6*b^12 - 948*B*C*a^7*b^11 + 960*B

*C*a^8*b^10 + 1306*B*C*a^9*b^9 - 1280*B*C*a^10*b^8 - 1128*B*C*a^11*b^7 + 960*B*C*a^12*b^6 + 592*B*C*a^13*b^5 -

384*B*C*a^14*b^4 - 160*B*C*a^15*b^3 + 64*B*C*a^16*b^2))/(a^20*b + a^21 - a^10*b^11 - a^11*b^10 + 5*a^12*b^9 +

5*a^13*b^8 - 10*a^14*b^7 - 10*a^15*b^6 + 10*a^16*b^5 + 10*a^17*b^4 - 5*a^18*b^3 - 5*a^19*b^2) - (b*((a + b)^7

*(a - b)^7)^(1/2)*((4*(4*A*a^27 + 8*C*a^27 - 80*A*a^12*b^15 + 40*A*a^13*b^14 + 516*A*a^14*b^13 - 248*A*a^15*b^

12 - 1404*A*a^16*b^11 + 640*A*a^17*b^10 + 2076*A*a^18*b^9 - 896*A*a^19*b^8 - 1764*A*a^20*b^7 + 724*A*a^21*b^6

+ 816*A*a^22*b^5 - 316*A*a^23*b^4 - 160*A*a^24*b^3 + 52*A*a^25*b^2 + 32*B*a^13*b^14 - 16*B*a^14*b^13 - 208*B*a

^15*b^12 + 100*B*a^16*b^11 + 572*B*a^17*b^10 - 252*B*a^18*b^9 - 868*B*a^19*b^8 + 348*B*a^20*b^7 + 772*B*a^21*b

^6 - 292*B*a^22*b^5 - 380*B*a^23*b^4 + 144*B*a^24*b^3 + 80*B*a^25*b^2 - 8*C*a^14*b^13 + 4*C*a^15*b^12 + 52*C*a

^16*b^11 - 28*C*a^17*b^10 - 140*C*a^18*b^9 + 60*C*a^19*b^8 + 220*C*a^20*b^7 - 60*C*a^21*b^6 - 220*C*a^22*b^5 +

40*C*a^23*b^4 + 128*C*a^24*b^3 - 24*C*a^25*b^2 - 32*B*a^26*b - 32*C*a^26*b))/(a^25*b + a^26 - a^15*b^11 - a^1

6*b^10 + 5*a^17*b^9 + 5*a^18*b^8 - 10*a^19*b^7 - 10*a^20*b^6 + 10*a^21*b^5 + 10*a^22*b^4 - 5*a^23*b^3 - 5*a^24

*b^2) + (4*b*tan(c/2 + (d*x)/2)*((a + b)^7*(a - b)^7)^(1/2)*(20*A*b^8 - 8*C*a^8 - 69*A*a^2*b^6 + 84*A*a^4*b^4

- 40*A*a^6*b^2 + 28*B*a^3*b^5 - 35*B*a^5*b^3 + 2*C*a^2*b^6 - 7*C*a^4*b^4 + 8*C*a^6*b^2 - 8*B*a*b^7 + 20*B*a^7*

b)*(8*a^25*b - 8*a^12*b^14 + 8*a^13*b^13 + 48*a^14*b^12 - 48*a^15*b^11 - 120*a^16*b^10 + 120*a^17*b^9 + 160*a^

18*b^8 - 160*a^19*b^7 - 120*a^20*b^6 + 120*a^21*b^5 + 48*a^22*b^4 - 48*a^23*b^3 - 8*a^24*b^2))/((a^20 - a^6*b^

14 + 7*a^8*b^12 - 21*a^10*b^10 + 35*a^12*b^8 - 35*a^14*b^6 + 21*a^16*b^4 - 7*a^18*b^2)*(a^20*b + a^21 - a^10*b

^11 - a^11*b^10 + 5*a^12*b^9 + 5*a^13*b^8 - 10*a^14*b^7 - 10*a^15*b^6 + 10*a^16*b^5 + 10*a^17*b^4 - 5*a^18*b^3

- 5*a^19*b^2)))*(20*A*b^8 - 8*C*a^8 - 69*A*a^2*b^6 + 84*A*a^4*b^4 - 40*A*a^6*b^2 + 28*B*a^3*b^5 - 35*B*a^5*b^

3 + 2*C*a^2*b^6 - 7*C*a^4*b^4 + 8*C*a^6*b^2 - 8*B*a*b^7 + 20*B*a^7*b))/(2*(a^20 - a^6*b^14 + 7*a^8*b^12 - 21*a

^10*b^10 + 35*a^12*b^8 - 35*a^14*b^6 + 21*a^16*b^4 - 7*a^18*b^2)))*((a + b)^7*(a - b)^7)^(1/2)*(20*A*b^8 - 8*C

*a^8 - 69*A*a^2*b^6 + 84*A*a^4*b^4 - 40*A*a^6*b^2 + 28*B*a^3*b^5 - 35*B*a^5*b^3 + 2*C*a^2*b^6 - 7*C*a^4*b^4 +

8*C*a^6*b^2 - 8*B*a*b^7 + 20*B*a^7*b)*1i)/(2*(a^20 - a^6*b^14 + 7*a^8*b^12 - 21*a^10*b^10 + 35*a^12*b^8 - 35*a

^14*b^6 + 21*a^16*b^4 - 7*a^18*b^2)))/((8*(8000*A^3*b^19 - 4000*A^3*a*b^18 + 32*C^3*a^18*b - 50800*A^3*a^2*b^1

7 + 24400*A^3*a^3*b^16 + 135260*A^3*a^4*b^15 - 62030*A^3*a^5*b^14 - 193689*A^3*a^6*b^13 + 82337*A^3*a^7*b^12 +

155991*A^3*a^8*b^11 - 57345*A^3*a^9*b^10 - 64479*A^3*a^10*b^9 + 16999*A^3*a^11*b^8 + 8281*A^3*a^12*b^7 + 204*

A^3*a^13*b^6 + 1396*A^3*a^14*b^5 - 40*A^3*a^15*b^4 + 40*A^3*a^16*b^3 - 512*B^3*a^3*b^16 + 256*B^3*a^4*b^15 + 3

328*B^3*a^5*b^14 - 1600*B^3*a^6*b^13 - 9152*B^3*a^7*b^12 + 4352*B^3*a^8*b^11 + 13888*B^3*a^9*b^10 - 6408*B^3*a

^10*b^9 - 12352*B^3*a^11*b^8 + 5120*B^3*a^12*b^7 + 6080*B^3*a^13*b^6 - 1920*B^3*a^14*b^5 - 1280*B^3*a^15*b^4 +

8*C^3*a^6*b^13 - 4*C^3*a^7*b^12 - 52*C^3*a^8*b^11 + 22*C^3*a^9*b^10 + 140*C^3*a^10*b^9 - 68*C^3*a^11*b^8 - 22

0*C^3*a^12*b^7 + 132*C^3*a^13*b^6 + 220*C^3*a^14*b^5 - 128*C^3*a^15*b^4 - 128*C^3*a^16*b^3 + 96*C^3*a^17*b^2 -

9600*A^2*B*a*b^18 + 32*A*C^2*a^18*b + 8*A^2*C*a^18*b + 3840*A*B^2*a^2*b^17 - 1920*A*B^2*a^3*b^16 - 24768*A*B^

2*a^4*b^15 + 11904*A*B^2*a^5*b^14 + 67392*A*B^2*a^6*b^13 - 31680*A*B^2*a^7*b^12 - 100368*A*B^2*a^8*b^11 + 4514

8*A*B^2*a^9*b^10 + 86512*A*B^2*a^10*b^9 - 34567*A*B^2*a^11*b^8 - 40368*A*B^2*a^12*b^7 + 11960*A*B^2*a^13*b^6 +

7440*A*B^2*a^14*b^5 + 80*A*B^2*a^15*b^4 + 320*A*B^2*a^16*b^3 + 4800*A^2*B*a^2*b^17 + 61440*A^2*B*a^3*b^16 - 2

9520*A^2*B*a^4*b^15 - 165384*A^2*B*a^5*b^14 + 76812*A^2*B*a^6*b^13 + 241596*A^2*B*a^7*b^12 - 105755*A^2*B*a^8*

b^11 - 201479*A^2*B*a^9*b^10 + 77359*A^2*B*a^10*b^9 + 88721*A^2*B*a^11*b^8 - 24711*A^2*B*a^12*b^7 - 13929*A^2*

B*a^13*b^6 - 255*A^2*B*a^14*b^5 - 1345*A^2*B*a^15*b^4 + 20*A^2*B*a^16*b^3 - 20*A^2*B*a^17*b^2 + 240*A*C^2*a^4*

b^15 - 120*A*C^2*a^5*b^14 - 1548*A*C^2*a^6*b^13 + 684*A*C^2*a^7*b^12 + 4152*A*C^2*a^8*b^11 - 1983*A*C^2*a^9*b^

10 - 6336*A*C^2*a^10*b^9 + 3448*A*C^2*a^11*b^8 + 5944*A*C^2*a^12*b^7 - 3196*A*C^2*a^13*b^6 - 3156*A*C^2*a^14*b

^5 + 1760*A*C^2*a^15*b^4 + 672*A*C^2*a^16*b^3 + 32*A*C^2*a^17*b^2 + 2400*A^2*C*a^2*b^17 - 1200*A^2*C*a^3*b^16

- 15360*A^2*C*a^4*b^15 + 7080*A^2*C*a^5*b^14 + 41046*A^2*C*a^6*b^13 - 19233*A^2*C*a^7*b^12 - 60729*A^2*C*a^8*b

^11 + 29513*A^2*C*a^9*b^10 + 53039*A^2*C*a^10*b^9 - 24901*A^2*C*a^11*b^8 - 25211*A^2*C*a^12*b^7 + 9657*A^2*C*a

^13*b^6 + 4359*A^2*C*a^14*b^5 + 192*A^2*C*a^15*b^4 + 448*A^2*C*a^16*b^3 - 8*A^2*C*a^17*b^2 - 96*B*C^2*a^5*b^14

+ 48*B*C^2*a^6*b^13 + 624*B*C^2*a^7*b^12 - 276*B*C^2*a^8*b^11 - 1692*B*C^2*a^9*b^10 + 816*B*C^2*a^10*b^9 + 26

28*B*C^2*a^11*b^8 - 1452*B*C^2*a^12*b^7 - 2532*B*C^2*a^13*b^6 + 1380*B*C^2*a^14*b^5 + 1404*B*C^2*a^15*b^4 - 81

6*B*C^2*a^16*b^3 - 336*B*C^2*a^17*b^2 + 384*B^2*C*a^4*b^15 - 192*B^2*C*a^5*b^14 - 2496*B^2*C*a^6*b^13 + 1152*B

^2*C*a^7*b^12 + 6816*B^2*C*a^8*b^11 - 3264*B^2*C*a^9*b^10 - 10464*B^2*C*a^10*b^9 + 5298*B^2*C*a^11*b^8 + 9696*

B^2*C*a^12*b^7 - 4752*B^2*C*a^13*b^6 - 5088*B^2*C*a^14*b^5 + 2208*B^2*C*a^15*b^4 + 1152*B^2*C*a^16*b^3 - 1920*

A*B*C*a^3*b^16 + 960*A*B*C*a^4*b^15 + 12384*A*B*C*a^5*b^14 - 5712*A*B*C*a^6*b^13 - 33456*A*B*C*a^7*b^12 + 1585

2*A*B*C*a^8*b^11 + 50436*A*B*C*a^9*b^10 - 25034*A*B*C*a^10*b^9 - 45404*A*B*C*a^11*b^8 + 21788*A*B*C*a^12*b^7 +

22716*A*B*C*a^13*b^6 - 9292*A*B*C*a^14*b^5 - 4548*A*B*C*a^15*b^4 - 112*A*B*C*a^16*b^3 - 208*A*B*C*a^17*b^2))/

(a^25*b + a^26 - a^15*b^11 - a^16*b^10 + 5*a^17*b^9 + 5*a^18*b^8 - 10*a^19*b^7 - 10*a^20*b^6 + 10*a^21*b^5 + 1

0*a^22*b^4 - 5*a^23*b^3 - 5*a^24*b^2) - (b*((8*tan(c/2 + (d*x)/2)*(A^2*a^18 + 800*A^2*b^18 + 4*C^2*a^18 - 800*

A^2*a*b^17 - 2*A^2*a^17*b - 8*C^2*a^17*b - 4720*A^2*a^2*b^16 + 4720*A^2*a^3*b^15 + 11522*A^2*a^4*b^14 - 11522*

A^2*a^5*b^13 - 14837*A^2*a^6*b^12 + 14812*A^2*a^7*b^11 + 10385*A^2*a^8*b^10 - 10430*A^2*a^9*b^9 - 3325*A^2*a^1

0*b^8 + 3640*A^2*a^11*b^7 - 45*A^2*a^12*b^6 - 350*A^2*a^13*b^5 + 209*A^2*a^14*b^4 - 68*A^2*a^15*b^3 + 35*A^2*a

^16*b^2 + 128*B^2*a^2*b^16 - 128*B^2*a^3*b^15 - 768*B^2*a^4*b^14 + 768*B^2*a^5*b^13 + 1920*B^2*a^6*b^12 - 1920

*B^2*a^7*b^11 - 2600*B^2*a^8*b^10 + 2560*B^2*a^9*b^9 + 2025*B^2*a^10*b^8 - 1920*B^2*a^11*b^7 - 824*B^2*a^12*b^

6 + 768*B^2*a^13*b^5 + 80*B^2*a^14*b^4 - 128*B^2*a^15*b^3 + 64*B^2*a^16*b^2 + 8*C^2*a^4*b^14 - 8*C^2*a^5*b^13

- 48*C^2*a^6*b^12 + 48*C^2*a^7*b^11 + 117*C^2*a^8*b^10 - 120*C^2*a^9*b^9 - 164*C^2*a^10*b^8 + 160*C^2*a^11*b^7

+ 156*C^2*a^12*b^6 - 120*C^2*a^13*b^5 - 92*C^2*a^14*b^4 + 48*C^2*a^15*b^3 + 44*C^2*a^16*b^2 + 4*A*C*a^18 - 64

0*A*B*a*b^17 - 16*A*B*a^17*b - 8*A*C*a^17*b - 32*B*C*a^17*b + 640*A*B*a^2*b^16 + 3808*A*B*a^3*b^15 - 3808*A*B*

a^4*b^14 - 9408*A*B*a^5*b^13 + 9408*A*B*a^6*b^12 + 12430*A*B*a^7*b^11 - 12320*A*B*a^8*b^10 - 9200*A*B*a^9*b^9

+ 8960*A*B*a^10*b^8 + 3360*A*B*a^11*b^7 - 3360*A*B*a^12*b^6 - 144*A*B*a^13*b^5 + 448*A*B*a^14*b^4 - 240*A*B*a^

15*b^3 + 32*A*B*a^16*b^2 + 160*A*C*a^2*b^16 - 160*A*C*a^3*b^15 - 952*A*C*a^4*b^14 + 952*A*C*a^5*b^13 + 2322*A*

C*a^6*b^12 - 2352*A*C*a^7*b^11 - 3124*A*C*a^8*b^10 + 3080*A*C*a^9*b^9 + 2588*A*C*a^10*b^8 - 2240*A*C*a^11*b^7

- 1284*A*C*a^12*b^6 + 840*A*C*a^13*b^5 + 276*A*C*a^14*b^4 - 112*A*C*a^15*b^3 + 60*A*C*a^16*b^2 - 64*B*C*a^3*b^

15 + 64*B*C*a^4*b^14 + 384*B*C*a^5*b^13 - 384*B*C*a^6*b^12 - 948*B*C*a^7*b^11 + 960*B*C*a^8*b^10 + 1306*B*C*a^

9*b^9 - 1280*B*C*a^10*b^8 - 1128*B*C*a^11*b^7 + 960*B*C*a^12*b^6 + 592*B*C*a^13*b^5 - 384*B*C*a^14*b^4 - 160*B

*C*a^15*b^3 + 64*B*C*a^16*b^2))/(a^20*b + a^21 - a^10*b^11 - a^11*b^10 + 5*a^12*b^9 + 5*a^13*b^8 - 10*a^14*b^7

- 10*a^15*b^6 + 10*a^16*b^5 + 10*a^17*b^4 - 5*a^18*b^3 - 5*a^19*b^2) + (b*((a + b)^7*(a - b)^7)^(1/2)*((4*(4*

A*a^27 + 8*C*a^27 - 80*A*a^12*b^15 + 40*A*a^13*b^14 + 516*A*a^14*b^13 - 248*A*a^15*b^12 - 1404*A*a^16*b^11 + 6

40*A*a^17*b^10 + 2076*A*a^18*b^9 - 896*A*a^19*b^8 - 1764*A*a^20*b^7 + 724*A*a^21*b^6 + 816*A*a^22*b^5 - 316*A*

a^23*b^4 - 160*A*a^24*b^3 + 52*A*a^25*b^2 + 32*B*a^13*b^14 - 16*B*a^14*b^13 - 208*B*a^15*b^12 + 100*B*a^16*b^1

1 + 572*B*a^17*b^10 - 252*B*a^18*b^9 - 868*B*a^19*b^8 + 348*B*a^20*b^7 + 772*B*a^21*b^6 - 292*B*a^22*b^5 - 380

*B*a^23*b^4 + 144*B*a^24*b^3 + 80*B*a^25*b^2 - 8*C*a^14*b^13 + 4*C*a^15*b^12 + 52*C*a^16*b^11 - 28*C*a^17*b^10

- 140*C*a^18*b^9 + 60*C*a^19*b^8 + 220*C*a^20*b^7 - 60*C*a^21*b^6 - 220*C*a^22*b^5 + 40*C*a^23*b^4 + 128*C*a^

24*b^3 - 24*C*a^25*b^2 - 32*B*a^26*b - 32*C*a^26*b))/(a^25*b + a^26 - a^15*b^11 - a^16*b^10 + 5*a^17*b^9 + 5*a

^18*b^8 - 10*a^19*b^7 - 10*a^20*b^6 + 10*a^21*b^5 + 10*a^22*b^4 - 5*a^23*b^3 - 5*a^24*b^2) - (4*b*tan(c/2 + (d

*x)/2)*((a + b)^7*(a - b)^7)^(1/2)*(20*A*b^8 - 8*C*a^8 - 69*A*a^2*b^6 + 84*A*a^4*b^4 - 40*A*a^6*b^2 + 28*B*a^3

*b^5 - 35*B*a^5*b^3 + 2*C*a^2*b^6 - 7*C*a^4*b^4 + 8*C*a^6*b^2 - 8*B*a*b^7 + 20*B*a^7*b)*(8*a^25*b - 8*a^12*b^1

4 + 8*a^13*b^13 + 48*a^14*b^12 - 48*a^15*b^11 - 120*a^16*b^10 + 120*a^17*b^9 + 160*a^18*b^8 - 160*a^19*b^7 - 1

20*a^20*b^6 + 120*a^21*b^5 + 48*a^22*b^4 - 48*a^23*b^3 - 8*a^24*b^2))/((a^20 - a^6*b^14 + 7*a^8*b^12 - 21*a^10

*b^10 + 35*a^12*b^8 - 35*a^14*b^6 + 21*a^16*b^4 - 7*a^18*b^2)*(a^20*b + a^21 - a^10*b^11 - a^11*b^10 + 5*a^12*

b^9 + 5*a^13*b^8 - 10*a^14*b^7 - 10*a^15*b^6 + 10*a^16*b^5 + 10*a^17*b^4 - 5*a^18*b^3 - 5*a^19*b^2)))*(20*A*b^

8 - 8*C*a^8 - 69*A*a^2*b^6 + 84*A*a^4*b^4 - 40*A*a^6*b^2 + 28*B*a^3*b^5 - 35*B*a^5*b^3 + 2*C*a^2*b^6 - 7*C*a^4

*b^4 + 8*C*a^6*b^2 - 8*B*a*b^7 + 20*B*a^7*b))/(2*(a^20 - a^6*b^14 + 7*a^8*b^12 - 21*a^10*b^10 + 35*a^12*b^8 -

35*a^14*b^6 + 21*a^16*b^4 - 7*a^18*b^2)))*((a + b)^7*(a - b)^7)^(1/2)*(20*A*b^8 - 8*C*a^8 - 69*A*a^2*b^6 + 84*

A*a^4*b^4 - 40*A*a^6*b^2 + 28*B*a^3*b^5 - 35*B*a^5*b^3 + 2*C*a^2*b^6 - 7*C*a^4*b^4 + 8*C*a^6*b^2 - 8*B*a*b^7 +

20*B*a^7*b))/(2*(a^20 - a^6*b^14 + 7*a^8*b^12 - 21*a^10*b^10 + 35*a^12*b^8 - 35*a^14*b^6 + 21*a^16*b^4 - 7*a^

18*b^2)) + (b*((8*tan(c/2 + (d*x)/2)*(A^2*a^18 + 800*A^2*b^18 + 4*C^2*a^18 - 800*A^2*a*b^17 - 2*A^2*a^17*b - 8