3.1008 \(\int \frac {(A+B \cos (c+d x)+C \cos ^2(c+d x)) \sec ^3(c+d x)}{(a+b \cos (c+d x))^4} \, dx\)

Optimal. Leaf size=657 \[ \frac {\tan (c+d x) \sec (c+d x) \left (A b^2-a (b B-a C)\right )}{3 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^3}+\frac {\left (a^2 (A+2 C)-8 a b B+20 A b^2\right ) \tanh ^{-1}(\sin (c+d x))}{2 a^6 d}-\frac {\tan (c+d x) \sec (c+d x) \left (-4 a^4 C+7 a^3 b B-a^2 b^2 (10 A+C)-2 a b^3 B+5 A b^4\right )}{6 a^2 d \left (a^2-b^2\right )^2 (a+b \cos (c+d x))^2}-\frac {\tan (c+d x) \sec (c+d x) \left (-\left (a^6 (A-6 C)\right )-12 a^5 b B+a^4 b^2 (23 A-2 C)+11 a^3 b^3 B-a^2 b^4 (27 A-C)-4 a b^5 B+10 A b^6\right )}{2 a^4 d \left (a^2-b^2\right )^3}+\frac {\tan (c+d x) \sec (c+d x) \left (12 a^6 C-27 a^5 b B+a^4 b^2 (48 A+C)+20 a^3 b^3 B-a^2 b^4 (53 A-2 C)-8 a b^5 B+20 A b^6\right )}{6 a^3 d \left (a^2-b^2\right )^3 (a+b \cos (c+d x))}+\frac {\tan (c+d x) \left (6 a^7 B-a^6 (24 A b-26 b C)-65 a^5 b^2 B+a^4 b^3 (146 A-17 C)+68 a^3 b^4 B-a^2 b^5 (167 A-6 C)-24 a b^6 B+60 A b^7\right )}{6 a^5 d \left (a^2-b^2\right )^3}+\frac {b \left (-8 a^8 C+20 a^7 b B-8 a^6 b^2 (5 A-C)-35 a^5 b^3 B+7 a^4 b^4 (12 A-C)+28 a^3 b^5 B-a^2 b^6 (69 A-2 C)-8 a b^7 B+20 A b^8\right ) \tan ^{-1}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{a^6 d \sqrt {a-b} \sqrt {a+b} \left (a^2-b^2\right )^3} \]

[Out]

1/2*(20*A*b^2-8*a*b*B+a^2*(A+2*C))*arctanh(sin(d*x+c))/a^6/d+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)*arctan((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)+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*(146*A-17*C)-a^2*b^5*(167*A-6*C)-a^6*(24*A*b-26*C*b))*tan(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))*sec(d*x+c)*tan(d*x+c)/a^4
/(a^2-b^2)^3/d+1/3*(A*b^2-a*(B*b-C*a))*sec(d*x+c)*tan(d*x+c)/a/(a^2-b^2)/d/(a+b*cos(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))*sec(d*x+c)*tan(d*x+c)/a^2/(a^2-b^2)^2/d/(a+b*cos(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))*sec(d*x+c)*tan(d*x+c)/a^
3/(a^2-b^2)^3/d/(a+b*cos(d*x+c))

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Rubi [A]  time = 12.86, antiderivative size = 657, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 5, integrand size = 41, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.122, Rules used = {3055, 3001, 3770, 2659, 205} \[ \frac {b \left (-8 a^6 b^2 (5 A-C)+7 a^4 b^4 (12 A-C)-a^2 b^6 (69 A-2 C)-35 a^5 b^3 B+28 a^3 b^5 B+20 a^7 b B-8 a^8 C-8 a b^7 B+20 A b^8\right ) \tan ^{-1}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{a^6 d \sqrt {a-b} \sqrt {a+b} \left (a^2-b^2\right )^3}+\frac {\tan (c+d x) \left (a^4 b^3 (146 A-17 C)-a^2 b^5 (167 A-6 C)-a^6 (24 A b-26 b C)-65 a^5 b^2 B+68 a^3 b^4 B+6 a^7 B-24 a b^6 B+60 A b^7\right )}{6 a^5 d \left (a^2-b^2\right )^3}+\frac {\left (a^2 (A+2 C)-8 a b B+20 A b^2\right ) \tanh ^{-1}(\sin (c+d x))}{2 a^6 d}-\frac {\tan (c+d x) \sec (c+d x) \left (a^4 b^2 (23 A-2 C)-a^2 b^4 (27 A-C)+a^6 (-(A-6 C))+11 a^3 b^3 B-12 a^5 b B-4 a b^5 B+10 A b^6\right )}{2 a^4 d \left (a^2-b^2\right )^3}+\frac {\tan (c+d x) \sec (c+d x) \left (a^4 b^2 (48 A+C)-a^2 b^4 (53 A-2 C)+20 a^3 b^3 B-27 a^5 b B+12 a^6 C-8 a b^5 B+20 A b^6\right )}{6 a^3 d \left (a^2-b^2\right )^3 (a+b \cos (c+d x))}-\frac {\tan (c+d x) \sec (c+d x) \left (-a^2 b^2 (10 A+C)+7 a^3 b B-4 a^4 C-2 a b^3 B+5 A b^4\right )}{6 a^2 d \left (a^2-b^2\right )^2 (a+b \cos (c+d x))^2}+\frac {\tan (c+d x) \sec (c+d x) \left (A b^2-a (b B-a C)\right )}{3 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^3} \]

Antiderivative was successfully verified.

[In]

Int[((A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^3)/(a + b*Cos[c + d*x])^4,x]

[Out]

(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)*ArcTan[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/(a^6*Sqrt[a - b]*Sqrt
[a + b]*(a^2 - b^2)^3*d) + ((20*A*b^2 - 8*a*b*B + a^2*(A + 2*C))*ArcTanh[Sin[c + d*x]])/(2*a^6*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))*Tan[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))*Sec[c + d*x]*Tan[c + d*x])/(2*a^4*(a^2 - b^2)^3*d)
+ ((A*b^2 - a*(b*B - a*C))*Sec[c + d*x]*Tan[c + d*x])/(3*a*(a^2 - b^2)*d*(a + b*Cos[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))*Sec[c + d*x]*Tan[c + d*x])/(6*a^2*(a^2 - b^2)^2*d*(a +
b*Cos[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))*Sec[c + d*x]*Tan[c + d*x])/(6*a^3*(a^2 - b^2)^3*d*(a + b*Cos[c + d*x]))

Rule 205

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]*ArcTan[x/Rt[a/b, 2]])/a, x] /; FreeQ[{a, b}, x]
&& PosQ[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 3001

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

Rule 3055

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((A_.) + (B_.)*s
in[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> -Simp[((A*b^2 - a*b*B + a^2*C)*Cos[e +
f*x]*(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^(n + 1))/(f*(m + 1)*(b*c - a*d)*(a^2 - b^2)), x] + Dis
t[1/((m + 1)*(b*c - a*d)*(a^2 - b^2)), Int[(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^n*Simp[(m + 1)*(b
*c - a*d)*(a*A - b*B + a*C) + d*(A*b^2 - a*b*B + a^2*C)*(m + n + 2) - (c*(A*b^2 - a*b*B + a^2*C) + (m + 1)*(b*
c - a*d)*(A*b - a*B + b*C))*Sin[e + f*x] - d*(A*b^2 - a*b*B + a^2*C)*(m + n + 3)*Sin[e + f*x]^2, x], x], x] /;
FreeQ[{a, b, c, d, e, f, A, B, C, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && Lt
Q[m, -1] && ((EqQ[a, 0] && IntegerQ[m] &&  !IntegerQ[n]) ||  !(IntegerQ[2*n] && LtQ[n, -1] && ((IntegerQ[n] &&
!IntegerQ[m]) || EqQ[a, 0])))

Rule 3770

Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> -Simp[ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]

Rubi steps

\begin {align*} \int \frac {\left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^3(c+d x)}{(a+b \cos (c+d x))^4} \, dx &=\frac {\left (A b^2-a (b B-a C)\right ) \sec (c+d x) \tan (c+d x)}{3 a \left (a^2-b^2\right ) d (a+b \cos (c+d x))^3}+\frac {\int \frac {\left (-5 A b^2+2 a b B+a^2 (3 A-2 C)-3 a (A b-a B+b C) \cos (c+d x)+4 \left (A b^2-a (b B-a C)\right ) \cos ^2(c+d x)\right ) \sec ^3(c+d x)}{(a+b \cos (c+d x))^3} \, dx}{3 a \left (a^2-b^2\right )}\\ &=\frac {\left (A b^2-a (b B-a C)\right ) \sec (c+d x) \tan (c+d x)}{3 a \left (a^2-b^2\right ) d (a+b \cos (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 ) \sec (c+d x) \tan (c+d x)}{6 a^2 \left (a^2-b^2\right )^2 d (a+b \cos (c+d x))^2}+\frac {\int \frac {\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 ) \cos (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 ) \cos ^2(c+d x)\right ) \sec ^3(c+d x)}{(a+b \cos (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 ) \sec (c+d x) \tan (c+d x)}{3 a \left (a^2-b^2\right ) d (a+b \cos (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 ) \sec (c+d x) \tan (c+d x)}{6 a^2 \left (a^2-b^2\right )^2 d (a+b \cos (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 ) \sec (c+d x) \tan (c+d x)}{6 a^3 \left (a^2-b^2\right )^3 d (a+b \cos (c+d x))}+\frac {\int \frac {\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 ) \cos (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 ) \cos ^2(c+d x)\right ) \sec ^3(c+d x)}{a+b \cos (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 ) \sec (c+d x) \tan (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 ) \sec (c+d x) \tan (c+d x)}{3 a \left (a^2-b^2\right ) d (a+b \cos (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 ) \sec (c+d x) \tan (c+d x)}{6 a^2 \left (a^2-b^2\right )^2 d (a+b \cos (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 ) \sec (c+d x) \tan (c+d x)}{6 a^3 \left (a^2-b^2\right )^3 d (a+b \cos (c+d x))}+\frac {\int \frac {\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 ) \cos (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 ) \cos ^2(c+d x)\right ) \sec ^2(c+d x)}{a+b \cos (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 ) \tan (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 ) \sec (c+d x) \tan (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 ) \sec (c+d x) \tan (c+d x)}{3 a \left (a^2-b^2\right ) d (a+b \cos (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 ) \sec (c+d x) \tan (c+d x)}{6 a^2 \left (a^2-b^2\right )^2 d (a+b \cos (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 ) \sec (c+d x) \tan (c+d x)}{6 a^3 \left (a^2-b^2\right )^3 d (a+b \cos (c+d x))}+\frac {\int \frac {\left (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 ) \cos (c+d x)\right ) \sec (c+d x)}{a+b \cos (c+d x)} \, dx}{12 a^5 \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 ) \tan (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 ) \sec (c+d x) \tan (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 ) \sec (c+d x) \tan (c+d x)}{3 a \left (a^2-b^2\right ) d (a+b \cos (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 ) \sec (c+d x) \tan (c+d x)}{6 a^2 \left (a^2-b^2\right )^2 d (a+b \cos (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 ) \sec (c+d x) \tan (c+d x)}{6 a^3 \left (a^2-b^2\right )^3 d (a+b \cos (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 {1}{a+b \cos (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 ) \int \sec (c+d x) \, dx}{2 a^6}\\ &=\frac {\left (20 A b^2-8 a b B+a^2 (A+2 C)\right ) \tanh ^{-1}(\sin (c+d x))}{2 a^6 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 ) \tan (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 ) \sec (c+d x) \tan (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 ) \sec (c+d x) \tan (c+d x)}{3 a \left (a^2-b^2\right ) d (a+b \cos (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 ) \sec (c+d x) \tan (c+d x)}{6 a^2 \left (a^2-b^2\right )^2 d (a+b \cos (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 ) \sec (c+d x) \tan (c+d x)}{6 a^3 \left (a^2-b^2\right )^3 d (a+b \cos (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 ) \operatorname {Subst}\left (\int \frac {1}{a+b+(a-b) 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 {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 ) \tan ^{-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 (20 A b^2-8 a b B+a^2 (A+2 C)\right ) \tanh ^{-1}(\sin (c+d x))}{2 a^6 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 ) \tan (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 ) \sec (c+d x) \tan (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 ) \sec (c+d x) \tan (c+d x)}{3 a \left (a^2-b^2\right ) d (a+b \cos (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 ) \sec (c+d x) \tan (c+d x)}{6 a^2 \left (a^2-b^2\right )^2 d (a+b \cos (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 ) \sec (c+d x) \tan (c+d x)}{6 a^3 \left (a^2-b^2\right )^3 d (a+b \cos (c+d x))}\\ \end {align*}

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Mathematica [A]  time = 6.50, size = 686, normalized size = 1.04 \[ \frac {\sec (c+d x) (a B \sin (c+d x)-4 A b \sin (c+d x))}{a^5 d}+\frac {A \tan (c+d x) \sec (c+d x)}{2 a^4 d}+\frac {\left (a^2 (-A)-2 a^2 C+8 a b B-20 A b^2\right ) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )}{2 a^6 d}+\frac {\left (a^2 A+2 a^2 C-8 a b B+20 A b^2\right ) \log \left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )}{2 a^6 d}+\frac {a^2 b^2 C \sin (c+d x)-a b^3 B \sin (c+d x)+A b^4 \sin (c+d x)}{3 a^3 d \left (a^2-b^2\right ) (a+b \cos (c+d x))^3}+\frac {8 a^4 b^2 C \sin (c+d x)-11 a^3 b^3 B \sin (c+d x)+14 a^2 A b^4 \sin (c+d x)-3 a^2 b^4 C \sin (c+d x)+6 a b^5 B \sin (c+d x)-9 A b^6 \sin (c+d x)}{6 a^4 d \left (a^2-b^2\right )^2 (a+b \cos (c+d x))^2}+\frac {26 a^6 b^2 C \sin (c+d x)-47 a^5 b^3 B \sin (c+d x)+74 a^4 A b^4 \sin (c+d x)-17 a^4 b^4 C \sin (c+d x)+50 a^3 b^5 B \sin (c+d x)-95 a^2 A b^6 \sin (c+d x)+6 a^2 b^6 C \sin (c+d x)-18 a b^7 B \sin (c+d x)+36 A b^8 \sin (c+d x)}{6 a^5 d \left (a^2-b^2\right )^3 (a+b \cos (c+d x))}+\frac {b \left (8 a^8 C-20 a^7 b B+40 a^6 A b^2-8 a^6 b^2 C+35 a^5 b^3 B-84 a^4 A b^4+7 a^4 b^4 C-28 a^3 b^5 B+69 a^2 A b^6-2 a^2 b^6 C+8 a b^7 B-20 A b^8\right ) \tanh ^{-1}\left (\frac {(a-b) \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {b^2-a^2}}\right )}{a^6 d \left (a^2-b^2\right )^3 \sqrt {b^2-a^2}} \]

Antiderivative was successfully verified.

[In]

Integrate[((A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^3)/(a + b*Cos[c + d*x])^4,x]

[Out]

(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*(a^2 - b^2)^3*Sqrt[-a^2 + b^2]*d) + ((-(a^2*A) - 20*A*b^2 + 8*a*b*B - 2*a^2*C)*Log[Cos[(c + d*x)/2] - Sin
[(c + d*x)/2]])/(2*a^6*d) + ((a^2*A + 20*A*b^2 - 8*a*b*B + 2*a^2*C)*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]])/
(2*a^6*d) + (Sec[c + d*x]*(-4*A*b*Sin[c + d*x] + a*B*Sin[c + d*x]))/(a^5*d) + (A*b^4*Sin[c + d*x] - a*b^3*B*Si
n[c + d*x] + a^2*b^2*C*Sin[c + d*x])/(3*a^3*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^3) + (14*a^2*A*b^4*Sin[c + d*x]
- 9*A*b^6*Sin[c + d*x] - 11*a^3*b^3*B*Sin[c + d*x] + 6*a*b^5*B*Sin[c + d*x] + 8*a^4*b^2*C*Sin[c + d*x] - 3*a^
2*b^4*C*Sin[c + d*x])/(6*a^4*(a^2 - b^2)^2*d*(a + b*Cos[c + d*x])^2) + (74*a^4*A*b^4*Sin[c + d*x] - 95*a^2*A*b
^6*Sin[c + d*x] + 36*A*b^8*Sin[c + d*x] - 47*a^5*b^3*B*Sin[c + d*x] + 50*a^3*b^5*B*Sin[c + d*x] - 18*a*b^7*B*S
in[c + d*x] + 26*a^6*b^2*C*Sin[c + d*x] - 17*a^4*b^4*C*Sin[c + d*x] + 6*a^2*b^6*C*Sin[c + d*x])/(6*a^5*(a^2 -
b^2)^3*d*(a + b*Cos[c + d*x])) + (A*Sec[c + d*x]*Tan[c + d*x])/(2*a^4*d)

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fricas [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((A+B*cos(d*x+c)+C*cos(d*x+c)^2)*sec(d*x+c)^3/(a+b*cos(d*x+c))^4,x, algorithm="fricas")

[Out]

Timed out

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giac [B]  time = 0.40, size = 1482, normalized size = 2.26 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((A+B*cos(d*x+c)+C*cos(d*x+c)^2)*sec(d*x+c)^3/(a+b*cos(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 - 28
*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*tan(
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*t
an(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) + 11
7*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)*log(abs(tan(1/
2*d*x + 1/2*c) + 1))/a^6 - 3*(A*a^2 + 2*C*a^2 - 8*B*a*b + 20*A*b^2)*log(abs(tan(1/2*d*x + 1/2*c) - 1))/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))/d

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maple [B]  time = 0.34, size = 4436, normalized size = 6.75 \[ \text {output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((A+B*cos(d*x+c)+C*cos(d*x+c)^2)*sec(d*x+c)^3/(a+b*cos(d*x+c))^4,x)

[Out]

12/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*a*b^2+12/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*a*b^2+24/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)*t
an(1/2*d*x+1/2*c)^3*C*a*b^2-7/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)*arctan(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)*arctan(tan(1
/2*d*x+1/2*c)*(a-b)/((a-b)*(a+b))^(1/2))*A-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)*arctan
(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
)*arctan(tan(1/2*d*x+1/2*c)*(a-b)/((a-b)*(a+b))^(1/2))*B+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)*arctan(tan(1/2*d*x+1/2*c)*(a-b)/((a-b)*(a+b))^(1/2))*C-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)*arctan(tan(1/2*d*x+1/2*c)*(a-b)/((a-b)*(a+b))^(1/2))*B-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)*b^3*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)^5*b^3*C-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-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+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-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+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)*ta
n(1/2*d*x+1/2*c)*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+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^4-34/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^6-212/3/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+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-t
an(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^5-5/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+18/d/a^2/(a*t
an(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+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+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)*B-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+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+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-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-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+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+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+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+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)*t
an(1/2*d*x+1/2*c)^3*B-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-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-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-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+1/2/d*A/a^4/(tan(1/2*d*x+1
/2*c)-1)+1/2/d*A/a^4/(tan(1/2*d*x+1/2*c)+1)-1/2/d*A/a^4*ln(tan(1/2*d*x+1/2*c)-1)+1/2/d*A/a^4*ln(tan(1/2*d*x+1/
2*c)+1)-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-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-8/d*b/(a^6-3*a^4*b^2+3*a^2*b^4-b^6)/((a-b)*(a+b))^(1/2)*arctan(tan(1/2*d*x+1/2*c)*(a-b)/
((a-b)*(a+b))^(1/2))*C*a^2+20/d*b^2/(a^6-3*a^4*b^2+3*a^2*b^4-b^6)/((a-b)*(a+b))^(1/2)*arctan(tan(1/2*d*x+1/2*c
)*(a-b)/((a-b)*(a+b))^(1/2))*a*B+8/d/(a^6-3*a^4*b^2+3*a^2*b^4-b^6)/((a-b)*(a+b))^(1/2)*arctan(tan(1/2*d*x+1/2*
c)*(a-b)/((a-b)*(a+b))^(1/2))*b^3*C-1/d/a^4/(tan(1/2*d*x+1/2*c)+1)*B+1/d/a^4*ln(tan(1/2*d*x+1/2*c)+1)*C+1/2/d*
A/a^4/(tan(1/2*d*x+1/2*c)-1)^2-1/d/a^4/(tan(1/2*d*x+1/2*c)-1)*B-1/d/a^4*ln(tan(1/2*d*x+1/2*c)-1)*C-1/2/d*A/a^4
/(tan(1/2*d*x+1/2*c)+1)^2-69/d/a^4/(a^6-3*a^4*b^2+3*a^2*b^4-b^6)/((a-b)*(a+b))^(1/2)*arctan(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)*arctan(tan(1/2*d*
x+1/2*c)*(a-b)/((a-b)*(a+b))^(1/2))*A*b^5-40/d*b^3/(a^6-3*a^4*b^2+3*a^2*b^4-b^6)/((a-b)*(a+b))^(1/2)*arctan(ta
n(1/2*d*x+1/2*c)*(a-b)/((a-b)*(a+b))^(1/2))*A+4/d/a^5/(tan(1/2*d*x+1/2*c)-1)*A*b-10/d/a^6*ln(tan(1/2*d*x+1/2*c
)-1)*A*b^2+4/d/a^5*ln(tan(1/2*d*x+1/2*c)-1)*B*b+4/d/a^5/(tan(1/2*d*x+1/2*c)+1)*A*b+10/d/a^6*ln(tan(1/2*d*x+1/2
*c)+1)*A*b^2-4/d/a^5*ln(tan(1/2*d*x+1/2*c)+1)*B*b

________________________________________________________________________________________

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((A+B*cos(d*x+c)+C*cos(d*x+c)^2)*sec(d*x+c)^3/(a+b*cos(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*b^2-4*a^2>0)', see `assume?`
for more details)Is 4*b^2-4*a^2 positive or negative?

________________________________________________________________________________________

mupad [B]  time = 22.19, size = 21844, normalized size = 33.25 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((A + B*cos(c + d*x) + C*cos(c + d*x)^2)/(cos(c + d*x)^3*(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)^4*(6*a*b^2 - 6*a^2*b - 2*a^3 + 10*b^3) - 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)^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)*(10*A*b^2 + a^2*(A/2 + C) - 4*B*a*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^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)))*(10*A
*b^2 + a^2*(A/2 + C) - 4*B*a*b))/a^6)*(10*A*b^2 + a^2*(A/2 + C) - 4*B*a*b)*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^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 + 20
9*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^1
0*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 - 95
2*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) - (((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 + 1
00*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 - 2
8*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*t
an(c/2 + (d*x)/2)*(10*A*b^2 + a^2*(A/2 + C) - 4*B*a*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^6*(a^20*b + a^21 - a^10*b^11 - a^11*b^10 + 5*a^12*b^9 + 5*a^13*b^8 - 1
0*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)))*(10*A*b^2 + a^2*(A/2 + C) -
4*B*a*b))/a^6)*(10*A*b^2 + a^2*(A/2 + C) - 4*B*a*b)*1i)/a^6)/((8*(8000*A^3*b^19 - 4000*A^3*a*b^18 + 32*C^3*a^1
8*b - 50800*A^3*a^2*b^17 + 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 + 8
281*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 + 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 - 10036
8*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 + 6
1440*A^2*B*a^3*b^16 - 29520*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 + 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 + 1
404*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 - 3345
6*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 - 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^2
0*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^1
8 + 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^16 - 128*B^2*a^3*b^15 - 768*B^2*a^4*b^14 + 768*B^2*a^5*b^13 + 192
0*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^1
0 - 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^1
6*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) + (((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)*(10*
A*b^2 + a^2*(A/2 + C) - 4*B*a*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^1
6*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)))*(10*A*b^2 + a^2*(A/2 + C) - 4*B*a*b))/a^6)*(10*A*b
^2 + a^2*(A/2 + C) - 4*B*a*b))/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^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 + 7
68*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 + 15
6*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 + 896
0*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 - 128
4*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) - (((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)*(10*A*b^2 + a^2*(A/2 + C) - 4*
B*a*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 + 16
0*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)))*(10*A*b^2 + a^2*(A/2 + C) - 4*B*a*b))/a^6)*(10*A*b^2 + a^2*(A/2 + C) - 4*B*a
*b))/a^6))*(10*A*b^2 + a^2*(A/2 + C) - 4*B*a*b)*2i)/(a^6*d) - (b*atan(((b*((8*tan(c/2 + (d*x)/2)*(A^2*a^18 + 8
00*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^1
5 + 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 - 1
0430*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 - 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^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^2
2*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^1
1 - 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 + 2
0*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^1
2 - 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^1
8 - 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^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^1
2 - 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 - 3
808*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 - 24
0*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 + 130
6*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^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)))/((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 + 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 + 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 - 1235
2*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^1
3 - 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 + 1
1904*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 - 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 - 2
55*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 + 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 - 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 + 10*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^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 - 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^1
2*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 - 940
8*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^1
0*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 - 2
352*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^1
2*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^1
0 + 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 - 16
0*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^1
7*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^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
))/(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 + 11522*A^2*a^4*b^14 - 11522*A^2*a^5*b^13 - 14837*A^2*a^6*b^12 + 1481
2*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 + 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 + 11
7*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^1
7*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^1
1*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^2
5*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 + 1
0*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))/(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)/(d*(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))

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sympy [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((A+B*cos(d*x+c)+C*cos(d*x+c)**2)*sec(d*x+c)**3/(a+b*cos(d*x+c))**4,x)

[Out]

Timed out

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