3.10.29 \(\int \frac {\cos ^2(c+d x) (A+B \sec (c+d x)+C \sec ^2(c+d x))}{(a+b \sec (c+d x))^4} \, dx\) [929]

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

[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 = 10.81, 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 = {4185, 4189, 4004, 3916, 2738, 214} \begin {gather*} \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} \end {gather*}

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 214

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

Rule 2738

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 3916

Int[csc[(e_.) + (f_.)*(x_)]/(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)), x_Symbol] :> Dist[1/b, Int[1/(1 + (a/b)*Si
n[e + f*x]), x], x] /; FreeQ[{a, b, e, f}, x] && NeQ[a^2 - b^2, 0]

Rule 4004

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 4185

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 4189

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 ) \text {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] Result contains complex when optimal does not.
time = 7.05, size = 658, normalized size = 1.02 \begin {gather*} \frac {\left (a^2 A+20 A b^2-8 a b B+2 a^2 C\right ) (c+d x)}{2 a^6 d}+\frac {b \left (-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\right ) \tanh ^{-1}\left (\frac {(-a+b) \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )}{a^6 \sqrt {a^2-b^2} \left (-a^2+b^2\right )^3 d}+\frac {(4 A b-a B) \left (-\frac {i \cos (c+d x)}{2 a^5}-\frac {\sin (c+d x)}{2 a^5}\right )}{d}+\frac {(4 A b-a B) \left (\frac {i \cos (c+d x)}{2 a^5}-\frac {\sin (c+d x)}{2 a^5}\right )}{d}+\frac {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 \left (a^2-b^2\right ) d (b+a \cos (c+d x))^3}+\frac {-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 \left (a^2-b^2\right )^2 d (b+a \cos (c+d x))^2}+\frac {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 \left (a^2-b^2\right )^3 d (b+a \cos (c+d x))}+\frac {A \sin (2 (c+d x))}{4 a^4 d} \end {gather*}

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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Maple [A]
time = 0.73, size = 771, normalized size = 1.19

method	result	size
derivativedivides	\(\frac {\frac {\frac {2 \left (\left (-\frac {1}{2} A \,a^{2}-4 a A b +a^{2} B \right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (\frac {1}{2} A \,a^{2}-4 a A b +a^{2} B \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{2}}+\left (A \,a^{2}+20 A \,b^{2}-8 a b B +2 a^{2} C \right ) \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a^{6}}+\frac {2 b \left (\frac {-\frac {\left (30 A \,a^{4} b^{2}+6 A \,a^{3} b^{3}-34 a^{2} A \,b^{4}-3 A a \,b^{5}+12 A \,b^{6}-20 a^{5} b B -5 B \,a^{4} b^{2}+18 a^{3} b^{3} B +2 B \,a^{2} b^{4}-6 a \,b^{5} B +12 a^{6} C +4 C \,a^{5} b -6 a^{4} b^{2} C -C \,a^{3} b^{3}+2 C \,a^{2} b^{4}\right ) a b \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 \left (a -b \right ) \left (a^{3}+3 a^{2} b +3 b^{2} a +b^{3}\right )}+\frac {2 \left (45 A \,a^{4} b^{2}-53 a^{2} A \,b^{4}+18 A \,b^{6}-30 a^{5} b B +29 a^{3} b^{3} B -9 a \,b^{5} B +18 a^{6} C -11 a^{4} b^{2} C +3 C \,a^{2} b^{4}\right ) a b \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 \left (a^{2}-2 a b +b^{2}\right ) \left (a^{2}+2 a b +b^{2}\right )}-\frac {\left (30 A \,a^{4} b^{2}-6 A \,a^{3} b^{3}-34 a^{2} A \,b^{4}+3 A a \,b^{5}+12 A \,b^{6}-20 a^{5} b B +5 B \,a^{4} b^{2}+18 a^{3} b^{3} B -2 B \,a^{2} b^{4}-6 a \,b^{5} B +12 a^{6} C -4 C \,a^{5} b -6 a^{4} b^{2} C +C \,a^{3} b^{3}+2 C \,a^{2} b^{4}\right ) a b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 \left (a +b \right ) \left (a^{3}-3 a^{2} b +3 b^{2} a -b^{3}\right )}}{\left (a \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-b \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-a -b \right )^{3}}-\frac {\left (40 A \,a^{6} 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 C \,a^{2} b^{6}\right ) \arctanh \left (\frac {\left (a -b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (a +b \right ) \left (a -b \right )}}\right )}{2 \left (a^{6}-3 a^{4} b^{2}+3 a^{2} b^{4}-b^{6}\right ) \sqrt {\left (a +b \right ) \left (a -b \right )}}\right )}{a^{6}}}{d}\)	\(771\)
default	\(\frac {\frac {\frac {2 \left (\left (-\frac {1}{2} A \,a^{2}-4 a A b +a^{2} B \right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (\frac {1}{2} A \,a^{2}-4 a A b +a^{2} B \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{2}}+\left (A \,a^{2}+20 A \,b^{2}-8 a b B +2 a^{2} C \right ) \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a^{6}}+\frac {2 b \left (\frac {-\frac {\left (30 A \,a^{4} b^{2}+6 A \,a^{3} b^{3}-34 a^{2} A \,b^{4}-3 A a \,b^{5}+12 A \,b^{6}-20 a^{5} b B -5 B \,a^{4} b^{2}+18 a^{3} b^{3} B +2 B \,a^{2} b^{4}-6 a \,b^{5} B +12 a^{6} C +4 C \,a^{5} b -6 a^{4} b^{2} C -C \,a^{3} b^{3}+2 C \,a^{2} b^{4}\right ) a b \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 \left (a -b \right ) \left (a^{3}+3 a^{2} b +3 b^{2} a +b^{3}\right )}+\frac {2 \left (45 A \,a^{4} b^{2}-53 a^{2} A \,b^{4}+18 A \,b^{6}-30 a^{5} b B +29 a^{3} b^{3} B -9 a \,b^{5} B +18 a^{6} C -11 a^{4} b^{2} C +3 C \,a^{2} b^{4}\right ) a b \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 \left (a^{2}-2 a b +b^{2}\right ) \left (a^{2}+2 a b +b^{2}\right )}-\frac {\left (30 A \,a^{4} b^{2}-6 A \,a^{3} b^{3}-34 a^{2} A \,b^{4}+3 A a \,b^{5}+12 A \,b^{6}-20 a^{5} b B +5 B \,a^{4} b^{2}+18 a^{3} b^{3} B -2 B \,a^{2} b^{4}-6 a \,b^{5} B +12 a^{6} C -4 C \,a^{5} b -6 a^{4} b^{2} C +C \,a^{3} b^{3}+2 C \,a^{2} b^{4}\right ) a b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 \left (a +b \right ) \left (a^{3}-3 a^{2} b +3 b^{2} a -b^{3}\right )}}{\left (a \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-b \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-a -b \right )^{3}}-\frac {\left (40 A \,a^{6} 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 C \,a^{2} b^{6}\right ) \arctanh \left (\frac {\left (a -b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (a +b \right ) \left (a -b \right )}}\right )}{2 \left (a^{6}-3 a^{4} b^{2}+3 a^{2} b^{4}-b^{6}\right ) \sqrt {\left (a +b \right ) \left (a -b \right )}}\right )}{a^{6}}}{d}\)	\(771\)
risch	\(\text {Expression too large to display}\)	\(3314\)

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,method=_RETURNVERBOSE)

[Out]

1/d*(2/a^6*(((-1/2*A*a^2-4*a*A*b+a^2*B)*tan(1/2*d*x+1/2*c)^3+(1/2*A*a^2-4*a*A*b+a^2*B)*tan(1/2*d*x+1/2*c))/(1+
tan(1/2*d*x+1/2*c)^2)^2+1/2*(A*a^2+20*A*b^2-8*B*a*b+2*C*a^2)*arctan(tan(1/2*d*x+1/2*c)))+2*b/a^6*((-1/2*(30*A*
a^4*b^2+6*A*a^3*b^3-34*A*a^2*b^4-3*A*a*b^5+12*A*b^6-20*B*a^5*b-5*B*a^4*b^2+18*B*a^3*b^3+2*B*a^2*b^4-6*B*a*b^5+
12*C*a^6+4*C*a^5*b-6*C*a^4*b^2-C*a^3*b^3+2*C*a^2*b^4)*a*b/(a-b)/(a^3+3*a^2*b+3*a*b^2+b^3)*tan(1/2*d*x+1/2*c)^5
+2/3*(45*A*a^4*b^2-53*A*a^2*b^4+18*A*b^6-30*B*a^5*b+29*B*a^3*b^3-9*B*a*b^5+18*C*a^6-11*C*a^4*b^2+3*C*a^2*b^4)*
a*b/(a^2-2*a*b+b^2)/(a^2+2*a*b+b^2)*tan(1/2*d*x+1/2*c)^3-1/2*(30*A*a^4*b^2-6*A*a^3*b^3-34*A*a^2*b^4+3*A*a*b^5+
12*A*b^6-20*B*a^5*b+5*B*a^4*b^2+18*B*a^3*b^3-2*B*a^2*b^4-6*B*a*b^5+12*C*a^6-4*C*a^5*b-6*C*a^4*b^2+C*a^3*b^3+2*
C*a^2*b^4)*a*b/(a+b)/(a^3-3*a^2*b+3*a*b^2-b^3)*tan(1/2*d*x+1/2*c))/(a*tan(1/2*d*x+1/2*c)^2-b*tan(1/2*d*x+1/2*c
)^2-a-b)^3-1/2*(40*A*a^6*b^2-84*A*a^4*b^4+69*A*a^2*b^6-20*A*b^8-20*B*a^7*b+35*B*a^5*b^3-28*B*a^3*b^5+8*B*a*b^7
+8*C*a^8-8*C*a^6*b^2+7*C*a^4*b^4-2*C*a^2*b^6)/(a^6-3*a^4*b^2+3*a^2*b^4-b^6)/((a+b)*(a-b))^(1/2)*arctanh((a-b)*
tan(1/2*d*x+1/2*c)/((a+b)*(a-b))^(1/2))))

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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}

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 de

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 1698 vs. \(2 (624) = 1248\).
time = 3.58, size = 3454, normalized size = 5.33 \begin {gather*} \text {Too large to display} \end {gather*}

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^...

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (A + B \sec {\left (c + d x \right )} + C \sec ^{2}{\left (c + d x \right )}\right ) \cos ^{2}{\left (c + d x \right )}}{\left (a + b \sec {\left (c + d x \right )}\right )^{4}}\, dx \end {gather*}

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)

[Out]

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

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 1438 vs. \(2 (624) = 1248\).
time = 0.57, size = 1438, normalized size = 2.22 \begin {gather*} \text {Too large to display} \end {gather*}

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))
/d

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Mupad [B]
time = 27.99, size = 2500, normalized size = 3.86 \begin {gather*} \text {Too large to display} \end {gather*}

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...

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