\(\int \frac {\cos (c+d x) (A+B \sec (c+d x)+C \sec ^2(c+d x))}{(a+b \sec (c+d x))^4} \, dx\) [928]

Optimal result

Integrand size = 39, antiderivative size = 471 \[ \int \frac {\cos (c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{(a+b \sec (c+d x))^4} \, dx=-\frac {(4 A b-a B) x}{a^5}-\frac {\left (35 a^4 A b^4-28 a^2 A b^6+8 A b^8+8 a^7 b B-8 a^5 b^3 B+7 a^3 b^5 B-2 a b^7 B-2 a^8 C-a^6 b^2 (20 A+3 C)\right ) \text {arctanh}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{a^5 (a-b)^{7/2} (a+b)^{7/2} d}+\frac {\left (68 a^2 A b^4-24 A b^6+26 a^5 b B-17 a^3 b^3 B+6 a b^5 B+a^6 (6 A-11 C)-a^4 b^2 (65 A+4 C)\right ) \sin (c+d x)}{6 a^4 \left (a^2-b^2\right )^3 d}+\frac {\left (A b^2-a (b B-a C)\right ) \sin (c+d x)}{3 a \left (a^2-b^2\right ) d (a+b \sec (c+d x))^3}-\frac {\left (4 A b^4+6 a^3 b B-a b^3 B-3 a^4 C-a^2 b^2 (9 A+2 C)\right ) \sin (c+d x)}{6 a^2 \left (a^2-b^2\right )^2 d (a+b \sec (c+d x))^2}-\frac {\left (11 a^2 A b^4-4 A b^6+6 a^5 b B-2 a^3 b^3 B+a b^5 B-2 a^6 C-3 a^4 b^2 (4 A+C)\right ) \sin (c+d x)}{2 a^3 \left (a^2-b^2\right )^3 d (a+b \sec (c+d x))} \]

[Out]

-(4*A*b-B*a)*x/a^5-(35*a^4*A*b^4-28*a^2*A*b^6+8*A*b^8+8*a^7*b*B-8*a^5*b^3*B+7*a^3*b^5*B-2*a*b^7*B-2*a^8*C-a^6*
b^2*(20*A+3*C))*arctanh((a-b)^(1/2)*tan(1/2*d*x+1/2*c)/(a+b)^(1/2))/a^5/(a-b)^(7/2)/(a+b)^(7/2)/d+1/6*(68*a^2*
A*b^4-24*A*b^6+26*a^5*b*B-17*a^3*b^3*B+6*a*b^5*B+a^6*(6*A-11*C)-a^4*b^2*(65*A+4*C))*sin(d*x+c)/a^4/(a^2-b^2)^3
/d+1/3*(A*b^2-a*(B*b-C*a))*sin(d*x+c)/a/(a^2-b^2)/d/(a+b*sec(d*x+c))^3-1/6*(4*A*b^4+6*B*a^3*b-B*a*b^3-3*a^4*C-
a^2*b^2*(9*A+2*C))*sin(d*x+c)/a^2/(a^2-b^2)^2/d/(a+b*sec(d*x+c))^2-1/2*(11*a^2*A*b^4-4*A*b^6+6*a^5*b*B-2*a^3*b
^3*B+a*b^5*B-2*a^6*C-3*a^4*b^2*(4*A+C))*sin(d*x+c)/a^3/(a^2-b^2)^3/d/(a+b*sec(d*x+c))

Rubi [A] (verified)

Time = 10.24 (sec) , antiderivative size = 471, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {4185, 4189, 4004, 3916, 2738, 214} \[ \int \frac {\cos (c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{(a+b \sec (c+d x))^4} \, dx=-\frac {x (4 A b-a B)}{a^5}+\frac {\sin (c+d x) \left (A b^2-a (b B-a C)\right )}{3 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^3}-\frac {\sin (c+d x) \left (-3 a^4 C+6 a^3 b B-a^2 b^2 (9 A+2 C)-a b^3 B+4 A b^4\right )}{6 a^2 d \left (a^2-b^2\right )^2 (a+b \sec (c+d x))^2}+\frac {\sin (c+d x) \left (a^6 (6 A-11 C)+26 a^5 b B-a^4 b^2 (65 A+4 C)-17 a^3 b^3 B+68 a^2 A b^4+6 a b^5 B-24 A b^6\right )}{6 a^4 d \left (a^2-b^2\right )^3}-\frac {\sin (c+d x) \left (-2 a^6 C+6 a^5 b B-3 a^4 b^2 (4 A+C)-2 a^3 b^3 B+11 a^2 A b^4+a b^5 B-4 A b^6\right )}{2 a^3 d \left (a^2-b^2\right )^3 (a+b \sec (c+d x))}-\frac {\left (-2 a^8 C+8 a^7 b B-a^6 b^2 (20 A+3 C)-8 a^5 b^3 B+35 a^4 A b^4+7 a^3 b^5 B-28 a^2 A b^6-2 a b^7 B+8 A b^8\right ) \text {arctanh}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{a^5 d (a-b)^{7/2} (a+b)^{7/2}} \]

[In]

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

[Out]

-(((4*A*b - a*B)*x)/a^5) - ((35*a^4*A*b^4 - 28*a^2*A*b^6 + 8*A*b^8 + 8*a^7*b*B - 8*a^5*b^3*B + 7*a^3*b^5*B - 2
*a*b^7*B - 2*a^8*C - a^6*b^2*(20*A + 3*C))*ArcTanh[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/(a^5*(a - b)^(
7/2)*(a + b)^(7/2)*d) + ((68*a^2*A*b^4 - 24*A*b^6 + 26*a^5*b*B - 17*a^3*b^3*B + 6*a*b^5*B + a^6*(6*A - 11*C) -
a^4*b^2*(65*A + 4*C))*Sin[c + d*x])/(6*a^4*(a^2 - b^2)^3*d) + ((A*b^2 - a*(b*B - a*C))*Sin[c + d*x])/(3*a*(a^
2 - b^2)*d*(a + b*Sec[c + d*x])^3) - ((4*A*b^4 + 6*a^3*b*B - a*b^3*B - 3*a^4*C - a^2*b^2*(9*A + 2*C))*Sin[c +
d*x])/(6*a^2*(a^2 - b^2)^2*d*(a + b*Sec[c + d*x])^2) - ((11*a^2*A*b^4 - 4*A*b^6 + 6*a^5*b*B - 2*a^3*b^3*B + a*
b^5*B - 2*a^6*C - 3*a^4*b^2*(4*A + C))*Sin[c + d*x])/(2*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*} \text {integral}& = \frac {\left (A b^2-a (b B-a C)\right ) \sin (c+d x)}{3 a \left (a^2-b^2\right ) d (a+b \sec (c+d x))^3}-\frac {\int \frac {\cos (c+d x) \left (4 A b^2-a b B-a^2 (3 A-C)+3 a (A b-a B+b C) \sec (c+d x)-3 \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 ) \sin (c+d x)}{3 a \left (a^2-b^2\right ) d (a+b \sec (c+d x))^3}-\frac {\left (4 A b^4+6 a^3 b B-a b^3 B-3 a^4 C-a^2 b^2 (9 A+2 C)\right ) \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 (c+d x) \left (-23 a^2 A b^2+12 A b^4+8 a^3 b B-3 a b^3 B+a^4 (6 A-5 C)+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)-2 \left (4 A b^4+6 a^3 b B-a b^3 B-3 a^4 C-a^2 b^2 (9 A+2 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 ) \sin (c+d x)}{3 a \left (a^2-b^2\right ) d (a+b \sec (c+d x))^3}-\frac {\left (4 A b^4+6 a^3 b B-a b^3 B-3 a^4 C-a^2 b^2 (9 A+2 C)\right ) \sin (c+d x)}{6 a^2 \left (a^2-b^2\right )^2 d (a+b \sec (c+d x))^2}-\frac {\left (11 a^2 A b^4-4 A b^6+6 a^5 b B-2 a^3 b^3 B+a b^5 B-2 a^6 C-3 a^4 b^2 (4 A+C)\right ) \sin (c+d x)}{2 a^3 \left (a^2-b^2\right )^3 d (a+b \sec (c+d x))}-\frac {\int \frac {\cos (c+d x) \left (-68 a^2 A b^4+24 A b^6-26 a^5 b B+17 a^3 b^3 B-6 a b^5 B-a^6 (6 A-11 C)+a^4 b^2 (65 A+4 C)+a \left (4 A b^5-6 a^5 B-8 a^3 b^2 B-a b^4 B-a^2 b^3 (7 A-4 C)+a^4 b (18 A+11 C)\right ) \sec (c+d x)+3 \left (11 a^2 A b^4-4 A b^6+6 a^5 b B-2 a^3 b^3 B+a b^5 B-2 a^6 C-3 a^4 b^2 (4 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 (68 a^2 A b^4-24 A b^6+26 a^5 b B-17 a^3 b^3 B+6 a b^5 B+a^6 (6 A-11 C)-a^4 b^2 (65 A+4 C)\right ) \sin (c+d x)}{6 a^4 \left (a^2-b^2\right )^3 d}+\frac {\left (A b^2-a (b B-a C)\right ) \sin (c+d x)}{3 a \left (a^2-b^2\right ) d (a+b \sec (c+d x))^3}-\frac {\left (4 A b^4+6 a^3 b B-a b^3 B-3 a^4 C-a^2 b^2 (9 A+2 C)\right ) \sin (c+d x)}{6 a^2 \left (a^2-b^2\right )^2 d (a+b \sec (c+d x))^2}-\frac {\left (11 a^2 A b^4-4 A b^6+6 a^5 b B-2 a^3 b^3 B+a b^5 B-2 a^6 C-3 a^4 b^2 (4 A+C)\right ) \sin (c+d x)}{2 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 (4 A b-a B)-3 a \left (11 a^2 A b^4-4 A b^6+6 a^5 b B-2 a^3 b^3 B+a b^5 B-2 a^6 C-3 a^4 b^2 (4 A+C)\right ) \sec (c+d x)}{a+b \sec (c+d x)} \, dx}{6 a^4 \left (a^2-b^2\right )^3} \\ & = -\frac {(4 A b-a B) x}{a^5}+\frac {\left (68 a^2 A b^4-24 A b^6+26 a^5 b B-17 a^3 b^3 B+6 a b^5 B+a^6 (6 A-11 C)-a^4 b^2 (65 A+4 C)\right ) \sin (c+d x)}{6 a^4 \left (a^2-b^2\right )^3 d}+\frac {\left (A b^2-a (b B-a C)\right ) \sin (c+d x)}{3 a \left (a^2-b^2\right ) d (a+b \sec (c+d x))^3}-\frac {\left (4 A b^4+6 a^3 b B-a b^3 B-3 a^4 C-a^2 b^2 (9 A+2 C)\right ) \sin (c+d x)}{6 a^2 \left (a^2-b^2\right )^2 d (a+b \sec (c+d x))^2}-\frac {\left (11 a^2 A b^4-4 A b^6+6 a^5 b B-2 a^3 b^3 B+a b^5 B-2 a^6 C-3 a^4 b^2 (4 A+C)\right ) \sin (c+d x)}{2 a^3 \left (a^2-b^2\right )^3 d (a+b \sec (c+d x))}-\frac {\left (35 a^4 A b^4-28 a^2 A b^6+8 A b^8+8 a^7 b B-8 a^5 b^3 B+7 a^3 b^5 B-2 a b^7 B-2 a^8 C-a^6 b^2 (20 A+3 C)\right ) \int \frac {\sec (c+d x)}{a+b \sec (c+d x)} \, dx}{2 a^5 \left (a^2-b^2\right )^3} \\ & = -\frac {(4 A b-a B) x}{a^5}+\frac {\left (68 a^2 A b^4-24 A b^6+26 a^5 b B-17 a^3 b^3 B+6 a b^5 B+a^6 (6 A-11 C)-a^4 b^2 (65 A+4 C)\right ) \sin (c+d x)}{6 a^4 \left (a^2-b^2\right )^3 d}+\frac {\left (A b^2-a (b B-a C)\right ) \sin (c+d x)}{3 a \left (a^2-b^2\right ) d (a+b \sec (c+d x))^3}-\frac {\left (4 A b^4+6 a^3 b B-a b^3 B-3 a^4 C-a^2 b^2 (9 A+2 C)\right ) \sin (c+d x)}{6 a^2 \left (a^2-b^2\right )^2 d (a+b \sec (c+d x))^2}-\frac {\left (11 a^2 A b^4-4 A b^6+6 a^5 b B-2 a^3 b^3 B+a b^5 B-2 a^6 C-3 a^4 b^2 (4 A+C)\right ) \sin (c+d x)}{2 a^3 \left (a^2-b^2\right )^3 d (a+b \sec (c+d x))}-\frac {\left (35 a^4 A b^4-28 a^2 A b^6+8 A b^8+8 a^7 b B-8 a^5 b^3 B+7 a^3 b^5 B-2 a b^7 B-2 a^8 C-a^6 b^2 (20 A+3 C)\right ) \int \frac {1}{1+\frac {a \cos (c+d x)}{b}} \, dx}{2 a^5 b \left (a^2-b^2\right )^3} \\ & = -\frac {(4 A b-a B) x}{a^5}+\frac {\left (68 a^2 A b^4-24 A b^6+26 a^5 b B-17 a^3 b^3 B+6 a b^5 B+a^6 (6 A-11 C)-a^4 b^2 (65 A+4 C)\right ) \sin (c+d x)}{6 a^4 \left (a^2-b^2\right )^3 d}+\frac {\left (A b^2-a (b B-a C)\right ) \sin (c+d x)}{3 a \left (a^2-b^2\right ) d (a+b \sec (c+d x))^3}-\frac {\left (4 A b^4+6 a^3 b B-a b^3 B-3 a^4 C-a^2 b^2 (9 A+2 C)\right ) \sin (c+d x)}{6 a^2 \left (a^2-b^2\right )^2 d (a+b \sec (c+d x))^2}-\frac {\left (11 a^2 A b^4-4 A b^6+6 a^5 b B-2 a^3 b^3 B+a b^5 B-2 a^6 C-3 a^4 b^2 (4 A+C)\right ) \sin (c+d x)}{2 a^3 \left (a^2-b^2\right )^3 d (a+b \sec (c+d x))}-\frac {\left (35 a^4 A b^4-28 a^2 A b^6+8 A b^8+8 a^7 b B-8 a^5 b^3 B+7 a^3 b^5 B-2 a b^7 B-2 a^8 C-a^6 b^2 (20 A+3 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^5 b \left (a^2-b^2\right )^3 d} \\ & = -\frac {(4 A b-a B) x}{a^5}+\frac {\left (20 a^6 A b^2-35 a^4 A b^4+28 a^2 A b^6-8 A b^8-8 a^7 b B+8 a^5 b^3 B-7 a^3 b^5 B+2 a b^7 B+2 a^8 C+3 a^6 b^2 C\right ) \text {arctanh}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{a^5 (a-b)^{7/2} (a+b)^{7/2} d}+\frac {\left (68 a^2 A b^4-24 A b^6+26 a^5 b B-17 a^3 b^3 B+6 a b^5 B+a^6 (6 A-11 C)-a^4 b^2 (65 A+4 C)\right ) \sin (c+d x)}{6 a^4 \left (a^2-b^2\right )^3 d}+\frac {\left (A b^2-a (b B-a C)\right ) \sin (c+d x)}{3 a \left (a^2-b^2\right ) d (a+b \sec (c+d x))^3}-\frac {\left (4 A b^4+6 a^3 b B-a b^3 B-3 a^4 C-a^2 b^2 (9 A+2 C)\right ) \sin (c+d x)}{6 a^2 \left (a^2-b^2\right )^2 d (a+b \sec (c+d x))^2}-\frac {\left (11 a^2 A b^4-4 A b^6+6 a^5 b B-2 a^3 b^3 B+a b^5 B-2 a^6 C-3 a^4 b^2 (4 A+C)\right ) \sin (c+d x)}{2 a^3 \left (a^2-b^2\right )^3 d (a+b \sec (c+d x))} \\ \end{align*}

Mathematica [C] (warning: unable to verify)

Result contains complex when optimal does not.

Time = 11.98 (sec) , antiderivative size = 1367, normalized size of antiderivative = 2.90 \[ \int \frac {\cos (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 {2 (4 A b-a B) x (b+a \cos (c+d x))^4 \sec ^2(c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{a^5 (A+2 C+2 B \cos (c+d x)+A \cos (2 c+2 d x)) (a+b \sec (c+d x))^4}+\frac {\left (-20 a^6 A b^2+35 a^4 A b^4-28 a^2 A b^6+8 A b^8+8 a^7 b B-8 a^5 b^3 B+7 a^3 b^5 B-2 a b^7 B-2 a^8 C-3 a^6 b^2 C\right ) (b+a \cos (c+d x))^4 \sec ^2(c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \left (-\frac {2 i \arctan \left (\sec \left (\frac {d x}{2}\right ) \left (\frac {\cos (c)}{\sqrt {a^2-b^2} \sqrt {\cos (2 c)-i \sin (2 c)}}-\frac {i \sin (c)}{\sqrt {a^2-b^2} \sqrt {\cos (2 c)-i \sin (2 c)}}\right ) \left (-i b \sin \left (\frac {d x}{2}\right )+i a \sin \left (c+\frac {d x}{2}\right )\right )\right ) \cos (c)}{a^5 \sqrt {a^2-b^2} d \sqrt {\cos (2 c)-i \sin (2 c)}}-\frac {2 \arctan \left (\sec \left (\frac {d x}{2}\right ) \left (\frac {\cos (c)}{\sqrt {a^2-b^2} \sqrt {\cos (2 c)-i \sin (2 c)}}-\frac {i \sin (c)}{\sqrt {a^2-b^2} \sqrt {\cos (2 c)-i \sin (2 c)}}\right ) \left (-i b \sin \left (\frac {d x}{2}\right )+i a \sin \left (c+\frac {d x}{2}\right )\right )\right ) \sin (c)}{a^5 \sqrt {a^2-b^2} d \sqrt {\cos (2 c)-i \sin (2 c)}}\right )}{\left (-a^2+b^2\right )^3 (A+2 C+2 B \cos (c+d x)+A \cos (2 c+2 d x)) (a+b \sec (c+d x))^4}+\frac {2 (b+a \cos (c+d x)) \sec (c) \sec ^2(c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \left (A b^6 \sin (c)-a b^5 B \sin (c)+a^2 b^4 C \sin (c)-a A b^5 \sin (d x)+a^2 b^4 B \sin (d x)-a^3 b^3 C \sin (d x)\right )}{3 a^5 \left (a^2-b^2\right ) d (A+2 C+2 B \cos (c+d x)+A \cos (2 c+2 d x)) (a+b \sec (c+d x))^4}+\frac {(b+a \cos (c+d x))^2 \sec (c) \sec ^2(c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \left (-17 a^2 A b^5 \sin (c)+12 A b^7 \sin (c)+14 a^3 b^4 B \sin (c)-9 a b^6 B \sin (c)-11 a^4 b^3 C \sin (c)+6 a^2 b^5 C \sin (c)+15 a^3 A b^4 \sin (d x)-10 a A b^6 \sin (d x)-12 a^4 b^3 B \sin (d x)+7 a^2 b^5 B \sin (d x)+9 a^5 b^2 C \sin (d x)-4 a^3 b^4 C \sin (d x)\right )}{3 a^5 \left (a^2-b^2\right )^2 d (A+2 C+2 B \cos (c+d x)+A \cos (2 c+2 d x)) (a+b \sec (c+d x))^4}+\frac {(b+a \cos (c+d x))^3 \sec (c) \sec ^2(c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \left (75 a^4 A b^4 \sin (c)-96 a^2 A b^6 \sin (c)+36 A b^8 \sin (c)-48 a^5 b^3 B \sin (c)+51 a^3 b^5 B \sin (c)-18 a b^7 B \sin (c)+27 a^6 b^2 C \sin (c)-18 a^4 b^4 C \sin (c)+6 a^2 b^6 C \sin (c)-60 a^5 A b^3 \sin (d x)+71 a^3 A b^5 \sin (d x)-26 a A b^7 \sin (d x)+36 a^6 b^2 B \sin (d x)-32 a^4 b^4 B \sin (d x)+11 a^2 b^6 B \sin (d x)-18 a^7 b C \sin (d x)+5 a^5 b^3 C \sin (d x)-2 a^3 b^5 C \sin (d x)\right )}{3 a^5 \left (a^2-b^2\right )^3 d (A+2 C+2 B \cos (c+d x)+A \cos (2 c+2 d x)) (a+b \sec (c+d x))^4}+\frac {2 A (b+a \cos (c+d x))^4 \sec (c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \tan (c+d x)}{a^4 d (A+2 C+2 B \cos (c+d x)+A \cos (2 c+2 d x)) (a+b \sec (c+d x))^4} \]

[In]

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

[Out]

(-2*(4*A*b - a*B)*x*(b + a*Cos[c + d*x])^4*Sec[c + d*x]^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(a^5*(A + 2
*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + b*Sec[c + d*x])^4) + ((-20*a^6*A*b^2 + 35*a^4*A*b^4 - 28*a^2*
A*b^6 + 8*A*b^8 + 8*a^7*b*B - 8*a^5*b^3*B + 7*a^3*b^5*B - 2*a*b^7*B - 2*a^8*C - 3*a^6*b^2*C)*(b + a*Cos[c + d*
x])^4*Sec[c + d*x]^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*(((-2*I)*ArcTan[Sec[(d*x)/2]*(Cos[c]/(Sqrt[a^2 -
b^2]*Sqrt[Cos[2*c] - I*Sin[2*c]]) - (I*Sin[c])/(Sqrt[a^2 - b^2]*Sqrt[Cos[2*c] - I*Sin[2*c]]))*((-I)*b*Sin[(d*x
)/2] + I*a*Sin[c + (d*x)/2])]*Cos[c])/(a^5*Sqrt[a^2 - b^2]*d*Sqrt[Cos[2*c] - I*Sin[2*c]]) - (2*ArcTan[Sec[(d*x
)/2]*(Cos[c]/(Sqrt[a^2 - b^2]*Sqrt[Cos[2*c] - I*Sin[2*c]]) - (I*Sin[c])/(Sqrt[a^2 - b^2]*Sqrt[Cos[2*c] - I*Sin
[2*c]]))*((-I)*b*Sin[(d*x)/2] + I*a*Sin[c + (d*x)/2])]*Sin[c])/(a^5*Sqrt[a^2 - b^2]*d*Sqrt[Cos[2*c] - I*Sin[2*
c]])))/((-a^2 + b^2)^3*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + b*Sec[c + d*x])^4) + (2*(b + a*C
os[c + d*x])*Sec[c]*Sec[c + d*x]^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*(A*b^6*Sin[c] - a*b^5*B*Sin[c] + a^
2*b^4*C*Sin[c] - a*A*b^5*Sin[d*x] + a^2*b^4*B*Sin[d*x] - a^3*b^3*C*Sin[d*x]))/(3*a^5*(a^2 - b^2)*d*(A + 2*C +
2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + b*Sec[c + d*x])^4) + ((b + a*Cos[c + d*x])^2*Sec[c]*Sec[c + d*x]^2
*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*(-17*a^2*A*b^5*Sin[c] + 12*A*b^7*Sin[c] + 14*a^3*b^4*B*Sin[c] - 9*a*b
^6*B*Sin[c] - 11*a^4*b^3*C*Sin[c] + 6*a^2*b^5*C*Sin[c] + 15*a^3*A*b^4*Sin[d*x] - 10*a*A*b^6*Sin[d*x] - 12*a^4*
b^3*B*Sin[d*x] + 7*a^2*b^5*B*Sin[d*x] + 9*a^5*b^2*C*Sin[d*x] - 4*a^3*b^4*C*Sin[d*x]))/(3*a^5*(a^2 - b^2)^2*d*(
A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + b*Sec[c + d*x])^4) + ((b + a*Cos[c + d*x])^3*Sec[c]*Sec[
c + d*x]^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*(75*a^4*A*b^4*Sin[c] - 96*a^2*A*b^6*Sin[c] + 36*A*b^8*Sin[c
] - 48*a^5*b^3*B*Sin[c] + 51*a^3*b^5*B*Sin[c] - 18*a*b^7*B*Sin[c] + 27*a^6*b^2*C*Sin[c] - 18*a^4*b^4*C*Sin[c]
+ 6*a^2*b^6*C*Sin[c] - 60*a^5*A*b^3*Sin[d*x] + 71*a^3*A*b^5*Sin[d*x] - 26*a*A*b^7*Sin[d*x] + 36*a^6*b^2*B*Sin[
d*x] - 32*a^4*b^4*B*Sin[d*x] + 11*a^2*b^6*B*Sin[d*x] - 18*a^7*b*C*Sin[d*x] + 5*a^5*b^3*C*Sin[d*x] - 2*a^3*b^5*
C*Sin[d*x]))/(3*a^5*(a^2 - b^2)^3*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + b*Sec[c + d*x])^4)
+ (2*A*(b + a*Cos[c + d*x])^4*Sec[c + d*x]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Tan[c + d*x])/(a^4*d*(A + 2
*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + b*Sec[c + d*x])^4)

Maple [A] (verified)

Time = 1.26 (sec) , antiderivative size = 646, normalized size of antiderivative = 1.37

method	result	size
derivativedivides	\(\frac {-\frac {2 \left (-\frac {A a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}+\left (4 A b -a B \right ) \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )\right )}{a^{5}}-\frac {2 \left (\frac {-\frac {\left (20 A \,a^{4} b^{2}+5 A \,a^{3} b^{3}-18 a^{2} A \,b^{4}-2 A a \,b^{5}+6 A \,b^{6}-12 a^{5} b B -4 B \,a^{4} b^{2}+6 a^{3} b^{3} B +B \,a^{2} b^{4}-2 a \,b^{5} B +6 a^{6} C +3 a^{5} C b +2 a^{4} b^{2} C \right ) a b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{2 \left (a -b \right ) \left (a^{3}+3 a^{2} b +3 a \,b^{2}+b^{3}\right )}+\frac {2 \left (30 A \,a^{4} b^{2}-29 a^{2} A \,b^{4}+9 A \,b^{6}-18 a^{5} b B +11 a^{3} b^{3} B -3 a \,b^{5} B +9 a^{6} C +a^{4} b^{2} C \right ) a b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{3 \left (a^{2}-2 a b +b^{2}\right ) \left (a^{2}+2 a b +b^{2}\right )}-\frac {\left (20 A \,a^{4} b^{2}-5 A \,a^{3} b^{3}-18 a^{2} A \,b^{4}+2 A a \,b^{5}+6 A \,b^{6}-12 a^{5} b B +4 B \,a^{4} b^{2}+6 a^{3} b^{3} B -B \,a^{2} b^{4}-2 a \,b^{5} B +6 a^{6} C -3 a^{5} C b +2 a^{4} b^{2} C \right ) a b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 \left (a +b \right ) \left (a^{3}-3 a^{2} b +3 a \,b^{2}-b^{3}\right )}}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a -\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} b -a -b \right )^{3}}-\frac {\left (20 A \,a^{6} b^{2}-35 a^{4} A \,b^{4}+28 a^{2} A \,b^{6}-8 A \,b^{8}-8 a^{7} b B +8 a^{5} b^{3} B -7 a^{3} b^{5} B +2 a \,b^{7} B +2 a^{8} C +3 a^{6} b^{2} C \right ) \operatorname {arctanh}\left (\frac {\left (a -b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (a +b \right ) \left (a -b \right )}}\right )}{2 \left (a^{6}-3 a^{4} b^{2}+3 a^{2} b^{4}-b^{6}\right ) \sqrt {\left (a +b \right ) \left (a -b \right )}}\right )}{a^{5}}}{d}\)	\(646\)
default	\(\frac {-\frac {2 \left (-\frac {A a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}+\left (4 A b -a B \right ) \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )\right )}{a^{5}}-\frac {2 \left (\frac {-\frac {\left (20 A \,a^{4} b^{2}+5 A \,a^{3} b^{3}-18 a^{2} A \,b^{4}-2 A a \,b^{5}+6 A \,b^{6}-12 a^{5} b B -4 B \,a^{4} b^{2}+6 a^{3} b^{3} B +B \,a^{2} b^{4}-2 a \,b^{5} B +6 a^{6} C +3 a^{5} C b +2 a^{4} b^{2} C \right ) a b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{2 \left (a -b \right ) \left (a^{3}+3 a^{2} b +3 a \,b^{2}+b^{3}\right )}+\frac {2 \left (30 A \,a^{4} b^{2}-29 a^{2} A \,b^{4}+9 A \,b^{6}-18 a^{5} b B +11 a^{3} b^{3} B -3 a \,b^{5} B +9 a^{6} C +a^{4} b^{2} C \right ) a b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{3 \left (a^{2}-2 a b +b^{2}\right ) \left (a^{2}+2 a b +b^{2}\right )}-\frac {\left (20 A \,a^{4} b^{2}-5 A \,a^{3} b^{3}-18 a^{2} A \,b^{4}+2 A a \,b^{5}+6 A \,b^{6}-12 a^{5} b B +4 B \,a^{4} b^{2}+6 a^{3} b^{3} B -B \,a^{2} b^{4}-2 a \,b^{5} B +6 a^{6} C -3 a^{5} C b +2 a^{4} b^{2} C \right ) a b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 \left (a +b \right ) \left (a^{3}-3 a^{2} b +3 a \,b^{2}-b^{3}\right )}}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a -\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} b -a -b \right )^{3}}-\frac {\left (20 A \,a^{6} b^{2}-35 a^{4} A \,b^{4}+28 a^{2} A \,b^{6}-8 A \,b^{8}-8 a^{7} b B +8 a^{5} b^{3} B -7 a^{3} b^{5} B +2 a \,b^{7} B +2 a^{8} C +3 a^{6} b^{2} C \right ) \operatorname {arctanh}\left (\frac {\left (a -b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (a +b \right ) \left (a -b \right )}}\right )}{2 \left (a^{6}-3 a^{4} b^{2}+3 a^{2} b^{4}-b^{6}\right ) \sqrt {\left (a +b \right ) \left (a -b \right )}}\right )}{a^{5}}}{d}\)	\(646\)
risch	\(\text {Expression too large to display}\)	\(2840\)

[In]

int(cos(d*x+c)*(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^5*(-A*a*tan(1/2*d*x+1/2*c)/(1+tan(1/2*d*x+1/2*c)^2)+(4*A*b-B*a)*arctan(tan(1/2*d*x+1/2*c)))-2/a^5*((
-1/2*(20*A*a^4*b^2+5*A*a^3*b^3-18*A*a^2*b^4-2*A*a*b^5+6*A*b^6-12*B*a^5*b-4*B*a^4*b^2+6*B*a^3*b^3+B*a^2*b^4-2*B
*a*b^5+6*C*a^6+3*C*a^5*b+2*C*a^4*b^2)*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*(30*A*a^4*b
^2-29*A*a^2*b^4+9*A*b^6-18*B*a^5*b+11*B*a^3*b^3-3*B*a*b^5+9*C*a^6+C*a^4*b^2)*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*(20*A*a^4*b^2-5*A*a^3*b^3-18*A*a^2*b^4+2*A*a*b^5+6*A*b^6-12*B*a^5*b+4*B*a^4*b^2+6*
B*a^3*b^3-B*a^2*b^4-2*B*a*b^5+6*C*a^6-3*C*a^5*b+2*C*a^4*b^2)*a*b/(a+b)/(a^3-3*a^2*b+3*a*b^2-b^3)*tan(1/2*d*x+1
/2*c))/(tan(1/2*d*x+1/2*c)^2*a-tan(1/2*d*x+1/2*c)^2*b-a-b)^3-1/2*(20*A*a^6*b^2-35*A*a^4*b^4+28*A*a^2*b^6-8*A*b
^8-8*B*a^7*b+8*B*a^5*b^3-7*B*a^3*b^5+2*B*a*b^7+2*C*a^8+3*C*a^6*b^2)/(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))))

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1354 vs. \(2 (448) = 896\).

Time = 0.55 (sec) , antiderivative size = 2766, normalized size of antiderivative = 5.87 \[ \int \frac {\cos (c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{(a+b \sec (c+d x))^4} \, dx=\text {Too large to display} \]

[In]

integrate(cos(d*x+c)*(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*(12*(B*a^12 - 4*A*a^11*b - 4*B*a^10*b^2 + 16*A*a^9*b^3 + 6*B*a^8*b^4 - 24*A*a^7*b^5 - 4*B*a^6*b^6 + 16*A
*a^5*b^7 + B*a^4*b^8 - 4*A*a^3*b^9)*d*x*cos(d*x + c)^3 + 36*(B*a^11*b - 4*A*a^10*b^2 - 4*B*a^9*b^3 + 16*A*a^8*
b^4 + 6*B*a^7*b^5 - 24*A*a^6*b^6 - 4*B*a^5*b^7 + 16*A*a^4*b^8 + B*a^3*b^9 - 4*A*a^2*b^10)*d*x*cos(d*x + c)^2 +
36*(B*a^10*b^2 - 4*A*a^9*b^3 - 4*B*a^8*b^4 + 16*A*a^7*b^5 + 6*B*a^6*b^6 - 24*A*a^5*b^7 - 4*B*a^4*b^8 + 16*A*a
^3*b^9 + B*a^2*b^10 - 4*A*a*b^11)*d*x*cos(d*x + c) + 12*(B*a^9*b^3 - 4*A*a^8*b^4 - 4*B*a^7*b^5 + 16*A*a^6*b^6
+ 6*B*a^5*b^7 - 24*A*a^4*b^8 - 4*B*a^3*b^9 + 16*A*a^2*b^10 + B*a*b^11 - 4*A*b^12)*d*x + 3*(2*C*a^8*b^3 - 8*B*a
^7*b^4 + (20*A + 3*C)*a^6*b^5 + 8*B*a^5*b^6 - 35*A*a^4*b^7 - 7*B*a^3*b^8 + 28*A*a^2*b^9 + 2*B*a*b^10 - 8*A*b^1
1 + (2*C*a^11 - 8*B*a^10*b + (20*A + 3*C)*a^9*b^2 + 8*B*a^8*b^3 - 35*A*a^7*b^4 - 7*B*a^6*b^5 + 28*A*a^5*b^6 +
2*B*a^4*b^7 - 8*A*a^3*b^8)*cos(d*x + c)^3 + 3*(2*C*a^10*b - 8*B*a^9*b^2 + (20*A + 3*C)*a^8*b^3 + 8*B*a^7*b^4 -
35*A*a^6*b^5 - 7*B*a^5*b^6 + 28*A*a^4*b^7 + 2*B*a^3*b^8 - 8*A*a^2*b^9)*cos(d*x + c)^2 + 3*(2*C*a^9*b^2 - 8*B*
a^8*b^3 + (20*A + 3*C)*a^7*b^4 + 8*B*a^6*b^5 - 35*A*a^5*b^6 - 7*B*a^4*b^7 + 28*A*a^3*b^8 + 2*B*a^2*b^9 - 8*A*a
*b^10)*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*A
- 11*C)*a^9*b^3 + 26*B*a^8*b^4 - (71*A - 7*C)*a^7*b^5 - 43*B*a^6*b^6 + (133*A + 4*C)*a^5*b^7 + 23*B*a^4*b^8 -
92*A*a^3*b^9 - 6*B*a^2*b^10 + 24*A*a*b^11 + 6*(A*a^12 - 4*A*a^10*b^2 + 6*A*a^8*b^4 - 4*A*a^6*b^6 + A*a^4*b^8)*
cos(d*x + c)^3 + (18*(A - C)*a^11*b + 36*B*a^10*b^2 - (132*A - 23*C)*a^9*b^3 - 68*B*a^8*b^4 + (239*A - 7*C)*a^
7*b^5 + 43*B*a^6*b^6 - (169*A - 2*C)*a^5*b^7 - 11*B*a^4*b^8 + 44*A*a^3*b^9)*cos(d*x + c)^2 + 3*(3*(2*A - 3*C)*
a^10*b^2 + 20*B*a^9*b^3 - (59*A - 8*C)*a^8*b^4 - 35*B*a^7*b^5 + (110*A + C)*a^6*b^6 + 20*B*a^5*b^7 - 77*A*a^4*
b^8 - 5*B*a^3*b^9 + 20*A*a^2*b^10)*cos(d*x + c))*sin(d*x + c))/((a^16 - 4*a^14*b^2 + 6*a^12*b^4 - 4*a^10*b^6 +
a^8*b^8)*d*cos(d*x + c)^3 + 3*(a^15*b - 4*a^13*b^3 + 6*a^11*b^5 - 4*a^9*b^7 + a^7*b^9)*d*cos(d*x + c)^2 + 3*(
a^14*b^2 - 4*a^12*b^4 + 6*a^10*b^6 - 4*a^8*b^8 + a^6*b^10)*d*cos(d*x + c) + (a^13*b^3 - 4*a^11*b^5 + 6*a^9*b^7
- 4*a^7*b^9 + a^5*b^11)*d), 1/6*(6*(B*a^12 - 4*A*a^11*b - 4*B*a^10*b^2 + 16*A*a^9*b^3 + 6*B*a^8*b^4 - 24*A*a^
7*b^5 - 4*B*a^6*b^6 + 16*A*a^5*b^7 + B*a^4*b^8 - 4*A*a^3*b^9)*d*x*cos(d*x + c)^3 + 18*(B*a^11*b - 4*A*a^10*b^2
- 4*B*a^9*b^3 + 16*A*a^8*b^4 + 6*B*a^7*b^5 - 24*A*a^6*b^6 - 4*B*a^5*b^7 + 16*A*a^4*b^8 + B*a^3*b^9 - 4*A*a^2*
b^10)*d*x*cos(d*x + c)^2 + 18*(B*a^10*b^2 - 4*A*a^9*b^3 - 4*B*a^8*b^4 + 16*A*a^7*b^5 + 6*B*a^6*b^6 - 24*A*a^5*
b^7 - 4*B*a^4*b^8 + 16*A*a^3*b^9 + B*a^2*b^10 - 4*A*a*b^11)*d*x*cos(d*x + c) + 6*(B*a^9*b^3 - 4*A*a^8*b^4 - 4*
B*a^7*b^5 + 16*A*a^6*b^6 + 6*B*a^5*b^7 - 24*A*a^4*b^8 - 4*B*a^3*b^9 + 16*A*a^2*b^10 + B*a*b^11 - 4*A*b^12)*d*x
+ 3*(2*C*a^8*b^3 - 8*B*a^7*b^4 + (20*A + 3*C)*a^6*b^5 + 8*B*a^5*b^6 - 35*A*a^4*b^7 - 7*B*a^3*b^8 + 28*A*a^2*b
^9 + 2*B*a*b^10 - 8*A*b^11 + (2*C*a^11 - 8*B*a^10*b + (20*A + 3*C)*a^9*b^2 + 8*B*a^8*b^3 - 35*A*a^7*b^4 - 7*B*
a^6*b^5 + 28*A*a^5*b^6 + 2*B*a^4*b^7 - 8*A*a^3*b^8)*cos(d*x + c)^3 + 3*(2*C*a^10*b - 8*B*a^9*b^2 + (20*A + 3*C
)*a^8*b^3 + 8*B*a^7*b^4 - 35*A*a^6*b^5 - 7*B*a^5*b^6 + 28*A*a^4*b^7 + 2*B*a^3*b^8 - 8*A*a^2*b^9)*cos(d*x + c)^
2 + 3*(2*C*a^9*b^2 - 8*B*a^8*b^3 + (20*A + 3*C)*a^7*b^4 + 8*B*a^6*b^5 - 35*A*a^5*b^6 - 7*B*a^4*b^7 + 28*A*a^3*
b^8 + 2*B*a^2*b^9 - 8*A*a*b^10)*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*A - 11*C)*a^9*b^3 + 26*B*a^8*b^4 - (71*A - 7*C)*a^7*b^5 - 43*B*a^6*b^6 + (133
*A + 4*C)*a^5*b^7 + 23*B*a^4*b^8 - 92*A*a^3*b^9 - 6*B*a^2*b^10 + 24*A*a*b^11 + 6*(A*a^12 - 4*A*a^10*b^2 + 6*A*
a^8*b^4 - 4*A*a^6*b^6 + A*a^4*b^8)*cos(d*x + c)^3 + (18*(A - C)*a^11*b + 36*B*a^10*b^2 - (132*A - 23*C)*a^9*b^
3 - 68*B*a^8*b^4 + (239*A - 7*C)*a^7*b^5 + 43*B*a^6*b^6 - (169*A - 2*C)*a^5*b^7 - 11*B*a^4*b^8 + 44*A*a^3*b^9)
*cos(d*x + c)^2 + 3*(3*(2*A - 3*C)*a^10*b^2 + 20*B*a^9*b^3 - (59*A - 8*C)*a^8*b^4 - 35*B*a^7*b^5 + (110*A + C)
*a^6*b^6 + 20*B*a^5*b^7 - 77*A*a^4*b^8 - 5*B*a^3*b^9 + 20*A*a^2*b^10)*cos(d*x + c))*sin(d*x + c))/((a^16 - 4*a
^14*b^2 + 6*a^12*b^4 - 4*a^10*b^6 + a^8*b^8)*d*cos(d*x + c)^3 + 3*(a^15*b - 4*a^13*b^3 + 6*a^11*b^5 - 4*a^9*b^
7 + a^7*b^9)*d*cos(d*x + c)^2 + 3*(a^14*b^2 - 4*a^12*b^4 + 6*a^10*b^6 - 4*a^8*b^8 + a^6*b^10)*d*cos(d*x + c) +
(a^13*b^3 - 4*a^11*b^5 + 6*a^9*b^7 - 4*a^7*b^9 + a^5*b^11)*d)]

Sympy [F]

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

[In]

integrate(cos(d*x+c)*(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)/(a + b*sec(c + d*x))**4, x)

Maxima [F(-2)]

Exception generated. \[ \int \frac {\cos (c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{(a+b \sec (c+d x))^4} \, dx=\text {Exception raised: ValueError} \]

[In]

integrate(cos(d*x+c)*(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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1225 vs. \(2 (448) = 896\).

Time = 0.41 (sec) , antiderivative size = 1225, normalized size of antiderivative = 2.60 \[ \int \frac {\cos (c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{(a+b \sec (c+d x))^4} \, dx=\text {Too large to display} \]

[In]

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

[Out]

1/3*(3*(2*C*a^8 - 8*B*a^7*b + 20*A*a^6*b^2 + 3*C*a^6*b^2 + 8*B*a^5*b^3 - 35*A*a^4*b^4 - 7*B*a^3*b^5 + 28*A*a^2
*b^6 + 2*B*a*b^7 - 8*A*b^8)*(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^11 - 3*a^9*b^2 + 3*a^7*b^4 - a^5*b^6)*sqrt(-a^2 + b^2)) + (
18*C*a^8*b*tan(1/2*d*x + 1/2*c)^5 - 36*B*a^7*b^2*tan(1/2*d*x + 1/2*c)^5 - 27*C*a^7*b^2*tan(1/2*d*x + 1/2*c)^5
+ 60*A*a^6*b^3*tan(1/2*d*x + 1/2*c)^5 + 60*B*a^6*b^3*tan(1/2*d*x + 1/2*c)^5 + 6*C*a^6*b^3*tan(1/2*d*x + 1/2*c)
^5 - 105*A*a^5*b^4*tan(1/2*d*x + 1/2*c)^5 + 6*B*a^5*b^4*tan(1/2*d*x + 1/2*c)^5 - 3*C*a^5*b^4*tan(1/2*d*x + 1/2
*c)^5 - 24*A*a^4*b^5*tan(1/2*d*x + 1/2*c)^5 - 45*B*a^4*b^5*tan(1/2*d*x + 1/2*c)^5 + 6*C*a^4*b^5*tan(1/2*d*x +
1/2*c)^5 + 117*A*a^3*b^6*tan(1/2*d*x + 1/2*c)^5 + 6*B*a^3*b^6*tan(1/2*d*x + 1/2*c)^5 - 24*A*a^2*b^7*tan(1/2*d*
x + 1/2*c)^5 + 15*B*a^2*b^7*tan(1/2*d*x + 1/2*c)^5 - 42*A*a*b^8*tan(1/2*d*x + 1/2*c)^5 - 6*B*a*b^8*tan(1/2*d*x
+ 1/2*c)^5 + 18*A*b^9*tan(1/2*d*x + 1/2*c)^5 - 36*C*a^8*b*tan(1/2*d*x + 1/2*c)^3 + 72*B*a^7*b^2*tan(1/2*d*x +
1/2*c)^3 - 120*A*a^6*b^3*tan(1/2*d*x + 1/2*c)^3 + 32*C*a^6*b^3*tan(1/2*d*x + 1/2*c)^3 - 116*B*a^5*b^4*tan(1/2
*d*x + 1/2*c)^3 + 236*A*a^4*b^5*tan(1/2*d*x + 1/2*c)^3 + 4*C*a^4*b^5*tan(1/2*d*x + 1/2*c)^3 + 56*B*a^3*b^6*tan
(1/2*d*x + 1/2*c)^3 - 152*A*a^2*b^7*tan(1/2*d*x + 1/2*c)^3 - 12*B*a*b^8*tan(1/2*d*x + 1/2*c)^3 + 36*A*b^9*tan(
1/2*d*x + 1/2*c)^3 + 18*C*a^8*b*tan(1/2*d*x + 1/2*c) - 36*B*a^7*b^2*tan(1/2*d*x + 1/2*c) + 27*C*a^7*b^2*tan(1/
2*d*x + 1/2*c) + 60*A*a^6*b^3*tan(1/2*d*x + 1/2*c) - 60*B*a^6*b^3*tan(1/2*d*x + 1/2*c) + 6*C*a^6*b^3*tan(1/2*d
*x + 1/2*c) + 105*A*a^5*b^4*tan(1/2*d*x + 1/2*c) + 6*B*a^5*b^4*tan(1/2*d*x + 1/2*c) + 3*C*a^5*b^4*tan(1/2*d*x
+ 1/2*c) - 24*A*a^4*b^5*tan(1/2*d*x + 1/2*c) + 45*B*a^4*b^5*tan(1/2*d*x + 1/2*c) + 6*C*a^4*b^5*tan(1/2*d*x + 1
/2*c) - 117*A*a^3*b^6*tan(1/2*d*x + 1/2*c) + 6*B*a^3*b^6*tan(1/2*d*x + 1/2*c) - 24*A*a^2*b^7*tan(1/2*d*x + 1/2
*c) - 15*B*a^2*b^7*tan(1/2*d*x + 1/2*c) + 42*A*a*b^8*tan(1/2*d*x + 1/2*c) - 6*B*a*b^8*tan(1/2*d*x + 1/2*c) + 1
8*A*b^9*tan(1/2*d*x + 1/2*c))/((a^10 - 3*a^8*b^2 + 3*a^6*b^4 - a^4*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*(B*a - 4*A*b)*(d*x + c)/a^5 + 6*A*tan(1/2*d*x + 1/2*c)/((tan(1/2*d*x + 1/2*c)^2
+ 1)*a^4))/d

Mupad [B] (verification not implemented)

Time = 27.14 (sec) , antiderivative size = 9463, normalized size of antiderivative = 20.09 \[ \int \frac {\cos (c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{(a+b \sec (c+d x))^4} \, dx=\text {Too large to display} \]

[In]

int((cos(c + d*x)*(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)^7*(2*A*a^7 + 8*A*b^7 - 24*A*a^2*b^5 + 11*A*a^3*b^4 + 26*A*a^4*b^3 - 6*A*a^5*b^2 + B*a^2*b
^5 + 6*B*a^3*b^4 - 4*B*a^4*b^3 - 12*B*a^5*b^2 + 2*C*a^4*b^3 + 3*C*a^5*b^2 - 4*A*a*b^6 - 2*A*a^6*b - 2*B*a*b^6
+ 6*C*a^6*b))/((a^4*b - a^5)*(a + b)^3) - (tan(c/2 + (d*x)/2)*(2*A*a^7 - 8*A*b^7 + 24*A*a^2*b^5 + 11*A*a^3*b^4
- 26*A*a^4*b^3 - 6*A*a^5*b^2 + B*a^2*b^5 - 6*B*a^3*b^4 - 4*B*a^4*b^3 + 12*B*a^5*b^2 - 2*C*a^4*b^3 + 3*C*a^5*b
^2 - 4*A*a*b^6 + 2*A*a^6*b + 2*B*a*b^6 - 6*C*a^6*b))/((a + b)*(3*a^6*b - a^7 + a^4*b^3 - 3*a^5*b^2)) + (tan(c/
2 + (d*x)/2)^3*(18*A*a^8 + 72*A*b^8 - 236*A*a^2*b^6 + 47*A*a^3*b^5 + 273*A*a^4*b^4 - 60*A*a^5*b^3 - 72*A*a^6*b
^2 + 3*B*a^2*b^6 + 59*B*a^3*b^5 - 14*B*a^4*b^4 - 96*B*a^5*b^3 + 36*B*a^6*b^2 + 10*C*a^4*b^4 - 7*C*a^5*b^3 + 45
*C*a^6*b^2 - 12*A*a*b^7 - 18*B*a*b^7 - 18*C*a^7*b))/(3*(a + b)^2*(3*a^6*b - a^7 + a^4*b^3 - 3*a^5*b^2)) - (tan
(c/2 + (d*x)/2)^5*(18*A*a^8 + 72*A*b^8 - 236*A*a^2*b^6 - 47*A*a^3*b^5 + 273*A*a^4*b^4 + 60*A*a^5*b^3 - 72*A*a^
6*b^2 - 3*B*a^2*b^6 + 59*B*a^3*b^5 + 14*B*a^4*b^4 - 96*B*a^5*b^3 - 36*B*a^6*b^2 + 10*C*a^4*b^4 + 7*C*a^5*b^3 +
45*C*a^6*b^2 + 12*A*a*b^7 - 18*B*a*b^7 + 18*C*a^7*b))/(3*(a^4*b - a^5)*(a + b)^3*(a - b)))/(d*(3*a*b^2 + 3*a^
2*b - tan(c/2 + (d*x)/2)^4*(6*a^2*b - 6*b^3) + tan(c/2 + (d*x)/2)^2*(6*a*b^2 - 2*a^3 + 4*b^3) + tan(c/2 + (d*x
)/2)^6*(2*a^3 - 6*a*b^2 + 4*b^3) + a^3 + b^3 - tan(c/2 + (d*x)/2)^8*(3*a*b^2 - 3*a^2*b + a^3 - b^3))) + (log(t
an(c/2 + (d*x)/2) - 1i)*(4*A*b - B*a)*1i)/(a^5*d) - (log(tan(c/2 + (d*x)/2) + 1i)*(A*b*4i - B*a*1i))/(a^5*d) -
(atan(((((8*tan(c/2 + (d*x)/2)*(128*A^2*b^16 + 4*B^2*a^16 + 4*C^2*a^16 - 128*A^2*a*b^15 - 8*B^2*a^15*b - 768*
A^2*a^2*b^14 + 768*A^2*a^3*b^13 + 1920*A^2*a^4*b^12 - 1920*A^2*a^5*b^11 - 2600*A^2*a^6*b^10 + 2560*A^2*a^7*b^9
+ 2025*A^2*a^8*b^8 - 1920*A^2*a^9*b^7 - 824*A^2*a^10*b^6 + 768*A^2*a^11*b^5 + 80*A^2*a^12*b^4 - 128*A^2*a^13*
b^3 + 64*A^2*a^14*b^2 + 8*B^2*a^2*b^14 - 8*B^2*a^3*b^13 - 48*B^2*a^4*b^12 + 48*B^2*a^5*b^11 + 117*B^2*a^6*b^10
- 120*B^2*a^7*b^9 - 164*B^2*a^8*b^8 + 160*B^2*a^9*b^7 + 156*B^2*a^10*b^6 - 120*B^2*a^11*b^5 - 92*B^2*a^12*b^4
+ 48*B^2*a^13*b^3 + 44*B^2*a^14*b^2 + 9*C^2*a^12*b^4 + 12*C^2*a^14*b^2 - 64*A*B*a*b^15 - 32*A*B*a^15*b - 32*B
*C*a^15*b + 64*A*B*a^2*b^14 + 384*A*B*a^3*b^13 - 384*A*B*a^4*b^12 - 948*A*B*a^5*b^11 + 960*A*B*a^6*b^10 + 1306
*A*B*a^7*b^9 - 1280*A*B*a^8*b^8 - 1128*A*B*a^9*b^7 + 960*A*B*a^10*b^6 + 592*A*B*a^11*b^5 - 384*A*B*a^12*b^4 -
160*A*B*a^13*b^3 + 64*A*B*a^14*b^2 - 48*A*C*a^6*b^10 + 136*A*C*a^8*b^8 - 98*A*C*a^10*b^6 - 20*A*C*a^12*b^4 + 8
0*A*C*a^14*b^2 + 12*B*C*a^7*b^9 - 34*B*C*a^9*b^7 + 20*B*C*a^11*b^5 - 16*B*C*a^13*b^3))/(a^18*b + a^19 - a^8*b^
11 - a^9*b^10 + 5*a^10*b^9 + 5*a^11*b^8 - 10*a^12*b^7 - 10*a^13*b^6 + 10*a^14*b^5 + 10*a^15*b^4 - 5*a^16*b^3 -
5*a^17*b^2) + (((8*(4*B*a^24 + 4*C*a^24 + 16*A*a^10*b^14 - 8*A*a^11*b^13 - 104*A*a^12*b^12 + 50*A*a^13*b^11 +
286*A*a^14*b^10 - 126*A*a^15*b^9 - 434*A*a^16*b^8 + 174*A*a^17*b^7 + 386*A*a^18*b^6 - 146*A*a^19*b^5 - 190*A*
a^20*b^4 + 72*A*a^21*b^3 + 40*A*a^22*b^2 - 4*B*a^11*b^13 + 2*B*a^12*b^12 + 26*B*a^13*b^11 - 14*B*a^14*b^10 - 7
0*B*a^15*b^9 + 30*B*a^16*b^8 + 110*B*a^17*b^7 - 30*B*a^18*b^6 - 110*B*a^19*b^5 + 20*B*a^20*b^4 + 64*B*a^21*b^3
- 12*B*a^22*b^2 + 6*C*a^15*b^9 - 6*C*a^16*b^8 - 14*C*a^17*b^7 + 14*C*a^18*b^6 + 6*C*a^19*b^5 - 6*C*a^20*b^4 +
6*C*a^21*b^3 - 6*C*a^22*b^2 - 16*A*a^23*b - 16*B*a^23*b - 4*C*a^23*b))/(a^22*b + a^23 - a^12*b^11 - a^13*b^10
+ 5*a^14*b^9 + 5*a^15*b^8 - 10*a^16*b^7 - 10*a^17*b^6 + 10*a^18*b^5 + 10*a^19*b^4 - 5*a^20*b^3 - 5*a^21*b^2)
- (4*tan(c/2 + (d*x)/2)*((a + b)^7*(a - b)^7)^(1/2)*(2*C*a^8 - 8*A*b^8 + 28*A*a^2*b^6 - 35*A*a^4*b^4 + 20*A*a^
6*b^2 - 7*B*a^3*b^5 + 8*B*a^5*b^3 + 3*C*a^6*b^2 + 2*B*a*b^7 - 8*B*a^7*b)*(8*a^23*b - 8*a^10*b^14 + 8*a^11*b^13
+ 48*a^12*b^12 - 48*a^13*b^11 - 120*a^14*b^10 + 120*a^15*b^9 + 160*a^16*b^8 - 160*a^17*b^7 - 120*a^18*b^6 + 1
20*a^19*b^5 + 48*a^20*b^4 - 48*a^21*b^3 - 8*a^22*b^2))/((a^19 - a^5*b^14 + 7*a^7*b^12 - 21*a^9*b^10 + 35*a^11*
b^8 - 35*a^13*b^6 + 21*a^15*b^4 - 7*a^17*b^2)*(a^18*b + a^19 - a^8*b^11 - a^9*b^10 + 5*a^10*b^9 + 5*a^11*b^8 -
10*a^12*b^7 - 10*a^13*b^6 + 10*a^14*b^5 + 10*a^15*b^4 - 5*a^16*b^3 - 5*a^17*b^2)))*((a + b)^7*(a - b)^7)^(1/2
)*(2*C*a^8 - 8*A*b^8 + 28*A*a^2*b^6 - 35*A*a^4*b^4 + 20*A*a^6*b^2 - 7*B*a^3*b^5 + 8*B*a^5*b^3 + 3*C*a^6*b^2 +
2*B*a*b^7 - 8*B*a^7*b))/(2*(a^19 - a^5*b^14 + 7*a^7*b^12 - 21*a^9*b^10 + 35*a^11*b^8 - 35*a^13*b^6 + 21*a^15*b
^4 - 7*a^17*b^2)))*((a + b)^7*(a - b)^7)^(1/2)*(2*C*a^8 - 8*A*b^8 + 28*A*a^2*b^6 - 35*A*a^4*b^4 + 20*A*a^6*b^2
- 7*B*a^3*b^5 + 8*B*a^5*b^3 + 3*C*a^6*b^2 + 2*B*a*b^7 - 8*B*a^7*b)*1i)/(2*(a^19 - a^5*b^14 + 7*a^7*b^12 - 21*
a^9*b^10 + 35*a^11*b^8 - 35*a^13*b^6 + 21*a^15*b^4 - 7*a^17*b^2)) + (((8*tan(c/2 + (d*x)/2)*(128*A^2*b^16 + 4*
B^2*a^16 + 4*C^2*a^16 - 128*A^2*a*b^15 - 8*B^2*a^15*b - 768*A^2*a^2*b^14 + 768*A^2*a^3*b^13 + 1920*A^2*a^4*b^1
2 - 1920*A^2*a^5*b^11 - 2600*A^2*a^6*b^10 + 2560*A^2*a^7*b^9 + 2025*A^2*a^8*b^8 - 1920*A^2*a^9*b^7 - 824*A^2*a
^10*b^6 + 768*A^2*a^11*b^5 + 80*A^2*a^12*b^4 - 128*A^2*a^13*b^3 + 64*A^2*a^14*b^2 + 8*B^2*a^2*b^14 - 8*B^2*a^3
*b^13 - 48*B^2*a^4*b^12 + 48*B^2*a^5*b^11 + 117*B^2*a^6*b^10 - 120*B^2*a^7*b^9 - 164*B^2*a^8*b^8 + 160*B^2*a^9
*b^7 + 156*B^2*a^10*b^6 - 120*B^2*a^11*b^5 - 92*B^2*a^12*b^4 + 48*B^2*a^13*b^3 + 44*B^2*a^14*b^2 + 9*C^2*a^12*
b^4 + 12*C^2*a^14*b^2 - 64*A*B*a*b^15 - 32*A*B*a^15*b - 32*B*C*a^15*b + 64*A*B*a^2*b^14 + 384*A*B*a^3*b^13 - 3
84*A*B*a^4*b^12 - 948*A*B*a^5*b^11 + 960*A*B*a^6*b^10 + 1306*A*B*a^7*b^9 - 1280*A*B*a^8*b^8 - 1128*A*B*a^9*b^7
+ 960*A*B*a^10*b^6 + 592*A*B*a^11*b^5 - 384*A*B*a^12*b^4 - 160*A*B*a^13*b^3 + 64*A*B*a^14*b^2 - 48*A*C*a^6*b^
10 + 136*A*C*a^8*b^8 - 98*A*C*a^10*b^6 - 20*A*C*a^12*b^4 + 80*A*C*a^14*b^2 + 12*B*C*a^7*b^9 - 34*B*C*a^9*b^7 +
20*B*C*a^11*b^5 - 16*B*C*a^13*b^3))/(a^18*b + a^19 - a^8*b^11 - a^9*b^10 + 5*a^10*b^9 + 5*a^11*b^8 - 10*a^12*
b^7 - 10*a^13*b^6 + 10*a^14*b^5 + 10*a^15*b^4 - 5*a^16*b^3 - 5*a^17*b^2) - (((8*(4*B*a^24 + 4*C*a^24 + 16*A*a^
10*b^14 - 8*A*a^11*b^13 - 104*A*a^12*b^12 + 50*A*a^13*b^11 + 286*A*a^14*b^10 - 126*A*a^15*b^9 - 434*A*a^16*b^8
+ 174*A*a^17*b^7 + 386*A*a^18*b^6 - 146*A*a^19*b^5 - 190*A*a^20*b^4 + 72*A*a^21*b^3 + 40*A*a^22*b^2 - 4*B*a^1
1*b^13 + 2*B*a^12*b^12 + 26*B*a^13*b^11 - 14*B*a^14*b^10 - 70*B*a^15*b^9 + 30*B*a^16*b^8 + 110*B*a^17*b^7 - 30
*B*a^18*b^6 - 110*B*a^19*b^5 + 20*B*a^20*b^4 + 64*B*a^21*b^3 - 12*B*a^22*b^2 + 6*C*a^15*b^9 - 6*C*a^16*b^8 - 1
4*C*a^17*b^7 + 14*C*a^18*b^6 + 6*C*a^19*b^5 - 6*C*a^20*b^4 + 6*C*a^21*b^3 - 6*C*a^22*b^2 - 16*A*a^23*b - 16*B*
a^23*b - 4*C*a^23*b))/(a^22*b + a^23 - a^12*b^11 - a^13*b^10 + 5*a^14*b^9 + 5*a^15*b^8 - 10*a^16*b^7 - 10*a^17
*b^6 + 10*a^18*b^5 + 10*a^19*b^4 - 5*a^20*b^3 - 5*a^21*b^2) + (4*tan(c/2 + (d*x)/2)*((a + b)^7*(a - b)^7)^(1/2
)*(2*C*a^8 - 8*A*b^8 + 28*A*a^2*b^6 - 35*A*a^4*b^4 + 20*A*a^6*b^2 - 7*B*a^3*b^5 + 8*B*a^5*b^3 + 3*C*a^6*b^2 +
2*B*a*b^7 - 8*B*a^7*b)*(8*a^23*b - 8*a^10*b^14 + 8*a^11*b^13 + 48*a^12*b^12 - 48*a^13*b^11 - 120*a^14*b^10 + 1
20*a^15*b^9 + 160*a^16*b^8 - 160*a^17*b^7 - 120*a^18*b^6 + 120*a^19*b^5 + 48*a^20*b^4 - 48*a^21*b^3 - 8*a^22*b
^2))/((a^19 - a^5*b^14 + 7*a^7*b^12 - 21*a^9*b^10 + 35*a^11*b^8 - 35*a^13*b^6 + 21*a^15*b^4 - 7*a^17*b^2)*(a^1
8*b + a^19 - a^8*b^11 - a^9*b^10 + 5*a^10*b^9 + 5*a^11*b^8 - 10*a^12*b^7 - 10*a^13*b^6 + 10*a^14*b^5 + 10*a^15
*b^4 - 5*a^16*b^3 - 5*a^17*b^2)))*((a + b)^7*(a - b)^7)^(1/2)*(2*C*a^8 - 8*A*b^8 + 28*A*a^2*b^6 - 35*A*a^4*b^4
+ 20*A*a^6*b^2 - 7*B*a^3*b^5 + 8*B*a^5*b^3 + 3*C*a^6*b^2 + 2*B*a*b^7 - 8*B*a^7*b))/(2*(a^19 - a^5*b^14 + 7*a^
7*b^12 - 21*a^9*b^10 + 35*a^11*b^8 - 35*a^13*b^6 + 21*a^15*b^4 - 7*a^17*b^2)))*((a + b)^7*(a - b)^7)^(1/2)*(2*
C*a^8 - 8*A*b^8 + 28*A*a^2*b^6 - 35*A*a^4*b^4 + 20*A*a^6*b^2 - 7*B*a^3*b^5 + 8*B*a^5*b^3 + 3*C*a^6*b^2 + 2*B*a
*b^7 - 8*B*a^7*b)*1i)/(2*(a^19 - a^5*b^14 + 7*a^7*b^12 - 21*a^9*b^10 + 35*a^11*b^8 - 35*a^13*b^6 + 21*a^15*b^4
- 7*a^17*b^2)))/((16*(256*A^3*b^16 - 4*B*C^2*a^16 + 4*B^2*C*a^16 - 128*A^3*a*b^15 - 16*B^3*a^15*b - 1664*A^3*
a^2*b^14 + 800*A^3*a^3*b^13 + 4576*A^3*a^4*b^12 - 2176*A^3*a^5*b^11 - 6944*A^3*a^6*b^10 + 3204*A^3*a^7*b^9 + 6
176*A^3*a^8*b^8 - 2560*A^3*a^9*b^7 - 3040*A^3*a^10*b^6 + 960*A^3*a^11*b^5 + 640*A^3*a^12*b^4 - 4*B^3*a^3*b^13
+ 2*B^3*a^4*b^12 + 26*B^3*a^5*b^11 - 11*B^3*a^6*b^10 - 70*B^3*a^7*b^9 + 34*B^3*a^8*b^8 + 110*B^3*a^9*b^7 - 66*
B^3*a^10*b^6 - 110*B^3*a^11*b^5 + 64*B^3*a^12*b^4 + 64*B^3*a^13*b^3 - 48*B^3*a^14*b^2 - 192*A^2*B*a*b^15 + 16*
A*C^2*a^15*b + 28*B^2*C*a^15*b + 48*A*B^2*a^2*b^14 - 24*A*B^2*a^3*b^13 - 312*A*B^2*a^4*b^12 + 138*A*B^2*a^5*b^
11 + 846*A*B^2*a^6*b^10 - 408*A*B^2*a^7*b^9 - 1314*A*B^2*a^8*b^8 + 726*A*B^2*a^9*b^7 + 1266*A*B^2*a^10*b^6 - 6
90*A*B^2*a^11*b^5 - 702*A*B^2*a^12*b^4 + 408*A*B^2*a^13*b^3 + 168*A*B^2*a^14*b^2 + 96*A^2*B*a^2*b^14 + 1248*A^
2*B*a^3*b^13 - 576*A^2*B*a^4*b^12 - 3408*A^2*B*a^5*b^11 + 1632*A^2*B*a^6*b^10 + 5232*A^2*B*a^7*b^9 - 2649*A^2*
B*a^8*b^8 - 4848*A^2*B*a^9*b^7 + 2376*A^2*B*a^10*b^6 + 2544*A^2*B*a^11*b^5 - 1104*A^2*B*a^12*b^4 - 576*A^2*B*a
^13*b^3 + 36*A*C^2*a^11*b^5 + 48*A*C^2*a^13*b^3 - 96*A^2*C*a^5*b^11 - 96*A^2*C*a^6*b^10 + 320*A^2*C*a^7*b^9 +
224*A^2*C*a^8*b^8 - 296*A^2*C*a^9*b^7 - 96*A^2*C*a^10*b^6 + 16*A^2*C*a^11*b^5 - 96*A^2*C*a^12*b^4 + 256*A^2*C*
a^13*b^3 + 64*A^2*C*a^14*b^2 - 9*B*C^2*a^12*b^4 - 12*B*C^2*a^14*b^2 - 6*B^2*C*a^7*b^9 - 6*B^2*C*a^8*b^8 + 20*B
^2*C*a^9*b^7 + 14*B^2*C*a^10*b^6 - 14*B^2*C*a^11*b^5 - 6*B^2*C*a^12*b^4 + 22*B^2*C*a^13*b^3 - 6*B^2*C*a^14*b^2
- 32*A*B*C*a^15*b + 48*A*B*C*a^6*b^10 + 48*A*B*C*a^7*b^9 - 160*A*B*C*a^8*b^8 - 112*A*B*C*a^9*b^7 + 130*A*B*C*
a^10*b^6 + 48*A*B*C*a^11*b^5 - 92*A*B*C*a^12*b^4 + 48*A*B*C*a^13*b^3 - 176*A*B*C*a^14*b^2))/(a^22*b + a^23 - a
^12*b^11 - a^13*b^10 + 5*a^14*b^9 + 5*a^15*b^8 - 10*a^16*b^7 - 10*a^17*b^6 + 10*a^18*b^5 + 10*a^19*b^4 - 5*a^2
0*b^3 - 5*a^21*b^2) + (((8*tan(c/2 + (d*x)/2)*(128*A^2*b^16 + 4*B^2*a^16 + 4*C^2*a^16 - 128*A^2*a*b^15 - 8*B^2
*a^15*b - 768*A^2*a^2*b^14 + 768*A^2*a^3*b^13 + 1920*A^2*a^4*b^12 - 1920*A^2*a^5*b^11 - 2600*A^2*a^6*b^10 + 25
60*A^2*a^7*b^9 + 2025*A^2*a^8*b^8 - 1920*A^2*a^9*b^7 - 824*A^2*a^10*b^6 + 768*A^2*a^11*b^5 + 80*A^2*a^12*b^4 -
128*A^2*a^13*b^3 + 64*A^2*a^14*b^2 + 8*B^2*a^2*b^14 - 8*B^2*a^3*b^13 - 48*B^2*a^4*b^12 + 48*B^2*a^5*b^11 + 11
7*B^2*a^6*b^10 - 120*B^2*a^7*b^9 - 164*B^2*a^8*b^8 + 160*B^2*a^9*b^7 + 156*B^2*a^10*b^6 - 120*B^2*a^11*b^5 - 9
2*B^2*a^12*b^4 + 48*B^2*a^13*b^3 + 44*B^2*a^14*b^2 + 9*C^2*a^12*b^4 + 12*C^2*a^14*b^2 - 64*A*B*a*b^15 - 32*A*B
*a^15*b - 32*B*C*a^15*b + 64*A*B*a^2*b^14 + 384*A*B*a^3*b^13 - 384*A*B*a^4*b^12 - 948*A*B*a^5*b^11 + 960*A*B*a
^6*b^10 + 1306*A*B*a^7*b^9 - 1280*A*B*a^8*b^8 - 1128*A*B*a^9*b^7 + 960*A*B*a^10*b^6 + 592*A*B*a^11*b^5 - 384*A
*B*a^12*b^4 - 160*A*B*a^13*b^3 + 64*A*B*a^14*b^2 - 48*A*C*a^6*b^10 + 136*A*C*a^8*b^8 - 98*A*C*a^10*b^6 - 20*A*
C*a^12*b^4 + 80*A*C*a^14*b^2 + 12*B*C*a^7*b^9 - 34*B*C*a^9*b^7 + 20*B*C*a^11*b^5 - 16*B*C*a^13*b^3))/(a^18*b +
a^19 - a^8*b^11 - a^9*b^10 + 5*a^10*b^9 + 5*a^11*b^8 - 10*a^12*b^7 - 10*a^13*b^6 + 10*a^14*b^5 + 10*a^15*b^4
- 5*a^16*b^3 - 5*a^17*b^2) + (((8*(4*B*a^24 + 4*C*a^24 + 16*A*a^10*b^14 - 8*A*a^11*b^13 - 104*A*a^12*b^12 + 50
*A*a^13*b^11 + 286*A*a^14*b^10 - 126*A*a^15*b^9 - 434*A*a^16*b^8 + 174*A*a^17*b^7 + 386*A*a^18*b^6 - 146*A*a^1
9*b^5 - 190*A*a^20*b^4 + 72*A*a^21*b^3 + 40*A*a^22*b^2 - 4*B*a^11*b^13 + 2*B*a^12*b^12 + 26*B*a^13*b^11 - 14*B
*a^14*b^10 - 70*B*a^15*b^9 + 30*B*a^16*b^8 + 110*B*a^17*b^7 - 30*B*a^18*b^6 - 110*B*a^19*b^5 + 20*B*a^20*b^4 +
64*B*a^21*b^3 - 12*B*a^22*b^2 + 6*C*a^15*b^9 - 6*C*a^16*b^8 - 14*C*a^17*b^7 + 14*C*a^18*b^6 + 6*C*a^19*b^5 -
6*C*a^20*b^4 + 6*C*a^21*b^3 - 6*C*a^22*b^2 - 16*A*a^23*b - 16*B*a^23*b - 4*C*a^23*b))/(a^22*b + a^23 - a^12*b^
11 - a^13*b^10 + 5*a^14*b^9 + 5*a^15*b^8 - 10*a^16*b^7 - 10*a^17*b^6 + 10*a^18*b^5 + 10*a^19*b^4 - 5*a^20*b^3
- 5*a^21*b^2) - (4*tan(c/2 + (d*x)/2)*((a + b)^7*(a - b)^7)^(1/2)*(2*C*a^8 - 8*A*b^8 + 28*A*a^2*b^6 - 35*A*a^4
*b^4 + 20*A*a^6*b^2 - 7*B*a^3*b^5 + 8*B*a^5*b^3 + 3*C*a^6*b^2 + 2*B*a*b^7 - 8*B*a^7*b)*(8*a^23*b - 8*a^10*b^14
+ 8*a^11*b^13 + 48*a^12*b^12 - 48*a^13*b^11 - 120*a^14*b^10 + 120*a^15*b^9 + 160*a^16*b^8 - 160*a^17*b^7 - 12
0*a^18*b^6 + 120*a^19*b^5 + 48*a^20*b^4 - 48*a^21*b^3 - 8*a^22*b^2))/((a^19 - a^5*b^14 + 7*a^7*b^12 - 21*a^9*b
^10 + 35*a^11*b^8 - 35*a^13*b^6 + 21*a^15*b^4 - 7*a^17*b^2)*(a^18*b + a^19 - a^8*b^11 - a^9*b^10 + 5*a^10*b^9
+ 5*a^11*b^8 - 10*a^12*b^7 - 10*a^13*b^6 + 10*a^14*b^5 + 10*a^15*b^4 - 5*a^16*b^3 - 5*a^17*b^2)))*((a + b)^7*(
a - b)^7)^(1/2)*(2*C*a^8 - 8*A*b^8 + 28*A*a^2*b^6 - 35*A*a^4*b^4 + 20*A*a^6*b^2 - 7*B*a^3*b^5 + 8*B*a^5*b^3 +
3*C*a^6*b^2 + 2*B*a*b^7 - 8*B*a^7*b))/(2*(a^19 - a^5*b^14 + 7*a^7*b^12 - 21*a^9*b^10 + 35*a^11*b^8 - 35*a^13*b
^6 + 21*a^15*b^4 - 7*a^17*b^2)))*((a + b)^7*(a - b)^7)^(1/2)*(2*C*a^8 - 8*A*b^8 + 28*A*a^2*b^6 - 35*A*a^4*b^4
+ 20*A*a^6*b^2 - 7*B*a^3*b^5 + 8*B*a^5*b^3 + 3*C*a^6*b^2 + 2*B*a*b^7 - 8*B*a^7*b))/(2*(a^19 - a^5*b^14 + 7*a^7
*b^12 - 21*a^9*b^10 + 35*a^11*b^8 - 35*a^13*b^6 + 21*a^15*b^4 - 7*a^17*b^2)) - (((8*tan(c/2 + (d*x)/2)*(128*A^
2*b^16 + 4*B^2*a^16 + 4*C^2*a^16 - 128*A^2*a*b^15 - 8*B^2*a^15*b - 768*A^2*a^2*b^14 + 768*A^2*a^3*b^13 + 1920*
A^2*a^4*b^12 - 1920*A^2*a^5*b^11 - 2600*A^2*a^6*b^10 + 2560*A^2*a^7*b^9 + 2025*A^2*a^8*b^8 - 1920*A^2*a^9*b^7
- 824*A^2*a^10*b^6 + 768*A^2*a^11*b^5 + 80*A^2*a^12*b^4 - 128*A^2*a^13*b^3 + 64*A^2*a^14*b^2 + 8*B^2*a^2*b^14
- 8*B^2*a^3*b^13 - 48*B^2*a^4*b^12 + 48*B^2*a^5*b^11 + 117*B^2*a^6*b^10 - 120*B^2*a^7*b^9 - 164*B^2*a^8*b^8 +
160*B^2*a^9*b^7 + 156*B^2*a^10*b^6 - 120*B^2*a^11*b^5 - 92*B^2*a^12*b^4 + 48*B^2*a^13*b^3 + 44*B^2*a^14*b^2 +
9*C^2*a^12*b^4 + 12*C^2*a^14*b^2 - 64*A*B*a*b^15 - 32*A*B*a^15*b - 32*B*C*a^15*b + 64*A*B*a^2*b^14 + 384*A*B*a
^3*b^13 - 384*A*B*a^4*b^12 - 948*A*B*a^5*b^11 + 960*A*B*a^6*b^10 + 1306*A*B*a^7*b^9 - 1280*A*B*a^8*b^8 - 1128*
A*B*a^9*b^7 + 960*A*B*a^10*b^6 + 592*A*B*a^11*b^5 - 384*A*B*a^12*b^4 - 160*A*B*a^13*b^3 + 64*A*B*a^14*b^2 - 48
*A*C*a^6*b^10 + 136*A*C*a^8*b^8 - 98*A*C*a^10*b^6 - 20*A*C*a^12*b^4 + 80*A*C*a^14*b^2 + 12*B*C*a^7*b^9 - 34*B*
C*a^9*b^7 + 20*B*C*a^11*b^5 - 16*B*C*a^13*b^3))/(a^18*b + a^19 - a^8*b^11 - a^9*b^10 + 5*a^10*b^9 + 5*a^11*b^8
- 10*a^12*b^7 - 10*a^13*b^6 + 10*a^14*b^5 + 10*a^15*b^4 - 5*a^16*b^3 - 5*a^17*b^2) - (((8*(4*B*a^24 + 4*C*a^2
4 + 16*A*a^10*b^14 - 8*A*a^11*b^13 - 104*A*a^12*b^12 + 50*A*a^13*b^11 + 286*A*a^14*b^10 - 126*A*a^15*b^9 - 434
*A*a^16*b^8 + 174*A*a^17*b^7 + 386*A*a^18*b^6 - 146*A*a^19*b^5 - 190*A*a^20*b^4 + 72*A*a^21*b^3 + 40*A*a^22*b^
2 - 4*B*a^11*b^13 + 2*B*a^12*b^12 + 26*B*a^13*b^11 - 14*B*a^14*b^10 - 70*B*a^15*b^9 + 30*B*a^16*b^8 + 110*B*a^
17*b^7 - 30*B*a^18*b^6 - 110*B*a^19*b^5 + 20*B*a^20*b^4 + 64*B*a^21*b^3 - 12*B*a^22*b^2 + 6*C*a^15*b^9 - 6*C*a
^16*b^8 - 14*C*a^17*b^7 + 14*C*a^18*b^6 + 6*C*a^19*b^5 - 6*C*a^20*b^4 + 6*C*a^21*b^3 - 6*C*a^22*b^2 - 16*A*a^2
3*b - 16*B*a^23*b - 4*C*a^23*b))/(a^22*b + a^23 - a^12*b^11 - a^13*b^10 + 5*a^14*b^9 + 5*a^15*b^8 - 10*a^16*b^
7 - 10*a^17*b^6 + 10*a^18*b^5 + 10*a^19*b^4 - 5*a^20*b^3 - 5*a^21*b^2) + (4*tan(c/2 + (d*x)/2)*((a + b)^7*(a -
b)^7)^(1/2)*(2*C*a^8 - 8*A*b^8 + 28*A*a^2*b^6 - 35*A*a^4*b^4 + 20*A*a^6*b^2 - 7*B*a^3*b^5 + 8*B*a^5*b^3 + 3*C
*a^6*b^2 + 2*B*a*b^7 - 8*B*a^7*b)*(8*a^23*b - 8*a^10*b^14 + 8*a^11*b^13 + 48*a^12*b^12 - 48*a^13*b^11 - 120*a^
14*b^10 + 120*a^15*b^9 + 160*a^16*b^8 - 160*a^17*b^7 - 120*a^18*b^6 + 120*a^19*b^5 + 48*a^20*b^4 - 48*a^21*b^3
- 8*a^22*b^2))/((a^19 - a^5*b^14 + 7*a^7*b^12 - 21*a^9*b^10 + 35*a^11*b^8 - 35*a^13*b^6 + 21*a^15*b^4 - 7*a^1
7*b^2)*(a^18*b + a^19 - a^8*b^11 - a^9*b^10 + 5*a^10*b^9 + 5*a^11*b^8 - 10*a^12*b^7 - 10*a^13*b^6 + 10*a^14*b^
5 + 10*a^15*b^4 - 5*a^16*b^3 - 5*a^17*b^2)))*((a + b)^7*(a - b)^7)^(1/2)*(2*C*a^8 - 8*A*b^8 + 28*A*a^2*b^6 - 3
5*A*a^4*b^4 + 20*A*a^6*b^2 - 7*B*a^3*b^5 + 8*B*a^5*b^3 + 3*C*a^6*b^2 + 2*B*a*b^7 - 8*B*a^7*b))/(2*(a^19 - a^5*
b^14 + 7*a^7*b^12 - 21*a^9*b^10 + 35*a^11*b^8 - 35*a^13*b^6 + 21*a^15*b^4 - 7*a^17*b^2)))*((a + b)^7*(a - b)^7
)^(1/2)*(2*C*a^8 - 8*A*b^8 + 28*A*a^2*b^6 - 35*A*a^4*b^4 + 20*A*a^6*b^2 - 7*B*a^3*b^5 + 8*B*a^5*b^3 + 3*C*a^6*
b^2 + 2*B*a*b^7 - 8*B*a^7*b))/(2*(a^19 - a^5*b^14 + 7*a^7*b^12 - 21*a^9*b^10 + 35*a^11*b^8 - 35*a^13*b^6 + 21*
a^15*b^4 - 7*a^17*b^2))))*((a + b)^7*(a - b)^7)^(1/2)*(2*C*a^8 - 8*A*b^8 + 28*A*a^2*b^6 - 35*A*a^4*b^4 + 20*A*
a^6*b^2 - 7*B*a^3*b^5 + 8*B*a^5*b^3 + 3*C*a^6*b^2 + 2*B*a*b^7 - 8*B*a^7*b)*1i)/(d*(a^19 - a^5*b^14 + 7*a^7*b^1
2 - 21*a^9*b^10 + 35*a^11*b^8 - 35*a^13*b^6 + 21*a^15*b^4 - 7*a^17*b^2))