\(\int \frac {\cos ^3(c+d x) (A+B \sec (c+d x)+C \sec ^2(c+d x))}{(a+b \sec (c+d x))^2} \, dx\) [915]

Optimal result

Integrand size = 41, antiderivative size = 396 \[ \int \frac {\cos ^3(c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{(a+b \sec (c+d x))^2} \, dx=-\frac {\left (8 A b^3-a^3 B-6 a b^2 B+2 a^2 b (A+2 C)\right ) x}{2 a^5}+\frac {2 b^2 \left (5 a^2 A b^2-4 A b^4-4 a^3 b B+3 a b^3 B+3 a^4 C-2 a^2 b^2 C\right ) \text {arctanh}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{a^5 (a-b)^{3/2} (a+b)^{3/2} d}-\frac {\left (12 A b^4+6 a^3 b B-9 a b^3 B-a^2 b^2 (7 A-6 C)-a^4 (2 A+3 C)\right ) \sin (c+d x)}{3 a^4 \left (a^2-b^2\right ) d}+\frac {\left (4 A b^3+a^3 B-3 a b^2 B-2 a^2 b (A-C)\right ) \cos (c+d x) \sin (c+d x)}{2 a^3 \left (a^2-b^2\right ) d}-\frac {\left (4 A b^2-3 a b B-a^2 (A-3 C)\right ) \cos ^2(c+d x) \sin (c+d x)}{3 a^2 \left (a^2-b^2\right ) d}+\frac {\left (A b^2-a (b B-a C)\right ) \cos ^2(c+d x) \sin (c+d x)}{a \left (a^2-b^2\right ) d (a+b \sec (c+d x))} \]

[Out]

-1/2*(8*A*b^3-B*a^3-6*B*a*b^2+2*a^2*b*(A+2*C))*x/a^5+2*b^2*(5*A*a^2*b^2-4*A*b^4-4*B*a^3*b+3*B*a*b^3+3*C*a^4-2*
C*a^2*b^2)*arctanh((a-b)^(1/2)*tan(1/2*d*x+1/2*c)/(a+b)^(1/2))/a^5/(a-b)^(3/2)/(a+b)^(3/2)/d-1/3*(12*A*b^4+6*B
*a^3*b-9*B*a*b^3-a^2*b^2*(7*A-6*C)-a^4*(2*A+3*C))*sin(d*x+c)/a^4/(a^2-b^2)/d+1/2*(4*A*b^3+B*a^3-3*B*a*b^2-2*a^
2*b*(A-C))*cos(d*x+c)*sin(d*x+c)/a^3/(a^2-b^2)/d-1/3*(4*A*b^2-3*B*a*b-a^2*(A-3*C))*cos(d*x+c)^2*sin(d*x+c)/a^2
/(a^2-b^2)/d+(A*b^2-a*(B*b-C*a))*cos(d*x+c)^2*sin(d*x+c)/a/(a^2-b^2)/d/(a+b*sec(d*x+c))

Rubi [A] (verified)

Time = 2.04 (sec) , antiderivative size = 396, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.146, Rules used = {4185, 4189, 4004, 3916, 2738, 214} \[ \int \frac {\cos ^3(c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{(a+b \sec (c+d x))^2} \, dx=-\frac {\sin (c+d x) \cos ^2(c+d x) \left (-\left (a^2 (A-3 C)\right )-3 a b B+4 A b^2\right )}{3 a^2 d \left (a^2-b^2\right )}+\frac {\sin (c+d x) \cos ^2(c+d x) \left (A b^2-a (b B-a C)\right )}{a d \left (a^2-b^2\right ) (a+b \sec (c+d x))}+\frac {\sin (c+d x) \cos (c+d x) \left (a^3 B-2 a^2 b (A-C)-3 a b^2 B+4 A b^3\right )}{2 a^3 d \left (a^2-b^2\right )}-\frac {x \left (a^3 (-B)+2 a^2 b (A+2 C)-6 a b^2 B+8 A b^3\right )}{2 a^5}-\frac {\sin (c+d x) \left (-\left (a^4 (2 A+3 C)\right )+6 a^3 b B-a^2 b^2 (7 A-6 C)-9 a b^3 B+12 A b^4\right )}{3 a^4 d \left (a^2-b^2\right )}+\frac {2 b^2 \left (3 a^4 C-4 a^3 b B+5 a^2 A b^2-2 a^2 b^2 C+3 a b^3 B-4 A b^4\right ) \text {arctanh}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{a^5 d (a-b)^{3/2} (a+b)^{3/2}} \]

[In]

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

[Out]

-1/2*((8*A*b^3 - a^3*B - 6*a*b^2*B + 2*a^2*b*(A + 2*C))*x)/a^5 + (2*b^2*(5*a^2*A*b^2 - 4*A*b^4 - 4*a^3*b*B + 3
*a*b^3*B + 3*a^4*C - 2*a^2*b^2*C)*ArcTanh[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/(a^5*(a - b)^(3/2)*(a +
b)^(3/2)*d) - ((12*A*b^4 + 6*a^3*b*B - 9*a*b^3*B - a^2*b^2*(7*A - 6*C) - a^4*(2*A + 3*C))*Sin[c + d*x])/(3*a^
4*(a^2 - b^2)*d) + ((4*A*b^3 + a^3*B - 3*a*b^2*B - 2*a^2*b*(A - C))*Cos[c + d*x]*Sin[c + d*x])/(2*a^3*(a^2 - b
^2)*d) - ((4*A*b^2 - 3*a*b*B - a^2*(A - 3*C))*Cos[c + d*x]^2*Sin[c + d*x])/(3*a^2*(a^2 - b^2)*d) + ((A*b^2 - a
*(b*B - a*C))*Cos[c + d*x]^2*Sin[c + d*x])/(a*(a^2 - b^2)*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 ) \cos ^2(c+d x) \sin (c+d x)}{a \left (a^2-b^2\right ) d (a+b \sec (c+d x))}-\frac {\int \frac {\cos ^3(c+d x) \left (4 A b^2-3 a b B-a^2 (A-3 C)+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)} \, dx}{a \left (a^2-b^2\right )} \\ & = -\frac {\left (4 A b^2-3 a b B-a^2 (A-3 C)\right ) \cos ^2(c+d x) \sin (c+d x)}{3 a^2 \left (a^2-b^2\right ) d}+\frac {\left (A b^2-a (b B-a C)\right ) \cos ^2(c+d x) \sin (c+d x)}{a \left (a^2-b^2\right ) d (a+b \sec (c+d x))}+\frac {\int \frac {\cos ^2(c+d x) \left (3 \left (4 A b^3+a^3 B-3 a b^2 B-2 a^2 b (A-C)\right )+a \left (A b^2-3 a b B+a^2 (2 A+3 C)\right ) \sec (c+d x)-2 b \left (4 A b^2-3 a b B-a^2 (A-3 C)\right ) \sec ^2(c+d x)\right )}{a+b \sec (c+d x)} \, dx}{3 a^2 \left (a^2-b^2\right )} \\ & = \frac {\left (4 A b^3+a^3 B-3 a b^2 B-2 a^2 b (A-C)\right ) \cos (c+d x) \sin (c+d x)}{2 a^3 \left (a^2-b^2\right ) d}-\frac {\left (4 A b^2-3 a b B-a^2 (A-3 C)\right ) \cos ^2(c+d x) \sin (c+d x)}{3 a^2 \left (a^2-b^2\right ) d}+\frac {\left (A b^2-a (b B-a C)\right ) \cos ^2(c+d x) \sin (c+d x)}{a \left (a^2-b^2\right ) d (a+b \sec (c+d x))}-\frac {\int \frac {\cos (c+d x) \left (2 \left (12 A b^4+6 a^3 b B-9 a b^3 B-a^2 b^2 (7 A-6 C)-a^4 (2 A+3 C)\right )+a \left (4 A b^3-3 a^3 B-3 a b^2 B+2 a^2 b (A+3 C)\right ) \sec (c+d x)-3 b \left (4 A b^3+a^3 B-3 a b^2 B-2 a^2 b (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 )} \\ & = -\frac {\left (12 A b^4+6 a^3 b B-9 a b^3 B-a^2 b^2 (7 A-6 C)-a^4 (2 A+3 C)\right ) \sin (c+d x)}{3 a^4 \left (a^2-b^2\right ) d}+\frac {\left (4 A b^3+a^3 B-3 a b^2 B-2 a^2 b (A-C)\right ) \cos (c+d x) \sin (c+d x)}{2 a^3 \left (a^2-b^2\right ) d}-\frac {\left (4 A b^2-3 a b B-a^2 (A-3 C)\right ) \cos ^2(c+d x) \sin (c+d x)}{3 a^2 \left (a^2-b^2\right ) d}+\frac {\left (A b^2-a (b B-a C)\right ) \cos ^2(c+d x) \sin (c+d x)}{a \left (a^2-b^2\right ) d (a+b \sec (c+d x))}+\frac {\int \frac {-3 \left (a^2-b^2\right ) \left (8 A b^3-a^3 B-6 a b^2 B+2 a^2 b (A+2 C)\right )+3 a b \left (4 A b^3+a^3 B-3 a b^2 B-2 a^2 b (A-C)\right ) \sec (c+d x)}{a+b \sec (c+d x)} \, dx}{6 a^4 \left (a^2-b^2\right )} \\ & = -\frac {\left (8 A b^3-a^3 B-6 a b^2 B+2 a^2 b (A+2 C)\right ) x}{2 a^5}-\frac {\left (12 A b^4+6 a^3 b B-9 a b^3 B-a^2 b^2 (7 A-6 C)-a^4 (2 A+3 C)\right ) \sin (c+d x)}{3 a^4 \left (a^2-b^2\right ) d}+\frac {\left (4 A b^3+a^3 B-3 a b^2 B-2 a^2 b (A-C)\right ) \cos (c+d x) \sin (c+d x)}{2 a^3 \left (a^2-b^2\right ) d}-\frac {\left (4 A b^2-3 a b B-a^2 (A-3 C)\right ) \cos ^2(c+d x) \sin (c+d x)}{3 a^2 \left (a^2-b^2\right ) d}+\frac {\left (A b^2-a (b B-a C)\right ) \cos ^2(c+d x) \sin (c+d x)}{a \left (a^2-b^2\right ) d (a+b \sec (c+d x))}-\frac {\left (b^2 \left (4 A b^4+4 a^3 b B-3 a b^3 B-a^2 b^2 (5 A-2 C)-3 a^4 C\right )\right ) \int \frac {\sec (c+d x)}{a+b \sec (c+d x)} \, dx}{a^5 \left (a^2-b^2\right )} \\ & = -\frac {\left (8 A b^3-a^3 B-6 a b^2 B+2 a^2 b (A+2 C)\right ) x}{2 a^5}-\frac {\left (12 A b^4+6 a^3 b B-9 a b^3 B-a^2 b^2 (7 A-6 C)-a^4 (2 A+3 C)\right ) \sin (c+d x)}{3 a^4 \left (a^2-b^2\right ) d}+\frac {\left (4 A b^3+a^3 B-3 a b^2 B-2 a^2 b (A-C)\right ) \cos (c+d x) \sin (c+d x)}{2 a^3 \left (a^2-b^2\right ) d}-\frac {\left (4 A b^2-3 a b B-a^2 (A-3 C)\right ) \cos ^2(c+d x) \sin (c+d x)}{3 a^2 \left (a^2-b^2\right ) d}+\frac {\left (A b^2-a (b B-a C)\right ) \cos ^2(c+d x) \sin (c+d x)}{a \left (a^2-b^2\right ) d (a+b \sec (c+d x))}-\frac {\left (b \left (4 A b^4+4 a^3 b B-3 a b^3 B-a^2 b^2 (5 A-2 C)-3 a^4 C\right )\right ) \int \frac {1}{1+\frac {a \cos (c+d x)}{b}} \, dx}{a^5 \left (a^2-b^2\right )} \\ & = -\frac {\left (8 A b^3-a^3 B-6 a b^2 B+2 a^2 b (A+2 C)\right ) x}{2 a^5}-\frac {\left (12 A b^4+6 a^3 b B-9 a b^3 B-a^2 b^2 (7 A-6 C)-a^4 (2 A+3 C)\right ) \sin (c+d x)}{3 a^4 \left (a^2-b^2\right ) d}+\frac {\left (4 A b^3+a^3 B-3 a b^2 B-2 a^2 b (A-C)\right ) \cos (c+d x) \sin (c+d x)}{2 a^3 \left (a^2-b^2\right ) d}-\frac {\left (4 A b^2-3 a b B-a^2 (A-3 C)\right ) \cos ^2(c+d x) \sin (c+d x)}{3 a^2 \left (a^2-b^2\right ) d}+\frac {\left (A b^2-a (b B-a C)\right ) \cos ^2(c+d x) \sin (c+d x)}{a \left (a^2-b^2\right ) d (a+b \sec (c+d x))}-\frac {\left (2 b \left (4 A b^4+4 a^3 b B-3 a b^3 B-a^2 b^2 (5 A-2 C)-3 a^4 C\right )\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 \left (a^2-b^2\right ) d} \\ & = -\frac {\left (8 A b^3-a^3 B-6 a b^2 B+2 a^2 b (A+2 C)\right ) x}{2 a^5}+\frac {2 b^2 \left (5 a^2 A b^2-4 A b^4-4 a^3 b B+3 a b^3 B+3 a^4 C-2 a^2 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)^{3/2} (a+b)^{3/2} d}-\frac {\left (12 A b^4+6 a^3 b B-9 a b^3 B-a^2 b^2 (7 A-6 C)-a^4 (2 A+3 C)\right ) \sin (c+d x)}{3 a^4 \left (a^2-b^2\right ) d}+\frac {\left (4 A b^3+a^3 B-3 a b^2 B-2 a^2 b (A-C)\right ) \cos (c+d x) \sin (c+d x)}{2 a^3 \left (a^2-b^2\right ) d}-\frac {\left (4 A b^2-3 a b B-a^2 (A-3 C)\right ) \cos ^2(c+d x) \sin (c+d x)}{3 a^2 \left (a^2-b^2\right ) d}+\frac {\left (A b^2-a (b B-a C)\right ) \cos ^2(c+d x) \sin (c+d x)}{a \left (a^2-b^2\right ) d (a+b \sec (c+d x))} \\ \end{align*}

Mathematica [A] (verified)

Time = 5.93 (sec) , antiderivative size = 255, normalized size of antiderivative = 0.64 \[ \int \frac {\cos ^3(c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{(a+b \sec (c+d x))^2} \, dx=\frac {6 \left (-8 A b^3+a^3 B+6 a b^2 B-2 a^2 b (A+2 C)\right ) (c+d x)+\frac {24 b^2 \left (4 A b^4+4 a^3 b B-3 a b^3 B-3 a^4 C+a^2 b^2 (-5 A+2 C)\right ) \text {arctanh}\left (\frac {(-a+b) \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right )^{3/2}}+3 a \left (12 A b^2-8 a b B+a^2 (3 A+4 C)\right ) \sin (c+d x)-\frac {12 a b^3 \left (A b^2+a (-b B+a C)\right ) \sin (c+d x)}{(a-b) (a+b) (b+a \cos (c+d x))}+3 a^2 (-2 A b+a B) \sin (2 (c+d x))+a^3 A \sin (3 (c+d x))}{12 a^5 d} \]

[In]

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

[Out]

(6*(-8*A*b^3 + a^3*B + 6*a*b^2*B - 2*a^2*b*(A + 2*C))*(c + d*x) + (24*b^2*(4*A*b^4 + 4*a^3*b*B - 3*a*b^3*B - 3
*a^4*C + a^2*b^2*(-5*A + 2*C))*ArcTanh[((-a + b)*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(a^2 - b^2)^(3/2) + 3*a*(
12*A*b^2 - 8*a*b*B + a^2*(3*A + 4*C))*Sin[c + d*x] - (12*a*b^3*(A*b^2 + a*(-(b*B) + a*C))*Sin[c + d*x])/((a -
b)*(a + b)*(b + a*Cos[c + d*x])) + 3*a^2*(-2*A*b + a*B)*Sin[2*(c + d*x)] + a^3*A*Sin[3*(c + d*x)])/(12*a^5*d)

Maple [A] (verified)

Time = 0.80 (sec) , antiderivative size = 397, normalized size of antiderivative = 1.00

method	result	size
derivativedivides	\(\frac {-\frac {2 b^{2} \left (-\frac {a b \left (A \,b^{2}-B a b +C \,a^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{\left (a^{2}-b^{2}\right ) \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a -\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} b -a -b \right )}-\frac {\left (5 A \,a^{2} b^{2}-4 A \,b^{4}-4 B \,a^{3} b +3 B a \,b^{3}+3 a^{4} C -2 C \,a^{2} b^{2}\right ) \operatorname {arctanh}\left (\frac {\left (a -b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (a +b \right ) \left (a -b \right )}}\right )}{\left (a +b \right ) \left (a -b \right ) \sqrt {\left (a +b \right ) \left (a -b \right )}}\right )}{a^{5}}-\frac {2 \left (\frac {\left (-a^{3} A -A \,a^{2} b -3 a A \,b^{2}+\frac {1}{2} B \,a^{3}+2 B \,a^{2} b -a^{3} C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}+\left (-\frac {2}{3} a^{3} A -6 a A \,b^{2}+4 B \,a^{2} b -2 a^{3} C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}+\left (-a^{3} A -3 a A \,b^{2}+2 B \,a^{2} b -a^{3} C +A \,a^{2} b -\frac {1}{2} B \,a^{3}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{3}}+\frac {\left (2 A \,a^{2} b +8 A \,b^{3}-B \,a^{3}-6 B a \,b^{2}+4 a^{2} b C \right ) \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}\right )}{a^{5}}}{d}\)	\(397\)
default	\(\frac {-\frac {2 b^{2} \left (-\frac {a b \left (A \,b^{2}-B a b +C \,a^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{\left (a^{2}-b^{2}\right ) \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a -\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} b -a -b \right )}-\frac {\left (5 A \,a^{2} b^{2}-4 A \,b^{4}-4 B \,a^{3} b +3 B a \,b^{3}+3 a^{4} C -2 C \,a^{2} b^{2}\right ) \operatorname {arctanh}\left (\frac {\left (a -b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (a +b \right ) \left (a -b \right )}}\right )}{\left (a +b \right ) \left (a -b \right ) \sqrt {\left (a +b \right ) \left (a -b \right )}}\right )}{a^{5}}-\frac {2 \left (\frac {\left (-a^{3} A -A \,a^{2} b -3 a A \,b^{2}+\frac {1}{2} B \,a^{3}+2 B \,a^{2} b -a^{3} C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}+\left (-\frac {2}{3} a^{3} A -6 a A \,b^{2}+4 B \,a^{2} b -2 a^{3} C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}+\left (-a^{3} A -3 a A \,b^{2}+2 B \,a^{2} b -a^{3} C +A \,a^{2} b -\frac {1}{2} B \,a^{3}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{3}}+\frac {\left (2 A \,a^{2} b +8 A \,b^{3}-B \,a^{3}-6 B a \,b^{2}+4 a^{2} b C \right ) \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}\right )}{a^{5}}}{d}\)	\(397\)
risch	\(\text {Expression too large to display}\)	\(1436\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.38 (sec) , antiderivative size = 1347, normalized size of antiderivative = 3.40 \[ \int \frac {\cos ^3(c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{(a+b \sec (c+d x))^2} \, dx=\text {Too large to display} \]

[In]

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

[Out]

[1/6*(3*(B*a^8 - 2*(A + 2*C)*a^7*b + 4*B*a^6*b^2 - 4*(A - 2*C)*a^5*b^3 - 11*B*a^4*b^4 + 2*(7*A - 2*C)*a^3*b^5
+ 6*B*a^2*b^6 - 8*A*a*b^7)*d*x*cos(d*x + c) + 3*(B*a^7*b - 2*(A + 2*C)*a^6*b^2 + 4*B*a^5*b^3 - 4*(A - 2*C)*a^4
*b^4 - 11*B*a^3*b^5 + 2*(7*A - 2*C)*a^2*b^6 + 6*B*a*b^7 - 8*A*b^8)*d*x + 3*(3*C*a^4*b^3 - 4*B*a^3*b^4 + (5*A -
2*C)*a^2*b^5 + 3*B*a*b^6 - 4*A*b^7 + (3*C*a^5*b^2 - 4*B*a^4*b^3 + (5*A - 2*C)*a^3*b^4 + 3*B*a^2*b^5 - 4*A*a*b
^6)*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*(2*A + 3
*C)*a^7*b - 12*B*a^6*b^2 + 2*(5*A - 9*C)*a^5*b^3 + 30*B*a^4*b^4 - 2*(19*A - 6*C)*a^3*b^5 - 18*B*a^2*b^6 + 24*A
*a*b^7 + 2*(A*a^8 - 2*A*a^6*b^2 + A*a^4*b^4)*cos(d*x + c)^3 + (3*B*a^8 - 4*A*a^7*b - 6*B*a^6*b^2 + 8*A*a^5*b^3
+ 3*B*a^4*b^4 - 4*A*a^3*b^5)*cos(d*x + c)^2 + (2*(2*A + 3*C)*a^8 - 9*B*a^7*b + 4*(A - 3*C)*a^6*b^2 + 18*B*a^5
*b^3 - 2*(10*A - 3*C)*a^4*b^4 - 9*B*a^3*b^5 + 12*A*a^2*b^6)*cos(d*x + c))*sin(d*x + c))/((a^10 - 2*a^8*b^2 + a
^6*b^4)*d*cos(d*x + c) + (a^9*b - 2*a^7*b^3 + a^5*b^5)*d), 1/6*(3*(B*a^8 - 2*(A + 2*C)*a^7*b + 4*B*a^6*b^2 - 4
*(A - 2*C)*a^5*b^3 - 11*B*a^4*b^4 + 2*(7*A - 2*C)*a^3*b^5 + 6*B*a^2*b^6 - 8*A*a*b^7)*d*x*cos(d*x + c) + 3*(B*a
^7*b - 2*(A + 2*C)*a^6*b^2 + 4*B*a^5*b^3 - 4*(A - 2*C)*a^4*b^4 - 11*B*a^3*b^5 + 2*(7*A - 2*C)*a^2*b^6 + 6*B*a*
b^7 - 8*A*b^8)*d*x + 6*(3*C*a^4*b^3 - 4*B*a^3*b^4 + (5*A - 2*C)*a^2*b^5 + 3*B*a*b^6 - 4*A*b^7 + (3*C*a^5*b^2 -
4*B*a^4*b^3 + (5*A - 2*C)*a^3*b^4 + 3*B*a^2*b^5 - 4*A*a*b^6)*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))) + (2*(2*A + 3*C)*a^7*b - 12*B*a^6*b^2 + 2*(5*A - 9*C)
*a^5*b^3 + 30*B*a^4*b^4 - 2*(19*A - 6*C)*a^3*b^5 - 18*B*a^2*b^6 + 24*A*a*b^7 + 2*(A*a^8 - 2*A*a^6*b^2 + A*a^4*
b^4)*cos(d*x + c)^3 + (3*B*a^8 - 4*A*a^7*b - 6*B*a^6*b^2 + 8*A*a^5*b^3 + 3*B*a^4*b^4 - 4*A*a^3*b^5)*cos(d*x +
c)^2 + (2*(2*A + 3*C)*a^8 - 9*B*a^7*b + 4*(A - 3*C)*a^6*b^2 + 18*B*a^5*b^3 - 2*(10*A - 3*C)*a^4*b^4 - 9*B*a^3*
b^5 + 12*A*a^2*b^6)*cos(d*x + c))*sin(d*x + c))/((a^10 - 2*a^8*b^2 + a^6*b^4)*d*cos(d*x + c) + (a^9*b - 2*a^7*
b^3 + a^5*b^5)*d)]

Sympy [F]

\[ \int \frac {\cos ^3(c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{(a+b \sec (c+d x))^2} \, dx=\int \frac {\left (A + B \sec {\left (c + d x \right )} + C \sec ^{2}{\left (c + d x \right )}\right ) \cos ^{3}{\left (c + d x \right )}}{\left (a + b \sec {\left (c + d x \right )}\right )^{2}}\, dx \]

[In]

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

[Out]

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

Maxima [F(-2)]

Exception generated. \[ \int \frac {\cos ^3(c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{(a+b \sec (c+d x))^2} \, dx=\text {Exception raised: ValueError} \]

[In]

integrate(cos(d*x+c)^3*(A+B*sec(d*x+c)+C*sec(d*x+c)^2)/(a+b*sec(d*x+c))^2,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 [A] (verification not implemented)

none

Time = 0.33 (sec) , antiderivative size = 564, normalized size of antiderivative = 1.42 \[ \int \frac {\cos ^3(c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{(a+b \sec (c+d x))^2} \, dx=\frac {\frac {12 \, {\left (3 \, C a^{4} b^{2} - 4 \, B a^{3} b^{3} + 5 \, A a^{2} b^{4} - 2 \, C a^{2} b^{4} + 3 \, B a b^{5} - 4 \, A b^{6}\right )} {\left (\pi \left \lfloor \frac {d x + c}{2 \, \pi } + \frac {1}{2} \right \rfloor \mathrm {sgn}\left (-2 \, a + 2 \, b\right ) + \arctan \left (-\frac {a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{\sqrt {-a^{2} + b^{2}}}\right )\right )}}{{\left (a^{7} - a^{5} b^{2}\right )} \sqrt {-a^{2} + b^{2}}} + \frac {12 \, {\left (C a^{2} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - B a b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + A b^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (a^{6} - a^{4} b^{2}\right )} {\left (a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - a - b\right )}} + \frac {3 \, {\left (B a^{3} - 2 \, A a^{2} b - 4 \, C a^{2} b + 6 \, B a b^{2} - 8 \, A b^{3}\right )} {\left (d x + c\right )}}{a^{5}} + \frac {2 \, {\left (6 \, A a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 3 \, B a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 6 \, C a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 6 \, A a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 12 \, B a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 18 \, A b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 4 \, A a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 12 \, C a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 24 \, B a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 36 \, A b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 6 \, A a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 3 \, B a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 6 \, C a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 6 \, A a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 12 \, B a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 18 \, A b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{3} a^{4}}}{6 \, d} \]

[In]

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

[Out]

1/6*(12*(3*C*a^4*b^2 - 4*B*a^3*b^3 + 5*A*a^2*b^4 - 2*C*a^2*b^4 + 3*B*a*b^5 - 4*A*b^6)*(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^7
- a^5*b^2)*sqrt(-a^2 + b^2)) + 12*(C*a^2*b^3*tan(1/2*d*x + 1/2*c) - B*a*b^4*tan(1/2*d*x + 1/2*c) + A*b^5*tan(
1/2*d*x + 1/2*c))/((a^6 - a^4*b^2)*(a*tan(1/2*d*x + 1/2*c)^2 - b*tan(1/2*d*x + 1/2*c)^2 - a - b)) + 3*(B*a^3 -
2*A*a^2*b - 4*C*a^2*b + 6*B*a*b^2 - 8*A*b^3)*(d*x + c)/a^5 + 2*(6*A*a^2*tan(1/2*d*x + 1/2*c)^5 - 3*B*a^2*tan(
1/2*d*x + 1/2*c)^5 + 6*C*a^2*tan(1/2*d*x + 1/2*c)^5 + 6*A*a*b*tan(1/2*d*x + 1/2*c)^5 - 12*B*a*b*tan(1/2*d*x +
1/2*c)^5 + 18*A*b^2*tan(1/2*d*x + 1/2*c)^5 + 4*A*a^2*tan(1/2*d*x + 1/2*c)^3 + 12*C*a^2*tan(1/2*d*x + 1/2*c)^3
- 24*B*a*b*tan(1/2*d*x + 1/2*c)^3 + 36*A*b^2*tan(1/2*d*x + 1/2*c)^3 + 6*A*a^2*tan(1/2*d*x + 1/2*c) + 3*B*a^2*t
an(1/2*d*x + 1/2*c) + 6*C*a^2*tan(1/2*d*x + 1/2*c) - 6*A*a*b*tan(1/2*d*x + 1/2*c) - 12*B*a*b*tan(1/2*d*x + 1/2
*c) + 18*A*b^2*tan(1/2*d*x + 1/2*c))/((tan(1/2*d*x + 1/2*c)^2 + 1)^3*a^4))/d

Mupad [B] (verification not implemented)

Time = 29.66 (sec) , antiderivative size = 11743, normalized size of antiderivative = 29.65 \[ \int \frac {\cos ^3(c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{(a+b \sec (c+d x))^2} \, dx=\text {Too large to display} \]

[In]

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

[Out]

((tan(c/2 + (d*x)/2)*(2*A*a^5 - 8*A*b^5 + B*a^5 + 2*C*a^5 + 6*A*a^2*b^3 + 2*A*a^3*b^2 + 3*B*a^2*b^3 - 5*B*a^3*
b^2 - 4*C*a^2*b^3 - 2*C*a^3*b^2 - 4*A*a*b^4 + 6*B*a*b^4 - 3*B*a^4*b + 2*C*a^4*b))/(a^4*(a + b)*(a - b)) - (tan
(c/2 + (d*x)/2)^3*(2*A*a^5 + 72*A*b^5 + 3*B*a^5 - 6*C*a^5 - 38*A*a^2*b^3 - 14*A*a^3*b^2 - 9*B*a^2*b^3 + 33*B*a
^3*b^2 + 36*C*a^2*b^3 + 6*C*a^3*b^2 + 12*A*a*b^4 - 16*A*a^4*b - 54*B*a*b^4 + 9*B*a^4*b - 18*C*a^4*b))/(3*a^4*(
a + b)*(a - b)) + (tan(c/2 + (d*x)/2)^5*(2*A*a^5 - 72*A*b^5 - 3*B*a^5 - 6*C*a^5 + 38*A*a^2*b^3 - 14*A*a^3*b^2
- 9*B*a^2*b^3 - 33*B*a^3*b^2 - 36*C*a^2*b^3 + 6*C*a^3*b^2 + 12*A*a*b^4 + 16*A*a^4*b + 54*B*a*b^4 + 9*B*a^4*b +
18*C*a^4*b))/(3*a^4*(a + b)*(a - b)) - (tan(c/2 + (d*x)/2)^7*(2*A*a^5 + 8*A*b^5 - B*a^5 + 2*C*a^5 - 6*A*a^2*b
^3 + 2*A*a^3*b^2 + 3*B*a^2*b^3 + 5*B*a^3*b^2 + 4*C*a^2*b^3 - 2*C*a^3*b^2 - 4*A*a*b^4 - 6*B*a*b^4 - 3*B*a^4*b -
2*C*a^4*b))/(a^4*(a + b)*(a - b)))/(d*(a + b - tan(c/2 + (d*x)/2)^8*(a - b) + tan(c/2 + (d*x)/2)^2*(2*a + 4*b
) - tan(c/2 + (d*x)/2)^6*(2*a - 4*b) + 6*b*tan(c/2 + (d*x)/2)^4)) - (atan(((((((8*(2*B*a^18 + 16*A*a^10*b^8 -
8*A*a^11*b^7 - 36*A*a^12*b^6 + 16*A*a^13*b^5 + 20*A*a^14*b^4 - 4*A*a^15*b^3 - 12*B*a^11*b^7 + 6*B*a^12*b^6 + 2
8*B*a^13*b^5 - 14*B*a^14*b^4 - 16*B*a^15*b^3 + 6*B*a^16*b^2 + 8*C*a^12*b^6 - 4*C*a^13*b^5 - 20*C*a^14*b^4 + 12
*C*a^15*b^3 + 12*C*a^16*b^2 - 4*A*a^17*b - 8*C*a^17*b))/(a^14*b + a^15 - a^12*b^3 - a^13*b^2) - (8*tan(c/2 + (
d*x)/2)*(A*b^3*4i - (B*a^3*1i)/2 + a^2*(A*b*1i + C*b*2i) - B*a*b^2*3i)*(8*a^15*b - 8*a^10*b^6 + 8*a^11*b^5 + 1
6*a^12*b^4 - 16*a^13*b^3 - 8*a^14*b^2))/(a^5*(a^10*b + a^11 - a^8*b^3 - a^9*b^2)))*(A*b^3*4i - (B*a^3*1i)/2 +
a^2*(A*b*1i + C*b*2i) - B*a*b^2*3i))/a^5 + (8*tan(c/2 + (d*x)/2)*(128*A^2*b^12 + B^2*a^12 - 128*A^2*a*b^11 - 2
*B^2*a^11*b - 192*A^2*a^2*b^10 + 192*A^2*a^3*b^9 + 8*A^2*a^4*b^8 - 8*A^2*a^5*b^7 + 28*A^2*a^6*b^6 - 48*A^2*a^7
*b^5 + 28*A^2*a^8*b^4 - 8*A^2*a^9*b^3 + 4*A^2*a^10*b^2 + 72*B^2*a^2*b^10 - 72*B^2*a^3*b^9 - 120*B^2*a^4*b^8 +
120*B^2*a^5*b^7 + 17*B^2*a^6*b^6 - 26*B^2*a^7*b^5 + 23*B^2*a^8*b^4 - 20*B^2*a^9*b^3 + 11*B^2*a^10*b^2 + 32*C^2
*a^4*b^8 - 32*C^2*a^5*b^7 - 64*C^2*a^6*b^6 + 64*C^2*a^7*b^5 + 20*C^2*a^8*b^4 - 32*C^2*a^9*b^3 + 16*C^2*a^10*b^
2 - 192*A*B*a*b^11 - 4*A*B*a^11*b - 8*B*C*a^11*b + 192*A*B*a^2*b^10 + 304*A*B*a^3*b^9 - 304*A*B*a^4*b^8 - 28*A
*B*a^5*b^7 + 40*A*B*a^6*b^6 - 52*A*B*a^7*b^5 + 64*A*B*a^8*b^4 - 36*A*B*a^9*b^3 + 8*A*B*a^10*b^2 + 128*A*C*a^2*
b^10 - 128*A*C*a^3*b^9 - 224*A*C*a^4*b^8 + 224*A*C*a^5*b^7 + 40*A*C*a^6*b^6 - 64*A*C*a^7*b^5 + 48*A*C*a^8*b^4
- 32*A*C*a^9*b^3 + 16*A*C*a^10*b^2 - 96*B*C*a^3*b^9 + 96*B*C*a^4*b^8 + 176*B*C*a^5*b^7 - 176*B*C*a^6*b^6 - 40*
B*C*a^7*b^5 + 64*B*C*a^8*b^4 - 40*B*C*a^9*b^3 + 16*B*C*a^10*b^2))/(a^10*b + a^11 - a^8*b^3 - a^9*b^2))*(A*b^3*
4i - (B*a^3*1i)/2 + a^2*(A*b*1i + C*b*2i) - B*a*b^2*3i)*1i)/a^5 - (((((8*(2*B*a^18 + 16*A*a^10*b^8 - 8*A*a^11*
b^7 - 36*A*a^12*b^6 + 16*A*a^13*b^5 + 20*A*a^14*b^4 - 4*A*a^15*b^3 - 12*B*a^11*b^7 + 6*B*a^12*b^6 + 28*B*a^13*
b^5 - 14*B*a^14*b^4 - 16*B*a^15*b^3 + 6*B*a^16*b^2 + 8*C*a^12*b^6 - 4*C*a^13*b^5 - 20*C*a^14*b^4 + 12*C*a^15*b
^3 + 12*C*a^16*b^2 - 4*A*a^17*b - 8*C*a^17*b))/(a^14*b + a^15 - a^12*b^3 - a^13*b^2) + (8*tan(c/2 + (d*x)/2)*(
A*b^3*4i - (B*a^3*1i)/2 + a^2*(A*b*1i + C*b*2i) - B*a*b^2*3i)*(8*a^15*b - 8*a^10*b^6 + 8*a^11*b^5 + 16*a^12*b^
4 - 16*a^13*b^3 - 8*a^14*b^2))/(a^5*(a^10*b + a^11 - a^8*b^3 - a^9*b^2)))*(A*b^3*4i - (B*a^3*1i)/2 + a^2*(A*b*
1i + C*b*2i) - B*a*b^2*3i))/a^5 - (8*tan(c/2 + (d*x)/2)*(128*A^2*b^12 + B^2*a^12 - 128*A^2*a*b^11 - 2*B^2*a^11
*b - 192*A^2*a^2*b^10 + 192*A^2*a^3*b^9 + 8*A^2*a^4*b^8 - 8*A^2*a^5*b^7 + 28*A^2*a^6*b^6 - 48*A^2*a^7*b^5 + 28
*A^2*a^8*b^4 - 8*A^2*a^9*b^3 + 4*A^2*a^10*b^2 + 72*B^2*a^2*b^10 - 72*B^2*a^3*b^9 - 120*B^2*a^4*b^8 + 120*B^2*a
^5*b^7 + 17*B^2*a^6*b^6 - 26*B^2*a^7*b^5 + 23*B^2*a^8*b^4 - 20*B^2*a^9*b^3 + 11*B^2*a^10*b^2 + 32*C^2*a^4*b^8
- 32*C^2*a^5*b^7 - 64*C^2*a^6*b^6 + 64*C^2*a^7*b^5 + 20*C^2*a^8*b^4 - 32*C^2*a^9*b^3 + 16*C^2*a^10*b^2 - 192*A
*B*a*b^11 - 4*A*B*a^11*b - 8*B*C*a^11*b + 192*A*B*a^2*b^10 + 304*A*B*a^3*b^9 - 304*A*B*a^4*b^8 - 28*A*B*a^5*b^
7 + 40*A*B*a^6*b^6 - 52*A*B*a^7*b^5 + 64*A*B*a^8*b^4 - 36*A*B*a^9*b^3 + 8*A*B*a^10*b^2 + 128*A*C*a^2*b^10 - 12
8*A*C*a^3*b^9 - 224*A*C*a^4*b^8 + 224*A*C*a^5*b^7 + 40*A*C*a^6*b^6 - 64*A*C*a^7*b^5 + 48*A*C*a^8*b^4 - 32*A*C*
a^9*b^3 + 16*A*C*a^10*b^2 - 96*B*C*a^3*b^9 + 96*B*C*a^4*b^8 + 176*B*C*a^5*b^7 - 176*B*C*a^6*b^6 - 40*B*C*a^7*b
^5 + 64*B*C*a^8*b^4 - 40*B*C*a^9*b^3 + 16*B*C*a^10*b^2))/(a^10*b + a^11 - a^8*b^3 - a^9*b^2))*(A*b^3*4i - (B*a
^3*1i)/2 + a^2*(A*b*1i + C*b*2i) - B*a*b^2*3i)*1i)/a^5)/((16*(256*A^3*b^14 - 128*A^3*a*b^13 - 448*A^3*a^2*b^12
+ 192*A^3*a^3*b^11 + 48*A^3*a^4*b^10 - 24*A^3*a^5*b^9 + 124*A^3*a^6*b^8 - 20*A^3*a^7*b^7 + 20*A^3*a^8*b^6 - 1
08*B^3*a^3*b^11 + 54*B^3*a^4*b^10 + 216*B^3*a^5*b^9 - 81*B^3*a^6*b^8 - 63*B^3*a^7*b^7 + 9*B^3*a^8*b^6 - 41*B^3
*a^9*b^5 + 4*B^3*a^10*b^4 - 4*B^3*a^11*b^3 + 32*C^3*a^6*b^8 - 16*C^3*a^7*b^7 - 80*C^3*a^8*b^6 + 24*C^3*a^9*b^5
+ 48*C^3*a^10*b^4 - 576*A^2*B*a*b^13 + 432*A*B^2*a^2*b^12 - 216*A*B^2*a^3*b^11 - 828*A*B^2*a^4*b^10 + 324*A*B
^2*a^5*b^9 + 192*A*B^2*a^6*b^8 - 39*A*B^2*a^7*b^7 + 183*A*B^2*a^8*b^6 - 21*A*B^2*a^9*b^5 + 21*A*B^2*a^10*b^4 +
288*A^2*B*a^2*b^12 + 1056*A^2*B*a^3*b^11 - 432*A^2*B*a^4*b^10 - 180*A^2*B*a^5*b^9 + 54*A^2*B*a^6*b^8 - 264*A^
2*B*a^7*b^7 + 36*A^2*B*a^8*b^6 - 36*A^2*B*a^9*b^5 + 192*A*C^2*a^4*b^10 - 96*A*C^2*a^5*b^9 - 432*A*C^2*a^6*b^8
+ 144*A*C^2*a^7*b^7 + 192*A*C^2*a^8*b^6 - 12*A*C^2*a^9*b^5 + 48*A*C^2*a^10*b^4 + 384*A^2*C*a^2*b^12 - 192*A^2*
C*a^3*b^11 - 768*A^2*C*a^4*b^10 + 288*A^2*C*a^5*b^9 + 216*A^2*C*a^6*b^8 - 36*A^2*C*a^7*b^7 + 156*A^2*C*a^8*b^6
- 12*A^2*C*a^9*b^5 + 12*A^2*C*a^10*b^4 - 144*B*C^2*a^5*b^9 + 72*B*C^2*a^6*b^8 + 336*B*C^2*a^7*b^7 - 108*B*C^2
*a^8*b^6 - 168*B*C^2*a^9*b^5 + 6*B*C^2*a^10*b^4 - 24*B*C^2*a^11*b^3 + 216*B^2*C*a^4*b^10 - 108*B^2*C*a^5*b^9 -
468*B^2*C*a^6*b^8 + 162*B^2*C*a^7*b^7 + 186*B^2*C*a^8*b^6 - 15*B^2*C*a^9*b^5 + 63*B^2*C*a^10*b^4 - 3*B^2*C*a^
11*b^3 + 3*B^2*C*a^12*b^2 - 576*A*B*C*a^3*b^11 + 288*A*B*C*a^4*b^10 + 1200*A*B*C*a^5*b^9 - 432*A*B*C*a^6*b^8 -
408*A*B*C*a^7*b^7 + 48*A*B*C*a^8*b^6 - 204*A*B*C*a^9*b^5 + 12*A*B*C*a^10*b^4 - 12*A*B*C*a^11*b^3))/(a^14*b +
a^15 - a^12*b^3 - a^13*b^2) + (((((8*(2*B*a^18 + 16*A*a^10*b^8 - 8*A*a^11*b^7 - 36*A*a^12*b^6 + 16*A*a^13*b^5
+ 20*A*a^14*b^4 - 4*A*a^15*b^3 - 12*B*a^11*b^7 + 6*B*a^12*b^6 + 28*B*a^13*b^5 - 14*B*a^14*b^4 - 16*B*a^15*b^3
+ 6*B*a^16*b^2 + 8*C*a^12*b^6 - 4*C*a^13*b^5 - 20*C*a^14*b^4 + 12*C*a^15*b^3 + 12*C*a^16*b^2 - 4*A*a^17*b - 8*
C*a^17*b))/(a^14*b + a^15 - a^12*b^3 - a^13*b^2) - (8*tan(c/2 + (d*x)/2)*(A*b^3*4i - (B*a^3*1i)/2 + a^2*(A*b*1
i + C*b*2i) - B*a*b^2*3i)*(8*a^15*b - 8*a^10*b^6 + 8*a^11*b^5 + 16*a^12*b^4 - 16*a^13*b^3 - 8*a^14*b^2))/(a^5*
(a^10*b + a^11 - a^8*b^3 - a^9*b^2)))*(A*b^3*4i - (B*a^3*1i)/2 + a^2*(A*b*1i + C*b*2i) - B*a*b^2*3i))/a^5 + (8
*tan(c/2 + (d*x)/2)*(128*A^2*b^12 + B^2*a^12 - 128*A^2*a*b^11 - 2*B^2*a^11*b - 192*A^2*a^2*b^10 + 192*A^2*a^3*
b^9 + 8*A^2*a^4*b^8 - 8*A^2*a^5*b^7 + 28*A^2*a^6*b^6 - 48*A^2*a^7*b^5 + 28*A^2*a^8*b^4 - 8*A^2*a^9*b^3 + 4*A^2
*a^10*b^2 + 72*B^2*a^2*b^10 - 72*B^2*a^3*b^9 - 120*B^2*a^4*b^8 + 120*B^2*a^5*b^7 + 17*B^2*a^6*b^6 - 26*B^2*a^7
*b^5 + 23*B^2*a^8*b^4 - 20*B^2*a^9*b^3 + 11*B^2*a^10*b^2 + 32*C^2*a^4*b^8 - 32*C^2*a^5*b^7 - 64*C^2*a^6*b^6 +
64*C^2*a^7*b^5 + 20*C^2*a^8*b^4 - 32*C^2*a^9*b^3 + 16*C^2*a^10*b^2 - 192*A*B*a*b^11 - 4*A*B*a^11*b - 8*B*C*a^1
1*b + 192*A*B*a^2*b^10 + 304*A*B*a^3*b^9 - 304*A*B*a^4*b^8 - 28*A*B*a^5*b^7 + 40*A*B*a^6*b^6 - 52*A*B*a^7*b^5
+ 64*A*B*a^8*b^4 - 36*A*B*a^9*b^3 + 8*A*B*a^10*b^2 + 128*A*C*a^2*b^10 - 128*A*C*a^3*b^9 - 224*A*C*a^4*b^8 + 22
4*A*C*a^5*b^7 + 40*A*C*a^6*b^6 - 64*A*C*a^7*b^5 + 48*A*C*a^8*b^4 - 32*A*C*a^9*b^3 + 16*A*C*a^10*b^2 - 96*B*C*a
^3*b^9 + 96*B*C*a^4*b^8 + 176*B*C*a^5*b^7 - 176*B*C*a^6*b^6 - 40*B*C*a^7*b^5 + 64*B*C*a^8*b^4 - 40*B*C*a^9*b^3
+ 16*B*C*a^10*b^2))/(a^10*b + a^11 - a^8*b^3 - a^9*b^2))*(A*b^3*4i - (B*a^3*1i)/2 + a^2*(A*b*1i + C*b*2i) - B
*a*b^2*3i))/a^5 + (((((8*(2*B*a^18 + 16*A*a^10*b^8 - 8*A*a^11*b^7 - 36*A*a^12*b^6 + 16*A*a^13*b^5 + 20*A*a^14*
b^4 - 4*A*a^15*b^3 - 12*B*a^11*b^7 + 6*B*a^12*b^6 + 28*B*a^13*b^5 - 14*B*a^14*b^4 - 16*B*a^15*b^3 + 6*B*a^16*b
^2 + 8*C*a^12*b^6 - 4*C*a^13*b^5 - 20*C*a^14*b^4 + 12*C*a^15*b^3 + 12*C*a^16*b^2 - 4*A*a^17*b - 8*C*a^17*b))/(
a^14*b + a^15 - a^12*b^3 - a^13*b^2) + (8*tan(c/2 + (d*x)/2)*(A*b^3*4i - (B*a^3*1i)/2 + a^2*(A*b*1i + C*b*2i)
- B*a*b^2*3i)*(8*a^15*b - 8*a^10*b^6 + 8*a^11*b^5 + 16*a^12*b^4 - 16*a^13*b^3 - 8*a^14*b^2))/(a^5*(a^10*b + a^
11 - a^8*b^3 - a^9*b^2)))*(A*b^3*4i - (B*a^3*1i)/2 + a^2*(A*b*1i + C*b*2i) - B*a*b^2*3i))/a^5 - (8*tan(c/2 + (
d*x)/2)*(128*A^2*b^12 + B^2*a^12 - 128*A^2*a*b^11 - 2*B^2*a^11*b - 192*A^2*a^2*b^10 + 192*A^2*a^3*b^9 + 8*A^2*
a^4*b^8 - 8*A^2*a^5*b^7 + 28*A^2*a^6*b^6 - 48*A^2*a^7*b^5 + 28*A^2*a^8*b^4 - 8*A^2*a^9*b^3 + 4*A^2*a^10*b^2 +
72*B^2*a^2*b^10 - 72*B^2*a^3*b^9 - 120*B^2*a^4*b^8 + 120*B^2*a^5*b^7 + 17*B^2*a^6*b^6 - 26*B^2*a^7*b^5 + 23*B^
2*a^8*b^4 - 20*B^2*a^9*b^3 + 11*B^2*a^10*b^2 + 32*C^2*a^4*b^8 - 32*C^2*a^5*b^7 - 64*C^2*a^6*b^6 + 64*C^2*a^7*b
^5 + 20*C^2*a^8*b^4 - 32*C^2*a^9*b^3 + 16*C^2*a^10*b^2 - 192*A*B*a*b^11 - 4*A*B*a^11*b - 8*B*C*a^11*b + 192*A*
B*a^2*b^10 + 304*A*B*a^3*b^9 - 304*A*B*a^4*b^8 - 28*A*B*a^5*b^7 + 40*A*B*a^6*b^6 - 52*A*B*a^7*b^5 + 64*A*B*a^8
*b^4 - 36*A*B*a^9*b^3 + 8*A*B*a^10*b^2 + 128*A*C*a^2*b^10 - 128*A*C*a^3*b^9 - 224*A*C*a^4*b^8 + 224*A*C*a^5*b^
7 + 40*A*C*a^6*b^6 - 64*A*C*a^7*b^5 + 48*A*C*a^8*b^4 - 32*A*C*a^9*b^3 + 16*A*C*a^10*b^2 - 96*B*C*a^3*b^9 + 96*
B*C*a^4*b^8 + 176*B*C*a^5*b^7 - 176*B*C*a^6*b^6 - 40*B*C*a^7*b^5 + 64*B*C*a^8*b^4 - 40*B*C*a^9*b^3 + 16*B*C*a^
10*b^2))/(a^10*b + a^11 - a^8*b^3 - a^9*b^2))*(A*b^3*4i - (B*a^3*1i)/2 + a^2*(A*b*1i + C*b*2i) - B*a*b^2*3i))/
a^5))*(A*b^3*4i - (B*a^3*1i)/2 + a^2*(A*b*1i + C*b*2i) - B*a*b^2*3i)*2i)/(a^5*d) - (b^2*atan(((b^2*((8*tan(c/2
+ (d*x)/2)*(128*A^2*b^12 + B^2*a^12 - 128*A^2*a*b^11 - 2*B^2*a^11*b - 192*A^2*a^2*b^10 + 192*A^2*a^3*b^9 + 8*
A^2*a^4*b^8 - 8*A^2*a^5*b^7 + 28*A^2*a^6*b^6 - 48*A^2*a^7*b^5 + 28*A^2*a^8*b^4 - 8*A^2*a^9*b^3 + 4*A^2*a^10*b^
2 + 72*B^2*a^2*b^10 - 72*B^2*a^3*b^9 - 120*B^2*a^4*b^8 + 120*B^2*a^5*b^7 + 17*B^2*a^6*b^6 - 26*B^2*a^7*b^5 + 2
3*B^2*a^8*b^4 - 20*B^2*a^9*b^3 + 11*B^2*a^10*b^2 + 32*C^2*a^4*b^8 - 32*C^2*a^5*b^7 - 64*C^2*a^6*b^6 + 64*C^2*a
^7*b^5 + 20*C^2*a^8*b^4 - 32*C^2*a^9*b^3 + 16*C^2*a^10*b^2 - 192*A*B*a*b^11 - 4*A*B*a^11*b - 8*B*C*a^11*b + 19
2*A*B*a^2*b^10 + 304*A*B*a^3*b^9 - 304*A*B*a^4*b^8 - 28*A*B*a^5*b^7 + 40*A*B*a^6*b^6 - 52*A*B*a^7*b^5 + 64*A*B
*a^8*b^4 - 36*A*B*a^9*b^3 + 8*A*B*a^10*b^2 + 128*A*C*a^2*b^10 - 128*A*C*a^3*b^9 - 224*A*C*a^4*b^8 + 224*A*C*a^
5*b^7 + 40*A*C*a^6*b^6 - 64*A*C*a^7*b^5 + 48*A*C*a^8*b^4 - 32*A*C*a^9*b^3 + 16*A*C*a^10*b^2 - 96*B*C*a^3*b^9 +
96*B*C*a^4*b^8 + 176*B*C*a^5*b^7 - 176*B*C*a^6*b^6 - 40*B*C*a^7*b^5 + 64*B*C*a^8*b^4 - 40*B*C*a^9*b^3 + 16*B*
C*a^10*b^2))/(a^10*b + a^11 - a^8*b^3 - a^9*b^2) + (b^2*((8*(2*B*a^18 + 16*A*a^10*b^8 - 8*A*a^11*b^7 - 36*A*a^
12*b^6 + 16*A*a^13*b^5 + 20*A*a^14*b^4 - 4*A*a^15*b^3 - 12*B*a^11*b^7 + 6*B*a^12*b^6 + 28*B*a^13*b^5 - 14*B*a^
14*b^4 - 16*B*a^15*b^3 + 6*B*a^16*b^2 + 8*C*a^12*b^6 - 4*C*a^13*b^5 - 20*C*a^14*b^4 + 12*C*a^15*b^3 + 12*C*a^1
6*b^2 - 4*A*a^17*b - 8*C*a^17*b))/(a^14*b + a^15 - a^12*b^3 - a^13*b^2) - (8*b^2*tan(c/2 + (d*x)/2)*((a + b)^3
*(a - b)^3)^(1/2)*(8*a^15*b - 8*a^10*b^6 + 8*a^11*b^5 + 16*a^12*b^4 - 16*a^13*b^3 - 8*a^14*b^2)*(4*A*b^4 - 3*C
*a^4 - 5*A*a^2*b^2 + 2*C*a^2*b^2 - 3*B*a*b^3 + 4*B*a^3*b))/((a^10*b + a^11 - a^8*b^3 - a^9*b^2)*(a^11 - a^5*b^
6 + 3*a^7*b^4 - 3*a^9*b^2)))*((a + b)^3*(a - b)^3)^(1/2)*(4*A*b^4 - 3*C*a^4 - 5*A*a^2*b^2 + 2*C*a^2*b^2 - 3*B*
a*b^3 + 4*B*a^3*b))/(a^11 - a^5*b^6 + 3*a^7*b^4 - 3*a^9*b^2))*((a + b)^3*(a - b)^3)^(1/2)*(4*A*b^4 - 3*C*a^4 -
5*A*a^2*b^2 + 2*C*a^2*b^2 - 3*B*a*b^3 + 4*B*a^3*b)*1i)/(a^11 - a^5*b^6 + 3*a^7*b^4 - 3*a^9*b^2) + (b^2*((8*ta
n(c/2 + (d*x)/2)*(128*A^2*b^12 + B^2*a^12 - 128*A^2*a*b^11 - 2*B^2*a^11*b - 192*A^2*a^2*b^10 + 192*A^2*a^3*b^9
+ 8*A^2*a^4*b^8 - 8*A^2*a^5*b^7 + 28*A^2*a^6*b^6 - 48*A^2*a^7*b^5 + 28*A^2*a^8*b^4 - 8*A^2*a^9*b^3 + 4*A^2*a^
10*b^2 + 72*B^2*a^2*b^10 - 72*B^2*a^3*b^9 - 120*B^2*a^4*b^8 + 120*B^2*a^5*b^7 + 17*B^2*a^6*b^6 - 26*B^2*a^7*b^
5 + 23*B^2*a^8*b^4 - 20*B^2*a^9*b^3 + 11*B^2*a^10*b^2 + 32*C^2*a^4*b^8 - 32*C^2*a^5*b^7 - 64*C^2*a^6*b^6 + 64*
C^2*a^7*b^5 + 20*C^2*a^8*b^4 - 32*C^2*a^9*b^3 + 16*C^2*a^10*b^2 - 192*A*B*a*b^11 - 4*A*B*a^11*b - 8*B*C*a^11*b
+ 192*A*B*a^2*b^10 + 304*A*B*a^3*b^9 - 304*A*B*a^4*b^8 - 28*A*B*a^5*b^7 + 40*A*B*a^6*b^6 - 52*A*B*a^7*b^5 + 6
4*A*B*a^8*b^4 - 36*A*B*a^9*b^3 + 8*A*B*a^10*b^2 + 128*A*C*a^2*b^10 - 128*A*C*a^3*b^9 - 224*A*C*a^4*b^8 + 224*A
*C*a^5*b^7 + 40*A*C*a^6*b^6 - 64*A*C*a^7*b^5 + 48*A*C*a^8*b^4 - 32*A*C*a^9*b^3 + 16*A*C*a^10*b^2 - 96*B*C*a^3*
b^9 + 96*B*C*a^4*b^8 + 176*B*C*a^5*b^7 - 176*B*C*a^6*b^6 - 40*B*C*a^7*b^5 + 64*B*C*a^8*b^4 - 40*B*C*a^9*b^3 +
16*B*C*a^10*b^2))/(a^10*b + a^11 - a^8*b^3 - a^9*b^2) - (b^2*((8*(2*B*a^18 + 16*A*a^10*b^8 - 8*A*a^11*b^7 - 36
*A*a^12*b^6 + 16*A*a^13*b^5 + 20*A*a^14*b^4 - 4*A*a^15*b^3 - 12*B*a^11*b^7 + 6*B*a^12*b^6 + 28*B*a^13*b^5 - 14
*B*a^14*b^4 - 16*B*a^15*b^3 + 6*B*a^16*b^2 + 8*C*a^12*b^6 - 4*C*a^13*b^5 - 20*C*a^14*b^4 + 12*C*a^15*b^3 + 12*
C*a^16*b^2 - 4*A*a^17*b - 8*C*a^17*b))/(a^14*b + a^15 - a^12*b^3 - a^13*b^2) + (8*b^2*tan(c/2 + (d*x)/2)*((a +
b)^3*(a - b)^3)^(1/2)*(8*a^15*b - 8*a^10*b^6 + 8*a^11*b^5 + 16*a^12*b^4 - 16*a^13*b^3 - 8*a^14*b^2)*(4*A*b^4
- 3*C*a^4 - 5*A*a^2*b^2 + 2*C*a^2*b^2 - 3*B*a*b^3 + 4*B*a^3*b))/((a^10*b + a^11 - a^8*b^3 - a^9*b^2)*(a^11 - a
^5*b^6 + 3*a^7*b^4 - 3*a^9*b^2)))*((a + b)^3*(a - b)^3)^(1/2)*(4*A*b^4 - 3*C*a^4 - 5*A*a^2*b^2 + 2*C*a^2*b^2 -
3*B*a*b^3 + 4*B*a^3*b))/(a^11 - a^5*b^6 + 3*a^7*b^4 - 3*a^9*b^2))*((a + b)^3*(a - b)^3)^(1/2)*(4*A*b^4 - 3*C*
a^4 - 5*A*a^2*b^2 + 2*C*a^2*b^2 - 3*B*a*b^3 + 4*B*a^3*b)*1i)/(a^11 - a^5*b^6 + 3*a^7*b^4 - 3*a^9*b^2))/((16*(2
56*A^3*b^14 - 128*A^3*a*b^13 - 448*A^3*a^2*b^12 + 192*A^3*a^3*b^11 + 48*A^3*a^4*b^10 - 24*A^3*a^5*b^9 + 124*A^
3*a^6*b^8 - 20*A^3*a^7*b^7 + 20*A^3*a^8*b^6 - 108*B^3*a^3*b^11 + 54*B^3*a^4*b^10 + 216*B^3*a^5*b^9 - 81*B^3*a^
6*b^8 - 63*B^3*a^7*b^7 + 9*B^3*a^8*b^6 - 41*B^3*a^9*b^5 + 4*B^3*a^10*b^4 - 4*B^3*a^11*b^3 + 32*C^3*a^6*b^8 - 1
6*C^3*a^7*b^7 - 80*C^3*a^8*b^6 + 24*C^3*a^9*b^5 + 48*C^3*a^10*b^4 - 576*A^2*B*a*b^13 + 432*A*B^2*a^2*b^12 - 21
6*A*B^2*a^3*b^11 - 828*A*B^2*a^4*b^10 + 324*A*B^2*a^5*b^9 + 192*A*B^2*a^6*b^8 - 39*A*B^2*a^7*b^7 + 183*A*B^2*a
^8*b^6 - 21*A*B^2*a^9*b^5 + 21*A*B^2*a^10*b^4 + 288*A^2*B*a^2*b^12 + 1056*A^2*B*a^3*b^11 - 432*A^2*B*a^4*b^10
- 180*A^2*B*a^5*b^9 + 54*A^2*B*a^6*b^8 - 264*A^2*B*a^7*b^7 + 36*A^2*B*a^8*b^6 - 36*A^2*B*a^9*b^5 + 192*A*C^2*a
^4*b^10 - 96*A*C^2*a^5*b^9 - 432*A*C^2*a^6*b^8 + 144*A*C^2*a^7*b^7 + 192*A*C^2*a^8*b^6 - 12*A*C^2*a^9*b^5 + 48
*A*C^2*a^10*b^4 + 384*A^2*C*a^2*b^12 - 192*A^2*C*a^3*b^11 - 768*A^2*C*a^4*b^10 + 288*A^2*C*a^5*b^9 + 216*A^2*C
*a^6*b^8 - 36*A^2*C*a^7*b^7 + 156*A^2*C*a^8*b^6 - 12*A^2*C*a^9*b^5 + 12*A^2*C*a^10*b^4 - 144*B*C^2*a^5*b^9 + 7
2*B*C^2*a^6*b^8 + 336*B*C^2*a^7*b^7 - 108*B*C^2*a^8*b^6 - 168*B*C^2*a^9*b^5 + 6*B*C^2*a^10*b^4 - 24*B*C^2*a^11
*b^3 + 216*B^2*C*a^4*b^10 - 108*B^2*C*a^5*b^9 - 468*B^2*C*a^6*b^8 + 162*B^2*C*a^7*b^7 + 186*B^2*C*a^8*b^6 - 15
*B^2*C*a^9*b^5 + 63*B^2*C*a^10*b^4 - 3*B^2*C*a^11*b^3 + 3*B^2*C*a^12*b^2 - 576*A*B*C*a^3*b^11 + 288*A*B*C*a^4*
b^10 + 1200*A*B*C*a^5*b^9 - 432*A*B*C*a^6*b^8 - 408*A*B*C*a^7*b^7 + 48*A*B*C*a^8*b^6 - 204*A*B*C*a^9*b^5 + 12*
A*B*C*a^10*b^4 - 12*A*B*C*a^11*b^3))/(a^14*b + a^15 - a^12*b^3 - a^13*b^2) + (b^2*((8*tan(c/2 + (d*x)/2)*(128*
A^2*b^12 + B^2*a^12 - 128*A^2*a*b^11 - 2*B^2*a^11*b - 192*A^2*a^2*b^10 + 192*A^2*a^3*b^9 + 8*A^2*a^4*b^8 - 8*A
^2*a^5*b^7 + 28*A^2*a^6*b^6 - 48*A^2*a^7*b^5 + 28*A^2*a^8*b^4 - 8*A^2*a^9*b^3 + 4*A^2*a^10*b^2 + 72*B^2*a^2*b^
10 - 72*B^2*a^3*b^9 - 120*B^2*a^4*b^8 + 120*B^2*a^5*b^7 + 17*B^2*a^6*b^6 - 26*B^2*a^7*b^5 + 23*B^2*a^8*b^4 - 2
0*B^2*a^9*b^3 + 11*B^2*a^10*b^2 + 32*C^2*a^4*b^8 - 32*C^2*a^5*b^7 - 64*C^2*a^6*b^6 + 64*C^2*a^7*b^5 + 20*C^2*a
^8*b^4 - 32*C^2*a^9*b^3 + 16*C^2*a^10*b^2 - 192*A*B*a*b^11 - 4*A*B*a^11*b - 8*B*C*a^11*b + 192*A*B*a^2*b^10 +
304*A*B*a^3*b^9 - 304*A*B*a^4*b^8 - 28*A*B*a^5*b^7 + 40*A*B*a^6*b^6 - 52*A*B*a^7*b^5 + 64*A*B*a^8*b^4 - 36*A*B
*a^9*b^3 + 8*A*B*a^10*b^2 + 128*A*C*a^2*b^10 - 128*A*C*a^3*b^9 - 224*A*C*a^4*b^8 + 224*A*C*a^5*b^7 + 40*A*C*a^
6*b^6 - 64*A*C*a^7*b^5 + 48*A*C*a^8*b^4 - 32*A*C*a^9*b^3 + 16*A*C*a^10*b^2 - 96*B*C*a^3*b^9 + 96*B*C*a^4*b^8 +
176*B*C*a^5*b^7 - 176*B*C*a^6*b^6 - 40*B*C*a^7*b^5 + 64*B*C*a^8*b^4 - 40*B*C*a^9*b^3 + 16*B*C*a^10*b^2))/(a^1
0*b + a^11 - a^8*b^3 - a^9*b^2) + (b^2*((8*(2*B*a^18 + 16*A*a^10*b^8 - 8*A*a^11*b^7 - 36*A*a^12*b^6 + 16*A*a^1
3*b^5 + 20*A*a^14*b^4 - 4*A*a^15*b^3 - 12*B*a^11*b^7 + 6*B*a^12*b^6 + 28*B*a^13*b^5 - 14*B*a^14*b^4 - 16*B*a^1
5*b^3 + 6*B*a^16*b^2 + 8*C*a^12*b^6 - 4*C*a^13*b^5 - 20*C*a^14*b^4 + 12*C*a^15*b^3 + 12*C*a^16*b^2 - 4*A*a^17*
b - 8*C*a^17*b))/(a^14*b + a^15 - a^12*b^3 - a^13*b^2) - (8*b^2*tan(c/2 + (d*x)/2)*((a + b)^3*(a - b)^3)^(1/2)
*(8*a^15*b - 8*a^10*b^6 + 8*a^11*b^5 + 16*a^12*b^4 - 16*a^13*b^3 - 8*a^14*b^2)*(4*A*b^4 - 3*C*a^4 - 5*A*a^2*b^
2 + 2*C*a^2*b^2 - 3*B*a*b^3 + 4*B*a^3*b))/((a^10*b + a^11 - a^8*b^3 - a^9*b^2)*(a^11 - a^5*b^6 + 3*a^7*b^4 - 3
*a^9*b^2)))*((a + b)^3*(a - b)^3)^(1/2)*(4*A*b^4 - 3*C*a^4 - 5*A*a^2*b^2 + 2*C*a^2*b^2 - 3*B*a*b^3 + 4*B*a^3*b
))/(a^11 - a^5*b^6 + 3*a^7*b^4 - 3*a^9*b^2))*((a + b)^3*(a - b)^3)^(1/2)*(4*A*b^4 - 3*C*a^4 - 5*A*a^2*b^2 + 2*
C*a^2*b^2 - 3*B*a*b^3 + 4*B*a^3*b))/(a^11 - a^5*b^6 + 3*a^7*b^4 - 3*a^9*b^2) - (b^2*((8*tan(c/2 + (d*x)/2)*(12
8*A^2*b^12 + B^2*a^12 - 128*A^2*a*b^11 - 2*B^2*a^11*b - 192*A^2*a^2*b^10 + 192*A^2*a^3*b^9 + 8*A^2*a^4*b^8 - 8
*A^2*a^5*b^7 + 28*A^2*a^6*b^6 - 48*A^2*a^7*b^5 + 28*A^2*a^8*b^4 - 8*A^2*a^9*b^3 + 4*A^2*a^10*b^2 + 72*B^2*a^2*
b^10 - 72*B^2*a^3*b^9 - 120*B^2*a^4*b^8 + 120*B^2*a^5*b^7 + 17*B^2*a^6*b^6 - 26*B^2*a^7*b^5 + 23*B^2*a^8*b^4 -
20*B^2*a^9*b^3 + 11*B^2*a^10*b^2 + 32*C^2*a^4*b^8 - 32*C^2*a^5*b^7 - 64*C^2*a^6*b^6 + 64*C^2*a^7*b^5 + 20*C^2
*a^8*b^4 - 32*C^2*a^9*b^3 + 16*C^2*a^10*b^2 - 192*A*B*a*b^11 - 4*A*B*a^11*b - 8*B*C*a^11*b + 192*A*B*a^2*b^10
+ 304*A*B*a^3*b^9 - 304*A*B*a^4*b^8 - 28*A*B*a^5*b^7 + 40*A*B*a^6*b^6 - 52*A*B*a^7*b^5 + 64*A*B*a^8*b^4 - 36*A
*B*a^9*b^3 + 8*A*B*a^10*b^2 + 128*A*C*a^2*b^10 - 128*A*C*a^3*b^9 - 224*A*C*a^4*b^8 + 224*A*C*a^5*b^7 + 40*A*C*
a^6*b^6 - 64*A*C*a^7*b^5 + 48*A*C*a^8*b^4 - 32*A*C*a^9*b^3 + 16*A*C*a^10*b^2 - 96*B*C*a^3*b^9 + 96*B*C*a^4*b^8
+ 176*B*C*a^5*b^7 - 176*B*C*a^6*b^6 - 40*B*C*a^7*b^5 + 64*B*C*a^8*b^4 - 40*B*C*a^9*b^3 + 16*B*C*a^10*b^2))/(a
^10*b + a^11 - a^8*b^3 - a^9*b^2) - (b^2*((8*(2*B*a^18 + 16*A*a^10*b^8 - 8*A*a^11*b^7 - 36*A*a^12*b^6 + 16*A*a
^13*b^5 + 20*A*a^14*b^4 - 4*A*a^15*b^3 - 12*B*a^11*b^7 + 6*B*a^12*b^6 + 28*B*a^13*b^5 - 14*B*a^14*b^4 - 16*B*a
^15*b^3 + 6*B*a^16*b^2 + 8*C*a^12*b^6 - 4*C*a^13*b^5 - 20*C*a^14*b^4 + 12*C*a^15*b^3 + 12*C*a^16*b^2 - 4*A*a^1
7*b - 8*C*a^17*b))/(a^14*b + a^15 - a^12*b^3 - a^13*b^2) + (8*b^2*tan(c/2 + (d*x)/2)*((a + b)^3*(a - b)^3)^(1/
2)*(8*a^15*b - 8*a^10*b^6 + 8*a^11*b^5 + 16*a^12*b^4 - 16*a^13*b^3 - 8*a^14*b^2)*(4*A*b^4 - 3*C*a^4 - 5*A*a^2*
b^2 + 2*C*a^2*b^2 - 3*B*a*b^3 + 4*B*a^3*b))/((a^10*b + a^11 - a^8*b^3 - a^9*b^2)*(a^11 - a^5*b^6 + 3*a^7*b^4 -
3*a^9*b^2)))*((a + b)^3*(a - b)^3)^(1/2)*(4*A*b^4 - 3*C*a^4 - 5*A*a^2*b^2 + 2*C*a^2*b^2 - 3*B*a*b^3 + 4*B*a^3
*b))/(a^11 - a^5*b^6 + 3*a^7*b^4 - 3*a^9*b^2))*((a + b)^3*(a - b)^3)^(1/2)*(4*A*b^4 - 3*C*a^4 - 5*A*a^2*b^2 +
2*C*a^2*b^2 - 3*B*a*b^3 + 4*B*a^3*b))/(a^11 - a^5*b^6 + 3*a^7*b^4 - 3*a^9*b^2)))*((a + b)^3*(a - b)^3)^(1/2)*(
4*A*b^4 - 3*C*a^4 - 5*A*a^2*b^2 + 2*C*a^2*b^2 - 3*B*a*b^3 + 4*B*a^3*b)*2i)/(d*(a^11 - a^5*b^6 + 3*a^7*b^4 - 3*
a^9*b^2))