\(\int \frac {\sec ^4(c+d x) (A+B \sec (c+d x)+C \sec ^2(c+d x))}{(a+b \sec (c+d x))^2} \, dx\) [908]

Optimal result

Integrand size = 41, antiderivative size = 407 \[ \int \frac {\sec ^4(c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{(a+b \sec (c+d x))^2} \, dx=\frac {\left (6 a^2 b B+b^3 B-8 a^3 C-2 a b^2 (2 A+C)\right ) \text {arctanh}(\sin (c+d x))}{2 b^5 d}+\frac {2 a^2 \left (2 a^2 A b^2-3 A b^4-3 a^3 b B+4 a b^3 B+4 a^4 C-5 a^2 b^2 C\right ) \text {arctanh}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{(a-b)^{3/2} b^5 (a+b)^{3/2} d}-\frac {\left (9 a^3 b B-6 a b^3 B-a^2 b^2 (6 A-7 C)-12 a^4 C+b^4 (3 A+2 C)\right ) \tan (c+d x)}{3 b^4 \left (a^2-b^2\right ) d}+\frac {\left (3 a^2 b B-b^3 B-2 a b^2 (A-C)-4 a^3 C\right ) \sec (c+d x) \tan (c+d x)}{2 b^3 \left (a^2-b^2\right ) d}+\frac {\left (3 A b^2-3 a b B+4 a^2 C-b^2 C\right ) \sec ^2(c+d x) \tan (c+d x)}{3 b^2 \left (a^2-b^2\right ) d}-\frac {\left (A b^2-a (b B-a C)\right ) \sec ^3(c+d x) \tan (c+d x)}{b \left (a^2-b^2\right ) d (a+b \sec (c+d x))} \]

[Out]

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

Rubi [A] (verified)

Time = 2.04 (sec) , antiderivative size = 407, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.220, Rules used = {4183, 4187, 4177, 4167, 4083, 3855, 3916, 2738, 214} \[ \int \frac {\sec ^4(c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{(a+b \sec (c+d x))^2} \, dx=-\frac {\tan (c+d x) \sec ^3(c+d x) \left (A b^2-a (b B-a C)\right )}{b d \left (a^2-b^2\right ) (a+b \sec (c+d x))}+\frac {\tan (c+d x) \sec ^2(c+d x) \left (4 a^2 C-3 a b B+3 A b^2-b^2 C\right )}{3 b^2 d \left (a^2-b^2\right )}+\frac {\left (-8 a^3 C+6 a^2 b B-2 a b^2 (2 A+C)+b^3 B\right ) \text {arctanh}(\sin (c+d x))}{2 b^5 d}+\frac {\tan (c+d x) \sec (c+d x) \left (-4 a^3 C+3 a^2 b B-2 a b^2 (A-C)-b^3 B\right )}{2 b^3 d \left (a^2-b^2\right )}+\frac {2 a^2 \left (4 a^4 C-3 a^3 b B+2 a^2 A b^2-5 a^2 b^2 C+4 a b^3 B-3 A b^4\right ) \text {arctanh}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{b^5 d (a-b)^{3/2} (a+b)^{3/2}}-\frac {\tan (c+d x) \left (-12 a^4 C+9 a^3 b B-a^2 b^2 (6 A-7 C)-6 a b^3 B+b^4 (3 A+2 C)\right )}{3 b^4 d \left (a^2-b^2\right )} \]

[In]

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

[Out]

((6*a^2*b*B + b^3*B - 8*a^3*C - 2*a*b^2*(2*A + C))*ArcTanh[Sin[c + d*x]])/(2*b^5*d) + (2*a^2*(2*a^2*A*b^2 - 3*
A*b^4 - 3*a^3*b*B + 4*a*b^3*B + 4*a^4*C - 5*a^2*b^2*C)*ArcTanh[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/((
a - b)^(3/2)*b^5*(a + b)^(3/2)*d) - ((9*a^3*b*B - 6*a*b^3*B - a^2*b^2*(6*A - 7*C) - 12*a^4*C + b^4*(3*A + 2*C)
)*Tan[c + d*x])/(3*b^4*(a^2 - b^2)*d) + ((3*a^2*b*B - b^3*B - 2*a*b^2*(A - C) - 4*a^3*C)*Sec[c + d*x]*Tan[c +
d*x])/(2*b^3*(a^2 - b^2)*d) + ((3*A*b^2 - 3*a*b*B + 4*a^2*C - b^2*C)*Sec[c + d*x]^2*Tan[c + d*x])/(3*b^2*(a^2
- b^2)*d) - ((A*b^2 - a*(b*B - a*C))*Sec[c + d*x]^3*Tan[c + d*x])/(b*(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 3855

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

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 4083

Int[(csc[(e_.) + (f_.)*(x_)]*(csc[(e_.) + (f_.)*(x_)]*(B_.) + (A_)))/(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)), x
_Symbol] :> Dist[B/b, Int[Csc[e + f*x], x], x] + Dist[(A*b - a*B)/b, Int[Csc[e + f*x]/(a + b*Csc[e + f*x]), x]
, x] /; FreeQ[{a, b, e, f, A, B}, x] && NeQ[A*b - a*B, 0]

Rule 4167

Int[csc[(e_.) + (f_.)*(x_)]*((A_.) + csc[(e_.) + (f_.)*(x_)]*(B_.) + csc[(e_.) + (f_.)*(x_)]^2*(C_.))*(csc[(e_
.) + (f_.)*(x_)]*(b_.) + (a_))^(m_), x_Symbol] :> Simp[(-C)*Cot[e + f*x]*((a + b*Csc[e + f*x])^(m + 1)/(b*f*(m
+ 2))), x] + Dist[1/(b*(m + 2)), Int[Csc[e + f*x]*(a + b*Csc[e + f*x])^m*Simp[b*A*(m + 2) + b*C*(m + 1) + (b*
B*(m + 2) - a*C)*Csc[e + f*x], x], x], x] /; FreeQ[{a, b, e, f, A, B, C, m}, x] &&  !LtQ[m, -1]

Rule 4177

Int[csc[(e_.) + (f_.)*(x_)]^2*((A_.) + csc[(e_.) + (f_.)*(x_)]*(B_.) + csc[(e_.) + (f_.)*(x_)]^2*(C_.))*(csc[(
e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_), x_Symbol] :> Simp[(-C)*Csc[e + f*x]*Cot[e + f*x]*((a + b*Csc[e + f*x])^
(m + 1)/(b*f*(m + 3))), x] + Dist[1/(b*(m + 3)), Int[Csc[e + f*x]*(a + b*Csc[e + f*x])^m*Simp[a*C + b*(C*(m +
2) + A*(m + 3))*Csc[e + f*x] - (2*a*C - b*B*(m + 3))*Csc[e + f*x]^2, x], x], x] /; FreeQ[{a, b, e, f, A, B, C,
m}, x] && NeQ[a^2 - b^2, 0] &&  !LtQ[m, -1]

Rule 4183

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[(-d)*(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 - 1)/(b*f*(a^2 - b^2)*(m + 1))), x] + Dist[d/(b*(a^2 - b^2)*
(m + 1)), Int[(a + b*Csc[e + f*x])^(m + 1)*(d*Csc[e + f*x])^(n - 1)*Simp[A*b^2*(n - 1) - a*(b*B - a*C)*(n - 1)
+ b*(a*A - b*B + a*C)*(m + 1)*Csc[e + f*x] - (b*(A*b - a*B)*(m + n + 1) + C*(a^2*n + b^2*(m + 1)))*Csc[e + f*
x]^2, x], x], x] /; FreeQ[{a, b, d, e, f, A, B, C}, x] && NeQ[a^2 - b^2, 0] && LtQ[m, -1] && GtQ[n, 0]

Rule 4187

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[(-C)*d*Cot[e + f*x]*(a + b*Csc[e + f*x])^(
m + 1)*((d*Csc[e + f*x])^(n - 1)/(b*f*(m + n + 1))), x] + Dist[d/(b*(m + n + 1)), Int[(a + b*Csc[e + f*x])^m*(
d*Csc[e + f*x])^(n - 1)*Simp[a*C*(n - 1) + (A*b*(m + n + 1) + b*C*(m + n))*Csc[e + f*x] + (b*B*(m + n + 1) - a
*C*n)*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] && GtQ[n, 0]

Rubi steps \begin{align*} \text {integral}& = -\frac {\left (A b^2-a (b B-a C)\right ) \sec ^3(c+d x) \tan (c+d x)}{b \left (a^2-b^2\right ) d (a+b \sec (c+d x))}-\frac {\int \frac {\sec ^3(c+d x) \left (3 \left (A b^2-a (b B-a C)\right )+b (b B-a (A+C)) \sec (c+d x)-\left (3 A b^2-3 a b B+4 a^2 C-b^2 C\right ) \sec ^2(c+d x)\right )}{a+b \sec (c+d x)} \, dx}{b \left (a^2-b^2\right )} \\ & = \frac {\left (3 A b^2-3 a b B+4 a^2 C-b^2 C\right ) \sec ^2(c+d x) \tan (c+d x)}{3 b^2 \left (a^2-b^2\right ) d}-\frac {\left (A b^2-a (b B-a C)\right ) \sec ^3(c+d x) \tan (c+d x)}{b \left (a^2-b^2\right ) d (a+b \sec (c+d x))}-\frac {\int \frac {\sec ^2(c+d x) \left (-2 a \left (3 A b^2-3 a b B+4 a^2 C-b^2 C\right )+b \left (3 A b^2-3 a b B+a^2 C+2 b^2 C\right ) \sec (c+d x)-3 \left (3 a^2 b B-b^3 B-2 a b^2 (A-C)-4 a^3 C\right ) \sec ^2(c+d x)\right )}{a+b \sec (c+d x)} \, dx}{3 b^2 \left (a^2-b^2\right )} \\ & = \frac {\left (3 a^2 b B-b^3 B-2 a b^2 (A-C)-4 a^3 C\right ) \sec (c+d x) \tan (c+d x)}{2 b^3 \left (a^2-b^2\right ) d}+\frac {\left (3 A b^2-3 a b B+4 a^2 C-b^2 C\right ) \sec ^2(c+d x) \tan (c+d x)}{3 b^2 \left (a^2-b^2\right ) d}-\frac {\left (A b^2-a (b B-a C)\right ) \sec ^3(c+d x) \tan (c+d x)}{b \left (a^2-b^2\right ) d (a+b \sec (c+d x))}-\frac {\int \frac {\sec (c+d x) \left (-3 a \left (3 a^2 b B-b^3 B-2 a b^2 (A-C)-4 a^3 C\right )+b \left (3 a^2 b B+3 b^3 B-4 a^3 C-2 a b^2 (3 A+C)\right ) \sec (c+d x)+2 \left (9 a^3 b B-6 a b^3 B-a^2 b^2 (6 A-7 C)-12 a^4 C+b^4 (3 A+2 C)\right ) \sec ^2(c+d x)\right )}{a+b \sec (c+d x)} \, dx}{6 b^3 \left (a^2-b^2\right )} \\ & = -\frac {\left (9 a^3 b B-6 a b^3 B-a^2 b^2 (6 A-7 C)-12 a^4 C+b^4 (3 A+2 C)\right ) \tan (c+d x)}{3 b^4 \left (a^2-b^2\right ) d}+\frac {\left (3 a^2 b B-b^3 B-2 a b^2 (A-C)-4 a^3 C\right ) \sec (c+d x) \tan (c+d x)}{2 b^3 \left (a^2-b^2\right ) d}+\frac {\left (3 A b^2-3 a b B+4 a^2 C-b^2 C\right ) \sec ^2(c+d x) \tan (c+d x)}{3 b^2 \left (a^2-b^2\right ) d}-\frac {\left (A b^2-a (b B-a C)\right ) \sec ^3(c+d x) \tan (c+d x)}{b \left (a^2-b^2\right ) d (a+b \sec (c+d x))}-\frac {\int \frac {\sec (c+d x) \left (-3 a b \left (3 a^2 b B-b^3 B-2 a b^2 (A-C)-4 a^3 C\right )-3 \left (a^2-b^2\right ) \left (6 a^2 b B+b^3 B-8 a^3 C-2 a b^2 (2 A+C)\right ) \sec (c+d x)\right )}{a+b \sec (c+d x)} \, dx}{6 b^4 \left (a^2-b^2\right )} \\ & = -\frac {\left (9 a^3 b B-6 a b^3 B-a^2 b^2 (6 A-7 C)-12 a^4 C+b^4 (3 A+2 C)\right ) \tan (c+d x)}{3 b^4 \left (a^2-b^2\right ) d}+\frac {\left (3 a^2 b B-b^3 B-2 a b^2 (A-C)-4 a^3 C\right ) \sec (c+d x) \tan (c+d x)}{2 b^3 \left (a^2-b^2\right ) d}+\frac {\left (3 A b^2-3 a b B+4 a^2 C-b^2 C\right ) \sec ^2(c+d x) \tan (c+d x)}{3 b^2 \left (a^2-b^2\right ) d}-\frac {\left (A b^2-a (b B-a C)\right ) \sec ^3(c+d x) \tan (c+d x)}{b \left (a^2-b^2\right ) d (a+b \sec (c+d x))}-\frac {\left (a^2 \left (3 A b^4+3 a^3 b B-4 a b^3 B-a^2 b^2 (2 A-5 C)-4 a^4 C\right )\right ) \int \frac {\sec (c+d x)}{a+b \sec (c+d x)} \, dx}{b^5 \left (a^2-b^2\right )}+\frac {\left (6 a^2 b B+b^3 B-8 a^3 C-2 a b^2 (2 A+C)\right ) \int \sec (c+d x) \, dx}{2 b^5} \\ & = \frac {\left (6 a^2 b B+b^3 B-8 a^3 C-2 a b^2 (2 A+C)\right ) \text {arctanh}(\sin (c+d x))}{2 b^5 d}-\frac {\left (9 a^3 b B-6 a b^3 B-a^2 b^2 (6 A-7 C)-12 a^4 C+b^4 (3 A+2 C)\right ) \tan (c+d x)}{3 b^4 \left (a^2-b^2\right ) d}+\frac {\left (3 a^2 b B-b^3 B-2 a b^2 (A-C)-4 a^3 C\right ) \sec (c+d x) \tan (c+d x)}{2 b^3 \left (a^2-b^2\right ) d}+\frac {\left (3 A b^2-3 a b B+4 a^2 C-b^2 C\right ) \sec ^2(c+d x) \tan (c+d x)}{3 b^2 \left (a^2-b^2\right ) d}-\frac {\left (A b^2-a (b B-a C)\right ) \sec ^3(c+d x) \tan (c+d x)}{b \left (a^2-b^2\right ) d (a+b \sec (c+d x))}-\frac {\left (a^2 \left (3 A b^4+3 a^3 b B-4 a b^3 B-a^2 b^2 (2 A-5 C)-4 a^4 C\right )\right ) \int \frac {1}{1+\frac {a \cos (c+d x)}{b}} \, dx}{b^6 \left (a^2-b^2\right )} \\ & = \frac {\left (6 a^2 b B+b^3 B-8 a^3 C-2 a b^2 (2 A+C)\right ) \text {arctanh}(\sin (c+d x))}{2 b^5 d}-\frac {\left (9 a^3 b B-6 a b^3 B-a^2 b^2 (6 A-7 C)-12 a^4 C+b^4 (3 A+2 C)\right ) \tan (c+d x)}{3 b^4 \left (a^2-b^2\right ) d}+\frac {\left (3 a^2 b B-b^3 B-2 a b^2 (A-C)-4 a^3 C\right ) \sec (c+d x) \tan (c+d x)}{2 b^3 \left (a^2-b^2\right ) d}+\frac {\left (3 A b^2-3 a b B+4 a^2 C-b^2 C\right ) \sec ^2(c+d x) \tan (c+d x)}{3 b^2 \left (a^2-b^2\right ) d}-\frac {\left (A b^2-a (b B-a C)\right ) \sec ^3(c+d x) \tan (c+d x)}{b \left (a^2-b^2\right ) d (a+b \sec (c+d x))}-\frac {\left (2 a^2 \left (3 A b^4+3 a^3 b B-4 a b^3 B-a^2 b^2 (2 A-5 C)-4 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 )}{b^6 \left (a^2-b^2\right ) d} \\ & = \frac {\left (6 a^2 b B+b^3 B-8 a^3 C-2 a b^2 (2 A+C)\right ) \text {arctanh}(\sin (c+d x))}{2 b^5 d}+\frac {2 a^2 \left (2 a^2 A b^2-3 A b^4-3 a^3 b B+4 a b^3 B+4 a^4 C-5 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-b)^{3/2} b^5 (a+b)^{3/2} d}-\frac {\left (9 a^3 b B-6 a b^3 B-a^2 b^2 (6 A-7 C)-12 a^4 C+b^4 (3 A+2 C)\right ) \tan (c+d x)}{3 b^4 \left (a^2-b^2\right ) d}+\frac {\left (3 a^2 b B-b^3 B-2 a b^2 (A-C)-4 a^3 C\right ) \sec (c+d x) \tan (c+d x)}{2 b^3 \left (a^2-b^2\right ) d}+\frac {\left (3 A b^2-3 a b B+4 a^2 C-b^2 C\right ) \sec ^2(c+d x) \tan (c+d x)}{3 b^2 \left (a^2-b^2\right ) d}-\frac {\left (A b^2-a (b B-a C)\right ) \sec ^3(c+d x) \tan (c+d x)}{b \left (a^2-b^2\right ) d (a+b \sec (c+d x))} \\ \end{align*}

Mathematica [A] (warning: unable to verify)

Time = 8.45 (sec) , antiderivative size = 605, normalized size of antiderivative = 1.49 \[ \int \frac {\sec ^4(c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{(a+b \sec (c+d x))^2} \, dx=\frac {(b+a \cos (c+d x)) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \left (-\frac {24 a^2 \left (-3 A b^4-3 a^3 b B+4 a b^3 B+a^2 b^2 (2 A-5 C)+4 a^4 C\right ) \text {arctanh}\left (\frac {(-a+b) \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right ) (b+a \cos (c+d x))}{\left (a^2-b^2\right )^{3/2}}+6 \left (-6 a^2 b B-b^3 B+8 a^3 C+2 a b^2 (2 A+C)\right ) (b+a \cos (c+d x)) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )+6 \left (6 a^2 b B+b^3 B-8 a^3 C-2 a b^2 (2 A+C)\right ) (b+a \cos (c+d x)) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )+\frac {b \left (-6 a^2 A b^3+6 A b^5+9 a^3 b^2 B-9 a b^4 B-12 a^4 b C+4 a^2 b^3 C+8 b^5 C+\left (27 a^4 b B-24 a^2 b^3 B+6 b^5 B+a b^4 (9 A-2 C)-36 a^5 C+a^3 b^2 (-18 A+29 C)\right ) \cos (c+d x)+b \left (-a^2+b^2\right ) \left (6 A b^2-9 a b B+12 a^2 C+4 b^2 C\right ) \cos (2 (c+d x))-6 a^3 A b^2 \cos (3 (c+d x))+3 a A b^4 \cos (3 (c+d x))+9 a^4 b B \cos (3 (c+d x))-6 a^2 b^3 B \cos (3 (c+d x))-12 a^5 C \cos (3 (c+d x))+7 a^3 b^2 C \cos (3 (c+d x))+2 a b^4 C \cos (3 (c+d x))\right ) \sec ^2(c+d x) \tan (c+d x)}{-a^2+b^2}\right )}{6 b^5 d (A+2 C+2 B \cos (c+d x)+A \cos (2 (c+d x))) (a+b \sec (c+d x))^2} \]

[In]

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

[Out]

((b + a*Cos[c + d*x])*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((-24*a^2*(-3*A*b^4 - 3*a^3*b*B + 4*a*b^3*B + a^
2*b^2*(2*A - 5*C) + 4*a^4*C)*ArcTanh[((-a + b)*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]]*(b + a*Cos[c + d*x]))/(a^2 -
b^2)^(3/2) + 6*(-6*a^2*b*B - b^3*B + 8*a^3*C + 2*a*b^2*(2*A + C))*(b + a*Cos[c + d*x])*Log[Cos[(c + d*x)/2] -
Sin[(c + d*x)/2]] + 6*(6*a^2*b*B + b^3*B - 8*a^3*C - 2*a*b^2*(2*A + C))*(b + a*Cos[c + d*x])*Log[Cos[(c + d*x
)/2] + Sin[(c + d*x)/2]] + (b*(-6*a^2*A*b^3 + 6*A*b^5 + 9*a^3*b^2*B - 9*a*b^4*B - 12*a^4*b*C + 4*a^2*b^3*C + 8
*b^5*C + (27*a^4*b*B - 24*a^2*b^3*B + 6*b^5*B + a*b^4*(9*A - 2*C) - 36*a^5*C + a^3*b^2*(-18*A + 29*C))*Cos[c +
d*x] + b*(-a^2 + b^2)*(6*A*b^2 - 9*a*b*B + 12*a^2*C + 4*b^2*C)*Cos[2*(c + d*x)] - 6*a^3*A*b^2*Cos[3*(c + d*x)
] + 3*a*A*b^4*Cos[3*(c + d*x)] + 9*a^4*b*B*Cos[3*(c + d*x)] - 6*a^2*b^3*B*Cos[3*(c + d*x)] - 12*a^5*C*Cos[3*(c
+ d*x)] + 7*a^3*b^2*C*Cos[3*(c + d*x)] + 2*a*b^4*C*Cos[3*(c + d*x)])*Sec[c + d*x]^2*Tan[c + d*x])/(-a^2 + b^2
)))/(6*b^5*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*(c + d*x)])*(a + b*Sec[c + d*x])^2)

Maple [A] (verified)

Time = 1.02 (sec) , antiderivative size = 492, normalized size of antiderivative = 1.21

method	result	size
derivativedivides	\(\frac {-\frac {C}{3 b^{2} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{3}}-\frac {B b -2 C a -C b}{2 b^{3} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}+\frac {\left (-4 a A \,b^{2}+6 B \,a^{2} b +B \,b^{3}-8 a^{3} C -2 C a \,b^{2}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{2 b^{5}}-\frac {2 A \,b^{2}-4 B a b -B \,b^{2}+6 C \,a^{2}+2 C a b +2 C \,b^{2}}{2 b^{4} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}-\frac {2 a^{2} \left (\frac {a b \left (A \,b^{2}-B a b +C \,a^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{\left (a^{2}-b^{2}\right ) \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a -\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} b -a -b \right )}-\frac {\left (2 A \,a^{2} b^{2}-3 A \,b^{4}-3 B \,a^{3} b +4 B a \,b^{3}+4 a^{4} C -5 C \,a^{2} b^{2}\right ) \operatorname {arctanh}\left (\frac {\left (a -b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (a +b \right ) \left (a -b \right )}}\right )}{\left (a +b \right ) \left (a -b \right ) \sqrt {\left (a +b \right ) \left (a -b \right )}}\right )}{b^{5}}-\frac {C}{3 b^{2} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{3}}-\frac {-B b +2 C a +C b}{2 b^{3} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}+\frac {\left (4 a A \,b^{2}-6 B \,a^{2} b -B \,b^{3}+8 a^{3} C +2 C a \,b^{2}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{2 b^{5}}-\frac {2 A \,b^{2}-4 B a b -B \,b^{2}+6 C \,a^{2}+2 C a b +2 C \,b^{2}}{2 b^{4} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}}{d}\)	\(492\)
default	\(\frac {-\frac {C}{3 b^{2} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{3}}-\frac {B b -2 C a -C b}{2 b^{3} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}+\frac {\left (-4 a A \,b^{2}+6 B \,a^{2} b +B \,b^{3}-8 a^{3} C -2 C a \,b^{2}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{2 b^{5}}-\frac {2 A \,b^{2}-4 B a b -B \,b^{2}+6 C \,a^{2}+2 C a b +2 C \,b^{2}}{2 b^{4} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}-\frac {2 a^{2} \left (\frac {a b \left (A \,b^{2}-B a b +C \,a^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{\left (a^{2}-b^{2}\right ) \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a -\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} b -a -b \right )}-\frac {\left (2 A \,a^{2} b^{2}-3 A \,b^{4}-3 B \,a^{3} b +4 B a \,b^{3}+4 a^{4} C -5 C \,a^{2} b^{2}\right ) \operatorname {arctanh}\left (\frac {\left (a -b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (a +b \right ) \left (a -b \right )}}\right )}{\left (a +b \right ) \left (a -b \right ) \sqrt {\left (a +b \right ) \left (a -b \right )}}\right )}{b^{5}}-\frac {C}{3 b^{2} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{3}}-\frac {-B b +2 C a +C b}{2 b^{3} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}+\frac {\left (4 a A \,b^{2}-6 B \,a^{2} b -B \,b^{3}+8 a^{3} C +2 C a \,b^{2}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{2 b^{5}}-\frac {2 A \,b^{2}-4 B a b -B \,b^{2}+6 C \,a^{2}+2 C a b +2 C \,b^{2}}{2 b^{4} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}}{d}\)	\(492\)
risch	\(\text {Expression too large to display}\)	\(2208\)

[In]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 889 vs. \(2 (389) = 778\).

Time = 48.41 (sec) , antiderivative size = 1835, normalized size of antiderivative = 4.51 \[ \int \frac {\sec ^4(c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{(a+b \sec (c+d x))^2} \, dx=\text {Too large to display} \]

[In]

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

[Out]

[1/12*(6*((4*C*a^7 - 3*B*a^6*b + (2*A - 5*C)*a^5*b^2 + 4*B*a^4*b^3 - 3*A*a^3*b^4)*cos(d*x + c)^4 + (4*C*a^6*b
- 3*B*a^5*b^2 + (2*A - 5*C)*a^4*b^3 + 4*B*a^3*b^4 - 3*A*a^2*b^5)*cos(d*x + c)^3)*sqrt(a^2 - b^2)*log((2*a*b*co
s(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)) - 3*((8*C*a^8 - 6*B*a^7*b + 2*(2*A - 7*C)*a^6*b^2 + 11*B*a^5
*b^3 - 4*(2*A - C)*a^4*b^4 - 4*B*a^3*b^5 + 2*(2*A + C)*a^2*b^6 - B*a*b^7)*cos(d*x + c)^4 + (8*C*a^7*b - 6*B*a^
6*b^2 + 2*(2*A - 7*C)*a^5*b^3 + 11*B*a^4*b^4 - 4*(2*A - C)*a^3*b^5 - 4*B*a^2*b^6 + 2*(2*A + C)*a*b^7 - B*b^8)*
cos(d*x + c)^3)*log(sin(d*x + c) + 1) + 3*((8*C*a^8 - 6*B*a^7*b + 2*(2*A - 7*C)*a^6*b^2 + 11*B*a^5*b^3 - 4*(2*
A - C)*a^4*b^4 - 4*B*a^3*b^5 + 2*(2*A + C)*a^2*b^6 - B*a*b^7)*cos(d*x + c)^4 + (8*C*a^7*b - 6*B*a^6*b^2 + 2*(2
*A - 7*C)*a^5*b^3 + 11*B*a^4*b^4 - 4*(2*A - C)*a^3*b^5 - 4*B*a^2*b^6 + 2*(2*A + C)*a*b^7 - B*b^8)*cos(d*x + c)
^3)*log(-sin(d*x + c) + 1) + 2*(2*C*a^4*b^4 - 4*C*a^2*b^6 + 2*C*b^8 + 2*(12*C*a^7*b - 9*B*a^6*b^2 + (6*A - 19*
C)*a^5*b^3 + 15*B*a^4*b^4 - (9*A - 5*C)*a^3*b^5 - 6*B*a^2*b^6 + (3*A + 2*C)*a*b^7)*cos(d*x + c)^3 + (12*C*a^6*
b^2 - 9*B*a^5*b^3 + 2*(3*A - 10*C)*a^4*b^4 + 18*B*a^3*b^5 - 4*(3*A - C)*a^2*b^6 - 9*B*a*b^7 + 2*(3*A + 2*C)*b^
8)*cos(d*x + c)^2 - (4*C*a^5*b^3 - 3*B*a^4*b^4 - 8*C*a^3*b^5 + 6*B*a^2*b^6 + 4*C*a*b^7 - 3*B*b^8)*cos(d*x + c)
)*sin(d*x + c))/((a^5*b^5 - 2*a^3*b^7 + a*b^9)*d*cos(d*x + c)^4 + (a^4*b^6 - 2*a^2*b^8 + b^10)*d*cos(d*x + c)^
3), 1/12*(12*((4*C*a^7 - 3*B*a^6*b + (2*A - 5*C)*a^5*b^2 + 4*B*a^4*b^3 - 3*A*a^3*b^4)*cos(d*x + c)^4 + (4*C*a^
6*b - 3*B*a^5*b^2 + (2*A - 5*C)*a^4*b^3 + 4*B*a^3*b^4 - 3*A*a^2*b^5)*cos(d*x + c)^3)*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))) - 3*((8*C*a^8 - 6*B*a^7*b + 2*(2*A - 7*C)*a^
6*b^2 + 11*B*a^5*b^3 - 4*(2*A - C)*a^4*b^4 - 4*B*a^3*b^5 + 2*(2*A + C)*a^2*b^6 - B*a*b^7)*cos(d*x + c)^4 + (8*
C*a^7*b - 6*B*a^6*b^2 + 2*(2*A - 7*C)*a^5*b^3 + 11*B*a^4*b^4 - 4*(2*A - C)*a^3*b^5 - 4*B*a^2*b^6 + 2*(2*A + C)
*a*b^7 - B*b^8)*cos(d*x + c)^3)*log(sin(d*x + c) + 1) + 3*((8*C*a^8 - 6*B*a^7*b + 2*(2*A - 7*C)*a^6*b^2 + 11*B
*a^5*b^3 - 4*(2*A - C)*a^4*b^4 - 4*B*a^3*b^5 + 2*(2*A + C)*a^2*b^6 - B*a*b^7)*cos(d*x + c)^4 + (8*C*a^7*b - 6*
B*a^6*b^2 + 2*(2*A - 7*C)*a^5*b^3 + 11*B*a^4*b^4 - 4*(2*A - C)*a^3*b^5 - 4*B*a^2*b^6 + 2*(2*A + C)*a*b^7 - B*b
^8)*cos(d*x + c)^3)*log(-sin(d*x + c) + 1) + 2*(2*C*a^4*b^4 - 4*C*a^2*b^6 + 2*C*b^8 + 2*(12*C*a^7*b - 9*B*a^6*
b^2 + (6*A - 19*C)*a^5*b^3 + 15*B*a^4*b^4 - (9*A - 5*C)*a^3*b^5 - 6*B*a^2*b^6 + (3*A + 2*C)*a*b^7)*cos(d*x + c
)^3 + (12*C*a^6*b^2 - 9*B*a^5*b^3 + 2*(3*A - 10*C)*a^4*b^4 + 18*B*a^3*b^5 - 4*(3*A - C)*a^2*b^6 - 9*B*a*b^7 +
2*(3*A + 2*C)*b^8)*cos(d*x + c)^2 - (4*C*a^5*b^3 - 3*B*a^4*b^4 - 8*C*a^3*b^5 + 6*B*a^2*b^6 + 4*C*a*b^7 - 3*B*b
^8)*cos(d*x + c))*sin(d*x + c))/((a^5*b^5 - 2*a^3*b^7 + a*b^9)*d*cos(d*x + c)^4 + (a^4*b^6 - 2*a^2*b^8 + b^10)
*d*cos(d*x + c)^3)]

Sympy [F]

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

[In]

integrate(sec(d*x+c)**4*(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)*sec(c + d*x)**4/(a + b*sec(c + d*x))**2, x)

Maxima [F(-2)]

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

[In]

integrate(sec(d*x+c)^4*(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 = 0.36 (sec) , antiderivative size = 627, normalized size of antiderivative = 1.54 \[ \int \frac {\sec ^4(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 (4 \, C a^{6} - 3 \, B a^{5} b + 2 \, A a^{4} b^{2} - 5 \, C a^{4} b^{2} + 4 \, B a^{3} b^{3} - 3 \, A a^{2} b^{4}\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^{2} b^{5} - b^{7}\right )} \sqrt {-a^{2} + b^{2}}} - \frac {12 \, {\left (C a^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - B a^{4} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + A a^{3} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (a^{2} b^{4} - b^{6}\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 (8 \, C a^{3} - 6 \, B a^{2} b + 4 \, A a b^{2} + 2 \, C a b^{2} - B b^{3}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right )}{b^{5}} + \frac {3 \, {\left (8 \, C a^{3} - 6 \, B a^{2} b + 4 \, A a b^{2} + 2 \, C a b^{2} - B b^{3}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right )}{b^{5}} - \frac {2 \, {\left (18 \, C a^{2} \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} + 6 \, C a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 6 \, A b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 3 \, B b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 6 \, C b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 36 \, 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} - 12 \, A b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 4 \, C b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 18 \, C a^{2} \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 ) - 6 \, C a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 6 \, A b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 3 \, B b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 6 \, C 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} b^{4}}}{6 \, d} \]

[In]

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

Mupad [B] (verification not implemented)

Time = 31.05 (sec) , antiderivative size = 11687, normalized size of antiderivative = 28.71 \[ \int \frac {\sec ^4(c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{(a+b \sec (c+d x))^2} \, dx=\text {Too large to display} \]

[In]

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

[Out]

(atan(((((((8*(2*B*b^18 + 12*A*a^2*b^16 + 12*A*a^3*b^15 - 20*A*a^4*b^14 - 4*A*a^5*b^13 + 8*A*a^6*b^12 + 6*B*a^
2*b^16 - 16*B*a^3*b^15 - 14*B*a^4*b^14 + 28*B*a^5*b^13 + 6*B*a^6*b^12 - 12*B*a^7*b^11 - 4*C*a^3*b^15 + 20*C*a^
4*b^14 + 16*C*a^5*b^13 - 36*C*a^6*b^12 - 8*C*a^7*b^11 + 16*C*a^8*b^10 - 8*A*a*b^17 - 4*C*a*b^17))/(a*b^14 + b^
15 - a^2*b^13 - a^3*b^12) - (8*tan(c/2 + (d*x)/2)*((B*b^3)/2 - 4*C*a^3 - b^2*(2*A*a + C*a) + 3*B*a^2*b)*(8*a*b
^15 - 8*a^2*b^14 - 16*a^3*b^13 + 16*a^4*b^12 + 8*a^5*b^11 - 8*a^6*b^10))/(b^5*(a*b^10 + b^11 - a^2*b^9 - a^3*b
^8)))*((B*b^3)/2 - 4*C*a^3 - b^2*(2*A*a + C*a) + 3*B*a^2*b))/b^5 - (8*tan(c/2 + (d*x)/2)*(B^2*b^12 + 128*C^2*a
^12 - 2*B^2*a*b^11 - 128*C^2*a^11*b + 16*A^2*a^2*b^10 - 32*A^2*a^3*b^9 + 20*A^2*a^4*b^8 + 64*A^2*a^5*b^7 - 64*
A^2*a^6*b^6 - 32*A^2*a^7*b^5 + 32*A^2*a^8*b^4 + 11*B^2*a^2*b^10 - 20*B^2*a^3*b^9 + 23*B^2*a^4*b^8 - 26*B^2*a^5
*b^7 + 17*B^2*a^6*b^6 + 120*B^2*a^7*b^5 - 120*B^2*a^8*b^4 - 72*B^2*a^9*b^3 + 72*B^2*a^10*b^2 + 4*C^2*a^2*b^10
- 8*C^2*a^3*b^9 + 28*C^2*a^4*b^8 - 48*C^2*a^5*b^7 + 28*C^2*a^6*b^6 - 8*C^2*a^7*b^5 + 8*C^2*a^8*b^4 + 192*C^2*a
^9*b^3 - 192*C^2*a^10*b^2 - 8*A*B*a*b^11 - 4*B*C*a*b^11 - 192*B*C*a^11*b + 16*A*B*a^2*b^10 - 40*A*B*a^3*b^9 +
64*A*B*a^4*b^8 - 40*A*B*a^5*b^7 - 176*A*B*a^6*b^6 + 176*A*B*a^7*b^5 + 96*A*B*a^8*b^4 - 96*A*B*a^9*b^3 + 16*A*C
*a^2*b^10 - 32*A*C*a^3*b^9 + 48*A*C*a^4*b^8 - 64*A*C*a^5*b^7 + 40*A*C*a^6*b^6 + 224*A*C*a^7*b^5 - 224*A*C*a^8*
b^4 - 128*A*C*a^9*b^3 + 128*A*C*a^10*b^2 + 8*B*C*a^2*b^10 - 36*B*C*a^3*b^9 + 64*B*C*a^4*b^8 - 52*B*C*a^5*b^7 +
40*B*C*a^6*b^6 - 28*B*C*a^7*b^5 - 304*B*C*a^8*b^4 + 304*B*C*a^9*b^3 + 192*B*C*a^10*b^2))/(a*b^10 + b^11 - a^2
*b^9 - a^3*b^8))*((B*b^3)/2 - 4*C*a^3 - b^2*(2*A*a + C*a) + 3*B*a^2*b)*1i)/b^5 - (((((8*(2*B*b^18 + 12*A*a^2*b
^16 + 12*A*a^3*b^15 - 20*A*a^4*b^14 - 4*A*a^5*b^13 + 8*A*a^6*b^12 + 6*B*a^2*b^16 - 16*B*a^3*b^15 - 14*B*a^4*b^
14 + 28*B*a^5*b^13 + 6*B*a^6*b^12 - 12*B*a^7*b^11 - 4*C*a^3*b^15 + 20*C*a^4*b^14 + 16*C*a^5*b^13 - 36*C*a^6*b^
12 - 8*C*a^7*b^11 + 16*C*a^8*b^10 - 8*A*a*b^17 - 4*C*a*b^17))/(a*b^14 + b^15 - a^2*b^13 - a^3*b^12) + (8*tan(c
/2 + (d*x)/2)*((B*b^3)/2 - 4*C*a^3 - b^2*(2*A*a + C*a) + 3*B*a^2*b)*(8*a*b^15 - 8*a^2*b^14 - 16*a^3*b^13 + 16*
a^4*b^12 + 8*a^5*b^11 - 8*a^6*b^10))/(b^5*(a*b^10 + b^11 - a^2*b^9 - a^3*b^8)))*((B*b^3)/2 - 4*C*a^3 - b^2*(2*
A*a + C*a) + 3*B*a^2*b))/b^5 + (8*tan(c/2 + (d*x)/2)*(B^2*b^12 + 128*C^2*a^12 - 2*B^2*a*b^11 - 128*C^2*a^11*b
+ 16*A^2*a^2*b^10 - 32*A^2*a^3*b^9 + 20*A^2*a^4*b^8 + 64*A^2*a^5*b^7 - 64*A^2*a^6*b^6 - 32*A^2*a^7*b^5 + 32*A^
2*a^8*b^4 + 11*B^2*a^2*b^10 - 20*B^2*a^3*b^9 + 23*B^2*a^4*b^8 - 26*B^2*a^5*b^7 + 17*B^2*a^6*b^6 + 120*B^2*a^7*
b^5 - 120*B^2*a^8*b^4 - 72*B^2*a^9*b^3 + 72*B^2*a^10*b^2 + 4*C^2*a^2*b^10 - 8*C^2*a^3*b^9 + 28*C^2*a^4*b^8 - 4
8*C^2*a^5*b^7 + 28*C^2*a^6*b^6 - 8*C^2*a^7*b^5 + 8*C^2*a^8*b^4 + 192*C^2*a^9*b^3 - 192*C^2*a^10*b^2 - 8*A*B*a*
b^11 - 4*B*C*a*b^11 - 192*B*C*a^11*b + 16*A*B*a^2*b^10 - 40*A*B*a^3*b^9 + 64*A*B*a^4*b^8 - 40*A*B*a^5*b^7 - 17
6*A*B*a^6*b^6 + 176*A*B*a^7*b^5 + 96*A*B*a^8*b^4 - 96*A*B*a^9*b^3 + 16*A*C*a^2*b^10 - 32*A*C*a^3*b^9 + 48*A*C*
a^4*b^8 - 64*A*C*a^5*b^7 + 40*A*C*a^6*b^6 + 224*A*C*a^7*b^5 - 224*A*C*a^8*b^4 - 128*A*C*a^9*b^3 + 128*A*C*a^10
*b^2 + 8*B*C*a^2*b^10 - 36*B*C*a^3*b^9 + 64*B*C*a^4*b^8 - 52*B*C*a^5*b^7 + 40*B*C*a^6*b^6 - 28*B*C*a^7*b^5 - 3
04*B*C*a^8*b^4 + 304*B*C*a^9*b^3 + 192*B*C*a^10*b^2))/(a*b^10 + b^11 - a^2*b^9 - a^3*b^8))*((B*b^3)/2 - 4*C*a^
3 - b^2*(2*A*a + C*a) + 3*B*a^2*b)*1i)/b^5)/((((((8*(2*B*b^18 + 12*A*a^2*b^16 + 12*A*a^3*b^15 - 20*A*a^4*b^14
- 4*A*a^5*b^13 + 8*A*a^6*b^12 + 6*B*a^2*b^16 - 16*B*a^3*b^15 - 14*B*a^4*b^14 + 28*B*a^5*b^13 + 6*B*a^6*b^12 -
12*B*a^7*b^11 - 4*C*a^3*b^15 + 20*C*a^4*b^14 + 16*C*a^5*b^13 - 36*C*a^6*b^12 - 8*C*a^7*b^11 + 16*C*a^8*b^10 -
8*A*a*b^17 - 4*C*a*b^17))/(a*b^14 + b^15 - a^2*b^13 - a^3*b^12) - (8*tan(c/2 + (d*x)/2)*((B*b^3)/2 - 4*C*a^3 -
b^2*(2*A*a + C*a) + 3*B*a^2*b)*(8*a*b^15 - 8*a^2*b^14 - 16*a^3*b^13 + 16*a^4*b^12 + 8*a^5*b^11 - 8*a^6*b^10))
/(b^5*(a*b^10 + b^11 - a^2*b^9 - a^3*b^8)))*((B*b^3)/2 - 4*C*a^3 - b^2*(2*A*a + C*a) + 3*B*a^2*b))/b^5 - (8*ta
n(c/2 + (d*x)/2)*(B^2*b^12 + 128*C^2*a^12 - 2*B^2*a*b^11 - 128*C^2*a^11*b + 16*A^2*a^2*b^10 - 32*A^2*a^3*b^9 +
20*A^2*a^4*b^8 + 64*A^2*a^5*b^7 - 64*A^2*a^6*b^6 - 32*A^2*a^7*b^5 + 32*A^2*a^8*b^4 + 11*B^2*a^2*b^10 - 20*B^2
*a^3*b^9 + 23*B^2*a^4*b^8 - 26*B^2*a^5*b^7 + 17*B^2*a^6*b^6 + 120*B^2*a^7*b^5 - 120*B^2*a^8*b^4 - 72*B^2*a^9*b
^3 + 72*B^2*a^10*b^2 + 4*C^2*a^2*b^10 - 8*C^2*a^3*b^9 + 28*C^2*a^4*b^8 - 48*C^2*a^5*b^7 + 28*C^2*a^6*b^6 - 8*C
^2*a^7*b^5 + 8*C^2*a^8*b^4 + 192*C^2*a^9*b^3 - 192*C^2*a^10*b^2 - 8*A*B*a*b^11 - 4*B*C*a*b^11 - 192*B*C*a^11*b
+ 16*A*B*a^2*b^10 - 40*A*B*a^3*b^9 + 64*A*B*a^4*b^8 - 40*A*B*a^5*b^7 - 176*A*B*a^6*b^6 + 176*A*B*a^7*b^5 + 96
*A*B*a^8*b^4 - 96*A*B*a^9*b^3 + 16*A*C*a^2*b^10 - 32*A*C*a^3*b^9 + 48*A*C*a^4*b^8 - 64*A*C*a^5*b^7 + 40*A*C*a^
6*b^6 + 224*A*C*a^7*b^5 - 224*A*C*a^8*b^4 - 128*A*C*a^9*b^3 + 128*A*C*a^10*b^2 + 8*B*C*a^2*b^10 - 36*B*C*a^3*b
^9 + 64*B*C*a^4*b^8 - 52*B*C*a^5*b^7 + 40*B*C*a^6*b^6 - 28*B*C*a^7*b^5 - 304*B*C*a^8*b^4 + 304*B*C*a^9*b^3 + 1
92*B*C*a^10*b^2))/(a*b^10 + b^11 - a^2*b^9 - a^3*b^8))*((B*b^3)/2 - 4*C*a^3 - b^2*(2*A*a + C*a) + 3*B*a^2*b))/
b^5 - (16*(256*C^3*a^14 - 128*C^3*a^13*b + 48*A^3*a^4*b^10 + 24*A^3*a^5*b^9 - 80*A^3*a^6*b^8 - 16*A^3*a^7*b^7
+ 32*A^3*a^8*b^6 - 4*B^3*a^3*b^11 + 4*B^3*a^4*b^10 - 41*B^3*a^5*b^9 + 9*B^3*a^6*b^8 - 63*B^3*a^7*b^7 - 81*B^3*
a^8*b^6 + 216*B^3*a^9*b^5 + 54*B^3*a^10*b^4 - 108*B^3*a^11*b^3 + 20*C^3*a^6*b^8 - 20*C^3*a^7*b^7 + 124*C^3*a^8
*b^6 - 24*C^3*a^9*b^5 + 48*C^3*a^10*b^4 + 192*C^3*a^11*b^3 - 448*C^3*a^12*b^2 - 576*B*C^2*a^13*b + 3*A*B^2*a^2
*b^12 - 3*A*B^2*a^3*b^11 + 63*A*B^2*a^4*b^10 - 15*A*B^2*a^5*b^9 + 186*A*B^2*a^6*b^8 + 162*A*B^2*a^7*b^7 - 468*
A*B^2*a^8*b^6 - 108*A*B^2*a^9*b^5 + 216*A*B^2*a^10*b^4 - 24*A^2*B*a^3*b^11 + 6*A^2*B*a^4*b^10 - 168*A^2*B*a^5*
b^9 - 108*A^2*B*a^6*b^8 + 336*A^2*B*a^7*b^7 + 72*A^2*B*a^8*b^6 - 144*A^2*B*a^9*b^5 + 12*A*C^2*a^4*b^10 - 12*A*
C^2*a^5*b^9 + 156*A*C^2*a^6*b^8 - 36*A*C^2*a^7*b^7 + 216*A*C^2*a^8*b^6 + 288*A*C^2*a^9*b^5 - 768*A*C^2*a^10*b^
4 - 192*A*C^2*a^11*b^3 + 384*A*C^2*a^12*b^2 + 48*A^2*C*a^4*b^10 - 12*A^2*C*a^5*b^9 + 192*A^2*C*a^6*b^8 + 144*A
^2*C*a^7*b^7 - 432*A^2*C*a^8*b^6 - 96*A^2*C*a^9*b^5 + 192*A^2*C*a^10*b^4 - 36*B*C^2*a^5*b^9 + 36*B*C^2*a^6*b^8
- 264*B*C^2*a^7*b^7 + 54*B*C^2*a^8*b^6 - 180*B*C^2*a^9*b^5 - 432*B*C^2*a^10*b^4 + 1056*B*C^2*a^11*b^3 + 288*B
*C^2*a^12*b^2 + 21*B^2*C*a^4*b^10 - 21*B^2*C*a^5*b^9 + 183*B^2*C*a^6*b^8 - 39*B^2*C*a^7*b^7 + 192*B^2*C*a^8*b^
6 + 324*B^2*C*a^9*b^5 - 828*B^2*C*a^10*b^4 - 216*B^2*C*a^11*b^3 + 432*B^2*C*a^12*b^2 - 12*A*B*C*a^3*b^11 + 12*
A*B*C*a^4*b^10 - 204*A*B*C*a^5*b^9 + 48*A*B*C*a^6*b^8 - 408*A*B*C*a^7*b^7 - 432*A*B*C*a^8*b^6 + 1200*A*B*C*a^9
*b^5 + 288*A*B*C*a^10*b^4 - 576*A*B*C*a^11*b^3))/(a*b^14 + b^15 - a^2*b^13 - a^3*b^12) + (((((8*(2*B*b^18 + 12
*A*a^2*b^16 + 12*A*a^3*b^15 - 20*A*a^4*b^14 - 4*A*a^5*b^13 + 8*A*a^6*b^12 + 6*B*a^2*b^16 - 16*B*a^3*b^15 - 14*
B*a^4*b^14 + 28*B*a^5*b^13 + 6*B*a^6*b^12 - 12*B*a^7*b^11 - 4*C*a^3*b^15 + 20*C*a^4*b^14 + 16*C*a^5*b^13 - 36*
C*a^6*b^12 - 8*C*a^7*b^11 + 16*C*a^8*b^10 - 8*A*a*b^17 - 4*C*a*b^17))/(a*b^14 + b^15 - a^2*b^13 - a^3*b^12) +
(8*tan(c/2 + (d*x)/2)*((B*b^3)/2 - 4*C*a^3 - b^2*(2*A*a + C*a) + 3*B*a^2*b)*(8*a*b^15 - 8*a^2*b^14 - 16*a^3*b^
13 + 16*a^4*b^12 + 8*a^5*b^11 - 8*a^6*b^10))/(b^5*(a*b^10 + b^11 - a^2*b^9 - a^3*b^8)))*((B*b^3)/2 - 4*C*a^3 -
b^2*(2*A*a + C*a) + 3*B*a^2*b))/b^5 + (8*tan(c/2 + (d*x)/2)*(B^2*b^12 + 128*C^2*a^12 - 2*B^2*a*b^11 - 128*C^2
*a^11*b + 16*A^2*a^2*b^10 - 32*A^2*a^3*b^9 + 20*A^2*a^4*b^8 + 64*A^2*a^5*b^7 - 64*A^2*a^6*b^6 - 32*A^2*a^7*b^5
+ 32*A^2*a^8*b^4 + 11*B^2*a^2*b^10 - 20*B^2*a^3*b^9 + 23*B^2*a^4*b^8 - 26*B^2*a^5*b^7 + 17*B^2*a^6*b^6 + 120*
B^2*a^7*b^5 - 120*B^2*a^8*b^4 - 72*B^2*a^9*b^3 + 72*B^2*a^10*b^2 + 4*C^2*a^2*b^10 - 8*C^2*a^3*b^9 + 28*C^2*a^4
*b^8 - 48*C^2*a^5*b^7 + 28*C^2*a^6*b^6 - 8*C^2*a^7*b^5 + 8*C^2*a^8*b^4 + 192*C^2*a^9*b^3 - 192*C^2*a^10*b^2 -
8*A*B*a*b^11 - 4*B*C*a*b^11 - 192*B*C*a^11*b + 16*A*B*a^2*b^10 - 40*A*B*a^3*b^9 + 64*A*B*a^4*b^8 - 40*A*B*a^5*
b^7 - 176*A*B*a^6*b^6 + 176*A*B*a^7*b^5 + 96*A*B*a^8*b^4 - 96*A*B*a^9*b^3 + 16*A*C*a^2*b^10 - 32*A*C*a^3*b^9 +
48*A*C*a^4*b^8 - 64*A*C*a^5*b^7 + 40*A*C*a^6*b^6 + 224*A*C*a^7*b^5 - 224*A*C*a^8*b^4 - 128*A*C*a^9*b^3 + 128*
A*C*a^10*b^2 + 8*B*C*a^2*b^10 - 36*B*C*a^3*b^9 + 64*B*C*a^4*b^8 - 52*B*C*a^5*b^7 + 40*B*C*a^6*b^6 - 28*B*C*a^7
*b^5 - 304*B*C*a^8*b^4 + 304*B*C*a^9*b^3 + 192*B*C*a^10*b^2))/(a*b^10 + b^11 - a^2*b^9 - a^3*b^8))*((B*b^3)/2
- 4*C*a^3 - b^2*(2*A*a + C*a) + 3*B*a^2*b))/b^5))*((B*b^3)/2 - 4*C*a^3 - b^2*(2*A*a + C*a) + 3*B*a^2*b)*2i)/(b
^5*d) - ((tan(c/2 + (d*x)/2)^5*(2*C*b^5 - 3*B*b^5 - 72*C*a^5 - 6*A*b^5 + 6*A*a^2*b^3 - 36*A*a^3*b^2 - 33*B*a^2
*b^3 - 9*B*a^3*b^2 - 14*C*a^2*b^3 + 38*C*a^3*b^2 + 18*A*a*b^4 + 9*B*a*b^4 + 54*B*a^4*b + 16*C*a*b^4 + 12*C*a^4
*b))/(3*(a*b^4 - b^5)*(a + b)) + (tan(c/2 + (d*x)/2)^7*(2*A*b^5 - B*b^5 + 8*C*a^5 + 2*C*b^5 - 2*A*a^2*b^3 + 4*
A*a^3*b^2 + 5*B*a^2*b^3 + 3*B*a^3*b^2 + 2*C*a^2*b^3 - 6*C*a^3*b^2 - 2*A*a*b^4 - 3*B*a*b^4 - 6*B*a^4*b - 4*C*a^
4*b))/((a*b^4 - b^5)*(a + b)) + (tan(c/2 + (d*x)/2)*(2*A*b^5 + B*b^5 - 8*C*a^5 + 2*C*b^5 - 2*A*a^2*b^3 - 4*A*a
^3*b^2 - 5*B*a^2*b^3 + 3*B*a^3*b^2 + 2*C*a^2*b^3 + 6*C*a^3*b^2 + 2*A*a*b^4 - 3*B*a*b^4 + 6*B*a^4*b - 4*C*a^4*b
))/((a*b^4 - b^5)*(a + b)) + (tan(c/2 + (d*x)/2)^3*(3*B*b^5 - 6*A*b^5 + 72*C*a^5 + 2*C*b^5 + 6*A*a^2*b^3 + 36*
A*a^3*b^2 + 33*B*a^2*b^3 - 9*B*a^3*b^2 - 14*C*a^2*b^3 - 38*C*a^3*b^2 - 18*A*a*b^4 + 9*B*a*b^4 - 54*B*a^4*b - 1
6*C*a*b^4 + 12*C*a^4*b))/(3*b^4*(a + b)*(a - b)))/(d*(a + b + tan(c/2 + (d*x)/2)^8*(a - b) - tan(c/2 + (d*x)/2
)^2*(4*a + 2*b) - tan(c/2 + (d*x)/2)^6*(4*a - 2*b) + 6*a*tan(c/2 + (d*x)/2)^4)) + (a^2*atan(((a^2*((8*tan(c/2
+ (d*x)/2)*(B^2*b^12 + 128*C^2*a^12 - 2*B^2*a*b^11 - 128*C^2*a^11*b + 16*A^2*a^2*b^10 - 32*A^2*a^3*b^9 + 20*A^
2*a^4*b^8 + 64*A^2*a^5*b^7 - 64*A^2*a^6*b^6 - 32*A^2*a^7*b^5 + 32*A^2*a^8*b^4 + 11*B^2*a^2*b^10 - 20*B^2*a^3*b
^9 + 23*B^2*a^4*b^8 - 26*B^2*a^5*b^7 + 17*B^2*a^6*b^6 + 120*B^2*a^7*b^5 - 120*B^2*a^8*b^4 - 72*B^2*a^9*b^3 + 7
2*B^2*a^10*b^2 + 4*C^2*a^2*b^10 - 8*C^2*a^3*b^9 + 28*C^2*a^4*b^8 - 48*C^2*a^5*b^7 + 28*C^2*a^6*b^6 - 8*C^2*a^7
*b^5 + 8*C^2*a^8*b^4 + 192*C^2*a^9*b^3 - 192*C^2*a^10*b^2 - 8*A*B*a*b^11 - 4*B*C*a*b^11 - 192*B*C*a^11*b + 16*
A*B*a^2*b^10 - 40*A*B*a^3*b^9 + 64*A*B*a^4*b^8 - 40*A*B*a^5*b^7 - 176*A*B*a^6*b^6 + 176*A*B*a^7*b^5 + 96*A*B*a
^8*b^4 - 96*A*B*a^9*b^3 + 16*A*C*a^2*b^10 - 32*A*C*a^3*b^9 + 48*A*C*a^4*b^8 - 64*A*C*a^5*b^7 + 40*A*C*a^6*b^6
+ 224*A*C*a^7*b^5 - 224*A*C*a^8*b^4 - 128*A*C*a^9*b^3 + 128*A*C*a^10*b^2 + 8*B*C*a^2*b^10 - 36*B*C*a^3*b^9 + 6
4*B*C*a^4*b^8 - 52*B*C*a^5*b^7 + 40*B*C*a^6*b^6 - 28*B*C*a^7*b^5 - 304*B*C*a^8*b^4 + 304*B*C*a^9*b^3 + 192*B*C
*a^10*b^2))/(a*b^10 + b^11 - a^2*b^9 - a^3*b^8) + (a^2*((8*(2*B*b^18 + 12*A*a^2*b^16 + 12*A*a^3*b^15 - 20*A*a^
4*b^14 - 4*A*a^5*b^13 + 8*A*a^6*b^12 + 6*B*a^2*b^16 - 16*B*a^3*b^15 - 14*B*a^4*b^14 + 28*B*a^5*b^13 + 6*B*a^6*
b^12 - 12*B*a^7*b^11 - 4*C*a^3*b^15 + 20*C*a^4*b^14 + 16*C*a^5*b^13 - 36*C*a^6*b^12 - 8*C*a^7*b^11 + 16*C*a^8*
b^10 - 8*A*a*b^17 - 4*C*a*b^17))/(a*b^14 + b^15 - a^2*b^13 - a^3*b^12) + (8*a^2*tan(c/2 + (d*x)/2)*((a + b)^3*
(a - b)^3)^(1/2)*(8*a*b^15 - 8*a^2*b^14 - 16*a^3*b^13 + 16*a^4*b^12 + 8*a^5*b^11 - 8*a^6*b^10)*(3*A*b^4 - 4*C*
a^4 - 2*A*a^2*b^2 + 5*C*a^2*b^2 - 4*B*a*b^3 + 3*B*a^3*b))/((a*b^10 + b^11 - a^2*b^9 - a^3*b^8)*(b^11 - 3*a^2*b
^9 + 3*a^4*b^7 - a^6*b^5)))*((a + b)^3*(a - b)^3)^(1/2)*(3*A*b^4 - 4*C*a^4 - 2*A*a^2*b^2 + 5*C*a^2*b^2 - 4*B*a
*b^3 + 3*B*a^3*b))/(b^11 - 3*a^2*b^9 + 3*a^4*b^7 - a^6*b^5))*((a + b)^3*(a - b)^3)^(1/2)*(3*A*b^4 - 4*C*a^4 -
2*A*a^2*b^2 + 5*C*a^2*b^2 - 4*B*a*b^3 + 3*B*a^3*b)*1i)/(b^11 - 3*a^2*b^9 + 3*a^4*b^7 - a^6*b^5) + (a^2*((8*tan
(c/2 + (d*x)/2)*(B^2*b^12 + 128*C^2*a^12 - 2*B^2*a*b^11 - 128*C^2*a^11*b + 16*A^2*a^2*b^10 - 32*A^2*a^3*b^9 +
20*A^2*a^4*b^8 + 64*A^2*a^5*b^7 - 64*A^2*a^6*b^6 - 32*A^2*a^7*b^5 + 32*A^2*a^8*b^4 + 11*B^2*a^2*b^10 - 20*B^2*
a^3*b^9 + 23*B^2*a^4*b^8 - 26*B^2*a^5*b^7 + 17*B^2*a^6*b^6 + 120*B^2*a^7*b^5 - 120*B^2*a^8*b^4 - 72*B^2*a^9*b^
3 + 72*B^2*a^10*b^2 + 4*C^2*a^2*b^10 - 8*C^2*a^3*b^9 + 28*C^2*a^4*b^8 - 48*C^2*a^5*b^7 + 28*C^2*a^6*b^6 - 8*C^
2*a^7*b^5 + 8*C^2*a^8*b^4 + 192*C^2*a^9*b^3 - 192*C^2*a^10*b^2 - 8*A*B*a*b^11 - 4*B*C*a*b^11 - 192*B*C*a^11*b
+ 16*A*B*a^2*b^10 - 40*A*B*a^3*b^9 + 64*A*B*a^4*b^8 - 40*A*B*a^5*b^7 - 176*A*B*a^6*b^6 + 176*A*B*a^7*b^5 + 96*
A*B*a^8*b^4 - 96*A*B*a^9*b^3 + 16*A*C*a^2*b^10 - 32*A*C*a^3*b^9 + 48*A*C*a^4*b^8 - 64*A*C*a^5*b^7 + 40*A*C*a^6
*b^6 + 224*A*C*a^7*b^5 - 224*A*C*a^8*b^4 - 128*A*C*a^9*b^3 + 128*A*C*a^10*b^2 + 8*B*C*a^2*b^10 - 36*B*C*a^3*b^
9 + 64*B*C*a^4*b^8 - 52*B*C*a^5*b^7 + 40*B*C*a^6*b^6 - 28*B*C*a^7*b^5 - 304*B*C*a^8*b^4 + 304*B*C*a^9*b^3 + 19
2*B*C*a^10*b^2))/(a*b^10 + b^11 - a^2*b^9 - a^3*b^8) - (a^2*((8*(2*B*b^18 + 12*A*a^2*b^16 + 12*A*a^3*b^15 - 20
*A*a^4*b^14 - 4*A*a^5*b^13 + 8*A*a^6*b^12 + 6*B*a^2*b^16 - 16*B*a^3*b^15 - 14*B*a^4*b^14 + 28*B*a^5*b^13 + 6*B
*a^6*b^12 - 12*B*a^7*b^11 - 4*C*a^3*b^15 + 20*C*a^4*b^14 + 16*C*a^5*b^13 - 36*C*a^6*b^12 - 8*C*a^7*b^11 + 16*C
*a^8*b^10 - 8*A*a*b^17 - 4*C*a*b^17))/(a*b^14 + b^15 - a^2*b^13 - a^3*b^12) - (8*a^2*tan(c/2 + (d*x)/2)*((a +
b)^3*(a - b)^3)^(1/2)*(8*a*b^15 - 8*a^2*b^14 - 16*a^3*b^13 + 16*a^4*b^12 + 8*a^5*b^11 - 8*a^6*b^10)*(3*A*b^4 -
4*C*a^4 - 2*A*a^2*b^2 + 5*C*a^2*b^2 - 4*B*a*b^3 + 3*B*a^3*b))/((a*b^10 + b^11 - a^2*b^9 - a^3*b^8)*(b^11 - 3*
a^2*b^9 + 3*a^4*b^7 - a^6*b^5)))*((a + b)^3*(a - b)^3)^(1/2)*(3*A*b^4 - 4*C*a^4 - 2*A*a^2*b^2 + 5*C*a^2*b^2 -
4*B*a*b^3 + 3*B*a^3*b))/(b^11 - 3*a^2*b^9 + 3*a^4*b^7 - a^6*b^5))*((a + b)^3*(a - b)^3)^(1/2)*(3*A*b^4 - 4*C*a
^4 - 2*A*a^2*b^2 + 5*C*a^2*b^2 - 4*B*a*b^3 + 3*B*a^3*b)*1i)/(b^11 - 3*a^2*b^9 + 3*a^4*b^7 - a^6*b^5))/((16*(25
6*C^3*a^14 - 128*C^3*a^13*b + 48*A^3*a^4*b^10 + 24*A^3*a^5*b^9 - 80*A^3*a^6*b^8 - 16*A^3*a^7*b^7 + 32*A^3*a^8*
b^6 - 4*B^3*a^3*b^11 + 4*B^3*a^4*b^10 - 41*B^3*a^5*b^9 + 9*B^3*a^6*b^8 - 63*B^3*a^7*b^7 - 81*B^3*a^8*b^6 + 216
*B^3*a^9*b^5 + 54*B^3*a^10*b^4 - 108*B^3*a^11*b^3 + 20*C^3*a^6*b^8 - 20*C^3*a^7*b^7 + 124*C^3*a^8*b^6 - 24*C^3
*a^9*b^5 + 48*C^3*a^10*b^4 + 192*C^3*a^11*b^3 - 448*C^3*a^12*b^2 - 576*B*C^2*a^13*b + 3*A*B^2*a^2*b^12 - 3*A*B
^2*a^3*b^11 + 63*A*B^2*a^4*b^10 - 15*A*B^2*a^5*b^9 + 186*A*B^2*a^6*b^8 + 162*A*B^2*a^7*b^7 - 468*A*B^2*a^8*b^6
- 108*A*B^2*a^9*b^5 + 216*A*B^2*a^10*b^4 - 24*A^2*B*a^3*b^11 + 6*A^2*B*a^4*b^10 - 168*A^2*B*a^5*b^9 - 108*A^2
*B*a^6*b^8 + 336*A^2*B*a^7*b^7 + 72*A^2*B*a^8*b^6 - 144*A^2*B*a^9*b^5 + 12*A*C^2*a^4*b^10 - 12*A*C^2*a^5*b^9 +
156*A*C^2*a^6*b^8 - 36*A*C^2*a^7*b^7 + 216*A*C^2*a^8*b^6 + 288*A*C^2*a^9*b^5 - 768*A*C^2*a^10*b^4 - 192*A*C^2
*a^11*b^3 + 384*A*C^2*a^12*b^2 + 48*A^2*C*a^4*b^10 - 12*A^2*C*a^5*b^9 + 192*A^2*C*a^6*b^8 + 144*A^2*C*a^7*b^7
- 432*A^2*C*a^8*b^6 - 96*A^2*C*a^9*b^5 + 192*A^2*C*a^10*b^4 - 36*B*C^2*a^5*b^9 + 36*B*C^2*a^6*b^8 - 264*B*C^2*
a^7*b^7 + 54*B*C^2*a^8*b^6 - 180*B*C^2*a^9*b^5 - 432*B*C^2*a^10*b^4 + 1056*B*C^2*a^11*b^3 + 288*B*C^2*a^12*b^2
+ 21*B^2*C*a^4*b^10 - 21*B^2*C*a^5*b^9 + 183*B^2*C*a^6*b^8 - 39*B^2*C*a^7*b^7 + 192*B^2*C*a^8*b^6 + 324*B^2*C
*a^9*b^5 - 828*B^2*C*a^10*b^4 - 216*B^2*C*a^11*b^3 + 432*B^2*C*a^12*b^2 - 12*A*B*C*a^3*b^11 + 12*A*B*C*a^4*b^1
0 - 204*A*B*C*a^5*b^9 + 48*A*B*C*a^6*b^8 - 408*A*B*C*a^7*b^7 - 432*A*B*C*a^8*b^6 + 1200*A*B*C*a^9*b^5 + 288*A*
B*C*a^10*b^4 - 576*A*B*C*a^11*b^3))/(a*b^14 + b^15 - a^2*b^13 - a^3*b^12) - (a^2*((8*tan(c/2 + (d*x)/2)*(B^2*b
^12 + 128*C^2*a^12 - 2*B^2*a*b^11 - 128*C^2*a^11*b + 16*A^2*a^2*b^10 - 32*A^2*a^3*b^9 + 20*A^2*a^4*b^8 + 64*A^
2*a^5*b^7 - 64*A^2*a^6*b^6 - 32*A^2*a^7*b^5 + 32*A^2*a^8*b^4 + 11*B^2*a^2*b^10 - 20*B^2*a^3*b^9 + 23*B^2*a^4*b
^8 - 26*B^2*a^5*b^7 + 17*B^2*a^6*b^6 + 120*B^2*a^7*b^5 - 120*B^2*a^8*b^4 - 72*B^2*a^9*b^3 + 72*B^2*a^10*b^2 +
4*C^2*a^2*b^10 - 8*C^2*a^3*b^9 + 28*C^2*a^4*b^8 - 48*C^2*a^5*b^7 + 28*C^2*a^6*b^6 - 8*C^2*a^7*b^5 + 8*C^2*a^8*
b^4 + 192*C^2*a^9*b^3 - 192*C^2*a^10*b^2 - 8*A*B*a*b^11 - 4*B*C*a*b^11 - 192*B*C*a^11*b + 16*A*B*a^2*b^10 - 40
*A*B*a^3*b^9 + 64*A*B*a^4*b^8 - 40*A*B*a^5*b^7 - 176*A*B*a^6*b^6 + 176*A*B*a^7*b^5 + 96*A*B*a^8*b^4 - 96*A*B*a
^9*b^3 + 16*A*C*a^2*b^10 - 32*A*C*a^3*b^9 + 48*A*C*a^4*b^8 - 64*A*C*a^5*b^7 + 40*A*C*a^6*b^6 + 224*A*C*a^7*b^5
- 224*A*C*a^8*b^4 - 128*A*C*a^9*b^3 + 128*A*C*a^10*b^2 + 8*B*C*a^2*b^10 - 36*B*C*a^3*b^9 + 64*B*C*a^4*b^8 - 5
2*B*C*a^5*b^7 + 40*B*C*a^6*b^6 - 28*B*C*a^7*b^5 - 304*B*C*a^8*b^4 + 304*B*C*a^9*b^3 + 192*B*C*a^10*b^2))/(a*b^
10 + b^11 - a^2*b^9 - a^3*b^8) + (a^2*((8*(2*B*b^18 + 12*A*a^2*b^16 + 12*A*a^3*b^15 - 20*A*a^4*b^14 - 4*A*a^5*
b^13 + 8*A*a^6*b^12 + 6*B*a^2*b^16 - 16*B*a^3*b^15 - 14*B*a^4*b^14 + 28*B*a^5*b^13 + 6*B*a^6*b^12 - 12*B*a^7*b
^11 - 4*C*a^3*b^15 + 20*C*a^4*b^14 + 16*C*a^5*b^13 - 36*C*a^6*b^12 - 8*C*a^7*b^11 + 16*C*a^8*b^10 - 8*A*a*b^17
- 4*C*a*b^17))/(a*b^14 + b^15 - a^2*b^13 - a^3*b^12) + (8*a^2*tan(c/2 + (d*x)/2)*((a + b)^3*(a - b)^3)^(1/2)*
(8*a*b^15 - 8*a^2*b^14 - 16*a^3*b^13 + 16*a^4*b^12 + 8*a^5*b^11 - 8*a^6*b^10)*(3*A*b^4 - 4*C*a^4 - 2*A*a^2*b^2
+ 5*C*a^2*b^2 - 4*B*a*b^3 + 3*B*a^3*b))/((a*b^10 + b^11 - a^2*b^9 - a^3*b^8)*(b^11 - 3*a^2*b^9 + 3*a^4*b^7 -
a^6*b^5)))*((a + b)^3*(a - b)^3)^(1/2)*(3*A*b^4 - 4*C*a^4 - 2*A*a^2*b^2 + 5*C*a^2*b^2 - 4*B*a*b^3 + 3*B*a^3*b)
)/(b^11 - 3*a^2*b^9 + 3*a^4*b^7 - a^6*b^5))*((a + b)^3*(a - b)^3)^(1/2)*(3*A*b^4 - 4*C*a^4 - 2*A*a^2*b^2 + 5*C
*a^2*b^2 - 4*B*a*b^3 + 3*B*a^3*b))/(b^11 - 3*a^2*b^9 + 3*a^4*b^7 - a^6*b^5) + (a^2*((8*tan(c/2 + (d*x)/2)*(B^2
*b^12 + 128*C^2*a^12 - 2*B^2*a*b^11 - 128*C^2*a^11*b + 16*A^2*a^2*b^10 - 32*A^2*a^3*b^9 + 20*A^2*a^4*b^8 + 64*
A^2*a^5*b^7 - 64*A^2*a^6*b^6 - 32*A^2*a^7*b^5 + 32*A^2*a^8*b^4 + 11*B^2*a^2*b^10 - 20*B^2*a^3*b^9 + 23*B^2*a^4
*b^8 - 26*B^2*a^5*b^7 + 17*B^2*a^6*b^6 + 120*B^2*a^7*b^5 - 120*B^2*a^8*b^4 - 72*B^2*a^9*b^3 + 72*B^2*a^10*b^2
+ 4*C^2*a^2*b^10 - 8*C^2*a^3*b^9 + 28*C^2*a^4*b^8 - 48*C^2*a^5*b^7 + 28*C^2*a^6*b^6 - 8*C^2*a^7*b^5 + 8*C^2*a^
8*b^4 + 192*C^2*a^9*b^3 - 192*C^2*a^10*b^2 - 8*A*B*a*b^11 - 4*B*C*a*b^11 - 192*B*C*a^11*b + 16*A*B*a^2*b^10 -
40*A*B*a^3*b^9 + 64*A*B*a^4*b^8 - 40*A*B*a^5*b^7 - 176*A*B*a^6*b^6 + 176*A*B*a^7*b^5 + 96*A*B*a^8*b^4 - 96*A*B
*a^9*b^3 + 16*A*C*a^2*b^10 - 32*A*C*a^3*b^9 + 48*A*C*a^4*b^8 - 64*A*C*a^5*b^7 + 40*A*C*a^6*b^6 + 224*A*C*a^7*b
^5 - 224*A*C*a^8*b^4 - 128*A*C*a^9*b^3 + 128*A*C*a^10*b^2 + 8*B*C*a^2*b^10 - 36*B*C*a^3*b^9 + 64*B*C*a^4*b^8 -
52*B*C*a^5*b^7 + 40*B*C*a^6*b^6 - 28*B*C*a^7*b^5 - 304*B*C*a^8*b^4 + 304*B*C*a^9*b^3 + 192*B*C*a^10*b^2))/(a*
b^10 + b^11 - a^2*b^9 - a^3*b^8) - (a^2*((8*(2*B*b^18 + 12*A*a^2*b^16 + 12*A*a^3*b^15 - 20*A*a^4*b^14 - 4*A*a^
5*b^13 + 8*A*a^6*b^12 + 6*B*a^2*b^16 - 16*B*a^3*b^15 - 14*B*a^4*b^14 + 28*B*a^5*b^13 + 6*B*a^6*b^12 - 12*B*a^7
*b^11 - 4*C*a^3*b^15 + 20*C*a^4*b^14 + 16*C*a^5*b^13 - 36*C*a^6*b^12 - 8*C*a^7*b^11 + 16*C*a^8*b^10 - 8*A*a*b^
17 - 4*C*a*b^17))/(a*b^14 + b^15 - a^2*b^13 - a^3*b^12) - (8*a^2*tan(c/2 + (d*x)/2)*((a + b)^3*(a - b)^3)^(1/2
)*(8*a*b^15 - 8*a^2*b^14 - 16*a^3*b^13 + 16*a^4*b^12 + 8*a^5*b^11 - 8*a^6*b^10)*(3*A*b^4 - 4*C*a^4 - 2*A*a^2*b
^2 + 5*C*a^2*b^2 - 4*B*a*b^3 + 3*B*a^3*b))/((a*b^10 + b^11 - a^2*b^9 - a^3*b^8)*(b^11 - 3*a^2*b^9 + 3*a^4*b^7
- a^6*b^5)))*((a + b)^3*(a - b)^3)^(1/2)*(3*A*b^4 - 4*C*a^4 - 2*A*a^2*b^2 + 5*C*a^2*b^2 - 4*B*a*b^3 + 3*B*a^3*
b))/(b^11 - 3*a^2*b^9 + 3*a^4*b^7 - a^6*b^5))*((a + b)^3*(a - b)^3)^(1/2)*(3*A*b^4 - 4*C*a^4 - 2*A*a^2*b^2 + 5
*C*a^2*b^2 - 4*B*a*b^3 + 3*B*a^3*b))/(b^11 - 3*a^2*b^9 + 3*a^4*b^7 - a^6*b^5)))*((a + b)^3*(a - b)^3)^(1/2)*(3
*A*b^4 - 4*C*a^4 - 2*A*a^2*b^2 + 5*C*a^2*b^2 - 4*B*a*b^3 + 3*B*a^3*b)*2i)/(d*(b^11 - 3*a^2*b^9 + 3*a^4*b^7 - a
^6*b^5))