3.8.4 \(\int \frac {\cos ^2(c+d x) (A+C \sec ^2(c+d x))}{(a+b \sec (c+d x))^4} \, dx\) [704]

Optimal. Leaf size=513 \[ \frac {\left (20 A b^2+a^2 (A+2 C)\right ) x}{2 a^6}+\frac {\left (20 A b^9-a^2 b^7 (69 A-2 C)-8 a^6 b^3 (5 A-C)+7 a^4 b^5 (12 A-C)-8 a^8 b C\right ) \tanh ^{-1}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{a^6 \sqrt {a-b} \sqrt {a+b} \left (a^2-b^2\right )^3 d}+\frac {b \left (60 A b^6-a^6 (24 A-26 C)+a^4 b^2 (146 A-17 C)-a^2 b^4 (167 A-6 C)\right ) \sin (c+d x)}{6 a^5 \left (a^2-b^2\right )^3 d}-\frac {\left (10 A b^6-a^6 (A-6 C)+a^4 b^2 (23 A-2 C)-a^2 b^4 (27 A-C)\right ) \cos (c+d x) \sin (c+d x)}{2 a^4 \left (a^2-b^2\right )^3 d}+\frac {\left (A b^2+a^2 C\right ) \cos (c+d x) \sin (c+d x)}{3 a \left (a^2-b^2\right ) d (a+b \sec (c+d x))^3}-\frac {\left (5 A b^4-4 a^4 C-a^2 b^2 (10 A+C)\right ) \cos (c+d x) \sin (c+d x)}{6 a^2 \left (a^2-b^2\right )^2 d (a+b \sec (c+d x))^2}+\frac {\left (20 A b^6-a^2 b^4 (53 A-2 C)+12 a^6 C+a^4 b^2 (48 A+C)\right ) \cos (c+d x) \sin (c+d x)}{6 a^3 \left (a^2-b^2\right )^3 d (a+b \sec (c+d x))} \]

[Out]

1/2*(20*A*b^2+a^2*(A+2*C))*x/a^6+1/6*b*(60*A*b^6-a^6*(24*A-26*C)+a^4*b^2*(146*A-17*C)-a^2*b^4*(167*A-6*C))*sin
(d*x+c)/a^5/(a^2-b^2)^3/d-1/2*(10*A*b^6-a^6*(A-6*C)+a^4*b^2*(23*A-2*C)-a^2*b^4*(27*A-C))*cos(d*x+c)*sin(d*x+c)
/a^4/(a^2-b^2)^3/d+1/3*(A*b^2+C*a^2)*cos(d*x+c)*sin(d*x+c)/a/(a^2-b^2)/d/(a+b*sec(d*x+c))^3-1/6*(5*A*b^4-4*a^4
*C-a^2*b^2*(10*A+C))*cos(d*x+c)*sin(d*x+c)/a^2/(a^2-b^2)^2/d/(a+b*sec(d*x+c))^2+1/6*(20*A*b^6-a^2*b^4*(53*A-2*
C)+12*a^6*C+a^4*b^2*(48*A+C))*cos(d*x+c)*sin(d*x+c)/a^3/(a^2-b^2)^3/d/(a+b*sec(d*x+c))+(20*A*b^9-a^2*b^7*(69*A
-2*C)-8*a^6*b^3*(5*A-C)+7*a^4*b^5*(12*A-C)-8*a^8*b*C)*arctanh((a-b)^(1/2)*tan(1/2*d*x+1/2*c)/(a+b)^(1/2))/a^6/
(a^2-b^2)^3/d/(a-b)^(1/2)/(a+b)^(1/2)

________________________________________________________________________________________

Rubi [A]
time = 1.64, antiderivative size = 513, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 7, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.212, Rules used = {4186, 4185, 4189, 4004, 3916, 2738, 214} \begin {gather*} \frac {\left (a^2 C+A b^2\right ) \sin (c+d x) \cos (c+d x)}{3 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^3}+\frac {x \left (a^2 (A+2 C)+20 A b^2\right )}{2 a^6}-\frac {\left (-4 a^4 C-a^2 b^2 (10 A+C)+5 A b^4\right ) \sin (c+d x) \cos (c+d x)}{6 a^2 d \left (a^2-b^2\right )^2 (a+b \sec (c+d x))^2}-\frac {\left (-\left (a^6 (A-6 C)\right )+a^4 b^2 (23 A-2 C)-a^2 b^4 (27 A-C)+10 A b^6\right ) \sin (c+d x) \cos (c+d x)}{2 a^4 d \left (a^2-b^2\right )^3}+\frac {\left (-8 a^8 b C-8 a^6 b^3 (5 A-C)+7 a^4 b^5 (12 A-C)-a^2 b^7 (69 A-2 C)+20 A b^9\right ) \tanh ^{-1}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{a^6 d \sqrt {a-b} \sqrt {a+b} \left (a^2-b^2\right )^3}+\frac {b \left (-\left (a^6 (24 A-26 C)\right )+a^4 b^2 (146 A-17 C)-a^2 b^4 (167 A-6 C)+60 A b^6\right ) \sin (c+d x)}{6 a^5 d \left (a^2-b^2\right )^3}+\frac {\left (12 a^6 C+a^4 b^2 (48 A+C)-a^2 b^4 (53 A-2 C)+20 A b^6\right ) \sin (c+d x) \cos (c+d x)}{6 a^3 d \left (a^2-b^2\right )^3 (a+b \sec (c+d x))} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((20*A*b^2 + a^2*(A + 2*C))*x)/(2*a^6) + ((20*A*b^9 - a^2*b^7*(69*A - 2*C) - 8*a^6*b^3*(5*A - C) + 7*a^4*b^5*(
12*A - C) - 8*a^8*b*C)*ArcTanh[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/(a^6*Sqrt[a - b]*Sqrt[a + b]*(a^2
- b^2)^3*d) + (b*(60*A*b^6 - a^6*(24*A - 26*C) + a^4*b^2*(146*A - 17*C) - a^2*b^4*(167*A - 6*C))*Sin[c + d*x])
/(6*a^5*(a^2 - b^2)^3*d) - ((10*A*b^6 - a^6*(A - 6*C) + a^4*b^2*(23*A - 2*C) - a^2*b^4*(27*A - C))*Cos[c + d*x
]*Sin[c + d*x])/(2*a^4*(a^2 - b^2)^3*d) + ((A*b^2 + a^2*C)*Cos[c + d*x]*Sin[c + d*x])/(3*a*(a^2 - b^2)*d*(a +
b*Sec[c + d*x])^3) - ((5*A*b^4 - 4*a^4*C - a^2*b^2*(10*A + C))*Cos[c + d*x]*Sin[c + d*x])/(6*a^2*(a^2 - b^2)^2
*d*(a + b*Sec[c + d*x])^2) + ((20*A*b^6 - a^2*b^4*(53*A - 2*C) + 12*a^6*C + a^4*b^2*(48*A + C))*Cos[c + d*x]*S
in[c + d*x])/(6*a^3*(a^2 - b^2)^3*d*(a + b*Sec[c + d*x]))

Rule 214

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

Rule 2738

Int[((a_) + (b_.)*sin[Pi/2 + (c_.) + (d_.)*(x_)])^(-1), x_Symbol] :> With[{e = FreeFactors[Tan[(c + d*x)/2], x
]}, Dist[2*(e/d), Subst[Int[1/(a + b + (a - b)*e^2*x^2), x], x, Tan[(c + d*x)/2]/e], x]] /; FreeQ[{a, b, c, d}
, x] && NeQ[a^2 - b^2, 0]

Rule 3916

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

Rule 4004

Int[(csc[(e_.) + (f_.)*(x_)]*(d_.) + (c_))/(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)), x_Symbol] :> Simp[c*(x/a),
x] - Dist[(b*c - a*d)/a, Int[Csc[e + f*x]/(a + b*Csc[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[
b*c - a*d, 0]

Rule 4185

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

Rule 4186

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

Rule 4189

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

Rubi steps

\begin {align*} \int \frac {\cos ^2(c+d x) \left (A+C \sec ^2(c+d x)\right )}{(a+b \sec (c+d x))^4} \, dx &=\frac {\left (A b^2+a^2 C\right ) \cos (c+d x) \sin (c+d x)}{3 a \left (a^2-b^2\right ) d (a+b \sec (c+d x))^3}-\frac {\int \frac {\cos ^2(c+d x) \left (5 A b^2-a^2 (3 A-2 C)+3 a b (A+C) \sec (c+d x)-4 \left (A b^2+a^2 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^2 C\right ) \cos (c+d x) \sin (c+d x)}{3 a \left (a^2-b^2\right ) d (a+b \sec (c+d x))^3}-\frac {\left (5 A b^4-4 a^4 C-a^2 b^2 (10 A+C)\right ) \cos (c+d x) \sin (c+d x)}{6 a^2 \left (a^2-b^2\right )^2 d (a+b \sec (c+d x))^2}+\frac {\int \frac {\cos ^2(c+d x) \left (2 \left (10 A b^4+3 a^4 (A-2 C)-a^2 b^2 (18 A-C)\right )+2 a b \left (A b^2-a^2 (6 A+5 C)\right ) \sec (c+d x)-3 \left (5 A b^4-4 a^4 C-a^2 b^2 (10 A+C)\right ) \sec ^2(c+d x)\right )}{(a+b \sec (c+d x))^2} \, dx}{6 a^2 \left (a^2-b^2\right )^2}\\ &=\frac {\left (A b^2+a^2 C\right ) \cos (c+d x) \sin (c+d x)}{3 a \left (a^2-b^2\right ) d (a+b \sec (c+d x))^3}-\frac {\left (5 A b^4-4 a^4 C-a^2 b^2 (10 A+C)\right ) \cos (c+d x) \sin (c+d x)}{6 a^2 \left (a^2-b^2\right )^2 d (a+b \sec (c+d x))^2}+\frac {\left (20 A b^6-a^2 b^4 (53 A-2 C)+12 a^6 C+a^4 b^2 (48 A+C)\right ) \cos (c+d x) \sin (c+d x)}{6 a^3 \left (a^2-b^2\right )^3 d (a+b \sec (c+d x))}-\frac {\int \frac {\cos ^2(c+d x) \left (6 \left (10 A b^6-a^6 (A-6 C)+a^4 b^2 (23 A-2 C)-a^2 b^4 (27 A-C)\right )+a b \left (5 A b^4-a^2 b^2 (8 A-5 C)+2 a^4 (9 A+5 C)\right ) \sec (c+d x)-2 \left (20 A b^6-a^2 b^4 (53 A-2 C)+12 a^6 C+a^4 b^2 (48 A+C)\right ) \sec ^2(c+d x)\right )}{a+b \sec (c+d x)} \, dx}{6 a^3 \left (a^2-b^2\right )^3}\\ &=-\frac {\left (10 A b^6-a^6 (A-6 C)+a^4 b^2 (23 A-2 C)-a^2 b^4 (27 A-C)\right ) \cos (c+d x) \sin (c+d x)}{2 a^4 \left (a^2-b^2\right )^3 d}+\frac {\left (A b^2+a^2 C\right ) \cos (c+d x) \sin (c+d x)}{3 a \left (a^2-b^2\right ) d (a+b \sec (c+d x))^3}-\frac {\left (5 A b^4-4 a^4 C-a^2 b^2 (10 A+C)\right ) \cos (c+d x) \sin (c+d x)}{6 a^2 \left (a^2-b^2\right )^2 d (a+b \sec (c+d x))^2}+\frac {\left (20 A b^6-a^2 b^4 (53 A-2 C)+12 a^6 C+a^4 b^2 (48 A+C)\right ) \cos (c+d x) \sin (c+d x)}{6 a^3 \left (a^2-b^2\right )^3 d (a+b \sec (c+d x))}+\frac {\int \frac {\cos (c+d x) \left (2 \left (60 A b^7+a^4 b^3 (146 A-17 C)-a^2 b^5 (167 A-6 C)-a^6 (24 A b-26 b C)\right )+2 a \left (10 A b^6-a^2 b^4 (25 A-C)+3 a^6 (A+2 C)+a^4 b^2 (27 A+8 C)\right ) \sec (c+d x)-6 b \left (10 A b^6-a^6 (A-6 C)+a^4 b^2 (23 A-2 C)-a^2 b^4 (27 A-C)\right ) \sec ^2(c+d x)\right )}{a+b \sec (c+d x)} \, dx}{12 a^4 \left (a^2-b^2\right )^3}\\ &=\frac {b \left (60 A b^6-a^6 (24 A-26 C)+a^4 b^2 (146 A-17 C)-a^2 b^4 (167 A-6 C)\right ) \sin (c+d x)}{6 a^5 \left (a^2-b^2\right )^3 d}-\frac {\left (10 A b^6-a^6 (A-6 C)+a^4 b^2 (23 A-2 C)-a^2 b^4 (27 A-C)\right ) \cos (c+d x) \sin (c+d x)}{2 a^4 \left (a^2-b^2\right )^3 d}+\frac {\left (A b^2+a^2 C\right ) \cos (c+d x) \sin (c+d x)}{3 a \left (a^2-b^2\right ) d (a+b \sec (c+d x))^3}-\frac {\left (5 A b^4-4 a^4 C-a^2 b^2 (10 A+C)\right ) \cos (c+d x) \sin (c+d x)}{6 a^2 \left (a^2-b^2\right )^2 d (a+b \sec (c+d x))^2}+\frac {\left (20 A b^6-a^2 b^4 (53 A-2 C)+12 a^6 C+a^4 b^2 (48 A+C)\right ) \cos (c+d x) \sin (c+d x)}{6 a^3 \left (a^2-b^2\right )^3 d (a+b \sec (c+d x))}-\frac {\int \frac {-6 \left (a^2-b^2\right )^3 \left (20 A b^2+a^2 (A+2 C)\right )+6 a b \left (10 A b^6-a^6 (A-6 C)+a^4 b^2 (23 A-2 C)-a^2 b^4 (27 A-C)\right ) \sec (c+d x)}{a+b \sec (c+d x)} \, dx}{12 a^5 \left (a^2-b^2\right )^3}\\ &=\frac {\left (20 A b^2+a^2 (A+2 C)\right ) x}{2 a^6}+\frac {b \left (60 A b^6-a^6 (24 A-26 C)+a^4 b^2 (146 A-17 C)-a^2 b^4 (167 A-6 C)\right ) \sin (c+d x)}{6 a^5 \left (a^2-b^2\right )^3 d}-\frac {\left (10 A b^6-a^6 (A-6 C)+a^4 b^2 (23 A-2 C)-a^2 b^4 (27 A-C)\right ) \cos (c+d x) \sin (c+d x)}{2 a^4 \left (a^2-b^2\right )^3 d}+\frac {\left (A b^2+a^2 C\right ) \cos (c+d x) \sin (c+d x)}{3 a \left (a^2-b^2\right ) d (a+b \sec (c+d x))^3}-\frac {\left (5 A b^4-4 a^4 C-a^2 b^2 (10 A+C)\right ) \cos (c+d x) \sin (c+d x)}{6 a^2 \left (a^2-b^2\right )^2 d (a+b \sec (c+d x))^2}+\frac {\left (20 A b^6-a^2 b^4 (53 A-2 C)+12 a^6 C+a^4 b^2 (48 A+C)\right ) \cos (c+d x) \sin (c+d x)}{6 a^3 \left (a^2-b^2\right )^3 d (a+b \sec (c+d x))}+\frac {\left (20 A b^9-a^2 b^7 (69 A-2 C)-8 a^6 b^3 (5 A-C)+7 a^4 b^5 (12 A-C)-8 a^8 b C\right ) \int \frac {\sec (c+d x)}{a+b \sec (c+d x)} \, dx}{2 a^6 \left (a^2-b^2\right )^3}\\ &=\frac {\left (20 A b^2+a^2 (A+2 C)\right ) x}{2 a^6}+\frac {b \left (60 A b^6-a^6 (24 A-26 C)+a^4 b^2 (146 A-17 C)-a^2 b^4 (167 A-6 C)\right ) \sin (c+d x)}{6 a^5 \left (a^2-b^2\right )^3 d}-\frac {\left (10 A b^6-a^6 (A-6 C)+a^4 b^2 (23 A-2 C)-a^2 b^4 (27 A-C)\right ) \cos (c+d x) \sin (c+d x)}{2 a^4 \left (a^2-b^2\right )^3 d}+\frac {\left (A b^2+a^2 C\right ) \cos (c+d x) \sin (c+d x)}{3 a \left (a^2-b^2\right ) d (a+b \sec (c+d x))^3}-\frac {\left (5 A b^4-4 a^4 C-a^2 b^2 (10 A+C)\right ) \cos (c+d x) \sin (c+d x)}{6 a^2 \left (a^2-b^2\right )^2 d (a+b \sec (c+d x))^2}+\frac {\left (20 A b^6-a^2 b^4 (53 A-2 C)+12 a^6 C+a^4 b^2 (48 A+C)\right ) \cos (c+d x) \sin (c+d x)}{6 a^3 \left (a^2-b^2\right )^3 d (a+b \sec (c+d x))}+\frac {\left (20 A b^8-a^2 b^6 (69 A-2 C)-8 a^6 b^2 (5 A-C)+7 a^4 b^4 (12 A-C)-8 a^8 C\right ) \int \frac {1}{1+\frac {a \cos (c+d x)}{b}} \, dx}{2 a^6 \left (a^2-b^2\right )^3}\\ &=\frac {\left (20 A b^2+a^2 (A+2 C)\right ) x}{2 a^6}+\frac {b \left (60 A b^6-a^6 (24 A-26 C)+a^4 b^2 (146 A-17 C)-a^2 b^4 (167 A-6 C)\right ) \sin (c+d x)}{6 a^5 \left (a^2-b^2\right )^3 d}-\frac {\left (10 A b^6-a^6 (A-6 C)+a^4 b^2 (23 A-2 C)-a^2 b^4 (27 A-C)\right ) \cos (c+d x) \sin (c+d x)}{2 a^4 \left (a^2-b^2\right )^3 d}+\frac {\left (A b^2+a^2 C\right ) \cos (c+d x) \sin (c+d x)}{3 a \left (a^2-b^2\right ) d (a+b \sec (c+d x))^3}-\frac {\left (5 A b^4-4 a^4 C-a^2 b^2 (10 A+C)\right ) \cos (c+d x) \sin (c+d x)}{6 a^2 \left (a^2-b^2\right )^2 d (a+b \sec (c+d x))^2}+\frac {\left (20 A b^6-a^2 b^4 (53 A-2 C)+12 a^6 C+a^4 b^2 (48 A+C)\right ) \cos (c+d x) \sin (c+d x)}{6 a^3 \left (a^2-b^2\right )^3 d (a+b \sec (c+d x))}+\frac {\left (20 A b^8-a^2 b^6 (69 A-2 C)-8 a^6 b^2 (5 A-C)+7 a^4 b^4 (12 A-C)-8 a^8 C\right ) \text {Subst}\left (\int \frac {1}{1+\frac {a}{b}+\left (1-\frac {a}{b}\right ) x^2} \, dx,x,\tan \left (\frac {1}{2} (c+d x)\right )\right )}{a^6 \left (a^2-b^2\right )^3 d}\\ &=\frac {\left (20 A b^2+a^2 (A+2 C)\right ) x}{2 a^6}+\frac {b \left (20 A b^8-a^2 b^6 (69 A-2 C)-8 a^6 b^2 (5 A-C)+7 a^4 b^4 (12 A-C)-8 a^8 C\right ) \tanh ^{-1}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{a^6 \sqrt {a-b} \sqrt {a+b} \left (a^2-b^2\right )^3 d}+\frac {b \left (60 A b^6-a^6 (24 A-26 C)+a^4 b^2 (146 A-17 C)-a^2 b^4 (167 A-6 C)\right ) \sin (c+d x)}{6 a^5 \left (a^2-b^2\right )^3 d}-\frac {\left (10 A b^6-a^6 (A-6 C)+a^4 b^2 (23 A-2 C)-a^2 b^4 (27 A-C)\right ) \cos (c+d x) \sin (c+d x)}{2 a^4 \left (a^2-b^2\right )^3 d}+\frac {\left (A b^2+a^2 C\right ) \cos (c+d x) \sin (c+d x)}{3 a \left (a^2-b^2\right ) d (a+b \sec (c+d x))^3}-\frac {\left (5 A b^4-4 a^4 C-a^2 b^2 (10 A+C)\right ) \cos (c+d x) \sin (c+d x)}{6 a^2 \left (a^2-b^2\right )^2 d (a+b \sec (c+d x))^2}+\frac {\left (20 A b^6-a^2 b^4 (53 A-2 C)+12 a^6 C+a^4 b^2 (48 A+C)\right ) \cos (c+d x) \sin (c+d x)}{6 a^3 \left (a^2-b^2\right )^3 d (a+b \sec (c+d x))}\\ \end {align*}

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(1314\) vs. \(2(513)=1026\).
time = 5.36, size = 1314, normalized size = 2.56 \begin {gather*} \frac {-\frac {96 b \left (20 A b^8+7 a^4 b^4 (12 A-C)-8 a^8 C+8 a^6 b^2 (-5 A+C)+a^2 b^6 (-69 A+2 C)\right ) \tanh ^{-1}\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 )^{7/2}}+\frac {72 a^{10} A b c+1272 a^8 A b^3 c-3288 a^6 A b^5 c+1512 a^4 A b^7 c+1392 a^2 A b^9 c-960 A b^{11} c+144 a^{10} b c C-336 a^8 b^3 c C+144 a^6 b^5 c C+144 a^4 b^7 c C-96 a^2 b^9 c C+72 a^{10} A b d x+1272 a^8 A b^3 d x-3288 a^6 A b^5 d x+1512 a^4 A b^7 d x+1392 a^2 A b^9 d x-960 A b^{11} d x+144 a^{10} b C d x-336 a^8 b^3 C d x+144 a^6 b^5 C d x+144 a^4 b^7 C d x-96 a^2 b^9 C d x+36 a \left (a^2-b^2\right )^3 \left (a^2+4 b^2\right ) \left (20 A b^2+a^2 (A+2 C)\right ) (c+d x) \cos (c+d x)+72 a^2 b \left (a^2-b^2\right )^3 \left (20 A b^2+a^2 (A+2 C)\right ) (c+d x) \cos (2 (c+d x))+12 a^{11} A c \cos (3 (c+d x))+204 a^9 A b^2 c \cos (3 (c+d x))-684 a^7 A b^4 c \cos (3 (c+d x))+708 a^5 A b^6 c \cos (3 (c+d x))-240 a^3 A b^8 c \cos (3 (c+d x))+24 a^{11} c C \cos (3 (c+d x))-72 a^9 b^2 c C \cos (3 (c+d x))+72 a^7 b^4 c C \cos (3 (c+d x))-24 a^5 b^6 c C \cos (3 (c+d x))+12 a^{11} A d x \cos (3 (c+d x))+204 a^9 A b^2 d x \cos (3 (c+d x))-684 a^7 A b^4 d x \cos (3 (c+d x))+708 a^5 A b^6 d x \cos (3 (c+d x))-240 a^3 A b^8 d x \cos (3 (c+d x))+24 a^{11} C d x \cos (3 (c+d x))-72 a^9 b^2 C d x \cos (3 (c+d x))+72 a^7 b^4 C d x \cos (3 (c+d x))-24 a^5 b^6 C d x \cos (3 (c+d x))+6 a^{11} A \sin (c+d x)-270 a^9 A b^2 \sin (c+d x)+750 a^7 A b^4 \sin (c+d x)+1086 a^5 A b^6 \sin (c+d x)-2232 a^3 A b^8 \sin (c+d x)+960 a A b^{10} \sin (c+d x)+144 a^9 b^2 C \sin (c+d x)+288 a^7 b^4 C \sin (c+d x)-228 a^5 b^6 C \sin (c+d x)+96 a^3 b^8 C \sin (c+d x)-60 a^{10} A b \sin (2 (c+d x))-372 a^8 A b^3 \sin (2 (c+d x))+2772 a^6 A b^5 \sin (2 (c+d x))-3300 a^4 A b^7 \sin (2 (c+d x))+1200 a^2 A b^9 \sin (2 (c+d x))+480 a^8 b^3 C \sin (2 (c+d x))-360 a^6 b^5 C \sin (2 (c+d x))+120 a^4 b^7 C \sin (2 (c+d x))+9 a^{11} A \sin (3 (c+d x))-279 a^9 A b^2 \sin (3 (c+d x))+1143 a^7 A b^4 \sin (3 (c+d x))-1253 a^5 A b^6 \sin (3 (c+d x))+440 a^3 A b^8 \sin (3 (c+d x))+144 a^9 b^2 C \sin (3 (c+d x))-128 a^7 b^4 C \sin (3 (c+d x))+44 a^5 b^6 C \sin (3 (c+d x))-30 a^{10} A b \sin (4 (c+d x))+90 a^8 A b^3 \sin (4 (c+d x))-90 a^6 A b^5 \sin (4 (c+d x))+30 a^4 A b^7 \sin (4 (c+d x))+3 a^{11} A \sin (5 (c+d x))-9 a^9 A b^2 \sin (5 (c+d x))+9 a^7 A b^4 \sin (5 (c+d x))-3 a^5 A b^6 \sin (5 (c+d x))}{\left (a^2-b^2\right )^3 (b+a \cos (c+d x))^3}}{96 a^6 d} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((-96*b*(20*A*b^8 + 7*a^4*b^4*(12*A - C) - 8*a^8*C + 8*a^6*b^2*(-5*A + C) + a^2*b^6*(-69*A + 2*C))*ArcTanh[((-
a + b)*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(a^2 - b^2)^(7/2) + (72*a^10*A*b*c + 1272*a^8*A*b^3*c - 3288*a^6*A*
b^5*c + 1512*a^4*A*b^7*c + 1392*a^2*A*b^9*c - 960*A*b^11*c + 144*a^10*b*c*C - 336*a^8*b^3*c*C + 144*a^6*b^5*c*
C + 144*a^4*b^7*c*C - 96*a^2*b^9*c*C + 72*a^10*A*b*d*x + 1272*a^8*A*b^3*d*x - 3288*a^6*A*b^5*d*x + 1512*a^4*A*
b^7*d*x + 1392*a^2*A*b^9*d*x - 960*A*b^11*d*x + 144*a^10*b*C*d*x - 336*a^8*b^3*C*d*x + 144*a^6*b^5*C*d*x + 144
*a^4*b^7*C*d*x - 96*a^2*b^9*C*d*x + 36*a*(a^2 - b^2)^3*(a^2 + 4*b^2)*(20*A*b^2 + a^2*(A + 2*C))*(c + d*x)*Cos[
c + d*x] + 72*a^2*b*(a^2 - b^2)^3*(20*A*b^2 + a^2*(A + 2*C))*(c + d*x)*Cos[2*(c + d*x)] + 12*a^11*A*c*Cos[3*(c
+ d*x)] + 204*a^9*A*b^2*c*Cos[3*(c + d*x)] - 684*a^7*A*b^4*c*Cos[3*(c + d*x)] + 708*a^5*A*b^6*c*Cos[3*(c + d*
x)] - 240*a^3*A*b^8*c*Cos[3*(c + d*x)] + 24*a^11*c*C*Cos[3*(c + d*x)] - 72*a^9*b^2*c*C*Cos[3*(c + d*x)] + 72*a
^7*b^4*c*C*Cos[3*(c + d*x)] - 24*a^5*b^6*c*C*Cos[3*(c + d*x)] + 12*a^11*A*d*x*Cos[3*(c + d*x)] + 204*a^9*A*b^2
*d*x*Cos[3*(c + d*x)] - 684*a^7*A*b^4*d*x*Cos[3*(c + d*x)] + 708*a^5*A*b^6*d*x*Cos[3*(c + d*x)] - 240*a^3*A*b^
8*d*x*Cos[3*(c + d*x)] + 24*a^11*C*d*x*Cos[3*(c + d*x)] - 72*a^9*b^2*C*d*x*Cos[3*(c + d*x)] + 72*a^7*b^4*C*d*x
*Cos[3*(c + d*x)] - 24*a^5*b^6*C*d*x*Cos[3*(c + d*x)] + 6*a^11*A*Sin[c + d*x] - 270*a^9*A*b^2*Sin[c + d*x] + 7
50*a^7*A*b^4*Sin[c + d*x] + 1086*a^5*A*b^6*Sin[c + d*x] - 2232*a^3*A*b^8*Sin[c + d*x] + 960*a*A*b^10*Sin[c + d
*x] + 144*a^9*b^2*C*Sin[c + d*x] + 288*a^7*b^4*C*Sin[c + d*x] - 228*a^5*b^6*C*Sin[c + d*x] + 96*a^3*b^8*C*Sin[
c + d*x] - 60*a^10*A*b*Sin[2*(c + d*x)] - 372*a^8*A*b^3*Sin[2*(c + d*x)] + 2772*a^6*A*b^5*Sin[2*(c + d*x)] - 3
300*a^4*A*b^7*Sin[2*(c + d*x)] + 1200*a^2*A*b^9*Sin[2*(c + d*x)] + 480*a^8*b^3*C*Sin[2*(c + d*x)] - 360*a^6*b^
5*C*Sin[2*(c + d*x)] + 120*a^4*b^7*C*Sin[2*(c + d*x)] + 9*a^11*A*Sin[3*(c + d*x)] - 279*a^9*A*b^2*Sin[3*(c + d
*x)] + 1143*a^7*A*b^4*Sin[3*(c + d*x)] - 1253*a^5*A*b^6*Sin[3*(c + d*x)] + 440*a^3*A*b^8*Sin[3*(c + d*x)] + 14
4*a^9*b^2*C*Sin[3*(c + d*x)] - 128*a^7*b^4*C*Sin[3*(c + d*x)] + 44*a^5*b^6*C*Sin[3*(c + d*x)] - 30*a^10*A*b*Si
n[4*(c + d*x)] + 90*a^8*A*b^3*Sin[4*(c + d*x)] - 90*a^6*A*b^5*Sin[4*(c + d*x)] + 30*a^4*A*b^7*Sin[4*(c + d*x)]
+ 3*a^11*A*Sin[5*(c + d*x)] - 9*a^9*A*b^2*Sin[5*(c + d*x)] + 9*a^7*A*b^4*Sin[5*(c + d*x)] - 3*a^5*A*b^6*Sin[5
*(c + d*x)])/((a^2 - b^2)^3*(b + a*Cos[c + d*x])^3))/(96*a^6*d)

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Maple [A]
time = 0.62, size = 619, normalized size = 1.21

method	result	size
derivativedivides	\(\frac {\frac {2 b \left (\frac {-\frac {\left (30 A \,a^{4} b^{2}+6 A \,a^{3} b^{3}-34 a^{2} A \,b^{4}-3 A a \,b^{5}+12 A \,b^{6}+12 a^{6} C +4 C \,a^{5} b -6 a^{4} b^{2} C -C \,a^{3} b^{3}+2 C \,a^{2} b^{4}\right ) a b \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 \left (a -b \right ) \left (a^{3}+3 a^{2} b +3 b^{2} a +b^{3}\right )}+\frac {2 \left (45 A \,a^{4} b^{2}-53 a^{2} A \,b^{4}+18 A \,b^{6}+18 a^{6} C -11 a^{4} b^{2} C +3 C \,a^{2} b^{4}\right ) a b \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 \left (a^{2}-2 a b +b^{2}\right ) \left (a^{2}+2 a b +b^{2}\right )}-\frac {\left (30 A \,a^{4} b^{2}-6 A \,a^{3} b^{3}-34 a^{2} A \,b^{4}+3 A a \,b^{5}+12 A \,b^{6}+12 a^{6} C -4 C \,a^{5} b -6 a^{4} b^{2} C +C \,a^{3} b^{3}+2 C \,a^{2} b^{4}\right ) a b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 \left (a +b \right ) \left (a^{3}-3 a^{2} b +3 b^{2} a -b^{3}\right )}}{\left (a \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-b \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-a -b \right )^{3}}-\frac {\left (40 A \,a^{6} b^{2}-84 a^{4} A \,b^{4}+69 a^{2} A \,b^{6}-20 A \,b^{8}+8 a^{8} C -8 a^{6} b^{2} C +7 a^{4} b^{4} C -2 C \,a^{2} b^{6}\right ) \arctanh \left (\frac {\left (a -b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (a +b \right ) \left (a -b \right )}}\right )}{2 \left (a^{6}-3 a^{4} b^{2}+3 a^{2} b^{4}-b^{6}\right ) \sqrt {\left (a +b \right ) \left (a -b \right )}}\right )}{a^{6}}+\frac {\frac {2 \left (\left (-\frac {1}{2} A \,a^{2}-4 a A b \right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (\frac {1}{2} A \,a^{2}-4 a A b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{2}}+\left (A \,a^{2}+20 A \,b^{2}+2 a^{2} C \right ) \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a^{6}}}{d}\)	\(619\)
default	\(\frac {\frac {2 b \left (\frac {-\frac {\left (30 A \,a^{4} b^{2}+6 A \,a^{3} b^{3}-34 a^{2} A \,b^{4}-3 A a \,b^{5}+12 A \,b^{6}+12 a^{6} C +4 C \,a^{5} b -6 a^{4} b^{2} C -C \,a^{3} b^{3}+2 C \,a^{2} b^{4}\right ) a b \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 \left (a -b \right ) \left (a^{3}+3 a^{2} b +3 b^{2} a +b^{3}\right )}+\frac {2 \left (45 A \,a^{4} b^{2}-53 a^{2} A \,b^{4}+18 A \,b^{6}+18 a^{6} C -11 a^{4} b^{2} C +3 C \,a^{2} b^{4}\right ) a b \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 \left (a^{2}-2 a b +b^{2}\right ) \left (a^{2}+2 a b +b^{2}\right )}-\frac {\left (30 A \,a^{4} b^{2}-6 A \,a^{3} b^{3}-34 a^{2} A \,b^{4}+3 A a \,b^{5}+12 A \,b^{6}+12 a^{6} C -4 C \,a^{5} b -6 a^{4} b^{2} C +C \,a^{3} b^{3}+2 C \,a^{2} b^{4}\right ) a b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 \left (a +b \right ) \left (a^{3}-3 a^{2} b +3 b^{2} a -b^{3}\right )}}{\left (a \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-b \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-a -b \right )^{3}}-\frac {\left (40 A \,a^{6} b^{2}-84 a^{4} A \,b^{4}+69 a^{2} A \,b^{6}-20 A \,b^{8}+8 a^{8} C -8 a^{6} b^{2} C +7 a^{4} b^{4} C -2 C \,a^{2} b^{6}\right ) \arctanh \left (\frac {\left (a -b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (a +b \right ) \left (a -b \right )}}\right )}{2 \left (a^{6}-3 a^{4} b^{2}+3 a^{2} b^{4}-b^{6}\right ) \sqrt {\left (a +b \right ) \left (a -b \right )}}\right )}{a^{6}}+\frac {\frac {2 \left (\left (-\frac {1}{2} A \,a^{2}-4 a A b \right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (\frac {1}{2} A \,a^{2}-4 a A b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{2}}+\left (A \,a^{2}+20 A \,b^{2}+2 a^{2} C \right ) \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a^{6}}}{d}\)	\(619\)
risch	\(\text {Expression too large to display}\)	\(2209\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(d*x+c)^2*(A+C*sec(d*x+c)^2)/(a+b*sec(d*x+c))^4,x,method=_RETURNVERBOSE)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(4*a^2-4*b^2>0)', see `assume?`
for more de

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 1204 vs. \(2 (490) = 980\).
time = 3.61, size = 2465, normalized size = 4.81 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/12*(6*((A + 2*C)*a^13 + 8*(2*A - C)*a^11*b^2 - 2*(37*A - 6*C)*a^9*b^4 + 4*(29*A - 2*C)*a^7*b^6 - (79*A - 2*
C)*a^5*b^8 + 20*A*a^3*b^10)*d*x*cos(d*x + c)^3 + 18*((A + 2*C)*a^12*b + 8*(2*A - C)*a^10*b^3 - 2*(37*A - 6*C)*
a^8*b^5 + 4*(29*A - 2*C)*a^6*b^7 - (79*A - 2*C)*a^4*b^9 + 20*A*a^2*b^11)*d*x*cos(d*x + c)^2 + 18*((A + 2*C)*a^
11*b^2 + 8*(2*A - C)*a^9*b^4 - 2*(37*A - 6*C)*a^7*b^6 + 4*(29*A - 2*C)*a^5*b^8 - (79*A - 2*C)*a^3*b^10 + 20*A*
a*b^12)*d*x*cos(d*x + c) + 6*((A + 2*C)*a^10*b^3 + 8*(2*A - C)*a^8*b^5 - 2*(37*A - 6*C)*a^6*b^7 + 4*(29*A - 2*
C)*a^4*b^9 - (79*A - 2*C)*a^2*b^11 + 20*A*b^13)*d*x + 3*(8*C*a^8*b^4 + 8*(5*A - C)*a^6*b^6 - 7*(12*A - C)*a^4*
b^8 + (69*A - 2*C)*a^2*b^10 - 20*A*b^12 + (8*C*a^11*b + 8*(5*A - C)*a^9*b^3 - 7*(12*A - C)*a^7*b^5 + (69*A - 2
*C)*a^5*b^7 - 20*A*a^3*b^9)*cos(d*x + c)^3 + 3*(8*C*a^10*b^2 + 8*(5*A - C)*a^8*b^4 - 7*(12*A - C)*a^6*b^6 + (6
9*A - 2*C)*a^4*b^8 - 20*A*a^2*b^10)*cos(d*x + c)^2 + 3*(8*C*a^9*b^3 + 8*(5*A - C)*a^7*b^5 - 7*(12*A - C)*a^5*b
^7 + (69*A - 2*C)*a^3*b^9 - 20*A*a*b^11)*cos(d*x + c))*sqrt(a^2 - b^2)*log((2*a*b*cos(d*x + c) - (a^2 - 2*b^2)
*cos(d*x + c)^2 - 2*sqrt(a^2 - b^2)*(b*cos(d*x + c) + a)*sin(d*x + c) + 2*a^2 - b^2)/(a^2*cos(d*x + c)^2 + 2*a
*b*cos(d*x + c) + b^2)) - 2*(2*(12*A - 13*C)*a^9*b^4 - (170*A - 43*C)*a^7*b^6 + (313*A - 23*C)*a^5*b^8 - (227*
A - 6*C)*a^3*b^10 + 60*A*a*b^12 - 3*(A*a^13 - 4*A*a^11*b^2 + 6*A*a^9*b^4 - 4*A*a^7*b^6 + A*a^5*b^8)*cos(d*x +
c)^4 + 15*(A*a^12*b - 4*A*a^10*b^3 + 6*A*a^8*b^5 - 4*A*a^6*b^7 + A*a^4*b^9)*cos(d*x + c)^3 + (9*(7*A - 4*C)*a^
11*b^2 - 2*(171*A - 34*C)*a^9*b^4 + (590*A - 43*C)*a^7*b^6 - (421*A - 11*C)*a^5*b^8 + 110*A*a^3*b^10)*cos(d*x
+ c)^2 + 3*((23*A - 20*C)*a^10*b^3 - (146*A - 35*C)*a^8*b^5 + (263*A - 20*C)*a^6*b^7 - 5*(38*A - C)*a^4*b^9 +
50*A*a^2*b^11)*cos(d*x + c))*sin(d*x + c))/((a^17 - 4*a^15*b^2 + 6*a^13*b^4 - 4*a^11*b^6 + a^9*b^8)*d*cos(d*x
+ c)^3 + 3*(a^16*b - 4*a^14*b^3 + 6*a^12*b^5 - 4*a^10*b^7 + a^8*b^9)*d*cos(d*x + c)^2 + 3*(a^15*b^2 - 4*a^13*b
^4 + 6*a^11*b^6 - 4*a^9*b^8 + a^7*b^10)*d*cos(d*x + c) + (a^14*b^3 - 4*a^12*b^5 + 6*a^10*b^7 - 4*a^8*b^9 + a^6
*b^11)*d), 1/6*(3*((A + 2*C)*a^13 + 8*(2*A - C)*a^11*b^2 - 2*(37*A - 6*C)*a^9*b^4 + 4*(29*A - 2*C)*a^7*b^6 - (
79*A - 2*C)*a^5*b^8 + 20*A*a^3*b^10)*d*x*cos(d*x + c)^3 + 9*((A + 2*C)*a^12*b + 8*(2*A - C)*a^10*b^3 - 2*(37*A
- 6*C)*a^8*b^5 + 4*(29*A - 2*C)*a^6*b^7 - (79*A - 2*C)*a^4*b^9 + 20*A*a^2*b^11)*d*x*cos(d*x + c)^2 + 9*((A +
2*C)*a^11*b^2 + 8*(2*A - C)*a^9*b^4 - 2*(37*A - 6*C)*a^7*b^6 + 4*(29*A - 2*C)*a^5*b^8 - (79*A - 2*C)*a^3*b^10
+ 20*A*a*b^12)*d*x*cos(d*x + c) + 3*((A + 2*C)*a^10*b^3 + 8*(2*A - C)*a^8*b^5 - 2*(37*A - 6*C)*a^6*b^7 + 4*(29
*A - 2*C)*a^4*b^9 - (79*A - 2*C)*a^2*b^11 + 20*A*b^13)*d*x - 3*(8*C*a^8*b^4 + 8*(5*A - C)*a^6*b^6 - 7*(12*A -
C)*a^4*b^8 + (69*A - 2*C)*a^2*b^10 - 20*A*b^12 + (8*C*a^11*b + 8*(5*A - C)*a^9*b^3 - 7*(12*A - C)*a^7*b^5 + (6
9*A - 2*C)*a^5*b^7 - 20*A*a^3*b^9)*cos(d*x + c)^3 + 3*(8*C*a^10*b^2 + 8*(5*A - C)*a^8*b^4 - 7*(12*A - C)*a^6*b
^6 + (69*A - 2*C)*a^4*b^8 - 20*A*a^2*b^10)*cos(d*x + c)^2 + 3*(8*C*a^9*b^3 + 8*(5*A - C)*a^7*b^5 - 7*(12*A - C
)*a^5*b^7 + (69*A - 2*C)*a^3*b^9 - 20*A*a*b^11)*cos(d*x + c))*sqrt(-a^2 + b^2)*arctan(-sqrt(-a^2 + b^2)*(b*cos
(d*x + c) + a)/((a^2 - b^2)*sin(d*x + c))) - (2*(12*A - 13*C)*a^9*b^4 - (170*A - 43*C)*a^7*b^6 + (313*A - 23*C
)*a^5*b^8 - (227*A - 6*C)*a^3*b^10 + 60*A*a*b^12 - 3*(A*a^13 - 4*A*a^11*b^2 + 6*A*a^9*b^4 - 4*A*a^7*b^6 + A*a^
5*b^8)*cos(d*x + c)^4 + 15*(A*a^12*b - 4*A*a^10*b^3 + 6*A*a^8*b^5 - 4*A*a^6*b^7 + A*a^4*b^9)*cos(d*x + c)^3 +
(9*(7*A - 4*C)*a^11*b^2 - 2*(171*A - 34*C)*a^9*b^4 + (590*A - 43*C)*a^7*b^6 - (421*A - 11*C)*a^5*b^8 + 110*A*a
^3*b^10)*cos(d*x + c)^2 + 3*((23*A - 20*C)*a^10*b^3 - (146*A - 35*C)*a^8*b^5 + (263*A - 20*C)*a^6*b^7 - 5*(38*
A - C)*a^4*b^9 + 50*A*a^2*b^11)*cos(d*x + c))*sin(d*x + c))/((a^17 - 4*a^15*b^2 + 6*a^13*b^4 - 4*a^11*b^6 + a^
9*b^8)*d*cos(d*x + c)^3 + 3*(a^16*b - 4*a^14*b^3 + 6*a^12*b^5 - 4*a^10*b^7 + a^8*b^9)*d*cos(d*x + c)^2 + 3*(a^
15*b^2 - 4*a^13*b^4 + 6*a^11*b^6 - 4*a^9*b^8 + a^7*b^10)*d*cos(d*x + c) + (a^14*b^3 - 4*a^12*b^5 + 6*a^10*b^7
- 4*a^8*b^9 + a^6*b^11)*d)]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 1031 vs. \(2 (490) = 980\).
time = 0.58, size = 1031, normalized size = 2.01 \begin {gather*} -\frac {\frac {6 \, {\left (8 \, C a^{8} b + 40 \, A a^{6} b^{3} - 8 \, C a^{6} b^{3} - 84 \, A a^{4} b^{5} + 7 \, C a^{4} b^{5} + 69 \, A a^{2} b^{7} - 2 \, C a^{2} b^{7} - 20 \, A b^{9}\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^{12} - 3 \, a^{10} b^{2} + 3 \, a^{8} b^{4} - a^{6} b^{6}\right )} \sqrt {-a^{2} + b^{2}}} + \frac {2 \, {\left (36 \, C a^{8} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 60 \, C a^{7} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 90 \, A a^{6} b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 6 \, C a^{6} b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 162 \, A a^{5} b^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 45 \, C a^{5} b^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 48 \, A a^{4} b^{6} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 6 \, C a^{4} b^{6} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 213 \, A a^{3} b^{7} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 15 \, C a^{3} b^{7} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 48 \, A a^{2} b^{8} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 6 \, C a^{2} b^{8} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 81 \, A a b^{9} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 36 \, A b^{10} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 72 \, C a^{8} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 180 \, A a^{6} b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 116 \, C a^{6} b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 392 \, A a^{4} b^{6} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 56 \, C a^{4} b^{6} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 284 \, A a^{2} b^{8} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 12 \, C a^{2} b^{8} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 72 \, A b^{10} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 36 \, C a^{8} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 60 \, C a^{7} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 90 \, A a^{6} b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 6 \, C a^{6} b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 162 \, A a^{5} b^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 45 \, C a^{5} b^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 48 \, A a^{4} b^{6} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 6 \, C a^{4} b^{6} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 213 \, A a^{3} b^{7} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 15 \, C a^{3} b^{7} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 48 \, A a^{2} b^{8} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 6 \, C a^{2} b^{8} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 81 \, A a b^{9} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 36 \, A b^{10} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (a^{11} - 3 \, a^{9} b^{2} + 3 \, a^{7} b^{4} - a^{5} 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 )}^{3}} - \frac {3 \, {\left (A a^{2} + 2 \, C a^{2} + 20 \, A b^{2}\right )} {\left (d x + c\right )}}{a^{6}} + \frac {6 \, {\left (A a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 8 \, A b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - A a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 8 \, A b \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 )}^{2} a^{5}}}{6 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/6*(6*(8*C*a^8*b + 40*A*a^6*b^3 - 8*C*a^6*b^3 - 84*A*a^4*b^5 + 7*C*a^4*b^5 + 69*A*a^2*b^7 - 2*C*a^2*b^7 - 20
*A*b^9)*(pi*floor(1/2*(d*x + c)/pi + 1/2)*sgn(-2*a + 2*b) + arctan(-(a*tan(1/2*d*x + 1/2*c) - b*tan(1/2*d*x +
1/2*c))/sqrt(-a^2 + b^2)))/((a^12 - 3*a^10*b^2 + 3*a^8*b^4 - a^6*b^6)*sqrt(-a^2 + b^2)) + 2*(36*C*a^8*b^2*tan(
1/2*d*x + 1/2*c)^5 - 60*C*a^7*b^3*tan(1/2*d*x + 1/2*c)^5 + 90*A*a^6*b^4*tan(1/2*d*x + 1/2*c)^5 - 6*C*a^6*b^4*t
an(1/2*d*x + 1/2*c)^5 - 162*A*a^5*b^5*tan(1/2*d*x + 1/2*c)^5 + 45*C*a^5*b^5*tan(1/2*d*x + 1/2*c)^5 - 48*A*a^4*
b^6*tan(1/2*d*x + 1/2*c)^5 - 6*C*a^4*b^6*tan(1/2*d*x + 1/2*c)^5 + 213*A*a^3*b^7*tan(1/2*d*x + 1/2*c)^5 - 15*C*
a^3*b^7*tan(1/2*d*x + 1/2*c)^5 - 48*A*a^2*b^8*tan(1/2*d*x + 1/2*c)^5 + 6*C*a^2*b^8*tan(1/2*d*x + 1/2*c)^5 - 81
*A*a*b^9*tan(1/2*d*x + 1/2*c)^5 + 36*A*b^10*tan(1/2*d*x + 1/2*c)^5 - 72*C*a^8*b^2*tan(1/2*d*x + 1/2*c)^3 - 180
*A*a^6*b^4*tan(1/2*d*x + 1/2*c)^3 + 116*C*a^6*b^4*tan(1/2*d*x + 1/2*c)^3 + 392*A*a^4*b^6*tan(1/2*d*x + 1/2*c)^
3 - 56*C*a^4*b^6*tan(1/2*d*x + 1/2*c)^3 - 284*A*a^2*b^8*tan(1/2*d*x + 1/2*c)^3 + 12*C*a^2*b^8*tan(1/2*d*x + 1/
2*c)^3 + 72*A*b^10*tan(1/2*d*x + 1/2*c)^3 + 36*C*a^8*b^2*tan(1/2*d*x + 1/2*c) + 60*C*a^7*b^3*tan(1/2*d*x + 1/2
*c) + 90*A*a^6*b^4*tan(1/2*d*x + 1/2*c) - 6*C*a^6*b^4*tan(1/2*d*x + 1/2*c) + 162*A*a^5*b^5*tan(1/2*d*x + 1/2*c
) - 45*C*a^5*b^5*tan(1/2*d*x + 1/2*c) - 48*A*a^4*b^6*tan(1/2*d*x + 1/2*c) - 6*C*a^4*b^6*tan(1/2*d*x + 1/2*c) -
213*A*a^3*b^7*tan(1/2*d*x + 1/2*c) + 15*C*a^3*b^7*tan(1/2*d*x + 1/2*c) - 48*A*a^2*b^8*tan(1/2*d*x + 1/2*c) +
6*C*a^2*b^8*tan(1/2*d*x + 1/2*c) + 81*A*a*b^9*tan(1/2*d*x + 1/2*c) + 36*A*b^10*tan(1/2*d*x + 1/2*c))/((a^11 -
3*a^9*b^2 + 3*a^7*b^4 - a^5*b^6)*(a*tan(1/2*d*x + 1/2*c)^2 - b*tan(1/2*d*x + 1/2*c)^2 - a - b)^3) - 3*(A*a^2 +
2*C*a^2 + 20*A*b^2)*(d*x + c)/a^6 + 6*(A*a*tan(1/2*d*x + 1/2*c)^3 + 8*A*b*tan(1/2*d*x + 1/2*c)^3 - A*a*tan(1/
2*d*x + 1/2*c) + 8*A*b*tan(1/2*d*x + 1/2*c))/((tan(1/2*d*x + 1/2*c)^2 + 1)^2*a^5))/d

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Mupad [B]
time = 23.99, size = 2500, normalized size = 4.87 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((tan(c/2 + (d*x)/2)*(A*a^8 + 20*A*b^8 - 59*A*a^2*b^6 - 27*A*a^3*b^5 + 57*A*a^4*b^4 + 21*A*a^5*b^3 - 11*A*a^6*
b^2 + 2*C*a^2*b^6 + C*a^3*b^5 - 6*C*a^4*b^4 - 4*C*a^5*b^3 + 12*C*a^6*b^2 + 10*A*a*b^7 - 7*A*a^7*b))/(a^5*(a +
b)*(a - b)^3) + (tan(c/2 + (d*x)/2)^9*(A*a^8 + 20*A*b^8 - 59*A*a^2*b^6 + 27*A*a^3*b^5 + 57*A*a^4*b^4 - 21*A*a^
5*b^3 - 11*A*a^6*b^2 + 2*C*a^2*b^6 - C*a^3*b^5 - 6*C*a^4*b^4 + 4*C*a^5*b^3 + 12*C*a^6*b^2 - 10*A*a*b^7 + 7*A*a
^7*b))/(a^5*(a + b)^3*(a - b)) + (2*tan(c/2 + (d*x)/2)^3*(120*A*b^9 - 6*A*a^9 - 364*A*a^2*b^7 - 71*A*a^3*b^6 +
369*A*a^4*b^5 + 45*A*a^5*b^4 - 111*A*a^6*b^3 - 3*A*a^7*b^2 + 12*C*a^2*b^7 + 3*C*a^3*b^6 - 37*C*a^4*b^5 - 8*C*
a^5*b^4 + 60*C*a^6*b^3 + 30*A*a*b^8 + 21*A*a^8*b))/(3*a^5*(a + b)^2*(a - b)^3) - (2*tan(c/2 + (d*x)/2)^7*(6*A*
a^9 + 120*A*b^9 - 364*A*a^2*b^7 + 71*A*a^3*b^6 + 369*A*a^4*b^5 - 45*A*a^5*b^4 - 111*A*a^6*b^3 + 3*A*a^7*b^2 +
12*C*a^2*b^7 - 3*C*a^3*b^6 - 37*C*a^4*b^5 + 8*C*a^5*b^4 + 60*C*a^6*b^3 - 30*A*a*b^8 + 21*A*a^8*b))/(3*a^5*(a +
b)^3*(a - b)^2) + (2*tan(c/2 + (d*x)/2)^5*(9*A*a^10 + 180*A*b^10 - 611*A*a^2*b^8 + 740*A*a^4*b^6 - 324*A*a^6*
b^4 + 36*A*a^8*b^2 + 18*C*a^2*b^8 - 62*C*a^4*b^6 + 110*C*a^6*b^4 - 36*C*a^8*b^2))/(3*a^5*(a + b)^3*(a - b)^3))
/(d*(tan(c/2 + (d*x)/2)^2*(9*a*b^2 + 3*a^2*b - a^3 + 5*b^3) + tan(c/2 + (d*x)/2)^4*(6*a*b^2 - 6*a^2*b - 2*a^3
+ 10*b^3) - tan(c/2 + (d*x)/2)^6*(6*a*b^2 + 6*a^2*b - 2*a^3 - 10*b^3) + 3*a*b^2 + 3*a^2*b + a^3 + b^3 - tan(c/
2 + (d*x)/2)^10*(3*a*b^2 - 3*a^2*b + a^3 - b^3) + tan(c/2 + (d*x)/2)^8*(3*a^2*b - 9*a*b^2 + a^3 + 5*b^3))) - (
atan((((A*b^2*10i + a^2*((A*1i)/2 + C*1i))*(((A*b^2*10i + a^2*((A*1i)/2 + C*1i))*((4*(4*A*a^27 + 8*C*a^27 - 80
*A*a^12*b^15 + 40*A*a^13*b^14 + 516*A*a^14*b^13 - 248*A*a^15*b^12 - 1404*A*a^16*b^11 + 640*A*a^17*b^10 + 2076*
A*a^18*b^9 - 896*A*a^19*b^8 - 1764*A*a^20*b^7 + 724*A*a^21*b^6 + 816*A*a^22*b^5 - 316*A*a^23*b^4 - 160*A*a^24*
b^3 + 52*A*a^25*b^2 - 8*C*a^14*b^13 + 4*C*a^15*b^12 + 52*C*a^16*b^11 - 28*C*a^17*b^10 - 140*C*a^18*b^9 + 60*C*
a^19*b^8 + 220*C*a^20*b^7 - 60*C*a^21*b^6 - 220*C*a^22*b^5 + 40*C*a^23*b^4 + 128*C*a^24*b^3 - 24*C*a^25*b^2 -
32*C*a^26*b))/(a^25*b + a^26 - a^15*b^11 - a^16*b^10 + 5*a^17*b^9 + 5*a^18*b^8 - 10*a^19*b^7 - 10*a^20*b^6 + 1
0*a^21*b^5 + 10*a^22*b^4 - 5*a^23*b^3 - 5*a^24*b^2) - (8*tan(c/2 + (d*x)/2)*(A*b^2*10i + a^2*((A*1i)/2 + C*1i)
)*(8*a^25*b - 8*a^12*b^14 + 8*a^13*b^13 + 48*a^14*b^12 - 48*a^15*b^11 - 120*a^16*b^10 + 120*a^17*b^9 + 160*a^1
8*b^8 - 160*a^19*b^7 - 120*a^20*b^6 + 120*a^21*b^5 + 48*a^22*b^4 - 48*a^23*b^3 - 8*a^24*b^2))/(a^6*(a^20*b + a
^21 - a^10*b^11 - a^11*b^10 + 5*a^12*b^9 + 5*a^13*b^8 - 10*a^14*b^7 - 10*a^15*b^6 + 10*a^16*b^5 + 10*a^17*b^4
- 5*a^18*b^3 - 5*a^19*b^2))))/a^6 + (8*tan(c/2 + (d*x)/2)*(A^2*a^18 + 800*A^2*b^18 + 4*C^2*a^18 - 800*A^2*a*b^
17 - 2*A^2*a^17*b - 8*C^2*a^17*b - 4720*A^2*a^2*b^16 + 4720*A^2*a^3*b^15 + 11522*A^2*a^4*b^14 - 11522*A^2*a^5*
b^13 - 14837*A^2*a^6*b^12 + 14812*A^2*a^7*b^11 + 10385*A^2*a^8*b^10 - 10430*A^2*a^9*b^9 - 3325*A^2*a^10*b^8 +
3640*A^2*a^11*b^7 - 45*A^2*a^12*b^6 - 350*A^2*a^13*b^5 + 209*A^2*a^14*b^4 - 68*A^2*a^15*b^3 + 35*A^2*a^16*b^2
+ 8*C^2*a^4*b^14 - 8*C^2*a^5*b^13 - 48*C^2*a^6*b^12 + 48*C^2*a^7*b^11 + 117*C^2*a^8*b^10 - 120*C^2*a^9*b^9 - 1
64*C^2*a^10*b^8 + 160*C^2*a^11*b^7 + 156*C^2*a^12*b^6 - 120*C^2*a^13*b^5 - 92*C^2*a^14*b^4 + 48*C^2*a^15*b^3 +
44*C^2*a^16*b^2 + 4*A*C*a^18 - 8*A*C*a^17*b + 160*A*C*a^2*b^16 - 160*A*C*a^3*b^15 - 952*A*C*a^4*b^14 + 952*A*
C*a^5*b^13 + 2322*A*C*a^6*b^12 - 2352*A*C*a^7*b^11 - 3124*A*C*a^8*b^10 + 3080*A*C*a^9*b^9 + 2588*A*C*a^10*b^8
- 2240*A*C*a^11*b^7 - 1284*A*C*a^12*b^6 + 840*A*C*a^13*b^5 + 276*A*C*a^14*b^4 - 112*A*C*a^15*b^3 + 60*A*C*a^16
*b^2))/(a^20*b + a^21 - a^10*b^11 - a^11*b^10 + 5*a^12*b^9 + 5*a^13*b^8 - 10*a^14*b^7 - 10*a^15*b^6 + 10*a^16*
b^5 + 10*a^17*b^4 - 5*a^18*b^3 - 5*a^19*b^2))*1i)/a^6 - ((A*b^2*10i + a^2*((A*1i)/2 + C*1i))*(((A*b^2*10i + a^
2*((A*1i)/2 + C*1i))*((4*(4*A*a^27 + 8*C*a^27 - 80*A*a^12*b^15 + 40*A*a^13*b^14 + 516*A*a^14*b^13 - 248*A*a^15
*b^12 - 1404*A*a^16*b^11 + 640*A*a^17*b^10 + 2076*A*a^18*b^9 - 896*A*a^19*b^8 - 1764*A*a^20*b^7 + 724*A*a^21*b
^6 + 816*A*a^22*b^5 - 316*A*a^23*b^4 - 160*A*a^24*b^3 + 52*A*a^25*b^2 - 8*C*a^14*b^13 + 4*C*a^15*b^12 + 52*C*a
^16*b^11 - 28*C*a^17*b^10 - 140*C*a^18*b^9 + 60*C*a^19*b^8 + 220*C*a^20*b^7 - 60*C*a^21*b^6 - 220*C*a^22*b^5 +
40*C*a^23*b^4 + 128*C*a^24*b^3 - 24*C*a^25*b^2 - 32*C*a^26*b))/(a^25*b + a^26 - a^15*b^11 - a^16*b^10 + 5*a^1
7*b^9 + 5*a^18*b^8 - 10*a^19*b^7 - 10*a^20*b^6 + 10*a^21*b^5 + 10*a^22*b^4 - 5*a^23*b^3 - 5*a^24*b^2) + (8*tan
(c/2 + (d*x)/2)*(A*b^2*10i + a^2*((A*1i)/2 + C*1i))*(8*a^25*b - 8*a^12*b^14 + 8*a^13*b^13 + 48*a^14*b^12 - 48*
a^15*b^11 - 120*a^16*b^10 + 120*a^17*b^9 + 160*a^18*b^8 - 160*a^19*b^7 - 120*a^20*b^6 + 120*a^21*b^5 + 48*a^22
*b^4 - 48*a^23*b^3 - 8*a^24*b^2))/(a^6*(a^20*b + a^21 - a^10*b^11 - a^11*b^10 + 5*a^12*b^9 + 5*a^13*b^8 - 10*a
^14*b^7 - 10*a^15*b^6 + 10*a^16*b^5 + 10*a^17*b^4 - 5*a^18*b^3 - 5*a^19*b^2))))/a^6 - (8*tan(c/2 + (d*x)/2)*(A
^2*a^18 + 800*A^2*b^18 + 4*C^2*a^18 - 800*A^2*a...

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