3.922 \(\int \frac{\cos ^2(c+d x) (A+B \sec (c+d x)+C \sec ^2(c+d x))}{(a+b \sec (c+d x))^3} \, dx\)
Optimal. Leaf size=453 \[ -\frac{\sin (c+d x) \left (-a^2 b^3 (21 A-2 C)+a^4 b (6 A-5 C)+11 a^3 b^2 B-2 a^5 B-6 a b^4 B+12 A b^5\right )}{2 a^4 d \left (a^2-b^2\right )^2}+\frac{\sin (c+d x) \cos (c+d x) \left (-a^2 b^2 (10 A-C)+a^4 (A-4 C)+6 a^3 b B-3 a b^3 B+6 A b^4\right )}{2 a^3 d \left (a^2-b^2\right )^2}-\frac{b \left (5 a^4 b^2 (4 A-C)-a^2 b^4 (29 A-2 C)+15 a^3 b^3 B-12 a^5 b B+6 a^6 C-6 a b^5 B+12 A b^6\right ) \tanh ^{-1}\left (\frac{\sqrt{a-b} \tan \left (\frac{1}{2} (c+d x)\right )}{\sqrt{a+b}}\right )}{a^5 d (a-b)^{5/2} (a+b)^{5/2}}+\frac{\sin (c+d x) \cos (c+d x) \left (7 a^2 A b^2-5 a^3 b B+3 a^4 C+2 a b^3 B-4 A b^4\right )}{2 a^2 d \left (a^2-b^2\right )^2 (a+b \sec (c+d x))}+\frac{\sin (c+d x) \cos (c+d x) \left (A b^2-a (b B-a C)\right )}{2 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}+\frac{x \left (a^2 (A+2 C)-6 a b B+12 A b^2\right )}{2 a^5} \]
[Out]
((12*A*b^2 - 6*a*b*B + a^2*(A + 2*C))*x)/(2*a^5) - (b*(12*A*b^6 - 12*a^5*b*B + 15*a^3*b^3*B - 6*a*b^5*B - a^2*
b^4*(29*A - 2*C) + 5*a^4*b^2*(4*A - C) + 6*a^6*C)*ArcTanh[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/(a^5*(a
- b)^(5/2)*(a + b)^(5/2)*d) - ((12*A*b^5 - 2*a^5*B + 11*a^3*b^2*B - 6*a*b^4*B + a^4*b*(6*A - 5*C) - a^2*b^3*(
21*A - 2*C))*Sin[c + d*x])/(2*a^4*(a^2 - b^2)^2*d) + ((6*A*b^4 + 6*a^3*b*B - 3*a*b^3*B + a^4*(A - 4*C) - a^2*b
^2*(10*A - C))*Cos[c + d*x]*Sin[c + d*x])/(2*a^3*(a^2 - b^2)^2*d) + ((A*b^2 - a*(b*B - a*C))*Cos[c + d*x]*Sin[
c + d*x])/(2*a*(a^2 - b^2)*d*(a + b*Sec[c + d*x])^2) + ((7*a^2*A*b^2 - 4*A*b^4 - 5*a^3*b*B + 2*a*b^3*B + 3*a^4
*C)*Cos[c + d*x]*Sin[c + d*x])/(2*a^2*(a^2 - b^2)^2*d*(a + b*Sec[c + d*x]))
________________________________________________________________________________________
Rubi [A] time = 4.82448, antiderivative size = 453, normalized size of antiderivative = 1.,
number of steps used = 8, number of rules used = 6, integrand size = 41, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.146, Rules used =
{4100, 4104, 3919, 3831, 2659, 208} \[ -\frac{\sin (c+d x) \left (-a^2 b^3 (21 A-2 C)+a^4 b (6 A-5 C)+11 a^3 b^2 B-2 a^5 B-6 a b^4 B+12 A b^5\right )}{2 a^4 d \left (a^2-b^2\right )^2}+\frac{\sin (c+d x) \cos (c+d x) \left (-a^2 b^2 (10 A-C)+a^4 (A-4 C)+6 a^3 b B-3 a b^3 B+6 A b^4\right )}{2 a^3 d \left (a^2-b^2\right )^2}-\frac{b \left (5 a^4 b^2 (4 A-C)-a^2 b^4 (29 A-2 C)+15 a^3 b^3 B-12 a^5 b B+6 a^6 C-6 a b^5 B+12 A b^6\right ) \tanh ^{-1}\left (\frac{\sqrt{a-b} \tan \left (\frac{1}{2} (c+d x)\right )}{\sqrt{a+b}}\right )}{a^5 d (a-b)^{5/2} (a+b)^{5/2}}+\frac{\sin (c+d x) \cos (c+d x) \left (7 a^2 A b^2-5 a^3 b B+3 a^4 C+2 a b^3 B-4 A b^4\right )}{2 a^2 d \left (a^2-b^2\right )^2 (a+b \sec (c+d x))}+\frac{\sin (c+d x) \cos (c+d x) \left (A b^2-a (b B-a C)\right )}{2 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}+\frac{x \left (a^2 (A+2 C)-6 a b B+12 A b^2\right )}{2 a^5} \]
Antiderivative was successfully verified.
[In]
Int[(Cos[c + d*x]^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(a + b*Sec[c + d*x])^3,x]
[Out]
((12*A*b^2 - 6*a*b*B + a^2*(A + 2*C))*x)/(2*a^5) - (b*(12*A*b^6 - 12*a^5*b*B + 15*a^3*b^3*B - 6*a*b^5*B - a^2*
b^4*(29*A - 2*C) + 5*a^4*b^2*(4*A - C) + 6*a^6*C)*ArcTanh[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/(a^5*(a
- b)^(5/2)*(a + b)^(5/2)*d) - ((12*A*b^5 - 2*a^5*B + 11*a^3*b^2*B - 6*a*b^4*B + a^4*b*(6*A - 5*C) - a^2*b^3*(
21*A - 2*C))*Sin[c + d*x])/(2*a^4*(a^2 - b^2)^2*d) + ((6*A*b^4 + 6*a^3*b*B - 3*a*b^3*B + a^4*(A - 4*C) - a^2*b
^2*(10*A - C))*Cos[c + d*x]*Sin[c + d*x])/(2*a^3*(a^2 - b^2)^2*d) + ((A*b^2 - a*(b*B - a*C))*Cos[c + d*x]*Sin[
c + d*x])/(2*a*(a^2 - b^2)*d*(a + b*Sec[c + d*x])^2) + ((7*a^2*A*b^2 - 4*A*b^4 - 5*a^3*b*B + 2*a*b^3*B + 3*a^4
*C)*Cos[c + d*x]*Sin[c + d*x])/(2*a^2*(a^2 - b^2)^2*d*(a + b*Sec[c + d*x]))
Rule 4100
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 4104
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]
Rule 3919
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 3831
Int[csc[(e_.) + (f_.)*(x_)]/(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)), x_Symbol] :> Dist[1/b, Int[1/(1 + (a*Sin[e
+ f*x])/b), x], x] /; FreeQ[{a, b, e, f}, x] && NeQ[a^2 - b^2, 0]
Rule 2659
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 208
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-(a/b), 2]*ArcTanh[x/Rt[-(a/b), 2]])/a, x] /; FreeQ[{a,
b}, x] && NegQ[a/b]
Rubi steps
\begin{align*} \int \frac{\cos ^2(c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{(a+b \sec (c+d x))^3} \, dx &=\frac{\left (A b^2-a (b B-a C)\right ) \cos (c+d x) \sin (c+d x)}{2 a \left (a^2-b^2\right ) d (a+b \sec (c+d x))^2}-\frac{\int \frac{\cos ^2(c+d x) \left (2 \left (2 A b^2-a b B-a^2 (A-C)\right )+2 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))^2} \, dx}{2 a \left (a^2-b^2\right )}\\ &=\frac{\left (A b^2-a (b B-a C)\right ) \cos (c+d x) \sin (c+d x)}{2 a \left (a^2-b^2\right ) d (a+b \sec (c+d x))^2}+\frac{\left (7 a^2 A b^2-4 A b^4-5 a^3 b B+2 a b^3 B+3 a^4 C\right ) \cos (c+d x) \sin (c+d x)}{2 a^2 \left (a^2-b^2\right )^2 d (a+b \sec (c+d x))}+\frac{\int \frac{\cos ^2(c+d x) \left (2 \left (6 A b^4+6 a^3 b B-3 a b^3 B+a^4 (A-4 C)-a^2 b^2 (10 A-C)\right )+a \left (A b^3+2 a^3 B+a b^2 B-a^2 b (4 A+3 C)\right ) \sec (c+d x)+2 \left (7 a^2 A b^2-4 A b^4-5 a^3 b B+2 a b^3 B+3 a^4 C\right ) \sec ^2(c+d x)\right )}{a+b \sec (c+d x)} \, dx}{2 a^2 \left (a^2-b^2\right )^2}\\ &=\frac{\left (6 A b^4+6 a^3 b B-3 a b^3 B+a^4 (A-4 C)-a^2 b^2 (10 A-C)\right ) \cos (c+d x) \sin (c+d x)}{2 a^3 \left (a^2-b^2\right )^2 d}+\frac{\left (A b^2-a (b B-a C)\right ) \cos (c+d x) \sin (c+d x)}{2 a \left (a^2-b^2\right ) d (a+b \sec (c+d x))^2}+\frac{\left (7 a^2 A b^2-4 A b^4-5 a^3 b B+2 a b^3 B+3 a^4 C\right ) \cos (c+d x) \sin (c+d x)}{2 a^2 \left (a^2-b^2\right )^2 d (a+b \sec (c+d x))}-\frac{\int \frac{\cos (c+d x) \left (2 \left (12 A b^5-2 a^5 B+11 a^3 b^2 B-6 a b^4 B+a^4 b (6 A-5 C)-a^2 b^3 (21 A-2 C)\right )+2 a \left (2 A b^4+4 a^3 b B-a b^3 B-a^2 b^2 (4 A+C)-a^4 (A+2 C)\right ) \sec (c+d x)-2 b \left (6 A b^4+6 a^3 b B-3 a b^3 B+a^4 (A-4 C)-a^2 b^2 (10 A-C)\right ) \sec ^2(c+d x)\right )}{a+b \sec (c+d x)} \, dx}{4 a^3 \left (a^2-b^2\right )^2}\\ &=-\frac{\left (12 A b^5-2 a^5 B+11 a^3 b^2 B-6 a b^4 B+a^4 b (6 A-5 C)-a^2 b^3 (21 A-2 C)\right ) \sin (c+d x)}{2 a^4 \left (a^2-b^2\right )^2 d}+\frac{\left (6 A b^4+6 a^3 b B-3 a b^3 B+a^4 (A-4 C)-a^2 b^2 (10 A-C)\right ) \cos (c+d x) \sin (c+d x)}{2 a^3 \left (a^2-b^2\right )^2 d}+\frac{\left (A b^2-a (b B-a C)\right ) \cos (c+d x) \sin (c+d x)}{2 a \left (a^2-b^2\right ) d (a+b \sec (c+d x))^2}+\frac{\left (7 a^2 A b^2-4 A b^4-5 a^3 b B+2 a b^3 B+3 a^4 C\right ) \cos (c+d x) \sin (c+d x)}{2 a^2 \left (a^2-b^2\right )^2 d (a+b \sec (c+d x))}+\frac{\int \frac{2 \left (a^2-b^2\right )^2 \left (12 A b^2-6 a b B+a^2 (A+2 C)\right )+2 a b \left (6 A b^4+6 a^3 b B-3 a b^3 B+a^4 (A-4 C)-a^2 b^2 (10 A-C)\right ) \sec (c+d x)}{a+b \sec (c+d x)} \, dx}{4 a^4 \left (a^2-b^2\right )^2}\\ &=\frac{\left (12 A b^2-6 a b B+a^2 (A+2 C)\right ) x}{2 a^5}-\frac{\left (12 A b^5-2 a^5 B+11 a^3 b^2 B-6 a b^4 B+a^4 b (6 A-5 C)-a^2 b^3 (21 A-2 C)\right ) \sin (c+d x)}{2 a^4 \left (a^2-b^2\right )^2 d}+\frac{\left (6 A b^4+6 a^3 b B-3 a b^3 B+a^4 (A-4 C)-a^2 b^2 (10 A-C)\right ) \cos (c+d x) \sin (c+d x)}{2 a^3 \left (a^2-b^2\right )^2 d}+\frac{\left (A b^2-a (b B-a C)\right ) \cos (c+d x) \sin (c+d x)}{2 a \left (a^2-b^2\right ) d (a+b \sec (c+d x))^2}+\frac{\left (7 a^2 A b^2-4 A b^4-5 a^3 b B+2 a b^3 B+3 a^4 C\right ) \cos (c+d x) \sin (c+d x)}{2 a^2 \left (a^2-b^2\right )^2 d (a+b \sec (c+d x))}-\frac{\left (b \left (12 A b^6-12 a^5 b B+15 a^3 b^3 B-6 a b^5 B-a^2 b^4 (29 A-2 C)+5 a^4 b^2 (4 A-C)+6 a^6 C\right )\right ) \int \frac{\sec (c+d x)}{a+b \sec (c+d x)} \, dx}{2 a^5 \left (a^2-b^2\right )^2}\\ &=\frac{\left (12 A b^2-6 a b B+a^2 (A+2 C)\right ) x}{2 a^5}-\frac{\left (12 A b^5-2 a^5 B+11 a^3 b^2 B-6 a b^4 B+a^4 b (6 A-5 C)-a^2 b^3 (21 A-2 C)\right ) \sin (c+d x)}{2 a^4 \left (a^2-b^2\right )^2 d}+\frac{\left (6 A b^4+6 a^3 b B-3 a b^3 B+a^4 (A-4 C)-a^2 b^2 (10 A-C)\right ) \cos (c+d x) \sin (c+d x)}{2 a^3 \left (a^2-b^2\right )^2 d}+\frac{\left (A b^2-a (b B-a C)\right ) \cos (c+d x) \sin (c+d x)}{2 a \left (a^2-b^2\right ) d (a+b \sec (c+d x))^2}+\frac{\left (7 a^2 A b^2-4 A b^4-5 a^3 b B+2 a b^3 B+3 a^4 C\right ) \cos (c+d x) \sin (c+d x)}{2 a^2 \left (a^2-b^2\right )^2 d (a+b \sec (c+d x))}-\frac{\left (12 A b^6-12 a^5 b B+15 a^3 b^3 B-6 a b^5 B-a^2 b^4 (29 A-2 C)+5 a^4 b^2 (4 A-C)+6 a^6 C\right ) \int \frac{1}{1+\frac{a \cos (c+d x)}{b}} \, dx}{2 a^5 \left (a^2-b^2\right )^2}\\ &=\frac{\left (12 A b^2-6 a b B+a^2 (A+2 C)\right ) x}{2 a^5}-\frac{\left (12 A b^5-2 a^5 B+11 a^3 b^2 B-6 a b^4 B+a^4 b (6 A-5 C)-a^2 b^3 (21 A-2 C)\right ) \sin (c+d x)}{2 a^4 \left (a^2-b^2\right )^2 d}+\frac{\left (6 A b^4+6 a^3 b B-3 a b^3 B+a^4 (A-4 C)-a^2 b^2 (10 A-C)\right ) \cos (c+d x) \sin (c+d x)}{2 a^3 \left (a^2-b^2\right )^2 d}+\frac{\left (A b^2-a (b B-a C)\right ) \cos (c+d x) \sin (c+d x)}{2 a \left (a^2-b^2\right ) d (a+b \sec (c+d x))^2}+\frac{\left (7 a^2 A b^2-4 A b^4-5 a^3 b B+2 a b^3 B+3 a^4 C\right ) \cos (c+d x) \sin (c+d x)}{2 a^2 \left (a^2-b^2\right )^2 d (a+b \sec (c+d x))}-\frac{\left (12 A b^6-12 a^5 b B+15 a^3 b^3 B-6 a b^5 B-a^2 b^4 (29 A-2 C)+5 a^4 b^2 (4 A-C)+6 a^6 C\right ) \operatorname{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 )^2 d}\\ &=\frac{\left (12 A b^2-6 a b B+a^2 (A+2 C)\right ) x}{2 a^5}-\frac{b \left (20 a^4 A b^2-29 a^2 A b^4+12 A b^6-12 a^5 b B+15 a^3 b^3 B-6 a b^5 B+6 a^6 C-5 a^4 b^2 C+2 a^2 b^4 C\right ) \tanh ^{-1}\left (\frac{\sqrt{a-b} \tan \left (\frac{1}{2} (c+d x)\right )}{\sqrt{a+b}}\right )}{a^5 (a-b)^{5/2} (a+b)^{5/2} d}-\frac{\left (12 A b^5-2 a^5 B+11 a^3 b^2 B-6 a b^4 B+a^4 b (6 A-5 C)-a^2 b^3 (21 A-2 C)\right ) \sin (c+d x)}{2 a^4 \left (a^2-b^2\right )^2 d}+\frac{\left (6 A b^4+6 a^3 b B-3 a b^3 B+a^4 (A-4 C)-a^2 b^2 (10 A-C)\right ) \cos (c+d x) \sin (c+d x)}{2 a^3 \left (a^2-b^2\right )^2 d}+\frac{\left (A b^2-a (b B-a C)\right ) \cos (c+d x) \sin (c+d x)}{2 a \left (a^2-b^2\right ) d (a+b \sec (c+d x))^2}+\frac{\left (7 a^2 A b^2-4 A b^4-5 a^3 b B+2 a b^3 B+3 a^4 C\right ) \cos (c+d x) \sin (c+d x)}{2 a^2 \left (a^2-b^2\right )^2 d (a+b \sec (c+d x))}\\ \end{align*}
Mathematica [A] time = 5.22424, size = 881, normalized size = 1.94 \[ \frac{\frac{16 b \left (6 C a^6-12 b B a^5+5 b^2 (4 A-C) a^4+15 b^3 B a^3+b^4 (2 C-29 A) a^2-6 b^5 B a+12 A b^6\right ) \tanh ^{-1}\left (\frac{(b-a) \tan \left (\frac{1}{2} (c+d x)\right )}{\sqrt{a^2-b^2}}\right )}{\left (a^2-b^2\right )^{5/2}}+\frac{4 A c a^8+8 c C a^8+4 A d x a^8+8 C d x a^8+4 B \sin (c+d x) a^8+2 A \sin (2 (c+d x)) a^8+4 B \sin (3 (c+d x)) a^8+A \sin (4 (c+d x)) a^8-24 b B c a^7-24 b B d x a^7-8 A b \sin (c+d x) a^7+16 b B \sin (2 (c+d x)) a^7-8 A b \sin (3 (c+d x)) a^7+48 A b^2 c a^6+48 A b^2 d x a^6+8 b^2 B \sin (c+d x) a^6-48 A b^2 \sin (2 (c+d x)) a^6+24 b^2 C \sin (2 (c+d x)) a^6-8 b^2 B \sin (3 (c+d x)) a^6-2 A b^2 \sin (4 (c+d x)) a^6-32 A b^3 \sin (c+d x) a^5+40 b^3 C \sin (c+d x) a^5-64 b^3 B \sin (2 (c+d x)) a^5+16 A b^3 \sin (3 (c+d x)) a^5-12 A b^4 c a^4-24 b^4 c C a^4-12 A b^4 d x a^4-24 b^4 C d x a^4-84 b^4 B \sin (c+d x) a^4+130 A b^4 \sin (2 (c+d x)) a^4-12 b^4 C \sin (2 (c+d x)) a^4+4 b^4 B \sin (3 (c+d x)) a^4+A b^4 \sin (4 (c+d x)) a^4+72 b^5 B c a^3+72 b^5 B d x a^3+160 A b^5 \sin (c+d x) a^3-16 b^5 C \sin (c+d x) a^3+36 b^5 B \sin (2 (c+d x)) a^3-8 A b^5 \sin (3 (c+d x)) a^3-136 A b^6 c a^2+16 b^6 c C a^2-136 A b^6 d x a^2+16 b^6 C d x a^2+48 b^6 B \sin (c+d x) a^2-72 A b^6 \sin (2 (c+d x)) a^2-48 b^7 B c a-48 b^7 B d x a+16 b \left (a^2-b^2\right )^2 \left ((A+2 C) a^2-6 b B a+12 A b^2\right ) (c+d x) \cos (c+d x) a-96 A b^7 \sin (c+d x) a+96 A b^8 c+96 A b^8 d x+4 \left (a^3-a b^2\right )^2 \left ((A+2 C) a^2-6 b B a+12 A b^2\right ) (c+d x) \cos (2 (c+d x))}{\left (a^2-b^2\right )^2 (b+a \cos (c+d x))^2}}{16 a^5 d} \]
Antiderivative was successfully verified.
[In]
Integrate[(Cos[c + d*x]^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(a + b*Sec[c + d*x])^3,x]
[Out]
((16*b*(12*A*b^6 - 12*a^5*b*B + 15*a^3*b^3*B - 6*a*b^5*B + 5*a^4*b^2*(4*A - C) + 6*a^6*C + a^2*b^4*(-29*A + 2*
C))*ArcTanh[((-a + b)*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(a^2 - b^2)^(5/2) + (4*a^8*A*c + 48*a^6*A*b^2*c - 12
*a^4*A*b^4*c - 136*a^2*A*b^6*c + 96*A*b^8*c - 24*a^7*b*B*c + 72*a^3*b^5*B*c - 48*a*b^7*B*c + 8*a^8*c*C - 24*a^
4*b^4*c*C + 16*a^2*b^6*c*C + 4*a^8*A*d*x + 48*a^6*A*b^2*d*x - 12*a^4*A*b^4*d*x - 136*a^2*A*b^6*d*x + 96*A*b^8*
d*x - 24*a^7*b*B*d*x + 72*a^3*b^5*B*d*x - 48*a*b^7*B*d*x + 8*a^8*C*d*x - 24*a^4*b^4*C*d*x + 16*a^2*b^6*C*d*x +
16*a*b*(a^2 - b^2)^2*(12*A*b^2 - 6*a*b*B + a^2*(A + 2*C))*(c + d*x)*Cos[c + d*x] + 4*(a^3 - a*b^2)^2*(12*A*b^
2 - 6*a*b*B + a^2*(A + 2*C))*(c + d*x)*Cos[2*(c + d*x)] - 8*a^7*A*b*Sin[c + d*x] - 32*a^5*A*b^3*Sin[c + d*x] +
160*a^3*A*b^5*Sin[c + d*x] - 96*a*A*b^7*Sin[c + d*x] + 4*a^8*B*Sin[c + d*x] + 8*a^6*b^2*B*Sin[c + d*x] - 84*a
^4*b^4*B*Sin[c + d*x] + 48*a^2*b^6*B*Sin[c + d*x] + 40*a^5*b^3*C*Sin[c + d*x] - 16*a^3*b^5*C*Sin[c + d*x] + 2*
a^8*A*Sin[2*(c + d*x)] - 48*a^6*A*b^2*Sin[2*(c + d*x)] + 130*a^4*A*b^4*Sin[2*(c + d*x)] - 72*a^2*A*b^6*Sin[2*(
c + d*x)] + 16*a^7*b*B*Sin[2*(c + d*x)] - 64*a^5*b^3*B*Sin[2*(c + d*x)] + 36*a^3*b^5*B*Sin[2*(c + d*x)] + 24*a
^6*b^2*C*Sin[2*(c + d*x)] - 12*a^4*b^4*C*Sin[2*(c + d*x)] - 8*a^7*A*b*Sin[3*(c + d*x)] + 16*a^5*A*b^3*Sin[3*(c
+ d*x)] - 8*a^3*A*b^5*Sin[3*(c + d*x)] + 4*a^8*B*Sin[3*(c + d*x)] - 8*a^6*b^2*B*Sin[3*(c + d*x)] + 4*a^4*b^4*
B*Sin[3*(c + d*x)] + a^8*A*Sin[4*(c + d*x)] - 2*a^6*A*b^2*Sin[4*(c + d*x)] + a^4*A*b^4*Sin[4*(c + d*x)])/((a^2
- b^2)^2*(b + a*Cos[c + d*x])^2))/(16*a^5*d)
________________________________________________________________________________________
Maple [B] time = 0.164, size = 2206, normalized size = 4.9 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(cos(d*x+c)^2*(A+B*sec(d*x+c)+C*sec(d*x+c)^2)/(a+b*sec(d*x+c))^3,x)
[Out]
-4/d*b^5/a^3/(tan(1/2*d*x+1/2*c)^2*a-tan(1/2*d*x+1/2*c)^2*b-a-b)^2/(a-b)/(a^2+2*a*b+b^2)*tan(1/2*d*x+1/2*c)^3*
B-1/d*b^3/a/(tan(1/2*d*x+1/2*c)^2*a-tan(1/2*d*x+1/2*c)^2*b-a-b)^2/(a-b)/(a^2+2*a*b+b^2)*tan(1/2*d*x+1/2*c)^3*C
+1/d*A/a^3*arctan(tan(1/2*d*x+1/2*c))+12/d*b^2/(a^4-2*a^2*b^2+b^4)/((a+b)*(a-b))^(1/2)*arctanh((a-b)*tan(1/2*d
*x+1/2*c)/((a+b)*(a-b))^(1/2))*B-15/d*b^4/a^2/(a^4-2*a^2*b^2+b^4)/((a+b)*(a-b))^(1/2)*arctanh((a-b)*tan(1/2*d*
x+1/2*c)/((a+b)*(a-b))^(1/2))*B+6/d*b^6/a^4/(a^4-2*a^2*b^2+b^4)/((a+b)*(a-b))^(1/2)*arctanh((a-b)*tan(1/2*d*x+
1/2*c)/((a+b)*(a-b))^(1/2))*B+5/d*b^3/a/(a^4-2*a^2*b^2+b^4)/((a+b)*(a-b))^(1/2)*arctanh((a-b)*tan(1/2*d*x+1/2*
c)/((a+b)*(a-b))^(1/2))*C-2/d*b^5/a^3/(a^4-2*a^2*b^2+b^4)/((a+b)*(a-b))^(1/2)*arctanh((a-b)*tan(1/2*d*x+1/2*c)
/((a+b)*(a-b))^(1/2))*C+10/d/a^2/(tan(1/2*d*x+1/2*c)^2*a-tan(1/2*d*x+1/2*c)^2*b-a-b)^2/(a+b)/(a-b)^2*tan(1/2*d
*x+1/2*c)*A*b^4-10/d/a^2/(tan(1/2*d*x+1/2*c)^2*a-tan(1/2*d*x+1/2*c)^2*b-a-b)^2/(a-b)/(a^2+2*a*b+b^2)*tan(1/2*d
*x+1/2*c)^3*A*b^4+1/d/a^2/(tan(1/2*d*x+1/2*c)^2*a-tan(1/2*d*x+1/2*c)^2*b-a-b)^2*b^4/(a+b)/(a-b)^2*tan(1/2*d*x+
1/2*c)*B-8/d/a/(tan(1/2*d*x+1/2*c)^2*a-tan(1/2*d*x+1/2*c)^2*b-a-b)^2*b^3/(a+b)/(a-b)^2*tan(1/2*d*x+1/2*c)*B-1/
d/a^3/(tan(1/2*d*x+1/2*c)^2*a-tan(1/2*d*x+1/2*c)^2*b-a-b)^2*b^5/(a-b)/(a^2+2*a*b+b^2)*tan(1/2*d*x+1/2*c)^3*A+8
/d/a/(tan(1/2*d*x+1/2*c)^2*a-tan(1/2*d*x+1/2*c)^2*b-a-b)^2*b^3/(a-b)/(a^2+2*a*b+b^2)*tan(1/2*d*x+1/2*c)^3*B+1/
d/a^2/(tan(1/2*d*x+1/2*c)^2*a-tan(1/2*d*x+1/2*c)^2*b-a-b)^2*b^4/(a-b)/(a^2+2*a*b+b^2)*tan(1/2*d*x+1/2*c)^3*B-1
/d/a^3/(tan(1/2*d*x+1/2*c)^2*a-tan(1/2*d*x+1/2*c)^2*b-a-b)^2*b^5/(a+b)/(a-b)^2*tan(1/2*d*x+1/2*c)*A-6/d/a^4/(1
+tan(1/2*d*x+1/2*c)^2)^2*tan(1/2*d*x+1/2*c)^3*A*b-6/d/a^4/(1+tan(1/2*d*x+1/2*c)^2)^2*tan(1/2*d*x+1/2*c)*A*b-2/
d*b^4/a^2/(tan(1/2*d*x+1/2*c)^2*a-tan(1/2*d*x+1/2*c)^2*b-a-b)^2/(a+b)/(a-b)^2*tan(1/2*d*x+1/2*c)*C+2/d*b^4/a^2
/(tan(1/2*d*x+1/2*c)^2*a-tan(1/2*d*x+1/2*c)^2*b-a-b)^2/(a-b)/(a^2+2*a*b+b^2)*tan(1/2*d*x+1/2*c)^3*C-6/d*b^6/a^
4/(tan(1/2*d*x+1/2*c)^2*a-tan(1/2*d*x+1/2*c)^2*b-a-b)^2/(a+b)/(a-b)^2*tan(1/2*d*x+1/2*c)*A-1/d*b^3/a/(tan(1/2*
d*x+1/2*c)^2*a-tan(1/2*d*x+1/2*c)^2*b-a-b)^2/(a+b)/(a-b)^2*tan(1/2*d*x+1/2*c)*C+4/d*b^5/a^3/(tan(1/2*d*x+1/2*c
)^2*a-tan(1/2*d*x+1/2*c)^2*b-a-b)^2/(a+b)/(a-b)^2*tan(1/2*d*x+1/2*c)*B+6/d*b^6/a^4/(tan(1/2*d*x+1/2*c)^2*a-tan
(1/2*d*x+1/2*c)^2*b-a-b)^2/(a-b)/(a^2+2*a*b+b^2)*tan(1/2*d*x+1/2*c)^3*A+6/d/(tan(1/2*d*x+1/2*c)^2*a-tan(1/2*d*
x+1/2*c)^2*b-a-b)^2/(a+b)/(a-b)^2*tan(1/2*d*x+1/2*c)*C*b^2-20/d/a/(a^4-2*a^2*b^2+b^4)/((a+b)*(a-b))^(1/2)*arct
anh((a-b)*tan(1/2*d*x+1/2*c)/((a+b)*(a-b))^(1/2))*A*b^3+29/d/a^3/(a^4-2*a^2*b^2+b^4)/((a+b)*(a-b))^(1/2)*arcta
nh((a-b)*tan(1/2*d*x+1/2*c)/((a+b)*(a-b))^(1/2))*A*b^5+2/d/a^3*arctan(tan(1/2*d*x+1/2*c))*C+2/d/a^3/(1+tan(1/2
*d*x+1/2*c)^2)^2*tan(1/2*d*x+1/2*c)^3*B+1/d/a^3/(1+tan(1/2*d*x+1/2*c)^2)^2*tan(1/2*d*x+1/2*c)*A+2/d/a^3/(1+tan
(1/2*d*x+1/2*c)^2)^2*tan(1/2*d*x+1/2*c)*B+12/d/a^5*arctan(tan(1/2*d*x+1/2*c))*A*b^2-6/d/a^4*arctan(tan(1/2*d*x
+1/2*c))*B*b-12/d*b^7/a^5/(a^4-2*a^2*b^2+b^4)/((a+b)*(a-b))^(1/2)*arctanh((a-b)*tan(1/2*d*x+1/2*c)/((a+b)*(a-b
))^(1/2))*A-6/d*b/(a^4-2*a^2*b^2+b^4)/((a+b)*(a-b))^(1/2)*arctanh((a-b)*tan(1/2*d*x+1/2*c)/((a+b)*(a-b))^(1/2)
)*C*a-1/d/a^3/(1+tan(1/2*d*x+1/2*c)^2)^2*tan(1/2*d*x+1/2*c)^3*A-6/d/(tan(1/2*d*x+1/2*c)^2*a-tan(1/2*d*x+1/2*c)
^2*b-a-b)^2/(a-b)/(a^2+2*a*b+b^2)*tan(1/2*d*x+1/2*c)^3*C*b^2
________________________________________________________________________________________
Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(cos(d*x+c)^2*(A+B*sec(d*x+c)+C*sec(d*x+c)^2)/(a+b*sec(d*x+c))^3,x, algorithm="maxima")
[Out]
Exception raised: ValueError
________________________________________________________________________________________
Fricas [B] time = 1.17964, size = 4668, normalized size = 10.3 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(cos(d*x+c)^2*(A+B*sec(d*x+c)+C*sec(d*x+c)^2)/(a+b*sec(d*x+c))^3,x, algorithm="fricas")
[Out]
[1/4*(2*((A + 2*C)*a^10 - 6*B*a^9*b + 3*(3*A - 2*C)*a^8*b^2 + 18*B*a^7*b^3 - 3*(11*A - 2*C)*a^6*b^4 - 18*B*a^5
*b^5 + (35*A - 2*C)*a^4*b^6 + 6*B*a^3*b^7 - 12*A*a^2*b^8)*d*x*cos(d*x + c)^2 + 4*((A + 2*C)*a^9*b - 6*B*a^8*b^
2 + 3*(3*A - 2*C)*a^7*b^3 + 18*B*a^6*b^4 - 3*(11*A - 2*C)*a^5*b^5 - 18*B*a^4*b^6 + (35*A - 2*C)*a^3*b^7 + 6*B*
a^2*b^8 - 12*A*a*b^9)*d*x*cos(d*x + c) + 2*((A + 2*C)*a^8*b^2 - 6*B*a^7*b^3 + 3*(3*A - 2*C)*a^6*b^4 + 18*B*a^5
*b^5 - 3*(11*A - 2*C)*a^4*b^6 - 18*B*a^3*b^7 + (35*A - 2*C)*a^2*b^8 + 6*B*a*b^9 - 12*A*b^10)*d*x + (6*C*a^6*b^
3 - 12*B*a^5*b^4 + 5*(4*A - C)*a^4*b^5 + 15*B*a^3*b^6 - (29*A - 2*C)*a^2*b^7 - 6*B*a*b^8 + 12*A*b^9 + (6*C*a^8
*b - 12*B*a^7*b^2 + 5*(4*A - C)*a^6*b^3 + 15*B*a^5*b^4 - (29*A - 2*C)*a^4*b^5 - 6*B*a^3*b^6 + 12*A*a^2*b^7)*co
s(d*x + c)^2 + 2*(6*C*a^7*b^2 - 12*B*a^6*b^3 + 5*(4*A - C)*a^5*b^4 + 15*B*a^4*b^5 - (29*A - 2*C)*a^3*b^6 - 6*B
*a^2*b^7 + 12*A*a*b^8)*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*B*a^8*b^2 - (6*A - 5*C)*a^7*b^3 - 13*B*a^6*b^4 + (27*A - 7*C)*a^5*b^5 + 17*B*a^4*b^6 - (33*A - 2*
C)*a^3*b^7 - 6*B*a^2*b^8 + 12*A*a*b^9 + (A*a^10 - 3*A*a^8*b^2 + 3*A*a^6*b^4 - A*a^4*b^6)*cos(d*x + c)^3 + 2*(B
*a^10 - 2*A*a^9*b - 3*B*a^8*b^2 + 6*A*a^7*b^3 + 3*B*a^6*b^4 - 6*A*a^5*b^5 - B*a^4*b^6 + 2*A*a^3*b^7)*cos(d*x +
c)^2 + (4*B*a^9*b - (11*A - 6*C)*a^8*b^2 - 20*B*a^7*b^3 + (43*A - 9*C)*a^6*b^4 + 25*B*a^5*b^5 - (50*A - 3*C)*
a^4*b^6 - 9*B*a^3*b^7 + 18*A*a^2*b^8)*cos(d*x + c))*sin(d*x + c))/((a^13 - 3*a^11*b^2 + 3*a^9*b^4 - a^7*b^6)*d
*cos(d*x + c)^2 + 2*(a^12*b - 3*a^10*b^3 + 3*a^8*b^5 - a^6*b^7)*d*cos(d*x + c) + (a^11*b^2 - 3*a^9*b^4 + 3*a^7
*b^6 - a^5*b^8)*d), 1/2*(((A + 2*C)*a^10 - 6*B*a^9*b + 3*(3*A - 2*C)*a^8*b^2 + 18*B*a^7*b^3 - 3*(11*A - 2*C)*a
^6*b^4 - 18*B*a^5*b^5 + (35*A - 2*C)*a^4*b^6 + 6*B*a^3*b^7 - 12*A*a^2*b^8)*d*x*cos(d*x + c)^2 + 2*((A + 2*C)*a
^9*b - 6*B*a^8*b^2 + 3*(3*A - 2*C)*a^7*b^3 + 18*B*a^6*b^4 - 3*(11*A - 2*C)*a^5*b^5 - 18*B*a^4*b^6 + (35*A - 2*
C)*a^3*b^7 + 6*B*a^2*b^8 - 12*A*a*b^9)*d*x*cos(d*x + c) + ((A + 2*C)*a^8*b^2 - 6*B*a^7*b^3 + 3*(3*A - 2*C)*a^6
*b^4 + 18*B*a^5*b^5 - 3*(11*A - 2*C)*a^4*b^6 - 18*B*a^3*b^7 + (35*A - 2*C)*a^2*b^8 + 6*B*a*b^9 - 12*A*b^10)*d*
x - (6*C*a^6*b^3 - 12*B*a^5*b^4 + 5*(4*A - C)*a^4*b^5 + 15*B*a^3*b^6 - (29*A - 2*C)*a^2*b^7 - 6*B*a*b^8 + 12*A
*b^9 + (6*C*a^8*b - 12*B*a^7*b^2 + 5*(4*A - C)*a^6*b^3 + 15*B*a^5*b^4 - (29*A - 2*C)*a^4*b^5 - 6*B*a^3*b^6 + 1
2*A*a^2*b^7)*cos(d*x + c)^2 + 2*(6*C*a^7*b^2 - 12*B*a^6*b^3 + 5*(4*A - C)*a^5*b^4 + 15*B*a^4*b^5 - (29*A - 2*C
)*a^3*b^6 - 6*B*a^2*b^7 + 12*A*a*b^8)*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*B*a^8*b^2 - (6*A - 5*C)*a^7*b^3 - 13*B*a^6*b^4 + (27*A - 7*C)*a^5*b^5 +
17*B*a^4*b^6 - (33*A - 2*C)*a^3*b^7 - 6*B*a^2*b^8 + 12*A*a*b^9 + (A*a^10 - 3*A*a^8*b^2 + 3*A*a^6*b^4 - A*a^4*b
^6)*cos(d*x + c)^3 + 2*(B*a^10 - 2*A*a^9*b - 3*B*a^8*b^2 + 6*A*a^7*b^3 + 3*B*a^6*b^4 - 6*A*a^5*b^5 - B*a^4*b^6
+ 2*A*a^3*b^7)*cos(d*x + c)^2 + (4*B*a^9*b - (11*A - 6*C)*a^8*b^2 - 20*B*a^7*b^3 + (43*A - 9*C)*a^6*b^4 + 25*
B*a^5*b^5 - (50*A - 3*C)*a^4*b^6 - 9*B*a^3*b^7 + 18*A*a^2*b^8)*cos(d*x + c))*sin(d*x + c))/((a^13 - 3*a^11*b^2
+ 3*a^9*b^4 - a^7*b^6)*d*cos(d*x + c)^2 + 2*(a^12*b - 3*a^10*b^3 + 3*a^8*b^5 - a^6*b^7)*d*cos(d*x + c) + (a^1
1*b^2 - 3*a^9*b^4 + 3*a^7*b^6 - a^5*b^8)*d)]
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(cos(d*x+c)**2*(A+B*sec(d*x+c)+C*sec(d*x+c)**2)/(a+b*sec(d*x+c))**3,x)
[Out]
Timed out
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Giac [B] time = 1.52141, size = 2291, normalized size = 5.06 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(cos(d*x+c)^2*(A+B*sec(d*x+c)+C*sec(d*x+c)^2)/(a+b*sec(d*x+c))^3,x, algorithm="giac")
[Out]
-1/2*(2*(6*C*a^6*b - 12*B*a^5*b^2 + 20*A*a^4*b^3 - 5*C*a^4*b^3 + 15*B*a^3*b^4 - 29*A*a^2*b^5 + 2*C*a^2*b^5 - 6
*B*a*b^6 + 12*A*b^7)*(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*t
an(1/2*d*x + 1/2*c))/sqrt(-a^2 + b^2)))/((a^9 - 2*a^7*b^2 + a^5*b^4)*sqrt(-a^2 + b^2)) + 2*(A*a^7*tan(1/2*d*x
+ 1/2*c)^7 - 2*B*a^7*tan(1/2*d*x + 1/2*c)^7 + 4*A*a^6*b*tan(1/2*d*x + 1/2*c)^7 + 4*B*a^6*b*tan(1/2*d*x + 1/2*c
)^7 - 13*A*a^5*b^2*tan(1/2*d*x + 1/2*c)^7 + 2*B*a^5*b^2*tan(1/2*d*x + 1/2*c)^7 + 6*C*a^5*b^2*tan(1/2*d*x + 1/2
*c)^7 - 2*A*a^4*b^3*tan(1/2*d*x + 1/2*c)^7 - 16*B*a^4*b^3*tan(1/2*d*x + 1/2*c)^7 - 5*C*a^4*b^3*tan(1/2*d*x + 1
/2*c)^7 + 33*A*a^3*b^4*tan(1/2*d*x + 1/2*c)^7 + 9*B*a^3*b^4*tan(1/2*d*x + 1/2*c)^7 - 3*C*a^3*b^4*tan(1/2*d*x +
1/2*c)^7 - 17*A*a^2*b^5*tan(1/2*d*x + 1/2*c)^7 + 9*B*a^2*b^5*tan(1/2*d*x + 1/2*c)^7 + 2*C*a^2*b^5*tan(1/2*d*x
+ 1/2*c)^7 - 18*A*a*b^6*tan(1/2*d*x + 1/2*c)^7 - 6*B*a*b^6*tan(1/2*d*x + 1/2*c)^7 + 12*A*b^7*tan(1/2*d*x + 1/
2*c)^7 - 3*A*a^7*tan(1/2*d*x + 1/2*c)^5 + 2*B*a^7*tan(1/2*d*x + 1/2*c)^5 - 4*A*a^6*b*tan(1/2*d*x + 1/2*c)^5 +
4*B*a^6*b*tan(1/2*d*x + 1/2*c)^5 - 5*A*a^5*b^2*tan(1/2*d*x + 1/2*c)^5 - 10*B*a^5*b^2*tan(1/2*d*x + 1/2*c)^5 +
6*C*a^5*b^2*tan(1/2*d*x + 1/2*c)^5 + 26*A*a^4*b^3*tan(1/2*d*x + 1/2*c)^5 - 16*B*a^4*b^3*tan(1/2*d*x + 1/2*c)^5
- 15*C*a^4*b^3*tan(1/2*d*x + 1/2*c)^5 + 29*A*a^3*b^4*tan(1/2*d*x + 1/2*c)^5 + 35*B*a^3*b^4*tan(1/2*d*x + 1/2*
c)^5 - 3*C*a^3*b^4*tan(1/2*d*x + 1/2*c)^5 - 67*A*a^2*b^5*tan(1/2*d*x + 1/2*c)^5 + 9*B*a^2*b^5*tan(1/2*d*x + 1/
2*c)^5 + 6*C*a^2*b^5*tan(1/2*d*x + 1/2*c)^5 - 18*A*a*b^6*tan(1/2*d*x + 1/2*c)^5 - 18*B*a*b^6*tan(1/2*d*x + 1/2
*c)^5 + 36*A*b^7*tan(1/2*d*x + 1/2*c)^5 + 3*A*a^7*tan(1/2*d*x + 1/2*c)^3 + 2*B*a^7*tan(1/2*d*x + 1/2*c)^3 - 4*
A*a^6*b*tan(1/2*d*x + 1/2*c)^3 - 4*B*a^6*b*tan(1/2*d*x + 1/2*c)^3 + 5*A*a^5*b^2*tan(1/2*d*x + 1/2*c)^3 - 10*B*
a^5*b^2*tan(1/2*d*x + 1/2*c)^3 - 6*C*a^5*b^2*tan(1/2*d*x + 1/2*c)^3 + 26*A*a^4*b^3*tan(1/2*d*x + 1/2*c)^3 + 16
*B*a^4*b^3*tan(1/2*d*x + 1/2*c)^3 - 15*C*a^4*b^3*tan(1/2*d*x + 1/2*c)^3 - 29*A*a^3*b^4*tan(1/2*d*x + 1/2*c)^3
+ 35*B*a^3*b^4*tan(1/2*d*x + 1/2*c)^3 + 3*C*a^3*b^4*tan(1/2*d*x + 1/2*c)^3 - 67*A*a^2*b^5*tan(1/2*d*x + 1/2*c)
^3 - 9*B*a^2*b^5*tan(1/2*d*x + 1/2*c)^3 + 6*C*a^2*b^5*tan(1/2*d*x + 1/2*c)^3 + 18*A*a*b^6*tan(1/2*d*x + 1/2*c)
^3 - 18*B*a*b^6*tan(1/2*d*x + 1/2*c)^3 + 36*A*b^7*tan(1/2*d*x + 1/2*c)^3 - A*a^7*tan(1/2*d*x + 1/2*c) - 2*B*a^
7*tan(1/2*d*x + 1/2*c) + 4*A*a^6*b*tan(1/2*d*x + 1/2*c) - 4*B*a^6*b*tan(1/2*d*x + 1/2*c) + 13*A*a^5*b^2*tan(1/
2*d*x + 1/2*c) + 2*B*a^5*b^2*tan(1/2*d*x + 1/2*c) - 6*C*a^5*b^2*tan(1/2*d*x + 1/2*c) - 2*A*a^4*b^3*tan(1/2*d*x
+ 1/2*c) + 16*B*a^4*b^3*tan(1/2*d*x + 1/2*c) - 5*C*a^4*b^3*tan(1/2*d*x + 1/2*c) - 33*A*a^3*b^4*tan(1/2*d*x +
1/2*c) + 9*B*a^3*b^4*tan(1/2*d*x + 1/2*c) + 3*C*a^3*b^4*tan(1/2*d*x + 1/2*c) - 17*A*a^2*b^5*tan(1/2*d*x + 1/2*
c) - 9*B*a^2*b^5*tan(1/2*d*x + 1/2*c) + 2*C*a^2*b^5*tan(1/2*d*x + 1/2*c) + 18*A*a*b^6*tan(1/2*d*x + 1/2*c) - 6
*B*a*b^6*tan(1/2*d*x + 1/2*c) + 12*A*b^7*tan(1/2*d*x + 1/2*c))/((a^8 - 2*a^6*b^2 + a^4*b^4)*(a*tan(1/2*d*x + 1
/2*c)^4 - b*tan(1/2*d*x + 1/2*c)^4 - 2*b*tan(1/2*d*x + 1/2*c)^2 - a - b)^2) - (A*a^2 + 2*C*a^2 - 6*B*a*b + 12*
A*b^2)*(d*x + c)/a^5)/d