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

Optimal. Leaf size=470 \[ -\frac{\tan (c+d x) \left (23 a^2 b^2 C+3 a^3 b B-12 a^4 C-8 a b^3 B+5 A b^4-6 b^4 C\right )}{6 b^4 d \left (a^2-b^2\right )^2}-\frac{\left (a^2 b^6 (3 A+20 C)-7 a^5 b^3 B+8 a^3 b^5 B+28 a^6 b^2 C-35 a^4 b^4 C+2 a^7 b B-8 a^8 C-8 a b^7 B+2 A b^8\right ) \tanh ^{-1}\left (\frac{\sqrt{a-b} \tan \left (\frac{1}{2} (c+d x)\right )}{\sqrt{a+b}}\right )}{b^5 d (a-b)^{7/2} (a+b)^{7/2}}-\frac{\tan (c+d x) \sec ^3(c+d x) \left (A b^2-a (b B-a C)\right )}{3 b d \left (a^2-b^2\right ) (a+b \sec (c+d x))^3}+\frac{\tan (c+d x) \sec ^2(c+d x) \left (a^2 b^2 (2 A+9 C)+a^3 b B-4 a^4 C-6 a b^3 B+3 A b^4\right )}{6 b^2 d \left (a^2-b^2\right )^2 (a+b \sec (c+d x))^2}+\frac{a \tan (c+d x) \left (3 a^2 b^4 (A+4 C)+2 a^3 b^3 B-11 a^4 b^2 C-a^5 b B+4 a^6 C-6 a b^5 B+2 A b^6\right )}{2 b^4 d \left (a^2-b^2\right )^3 (a+b \sec (c+d x))}+\frac{(b B-4 a C) \tanh ^{-1}(\sin (c+d x))}{b^5 d} \]

[Out]

((b*B - 4*a*C)*ArcTanh[Sin[c + d*x]])/(b^5*d) - ((2*A*b^8 + 2*a^7*b*B - 7*a^5*b^3*B + 8*a^3*b^5*B - 8*a*b^7*B
- 8*a^8*C + 28*a^6*b^2*C - 35*a^4*b^4*C + a^2*b^6*(3*A + 20*C))*ArcTanh[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a
+ b]])/((a - b)^(7/2)*b^5*(a + b)^(7/2)*d) - ((5*A*b^4 + 3*a^3*b*B - 8*a*b^3*B - 12*a^4*C + 23*a^2*b^2*C - 6*b
^4*C)*Tan[c + d*x])/(6*b^4*(a^2 - b^2)^2*d) - ((A*b^2 - a*(b*B - a*C))*Sec[c + d*x]^3*Tan[c + d*x])/(3*b*(a^2
- b^2)*d*(a + b*Sec[c + d*x])^3) + ((3*A*b^4 + a^3*b*B - 6*a*b^3*B - 4*a^4*C + a^2*b^2*(2*A + 9*C))*Sec[c + d*
x]^2*Tan[c + d*x])/(6*b^2*(a^2 - b^2)^2*d*(a + b*Sec[c + d*x])^2) + (a*(2*A*b^6 - a^5*b*B + 2*a^3*b^3*B - 6*a*
b^5*B + 4*a^6*C - 11*a^4*b^2*C + 3*a^2*b^4*(A + 4*C))*Tan[c + d*x])/(2*b^4*(a^2 - b^2)^3*d*(a + b*Sec[c + d*x]
))

________________________________________________________________________________________

Rubi [A]  time = 9.91341, antiderivative size = 470, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 8, integrand size = 41, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.195, Rules used = {4098, 4090, 4082, 3998, 3770, 3831, 2659, 208} \[ -\frac{\tan (c+d x) \left (23 a^2 b^2 C+3 a^3 b B-12 a^4 C-8 a b^3 B+5 A b^4-6 b^4 C\right )}{6 b^4 d \left (a^2-b^2\right )^2}-\frac{\left (a^2 b^6 (3 A+20 C)-7 a^5 b^3 B+8 a^3 b^5 B+28 a^6 b^2 C-35 a^4 b^4 C+2 a^7 b B-8 a^8 C-8 a b^7 B+2 A b^8\right ) \tanh ^{-1}\left (\frac{\sqrt{a-b} \tan \left (\frac{1}{2} (c+d x)\right )}{\sqrt{a+b}}\right )}{b^5 d (a-b)^{7/2} (a+b)^{7/2}}-\frac{\tan (c+d x) \sec ^3(c+d x) \left (A b^2-a (b B-a C)\right )}{3 b d \left (a^2-b^2\right ) (a+b \sec (c+d x))^3}+\frac{\tan (c+d x) \sec ^2(c+d x) \left (a^2 b^2 (2 A+9 C)+a^3 b B-4 a^4 C-6 a b^3 B+3 A b^4\right )}{6 b^2 d \left (a^2-b^2\right )^2 (a+b \sec (c+d x))^2}+\frac{a \tan (c+d x) \left (3 a^2 b^4 (A+4 C)+2 a^3 b^3 B-11 a^4 b^2 C-a^5 b B+4 a^6 C-6 a b^5 B+2 A b^6\right )}{2 b^4 d \left (a^2-b^2\right )^3 (a+b \sec (c+d x))}+\frac{(b B-4 a C) \tanh ^{-1}(\sin (c+d x))}{b^5 d} \]

Antiderivative was successfully verified.

[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])^4,x]

[Out]

((b*B - 4*a*C)*ArcTanh[Sin[c + d*x]])/(b^5*d) - ((2*A*b^8 + 2*a^7*b*B - 7*a^5*b^3*B + 8*a^3*b^5*B - 8*a*b^7*B
- 8*a^8*C + 28*a^6*b^2*C - 35*a^4*b^4*C + a^2*b^6*(3*A + 20*C))*ArcTanh[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a
+ b]])/((a - b)^(7/2)*b^5*(a + b)^(7/2)*d) - ((5*A*b^4 + 3*a^3*b*B - 8*a*b^3*B - 12*a^4*C + 23*a^2*b^2*C - 6*b
^4*C)*Tan[c + d*x])/(6*b^4*(a^2 - b^2)^2*d) - ((A*b^2 - a*(b*B - a*C))*Sec[c + d*x]^3*Tan[c + d*x])/(3*b*(a^2
- b^2)*d*(a + b*Sec[c + d*x])^3) + ((3*A*b^4 + a^3*b*B - 6*a*b^3*B - 4*a^4*C + a^2*b^2*(2*A + 9*C))*Sec[c + d*
x]^2*Tan[c + d*x])/(6*b^2*(a^2 - b^2)^2*d*(a + b*Sec[c + d*x])^2) + (a*(2*A*b^6 - a^5*b*B + 2*a^3*b^3*B - 6*a*
b^5*B + 4*a^6*C - 11*a^4*b^2*C + 3*a^2*b^4*(A + 4*C))*Tan[c + d*x])/(2*b^4*(a^2 - b^2)^3*d*(a + b*Sec[c + d*x]
))

Rule 4098

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 4090

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

Rule 4082

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 3998

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 3770

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

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{\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))^4} \, dx &=-\frac{\left (A b^2-a (b B-a C)\right ) \sec ^3(c+d x) \tan (c+d x)}{3 b \left (a^2-b^2\right ) d (a+b \sec (c+d x))^3}-\frac{\int \frac{\sec ^3(c+d x) \left (3 \left (A b^2-a (b B-a C)\right )+3 b (b B-a (A+C)) \sec (c+d x)-\left (A b^2-a b B+4 a^2 C-3 b^2 C\right ) \sec ^2(c+d x)\right )}{(a+b \sec (c+d x))^3} \, dx}{3 b \left (a^2-b^2\right )}\\ &=-\frac{\left (A b^2-a (b B-a C)\right ) \sec ^3(c+d x) \tan (c+d x)}{3 b \left (a^2-b^2\right ) d (a+b \sec (c+d x))^3}+\frac{\left (3 A b^4+a^3 b B-6 a b^3 B-4 a^4 C+a^2 b^2 (2 A+9 C)\right ) \sec ^2(c+d x) \tan (c+d x)}{6 b^2 \left (a^2-b^2\right )^2 d (a+b \sec (c+d x))^2}+\frac{\int \frac{\sec ^2(c+d x) \left (2 \left (3 A b^4+a^3 b B-6 a b^3 B-4 a^4 C+a^2 b^2 (2 A+9 C)\right )+2 b \left (2 a^2 b B+3 b^3 B+a^3 C-a b^2 (5 A+6 C)\right ) \sec (c+d x)-\left (5 A b^4+3 a^3 b B-8 a b^3 B-12 a^4 C+23 a^2 b^2 C-6 b^4 C\right ) \sec ^2(c+d x)\right )}{(a+b \sec (c+d x))^2} \, dx}{6 b^2 \left (a^2-b^2\right )^2}\\ &=-\frac{\left (A b^2-a (b B-a C)\right ) \sec ^3(c+d x) \tan (c+d x)}{3 b \left (a^2-b^2\right ) d (a+b \sec (c+d x))^3}+\frac{\left (3 A b^4+a^3 b B-6 a b^3 B-4 a^4 C+a^2 b^2 (2 A+9 C)\right ) \sec ^2(c+d x) \tan (c+d x)}{6 b^2 \left (a^2-b^2\right )^2 d (a+b \sec (c+d x))^2}+\frac{a \left (2 A b^6-a^5 b B+2 a^3 b^3 B-6 a b^5 B+4 a^6 C-11 a^4 b^2 C+3 a^2 b^4 (A+4 C)\right ) \tan (c+d x)}{2 b^4 \left (a^2-b^2\right )^3 d (a+b \sec (c+d x))}+\frac{\int \frac{\sec (c+d x) \left (-3 b \left (2 A b^6-a^5 b B+2 a^3 b^3 B-6 a b^5 B+4 a^6 C-11 a^4 b^2 C+3 a^2 b^4 (A+4 C)\right )+\left (a^2-b^2\right ) \left (3 a^4 b B-4 a^2 b^3 B+6 b^5 B-12 a^5 C+25 a^3 b^2 C-a b^4 (5 A+18 C)\right ) \sec (c+d x)-b \left (a^2-b^2\right ) \left (5 A b^4+3 a^3 b B-8 a b^3 B-12 a^4 C+23 a^2 b^2 C-6 b^4 C\right ) \sec ^2(c+d x)\right )}{a+b \sec (c+d x)} \, dx}{6 b^4 \left (a^2-b^2\right )^3}\\ &=-\frac{\left (5 A b^4+3 a^3 b B-8 a b^3 B-12 a^4 C+23 a^2 b^2 C-6 b^4 C\right ) \tan (c+d x)}{6 b^4 \left (a^2-b^2\right )^2 d}-\frac{\left (A b^2-a (b B-a C)\right ) \sec ^3(c+d x) \tan (c+d x)}{3 b \left (a^2-b^2\right ) d (a+b \sec (c+d x))^3}+\frac{\left (3 A b^4+a^3 b B-6 a b^3 B-4 a^4 C+a^2 b^2 (2 A+9 C)\right ) \sec ^2(c+d x) \tan (c+d x)}{6 b^2 \left (a^2-b^2\right )^2 d (a+b \sec (c+d x))^2}+\frac{a \left (2 A b^6-a^5 b B+2 a^3 b^3 B-6 a b^5 B+4 a^6 C-11 a^4 b^2 C+3 a^2 b^4 (A+4 C)\right ) \tan (c+d x)}{2 b^4 \left (a^2-b^2\right )^3 d (a+b \sec (c+d x))}+\frac{\int \frac{\sec (c+d x) \left (-3 b^2 \left (2 A b^6-a^5 b B+2 a^3 b^3 B-6 a b^5 B+4 a^6 C-11 a^4 b^2 C+3 a^2 b^4 (A+4 C)\right )+6 b \left (a^2-b^2\right )^3 (b B-4 a C) \sec (c+d x)\right )}{a+b \sec (c+d x)} \, dx}{6 b^5 \left (a^2-b^2\right )^3}\\ &=-\frac{\left (5 A b^4+3 a^3 b B-8 a b^3 B-12 a^4 C+23 a^2 b^2 C-6 b^4 C\right ) \tan (c+d x)}{6 b^4 \left (a^2-b^2\right )^2 d}-\frac{\left (A b^2-a (b B-a C)\right ) \sec ^3(c+d x) \tan (c+d x)}{3 b \left (a^2-b^2\right ) d (a+b \sec (c+d x))^3}+\frac{\left (3 A b^4+a^3 b B-6 a b^3 B-4 a^4 C+a^2 b^2 (2 A+9 C)\right ) \sec ^2(c+d x) \tan (c+d x)}{6 b^2 \left (a^2-b^2\right )^2 d (a+b \sec (c+d x))^2}+\frac{a \left (2 A b^6-a^5 b B+2 a^3 b^3 B-6 a b^5 B+4 a^6 C-11 a^4 b^2 C+3 a^2 b^4 (A+4 C)\right ) \tan (c+d x)}{2 b^4 \left (a^2-b^2\right )^3 d (a+b \sec (c+d x))}+\frac{(b B-4 a C) \int \sec (c+d x) \, dx}{b^5}-\frac{\left (2 A b^8+2 a^7 b B-7 a^5 b^3 B+8 a^3 b^5 B-8 a b^7 B-8 a^8 C+28 a^6 b^2 C-35 a^4 b^4 C+a^2 b^6 (3 A+20 C)\right ) \int \frac{\sec (c+d x)}{a+b \sec (c+d x)} \, dx}{2 b^5 \left (a^2-b^2\right )^3}\\ &=\frac{(b B-4 a C) \tanh ^{-1}(\sin (c+d x))}{b^5 d}-\frac{\left (5 A b^4+3 a^3 b B-8 a b^3 B-12 a^4 C+23 a^2 b^2 C-6 b^4 C\right ) \tan (c+d x)}{6 b^4 \left (a^2-b^2\right )^2 d}-\frac{\left (A b^2-a (b B-a C)\right ) \sec ^3(c+d x) \tan (c+d x)}{3 b \left (a^2-b^2\right ) d (a+b \sec (c+d x))^3}+\frac{\left (3 A b^4+a^3 b B-6 a b^3 B-4 a^4 C+a^2 b^2 (2 A+9 C)\right ) \sec ^2(c+d x) \tan (c+d x)}{6 b^2 \left (a^2-b^2\right )^2 d (a+b \sec (c+d x))^2}+\frac{a \left (2 A b^6-a^5 b B+2 a^3 b^3 B-6 a b^5 B+4 a^6 C-11 a^4 b^2 C+3 a^2 b^4 (A+4 C)\right ) \tan (c+d x)}{2 b^4 \left (a^2-b^2\right )^3 d (a+b \sec (c+d x))}-\frac{\left (2 A b^8+2 a^7 b B-7 a^5 b^3 B+8 a^3 b^5 B-8 a b^7 B-8 a^8 C+28 a^6 b^2 C-35 a^4 b^4 C+a^2 b^6 (3 A+20 C)\right ) \int \frac{1}{1+\frac{a \cos (c+d x)}{b}} \, dx}{2 b^6 \left (a^2-b^2\right )^3}\\ &=\frac{(b B-4 a C) \tanh ^{-1}(\sin (c+d x))}{b^5 d}-\frac{\left (5 A b^4+3 a^3 b B-8 a b^3 B-12 a^4 C+23 a^2 b^2 C-6 b^4 C\right ) \tan (c+d x)}{6 b^4 \left (a^2-b^2\right )^2 d}-\frac{\left (A b^2-a (b B-a C)\right ) \sec ^3(c+d x) \tan (c+d x)}{3 b \left (a^2-b^2\right ) d (a+b \sec (c+d x))^3}+\frac{\left (3 A b^4+a^3 b B-6 a b^3 B-4 a^4 C+a^2 b^2 (2 A+9 C)\right ) \sec ^2(c+d x) \tan (c+d x)}{6 b^2 \left (a^2-b^2\right )^2 d (a+b \sec (c+d x))^2}+\frac{a \left (2 A b^6-a^5 b B+2 a^3 b^3 B-6 a b^5 B+4 a^6 C-11 a^4 b^2 C+3 a^2 b^4 (A+4 C)\right ) \tan (c+d x)}{2 b^4 \left (a^2-b^2\right )^3 d (a+b \sec (c+d x))}-\frac{\left (2 A b^8+2 a^7 b B-7 a^5 b^3 B+8 a^3 b^5 B-8 a b^7 B-8 a^8 C+28 a^6 b^2 C-35 a^4 b^4 C+a^2 b^6 (3 A+20 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 )}{b^6 \left (a^2-b^2\right )^3 d}\\ &=\frac{(b B-4 a C) \tanh ^{-1}(\sin (c+d x))}{b^5 d}-\frac{\left (3 a^2 A b^6+2 A b^8+2 a^7 b B-7 a^5 b^3 B+8 a^3 b^5 B-8 a b^7 B-8 a^8 C+28 a^6 b^2 C-35 a^4 b^4 C+20 a^2 b^6 C\right ) \tanh ^{-1}\left (\frac{\sqrt{a-b} \tan \left (\frac{1}{2} (c+d x)\right )}{\sqrt{a+b}}\right )}{(a-b)^{7/2} b^5 (a+b)^{7/2} d}-\frac{\left (5 A b^4+3 a^3 b B-8 a b^3 B-12 a^4 C+23 a^2 b^2 C-6 b^4 C\right ) \tan (c+d x)}{6 b^4 \left (a^2-b^2\right )^2 d}-\frac{\left (A b^2-a (b B-a C)\right ) \sec ^3(c+d x) \tan (c+d x)}{3 b \left (a^2-b^2\right ) d (a+b \sec (c+d x))^3}+\frac{\left (3 A b^4+a^3 b B-6 a b^3 B-4 a^4 C+a^2 b^2 (2 A+9 C)\right ) \sec ^2(c+d x) \tan (c+d x)}{6 b^2 \left (a^2-b^2\right )^2 d (a+b \sec (c+d x))^2}+\frac{a \left (2 A b^6-a^5 b B+2 a^3 b^3 B-6 a b^5 B+4 a^6 C-11 a^4 b^2 C+3 a^2 b^4 (A+4 C)\right ) \tan (c+d x)}{2 b^4 \left (a^2-b^2\right )^3 d (a+b \sec (c+d x))}\\ \end{align*}

Mathematica [B]  time = 6.48735, size = 1197, normalized size = 2.55 \[ -\frac{2 (b B-4 a C) \log \left (\cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )\right ) \sec ^2(c+d x) \left (C \sec ^2(c+d x)+B \sec (c+d x)+A\right ) (b+a \cos (c+d x))^4}{b^5 d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) (a+b \sec (c+d x))^4}+\frac{2 (b B-4 a C) \log \left (\cos \left (\frac{1}{2} (c+d x)\right )+\sin \left (\frac{1}{2} (c+d x)\right )\right ) \sec ^2(c+d x) \left (C \sec ^2(c+d x)+B \sec (c+d x)+A\right ) (b+a \cos (c+d x))^4}{b^5 d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) (a+b \sec (c+d x))^4}-\frac{2 \left (-8 C a^8+2 b B a^7+28 b^2 C a^6-7 b^3 B a^5-35 b^4 C a^4+8 b^5 B a^3+3 A b^6 a^2+20 b^6 C a^2-8 b^7 B a+2 A b^8\right ) \tanh ^{-1}\left (\frac{(b-a) \tan \left (\frac{1}{2} (c+d x)\right )}{\sqrt{a^2-b^2}}\right ) \sec ^2(c+d x) \left (C \sec ^2(c+d x)+B \sec (c+d x)+A\right ) (b+a \cos (c+d x))^4}{b^5 \sqrt{a^2-b^2} \left (b^2-a^2\right )^3 d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) (a+b \sec (c+d x))^4}+\frac{\sec ^3(c+d x) \left (C \sec ^2(c+d x)+B \sec (c+d x)+A\right ) \left (-48 C \sin (2 (c+d x)) a^9-24 C \sin (4 (c+d x)) a^9-120 b C \sin (c+d x) a^8+12 b B \sin (2 (c+d x)) a^8-120 b C \sin (3 (c+d x)) a^8+6 b B \sin (4 (c+d x)) a^8+30 b^2 B \sin (c+d x) a^7-40 b^2 C \sin (2 (c+d x)) a^7+30 b^2 B \sin (3 (c+d x)) a^7+68 b^2 C \sin (4 (c+d x)) a^7+294 b^3 C \sin (c+d x) a^6+10 b^3 B \sin (2 (c+d x)) a^6+342 b^3 C \sin (3 (c+d x)) a^6-17 b^3 B \sin (4 (c+d x)) a^6-90 b^4 B \sin (c+d x) a^5-16 A b^4 \sin (2 (c+d x)) a^5+370 b^4 C \sin (2 (c+d x)) a^5-90 b^4 B \sin (3 (c+d x)) a^5-4 A b^4 \sin (4 (c+d x)) a^5-65 b^4 C \sin (4 (c+d x)) a^5-6 A b^5 \sin (c+d x) a^4-174 b^5 C \sin (c+d x) a^4-76 b^5 B \sin (2 (c+d x)) a^4-6 A b^5 \sin (3 (c+d x)) a^4-318 b^5 C \sin (3 (c+d x)) a^4+26 b^5 B \sin (4 (c+d x)) a^4+120 b^6 B \sin (c+d x) a^3-2 A b^6 \sin (2 (c+d x)) a^3-444 b^6 C \sin (2 (c+d x)) a^3+120 b^6 B \sin (3 (c+d x)) a^3-11 A b^6 \sin (4 (c+d x)) a^3+6 b^6 C \sin (4 (c+d x)) a^3-54 A b^7 \sin (c+d x) a^2-108 b^7 C \sin (c+d x) a^2+144 b^7 B \sin (2 (c+d x)) a^2-54 A b^7 \sin (3 (c+d x)) a^2+36 b^7 C \sin (3 (c+d x)) a^2-72 A b^8 \sin (2 (c+d x)) a+72 b^8 C \sin (2 (c+d x)) a+48 b^9 C \sin (c+d x)\right ) (b+a \cos (c+d x))}{24 b^4 \left (b^2-a^2\right )^3 d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) (a+b \sec (c+d x))^4} \]

Antiderivative was successfully verified.

[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])^4,x]

[Out]

(-2*(3*a^2*A*b^6 + 2*A*b^8 + 2*a^7*b*B - 7*a^5*b^3*B + 8*a^3*b^5*B - 8*a*b^7*B - 8*a^8*C + 28*a^6*b^2*C - 35*a
^4*b^4*C + 20*a^2*b^6*C)*ArcTanh[((-a + b)*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]]*(b + a*Cos[c + d*x])^4*Sec[c + d
*x]^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(b^5*Sqrt[a^2 - b^2]*(-a^2 + b^2)^3*d*(A + 2*C + 2*B*Cos[c + d*
x] + A*Cos[2*c + 2*d*x])*(a + b*Sec[c + d*x])^4) - (2*(b*B - 4*a*C)*(b + a*Cos[c + d*x])^4*Log[Cos[(c + d*x)/2
] - Sin[(c + d*x)/2]]*Sec[c + d*x]^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(b^5*d*(A + 2*C + 2*B*Cos[c + d*
x] + A*Cos[2*c + 2*d*x])*(a + b*Sec[c + d*x])^4) + (2*(b*B - 4*a*C)*(b + a*Cos[c + d*x])^4*Log[Cos[(c + d*x)/2
] + Sin[(c + d*x)/2]]*Sec[c + d*x]^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(b^5*d*(A + 2*C + 2*B*Cos[c + d*
x] + A*Cos[2*c + 2*d*x])*(a + b*Sec[c + d*x])^4) + ((b + a*Cos[c + d*x])*Sec[c + d*x]^3*(A + B*Sec[c + d*x] +
C*Sec[c + d*x]^2)*(-6*a^4*A*b^5*Sin[c + d*x] - 54*a^2*A*b^7*Sin[c + d*x] + 30*a^7*b^2*B*Sin[c + d*x] - 90*a^5*
b^4*B*Sin[c + d*x] + 120*a^3*b^6*B*Sin[c + d*x] - 120*a^8*b*C*Sin[c + d*x] + 294*a^6*b^3*C*Sin[c + d*x] - 174*
a^4*b^5*C*Sin[c + d*x] - 108*a^2*b^7*C*Sin[c + d*x] + 48*b^9*C*Sin[c + d*x] - 16*a^5*A*b^4*Sin[2*(c + d*x)] -
2*a^3*A*b^6*Sin[2*(c + d*x)] - 72*a*A*b^8*Sin[2*(c + d*x)] + 12*a^8*b*B*Sin[2*(c + d*x)] + 10*a^6*b^3*B*Sin[2*
(c + d*x)] - 76*a^4*b^5*B*Sin[2*(c + d*x)] + 144*a^2*b^7*B*Sin[2*(c + d*x)] - 48*a^9*C*Sin[2*(c + d*x)] - 40*a
^7*b^2*C*Sin[2*(c + d*x)] + 370*a^5*b^4*C*Sin[2*(c + d*x)] - 444*a^3*b^6*C*Sin[2*(c + d*x)] + 72*a*b^8*C*Sin[2
*(c + d*x)] - 6*a^4*A*b^5*Sin[3*(c + d*x)] - 54*a^2*A*b^7*Sin[3*(c + d*x)] + 30*a^7*b^2*B*Sin[3*(c + d*x)] - 9
0*a^5*b^4*B*Sin[3*(c + d*x)] + 120*a^3*b^6*B*Sin[3*(c + d*x)] - 120*a^8*b*C*Sin[3*(c + d*x)] + 342*a^6*b^3*C*S
in[3*(c + d*x)] - 318*a^4*b^5*C*Sin[3*(c + d*x)] + 36*a^2*b^7*C*Sin[3*(c + d*x)] - 4*a^5*A*b^4*Sin[4*(c + d*x)
] - 11*a^3*A*b^6*Sin[4*(c + d*x)] + 6*a^8*b*B*Sin[4*(c + d*x)] - 17*a^6*b^3*B*Sin[4*(c + d*x)] + 26*a^4*b^5*B*
Sin[4*(c + d*x)] - 24*a^9*C*Sin[4*(c + d*x)] + 68*a^7*b^2*C*Sin[4*(c + d*x)] - 65*a^5*b^4*C*Sin[4*(c + d*x)] +
6*a^3*b^6*C*Sin[4*(c + d*x)]))/(24*b^4*(-a^2 + b^2)^3*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a
+ b*Sec[c + d*x])^4)

________________________________________________________________________________________

Maple [B]  time = 0.117, size = 3764, normalized size = 8. \begin{align*} \text{output too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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))^4,x)

[Out]

8/d*b^2/(a^6-3*a^4*b^2+3*a^2*b^4-b^6)/((a+b)*(a-b))^(1/2)*arctanh((a-b)*tan(1/2*d*x+1/2*c)/((a+b)*(a-b))^(1/2)
)*B*a-28/d/b^3/(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))*a^6*C+35/d/b/(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))*a^4*C+12/d*b/(tan(1/2*d*x+1/2*c)^2*a-tan(1/2*d*x+1/2*c)^2*b-a-b)^3*a^2/(a-b)/(a^3+3*a^2*b+3*a*
b^2+b^3)*tan(1/2*d*x+1/2*c)^5*B-3/d*b/(tan(1/2*d*x+1/2*c)^2*a-tan(1/2*d*x+1/2*c)^2*b-a-b)^3*a^2/(a-b)/(a^3+3*a
^2*b+3*a*b^2+b^3)*tan(1/2*d*x+1/2*c)^5*A-6/d*b^2/(tan(1/2*d*x+1/2*c)^2*a-tan(1/2*d*x+1/2*c)^2*b-a-b)^3*a/(a-b)
/(a^3+3*a^2*b+3*a*b^2+b^3)*tan(1/2*d*x+1/2*c)^5*A+2/d/b^3/(tan(1/2*d*x+1/2*c)^2*a-tan(1/2*d*x+1/2*c)^2*b-a-b)^
3*a^6/(a-b)/(a^3+3*a^2*b+3*a*b^2+b^3)*tan(1/2*d*x+1/2*c)^5*B+44/3/d/b/(tan(1/2*d*x+1/2*c)^2*a-tan(1/2*d*x+1/2*
c)^2*b-a-b)^3*a^4/(a^2-2*a*b+b^2)/(a^2+2*a*b+b^2)*tan(1/2*d*x+1/2*c)^3*B-24/d*b/(tan(1/2*d*x+1/2*c)^2*a-tan(1/
2*d*x+1/2*c)^2*b-a-b)^3*a^2/(a^2-2*a*b+b^2)/(a^2+2*a*b+b^2)*tan(1/2*d*x+1/2*c)^3*B+12/d/b^4/(tan(1/2*d*x+1/2*c
)^2*a-tan(1/2*d*x+1/2*c)^2*b-a-b)^3*a^7/(a^2-2*a*b+b^2)/(a^2+2*a*b+b^2)*tan(1/2*d*x+1/2*c)^3*C+1/d/b^2/(tan(1/
2*d*x+1/2*c)^2*a-tan(1/2*d*x+1/2*c)^2*b-a-b)^3*a^5/(a+b)/(a^3-3*a^2*b+3*a*b^2-b^3)*tan(1/2*d*x+1/2*c)*B+18/d/b
^2/(tan(1/2*d*x+1/2*c)^2*a-tan(1/2*d*x+1/2*c)^2*b-a-b)^3*a^5/(a-b)/(a^3+3*a^2*b+3*a*b^2+b^3)*tan(1/2*d*x+1/2*c
)^5*C-5/d/b/(tan(1/2*d*x+1/2*c)^2*a-tan(1/2*d*x+1/2*c)^2*b-a-b)^3*a^4/(a-b)/(a^3+3*a^2*b+3*a*b^2+b^3)*tan(1/2*
d*x+1/2*c)^5*C-1/d/b^2/(tan(1/2*d*x+1/2*c)^2*a-tan(1/2*d*x+1/2*c)^2*b-a-b)^3*a^5/(a-b)/(a^3+3*a^2*b+3*a*b^2+b^
3)*tan(1/2*d*x+1/2*c)^5*B-6/d/b/(tan(1/2*d*x+1/2*c)^2*a-tan(1/2*d*x+1/2*c)^2*b-a-b)^3*a^4/(a-b)/(a^3+3*a^2*b+3
*a*b^2+b^3)*tan(1/2*d*x+1/2*c)^5*B-6/d/b/(tan(1/2*d*x+1/2*c)^2*a-tan(1/2*d*x+1/2*c)^2*b-a-b)^3*a^4/(a+b)/(a^3-
3*a^2*b+3*a*b^2-b^3)*tan(1/2*d*x+1/2*c)*B-2/d/b^3/(tan(1/2*d*x+1/2*c)^2*a-tan(1/2*d*x+1/2*c)^2*b-a-b)^3*a^6/(a
+b)/(a^3-3*a^2*b+3*a*b^2-b^3)*tan(1/2*d*x+1/2*c)*C+18/d/b^2/(tan(1/2*d*x+1/2*c)^2*a-tan(1/2*d*x+1/2*c)^2*b-a-b
)^3*a^5/(a+b)/(a^3-3*a^2*b+3*a*b^2-b^3)*tan(1/2*d*x+1/2*c)*C+5/d/b/(tan(1/2*d*x+1/2*c)^2*a-tan(1/2*d*x+1/2*c)^
2*b-a-b)^3*a^4/(a+b)/(a^3-3*a^2*b+3*a*b^2-b^3)*tan(1/2*d*x+1/2*c)*C+12/d*b/(tan(1/2*d*x+1/2*c)^2*a-tan(1/2*d*x
+1/2*c)^2*b-a-b)^3*a^2/(a+b)/(a^3-3*a^2*b+3*a*b^2-b^3)*tan(1/2*d*x+1/2*c)*B-6/d*b^2/(tan(1/2*d*x+1/2*c)^2*a-ta
n(1/2*d*x+1/2*c)^2*b-a-b)^3*a/(a+b)/(a^3-3*a^2*b+3*a*b^2-b^3)*tan(1/2*d*x+1/2*c)*A+3/d*b/(tan(1/2*d*x+1/2*c)^2
*a-tan(1/2*d*x+1/2*c)^2*b-a-b)^3*a^2/(a+b)/(a^3-3*a^2*b+3*a*b^2-b^3)*tan(1/2*d*x+1/2*c)*A-6/d/b^4/(tan(1/2*d*x
+1/2*c)^2*a-tan(1/2*d*x+1/2*c)^2*b-a-b)^3*a^7/(a+b)/(a^3-3*a^2*b+3*a*b^2-b^3)*tan(1/2*d*x+1/2*c)*C-116/3/d/b^2
/(tan(1/2*d*x+1/2*c)^2*a-tan(1/2*d*x+1/2*c)^2*b-a-b)^3*a^5/(a^2-2*a*b+b^2)/(a^2+2*a*b+b^2)*tan(1/2*d*x+1/2*c)^
3*C-6/d/b^4/(tan(1/2*d*x+1/2*c)^2*a-tan(1/2*d*x+1/2*c)^2*b-a-b)^3*a^7/(a-b)/(a^3+3*a^2*b+3*a*b^2+b^3)*tan(1/2*
d*x+1/2*c)^5*C+2/d/b^3/(tan(1/2*d*x+1/2*c)^2*a-tan(1/2*d*x+1/2*c)^2*b-a-b)^3*a^6/(a-b)/(a^3+3*a^2*b+3*a*b^2+b^
3)*tan(1/2*d*x+1/2*c)^5*C+2/d/b^3/(tan(1/2*d*x+1/2*c)^2*a-tan(1/2*d*x+1/2*c)^2*b-a-b)^3*a^6/(a+b)/(a^3-3*a^2*b
+3*a*b^2-b^3)*tan(1/2*d*x+1/2*c)*B+12/d*b^2/(tan(1/2*d*x+1/2*c)^2*a-tan(1/2*d*x+1/2*c)^2*b-a-b)^3*a/(a^2-2*a*b
+b^2)/(a^2+2*a*b+b^2)*tan(1/2*d*x+1/2*c)^3*A-4/d/b^3/(tan(1/2*d*x+1/2*c)^2*a-tan(1/2*d*x+1/2*c)^2*b-a-b)^3*a^6
/(a^2-2*a*b+b^2)/(a^2+2*a*b+b^2)*tan(1/2*d*x+1/2*c)^3*B+8/d/b^5/(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))*a^8*C-3/d*b/(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))*A*a^2-2/d/(tan(1/2*d*x+1/2*c)^2*a-tan(1/2*d*x+
1/2*c)^2*b-a-b)^3*a^3/(a-b)/(a^3+3*a^2*b+3*a*b^2+b^3)*tan(1/2*d*x+1/2*c)^5*A+4/d/(tan(1/2*d*x+1/2*c)^2*a-tan(1
/2*d*x+1/2*c)^2*b-a-b)^3*a^3/(a-b)/(a^3+3*a^2*b+3*a*b^2+b^3)*tan(1/2*d*x+1/2*c)^5*B-20/d/(tan(1/2*d*x+1/2*c)^2
*a-tan(1/2*d*x+1/2*c)^2*b-a-b)^3*a^3/(a+b)/(a^3-3*a^2*b+3*a*b^2-b^3)*tan(1/2*d*x+1/2*c)*C-20/d/(tan(1/2*d*x+1/
2*c)^2*a-tan(1/2*d*x+1/2*c)^2*b-a-b)^3*a^3/(a-b)/(a^3+3*a^2*b+3*a*b^2+b^3)*tan(1/2*d*x+1/2*c)^5*C+4/3/d/(tan(1
/2*d*x+1/2*c)^2*a-tan(1/2*d*x+1/2*c)^2*b-a-b)^3*a^3/(a^2-2*a*b+b^2)/(a^2+2*a*b+b^2)*tan(1/2*d*x+1/2*c)^3*A-4/d
/(tan(1/2*d*x+1/2*c)^2*a-tan(1/2*d*x+1/2*c)^2*b-a-b)^3*a^3/(a+b)/(a^3-3*a^2*b+3*a*b^2-b^3)*tan(1/2*d*x+1/2*c)*
B-1/d/b^4*ln(tan(1/2*d*x+1/2*c)-1)*B-1/d*C/b^4/(tan(1/2*d*x+1/2*c)+1)+1/d/b^4*ln(tan(1/2*d*x+1/2*c)+1)*B-1/d*C
/b^4/(tan(1/2*d*x+1/2*c)-1)-20/d*b/(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))*C*a^2+40/d/(tan(1/2*d*x+1/2*c)^2*a-tan(1/2*d*x+1/2*c)^2*b-a-b)^3*a^3/(a^2-2*a*b+b
^2)/(a^2+2*a*b+b^2)*tan(1/2*d*x+1/2*c)^3*C-2/d/(tan(1/2*d*x+1/2*c)^2*a-tan(1/2*d*x+1/2*c)^2*b-a-b)^3*a^3/(a+b)
/(a^3-3*a^2*b+3*a*b^2-b^3)*tan(1/2*d*x+1/2*c)*A-2/d/b^4/(a^6-3*a^4*b^2+3*a^2*b^4-b^6)/((a+b)*(a-b))^(1/2)*arct
anh((a-b)*tan(1/2*d*x+1/2*c)/((a+b)*(a-b))^(1/2))*B*a^7+7/d/b^2/(a^6-3*a^4*b^2+3*a^2*b^4-b^6)/((a+b)*(a-b))^(1
/2)*arctanh((a-b)*tan(1/2*d*x+1/2*c)/((a+b)*(a-b))^(1/2))*B*a^5-4/d/b^5*ln(tan(1/2*d*x+1/2*c)+1)*a*C-2/d*b^3/(
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))*A-8/d/(
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))*B*a^3+4
/d/b^5*ln(tan(1/2*d*x+1/2*c)-1)*a*C

________________________________________________________________________________________

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

[Out]

Exception raised: ValueError

________________________________________________________________________________________

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

[Out]

Timed out

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \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 )^{4}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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))**4,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))**4, x)

________________________________________________________________________________________

Giac [B]  time = 1.53547, size = 1706, normalized size = 3.63 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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))^4,x, algorithm="giac")

[Out]

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