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

[Out]

(B*b-4*C*a)*arctanh(sin(d*x+c))/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((a-b)^(1/2)*tan(1/2*d*x+1/2*c)/(a+b)^(1/2))/(a-b)^(7/2)/b^5/(a+b)^(
7/2)/d-1/6*(5*A*b^4+3*B*a^3*b-8*B*a*b^3-12*C*a^4+23*C*a^2*b^2-6*C*b^4)*tan(d*x+c)/b^4/(a^2-b^2)^2/d-1/3*(A*b^2
-a*(B*b-C*a))*sec(d*x+c)^3*tan(d*x+c)/b/(a^2-b^2)/d/(a+b*sec(d*x+c))^3+1/6*(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(d*x+c)^2*tan(d*x+c)/b^2/(a^2-b^2)^2/d/(a+b*sec(d*x+c))^2+1/2*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(d*x+c)/b^4/(a^2-b^2)^3/d/(a+b*sec(d*x+c))

________________________________________________________________________________________

Rubi [A]  time = 9.91, antiderivative size = 470, normalized size of antiderivative = 1.00, 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 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]

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 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 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 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 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 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]

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*}

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Mathematica [B]  time = 6.49, 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)

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fricas [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

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

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giac [B]  time = 0.50, size = 1264, normalized size = 2.69 \[ \text {result too large to display} \]

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

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maple [B]  time = 0.61, size = 3764, normalized size = 8.01 \[ \text {output too large to display} \]

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

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maxima [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: ValueError} \]

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 >> 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 details)Is 4*a^2-4*b^2 positive or negative?

________________________________________________________________________________________

mupad [B]  time = 20.92, size = 15959, normalized size = 33.96 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(atan(((((8*tan(c/2 + (d*x)/2)*(4*A^2*b^16 + 4*B^2*b^16 + 128*C^2*a^16 - 8*B^2*a*b^15 - 128*C^2*a^15*b + 12*A^
2*a^2*b^14 + 9*A^2*a^4*b^12 + 44*B^2*a^2*b^14 + 48*B^2*a^3*b^13 - 92*B^2*a^4*b^12 - 120*B^2*a^5*b^11 + 156*B^2
*a^6*b^10 + 160*B^2*a^7*b^9 - 164*B^2*a^8*b^8 - 120*B^2*a^9*b^7 + 117*B^2*a^10*b^6 + 48*B^2*a^11*b^5 - 48*B^2*
a^12*b^4 - 8*B^2*a^13*b^3 + 8*B^2*a^14*b^2 + 64*C^2*a^2*b^14 - 128*C^2*a^3*b^13 + 80*C^2*a^4*b^12 + 768*C^2*a^
5*b^11 - 824*C^2*a^6*b^10 - 1920*C^2*a^7*b^9 + 2025*C^2*a^8*b^8 + 2560*C^2*a^9*b^7 - 2600*C^2*a^10*b^6 - 1920*
C^2*a^11*b^5 + 1920*C^2*a^12*b^4 + 768*C^2*a^13*b^3 - 768*C^2*a^14*b^2 - 32*A*B*a*b^15 - 32*B*C*a*b^15 - 64*B*
C*a^15*b - 16*A*B*a^3*b^13 + 20*A*B*a^5*b^11 - 34*A*B*a^7*b^9 + 12*A*B*a^9*b^7 + 80*A*C*a^2*b^14 - 20*A*C*a^4*
b^12 - 98*A*C*a^6*b^10 + 136*A*C*a^8*b^8 - 48*A*C*a^10*b^6 + 64*B*C*a^2*b^14 - 160*B*C*a^3*b^13 - 384*B*C*a^4*
b^12 + 592*B*C*a^5*b^11 + 960*B*C*a^6*b^10 - 1128*B*C*a^7*b^9 - 1280*B*C*a^8*b^8 + 1306*B*C*a^9*b^7 + 960*B*C*
a^10*b^6 - 948*B*C*a^11*b^5 - 384*B*C*a^12*b^4 + 384*B*C*a^13*b^3 + 64*B*C*a^14*b^2))/(a*b^18 + b^19 - 5*a^2*b
^17 - 5*a^3*b^16 + 10*a^4*b^15 + 10*a^5*b^14 - 10*a^6*b^13 - 10*a^7*b^12 + 5*a^8*b^11 + 5*a^9*b^10 - a^10*b^9
- a^11*b^8) + (((8*(4*A*b^24 + 4*B*b^24 - 6*A*a^2*b^22 + 6*A*a^3*b^21 - 6*A*a^4*b^20 + 6*A*a^5*b^19 + 14*A*a^6
*b^18 - 14*A*a^7*b^17 - 6*A*a^8*b^16 + 6*A*a^9*b^15 - 12*B*a^2*b^22 + 64*B*a^3*b^21 + 20*B*a^4*b^20 - 110*B*a^
5*b^19 - 30*B*a^6*b^18 + 110*B*a^7*b^17 + 30*B*a^8*b^16 - 70*B*a^9*b^15 - 14*B*a^10*b^14 + 26*B*a^11*b^13 + 2*
B*a^12*b^12 - 4*B*a^13*b^11 + 40*C*a^2*b^22 + 72*C*a^3*b^21 - 190*C*a^4*b^20 - 146*C*a^5*b^19 + 386*C*a^6*b^18
 + 174*C*a^7*b^17 - 434*C*a^8*b^16 - 126*C*a^9*b^15 + 286*C*a^10*b^14 + 50*C*a^11*b^13 - 104*C*a^12*b^12 - 8*C
*a^13*b^11 + 16*C*a^14*b^10 - 4*A*a*b^23 - 16*B*a*b^23 - 16*C*a*b^23))/(a*b^22 + b^23 - 5*a^2*b^21 - 5*a^3*b^2
0 + 10*a^4*b^19 + 10*a^5*b^18 - 10*a^6*b^17 - 10*a^7*b^16 + 5*a^8*b^15 + 5*a^9*b^14 - a^10*b^13 - a^11*b^12) +
 (8*tan(c/2 + (d*x)/2)*(B*b - 4*C*a)*(8*a*b^23 - 8*a^2*b^22 - 48*a^3*b^21 + 48*a^4*b^20 + 120*a^5*b^19 - 120*a
^6*b^18 - 160*a^7*b^17 + 160*a^8*b^16 + 120*a^9*b^15 - 120*a^10*b^14 - 48*a^11*b^13 + 48*a^12*b^12 + 8*a^13*b^
11 - 8*a^14*b^10))/(b^5*(a*b^18 + b^19 - 5*a^2*b^17 - 5*a^3*b^16 + 10*a^4*b^15 + 10*a^5*b^14 - 10*a^6*b^13 - 1
0*a^7*b^12 + 5*a^8*b^11 + 5*a^9*b^10 - a^10*b^9 - a^11*b^8)))*(B*b - 4*C*a))/b^5)*(B*b - 4*C*a)*1i)/b^5 + (((8
*tan(c/2 + (d*x)/2)*(4*A^2*b^16 + 4*B^2*b^16 + 128*C^2*a^16 - 8*B^2*a*b^15 - 128*C^2*a^15*b + 12*A^2*a^2*b^14
+ 9*A^2*a^4*b^12 + 44*B^2*a^2*b^14 + 48*B^2*a^3*b^13 - 92*B^2*a^4*b^12 - 120*B^2*a^5*b^11 + 156*B^2*a^6*b^10 +
 160*B^2*a^7*b^9 - 164*B^2*a^8*b^8 - 120*B^2*a^9*b^7 + 117*B^2*a^10*b^6 + 48*B^2*a^11*b^5 - 48*B^2*a^12*b^4 -
8*B^2*a^13*b^3 + 8*B^2*a^14*b^2 + 64*C^2*a^2*b^14 - 128*C^2*a^3*b^13 + 80*C^2*a^4*b^12 + 768*C^2*a^5*b^11 - 82
4*C^2*a^6*b^10 - 1920*C^2*a^7*b^9 + 2025*C^2*a^8*b^8 + 2560*C^2*a^9*b^7 - 2600*C^2*a^10*b^6 - 1920*C^2*a^11*b^
5 + 1920*C^2*a^12*b^4 + 768*C^2*a^13*b^3 - 768*C^2*a^14*b^2 - 32*A*B*a*b^15 - 32*B*C*a*b^15 - 64*B*C*a^15*b -
16*A*B*a^3*b^13 + 20*A*B*a^5*b^11 - 34*A*B*a^7*b^9 + 12*A*B*a^9*b^7 + 80*A*C*a^2*b^14 - 20*A*C*a^4*b^12 - 98*A
*C*a^6*b^10 + 136*A*C*a^8*b^8 - 48*A*C*a^10*b^6 + 64*B*C*a^2*b^14 - 160*B*C*a^3*b^13 - 384*B*C*a^4*b^12 + 592*
B*C*a^5*b^11 + 960*B*C*a^6*b^10 - 1128*B*C*a^7*b^9 - 1280*B*C*a^8*b^8 + 1306*B*C*a^9*b^7 + 960*B*C*a^10*b^6 -
948*B*C*a^11*b^5 - 384*B*C*a^12*b^4 + 384*B*C*a^13*b^3 + 64*B*C*a^14*b^2))/(a*b^18 + b^19 - 5*a^2*b^17 - 5*a^3
*b^16 + 10*a^4*b^15 + 10*a^5*b^14 - 10*a^6*b^13 - 10*a^7*b^12 + 5*a^8*b^11 + 5*a^9*b^10 - a^10*b^9 - a^11*b^8)
 - (((8*(4*A*b^24 + 4*B*b^24 - 6*A*a^2*b^22 + 6*A*a^3*b^21 - 6*A*a^4*b^20 + 6*A*a^5*b^19 + 14*A*a^6*b^18 - 14*
A*a^7*b^17 - 6*A*a^8*b^16 + 6*A*a^9*b^15 - 12*B*a^2*b^22 + 64*B*a^3*b^21 + 20*B*a^4*b^20 - 110*B*a^5*b^19 - 30
*B*a^6*b^18 + 110*B*a^7*b^17 + 30*B*a^8*b^16 - 70*B*a^9*b^15 - 14*B*a^10*b^14 + 26*B*a^11*b^13 + 2*B*a^12*b^12
 - 4*B*a^13*b^11 + 40*C*a^2*b^22 + 72*C*a^3*b^21 - 190*C*a^4*b^20 - 146*C*a^5*b^19 + 386*C*a^6*b^18 + 174*C*a^
7*b^17 - 434*C*a^8*b^16 - 126*C*a^9*b^15 + 286*C*a^10*b^14 + 50*C*a^11*b^13 - 104*C*a^12*b^12 - 8*C*a^13*b^11
+ 16*C*a^14*b^10 - 4*A*a*b^23 - 16*B*a*b^23 - 16*C*a*b^23))/(a*b^22 + b^23 - 5*a^2*b^21 - 5*a^3*b^20 + 10*a^4*
b^19 + 10*a^5*b^18 - 10*a^6*b^17 - 10*a^7*b^16 + 5*a^8*b^15 + 5*a^9*b^14 - a^10*b^13 - a^11*b^12) - (8*tan(c/2
 + (d*x)/2)*(B*b - 4*C*a)*(8*a*b^23 - 8*a^2*b^22 - 48*a^3*b^21 + 48*a^4*b^20 + 120*a^5*b^19 - 120*a^6*b^18 - 1
60*a^7*b^17 + 160*a^8*b^16 + 120*a^9*b^15 - 120*a^10*b^14 - 48*a^11*b^13 + 48*a^12*b^12 + 8*a^13*b^11 - 8*a^14
*b^10))/(b^5*(a*b^18 + b^19 - 5*a^2*b^17 - 5*a^3*b^16 + 10*a^4*b^15 + 10*a^5*b^14 - 10*a^6*b^13 - 10*a^7*b^12
+ 5*a^8*b^11 + 5*a^9*b^10 - a^10*b^9 - a^11*b^8)))*(B*b - 4*C*a))/b^5)*(B*b - 4*C*a)*1i)/b^5)/((16*(256*C^3*a^
16 + 4*A*B^2*b^16 - 4*A^2*B*b^16 - 16*B^3*a*b^15 - 128*C^3*a^15*b - 48*B^3*a^2*b^14 + 64*B^3*a^3*b^13 + 64*B^3
*a^4*b^12 - 110*B^3*a^5*b^11 - 66*B^3*a^6*b^10 + 110*B^3*a^7*b^9 + 34*B^3*a^8*b^8 - 70*B^3*a^9*b^7 - 11*B^3*a^
10*b^6 + 26*B^3*a^11*b^5 + 2*B^3*a^12*b^4 - 4*B^3*a^13*b^3 + 640*C^3*a^4*b^12 + 960*C^3*a^5*b^11 - 3040*C^3*a^
6*b^10 - 2560*C^3*a^7*b^9 + 6176*C^3*a^8*b^8 + 3204*C^3*a^9*b^7 - 6944*C^3*a^10*b^6 - 2176*C^3*a^11*b^5 + 4576
*C^3*a^12*b^4 + 800*C^3*a^13*b^3 - 1664*C^3*a^14*b^2 + 28*A*B^2*a*b^15 + 16*A^2*C*a*b^15 - 192*B*C^2*a^15*b -
6*A*B^2*a^2*b^14 + 22*A*B^2*a^3*b^13 - 6*A*B^2*a^4*b^12 - 14*A*B^2*a^5*b^11 + 14*A*B^2*a^6*b^10 + 20*A*B^2*a^7
*b^9 - 6*A*B^2*a^8*b^8 - 6*A*B^2*a^9*b^7 - 12*A^2*B*a^2*b^14 - 9*A^2*B*a^4*b^12 + 64*A*C^2*a^2*b^14 + 256*A*C^
2*a^3*b^13 - 96*A*C^2*a^4*b^12 + 16*A*C^2*a^5*b^11 - 96*A*C^2*a^6*b^10 - 296*A*C^2*a^7*b^9 + 224*A*C^2*a^8*b^8
 + 320*A*C^2*a^9*b^7 - 96*A*C^2*a^10*b^6 - 96*A*C^2*a^11*b^5 + 48*A^2*C*a^3*b^13 + 36*A^2*C*a^5*b^11 - 576*B*C
^2*a^3*b^13 - 1104*B*C^2*a^4*b^12 + 2544*B*C^2*a^5*b^11 + 2376*B*C^2*a^6*b^10 - 4848*B*C^2*a^7*b^9 - 2649*B*C^
2*a^8*b^8 + 5232*B*C^2*a^9*b^7 + 1632*B*C^2*a^10*b^6 - 3408*B*C^2*a^11*b^5 - 576*B*C^2*a^12*b^4 + 1248*B*C^2*a
^13*b^3 + 96*B*C^2*a^14*b^2 + 168*B^2*C*a^2*b^14 + 408*B^2*C*a^3*b^13 - 702*B^2*C*a^4*b^12 - 690*B^2*C*a^5*b^1
1 + 1266*B^2*C*a^6*b^10 + 726*B^2*C*a^7*b^9 - 1314*B^2*C*a^8*b^8 - 408*B^2*C*a^9*b^7 + 846*B^2*C*a^10*b^6 + 13
8*B^2*C*a^11*b^5 - 312*B^2*C*a^12*b^4 - 24*B^2*C*a^13*b^3 + 48*B^2*C*a^14*b^2 - 32*A*B*C*a*b^15 - 176*A*B*C*a^
2*b^14 + 48*A*B*C*a^3*b^13 - 92*A*B*C*a^4*b^12 + 48*A*B*C*a^5*b^11 + 130*A*B*C*a^6*b^10 - 112*A*B*C*a^7*b^9 -
160*A*B*C*a^8*b^8 + 48*A*B*C*a^9*b^7 + 48*A*B*C*a^10*b^6))/(a*b^22 + b^23 - 5*a^2*b^21 - 5*a^3*b^20 + 10*a^4*b
^19 + 10*a^5*b^18 - 10*a^6*b^17 - 10*a^7*b^16 + 5*a^8*b^15 + 5*a^9*b^14 - a^10*b^13 - a^11*b^12) - (((8*tan(c/
2 + (d*x)/2)*(4*A^2*b^16 + 4*B^2*b^16 + 128*C^2*a^16 - 8*B^2*a*b^15 - 128*C^2*a^15*b + 12*A^2*a^2*b^14 + 9*A^2
*a^4*b^12 + 44*B^2*a^2*b^14 + 48*B^2*a^3*b^13 - 92*B^2*a^4*b^12 - 120*B^2*a^5*b^11 + 156*B^2*a^6*b^10 + 160*B^
2*a^7*b^9 - 164*B^2*a^8*b^8 - 120*B^2*a^9*b^7 + 117*B^2*a^10*b^6 + 48*B^2*a^11*b^5 - 48*B^2*a^12*b^4 - 8*B^2*a
^13*b^3 + 8*B^2*a^14*b^2 + 64*C^2*a^2*b^14 - 128*C^2*a^3*b^13 + 80*C^2*a^4*b^12 + 768*C^2*a^5*b^11 - 824*C^2*a
^6*b^10 - 1920*C^2*a^7*b^9 + 2025*C^2*a^8*b^8 + 2560*C^2*a^9*b^7 - 2600*C^2*a^10*b^6 - 1920*C^2*a^11*b^5 + 192
0*C^2*a^12*b^4 + 768*C^2*a^13*b^3 - 768*C^2*a^14*b^2 - 32*A*B*a*b^15 - 32*B*C*a*b^15 - 64*B*C*a^15*b - 16*A*B*
a^3*b^13 + 20*A*B*a^5*b^11 - 34*A*B*a^7*b^9 + 12*A*B*a^9*b^7 + 80*A*C*a^2*b^14 - 20*A*C*a^4*b^12 - 98*A*C*a^6*
b^10 + 136*A*C*a^8*b^8 - 48*A*C*a^10*b^6 + 64*B*C*a^2*b^14 - 160*B*C*a^3*b^13 - 384*B*C*a^4*b^12 + 592*B*C*a^5
*b^11 + 960*B*C*a^6*b^10 - 1128*B*C*a^7*b^9 - 1280*B*C*a^8*b^8 + 1306*B*C*a^9*b^7 + 960*B*C*a^10*b^6 - 948*B*C
*a^11*b^5 - 384*B*C*a^12*b^4 + 384*B*C*a^13*b^3 + 64*B*C*a^14*b^2))/(a*b^18 + b^19 - 5*a^2*b^17 - 5*a^3*b^16 +
 10*a^4*b^15 + 10*a^5*b^14 - 10*a^6*b^13 - 10*a^7*b^12 + 5*a^8*b^11 + 5*a^9*b^10 - a^10*b^9 - a^11*b^8) + (((8
*(4*A*b^24 + 4*B*b^24 - 6*A*a^2*b^22 + 6*A*a^3*b^21 - 6*A*a^4*b^20 + 6*A*a^5*b^19 + 14*A*a^6*b^18 - 14*A*a^7*b
^17 - 6*A*a^8*b^16 + 6*A*a^9*b^15 - 12*B*a^2*b^22 + 64*B*a^3*b^21 + 20*B*a^4*b^20 - 110*B*a^5*b^19 - 30*B*a^6*
b^18 + 110*B*a^7*b^17 + 30*B*a^8*b^16 - 70*B*a^9*b^15 - 14*B*a^10*b^14 + 26*B*a^11*b^13 + 2*B*a^12*b^12 - 4*B*
a^13*b^11 + 40*C*a^2*b^22 + 72*C*a^3*b^21 - 190*C*a^4*b^20 - 146*C*a^5*b^19 + 386*C*a^6*b^18 + 174*C*a^7*b^17
- 434*C*a^8*b^16 - 126*C*a^9*b^15 + 286*C*a^10*b^14 + 50*C*a^11*b^13 - 104*C*a^12*b^12 - 8*C*a^13*b^11 + 16*C*
a^14*b^10 - 4*A*a*b^23 - 16*B*a*b^23 - 16*C*a*b^23))/(a*b^22 + b^23 - 5*a^2*b^21 - 5*a^3*b^20 + 10*a^4*b^19 +
10*a^5*b^18 - 10*a^6*b^17 - 10*a^7*b^16 + 5*a^8*b^15 + 5*a^9*b^14 - a^10*b^13 - a^11*b^12) + (8*tan(c/2 + (d*x
)/2)*(B*b - 4*C*a)*(8*a*b^23 - 8*a^2*b^22 - 48*a^3*b^21 + 48*a^4*b^20 + 120*a^5*b^19 - 120*a^6*b^18 - 160*a^7*
b^17 + 160*a^8*b^16 + 120*a^9*b^15 - 120*a^10*b^14 - 48*a^11*b^13 + 48*a^12*b^12 + 8*a^13*b^11 - 8*a^14*b^10))
/(b^5*(a*b^18 + b^19 - 5*a^2*b^17 - 5*a^3*b^16 + 10*a^4*b^15 + 10*a^5*b^14 - 10*a^6*b^13 - 10*a^7*b^12 + 5*a^8
*b^11 + 5*a^9*b^10 - a^10*b^9 - a^11*b^8)))*(B*b - 4*C*a))/b^5)*(B*b - 4*C*a))/b^5 + (((8*tan(c/2 + (d*x)/2)*(
4*A^2*b^16 + 4*B^2*b^16 + 128*C^2*a^16 - 8*B^2*a*b^15 - 128*C^2*a^15*b + 12*A^2*a^2*b^14 + 9*A^2*a^4*b^12 + 44
*B^2*a^2*b^14 + 48*B^2*a^3*b^13 - 92*B^2*a^4*b^12 - 120*B^2*a^5*b^11 + 156*B^2*a^6*b^10 + 160*B^2*a^7*b^9 - 16
4*B^2*a^8*b^8 - 120*B^2*a^9*b^7 + 117*B^2*a^10*b^6 + 48*B^2*a^11*b^5 - 48*B^2*a^12*b^4 - 8*B^2*a^13*b^3 + 8*B^
2*a^14*b^2 + 64*C^2*a^2*b^14 - 128*C^2*a^3*b^13 + 80*C^2*a^4*b^12 + 768*C^2*a^5*b^11 - 824*C^2*a^6*b^10 - 1920
*C^2*a^7*b^9 + 2025*C^2*a^8*b^8 + 2560*C^2*a^9*b^7 - 2600*C^2*a^10*b^6 - 1920*C^2*a^11*b^5 + 1920*C^2*a^12*b^4
 + 768*C^2*a^13*b^3 - 768*C^2*a^14*b^2 - 32*A*B*a*b^15 - 32*B*C*a*b^15 - 64*B*C*a^15*b - 16*A*B*a^3*b^13 + 20*
A*B*a^5*b^11 - 34*A*B*a^7*b^9 + 12*A*B*a^9*b^7 + 80*A*C*a^2*b^14 - 20*A*C*a^4*b^12 - 98*A*C*a^6*b^10 + 136*A*C
*a^8*b^8 - 48*A*C*a^10*b^6 + 64*B*C*a^2*b^14 - 160*B*C*a^3*b^13 - 384*B*C*a^4*b^12 + 592*B*C*a^5*b^11 + 960*B*
C*a^6*b^10 - 1128*B*C*a^7*b^9 - 1280*B*C*a^8*b^8 + 1306*B*C*a^9*b^7 + 960*B*C*a^10*b^6 - 948*B*C*a^11*b^5 - 38
4*B*C*a^12*b^4 + 384*B*C*a^13*b^3 + 64*B*C*a^14*b^2))/(a*b^18 + b^19 - 5*a^2*b^17 - 5*a^3*b^16 + 10*a^4*b^15 +
 10*a^5*b^14 - 10*a^6*b^13 - 10*a^7*b^12 + 5*a^8*b^11 + 5*a^9*b^10 - a^10*b^9 - a^11*b^8) - (((8*(4*A*b^24 + 4
*B*b^24 - 6*A*a^2*b^22 + 6*A*a^3*b^21 - 6*A*a^4*b^20 + 6*A*a^5*b^19 + 14*A*a^6*b^18 - 14*A*a^7*b^17 - 6*A*a^8*
b^16 + 6*A*a^9*b^15 - 12*B*a^2*b^22 + 64*B*a^3*b^21 + 20*B*a^4*b^20 - 110*B*a^5*b^19 - 30*B*a^6*b^18 + 110*B*a
^7*b^17 + 30*B*a^8*b^16 - 70*B*a^9*b^15 - 14*B*a^10*b^14 + 26*B*a^11*b^13 + 2*B*a^12*b^12 - 4*B*a^13*b^11 + 40
*C*a^2*b^22 + 72*C*a^3*b^21 - 190*C*a^4*b^20 - 146*C*a^5*b^19 + 386*C*a^6*b^18 + 174*C*a^7*b^17 - 434*C*a^8*b^
16 - 126*C*a^9*b^15 + 286*C*a^10*b^14 + 50*C*a^11*b^13 - 104*C*a^12*b^12 - 8*C*a^13*b^11 + 16*C*a^14*b^10 - 4*
A*a*b^23 - 16*B*a*b^23 - 16*C*a*b^23))/(a*b^22 + b^23 - 5*a^2*b^21 - 5*a^3*b^20 + 10*a^4*b^19 + 10*a^5*b^18 -
10*a^6*b^17 - 10*a^7*b^16 + 5*a^8*b^15 + 5*a^9*b^14 - a^10*b^13 - a^11*b^12) - (8*tan(c/2 + (d*x)/2)*(B*b - 4*
C*a)*(8*a*b^23 - 8*a^2*b^22 - 48*a^3*b^21 + 48*a^4*b^20 + 120*a^5*b^19 - 120*a^6*b^18 - 160*a^7*b^17 + 160*a^8
*b^16 + 120*a^9*b^15 - 120*a^10*b^14 - 48*a^11*b^13 + 48*a^12*b^12 + 8*a^13*b^11 - 8*a^14*b^10))/(b^5*(a*b^18
+ b^19 - 5*a^2*b^17 - 5*a^3*b^16 + 10*a^4*b^15 + 10*a^5*b^14 - 10*a^6*b^13 - 10*a^7*b^12 + 5*a^8*b^11 + 5*a^9*
b^10 - a^10*b^9 - a^11*b^8)))*(B*b - 4*C*a))/b^5)*(B*b - 4*C*a))/b^5))*(B*b - 4*C*a)*2i)/(b^5*d) - ((tan(c/2 +
 (d*x)/2)^7*(8*C*a^7 + 2*C*b^7 + 3*A*a^2*b^5 + 2*A*a^3*b^4 - 12*B*a^2*b^5 - 4*B*a^3*b^4 + 6*B*a^4*b^3 + B*a^5*
b^2 - 6*C*a^2*b^5 + 26*C*a^3*b^4 + 11*C*a^4*b^3 - 24*C*a^5*b^2 + 6*A*a*b^6 - 2*B*a^6*b - 2*C*a*b^6 - 4*C*a^6*b
))/(b^4*(a + b)^3*(a - b)) - (tan(c/2 + (d*x)/2)*(8*C*a^7 - 2*C*b^7 - 3*A*a^2*b^5 + 2*A*a^3*b^4 - 12*B*a^2*b^5
 + 4*B*a^3*b^4 + 6*B*a^4*b^3 - B*a^5*b^2 + 6*C*a^2*b^5 + 26*C*a^3*b^4 - 11*C*a^4*b^3 - 24*C*a^5*b^2 + 6*A*a*b^
6 - 2*B*a^6*b - 2*C*a*b^6 + 4*C*a^6*b))/(b^4*(a + b)*(a - b)^3) + (tan(c/2 + (d*x)/2)^3*(72*C*a^8 + 18*C*b^8 +
 45*A*a^2*b^6 - 7*A*a^3*b^5 + 10*A*a^4*b^4 + 36*B*a^2*b^6 - 96*B*a^3*b^5 - 14*B*a^4*b^4 + 59*B*a^5*b^3 + 3*B*a
^6*b^2 - 72*C*a^2*b^6 - 60*C*a^3*b^5 + 273*C*a^4*b^4 + 47*C*a^5*b^3 - 236*C*a^6*b^2 - 18*A*a*b^7 - 18*B*a^7*b
- 12*C*a^7*b))/(3*b^4*(a + b)^2*(a - b)^3) - (tan(c/2 + (d*x)/2)^5*(72*C*a^8 + 18*C*b^8 + 45*A*a^2*b^6 + 7*A*a
^3*b^5 + 10*A*a^4*b^4 - 36*B*a^2*b^6 - 96*B*a^3*b^5 + 14*B*a^4*b^4 + 59*B*a^5*b^3 - 3*B*a^6*b^2 - 72*C*a^2*b^6
 + 60*C*a^3*b^5 + 273*C*a^4*b^4 - 47*C*a^5*b^3 - 236*C*a^6*b^2 + 18*A*a*b^7 - 18*B*a^7*b + 12*C*a^7*b))/(3*b^4
*(a + b)^3*(a - b)^2))/(d*(3*a*b^2 + 3*a^2*b - tan(c/2 + (d*x)/2)^4*(6*a*b^2 - 6*a^3) - tan(c/2 + (d*x)/2)^2*(
6*a^2*b + 4*a^3 - 2*b^3) - tan(c/2 + (d*x)/2)^6*(4*a^3 - 6*a^2*b + 2*b^3) + a^3 + b^3 + tan(c/2 + (d*x)/2)^8*(
3*a*b^2 - 3*a^2*b + a^3 - b^3))) + (atan(((((8*tan(c/2 + (d*x)/2)*(4*A^2*b^16 + 4*B^2*b^16 + 128*C^2*a^16 - 8*
B^2*a*b^15 - 128*C^2*a^15*b + 12*A^2*a^2*b^14 + 9*A^2*a^4*b^12 + 44*B^2*a^2*b^14 + 48*B^2*a^3*b^13 - 92*B^2*a^
4*b^12 - 120*B^2*a^5*b^11 + 156*B^2*a^6*b^10 + 160*B^2*a^7*b^9 - 164*B^2*a^8*b^8 - 120*B^2*a^9*b^7 + 117*B^2*a
^10*b^6 + 48*B^2*a^11*b^5 - 48*B^2*a^12*b^4 - 8*B^2*a^13*b^3 + 8*B^2*a^14*b^2 + 64*C^2*a^2*b^14 - 128*C^2*a^3*
b^13 + 80*C^2*a^4*b^12 + 768*C^2*a^5*b^11 - 824*C^2*a^6*b^10 - 1920*C^2*a^7*b^9 + 2025*C^2*a^8*b^8 + 2560*C^2*
a^9*b^7 - 2600*C^2*a^10*b^6 - 1920*C^2*a^11*b^5 + 1920*C^2*a^12*b^4 + 768*C^2*a^13*b^3 - 768*C^2*a^14*b^2 - 32
*A*B*a*b^15 - 32*B*C*a*b^15 - 64*B*C*a^15*b - 16*A*B*a^3*b^13 + 20*A*B*a^5*b^11 - 34*A*B*a^7*b^9 + 12*A*B*a^9*
b^7 + 80*A*C*a^2*b^14 - 20*A*C*a^4*b^12 - 98*A*C*a^6*b^10 + 136*A*C*a^8*b^8 - 48*A*C*a^10*b^6 + 64*B*C*a^2*b^1
4 - 160*B*C*a^3*b^13 - 384*B*C*a^4*b^12 + 592*B*C*a^5*b^11 + 960*B*C*a^6*b^10 - 1128*B*C*a^7*b^9 - 1280*B*C*a^
8*b^8 + 1306*B*C*a^9*b^7 + 960*B*C*a^10*b^6 - 948*B*C*a^11*b^5 - 384*B*C*a^12*b^4 + 384*B*C*a^13*b^3 + 64*B*C*
a^14*b^2))/(a*b^18 + b^19 - 5*a^2*b^17 - 5*a^3*b^16 + 10*a^4*b^15 + 10*a^5*b^14 - 10*a^6*b^13 - 10*a^7*b^12 +
5*a^8*b^11 + 5*a^9*b^10 - a^10*b^9 - a^11*b^8) - (((8*(4*A*b^24 + 4*B*b^24 - 6*A*a^2*b^22 + 6*A*a^3*b^21 - 6*A
*a^4*b^20 + 6*A*a^5*b^19 + 14*A*a^6*b^18 - 14*A*a^7*b^17 - 6*A*a^8*b^16 + 6*A*a^9*b^15 - 12*B*a^2*b^22 + 64*B*
a^3*b^21 + 20*B*a^4*b^20 - 110*B*a^5*b^19 - 30*B*a^6*b^18 + 110*B*a^7*b^17 + 30*B*a^8*b^16 - 70*B*a^9*b^15 - 1
4*B*a^10*b^14 + 26*B*a^11*b^13 + 2*B*a^12*b^12 - 4*B*a^13*b^11 + 40*C*a^2*b^22 + 72*C*a^3*b^21 - 190*C*a^4*b^2
0 - 146*C*a^5*b^19 + 386*C*a^6*b^18 + 174*C*a^7*b^17 - 434*C*a^8*b^16 - 126*C*a^9*b^15 + 286*C*a^10*b^14 + 50*
C*a^11*b^13 - 104*C*a^12*b^12 - 8*C*a^13*b^11 + 16*C*a^14*b^10 - 4*A*a*b^23 - 16*B*a*b^23 - 16*C*a*b^23))/(a*b
^22 + b^23 - 5*a^2*b^21 - 5*a^3*b^20 + 10*a^4*b^19 + 10*a^5*b^18 - 10*a^6*b^17 - 10*a^7*b^16 + 5*a^8*b^15 + 5*
a^9*b^14 - a^10*b^13 - a^11*b^12) - (4*tan(c/2 + (d*x)/2)*((a + b)^7*(a - b)^7)^(1/2)*(2*A*b^8 - 8*C*a^8 + 3*A
*a^2*b^6 + 8*B*a^3*b^5 - 7*B*a^5*b^3 + 20*C*a^2*b^6 - 35*C*a^4*b^4 + 28*C*a^6*b^2 - 8*B*a*b^7 + 2*B*a^7*b)*(8*
a*b^23 - 8*a^2*b^22 - 48*a^3*b^21 + 48*a^4*b^20 + 120*a^5*b^19 - 120*a^6*b^18 - 160*a^7*b^17 + 160*a^8*b^16 +
120*a^9*b^15 - 120*a^10*b^14 - 48*a^11*b^13 + 48*a^12*b^12 + 8*a^13*b^11 - 8*a^14*b^10))/((b^19 - 7*a^2*b^17 +
 21*a^4*b^15 - 35*a^6*b^13 + 35*a^8*b^11 - 21*a^10*b^9 + 7*a^12*b^7 - a^14*b^5)*(a*b^18 + b^19 - 5*a^2*b^17 -
5*a^3*b^16 + 10*a^4*b^15 + 10*a^5*b^14 - 10*a^6*b^13 - 10*a^7*b^12 + 5*a^8*b^11 + 5*a^9*b^10 - a^10*b^9 - a^11
*b^8)))*((a + b)^7*(a - b)^7)^(1/2)*(2*A*b^8 - 8*C*a^8 + 3*A*a^2*b^6 + 8*B*a^3*b^5 - 7*B*a^5*b^3 + 20*C*a^2*b^
6 - 35*C*a^4*b^4 + 28*C*a^6*b^2 - 8*B*a*b^7 + 2*B*a^7*b))/(2*(b^19 - 7*a^2*b^17 + 21*a^4*b^15 - 35*a^6*b^13 +
35*a^8*b^11 - 21*a^10*b^9 + 7*a^12*b^7 - a^14*b^5)))*((a + b)^7*(a - b)^7)^(1/2)*(2*A*b^8 - 8*C*a^8 + 3*A*a^2*
b^6 + 8*B*a^3*b^5 - 7*B*a^5*b^3 + 20*C*a^2*b^6 - 35*C*a^4*b^4 + 28*C*a^6*b^2 - 8*B*a*b^7 + 2*B*a^7*b)*1i)/(2*(
b^19 - 7*a^2*b^17 + 21*a^4*b^15 - 35*a^6*b^13 + 35*a^8*b^11 - 21*a^10*b^9 + 7*a^12*b^7 - a^14*b^5)) + (((8*tan
(c/2 + (d*x)/2)*(4*A^2*b^16 + 4*B^2*b^16 + 128*C^2*a^16 - 8*B^2*a*b^15 - 128*C^2*a^15*b + 12*A^2*a^2*b^14 + 9*
A^2*a^4*b^12 + 44*B^2*a^2*b^14 + 48*B^2*a^3*b^13 - 92*B^2*a^4*b^12 - 120*B^2*a^5*b^11 + 156*B^2*a^6*b^10 + 160
*B^2*a^7*b^9 - 164*B^2*a^8*b^8 - 120*B^2*a^9*b^7 + 117*B^2*a^10*b^6 + 48*B^2*a^11*b^5 - 48*B^2*a^12*b^4 - 8*B^
2*a^13*b^3 + 8*B^2*a^14*b^2 + 64*C^2*a^2*b^14 - 128*C^2*a^3*b^13 + 80*C^2*a^4*b^12 + 768*C^2*a^5*b^11 - 824*C^
2*a^6*b^10 - 1920*C^2*a^7*b^9 + 2025*C^2*a^8*b^8 + 2560*C^2*a^9*b^7 - 2600*C^2*a^10*b^6 - 1920*C^2*a^11*b^5 +
1920*C^2*a^12*b^4 + 768*C^2*a^13*b^3 - 768*C^2*a^14*b^2 - 32*A*B*a*b^15 - 32*B*C*a*b^15 - 64*B*C*a^15*b - 16*A
*B*a^3*b^13 + 20*A*B*a^5*b^11 - 34*A*B*a^7*b^9 + 12*A*B*a^9*b^7 + 80*A*C*a^2*b^14 - 20*A*C*a^4*b^12 - 98*A*C*a
^6*b^10 + 136*A*C*a^8*b^8 - 48*A*C*a^10*b^6 + 64*B*C*a^2*b^14 - 160*B*C*a^3*b^13 - 384*B*C*a^4*b^12 + 592*B*C*
a^5*b^11 + 960*B*C*a^6*b^10 - 1128*B*C*a^7*b^9 - 1280*B*C*a^8*b^8 + 1306*B*C*a^9*b^7 + 960*B*C*a^10*b^6 - 948*
B*C*a^11*b^5 - 384*B*C*a^12*b^4 + 384*B*C*a^13*b^3 + 64*B*C*a^14*b^2))/(a*b^18 + b^19 - 5*a^2*b^17 - 5*a^3*b^1
6 + 10*a^4*b^15 + 10*a^5*b^14 - 10*a^6*b^13 - 10*a^7*b^12 + 5*a^8*b^11 + 5*a^9*b^10 - a^10*b^9 - a^11*b^8) + (
((8*(4*A*b^24 + 4*B*b^24 - 6*A*a^2*b^22 + 6*A*a^3*b^21 - 6*A*a^4*b^20 + 6*A*a^5*b^19 + 14*A*a^6*b^18 - 14*A*a^
7*b^17 - 6*A*a^8*b^16 + 6*A*a^9*b^15 - 12*B*a^2*b^22 + 64*B*a^3*b^21 + 20*B*a^4*b^20 - 110*B*a^5*b^19 - 30*B*a
^6*b^18 + 110*B*a^7*b^17 + 30*B*a^8*b^16 - 70*B*a^9*b^15 - 14*B*a^10*b^14 + 26*B*a^11*b^13 + 2*B*a^12*b^12 - 4
*B*a^13*b^11 + 40*C*a^2*b^22 + 72*C*a^3*b^21 - 190*C*a^4*b^20 - 146*C*a^5*b^19 + 386*C*a^6*b^18 + 174*C*a^7*b^
17 - 434*C*a^8*b^16 - 126*C*a^9*b^15 + 286*C*a^10*b^14 + 50*C*a^11*b^13 - 104*C*a^12*b^12 - 8*C*a^13*b^11 + 16
*C*a^14*b^10 - 4*A*a*b^23 - 16*B*a*b^23 - 16*C*a*b^23))/(a*b^22 + b^23 - 5*a^2*b^21 - 5*a^3*b^20 + 10*a^4*b^19
 + 10*a^5*b^18 - 10*a^6*b^17 - 10*a^7*b^16 + 5*a^8*b^15 + 5*a^9*b^14 - a^10*b^13 - a^11*b^12) + (4*tan(c/2 + (
d*x)/2)*((a + b)^7*(a - b)^7)^(1/2)*(2*A*b^8 - 8*C*a^8 + 3*A*a^2*b^6 + 8*B*a^3*b^5 - 7*B*a^5*b^3 + 20*C*a^2*b^
6 - 35*C*a^4*b^4 + 28*C*a^6*b^2 - 8*B*a*b^7 + 2*B*a^7*b)*(8*a*b^23 - 8*a^2*b^22 - 48*a^3*b^21 + 48*a^4*b^20 +
120*a^5*b^19 - 120*a^6*b^18 - 160*a^7*b^17 + 160*a^8*b^16 + 120*a^9*b^15 - 120*a^10*b^14 - 48*a^11*b^13 + 48*a
^12*b^12 + 8*a^13*b^11 - 8*a^14*b^10))/((b^19 - 7*a^2*b^17 + 21*a^4*b^15 - 35*a^6*b^13 + 35*a^8*b^11 - 21*a^10
*b^9 + 7*a^12*b^7 - a^14*b^5)*(a*b^18 + b^19 - 5*a^2*b^17 - 5*a^3*b^16 + 10*a^4*b^15 + 10*a^5*b^14 - 10*a^6*b^
13 - 10*a^7*b^12 + 5*a^8*b^11 + 5*a^9*b^10 - a^10*b^9 - a^11*b^8)))*((a + b)^7*(a - b)^7)^(1/2)*(2*A*b^8 - 8*C
*a^8 + 3*A*a^2*b^6 + 8*B*a^3*b^5 - 7*B*a^5*b^3 + 20*C*a^2*b^6 - 35*C*a^4*b^4 + 28*C*a^6*b^2 - 8*B*a*b^7 + 2*B*
a^7*b))/(2*(b^19 - 7*a^2*b^17 + 21*a^4*b^15 - 35*a^6*b^13 + 35*a^8*b^11 - 21*a^10*b^9 + 7*a^12*b^7 - a^14*b^5)
))*((a + b)^7*(a - b)^7)^(1/2)*(2*A*b^8 - 8*C*a^8 + 3*A*a^2*b^6 + 8*B*a^3*b^5 - 7*B*a^5*b^3 + 20*C*a^2*b^6 - 3
5*C*a^4*b^4 + 28*C*a^6*b^2 - 8*B*a*b^7 + 2*B*a^7*b)*1i)/(2*(b^19 - 7*a^2*b^17 + 21*a^4*b^15 - 35*a^6*b^13 + 35
*a^8*b^11 - 21*a^10*b^9 + 7*a^12*b^7 - a^14*b^5)))/((16*(256*C^3*a^16 + 4*A*B^2*b^16 - 4*A^2*B*b^16 - 16*B^3*a
*b^15 - 128*C^3*a^15*b - 48*B^3*a^2*b^14 + 64*B^3*a^3*b^13 + 64*B^3*a^4*b^12 - 110*B^3*a^5*b^11 - 66*B^3*a^6*b
^10 + 110*B^3*a^7*b^9 + 34*B^3*a^8*b^8 - 70*B^3*a^9*b^7 - 11*B^3*a^10*b^6 + 26*B^3*a^11*b^5 + 2*B^3*a^12*b^4 -
 4*B^3*a^13*b^3 + 640*C^3*a^4*b^12 + 960*C^3*a^5*b^11 - 3040*C^3*a^6*b^10 - 2560*C^3*a^7*b^9 + 6176*C^3*a^8*b^
8 + 3204*C^3*a^9*b^7 - 6944*C^3*a^10*b^6 - 2176*C^3*a^11*b^5 + 4576*C^3*a^12*b^4 + 800*C^3*a^13*b^3 - 1664*C^3
*a^14*b^2 + 28*A*B^2*a*b^15 + 16*A^2*C*a*b^15 - 192*B*C^2*a^15*b - 6*A*B^2*a^2*b^14 + 22*A*B^2*a^3*b^13 - 6*A*
B^2*a^4*b^12 - 14*A*B^2*a^5*b^11 + 14*A*B^2*a^6*b^10 + 20*A*B^2*a^7*b^9 - 6*A*B^2*a^8*b^8 - 6*A*B^2*a^9*b^7 -
12*A^2*B*a^2*b^14 - 9*A^2*B*a^4*b^12 + 64*A*C^2*a^2*b^14 + 256*A*C^2*a^3*b^13 - 96*A*C^2*a^4*b^12 + 16*A*C^2*a
^5*b^11 - 96*A*C^2*a^6*b^10 - 296*A*C^2*a^7*b^9 + 224*A*C^2*a^8*b^8 + 320*A*C^2*a^9*b^7 - 96*A*C^2*a^10*b^6 -
96*A*C^2*a^11*b^5 + 48*A^2*C*a^3*b^13 + 36*A^2*C*a^5*b^11 - 576*B*C^2*a^3*b^13 - 1104*B*C^2*a^4*b^12 + 2544*B*
C^2*a^5*b^11 + 2376*B*C^2*a^6*b^10 - 4848*B*C^2*a^7*b^9 - 2649*B*C^2*a^8*b^8 + 5232*B*C^2*a^9*b^7 + 1632*B*C^2
*a^10*b^6 - 3408*B*C^2*a^11*b^5 - 576*B*C^2*a^12*b^4 + 1248*B*C^2*a^13*b^3 + 96*B*C^2*a^14*b^2 + 168*B^2*C*a^2
*b^14 + 408*B^2*C*a^3*b^13 - 702*B^2*C*a^4*b^12 - 690*B^2*C*a^5*b^11 + 1266*B^2*C*a^6*b^10 + 726*B^2*C*a^7*b^9
 - 1314*B^2*C*a^8*b^8 - 408*B^2*C*a^9*b^7 + 846*B^2*C*a^10*b^6 + 138*B^2*C*a^11*b^5 - 312*B^2*C*a^12*b^4 - 24*
B^2*C*a^13*b^3 + 48*B^2*C*a^14*b^2 - 32*A*B*C*a*b^15 - 176*A*B*C*a^2*b^14 + 48*A*B*C*a^3*b^13 - 92*A*B*C*a^4*b
^12 + 48*A*B*C*a^5*b^11 + 130*A*B*C*a^6*b^10 - 112*A*B*C*a^7*b^9 - 160*A*B*C*a^8*b^8 + 48*A*B*C*a^9*b^7 + 48*A
*B*C*a^10*b^6))/(a*b^22 + b^23 - 5*a^2*b^21 - 5*a^3*b^20 + 10*a^4*b^19 + 10*a^5*b^18 - 10*a^6*b^17 - 10*a^7*b^
16 + 5*a^8*b^15 + 5*a^9*b^14 - a^10*b^13 - a^11*b^12) + (((8*tan(c/2 + (d*x)/2)*(4*A^2*b^16 + 4*B^2*b^16 + 128
*C^2*a^16 - 8*B^2*a*b^15 - 128*C^2*a^15*b + 12*A^2*a^2*b^14 + 9*A^2*a^4*b^12 + 44*B^2*a^2*b^14 + 48*B^2*a^3*b^
13 - 92*B^2*a^4*b^12 - 120*B^2*a^5*b^11 + 156*B^2*a^6*b^10 + 160*B^2*a^7*b^9 - 164*B^2*a^8*b^8 - 120*B^2*a^9*b
^7 + 117*B^2*a^10*b^6 + 48*B^2*a^11*b^5 - 48*B^2*a^12*b^4 - 8*B^2*a^13*b^3 + 8*B^2*a^14*b^2 + 64*C^2*a^2*b^14
- 128*C^2*a^3*b^13 + 80*C^2*a^4*b^12 + 768*C^2*a^5*b^11 - 824*C^2*a^6*b^10 - 1920*C^2*a^7*b^9 + 2025*C^2*a^8*b
^8 + 2560*C^2*a^9*b^7 - 2600*C^2*a^10*b^6 - 1920*C^2*a^11*b^5 + 1920*C^2*a^12*b^4 + 768*C^2*a^13*b^3 - 768*C^2
*a^14*b^2 - 32*A*B*a*b^15 - 32*B*C*a*b^15 - 64*B*C*a^15*b - 16*A*B*a^3*b^13 + 20*A*B*a^5*b^11 - 34*A*B*a^7*b^9
 + 12*A*B*a^9*b^7 + 80*A*C*a^2*b^14 - 20*A*C*a^4*b^12 - 98*A*C*a^6*b^10 + 136*A*C*a^8*b^8 - 48*A*C*a^10*b^6 +
64*B*C*a^2*b^14 - 160*B*C*a^3*b^13 - 384*B*C*a^4*b^12 + 592*B*C*a^5*b^11 + 960*B*C*a^6*b^10 - 1128*B*C*a^7*b^9
 - 1280*B*C*a^8*b^8 + 1306*B*C*a^9*b^7 + 960*B*C*a^10*b^6 - 948*B*C*a^11*b^5 - 384*B*C*a^12*b^4 + 384*B*C*a^13
*b^3 + 64*B*C*a^14*b^2))/(a*b^18 + b^19 - 5*a^2*b^17 - 5*a^3*b^16 + 10*a^4*b^15 + 10*a^5*b^14 - 10*a^6*b^13 -
10*a^7*b^12 + 5*a^8*b^11 + 5*a^9*b^10 - a^10*b^9 - a^11*b^8) - (((8*(4*A*b^24 + 4*B*b^24 - 6*A*a^2*b^22 + 6*A*
a^3*b^21 - 6*A*a^4*b^20 + 6*A*a^5*b^19 + 14*A*a^6*b^18 - 14*A*a^7*b^17 - 6*A*a^8*b^16 + 6*A*a^9*b^15 - 12*B*a^
2*b^22 + 64*B*a^3*b^21 + 20*B*a^4*b^20 - 110*B*a^5*b^19 - 30*B*a^6*b^18 + 110*B*a^7*b^17 + 30*B*a^8*b^16 - 70*
B*a^9*b^15 - 14*B*a^10*b^14 + 26*B*a^11*b^13 + 2*B*a^12*b^12 - 4*B*a^13*b^11 + 40*C*a^2*b^22 + 72*C*a^3*b^21 -
 190*C*a^4*b^20 - 146*C*a^5*b^19 + 386*C*a^6*b^18 + 174*C*a^7*b^17 - 434*C*a^8*b^16 - 126*C*a^9*b^15 + 286*C*a
^10*b^14 + 50*C*a^11*b^13 - 104*C*a^12*b^12 - 8*C*a^13*b^11 + 16*C*a^14*b^10 - 4*A*a*b^23 - 16*B*a*b^23 - 16*C
*a*b^23))/(a*b^22 + b^23 - 5*a^2*b^21 - 5*a^3*b^20 + 10*a^4*b^19 + 10*a^5*b^18 - 10*a^6*b^17 - 10*a^7*b^16 + 5
*a^8*b^15 + 5*a^9*b^14 - a^10*b^13 - a^11*b^12) - (4*tan(c/2 + (d*x)/2)*((a + b)^7*(a - b)^7)^(1/2)*(2*A*b^8 -
 8*C*a^8 + 3*A*a^2*b^6 + 8*B*a^3*b^5 - 7*B*a^5*b^3 + 20*C*a^2*b^6 - 35*C*a^4*b^4 + 28*C*a^6*b^2 - 8*B*a*b^7 +
2*B*a^7*b)*(8*a*b^23 - 8*a^2*b^22 - 48*a^3*b^21 + 48*a^4*b^20 + 120*a^5*b^19 - 120*a^6*b^18 - 160*a^7*b^17 + 1
60*a^8*b^16 + 120*a^9*b^15 - 120*a^10*b^14 - 48*a^11*b^13 + 48*a^12*b^12 + 8*a^13*b^11 - 8*a^14*b^10))/((b^19
- 7*a^2*b^17 + 21*a^4*b^15 - 35*a^6*b^13 + 35*a^8*b^11 - 21*a^10*b^9 + 7*a^12*b^7 - a^14*b^5)*(a*b^18 + b^19 -
 5*a^2*b^17 - 5*a^3*b^16 + 10*a^4*b^15 + 10*a^5*b^14 - 10*a^6*b^13 - 10*a^7*b^12 + 5*a^8*b^11 + 5*a^9*b^10 - a
^10*b^9 - a^11*b^8)))*((a + b)^7*(a - b)^7)^(1/2)*(2*A*b^8 - 8*C*a^8 + 3*A*a^2*b^6 + 8*B*a^3*b^5 - 7*B*a^5*b^3
 + 20*C*a^2*b^6 - 35*C*a^4*b^4 + 28*C*a^6*b^2 - 8*B*a*b^7 + 2*B*a^7*b))/(2*(b^19 - 7*a^2*b^17 + 21*a^4*b^15 -
35*a^6*b^13 + 35*a^8*b^11 - 21*a^10*b^9 + 7*a^12*b^7 - a^14*b^5)))*((a + b)^7*(a - b)^7)^(1/2)*(2*A*b^8 - 8*C*
a^8 + 3*A*a^2*b^6 + 8*B*a^3*b^5 - 7*B*a^5*b^3 + 20*C*a^2*b^6 - 35*C*a^4*b^4 + 28*C*a^6*b^2 - 8*B*a*b^7 + 2*B*a
^7*b))/(2*(b^19 - 7*a^2*b^17 + 21*a^4*b^15 - 35*a^6*b^13 + 35*a^8*b^11 - 21*a^10*b^9 + 7*a^12*b^7 - a^14*b^5))
 - (((8*tan(c/2 + (d*x)/2)*(4*A^2*b^16 + 4*B^2*b^16 + 128*C^2*a^16 - 8*B^2*a*b^15 - 128*C^2*a^15*b + 12*A^2*a^
2*b^14 + 9*A^2*a^4*b^12 + 44*B^2*a^2*b^14 + 48*B^2*a^3*b^13 - 92*B^2*a^4*b^12 - 120*B^2*a^5*b^11 + 156*B^2*a^6
*b^10 + 160*B^2*a^7*b^9 - 164*B^2*a^8*b^8 - 120*B^2*a^9*b^7 + 117*B^2*a^10*b^6 + 48*B^2*a^11*b^5 - 48*B^2*a^12
*b^4 - 8*B^2*a^13*b^3 + 8*B^2*a^14*b^2 + 64*C^2*a^2*b^14 - 128*C^2*a^3*b^13 + 80*C^2*a^4*b^12 + 768*C^2*a^5*b^
11 - 824*C^2*a^6*b^10 - 1920*C^2*a^7*b^9 + 2025*C^2*a^8*b^8 + 2560*C^2*a^9*b^7 - 2600*C^2*a^10*b^6 - 1920*C^2*
a^11*b^5 + 1920*C^2*a^12*b^4 + 768*C^2*a^13*b^3 - 768*C^2*a^14*b^2 - 32*A*B*a*b^15 - 32*B*C*a*b^15 - 64*B*C*a^
15*b - 16*A*B*a^3*b^13 + 20*A*B*a^5*b^11 - 34*A*B*a^7*b^9 + 12*A*B*a^9*b^7 + 80*A*C*a^2*b^14 - 20*A*C*a^4*b^12
 - 98*A*C*a^6*b^10 + 136*A*C*a^8*b^8 - 48*A*C*a^10*b^6 + 64*B*C*a^2*b^14 - 160*B*C*a^3*b^13 - 384*B*C*a^4*b^12
 + 592*B*C*a^5*b^11 + 960*B*C*a^6*b^10 - 1128*B*C*a^7*b^9 - 1280*B*C*a^8*b^8 + 1306*B*C*a^9*b^7 + 960*B*C*a^10
*b^6 - 948*B*C*a^11*b^5 - 384*B*C*a^12*b^4 + 384*B*C*a^13*b^3 + 64*B*C*a^14*b^2))/(a*b^18 + b^19 - 5*a^2*b^17
- 5*a^3*b^16 + 10*a^4*b^15 + 10*a^5*b^14 - 10*a^6*b^13 - 10*a^7*b^12 + 5*a^8*b^11 + 5*a^9*b^10 - a^10*b^9 - a^
11*b^8) + (((8*(4*A*b^24 + 4*B*b^24 - 6*A*a^2*b^22 + 6*A*a^3*b^21 - 6*A*a^4*b^20 + 6*A*a^5*b^19 + 14*A*a^6*b^1
8 - 14*A*a^7*b^17 - 6*A*a^8*b^16 + 6*A*a^9*b^15 - 12*B*a^2*b^22 + 64*B*a^3*b^21 + 20*B*a^4*b^20 - 110*B*a^5*b^
19 - 30*B*a^6*b^18 + 110*B*a^7*b^17 + 30*B*a^8*b^16 - 70*B*a^9*b^15 - 14*B*a^10*b^14 + 26*B*a^11*b^13 + 2*B*a^
12*b^12 - 4*B*a^13*b^11 + 40*C*a^2*b^22 + 72*C*a^3*b^21 - 190*C*a^4*b^20 - 146*C*a^5*b^19 + 386*C*a^6*b^18 + 1
74*C*a^7*b^17 - 434*C*a^8*b^16 - 126*C*a^9*b^15 + 286*C*a^10*b^14 + 50*C*a^11*b^13 - 104*C*a^12*b^12 - 8*C*a^1
3*b^11 + 16*C*a^14*b^10 - 4*A*a*b^23 - 16*B*a*b^23 - 16*C*a*b^23))/(a*b^22 + b^23 - 5*a^2*b^21 - 5*a^3*b^20 +
10*a^4*b^19 + 10*a^5*b^18 - 10*a^6*b^17 - 10*a^7*b^16 + 5*a^8*b^15 + 5*a^9*b^14 - a^10*b^13 - a^11*b^12) + (4*
tan(c/2 + (d*x)/2)*((a + b)^7*(a - b)^7)^(1/2)*(2*A*b^8 - 8*C*a^8 + 3*A*a^2*b^6 + 8*B*a^3*b^5 - 7*B*a^5*b^3 +
20*C*a^2*b^6 - 35*C*a^4*b^4 + 28*C*a^6*b^2 - 8*B*a*b^7 + 2*B*a^7*b)*(8*a*b^23 - 8*a^2*b^22 - 48*a^3*b^21 + 48*
a^4*b^20 + 120*a^5*b^19 - 120*a^6*b^18 - 160*a^7*b^17 + 160*a^8*b^16 + 120*a^9*b^15 - 120*a^10*b^14 - 48*a^11*
b^13 + 48*a^12*b^12 + 8*a^13*b^11 - 8*a^14*b^10))/((b^19 - 7*a^2*b^17 + 21*a^4*b^15 - 35*a^6*b^13 + 35*a^8*b^1
1 - 21*a^10*b^9 + 7*a^12*b^7 - a^14*b^5)*(a*b^18 + b^19 - 5*a^2*b^17 - 5*a^3*b^16 + 10*a^4*b^15 + 10*a^5*b^14
- 10*a^6*b^13 - 10*a^7*b^12 + 5*a^8*b^11 + 5*a^9*b^10 - a^10*b^9 - a^11*b^8)))*((a + b)^7*(a - b)^7)^(1/2)*(2*
A*b^8 - 8*C*a^8 + 3*A*a^2*b^6 + 8*B*a^3*b^5 - 7*B*a^5*b^3 + 20*C*a^2*b^6 - 35*C*a^4*b^4 + 28*C*a^6*b^2 - 8*B*a
*b^7 + 2*B*a^7*b))/(2*(b^19 - 7*a^2*b^17 + 21*a^4*b^15 - 35*a^6*b^13 + 35*a^8*b^11 - 21*a^10*b^9 + 7*a^12*b^7
- a^14*b^5)))*((a + b)^7*(a - b)^7)^(1/2)*(2*A*b^8 - 8*C*a^8 + 3*A*a^2*b^6 + 8*B*a^3*b^5 - 7*B*a^5*b^3 + 20*C*
a^2*b^6 - 35*C*a^4*b^4 + 28*C*a^6*b^2 - 8*B*a*b^7 + 2*B*a^7*b))/(2*(b^19 - 7*a^2*b^17 + 21*a^4*b^15 - 35*a^6*b
^13 + 35*a^8*b^11 - 21*a^10*b^9 + 7*a^12*b^7 - a^14*b^5))))*((a + b)^7*(a - b)^7)^(1/2)*(2*A*b^8 - 8*C*a^8 + 3
*A*a^2*b^6 + 8*B*a^3*b^5 - 7*B*a^5*b^3 + 20*C*a^2*b^6 - 35*C*a^4*b^4 + 28*C*a^6*b^2 - 8*B*a*b^7 + 2*B*a^7*b)*1
i)/(d*(b^19 - 7*a^2*b^17 + 21*a^4*b^15 - 35*a^6*b^13 + 35*a^8*b^11 - 21*a^10*b^9 + 7*a^12*b^7 - a^14*b^5))

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

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)

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