\(\int (a+b \sec (c+d x))^4 (A+B \sec (c+d x)+C \sec ^2(c+d x)) \, dx\) [889]

Optimal result

Integrand size = 33, antiderivative size = 290 \[ \int (a+b \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=a^4 A x+\frac {\left (8 a^4 B+24 a^2 b^2 B+3 b^4 B+16 a^3 b (2 A+C)+4 a b^3 (4 A+3 C)\right ) \text {arctanh}(\sin (c+d x))}{8 d}+\frac {\left (95 a^3 b B+80 a b^3 B+12 a^4 C+4 b^4 (5 A+4 C)+2 a^2 b^2 (85 A+56 C)\right ) \tan (c+d x)}{30 d}+\frac {b \left (130 a^2 b B+45 b^3 B+24 a^3 C+4 a b^2 (40 A+29 C)\right ) \sec (c+d x) \tan (c+d x)}{120 d}+\frac {\left (20 A b^2+35 a b B+12 a^2 C+16 b^2 C\right ) (a+b \sec (c+d x))^2 \tan (c+d x)}{60 d}+\frac {(5 b B+4 a C) (a+b \sec (c+d x))^3 \tan (c+d x)}{20 d}+\frac {C (a+b \sec (c+d x))^4 \tan (c+d x)}{5 d} \]

[Out]

a^4*A*x+1/8*(8*B*a^4+24*B*a^2*b^2+3*B*b^4+16*a^3*b*(2*A+C)+4*a*b^3*(4*A+3*C))*arctanh(sin(d*x+c))/d+1/30*(95*B
*a^3*b+80*B*a*b^3+12*a^4*C+4*b^4*(5*A+4*C)+2*a^2*b^2*(85*A+56*C))*tan(d*x+c)/d+1/120*b*(130*B*a^2*b+45*B*b^3+2
4*a^3*C+4*a*b^2*(40*A+29*C))*sec(d*x+c)*tan(d*x+c)/d+1/60*(20*A*b^2+35*B*a*b+12*C*a^2+16*C*b^2)*(a+b*sec(d*x+c
))^2*tan(d*x+c)/d+1/20*(5*B*b+4*C*a)*(a+b*sec(d*x+c))^3*tan(d*x+c)/d+1/5*C*(a+b*sec(d*x+c))^4*tan(d*x+c)/d

Rubi [A] (verified)

Time = 0.65 (sec) , antiderivative size = 290, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.152, Rules used = {4141, 4133, 3855, 3852, 8} \[ \int (a+b \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=a^4 A x+\frac {\tan (c+d x) \left (12 a^2 C+35 a b B+20 A b^2+16 b^2 C\right ) (a+b \sec (c+d x))^2}{60 d}+\frac {b \tan (c+d x) \sec (c+d x) \left (24 a^3 C+130 a^2 b B+4 a b^2 (40 A+29 C)+45 b^3 B\right )}{120 d}+\frac {\left (8 a^4 B+16 a^3 b (2 A+C)+24 a^2 b^2 B+4 a b^3 (4 A+3 C)+3 b^4 B\right ) \text {arctanh}(\sin (c+d x))}{8 d}+\frac {\tan (c+d x) \left (12 a^4 C+95 a^3 b B+2 a^2 b^2 (85 A+56 C)+80 a b^3 B+4 b^4 (5 A+4 C)\right )}{30 d}+\frac {(4 a C+5 b B) \tan (c+d x) (a+b \sec (c+d x))^3}{20 d}+\frac {C \tan (c+d x) (a+b \sec (c+d x))^4}{5 d} \]

[In]

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

[Out]

a^4*A*x + ((8*a^4*B + 24*a^2*b^2*B + 3*b^4*B + 16*a^3*b*(2*A + C) + 4*a*b^3*(4*A + 3*C))*ArcTanh[Sin[c + d*x]]
)/(8*d) + ((95*a^3*b*B + 80*a*b^3*B + 12*a^4*C + 4*b^4*(5*A + 4*C) + 2*a^2*b^2*(85*A + 56*C))*Tan[c + d*x])/(3
0*d) + (b*(130*a^2*b*B + 45*b^3*B + 24*a^3*C + 4*a*b^2*(40*A + 29*C))*Sec[c + d*x]*Tan[c + d*x])/(120*d) + ((2
0*A*b^2 + 35*a*b*B + 12*a^2*C + 16*b^2*C)*(a + b*Sec[c + d*x])^2*Tan[c + d*x])/(60*d) + ((5*b*B + 4*a*C)*(a +
b*Sec[c + d*x])^3*Tan[c + d*x])/(20*d) + (C*(a + b*Sec[c + d*x])^4*Tan[c + d*x])/(5*d)

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rule 3852

Int[csc[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> Dist[-d^(-1), Subst[Int[ExpandIntegrand[(1 + x^2)^(n/2 - 1), x]
, x], x, Cot[c + d*x]], x] /; FreeQ[{c, d}, x] && IGtQ[n/2, 0]

Rule 3855

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

Rule 4133

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

Rule 4141

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

Rubi steps \begin{align*} \text {integral}& = \frac {C (a+b \sec (c+d x))^4 \tan (c+d x)}{5 d}+\frac {1}{5} \int (a+b \sec (c+d x))^3 \left (5 a A+(5 A b+5 a B+4 b C) \sec (c+d x)+(5 b B+4 a C) \sec ^2(c+d x)\right ) \, dx \\ & = \frac {(5 b B+4 a C) (a+b \sec (c+d x))^3 \tan (c+d x)}{20 d}+\frac {C (a+b \sec (c+d x))^4 \tan (c+d x)}{5 d}+\frac {1}{20} \int (a+b \sec (c+d x))^2 \left (20 a^2 A+\left (40 a A b+20 a^2 B+15 b^2 B+28 a b C\right ) \sec (c+d x)+\left (20 A b^2+35 a b B+12 a^2 C+16 b^2 C\right ) \sec ^2(c+d x)\right ) \, dx \\ & = \frac {\left (20 A b^2+35 a b B+12 a^2 C+16 b^2 C\right ) (a+b \sec (c+d x))^2 \tan (c+d x)}{60 d}+\frac {(5 b B+4 a C) (a+b \sec (c+d x))^3 \tan (c+d x)}{20 d}+\frac {C (a+b \sec (c+d x))^4 \tan (c+d x)}{5 d}+\frac {1}{60} \int (a+b \sec (c+d x)) \left (60 a^3 A+\left (60 a^3 B+115 a b^2 B+36 a^2 b (5 A+3 C)+8 b^3 (5 A+4 C)\right ) \sec (c+d x)+\left (130 a^2 b B+45 b^3 B+24 a^3 C+4 a b^2 (40 A+29 C)\right ) \sec ^2(c+d x)\right ) \, dx \\ & = \frac {b \left (130 a^2 b B+45 b^3 B+24 a^3 C+4 a b^2 (40 A+29 C)\right ) \sec (c+d x) \tan (c+d x)}{120 d}+\frac {\left (20 A b^2+35 a b B+12 a^2 C+16 b^2 C\right ) (a+b \sec (c+d x))^2 \tan (c+d x)}{60 d}+\frac {(5 b B+4 a C) (a+b \sec (c+d x))^3 \tan (c+d x)}{20 d}+\frac {C (a+b \sec (c+d x))^4 \tan (c+d x)}{5 d}+\frac {1}{120} \int \left (120 a^4 A+15 \left (8 a^4 B+24 a^2 b^2 B+3 b^4 B+16 a^3 b (2 A+C)+4 a b^3 (4 A+3 C)\right ) \sec (c+d x)+4 \left (95 a^3 b B+80 a b^3 B+12 a^4 C+4 b^4 (5 A+4 C)+2 a^2 b^2 (85 A+56 C)\right ) \sec ^2(c+d x)\right ) \, dx \\ & = a^4 A x+\frac {b \left (130 a^2 b B+45 b^3 B+24 a^3 C+4 a b^2 (40 A+29 C)\right ) \sec (c+d x) \tan (c+d x)}{120 d}+\frac {\left (20 A b^2+35 a b B+12 a^2 C+16 b^2 C\right ) (a+b \sec (c+d x))^2 \tan (c+d x)}{60 d}+\frac {(5 b B+4 a C) (a+b \sec (c+d x))^3 \tan (c+d x)}{20 d}+\frac {C (a+b \sec (c+d x))^4 \tan (c+d x)}{5 d}+\frac {1}{8} \left (8 a^4 B+24 a^2 b^2 B+3 b^4 B+16 a^3 b (2 A+C)+4 a b^3 (4 A+3 C)\right ) \int \sec (c+d x) \, dx+\frac {1}{30} \left (95 a^3 b B+80 a b^3 B+12 a^4 C+4 b^4 (5 A+4 C)+2 a^2 b^2 (85 A+56 C)\right ) \int \sec ^2(c+d x) \, dx \\ & = a^4 A x+\frac {\left (8 a^4 B+24 a^2 b^2 B+3 b^4 B+16 a^3 b (2 A+C)+4 a b^3 (4 A+3 C)\right ) \text {arctanh}(\sin (c+d x))}{8 d}+\frac {b \left (130 a^2 b B+45 b^3 B+24 a^3 C+4 a b^2 (40 A+29 C)\right ) \sec (c+d x) \tan (c+d x)}{120 d}+\frac {\left (20 A b^2+35 a b B+12 a^2 C+16 b^2 C\right ) (a+b \sec (c+d x))^2 \tan (c+d x)}{60 d}+\frac {(5 b B+4 a C) (a+b \sec (c+d x))^3 \tan (c+d x)}{20 d}+\frac {C (a+b \sec (c+d x))^4 \tan (c+d x)}{5 d}-\frac {\left (95 a^3 b B+80 a b^3 B+12 a^4 C+4 b^4 (5 A+4 C)+2 a^2 b^2 (85 A+56 C)\right ) \text {Subst}(\int 1 \, dx,x,-\tan (c+d x))}{30 d} \\ & = a^4 A x+\frac {\left (8 a^4 B+24 a^2 b^2 B+3 b^4 B+16 a^3 b (2 A+C)+4 a b^3 (4 A+3 C)\right ) \text {arctanh}(\sin (c+d x))}{8 d}+\frac {\left (95 a^3 b B+80 a b^3 B+12 a^4 C+4 b^4 (5 A+4 C)+2 a^2 b^2 (85 A+56 C)\right ) \tan (c+d x)}{30 d}+\frac {b \left (130 a^2 b B+45 b^3 B+24 a^3 C+4 a b^2 (40 A+29 C)\right ) \sec (c+d x) \tan (c+d x)}{120 d}+\frac {\left (20 A b^2+35 a b B+12 a^2 C+16 b^2 C\right ) (a+b \sec (c+d x))^2 \tan (c+d x)}{60 d}+\frac {(5 b B+4 a C) (a+b \sec (c+d x))^3 \tan (c+d x)}{20 d}+\frac {C (a+b \sec (c+d x))^4 \tan (c+d x)}{5 d} \\ \end{align*}

Mathematica [A] (verified)

Time = 6.30 (sec) , antiderivative size = 232, normalized size of antiderivative = 0.80 \[ \int (a+b \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {120 a^4 A d x+15 \left (8 a^4 B+24 a^2 b^2 B+3 b^4 B+16 a^3 b (2 A+C)+4 a b^3 (4 A+3 C)\right ) \text {arctanh}(\sin (c+d x))+15 \left (8 \left (4 a^3 b B+4 a b^3 B+a^4 C+6 a^2 b^2 (A+C)+b^4 (A+C)\right )+b \left (24 a^2 b B+3 b^3 B+16 a^3 C+4 a b^2 (4 A+3 C)\right ) \sec (c+d x)+2 b^3 (b B+4 a C) \sec ^3(c+d x)\right ) \tan (c+d x)+40 b^2 \left (A b^2+4 a b B+6 a^2 C+2 b^2 C\right ) \tan ^3(c+d x)+24 b^4 C \tan ^5(c+d x)}{120 d} \]

[In]

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

[Out]

(120*a^4*A*d*x + 15*(8*a^4*B + 24*a^2*b^2*B + 3*b^4*B + 16*a^3*b*(2*A + C) + 4*a*b^3*(4*A + 3*C))*ArcTanh[Sin[
c + d*x]] + 15*(8*(4*a^3*b*B + 4*a*b^3*B + a^4*C + 6*a^2*b^2*(A + C) + b^4*(A + C)) + b*(24*a^2*b*B + 3*b^3*B
+ 16*a^3*C + 4*a*b^2*(4*A + 3*C))*Sec[c + d*x] + 2*b^3*(b*B + 4*a*C)*Sec[c + d*x]^3)*Tan[c + d*x] + 40*b^2*(A*
b^2 + 4*a*b*B + 6*a^2*C + 2*b^2*C)*Tan[c + d*x]^3 + 24*b^4*C*Tan[c + d*x]^5)/(120*d)

Maple [A] (verified)

Time = 1.70 (sec) , antiderivative size = 273, normalized size of antiderivative = 0.94

method	result	size
parts	\(a^{4} A x +\frac {\left (4 A \,a^{3} b +B \,a^{4}\right ) \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{d}+\frac {\left (B \,b^{4}+4 C a \,b^{3}\right ) \left (-\left (-\frac {\sec \left (d x +c \right )^{3}}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )}{d}-\frac {\left (A \,b^{4}+4 B a \,b^{3}+6 C \,a^{2} b^{2}\right ) \left (-\frac {2}{3}-\frac {\sec \left (d x +c \right )^{2}}{3}\right ) \tan \left (d x +c \right )}{d}+\frac {\left (4 a A \,b^{3}+6 B \,a^{2} b^{2}+4 a^{3} b C \right ) \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )}{d}+\frac {\left (6 A \,a^{2} b^{2}+4 B \,a^{3} b +a^{4} C \right ) \tan \left (d x +c \right )}{d}-\frac {C \,b^{4} \left (-\frac {8}{15}-\frac {\sec \left (d x +c \right )^{4}}{5}-\frac {4 \sec \left (d x +c \right )^{2}}{15}\right ) \tan \left (d x +c \right )}{d}\)	\(273\)
derivativedivides	\(\frac {a^{4} A \left (d x +c \right )+B \,a^{4} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+a^{4} C \tan \left (d x +c \right )+4 A \,a^{3} b \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+4 B \tan \left (d x +c \right ) a^{3} b +4 a^{3} b C \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )+6 A \tan \left (d x +c \right ) a^{2} b^{2}+6 B \,a^{2} b^{2} \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )-6 C \,a^{2} b^{2} \left (-\frac {2}{3}-\frac {\sec \left (d x +c \right )^{2}}{3}\right ) \tan \left (d x +c \right )+4 a A \,b^{3} \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )-4 B a \,b^{3} \left (-\frac {2}{3}-\frac {\sec \left (d x +c \right )^{2}}{3}\right ) \tan \left (d x +c \right )+4 C a \,b^{3} \left (-\left (-\frac {\sec \left (d x +c \right )^{3}}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )-A \,b^{4} \left (-\frac {2}{3}-\frac {\sec \left (d x +c \right )^{2}}{3}\right ) \tan \left (d x +c \right )+B \,b^{4} \left (-\left (-\frac {\sec \left (d x +c \right )^{3}}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )-C \,b^{4} \left (-\frac {8}{15}-\frac {\sec \left (d x +c \right )^{4}}{5}-\frac {4 \sec \left (d x +c \right )^{2}}{15}\right ) \tan \left (d x +c \right )}{d}\)	\(421\)
default	\(\frac {a^{4} A \left (d x +c \right )+B \,a^{4} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+a^{4} C \tan \left (d x +c \right )+4 A \,a^{3} b \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+4 B \tan \left (d x +c \right ) a^{3} b +4 a^{3} b C \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )+6 A \tan \left (d x +c \right ) a^{2} b^{2}+6 B \,a^{2} b^{2} \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )-6 C \,a^{2} b^{2} \left (-\frac {2}{3}-\frac {\sec \left (d x +c \right )^{2}}{3}\right ) \tan \left (d x +c \right )+4 a A \,b^{3} \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )-4 B a \,b^{3} \left (-\frac {2}{3}-\frac {\sec \left (d x +c \right )^{2}}{3}\right ) \tan \left (d x +c \right )+4 C a \,b^{3} \left (-\left (-\frac {\sec \left (d x +c \right )^{3}}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )-A \,b^{4} \left (-\frac {2}{3}-\frac {\sec \left (d x +c \right )^{2}}{3}\right ) \tan \left (d x +c \right )+B \,b^{4} \left (-\left (-\frac {\sec \left (d x +c \right )^{3}}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )-C \,b^{4} \left (-\frac {8}{15}-\frac {\sec \left (d x +c \right )^{4}}{5}-\frac {4 \sec \left (d x +c \right )^{2}}{15}\right ) \tan \left (d x +c \right )}{d}\)	\(421\)
parallelrisch	\(\frac {-2400 \left (\frac {3 B \,b^{4}}{32}+\frac {a \left (A +\frac {3 C}{4}\right ) b^{3}}{2}+\frac {3 B \,a^{2} b^{2}}{4}+a^{3} \left (A +\frac {C}{2}\right ) b +\frac {B \,a^{4}}{4}\right ) \left (\frac {\cos \left (5 d x +5 c \right )}{5}+\cos \left (3 d x +3 c \right )+2 \cos \left (d x +c \right )\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )+2400 \left (\frac {3 B \,b^{4}}{32}+\frac {a \left (A +\frac {3 C}{4}\right ) b^{3}}{2}+\frac {3 B \,a^{2} b^{2}}{4}+a^{3} \left (A +\frac {C}{2}\right ) b +\frac {B \,a^{4}}{4}\right ) \left (\frac {\cos \left (5 d x +5 c \right )}{5}+\cos \left (3 d x +3 c \right )+2 \cos \left (d x +c \right )\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )+600 a^{4} A x d \cos \left (3 d x +3 c \right )+120 a^{4} A x d \cos \left (5 d x +5 c \right )+\left (\left (400 A +320 C \right ) b^{4}+1600 B a \,b^{3}+2160 \left (A +\frac {10 C}{9}\right ) a^{2} b^{2}+1440 B \,a^{3} b +360 a^{4} C \right ) \sin \left (3 d x +3 c \right )+\left (\left (80 A +64 C \right ) b^{4}+320 B a \,b^{3}+720 a^{2} \left (A +\frac {2 C}{3}\right ) b^{2}+480 B \,a^{3} b +120 a^{4} C \right ) \sin \left (5 d x +5 c \right )+960 b \left (\frac {7 B \,b^{3}}{16}+b^{2} \left (A +\frac {7 C}{4}\right ) a +\frac {3 B \,a^{2} b}{2}+a^{3} C \right ) \sin \left (2 d x +2 c \right )+480 \left (\frac {3 B \,b^{3}}{16}+\left (A +\frac {3 C}{4}\right ) b^{2} a +\frac {3 B \,a^{2} b}{2}+a^{3} C \right ) b \sin \left (4 d x +4 c \right )+1200 a^{4} A x d \cos \left (d x +c \right )+1440 \left (\left (\frac {2 A}{9}+\frac {4 C}{9}\right ) b^{4}+\frac {8 B a \,b^{3}}{9}+b^{2} \left (A +\frac {4 C}{3}\right ) a^{2}+\frac {2 B \,a^{3} b}{3}+\frac {a^{4} C}{6}\right ) \sin \left (d x +c \right )}{600 d \left (\frac {\cos \left (5 d x +5 c \right )}{5}+\cos \left (3 d x +3 c \right )+2 \cos \left (d x +c \right )\right )}\)	\(502\)
norman	\(\frac {a^{4} A x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{10}-a^{4} A x -\frac {4 \left (270 A \,a^{2} b^{2}+25 A \,b^{4}+180 B \,a^{3} b +100 B a \,b^{3}+45 a^{4} C +150 C \,a^{2} b^{2}+29 C \,b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{15 d}-\frac {\left (48 A \,a^{2} b^{2}-16 a A \,b^{3}+8 A \,b^{4}+32 B \,a^{3} b -24 B \,a^{2} b^{2}+32 B a \,b^{3}-5 B \,b^{4}+8 a^{4} C -16 a^{3} b C +48 C \,a^{2} b^{2}-20 C a \,b^{3}+8 C \,b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}}{4 d}-\frac {\left (48 A \,a^{2} b^{2}+16 a A \,b^{3}+8 A \,b^{4}+32 B \,a^{3} b +24 B \,a^{2} b^{2}+32 B a \,b^{3}+5 B \,b^{4}+8 a^{4} C +16 a^{3} b C +48 C \,a^{2} b^{2}+20 C a \,b^{3}+8 C \,b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{4 d}+\frac {\left (288 A \,a^{2} b^{2}-48 a A \,b^{3}+32 A \,b^{4}+192 B \,a^{3} b -72 B \,a^{2} b^{2}+128 B a \,b^{3}-3 B \,b^{4}+48 a^{4} C -48 a^{3} b C +192 C \,a^{2} b^{2}-12 C a \,b^{3}+16 C \,b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{6 d}+\frac {\left (288 A \,a^{2} b^{2}+48 a A \,b^{3}+32 A \,b^{4}+192 B \,a^{3} b +72 B \,a^{2} b^{2}+128 B a \,b^{3}+3 B \,b^{4}+48 a^{4} C +48 a^{3} b C +192 C \,a^{2} b^{2}+12 C a \,b^{3}+16 C \,b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{6 d}+5 a^{4} A x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-10 a^{4} A x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}+10 a^{4} A x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}-5 a^{4} A x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1\right )^{5}}-\frac {\left (32 A \,a^{3} b +16 a A \,b^{3}+8 B \,a^{4}+24 B \,a^{2} b^{2}+3 B \,b^{4}+16 a^{3} b C +12 C a \,b^{3}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{8 d}+\frac {\left (32 A \,a^{3} b +16 a A \,b^{3}+8 B \,a^{4}+24 B \,a^{2} b^{2}+3 B \,b^{4}+16 a^{3} b C +12 C a \,b^{3}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{8 d}\)	\(726\)
risch	\(\text {Expression too large to display}\)	\(1096\)

[In]

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

[Out]

a^4*A*x+(4*A*a^3*b+B*a^4)/d*ln(sec(d*x+c)+tan(d*x+c))+(B*b^4+4*C*a*b^3)/d*(-(-1/4*sec(d*x+c)^3-3/8*sec(d*x+c))
*tan(d*x+c)+3/8*ln(sec(d*x+c)+tan(d*x+c)))-(A*b^4+4*B*a*b^3+6*C*a^2*b^2)/d*(-2/3-1/3*sec(d*x+c)^2)*tan(d*x+c)+
(4*A*a*b^3+6*B*a^2*b^2+4*C*a^3*b)/d*(1/2*sec(d*x+c)*tan(d*x+c)+1/2*ln(sec(d*x+c)+tan(d*x+c)))+(6*A*a^2*b^2+4*B
*a^3*b+C*a^4)/d*tan(d*x+c)-C*b^4/d*(-8/15-1/5*sec(d*x+c)^4-4/15*sec(d*x+c)^2)*tan(d*x+c)

Fricas [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 340, normalized size of antiderivative = 1.17 \[ \int (a+b \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {240 \, A a^{4} d x \cos \left (d x + c\right )^{5} + 15 \, {\left (8 \, B a^{4} + 16 \, {\left (2 \, A + C\right )} a^{3} b + 24 \, B a^{2} b^{2} + 4 \, {\left (4 \, A + 3 \, C\right )} a b^{3} + 3 \, B b^{4}\right )} \cos \left (d x + c\right )^{5} \log \left (\sin \left (d x + c\right ) + 1\right ) - 15 \, {\left (8 \, B a^{4} + 16 \, {\left (2 \, A + C\right )} a^{3} b + 24 \, B a^{2} b^{2} + 4 \, {\left (4 \, A + 3 \, C\right )} a b^{3} + 3 \, B b^{4}\right )} \cos \left (d x + c\right )^{5} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \, {\left (24 \, C b^{4} + 8 \, {\left (15 \, C a^{4} + 60 \, B a^{3} b + 30 \, {\left (3 \, A + 2 \, C\right )} a^{2} b^{2} + 40 \, B a b^{3} + 2 \, {\left (5 \, A + 4 \, C\right )} b^{4}\right )} \cos \left (d x + c\right )^{4} + 15 \, {\left (16 \, C a^{3} b + 24 \, B a^{2} b^{2} + 4 \, {\left (4 \, A + 3 \, C\right )} a b^{3} + 3 \, B b^{4}\right )} \cos \left (d x + c\right )^{3} + 8 \, {\left (30 \, C a^{2} b^{2} + 20 \, B a b^{3} + {\left (5 \, A + 4 \, C\right )} b^{4}\right )} \cos \left (d x + c\right )^{2} + 30 \, {\left (4 \, C a b^{3} + B b^{4}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{240 \, d \cos \left (d x + c\right )^{5}} \]

[In]

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

[Out]

1/240*(240*A*a^4*d*x*cos(d*x + c)^5 + 15*(8*B*a^4 + 16*(2*A + C)*a^3*b + 24*B*a^2*b^2 + 4*(4*A + 3*C)*a*b^3 +
3*B*b^4)*cos(d*x + c)^5*log(sin(d*x + c) + 1) - 15*(8*B*a^4 + 16*(2*A + C)*a^3*b + 24*B*a^2*b^2 + 4*(4*A + 3*C
)*a*b^3 + 3*B*b^4)*cos(d*x + c)^5*log(-sin(d*x + c) + 1) + 2*(24*C*b^4 + 8*(15*C*a^4 + 60*B*a^3*b + 30*(3*A +
2*C)*a^2*b^2 + 40*B*a*b^3 + 2*(5*A + 4*C)*b^4)*cos(d*x + c)^4 + 15*(16*C*a^3*b + 24*B*a^2*b^2 + 4*(4*A + 3*C)*
a*b^3 + 3*B*b^4)*cos(d*x + c)^3 + 8*(30*C*a^2*b^2 + 20*B*a*b^3 + (5*A + 4*C)*b^4)*cos(d*x + c)^2 + 30*(4*C*a*b
^3 + B*b^4)*cos(d*x + c))*sin(d*x + c))/(d*cos(d*x + c)^5)

Sympy [F]

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

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 497, normalized size of antiderivative = 1.71 \[ \int (a+b \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {240 \, {\left (d x + c\right )} A a^{4} + 480 \, {\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} C a^{2} b^{2} + 320 \, {\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} B a b^{3} + 80 \, {\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} A b^{4} + 16 \, {\left (3 \, \tan \left (d x + c\right )^{5} + 10 \, \tan \left (d x + c\right )^{3} + 15 \, \tan \left (d x + c\right )\right )} C b^{4} - 60 \, C a b^{3} {\left (\frac {2 \, {\left (3 \, \sin \left (d x + c\right )^{3} - 5 \, \sin \left (d x + c\right )\right )}}{\sin \left (d x + c\right )^{4} - 2 \, \sin \left (d x + c\right )^{2} + 1} - 3 \, \log \left (\sin \left (d x + c\right ) + 1\right ) + 3 \, \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - 15 \, B b^{4} {\left (\frac {2 \, {\left (3 \, \sin \left (d x + c\right )^{3} - 5 \, \sin \left (d x + c\right )\right )}}{\sin \left (d x + c\right )^{4} - 2 \, \sin \left (d x + c\right )^{2} + 1} - 3 \, \log \left (\sin \left (d x + c\right ) + 1\right ) + 3 \, \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - 240 \, C a^{3} b {\left (\frac {2 \, \sin \left (d x + c\right )}{\sin \left (d x + c\right )^{2} - 1} - \log \left (\sin \left (d x + c\right ) + 1\right ) + \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - 360 \, B a^{2} b^{2} {\left (\frac {2 \, \sin \left (d x + c\right )}{\sin \left (d x + c\right )^{2} - 1} - \log \left (\sin \left (d x + c\right ) + 1\right ) + \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - 240 \, A a b^{3} {\left (\frac {2 \, \sin \left (d x + c\right )}{\sin \left (d x + c\right )^{2} - 1} - \log \left (\sin \left (d x + c\right ) + 1\right ) + \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 240 \, B a^{4} \log \left (\sec \left (d x + c\right ) + \tan \left (d x + c\right )\right ) + 960 \, A a^{3} b \log \left (\sec \left (d x + c\right ) + \tan \left (d x + c\right )\right ) + 240 \, C a^{4} \tan \left (d x + c\right ) + 960 \, B a^{3} b \tan \left (d x + c\right ) + 1440 \, A a^{2} b^{2} \tan \left (d x + c\right )}{240 \, d} \]

[In]

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

[Out]

1/240*(240*(d*x + c)*A*a^4 + 480*(tan(d*x + c)^3 + 3*tan(d*x + c))*C*a^2*b^2 + 320*(tan(d*x + c)^3 + 3*tan(d*x
+ c))*B*a*b^3 + 80*(tan(d*x + c)^3 + 3*tan(d*x + c))*A*b^4 + 16*(3*tan(d*x + c)^5 + 10*tan(d*x + c)^3 + 15*ta
n(d*x + c))*C*b^4 - 60*C*a*b^3*(2*(3*sin(d*x + c)^3 - 5*sin(d*x + c))/(sin(d*x + c)^4 - 2*sin(d*x + c)^2 + 1)
- 3*log(sin(d*x + c) + 1) + 3*log(sin(d*x + c) - 1)) - 15*B*b^4*(2*(3*sin(d*x + c)^3 - 5*sin(d*x + c))/(sin(d*
x + c)^4 - 2*sin(d*x + c)^2 + 1) - 3*log(sin(d*x + c) + 1) + 3*log(sin(d*x + c) - 1)) - 240*C*a^3*b*(2*sin(d*x
+ c)/(sin(d*x + c)^2 - 1) - log(sin(d*x + c) + 1) + log(sin(d*x + c) - 1)) - 360*B*a^2*b^2*(2*sin(d*x + c)/(s
in(d*x + c)^2 - 1) - log(sin(d*x + c) + 1) + log(sin(d*x + c) - 1)) - 240*A*a*b^3*(2*sin(d*x + c)/(sin(d*x + c
)^2 - 1) - log(sin(d*x + c) + 1) + log(sin(d*x + c) - 1)) + 240*B*a^4*log(sec(d*x + c) + tan(d*x + c)) + 960*A
*a^3*b*log(sec(d*x + c) + tan(d*x + c)) + 240*C*a^4*tan(d*x + c) + 960*B*a^3*b*tan(d*x + c) + 1440*A*a^2*b^2*t
an(d*x + c))/d

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1140 vs. \(2 (278) = 556\).

Time = 0.39 (sec) , antiderivative size = 1140, normalized size of antiderivative = 3.93 \[ \int (a+b \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\text {Too large to display} \]

[In]

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

[Out]

1/120*(120*(d*x + c)*A*a^4 + 15*(8*B*a^4 + 32*A*a^3*b + 16*C*a^3*b + 24*B*a^2*b^2 + 16*A*a*b^3 + 12*C*a*b^3 +
3*B*b^4)*log(abs(tan(1/2*d*x + 1/2*c) + 1)) - 15*(8*B*a^4 + 32*A*a^3*b + 16*C*a^3*b + 24*B*a^2*b^2 + 16*A*a*b^
3 + 12*C*a*b^3 + 3*B*b^4)*log(abs(tan(1/2*d*x + 1/2*c) - 1)) - 2*(120*C*a^4*tan(1/2*d*x + 1/2*c)^9 + 480*B*a^3
*b*tan(1/2*d*x + 1/2*c)^9 - 240*C*a^3*b*tan(1/2*d*x + 1/2*c)^9 + 720*A*a^2*b^2*tan(1/2*d*x + 1/2*c)^9 - 360*B*
a^2*b^2*tan(1/2*d*x + 1/2*c)^9 + 720*C*a^2*b^2*tan(1/2*d*x + 1/2*c)^9 - 240*A*a*b^3*tan(1/2*d*x + 1/2*c)^9 + 4
80*B*a*b^3*tan(1/2*d*x + 1/2*c)^9 - 300*C*a*b^3*tan(1/2*d*x + 1/2*c)^9 + 120*A*b^4*tan(1/2*d*x + 1/2*c)^9 - 75
*B*b^4*tan(1/2*d*x + 1/2*c)^9 + 120*C*b^4*tan(1/2*d*x + 1/2*c)^9 - 480*C*a^4*tan(1/2*d*x + 1/2*c)^7 - 1920*B*a
^3*b*tan(1/2*d*x + 1/2*c)^7 + 480*C*a^3*b*tan(1/2*d*x + 1/2*c)^7 - 2880*A*a^2*b^2*tan(1/2*d*x + 1/2*c)^7 + 720
*B*a^2*b^2*tan(1/2*d*x + 1/2*c)^7 - 1920*C*a^2*b^2*tan(1/2*d*x + 1/2*c)^7 + 480*A*a*b^3*tan(1/2*d*x + 1/2*c)^7
- 1280*B*a*b^3*tan(1/2*d*x + 1/2*c)^7 + 120*C*a*b^3*tan(1/2*d*x + 1/2*c)^7 - 320*A*b^4*tan(1/2*d*x + 1/2*c)^7
+ 30*B*b^4*tan(1/2*d*x + 1/2*c)^7 - 160*C*b^4*tan(1/2*d*x + 1/2*c)^7 + 720*C*a^4*tan(1/2*d*x + 1/2*c)^5 + 288
0*B*a^3*b*tan(1/2*d*x + 1/2*c)^5 + 4320*A*a^2*b^2*tan(1/2*d*x + 1/2*c)^5 + 2400*C*a^2*b^2*tan(1/2*d*x + 1/2*c)
^5 + 1600*B*a*b^3*tan(1/2*d*x + 1/2*c)^5 + 400*A*b^4*tan(1/2*d*x + 1/2*c)^5 + 464*C*b^4*tan(1/2*d*x + 1/2*c)^5
- 480*C*a^4*tan(1/2*d*x + 1/2*c)^3 - 1920*B*a^3*b*tan(1/2*d*x + 1/2*c)^3 - 480*C*a^3*b*tan(1/2*d*x + 1/2*c)^3
- 2880*A*a^2*b^2*tan(1/2*d*x + 1/2*c)^3 - 720*B*a^2*b^2*tan(1/2*d*x + 1/2*c)^3 - 1920*C*a^2*b^2*tan(1/2*d*x +
1/2*c)^3 - 480*A*a*b^3*tan(1/2*d*x + 1/2*c)^3 - 1280*B*a*b^3*tan(1/2*d*x + 1/2*c)^3 - 120*C*a*b^3*tan(1/2*d*x
+ 1/2*c)^3 - 320*A*b^4*tan(1/2*d*x + 1/2*c)^3 - 30*B*b^4*tan(1/2*d*x + 1/2*c)^3 - 160*C*b^4*tan(1/2*d*x + 1/2
*c)^3 + 120*C*a^4*tan(1/2*d*x + 1/2*c) + 480*B*a^3*b*tan(1/2*d*x + 1/2*c) + 240*C*a^3*b*tan(1/2*d*x + 1/2*c) +
720*A*a^2*b^2*tan(1/2*d*x + 1/2*c) + 360*B*a^2*b^2*tan(1/2*d*x + 1/2*c) + 720*C*a^2*b^2*tan(1/2*d*x + 1/2*c)
+ 240*A*a*b^3*tan(1/2*d*x + 1/2*c) + 480*B*a*b^3*tan(1/2*d*x + 1/2*c) + 300*C*a*b^3*tan(1/2*d*x + 1/2*c) + 120
*A*b^4*tan(1/2*d*x + 1/2*c) + 75*B*b^4*tan(1/2*d*x + 1/2*c) + 120*C*b^4*tan(1/2*d*x + 1/2*c))/(tan(1/2*d*x + 1
/2*c)^2 - 1)^5)/d

Mupad [B] (verification not implemented)

Time = 20.10 (sec) , antiderivative size = 4068, normalized size of antiderivative = 14.03 \[ \int (a+b \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\text {Too large to display} \]

[In]

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

[Out]

(atan(((tan(c/2 + (d*x)/2)*(32*A^2*a^8 + 32*B^2*a^8 + (9*B^2*b^8)/2 + 128*A^2*a^2*b^6 + 512*A^2*a^4*b^4 + 512*
A^2*a^6*b^2 + 72*B^2*a^2*b^6 + 312*B^2*a^4*b^4 + 192*B^2*a^6*b^2 + 72*C^2*a^2*b^6 + 192*C^2*a^4*b^4 + 128*C^2*
a^6*b^2 + 48*A*B*a*b^7 + 256*A*B*a^7*b + 36*B*C*a*b^7 + 128*B*C*a^7*b + 480*A*B*a^3*b^5 + 896*A*B*a^5*b^3 + 19
2*A*C*a^2*b^6 + 640*A*C*a^4*b^4 + 512*A*C*a^6*b^2 + 336*B*C*a^3*b^5 + 480*B*C*a^5*b^3) + (B*a^4 + (3*B*b^4)/8
+ 3*B*a^2*b^2 + 2*A*a*b^3 + 4*A*a^3*b + (3*C*a*b^3)/2 + 2*C*a^3*b)*(32*A*a^4 + 32*B*a^4 + 12*B*b^4 + 96*B*a^2*
b^2 + 64*A*a*b^3 + 128*A*a^3*b + 48*C*a*b^3 + 64*C*a^3*b))*(B*a^4 + (3*B*b^4)/8 + 3*B*a^2*b^2 + 2*A*a*b^3 + 4*
A*a^3*b + (3*C*a*b^3)/2 + 2*C*a^3*b)*1i + (tan(c/2 + (d*x)/2)*(32*A^2*a^8 + 32*B^2*a^8 + (9*B^2*b^8)/2 + 128*A
^2*a^2*b^6 + 512*A^2*a^4*b^4 + 512*A^2*a^6*b^2 + 72*B^2*a^2*b^6 + 312*B^2*a^4*b^4 + 192*B^2*a^6*b^2 + 72*C^2*a
^2*b^6 + 192*C^2*a^4*b^4 + 128*C^2*a^6*b^2 + 48*A*B*a*b^7 + 256*A*B*a^7*b + 36*B*C*a*b^7 + 128*B*C*a^7*b + 480
*A*B*a^3*b^5 + 896*A*B*a^5*b^3 + 192*A*C*a^2*b^6 + 640*A*C*a^4*b^4 + 512*A*C*a^6*b^2 + 336*B*C*a^3*b^5 + 480*B
*C*a^5*b^3) - (B*a^4 + (3*B*b^4)/8 + 3*B*a^2*b^2 + 2*A*a*b^3 + 4*A*a^3*b + (3*C*a*b^3)/2 + 2*C*a^3*b)*(32*A*a^
4 + 32*B*a^4 + 12*B*b^4 + 96*B*a^2*b^2 + 64*A*a*b^3 + 128*A*a^3*b + 48*C*a*b^3 + 64*C*a^3*b))*(B*a^4 + (3*B*b^
4)/8 + 3*B*a^2*b^2 + 2*A*a*b^3 + 4*A*a^3*b + (3*C*a*b^3)/2 + 2*C*a^3*b)*1i)/((tan(c/2 + (d*x)/2)*(32*A^2*a^8 +
32*B^2*a^8 + (9*B^2*b^8)/2 + 128*A^2*a^2*b^6 + 512*A^2*a^4*b^4 + 512*A^2*a^6*b^2 + 72*B^2*a^2*b^6 + 312*B^2*a
^4*b^4 + 192*B^2*a^6*b^2 + 72*C^2*a^2*b^6 + 192*C^2*a^4*b^4 + 128*C^2*a^6*b^2 + 48*A*B*a*b^7 + 256*A*B*a^7*b +
36*B*C*a*b^7 + 128*B*C*a^7*b + 480*A*B*a^3*b^5 + 896*A*B*a^5*b^3 + 192*A*C*a^2*b^6 + 640*A*C*a^4*b^4 + 512*A*
C*a^6*b^2 + 336*B*C*a^3*b^5 + 480*B*C*a^5*b^3) - (B*a^4 + (3*B*b^4)/8 + 3*B*a^2*b^2 + 2*A*a*b^3 + 4*A*a^3*b +
(3*C*a*b^3)/2 + 2*C*a^3*b)*(32*A*a^4 + 32*B*a^4 + 12*B*b^4 + 96*B*a^2*b^2 + 64*A*a*b^3 + 128*A*a^3*b + 48*C*a*
b^3 + 64*C*a^3*b))*(B*a^4 + (3*B*b^4)/8 + 3*B*a^2*b^2 + 2*A*a*b^3 + 4*A*a^3*b + (3*C*a*b^3)/2 + 2*C*a^3*b) - (
tan(c/2 + (d*x)/2)*(32*A^2*a^8 + 32*B^2*a^8 + (9*B^2*b^8)/2 + 128*A^2*a^2*b^6 + 512*A^2*a^4*b^4 + 512*A^2*a^6*
b^2 + 72*B^2*a^2*b^6 + 312*B^2*a^4*b^4 + 192*B^2*a^6*b^2 + 72*C^2*a^2*b^6 + 192*C^2*a^4*b^4 + 128*C^2*a^6*b^2
+ 48*A*B*a*b^7 + 256*A*B*a^7*b + 36*B*C*a*b^7 + 128*B*C*a^7*b + 480*A*B*a^3*b^5 + 896*A*B*a^5*b^3 + 192*A*C*a^
2*b^6 + 640*A*C*a^4*b^4 + 512*A*C*a^6*b^2 + 336*B*C*a^3*b^5 + 480*B*C*a^5*b^3) + (B*a^4 + (3*B*b^4)/8 + 3*B*a^
2*b^2 + 2*A*a*b^3 + 4*A*a^3*b + (3*C*a*b^3)/2 + 2*C*a^3*b)*(32*A*a^4 + 32*B*a^4 + 12*B*b^4 + 96*B*a^2*b^2 + 64
*A*a*b^3 + 128*A*a^3*b + 48*C*a*b^3 + 64*C*a^3*b))*(B*a^4 + (3*B*b^4)/8 + 3*B*a^2*b^2 + 2*A*a*b^3 + 4*A*a^3*b
+ (3*C*a*b^3)/2 + 2*C*a^3*b) + 64*A*B^2*a^12 - 64*A^2*B*a^12 - 256*A^3*a^11*b + 256*A^3*a^6*b^6 + 1024*A^3*a^8
*b^4 - 128*A^3*a^9*b^3 + 1024*A^3*a^10*b^2 + 512*A^2*B*a^11*b - 128*A^2*C*a^11*b + 9*A*B^2*a^4*b^8 + 144*A*B^2
*a^6*b^6 + 624*A*B^2*a^8*b^4 + 384*A*B^2*a^10*b^2 + 96*A^2*B*a^5*b^7 + 960*A^2*B*a^7*b^5 - 24*A^2*B*a^8*b^4 +
1792*A^2*B*a^9*b^3 - 192*A^2*B*a^10*b^2 + 144*A*C^2*a^6*b^6 + 384*A*C^2*a^8*b^4 + 256*A*C^2*a^10*b^2 + 384*A^2
*C*a^6*b^6 + 1280*A^2*C*a^8*b^4 - 96*A^2*C*a^9*b^3 + 1024*A^2*C*a^10*b^2 + 256*A*B*C*a^11*b + 72*A*B*C*a^5*b^7
+ 672*A*B*C*a^7*b^5 + 960*A*B*C*a^9*b^3))*(B*a^4*2i + (B*b^4*3i)/4 + B*a^2*b^2*6i + A*a*b^3*4i + A*a^3*b*8i +
C*a*b^3*3i + C*a^3*b*4i))/d - (tan(c/2 + (d*x)/2)*(2*A*b^4 + (5*B*b^4)/4 + 2*C*a^4 + 2*C*b^4 + 12*A*a^2*b^2 +
6*B*a^2*b^2 + 12*C*a^2*b^2 + 4*A*a*b^3 + 8*B*a*b^3 + 8*B*a^3*b + 5*C*a*b^3 + 4*C*a^3*b) + tan(c/2 + (d*x)/2)^
9*(2*A*b^4 - (5*B*b^4)/4 + 2*C*a^4 + 2*C*b^4 + 12*A*a^2*b^2 - 6*B*a^2*b^2 + 12*C*a^2*b^2 - 4*A*a*b^3 + 8*B*a*b
^3 + 8*B*a^3*b - 5*C*a*b^3 - 4*C*a^3*b) - tan(c/2 + (d*x)/2)^3*((16*A*b^4)/3 + (B*b^4)/2 + 8*C*a^4 + (8*C*b^4)
/3 + 48*A*a^2*b^2 + 12*B*a^2*b^2 + 32*C*a^2*b^2 + 8*A*a*b^3 + (64*B*a*b^3)/3 + 32*B*a^3*b + 2*C*a*b^3 + 8*C*a^
3*b) - tan(c/2 + (d*x)/2)^7*((16*A*b^4)/3 - (B*b^4)/2 + 8*C*a^4 + (8*C*b^4)/3 + 48*A*a^2*b^2 - 12*B*a^2*b^2 +
32*C*a^2*b^2 - 8*A*a*b^3 + (64*B*a*b^3)/3 + 32*B*a^3*b - 2*C*a*b^3 - 8*C*a^3*b) + tan(c/2 + (d*x)/2)^5*((20*A*
b^4)/3 + 12*C*a^4 + (116*C*b^4)/15 + 72*A*a^2*b^2 + 40*C*a^2*b^2 + (80*B*a*b^3)/3 + 48*B*a^3*b))/(d*(5*tan(c/2
+ (d*x)/2)^2 - 10*tan(c/2 + (d*x)/2)^4 + 10*tan(c/2 + (d*x)/2)^6 - 5*tan(c/2 + (d*x)/2)^8 + tan(c/2 + (d*x)/2
)^10 - 1)) + (2*A*a^4*atan((A*a^4*(tan(c/2 + (d*x)/2)*(32*A^2*a^8 + 32*B^2*a^8 + (9*B^2*b^8)/2 + 128*A^2*a^2*b
^6 + 512*A^2*a^4*b^4 + 512*A^2*a^6*b^2 + 72*B^2*a^2*b^6 + 312*B^2*a^4*b^4 + 192*B^2*a^6*b^2 + 72*C^2*a^2*b^6 +
192*C^2*a^4*b^4 + 128*C^2*a^6*b^2 + 48*A*B*a*b^7 + 256*A*B*a^7*b + 36*B*C*a*b^7 + 128*B*C*a^7*b + 480*A*B*a^3
*b^5 + 896*A*B*a^5*b^3 + 192*A*C*a^2*b^6 + 640*A*C*a^4*b^4 + 512*A*C*a^6*b^2 + 336*B*C*a^3*b^5 + 480*B*C*a^5*b
^3) - A*a^4*(32*A*a^4 + 32*B*a^4 + 12*B*b^4 + 96*B*a^2*b^2 + 64*A*a*b^3 + 128*A*a^3*b + 48*C*a*b^3 + 64*C*a^3*
b)*1i) + A*a^4*(tan(c/2 + (d*x)/2)*(32*A^2*a^8 + 32*B^2*a^8 + (9*B^2*b^8)/2 + 128*A^2*a^2*b^6 + 512*A^2*a^4*b^
4 + 512*A^2*a^6*b^2 + 72*B^2*a^2*b^6 + 312*B^2*a^4*b^4 + 192*B^2*a^6*b^2 + 72*C^2*a^2*b^6 + 192*C^2*a^4*b^4 +
128*C^2*a^6*b^2 + 48*A*B*a*b^7 + 256*A*B*a^7*b + 36*B*C*a*b^7 + 128*B*C*a^7*b + 480*A*B*a^3*b^5 + 896*A*B*a^5*
b^3 + 192*A*C*a^2*b^6 + 640*A*C*a^4*b^4 + 512*A*C*a^6*b^2 + 336*B*C*a^3*b^5 + 480*B*C*a^5*b^3) + A*a^4*(32*A*a
^4 + 32*B*a^4 + 12*B*b^4 + 96*B*a^2*b^2 + 64*A*a*b^3 + 128*A*a^3*b + 48*C*a*b^3 + 64*C*a^3*b)*1i))/(64*A*B^2*a
^12 - 64*A^2*B*a^12 - 256*A^3*a^11*b + A*a^4*(tan(c/2 + (d*x)/2)*(32*A^2*a^8 + 32*B^2*a^8 + (9*B^2*b^8)/2 + 12
8*A^2*a^2*b^6 + 512*A^2*a^4*b^4 + 512*A^2*a^6*b^2 + 72*B^2*a^2*b^6 + 312*B^2*a^4*b^4 + 192*B^2*a^6*b^2 + 72*C^
2*a^2*b^6 + 192*C^2*a^4*b^4 + 128*C^2*a^6*b^2 + 48*A*B*a*b^7 + 256*A*B*a^7*b + 36*B*C*a*b^7 + 128*B*C*a^7*b +
480*A*B*a^3*b^5 + 896*A*B*a^5*b^3 + 192*A*C*a^2*b^6 + 640*A*C*a^4*b^4 + 512*A*C*a^6*b^2 + 336*B*C*a^3*b^5 + 48
0*B*C*a^5*b^3) - A*a^4*(32*A*a^4 + 32*B*a^4 + 12*B*b^4 + 96*B*a^2*b^2 + 64*A*a*b^3 + 128*A*a^3*b + 48*C*a*b^3
+ 64*C*a^3*b)*1i)*1i - A*a^4*(tan(c/2 + (d*x)/2)*(32*A^2*a^8 + 32*B^2*a^8 + (9*B^2*b^8)/2 + 128*A^2*a^2*b^6 +
512*A^2*a^4*b^4 + 512*A^2*a^6*b^2 + 72*B^2*a^2*b^6 + 312*B^2*a^4*b^4 + 192*B^2*a^6*b^2 + 72*C^2*a^2*b^6 + 192*
C^2*a^4*b^4 + 128*C^2*a^6*b^2 + 48*A*B*a*b^7 + 256*A*B*a^7*b + 36*B*C*a*b^7 + 128*B*C*a^7*b + 480*A*B*a^3*b^5
+ 896*A*B*a^5*b^3 + 192*A*C*a^2*b^6 + 640*A*C*a^4*b^4 + 512*A*C*a^6*b^2 + 336*B*C*a^3*b^5 + 480*B*C*a^5*b^3) +
A*a^4*(32*A*a^4 + 32*B*a^4 + 12*B*b^4 + 96*B*a^2*b^2 + 64*A*a*b^3 + 128*A*a^3*b + 48*C*a*b^3 + 64*C*a^3*b)*1i
)*1i + 256*A^3*a^6*b^6 + 1024*A^3*a^8*b^4 - 128*A^3*a^9*b^3 + 1024*A^3*a^10*b^2 + 512*A^2*B*a^11*b - 128*A^2*C
*a^11*b + 9*A*B^2*a^4*b^8 + 144*A*B^2*a^6*b^6 + 624*A*B^2*a^8*b^4 + 384*A*B^2*a^10*b^2 + 96*A^2*B*a^5*b^7 + 96
0*A^2*B*a^7*b^5 - 24*A^2*B*a^8*b^4 + 1792*A^2*B*a^9*b^3 - 192*A^2*B*a^10*b^2 + 144*A*C^2*a^6*b^6 + 384*A*C^2*a
^8*b^4 + 256*A*C^2*a^10*b^2 + 384*A^2*C*a^6*b^6 + 1280*A^2*C*a^8*b^4 - 96*A^2*C*a^9*b^3 + 1024*A^2*C*a^10*b^2
+ 256*A*B*C*a^11*b + 72*A*B*C*a^5*b^7 + 672*A*B*C*a^7*b^5 + 960*A*B*C*a^9*b^3)))/d