\(\int (a+b \cos (c+d x))^4 (A+B \cos (c+d x)+C \cos ^2(c+d x)) \sec ^3(c+d x) \, dx\) [968]

Optimal result

Integrand size = 41, antiderivative size = 274 \[ \int (a+b \cos (c+d x))^4 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^3(c+d x) \, dx=\frac {1}{2} b \left (12 a^2 b B+b^3 B+8 a^3 C+4 a b^2 (2 A+C)\right ) x+\frac {a^2 \left (12 A b^2+8 a b B+a^2 (A+2 C)\right ) \text {arctanh}(\sin (c+d x))}{2 d}-\frac {b \left (12 a^3 B-24 a b^2 B+a^2 b (39 A-34 C)-2 b^3 (3 A+2 C)\right ) \sin (c+d x)}{6 d}-\frac {b^2 \left (6 a^2 B-3 b^2 B+2 a b (9 A-4 C)\right ) \cos (c+d x) \sin (c+d x)}{6 d}-\frac {b (15 A b+6 a B-2 b C) (a+b \cos (c+d x))^2 \sin (c+d x)}{6 d}+\frac {(2 A b+a B) (a+b \cos (c+d x))^3 \tan (c+d x)}{d}+\frac {A (a+b \cos (c+d x))^4 \sec (c+d x) \tan (c+d x)}{2 d} \]

[Out]

1/2*b*(12*B*a^2*b+B*b^3+8*a^3*C+4*a*b^2*(2*A+C))*x+1/2*a^2*(12*A*b^2+8*B*a*b+a^2*(A+2*C))*arctanh(sin(d*x+c))/
d-1/6*b*(12*B*a^3-24*B*a*b^2+a^2*b*(39*A-34*C)-2*b^3*(3*A+2*C))*sin(d*x+c)/d-1/6*b^2*(6*B*a^2-3*B*b^2+2*a*b*(9
*A-4*C))*cos(d*x+c)*sin(d*x+c)/d-1/6*b*(15*A*b+6*B*a-2*C*b)*(a+b*cos(d*x+c))^2*sin(d*x+c)/d+(2*A*b+B*a)*(a+b*c
os(d*x+c))^3*tan(d*x+c)/d+1/2*A*(a+b*cos(d*x+c))^4*sec(d*x+c)*tan(d*x+c)/d

Rubi [A] (verified)

Time = 1.07 (sec) , antiderivative size = 274, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.146, Rules used = {3126, 3128, 3112, 3102, 2814, 3855} \[ \int (a+b \cos (c+d x))^4 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^3(c+d x) \, dx=\frac {a^2 \left (a^2 (A+2 C)+8 a b B+12 A b^2\right ) \text {arctanh}(\sin (c+d x))}{2 d}-\frac {b^2 \sin (c+d x) \cos (c+d x) \left (6 a^2 B+2 a b (9 A-4 C)-3 b^2 B\right )}{6 d}-\frac {b \sin (c+d x) \left (12 a^3 B+a^2 b (39 A-34 C)-24 a b^2 B-2 b^3 (3 A+2 C)\right )}{6 d}+\frac {1}{2} b x \left (8 a^3 C+12 a^2 b B+4 a b^2 (2 A+C)+b^3 B\right )-\frac {b \sin (c+d x) (6 a B+15 A b-2 b C) (a+b \cos (c+d x))^2}{6 d}+\frac {(a B+2 A b) \tan (c+d x) (a+b \cos (c+d x))^3}{d}+\frac {A \tan (c+d x) \sec (c+d x) (a+b \cos (c+d x))^4}{2 d} \]

[In]

Int[(a + b*Cos[c + d*x])^4*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^3,x]

[Out]

(b*(12*a^2*b*B + b^3*B + 8*a^3*C + 4*a*b^2*(2*A + C))*x)/2 + (a^2*(12*A*b^2 + 8*a*b*B + a^2*(A + 2*C))*ArcTanh
[Sin[c + d*x]])/(2*d) - (b*(12*a^3*B - 24*a*b^2*B + a^2*b*(39*A - 34*C) - 2*b^3*(3*A + 2*C))*Sin[c + d*x])/(6*
d) - (b^2*(6*a^2*B - 3*b^2*B + 2*a*b*(9*A - 4*C))*Cos[c + d*x]*Sin[c + d*x])/(6*d) - (b*(15*A*b + 6*a*B - 2*b*
C)*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(6*d) + ((2*A*b + a*B)*(a + b*Cos[c + d*x])^3*Tan[c + d*x])/d + (A*(a
+ b*Cos[c + d*x])^4*Sec[c + d*x]*Tan[c + d*x])/(2*d)

Rule 2814

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])/((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[b*(x/d)
, x] - Dist[(b*c - a*d)/d, Int[1/(c + d*Sin[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d
, 0]

Rule 3102

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (
f_.)*(x_)]^2), x_Symbol] :> Simp[(-C)*Cos[e + f*x]*((a + b*Sin[e + f*x])^(m + 1)/(b*f*(m + 2))), x] + Dist[1/(
b*(m + 2)), Int[(a + b*Sin[e + f*x])^m*Simp[A*b*(m + 2) + b*C*(m + 1) + (b*B*(m + 2) - a*C)*Sin[e + f*x], x],
x], x] /; FreeQ[{a, b, e, f, A, B, C, m}, x] &&  !LtQ[m, -1]

Rule 3112

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((c_) + (d_.)*sin[(e_.) + (f_.)*(x_)])*((A_.) + (B_.)*sin[(e
_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(-C)*d*Cos[e + f*x]*Sin[e + f*x]*((a +
b*Sin[e + f*x])^(m + 1)/(b*f*(m + 3))), x] + Dist[1/(b*(m + 3)), Int[(a + b*Sin[e + f*x])^m*Simp[a*C*d + A*b*
c*(m + 3) + b*(B*c*(m + 3) + d*(C*(m + 2) + A*(m + 3)))*Sin[e + f*x] - (2*a*C*d - b*(c*C + B*d)*(m + 3))*Sin[e
+ f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C, m}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] &&
!LtQ[m, -1]

Rule 3126

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((A_.) + (B_.)*s
in[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(-(c^2*C - B*c*d + A*d^2))*Cos[e
+ f*x]*(a + b*Sin[e + f*x])^m*((c + d*Sin[e + f*x])^(n + 1)/(d*f*(n + 1)*(c^2 - d^2))), x] + Dist[1/(d*(n + 1)
*(c^2 - d^2)), Int[(a + b*Sin[e + f*x])^(m - 1)*(c + d*Sin[e + f*x])^(n + 1)*Simp[A*d*(b*d*m + a*c*(n + 1)) +
(c*C - B*d)*(b*c*m + a*d*(n + 1)) - (d*(A*(a*d*(n + 2) - b*c*(n + 1)) + B*(b*d*(n + 1) - a*c*(n + 2))) - C*(b*
c*d*(n + 1) - a*(c^2 + d^2*(n + 1))))*Sin[e + f*x] + b*(d*(B*c - A*d)*(m + n + 2) - C*(c^2*(m + 1) + d^2*(n +
1)))*Sin[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2
, 0] && NeQ[c^2 - d^2, 0] && GtQ[m, 0] && LtQ[n, -1]

Rule 3128

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_.)*((A_.) + (B_.)
*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(-C)*Cos[e + f*x]*(a + b*Sin[e
+ f*x])^m*((c + d*Sin[e + f*x])^(n + 1)/(d*f*(m + n + 2))), x] + Dist[1/(d*(m + n + 2)), Int[(a + b*Sin[e + f*
x])^(m - 1)*(c + d*Sin[e + f*x])^n*Simp[a*A*d*(m + n + 2) + C*(b*c*m + a*d*(n + 1)) + (d*(A*b + a*B)*(m + n +
2) - C*(a*c - b*d*(m + n + 1)))*Sin[e + f*x] + (C*(a*d*m - b*c*(m + 1)) + b*B*d*(m + n + 2))*Sin[e + f*x]^2, x
], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d
^2, 0] && GtQ[m, 0] &&  !(IGtQ[n, 0] && ( !IntegerQ[m] || (EqQ[a, 0] && NeQ[c, 0])))

Rule 3855

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

Rubi steps \begin{align*} \text {integral}& = \frac {A (a+b \cos (c+d x))^4 \sec (c+d x) \tan (c+d x)}{2 d}+\frac {1}{2} \int (a+b \cos (c+d x))^3 \left (2 (2 A b+a B)+(2 b B+a (A+2 C)) \cos (c+d x)-b (3 A-2 C) \cos ^2(c+d x)\right ) \sec ^2(c+d x) \, dx \\ & = \frac {(2 A b+a B) (a+b \cos (c+d x))^3 \tan (c+d x)}{d}+\frac {A (a+b \cos (c+d x))^4 \sec (c+d x) \tan (c+d x)}{2 d}+\frac {1}{2} \int (a+b \cos (c+d x))^2 \left (12 A b^2+8 a b B+a^2 (A+2 C)-2 b (a A-b B-2 a C) \cos (c+d x)-b (15 A b+6 a B-2 b C) \cos ^2(c+d x)\right ) \sec (c+d x) \, dx \\ & = -\frac {b (15 A b+6 a B-2 b C) (a+b \cos (c+d x))^2 \sin (c+d x)}{6 d}+\frac {(2 A b+a B) (a+b \cos (c+d x))^3 \tan (c+d x)}{d}+\frac {A (a+b \cos (c+d x))^4 \sec (c+d x) \tan (c+d x)}{2 d}+\frac {1}{6} \int (a+b \cos (c+d x)) \left (3 a \left (12 A b^2+8 a b B+a^2 (A+2 C)\right )+b \left (18 a b B-3 a^2 (A-6 C)+2 b^2 (3 A+2 C)\right ) \cos (c+d x)-2 b \left (18 a A b+6 a^2 B-3 b^2 B-8 a b C\right ) \cos ^2(c+d x)\right ) \sec (c+d x) \, dx \\ & = -\frac {b^2 \left (6 a^2 B-3 b^2 B+2 a b (9 A-4 C)\right ) \cos (c+d x) \sin (c+d x)}{6 d}-\frac {b (15 A b+6 a B-2 b C) (a+b \cos (c+d x))^2 \sin (c+d x)}{6 d}+\frac {(2 A b+a B) (a+b \cos (c+d x))^3 \tan (c+d x)}{d}+\frac {A (a+b \cos (c+d x))^4 \sec (c+d x) \tan (c+d x)}{2 d}+\frac {1}{12} \int \left (6 a^2 \left (12 A b^2+8 a b B+a^2 (A+2 C)\right )+6 b \left (12 a^2 b B+b^3 B+8 a^3 C+4 a b^2 (2 A+C)\right ) \cos (c+d x)-2 b \left (12 a^3 B-24 a b^2 B-2 b^3 (3 A+2 C)+a^2 (39 A b-34 b C)\right ) \cos ^2(c+d x)\right ) \sec (c+d x) \, dx \\ & = -\frac {b \left (12 a^3 B-24 a b^2 B+a^2 b (39 A-34 C)-2 b^3 (3 A+2 C)\right ) \sin (c+d x)}{6 d}-\frac {b^2 \left (6 a^2 B-3 b^2 B+2 a b (9 A-4 C)\right ) \cos (c+d x) \sin (c+d x)}{6 d}-\frac {b (15 A b+6 a B-2 b C) (a+b \cos (c+d x))^2 \sin (c+d x)}{6 d}+\frac {(2 A b+a B) (a+b \cos (c+d x))^3 \tan (c+d x)}{d}+\frac {A (a+b \cos (c+d x))^4 \sec (c+d x) \tan (c+d x)}{2 d}+\frac {1}{12} \int \left (6 a^2 \left (12 A b^2+8 a b B+a^2 (A+2 C)\right )+6 b \left (12 a^2 b B+b^3 B+8 a^3 C+4 a b^2 (2 A+C)\right ) \cos (c+d x)\right ) \sec (c+d x) \, dx \\ & = \frac {1}{2} b \left (12 a^2 b B+b^3 B+8 a^3 C+4 a b^2 (2 A+C)\right ) x-\frac {b \left (12 a^3 B-24 a b^2 B+a^2 b (39 A-34 C)-2 b^3 (3 A+2 C)\right ) \sin (c+d x)}{6 d}-\frac {b^2 \left (6 a^2 B-3 b^2 B+2 a b (9 A-4 C)\right ) \cos (c+d x) \sin (c+d x)}{6 d}-\frac {b (15 A b+6 a B-2 b C) (a+b \cos (c+d x))^2 \sin (c+d x)}{6 d}+\frac {(2 A b+a B) (a+b \cos (c+d x))^3 \tan (c+d x)}{d}+\frac {A (a+b \cos (c+d x))^4 \sec (c+d x) \tan (c+d x)}{2 d}+\frac {1}{2} \left (a^2 \left (12 A b^2+8 a b B+a^2 (A+2 C)\right )\right ) \int \sec (c+d x) \, dx \\ & = \frac {1}{2} b \left (12 a^2 b B+b^3 B+8 a^3 C+4 a b^2 (2 A+C)\right ) x+\frac {a^2 \left (12 A b^2+8 a b B+a^2 (A+2 C)\right ) \text {arctanh}(\sin (c+d x))}{2 d}-\frac {b \left (12 a^3 B-24 a b^2 B+a^2 b (39 A-34 C)-2 b^3 (3 A+2 C)\right ) \sin (c+d x)}{6 d}-\frac {b^2 \left (6 a^2 B-3 b^2 B+2 a b (9 A-4 C)\right ) \cos (c+d x) \sin (c+d x)}{6 d}-\frac {b (15 A b+6 a B-2 b C) (a+b \cos (c+d x))^2 \sin (c+d x)}{6 d}+\frac {(2 A b+a B) (a+b \cos (c+d x))^3 \tan (c+d x)}{d}+\frac {A (a+b \cos (c+d x))^4 \sec (c+d x) \tan (c+d x)}{2 d} \\ \end{align*}

Mathematica [A] (verified)

Time = 9.25 (sec) , antiderivative size = 367, normalized size of antiderivative = 1.34 \[ \int (a+b \cos (c+d x))^4 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^3(c+d x) \, dx=\frac {6 b \left (12 a^2 b B+b^3 B+8 a^3 C+4 a b^2 (2 A+C)\right ) (c+d x)-6 a^2 \left (12 A b^2+8 a b B+a^2 (A+2 C)\right ) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )+6 a^2 \left (12 A b^2+8 a b B+a^2 (A+2 C)\right ) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )+\frac {3 a^4 A}{\left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )^2}+\frac {12 a^3 (4 A b+a B) \sin \left (\frac {1}{2} (c+d x)\right )}{\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )}-\frac {3 a^4 A}{\left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^2}+\frac {12 a^3 (4 A b+a B) \sin \left (\frac {1}{2} (c+d x)\right )}{\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )}+3 b^2 \left (4 A b^2+16 a b B+24 a^2 C+3 b^2 C\right ) \sin (c+d x)+3 b^3 (b B+4 a C) \sin (2 (c+d x))+b^4 C \sin (3 (c+d x))}{12 d} \]

[In]

Integrate[(a + b*Cos[c + d*x])^4*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^3,x]

[Out]

(6*b*(12*a^2*b*B + b^3*B + 8*a^3*C + 4*a*b^2*(2*A + C))*(c + d*x) - 6*a^2*(12*A*b^2 + 8*a*b*B + a^2*(A + 2*C))
*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 6*a^2*(12*A*b^2 + 8*a*b*B + a^2*(A + 2*C))*Log[Cos[(c + d*x)/2] +
Sin[(c + d*x)/2]] + (3*a^4*A)/(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^2 + (12*a^3*(4*A*b + a*B)*Sin[(c + d*x)/2]
)/(Cos[(c + d*x)/2] - Sin[(c + d*x)/2]) - (3*a^4*A)/(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^2 + (12*a^3*(4*A*b +
a*B)*Sin[(c + d*x)/2])/(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]) + 3*b^2*(4*A*b^2 + 16*a*b*B + 24*a^2*C + 3*b^2*C
)*Sin[c + d*x] + 3*b^3*(b*B + 4*a*C)*Sin[2*(c + d*x)] + b^4*C*Sin[3*(c + d*x)])/(12*d)

Maple [A] (verified)

Time = 0.82 (sec) , antiderivative size = 233, normalized size of antiderivative = 0.85

method	result	size
parts	\(\frac {A \,a^{4} \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )}{d}+\frac {\left (4 A \,a^{3} b +B \,a^{4}\right ) \tan \left (d x +c \right )}{d}+\frac {\left (B \,b^{4}+4 C a \,b^{3}\right ) \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )}{d}+\frac {\left (A \,b^{4}+4 B a \,b^{3}+6 C \,a^{2} b^{2}\right ) \sin \left (d x +c \right )}{d}+\frac {\left (4 a A \,b^{3}+6 B \,a^{2} b^{2}+4 a^{3} b C \right ) \left (d x +c \right )}{d}+\frac {\left (6 A \,a^{2} b^{2}+4 B \,a^{3} b +a^{4} C \right ) \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{d}+\frac {C \,b^{4} \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3 d}\)	\(233\)
derivativedivides	\(\frac {A \,a^{4} \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )+B \,a^{4} \tan \left (d x +c \right )+a^{4} C \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+4 A \,a^{3} b \tan \left (d x +c \right )+4 B \,a^{3} b \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+4 a^{3} b C \left (d x +c \right )+6 A \,a^{2} b^{2} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+6 B \,a^{2} b^{2} \left (d x +c \right )+6 C \sin \left (d x +c \right ) a^{2} b^{2}+4 a A \,b^{3} \left (d x +c \right )+4 B \sin \left (d x +c \right ) a \,b^{3}+4 C a \,b^{3} \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+A \sin \left (d x +c \right ) b^{4}+B \,b^{4} \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+\frac {C \,b^{4} \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3}}{d}\)	\(284\)
default	\(\frac {A \,a^{4} \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )+B \,a^{4} \tan \left (d x +c \right )+a^{4} C \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+4 A \,a^{3} b \tan \left (d x +c \right )+4 B \,a^{3} b \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+4 a^{3} b C \left (d x +c \right )+6 A \,a^{2} b^{2} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+6 B \,a^{2} b^{2} \left (d x +c \right )+6 C \sin \left (d x +c \right ) a^{2} b^{2}+4 a A \,b^{3} \left (d x +c \right )+4 B \sin \left (d x +c \right ) a \,b^{3}+4 C a \,b^{3} \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+A \sin \left (d x +c \right ) b^{4}+B \,b^{4} \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+\frac {C \,b^{4} \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3}}{d}\)	\(284\)
parallelrisch	\(\frac {-12 \left (1+\cos \left (2 d x +2 c \right )\right ) a^{2} \left (12 A \,b^{2}+8 B a b +a^{2} \left (A +2 C \right )\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )+12 \left (1+\cos \left (2 d x +2 c \right )\right ) a^{2} \left (12 A \,b^{2}+8 B a b +a^{2} \left (A +2 C \right )\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )+96 \left (a^{3} C +\frac {3 B \,a^{2} b}{2}+a \left (A +\frac {C}{2}\right ) b^{2}+\frac {B \,b^{3}}{8}\right ) x d b \cos \left (2 d x +2 c \right )+6 \left (16 A \,a^{3} b +4 B \,a^{4}+B \,b^{4}+4 C a \,b^{3}\right ) \sin \left (2 d x +2 c \right )+12 \left (6 a^{2} C +4 B a b +b^{2} \left (A +\frac {11 C}{12}\right )\right ) b^{2} \sin \left (3 d x +3 c \right )+3 \left (B \,b^{4}+4 C a \,b^{3}\right ) \sin \left (4 d x +4 c \right )+C \sin \left (5 d x +5 c \right ) b^{4}+12 \left (2 A \,a^{4}+6 C \,a^{2} b^{2}+4 B a \,b^{3}+\left (A +\frac {5 C}{6}\right ) b^{4}\right ) \sin \left (d x +c \right )+96 \left (a^{3} C +\frac {3 B \,a^{2} b}{2}+a \left (A +\frac {C}{2}\right ) b^{2}+\frac {B \,b^{3}}{8}\right ) x d b}{24 d \left (1+\cos \left (2 d x +2 c \right )\right )}\)	\(345\)
risch	\(\frac {x B \,b^{4}}{2}+2 a \,b^{3} C x +6 x B \,a^{2} b^{2}+4 x \,a^{3} b C +4 x a A \,b^{3}-\frac {2 i {\mathrm e}^{i \left (d x +c \right )} B a \,b^{3}}{d}+\frac {3 i {\mathrm e}^{-i \left (d x +c \right )} C \,a^{2} b^{2}}{d}+\frac {4 a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) B b}{d}-\frac {6 a^{2} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) A \,b^{2}}{d}-\frac {4 a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) B b}{d}+\frac {i C \,b^{4} {\mathrm e}^{-3 i \left (d x +c \right )}}{24 d}-\frac {i a^{3} \left (A a \,{\mathrm e}^{3 i \left (d x +c \right )}-8 A b \,{\mathrm e}^{2 i \left (d x +c \right )}-2 B a \,{\mathrm e}^{2 i \left (d x +c \right )}-a A \,{\mathrm e}^{i \left (d x +c \right )}-8 A b -2 B a \right )}{d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{2}}+\frac {3 i {\mathrm e}^{-i \left (d x +c \right )} C \,b^{4}}{8 d}+\frac {i {\mathrm e}^{-2 i \left (d x +c \right )} B \,b^{4}}{8 d}-\frac {i {\mathrm e}^{2 i \left (d x +c \right )} C a \,b^{3}}{2 d}-\frac {3 i {\mathrm e}^{i \left (d x +c \right )} C \,a^{2} b^{2}}{d}+\frac {2 i {\mathrm e}^{-i \left (d x +c \right )} B a \,b^{3}}{d}+\frac {i {\mathrm e}^{-2 i \left (d x +c \right )} C a \,b^{3}}{2 d}-\frac {a^{4} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) C}{d}+\frac {a^{4} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) C}{d}+\frac {A \,a^{4} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}{2 d}-\frac {A \,a^{4} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )}{2 d}-\frac {i C \,b^{4} {\mathrm e}^{3 i \left (d x +c \right )}}{24 d}-\frac {i {\mathrm e}^{2 i \left (d x +c \right )} B \,b^{4}}{8 d}-\frac {i {\mathrm e}^{i \left (d x +c \right )} A \,b^{4}}{2 d}-\frac {3 i {\mathrm e}^{i \left (d x +c \right )} C \,b^{4}}{8 d}+\frac {6 a^{2} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) A \,b^{2}}{d}+\frac {i {\mathrm e}^{-i \left (d x +c \right )} A \,b^{4}}{2 d}\)	\(584\)
norman	\(\text {Expression too large to display}\)	\(1168\)

[In]

int((a+b*cos(d*x+c))^4*(A+B*cos(d*x+c)+C*cos(d*x+c)^2)*sec(d*x+c)^3,x,method=_RETURNVERBOSE)

[Out]

A*a^4/d*(1/2*sec(d*x+c)*tan(d*x+c)+1/2*ln(sec(d*x+c)+tan(d*x+c)))+(4*A*a^3*b+B*a^4)/d*tan(d*x+c)+(B*b^4+4*C*a*
b^3)/d*(1/2*cos(d*x+c)*sin(d*x+c)+1/2*d*x+1/2*c)+(A*b^4+4*B*a*b^3+6*C*a^2*b^2)/d*sin(d*x+c)+(4*A*a*b^3+6*B*a^2
*b^2+4*C*a^3*b)/d*(d*x+c)+(6*A*a^2*b^2+4*B*a^3*b+C*a^4)/d*ln(sec(d*x+c)+tan(d*x+c))+1/3*C*b^4/d*(2+cos(d*x+c)^
2)*sin(d*x+c)

Fricas [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 262, normalized size of antiderivative = 0.96 \[ \int (a+b \cos (c+d x))^4 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^3(c+d x) \, dx=\frac {6 \, {\left (8 \, C a^{3} b + 12 \, B a^{2} b^{2} + 4 \, {\left (2 \, A + C\right )} a b^{3} + B b^{4}\right )} d x \cos \left (d x + c\right )^{2} + 3 \, {\left ({\left (A + 2 \, C\right )} a^{4} + 8 \, B a^{3} b + 12 \, A a^{2} b^{2}\right )} \cos \left (d x + c\right )^{2} \log \left (\sin \left (d x + c\right ) + 1\right ) - 3 \, {\left ({\left (A + 2 \, C\right )} a^{4} + 8 \, B a^{3} b + 12 \, A a^{2} b^{2}\right )} \cos \left (d x + c\right )^{2} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \, {\left (2 \, C b^{4} \cos \left (d x + c\right )^{4} + 3 \, A a^{4} + 3 \, {\left (4 \, C a b^{3} + B b^{4}\right )} \cos \left (d x + c\right )^{3} + 2 \, {\left (18 \, C a^{2} b^{2} + 12 \, B a b^{3} + {\left (3 \, A + 2 \, C\right )} b^{4}\right )} \cos \left (d x + c\right )^{2} + 6 \, {\left (B a^{4} + 4 \, A a^{3} b\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{12 \, d \cos \left (d x + c\right )^{2}} \]

[In]

integrate((a+b*cos(d*x+c))^4*(A+B*cos(d*x+c)+C*cos(d*x+c)^2)*sec(d*x+c)^3,x, algorithm="fricas")

[Out]

1/12*(6*(8*C*a^3*b + 12*B*a^2*b^2 + 4*(2*A + C)*a*b^3 + B*b^4)*d*x*cos(d*x + c)^2 + 3*((A + 2*C)*a^4 + 8*B*a^3
*b + 12*A*a^2*b^2)*cos(d*x + c)^2*log(sin(d*x + c) + 1) - 3*((A + 2*C)*a^4 + 8*B*a^3*b + 12*A*a^2*b^2)*cos(d*x
+ c)^2*log(-sin(d*x + c) + 1) + 2*(2*C*b^4*cos(d*x + c)^4 + 3*A*a^4 + 3*(4*C*a*b^3 + B*b^4)*cos(d*x + c)^3 +
2*(18*C*a^2*b^2 + 12*B*a*b^3 + (3*A + 2*C)*b^4)*cos(d*x + c)^2 + 6*(B*a^4 + 4*A*a^3*b)*cos(d*x + c))*sin(d*x +
c))/(d*cos(d*x + c)^2)

Sympy [F(-1)]

Timed out. \[ \int (a+b \cos (c+d x))^4 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^3(c+d x) \, dx=\text {Timed out} \]

[In]

integrate((a+b*cos(d*x+c))**4*(A+B*cos(d*x+c)+C*cos(d*x+c)**2)*sec(d*x+c)**3,x)

[Out]

Timed out

Maxima [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 311, normalized size of antiderivative = 1.14 \[ \int (a+b \cos (c+d x))^4 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^3(c+d x) \, dx=\frac {48 \, {\left (d x + c\right )} C a^{3} b + 72 \, {\left (d x + c\right )} B a^{2} b^{2} + 48 \, {\left (d x + c\right )} A a b^{3} + 12 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} C a b^{3} + 3 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} B b^{4} - 4 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} C b^{4} - 3 \, A a^{4} {\left (\frac {2 \, \sin \left (d x + c\right )}{\sin \left (d x + c\right )^{2} - 1} - \log \left (\sin \left (d x + c\right ) + 1\right ) + \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 6 \, C a^{4} {\left (\log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 24 \, B a^{3} b {\left (\log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 36 \, A a^{2} b^{2} {\left (\log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 72 \, C a^{2} b^{2} \sin \left (d x + c\right ) + 48 \, B a b^{3} \sin \left (d x + c\right ) + 12 \, A b^{4} \sin \left (d x + c\right ) + 12 \, B a^{4} \tan \left (d x + c\right ) + 48 \, A a^{3} b \tan \left (d x + c\right )}{12 \, d} \]

[In]

integrate((a+b*cos(d*x+c))^4*(A+B*cos(d*x+c)+C*cos(d*x+c)^2)*sec(d*x+c)^3,x, algorithm="maxima")

[Out]

1/12*(48*(d*x + c)*C*a^3*b + 72*(d*x + c)*B*a^2*b^2 + 48*(d*x + c)*A*a*b^3 + 12*(2*d*x + 2*c + sin(2*d*x + 2*c
))*C*a*b^3 + 3*(2*d*x + 2*c + sin(2*d*x + 2*c))*B*b^4 - 4*(sin(d*x + c)^3 - 3*sin(d*x + c))*C*b^4 - 3*A*a^4*(2
*sin(d*x + c)/(sin(d*x + c)^2 - 1) - log(sin(d*x + c) + 1) + log(sin(d*x + c) - 1)) + 6*C*a^4*(log(sin(d*x + c
) + 1) - log(sin(d*x + c) - 1)) + 24*B*a^3*b*(log(sin(d*x + c) + 1) - log(sin(d*x + c) - 1)) + 36*A*a^2*b^2*(l
og(sin(d*x + c) + 1) - log(sin(d*x + c) - 1)) + 72*C*a^2*b^2*sin(d*x + c) + 48*B*a*b^3*sin(d*x + c) + 12*A*b^4
*sin(d*x + c) + 12*B*a^4*tan(d*x + c) + 48*A*a^3*b*tan(d*x + c))/d

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 541 vs. \(2 (262) = 524\).

Time = 0.39 (sec) , antiderivative size = 541, normalized size of antiderivative = 1.97 \[ \int (a+b \cos (c+d x))^4 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^3(c+d x) \, dx=\frac {3 \, {\left (8 \, C a^{3} b + 12 \, B a^{2} b^{2} + 8 \, A a b^{3} + 4 \, C a b^{3} + B b^{4}\right )} {\left (d x + c\right )} + 3 \, {\left (A a^{4} + 2 \, C a^{4} + 8 \, B a^{3} b + 12 \, A a^{2} b^{2}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right ) - 3 \, {\left (A a^{4} + 2 \, C a^{4} + 8 \, B a^{3} b + 12 \, A a^{2} b^{2}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right ) + \frac {6 \, {\left (A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 2 \, B a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 8 \, A a^{3} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 2 \, B a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 8 \, A a^{3} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )}^{2}} + \frac {2 \, {\left (36 \, C a^{2} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 24 \, B a b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 12 \, C a b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 6 \, A b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 3 \, B b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 6 \, C b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 72 \, C a^{2} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 48 \, B a b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 12 \, A b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 4 \, C b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 36 \, C a^{2} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 24 \, B a b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 12 \, C a b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 6 \, A b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 3 \, B b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 6 \, C b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{3}}}{6 \, d} \]

[In]

integrate((a+b*cos(d*x+c))^4*(A+B*cos(d*x+c)+C*cos(d*x+c)^2)*sec(d*x+c)^3,x, algorithm="giac")

[Out]

1/6*(3*(8*C*a^3*b + 12*B*a^2*b^2 + 8*A*a*b^3 + 4*C*a*b^3 + B*b^4)*(d*x + c) + 3*(A*a^4 + 2*C*a^4 + 8*B*a^3*b +
12*A*a^2*b^2)*log(abs(tan(1/2*d*x + 1/2*c) + 1)) - 3*(A*a^4 + 2*C*a^4 + 8*B*a^3*b + 12*A*a^2*b^2)*log(abs(tan
(1/2*d*x + 1/2*c) - 1)) + 6*(A*a^4*tan(1/2*d*x + 1/2*c)^3 - 2*B*a^4*tan(1/2*d*x + 1/2*c)^3 - 8*A*a^3*b*tan(1/2
*d*x + 1/2*c)^3 + A*a^4*tan(1/2*d*x + 1/2*c) + 2*B*a^4*tan(1/2*d*x + 1/2*c) + 8*A*a^3*b*tan(1/2*d*x + 1/2*c))/
(tan(1/2*d*x + 1/2*c)^2 - 1)^2 + 2*(36*C*a^2*b^2*tan(1/2*d*x + 1/2*c)^5 + 24*B*a*b^3*tan(1/2*d*x + 1/2*c)^5 -
12*C*a*b^3*tan(1/2*d*x + 1/2*c)^5 + 6*A*b^4*tan(1/2*d*x + 1/2*c)^5 - 3*B*b^4*tan(1/2*d*x + 1/2*c)^5 + 6*C*b^4*
tan(1/2*d*x + 1/2*c)^5 + 72*C*a^2*b^2*tan(1/2*d*x + 1/2*c)^3 + 48*B*a*b^3*tan(1/2*d*x + 1/2*c)^3 + 12*A*b^4*ta
n(1/2*d*x + 1/2*c)^3 + 4*C*b^4*tan(1/2*d*x + 1/2*c)^3 + 36*C*a^2*b^2*tan(1/2*d*x + 1/2*c) + 24*B*a*b^3*tan(1/2
*d*x + 1/2*c) + 12*C*a*b^3*tan(1/2*d*x + 1/2*c) + 6*A*b^4*tan(1/2*d*x + 1/2*c) + 3*B*b^4*tan(1/2*d*x + 1/2*c)
+ 6*C*b^4*tan(1/2*d*x + 1/2*c))/(tan(1/2*d*x + 1/2*c)^2 + 1)^3)/d

Mupad [B] (verification not implemented)

Time = 6.72 (sec) , antiderivative size = 4837, normalized size of antiderivative = 17.65 \[ \int (a+b \cos (c+d x))^4 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^3(c+d x) \, dx=\text {Too large to display} \]

[In]

int(((a + b*cos(c + d*x))^4*(A + B*cos(c + d*x) + C*cos(c + d*x)^2))/cos(c + d*x)^3,x)

[Out]

(tan(c/2 + (d*x)/2)^3*(4*A*a^4 + 4*B*a^4 - 2*B*b^4 - (8*C*b^4)/3 + 16*A*a^3*b - 8*C*a*b^3) + tan(c/2 + (d*x)/2
)^7*(4*A*a^4 - 4*B*a^4 + 2*B*b^4 - (8*C*b^4)/3 - 16*A*a^3*b + 8*C*a*b^3) + tan(c/2 + (d*x)/2)^9*(A*a^4 + 2*A*b
^4 - 2*B*a^4 - B*b^4 + 2*C*b^4 + 12*C*a^2*b^2 - 8*A*a^3*b + 8*B*a*b^3 - 4*C*a*b^3) + tan(c/2 + (d*x)/2)*(A*a^4
+ 2*A*b^4 + 2*B*a^4 + B*b^4 + 2*C*b^4 + 12*C*a^2*b^2 + 8*A*a^3*b + 8*B*a*b^3 + 4*C*a*b^3) - tan(c/2 + (d*x)/2
)^5*(4*A*b^4 - 6*A*a^4 - (4*C*b^4)/3 + 24*C*a^2*b^2 + 16*B*a*b^3))/(d*(tan(c/2 + (d*x)/2)^2 - 2*tan(c/2 + (d*x
)/2)^4 - 2*tan(c/2 + (d*x)/2)^6 + tan(c/2 + (d*x)/2)^8 + tan(c/2 + (d*x)/2)^10 + 1)) - (atan(((((A*a^4)/2 + C*
a^4 + 6*A*a^2*b^2 + 4*B*a^3*b)*(16*A*a^4 + 16*B*b^4 + 32*C*a^4 + 192*A*a^2*b^2 + 192*B*a^2*b^2 + 128*A*a*b^3 +
128*B*a^3*b + 64*C*a*b^3 + 128*C*a^3*b) + tan(c/2 + (d*x)/2)*(8*A^2*a^8 + 8*B^2*b^8 + 32*C^2*a^8 + 512*A^2*a^
2*b^6 + 1152*A^2*a^4*b^4 + 192*A^2*a^6*b^2 + 192*B^2*a^2*b^6 + 1152*B^2*a^4*b^4 + 512*B^2*a^6*b^2 + 128*C^2*a^
2*b^6 + 512*C^2*a^4*b^4 + 512*C^2*a^6*b^2 + 32*A*C*a^8 + 128*A*B*a*b^7 + 128*A*B*a^7*b + 64*B*C*a*b^7 + 256*B*
C*a^7*b + 1536*A*B*a^3*b^5 + 1536*A*B*a^5*b^3 + 512*A*C*a^2*b^6 + 1024*A*C*a^4*b^4 + 384*A*C*a^6*b^2 + 896*B*C
*a^3*b^5 + 1536*B*C*a^5*b^3))*((A*a^4)/2 + C*a^4 + 6*A*a^2*b^2 + 4*B*a^3*b)*1i - (((A*a^4)/2 + C*a^4 + 6*A*a^2
*b^2 + 4*B*a^3*b)*(16*A*a^4 + 16*B*b^4 + 32*C*a^4 + 192*A*a^2*b^2 + 192*B*a^2*b^2 + 128*A*a*b^3 + 128*B*a^3*b
+ 64*C*a*b^3 + 128*C*a^3*b) - tan(c/2 + (d*x)/2)*(8*A^2*a^8 + 8*B^2*b^8 + 32*C^2*a^8 + 512*A^2*a^2*b^6 + 1152*
A^2*a^4*b^4 + 192*A^2*a^6*b^2 + 192*B^2*a^2*b^6 + 1152*B^2*a^4*b^4 + 512*B^2*a^6*b^2 + 128*C^2*a^2*b^6 + 512*C
^2*a^4*b^4 + 512*C^2*a^6*b^2 + 32*A*C*a^8 + 128*A*B*a*b^7 + 128*A*B*a^7*b + 64*B*C*a*b^7 + 256*B*C*a^7*b + 153
6*A*B*a^3*b^5 + 1536*A*B*a^5*b^3 + 512*A*C*a^2*b^6 + 1024*A*C*a^4*b^4 + 384*A*C*a^6*b^2 + 896*B*C*a^3*b^5 + 15
36*B*C*a^5*b^3))*((A*a^4)/2 + C*a^4 + 6*A*a^2*b^2 + 4*B*a^3*b)*1i)/((((A*a^4)/2 + C*a^4 + 6*A*a^2*b^2 + 4*B*a^
3*b)*(16*A*a^4 + 16*B*b^4 + 32*C*a^4 + 192*A*a^2*b^2 + 192*B*a^2*b^2 + 128*A*a*b^3 + 128*B*a^3*b + 64*C*a*b^3
+ 128*C*a^3*b) + tan(c/2 + (d*x)/2)*(8*A^2*a^8 + 8*B^2*b^8 + 32*C^2*a^8 + 512*A^2*a^2*b^6 + 1152*A^2*a^4*b^4 +
192*A^2*a^6*b^2 + 192*B^2*a^2*b^6 + 1152*B^2*a^4*b^4 + 512*B^2*a^6*b^2 + 128*C^2*a^2*b^6 + 512*C^2*a^4*b^4 +
512*C^2*a^6*b^2 + 32*A*C*a^8 + 128*A*B*a*b^7 + 128*A*B*a^7*b + 64*B*C*a*b^7 + 256*B*C*a^7*b + 1536*A*B*a^3*b^5
+ 1536*A*B*a^5*b^3 + 512*A*C*a^2*b^6 + 1024*A*C*a^4*b^4 + 384*A*C*a^6*b^2 + 896*B*C*a^3*b^5 + 1536*B*C*a^5*b^
3))*((A*a^4)/2 + C*a^4 + 6*A*a^2*b^2 + 4*B*a^3*b) + (((A*a^4)/2 + C*a^4 + 6*A*a^2*b^2 + 4*B*a^3*b)*(16*A*a^4 +
16*B*b^4 + 32*C*a^4 + 192*A*a^2*b^2 + 192*B*a^2*b^2 + 128*A*a*b^3 + 128*B*a^3*b + 64*C*a*b^3 + 128*C*a^3*b) -
tan(c/2 + (d*x)/2)*(8*A^2*a^8 + 8*B^2*b^8 + 32*C^2*a^8 + 512*A^2*a^2*b^6 + 1152*A^2*a^4*b^4 + 192*A^2*a^6*b^2
+ 192*B^2*a^2*b^6 + 1152*B^2*a^4*b^4 + 512*B^2*a^6*b^2 + 128*C^2*a^2*b^6 + 512*C^2*a^4*b^4 + 512*C^2*a^6*b^2
+ 32*A*C*a^8 + 128*A*B*a*b^7 + 128*A*B*a^7*b + 64*B*C*a*b^7 + 256*B*C*a^7*b + 1536*A*B*a^3*b^5 + 1536*A*B*a^5*
b^3 + 512*A*C*a^2*b^6 + 1024*A*C*a^4*b^4 + 384*A*C*a^6*b^2 + 896*B*C*a^3*b^5 + 1536*B*C*a^5*b^3))*((A*a^4)/2 +
C*a^4 + 6*A*a^2*b^2 + 4*B*a^3*b) - 256*C^3*a^11*b + 6144*A^3*a^4*b^8 - 9216*A^3*a^5*b^7 + 512*A^3*a^6*b^6 - 1
536*A^3*a^7*b^5 - 64*A^3*a^9*b^3 + 64*B^3*a^3*b^9 + 1536*B^3*a^5*b^7 - 512*B^3*a^6*b^6 + 9216*B^3*a^7*b^5 - 61
44*B^3*a^8*b^4 + 256*C^3*a^6*b^6 + 1024*C^3*a^8*b^4 - 128*C^3*a^9*b^3 + 1024*C^3*a^10*b^2 - 256*A*C^2*a^11*b -
64*A^2*C*a^11*b + 96*A*B^2*a^2*b^10 + 3336*A*B^2*a^4*b^8 - 1536*A*B^2*a^5*b^7 + 26304*A*B^2*a^6*b^6 - 22656*A
*B^2*a^7*b^5 + 1152*A*B^2*a^8*b^4 - 1536*A*B^2*a^9*b^3 + 1536*A^2*B*a^3*b^9 - 1152*A^2*B*a^4*b^8 + 22656*A^2*B
*a^5*b^7 - 26304*A^2*B*a^6*b^6 + 1536*A^2*B*a^7*b^5 - 3336*A^2*B*a^8*b^4 - 96*A^2*B*a^10*b^2 + 1536*A*C^2*a^4*
b^8 + 7296*A*C^2*a^6*b^6 - 1536*A*C^2*a^7*b^5 + 8704*A*C^2*a^8*b^4 - 3456*A*C^2*a^9*b^3 + 512*A*C^2*a^10*b^2 +
6144*A^2*C*a^4*b^8 - 4608*A^2*C*a^5*b^7 + 13824*A^2*C*a^6*b^6 - 13056*A^2*C*a^7*b^5 + 1024*A^2*C*a^8*b^4 - 18
24*A^2*C*a^9*b^3 + 1152*B*C^2*a^5*b^7 + 5888*B*C^2*a^7*b^5 - 1056*B*C^2*a^8*b^4 + 7168*B*C^2*a^9*b^3 - 2432*B*
C^2*a^10*b^2 + 528*B^2*C*a^4*b^8 + 7552*B^2*C*a^6*b^6 - 2304*B^2*C*a^7*b^5 + 14592*B^2*C*a^8*b^4 - 7168*B^2*C*
a^9*b^3 + 768*A*B*C*a^3*b^9 + 15168*A*B*C*a^5*b^7 - 6528*A*B*C*a^6*b^6 + 30592*A*B*C*a^7*b^5 - 19488*A*B*C*a^8
*b^4 + 1536*A*B*C*a^9*b^3 - 1408*A*B*C*a^10*b^2))*(A*a^4*1i + C*a^4*2i + A*a^2*b^2*12i + B*a^3*b*8i))/d - (b*a
tan(((b*(tan(c/2 + (d*x)/2)*(8*A^2*a^8 + 8*B^2*b^8 + 32*C^2*a^8 + 512*A^2*a^2*b^6 + 1152*A^2*a^4*b^4 + 192*A^2
*a^6*b^2 + 192*B^2*a^2*b^6 + 1152*B^2*a^4*b^4 + 512*B^2*a^6*b^2 + 128*C^2*a^2*b^6 + 512*C^2*a^4*b^4 + 512*C^2*
a^6*b^2 + 32*A*C*a^8 + 128*A*B*a*b^7 + 128*A*B*a^7*b + 64*B*C*a*b^7 + 256*B*C*a^7*b + 1536*A*B*a^3*b^5 + 1536*
A*B*a^5*b^3 + 512*A*C*a^2*b^6 + 1024*A*C*a^4*b^4 + 384*A*C*a^6*b^2 + 896*B*C*a^3*b^5 + 1536*B*C*a^5*b^3) - (b*
(B*b^3 + 8*C*a^3 + 8*A*a*b^2 + 12*B*a^2*b + 4*C*a*b^2)*(16*A*a^4 + 16*B*b^4 + 32*C*a^4 + 192*A*a^2*b^2 + 192*B
*a^2*b^2 + 128*A*a*b^3 + 128*B*a^3*b + 64*C*a*b^3 + 128*C*a^3*b)*1i)/2)*(B*b^3 + 8*C*a^3 + 8*A*a*b^2 + 12*B*a^
2*b + 4*C*a*b^2))/2 + (b*(tan(c/2 + (d*x)/2)*(8*A^2*a^8 + 8*B^2*b^8 + 32*C^2*a^8 + 512*A^2*a^2*b^6 + 1152*A^2*
a^4*b^4 + 192*A^2*a^6*b^2 + 192*B^2*a^2*b^6 + 1152*B^2*a^4*b^4 + 512*B^2*a^6*b^2 + 128*C^2*a^2*b^6 + 512*C^2*a
^4*b^4 + 512*C^2*a^6*b^2 + 32*A*C*a^8 + 128*A*B*a*b^7 + 128*A*B*a^7*b + 64*B*C*a*b^7 + 256*B*C*a^7*b + 1536*A*
B*a^3*b^5 + 1536*A*B*a^5*b^3 + 512*A*C*a^2*b^6 + 1024*A*C*a^4*b^4 + 384*A*C*a^6*b^2 + 896*B*C*a^3*b^5 + 1536*B
*C*a^5*b^3) + (b*(B*b^3 + 8*C*a^3 + 8*A*a*b^2 + 12*B*a^2*b + 4*C*a*b^2)*(16*A*a^4 + 16*B*b^4 + 32*C*a^4 + 192*
A*a^2*b^2 + 192*B*a^2*b^2 + 128*A*a*b^3 + 128*B*a^3*b + 64*C*a*b^3 + 128*C*a^3*b)*1i)/2)*(B*b^3 + 8*C*a^3 + 8*
A*a*b^2 + 12*B*a^2*b + 4*C*a*b^2))/2)/(6144*A^3*a^4*b^8 - 256*C^3*a^11*b - 9216*A^3*a^5*b^7 + 512*A^3*a^6*b^6
- 1536*A^3*a^7*b^5 - 64*A^3*a^9*b^3 + 64*B^3*a^3*b^9 + 1536*B^3*a^5*b^7 - 512*B^3*a^6*b^6 + 9216*B^3*a^7*b^5 -
6144*B^3*a^8*b^4 + 256*C^3*a^6*b^6 + 1024*C^3*a^8*b^4 - 128*C^3*a^9*b^3 + 1024*C^3*a^10*b^2 - (b*(tan(c/2 + (
d*x)/2)*(8*A^2*a^8 + 8*B^2*b^8 + 32*C^2*a^8 + 512*A^2*a^2*b^6 + 1152*A^2*a^4*b^4 + 192*A^2*a^6*b^2 + 192*B^2*a
^2*b^6 + 1152*B^2*a^4*b^4 + 512*B^2*a^6*b^2 + 128*C^2*a^2*b^6 + 512*C^2*a^4*b^4 + 512*C^2*a^6*b^2 + 32*A*C*a^8
+ 128*A*B*a*b^7 + 128*A*B*a^7*b + 64*B*C*a*b^7 + 256*B*C*a^7*b + 1536*A*B*a^3*b^5 + 1536*A*B*a^5*b^3 + 512*A*
C*a^2*b^6 + 1024*A*C*a^4*b^4 + 384*A*C*a^6*b^2 + 896*B*C*a^3*b^5 + 1536*B*C*a^5*b^3) - (b*(B*b^3 + 8*C*a^3 + 8
*A*a*b^2 + 12*B*a^2*b + 4*C*a*b^2)*(16*A*a^4 + 16*B*b^4 + 32*C*a^4 + 192*A*a^2*b^2 + 192*B*a^2*b^2 + 128*A*a*b
^3 + 128*B*a^3*b + 64*C*a*b^3 + 128*C*a^3*b)*1i)/2)*(B*b^3 + 8*C*a^3 + 8*A*a*b^2 + 12*B*a^2*b + 4*C*a*b^2)*1i)
/2 + (b*(tan(c/2 + (d*x)/2)*(8*A^2*a^8 + 8*B^2*b^8 + 32*C^2*a^8 + 512*A^2*a^2*b^6 + 1152*A^2*a^4*b^4 + 192*A^2
*a^6*b^2 + 192*B^2*a^2*b^6 + 1152*B^2*a^4*b^4 + 512*B^2*a^6*b^2 + 128*C^2*a^2*b^6 + 512*C^2*a^4*b^4 + 512*C^2*
a^6*b^2 + 32*A*C*a^8 + 128*A*B*a*b^7 + 128*A*B*a^7*b + 64*B*C*a*b^7 + 256*B*C*a^7*b + 1536*A*B*a^3*b^5 + 1536*
A*B*a^5*b^3 + 512*A*C*a^2*b^6 + 1024*A*C*a^4*b^4 + 384*A*C*a^6*b^2 + 896*B*C*a^3*b^5 + 1536*B*C*a^5*b^3) + (b*
(B*b^3 + 8*C*a^3 + 8*A*a*b^2 + 12*B*a^2*b + 4*C*a*b^2)*(16*A*a^4 + 16*B*b^4 + 32*C*a^4 + 192*A*a^2*b^2 + 192*B
*a^2*b^2 + 128*A*a*b^3 + 128*B*a^3*b + 64*C*a*b^3 + 128*C*a^3*b)*1i)/2)*(B*b^3 + 8*C*a^3 + 8*A*a*b^2 + 12*B*a^
2*b + 4*C*a*b^2)*1i)/2 - 256*A*C^2*a^11*b - 64*A^2*C*a^11*b + 96*A*B^2*a^2*b^10 + 3336*A*B^2*a^4*b^8 - 1536*A*
B^2*a^5*b^7 + 26304*A*B^2*a^6*b^6 - 22656*A*B^2*a^7*b^5 + 1152*A*B^2*a^8*b^4 - 1536*A*B^2*a^9*b^3 + 1536*A^2*B
*a^3*b^9 - 1152*A^2*B*a^4*b^8 + 22656*A^2*B*a^5*b^7 - 26304*A^2*B*a^6*b^6 + 1536*A^2*B*a^7*b^5 - 3336*A^2*B*a^
8*b^4 - 96*A^2*B*a^10*b^2 + 1536*A*C^2*a^4*b^8 + 7296*A*C^2*a^6*b^6 - 1536*A*C^2*a^7*b^5 + 8704*A*C^2*a^8*b^4
- 3456*A*C^2*a^9*b^3 + 512*A*C^2*a^10*b^2 + 6144*A^2*C*a^4*b^8 - 4608*A^2*C*a^5*b^7 + 13824*A^2*C*a^6*b^6 - 13
056*A^2*C*a^7*b^5 + 1024*A^2*C*a^8*b^4 - 1824*A^2*C*a^9*b^3 + 1152*B*C^2*a^5*b^7 + 5888*B*C^2*a^7*b^5 - 1056*B
*C^2*a^8*b^4 + 7168*B*C^2*a^9*b^3 - 2432*B*C^2*a^10*b^2 + 528*B^2*C*a^4*b^8 + 7552*B^2*C*a^6*b^6 - 2304*B^2*C*
a^7*b^5 + 14592*B^2*C*a^8*b^4 - 7168*B^2*C*a^9*b^3 + 768*A*B*C*a^3*b^9 + 15168*A*B*C*a^5*b^7 - 6528*A*B*C*a^6*
b^6 + 30592*A*B*C*a^7*b^5 - 19488*A*B*C*a^8*b^4 + 1536*A*B*C*a^9*b^3 - 1408*A*B*C*a^10*b^2))*(B*b^3 + 8*C*a^3
+ 8*A*a*b^2 + 12*B*a^2*b + 4*C*a*b^2))/d