3.9.90 \(\int \cos (c+d x) (a+b \sec (c+d x))^4 (A+B \sec (c+d x)+C \sec ^2(c+d x)) \, dx\) [890]

Optimal. Leaf size=273 \[ a^3 (4 A b+a B) x+\frac {\left (32 a^3 b B+16 a b^3 B+8 a^4 C+24 a^2 b^2 (2 A+C)+b^4 (4 A+3 C)\right ) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac {A (a+b \sec (c+d x))^4 \sin (c+d x)}{d}+\frac {b \left (34 a^2 b B+4 b^3 B-a^3 (12 A-19 C)+8 a b^2 (3 A+2 C)\right ) \tan (c+d x)}{6 d}+\frac {b^2 \left (32 a b B-a^2 (24 A-26 C)+3 b^2 (4 A+3 C)\right ) \sec (c+d x) \tan (c+d x)}{24 d}-\frac {b (12 a A-4 b B-7 a C) (a+b \sec (c+d x))^2 \tan (c+d x)}{12 d}-\frac {b (4 A-C) (a+b \sec (c+d x))^3 \tan (c+d x)}{4 d} \]

[Out]

a^3*(4*A*b+B*a)*x+1/8*(32*a^3*b*B+16*a*b^3*B+8*a^4*C+24*a^2*b^2*(2*A+C)+b^4*(4*A+3*C))*arctanh(sin(d*x+c))/d+A
*(a+b*sec(d*x+c))^4*sin(d*x+c)/d+1/6*b*(34*a^2*b*B+4*b^3*B-a^3*(12*A-19*C)+8*a*b^2*(3*A+2*C))*tan(d*x+c)/d+1/2
4*b^2*(32*a*b*B-a^2*(24*A-26*C)+3*b^2*(4*A+3*C))*sec(d*x+c)*tan(d*x+c)/d-1/12*b*(12*A*a-4*B*b-7*C*a)*(a+b*sec(
d*x+c))^2*tan(d*x+c)/d-1/4*b*(4*A-C)*(a+b*sec(d*x+c))^3*tan(d*x+c)/d

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Rubi [A]
time = 0.42, antiderivative size = 273, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, integrand size = 39, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {4179, 4141, 4133, 3855, 3852, 8} \begin {gather*} a^3 x (a B+4 A b)+\frac {b^2 \tan (c+d x) \sec (c+d x) \left (-\left (a^2 (24 A-26 C)\right )+32 a b B+3 b^2 (4 A+3 C)\right )}{24 d}+\frac {b \tan (c+d x) \left (-\left (a^3 (12 A-19 C)\right )+34 a^2 b B+8 a b^2 (3 A+2 C)+4 b^3 B\right )}{6 d}+\frac {\left (8 a^4 C+32 a^3 b B+24 a^2 b^2 (2 A+C)+16 a b^3 B+b^4 (4 A+3 C)\right ) \tanh ^{-1}(\sin (c+d x))}{8 d}-\frac {b \tan (c+d x) (12 a A-7 a C-4 b B) (a+b \sec (c+d x))^2}{12 d}-\frac {b (4 A-C) \tan (c+d x) (a+b \sec (c+d x))^3}{4 d}+\frac {A \sin (c+d x) (a+b \sec (c+d x))^4}{d} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

a^3*(4*A*b + a*B)*x + ((32*a^3*b*B + 16*a*b^3*B + 8*a^4*C + 24*a^2*b^2*(2*A + C) + b^4*(4*A + 3*C))*ArcTanh[Si
n[c + d*x]])/(8*d) + (A*(a + b*Sec[c + d*x])^4*Sin[c + d*x])/d + (b*(34*a^2*b*B + 4*b^3*B - a^3*(12*A - 19*C)
+ 8*a*b^2*(3*A + 2*C))*Tan[c + d*x])/(6*d) + (b^2*(32*a*b*B - a^2*(24*A - 26*C) + 3*b^2*(4*A + 3*C))*Sec[c + d
*x]*Tan[c + d*x])/(24*d) - (b*(12*a*A - 4*b*B - 7*a*C)*(a + b*Sec[c + d*x])^2*Tan[c + d*x])/(12*d) - (b*(4*A -
C)*(a + b*Sec[c + d*x])^3*Tan[c + d*x])/(4*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]

Rule 4179

Int[((A_.) + csc[(e_.) + (f_.)*(x_)]*(B_.) + csc[(e_.) + (f_.)*(x_)]^2*(C_.))*(csc[(e_.) + (f_.)*(x_)]*(d_.))^
(n_)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_), x_Symbol] :> Simp[A*Cot[e + f*x]*(a + b*Csc[e + f*x])^m*((d*
Csc[e + f*x])^n/(f*n)), x] - Dist[1/(d*n), Int[(a + b*Csc[e + f*x])^(m - 1)*(d*Csc[e + f*x])^(n + 1)*Simp[A*b*
m - a*B*n - (b*B*n + a*(C*n + A*(n + 1)))*Csc[e + f*x] - b*(C*n + A*(m + n + 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] && GtQ[m, 0] && LeQ[n, -1]

Rubi steps

\begin {align*} \int \cos (c+d x) (a+b \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx &=\frac {A (a+b \sec (c+d x))^4 \sin (c+d x)}{d}+\int (a+b \sec (c+d x))^3 \left (4 A b+a B+(b B+a C) \sec (c+d x)-b (4 A-C) \sec ^2(c+d x)\right ) \, dx\\ &=\frac {A (a+b \sec (c+d x))^4 \sin (c+d x)}{d}-\frac {b (4 A-C) (a+b \sec (c+d x))^3 \tan (c+d x)}{4 d}+\frac {1}{4} \int (a+b \sec (c+d x))^2 \left (4 a (4 A b+a B)+\left (4 A b^2+8 a b B+4 a^2 C+3 b^2 C\right ) \sec (c+d x)-b (12 a A-4 b B-7 a C) \sec ^2(c+d x)\right ) \, dx\\ &=\frac {A (a+b \sec (c+d x))^4 \sin (c+d x)}{d}-\frac {b (12 a A-4 b B-7 a C) (a+b \sec (c+d x))^2 \tan (c+d x)}{12 d}-\frac {b (4 A-C) (a+b \sec (c+d x))^3 \tan (c+d x)}{4 d}+\frac {1}{12} \int (a+b \sec (c+d x)) \left (12 a^2 (4 A b+a B)+\left (36 a^2 b B+8 b^3 B+12 a^3 C+a b^2 (36 A+23 C)\right ) \sec (c+d x)+b \left (32 a b B-a^2 (24 A-26 C)+3 b^2 (4 A+3 C)\right ) \sec ^2(c+d x)\right ) \, dx\\ &=\frac {A (a+b \sec (c+d x))^4 \sin (c+d x)}{d}+\frac {b^2 \left (32 a b B-a^2 (24 A-26 C)+3 b^2 (4 A+3 C)\right ) \sec (c+d x) \tan (c+d x)}{24 d}-\frac {b (12 a A-4 b B-7 a C) (a+b \sec (c+d x))^2 \tan (c+d x)}{12 d}-\frac {b (4 A-C) (a+b \sec (c+d x))^3 \tan (c+d x)}{4 d}+\frac {1}{24} \int \left (24 a^3 (4 A b+a B)+3 \left (32 a^3 b B+16 a b^3 B+8 a^4 C+24 a^2 b^2 (2 A+C)+b^4 (4 A+3 C)\right ) \sec (c+d x)+4 b \left (34 a^2 b B+4 b^3 B-a^3 (12 A-19 C)+8 a b^2 (3 A+2 C)\right ) \sec ^2(c+d x)\right ) \, dx\\ &=a^3 (4 A b+a B) x+\frac {A (a+b \sec (c+d x))^4 \sin (c+d x)}{d}+\frac {b^2 \left (32 a b B-a^2 (24 A-26 C)+3 b^2 (4 A+3 C)\right ) \sec (c+d x) \tan (c+d x)}{24 d}-\frac {b (12 a A-4 b B-7 a C) (a+b \sec (c+d x))^2 \tan (c+d x)}{12 d}-\frac {b (4 A-C) (a+b \sec (c+d x))^3 \tan (c+d x)}{4 d}+\frac {1}{6} \left (b \left (34 a^2 b B+4 b^3 B-a^3 (12 A-19 C)+8 a b^2 (3 A+2 C)\right )\right ) \int \sec ^2(c+d x) \, dx+\frac {1}{8} \left (32 a^3 b B+16 a b^3 B+8 a^4 C+24 a^2 b^2 (2 A+C)+b^4 (4 A+3 C)\right ) \int \sec (c+d x) \, dx\\ &=a^3 (4 A b+a B) x+\frac {\left (32 a^3 b B+16 a b^3 B+8 a^4 C+24 a^2 b^2 (2 A+C)+b^4 (4 A+3 C)\right ) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac {A (a+b \sec (c+d x))^4 \sin (c+d x)}{d}+\frac {b^2 \left (32 a b B-a^2 (24 A-26 C)+3 b^2 (4 A+3 C)\right ) \sec (c+d x) \tan (c+d x)}{24 d}-\frac {b (12 a A-4 b B-7 a C) (a+b \sec (c+d x))^2 \tan (c+d x)}{12 d}-\frac {b (4 A-C) (a+b \sec (c+d x))^3 \tan (c+d x)}{4 d}-\frac {\left (b \left (34 a^2 b B+4 b^3 B-a^3 (12 A-19 C)+8 a b^2 (3 A+2 C)\right )\right ) \text {Subst}(\int 1 \, dx,x,-\tan (c+d x))}{6 d}\\ &=a^3 (4 A b+a B) x+\frac {\left (32 a^3 b B+16 a b^3 B+8 a^4 C+24 a^2 b^2 (2 A+C)+b^4 (4 A+3 C)\right ) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac {A (a+b \sec (c+d x))^4 \sin (c+d x)}{d}+\frac {b \left (34 a^2 b B+4 b^3 B-a^3 (12 A-19 C)+8 a b^2 (3 A+2 C)\right ) \tan (c+d x)}{6 d}+\frac {b^2 \left (32 a b B-a^2 (24 A-26 C)+3 b^2 (4 A+3 C)\right ) \sec (c+d x) \tan (c+d x)}{24 d}-\frac {b (12 a A-4 b B-7 a C) (a+b \sec (c+d x))^2 \tan (c+d x)}{12 d}-\frac {b (4 A-C) (a+b \sec (c+d x))^3 \tan (c+d x)}{4 d}\\ \end {align*}

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(813\) vs. \(2(273)=546\).
time = 6.90, size = 813, normalized size = 2.98 \begin {gather*} \frac {\left (-48 a^2 A b^2-4 A b^4-32 a^3 b B-16 a b^3 B-8 a^4 C-24 a^2 b^2 C-3 b^4 C\right ) \cos ^6(c+d x) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right ) (a+b \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{4 d (b+a \cos (c+d x))^4 (A+2 C+2 B \cos (c+d x)+A \cos (2 c+2 d x))}+\frac {\left (48 a^2 A b^2+4 A b^4+32 a^3 b B+16 a b^3 B+8 a^4 C+24 a^2 b^2 C+3 b^4 C\right ) \cos ^6(c+d x) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right ) (a+b \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{4 d (b+a \cos (c+d x))^4 (A+2 C+2 B \cos (c+d x)+A \cos (2 c+2 d x))}+\frac {\cos ^2(c+d x) (a+b \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \left (144 a^3 A b (c+d x)+36 a^4 B (c+d x)+192 a^3 A b (c+d x) \cos (2 (c+d x))+48 a^4 B (c+d x) \cos (2 (c+d x))+48 a^3 A b (c+d x) \cos (4 (c+d x))+12 a^4 B (c+d x) \cos (4 (c+d x))+12 a^4 A \sin (c+d x)+12 A b^4 \sin (c+d x)+48 a b^3 B \sin (c+d x)+72 a^2 b^2 C \sin (c+d x)+33 b^4 C \sin (c+d x)+96 a A b^3 \sin (2 (c+d x))+144 a^2 b^2 B \sin (2 (c+d x))+32 b^4 B \sin (2 (c+d x))+96 a^3 b C \sin (2 (c+d x))+128 a b^3 C \sin (2 (c+d x))+18 a^4 A \sin (3 (c+d x))+12 A b^4 \sin (3 (c+d x))+48 a b^3 B \sin (3 (c+d x))+72 a^2 b^2 C \sin (3 (c+d x))+9 b^4 C \sin (3 (c+d x))+48 a A b^3 \sin (4 (c+d x))+72 a^2 b^2 B \sin (4 (c+d x))+8 b^4 B \sin (4 (c+d x))+48 a^3 b C \sin (4 (c+d x))+32 a b^3 C \sin (4 (c+d x))+6 a^4 A \sin (5 (c+d x))\right )}{48 d (b+a \cos (c+d x))^4 (A+2 C+2 B \cos (c+d x)+A \cos (2 c+2 d x))} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((-48*a^2*A*b^2 - 4*A*b^4 - 32*a^3*b*B - 16*a*b^3*B - 8*a^4*C - 24*a^2*b^2*C - 3*b^4*C)*Cos[c + d*x]^6*Log[Cos
[(c + d*x)/2] - Sin[(c + d*x)/2]]*(a + b*Sec[c + d*x])^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(4*d*(b + a*
Cos[c + d*x])^4*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])) + ((48*a^2*A*b^2 + 4*A*b^4 + 32*a^3*b*B + 1
6*a*b^3*B + 8*a^4*C + 24*a^2*b^2*C + 3*b^4*C)*Cos[c + d*x]^6*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]*(a + b*S
ec[c + d*x])^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(4*d*(b + a*Cos[c + d*x])^4*(A + 2*C + 2*B*Cos[c + d*x
] + A*Cos[2*c + 2*d*x])) + (Cos[c + d*x]^2*(a + b*Sec[c + d*x])^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*(144
*a^3*A*b*(c + d*x) + 36*a^4*B*(c + d*x) + 192*a^3*A*b*(c + d*x)*Cos[2*(c + d*x)] + 48*a^4*B*(c + d*x)*Cos[2*(c
+ d*x)] + 48*a^3*A*b*(c + d*x)*Cos[4*(c + d*x)] + 12*a^4*B*(c + d*x)*Cos[4*(c + d*x)] + 12*a^4*A*Sin[c + d*x]
+ 12*A*b^4*Sin[c + d*x] + 48*a*b^3*B*Sin[c + d*x] + 72*a^2*b^2*C*Sin[c + d*x] + 33*b^4*C*Sin[c + d*x] + 96*a*
A*b^3*Sin[2*(c + d*x)] + 144*a^2*b^2*B*Sin[2*(c + d*x)] + 32*b^4*B*Sin[2*(c + d*x)] + 96*a^3*b*C*Sin[2*(c + d*
x)] + 128*a*b^3*C*Sin[2*(c + d*x)] + 18*a^4*A*Sin[3*(c + d*x)] + 12*A*b^4*Sin[3*(c + d*x)] + 48*a*b^3*B*Sin[3*
(c + d*x)] + 72*a^2*b^2*C*Sin[3*(c + d*x)] + 9*b^4*C*Sin[3*(c + d*x)] + 48*a*A*b^3*Sin[4*(c + d*x)] + 72*a^2*b
^2*B*Sin[4*(c + d*x)] + 8*b^4*B*Sin[4*(c + d*x)] + 48*a^3*b*C*Sin[4*(c + d*x)] + 32*a*b^3*C*Sin[4*(c + d*x)] +
6*a^4*A*Sin[5*(c + d*x)]))/(48*d*(b + a*Cos[c + d*x])^4*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x]))

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Maple [A]
time = 0.17, size = 355, normalized size = 1.30 Too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.29, size = 431, normalized size = 1.58 \begin {gather*} \frac {48 \, {\left (d x + c\right )} B a^{4} + 192 \, {\left (d x + c\right )} A a^{3} b + 64 \, {\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} C a b^{3} + 16 \, {\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} B b^{4} - 3 \, C 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 )} - 72 \, C 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 )} - 48 \, B 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 )} - 12 \, A b^{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 )} + 24 \, 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 )} + 96 \, 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 )} + 144 \, 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 )} + 48 \, A a^{4} \sin \left (d x + c\right ) + 192 \, C a^{3} b \tan \left (d x + c\right ) + 288 \, B a^{2} b^{2} \tan \left (d x + c\right ) + 192 \, A a b^{3} \tan \left (d x + c\right )}{48 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/48*(48*(d*x + c)*B*a^4 + 192*(d*x + c)*A*a^3*b + 64*(tan(d*x + c)^3 + 3*tan(d*x + c))*C*a*b^3 + 16*(tan(d*x
+ c)^3 + 3*tan(d*x + c))*B*b^4 - 3*C*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)) - 72*C*a^2*b^2*(2*sin(d*x + c)/(sin(d*x + c)^2
- 1) - log(sin(d*x + c) + 1) + log(sin(d*x + c) - 1)) - 48*B*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)) - 12*A*b^4*(2*sin(d*x + c)/(sin(d*x + c)^2 - 1) - log(sin(d*x + c)
+ 1) + log(sin(d*x + c) - 1)) + 24*C*a^4*(log(sin(d*x + c) + 1) - log(sin(d*x + c) - 1)) + 96*B*a^3*b*(log(sin
(d*x + c) + 1) - log(sin(d*x + c) - 1)) + 144*A*a^2*b^2*(log(sin(d*x + c) + 1) - log(sin(d*x + c) - 1)) + 48*A
*a^4*sin(d*x + c) + 192*C*a^3*b*tan(d*x + c) + 288*B*a^2*b^2*tan(d*x + c) + 192*A*a*b^3*tan(d*x + c))/d

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Fricas [A]
time = 4.67, size = 302, normalized size = 1.11 \begin {gather*} \frac {48 \, {\left (B a^{4} + 4 \, A a^{3} b\right )} d x \cos \left (d x + c\right )^{4} + 3 \, {\left (8 \, C a^{4} + 32 \, B a^{3} b + 24 \, {\left (2 \, A + C\right )} a^{2} b^{2} + 16 \, B a b^{3} + {\left (4 \, A + 3 \, C\right )} b^{4}\right )} \cos \left (d x + c\right )^{4} \log \left (\sin \left (d x + c\right ) + 1\right ) - 3 \, {\left (8 \, C a^{4} + 32 \, B a^{3} b + 24 \, {\left (2 \, A + C\right )} a^{2} b^{2} + 16 \, B a b^{3} + {\left (4 \, A + 3 \, C\right )} b^{4}\right )} \cos \left (d x + c\right )^{4} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \, {\left (24 \, A a^{4} \cos \left (d x + c\right )^{4} + 6 \, C b^{4} + 16 \, {\left (6 \, C a^{3} b + 9 \, B a^{2} b^{2} + 2 \, {\left (3 \, A + 2 \, C\right )} a b^{3} + B b^{4}\right )} \cos \left (d x + c\right )^{3} + 3 \, {\left (24 \, C a^{2} b^{2} + 16 \, B a b^{3} + {\left (4 \, A + 3 \, C\right )} b^{4}\right )} \cos \left (d x + c\right )^{2} + 8 \, {\left (4 \, C a b^{3} + B b^{4}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{48 \, d \cos \left (d x + c\right )^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/48*(48*(B*a^4 + 4*A*a^3*b)*d*x*cos(d*x + c)^4 + 3*(8*C*a^4 + 32*B*a^3*b + 24*(2*A + C)*a^2*b^2 + 16*B*a*b^3
+ (4*A + 3*C)*b^4)*cos(d*x + c)^4*log(sin(d*x + c) + 1) - 3*(8*C*a^4 + 32*B*a^3*b + 24*(2*A + C)*a^2*b^2 + 16*
B*a*b^3 + (4*A + 3*C)*b^4)*cos(d*x + c)^4*log(-sin(d*x + c) + 1) + 2*(24*A*a^4*cos(d*x + c)^4 + 6*C*b^4 + 16*(
6*C*a^3*b + 9*B*a^2*b^2 + 2*(3*A + 2*C)*a*b^3 + B*b^4)*cos(d*x + c)^3 + 3*(24*C*a^2*b^2 + 16*B*a*b^3 + (4*A +
3*C)*b^4)*cos(d*x + c)^2 + 8*(4*C*a*b^3 + B*b^4)*cos(d*x + c))*sin(d*x + c))/(d*cos(d*x + c)^4)

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 840 vs. \(2 (262) = 524\).
time = 0.59, size = 840, normalized size = 3.08 \begin {gather*} \frac {\frac {48 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1} + 24 \, {\left (B a^{4} + 4 \, A a^{3} b\right )} {\left (d x + c\right )} + 3 \, {\left (8 \, C a^{4} + 32 \, B a^{3} b + 48 \, A a^{2} b^{2} + 24 \, C a^{2} b^{2} + 16 \, B a b^{3} + 4 \, A b^{4} + 3 \, C b^{4}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right ) - 3 \, {\left (8 \, C a^{4} + 32 \, B a^{3} b + 48 \, A a^{2} b^{2} + 24 \, C a^{2} b^{2} + 16 \, B a b^{3} + 4 \, A b^{4} + 3 \, C b^{4}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right ) - \frac {2 \, {\left (96 \, C a^{3} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 144 \, B a^{2} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 72 \, C a^{2} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 96 \, A a b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 48 \, B a b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 96 \, C a b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 12 \, A b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 24 \, B b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 15 \, C b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 288 \, C a^{3} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 432 \, B a^{2} b^{2} \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 )^{5} - 288 \, A a b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 48 \, B a b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 160 \, C a b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 12 \, A b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 40 \, B b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 9 \, C b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 288 \, C a^{3} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 432 \, B a^{2} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 72 \, C a^{2} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 288 \, A a b^{3} \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} + 160 \, C 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} + 40 \, B b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 9 \, C b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 96 \, C a^{3} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 144 \, B a^{2} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 72 \, C a^{2} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 96 \, A a b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 48 \, B a b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 96 \, C a b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 12 \, A b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 24 \, B b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 15 \, 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 )}^{4}}}{24 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/24*(48*A*a^4*tan(1/2*d*x + 1/2*c)/(tan(1/2*d*x + 1/2*c)^2 + 1) + 24*(B*a^4 + 4*A*a^3*b)*(d*x + c) + 3*(8*C*a
^4 + 32*B*a^3*b + 48*A*a^2*b^2 + 24*C*a^2*b^2 + 16*B*a*b^3 + 4*A*b^4 + 3*C*b^4)*log(abs(tan(1/2*d*x + 1/2*c) +
1)) - 3*(8*C*a^4 + 32*B*a^3*b + 48*A*a^2*b^2 + 24*C*a^2*b^2 + 16*B*a*b^3 + 4*A*b^4 + 3*C*b^4)*log(abs(tan(1/2
*d*x + 1/2*c) - 1)) - 2*(96*C*a^3*b*tan(1/2*d*x + 1/2*c)^7 + 144*B*a^2*b^2*tan(1/2*d*x + 1/2*c)^7 - 72*C*a^2*b
^2*tan(1/2*d*x + 1/2*c)^7 + 96*A*a*b^3*tan(1/2*d*x + 1/2*c)^7 - 48*B*a*b^3*tan(1/2*d*x + 1/2*c)^7 + 96*C*a*b^3
*tan(1/2*d*x + 1/2*c)^7 - 12*A*b^4*tan(1/2*d*x + 1/2*c)^7 + 24*B*b^4*tan(1/2*d*x + 1/2*c)^7 - 15*C*b^4*tan(1/2
*d*x + 1/2*c)^7 - 288*C*a^3*b*tan(1/2*d*x + 1/2*c)^5 - 432*B*a^2*b^2*tan(1/2*d*x + 1/2*c)^5 + 72*C*a^2*b^2*tan
(1/2*d*x + 1/2*c)^5 - 288*A*a*b^3*tan(1/2*d*x + 1/2*c)^5 + 48*B*a*b^3*tan(1/2*d*x + 1/2*c)^5 - 160*C*a*b^3*tan
(1/2*d*x + 1/2*c)^5 + 12*A*b^4*tan(1/2*d*x + 1/2*c)^5 - 40*B*b^4*tan(1/2*d*x + 1/2*c)^5 - 9*C*b^4*tan(1/2*d*x
+ 1/2*c)^5 + 288*C*a^3*b*tan(1/2*d*x + 1/2*c)^3 + 432*B*a^2*b^2*tan(1/2*d*x + 1/2*c)^3 + 72*C*a^2*b^2*tan(1/2*
d*x + 1/2*c)^3 + 288*A*a*b^3*tan(1/2*d*x + 1/2*c)^3 + 48*B*a*b^3*tan(1/2*d*x + 1/2*c)^3 + 160*C*a*b^3*tan(1/2*
d*x + 1/2*c)^3 + 12*A*b^4*tan(1/2*d*x + 1/2*c)^3 + 40*B*b^4*tan(1/2*d*x + 1/2*c)^3 - 9*C*b^4*tan(1/2*d*x + 1/2
*c)^3 - 96*C*a^3*b*tan(1/2*d*x + 1/2*c) - 144*B*a^2*b^2*tan(1/2*d*x + 1/2*c) - 72*C*a^2*b^2*tan(1/2*d*x + 1/2*
c) - 96*A*a*b^3*tan(1/2*d*x + 1/2*c) - 48*B*a*b^3*tan(1/2*d*x + 1/2*c) - 96*C*a*b^3*tan(1/2*d*x + 1/2*c) - 12*
A*b^4*tan(1/2*d*x + 1/2*c) - 24*B*b^4*tan(1/2*d*x + 1/2*c) - 15*C*b^4*tan(1/2*d*x + 1/2*c))/(tan(1/2*d*x + 1/2
*c)^2 - 1)^4)/d

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Mupad [B]
time = 8.81, size = 2500, normalized size = 9.16 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(tan(c/2 + (d*x)/2)*(2*A*a^4 + A*b^4 + 2*B*b^4 + (5*C*b^4)/4 + 12*B*a^2*b^2 + 6*C*a^2*b^2 + 8*A*a*b^3 + 4*B*a*
b^3 + 8*C*a*b^3 + 8*C*a^3*b) - tan(c/2 + (d*x)/2)^3*(8*A*a^4 + (4*B*b^4)/3 - 2*C*b^4 + 24*B*a^2*b^2 + 16*A*a*b
^3 + (16*C*a*b^3)/3 + 16*C*a^3*b) + tan(c/2 + (d*x)/2)^7*((4*B*b^4)/3 - 8*A*a^4 + 2*C*b^4 + 24*B*a^2*b^2 + 16*
A*a*b^3 + (16*C*a*b^3)/3 + 16*C*a^3*b) + tan(c/2 + (d*x)/2)^9*(2*A*a^4 + A*b^4 - 2*B*b^4 + (5*C*b^4)/4 - 12*B*
a^2*b^2 + 6*C*a^2*b^2 - 8*A*a*b^3 + 4*B*a*b^3 - 8*C*a*b^3 - 8*C*a^3*b) - tan(c/2 + (d*x)/2)^5*(2*A*b^4 - 12*A*
a^4 - (3*C*b^4)/2 + 12*C*a^2*b^2 + 8*B*a*b^3))/(d*(2*tan(c/2 + (d*x)/2)^4 - 3*tan(c/2 + (d*x)/2)^2 + 2*tan(c/2
+ (d*x)/2)^6 - 3*tan(c/2 + (d*x)/2)^8 + tan(c/2 + (d*x)/2)^10 + 1)) + (atan(((((A*b^4)/2 + C*a^4 + (3*C*b^4)/
8 + 6*A*a^2*b^2 + 3*C*a^2*b^2 + 2*B*a*b^3 + 4*B*a^3*b)*(16*A*b^4 + 32*B*a^4 + 32*C*a^4 + 12*C*b^4 + 192*A*a^2*
b^2 + 96*C*a^2*b^2 + 128*A*a^3*b + 64*B*a*b^3 + 128*B*a^3*b) + tan(c/2 + (d*x)/2)*(8*A^2*b^8 + 32*B^2*a^8 + 32
*C^2*a^8 + (9*C^2*b^8)/2 + 192*A^2*a^2*b^6 + 1152*A^2*a^4*b^4 + 512*A^2*a^6*b^2 + 128*B^2*a^2*b^6 + 512*B^2*a^
4*b^4 + 512*B^2*a^6*b^2 + 72*C^2*a^2*b^6 + 312*C^2*a^4*b^4 + 192*C^2*a^6*b^2 + 12*A*C*b^8 + 64*A*B*a*b^7 + 256
*A*B*a^7*b + 48*B*C*a*b^7 + 256*B*C*a^7*b + 896*A*B*a^3*b^5 + 1536*A*B*a^5*b^3 + 240*A*C*a^2*b^6 + 1184*A*C*a^
4*b^4 + 384*A*C*a^6*b^2 + 480*B*C*a^3*b^5 + 896*B*C*a^5*b^3))*((A*b^4)/2 + C*a^4 + (3*C*b^4)/8 + 6*A*a^2*b^2 +
3*C*a^2*b^2 + 2*B*a*b^3 + 4*B*a^3*b)*1i - (((A*b^4)/2 + C*a^4 + (3*C*b^4)/8 + 6*A*a^2*b^2 + 3*C*a^2*b^2 + 2*B
*a*b^3 + 4*B*a^3*b)*(16*A*b^4 + 32*B*a^4 + 32*C*a^4 + 12*C*b^4 + 192*A*a^2*b^2 + 96*C*a^2*b^2 + 128*A*a^3*b +
64*B*a*b^3 + 128*B*a^3*b) - tan(c/2 + (d*x)/2)*(8*A^2*b^8 + 32*B^2*a^8 + 32*C^2*a^8 + (9*C^2*b^8)/2 + 192*A^2*
a^2*b^6 + 1152*A^2*a^4*b^4 + 512*A^2*a^6*b^2 + 128*B^2*a^2*b^6 + 512*B^2*a^4*b^4 + 512*B^2*a^6*b^2 + 72*C^2*a^
2*b^6 + 312*C^2*a^4*b^4 + 192*C^2*a^6*b^2 + 12*A*C*b^8 + 64*A*B*a*b^7 + 256*A*B*a^7*b + 48*B*C*a*b^7 + 256*B*C
*a^7*b + 896*A*B*a^3*b^5 + 1536*A*B*a^5*b^3 + 240*A*C*a^2*b^6 + 1184*A*C*a^4*b^4 + 384*A*C*a^6*b^2 + 480*B*C*a
^3*b^5 + 896*B*C*a^5*b^3))*((A*b^4)/2 + C*a^4 + (3*C*b^4)/8 + 6*A*a^2*b^2 + 3*C*a^2*b^2 + 2*B*a*b^3 + 4*B*a^3*
b)*1i)/(64*B*C^2*a^12 - (((A*b^4)/2 + C*a^4 + (3*C*b^4)/8 + 6*A*a^2*b^2 + 3*C*a^2*b^2 + 2*B*a*b^3 + 4*B*a^3*b)
*(16*A*b^4 + 32*B*a^4 + 32*C*a^4 + 12*C*b^4 + 192*A*a^2*b^2 + 96*C*a^2*b^2 + 128*A*a^3*b + 64*B*a*b^3 + 128*B*
a^3*b) - tan(c/2 + (d*x)/2)*(8*A^2*b^8 + 32*B^2*a^8 + 32*C^2*a^8 + (9*C^2*b^8)/2 + 192*A^2*a^2*b^6 + 1152*A^2*
a^4*b^4 + 512*A^2*a^6*b^2 + 128*B^2*a^2*b^6 + 512*B^2*a^4*b^4 + 512*B^2*a^6*b^2 + 72*C^2*a^2*b^6 + 312*C^2*a^4
*b^4 + 192*C^2*a^6*b^2 + 12*A*C*b^8 + 64*A*B*a*b^7 + 256*A*B*a^7*b + 48*B*C*a*b^7 + 256*B*C*a^7*b + 896*A*B*a^
3*b^5 + 1536*A*B*a^5*b^3 + 240*A*C*a^2*b^6 + 1184*A*C*a^4*b^4 + 384*A*C*a^6*b^2 + 480*B*C*a^3*b^5 + 896*B*C*a^
5*b^3))*((A*b^4)/2 + C*a^4 + (3*C*b^4)/8 + 6*A*a^2*b^2 + 3*C*a^2*b^2 + 2*B*a*b^3 + 4*B*a^3*b) - (((A*b^4)/2 +
C*a^4 + (3*C*b^4)/8 + 6*A*a^2*b^2 + 3*C*a^2*b^2 + 2*B*a*b^3 + 4*B*a^3*b)*(16*A*b^4 + 32*B*a^4 + 32*C*a^4 + 12*
C*b^4 + 192*A*a^2*b^2 + 96*C*a^2*b^2 + 128*A*a^3*b + 64*B*a*b^3 + 128*B*a^3*b) + tan(c/2 + (d*x)/2)*(8*A^2*b^8
+ 32*B^2*a^8 + 32*C^2*a^8 + (9*C^2*b^8)/2 + 192*A^2*a^2*b^6 + 1152*A^2*a^4*b^4 + 512*A^2*a^6*b^2 + 128*B^2*a^
2*b^6 + 512*B^2*a^4*b^4 + 512*B^2*a^6*b^2 + 72*C^2*a^2*b^6 + 312*C^2*a^4*b^4 + 192*C^2*a^6*b^2 + 12*A*C*b^8 +
64*A*B*a*b^7 + 256*A*B*a^7*b + 48*B*C*a*b^7 + 256*B*C*a^7*b + 896*A*B*a^3*b^5 + 1536*A*B*a^5*b^3 + 240*A*C*a^2
*b^6 + 1184*A*C*a^4*b^4 + 384*A*C*a^6*b^2 + 480*B*C*a^3*b^5 + 896*B*C*a^5*b^3))*((A*b^4)/2 + C*a^4 + (3*C*b^4)
/8 + 6*A*a^2*b^2 + 3*C*a^2*b^2 + 2*B*a*b^3 + 4*B*a^3*b) - 64*B^2*C*a^12 - 256*B^3*a^11*b + 64*A^3*a^3*b^9 + 15
36*A^3*a^5*b^7 - 512*A^3*a^6*b^6 + 9216*A^3*a^7*b^5 - 6144*A^3*a^8*b^4 + 256*B^3*a^6*b^6 + 1024*B^3*a^8*b^4 -
128*B^3*a^9*b^3 + 1024*B^3*a^10*b^2 + 256*A*C^2*a^11*b + 512*B^2*C*a^11*b + 1152*A*B^2*a^5*b^7 + 5888*A*B^2*a^
7*b^5 - 1056*A*B^2*a^8*b^4 + 7168*A*B^2*a^9*b^3 - 2432*A*B^2*a^10*b^2 + 528*A^2*B*a^4*b^8 + 7552*A^2*B*a^6*b^6
- 2304*A^2*B*a^7*b^5 + 14592*A^2*B*a^8*b^4 - 7168*A^2*B*a^9*b^3 + 36*A*C^2*a^3*b^9 + 576*A*C^2*a^5*b^7 + 2496
*A*C^2*a^7*b^5 + 1536*A*C^2*a^9*b^3 + 96*A^2*C*a^3*b^9 + 1920*A^2*C*a^5*b^7 - 384*A^2*C*a^6*b^6 + 9472*A^2*C*a
^7*b^5 - 3072*A^2*C*a^8*b^4 + 3072*A^2*C*a^9*b^3 - 1024*A^2*C*a^10*b^2 + 9*B*C^2*a^4*b^8 + 144*B*C^2*a^6*b^6 +
624*B*C^2*a^8*b^4 + 384*B*C^2*a^10*b^2 + 96*B^2*C*a^5*b^7 + 960*B^2*C*a^7*b^5 - 24*B^2*C*a^8*b^4 + 1792*B^2*C
*a^9*b^3 - 192*B^2*C*a^10*b^2 - 512*A*B*C*a^11*b + 408*A*B*C*a^4*b^8 + 4320*A*B*C*a^6*b^6 - 192*A*B*C*a^7*b^5
+ 9536*A*B*C*a^8*b^4 - 1536*A*B*C*a^9*b^3 + 2816*A*B*C*a^10*b^2))*(A*b^4*1i + C*a^4*2i + (C*b^4*3i)/4 + A*a^2*
b^2*12i + C*a^2*b^2*6i + B*a*b^3*4i + B*a^3*b*8i))/d + (2*a^3*atan((a^3*(tan(c/2 + (d*x)/2)*(8*A^2*b^8 + 32*B^
2*a^8 + 32*C^2*a^8 + (9*C^2*b^8)/2 + 192*A^2*a^2*b^6 + 1152*A^2*a^4*b^4 + 512*A^2*a^6*b^2 + 128*B^2*a^2*b^6 +
512*B^2*a^4*b^4 + 512*B^2*a^6*b^2 + 72*C^2*a^2*...

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