3.1007 \(\int \frac {(A+B \cos (c+d x)+C \cos ^2(c+d x)) \sec ^2(c+d x)}{(a+b \cos (c+d x))^4} \, dx\)
Optimal. Leaf size=480 \[ -\frac {(4 A b-a B) \tanh ^{-1}(\sin (c+d x))}{a^5 d}+\frac {\tan (c+d x) \left (A b^2-a (b B-a C)\right )}{3 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^3}-\frac {\tan (c+d x) \left (-3 a^4 C+6 a^3 b B-a^2 b^2 (9 A+2 C)-a b^3 B+4 A b^4\right )}{6 a^2 d \left (a^2-b^2\right )^2 (a+b \cos (c+d x))^2}+\frac {\tan (c+d x) \left (a^6 (6 A-11 C)+26 a^5 b B-a^4 b^2 (65 A+4 C)-17 a^3 b^3 B+68 a^2 A b^4+6 a b^5 B-24 A b^6\right )}{6 a^4 d \left (a^2-b^2\right )^3}-\frac {\tan (c+d x) \left (-2 a^6 C+6 a^5 b B-3 a^4 b^2 (4 A+C)-2 a^3 b^3 B+11 a^2 A b^4+a b^5 B-4 A b^6\right )}{2 a^3 d \left (a^2-b^2\right )^3 (a+b \cos (c+d x))}-\frac {\left (-2 a^8 C+8 a^7 b B-a^6 b^2 (20 A+3 C)-8 a^5 b^3 B+35 a^4 A b^4+7 a^3 b^5 B-28 a^2 A b^6-2 a b^7 B+8 A b^8\right ) \tan ^{-1}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{a^5 d (a-b)^{7/2} (a+b)^{7/2}} \]
[Out]
-(35*a^4*A*b^4-28*a^2*A*b^6+8*A*b^8+8*a^7*b*B-8*a^5*b^3*B+7*a^3*b^5*B-2*a*b^7*B-2*a^8*C-a^6*b^2*(20*A+3*C))*ar
ctan((a-b)^(1/2)*tan(1/2*d*x+1/2*c)/(a+b)^(1/2))/a^5/(a-b)^(7/2)/(a+b)^(7/2)/d-(4*A*b-B*a)*arctanh(sin(d*x+c))
/a^5/d+1/6*(68*a^2*A*b^4-24*A*b^6+26*a^5*b*B-17*a^3*b^3*B+6*a*b^5*B+a^6*(6*A-11*C)-a^4*b^2*(65*A+4*C))*tan(d*x
+c)/a^4/(a^2-b^2)^3/d+1/3*(A*b^2-a*(B*b-C*a))*tan(d*x+c)/a/(a^2-b^2)/d/(a+b*cos(d*x+c))^3-1/6*(4*A*b^4+6*a^3*b
*B-a*b^3*B-3*a^4*C-a^2*b^2*(9*A+2*C))*tan(d*x+c)/a^2/(a^2-b^2)^2/d/(a+b*cos(d*x+c))^2-1/2*(11*a^2*A*b^4-4*A*b^
6+6*a^5*b*B-2*a^3*b^3*B+a*b^5*B-2*a^6*C-3*a^4*b^2*(4*A+C))*tan(d*x+c)/a^3/(a^2-b^2)^3/d/(a+b*cos(d*x+c))
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Rubi [A] time = 10.54, antiderivative size = 480, normalized size of antiderivative = 1.00,
number of steps used = 8, number of rules used = 5, integrand size = 41, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.122, Rules used =
{3055, 3001, 3770, 2659, 205} \[ -\frac {\left (-a^6 b^2 (20 A+3 C)+35 a^4 A b^4-28 a^2 A b^6-8 a^5 b^3 B+7 a^3 b^5 B+8 a^7 b B-2 a^8 C-2 a b^7 B+8 A b^8\right ) \tan ^{-1}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{a^5 d (a-b)^{7/2} (a+b)^{7/2}}+\frac {\tan (c+d x) \left (-a^4 b^2 (65 A+4 C)+68 a^2 A b^4+a^6 (6 A-11 C)-17 a^3 b^3 B+26 a^5 b B+6 a b^5 B-24 A b^6\right )}{6 a^4 d \left (a^2-b^2\right )^3}-\frac {\tan (c+d x) \left (-3 a^4 b^2 (4 A+C)+11 a^2 A b^4-2 a^3 b^3 B+6 a^5 b B-2 a^6 C+a b^5 B-4 A b^6\right )}{2 a^3 d \left (a^2-b^2\right )^3 (a+b \cos (c+d x))}-\frac {\tan (c+d x) \left (-a^2 b^2 (9 A+2 C)+6 a^3 b B-3 a^4 C-a b^3 B+4 A b^4\right )}{6 a^2 d \left (a^2-b^2\right )^2 (a+b \cos (c+d x))^2}+\frac {\tan (c+d x) \left (A b^2-a (b B-a C)\right )}{3 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^3}-\frac {(4 A b-a B) \tanh ^{-1}(\sin (c+d x))}{a^5 d} \]
Antiderivative was successfully verified.
[In]
Int[((A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^2)/(a + b*Cos[c + d*x])^4,x]
[Out]
-(((35*a^4*A*b^4 - 28*a^2*A*b^6 + 8*A*b^8 + 8*a^7*b*B - 8*a^5*b^3*B + 7*a^3*b^5*B - 2*a*b^7*B - 2*a^8*C - a^6*
b^2*(20*A + 3*C))*ArcTan[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/(a^5*(a - b)^(7/2)*(a + b)^(7/2)*d)) - (
(4*A*b - a*B)*ArcTanh[Sin[c + d*x]])/(a^5*d) + ((68*a^2*A*b^4 - 24*A*b^6 + 26*a^5*b*B - 17*a^3*b^3*B + 6*a*b^5
*B + a^6*(6*A - 11*C) - a^4*b^2*(65*A + 4*C))*Tan[c + d*x])/(6*a^4*(a^2 - b^2)^3*d) + ((A*b^2 - a*(b*B - a*C))
*Tan[c + d*x])/(3*a*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^3) - ((4*A*b^4 + 6*a^3*b*B - a*b^3*B - 3*a^4*C - a^2*b^
2*(9*A + 2*C))*Tan[c + d*x])/(6*a^2*(a^2 - b^2)^2*d*(a + b*Cos[c + d*x])^2) - ((11*a^2*A*b^4 - 4*A*b^6 + 6*a^5
*b*B - 2*a^3*b^3*B + a*b^5*B - 2*a^6*C - 3*a^4*b^2*(4*A + C))*Tan[c + d*x])/(2*a^3*(a^2 - b^2)^3*d*(a + b*Cos[
c + d*x]))
Rule 205
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]*ArcTan[x/Rt[a/b, 2]])/a, x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]
Rule 2659
Int[((a_) + (b_.)*sin[Pi/2 + (c_.) + (d_.)*(x_)])^(-1), x_Symbol] :> With[{e = FreeFactors[Tan[(c + d*x)/2], x
]}, Dist[(2*e)/d, Subst[Int[1/(a + b + (a - b)*e^2*x^2), x], x, Tan[(c + d*x)/2]/e], x]] /; FreeQ[{a, b, c, d}
, x] && NeQ[a^2 - b^2, 0]
Rule 3001
Int[((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])/(((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*sin[(e_.)
+ (f_.)*(x_)])), x_Symbol] :> Dist[(A*b - a*B)/(b*c - a*d), Int[1/(a + b*Sin[e + f*x]), x], x] + Dist[(B*c - A
*d)/(b*c - a*d), Int[1/(c + d*Sin[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f, A, B}, x] && NeQ[b*c - a*d, 0]
&& NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0]
Rule 3055
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[((A*b^2 - a*b*B + a^2*C)*Cos[e +
f*x]*(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^(n + 1))/(f*(m + 1)*(b*c - a*d)*(a^2 - b^2)), x] + Dis
t[1/((m + 1)*(b*c - a*d)*(a^2 - b^2)), Int[(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^n*Simp[(m + 1)*(b
*c - a*d)*(a*A - b*B + a*C) + d*(A*b^2 - a*b*B + a^2*C)*(m + n + 2) - (c*(A*b^2 - a*b*B + a^2*C) + (m + 1)*(b*
c - a*d)*(A*b - a*B + b*C))*Sin[e + f*x] - d*(A*b^2 - a*b*B + a^2*C)*(m + n + 3)*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] && Lt
Q[m, -1] && ((EqQ[a, 0] && IntegerQ[m] && !IntegerQ[n]) || !(IntegerQ[2*n] && LtQ[n, -1] && ((IntegerQ[n] &&
!IntegerQ[m]) || EqQ[a, 0])))
Rule 3770
Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> -Simp[ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]
Rubi steps
\begin {align*} \int \frac {\left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^2(c+d x)}{(a+b \cos (c+d x))^4} \, dx &=\frac {\left (A b^2-a (b B-a C)\right ) \tan (c+d x)}{3 a \left (a^2-b^2\right ) d (a+b \cos (c+d x))^3}+\frac {\int \frac {\left (-4 A b^2+a b B+a^2 (3 A-C)-3 a (A b-a B+b C) \cos (c+d x)+3 \left (A b^2-a (b B-a C)\right ) \cos ^2(c+d x)\right ) \sec ^2(c+d x)}{(a+b \cos (c+d x))^3} \, dx}{3 a \left (a^2-b^2\right )}\\ &=\frac {\left (A b^2-a (b B-a C)\right ) \tan (c+d x)}{3 a \left (a^2-b^2\right ) d (a+b \cos (c+d x))^3}-\frac {\left (4 A b^4+6 a^3 b B-a b^3 B-3 a^4 C-a^2 b^2 (9 A+2 C)\right ) \tan (c+d x)}{6 a^2 \left (a^2-b^2\right )^2 d (a+b \cos (c+d x))^2}+\frac {\int \frac {\left (-23 a^2 A b^2+12 A b^4+8 a^3 b B-3 a b^3 B+a^4 (6 A-5 C)+2 a \left (A b^3+3 a^3 B+2 a b^2 B-a^2 b (6 A+5 C)\right ) \cos (c+d x)-2 \left (4 A b^4+6 a^3 b B-a b^3 B-3 a^4 C-a^2 b^2 (9 A+2 C)\right ) \cos ^2(c+d x)\right ) \sec ^2(c+d x)}{(a+b \cos (c+d x))^2} \, dx}{6 a^2 \left (a^2-b^2\right )^2}\\ &=\frac {\left (A b^2-a (b B-a C)\right ) \tan (c+d x)}{3 a \left (a^2-b^2\right ) d (a+b \cos (c+d x))^3}-\frac {\left (4 A b^4+6 a^3 b B-a b^3 B-3 a^4 C-a^2 b^2 (9 A+2 C)\right ) \tan (c+d x)}{6 a^2 \left (a^2-b^2\right )^2 d (a+b \cos (c+d x))^2}-\frac {\left (11 a^2 A b^4-4 A b^6+6 a^5 b B-2 a^3 b^3 B+a b^5 B-2 a^6 C-3 a^4 b^2 (4 A+C)\right ) \tan (c+d x)}{2 a^3 \left (a^2-b^2\right )^3 d (a+b \cos (c+d x))}+\frac {\int \frac {\left (68 a^2 A b^4-24 A b^6+26 a^5 b B-17 a^3 b^3 B+6 a b^5 B+a^6 (6 A-11 C)-a^4 b^2 (65 A+4 C)-a \left (4 A b^5-6 a^5 B-8 a^3 b^2 B-a b^4 B-a^2 b^3 (7 A-4 C)+a^4 b (18 A+11 C)\right ) \cos (c+d x)-3 \left (11 a^2 A b^4-4 A b^6+6 a^5 b B-2 a^3 b^3 B+a b^5 B-2 a^6 C-3 a^4 b^2 (4 A+C)\right ) \cos ^2(c+d x)\right ) \sec ^2(c+d x)}{a+b \cos (c+d x)} \, dx}{6 a^3 \left (a^2-b^2\right )^3}\\ &=\frac {\left (68 a^2 A b^4-24 A b^6+26 a^5 b B-17 a^3 b^3 B+6 a b^5 B+a^6 (6 A-11 C)-a^4 b^2 (65 A+4 C)\right ) \tan (c+d x)}{6 a^4 \left (a^2-b^2\right )^3 d}+\frac {\left (A b^2-a (b B-a C)\right ) \tan (c+d x)}{3 a \left (a^2-b^2\right ) d (a+b \cos (c+d x))^3}-\frac {\left (4 A b^4+6 a^3 b B-a b^3 B-3 a^4 C-a^2 b^2 (9 A+2 C)\right ) \tan (c+d x)}{6 a^2 \left (a^2-b^2\right )^2 d (a+b \cos (c+d x))^2}-\frac {\left (11 a^2 A b^4-4 A b^6+6 a^5 b B-2 a^3 b^3 B+a b^5 B-2 a^6 C-3 a^4 b^2 (4 A+C)\right ) \tan (c+d x)}{2 a^3 \left (a^2-b^2\right )^3 d (a+b \cos (c+d x))}+\frac {\int \frac {\left (-6 \left (a^2-b^2\right )^3 (4 A b-a B)-3 a \left (11 a^2 A b^4-4 A b^6+6 a^5 b B-2 a^3 b^3 B+a b^5 B-2 a^6 C-3 a^4 b^2 (4 A+C)\right ) \cos (c+d x)\right ) \sec (c+d x)}{a+b \cos (c+d x)} \, dx}{6 a^4 \left (a^2-b^2\right )^3}\\ &=\frac {\left (68 a^2 A b^4-24 A b^6+26 a^5 b B-17 a^3 b^3 B+6 a b^5 B+a^6 (6 A-11 C)-a^4 b^2 (65 A+4 C)\right ) \tan (c+d x)}{6 a^4 \left (a^2-b^2\right )^3 d}+\frac {\left (A b^2-a (b B-a C)\right ) \tan (c+d x)}{3 a \left (a^2-b^2\right ) d (a+b \cos (c+d x))^3}-\frac {\left (4 A b^4+6 a^3 b B-a b^3 B-3 a^4 C-a^2 b^2 (9 A+2 C)\right ) \tan (c+d x)}{6 a^2 \left (a^2-b^2\right )^2 d (a+b \cos (c+d x))^2}-\frac {\left (11 a^2 A b^4-4 A b^6+6 a^5 b B-2 a^3 b^3 B+a b^5 B-2 a^6 C-3 a^4 b^2 (4 A+C)\right ) \tan (c+d x)}{2 a^3 \left (a^2-b^2\right )^3 d (a+b \cos (c+d x))}-\frac {(4 A b-a B) \int \sec (c+d x) \, dx}{a^5}-\frac {\left (35 a^4 A b^4-28 a^2 A b^6+8 A b^8+8 a^7 b B-8 a^5 b^3 B+7 a^3 b^5 B-2 a b^7 B-2 a^8 C-a^6 b^2 (20 A+3 C)\right ) \int \frac {1}{a+b \cos (c+d x)} \, dx}{2 a^5 \left (a^2-b^2\right )^3}\\ &=-\frac {(4 A b-a B) \tanh ^{-1}(\sin (c+d x))}{a^5 d}+\frac {\left (68 a^2 A b^4-24 A b^6+26 a^5 b B-17 a^3 b^3 B+6 a b^5 B+a^6 (6 A-11 C)-a^4 b^2 (65 A+4 C)\right ) \tan (c+d x)}{6 a^4 \left (a^2-b^2\right )^3 d}+\frac {\left (A b^2-a (b B-a C)\right ) \tan (c+d x)}{3 a \left (a^2-b^2\right ) d (a+b \cos (c+d x))^3}-\frac {\left (4 A b^4+6 a^3 b B-a b^3 B-3 a^4 C-a^2 b^2 (9 A+2 C)\right ) \tan (c+d x)}{6 a^2 \left (a^2-b^2\right )^2 d (a+b \cos (c+d x))^2}-\frac {\left (11 a^2 A b^4-4 A b^6+6 a^5 b B-2 a^3 b^3 B+a b^5 B-2 a^6 C-3 a^4 b^2 (4 A+C)\right ) \tan (c+d x)}{2 a^3 \left (a^2-b^2\right )^3 d (a+b \cos (c+d x))}-\frac {\left (35 a^4 A b^4-28 a^2 A b^6+8 A b^8+8 a^7 b B-8 a^5 b^3 B+7 a^3 b^5 B-2 a b^7 B-2 a^8 C-a^6 b^2 (20 A+3 C)\right ) \operatorname {Subst}\left (\int \frac {1}{a+b+(a-b) x^2} \, dx,x,\tan \left (\frac {1}{2} (c+d x)\right )\right )}{a^5 \left (a^2-b^2\right )^3 d}\\ &=\frac {\left (20 a^6 A b^2-35 a^4 A b^4+28 a^2 A b^6-8 A b^8-8 a^7 b B+8 a^5 b^3 B-7 a^3 b^5 B+2 a b^7 B+2 a^8 C+3 a^6 b^2 C\right ) \tan ^{-1}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{a^5 (a-b)^{7/2} (a+b)^{7/2} d}-\frac {(4 A b-a B) \tanh ^{-1}(\sin (c+d x))}{a^5 d}+\frac {\left (68 a^2 A b^4-24 A b^6+26 a^5 b B-17 a^3 b^3 B+6 a b^5 B+a^6 (6 A-11 C)-a^4 b^2 (65 A+4 C)\right ) \tan (c+d x)}{6 a^4 \left (a^2-b^2\right )^3 d}+\frac {\left (A b^2-a (b B-a C)\right ) \tan (c+d x)}{3 a \left (a^2-b^2\right ) d (a+b \cos (c+d x))^3}-\frac {\left (4 A b^4+6 a^3 b B-a b^3 B-3 a^4 C-a^2 b^2 (9 A+2 C)\right ) \tan (c+d x)}{6 a^2 \left (a^2-b^2\right )^2 d (a+b \cos (c+d x))^2}-\frac {\left (11 a^2 A b^4-4 A b^6+6 a^5 b B-2 a^3 b^3 B+a b^5 B-2 a^6 C-3 a^4 b^2 (4 A+C)\right ) \tan (c+d x)}{2 a^3 \left (a^2-b^2\right )^3 d (a+b \cos (c+d x))}\\ \end {align*}
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Mathematica [A] time = 4.99, size = 709, normalized size = 1.48 \[ \frac {\cos (c+d x) \left (A \sec ^2(c+d x)+B \sec (c+d x)+C\right ) \left (\frac {48 \cos (c+d x) \left (2 a^8 C-8 a^7 b B+a^6 b^2 (20 A+3 C)+8 a^5 b^3 B-35 a^4 A b^4-7 a^3 b^5 B+28 a^2 A b^6+2 a b^7 B-8 A b^8\right ) \tanh ^{-1}\left (\frac {(a-b) \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {b^2-a^2}}\right )}{\left (b^2-a^2\right )^{7/2}}+\frac {2 a \sin (c+d x) \left (24 a^9 A-36 a^7 A b^2-54 a^7 b^2 C+6 a^6 A b^3 \cos (3 (c+d x))+120 a^6 b^3 B-11 a^6 b^3 C \cos (3 (c+d x))-246 a^5 A b^4+26 a^5 b^4 B \cos (3 (c+d x))-6 a^5 b^4 C-65 a^4 A b^5 \cos (3 (c+d x))-90 a^4 b^5 B-4 a^4 b^5 C \cos (3 (c+d x))+318 a^3 A b^6-17 a^3 b^6 B \cos (3 (c+d x))+68 a^2 A b^7 \cos (3 (c+d x))+30 a^2 b^7 B+6 a b^2 \cos (2 (c+d x)) \left (a^6 (6 A-9 C)+20 a^5 b B-a^4 b^2 (53 A+C)-15 a^3 b^3 B+57 a^2 A b^4+5 a b^5 B-20 A b^6\right )-b \cos (c+d x) \left (-72 a^8 (A-C)-144 a^7 b B+a^6 b^2 (438 A+13 C)+50 a^5 b^3 B-5 a^4 b^4 (61 A-4 C)+7 a^3 b^5 B-28 a^2 A b^6-18 a b^7 B+72 A b^8\right )-120 a A b^8+6 a b^8 B \cos (3 (c+d x))-24 A b^9 \cos (3 (c+d x))\right )}{\left (a^2-b^2\right )^3 (a+b \cos (c+d x))^3}+48 (4 A b-a B) \cos (c+d x) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )+48 (a B-4 A b) \cos (c+d x) \log \left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )\right )}{24 a^5 d (2 A+2 B \cos (c+d x)+C \cos (2 (c+d x))+C)} \]
Warning: Unable to verify antiderivative.
[In]
Integrate[((A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^2)/(a + b*Cos[c + d*x])^4,x]
[Out]
(Cos[c + d*x]*(C + B*Sec[c + d*x] + A*Sec[c + d*x]^2)*((48*(-35*a^4*A*b^4 + 28*a^2*A*b^6 - 8*A*b^8 - 8*a^7*b*B
+ 8*a^5*b^3*B - 7*a^3*b^5*B + 2*a*b^7*B + 2*a^8*C + a^6*b^2*(20*A + 3*C))*ArcTanh[((a - b)*Tan[(c + d*x)/2])/
Sqrt[-a^2 + b^2]]*Cos[c + d*x])/(-a^2 + b^2)^(7/2) + 48*(4*A*b - a*B)*Cos[c + d*x]*Log[Cos[(c + d*x)/2] - Sin[
(c + d*x)/2]] + 48*(-4*A*b + a*B)*Cos[c + d*x]*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + (2*a*(24*a^9*A - 36*
a^7*A*b^2 - 246*a^5*A*b^4 + 318*a^3*A*b^6 - 120*a*A*b^8 + 120*a^6*b^3*B - 90*a^4*b^5*B + 30*a^2*b^7*B - 54*a^7
*b^2*C - 6*a^5*b^4*C - b*(-28*a^2*A*b^6 + 72*A*b^8 - 144*a^7*b*B + 50*a^5*b^3*B + 7*a^3*b^5*B - 18*a*b^7*B - 5
*a^4*b^4*(61*A - 4*C) - 72*a^8*(A - C) + a^6*b^2*(438*A + 13*C))*Cos[c + d*x] + 6*a*b^2*(57*a^2*A*b^4 - 20*A*b
^6 + 20*a^5*b*B - 15*a^3*b^3*B + 5*a*b^5*B + a^6*(6*A - 9*C) - a^4*b^2*(53*A + C))*Cos[2*(c + d*x)] + 6*a^6*A*
b^3*Cos[3*(c + d*x)] - 65*a^4*A*b^5*Cos[3*(c + d*x)] + 68*a^2*A*b^7*Cos[3*(c + d*x)] - 24*A*b^9*Cos[3*(c + d*x
)] + 26*a^5*b^4*B*Cos[3*(c + d*x)] - 17*a^3*b^6*B*Cos[3*(c + d*x)] + 6*a*b^8*B*Cos[3*(c + d*x)] - 11*a^6*b^3*C
*Cos[3*(c + d*x)] - 4*a^4*b^5*C*Cos[3*(c + d*x)])*Sin[c + d*x])/((a^2 - b^2)^3*(a + b*Cos[c + d*x])^3)))/(24*a
^5*d*(2*A + C + 2*B*Cos[c + d*x] + C*Cos[2*(c + d*x)]))
________________________________________________________________________________________
fricas [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((A+B*cos(d*x+c)+C*cos(d*x+c)^2)*sec(d*x+c)^2/(a+b*cos(d*x+c))^4,x, algorithm="fricas")
[Out]
Timed out
________________________________________________________________________________________
giac [B] time = 0.38, size = 1255, normalized size = 2.61 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((A+B*cos(d*x+c)+C*cos(d*x+c)^2)*sec(d*x+c)^2/(a+b*cos(d*x+c))^4,x, algorithm="giac")
[Out]
-1/3*(3*(2*C*a^8 - 8*B*a^7*b + 20*A*a^6*b^2 + 3*C*a^6*b^2 + 8*B*a^5*b^3 - 35*A*a^4*b^4 - 7*B*a^3*b^5 + 28*A*a^
2*b^6 + 2*B*a*b^7 - 8*A*b^8)*(pi*floor(1/2*(d*x + c)/pi + 1/2)*sgn(-2*a + 2*b) + arctan(-(a*tan(1/2*d*x + 1/2*
c) - b*tan(1/2*d*x + 1/2*c))/sqrt(a^2 - b^2)))/((a^11 - 3*a^9*b^2 + 3*a^7*b^4 - a^5*b^6)*sqrt(a^2 - b^2)) + (1
8*C*a^8*b*tan(1/2*d*x + 1/2*c)^5 - 36*B*a^7*b^2*tan(1/2*d*x + 1/2*c)^5 - 27*C*a^7*b^2*tan(1/2*d*x + 1/2*c)^5 +
60*A*a^6*b^3*tan(1/2*d*x + 1/2*c)^5 + 60*B*a^6*b^3*tan(1/2*d*x + 1/2*c)^5 + 6*C*a^6*b^3*tan(1/2*d*x + 1/2*c)^
5 - 105*A*a^5*b^4*tan(1/2*d*x + 1/2*c)^5 + 6*B*a^5*b^4*tan(1/2*d*x + 1/2*c)^5 - 3*C*a^5*b^4*tan(1/2*d*x + 1/2*
c)^5 - 24*A*a^4*b^5*tan(1/2*d*x + 1/2*c)^5 - 45*B*a^4*b^5*tan(1/2*d*x + 1/2*c)^5 + 6*C*a^4*b^5*tan(1/2*d*x + 1
/2*c)^5 + 117*A*a^3*b^6*tan(1/2*d*x + 1/2*c)^5 + 6*B*a^3*b^6*tan(1/2*d*x + 1/2*c)^5 - 24*A*a^2*b^7*tan(1/2*d*x
+ 1/2*c)^5 + 15*B*a^2*b^7*tan(1/2*d*x + 1/2*c)^5 - 42*A*a*b^8*tan(1/2*d*x + 1/2*c)^5 - 6*B*a*b^8*tan(1/2*d*x
+ 1/2*c)^5 + 18*A*b^9*tan(1/2*d*x + 1/2*c)^5 + 36*C*a^8*b*tan(1/2*d*x + 1/2*c)^3 - 72*B*a^7*b^2*tan(1/2*d*x +
1/2*c)^3 + 120*A*a^6*b^3*tan(1/2*d*x + 1/2*c)^3 - 32*C*a^6*b^3*tan(1/2*d*x + 1/2*c)^3 + 116*B*a^5*b^4*tan(1/2*
d*x + 1/2*c)^3 - 236*A*a^4*b^5*tan(1/2*d*x + 1/2*c)^3 - 4*C*a^4*b^5*tan(1/2*d*x + 1/2*c)^3 - 56*B*a^3*b^6*tan(
1/2*d*x + 1/2*c)^3 + 152*A*a^2*b^7*tan(1/2*d*x + 1/2*c)^3 + 12*B*a*b^8*tan(1/2*d*x + 1/2*c)^3 - 36*A*b^9*tan(1
/2*d*x + 1/2*c)^3 + 18*C*a^8*b*tan(1/2*d*x + 1/2*c) - 36*B*a^7*b^2*tan(1/2*d*x + 1/2*c) + 27*C*a^7*b^2*tan(1/2
*d*x + 1/2*c) + 60*A*a^6*b^3*tan(1/2*d*x + 1/2*c) - 60*B*a^6*b^3*tan(1/2*d*x + 1/2*c) + 6*C*a^6*b^3*tan(1/2*d*
x + 1/2*c) + 105*A*a^5*b^4*tan(1/2*d*x + 1/2*c) + 6*B*a^5*b^4*tan(1/2*d*x + 1/2*c) + 3*C*a^5*b^4*tan(1/2*d*x +
1/2*c) - 24*A*a^4*b^5*tan(1/2*d*x + 1/2*c) + 45*B*a^4*b^5*tan(1/2*d*x + 1/2*c) + 6*C*a^4*b^5*tan(1/2*d*x + 1/
2*c) - 117*A*a^3*b^6*tan(1/2*d*x + 1/2*c) + 6*B*a^3*b^6*tan(1/2*d*x + 1/2*c) - 24*A*a^2*b^7*tan(1/2*d*x + 1/2*
c) - 15*B*a^2*b^7*tan(1/2*d*x + 1/2*c) + 42*A*a*b^8*tan(1/2*d*x + 1/2*c) - 6*B*a*b^8*tan(1/2*d*x + 1/2*c) + 18
*A*b^9*tan(1/2*d*x + 1/2*c))/((a^10 - 3*a^8*b^2 + 3*a^6*b^4 - a^4*b^6)*(a*tan(1/2*d*x + 1/2*c)^2 - b*tan(1/2*d
*x + 1/2*c)^2 + a + b)^3) - 3*(B*a - 4*A*b)*log(abs(tan(1/2*d*x + 1/2*c) + 1))/a^5 + 3*(B*a - 4*A*b)*log(abs(t
an(1/2*d*x + 1/2*c) - 1))/a^5 + 6*A*tan(1/2*d*x + 1/2*c)/((tan(1/2*d*x + 1/2*c)^2 - 1)*a^4))/d
________________________________________________________________________________________
maple [B] time = 0.30, size = 3628, normalized size = 7.56 \[ \text {output too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((A+B*cos(d*x+c)+C*cos(d*x+c)^2)*sec(d*x+c)^2/(a+b*cos(d*x+c))^4,x)
[Out]
28/d/a^3/(a^6-3*a^4*b^2+3*a^2*b^4-b^6)/((a-b)*(a+b))^(1/2)*arctan(tan(1/2*d*x+1/2*c)*(a-b)/((a-b)*(a+b))^(1/2)
)*A*b^6+3/d/(a*tan(1/2*d*x+1/2*c)^2-tan(1/2*d*x+1/2*c)^2*b+a+b)^3/(a+b)/(a^3-3*a^2*b+3*a*b^2-b^3)*tan(1/2*d*x+
1/2*c)*C*a*b^2-3/d/(a*tan(1/2*d*x+1/2*c)^2-tan(1/2*d*x+1/2*c)^2*b+a+b)^3/(a-b)/(a^3+3*a^2*b+3*a*b^2+b^3)*tan(1
/2*d*x+1/2*c)^5*C*a*b^2+4/d/a^5*ln(tan(1/2*d*x+1/2*c)-1)*A*b-4/d/a^5*ln(tan(1/2*d*x+1/2*c)+1)*A*b-20/d*b^3/(a*
tan(1/2*d*x+1/2*c)^2-tan(1/2*d*x+1/2*c)^2*b+a+b)^3/(a-b)/(a^3+3*a^2*b+3*a*b^2+b^3)*tan(1/2*d*x+1/2*c)^5*A-2/d/
(a*tan(1/2*d*x+1/2*c)^2-tan(1/2*d*x+1/2*c)^2*b+a+b)^3/(a+b)/(a^3-3*a^2*b+3*a*b^2-b^3)*tan(1/2*d*x+1/2*c)*b^3*C
-4/3/d/(a*tan(1/2*d*x+1/2*c)^2-tan(1/2*d*x+1/2*c)^2*b+a+b)^3/(a^2-2*a*b+b^2)/(a^2+2*a*b+b^2)*tan(1/2*d*x+1/2*c
)^3*b^3*C-2/d/(a*tan(1/2*d*x+1/2*c)^2-tan(1/2*d*x+1/2*c)^2*b+a+b)^3/(a-b)/(a^3+3*a^2*b+3*a*b^2+b^3)*tan(1/2*d*
x+1/2*c)^5*b^3*C+4/d/(a*tan(1/2*d*x+1/2*c)^2-tan(1/2*d*x+1/2*c)^2*b+a+b)^3/(a-b)/(a^3+3*a^2*b+3*a*b^2+b^3)*tan
(1/2*d*x+1/2*c)^5*b^3*B-6/d/a^4/(a*tan(1/2*d*x+1/2*c)^2-tan(1/2*d*x+1/2*c)^2*b+a+b)^3*b^7/(a-b)/(a^3+3*a^2*b+3
*a*b^2+b^3)*tan(1/2*d*x+1/2*c)^5*A+2/d/a^3/(a*tan(1/2*d*x+1/2*c)^2-tan(1/2*d*x+1/2*c)^2*b+a+b)^3*b^6/(a-b)/(a^
3+3*a^2*b+3*a*b^2+b^3)*tan(1/2*d*x+1/2*c)^5*B-12/d/a^4/(a*tan(1/2*d*x+1/2*c)^2-tan(1/2*d*x+1/2*c)^2*b+a+b)^3*b
^7/(a^2+2*a*b+b^2)/(a^2-2*a*b+b^2)*tan(1/2*d*x+1/2*c)^3*A+2/d/a^3/(a*tan(1/2*d*x+1/2*c)^2-tan(1/2*d*x+1/2*c)^2
*b+a+b)^3/(a-b)/(a^3+3*a^2*b+3*a*b^2+b^3)*tan(1/2*d*x+1/2*c)^5*A*b^6+5/d/a/(a*tan(1/2*d*x+1/2*c)^2-tan(1/2*d*x
+1/2*c)^2*b+a+b)^3/(a+b)/(a^3-3*a^2*b+3*a*b^2-b^3)*tan(1/2*d*x+1/2*c)*A*b^4+18/d/a^2/(a*tan(1/2*d*x+1/2*c)^2-t
an(1/2*d*x+1/2*c)^2*b+a+b)^3/(a+b)/(a^3-3*a^2*b+3*a*b^2-b^3)*tan(1/2*d*x+1/2*c)*A*b^5-2/d/a^3/(a*tan(1/2*d*x+1
/2*c)^2-tan(1/2*d*x+1/2*c)^2*b+a+b)^3/(a+b)/(a^3-3*a^2*b+3*a*b^2-b^3)*tan(1/2*d*x+1/2*c)*A*b^6-5/d/a/(a*tan(1/
2*d*x+1/2*c)^2-tan(1/2*d*x+1/2*c)^2*b+a+b)^3/(a-b)/(a^3+3*a^2*b+3*a*b^2+b^3)*tan(1/2*d*x+1/2*c)^5*A*b^4+18/d/a
^2/(a*tan(1/2*d*x+1/2*c)^2-tan(1/2*d*x+1/2*c)^2*b+a+b)^3/(a-b)/(a^3+3*a^2*b+3*a*b^2+b^3)*tan(1/2*d*x+1/2*c)^5*
A*b^5+3/d*b^2*a/(a^6-3*a^4*b^2+3*a^2*b^4-b^6)/((a-b)*(a+b))^(1/2)*arctan(tan(1/2*d*x+1/2*c)*(a-b)/((a-b)*(a+b)
)^(1/2))*C-6/d/a/(a*tan(1/2*d*x+1/2*c)^2-tan(1/2*d*x+1/2*c)^2*b+a+b)^3*b^4/(a-b)/(a^3+3*a^2*b+3*a*b^2+b^3)*tan
(1/2*d*x+1/2*c)^5*B-1/d/a^2/(a*tan(1/2*d*x+1/2*c)^2-tan(1/2*d*x+1/2*c)^2*b+a+b)^3*b^5/(a-b)/(a^3+3*a^2*b+3*a*b
^2+b^3)*tan(1/2*d*x+1/2*c)^5*B+2/d/a^3/(a*tan(1/2*d*x+1/2*c)^2-tan(1/2*d*x+1/2*c)^2*b+a+b)^3*b^6/(a+b)/(a^3-3*
a^2*b+3*a*b^2-b^3)*tan(1/2*d*x+1/2*c)*B-6/d/a^4/(a*tan(1/2*d*x+1/2*c)^2-tan(1/2*d*x+1/2*c)^2*b+a+b)^3*b^7/(a+b
)/(a^3-3*a^2*b+3*a*b^2-b^3)*tan(1/2*d*x+1/2*c)*A-44/3/d/a/(a*tan(1/2*d*x+1/2*c)^2-tan(1/2*d*x+1/2*c)^2*b+a+b)^
3*b^4/(a^2+2*a*b+b^2)/(a^2-2*a*b+b^2)*tan(1/2*d*x+1/2*c)^3*B+4/d/a^3/(a*tan(1/2*d*x+1/2*c)^2-tan(1/2*d*x+1/2*c
)^2*b+a+b)^3*b^6/(a^2+2*a*b+b^2)/(a^2-2*a*b+b^2)*tan(1/2*d*x+1/2*c)^3*B+24/d*b^2/(a*tan(1/2*d*x+1/2*c)^2-tan(1
/2*d*x+1/2*c)^2*b+a+b)^3/(a^2+2*a*b+b^2)/(a^2-2*a*b+b^2)*tan(1/2*d*x+1/2*c)^3*a*B-12/d*b/(a*tan(1/2*d*x+1/2*c)
^2-tan(1/2*d*x+1/2*c)^2*b+a+b)^3/(a^2+2*a*b+b^2)/(a^2-2*a*b+b^2)*tan(1/2*d*x+1/2*c)^3*C*a^2-6/d/a/(a*tan(1/2*d
*x+1/2*c)^2-tan(1/2*d*x+1/2*c)^2*b+a+b)^3*b^4/(a+b)/(a^3-3*a^2*b+3*a*b^2-b^3)*tan(1/2*d*x+1/2*c)*B+1/d/a^2/(a*
tan(1/2*d*x+1/2*c)^2-tan(1/2*d*x+1/2*c)^2*b+a+b)^3*b^5/(a+b)/(a^3-3*a^2*b+3*a*b^2-b^3)*tan(1/2*d*x+1/2*c)*B+11
6/3/d/a^2/(a*tan(1/2*d*x+1/2*c)^2-tan(1/2*d*x+1/2*c)^2*b+a+b)^3*b^5/(a^2+2*a*b+b^2)/(a^2-2*a*b+b^2)*tan(1/2*d*
x+1/2*c)^3*A+12/d*b^2/(a*tan(1/2*d*x+1/2*c)^2-tan(1/2*d*x+1/2*c)^2*b+a+b)^3/(a+b)/(a^3-3*a^2*b+3*a*b^2-b^3)*ta
n(1/2*d*x+1/2*c)*a*B+12/d*b^2/(a*tan(1/2*d*x+1/2*c)^2-tan(1/2*d*x+1/2*c)^2*b+a+b)^3/(a-b)/(a^3+3*a^2*b+3*a*b^2
+b^3)*tan(1/2*d*x+1/2*c)^5*a*B-6/d*b/(a*tan(1/2*d*x+1/2*c)^2-tan(1/2*d*x+1/2*c)^2*b+a+b)^3/(a+b)/(a^3-3*a^2*b+
3*a*b^2-b^3)*tan(1/2*d*x+1/2*c)*C*a^2-6/d*b/(a*tan(1/2*d*x+1/2*c)^2-tan(1/2*d*x+1/2*c)^2*b+a+b)^3/(a-b)/(a^3+3
*a^2*b+3*a*b^2+b^3)*tan(1/2*d*x+1/2*c)^5*C*a^2-7/d/a^2/(a^6-3*a^4*b^2+3*a^2*b^4-b^6)/((a-b)*(a+b))^(1/2)*arcta
n(tan(1/2*d*x+1/2*c)*(a-b)/((a-b)*(a+b))^(1/2))*b^5*B-1/d*A/a^4/(tan(1/2*d*x+1/2*c)-1)-1/d/a^4*ln(tan(1/2*d*x+
1/2*c)-1)*B-1/d*A/a^4/(tan(1/2*d*x+1/2*c)+1)+1/d/a^4*ln(tan(1/2*d*x+1/2*c)+1)*B+2/d*a^3/(a^6-3*a^4*b^2+3*a^2*b
^4-b^6)/((a-b)*(a+b))^(1/2)*arctan(tan(1/2*d*x+1/2*c)*(a-b)/((a-b)*(a+b))^(1/2))*C+2/d/a^4/(a^6-3*a^4*b^2+3*a^
2*b^4-b^6)/((a-b)*(a+b))^(1/2)*arctan(tan(1/2*d*x+1/2*c)*(a-b)/((a-b)*(a+b))^(1/2))*b^7*B-35/d/a/(a^6-3*a^4*b^
2+3*a^2*b^4-b^6)/((a-b)*(a+b))^(1/2)*arctan(tan(1/2*d*x+1/2*c)*(a-b)/((a-b)*(a+b))^(1/2))*A*b^4-4/d/(a*tan(1/2
*d*x+1/2*c)^2-tan(1/2*d*x+1/2*c)^2*b+a+b)^3/(a+b)/(a^3-3*a^2*b+3*a*b^2-b^3)*tan(1/2*d*x+1/2*c)*b^3*B+20/d*a*b^
2/(a^6-3*a^4*b^2+3*a^2*b^4-b^6)/((a-b)*(a+b))^(1/2)*arctan(tan(1/2*d*x+1/2*c)*(a-b)/((a-b)*(a+b))^(1/2))*A-20/
d*b^3/(a*tan(1/2*d*x+1/2*c)^2-tan(1/2*d*x+1/2*c)^2*b+a+b)^3/(a+b)/(a^3-3*a^2*b+3*a*b^2-b^3)*tan(1/2*d*x+1/2*c)
*A-40/d*b^3/(a*tan(1/2*d*x+1/2*c)^2-tan(1/2*d*x+1/2*c)^2*b+a+b)^3/(a^2+2*a*b+b^2)/(a^2-2*a*b+b^2)*tan(1/2*d*x+
1/2*c)^3*A-8/d*a^2*b/(a^6-3*a^4*b^2+3*a^2*b^4-b^6)/((a-b)*(a+b))^(1/2)*arctan(tan(1/2*d*x+1/2*c)*(a-b)/((a-b)*
(a+b))^(1/2))*B+8/d*b^3/(a^6-3*a^4*b^2+3*a^2*b^4-b^6)/((a-b)*(a+b))^(1/2)*arctan(tan(1/2*d*x+1/2*c)*(a-b)/((a-
b)*(a+b))^(1/2))*B-8/d/a^5/(a^6-3*a^4*b^2+3*a^2*b^4-b^6)/((a-b)*(a+b))^(1/2)*arctan(tan(1/2*d*x+1/2*c)*(a-b)/(
(a-b)*(a+b))^(1/2))*A*b^8
________________________________________________________________________________________
maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((A+B*cos(d*x+c)+C*cos(d*x+c)^2)*sec(d*x+c)^2/(a+b*cos(d*x+c))^4,x, algorithm="maxima")
[Out]
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(4*b^2-4*a^2>0)', see `assume?`
for more details)Is 4*b^2-4*a^2 positive or negative?
________________________________________________________________________________________
mupad [B] time = 15.87, size = 15980, normalized size = 33.29 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((A + B*cos(c + d*x) + C*cos(c + d*x)^2)/(cos(c + d*x)^2*(a + b*cos(c + d*x))^4),x)
[Out]
((tan(c/2 + (d*x)/2)^7*(2*A*a^7 + 8*A*b^7 - 24*A*a^2*b^5 + 11*A*a^3*b^4 + 26*A*a^4*b^3 - 6*A*a^5*b^2 + B*a^2*b
^5 + 6*B*a^3*b^4 - 4*B*a^4*b^3 - 12*B*a^5*b^2 + 2*C*a^4*b^3 + 3*C*a^5*b^2 - 4*A*a*b^6 - 2*A*a^6*b - 2*B*a*b^6
+ 6*C*a^6*b))/(a^4*(a + b)^3*(a - b)) + (tan(c/2 + (d*x)/2)*(2*A*a^7 - 8*A*b^7 + 24*A*a^2*b^5 + 11*A*a^3*b^4 -
26*A*a^4*b^3 - 6*A*a^5*b^2 + B*a^2*b^5 - 6*B*a^3*b^4 - 4*B*a^4*b^3 + 12*B*a^5*b^2 - 2*C*a^4*b^3 + 3*C*a^5*b^2
- 4*A*a*b^6 + 2*A*a^6*b + 2*B*a*b^6 - 6*C*a^6*b))/(a^4*(a + b)*(a - b)^3) + (tan(c/2 + (d*x)/2)^3*(18*A*a^8 +
72*A*b^8 - 236*A*a^2*b^6 + 47*A*a^3*b^5 + 273*A*a^4*b^4 - 60*A*a^5*b^3 - 72*A*a^6*b^2 + 3*B*a^2*b^6 + 59*B*a^
3*b^5 - 14*B*a^4*b^4 - 96*B*a^5*b^3 + 36*B*a^6*b^2 + 10*C*a^4*b^4 - 7*C*a^5*b^3 + 45*C*a^6*b^2 - 12*A*a*b^7 -
18*B*a*b^7 - 18*C*a^7*b))/(3*a^4*(a + b)^2*(a - b)^3) + (tan(c/2 + (d*x)/2)^5*(18*A*a^8 + 72*A*b^8 - 236*A*a^2
*b^6 - 47*A*a^3*b^5 + 273*A*a^4*b^4 + 60*A*a^5*b^3 - 72*A*a^6*b^2 - 3*B*a^2*b^6 + 59*B*a^3*b^5 + 14*B*a^4*b^4
- 96*B*a^5*b^3 - 36*B*a^6*b^2 + 10*C*a^4*b^4 + 7*C*a^5*b^3 + 45*C*a^6*b^2 + 12*A*a*b^7 - 18*B*a*b^7 + 18*C*a^7
*b))/(3*a^4*(a + b)^3*(a - b)^2))/(d*(3*a*b^2 + 3*a^2*b - tan(c/2 + (d*x)/2)^4*(6*a^2*b - 6*b^3) - tan(c/2 + (
d*x)/2)^2*(6*a*b^2 - 2*a^3 + 4*b^3) - tan(c/2 + (d*x)/2)^6*(2*a^3 - 6*a*b^2 + 4*b^3) + a^3 + b^3 - tan(c/2 + (
d*x)/2)^8*(3*a*b^2 - 3*a^2*b + a^3 - b^3))) + (atan((((4*A*b - B*a)*(((4*A*b - B*a)*((8*(4*B*a^24 + 4*C*a^24 +
16*A*a^10*b^14 - 8*A*a^11*b^13 - 104*A*a^12*b^12 + 50*A*a^13*b^11 + 286*A*a^14*b^10 - 126*A*a^15*b^9 - 434*A*
a^16*b^8 + 174*A*a^17*b^7 + 386*A*a^18*b^6 - 146*A*a^19*b^5 - 190*A*a^20*b^4 + 72*A*a^21*b^3 + 40*A*a^22*b^2 -
4*B*a^11*b^13 + 2*B*a^12*b^12 + 26*B*a^13*b^11 - 14*B*a^14*b^10 - 70*B*a^15*b^9 + 30*B*a^16*b^8 + 110*B*a^17*
b^7 - 30*B*a^18*b^6 - 110*B*a^19*b^5 + 20*B*a^20*b^4 + 64*B*a^21*b^3 - 12*B*a^22*b^2 + 6*C*a^15*b^9 - 6*C*a^16
*b^8 - 14*C*a^17*b^7 + 14*C*a^18*b^6 + 6*C*a^19*b^5 - 6*C*a^20*b^4 + 6*C*a^21*b^3 - 6*C*a^22*b^2 - 16*A*a^23*b
- 16*B*a^23*b - 4*C*a^23*b))/(a^22*b + a^23 - a^12*b^11 - a^13*b^10 + 5*a^14*b^9 + 5*a^15*b^8 - 10*a^16*b^7 -
10*a^17*b^6 + 10*a^18*b^5 + 10*a^19*b^4 - 5*a^20*b^3 - 5*a^21*b^2) - (8*tan(c/2 + (d*x)/2)*(4*A*b - B*a)*(8*a
^23*b - 8*a^10*b^14 + 8*a^11*b^13 + 48*a^12*b^12 - 48*a^13*b^11 - 120*a^14*b^10 + 120*a^15*b^9 + 160*a^16*b^8
- 160*a^17*b^7 - 120*a^18*b^6 + 120*a^19*b^5 + 48*a^20*b^4 - 48*a^21*b^3 - 8*a^22*b^2))/(a^5*(a^18*b + a^19 -
a^8*b^11 - a^9*b^10 + 5*a^10*b^9 + 5*a^11*b^8 - 10*a^12*b^7 - 10*a^13*b^6 + 10*a^14*b^5 + 10*a^15*b^4 - 5*a^16
*b^3 - 5*a^17*b^2))))/a^5 - (8*tan(c/2 + (d*x)/2)*(128*A^2*b^16 + 4*B^2*a^16 + 4*C^2*a^16 - 128*A^2*a*b^15 - 8
*B^2*a^15*b - 768*A^2*a^2*b^14 + 768*A^2*a^3*b^13 + 1920*A^2*a^4*b^12 - 1920*A^2*a^5*b^11 - 2600*A^2*a^6*b^10
+ 2560*A^2*a^7*b^9 + 2025*A^2*a^8*b^8 - 1920*A^2*a^9*b^7 - 824*A^2*a^10*b^6 + 768*A^2*a^11*b^5 + 80*A^2*a^12*b
^4 - 128*A^2*a^13*b^3 + 64*A^2*a^14*b^2 + 8*B^2*a^2*b^14 - 8*B^2*a^3*b^13 - 48*B^2*a^4*b^12 + 48*B^2*a^5*b^11
+ 117*B^2*a^6*b^10 - 120*B^2*a^7*b^9 - 164*B^2*a^8*b^8 + 160*B^2*a^9*b^7 + 156*B^2*a^10*b^6 - 120*B^2*a^11*b^5
- 92*B^2*a^12*b^4 + 48*B^2*a^13*b^3 + 44*B^2*a^14*b^2 + 9*C^2*a^12*b^4 + 12*C^2*a^14*b^2 - 64*A*B*a*b^15 - 32
*A*B*a^15*b - 32*B*C*a^15*b + 64*A*B*a^2*b^14 + 384*A*B*a^3*b^13 - 384*A*B*a^4*b^12 - 948*A*B*a^5*b^11 + 960*A
*B*a^6*b^10 + 1306*A*B*a^7*b^9 - 1280*A*B*a^8*b^8 - 1128*A*B*a^9*b^7 + 960*A*B*a^10*b^6 + 592*A*B*a^11*b^5 - 3
84*A*B*a^12*b^4 - 160*A*B*a^13*b^3 + 64*A*B*a^14*b^2 - 48*A*C*a^6*b^10 + 136*A*C*a^8*b^8 - 98*A*C*a^10*b^6 - 2
0*A*C*a^12*b^4 + 80*A*C*a^14*b^2 + 12*B*C*a^7*b^9 - 34*B*C*a^9*b^7 + 20*B*C*a^11*b^5 - 16*B*C*a^13*b^3))/(a^18
*b + a^19 - a^8*b^11 - a^9*b^10 + 5*a^10*b^9 + 5*a^11*b^8 - 10*a^12*b^7 - 10*a^13*b^6 + 10*a^14*b^5 + 10*a^15*
b^4 - 5*a^16*b^3 - 5*a^17*b^2))*1i)/a^5 - ((4*A*b - B*a)*(((4*A*b - B*a)*((8*(4*B*a^24 + 4*C*a^24 + 16*A*a^10*
b^14 - 8*A*a^11*b^13 - 104*A*a^12*b^12 + 50*A*a^13*b^11 + 286*A*a^14*b^10 - 126*A*a^15*b^9 - 434*A*a^16*b^8 +
174*A*a^17*b^7 + 386*A*a^18*b^6 - 146*A*a^19*b^5 - 190*A*a^20*b^4 + 72*A*a^21*b^3 + 40*A*a^22*b^2 - 4*B*a^11*b
^13 + 2*B*a^12*b^12 + 26*B*a^13*b^11 - 14*B*a^14*b^10 - 70*B*a^15*b^9 + 30*B*a^16*b^8 + 110*B*a^17*b^7 - 30*B*
a^18*b^6 - 110*B*a^19*b^5 + 20*B*a^20*b^4 + 64*B*a^21*b^3 - 12*B*a^22*b^2 + 6*C*a^15*b^9 - 6*C*a^16*b^8 - 14*C
*a^17*b^7 + 14*C*a^18*b^6 + 6*C*a^19*b^5 - 6*C*a^20*b^4 + 6*C*a^21*b^3 - 6*C*a^22*b^2 - 16*A*a^23*b - 16*B*a^2
3*b - 4*C*a^23*b))/(a^22*b + a^23 - a^12*b^11 - a^13*b^10 + 5*a^14*b^9 + 5*a^15*b^8 - 10*a^16*b^7 - 10*a^17*b^
6 + 10*a^18*b^5 + 10*a^19*b^4 - 5*a^20*b^3 - 5*a^21*b^2) + (8*tan(c/2 + (d*x)/2)*(4*A*b - B*a)*(8*a^23*b - 8*a
^10*b^14 + 8*a^11*b^13 + 48*a^12*b^12 - 48*a^13*b^11 - 120*a^14*b^10 + 120*a^15*b^9 + 160*a^16*b^8 - 160*a^17*
b^7 - 120*a^18*b^6 + 120*a^19*b^5 + 48*a^20*b^4 - 48*a^21*b^3 - 8*a^22*b^2))/(a^5*(a^18*b + a^19 - a^8*b^11 -
a^9*b^10 + 5*a^10*b^9 + 5*a^11*b^8 - 10*a^12*b^7 - 10*a^13*b^6 + 10*a^14*b^5 + 10*a^15*b^4 - 5*a^16*b^3 - 5*a^
17*b^2))))/a^5 + (8*tan(c/2 + (d*x)/2)*(128*A^2*b^16 + 4*B^2*a^16 + 4*C^2*a^16 - 128*A^2*a*b^15 - 8*B^2*a^15*b
- 768*A^2*a^2*b^14 + 768*A^2*a^3*b^13 + 1920*A^2*a^4*b^12 - 1920*A^2*a^5*b^11 - 2600*A^2*a^6*b^10 + 2560*A^2*
a^7*b^9 + 2025*A^2*a^8*b^8 - 1920*A^2*a^9*b^7 - 824*A^2*a^10*b^6 + 768*A^2*a^11*b^5 + 80*A^2*a^12*b^4 - 128*A^
2*a^13*b^3 + 64*A^2*a^14*b^2 + 8*B^2*a^2*b^14 - 8*B^2*a^3*b^13 - 48*B^2*a^4*b^12 + 48*B^2*a^5*b^11 + 117*B^2*a
^6*b^10 - 120*B^2*a^7*b^9 - 164*B^2*a^8*b^8 + 160*B^2*a^9*b^7 + 156*B^2*a^10*b^6 - 120*B^2*a^11*b^5 - 92*B^2*a
^12*b^4 + 48*B^2*a^13*b^3 + 44*B^2*a^14*b^2 + 9*C^2*a^12*b^4 + 12*C^2*a^14*b^2 - 64*A*B*a*b^15 - 32*A*B*a^15*b
- 32*B*C*a^15*b + 64*A*B*a^2*b^14 + 384*A*B*a^3*b^13 - 384*A*B*a^4*b^12 - 948*A*B*a^5*b^11 + 960*A*B*a^6*b^10
+ 1306*A*B*a^7*b^9 - 1280*A*B*a^8*b^8 - 1128*A*B*a^9*b^7 + 960*A*B*a^10*b^6 + 592*A*B*a^11*b^5 - 384*A*B*a^12
*b^4 - 160*A*B*a^13*b^3 + 64*A*B*a^14*b^2 - 48*A*C*a^6*b^10 + 136*A*C*a^8*b^8 - 98*A*C*a^10*b^6 - 20*A*C*a^12*
b^4 + 80*A*C*a^14*b^2 + 12*B*C*a^7*b^9 - 34*B*C*a^9*b^7 + 20*B*C*a^11*b^5 - 16*B*C*a^13*b^3))/(a^18*b + a^19 -
a^8*b^11 - a^9*b^10 + 5*a^10*b^9 + 5*a^11*b^8 - 10*a^12*b^7 - 10*a^13*b^6 + 10*a^14*b^5 + 10*a^15*b^4 - 5*a^1
6*b^3 - 5*a^17*b^2))*1i)/a^5)/(((4*A*b - B*a)*(((4*A*b - B*a)*((8*(4*B*a^24 + 4*C*a^24 + 16*A*a^10*b^14 - 8*A*
a^11*b^13 - 104*A*a^12*b^12 + 50*A*a^13*b^11 + 286*A*a^14*b^10 - 126*A*a^15*b^9 - 434*A*a^16*b^8 + 174*A*a^17*
b^7 + 386*A*a^18*b^6 - 146*A*a^19*b^5 - 190*A*a^20*b^4 + 72*A*a^21*b^3 + 40*A*a^22*b^2 - 4*B*a^11*b^13 + 2*B*a
^12*b^12 + 26*B*a^13*b^11 - 14*B*a^14*b^10 - 70*B*a^15*b^9 + 30*B*a^16*b^8 + 110*B*a^17*b^7 - 30*B*a^18*b^6 -
110*B*a^19*b^5 + 20*B*a^20*b^4 + 64*B*a^21*b^3 - 12*B*a^22*b^2 + 6*C*a^15*b^9 - 6*C*a^16*b^8 - 14*C*a^17*b^7 +
14*C*a^18*b^6 + 6*C*a^19*b^5 - 6*C*a^20*b^4 + 6*C*a^21*b^3 - 6*C*a^22*b^2 - 16*A*a^23*b - 16*B*a^23*b - 4*C*a
^23*b))/(a^22*b + a^23 - a^12*b^11 - a^13*b^10 + 5*a^14*b^9 + 5*a^15*b^8 - 10*a^16*b^7 - 10*a^17*b^6 + 10*a^18
*b^5 + 10*a^19*b^4 - 5*a^20*b^3 - 5*a^21*b^2) - (8*tan(c/2 + (d*x)/2)*(4*A*b - B*a)*(8*a^23*b - 8*a^10*b^14 +
8*a^11*b^13 + 48*a^12*b^12 - 48*a^13*b^11 - 120*a^14*b^10 + 120*a^15*b^9 + 160*a^16*b^8 - 160*a^17*b^7 - 120*a
^18*b^6 + 120*a^19*b^5 + 48*a^20*b^4 - 48*a^21*b^3 - 8*a^22*b^2))/(a^5*(a^18*b + a^19 - a^8*b^11 - a^9*b^10 +
5*a^10*b^9 + 5*a^11*b^8 - 10*a^12*b^7 - 10*a^13*b^6 + 10*a^14*b^5 + 10*a^15*b^4 - 5*a^16*b^3 - 5*a^17*b^2))))/
a^5 - (8*tan(c/2 + (d*x)/2)*(128*A^2*b^16 + 4*B^2*a^16 + 4*C^2*a^16 - 128*A^2*a*b^15 - 8*B^2*a^15*b - 768*A^2*
a^2*b^14 + 768*A^2*a^3*b^13 + 1920*A^2*a^4*b^12 - 1920*A^2*a^5*b^11 - 2600*A^2*a^6*b^10 + 2560*A^2*a^7*b^9 + 2
025*A^2*a^8*b^8 - 1920*A^2*a^9*b^7 - 824*A^2*a^10*b^6 + 768*A^2*a^11*b^5 + 80*A^2*a^12*b^4 - 128*A^2*a^13*b^3
+ 64*A^2*a^14*b^2 + 8*B^2*a^2*b^14 - 8*B^2*a^3*b^13 - 48*B^2*a^4*b^12 + 48*B^2*a^5*b^11 + 117*B^2*a^6*b^10 - 1
20*B^2*a^7*b^9 - 164*B^2*a^8*b^8 + 160*B^2*a^9*b^7 + 156*B^2*a^10*b^6 - 120*B^2*a^11*b^5 - 92*B^2*a^12*b^4 + 4
8*B^2*a^13*b^3 + 44*B^2*a^14*b^2 + 9*C^2*a^12*b^4 + 12*C^2*a^14*b^2 - 64*A*B*a*b^15 - 32*A*B*a^15*b - 32*B*C*a
^15*b + 64*A*B*a^2*b^14 + 384*A*B*a^3*b^13 - 384*A*B*a^4*b^12 - 948*A*B*a^5*b^11 + 960*A*B*a^6*b^10 + 1306*A*B
*a^7*b^9 - 1280*A*B*a^8*b^8 - 1128*A*B*a^9*b^7 + 960*A*B*a^10*b^6 + 592*A*B*a^11*b^5 - 384*A*B*a^12*b^4 - 160*
A*B*a^13*b^3 + 64*A*B*a^14*b^2 - 48*A*C*a^6*b^10 + 136*A*C*a^8*b^8 - 98*A*C*a^10*b^6 - 20*A*C*a^12*b^4 + 80*A*
C*a^14*b^2 + 12*B*C*a^7*b^9 - 34*B*C*a^9*b^7 + 20*B*C*a^11*b^5 - 16*B*C*a^13*b^3))/(a^18*b + a^19 - a^8*b^11 -
a^9*b^10 + 5*a^10*b^9 + 5*a^11*b^8 - 10*a^12*b^7 - 10*a^13*b^6 + 10*a^14*b^5 + 10*a^15*b^4 - 5*a^16*b^3 - 5*a
^17*b^2)))/a^5 - (16*(256*A^3*b^16 - 4*B*C^2*a^16 + 4*B^2*C*a^16 - 128*A^3*a*b^15 - 16*B^3*a^15*b - 1664*A^3*a
^2*b^14 + 800*A^3*a^3*b^13 + 4576*A^3*a^4*b^12 - 2176*A^3*a^5*b^11 - 6944*A^3*a^6*b^10 + 3204*A^3*a^7*b^9 + 61
76*A^3*a^8*b^8 - 2560*A^3*a^9*b^7 - 3040*A^3*a^10*b^6 + 960*A^3*a^11*b^5 + 640*A^3*a^12*b^4 - 4*B^3*a^3*b^13 +
2*B^3*a^4*b^12 + 26*B^3*a^5*b^11 - 11*B^3*a^6*b^10 - 70*B^3*a^7*b^9 + 34*B^3*a^8*b^8 + 110*B^3*a^9*b^7 - 66*B
^3*a^10*b^6 - 110*B^3*a^11*b^5 + 64*B^3*a^12*b^4 + 64*B^3*a^13*b^3 - 48*B^3*a^14*b^2 - 192*A^2*B*a*b^15 + 16*A
*C^2*a^15*b + 28*B^2*C*a^15*b + 48*A*B^2*a^2*b^14 - 24*A*B^2*a^3*b^13 - 312*A*B^2*a^4*b^12 + 138*A*B^2*a^5*b^1
1 + 846*A*B^2*a^6*b^10 - 408*A*B^2*a^7*b^9 - 1314*A*B^2*a^8*b^8 + 726*A*B^2*a^9*b^7 + 1266*A*B^2*a^10*b^6 - 69
0*A*B^2*a^11*b^5 - 702*A*B^2*a^12*b^4 + 408*A*B^2*a^13*b^3 + 168*A*B^2*a^14*b^2 + 96*A^2*B*a^2*b^14 + 1248*A^2
*B*a^3*b^13 - 576*A^2*B*a^4*b^12 - 3408*A^2*B*a^5*b^11 + 1632*A^2*B*a^6*b^10 + 5232*A^2*B*a^7*b^9 - 2649*A^2*B
*a^8*b^8 - 4848*A^2*B*a^9*b^7 + 2376*A^2*B*a^10*b^6 + 2544*A^2*B*a^11*b^5 - 1104*A^2*B*a^12*b^4 - 576*A^2*B*a^
13*b^3 + 36*A*C^2*a^11*b^5 + 48*A*C^2*a^13*b^3 - 96*A^2*C*a^5*b^11 - 96*A^2*C*a^6*b^10 + 320*A^2*C*a^7*b^9 + 2
24*A^2*C*a^8*b^8 - 296*A^2*C*a^9*b^7 - 96*A^2*C*a^10*b^6 + 16*A^2*C*a^11*b^5 - 96*A^2*C*a^12*b^4 + 256*A^2*C*a
^13*b^3 + 64*A^2*C*a^14*b^2 - 9*B*C^2*a^12*b^4 - 12*B*C^2*a^14*b^2 - 6*B^2*C*a^7*b^9 - 6*B^2*C*a^8*b^8 + 20*B^
2*C*a^9*b^7 + 14*B^2*C*a^10*b^6 - 14*B^2*C*a^11*b^5 - 6*B^2*C*a^12*b^4 + 22*B^2*C*a^13*b^3 - 6*B^2*C*a^14*b^2
- 32*A*B*C*a^15*b + 48*A*B*C*a^6*b^10 + 48*A*B*C*a^7*b^9 - 160*A*B*C*a^8*b^8 - 112*A*B*C*a^9*b^7 + 130*A*B*C*a
^10*b^6 + 48*A*B*C*a^11*b^5 - 92*A*B*C*a^12*b^4 + 48*A*B*C*a^13*b^3 - 176*A*B*C*a^14*b^2))/(a^22*b + a^23 - a^
12*b^11 - a^13*b^10 + 5*a^14*b^9 + 5*a^15*b^8 - 10*a^16*b^7 - 10*a^17*b^6 + 10*a^18*b^5 + 10*a^19*b^4 - 5*a^20
*b^3 - 5*a^21*b^2) + ((4*A*b - B*a)*(((4*A*b - B*a)*((8*(4*B*a^24 + 4*C*a^24 + 16*A*a^10*b^14 - 8*A*a^11*b^13
- 104*A*a^12*b^12 + 50*A*a^13*b^11 + 286*A*a^14*b^10 - 126*A*a^15*b^9 - 434*A*a^16*b^8 + 174*A*a^17*b^7 + 386*
A*a^18*b^6 - 146*A*a^19*b^5 - 190*A*a^20*b^4 + 72*A*a^21*b^3 + 40*A*a^22*b^2 - 4*B*a^11*b^13 + 2*B*a^12*b^12 +
26*B*a^13*b^11 - 14*B*a^14*b^10 - 70*B*a^15*b^9 + 30*B*a^16*b^8 + 110*B*a^17*b^7 - 30*B*a^18*b^6 - 110*B*a^19
*b^5 + 20*B*a^20*b^4 + 64*B*a^21*b^3 - 12*B*a^22*b^2 + 6*C*a^15*b^9 - 6*C*a^16*b^8 - 14*C*a^17*b^7 + 14*C*a^18
*b^6 + 6*C*a^19*b^5 - 6*C*a^20*b^4 + 6*C*a^21*b^3 - 6*C*a^22*b^2 - 16*A*a^23*b - 16*B*a^23*b - 4*C*a^23*b))/(a
^22*b + a^23 - a^12*b^11 - a^13*b^10 + 5*a^14*b^9 + 5*a^15*b^8 - 10*a^16*b^7 - 10*a^17*b^6 + 10*a^18*b^5 + 10*
a^19*b^4 - 5*a^20*b^3 - 5*a^21*b^2) + (8*tan(c/2 + (d*x)/2)*(4*A*b - B*a)*(8*a^23*b - 8*a^10*b^14 + 8*a^11*b^1
3 + 48*a^12*b^12 - 48*a^13*b^11 - 120*a^14*b^10 + 120*a^15*b^9 + 160*a^16*b^8 - 160*a^17*b^7 - 120*a^18*b^6 +
120*a^19*b^5 + 48*a^20*b^4 - 48*a^21*b^3 - 8*a^22*b^2))/(a^5*(a^18*b + a^19 - a^8*b^11 - a^9*b^10 + 5*a^10*b^9
+ 5*a^11*b^8 - 10*a^12*b^7 - 10*a^13*b^6 + 10*a^14*b^5 + 10*a^15*b^4 - 5*a^16*b^3 - 5*a^17*b^2))))/a^5 + (8*t
an(c/2 + (d*x)/2)*(128*A^2*b^16 + 4*B^2*a^16 + 4*C^2*a^16 - 128*A^2*a*b^15 - 8*B^2*a^15*b - 768*A^2*a^2*b^14 +
768*A^2*a^3*b^13 + 1920*A^2*a^4*b^12 - 1920*A^2*a^5*b^11 - 2600*A^2*a^6*b^10 + 2560*A^2*a^7*b^9 + 2025*A^2*a^
8*b^8 - 1920*A^2*a^9*b^7 - 824*A^2*a^10*b^6 + 768*A^2*a^11*b^5 + 80*A^2*a^12*b^4 - 128*A^2*a^13*b^3 + 64*A^2*a
^14*b^2 + 8*B^2*a^2*b^14 - 8*B^2*a^3*b^13 - 48*B^2*a^4*b^12 + 48*B^2*a^5*b^11 + 117*B^2*a^6*b^10 - 120*B^2*a^7
*b^9 - 164*B^2*a^8*b^8 + 160*B^2*a^9*b^7 + 156*B^2*a^10*b^6 - 120*B^2*a^11*b^5 - 92*B^2*a^12*b^4 + 48*B^2*a^13
*b^3 + 44*B^2*a^14*b^2 + 9*C^2*a^12*b^4 + 12*C^2*a^14*b^2 - 64*A*B*a*b^15 - 32*A*B*a^15*b - 32*B*C*a^15*b + 64
*A*B*a^2*b^14 + 384*A*B*a^3*b^13 - 384*A*B*a^4*b^12 - 948*A*B*a^5*b^11 + 960*A*B*a^6*b^10 + 1306*A*B*a^7*b^9 -
1280*A*B*a^8*b^8 - 1128*A*B*a^9*b^7 + 960*A*B*a^10*b^6 + 592*A*B*a^11*b^5 - 384*A*B*a^12*b^4 - 160*A*B*a^13*b
^3 + 64*A*B*a^14*b^2 - 48*A*C*a^6*b^10 + 136*A*C*a^8*b^8 - 98*A*C*a^10*b^6 - 20*A*C*a^12*b^4 + 80*A*C*a^14*b^2
+ 12*B*C*a^7*b^9 - 34*B*C*a^9*b^7 + 20*B*C*a^11*b^5 - 16*B*C*a^13*b^3))/(a^18*b + a^19 - a^8*b^11 - a^9*b^10
+ 5*a^10*b^9 + 5*a^11*b^8 - 10*a^12*b^7 - 10*a^13*b^6 + 10*a^14*b^5 + 10*a^15*b^4 - 5*a^16*b^3 - 5*a^17*b^2)))
/a^5))*(4*A*b - B*a)*2i)/(a^5*d) + (atan(((((8*tan(c/2 + (d*x)/2)*(128*A^2*b^16 + 4*B^2*a^16 + 4*C^2*a^16 - 12
8*A^2*a*b^15 - 8*B^2*a^15*b - 768*A^2*a^2*b^14 + 768*A^2*a^3*b^13 + 1920*A^2*a^4*b^12 - 1920*A^2*a^5*b^11 - 26
00*A^2*a^6*b^10 + 2560*A^2*a^7*b^9 + 2025*A^2*a^8*b^8 - 1920*A^2*a^9*b^7 - 824*A^2*a^10*b^6 + 768*A^2*a^11*b^5
+ 80*A^2*a^12*b^4 - 128*A^2*a^13*b^3 + 64*A^2*a^14*b^2 + 8*B^2*a^2*b^14 - 8*B^2*a^3*b^13 - 48*B^2*a^4*b^12 +
48*B^2*a^5*b^11 + 117*B^2*a^6*b^10 - 120*B^2*a^7*b^9 - 164*B^2*a^8*b^8 + 160*B^2*a^9*b^7 + 156*B^2*a^10*b^6 -
120*B^2*a^11*b^5 - 92*B^2*a^12*b^4 + 48*B^2*a^13*b^3 + 44*B^2*a^14*b^2 + 9*C^2*a^12*b^4 + 12*C^2*a^14*b^2 - 64
*A*B*a*b^15 - 32*A*B*a^15*b - 32*B*C*a^15*b + 64*A*B*a^2*b^14 + 384*A*B*a^3*b^13 - 384*A*B*a^4*b^12 - 948*A*B*
a^5*b^11 + 960*A*B*a^6*b^10 + 1306*A*B*a^7*b^9 - 1280*A*B*a^8*b^8 - 1128*A*B*a^9*b^7 + 960*A*B*a^10*b^6 + 592*
A*B*a^11*b^5 - 384*A*B*a^12*b^4 - 160*A*B*a^13*b^3 + 64*A*B*a^14*b^2 - 48*A*C*a^6*b^10 + 136*A*C*a^8*b^8 - 98*
A*C*a^10*b^6 - 20*A*C*a^12*b^4 + 80*A*C*a^14*b^2 + 12*B*C*a^7*b^9 - 34*B*C*a^9*b^7 + 20*B*C*a^11*b^5 - 16*B*C*
a^13*b^3))/(a^18*b + a^19 - a^8*b^11 - a^9*b^10 + 5*a^10*b^9 + 5*a^11*b^8 - 10*a^12*b^7 - 10*a^13*b^6 + 10*a^1
4*b^5 + 10*a^15*b^4 - 5*a^16*b^3 - 5*a^17*b^2) - (((8*(4*B*a^24 + 4*C*a^24 + 16*A*a^10*b^14 - 8*A*a^11*b^13 -
104*A*a^12*b^12 + 50*A*a^13*b^11 + 286*A*a^14*b^10 - 126*A*a^15*b^9 - 434*A*a^16*b^8 + 174*A*a^17*b^7 + 386*A*
a^18*b^6 - 146*A*a^19*b^5 - 190*A*a^20*b^4 + 72*A*a^21*b^3 + 40*A*a^22*b^2 - 4*B*a^11*b^13 + 2*B*a^12*b^12 + 2
6*B*a^13*b^11 - 14*B*a^14*b^10 - 70*B*a^15*b^9 + 30*B*a^16*b^8 + 110*B*a^17*b^7 - 30*B*a^18*b^6 - 110*B*a^19*b
^5 + 20*B*a^20*b^4 + 64*B*a^21*b^3 - 12*B*a^22*b^2 + 6*C*a^15*b^9 - 6*C*a^16*b^8 - 14*C*a^17*b^7 + 14*C*a^18*b
^6 + 6*C*a^19*b^5 - 6*C*a^20*b^4 + 6*C*a^21*b^3 - 6*C*a^22*b^2 - 16*A*a^23*b - 16*B*a^23*b - 4*C*a^23*b))/(a^2
2*b + a^23 - a^12*b^11 - a^13*b^10 + 5*a^14*b^9 + 5*a^15*b^8 - 10*a^16*b^7 - 10*a^17*b^6 + 10*a^18*b^5 + 10*a^
19*b^4 - 5*a^20*b^3 - 5*a^21*b^2) - (4*tan(c/2 + (d*x)/2)*(-(a + b)^7*(a - b)^7)^(1/2)*(2*C*a^8 - 8*A*b^8 + 28
*A*a^2*b^6 - 35*A*a^4*b^4 + 20*A*a^6*b^2 - 7*B*a^3*b^5 + 8*B*a^5*b^3 + 3*C*a^6*b^2 + 2*B*a*b^7 - 8*B*a^7*b)*(8
*a^23*b - 8*a^10*b^14 + 8*a^11*b^13 + 48*a^12*b^12 - 48*a^13*b^11 - 120*a^14*b^10 + 120*a^15*b^9 + 160*a^16*b^
8 - 160*a^17*b^7 - 120*a^18*b^6 + 120*a^19*b^5 + 48*a^20*b^4 - 48*a^21*b^3 - 8*a^22*b^2))/((a^19 - a^5*b^14 +
7*a^7*b^12 - 21*a^9*b^10 + 35*a^11*b^8 - 35*a^13*b^6 + 21*a^15*b^4 - 7*a^17*b^2)*(a^18*b + a^19 - a^8*b^11 - a
^9*b^10 + 5*a^10*b^9 + 5*a^11*b^8 - 10*a^12*b^7 - 10*a^13*b^6 + 10*a^14*b^5 + 10*a^15*b^4 - 5*a^16*b^3 - 5*a^1
7*b^2)))*(-(a + b)^7*(a - b)^7)^(1/2)*(2*C*a^8 - 8*A*b^8 + 28*A*a^2*b^6 - 35*A*a^4*b^4 + 20*A*a^6*b^2 - 7*B*a^
3*b^5 + 8*B*a^5*b^3 + 3*C*a^6*b^2 + 2*B*a*b^7 - 8*B*a^7*b))/(2*(a^19 - a^5*b^14 + 7*a^7*b^12 - 21*a^9*b^10 + 3
5*a^11*b^8 - 35*a^13*b^6 + 21*a^15*b^4 - 7*a^17*b^2)))*(-(a + b)^7*(a - b)^7)^(1/2)*(2*C*a^8 - 8*A*b^8 + 28*A*
a^2*b^6 - 35*A*a^4*b^4 + 20*A*a^6*b^2 - 7*B*a^3*b^5 + 8*B*a^5*b^3 + 3*C*a^6*b^2 + 2*B*a*b^7 - 8*B*a^7*b)*1i)/(
2*(a^19 - a^5*b^14 + 7*a^7*b^12 - 21*a^9*b^10 + 35*a^11*b^8 - 35*a^13*b^6 + 21*a^15*b^4 - 7*a^17*b^2)) + (((8*
tan(c/2 + (d*x)/2)*(128*A^2*b^16 + 4*B^2*a^16 + 4*C^2*a^16 - 128*A^2*a*b^15 - 8*B^2*a^15*b - 768*A^2*a^2*b^14
+ 768*A^2*a^3*b^13 + 1920*A^2*a^4*b^12 - 1920*A^2*a^5*b^11 - 2600*A^2*a^6*b^10 + 2560*A^2*a^7*b^9 + 2025*A^2*a
^8*b^8 - 1920*A^2*a^9*b^7 - 824*A^2*a^10*b^6 + 768*A^2*a^11*b^5 + 80*A^2*a^12*b^4 - 128*A^2*a^13*b^3 + 64*A^2*
a^14*b^2 + 8*B^2*a^2*b^14 - 8*B^2*a^3*b^13 - 48*B^2*a^4*b^12 + 48*B^2*a^5*b^11 + 117*B^2*a^6*b^10 - 120*B^2*a^
7*b^9 - 164*B^2*a^8*b^8 + 160*B^2*a^9*b^7 + 156*B^2*a^10*b^6 - 120*B^2*a^11*b^5 - 92*B^2*a^12*b^4 + 48*B^2*a^1
3*b^3 + 44*B^2*a^14*b^2 + 9*C^2*a^12*b^4 + 12*C^2*a^14*b^2 - 64*A*B*a*b^15 - 32*A*B*a^15*b - 32*B*C*a^15*b + 6
4*A*B*a^2*b^14 + 384*A*B*a^3*b^13 - 384*A*B*a^4*b^12 - 948*A*B*a^5*b^11 + 960*A*B*a^6*b^10 + 1306*A*B*a^7*b^9
- 1280*A*B*a^8*b^8 - 1128*A*B*a^9*b^7 + 960*A*B*a^10*b^6 + 592*A*B*a^11*b^5 - 384*A*B*a^12*b^4 - 160*A*B*a^13*
b^3 + 64*A*B*a^14*b^2 - 48*A*C*a^6*b^10 + 136*A*C*a^8*b^8 - 98*A*C*a^10*b^6 - 20*A*C*a^12*b^4 + 80*A*C*a^14*b^
2 + 12*B*C*a^7*b^9 - 34*B*C*a^9*b^7 + 20*B*C*a^11*b^5 - 16*B*C*a^13*b^3))/(a^18*b + a^19 - a^8*b^11 - a^9*b^10
+ 5*a^10*b^9 + 5*a^11*b^8 - 10*a^12*b^7 - 10*a^13*b^6 + 10*a^14*b^5 + 10*a^15*b^4 - 5*a^16*b^3 - 5*a^17*b^2)
+ (((8*(4*B*a^24 + 4*C*a^24 + 16*A*a^10*b^14 - 8*A*a^11*b^13 - 104*A*a^12*b^12 + 50*A*a^13*b^11 + 286*A*a^14*b
^10 - 126*A*a^15*b^9 - 434*A*a^16*b^8 + 174*A*a^17*b^7 + 386*A*a^18*b^6 - 146*A*a^19*b^5 - 190*A*a^20*b^4 + 72
*A*a^21*b^3 + 40*A*a^22*b^2 - 4*B*a^11*b^13 + 2*B*a^12*b^12 + 26*B*a^13*b^11 - 14*B*a^14*b^10 - 70*B*a^15*b^9
+ 30*B*a^16*b^8 + 110*B*a^17*b^7 - 30*B*a^18*b^6 - 110*B*a^19*b^5 + 20*B*a^20*b^4 + 64*B*a^21*b^3 - 12*B*a^22*
b^2 + 6*C*a^15*b^9 - 6*C*a^16*b^8 - 14*C*a^17*b^7 + 14*C*a^18*b^6 + 6*C*a^19*b^5 - 6*C*a^20*b^4 + 6*C*a^21*b^3
- 6*C*a^22*b^2 - 16*A*a^23*b - 16*B*a^23*b - 4*C*a^23*b))/(a^22*b + a^23 - a^12*b^11 - a^13*b^10 + 5*a^14*b^9
+ 5*a^15*b^8 - 10*a^16*b^7 - 10*a^17*b^6 + 10*a^18*b^5 + 10*a^19*b^4 - 5*a^20*b^3 - 5*a^21*b^2) + (4*tan(c/2
+ (d*x)/2)*(-(a + b)^7*(a - b)^7)^(1/2)*(2*C*a^8 - 8*A*b^8 + 28*A*a^2*b^6 - 35*A*a^4*b^4 + 20*A*a^6*b^2 - 7*B*
a^3*b^5 + 8*B*a^5*b^3 + 3*C*a^6*b^2 + 2*B*a*b^7 - 8*B*a^7*b)*(8*a^23*b - 8*a^10*b^14 + 8*a^11*b^13 + 48*a^12*b
^12 - 48*a^13*b^11 - 120*a^14*b^10 + 120*a^15*b^9 + 160*a^16*b^8 - 160*a^17*b^7 - 120*a^18*b^6 + 120*a^19*b^5
+ 48*a^20*b^4 - 48*a^21*b^3 - 8*a^22*b^2))/((a^19 - a^5*b^14 + 7*a^7*b^12 - 21*a^9*b^10 + 35*a^11*b^8 - 35*a^1
3*b^6 + 21*a^15*b^4 - 7*a^17*b^2)*(a^18*b + a^19 - a^8*b^11 - a^9*b^10 + 5*a^10*b^9 + 5*a^11*b^8 - 10*a^12*b^7
- 10*a^13*b^6 + 10*a^14*b^5 + 10*a^15*b^4 - 5*a^16*b^3 - 5*a^17*b^2)))*(-(a + b)^7*(a - b)^7)^(1/2)*(2*C*a^8
- 8*A*b^8 + 28*A*a^2*b^6 - 35*A*a^4*b^4 + 20*A*a^6*b^2 - 7*B*a^3*b^5 + 8*B*a^5*b^3 + 3*C*a^6*b^2 + 2*B*a*b^7 -
8*B*a^7*b))/(2*(a^19 - a^5*b^14 + 7*a^7*b^12 - 21*a^9*b^10 + 35*a^11*b^8 - 35*a^13*b^6 + 21*a^15*b^4 - 7*a^17
*b^2)))*(-(a + b)^7*(a - b)^7)^(1/2)*(2*C*a^8 - 8*A*b^8 + 28*A*a^2*b^6 - 35*A*a^4*b^4 + 20*A*a^6*b^2 - 7*B*a^3
*b^5 + 8*B*a^5*b^3 + 3*C*a^6*b^2 + 2*B*a*b^7 - 8*B*a^7*b)*1i)/(2*(a^19 - a^5*b^14 + 7*a^7*b^12 - 21*a^9*b^10 +
35*a^11*b^8 - 35*a^13*b^6 + 21*a^15*b^4 - 7*a^17*b^2)))/((16*(256*A^3*b^16 - 4*B*C^2*a^16 + 4*B^2*C*a^16 - 12
8*A^3*a*b^15 - 16*B^3*a^15*b - 1664*A^3*a^2*b^14 + 800*A^3*a^3*b^13 + 4576*A^3*a^4*b^12 - 2176*A^3*a^5*b^11 -
6944*A^3*a^6*b^10 + 3204*A^3*a^7*b^9 + 6176*A^3*a^8*b^8 - 2560*A^3*a^9*b^7 - 3040*A^3*a^10*b^6 + 960*A^3*a^11*
b^5 + 640*A^3*a^12*b^4 - 4*B^3*a^3*b^13 + 2*B^3*a^4*b^12 + 26*B^3*a^5*b^11 - 11*B^3*a^6*b^10 - 70*B^3*a^7*b^9
+ 34*B^3*a^8*b^8 + 110*B^3*a^9*b^7 - 66*B^3*a^10*b^6 - 110*B^3*a^11*b^5 + 64*B^3*a^12*b^4 + 64*B^3*a^13*b^3 -
48*B^3*a^14*b^2 - 192*A^2*B*a*b^15 + 16*A*C^2*a^15*b + 28*B^2*C*a^15*b + 48*A*B^2*a^2*b^14 - 24*A*B^2*a^3*b^13
- 312*A*B^2*a^4*b^12 + 138*A*B^2*a^5*b^11 + 846*A*B^2*a^6*b^10 - 408*A*B^2*a^7*b^9 - 1314*A*B^2*a^8*b^8 + 726
*A*B^2*a^9*b^7 + 1266*A*B^2*a^10*b^6 - 690*A*B^2*a^11*b^5 - 702*A*B^2*a^12*b^4 + 408*A*B^2*a^13*b^3 + 168*A*B^
2*a^14*b^2 + 96*A^2*B*a^2*b^14 + 1248*A^2*B*a^3*b^13 - 576*A^2*B*a^4*b^12 - 3408*A^2*B*a^5*b^11 + 1632*A^2*B*a
^6*b^10 + 5232*A^2*B*a^7*b^9 - 2649*A^2*B*a^8*b^8 - 4848*A^2*B*a^9*b^7 + 2376*A^2*B*a^10*b^6 + 2544*A^2*B*a^11
*b^5 - 1104*A^2*B*a^12*b^4 - 576*A^2*B*a^13*b^3 + 36*A*C^2*a^11*b^5 + 48*A*C^2*a^13*b^3 - 96*A^2*C*a^5*b^11 -
96*A^2*C*a^6*b^10 + 320*A^2*C*a^7*b^9 + 224*A^2*C*a^8*b^8 - 296*A^2*C*a^9*b^7 - 96*A^2*C*a^10*b^6 + 16*A^2*C*a
^11*b^5 - 96*A^2*C*a^12*b^4 + 256*A^2*C*a^13*b^3 + 64*A^2*C*a^14*b^2 - 9*B*C^2*a^12*b^4 - 12*B*C^2*a^14*b^2 -
6*B^2*C*a^7*b^9 - 6*B^2*C*a^8*b^8 + 20*B^2*C*a^9*b^7 + 14*B^2*C*a^10*b^6 - 14*B^2*C*a^11*b^5 - 6*B^2*C*a^12*b^
4 + 22*B^2*C*a^13*b^3 - 6*B^2*C*a^14*b^2 - 32*A*B*C*a^15*b + 48*A*B*C*a^6*b^10 + 48*A*B*C*a^7*b^9 - 160*A*B*C*
a^8*b^8 - 112*A*B*C*a^9*b^7 + 130*A*B*C*a^10*b^6 + 48*A*B*C*a^11*b^5 - 92*A*B*C*a^12*b^4 + 48*A*B*C*a^13*b^3 -
176*A*B*C*a^14*b^2))/(a^22*b + a^23 - a^12*b^11 - a^13*b^10 + 5*a^14*b^9 + 5*a^15*b^8 - 10*a^16*b^7 - 10*a^17
*b^6 + 10*a^18*b^5 + 10*a^19*b^4 - 5*a^20*b^3 - 5*a^21*b^2) + (((8*tan(c/2 + (d*x)/2)*(128*A^2*b^16 + 4*B^2*a^
16 + 4*C^2*a^16 - 128*A^2*a*b^15 - 8*B^2*a^15*b - 768*A^2*a^2*b^14 + 768*A^2*a^3*b^13 + 1920*A^2*a^4*b^12 - 19
20*A^2*a^5*b^11 - 2600*A^2*a^6*b^10 + 2560*A^2*a^7*b^9 + 2025*A^2*a^8*b^8 - 1920*A^2*a^9*b^7 - 824*A^2*a^10*b^
6 + 768*A^2*a^11*b^5 + 80*A^2*a^12*b^4 - 128*A^2*a^13*b^3 + 64*A^2*a^14*b^2 + 8*B^2*a^2*b^14 - 8*B^2*a^3*b^13
- 48*B^2*a^4*b^12 + 48*B^2*a^5*b^11 + 117*B^2*a^6*b^10 - 120*B^2*a^7*b^9 - 164*B^2*a^8*b^8 + 160*B^2*a^9*b^7 +
156*B^2*a^10*b^6 - 120*B^2*a^11*b^5 - 92*B^2*a^12*b^4 + 48*B^2*a^13*b^3 + 44*B^2*a^14*b^2 + 9*C^2*a^12*b^4 +
12*C^2*a^14*b^2 - 64*A*B*a*b^15 - 32*A*B*a^15*b - 32*B*C*a^15*b + 64*A*B*a^2*b^14 + 384*A*B*a^3*b^13 - 384*A*B
*a^4*b^12 - 948*A*B*a^5*b^11 + 960*A*B*a^6*b^10 + 1306*A*B*a^7*b^9 - 1280*A*B*a^8*b^8 - 1128*A*B*a^9*b^7 + 960
*A*B*a^10*b^6 + 592*A*B*a^11*b^5 - 384*A*B*a^12*b^4 - 160*A*B*a^13*b^3 + 64*A*B*a^14*b^2 - 48*A*C*a^6*b^10 + 1
36*A*C*a^8*b^8 - 98*A*C*a^10*b^6 - 20*A*C*a^12*b^4 + 80*A*C*a^14*b^2 + 12*B*C*a^7*b^9 - 34*B*C*a^9*b^7 + 20*B*
C*a^11*b^5 - 16*B*C*a^13*b^3))/(a^18*b + a^19 - a^8*b^11 - a^9*b^10 + 5*a^10*b^9 + 5*a^11*b^8 - 10*a^12*b^7 -
10*a^13*b^6 + 10*a^14*b^5 + 10*a^15*b^4 - 5*a^16*b^3 - 5*a^17*b^2) - (((8*(4*B*a^24 + 4*C*a^24 + 16*A*a^10*b^1
4 - 8*A*a^11*b^13 - 104*A*a^12*b^12 + 50*A*a^13*b^11 + 286*A*a^14*b^10 - 126*A*a^15*b^9 - 434*A*a^16*b^8 + 174
*A*a^17*b^7 + 386*A*a^18*b^6 - 146*A*a^19*b^5 - 190*A*a^20*b^4 + 72*A*a^21*b^3 + 40*A*a^22*b^2 - 4*B*a^11*b^13
+ 2*B*a^12*b^12 + 26*B*a^13*b^11 - 14*B*a^14*b^10 - 70*B*a^15*b^9 + 30*B*a^16*b^8 + 110*B*a^17*b^7 - 30*B*a^1
8*b^6 - 110*B*a^19*b^5 + 20*B*a^20*b^4 + 64*B*a^21*b^3 - 12*B*a^22*b^2 + 6*C*a^15*b^9 - 6*C*a^16*b^8 - 14*C*a^
17*b^7 + 14*C*a^18*b^6 + 6*C*a^19*b^5 - 6*C*a^20*b^4 + 6*C*a^21*b^3 - 6*C*a^22*b^2 - 16*A*a^23*b - 16*B*a^23*b
- 4*C*a^23*b))/(a^22*b + a^23 - a^12*b^11 - a^13*b^10 + 5*a^14*b^9 + 5*a^15*b^8 - 10*a^16*b^7 - 10*a^17*b^6 +
10*a^18*b^5 + 10*a^19*b^4 - 5*a^20*b^3 - 5*a^21*b^2) - (4*tan(c/2 + (d*x)/2)*(-(a + b)^7*(a - b)^7)^(1/2)*(2*
C*a^8 - 8*A*b^8 + 28*A*a^2*b^6 - 35*A*a^4*b^4 + 20*A*a^6*b^2 - 7*B*a^3*b^5 + 8*B*a^5*b^3 + 3*C*a^6*b^2 + 2*B*a
*b^7 - 8*B*a^7*b)*(8*a^23*b - 8*a^10*b^14 + 8*a^11*b^13 + 48*a^12*b^12 - 48*a^13*b^11 - 120*a^14*b^10 + 120*a^
15*b^9 + 160*a^16*b^8 - 160*a^17*b^7 - 120*a^18*b^6 + 120*a^19*b^5 + 48*a^20*b^4 - 48*a^21*b^3 - 8*a^22*b^2))/
((a^19 - a^5*b^14 + 7*a^7*b^12 - 21*a^9*b^10 + 35*a^11*b^8 - 35*a^13*b^6 + 21*a^15*b^4 - 7*a^17*b^2)*(a^18*b +
a^19 - a^8*b^11 - a^9*b^10 + 5*a^10*b^9 + 5*a^11*b^8 - 10*a^12*b^7 - 10*a^13*b^6 + 10*a^14*b^5 + 10*a^15*b^4
- 5*a^16*b^3 - 5*a^17*b^2)))*(-(a + b)^7*(a - b)^7)^(1/2)*(2*C*a^8 - 8*A*b^8 + 28*A*a^2*b^6 - 35*A*a^4*b^4 + 2
0*A*a^6*b^2 - 7*B*a^3*b^5 + 8*B*a^5*b^3 + 3*C*a^6*b^2 + 2*B*a*b^7 - 8*B*a^7*b))/(2*(a^19 - a^5*b^14 + 7*a^7*b^
12 - 21*a^9*b^10 + 35*a^11*b^8 - 35*a^13*b^6 + 21*a^15*b^4 - 7*a^17*b^2)))*(-(a + b)^7*(a - b)^7)^(1/2)*(2*C*a
^8 - 8*A*b^8 + 28*A*a^2*b^6 - 35*A*a^4*b^4 + 20*A*a^6*b^2 - 7*B*a^3*b^5 + 8*B*a^5*b^3 + 3*C*a^6*b^2 + 2*B*a*b^
7 - 8*B*a^7*b))/(2*(a^19 - a^5*b^14 + 7*a^7*b^12 - 21*a^9*b^10 + 35*a^11*b^8 - 35*a^13*b^6 + 21*a^15*b^4 - 7*a
^17*b^2)) - (((8*tan(c/2 + (d*x)/2)*(128*A^2*b^16 + 4*B^2*a^16 + 4*C^2*a^16 - 128*A^2*a*b^15 - 8*B^2*a^15*b -
768*A^2*a^2*b^14 + 768*A^2*a^3*b^13 + 1920*A^2*a^4*b^12 - 1920*A^2*a^5*b^11 - 2600*A^2*a^6*b^10 + 2560*A^2*a^7
*b^9 + 2025*A^2*a^8*b^8 - 1920*A^2*a^9*b^7 - 824*A^2*a^10*b^6 + 768*A^2*a^11*b^5 + 80*A^2*a^12*b^4 - 128*A^2*a
^13*b^3 + 64*A^2*a^14*b^2 + 8*B^2*a^2*b^14 - 8*B^2*a^3*b^13 - 48*B^2*a^4*b^12 + 48*B^2*a^5*b^11 + 117*B^2*a^6*
b^10 - 120*B^2*a^7*b^9 - 164*B^2*a^8*b^8 + 160*B^2*a^9*b^7 + 156*B^2*a^10*b^6 - 120*B^2*a^11*b^5 - 92*B^2*a^12
*b^4 + 48*B^2*a^13*b^3 + 44*B^2*a^14*b^2 + 9*C^2*a^12*b^4 + 12*C^2*a^14*b^2 - 64*A*B*a*b^15 - 32*A*B*a^15*b -
32*B*C*a^15*b + 64*A*B*a^2*b^14 + 384*A*B*a^3*b^13 - 384*A*B*a^4*b^12 - 948*A*B*a^5*b^11 + 960*A*B*a^6*b^10 +
1306*A*B*a^7*b^9 - 1280*A*B*a^8*b^8 - 1128*A*B*a^9*b^7 + 960*A*B*a^10*b^6 + 592*A*B*a^11*b^5 - 384*A*B*a^12*b^
4 - 160*A*B*a^13*b^3 + 64*A*B*a^14*b^2 - 48*A*C*a^6*b^10 + 136*A*C*a^8*b^8 - 98*A*C*a^10*b^6 - 20*A*C*a^12*b^4
+ 80*A*C*a^14*b^2 + 12*B*C*a^7*b^9 - 34*B*C*a^9*b^7 + 20*B*C*a^11*b^5 - 16*B*C*a^13*b^3))/(a^18*b + a^19 - a^
8*b^11 - a^9*b^10 + 5*a^10*b^9 + 5*a^11*b^8 - 10*a^12*b^7 - 10*a^13*b^6 + 10*a^14*b^5 + 10*a^15*b^4 - 5*a^16*b
^3 - 5*a^17*b^2) + (((8*(4*B*a^24 + 4*C*a^24 + 16*A*a^10*b^14 - 8*A*a^11*b^13 - 104*A*a^12*b^12 + 50*A*a^13*b^
11 + 286*A*a^14*b^10 - 126*A*a^15*b^9 - 434*A*a^16*b^8 + 174*A*a^17*b^7 + 386*A*a^18*b^6 - 146*A*a^19*b^5 - 19
0*A*a^20*b^4 + 72*A*a^21*b^3 + 40*A*a^22*b^2 - 4*B*a^11*b^13 + 2*B*a^12*b^12 + 26*B*a^13*b^11 - 14*B*a^14*b^10
- 70*B*a^15*b^9 + 30*B*a^16*b^8 + 110*B*a^17*b^7 - 30*B*a^18*b^6 - 110*B*a^19*b^5 + 20*B*a^20*b^4 + 64*B*a^21
*b^3 - 12*B*a^22*b^2 + 6*C*a^15*b^9 - 6*C*a^16*b^8 - 14*C*a^17*b^7 + 14*C*a^18*b^6 + 6*C*a^19*b^5 - 6*C*a^20*b
^4 + 6*C*a^21*b^3 - 6*C*a^22*b^2 - 16*A*a^23*b - 16*B*a^23*b - 4*C*a^23*b))/(a^22*b + a^23 - a^12*b^11 - a^13*
b^10 + 5*a^14*b^9 + 5*a^15*b^8 - 10*a^16*b^7 - 10*a^17*b^6 + 10*a^18*b^5 + 10*a^19*b^4 - 5*a^20*b^3 - 5*a^21*b
^2) + (4*tan(c/2 + (d*x)/2)*(-(a + b)^7*(a - b)^7)^(1/2)*(2*C*a^8 - 8*A*b^8 + 28*A*a^2*b^6 - 35*A*a^4*b^4 + 20
*A*a^6*b^2 - 7*B*a^3*b^5 + 8*B*a^5*b^3 + 3*C*a^6*b^2 + 2*B*a*b^7 - 8*B*a^7*b)*(8*a^23*b - 8*a^10*b^14 + 8*a^11
*b^13 + 48*a^12*b^12 - 48*a^13*b^11 - 120*a^14*b^10 + 120*a^15*b^9 + 160*a^16*b^8 - 160*a^17*b^7 - 120*a^18*b^
6 + 120*a^19*b^5 + 48*a^20*b^4 - 48*a^21*b^3 - 8*a^22*b^2))/((a^19 - a^5*b^14 + 7*a^7*b^12 - 21*a^9*b^10 + 35*
a^11*b^8 - 35*a^13*b^6 + 21*a^15*b^4 - 7*a^17*b^2)*(a^18*b + a^19 - a^8*b^11 - a^9*b^10 + 5*a^10*b^9 + 5*a^11*
b^8 - 10*a^12*b^7 - 10*a^13*b^6 + 10*a^14*b^5 + 10*a^15*b^4 - 5*a^16*b^3 - 5*a^17*b^2)))*(-(a + b)^7*(a - b)^7
)^(1/2)*(2*C*a^8 - 8*A*b^8 + 28*A*a^2*b^6 - 35*A*a^4*b^4 + 20*A*a^6*b^2 - 7*B*a^3*b^5 + 8*B*a^5*b^3 + 3*C*a^6*
b^2 + 2*B*a*b^7 - 8*B*a^7*b))/(2*(a^19 - a^5*b^14 + 7*a^7*b^12 - 21*a^9*b^10 + 35*a^11*b^8 - 35*a^13*b^6 + 21*
a^15*b^4 - 7*a^17*b^2)))*(-(a + b)^7*(a - b)^7)^(1/2)*(2*C*a^8 - 8*A*b^8 + 28*A*a^2*b^6 - 35*A*a^4*b^4 + 20*A*
a^6*b^2 - 7*B*a^3*b^5 + 8*B*a^5*b^3 + 3*C*a^6*b^2 + 2*B*a*b^7 - 8*B*a^7*b))/(2*(a^19 - a^5*b^14 + 7*a^7*b^12 -
21*a^9*b^10 + 35*a^11*b^8 - 35*a^13*b^6 + 21*a^15*b^4 - 7*a^17*b^2))))*(-(a + b)^7*(a - b)^7)^(1/2)*(2*C*a^8
- 8*A*b^8 + 28*A*a^2*b^6 - 35*A*a^4*b^4 + 20*A*a^6*b^2 - 7*B*a^3*b^5 + 8*B*a^5*b^3 + 3*C*a^6*b^2 + 2*B*a*b^7 -
8*B*a^7*b)*1i)/(d*(a^19 - a^5*b^14 + 7*a^7*b^12 - 21*a^9*b^10 + 35*a^11*b^8 - 35*a^13*b^6 + 21*a^15*b^4 - 7*a
^17*b^2))
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((A+B*cos(d*x+c)+C*cos(d*x+c)**2)*sec(d*x+c)**2/(a+b*cos(d*x+c))**4,x)
[Out]
Timed out
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