3.1003 \(\int \frac {\cos ^2(c+d x) (A+B \cos (c+d x)+C \cos ^2(c+d x))}{(a+b \cos (c+d x))^4} \, dx\)

Optimal. Leaf size=349 \[ -\frac {\sin (c+d x) \cos ^2(c+d x) \left (A b^2-a (b B-a C)\right )}{3 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^3}-\frac {a \sin (c+d x) \left (-3 a^4 C+a^2 b^2 (3 A+8 C)-5 a b^3 B+2 A b^4\right )}{6 b^3 d \left (a^2-b^2\right )^2 (a+b \cos (c+d x))^2}-\frac {\left (2 a^7 C-7 a^5 b^2 C-a^3 b^4 (A-8 C)+3 a^2 b^5 B-4 a b^6 (A+2 C)+2 b^7 B\right ) \tan ^{-1}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{b^4 d (a-b)^{7/2} (a+b)^{7/2}}-\frac {\sin (c+d x) \left (9 a^6 C-a^4 b^2 (3 A+28 C)+a^3 b^3 B+2 a^2 b^4 (7 A+17 C)-16 a b^5 B+4 A b^6\right )}{6 b^3 d \left (a^2-b^2\right )^3 (a+b \cos (c+d x))}+\frac {C x}{b^4} \]

[Out]

C*x/b^4-(3*a^2*b^5*B+2*b^7*B-a^3*b^4*(A-8*C)+2*a^7*C-7*a^5*b^2*C-4*a*b^6*(A+2*C))*arctan((a-b)^(1/2)*tan(1/2*d
*x+1/2*c)/(a+b)^(1/2))/(a-b)^(7/2)/b^4/(a+b)^(7/2)/d-1/3*(A*b^2-a*(B*b-C*a))*cos(d*x+c)^2*sin(d*x+c)/b/(a^2-b^
2)/d/(a+b*cos(d*x+c))^3-1/6*a*(2*A*b^4-5*a*b^3*B-3*a^4*C+a^2*b^2*(3*A+8*C))*sin(d*x+c)/b^3/(a^2-b^2)^2/d/(a+b*
cos(d*x+c))^2-1/6*(4*A*b^6+a^3*b^3*B-16*a*b^5*B+9*a^6*C+2*a^2*b^4*(7*A+17*C)-a^4*b^2*(3*A+28*C))*sin(d*x+c)/b^
3/(a^2-b^2)^3/d/(a+b*cos(d*x+c))

________________________________________________________________________________________

Rubi [A]  time = 2.39, antiderivative size = 349, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 41, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.146, Rules used = {3047, 3031, 3021, 2735, 2659, 205} \[ -\frac {\left (-a^3 b^4 (A-8 C)+3 a^2 b^5 B-7 a^5 b^2 C+2 a^7 C-4 a b^6 (A+2 C)+2 b^7 B\right ) \tan ^{-1}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{b^4 d (a-b)^{7/2} (a+b)^{7/2}}-\frac {\sin (c+d x) \cos ^2(c+d x) \left (A b^2-a (b B-a C)\right )}{3 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^3}-\frac {\sin (c+d x) \left (-a^4 b^2 (3 A+28 C)+2 a^2 b^4 (7 A+17 C)+a^3 b^3 B+9 a^6 C-16 a b^5 B+4 A b^6\right )}{6 b^3 d \left (a^2-b^2\right )^3 (a+b \cos (c+d x))}-\frac {a \sin (c+d x) \left (a^2 b^2 (3 A+8 C)-3 a^4 C-5 a b^3 B+2 A b^4\right )}{6 b^3 d \left (a^2-b^2\right )^2 (a+b \cos (c+d x))^2}+\frac {C x}{b^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(C*x)/b^4 - ((3*a^2*b^5*B + 2*b^7*B - a^3*b^4*(A - 8*C) + 2*a^7*C - 7*a^5*b^2*C - 4*a*b^6*(A + 2*C))*ArcTan[(S
qrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/((a - b)^(7/2)*b^4*(a + b)^(7/2)*d) - ((A*b^2 - a*(b*B - a*C))*Cos[
c + d*x]^2*Sin[c + d*x])/(3*b*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^3) - (a*(2*A*b^4 - 5*a*b^3*B - 3*a^4*C + a^2*
b^2*(3*A + 8*C))*Sin[c + d*x])/(6*b^3*(a^2 - b^2)^2*d*(a + b*Cos[c + d*x])^2) - ((4*A*b^6 + a^3*b^3*B - 16*a*b
^5*B + 9*a^6*C + 2*a^2*b^4*(7*A + 17*C) - a^4*b^2*(3*A + 28*C))*Sin[c + d*x])/(6*b^3*(a^2 - b^2)^3*d*(a + b*Co
s[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 2735

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 3021

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*sin[(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))/(b*f*(m +
1)*(a^2 - b^2)), x] + Dist[1/(b*(m + 1)*(a^2 - b^2)), Int[(a + b*Sin[e + f*x])^(m + 1)*Simp[b*(a*A - b*B + a*
C)*(m + 1) - (A*b^2 - a*b*B + a^2*C + b*(A*b - a*B + b*C)*(m + 1))*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, e,
f, A, B, C}, x] && LtQ[m, -1] && NeQ[a^2 - b^2, 0]

Rule 3031

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[((b*c - a*d)*(A*b^2 - a*b*B + a^2*C)*
Cos[e + f*x]*(a + b*Sin[e + f*x])^(m + 1))/(b^2*f*(m + 1)*(a^2 - b^2)), x] - Dist[1/(b^2*(m + 1)*(a^2 - b^2)),
Int[(a + b*Sin[e + f*x])^(m + 1)*Simp[b*(m + 1)*((b*B - a*C)*(b*c - a*d) - A*b*(a*c - b*d)) + (b*B*(a^2*d + b
^2*d*(m + 1) - a*b*c*(m + 2)) + (b*c - a*d)*(A*b^2*(m + 2) + C*(a^2 + b^2*(m + 1))))*Sin[e + f*x] - b*C*d*(m +
1)*(a^2 - b^2)*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] && Ne
Q[a^2 - b^2, 0] && LtQ[m, -1]

Rule 3047

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]

Rubi steps

\begin {align*} \int \frac {\cos ^2(c+d x) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{(a+b \cos (c+d x))^4} \, dx &=-\frac {\left (A b^2-a (b B-a C)\right ) \cos ^2(c+d x) \sin (c+d x)}{3 b \left (a^2-b^2\right ) d (a+b \cos (c+d x))^3}-\frac {\int \frac {\cos (c+d x) \left (2 \left (A b^2-a (b B-a C)\right )+3 b (b B-a (A+C)) \cos (c+d x)-3 \left (a^2-b^2\right ) C \cos ^2(c+d x)\right )}{(a+b \cos (c+d x))^3} \, dx}{3 b \left (a^2-b^2\right )}\\ &=-\frac {\left (A b^2-a (b B-a C)\right ) \cos ^2(c+d x) \sin (c+d x)}{3 b \left (a^2-b^2\right ) d (a+b \cos (c+d x))^3}-\frac {a \left (2 A b^4-5 a b^3 B-3 a^4 C+a^2 b^2 (3 A+8 C)\right ) \sin (c+d x)}{6 b^3 \left (a^2-b^2\right )^2 d (a+b \cos (c+d x))^2}-\frac {\int \frac {-2 b \left (2 A b^4-5 a b^3 B-3 a^4 C+a^2 b^2 (3 A+8 C)\right )+\left (a^2 b^3 B-6 b^5 B+3 a^5 C+4 a b^4 (2 A+3 C)-a^3 b^2 (3 A+10 C)\right ) \cos (c+d x)-6 b \left (a^2-b^2\right )^2 C \cos ^2(c+d x)}{(a+b \cos (c+d x))^2} \, dx}{6 b^3 \left (a^2-b^2\right )^2}\\ &=-\frac {\left (A b^2-a (b B-a C)\right ) \cos ^2(c+d x) \sin (c+d x)}{3 b \left (a^2-b^2\right ) d (a+b \cos (c+d x))^3}-\frac {a \left (2 A b^4-5 a b^3 B-3 a^4 C+a^2 b^2 (3 A+8 C)\right ) \sin (c+d x)}{6 b^3 \left (a^2-b^2\right )^2 d (a+b \cos (c+d x))^2}-\frac {\left (4 A b^6+a^3 b^3 B-16 a b^5 B+9 a^6 C+2 a^2 b^4 (7 A+17 C)-a^4 b^2 (3 A+28 C)\right ) \sin (c+d x)}{6 b^3 \left (a^2-b^2\right )^3 d (a+b \cos (c+d x))}+\frac {\int \frac {-3 \left (3 a^2 b^5 B+2 b^7 B-a^3 b^4 (A-2 C)-a^5 b^2 C-2 a b^6 (2 A+3 C)\right )+6 b \left (a^2-b^2\right )^3 C \cos (c+d x)}{a+b \cos (c+d x)} \, dx}{6 b^4 \left (a^2-b^2\right )^3}\\ &=\frac {C x}{b^4}-\frac {\left (A b^2-a (b B-a C)\right ) \cos ^2(c+d x) \sin (c+d x)}{3 b \left (a^2-b^2\right ) d (a+b \cos (c+d x))^3}-\frac {a \left (2 A b^4-5 a b^3 B-3 a^4 C+a^2 b^2 (3 A+8 C)\right ) \sin (c+d x)}{6 b^3 \left (a^2-b^2\right )^2 d (a+b \cos (c+d x))^2}-\frac {\left (4 A b^6+a^3 b^3 B-16 a b^5 B+9 a^6 C+2 a^2 b^4 (7 A+17 C)-a^4 b^2 (3 A+28 C)\right ) \sin (c+d x)}{6 b^3 \left (a^2-b^2\right )^3 d (a+b \cos (c+d x))}-\frac {\left (3 a^2 b^5 B+2 b^7 B-a^3 b^4 (A-8 C)+2 a^7 C-7 a^5 b^2 C-4 a b^6 (A+2 C)\right ) \int \frac {1}{a+b \cos (c+d x)} \, dx}{2 b^4 \left (a^2-b^2\right )^3}\\ &=\frac {C x}{b^4}-\frac {\left (A b^2-a (b B-a C)\right ) \cos ^2(c+d x) \sin (c+d x)}{3 b \left (a^2-b^2\right ) d (a+b \cos (c+d x))^3}-\frac {a \left (2 A b^4-5 a b^3 B-3 a^4 C+a^2 b^2 (3 A+8 C)\right ) \sin (c+d x)}{6 b^3 \left (a^2-b^2\right )^2 d (a+b \cos (c+d x))^2}-\frac {\left (4 A b^6+a^3 b^3 B-16 a b^5 B+9 a^6 C+2 a^2 b^4 (7 A+17 C)-a^4 b^2 (3 A+28 C)\right ) \sin (c+d x)}{6 b^3 \left (a^2-b^2\right )^3 d (a+b \cos (c+d x))}-\frac {\left (3 a^2 b^5 B+2 b^7 B-a^3 b^4 (A-8 C)+2 a^7 C-7 a^5 b^2 C-4 a b^6 (A+2 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 )}{b^4 \left (a^2-b^2\right )^3 d}\\ &=\frac {C x}{b^4}+\frac {\left (a^3 A b^4+4 a A b^6-3 a^2 b^5 B-2 b^7 B-2 a^7 C+7 a^5 b^2 C-8 a^3 b^4 C+8 a b^6 C\right ) \tan ^{-1}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{(a-b)^{7/2} b^4 (a+b)^{7/2} d}-\frac {\left (A b^2-a (b B-a C)\right ) \cos ^2(c+d x) \sin (c+d x)}{3 b \left (a^2-b^2\right ) d (a+b \cos (c+d x))^3}-\frac {a \left (2 A b^4-5 a b^3 B-3 a^4 C+a^2 b^2 (3 A+8 C)\right ) \sin (c+d x)}{6 b^3 \left (a^2-b^2\right )^2 d (a+b \cos (c+d x))^2}-\frac {\left (4 A b^6+a^3 b^3 B-16 a b^5 B+9 a^6 C+2 a^2 b^4 (7 A+17 C)-a^4 b^2 (3 A+28 C)\right ) \sin (c+d x)}{6 b^3 \left (a^2-b^2\right )^3 d (a+b \cos (c+d x))}\\ \end {align*}

________________________________________________________________________________________

Mathematica [B]  time = 4.45, size = 863, normalized size = 2.47 \[ \frac {\frac {24 c C a^9+24 C d x a^9-24 b C \sin (c+d x) a^8-36 b^2 c C a^7-36 b^2 C d x a^7-30 b^2 C \sin (2 (c+d x)) a^7+6 b^3 c C \cos (3 (c+d x)) a^6+6 b^3 C d x \cos (3 (c+d x)) a^6+57 b^3 C \sin (c+d x) a^6-11 b^3 C \sin (3 (c+d x)) a^6-36 b^4 c C a^5-36 b^4 C d x a^5+18 b^4 B \sin (c+d x) a^5+6 A b^4 \sin (2 (c+d x)) a^5+90 b^4 C \sin (2 (c+d x)) a^5+2 b^4 B \sin (3 (c+d x)) a^5-18 b^5 c C \cos (3 (c+d x)) a^4-18 b^5 C d x \cos (3 (c+d x)) a^4-51 A b^5 \sin (c+d x) a^4-72 b^5 C \sin (c+d x) a^4+6 b^5 B \sin (2 (c+d x)) a^4+A b^5 \sin (3 (c+d x)) a^4+32 b^5 C \sin (3 (c+d x)) a^4+84 b^6 c C a^3+84 b^6 C d x a^3+39 b^6 B \sin (c+d x) a^3-54 A b^6 \sin (2 (c+d x)) a^3-120 b^6 C \sin (2 (c+d x)) a^3-5 b^6 B \sin (3 (c+d x)) a^3+18 b^7 c C \cos (3 (c+d x)) a^2+18 b^7 C d x \cos (3 (c+d x)) a^2-18 A b^7 \sin (c+d x) a^2-36 b^7 C \sin (c+d x) a^2+54 b^7 B \sin (2 (c+d x)) a^2-10 A b^7 \sin (3 (c+d x)) a^2-36 b^7 C \sin (3 (c+d x)) a^2-36 b^8 c C a-36 b^8 C d x a+36 b^2 \left (a^2-b^2\right )^3 C (c+d x) \cos (2 (c+d x)) a+18 b^8 B \sin (c+d x) a-12 A b^8 \sin (2 (c+d x)) a+18 b^8 B \sin (3 (c+d x)) a+18 b \left (a^2-b^2\right )^3 \left (4 a^2+b^2\right ) C (c+d x) \cos (c+d x)-6 b^9 c C \cos (3 (c+d x))-6 b^9 C d x \cos (3 (c+d x))-6 A b^9 \sin (c+d x)-6 A b^9 \sin (3 (c+d x))}{\left (a^2-b^2\right )^3 (a+b \cos (c+d x))^3}-\frac {24 \left (2 C a^7-7 b^2 C a^5-b^4 (A-8 C) a^3+3 b^5 B a^2-4 b^6 (A+2 C) a+2 b^7 B\right ) \tanh ^{-1}\left (\frac {(a-b) \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {b^2-a^2}}\right )}{\left (b^2-a^2\right )^{7/2}}}{24 b^4 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((-24*(3*a^2*b^5*B + 2*b^7*B - a^3*b^4*(A - 8*C) + 2*a^7*C - 7*a^5*b^2*C - 4*a*b^6*(A + 2*C))*ArcTanh[((a - b)
*Tan[(c + d*x)/2])/Sqrt[-a^2 + b^2]])/(-a^2 + b^2)^(7/2) + (24*a^9*c*C - 36*a^7*b^2*c*C - 36*a^5*b^4*c*C + 84*
a^3*b^6*c*C - 36*a*b^8*c*C + 24*a^9*C*d*x - 36*a^7*b^2*C*d*x - 36*a^5*b^4*C*d*x + 84*a^3*b^6*C*d*x - 36*a*b^8*
C*d*x + 18*b*(a^2 - b^2)^3*(4*a^2 + b^2)*C*(c + d*x)*Cos[c + d*x] + 36*a*b^2*(a^2 - b^2)^3*C*(c + d*x)*Cos[2*(
c + d*x)] + 6*a^6*b^3*c*C*Cos[3*(c + d*x)] - 18*a^4*b^5*c*C*Cos[3*(c + d*x)] + 18*a^2*b^7*c*C*Cos[3*(c + d*x)]
- 6*b^9*c*C*Cos[3*(c + d*x)] + 6*a^6*b^3*C*d*x*Cos[3*(c + d*x)] - 18*a^4*b^5*C*d*x*Cos[3*(c + d*x)] + 18*a^2*
b^7*C*d*x*Cos[3*(c + d*x)] - 6*b^9*C*d*x*Cos[3*(c + d*x)] - 51*a^4*A*b^5*Sin[c + d*x] - 18*a^2*A*b^7*Sin[c + d
*x] - 6*A*b^9*Sin[c + d*x] + 18*a^5*b^4*B*Sin[c + d*x] + 39*a^3*b^6*B*Sin[c + d*x] + 18*a*b^8*B*Sin[c + d*x] -
24*a^8*b*C*Sin[c + d*x] + 57*a^6*b^3*C*Sin[c + d*x] - 72*a^4*b^5*C*Sin[c + d*x] - 36*a^2*b^7*C*Sin[c + d*x] +
6*a^5*A*b^4*Sin[2*(c + d*x)] - 54*a^3*A*b^6*Sin[2*(c + d*x)] - 12*a*A*b^8*Sin[2*(c + d*x)] + 6*a^4*b^5*B*Sin[
2*(c + d*x)] + 54*a^2*b^7*B*Sin[2*(c + d*x)] - 30*a^7*b^2*C*Sin[2*(c + d*x)] + 90*a^5*b^4*C*Sin[2*(c + d*x)] -
120*a^3*b^6*C*Sin[2*(c + d*x)] + a^4*A*b^5*Sin[3*(c + d*x)] - 10*a^2*A*b^7*Sin[3*(c + d*x)] - 6*A*b^9*Sin[3*(
c + d*x)] + 2*a^5*b^4*B*Sin[3*(c + d*x)] - 5*a^3*b^6*B*Sin[3*(c + d*x)] + 18*a*b^8*B*Sin[3*(c + d*x)] - 11*a^6
*b^3*C*Sin[3*(c + d*x)] + 32*a^4*b^5*C*Sin[3*(c + d*x)] - 36*a^2*b^7*C*Sin[3*(c + d*x)])/((a^2 - b^2)^3*(a + b
*Cos[c + d*x])^3))/(24*b^4*d)

________________________________________________________________________________________

fricas [B]  time = 1.41, size = 2045, normalized size = 5.86 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/12*(12*(C*a^8*b^3 - 4*C*a^6*b^5 + 6*C*a^4*b^7 - 4*C*a^2*b^9 + C*b^11)*d*x*cos(d*x + c)^3 + 36*(C*a^9*b^2 -
4*C*a^7*b^4 + 6*C*a^5*b^6 - 4*C*a^3*b^8 + C*a*b^10)*d*x*cos(d*x + c)^2 + 36*(C*a^10*b - 4*C*a^8*b^3 + 6*C*a^6*
b^5 - 4*C*a^4*b^7 + C*a^2*b^9)*d*x*cos(d*x + c) + 12*(C*a^11 - 4*C*a^9*b^2 + 6*C*a^7*b^4 - 4*C*a^5*b^6 + C*a^3
*b^8)*d*x + 3*(2*C*a^10 - 7*C*a^8*b^2 - (A - 8*C)*a^6*b^4 + 3*B*a^5*b^5 - 4*(A + 2*C)*a^4*b^6 + 2*B*a^3*b^7 +
(2*C*a^7*b^3 - 7*C*a^5*b^5 - (A - 8*C)*a^3*b^7 + 3*B*a^2*b^8 - 4*(A + 2*C)*a*b^9 + 2*B*b^10)*cos(d*x + c)^3 +
3*(2*C*a^8*b^2 - 7*C*a^6*b^4 - (A - 8*C)*a^4*b^6 + 3*B*a^3*b^7 - 4*(A + 2*C)*a^2*b^8 + 2*B*a*b^9)*cos(d*x + c)
^2 + 3*(2*C*a^9*b - 7*C*a^7*b^3 - (A - 8*C)*a^5*b^5 + 3*B*a^4*b^6 - 4*(A + 2*C)*a^3*b^7 + 2*B*a^2*b^8)*cos(d*x
+ c))*sqrt(-a^2 + b^2)*log((2*a*b*cos(d*x + c) + (2*a^2 - b^2)*cos(d*x + c)^2 + 2*sqrt(-a^2 + b^2)*(a*cos(d*x
+ c) + b)*sin(d*x + c) - a^2 + 2*b^2)/(b^2*cos(d*x + c)^2 + 2*a*b*cos(d*x + c) + a^2)) - 2*(6*C*a^10*b - 23*C
*a^8*b^3 - 4*B*a^7*b^4 + (13*A + 43*C)*a^6*b^5 - 7*B*a^5*b^6 - (11*A + 26*C)*a^4*b^7 + 11*B*a^3*b^8 - 2*A*a^2*
b^9 + (11*C*a^8*b^3 - 2*B*a^7*b^4 - (A + 43*C)*a^6*b^5 + 7*B*a^5*b^6 + (11*A + 68*C)*a^4*b^7 - 23*B*a^3*b^8 -
4*(A + 9*C)*a^2*b^9 + 18*B*a*b^10 - 6*A*b^11)*cos(d*x + c)^2 + 3*(5*C*a^9*b^2 - (A + 20*C)*a^7*b^4 - B*a^6*b^5
+ 5*(2*A + 7*C)*a^5*b^6 - 8*B*a^4*b^7 - (7*A + 20*C)*a^3*b^8 + 9*B*a^2*b^9 - 2*A*a*b^10)*cos(d*x + c))*sin(d*
x + c))/((a^8*b^7 - 4*a^6*b^9 + 6*a^4*b^11 - 4*a^2*b^13 + b^15)*d*cos(d*x + c)^3 + 3*(a^9*b^6 - 4*a^7*b^8 + 6*
a^5*b^10 - 4*a^3*b^12 + a*b^14)*d*cos(d*x + c)^2 + 3*(a^10*b^5 - 4*a^8*b^7 + 6*a^6*b^9 - 4*a^4*b^11 + a^2*b^13
)*d*cos(d*x + c) + (a^11*b^4 - 4*a^9*b^6 + 6*a^7*b^8 - 4*a^5*b^10 + a^3*b^12)*d), 1/6*(6*(C*a^8*b^3 - 4*C*a^6*
b^5 + 6*C*a^4*b^7 - 4*C*a^2*b^9 + C*b^11)*d*x*cos(d*x + c)^3 + 18*(C*a^9*b^2 - 4*C*a^7*b^4 + 6*C*a^5*b^6 - 4*C
*a^3*b^8 + C*a*b^10)*d*x*cos(d*x + c)^2 + 18*(C*a^10*b - 4*C*a^8*b^3 + 6*C*a^6*b^5 - 4*C*a^4*b^7 + C*a^2*b^9)*
d*x*cos(d*x + c) + 6*(C*a^11 - 4*C*a^9*b^2 + 6*C*a^7*b^4 - 4*C*a^5*b^6 + C*a^3*b^8)*d*x - 3*(2*C*a^10 - 7*C*a^
8*b^2 - (A - 8*C)*a^6*b^4 + 3*B*a^5*b^5 - 4*(A + 2*C)*a^4*b^6 + 2*B*a^3*b^7 + (2*C*a^7*b^3 - 7*C*a^5*b^5 - (A
- 8*C)*a^3*b^7 + 3*B*a^2*b^8 - 4*(A + 2*C)*a*b^9 + 2*B*b^10)*cos(d*x + c)^3 + 3*(2*C*a^8*b^2 - 7*C*a^6*b^4 - (
A - 8*C)*a^4*b^6 + 3*B*a^3*b^7 - 4*(A + 2*C)*a^2*b^8 + 2*B*a*b^9)*cos(d*x + c)^2 + 3*(2*C*a^9*b - 7*C*a^7*b^3
- (A - 8*C)*a^5*b^5 + 3*B*a^4*b^6 - 4*(A + 2*C)*a^3*b^7 + 2*B*a^2*b^8)*cos(d*x + c))*sqrt(a^2 - b^2)*arctan(-(
a*cos(d*x + c) + b)/(sqrt(a^2 - b^2)*sin(d*x + c))) - (6*C*a^10*b - 23*C*a^8*b^3 - 4*B*a^7*b^4 + (13*A + 43*C)
*a^6*b^5 - 7*B*a^5*b^6 - (11*A + 26*C)*a^4*b^7 + 11*B*a^3*b^8 - 2*A*a^2*b^9 + (11*C*a^8*b^3 - 2*B*a^7*b^4 - (A
+ 43*C)*a^6*b^5 + 7*B*a^5*b^6 + (11*A + 68*C)*a^4*b^7 - 23*B*a^3*b^8 - 4*(A + 9*C)*a^2*b^9 + 18*B*a*b^10 - 6*
A*b^11)*cos(d*x + c)^2 + 3*(5*C*a^9*b^2 - (A + 20*C)*a^7*b^4 - B*a^6*b^5 + 5*(2*A + 7*C)*a^5*b^6 - 8*B*a^4*b^7
- (7*A + 20*C)*a^3*b^8 + 9*B*a^2*b^9 - 2*A*a*b^10)*cos(d*x + c))*sin(d*x + c))/((a^8*b^7 - 4*a^6*b^9 + 6*a^4*
b^11 - 4*a^2*b^13 + b^15)*d*cos(d*x + c)^3 + 3*(a^9*b^6 - 4*a^7*b^8 + 6*a^5*b^10 - 4*a^3*b^12 + a*b^14)*d*cos(
d*x + c)^2 + 3*(a^10*b^5 - 4*a^8*b^7 + 6*a^6*b^9 - 4*a^4*b^11 + a^2*b^13)*d*cos(d*x + c) + (a^11*b^4 - 4*a^9*b
^6 + 6*a^7*b^8 - 4*a^5*b^10 + a^3*b^12)*d)]

________________________________________________________________________________________

giac [B]  time = 0.31, size = 1104, normalized size = 3.16 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*(3*(2*C*a^7 - 7*C*a^5*b^2 - A*a^3*b^4 + 8*C*a^3*b^4 + 3*B*a^2*b^5 - 4*A*a*b^6 - 8*C*a*b^6 + 2*B*b^7)*(pi*f
loor(1/2*(d*x + c)/pi + 1/2)*sgn(-2*a + 2*b) + arctan(-(a*tan(1/2*d*x + 1/2*c) - b*tan(1/2*d*x + 1/2*c))/sqrt(
a^2 - b^2)))/((a^6*b^4 - 3*a^4*b^6 + 3*a^2*b^8 - b^10)*sqrt(a^2 - b^2)) + 3*(d*x + c)*C/b^4 - (6*C*a^8*tan(1/2
*d*x + 1/2*c)^5 - 15*C*a^7*b*tan(1/2*d*x + 1/2*c)^5 - 6*C*a^6*b^2*tan(1/2*d*x + 1/2*c)^5 + 3*A*a^5*b^3*tan(1/2
*d*x + 1/2*c)^5 - 6*B*a^5*b^3*tan(1/2*d*x + 1/2*c)^5 + 45*C*a^5*b^3*tan(1/2*d*x + 1/2*c)^5 + 12*A*a^4*b^4*tan(
1/2*d*x + 1/2*c)^5 + 3*B*a^4*b^4*tan(1/2*d*x + 1/2*c)^5 - 6*C*a^4*b^4*tan(1/2*d*x + 1/2*c)^5 - 27*A*a^3*b^5*ta
n(1/2*d*x + 1/2*c)^5 - 6*B*a^3*b^5*tan(1/2*d*x + 1/2*c)^5 - 60*C*a^3*b^5*tan(1/2*d*x + 1/2*c)^5 + 12*A*a^2*b^6
*tan(1/2*d*x + 1/2*c)^5 + 27*B*a^2*b^6*tan(1/2*d*x + 1/2*c)^5 + 36*C*a^2*b^6*tan(1/2*d*x + 1/2*c)^5 - 6*A*a*b^
7*tan(1/2*d*x + 1/2*c)^5 - 18*B*a*b^7*tan(1/2*d*x + 1/2*c)^5 + 6*A*b^8*tan(1/2*d*x + 1/2*c)^5 + 12*C*a^8*tan(1
/2*d*x + 1/2*c)^3 - 56*C*a^6*b^2*tan(1/2*d*x + 1/2*c)^3 - 4*B*a^5*b^3*tan(1/2*d*x + 1/2*c)^3 + 28*A*a^4*b^4*ta
n(1/2*d*x + 1/2*c)^3 + 116*C*a^4*b^4*tan(1/2*d*x + 1/2*c)^3 - 32*B*a^3*b^5*tan(1/2*d*x + 1/2*c)^3 - 16*A*a^2*b
^6*tan(1/2*d*x + 1/2*c)^3 - 72*C*a^2*b^6*tan(1/2*d*x + 1/2*c)^3 + 36*B*a*b^7*tan(1/2*d*x + 1/2*c)^3 - 12*A*b^8
*tan(1/2*d*x + 1/2*c)^3 + 6*C*a^8*tan(1/2*d*x + 1/2*c) + 15*C*a^7*b*tan(1/2*d*x + 1/2*c) - 6*C*a^6*b^2*tan(1/2
*d*x + 1/2*c) - 3*A*a^5*b^3*tan(1/2*d*x + 1/2*c) - 6*B*a^5*b^3*tan(1/2*d*x + 1/2*c) - 45*C*a^5*b^3*tan(1/2*d*x
+ 1/2*c) + 12*A*a^4*b^4*tan(1/2*d*x + 1/2*c) - 3*B*a^4*b^4*tan(1/2*d*x + 1/2*c) - 6*C*a^4*b^4*tan(1/2*d*x + 1
/2*c) + 27*A*a^3*b^5*tan(1/2*d*x + 1/2*c) - 6*B*a^3*b^5*tan(1/2*d*x + 1/2*c) + 60*C*a^3*b^5*tan(1/2*d*x + 1/2*
c) + 12*A*a^2*b^6*tan(1/2*d*x + 1/2*c) - 27*B*a^2*b^6*tan(1/2*d*x + 1/2*c) + 36*C*a^2*b^6*tan(1/2*d*x + 1/2*c)
+ 6*A*a*b^7*tan(1/2*d*x + 1/2*c) - 18*B*a*b^7*tan(1/2*d*x + 1/2*c) + 6*A*b^8*tan(1/2*d*x + 1/2*c))/((a^6*b^3
- 3*a^4*b^5 + 3*a^2*b^7 - b^9)*(a*tan(1/2*d*x + 1/2*c)^2 - b*tan(1/2*d*x + 1/2*c)^2 + a + b)^3))/d

________________________________________________________________________________________

maple [B]  time = 0.13, size = 3098, normalized size = 8.88 \[ \text {output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/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-2/d*b^3/(a*tan(1/2*d*x+1/2*c)^2-tan(1/2*d*x+1/2*c)^2*b+a+b)^3/(a-b)/(a^3+3*a^2*b+3*a*b^2+b^3)*tan(1/2*d*x+1
/2*c)^5*A-4/d/(a*tan(1/2*d*x+1/2*c)^2-tan(1/2*d*x+1/2*c)^2*b+a+b)^3*a^3/(a-b)/(a^3+3*a^2*b+3*a*b^2+b^3)*tan(1/
2*d*x+1/2*c)^5*C+8/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-2/d*b^2/(a*tan(1/2*d*x+1/2*c)^2-tan(1/2*d*x+1/2*c)^2*b+a+b)^3*a/(a-b)/(a^3+3*a^2*b+3*a*b
^2+b^3)*tan(1/2*d*x+1/2*c)^5*A-3/d*b/(a*tan(1/2*d*x+1/2*c)^2-tan(1/2*d*x+1/2*c)^2*b+a+b)^3*a^2/(a+b)/(a^3-3*a^
2*b+3*a*b^2-b^3)*tan(1/2*d*x+1/2*c)*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^2+2*a*
b+b^2)/(a^2-2*a*b+b^2)*tan(1/2*d*x+1/2*c)^3*a*B+2/d*b^2/(a*tan(1/2*d*x+1/2*c)^2-tan(1/2*d*x+1/2*c)^2*b+a+b)^3*
a/(a+b)/(a^3-3*a^2*b+3*a*b^2-b^3)*tan(1/2*d*x+1/2*c)*A-24/d*b/(a*tan(1/2*d*x+1/2*c)^2-tan(1/2*d*x+1/2*c)^2*b+a
+b)^3/(a^2+2*a*b+b^2)/(a^2-2*a*b+b^2)*tan(1/2*d*x+1/2*c)^3*C*a^2+3/d*b/(a*tan(1/2*d*x+1/2*c)^2-tan(1/2*d*x+1/2
*c)^2*b+a+b)^3*a^2/(a-b)/(a^3+3*a^2*b+3*a*b^2+b^3)*tan(1/2*d*x+1/2*c)^5*B+1/d*a^5/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*C-2/d*a^6/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*C-6/d*a^2*b/(a*t
an(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+6/d*a
^4/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-28/3/d*a^2*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*A-6/d*a^2*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)*A-2/d*a^6/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)*C-1/d*a^5/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)*C+6/d*a^4/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-4/d*a^6/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*C+44/3/d*a^4/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+6/d*b^2/(a*tan(1/2*d*x+1/2*c)^2-tan(1/2
*d*x+1/2*c)^2*b+a+b)^3/(a+b)/(a^3-3*a^2*b+3*a*b^2-b^3)*tan(1/2*d*x+1/2*c)*a*B+6/d*b^2/(a*tan(1/2*d*x+1/2*c)^2-
tan(1/2*d*x+1/2*c)^2*b+a+b)^3/(a-b)/(a^3+3*a^2*b+3*a*b^2+b^3)*tan(1/2*d*x+1/2*c)^5*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+b)/(a^3-3*a^2*b+3*a*b^2-b^3)*tan(1/2*d*x+1/2*c)*C*a^2-12/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-b)/(a^3+3*a^2*b+3*a*b^2+b^3)*tan(1/2*d*x+1/2*c)^5*C*a^2-8/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^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)
*B+7/d*a^5/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))*C+4/3/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^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*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-2/d*b^3/(a*tan(1/2*d*x+1/2*c)^2-tan(1/2*d*x+1/2*c)^2*b+a+b)^3/(a+b)/(a^3-3*a^2*b+3*a*b
^2-b^3)*tan(1/2*d*x+1/2*c)*A-4/d*b^3/(a*tan(1/2*d*x+1/2*c)^2-tan(1/2*d*x+1/2*c)^2*b+a+b)^3/(a^2+2*a*b+b^2)/(a^
2-2*a*b+b^2)*tan(1/2*d*x+1/2*c)^3*A+4/d/(a*tan(1/2*d*x+1/2*c)^2-tan(1/2*d*x+1/2*c)^2*b+a+b)^3*a^3/(a+b)/(a^3-3
*a^2*b+3*a*b^2-b^3)*tan(1/2*d*x+1/2*c)*C+2/d/b^4*arctan(tan(1/2*d*x+1/2*c))*C+1/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-1/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+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*B-2/d*a^7/b^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))*C-3/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-2
/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

________________________________________________________________________________________

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(cos(d*x+c)^2*(A+B*cos(d*x+c)+C*cos(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 = 14.67, size = 11947, normalized size = 34.23 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(2*C*atan(((C*((8*tan(c/2 + (d*x)/2)*(4*B^2*b^14 + 8*C^2*a^14 + 4*C^2*b^14 - 8*C^2*a*b^13 - 8*C^2*a^13*b + 16*
A^2*a^2*b^12 + 8*A^2*a^4*b^10 + A^2*a^6*b^8 + 12*B^2*a^2*b^12 + 9*B^2*a^4*b^10 + 44*C^2*a^2*b^12 + 48*C^2*a^3*
b^11 - 92*C^2*a^4*b^10 - 120*C^2*a^5*b^9 + 156*C^2*a^6*b^8 + 160*C^2*a^7*b^7 - 164*C^2*a^8*b^6 - 120*C^2*a^9*b
^5 + 117*C^2*a^10*b^4 + 48*C^2*a^11*b^3 - 48*C^2*a^12*b^2 - 16*A*B*a*b^13 - 32*B*C*a*b^13 - 28*A*B*a^3*b^11 -
6*A*B*a^5*b^9 + 64*A*C*a^2*b^12 - 48*A*C*a^4*b^10 + 40*A*C*a^6*b^8 - 2*A*C*a^8*b^6 - 4*A*C*a^10*b^4 - 16*B*C*a
^3*b^11 + 20*B*C*a^5*b^9 - 34*B*C*a^7*b^7 + 12*B*C*a^9*b^5))/(a*b^16 + b^17 - 5*a^2*b^15 - 5*a^3*b^14 + 10*a^4
*b^13 + 10*a^5*b^12 - 10*a^6*b^11 - 10*a^7*b^10 + 5*a^8*b^9 + 5*a^9*b^8 - a^10*b^7 - a^11*b^6) + (C*((8*(4*B*b
^21 + 4*C*b^21 + 8*A*a^2*b^19 + 22*A*a^3*b^18 - 22*A*a^4*b^17 - 18*A*a^5*b^16 + 18*A*a^6*b^15 + 2*A*a^7*b^14 -
2*A*a^8*b^13 + 2*A*a^9*b^12 - 2*A*a^10*b^11 - 6*B*a^2*b^19 + 6*B*a^3*b^18 - 6*B*a^4*b^17 + 6*B*a^5*b^16 + 14*
B*a^6*b^15 - 14*B*a^7*b^14 - 6*B*a^8*b^13 + 6*B*a^9*b^12 - 12*C*a^2*b^19 + 64*C*a^3*b^18 + 20*C*a^4*b^17 - 110
*C*a^5*b^16 - 30*C*a^6*b^15 + 110*C*a^7*b^14 + 30*C*a^8*b^13 - 70*C*a^9*b^12 - 14*C*a^10*b^11 + 26*C*a^11*b^10
+ 2*C*a^12*b^9 - 4*C*a^13*b^8 - 8*A*a*b^20 - 4*B*a*b^20 - 16*C*a*b^20))/(a*b^19 + b^20 - 5*a^2*b^18 - 5*a^3*b
^17 + 10*a^4*b^16 + 10*a^5*b^15 - 10*a^6*b^14 - 10*a^7*b^13 + 5*a^8*b^12 + 5*a^9*b^11 - a^10*b^10 - a^11*b^9)
- (C*tan(c/2 + (d*x)/2)*(8*a*b^21 - 8*a^2*b^20 - 48*a^3*b^19 + 48*a^4*b^18 + 120*a^5*b^17 - 120*a^6*b^16 - 160
*a^7*b^15 + 160*a^8*b^14 + 120*a^9*b^13 - 120*a^10*b^12 - 48*a^11*b^11 + 48*a^12*b^10 + 8*a^13*b^9 - 8*a^14*b^
8)*8i)/(b^4*(a*b^16 + b^17 - 5*a^2*b^15 - 5*a^3*b^14 + 10*a^4*b^13 + 10*a^5*b^12 - 10*a^6*b^11 - 10*a^7*b^10 +
5*a^8*b^9 + 5*a^9*b^8 - a^10*b^7 - a^11*b^6)))*1i)/b^4))/b^4 + (C*((8*tan(c/2 + (d*x)/2)*(4*B^2*b^14 + 8*C^2*
a^14 + 4*C^2*b^14 - 8*C^2*a*b^13 - 8*C^2*a^13*b + 16*A^2*a^2*b^12 + 8*A^2*a^4*b^10 + A^2*a^6*b^8 + 12*B^2*a^2*
b^12 + 9*B^2*a^4*b^10 + 44*C^2*a^2*b^12 + 48*C^2*a^3*b^11 - 92*C^2*a^4*b^10 - 120*C^2*a^5*b^9 + 156*C^2*a^6*b^
8 + 160*C^2*a^7*b^7 - 164*C^2*a^8*b^6 - 120*C^2*a^9*b^5 + 117*C^2*a^10*b^4 + 48*C^2*a^11*b^3 - 48*C^2*a^12*b^2
- 16*A*B*a*b^13 - 32*B*C*a*b^13 - 28*A*B*a^3*b^11 - 6*A*B*a^5*b^9 + 64*A*C*a^2*b^12 - 48*A*C*a^4*b^10 + 40*A*
C*a^6*b^8 - 2*A*C*a^8*b^6 - 4*A*C*a^10*b^4 - 16*B*C*a^3*b^11 + 20*B*C*a^5*b^9 - 34*B*C*a^7*b^7 + 12*B*C*a^9*b^
5))/(a*b^16 + b^17 - 5*a^2*b^15 - 5*a^3*b^14 + 10*a^4*b^13 + 10*a^5*b^12 - 10*a^6*b^11 - 10*a^7*b^10 + 5*a^8*b
^9 + 5*a^9*b^8 - a^10*b^7 - a^11*b^6) - (C*((8*(4*B*b^21 + 4*C*b^21 + 8*A*a^2*b^19 + 22*A*a^3*b^18 - 22*A*a^4*
b^17 - 18*A*a^5*b^16 + 18*A*a^6*b^15 + 2*A*a^7*b^14 - 2*A*a^8*b^13 + 2*A*a^9*b^12 - 2*A*a^10*b^11 - 6*B*a^2*b^
19 + 6*B*a^3*b^18 - 6*B*a^4*b^17 + 6*B*a^5*b^16 + 14*B*a^6*b^15 - 14*B*a^7*b^14 - 6*B*a^8*b^13 + 6*B*a^9*b^12
- 12*C*a^2*b^19 + 64*C*a^3*b^18 + 20*C*a^4*b^17 - 110*C*a^5*b^16 - 30*C*a^6*b^15 + 110*C*a^7*b^14 + 30*C*a^8*b
^13 - 70*C*a^9*b^12 - 14*C*a^10*b^11 + 26*C*a^11*b^10 + 2*C*a^12*b^9 - 4*C*a^13*b^8 - 8*A*a*b^20 - 4*B*a*b^20
- 16*C*a*b^20))/(a*b^19 + b^20 - 5*a^2*b^18 - 5*a^3*b^17 + 10*a^4*b^16 + 10*a^5*b^15 - 10*a^6*b^14 - 10*a^7*b^
13 + 5*a^8*b^12 + 5*a^9*b^11 - a^10*b^10 - a^11*b^9) + (C*tan(c/2 + (d*x)/2)*(8*a*b^21 - 8*a^2*b^20 - 48*a^3*b
^19 + 48*a^4*b^18 + 120*a^5*b^17 - 120*a^6*b^16 - 160*a^7*b^15 + 160*a^8*b^14 + 120*a^9*b^13 - 120*a^10*b^12 -
48*a^11*b^11 + 48*a^12*b^10 + 8*a^13*b^9 - 8*a^14*b^8)*8i)/(b^4*(a*b^16 + b^17 - 5*a^2*b^15 - 5*a^3*b^14 + 10
*a^4*b^13 + 10*a^5*b^12 - 10*a^6*b^11 - 10*a^7*b^10 + 5*a^8*b^9 + 5*a^9*b^8 - a^10*b^7 - a^11*b^6)))*1i)/b^4))
/b^4)/((16*(4*C^3*a^13 - 4*B*C^2*b^13 + 4*B^2*C*b^13 + 16*C^3*a*b^12 - 2*C^3*a^12*b + 48*C^3*a^2*b^11 - 64*C^3
*a^3*b^10 - 64*C^3*a^4*b^9 + 110*C^3*a^5*b^8 + 66*C^3*a^6*b^7 - 110*C^3*a^7*b^6 - 34*C^3*a^8*b^5 + 70*C^3*a^9*
b^4 + 11*C^3*a^10*b^3 - 26*C^3*a^11*b^2 + 8*A*C^2*a*b^12 - 28*B*C^2*a*b^12 + 56*A*C^2*a^2*b^11 - 22*A*C^2*a^3*
b^10 - 26*A*C^2*a^4*b^9 + 18*A*C^2*a^5*b^8 + 22*A*C^2*a^6*b^7 - 2*A*C^2*a^7*b^6 - 2*A*C^2*a^9*b^4 - 2*A*C^2*a^
10*b^3 + 16*A^2*C*a^2*b^11 + 8*A^2*C*a^4*b^9 + A^2*C*a^6*b^7 + 6*B*C^2*a^2*b^11 - 22*B*C^2*a^3*b^10 + 6*B*C^2*
a^4*b^9 + 14*B*C^2*a^5*b^8 - 14*B*C^2*a^6*b^7 - 20*B*C^2*a^7*b^6 + 6*B*C^2*a^8*b^5 + 6*B*C^2*a^9*b^4 + 12*B^2*
C*a^2*b^11 + 9*B^2*C*a^4*b^9 - 16*A*B*C*a*b^12 - 28*A*B*C*a^3*b^10 - 6*A*B*C*a^5*b^8))/(a*b^19 + b^20 - 5*a^2*
b^18 - 5*a^3*b^17 + 10*a^4*b^16 + 10*a^5*b^15 - 10*a^6*b^14 - 10*a^7*b^13 + 5*a^8*b^12 + 5*a^9*b^11 - a^10*b^1
0 - a^11*b^9) - (C*((8*tan(c/2 + (d*x)/2)*(4*B^2*b^14 + 8*C^2*a^14 + 4*C^2*b^14 - 8*C^2*a*b^13 - 8*C^2*a^13*b
+ 16*A^2*a^2*b^12 + 8*A^2*a^4*b^10 + A^2*a^6*b^8 + 12*B^2*a^2*b^12 + 9*B^2*a^4*b^10 + 44*C^2*a^2*b^12 + 48*C^2
*a^3*b^11 - 92*C^2*a^4*b^10 - 120*C^2*a^5*b^9 + 156*C^2*a^6*b^8 + 160*C^2*a^7*b^7 - 164*C^2*a^8*b^6 - 120*C^2*
a^9*b^5 + 117*C^2*a^10*b^4 + 48*C^2*a^11*b^3 - 48*C^2*a^12*b^2 - 16*A*B*a*b^13 - 32*B*C*a*b^13 - 28*A*B*a^3*b^
11 - 6*A*B*a^5*b^9 + 64*A*C*a^2*b^12 - 48*A*C*a^4*b^10 + 40*A*C*a^6*b^8 - 2*A*C*a^8*b^6 - 4*A*C*a^10*b^4 - 16*
B*C*a^3*b^11 + 20*B*C*a^5*b^9 - 34*B*C*a^7*b^7 + 12*B*C*a^9*b^5))/(a*b^16 + b^17 - 5*a^2*b^15 - 5*a^3*b^14 + 1
0*a^4*b^13 + 10*a^5*b^12 - 10*a^6*b^11 - 10*a^7*b^10 + 5*a^8*b^9 + 5*a^9*b^8 - a^10*b^7 - a^11*b^6) + (C*((8*(
4*B*b^21 + 4*C*b^21 + 8*A*a^2*b^19 + 22*A*a^3*b^18 - 22*A*a^4*b^17 - 18*A*a^5*b^16 + 18*A*a^6*b^15 + 2*A*a^7*b
^14 - 2*A*a^8*b^13 + 2*A*a^9*b^12 - 2*A*a^10*b^11 - 6*B*a^2*b^19 + 6*B*a^3*b^18 - 6*B*a^4*b^17 + 6*B*a^5*b^16
+ 14*B*a^6*b^15 - 14*B*a^7*b^14 - 6*B*a^8*b^13 + 6*B*a^9*b^12 - 12*C*a^2*b^19 + 64*C*a^3*b^18 + 20*C*a^4*b^17
- 110*C*a^5*b^16 - 30*C*a^6*b^15 + 110*C*a^7*b^14 + 30*C*a^8*b^13 - 70*C*a^9*b^12 - 14*C*a^10*b^11 + 26*C*a^11
*b^10 + 2*C*a^12*b^9 - 4*C*a^13*b^8 - 8*A*a*b^20 - 4*B*a*b^20 - 16*C*a*b^20))/(a*b^19 + b^20 - 5*a^2*b^18 - 5*
a^3*b^17 + 10*a^4*b^16 + 10*a^5*b^15 - 10*a^6*b^14 - 10*a^7*b^13 + 5*a^8*b^12 + 5*a^9*b^11 - a^10*b^10 - a^11*
b^9) - (C*tan(c/2 + (d*x)/2)*(8*a*b^21 - 8*a^2*b^20 - 48*a^3*b^19 + 48*a^4*b^18 + 120*a^5*b^17 - 120*a^6*b^16
- 160*a^7*b^15 + 160*a^8*b^14 + 120*a^9*b^13 - 120*a^10*b^12 - 48*a^11*b^11 + 48*a^12*b^10 + 8*a^13*b^9 - 8*a^
14*b^8)*8i)/(b^4*(a*b^16 + b^17 - 5*a^2*b^15 - 5*a^3*b^14 + 10*a^4*b^13 + 10*a^5*b^12 - 10*a^6*b^11 - 10*a^7*b
^10 + 5*a^8*b^9 + 5*a^9*b^8 - a^10*b^7 - a^11*b^6)))*1i)/b^4)*1i)/b^4 + (C*((8*tan(c/2 + (d*x)/2)*(4*B^2*b^14
+ 8*C^2*a^14 + 4*C^2*b^14 - 8*C^2*a*b^13 - 8*C^2*a^13*b + 16*A^2*a^2*b^12 + 8*A^2*a^4*b^10 + A^2*a^6*b^8 + 12*
B^2*a^2*b^12 + 9*B^2*a^4*b^10 + 44*C^2*a^2*b^12 + 48*C^2*a^3*b^11 - 92*C^2*a^4*b^10 - 120*C^2*a^5*b^9 + 156*C^
2*a^6*b^8 + 160*C^2*a^7*b^7 - 164*C^2*a^8*b^6 - 120*C^2*a^9*b^5 + 117*C^2*a^10*b^4 + 48*C^2*a^11*b^3 - 48*C^2*
a^12*b^2 - 16*A*B*a*b^13 - 32*B*C*a*b^13 - 28*A*B*a^3*b^11 - 6*A*B*a^5*b^9 + 64*A*C*a^2*b^12 - 48*A*C*a^4*b^10
+ 40*A*C*a^6*b^8 - 2*A*C*a^8*b^6 - 4*A*C*a^10*b^4 - 16*B*C*a^3*b^11 + 20*B*C*a^5*b^9 - 34*B*C*a^7*b^7 + 12*B*
C*a^9*b^5))/(a*b^16 + b^17 - 5*a^2*b^15 - 5*a^3*b^14 + 10*a^4*b^13 + 10*a^5*b^12 - 10*a^6*b^11 - 10*a^7*b^10 +
5*a^8*b^9 + 5*a^9*b^8 - a^10*b^7 - a^11*b^6) - (C*((8*(4*B*b^21 + 4*C*b^21 + 8*A*a^2*b^19 + 22*A*a^3*b^18 - 2
2*A*a^4*b^17 - 18*A*a^5*b^16 + 18*A*a^6*b^15 + 2*A*a^7*b^14 - 2*A*a^8*b^13 + 2*A*a^9*b^12 - 2*A*a^10*b^11 - 6*
B*a^2*b^19 + 6*B*a^3*b^18 - 6*B*a^4*b^17 + 6*B*a^5*b^16 + 14*B*a^6*b^15 - 14*B*a^7*b^14 - 6*B*a^8*b^13 + 6*B*a
^9*b^12 - 12*C*a^2*b^19 + 64*C*a^3*b^18 + 20*C*a^4*b^17 - 110*C*a^5*b^16 - 30*C*a^6*b^15 + 110*C*a^7*b^14 + 30
*C*a^8*b^13 - 70*C*a^9*b^12 - 14*C*a^10*b^11 + 26*C*a^11*b^10 + 2*C*a^12*b^9 - 4*C*a^13*b^8 - 8*A*a*b^20 - 4*B
*a*b^20 - 16*C*a*b^20))/(a*b^19 + b^20 - 5*a^2*b^18 - 5*a^3*b^17 + 10*a^4*b^16 + 10*a^5*b^15 - 10*a^6*b^14 - 1
0*a^7*b^13 + 5*a^8*b^12 + 5*a^9*b^11 - a^10*b^10 - a^11*b^9) + (C*tan(c/2 + (d*x)/2)*(8*a*b^21 - 8*a^2*b^20 -
48*a^3*b^19 + 48*a^4*b^18 + 120*a^5*b^17 - 120*a^6*b^16 - 160*a^7*b^15 + 160*a^8*b^14 + 120*a^9*b^13 - 120*a^1
0*b^12 - 48*a^11*b^11 + 48*a^12*b^10 + 8*a^13*b^9 - 8*a^14*b^8)*8i)/(b^4*(a*b^16 + b^17 - 5*a^2*b^15 - 5*a^3*b
^14 + 10*a^4*b^13 + 10*a^5*b^12 - 10*a^6*b^11 - 10*a^7*b^10 + 5*a^8*b^9 + 5*a^9*b^8 - a^10*b^7 - a^11*b^6)))*1
i)/b^4)*1i)/b^4)))/(b^4*d) - ((tan(c/2 + (d*x)/2)*(2*A*b^6 + 2*C*a^6 + 6*A*a^2*b^4 - A*a^3*b^3 + 3*B*a^2*b^4 -
2*B*a^3*b^3 + 12*C*a^2*b^4 - 4*C*a^3*b^3 - 6*C*a^4*b^2 - 2*A*a*b^5 - 6*B*a*b^5 + C*a^5*b))/((a + b)*(3*a*b^5
- b^6 - 3*a^2*b^4 + a^3*b^3)) + (4*tan(c/2 + (d*x)/2)^3*(3*A*b^6 + 3*C*a^6 + 7*A*a^2*b^4 - B*a^3*b^3 + 18*C*a^
2*b^4 - 11*C*a^4*b^2 - 9*B*a*b^5))/(3*(a + b)^2*(b^5 - 2*a*b^4 + a^2*b^3)) + (tan(c/2 + (d*x)/2)^5*(2*A*b^6 +
2*C*a^6 + 6*A*a^2*b^4 + A*a^3*b^3 - 3*B*a^2*b^4 - 2*B*a^3*b^3 + 12*C*a^2*b^4 + 4*C*a^3*b^3 - 6*C*a^4*b^2 + 2*A
*a*b^5 - 6*B*a*b^5 - C*a^5*b))/((a*b^3 - b^4)*(a + b)^3))/(d*(3*a*b^2 - tan(c/2 + (d*x)/2)^4*(3*a*b^2 + 3*a^2*
b - 3*a^3 - 3*b^3) - tan(c/2 + (d*x)/2)^2*(3*a*b^2 - 3*a^2*b - 3*a^3 + 3*b^3) + 3*a^2*b + a^3 + b^3 + tan(c/2
+ (d*x)/2)^6*(3*a*b^2 - 3*a^2*b + a^3 - b^3))) + (atan(((((8*tan(c/2 + (d*x)/2)*(4*B^2*b^14 + 8*C^2*a^14 + 4*C
^2*b^14 - 8*C^2*a*b^13 - 8*C^2*a^13*b + 16*A^2*a^2*b^12 + 8*A^2*a^4*b^10 + A^2*a^6*b^8 + 12*B^2*a^2*b^12 + 9*B
^2*a^4*b^10 + 44*C^2*a^2*b^12 + 48*C^2*a^3*b^11 - 92*C^2*a^4*b^10 - 120*C^2*a^5*b^9 + 156*C^2*a^6*b^8 + 160*C^
2*a^7*b^7 - 164*C^2*a^8*b^6 - 120*C^2*a^9*b^5 + 117*C^2*a^10*b^4 + 48*C^2*a^11*b^3 - 48*C^2*a^12*b^2 - 16*A*B*
a*b^13 - 32*B*C*a*b^13 - 28*A*B*a^3*b^11 - 6*A*B*a^5*b^9 + 64*A*C*a^2*b^12 - 48*A*C*a^4*b^10 + 40*A*C*a^6*b^8
- 2*A*C*a^8*b^6 - 4*A*C*a^10*b^4 - 16*B*C*a^3*b^11 + 20*B*C*a^5*b^9 - 34*B*C*a^7*b^7 + 12*B*C*a^9*b^5))/(a*b^1
6 + b^17 - 5*a^2*b^15 - 5*a^3*b^14 + 10*a^4*b^13 + 10*a^5*b^12 - 10*a^6*b^11 - 10*a^7*b^10 + 5*a^8*b^9 + 5*a^9
*b^8 - a^10*b^7 - a^11*b^6) + (((8*(4*B*b^21 + 4*C*b^21 + 8*A*a^2*b^19 + 22*A*a^3*b^18 - 22*A*a^4*b^17 - 18*A*
a^5*b^16 + 18*A*a^6*b^15 + 2*A*a^7*b^14 - 2*A*a^8*b^13 + 2*A*a^9*b^12 - 2*A*a^10*b^11 - 6*B*a^2*b^19 + 6*B*a^3
*b^18 - 6*B*a^4*b^17 + 6*B*a^5*b^16 + 14*B*a^6*b^15 - 14*B*a^7*b^14 - 6*B*a^8*b^13 + 6*B*a^9*b^12 - 12*C*a^2*b
^19 + 64*C*a^3*b^18 + 20*C*a^4*b^17 - 110*C*a^5*b^16 - 30*C*a^6*b^15 + 110*C*a^7*b^14 + 30*C*a^8*b^13 - 70*C*a
^9*b^12 - 14*C*a^10*b^11 + 26*C*a^11*b^10 + 2*C*a^12*b^9 - 4*C*a^13*b^8 - 8*A*a*b^20 - 4*B*a*b^20 - 16*C*a*b^2
0))/(a*b^19 + b^20 - 5*a^2*b^18 - 5*a^3*b^17 + 10*a^4*b^16 + 10*a^5*b^15 - 10*a^6*b^14 - 10*a^7*b^13 + 5*a^8*b
^12 + 5*a^9*b^11 - a^10*b^10 - a^11*b^9) - (4*tan(c/2 + (d*x)/2)*(-(a + b)^7*(a - b)^7)^(1/2)*(2*B*b^7 + 2*C*a
^7 - A*a^3*b^4 + 3*B*a^2*b^5 + 8*C*a^3*b^4 - 7*C*a^5*b^2 - 4*A*a*b^6 - 8*C*a*b^6)*(8*a*b^21 - 8*a^2*b^20 - 48*
a^3*b^19 + 48*a^4*b^18 + 120*a^5*b^17 - 120*a^6*b^16 - 160*a^7*b^15 + 160*a^8*b^14 + 120*a^9*b^13 - 120*a^10*b
^12 - 48*a^11*b^11 + 48*a^12*b^10 + 8*a^13*b^9 - 8*a^14*b^8))/((b^18 - 7*a^2*b^16 + 21*a^4*b^14 - 35*a^6*b^12
+ 35*a^8*b^10 - 21*a^10*b^8 + 7*a^12*b^6 - a^14*b^4)*(a*b^16 + b^17 - 5*a^2*b^15 - 5*a^3*b^14 + 10*a^4*b^13 +
10*a^5*b^12 - 10*a^6*b^11 - 10*a^7*b^10 + 5*a^8*b^9 + 5*a^9*b^8 - a^10*b^7 - a^11*b^6)))*(-(a + b)^7*(a - b)^7
)^(1/2)*(2*B*b^7 + 2*C*a^7 - A*a^3*b^4 + 3*B*a^2*b^5 + 8*C*a^3*b^4 - 7*C*a^5*b^2 - 4*A*a*b^6 - 8*C*a*b^6))/(2*
(b^18 - 7*a^2*b^16 + 21*a^4*b^14 - 35*a^6*b^12 + 35*a^8*b^10 - 21*a^10*b^8 + 7*a^12*b^6 - a^14*b^4)))*(-(a + b
)^7*(a - b)^7)^(1/2)*(2*B*b^7 + 2*C*a^7 - A*a^3*b^4 + 3*B*a^2*b^5 + 8*C*a^3*b^4 - 7*C*a^5*b^2 - 4*A*a*b^6 - 8*
C*a*b^6)*1i)/(2*(b^18 - 7*a^2*b^16 + 21*a^4*b^14 - 35*a^6*b^12 + 35*a^8*b^10 - 21*a^10*b^8 + 7*a^12*b^6 - a^14
*b^4)) + (((8*tan(c/2 + (d*x)/2)*(4*B^2*b^14 + 8*C^2*a^14 + 4*C^2*b^14 - 8*C^2*a*b^13 - 8*C^2*a^13*b + 16*A^2*
a^2*b^12 + 8*A^2*a^4*b^10 + A^2*a^6*b^8 + 12*B^2*a^2*b^12 + 9*B^2*a^4*b^10 + 44*C^2*a^2*b^12 + 48*C^2*a^3*b^11
- 92*C^2*a^4*b^10 - 120*C^2*a^5*b^9 + 156*C^2*a^6*b^8 + 160*C^2*a^7*b^7 - 164*C^2*a^8*b^6 - 120*C^2*a^9*b^5 +
117*C^2*a^10*b^4 + 48*C^2*a^11*b^3 - 48*C^2*a^12*b^2 - 16*A*B*a*b^13 - 32*B*C*a*b^13 - 28*A*B*a^3*b^11 - 6*A*
B*a^5*b^9 + 64*A*C*a^2*b^12 - 48*A*C*a^4*b^10 + 40*A*C*a^6*b^8 - 2*A*C*a^8*b^6 - 4*A*C*a^10*b^4 - 16*B*C*a^3*b
^11 + 20*B*C*a^5*b^9 - 34*B*C*a^7*b^7 + 12*B*C*a^9*b^5))/(a*b^16 + b^17 - 5*a^2*b^15 - 5*a^3*b^14 + 10*a^4*b^1
3 + 10*a^5*b^12 - 10*a^6*b^11 - 10*a^7*b^10 + 5*a^8*b^9 + 5*a^9*b^8 - a^10*b^7 - a^11*b^6) - (((8*(4*B*b^21 +
4*C*b^21 + 8*A*a^2*b^19 + 22*A*a^3*b^18 - 22*A*a^4*b^17 - 18*A*a^5*b^16 + 18*A*a^6*b^15 + 2*A*a^7*b^14 - 2*A*a
^8*b^13 + 2*A*a^9*b^12 - 2*A*a^10*b^11 - 6*B*a^2*b^19 + 6*B*a^3*b^18 - 6*B*a^4*b^17 + 6*B*a^5*b^16 + 14*B*a^6*
b^15 - 14*B*a^7*b^14 - 6*B*a^8*b^13 + 6*B*a^9*b^12 - 12*C*a^2*b^19 + 64*C*a^3*b^18 + 20*C*a^4*b^17 - 110*C*a^5
*b^16 - 30*C*a^6*b^15 + 110*C*a^7*b^14 + 30*C*a^8*b^13 - 70*C*a^9*b^12 - 14*C*a^10*b^11 + 26*C*a^11*b^10 + 2*C
*a^12*b^9 - 4*C*a^13*b^8 - 8*A*a*b^20 - 4*B*a*b^20 - 16*C*a*b^20))/(a*b^19 + b^20 - 5*a^2*b^18 - 5*a^3*b^17 +
10*a^4*b^16 + 10*a^5*b^15 - 10*a^6*b^14 - 10*a^7*b^13 + 5*a^8*b^12 + 5*a^9*b^11 - a^10*b^10 - a^11*b^9) + (4*t
an(c/2 + (d*x)/2)*(-(a + b)^7*(a - b)^7)^(1/2)*(2*B*b^7 + 2*C*a^7 - A*a^3*b^4 + 3*B*a^2*b^5 + 8*C*a^3*b^4 - 7*
C*a^5*b^2 - 4*A*a*b^6 - 8*C*a*b^6)*(8*a*b^21 - 8*a^2*b^20 - 48*a^3*b^19 + 48*a^4*b^18 + 120*a^5*b^17 - 120*a^6
*b^16 - 160*a^7*b^15 + 160*a^8*b^14 + 120*a^9*b^13 - 120*a^10*b^12 - 48*a^11*b^11 + 48*a^12*b^10 + 8*a^13*b^9
- 8*a^14*b^8))/((b^18 - 7*a^2*b^16 + 21*a^4*b^14 - 35*a^6*b^12 + 35*a^8*b^10 - 21*a^10*b^8 + 7*a^12*b^6 - a^14
*b^4)*(a*b^16 + b^17 - 5*a^2*b^15 - 5*a^3*b^14 + 10*a^4*b^13 + 10*a^5*b^12 - 10*a^6*b^11 - 10*a^7*b^10 + 5*a^8
*b^9 + 5*a^9*b^8 - a^10*b^7 - a^11*b^6)))*(-(a + b)^7*(a - b)^7)^(1/2)*(2*B*b^7 + 2*C*a^7 - A*a^3*b^4 + 3*B*a^
2*b^5 + 8*C*a^3*b^4 - 7*C*a^5*b^2 - 4*A*a*b^6 - 8*C*a*b^6))/(2*(b^18 - 7*a^2*b^16 + 21*a^4*b^14 - 35*a^6*b^12
+ 35*a^8*b^10 - 21*a^10*b^8 + 7*a^12*b^6 - a^14*b^4)))*(-(a + b)^7*(a - b)^7)^(1/2)*(2*B*b^7 + 2*C*a^7 - A*a^3
*b^4 + 3*B*a^2*b^5 + 8*C*a^3*b^4 - 7*C*a^5*b^2 - 4*A*a*b^6 - 8*C*a*b^6)*1i)/(2*(b^18 - 7*a^2*b^16 + 21*a^4*b^1
4 - 35*a^6*b^12 + 35*a^8*b^10 - 21*a^10*b^8 + 7*a^12*b^6 - a^14*b^4)))/((16*(4*C^3*a^13 - 4*B*C^2*b^13 + 4*B^2
*C*b^13 + 16*C^3*a*b^12 - 2*C^3*a^12*b + 48*C^3*a^2*b^11 - 64*C^3*a^3*b^10 - 64*C^3*a^4*b^9 + 110*C^3*a^5*b^8
+ 66*C^3*a^6*b^7 - 110*C^3*a^7*b^6 - 34*C^3*a^8*b^5 + 70*C^3*a^9*b^4 + 11*C^3*a^10*b^3 - 26*C^3*a^11*b^2 + 8*A
*C^2*a*b^12 - 28*B*C^2*a*b^12 + 56*A*C^2*a^2*b^11 - 22*A*C^2*a^3*b^10 - 26*A*C^2*a^4*b^9 + 18*A*C^2*a^5*b^8 +
22*A*C^2*a^6*b^7 - 2*A*C^2*a^7*b^6 - 2*A*C^2*a^9*b^4 - 2*A*C^2*a^10*b^3 + 16*A^2*C*a^2*b^11 + 8*A^2*C*a^4*b^9
+ A^2*C*a^6*b^7 + 6*B*C^2*a^2*b^11 - 22*B*C^2*a^3*b^10 + 6*B*C^2*a^4*b^9 + 14*B*C^2*a^5*b^8 - 14*B*C^2*a^6*b^7
- 20*B*C^2*a^7*b^6 + 6*B*C^2*a^8*b^5 + 6*B*C^2*a^9*b^4 + 12*B^2*C*a^2*b^11 + 9*B^2*C*a^4*b^9 - 16*A*B*C*a*b^1
2 - 28*A*B*C*a^3*b^10 - 6*A*B*C*a^5*b^8))/(a*b^19 + b^20 - 5*a^2*b^18 - 5*a^3*b^17 + 10*a^4*b^16 + 10*a^5*b^15
- 10*a^6*b^14 - 10*a^7*b^13 + 5*a^8*b^12 + 5*a^9*b^11 - a^10*b^10 - a^11*b^9) - (((8*tan(c/2 + (d*x)/2)*(4*B^
2*b^14 + 8*C^2*a^14 + 4*C^2*b^14 - 8*C^2*a*b^13 - 8*C^2*a^13*b + 16*A^2*a^2*b^12 + 8*A^2*a^4*b^10 + A^2*a^6*b^
8 + 12*B^2*a^2*b^12 + 9*B^2*a^4*b^10 + 44*C^2*a^2*b^12 + 48*C^2*a^3*b^11 - 92*C^2*a^4*b^10 - 120*C^2*a^5*b^9 +
156*C^2*a^6*b^8 + 160*C^2*a^7*b^7 - 164*C^2*a^8*b^6 - 120*C^2*a^9*b^5 + 117*C^2*a^10*b^4 + 48*C^2*a^11*b^3 -
48*C^2*a^12*b^2 - 16*A*B*a*b^13 - 32*B*C*a*b^13 - 28*A*B*a^3*b^11 - 6*A*B*a^5*b^9 + 64*A*C*a^2*b^12 - 48*A*C*a
^4*b^10 + 40*A*C*a^6*b^8 - 2*A*C*a^8*b^6 - 4*A*C*a^10*b^4 - 16*B*C*a^3*b^11 + 20*B*C*a^5*b^9 - 34*B*C*a^7*b^7
+ 12*B*C*a^9*b^5))/(a*b^16 + b^17 - 5*a^2*b^15 - 5*a^3*b^14 + 10*a^4*b^13 + 10*a^5*b^12 - 10*a^6*b^11 - 10*a^7
*b^10 + 5*a^8*b^9 + 5*a^9*b^8 - a^10*b^7 - a^11*b^6) + (((8*(4*B*b^21 + 4*C*b^21 + 8*A*a^2*b^19 + 22*A*a^3*b^1
8 - 22*A*a^4*b^17 - 18*A*a^5*b^16 + 18*A*a^6*b^15 + 2*A*a^7*b^14 - 2*A*a^8*b^13 + 2*A*a^9*b^12 - 2*A*a^10*b^11
- 6*B*a^2*b^19 + 6*B*a^3*b^18 - 6*B*a^4*b^17 + 6*B*a^5*b^16 + 14*B*a^6*b^15 - 14*B*a^7*b^14 - 6*B*a^8*b^13 +
6*B*a^9*b^12 - 12*C*a^2*b^19 + 64*C*a^3*b^18 + 20*C*a^4*b^17 - 110*C*a^5*b^16 - 30*C*a^6*b^15 + 110*C*a^7*b^14
+ 30*C*a^8*b^13 - 70*C*a^9*b^12 - 14*C*a^10*b^11 + 26*C*a^11*b^10 + 2*C*a^12*b^9 - 4*C*a^13*b^8 - 8*A*a*b^20
- 4*B*a*b^20 - 16*C*a*b^20))/(a*b^19 + b^20 - 5*a^2*b^18 - 5*a^3*b^17 + 10*a^4*b^16 + 10*a^5*b^15 - 10*a^6*b^1
4 - 10*a^7*b^13 + 5*a^8*b^12 + 5*a^9*b^11 - a^10*b^10 - a^11*b^9) - (4*tan(c/2 + (d*x)/2)*(-(a + b)^7*(a - b)^
7)^(1/2)*(2*B*b^7 + 2*C*a^7 - A*a^3*b^4 + 3*B*a^2*b^5 + 8*C*a^3*b^4 - 7*C*a^5*b^2 - 4*A*a*b^6 - 8*C*a*b^6)*(8*
a*b^21 - 8*a^2*b^20 - 48*a^3*b^19 + 48*a^4*b^18 + 120*a^5*b^17 - 120*a^6*b^16 - 160*a^7*b^15 + 160*a^8*b^14 +
120*a^9*b^13 - 120*a^10*b^12 - 48*a^11*b^11 + 48*a^12*b^10 + 8*a^13*b^9 - 8*a^14*b^8))/((b^18 - 7*a^2*b^16 + 2
1*a^4*b^14 - 35*a^6*b^12 + 35*a^8*b^10 - 21*a^10*b^8 + 7*a^12*b^6 - a^14*b^4)*(a*b^16 + b^17 - 5*a^2*b^15 - 5*
a^3*b^14 + 10*a^4*b^13 + 10*a^5*b^12 - 10*a^6*b^11 - 10*a^7*b^10 + 5*a^8*b^9 + 5*a^9*b^8 - a^10*b^7 - a^11*b^6
)))*(-(a + b)^7*(a - b)^7)^(1/2)*(2*B*b^7 + 2*C*a^7 - A*a^3*b^4 + 3*B*a^2*b^5 + 8*C*a^3*b^4 - 7*C*a^5*b^2 - 4*
A*a*b^6 - 8*C*a*b^6))/(2*(b^18 - 7*a^2*b^16 + 21*a^4*b^14 - 35*a^6*b^12 + 35*a^8*b^10 - 21*a^10*b^8 + 7*a^12*b
^6 - a^14*b^4)))*(-(a + b)^7*(a - b)^7)^(1/2)*(2*B*b^7 + 2*C*a^7 - A*a^3*b^4 + 3*B*a^2*b^5 + 8*C*a^3*b^4 - 7*C
*a^5*b^2 - 4*A*a*b^6 - 8*C*a*b^6))/(2*(b^18 - 7*a^2*b^16 + 21*a^4*b^14 - 35*a^6*b^12 + 35*a^8*b^10 - 21*a^10*b
^8 + 7*a^12*b^6 - a^14*b^4)) + (((8*tan(c/2 + (d*x)/2)*(4*B^2*b^14 + 8*C^2*a^14 + 4*C^2*b^14 - 8*C^2*a*b^13 -
8*C^2*a^13*b + 16*A^2*a^2*b^12 + 8*A^2*a^4*b^10 + A^2*a^6*b^8 + 12*B^2*a^2*b^12 + 9*B^2*a^4*b^10 + 44*C^2*a^2*
b^12 + 48*C^2*a^3*b^11 - 92*C^2*a^4*b^10 - 120*C^2*a^5*b^9 + 156*C^2*a^6*b^8 + 160*C^2*a^7*b^7 - 164*C^2*a^8*b
^6 - 120*C^2*a^9*b^5 + 117*C^2*a^10*b^4 + 48*C^2*a^11*b^3 - 48*C^2*a^12*b^2 - 16*A*B*a*b^13 - 32*B*C*a*b^13 -
28*A*B*a^3*b^11 - 6*A*B*a^5*b^9 + 64*A*C*a^2*b^12 - 48*A*C*a^4*b^10 + 40*A*C*a^6*b^8 - 2*A*C*a^8*b^6 - 4*A*C*a
^10*b^4 - 16*B*C*a^3*b^11 + 20*B*C*a^5*b^9 - 34*B*C*a^7*b^7 + 12*B*C*a^9*b^5))/(a*b^16 + b^17 - 5*a^2*b^15 - 5
*a^3*b^14 + 10*a^4*b^13 + 10*a^5*b^12 - 10*a^6*b^11 - 10*a^7*b^10 + 5*a^8*b^9 + 5*a^9*b^8 - a^10*b^7 - a^11*b^
6) - (((8*(4*B*b^21 + 4*C*b^21 + 8*A*a^2*b^19 + 22*A*a^3*b^18 - 22*A*a^4*b^17 - 18*A*a^5*b^16 + 18*A*a^6*b^15
+ 2*A*a^7*b^14 - 2*A*a^8*b^13 + 2*A*a^9*b^12 - 2*A*a^10*b^11 - 6*B*a^2*b^19 + 6*B*a^3*b^18 - 6*B*a^4*b^17 + 6*
B*a^5*b^16 + 14*B*a^6*b^15 - 14*B*a^7*b^14 - 6*B*a^8*b^13 + 6*B*a^9*b^12 - 12*C*a^2*b^19 + 64*C*a^3*b^18 + 20*
C*a^4*b^17 - 110*C*a^5*b^16 - 30*C*a^6*b^15 + 110*C*a^7*b^14 + 30*C*a^8*b^13 - 70*C*a^9*b^12 - 14*C*a^10*b^11
+ 26*C*a^11*b^10 + 2*C*a^12*b^9 - 4*C*a^13*b^8 - 8*A*a*b^20 - 4*B*a*b^20 - 16*C*a*b^20))/(a*b^19 + b^20 - 5*a^
2*b^18 - 5*a^3*b^17 + 10*a^4*b^16 + 10*a^5*b^15 - 10*a^6*b^14 - 10*a^7*b^13 + 5*a^8*b^12 + 5*a^9*b^11 - a^10*b
^10 - a^11*b^9) + (4*tan(c/2 + (d*x)/2)*(-(a + b)^7*(a - b)^7)^(1/2)*(2*B*b^7 + 2*C*a^7 - A*a^3*b^4 + 3*B*a^2*
b^5 + 8*C*a^3*b^4 - 7*C*a^5*b^2 - 4*A*a*b^6 - 8*C*a*b^6)*(8*a*b^21 - 8*a^2*b^20 - 48*a^3*b^19 + 48*a^4*b^18 +
120*a^5*b^17 - 120*a^6*b^16 - 160*a^7*b^15 + 160*a^8*b^14 + 120*a^9*b^13 - 120*a^10*b^12 - 48*a^11*b^11 + 48*a
^12*b^10 + 8*a^13*b^9 - 8*a^14*b^8))/((b^18 - 7*a^2*b^16 + 21*a^4*b^14 - 35*a^6*b^12 + 35*a^8*b^10 - 21*a^10*b
^8 + 7*a^12*b^6 - a^14*b^4)*(a*b^16 + b^17 - 5*a^2*b^15 - 5*a^3*b^14 + 10*a^4*b^13 + 10*a^5*b^12 - 10*a^6*b^11
- 10*a^7*b^10 + 5*a^8*b^9 + 5*a^9*b^8 - a^10*b^7 - a^11*b^6)))*(-(a + b)^7*(a - b)^7)^(1/2)*(2*B*b^7 + 2*C*a^
7 - A*a^3*b^4 + 3*B*a^2*b^5 + 8*C*a^3*b^4 - 7*C*a^5*b^2 - 4*A*a*b^6 - 8*C*a*b^6))/(2*(b^18 - 7*a^2*b^16 + 21*a
^4*b^14 - 35*a^6*b^12 + 35*a^8*b^10 - 21*a^10*b^8 + 7*a^12*b^6 - a^14*b^4)))*(-(a + b)^7*(a - b)^7)^(1/2)*(2*B
*b^7 + 2*C*a^7 - A*a^3*b^4 + 3*B*a^2*b^5 + 8*C*a^3*b^4 - 7*C*a^5*b^2 - 4*A*a*b^6 - 8*C*a*b^6))/(2*(b^18 - 7*a^
2*b^16 + 21*a^4*b^14 - 35*a^6*b^12 + 35*a^8*b^10 - 21*a^10*b^8 + 7*a^12*b^6 - a^14*b^4))))*(-(a + b)^7*(a - b)
^7)^(1/2)*(2*B*b^7 + 2*C*a^7 - A*a^3*b^4 + 3*B*a^2*b^5 + 8*C*a^3*b^4 - 7*C*a^5*b^2 - 4*A*a*b^6 - 8*C*a*b^6)*1i
)/(d*(b^18 - 7*a^2*b^16 + 21*a^4*b^14 - 35*a^6*b^12 + 35*a^8*b^10 - 21*a^10*b^8 + 7*a^12*b^6 - a^14*b^4))

________________________________________________________________________________________

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

[Out]

Timed out

________________________________________________________________________________________