\(\int \frac {\cos ^3(c+d x) (A+B \cos (c+d x)+C \cos ^2(c+d x))}{(a+b \cos (c+d x))^4} \, dx\) [1002]

Optimal result

Integrand size = 41, antiderivative size = 461 \[ \int \frac {\cos ^3(c+d x) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{(a+b \cos (c+d x))^4} \, dx=\frac {(b B-4 a C) x}{b^5}-\frac {\left (2 A b^8+2 a^7 b B-7 a^5 b^3 B+8 a^3 b^5 B-8 a b^7 B-8 a^8 C+28 a^6 b^2 C-35 a^4 b^4 C+a^2 b^6 (3 A+20 C)\right ) \arctan \left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{(a-b)^{7/2} b^5 (a+b)^{7/2} d}-\frac {\left (5 A b^4+3 a^3 b B-8 a b^3 B-12 a^4 C+23 a^2 b^2 C-6 b^4 C\right ) \sin (c+d x)}{6 b^4 \left (a^2-b^2\right )^2 d}-\frac {\left (A b^2-a (b B-a C)\right ) \cos ^3(c+d x) \sin (c+d x)}{3 b \left (a^2-b^2\right ) d (a+b \cos (c+d x))^3}+\frac {\left (3 A b^4+a^3 b B-6 a b^3 B-4 a^4 C+a^2 b^2 (2 A+9 C)\right ) \cos ^2(c+d x) \sin (c+d x)}{6 b^2 \left (a^2-b^2\right )^2 d (a+b \cos (c+d x))^2}+\frac {a \left (2 A b^6-a^5 b B+2 a^3 b^3 B-6 a b^5 B+4 a^6 C-11 a^4 b^2 C+3 a^2 b^4 (A+4 C)\right ) \sin (c+d x)}{2 b^4 \left (a^2-b^2\right )^3 d (a+b \cos (c+d x))} \]

[Out]

(B*b-4*C*a)*x/b^5-(2*A*b^8+2*a^7*b*B-7*a^5*b^3*B+8*a^3*b^5*B-8*a*b^7*B-8*a^8*C+28*a^6*b^2*C-35*a^4*b^4*C+a^2*b
^6*(3*A+20*C))*arctan((a-b)^(1/2)*tan(1/2*d*x+1/2*c)/(a+b)^(1/2))/(a-b)^(7/2)/b^5/(a+b)^(7/2)/d-1/6*(5*A*b^4+3
*B*a^3*b-8*B*a*b^3-12*C*a^4+23*C*a^2*b^2-6*C*b^4)*sin(d*x+c)/b^4/(a^2-b^2)^2/d-1/3*(A*b^2-a*(B*b-C*a))*cos(d*x
+c)^3*sin(d*x+c)/b/(a^2-b^2)/d/(a+b*cos(d*x+c))^3+1/6*(3*A*b^4+B*a^3*b-6*B*a*b^3-4*a^4*C+a^2*b^2*(2*A+9*C))*co
s(d*x+c)^2*sin(d*x+c)/b^2/(a^2-b^2)^2/d/(a+b*cos(d*x+c))^2+1/2*a*(2*A*b^6-B*a^5*b+2*B*a^3*b^3-6*B*a*b^5+4*a^6*
C-11*a^4*b^2*C+3*a^2*b^4*(A+4*C))*sin(d*x+c)/b^4/(a^2-b^2)^3/d/(a+b*cos(d*x+c))

Rubi [A] (verified)

Time = 9.85 (sec) , antiderivative size = 461, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.146, Rules used = {3126, 3110, 3102, 2814, 2738, 211} \[ \int \frac {\cos ^3(c+d x) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{(a+b \cos (c+d x))^4} \, dx=-\frac {\sin (c+d x) \cos ^3(c+d x) \left (A b^2-a (b B-a C)\right )}{3 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^3}-\frac {\sin (c+d x) \left (-12 a^4 C+3 a^3 b B+23 a^2 b^2 C-8 a b^3 B+5 A b^4-6 b^4 C\right )}{6 b^4 d \left (a^2-b^2\right )^2}+\frac {\sin (c+d x) \cos ^2(c+d x) \left (-4 a^4 C+a^3 b B+a^2 b^2 (2 A+9 C)-6 a b^3 B+3 A b^4\right )}{6 b^2 d \left (a^2-b^2\right )^2 (a+b \cos (c+d x))^2}+\frac {a \sin (c+d x) \left (4 a^6 C-a^5 b B-11 a^4 b^2 C+2 a^3 b^3 B+3 a^2 b^4 (A+4 C)-6 a b^5 B+2 A b^6\right )}{2 b^4 d \left (a^2-b^2\right )^3 (a+b \cos (c+d x))}-\frac {\left (-8 a^8 C+2 a^7 b B+28 a^6 b^2 C-7 a^5 b^3 B-35 a^4 b^4 C+8 a^3 b^5 B+a^2 b^6 (3 A+20 C)-8 a b^7 B+2 A b^8\right ) \arctan \left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{b^5 d (a-b)^{7/2} (a+b)^{7/2}}+\frac {x (b B-4 a C)}{b^5} \]

[In]

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

[Out]

((b*B - 4*a*C)*x)/b^5 - ((2*A*b^8 + 2*a^7*b*B - 7*a^5*b^3*B + 8*a^3*b^5*B - 8*a*b^7*B - 8*a^8*C + 28*a^6*b^2*C
- 35*a^4*b^4*C + a^2*b^6*(3*A + 20*C))*ArcTan[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/((a - b)^(7/2)*b^5
*(a + b)^(7/2)*d) - ((5*A*b^4 + 3*a^3*b*B - 8*a*b^3*B - 12*a^4*C + 23*a^2*b^2*C - 6*b^4*C)*Sin[c + d*x])/(6*b^
4*(a^2 - b^2)^2*d) - ((A*b^2 - a*(b*B - a*C))*Cos[c + d*x]^3*Sin[c + d*x])/(3*b*(a^2 - b^2)*d*(a + b*Cos[c + d
*x])^3) + ((3*A*b^4 + a^3*b*B - 6*a*b^3*B - 4*a^4*C + a^2*b^2*(2*A + 9*C))*Cos[c + d*x]^2*Sin[c + d*x])/(6*b^2
*(a^2 - b^2)^2*d*(a + b*Cos[c + d*x])^2) + (a*(2*A*b^6 - a^5*b*B + 2*a^3*b^3*B - 6*a*b^5*B + 4*a^6*C - 11*a^4*
b^2*C + 3*a^2*b^4*(A + 4*C))*Sin[c + d*x])/(2*b^4*(a^2 - b^2)^3*d*(a + b*Cos[c + d*x]))

Rule 211

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/Rt[a/b, 2]], x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]

Rule 2738

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 2814

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

Rule 3102

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

Rule 3110

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] &&
NeQ[a^2 - b^2, 0] && LtQ[m, -1]

Rule 3126

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

Rubi steps \begin{align*} \text {integral}& = -\frac {\left (A b^2-a (b B-a C)\right ) \cos ^3(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 ^2(c+d x) \left (3 \left (A b^2-a (b B-a C)\right )+3 b (b B-a (A+C)) \cos (c+d x)-\left (A b^2-a b B+4 a^2 C-3 b^2 C\right ) \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 ^3(c+d x) \sin (c+d x)}{3 b \left (a^2-b^2\right ) d (a+b \cos (c+d x))^3}+\frac {\left (3 A b^4+a^3 b B-6 a b^3 B-4 a^4 C+a^2 b^2 (2 A+9 C)\right ) \cos ^2(c+d x) \sin (c+d x)}{6 b^2 \left (a^2-b^2\right )^2 d (a+b \cos (c+d x))^2}+\frac {\int \frac {\cos (c+d x) \left (2 \left (3 A b^4+a^3 b B-6 a b^3 B-4 a^4 C+a^2 b^2 (2 A+9 C)\right )+2 b \left (2 a^2 b B+3 b^3 B+a^3 C-a b^2 (5 A+6 C)\right ) \cos (c+d x)-\left (5 A b^4+3 a^3 b B-8 a b^3 B-12 a^4 C+23 a^2 b^2 C-6 b^4 C\right ) \cos ^2(c+d x)\right )}{(a+b \cos (c+d x))^2} \, dx}{6 b^2 \left (a^2-b^2\right )^2} \\ & = -\frac {\left (A b^2-a (b B-a C)\right ) \cos ^3(c+d x) \sin (c+d x)}{3 b \left (a^2-b^2\right ) d (a+b \cos (c+d x))^3}+\frac {\left (3 A b^4+a^3 b B-6 a b^3 B-4 a^4 C+a^2 b^2 (2 A+9 C)\right ) \cos ^2(c+d x) \sin (c+d x)}{6 b^2 \left (a^2-b^2\right )^2 d (a+b \cos (c+d x))^2}+\frac {a \left (2 A b^6-a^5 b B+2 a^3 b^3 B-6 a b^5 B+4 a^6 C-11 a^4 b^2 C+3 a^2 b^4 (A+4 C)\right ) \sin (c+d x)}{2 b^4 \left (a^2-b^2\right )^3 d (a+b \cos (c+d x))}+\frac {\int \frac {-3 b \left (2 A b^6-a^5 b B+2 a^3 b^3 B-6 a b^5 B+4 a^6 C-11 a^4 b^2 C+3 a^2 b^4 (A+4 C)\right )+\left (a^2-b^2\right ) \left (3 a^4 b B-4 a^2 b^3 B+6 b^5 B-12 a^5 C+25 a^3 b^2 C-a b^4 (5 A+18 C)\right ) \cos (c+d x)-b \left (a^2-b^2\right ) \left (5 A b^4+3 a^3 b B-8 a b^3 B-12 a^4 C+23 a^2 b^2 C-6 b^4 C\right ) \cos ^2(c+d x)}{a+b \cos (c+d x)} \, dx}{6 b^4 \left (a^2-b^2\right )^3} \\ & = -\frac {\left (5 A b^4+3 a^3 b B-8 a b^3 B-12 a^4 C+23 a^2 b^2 C-6 b^4 C\right ) \sin (c+d x)}{6 b^4 \left (a^2-b^2\right )^2 d}-\frac {\left (A b^2-a (b B-a C)\right ) \cos ^3(c+d x) \sin (c+d x)}{3 b \left (a^2-b^2\right ) d (a+b \cos (c+d x))^3}+\frac {\left (3 A b^4+a^3 b B-6 a b^3 B-4 a^4 C+a^2 b^2 (2 A+9 C)\right ) \cos ^2(c+d x) \sin (c+d x)}{6 b^2 \left (a^2-b^2\right )^2 d (a+b \cos (c+d x))^2}+\frac {a \left (2 A b^6-a^5 b B+2 a^3 b^3 B-6 a b^5 B+4 a^6 C-11 a^4 b^2 C+3 a^2 b^4 (A+4 C)\right ) \sin (c+d x)}{2 b^4 \left (a^2-b^2\right )^3 d (a+b \cos (c+d x))}+\frac {\int \frac {-3 b^2 \left (2 A b^6-a^5 b B+2 a^3 b^3 B-6 a b^5 B+4 a^6 C-11 a^4 b^2 C+3 a^2 b^4 (A+4 C)\right )+6 b \left (a^2-b^2\right )^3 (b B-4 a C) \cos (c+d x)}{a+b \cos (c+d x)} \, dx}{6 b^5 \left (a^2-b^2\right )^3} \\ & = \frac {(b B-4 a C) x}{b^5}-\frac {\left (5 A b^4+3 a^3 b B-8 a b^3 B-12 a^4 C+23 a^2 b^2 C-6 b^4 C\right ) \sin (c+d x)}{6 b^4 \left (a^2-b^2\right )^2 d}-\frac {\left (A b^2-a (b B-a C)\right ) \cos ^3(c+d x) \sin (c+d x)}{3 b \left (a^2-b^2\right ) d (a+b \cos (c+d x))^3}+\frac {\left (3 A b^4+a^3 b B-6 a b^3 B-4 a^4 C+a^2 b^2 (2 A+9 C)\right ) \cos ^2(c+d x) \sin (c+d x)}{6 b^2 \left (a^2-b^2\right )^2 d (a+b \cos (c+d x))^2}+\frac {a \left (2 A b^6-a^5 b B+2 a^3 b^3 B-6 a b^5 B+4 a^6 C-11 a^4 b^2 C+3 a^2 b^4 (A+4 C)\right ) \sin (c+d x)}{2 b^4 \left (a^2-b^2\right )^3 d (a+b \cos (c+d x))}-\frac {\left (2 A b^8+2 a^7 b B-7 a^5 b^3 B+8 a^3 b^5 B-8 a b^7 B-8 a^8 C+28 a^6 b^2 C-35 a^4 b^4 C+a^2 b^6 (3 A+20 C)\right ) \int \frac {1}{a+b \cos (c+d x)} \, dx}{2 b^5 \left (a^2-b^2\right )^3} \\ & = \frac {(b B-4 a C) x}{b^5}-\frac {\left (5 A b^4+3 a^3 b B-8 a b^3 B-12 a^4 C+23 a^2 b^2 C-6 b^4 C\right ) \sin (c+d x)}{6 b^4 \left (a^2-b^2\right )^2 d}-\frac {\left (A b^2-a (b B-a C)\right ) \cos ^3(c+d x) \sin (c+d x)}{3 b \left (a^2-b^2\right ) d (a+b \cos (c+d x))^3}+\frac {\left (3 A b^4+a^3 b B-6 a b^3 B-4 a^4 C+a^2 b^2 (2 A+9 C)\right ) \cos ^2(c+d x) \sin (c+d x)}{6 b^2 \left (a^2-b^2\right )^2 d (a+b \cos (c+d x))^2}+\frac {a \left (2 A b^6-a^5 b B+2 a^3 b^3 B-6 a b^5 B+4 a^6 C-11 a^4 b^2 C+3 a^2 b^4 (A+4 C)\right ) \sin (c+d x)}{2 b^4 \left (a^2-b^2\right )^3 d (a+b \cos (c+d x))}-\frac {\left (2 A b^8+2 a^7 b B-7 a^5 b^3 B+8 a^3 b^5 B-8 a b^7 B-8 a^8 C+28 a^6 b^2 C-35 a^4 b^4 C+a^2 b^6 (3 A+20 C)\right ) \text {Subst}\left (\int \frac {1}{a+b+(a-b) x^2} \, dx,x,\tan \left (\frac {1}{2} (c+d x)\right )\right )}{b^5 \left (a^2-b^2\right )^3 d} \\ & = \frac {(b B-4 a C) x}{b^5}-\frac {\left (3 a^2 A b^6+2 A b^8+2 a^7 b B-7 a^5 b^3 B+8 a^3 b^5 B-8 a b^7 B-8 a^8 C+28 a^6 b^2 C-35 a^4 b^4 C+20 a^2 b^6 C\right ) \arctan \left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{(a-b)^{7/2} b^5 (a+b)^{7/2} d}-\frac {\left (5 A b^4+3 a^3 b B-8 a b^3 B-12 a^4 C+23 a^2 b^2 C-6 b^4 C\right ) \sin (c+d x)}{6 b^4 \left (a^2-b^2\right )^2 d}-\frac {\left (A b^2-a (b B-a C)\right ) \cos ^3(c+d x) \sin (c+d x)}{3 b \left (a^2-b^2\right ) d (a+b \cos (c+d x))^3}+\frac {\left (3 A b^4+a^3 b B-6 a b^3 B-4 a^4 C+a^2 b^2 (2 A+9 C)\right ) \cos ^2(c+d x) \sin (c+d x)}{6 b^2 \left (a^2-b^2\right )^2 d (a+b \cos (c+d x))^2}+\frac {a \left (2 A b^6-a^5 b B+2 a^3 b^3 B-6 a b^5 B+4 a^6 C-11 a^4 b^2 C+3 a^2 b^4 (A+4 C)\right ) \sin (c+d x)}{2 b^4 \left (a^2-b^2\right )^3 d (a+b \cos (c+d x))} \\ \end{align*}

Mathematica [A] (verified)

Time = 10.08 (sec) , antiderivative size = 532, normalized size of antiderivative = 1.15 \[ \int \frac {\cos ^3(c+d x) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{(a+b \cos (c+d x))^4} \, dx=\frac {(b B-4 a C) (c+d x)}{b^5 d}-\frac {\left (-3 a^2 A b^6-2 A b^8-2 a^7 b B+7 a^5 b^3 B-8 a^3 b^5 B+8 a b^7 B+8 a^8 C-28 a^6 b^2 C+35 a^4 b^4 C-20 a^2 b^6 C\right ) \text {arctanh}\left (\frac {(a-b) \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {-a^2+b^2}}\right )}{b^5 \left (a^2-b^2\right )^3 \sqrt {-a^2+b^2} d}+\frac {C \sin (c+d x)}{b^4 d}+\frac {-a^3 A b^2 \sin (c+d x)+a^4 b B \sin (c+d x)-a^5 C \sin (c+d x)}{3 b^4 \left (-a^2+b^2\right ) d (a+b \cos (c+d x))^3}+\frac {-4 a^4 A b^2 \sin (c+d x)+9 a^2 A b^4 \sin (c+d x)+7 a^5 b B \sin (c+d x)-12 a^3 b^3 B \sin (c+d x)-10 a^6 C \sin (c+d x)+15 a^4 b^2 C \sin (c+d x)}{6 b^4 \left (-a^2+b^2\right )^2 d (a+b \cos (c+d x))^2}+\frac {-2 a^5 A b^2 \sin (c+d x)+5 a^3 A b^4 \sin (c+d x)-18 a A b^6 \sin (c+d x)+11 a^6 b B \sin (c+d x)-32 a^4 b^3 B \sin (c+d x)+36 a^2 b^5 B \sin (c+d x)-26 a^7 C \sin (c+d x)+71 a^5 b^2 C \sin (c+d x)-60 a^3 b^4 C \sin (c+d x)}{6 b^4 \left (-a^2+b^2\right )^3 d (a+b \cos (c+d x))} \]

[In]

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

[Out]

((b*B - 4*a*C)*(c + d*x))/(b^5*d) - ((-3*a^2*A*b^6 - 2*A*b^8 - 2*a^7*b*B + 7*a^5*b^3*B - 8*a^3*b^5*B + 8*a*b^7
*B + 8*a^8*C - 28*a^6*b^2*C + 35*a^4*b^4*C - 20*a^2*b^6*C)*ArcTanh[((a - b)*Tan[(c + d*x)/2])/Sqrt[-a^2 + b^2]
])/(b^5*(a^2 - b^2)^3*Sqrt[-a^2 + b^2]*d) + (C*Sin[c + d*x])/(b^4*d) + (-(a^3*A*b^2*Sin[c + d*x]) + a^4*b*B*Si
n[c + d*x] - a^5*C*Sin[c + d*x])/(3*b^4*(-a^2 + b^2)*d*(a + b*Cos[c + d*x])^3) + (-4*a^4*A*b^2*Sin[c + d*x] +
9*a^2*A*b^4*Sin[c + d*x] + 7*a^5*b*B*Sin[c + d*x] - 12*a^3*b^3*B*Sin[c + d*x] - 10*a^6*C*Sin[c + d*x] + 15*a^4
*b^2*C*Sin[c + d*x])/(6*b^4*(-a^2 + b^2)^2*d*(a + b*Cos[c + d*x])^2) + (-2*a^5*A*b^2*Sin[c + d*x] + 5*a^3*A*b^
4*Sin[c + d*x] - 18*a*A*b^6*Sin[c + d*x] + 11*a^6*b*B*Sin[c + d*x] - 32*a^4*b^3*B*Sin[c + d*x] + 36*a^2*b^5*B*
Sin[c + d*x] - 26*a^7*C*Sin[c + d*x] + 71*a^5*b^2*C*Sin[c + d*x] - 60*a^3*b^4*C*Sin[c + d*x])/(6*b^4*(-a^2 + b
^2)^3*d*(a + b*Cos[c + d*x]))

Maple [A] (verified)

Time = 1.27 (sec) , antiderivative size = 640, normalized size of antiderivative = 1.39

method	result	size
derivativedivides	\(\frac {\frac {\frac {2 C b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )}+2 \left (B b -4 C a \right ) \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{b^{5}}-\frac {2 \left (\frac {-\frac {\left (2 a^{2} A \,b^{4}+3 A a \,b^{5}+6 A \,b^{6}-2 B \,a^{5} b +B \,a^{4} b^{2}+6 B \,a^{3} b^{3}-4 B \,a^{2} b^{4}-12 B a \,b^{5}+6 a^{6} C -2 C \,a^{5} b -18 a^{4} b^{2} C +5 C \,a^{3} b^{3}+20 a^{2} C \,b^{4}\right ) a b \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 \left (a -b \right ) \left (a^{3}+3 a^{2} b +3 b^{2} a +b^{3}\right )}-\frac {2 \left (a^{2} A \,b^{4}+9 A \,b^{6}-3 B \,a^{5} b +11 B \,a^{3} b^{3}-18 B a \,b^{5}+9 a^{6} C -29 a^{4} b^{2} C +30 a^{2} C \,b^{4}\right ) a b \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 \left (a^{2}+2 a b +b^{2}\right ) \left (a^{2}-2 a b +b^{2}\right )}-\frac {\left (2 a^{2} A \,b^{4}-3 A a \,b^{5}+6 A \,b^{6}-2 B \,a^{5} b -B \,a^{4} b^{2}+6 B \,a^{3} b^{3}+4 B \,a^{2} b^{4}-12 B a \,b^{5}+6 a^{6} C +2 C \,a^{5} b -18 a^{4} b^{2} C -5 C \,a^{3} b^{3}+20 a^{2} C \,b^{4}\right ) a b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 \left (a +b \right ) \left (a^{3}-3 a^{2} b +3 b^{2} a -b^{3}\right )}}{{\left (\left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a -\left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) b +a +b \right )}^{3}}+\frac {\left (3 a^{2} A \,b^{6}+2 A \,b^{8}+2 a^{7} b B -7 a^{5} b^{3} B +8 a^{3} b^{5} B -8 a \,b^{7} B -8 a^{8} C +28 a^{6} b^{2} C -35 a^{4} b^{4} C +20 C \,a^{2} b^{6}\right ) \arctan \left (\frac {\left (a -b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (a -b \right ) \left (a +b \right )}}\right )}{2 \left (a^{6}-3 a^{4} b^{2}+3 a^{2} b^{4}-b^{6}\right ) \sqrt {\left (a -b \right ) \left (a +b \right )}}\right )}{b^{5}}}{d}\)	\(640\)
default	\(\frac {\frac {\frac {2 C b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )}+2 \left (B b -4 C a \right ) \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{b^{5}}-\frac {2 \left (\frac {-\frac {\left (2 a^{2} A \,b^{4}+3 A a \,b^{5}+6 A \,b^{6}-2 B \,a^{5} b +B \,a^{4} b^{2}+6 B \,a^{3} b^{3}-4 B \,a^{2} b^{4}-12 B a \,b^{5}+6 a^{6} C -2 C \,a^{5} b -18 a^{4} b^{2} C +5 C \,a^{3} b^{3}+20 a^{2} C \,b^{4}\right ) a b \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 \left (a -b \right ) \left (a^{3}+3 a^{2} b +3 b^{2} a +b^{3}\right )}-\frac {2 \left (a^{2} A \,b^{4}+9 A \,b^{6}-3 B \,a^{5} b +11 B \,a^{3} b^{3}-18 B a \,b^{5}+9 a^{6} C -29 a^{4} b^{2} C +30 a^{2} C \,b^{4}\right ) a b \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 \left (a^{2}+2 a b +b^{2}\right ) \left (a^{2}-2 a b +b^{2}\right )}-\frac {\left (2 a^{2} A \,b^{4}-3 A a \,b^{5}+6 A \,b^{6}-2 B \,a^{5} b -B \,a^{4} b^{2}+6 B \,a^{3} b^{3}+4 B \,a^{2} b^{4}-12 B a \,b^{5}+6 a^{6} C +2 C \,a^{5} b -18 a^{4} b^{2} C -5 C \,a^{3} b^{3}+20 a^{2} C \,b^{4}\right ) a b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 \left (a +b \right ) \left (a^{3}-3 a^{2} b +3 b^{2} a -b^{3}\right )}}{{\left (\left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a -\left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) b +a +b \right )}^{3}}+\frac {\left (3 a^{2} A \,b^{6}+2 A \,b^{8}+2 a^{7} b B -7 a^{5} b^{3} B +8 a^{3} b^{5} B -8 a \,b^{7} B -8 a^{8} C +28 a^{6} b^{2} C -35 a^{4} b^{4} C +20 C \,a^{2} b^{6}\right ) \arctan \left (\frac {\left (a -b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (a -b \right ) \left (a +b \right )}}\right )}{2 \left (a^{6}-3 a^{4} b^{2}+3 a^{2} b^{4}-b^{6}\right ) \sqrt {\left (a -b \right ) \left (a +b \right )}}\right )}{b^{5}}}{d}\)	\(640\)
risch	\(\text {Expression too large to display}\)	\(2860\)

[In]

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

[Out]

1/d*(2/b^5*(C*b*tan(1/2*d*x+1/2*c)/(1+tan(1/2*d*x+1/2*c)^2)+(B*b-4*C*a)*arctan(tan(1/2*d*x+1/2*c)))-2/b^5*((-1
/2*(2*A*a^2*b^4+3*A*a*b^5+6*A*b^6-2*B*a^5*b+B*a^4*b^2+6*B*a^3*b^3-4*B*a^2*b^4-12*B*a*b^5+6*C*a^6-2*C*a^5*b-18*
C*a^4*b^2+5*C*a^3*b^3+20*C*a^2*b^4)*a*b/(a-b)/(a^3+3*a^2*b+3*a*b^2+b^3)*tan(1/2*d*x+1/2*c)^5-2/3*(A*a^2*b^4+9*
A*b^6-3*B*a^5*b+11*B*a^3*b^3-18*B*a*b^5+9*C*a^6-29*C*a^4*b^2+30*C*a^2*b^4)*a*b/(a^2+2*a*b+b^2)/(a^2-2*a*b+b^2)
*tan(1/2*d*x+1/2*c)^3-1/2*(2*A*a^2*b^4-3*A*a*b^5+6*A*b^6-2*B*a^5*b-B*a^4*b^2+6*B*a^3*b^3+4*B*a^2*b^4-12*B*a*b^
5+6*C*a^6+2*C*a^5*b-18*C*a^4*b^2-5*C*a^3*b^3+20*C*a^2*b^4)*a*b/(a+b)/(a^3-3*a^2*b+3*a*b^2-b^3)*tan(1/2*d*x+1/2
*c))/(tan(1/2*d*x+1/2*c)^2*a-tan(1/2*d*x+1/2*c)^2*b+a+b)^3+1/2*(3*A*a^2*b^6+2*A*b^8+2*B*a^7*b-7*B*a^5*b^3+8*B*
a^3*b^5-8*B*a*b^7-8*C*a^8+28*C*a^6*b^2-35*C*a^4*b^4+20*C*a^2*b^6)/(a^6-3*a^4*b^2+3*a^2*b^4-b^6)/((a-b)*(a+b))^
(1/2)*arctan((a-b)*tan(1/2*d*x+1/2*c)/((a-b)*(a+b))^(1/2))))

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1354 vs. \(2 (447) = 894\).

Time = 0.56 (sec) , antiderivative size = 2777, normalized size of antiderivative = 6.02 \[ \int \frac {\cos ^3(c+d x) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{(a+b \cos (c+d x))^4} \, dx=\text {Too large to display} \]

[In]

integrate(cos(d*x+c)^3*(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*(4*C*a^9*b^3 - B*a^8*b^4 - 16*C*a^7*b^5 + 4*B*a^6*b^6 + 24*C*a^5*b^7 - 6*B*a^4*b^8 - 16*C*a^3*b^9 +
4*B*a^2*b^10 + 4*C*a*b^11 - B*b^12)*d*x*cos(d*x + c)^3 + 36*(4*C*a^10*b^2 - B*a^9*b^3 - 16*C*a^8*b^4 + 4*B*a^
7*b^5 + 24*C*a^6*b^6 - 6*B*a^5*b^7 - 16*C*a^4*b^8 + 4*B*a^3*b^9 + 4*C*a^2*b^10 - B*a*b^11)*d*x*cos(d*x + c)^2
+ 36*(4*C*a^11*b - B*a^10*b^2 - 16*C*a^9*b^3 + 4*B*a^8*b^4 + 24*C*a^7*b^5 - 6*B*a^6*b^6 - 16*C*a^5*b^7 + 4*B*a
^4*b^8 + 4*C*a^3*b^9 - B*a^2*b^10)*d*x*cos(d*x + c) + 12*(4*C*a^12 - B*a^11*b - 16*C*a^10*b^2 + 4*B*a^9*b^3 +
24*C*a^8*b^4 - 6*B*a^7*b^5 - 16*C*a^6*b^6 + 4*B*a^5*b^7 + 4*C*a^4*b^8 - B*a^3*b^9)*d*x + 3*(8*C*a^11 - 2*B*a^1
0*b - 28*C*a^9*b^2 + 7*B*a^8*b^3 + 35*C*a^7*b^4 - 8*B*a^6*b^5 - (3*A + 20*C)*a^5*b^6 + 8*B*a^4*b^7 - 2*A*a^3*b
^8 + (8*C*a^8*b^3 - 2*B*a^7*b^4 - 28*C*a^6*b^5 + 7*B*a^5*b^6 + 35*C*a^4*b^7 - 8*B*a^3*b^8 - (3*A + 20*C)*a^2*b
^9 + 8*B*a*b^10 - 2*A*b^11)*cos(d*x + c)^3 + 3*(8*C*a^9*b^2 - 2*B*a^8*b^3 - 28*C*a^7*b^4 + 7*B*a^6*b^5 + 35*C*
a^5*b^6 - 8*B*a^4*b^7 - (3*A + 20*C)*a^3*b^8 + 8*B*a^2*b^9 - 2*A*a*b^10)*cos(d*x + c)^2 + 3*(8*C*a^10*b - 2*B*
a^9*b^2 - 28*C*a^8*b^3 + 7*B*a^7*b^4 + 35*C*a^6*b^5 - 8*B*a^5*b^6 - (3*A + 20*C)*a^4*b^7 + 8*B*a^3*b^8 - 2*A*a
^2*b^9)*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*(24
*C*a^11*b - 6*B*a^10*b^2 - 92*C*a^9*b^3 + 23*B*a^8*b^4 + (4*A + 133*C)*a^7*b^5 - 43*B*a^6*b^6 + (7*A - 71*C)*a
^5*b^7 + 26*B*a^4*b^8 - (11*A - 6*C)*a^3*b^9 + 6*(C*a^8*b^4 - 4*C*a^6*b^6 + 6*C*a^4*b^8 - 4*C*a^2*b^10 + C*b^1
2)*cos(d*x + c)^3 + (44*C*a^9*b^3 - 11*B*a^8*b^4 + (2*A - 169*C)*a^7*b^5 + 43*B*a^6*b^6 - (7*A - 239*C)*a^5*b^
7 - 68*B*a^4*b^8 + (23*A - 132*C)*a^3*b^9 + 36*B*a^2*b^10 - 18*(A - C)*a*b^11)*cos(d*x + c)^2 + 3*(20*C*a^10*b
^2 - 5*B*a^9*b^3 - 77*C*a^8*b^4 + 20*B*a^7*b^5 + (A + 110*C)*a^6*b^6 - 35*B*a^5*b^7 + (8*A - 59*C)*a^4*b^8 + 2
0*B*a^3*b^9 - 3*(3*A - 2*C)*a^2*b^10)*cos(d*x + c))*sin(d*x + c))/((a^8*b^8 - 4*a^6*b^10 + 6*a^4*b^12 - 4*a^2*
b^14 + b^16)*d*cos(d*x + c)^3 + 3*(a^9*b^7 - 4*a^7*b^9 + 6*a^5*b^11 - 4*a^3*b^13 + a*b^15)*d*cos(d*x + c)^2 +
3*(a^10*b^6 - 4*a^8*b^8 + 6*a^6*b^10 - 4*a^4*b^12 + a^2*b^14)*d*cos(d*x + c) + (a^11*b^5 - 4*a^9*b^7 + 6*a^7*b
^9 - 4*a^5*b^11 + a^3*b^13)*d), -1/6*(6*(4*C*a^9*b^3 - B*a^8*b^4 - 16*C*a^7*b^5 + 4*B*a^6*b^6 + 24*C*a^5*b^7 -
6*B*a^4*b^8 - 16*C*a^3*b^9 + 4*B*a^2*b^10 + 4*C*a*b^11 - B*b^12)*d*x*cos(d*x + c)^3 + 18*(4*C*a^10*b^2 - B*a^
9*b^3 - 16*C*a^8*b^4 + 4*B*a^7*b^5 + 24*C*a^6*b^6 - 6*B*a^5*b^7 - 16*C*a^4*b^8 + 4*B*a^3*b^9 + 4*C*a^2*b^10 -
B*a*b^11)*d*x*cos(d*x + c)^2 + 18*(4*C*a^11*b - B*a^10*b^2 - 16*C*a^9*b^3 + 4*B*a^8*b^4 + 24*C*a^7*b^5 - 6*B*a
^6*b^6 - 16*C*a^5*b^7 + 4*B*a^4*b^8 + 4*C*a^3*b^9 - B*a^2*b^10)*d*x*cos(d*x + c) + 6*(4*C*a^12 - B*a^11*b - 16
*C*a^10*b^2 + 4*B*a^9*b^3 + 24*C*a^8*b^4 - 6*B*a^7*b^5 - 16*C*a^6*b^6 + 4*B*a^5*b^7 + 4*C*a^4*b^8 - B*a^3*b^9)
*d*x - 3*(8*C*a^11 - 2*B*a^10*b - 28*C*a^9*b^2 + 7*B*a^8*b^3 + 35*C*a^7*b^4 - 8*B*a^6*b^5 - (3*A + 20*C)*a^5*b
^6 + 8*B*a^4*b^7 - 2*A*a^3*b^8 + (8*C*a^8*b^3 - 2*B*a^7*b^4 - 28*C*a^6*b^5 + 7*B*a^5*b^6 + 35*C*a^4*b^7 - 8*B*
a^3*b^8 - (3*A + 20*C)*a^2*b^9 + 8*B*a*b^10 - 2*A*b^11)*cos(d*x + c)^3 + 3*(8*C*a^9*b^2 - 2*B*a^8*b^3 - 28*C*a
^7*b^4 + 7*B*a^6*b^5 + 35*C*a^5*b^6 - 8*B*a^4*b^7 - (3*A + 20*C)*a^3*b^8 + 8*B*a^2*b^9 - 2*A*a*b^10)*cos(d*x +
c)^2 + 3*(8*C*a^10*b - 2*B*a^9*b^2 - 28*C*a^8*b^3 + 7*B*a^7*b^4 + 35*C*a^6*b^5 - 8*B*a^5*b^6 - (3*A + 20*C)*a
^4*b^7 + 8*B*a^3*b^8 - 2*A*a^2*b^9)*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))) - (24*C*a^11*b - 6*B*a^10*b^2 - 92*C*a^9*b^3 + 23*B*a^8*b^4 + (4*A + 133*C)*a^7*b^5 - 43*B*a^
6*b^6 + (7*A - 71*C)*a^5*b^7 + 26*B*a^4*b^8 - (11*A - 6*C)*a^3*b^9 + 6*(C*a^8*b^4 - 4*C*a^6*b^6 + 6*C*a^4*b^8
- 4*C*a^2*b^10 + C*b^12)*cos(d*x + c)^3 + (44*C*a^9*b^3 - 11*B*a^8*b^4 + (2*A - 169*C)*a^7*b^5 + 43*B*a^6*b^6
- (7*A - 239*C)*a^5*b^7 - 68*B*a^4*b^8 + (23*A - 132*C)*a^3*b^9 + 36*B*a^2*b^10 - 18*(A - C)*a*b^11)*cos(d*x +
c)^2 + 3*(20*C*a^10*b^2 - 5*B*a^9*b^3 - 77*C*a^8*b^4 + 20*B*a^7*b^5 + (A + 110*C)*a^6*b^6 - 35*B*a^5*b^7 + (8
*A - 59*C)*a^4*b^8 + 20*B*a^3*b^9 - 3*(3*A - 2*C)*a^2*b^10)*cos(d*x + c))*sin(d*x + c))/((a^8*b^8 - 4*a^6*b^10
+ 6*a^4*b^12 - 4*a^2*b^14 + b^16)*d*cos(d*x + c)^3 + 3*(a^9*b^7 - 4*a^7*b^9 + 6*a^5*b^11 - 4*a^3*b^13 + a*b^1
5)*d*cos(d*x + c)^2 + 3*(a^10*b^6 - 4*a^8*b^8 + 6*a^6*b^10 - 4*a^4*b^12 + a^2*b^14)*d*cos(d*x + c) + (a^11*b^5
- 4*a^9*b^7 + 6*a^7*b^9 - 4*a^5*b^11 + a^3*b^13)*d)]

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [F(-2)]

Exception generated. \[ \int \frac {\cos ^3(c+d x) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{(a+b \cos (c+d x))^4} \, dx=\text {Exception raised: ValueError} \]

[In]

integrate(cos(d*x+c)^3*(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 de

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1225 vs. \(2 (447) = 894\).

Time = 0.43 (sec) , antiderivative size = 1225, normalized size of antiderivative = 2.66 \[ \int \frac {\cos ^3(c+d x) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{(a+b \cos (c+d x))^4} \, dx=\text {Too large to display} \]

[In]

integrate(cos(d*x+c)^3*(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*(8*C*a^8 - 2*B*a^7*b - 28*C*a^6*b^2 + 7*B*a^5*b^3 + 35*C*a^4*b^4 - 8*B*a^3*b^5 - 3*A*a^2*b^6 - 20*C*a^
2*b^6 + 8*B*a*b^7 - 2*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^6*b^5 - 3*a^4*b^7 + 3*a^2*b^9 - b^11)*sqrt(a^2 - b^2)) - (1
8*C*a^9*tan(1/2*d*x + 1/2*c)^5 - 6*B*a^8*b*tan(1/2*d*x + 1/2*c)^5 - 42*C*a^8*b*tan(1/2*d*x + 1/2*c)^5 + 15*B*a
^7*b^2*tan(1/2*d*x + 1/2*c)^5 - 24*C*a^7*b^2*tan(1/2*d*x + 1/2*c)^5 + 6*B*a^6*b^3*tan(1/2*d*x + 1/2*c)^5 + 117
*C*a^6*b^3*tan(1/2*d*x + 1/2*c)^5 + 6*A*a^5*b^4*tan(1/2*d*x + 1/2*c)^5 - 45*B*a^5*b^4*tan(1/2*d*x + 1/2*c)^5 -
24*C*a^5*b^4*tan(1/2*d*x + 1/2*c)^5 - 3*A*a^4*b^5*tan(1/2*d*x + 1/2*c)^5 + 6*B*a^4*b^5*tan(1/2*d*x + 1/2*c)^5
- 105*C*a^4*b^5*tan(1/2*d*x + 1/2*c)^5 + 6*A*a^3*b^6*tan(1/2*d*x + 1/2*c)^5 + 60*B*a^3*b^6*tan(1/2*d*x + 1/2*
c)^5 + 60*C*a^3*b^6*tan(1/2*d*x + 1/2*c)^5 - 27*A*a^2*b^7*tan(1/2*d*x + 1/2*c)^5 - 36*B*a^2*b^7*tan(1/2*d*x +
1/2*c)^5 + 18*A*a*b^8*tan(1/2*d*x + 1/2*c)^5 + 36*C*a^9*tan(1/2*d*x + 1/2*c)^3 - 12*B*a^8*b*tan(1/2*d*x + 1/2*
c)^3 - 152*C*a^7*b^2*tan(1/2*d*x + 1/2*c)^3 + 56*B*a^6*b^3*tan(1/2*d*x + 1/2*c)^3 + 4*A*a^5*b^4*tan(1/2*d*x +
1/2*c)^3 + 236*C*a^5*b^4*tan(1/2*d*x + 1/2*c)^3 - 116*B*a^4*b^5*tan(1/2*d*x + 1/2*c)^3 + 32*A*a^3*b^6*tan(1/2*
d*x + 1/2*c)^3 - 120*C*a^3*b^6*tan(1/2*d*x + 1/2*c)^3 + 72*B*a^2*b^7*tan(1/2*d*x + 1/2*c)^3 - 36*A*a*b^8*tan(1
/2*d*x + 1/2*c)^3 + 18*C*a^9*tan(1/2*d*x + 1/2*c) - 6*B*a^8*b*tan(1/2*d*x + 1/2*c) + 42*C*a^8*b*tan(1/2*d*x +
1/2*c) - 15*B*a^7*b^2*tan(1/2*d*x + 1/2*c) - 24*C*a^7*b^2*tan(1/2*d*x + 1/2*c) + 6*B*a^6*b^3*tan(1/2*d*x + 1/2
*c) - 117*C*a^6*b^3*tan(1/2*d*x + 1/2*c) + 6*A*a^5*b^4*tan(1/2*d*x + 1/2*c) + 45*B*a^5*b^4*tan(1/2*d*x + 1/2*c
) - 24*C*a^5*b^4*tan(1/2*d*x + 1/2*c) + 3*A*a^4*b^5*tan(1/2*d*x + 1/2*c) + 6*B*a^4*b^5*tan(1/2*d*x + 1/2*c) +
105*C*a^4*b^5*tan(1/2*d*x + 1/2*c) + 6*A*a^3*b^6*tan(1/2*d*x + 1/2*c) - 60*B*a^3*b^6*tan(1/2*d*x + 1/2*c) + 60
*C*a^3*b^6*tan(1/2*d*x + 1/2*c) + 27*A*a^2*b^7*tan(1/2*d*x + 1/2*c) - 36*B*a^2*b^7*tan(1/2*d*x + 1/2*c) + 18*A
*a*b^8*tan(1/2*d*x + 1/2*c))/((a^6*b^4 - 3*a^4*b^6 + 3*a^2*b^8 - b^10)*(a*tan(1/2*d*x + 1/2*c)^2 - b*tan(1/2*d
*x + 1/2*c)^2 + a + b)^3) + 3*(4*C*a - B*b)*(d*x + c)/b^5 - 6*C*tan(1/2*d*x + 1/2*c)/((tan(1/2*d*x + 1/2*c)^2
+ 1)*b^4))/d

Mupad [B] (verification not implemented)

Time = 10.69 (sec) , antiderivative size = 9423, normalized size of antiderivative = 20.44 \[ \int \frac {\cos ^3(c+d x) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{(a+b \cos (c+d x))^4} \, dx=\text {Too large to display} \]

[In]

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

[Out]

((tan(c/2 + (d*x)/2)^7*(8*C*a^7 + 2*C*b^7 + 3*A*a^2*b^5 + 2*A*a^3*b^4 - 12*B*a^2*b^5 - 4*B*a^3*b^4 + 6*B*a^4*b
^3 + B*a^5*b^2 - 6*C*a^2*b^5 + 26*C*a^3*b^4 + 11*C*a^4*b^3 - 24*C*a^5*b^2 + 6*A*a*b^6 - 2*B*a^6*b - 2*C*a*b^6
- 4*C*a^6*b))/(b^4*(a + b)^3*(a - b)) + (tan(c/2 + (d*x)/2)*(8*C*a^7 - 2*C*b^7 - 3*A*a^2*b^5 + 2*A*a^3*b^4 - 1
2*B*a^2*b^5 + 4*B*a^3*b^4 + 6*B*a^4*b^3 - B*a^5*b^2 + 6*C*a^2*b^5 + 26*C*a^3*b^4 - 11*C*a^4*b^3 - 24*C*a^5*b^2
+ 6*A*a*b^6 - 2*B*a^6*b - 2*C*a*b^6 + 4*C*a^6*b))/(b^4*(a + b)*(a - b)^3) + (tan(c/2 + (d*x)/2)^3*(72*C*a^8 +
18*C*b^8 + 45*A*a^2*b^6 - 7*A*a^3*b^5 + 10*A*a^4*b^4 + 36*B*a^2*b^6 - 96*B*a^3*b^5 - 14*B*a^4*b^4 + 59*B*a^5*
b^3 + 3*B*a^6*b^2 - 72*C*a^2*b^6 - 60*C*a^3*b^5 + 273*C*a^4*b^4 + 47*C*a^5*b^3 - 236*C*a^6*b^2 - 18*A*a*b^7 -
18*B*a^7*b - 12*C*a^7*b))/(3*b^4*(a + b)^2*(a - b)^3) + (tan(c/2 + (d*x)/2)^5*(72*C*a^8 + 18*C*b^8 + 45*A*a^2*
b^6 + 7*A*a^3*b^5 + 10*A*a^4*b^4 - 36*B*a^2*b^6 - 96*B*a^3*b^5 + 14*B*a^4*b^4 + 59*B*a^5*b^3 - 3*B*a^6*b^2 - 7
2*C*a^2*b^6 + 60*C*a^3*b^5 + 273*C*a^4*b^4 - 47*C*a^5*b^3 - 236*C*a^6*b^2 + 18*A*a*b^7 - 18*B*a^7*b + 12*C*a^7
*b))/(3*b^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*b^2 - 6*a^3) + tan(c/2 + (
d*x)/2)^2*(6*a^2*b + 4*a^3 - 2*b^3) + tan(c/2 + (d*x)/2)^6*(4*a^3 - 6*a^2*b + 2*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))) + (log(tan(c/2 + (d*x)/2) + 1i)*(B*b - 4*C*a)*1i)/(b^5*d) - (log(t
an(c/2 + (d*x)/2) - 1i)*(B*b*1i - C*a*4i))/(b^5*d) - (atan(((((8*tan(c/2 + (d*x)/2)*(4*A^2*b^16 + 4*B^2*b^16 +
128*C^2*a^16 - 8*B^2*a*b^15 - 128*C^2*a^15*b + 12*A^2*a^2*b^14 + 9*A^2*a^4*b^12 + 44*B^2*a^2*b^14 + 48*B^2*a^
3*b^13 - 92*B^2*a^4*b^12 - 120*B^2*a^5*b^11 + 156*B^2*a^6*b^10 + 160*B^2*a^7*b^9 - 164*B^2*a^8*b^8 - 120*B^2*a
^9*b^7 + 117*B^2*a^10*b^6 + 48*B^2*a^11*b^5 - 48*B^2*a^12*b^4 - 8*B^2*a^13*b^3 + 8*B^2*a^14*b^2 + 64*C^2*a^2*b
^14 - 128*C^2*a^3*b^13 + 80*C^2*a^4*b^12 + 768*C^2*a^5*b^11 - 824*C^2*a^6*b^10 - 1920*C^2*a^7*b^9 + 2025*C^2*a
^8*b^8 + 2560*C^2*a^9*b^7 - 2600*C^2*a^10*b^6 - 1920*C^2*a^11*b^5 + 1920*C^2*a^12*b^4 + 768*C^2*a^13*b^3 - 768
*C^2*a^14*b^2 - 32*A*B*a*b^15 - 32*B*C*a*b^15 - 64*B*C*a^15*b - 16*A*B*a^3*b^13 + 20*A*B*a^5*b^11 - 34*A*B*a^7
*b^9 + 12*A*B*a^9*b^7 + 80*A*C*a^2*b^14 - 20*A*C*a^4*b^12 - 98*A*C*a^6*b^10 + 136*A*C*a^8*b^8 - 48*A*C*a^10*b^
6 + 64*B*C*a^2*b^14 - 160*B*C*a^3*b^13 - 384*B*C*a^4*b^12 + 592*B*C*a^5*b^11 + 960*B*C*a^6*b^10 - 1128*B*C*a^7
*b^9 - 1280*B*C*a^8*b^8 + 1306*B*C*a^9*b^7 + 960*B*C*a^10*b^6 - 948*B*C*a^11*b^5 - 384*B*C*a^12*b^4 + 384*B*C*
a^13*b^3 + 64*B*C*a^14*b^2))/(a*b^18 + b^19 - 5*a^2*b^17 - 5*a^3*b^16 + 10*a^4*b^15 + 10*a^5*b^14 - 10*a^6*b^1
3 - 10*a^7*b^12 + 5*a^8*b^11 + 5*a^9*b^10 - a^10*b^9 - a^11*b^8) + (((8*(4*A*b^24 + 4*B*b^24 - 6*A*a^2*b^22 +
6*A*a^3*b^21 - 6*A*a^4*b^20 + 6*A*a^5*b^19 + 14*A*a^6*b^18 - 14*A*a^7*b^17 - 6*A*a^8*b^16 + 6*A*a^9*b^15 - 12*
B*a^2*b^22 + 64*B*a^3*b^21 + 20*B*a^4*b^20 - 110*B*a^5*b^19 - 30*B*a^6*b^18 + 110*B*a^7*b^17 + 30*B*a^8*b^16 -
70*B*a^9*b^15 - 14*B*a^10*b^14 + 26*B*a^11*b^13 + 2*B*a^12*b^12 - 4*B*a^13*b^11 + 40*C*a^2*b^22 + 72*C*a^3*b^
21 - 190*C*a^4*b^20 - 146*C*a^5*b^19 + 386*C*a^6*b^18 + 174*C*a^7*b^17 - 434*C*a^8*b^16 - 126*C*a^9*b^15 + 286
*C*a^10*b^14 + 50*C*a^11*b^13 - 104*C*a^12*b^12 - 8*C*a^13*b^11 + 16*C*a^14*b^10 - 4*A*a*b^23 - 16*B*a*b^23 -
16*C*a*b^23))/(a*b^22 + b^23 - 5*a^2*b^21 - 5*a^3*b^20 + 10*a^4*b^19 + 10*a^5*b^18 - 10*a^6*b^17 - 10*a^7*b^16
+ 5*a^8*b^15 + 5*a^9*b^14 - a^10*b^13 - a^11*b^12) - (4*tan(c/2 + (d*x)/2)*(-(a + b)^7*(a - b)^7)^(1/2)*(2*A*
b^8 - 8*C*a^8 + 3*A*a^2*b^6 + 8*B*a^3*b^5 - 7*B*a^5*b^3 + 20*C*a^2*b^6 - 35*C*a^4*b^4 + 28*C*a^6*b^2 - 8*B*a*b
^7 + 2*B*a^7*b)*(8*a*b^23 - 8*a^2*b^22 - 48*a^3*b^21 + 48*a^4*b^20 + 120*a^5*b^19 - 120*a^6*b^18 - 160*a^7*b^1
7 + 160*a^8*b^16 + 120*a^9*b^15 - 120*a^10*b^14 - 48*a^11*b^13 + 48*a^12*b^12 + 8*a^13*b^11 - 8*a^14*b^10))/((
b^19 - 7*a^2*b^17 + 21*a^4*b^15 - 35*a^6*b^13 + 35*a^8*b^11 - 21*a^10*b^9 + 7*a^12*b^7 - a^14*b^5)*(a*b^18 + b
^19 - 5*a^2*b^17 - 5*a^3*b^16 + 10*a^4*b^15 + 10*a^5*b^14 - 10*a^6*b^13 - 10*a^7*b^12 + 5*a^8*b^11 + 5*a^9*b^1
0 - a^10*b^9 - a^11*b^8)))*(-(a + b)^7*(a - b)^7)^(1/2)*(2*A*b^8 - 8*C*a^8 + 3*A*a^2*b^6 + 8*B*a^3*b^5 - 7*B*a
^5*b^3 + 20*C*a^2*b^6 - 35*C*a^4*b^4 + 28*C*a^6*b^2 - 8*B*a*b^7 + 2*B*a^7*b))/(2*(b^19 - 7*a^2*b^17 + 21*a^4*b
^15 - 35*a^6*b^13 + 35*a^8*b^11 - 21*a^10*b^9 + 7*a^12*b^7 - a^14*b^5)))*(-(a + b)^7*(a - b)^7)^(1/2)*(2*A*b^8
- 8*C*a^8 + 3*A*a^2*b^6 + 8*B*a^3*b^5 - 7*B*a^5*b^3 + 20*C*a^2*b^6 - 35*C*a^4*b^4 + 28*C*a^6*b^2 - 8*B*a*b^7
+ 2*B*a^7*b)*1i)/(2*(b^19 - 7*a^2*b^17 + 21*a^4*b^15 - 35*a^6*b^13 + 35*a^8*b^11 - 21*a^10*b^9 + 7*a^12*b^7 -
a^14*b^5)) + (((8*tan(c/2 + (d*x)/2)*(4*A^2*b^16 + 4*B^2*b^16 + 128*C^2*a^16 - 8*B^2*a*b^15 - 128*C^2*a^15*b +
12*A^2*a^2*b^14 + 9*A^2*a^4*b^12 + 44*B^2*a^2*b^14 + 48*B^2*a^3*b^13 - 92*B^2*a^4*b^12 - 120*B^2*a^5*b^11 + 1
56*B^2*a^6*b^10 + 160*B^2*a^7*b^9 - 164*B^2*a^8*b^8 - 120*B^2*a^9*b^7 + 117*B^2*a^10*b^6 + 48*B^2*a^11*b^5 - 4
8*B^2*a^12*b^4 - 8*B^2*a^13*b^3 + 8*B^2*a^14*b^2 + 64*C^2*a^2*b^14 - 128*C^2*a^3*b^13 + 80*C^2*a^4*b^12 + 768*
C^2*a^5*b^11 - 824*C^2*a^6*b^10 - 1920*C^2*a^7*b^9 + 2025*C^2*a^8*b^8 + 2560*C^2*a^9*b^7 - 2600*C^2*a^10*b^6 -
1920*C^2*a^11*b^5 + 1920*C^2*a^12*b^4 + 768*C^2*a^13*b^3 - 768*C^2*a^14*b^2 - 32*A*B*a*b^15 - 32*B*C*a*b^15 -
64*B*C*a^15*b - 16*A*B*a^3*b^13 + 20*A*B*a^5*b^11 - 34*A*B*a^7*b^9 + 12*A*B*a^9*b^7 + 80*A*C*a^2*b^14 - 20*A*
C*a^4*b^12 - 98*A*C*a^6*b^10 + 136*A*C*a^8*b^8 - 48*A*C*a^10*b^6 + 64*B*C*a^2*b^14 - 160*B*C*a^3*b^13 - 384*B*
C*a^4*b^12 + 592*B*C*a^5*b^11 + 960*B*C*a^6*b^10 - 1128*B*C*a^7*b^9 - 1280*B*C*a^8*b^8 + 1306*B*C*a^9*b^7 + 96
0*B*C*a^10*b^6 - 948*B*C*a^11*b^5 - 384*B*C*a^12*b^4 + 384*B*C*a^13*b^3 + 64*B*C*a^14*b^2))/(a*b^18 + b^19 - 5
*a^2*b^17 - 5*a^3*b^16 + 10*a^4*b^15 + 10*a^5*b^14 - 10*a^6*b^13 - 10*a^7*b^12 + 5*a^8*b^11 + 5*a^9*b^10 - a^1
0*b^9 - a^11*b^8) - (((8*(4*A*b^24 + 4*B*b^24 - 6*A*a^2*b^22 + 6*A*a^3*b^21 - 6*A*a^4*b^20 + 6*A*a^5*b^19 + 14
*A*a^6*b^18 - 14*A*a^7*b^17 - 6*A*a^8*b^16 + 6*A*a^9*b^15 - 12*B*a^2*b^22 + 64*B*a^3*b^21 + 20*B*a^4*b^20 - 11
0*B*a^5*b^19 - 30*B*a^6*b^18 + 110*B*a^7*b^17 + 30*B*a^8*b^16 - 70*B*a^9*b^15 - 14*B*a^10*b^14 + 26*B*a^11*b^1
3 + 2*B*a^12*b^12 - 4*B*a^13*b^11 + 40*C*a^2*b^22 + 72*C*a^3*b^21 - 190*C*a^4*b^20 - 146*C*a^5*b^19 + 386*C*a^
6*b^18 + 174*C*a^7*b^17 - 434*C*a^8*b^16 - 126*C*a^9*b^15 + 286*C*a^10*b^14 + 50*C*a^11*b^13 - 104*C*a^12*b^12
- 8*C*a^13*b^11 + 16*C*a^14*b^10 - 4*A*a*b^23 - 16*B*a*b^23 - 16*C*a*b^23))/(a*b^22 + b^23 - 5*a^2*b^21 - 5*a
^3*b^20 + 10*a^4*b^19 + 10*a^5*b^18 - 10*a^6*b^17 - 10*a^7*b^16 + 5*a^8*b^15 + 5*a^9*b^14 - a^10*b^13 - a^11*b
^12) + (4*tan(c/2 + (d*x)/2)*(-(a + b)^7*(a - b)^7)^(1/2)*(2*A*b^8 - 8*C*a^8 + 3*A*a^2*b^6 + 8*B*a^3*b^5 - 7*B
*a^5*b^3 + 20*C*a^2*b^6 - 35*C*a^4*b^4 + 28*C*a^6*b^2 - 8*B*a*b^7 + 2*B*a^7*b)*(8*a*b^23 - 8*a^2*b^22 - 48*a^3
*b^21 + 48*a^4*b^20 + 120*a^5*b^19 - 120*a^6*b^18 - 160*a^7*b^17 + 160*a^8*b^16 + 120*a^9*b^15 - 120*a^10*b^14
- 48*a^11*b^13 + 48*a^12*b^12 + 8*a^13*b^11 - 8*a^14*b^10))/((b^19 - 7*a^2*b^17 + 21*a^4*b^15 - 35*a^6*b^13 +
35*a^8*b^11 - 21*a^10*b^9 + 7*a^12*b^7 - a^14*b^5)*(a*b^18 + b^19 - 5*a^2*b^17 - 5*a^3*b^16 + 10*a^4*b^15 + 1
0*a^5*b^14 - 10*a^6*b^13 - 10*a^7*b^12 + 5*a^8*b^11 + 5*a^9*b^10 - a^10*b^9 - a^11*b^8)))*(-(a + b)^7*(a - b)^
7)^(1/2)*(2*A*b^8 - 8*C*a^8 + 3*A*a^2*b^6 + 8*B*a^3*b^5 - 7*B*a^5*b^3 + 20*C*a^2*b^6 - 35*C*a^4*b^4 + 28*C*a^6
*b^2 - 8*B*a*b^7 + 2*B*a^7*b))/(2*(b^19 - 7*a^2*b^17 + 21*a^4*b^15 - 35*a^6*b^13 + 35*a^8*b^11 - 21*a^10*b^9 +
7*a^12*b^7 - a^14*b^5)))*(-(a + b)^7*(a - b)^7)^(1/2)*(2*A*b^8 - 8*C*a^8 + 3*A*a^2*b^6 + 8*B*a^3*b^5 - 7*B*a^
5*b^3 + 20*C*a^2*b^6 - 35*C*a^4*b^4 + 28*C*a^6*b^2 - 8*B*a*b^7 + 2*B*a^7*b)*1i)/(2*(b^19 - 7*a^2*b^17 + 21*a^4
*b^15 - 35*a^6*b^13 + 35*a^8*b^11 - 21*a^10*b^9 + 7*a^12*b^7 - a^14*b^5)))/((16*(256*C^3*a^16 + 4*A*B^2*b^16 -
4*A^2*B*b^16 - 16*B^3*a*b^15 - 128*C^3*a^15*b - 48*B^3*a^2*b^14 + 64*B^3*a^3*b^13 + 64*B^3*a^4*b^12 - 110*B^3
*a^5*b^11 - 66*B^3*a^6*b^10 + 110*B^3*a^7*b^9 + 34*B^3*a^8*b^8 - 70*B^3*a^9*b^7 - 11*B^3*a^10*b^6 + 26*B^3*a^1
1*b^5 + 2*B^3*a^12*b^4 - 4*B^3*a^13*b^3 + 640*C^3*a^4*b^12 + 960*C^3*a^5*b^11 - 3040*C^3*a^6*b^10 - 2560*C^3*a
^7*b^9 + 6176*C^3*a^8*b^8 + 3204*C^3*a^9*b^7 - 6944*C^3*a^10*b^6 - 2176*C^3*a^11*b^5 + 4576*C^3*a^12*b^4 + 800
*C^3*a^13*b^3 - 1664*C^3*a^14*b^2 + 28*A*B^2*a*b^15 + 16*A^2*C*a*b^15 - 192*B*C^2*a^15*b - 6*A*B^2*a^2*b^14 +
22*A*B^2*a^3*b^13 - 6*A*B^2*a^4*b^12 - 14*A*B^2*a^5*b^11 + 14*A*B^2*a^6*b^10 + 20*A*B^2*a^7*b^9 - 6*A*B^2*a^8*
b^8 - 6*A*B^2*a^9*b^7 - 12*A^2*B*a^2*b^14 - 9*A^2*B*a^4*b^12 + 64*A*C^2*a^2*b^14 + 256*A*C^2*a^3*b^13 - 96*A*C
^2*a^4*b^12 + 16*A*C^2*a^5*b^11 - 96*A*C^2*a^6*b^10 - 296*A*C^2*a^7*b^9 + 224*A*C^2*a^8*b^8 + 320*A*C^2*a^9*b^
7 - 96*A*C^2*a^10*b^6 - 96*A*C^2*a^11*b^5 + 48*A^2*C*a^3*b^13 + 36*A^2*C*a^5*b^11 - 576*B*C^2*a^3*b^13 - 1104*
B*C^2*a^4*b^12 + 2544*B*C^2*a^5*b^11 + 2376*B*C^2*a^6*b^10 - 4848*B*C^2*a^7*b^9 - 2649*B*C^2*a^8*b^8 + 5232*B*
C^2*a^9*b^7 + 1632*B*C^2*a^10*b^6 - 3408*B*C^2*a^11*b^5 - 576*B*C^2*a^12*b^4 + 1248*B*C^2*a^13*b^3 + 96*B*C^2*
a^14*b^2 + 168*B^2*C*a^2*b^14 + 408*B^2*C*a^3*b^13 - 702*B^2*C*a^4*b^12 - 690*B^2*C*a^5*b^11 + 1266*B^2*C*a^6*
b^10 + 726*B^2*C*a^7*b^9 - 1314*B^2*C*a^8*b^8 - 408*B^2*C*a^9*b^7 + 846*B^2*C*a^10*b^6 + 138*B^2*C*a^11*b^5 -
312*B^2*C*a^12*b^4 - 24*B^2*C*a^13*b^3 + 48*B^2*C*a^14*b^2 - 32*A*B*C*a*b^15 - 176*A*B*C*a^2*b^14 + 48*A*B*C*a
^3*b^13 - 92*A*B*C*a^4*b^12 + 48*A*B*C*a^5*b^11 + 130*A*B*C*a^6*b^10 - 112*A*B*C*a^7*b^9 - 160*A*B*C*a^8*b^8 +
48*A*B*C*a^9*b^7 + 48*A*B*C*a^10*b^6))/(a*b^22 + b^23 - 5*a^2*b^21 - 5*a^3*b^20 + 10*a^4*b^19 + 10*a^5*b^18 -
10*a^6*b^17 - 10*a^7*b^16 + 5*a^8*b^15 + 5*a^9*b^14 - a^10*b^13 - a^11*b^12) + (((8*tan(c/2 + (d*x)/2)*(4*A^2
*b^16 + 4*B^2*b^16 + 128*C^2*a^16 - 8*B^2*a*b^15 - 128*C^2*a^15*b + 12*A^2*a^2*b^14 + 9*A^2*a^4*b^12 + 44*B^2*
a^2*b^14 + 48*B^2*a^3*b^13 - 92*B^2*a^4*b^12 - 120*B^2*a^5*b^11 + 156*B^2*a^6*b^10 + 160*B^2*a^7*b^9 - 164*B^2
*a^8*b^8 - 120*B^2*a^9*b^7 + 117*B^2*a^10*b^6 + 48*B^2*a^11*b^5 - 48*B^2*a^12*b^4 - 8*B^2*a^13*b^3 + 8*B^2*a^1
4*b^2 + 64*C^2*a^2*b^14 - 128*C^2*a^3*b^13 + 80*C^2*a^4*b^12 + 768*C^2*a^5*b^11 - 824*C^2*a^6*b^10 - 1920*C^2*
a^7*b^9 + 2025*C^2*a^8*b^8 + 2560*C^2*a^9*b^7 - 2600*C^2*a^10*b^6 - 1920*C^2*a^11*b^5 + 1920*C^2*a^12*b^4 + 76
8*C^2*a^13*b^3 - 768*C^2*a^14*b^2 - 32*A*B*a*b^15 - 32*B*C*a*b^15 - 64*B*C*a^15*b - 16*A*B*a^3*b^13 + 20*A*B*a
^5*b^11 - 34*A*B*a^7*b^9 + 12*A*B*a^9*b^7 + 80*A*C*a^2*b^14 - 20*A*C*a^4*b^12 - 98*A*C*a^6*b^10 + 136*A*C*a^8*
b^8 - 48*A*C*a^10*b^6 + 64*B*C*a^2*b^14 - 160*B*C*a^3*b^13 - 384*B*C*a^4*b^12 + 592*B*C*a^5*b^11 + 960*B*C*a^6
*b^10 - 1128*B*C*a^7*b^9 - 1280*B*C*a^8*b^8 + 1306*B*C*a^9*b^7 + 960*B*C*a^10*b^6 - 948*B*C*a^11*b^5 - 384*B*C
*a^12*b^4 + 384*B*C*a^13*b^3 + 64*B*C*a^14*b^2))/(a*b^18 + b^19 - 5*a^2*b^17 - 5*a^3*b^16 + 10*a^4*b^15 + 10*a
^5*b^14 - 10*a^6*b^13 - 10*a^7*b^12 + 5*a^8*b^11 + 5*a^9*b^10 - a^10*b^9 - a^11*b^8) + (((8*(4*A*b^24 + 4*B*b^
24 - 6*A*a^2*b^22 + 6*A*a^3*b^21 - 6*A*a^4*b^20 + 6*A*a^5*b^19 + 14*A*a^6*b^18 - 14*A*a^7*b^17 - 6*A*a^8*b^16
+ 6*A*a^9*b^15 - 12*B*a^2*b^22 + 64*B*a^3*b^21 + 20*B*a^4*b^20 - 110*B*a^5*b^19 - 30*B*a^6*b^18 + 110*B*a^7*b^
17 + 30*B*a^8*b^16 - 70*B*a^9*b^15 - 14*B*a^10*b^14 + 26*B*a^11*b^13 + 2*B*a^12*b^12 - 4*B*a^13*b^11 + 40*C*a^
2*b^22 + 72*C*a^3*b^21 - 190*C*a^4*b^20 - 146*C*a^5*b^19 + 386*C*a^6*b^18 + 174*C*a^7*b^17 - 434*C*a^8*b^16 -
126*C*a^9*b^15 + 286*C*a^10*b^14 + 50*C*a^11*b^13 - 104*C*a^12*b^12 - 8*C*a^13*b^11 + 16*C*a^14*b^10 - 4*A*a*b
^23 - 16*B*a*b^23 - 16*C*a*b^23))/(a*b^22 + b^23 - 5*a^2*b^21 - 5*a^3*b^20 + 10*a^4*b^19 + 10*a^5*b^18 - 10*a^
6*b^17 - 10*a^7*b^16 + 5*a^8*b^15 + 5*a^9*b^14 - a^10*b^13 - a^11*b^12) - (4*tan(c/2 + (d*x)/2)*(-(a + b)^7*(a
- b)^7)^(1/2)*(2*A*b^8 - 8*C*a^8 + 3*A*a^2*b^6 + 8*B*a^3*b^5 - 7*B*a^5*b^3 + 20*C*a^2*b^6 - 35*C*a^4*b^4 + 28
*C*a^6*b^2 - 8*B*a*b^7 + 2*B*a^7*b)*(8*a*b^23 - 8*a^2*b^22 - 48*a^3*b^21 + 48*a^4*b^20 + 120*a^5*b^19 - 120*a^
6*b^18 - 160*a^7*b^17 + 160*a^8*b^16 + 120*a^9*b^15 - 120*a^10*b^14 - 48*a^11*b^13 + 48*a^12*b^12 + 8*a^13*b^1
1 - 8*a^14*b^10))/((b^19 - 7*a^2*b^17 + 21*a^4*b^15 - 35*a^6*b^13 + 35*a^8*b^11 - 21*a^10*b^9 + 7*a^12*b^7 - a
^14*b^5)*(a*b^18 + b^19 - 5*a^2*b^17 - 5*a^3*b^16 + 10*a^4*b^15 + 10*a^5*b^14 - 10*a^6*b^13 - 10*a^7*b^12 + 5*
a^8*b^11 + 5*a^9*b^10 - a^10*b^9 - a^11*b^8)))*(-(a + b)^7*(a - b)^7)^(1/2)*(2*A*b^8 - 8*C*a^8 + 3*A*a^2*b^6 +
8*B*a^3*b^5 - 7*B*a^5*b^3 + 20*C*a^2*b^6 - 35*C*a^4*b^4 + 28*C*a^6*b^2 - 8*B*a*b^7 + 2*B*a^7*b))/(2*(b^19 - 7
*a^2*b^17 + 21*a^4*b^15 - 35*a^6*b^13 + 35*a^8*b^11 - 21*a^10*b^9 + 7*a^12*b^7 - a^14*b^5)))*(-(a + b)^7*(a -
b)^7)^(1/2)*(2*A*b^8 - 8*C*a^8 + 3*A*a^2*b^6 + 8*B*a^3*b^5 - 7*B*a^5*b^3 + 20*C*a^2*b^6 - 35*C*a^4*b^4 + 28*C*
a^6*b^2 - 8*B*a*b^7 + 2*B*a^7*b))/(2*(b^19 - 7*a^2*b^17 + 21*a^4*b^15 - 35*a^6*b^13 + 35*a^8*b^11 - 21*a^10*b^
9 + 7*a^12*b^7 - a^14*b^5)) - (((8*tan(c/2 + (d*x)/2)*(4*A^2*b^16 + 4*B^2*b^16 + 128*C^2*a^16 - 8*B^2*a*b^15 -
128*C^2*a^15*b + 12*A^2*a^2*b^14 + 9*A^2*a^4*b^12 + 44*B^2*a^2*b^14 + 48*B^2*a^3*b^13 - 92*B^2*a^4*b^12 - 120
*B^2*a^5*b^11 + 156*B^2*a^6*b^10 + 160*B^2*a^7*b^9 - 164*B^2*a^8*b^8 - 120*B^2*a^9*b^7 + 117*B^2*a^10*b^6 + 48
*B^2*a^11*b^5 - 48*B^2*a^12*b^4 - 8*B^2*a^13*b^3 + 8*B^2*a^14*b^2 + 64*C^2*a^2*b^14 - 128*C^2*a^3*b^13 + 80*C^
2*a^4*b^12 + 768*C^2*a^5*b^11 - 824*C^2*a^6*b^10 - 1920*C^2*a^7*b^9 + 2025*C^2*a^8*b^8 + 2560*C^2*a^9*b^7 - 26
00*C^2*a^10*b^6 - 1920*C^2*a^11*b^5 + 1920*C^2*a^12*b^4 + 768*C^2*a^13*b^3 - 768*C^2*a^14*b^2 - 32*A*B*a*b^15
- 32*B*C*a*b^15 - 64*B*C*a^15*b - 16*A*B*a^3*b^13 + 20*A*B*a^5*b^11 - 34*A*B*a^7*b^9 + 12*A*B*a^9*b^7 + 80*A*C
*a^2*b^14 - 20*A*C*a^4*b^12 - 98*A*C*a^6*b^10 + 136*A*C*a^8*b^8 - 48*A*C*a^10*b^6 + 64*B*C*a^2*b^14 - 160*B*C*
a^3*b^13 - 384*B*C*a^4*b^12 + 592*B*C*a^5*b^11 + 960*B*C*a^6*b^10 - 1128*B*C*a^7*b^9 - 1280*B*C*a^8*b^8 + 1306
*B*C*a^9*b^7 + 960*B*C*a^10*b^6 - 948*B*C*a^11*b^5 - 384*B*C*a^12*b^4 + 384*B*C*a^13*b^3 + 64*B*C*a^14*b^2))/(
a*b^18 + b^19 - 5*a^2*b^17 - 5*a^3*b^16 + 10*a^4*b^15 + 10*a^5*b^14 - 10*a^6*b^13 - 10*a^7*b^12 + 5*a^8*b^11 +
5*a^9*b^10 - a^10*b^9 - a^11*b^8) - (((8*(4*A*b^24 + 4*B*b^24 - 6*A*a^2*b^22 + 6*A*a^3*b^21 - 6*A*a^4*b^20 +
6*A*a^5*b^19 + 14*A*a^6*b^18 - 14*A*a^7*b^17 - 6*A*a^8*b^16 + 6*A*a^9*b^15 - 12*B*a^2*b^22 + 64*B*a^3*b^21 + 2
0*B*a^4*b^20 - 110*B*a^5*b^19 - 30*B*a^6*b^18 + 110*B*a^7*b^17 + 30*B*a^8*b^16 - 70*B*a^9*b^15 - 14*B*a^10*b^1
4 + 26*B*a^11*b^13 + 2*B*a^12*b^12 - 4*B*a^13*b^11 + 40*C*a^2*b^22 + 72*C*a^3*b^21 - 190*C*a^4*b^20 - 146*C*a^
5*b^19 + 386*C*a^6*b^18 + 174*C*a^7*b^17 - 434*C*a^8*b^16 - 126*C*a^9*b^15 + 286*C*a^10*b^14 + 50*C*a^11*b^13
- 104*C*a^12*b^12 - 8*C*a^13*b^11 + 16*C*a^14*b^10 - 4*A*a*b^23 - 16*B*a*b^23 - 16*C*a*b^23))/(a*b^22 + b^23 -
5*a^2*b^21 - 5*a^3*b^20 + 10*a^4*b^19 + 10*a^5*b^18 - 10*a^6*b^17 - 10*a^7*b^16 + 5*a^8*b^15 + 5*a^9*b^14 - a
^10*b^13 - a^11*b^12) + (4*tan(c/2 + (d*x)/2)*(-(a + b)^7*(a - b)^7)^(1/2)*(2*A*b^8 - 8*C*a^8 + 3*A*a^2*b^6 +
8*B*a^3*b^5 - 7*B*a^5*b^3 + 20*C*a^2*b^6 - 35*C*a^4*b^4 + 28*C*a^6*b^2 - 8*B*a*b^7 + 2*B*a^7*b)*(8*a*b^23 - 8*
a^2*b^22 - 48*a^3*b^21 + 48*a^4*b^20 + 120*a^5*b^19 - 120*a^6*b^18 - 160*a^7*b^17 + 160*a^8*b^16 + 120*a^9*b^1
5 - 120*a^10*b^14 - 48*a^11*b^13 + 48*a^12*b^12 + 8*a^13*b^11 - 8*a^14*b^10))/((b^19 - 7*a^2*b^17 + 21*a^4*b^1
5 - 35*a^6*b^13 + 35*a^8*b^11 - 21*a^10*b^9 + 7*a^12*b^7 - a^14*b^5)*(a*b^18 + b^19 - 5*a^2*b^17 - 5*a^3*b^16
+ 10*a^4*b^15 + 10*a^5*b^14 - 10*a^6*b^13 - 10*a^7*b^12 + 5*a^8*b^11 + 5*a^9*b^10 - a^10*b^9 - a^11*b^8)))*(-(
a + b)^7*(a - b)^7)^(1/2)*(2*A*b^8 - 8*C*a^8 + 3*A*a^2*b^6 + 8*B*a^3*b^5 - 7*B*a^5*b^3 + 20*C*a^2*b^6 - 35*C*a
^4*b^4 + 28*C*a^6*b^2 - 8*B*a*b^7 + 2*B*a^7*b))/(2*(b^19 - 7*a^2*b^17 + 21*a^4*b^15 - 35*a^6*b^13 + 35*a^8*b^1
1 - 21*a^10*b^9 + 7*a^12*b^7 - a^14*b^5)))*(-(a + b)^7*(a - b)^7)^(1/2)*(2*A*b^8 - 8*C*a^8 + 3*A*a^2*b^6 + 8*B
*a^3*b^5 - 7*B*a^5*b^3 + 20*C*a^2*b^6 - 35*C*a^4*b^4 + 28*C*a^6*b^2 - 8*B*a*b^7 + 2*B*a^7*b))/(2*(b^19 - 7*a^2
*b^17 + 21*a^4*b^15 - 35*a^6*b^13 + 35*a^8*b^11 - 21*a^10*b^9 + 7*a^12*b^7 - a^14*b^5))))*(-(a + b)^7*(a - b)^
7)^(1/2)*(2*A*b^8 - 8*C*a^8 + 3*A*a^2*b^6 + 8*B*a^3*b^5 - 7*B*a^5*b^3 + 20*C*a^2*b^6 - 35*C*a^4*b^4 + 28*C*a^6
*b^2 - 8*B*a*b^7 + 2*B*a^7*b)*1i)/(d*(b^19 - 7*a^2*b^17 + 21*a^4*b^15 - 35*a^6*b^13 + 35*a^8*b^11 - 21*a^10*b^
9 + 7*a^12*b^7 - a^14*b^5))