∫ cos3 (c+dx)(A+B cos(c+dx)+C cos2(c+dx))______________________________

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

Optimal. Leaf size=398 \[ -\frac {\sin (c+d x) \cos ^3(c+d x) \left (A b^2-a (b B-a C)\right )}{b d \left (a^2-b^2\right ) (a+b \cos (c+d x))}+\frac {\sin (c+d x) \cos ^2(c+d x) \left (4 a^2 C-3 a b B+3 A b^2-b^2 C\right )}{3 b^2 d \left (a^2-b^2\right )}+\frac {\sin (c+d x) \cos (c+d x) \left (-4 a^3 C+3 a^2 b B-2 a b^2 (A-C)-b^3 B\right )}{2 b^3 d \left (a^2-b^2\right )}+\frac {x \left (-8 a^3 C+6 a^2 b B-2 a b^2 (2 A+C)+b^3 B\right )}{2 b^5}-\frac {\sin (c+d x) \left (-12 a^4 C+9 a^3 b B-a^2 b^2 (6 A-7 C)-6 a b^3 B+b^4 (3 A+2 C)\right )}{3 b^4 d \left (a^2-b^2\right )}+\frac {2 a^2 \left (4 a^4 C-3 a^3 b B+2 a^2 A b^2-5 a^2 b^2 C+4 a b^3 B-3 A b^4\right ) \tan ^{-1}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{b^5 d (a-b)^{3/2} (a+b)^{3/2}} \]

[Out]

1/2*(6*a^2*b*B+b^3*B-8*a^3*C-2*a*b^2*(2*A+C))*x/b^5+2*a^2*(2*A*a^2*b^2-3*A*b^4-3*B*a^3*b+4*B*a*b^3+4*C*a^4-5*C

*a^2*b^2)*arctan((a-b)^(1/2)*tan(1/2*d*x+1/2*c)/(a+b)^(1/2))/(a-b)^(3/2)/b^5/(a+b)^(3/2)/d-1/3*(9*a^3*b*B-6*a*

b^3*B-a^2*b^2*(6*A-7*C)-12*a^4*C+b^4*(3*A+2*C))*sin(d*x+c)/b^4/(a^2-b^2)/d+1/2*(3*a^2*b*B-b^3*B-2*a*b^2*(A-C)-

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

/(a^2-b^2)/d-(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))

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Rubi [A] time = 1.60, antiderivative size = 398, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 41, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.146, Rules used = {3047, 3049, 3023, 2735, 2659, 205} \[ -\frac {\sin (c+d x) \left (-a^2 b^2 (6 A-7 C)+9 a^3 b B-12 a^4 C-6 a b^3 B+b^4 (3 A+2 C)\right )}{3 b^4 d \left (a^2-b^2\right )}+\frac {2 a^2 \left (2 a^2 A b^2-5 a^2 b^2 C-3 a^3 b B+4 a^4 C+4 a b^3 B-3 A b^4\right ) \tan ^{-1}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{b^5 d (a-b)^{3/2} (a+b)^{3/2}}-\frac {\sin (c+d x) \cos ^3(c+d x) \left (A b^2-a (b B-a C)\right )}{b d \left (a^2-b^2\right ) (a+b \cos (c+d x))}+\frac {\sin (c+d x) \cos ^2(c+d x) \left (4 a^2 C-3 a b B+3 A b^2-b^2 C\right )}{3 b^2 d \left (a^2-b^2\right )}+\frac {\sin (c+d x) \cos (c+d x) \left (3 a^2 b B-4 a^3 C-2 a b^2 (A-C)-b^3 B\right )}{2 b^3 d \left (a^2-b^2\right )}+\frac {x \left (6 a^2 b B-8 a^3 C-2 a b^2 (2 A+C)+b^3 B\right )}{2 b^5} \]

Antiderivative was successfully veriﬁed.

[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])^2,x]

[Out]

((6*a^2*b*B + b^3*B - 8*a^3*C - 2*a*b^2*(2*A + C))*x)/(2*b^5) + (2*a^2*(2*a^2*A*b^2 - 3*A*b^4 - 3*a^3*b*B + 4*

a*b^3*B + 4*a^4*C - 5*a^2*b^2*C)*ArcTan[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/((a - b)^(3/2)*b^5*(a + b

)^(3/2)*d) - ((9*a^3*b*B - 6*a*b^3*B - a^2*b^2*(6*A - 7*C) - 12*a^4*C + b^4*(3*A + 2*C))*Sin[c + d*x])/(3*b^4*

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

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

*(b*B - a*C))*Cos[c + d*x]^3*Sin[c + d*x])/(b*(a^2 - b^2)*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 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 3023

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 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]

Rule 3049

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_.)*((A_.) + (B_.)

*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> -Simp[(C*Cos[e + f*x]*(a + b*Sin[e +

f*x])^m*(c + d*Sin[e + f*x])^(n + 1))/(d*f*(m + n + 2)), x] + Dist[1/(d*(m + n + 2)), Int[(a + b*Sin[e + f*x]

)^(m - 1)*(c + d*Sin[e + f*x])^n*Simp[a*A*d*(m + n + 2) + C*(b*c*m + a*d*(n + 1)) + (d*(A*b + a*B)*(m + n + 2)

- C*(a*c - b*d*(m + n + 1)))*Sin[e + f*x] + (C*(a*d*m - b*c*(m + 1)) + b*B*d*(m + n + 2))*Sin[e + f*x]^2, x],

x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2

, 0] && GtQ[m, 0] && !(IGtQ[n, 0] && ( !IntegerQ[m] || (EqQ[a, 0] && NeQ[c, 0])))

Rubi steps

\begin {align*} \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))^2} \, dx &=-\frac {\left (A b^2-a (b B-a C)\right ) \cos ^3(c+d x) \sin (c+d x)}{b \left (a^2-b^2\right ) d (a+b \cos (c+d x))}-\frac {\int \frac {\cos ^2(c+d x) \left (3 \left (A b^2-a (b B-a C)\right )+b (b B-a (A+C)) \cos (c+d x)-\left (3 A b^2-3 a b B+4 a^2 C-b^2 C\right ) \cos ^2(c+d x)\right )}{a+b \cos (c+d x)} \, dx}{b \left (a^2-b^2\right )}\\ &=\frac {\left (3 A b^2-3 a b B+4 a^2 C-b^2 C\right ) \cos ^2(c+d x) \sin (c+d x)}{3 b^2 \left (a^2-b^2\right ) d}-\frac {\left (A b^2-a (b B-a C)\right ) \cos ^3(c+d x) \sin (c+d x)}{b \left (a^2-b^2\right ) d (a+b \cos (c+d x))}-\frac {\int \frac {\cos (c+d x) \left (-2 a \left (3 A b^2-3 a b B+4 a^2 C-b^2 C\right )+b \left (3 A b^2-3 a b B+a^2 C+2 b^2 C\right ) \cos (c+d x)-3 \left (3 a^2 b B-b^3 B-2 a b^2 (A-C)-4 a^3 C\right ) \cos ^2(c+d x)\right )}{a+b \cos (c+d x)} \, dx}{3 b^2 \left (a^2-b^2\right )}\\ &=\frac {\left (3 a^2 b B-b^3 B-2 a b^2 (A-C)-4 a^3 C\right ) \cos (c+d x) \sin (c+d x)}{2 b^3 \left (a^2-b^2\right ) d}+\frac {\left (3 A b^2-3 a b B+4 a^2 C-b^2 C\right ) \cos ^2(c+d x) \sin (c+d x)}{3 b^2 \left (a^2-b^2\right ) d}-\frac {\left (A b^2-a (b B-a C)\right ) \cos ^3(c+d x) \sin (c+d x)}{b \left (a^2-b^2\right ) d (a+b \cos (c+d x))}-\frac {\int \frac {-3 a \left (3 a^2 b B-b^3 B-2 a b^2 (A-C)-4 a^3 C\right )+b \left (3 a^2 b B+3 b^3 B-4 a^3 C-2 a b^2 (3 A+C)\right ) \cos (c+d x)+2 \left (9 a^3 b B-6 a b^3 B-a^2 b^2 (6 A-7 C)-12 a^4 C+b^4 (3 A+2 C)\right ) \cos ^2(c+d x)}{a+b \cos (c+d x)} \, dx}{6 b^3 \left (a^2-b^2\right )}\\ &=-\frac {\left (9 a^3 b B-6 a b^3 B-a^2 b^2 (6 A-7 C)-12 a^4 C+b^4 (3 A+2 C)\right ) \sin (c+d x)}{3 b^4 \left (a^2-b^2\right ) d}+\frac {\left (3 a^2 b B-b^3 B-2 a b^2 (A-C)-4 a^3 C\right ) \cos (c+d x) \sin (c+d x)}{2 b^3 \left (a^2-b^2\right ) d}+\frac {\left (3 A b^2-3 a b B+4 a^2 C-b^2 C\right ) \cos ^2(c+d x) \sin (c+d x)}{3 b^2 \left (a^2-b^2\right ) d}-\frac {\left (A b^2-a (b B-a C)\right ) \cos ^3(c+d x) \sin (c+d x)}{b \left (a^2-b^2\right ) d (a+b \cos (c+d x))}-\frac {\int \frac {-3 a b \left (3 a^2 b B-b^3 B-2 a b^2 (A-C)-4 a^3 C\right )-3 \left (a^2-b^2\right ) \left (6 a^2 b B+b^3 B-8 a^3 C-2 a b^2 (2 A+C)\right ) \cos (c+d x)}{a+b \cos (c+d x)} \, dx}{6 b^4 \left (a^2-b^2\right )}\\ &=\frac {\left (6 a^2 b B+b^3 B-8 a^3 C-2 a b^2 (2 A+C)\right ) x}{2 b^5}-\frac {\left (9 a^3 b B-6 a b^3 B-a^2 b^2 (6 A-7 C)-12 a^4 C+b^4 (3 A+2 C)\right ) \sin (c+d x)}{3 b^4 \left (a^2-b^2\right ) d}+\frac {\left (3 a^2 b B-b^3 B-2 a b^2 (A-C)-4 a^3 C\right ) \cos (c+d x) \sin (c+d x)}{2 b^3 \left (a^2-b^2\right ) d}+\frac {\left (3 A b^2-3 a b B+4 a^2 C-b^2 C\right ) \cos ^2(c+d x) \sin (c+d x)}{3 b^2 \left (a^2-b^2\right ) d}-\frac {\left (A b^2-a (b B-a C)\right ) \cos ^3(c+d x) \sin (c+d x)}{b \left (a^2-b^2\right ) d (a+b \cos (c+d x))}-\frac {\left (a^2 \left (3 A b^4+3 a^3 b B-4 a b^3 B-a^2 b^2 (2 A-5 C)-4 a^4 C\right )\right ) \int \frac {1}{a+b \cos (c+d x)} \, dx}{b^5 \left (a^2-b^2\right )}\\ &=\frac {\left (6 a^2 b B+b^3 B-8 a^3 C-2 a b^2 (2 A+C)\right ) x}{2 b^5}-\frac {\left (9 a^3 b B-6 a b^3 B-a^2 b^2 (6 A-7 C)-12 a^4 C+b^4 (3 A+2 C)\right ) \sin (c+d x)}{3 b^4 \left (a^2-b^2\right ) d}+\frac {\left (3 a^2 b B-b^3 B-2 a b^2 (A-C)-4 a^3 C\right ) \cos (c+d x) \sin (c+d x)}{2 b^3 \left (a^2-b^2\right ) d}+\frac {\left (3 A b^2-3 a b B+4 a^2 C-b^2 C\right ) \cos ^2(c+d x) \sin (c+d x)}{3 b^2 \left (a^2-b^2\right ) d}-\frac {\left (A b^2-a (b B-a C)\right ) \cos ^3(c+d x) \sin (c+d x)}{b \left (a^2-b^2\right ) d (a+b \cos (c+d x))}-\frac {\left (2 a^2 \left (3 A b^4+3 a^3 b B-4 a b^3 B-a^2 b^2 (2 A-5 C)-4 a^4 C\right )\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^5 \left (a^2-b^2\right ) d}\\ &=\frac {\left (6 a^2 b B+b^3 B-8 a^3 C-2 a b^2 (2 A+C)\right ) x}{2 b^5}+\frac {2 a^2 \left (2 a^2 A b^2-3 A b^4-3 a^3 b B+4 a b^3 B+4 a^4 C-5 a^2 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-b)^{3/2} b^5 (a+b)^{3/2} d}-\frac {\left (9 a^3 b B-6 a b^3 B-a^2 b^2 (6 A-7 C)-12 a^4 C+b^4 (3 A+2 C)\right ) \sin (c+d x)}{3 b^4 \left (a^2-b^2\right ) d}+\frac {\left (3 a^2 b B-b^3 B-2 a b^2 (A-C)-4 a^3 C\right ) \cos (c+d x) \sin (c+d x)}{2 b^3 \left (a^2-b^2\right ) d}+\frac {\left (3 A b^2-3 a b B+4 a^2 C-b^2 C\right ) \cos ^2(c+d x) \sin (c+d x)}{3 b^2 \left (a^2-b^2\right ) d}-\frac {\left (A b^2-a (b B-a C)\right ) \cos ^3(c+d x) \sin (c+d x)}{b \left (a^2-b^2\right ) d (a+b \cos (c+d x))}\\ \end {align*}

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Mathematica [A] time = 1.85, size = 256, normalized size = 0.64 \[ \frac {\frac {12 a^3 b \sin (c+d x) \left (a (a C-b B)+A b^2\right )}{(a-b) (a+b) (a+b \cos (c+d x))}+3 b \sin (c+d x) \left (12 a^2 C-8 a b B+4 A b^2+3 b^2 C\right )+6 (c+d x) \left (-8 a^3 C+6 a^2 b B-2 a b^2 (2 A+C)+b^3 B\right )+\frac {24 a^2 \left (4 a^4 C-3 a^3 b B+a^2 b^2 (2 A-5 C)+4 a b^3 B-3 A b^4\right ) \tanh ^{-1}\left (\frac {(a-b) \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {b^2-a^2}}\right )}{\left (b^2-a^2\right )^{3/2}}+3 b^2 (b B-2 a C) \sin (2 (c+d x))+b^3 C \sin (3 (c+d x))}{12 b^5 d} \]

Antiderivative was successfully veriﬁed.

[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])^2,x]

[Out]

(6*(6*a^2*b*B + b^3*B - 8*a^3*C - 2*a*b^2*(2*A + C))*(c + d*x) + (24*a^2*(-3*A*b^4 - 3*a^3*b*B + 4*a*b^3*B + a

^2*b^2*(2*A - 5*C) + 4*a^4*C)*ArcTanh[((a - b)*Tan[(c + d*x)/2])/Sqrt[-a^2 + b^2]])/(-a^2 + b^2)^(3/2) + 3*b*(

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

- b)*(a + b)*(a + b*Cos[c + d*x])) + 3*b^2*(b*B - 2*a*C)*Sin[2*(c + d*x)] + b^3*C*Sin[3*(c + d*x)])/(12*b^5*d)

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fricas [A] time = 0.70, size = 1357, normalized size = 3.41 \[ \text {result too large to display} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[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))^2,x, algorithm="fricas")

[Out]

[-1/6*(3*(8*C*a^7*b - 6*B*a^6*b^2 + 2*(2*A - 7*C)*a^5*b^3 + 11*B*a^4*b^4 - 4*(2*A - C)*a^3*b^5 - 4*B*a^2*b^6 +

2*(2*A + C)*a*b^7 - B*b^8)*d*x*cos(d*x + c) + 3*(8*C*a^8 - 6*B*a^7*b + 2*(2*A - 7*C)*a^6*b^2 + 11*B*a^5*b^3 -

4*(2*A - C)*a^4*b^4 - 4*B*a^3*b^5 + 2*(2*A + C)*a^2*b^6 - B*a*b^7)*d*x + 3*(4*C*a^7 - 3*B*a^6*b + (2*A - 5*C)

*a^5*b^2 + 4*B*a^4*b^3 - 3*A*a^3*b^4 + (4*C*a^6*b - 3*B*a^5*b^2 + (2*A - 5*C)*a^4*b^3 + 4*B*a^3*b^4 - 3*A*a^2*

b^5)*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)) - (24*C*a^

7*b - 18*B*a^6*b^2 + 2*(6*A - 19*C)*a^5*b^3 + 30*B*a^4*b^4 - 2*(9*A - 5*C)*a^3*b^5 - 12*B*a^2*b^6 + 2*(3*A + 2

*C)*a*b^7 + 2*(C*a^4*b^4 - 2*C*a^2*b^6 + C*b^8)*cos(d*x + c)^3 - (4*C*a^5*b^3 - 3*B*a^4*b^4 - 8*C*a^3*b^5 + 6*

B*a^2*b^6 + 4*C*a*b^7 - 3*B*b^8)*cos(d*x + c)^2 + (12*C*a^6*b^2 - 9*B*a^5*b^3 + 2*(3*A - 10*C)*a^4*b^4 + 18*B*

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

^8 + b^10)*d*cos(d*x + c) + (a^5*b^5 - 2*a^3*b^7 + a*b^9)*d), -1/6*(3*(8*C*a^7*b - 6*B*a^6*b^2 + 2*(2*A - 7*C)

*a^5*b^3 + 11*B*a^4*b^4 - 4*(2*A - C)*a^3*b^5 - 4*B*a^2*b^6 + 2*(2*A + C)*a*b^7 - B*b^8)*d*x*cos(d*x + c) + 3*

(8*C*a^8 - 6*B*a^7*b + 2*(2*A - 7*C)*a^6*b^2 + 11*B*a^5*b^3 - 4*(2*A - C)*a^4*b^4 - 4*B*a^3*b^5 + 2*(2*A + C)*

a^2*b^6 - B*a*b^7)*d*x - 6*(4*C*a^7 - 3*B*a^6*b + (2*A - 5*C)*a^5*b^2 + 4*B*a^4*b^3 - 3*A*a^3*b^4 + (4*C*a^6*b

- 3*B*a^5*b^2 + (2*A - 5*C)*a^4*b^3 + 4*B*a^3*b^4 - 3*A*a^2*b^5)*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^7*b - 18*B*a^6*b^2 + 2*(6*A - 19*C)*a^5*b^3 + 30*B*a^

4*b^4 - 2*(9*A - 5*C)*a^3*b^5 - 12*B*a^2*b^6 + 2*(3*A + 2*C)*a*b^7 + 2*(C*a^4*b^4 - 2*C*a^2*b^6 + C*b^8)*cos(d

*x + c)^3 - (4*C*a^5*b^3 - 3*B*a^4*b^4 - 8*C*a^3*b^5 + 6*B*a^2*b^6 + 4*C*a*b^7 - 3*B*b^8)*cos(d*x + c)^2 + (12

*C*a^6*b^2 - 9*B*a^5*b^3 + 2*(3*A - 10*C)*a^4*b^4 + 18*B*a^3*b^5 - 4*(3*A - C)*a^2*b^6 - 9*B*a*b^7 + 2*(3*A +

2*C)*b^8)*cos(d*x + c))*sin(d*x + c))/((a^4*b^6 - 2*a^2*b^8 + b^10)*d*cos(d*x + c) + (a^5*b^5 - 2*a^3*b^7 + a*

b^9)*d)]

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giac [A] time = 0.25, size = 563, normalized size = 1.41 \[ -\frac {\frac {12 \, {\left (4 \, C a^{6} - 3 \, B a^{5} b + 2 \, A a^{4} b^{2} - 5 \, C a^{4} b^{2} + 4 \, B a^{3} b^{3} - 3 \, A a^{2} b^{4}\right )} {\left (\pi \left \lfloor \frac {d x + c}{2 \, \pi } + \frac {1}{2} \right \rfloor \mathrm {sgn}\left (-2 \, a + 2 \, b\right ) + \arctan \left (-\frac {a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{\sqrt {a^{2} - b^{2}}}\right )\right )}}{{\left (a^{2} b^{5} - b^{7}\right )} \sqrt {a^{2} - b^{2}}} - \frac {12 \, {\left (C a^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - B a^{4} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + A a^{3} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (a^{2} b^{4} - b^{6}\right )} {\left (a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a + b\right )}} + \frac {3 \, {\left (8 \, C a^{3} - 6 \, B a^{2} b + 4 \, A a b^{2} + 2 \, C a b^{2} - B b^{3}\right )} {\left (d x + c\right )}}{b^{5}} - \frac {2 \, {\left (18 \, C a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 12 \, B a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 6 \, C a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 6 \, A b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 3 \, B b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 6 \, C b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 36 \, C a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 24 \, B a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 12 \, A b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 4 \, C b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 18 \, C a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 12 \, B a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 6 \, C a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 6 \, A b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 3 \, B b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 6 \, C b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{3} b^{4}}}{6 \, d} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[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))^2,x, algorithm="giac")

[Out]

-1/6*(12*(4*C*a^6 - 3*B*a^5*b + 2*A*a^4*b^2 - 5*C*a^4*b^2 + 4*B*a^3*b^3 - 3*A*a^2*b^4)*(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^2

*b^5 - b^7)*sqrt(a^2 - b^2)) - 12*(C*a^5*tan(1/2*d*x + 1/2*c) - B*a^4*b*tan(1/2*d*x + 1/2*c) + A*a^3*b^2*tan(1

/2*d*x + 1/2*c))/((a^2*b^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*(8*C*a^3

- 6*B*a^2*b + 4*A*a*b^2 + 2*C*a*b^2 - B*b^3)*(d*x + c)/b^5 - 2*(18*C*a^2*tan(1/2*d*x + 1/2*c)^5 - 12*B*a*b*tan

(1/2*d*x + 1/2*c)^5 + 6*C*a*b*tan(1/2*d*x + 1/2*c)^5 + 6*A*b^2*tan(1/2*d*x + 1/2*c)^5 - 3*B*b^2*tan(1/2*d*x +

1/2*c)^5 + 6*C*b^2*tan(1/2*d*x + 1/2*c)^5 + 36*C*a^2*tan(1/2*d*x + 1/2*c)^3 - 24*B*a*b*tan(1/2*d*x + 1/2*c)^3

+ 12*A*b^2*tan(1/2*d*x + 1/2*c)^3 + 4*C*b^2*tan(1/2*d*x + 1/2*c)^3 + 18*C*a^2*tan(1/2*d*x + 1/2*c) - 12*B*a*b*

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

*c) + 6*C*b^2*tan(1/2*d*x + 1/2*c))/((tan(1/2*d*x + 1/2*c)^2 + 1)^3*b^4))/d

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maple [B] time = 0.13, size = 1229, normalized size = 3.09 \[ \text {result too large to display} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[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))^2,x)

[Out]

-6/d*a^2/b/(a-b)/(a+b)/((a-b)*(a+b))^(1/2)*arctan(tan(1/2*d*x+1/2*c)*(a-b)/((a-b)*(a+b))^(1/2))*A+8/d*a^6/b^5/

(a-b)/(a+b)/((a-b)*(a+b))^(1/2)*arctan(tan(1/2*d*x+1/2*c)*(a-b)/((a-b)*(a+b))^(1/2))*C-10/d*a^4/b^3/(a-b)/(a+b

)/((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^5/b^4/(a-b)/(a+b)/((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^3/b^2/(a-b)/(a+b)/((a-b)*(a+b))^(1/2)*

arctan(tan(1/2*d*x+1/2*c)*(a-b)/((a-b)*(a+b))^(1/2))*B+4/d*a^4/b^3/(a-b)/(a+b)/((a-b)*(a+b))^(1/2)*arctan(tan(

1/2*d*x+1/2*c)*(a-b)/((a-b)*(a+b))^(1/2))*A-8/d/b^5*arctan(tan(1/2*d*x+1/2*c))*C*a^3-2/d/b^3*arctan(tan(1/2*d*

x+1/2*c))*C*a+2/d/b^2/(1+tan(1/2*d*x+1/2*c)^2)^3*tan(1/2*d*x+1/2*c)^5*C+4/d/b^2/(1+tan(1/2*d*x+1/2*c)^2)^3*tan

(1/2*d*x+1/2*c)^3*A+1/d/b^2*arctan(tan(1/2*d*x+1/2*c))*B+4/3/d/b^2/(1+tan(1/2*d*x+1/2*c)^2)^3*tan(1/2*d*x+1/2*

c)^3*C+2/d/b^2/(1+tan(1/2*d*x+1/2*c)^2)^3*tan(1/2*d*x+1/2*c)*C+2/d/b^2/(1+tan(1/2*d*x+1/2*c)^2)^3*tan(1/2*d*x+

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

x+1/2*c)^2)^3*tan(1/2*d*x+1/2*c)^5*B+2/d/b^2/(1+tan(1/2*d*x+1/2*c)^2)^3*tan(1/2*d*x+1/2*c)^5*A+1/d/b^2/(1+tan(

1/2*d*x+1/2*c)^2)^3*tan(1/2*d*x+1/2*c)*B+2/d*a^5/b^4/(a^2-b^2)*tan(1/2*d*x+1/2*c)/(a*tan(1/2*d*x+1/2*c)^2-tan(

1/2*d*x+1/2*c)^2*b+a+b)*C+2/d*a^3/b^2/(a^2-b^2)*tan(1/2*d*x+1/2*c)/(a*tan(1/2*d*x+1/2*c)^2-tan(1/2*d*x+1/2*c)^

2*b+a+b)*A-2/d*a^4/b^3/(a^2-b^2)*tan(1/2*d*x+1/2*c)/(a*tan(1/2*d*x+1/2*c)^2-tan(1/2*d*x+1/2*c)^2*b+a+b)*B+12/d

/b^4/(1+tan(1/2*d*x+1/2*c)^2)^3*tan(1/2*d*x+1/2*c)^3*C*a^2-4/d/b^3/(1+tan(1/2*d*x+1/2*c)^2)^3*tan(1/2*d*x+1/2*

c)*B*a+2/d/b^3/(1+tan(1/2*d*x+1/2*c)^2)^3*tan(1/2*d*x+1/2*c)^5*C*a-4/d/b^3/(1+tan(1/2*d*x+1/2*c)^2)^3*tan(1/2*

d*x+1/2*c)^5*B*a+6/d/b^4/(1+tan(1/2*d*x+1/2*c)^2)^3*tan(1/2*d*x+1/2*c)^5*C*a^2+6/d/b^4/(1+tan(1/2*d*x+1/2*c)^2

)^3*tan(1/2*d*x+1/2*c)*C*a^2-2/d/b^3/(1+tan(1/2*d*x+1/2*c)^2)^3*tan(1/2*d*x+1/2*c)*C*a-8/d/b^3/(1+tan(1/2*d*x+

1/2*c)^2)^3*tan(1/2*d*x+1/2*c)^3*B*a

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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)^3*(A+B*cos(d*x+c)+C*cos(d*x+c)^2)/(a+b*cos(d*x+c))^2,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 = 13.28, size = 11768, normalized size = 29.57 \[ \text {result too large to display} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[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))^2,x)

[Out]

- ((tan(c/2 + (d*x)/2)*(2*A*b^5 + B*b^5 - 8*C*a^5 + 2*C*b^5 - 2*A*a^2*b^3 - 4*A*a^3*b^2 - 5*B*a^2*b^3 + 3*B*a^

3*b^2 + 2*C*a^2*b^3 + 6*C*a^3*b^2 + 2*A*a*b^4 - 3*B*a*b^4 + 6*B*a^4*b - 4*C*a^4*b))/(b^4*(a + b)*(a - b)) - (t

an(c/2 + (d*x)/2)^3*(3*B*b^5 - 6*A*b^5 + 72*C*a^5 + 2*C*b^5 + 6*A*a^2*b^3 + 36*A*a^3*b^2 + 33*B*a^2*b^3 - 9*B*

a^3*b^2 - 14*C*a^2*b^3 - 38*C*a^3*b^2 - 18*A*a*b^4 + 9*B*a*b^4 - 54*B*a^4*b - 16*C*a*b^4 + 12*C*a^4*b))/(3*b^4

*(a + b)*(a - b)) + (tan(c/2 + (d*x)/2)^5*(2*C*b^5 - 3*B*b^5 - 72*C*a^5 - 6*A*b^5 + 6*A*a^2*b^3 - 36*A*a^3*b^2

- 33*B*a^2*b^3 - 9*B*a^3*b^2 - 14*C*a^2*b^3 + 38*C*a^3*b^2 + 18*A*a*b^4 + 9*B*a*b^4 + 54*B*a^4*b + 16*C*a*b^4

+ 12*C*a^4*b))/(3*b^4*(a + b)*(a - b)) - (tan(c/2 + (d*x)/2)^7*(2*A*b^5 - B*b^5 + 8*C*a^5 + 2*C*b^5 - 2*A*a^2

*b^3 + 4*A*a^3*b^2 + 5*B*a^2*b^3 + 3*B*a^3*b^2 + 2*C*a^2*b^3 - 6*C*a^3*b^2 - 2*A*a*b^4 - 3*B*a*b^4 - 6*B*a^4*b

- 4*C*a^4*b))/(b^4*(a + b)*(a - b)))/(d*(a + b + tan(c/2 + (d*x)/2)^8*(a - b) + tan(c/2 + (d*x)/2)^2*(4*a + 2

*b) + tan(c/2 + (d*x)/2)^6*(4*a - 2*b) + 6*a*tan(c/2 + (d*x)/2)^4)) - (atan(((((((8*(2*B*b^18 + 12*A*a^2*b^16

+ 12*A*a^3*b^15 - 20*A*a^4*b^14 - 4*A*a^5*b^13 + 8*A*a^6*b^12 + 6*B*a^2*b^16 - 16*B*a^3*b^15 - 14*B*a^4*b^14 +

28*B*a^5*b^13 + 6*B*a^6*b^12 - 12*B*a^7*b^11 - 4*C*a^3*b^15 + 20*C*a^4*b^14 + 16*C*a^5*b^13 - 36*C*a^6*b^12 -

8*C*a^7*b^11 + 16*C*a^8*b^10 - 8*A*a*b^17 - 4*C*a*b^17))/(a*b^14 + b^15 - a^2*b^13 - a^3*b^12) - (8*tan(c/2 +

(d*x)/2)*((B*b^3*1i)/2 - C*a^3*4i - b^2*(A*a*2i + C*a*1i) + B*a^2*b*3i)*(8*a*b^15 - 8*a^2*b^14 - 16*a^3*b^13

+ 16*a^4*b^12 + 8*a^5*b^11 - 8*a^6*b^10))/(b^5*(a*b^10 + b^11 - a^2*b^9 - a^3*b^8)))*((B*b^3*1i)/2 - C*a^3*4i

- b^2*(A*a*2i + C*a*1i) + B*a^2*b*3i))/b^5 + (8*tan(c/2 + (d*x)/2)*(B^2*b^12 + 128*C^2*a^12 - 2*B^2*a*b^11 - 1

28*C^2*a^11*b + 16*A^2*a^2*b^10 - 32*A^2*a^3*b^9 + 20*A^2*a^4*b^8 + 64*A^2*a^5*b^7 - 64*A^2*a^6*b^6 - 32*A^2*a

^7*b^5 + 32*A^2*a^8*b^4 + 11*B^2*a^2*b^10 - 20*B^2*a^3*b^9 + 23*B^2*a^4*b^8 - 26*B^2*a^5*b^7 + 17*B^2*a^6*b^6

+ 120*B^2*a^7*b^5 - 120*B^2*a^8*b^4 - 72*B^2*a^9*b^3 + 72*B^2*a^10*b^2 + 4*C^2*a^2*b^10 - 8*C^2*a^3*b^9 + 28*C

^2*a^4*b^8 - 48*C^2*a^5*b^7 + 28*C^2*a^6*b^6 - 8*C^2*a^7*b^5 + 8*C^2*a^8*b^4 + 192*C^2*a^9*b^3 - 192*C^2*a^10*

b^2 - 8*A*B*a*b^11 - 4*B*C*a*b^11 - 192*B*C*a^11*b + 16*A*B*a^2*b^10 - 40*A*B*a^3*b^9 + 64*A*B*a^4*b^8 - 40*A*

B*a^5*b^7 - 176*A*B*a^6*b^6 + 176*A*B*a^7*b^5 + 96*A*B*a^8*b^4 - 96*A*B*a^9*b^3 + 16*A*C*a^2*b^10 - 32*A*C*a^3

*b^9 + 48*A*C*a^4*b^8 - 64*A*C*a^5*b^7 + 40*A*C*a^6*b^6 + 224*A*C*a^7*b^5 - 224*A*C*a^8*b^4 - 128*A*C*a^9*b^3

+ 128*A*C*a^10*b^2 + 8*B*C*a^2*b^10 - 36*B*C*a^3*b^9 + 64*B*C*a^4*b^8 - 52*B*C*a^5*b^7 + 40*B*C*a^6*b^6 - 28*B

*C*a^7*b^5 - 304*B*C*a^8*b^4 + 304*B*C*a^9*b^3 + 192*B*C*a^10*b^2))/(a*b^10 + b^11 - a^2*b^9 - a^3*b^8))*((B*b

^3*1i)/2 - C*a^3*4i - b^2*(A*a*2i + C*a*1i) + B*a^2*b*3i)*1i)/b^5 - (((((8*(2*B*b^18 + 12*A*a^2*b^16 + 12*A*a^

3*b^15 - 20*A*a^4*b^14 - 4*A*a^5*b^13 + 8*A*a^6*b^12 + 6*B*a^2*b^16 - 16*B*a^3*b^15 - 14*B*a^4*b^14 + 28*B*a^5

*b^13 + 6*B*a^6*b^12 - 12*B*a^7*b^11 - 4*C*a^3*b^15 + 20*C*a^4*b^14 + 16*C*a^5*b^13 - 36*C*a^6*b^12 - 8*C*a^7*

b^11 + 16*C*a^8*b^10 - 8*A*a*b^17 - 4*C*a*b^17))/(a*b^14 + b^15 - a^2*b^13 - a^3*b^12) + (8*tan(c/2 + (d*x)/2)

*((B*b^3*1i)/2 - C*a^3*4i - b^2*(A*a*2i + C*a*1i) + B*a^2*b*3i)*(8*a*b^15 - 8*a^2*b^14 - 16*a^3*b^13 + 16*a^4*

b^12 + 8*a^5*b^11 - 8*a^6*b^10))/(b^5*(a*b^10 + b^11 - a^2*b^9 - a^3*b^8)))*((B*b^3*1i)/2 - C*a^3*4i - b^2*(A*

a*2i + C*a*1i) + B*a^2*b*3i))/b^5 - (8*tan(c/2 + (d*x)/2)*(B^2*b^12 + 128*C^2*a^12 - 2*B^2*a*b^11 - 128*C^2*a^

11*b + 16*A^2*a^2*b^10 - 32*A^2*a^3*b^9 + 20*A^2*a^4*b^8 + 64*A^2*a^5*b^7 - 64*A^2*a^6*b^6 - 32*A^2*a^7*b^5 +

32*A^2*a^8*b^4 + 11*B^2*a^2*b^10 - 20*B^2*a^3*b^9 + 23*B^2*a^4*b^8 - 26*B^2*a^5*b^7 + 17*B^2*a^6*b^6 + 120*B^2

*a^7*b^5 - 120*B^2*a^8*b^4 - 72*B^2*a^9*b^3 + 72*B^2*a^10*b^2 + 4*C^2*a^2*b^10 - 8*C^2*a^3*b^9 + 28*C^2*a^4*b^

8 - 48*C^2*a^5*b^7 + 28*C^2*a^6*b^6 - 8*C^2*a^7*b^5 + 8*C^2*a^8*b^4 + 192*C^2*a^9*b^3 - 192*C^2*a^10*b^2 - 8*A

*B*a*b^11 - 4*B*C*a*b^11 - 192*B*C*a^11*b + 16*A*B*a^2*b^10 - 40*A*B*a^3*b^9 + 64*A*B*a^4*b^8 - 40*A*B*a^5*b^7

- 176*A*B*a^6*b^6 + 176*A*B*a^7*b^5 + 96*A*B*a^8*b^4 - 96*A*B*a^9*b^3 + 16*A*C*a^2*b^10 - 32*A*C*a^3*b^9 + 48

*A*C*a^4*b^8 - 64*A*C*a^5*b^7 + 40*A*C*a^6*b^6 + 224*A*C*a^7*b^5 - 224*A*C*a^8*b^4 - 128*A*C*a^9*b^3 + 128*A*C

*a^10*b^2 + 8*B*C*a^2*b^10 - 36*B*C*a^3*b^9 + 64*B*C*a^4*b^8 - 52*B*C*a^5*b^7 + 40*B*C*a^6*b^6 - 28*B*C*a^7*b^

5 - 304*B*C*a^8*b^4 + 304*B*C*a^9*b^3 + 192*B*C*a^10*b^2))/(a*b^10 + b^11 - a^2*b^9 - a^3*b^8))*((B*b^3*1i)/2

- C*a^3*4i - b^2*(A*a*2i + C*a*1i) + B*a^2*b*3i)*1i)/b^5)/((16*(256*C^3*a^14 - 128*C^3*a^13*b + 48*A^3*a^4*b^1

0 + 24*A^3*a^5*b^9 - 80*A^3*a^6*b^8 - 16*A^3*a^7*b^7 + 32*A^3*a^8*b^6 - 4*B^3*a^3*b^11 + 4*B^3*a^4*b^10 - 41*B

^3*a^5*b^9 + 9*B^3*a^6*b^8 - 63*B^3*a^7*b^7 - 81*B^3*a^8*b^6 + 216*B^3*a^9*b^5 + 54*B^3*a^10*b^4 - 108*B^3*a^1

1*b^3 + 20*C^3*a^6*b^8 - 20*C^3*a^7*b^7 + 124*C^3*a^8*b^6 - 24*C^3*a^9*b^5 + 48*C^3*a^10*b^4 + 192*C^3*a^11*b^

3 - 448*C^3*a^12*b^2 - 576*B*C^2*a^13*b + 3*A*B^2*a^2*b^12 - 3*A*B^2*a^3*b^11 + 63*A*B^2*a^4*b^10 - 15*A*B^2*a

^5*b^9 + 186*A*B^2*a^6*b^8 + 162*A*B^2*a^7*b^7 - 468*A*B^2*a^8*b^6 - 108*A*B^2*a^9*b^5 + 216*A*B^2*a^10*b^4 -

24*A^2*B*a^3*b^11 + 6*A^2*B*a^4*b^10 - 168*A^2*B*a^5*b^9 - 108*A^2*B*a^6*b^8 + 336*A^2*B*a^7*b^7 + 72*A^2*B*a^

8*b^6 - 144*A^2*B*a^9*b^5 + 12*A*C^2*a^4*b^10 - 12*A*C^2*a^5*b^9 + 156*A*C^2*a^6*b^8 - 36*A*C^2*a^7*b^7 + 216*

A*C^2*a^8*b^6 + 288*A*C^2*a^9*b^5 - 768*A*C^2*a^10*b^4 - 192*A*C^2*a^11*b^3 + 384*A*C^2*a^12*b^2 + 48*A^2*C*a^

4*b^10 - 12*A^2*C*a^5*b^9 + 192*A^2*C*a^6*b^8 + 144*A^2*C*a^7*b^7 - 432*A^2*C*a^8*b^6 - 96*A^2*C*a^9*b^5 + 192

*A^2*C*a^10*b^4 - 36*B*C^2*a^5*b^9 + 36*B*C^2*a^6*b^8 - 264*B*C^2*a^7*b^7 + 54*B*C^2*a^8*b^6 - 180*B*C^2*a^9*b

^5 - 432*B*C^2*a^10*b^4 + 1056*B*C^2*a^11*b^3 + 288*B*C^2*a^12*b^2 + 21*B^2*C*a^4*b^10 - 21*B^2*C*a^5*b^9 + 18

3*B^2*C*a^6*b^8 - 39*B^2*C*a^7*b^7 + 192*B^2*C*a^8*b^6 + 324*B^2*C*a^9*b^5 - 828*B^2*C*a^10*b^4 - 216*B^2*C*a^

11*b^3 + 432*B^2*C*a^12*b^2 - 12*A*B*C*a^3*b^11 + 12*A*B*C*a^4*b^10 - 204*A*B*C*a^5*b^9 + 48*A*B*C*a^6*b^8 - 4

08*A*B*C*a^7*b^7 - 432*A*B*C*a^8*b^6 + 1200*A*B*C*a^9*b^5 + 288*A*B*C*a^10*b^4 - 576*A*B*C*a^11*b^3))/(a*b^14

+ b^15 - a^2*b^13 - a^3*b^12) + (((((8*(2*B*b^18 + 12*A*a^2*b^16 + 12*A*a^3*b^15 - 20*A*a^4*b^14 - 4*A*a^5*b^1

3 + 8*A*a^6*b^12 + 6*B*a^2*b^16 - 16*B*a^3*b^15 - 14*B*a^4*b^14 + 28*B*a^5*b^13 + 6*B*a^6*b^12 - 12*B*a^7*b^11

- 4*C*a^3*b^15 + 20*C*a^4*b^14 + 16*C*a^5*b^13 - 36*C*a^6*b^12 - 8*C*a^7*b^11 + 16*C*a^8*b^10 - 8*A*a*b^17 -

4*C*a*b^17))/(a*b^14 + b^15 - a^2*b^13 - a^3*b^12) - (8*tan(c/2 + (d*x)/2)*((B*b^3*1i)/2 - C*a^3*4i - b^2*(A*a

*2i + C*a*1i) + B*a^2*b*3i)*(8*a*b^15 - 8*a^2*b^14 - 16*a^3*b^13 + 16*a^4*b^12 + 8*a^5*b^11 - 8*a^6*b^10))/(b^

5*(a*b^10 + b^11 - a^2*b^9 - a^3*b^8)))*((B*b^3*1i)/2 - C*a^3*4i - b^2*(A*a*2i + C*a*1i) + B*a^2*b*3i))/b^5 +

(8*tan(c/2 + (d*x)/2)*(B^2*b^12 + 128*C^2*a^12 - 2*B^2*a*b^11 - 128*C^2*a^11*b + 16*A^2*a^2*b^10 - 32*A^2*a^3*

b^9 + 20*A^2*a^4*b^8 + 64*A^2*a^5*b^7 - 64*A^2*a^6*b^6 - 32*A^2*a^7*b^5 + 32*A^2*a^8*b^4 + 11*B^2*a^2*b^10 - 2

0*B^2*a^3*b^9 + 23*B^2*a^4*b^8 - 26*B^2*a^5*b^7 + 17*B^2*a^6*b^6 + 120*B^2*a^7*b^5 - 120*B^2*a^8*b^4 - 72*B^2*

a^9*b^3 + 72*B^2*a^10*b^2 + 4*C^2*a^2*b^10 - 8*C^2*a^3*b^9 + 28*C^2*a^4*b^8 - 48*C^2*a^5*b^7 + 28*C^2*a^6*b^6

- 8*C^2*a^7*b^5 + 8*C^2*a^8*b^4 + 192*C^2*a^9*b^3 - 192*C^2*a^10*b^2 - 8*A*B*a*b^11 - 4*B*C*a*b^11 - 192*B*C*a

^11*b + 16*A*B*a^2*b^10 - 40*A*B*a^3*b^9 + 64*A*B*a^4*b^8 - 40*A*B*a^5*b^7 - 176*A*B*a^6*b^6 + 176*A*B*a^7*b^5

+ 96*A*B*a^8*b^4 - 96*A*B*a^9*b^3 + 16*A*C*a^2*b^10 - 32*A*C*a^3*b^9 + 48*A*C*a^4*b^8 - 64*A*C*a^5*b^7 + 40*A

*C*a^6*b^6 + 224*A*C*a^7*b^5 - 224*A*C*a^8*b^4 - 128*A*C*a^9*b^3 + 128*A*C*a^10*b^2 + 8*B*C*a^2*b^10 - 36*B*C*

a^3*b^9 + 64*B*C*a^4*b^8 - 52*B*C*a^5*b^7 + 40*B*C*a^6*b^6 - 28*B*C*a^7*b^5 - 304*B*C*a^8*b^4 + 304*B*C*a^9*b^

3 + 192*B*C*a^10*b^2))/(a*b^10 + b^11 - a^2*b^9 - a^3*b^8))*((B*b^3*1i)/2 - C*a^3*4i - b^2*(A*a*2i + C*a*1i) +

B*a^2*b*3i))/b^5 + (((((8*(2*B*b^18 + 12*A*a^2*b^16 + 12*A*a^3*b^15 - 20*A*a^4*b^14 - 4*A*a^5*b^13 + 8*A*a^6*

b^12 + 6*B*a^2*b^16 - 16*B*a^3*b^15 - 14*B*a^4*b^14 + 28*B*a^5*b^13 + 6*B*a^6*b^12 - 12*B*a^7*b^11 - 4*C*a^3*b

^15 + 20*C*a^4*b^14 + 16*C*a^5*b^13 - 36*C*a^6*b^12 - 8*C*a^7*b^11 + 16*C*a^8*b^10 - 8*A*a*b^17 - 4*C*a*b^17))

/(a*b^14 + b^15 - a^2*b^13 - a^3*b^12) + (8*tan(c/2 + (d*x)/2)*((B*b^3*1i)/2 - C*a^3*4i - b^2*(A*a*2i + C*a*1i

) + B*a^2*b*3i)*(8*a*b^15 - 8*a^2*b^14 - 16*a^3*b^13 + 16*a^4*b^12 + 8*a^5*b^11 - 8*a^6*b^10))/(b^5*(a*b^10 +

b^11 - a^2*b^9 - a^3*b^8)))*((B*b^3*1i)/2 - C*a^3*4i - b^2*(A*a*2i + C*a*1i) + B*a^2*b*3i))/b^5 - (8*tan(c/2 +

(d*x)/2)*(B^2*b^12 + 128*C^2*a^12 - 2*B^2*a*b^11 - 128*C^2*a^11*b + 16*A^2*a^2*b^10 - 32*A^2*a^3*b^9 + 20*A^2

*a^4*b^8 + 64*A^2*a^5*b^7 - 64*A^2*a^6*b^6 - 32*A^2*a^7*b^5 + 32*A^2*a^8*b^4 + 11*B^2*a^2*b^10 - 20*B^2*a^3*b^

9 + 23*B^2*a^4*b^8 - 26*B^2*a^5*b^7 + 17*B^2*a^6*b^6 + 120*B^2*a^7*b^5 - 120*B^2*a^8*b^4 - 72*B^2*a^9*b^3 + 72

*B^2*a^10*b^2 + 4*C^2*a^2*b^10 - 8*C^2*a^3*b^9 + 28*C^2*a^4*b^8 - 48*C^2*a^5*b^7 + 28*C^2*a^6*b^6 - 8*C^2*a^7*

b^5 + 8*C^2*a^8*b^4 + 192*C^2*a^9*b^3 - 192*C^2*a^10*b^2 - 8*A*B*a*b^11 - 4*B*C*a*b^11 - 192*B*C*a^11*b + 16*A

*B*a^2*b^10 - 40*A*B*a^3*b^9 + 64*A*B*a^4*b^8 - 40*A*B*a^5*b^7 - 176*A*B*a^6*b^6 + 176*A*B*a^7*b^5 + 96*A*B*a^

8*b^4 - 96*A*B*a^9*b^3 + 16*A*C*a^2*b^10 - 32*A*C*a^3*b^9 + 48*A*C*a^4*b^8 - 64*A*C*a^5*b^7 + 40*A*C*a^6*b^6 +

224*A*C*a^7*b^5 - 224*A*C*a^8*b^4 - 128*A*C*a^9*b^3 + 128*A*C*a^10*b^2 + 8*B*C*a^2*b^10 - 36*B*C*a^3*b^9 + 64

*B*C*a^4*b^8 - 52*B*C*a^5*b^7 + 40*B*C*a^6*b^6 - 28*B*C*a^7*b^5 - 304*B*C*a^8*b^4 + 304*B*C*a^9*b^3 + 192*B*C*

a^10*b^2))/(a*b^10 + b^11 - a^2*b^9 - a^3*b^8))*((B*b^3*1i)/2 - C*a^3*4i - b^2*(A*a*2i + C*a*1i) + B*a^2*b*3i)

)/b^5))*((B*b^3*1i)/2 - C*a^3*4i - b^2*(A*a*2i + C*a*1i) + B*a^2*b*3i)*2i)/(b^5*d) - (a^2*atan(((a^2*((8*tan(c

/2 + (d*x)/2)*(B^2*b^12 + 128*C^2*a^12 - 2*B^2*a*b^11 - 128*C^2*a^11*b + 16*A^2*a^2*b^10 - 32*A^2*a^3*b^9 + 20

*A^2*a^4*b^8 + 64*A^2*a^5*b^7 - 64*A^2*a^6*b^6 - 32*A^2*a^7*b^5 + 32*A^2*a^8*b^4 + 11*B^2*a^2*b^10 - 20*B^2*a^

3*b^9 + 23*B^2*a^4*b^8 - 26*B^2*a^5*b^7 + 17*B^2*a^6*b^6 + 120*B^2*a^7*b^5 - 120*B^2*a^8*b^4 - 72*B^2*a^9*b^3

+ 72*B^2*a^10*b^2 + 4*C^2*a^2*b^10 - 8*C^2*a^3*b^9 + 28*C^2*a^4*b^8 - 48*C^2*a^5*b^7 + 28*C^2*a^6*b^6 - 8*C^2*

a^7*b^5 + 8*C^2*a^8*b^4 + 192*C^2*a^9*b^3 - 192*C^2*a^10*b^2 - 8*A*B*a*b^11 - 4*B*C*a*b^11 - 192*B*C*a^11*b +

16*A*B*a^2*b^10 - 40*A*B*a^3*b^9 + 64*A*B*a^4*b^8 - 40*A*B*a^5*b^7 - 176*A*B*a^6*b^6 + 176*A*B*a^7*b^5 + 96*A*

B*a^8*b^4 - 96*A*B*a^9*b^3 + 16*A*C*a^2*b^10 - 32*A*C*a^3*b^9 + 48*A*C*a^4*b^8 - 64*A*C*a^5*b^7 + 40*A*C*a^6*b

^6 + 224*A*C*a^7*b^5 - 224*A*C*a^8*b^4 - 128*A*C*a^9*b^3 + 128*A*C*a^10*b^2 + 8*B*C*a^2*b^10 - 36*B*C*a^3*b^9

+ 64*B*C*a^4*b^8 - 52*B*C*a^5*b^7 + 40*B*C*a^6*b^6 - 28*B*C*a^7*b^5 - 304*B*C*a^8*b^4 + 304*B*C*a^9*b^3 + 192*

B*C*a^10*b^2))/(a*b^10 + b^11 - a^2*b^9 - a^3*b^8) + (a^2*((8*(2*B*b^18 + 12*A*a^2*b^16 + 12*A*a^3*b^15 - 20*A

*a^4*b^14 - 4*A*a^5*b^13 + 8*A*a^6*b^12 + 6*B*a^2*b^16 - 16*B*a^3*b^15 - 14*B*a^4*b^14 + 28*B*a^5*b^13 + 6*B*a

^6*b^12 - 12*B*a^7*b^11 - 4*C*a^3*b^15 + 20*C*a^4*b^14 + 16*C*a^5*b^13 - 36*C*a^6*b^12 - 8*C*a^7*b^11 + 16*C*a

^8*b^10 - 8*A*a*b^17 - 4*C*a*b^17))/(a*b^14 + b^15 - a^2*b^13 - a^3*b^12) - (8*a^2*tan(c/2 + (d*x)/2)*(-(a + b

)^3*(a - b)^3)^(1/2)*(8*a*b^15 - 8*a^2*b^14 - 16*a^3*b^13 + 16*a^4*b^12 + 8*a^5*b^11 - 8*a^6*b^10)*(3*A*b^4 -

4*C*a^4 - 2*A*a^2*b^2 + 5*C*a^2*b^2 - 4*B*a*b^3 + 3*B*a^3*b))/((a*b^10 + b^11 - a^2*b^9 - a^3*b^8)*(b^11 - 3*a

^2*b^9 + 3*a^4*b^7 - a^6*b^5)))*(-(a + b)^3*(a - b)^3)^(1/2)*(3*A*b^4 - 4*C*a^4 - 2*A*a^2*b^2 + 5*C*a^2*b^2 -

4*B*a*b^3 + 3*B*a^3*b))/(b^11 - 3*a^2*b^9 + 3*a^4*b^7 - a^6*b^5))*(-(a + b)^3*(a - b)^3)^(1/2)*(3*A*b^4 - 4*C*

a^4 - 2*A*a^2*b^2 + 5*C*a^2*b^2 - 4*B*a*b^3 + 3*B*a^3*b)*1i)/(b^11 - 3*a^2*b^9 + 3*a^4*b^7 - a^6*b^5) + (a^2*(