∫ cos2 (c+dx)(A+C cos2(c+dx))____________________

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

Optimal. Leaf size=262 \[ -\frac {\left (a^2 C+A b^2\right ) \sin (c+d x) \cos ^2(c+d x)}{b d \left (a^2-b^2\right ) (a+b \cos (c+d x))}+\frac {\left (3 a^2 C+2 A b^2-b^2 C\right ) \sin (c+d x) \cos (c+d x)}{2 b^2 d \left (a^2-b^2\right )}+\frac {x \left (C \left (6 a^2+b^2\right )+2 A b^2\right )}{2 b^4}-\frac {a \left (3 a^2 C+A b^2-2 b^2 C\right ) \sin (c+d x)}{b^3 d \left (a^2-b^2\right )}-\frac {2 a \left (3 a^4 C+a^2 A b^2-4 a^2 b^2 C-2 A b^4\right ) \tan ^{-1}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{b^4 d (a-b)^{3/2} (a+b)^{3/2}} \]

[Out]

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

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

(a+b*cos(d*x+c))

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Rubi [A] time = 0.69, antiderivative size = 262, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {3048, 3049, 3023, 2735, 2659, 205} \[ -\frac {a \left (3 a^2 C+A b^2-2 b^2 C\right ) \sin (c+d x)}{b^3 d \left (a^2-b^2\right )}-\frac {2 a \left (a^2 A b^2-4 a^2 b^2 C+3 a^4 C-2 A b^4\right ) \tan ^{-1}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{b^4 d (a-b)^{3/2} (a+b)^{3/2}}-\frac {\left (a^2 C+A b^2\right ) \sin (c+d x) \cos ^2(c+d x)}{b d \left (a^2-b^2\right ) (a+b \cos (c+d x))}+\frac {\left (3 a^2 C+2 A b^2-b^2 C\right ) \sin (c+d x) \cos (c+d x)}{2 b^2 d \left (a^2-b^2\right )}+\frac {x \left (C \left (6 a^2+b^2\right )+2 A b^2\right )}{2 b^4} \]

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

((2*A*b^2 + (6*a^2 + b^2)*C)*x)/(2*b^4) - (2*a*(a^2*A*b^2 - 2*A*b^4 + 3*a^4*C - 4*a^2*b^2*C)*ArcTan[(Sqrt[a -

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

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

((A*b^2 + a^2*C)*Cos[c + d*x]^2*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 3048

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

in[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> -Simp[((c^2*C + 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*c*m + a*d*(n + 1)) - (A*d*(a*d*(n +

2) - b*c*(n + 1)) - C*(b*c*d*(n + 1) - a*(c^2 + d^2*(n + 1))))*Sin[e + f*x] - b*(A*d^2*(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, 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 ^2(c+d x) \left (A+C \cos ^2(c+d x)\right )}{(a+b \cos (c+d x))^2} \, dx &=-\frac {\left (A b^2+a^2 C\right ) \cos ^2(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 \left (A b^2+a^2 C\right )-a b (A+C) \cos (c+d x)-\left (2 A b^2+3 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 (2 A b^2+3 a^2 C-b^2 C\right ) \cos (c+d x) \sin (c+d x)}{2 b^2 \left (a^2-b^2\right ) d}-\frac {\left (A b^2+a^2 C\right ) \cos ^2(c+d x) \sin (c+d x)}{b \left (a^2-b^2\right ) d (a+b \cos (c+d x))}-\frac {\int \frac {-a \left (2 A b^2+\left (3 a^2-b^2\right ) C\right )+b \left (2 A b^2+\left (a^2+b^2\right ) C\right ) \cos (c+d x)+2 a \left (A b^2+3 a^2 C-2 b^2 C\right ) \cos ^2(c+d x)}{a+b \cos (c+d x)} \, dx}{2 b^2 \left (a^2-b^2\right )}\\ &=-\frac {a \left (A b^2+3 a^2 C-2 b^2 C\right ) \sin (c+d x)}{b^3 \left (a^2-b^2\right ) d}+\frac {\left (2 A b^2+3 a^2 C-b^2 C\right ) \cos (c+d x) \sin (c+d x)}{2 b^2 \left (a^2-b^2\right ) d}-\frac {\left (A b^2+a^2 C\right ) \cos ^2(c+d x) \sin (c+d x)}{b \left (a^2-b^2\right ) d (a+b \cos (c+d x))}-\frac {\int \frac {-a b \left (b^2 (2 A-C)+3 a^2 C\right )-\left (a^2-b^2\right ) \left (2 A b^2+6 a^2 C+b^2 C\right ) \cos (c+d x)}{a+b \cos (c+d x)} \, dx}{2 b^3 \left (a^2-b^2\right )}\\ &=\frac {\left (2 A b^2+\left (6 a^2+b^2\right ) C\right ) x}{2 b^4}-\frac {a \left (A b^2+3 a^2 C-2 b^2 C\right ) \sin (c+d x)}{b^3 \left (a^2-b^2\right ) d}+\frac {\left (2 A b^2+3 a^2 C-b^2 C\right ) \cos (c+d x) \sin (c+d x)}{2 b^2 \left (a^2-b^2\right ) d}-\frac {\left (A b^2+a^2 C\right ) \cos ^2(c+d x) \sin (c+d x)}{b \left (a^2-b^2\right ) d (a+b \cos (c+d x))}+\frac {\left (a \left (2 A b^4-a^2 b^2 (A-4 C)-3 a^4 C\right )\right ) \int \frac {1}{a+b \cos (c+d x)} \, dx}{b^4 \left (a^2-b^2\right )}\\ &=\frac {\left (2 A b^2+\left (6 a^2+b^2\right ) C\right ) x}{2 b^4}-\frac {a \left (A b^2+3 a^2 C-2 b^2 C\right ) \sin (c+d x)}{b^3 \left (a^2-b^2\right ) d}+\frac {\left (2 A b^2+3 a^2 C-b^2 C\right ) \cos (c+d x) \sin (c+d x)}{2 b^2 \left (a^2-b^2\right ) d}-\frac {\left (A b^2+a^2 C\right ) \cos ^2(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 \left (2 A b^4-a^2 b^2 (A-4 C)-3 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^4 \left (a^2-b^2\right ) d}\\ &=\frac {\left (2 A b^2+\left (6 a^2+b^2\right ) C\right ) x}{2 b^4}-\frac {2 a \left (a^2 A b^2-2 A b^4+3 a^4 C-4 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^4 (a+b)^{3/2} d}-\frac {a \left (A b^2+3 a^2 C-2 b^2 C\right ) \sin (c+d x)}{b^3 \left (a^2-b^2\right ) d}+\frac {\left (2 A b^2+3 a^2 C-b^2 C\right ) \cos (c+d x) \sin (c+d x)}{2 b^2 \left (a^2-b^2\right ) d}-\frac {\left (A b^2+a^2 C\right ) \cos ^2(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 = 0.99, size = 178, normalized size = 0.68 \[ \frac {2 (c+d x) \left (C \left (6 a^2+b^2\right )+2 A b^2\right )-\frac {4 a^2 b \left (a^2 C+A b^2\right ) \sin (c+d x)}{(a-b) (a+b) (a+b \cos (c+d x))}-\frac {8 a \left (3 a^4 C+a^2 b^2 (A-4 C)-2 A b^4\right ) \tanh ^{-1}\left (\frac {(a-b) \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {b^2-a^2}}\right )}{\left (b^2-a^2\right )^{3/2}}-8 a b C \sin (c+d x)+b^2 C \sin (2 (c+d x))}{4 b^4 d} \]

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(2*(2*A*b^2 + (6*a^2 + b^2)*C)*(c + d*x) - (8*a*(-2*A*b^4 + a^2*b^2*(A - 4*C) + 3*a^4*C)*ArcTanh[((a - b)*Tan[

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

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

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fricas [A] time = 0.81, size = 809, normalized size = 3.09 \[ \left [\frac {{\left (6 \, C a^{6} b + {\left (2 \, A - 11 \, C\right )} a^{4} b^{3} - 4 \, {\left (A - C\right )} a^{2} b^{5} + {\left (2 \, A + C\right )} b^{7}\right )} d x \cos \left (d x + c\right ) + {\left (6 \, C a^{7} + {\left (2 \, A - 11 \, C\right )} a^{5} b^{2} - 4 \, {\left (A - C\right )} a^{3} b^{4} + {\left (2 \, A + C\right )} a b^{6}\right )} d x - {\left (3 \, C a^{6} + {\left (A - 4 \, C\right )} a^{4} b^{2} - 2 \, A a^{2} b^{4} + {\left (3 \, C a^{5} b + {\left (A - 4 \, C\right )} a^{3} b^{3} - 2 \, A a b^{5}\right )} \cos \left (d x + c\right )\right )} \sqrt {-a^{2} + b^{2}} \log \left (\frac {2 \, a b \cos \left (d x + c\right ) + {\left (2 \, a^{2} - b^{2}\right )} \cos \left (d x + c\right )^{2} - 2 \, \sqrt {-a^{2} + b^{2}} {\left (a \cos \left (d x + c\right ) + b\right )} \sin \left (d x + c\right ) - a^{2} + 2 \, b^{2}}{b^{2} \cos \left (d x + c\right )^{2} + 2 \, a b \cos \left (d x + c\right ) + a^{2}}\right ) - {\left (6 \, C a^{6} b + 2 \, {\left (A - 5 \, C\right )} a^{4} b^{3} - 2 \, {\left (A - 2 \, C\right )} a^{2} b^{5} - {\left (C a^{4} b^{3} - 2 \, C a^{2} b^{5} + C b^{7}\right )} \cos \left (d x + c\right )^{2} + 3 \, {\left (C a^{5} b^{2} - 2 \, C a^{3} b^{4} + C a b^{6}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{2 \, {\left ({\left (a^{4} b^{5} - 2 \, a^{2} b^{7} + b^{9}\right )} d \cos \left (d x + c\right ) + {\left (a^{5} b^{4} - 2 \, a^{3} b^{6} + a b^{8}\right )} d\right )}}, \frac {{\left (6 \, C a^{6} b + {\left (2 \, A - 11 \, C\right )} a^{4} b^{3} - 4 \, {\left (A - C\right )} a^{2} b^{5} + {\left (2 \, A + C\right )} b^{7}\right )} d x \cos \left (d x + c\right ) + {\left (6 \, C a^{7} + {\left (2 \, A - 11 \, C\right )} a^{5} b^{2} - 4 \, {\left (A - C\right )} a^{3} b^{4} + {\left (2 \, A + C\right )} a b^{6}\right )} d x - 2 \, {\left (3 \, C a^{6} + {\left (A - 4 \, C\right )} a^{4} b^{2} - 2 \, A a^{2} b^{4} + {\left (3 \, C a^{5} b + {\left (A - 4 \, C\right )} a^{3} b^{3} - 2 \, A a b^{5}\right )} \cos \left (d x + c\right )\right )} \sqrt {a^{2} - b^{2}} \arctan \left (-\frac {a \cos \left (d x + c\right ) + b}{\sqrt {a^{2} - b^{2}} \sin \left (d x + c\right )}\right ) - {\left (6 \, C a^{6} b + 2 \, {\left (A - 5 \, C\right )} a^{4} b^{3} - 2 \, {\left (A - 2 \, C\right )} a^{2} b^{5} - {\left (C a^{4} b^{3} - 2 \, C a^{2} b^{5} + C b^{7}\right )} \cos \left (d x + c\right )^{2} + 3 \, {\left (C a^{5} b^{2} - 2 \, C a^{3} b^{4} + C a b^{6}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{2 \, {\left ({\left (a^{4} b^{5} - 2 \, a^{2} b^{7} + b^{9}\right )} d \cos \left (d x + c\right ) + {\left (a^{5} b^{4} - 2 \, a^{3} b^{6} + a b^{8}\right )} d\right )}}\right ] \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/2*((6*C*a^6*b + (2*A - 11*C)*a^4*b^3 - 4*(A - C)*a^2*b^5 + (2*A + C)*b^7)*d*x*cos(d*x + c) + (6*C*a^7 + (2*

A - 11*C)*a^5*b^2 - 4*(A - C)*a^3*b^4 + (2*A + C)*a*b^6)*d*x - (3*C*a^6 + (A - 4*C)*a^4*b^2 - 2*A*a^2*b^4 + (3

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

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

*b^7 + b^9)*d*cos(d*x + c) + (a^5*b^4 - 2*a^3*b^6 + a*b^8)*d), 1/2*((6*C*a^6*b + (2*A - 11*C)*a^4*b^3 - 4*(A -

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

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

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

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

+ a*b^8)*d)]

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giac [A] time = 0.64, size = 311, normalized size = 1.19 \[ \frac {\frac {4 \, {\left (3 \, C a^{5} + A a^{3} b^{2} - 4 \, C a^{3} b^{2} - 2 \, A a b^{4}\right )} {\left (\pi \left \lfloor \frac {d x + c}{2 \, \pi } + \frac {1}{2} \right \rfloor \mathrm {sgn}\left (-2 \, a + 2 \, b\right ) + \arctan \left (-\frac {a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{\sqrt {a^{2} - b^{2}}}\right )\right )}}{{\left (a^{2} b^{4} - b^{6}\right )} \sqrt {a^{2} - b^{2}}} - \frac {4 \, {\left (C a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + A a^{2} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (a^{2} b^{3} - b^{5}\right )} {\left (a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a + b\right )}} + \frac {{\left (6 \, C a^{2} + 2 \, A b^{2} + C b^{2}\right )} {\left (d x + c\right )}}{b^{4}} - \frac {2 \, {\left (4 \, C a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + C b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 4 \, C a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - C b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{2} b^{3}}}{2 \, d} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(4*(3*C*a^5 + A*a^3*b^2 - 4*C*a^3*b^2 - 2*A*a*b^4)*(pi*floor(1/2*(d*x + c)/pi + 1/2)*sgn(-2*a + 2*b) + arc

tan(-(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^4 - b^6)*sqrt(a^2 - b^2)) - 4

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

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

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

2 + 1)^2*b^3))/d

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maple [B] time = 0.12, size = 569, normalized size = 2.17 \[ -\frac {2 a^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) A}{d b \left (a^{2}-b^{2}\right ) \left (a \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) b +a +b \right )}-\frac {2 a^{4} \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) C}{d \,b^{3} \left (a^{2}-b^{2}\right ) \left (a \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) b +a +b \right )}-\frac {2 a^{3} \arctan \left (\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (a -b \right )}{\sqrt {\left (a -b \right ) \left (a +b \right )}}\right ) A}{d \,b^{2} \left (a -b \right ) \left (a +b \right ) \sqrt {\left (a -b \right ) \left (a +b \right )}}+\frac {4 a \arctan \left (\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (a -b \right )}{\sqrt {\left (a -b \right ) \left (a +b \right )}}\right ) A}{d \left (a -b \right ) \left (a +b \right ) \sqrt {\left (a -b \right ) \left (a +b \right )}}-\frac {6 a^{5} \arctan \left (\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (a -b \right )}{\sqrt {\left (a -b \right ) \left (a +b \right )}}\right ) C}{d \,b^{4} \left (a -b \right ) \left (a +b \right ) \sqrt {\left (a -b \right ) \left (a +b \right )}}+\frac {8 a^{3} \arctan \left (\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (a -b \right )}{\sqrt {\left (a -b \right ) \left (a +b \right )}}\right ) C}{d \,b^{2} \left (a -b \right ) \left (a +b \right ) \sqrt {\left (a -b \right ) \left (a +b \right )}}-\frac {4 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) C a}{d \,b^{3} \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{2}}-\frac {\left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) C}{d \,b^{2} \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{2}}-\frac {4 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) C a}{d \,b^{3} \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{2}}+\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right ) C}{d \,b^{2} \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{2}}+\frac {2 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) A}{d \,b^{2}}+\frac {6 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{2} C}{d \,b^{4}}+\frac {\arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) C}{d \,b^{2}} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/d*a^2/b/(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)*C-2/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))*A+4/d*a/(a-b)/(a+b)/((a-b)*(a+b))^(1/2)*arct

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

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

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

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

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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)^2*(A+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?

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mupad [B] time = 10.11, size = 6546, normalized size = 24.98 \[ \text {result too large to display} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(atan((((C*a^2*3i + b^2*(A*1i + (C*1i)/2))*((8*tan(c/2 + (d*x)/2)*(4*A^2*b^10 + 72*C^2*a^10 + C^2*b^10 - 8*A^2

*a*b^9 - 2*C^2*a*b^9 - 72*C^2*a^9*b + 12*A^2*a^2*b^8 + 16*A^2*a^3*b^7 - 20*A^2*a^4*b^6 - 8*A^2*a^5*b^5 + 8*A^2

*a^6*b^4 + 11*C^2*a^2*b^8 - 20*C^2*a^3*b^7 + 23*C^2*a^4*b^6 - 26*C^2*a^5*b^5 + 17*C^2*a^6*b^4 + 120*C^2*a^7*b^

3 - 120*C^2*a^8*b^2 + 4*A*C*b^10 - 8*A*C*a*b^9 + 20*A*C*a^2*b^8 - 32*A*C*a^3*b^7 + 36*A*C*a^4*b^6 + 88*A*C*a^5

*b^5 - 100*A*C*a^6*b^4 - 48*A*C*a^7*b^3 + 48*A*C*a^8*b^2))/(a*b^8 + b^9 - a^2*b^7 - a^3*b^6) + (((8*(4*A*b^15

+ 2*C*b^15 - 4*A*a^2*b^13 + 12*A*a^3*b^12 - 4*A*a^5*b^10 + 6*C*a^2*b^13 - 16*C*a^3*b^12 - 14*C*a^4*b^11 + 28*C

*a^5*b^10 + 6*C*a^6*b^9 - 12*C*a^7*b^8 - 8*A*a*b^14))/(a*b^11 + b^12 - a^2*b^10 - a^3*b^9) - (8*tan(c/2 + (d*x

)/2)*(C*a^2*3i + b^2*(A*1i + (C*1i)/2))*(8*a*b^13 - 8*a^2*b^12 - 16*a^3*b^11 + 16*a^4*b^10 + 8*a^5*b^9 - 8*a^6

*b^8))/(b^4*(a*b^8 + b^9 - a^2*b^7 - a^3*b^6)))*(C*a^2*3i + b^2*(A*1i + (C*1i)/2)))/b^4)*1i)/b^4 + ((C*a^2*3i

+ b^2*(A*1i + (C*1i)/2))*((8*tan(c/2 + (d*x)/2)*(4*A^2*b^10 + 72*C^2*a^10 + C^2*b^10 - 8*A^2*a*b^9 - 2*C^2*a*b

^9 - 72*C^2*a^9*b + 12*A^2*a^2*b^8 + 16*A^2*a^3*b^7 - 20*A^2*a^4*b^6 - 8*A^2*a^5*b^5 + 8*A^2*a^6*b^4 + 11*C^2*

a^2*b^8 - 20*C^2*a^3*b^7 + 23*C^2*a^4*b^6 - 26*C^2*a^5*b^5 + 17*C^2*a^6*b^4 + 120*C^2*a^7*b^3 - 120*C^2*a^8*b^

2 + 4*A*C*b^10 - 8*A*C*a*b^9 + 20*A*C*a^2*b^8 - 32*A*C*a^3*b^7 + 36*A*C*a^4*b^6 + 88*A*C*a^5*b^5 - 100*A*C*a^6

*b^4 - 48*A*C*a^7*b^3 + 48*A*C*a^8*b^2))/(a*b^8 + b^9 - a^2*b^7 - a^3*b^6) - (((8*(4*A*b^15 + 2*C*b^15 - 4*A*a

^2*b^13 + 12*A*a^3*b^12 - 4*A*a^5*b^10 + 6*C*a^2*b^13 - 16*C*a^3*b^12 - 14*C*a^4*b^11 + 28*C*a^5*b^10 + 6*C*a^

6*b^9 - 12*C*a^7*b^8 - 8*A*a*b^14))/(a*b^11 + b^12 - a^2*b^10 - a^3*b^9) + (8*tan(c/2 + (d*x)/2)*(C*a^2*3i + b

^2*(A*1i + (C*1i)/2))*(8*a*b^13 - 8*a^2*b^12 - 16*a^3*b^11 + 16*a^4*b^10 + 8*a^5*b^9 - 8*a^6*b^8))/(b^4*(a*b^8

+ b^9 - a^2*b^7 - a^3*b^6)))*(C*a^2*3i + b^2*(A*1i + (C*1i)/2)))/b^4)*1i)/b^4)/((16*(108*C^3*a^11 + 8*A^3*a*b

^10 - 54*C^3*a^10*b + 8*A^3*a^2*b^9 - 12*A^3*a^3*b^8 - 4*A^3*a^4*b^7 + 4*A^3*a^5*b^6 + 4*C^3*a^3*b^8 - 4*C^3*a

^4*b^7 + 41*C^3*a^5*b^6 - 9*C^3*a^6*b^5 + 63*C^3*a^7*b^4 + 81*C^3*a^8*b^3 - 216*C^3*a^9*b^2 + 2*A*C^2*a*b^10 +

8*A^2*C*a*b^10 - 2*A*C^2*a^2*b^9 + 37*A*C^2*a^3*b^8 - 5*A*C^2*a^4*b^7 + 105*A*C^2*a^5*b^6 + 111*A*C^2*a^6*b^5

- 252*A*C^2*a^7*b^4 - 72*A*C^2*a^8*b^3 + 108*A*C^2*a^9*b^2 + 52*A^2*C*a^3*b^8 + 52*A^2*C*a^4*b^7 - 96*A^2*C*a

^5*b^6 - 30*A^2*C*a^6*b^5 + 36*A^2*C*a^7*b^4))/(a*b^11 + b^12 - a^2*b^10 - a^3*b^9) - ((C*a^2*3i + b^2*(A*1i +

(C*1i)/2))*((8*tan(c/2 + (d*x)/2)*(4*A^2*b^10 + 72*C^2*a^10 + C^2*b^10 - 8*A^2*a*b^9 - 2*C^2*a*b^9 - 72*C^2*a

^9*b + 12*A^2*a^2*b^8 + 16*A^2*a^3*b^7 - 20*A^2*a^4*b^6 - 8*A^2*a^5*b^5 + 8*A^2*a^6*b^4 + 11*C^2*a^2*b^8 - 20*

C^2*a^3*b^7 + 23*C^2*a^4*b^6 - 26*C^2*a^5*b^5 + 17*C^2*a^6*b^4 + 120*C^2*a^7*b^3 - 120*C^2*a^8*b^2 + 4*A*C*b^1

0 - 8*A*C*a*b^9 + 20*A*C*a^2*b^8 - 32*A*C*a^3*b^7 + 36*A*C*a^4*b^6 + 88*A*C*a^5*b^5 - 100*A*C*a^6*b^4 - 48*A*C

*a^7*b^3 + 48*A*C*a^8*b^2))/(a*b^8 + b^9 - a^2*b^7 - a^3*b^6) + (((8*(4*A*b^15 + 2*C*b^15 - 4*A*a^2*b^13 + 12*

A*a^3*b^12 - 4*A*a^5*b^10 + 6*C*a^2*b^13 - 16*C*a^3*b^12 - 14*C*a^4*b^11 + 28*C*a^5*b^10 + 6*C*a^6*b^9 - 12*C*

a^7*b^8 - 8*A*a*b^14))/(a*b^11 + b^12 - a^2*b^10 - a^3*b^9) - (8*tan(c/2 + (d*x)/2)*(C*a^2*3i + b^2*(A*1i + (C

*1i)/2))*(8*a*b^13 - 8*a^2*b^12 - 16*a^3*b^11 + 16*a^4*b^10 + 8*a^5*b^9 - 8*a^6*b^8))/(b^4*(a*b^8 + b^9 - a^2*

b^7 - a^3*b^6)))*(C*a^2*3i + b^2*(A*1i + (C*1i)/2)))/b^4))/b^4 + ((C*a^2*3i + b^2*(A*1i + (C*1i)/2))*((8*tan(c

/2 + (d*x)/2)*(4*A^2*b^10 + 72*C^2*a^10 + C^2*b^10 - 8*A^2*a*b^9 - 2*C^2*a*b^9 - 72*C^2*a^9*b + 12*A^2*a^2*b^8

+ 16*A^2*a^3*b^7 - 20*A^2*a^4*b^6 - 8*A^2*a^5*b^5 + 8*A^2*a^6*b^4 + 11*C^2*a^2*b^8 - 20*C^2*a^3*b^7 + 23*C^2*

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

A*C*a^2*b^8 - 32*A*C*a^3*b^7 + 36*A*C*a^4*b^6 + 88*A*C*a^5*b^5 - 100*A*C*a^6*b^4 - 48*A*C*a^7*b^3 + 48*A*C*a^8

*b^2))/(a*b^8 + b^9 - a^2*b^7 - a^3*b^6) - (((8*(4*A*b^15 + 2*C*b^15 - 4*A*a^2*b^13 + 12*A*a^3*b^12 - 4*A*a^5*

b^10 + 6*C*a^2*b^13 - 16*C*a^3*b^12 - 14*C*a^4*b^11 + 28*C*a^5*b^10 + 6*C*a^6*b^9 - 12*C*a^7*b^8 - 8*A*a*b^14)

)/(a*b^11 + b^12 - a^2*b^10 - a^3*b^9) + (8*tan(c/2 + (d*x)/2)*(C*a^2*3i + b^2*(A*1i + (C*1i)/2))*(8*a*b^13 -

8*a^2*b^12 - 16*a^3*b^11 + 16*a^4*b^10 + 8*a^5*b^9 - 8*a^6*b^8))/(b^4*(a*b^8 + b^9 - a^2*b^7 - a^3*b^6)))*(C*a

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

/2)*(6*C*a^4 + C*b^4 + 2*A*a^2*b^2 - 5*C*a^2*b^2 - 3*C*a*b^3 + 3*C*a^3*b))/((a*b^3 - b^4)*(a + b)) + (tan(c/2

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

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

tan(c/2 + (d*x)/2)^2*(3*a + b) + tan(c/2 + (d*x)/2)^6*(a - b) + tan(c/2 + (d*x)/2)^4*(3*a - b))) + (a*atan(((a

*((8*tan(c/2 + (d*x)/2)*(4*A^2*b^10 + 72*C^2*a^10 + C^2*b^10 - 8*A^2*a*b^9 - 2*C^2*a*b^9 - 72*C^2*a^9*b + 12*A

^2*a^2*b^8 + 16*A^2*a^3*b^7 - 20*A^2*a^4*b^6 - 8*A^2*a^5*b^5 + 8*A^2*a^6*b^4 + 11*C^2*a^2*b^8 - 20*C^2*a^3*b^7

+ 23*C^2*a^4*b^6 - 26*C^2*a^5*b^5 + 17*C^2*a^6*b^4 + 120*C^2*a^7*b^3 - 120*C^2*a^8*b^2 + 4*A*C*b^10 - 8*A*C*a

*b^9 + 20*A*C*a^2*b^8 - 32*A*C*a^3*b^7 + 36*A*C*a^4*b^6 + 88*A*C*a^5*b^5 - 100*A*C*a^6*b^4 - 48*A*C*a^7*b^3 +

48*A*C*a^8*b^2))/(a*b^8 + b^9 - a^2*b^7 - a^3*b^6) + (a*(-(a + b)^3*(a - b)^3)^(1/2)*((8*(4*A*b^15 + 2*C*b^15

- 4*A*a^2*b^13 + 12*A*a^3*b^12 - 4*A*a^5*b^10 + 6*C*a^2*b^13 - 16*C*a^3*b^12 - 14*C*a^4*b^11 + 28*C*a^5*b^10 +

6*C*a^6*b^9 - 12*C*a^7*b^8 - 8*A*a*b^14))/(a*b^11 + b^12 - a^2*b^10 - a^3*b^9) - (8*a*tan(c/2 + (d*x)/2)*(-(a

+ b)^3*(a - b)^3)^(1/2)*(2*A*b^4 - 3*C*a^4 - A*a^2*b^2 + 4*C*a^2*b^2)*(8*a*b^13 - 8*a^2*b^12 - 16*a^3*b^11 +

16*a^4*b^10 + 8*a^5*b^9 - 8*a^6*b^8))/((a*b^8 + b^9 - a^2*b^7 - a^3*b^6)*(b^10 - 3*a^2*b^8 + 3*a^4*b^6 - a^6*b

^4)))*(2*A*b^4 - 3*C*a^4 - A*a^2*b^2 + 4*C*a^2*b^2))/(b^10 - 3*a^2*b^8 + 3*a^4*b^6 - a^6*b^4))*(-(a + b)^3*(a

- b)^3)^(1/2)*(2*A*b^4 - 3*C*a^4 - A*a^2*b^2 + 4*C*a^2*b^2)*1i)/(b^10 - 3*a^2*b^8 + 3*a^4*b^6 - a^6*b^4) + (a*

((8*tan(c/2 + (d*x)/2)*(4*A^2*b^10 + 72*C^2*a^10 + C^2*b^10 - 8*A^2*a*b^9 - 2*C^2*a*b^9 - 72*C^2*a^9*b + 12*A^

2*a^2*b^8 + 16*A^2*a^3*b^7 - 20*A^2*a^4*b^6 - 8*A^2*a^5*b^5 + 8*A^2*a^6*b^4 + 11*C^2*a^2*b^8 - 20*C^2*a^3*b^7

+ 23*C^2*a^4*b^6 - 26*C^2*a^5*b^5 + 17*C^2*a^6*b^4 + 120*C^2*a^7*b^3 - 120*C^2*a^8*b^2 + 4*A*C*b^10 - 8*A*C*a*

b^9 + 20*A*C*a^2*b^8 - 32*A*C*a^3*b^7 + 36*A*C*a^4*b^6 + 88*A*C*a^5*b^5 - 100*A*C*a^6*b^4 - 48*A*C*a^7*b^3 + 4

8*A*C*a^8*b^2))/(a*b^8 + b^9 - a^2*b^7 - a^3*b^6) - (a*(-(a + b)^3*(a - b)^3)^(1/2)*((8*(4*A*b^15 + 2*C*b^15 -

4*A*a^2*b^13 + 12*A*a^3*b^12 - 4*A*a^5*b^10 + 6*C*a^2*b^13 - 16*C*a^3*b^12 - 14*C*a^4*b^11 + 28*C*a^5*b^10 +

6*C*a^6*b^9 - 12*C*a^7*b^8 - 8*A*a*b^14))/(a*b^11 + b^12 - a^2*b^10 - a^3*b^9) + (8*a*tan(c/2 + (d*x)/2)*(-(a

+ b)^3*(a - b)^3)^(1/2)*(2*A*b^4 - 3*C*a^4 - A*a^2*b^2 + 4*C*a^2*b^2)*(8*a*b^13 - 8*a^2*b^12 - 16*a^3*b^11 + 1

6*a^4*b^10 + 8*a^5*b^9 - 8*a^6*b^8))/((a*b^8 + b^9 - a^2*b^7 - a^3*b^6)*(b^10 - 3*a^2*b^8 + 3*a^4*b^6 - a^6*b^

4)))*(2*A*b^4 - 3*C*a^4 - A*a^2*b^2 + 4*C*a^2*b^2))/(b^10 - 3*a^2*b^8 + 3*a^4*b^6 - a^6*b^4))*(-(a + b)^3*(a -

b)^3)^(1/2)*(2*A*b^4 - 3*C*a^4 - A*a^2*b^2 + 4*C*a^2*b^2)*1i)/(b^10 - 3*a^2*b^8 + 3*a^4*b^6 - a^6*b^4))/((16*

(108*C^3*a^11 + 8*A^3*a*b^10 - 54*C^3*a^10*b + 8*A^3*a^2*b^9 - 12*A^3*a^3*b^8 - 4*A^3*a^4*b^7 + 4*A^3*a^5*b^6

+ 4*C^3*a^3*b^8 - 4*C^3*a^4*b^7 + 41*C^3*a^5*b^6 - 9*C^3*a^6*b^5 + 63*C^3*a^7*b^4 + 81*C^3*a^8*b^3 - 216*C^3*a

^9*b^2 + 2*A*C^2*a*b^10 + 8*A^2*C*a*b^10 - 2*A*C^2*a^2*b^9 + 37*A*C^2*a^3*b^8 - 5*A*C^2*a^4*b^7 + 105*A*C^2*a^

5*b^6 + 111*A*C^2*a^6*b^5 - 252*A*C^2*a^7*b^4 - 72*A*C^2*a^8*b^3 + 108*A*C^2*a^9*b^2 + 52*A^2*C*a^3*b^8 + 52*A

^2*C*a^4*b^7 - 96*A^2*C*a^5*b^6 - 30*A^2*C*a^6*b^5 + 36*A^2*C*a^7*b^4))/(a*b^11 + b^12 - a^2*b^10 - a^3*b^9) -

(a*((8*tan(c/2 + (d*x)/2)*(4*A^2*b^10 + 72*C^2*a^10 + C^2*b^10 - 8*A^2*a*b^9 - 2*C^2*a*b^9 - 72*C^2*a^9*b + 1

2*A^2*a^2*b^8 + 16*A^2*a^3*b^7 - 20*A^2*a^4*b^6 - 8*A^2*a^5*b^5 + 8*A^2*a^6*b^4 + 11*C^2*a^2*b^8 - 20*C^2*a^3*

b^7 + 23*C^2*a^4*b^6 - 26*C^2*a^5*b^5 + 17*C^2*a^6*b^4 + 120*C^2*a^7*b^3 - 120*C^2*a^8*b^2 + 4*A*C*b^10 - 8*A*

C*a*b^9 + 20*A*C*a^2*b^8 - 32*A*C*a^3*b^7 + 36*A*C*a^4*b^6 + 88*A*C*a^5*b^5 - 100*A*C*a^6*b^4 - 48*A*C*a^7*b^3

+ 48*A*C*a^8*b^2))/(a*b^8 + b^9 - a^2*b^7 - a^3*b^6) + (a*(-(a + b)^3*(a - b)^3)^(1/2)*((8*(4*A*b^15 + 2*C*b^

15 - 4*A*a^2*b^13 + 12*A*a^3*b^12 - 4*A*a^5*b^10 + 6*C*a^2*b^13 - 16*C*a^3*b^12 - 14*C*a^4*b^11 + 28*C*a^5*b^1

0 + 6*C*a^6*b^9 - 12*C*a^7*b^8 - 8*A*a*b^14))/(a*b^11 + b^12 - a^2*b^10 - a^3*b^9) - (8*a*tan(c/2 + (d*x)/2)*(

-(a + b)^3*(a - b)^3)^(1/2)*(2*A*b^4 - 3*C*a^4 - A*a^2*b^2 + 4*C*a^2*b^2)*(8*a*b^13 - 8*a^2*b^12 - 16*a^3*b^11

+ 16*a^4*b^10 + 8*a^5*b^9 - 8*a^6*b^8))/((a*b^8 + b^9 - a^2*b^7 - a^3*b^6)*(b^10 - 3*a^2*b^8 + 3*a^4*b^6 - a^

6*b^4)))*(2*A*b^4 - 3*C*a^4 - A*a^2*b^2 + 4*C*a^2*b^2))/(b^10 - 3*a^2*b^8 + 3*a^4*b^6 - a^6*b^4))*(-(a + b)^3*

(a - b)^3)^(1/2)*(2*A*b^4 - 3*C*a^4 - A*a^2*b^2 + 4*C*a^2*b^2))/(b^10 - 3*a^2*b^8 + 3*a^4*b^6 - a^6*b^4) + (a*