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

Optimal. Leaf size=405 \[ -\frac {\tan (c+d x) \sec ^2(c+d x) \left (-\left (a^2 (A-3 C)\right )-3 a b B+4 A b^2\right )}{3 a^2 d \left (a^2-b^2\right )}+\frac {\tan (c+d x) \sec ^2(c+d x) \left (A b^2-a (b B-a C)\right )}{a d \left (a^2-b^2\right ) (a+b \cos (c+d x))}+\frac {\tan (c+d x) \sec (c+d x) \left (a^3 B-2 a^2 b (A-C)-3 a b^2 B+4 A b^3\right )}{2 a^3 d \left (a^2-b^2\right )}-\frac {\left (a^3 (-B)+2 a^2 b (A+2 C)-6 a b^2 B+8 A b^3\right ) \tanh ^{-1}(\sin (c+d x))}{2 a^5 d}-\frac {\tan (c+d x) \left (-\left (a^4 (2 A+3 C)\right )+6 a^3 b B-a^2 b^2 (7 A-6 C)-9 a b^3 B+12 A b^4\right )}{3 a^4 d \left (a^2-b^2\right )}+\frac {2 b^2 \left (3 a^4 C-4 a^3 b B+5 a^2 A b^2-2 a^2 b^2 C+3 a b^3 B-4 A b^4\right ) \tan ^{-1}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{a^5 d (a-b)^{3/2} (a+b)^{3/2}} \]

[Out]

2*b^2*(5*A*a^2*b^2-4*A*b^4-4*B*a^3*b+3*B*a*b^3+3*C*a^4-2*C*a^2*b^2)*arctan((a-b)^(1/2)*tan(1/2*d*x+1/2*c)/(a+b
)^(1/2))/a^5/(a-b)^(3/2)/(a+b)^(3/2)/d-1/2*(8*A*b^3-a^3*B-6*a*b^2*B+2*a^2*b*(A+2*C))*arctanh(sin(d*x+c))/a^5/d
-1/3*(12*A*b^4+6*a^3*b*B-9*a*b^3*B-a^2*b^2*(7*A-6*C)-a^4*(2*A+3*C))*tan(d*x+c)/a^4/(a^2-b^2)/d+1/2*(4*A*b^3+a^
3*B-3*a*b^2*B-2*a^2*b*(A-C))*sec(d*x+c)*tan(d*x+c)/a^3/(a^2-b^2)/d-1/3*(4*A*b^2-3*a*b*B-a^2*(A-3*C))*sec(d*x+c
)^2*tan(d*x+c)/a^2/(a^2-b^2)/d+(A*b^2-a*(B*b-C*a))*sec(d*x+c)^2*tan(d*x+c)/a/(a^2-b^2)/d/(a+b*cos(d*x+c))

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Rubi [A]  time = 2.00, antiderivative size = 405, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 5, integrand size = 41, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.122, Rules used = {3055, 3001, 3770, 2659, 205} \[ \frac {2 b^2 \left (5 a^2 A b^2-2 a^2 b^2 C-4 a^3 b B+3 a^4 C+3 a b^3 B-4 A b^4\right ) \tan ^{-1}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{a^5 d (a-b)^{3/2} (a+b)^{3/2}}-\frac {\tan (c+d x) \left (-a^2 b^2 (7 A-6 C)+a^4 (-(2 A+3 C))+6 a^3 b B-9 a b^3 B+12 A b^4\right )}{3 a^4 d \left (a^2-b^2\right )}-\frac {\left (2 a^2 b (A+2 C)+a^3 (-B)-6 a b^2 B+8 A b^3\right ) \tanh ^{-1}(\sin (c+d x))}{2 a^5 d}-\frac {\tan (c+d x) \sec ^2(c+d x) \left (a^2 (-(A-3 C))-3 a b B+4 A b^2\right )}{3 a^2 d \left (a^2-b^2\right )}+\frac {\tan (c+d x) \sec (c+d x) \left (-2 a^2 b (A-C)+a^3 B-3 a b^2 B+4 A b^3\right )}{2 a^3 d \left (a^2-b^2\right )}+\frac {\tan (c+d x) \sec ^2(c+d x) \left (A b^2-a (b B-a C)\right )}{a d \left (a^2-b^2\right ) (a+b \cos (c+d x))} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*b^2*(5*a^2*A*b^2 - 4*A*b^4 - 4*a^3*b*B + 3*a*b^3*B + 3*a^4*C - 2*a^2*b^2*C)*ArcTan[(Sqrt[a - b]*Tan[(c + d*
x)/2])/Sqrt[a + b]])/(a^5*(a - b)^(3/2)*(a + b)^(3/2)*d) - ((8*A*b^3 - a^3*B - 6*a*b^2*B + 2*a^2*b*(A + 2*C))*
ArcTanh[Sin[c + d*x]])/(2*a^5*d) - ((12*A*b^4 + 6*a^3*b*B - 9*a*b^3*B - a^2*b^2*(7*A - 6*C) - a^4*(2*A + 3*C))
*Tan[c + d*x])/(3*a^4*(a^2 - b^2)*d) + ((4*A*b^3 + a^3*B - 3*a*b^2*B - 2*a^2*b*(A - C))*Sec[c + d*x]*Tan[c + d
*x])/(2*a^3*(a^2 - b^2)*d) - ((4*A*b^2 - 3*a*b*B - a^2*(A - 3*C))*Sec[c + d*x]^2*Tan[c + d*x])/(3*a^2*(a^2 - b
^2)*d) + ((A*b^2 - a*(b*B - a*C))*Sec[c + d*x]^2*Tan[c + d*x])/(a*(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 3001

Int[((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])/(((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*sin[(e_.)
+ (f_.)*(x_)])), x_Symbol] :> Dist[(A*b - a*B)/(b*c - a*d), Int[1/(a + b*Sin[e + f*x]), x], x] + Dist[(B*c - A
*d)/(b*c - a*d), Int[1/(c + d*Sin[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f, A, B}, x] && NeQ[b*c - a*d, 0]
 && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0]

Rule 3055

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((A_.) + (B_.)*s
in[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> -Simp[((A*b^2 - a*b*B + a^2*C)*Cos[e +
 f*x]*(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^(n + 1))/(f*(m + 1)*(b*c - a*d)*(a^2 - b^2)), x] + Dis
t[1/((m + 1)*(b*c - a*d)*(a^2 - b^2)), Int[(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^n*Simp[(m + 1)*(b
*c - a*d)*(a*A - b*B + a*C) + d*(A*b^2 - a*b*B + a^2*C)*(m + n + 2) - (c*(A*b^2 - a*b*B + a^2*C) + (m + 1)*(b*
c - a*d)*(A*b - a*B + b*C))*Sin[e + f*x] - d*(A*b^2 - a*b*B + a^2*C)*(m + n + 3)*Sin[e + f*x]^2, x], x], x] /;
 FreeQ[{a, b, c, d, e, f, A, B, C, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && Lt
Q[m, -1] && ((EqQ[a, 0] && IntegerQ[m] &&  !IntegerQ[n]) ||  !(IntegerQ[2*n] && LtQ[n, -1] && ((IntegerQ[n] &&
  !IntegerQ[m]) || EqQ[a, 0])))

Rule 3770

Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> -Simp[ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]

Rubi steps

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

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Mathematica [A]  time = 3.32, size = 519, normalized size = 1.28 \[ \frac {6 \left (a^3 (-B)+2 a^2 b (A+2 C)-6 a b^2 B+8 A b^3\right ) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )+6 \left (a^3 B-2 a^2 b (A+2 C)+6 a b^2 B-8 A b^3\right ) \log \left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )-\frac {24 b^2 \left (-3 a^4 C+4 a^3 b B+a^2 b^2 (2 C-5 A)-3 a b^3 B+4 A b^4\right ) \tanh ^{-1}\left (\frac {(a-b) \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {b^2-a^2}}\right )}{\left (b^2-a^2\right )^{3/2}}+\frac {a \tan (c+d x) \sec ^2(c+d x) \left (8 a^5 A+6 a^5 C+2 a^4 A b \cos (3 (c+d x))-9 a^4 b B+3 a^4 b C \cos (3 (c+d x))+4 a^3 A b^2-6 a^3 b^2 B \cos (3 (c+d x))-6 a^3 b^2 C+7 a^2 A b^3 \cos (3 (c+d x))+a \left (a^2-b^2\right ) \cos (2 (c+d x)) \left (a^2 (4 A+6 C)-9 a b B+12 A b^2\right )+9 a^2 b^3 B-6 a^2 b^3 C \cos (3 (c+d x))+\cos (c+d x) \left (6 a^5 B+a^4 (9 b C-2 A b)-24 a^3 b^2 B+a^2 b^3 (29 A-18 C)+27 a b^4 B-36 A b^5\right )-12 a A b^4+9 a b^4 B \cos (3 (c+d x))-12 A b^5 \cos (3 (c+d x))\right )}{\left (a^2-b^2\right ) (a+b \cos (c+d x))}}{12 a^5 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((-24*b^2*(4*A*b^4 + 4*a^3*b*B - 3*a*b^3*B - 3*a^4*C + a^2*b^2*(-5*A + 2*C))*ArcTanh[((a - b)*Tan[(c + d*x)/2]
)/Sqrt[-a^2 + b^2]])/(-a^2 + b^2)^(3/2) + 6*(8*A*b^3 - a^3*B - 6*a*b^2*B + 2*a^2*b*(A + 2*C))*Log[Cos[(c + d*x
)/2] - Sin[(c + d*x)/2]] + 6*(-8*A*b^3 + a^3*B + 6*a*b^2*B - 2*a^2*b*(A + 2*C))*Log[Cos[(c + d*x)/2] + Sin[(c
+ d*x)/2]] + (a*(8*a^5*A + 4*a^3*A*b^2 - 12*a*A*b^4 - 9*a^4*b*B + 9*a^2*b^3*B + 6*a^5*C - 6*a^3*b^2*C + (-36*A
*b^5 + 6*a^5*B - 24*a^3*b^2*B + 27*a*b^4*B + a^2*b^3*(29*A - 18*C) + a^4*(-2*A*b + 9*b*C))*Cos[c + d*x] + a*(a
^2 - b^2)*(12*A*b^2 - 9*a*b*B + a^2*(4*A + 6*C))*Cos[2*(c + d*x)] + 2*a^4*A*b*Cos[3*(c + d*x)] + 7*a^2*A*b^3*C
os[3*(c + d*x)] - 12*A*b^5*Cos[3*(c + d*x)] - 6*a^3*b^2*B*Cos[3*(c + d*x)] + 9*a*b^4*B*Cos[3*(c + d*x)] + 3*a^
4*b*C*Cos[3*(c + d*x)] - 6*a^2*b^3*C*Cos[3*(c + d*x)])*Sec[c + d*x]^2*Tan[c + d*x])/((a^2 - b^2)*(a + b*Cos[c
+ d*x])))/(12*a^5*d)

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fricas [B]  time = 121.51, size = 1795, normalized size = 4.43 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[-1/12*(6*((3*C*a^4*b^3 - 4*B*a^3*b^4 + (5*A - 2*C)*a^2*b^5 + 3*B*a*b^6 - 4*A*b^7)*cos(d*x + c)^4 + (3*C*a^5*b
^2 - 4*B*a^4*b^3 + (5*A - 2*C)*a^3*b^4 + 3*B*a^2*b^5 - 4*A*a*b^6)*cos(d*x + c)^3)*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)) - 3*((B*a^7*b - 2*(A + 2*C)*a^6*b^2 + 4*B*a^5*b^3 - 4*(A
- 2*C)*a^4*b^4 - 11*B*a^3*b^5 + 2*(7*A - 2*C)*a^2*b^6 + 6*B*a*b^7 - 8*A*b^8)*cos(d*x + c)^4 + (B*a^8 - 2*(A +
2*C)*a^7*b + 4*B*a^6*b^2 - 4*(A - 2*C)*a^5*b^3 - 11*B*a^4*b^4 + 2*(7*A - 2*C)*a^3*b^5 + 6*B*a^2*b^6 - 8*A*a*b^
7)*cos(d*x + c)^3)*log(sin(d*x + c) + 1) + 3*((B*a^7*b - 2*(A + 2*C)*a^6*b^2 + 4*B*a^5*b^3 - 4*(A - 2*C)*a^4*b
^4 - 11*B*a^3*b^5 + 2*(7*A - 2*C)*a^2*b^6 + 6*B*a*b^7 - 8*A*b^8)*cos(d*x + c)^4 + (B*a^8 - 2*(A + 2*C)*a^7*b +
 4*B*a^6*b^2 - 4*(A - 2*C)*a^5*b^3 - 11*B*a^4*b^4 + 2*(7*A - 2*C)*a^3*b^5 + 6*B*a^2*b^6 - 8*A*a*b^7)*cos(d*x +
 c)^3)*log(-sin(d*x + c) + 1) - 2*(2*A*a^8 - 4*A*a^6*b^2 + 2*A*a^4*b^4 + 2*((2*A + 3*C)*a^7*b - 6*B*a^6*b^2 +
(5*A - 9*C)*a^5*b^3 + 15*B*a^4*b^4 - (19*A - 6*C)*a^3*b^5 - 9*B*a^2*b^6 + 12*A*a*b^7)*cos(d*x + c)^3 + (2*(2*A
 + 3*C)*a^8 - 9*B*a^7*b + 4*(A - 3*C)*a^6*b^2 + 18*B*a^5*b^3 - 2*(10*A - 3*C)*a^4*b^4 - 9*B*a^3*b^5 + 12*A*a^2
*b^6)*cos(d*x + c)^2 + (3*B*a^8 - 4*A*a^7*b - 6*B*a^6*b^2 + 8*A*a^5*b^3 + 3*B*a^4*b^4 - 4*A*a^3*b^5)*cos(d*x +
 c))*sin(d*x + c))/((a^9*b - 2*a^7*b^3 + a^5*b^5)*d*cos(d*x + c)^4 + (a^10 - 2*a^8*b^2 + a^6*b^4)*d*cos(d*x +
c)^3), 1/12*(12*((3*C*a^4*b^3 - 4*B*a^3*b^4 + (5*A - 2*C)*a^2*b^5 + 3*B*a*b^6 - 4*A*b^7)*cos(d*x + c)^4 + (3*C
*a^5*b^2 - 4*B*a^4*b^3 + (5*A - 2*C)*a^3*b^4 + 3*B*a^2*b^5 - 4*A*a*b^6)*cos(d*x + c)^3)*sqrt(a^2 - b^2)*arctan
(-(a*cos(d*x + c) + b)/(sqrt(a^2 - b^2)*sin(d*x + c))) + 3*((B*a^7*b - 2*(A + 2*C)*a^6*b^2 + 4*B*a^5*b^3 - 4*(
A - 2*C)*a^4*b^4 - 11*B*a^3*b^5 + 2*(7*A - 2*C)*a^2*b^6 + 6*B*a*b^7 - 8*A*b^8)*cos(d*x + c)^4 + (B*a^8 - 2*(A
+ 2*C)*a^7*b + 4*B*a^6*b^2 - 4*(A - 2*C)*a^5*b^3 - 11*B*a^4*b^4 + 2*(7*A - 2*C)*a^3*b^5 + 6*B*a^2*b^6 - 8*A*a*
b^7)*cos(d*x + c)^3)*log(sin(d*x + c) + 1) - 3*((B*a^7*b - 2*(A + 2*C)*a^6*b^2 + 4*B*a^5*b^3 - 4*(A - 2*C)*a^4
*b^4 - 11*B*a^3*b^5 + 2*(7*A - 2*C)*a^2*b^6 + 6*B*a*b^7 - 8*A*b^8)*cos(d*x + c)^4 + (B*a^8 - 2*(A + 2*C)*a^7*b
 + 4*B*a^6*b^2 - 4*(A - 2*C)*a^5*b^3 - 11*B*a^4*b^4 + 2*(7*A - 2*C)*a^3*b^5 + 6*B*a^2*b^6 - 8*A*a*b^7)*cos(d*x
 + c)^3)*log(-sin(d*x + c) + 1) + 2*(2*A*a^8 - 4*A*a^6*b^2 + 2*A*a^4*b^4 + 2*((2*A + 3*C)*a^7*b - 6*B*a^6*b^2
+ (5*A - 9*C)*a^5*b^3 + 15*B*a^4*b^4 - (19*A - 6*C)*a^3*b^5 - 9*B*a^2*b^6 + 12*A*a*b^7)*cos(d*x + c)^3 + (2*(2
*A + 3*C)*a^8 - 9*B*a^7*b + 4*(A - 3*C)*a^6*b^2 + 18*B*a^5*b^3 - 2*(10*A - 3*C)*a^4*b^4 - 9*B*a^3*b^5 + 12*A*a
^2*b^6)*cos(d*x + c)^2 + (3*B*a^8 - 4*A*a^7*b - 6*B*a^6*b^2 + 8*A*a^5*b^3 + 3*B*a^4*b^4 - 4*A*a^3*b^5)*cos(d*x
 + c))*sin(d*x + c))/((a^9*b - 2*a^7*b^3 + a^5*b^5)*d*cos(d*x + c)^4 + (a^10 - 2*a^8*b^2 + a^6*b^4)*d*cos(d*x
+ c)^3)]

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giac [A]  time = 0.32, size = 619, normalized size = 1.53 \[ -\frac {\frac {12 \, {\left (3 \, C a^{4} b^{2} - 4 \, B a^{3} b^{3} + 5 \, A a^{2} b^{4} - 2 \, C a^{2} b^{4} + 3 \, B a b^{5} - 4 \, A b^{6}\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^{7} - a^{5} b^{2}\right )} \sqrt {a^{2} - b^{2}}} + \frac {12 \, {\left (C a^{2} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - B a b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + A b^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (a^{6} - a^{4} b^{2}\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 (B a^{3} - 2 \, A a^{2} b - 4 \, C a^{2} b + 6 \, B a b^{2} - 8 \, A b^{3}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right )}{a^{5}} + \frac {3 \, {\left (B a^{3} - 2 \, A a^{2} b - 4 \, C a^{2} b + 6 \, B a b^{2} - 8 \, A b^{3}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right )}{a^{5}} + \frac {2 \, {\left (6 \, A a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 3 \, B a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 6 \, C a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 6 \, A a b \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} + 18 \, A b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 4 \, A a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 12 \, 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} - 36 \, A b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 6 \, A a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 3 \, B a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 6 \, C a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 6 \, A a b \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 ) + 18 \, A 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} a^{4}}}{6 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/6*(12*(3*C*a^4*b^2 - 4*B*a^3*b^3 + 5*A*a^2*b^4 - 2*C*a^2*b^4 + 3*B*a*b^5 - 4*A*b^6)*(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^7
 - a^5*b^2)*sqrt(a^2 - b^2)) + 12*(C*a^2*b^3*tan(1/2*d*x + 1/2*c) - B*a*b^4*tan(1/2*d*x + 1/2*c) + A*b^5*tan(1
/2*d*x + 1/2*c))/((a^6 - a^4*b^2)*(a*tan(1/2*d*x + 1/2*c)^2 - b*tan(1/2*d*x + 1/2*c)^2 + a + b)) - 3*(B*a^3 -
2*A*a^2*b - 4*C*a^2*b + 6*B*a*b^2 - 8*A*b^3)*log(abs(tan(1/2*d*x + 1/2*c) + 1))/a^5 + 3*(B*a^3 - 2*A*a^2*b - 4
*C*a^2*b + 6*B*a*b^2 - 8*A*b^3)*log(abs(tan(1/2*d*x + 1/2*c) - 1))/a^5 + 2*(6*A*a^2*tan(1/2*d*x + 1/2*c)^5 - 3
*B*a^2*tan(1/2*d*x + 1/2*c)^5 + 6*C*a^2*tan(1/2*d*x + 1/2*c)^5 + 6*A*a*b*tan(1/2*d*x + 1/2*c)^5 - 12*B*a*b*tan
(1/2*d*x + 1/2*c)^5 + 18*A*b^2*tan(1/2*d*x + 1/2*c)^5 - 4*A*a^2*tan(1/2*d*x + 1/2*c)^3 - 12*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 - 36*A*b^2*tan(1/2*d*x + 1/2*c)^3 + 6*A*a^2*tan(1/2*d*x + 1/2*c)
+ 3*B*a^2*tan(1/2*d*x + 1/2*c) + 6*C*a^2*tan(1/2*d*x + 1/2*c) - 6*A*a*b*tan(1/2*d*x + 1/2*c) - 12*B*a*b*tan(1/
2*d*x + 1/2*c) + 18*A*b^2*tan(1/2*d*x + 1/2*c))/((tan(1/2*d*x + 1/2*c)^2 - 1)^3*a^4))/d

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maple [B]  time = 0.28, size = 1242, normalized size = 3.07 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

10/d/a^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*b^4-8/d/a^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*b^3-1/2/d/a^2/(tan(1/2*d*
x+1/2*c)+1)^2*B+1/2/d/a^2*ln(tan(1/2*d*x+1/2*c)+1)*B-1/d/a^2/(tan(1/2*d*x+1/2*c)-1)*C-1/d/a^2/(tan(1/2*d*x+1/2
*c)+1)*C+1/2/d/a^2/(tan(1/2*d*x+1/2*c)-1)*B-2/d*b^5/a^4/(a^2-b^2)*tan(1/2*d*x+1/2*c)/(a*tan(1/2*d*x+1/2*c)^2-t
an(1/2*d*x+1/2*c)^2*b+a+b)*A-2/d*b^3/a^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)*C-1/2/d/a^2*A/(tan(1/2*d*x+1/2*c)-1)^2+1/2/d/a^2/(tan(1/2*d*x+1/2*c)+1)*B+1/2/d/a^2*A/(tan(1/2*d*x
+1/2*c)+1)^2-1/d/a^2*A/(tan(1/2*d*x+1/2*c)-1)+6/d*b^5/a^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+6/d*b^2/a/(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^4/a^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-8/d*b^6/a^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))*A-
1/3/d/a^2*A/(tan(1/2*d*x+1/2*c)+1)^3-1/d/a^2*A/(tan(1/2*d*x+1/2*c)+1)-1/3/d/a^2*A/(tan(1/2*d*x+1/2*c)-1)^3-1/2
/d/a^2*ln(tan(1/2*d*x+1/2*c)-1)*B+1/2/d/a^2/(tan(1/2*d*x+1/2*c)-1)^2*B+2/d*b^4/a^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-4/d/a^5*ln(tan(1/2*d*x+1/2*c)+1)*A*b^3+3/d/a^4*ln(tan(
1/2*d*x+1/2*c)+1)*B*b^2-2/d/a^3*ln(tan(1/2*d*x+1/2*c)+1)*C*b-3/d/a^4/(tan(1/2*d*x+1/2*c)+1)*A*b^2+2/d/a^3/(tan
(1/2*d*x+1/2*c)+1)*B*b-1/d/a^3/(tan(1/2*d*x+1/2*c)-1)^2*A*b+4/d/a^5*ln(tan(1/2*d*x+1/2*c)-1)*A*b^3-3/d/a^4*ln(
tan(1/2*d*x+1/2*c)-1)*B*b^2+2/d/a^3*ln(tan(1/2*d*x+1/2*c)-1)*C*b-3/d/a^4/(tan(1/2*d*x+1/2*c)-1)*A*b^2+2/d/a^3/
(tan(1/2*d*x+1/2*c)-1)*B*b+1/d/a^3/(tan(1/2*d*x+1/2*c)+1)^2*A*b-1/d/a^3/(tan(1/2*d*x+1/2*c)-1)*A*b-1/d/a^3/(ta
n(1/2*d*x+1/2*c)+1)*A*b+1/d/a^3*ln(tan(1/2*d*x+1/2*c)-1)*A*b-1/d/a^3*ln(tan(1/2*d*x+1/2*c)+1)*A*b

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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((A+B*cos(d*x+c)+C*cos(d*x+c)^2)*sec(d*x+c)^4/(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 = 13.53, size = 11677, normalized size = 28.83 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((tan(c/2 + (d*x)/2)*(2*A*a^5 - 8*A*b^5 + B*a^5 + 2*C*a^5 + 6*A*a^2*b^3 + 2*A*a^3*b^2 + 3*B*a^2*b^3 - 5*B*a^3*
b^2 - 4*C*a^2*b^3 - 2*C*a^3*b^2 - 4*A*a*b^4 + 6*B*a*b^4 - 3*B*a^4*b + 2*C*a^4*b))/(a^4*(a + b)*(a - b)) + (tan
(c/2 + (d*x)/2)^3*(2*A*a^5 + 72*A*b^5 + 3*B*a^5 - 6*C*a^5 - 38*A*a^2*b^3 - 14*A*a^3*b^2 - 9*B*a^2*b^3 + 33*B*a
^3*b^2 + 36*C*a^2*b^3 + 6*C*a^3*b^2 + 12*A*a*b^4 - 16*A*a^4*b - 54*B*a*b^4 + 9*B*a^4*b - 18*C*a^4*b))/(3*a^4*(
a + b)*(a - b)) + (tan(c/2 + (d*x)/2)^5*(2*A*a^5 - 72*A*b^5 - 3*B*a^5 - 6*C*a^5 + 38*A*a^2*b^3 - 14*A*a^3*b^2
- 9*B*a^2*b^3 - 33*B*a^3*b^2 - 36*C*a^2*b^3 + 6*C*a^3*b^2 + 12*A*a*b^4 + 16*A*a^4*b + 54*B*a*b^4 + 9*B*a^4*b +
 18*C*a^4*b))/(3*a^4*(a + b)*(a - b)) + (tan(c/2 + (d*x)/2)^7*(2*A*a^5 + 8*A*b^5 - B*a^5 + 2*C*a^5 - 6*A*a^2*b
^3 + 2*A*a^3*b^2 + 3*B*a^2*b^3 + 5*B*a^3*b^2 + 4*C*a^2*b^3 - 2*C*a^3*b^2 - 4*A*a*b^4 - 6*B*a*b^4 - 3*B*a^4*b -
 2*C*a^4*b))/(a^4*(a + b)*(a - b)))/(d*(a + b - tan(c/2 + (d*x)/2)^8*(a - b) - tan(c/2 + (d*x)/2)^2*(2*a + 4*b
) + tan(c/2 + (d*x)/2)^6*(2*a - 4*b) + 6*b*tan(c/2 + (d*x)/2)^4)) + (atan(((((((8*(2*B*a^18 + 16*A*a^10*b^8 -
8*A*a^11*b^7 - 36*A*a^12*b^6 + 16*A*a^13*b^5 + 20*A*a^14*b^4 - 4*A*a^15*b^3 - 12*B*a^11*b^7 + 6*B*a^12*b^6 + 2
8*B*a^13*b^5 - 14*B*a^14*b^4 - 16*B*a^15*b^3 + 6*B*a^16*b^2 + 8*C*a^12*b^6 - 4*C*a^13*b^5 - 20*C*a^14*b^4 + 12
*C*a^15*b^3 + 12*C*a^16*b^2 - 4*A*a^17*b - 8*C*a^17*b))/(a^14*b + a^15 - a^12*b^3 - a^13*b^2) - (8*tan(c/2 + (
d*x)/2)*(4*A*b^3 - (B*a^3)/2 + a^2*(A*b + 2*C*b) - 3*B*a*b^2)*(8*a^15*b - 8*a^10*b^6 + 8*a^11*b^5 + 16*a^12*b^
4 - 16*a^13*b^3 - 8*a^14*b^2))/(a^5*(a^10*b + a^11 - a^8*b^3 - a^9*b^2)))*(4*A*b^3 - (B*a^3)/2 + a^2*(A*b + 2*
C*b) - 3*B*a*b^2))/a^5 - (8*tan(c/2 + (d*x)/2)*(128*A^2*b^12 + B^2*a^12 - 128*A^2*a*b^11 - 2*B^2*a^11*b - 192*
A^2*a^2*b^10 + 192*A^2*a^3*b^9 + 8*A^2*a^4*b^8 - 8*A^2*a^5*b^7 + 28*A^2*a^6*b^6 - 48*A^2*a^7*b^5 + 28*A^2*a^8*
b^4 - 8*A^2*a^9*b^3 + 4*A^2*a^10*b^2 + 72*B^2*a^2*b^10 - 72*B^2*a^3*b^9 - 120*B^2*a^4*b^8 + 120*B^2*a^5*b^7 +
17*B^2*a^6*b^6 - 26*B^2*a^7*b^5 + 23*B^2*a^8*b^4 - 20*B^2*a^9*b^3 + 11*B^2*a^10*b^2 + 32*C^2*a^4*b^8 - 32*C^2*
a^5*b^7 - 64*C^2*a^6*b^6 + 64*C^2*a^7*b^5 + 20*C^2*a^8*b^4 - 32*C^2*a^9*b^3 + 16*C^2*a^10*b^2 - 192*A*B*a*b^11
 - 4*A*B*a^11*b - 8*B*C*a^11*b + 192*A*B*a^2*b^10 + 304*A*B*a^3*b^9 - 304*A*B*a^4*b^8 - 28*A*B*a^5*b^7 + 40*A*
B*a^6*b^6 - 52*A*B*a^7*b^5 + 64*A*B*a^8*b^4 - 36*A*B*a^9*b^3 + 8*A*B*a^10*b^2 + 128*A*C*a^2*b^10 - 128*A*C*a^3
*b^9 - 224*A*C*a^4*b^8 + 224*A*C*a^5*b^7 + 40*A*C*a^6*b^6 - 64*A*C*a^7*b^5 + 48*A*C*a^8*b^4 - 32*A*C*a^9*b^3 +
 16*A*C*a^10*b^2 - 96*B*C*a^3*b^9 + 96*B*C*a^4*b^8 + 176*B*C*a^5*b^7 - 176*B*C*a^6*b^6 - 40*B*C*a^7*b^5 + 64*B
*C*a^8*b^4 - 40*B*C*a^9*b^3 + 16*B*C*a^10*b^2))/(a^10*b + a^11 - a^8*b^3 - a^9*b^2))*(4*A*b^3 - (B*a^3)/2 + a^
2*(A*b + 2*C*b) - 3*B*a*b^2)*1i)/a^5 - (((((8*(2*B*a^18 + 16*A*a^10*b^8 - 8*A*a^11*b^7 - 36*A*a^12*b^6 + 16*A*
a^13*b^5 + 20*A*a^14*b^4 - 4*A*a^15*b^3 - 12*B*a^11*b^7 + 6*B*a^12*b^6 + 28*B*a^13*b^5 - 14*B*a^14*b^4 - 16*B*
a^15*b^3 + 6*B*a^16*b^2 + 8*C*a^12*b^6 - 4*C*a^13*b^5 - 20*C*a^14*b^4 + 12*C*a^15*b^3 + 12*C*a^16*b^2 - 4*A*a^
17*b - 8*C*a^17*b))/(a^14*b + a^15 - a^12*b^3 - a^13*b^2) + (8*tan(c/2 + (d*x)/2)*(4*A*b^3 - (B*a^3)/2 + a^2*(
A*b + 2*C*b) - 3*B*a*b^2)*(8*a^15*b - 8*a^10*b^6 + 8*a^11*b^5 + 16*a^12*b^4 - 16*a^13*b^3 - 8*a^14*b^2))/(a^5*
(a^10*b + a^11 - a^8*b^3 - a^9*b^2)))*(4*A*b^3 - (B*a^3)/2 + a^2*(A*b + 2*C*b) - 3*B*a*b^2))/a^5 + (8*tan(c/2
+ (d*x)/2)*(128*A^2*b^12 + B^2*a^12 - 128*A^2*a*b^11 - 2*B^2*a^11*b - 192*A^2*a^2*b^10 + 192*A^2*a^3*b^9 + 8*A
^2*a^4*b^8 - 8*A^2*a^5*b^7 + 28*A^2*a^6*b^6 - 48*A^2*a^7*b^5 + 28*A^2*a^8*b^4 - 8*A^2*a^9*b^3 + 4*A^2*a^10*b^2
 + 72*B^2*a^2*b^10 - 72*B^2*a^3*b^9 - 120*B^2*a^4*b^8 + 120*B^2*a^5*b^7 + 17*B^2*a^6*b^6 - 26*B^2*a^7*b^5 + 23
*B^2*a^8*b^4 - 20*B^2*a^9*b^3 + 11*B^2*a^10*b^2 + 32*C^2*a^4*b^8 - 32*C^2*a^5*b^7 - 64*C^2*a^6*b^6 + 64*C^2*a^
7*b^5 + 20*C^2*a^8*b^4 - 32*C^2*a^9*b^3 + 16*C^2*a^10*b^2 - 192*A*B*a*b^11 - 4*A*B*a^11*b - 8*B*C*a^11*b + 192
*A*B*a^2*b^10 + 304*A*B*a^3*b^9 - 304*A*B*a^4*b^8 - 28*A*B*a^5*b^7 + 40*A*B*a^6*b^6 - 52*A*B*a^7*b^5 + 64*A*B*
a^8*b^4 - 36*A*B*a^9*b^3 + 8*A*B*a^10*b^2 + 128*A*C*a^2*b^10 - 128*A*C*a^3*b^9 - 224*A*C*a^4*b^8 + 224*A*C*a^5
*b^7 + 40*A*C*a^6*b^6 - 64*A*C*a^7*b^5 + 48*A*C*a^8*b^4 - 32*A*C*a^9*b^3 + 16*A*C*a^10*b^2 - 96*B*C*a^3*b^9 +
96*B*C*a^4*b^8 + 176*B*C*a^5*b^7 - 176*B*C*a^6*b^6 - 40*B*C*a^7*b^5 + 64*B*C*a^8*b^4 - 40*B*C*a^9*b^3 + 16*B*C
*a^10*b^2))/(a^10*b + a^11 - a^8*b^3 - a^9*b^2))*(4*A*b^3 - (B*a^3)/2 + a^2*(A*b + 2*C*b) - 3*B*a*b^2)*1i)/a^5
)/((((((8*(2*B*a^18 + 16*A*a^10*b^8 - 8*A*a^11*b^7 - 36*A*a^12*b^6 + 16*A*a^13*b^5 + 20*A*a^14*b^4 - 4*A*a^15*
b^3 - 12*B*a^11*b^7 + 6*B*a^12*b^6 + 28*B*a^13*b^5 - 14*B*a^14*b^4 - 16*B*a^15*b^3 + 6*B*a^16*b^2 + 8*C*a^12*b
^6 - 4*C*a^13*b^5 - 20*C*a^14*b^4 + 12*C*a^15*b^3 + 12*C*a^16*b^2 - 4*A*a^17*b - 8*C*a^17*b))/(a^14*b + a^15 -
 a^12*b^3 - a^13*b^2) - (8*tan(c/2 + (d*x)/2)*(4*A*b^3 - (B*a^3)/2 + a^2*(A*b + 2*C*b) - 3*B*a*b^2)*(8*a^15*b
- 8*a^10*b^6 + 8*a^11*b^5 + 16*a^12*b^4 - 16*a^13*b^3 - 8*a^14*b^2))/(a^5*(a^10*b + a^11 - a^8*b^3 - a^9*b^2))
)*(4*A*b^3 - (B*a^3)/2 + a^2*(A*b + 2*C*b) - 3*B*a*b^2))/a^5 - (8*tan(c/2 + (d*x)/2)*(128*A^2*b^12 + B^2*a^12
- 128*A^2*a*b^11 - 2*B^2*a^11*b - 192*A^2*a^2*b^10 + 192*A^2*a^3*b^9 + 8*A^2*a^4*b^8 - 8*A^2*a^5*b^7 + 28*A^2*
a^6*b^6 - 48*A^2*a^7*b^5 + 28*A^2*a^8*b^4 - 8*A^2*a^9*b^3 + 4*A^2*a^10*b^2 + 72*B^2*a^2*b^10 - 72*B^2*a^3*b^9
- 120*B^2*a^4*b^8 + 120*B^2*a^5*b^7 + 17*B^2*a^6*b^6 - 26*B^2*a^7*b^5 + 23*B^2*a^8*b^4 - 20*B^2*a^9*b^3 + 11*B
^2*a^10*b^2 + 32*C^2*a^4*b^8 - 32*C^2*a^5*b^7 - 64*C^2*a^6*b^6 + 64*C^2*a^7*b^5 + 20*C^2*a^8*b^4 - 32*C^2*a^9*
b^3 + 16*C^2*a^10*b^2 - 192*A*B*a*b^11 - 4*A*B*a^11*b - 8*B*C*a^11*b + 192*A*B*a^2*b^10 + 304*A*B*a^3*b^9 - 30
4*A*B*a^4*b^8 - 28*A*B*a^5*b^7 + 40*A*B*a^6*b^6 - 52*A*B*a^7*b^5 + 64*A*B*a^8*b^4 - 36*A*B*a^9*b^3 + 8*A*B*a^1
0*b^2 + 128*A*C*a^2*b^10 - 128*A*C*a^3*b^9 - 224*A*C*a^4*b^8 + 224*A*C*a^5*b^7 + 40*A*C*a^6*b^6 - 64*A*C*a^7*b
^5 + 48*A*C*a^8*b^4 - 32*A*C*a^9*b^3 + 16*A*C*a^10*b^2 - 96*B*C*a^3*b^9 + 96*B*C*a^4*b^8 + 176*B*C*a^5*b^7 - 1
76*B*C*a^6*b^6 - 40*B*C*a^7*b^5 + 64*B*C*a^8*b^4 - 40*B*C*a^9*b^3 + 16*B*C*a^10*b^2))/(a^10*b + a^11 - a^8*b^3
 - a^9*b^2))*(4*A*b^3 - (B*a^3)/2 + a^2*(A*b + 2*C*b) - 3*B*a*b^2))/a^5 - (16*(256*A^3*b^14 - 128*A^3*a*b^13 -
 448*A^3*a^2*b^12 + 192*A^3*a^3*b^11 + 48*A^3*a^4*b^10 - 24*A^3*a^5*b^9 + 124*A^3*a^6*b^8 - 20*A^3*a^7*b^7 + 2
0*A^3*a^8*b^6 - 108*B^3*a^3*b^11 + 54*B^3*a^4*b^10 + 216*B^3*a^5*b^9 - 81*B^3*a^6*b^8 - 63*B^3*a^7*b^7 + 9*B^3
*a^8*b^6 - 41*B^3*a^9*b^5 + 4*B^3*a^10*b^4 - 4*B^3*a^11*b^3 + 32*C^3*a^6*b^8 - 16*C^3*a^7*b^7 - 80*C^3*a^8*b^6
 + 24*C^3*a^9*b^5 + 48*C^3*a^10*b^4 - 576*A^2*B*a*b^13 + 432*A*B^2*a^2*b^12 - 216*A*B^2*a^3*b^11 - 828*A*B^2*a
^4*b^10 + 324*A*B^2*a^5*b^9 + 192*A*B^2*a^6*b^8 - 39*A*B^2*a^7*b^7 + 183*A*B^2*a^8*b^6 - 21*A*B^2*a^9*b^5 + 21
*A*B^2*a^10*b^4 + 288*A^2*B*a^2*b^12 + 1056*A^2*B*a^3*b^11 - 432*A^2*B*a^4*b^10 - 180*A^2*B*a^5*b^9 + 54*A^2*B
*a^6*b^8 - 264*A^2*B*a^7*b^7 + 36*A^2*B*a^8*b^6 - 36*A^2*B*a^9*b^5 + 192*A*C^2*a^4*b^10 - 96*A*C^2*a^5*b^9 - 4
32*A*C^2*a^6*b^8 + 144*A*C^2*a^7*b^7 + 192*A*C^2*a^8*b^6 - 12*A*C^2*a^9*b^5 + 48*A*C^2*a^10*b^4 + 384*A^2*C*a^
2*b^12 - 192*A^2*C*a^3*b^11 - 768*A^2*C*a^4*b^10 + 288*A^2*C*a^5*b^9 + 216*A^2*C*a^6*b^8 - 36*A^2*C*a^7*b^7 +
156*A^2*C*a^8*b^6 - 12*A^2*C*a^9*b^5 + 12*A^2*C*a^10*b^4 - 144*B*C^2*a^5*b^9 + 72*B*C^2*a^6*b^8 + 336*B*C^2*a^
7*b^7 - 108*B*C^2*a^8*b^6 - 168*B*C^2*a^9*b^5 + 6*B*C^2*a^10*b^4 - 24*B*C^2*a^11*b^3 + 216*B^2*C*a^4*b^10 - 10
8*B^2*C*a^5*b^9 - 468*B^2*C*a^6*b^8 + 162*B^2*C*a^7*b^7 + 186*B^2*C*a^8*b^6 - 15*B^2*C*a^9*b^5 + 63*B^2*C*a^10
*b^4 - 3*B^2*C*a^11*b^3 + 3*B^2*C*a^12*b^2 - 576*A*B*C*a^3*b^11 + 288*A*B*C*a^4*b^10 + 1200*A*B*C*a^5*b^9 - 43
2*A*B*C*a^6*b^8 - 408*A*B*C*a^7*b^7 + 48*A*B*C*a^8*b^6 - 204*A*B*C*a^9*b^5 + 12*A*B*C*a^10*b^4 - 12*A*B*C*a^11
*b^3))/(a^14*b + a^15 - a^12*b^3 - a^13*b^2) + (((((8*(2*B*a^18 + 16*A*a^10*b^8 - 8*A*a^11*b^7 - 36*A*a^12*b^6
 + 16*A*a^13*b^5 + 20*A*a^14*b^4 - 4*A*a^15*b^3 - 12*B*a^11*b^7 + 6*B*a^12*b^6 + 28*B*a^13*b^5 - 14*B*a^14*b^4
 - 16*B*a^15*b^3 + 6*B*a^16*b^2 + 8*C*a^12*b^6 - 4*C*a^13*b^5 - 20*C*a^14*b^4 + 12*C*a^15*b^3 + 12*C*a^16*b^2
- 4*A*a^17*b - 8*C*a^17*b))/(a^14*b + a^15 - a^12*b^3 - a^13*b^2) + (8*tan(c/2 + (d*x)/2)*(4*A*b^3 - (B*a^3)/2
 + a^2*(A*b + 2*C*b) - 3*B*a*b^2)*(8*a^15*b - 8*a^10*b^6 + 8*a^11*b^5 + 16*a^12*b^4 - 16*a^13*b^3 - 8*a^14*b^2
))/(a^5*(a^10*b + a^11 - a^8*b^3 - a^9*b^2)))*(4*A*b^3 - (B*a^3)/2 + a^2*(A*b + 2*C*b) - 3*B*a*b^2))/a^5 + (8*
tan(c/2 + (d*x)/2)*(128*A^2*b^12 + B^2*a^12 - 128*A^2*a*b^11 - 2*B^2*a^11*b - 192*A^2*a^2*b^10 + 192*A^2*a^3*b
^9 + 8*A^2*a^4*b^8 - 8*A^2*a^5*b^7 + 28*A^2*a^6*b^6 - 48*A^2*a^7*b^5 + 28*A^2*a^8*b^4 - 8*A^2*a^9*b^3 + 4*A^2*
a^10*b^2 + 72*B^2*a^2*b^10 - 72*B^2*a^3*b^9 - 120*B^2*a^4*b^8 + 120*B^2*a^5*b^7 + 17*B^2*a^6*b^6 - 26*B^2*a^7*
b^5 + 23*B^2*a^8*b^4 - 20*B^2*a^9*b^3 + 11*B^2*a^10*b^2 + 32*C^2*a^4*b^8 - 32*C^2*a^5*b^7 - 64*C^2*a^6*b^6 + 6
4*C^2*a^7*b^5 + 20*C^2*a^8*b^4 - 32*C^2*a^9*b^3 + 16*C^2*a^10*b^2 - 192*A*B*a*b^11 - 4*A*B*a^11*b - 8*B*C*a^11
*b + 192*A*B*a^2*b^10 + 304*A*B*a^3*b^9 - 304*A*B*a^4*b^8 - 28*A*B*a^5*b^7 + 40*A*B*a^6*b^6 - 52*A*B*a^7*b^5 +
 64*A*B*a^8*b^4 - 36*A*B*a^9*b^3 + 8*A*B*a^10*b^2 + 128*A*C*a^2*b^10 - 128*A*C*a^3*b^9 - 224*A*C*a^4*b^8 + 224
*A*C*a^5*b^7 + 40*A*C*a^6*b^6 - 64*A*C*a^7*b^5 + 48*A*C*a^8*b^4 - 32*A*C*a^9*b^3 + 16*A*C*a^10*b^2 - 96*B*C*a^
3*b^9 + 96*B*C*a^4*b^8 + 176*B*C*a^5*b^7 - 176*B*C*a^6*b^6 - 40*B*C*a^7*b^5 + 64*B*C*a^8*b^4 - 40*B*C*a^9*b^3
+ 16*B*C*a^10*b^2))/(a^10*b + a^11 - a^8*b^3 - a^9*b^2))*(4*A*b^3 - (B*a^3)/2 + a^2*(A*b + 2*C*b) - 3*B*a*b^2)
)/a^5))*(4*A*b^3 - (B*a^3)/2 + a^2*(A*b + 2*C*b) - 3*B*a*b^2)*2i)/(a^5*d) + (b^2*atan(((b^2*((8*tan(c/2 + (d*x
)/2)*(128*A^2*b^12 + B^2*a^12 - 128*A^2*a*b^11 - 2*B^2*a^11*b - 192*A^2*a^2*b^10 + 192*A^2*a^3*b^9 + 8*A^2*a^4
*b^8 - 8*A^2*a^5*b^7 + 28*A^2*a^6*b^6 - 48*A^2*a^7*b^5 + 28*A^2*a^8*b^4 - 8*A^2*a^9*b^3 + 4*A^2*a^10*b^2 + 72*
B^2*a^2*b^10 - 72*B^2*a^3*b^9 - 120*B^2*a^4*b^8 + 120*B^2*a^5*b^7 + 17*B^2*a^6*b^6 - 26*B^2*a^7*b^5 + 23*B^2*a
^8*b^4 - 20*B^2*a^9*b^3 + 11*B^2*a^10*b^2 + 32*C^2*a^4*b^8 - 32*C^2*a^5*b^7 - 64*C^2*a^6*b^6 + 64*C^2*a^7*b^5
+ 20*C^2*a^8*b^4 - 32*C^2*a^9*b^3 + 16*C^2*a^10*b^2 - 192*A*B*a*b^11 - 4*A*B*a^11*b - 8*B*C*a^11*b + 192*A*B*a
^2*b^10 + 304*A*B*a^3*b^9 - 304*A*B*a^4*b^8 - 28*A*B*a^5*b^7 + 40*A*B*a^6*b^6 - 52*A*B*a^7*b^5 + 64*A*B*a^8*b^
4 - 36*A*B*a^9*b^3 + 8*A*B*a^10*b^2 + 128*A*C*a^2*b^10 - 128*A*C*a^3*b^9 - 224*A*C*a^4*b^8 + 224*A*C*a^5*b^7 +
 40*A*C*a^6*b^6 - 64*A*C*a^7*b^5 + 48*A*C*a^8*b^4 - 32*A*C*a^9*b^3 + 16*A*C*a^10*b^2 - 96*B*C*a^3*b^9 + 96*B*C
*a^4*b^8 + 176*B*C*a^5*b^7 - 176*B*C*a^6*b^6 - 40*B*C*a^7*b^5 + 64*B*C*a^8*b^4 - 40*B*C*a^9*b^3 + 16*B*C*a^10*
b^2))/(a^10*b + a^11 - a^8*b^3 - a^9*b^2) + (b^2*((8*(2*B*a^18 + 16*A*a^10*b^8 - 8*A*a^11*b^7 - 36*A*a^12*b^6
+ 16*A*a^13*b^5 + 20*A*a^14*b^4 - 4*A*a^15*b^3 - 12*B*a^11*b^7 + 6*B*a^12*b^6 + 28*B*a^13*b^5 - 14*B*a^14*b^4
- 16*B*a^15*b^3 + 6*B*a^16*b^2 + 8*C*a^12*b^6 - 4*C*a^13*b^5 - 20*C*a^14*b^4 + 12*C*a^15*b^3 + 12*C*a^16*b^2 -
 4*A*a^17*b - 8*C*a^17*b))/(a^14*b + a^15 - a^12*b^3 - a^13*b^2) + (8*b^2*tan(c/2 + (d*x)/2)*(-(a + b)^3*(a -
b)^3)^(1/2)*(8*a^15*b - 8*a^10*b^6 + 8*a^11*b^5 + 16*a^12*b^4 - 16*a^13*b^3 - 8*a^14*b^2)*(4*A*b^4 - 3*C*a^4 -
 5*A*a^2*b^2 + 2*C*a^2*b^2 - 3*B*a*b^3 + 4*B*a^3*b))/((a^10*b + a^11 - a^8*b^3 - a^9*b^2)*(a^11 - a^5*b^6 + 3*
a^7*b^4 - 3*a^9*b^2)))*(-(a + b)^3*(a - b)^3)^(1/2)*(4*A*b^4 - 3*C*a^4 - 5*A*a^2*b^2 + 2*C*a^2*b^2 - 3*B*a*b^3
 + 4*B*a^3*b))/(a^11 - a^5*b^6 + 3*a^7*b^4 - 3*a^9*b^2))*(-(a + b)^3*(a - b)^3)^(1/2)*(4*A*b^4 - 3*C*a^4 - 5*A
*a^2*b^2 + 2*C*a^2*b^2 - 3*B*a*b^3 + 4*B*a^3*b)*1i)/(a^11 - a^5*b^6 + 3*a^7*b^4 - 3*a^9*b^2) + (b^2*((8*tan(c/
2 + (d*x)/2)*(128*A^2*b^12 + B^2*a^12 - 128*A^2*a*b^11 - 2*B^2*a^11*b - 192*A^2*a^2*b^10 + 192*A^2*a^3*b^9 + 8
*A^2*a^4*b^8 - 8*A^2*a^5*b^7 + 28*A^2*a^6*b^6 - 48*A^2*a^7*b^5 + 28*A^2*a^8*b^4 - 8*A^2*a^9*b^3 + 4*A^2*a^10*b
^2 + 72*B^2*a^2*b^10 - 72*B^2*a^3*b^9 - 120*B^2*a^4*b^8 + 120*B^2*a^5*b^7 + 17*B^2*a^6*b^6 - 26*B^2*a^7*b^5 +
23*B^2*a^8*b^4 - 20*B^2*a^9*b^3 + 11*B^2*a^10*b^2 + 32*C^2*a^4*b^8 - 32*C^2*a^5*b^7 - 64*C^2*a^6*b^6 + 64*C^2*
a^7*b^5 + 20*C^2*a^8*b^4 - 32*C^2*a^9*b^3 + 16*C^2*a^10*b^2 - 192*A*B*a*b^11 - 4*A*B*a^11*b - 8*B*C*a^11*b + 1
92*A*B*a^2*b^10 + 304*A*B*a^3*b^9 - 304*A*B*a^4*b^8 - 28*A*B*a^5*b^7 + 40*A*B*a^6*b^6 - 52*A*B*a^7*b^5 + 64*A*
B*a^8*b^4 - 36*A*B*a^9*b^3 + 8*A*B*a^10*b^2 + 128*A*C*a^2*b^10 - 128*A*C*a^3*b^9 - 224*A*C*a^4*b^8 + 224*A*C*a
^5*b^7 + 40*A*C*a^6*b^6 - 64*A*C*a^7*b^5 + 48*A*C*a^8*b^4 - 32*A*C*a^9*b^3 + 16*A*C*a^10*b^2 - 96*B*C*a^3*b^9
+ 96*B*C*a^4*b^8 + 176*B*C*a^5*b^7 - 176*B*C*a^6*b^6 - 40*B*C*a^7*b^5 + 64*B*C*a^8*b^4 - 40*B*C*a^9*b^3 + 16*B
*C*a^10*b^2))/(a^10*b + a^11 - a^8*b^3 - a^9*b^2) - (b^2*((8*(2*B*a^18 + 16*A*a^10*b^8 - 8*A*a^11*b^7 - 36*A*a
^12*b^6 + 16*A*a^13*b^5 + 20*A*a^14*b^4 - 4*A*a^15*b^3 - 12*B*a^11*b^7 + 6*B*a^12*b^6 + 28*B*a^13*b^5 - 14*B*a
^14*b^4 - 16*B*a^15*b^3 + 6*B*a^16*b^2 + 8*C*a^12*b^6 - 4*C*a^13*b^5 - 20*C*a^14*b^4 + 12*C*a^15*b^3 + 12*C*a^
16*b^2 - 4*A*a^17*b - 8*C*a^17*b))/(a^14*b + a^15 - a^12*b^3 - a^13*b^2) - (8*b^2*tan(c/2 + (d*x)/2)*(-(a + b)
^3*(a - b)^3)^(1/2)*(8*a^15*b - 8*a^10*b^6 + 8*a^11*b^5 + 16*a^12*b^4 - 16*a^13*b^3 - 8*a^14*b^2)*(4*A*b^4 - 3
*C*a^4 - 5*A*a^2*b^2 + 2*C*a^2*b^2 - 3*B*a*b^3 + 4*B*a^3*b))/((a^10*b + a^11 - a^8*b^3 - a^9*b^2)*(a^11 - a^5*
b^6 + 3*a^7*b^4 - 3*a^9*b^2)))*(-(a + b)^3*(a - b)^3)^(1/2)*(4*A*b^4 - 3*C*a^4 - 5*A*a^2*b^2 + 2*C*a^2*b^2 - 3
*B*a*b^3 + 4*B*a^3*b))/(a^11 - a^5*b^6 + 3*a^7*b^4 - 3*a^9*b^2))*(-(a + b)^3*(a - b)^3)^(1/2)*(4*A*b^4 - 3*C*a
^4 - 5*A*a^2*b^2 + 2*C*a^2*b^2 - 3*B*a*b^3 + 4*B*a^3*b)*1i)/(a^11 - a^5*b^6 + 3*a^7*b^4 - 3*a^9*b^2))/((16*(25
6*A^3*b^14 - 128*A^3*a*b^13 - 448*A^3*a^2*b^12 + 192*A^3*a^3*b^11 + 48*A^3*a^4*b^10 - 24*A^3*a^5*b^9 + 124*A^3
*a^6*b^8 - 20*A^3*a^7*b^7 + 20*A^3*a^8*b^6 - 108*B^3*a^3*b^11 + 54*B^3*a^4*b^10 + 216*B^3*a^5*b^9 - 81*B^3*a^6
*b^8 - 63*B^3*a^7*b^7 + 9*B^3*a^8*b^6 - 41*B^3*a^9*b^5 + 4*B^3*a^10*b^4 - 4*B^3*a^11*b^3 + 32*C^3*a^6*b^8 - 16
*C^3*a^7*b^7 - 80*C^3*a^8*b^6 + 24*C^3*a^9*b^5 + 48*C^3*a^10*b^4 - 576*A^2*B*a*b^13 + 432*A*B^2*a^2*b^12 - 216
*A*B^2*a^3*b^11 - 828*A*B^2*a^4*b^10 + 324*A*B^2*a^5*b^9 + 192*A*B^2*a^6*b^8 - 39*A*B^2*a^7*b^7 + 183*A*B^2*a^
8*b^6 - 21*A*B^2*a^9*b^5 + 21*A*B^2*a^10*b^4 + 288*A^2*B*a^2*b^12 + 1056*A^2*B*a^3*b^11 - 432*A^2*B*a^4*b^10 -
 180*A^2*B*a^5*b^9 + 54*A^2*B*a^6*b^8 - 264*A^2*B*a^7*b^7 + 36*A^2*B*a^8*b^6 - 36*A^2*B*a^9*b^5 + 192*A*C^2*a^
4*b^10 - 96*A*C^2*a^5*b^9 - 432*A*C^2*a^6*b^8 + 144*A*C^2*a^7*b^7 + 192*A*C^2*a^8*b^6 - 12*A*C^2*a^9*b^5 + 48*
A*C^2*a^10*b^4 + 384*A^2*C*a^2*b^12 - 192*A^2*C*a^3*b^11 - 768*A^2*C*a^4*b^10 + 288*A^2*C*a^5*b^9 + 216*A^2*C*
a^6*b^8 - 36*A^2*C*a^7*b^7 + 156*A^2*C*a^8*b^6 - 12*A^2*C*a^9*b^5 + 12*A^2*C*a^10*b^4 - 144*B*C^2*a^5*b^9 + 72
*B*C^2*a^6*b^8 + 336*B*C^2*a^7*b^7 - 108*B*C^2*a^8*b^6 - 168*B*C^2*a^9*b^5 + 6*B*C^2*a^10*b^4 - 24*B*C^2*a^11*
b^3 + 216*B^2*C*a^4*b^10 - 108*B^2*C*a^5*b^9 - 468*B^2*C*a^6*b^8 + 162*B^2*C*a^7*b^7 + 186*B^2*C*a^8*b^6 - 15*
B^2*C*a^9*b^5 + 63*B^2*C*a^10*b^4 - 3*B^2*C*a^11*b^3 + 3*B^2*C*a^12*b^2 - 576*A*B*C*a^3*b^11 + 288*A*B*C*a^4*b
^10 + 1200*A*B*C*a^5*b^9 - 432*A*B*C*a^6*b^8 - 408*A*B*C*a^7*b^7 + 48*A*B*C*a^8*b^6 - 204*A*B*C*a^9*b^5 + 12*A
*B*C*a^10*b^4 - 12*A*B*C*a^11*b^3))/(a^14*b + a^15 - a^12*b^3 - a^13*b^2) - (b^2*((8*tan(c/2 + (d*x)/2)*(128*A
^2*b^12 + B^2*a^12 - 128*A^2*a*b^11 - 2*B^2*a^11*b - 192*A^2*a^2*b^10 + 192*A^2*a^3*b^9 + 8*A^2*a^4*b^8 - 8*A^
2*a^5*b^7 + 28*A^2*a^6*b^6 - 48*A^2*a^7*b^5 + 28*A^2*a^8*b^4 - 8*A^2*a^9*b^3 + 4*A^2*a^10*b^2 + 72*B^2*a^2*b^1
0 - 72*B^2*a^3*b^9 - 120*B^2*a^4*b^8 + 120*B^2*a^5*b^7 + 17*B^2*a^6*b^6 - 26*B^2*a^7*b^5 + 23*B^2*a^8*b^4 - 20
*B^2*a^9*b^3 + 11*B^2*a^10*b^2 + 32*C^2*a^4*b^8 - 32*C^2*a^5*b^7 - 64*C^2*a^6*b^6 + 64*C^2*a^7*b^5 + 20*C^2*a^
8*b^4 - 32*C^2*a^9*b^3 + 16*C^2*a^10*b^2 - 192*A*B*a*b^11 - 4*A*B*a^11*b - 8*B*C*a^11*b + 192*A*B*a^2*b^10 + 3
04*A*B*a^3*b^9 - 304*A*B*a^4*b^8 - 28*A*B*a^5*b^7 + 40*A*B*a^6*b^6 - 52*A*B*a^7*b^5 + 64*A*B*a^8*b^4 - 36*A*B*
a^9*b^3 + 8*A*B*a^10*b^2 + 128*A*C*a^2*b^10 - 128*A*C*a^3*b^9 - 224*A*C*a^4*b^8 + 224*A*C*a^5*b^7 + 40*A*C*a^6
*b^6 - 64*A*C*a^7*b^5 + 48*A*C*a^8*b^4 - 32*A*C*a^9*b^3 + 16*A*C*a^10*b^2 - 96*B*C*a^3*b^9 + 96*B*C*a^4*b^8 +
176*B*C*a^5*b^7 - 176*B*C*a^6*b^6 - 40*B*C*a^7*b^5 + 64*B*C*a^8*b^4 - 40*B*C*a^9*b^3 + 16*B*C*a^10*b^2))/(a^10
*b + a^11 - a^8*b^3 - a^9*b^2) + (b^2*((8*(2*B*a^18 + 16*A*a^10*b^8 - 8*A*a^11*b^7 - 36*A*a^12*b^6 + 16*A*a^13
*b^5 + 20*A*a^14*b^4 - 4*A*a^15*b^3 - 12*B*a^11*b^7 + 6*B*a^12*b^6 + 28*B*a^13*b^5 - 14*B*a^14*b^4 - 16*B*a^15
*b^3 + 6*B*a^16*b^2 + 8*C*a^12*b^6 - 4*C*a^13*b^5 - 20*C*a^14*b^4 + 12*C*a^15*b^3 + 12*C*a^16*b^2 - 4*A*a^17*b
 - 8*C*a^17*b))/(a^14*b + a^15 - a^12*b^3 - a^13*b^2) + (8*b^2*tan(c/2 + (d*x)/2)*(-(a + b)^3*(a - b)^3)^(1/2)
*(8*a^15*b - 8*a^10*b^6 + 8*a^11*b^5 + 16*a^12*b^4 - 16*a^13*b^3 - 8*a^14*b^2)*(4*A*b^4 - 3*C*a^4 - 5*A*a^2*b^
2 + 2*C*a^2*b^2 - 3*B*a*b^3 + 4*B*a^3*b))/((a^10*b + a^11 - a^8*b^3 - a^9*b^2)*(a^11 - a^5*b^6 + 3*a^7*b^4 - 3
*a^9*b^2)))*(-(a + b)^3*(a - b)^3)^(1/2)*(4*A*b^4 - 3*C*a^4 - 5*A*a^2*b^2 + 2*C*a^2*b^2 - 3*B*a*b^3 + 4*B*a^3*
b))/(a^11 - a^5*b^6 + 3*a^7*b^4 - 3*a^9*b^2))*(-(a + b)^3*(a - b)^3)^(1/2)*(4*A*b^4 - 3*C*a^4 - 5*A*a^2*b^2 +
2*C*a^2*b^2 - 3*B*a*b^3 + 4*B*a^3*b))/(a^11 - a^5*b^6 + 3*a^7*b^4 - 3*a^9*b^2) + (b^2*((8*tan(c/2 + (d*x)/2)*(
128*A^2*b^12 + B^2*a^12 - 128*A^2*a*b^11 - 2*B^2*a^11*b - 192*A^2*a^2*b^10 + 192*A^2*a^3*b^9 + 8*A^2*a^4*b^8 -
 8*A^2*a^5*b^7 + 28*A^2*a^6*b^6 - 48*A^2*a^7*b^5 + 28*A^2*a^8*b^4 - 8*A^2*a^9*b^3 + 4*A^2*a^10*b^2 + 72*B^2*a^
2*b^10 - 72*B^2*a^3*b^9 - 120*B^2*a^4*b^8 + 120*B^2*a^5*b^7 + 17*B^2*a^6*b^6 - 26*B^2*a^7*b^5 + 23*B^2*a^8*b^4
 - 20*B^2*a^9*b^3 + 11*B^2*a^10*b^2 + 32*C^2*a^4*b^8 - 32*C^2*a^5*b^7 - 64*C^2*a^6*b^6 + 64*C^2*a^7*b^5 + 20*C
^2*a^8*b^4 - 32*C^2*a^9*b^3 + 16*C^2*a^10*b^2 - 192*A*B*a*b^11 - 4*A*B*a^11*b - 8*B*C*a^11*b + 192*A*B*a^2*b^1
0 + 304*A*B*a^3*b^9 - 304*A*B*a^4*b^8 - 28*A*B*a^5*b^7 + 40*A*B*a^6*b^6 - 52*A*B*a^7*b^5 + 64*A*B*a^8*b^4 - 36
*A*B*a^9*b^3 + 8*A*B*a^10*b^2 + 128*A*C*a^2*b^10 - 128*A*C*a^3*b^9 - 224*A*C*a^4*b^8 + 224*A*C*a^5*b^7 + 40*A*
C*a^6*b^6 - 64*A*C*a^7*b^5 + 48*A*C*a^8*b^4 - 32*A*C*a^9*b^3 + 16*A*C*a^10*b^2 - 96*B*C*a^3*b^9 + 96*B*C*a^4*b
^8 + 176*B*C*a^5*b^7 - 176*B*C*a^6*b^6 - 40*B*C*a^7*b^5 + 64*B*C*a^8*b^4 - 40*B*C*a^9*b^3 + 16*B*C*a^10*b^2))/
(a^10*b + a^11 - a^8*b^3 - a^9*b^2) - (b^2*((8*(2*B*a^18 + 16*A*a^10*b^8 - 8*A*a^11*b^7 - 36*A*a^12*b^6 + 16*A
*a^13*b^5 + 20*A*a^14*b^4 - 4*A*a^15*b^3 - 12*B*a^11*b^7 + 6*B*a^12*b^6 + 28*B*a^13*b^5 - 14*B*a^14*b^4 - 16*B
*a^15*b^3 + 6*B*a^16*b^2 + 8*C*a^12*b^6 - 4*C*a^13*b^5 - 20*C*a^14*b^4 + 12*C*a^15*b^3 + 12*C*a^16*b^2 - 4*A*a
^17*b - 8*C*a^17*b))/(a^14*b + a^15 - a^12*b^3 - a^13*b^2) - (8*b^2*tan(c/2 + (d*x)/2)*(-(a + b)^3*(a - b)^3)^
(1/2)*(8*a^15*b - 8*a^10*b^6 + 8*a^11*b^5 + 16*a^12*b^4 - 16*a^13*b^3 - 8*a^14*b^2)*(4*A*b^4 - 3*C*a^4 - 5*A*a
^2*b^2 + 2*C*a^2*b^2 - 3*B*a*b^3 + 4*B*a^3*b))/((a^10*b + a^11 - a^8*b^3 - a^9*b^2)*(a^11 - a^5*b^6 + 3*a^7*b^
4 - 3*a^9*b^2)))*(-(a + b)^3*(a - b)^3)^(1/2)*(4*A*b^4 - 3*C*a^4 - 5*A*a^2*b^2 + 2*C*a^2*b^2 - 3*B*a*b^3 + 4*B
*a^3*b))/(a^11 - a^5*b^6 + 3*a^7*b^4 - 3*a^9*b^2))*(-(a + b)^3*(a - b)^3)^(1/2)*(4*A*b^4 - 3*C*a^4 - 5*A*a^2*b
^2 + 2*C*a^2*b^2 - 3*B*a*b^3 + 4*B*a^3*b))/(a^11 - a^5*b^6 + 3*a^7*b^4 - 3*a^9*b^2)))*(-(a + b)^3*(a - b)^3)^(
1/2)*(4*A*b^4 - 3*C*a^4 - 5*A*a^2*b^2 + 2*C*a^2*b^2 - 3*B*a*b^3 + 4*B*a^3*b)*2i)/(d*(a^11 - a^5*b^6 + 3*a^7*b^
4 - 3*a^9*b^2))

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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