\(\int \frac {(A+B \cos (c+d x)+C \cos ^2(c+d x)) \sec ^5(c+d x)}{a+b \cos (c+d x)} \, dx\) [985]

Optimal result

Integrand size = 41, antiderivative size = 285 \[ \int \frac {\left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^5(c+d x)}{a+b \cos (c+d x)} \, dx=-\frac {2 b^3 \left (A b^2-a (b B-a C)\right ) \arctan \left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{a^5 \sqrt {a-b} \sqrt {a+b} d}+\frac {\left (8 A b^4-4 a^3 b B-8 a b^3 B+4 a^2 b^2 (A+2 C)+a^4 (3 A+4 C)\right ) \text {arctanh}(\sin (c+d x))}{8 a^5 d}-\frac {\left (3 A b^3-2 a^3 B-3 a b^2 B+a^2 b (2 A+3 C)\right ) \tan (c+d x)}{3 a^4 d}+\frac {\left (4 A b^2-4 a b B+a^2 (3 A+4 C)\right ) \sec (c+d x) \tan (c+d x)}{8 a^3 d}-\frac {(A b-a B) \sec ^2(c+d x) \tan (c+d x)}{3 a^2 d}+\frac {A \sec ^3(c+d x) \tan (c+d x)}{4 a d} \]

[Out]

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

Rubi [A] (verified)

Time = 1.48 (sec) , antiderivative size = 285, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.122, Rules used = {3134, 3080, 3855, 2738, 211} \[ \int \frac {\left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^5(c+d x)}{a+b \cos (c+d x)} \, dx=-\frac {2 b^3 \left (A b^2-a (b B-a C)\right ) \arctan \left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{a^5 d \sqrt {a-b} \sqrt {a+b}}-\frac {(A b-a B) \tan (c+d x) \sec ^2(c+d x)}{3 a^2 d}+\frac {\tan (c+d x) \sec (c+d x) \left (a^2 (3 A+4 C)-4 a b B+4 A b^2\right )}{8 a^3 d}-\frac {\tan (c+d x) \left (-2 a^3 B+a^2 b (2 A+3 C)-3 a b^2 B+3 A b^3\right )}{3 a^4 d}+\frac {\left (a^4 (3 A+4 C)-4 a^3 b B+4 a^2 b^2 (A+2 C)-8 a b^3 B+8 A b^4\right ) \text {arctanh}(\sin (c+d x))}{8 a^5 d}+\frac {A \tan (c+d x) \sec ^3(c+d x)}{4 a d} \]

[In]

Int[((A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^5)/(a + b*Cos[c + d*x]),x]

[Out]

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

Rule 211

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

Rule 2738

Int[((a_) + (b_.)*sin[Pi/2 + (c_.) + (d_.)*(x_)])^(-1), x_Symbol] :> With[{e = FreeFactors[Tan[(c + d*x)/2], x
]}, Dist[2*(e/d), Subst[Int[1/(a + b + (a - b)*e^2*x^2), x], x, Tan[(c + d*x)/2]/e], x]] /; FreeQ[{a, b, c, d}
, x] && NeQ[a^2 - b^2, 0]

Rule 3080

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 3134

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] + D
ist[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] &&
LtQ[m, -1] && ((EqQ[a, 0] && IntegerQ[m] &&  !IntegerQ[n]) ||  !(IntegerQ[2*n] && LtQ[n, -1] && ((IntegerQ[n]
&&  !IntegerQ[m]) || EqQ[a, 0])))

Rule 3855

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

Rubi steps \begin{align*} \text {integral}& = \frac {A \sec ^3(c+d x) \tan (c+d x)}{4 a d}+\frac {\int \frac {\left (-4 (A b-a B)+a (3 A+4 C) \cos (c+d x)+3 A b \cos ^2(c+d x)\right ) \sec ^4(c+d x)}{a+b \cos (c+d x)} \, dx}{4 a} \\ & = -\frac {(A b-a B) \sec ^2(c+d x) \tan (c+d x)}{3 a^2 d}+\frac {A \sec ^3(c+d x) \tan (c+d x)}{4 a d}+\frac {\int \frac {\left (3 \left (4 b (A b-a B)+a^2 (3 A+4 C)\right )+a (A b+8 a B) \cos (c+d x)-8 b (A b-a B) \cos ^2(c+d x)\right ) \sec ^3(c+d x)}{a+b \cos (c+d x)} \, dx}{12 a^2} \\ & = \frac {\left (4 A b^2-4 a b B+a^2 (3 A+4 C)\right ) \sec (c+d x) \tan (c+d x)}{8 a^3 d}-\frac {(A b-a B) \sec ^2(c+d x) \tan (c+d x)}{3 a^2 d}+\frac {A \sec ^3(c+d x) \tan (c+d x)}{4 a d}+\frac {\int \frac {\left (-8 \left (3 A b^3-2 a^3 B-3 a b^2 B+a^2 b (2 A+3 C)\right )-a \left (4 A b^2-4 a b B-3 a^2 (3 A+4 C)\right ) \cos (c+d x)+3 b \left (4 A b^2-4 a b B+a^2 (3 A+4 C)\right ) \cos ^2(c+d x)\right ) \sec ^2(c+d x)}{a+b \cos (c+d x)} \, dx}{24 a^3} \\ & = -\frac {\left (3 A b^3-2 a^3 B-3 a b^2 B+a^2 b (2 A+3 C)\right ) \tan (c+d x)}{3 a^4 d}+\frac {\left (4 A b^2-4 a b B+a^2 (3 A+4 C)\right ) \sec (c+d x) \tan (c+d x)}{8 a^3 d}-\frac {(A b-a B) \sec ^2(c+d x) \tan (c+d x)}{3 a^2 d}+\frac {A \sec ^3(c+d x) \tan (c+d x)}{4 a d}+\frac {\int \frac {\left (3 \left (8 A b^4-4 a^3 b B-8 a b^3 B+4 a^2 b^2 (A+2 C)+a^4 (3 A+4 C)\right )+3 a b \left (4 A b^2-4 a b B+a^2 (3 A+4 C)\right ) \cos (c+d x)\right ) \sec (c+d x)}{a+b \cos (c+d x)} \, dx}{24 a^4} \\ & = -\frac {\left (3 A b^3-2 a^3 B-3 a b^2 B+a^2 b (2 A+3 C)\right ) \tan (c+d x)}{3 a^4 d}+\frac {\left (4 A b^2-4 a b B+a^2 (3 A+4 C)\right ) \sec (c+d x) \tan (c+d x)}{8 a^3 d}-\frac {(A b-a B) \sec ^2(c+d x) \tan (c+d x)}{3 a^2 d}+\frac {A \sec ^3(c+d x) \tan (c+d x)}{4 a d}+\frac {\left (8 A b^4-4 a^3 b B-8 a b^3 B+4 a^2 b^2 (A+2 C)+a^4 (3 A+4 C)\right ) \int \sec (c+d x) \, dx}{8 a^5}-\frac {\left (b^3 \left (A b^2-a (b B-a C)\right )\right ) \int \frac {1}{a+b \cos (c+d x)} \, dx}{a^5} \\ & = \frac {\left (8 A b^4-4 a^3 b B-8 a b^3 B+4 a^2 b^2 (A+2 C)+a^4 (3 A+4 C)\right ) \text {arctanh}(\sin (c+d x))}{8 a^5 d}-\frac {\left (3 A b^3-2 a^3 B-3 a b^2 B+a^2 b (2 A+3 C)\right ) \tan (c+d x)}{3 a^4 d}+\frac {\left (4 A b^2-4 a b B+a^2 (3 A+4 C)\right ) \sec (c+d x) \tan (c+d x)}{8 a^3 d}-\frac {(A b-a B) \sec ^2(c+d x) \tan (c+d x)}{3 a^2 d}+\frac {A \sec ^3(c+d x) \tan (c+d x)}{4 a d}-\frac {\left (2 b^3 \left (A b^2-a (b B-a C)\right )\right ) \text {Subst}\left (\int \frac {1}{a+b+(a-b) x^2} \, dx,x,\tan \left (\frac {1}{2} (c+d x)\right )\right )}{a^5 d} \\ & = -\frac {2 b^3 \left (A b^2-a (b B-a C)\right ) \arctan \left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{a^5 \sqrt {a-b} \sqrt {a+b} d}+\frac {\left (8 A b^4-4 a^3 b B-8 a b^3 B+4 a^2 b^2 (A+2 C)+a^4 (3 A+4 C)\right ) \text {arctanh}(\sin (c+d x))}{8 a^5 d}-\frac {\left (3 A b^3-2 a^3 B-3 a b^2 B+a^2 b (2 A+3 C)\right ) \tan (c+d x)}{3 a^4 d}+\frac {\left (4 A b^2-4 a b B+a^2 (3 A+4 C)\right ) \sec (c+d x) \tan (c+d x)}{8 a^3 d}-\frac {(A b-a B) \sec ^2(c+d x) \tan (c+d x)}{3 a^2 d}+\frac {A \sec ^3(c+d x) \tan (c+d x)}{4 a d} \\ \end{align*}

Mathematica [A] (verified)

Time = 2.62 (sec) , antiderivative size = 406, normalized size of antiderivative = 1.42 \[ \int \frac {\left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^5(c+d x)}{a+b \cos (c+d x)} \, dx=\frac {\frac {96 b^3 \left (A b^2+a (-b B+a C)\right ) \text {arctanh}\left (\frac {(a-b) \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {-a^2+b^2}}\right )}{\sqrt {-a^2+b^2}}-6 \left (8 A b^4-4 a^3 b B-8 a b^3 B+4 a^2 b^2 (A+2 C)+a^4 (3 A+4 C)\right ) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )+6 \left (8 A b^4-4 a^3 b B-8 a b^3 B+4 a^2 b^2 (A+2 C)+a^4 (3 A+4 C)\right ) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )+a \left (21 a^3 A+12 a A b^2-12 a^2 b B+12 a^3 C+4 \left (-9 A b^3+10 a^3 B+9 a b^2 B-a^2 b (10 A+9 C)\right ) \cos (c+d x)+3 a \left (4 A b^2-4 a b B+a^2 (3 A+4 C)\right ) \cos (2 (c+d x))-8 a^2 A b \cos (3 (c+d x))-12 A b^3 \cos (3 (c+d x))+8 a^3 B \cos (3 (c+d x))+12 a b^2 B \cos (3 (c+d x))-12 a^2 b C \cos (3 (c+d x))\right ) \sec ^3(c+d x) \tan (c+d x)}{48 a^5 d} \]

[In]

Integrate[((A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^5)/(a + b*Cos[c + d*x]),x]

[Out]

((96*b^3*(A*b^2 + a*(-(b*B) + a*C))*ArcTanh[((a - b)*Tan[(c + d*x)/2])/Sqrt[-a^2 + b^2]])/Sqrt[-a^2 + b^2] - 6
*(8*A*b^4 - 4*a^3*b*B - 8*a*b^3*B + 4*a^2*b^2*(A + 2*C) + a^4*(3*A + 4*C))*Log[Cos[(c + d*x)/2] - Sin[(c + d*x
)/2]] + 6*(8*A*b^4 - 4*a^3*b*B - 8*a*b^3*B + 4*a^2*b^2*(A + 2*C) + a^4*(3*A + 4*C))*Log[Cos[(c + d*x)/2] + Sin
[(c + d*x)/2]] + a*(21*a^3*A + 12*a*A*b^2 - 12*a^2*b*B + 12*a^3*C + 4*(-9*A*b^3 + 10*a^3*B + 9*a*b^2*B - a^2*b
*(10*A + 9*C))*Cos[c + d*x] + 3*a*(4*A*b^2 - 4*a*b*B + a^2*(3*A + 4*C))*Cos[2*(c + d*x)] - 8*a^2*A*b*Cos[3*(c
+ d*x)] - 12*A*b^3*Cos[3*(c + d*x)] + 8*a^3*B*Cos[3*(c + d*x)] + 12*a*b^2*B*Cos[3*(c + d*x)] - 12*a^2*b*C*Cos[
3*(c + d*x)])*Sec[c + d*x]^3*Tan[c + d*x])/(48*a^5*d)

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(565\) vs. \(2(266)=532\).

Time = 0.78 (sec) , antiderivative size = 566, normalized size of antiderivative = 1.99

method	result	size
derivativedivides	\(\frac {-\frac {A}{4 a \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{4}}-\frac {-3 a A -2 A b +2 B a}{6 a^{2} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{3}}-\frac {7 A \,a^{2}+4 a A b +4 A \,b^{2}-4 B \,a^{2}-4 B a b +4 a^{2} C}{8 a^{3} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}+\frac {\left (3 A \,a^{4}+4 A \,a^{2} b^{2}+8 A \,b^{4}-4 B \,a^{3} b -8 B a \,b^{3}+4 a^{4} C +8 C \,a^{2} b^{2}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{8 a^{5}}-\frac {-5 A \,a^{3}-8 A \,a^{2} b -4 a A \,b^{2}-8 A \,b^{3}+8 B \,a^{3}+4 B \,a^{2} b +8 B a \,b^{2}-4 a^{3} C -8 a^{2} b C}{8 a^{4} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}+\frac {A}{4 a \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{4}}-\frac {-3 a A -2 A b +2 B a}{6 a^{2} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{3}}-\frac {-7 A \,a^{2}-4 a A b -4 A \,b^{2}+4 B \,a^{2}+4 B a b -4 a^{2} C}{8 a^{3} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}+\frac {\left (-3 A \,a^{4}-4 A \,a^{2} b^{2}-8 A \,b^{4}+4 B \,a^{3} b +8 B a \,b^{3}-4 a^{4} C -8 C \,a^{2} b^{2}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{8 a^{5}}-\frac {-5 A \,a^{3}-8 A \,a^{2} b -4 a A \,b^{2}-8 A \,b^{3}+8 B \,a^{3}+4 B \,a^{2} b +8 B a \,b^{2}-4 a^{3} C -8 a^{2} b C}{8 a^{4} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}-\frac {2 b^{3} \left (A \,b^{2}-B a b +a^{2} C \right ) \arctan \left (\frac {\left (a -b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (a -b \right ) \left (a +b \right )}}\right )}{a^{5} \sqrt {\left (a -b \right ) \left (a +b \right )}}}{d}\)	\(566\)
default	\(\frac {-\frac {A}{4 a \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{4}}-\frac {-3 a A -2 A b +2 B a}{6 a^{2} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{3}}-\frac {7 A \,a^{2}+4 a A b +4 A \,b^{2}-4 B \,a^{2}-4 B a b +4 a^{2} C}{8 a^{3} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}+\frac {\left (3 A \,a^{4}+4 A \,a^{2} b^{2}+8 A \,b^{4}-4 B \,a^{3} b -8 B a \,b^{3}+4 a^{4} C +8 C \,a^{2} b^{2}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{8 a^{5}}-\frac {-5 A \,a^{3}-8 A \,a^{2} b -4 a A \,b^{2}-8 A \,b^{3}+8 B \,a^{3}+4 B \,a^{2} b +8 B a \,b^{2}-4 a^{3} C -8 a^{2} b C}{8 a^{4} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}+\frac {A}{4 a \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{4}}-\frac {-3 a A -2 A b +2 B a}{6 a^{2} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{3}}-\frac {-7 A \,a^{2}-4 a A b -4 A \,b^{2}+4 B \,a^{2}+4 B a b -4 a^{2} C}{8 a^{3} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}+\frac {\left (-3 A \,a^{4}-4 A \,a^{2} b^{2}-8 A \,b^{4}+4 B \,a^{3} b +8 B a \,b^{3}-4 a^{4} C -8 C \,a^{2} b^{2}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{8 a^{5}}-\frac {-5 A \,a^{3}-8 A \,a^{2} b -4 a A \,b^{2}-8 A \,b^{3}+8 B \,a^{3}+4 B \,a^{2} b +8 B a \,b^{2}-4 a^{3} C -8 a^{2} b C}{8 a^{4} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}-\frac {2 b^{3} \left (A \,b^{2}-B a b +a^{2} C \right ) \arctan \left (\frac {\left (a -b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (a -b \right ) \left (a +b \right )}}\right )}{a^{5} \sqrt {\left (a -b \right ) \left (a +b \right )}}}{d}\)	\(566\)
risch	\(\text {Expression too large to display}\)	\(1293\)

[In]

int((A+B*cos(d*x+c)+C*cos(d*x+c)^2)*sec(d*x+c)^5/(a+b*cos(d*x+c)),x,method=_RETURNVERBOSE)

[Out]

1/d*(-1/4*A/a/(tan(1/2*d*x+1/2*c)+1)^4-1/6*(-3*A*a-2*A*b+2*B*a)/a^2/(tan(1/2*d*x+1/2*c)+1)^3-1/8*(7*A*a^2+4*A*
a*b+4*A*b^2-4*B*a^2-4*B*a*b+4*C*a^2)/a^3/(tan(1/2*d*x+1/2*c)+1)^2+1/8*(3*A*a^4+4*A*a^2*b^2+8*A*b^4-4*B*a^3*b-8
*B*a*b^3+4*C*a^4+8*C*a^2*b^2)/a^5*ln(tan(1/2*d*x+1/2*c)+1)-1/8*(-5*A*a^3-8*A*a^2*b-4*A*a*b^2-8*A*b^3+8*B*a^3+4
*B*a^2*b+8*B*a*b^2-4*C*a^3-8*C*a^2*b)/a^4/(tan(1/2*d*x+1/2*c)+1)+1/4*A/a/(tan(1/2*d*x+1/2*c)-1)^4-1/6*(-3*A*a-
2*A*b+2*B*a)/a^2/(tan(1/2*d*x+1/2*c)-1)^3-1/8*(-7*A*a^2-4*A*a*b-4*A*b^2+4*B*a^2+4*B*a*b-4*C*a^2)/a^3/(tan(1/2*
d*x+1/2*c)-1)^2+1/8/a^5*(-3*A*a^4-4*A*a^2*b^2-8*A*b^4+4*B*a^3*b+8*B*a*b^3-4*C*a^4-8*C*a^2*b^2)*ln(tan(1/2*d*x+
1/2*c)-1)-1/8*(-5*A*a^3-8*A*a^2*b-4*A*a*b^2-8*A*b^3+8*B*a^3+4*B*a^2*b+8*B*a*b^2-4*C*a^3-8*C*a^2*b)/a^4/(tan(1/
2*d*x+1/2*c)-1)-2*b^3*(A*b^2-B*a*b+C*a^2)/a^5/((a-b)*(a+b))^(1/2)*arctan((a-b)*tan(1/2*d*x+1/2*c)/((a-b)*(a+b)
)^(1/2)))

Fricas [A] (verification not implemented)

none

Time = 28.22 (sec) , antiderivative size = 993, normalized size of antiderivative = 3.48 \[ \int \frac {\left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^5(c+d x)}{a+b \cos (c+d x)} \, dx=\text {Too large to display} \]

[In]

integrate((A+B*cos(d*x+c)+C*cos(d*x+c)^2)*sec(d*x+c)^5/(a+b*cos(d*x+c)),x, algorithm="fricas")

[Out]

[-1/48*(24*(C*a^2*b^3 - B*a*b^4 + A*b^5)*sqrt(-a^2 + b^2)*cos(d*x + c)^4*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*((3*A + 4*C)*a^6 - 4*B*a^5*b + (A + 4*C)*a^4*b^2 - 4*B*a^3*b^3 + 4*(A - 2*C)*a^
2*b^4 + 8*B*a*b^5 - 8*A*b^6)*cos(d*x + c)^4*log(sin(d*x + c) + 1) + 3*((3*A + 4*C)*a^6 - 4*B*a^5*b + (A + 4*C)
*a^4*b^2 - 4*B*a^3*b^3 + 4*(A - 2*C)*a^2*b^4 + 8*B*a*b^5 - 8*A*b^6)*cos(d*x + c)^4*log(-sin(d*x + c) + 1) - 2*
(6*A*a^6 - 6*A*a^4*b^2 + 8*(2*B*a^6 - (2*A + 3*C)*a^5*b + B*a^4*b^2 - (A - 3*C)*a^3*b^3 - 3*B*a^2*b^4 + 3*A*a*
b^5)*cos(d*x + c)^3 + 3*((3*A + 4*C)*a^6 - 4*B*a^5*b + (A - 4*C)*a^4*b^2 + 4*B*a^3*b^3 - 4*A*a^2*b^4)*cos(d*x
+ c)^2 + 8*(B*a^6 - A*a^5*b - B*a^4*b^2 + A*a^3*b^3)*cos(d*x + c))*sin(d*x + c))/((a^7 - a^5*b^2)*d*cos(d*x +
c)^4), -1/48*(48*(C*a^2*b^3 - B*a*b^4 + A*b^5)*sqrt(a^2 - b^2)*arctan(-(a*cos(d*x + c) + b)/(sqrt(a^2 - b^2)*s
in(d*x + c)))*cos(d*x + c)^4 - 3*((3*A + 4*C)*a^6 - 4*B*a^5*b + (A + 4*C)*a^4*b^2 - 4*B*a^3*b^3 + 4*(A - 2*C)*
a^2*b^4 + 8*B*a*b^5 - 8*A*b^6)*cos(d*x + c)^4*log(sin(d*x + c) + 1) + 3*((3*A + 4*C)*a^6 - 4*B*a^5*b + (A + 4*
C)*a^4*b^2 - 4*B*a^3*b^3 + 4*(A - 2*C)*a^2*b^4 + 8*B*a*b^5 - 8*A*b^6)*cos(d*x + c)^4*log(-sin(d*x + c) + 1) -
2*(6*A*a^6 - 6*A*a^4*b^2 + 8*(2*B*a^6 - (2*A + 3*C)*a^5*b + B*a^4*b^2 - (A - 3*C)*a^3*b^3 - 3*B*a^2*b^4 + 3*A*
a*b^5)*cos(d*x + c)^3 + 3*((3*A + 4*C)*a^6 - 4*B*a^5*b + (A - 4*C)*a^4*b^2 + 4*B*a^3*b^3 - 4*A*a^2*b^4)*cos(d*
x + c)^2 + 8*(B*a^6 - A*a^5*b - B*a^4*b^2 + A*a^3*b^3)*cos(d*x + c))*sin(d*x + c))/((a^7 - a^5*b^2)*d*cos(d*x
+ c)^4)]

Sympy [F]

\[ \int \frac {\left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^5(c+d x)}{a+b \cos (c+d x)} \, dx=\int \frac {\left (A + B \cos {\left (c + d x \right )} + C \cos ^{2}{\left (c + d x \right )}\right ) \sec ^{5}{\left (c + d x \right )}}{a + b \cos {\left (c + d x \right )}}\, dx \]

[In]

integrate((A+B*cos(d*x+c)+C*cos(d*x+c)**2)*sec(d*x+c)**5/(a+b*cos(d*x+c)),x)

[Out]

Integral((A + B*cos(c + d*x) + C*cos(c + d*x)**2)*sec(c + d*x)**5/(a + b*cos(c + d*x)), x)

Maxima [F(-2)]

Exception generated. \[ \int \frac {\left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^5(c+d x)}{a+b \cos (c+d x)} \, dx=\text {Exception raised: ValueError} \]

[In]

integrate((A+B*cos(d*x+c)+C*cos(d*x+c)^2)*sec(d*x+c)^5/(a+b*cos(d*x+c)),x, algorithm="maxima")

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(4*b^2-4*a^2>0)', see `assume?`
for more de

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 878 vs. \(2 (266) = 532\).

Time = 0.36 (sec) , antiderivative size = 878, normalized size of antiderivative = 3.08 \[ \int \frac {\left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^5(c+d x)}{a+b \cos (c+d x)} \, dx=\text {Too large to display} \]

[In]

integrate((A+B*cos(d*x+c)+C*cos(d*x+c)^2)*sec(d*x+c)^5/(a+b*cos(d*x+c)),x, algorithm="giac")

[Out]

1/24*(3*(3*A*a^4 + 4*C*a^4 - 4*B*a^3*b + 4*A*a^2*b^2 + 8*C*a^2*b^2 - 8*B*a*b^3 + 8*A*b^4)*log(abs(tan(1/2*d*x
+ 1/2*c) + 1))/a^5 - 3*(3*A*a^4 + 4*C*a^4 - 4*B*a^3*b + 4*A*a^2*b^2 + 8*C*a^2*b^2 - 8*B*a*b^3 + 8*A*b^4)*log(a
bs(tan(1/2*d*x + 1/2*c) - 1))/a^5 + 48*(C*a^2*b^3 - B*a*b^4 + A*b^5)*(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)))/(sqrt(a^2 - b^2)*a^5) +
2*(15*A*a^3*tan(1/2*d*x + 1/2*c)^7 - 24*B*a^3*tan(1/2*d*x + 1/2*c)^7 + 12*C*a^3*tan(1/2*d*x + 1/2*c)^7 + 24*A
*a^2*b*tan(1/2*d*x + 1/2*c)^7 - 12*B*a^2*b*tan(1/2*d*x + 1/2*c)^7 + 24*C*a^2*b*tan(1/2*d*x + 1/2*c)^7 + 12*A*a
*b^2*tan(1/2*d*x + 1/2*c)^7 - 24*B*a*b^2*tan(1/2*d*x + 1/2*c)^7 + 24*A*b^3*tan(1/2*d*x + 1/2*c)^7 + 9*A*a^3*ta
n(1/2*d*x + 1/2*c)^5 + 40*B*a^3*tan(1/2*d*x + 1/2*c)^5 - 12*C*a^3*tan(1/2*d*x + 1/2*c)^5 - 40*A*a^2*b*tan(1/2*
d*x + 1/2*c)^5 + 12*B*a^2*b*tan(1/2*d*x + 1/2*c)^5 - 72*C*a^2*b*tan(1/2*d*x + 1/2*c)^5 - 12*A*a*b^2*tan(1/2*d*
x + 1/2*c)^5 + 72*B*a*b^2*tan(1/2*d*x + 1/2*c)^5 - 72*A*b^3*tan(1/2*d*x + 1/2*c)^5 + 9*A*a^3*tan(1/2*d*x + 1/2
*c)^3 - 40*B*a^3*tan(1/2*d*x + 1/2*c)^3 - 12*C*a^3*tan(1/2*d*x + 1/2*c)^3 + 40*A*a^2*b*tan(1/2*d*x + 1/2*c)^3
+ 12*B*a^2*b*tan(1/2*d*x + 1/2*c)^3 + 72*C*a^2*b*tan(1/2*d*x + 1/2*c)^3 - 12*A*a*b^2*tan(1/2*d*x + 1/2*c)^3 -
72*B*a*b^2*tan(1/2*d*x + 1/2*c)^3 + 72*A*b^3*tan(1/2*d*x + 1/2*c)^3 + 15*A*a^3*tan(1/2*d*x + 1/2*c) + 24*B*a^3
*tan(1/2*d*x + 1/2*c) + 12*C*a^3*tan(1/2*d*x + 1/2*c) - 24*A*a^2*b*tan(1/2*d*x + 1/2*c) - 12*B*a^2*b*tan(1/2*d
*x + 1/2*c) - 24*C*a^2*b*tan(1/2*d*x + 1/2*c) + 12*A*a*b^2*tan(1/2*d*x + 1/2*c) + 24*B*a*b^2*tan(1/2*d*x + 1/2
*c) - 24*A*b^3*tan(1/2*d*x + 1/2*c))/((tan(1/2*d*x + 1/2*c)^2 - 1)^4*a^4))/d

Mupad [B] (verification not implemented)

Time = 12.55 (sec) , antiderivative size = 9543, normalized size of antiderivative = 33.48 \[ \int \frac {\left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^5(c+d x)}{a+b \cos (c+d x)} \, dx=\text {Too large to display} \]

[In]

int((A + B*cos(c + d*x) + C*cos(c + d*x)^2)/(cos(c + d*x)^5*(a + b*cos(c + d*x))),x)

[Out]

((tan(c/2 + (d*x)/2)^7*(5*A*a^3 + 8*A*b^3 - 8*B*a^3 + 4*C*a^3 + 4*A*a*b^2 + 8*A*a^2*b - 8*B*a*b^2 - 4*B*a^2*b
+ 8*C*a^2*b))/(4*a^4) + (tan(c/2 + (d*x)/2)^3*(9*A*a^3 + 72*A*b^3 - 40*B*a^3 - 12*C*a^3 - 12*A*a*b^2 + 40*A*a^
2*b - 72*B*a*b^2 + 12*B*a^2*b + 72*C*a^2*b))/(12*a^4) - (tan(c/2 + (d*x)/2)^5*(72*A*b^3 - 9*A*a^3 - 40*B*a^3 +
12*C*a^3 + 12*A*a*b^2 + 40*A*a^2*b - 72*B*a*b^2 - 12*B*a^2*b + 72*C*a^2*b))/(12*a^4) + (tan(c/2 + (d*x)/2)*(5
*A*a^3 - 8*A*b^3 + 8*B*a^3 + 4*C*a^3 + 4*A*a*b^2 - 8*A*a^2*b + 8*B*a*b^2 - 4*B*a^2*b - 8*C*a^2*b))/(4*a^4))/(d
*(6*tan(c/2 + (d*x)/2)^4 - 4*tan(c/2 + (d*x)/2)^2 - 4*tan(c/2 + (d*x)/2)^6 + tan(c/2 + (d*x)/2)^8 + 1)) + (ata
n(((((tan(c/2 + (d*x)/2)*(9*A^2*a^11 - 128*A^2*b^11 + 16*C^2*a^11 + 256*A^2*a*b^10 - 27*A^2*a^10*b - 48*C^2*a^
10*b - 256*A^2*a^2*b^9 + 256*A^2*a^3*b^8 - 256*A^2*a^4*b^7 + 256*A^2*a^5*b^6 - 216*A^2*a^6*b^5 + 136*A^2*a^7*b
^4 - 81*A^2*a^8*b^3 + 51*A^2*a^9*b^2 - 128*B^2*a^2*b^9 + 256*B^2*a^3*b^8 - 256*B^2*a^4*b^7 + 256*B^2*a^5*b^6 -
208*B^2*a^6*b^5 + 112*B^2*a^7*b^4 - 48*B^2*a^8*b^3 + 16*B^2*a^9*b^2 - 128*C^2*a^4*b^7 + 256*C^2*a^5*b^6 - 256
*C^2*a^6*b^5 + 256*C^2*a^7*b^4 - 208*C^2*a^8*b^3 + 112*C^2*a^9*b^2 + 24*A*C*a^11 + 256*A*B*a*b^10 - 24*A*B*a^1
0*b - 72*A*C*a^10*b - 32*B*C*a^10*b - 512*A*B*a^2*b^9 + 512*A*B*a^3*b^8 - 512*A*B*a^4*b^7 + 464*A*B*a^5*b^6 -
368*A*B*a^6*b^5 + 264*A*B*a^7*b^4 - 152*A*B*a^8*b^3 + 72*A*B*a^9*b^2 - 256*A*C*a^2*b^9 + 512*A*C*a^3*b^8 - 512
*A*C*a^4*b^7 + 512*A*C*a^5*b^6 - 464*A*C*a^6*b^5 + 368*A*C*a^7*b^4 - 264*A*C*a^8*b^3 + 152*A*C*a^9*b^2 + 256*B
*C*a^3*b^8 - 512*B*C*a^4*b^7 + 512*B*C*a^5*b^6 - 512*B*C*a^6*b^5 + 416*B*C*a^7*b^4 - 224*B*C*a^8*b^3 + 96*B*C*
a^9*b^2))/(2*a^8) + (((12*A*a^16 + 16*C*a^16 + 32*A*a^10*b^6 - 48*A*a^11*b^5 + 16*A*a^12*b^4 - 4*A*a^13*b^3 +
4*A*a^14*b^2 - 32*B*a^11*b^5 + 48*B*a^12*b^4 - 16*B*a^13*b^3 + 16*B*a^14*b^2 + 32*C*a^12*b^4 - 48*C*a^13*b^3 +
16*C*a^14*b^2 - 12*A*a^15*b - 16*B*a^15*b - 16*C*a^15*b)/a^12 + (tan(c/2 + (d*x)/2)*(128*a^12*b + 128*a^10*b^
3 - 256*a^11*b^2)*(a^2*((A*b^2)/2 + C*b^2) + A*b^4 + a^4*((3*A)/8 + C/2) - B*a*b^3 - (B*a^3*b)/2))/(2*a^13))*(
a^2*((A*b^2)/2 + C*b^2) + A*b^4 + a^4*((3*A)/8 + C/2) - B*a*b^3 - (B*a^3*b)/2))/a^5)*(a^2*((A*b^2)/2 + C*b^2)
+ A*b^4 + a^4*((3*A)/8 + C/2) - B*a*b^3 - (B*a^3*b)/2)*1i)/a^5 + (((tan(c/2 + (d*x)/2)*(9*A^2*a^11 - 128*A^2*b
^11 + 16*C^2*a^11 + 256*A^2*a*b^10 - 27*A^2*a^10*b - 48*C^2*a^10*b - 256*A^2*a^2*b^9 + 256*A^2*a^3*b^8 - 256*A
^2*a^4*b^7 + 256*A^2*a^5*b^6 - 216*A^2*a^6*b^5 + 136*A^2*a^7*b^4 - 81*A^2*a^8*b^3 + 51*A^2*a^9*b^2 - 128*B^2*a
^2*b^9 + 256*B^2*a^3*b^8 - 256*B^2*a^4*b^7 + 256*B^2*a^5*b^6 - 208*B^2*a^6*b^5 + 112*B^2*a^7*b^4 - 48*B^2*a^8*
b^3 + 16*B^2*a^9*b^2 - 128*C^2*a^4*b^7 + 256*C^2*a^5*b^6 - 256*C^2*a^6*b^5 + 256*C^2*a^7*b^4 - 208*C^2*a^8*b^3
+ 112*C^2*a^9*b^2 + 24*A*C*a^11 + 256*A*B*a*b^10 - 24*A*B*a^10*b - 72*A*C*a^10*b - 32*B*C*a^10*b - 512*A*B*a^
2*b^9 + 512*A*B*a^3*b^8 - 512*A*B*a^4*b^7 + 464*A*B*a^5*b^6 - 368*A*B*a^6*b^5 + 264*A*B*a^7*b^4 - 152*A*B*a^8*
b^3 + 72*A*B*a^9*b^2 - 256*A*C*a^2*b^9 + 512*A*C*a^3*b^8 - 512*A*C*a^4*b^7 + 512*A*C*a^5*b^6 - 464*A*C*a^6*b^5
+ 368*A*C*a^7*b^4 - 264*A*C*a^8*b^3 + 152*A*C*a^9*b^2 + 256*B*C*a^3*b^8 - 512*B*C*a^4*b^7 + 512*B*C*a^5*b^6 -
512*B*C*a^6*b^5 + 416*B*C*a^7*b^4 - 224*B*C*a^8*b^3 + 96*B*C*a^9*b^2))/(2*a^8) - (((12*A*a^16 + 16*C*a^16 + 3
2*A*a^10*b^6 - 48*A*a^11*b^5 + 16*A*a^12*b^4 - 4*A*a^13*b^3 + 4*A*a^14*b^2 - 32*B*a^11*b^5 + 48*B*a^12*b^4 - 1
6*B*a^13*b^3 + 16*B*a^14*b^2 + 32*C*a^12*b^4 - 48*C*a^13*b^3 + 16*C*a^14*b^2 - 12*A*a^15*b - 16*B*a^15*b - 16*
C*a^15*b)/a^12 - (tan(c/2 + (d*x)/2)*(128*a^12*b + 128*a^10*b^3 - 256*a^11*b^2)*(a^2*((A*b^2)/2 + C*b^2) + A*b
^4 + a^4*((3*A)/8 + C/2) - B*a*b^3 - (B*a^3*b)/2))/(2*a^13))*(a^2*((A*b^2)/2 + C*b^2) + A*b^4 + a^4*((3*A)/8 +
C/2) - B*a*b^3 - (B*a^3*b)/2))/a^5)*(a^2*((A*b^2)/2 + C*b^2) + A*b^4 + a^4*((3*A)/8 + C/2) - B*a*b^3 - (B*a^3
*b)/2)*1i)/a^5)/((64*A^3*b^14 - 96*A^3*a*b^13 + 96*A^3*a^2*b^12 - 104*A^3*a^3*b^11 + 104*A^3*a^4*b^10 - 88*A^3
*a^5*b^9 + 48*A^3*a^6*b^8 - 33*A^3*a^7*b^7 + 18*A^3*a^8*b^6 - 9*A^3*a^9*b^5 - 64*B^3*a^3*b^11 + 96*B^3*a^4*b^1
0 - 96*B^3*a^5*b^9 + 80*B^3*a^6*b^8 - 32*B^3*a^7*b^7 + 16*B^3*a^8*b^6 + 64*C^3*a^6*b^8 - 96*C^3*a^7*b^7 + 96*C
^3*a^8*b^6 - 80*C^3*a^9*b^5 + 32*C^3*a^10*b^4 - 16*C^3*a^11*b^3 - 192*A^2*B*a*b^13 + 192*A*B^2*a^2*b^12 - 288*
A*B^2*a^3*b^11 + 288*A*B^2*a^4*b^10 - 264*A*B^2*a^5*b^9 + 168*A*B^2*a^6*b^8 - 120*A*B^2*a^7*b^7 + 48*A*B^2*a^8
*b^6 - 24*A*B^2*a^9*b^5 + 288*A^2*B*a^2*b^12 - 288*A^2*B*a^3*b^11 + 288*A^2*B*a^4*b^10 - 240*A^2*B*a^5*b^9 + 1
92*A^2*B*a^6*b^8 - 96*A^2*B*a^7*b^7 + 57*A^2*B*a^8*b^6 - 18*A^2*B*a^9*b^5 + 9*A^2*B*a^10*b^4 + 192*A*C^2*a^4*b
^10 - 288*A*C^2*a^5*b^9 + 288*A*C^2*a^6*b^8 - 264*A*C^2*a^7*b^7 + 168*A*C^2*a^8*b^6 - 120*A*C^2*a^9*b^5 + 48*A
*C^2*a^10*b^4 - 24*A*C^2*a^11*b^3 + 192*A^2*C*a^2*b^12 - 288*A^2*C*a^3*b^11 + 288*A^2*C*a^4*b^10 - 288*A^2*C*a
^5*b^9 + 240*A^2*C*a^6*b^8 - 192*A^2*C*a^7*b^7 + 96*A^2*C*a^8*b^6 - 57*A^2*C*a^9*b^5 + 18*A^2*C*a^10*b^4 - 9*A
^2*C*a^11*b^3 - 192*B*C^2*a^5*b^9 + 288*B*C^2*a^6*b^8 - 288*B*C^2*a^7*b^7 + 240*B*C^2*a^8*b^6 - 96*B*C^2*a^9*b
^5 + 48*B*C^2*a^10*b^4 + 192*B^2*C*a^4*b^10 - 288*B^2*C*a^5*b^9 + 288*B^2*C*a^6*b^8 - 240*B^2*C*a^7*b^7 + 96*B
^2*C*a^8*b^6 - 48*B^2*C*a^9*b^5 - 384*A*B*C*a^3*b^11 + 576*A*B*C*a^4*b^10 - 576*A*B*C*a^5*b^9 + 528*A*B*C*a^6*
b^8 - 336*A*B*C*a^7*b^7 + 240*A*B*C*a^8*b^6 - 96*A*B*C*a^9*b^5 + 48*A*B*C*a^10*b^4)/a^12 - (((tan(c/2 + (d*x)/
2)*(9*A^2*a^11 - 128*A^2*b^11 + 16*C^2*a^11 + 256*A^2*a*b^10 - 27*A^2*a^10*b - 48*C^2*a^10*b - 256*A^2*a^2*b^9
+ 256*A^2*a^3*b^8 - 256*A^2*a^4*b^7 + 256*A^2*a^5*b^6 - 216*A^2*a^6*b^5 + 136*A^2*a^7*b^4 - 81*A^2*a^8*b^3 +
51*A^2*a^9*b^2 - 128*B^2*a^2*b^9 + 256*B^2*a^3*b^8 - 256*B^2*a^4*b^7 + 256*B^2*a^5*b^6 - 208*B^2*a^6*b^5 + 112
*B^2*a^7*b^4 - 48*B^2*a^8*b^3 + 16*B^2*a^9*b^2 - 128*C^2*a^4*b^7 + 256*C^2*a^5*b^6 - 256*C^2*a^6*b^5 + 256*C^2
*a^7*b^4 - 208*C^2*a^8*b^3 + 112*C^2*a^9*b^2 + 24*A*C*a^11 + 256*A*B*a*b^10 - 24*A*B*a^10*b - 72*A*C*a^10*b -
32*B*C*a^10*b - 512*A*B*a^2*b^9 + 512*A*B*a^3*b^8 - 512*A*B*a^4*b^7 + 464*A*B*a^5*b^6 - 368*A*B*a^6*b^5 + 264*
A*B*a^7*b^4 - 152*A*B*a^8*b^3 + 72*A*B*a^9*b^2 - 256*A*C*a^2*b^9 + 512*A*C*a^3*b^8 - 512*A*C*a^4*b^7 + 512*A*C
*a^5*b^6 - 464*A*C*a^6*b^5 + 368*A*C*a^7*b^4 - 264*A*C*a^8*b^3 + 152*A*C*a^9*b^2 + 256*B*C*a^3*b^8 - 512*B*C*a
^4*b^7 + 512*B*C*a^5*b^6 - 512*B*C*a^6*b^5 + 416*B*C*a^7*b^4 - 224*B*C*a^8*b^3 + 96*B*C*a^9*b^2))/(2*a^8) + ((
(12*A*a^16 + 16*C*a^16 + 32*A*a^10*b^6 - 48*A*a^11*b^5 + 16*A*a^12*b^4 - 4*A*a^13*b^3 + 4*A*a^14*b^2 - 32*B*a^
11*b^5 + 48*B*a^12*b^4 - 16*B*a^13*b^3 + 16*B*a^14*b^2 + 32*C*a^12*b^4 - 48*C*a^13*b^3 + 16*C*a^14*b^2 - 12*A*
a^15*b - 16*B*a^15*b - 16*C*a^15*b)/a^12 + (tan(c/2 + (d*x)/2)*(128*a^12*b + 128*a^10*b^3 - 256*a^11*b^2)*(a^2
*((A*b^2)/2 + C*b^2) + A*b^4 + a^4*((3*A)/8 + C/2) - B*a*b^3 - (B*a^3*b)/2))/(2*a^13))*(a^2*((A*b^2)/2 + C*b^2
) + A*b^4 + a^4*((3*A)/8 + C/2) - B*a*b^3 - (B*a^3*b)/2))/a^5)*(a^2*((A*b^2)/2 + C*b^2) + A*b^4 + a^4*((3*A)/8
+ C/2) - B*a*b^3 - (B*a^3*b)/2))/a^5 + (((tan(c/2 + (d*x)/2)*(9*A^2*a^11 - 128*A^2*b^11 + 16*C^2*a^11 + 256*A
^2*a*b^10 - 27*A^2*a^10*b - 48*C^2*a^10*b - 256*A^2*a^2*b^9 + 256*A^2*a^3*b^8 - 256*A^2*a^4*b^7 + 256*A^2*a^5*
b^6 - 216*A^2*a^6*b^5 + 136*A^2*a^7*b^4 - 81*A^2*a^8*b^3 + 51*A^2*a^9*b^2 - 128*B^2*a^2*b^9 + 256*B^2*a^3*b^8
- 256*B^2*a^4*b^7 + 256*B^2*a^5*b^6 - 208*B^2*a^6*b^5 + 112*B^2*a^7*b^4 - 48*B^2*a^8*b^3 + 16*B^2*a^9*b^2 - 12
8*C^2*a^4*b^7 + 256*C^2*a^5*b^6 - 256*C^2*a^6*b^5 + 256*C^2*a^7*b^4 - 208*C^2*a^8*b^3 + 112*C^2*a^9*b^2 + 24*A
*C*a^11 + 256*A*B*a*b^10 - 24*A*B*a^10*b - 72*A*C*a^10*b - 32*B*C*a^10*b - 512*A*B*a^2*b^9 + 512*A*B*a^3*b^8 -
512*A*B*a^4*b^7 + 464*A*B*a^5*b^6 - 368*A*B*a^6*b^5 + 264*A*B*a^7*b^4 - 152*A*B*a^8*b^3 + 72*A*B*a^9*b^2 - 25
6*A*C*a^2*b^9 + 512*A*C*a^3*b^8 - 512*A*C*a^4*b^7 + 512*A*C*a^5*b^6 - 464*A*C*a^6*b^5 + 368*A*C*a^7*b^4 - 264*
A*C*a^8*b^3 + 152*A*C*a^9*b^2 + 256*B*C*a^3*b^8 - 512*B*C*a^4*b^7 + 512*B*C*a^5*b^6 - 512*B*C*a^6*b^5 + 416*B*
C*a^7*b^4 - 224*B*C*a^8*b^3 + 96*B*C*a^9*b^2))/(2*a^8) - (((12*A*a^16 + 16*C*a^16 + 32*A*a^10*b^6 - 48*A*a^11*
b^5 + 16*A*a^12*b^4 - 4*A*a^13*b^3 + 4*A*a^14*b^2 - 32*B*a^11*b^5 + 48*B*a^12*b^4 - 16*B*a^13*b^3 + 16*B*a^14*
b^2 + 32*C*a^12*b^4 - 48*C*a^13*b^3 + 16*C*a^14*b^2 - 12*A*a^15*b - 16*B*a^15*b - 16*C*a^15*b)/a^12 - (tan(c/2
+ (d*x)/2)*(128*a^12*b + 128*a^10*b^3 - 256*a^11*b^2)*(a^2*((A*b^2)/2 + C*b^2) + A*b^4 + a^4*((3*A)/8 + C/2)
- B*a*b^3 - (B*a^3*b)/2))/(2*a^13))*(a^2*((A*b^2)/2 + C*b^2) + A*b^4 + a^4*((3*A)/8 + C/2) - B*a*b^3 - (B*a^3*
b)/2))/a^5)*(a^2*((A*b^2)/2 + C*b^2) + A*b^4 + a^4*((3*A)/8 + C/2) - B*a*b^3 - (B*a^3*b)/2))/a^5))*(a^2*((A*b^
2)/2 + C*b^2) + A*b^4 + a^4*((3*A)/8 + C/2) - B*a*b^3 - (B*a^3*b)/2)*2i)/(a^5*d) + (b^3*atan(((b^3*(-(a + b)*(
a - b))^(1/2)*((tan(c/2 + (d*x)/2)*(9*A^2*a^11 - 128*A^2*b^11 + 16*C^2*a^11 + 256*A^2*a*b^10 - 27*A^2*a^10*b -
48*C^2*a^10*b - 256*A^2*a^2*b^9 + 256*A^2*a^3*b^8 - 256*A^2*a^4*b^7 + 256*A^2*a^5*b^6 - 216*A^2*a^6*b^5 + 136
*A^2*a^7*b^4 - 81*A^2*a^8*b^3 + 51*A^2*a^9*b^2 - 128*B^2*a^2*b^9 + 256*B^2*a^3*b^8 - 256*B^2*a^4*b^7 + 256*B^2
*a^5*b^6 - 208*B^2*a^6*b^5 + 112*B^2*a^7*b^4 - 48*B^2*a^8*b^3 + 16*B^2*a^9*b^2 - 128*C^2*a^4*b^7 + 256*C^2*a^5
*b^6 - 256*C^2*a^6*b^5 + 256*C^2*a^7*b^4 - 208*C^2*a^8*b^3 + 112*C^2*a^9*b^2 + 24*A*C*a^11 + 256*A*B*a*b^10 -
24*A*B*a^10*b - 72*A*C*a^10*b - 32*B*C*a^10*b - 512*A*B*a^2*b^9 + 512*A*B*a^3*b^8 - 512*A*B*a^4*b^7 + 464*A*B*
a^5*b^6 - 368*A*B*a^6*b^5 + 264*A*B*a^7*b^4 - 152*A*B*a^8*b^3 + 72*A*B*a^9*b^2 - 256*A*C*a^2*b^9 + 512*A*C*a^3
*b^8 - 512*A*C*a^4*b^7 + 512*A*C*a^5*b^6 - 464*A*C*a^6*b^5 + 368*A*C*a^7*b^4 - 264*A*C*a^8*b^3 + 152*A*C*a^9*b
^2 + 256*B*C*a^3*b^8 - 512*B*C*a^4*b^7 + 512*B*C*a^5*b^6 - 512*B*C*a^6*b^5 + 416*B*C*a^7*b^4 - 224*B*C*a^8*b^3
+ 96*B*C*a^9*b^2))/(2*a^8) + (b^3*(-(a + b)*(a - b))^(1/2)*((12*A*a^16 + 16*C*a^16 + 32*A*a^10*b^6 - 48*A*a^1
1*b^5 + 16*A*a^12*b^4 - 4*A*a^13*b^3 + 4*A*a^14*b^2 - 32*B*a^11*b^5 + 48*B*a^12*b^4 - 16*B*a^13*b^3 + 16*B*a^1
4*b^2 + 32*C*a^12*b^4 - 48*C*a^13*b^3 + 16*C*a^14*b^2 - 12*A*a^15*b - 16*B*a^15*b - 16*C*a^15*b)/a^12 + (b^3*t
an(c/2 + (d*x)/2)*(-(a + b)*(a - b))^(1/2)*(A*b^2 + C*a^2 - B*a*b)*(128*a^12*b + 128*a^10*b^3 - 256*a^11*b^2))
/(2*a^8*(a^7 - a^5*b^2)))*(A*b^2 + C*a^2 - B*a*b))/(a^7 - a^5*b^2))*(A*b^2 + C*a^2 - B*a*b)*1i)/(a^7 - a^5*b^2
) + (b^3*(-(a + b)*(a - b))^(1/2)*((tan(c/2 + (d*x)/2)*(9*A^2*a^11 - 128*A^2*b^11 + 16*C^2*a^11 + 256*A^2*a*b^
10 - 27*A^2*a^10*b - 48*C^2*a^10*b - 256*A^2*a^2*b^9 + 256*A^2*a^3*b^8 - 256*A^2*a^4*b^7 + 256*A^2*a^5*b^6 - 2
16*A^2*a^6*b^5 + 136*A^2*a^7*b^4 - 81*A^2*a^8*b^3 + 51*A^2*a^9*b^2 - 128*B^2*a^2*b^9 + 256*B^2*a^3*b^8 - 256*B
^2*a^4*b^7 + 256*B^2*a^5*b^6 - 208*B^2*a^6*b^5 + 112*B^2*a^7*b^4 - 48*B^2*a^8*b^3 + 16*B^2*a^9*b^2 - 128*C^2*a
^4*b^7 + 256*C^2*a^5*b^6 - 256*C^2*a^6*b^5 + 256*C^2*a^7*b^4 - 208*C^2*a^8*b^3 + 112*C^2*a^9*b^2 + 24*A*C*a^11
+ 256*A*B*a*b^10 - 24*A*B*a^10*b - 72*A*C*a^10*b - 32*B*C*a^10*b - 512*A*B*a^2*b^9 + 512*A*B*a^3*b^8 - 512*A*
B*a^4*b^7 + 464*A*B*a^5*b^6 - 368*A*B*a^6*b^5 + 264*A*B*a^7*b^4 - 152*A*B*a^8*b^3 + 72*A*B*a^9*b^2 - 256*A*C*a
^2*b^9 + 512*A*C*a^3*b^8 - 512*A*C*a^4*b^7 + 512*A*C*a^5*b^6 - 464*A*C*a^6*b^5 + 368*A*C*a^7*b^4 - 264*A*C*a^8
*b^3 + 152*A*C*a^9*b^2 + 256*B*C*a^3*b^8 - 512*B*C*a^4*b^7 + 512*B*C*a^5*b^6 - 512*B*C*a^6*b^5 + 416*B*C*a^7*b
^4 - 224*B*C*a^8*b^3 + 96*B*C*a^9*b^2))/(2*a^8) - (b^3*(-(a + b)*(a - b))^(1/2)*((12*A*a^16 + 16*C*a^16 + 32*A
*a^10*b^6 - 48*A*a^11*b^5 + 16*A*a^12*b^4 - 4*A*a^13*b^3 + 4*A*a^14*b^2 - 32*B*a^11*b^5 + 48*B*a^12*b^4 - 16*B
*a^13*b^3 + 16*B*a^14*b^2 + 32*C*a^12*b^4 - 48*C*a^13*b^3 + 16*C*a^14*b^2 - 12*A*a^15*b - 16*B*a^15*b - 16*C*a
^15*b)/a^12 - (b^3*tan(c/2 + (d*x)/2)*(-(a + b)*(a - b))^(1/2)*(A*b^2 + C*a^2 - B*a*b)*(128*a^12*b + 128*a^10*
b^3 - 256*a^11*b^2))/(2*a^8*(a^7 - a^5*b^2)))*(A*b^2 + C*a^2 - B*a*b))/(a^7 - a^5*b^2))*(A*b^2 + C*a^2 - B*a*b
)*1i)/(a^7 - a^5*b^2))/((64*A^3*b^14 - 96*A^3*a*b^13 + 96*A^3*a^2*b^12 - 104*A^3*a^3*b^11 + 104*A^3*a^4*b^10 -
88*A^3*a^5*b^9 + 48*A^3*a^6*b^8 - 33*A^3*a^7*b^7 + 18*A^3*a^8*b^6 - 9*A^3*a^9*b^5 - 64*B^3*a^3*b^11 + 96*B^3*
a^4*b^10 - 96*B^3*a^5*b^9 + 80*B^3*a^6*b^8 - 32*B^3*a^7*b^7 + 16*B^3*a^8*b^6 + 64*C^3*a^6*b^8 - 96*C^3*a^7*b^7
+ 96*C^3*a^8*b^6 - 80*C^3*a^9*b^5 + 32*C^3*a^10*b^4 - 16*C^3*a^11*b^3 - 192*A^2*B*a*b^13 + 192*A*B^2*a^2*b^12
- 288*A*B^2*a^3*b^11 + 288*A*B^2*a^4*b^10 - 264*A*B^2*a^5*b^9 + 168*A*B^2*a^6*b^8 - 120*A*B^2*a^7*b^7 + 48*A*
B^2*a^8*b^6 - 24*A*B^2*a^9*b^5 + 288*A^2*B*a^2*b^12 - 288*A^2*B*a^3*b^11 + 288*A^2*B*a^4*b^10 - 240*A^2*B*a^5*
b^9 + 192*A^2*B*a^6*b^8 - 96*A^2*B*a^7*b^7 + 57*A^2*B*a^8*b^6 - 18*A^2*B*a^9*b^5 + 9*A^2*B*a^10*b^4 + 192*A*C^
2*a^4*b^10 - 288*A*C^2*a^5*b^9 + 288*A*C^2*a^6*b^8 - 264*A*C^2*a^7*b^7 + 168*A*C^2*a^8*b^6 - 120*A*C^2*a^9*b^5
+ 48*A*C^2*a^10*b^4 - 24*A*C^2*a^11*b^3 + 192*A^2*C*a^2*b^12 - 288*A^2*C*a^3*b^11 + 288*A^2*C*a^4*b^10 - 288*
A^2*C*a^5*b^9 + 240*A^2*C*a^6*b^8 - 192*A^2*C*a^7*b^7 + 96*A^2*C*a^8*b^6 - 57*A^2*C*a^9*b^5 + 18*A^2*C*a^10*b^
4 - 9*A^2*C*a^11*b^3 - 192*B*C^2*a^5*b^9 + 288*B*C^2*a^6*b^8 - 288*B*C^2*a^7*b^7 + 240*B*C^2*a^8*b^6 - 96*B*C^
2*a^9*b^5 + 48*B*C^2*a^10*b^4 + 192*B^2*C*a^4*b^10 - 288*B^2*C*a^5*b^9 + 288*B^2*C*a^6*b^8 - 240*B^2*C*a^7*b^7
+ 96*B^2*C*a^8*b^6 - 48*B^2*C*a^9*b^5 - 384*A*B*C*a^3*b^11 + 576*A*B*C*a^4*b^10 - 576*A*B*C*a^5*b^9 + 528*A*B
*C*a^6*b^8 - 336*A*B*C*a^7*b^7 + 240*A*B*C*a^8*b^6 - 96*A*B*C*a^9*b^5 + 48*A*B*C*a^10*b^4)/a^12 - (b^3*(-(a +
b)*(a - b))^(1/2)*((tan(c/2 + (d*x)/2)*(9*A^2*a^11 - 128*A^2*b^11 + 16*C^2*a^11 + 256*A^2*a*b^10 - 27*A^2*a^10
*b - 48*C^2*a^10*b - 256*A^2*a^2*b^9 + 256*A^2*a^3*b^8 - 256*A^2*a^4*b^7 + 256*A^2*a^5*b^6 - 216*A^2*a^6*b^5 +
136*A^2*a^7*b^4 - 81*A^2*a^8*b^3 + 51*A^2*a^9*b^2 - 128*B^2*a^2*b^9 + 256*B^2*a^3*b^8 - 256*B^2*a^4*b^7 + 256
*B^2*a^5*b^6 - 208*B^2*a^6*b^5 + 112*B^2*a^7*b^4 - 48*B^2*a^8*b^3 + 16*B^2*a^9*b^2 - 128*C^2*a^4*b^7 + 256*C^2
*a^5*b^6 - 256*C^2*a^6*b^5 + 256*C^2*a^7*b^4 - 208*C^2*a^8*b^3 + 112*C^2*a^9*b^2 + 24*A*C*a^11 + 256*A*B*a*b^1
0 - 24*A*B*a^10*b - 72*A*C*a^10*b - 32*B*C*a^10*b - 512*A*B*a^2*b^9 + 512*A*B*a^3*b^8 - 512*A*B*a^4*b^7 + 464*
A*B*a^5*b^6 - 368*A*B*a^6*b^5 + 264*A*B*a^7*b^4 - 152*A*B*a^8*b^3 + 72*A*B*a^9*b^2 - 256*A*C*a^2*b^9 + 512*A*C
*a^3*b^8 - 512*A*C*a^4*b^7 + 512*A*C*a^5*b^6 - 464*A*C*a^6*b^5 + 368*A*C*a^7*b^4 - 264*A*C*a^8*b^3 + 152*A*C*a
^9*b^2 + 256*B*C*a^3*b^8 - 512*B*C*a^4*b^7 + 512*B*C*a^5*b^6 - 512*B*C*a^6*b^5 + 416*B*C*a^7*b^4 - 224*B*C*a^8
*b^3 + 96*B*C*a^9*b^2))/(2*a^8) + (b^3*(-(a + b)*(a - b))^(1/2)*((12*A*a^16 + 16*C*a^16 + 32*A*a^10*b^6 - 48*A
*a^11*b^5 + 16*A*a^12*b^4 - 4*A*a^13*b^3 + 4*A*a^14*b^2 - 32*B*a^11*b^5 + 48*B*a^12*b^4 - 16*B*a^13*b^3 + 16*B
*a^14*b^2 + 32*C*a^12*b^4 - 48*C*a^13*b^3 + 16*C*a^14*b^2 - 12*A*a^15*b - 16*B*a^15*b - 16*C*a^15*b)/a^12 + (b
^3*tan(c/2 + (d*x)/2)*(-(a + b)*(a - b))^(1/2)*(A*b^2 + C*a^2 - B*a*b)*(128*a^12*b + 128*a^10*b^3 - 256*a^11*b
^2))/(2*a^8*(a^7 - a^5*b^2)))*(A*b^2 + C*a^2 - B*a*b))/(a^7 - a^5*b^2))*(A*b^2 + C*a^2 - B*a*b))/(a^7 - a^5*b^
2) + (b^3*(-(a + b)*(a - b))^(1/2)*((tan(c/2 + (d*x)/2)*(9*A^2*a^11 - 128*A^2*b^11 + 16*C^2*a^11 + 256*A^2*a*b
^10 - 27*A^2*a^10*b - 48*C^2*a^10*b - 256*A^2*a^2*b^9 + 256*A^2*a^3*b^8 - 256*A^2*a^4*b^7 + 256*A^2*a^5*b^6 -
216*A^2*a^6*b^5 + 136*A^2*a^7*b^4 - 81*A^2*a^8*b^3 + 51*A^2*a^9*b^2 - 128*B^2*a^2*b^9 + 256*B^2*a^3*b^8 - 256*
B^2*a^4*b^7 + 256*B^2*a^5*b^6 - 208*B^2*a^6*b^5 + 112*B^2*a^7*b^4 - 48*B^2*a^8*b^3 + 16*B^2*a^9*b^2 - 128*C^2*
a^4*b^7 + 256*C^2*a^5*b^6 - 256*C^2*a^6*b^5 + 256*C^2*a^7*b^4 - 208*C^2*a^8*b^3 + 112*C^2*a^9*b^2 + 24*A*C*a^1
1 + 256*A*B*a*b^10 - 24*A*B*a^10*b - 72*A*C*a^10*b - 32*B*C*a^10*b - 512*A*B*a^2*b^9 + 512*A*B*a^3*b^8 - 512*A
*B*a^4*b^7 + 464*A*B*a^5*b^6 - 368*A*B*a^6*b^5 + 264*A*B*a^7*b^4 - 152*A*B*a^8*b^3 + 72*A*B*a^9*b^2 - 256*A*C*
a^2*b^9 + 512*A*C*a^3*b^8 - 512*A*C*a^4*b^7 + 512*A*C*a^5*b^6 - 464*A*C*a^6*b^5 + 368*A*C*a^7*b^4 - 264*A*C*a^
8*b^3 + 152*A*C*a^9*b^2 + 256*B*C*a^3*b^8 - 512*B*C*a^4*b^7 + 512*B*C*a^5*b^6 - 512*B*C*a^6*b^5 + 416*B*C*a^7*
b^4 - 224*B*C*a^8*b^3 + 96*B*C*a^9*b^2))/(2*a^8) - (b^3*(-(a + b)*(a - b))^(1/2)*((12*A*a^16 + 16*C*a^16 + 32*
A*a^10*b^6 - 48*A*a^11*b^5 + 16*A*a^12*b^4 - 4*A*a^13*b^3 + 4*A*a^14*b^2 - 32*B*a^11*b^5 + 48*B*a^12*b^4 - 16*
B*a^13*b^3 + 16*B*a^14*b^2 + 32*C*a^12*b^4 - 48*C*a^13*b^3 + 16*C*a^14*b^2 - 12*A*a^15*b - 16*B*a^15*b - 16*C*
a^15*b)/a^12 - (b^3*tan(c/2 + (d*x)/2)*(-(a + b)*(a - b))^(1/2)*(A*b^2 + C*a^2 - B*a*b)*(128*a^12*b + 128*a^10
*b^3 - 256*a^11*b^2))/(2*a^8*(a^7 - a^5*b^2)))*(A*b^2 + C*a^2 - B*a*b))/(a^7 - a^5*b^2))*(A*b^2 + C*a^2 - B*a*
b))/(a^7 - a^5*b^2)))*(-(a + b)*(a - b))^(1/2)*(A*b^2 + C*a^2 - B*a*b)*2i)/(d*(a^7 - a^5*b^2))