\(\int \frac {(A+B \cos (c+d x)+C \cos ^2(c+d x)) \sec ^2(c+d x)}{(a+b \cos (c+d x))^2} \, dx\) [991]

Optimal result

Integrand size = 41, antiderivative size = 211 \[ \int \frac {\left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^2(c+d x)}{(a+b \cos (c+d x))^2} \, dx=\frac {2 \left (3 a^2 A b^2-2 A b^4-2 a^3 b B+a b^3 B+a^4 C\right ) \arctan \left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{a^3 (a-b)^{3/2} (a+b)^{3/2} d}-\frac {(2 A b-a B) \text {arctanh}(\sin (c+d x))}{a^3 d}-\frac {\left (2 A b^2-a b B-a^2 (A-C)\right ) \tan (c+d x)}{a^2 \left (a^2-b^2\right ) d}+\frac {\left (A b^2-a (b B-a C)\right ) \tan (c+d x)}{a \left (a^2-b^2\right ) d (a+b \cos (c+d x))} \]

[Out]

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

Rubi [A] (verified)

Time = 0.81 (sec) , antiderivative size = 211, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.122, Rules used = {3134, 3080, 3855, 2738, 211} \[ \int \frac {\left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^2(c+d x)}{(a+b \cos (c+d x))^2} \, dx=-\frac {(2 A b-a B) \text {arctanh}(\sin (c+d x))}{a^3 d}-\frac {\tan (c+d x) \left (-\left (a^2 (A-C)\right )-a b B+2 A b^2\right )}{a^2 d \left (a^2-b^2\right )}+\frac {\tan (c+d x) \left (A b^2-a (b B-a C)\right )}{a d \left (a^2-b^2\right ) (a+b \cos (c+d x))}+\frac {2 \left (a^4 C-2 a^3 b B+3 a^2 A b^2+a b^3 B-2 A b^4\right ) \arctan \left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{a^3 d (a-b)^{3/2} (a+b)^{3/2}} \]

[In]

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

[Out]

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

Rule 211

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

Rule 2738

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

Rule 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 {\left (A b^2-a (b B-a C)\right ) \tan (c+d x)}{a \left (a^2-b^2\right ) d (a+b \cos (c+d x))}+\frac {\int \frac {\left (-2 A b^2+a b B+a^2 (A-C)-a (A b-a B+b C) \cos (c+d x)+\left (A b^2-a (b B-a C)\right ) \cos ^2(c+d x)\right ) \sec ^2(c+d x)}{a+b \cos (c+d x)} \, dx}{a \left (a^2-b^2\right )} \\ & = -\frac {\left (2 A b^2-a b B-a^2 (A-C)\right ) \tan (c+d x)}{a^2 \left (a^2-b^2\right ) d}+\frac {\left (A b^2-a (b B-a C)\right ) \tan (c+d x)}{a \left (a^2-b^2\right ) d (a+b \cos (c+d x))}+\frac {\int \frac {\left (-\left (\left (a^2-b^2\right ) (2 A b-a B)\right )+a \left (A b^2-a (b B-a C)\right ) \cos (c+d x)\right ) \sec (c+d x)}{a+b \cos (c+d x)} \, dx}{a^2 \left (a^2-b^2\right )} \\ & = -\frac {\left (2 A b^2-a b B-a^2 (A-C)\right ) \tan (c+d x)}{a^2 \left (a^2-b^2\right ) d}+\frac {\left (A b^2-a (b B-a C)\right ) \tan (c+d x)}{a \left (a^2-b^2\right ) d (a+b \cos (c+d x))}-\frac {(2 A b-a B) \int \sec (c+d x) \, dx}{a^3}+\frac {\left (3 a^2 A b^2-2 A b^4-2 a^3 b B+a b^3 B+a^4 C\right ) \int \frac {1}{a+b \cos (c+d x)} \, dx}{a^3 \left (a^2-b^2\right )} \\ & = -\frac {(2 A b-a B) \text {arctanh}(\sin (c+d x))}{a^3 d}-\frac {\left (2 A b^2-a b B-a^2 (A-C)\right ) \tan (c+d x)}{a^2 \left (a^2-b^2\right ) d}+\frac {\left (A b^2-a (b B-a C)\right ) \tan (c+d x)}{a \left (a^2-b^2\right ) d (a+b \cos (c+d x))}+\frac {\left (2 \left (3 a^2 A b^2-2 A b^4-2 a^3 b B+a b^3 B+a^4 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^3 \left (a^2-b^2\right ) d} \\ & = \frac {2 \left (3 a^2 A b^2-2 A b^4-2 a^3 b B+a b^3 B+a^4 C\right ) \arctan \left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{a^3 (a-b)^{3/2} (a+b)^{3/2} d}-\frac {(2 A b-a B) \text {arctanh}(\sin (c+d x))}{a^3 d}-\frac {\left (2 A b^2-a b B-a^2 (A-C)\right ) \tan (c+d x)}{a^2 \left (a^2-b^2\right ) d}+\frac {\left (A b^2-a (b B-a C)\right ) \tan (c+d x)}{a \left (a^2-b^2\right ) d (a+b \cos (c+d x))} \\ \end{align*}

Mathematica [A] (warning: unable to verify)

Time = 4.55 (sec) , antiderivative size = 331, normalized size of antiderivative = 1.57 \[ \int \frac {\left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^2(c+d x)}{(a+b \cos (c+d x))^2} \, dx=\frac {2 \cos ^2(c+d x) \left (C+B \sec (c+d x)+A \sec ^2(c+d x)\right ) \left (\frac {2 \left (3 a^2 A b^2-2 A b^4-2 a^3 b B+a b^3 B+a^4 C\right ) \text {arctanh}\left (\frac {(a-b) \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {-a^2+b^2}}\right )}{\left (-a^2+b^2\right )^{3/2}}+(2 A b-a B) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )+(-2 A b+a B) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )+\frac {a A \sin \left (\frac {1}{2} (c+d x)\right )}{\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )}+\frac {a A \sin \left (\frac {1}{2} (c+d x)\right )}{\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )}-\frac {a b \left (A b^2+a (-b B+a C)\right ) \sin (c+d x)}{(a-b) (a+b) (a+b \cos (c+d x))}\right )}{a^3 d (2 A+C+2 B \cos (c+d x)+C \cos (2 (c+d x)))} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 0.59 (sec) , antiderivative size = 256, normalized size of antiderivative = 1.21

method	result	size
derivativedivides	\(\frac {-\frac {A}{a^{2} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}+\frac {\left (-2 A b +B a \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{a^{3}}-\frac {A}{a^{2} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}+\frac {\left (2 A b -B a \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{a^{3}}+\frac {-\frac {2 a \left (A \,b^{2}-B a b +a^{2} C \right ) b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{\left (a^{2}-b^{2}\right ) \left (\left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a -\left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) b +a +b \right )}+\frac {2 \left (3 A \,a^{2} b^{2}-2 A \,b^{4}-2 B \,a^{3} b +B a \,b^{3}+a^{4} C \right ) \arctan \left (\frac {\left (a -b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (a -b \right ) \left (a +b \right )}}\right )}{\left (a -b \right ) \left (a +b \right ) \sqrt {\left (a -b \right ) \left (a +b \right )}}}{a^{3}}}{d}\)	\(256\)
default	\(\frac {-\frac {A}{a^{2} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}+\frac {\left (-2 A b +B a \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{a^{3}}-\frac {A}{a^{2} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}+\frac {\left (2 A b -B a \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{a^{3}}+\frac {-\frac {2 a \left (A \,b^{2}-B a b +a^{2} C \right ) b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{\left (a^{2}-b^{2}\right ) \left (\left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a -\left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) b +a +b \right )}+\frac {2 \left (3 A \,a^{2} b^{2}-2 A \,b^{4}-2 B \,a^{3} b +B a \,b^{3}+a^{4} C \right ) \arctan \left (\frac {\left (a -b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (a -b \right ) \left (a +b \right )}}\right )}{\left (a -b \right ) \left (a +b \right ) \sqrt {\left (a -b \right ) \left (a +b \right )}}}{a^{3}}}{d}\)	\(256\)
risch	\(\text {Expression too large to display}\)	\(1204\)

[In]

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

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 529 vs. \(2 (196) = 392\).

Time = 16.73 (sec) , antiderivative size = 1126, normalized size of antiderivative = 5.34 \[ \int \frac {\left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^2(c+d x)}{(a+b \cos (c+d x))^2} \, dx=\text {Too large to display} \]

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F(-2)]

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

[In]

integrate((A+B*cos(d*x+c)+C*cos(d*x+c)^2)*sec(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 de

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 442 vs. \(2 (196) = 392\).

Time = 0.35 (sec) , antiderivative size = 442, normalized size of antiderivative = 2.09 \[ \int \frac {\left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^2(c+d x)}{(a+b \cos (c+d x))^2} \, dx=-\frac {\frac {2 \, {\left (C a^{4} - 2 \, B a^{3} b + 3 \, A a^{2} b^{2} + B a b^{3} - 2 \, 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^{5} - a^{3} b^{2}\right )} \sqrt {a^{2} - b^{2}}} + \frac {2 \, {\left (A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - A a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + C a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - A a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - B a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 2 \, A b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + A a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - C a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - A a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + B a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 2 \, A b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 2 \, b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - a - b\right )} {\left (a^{4} - a^{2} b^{2}\right )}} - \frac {{\left (B a - 2 \, A b\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right )}{a^{3}} + \frac {{\left (B a - 2 \, A b\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right )}{a^{3}}}{d} \]

[In]

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

[Out]

-(2*(C*a^4 - 2*B*a^3*b + 3*A*a^2*b^2 + B*a*b^3 - 2*A*b^4)*(pi*floor(1/2*(d*x + c)/pi + 1/2)*sgn(-2*a + 2*b) +
arctan(-(a*tan(1/2*d*x + 1/2*c) - b*tan(1/2*d*x + 1/2*c))/sqrt(a^2 - b^2)))/((a^5 - a^3*b^2)*sqrt(a^2 - b^2))
+ 2*(A*a^3*tan(1/2*d*x + 1/2*c)^3 - A*a^2*b*tan(1/2*d*x + 1/2*c)^3 + C*a^2*b*tan(1/2*d*x + 1/2*c)^3 - A*a*b^2*
tan(1/2*d*x + 1/2*c)^3 - B*a*b^2*tan(1/2*d*x + 1/2*c)^3 + 2*A*b^3*tan(1/2*d*x + 1/2*c)^3 + A*a^3*tan(1/2*d*x +
1/2*c) + A*a^2*b*tan(1/2*d*x + 1/2*c) - C*a^2*b*tan(1/2*d*x + 1/2*c) - A*a*b^2*tan(1/2*d*x + 1/2*c) + B*a*b^2
*tan(1/2*d*x + 1/2*c) - 2*A*b^3*tan(1/2*d*x + 1/2*c))/((a*tan(1/2*d*x + 1/2*c)^4 - b*tan(1/2*d*x + 1/2*c)^4 +
2*b*tan(1/2*d*x + 1/2*c)^2 - a - b)*(a^4 - a^2*b^2)) - (B*a - 2*A*b)*log(abs(tan(1/2*d*x + 1/2*c) + 1))/a^3 +
(B*a - 2*A*b)*log(abs(tan(1/2*d*x + 1/2*c) - 1))/a^3)/d

Mupad [B] (verification not implemented)

Time = 11.79 (sec) , antiderivative size = 6450, normalized size of antiderivative = 30.57 \[ \int \frac {\left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^2(c+d x)}{(a+b \cos (c+d x))^2} \, dx=\text {Too large to display} \]

[In]

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

[Out]

((2*tan(c/2 + (d*x)/2)*(A*a^3 - 2*A*b^3 - A*a*b^2 + A*a^2*b + B*a*b^2 - C*a^2*b))/(a^2*(a + b)*(a - b)) + (2*t
an(c/2 + (d*x)/2)^3*(A*a^3 + 2*A*b^3 - A*a*b^2 - A*a^2*b - B*a*b^2 + C*a^2*b))/(a^2*(a + b)*(a - b)))/(d*(a +
b - tan(c/2 + (d*x)/2)^4*(a - b) - 2*b*tan(c/2 + (d*x)/2)^2)) + (atan((((2*A*b - B*a)*((32*tan(c/2 + (d*x)/2)*
(8*A^2*b^8 + B^2*a^8 + C^2*a^8 - 8*A^2*a*b^7 - 2*B^2*a^7*b - 16*A^2*a^2*b^6 + 16*A^2*a^3*b^5 + 5*A^2*a^4*b^4 -
8*A^2*a^5*b^3 + 4*A^2*a^6*b^2 + 2*B^2*a^2*b^6 - 2*B^2*a^3*b^5 - 5*B^2*a^4*b^4 + 4*B^2*a^5*b^3 + 3*B^2*a^6*b^2
- 8*A*B*a*b^7 - 4*A*B*a^7*b - 4*B*C*a^7*b + 8*A*B*a^2*b^6 + 18*A*B*a^3*b^5 - 16*A*B*a^4*b^4 - 8*A*B*a^5*b^3 +
8*A*B*a^6*b^2 - 4*A*C*a^4*b^4 + 6*A*C*a^6*b^2 + 2*B*C*a^5*b^3))/(a^6*b + a^7 - a^4*b^3 - a^5*b^2) + ((2*A*b -
B*a)*((32*(A*a^7*b^5 - C*a^12 - 2*A*a^6*b^6 - B*a^12 + 5*A*a^8*b^4 - 3*A*a^9*b^3 - 3*A*a^10*b^2 + B*a^7*b^5 -
3*B*a^9*b^3 + B*a^10*b^2 - C*a^9*b^3 + C*a^10*b^2 + 2*A*a^11*b + 2*B*a^11*b + C*a^11*b))/(a^8*b + a^9 - a^6*b
^3 - a^7*b^2) + (32*tan(c/2 + (d*x)/2)*(2*A*b - B*a)*(2*a^11*b - 2*a^6*b^6 + 2*a^7*b^5 + 4*a^8*b^4 - 4*a^9*b^3
- 2*a^10*b^2))/(a^3*(a^6*b + a^7 - a^4*b^3 - a^5*b^2))))/a^3)*1i)/a^3 + ((2*A*b - B*a)*((32*tan(c/2 + (d*x)/2
)*(8*A^2*b^8 + B^2*a^8 + C^2*a^8 - 8*A^2*a*b^7 - 2*B^2*a^7*b - 16*A^2*a^2*b^6 + 16*A^2*a^3*b^5 + 5*A^2*a^4*b^4
- 8*A^2*a^5*b^3 + 4*A^2*a^6*b^2 + 2*B^2*a^2*b^6 - 2*B^2*a^3*b^5 - 5*B^2*a^4*b^4 + 4*B^2*a^5*b^3 + 3*B^2*a^6*b
^2 - 8*A*B*a*b^7 - 4*A*B*a^7*b - 4*B*C*a^7*b + 8*A*B*a^2*b^6 + 18*A*B*a^3*b^5 - 16*A*B*a^4*b^4 - 8*A*B*a^5*b^3
+ 8*A*B*a^6*b^2 - 4*A*C*a^4*b^4 + 6*A*C*a^6*b^2 + 2*B*C*a^5*b^3))/(a^6*b + a^7 - a^4*b^3 - a^5*b^2) - ((2*A*b
- B*a)*((32*(A*a^7*b^5 - C*a^12 - 2*A*a^6*b^6 - B*a^12 + 5*A*a^8*b^4 - 3*A*a^9*b^3 - 3*A*a^10*b^2 + B*a^7*b^5
- 3*B*a^9*b^3 + B*a^10*b^2 - C*a^9*b^3 + C*a^10*b^2 + 2*A*a^11*b + 2*B*a^11*b + C*a^11*b))/(a^8*b + a^9 - a^6
*b^3 - a^7*b^2) - (32*tan(c/2 + (d*x)/2)*(2*A*b - B*a)*(2*a^11*b - 2*a^6*b^6 + 2*a^7*b^5 + 4*a^8*b^4 - 4*a^9*b
^3 - 2*a^10*b^2))/(a^3*(a^6*b + a^7 - a^4*b^3 - a^5*b^2))))/a^3)*1i)/a^3)/((64*(8*A^3*b^8 - B*C^2*a^8 + B^2*C*
a^8 - 4*A^3*a*b^7 - 2*B^3*a^7*b - 20*A^3*a^2*b^6 + 6*A^3*a^3*b^5 + 12*A^3*a^4*b^4 - B^3*a^3*b^5 + B^3*a^4*b^4
+ 3*B^3*a^5*b^3 - 2*B^3*a^6*b^2 - 12*A^2*B*a*b^7 + 2*A*C^2*a^7*b + 3*B^2*C*a^7*b + 6*A*B^2*a^2*b^6 - 5*A*B^2*a
^3*b^5 - 17*A*B^2*a^4*b^4 + 9*A*B^2*a^5*b^3 + 11*A*B^2*a^6*b^2 + 8*A^2*B*a^2*b^6 + 32*A^2*B*a^3*b^5 - 13*A^2*B
*a^4*b^4 - 20*A^2*B*a^5*b^3 - 4*A^2*C*a^3*b^5 - 4*A^2*C*a^4*b^4 + 8*A^2*C*a^5*b^3 + 4*A^2*C*a^6*b^2 - B^2*C*a^
5*b^3 - B^2*C*a^6*b^2 - 4*A*B*C*a^7*b + 4*A*B*C*a^4*b^4 + 4*A*B*C*a^5*b^3 - 10*A*B*C*a^6*b^2))/(a^8*b + a^9 -
a^6*b^3 - a^7*b^2) + ((2*A*b - B*a)*((32*tan(c/2 + (d*x)/2)*(8*A^2*b^8 + B^2*a^8 + C^2*a^8 - 8*A^2*a*b^7 - 2*B
^2*a^7*b - 16*A^2*a^2*b^6 + 16*A^2*a^3*b^5 + 5*A^2*a^4*b^4 - 8*A^2*a^5*b^3 + 4*A^2*a^6*b^2 + 2*B^2*a^2*b^6 - 2
*B^2*a^3*b^5 - 5*B^2*a^4*b^4 + 4*B^2*a^5*b^3 + 3*B^2*a^6*b^2 - 8*A*B*a*b^7 - 4*A*B*a^7*b - 4*B*C*a^7*b + 8*A*B
*a^2*b^6 + 18*A*B*a^3*b^5 - 16*A*B*a^4*b^4 - 8*A*B*a^5*b^3 + 8*A*B*a^6*b^2 - 4*A*C*a^4*b^4 + 6*A*C*a^6*b^2 + 2
*B*C*a^5*b^3))/(a^6*b + a^7 - a^4*b^3 - a^5*b^2) + ((2*A*b - B*a)*((32*(A*a^7*b^5 - C*a^12 - 2*A*a^6*b^6 - B*a
^12 + 5*A*a^8*b^4 - 3*A*a^9*b^3 - 3*A*a^10*b^2 + B*a^7*b^5 - 3*B*a^9*b^3 + B*a^10*b^2 - C*a^9*b^3 + C*a^10*b^2
+ 2*A*a^11*b + 2*B*a^11*b + C*a^11*b))/(a^8*b + a^9 - a^6*b^3 - a^7*b^2) + (32*tan(c/2 + (d*x)/2)*(2*A*b - B*
a)*(2*a^11*b - 2*a^6*b^6 + 2*a^7*b^5 + 4*a^8*b^4 - 4*a^9*b^3 - 2*a^10*b^2))/(a^3*(a^6*b + a^7 - a^4*b^3 - a^5*
b^2))))/a^3))/a^3 - ((2*A*b - B*a)*((32*tan(c/2 + (d*x)/2)*(8*A^2*b^8 + B^2*a^8 + C^2*a^8 - 8*A^2*a*b^7 - 2*B^
2*a^7*b - 16*A^2*a^2*b^6 + 16*A^2*a^3*b^5 + 5*A^2*a^4*b^4 - 8*A^2*a^5*b^3 + 4*A^2*a^6*b^2 + 2*B^2*a^2*b^6 - 2*
B^2*a^3*b^5 - 5*B^2*a^4*b^4 + 4*B^2*a^5*b^3 + 3*B^2*a^6*b^2 - 8*A*B*a*b^7 - 4*A*B*a^7*b - 4*B*C*a^7*b + 8*A*B*
a^2*b^6 + 18*A*B*a^3*b^5 - 16*A*B*a^4*b^4 - 8*A*B*a^5*b^3 + 8*A*B*a^6*b^2 - 4*A*C*a^4*b^4 + 6*A*C*a^6*b^2 + 2*
B*C*a^5*b^3))/(a^6*b + a^7 - a^4*b^3 - a^5*b^2) - ((2*A*b - B*a)*((32*(A*a^7*b^5 - C*a^12 - 2*A*a^6*b^6 - B*a^
12 + 5*A*a^8*b^4 - 3*A*a^9*b^3 - 3*A*a^10*b^2 + B*a^7*b^5 - 3*B*a^9*b^3 + B*a^10*b^2 - C*a^9*b^3 + C*a^10*b^2
+ 2*A*a^11*b + 2*B*a^11*b + C*a^11*b))/(a^8*b + a^9 - a^6*b^3 - a^7*b^2) - (32*tan(c/2 + (d*x)/2)*(2*A*b - B*a
)*(2*a^11*b - 2*a^6*b^6 + 2*a^7*b^5 + 4*a^8*b^4 - 4*a^9*b^3 - 2*a^10*b^2))/(a^3*(a^6*b + a^7 - a^4*b^3 - a^5*b
^2))))/a^3))/a^3))*(2*A*b - B*a)*2i)/(a^3*d) + (atan(((((32*tan(c/2 + (d*x)/2)*(8*A^2*b^8 + B^2*a^8 + C^2*a^8
- 8*A^2*a*b^7 - 2*B^2*a^7*b - 16*A^2*a^2*b^6 + 16*A^2*a^3*b^5 + 5*A^2*a^4*b^4 - 8*A^2*a^5*b^3 + 4*A^2*a^6*b^2
+ 2*B^2*a^2*b^6 - 2*B^2*a^3*b^5 - 5*B^2*a^4*b^4 + 4*B^2*a^5*b^3 + 3*B^2*a^6*b^2 - 8*A*B*a*b^7 - 4*A*B*a^7*b -
4*B*C*a^7*b + 8*A*B*a^2*b^6 + 18*A*B*a^3*b^5 - 16*A*B*a^4*b^4 - 8*A*B*a^5*b^3 + 8*A*B*a^6*b^2 - 4*A*C*a^4*b^4
+ 6*A*C*a^6*b^2 + 2*B*C*a^5*b^3))/(a^6*b + a^7 - a^4*b^3 - a^5*b^2) + (((32*(A*a^7*b^5 - C*a^12 - 2*A*a^6*b^6
- B*a^12 + 5*A*a^8*b^4 - 3*A*a^9*b^3 - 3*A*a^10*b^2 + B*a^7*b^5 - 3*B*a^9*b^3 + B*a^10*b^2 - C*a^9*b^3 + C*a^1
0*b^2 + 2*A*a^11*b + 2*B*a^11*b + C*a^11*b))/(a^8*b + a^9 - a^6*b^3 - a^7*b^2) + (32*tan(c/2 + (d*x)/2)*(-(a +
b)^3*(a - b)^3)^(1/2)*(C*a^4 - 2*A*b^4 + 3*A*a^2*b^2 + B*a*b^3 - 2*B*a^3*b)*(2*a^11*b - 2*a^6*b^6 + 2*a^7*b^5
+ 4*a^8*b^4 - 4*a^9*b^3 - 2*a^10*b^2))/((a^6*b + a^7 - a^4*b^3 - a^5*b^2)*(a^9 - a^3*b^6 + 3*a^5*b^4 - 3*a^7*
b^2)))*(-(a + b)^3*(a - b)^3)^(1/2)*(C*a^4 - 2*A*b^4 + 3*A*a^2*b^2 + B*a*b^3 - 2*B*a^3*b))/(a^9 - a^3*b^6 + 3*
a^5*b^4 - 3*a^7*b^2))*(-(a + b)^3*(a - b)^3)^(1/2)*(C*a^4 - 2*A*b^4 + 3*A*a^2*b^2 + B*a*b^3 - 2*B*a^3*b)*1i)/(
a^9 - a^3*b^6 + 3*a^5*b^4 - 3*a^7*b^2) + (((32*tan(c/2 + (d*x)/2)*(8*A^2*b^8 + B^2*a^8 + C^2*a^8 - 8*A^2*a*b^7
- 2*B^2*a^7*b - 16*A^2*a^2*b^6 + 16*A^2*a^3*b^5 + 5*A^2*a^4*b^4 - 8*A^2*a^5*b^3 + 4*A^2*a^6*b^2 + 2*B^2*a^2*b
^6 - 2*B^2*a^3*b^5 - 5*B^2*a^4*b^4 + 4*B^2*a^5*b^3 + 3*B^2*a^6*b^2 - 8*A*B*a*b^7 - 4*A*B*a^7*b - 4*B*C*a^7*b +
8*A*B*a^2*b^6 + 18*A*B*a^3*b^5 - 16*A*B*a^4*b^4 - 8*A*B*a^5*b^3 + 8*A*B*a^6*b^2 - 4*A*C*a^4*b^4 + 6*A*C*a^6*b
^2 + 2*B*C*a^5*b^3))/(a^6*b + a^7 - a^4*b^3 - a^5*b^2) - (((32*(A*a^7*b^5 - C*a^12 - 2*A*a^6*b^6 - B*a^12 + 5*
A*a^8*b^4 - 3*A*a^9*b^3 - 3*A*a^10*b^2 + B*a^7*b^5 - 3*B*a^9*b^3 + B*a^10*b^2 - C*a^9*b^3 + C*a^10*b^2 + 2*A*a
^11*b + 2*B*a^11*b + C*a^11*b))/(a^8*b + a^9 - a^6*b^3 - a^7*b^2) - (32*tan(c/2 + (d*x)/2)*(-(a + b)^3*(a - b)
^3)^(1/2)*(C*a^4 - 2*A*b^4 + 3*A*a^2*b^2 + B*a*b^3 - 2*B*a^3*b)*(2*a^11*b - 2*a^6*b^6 + 2*a^7*b^5 + 4*a^8*b^4
- 4*a^9*b^3 - 2*a^10*b^2))/((a^6*b + a^7 - a^4*b^3 - a^5*b^2)*(a^9 - a^3*b^6 + 3*a^5*b^4 - 3*a^7*b^2)))*(-(a +
b)^3*(a - b)^3)^(1/2)*(C*a^4 - 2*A*b^4 + 3*A*a^2*b^2 + B*a*b^3 - 2*B*a^3*b))/(a^9 - a^3*b^6 + 3*a^5*b^4 - 3*a
^7*b^2))*(-(a + b)^3*(a - b)^3)^(1/2)*(C*a^4 - 2*A*b^4 + 3*A*a^2*b^2 + B*a*b^3 - 2*B*a^3*b)*1i)/(a^9 - a^3*b^6
+ 3*a^5*b^4 - 3*a^7*b^2))/((64*(8*A^3*b^8 - B*C^2*a^8 + B^2*C*a^8 - 4*A^3*a*b^7 - 2*B^3*a^7*b - 20*A^3*a^2*b^
6 + 6*A^3*a^3*b^5 + 12*A^3*a^4*b^4 - B^3*a^3*b^5 + B^3*a^4*b^4 + 3*B^3*a^5*b^3 - 2*B^3*a^6*b^2 - 12*A^2*B*a*b^
7 + 2*A*C^2*a^7*b + 3*B^2*C*a^7*b + 6*A*B^2*a^2*b^6 - 5*A*B^2*a^3*b^5 - 17*A*B^2*a^4*b^4 + 9*A*B^2*a^5*b^3 + 1
1*A*B^2*a^6*b^2 + 8*A^2*B*a^2*b^6 + 32*A^2*B*a^3*b^5 - 13*A^2*B*a^4*b^4 - 20*A^2*B*a^5*b^3 - 4*A^2*C*a^3*b^5 -
4*A^2*C*a^4*b^4 + 8*A^2*C*a^5*b^3 + 4*A^2*C*a^6*b^2 - B^2*C*a^5*b^3 - B^2*C*a^6*b^2 - 4*A*B*C*a^7*b + 4*A*B*C
*a^4*b^4 + 4*A*B*C*a^5*b^3 - 10*A*B*C*a^6*b^2))/(a^8*b + a^9 - a^6*b^3 - a^7*b^2) + (((32*tan(c/2 + (d*x)/2)*(
8*A^2*b^8 + B^2*a^8 + C^2*a^8 - 8*A^2*a*b^7 - 2*B^2*a^7*b - 16*A^2*a^2*b^6 + 16*A^2*a^3*b^5 + 5*A^2*a^4*b^4 -
8*A^2*a^5*b^3 + 4*A^2*a^6*b^2 + 2*B^2*a^2*b^6 - 2*B^2*a^3*b^5 - 5*B^2*a^4*b^4 + 4*B^2*a^5*b^3 + 3*B^2*a^6*b^2
- 8*A*B*a*b^7 - 4*A*B*a^7*b - 4*B*C*a^7*b + 8*A*B*a^2*b^6 + 18*A*B*a^3*b^5 - 16*A*B*a^4*b^4 - 8*A*B*a^5*b^3 +
8*A*B*a^6*b^2 - 4*A*C*a^4*b^4 + 6*A*C*a^6*b^2 + 2*B*C*a^5*b^3))/(a^6*b + a^7 - a^4*b^3 - a^5*b^2) + (((32*(A*a
^7*b^5 - C*a^12 - 2*A*a^6*b^6 - B*a^12 + 5*A*a^8*b^4 - 3*A*a^9*b^3 - 3*A*a^10*b^2 + B*a^7*b^5 - 3*B*a^9*b^3 +
B*a^10*b^2 - C*a^9*b^3 + C*a^10*b^2 + 2*A*a^11*b + 2*B*a^11*b + C*a^11*b))/(a^8*b + a^9 - a^6*b^3 - a^7*b^2) +
(32*tan(c/2 + (d*x)/2)*(-(a + b)^3*(a - b)^3)^(1/2)*(C*a^4 - 2*A*b^4 + 3*A*a^2*b^2 + B*a*b^3 - 2*B*a^3*b)*(2*
a^11*b - 2*a^6*b^6 + 2*a^7*b^5 + 4*a^8*b^4 - 4*a^9*b^3 - 2*a^10*b^2))/((a^6*b + a^7 - a^4*b^3 - a^5*b^2)*(a^9
- a^3*b^6 + 3*a^5*b^4 - 3*a^7*b^2)))*(-(a + b)^3*(a - b)^3)^(1/2)*(C*a^4 - 2*A*b^4 + 3*A*a^2*b^2 + B*a*b^3 - 2
*B*a^3*b))/(a^9 - a^3*b^6 + 3*a^5*b^4 - 3*a^7*b^2))*(-(a + b)^3*(a - b)^3)^(1/2)*(C*a^4 - 2*A*b^4 + 3*A*a^2*b^
2 + B*a*b^3 - 2*B*a^3*b))/(a^9 - a^3*b^6 + 3*a^5*b^4 - 3*a^7*b^2) - (((32*tan(c/2 + (d*x)/2)*(8*A^2*b^8 + B^2*
a^8 + C^2*a^8 - 8*A^2*a*b^7 - 2*B^2*a^7*b - 16*A^2*a^2*b^6 + 16*A^2*a^3*b^5 + 5*A^2*a^4*b^4 - 8*A^2*a^5*b^3 +
4*A^2*a^6*b^2 + 2*B^2*a^2*b^6 - 2*B^2*a^3*b^5 - 5*B^2*a^4*b^4 + 4*B^2*a^5*b^3 + 3*B^2*a^6*b^2 - 8*A*B*a*b^7 -
4*A*B*a^7*b - 4*B*C*a^7*b + 8*A*B*a^2*b^6 + 18*A*B*a^3*b^5 - 16*A*B*a^4*b^4 - 8*A*B*a^5*b^3 + 8*A*B*a^6*b^2 -
4*A*C*a^4*b^4 + 6*A*C*a^6*b^2 + 2*B*C*a^5*b^3))/(a^6*b + a^7 - a^4*b^3 - a^5*b^2) - (((32*(A*a^7*b^5 - C*a^12
- 2*A*a^6*b^6 - B*a^12 + 5*A*a^8*b^4 - 3*A*a^9*b^3 - 3*A*a^10*b^2 + B*a^7*b^5 - 3*B*a^9*b^3 + B*a^10*b^2 - C*a
^9*b^3 + C*a^10*b^2 + 2*A*a^11*b + 2*B*a^11*b + C*a^11*b))/(a^8*b + a^9 - a^6*b^3 - a^7*b^2) - (32*tan(c/2 + (
d*x)/2)*(-(a + b)^3*(a - b)^3)^(1/2)*(C*a^4 - 2*A*b^4 + 3*A*a^2*b^2 + B*a*b^3 - 2*B*a^3*b)*(2*a^11*b - 2*a^6*b
^6 + 2*a^7*b^5 + 4*a^8*b^4 - 4*a^9*b^3 - 2*a^10*b^2))/((a^6*b + a^7 - a^4*b^3 - a^5*b^2)*(a^9 - a^3*b^6 + 3*a^
5*b^4 - 3*a^7*b^2)))*(-(a + b)^3*(a - b)^3)^(1/2)*(C*a^4 - 2*A*b^4 + 3*A*a^2*b^2 + B*a*b^3 - 2*B*a^3*b))/(a^9
- a^3*b^6 + 3*a^5*b^4 - 3*a^7*b^2))*(-(a + b)^3*(a - b)^3)^(1/2)*(C*a^4 - 2*A*b^4 + 3*A*a^2*b^2 + B*a*b^3 - 2*
B*a^3*b))/(a^9 - a^3*b^6 + 3*a^5*b^4 - 3*a^7*b^2)))*(-(a + b)^3*(a - b)^3)^(1/2)*(C*a^4 - 2*A*b^4 + 3*A*a^2*b^
2 + B*a*b^3 - 2*B*a^3*b)*2i)/(d*(a^9 - a^3*b^6 + 3*a^5*b^4 - 3*a^7*b^2))