\(\int \frac {A+C \sec ^2(c+d x)}{(a+b \sec (c+d x))^3} \, dx\) [695]

Optimal result

Integrand size = 25, antiderivative size = 202 \[ \int \frac {A+C \sec ^2(c+d x)}{(a+b \sec (c+d x))^3} \, dx=\frac {A x}{a^3}+\frac {b \left (5 a^2 A b^2-2 A b^4-3 a^4 (2 A+C)\right ) \text {arctanh}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{a^3 (a-b)^{5/2} (a+b)^{5/2} d}+\frac {\left (A b^2+a^2 C\right ) \tan (c+d x)}{2 a \left (a^2-b^2\right ) d (a+b \sec (c+d x))^2}-\frac {\left (2 A b^4-a^4 C-a^2 b^2 (5 A+2 C)\right ) \tan (c+d x)}{2 a^2 \left (a^2-b^2\right )^2 d (a+b \sec (c+d x))} \]

[Out]

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

Rubi [A] (verified)

Time = 0.54 (sec) , antiderivative size = 202, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.240, Rules used = {4146, 4145, 4004, 3916, 2738, 214} \[ \int \frac {A+C \sec ^2(c+d x)}{(a+b \sec (c+d x))^3} \, dx=\frac {A x}{a^3}+\frac {\left (a^2 C+A b^2\right ) \tan (c+d x)}{2 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}-\frac {\left (a^4 (-C)-a^2 b^2 (5 A+2 C)+2 A b^4\right ) \tan (c+d x)}{2 a^2 d \left (a^2-b^2\right )^2 (a+b \sec (c+d x))}+\frac {b \left (-3 a^4 (2 A+C)+5 a^2 A b^2-2 A b^4\right ) \text {arctanh}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{a^3 d (a-b)^{5/2} (a+b)^{5/2}} \]

[In]

Int[(A + C*Sec[c + d*x]^2)/(a + b*Sec[c + d*x])^3,x]

[Out]

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

Rule 214

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x/Rt[-a/b, 2]], x] /; FreeQ[{a, b},
x] && NegQ[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 3916

Int[csc[(e_.) + (f_.)*(x_)]/(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)), x_Symbol] :> Dist[1/b, Int[1/(1 + (a/b)*Si
n[e + f*x]), x], x] /; FreeQ[{a, b, e, f}, x] && NeQ[a^2 - b^2, 0]

Rule 4004

Int[(csc[(e_.) + (f_.)*(x_)]*(d_.) + (c_))/(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)), x_Symbol] :> Simp[c*(x/a),
x] - Dist[(b*c - a*d)/a, Int[Csc[e + f*x]/(a + b*Csc[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[
b*c - a*d, 0]

Rule 4145

Int[((A_.) + csc[(e_.) + (f_.)*(x_)]*(B_.) + csc[(e_.) + (f_.)*(x_)]^2*(C_.))*(csc[(e_.) + (f_.)*(x_)]*(b_.) +
(a_))^(m_), x_Symbol] :> Simp[(A*b^2 - a*b*B + a^2*C)*Cot[e + f*x]*((a + b*Csc[e + f*x])^(m + 1)/(a*f*(m + 1)
*(a^2 - b^2))), x] + Dist[1/(a*(m + 1)*(a^2 - b^2)), Int[(a + b*Csc[e + f*x])^(m + 1)*Simp[A*(a^2 - b^2)*(m +
1) - a*(A*b - a*B + b*C)*(m + 1)*Csc[e + f*x] + (A*b^2 - a*b*B + a^2*C)*(m + 2)*Csc[e + f*x]^2, x], x], x] /;
FreeQ[{a, b, e, f, A, B, C}, x] && NeQ[a^2 - b^2, 0] && LtQ[m, -1]

Rule 4146

Int[((A_.) + csc[(e_.) + (f_.)*(x_)]^2*(C_.))*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_), x_Symbol] :> Simp[(
A*b^2 + a^2*C)*Cot[e + f*x]*((a + b*Csc[e + f*x])^(m + 1)/(a*f*(m + 1)*(a^2 - b^2))), x] + Dist[1/(a*(m + 1)*(
a^2 - b^2)), Int[(a + b*Csc[e + f*x])^(m + 1)*Simp[A*(a^2 - b^2)*(m + 1) - a*b*(A + C)*(m + 1)*Csc[e + f*x] +
(A*b^2 + a^2*C)*(m + 2)*Csc[e + f*x]^2, x], x], x] /; FreeQ[{a, b, e, f, A, C}, x] && NeQ[a^2 - b^2, 0] && Int
egerQ[2*m] && LtQ[m, -1]

Rubi steps \begin{align*} \text {integral}& = \frac {\left (A b^2+a^2 C\right ) \tan (c+d x)}{2 a \left (a^2-b^2\right ) d (a+b \sec (c+d x))^2}-\frac {\int \frac {-2 A \left (a^2-b^2\right )+2 a b (A+C) \sec (c+d x)-\left (A b^2+a^2 C\right ) \sec ^2(c+d x)}{(a+b \sec (c+d x))^2} \, dx}{2 a \left (a^2-b^2\right )} \\ & = \frac {\left (A b^2+a^2 C\right ) \tan (c+d x)}{2 a \left (a^2-b^2\right ) d (a+b \sec (c+d x))^2}-\frac {\left (2 A b^4-a^4 C-a^2 b^2 (5 A+2 C)\right ) \tan (c+d x)}{2 a^2 \left (a^2-b^2\right )^2 d (a+b \sec (c+d x))}+\frac {\int \frac {2 A \left (a^2-b^2\right )^2+a b \left (A b^2-a^2 (4 A+3 C)\right ) \sec (c+d x)}{a+b \sec (c+d x)} \, dx}{2 a^2 \left (a^2-b^2\right )^2} \\ & = \frac {A x}{a^3}+\frac {\left (A b^2+a^2 C\right ) \tan (c+d x)}{2 a \left (a^2-b^2\right ) d (a+b \sec (c+d x))^2}-\frac {\left (2 A b^4-a^4 C-a^2 b^2 (5 A+2 C)\right ) \tan (c+d x)}{2 a^2 \left (a^2-b^2\right )^2 d (a+b \sec (c+d x))}+\frac {\left (b \left (5 a^2 A b^2-2 A b^4-3 a^4 (2 A+C)\right )\right ) \int \frac {\sec (c+d x)}{a+b \sec (c+d x)} \, dx}{2 a^3 \left (a^2-b^2\right )^2} \\ & = \frac {A x}{a^3}+\frac {\left (A b^2+a^2 C\right ) \tan (c+d x)}{2 a \left (a^2-b^2\right ) d (a+b \sec (c+d x))^2}-\frac {\left (2 A b^4-a^4 C-a^2 b^2 (5 A+2 C)\right ) \tan (c+d x)}{2 a^2 \left (a^2-b^2\right )^2 d (a+b \sec (c+d x))}+\frac {\left (5 a^2 A b^2-2 A b^4-3 a^4 (2 A+C)\right ) \int \frac {1}{1+\frac {a \cos (c+d x)}{b}} \, dx}{2 a^3 \left (a^2-b^2\right )^2} \\ & = \frac {A x}{a^3}+\frac {\left (A b^2+a^2 C\right ) \tan (c+d x)}{2 a \left (a^2-b^2\right ) d (a+b \sec (c+d x))^2}-\frac {\left (2 A b^4-a^4 C-a^2 b^2 (5 A+2 C)\right ) \tan (c+d x)}{2 a^2 \left (a^2-b^2\right )^2 d (a+b \sec (c+d x))}+\frac {\left (5 a^2 A b^2-2 A b^4-3 a^4 (2 A+C)\right ) \text {Subst}\left (\int \frac {1}{1+\frac {a}{b}+\left (1-\frac {a}{b}\right ) x^2} \, dx,x,\tan \left (\frac {1}{2} (c+d x)\right )\right )}{a^3 \left (a^2-b^2\right )^2 d} \\ & = \frac {A x}{a^3}-\frac {b \left (6 a^4 A-5 a^2 A b^2+2 A b^4+3 a^4 C\right ) \text {arctanh}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{a^3 (a-b)^{5/2} (a+b)^{5/2} d}+\frac {\left (A b^2+a^2 C\right ) \tan (c+d x)}{2 a \left (a^2-b^2\right ) d (a+b \sec (c+d x))^2}-\frac {\left (2 A b^4-a^4 C-a^2 b^2 (5 A+2 C)\right ) \tan (c+d x)}{2 a^2 \left (a^2-b^2\right )^2 d (a+b \sec (c+d x))} \\ \end{align*}

Mathematica [C] (warning: unable to verify)

Result contains complex when optimal does not.

Time = 4.12 (sec) , antiderivative size = 642, normalized size of antiderivative = 3.18 \[ \int \frac {A+C \sec ^2(c+d x)}{(a+b \sec (c+d x))^3} \, dx=\frac {(b+a \cos (c+d x)) \sec (c+d x) \left (A+C \sec ^2(c+d x)\right ) \left (\frac {4 b \left (-5 a^2 A b^2+2 A b^4+3 a^4 (2 A+C)\right ) \arctan \left (\frac {(i \cos (c)+\sin (c)) \left (a \sin (c)+(-b+a \cos (c)) \tan \left (\frac {d x}{2}\right )\right )}{\sqrt {a^2-b^2} \sqrt {(\cos (c)-i \sin (c))^2}}\right ) (b+a \cos (c+d x))^2 (i \cos (c)+\sin (c))}{\left (a^2-b^2\right )^{5/2} \sqrt {(\cos (c)-i \sin (c))^2}}+\frac {\sec (c) \left (2 A \left (a^2-b^2\right )^2 \left (a^2+2 b^2\right ) d x \cos (c)+4 a A b \left (a^2-b^2\right )^2 d x \cos (d x)+4 a^5 A b d x \cos (2 c+d x)-8 a^3 A b^3 d x \cos (2 c+d x)+4 a A b^5 d x \cos (2 c+d x)+a^6 A d x \cos (c+2 d x)-2 a^4 A b^2 d x \cos (c+2 d x)+a^2 A b^4 d x \cos (c+2 d x)+a^6 A d x \cos (3 c+2 d x)-2 a^4 A b^2 d x \cos (3 c+2 d x)+a^2 A b^4 d x \cos (3 c+2 d x)-6 a^4 A b^2 \sin (c)-9 a^2 A b^4 \sin (c)+6 A b^6 \sin (c)-2 a^6 C \sin (c)-5 a^4 b^2 C \sin (c)-2 a^2 b^4 C \sin (c)+17 a^3 A b^3 \sin (d x)-8 a A b^5 \sin (d x)+5 a^5 b C \sin (d x)+4 a^3 b^3 C \sin (d x)-7 a^3 A b^3 \sin (2 c+d x)+4 a A b^5 \sin (2 c+d x)-3 a^5 b C \sin (2 c+d x)+6 a^4 A b^2 \sin (c+2 d x)-3 a^2 A b^4 \sin (c+2 d x)+2 a^6 C \sin (c+2 d x)+a^4 b^2 C \sin (c+2 d x)\right )}{\left (a^2-b^2\right )^2}\right )}{2 a^3 d (A+2 C+A \cos (2 (c+d x))) (a+b \sec (c+d x))^3} \]

[In]

Integrate[(A + C*Sec[c + d*x]^2)/(a + b*Sec[c + d*x])^3,x]

[Out]

((b + a*Cos[c + d*x])*Sec[c + d*x]*(A + C*Sec[c + d*x]^2)*((4*b*(-5*a^2*A*b^2 + 2*A*b^4 + 3*a^4*(2*A + C))*Arc
Tan[((I*Cos[c] + Sin[c])*(a*Sin[c] + (-b + a*Cos[c])*Tan[(d*x)/2]))/(Sqrt[a^2 - b^2]*Sqrt[(Cos[c] - I*Sin[c])^
2])]*(b + a*Cos[c + d*x])^2*(I*Cos[c] + Sin[c]))/((a^2 - b^2)^(5/2)*Sqrt[(Cos[c] - I*Sin[c])^2]) + (Sec[c]*(2*
A*(a^2 - b^2)^2*(a^2 + 2*b^2)*d*x*Cos[c] + 4*a*A*b*(a^2 - b^2)^2*d*x*Cos[d*x] + 4*a^5*A*b*d*x*Cos[2*c + d*x] -
8*a^3*A*b^3*d*x*Cos[2*c + d*x] + 4*a*A*b^5*d*x*Cos[2*c + d*x] + a^6*A*d*x*Cos[c + 2*d*x] - 2*a^4*A*b^2*d*x*Co
s[c + 2*d*x] + a^2*A*b^4*d*x*Cos[c + 2*d*x] + a^6*A*d*x*Cos[3*c + 2*d*x] - 2*a^4*A*b^2*d*x*Cos[3*c + 2*d*x] +
a^2*A*b^4*d*x*Cos[3*c + 2*d*x] - 6*a^4*A*b^2*Sin[c] - 9*a^2*A*b^4*Sin[c] + 6*A*b^6*Sin[c] - 2*a^6*C*Sin[c] - 5
*a^4*b^2*C*Sin[c] - 2*a^2*b^4*C*Sin[c] + 17*a^3*A*b^3*Sin[d*x] - 8*a*A*b^5*Sin[d*x] + 5*a^5*b*C*Sin[d*x] + 4*a
^3*b^3*C*Sin[d*x] - 7*a^3*A*b^3*Sin[2*c + d*x] + 4*a*A*b^5*Sin[2*c + d*x] - 3*a^5*b*C*Sin[2*c + d*x] + 6*a^4*A
*b^2*Sin[c + 2*d*x] - 3*a^2*A*b^4*Sin[c + 2*d*x] + 2*a^6*C*Sin[c + 2*d*x] + a^4*b^2*C*Sin[c + 2*d*x]))/(a^2 -
b^2)^2))/(2*a^3*d*(A + 2*C + A*Cos[2*(c + d*x)])*(a + b*Sec[c + d*x])^3)

Maple [A] (verified)

Time = 0.40 (sec) , antiderivative size = 298, normalized size of antiderivative = 1.48

method	result	size
derivativedivides	\(\frac {\frac {2 A \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a^{3}}+\frac {\frac {2 \left (-\frac {\left (6 A \,a^{2} b^{2}+a A \,b^{3}-2 A \,b^{4}+2 a^{4} C +a^{3} b C +2 C \,a^{2} b^{2}\right ) a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{2 \left (a -b \right ) \left (a^{2}+2 a b +b^{2}\right )}+\frac {a \left (6 A \,a^{2} b^{2}-a A \,b^{3}-2 A \,b^{4}+2 a^{4} C -a^{3} b C +2 C \,a^{2} b^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 \left (a +b \right ) \left (a -b \right )^{2}}\right )}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a -\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} b -a -b \right )^{2}}-\frac {b \left (6 a^{4} A -5 A \,a^{2} b^{2}+2 A \,b^{4}+3 a^{4} C \right ) \operatorname {arctanh}\left (\frac {\left (a -b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (a +b \right ) \left (a -b \right )}}\right )}{\left (a^{4}-2 a^{2} b^{2}+b^{4}\right ) \sqrt {\left (a +b \right ) \left (a -b \right )}}}{a^{3}}}{d}\)	\(298\)
default	\(\frac {\frac {2 A \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a^{3}}+\frac {\frac {2 \left (-\frac {\left (6 A \,a^{2} b^{2}+a A \,b^{3}-2 A \,b^{4}+2 a^{4} C +a^{3} b C +2 C \,a^{2} b^{2}\right ) a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{2 \left (a -b \right ) \left (a^{2}+2 a b +b^{2}\right )}+\frac {a \left (6 A \,a^{2} b^{2}-a A \,b^{3}-2 A \,b^{4}+2 a^{4} C -a^{3} b C +2 C \,a^{2} b^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 \left (a +b \right ) \left (a -b \right )^{2}}\right )}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a -\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} b -a -b \right )^{2}}-\frac {b \left (6 a^{4} A -5 A \,a^{2} b^{2}+2 A \,b^{4}+3 a^{4} C \right ) \operatorname {arctanh}\left (\frac {\left (a -b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (a +b \right ) \left (a -b \right )}}\right )}{\left (a^{4}-2 a^{2} b^{2}+b^{4}\right ) \sqrt {\left (a +b \right ) \left (a -b \right )}}}{a^{3}}}{d}\)	\(298\)
risch	\(\frac {A x}{a^{3}}-\frac {i \left (-7 A \,a^{3} b^{3} {\mathrm e}^{3 i \left (d x +c \right )}+4 A a \,b^{5} {\mathrm e}^{3 i \left (d x +c \right )}-3 C \,a^{5} b \,{\mathrm e}^{3 i \left (d x +c \right )}-6 A \,a^{4} b^{2} {\mathrm e}^{2 i \left (d x +c \right )}-9 A \,a^{2} b^{4} {\mathrm e}^{2 i \left (d x +c \right )}+6 A \,b^{6} {\mathrm e}^{2 i \left (d x +c \right )}-2 C \,a^{6} {\mathrm e}^{2 i \left (d x +c \right )}-5 C \,a^{4} b^{2} {\mathrm e}^{2 i \left (d x +c \right )}-2 C \,a^{2} b^{4} {\mathrm e}^{2 i \left (d x +c \right )}-17 b^{3} A \,a^{3} {\mathrm e}^{i \left (d x +c \right )}+8 b^{5} A a \,{\mathrm e}^{i \left (d x +c \right )}-5 C \,a^{5} b \,{\mathrm e}^{i \left (d x +c \right )}-4 C \,a^{3} b^{3} {\mathrm e}^{i \left (d x +c \right )}-6 A \,a^{4} b^{2}+3 a^{2} A \,b^{4}-2 a^{6} C -a^{4} b^{2} C \right )}{a^{3} \left (-a^{2}+b^{2}\right )^{2} d \left (a \,{\mathrm e}^{2 i \left (d x +c \right )}+2 b \,{\mathrm e}^{i \left (d x +c \right )}+a \right )^{2}}+\frac {3 b a \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {-i a^{2}+i b^{2}+\sqrt {a^{2}-b^{2}}\, b}{a \sqrt {a^{2}-b^{2}}}\right ) A}{\sqrt {a^{2}-b^{2}}\, \left (a +b \right )^{2} \left (a -b \right )^{2} d}-\frac {5 b^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {-i a^{2}+i b^{2}+\sqrt {a^{2}-b^{2}}\, b}{a \sqrt {a^{2}-b^{2}}}\right ) A}{2 \sqrt {a^{2}-b^{2}}\, \left (a +b \right )^{2} \left (a -b \right )^{2} d a}+\frac {b^{5} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {-i a^{2}+i b^{2}+\sqrt {a^{2}-b^{2}}\, b}{a \sqrt {a^{2}-b^{2}}}\right ) A}{\sqrt {a^{2}-b^{2}}\, \left (a +b \right )^{2} \left (a -b \right )^{2} d \,a^{3}}+\frac {3 b a \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {-i a^{2}+i b^{2}+\sqrt {a^{2}-b^{2}}\, b}{a \sqrt {a^{2}-b^{2}}}\right ) C}{2 \sqrt {a^{2}-b^{2}}\, \left (a +b \right )^{2} \left (a -b \right )^{2} d}-\frac {3 a b \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i a^{2}-i b^{2}+\sqrt {a^{2}-b^{2}}\, b}{a \sqrt {a^{2}-b^{2}}}\right ) A}{\sqrt {a^{2}-b^{2}}\, \left (a +b \right )^{2} \left (a -b \right )^{2} d}+\frac {5 b^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i a^{2}-i b^{2}+\sqrt {a^{2}-b^{2}}\, b}{a \sqrt {a^{2}-b^{2}}}\right ) A}{2 \sqrt {a^{2}-b^{2}}\, \left (a +b \right )^{2} \left (a -b \right )^{2} d a}-\frac {b^{5} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i a^{2}-i b^{2}+\sqrt {a^{2}-b^{2}}\, b}{a \sqrt {a^{2}-b^{2}}}\right ) A}{\sqrt {a^{2}-b^{2}}\, \left (a +b \right )^{2} \left (a -b \right )^{2} d \,a^{3}}-\frac {3 a b \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i a^{2}-i b^{2}+\sqrt {a^{2}-b^{2}}\, b}{a \sqrt {a^{2}-b^{2}}}\right ) C}{2 \sqrt {a^{2}-b^{2}}\, \left (a +b \right )^{2} \left (a -b \right )^{2} d}\)	\(988\)

[In]

int((A+C*sec(d*x+c)^2)/(a+b*sec(d*x+c))^3,x,method=_RETURNVERBOSE)

[Out]

1/d*(2*A/a^3*arctan(tan(1/2*d*x+1/2*c))+2/a^3*((-1/2*(6*A*a^2*b^2+A*a*b^3-2*A*b^4+2*C*a^4+C*a^3*b+2*C*a^2*b^2)
*a/(a-b)/(a^2+2*a*b+b^2)*tan(1/2*d*x+1/2*c)^3+1/2*a*(6*A*a^2*b^2-A*a*b^3-2*A*b^4+2*C*a^4-C*a^3*b+2*C*a^2*b^2)/
(a+b)/(a-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)^2-1/2*b*(6*A*a^4-5*A*a^2
*b^2+2*A*b^4+3*C*a^4)/(a^4-2*a^2*b^2+b^4)/((a+b)*(a-b))^(1/2)*arctanh((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. 501 vs. \(2 (188) = 376\).

Time = 0.33 (sec) , antiderivative size = 1059, normalized size of antiderivative = 5.24 \[ \int \frac {A+C \sec ^2(c+d x)}{(a+b \sec (c+d x))^3} \, dx=\text {Too large to display} \]

[In]

integrate((A+C*sec(d*x+c)^2)/(a+b*sec(d*x+c))^3,x, algorithm="fricas")

[Out]

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

Sympy [F]

\[ \int \frac {A+C \sec ^2(c+d x)}{(a+b \sec (c+d x))^3} \, dx=\int \frac {A + C \sec ^{2}{\left (c + d x \right )}}{\left (a + b \sec {\left (c + d x \right )}\right )^{3}}\, dx \]

[In]

integrate((A+C*sec(d*x+c)**2)/(a+b*sec(d*x+c))**3,x)

[Out]

Integral((A + C*sec(c + d*x)**2)/(a + b*sec(c + d*x))**3, x)

Maxima [F(-2)]

Exception generated. \[ \int \frac {A+C \sec ^2(c+d x)}{(a+b \sec (c+d x))^3} \, dx=\text {Exception raised: ValueError} \]

[In]

integrate((A+C*sec(d*x+c)^2)/(a+b*sec(d*x+c))^3,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*a^2-4*b^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. 484 vs. \(2 (188) = 376\).

Time = 0.37 (sec) , antiderivative size = 484, normalized size of antiderivative = 2.40 \[ \int \frac {A+C \sec ^2(c+d x)}{(a+b \sec (c+d x))^3} \, dx=-\frac {\frac {{\left (6 \, A a^{4} b + 3 \, C a^{4} b - 5 \, A a^{2} b^{3} + 2 \, A b^{5}\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} - 2 \, a^{5} b^{2} + a^{3} b^{4}\right )} \sqrt {-a^{2} + b^{2}}} - \frac {{\left (d x + c\right )} A}{a^{3}} + \frac {2 \, C a^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - C a^{4} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 6 \, A a^{3} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + C a^{3} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 5 \, A a^{2} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 2 \, C a^{2} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 3 \, A a b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 2 \, A b^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 2 \, C a^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - C a^{4} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 6 \, A a^{3} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - C a^{3} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 5 \, A a^{2} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 2 \, C a^{2} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 3 \, A a b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 2 \, A b^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{{\left (a^{6} - 2 \, a^{4} b^{2} + a^{2} b^{4}\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 )}^{2}}}{d} \]

[In]

integrate((A+C*sec(d*x+c)^2)/(a+b*sec(d*x+c))^3,x, algorithm="giac")

[Out]

-((6*A*a^4*b + 3*C*a^4*b - 5*A*a^2*b^3 + 2*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)))/((a^7 - 2*a^5*b^2 + a^3*b^4)*sqrt(-a^2 +
b^2)) - (d*x + c)*A/a^3 + (2*C*a^5*tan(1/2*d*x + 1/2*c)^3 - C*a^4*b*tan(1/2*d*x + 1/2*c)^3 + 6*A*a^3*b^2*tan(1
/2*d*x + 1/2*c)^3 + C*a^3*b^2*tan(1/2*d*x + 1/2*c)^3 - 5*A*a^2*b^3*tan(1/2*d*x + 1/2*c)^3 - 2*C*a^2*b^3*tan(1/
2*d*x + 1/2*c)^3 - 3*A*a*b^4*tan(1/2*d*x + 1/2*c)^3 + 2*A*b^5*tan(1/2*d*x + 1/2*c)^3 - 2*C*a^5*tan(1/2*d*x + 1
/2*c) - C*a^4*b*tan(1/2*d*x + 1/2*c) - 6*A*a^3*b^2*tan(1/2*d*x + 1/2*c) - C*a^3*b^2*tan(1/2*d*x + 1/2*c) - 5*A
*a^2*b^3*tan(1/2*d*x + 1/2*c) - 2*C*a^2*b^3*tan(1/2*d*x + 1/2*c) + 3*A*a*b^4*tan(1/2*d*x + 1/2*c) + 2*A*b^5*ta
n(1/2*d*x + 1/2*c))/((a^6 - 2*a^4*b^2 + a^2*b^4)*(a*tan(1/2*d*x + 1/2*c)^2 - b*tan(1/2*d*x + 1/2*c)^2 - a - b)
^2))/d

Mupad [B] (verification not implemented)

Time = 26.24 (sec) , antiderivative size = 6574, normalized size of antiderivative = 32.54 \[ \int \frac {A+C \sec ^2(c+d x)}{(a+b \sec (c+d x))^3} \, dx=\text {Too large to display} \]

[In]

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

[Out]

((tan(c/2 + (d*x)/2)^3*(2*C*a^4 - 2*A*b^4 + 6*A*a^2*b^2 + 2*C*a^2*b^2 + A*a*b^3 + C*a^3*b))/((a^2*b - a^3)*(a
+ b)^2) - (tan(c/2 + (d*x)/2)*(2*A*b^4 - 2*C*a^4 - 6*A*a^2*b^2 - 2*C*a^2*b^2 + A*a*b^3 + C*a^3*b))/((a + b)*(a
^4 - 2*a^3*b + a^2*b^2)))/(d*(2*a*b - tan(c/2 + (d*x)/2)^2*(2*a^2 - 2*b^2) + tan(c/2 + (d*x)/2)^4*(a^2 - 2*a*b
+ b^2) + a^2 + b^2)) - (2*A*atan(((A*((A*((8*(4*A*a^15 - 4*A*a^6*b^9 + 2*A*a^7*b^8 + 18*A*a^8*b^7 - 4*A*a^9*b
^6 - 36*A*a^10*b^5 + 6*A*a^11*b^4 + 34*A*a^12*b^3 - 8*A*a^13*b^2 + 6*C*a^9*b^6 - 6*C*a^10*b^5 - 12*C*a^11*b^4
+ 12*C*a^12*b^3 + 6*C*a^13*b^2 - 12*A*a^14*b - 6*C*a^14*b))/(a^12*b + a^13 - a^6*b^7 - a^7*b^6 + 3*a^8*b^5 + 3
*a^9*b^4 - 3*a^10*b^3 - 3*a^11*b^2) - (A*tan(c/2 + (d*x)/2)*(8*a^15*b - 8*a^6*b^10 + 8*a^7*b^9 + 32*a^8*b^8 -
32*a^9*b^7 - 48*a^10*b^6 + 48*a^11*b^5 + 32*a^12*b^4 - 32*a^13*b^3 - 8*a^14*b^2)*8i)/(a^3*(a^10*b + a^11 - a^4
*b^7 - a^5*b^6 + 3*a^6*b^5 + 3*a^7*b^4 - 3*a^8*b^3 - 3*a^9*b^2)))*1i)/a^3 + (8*tan(c/2 + (d*x)/2)*(4*A^2*a^10
+ 8*A^2*b^10 - 8*A^2*a*b^9 - 8*A^2*a^9*b - 32*A^2*a^2*b^8 + 32*A^2*a^3*b^7 + 57*A^2*a^4*b^6 - 48*A^2*a^5*b^5 -
52*A^2*a^6*b^4 + 32*A^2*a^7*b^3 + 24*A^2*a^8*b^2 + 9*C^2*a^8*b^2 + 12*A*C*a^4*b^6 - 30*A*C*a^6*b^4 + 36*A*C*a
^8*b^2))/(a^10*b + a^11 - a^4*b^7 - a^5*b^6 + 3*a^6*b^5 + 3*a^7*b^4 - 3*a^8*b^3 - 3*a^9*b^2)))/a^3 - (A*((A*((
8*(4*A*a^15 - 4*A*a^6*b^9 + 2*A*a^7*b^8 + 18*A*a^8*b^7 - 4*A*a^9*b^6 - 36*A*a^10*b^5 + 6*A*a^11*b^4 + 34*A*a^1
2*b^3 - 8*A*a^13*b^2 + 6*C*a^9*b^6 - 6*C*a^10*b^5 - 12*C*a^11*b^4 + 12*C*a^12*b^3 + 6*C*a^13*b^2 - 12*A*a^14*b
- 6*C*a^14*b))/(a^12*b + a^13 - a^6*b^7 - a^7*b^6 + 3*a^8*b^5 + 3*a^9*b^4 - 3*a^10*b^3 - 3*a^11*b^2) + (A*tan
(c/2 + (d*x)/2)*(8*a^15*b - 8*a^6*b^10 + 8*a^7*b^9 + 32*a^8*b^8 - 32*a^9*b^7 - 48*a^10*b^6 + 48*a^11*b^5 + 32*
a^12*b^4 - 32*a^13*b^3 - 8*a^14*b^2)*8i)/(a^3*(a^10*b + a^11 - a^4*b^7 - a^5*b^6 + 3*a^6*b^5 + 3*a^7*b^4 - 3*a
^8*b^3 - 3*a^9*b^2)))*1i)/a^3 - (8*tan(c/2 + (d*x)/2)*(4*A^2*a^10 + 8*A^2*b^10 - 8*A^2*a*b^9 - 8*A^2*a^9*b - 3
2*A^2*a^2*b^8 + 32*A^2*a^3*b^7 + 57*A^2*a^4*b^6 - 48*A^2*a^5*b^5 - 52*A^2*a^6*b^4 + 32*A^2*a^7*b^3 + 24*A^2*a^
8*b^2 + 9*C^2*a^8*b^2 + 12*A*C*a^4*b^6 - 30*A*C*a^6*b^4 + 36*A*C*a^8*b^2))/(a^10*b + a^11 - a^4*b^7 - a^5*b^6
+ 3*a^6*b^5 + 3*a^7*b^4 - 3*a^8*b^3 - 3*a^9*b^2)))/a^3)/((A*((A*((8*(4*A*a^15 - 4*A*a^6*b^9 + 2*A*a^7*b^8 + 18
*A*a^8*b^7 - 4*A*a^9*b^6 - 36*A*a^10*b^5 + 6*A*a^11*b^4 + 34*A*a^12*b^3 - 8*A*a^13*b^2 + 6*C*a^9*b^6 - 6*C*a^1
0*b^5 - 12*C*a^11*b^4 + 12*C*a^12*b^3 + 6*C*a^13*b^2 - 12*A*a^14*b - 6*C*a^14*b))/(a^12*b + a^13 - a^6*b^7 - a
^7*b^6 + 3*a^8*b^5 + 3*a^9*b^4 - 3*a^10*b^3 - 3*a^11*b^2) - (A*tan(c/2 + (d*x)/2)*(8*a^15*b - 8*a^6*b^10 + 8*a
^7*b^9 + 32*a^8*b^8 - 32*a^9*b^7 - 48*a^10*b^6 + 48*a^11*b^5 + 32*a^12*b^4 - 32*a^13*b^3 - 8*a^14*b^2)*8i)/(a^
3*(a^10*b + a^11 - a^4*b^7 - a^5*b^6 + 3*a^6*b^5 + 3*a^7*b^4 - 3*a^8*b^3 - 3*a^9*b^2)))*1i)/a^3 + (8*tan(c/2 +
(d*x)/2)*(4*A^2*a^10 + 8*A^2*b^10 - 8*A^2*a*b^9 - 8*A^2*a^9*b - 32*A^2*a^2*b^8 + 32*A^2*a^3*b^7 + 57*A^2*a^4*
b^6 - 48*A^2*a^5*b^5 - 52*A^2*a^6*b^4 + 32*A^2*a^7*b^3 + 24*A^2*a^8*b^2 + 9*C^2*a^8*b^2 + 12*A*C*a^4*b^6 - 30*
A*C*a^6*b^4 + 36*A*C*a^8*b^2))/(a^10*b + a^11 - a^4*b^7 - a^5*b^6 + 3*a^6*b^5 + 3*a^7*b^4 - 3*a^8*b^3 - 3*a^9*
b^2))*1i)/a^3 - (16*(4*A^3*b^9 - 2*A^3*a*b^8 + 12*A^3*a^8*b - 18*A^3*a^2*b^7 + 13*A^3*a^3*b^6 + 36*A^3*a^4*b^5
- 26*A^3*a^5*b^4 - 34*A^3*a^6*b^3 + 24*A^3*a^7*b^2 + 6*A^2*C*a^8*b + 9*A*C^2*a^7*b^2 + 6*A^2*C*a^3*b^6 + 6*A^
2*C*a^4*b^5 - 18*A^2*C*a^5*b^4 - 12*A^2*C*a^6*b^3 + 30*A^2*C*a^7*b^2))/(a^12*b + a^13 - a^6*b^7 - a^7*b^6 + 3*
a^8*b^5 + 3*a^9*b^4 - 3*a^10*b^3 - 3*a^11*b^2) + (A*((A*((8*(4*A*a^15 - 4*A*a^6*b^9 + 2*A*a^7*b^8 + 18*A*a^8*b
^7 - 4*A*a^9*b^6 - 36*A*a^10*b^5 + 6*A*a^11*b^4 + 34*A*a^12*b^3 - 8*A*a^13*b^2 + 6*C*a^9*b^6 - 6*C*a^10*b^5 -
12*C*a^11*b^4 + 12*C*a^12*b^3 + 6*C*a^13*b^2 - 12*A*a^14*b - 6*C*a^14*b))/(a^12*b + a^13 - a^6*b^7 - a^7*b^6 +
3*a^8*b^5 + 3*a^9*b^4 - 3*a^10*b^3 - 3*a^11*b^2) + (A*tan(c/2 + (d*x)/2)*(8*a^15*b - 8*a^6*b^10 + 8*a^7*b^9 +
32*a^8*b^8 - 32*a^9*b^7 - 48*a^10*b^6 + 48*a^11*b^5 + 32*a^12*b^4 - 32*a^13*b^3 - 8*a^14*b^2)*8i)/(a^3*(a^10*
b + a^11 - a^4*b^7 - a^5*b^6 + 3*a^6*b^5 + 3*a^7*b^4 - 3*a^8*b^3 - 3*a^9*b^2)))*1i)/a^3 - (8*tan(c/2 + (d*x)/2
)*(4*A^2*a^10 + 8*A^2*b^10 - 8*A^2*a*b^9 - 8*A^2*a^9*b - 32*A^2*a^2*b^8 + 32*A^2*a^3*b^7 + 57*A^2*a^4*b^6 - 48
*A^2*a^5*b^5 - 52*A^2*a^6*b^4 + 32*A^2*a^7*b^3 + 24*A^2*a^8*b^2 + 9*C^2*a^8*b^2 + 12*A*C*a^4*b^6 - 30*A*C*a^6*
b^4 + 36*A*C*a^8*b^2))/(a^10*b + a^11 - a^4*b^7 - a^5*b^6 + 3*a^6*b^5 + 3*a^7*b^4 - 3*a^8*b^3 - 3*a^9*b^2))*1i
)/a^3)))/(a^3*d) + (b*atan(((b*((8*tan(c/2 + (d*x)/2)*(4*A^2*a^10 + 8*A^2*b^10 - 8*A^2*a*b^9 - 8*A^2*a^9*b - 3
2*A^2*a^2*b^8 + 32*A^2*a^3*b^7 + 57*A^2*a^4*b^6 - 48*A^2*a^5*b^5 - 52*A^2*a^6*b^4 + 32*A^2*a^7*b^3 + 24*A^2*a^
8*b^2 + 9*C^2*a^8*b^2 + 12*A*C*a^4*b^6 - 30*A*C*a^6*b^4 + 36*A*C*a^8*b^2))/(a^10*b + a^11 - a^4*b^7 - a^5*b^6
+ 3*a^6*b^5 + 3*a^7*b^4 - 3*a^8*b^3 - 3*a^9*b^2) + (b*((8*(4*A*a^15 - 4*A*a^6*b^9 + 2*A*a^7*b^8 + 18*A*a^8*b^7
- 4*A*a^9*b^6 - 36*A*a^10*b^5 + 6*A*a^11*b^4 + 34*A*a^12*b^3 - 8*A*a^13*b^2 + 6*C*a^9*b^6 - 6*C*a^10*b^5 - 12
*C*a^11*b^4 + 12*C*a^12*b^3 + 6*C*a^13*b^2 - 12*A*a^14*b - 6*C*a^14*b))/(a^12*b + a^13 - a^6*b^7 - a^7*b^6 + 3
*a^8*b^5 + 3*a^9*b^4 - 3*a^10*b^3 - 3*a^11*b^2) - (4*b*tan(c/2 + (d*x)/2)*((a + b)^5*(a - b)^5)^(1/2)*(6*A*a^4
+ 2*A*b^4 + 3*C*a^4 - 5*A*a^2*b^2)*(8*a^15*b - 8*a^6*b^10 + 8*a^7*b^9 + 32*a^8*b^8 - 32*a^9*b^7 - 48*a^10*b^6
+ 48*a^11*b^5 + 32*a^12*b^4 - 32*a^13*b^3 - 8*a^14*b^2))/((a^13 - a^3*b^10 + 5*a^5*b^8 - 10*a^7*b^6 + 10*a^9*
b^4 - 5*a^11*b^2)*(a^10*b + a^11 - a^4*b^7 - a^5*b^6 + 3*a^6*b^5 + 3*a^7*b^4 - 3*a^8*b^3 - 3*a^9*b^2)))*((a +
b)^5*(a - b)^5)^(1/2)*(6*A*a^4 + 2*A*b^4 + 3*C*a^4 - 5*A*a^2*b^2))/(2*(a^13 - a^3*b^10 + 5*a^5*b^8 - 10*a^7*b^
6 + 10*a^9*b^4 - 5*a^11*b^2)))*((a + b)^5*(a - b)^5)^(1/2)*(6*A*a^4 + 2*A*b^4 + 3*C*a^4 - 5*A*a^2*b^2)*1i)/(2*
(a^13 - a^3*b^10 + 5*a^5*b^8 - 10*a^7*b^6 + 10*a^9*b^4 - 5*a^11*b^2)) + (b*((8*tan(c/2 + (d*x)/2)*(4*A^2*a^10
+ 8*A^2*b^10 - 8*A^2*a*b^9 - 8*A^2*a^9*b - 32*A^2*a^2*b^8 + 32*A^2*a^3*b^7 + 57*A^2*a^4*b^6 - 48*A^2*a^5*b^5 -
52*A^2*a^6*b^4 + 32*A^2*a^7*b^3 + 24*A^2*a^8*b^2 + 9*C^2*a^8*b^2 + 12*A*C*a^4*b^6 - 30*A*C*a^6*b^4 + 36*A*C*a
^8*b^2))/(a^10*b + a^11 - a^4*b^7 - a^5*b^6 + 3*a^6*b^5 + 3*a^7*b^4 - 3*a^8*b^3 - 3*a^9*b^2) - (b*((8*(4*A*a^1
5 - 4*A*a^6*b^9 + 2*A*a^7*b^8 + 18*A*a^8*b^7 - 4*A*a^9*b^6 - 36*A*a^10*b^5 + 6*A*a^11*b^4 + 34*A*a^12*b^3 - 8*
A*a^13*b^2 + 6*C*a^9*b^6 - 6*C*a^10*b^5 - 12*C*a^11*b^4 + 12*C*a^12*b^3 + 6*C*a^13*b^2 - 12*A*a^14*b - 6*C*a^1
4*b))/(a^12*b + a^13 - a^6*b^7 - a^7*b^6 + 3*a^8*b^5 + 3*a^9*b^4 - 3*a^10*b^3 - 3*a^11*b^2) + (4*b*tan(c/2 + (
d*x)/2)*((a + b)^5*(a - b)^5)^(1/2)*(6*A*a^4 + 2*A*b^4 + 3*C*a^4 - 5*A*a^2*b^2)*(8*a^15*b - 8*a^6*b^10 + 8*a^7
*b^9 + 32*a^8*b^8 - 32*a^9*b^7 - 48*a^10*b^6 + 48*a^11*b^5 + 32*a^12*b^4 - 32*a^13*b^3 - 8*a^14*b^2))/((a^13 -
a^3*b^10 + 5*a^5*b^8 - 10*a^7*b^6 + 10*a^9*b^4 - 5*a^11*b^2)*(a^10*b + a^11 - a^4*b^7 - a^5*b^6 + 3*a^6*b^5 +
3*a^7*b^4 - 3*a^8*b^3 - 3*a^9*b^2)))*((a + b)^5*(a - b)^5)^(1/2)*(6*A*a^4 + 2*A*b^4 + 3*C*a^4 - 5*A*a^2*b^2))
/(2*(a^13 - a^3*b^10 + 5*a^5*b^8 - 10*a^7*b^6 + 10*a^9*b^4 - 5*a^11*b^2)))*((a + b)^5*(a - b)^5)^(1/2)*(6*A*a^
4 + 2*A*b^4 + 3*C*a^4 - 5*A*a^2*b^2)*1i)/(2*(a^13 - a^3*b^10 + 5*a^5*b^8 - 10*a^7*b^6 + 10*a^9*b^4 - 5*a^11*b^
2)))/((16*(4*A^3*b^9 - 2*A^3*a*b^8 + 12*A^3*a^8*b - 18*A^3*a^2*b^7 + 13*A^3*a^3*b^6 + 36*A^3*a^4*b^5 - 26*A^3*
a^5*b^4 - 34*A^3*a^6*b^3 + 24*A^3*a^7*b^2 + 6*A^2*C*a^8*b + 9*A*C^2*a^7*b^2 + 6*A^2*C*a^3*b^6 + 6*A^2*C*a^4*b^
5 - 18*A^2*C*a^5*b^4 - 12*A^2*C*a^6*b^3 + 30*A^2*C*a^7*b^2))/(a^12*b + a^13 - a^6*b^7 - a^7*b^6 + 3*a^8*b^5 +
3*a^9*b^4 - 3*a^10*b^3 - 3*a^11*b^2) - (b*((8*tan(c/2 + (d*x)/2)*(4*A^2*a^10 + 8*A^2*b^10 - 8*A^2*a*b^9 - 8*A^
2*a^9*b - 32*A^2*a^2*b^8 + 32*A^2*a^3*b^7 + 57*A^2*a^4*b^6 - 48*A^2*a^5*b^5 - 52*A^2*a^6*b^4 + 32*A^2*a^7*b^3
+ 24*A^2*a^8*b^2 + 9*C^2*a^8*b^2 + 12*A*C*a^4*b^6 - 30*A*C*a^6*b^4 + 36*A*C*a^8*b^2))/(a^10*b + a^11 - a^4*b^7
- a^5*b^6 + 3*a^6*b^5 + 3*a^7*b^4 - 3*a^8*b^3 - 3*a^9*b^2) + (b*((8*(4*A*a^15 - 4*A*a^6*b^9 + 2*A*a^7*b^8 + 1
8*A*a^8*b^7 - 4*A*a^9*b^6 - 36*A*a^10*b^5 + 6*A*a^11*b^4 + 34*A*a^12*b^3 - 8*A*a^13*b^2 + 6*C*a^9*b^6 - 6*C*a^
10*b^5 - 12*C*a^11*b^4 + 12*C*a^12*b^3 + 6*C*a^13*b^2 - 12*A*a^14*b - 6*C*a^14*b))/(a^12*b + a^13 - a^6*b^7 -
a^7*b^6 + 3*a^8*b^5 + 3*a^9*b^4 - 3*a^10*b^3 - 3*a^11*b^2) - (4*b*tan(c/2 + (d*x)/2)*((a + b)^5*(a - b)^5)^(1/
2)*(6*A*a^4 + 2*A*b^4 + 3*C*a^4 - 5*A*a^2*b^2)*(8*a^15*b - 8*a^6*b^10 + 8*a^7*b^9 + 32*a^8*b^8 - 32*a^9*b^7 -
48*a^10*b^6 + 48*a^11*b^5 + 32*a^12*b^4 - 32*a^13*b^3 - 8*a^14*b^2))/((a^13 - a^3*b^10 + 5*a^5*b^8 - 10*a^7*b^
6 + 10*a^9*b^4 - 5*a^11*b^2)*(a^10*b + a^11 - a^4*b^7 - a^5*b^6 + 3*a^6*b^5 + 3*a^7*b^4 - 3*a^8*b^3 - 3*a^9*b^
2)))*((a + b)^5*(a - b)^5)^(1/2)*(6*A*a^4 + 2*A*b^4 + 3*C*a^4 - 5*A*a^2*b^2))/(2*(a^13 - a^3*b^10 + 5*a^5*b^8
- 10*a^7*b^6 + 10*a^9*b^4 - 5*a^11*b^2)))*((a + b)^5*(a - b)^5)^(1/2)*(6*A*a^4 + 2*A*b^4 + 3*C*a^4 - 5*A*a^2*b
^2))/(2*(a^13 - a^3*b^10 + 5*a^5*b^8 - 10*a^7*b^6 + 10*a^9*b^4 - 5*a^11*b^2)) + (b*((8*tan(c/2 + (d*x)/2)*(4*A
^2*a^10 + 8*A^2*b^10 - 8*A^2*a*b^9 - 8*A^2*a^9*b - 32*A^2*a^2*b^8 + 32*A^2*a^3*b^7 + 57*A^2*a^4*b^6 - 48*A^2*a
^5*b^5 - 52*A^2*a^6*b^4 + 32*A^2*a^7*b^3 + 24*A^2*a^8*b^2 + 9*C^2*a^8*b^2 + 12*A*C*a^4*b^6 - 30*A*C*a^6*b^4 +
36*A*C*a^8*b^2))/(a^10*b + a^11 - a^4*b^7 - a^5*b^6 + 3*a^6*b^5 + 3*a^7*b^4 - 3*a^8*b^3 - 3*a^9*b^2) - (b*((8*
(4*A*a^15 - 4*A*a^6*b^9 + 2*A*a^7*b^8 + 18*A*a^8*b^7 - 4*A*a^9*b^6 - 36*A*a^10*b^5 + 6*A*a^11*b^4 + 34*A*a^12*
b^3 - 8*A*a^13*b^2 + 6*C*a^9*b^6 - 6*C*a^10*b^5 - 12*C*a^11*b^4 + 12*C*a^12*b^3 + 6*C*a^13*b^2 - 12*A*a^14*b -
6*C*a^14*b))/(a^12*b + a^13 - a^6*b^7 - a^7*b^6 + 3*a^8*b^5 + 3*a^9*b^4 - 3*a^10*b^3 - 3*a^11*b^2) + (4*b*tan
(c/2 + (d*x)/2)*((a + b)^5*(a - b)^5)^(1/2)*(6*A*a^4 + 2*A*b^4 + 3*C*a^4 - 5*A*a^2*b^2)*(8*a^15*b - 8*a^6*b^10
+ 8*a^7*b^9 + 32*a^8*b^8 - 32*a^9*b^7 - 48*a^10*b^6 + 48*a^11*b^5 + 32*a^12*b^4 - 32*a^13*b^3 - 8*a^14*b^2))/
((a^13 - a^3*b^10 + 5*a^5*b^8 - 10*a^7*b^6 + 10*a^9*b^4 - 5*a^11*b^2)*(a^10*b + a^11 - a^4*b^7 - a^5*b^6 + 3*a
^6*b^5 + 3*a^7*b^4 - 3*a^8*b^3 - 3*a^9*b^2)))*((a + b)^5*(a - b)^5)^(1/2)*(6*A*a^4 + 2*A*b^4 + 3*C*a^4 - 5*A*a
^2*b^2))/(2*(a^13 - a^3*b^10 + 5*a^5*b^8 - 10*a^7*b^6 + 10*a^9*b^4 - 5*a^11*b^2)))*((a + b)^5*(a - b)^5)^(1/2)
*(6*A*a^4 + 2*A*b^4 + 3*C*a^4 - 5*A*a^2*b^2))/(2*(a^13 - a^3*b^10 + 5*a^5*b^8 - 10*a^7*b^6 + 10*a^9*b^4 - 5*a^
11*b^2))))*((a + b)^5*(a - b)^5)^(1/2)*(6*A*a^4 + 2*A*b^4 + 3*C*a^4 - 5*A*a^2*b^2)*1i)/(d*(a^13 - a^3*b^10 + 5
*a^5*b^8 - 10*a^7*b^6 + 10*a^9*b^4 - 5*a^11*b^2))