\(\int \frac {\sec (e+f x)}{(a+b \sec (e+f x))^2 (c+d \sec (e+f x))} \, dx\) [263]

Optimal result

Integrand size = 31, antiderivative size = 186 \[ \int \frac {\sec (e+f x)}{(a+b \sec (e+f x))^2 (c+d \sec (e+f x))} \, dx=\frac {2 b \left (a b c-2 a^2 d+b^2 d\right ) \text {arctanh}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (e+f x)\right )}{\sqrt {a+b}}\right )}{(a-b)^{3/2} (a+b)^{3/2} (b c-a d)^2 f}+\frac {2 d^2 \text {arctanh}\left (\frac {\sqrt {c-d} \tan \left (\frac {1}{2} (e+f x)\right )}{\sqrt {c+d}}\right )}{\sqrt {c-d} \sqrt {c+d} (b c-a d)^2 f}-\frac {b^2 \sin (e+f x)}{\left (a^2-b^2\right ) (b c-a d) f (b+a \cos (e+f x))} \]

[Out]

2*b*(-2*a^2*d+a*b*c+b^2*d)*arctanh((a-b)^(1/2)*tan(1/2*f*x+1/2*e)/(a+b)^(1/2))/(a-b)^(3/2)/(a+b)^(3/2)/(-a*d+b
*c)^2/f-b^2*sin(f*x+e)/(a^2-b^2)/(-a*d+b*c)/f/(b+a*cos(f*x+e))+2*d^2*arctanh((c-d)^(1/2)*tan(1/2*f*x+1/2*e)/(c
+d)^(1/2))/(-a*d+b*c)^2/f/(c-d)^(1/2)/(c+d)^(1/2)

Rubi [A] (verified)

Time = 0.93 (sec) , antiderivative size = 186, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.161, Rules used = {4073, 3135, 3080, 2738, 214} \[ \int \frac {\sec (e+f x)}{(a+b \sec (e+f x))^2 (c+d \sec (e+f x))} \, dx=\frac {2 b \left (-2 a^2 d+a b c+b^2 d\right ) \text {arctanh}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (e+f x)\right )}{\sqrt {a+b}}\right )}{f (a-b)^{3/2} (a+b)^{3/2} (b c-a d)^2}-\frac {b^2 \sin (e+f x)}{f \left (a^2-b^2\right ) (b c-a d) (a \cos (e+f x)+b)}+\frac {2 d^2 \text {arctanh}\left (\frac {\sqrt {c-d} \tan \left (\frac {1}{2} (e+f x)\right )}{\sqrt {c+d}}\right )}{f \sqrt {c-d} \sqrt {c+d} (b c-a d)^2} \]

[In]

Int[Sec[e + f*x]/((a + b*Sec[e + f*x])^2*(c + d*Sec[e + f*x])),x]

[Out]

(2*b*(a*b*c - 2*a^2*d + b^2*d)*ArcTanh[(Sqrt[a - b]*Tan[(e + f*x)/2])/Sqrt[a + b]])/((a - b)^(3/2)*(a + b)^(3/
2)*(b*c - a*d)^2*f) + (2*d^2*ArcTanh[(Sqrt[c - d]*Tan[(e + f*x)/2])/Sqrt[c + d]])/(Sqrt[c - d]*Sqrt[c + d]*(b*
c - a*d)^2*f) - (b^2*Sin[e + f*x])/((a^2 - b^2)*(b*c - a*d)*f*(b + a*Cos[e + f*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 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 3135

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

Rule 4073

Int[(csc[(e_.) + (f_.)*(x_)]*(g_.))^(p_.)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_)*(csc[(e_.) + (f_.)*(x_)]
*(d_.) + (c_))^(n_), x_Symbol] :> Dist[1/g^(m + n), Int[(g*Csc[e + f*x])^(m + n + p)*(b + a*Sin[e + f*x])^m*(d
+ c*Sin[e + f*x])^n, x], x] /; FreeQ[{a, b, c, d, e, f, g, p}, x] && NeQ[b*c - a*d, 0] && IntegerQ[m] && Inte
gerQ[n]

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

Mathematica [A] (verified)

Time = 1.59 (sec) , antiderivative size = 176, normalized size of antiderivative = 0.95 \[ \int \frac {\sec (e+f x)}{(a+b \sec (e+f x))^2 (c+d \sec (e+f x))} \, dx=\frac {\frac {2 b \left (a b c-2 a^2 d+b^2 d\right ) \text {arctanh}\left (\frac {(-a+b) \tan \left (\frac {1}{2} (e+f x)\right )}{\sqrt {a^2-b^2}}\right )}{\sqrt {a^2-b^2}}+\frac {2 \left (a^2-b^2\right ) d^2 \text {arctanh}\left (\frac {(-c+d) \tan \left (\frac {1}{2} (e+f x)\right )}{\sqrt {c^2-d^2}}\right )}{\sqrt {c^2-d^2}}+\frac {b^2 (b c-a d) \sin (e+f x)}{b+a \cos (e+f x)}}{(-a+b) (a+b) (b c-a d)^2 f} \]

[In]

Integrate[Sec[e + f*x]/((a + b*Sec[e + f*x])^2*(c + d*Sec[e + f*x])),x]

[Out]

((2*b*(a*b*c - 2*a^2*d + b^2*d)*ArcTanh[((-a + b)*Tan[(e + f*x)/2])/Sqrt[a^2 - b^2]])/Sqrt[a^2 - b^2] + (2*(a^
2 - b^2)*d^2*ArcTanh[((-c + d)*Tan[(e + f*x)/2])/Sqrt[c^2 - d^2]])/Sqrt[c^2 - d^2] + (b^2*(b*c - a*d)*Sin[e +
f*x])/(b + a*Cos[e + f*x]))/((-a + b)*(a + b)*(b*c - a*d)^2*f)

Maple [A] (verified)

Time = 1.45 (sec) , antiderivative size = 210, normalized size of antiderivative = 1.13

method	result	size
derivativedivides	\(\frac {\frac {2 b \left (-\frac {b \left (a d -b c \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{\left (a^{2}-b^{2}\right ) \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{2} a -\tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{2} b -a -b \right )}-\frac {\left (2 a^{2} d -a b c -d \,b^{2}\right ) \operatorname {arctanh}\left (\frac {\left (a -b \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{\sqrt {\left (a -b \right ) \left (a +b \right )}}\right )}{\left (a -b \right ) \left (a +b \right ) \sqrt {\left (a -b \right ) \left (a +b \right )}}\right )}{\left (a d -b c \right )^{2}}+\frac {2 d^{2} \operatorname {arctanh}\left (\frac {\left (c -d \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{\sqrt {\left (c +d \right ) \left (c -d \right )}}\right )}{\left (a d -b c \right )^{2} \sqrt {\left (c +d \right ) \left (c -d \right )}}}{f}\)	\(210\)
default	\(\frac {\frac {2 b \left (-\frac {b \left (a d -b c \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{\left (a^{2}-b^{2}\right ) \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{2} a -\tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{2} b -a -b \right )}-\frac {\left (2 a^{2} d -a b c -d \,b^{2}\right ) \operatorname {arctanh}\left (\frac {\left (a -b \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{\sqrt {\left (a -b \right ) \left (a +b \right )}}\right )}{\left (a -b \right ) \left (a +b \right ) \sqrt {\left (a -b \right ) \left (a +b \right )}}\right )}{\left (a d -b c \right )^{2}}+\frac {2 d^{2} \operatorname {arctanh}\left (\frac {\left (c -d \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{\sqrt {\left (c +d \right ) \left (c -d \right )}}\right )}{\left (a d -b c \right )^{2} \sqrt {\left (c +d \right ) \left (c -d \right )}}}{f}\)	\(210\)
risch	\(\frac {2 i b^{2} \left ({\mathrm e}^{i \left (f x +e \right )} b +a \right )}{a \left (a^{2}-b^{2}\right ) \left (a d -b c \right ) f \left ({\mathrm e}^{2 i \left (f x +e \right )} a +2 \,{\mathrm e}^{i \left (f x +e \right )} b +a \right )}+\frac {2 b \ln \left ({\mathrm e}^{i \left (f x +e \right )}-\frac {i a^{2}-i b^{2}-b \sqrt {a^{2}-b^{2}}}{\sqrt {a^{2}-b^{2}}\, a}\right ) a^{2} d}{\sqrt {a^{2}-b^{2}}\, \left (a d -b c \right )^{2} \left (a +b \right ) \left (a -b \right ) f}-\frac {b^{2} \ln \left ({\mathrm e}^{i \left (f x +e \right )}-\frac {i a^{2}-i b^{2}-b \sqrt {a^{2}-b^{2}}}{\sqrt {a^{2}-b^{2}}\, a}\right ) a c}{\sqrt {a^{2}-b^{2}}\, \left (a d -b c \right )^{2} \left (a +b \right ) \left (a -b \right ) f}-\frac {b^{3} \ln \left ({\mathrm e}^{i \left (f x +e \right )}-\frac {i a^{2}-i b^{2}-b \sqrt {a^{2}-b^{2}}}{\sqrt {a^{2}-b^{2}}\, a}\right ) d}{\sqrt {a^{2}-b^{2}}\, \left (a d -b c \right )^{2} \left (a +b \right ) \left (a -b \right ) f}-\frac {2 b \ln \left ({\mathrm e}^{i \left (f x +e \right )}+\frac {i a^{2}-i b^{2}+b \sqrt {a^{2}-b^{2}}}{\sqrt {a^{2}-b^{2}}\, a}\right ) a^{2} d}{\sqrt {a^{2}-b^{2}}\, \left (a d -b c \right )^{2} \left (a +b \right ) \left (a -b \right ) f}+\frac {b^{2} \ln \left ({\mathrm e}^{i \left (f x +e \right )}+\frac {i a^{2}-i b^{2}+b \sqrt {a^{2}-b^{2}}}{\sqrt {a^{2}-b^{2}}\, a}\right ) a c}{\sqrt {a^{2}-b^{2}}\, \left (a d -b c \right )^{2} \left (a +b \right ) \left (a -b \right ) f}+\frac {b^{3} \ln \left ({\mathrm e}^{i \left (f x +e \right )}+\frac {i a^{2}-i b^{2}+b \sqrt {a^{2}-b^{2}}}{\sqrt {a^{2}-b^{2}}\, a}\right ) d}{\sqrt {a^{2}-b^{2}}\, \left (a d -b c \right )^{2} \left (a +b \right ) \left (a -b \right ) f}+\frac {d^{2} \ln \left ({\mathrm e}^{i \left (f x +e \right )}+\frac {i c^{2}-i d^{2}+\sqrt {c^{2}-d^{2}}\, d}{\sqrt {c^{2}-d^{2}}\, c}\right )}{\sqrt {c^{2}-d^{2}}\, \left (a d -b c \right )^{2} f}-\frac {d^{2} \ln \left ({\mathrm e}^{i \left (f x +e \right )}-\frac {i c^{2}-i d^{2}-\sqrt {c^{2}-d^{2}}\, d}{\sqrt {c^{2}-d^{2}}\, c}\right )}{\sqrt {c^{2}-d^{2}}\, \left (a d -b c \right )^{2} f}\)	\(810\)

[In]

int(sec(f*x+e)/(a+b*sec(f*x+e))^2/(c+d*sec(f*x+e)),x,method=_RETURNVERBOSE)

[Out]

1/f*(2*b/(a*d-b*c)^2*(-b*(a*d-b*c)/(a^2-b^2)*tan(1/2*f*x+1/2*e)/(tan(1/2*f*x+1/2*e)^2*a-tan(1/2*f*x+1/2*e)^2*b
-a-b)-(2*a^2*d-a*b*c-b^2*d)/(a-b)/(a+b)/((a-b)*(a+b))^(1/2)*arctanh((a-b)*tan(1/2*f*x+1/2*e)/((a-b)*(a+b))^(1/
2)))+2*d^2/(a*d-b*c)^2/((c+d)*(c-d))^(1/2)*arctanh((c-d)*tan(1/2*f*x+1/2*e)/((c+d)*(c-d))^(1/2)))

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 655 vs. \(2 (168) = 336\).

Time = 60.79 (sec) , antiderivative size = 2852, normalized size of antiderivative = 15.33 \[ \int \frac {\sec (e+f x)}{(a+b \sec (e+f x))^2 (c+d \sec (e+f x))} \, dx=\text {Too large to display} \]

[In]

integrate(sec(f*x+e)/(a+b*sec(f*x+e))^2/(c+d*sec(f*x+e)),x, algorithm="fricas")

[Out]

[-1/2*((a*b^3*c^3 - a*b^3*c*d^2 - (2*a^2*b^2 - b^4)*c^2*d + (2*a^2*b^2 - b^4)*d^3 + (a^2*b^2*c^3 - a^2*b^2*c*d
^2 - (2*a^3*b - a*b^3)*c^2*d + (2*a^3*b - a*b^3)*d^3)*cos(f*x + e))*sqrt(a^2 - b^2)*log((2*a*b*cos(f*x + e) -
(a^2 - 2*b^2)*cos(f*x + e)^2 - 2*sqrt(a^2 - b^2)*(b*cos(f*x + e) + a)*sin(f*x + e) + 2*a^2 - b^2)/(a^2*cos(f*x
+ e)^2 + 2*a*b*cos(f*x + e) + b^2)) - ((a^5 - 2*a^3*b^2 + a*b^4)*d^2*cos(f*x + e) + (a^4*b - 2*a^2*b^3 + b^5)
*d^2)*sqrt(c^2 - d^2)*log((2*c*d*cos(f*x + e) - (c^2 - 2*d^2)*cos(f*x + e)^2 + 2*sqrt(c^2 - d^2)*(d*cos(f*x +
e) + c)*sin(f*x + e) + 2*c^2 - d^2)/(c^2*cos(f*x + e)^2 + 2*c*d*cos(f*x + e) + d^2)) + 2*((a^2*b^3 - b^5)*c^3
- (a^3*b^2 - a*b^4)*c^2*d - (a^2*b^3 - b^5)*c*d^2 + (a^3*b^2 - a*b^4)*d^3)*sin(f*x + e))/(((a^5*b^2 - 2*a^3*b^
4 + a*b^6)*c^4 - 2*(a^6*b - 2*a^4*b^3 + a^2*b^5)*c^3*d + (a^7 - 3*a^5*b^2 + 3*a^3*b^4 - a*b^6)*c^2*d^2 + 2*(a^
6*b - 2*a^4*b^3 + a^2*b^5)*c*d^3 - (a^7 - 2*a^5*b^2 + a^3*b^4)*d^4)*f*cos(f*x + e) + ((a^4*b^3 - 2*a^2*b^5 + b
^7)*c^4 - 2*(a^5*b^2 - 2*a^3*b^4 + a*b^6)*c^3*d + (a^6*b - 3*a^4*b^3 + 3*a^2*b^5 - b^7)*c^2*d^2 + 2*(a^5*b^2 -
2*a^3*b^4 + a*b^6)*c*d^3 - (a^6*b - 2*a^4*b^3 + a^2*b^5)*d^4)*f), 1/2*(2*((a^5 - 2*a^3*b^2 + a*b^4)*d^2*cos(f
*x + e) + (a^4*b - 2*a^2*b^3 + b^5)*d^2)*sqrt(-c^2 + d^2)*arctan(-sqrt(-c^2 + d^2)*(d*cos(f*x + e) + c)/((c^2
- d^2)*sin(f*x + e))) - (a*b^3*c^3 - a*b^3*c*d^2 - (2*a^2*b^2 - b^4)*c^2*d + (2*a^2*b^2 - b^4)*d^3 + (a^2*b^2*
c^3 - a^2*b^2*c*d^2 - (2*a^3*b - a*b^3)*c^2*d + (2*a^3*b - a*b^3)*d^3)*cos(f*x + e))*sqrt(a^2 - b^2)*log((2*a*
b*cos(f*x + e) - (a^2 - 2*b^2)*cos(f*x + e)^2 - 2*sqrt(a^2 - b^2)*(b*cos(f*x + e) + a)*sin(f*x + e) + 2*a^2 -
b^2)/(a^2*cos(f*x + e)^2 + 2*a*b*cos(f*x + e) + b^2)) - 2*((a^2*b^3 - b^5)*c^3 - (a^3*b^2 - a*b^4)*c^2*d - (a^
2*b^3 - b^5)*c*d^2 + (a^3*b^2 - a*b^4)*d^3)*sin(f*x + e))/(((a^5*b^2 - 2*a^3*b^4 + a*b^6)*c^4 - 2*(a^6*b - 2*a
^4*b^3 + a^2*b^5)*c^3*d + (a^7 - 3*a^5*b^2 + 3*a^3*b^4 - a*b^6)*c^2*d^2 + 2*(a^6*b - 2*a^4*b^3 + a^2*b^5)*c*d^
3 - (a^7 - 2*a^5*b^2 + a^3*b^4)*d^4)*f*cos(f*x + e) + ((a^4*b^3 - 2*a^2*b^5 + b^7)*c^4 - 2*(a^5*b^2 - 2*a^3*b^
4 + a*b^6)*c^3*d + (a^6*b - 3*a^4*b^3 + 3*a^2*b^5 - b^7)*c^2*d^2 + 2*(a^5*b^2 - 2*a^3*b^4 + a*b^6)*c*d^3 - (a^
6*b - 2*a^4*b^3 + a^2*b^5)*d^4)*f), 1/2*(2*(a*b^3*c^3 - a*b^3*c*d^2 - (2*a^2*b^2 - b^4)*c^2*d + (2*a^2*b^2 - b
^4)*d^3 + (a^2*b^2*c^3 - a^2*b^2*c*d^2 - (2*a^3*b - a*b^3)*c^2*d + (2*a^3*b - a*b^3)*d^3)*cos(f*x + e))*sqrt(-
a^2 + b^2)*arctan(-sqrt(-a^2 + b^2)*(b*cos(f*x + e) + a)/((a^2 - b^2)*sin(f*x + e))) + ((a^5 - 2*a^3*b^2 + a*b
^4)*d^2*cos(f*x + e) + (a^4*b - 2*a^2*b^3 + b^5)*d^2)*sqrt(c^2 - d^2)*log((2*c*d*cos(f*x + e) - (c^2 - 2*d^2)*
cos(f*x + e)^2 + 2*sqrt(c^2 - d^2)*(d*cos(f*x + e) + c)*sin(f*x + e) + 2*c^2 - d^2)/(c^2*cos(f*x + e)^2 + 2*c*
d*cos(f*x + e) + d^2)) - 2*((a^2*b^3 - b^5)*c^3 - (a^3*b^2 - a*b^4)*c^2*d - (a^2*b^3 - b^5)*c*d^2 + (a^3*b^2 -
a*b^4)*d^3)*sin(f*x + e))/(((a^5*b^2 - 2*a^3*b^4 + a*b^6)*c^4 - 2*(a^6*b - 2*a^4*b^3 + a^2*b^5)*c^3*d + (a^7
- 3*a^5*b^2 + 3*a^3*b^4 - a*b^6)*c^2*d^2 + 2*(a^6*b - 2*a^4*b^3 + a^2*b^5)*c*d^3 - (a^7 - 2*a^5*b^2 + a^3*b^4)
*d^4)*f*cos(f*x + e) + ((a^4*b^3 - 2*a^2*b^5 + b^7)*c^4 - 2*(a^5*b^2 - 2*a^3*b^4 + a*b^6)*c^3*d + (a^6*b - 3*a
^4*b^3 + 3*a^2*b^5 - b^7)*c^2*d^2 + 2*(a^5*b^2 - 2*a^3*b^4 + a*b^6)*c*d^3 - (a^6*b - 2*a^4*b^3 + a^2*b^5)*d^4)
*f), ((a*b^3*c^3 - a*b^3*c*d^2 - (2*a^2*b^2 - b^4)*c^2*d + (2*a^2*b^2 - b^4)*d^3 + (a^2*b^2*c^3 - a^2*b^2*c*d^
2 - (2*a^3*b - a*b^3)*c^2*d + (2*a^3*b - a*b^3)*d^3)*cos(f*x + e))*sqrt(-a^2 + b^2)*arctan(-sqrt(-a^2 + b^2)*(
b*cos(f*x + e) + a)/((a^2 - b^2)*sin(f*x + e))) + ((a^5 - 2*a^3*b^2 + a*b^4)*d^2*cos(f*x + e) + (a^4*b - 2*a^2
*b^3 + b^5)*d^2)*sqrt(-c^2 + d^2)*arctan(-sqrt(-c^2 + d^2)*(d*cos(f*x + e) + c)/((c^2 - d^2)*sin(f*x + e))) -
((a^2*b^3 - b^5)*c^3 - (a^3*b^2 - a*b^4)*c^2*d - (a^2*b^3 - b^5)*c*d^2 + (a^3*b^2 - a*b^4)*d^3)*sin(f*x + e))/
(((a^5*b^2 - 2*a^3*b^4 + a*b^6)*c^4 - 2*(a^6*b - 2*a^4*b^3 + a^2*b^5)*c^3*d + (a^7 - 3*a^5*b^2 + 3*a^3*b^4 - a
*b^6)*c^2*d^2 + 2*(a^6*b - 2*a^4*b^3 + a^2*b^5)*c*d^3 - (a^7 - 2*a^5*b^2 + a^3*b^4)*d^4)*f*cos(f*x + e) + ((a^
4*b^3 - 2*a^2*b^5 + b^7)*c^4 - 2*(a^5*b^2 - 2*a^3*b^4 + a*b^6)*c^3*d + (a^6*b - 3*a^4*b^3 + 3*a^2*b^5 - b^7)*c
^2*d^2 + 2*(a^5*b^2 - 2*a^3*b^4 + a*b^6)*c*d^3 - (a^6*b - 2*a^4*b^3 + a^2*b^5)*d^4)*f)]

Sympy [F]

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

[In]

integrate(sec(f*x+e)/(a+b*sec(f*x+e))**2/(c+d*sec(f*x+e)),x)

[Out]

Integral(sec(e + f*x)/((a + b*sec(e + f*x))**2*(c + d*sec(e + f*x))), x)

Maxima [F(-2)]

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

[In]

integrate(sec(f*x+e)/(a+b*sec(f*x+e))^2/(c+d*sec(f*x+e)),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*c^2-4*d^2>0)', see `assume?`
for more de

Giac [A] (verification not implemented)

none

Time = 0.36 (sec) , antiderivative size = 330, normalized size of antiderivative = 1.77 \[ \int \frac {\sec (e+f x)}{(a+b \sec (e+f x))^2 (c+d \sec (e+f x))} \, dx=\frac {2 \, {\left (\frac {{\left (\pi \left \lfloor \frac {f x + e}{2 \, \pi } + \frac {1}{2} \right \rfloor \mathrm {sgn}\left (-2 \, c + 2 \, d\right ) + \arctan \left (-\frac {c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )}{\sqrt {-c^{2} + d^{2}}}\right )\right )} d^{2}}{{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \sqrt {-c^{2} + d^{2}}} + \frac {b^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )}{{\left (a^{2} b c - b^{3} c - a^{3} d + a b^{2} d\right )} {\left (a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - b \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - a - b\right )}} - \frac {{\left (a b^{2} c - 2 \, a^{2} b d + b^{3} d\right )} {\left (\pi \left \lfloor \frac {f x + e}{2 \, \pi } + \frac {1}{2} \right \rfloor \mathrm {sgn}\left (2 \, a - 2 \, b\right ) + \arctan \left (\frac {a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - b \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )}{\sqrt {-a^{2} + b^{2}}}\right )\right )}}{{\left (a^{2} b^{2} c^{2} - b^{4} c^{2} - 2 \, a^{3} b c d + 2 \, a b^{3} c d + a^{4} d^{2} - a^{2} b^{2} d^{2}\right )} \sqrt {-a^{2} + b^{2}}}\right )}}{f} \]

[In]

integrate(sec(f*x+e)/(a+b*sec(f*x+e))^2/(c+d*sec(f*x+e)),x, algorithm="giac")

[Out]

2*((pi*floor(1/2*(f*x + e)/pi + 1/2)*sgn(-2*c + 2*d) + arctan(-(c*tan(1/2*f*x + 1/2*e) - d*tan(1/2*f*x + 1/2*e
))/sqrt(-c^2 + d^2)))*d^2/((b^2*c^2 - 2*a*b*c*d + a^2*d^2)*sqrt(-c^2 + d^2)) + b^2*tan(1/2*f*x + 1/2*e)/((a^2*
b*c - b^3*c - a^3*d + a*b^2*d)*(a*tan(1/2*f*x + 1/2*e)^2 - b*tan(1/2*f*x + 1/2*e)^2 - a - b)) - (a*b^2*c - 2*a
^2*b*d + b^3*d)*(pi*floor(1/2*(f*x + e)/pi + 1/2)*sgn(2*a - 2*b) + arctan((a*tan(1/2*f*x + 1/2*e) - b*tan(1/2*
f*x + 1/2*e))/sqrt(-a^2 + b^2)))/((a^2*b^2*c^2 - b^4*c^2 - 2*a^3*b*c*d + 2*a*b^3*c*d + a^4*d^2 - a^2*b^2*d^2)*
sqrt(-a^2 + b^2)))/f

Mupad [B] (verification not implemented)

Time = 27.48 (sec) , antiderivative size = 20827, normalized size of antiderivative = 111.97 \[ \int \frac {\sec (e+f x)}{(a+b \sec (e+f x))^2 (c+d \sec (e+f x))} \, dx=\text {Too large to display} \]

[In]

int(1/(cos(e + f*x)*(a + b/cos(e + f*x))^2*(c + d/cos(e + f*x))),x)

[Out]

(d^2*atan(((d^2*(c^2 - d^2)^(1/2)*((32*tan(e/2 + (f*x)/2)*(a^6*d^5 + 2*b^6*d^5 - 2*a*b^5*d^5 - 2*a^5*b*d^5 - a
^6*c*d^4 - 4*b^6*c*d^4 - a^2*b^4*c^5 - 5*a^2*b^4*d^5 + 4*a^3*b^3*d^5 + 3*a^4*b^2*d^5 + 3*b^6*c^2*d^3 - b^6*c^3
*d^2 - 6*a*b^5*c^2*d^3 + 6*a*b^5*c^3*d^2 + 13*a^2*b^4*c*d^4 + 3*a^2*b^4*c^4*d - 8*a^3*b^3*c*d^4 + 4*a^3*b^3*c^
4*d - 11*a^4*b^2*c*d^4 - 11*a^2*b^4*c^2*d^3 + a^2*b^4*c^3*d^2 + 12*a^3*b^3*c^2*d^3 - 12*a^3*b^3*c^3*d^2 + 12*a
^4*b^2*c^2*d^3 - 4*a^4*b^2*c^3*d^2 + 4*a*b^5*c*d^4 - 2*a*b^5*c^4*d + 2*a^5*b*c*d^4))/(a^5*d^2 - b^5*c^2 - a*b^
4*c^2 + a^4*b*d^2 + a^2*b^3*c^2 + a^3*b^2*c^2 - a^2*b^3*d^2 - a^3*b^2*d^2 + 2*a*b^4*c*d - 2*a^4*b*c*d + 2*a^2*
b^3*c*d - 2*a^3*b^2*c*d) + (d^2*(c^2 - d^2)^(1/2)*((32*(a*b^8*c^7 - a^9*d^7 + 2*a^8*b*d^7 + 2*a^9*c*d^6 + b^9*
c^6*d - a^2*b^7*c^7 - a^3*b^6*c^7 + a^4*b^5*c^7 + a^4*b^5*d^7 - 3*a^6*b^3*d^7 + a^7*b^2*d^7 - a^9*c^2*d^5 + b^
9*c^4*d^3 - 2*b^9*c^5*d^2 - 4*a*b^8*c^3*d^4 + 8*a*b^8*c^4*d^3 - 3*a*b^8*c^5*d^2 - 5*a^2*b^7*c^6*d - 4*a^3*b^6*
c*d^6 + 7*a^3*b^6*c^6*d - 2*a^4*b^5*c*d^6 + 4*a^4*b^5*c^6*d + 13*a^5*b^4*c*d^6 - 5*a^5*b^4*c^6*d + a^6*b^3*c*d
^6 - 11*a^7*b^2*c*d^6 - 8*a^8*b*c^2*d^5 + 5*a^8*b*c^3*d^4 + 6*a^2*b^7*c^2*d^5 - 12*a^2*b^7*c^3*d^4 - a^2*b^7*c
^4*d^3 + 13*a^2*b^7*c^5*d^2 + 8*a^3*b^6*c^2*d^5 + 14*a^3*b^6*c^3*d^4 - 31*a^3*b^6*c^4*d^3 + 7*a^3*b^6*c^5*d^2
- 21*a^4*b^5*c^2*d^5 + 34*a^4*b^5*c^3*d^4 + 4*a^4*b^5*c^4*d^3 - 21*a^4*b^5*c^5*d^2 - 16*a^5*b^4*c^2*d^5 - 21*a
^5*b^4*c^3*d^4 + 33*a^5*b^4*c^4*d^3 - 4*a^5*b^4*c^5*d^2 + 23*a^6*b^3*c^2*d^5 - 27*a^6*b^3*c^3*d^4 - 4*a^6*b^3*
c^4*d^3 + 10*a^6*b^3*c^5*d^2 + 9*a^7*b^2*c^2*d^5 + 11*a^7*b^2*c^3*d^4 - 10*a^7*b^2*c^4*d^3 - 2*a*b^8*c^6*d + a
^8*b*c*d^6))/(a^6*d^3 + b^6*c^3 + a*b^5*c^3 + a^5*b*d^3 - a^2*b^4*c^3 - a^3*b^3*c^3 - a^3*b^3*d^3 - a^4*b^2*d^
3 + 3*a^2*b^4*c*d^2 - 3*a^2*b^4*c^2*d + 3*a^3*b^3*c*d^2 + 3*a^3*b^3*c^2*d - 3*a^4*b^2*c*d^2 + 3*a^4*b^2*c^2*d
- 3*a*b^5*c^2*d - 3*a^5*b*c*d^2) + (32*d^2*tan(e/2 + (f*x)/2)*(c^2 - d^2)^(1/2)*(2*a^10*c*d^6 - 2*a^9*b*d^7 -
2*a*b^9*c^7 + 2*b^10*c^6*d + 2*a^2*b^8*c^7 + 4*a^3*b^7*c^7 - 4*a^4*b^6*c^7 - 2*a^5*b^5*c^7 + 2*a^6*b^4*c^7 + 2
*a^4*b^6*d^7 - 2*a^5*b^5*d^7 - 4*a^6*b^4*d^7 + 4*a^7*b^3*d^7 + 2*a^8*b^2*d^7 - 4*a^10*c^2*d^5 + 2*a^10*c^3*d^4
+ 2*b^10*c^4*d^3 - 4*b^10*c^5*d^2 - 8*a*b^9*c^3*d^4 + 14*a*b^9*c^4*d^3 - 6*a*b^9*c^5*d^2 - 8*a^3*b^7*c*d^6 -
12*a^3*b^7*c^6*d + 4*a^4*b^6*c*d^6 - 6*a^4*b^6*c^6*d + 18*a^5*b^5*c*d^6 + 18*a^5*b^5*c^6*d - 6*a^6*b^4*c*d^6 +
4*a^6*b^4*c^6*d - 12*a^7*b^3*c*d^6 - 8*a^7*b^3*c^6*d - 6*a^9*b*c^2*d^5 + 14*a^9*b*c^3*d^4 - 8*a^9*b*c^4*d^3 +
12*a^2*b^8*c^2*d^5 - 16*a^2*b^8*c^3*d^4 + 2*a^2*b^8*c^5*d^2 + 4*a^3*b^7*c^2*d^5 + 20*a^3*b^7*c^3*d^4 - 24*a^3
*b^7*c^4*d^3 + 16*a^3*b^7*c^5*d^2 - 30*a^4*b^6*c^2*d^5 + 36*a^4*b^6*c^3*d^4 - 22*a^4*b^6*c^4*d^3 + 20*a^4*b^6*
c^5*d^2 - 14*a^5*b^5*c^2*d^5 - 2*a^5*b^5*c^3*d^4 - 2*a^5*b^5*c^4*d^3 - 14*a^5*b^5*c^5*d^2 + 20*a^6*b^4*c^2*d^5
- 22*a^6*b^4*c^3*d^4 + 36*a^6*b^4*c^4*d^3 - 30*a^6*b^4*c^5*d^2 + 16*a^7*b^3*c^2*d^5 - 24*a^7*b^3*c^3*d^4 + 20
*a^7*b^3*c^4*d^3 + 4*a^7*b^3*c^5*d^2 + 2*a^8*b^2*c^2*d^5 - 16*a^8*b^2*c^4*d^3 + 12*a^8*b^2*c^5*d^2 + 2*a*b^9*c
^6*d + 2*a^9*b*c*d^6))/((a^2*d^4 - b^2*c^4 - a^2*c^2*d^2 + b^2*c^2*d^2 - 2*a*b*c*d^3 + 2*a*b*c^3*d)*(a^5*d^2 -
b^5*c^2 - a*b^4*c^2 + a^4*b*d^2 + a^2*b^3*c^2 + a^3*b^2*c^2 - a^2*b^3*d^2 - a^3*b^2*d^2 + 2*a*b^4*c*d - 2*a^4
*b*c*d + 2*a^2*b^3*c*d - 2*a^3*b^2*c*d))))/(a^2*d^4 - b^2*c^4 - a^2*c^2*d^2 + b^2*c^2*d^2 - 2*a*b*c*d^3 + 2*a*
b*c^3*d))*1i)/(a^2*d^4 - b^2*c^4 - a^2*c^2*d^2 + b^2*c^2*d^2 - 2*a*b*c*d^3 + 2*a*b*c^3*d) + (d^2*(c^2 - d^2)^(
1/2)*((32*tan(e/2 + (f*x)/2)*(a^6*d^5 + 2*b^6*d^5 - 2*a*b^5*d^5 - 2*a^5*b*d^5 - a^6*c*d^4 - 4*b^6*c*d^4 - a^2*
b^4*c^5 - 5*a^2*b^4*d^5 + 4*a^3*b^3*d^5 + 3*a^4*b^2*d^5 + 3*b^6*c^2*d^3 - b^6*c^3*d^2 - 6*a*b^5*c^2*d^3 + 6*a*
b^5*c^3*d^2 + 13*a^2*b^4*c*d^4 + 3*a^2*b^4*c^4*d - 8*a^3*b^3*c*d^4 + 4*a^3*b^3*c^4*d - 11*a^4*b^2*c*d^4 - 11*a
^2*b^4*c^2*d^3 + a^2*b^4*c^3*d^2 + 12*a^3*b^3*c^2*d^3 - 12*a^3*b^3*c^3*d^2 + 12*a^4*b^2*c^2*d^3 - 4*a^4*b^2*c^
3*d^2 + 4*a*b^5*c*d^4 - 2*a*b^5*c^4*d + 2*a^5*b*c*d^4))/(a^5*d^2 - b^5*c^2 - a*b^4*c^2 + a^4*b*d^2 + a^2*b^3*c
^2 + a^3*b^2*c^2 - a^2*b^3*d^2 - a^3*b^2*d^2 + 2*a*b^4*c*d - 2*a^4*b*c*d + 2*a^2*b^3*c*d - 2*a^3*b^2*c*d) - (d
^2*(c^2 - d^2)^(1/2)*((32*(a*b^8*c^7 - a^9*d^7 + 2*a^8*b*d^7 + 2*a^9*c*d^6 + b^9*c^6*d - a^2*b^7*c^7 - a^3*b^6
*c^7 + a^4*b^5*c^7 + a^4*b^5*d^7 - 3*a^6*b^3*d^7 + a^7*b^2*d^7 - a^9*c^2*d^5 + b^9*c^4*d^3 - 2*b^9*c^5*d^2 - 4
*a*b^8*c^3*d^4 + 8*a*b^8*c^4*d^3 - 3*a*b^8*c^5*d^2 - 5*a^2*b^7*c^6*d - 4*a^3*b^6*c*d^6 + 7*a^3*b^6*c^6*d - 2*a
^4*b^5*c*d^6 + 4*a^4*b^5*c^6*d + 13*a^5*b^4*c*d^6 - 5*a^5*b^4*c^6*d + a^6*b^3*c*d^6 - 11*a^7*b^2*c*d^6 - 8*a^8
*b*c^2*d^5 + 5*a^8*b*c^3*d^4 + 6*a^2*b^7*c^2*d^5 - 12*a^2*b^7*c^3*d^4 - a^2*b^7*c^4*d^3 + 13*a^2*b^7*c^5*d^2 +
8*a^3*b^6*c^2*d^5 + 14*a^3*b^6*c^3*d^4 - 31*a^3*b^6*c^4*d^3 + 7*a^3*b^6*c^5*d^2 - 21*a^4*b^5*c^2*d^5 + 34*a^4
*b^5*c^3*d^4 + 4*a^4*b^5*c^4*d^3 - 21*a^4*b^5*c^5*d^2 - 16*a^5*b^4*c^2*d^5 - 21*a^5*b^4*c^3*d^4 + 33*a^5*b^4*c
^4*d^3 - 4*a^5*b^4*c^5*d^2 + 23*a^6*b^3*c^2*d^5 - 27*a^6*b^3*c^3*d^4 - 4*a^6*b^3*c^4*d^3 + 10*a^6*b^3*c^5*d^2
+ 9*a^7*b^2*c^2*d^5 + 11*a^7*b^2*c^3*d^4 - 10*a^7*b^2*c^4*d^3 - 2*a*b^8*c^6*d + a^8*b*c*d^6))/(a^6*d^3 + b^6*c
^3 + a*b^5*c^3 + a^5*b*d^3 - a^2*b^4*c^3 - a^3*b^3*c^3 - a^3*b^3*d^3 - a^4*b^2*d^3 + 3*a^2*b^4*c*d^2 - 3*a^2*b
^4*c^2*d + 3*a^3*b^3*c*d^2 + 3*a^3*b^3*c^2*d - 3*a^4*b^2*c*d^2 + 3*a^4*b^2*c^2*d - 3*a*b^5*c^2*d - 3*a^5*b*c*d
^2) - (32*d^2*tan(e/2 + (f*x)/2)*(c^2 - d^2)^(1/2)*(2*a^10*c*d^6 - 2*a^9*b*d^7 - 2*a*b^9*c^7 + 2*b^10*c^6*d +
2*a^2*b^8*c^7 + 4*a^3*b^7*c^7 - 4*a^4*b^6*c^7 - 2*a^5*b^5*c^7 + 2*a^6*b^4*c^7 + 2*a^4*b^6*d^7 - 2*a^5*b^5*d^7
- 4*a^6*b^4*d^7 + 4*a^7*b^3*d^7 + 2*a^8*b^2*d^7 - 4*a^10*c^2*d^5 + 2*a^10*c^3*d^4 + 2*b^10*c^4*d^3 - 4*b^10*c^
5*d^2 - 8*a*b^9*c^3*d^4 + 14*a*b^9*c^4*d^3 - 6*a*b^9*c^5*d^2 - 8*a^3*b^7*c*d^6 - 12*a^3*b^7*c^6*d + 4*a^4*b^6*
c*d^6 - 6*a^4*b^6*c^6*d + 18*a^5*b^5*c*d^6 + 18*a^5*b^5*c^6*d - 6*a^6*b^4*c*d^6 + 4*a^6*b^4*c^6*d - 12*a^7*b^3
*c*d^6 - 8*a^7*b^3*c^6*d - 6*a^9*b*c^2*d^5 + 14*a^9*b*c^3*d^4 - 8*a^9*b*c^4*d^3 + 12*a^2*b^8*c^2*d^5 - 16*a^2*
b^8*c^3*d^4 + 2*a^2*b^8*c^5*d^2 + 4*a^3*b^7*c^2*d^5 + 20*a^3*b^7*c^3*d^4 - 24*a^3*b^7*c^4*d^3 + 16*a^3*b^7*c^5
*d^2 - 30*a^4*b^6*c^2*d^5 + 36*a^4*b^6*c^3*d^4 - 22*a^4*b^6*c^4*d^3 + 20*a^4*b^6*c^5*d^2 - 14*a^5*b^5*c^2*d^5
- 2*a^5*b^5*c^3*d^4 - 2*a^5*b^5*c^4*d^3 - 14*a^5*b^5*c^5*d^2 + 20*a^6*b^4*c^2*d^5 - 22*a^6*b^4*c^3*d^4 + 36*a^
6*b^4*c^4*d^3 - 30*a^6*b^4*c^5*d^2 + 16*a^7*b^3*c^2*d^5 - 24*a^7*b^3*c^3*d^4 + 20*a^7*b^3*c^4*d^3 + 4*a^7*b^3*
c^5*d^2 + 2*a^8*b^2*c^2*d^5 - 16*a^8*b^2*c^4*d^3 + 12*a^8*b^2*c^5*d^2 + 2*a*b^9*c^6*d + 2*a^9*b*c*d^6))/((a^2*
d^4 - b^2*c^4 - a^2*c^2*d^2 + b^2*c^2*d^2 - 2*a*b*c*d^3 + 2*a*b*c^3*d)*(a^5*d^2 - b^5*c^2 - a*b^4*c^2 + a^4*b*
d^2 + a^2*b^3*c^2 + a^3*b^2*c^2 - a^2*b^3*d^2 - a^3*b^2*d^2 + 2*a*b^4*c*d - 2*a^4*b*c*d + 2*a^2*b^3*c*d - 2*a^
3*b^2*c*d))))/(a^2*d^4 - b^2*c^4 - a^2*c^2*d^2 + b^2*c^2*d^2 - 2*a*b*c*d^3 + 2*a*b*c^3*d))*1i)/(a^2*d^4 - b^2*
c^4 - a^2*c^2*d^2 + b^2*c^2*d^2 - 2*a*b*c*d^3 + 2*a*b*c^3*d))/((64*(b^5*d^5 - a*b^4*d^5 + 2*a^4*b*d^5 - b^5*c*
d^4 - 3*a^2*b^3*d^5 + 2*a^3*b^2*d^5 - 2*a*b^4*c^2*d^3 + 2*a^2*b^3*c*d^4 - 5*a^3*b^2*c*d^4 + 2*a^2*b^3*c^2*d^3
- a^2*b^3*c^3*d^2 + 3*a^3*b^2*c^2*d^3 + 3*a*b^4*c*d^4 - 2*a^4*b*c*d^4))/(a^6*d^3 + b^6*c^3 + a*b^5*c^3 + a^5*b
*d^3 - a^2*b^4*c^3 - a^3*b^3*c^3 - a^3*b^3*d^3 - a^4*b^2*d^3 + 3*a^2*b^4*c*d^2 - 3*a^2*b^4*c^2*d + 3*a^3*b^3*c
*d^2 + 3*a^3*b^3*c^2*d - 3*a^4*b^2*c*d^2 + 3*a^4*b^2*c^2*d - 3*a*b^5*c^2*d - 3*a^5*b*c*d^2) + (d^2*(c^2 - d^2)
^(1/2)*((32*tan(e/2 + (f*x)/2)*(a^6*d^5 + 2*b^6*d^5 - 2*a*b^5*d^5 - 2*a^5*b*d^5 - a^6*c*d^4 - 4*b^6*c*d^4 - a^
2*b^4*c^5 - 5*a^2*b^4*d^5 + 4*a^3*b^3*d^5 + 3*a^4*b^2*d^5 + 3*b^6*c^2*d^3 - b^6*c^3*d^2 - 6*a*b^5*c^2*d^3 + 6*
a*b^5*c^3*d^2 + 13*a^2*b^4*c*d^4 + 3*a^2*b^4*c^4*d - 8*a^3*b^3*c*d^4 + 4*a^3*b^3*c^4*d - 11*a^4*b^2*c*d^4 - 11
*a^2*b^4*c^2*d^3 + a^2*b^4*c^3*d^2 + 12*a^3*b^3*c^2*d^3 - 12*a^3*b^3*c^3*d^2 + 12*a^4*b^2*c^2*d^3 - 4*a^4*b^2*
c^3*d^2 + 4*a*b^5*c*d^4 - 2*a*b^5*c^4*d + 2*a^5*b*c*d^4))/(a^5*d^2 - b^5*c^2 - a*b^4*c^2 + a^4*b*d^2 + a^2*b^3
*c^2 + a^3*b^2*c^2 - a^2*b^3*d^2 - a^3*b^2*d^2 + 2*a*b^4*c*d - 2*a^4*b*c*d + 2*a^2*b^3*c*d - 2*a^3*b^2*c*d) +
(d^2*(c^2 - d^2)^(1/2)*((32*(a*b^8*c^7 - a^9*d^7 + 2*a^8*b*d^7 + 2*a^9*c*d^6 + b^9*c^6*d - a^2*b^7*c^7 - a^3*b
^6*c^7 + a^4*b^5*c^7 + a^4*b^5*d^7 - 3*a^6*b^3*d^7 + a^7*b^2*d^7 - a^9*c^2*d^5 + b^9*c^4*d^3 - 2*b^9*c^5*d^2 -
4*a*b^8*c^3*d^4 + 8*a*b^8*c^4*d^3 - 3*a*b^8*c^5*d^2 - 5*a^2*b^7*c^6*d - 4*a^3*b^6*c*d^6 + 7*a^3*b^6*c^6*d - 2
*a^4*b^5*c*d^6 + 4*a^4*b^5*c^6*d + 13*a^5*b^4*c*d^6 - 5*a^5*b^4*c^6*d + a^6*b^3*c*d^6 - 11*a^7*b^2*c*d^6 - 8*a
^8*b*c^2*d^5 + 5*a^8*b*c^3*d^4 + 6*a^2*b^7*c^2*d^5 - 12*a^2*b^7*c^3*d^4 - a^2*b^7*c^4*d^3 + 13*a^2*b^7*c^5*d^2
+ 8*a^3*b^6*c^2*d^5 + 14*a^3*b^6*c^3*d^4 - 31*a^3*b^6*c^4*d^3 + 7*a^3*b^6*c^5*d^2 - 21*a^4*b^5*c^2*d^5 + 34*a
^4*b^5*c^3*d^4 + 4*a^4*b^5*c^4*d^3 - 21*a^4*b^5*c^5*d^2 - 16*a^5*b^4*c^2*d^5 - 21*a^5*b^4*c^3*d^4 + 33*a^5*b^4
*c^4*d^3 - 4*a^5*b^4*c^5*d^2 + 23*a^6*b^3*c^2*d^5 - 27*a^6*b^3*c^3*d^4 - 4*a^6*b^3*c^4*d^3 + 10*a^6*b^3*c^5*d^
2 + 9*a^7*b^2*c^2*d^5 + 11*a^7*b^2*c^3*d^4 - 10*a^7*b^2*c^4*d^3 - 2*a*b^8*c^6*d + a^8*b*c*d^6))/(a^6*d^3 + b^6
*c^3 + a*b^5*c^3 + a^5*b*d^3 - a^2*b^4*c^3 - a^3*b^3*c^3 - a^3*b^3*d^3 - a^4*b^2*d^3 + 3*a^2*b^4*c*d^2 - 3*a^2
*b^4*c^2*d + 3*a^3*b^3*c*d^2 + 3*a^3*b^3*c^2*d - 3*a^4*b^2*c*d^2 + 3*a^4*b^2*c^2*d - 3*a*b^5*c^2*d - 3*a^5*b*c
*d^2) + (32*d^2*tan(e/2 + (f*x)/2)*(c^2 - d^2)^(1/2)*(2*a^10*c*d^6 - 2*a^9*b*d^7 - 2*a*b^9*c^7 + 2*b^10*c^6*d
+ 2*a^2*b^8*c^7 + 4*a^3*b^7*c^7 - 4*a^4*b^6*c^7 - 2*a^5*b^5*c^7 + 2*a^6*b^4*c^7 + 2*a^4*b^6*d^7 - 2*a^5*b^5*d^
7 - 4*a^6*b^4*d^7 + 4*a^7*b^3*d^7 + 2*a^8*b^2*d^7 - 4*a^10*c^2*d^5 + 2*a^10*c^3*d^4 + 2*b^10*c^4*d^3 - 4*b^10*
c^5*d^2 - 8*a*b^9*c^3*d^4 + 14*a*b^9*c^4*d^3 - 6*a*b^9*c^5*d^2 - 8*a^3*b^7*c*d^6 - 12*a^3*b^7*c^6*d + 4*a^4*b^
6*c*d^6 - 6*a^4*b^6*c^6*d + 18*a^5*b^5*c*d^6 + 18*a^5*b^5*c^6*d - 6*a^6*b^4*c*d^6 + 4*a^6*b^4*c^6*d - 12*a^7*b
^3*c*d^6 - 8*a^7*b^3*c^6*d - 6*a^9*b*c^2*d^5 + 14*a^9*b*c^3*d^4 - 8*a^9*b*c^4*d^3 + 12*a^2*b^8*c^2*d^5 - 16*a^
2*b^8*c^3*d^4 + 2*a^2*b^8*c^5*d^2 + 4*a^3*b^7*c^2*d^5 + 20*a^3*b^7*c^3*d^4 - 24*a^3*b^7*c^4*d^3 + 16*a^3*b^7*c
^5*d^2 - 30*a^4*b^6*c^2*d^5 + 36*a^4*b^6*c^3*d^4 - 22*a^4*b^6*c^4*d^3 + 20*a^4*b^6*c^5*d^2 - 14*a^5*b^5*c^2*d^
5 - 2*a^5*b^5*c^3*d^4 - 2*a^5*b^5*c^4*d^3 - 14*a^5*b^5*c^5*d^2 + 20*a^6*b^4*c^2*d^5 - 22*a^6*b^4*c^3*d^4 + 36*
a^6*b^4*c^4*d^3 - 30*a^6*b^4*c^5*d^2 + 16*a^7*b^3*c^2*d^5 - 24*a^7*b^3*c^3*d^4 + 20*a^7*b^3*c^4*d^3 + 4*a^7*b^
3*c^5*d^2 + 2*a^8*b^2*c^2*d^5 - 16*a^8*b^2*c^4*d^3 + 12*a^8*b^2*c^5*d^2 + 2*a*b^9*c^6*d + 2*a^9*b*c*d^6))/((a^
2*d^4 - b^2*c^4 - a^2*c^2*d^2 + b^2*c^2*d^2 - 2*a*b*c*d^3 + 2*a*b*c^3*d)*(a^5*d^2 - b^5*c^2 - a*b^4*c^2 + a^4*
b*d^2 + a^2*b^3*c^2 + a^3*b^2*c^2 - a^2*b^3*d^2 - a^3*b^2*d^2 + 2*a*b^4*c*d - 2*a^4*b*c*d + 2*a^2*b^3*c*d - 2*
a^3*b^2*c*d))))/(a^2*d^4 - b^2*c^4 - a^2*c^2*d^2 + b^2*c^2*d^2 - 2*a*b*c*d^3 + 2*a*b*c^3*d)))/(a^2*d^4 - b^2*c
^4 - a^2*c^2*d^2 + b^2*c^2*d^2 - 2*a*b*c*d^3 + 2*a*b*c^3*d) - (d^2*(c^2 - d^2)^(1/2)*((32*tan(e/2 + (f*x)/2)*(
a^6*d^5 + 2*b^6*d^5 - 2*a*b^5*d^5 - 2*a^5*b*d^5 - a^6*c*d^4 - 4*b^6*c*d^4 - a^2*b^4*c^5 - 5*a^2*b^4*d^5 + 4*a^
3*b^3*d^5 + 3*a^4*b^2*d^5 + 3*b^6*c^2*d^3 - b^6*c^3*d^2 - 6*a*b^5*c^2*d^3 + 6*a*b^5*c^3*d^2 + 13*a^2*b^4*c*d^4
+ 3*a^2*b^4*c^4*d - 8*a^3*b^3*c*d^4 + 4*a^3*b^3*c^4*d - 11*a^4*b^2*c*d^4 - 11*a^2*b^4*c^2*d^3 + a^2*b^4*c^3*d
^2 + 12*a^3*b^3*c^2*d^3 - 12*a^3*b^3*c^3*d^2 + 12*a^4*b^2*c^2*d^3 - 4*a^4*b^2*c^3*d^2 + 4*a*b^5*c*d^4 - 2*a*b^
5*c^4*d + 2*a^5*b*c*d^4))/(a^5*d^2 - b^5*c^2 - a*b^4*c^2 + a^4*b*d^2 + a^2*b^3*c^2 + a^3*b^2*c^2 - a^2*b^3*d^2
- a^3*b^2*d^2 + 2*a*b^4*c*d - 2*a^4*b*c*d + 2*a^2*b^3*c*d - 2*a^3*b^2*c*d) - (d^2*(c^2 - d^2)^(1/2)*((32*(a*b
^8*c^7 - a^9*d^7 + 2*a^8*b*d^7 + 2*a^9*c*d^6 + b^9*c^6*d - a^2*b^7*c^7 - a^3*b^6*c^7 + a^4*b^5*c^7 + a^4*b^5*d
^7 - 3*a^6*b^3*d^7 + a^7*b^2*d^7 - a^9*c^2*d^5 + b^9*c^4*d^3 - 2*b^9*c^5*d^2 - 4*a*b^8*c^3*d^4 + 8*a*b^8*c^4*d
^3 - 3*a*b^8*c^5*d^2 - 5*a^2*b^7*c^6*d - 4*a^3*b^6*c*d^6 + 7*a^3*b^6*c^6*d - 2*a^4*b^5*c*d^6 + 4*a^4*b^5*c^6*d
+ 13*a^5*b^4*c*d^6 - 5*a^5*b^4*c^6*d + a^6*b^3*c*d^6 - 11*a^7*b^2*c*d^6 - 8*a^8*b*c^2*d^5 + 5*a^8*b*c^3*d^4 +
6*a^2*b^7*c^2*d^5 - 12*a^2*b^7*c^3*d^4 - a^2*b^7*c^4*d^3 + 13*a^2*b^7*c^5*d^2 + 8*a^3*b^6*c^2*d^5 + 14*a^3*b^
6*c^3*d^4 - 31*a^3*b^6*c^4*d^3 + 7*a^3*b^6*c^5*d^2 - 21*a^4*b^5*c^2*d^5 + 34*a^4*b^5*c^3*d^4 + 4*a^4*b^5*c^4*d
^3 - 21*a^4*b^5*c^5*d^2 - 16*a^5*b^4*c^2*d^5 - 21*a^5*b^4*c^3*d^4 + 33*a^5*b^4*c^4*d^3 - 4*a^5*b^4*c^5*d^2 + 2
3*a^6*b^3*c^2*d^5 - 27*a^6*b^3*c^3*d^4 - 4*a^6*b^3*c^4*d^3 + 10*a^6*b^3*c^5*d^2 + 9*a^7*b^2*c^2*d^5 + 11*a^7*b
^2*c^3*d^4 - 10*a^7*b^2*c^4*d^3 - 2*a*b^8*c^6*d + a^8*b*c*d^6))/(a^6*d^3 + b^6*c^3 + a*b^5*c^3 + a^5*b*d^3 - a
^2*b^4*c^3 - a^3*b^3*c^3 - a^3*b^3*d^3 - a^4*b^2*d^3 + 3*a^2*b^4*c*d^2 - 3*a^2*b^4*c^2*d + 3*a^3*b^3*c*d^2 + 3
*a^3*b^3*c^2*d - 3*a^4*b^2*c*d^2 + 3*a^4*b^2*c^2*d - 3*a*b^5*c^2*d - 3*a^5*b*c*d^2) - (32*d^2*tan(e/2 + (f*x)/
2)*(c^2 - d^2)^(1/2)*(2*a^10*c*d^6 - 2*a^9*b*d^7 - 2*a*b^9*c^7 + 2*b^10*c^6*d + 2*a^2*b^8*c^7 + 4*a^3*b^7*c^7
- 4*a^4*b^6*c^7 - 2*a^5*b^5*c^7 + 2*a^6*b^4*c^7 + 2*a^4*b^6*d^7 - 2*a^5*b^5*d^7 - 4*a^6*b^4*d^7 + 4*a^7*b^3*d^
7 + 2*a^8*b^2*d^7 - 4*a^10*c^2*d^5 + 2*a^10*c^3*d^4 + 2*b^10*c^4*d^3 - 4*b^10*c^5*d^2 - 8*a*b^9*c^3*d^4 + 14*a
*b^9*c^4*d^3 - 6*a*b^9*c^5*d^2 - 8*a^3*b^7*c*d^6 - 12*a^3*b^7*c^6*d + 4*a^4*b^6*c*d^6 - 6*a^4*b^6*c^6*d + 18*a
^5*b^5*c*d^6 + 18*a^5*b^5*c^6*d - 6*a^6*b^4*c*d^6 + 4*a^6*b^4*c^6*d - 12*a^7*b^3*c*d^6 - 8*a^7*b^3*c^6*d - 6*a
^9*b*c^2*d^5 + 14*a^9*b*c^3*d^4 - 8*a^9*b*c^4*d^3 + 12*a^2*b^8*c^2*d^5 - 16*a^2*b^8*c^3*d^4 + 2*a^2*b^8*c^5*d^
2 + 4*a^3*b^7*c^2*d^5 + 20*a^3*b^7*c^3*d^4 - 24*a^3*b^7*c^4*d^3 + 16*a^3*b^7*c^5*d^2 - 30*a^4*b^6*c^2*d^5 + 36
*a^4*b^6*c^3*d^4 - 22*a^4*b^6*c^4*d^3 + 20*a^4*b^6*c^5*d^2 - 14*a^5*b^5*c^2*d^5 - 2*a^5*b^5*c^3*d^4 - 2*a^5*b^
5*c^4*d^3 - 14*a^5*b^5*c^5*d^2 + 20*a^6*b^4*c^2*d^5 - 22*a^6*b^4*c^3*d^4 + 36*a^6*b^4*c^4*d^3 - 30*a^6*b^4*c^5
*d^2 + 16*a^7*b^3*c^2*d^5 - 24*a^7*b^3*c^3*d^4 + 20*a^7*b^3*c^4*d^3 + 4*a^7*b^3*c^5*d^2 + 2*a^8*b^2*c^2*d^5 -
16*a^8*b^2*c^4*d^3 + 12*a^8*b^2*c^5*d^2 + 2*a*b^9*c^6*d + 2*a^9*b*c*d^6))/((a^2*d^4 - b^2*c^4 - a^2*c^2*d^2 +
b^2*c^2*d^2 - 2*a*b*c*d^3 + 2*a*b*c^3*d)*(a^5*d^2 - b^5*c^2 - a*b^4*c^2 + a^4*b*d^2 + a^2*b^3*c^2 + a^3*b^2*c^
2 - a^2*b^3*d^2 - a^3*b^2*d^2 + 2*a*b^4*c*d - 2*a^4*b*c*d + 2*a^2*b^3*c*d - 2*a^3*b^2*c*d))))/(a^2*d^4 - b^2*c
^4 - a^2*c^2*d^2 + b^2*c^2*d^2 - 2*a*b*c*d^3 + 2*a*b*c^3*d)))/(a^2*d^4 - b^2*c^4 - a^2*c^2*d^2 + b^2*c^2*d^2 -
2*a*b*c*d^3 + 2*a*b*c^3*d)))*(c^2 - d^2)^(1/2)*2i)/(f*(a^2*d^4 - b^2*c^4 - a^2*c^2*d^2 + b^2*c^2*d^2 - 2*a*b*
c*d^3 + 2*a*b*c^3*d)) + (2*b^2*tan(e/2 + (f*x)/2))/(f*(a + b)*(a + b - tan(e/2 + (f*x)/2)^2*(a - b))*(a^2*d +
b^2*c - a*b*c - a*b*d)) + (b*atan(((b*((32*tan(e/2 + (f*x)/2)*(a^6*d^5 + 2*b^6*d^5 - 2*a*b^5*d^5 - 2*a^5*b*d^5
- a^6*c*d^4 - 4*b^6*c*d^4 - a^2*b^4*c^5 - 5*a^2*b^4*d^5 + 4*a^3*b^3*d^5 + 3*a^4*b^2*d^5 + 3*b^6*c^2*d^3 - b^6
*c^3*d^2 - 6*a*b^5*c^2*d^3 + 6*a*b^5*c^3*d^2 + 13*a^2*b^4*c*d^4 + 3*a^2*b^4*c^4*d - 8*a^3*b^3*c*d^4 + 4*a^3*b^
3*c^4*d - 11*a^4*b^2*c*d^4 - 11*a^2*b^4*c^2*d^3 + a^2*b^4*c^3*d^2 + 12*a^3*b^3*c^2*d^3 - 12*a^3*b^3*c^3*d^2 +
12*a^4*b^2*c^2*d^3 - 4*a^4*b^2*c^3*d^2 + 4*a*b^5*c*d^4 - 2*a*b^5*c^4*d + 2*a^5*b*c*d^4))/(a^5*d^2 - b^5*c^2 -
a*b^4*c^2 + a^4*b*d^2 + a^2*b^3*c^2 + a^3*b^2*c^2 - a^2*b^3*d^2 - a^3*b^2*d^2 + 2*a*b^4*c*d - 2*a^4*b*c*d + 2*
a^2*b^3*c*d - 2*a^3*b^2*c*d) + (b*((32*(a*b^8*c^7 - a^9*d^7 + 2*a^8*b*d^7 + 2*a^9*c*d^6 + b^9*c^6*d - a^2*b^7*
c^7 - a^3*b^6*c^7 + a^4*b^5*c^7 + a^4*b^5*d^7 - 3*a^6*b^3*d^7 + a^7*b^2*d^7 - a^9*c^2*d^5 + b^9*c^4*d^3 - 2*b^
9*c^5*d^2 - 4*a*b^8*c^3*d^4 + 8*a*b^8*c^4*d^3 - 3*a*b^8*c^5*d^2 - 5*a^2*b^7*c^6*d - 4*a^3*b^6*c*d^6 + 7*a^3*b^
6*c^6*d - 2*a^4*b^5*c*d^6 + 4*a^4*b^5*c^6*d + 13*a^5*b^4*c*d^6 - 5*a^5*b^4*c^6*d + a^6*b^3*c*d^6 - 11*a^7*b^2*
c*d^6 - 8*a^8*b*c^2*d^5 + 5*a^8*b*c^3*d^4 + 6*a^2*b^7*c^2*d^5 - 12*a^2*b^7*c^3*d^4 - a^2*b^7*c^4*d^3 + 13*a^2*
b^7*c^5*d^2 + 8*a^3*b^6*c^2*d^5 + 14*a^3*b^6*c^3*d^4 - 31*a^3*b^6*c^4*d^3 + 7*a^3*b^6*c^5*d^2 - 21*a^4*b^5*c^2
*d^5 + 34*a^4*b^5*c^3*d^4 + 4*a^4*b^5*c^4*d^3 - 21*a^4*b^5*c^5*d^2 - 16*a^5*b^4*c^2*d^5 - 21*a^5*b^4*c^3*d^4 +
33*a^5*b^4*c^4*d^3 - 4*a^5*b^4*c^5*d^2 + 23*a^6*b^3*c^2*d^5 - 27*a^6*b^3*c^3*d^4 - 4*a^6*b^3*c^4*d^3 + 10*a^6
*b^3*c^5*d^2 + 9*a^7*b^2*c^2*d^5 + 11*a^7*b^2*c^3*d^4 - 10*a^7*b^2*c^4*d^3 - 2*a*b^8*c^6*d + a^8*b*c*d^6))/(a^
6*d^3 + b^6*c^3 + a*b^5*c^3 + a^5*b*d^3 - a^2*b^4*c^3 - a^3*b^3*c^3 - a^3*b^3*d^3 - a^4*b^2*d^3 + 3*a^2*b^4*c*
d^2 - 3*a^2*b^4*c^2*d + 3*a^3*b^3*c*d^2 + 3*a^3*b^3*c^2*d - 3*a^4*b^2*c*d^2 + 3*a^4*b^2*c^2*d - 3*a*b^5*c^2*d
- 3*a^5*b*c*d^2) + (32*b*tan(e/2 + (f*x)/2)*((a + b)^3*(a - b)^3)^(1/2)*(b^2*d - 2*a^2*d + a*b*c)*(2*a^10*c*d^
6 - 2*a^9*b*d^7 - 2*a*b^9*c^7 + 2*b^10*c^6*d + 2*a^2*b^8*c^7 + 4*a^3*b^7*c^7 - 4*a^4*b^6*c^7 - 2*a^5*b^5*c^7 +
2*a^6*b^4*c^7 + 2*a^4*b^6*d^7 - 2*a^5*b^5*d^7 - 4*a^6*b^4*d^7 + 4*a^7*b^3*d^7 + 2*a^8*b^2*d^7 - 4*a^10*c^2*d^
5 + 2*a^10*c^3*d^4 + 2*b^10*c^4*d^3 - 4*b^10*c^5*d^2 - 8*a*b^9*c^3*d^4 + 14*a*b^9*c^4*d^3 - 6*a*b^9*c^5*d^2 -
8*a^3*b^7*c*d^6 - 12*a^3*b^7*c^6*d + 4*a^4*b^6*c*d^6 - 6*a^4*b^6*c^6*d + 18*a^5*b^5*c*d^6 + 18*a^5*b^5*c^6*d -
6*a^6*b^4*c*d^6 + 4*a^6*b^4*c^6*d - 12*a^7*b^3*c*d^6 - 8*a^7*b^3*c^6*d - 6*a^9*b*c^2*d^5 + 14*a^9*b*c^3*d^4 -
8*a^9*b*c^4*d^3 + 12*a^2*b^8*c^2*d^5 - 16*a^2*b^8*c^3*d^4 + 2*a^2*b^8*c^5*d^2 + 4*a^3*b^7*c^2*d^5 + 20*a^3*b^
7*c^3*d^4 - 24*a^3*b^7*c^4*d^3 + 16*a^3*b^7*c^5*d^2 - 30*a^4*b^6*c^2*d^5 + 36*a^4*b^6*c^3*d^4 - 22*a^4*b^6*c^4
*d^3 + 20*a^4*b^6*c^5*d^2 - 14*a^5*b^5*c^2*d^5 - 2*a^5*b^5*c^3*d^4 - 2*a^5*b^5*c^4*d^3 - 14*a^5*b^5*c^5*d^2 +
20*a^6*b^4*c^2*d^5 - 22*a^6*b^4*c^3*d^4 + 36*a^6*b^4*c^4*d^3 - 30*a^6*b^4*c^5*d^2 + 16*a^7*b^3*c^2*d^5 - 24*a^
7*b^3*c^3*d^4 + 20*a^7*b^3*c^4*d^3 + 4*a^7*b^3*c^5*d^2 + 2*a^8*b^2*c^2*d^5 - 16*a^8*b^2*c^4*d^3 + 12*a^8*b^2*c
^5*d^2 + 2*a*b^9*c^6*d + 2*a^9*b*c*d^6))/((a^5*d^2 - b^5*c^2 - a*b^4*c^2 + a^4*b*d^2 + a^2*b^3*c^2 + a^3*b^2*c
^2 - a^2*b^3*d^2 - a^3*b^2*d^2 + 2*a*b^4*c*d - 2*a^4*b*c*d + 2*a^2*b^3*c*d - 2*a^3*b^2*c*d)*(a^8*d^2 - b^8*c^2
+ 3*a^2*b^6*c^2 - 3*a^4*b^4*c^2 + a^6*b^2*c^2 - a^2*b^6*d^2 + 3*a^4*b^4*d^2 - 3*a^6*b^2*d^2 + 2*a*b^7*c*d - 2
*a^7*b*c*d - 6*a^3*b^5*c*d + 6*a^5*b^3*c*d)))*((a + b)^3*(a - b)^3)^(1/2)*(b^2*d - 2*a^2*d + a*b*c))/(a^8*d^2
- b^8*c^2 + 3*a^2*b^6*c^2 - 3*a^4*b^4*c^2 + a^6*b^2*c^2 - a^2*b^6*d^2 + 3*a^4*b^4*d^2 - 3*a^6*b^2*d^2 + 2*a*b^
7*c*d - 2*a^7*b*c*d - 6*a^3*b^5*c*d + 6*a^5*b^3*c*d))*((a + b)^3*(a - b)^3)^(1/2)*(b^2*d - 2*a^2*d + a*b*c)*1i
)/(a^8*d^2 - b^8*c^2 + 3*a^2*b^6*c^2 - 3*a^4*b^4*c^2 + a^6*b^2*c^2 - a^2*b^6*d^2 + 3*a^4*b^4*d^2 - 3*a^6*b^2*d
^2 + 2*a*b^7*c*d - 2*a^7*b*c*d - 6*a^3*b^5*c*d + 6*a^5*b^3*c*d) + (b*((32*tan(e/2 + (f*x)/2)*(a^6*d^5 + 2*b^6*
d^5 - 2*a*b^5*d^5 - 2*a^5*b*d^5 - a^6*c*d^4 - 4*b^6*c*d^4 - a^2*b^4*c^5 - 5*a^2*b^4*d^5 + 4*a^3*b^3*d^5 + 3*a^
4*b^2*d^5 + 3*b^6*c^2*d^3 - b^6*c^3*d^2 - 6*a*b^5*c^2*d^3 + 6*a*b^5*c^3*d^2 + 13*a^2*b^4*c*d^4 + 3*a^2*b^4*c^4
*d - 8*a^3*b^3*c*d^4 + 4*a^3*b^3*c^4*d - 11*a^4*b^2*c*d^4 - 11*a^2*b^4*c^2*d^3 + a^2*b^4*c^3*d^2 + 12*a^3*b^3*
c^2*d^3 - 12*a^3*b^3*c^3*d^2 + 12*a^4*b^2*c^2*d^3 - 4*a^4*b^2*c^3*d^2 + 4*a*b^5*c*d^4 - 2*a*b^5*c^4*d + 2*a^5*
b*c*d^4))/(a^5*d^2 - b^5*c^2 - a*b^4*c^2 + a^4*b*d^2 + a^2*b^3*c^2 + a^3*b^2*c^2 - a^2*b^3*d^2 - a^3*b^2*d^2 +
2*a*b^4*c*d - 2*a^4*b*c*d + 2*a^2*b^3*c*d - 2*a^3*b^2*c*d) - (b*((32*(a*b^8*c^7 - a^9*d^7 + 2*a^8*b*d^7 + 2*a
^9*c*d^6 + b^9*c^6*d - a^2*b^7*c^7 - a^3*b^6*c^7 + a^4*b^5*c^7 + a^4*b^5*d^7 - 3*a^6*b^3*d^7 + a^7*b^2*d^7 - a
^9*c^2*d^5 + b^9*c^4*d^3 - 2*b^9*c^5*d^2 - 4*a*b^8*c^3*d^4 + 8*a*b^8*c^4*d^3 - 3*a*b^8*c^5*d^2 - 5*a^2*b^7*c^6
*d - 4*a^3*b^6*c*d^6 + 7*a^3*b^6*c^6*d - 2*a^4*b^5*c*d^6 + 4*a^4*b^5*c^6*d + 13*a^5*b^4*c*d^6 - 5*a^5*b^4*c^6*
d + a^6*b^3*c*d^6 - 11*a^7*b^2*c*d^6 - 8*a^8*b*c^2*d^5 + 5*a^8*b*c^3*d^4 + 6*a^2*b^7*c^2*d^5 - 12*a^2*b^7*c^3*
d^4 - a^2*b^7*c^4*d^3 + 13*a^2*b^7*c^5*d^2 + 8*a^3*b^6*c^2*d^5 + 14*a^3*b^6*c^3*d^4 - 31*a^3*b^6*c^4*d^3 + 7*a
^3*b^6*c^5*d^2 - 21*a^4*b^5*c^2*d^5 + 34*a^4*b^5*c^3*d^4 + 4*a^4*b^5*c^4*d^3 - 21*a^4*b^5*c^5*d^2 - 16*a^5*b^4
*c^2*d^5 - 21*a^5*b^4*c^3*d^4 + 33*a^5*b^4*c^4*d^3 - 4*a^5*b^4*c^5*d^2 + 23*a^6*b^3*c^2*d^5 - 27*a^6*b^3*c^3*d
^4 - 4*a^6*b^3*c^4*d^3 + 10*a^6*b^3*c^5*d^2 + 9*a^7*b^2*c^2*d^5 + 11*a^7*b^2*c^3*d^4 - 10*a^7*b^2*c^4*d^3 - 2*
a*b^8*c^6*d + a^8*b*c*d^6))/(a^6*d^3 + b^6*c^3 + a*b^5*c^3 + a^5*b*d^3 - a^2*b^4*c^3 - a^3*b^3*c^3 - a^3*b^3*d
^3 - a^4*b^2*d^3 + 3*a^2*b^4*c*d^2 - 3*a^2*b^4*c^2*d + 3*a^3*b^3*c*d^2 + 3*a^3*b^3*c^2*d - 3*a^4*b^2*c*d^2 + 3
*a^4*b^2*c^2*d - 3*a*b^5*c^2*d - 3*a^5*b*c*d^2) - (32*b*tan(e/2 + (f*x)/2)*((a + b)^3*(a - b)^3)^(1/2)*(b^2*d
- 2*a^2*d + a*b*c)*(2*a^10*c*d^6 - 2*a^9*b*d^7 - 2*a*b^9*c^7 + 2*b^10*c^6*d + 2*a^2*b^8*c^7 + 4*a^3*b^7*c^7 -
4*a^4*b^6*c^7 - 2*a^5*b^5*c^7 + 2*a^6*b^4*c^7 + 2*a^4*b^6*d^7 - 2*a^5*b^5*d^7 - 4*a^6*b^4*d^7 + 4*a^7*b^3*d^7
+ 2*a^8*b^2*d^7 - 4*a^10*c^2*d^5 + 2*a^10*c^3*d^4 + 2*b^10*c^4*d^3 - 4*b^10*c^5*d^2 - 8*a*b^9*c^3*d^4 + 14*a*b
^9*c^4*d^3 - 6*a*b^9*c^5*d^2 - 8*a^3*b^7*c*d^6 - 12*a^3*b^7*c^6*d + 4*a^4*b^6*c*d^6 - 6*a^4*b^6*c^6*d + 18*a^5
*b^5*c*d^6 + 18*a^5*b^5*c^6*d - 6*a^6*b^4*c*d^6 + 4*a^6*b^4*c^6*d - 12*a^7*b^3*c*d^6 - 8*a^7*b^3*c^6*d - 6*a^9
*b*c^2*d^5 + 14*a^9*b*c^3*d^4 - 8*a^9*b*c^4*d^3 + 12*a^2*b^8*c^2*d^5 - 16*a^2*b^8*c^3*d^4 + 2*a^2*b^8*c^5*d^2
+ 4*a^3*b^7*c^2*d^5 + 20*a^3*b^7*c^3*d^4 - 24*a^3*b^7*c^4*d^3 + 16*a^3*b^7*c^5*d^2 - 30*a^4*b^6*c^2*d^5 + 36*a
^4*b^6*c^3*d^4 - 22*a^4*b^6*c^4*d^3 + 20*a^4*b^6*c^5*d^2 - 14*a^5*b^5*c^2*d^5 - 2*a^5*b^5*c^3*d^4 - 2*a^5*b^5*
c^4*d^3 - 14*a^5*b^5*c^5*d^2 + 20*a^6*b^4*c^2*d^5 - 22*a^6*b^4*c^3*d^4 + 36*a^6*b^4*c^4*d^3 - 30*a^6*b^4*c^5*d
^2 + 16*a^7*b^3*c^2*d^5 - 24*a^7*b^3*c^3*d^4 + 20*a^7*b^3*c^4*d^3 + 4*a^7*b^3*c^5*d^2 + 2*a^8*b^2*c^2*d^5 - 16
*a^8*b^2*c^4*d^3 + 12*a^8*b^2*c^5*d^2 + 2*a*b^9*c^6*d + 2*a^9*b*c*d^6))/((a^5*d^2 - b^5*c^2 - a*b^4*c^2 + a^4*
b*d^2 + a^2*b^3*c^2 + a^3*b^2*c^2 - a^2*b^3*d^2 - a^3*b^2*d^2 + 2*a*b^4*c*d - 2*a^4*b*c*d + 2*a^2*b^3*c*d - 2*
a^3*b^2*c*d)*(a^8*d^2 - b^8*c^2 + 3*a^2*b^6*c^2 - 3*a^4*b^4*c^2 + a^6*b^2*c^2 - a^2*b^6*d^2 + 3*a^4*b^4*d^2 -
3*a^6*b^2*d^2 + 2*a*b^7*c*d - 2*a^7*b*c*d - 6*a^3*b^5*c*d + 6*a^5*b^3*c*d)))*((a + b)^3*(a - b)^3)^(1/2)*(b^2*
d - 2*a^2*d + a*b*c))/(a^8*d^2 - b^8*c^2 + 3*a^2*b^6*c^2 - 3*a^4*b^4*c^2 + a^6*b^2*c^2 - a^2*b^6*d^2 + 3*a^4*b
^4*d^2 - 3*a^6*b^2*d^2 + 2*a*b^7*c*d - 2*a^7*b*c*d - 6*a^3*b^5*c*d + 6*a^5*b^3*c*d))*((a + b)^3*(a - b)^3)^(1/
2)*(b^2*d - 2*a^2*d + a*b*c)*1i)/(a^8*d^2 - b^8*c^2 + 3*a^2*b^6*c^2 - 3*a^4*b^4*c^2 + a^6*b^2*c^2 - a^2*b^6*d^
2 + 3*a^4*b^4*d^2 - 3*a^6*b^2*d^2 + 2*a*b^7*c*d - 2*a^7*b*c*d - 6*a^3*b^5*c*d + 6*a^5*b^3*c*d))/((64*(b^5*d^5
- a*b^4*d^5 + 2*a^4*b*d^5 - b^5*c*d^4 - 3*a^2*b^3*d^5 + 2*a^3*b^2*d^5 - 2*a*b^4*c^2*d^3 + 2*a^2*b^3*c*d^4 - 5*
a^3*b^2*c*d^4 + 2*a^2*b^3*c^2*d^3 - a^2*b^3*c^3*d^2 + 3*a^3*b^2*c^2*d^3 + 3*a*b^4*c*d^4 - 2*a^4*b*c*d^4))/(a^6
*d^3 + b^6*c^3 + a*b^5*c^3 + a^5*b*d^3 - a^2*b^4*c^3 - a^3*b^3*c^3 - a^3*b^3*d^3 - a^4*b^2*d^3 + 3*a^2*b^4*c*d
^2 - 3*a^2*b^4*c^2*d + 3*a^3*b^3*c*d^2 + 3*a^3*b^3*c^2*d - 3*a^4*b^2*c*d^2 + 3*a^4*b^2*c^2*d - 3*a*b^5*c^2*d -
3*a^5*b*c*d^2) + (b*((32*tan(e/2 + (f*x)/2)*(a^6*d^5 + 2*b^6*d^5 - 2*a*b^5*d^5 - 2*a^5*b*d^5 - a^6*c*d^4 - 4*
b^6*c*d^4 - a^2*b^4*c^5 - 5*a^2*b^4*d^5 + 4*a^3*b^3*d^5 + 3*a^4*b^2*d^5 + 3*b^6*c^2*d^3 - b^6*c^3*d^2 - 6*a*b^
5*c^2*d^3 + 6*a*b^5*c^3*d^2 + 13*a^2*b^4*c*d^4 + 3*a^2*b^4*c^4*d - 8*a^3*b^3*c*d^4 + 4*a^3*b^3*c^4*d - 11*a^4*
b^2*c*d^4 - 11*a^2*b^4*c^2*d^3 + a^2*b^4*c^3*d^2 + 12*a^3*b^3*c^2*d^3 - 12*a^3*b^3*c^3*d^2 + 12*a^4*b^2*c^2*d^
3 - 4*a^4*b^2*c^3*d^2 + 4*a*b^5*c*d^4 - 2*a*b^5*c^4*d + 2*a^5*b*c*d^4))/(a^5*d^2 - b^5*c^2 - a*b^4*c^2 + a^4*b
*d^2 + a^2*b^3*c^2 + a^3*b^2*c^2 - a^2*b^3*d^2 - a^3*b^2*d^2 + 2*a*b^4*c*d - 2*a^4*b*c*d + 2*a^2*b^3*c*d - 2*a
^3*b^2*c*d) + (b*((32*(a*b^8*c^7 - a^9*d^7 + 2*a^8*b*d^7 + 2*a^9*c*d^6 + b^9*c^6*d - a^2*b^7*c^7 - a^3*b^6*c^7
+ a^4*b^5*c^7 + a^4*b^5*d^7 - 3*a^6*b^3*d^7 + a^7*b^2*d^7 - a^9*c^2*d^5 + b^9*c^4*d^3 - 2*b^9*c^5*d^2 - 4*a*b
^8*c^3*d^4 + 8*a*b^8*c^4*d^3 - 3*a*b^8*c^5*d^2 - 5*a^2*b^7*c^6*d - 4*a^3*b^6*c*d^6 + 7*a^3*b^6*c^6*d - 2*a^4*b
^5*c*d^6 + 4*a^4*b^5*c^6*d + 13*a^5*b^4*c*d^6 - 5*a^5*b^4*c^6*d + a^6*b^3*c*d^6 - 11*a^7*b^2*c*d^6 - 8*a^8*b*c
^2*d^5 + 5*a^8*b*c^3*d^4 + 6*a^2*b^7*c^2*d^5 - 12*a^2*b^7*c^3*d^4 - a^2*b^7*c^4*d^3 + 13*a^2*b^7*c^5*d^2 + 8*a
^3*b^6*c^2*d^5 + 14*a^3*b^6*c^3*d^4 - 31*a^3*b^6*c^4*d^3 + 7*a^3*b^6*c^5*d^2 - 21*a^4*b^5*c^2*d^5 + 34*a^4*b^5
*c^3*d^4 + 4*a^4*b^5*c^4*d^3 - 21*a^4*b^5*c^5*d^2 - 16*a^5*b^4*c^2*d^5 - 21*a^5*b^4*c^3*d^4 + 33*a^5*b^4*c^4*d
^3 - 4*a^5*b^4*c^5*d^2 + 23*a^6*b^3*c^2*d^5 - 27*a^6*b^3*c^3*d^4 - 4*a^6*b^3*c^4*d^3 + 10*a^6*b^3*c^5*d^2 + 9*
a^7*b^2*c^2*d^5 + 11*a^7*b^2*c^3*d^4 - 10*a^7*b^2*c^4*d^3 - 2*a*b^8*c^6*d + a^8*b*c*d^6))/(a^6*d^3 + b^6*c^3 +
a*b^5*c^3 + a^5*b*d^3 - a^2*b^4*c^3 - a^3*b^3*c^3 - a^3*b^3*d^3 - a^4*b^2*d^3 + 3*a^2*b^4*c*d^2 - 3*a^2*b^4*c
^2*d + 3*a^3*b^3*c*d^2 + 3*a^3*b^3*c^2*d - 3*a^4*b^2*c*d^2 + 3*a^4*b^2*c^2*d - 3*a*b^5*c^2*d - 3*a^5*b*c*d^2)
+ (32*b*tan(e/2 + (f*x)/2)*((a + b)^3*(a - b)^3)^(1/2)*(b^2*d - 2*a^2*d + a*b*c)*(2*a^10*c*d^6 - 2*a^9*b*d^7 -
2*a*b^9*c^7 + 2*b^10*c^6*d + 2*a^2*b^8*c^7 + 4*a^3*b^7*c^7 - 4*a^4*b^6*c^7 - 2*a^5*b^5*c^7 + 2*a^6*b^4*c^7 +
2*a^4*b^6*d^7 - 2*a^5*b^5*d^7 - 4*a^6*b^4*d^7 + 4*a^7*b^3*d^7 + 2*a^8*b^2*d^7 - 4*a^10*c^2*d^5 + 2*a^10*c^3*d^
4 + 2*b^10*c^4*d^3 - 4*b^10*c^5*d^2 - 8*a*b^9*c^3*d^4 + 14*a*b^9*c^4*d^3 - 6*a*b^9*c^5*d^2 - 8*a^3*b^7*c*d^6 -
12*a^3*b^7*c^6*d + 4*a^4*b^6*c*d^6 - 6*a^4*b^6*c^6*d + 18*a^5*b^5*c*d^6 + 18*a^5*b^5*c^6*d - 6*a^6*b^4*c*d^6
+ 4*a^6*b^4*c^6*d - 12*a^7*b^3*c*d^6 - 8*a^7*b^3*c^6*d - 6*a^9*b*c^2*d^5 + 14*a^9*b*c^3*d^4 - 8*a^9*b*c^4*d^3
+ 12*a^2*b^8*c^2*d^5 - 16*a^2*b^8*c^3*d^4 + 2*a^2*b^8*c^5*d^2 + 4*a^3*b^7*c^2*d^5 + 20*a^3*b^7*c^3*d^4 - 24*a^
3*b^7*c^4*d^3 + 16*a^3*b^7*c^5*d^2 - 30*a^4*b^6*c^2*d^5 + 36*a^4*b^6*c^3*d^4 - 22*a^4*b^6*c^4*d^3 + 20*a^4*b^6
*c^5*d^2 - 14*a^5*b^5*c^2*d^5 - 2*a^5*b^5*c^3*d^4 - 2*a^5*b^5*c^4*d^3 - 14*a^5*b^5*c^5*d^2 + 20*a^6*b^4*c^2*d^
5 - 22*a^6*b^4*c^3*d^4 + 36*a^6*b^4*c^4*d^3 - 30*a^6*b^4*c^5*d^2 + 16*a^7*b^3*c^2*d^5 - 24*a^7*b^3*c^3*d^4 + 2
0*a^7*b^3*c^4*d^3 + 4*a^7*b^3*c^5*d^2 + 2*a^8*b^2*c^2*d^5 - 16*a^8*b^2*c^4*d^3 + 12*a^8*b^2*c^5*d^2 + 2*a*b^9*
c^6*d + 2*a^9*b*c*d^6))/((a^5*d^2 - b^5*c^2 - a*b^4*c^2 + a^4*b*d^2 + a^2*b^3*c^2 + a^3*b^2*c^2 - a^2*b^3*d^2
- a^3*b^2*d^2 + 2*a*b^4*c*d - 2*a^4*b*c*d + 2*a^2*b^3*c*d - 2*a^3*b^2*c*d)*(a^8*d^2 - b^8*c^2 + 3*a^2*b^6*c^2
- 3*a^4*b^4*c^2 + a^6*b^2*c^2 - a^2*b^6*d^2 + 3*a^4*b^4*d^2 - 3*a^6*b^2*d^2 + 2*a*b^7*c*d - 2*a^7*b*c*d - 6*a^
3*b^5*c*d + 6*a^5*b^3*c*d)))*((a + b)^3*(a - b)^3)^(1/2)*(b^2*d - 2*a^2*d + a*b*c))/(a^8*d^2 - b^8*c^2 + 3*a^2
*b^6*c^2 - 3*a^4*b^4*c^2 + a^6*b^2*c^2 - a^2*b^6*d^2 + 3*a^4*b^4*d^2 - 3*a^6*b^2*d^2 + 2*a*b^7*c*d - 2*a^7*b*c
*d - 6*a^3*b^5*c*d + 6*a^5*b^3*c*d))*((a + b)^3*(a - b)^3)^(1/2)*(b^2*d - 2*a^2*d + a*b*c))/(a^8*d^2 - b^8*c^2
+ 3*a^2*b^6*c^2 - 3*a^4*b^4*c^2 + a^6*b^2*c^2 - a^2*b^6*d^2 + 3*a^4*b^4*d^2 - 3*a^6*b^2*d^2 + 2*a*b^7*c*d - 2
*a^7*b*c*d - 6*a^3*b^5*c*d + 6*a^5*b^3*c*d) - (b*((32*tan(e/2 + (f*x)/2)*(a^6*d^5 + 2*b^6*d^5 - 2*a*b^5*d^5 -
2*a^5*b*d^5 - a^6*c*d^4 - 4*b^6*c*d^4 - a^2*b^4*c^5 - 5*a^2*b^4*d^5 + 4*a^3*b^3*d^5 + 3*a^4*b^2*d^5 + 3*b^6*c^
2*d^3 - b^6*c^3*d^2 - 6*a*b^5*c^2*d^3 + 6*a*b^5*c^3*d^2 + 13*a^2*b^4*c*d^4 + 3*a^2*b^4*c^4*d - 8*a^3*b^3*c*d^4
+ 4*a^3*b^3*c^4*d - 11*a^4*b^2*c*d^4 - 11*a^2*b^4*c^2*d^3 + a^2*b^4*c^3*d^2 + 12*a^3*b^3*c^2*d^3 - 12*a^3*b^3
*c^3*d^2 + 12*a^4*b^2*c^2*d^3 - 4*a^4*b^2*c^3*d^2 + 4*a*b^5*c*d^4 - 2*a*b^5*c^4*d + 2*a^5*b*c*d^4))/(a^5*d^2 -
b^5*c^2 - a*b^4*c^2 + a^4*b*d^2 + a^2*b^3*c^2 + a^3*b^2*c^2 - a^2*b^3*d^2 - a^3*b^2*d^2 + 2*a*b^4*c*d - 2*a^4
*b*c*d + 2*a^2*b^3*c*d - 2*a^3*b^2*c*d) - (b*((32*(a*b^8*c^7 - a^9*d^7 + 2*a^8*b*d^7 + 2*a^9*c*d^6 + b^9*c^6*d
- a^2*b^7*c^7 - a^3*b^6*c^7 + a^4*b^5*c^7 + a^4*b^5*d^7 - 3*a^6*b^3*d^7 + a^7*b^2*d^7 - a^9*c^2*d^5 + b^9*c^4
*d^3 - 2*b^9*c^5*d^2 - 4*a*b^8*c^3*d^4 + 8*a*b^8*c^4*d^3 - 3*a*b^8*c^5*d^2 - 5*a^2*b^7*c^6*d - 4*a^3*b^6*c*d^6
+ 7*a^3*b^6*c^6*d - 2*a^4*b^5*c*d^6 + 4*a^4*b^5*c^6*d + 13*a^5*b^4*c*d^6 - 5*a^5*b^4*c^6*d + a^6*b^3*c*d^6 -
11*a^7*b^2*c*d^6 - 8*a^8*b*c^2*d^5 + 5*a^8*b*c^3*d^4 + 6*a^2*b^7*c^2*d^5 - 12*a^2*b^7*c^3*d^4 - a^2*b^7*c^4*d^
3 + 13*a^2*b^7*c^5*d^2 + 8*a^3*b^6*c^2*d^5 + 14*a^3*b^6*c^3*d^4 - 31*a^3*b^6*c^4*d^3 + 7*a^3*b^6*c^5*d^2 - 21*
a^4*b^5*c^2*d^5 + 34*a^4*b^5*c^3*d^4 + 4*a^4*b^5*c^4*d^3 - 21*a^4*b^5*c^5*d^2 - 16*a^5*b^4*c^2*d^5 - 21*a^5*b^
4*c^3*d^4 + 33*a^5*b^4*c^4*d^3 - 4*a^5*b^4*c^5*d^2 + 23*a^6*b^3*c^2*d^5 - 27*a^6*b^3*c^3*d^4 - 4*a^6*b^3*c^4*d
^3 + 10*a^6*b^3*c^5*d^2 + 9*a^7*b^2*c^2*d^5 + 11*a^7*b^2*c^3*d^4 - 10*a^7*b^2*c^4*d^3 - 2*a*b^8*c^6*d + a^8*b*
c*d^6))/(a^6*d^3 + b^6*c^3 + a*b^5*c^3 + a^5*b*d^3 - a^2*b^4*c^3 - a^3*b^3*c^3 - a^3*b^3*d^3 - a^4*b^2*d^3 + 3
*a^2*b^4*c*d^2 - 3*a^2*b^4*c^2*d + 3*a^3*b^3*c*d^2 + 3*a^3*b^3*c^2*d - 3*a^4*b^2*c*d^2 + 3*a^4*b^2*c^2*d - 3*a
*b^5*c^2*d - 3*a^5*b*c*d^2) - (32*b*tan(e/2 + (f*x)/2)*((a + b)^3*(a - b)^3)^(1/2)*(b^2*d - 2*a^2*d + a*b*c)*(
2*a^10*c*d^6 - 2*a^9*b*d^7 - 2*a*b^9*c^7 + 2*b^10*c^6*d + 2*a^2*b^8*c^7 + 4*a^3*b^7*c^7 - 4*a^4*b^6*c^7 - 2*a^
5*b^5*c^7 + 2*a^6*b^4*c^7 + 2*a^4*b^6*d^7 - 2*a^5*b^5*d^7 - 4*a^6*b^4*d^7 + 4*a^7*b^3*d^7 + 2*a^8*b^2*d^7 - 4*
a^10*c^2*d^5 + 2*a^10*c^3*d^4 + 2*b^10*c^4*d^3 - 4*b^10*c^5*d^2 - 8*a*b^9*c^3*d^4 + 14*a*b^9*c^4*d^3 - 6*a*b^9
*c^5*d^2 - 8*a^3*b^7*c*d^6 - 12*a^3*b^7*c^6*d + 4*a^4*b^6*c*d^6 - 6*a^4*b^6*c^6*d + 18*a^5*b^5*c*d^6 + 18*a^5*
b^5*c^6*d - 6*a^6*b^4*c*d^6 + 4*a^6*b^4*c^6*d - 12*a^7*b^3*c*d^6 - 8*a^7*b^3*c^6*d - 6*a^9*b*c^2*d^5 + 14*a^9*
b*c^3*d^4 - 8*a^9*b*c^4*d^3 + 12*a^2*b^8*c^2*d^5 - 16*a^2*b^8*c^3*d^4 + 2*a^2*b^8*c^5*d^2 + 4*a^3*b^7*c^2*d^5
+ 20*a^3*b^7*c^3*d^4 - 24*a^3*b^7*c^4*d^3 + 16*a^3*b^7*c^5*d^2 - 30*a^4*b^6*c^2*d^5 + 36*a^4*b^6*c^3*d^4 - 22*
a^4*b^6*c^4*d^3 + 20*a^4*b^6*c^5*d^2 - 14*a^5*b^5*c^2*d^5 - 2*a^5*b^5*c^3*d^4 - 2*a^5*b^5*c^4*d^3 - 14*a^5*b^5
*c^5*d^2 + 20*a^6*b^4*c^2*d^5 - 22*a^6*b^4*c^3*d^4 + 36*a^6*b^4*c^4*d^3 - 30*a^6*b^4*c^5*d^2 + 16*a^7*b^3*c^2*
d^5 - 24*a^7*b^3*c^3*d^4 + 20*a^7*b^3*c^4*d^3 + 4*a^7*b^3*c^5*d^2 + 2*a^8*b^2*c^2*d^5 - 16*a^8*b^2*c^4*d^3 + 1
2*a^8*b^2*c^5*d^2 + 2*a*b^9*c^6*d + 2*a^9*b*c*d^6))/((a^5*d^2 - b^5*c^2 - a*b^4*c^2 + a^4*b*d^2 + a^2*b^3*c^2
+ a^3*b^2*c^2 - a^2*b^3*d^2 - a^3*b^2*d^2 + 2*a*b^4*c*d - 2*a^4*b*c*d + 2*a^2*b^3*c*d - 2*a^3*b^2*c*d)*(a^8*d^
2 - b^8*c^2 + 3*a^2*b^6*c^2 - 3*a^4*b^4*c^2 + a^6*b^2*c^2 - a^2*b^6*d^2 + 3*a^4*b^4*d^2 - 3*a^6*b^2*d^2 + 2*a*
b^7*c*d - 2*a^7*b*c*d - 6*a^3*b^5*c*d + 6*a^5*b^3*c*d)))*((a + b)^3*(a - b)^3)^(1/2)*(b^2*d - 2*a^2*d + a*b*c)
)/(a^8*d^2 - b^8*c^2 + 3*a^2*b^6*c^2 - 3*a^4*b^4*c^2 + a^6*b^2*c^2 - a^2*b^6*d^2 + 3*a^4*b^4*d^2 - 3*a^6*b^2*d
^2 + 2*a*b^7*c*d - 2*a^7*b*c*d - 6*a^3*b^5*c*d + 6*a^5*b^3*c*d))*((a + b)^3*(a - b)^3)^(1/2)*(b^2*d - 2*a^2*d
+ a*b*c))/(a^8*d^2 - b^8*c^2 + 3*a^2*b^6*c^2 - 3*a^4*b^4*c^2 + a^6*b^2*c^2 - a^2*b^6*d^2 + 3*a^4*b^4*d^2 - 3*a
^6*b^2*d^2 + 2*a*b^7*c*d - 2*a^7*b*c*d - 6*a^3*b^5*c*d + 6*a^5*b^3*c*d)))*((a + b)^3*(a - b)^3)^(1/2)*(b^2*d -
2*a^2*d + a*b*c)*2i)/(f*(a^8*d^2 - b^8*c^2 + 3*a^2*b^6*c^2 - 3*a^4*b^4*c^2 + a^6*b^2*c^2 - a^2*b^6*d^2 + 3*a^
4*b^4*d^2 - 3*a^6*b^2*d^2 + 2*a*b^7*c*d - 2*a^7*b*c*d - 6*a^3*b^5*c*d + 6*a^5*b^3*c*d))