\(\int \frac {\sec (e+f x) (c+d \sec (e+f x))^5}{(a+b \sec (e+f x))^2} \, dx\) [258]

Optimal result

Integrand size = 31, antiderivative size = 379 \[ \int \frac {\sec (e+f x) (c+d \sec (e+f x))^5}{(a+b \sec (e+f x))^2} \, dx=\frac {d^4 (5 b c-2 a d) \text {arctanh}(\sin (e+f x))}{2 b^3 f}+\frac {d^2 \left (10 b^3 c^3-20 a b^2 c^2 d+15 a^2 b c d^2-4 a^3 d^3\right ) \text {arctanh}(\sin (e+f x))}{b^5 f}+\frac {2 (b c-a d)^5 \text {arctanh}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (e+f x)\right )}{\sqrt {a+b}}\right )}{a (a-b)^{3/2} b^3 (a+b)^{3/2} f}+\frac {2 (b c-a d)^4 (b c+4 a d) \text {arctanh}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (e+f x)\right )}{\sqrt {a+b}}\right )}{a \sqrt {a-b} b^5 \sqrt {a+b} f}-\frac {(b c-a d)^5 \sin (e+f x)}{b^4 \left (a^2-b^2\right ) f (b+a \cos (e+f x))}+\frac {d^5 \tan (e+f x)}{b^2 f}+\frac {d^3 \left (10 b^2 c^2-10 a b c d+3 a^2 d^2\right ) \tan (e+f x)}{b^4 f}+\frac {d^4 (5 b c-2 a d) \sec (e+f x) \tan (e+f x)}{2 b^3 f}+\frac {d^5 \tan ^3(e+f x)}{3 b^2 f} \]

[Out]

1/2*d^4*(-2*a*d+5*b*c)*arctanh(sin(f*x+e))/b^3/f+d^2*(-4*a^3*d^3+15*a^2*b*c*d^2-20*a*b^2*c^2*d+10*b^3*c^3)*arc
tanh(sin(f*x+e))/b^5/f+2*(-a*d+b*c)^5*arctanh((a-b)^(1/2)*tan(1/2*f*x+1/2*e)/(a+b)^(1/2))/a/(a-b)^(3/2)/b^3/(a
+b)^(3/2)/f-(-a*d+b*c)^5*sin(f*x+e)/b^4/(a^2-b^2)/f/(b+a*cos(f*x+e))+2*(-a*d+b*c)^4*(4*a*d+b*c)*arctanh((a-b)^
(1/2)*tan(1/2*f*x+1/2*e)/(a+b)^(1/2))/a/b^5/f/(a-b)^(1/2)/(a+b)^(1/2)+d^5*tan(f*x+e)/b^2/f+d^3*(3*a^2*d^2-10*a
*b*c*d+10*b^2*c^2)*tan(f*x+e)/b^4/f+1/2*d^4*(-2*a*d+5*b*c)*sec(f*x+e)*tan(f*x+e)/b^3/f+1/3*d^5*tan(f*x+e)^3/b^
2/f

Rubi [A] (verified)

Time = 0.80 (sec) , antiderivative size = 379, normalized size of antiderivative = 1.00, number of steps used = 16, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.323, Rules used = {4073, 3031, 2743, 12, 2738, 214, 3855, 3852, 8, 3853} \[ \int \frac {\sec (e+f x) (c+d \sec (e+f x))^5}{(a+b \sec (e+f x))^2} \, dx=\frac {d^3 \left (3 a^2 d^2-10 a b c d+10 b^2 c^2\right ) \tan (e+f x)}{b^4 f}-\frac {(b c-a d)^5 \sin (e+f x)}{b^4 f \left (a^2-b^2\right ) (a \cos (e+f x)+b)}+\frac {d^2 \left (-4 a^3 d^3+15 a^2 b c d^2-20 a b^2 c^2 d+10 b^3 c^3\right ) \text {arctanh}(\sin (e+f x))}{b^5 f}+\frac {2 (b c-a d)^4 (4 a d+b c) \text {arctanh}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (e+f x)\right )}{\sqrt {a+b}}\right )}{a b^5 f \sqrt {a-b} \sqrt {a+b}}+\frac {d^4 (5 b c-2 a d) \text {arctanh}(\sin (e+f x))}{2 b^3 f}+\frac {2 (b c-a d)^5 \text {arctanh}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (e+f x)\right )}{\sqrt {a+b}}\right )}{a b^3 f (a-b)^{3/2} (a+b)^{3/2}}+\frac {d^4 (5 b c-2 a d) \tan (e+f x) \sec (e+f x)}{2 b^3 f}+\frac {d^5 \tan ^3(e+f x)}{3 b^2 f}+\frac {d^5 \tan (e+f x)}{b^2 f} \]

[In]

Int[(Sec[e + f*x]*(c + d*Sec[e + f*x])^5)/(a + b*Sec[e + f*x])^2,x]

[Out]

(d^4*(5*b*c - 2*a*d)*ArcTanh[Sin[e + f*x]])/(2*b^3*f) + (d^2*(10*b^3*c^3 - 20*a*b^2*c^2*d + 15*a^2*b*c*d^2 - 4
*a^3*d^3)*ArcTanh[Sin[e + f*x]])/(b^5*f) + (2*(b*c - a*d)^5*ArcTanh[(Sqrt[a - b]*Tan[(e + f*x)/2])/Sqrt[a + b]
])/(a*(a - b)^(3/2)*b^3*(a + b)^(3/2)*f) + (2*(b*c - a*d)^4*(b*c + 4*a*d)*ArcTanh[(Sqrt[a - b]*Tan[(e + f*x)/2
])/Sqrt[a + b]])/(a*Sqrt[a - b]*b^5*Sqrt[a + b]*f) - ((b*c - a*d)^5*Sin[e + f*x])/(b^4*(a^2 - b^2)*f*(b + a*Co
s[e + f*x])) + (d^5*Tan[e + f*x])/(b^2*f) + (d^3*(10*b^2*c^2 - 10*a*b*c*d + 3*a^2*d^2)*Tan[e + f*x])/(b^4*f) +
(d^4*(5*b*c - 2*a*d)*Sec[e + f*x]*Tan[e + f*x])/(2*b^3*f) + (d^5*Tan[e + f*x]^3)/(3*b^2*f)

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, 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 2743

Int[((a_) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(-b)*Cos[c + d*x]*((a + b*Sin[c + d*x])^(n
+ 1)/(d*(n + 1)*(a^2 - b^2))), x] + Dist[1/((n + 1)*(a^2 - b^2)), Int[(a + b*Sin[c + d*x])^(n + 1)*Simp[a*(n +
1) - b*(n + 2)*Sin[c + d*x], x], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && LtQ[n, -1] && Integ
erQ[2*n]

Rule 3031

Int[((g_.)*sin[(e_.) + (f_.)*(x_)])^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_) + (d_.)*sin[(e_.) +
(f_.)*(x_)])^(n_), x_Symbol] :> Int[ExpandTrig[(g*sin[e + f*x])^p*(a + b*sin[e + f*x])^m*(c + d*sin[e + f*x])
^n, x], x] /; FreeQ[{a, b, c, d, e, f, g, n, p}, x] && NeQ[b*c - a*d, 0] && (IntegersQ[m, n] || IntegersQ[m, p
] || IntegersQ[n, p]) && NeQ[p, 2]

Rule 3852

Int[csc[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> Dist[-d^(-1), Subst[Int[ExpandIntegrand[(1 + x^2)^(n/2 - 1), x]
, x], x, Cot[c + d*x]], x] /; FreeQ[{c, d}, x] && IGtQ[n/2, 0]

Rule 3853

Int[(csc[(c_.) + (d_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Simp[(-b)*Cos[c + d*x]*((b*Csc[c + d*x])^(n - 1)/(d*(n
- 1))), x] + Dist[b^2*((n - 2)/(n - 1)), Int[(b*Csc[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n,
1] && IntegerQ[2*n]

Rule 3855

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

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 {(d+c \cos (e+f x))^5 \sec ^4(e+f x)}{(b+a \cos (e+f x))^2} \, dx \\ & = \int \left (\frac {(-b c+a d)^5}{a b^4 (b+a \cos (e+f x))^2}+\frac {(-b c+a d)^4 (b c+4 a d)}{a b^5 (b+a \cos (e+f x))}+\frac {d^2 \left (10 b^3 c^3-20 a b^2 c^2 d+15 a^2 b c d^2-4 a^3 d^3\right ) \sec (e+f x)}{b^5}+\frac {d^3 \left (10 b^2 c^2-10 a b c d+3 a^2 d^2\right ) \sec ^2(e+f x)}{b^4}+\frac {d^4 (5 b c-2 a d) \sec ^3(e+f x)}{b^3}+\frac {d^5 \sec ^4(e+f x)}{b^2}\right ) \, dx \\ & = \frac {d^5 \int \sec ^4(e+f x) \, dx}{b^2}+\frac {\left (d^4 (5 b c-2 a d)\right ) \int \sec ^3(e+f x) \, dx}{b^3}-\frac {(b c-a d)^5 \int \frac {1}{(b+a \cos (e+f x))^2} \, dx}{a b^4}+\frac {\left ((b c-a d)^4 (b c+4 a d)\right ) \int \frac {1}{b+a \cos (e+f x)} \, dx}{a b^5}+\frac {\left (d^3 \left (10 b^2 c^2-10 a b c d+3 a^2 d^2\right )\right ) \int \sec ^2(e+f x) \, dx}{b^4}+\frac {\left (d^2 \left (10 b^3 c^3-20 a b^2 c^2 d+15 a^2 b c d^2-4 a^3 d^3\right )\right ) \int \sec (e+f x) \, dx}{b^5} \\ & = \frac {d^2 \left (10 b^3 c^3-20 a b^2 c^2 d+15 a^2 b c d^2-4 a^3 d^3\right ) \text {arctanh}(\sin (e+f x))}{b^5 f}-\frac {(b c-a d)^5 \sin (e+f x)}{b^4 \left (a^2-b^2\right ) f (b+a \cos (e+f x))}+\frac {d^4 (5 b c-2 a d) \sec (e+f x) \tan (e+f x)}{2 b^3 f}+\frac {\left (d^4 (5 b c-2 a d)\right ) \int \sec (e+f x) \, dx}{2 b^3}+\frac {(b c-a d)^5 \int \frac {b}{b+a \cos (e+f x)} \, dx}{a b^4 \left (a^2-b^2\right )}-\frac {d^5 \text {Subst}\left (\int \left (1+x^2\right ) \, dx,x,-\tan (e+f x)\right )}{b^2 f}+\frac {\left (2 (b c-a d)^4 (b c+4 a d)\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 )}{a b^5 f}-\frac {\left (d^3 \left (10 b^2 c^2-10 a b c d+3 a^2 d^2\right )\right ) \text {Subst}(\int 1 \, dx,x,-\tan (e+f x))}{b^4 f} \\ & = \frac {d^4 (5 b c-2 a d) \text {arctanh}(\sin (e+f x))}{2 b^3 f}+\frac {d^2 \left (10 b^3 c^3-20 a b^2 c^2 d+15 a^2 b c d^2-4 a^3 d^3\right ) \text {arctanh}(\sin (e+f x))}{b^5 f}+\frac {2 (b c-a d)^4 (b c+4 a d) \text {arctanh}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (e+f x)\right )}{\sqrt {a+b}}\right )}{a \sqrt {a-b} b^5 \sqrt {a+b} f}-\frac {(b c-a d)^5 \sin (e+f x)}{b^4 \left (a^2-b^2\right ) f (b+a \cos (e+f x))}+\frac {d^5 \tan (e+f x)}{b^2 f}+\frac {d^3 \left (10 b^2 c^2-10 a b c d+3 a^2 d^2\right ) \tan (e+f x)}{b^4 f}+\frac {d^4 (5 b c-2 a d) \sec (e+f x) \tan (e+f x)}{2 b^3 f}+\frac {d^5 \tan ^3(e+f x)}{3 b^2 f}+\frac {(b c-a d)^5 \int \frac {1}{b+a \cos (e+f x)} \, dx}{a b^3 \left (a^2-b^2\right )} \\ & = \frac {d^4 (5 b c-2 a d) \text {arctanh}(\sin (e+f x))}{2 b^3 f}+\frac {d^2 \left (10 b^3 c^3-20 a b^2 c^2 d+15 a^2 b c d^2-4 a^3 d^3\right ) \text {arctanh}(\sin (e+f x))}{b^5 f}+\frac {2 (b c-a d)^4 (b c+4 a d) \text {arctanh}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (e+f x)\right )}{\sqrt {a+b}}\right )}{a \sqrt {a-b} b^5 \sqrt {a+b} f}-\frac {(b c-a d)^5 \sin (e+f x)}{b^4 \left (a^2-b^2\right ) f (b+a \cos (e+f x))}+\frac {d^5 \tan (e+f x)}{b^2 f}+\frac {d^3 \left (10 b^2 c^2-10 a b c d+3 a^2 d^2\right ) \tan (e+f x)}{b^4 f}+\frac {d^4 (5 b c-2 a d) \sec (e+f x) \tan (e+f x)}{2 b^3 f}+\frac {d^5 \tan ^3(e+f x)}{3 b^2 f}+\frac {\left (2 (b c-a d)^5\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 )}{a b^3 \left (a^2-b^2\right ) f} \\ & = \frac {d^4 (5 b c-2 a d) \text {arctanh}(\sin (e+f x))}{2 b^3 f}+\frac {d^2 \left (10 b^3 c^3-20 a b^2 c^2 d+15 a^2 b c d^2-4 a^3 d^3\right ) \text {arctanh}(\sin (e+f x))}{b^5 f}+\frac {2 (b c-a d)^5 \text {arctanh}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (e+f x)\right )}{\sqrt {a+b}}\right )}{a (a-b)^{3/2} b^3 (a+b)^{3/2} f}+\frac {2 (b c-a d)^4 (b c+4 a d) \text {arctanh}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (e+f x)\right )}{\sqrt {a+b}}\right )}{a \sqrt {a-b} b^5 \sqrt {a+b} f}-\frac {(b c-a d)^5 \sin (e+f x)}{b^4 \left (a^2-b^2\right ) f (b+a \cos (e+f x))}+\frac {d^5 \tan (e+f x)}{b^2 f}+\frac {d^3 \left (10 b^2 c^2-10 a b c d+3 a^2 d^2\right ) \tan (e+f x)}{b^4 f}+\frac {d^4 (5 b c-2 a d) \sec (e+f x) \tan (e+f x)}{2 b^3 f}+\frac {d^5 \tan ^3(e+f x)}{3 b^2 f} \\ \end{align*}

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(784\) vs. \(2(379)=758\).

Time = 10.09 (sec) , antiderivative size = 784, normalized size of antiderivative = 2.07 \[ \int \frac {\sec (e+f x) (c+d \sec (e+f x))^5}{(a+b \sec (e+f x))^2} \, dx=\frac {(b+a \cos (e+f x)) (c+d \sec (e+f x))^5 \left (-\frac {24 (b c-a d)^4 \left (a b c+4 a^2 d-5 b^2 d\right ) \text {arctanh}\left (\frac {(-a+b) \tan \left (\frac {1}{2} (e+f x)\right )}{\sqrt {a^2-b^2}}\right ) \cos ^3(e+f x) (b+a \cos (e+f x))}{\left (a^2-b^2\right )^{3/2}}+6 d^2 \left (-30 a^2 b c d^2+8 a^3 d^3-5 b^3 c \left (4 c^2+d^2\right )+2 a b^2 d \left (20 c^2+d^2\right )\right ) \cos ^3(e+f x) (b+a \cos (e+f x)) \log \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )+6 d^2 \left (30 a^2 b c d^2-8 a^3 d^3+5 b^3 c \left (4 c^2+d^2\right )-2 a b^2 d \left (20 c^2+d^2\right )\right ) \cos ^3(e+f x) (b+a \cos (e+f x)) \log \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )+\frac {b \left (-60 a^2 b^3 c^2 d^3+60 b^5 c^2 d^3+45 a^3 b^2 c d^4-45 a b^4 c d^4-12 a^4 b d^5+4 a^2 b^3 d^5+8 b^5 d^5+\left (135 a^4 b c d^4-36 a^5 d^5+30 a^2 b^3 c d^2 \left (3 c^2-4 d^2\right )+a^3 b^2 d^3 \left (-180 c^2+29 d^2\right )+a b^4 d \left (-45 c^4+90 c^2 d^2-2 d^4\right )+b^5 \left (9 c^5+30 c d^4\right )\right ) \cos (e+f x)+b \left (-a^2+b^2\right ) d^3 \left (-45 a b c d+12 a^2 d^2+4 b^2 \left (15 c^2+d^2\right )\right ) \cos (2 (e+f x))+3 b^5 c^5 \cos (3 (e+f x))-15 a b^4 c^4 d \cos (3 (e+f x))+30 a^2 b^3 c^3 d^2 \cos (3 (e+f x))-60 a^3 b^2 c^2 d^3 \cos (3 (e+f x))+30 a b^4 c^2 d^3 \cos (3 (e+f x))+45 a^4 b c d^4 \cos (3 (e+f x))-30 a^2 b^3 c d^4 \cos (3 (e+f x))-12 a^5 d^5 \cos (3 (e+f x))+7 a^3 b^2 d^5 \cos (3 (e+f x))+2 a b^4 d^5 \cos (3 (e+f x))\right ) \sin (e+f x)}{-a^2+b^2}\right )}{12 b^5 f (d+c \cos (e+f x))^5 (a+b \sec (e+f x))^2} \]

[In]

Integrate[(Sec[e + f*x]*(c + d*Sec[e + f*x])^5)/(a + b*Sec[e + f*x])^2,x]

[Out]

((b + a*Cos[e + f*x])*(c + d*Sec[e + f*x])^5*((-24*(b*c - a*d)^4*(a*b*c + 4*a^2*d - 5*b^2*d)*ArcTanh[((-a + b)
*Tan[(e + f*x)/2])/Sqrt[a^2 - b^2]]*Cos[e + f*x]^3*(b + a*Cos[e + f*x]))/(a^2 - b^2)^(3/2) + 6*d^2*(-30*a^2*b*
c*d^2 + 8*a^3*d^3 - 5*b^3*c*(4*c^2 + d^2) + 2*a*b^2*d*(20*c^2 + d^2))*Cos[e + f*x]^3*(b + a*Cos[e + f*x])*Log[
Cos[(e + f*x)/2] - Sin[(e + f*x)/2]] + 6*d^2*(30*a^2*b*c*d^2 - 8*a^3*d^3 + 5*b^3*c*(4*c^2 + d^2) - 2*a*b^2*d*(
20*c^2 + d^2))*Cos[e + f*x]^3*(b + a*Cos[e + f*x])*Log[Cos[(e + f*x)/2] + Sin[(e + f*x)/2]] + (b*(-60*a^2*b^3*
c^2*d^3 + 60*b^5*c^2*d^3 + 45*a^3*b^2*c*d^4 - 45*a*b^4*c*d^4 - 12*a^4*b*d^5 + 4*a^2*b^3*d^5 + 8*b^5*d^5 + (135
*a^4*b*c*d^4 - 36*a^5*d^5 + 30*a^2*b^3*c*d^2*(3*c^2 - 4*d^2) + a^3*b^2*d^3*(-180*c^2 + 29*d^2) + a*b^4*d*(-45*
c^4 + 90*c^2*d^2 - 2*d^4) + b^5*(9*c^5 + 30*c*d^4))*Cos[e + f*x] + b*(-a^2 + b^2)*d^3*(-45*a*b*c*d + 12*a^2*d^
2 + 4*b^2*(15*c^2 + d^2))*Cos[2*(e + f*x)] + 3*b^5*c^5*Cos[3*(e + f*x)] - 15*a*b^4*c^4*d*Cos[3*(e + f*x)] + 30
*a^2*b^3*c^3*d^2*Cos[3*(e + f*x)] - 60*a^3*b^2*c^2*d^3*Cos[3*(e + f*x)] + 30*a*b^4*c^2*d^3*Cos[3*(e + f*x)] +
45*a^4*b*c*d^4*Cos[3*(e + f*x)] - 30*a^2*b^3*c*d^4*Cos[3*(e + f*x)] - 12*a^5*d^5*Cos[3*(e + f*x)] + 7*a^3*b^2*
d^5*Cos[3*(e + f*x)] + 2*a*b^4*d^5*Cos[3*(e + f*x)])*Sin[e + f*x])/(-a^2 + b^2)))/(12*b^5*f*(d + c*Cos[e + f*x
])^5*(a + b*Sec[e + f*x])^2)

Maple [A] (verified)

Time = 2.51 (sec) , antiderivative size = 689, normalized size of antiderivative = 1.82

method	result	size
derivativedivides	\(\frac {-\frac {d^{5}}{3 b^{2} \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{3}}-\frac {d^{2} \left (8 a^{3} d^{3}-30 a^{2} b c \,d^{2}+40 a \,b^{2} c^{2} d +2 a \,b^{2} d^{3}-20 b^{3} c^{3}-5 c \,d^{2} b^{3}\right ) \ln \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )}{2 b^{5}}-\frac {d^{3} \left (6 a^{2} d^{2}-20 a b c d +2 b \,d^{2} a +20 b^{2} c^{2}-5 c d \,b^{2}+2 b^{2} d^{2}\right )}{2 b^{4} \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )}+\frac {d^{4} \left (2 a d -5 b c +b d \right )}{2 b^{3} \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{2}}-\frac {d^{5}}{3 b^{2} \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{3}}+\frac {d^{2} \left (8 a^{3} d^{3}-30 a^{2} b c \,d^{2}+40 a \,b^{2} c^{2} d +2 a \,b^{2} d^{3}-20 b^{3} c^{3}-5 c \,d^{2} b^{3}\right ) \ln \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )}{2 b^{5}}-\frac {d^{3} \left (6 a^{2} d^{2}-20 a b c d +2 b \,d^{2} a +20 b^{2} c^{2}-5 c d \,b^{2}+2 b^{2} d^{2}\right )}{2 b^{4} \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )}-\frac {d^{4} \left (2 a d -5 b c +b d \right )}{2 b^{3} \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{2}}-\frac {2 \left (\frac {b \left (a^{5} d^{5}-5 a^{4} b c \,d^{4}+10 a^{3} b^{2} c^{2} d^{3}-10 a^{2} b^{3} c^{3} d^{2}+5 a \,b^{4} c^{4} d -b^{5} c^{5}\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 (4 a^{6} d^{5}-15 a^{5} b c \,d^{4}+20 a^{4} b^{2} c^{2} d^{3}-5 a^{4} b^{2} d^{5}-10 a^{3} b^{3} c^{3} d^{2}+20 a^{3} b^{3} c \,d^{4}-30 a^{2} b^{4} c^{2} d^{3}+b^{5} c^{5} a +20 a \,b^{5} c^{3} d^{2}-5 b^{6} c^{4} d \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 )}{b^{5}}}{f}\)	\(689\)
default	\(\frac {-\frac {d^{5}}{3 b^{2} \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{3}}-\frac {d^{2} \left (8 a^{3} d^{3}-30 a^{2} b c \,d^{2}+40 a \,b^{2} c^{2} d +2 a \,b^{2} d^{3}-20 b^{3} c^{3}-5 c \,d^{2} b^{3}\right ) \ln \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )}{2 b^{5}}-\frac {d^{3} \left (6 a^{2} d^{2}-20 a b c d +2 b \,d^{2} a +20 b^{2} c^{2}-5 c d \,b^{2}+2 b^{2} d^{2}\right )}{2 b^{4} \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )}+\frac {d^{4} \left (2 a d -5 b c +b d \right )}{2 b^{3} \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{2}}-\frac {d^{5}}{3 b^{2} \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{3}}+\frac {d^{2} \left (8 a^{3} d^{3}-30 a^{2} b c \,d^{2}+40 a \,b^{2} c^{2} d +2 a \,b^{2} d^{3}-20 b^{3} c^{3}-5 c \,d^{2} b^{3}\right ) \ln \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )}{2 b^{5}}-\frac {d^{3} \left (6 a^{2} d^{2}-20 a b c d +2 b \,d^{2} a +20 b^{2} c^{2}-5 c d \,b^{2}+2 b^{2} d^{2}\right )}{2 b^{4} \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )}-\frac {d^{4} \left (2 a d -5 b c +b d \right )}{2 b^{3} \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{2}}-\frac {2 \left (\frac {b \left (a^{5} d^{5}-5 a^{4} b c \,d^{4}+10 a^{3} b^{2} c^{2} d^{3}-10 a^{2} b^{3} c^{3} d^{2}+5 a \,b^{4} c^{4} d -b^{5} c^{5}\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 (4 a^{6} d^{5}-15 a^{5} b c \,d^{4}+20 a^{4} b^{2} c^{2} d^{3}-5 a^{4} b^{2} d^{5}-10 a^{3} b^{3} c^{3} d^{2}+20 a^{3} b^{3} c \,d^{4}-30 a^{2} b^{4} c^{2} d^{3}+b^{5} c^{5} a +20 a \,b^{5} c^{3} d^{2}-5 b^{6} c^{4} d \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 )}{b^{5}}}{f}\)	\(689\)
risch	\(\text {Expression too large to display}\)	\(3699\)

[In]

int(sec(f*x+e)*(c+d*sec(f*x+e))^5/(a+b*sec(f*x+e))^2,x,method=_RETURNVERBOSE)

[Out]

1/f*(-1/3*d^5/b^2/(tan(1/2*f*x+1/2*e)+1)^3-1/2*d^2*(8*a^3*d^3-30*a^2*b*c*d^2+40*a*b^2*c^2*d+2*a*b^2*d^3-20*b^3
*c^3-5*b^3*c*d^2)/b^5*ln(tan(1/2*f*x+1/2*e)+1)-1/2*d^3*(6*a^2*d^2-20*a*b*c*d+2*a*b*d^2+20*b^2*c^2-5*b^2*c*d+2*
b^2*d^2)/b^4/(tan(1/2*f*x+1/2*e)+1)+1/2*d^4*(2*a*d-5*b*c+b*d)/b^3/(tan(1/2*f*x+1/2*e)+1)^2-1/3*d^5/b^2/(tan(1/
2*f*x+1/2*e)-1)^3+1/2*d^2*(8*a^3*d^3-30*a^2*b*c*d^2+40*a*b^2*c^2*d+2*a*b^2*d^3-20*b^3*c^3-5*b^3*c*d^2)/b^5*ln(
tan(1/2*f*x+1/2*e)-1)-1/2*d^3*(6*a^2*d^2-20*a*b*c*d+2*a*b*d^2+20*b^2*c^2-5*b^2*c*d+2*b^2*d^2)/b^4/(tan(1/2*f*x
+1/2*e)-1)-1/2*d^4*(2*a*d-5*b*c+b*d)/b^3/(tan(1/2*f*x+1/2*e)-1)^2-2/b^5*(b*(a^5*d^5-5*a^4*b*c*d^4+10*a^3*b^2*c
^2*d^3-10*a^2*b^3*c^3*d^2+5*a*b^4*c^4*d-b^5*c^5)/(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)-(4*a^6*d^5-15*a^5*b*c*d^4+20*a^4*b^2*c^2*d^3-5*a^4*b^2*d^5-10*a^3*b^3*c^3*d^2+20*a^3*b^3*c
*d^4-30*a^2*b^4*c^2*d^3+a*b^5*c^5+20*a*b^5*c^3*d^2-5*b^6*c^4*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))))

Fricas [F(-1)]

Timed out. \[ \int \frac {\sec (e+f x) (c+d \sec (e+f x))^5}{(a+b \sec (e+f x))^2} \, dx=\text {Timed out} \]

[In]

integrate(sec(f*x+e)*(c+d*sec(f*x+e))^5/(a+b*sec(f*x+e))^2,x, algorithm="fricas")

[Out]

Timed out

Sympy [F]

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

[In]

integrate(sec(f*x+e)*(c+d*sec(f*x+e))**5/(a+b*sec(f*x+e))**2,x)

[Out]

Integral((c + d*sec(e + f*x))**5*sec(e + f*x)/(a + b*sec(e + f*x))**2, x)

Maxima [F(-2)]

Exception generated. \[ \int \frac {\sec (e+f x) (c+d \sec (e+f x))^5}{(a+b \sec (e+f x))^2} \, dx=\text {Exception raised: ValueError} \]

[In]

integrate(sec(f*x+e)*(c+d*sec(f*x+e))^5/(a+b*sec(f*x+e))^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*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. 857 vs. \(2 (355) = 710\).

Time = 0.45 (sec) , antiderivative size = 857, normalized size of antiderivative = 2.26 \[ \int \frac {\sec (e+f x) (c+d \sec (e+f x))^5}{(a+b \sec (e+f x))^2} \, dx=\text {Too large to display} \]

[In]

integrate(sec(f*x+e)*(c+d*sec(f*x+e))^5/(a+b*sec(f*x+e))^2,x, algorithm="giac")

[Out]

-1/6*(12*(a*b^5*c^5 - 5*b^6*c^4*d - 10*a^3*b^3*c^3*d^2 + 20*a*b^5*c^3*d^2 + 20*a^4*b^2*c^2*d^3 - 30*a^2*b^4*c^
2*d^3 - 15*a^5*b*c*d^4 + 20*a^3*b^3*c*d^4 + 4*a^6*d^5 - 5*a^4*b^2*d^5)*(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^5 - b^7)*sqrt(
-a^2 + b^2)) - 12*(b^5*c^5*tan(1/2*f*x + 1/2*e) - 5*a*b^4*c^4*d*tan(1/2*f*x + 1/2*e) + 10*a^2*b^3*c^3*d^2*tan(
1/2*f*x + 1/2*e) - 10*a^3*b^2*c^2*d^3*tan(1/2*f*x + 1/2*e) + 5*a^4*b*c*d^4*tan(1/2*f*x + 1/2*e) - a^5*d^5*tan(
1/2*f*x + 1/2*e))/((a^2*b^4 - b^6)*(a*tan(1/2*f*x + 1/2*e)^2 - b*tan(1/2*f*x + 1/2*e)^2 - a - b)) - 3*(20*b^3*
c^3*d^2 - 40*a*b^2*c^2*d^3 + 30*a^2*b*c*d^4 + 5*b^3*c*d^4 - 8*a^3*d^5 - 2*a*b^2*d^5)*log(abs(tan(1/2*f*x + 1/2
*e) + 1))/b^5 + 3*(20*b^3*c^3*d^2 - 40*a*b^2*c^2*d^3 + 30*a^2*b*c*d^4 + 5*b^3*c*d^4 - 8*a^3*d^5 - 2*a*b^2*d^5)
*log(abs(tan(1/2*f*x + 1/2*e) - 1))/b^5 + 2*(60*b^2*c^2*d^3*tan(1/2*f*x + 1/2*e)^5 - 60*a*b*c*d^4*tan(1/2*f*x
+ 1/2*e)^5 - 15*b^2*c*d^4*tan(1/2*f*x + 1/2*e)^5 + 18*a^2*d^5*tan(1/2*f*x + 1/2*e)^5 + 6*a*b*d^5*tan(1/2*f*x +
1/2*e)^5 + 6*b^2*d^5*tan(1/2*f*x + 1/2*e)^5 - 120*b^2*c^2*d^3*tan(1/2*f*x + 1/2*e)^3 + 120*a*b*c*d^4*tan(1/2*
f*x + 1/2*e)^3 - 36*a^2*d^5*tan(1/2*f*x + 1/2*e)^3 - 4*b^2*d^5*tan(1/2*f*x + 1/2*e)^3 + 60*b^2*c^2*d^3*tan(1/2
*f*x + 1/2*e) - 60*a*b*c*d^4*tan(1/2*f*x + 1/2*e) + 15*b^2*c*d^4*tan(1/2*f*x + 1/2*e) + 18*a^2*d^5*tan(1/2*f*x
+ 1/2*e) - 6*a*b*d^5*tan(1/2*f*x + 1/2*e) + 6*b^2*d^5*tan(1/2*f*x + 1/2*e))/((tan(1/2*f*x + 1/2*e)^2 - 1)^3*b
^4))/f

Mupad [B] (verification not implemented)

Time = 28.86 (sec) , antiderivative size = 17256, normalized size of antiderivative = 45.53 \[ \int \frac {\sec (e+f x) (c+d \sec (e+f x))^5}{(a+b \sec (e+f x))^2} \, dx=\text {Too large to display} \]

[In]

int((c + d/cos(e + f*x))^5/(cos(e + f*x)*(a + b/cos(e + f*x))^2),x)

[Out]

(atan(((((8*tan(e/2 + (f*x)/2)*(128*a^12*d^10 - 128*a^11*b*d^10 + 4*a^2*b^10*c^10 + 4*a^2*b^10*d^10 - 8*a^3*b^
9*d^10 + 28*a^4*b^8*d^10 - 48*a^5*b^7*d^10 + 28*a^6*b^6*d^10 - 8*a^7*b^5*d^10 + 8*a^8*b^4*d^10 + 192*a^9*b^3*d
^10 - 192*a^10*b^2*d^10 + 25*b^12*c^2*d^8 + 200*b^12*c^4*d^6 + 400*b^12*c^6*d^4 + 100*b^12*c^8*d^2 - 50*a*b^11
*c^2*d^8 - 480*a*b^11*c^3*d^7 - 400*a*b^11*c^4*d^6 - 1600*a*b^11*c^5*d^5 - 800*a*b^11*c^6*d^4 - 800*a*b^11*c^7
*d^3 + 40*a^2*b^10*c*d^9 - 180*a^3*b^9*c*d^9 + 320*a^4*b^8*c*d^9 - 260*a^5*b^7*c*d^9 + 200*a^6*b^6*c*d^9 - 140
*a^7*b^5*c*d^9 - 1520*a^8*b^4*c*d^9 + 1520*a^9*b^3*c*d^9 + 960*a^10*b^2*c*d^9 + 435*a^2*b^10*c^2*d^8 + 960*a^2
*b^10*c^3*d^7 + 2600*a^2*b^10*c^4*d^6 + 3200*a^2*b^10*c^5*d^5 + 2400*a^2*b^10*c^6*d^4 + 160*a^2*b^10*c^8*d^2 -
820*a^3*b^9*c^2*d^8 - 2240*a^3*b^9*c^3*d^7 - 4800*a^3*b^9*c^4*d^6 - 4000*a^3*b^9*c^5*d^5 + 1600*a^3*b^9*c^6*d
^4 + 160*a^3*b^9*c^7*d^3 + 1055*a^4*b^8*c^2*d^8 + 3520*a^4*b^8*c^3*d^7 + 4000*a^4*b^8*c^4*d^6 - 6400*a^4*b^8*c
^5*d^5 - 2640*a^4*b^8*c^6*d^4 - 80*a^4*b^8*c^8*d^2 - 1290*a^5*b^7*c^2*d^8 - 2400*a^5*b^7*c^3*d^7 + 10800*a^5*b
^7*c^4*d^6 + 7760*a^5*b^7*c^5*d^5 - 800*a^5*b^7*c^6*d^4 + 160*a^5*b^7*c^7*d^3 + 825*a^6*b^6*c^2*d^8 - 9920*a^6
*b^6*c^3*d^7 - 11560*a^6*b^6*c^4*d^6 + 3200*a^6*b^6*c^5*d^5 + 680*a^6*b^6*c^6*d^4 + 5240*a^7*b^5*c^2*d^8 + 100
80*a^7*b^5*c^3*d^7 - 5600*a^7*b^5*c^4*d^6 - 3168*a^7*b^5*c^5*d^5 - 5240*a^8*b^4*c^2*d^8 + 5440*a^8*b^4*c^3*d^7
+ 5600*a^8*b^4*c^4*d^6 - 3080*a^9*b^3*c^2*d^8 - 5440*a^9*b^3*c^3*d^7 + 3080*a^10*b^2*c^2*d^8 - 20*a*b^11*c*d^
9 - 40*a*b^11*c^9*d - 960*a^11*b*c*d^9))/(a*b^10 + b^11 - a^2*b^9 - a^3*b^8) + (((8*(4*a*b^17*c^5 + 4*a*b^17*d
^5 - 10*b^18*c*d^4 - 20*b^18*c^4*d - 4*a^2*b^16*c^5 - 4*a^3*b^15*c^5 + 4*a^4*b^14*c^5 + 4*a^3*b^15*d^5 - 20*a^
4*b^14*d^5 - 16*a^5*b^13*d^5 + 36*a^6*b^12*d^5 + 8*a^7*b^11*d^5 - 16*a^8*b^10*d^5 - 40*b^18*c^3*d^2 + 80*a*b^1
7*c^2*d^3 + 80*a*b^17*c^3*d^2 - 30*a^2*b^16*c*d^4 + 20*a^2*b^16*c^4*d + 80*a^3*b^15*c*d^4 - 20*a^3*b^15*c^4*d
+ 70*a^4*b^14*c*d^4 - 140*a^5*b^13*c*d^4 - 30*a^6*b^12*c*d^4 + 60*a^7*b^11*c*d^4 - 120*a^2*b^16*c^2*d^3 + 40*a
^2*b^16*c^3*d^2 - 120*a^3*b^15*c^2*d^3 - 120*a^3*b^15*c^3*d^2 + 200*a^4*b^14*c^2*d^3 + 40*a^5*b^13*c^2*d^3 + 4
0*a^5*b^13*c^3*d^2 - 80*a^6*b^12*c^2*d^3 + 20*a*b^17*c^4*d))/(a*b^14 + b^15 - a^2*b^13 - a^3*b^12) + (8*tan(e/
2 + (f*x)/2)*(b^2*(a*d^5 + 20*a*c^2*d^3) + 4*a^3*d^5 - b^3*((5*c*d^4)/2 + 10*c^3*d^2) - 15*a^2*b*c*d^4)*(8*a*b
^15 - 8*a^2*b^14 - 16*a^3*b^13 + 16*a^4*b^12 + 8*a^5*b^11 - 8*a^6*b^10))/(b^5*(a*b^10 + b^11 - a^2*b^9 - a^3*b
^8)))*(b^2*(a*d^5 + 20*a*c^2*d^3) + 4*a^3*d^5 - b^3*((5*c*d^4)/2 + 10*c^3*d^2) - 15*a^2*b*c*d^4))/b^5)*(b^2*(a
*d^5 + 20*a*c^2*d^3) + 4*a^3*d^5 - b^3*((5*c*d^4)/2 + 10*c^3*d^2) - 15*a^2*b*c*d^4)*1i)/b^5 + (((8*tan(e/2 + (
f*x)/2)*(128*a^12*d^10 - 128*a^11*b*d^10 + 4*a^2*b^10*c^10 + 4*a^2*b^10*d^10 - 8*a^3*b^9*d^10 + 28*a^4*b^8*d^1
0 - 48*a^5*b^7*d^10 + 28*a^6*b^6*d^10 - 8*a^7*b^5*d^10 + 8*a^8*b^4*d^10 + 192*a^9*b^3*d^10 - 192*a^10*b^2*d^10
+ 25*b^12*c^2*d^8 + 200*b^12*c^4*d^6 + 400*b^12*c^6*d^4 + 100*b^12*c^8*d^2 - 50*a*b^11*c^2*d^8 - 480*a*b^11*c
^3*d^7 - 400*a*b^11*c^4*d^6 - 1600*a*b^11*c^5*d^5 - 800*a*b^11*c^6*d^4 - 800*a*b^11*c^7*d^3 + 40*a^2*b^10*c*d^
9 - 180*a^3*b^9*c*d^9 + 320*a^4*b^8*c*d^9 - 260*a^5*b^7*c*d^9 + 200*a^6*b^6*c*d^9 - 140*a^7*b^5*c*d^9 - 1520*a
^8*b^4*c*d^9 + 1520*a^9*b^3*c*d^9 + 960*a^10*b^2*c*d^9 + 435*a^2*b^10*c^2*d^8 + 960*a^2*b^10*c^3*d^7 + 2600*a^
2*b^10*c^4*d^6 + 3200*a^2*b^10*c^5*d^5 + 2400*a^2*b^10*c^6*d^4 + 160*a^2*b^10*c^8*d^2 - 820*a^3*b^9*c^2*d^8 -
2240*a^3*b^9*c^3*d^7 - 4800*a^3*b^9*c^4*d^6 - 4000*a^3*b^9*c^5*d^5 + 1600*a^3*b^9*c^6*d^4 + 160*a^3*b^9*c^7*d^
3 + 1055*a^4*b^8*c^2*d^8 + 3520*a^4*b^8*c^3*d^7 + 4000*a^4*b^8*c^4*d^6 - 6400*a^4*b^8*c^5*d^5 - 2640*a^4*b^8*c
^6*d^4 - 80*a^4*b^8*c^8*d^2 - 1290*a^5*b^7*c^2*d^8 - 2400*a^5*b^7*c^3*d^7 + 10800*a^5*b^7*c^4*d^6 + 7760*a^5*b
^7*c^5*d^5 - 800*a^5*b^7*c^6*d^4 + 160*a^5*b^7*c^7*d^3 + 825*a^6*b^6*c^2*d^8 - 9920*a^6*b^6*c^3*d^7 - 11560*a^
6*b^6*c^4*d^6 + 3200*a^6*b^6*c^5*d^5 + 680*a^6*b^6*c^6*d^4 + 5240*a^7*b^5*c^2*d^8 + 10080*a^7*b^5*c^3*d^7 - 56
00*a^7*b^5*c^4*d^6 - 3168*a^7*b^5*c^5*d^5 - 5240*a^8*b^4*c^2*d^8 + 5440*a^8*b^4*c^3*d^7 + 5600*a^8*b^4*c^4*d^6
- 3080*a^9*b^3*c^2*d^8 - 5440*a^9*b^3*c^3*d^7 + 3080*a^10*b^2*c^2*d^8 - 20*a*b^11*c*d^9 - 40*a*b^11*c^9*d - 9
60*a^11*b*c*d^9))/(a*b^10 + b^11 - a^2*b^9 - a^3*b^8) - (((8*(4*a*b^17*c^5 + 4*a*b^17*d^5 - 10*b^18*c*d^4 - 20
*b^18*c^4*d - 4*a^2*b^16*c^5 - 4*a^3*b^15*c^5 + 4*a^4*b^14*c^5 + 4*a^3*b^15*d^5 - 20*a^4*b^14*d^5 - 16*a^5*b^1
3*d^5 + 36*a^6*b^12*d^5 + 8*a^7*b^11*d^5 - 16*a^8*b^10*d^5 - 40*b^18*c^3*d^2 + 80*a*b^17*c^2*d^3 + 80*a*b^17*c
^3*d^2 - 30*a^2*b^16*c*d^4 + 20*a^2*b^16*c^4*d + 80*a^3*b^15*c*d^4 - 20*a^3*b^15*c^4*d + 70*a^4*b^14*c*d^4 - 1
40*a^5*b^13*c*d^4 - 30*a^6*b^12*c*d^4 + 60*a^7*b^11*c*d^4 - 120*a^2*b^16*c^2*d^3 + 40*a^2*b^16*c^3*d^2 - 120*a
^3*b^15*c^2*d^3 - 120*a^3*b^15*c^3*d^2 + 200*a^4*b^14*c^2*d^3 + 40*a^5*b^13*c^2*d^3 + 40*a^5*b^13*c^3*d^2 - 80
*a^6*b^12*c^2*d^3 + 20*a*b^17*c^4*d))/(a*b^14 + b^15 - a^2*b^13 - a^3*b^12) - (8*tan(e/2 + (f*x)/2)*(b^2*(a*d^
5 + 20*a*c^2*d^3) + 4*a^3*d^5 - b^3*((5*c*d^4)/2 + 10*c^3*d^2) - 15*a^2*b*c*d^4)*(8*a*b^15 - 8*a^2*b^14 - 16*a
^3*b^13 + 16*a^4*b^12 + 8*a^5*b^11 - 8*a^6*b^10))/(b^5*(a*b^10 + b^11 - a^2*b^9 - a^3*b^8)))*(b^2*(a*d^5 + 20*
a*c^2*d^3) + 4*a^3*d^5 - b^3*((5*c*d^4)/2 + 10*c^3*d^2) - 15*a^2*b*c*d^4))/b^5)*(b^2*(a*d^5 + 20*a*c^2*d^3) +
4*a^3*d^5 - b^3*((5*c*d^4)/2 + 10*c^3*d^2) - 15*a^2*b*c*d^4)*1i)/b^5)/((16*(256*a^14*d^15 - 128*a^13*b*d^15 +
20*a^6*b^8*d^15 - 20*a^7*b^7*d^15 + 124*a^8*b^6*d^15 - 24*a^9*b^5*d^15 + 48*a^10*b^4*d^15 + 192*a^11*b^3*d^15
- 448*a^12*b^2*d^15 + 125*b^14*c^6*d^9 + 1000*b^14*c^8*d^7 - 250*b^14*c^9*d^6 + 2000*b^14*c^10*d^5 - 1000*b^14
*c^11*d^4 - 600*a*b^13*c^5*d^10 - 125*a*b^13*c^6*d^9 - 6425*a*b^13*c^7*d^8 + 1100*a*b^13*c^8*d^7 - 16200*a*b^1
3*c^9*d^6 + 8100*a*b^13*c^10*d^5 - 400*a*b^13*c^11*d^4 + 400*a*b^13*c^12*d^3 - 180*a^5*b^9*c*d^14 + 180*a^6*b^
8*c*d^14 - 1320*a^7*b^7*c*d^14 + 270*a^8*b^6*c*d^14 - 900*a^9*b^5*c*d^14 - 2160*a^10*b^4*c*d^14 + 5280*a^11*b^
3*c*d^14 + 1440*a^12*b^2*c*d^14 + 1170*a^2*b^12*c^4*d^11 + 600*a^2*b^12*c^5*d^10 + 17795*a^2*b^12*c^6*d^9 - 13
75*a^2*b^12*c^7*d^8 + 57480*a^2*b^12*c^8*d^7 - 29740*a^2*b^12*c^9*d^6 - 400*a^2*b^12*c^10*d^5 - 2010*a^2*b^12*
c^11*d^4 - 40*a^2*b^12*c^13*d^2 - 1180*a^3*b^11*c^3*d^12 - 1170*a^3*b^11*c^4*d^11 - 27754*a^3*b^11*c^5*d^10 -
995*a^3*b^11*c^6*d^9 - 117635*a^3*b^11*c^7*d^8 + 66680*a^3*b^11*c^8*d^7 + 17400*a^3*b^11*c^9*d^6 + 2604*a^3*b^
11*c^10*d^5 + 400*a^3*b^11*c^11*d^4 + 80*a^3*b^11*c^12*d^3 + 645*a^4*b^10*c^2*d^13 + 1180*a^4*b^10*c^3*d^12 +
26690*a^4*b^10*c^4*d^11 + 4654*a^4*b^10*c^5*d^10 + 153580*a^4*b^10*c^6*d^9 - 103805*a^4*b^10*c^7*d^8 - 79760*a
^4*b^10*c^8*d^7 + 5840*a^4*b^10*c^9*d^6 - 1600*a^4*b^10*c^10*d^5 + 340*a^4*b^10*c^11*d^4 - 645*a^5*b^9*c^2*d^1
3 - 16245*a^5*b^9*c^3*d^12 - 5690*a^5*b^9*c^4*d^11 - 133278*a^5*b^9*c^5*d^10 + 119980*a^5*b^9*c^6*d^9 + 188520
*a^5*b^9*c^7*d^8 - 28880*a^5*b^9*c^8*d^7 - 1200*a^5*b^9*c^9*d^6 - 1584*a^5*b^9*c^10*d^5 + 6135*a^6*b^8*c^2*d^1
3 + 3645*a^6*b^8*c^3*d^12 + 77460*a^6*b^8*c^4*d^11 - 105562*a^6*b^8*c^5*d^10 - 279820*a^6*b^8*c^6*d^9 + 57980*
a^6*b^8*c^7*d^8 + 21280*a^6*b^8*c^8*d^7 + 2800*a^6*b^8*c^9*d^6 - 1335*a^7*b^7*c^2*d^13 - 29515*a^7*b^7*c^3*d^1
2 + 69980*a^7*b^7*c^4*d^11 + 279768*a^7*b^7*c^5*d^10 - 74940*a^7*b^7*c^6*d^9 - 64460*a^7*b^7*c^7*d^8 - 2720*a^
7*b^7*c^8*d^7 + 6960*a^8*b^6*c^2*d^13 - 33645*a^8*b^6*c^3*d^12 - 192920*a^8*b^6*c^4*d^11 + 69104*a^8*b^6*c^5*d
^10 + 108320*a^8*b^6*c^6*d^9 + 1540*a^8*b^6*c^7*d^8 + 10980*a^9*b^5*c^2*d^13 + 91160*a^9*b^5*c^3*d^12 - 46520*
a^9*b^5*c^4*d^11 - 118136*a^9*b^5*c^5*d^10 - 480*a^9*b^5*c^6*d^9 - 28380*a^10*b^4*c^2*d^13 + 22430*a^10*b^4*c^
3*d^12 + 87600*a^10*b^4*c^4*d^11 + 64*a^10*b^4*c^5*d^10 - 7320*a^11*b^3*c^2*d^13 - 44220*a^11*b^3*c^3*d^12 + 1
4640*a^12*b^2*c^2*d^13 - 2880*a^13*b*c*d^14))/(a*b^14 + b^15 - a^2*b^13 - a^3*b^12) + (((8*tan(e/2 + (f*x)/2)*
(128*a^12*d^10 - 128*a^11*b*d^10 + 4*a^2*b^10*c^10 + 4*a^2*b^10*d^10 - 8*a^3*b^9*d^10 + 28*a^4*b^8*d^10 - 48*a
^5*b^7*d^10 + 28*a^6*b^6*d^10 - 8*a^7*b^5*d^10 + 8*a^8*b^4*d^10 + 192*a^9*b^3*d^10 - 192*a^10*b^2*d^10 + 25*b^
12*c^2*d^8 + 200*b^12*c^4*d^6 + 400*b^12*c^6*d^4 + 100*b^12*c^8*d^2 - 50*a*b^11*c^2*d^8 - 480*a*b^11*c^3*d^7 -
400*a*b^11*c^4*d^6 - 1600*a*b^11*c^5*d^5 - 800*a*b^11*c^6*d^4 - 800*a*b^11*c^7*d^3 + 40*a^2*b^10*c*d^9 - 180*
a^3*b^9*c*d^9 + 320*a^4*b^8*c*d^9 - 260*a^5*b^7*c*d^9 + 200*a^6*b^6*c*d^9 - 140*a^7*b^5*c*d^9 - 1520*a^8*b^4*c
*d^9 + 1520*a^9*b^3*c*d^9 + 960*a^10*b^2*c*d^9 + 435*a^2*b^10*c^2*d^8 + 960*a^2*b^10*c^3*d^7 + 2600*a^2*b^10*c
^4*d^6 + 3200*a^2*b^10*c^5*d^5 + 2400*a^2*b^10*c^6*d^4 + 160*a^2*b^10*c^8*d^2 - 820*a^3*b^9*c^2*d^8 - 2240*a^3
*b^9*c^3*d^7 - 4800*a^3*b^9*c^4*d^6 - 4000*a^3*b^9*c^5*d^5 + 1600*a^3*b^9*c^6*d^4 + 160*a^3*b^9*c^7*d^3 + 1055
*a^4*b^8*c^2*d^8 + 3520*a^4*b^8*c^3*d^7 + 4000*a^4*b^8*c^4*d^6 - 6400*a^4*b^8*c^5*d^5 - 2640*a^4*b^8*c^6*d^4 -
80*a^4*b^8*c^8*d^2 - 1290*a^5*b^7*c^2*d^8 - 2400*a^5*b^7*c^3*d^7 + 10800*a^5*b^7*c^4*d^6 + 7760*a^5*b^7*c^5*d
^5 - 800*a^5*b^7*c^6*d^4 + 160*a^5*b^7*c^7*d^3 + 825*a^6*b^6*c^2*d^8 - 9920*a^6*b^6*c^3*d^7 - 11560*a^6*b^6*c^
4*d^6 + 3200*a^6*b^6*c^5*d^5 + 680*a^6*b^6*c^6*d^4 + 5240*a^7*b^5*c^2*d^8 + 10080*a^7*b^5*c^3*d^7 - 5600*a^7*b
^5*c^4*d^6 - 3168*a^7*b^5*c^5*d^5 - 5240*a^8*b^4*c^2*d^8 + 5440*a^8*b^4*c^3*d^7 + 5600*a^8*b^4*c^4*d^6 - 3080*
a^9*b^3*c^2*d^8 - 5440*a^9*b^3*c^3*d^7 + 3080*a^10*b^2*c^2*d^8 - 20*a*b^11*c*d^9 - 40*a*b^11*c^9*d - 960*a^11*
b*c*d^9))/(a*b^10 + b^11 - a^2*b^9 - a^3*b^8) + (((8*(4*a*b^17*c^5 + 4*a*b^17*d^5 - 10*b^18*c*d^4 - 20*b^18*c^
4*d - 4*a^2*b^16*c^5 - 4*a^3*b^15*c^5 + 4*a^4*b^14*c^5 + 4*a^3*b^15*d^5 - 20*a^4*b^14*d^5 - 16*a^5*b^13*d^5 +
36*a^6*b^12*d^5 + 8*a^7*b^11*d^5 - 16*a^8*b^10*d^5 - 40*b^18*c^3*d^2 + 80*a*b^17*c^2*d^3 + 80*a*b^17*c^3*d^2 -
30*a^2*b^16*c*d^4 + 20*a^2*b^16*c^4*d + 80*a^3*b^15*c*d^4 - 20*a^3*b^15*c^4*d + 70*a^4*b^14*c*d^4 - 140*a^5*b
^13*c*d^4 - 30*a^6*b^12*c*d^4 + 60*a^7*b^11*c*d^4 - 120*a^2*b^16*c^2*d^3 + 40*a^2*b^16*c^3*d^2 - 120*a^3*b^15*
c^2*d^3 - 120*a^3*b^15*c^3*d^2 + 200*a^4*b^14*c^2*d^3 + 40*a^5*b^13*c^2*d^3 + 40*a^5*b^13*c^3*d^2 - 80*a^6*b^1
2*c^2*d^3 + 20*a*b^17*c^4*d))/(a*b^14 + b^15 - a^2*b^13 - a^3*b^12) + (8*tan(e/2 + (f*x)/2)*(b^2*(a*d^5 + 20*a
*c^2*d^3) + 4*a^3*d^5 - b^3*((5*c*d^4)/2 + 10*c^3*d^2) - 15*a^2*b*c*d^4)*(8*a*b^15 - 8*a^2*b^14 - 16*a^3*b^13
+ 16*a^4*b^12 + 8*a^5*b^11 - 8*a^6*b^10))/(b^5*(a*b^10 + b^11 - a^2*b^9 - a^3*b^8)))*(b^2*(a*d^5 + 20*a*c^2*d^
3) + 4*a^3*d^5 - b^3*((5*c*d^4)/2 + 10*c^3*d^2) - 15*a^2*b*c*d^4))/b^5)*(b^2*(a*d^5 + 20*a*c^2*d^3) + 4*a^3*d^
5 - b^3*((5*c*d^4)/2 + 10*c^3*d^2) - 15*a^2*b*c*d^4))/b^5 - (((8*tan(e/2 + (f*x)/2)*(128*a^12*d^10 - 128*a^11*
b*d^10 + 4*a^2*b^10*c^10 + 4*a^2*b^10*d^10 - 8*a^3*b^9*d^10 + 28*a^4*b^8*d^10 - 48*a^5*b^7*d^10 + 28*a^6*b^6*d
^10 - 8*a^7*b^5*d^10 + 8*a^8*b^4*d^10 + 192*a^9*b^3*d^10 - 192*a^10*b^2*d^10 + 25*b^12*c^2*d^8 + 200*b^12*c^4*
d^6 + 400*b^12*c^6*d^4 + 100*b^12*c^8*d^2 - 50*a*b^11*c^2*d^8 - 480*a*b^11*c^3*d^7 - 400*a*b^11*c^4*d^6 - 1600
*a*b^11*c^5*d^5 - 800*a*b^11*c^6*d^4 - 800*a*b^11*c^7*d^3 + 40*a^2*b^10*c*d^9 - 180*a^3*b^9*c*d^9 + 320*a^4*b^
8*c*d^9 - 260*a^5*b^7*c*d^9 + 200*a^6*b^6*c*d^9 - 140*a^7*b^5*c*d^9 - 1520*a^8*b^4*c*d^9 + 1520*a^9*b^3*c*d^9
+ 960*a^10*b^2*c*d^9 + 435*a^2*b^10*c^2*d^8 + 960*a^2*b^10*c^3*d^7 + 2600*a^2*b^10*c^4*d^6 + 3200*a^2*b^10*c^5
*d^5 + 2400*a^2*b^10*c^6*d^4 + 160*a^2*b^10*c^8*d^2 - 820*a^3*b^9*c^2*d^8 - 2240*a^3*b^9*c^3*d^7 - 4800*a^3*b^
9*c^4*d^6 - 4000*a^3*b^9*c^5*d^5 + 1600*a^3*b^9*c^6*d^4 + 160*a^3*b^9*c^7*d^3 + 1055*a^4*b^8*c^2*d^8 + 3520*a^
4*b^8*c^3*d^7 + 4000*a^4*b^8*c^4*d^6 - 6400*a^4*b^8*c^5*d^5 - 2640*a^4*b^8*c^6*d^4 - 80*a^4*b^8*c^8*d^2 - 1290
*a^5*b^7*c^2*d^8 - 2400*a^5*b^7*c^3*d^7 + 10800*a^5*b^7*c^4*d^6 + 7760*a^5*b^7*c^5*d^5 - 800*a^5*b^7*c^6*d^4 +
160*a^5*b^7*c^7*d^3 + 825*a^6*b^6*c^2*d^8 - 9920*a^6*b^6*c^3*d^7 - 11560*a^6*b^6*c^4*d^6 + 3200*a^6*b^6*c^5*d
^5 + 680*a^6*b^6*c^6*d^4 + 5240*a^7*b^5*c^2*d^8 + 10080*a^7*b^5*c^3*d^7 - 5600*a^7*b^5*c^4*d^6 - 3168*a^7*b^5*
c^5*d^5 - 5240*a^8*b^4*c^2*d^8 + 5440*a^8*b^4*c^3*d^7 + 5600*a^8*b^4*c^4*d^6 - 3080*a^9*b^3*c^2*d^8 - 5440*a^9
*b^3*c^3*d^7 + 3080*a^10*b^2*c^2*d^8 - 20*a*b^11*c*d^9 - 40*a*b^11*c^9*d - 960*a^11*b*c*d^9))/(a*b^10 + b^11 -
a^2*b^9 - a^3*b^8) - (((8*(4*a*b^17*c^5 + 4*a*b^17*d^5 - 10*b^18*c*d^4 - 20*b^18*c^4*d - 4*a^2*b^16*c^5 - 4*a
^3*b^15*c^5 + 4*a^4*b^14*c^5 + 4*a^3*b^15*d^5 - 20*a^4*b^14*d^5 - 16*a^5*b^13*d^5 + 36*a^6*b^12*d^5 + 8*a^7*b^
11*d^5 - 16*a^8*b^10*d^5 - 40*b^18*c^3*d^2 + 80*a*b^17*c^2*d^3 + 80*a*b^17*c^3*d^2 - 30*a^2*b^16*c*d^4 + 20*a^
2*b^16*c^4*d + 80*a^3*b^15*c*d^4 - 20*a^3*b^15*c^4*d + 70*a^4*b^14*c*d^4 - 140*a^5*b^13*c*d^4 - 30*a^6*b^12*c*
d^4 + 60*a^7*b^11*c*d^4 - 120*a^2*b^16*c^2*d^3 + 40*a^2*b^16*c^3*d^2 - 120*a^3*b^15*c^2*d^3 - 120*a^3*b^15*c^3
*d^2 + 200*a^4*b^14*c^2*d^3 + 40*a^5*b^13*c^2*d^3 + 40*a^5*b^13*c^3*d^2 - 80*a^6*b^12*c^2*d^3 + 20*a*b^17*c^4*
d))/(a*b^14 + b^15 - a^2*b^13 - a^3*b^12) - (8*tan(e/2 + (f*x)/2)*(b^2*(a*d^5 + 20*a*c^2*d^3) + 4*a^3*d^5 - b^
3*((5*c*d^4)/2 + 10*c^3*d^2) - 15*a^2*b*c*d^4)*(8*a*b^15 - 8*a^2*b^14 - 16*a^3*b^13 + 16*a^4*b^12 + 8*a^5*b^11
- 8*a^6*b^10))/(b^5*(a*b^10 + b^11 - a^2*b^9 - a^3*b^8)))*(b^2*(a*d^5 + 20*a*c^2*d^3) + 4*a^3*d^5 - b^3*((5*c
*d^4)/2 + 10*c^3*d^2) - 15*a^2*b*c*d^4))/b^5)*(b^2*(a*d^5 + 20*a*c^2*d^3) + 4*a^3*d^5 - b^3*((5*c*d^4)/2 + 10*
c^3*d^2) - 15*a^2*b*c*d^4))/b^5))*(b^2*(a*d^5 + 20*a*c^2*d^3) + 4*a^3*d^5 - b^3*((5*c*d^4)/2 + 10*c^3*d^2) - 1
5*a^2*b*c*d^4)*2i)/(b^5*f) - ((tan(e/2 + (f*x)/2)^5*(18*b^5*c^5 - 72*a^5*d^5 + 2*b^5*d^5 + 16*a*b^4*d^5 + 12*a
^4*b*d^5 - 15*b^5*c*d^4 - 14*a^2*b^3*d^5 + 38*a^3*b^2*d^5 - 60*b^5*c^2*d^3 + 180*a*b^4*c^2*d^3 - 165*a^2*b^3*c
*d^4 - 45*a^3*b^2*c*d^4 + 60*a^2*b^3*c^2*d^3 + 180*a^2*b^3*c^3*d^2 - 360*a^3*b^2*c^2*d^3 + 45*a*b^4*c*d^4 - 90
*a*b^4*c^4*d + 270*a^4*b*c*d^4))/(3*(a*b^4 - b^5)*(a + b)) - (tan(e/2 + (f*x)/2)^7*(2*b^5*c^5 - 8*a^5*d^5 - 2*
b^5*d^5 + 4*a^4*b*d^5 + 5*b^5*c*d^4 - 2*a^2*b^3*d^5 + 6*a^3*b^2*d^5 - 20*b^5*c^2*d^3 + 20*a*b^4*c^2*d^3 - 25*a
^2*b^3*c*d^4 - 15*a^3*b^2*c*d^4 + 20*a^2*b^3*c^2*d^3 + 20*a^2*b^3*c^3*d^2 - 40*a^3*b^2*c^2*d^3 + 15*a*b^4*c*d^
4 - 10*a*b^4*c^4*d + 30*a^4*b*c*d^4))/((a*b^4 - b^5)*(a + b)) + (tan(e/2 + (f*x)/2)*(2*b^5*c^5 - 8*a^5*d^5 + 2
*b^5*d^5 - 4*a^4*b*d^5 + 5*b^5*c*d^4 + 2*a^2*b^3*d^5 + 6*a^3*b^2*d^5 + 20*b^5*c^2*d^3 + 20*a*b^4*c^2*d^3 - 25*
a^2*b^3*c*d^4 + 15*a^3*b^2*c*d^4 - 20*a^2*b^3*c^2*d^3 + 20*a^2*b^3*c^3*d^2 - 40*a^3*b^2*c^2*d^3 - 15*a*b^4*c*d
^4 - 10*a*b^4*c^4*d + 30*a^4*b*c*d^4))/((a*b^4 - b^5)*(a + b)) + (tan(e/2 + (f*x)/2)^3*(72*a^5*d^5 - 18*b^5*c^
5 + 2*b^5*d^5 - 16*a*b^4*d^5 + 12*a^4*b*d^5 + 15*b^5*c*d^4 - 14*a^2*b^3*d^5 - 38*a^3*b^2*d^5 - 60*b^5*c^2*d^3
- 180*a*b^4*c^2*d^3 + 165*a^2*b^3*c*d^4 - 45*a^3*b^2*c*d^4 + 60*a^2*b^3*c^2*d^3 - 180*a^2*b^3*c^3*d^2 + 360*a^
3*b^2*c^2*d^3 + 45*a*b^4*c*d^4 + 90*a*b^4*c^4*d - 270*a^4*b*c*d^4))/(3*b^4*(a + b)*(a - b)))/(f*(a + b + tan(e
/2 + (f*x)/2)^8*(a - b) - tan(e/2 + (f*x)/2)^2*(4*a + 2*b) - tan(e/2 + (f*x)/2)^6*(4*a - 2*b) + 6*a*tan(e/2 +
(f*x)/2)^4)) + (atan((((a*d - b*c)^4*((8*tan(e/2 + (f*x)/2)*(128*a^12*d^10 - 128*a^11*b*d^10 + 4*a^2*b^10*c^10
+ 4*a^2*b^10*d^10 - 8*a^3*b^9*d^10 + 28*a^4*b^8*d^10 - 48*a^5*b^7*d^10 + 28*a^6*b^6*d^10 - 8*a^7*b^5*d^10 + 8
*a^8*b^4*d^10 + 192*a^9*b^3*d^10 - 192*a^10*b^2*d^10 + 25*b^12*c^2*d^8 + 200*b^12*c^4*d^6 + 400*b^12*c^6*d^4 +
100*b^12*c^8*d^2 - 50*a*b^11*c^2*d^8 - 480*a*b^11*c^3*d^7 - 400*a*b^11*c^4*d^6 - 1600*a*b^11*c^5*d^5 - 800*a*
b^11*c^6*d^4 - 800*a*b^11*c^7*d^3 + 40*a^2*b^10*c*d^9 - 180*a^3*b^9*c*d^9 + 320*a^4*b^8*c*d^9 - 260*a^5*b^7*c*
d^9 + 200*a^6*b^6*c*d^9 - 140*a^7*b^5*c*d^9 - 1520*a^8*b^4*c*d^9 + 1520*a^9*b^3*c*d^9 + 960*a^10*b^2*c*d^9 + 4
35*a^2*b^10*c^2*d^8 + 960*a^2*b^10*c^3*d^7 + 2600*a^2*b^10*c^4*d^6 + 3200*a^2*b^10*c^5*d^5 + 2400*a^2*b^10*c^6
*d^4 + 160*a^2*b^10*c^8*d^2 - 820*a^3*b^9*c^2*d^8 - 2240*a^3*b^9*c^3*d^7 - 4800*a^3*b^9*c^4*d^6 - 4000*a^3*b^9
*c^5*d^5 + 1600*a^3*b^9*c^6*d^4 + 160*a^3*b^9*c^7*d^3 + 1055*a^4*b^8*c^2*d^8 + 3520*a^4*b^8*c^3*d^7 + 4000*a^4
*b^8*c^4*d^6 - 6400*a^4*b^8*c^5*d^5 - 2640*a^4*b^8*c^6*d^4 - 80*a^4*b^8*c^8*d^2 - 1290*a^5*b^7*c^2*d^8 - 2400*
a^5*b^7*c^3*d^7 + 10800*a^5*b^7*c^4*d^6 + 7760*a^5*b^7*c^5*d^5 - 800*a^5*b^7*c^6*d^4 + 160*a^5*b^7*c^7*d^3 + 8
25*a^6*b^6*c^2*d^8 - 9920*a^6*b^6*c^3*d^7 - 11560*a^6*b^6*c^4*d^6 + 3200*a^6*b^6*c^5*d^5 + 680*a^6*b^6*c^6*d^4
+ 5240*a^7*b^5*c^2*d^8 + 10080*a^7*b^5*c^3*d^7 - 5600*a^7*b^5*c^4*d^6 - 3168*a^7*b^5*c^5*d^5 - 5240*a^8*b^4*c
^2*d^8 + 5440*a^8*b^4*c^3*d^7 + 5600*a^8*b^4*c^4*d^6 - 3080*a^9*b^3*c^2*d^8 - 5440*a^9*b^3*c^3*d^7 + 3080*a^10
*b^2*c^2*d^8 - 20*a*b^11*c*d^9 - 40*a*b^11*c^9*d - 960*a^11*b*c*d^9))/(a*b^10 + b^11 - a^2*b^9 - a^3*b^8) + ((
(8*(4*a*b^17*c^5 + 4*a*b^17*d^5 - 10*b^18*c*d^4 - 20*b^18*c^4*d - 4*a^2*b^16*c^5 - 4*a^3*b^15*c^5 + 4*a^4*b^14
*c^5 + 4*a^3*b^15*d^5 - 20*a^4*b^14*d^5 - 16*a^5*b^13*d^5 + 36*a^6*b^12*d^5 + 8*a^7*b^11*d^5 - 16*a^8*b^10*d^5
- 40*b^18*c^3*d^2 + 80*a*b^17*c^2*d^3 + 80*a*b^17*c^3*d^2 - 30*a^2*b^16*c*d^4 + 20*a^2*b^16*c^4*d + 80*a^3*b^
15*c*d^4 - 20*a^3*b^15*c^4*d + 70*a^4*b^14*c*d^4 - 140*a^5*b^13*c*d^4 - 30*a^6*b^12*c*d^4 + 60*a^7*b^11*c*d^4
- 120*a^2*b^16*c^2*d^3 + 40*a^2*b^16*c^3*d^2 - 120*a^3*b^15*c^2*d^3 - 120*a^3*b^15*c^3*d^2 + 200*a^4*b^14*c^2*
d^3 + 40*a^5*b^13*c^2*d^3 + 40*a^5*b^13*c^3*d^2 - 80*a^6*b^12*c^2*d^3 + 20*a*b^17*c^4*d))/(a*b^14 + b^15 - a^2
*b^13 - a^3*b^12) + (8*tan(e/2 + (f*x)/2)*(a*d - b*c)^4*((a + b)^3*(a - b)^3)^(1/2)*(4*a^2*d - 5*b^2*d + a*b*c
)*(8*a*b^15 - 8*a^2*b^14 - 16*a^3*b^13 + 16*a^4*b^12 + 8*a^5*b^11 - 8*a^6*b^10))/((a*b^10 + b^11 - a^2*b^9 - a
^3*b^8)*(b^11 - 3*a^2*b^9 + 3*a^4*b^7 - a^6*b^5)))*(a*d - b*c)^4*((a + b)^3*(a - b)^3)^(1/2)*(4*a^2*d - 5*b^2*
d + a*b*c))/(b^11 - 3*a^2*b^9 + 3*a^4*b^7 - a^6*b^5))*((a + b)^3*(a - b)^3)^(1/2)*(4*a^2*d - 5*b^2*d + a*b*c)*
1i)/(b^11 - 3*a^2*b^9 + 3*a^4*b^7 - a^6*b^5) + ((a*d - b*c)^4*((8*tan(e/2 + (f*x)/2)*(128*a^12*d^10 - 128*a^11
*b*d^10 + 4*a^2*b^10*c^10 + 4*a^2*b^10*d^10 - 8*a^3*b^9*d^10 + 28*a^4*b^8*d^10 - 48*a^5*b^7*d^10 + 28*a^6*b^6*
d^10 - 8*a^7*b^5*d^10 + 8*a^8*b^4*d^10 + 192*a^9*b^3*d^10 - 192*a^10*b^2*d^10 + 25*b^12*c^2*d^8 + 200*b^12*c^4
*d^6 + 400*b^12*c^6*d^4 + 100*b^12*c^8*d^2 - 50*a*b^11*c^2*d^8 - 480*a*b^11*c^3*d^7 - 400*a*b^11*c^4*d^6 - 160
0*a*b^11*c^5*d^5 - 800*a*b^11*c^6*d^4 - 800*a*b^11*c^7*d^3 + 40*a^2*b^10*c*d^9 - 180*a^3*b^9*c*d^9 + 320*a^4*b
^8*c*d^9 - 260*a^5*b^7*c*d^9 + 200*a^6*b^6*c*d^9 - 140*a^7*b^5*c*d^9 - 1520*a^8*b^4*c*d^9 + 1520*a^9*b^3*c*d^9
+ 960*a^10*b^2*c*d^9 + 435*a^2*b^10*c^2*d^8 + 960*a^2*b^10*c^3*d^7 + 2600*a^2*b^10*c^4*d^6 + 3200*a^2*b^10*c^
5*d^5 + 2400*a^2*b^10*c^6*d^4 + 160*a^2*b^10*c^8*d^2 - 820*a^3*b^9*c^2*d^8 - 2240*a^3*b^9*c^3*d^7 - 4800*a^3*b
^9*c^4*d^6 - 4000*a^3*b^9*c^5*d^5 + 1600*a^3*b^9*c^6*d^4 + 160*a^3*b^9*c^7*d^3 + 1055*a^4*b^8*c^2*d^8 + 3520*a
^4*b^8*c^3*d^7 + 4000*a^4*b^8*c^4*d^6 - 6400*a^4*b^8*c^5*d^5 - 2640*a^4*b^8*c^6*d^4 - 80*a^4*b^8*c^8*d^2 - 129
0*a^5*b^7*c^2*d^8 - 2400*a^5*b^7*c^3*d^7 + 10800*a^5*b^7*c^4*d^6 + 7760*a^5*b^7*c^5*d^5 - 800*a^5*b^7*c^6*d^4
+ 160*a^5*b^7*c^7*d^3 + 825*a^6*b^6*c^2*d^8 - 9920*a^6*b^6*c^3*d^7 - 11560*a^6*b^6*c^4*d^6 + 3200*a^6*b^6*c^5*
d^5 + 680*a^6*b^6*c^6*d^4 + 5240*a^7*b^5*c^2*d^8 + 10080*a^7*b^5*c^3*d^7 - 5600*a^7*b^5*c^4*d^6 - 3168*a^7*b^5
*c^5*d^5 - 5240*a^8*b^4*c^2*d^8 + 5440*a^8*b^4*c^3*d^7 + 5600*a^8*b^4*c^4*d^6 - 3080*a^9*b^3*c^2*d^8 - 5440*a^
9*b^3*c^3*d^7 + 3080*a^10*b^2*c^2*d^8 - 20*a*b^11*c*d^9 - 40*a*b^11*c^9*d - 960*a^11*b*c*d^9))/(a*b^10 + b^11
- a^2*b^9 - a^3*b^8) - (((8*(4*a*b^17*c^5 + 4*a*b^17*d^5 - 10*b^18*c*d^4 - 20*b^18*c^4*d - 4*a^2*b^16*c^5 - 4*
a^3*b^15*c^5 + 4*a^4*b^14*c^5 + 4*a^3*b^15*d^5 - 20*a^4*b^14*d^5 - 16*a^5*b^13*d^5 + 36*a^6*b^12*d^5 + 8*a^7*b
^11*d^5 - 16*a^8*b^10*d^5 - 40*b^18*c^3*d^2 + 80*a*b^17*c^2*d^3 + 80*a*b^17*c^3*d^2 - 30*a^2*b^16*c*d^4 + 20*a
^2*b^16*c^4*d + 80*a^3*b^15*c*d^4 - 20*a^3*b^15*c^4*d + 70*a^4*b^14*c*d^4 - 140*a^5*b^13*c*d^4 - 30*a^6*b^12*c
*d^4 + 60*a^7*b^11*c*d^4 - 120*a^2*b^16*c^2*d^3 + 40*a^2*b^16*c^3*d^2 - 120*a^3*b^15*c^2*d^3 - 120*a^3*b^15*c^
3*d^2 + 200*a^4*b^14*c^2*d^3 + 40*a^5*b^13*c^2*d^3 + 40*a^5*b^13*c^3*d^2 - 80*a^6*b^12*c^2*d^3 + 20*a*b^17*c^4
*d))/(a*b^14 + b^15 - a^2*b^13 - a^3*b^12) - (8*tan(e/2 + (f*x)/2)*(a*d - b*c)^4*((a + b)^3*(a - b)^3)^(1/2)*(
4*a^2*d - 5*b^2*d + a*b*c)*(8*a*b^15 - 8*a^2*b^14 - 16*a^3*b^13 + 16*a^4*b^12 + 8*a^5*b^11 - 8*a^6*b^10))/((a*
b^10 + b^11 - a^2*b^9 - a^3*b^8)*(b^11 - 3*a^2*b^9 + 3*a^4*b^7 - a^6*b^5)))*(a*d - b*c)^4*((a + b)^3*(a - b)^3
)^(1/2)*(4*a^2*d - 5*b^2*d + a*b*c))/(b^11 - 3*a^2*b^9 + 3*a^4*b^7 - a^6*b^5))*((a + b)^3*(a - b)^3)^(1/2)*(4*
a^2*d - 5*b^2*d + a*b*c)*1i)/(b^11 - 3*a^2*b^9 + 3*a^4*b^7 - a^6*b^5))/((16*(256*a^14*d^15 - 128*a^13*b*d^15 +
20*a^6*b^8*d^15 - 20*a^7*b^7*d^15 + 124*a^8*b^6*d^15 - 24*a^9*b^5*d^15 + 48*a^10*b^4*d^15 + 192*a^11*b^3*d^15
- 448*a^12*b^2*d^15 + 125*b^14*c^6*d^9 + 1000*b^14*c^8*d^7 - 250*b^14*c^9*d^6 + 2000*b^14*c^10*d^5 - 1000*b^1
4*c^11*d^4 - 600*a*b^13*c^5*d^10 - 125*a*b^13*c^6*d^9 - 6425*a*b^13*c^7*d^8 + 1100*a*b^13*c^8*d^7 - 16200*a*b^
13*c^9*d^6 + 8100*a*b^13*c^10*d^5 - 400*a*b^13*c^11*d^4 + 400*a*b^13*c^12*d^3 - 180*a^5*b^9*c*d^14 + 180*a^6*b
^8*c*d^14 - 1320*a^7*b^7*c*d^14 + 270*a^8*b^6*c*d^14 - 900*a^9*b^5*c*d^14 - 2160*a^10*b^4*c*d^14 + 5280*a^11*b
^3*c*d^14 + 1440*a^12*b^2*c*d^14 + 1170*a^2*b^12*c^4*d^11 + 600*a^2*b^12*c^5*d^10 + 17795*a^2*b^12*c^6*d^9 - 1
375*a^2*b^12*c^7*d^8 + 57480*a^2*b^12*c^8*d^7 - 29740*a^2*b^12*c^9*d^6 - 400*a^2*b^12*c^10*d^5 - 2010*a^2*b^12
*c^11*d^4 - 40*a^2*b^12*c^13*d^2 - 1180*a^3*b^11*c^3*d^12 - 1170*a^3*b^11*c^4*d^11 - 27754*a^3*b^11*c^5*d^10 -
995*a^3*b^11*c^6*d^9 - 117635*a^3*b^11*c^7*d^8 + 66680*a^3*b^11*c^8*d^7 + 17400*a^3*b^11*c^9*d^6 + 2604*a^3*b
^11*c^10*d^5 + 400*a^3*b^11*c^11*d^4 + 80*a^3*b^11*c^12*d^3 + 645*a^4*b^10*c^2*d^13 + 1180*a^4*b^10*c^3*d^12 +
26690*a^4*b^10*c^4*d^11 + 4654*a^4*b^10*c^5*d^10 + 153580*a^4*b^10*c^6*d^9 - 103805*a^4*b^10*c^7*d^8 - 79760*
a^4*b^10*c^8*d^7 + 5840*a^4*b^10*c^9*d^6 - 1600*a^4*b^10*c^10*d^5 + 340*a^4*b^10*c^11*d^4 - 645*a^5*b^9*c^2*d^
13 - 16245*a^5*b^9*c^3*d^12 - 5690*a^5*b^9*c^4*d^11 - 133278*a^5*b^9*c^5*d^10 + 119980*a^5*b^9*c^6*d^9 + 18852
0*a^5*b^9*c^7*d^8 - 28880*a^5*b^9*c^8*d^7 - 1200*a^5*b^9*c^9*d^6 - 1584*a^5*b^9*c^10*d^5 + 6135*a^6*b^8*c^2*d^
13 + 3645*a^6*b^8*c^3*d^12 + 77460*a^6*b^8*c^4*d^11 - 105562*a^6*b^8*c^5*d^10 - 279820*a^6*b^8*c^6*d^9 + 57980
*a^6*b^8*c^7*d^8 + 21280*a^6*b^8*c^8*d^7 + 2800*a^6*b^8*c^9*d^6 - 1335*a^7*b^7*c^2*d^13 - 29515*a^7*b^7*c^3*d^
12 + 69980*a^7*b^7*c^4*d^11 + 279768*a^7*b^7*c^5*d^10 - 74940*a^7*b^7*c^6*d^9 - 64460*a^7*b^7*c^7*d^8 - 2720*a
^7*b^7*c^8*d^7 + 6960*a^8*b^6*c^2*d^13 - 33645*a^8*b^6*c^3*d^12 - 192920*a^8*b^6*c^4*d^11 + 69104*a^8*b^6*c^5*
d^10 + 108320*a^8*b^6*c^6*d^9 + 1540*a^8*b^6*c^7*d^8 + 10980*a^9*b^5*c^2*d^13 + 91160*a^9*b^5*c^3*d^12 - 46520
*a^9*b^5*c^4*d^11 - 118136*a^9*b^5*c^5*d^10 - 480*a^9*b^5*c^6*d^9 - 28380*a^10*b^4*c^2*d^13 + 22430*a^10*b^4*c
^3*d^12 + 87600*a^10*b^4*c^4*d^11 + 64*a^10*b^4*c^5*d^10 - 7320*a^11*b^3*c^2*d^13 - 44220*a^11*b^3*c^3*d^12 +
14640*a^12*b^2*c^2*d^13 - 2880*a^13*b*c*d^14))/(a*b^14 + b^15 - a^2*b^13 - a^3*b^12) + ((a*d - b*c)^4*((8*tan(
e/2 + (f*x)/2)*(128*a^12*d^10 - 128*a^11*b*d^10 + 4*a^2*b^10*c^10 + 4*a^2*b^10*d^10 - 8*a^3*b^9*d^10 + 28*a^4*
b^8*d^10 - 48*a^5*b^7*d^10 + 28*a^6*b^6*d^10 - 8*a^7*b^5*d^10 + 8*a^8*b^4*d^10 + 192*a^9*b^3*d^10 - 192*a^10*b
^2*d^10 + 25*b^12*c^2*d^8 + 200*b^12*c^4*d^6 + 400*b^12*c^6*d^4 + 100*b^12*c^8*d^2 - 50*a*b^11*c^2*d^8 - 480*a
*b^11*c^3*d^7 - 400*a*b^11*c^4*d^6 - 1600*a*b^11*c^5*d^5 - 800*a*b^11*c^6*d^4 - 800*a*b^11*c^7*d^3 + 40*a^2*b^
10*c*d^9 - 180*a^3*b^9*c*d^9 + 320*a^4*b^8*c*d^9 - 260*a^5*b^7*c*d^9 + 200*a^6*b^6*c*d^9 - 140*a^7*b^5*c*d^9 -
1520*a^8*b^4*c*d^9 + 1520*a^9*b^3*c*d^9 + 960*a^10*b^2*c*d^9 + 435*a^2*b^10*c^2*d^8 + 960*a^2*b^10*c^3*d^7 +
2600*a^2*b^10*c^4*d^6 + 3200*a^2*b^10*c^5*d^5 + 2400*a^2*b^10*c^6*d^4 + 160*a^2*b^10*c^8*d^2 - 820*a^3*b^9*c^2
*d^8 - 2240*a^3*b^9*c^3*d^7 - 4800*a^3*b^9*c^4*d^6 - 4000*a^3*b^9*c^5*d^5 + 1600*a^3*b^9*c^6*d^4 + 160*a^3*b^9
*c^7*d^3 + 1055*a^4*b^8*c^2*d^8 + 3520*a^4*b^8*c^3*d^7 + 4000*a^4*b^8*c^4*d^6 - 6400*a^4*b^8*c^5*d^5 - 2640*a^
4*b^8*c^6*d^4 - 80*a^4*b^8*c^8*d^2 - 1290*a^5*b^7*c^2*d^8 - 2400*a^5*b^7*c^3*d^7 + 10800*a^5*b^7*c^4*d^6 + 776
0*a^5*b^7*c^5*d^5 - 800*a^5*b^7*c^6*d^4 + 160*a^5*b^7*c^7*d^3 + 825*a^6*b^6*c^2*d^8 - 9920*a^6*b^6*c^3*d^7 - 1
1560*a^6*b^6*c^4*d^6 + 3200*a^6*b^6*c^5*d^5 + 680*a^6*b^6*c^6*d^4 + 5240*a^7*b^5*c^2*d^8 + 10080*a^7*b^5*c^3*d
^7 - 5600*a^7*b^5*c^4*d^6 - 3168*a^7*b^5*c^5*d^5 - 5240*a^8*b^4*c^2*d^8 + 5440*a^8*b^4*c^3*d^7 + 5600*a^8*b^4*
c^4*d^6 - 3080*a^9*b^3*c^2*d^8 - 5440*a^9*b^3*c^3*d^7 + 3080*a^10*b^2*c^2*d^8 - 20*a*b^11*c*d^9 - 40*a*b^11*c^
9*d - 960*a^11*b*c*d^9))/(a*b^10 + b^11 - a^2*b^9 - a^3*b^8) + (((8*(4*a*b^17*c^5 + 4*a*b^17*d^5 - 10*b^18*c*d
^4 - 20*b^18*c^4*d - 4*a^2*b^16*c^5 - 4*a^3*b^15*c^5 + 4*a^4*b^14*c^5 + 4*a^3*b^15*d^5 - 20*a^4*b^14*d^5 - 16*
a^5*b^13*d^5 + 36*a^6*b^12*d^5 + 8*a^7*b^11*d^5 - 16*a^8*b^10*d^5 - 40*b^18*c^3*d^2 + 80*a*b^17*c^2*d^3 + 80*a
*b^17*c^3*d^2 - 30*a^2*b^16*c*d^4 + 20*a^2*b^16*c^4*d + 80*a^3*b^15*c*d^4 - 20*a^3*b^15*c^4*d + 70*a^4*b^14*c*
d^4 - 140*a^5*b^13*c*d^4 - 30*a^6*b^12*c*d^4 + 60*a^7*b^11*c*d^4 - 120*a^2*b^16*c^2*d^3 + 40*a^2*b^16*c^3*d^2
- 120*a^3*b^15*c^2*d^3 - 120*a^3*b^15*c^3*d^2 + 200*a^4*b^14*c^2*d^3 + 40*a^5*b^13*c^2*d^3 + 40*a^5*b^13*c^3*d
^2 - 80*a^6*b^12*c^2*d^3 + 20*a*b^17*c^4*d))/(a*b^14 + b^15 - a^2*b^13 - a^3*b^12) + (8*tan(e/2 + (f*x)/2)*(a*
d - b*c)^4*((a + b)^3*(a - b)^3)^(1/2)*(4*a^2*d - 5*b^2*d + a*b*c)*(8*a*b^15 - 8*a^2*b^14 - 16*a^3*b^13 + 16*a
^4*b^12 + 8*a^5*b^11 - 8*a^6*b^10))/((a*b^10 + b^11 - a^2*b^9 - a^3*b^8)*(b^11 - 3*a^2*b^9 + 3*a^4*b^7 - a^6*b
^5)))*(a*d - b*c)^4*((a + b)^3*(a - b)^3)^(1/2)*(4*a^2*d - 5*b^2*d + a*b*c))/(b^11 - 3*a^2*b^9 + 3*a^4*b^7 - a
^6*b^5))*((a + b)^3*(a - b)^3)^(1/2)*(4*a^2*d - 5*b^2*d + a*b*c))/(b^11 - 3*a^2*b^9 + 3*a^4*b^7 - a^6*b^5) - (
(a*d - b*c)^4*((8*tan(e/2 + (f*x)/2)*(128*a^12*d^10 - 128*a^11*b*d^10 + 4*a^2*b^10*c^10 + 4*a^2*b^10*d^10 - 8*
a^3*b^9*d^10 + 28*a^4*b^8*d^10 - 48*a^5*b^7*d^10 + 28*a^6*b^6*d^10 - 8*a^7*b^5*d^10 + 8*a^8*b^4*d^10 + 192*a^9
*b^3*d^10 - 192*a^10*b^2*d^10 + 25*b^12*c^2*d^8 + 200*b^12*c^4*d^6 + 400*b^12*c^6*d^4 + 100*b^12*c^8*d^2 - 50*
a*b^11*c^2*d^8 - 480*a*b^11*c^3*d^7 - 400*a*b^11*c^4*d^6 - 1600*a*b^11*c^5*d^5 - 800*a*b^11*c^6*d^4 - 800*a*b^
11*c^7*d^3 + 40*a^2*b^10*c*d^9 - 180*a^3*b^9*c*d^9 + 320*a^4*b^8*c*d^9 - 260*a^5*b^7*c*d^9 + 200*a^6*b^6*c*d^9
- 140*a^7*b^5*c*d^9 - 1520*a^8*b^4*c*d^9 + 1520*a^9*b^3*c*d^9 + 960*a^10*b^2*c*d^9 + 435*a^2*b^10*c^2*d^8 + 9
60*a^2*b^10*c^3*d^7 + 2600*a^2*b^10*c^4*d^6 + 3200*a^2*b^10*c^5*d^5 + 2400*a^2*b^10*c^6*d^4 + 160*a^2*b^10*c^8
*d^2 - 820*a^3*b^9*c^2*d^8 - 2240*a^3*b^9*c^3*d^7 - 4800*a^3*b^9*c^4*d^6 - 4000*a^3*b^9*c^5*d^5 + 1600*a^3*b^9
*c^6*d^4 + 160*a^3*b^9*c^7*d^3 + 1055*a^4*b^8*c^2*d^8 + 3520*a^4*b^8*c^3*d^7 + 4000*a^4*b^8*c^4*d^6 - 6400*a^4
*b^8*c^5*d^5 - 2640*a^4*b^8*c^6*d^4 - 80*a^4*b^8*c^8*d^2 - 1290*a^5*b^7*c^2*d^8 - 2400*a^5*b^7*c^3*d^7 + 10800
*a^5*b^7*c^4*d^6 + 7760*a^5*b^7*c^5*d^5 - 800*a^5*b^7*c^6*d^4 + 160*a^5*b^7*c^7*d^3 + 825*a^6*b^6*c^2*d^8 - 99
20*a^6*b^6*c^3*d^7 - 11560*a^6*b^6*c^4*d^6 + 3200*a^6*b^6*c^5*d^5 + 680*a^6*b^6*c^6*d^4 + 5240*a^7*b^5*c^2*d^8
+ 10080*a^7*b^5*c^3*d^7 - 5600*a^7*b^5*c^4*d^6 - 3168*a^7*b^5*c^5*d^5 - 5240*a^8*b^4*c^2*d^8 + 5440*a^8*b^4*c
^3*d^7 + 5600*a^8*b^4*c^4*d^6 - 3080*a^9*b^3*c^2*d^8 - 5440*a^9*b^3*c^3*d^7 + 3080*a^10*b^2*c^2*d^8 - 20*a*b^1
1*c*d^9 - 40*a*b^11*c^9*d - 960*a^11*b*c*d^9))/(a*b^10 + b^11 - a^2*b^9 - a^3*b^8) - (((8*(4*a*b^17*c^5 + 4*a*
b^17*d^5 - 10*b^18*c*d^4 - 20*b^18*c^4*d - 4*a^2*b^16*c^5 - 4*a^3*b^15*c^5 + 4*a^4*b^14*c^5 + 4*a^3*b^15*d^5 -
20*a^4*b^14*d^5 - 16*a^5*b^13*d^5 + 36*a^6*b^12*d^5 + 8*a^7*b^11*d^5 - 16*a^8*b^10*d^5 - 40*b^18*c^3*d^2 + 80
*a*b^17*c^2*d^3 + 80*a*b^17*c^3*d^2 - 30*a^2*b^16*c*d^4 + 20*a^2*b^16*c^4*d + 80*a^3*b^15*c*d^4 - 20*a^3*b^15*
c^4*d + 70*a^4*b^14*c*d^4 - 140*a^5*b^13*c*d^4 - 30*a^6*b^12*c*d^4 + 60*a^7*b^11*c*d^4 - 120*a^2*b^16*c^2*d^3
+ 40*a^2*b^16*c^3*d^2 - 120*a^3*b^15*c^2*d^3 - 120*a^3*b^15*c^3*d^2 + 200*a^4*b^14*c^2*d^3 + 40*a^5*b^13*c^2*d
^3 + 40*a^5*b^13*c^3*d^2 - 80*a^6*b^12*c^2*d^3 + 20*a*b^17*c^4*d))/(a*b^14 + b^15 - a^2*b^13 - a^3*b^12) - (8*
tan(e/2 + (f*x)/2)*(a*d - b*c)^4*((a + b)^3*(a - b)^3)^(1/2)*(4*a^2*d - 5*b^2*d + a*b*c)*(8*a*b^15 - 8*a^2*b^1
4 - 16*a^3*b^13 + 16*a^4*b^12 + 8*a^5*b^11 - 8*a^6*b^10))/((a*b^10 + b^11 - a^2*b^9 - a^3*b^8)*(b^11 - 3*a^2*b
^9 + 3*a^4*b^7 - a^6*b^5)))*(a*d - b*c)^4*((a + b)^3*(a - b)^3)^(1/2)*(4*a^2*d - 5*b^2*d + a*b*c))/(b^11 - 3*a
^2*b^9 + 3*a^4*b^7 - a^6*b^5))*((a + b)^3*(a - b)^3)^(1/2)*(4*a^2*d - 5*b^2*d + a*b*c))/(b^11 - 3*a^2*b^9 + 3*
a^4*b^7 - a^6*b^5)))*(a*d - b*c)^4*((a + b)^3*(a - b)^3)^(1/2)*(4*a^2*d - 5*b^2*d + a*b*c)*2i)/(f*(b^11 - 3*a^
2*b^9 + 3*a^4*b^7 - a^6*b^5))