∫ (c+d tan(e+fx))3∕2 ____________________________

3.1240 \(\int \frac {(c+d \tan (e+f x))^{3/2}}{(a+b \tan (e+f x))^3} \, dx\)

Optimal. Leaf size=341 \[ -\frac {\left (-3 a^2 d+8 a b c+5 b^2 d\right ) \sqrt {c+d \tan (e+f x)}}{4 f \left (a^2+b^2\right )^2 (a+b \tan (e+f x))}-\frac {(b c-a d) \sqrt {c+d \tan (e+f x)}}{2 f \left (a^2+b^2\right ) (a+b \tan (e+f x))^2}+\frac {\left (-3 a^4 d^2+24 a^3 b c d-2 a^2 b^2 \left (12 c^2-13 d^2\right )-40 a b^3 c d+b^4 \left (8 c^2-3 d^2\right )\right ) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d \tan (e+f x)}}{\sqrt {b c-a d}}\right )}{4 \sqrt {b} f \left (a^2+b^2\right )^3 \sqrt {b c-a d}}-\frac {(c-i d)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d}}\right )}{f (b+i a)^3}+\frac {(c+i d)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c+i d}}\right )}{f (-b+i a)^3} \]

[Out]

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

)^(1/2)/(c+I*d)^(1/2))/(I*a-b)^3/f+1/4*(24*a^3*b*c*d-40*a*b^3*c*d-3*a^4*d^2-2*a^2*b^2*(12*c^2-13*d^2)+b^4*(8*c

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

*(-a*d+b*c)*(c+d*tan(f*x+e))^(1/2)/(a^2+b^2)/f/(a+b*tan(f*x+e))^2-1/4*(-3*a^2*d+8*a*b*c+5*b^2*d)*(c+d*tan(f*x+

e))^(1/2)/(a^2+b^2)^2/f/(a+b*tan(f*x+e))

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Rubi [A] time = 1.87, antiderivative size = 341, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 8, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.296, Rules used = {3567, 3649, 3653, 3539, 3537, 63, 208, 3634} \[ \frac {\left (-2 a^2 b^2 \left (12 c^2-13 d^2\right )+24 a^3 b c d-3 a^4 d^2-40 a b^3 c d+b^4 \left (8 c^2-3 d^2\right )\right ) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d \tan (e+f x)}}{\sqrt {b c-a d}}\right )}{4 \sqrt {b} f \left (a^2+b^2\right )^3 \sqrt {b c-a d}}-\frac {\left (-3 a^2 d+8 a b c+5 b^2 d\right ) \sqrt {c+d \tan (e+f x)}}{4 f \left (a^2+b^2\right )^2 (a+b \tan (e+f x))}-\frac {(b c-a d) \sqrt {c+d \tan (e+f x)}}{2 f \left (a^2+b^2\right ) (a+b \tan (e+f x))^2}-\frac {(c-i d)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d}}\right )}{f (b+i a)^3}+\frac {(c+i d)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c+i d}}\right )}{f (-b+i a)^3} \]

Antiderivative was successfully veriﬁed.

[In]

Int[(c + d*Tan[e + f*x])^(3/2)/(a + b*Tan[e + f*x])^3,x]

[Out]

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

h[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c + I*d]])/((I*a - b)^3*f) + ((24*a^3*b*c*d - 40*a*b^3*c*d - 3*a^4*d^2 - 2*a^2

*b^2*(12*c^2 - 13*d^2) + b^4*(8*c^2 - 3*d^2))*ArcTanh[(Sqrt[b]*Sqrt[c + d*Tan[e + f*x]])/Sqrt[b*c - a*d]])/(4*

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

e + f*x])^2) - ((8*a*b*c - 3*a^2*d + 5*b^2*d)*Sqrt[c + d*Tan[e + f*x]])/(4*(a^2 + b^2)^2*f*(a + b*Tan[e + f*x]

))

Rule 63

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub

st[Int[x^(p*(m + 1) - 1)*(c - (a*d)/b + (d*x^p)/b)^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &

& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,

b, c, d, m, n, x]

Rule 208

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

b}, x] && NegQ[a/b]

Rule 3537

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_) + (d_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Dist[(c*

d)/f, Subst[Int[(a + (b*x)/d)^m/(d^2 + c*x), x], x, d*Tan[e + f*x]], x] /; FreeQ[{a, b, c, d, e, f, m}, x] &&

NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && EqQ[c^2 + d^2, 0]

Rule 3539

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Dist[(c

+ I*d)/2, Int[(a + b*Tan[e + f*x])^m*(1 - I*Tan[e + f*x]), x], x] + Dist[(c - I*d)/2, Int[(a + b*Tan[e + f*x]

)^m*(1 + I*Tan[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0]

&& NeQ[c^2 + d^2, 0] && !IntegerQ[m]

Rule 3567

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Si

mp[((b*c - a*d)*(a + b*Tan[e + f*x])^(m + 1)*(c + d*Tan[e + f*x])^(n - 1))/(f*(m + 1)*(a^2 + b^2)), x] + Dist[

1/((m + 1)*(a^2 + b^2)), Int[(a + b*Tan[e + f*x])^(m + 1)*(c + d*Tan[e + f*x])^(n - 2)*Simp[a*c^2*(m + 1) + a*

d^2*(n - 1) + b*c*d*(m - n + 2) - (b*c^2 - 2*a*c*d - b*d^2)*(m + 1)*Tan[e + f*x] - d*(b*c - a*d)*(m + n)*Tan[e

+ f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d

^2, 0] && LtQ[m, -1] && LtQ[1, n, 2] && IntegerQ[2*m]

Rule 3634

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_.)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_.)*((A_) + (C_.)*

tan[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Dist[A/f, Subst[Int[(a + b*x)^m*(c + d*x)^n, x], x, Tan[e + f*x]], x]

/; FreeQ[{a, b, c, d, e, f, A, C, m, n}, x] && EqQ[A, C]

Rule 3649

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_)*((A_.) + (B_.)*t

an[(e_.) + (f_.)*(x_)] + (C_.)*tan[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[((A*b^2 - a*(b*B - a*C))*(a + b*T

an[e + f*x])^(m + 1)*(c + d*Tan[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*Tan[e + f*x])^(m + 1)*(c + d*Tan[e + f*x])^n*Simp[A*(a*(b*c - a*d)*(m + 1)

- b^2*d*(m + n + 2)) + (b*B - a*C)*(b*c*(m + 1) + a*d*(n + 1)) - (m + 1)*(b*c - a*d)*(A*b - a*B - b*C)*Tan[e

+ f*x] - d*(A*b^2 - a*(b*B - a*C))*(m + n + 2)*Tan[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C,

n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && LtQ[m, -1] && !(ILtQ[n, -1] && ( !I

ntegerQ[m] || (EqQ[c, 0] && NeQ[a, 0])))

Rule 3653

Int[(((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_)*((A_.) + (B_.)*tan[(e_.) + (f_.)*(x_)] + (C_.)*tan[(e_.) + (

f_.)*(x_)]^2))/((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Dist[1/(a^2 + b^2), Int[(c + d*Tan[e + f*

x])^n*Simp[b*B + a*(A - C) + (a*B - b*(A - C))*Tan[e + f*x], x], x], x] + Dist[(A*b^2 - a*b*B + a^2*C)/(a^2 +

b^2), Int[((c + d*Tan[e + f*x])^n*(1 + Tan[e + f*x]^2))/(a + b*Tan[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e,

f, A, B, C, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && !GtQ[n, 0] && !LeQ[n, -

Rubi steps

\begin {align*} \int \frac {(c+d \tan (e+f x))^{3/2}}{(a+b \tan (e+f x))^3} \, dx &=-\frac {(b c-a d) \sqrt {c+d \tan (e+f x)}}{2 \left (a^2+b^2\right ) f (a+b \tan (e+f x))^2}-\frac {\int \frac {\frac {1}{2} \left (-5 b c d-a \left (4 c^2-d^2\right )\right )-2 \left (2 a c d-b \left (c^2-d^2\right )\right ) \tan (e+f x)+\frac {3}{2} d (b c-a d) \tan ^2(e+f x)}{(a+b \tan (e+f x))^2 \sqrt {c+d \tan (e+f x)}} \, dx}{2 \left (a^2+b^2\right )}\\ &=-\frac {(b c-a d) \sqrt {c+d \tan (e+f x)}}{2 \left (a^2+b^2\right ) f (a+b \tan (e+f x))^2}-\frac {\left (8 a b c-3 a^2 d+5 b^2 d\right ) \sqrt {c+d \tan (e+f x)}}{4 \left (a^2+b^2\right )^2 f (a+b \tan (e+f x))}+\frac {\int \frac {\frac {1}{4} (b c-a d) \left (8 a^2 c^2-8 b^2 c^2+24 a b c d-5 a^2 d^2+3 b^2 d^2\right )-4 (b c-a d)^2 (a c+b d) \tan (e+f x)-\frac {1}{4} d (b c-a d) \left (8 a b c-3 a^2 d+5 b^2 d\right ) \tan ^2(e+f x)}{(a+b \tan (e+f x)) \sqrt {c+d \tan (e+f x)}} \, dx}{2 \left (a^2+b^2\right )^2 (b c-a d)}\\ &=-\frac {(b c-a d) \sqrt {c+d \tan (e+f x)}}{2 \left (a^2+b^2\right ) f (a+b \tan (e+f x))^2}-\frac {\left (8 a b c-3 a^2 d+5 b^2 d\right ) \sqrt {c+d \tan (e+f x)}}{4 \left (a^2+b^2\right )^2 f (a+b \tan (e+f x))}+\frac {\int \frac {2 (b c-a d) \left (a^3 c^2-3 a b^2 c^2+6 a^2 b c d-2 b^3 c d-a^3 d^2+3 a b^2 d^2\right )-2 (b c-a d) \left (3 a^2 b c^2-b^3 c^2-2 a^3 c d+6 a b^2 c d-3 a^2 b d^2+b^3 d^2\right ) \tan (e+f x)}{\sqrt {c+d \tan (e+f x)}} \, dx}{2 \left (a^2+b^2\right )^3 (b c-a d)}-\frac {\left (24 a^3 b c d-40 a b^3 c d-3 a^4 d^2-2 a^2 b^2 \left (12 c^2-13 d^2\right )+b^4 \left (8 c^2-3 d^2\right )\right ) \int \frac {1+\tan ^2(e+f x)}{(a+b \tan (e+f x)) \sqrt {c+d \tan (e+f x)}} \, dx}{8 \left (a^2+b^2\right )^3}\\ &=-\frac {(b c-a d) \sqrt {c+d \tan (e+f x)}}{2 \left (a^2+b^2\right ) f (a+b \tan (e+f x))^2}-\frac {\left (8 a b c-3 a^2 d+5 b^2 d\right ) \sqrt {c+d \tan (e+f x)}}{4 \left (a^2+b^2\right )^2 f (a+b \tan (e+f x))}+\frac {(c-i d)^2 \int \frac {1+i \tan (e+f x)}{\sqrt {c+d \tan (e+f x)}} \, dx}{2 (a-i b)^3}+\frac {(c+i d)^2 \int \frac {1-i \tan (e+f x)}{\sqrt {c+d \tan (e+f x)}} \, dx}{2 (a+i b)^3}-\frac {\left (24 a^3 b c d-40 a b^3 c d-3 a^4 d^2-2 a^2 b^2 \left (12 c^2-13 d^2\right )+b^4 \left (8 c^2-3 d^2\right )\right ) \operatorname {Subst}\left (\int \frac {1}{(a+b x) \sqrt {c+d x}} \, dx,x,\tan (e+f x)\right )}{8 \left (a^2+b^2\right )^3 f}\\ &=-\frac {(b c-a d) \sqrt {c+d \tan (e+f x)}}{2 \left (a^2+b^2\right ) f (a+b \tan (e+f x))^2}-\frac {\left (8 a b c-3 a^2 d+5 b^2 d\right ) \sqrt {c+d \tan (e+f x)}}{4 \left (a^2+b^2\right )^2 f (a+b \tan (e+f x))}+\frac {(c-i d)^2 \operatorname {Subst}\left (\int \frac {1}{(-1+x) \sqrt {c-i d x}} \, dx,x,i \tan (e+f x)\right )}{2 (i a+b)^3 f}-\frac {(c+i d)^2 \operatorname {Subst}\left (\int \frac {1}{(-1+x) \sqrt {c+i d x}} \, dx,x,-i \tan (e+f x)\right )}{2 (i a-b)^3 f}-\frac {\left (24 a^3 b c d-40 a b^3 c d-3 a^4 d^2-2 a^2 b^2 \left (12 c^2-13 d^2\right )+b^4 \left (8 c^2-3 d^2\right )\right ) \operatorname {Subst}\left (\int \frac {1}{a-\frac {b c}{d}+\frac {b x^2}{d}} \, dx,x,\sqrt {c+d \tan (e+f x)}\right )}{4 \left (a^2+b^2\right )^3 d f}\\ &=\frac {\left (24 a^3 b c d-40 a b^3 c d-3 a^4 d^2-2 a^2 b^2 \left (12 c^2-13 d^2\right )+b^4 \left (8 c^2-3 d^2\right )\right ) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d \tan (e+f x)}}{\sqrt {b c-a d}}\right )}{4 \sqrt {b} \left (a^2+b^2\right )^3 \sqrt {b c-a d} f}-\frac {(b c-a d) \sqrt {c+d \tan (e+f x)}}{2 \left (a^2+b^2\right ) f (a+b \tan (e+f x))^2}-\frac {\left (8 a b c-3 a^2 d+5 b^2 d\right ) \sqrt {c+d \tan (e+f x)}}{4 \left (a^2+b^2\right )^2 f (a+b \tan (e+f x))}-\frac {(c-i d)^2 \operatorname {Subst}\left (\int \frac {1}{-1-\frac {i c}{d}+\frac {i x^2}{d}} \, dx,x,\sqrt {c+d \tan (e+f x)}\right )}{(a-i b)^3 d f}-\frac {(c+i d)^2 \operatorname {Subst}\left (\int \frac {1}{-1+\frac {i c}{d}-\frac {i x^2}{d}} \, dx,x,\sqrt {c+d \tan (e+f x)}\right )}{(a+i b)^3 d f}\\ &=-\frac {(c-i d)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d}}\right )}{(i a+b)^3 f}+\frac {(c+i d)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c+i d}}\right )}{(i a-b)^3 f}+\frac {\left (24 a^3 b c d-40 a b^3 c d-3 a^4 d^2-2 a^2 b^2 \left (12 c^2-13 d^2\right )+b^4 \left (8 c^2-3 d^2\right )\right ) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d \tan (e+f x)}}{\sqrt {b c-a d}}\right )}{4 \sqrt {b} \left (a^2+b^2\right )^3 \sqrt {b c-a d} f}-\frac {(b c-a d) \sqrt {c+d \tan (e+f x)}}{2 \left (a^2+b^2\right ) f (a+b \tan (e+f x))^2}-\frac {\left (8 a b c-3 a^2 d+5 b^2 d\right ) \sqrt {c+d \tan (e+f x)}}{4 \left (a^2+b^2\right )^2 f (a+b \tan (e+f x))}\\ \end {align*}

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Mathematica [B] time = 6.34, size = 2093, normalized size = 6.14 \[ \text {Result too large to show} \]

Antiderivative was successfully veriﬁed.

[In]

Integrate[(c + d*Tan[e + f*x])^(3/2)/(a + b*Tan[e + f*x])^3,x]

[Out]

-1/2*(b^2*(c + d*Tan[e + f*x])^(5/2))/((a^2 + b^2)*(b*c - a*d)*f*(a + b*Tan[e + f*x])^2) - (-((b*d*(c + d*Tan[

e + f*x])^(3/2))/(f*(a + b*Tan[e + f*x]))) + (2*(-1/2*(b*d*(b*c - a*d)*Sqrt[c + d*Tan[e + f*x]])/(f*(a + b*Tan

[e + f*x])) - (2*(-((((I*Sqrt[c - I*d]*(b*(b*c - a*d)*((3*b^3*d*(b*c - a*d)^2)/8 + (b^3*(b*c - a*d)*(4*a*c^2 +

5*b*c*d - a*d^2))/8 + (a*b^2*(b*c - a*d)*(b*c^2 - 2*a*c*d - b*d^2))/2) + a*((b^2*(b*c - a*d)*(4*a*c^2 + 5*b*c

*d - a*d^2)*((b^2*d)/2 - a*(b*c - a*d)))/8 + (-(b*c) + (a*d)/2)*((3*a*b^2*d*(b*c - a*d)^2)/8 - (b^3*(b*c - a*d

)*(b*c^2 - 2*a*c*d - b*d^2))/2) - (d*((b^4*(b*c - a*d)*(4*a*c^2 + 5*b*c*d - a*d^2))/8 - a*((3*a*b^2*d*(b*c - a

*d)^2)/8 - (b^3*(b*c - a*d)*(b*c^2 - 2*a*c*d - b*d^2))/2)))/2) - I*(a*(b*c - a*d)*((3*b^3*d*(b*c - a*d)^2)/8 +

(b^3*(b*c - a*d)*(4*a*c^2 + 5*b*c*d - a*d^2))/8 + (a*b^2*(b*c - a*d)*(b*c^2 - 2*a*c*d - b*d^2))/2) - b*((b^2*

(b*c - a*d)*(4*a*c^2 + 5*b*c*d - a*d^2)*((b^2*d)/2 - a*(b*c - a*d)))/8 + (-(b*c) + (a*d)/2)*((3*a*b^2*d*(b*c -

a*d)^2)/8 - (b^3*(b*c - a*d)*(b*c^2 - 2*a*c*d - b*d^2))/2) - (d*((b^4*(b*c - a*d)*(4*a*c^2 + 5*b*c*d - a*d^2)

)/8 - a*((3*a*b^2*d*(b*c - a*d)^2)/8 - (b^3*(b*c - a*d)*(b*c^2 - 2*a*c*d - b*d^2))/2)))/2)))*ArcTanh[Sqrt[c +

d*Tan[e + f*x]]/Sqrt[c - I*d]])/((-c + I*d)*f) - (I*Sqrt[c + I*d]*(b*(b*c - a*d)*((3*b^3*d*(b*c - a*d)^2)/8 +

(b^3*(b*c - a*d)*(4*a*c^2 + 5*b*c*d - a*d^2))/8 + (a*b^2*(b*c - a*d)*(b*c^2 - 2*a*c*d - b*d^2))/2) + a*((b^2*(

b*c - a*d)*(4*a*c^2 + 5*b*c*d - a*d^2)*((b^2*d)/2 - a*(b*c - a*d)))/8 + (-(b*c) + (a*d)/2)*((3*a*b^2*d*(b*c -

a*d)^2)/8 - (b^3*(b*c - a*d)*(b*c^2 - 2*a*c*d - b*d^2))/2) - (d*((b^4*(b*c - a*d)*(4*a*c^2 + 5*b*c*d - a*d^2))

/8 - a*((3*a*b^2*d*(b*c - a*d)^2)/8 - (b^3*(b*c - a*d)*(b*c^2 - 2*a*c*d - b*d^2))/2)))/2) + I*(a*(b*c - a*d)*(

(3*b^3*d*(b*c - a*d)^2)/8 + (b^3*(b*c - a*d)*(4*a*c^2 + 5*b*c*d - a*d^2))/8 + (a*b^2*(b*c - a*d)*(b*c^2 - 2*a*

c*d - b*d^2))/2) - b*((b^2*(b*c - a*d)*(4*a*c^2 + 5*b*c*d - a*d^2)*((b^2*d)/2 - a*(b*c - a*d)))/8 + (-(b*c) +

(a*d)/2)*((3*a*b^2*d*(b*c - a*d)^2)/8 - (b^3*(b*c - a*d)*(b*c^2 - 2*a*c*d - b*d^2))/2) - (d*((b^4*(b*c - a*d)*

(4*a*c^2 + 5*b*c*d - a*d^2))/8 - a*((3*a*b^2*d*(b*c - a*d)^2)/8 - (b^3*(b*c - a*d)*(b*c^2 - 2*a*c*d - b*d^2))/

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

(a*b*(b*c - a*d)*((3*b^3*d*(b*c - a*d)^2)/8 + (b^3*(b*c - a*d)*(4*a*c^2 + 5*b*c*d - a*d^2))/8 + (a*b^2*(b*c -

a*d)*(b*c^2 - 2*a*c*d - b*d^2))/2)) + (a^2*d*((b^4*(b*c - a*d)*(4*a*c^2 + 5*b*c*d - a*d^2))/8 - a*((3*a*b^2*d*

(b*c - a*d)^2)/8 - (b^3*(b*c - a*d)*(b*c^2 - 2*a*c*d - b*d^2))/2)))/2 + b^2*((b^2*(b*c - a*d)*(4*a*c^2 + 5*b*c

*d - a*d^2)*((b^2*d)/2 - a*(b*c - a*d)))/8 + (-(b*c) + (a*d)/2)*((3*a*b^2*d*(b*c - a*d)^2)/8 - (b^3*(b*c - a*d

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

b^2)*(-(b*c) + a*d)*f))/((a^2 + b^2)*(b*c - a*d))) - (((b^4*(b*c - a*d)*(4*a*c^2 + 5*b*c*d - a*d^2))/8 - a*((

3*a*b^2*d*(b*c - a*d)^2)/8 - (b^3*(b*c - a*d)*(b*c^2 - 2*a*c*d - b*d^2))/2))*Sqrt[c + d*Tan[e + f*x]])/((a^2 +

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

________________________________________________________________________________________

fricas [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((c+d*tan(f*x+e))^(3/2)/(a+b*tan(f*x+e))^3,x, algorithm="fricas")

[Out]

Timed out

________________________________________________________________________________________

giac [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((c+d*tan(f*x+e))^(3/2)/(a+b*tan(f*x+e))^3,x, algorithm="giac")

[Out]

Timed out

________________________________________________________________________________________

maple [B] time = 0.49, size = 4715, normalized size = 13.83 \[ \text {output too large to display} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int((c+d*tan(f*x+e))^(3/2)/(a+b*tan(f*x+e))^3,x)

[Out]

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

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

/2)-(2*(c^2+d^2)^(1/2)+2*c)^(1/2))/(2*(c^2+d^2)^(1/2)-2*c)^(1/2))*(c^2+d^2)^(1/2)*b^3*c+3/4/f/(a^2+b^2)^3*ln(d

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

2)^(1/2)+2*c)^(1/2)*a^2*b-3/4/f/(a^2+b^2)^3*ln(d*tan(f*x+e)+c+(c+d*tan(f*x+e))^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(

1/2)+(c^2+d^2)^(1/2))*(c^2+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a^2*b-3/f/(a^2+b^2)^3/(2*(c^2+d^2)^(1/2)-2

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

*c^2+3/2/f/(a^2+b^2)^3*ln(d*tan(f*x+e)+c+(c+d*tan(f*x+e))^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)+(c^2+d^2)^(1/2))

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

)^(1/2)+(2*(c^2+d^2)^(1/2)+2*c)^(1/2))/(2*(c^2+d^2)^(1/2)-2*c)^(1/2))*(c^2+d^2)^(1/2)*b^3*c+6/f/(a^2+b^2)^3/((

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

)^(1/2)-2*c)^(1/2)*arctan((2*(c+d*tan(f*x+e))^(1/2)-(2*(c^2+d^2)^(1/2)+2*c)^(1/2))/(2*(c^2+d^2)^(1/2)-2*c)^(1/

2))*a^2*b*c^2-3/4/f*d/(a^2+b^2)^3*ln(d*tan(f*x+e)+c-(c+d*tan(f*x+e))^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)+(c^2+

d^2)^(1/2))*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a*b^2+3/f*d^2/(a^2+b^2)^3/(2*(c^2+d^2)^(1/2)-2*c)^(1/2)*arctan((2*(c

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

2*(c^2+d^2)^(1/2)-2*c)^(1/2)*arctan((2*(c+d*tan(f*x+e))^(1/2)+(2*(c^2+d^2)^(1/2)+2*c)^(1/2))/(2*(c^2+d^2)^(1/2

)-2*c)^(1/2))*a^2*b+1/4/f/d/(a^2+b^2)^3*ln(d*tan(f*x+e)+c+(c+d*tan(f*x+e))^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)

+(c^2+d^2)^(1/2))*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a^3*c^2-1/f*d/(a^2+b^2)^3/(2*(c^2+d^2)^(1/2)-2*c)^(1/2)*arctan

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

/4/f/d/(a^2+b^2)^3*ln(d*tan(f*x+e)+c-(c+d*tan(f*x+e))^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)+(c^2+d^2)^(1/2))*(2*

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

(1/2)+(2*(c^2+d^2)^(1/2)+2*c)^(1/2))/(2*(c^2+d^2)^(1/2)-2*c)^(1/2))*(c^2+d^2)^(1/2)*a^3+2/f*d/(a^2+b^2)^3/(2*(

c^2+d^2)^(1/2)-2*c)^(1/2)*arctan((2*(c+d*tan(f*x+e))^(1/2)+(2*(c^2+d^2)^(1/2)+2*c)^(1/2))/(2*(c^2+d^2)^(1/2)-2

*c)^(1/2))*a^3*c+3/4/f*d/(a^2+b^2)^3*ln(d*tan(f*x+e)+c+(c+d*tan(f*x+e))^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)+(c

^2+d^2)^(1/2))*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a*b^2+3/4/f*d^2/(a^2+b^2)^3/((a*d-b*c)*b)^(1/2)*arctan((c+d*tan(f

*x+e))^(1/2)*b/((a*d-b*c)*b)^(1/2))*b^4-5/4/f*d^2/(a^2+b^2)^3/(tan(f*x+e)*b*d+d*a)^2*(c+d*tan(f*x+e))^(3/2)*b^

5-2/f/(a^2+b^2)^3/((a*d-b*c)*b)^(1/2)*arctan((c+d*tan(f*x+e))^(1/2)*b/((a*d-b*c)*b)^(1/2))*b^4*c^2+1/f/(a^2+b^

2)^3/(2*(c^2+d^2)^(1/2)-2*c)^(1/2)*arctan((2*(c+d*tan(f*x+e))^(1/2)-(2*(c^2+d^2)^(1/2)+2*c)^(1/2))/(2*(c^2+d^2

)^(1/2)-2*c)^(1/2))*b^3*c^2+3/4/f/d/(a^2+b^2)^3*ln(d*tan(f*x+e)+c+(c+d*tan(f*x+e))^(1/2)*(2*(c^2+d^2)^(1/2)+2*

c)^(1/2)+(c^2+d^2)^(1/2))*(c^2+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a*b^2*c-3/4/f/d/(a^2+b^2)^3*ln(d*tan(f

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

2)+2*c)^(1/2)*a*b^2*c+1/4/f/d/(a^2+b^2)^3*ln(d*tan(f*x+e)+c-(c+d*tan(f*x+e))^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/

2)+(c^2+d^2)^(1/2))*(c^2+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a^3*c-6/f*d/(a^2+b^2)^3/(2*(c^2+d^2)^(1/2)-2

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

*c+3/4/f/d/(a^2+b^2)^3*ln(d*tan(f*x+e)+c-(c+d*tan(f*x+e))^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)+(c^2+d^2)^(1/2))

*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a*b^2*c^2-6/f*d/(a^2+b^2)^3/((a*d-b*c)*b)^(1/2)*arctan((c+d*tan(f*x+e))^(1/2)*b

/((a*d-b*c)*b)^(1/2))*a^3*b*c+10/f*d/(a^2+b^2)^3/((a*d-b*c)*b)^(1/2)*arctan((c+d*tan(f*x+e))^(1/2)*b/((a*d-b*c

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

3/(tan(f*x+e)*b*d+d*a)^2*(c+d*tan(f*x+e))^(3/2)*a*c*b^4-5/2/f*d^2/(a^2+b^2)^3/(tan(f*x+e)*b*d+d*a)^2*(c+d*tan(

f*x+e))^(1/2)*a^2*b^3*c+2/f*d/(a^2+b^2)^3/(tan(f*x+e)*b*d+d*a)^2*(c+d*tan(f*x+e))^(1/2)*a*b^4*c^2-13/4/f*d^2/(

a^2+b^2)^3/(tan(f*x+e)*b*d+d*a)^2*(c+d*tan(f*x+e))^(1/2)*a^4*b*c-6/f*d/(a^2+b^2)^3/(2*(c^2+d^2)^(1/2)-2*c)^(1/

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

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

*(c^2+d^2)^(1/2)-2*c)^(1/2))*(c^2+d^2)^(1/2)*a*b^2+3/f/(a^2+b^2)^3/(2*(c^2+d^2)^(1/2)-2*c)^(1/2)*arctan((2*(c+

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

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

c^2+d^2)^(1/2)-2*c)^(1/2))*(c^2+d^2)^(1/2)*a^2*b*c-3/4/f/d/(a^2+b^2)^3*ln(d*tan(f*x+e)+c+(c+d*tan(f*x+e))^(1/2

)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)+(c^2+d^2)^(1/2))*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a*b^2*c^2+3/f*d/(a^2+b^2)^3/(2*

(c^2+d^2)^(1/2)-2*c)^(1/2)*arctan((2*(c+d*tan(f*x+e))^(1/2)+(2*(c^2+d^2)^(1/2)+2*c)^(1/2))/(2*(c^2+d^2)^(1/2)-

2*c)^(1/2))*(c^2+d^2)^(1/2)*a*b^2-1/4/f/d/(a^2+b^2)^3*ln(d*tan(f*x+e)+c+(c+d*tan(f*x+e))^(1/2)*(2*(c^2+d^2)^(1

/2)+2*c)^(1/2)+(c^2+d^2)^(1/2))*(c^2+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a^3*c+2/f*d/(a^2+b^2)^3/(tan(f*x

+e)*b*d+d*a)^2*(c+d*tan(f*x+e))^(1/2)*a^3*b^2*c^2+3/4/f*d^2/(a^2+b^2)^3/(tan(f*x+e)*b*d+d*a)^2*(c+d*tan(f*x+e)

)^(3/2)*a^4*b-1/2/f*d^2/(a^2+b^2)^3/(tan(f*x+e)*b*d+d*a)^2*(c+d*tan(f*x+e))^(3/2)*a^2*b^3+1/2/f*d^3/(a^2+b^2)^

3/(tan(f*x+e)*b*d+d*a)^2*(c+d*tan(f*x+e))^(1/2)*a^3*b^2-3/4/f*d^3/(a^2+b^2)^3/(tan(f*x+e)*b*d+d*a)^2*(c+d*tan(

f*x+e))^(1/2)*a*b^4+3/4/f*d^2/(a^2+b^2)^3/(tan(f*x+e)*b*d+d*a)^2*(c+d*tan(f*x+e))^(1/2)*b^5*c-13/2/f*d^2/(a^2+

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

c^2+d^2)^(1/2)-2*c)^(1/2)*arctan((2*(c+d*tan(f*x+e))^(1/2)-(2*(c^2+d^2)^(1/2)+2*c)^(1/2))/(2*(c^2+d^2)^(1/2)-2

*c)^(1/2))*a^3*c-1/4/f*d/(a^2+b^2)^3*ln(d*tan(f*x+e)+c+(c+d*tan(f*x+e))^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)+(c

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

*(c^2+d^2)^(1/2)+2*c)^(1/2)+(c^2+d^2)^(1/2))*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*b^3*c-1/4/f/(a^2+b^2)^3*ln(d*tan(f*

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

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

+d^2)^(1/2))*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*b^3*c+1/4/f/(a^2+b^2)^3*ln(d*tan(f*x+e)+c+(c+d*tan(f*x+e))^(1/2)*(2

*(c^2+d^2)^(1/2)+2*c)^(1/2)+(c^2+d^2)^(1/2))*(c^2+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*b^3+1/f/(a^2+b^2)^3

/(2*(c^2+d^2)^(1/2)-2*c)^(1/2)*arctan((2*(c+d*tan(f*x+e))^(1/2)+(2*(c^2+d^2)^(1/2)+2*c)^(1/2))/(2*(c^2+d^2)^(1

/2)-2*c)^(1/2))*b^3*c^2+5/4/f*d^3/(a^2+b^2)^3/(tan(f*x+e)*b*d+d*a)^2*(c+d*tan(f*x+e))^(1/2)*a^5+3/4/f*d^2/(a^2

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

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

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

*c)^(1/2))/(2*(c^2+d^2)^(1/2)-2*c)^(1/2))*b^3-1/f*d^2/(a^2+b^2)^3/(2*(c^2+d^2)^(1/2)-2*c)^(1/2)*arctan((2*(c+d

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

________________________________________________________________________________________

maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: ValueError} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((c+d*tan(f*x+e))^(3/2)/(a+b*tan(f*x+e))^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a

ssume' command before evaluation *may* help (example of legal syntax is 'assume(a*d-b*c>0)', see `assume?` for

more details)Is a*d-b*c positive or negative?

________________________________________________________________________________________

mupad [B] time = 36.70, size = 90146, normalized size = 264.36 \[ \text {result too large to display} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int((c + d*tan(e + f*x))^(3/2)/(a + b*tan(e + f*x))^3,x)

[Out]

(((c + d*tan(e + f*x))^(1/2)*(5*a^3*d^3 - 3*a*b^2*d^3 + 3*b^3*c*d^2 + 8*a*b^2*c^2*d - 13*a^2*b*c*d^2))/(4*(a^4

+ b^4 + 2*a^2*b^2)) - (b*(c + d*tan(e + f*x))^(3/2)*(5*b^2*d^2 - 3*a^2*d^2 + 8*a*b*c*d))/(4*(a^4 + b^4 + 2*a^

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

- 2*a*b*c*d*f) - atan(((((1036*a*b^15*d^15*f^2 - 36*a^15*b*d^15*f^2 - 604*b^16*c*d^14*f^2 - 8988*a^3*b^13*d^15

*f^2 + 6044*a^5*b^11*d^15*f^2 + 34388*a^7*b^9*d^15*f^2 + 10596*a^9*b^7*d^15*f^2 - 6676*a^11*b^5*d^15*f^2 + 101

2*a^13*b^3*d^15*f^2 + 932*b^16*c^3*d^12*f^2 + 1344*b^16*c^5*d^10*f^2 - 192*b^16*c^7*d^8*f^2 - 30836*a^2*b^14*c

^3*d^12*f^2 - 48000*a^2*b^14*c^5*d^10*f^2 + 5248*a^2*b^14*c^7*d^8*f^2 + 95076*a^3*b^13*c^2*d^13*f^2 + 57600*a^

3*b^13*c^4*d^11*f^2 - 46464*a^3*b^13*c^6*d^9*f^2 + 17172*a^4*b^12*c^3*d^12*f^2 + 69696*a^4*b^12*c^5*d^10*f^2 -

5696*a^4*b^12*c^7*d^8*f^2 + 47004*a^5*b^11*c^2*d^13*f^2 + 10944*a^5*b^11*c^4*d^11*f^2 - 30016*a^5*b^11*c^6*d^

9*f^2 + 85404*a^6*b^10*c^3*d^12*f^2 + 169344*a^6*b^10*c^5*d^10*f^2 - 17664*a^6*b^10*c^7*d^8*f^2 - 171180*a^7*b

^9*c^2*d^13*f^2 - 119808*a^7*b^9*c^4*d^11*f^2 + 85760*a^7*b^9*c^6*d^9*f^2 - 4308*a^8*b^8*c^3*d^12*f^2 - 49728*

a^8*b^8*c^5*d^10*f^2 + 3776*a^8*b^8*c^7*d^8*f^2 - 50972*a^9*b^7*c^2*d^13*f^2 - 24768*a^9*b^7*c^4*d^11*f^2 + 36

800*a^9*b^7*c^6*d^9*f^2 - 35356*a^10*b^6*c^3*d^12*f^2 - 80512*a^10*b^6*c^5*d^10*f^2 + 10368*a^10*b^6*c^7*d^8*f

^2 + 55916*a^11*b^5*c^2*d^13*f^2 + 37632*a^11*b^5*c^4*d^11*f^2 - 24960*a^11*b^5*c^6*d^9*f^2 + 6428*a^12*b^4*c^

3*d^12*f^2 + 19648*a^12*b^4*c^5*d^10*f^2 + 64*a^12*b^4*c^7*d^8*f^2 - 4876*a^13*b^3*c^2*d^13*f^2 - 6080*a^13*b^

3*c^4*d^11*f^2 - 192*a^13*b^3*c^6*d^9*f^2 + 1012*a^14*b^2*c^3*d^12*f^2 + 128*a^14*b^2*c^5*d^10*f^2 - 11380*a*b

^15*c^2*d^13*f^2 - 4672*a*b^15*c^4*d^11*f^2 + 7744*a*b^15*c^6*d^9*f^2 + 22412*a^2*b^14*c*d^14*f^2 - 58220*a^4*

b^12*c*d^14*f^2 - 101604*a^6*b^10*c*d^14*f^2 + 49196*a^8*b^8*c*d^14*f^2 + 55524*a^10*b^6*c*d^14*f^2 - 13156*a^

12*b^4*c*d^14*f^2 + 884*a^14*b^2*c*d^14*f^2 - 36*a^15*b*c^2*d^13*f^2)/(2*(a^16*f^5 + b^16*f^5 + 8*a^2*b^14*f^5

+ 28*a^4*b^12*f^5 + 56*a^6*b^10*f^5 + 70*a^8*b^8*f^5 + 56*a^10*b^6*f^5 + 28*a^12*b^4*f^5 + 8*a^14*b^2*f^5)) +

(((8448*a^4*b^18*d^12*f^4 - 640*a^2*b^20*d^12*f^4 - 384*b^22*d^12*f^4 + 44544*a^6*b^16*d^12*f^4 + 102144*a^8*

b^14*d^12*f^4 + 134400*a^10*b^12*d^12*f^4 + 107520*a^12*b^10*d^12*f^4 + 50688*a^14*b^8*d^12*f^4 + 11904*a^16*b

^6*d^12*f^4 + 384*a^18*b^4*d^12*f^4 - 256*a^20*b^2*d^12*f^4 + 384*b^22*c^2*d^10*f^4 + 768*b^22*c^4*d^8*f^4 + 4

224*a^2*b^20*c^2*d^10*f^4 + 4864*a^2*b^20*c^4*d^8*f^4 - 27136*a^3*b^19*c^3*d^9*f^4 + 19712*a^4*b^18*c^2*d^10*f

^4 + 11264*a^4*b^18*c^4*d^8*f^4 - 88064*a^5*b^17*c^3*d^9*f^4 + 51712*a^6*b^16*c^2*d^10*f^4 + 7168*a^6*b^16*c^4

*d^8*f^4 - 157696*a^7*b^15*c^3*d^9*f^4 + 84224*a^8*b^14*c^2*d^10*f^4 - 17920*a^8*b^14*c^4*d^8*f^4 - 164864*a^9

*b^13*c^3*d^9*f^4 + 87808*a^10*b^12*c^2*d^10*f^4 - 46592*a^10*b^12*c^4*d^8*f^4 - 93184*a^11*b^11*c^3*d^9*f^4 +

57344*a^12*b^10*c^2*d^10*f^4 - 50176*a^12*b^10*c^4*d^8*f^4 - 14336*a^13*b^9*c^3*d^9*f^4 + 20992*a^14*b^8*c^2*

d^10*f^4 - 29696*a^14*b^8*c^4*d^8*f^4 + 14336*a^15*b^7*c^3*d^9*f^4 + 2432*a^16*b^6*c^2*d^10*f^4 - 9472*a^16*b^

6*c^4*d^8*f^4 + 8704*a^17*b^5*c^3*d^9*f^4 - 896*a^18*b^4*c^2*d^10*f^4 - 1280*a^18*b^4*c^4*d^8*f^4 + 1536*a^19*

b^3*c^3*d^9*f^4 - 256*a^20*b^2*c^2*d^10*f^4 - 3584*a*b^21*c*d^11*f^4 - 3584*a*b^21*c^3*d^9*f^4 - 27136*a^3*b^1

9*c*d^11*f^4 - 88064*a^5*b^17*c*d^11*f^4 - 157696*a^7*b^15*c*d^11*f^4 - 164864*a^9*b^13*c*d^11*f^4 - 93184*a^1

1*b^11*c*d^11*f^4 - 14336*a^13*b^9*c*d^11*f^4 + 14336*a^15*b^7*c*d^11*f^4 + 8704*a^17*b^5*c*d^11*f^4 + 1536*a^

19*b^3*c*d^11*f^4)/(2*(a^16*f^5 + b^16*f^5 + 8*a^2*b^14*f^5 + 28*a^4*b^12*f^5 + 56*a^6*b^10*f^5 + 70*a^8*b^8*f

^5 + 56*a^10*b^6*f^5 + 28*a^12*b^4*f^5 + 8*a^14*b^2*f^5)) + ((c + d*tan(e + f*x))^(1/2)*(-(((8*a^6*c^3*f^2 - 8

*b^6*c^3*f^2 - 48*a*b^5*d^3*f^2 - 48*a^5*b*d^3*f^2 - 24*a^6*c*d^2*f^2 + 24*b^6*c*d^2*f^2 + 120*a^2*b^4*c^3*f^2

- 120*a^4*b^2*c^3*f^2 + 160*a^3*b^3*d^3*f^2 + 144*a*b^5*c^2*d*f^2 + 144*a^5*b*c^2*d*f^2 - 360*a^2*b^4*c*d^2*f

^2 - 480*a^3*b^3*c^2*d*f^2 + 360*a^4*b^2*c*d^2*f^2)^2/4 - (c^6 + d^6 + 3*c^2*d^4 + 3*c^4*d^2)*(16*a^12*f^4 + 1

6*b^12*f^4 + 96*a^2*b^10*f^4 + 240*a^4*b^8*f^4 + 320*a^6*b^6*f^4 + 240*a^8*b^4*f^4 + 96*a^10*b^2*f^4))^(1/2) +

4*a^6*c^3*f^2 - 4*b^6*c^3*f^2 - 24*a*b^5*d^3*f^2 - 24*a^5*b*d^3*f^2 - 12*a^6*c*d^2*f^2 + 12*b^6*c*d^2*f^2 + 6

0*a^2*b^4*c^3*f^2 - 60*a^4*b^2*c^3*f^2 + 80*a^3*b^3*d^3*f^2 + 72*a*b^5*c^2*d*f^2 + 72*a^5*b*c^2*d*f^2 - 180*a^

2*b^4*c*d^2*f^2 - 240*a^3*b^3*c^2*d*f^2 + 180*a^4*b^2*c*d^2*f^2)/(16*(a^12*f^4 + b^12*f^4 + 6*a^2*b^10*f^4 + 1

5*a^4*b^8*f^4 + 20*a^6*b^6*f^4 + 15*a^8*b^4*f^4 + 6*a^10*b^2*f^4)))^(1/2)*(512*b^25*d^10*f^4 + 4608*a^2*b^23*d

^10*f^4 + 17920*a^4*b^21*d^10*f^4 + 38400*a^6*b^19*d^10*f^4 + 46080*a^8*b^17*d^10*f^4 + 21504*a^10*b^15*d^10*f

^4 - 21504*a^12*b^13*d^10*f^4 - 46080*a^14*b^11*d^10*f^4 - 38400*a^16*b^9*d^10*f^4 - 17920*a^18*b^7*d^10*f^4 -

4608*a^20*b^5*d^10*f^4 - 512*a^22*b^3*d^10*f^4 + 768*b^25*c^2*d^8*f^4 + 7424*a^2*b^23*c^2*d^8*f^4 + 32000*a^4

*b^21*c^2*d^8*f^4 + 80640*a^6*b^19*c^2*d^8*f^4 + 130560*a^8*b^17*c^2*d^8*f^4 + 139776*a^10*b^15*c^2*d^8*f^4 +

96768*a^12*b^13*c^2*d^8*f^4 + 38400*a^14*b^11*c^2*d^8*f^4 + 3840*a^16*b^9*c^2*d^8*f^4 - 3840*a^18*b^7*c^2*d^8*

f^4 - 1792*a^20*b^5*c^2*d^8*f^4 - 256*a^22*b^3*c^2*d^8*f^4 + 256*a*b^24*c*d^9*f^4 + 2816*a^3*b^22*c*d^9*f^4 +

14080*a^5*b^20*c*d^9*f^4 + 42240*a^7*b^18*c*d^9*f^4 + 84480*a^9*b^16*c*d^9*f^4 + 118272*a^11*b^14*c*d^9*f^4 +

118272*a^13*b^12*c*d^9*f^4 + 84480*a^15*b^10*c*d^9*f^4 + 42240*a^17*b^8*c*d^9*f^4 + 14080*a^19*b^6*c*d^9*f^4 +

2816*a^21*b^4*c*d^9*f^4 + 256*a^23*b^2*c*d^9*f^4))/(a^16*f^4 + b^16*f^4 + 8*a^2*b^14*f^4 + 28*a^4*b^12*f^4 +

56*a^6*b^10*f^4 + 70*a^8*b^8*f^4 + 56*a^10*b^6*f^4 + 28*a^12*b^4*f^4 + 8*a^14*b^2*f^4))*(-(((8*a^6*c^3*f^2 - 8

*b^6*c^3*f^2 - 48*a*b^5*d^3*f^2 - 48*a^5*b*d^3*f^2 - 24*a^6*c*d^2*f^2 + 24*b^6*c*d^2*f^2 + 120*a^2*b^4*c^3*f^2

- 120*a^4*b^2*c^3*f^2 + 160*a^3*b^3*d^3*f^2 + 144*a*b^5*c^2*d*f^2 + 144*a^5*b*c^2*d*f^2 - 360*a^2*b^4*c*d^2*f

^2 - 480*a^3*b^3*c^2*d*f^2 + 360*a^4*b^2*c*d^2*f^2)^2/4 - (c^6 + d^6 + 3*c^2*d^4 + 3*c^4*d^2)*(16*a^12*f^4 + 1

6*b^12*f^4 + 96*a^2*b^10*f^4 + 240*a^4*b^8*f^4 + 320*a^6*b^6*f^4 + 240*a^8*b^4*f^4 + 96*a^10*b^2*f^4))^(1/2) +

4*a^6*c^3*f^2 - 4*b^6*c^3*f^2 - 24*a*b^5*d^3*f^2 - 24*a^5*b*d^3*f^2 - 12*a^6*c*d^2*f^2 + 12*b^6*c*d^2*f^2 + 6

0*a^2*b^4*c^3*f^2 - 60*a^4*b^2*c^3*f^2 + 80*a^3*b^3*d^3*f^2 + 72*a*b^5*c^2*d*f^2 + 72*a^5*b*c^2*d*f^2 - 180*a^

2*b^4*c*d^2*f^2 - 240*a^3*b^3*c^2*d*f^2 + 180*a^4*b^2*c*d^2*f^2)/(16*(a^12*f^4 + b^12*f^4 + 6*a^2*b^10*f^4 + 1

5*a^4*b^8*f^4 + 20*a^6*b^6*f^4 + 15*a^8*b^4*f^4 + 6*a^10*b^2*f^4)))^(1/2) - ((c + d*tan(e + f*x))^(1/2)*(1544*

a*b^18*d^13*f^2 - 668*b^19*c*d^12*f^2 + 64*a^3*b^16*d^13*f^2 - 7456*a^5*b^14*d^13*f^2 - 576*a^7*b^12*d^13*f^2

+ 19504*a^9*b^10*d^13*f^2 + 18880*a^11*b^8*d^13*f^2 + 3808*a^13*b^6*d^13*f^2 - 960*a^15*b^4*d^13*f^2 + 8*a^17*

b^2*d^13*f^2 - 1088*b^19*c^3*d^10*f^2 + 576*b^19*c^5*d^8*f^2 + 9600*a^2*b^17*c^3*d^10*f^2 - 3584*a^2*b^17*c^5*

d^8*f^2 + 12800*a^3*b^16*c^2*d^11*f^2 - 7680*a^3*b^16*c^4*d^9*f^2 + 43392*a^4*b^15*c^3*d^10*f^2 - 13056*a^4*b^

15*c^5*d^8*f^2 + 43264*a^5*b^14*c^2*d^11*f^2 + 34560*a^5*b^14*c^4*d^9*f^2 + 63360*a^6*b^13*c^3*d^10*f^2 - 7680

*a^6*b^13*c^5*d^8*f^2 + 19968*a^7*b^12*c^2*d^11*f^2 + 66048*a^7*b^12*c^4*d^9*f^2 + 45312*a^8*b^11*c^3*d^10*f^2

+ 8064*a^8*b^11*c^5*d^8*f^2 - 57728*a^9*b^10*c^2*d^11*f^2 + 35968*a^9*b^10*c^4*d^9*f^2 + 18560*a^10*b^9*c^3*d

^10*f^2 + 5632*a^10*b^9*c^5*d^8*f^2 - 74240*a^11*b^8*c^2*d^11*f^2 + 9728*a^11*b^8*c^4*d^9*f^2 - 1408*a^12*b^7*

c^3*d^10*f^2 - 3840*a^12*b^7*c^5*d^8*f^2 - 26368*a^13*b^6*c^2*d^11*f^2 + 11008*a^13*b^6*c^4*d^9*f^2 - 9600*a^1

4*b^5*c^3*d^10*f^2 - 2560*a^14*b^5*c^5*d^8*f^2 + 512*a^15*b^4*c^2*d^11*f^2 + 5632*a^15*b^4*c^4*d^9*f^2 - 4288*

a^16*b^3*c^3*d^10*f^2 + 64*a^16*b^3*c^5*d^8*f^2 + 960*a^17*b^2*c^2*d^11*f^2 - 64*a^17*b^2*c^4*d^9*f^2 - 36*a^1

8*b*c*d^12*f^2 - 1088*a*b^18*c^2*d^11*f^2 - 7744*a*b^18*c^4*d^9*f^2 + 8828*a^2*b^17*c*d^12*f^2 + 14416*a^4*b^1

5*c*d^12*f^2 - 33936*a^6*b^13*c*d^12*f^2 - 79368*a^8*b^11*c*d^12*f^2 - 36216*a^10*b^9*c*d^12*f^2 + 16016*a^12*

b^7*c*d^12*f^2 + 12208*a^14*b^5*c*d^12*f^2 + 452*a^16*b^3*c*d^12*f^2))/(a^16*f^4 + b^16*f^4 + 8*a^2*b^14*f^4 +

28*a^4*b^12*f^4 + 56*a^6*b^10*f^4 + 70*a^8*b^8*f^4 + 56*a^10*b^6*f^4 + 28*a^12*b^4*f^4 + 8*a^14*b^2*f^4))*(-(

((8*a^6*c^3*f^2 - 8*b^6*c^3*f^2 - 48*a*b^5*d^3*f^2 - 48*a^5*b*d^3*f^2 - 24*a^6*c*d^2*f^2 + 24*b^6*c*d^2*f^2 +

120*a^2*b^4*c^3*f^2 - 120*a^4*b^2*c^3*f^2 + 160*a^3*b^3*d^3*f^2 + 144*a*b^5*c^2*d*f^2 + 144*a^5*b*c^2*d*f^2 -

360*a^2*b^4*c*d^2*f^2 - 480*a^3*b^3*c^2*d*f^2 + 360*a^4*b^2*c*d^2*f^2)^2/4 - (c^6 + d^6 + 3*c^2*d^4 + 3*c^4*d^

2)*(16*a^12*f^4 + 16*b^12*f^4 + 96*a^2*b^10*f^4 + 240*a^4*b^8*f^4 + 320*a^6*b^6*f^4 + 240*a^8*b^4*f^4 + 96*a^1

0*b^2*f^4))^(1/2) + 4*a^6*c^3*f^2 - 4*b^6*c^3*f^2 - 24*a*b^5*d^3*f^2 - 24*a^5*b*d^3*f^2 - 12*a^6*c*d^2*f^2 + 1

2*b^6*c*d^2*f^2 + 60*a^2*b^4*c^3*f^2 - 60*a^4*b^2*c^3*f^2 + 80*a^3*b^3*d^3*f^2 + 72*a*b^5*c^2*d*f^2 + 72*a^5*b

*c^2*d*f^2 - 180*a^2*b^4*c*d^2*f^2 - 240*a^3*b^3*c^2*d*f^2 + 180*a^4*b^2*c*d^2*f^2)/(16*(a^12*f^4 + b^12*f^4 +

6*a^2*b^10*f^4 + 15*a^4*b^8*f^4 + 20*a^6*b^6*f^4 + 15*a^8*b^4*f^4 + 6*a^10*b^2*f^4)))^(1/2))*(-(((8*a^6*c^3*f

^2 - 8*b^6*c^3*f^2 - 48*a*b^5*d^3*f^2 - 48*a^5*b*d^3*f^2 - 24*a^6*c*d^2*f^2 + 24*b^6*c*d^2*f^2 + 120*a^2*b^4*c

^3*f^2 - 120*a^4*b^2*c^3*f^2 + 160*a^3*b^3*d^3*f^2 + 144*a*b^5*c^2*d*f^2 + 144*a^5*b*c^2*d*f^2 - 360*a^2*b^4*c

*d^2*f^2 - 480*a^3*b^3*c^2*d*f^2 + 360*a^4*b^2*c*d^2*f^2)^2/4 - (c^6 + d^6 + 3*c^2*d^4 + 3*c^4*d^2)*(16*a^12*f

^4 + 16*b^12*f^4 + 96*a^2*b^10*f^4 + 240*a^4*b^8*f^4 + 320*a^6*b^6*f^4 + 240*a^8*b^4*f^4 + 96*a^10*b^2*f^4))^(

1/2) + 4*a^6*c^3*f^2 - 4*b^6*c^3*f^2 - 24*a*b^5*d^3*f^2 - 24*a^5*b*d^3*f^2 - 12*a^6*c*d^2*f^2 + 12*b^6*c*d^2*f

^2 + 60*a^2*b^4*c^3*f^2 - 60*a^4*b^2*c^3*f^2 + 80*a^3*b^3*d^3*f^2 + 72*a*b^5*c^2*d*f^2 + 72*a^5*b*c^2*d*f^2 -

180*a^2*b^4*c*d^2*f^2 - 240*a^3*b^3*c^2*d*f^2 + 180*a^4*b^2*c*d^2*f^2)/(16*(a^12*f^4 + b^12*f^4 + 6*a^2*b^10*f

^4 + 15*a^4*b^8*f^4 + 20*a^6*b^6*f^4 + 15*a^8*b^4*f^4 + 6*a^10*b^2*f^4)))^(1/2) + ((c + d*tan(e + f*x))^(1/2)*

(41*b^13*d^16 + 9*a^12*b*d^16 - 82*a^2*b^11*d^16 + 1831*a^4*b^9*d^16 - 4348*a^6*b^7*d^16 + 1671*a^8*b^5*d^16 -

210*a^10*b^3*d^16 + 26*b^13*c^2*d^14 + 553*b^13*c^4*d^12 - 304*b^13*c^6*d^10 + 96*b^13*c^8*d^8 - 2992*a*b^12*

c^3*d^13 + 5872*a*b^12*c^5*d^11 - 1664*a*b^12*c^7*d^9 - 6304*a^3*b^10*c*d^15 + 28672*a^5*b^8*c*d^15 - 24896*a^

7*b^6*c*d^15 + 4992*a^9*b^4*c*d^15 - 288*a^11*b^2*c*d^15 - 54*a^12*b*c^2*d^14 + 9*a^12*b*c^4*d^12 + 8060*a^2*b

^11*c^2*d^14 - 28978*a^2*b^11*c^4*d^12 + 17808*a^2*b^11*c^6*d^10 - 640*a^2*b^11*c^8*d^8 + 66096*a^3*b^10*c^3*d

^13 - 83024*a^3*b^10*c^5*d^11 + 13312*a^3*b^10*c^7*d^9 - 64714*a^4*b^9*c^2*d^14 + 189607*a^4*b^9*c^4*d^12 - 80

224*a^4*b^9*c^6*d^10 + 3136*a^4*b^9*c^8*d^8 - 205664*a^5*b^8*c^3*d^13 + 210144*a^5*b^8*c^5*d^11 - 30976*a^5*b^

8*c^7*d^9 + 108936*a^6*b^7*c^2*d^14 - 255292*a^6*b^7*c^4*d^12 + 101024*a^6*b^7*c^6*d^10 - 3712*a^6*b^7*c^8*d^8

+ 153568*a^7*b^6*c^3*d^13 - 138784*a^7*b^6*c^5*d^11 + 18432*a^7*b^6*c^7*d^9 - 43354*a^8*b^5*c^2*d^14 + 92167*

a^8*b^5*c^4*d^12 - 29296*a^8*b^5*c^6*d^10 + 608*a^8*b^5*c^8*d^8 - 29424*a^9*b^4*c^3*d^13 + 20784*a^9*b^4*c^5*d

^11 - 1152*a^9*b^4*c^7*d^9 + 4284*a^10*b^3*c^2*d^14 - 6834*a^10*b^3*c^4*d^12 + 720*a^10*b^3*c^6*d^10 + 1008*a^

11*b^2*c^3*d^13 - 144*a^11*b^2*c^5*d^11 + 384*a*b^12*c*d^15))/(a^16*f^4 + b^16*f^4 + 8*a^2*b^14*f^4 + 28*a^4*b

^12*f^4 + 56*a^6*b^10*f^4 + 70*a^8*b^8*f^4 + 56*a^10*b^6*f^4 + 28*a^12*b^4*f^4 + 8*a^14*b^2*f^4))*(-(((8*a^6*c

^3*f^2 - 8*b^6*c^3*f^2 - 48*a*b^5*d^3*f^2 - 48*a^5*b*d^3*f^2 - 24*a^6*c*d^2*f^2 + 24*b^6*c*d^2*f^2 + 120*a^2*b

^4*c^3*f^2 - 120*a^4*b^2*c^3*f^2 + 160*a^3*b^3*d^3*f^2 + 144*a*b^5*c^2*d*f^2 + 144*a^5*b*c^2*d*f^2 - 360*a^2*b

^4*c*d^2*f^2 - 480*a^3*b^3*c^2*d*f^2 + 360*a^4*b^2*c*d^2*f^2)^2/4 - (c^6 + d^6 + 3*c^2*d^4 + 3*c^4*d^2)*(16*a^

12*f^4 + 16*b^12*f^4 + 96*a^2*b^10*f^4 + 240*a^4*b^8*f^4 + 320*a^6*b^6*f^4 + 240*a^8*b^4*f^4 + 96*a^10*b^2*f^4

))^(1/2) + 4*a^6*c^3*f^2 - 4*b^6*c^3*f^2 - 24*a*b^5*d^3*f^2 - 24*a^5*b*d^3*f^2 - 12*a^6*c*d^2*f^2 + 12*b^6*c*d

^2*f^2 + 60*a^2*b^4*c^3*f^2 - 60*a^4*b^2*c^3*f^2 + 80*a^3*b^3*d^3*f^2 + 72*a*b^5*c^2*d*f^2 + 72*a^5*b*c^2*d*f^

2 - 180*a^2*b^4*c*d^2*f^2 - 240*a^3*b^3*c^2*d*f^2 + 180*a^4*b^2*c*d^2*f^2)/(16*(a^12*f^4 + b^12*f^4 + 6*a^2*b^

10*f^4 + 15*a^4*b^8*f^4 + 20*a^6*b^6*f^4 + 15*a^8*b^4*f^4 + 6*a^10*b^2*f^4)))^(1/2)*1i - (((1036*a*b^15*d^15*f

^2 - 36*a^15*b*d^15*f^2 - 604*b^16*c*d^14*f^2 - 8988*a^3*b^13*d^15*f^2 + 6044*a^5*b^11*d^15*f^2 + 34388*a^7*b^

9*d^15*f^2 + 10596*a^9*b^7*d^15*f^2 - 6676*a^11*b^5*d^15*f^2 + 1012*a^13*b^3*d^15*f^2 + 932*b^16*c^3*d^12*f^2

+ 1344*b^16*c^5*d^10*f^2 - 192*b^16*c^7*d^8*f^2 - 30836*a^2*b^14*c^3*d^12*f^2 - 48000*a^2*b^14*c^5*d^10*f^2 +

5248*a^2*b^14*c^7*d^8*f^2 + 95076*a^3*b^13*c^2*d^13*f^2 + 57600*a^3*b^13*c^4*d^11*f^2 - 46464*a^3*b^13*c^6*d^9

*f^2 + 17172*a^4*b^12*c^3*d^12*f^2 + 69696*a^4*b^12*c^5*d^10*f^2 - 5696*a^4*b^12*c^7*d^8*f^2 + 47004*a^5*b^11*

c^2*d^13*f^2 + 10944*a^5*b^11*c^4*d^11*f^2 - 30016*a^5*b^11*c^6*d^9*f^2 + 85404*a^6*b^10*c^3*d^12*f^2 + 169344

*a^6*b^10*c^5*d^10*f^2 - 17664*a^6*b^10*c^7*d^8*f^2 - 171180*a^7*b^9*c^2*d^13*f^2 - 119808*a^7*b^9*c^4*d^11*f^

2 + 85760*a^7*b^9*c^6*d^9*f^2 - 4308*a^8*b^8*c^3*d^12*f^2 - 49728*a^8*b^8*c^5*d^10*f^2 + 3776*a^8*b^8*c^7*d^8*

f^2 - 50972*a^9*b^7*c^2*d^13*f^2 - 24768*a^9*b^7*c^4*d^11*f^2 + 36800*a^9*b^7*c^6*d^9*f^2 - 35356*a^10*b^6*c^3

*d^12*f^2 - 80512*a^10*b^6*c^5*d^10*f^2 + 10368*a^10*b^6*c^7*d^8*f^2 + 55916*a^11*b^5*c^2*d^13*f^2 + 37632*a^1

1*b^5*c^4*d^11*f^2 - 24960*a^11*b^5*c^6*d^9*f^2 + 6428*a^12*b^4*c^3*d^12*f^2 + 19648*a^12*b^4*c^5*d^10*f^2 + 6

4*a^12*b^4*c^7*d^8*f^2 - 4876*a^13*b^3*c^2*d^13*f^2 - 6080*a^13*b^3*c^4*d^11*f^2 - 192*a^13*b^3*c^6*d^9*f^2 +

1012*a^14*b^2*c^3*d^12*f^2 + 128*a^14*b^2*c^5*d^10*f^2 - 11380*a*b^15*c^2*d^13*f^2 - 4672*a*b^15*c^4*d^11*f^2

+ 7744*a*b^15*c^6*d^9*f^2 + 22412*a^2*b^14*c*d^14*f^2 - 58220*a^4*b^12*c*d^14*f^2 - 101604*a^6*b^10*c*d^14*f^2

+ 49196*a^8*b^8*c*d^14*f^2 + 55524*a^10*b^6*c*d^14*f^2 - 13156*a^12*b^4*c*d^14*f^2 + 884*a^14*b^2*c*d^14*f^2

- 36*a^15*b*c^2*d^13*f^2)/(2*(a^16*f^5 + b^16*f^5 + 8*a^2*b^14*f^5 + 28*a^4*b^12*f^5 + 56*a^6*b^10*f^5 + 70*a^

8*b^8*f^5 + 56*a^10*b^6*f^5 + 28*a^12*b^4*f^5 + 8*a^14*b^2*f^5)) + (((8448*a^4*b^18*d^12*f^4 - 640*a^2*b^20*d^

12*f^4 - 384*b^22*d^12*f^4 + 44544*a^6*b^16*d^12*f^4 + 102144*a^8*b^14*d^12*f^4 + 134400*a^10*b^12*d^12*f^4 +

107520*a^12*b^10*d^12*f^4 + 50688*a^14*b^8*d^12*f^4 + 11904*a^16*b^6*d^12*f^4 + 384*a^18*b^4*d^12*f^4 - 256*a^

20*b^2*d^12*f^4 + 384*b^22*c^2*d^10*f^4 + 768*b^22*c^4*d^8*f^4 + 4224*a^2*b^20*c^2*d^10*f^4 + 4864*a^2*b^20*c^

4*d^8*f^4 - 27136*a^3*b^19*c^3*d^9*f^4 + 19712*a^4*b^18*c^2*d^10*f^4 + 11264*a^4*b^18*c^4*d^8*f^4 - 88064*a^5*

b^17*c^3*d^9*f^4 + 51712*a^6*b^16*c^2*d^10*f^4 + 7168*a^6*b^16*c^4*d^8*f^4 - 157696*a^7*b^15*c^3*d^9*f^4 + 842

24*a^8*b^14*c^2*d^10*f^4 - 17920*a^8*b^14*c^4*d^8*f^4 - 164864*a^9*b^13*c^3*d^9*f^4 + 87808*a^10*b^12*c^2*d^10

*f^4 - 46592*a^10*b^12*c^4*d^8*f^4 - 93184*a^11*b^11*c^3*d^9*f^4 + 57344*a^12*b^10*c^2*d^10*f^4 - 50176*a^12*b

^10*c^4*d^8*f^4 - 14336*a^13*b^9*c^3*d^9*f^4 + 20992*a^14*b^8*c^2*d^10*f^4 - 29696*a^14*b^8*c^4*d^8*f^4 + 1433

6*a^15*b^7*c^3*d^9*f^4 + 2432*a^16*b^6*c^2*d^10*f^4 - 9472*a^16*b^6*c^4*d^8*f^4 + 8704*a^17*b^5*c^3*d^9*f^4 -

896*a^18*b^4*c^2*d^10*f^4 - 1280*a^18*b^4*c^4*d^8*f^4 + 1536*a^19*b^3*c^3*d^9*f^4 - 256*a^20*b^2*c^2*d^10*f^4

- 3584*a*b^21*c*d^11*f^4 - 3584*a*b^21*c^3*d^9*f^4 - 27136*a^3*b^19*c*d^11*f^4 - 88064*a^5*b^17*c*d^11*f^4 - 1

57696*a^7*b^15*c*d^11*f^4 - 164864*a^9*b^13*c*d^11*f^4 - 93184*a^11*b^11*c*d^11*f^4 - 14336*a^13*b^9*c*d^11*f^

4 + 14336*a^15*b^7*c*d^11*f^4 + 8704*a^17*b^5*c*d^11*f^4 + 1536*a^19*b^3*c*d^11*f^4)/(2*(a^16*f^5 + b^16*f^5 +

8*a^2*b^14*f^5 + 28*a^4*b^12*f^5 + 56*a^6*b^10*f^5 + 70*a^8*b^8*f^5 + 56*a^10*b^6*f^5 + 28*a^12*b^4*f^5 + 8*a

^14*b^2*f^5)) - ((c + d*tan(e + f*x))^(1/2)*(-(((8*a^6*c^3*f^2 - 8*b^6*c^3*f^2 - 48*a*b^5*d^3*f^2 - 48*a^5*b*d

^3*f^2 - 24*a^6*c*d^2*f^2 + 24*b^6*c*d^2*f^2 + 120*a^2*b^4*c^3*f^2 - 120*a^4*b^2*c^3*f^2 + 160*a^3*b^3*d^3*f^2

+ 144*a*b^5*c^2*d*f^2 + 144*a^5*b*c^2*d*f^2 - 360*a^2*b^4*c*d^2*f^2 - 480*a^3*b^3*c^2*d*f^2 + 360*a^4*b^2*c*d

^2*f^2)^2/4 - (c^6 + d^6 + 3*c^2*d^4 + 3*c^4*d^2)*(16*a^12*f^4 + 16*b^12*f^4 + 96*a^2*b^10*f^4 + 240*a^4*b^8*f

^4 + 320*a^6*b^6*f^4 + 240*a^8*b^4*f^4 + 96*a^10*b^2*f^4))^(1/2) + 4*a^6*c^3*f^2 - 4*b^6*c^3*f^2 - 24*a*b^5*d^

3*f^2 - 24*a^5*b*d^3*f^2 - 12*a^6*c*d^2*f^2 + 12*b^6*c*d^2*f^2 + 60*a^2*b^4*c^3*f^2 - 60*a^4*b^2*c^3*f^2 + 80*

a^3*b^3*d^3*f^2 + 72*a*b^5*c^2*d*f^2 + 72*a^5*b*c^2*d*f^2 - 180*a^2*b^4*c*d^2*f^2 - 240*a^3*b^3*c^2*d*f^2 + 18

0*a^4*b^2*c*d^2*f^2)/(16*(a^12*f^4 + b^12*f^4 + 6*a^2*b^10*f^4 + 15*a^4*b^8*f^4 + 20*a^6*b^6*f^4 + 15*a^8*b^4*

f^4 + 6*a^10*b^2*f^4)))^(1/2)*(512*b^25*d^10*f^4 + 4608*a^2*b^23*d^10*f^4 + 17920*a^4*b^21*d^10*f^4 + 38400*a^

6*b^19*d^10*f^4 + 46080*a^8*b^17*d^10*f^4 + 21504*a^10*b^15*d^10*f^4 - 21504*a^12*b^13*d^10*f^4 - 46080*a^14*b

^11*d^10*f^4 - 38400*a^16*b^9*d^10*f^4 - 17920*a^18*b^7*d^10*f^4 - 4608*a^20*b^5*d^10*f^4 - 512*a^22*b^3*d^10*

f^4 + 768*b^25*c^2*d^8*f^4 + 7424*a^2*b^23*c^2*d^8*f^4 + 32000*a^4*b^21*c^2*d^8*f^4 + 80640*a^6*b^19*c^2*d^8*f

^4 + 130560*a^8*b^17*c^2*d^8*f^4 + 139776*a^10*b^15*c^2*d^8*f^4 + 96768*a^12*b^13*c^2*d^8*f^4 + 38400*a^14*b^1

1*c^2*d^8*f^4 + 3840*a^16*b^9*c^2*d^8*f^4 - 3840*a^18*b^7*c^2*d^8*f^4 - 1792*a^20*b^5*c^2*d^8*f^4 - 256*a^22*b

^3*c^2*d^8*f^4 + 256*a*b^24*c*d^9*f^4 + 2816*a^3*b^22*c*d^9*f^4 + 14080*a^5*b^20*c*d^9*f^4 + 42240*a^7*b^18*c*

d^9*f^4 + 84480*a^9*b^16*c*d^9*f^4 + 118272*a^11*b^14*c*d^9*f^4 + 118272*a^13*b^12*c*d^9*f^4 + 84480*a^15*b^10

*c*d^9*f^4 + 42240*a^17*b^8*c*d^9*f^4 + 14080*a^19*b^6*c*d^9*f^4 + 2816*a^21*b^4*c*d^9*f^4 + 256*a^23*b^2*c*d^

9*f^4))/(a^16*f^4 + b^16*f^4 + 8*a^2*b^14*f^4 + 28*a^4*b^12*f^4 + 56*a^6*b^10*f^4 + 70*a^8*b^8*f^4 + 56*a^10*b

^6*f^4 + 28*a^12*b^4*f^4 + 8*a^14*b^2*f^4))*(-(((8*a^6*c^3*f^2 - 8*b^6*c^3*f^2 - 48*a*b^5*d^3*f^2 - 48*a^5*b*d

^3*f^2 - 24*a^6*c*d^2*f^2 + 24*b^6*c*d^2*f^2 + 120*a^2*b^4*c^3*f^2 - 120*a^4*b^2*c^3*f^2 + 160*a^3*b^3*d^3*f^2

+ 144*a*b^5*c^2*d*f^2 + 144*a^5*b*c^2*d*f^2 - 360*a^2*b^4*c*d^2*f^2 - 480*a^3*b^3*c^2*d*f^2 + 360*a^4*b^2*c*d

^2*f^2)^2/4 - (c^6 + d^6 + 3*c^2*d^4 + 3*c^4*d^2)*(16*a^12*f^4 + 16*b^12*f^4 + 96*a^2*b^10*f^4 + 240*a^4*b^8*f

^4 + 320*a^6*b^6*f^4 + 240*a^8*b^4*f^4 + 96*a^10*b^2*f^4))^(1/2) + 4*a^6*c^3*f^2 - 4*b^6*c^3*f^2 - 24*a*b^5*d^

3*f^2 - 24*a^5*b*d^3*f^2 - 12*a^6*c*d^2*f^2 + 12*b^6*c*d^2*f^2 + 60*a^2*b^4*c^3*f^2 - 60*a^4*b^2*c^3*f^2 + 80*

a^3*b^3*d^3*f^2 + 72*a*b^5*c^2*d*f^2 + 72*a^5*b*c^2*d*f^2 - 180*a^2*b^4*c*d^2*f^2 - 240*a^3*b^3*c^2*d*f^2 + 18

0*a^4*b^2*c*d^2*f^2)/(16*(a^12*f^4 + b^12*f^4 + 6*a^2*b^10*f^4 + 15*a^4*b^8*f^4 + 20*a^6*b^6*f^4 + 15*a^8*b^4*

f^4 + 6*a^10*b^2*f^4)))^(1/2) + ((c + d*tan(e + f*x))^(1/2)*(1544*a*b^18*d^13*f^2 - 668*b^19*c*d^12*f^2 + 64*a

^3*b^16*d^13*f^2 - 7456*a^5*b^14*d^13*f^2 - 576*a^7*b^12*d^13*f^2 + 19504*a^9*b^10*d^13*f^2 + 18880*a^11*b^8*d

^13*f^2 + 3808*a^13*b^6*d^13*f^2 - 960*a^15*b^4*d^13*f^2 + 8*a^17*b^2*d^13*f^2 - 1088*b^19*c^3*d^10*f^2 + 576*

b^19*c^5*d^8*f^2 + 9600*a^2*b^17*c^3*d^10*f^2 - 3584*a^2*b^17*c^5*d^8*f^2 + 12800*a^3*b^16*c^2*d^11*f^2 - 7680

*a^3*b^16*c^4*d^9*f^2 + 43392*a^4*b^15*c^3*d^10*f^2 - 13056*a^4*b^15*c^5*d^8*f^2 + 43264*a^5*b^14*c^2*d^11*f^2

+ 34560*a^5*b^14*c^4*d^9*f^2 + 63360*a^6*b^13*c^3*d^10*f^2 - 7680*a^6*b^13*c^5*d^8*f^2 + 19968*a^7*b^12*c^2*d

^11*f^2 + 66048*a^7*b^12*c^4*d^9*f^2 + 45312*a^8*b^11*c^3*d^10*f^2 + 8064*a^8*b^11*c^5*d^8*f^2 - 57728*a^9*b^1

0*c^2*d^11*f^2 + 35968*a^9*b^10*c^4*d^9*f^2 + 18560*a^10*b^9*c^3*d^10*f^2 + 5632*a^10*b^9*c^5*d^8*f^2 - 74240*

a^11*b^8*c^2*d^11*f^2 + 9728*a^11*b^8*c^4*d^9*f^2 - 1408*a^12*b^7*c^3*d^10*f^2 - 3840*a^12*b^7*c^5*d^8*f^2 - 2

6368*a^13*b^6*c^2*d^11*f^2 + 11008*a^13*b^6*c^4*d^9*f^2 - 9600*a^14*b^5*c^3*d^10*f^2 - 2560*a^14*b^5*c^5*d^8*f

^2 + 512*a^15*b^4*c^2*d^11*f^2 + 5632*a^15*b^4*c^4*d^9*f^2 - 4288*a^16*b^3*c^3*d^10*f^2 + 64*a^16*b^3*c^5*d^8*

f^2 + 960*a^17*b^2*c^2*d^11*f^2 - 64*a^17*b^2*c^4*d^9*f^2 - 36*a^18*b*c*d^12*f^2 - 1088*a*b^18*c^2*d^11*f^2 -

7744*a*b^18*c^4*d^9*f^2 + 8828*a^2*b^17*c*d^12*f^2 + 14416*a^4*b^15*c*d^12*f^2 - 33936*a^6*b^13*c*d^12*f^2 - 7

9368*a^8*b^11*c*d^12*f^2 - 36216*a^10*b^9*c*d^12*f^2 + 16016*a^12*b^7*c*d^12*f^2 + 12208*a^14*b^5*c*d^12*f^2 +

452*a^16*b^3*c*d^12*f^2))/(a^16*f^4 + b^16*f^4 + 8*a^2*b^14*f^4 + 28*a^4*b^12*f^4 + 56*a^6*b^10*f^4 + 70*a^8*

b^8*f^4 + 56*a^10*b^6*f^4 + 28*a^12*b^4*f^4 + 8*a^14*b^2*f^4))*(-(((8*a^6*c^3*f^2 - 8*b^6*c^3*f^2 - 48*a*b^5*d

^3*f^2 - 48*a^5*b*d^3*f^2 - 24*a^6*c*d^2*f^2 + 24*b^6*c*d^2*f^2 + 120*a^2*b^4*c^3*f^2 - 120*a^4*b^2*c^3*f^2 +

160*a^3*b^3*d^3*f^2 + 144*a*b^5*c^2*d*f^2 + 144*a^5*b*c^2*d*f^2 - 360*a^2*b^4*c*d^2*f^2 - 480*a^3*b^3*c^2*d*f^

2 + 360*a^4*b^2*c*d^2*f^2)^2/4 - (c^6 + d^6 + 3*c^2*d^4 + 3*c^4*d^2)*(16*a^12*f^4 + 16*b^12*f^4 + 96*a^2*b^10*

f^4 + 240*a^4*b^8*f^4 + 320*a^6*b^6*f^4 + 240*a^8*b^4*f^4 + 96*a^10*b^2*f^4))^(1/2) + 4*a^6*c^3*f^2 - 4*b^6*c^

3*f^2 - 24*a*b^5*d^3*f^2 - 24*a^5*b*d^3*f^2 - 12*a^6*c*d^2*f^2 + 12*b^6*c*d^2*f^2 + 60*a^2*b^4*c^3*f^2 - 60*a^

4*b^2*c^3*f^2 + 80*a^3*b^3*d^3*f^2 + 72*a*b^5*c^2*d*f^2 + 72*a^5*b*c^2*d*f^2 - 180*a^2*b^4*c*d^2*f^2 - 240*a^3

*b^3*c^2*d*f^2 + 180*a^4*b^2*c*d^2*f^2)/(16*(a^12*f^4 + b^12*f^4 + 6*a^2*b^10*f^4 + 15*a^4*b^8*f^4 + 20*a^6*b^

6*f^4 + 15*a^8*b^4*f^4 + 6*a^10*b^2*f^4)))^(1/2))*(-(((8*a^6*c^3*f^2 - 8*b^6*c^3*f^2 - 48*a*b^5*d^3*f^2 - 48*a

^5*b*d^3*f^2 - 24*a^6*c*d^2*f^2 + 24*b^6*c*d^2*f^2 + 120*a^2*b^4*c^3*f^2 - 120*a^4*b^2*c^3*f^2 + 160*a^3*b^3*d

^3*f^2 + 144*a*b^5*c^2*d*f^2 + 144*a^5*b*c^2*d*f^2 - 360*a^2*b^4*c*d^2*f^2 - 480*a^3*b^3*c^2*d*f^2 + 360*a^4*b

^2*c*d^2*f^2)^2/4 - (c^6 + d^6 + 3*c^2*d^4 + 3*c^4*d^2)*(16*a^12*f^4 + 16*b^12*f^4 + 96*a^2*b^10*f^4 + 240*a^4

*b^8*f^4 + 320*a^6*b^6*f^4 + 240*a^8*b^4*f^4 + 96*a^10*b^2*f^4))^(1/2) + 4*a^6*c^3*f^2 - 4*b^6*c^3*f^2 - 24*a*

b^5*d^3*f^2 - 24*a^5*b*d^3*f^2 - 12*a^6*c*d^2*f^2 + 12*b^6*c*d^2*f^2 + 60*a^2*b^4*c^3*f^2 - 60*a^4*b^2*c^3*f^2

+ 80*a^3*b^3*d^3*f^2 + 72*a*b^5*c^2*d*f^2 + 72*a^5*b*c^2*d*f^2 - 180*a^2*b^4*c*d^2*f^2 - 240*a^3*b^3*c^2*d*f^

2 + 180*a^4*b^2*c*d^2*f^2)/(16*(a^12*f^4 + b^12*f^4 + 6*a^2*b^10*f^4 + 15*a^4*b^8*f^4 + 20*a^6*b^6*f^4 + 15*a^

8*b^4*f^4 + 6*a^10*b^2*f^4)))^(1/2) - ((c + d*tan(e + f*x))^(1/2)*(41*b^13*d^16 + 9*a^12*b*d^16 - 82*a^2*b^11*

d^16 + 1831*a^4*b^9*d^16 - 4348*a^6*b^7*d^16 + 1671*a^8*b^5*d^16 - 210*a^10*b^3*d^16 + 26*b^13*c^2*d^14 + 553*

b^13*c^4*d^12 - 304*b^13*c^6*d^10 + 96*b^13*c^8*d^8 - 2992*a*b^12*c^3*d^13 + 5872*a*b^12*c^5*d^11 - 1664*a*b^1

2*c^7*d^9 - 6304*a^3*b^10*c*d^15 + 28672*a^5*b^8*c*d^15 - 24896*a^7*b^6*c*d^15 + 4992*a^9*b^4*c*d^15 - 288*a^1

1*b^2*c*d^15 - 54*a^12*b*c^2*d^14 + 9*a^12*b*c^4*d^12 + 8060*a^2*b^11*c^2*d^14 - 28978*a^2*b^11*c^4*d^12 + 178

08*a^2*b^11*c^6*d^10 - 640*a^2*b^11*c^8*d^8 + 66096*a^3*b^10*c^3*d^13 - 83024*a^3*b^10*c^5*d^11 + 13312*a^3*b^

10*c^7*d^9 - 64714*a^4*b^9*c^2*d^14 + 189607*a^4*b^9*c^4*d^12 - 80224*a^4*b^9*c^6*d^10 + 3136*a^4*b^9*c^8*d^8

- 205664*a^5*b^8*c^3*d^13 + 210144*a^5*b^8*c^5*d^11 - 30976*a^5*b^8*c^7*d^9 + 108936*a^6*b^7*c^2*d^14 - 255292

*a^6*b^7*c^4*d^12 + 101024*a^6*b^7*c^6*d^10 - 3712*a^6*b^7*c^8*d^8 + 153568*a^7*b^6*c^3*d^13 - 138784*a^7*b^6*

c^5*d^11 + 18432*a^7*b^6*c^7*d^9 - 43354*a^8*b^5*c^2*d^14 + 92167*a^8*b^5*c^4*d^12 - 29296*a^8*b^5*c^6*d^10 +

608*a^8*b^5*c^8*d^8 - 29424*a^9*b^4*c^3*d^13 + 20784*a^9*b^4*c^5*d^11 - 1152*a^9*b^4*c^7*d^9 + 4284*a^10*b^3*c

^2*d^14 - 6834*a^10*b^3*c^4*d^12 + 720*a^10*b^3*c^6*d^10 + 1008*a^11*b^2*c^3*d^13 - 144*a^11*b^2*c^5*d^11 + 38

4*a*b^12*c*d^15))/(a^16*f^4 + b^16*f^4 + 8*a^2*b^14*f^4 + 28*a^4*b^12*f^4 + 56*a^6*b^10*f^4 + 70*a^8*b^8*f^4 +

56*a^10*b^6*f^4 + 28*a^12*b^4*f^4 + 8*a^14*b^2*f^4))*(-(((8*a^6*c^3*f^2 - 8*b^6*c^3*f^2 - 48*a*b^5*d^3*f^2 -

48*a^5*b*d^3*f^2 - 24*a^6*c*d^2*f^2 + 24*b^6*c*d^2*f^2 + 120*a^2*b^4*c^3*f^2 - 120*a^4*b^2*c^3*f^2 + 160*a^3*b

^3*d^3*f^2 + 144*a*b^5*c^2*d*f^2 + 144*a^5*b*c^2*d*f^2 - 360*a^2*b^4*c*d^2*f^2 - 480*a^3*b^3*c^2*d*f^2 + 360*a

^4*b^2*c*d^2*f^2)^2/4 - (c^6 + d^6 + 3*c^2*d^4 + 3*c^4*d^2)*(16*a^12*f^4 + 16*b^12*f^4 + 96*a^2*b^10*f^4 + 240

*a^4*b^8*f^4 + 320*a^6*b^6*f^4 + 240*a^8*b^4*f^4 + 96*a^10*b^2*f^4))^(1/2) + 4*a^6*c^3*f^2 - 4*b^6*c^3*f^2 - 2

4*a*b^5*d^3*f^2 - 24*a^5*b*d^3*f^2 - 12*a^6*c*d^2*f^2 + 12*b^6*c*d^2*f^2 + 60*a^2*b^4*c^3*f^2 - 60*a^4*b^2*c^3

*f^2 + 80*a^3*b^3*d^3*f^2 + 72*a*b^5*c^2*d*f^2 + 72*a^5*b*c^2*d*f^2 - 180*a^2*b^4*c*d^2*f^2 - 240*a^3*b^3*c^2*

d*f^2 + 180*a^4*b^2*c*d^2*f^2)/(16*(a^12*f^4 + b^12*f^4 + 6*a^2*b^10*f^4 + 15*a^4*b^8*f^4 + 20*a^6*b^6*f^4 + 1

5*a^8*b^4*f^4 + 6*a^10*b^2*f^4)))^(1/2)*1i)/((((1036*a*b^15*d^15*f^2 - 36*a^15*b*d^15*f^2 - 604*b^16*c*d^14*f^

2 - 8988*a^3*b^13*d^15*f^2 + 6044*a^5*b^11*d^15*f^2 + 34388*a^7*b^9*d^15*f^2 + 10596*a^9*b^7*d^15*f^2 - 6676*a

^11*b^5*d^15*f^2 + 1012*a^13*b^3*d^15*f^2 + 932*b^16*c^3*d^12*f^2 + 1344*b^16*c^5*d^10*f^2 - 192*b^16*c^7*d^8*

f^2 - 30836*a^2*b^14*c^3*d^12*f^2 - 48000*a^2*b^14*c^5*d^10*f^2 + 5248*a^2*b^14*c^7*d^8*f^2 + 95076*a^3*b^13*c

^2*d^13*f^2 + 57600*a^3*b^13*c^4*d^11*f^2 - 46464*a^3*b^13*c^6*d^9*f^2 + 17172*a^4*b^12*c^3*d^12*f^2 + 69696*a

^4*b^12*c^5*d^10*f^2 - 5696*a^4*b^12*c^7*d^8*f^2 + 47004*a^5*b^11*c^2*d^13*f^2 + 10944*a^5*b^11*c^4*d^11*f^2 -

30016*a^5*b^11*c^6*d^9*f^2 + 85404*a^6*b^10*c^3*d^12*f^2 + 169344*a^6*b^10*c^5*d^10*f^2 - 17664*a^6*b^10*c^7*

d^8*f^2 - 171180*a^7*b^9*c^2*d^13*f^2 - 119808*a^7*b^9*c^4*d^11*f^2 + 85760*a^7*b^9*c^6*d^9*f^2 - 4308*a^8*b^8

*c^3*d^12*f^2 - 49728*a^8*b^8*c^5*d^10*f^2 + 3776*a^8*b^8*c^7*d^8*f^2 - 50972*a^9*b^7*c^2*d^13*f^2 - 24768*a^9

*b^7*c^4*d^11*f^2 + 36800*a^9*b^7*c^6*d^9*f^2 - 35356*a^10*b^6*c^3*d^12*f^2 - 80512*a^10*b^6*c^5*d^10*f^2 + 10

368*a^10*b^6*c^7*d^8*f^2 + 55916*a^11*b^5*c^2*d^13*f^2 + 37632*a^11*b^5*c^4*d^11*f^2 - 24960*a^11*b^5*c^6*d^9*

f^2 + 6428*a^12*b^4*c^3*d^12*f^2 + 19648*a^12*b^4*c^5*d^10*f^2 + 64*a^12*b^4*c^7*d^8*f^2 - 4876*a^13*b^3*c^2*d

^13*f^2 - 6080*a^13*b^3*c^4*d^11*f^2 - 192*a^13*b^3*c^6*d^9*f^2 + 1012*a^14*b^2*c^3*d^12*f^2 + 128*a^14*b^2*c^

5*d^10*f^2 - 11380*a*b^15*c^2*d^13*f^2 - 4672*a*b^15*c^4*d^11*f^2 + 7744*a*b^15*c^6*d^9*f^2 + 22412*a^2*b^14*c

*d^14*f^2 - 58220*a^4*b^12*c*d^14*f^2 - 101604*a^6*b^10*c*d^14*f^2 + 49196*a^8*b^8*c*d^14*f^2 + 55524*a^10*b^6

*c*d^14*f^2 - 13156*a^12*b^4*c*d^14*f^2 + 884*a^14*b^2*c*d^14*f^2 - 36*a^15*b*c^2*d^13*f^2)/(2*(a^16*f^5 + b^1

6*f^5 + 8*a^2*b^14*f^5 + 28*a^4*b^12*f^5 + 56*a^6*b^10*f^5 + 70*a^8*b^8*f^5 + 56*a^10*b^6*f^5 + 28*a^12*b^4*f^

5 + 8*a^14*b^2*f^5)) + (((8448*a^4*b^18*d^12*f^4 - 640*a^2*b^20*d^12*f^4 - 384*b^22*d^12*f^4 + 44544*a^6*b^16*

d^12*f^4 + 102144*a^8*b^14*d^12*f^4 + 134400*a^10*b^12*d^12*f^4 + 107520*a^12*b^10*d^12*f^4 + 50688*a^14*b^8*d

^12*f^4 + 11904*a^16*b^6*d^12*f^4 + 384*a^18*b^4*d^12*f^4 - 256*a^20*b^2*d^12*f^4 + 384*b^22*c^2*d^10*f^4 + 76

8*b^22*c^4*d^8*f^4 + 4224*a^2*b^20*c^2*d^10*f^4 + 4864*a^2*b^20*c^4*d^8*f^4 - 27136*a^3*b^19*c^3*d^9*f^4 + 197

12*a^4*b^18*c^2*d^10*f^4 + 11264*a^4*b^18*c^4*d^8*f^4 - 88064*a^5*b^17*c^3*d^9*f^4 + 51712*a^6*b^16*c^2*d^10*f

^4 + 7168*a^6*b^16*c^4*d^8*f^4 - 157696*a^7*b^15*c^3*d^9*f^4 + 84224*a^8*b^14*c^2*d^10*f^4 - 17920*a^8*b^14*c^

4*d^8*f^4 - 164864*a^9*b^13*c^3*d^9*f^4 + 87808*a^10*b^12*c^2*d^10*f^4 - 46592*a^10*b^12*c^4*d^8*f^4 - 93184*a

^11*b^11*c^3*d^9*f^4 + 57344*a^12*b^10*c^2*d^10*f^4 - 50176*a^12*b^10*c^4*d^8*f^4 - 14336*a^13*b^9*c^3*d^9*f^4

+ 20992*a^14*b^8*c^2*d^10*f^4 - 29696*a^14*b^8*c^4*d^8*f^4 + 14336*a^15*b^7*c^3*d^9*f^4 + 2432*a^16*b^6*c^2*d

^10*f^4 - 9472*a^16*b^6*c^4*d^8*f^4 + 8704*a^17*b^5*c^3*d^9*f^4 - 896*a^18*b^4*c^2*d^10*f^4 - 1280*a^18*b^4*c^

4*d^8*f^4 + 1536*a^19*b^3*c^3*d^9*f^4 - 256*a^20*b^2*c^2*d^10*f^4 - 3584*a*b^21*c*d^11*f^4 - 3584*a*b^21*c^3*d

^9*f^4 - 27136*a^3*b^19*c*d^11*f^4 - 88064*a^5*b^17*c*d^11*f^4 - 157696*a^7*b^15*c*d^11*f^4 - 164864*a^9*b^13*

c*d^11*f^4 - 93184*a^11*b^11*c*d^11*f^4 - 14336*a^13*b^9*c*d^11*f^4 + 14336*a^15*b^7*c*d^11*f^4 + 8704*a^17*b^

5*c*d^11*f^4 + 1536*a^19*b^3*c*d^11*f^4)/(2*(a^16*f^5 + b^16*f^5 + 8*a^2*b^14*f^5 + 28*a^4*b^12*f^5 + 56*a^6*b

^10*f^5 + 70*a^8*b^8*f^5 + 56*a^10*b^6*f^5 + 28*a^12*b^4*f^5 + 8*a^14*b^2*f^5)) + ((c + d*tan(e + f*x))^(1/2)*

(-(((8*a^6*c^3*f^2 - 8*b^6*c^3*f^2 - 48*a*b^5*d^3*f^2 - 48*a^5*b*d^3*f^2 - 24*a^6*c*d^2*f^2 + 24*b^6*c*d^2*f^2

+ 120*a^2*b^4*c^3*f^2 - 120*a^4*b^2*c^3*f^2 + 160*a^3*b^3*d^3*f^2 + 144*a*b^5*c^2*d*f^2 + 144*a^5*b*c^2*d*f^2

- 360*a^2*b^4*c*d^2*f^2 - 480*a^3*b^3*c^2*d*f^2 + 360*a^4*b^2*c*d^2*f^2)^2/4 - (c^6 + d^6 + 3*c^2*d^4 + 3*c^4

*d^2)*(16*a^12*f^4 + 16*b^12*f^4 + 96*a^2*b^10*f^4 + 240*a^4*b^8*f^4 + 320*a^6*b^6*f^4 + 240*a^8*b^4*f^4 + 96*

a^10*b^2*f^4))^(1/2) + 4*a^6*c^3*f^2 - 4*b^6*c^3*f^2 - 24*a*b^5*d^3*f^2 - 24*a^5*b*d^3*f^2 - 12*a^6*c*d^2*f^2

+ 12*b^6*c*d^2*f^2 + 60*a^2*b^4*c^3*f^2 - 60*a^4*b^2*c^3*f^2 + 80*a^3*b^3*d^3*f^2 + 72*a*b^5*c^2*d*f^2 + 72*a^

5*b*c^2*d*f^2 - 180*a^2*b^4*c*d^2*f^2 - 240*a^3*b^3*c^2*d*f^2 + 180*a^4*b^2*c*d^2*f^2)/(16*(a^12*f^4 + b^12*f^

4 + 6*a^2*b^10*f^4 + 15*a^4*b^8*f^4 + 20*a^6*b^6*f^4 + 15*a^8*b^4*f^4 + 6*a^10*b^2*f^4)))^(1/2)*(512*b^25*d^10

*f^4 + 4608*a^2*b^23*d^10*f^4 + 17920*a^4*b^21*d^10*f^4 + 38400*a^6*b^19*d^10*f^4 + 46080*a^8*b^17*d^10*f^4 +

21504*a^10*b^15*d^10*f^4 - 21504*a^12*b^13*d^10*f^4 - 46080*a^14*b^11*d^10*f^4 - 38400*a^16*b^9*d^10*f^4 - 179

20*a^18*b^7*d^10*f^4 - 4608*a^20*b^5*d^10*f^4 - 512*a^22*b^3*d^10*f^4 + 768*b^25*c^2*d^8*f^4 + 7424*a^2*b^23*c

^2*d^8*f^4 + 32000*a^4*b^21*c^2*d^8*f^4 + 80640*a^6*b^19*c^2*d^8*f^4 + 130560*a^8*b^17*c^2*d^8*f^4 + 139776*a^

10*b^15*c^2*d^8*f^4 + 96768*a^12*b^13*c^2*d^8*f^4 + 38400*a^14*b^11*c^2*d^8*f^4 + 3840*a^16*b^9*c^2*d^8*f^4 -

3840*a^18*b^7*c^2*d^8*f^4 - 1792*a^20*b^5*c^2*d^8*f^4 - 256*a^22*b^3*c^2*d^8*f^4 + 256*a*b^24*c*d^9*f^4 + 2816

*a^3*b^22*c*d^9*f^4 + 14080*a^5*b^20*c*d^9*f^4 + 42240*a^7*b^18*c*d^9*f^4 + 84480*a^9*b^16*c*d^9*f^4 + 118272*

a^11*b^14*c*d^9*f^4 + 118272*a^13*b^12*c*d^9*f^4 + 84480*a^15*b^10*c*d^9*f^4 + 42240*a^17*b^8*c*d^9*f^4 + 1408

0*a^19*b^6*c*d^9*f^4 + 2816*a^21*b^4*c*d^9*f^4 + 256*a^23*b^2*c*d^9*f^4))/(a^16*f^4 + b^16*f^4 + 8*a^2*b^14*f^

4 + 28*a^4*b^12*f^4 + 56*a^6*b^10*f^4 + 70*a^8*b^8*f^4 + 56*a^10*b^6*f^4 + 28*a^12*b^4*f^4 + 8*a^14*b^2*f^4))*

(-(((8*a^6*c^3*f^2 - 8*b^6*c^3*f^2 - 48*a*b^5*d^3*f^2 - 48*a^5*b*d^3*f^2 - 24*a^6*c*d^2*f^2 + 24*b^6*c*d^2*f^2

+ 120*a^2*b^4*c^3*f^2 - 120*a^4*b^2*c^3*f^2 + 160*a^3*b^3*d^3*f^2 + 144*a*b^5*c^2*d*f^2 + 144*a^5*b*c^2*d*f^2

- 360*a^2*b^4*c*d^2*f^2 - 480*a^3*b^3*c^2*d*f^2 + 360*a^4*b^2*c*d^2*f^2)^2/4 - (c^6 + d^6 + 3*c^2*d^4 + 3*c^4

*d^2)*(16*a^12*f^4 + 16*b^12*f^4 + 96*a^2*b^10*f^4 + 240*a^4*b^8*f^4 + 320*a^6*b^6*f^4 + 240*a^8*b^4*f^4 + 96*

a^10*b^2*f^4))^(1/2) + 4*a^6*c^3*f^2 - 4*b^6*c^3*f^2 - 24*a*b^5*d^3*f^2 - 24*a^5*b*d^3*f^2 - 12*a^6*c*d^2*f^2

+ 12*b^6*c*d^2*f^2 + 60*a^2*b^4*c^3*f^2 - 60*a^4*b^2*c^3*f^2 + 80*a^3*b^3*d^3*f^2 + 72*a*b^5*c^2*d*f^2 + 72*a^

5*b*c^2*d*f^2 - 180*a^2*b^4*c*d^2*f^2 - 240*a^3*b^3*c^2*d*f^2 + 180*a^4*b^2*c*d^2*f^2)/(16*(a^12*f^4 + b^12*f^

4 + 6*a^2*b^10*f^4 + 15*a^4*b^8*f^4 + 20*a^6*b^6*f^4 + 15*a^8*b^4*f^4 + 6*a^10*b^2*f^4)))^(1/2) - ((c + d*tan(

e + f*x))^(1/2)*(1544*a*b^18*d^13*f^2 - 668*b^19*c*d^12*f^2 + 64*a^3*b^16*d^13*f^2 - 7456*a^5*b^14*d^13*f^2 -

576*a^7*b^12*d^13*f^2 + 19504*a^9*b^10*d^13*f^2 + 18880*a^11*b^8*d^13*f^2 + 3808*a^13*b^6*d^13*f^2 - 960*a^15*

b^4*d^13*f^2 + 8*a^17*b^2*d^13*f^2 - 1088*b^19*c^3*d^10*f^2 + 576*b^19*c^5*d^8*f^2 + 9600*a^2*b^17*c^3*d^10*f^

2 - 3584*a^2*b^17*c^5*d^8*f^2 + 12800*a^3*b^16*c^2*d^11*f^2 - 7680*a^3*b^16*c^4*d^9*f^2 + 43392*a^4*b^15*c^3*d

^10*f^2 - 13056*a^4*b^15*c^5*d^8*f^2 + 43264*a^5*b^14*c^2*d^11*f^2 + 34560*a^5*b^14*c^4*d^9*f^2 + 63360*a^6*b^

13*c^3*d^10*f^2 - 7680*a^6*b^13*c^5*d^8*f^2 + 19968*a^7*b^12*c^2*d^11*f^2 + 66048*a^7*b^12*c^4*d^9*f^2 + 45312

*a^8*b^11*c^3*d^10*f^2 + 8064*a^8*b^11*c^5*d^8*f^2 - 57728*a^9*b^10*c^2*d^11*f^2 + 35968*a^9*b^10*c^4*d^9*f^2

+ 18560*a^10*b^9*c^3*d^10*f^2 + 5632*a^10*b^9*c^5*d^8*f^2 - 74240*a^11*b^8*c^2*d^11*f^2 + 9728*a^11*b^8*c^4*d^

9*f^2 - 1408*a^12*b^7*c^3*d^10*f^2 - 3840*a^12*b^7*c^5*d^8*f^2 - 26368*a^13*b^6*c^2*d^11*f^2 + 11008*a^13*b^6*

c^4*d^9*f^2 - 9600*a^14*b^5*c^3*d^10*f^2 - 2560*a^14*b^5*c^5*d^8*f^2 + 512*a^15*b^4*c^2*d^11*f^2 + 5632*a^15*b

^4*c^4*d^9*f^2 - 4288*a^16*b^3*c^3*d^10*f^2 + 64*a^16*b^3*c^5*d^8*f^2 + 960*a^17*b^2*c^2*d^11*f^2 - 64*a^17*b^

2*c^4*d^9*f^2 - 36*a^18*b*c*d^12*f^2 - 1088*a*b^18*c^2*d^11*f^2 - 7744*a*b^18*c^4*d^9*f^2 + 8828*a^2*b^17*c*d^

12*f^2 + 14416*a^4*b^15*c*d^12*f^2 - 33936*a^6*b^13*c*d^12*f^2 - 79368*a^8*b^11*c*d^12*f^2 - 36216*a^10*b^9*c*

d^12*f^2 + 16016*a^12*b^7*c*d^12*f^2 + 12208*a^14*b^5*c*d^12*f^2 + 452*a^16*b^3*c*d^12*f^2))/(a^16*f^4 + b^16*

f^4 + 8*a^2*b^14*f^4 + 28*a^4*b^12*f^4 + 56*a^6*b^10*f^4 + 70*a^8*b^8*f^4 + 56*a^10*b^6*f^4 + 28*a^12*b^4*f^4

+ 8*a^14*b^2*f^4))*(-(((8*a^6*c^3*f^2 - 8*b^6*c^3*f^2 - 48*a*b^5*d^3*f^2 - 48*a^5*b*d^3*f^2 - 24*a^6*c*d^2*f^2

+ 24*b^6*c*d^2*f^2 + 120*a^2*b^4*c^3*f^2 - 120*a^4*b^2*c^3*f^2 + 160*a^3*b^3*d^3*f^2 + 144*a*b^5*c^2*d*f^2 +

144*a^5*b*c^2*d*f^2 - 360*a^2*b^4*c*d^2*f^2 - 480*a^3*b^3*c^2*d*f^2 + 360*a^4*b^2*c*d^2*f^2)^2/4 - (c^6 + d^6

+ 3*c^2*d^4 + 3*c^4*d^2)*(16*a^12*f^4 + 16*b^12*f^4 + 96*a^2*b^10*f^4 + 240*a^4*b^8*f^4 + 320*a^6*b^6*f^4 + 24

0*a^8*b^4*f^4 + 96*a^10*b^2*f^4))^(1/2) + 4*a^6*c^3*f^2 - 4*b^6*c^3*f^2 - 24*a*b^5*d^3*f^2 - 24*a^5*b*d^3*f^2

- 12*a^6*c*d^2*f^2 + 12*b^6*c*d^2*f^2 + 60*a^2*b^4*c^3*f^2 - 60*a^4*b^2*c^3*f^2 + 80*a^3*b^3*d^3*f^2 + 72*a*b^

5*c^2*d*f^2 + 72*a^5*b*c^2*d*f^2 - 180*a^2*b^4*c*d^2*f^2 - 240*a^3*b^3*c^2*d*f^2 + 180*a^4*b^2*c*d^2*f^2)/(16*

(a^12*f^4 + b^12*f^4 + 6*a^2*b^10*f^4 + 15*a^4*b^8*f^4 + 20*a^6*b^6*f^4 + 15*a^8*b^4*f^4 + 6*a^10*b^2*f^4)))^(

1/2))*(-(((8*a^6*c^3*f^2 - 8*b^6*c^3*f^2 - 48*a*b^5*d^3*f^2 - 48*a^5*b*d^3*f^2 - 24*a^6*c*d^2*f^2 + 24*b^6*c*d

^2*f^2 + 120*a^2*b^4*c^3*f^2 - 120*a^4*b^2*c^3*f^2 + 160*a^3*b^3*d^3*f^2 + 144*a*b^5*c^2*d*f^2 + 144*a^5*b*c^2

*d*f^2 - 360*a^2*b^4*c*d^2*f^2 - 480*a^3*b^3*c^2*d*f^2 + 360*a^4*b^2*c*d^2*f^2)^2/4 - (c^6 + d^6 + 3*c^2*d^4 +

3*c^4*d^2)*(16*a^12*f^4 + 16*b^12*f^4 + 96*a^2*b^10*f^4 + 240*a^4*b^8*f^4 + 320*a^6*b^6*f^4 + 240*a^8*b^4*f^4

+ 96*a^10*b^2*f^4))^(1/2) + 4*a^6*c^3*f^2 - 4*b^6*c^3*f^2 - 24*a*b^5*d^3*f^2 - 24*a^5*b*d^3*f^2 - 12*a^6*c*d^

2*f^2 + 12*b^6*c*d^2*f^2 + 60*a^2*b^4*c^3*f^2 - 60*a^4*b^2*c^3*f^2 + 80*a^3*b^3*d^3*f^2 + 72*a*b^5*c^2*d*f^2 +

72*a^5*b*c^2*d*f^2 - 180*a^2*b^4*c*d^2*f^2 - 240*a^3*b^3*c^2*d*f^2 + 180*a^4*b^2*c*d^2*f^2)/(16*(a^12*f^4 + b

^12*f^4 + 6*a^2*b^10*f^4 + 15*a^4*b^8*f^4 + 20*a^6*b^6*f^4 + 15*a^8*b^4*f^4 + 6*a^10*b^2*f^4)))^(1/2) + ((c +

d*tan(e + f*x))^(1/2)*(41*b^13*d^16 + 9*a^12*b*d^16 - 82*a^2*b^11*d^16 + 1831*a^4*b^9*d^16 - 4348*a^6*b^7*d^16

+ 1671*a^8*b^5*d^16 - 210*a^10*b^3*d^16 + 26*b^13*c^2*d^14 + 553*b^13*c^4*d^12 - 304*b^13*c^6*d^10 + 96*b^13*

c^8*d^8 - 2992*a*b^12*c^3*d^13 + 5872*a*b^12*c^5*d^11 - 1664*a*b^12*c^7*d^9 - 6304*a^3*b^10*c*d^15 + 28672*a^5

*b^8*c*d^15 - 24896*a^7*b^6*c*d^15 + 4992*a^9*b^4*c*d^15 - 288*a^11*b^2*c*d^15 - 54*a^12*b*c^2*d^14 + 9*a^12*b

*c^4*d^12 + 8060*a^2*b^11*c^2*d^14 - 28978*a^2*b^11*c^4*d^12 + 17808*a^2*b^11*c^6*d^10 - 640*a^2*b^11*c^8*d^8

+ 66096*a^3*b^10*c^3*d^13 - 83024*a^3*b^10*c^5*d^11 + 13312*a^3*b^10*c^7*d^9 - 64714*a^4*b^9*c^2*d^14 + 189607

*a^4*b^9*c^4*d^12 - 80224*a^4*b^9*c^6*d^10 + 3136*a^4*b^9*c^8*d^8 - 205664*a^5*b^8*c^3*d^13 + 210144*a^5*b^8*c

^5*d^11 - 30976*a^5*b^8*c^7*d^9 + 108936*a^6*b^7*c^2*d^14 - 255292*a^6*b^7*c^4*d^12 + 101024*a^6*b^7*c^6*d^10

- 3712*a^6*b^7*c^8*d^8 + 153568*a^7*b^6*c^3*d^13 - 138784*a^7*b^6*c^5*d^11 + 18432*a^7*b^6*c^7*d^9 - 43354*a^8

*b^5*c^2*d^14 + 92167*a^8*b^5*c^4*d^12 - 29296*a^8*b^5*c^6*d^10 + 608*a^8*b^5*c^8*d^8 - 29424*a^9*b^4*c^3*d^13

+ 20784*a^9*b^4*c^5*d^11 - 1152*a^9*b^4*c^7*d^9 + 4284*a^10*b^3*c^2*d^14 - 6834*a^10*b^3*c^4*d^12 + 720*a^10*

b^3*c^6*d^10 + 1008*a^11*b^2*c^3*d^13 - 144*a^11*b^2*c^5*d^11 + 384*a*b^12*c*d^15))/(a^16*f^4 + b^16*f^4 + 8*a

^2*b^14*f^4 + 28*a^4*b^12*f^4 + 56*a^6*b^10*f^4 + 70*a^8*b^8*f^4 + 56*a^10*b^6*f^4 + 28*a^12*b^4*f^4 + 8*a^14*

b^2*f^4))*(-(((8*a^6*c^3*f^2 - 8*b^6*c^3*f^2 - 48*a*b^5*d^3*f^2 - 48*a^5*b*d^3*f^2 - 24*a^6*c*d^2*f^2 + 24*b^6

*c*d^2*f^2 + 120*a^2*b^4*c^3*f^2 - 120*a^4*b^2*c^3*f^2 + 160*a^3*b^3*d^3*f^2 + 144*a*b^5*c^2*d*f^2 + 144*a^5*b

*c^2*d*f^2 - 360*a^2*b^4*c*d^2*f^2 - 480*a^3*b^3*c^2*d*f^2 + 360*a^4*b^2*c*d^2*f^2)^2/4 - (c^6 + d^6 + 3*c^2*d

^4 + 3*c^4*d^2)*(16*a^12*f^4 + 16*b^12*f^4 + 96*a^2*b^10*f^4 + 240*a^4*b^8*f^4 + 320*a^6*b^6*f^4 + 240*a^8*b^4

*f^4 + 96*a^10*b^2*f^4))^(1/2) + 4*a^6*c^3*f^2 - 4*b^6*c^3*f^2 - 24*a*b^5*d^3*f^2 - 24*a^5*b*d^3*f^2 - 12*a^6*

c*d^2*f^2 + 12*b^6*c*d^2*f^2 + 60*a^2*b^4*c^3*f^2 - 60*a^4*b^2*c^3*f^2 + 80*a^3*b^3*d^3*f^2 + 72*a*b^5*c^2*d*f

^2 + 72*a^5*b*c^2*d*f^2 - 180*a^2*b^4*c*d^2*f^2 - 240*a^3*b^3*c^2*d*f^2 + 180*a^4*b^2*c*d^2*f^2)/(16*(a^12*f^4

+ b^12*f^4 + 6*a^2*b^10*f^4 + 15*a^4*b^8*f^4 + 20*a^6*b^6*f^4 + 15*a^8*b^4*f^4 + 6*a^10*b^2*f^4)))^(1/2) + ((

(1036*a*b^15*d^15*f^2 - 36*a^15*b*d^15*f^2 - 604*b^16*c*d^14*f^2 - 8988*a^3*b^13*d^15*f^2 + 6044*a^5*b^11*d^15

*f^2 + 34388*a^7*b^9*d^15*f^2 + 10596*a^9*b^7*d^15*f^2 - 6676*a^11*b^5*d^15*f^2 + 1012*a^13*b^3*d^15*f^2 + 932

*b^16*c^3*d^12*f^2 + 1344*b^16*c^5*d^10*f^2 - 192*b^16*c^7*d^8*f^2 - 30836*a^2*b^14*c^3*d^12*f^2 - 48000*a^2*b

^14*c^5*d^10*f^2 + 5248*a^2*b^14*c^7*d^8*f^2 + 95076*a^3*b^13*c^2*d^13*f^2 + 57600*a^3*b^13*c^4*d^11*f^2 - 464

64*a^3*b^13*c^6*d^9*f^2 + 17172*a^4*b^12*c^3*d^12*f^2 + 69696*a^4*b^12*c^5*d^10*f^2 - 5696*a^4*b^12*c^7*d^8*f^

2 + 47004*a^5*b^11*c^2*d^13*f^2 + 10944*a^5*b^11*c^4*d^11*f^2 - 30016*a^5*b^11*c^6*d^9*f^2 + 85404*a^6*b^10*c^

3*d^12*f^2 + 169344*a^6*b^10*c^5*d^10*f^2 - 17664*a^6*b^10*c^7*d^8*f^2 - 171180*a^7*b^9*c^2*d^13*f^2 - 119808*

a^7*b^9*c^4*d^11*f^2 + 85760*a^7*b^9*c^6*d^9*f^2 - 4308*a^8*b^8*c^3*d^12*f^2 - 49728*a^8*b^8*c^5*d^10*f^2 + 37

76*a^8*b^8*c^7*d^8*f^2 - 50972*a^9*b^7*c^2*d^13*f^2 - 24768*a^9*b^7*c^4*d^11*f^2 + 36800*a^9*b^7*c^6*d^9*f^2 -

35356*a^10*b^6*c^3*d^12*f^2 - 80512*a^10*b^6*c^5*d^10*f^2 + 10368*a^10*b^6*c^7*d^8*f^2 + 55916*a^11*b^5*c^2*d

^13*f^2 + 37632*a^11*b^5*c^4*d^11*f^2 - 24960*a^11*b^5*c^6*d^9*f^2 + 6428*a^12*b^4*c^3*d^12*f^2 + 19648*a^12*b

^4*c^5*d^10*f^2 + 64*a^12*b^4*c^7*d^8*f^2 - 4876*a^13*b^3*c^2*d^13*f^2 - 6080*a^13*b^3*c^4*d^11*f^2 - 192*a^13

*b^3*c^6*d^9*f^2 + 1012*a^14*b^2*c^3*d^12*f^2 + 128*a^14*b^2*c^5*d^10*f^2 - 11380*a*b^15*c^2*d^13*f^2 - 4672*a

*b^15*c^4*d^11*f^2 + 7744*a*b^15*c^6*d^9*f^2 + 22412*a^2*b^14*c*d^14*f^2 - 58220*a^4*b^12*c*d^14*f^2 - 101604*

a^6*b^10*c*d^14*f^2 + 49196*a^8*b^8*c*d^14*f^2 + 55524*a^10*b^6*c*d^14*f^2 - 13156*a^12*b^4*c*d^14*f^2 + 884*a

^14*b^2*c*d^14*f^2 - 36*a^15*b*c^2*d^13*f^2)/(2*(a^16*f^5 + b^16*f^5 + 8*a^2*b^14*f^5 + 28*a^4*b^12*f^5 + 56*a

^6*b^10*f^5 + 70*a^8*b^8*f^5 + 56*a^10*b^6*f^5 + 28*a^12*b^4*f^5 + 8*a^14*b^2*f^5)) + (((8448*a^4*b^18*d^12*f^

4 - 640*a^2*b^20*d^12*f^4 - 384*b^22*d^12*f^4 + 44544*a^6*b^16*d^12*f^4 + 102144*a^8*b^14*d^12*f^4 + 134400*a^

10*b^12*d^12*f^4 + 107520*a^12*b^10*d^12*f^4 + 50688*a^14*b^8*d^12*f^4 + 11904*a^16*b^6*d^12*f^4 + 384*a^18*b^

4*d^12*f^4 - 256*a^20*b^2*d^12*f^4 + 384*b^22*c^2*d^10*f^4 + 768*b^22*c^4*d^8*f^4 + 4224*a^2*b^20*c^2*d^10*f^4

+ 4864*a^2*b^20*c^4*d^8*f^4 - 27136*a^3*b^19*c^3*d^9*f^4 + 19712*a^4*b^18*c^2*d^10*f^4 + 11264*a^4*b^18*c^4*d

^8*f^4 - 88064*a^5*b^17*c^3*d^9*f^4 + 51712*a^6*b^16*c^2*d^10*f^4 + 7168*a^6*b^16*c^4*d^8*f^4 - 157696*a^7*b^1

5*c^3*d^9*f^4 + 84224*a^8*b^14*c^2*d^10*f^4 - 17920*a^8*b^14*c^4*d^8*f^4 - 164864*a^9*b^13*c^3*d^9*f^4 + 87808

*a^10*b^12*c^2*d^10*f^4 - 46592*a^10*b^12*c^4*d^8*f^4 - 93184*a^11*b^11*c^3*d^9*f^4 + 57344*a^12*b^10*c^2*d^10

*f^4 - 50176*a^12*b^10*c^4*d^8*f^4 - 14336*a^13*b^9*c^3*d^9*f^4 + 20992*a^14*b^8*c^2*d^10*f^4 - 29696*a^14*b^8

*c^4*d^8*f^4 + 14336*a^15*b^7*c^3*d^9*f^4 + 2432*a^16*b^6*c^2*d^10*f^4 - 9472*a^16*b^6*c^4*d^8*f^4 + 8704*a^17

*b^5*c^3*d^9*f^4 - 896*a^18*b^4*c^2*d^10*f^4 - 1280*a^18*b^4*c^4*d^8*f^4 + 1536*a^19*b^3*c^3*d^9*f^4 - 256*a^2

0*b^2*c^2*d^10*f^4 - 3584*a*b^21*c*d^11*f^4 - 3584*a*b^21*c^3*d^9*f^4 - 27136*a^3*b^19*c*d^11*f^4 - 88064*a^5*

b^17*c*d^11*f^4 - 157696*a^7*b^15*c*d^11*f^4 - 164864*a^9*b^13*c*d^11*f^4 - 93184*a^11*b^11*c*d^11*f^4 - 14336

*a^13*b^9*c*d^11*f^4 + 14336*a^15*b^7*c*d^11*f^4 + 8704*a^17*b^5*c*d^11*f^4 + 1536*a^19*b^3*c*d^11*f^4)/(2*(a^

16*f^5 + b^16*f^5 + 8*a^2*b^14*f^5 + 28*a^4*b^12*f^5 + 56*a^6*b^10*f^5 + 70*a^8*b^8*f^5 + 56*a^10*b^6*f^5 + 28

*a^12*b^4*f^5 + 8*a^14*b^2*f^5)) - ((c + d*tan(e + f*x))^(1/2)*(-(((8*a^6*c^3*f^2 - 8*b^6*c^3*f^2 - 48*a*b^5*d

^3*f^2 - 48*a^5*b*d^3*f^2 - 24*a^6*c*d^2*f^2 + 24*b^6*c*d^2*f^2 + 120*a^2*b^4*c^3*f^2 - 120*a^4*b^2*c^3*f^2 +

160*a^3*b^3*d^3*f^2 + 144*a*b^5*c^2*d*f^2 + 144*a^5*b*c^2*d*f^2 - 360*a^2*b^4*c*d^2*f^2 - 480*a^3*b^3*c^2*d*f^

2 + 360*a^4*b^2*c*d^2*f^2)^2/4 - (c^6 + d^6 + 3*c^2*d^4 + 3*c^4*d^2)*(16*a^12*f^4 + 16*b^12*f^4 + 96*a^2*b^10*

f^4 + 240*a^4*b^8*f^4 + 320*a^6*b^6*f^4 + 240*a^8*b^4*f^4 + 96*a^10*b^2*f^4))^(1/2) + 4*a^6*c^3*f^2 - 4*b^6*c^

3*f^2 - 24*a*b^5*d^3*f^2 - 24*a^5*b*d^3*f^2 - 12*a^6*c*d^2*f^2 + 12*b^6*c*d^2*f^2 + 60*a^2*b^4*c^3*f^2 - 60*a^

4*b^2*c^3*f^2 + 80*a^3*b^3*d^3*f^2 + 72*a*b^5*c^2*d*f^2 + 72*a^5*b*c^2*d*f^2 - 180*a^2*b^4*c*d^2*f^2 - 240*a^3

*b^3*c^2*d*f^2 + 180*a^4*b^2*c*d^2*f^2)/(16*(a^12*f^4 + b^12*f^4 + 6*a^2*b^10*f^4 + 15*a^4*b^8*f^4 + 20*a^6*b^

6*f^4 + 15*a^8*b^4*f^4 + 6*a^10*b^2*f^4)))^(1/2)*(512*b^25*d^10*f^4 + 4608*a^2*b^23*d^10*f^4 + 17920*a^4*b^21*

d^10*f^4 + 38400*a^6*b^19*d^10*f^4 + 46080*a^8*b^17*d^10*f^4 + 21504*a^10*b^15*d^10*f^4 - 21504*a^12*b^13*d^10

*f^4 - 46080*a^14*b^11*d^10*f^4 - 38400*a^16*b^9*d^10*f^4 - 17920*a^18*b^7*d^10*f^4 - 4608*a^20*b^5*d^10*f^4 -

512*a^22*b^3*d^10*f^4 + 768*b^25*c^2*d^8*f^4 + 7424*a^2*b^23*c^2*d^8*f^4 + 32000*a^4*b^21*c^2*d^8*f^4 + 80640

*a^6*b^19*c^2*d^8*f^4 + 130560*a^8*b^17*c^2*d^8*f^4 + 139776*a^10*b^15*c^2*d^8*f^4 + 96768*a^12*b^13*c^2*d^8*f

^4 + 38400*a^14*b^11*c^2*d^8*f^4 + 3840*a^16*b^9*c^2*d^8*f^4 - 3840*a^18*b^7*c^2*d^8*f^4 - 1792*a^20*b^5*c^2*d

^8*f^4 - 256*a^22*b^3*c^2*d^8*f^4 + 256*a*b^24*c*d^9*f^4 + 2816*a^3*b^22*c*d^9*f^4 + 14080*a^5*b^20*c*d^9*f^4

+ 42240*a^7*b^18*c*d^9*f^4 + 84480*a^9*b^16*c*d^9*f^4 + 118272*a^11*b^14*c*d^9*f^4 + 118272*a^13*b^12*c*d^9*f^

4 + 84480*a^15*b^10*c*d^9*f^4 + 42240*a^17*b^8*c*d^9*f^4 + 14080*a^19*b^6*c*d^9*f^4 + 2816*a^21*b^4*c*d^9*f^4

+ 256*a^23*b^2*c*d^9*f^4))/(a^16*f^4 + b^16*f^4 + 8*a^2*b^14*f^4 + 28*a^4*b^12*f^4 + 56*a^6*b^10*f^4 + 70*a^8*