3.1252 \(\int \frac {1}{(a+b \tan (e+f x))^2 \sqrt {c+d \tan (e+f x)}} \, dx\)

Optimal. Leaf size=244 \[ -\frac {b^2 \sqrt {c+d \tan (e+f x)}}{f \left (a^2+b^2\right ) (b c-a d) (a+b \tan (e+f x))}-\frac {b^{3/2} \left (-5 a^2 d+4 a b c-b^2 d\right ) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d \tan (e+f x)}}{\sqrt {b c-a d}}\right )}{f \left (a^2+b^2\right )^2 (b c-a d)^{3/2}}-\frac {i \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d}}\right )}{f (a-i b)^2 \sqrt {c-i d}}+\frac {i \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c+i d}}\right )}{f (a+i b)^2 \sqrt {c+i d}} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.85, antiderivative size = 244, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 7, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.259, Rules used = {3569, 3653, 3539, 3537, 63, 208, 3634} \[ -\frac {b^2 \sqrt {c+d \tan (e+f x)}}{f \left (a^2+b^2\right ) (b c-a d) (a+b \tan (e+f x))}-\frac {b^{3/2} \left (-5 a^2 d+4 a b c-b^2 d\right ) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d \tan (e+f x)}}{\sqrt {b c-a d}}\right )}{f \left (a^2+b^2\right )^2 (b c-a d)^{3/2}}-\frac {i \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d}}\right )}{f (a-i b)^2 \sqrt {c-i d}}+\frac {i \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c+i d}}\right )}{f (a+i b)^2 \sqrt {c+i d}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((-I)*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c - I*d]])/((a - I*b)^2*Sqrt[c - I*d]*f) + (I*ArcTanh[Sqrt[c + d*T
an[e + f*x]]/Sqrt[c + I*d]])/((a + I*b)^2*Sqrt[c + I*d]*f) - (b^(3/2)*(4*a*b*c - 5*a^2*d - b^2*d)*ArcTanh[(Sqr
t[b]*Sqrt[c + d*Tan[e + f*x]])/Sqrt[b*c - a*d]])/((a^2 + b^2)^2*(b*c - a*d)^(3/2)*f) - (b^2*Sqrt[c + d*Tan[e +
f*x]])/((a^2 + b^2)*(b*c - a*d)*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 3569

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Si
mp[(b^2*(a + b*Tan[e + f*x])^(m + 1)*(c + d*Tan[e + f*x])^(n + 1))/(f*(m + 1)*(a^2 + b^2)*(b*c - a*d)), x] + D
ist[1/((m + 1)*(a^2 + b^2)*(b*c - a*d)), Int[(a + b*Tan[e + f*x])^(m + 1)*(c + d*Tan[e + f*x])^n*Simp[a*(b*c -
a*d)*(m + 1) - b^2*d*(m + n + 2) - b*(b*c - a*d)*(m + 1)*Tan[e + f*x] - b^2*d*(m + n + 2)*Tan[e + f*x]^2, x],
x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && I
ntegerQ[2*m] && LtQ[m, -1] && (LtQ[n, 0] || IntegerQ[m]) &&  !(ILtQ[n, -1] && ( !IntegerQ[m] || (EqQ[c, 0] &&
NeQ[a, 0])))

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 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, -
1]

Rubi steps

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

________________________________________________________________________________________

Mathematica [A]  time = 2.39, size = 258, normalized size = 1.06 \[ \frac {-\frac {i \left (\frac {(a-i b)^2 (a d-b c) \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c+i d}}\right )}{\sqrt {c+i d}}+\frac {(a+i b)^2 (b c-a d) \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d}}\right )}{\sqrt {c-i d}}\right )}{a^2+b^2}+\frac {b^{3/2} \left (5 a^2 d-4 a b c+b^2 d\right ) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d \tan (e+f x)}}{\sqrt {b c-a d}}\right )}{\left (a^2+b^2\right ) \sqrt {b c-a d}}-\frac {b^2 \sqrt {c+d \tan (e+f x)}}{a+b \tan (e+f x)}}{f \left (a^2+b^2\right ) (b c-a d)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

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(1/(c+d*tan(f*x+e))^(1/2)/(a+b*tan(f*x+e))^2,x, algorithm="fricas")

[Out]

Timed out

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(c+d*tan(f*x+e))^(1/2)/(a+b*tan(f*x+e))^2,x, algorithm="giac")

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,x):;OUTPUT:Unab
le to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*p
i/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unab
le to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*p
i/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unab
le to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*p
i/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unab
le to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*p
i/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unab
le to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*p
i/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unab
le to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*p
i/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unab
le to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*p
i/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unab
le to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*p
i/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unab
le to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*p
i/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unab
le to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*p
i/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unab
le to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*p
i/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unab
le to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*p
i/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unab
le to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*p
i/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unab
le to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*p
i/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Warn
ing, need to choose a branch for the root of a polynomial with parameters. This might be wrong.The choice was
done assuming [a,b,f]=[32,-66]Warning, need to choose a branch for the root of a polynomial with parameters. T
his might be wrong.The choice was done assuming [a,b,f]=[39,65]sym2poly/r2sym(const gen & e,const index_m & i,
const vecteur & l) Error: Bad Argument ValueWarning, need to choose a branch for the root of a polynomial with
parameters. This might be wrong.The choice was done assuming [a,b,f]=[-23,-48]Warning, need to choose a branc
h for the root of a polynomial with parameters. This might be wrong.The choice was done assuming [a,b,f]=[63,-
67]sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument ValueWarning, choosin
g root of [1,0,0,0,%%%{-1,[0,0,2,0,0]%%%}+%%%{-1,[0,0,0,2,0]%%%}] at parameters values [70,22,42,56,-9]Warning
, choosing root of [1,0,0,0,%%%{-1,[0,0,2,0,0]%%%}+%%%{-1,[0,0,0,2,0]%%%}] at parameters values [-13,46,24,49,
-6]Warning, choosing root of [1,0,0,0,%%%{-1,[0,0,2,0,0]%%%}+%%%{-1,[0,0,0,2,0]%%%}] at parameters values [-70
,8,63,-64,2]Warning, choosing root of [1,0,0,0,%%%{-1,[0,0,2,0,0]%%%}+%%%{-1,[0,0,0,2,0]%%%}] at parameters va
lues [62,-37,-80,-23,65]Warning, choosing root of [1,0,0,0,%%%{-1,[0,0,2,0,0]%%%}+%%%{-1,[0,0,0,2,0]%%%}] at p
arameters values [-85,28,-44,-22,93]Warning, choosing root of [1,0,0,0,%%%{-1,[0,0,2,0,0]%%%}+%%%{-1,[0,0,0,2,
0]%%%}] at parameters values [91,31,-21,88,76]Warning, choosing root of [1,0,0,0,%%%{-1,[0,0,2,0,0]%%%}+%%%{-1
,[0,0,0,2,0]%%%}] at parameters values [-66,66,5,-23,79]Warning, choosing root of [1,0,0,0,%%%{-1,[0,0,2,0,0]%
%%}+%%%{-1,[0,0,0,2,0]%%%}] at parameters values [-88,9,6,-69,-8]Warning, need to choose a branch for the root
of a polynomial with parameters. This might be wrong.The choice was done assuming [tan(f*x+exp(1))]=[48,-92,3
0,41,55]Warning, need to choose a branch for the root of a polynomial with parameters. This might be wrong.The
choice was done assuming [tan(f*x+exp(1))]=[63,82,97,51,90]Warning, need to choose a branch for the root of a
polynomial with parameters. This might be wrong.The choice was done assuming [tan(f*x+exp(1))]=[-64,-40,-89,-
64,-67]sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument ValueWarning, nee
d to choose a branch for the root of a polynomial with parameters. This might be wrong.The choice was done ass
uming [tan(f*x+exp(1))]=[-78,63,-93,-25,61]Warning, need to choose a branch for the root of a polynomial with
parameters. This might be wrong.The choice was done assuming [tan(f*x+exp(1))]=[-92,-18,-23,24,16]Warning, nee
d to choose a branch for the root of a polynomial with parameters. This might be wrong.The choice was done ass
uming [tan(f*x+exp(1))]=[-12,0,62,-54,3]sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Erro
r: Bad Argument ValueEvaluation time: 170.46Done

________________________________________________________________________________________

maple [B]  time = 0.40, size = 6548, normalized size = 26.84 \[ \text {output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

result too large to display

________________________________________________________________________________________

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(1/(c+d*tan(f*x+e))^(1/2)/(a+b*tan(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(a*d-b*c>0)', see `assume?` for
more details)Is a*d-b*c positive or negative?

________________________________________________________________________________________

mupad [B]  time = 15.26, size = 51069, normalized size = 209.30 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(atan(((((((16*(2*b^13*d^11*f^2 - 24*a^2*b^11*d^11*f^2 - 196*a^4*b^9*d^11*f^2 - 120*a^6*b^7*d^11*f^2 + 50*a^8*
b^5*d^11*f^2 + 8*b^13*c^2*d^9*f^2 - 8*a^2*b^11*c^2*d^9*f^2 + 64*a^3*b^10*c^3*d^8*f^2 - 232*a^4*b^9*c^2*d^9*f^2
+ 96*a^5*b^8*c^3*d^8*f^2 - 216*a^6*b^7*c^2*d^9*f^2 - 32*a*b^12*c^3*d^8*f^2 + 208*a^3*b^10*c*d^10*f^2 + 288*a^
5*b^8*c*d^10*f^2 + 80*a^7*b^6*c*d^10*f^2))/(a^10*d^2*f^5 + b^10*c^2*f^5 + 4*a^2*b^8*c^2*f^5 + 6*a^4*b^6*c^2*f^
5 + 4*a^6*b^4*c^2*f^5 + a^8*b^2*c^2*f^5 + a^2*b^8*d^2*f^5 + 4*a^4*b^6*d^2*f^5 + 6*a^6*b^4*d^2*f^5 + 4*a^8*b^2*
d^2*f^5 - 2*a*b^9*c*d*f^5 - 2*a^9*b*c*d*f^5 - 8*a^3*b^7*c*d*f^5 - 12*a^5*b^5*c*d*f^5 - 8*a^7*b^3*c*d*f^5) + ((
(16*(c + d*tan(e + f*x))^(1/2)*(8*a*b^14*d^11*f^2 + 4*b^15*c*d^10*f^2 + 36*a^3*b^12*d^11*f^2 + 316*a^5*b^10*d^
11*f^2 + 552*a^7*b^8*d^11*f^2 + 256*a^9*b^6*d^11*f^2 - 12*a^11*b^4*d^11*f^2 - 4*a^13*b^2*d^11*f^2 - 20*b^15*c^
3*d^8*f^2 + 116*a^2*b^13*c^3*d^8*f^2 - 220*a^3*b^12*c^2*d^9*f^2 + 216*a^4*b^11*c^3*d^8*f^2 - 104*a^5*b^10*c^2*
d^9*f^2 + 8*a^6*b^9*c^3*d^8*f^2 + 232*a^7*b^8*c^2*d^9*f^2 - 68*a^8*b^7*c^3*d^8*f^2 + 156*a^9*b^6*c^2*d^9*f^2 +
4*a^10*b^5*c^3*d^8*f^2 - 12*a^11*b^4*c^2*d^9*f^2 - 52*a*b^14*c^2*d^9*f^2 + 80*a^2*b^13*c*d^10*f^2 - 156*a^4*b
^11*c*d^10*f^2 - 640*a^6*b^9*c*d^10*f^2 - 500*a^8*b^7*c*d^10*f^2 - 80*a^10*b^5*c*d^10*f^2 + 12*a^12*b^3*c*d^10
*f^2))/(a^10*d^2*f^4 + b^10*c^2*f^4 + 4*a^2*b^8*c^2*f^4 + 6*a^4*b^6*c^2*f^4 + 4*a^6*b^4*c^2*f^4 + a^8*b^2*c^2*
f^4 + a^2*b^8*d^2*f^4 + 4*a^4*b^6*d^2*f^4 + 6*a^6*b^4*d^2*f^4 + 4*a^8*b^2*d^2*f^4 - 2*a*b^9*c*d*f^4 - 2*a^9*b*
c*d*f^4 - 8*a^3*b^7*c*d*f^4 - 12*a^5*b^5*c*d*f^4 - 8*a^7*b^3*c*d*f^4) - (((16*(16*a*b^16*d^12*f^4 - 16*b^17*c*
d^11*f^4 + 136*a^3*b^14*d^12*f^4 + 432*a^5*b^12*d^12*f^4 + 680*a^7*b^10*d^12*f^4 + 560*a^9*b^8*d^12*f^4 + 216*
a^11*b^6*d^12*f^4 + 16*a^13*b^4*d^12*f^4 - 8*a^15*b^2*d^12*f^4 - 8*b^17*c^3*d^9*f^4 - 128*a^2*b^15*c^3*d^9*f^4
+ 352*a^3*b^14*c^2*d^10*f^4 + 160*a^3*b^14*c^4*d^8*f^4 - 520*a^4*b^13*c^3*d^9*f^4 + 920*a^5*b^12*c^2*d^10*f^4
+ 320*a^5*b^12*c^4*d^8*f^4 - 960*a^6*b^11*c^3*d^9*f^4 + 1280*a^7*b^10*c^2*d^10*f^4 + 320*a^7*b^10*c^4*d^8*f^4
- 920*a^8*b^9*c^3*d^9*f^4 + 1000*a^9*b^8*c^2*d^10*f^4 + 160*a^9*b^8*c^4*d^8*f^4 - 448*a^10*b^7*c^3*d^9*f^4 +
416*a^11*b^6*c^2*d^10*f^4 + 32*a^11*b^6*c^4*d^8*f^4 - 88*a^12*b^5*c^3*d^9*f^4 + 72*a^13*b^4*c^2*d^10*f^4 + 56*
a*b^16*c^2*d^10*f^4 + 32*a*b^16*c^4*d^8*f^4 - 184*a^2*b^15*c*d^11*f^4 - 688*a^4*b^13*c*d^11*f^4 - 1240*a^6*b^1
1*c*d^11*f^4 - 1200*a^8*b^9*c*d^11*f^4 - 616*a^10*b^7*c*d^11*f^4 - 144*a^12*b^5*c*d^11*f^4 - 8*a^14*b^3*c*d^11
*f^4))/(a^10*d^2*f^5 + b^10*c^2*f^5 + 4*a^2*b^8*c^2*f^5 + 6*a^4*b^6*c^2*f^5 + 4*a^6*b^4*c^2*f^5 + a^8*b^2*c^2*
f^5 + a^2*b^8*d^2*f^5 + 4*a^4*b^6*d^2*f^5 + 6*a^6*b^4*d^2*f^5 + 4*a^8*b^2*d^2*f^5 - 2*a*b^9*c*d*f^5 - 2*a^9*b*
c*d*f^5 - 8*a^3*b^7*c*d*f^5 - 12*a^5*b^5*c*d*f^5 - 8*a^7*b^3*c*d*f^5) + (16*(-(b^7*d^2 + 16*a^2*b^5*c^2 + 10*a
^2*b^5*d^2 + 25*a^4*b^3*d^2 - 8*a*b^6*c*d - 40*a^3*b^4*c*d)*(a^11*d^3*f^2 - b^11*c^3*f^2 - 4*a^2*b^9*c^3*f^2 -
6*a^4*b^7*c^3*f^2 - 4*a^6*b^5*c^3*f^2 - a^8*b^3*c^3*f^2 + a^3*b^8*d^3*f^2 + 4*a^5*b^6*d^3*f^2 + 6*a^7*b^4*d^3
*f^2 + 4*a^9*b^2*d^3*f^2 + 3*a*b^10*c^2*d*f^2 - 3*a^10*b*c*d^2*f^2 - 3*a^2*b^9*c*d^2*f^2 + 12*a^3*b^8*c^2*d*f^
2 - 12*a^4*b^7*c*d^2*f^2 + 18*a^5*b^6*c^2*d*f^2 - 18*a^6*b^5*c*d^2*f^2 + 12*a^7*b^4*c^2*d*f^2 - 12*a^8*b^3*c*d
^2*f^2 + 3*a^9*b^2*c^2*d*f^2))^(1/2)*(c + d*tan(e + f*x))^(1/2)*(32*a^2*b^17*d^12*f^4 + 160*a^4*b^15*d^12*f^4
+ 288*a^6*b^13*d^12*f^4 + 160*a^8*b^11*d^12*f^4 - 160*a^10*b^9*d^12*f^4 - 288*a^12*b^7*d^12*f^4 - 160*a^14*b^5
*d^12*f^4 - 32*a^16*b^3*d^12*f^4 + 32*b^19*c^2*d^10*f^4 + 48*b^19*c^4*d^8*f^4 + 176*a^2*b^17*c^2*d^10*f^4 + 27
2*a^2*b^17*c^4*d^8*f^4 - 432*a^3*b^16*c^3*d^9*f^4 + 336*a^4*b^15*c^2*d^10*f^4 + 624*a^4*b^15*c^4*d^8*f^4 - 912
*a^5*b^14*c^3*d^9*f^4 + 112*a^6*b^13*c^2*d^10*f^4 + 720*a^6*b^13*c^4*d^8*f^4 - 880*a^7*b^12*c^3*d^9*f^4 - 560*
a^8*b^11*c^2*d^10*f^4 + 400*a^8*b^11*c^4*d^8*f^4 - 240*a^9*b^10*c^3*d^9*f^4 - 1008*a^10*b^9*c^2*d^10*f^4 + 48*
a^10*b^9*c^4*d^8*f^4 + 240*a^11*b^8*c^3*d^9*f^4 - 784*a^12*b^7*c^2*d^10*f^4 - 48*a^12*b^7*c^4*d^8*f^4 + 208*a^
13*b^6*c^3*d^9*f^4 - 304*a^14*b^5*c^2*d^10*f^4 - 16*a^14*b^5*c^4*d^8*f^4 + 48*a^15*b^4*c^3*d^9*f^4 - 48*a^16*b
^3*c^2*d^10*f^4 - 64*a*b^18*c*d^11*f^4 - 80*a*b^18*c^3*d^9*f^4 - 304*a^3*b^16*c*d^11*f^4 - 464*a^5*b^14*c*d^11
*f^4 + 16*a^7*b^12*c*d^11*f^4 + 880*a^9*b^10*c*d^11*f^4 + 1136*a^11*b^8*c*d^11*f^4 + 656*a^13*b^6*c*d^11*f^4 +
176*a^15*b^4*c*d^11*f^4 + 16*a^17*b^2*c*d^11*f^4))/((b^9*(8*a^2*c^3*f^2 + 6*a^2*c*d^2*f^2) + b^3*(2*a^8*c^3*f
^2 + 24*a^8*c*d^2*f^2) + b^7*(12*a^4*c^3*f^2 + 24*a^4*c*d^2*f^2) + b^5*(8*a^6*c^3*f^2 + 36*a^6*c*d^2*f^2) - b^
2*(8*a^9*d^3*f^2 + 6*a^9*c^2*d*f^2) - b^8*(2*a^3*d^3*f^2 + 24*a^3*c^2*d*f^2) - b^4*(12*a^7*d^3*f^2 + 24*a^7*c^
2*d*f^2) - b^6*(8*a^5*d^3*f^2 + 36*a^5*c^2*d*f^2) - 2*a^11*d^3*f^2 + 2*b^11*c^3*f^2 - 6*a*b^10*c^2*d*f^2 + 6*a
^10*b*c*d^2*f^2)*(a^10*d^2*f^4 + b^10*c^2*f^4 + 4*a^2*b^8*c^2*f^4 + 6*a^4*b^6*c^2*f^4 + 4*a^6*b^4*c^2*f^4 + a^
8*b^2*c^2*f^4 + a^2*b^8*d^2*f^4 + 4*a^4*b^6*d^2*f^4 + 6*a^6*b^4*d^2*f^4 + 4*a^8*b^2*d^2*f^4 - 2*a*b^9*c*d*f^4
- 2*a^9*b*c*d*f^4 - 8*a^3*b^7*c*d*f^4 - 12*a^5*b^5*c*d*f^4 - 8*a^7*b^3*c*d*f^4)))*(-(b^7*d^2 + 16*a^2*b^5*c^2
+ 10*a^2*b^5*d^2 + 25*a^4*b^3*d^2 - 8*a*b^6*c*d - 40*a^3*b^4*c*d)*(a^11*d^3*f^2 - b^11*c^3*f^2 - 4*a^2*b^9*c^3
*f^2 - 6*a^4*b^7*c^3*f^2 - 4*a^6*b^5*c^3*f^2 - a^8*b^3*c^3*f^2 + a^3*b^8*d^3*f^2 + 4*a^5*b^6*d^3*f^2 + 6*a^7*b
^4*d^3*f^2 + 4*a^9*b^2*d^3*f^2 + 3*a*b^10*c^2*d*f^2 - 3*a^10*b*c*d^2*f^2 - 3*a^2*b^9*c*d^2*f^2 + 12*a^3*b^8*c^
2*d*f^2 - 12*a^4*b^7*c*d^2*f^2 + 18*a^5*b^6*c^2*d*f^2 - 18*a^6*b^5*c*d^2*f^2 + 12*a^7*b^4*c^2*d*f^2 - 12*a^8*b
^3*c*d^2*f^2 + 3*a^9*b^2*c^2*d*f^2))^(1/2))/(b^9*(8*a^2*c^3*f^2 + 6*a^2*c*d^2*f^2) + b^3*(2*a^8*c^3*f^2 + 24*a
^8*c*d^2*f^2) + b^7*(12*a^4*c^3*f^2 + 24*a^4*c*d^2*f^2) + b^5*(8*a^6*c^3*f^2 + 36*a^6*c*d^2*f^2) - b^2*(8*a^9*
d^3*f^2 + 6*a^9*c^2*d*f^2) - b^8*(2*a^3*d^3*f^2 + 24*a^3*c^2*d*f^2) - b^4*(12*a^7*d^3*f^2 + 24*a^7*c^2*d*f^2)
- b^6*(8*a^5*d^3*f^2 + 36*a^5*c^2*d*f^2) - 2*a^11*d^3*f^2 + 2*b^11*c^3*f^2 - 6*a*b^10*c^2*d*f^2 + 6*a^10*b*c*d
^2*f^2))*(-(b^7*d^2 + 16*a^2*b^5*c^2 + 10*a^2*b^5*d^2 + 25*a^4*b^3*d^2 - 8*a*b^6*c*d - 40*a^3*b^4*c*d)*(a^11*d
^3*f^2 - b^11*c^3*f^2 - 4*a^2*b^9*c^3*f^2 - 6*a^4*b^7*c^3*f^2 - 4*a^6*b^5*c^3*f^2 - a^8*b^3*c^3*f^2 + a^3*b^8*
d^3*f^2 + 4*a^5*b^6*d^3*f^2 + 6*a^7*b^4*d^3*f^2 + 4*a^9*b^2*d^3*f^2 + 3*a*b^10*c^2*d*f^2 - 3*a^10*b*c*d^2*f^2
- 3*a^2*b^9*c*d^2*f^2 + 12*a^3*b^8*c^2*d*f^2 - 12*a^4*b^7*c*d^2*f^2 + 18*a^5*b^6*c^2*d*f^2 - 18*a^6*b^5*c*d^2*
f^2 + 12*a^7*b^4*c^2*d*f^2 - 12*a^8*b^3*c*d^2*f^2 + 3*a^9*b^2*c^2*d*f^2))^(1/2))/(b^9*(8*a^2*c^3*f^2 + 6*a^2*c
*d^2*f^2) + b^3*(2*a^8*c^3*f^2 + 24*a^8*c*d^2*f^2) + b^7*(12*a^4*c^3*f^2 + 24*a^4*c*d^2*f^2) + b^5*(8*a^6*c^3*
f^2 + 36*a^6*c*d^2*f^2) - b^2*(8*a^9*d^3*f^2 + 6*a^9*c^2*d*f^2) - b^8*(2*a^3*d^3*f^2 + 24*a^3*c^2*d*f^2) - b^4
*(12*a^7*d^3*f^2 + 24*a^7*c^2*d*f^2) - b^6*(8*a^5*d^3*f^2 + 36*a^5*c^2*d*f^2) - 2*a^11*d^3*f^2 + 2*b^11*c^3*f^
2 - 6*a*b^10*c^2*d*f^2 + 6*a^10*b*c*d^2*f^2))*(-(b^7*d^2 + 16*a^2*b^5*c^2 + 10*a^2*b^5*d^2 + 25*a^4*b^3*d^2 -
8*a*b^6*c*d - 40*a^3*b^4*c*d)*(a^11*d^3*f^2 - b^11*c^3*f^2 - 4*a^2*b^9*c^3*f^2 - 6*a^4*b^7*c^3*f^2 - 4*a^6*b^5
*c^3*f^2 - a^8*b^3*c^3*f^2 + a^3*b^8*d^3*f^2 + 4*a^5*b^6*d^3*f^2 + 6*a^7*b^4*d^3*f^2 + 4*a^9*b^2*d^3*f^2 + 3*a
*b^10*c^2*d*f^2 - 3*a^10*b*c*d^2*f^2 - 3*a^2*b^9*c*d^2*f^2 + 12*a^3*b^8*c^2*d*f^2 - 12*a^4*b^7*c*d^2*f^2 + 18*
a^5*b^6*c^2*d*f^2 - 18*a^6*b^5*c*d^2*f^2 + 12*a^7*b^4*c^2*d*f^2 - 12*a^8*b^3*c*d^2*f^2 + 3*a^9*b^2*c^2*d*f^2))
^(1/2))/(b^9*(8*a^2*c^3*f^2 + 6*a^2*c*d^2*f^2) + b^3*(2*a^8*c^3*f^2 + 24*a^8*c*d^2*f^2) + b^7*(12*a^4*c^3*f^2
+ 24*a^4*c*d^2*f^2) + b^5*(8*a^6*c^3*f^2 + 36*a^6*c*d^2*f^2) - b^2*(8*a^9*d^3*f^2 + 6*a^9*c^2*d*f^2) - b^8*(2*
a^3*d^3*f^2 + 24*a^3*c^2*d*f^2) - b^4*(12*a^7*d^3*f^2 + 24*a^7*c^2*d*f^2) - b^6*(8*a^5*d^3*f^2 + 36*a^5*c^2*d*
f^2) - 2*a^11*d^3*f^2 + 2*b^11*c^3*f^2 - 6*a*b^10*c^2*d*f^2 + 6*a^10*b*c*d^2*f^2) + (16*(c + d*tan(e + f*x))^(
1/2)*(b^11*d^10 + 7*a^2*b^9*d^10 + 11*a^4*b^7*d^10 - 27*a^6*b^5*d^10 - 2*b^11*c^2*d^8 - 24*a^3*b^8*c*d^9 + 44*
a^5*b^6*c*d^9 + 12*a^2*b^9*c^2*d^8 - 18*a^4*b^7*c^2*d^8 - 4*a*b^10*c*d^9))/(a^10*d^2*f^4 + b^10*c^2*f^4 + 4*a^
2*b^8*c^2*f^4 + 6*a^4*b^6*c^2*f^4 + 4*a^6*b^4*c^2*f^4 + a^8*b^2*c^2*f^4 + a^2*b^8*d^2*f^4 + 4*a^4*b^6*d^2*f^4
+ 6*a^6*b^4*d^2*f^4 + 4*a^8*b^2*d^2*f^4 - 2*a*b^9*c*d*f^4 - 2*a^9*b*c*d*f^4 - 8*a^3*b^7*c*d*f^4 - 12*a^5*b^5*c
*d*f^4 - 8*a^7*b^3*c*d*f^4))*(-(b^7*d^2 + 16*a^2*b^5*c^2 + 10*a^2*b^5*d^2 + 25*a^4*b^3*d^2 - 8*a*b^6*c*d - 40*
a^3*b^4*c*d)*(a^11*d^3*f^2 - b^11*c^3*f^2 - 4*a^2*b^9*c^3*f^2 - 6*a^4*b^7*c^3*f^2 - 4*a^6*b^5*c^3*f^2 - a^8*b^
3*c^3*f^2 + a^3*b^8*d^3*f^2 + 4*a^5*b^6*d^3*f^2 + 6*a^7*b^4*d^3*f^2 + 4*a^9*b^2*d^3*f^2 + 3*a*b^10*c^2*d*f^2 -
3*a^10*b*c*d^2*f^2 - 3*a^2*b^9*c*d^2*f^2 + 12*a^3*b^8*c^2*d*f^2 - 12*a^4*b^7*c*d^2*f^2 + 18*a^5*b^6*c^2*d*f^2
- 18*a^6*b^5*c*d^2*f^2 + 12*a^7*b^4*c^2*d*f^2 - 12*a^8*b^3*c*d^2*f^2 + 3*a^9*b^2*c^2*d*f^2))^(1/2)*1i)/(b^9*(
8*a^2*c^3*f^2 + 6*a^2*c*d^2*f^2) + b^3*(2*a^8*c^3*f^2 + 24*a^8*c*d^2*f^2) + b^7*(12*a^4*c^3*f^2 + 24*a^4*c*d^2
*f^2) + b^5*(8*a^6*c^3*f^2 + 36*a^6*c*d^2*f^2) - b^2*(8*a^9*d^3*f^2 + 6*a^9*c^2*d*f^2) - b^8*(2*a^3*d^3*f^2 +
24*a^3*c^2*d*f^2) - b^4*(12*a^7*d^3*f^2 + 24*a^7*c^2*d*f^2) - b^6*(8*a^5*d^3*f^2 + 36*a^5*c^2*d*f^2) - 2*a^11*
d^3*f^2 + 2*b^11*c^3*f^2 - 6*a*b^10*c^2*d*f^2 + 6*a^10*b*c*d^2*f^2) - (((((16*(2*b^13*d^11*f^2 - 24*a^2*b^11*d
^11*f^2 - 196*a^4*b^9*d^11*f^2 - 120*a^6*b^7*d^11*f^2 + 50*a^8*b^5*d^11*f^2 + 8*b^13*c^2*d^9*f^2 - 8*a^2*b^11*
c^2*d^9*f^2 + 64*a^3*b^10*c^3*d^8*f^2 - 232*a^4*b^9*c^2*d^9*f^2 + 96*a^5*b^8*c^3*d^8*f^2 - 216*a^6*b^7*c^2*d^9
*f^2 - 32*a*b^12*c^3*d^8*f^2 + 208*a^3*b^10*c*d^10*f^2 + 288*a^5*b^8*c*d^10*f^2 + 80*a^7*b^6*c*d^10*f^2))/(a^1
0*d^2*f^5 + b^10*c^2*f^5 + 4*a^2*b^8*c^2*f^5 + 6*a^4*b^6*c^2*f^5 + 4*a^6*b^4*c^2*f^5 + a^8*b^2*c^2*f^5 + a^2*b
^8*d^2*f^5 + 4*a^4*b^6*d^2*f^5 + 6*a^6*b^4*d^2*f^5 + 4*a^8*b^2*d^2*f^5 - 2*a*b^9*c*d*f^5 - 2*a^9*b*c*d*f^5 - 8
*a^3*b^7*c*d*f^5 - 12*a^5*b^5*c*d*f^5 - 8*a^7*b^3*c*d*f^5) - (((16*(c + d*tan(e + f*x))^(1/2)*(8*a*b^14*d^11*f
^2 + 4*b^15*c*d^10*f^2 + 36*a^3*b^12*d^11*f^2 + 316*a^5*b^10*d^11*f^2 + 552*a^7*b^8*d^11*f^2 + 256*a^9*b^6*d^1
1*f^2 - 12*a^11*b^4*d^11*f^2 - 4*a^13*b^2*d^11*f^2 - 20*b^15*c^3*d^8*f^2 + 116*a^2*b^13*c^3*d^8*f^2 - 220*a^3*
b^12*c^2*d^9*f^2 + 216*a^4*b^11*c^3*d^8*f^2 - 104*a^5*b^10*c^2*d^9*f^2 + 8*a^6*b^9*c^3*d^8*f^2 + 232*a^7*b^8*c
^2*d^9*f^2 - 68*a^8*b^7*c^3*d^8*f^2 + 156*a^9*b^6*c^2*d^9*f^2 + 4*a^10*b^5*c^3*d^8*f^2 - 12*a^11*b^4*c^2*d^9*f
^2 - 52*a*b^14*c^2*d^9*f^2 + 80*a^2*b^13*c*d^10*f^2 - 156*a^4*b^11*c*d^10*f^2 - 640*a^6*b^9*c*d^10*f^2 - 500*a
^8*b^7*c*d^10*f^2 - 80*a^10*b^5*c*d^10*f^2 + 12*a^12*b^3*c*d^10*f^2))/(a^10*d^2*f^4 + b^10*c^2*f^4 + 4*a^2*b^8
*c^2*f^4 + 6*a^4*b^6*c^2*f^4 + 4*a^6*b^4*c^2*f^4 + a^8*b^2*c^2*f^4 + a^2*b^8*d^2*f^4 + 4*a^4*b^6*d^2*f^4 + 6*a
^6*b^4*d^2*f^4 + 4*a^8*b^2*d^2*f^4 - 2*a*b^9*c*d*f^4 - 2*a^9*b*c*d*f^4 - 8*a^3*b^7*c*d*f^4 - 12*a^5*b^5*c*d*f^
4 - 8*a^7*b^3*c*d*f^4) + (((16*(16*a*b^16*d^12*f^4 - 16*b^17*c*d^11*f^4 + 136*a^3*b^14*d^12*f^4 + 432*a^5*b^12
*d^12*f^4 + 680*a^7*b^10*d^12*f^4 + 560*a^9*b^8*d^12*f^4 + 216*a^11*b^6*d^12*f^4 + 16*a^13*b^4*d^12*f^4 - 8*a^
15*b^2*d^12*f^4 - 8*b^17*c^3*d^9*f^4 - 128*a^2*b^15*c^3*d^9*f^4 + 352*a^3*b^14*c^2*d^10*f^4 + 160*a^3*b^14*c^4
*d^8*f^4 - 520*a^4*b^13*c^3*d^9*f^4 + 920*a^5*b^12*c^2*d^10*f^4 + 320*a^5*b^12*c^4*d^8*f^4 - 960*a^6*b^11*c^3*
d^9*f^4 + 1280*a^7*b^10*c^2*d^10*f^4 + 320*a^7*b^10*c^4*d^8*f^4 - 920*a^8*b^9*c^3*d^9*f^4 + 1000*a^9*b^8*c^2*d
^10*f^4 + 160*a^9*b^8*c^4*d^8*f^4 - 448*a^10*b^7*c^3*d^9*f^4 + 416*a^11*b^6*c^2*d^10*f^4 + 32*a^11*b^6*c^4*d^8
*f^4 - 88*a^12*b^5*c^3*d^9*f^4 + 72*a^13*b^4*c^2*d^10*f^4 + 56*a*b^16*c^2*d^10*f^4 + 32*a*b^16*c^4*d^8*f^4 - 1
84*a^2*b^15*c*d^11*f^4 - 688*a^4*b^13*c*d^11*f^4 - 1240*a^6*b^11*c*d^11*f^4 - 1200*a^8*b^9*c*d^11*f^4 - 616*a^
10*b^7*c*d^11*f^4 - 144*a^12*b^5*c*d^11*f^4 - 8*a^14*b^3*c*d^11*f^4))/(a^10*d^2*f^5 + b^10*c^2*f^5 + 4*a^2*b^8
*c^2*f^5 + 6*a^4*b^6*c^2*f^5 + 4*a^6*b^4*c^2*f^5 + a^8*b^2*c^2*f^5 + a^2*b^8*d^2*f^5 + 4*a^4*b^6*d^2*f^5 + 6*a
^6*b^4*d^2*f^5 + 4*a^8*b^2*d^2*f^5 - 2*a*b^9*c*d*f^5 - 2*a^9*b*c*d*f^5 - 8*a^3*b^7*c*d*f^5 - 12*a^5*b^5*c*d*f^
5 - 8*a^7*b^3*c*d*f^5) - (16*(-(b^7*d^2 + 16*a^2*b^5*c^2 + 10*a^2*b^5*d^2 + 25*a^4*b^3*d^2 - 8*a*b^6*c*d - 40*
a^3*b^4*c*d)*(a^11*d^3*f^2 - b^11*c^3*f^2 - 4*a^2*b^9*c^3*f^2 - 6*a^4*b^7*c^3*f^2 - 4*a^6*b^5*c^3*f^2 - a^8*b^
3*c^3*f^2 + a^3*b^8*d^3*f^2 + 4*a^5*b^6*d^3*f^2 + 6*a^7*b^4*d^3*f^2 + 4*a^9*b^2*d^3*f^2 + 3*a*b^10*c^2*d*f^2 -
3*a^10*b*c*d^2*f^2 - 3*a^2*b^9*c*d^2*f^2 + 12*a^3*b^8*c^2*d*f^2 - 12*a^4*b^7*c*d^2*f^2 + 18*a^5*b^6*c^2*d*f^2
- 18*a^6*b^5*c*d^2*f^2 + 12*a^7*b^4*c^2*d*f^2 - 12*a^8*b^3*c*d^2*f^2 + 3*a^9*b^2*c^2*d*f^2))^(1/2)*(c + d*tan
(e + f*x))^(1/2)*(32*a^2*b^17*d^12*f^4 + 160*a^4*b^15*d^12*f^4 + 288*a^6*b^13*d^12*f^4 + 160*a^8*b^11*d^12*f^4
- 160*a^10*b^9*d^12*f^4 - 288*a^12*b^7*d^12*f^4 - 160*a^14*b^5*d^12*f^4 - 32*a^16*b^3*d^12*f^4 + 32*b^19*c^2*
d^10*f^4 + 48*b^19*c^4*d^8*f^4 + 176*a^2*b^17*c^2*d^10*f^4 + 272*a^2*b^17*c^4*d^8*f^4 - 432*a^3*b^16*c^3*d^9*f
^4 + 336*a^4*b^15*c^2*d^10*f^4 + 624*a^4*b^15*c^4*d^8*f^4 - 912*a^5*b^14*c^3*d^9*f^4 + 112*a^6*b^13*c^2*d^10*f
^4 + 720*a^6*b^13*c^4*d^8*f^4 - 880*a^7*b^12*c^3*d^9*f^4 - 560*a^8*b^11*c^2*d^10*f^4 + 400*a^8*b^11*c^4*d^8*f^
4 - 240*a^9*b^10*c^3*d^9*f^4 - 1008*a^10*b^9*c^2*d^10*f^4 + 48*a^10*b^9*c^4*d^8*f^4 + 240*a^11*b^8*c^3*d^9*f^4
- 784*a^12*b^7*c^2*d^10*f^4 - 48*a^12*b^7*c^4*d^8*f^4 + 208*a^13*b^6*c^3*d^9*f^4 - 304*a^14*b^5*c^2*d^10*f^4
- 16*a^14*b^5*c^4*d^8*f^4 + 48*a^15*b^4*c^3*d^9*f^4 - 48*a^16*b^3*c^2*d^10*f^4 - 64*a*b^18*c*d^11*f^4 - 80*a*b
^18*c^3*d^9*f^4 - 304*a^3*b^16*c*d^11*f^4 - 464*a^5*b^14*c*d^11*f^4 + 16*a^7*b^12*c*d^11*f^4 + 880*a^9*b^10*c*
d^11*f^4 + 1136*a^11*b^8*c*d^11*f^4 + 656*a^13*b^6*c*d^11*f^4 + 176*a^15*b^4*c*d^11*f^4 + 16*a^17*b^2*c*d^11*f
^4))/((b^9*(8*a^2*c^3*f^2 + 6*a^2*c*d^2*f^2) + b^3*(2*a^8*c^3*f^2 + 24*a^8*c*d^2*f^2) + b^7*(12*a^4*c^3*f^2 +
24*a^4*c*d^2*f^2) + b^5*(8*a^6*c^3*f^2 + 36*a^6*c*d^2*f^2) - b^2*(8*a^9*d^3*f^2 + 6*a^9*c^2*d*f^2) - b^8*(2*a^
3*d^3*f^2 + 24*a^3*c^2*d*f^2) - b^4*(12*a^7*d^3*f^2 + 24*a^7*c^2*d*f^2) - b^6*(8*a^5*d^3*f^2 + 36*a^5*c^2*d*f^
2) - 2*a^11*d^3*f^2 + 2*b^11*c^3*f^2 - 6*a*b^10*c^2*d*f^2 + 6*a^10*b*c*d^2*f^2)*(a^10*d^2*f^4 + b^10*c^2*f^4 +
4*a^2*b^8*c^2*f^4 + 6*a^4*b^6*c^2*f^4 + 4*a^6*b^4*c^2*f^4 + a^8*b^2*c^2*f^4 + a^2*b^8*d^2*f^4 + 4*a^4*b^6*d^2
*f^4 + 6*a^6*b^4*d^2*f^4 + 4*a^8*b^2*d^2*f^4 - 2*a*b^9*c*d*f^4 - 2*a^9*b*c*d*f^4 - 8*a^3*b^7*c*d*f^4 - 12*a^5*
b^5*c*d*f^4 - 8*a^7*b^3*c*d*f^4)))*(-(b^7*d^2 + 16*a^2*b^5*c^2 + 10*a^2*b^5*d^2 + 25*a^4*b^3*d^2 - 8*a*b^6*c*d
- 40*a^3*b^4*c*d)*(a^11*d^3*f^2 - b^11*c^3*f^2 - 4*a^2*b^9*c^3*f^2 - 6*a^4*b^7*c^3*f^2 - 4*a^6*b^5*c^3*f^2 -
a^8*b^3*c^3*f^2 + a^3*b^8*d^3*f^2 + 4*a^5*b^6*d^3*f^2 + 6*a^7*b^4*d^3*f^2 + 4*a^9*b^2*d^3*f^2 + 3*a*b^10*c^2*d
*f^2 - 3*a^10*b*c*d^2*f^2 - 3*a^2*b^9*c*d^2*f^2 + 12*a^3*b^8*c^2*d*f^2 - 12*a^4*b^7*c*d^2*f^2 + 18*a^5*b^6*c^2
*d*f^2 - 18*a^6*b^5*c*d^2*f^2 + 12*a^7*b^4*c^2*d*f^2 - 12*a^8*b^3*c*d^2*f^2 + 3*a^9*b^2*c^2*d*f^2))^(1/2))/(b^
9*(8*a^2*c^3*f^2 + 6*a^2*c*d^2*f^2) + b^3*(2*a^8*c^3*f^2 + 24*a^8*c*d^2*f^2) + b^7*(12*a^4*c^3*f^2 + 24*a^4*c*
d^2*f^2) + b^5*(8*a^6*c^3*f^2 + 36*a^6*c*d^2*f^2) - b^2*(8*a^9*d^3*f^2 + 6*a^9*c^2*d*f^2) - b^8*(2*a^3*d^3*f^2
+ 24*a^3*c^2*d*f^2) - b^4*(12*a^7*d^3*f^2 + 24*a^7*c^2*d*f^2) - b^6*(8*a^5*d^3*f^2 + 36*a^5*c^2*d*f^2) - 2*a^
11*d^3*f^2 + 2*b^11*c^3*f^2 - 6*a*b^10*c^2*d*f^2 + 6*a^10*b*c*d^2*f^2))*(-(b^7*d^2 + 16*a^2*b^5*c^2 + 10*a^2*b
^5*d^2 + 25*a^4*b^3*d^2 - 8*a*b^6*c*d - 40*a^3*b^4*c*d)*(a^11*d^3*f^2 - b^11*c^3*f^2 - 4*a^2*b^9*c^3*f^2 - 6*a
^4*b^7*c^3*f^2 - 4*a^6*b^5*c^3*f^2 - a^8*b^3*c^3*f^2 + a^3*b^8*d^3*f^2 + 4*a^5*b^6*d^3*f^2 + 6*a^7*b^4*d^3*f^2
+ 4*a^9*b^2*d^3*f^2 + 3*a*b^10*c^2*d*f^2 - 3*a^10*b*c*d^2*f^2 - 3*a^2*b^9*c*d^2*f^2 + 12*a^3*b^8*c^2*d*f^2 -
12*a^4*b^7*c*d^2*f^2 + 18*a^5*b^6*c^2*d*f^2 - 18*a^6*b^5*c*d^2*f^2 + 12*a^7*b^4*c^2*d*f^2 - 12*a^8*b^3*c*d^2*f
^2 + 3*a^9*b^2*c^2*d*f^2))^(1/2))/(b^9*(8*a^2*c^3*f^2 + 6*a^2*c*d^2*f^2) + b^3*(2*a^8*c^3*f^2 + 24*a^8*c*d^2*f
^2) + b^7*(12*a^4*c^3*f^2 + 24*a^4*c*d^2*f^2) + b^5*(8*a^6*c^3*f^2 + 36*a^6*c*d^2*f^2) - b^2*(8*a^9*d^3*f^2 +
6*a^9*c^2*d*f^2) - b^8*(2*a^3*d^3*f^2 + 24*a^3*c^2*d*f^2) - b^4*(12*a^7*d^3*f^2 + 24*a^7*c^2*d*f^2) - b^6*(8*a
^5*d^3*f^2 + 36*a^5*c^2*d*f^2) - 2*a^11*d^3*f^2 + 2*b^11*c^3*f^2 - 6*a*b^10*c^2*d*f^2 + 6*a^10*b*c*d^2*f^2))*(
-(b^7*d^2 + 16*a^2*b^5*c^2 + 10*a^2*b^5*d^2 + 25*a^4*b^3*d^2 - 8*a*b^6*c*d - 40*a^3*b^4*c*d)*(a^11*d^3*f^2 - b
^11*c^3*f^2 - 4*a^2*b^9*c^3*f^2 - 6*a^4*b^7*c^3*f^2 - 4*a^6*b^5*c^3*f^2 - a^8*b^3*c^3*f^2 + a^3*b^8*d^3*f^2 +
4*a^5*b^6*d^3*f^2 + 6*a^7*b^4*d^3*f^2 + 4*a^9*b^2*d^3*f^2 + 3*a*b^10*c^2*d*f^2 - 3*a^10*b*c*d^2*f^2 - 3*a^2*b^
9*c*d^2*f^2 + 12*a^3*b^8*c^2*d*f^2 - 12*a^4*b^7*c*d^2*f^2 + 18*a^5*b^6*c^2*d*f^2 - 18*a^6*b^5*c*d^2*f^2 + 12*a
^7*b^4*c^2*d*f^2 - 12*a^8*b^3*c*d^2*f^2 + 3*a^9*b^2*c^2*d*f^2))^(1/2))/(b^9*(8*a^2*c^3*f^2 + 6*a^2*c*d^2*f^2)
+ b^3*(2*a^8*c^3*f^2 + 24*a^8*c*d^2*f^2) + b^7*(12*a^4*c^3*f^2 + 24*a^4*c*d^2*f^2) + b^5*(8*a^6*c^3*f^2 + 36*a
^6*c*d^2*f^2) - b^2*(8*a^9*d^3*f^2 + 6*a^9*c^2*d*f^2) - b^8*(2*a^3*d^3*f^2 + 24*a^3*c^2*d*f^2) - b^4*(12*a^7*d
^3*f^2 + 24*a^7*c^2*d*f^2) - b^6*(8*a^5*d^3*f^2 + 36*a^5*c^2*d*f^2) - 2*a^11*d^3*f^2 + 2*b^11*c^3*f^2 - 6*a*b^
10*c^2*d*f^2 + 6*a^10*b*c*d^2*f^2) - (16*(c + d*tan(e + f*x))^(1/2)*(b^11*d^10 + 7*a^2*b^9*d^10 + 11*a^4*b^7*d
^10 - 27*a^6*b^5*d^10 - 2*b^11*c^2*d^8 - 24*a^3*b^8*c*d^9 + 44*a^5*b^6*c*d^9 + 12*a^2*b^9*c^2*d^8 - 18*a^4*b^7
*c^2*d^8 - 4*a*b^10*c*d^9))/(a^10*d^2*f^4 + b^10*c^2*f^4 + 4*a^2*b^8*c^2*f^4 + 6*a^4*b^6*c^2*f^4 + 4*a^6*b^4*c
^2*f^4 + a^8*b^2*c^2*f^4 + a^2*b^8*d^2*f^4 + 4*a^4*b^6*d^2*f^4 + 6*a^6*b^4*d^2*f^4 + 4*a^8*b^2*d^2*f^4 - 2*a*b
^9*c*d*f^4 - 2*a^9*b*c*d*f^4 - 8*a^3*b^7*c*d*f^4 - 12*a^5*b^5*c*d*f^4 - 8*a^7*b^3*c*d*f^4))*(-(b^7*d^2 + 16*a^
2*b^5*c^2 + 10*a^2*b^5*d^2 + 25*a^4*b^3*d^2 - 8*a*b^6*c*d - 40*a^3*b^4*c*d)*(a^11*d^3*f^2 - b^11*c^3*f^2 - 4*a
^2*b^9*c^3*f^2 - 6*a^4*b^7*c^3*f^2 - 4*a^6*b^5*c^3*f^2 - a^8*b^3*c^3*f^2 + a^3*b^8*d^3*f^2 + 4*a^5*b^6*d^3*f^2
+ 6*a^7*b^4*d^3*f^2 + 4*a^9*b^2*d^3*f^2 + 3*a*b^10*c^2*d*f^2 - 3*a^10*b*c*d^2*f^2 - 3*a^2*b^9*c*d^2*f^2 + 12*
a^3*b^8*c^2*d*f^2 - 12*a^4*b^7*c*d^2*f^2 + 18*a^5*b^6*c^2*d*f^2 - 18*a^6*b^5*c*d^2*f^2 + 12*a^7*b^4*c^2*d*f^2
- 12*a^8*b^3*c*d^2*f^2 + 3*a^9*b^2*c^2*d*f^2))^(1/2)*1i)/(b^9*(8*a^2*c^3*f^2 + 6*a^2*c*d^2*f^2) + b^3*(2*a^8*c
^3*f^2 + 24*a^8*c*d^2*f^2) + b^7*(12*a^4*c^3*f^2 + 24*a^4*c*d^2*f^2) + b^5*(8*a^6*c^3*f^2 + 36*a^6*c*d^2*f^2)
- b^2*(8*a^9*d^3*f^2 + 6*a^9*c^2*d*f^2) - b^8*(2*a^3*d^3*f^2 + 24*a^3*c^2*d*f^2) - b^4*(12*a^7*d^3*f^2 + 24*a^
7*c^2*d*f^2) - b^6*(8*a^5*d^3*f^2 + 36*a^5*c^2*d*f^2) - 2*a^11*d^3*f^2 + 2*b^11*c^3*f^2 - 6*a*b^10*c^2*d*f^2 +
6*a^10*b*c*d^2*f^2))/((((((16*(2*b^13*d^11*f^2 - 24*a^2*b^11*d^11*f^2 - 196*a^4*b^9*d^11*f^2 - 120*a^6*b^7*d^
11*f^2 + 50*a^8*b^5*d^11*f^2 + 8*b^13*c^2*d^9*f^2 - 8*a^2*b^11*c^2*d^9*f^2 + 64*a^3*b^10*c^3*d^8*f^2 - 232*a^4
*b^9*c^2*d^9*f^2 + 96*a^5*b^8*c^3*d^8*f^2 - 216*a^6*b^7*c^2*d^9*f^2 - 32*a*b^12*c^3*d^8*f^2 + 208*a^3*b^10*c*d
^10*f^2 + 288*a^5*b^8*c*d^10*f^2 + 80*a^7*b^6*c*d^10*f^2))/(a^10*d^2*f^5 + b^10*c^2*f^5 + 4*a^2*b^8*c^2*f^5 +
6*a^4*b^6*c^2*f^5 + 4*a^6*b^4*c^2*f^5 + a^8*b^2*c^2*f^5 + a^2*b^8*d^2*f^5 + 4*a^4*b^6*d^2*f^5 + 6*a^6*b^4*d^2*
f^5 + 4*a^8*b^2*d^2*f^5 - 2*a*b^9*c*d*f^5 - 2*a^9*b*c*d*f^5 - 8*a^3*b^7*c*d*f^5 - 12*a^5*b^5*c*d*f^5 - 8*a^7*b
^3*c*d*f^5) + (((16*(c + d*tan(e + f*x))^(1/2)*(8*a*b^14*d^11*f^2 + 4*b^15*c*d^10*f^2 + 36*a^3*b^12*d^11*f^2 +
316*a^5*b^10*d^11*f^2 + 552*a^7*b^8*d^11*f^2 + 256*a^9*b^6*d^11*f^2 - 12*a^11*b^4*d^11*f^2 - 4*a^13*b^2*d^11*
f^2 - 20*b^15*c^3*d^8*f^2 + 116*a^2*b^13*c^3*d^8*f^2 - 220*a^3*b^12*c^2*d^9*f^2 + 216*a^4*b^11*c^3*d^8*f^2 - 1
04*a^5*b^10*c^2*d^9*f^2 + 8*a^6*b^9*c^3*d^8*f^2 + 232*a^7*b^8*c^2*d^9*f^2 - 68*a^8*b^7*c^3*d^8*f^2 + 156*a^9*b
^6*c^2*d^9*f^2 + 4*a^10*b^5*c^3*d^8*f^2 - 12*a^11*b^4*c^2*d^9*f^2 - 52*a*b^14*c^2*d^9*f^2 + 80*a^2*b^13*c*d^10
*f^2 - 156*a^4*b^11*c*d^10*f^2 - 640*a^6*b^9*c*d^10*f^2 - 500*a^8*b^7*c*d^10*f^2 - 80*a^10*b^5*c*d^10*f^2 + 12
*a^12*b^3*c*d^10*f^2))/(a^10*d^2*f^4 + b^10*c^2*f^4 + 4*a^2*b^8*c^2*f^4 + 6*a^4*b^6*c^2*f^4 + 4*a^6*b^4*c^2*f^
4 + a^8*b^2*c^2*f^4 + a^2*b^8*d^2*f^4 + 4*a^4*b^6*d^2*f^4 + 6*a^6*b^4*d^2*f^4 + 4*a^8*b^2*d^2*f^4 - 2*a*b^9*c*
d*f^4 - 2*a^9*b*c*d*f^4 - 8*a^3*b^7*c*d*f^4 - 12*a^5*b^5*c*d*f^4 - 8*a^7*b^3*c*d*f^4) - (((16*(16*a*b^16*d^12*
f^4 - 16*b^17*c*d^11*f^4 + 136*a^3*b^14*d^12*f^4 + 432*a^5*b^12*d^12*f^4 + 680*a^7*b^10*d^12*f^4 + 560*a^9*b^8
*d^12*f^4 + 216*a^11*b^6*d^12*f^4 + 16*a^13*b^4*d^12*f^4 - 8*a^15*b^2*d^12*f^4 - 8*b^17*c^3*d^9*f^4 - 128*a^2*
b^15*c^3*d^9*f^4 + 352*a^3*b^14*c^2*d^10*f^4 + 160*a^3*b^14*c^4*d^8*f^4 - 520*a^4*b^13*c^3*d^9*f^4 + 920*a^5*b
^12*c^2*d^10*f^4 + 320*a^5*b^12*c^4*d^8*f^4 - 960*a^6*b^11*c^3*d^9*f^4 + 1280*a^7*b^10*c^2*d^10*f^4 + 320*a^7*
b^10*c^4*d^8*f^4 - 920*a^8*b^9*c^3*d^9*f^4 + 1000*a^9*b^8*c^2*d^10*f^4 + 160*a^9*b^8*c^4*d^8*f^4 - 448*a^10*b^
7*c^3*d^9*f^4 + 416*a^11*b^6*c^2*d^10*f^4 + 32*a^11*b^6*c^4*d^8*f^4 - 88*a^12*b^5*c^3*d^9*f^4 + 72*a^13*b^4*c^
2*d^10*f^4 + 56*a*b^16*c^2*d^10*f^4 + 32*a*b^16*c^4*d^8*f^4 - 184*a^2*b^15*c*d^11*f^4 - 688*a^4*b^13*c*d^11*f^
4 - 1240*a^6*b^11*c*d^11*f^4 - 1200*a^8*b^9*c*d^11*f^4 - 616*a^10*b^7*c*d^11*f^4 - 144*a^12*b^5*c*d^11*f^4 - 8
*a^14*b^3*c*d^11*f^4))/(a^10*d^2*f^5 + b^10*c^2*f^5 + 4*a^2*b^8*c^2*f^5 + 6*a^4*b^6*c^2*f^5 + 4*a^6*b^4*c^2*f^
5 + a^8*b^2*c^2*f^5 + a^2*b^8*d^2*f^5 + 4*a^4*b^6*d^2*f^5 + 6*a^6*b^4*d^2*f^5 + 4*a^8*b^2*d^2*f^5 - 2*a*b^9*c*
d*f^5 - 2*a^9*b*c*d*f^5 - 8*a^3*b^7*c*d*f^5 - 12*a^5*b^5*c*d*f^5 - 8*a^7*b^3*c*d*f^5) + (16*(-(b^7*d^2 + 16*a^
2*b^5*c^2 + 10*a^2*b^5*d^2 + 25*a^4*b^3*d^2 - 8*a*b^6*c*d - 40*a^3*b^4*c*d)*(a^11*d^3*f^2 - b^11*c^3*f^2 - 4*a
^2*b^9*c^3*f^2 - 6*a^4*b^7*c^3*f^2 - 4*a^6*b^5*c^3*f^2 - a^8*b^3*c^3*f^2 + a^3*b^8*d^3*f^2 + 4*a^5*b^6*d^3*f^2
+ 6*a^7*b^4*d^3*f^2 + 4*a^9*b^2*d^3*f^2 + 3*a*b^10*c^2*d*f^2 - 3*a^10*b*c*d^2*f^2 - 3*a^2*b^9*c*d^2*f^2 + 12*
a^3*b^8*c^2*d*f^2 - 12*a^4*b^7*c*d^2*f^2 + 18*a^5*b^6*c^2*d*f^2 - 18*a^6*b^5*c*d^2*f^2 + 12*a^7*b^4*c^2*d*f^2
- 12*a^8*b^3*c*d^2*f^2 + 3*a^9*b^2*c^2*d*f^2))^(1/2)*(c + d*tan(e + f*x))^(1/2)*(32*a^2*b^17*d^12*f^4 + 160*a^
4*b^15*d^12*f^4 + 288*a^6*b^13*d^12*f^4 + 160*a^8*b^11*d^12*f^4 - 160*a^10*b^9*d^12*f^4 - 288*a^12*b^7*d^12*f^
4 - 160*a^14*b^5*d^12*f^4 - 32*a^16*b^3*d^12*f^4 + 32*b^19*c^2*d^10*f^4 + 48*b^19*c^4*d^8*f^4 + 176*a^2*b^17*c
^2*d^10*f^4 + 272*a^2*b^17*c^4*d^8*f^4 - 432*a^3*b^16*c^3*d^9*f^4 + 336*a^4*b^15*c^2*d^10*f^4 + 624*a^4*b^15*c
^4*d^8*f^4 - 912*a^5*b^14*c^3*d^9*f^4 + 112*a^6*b^13*c^2*d^10*f^4 + 720*a^6*b^13*c^4*d^8*f^4 - 880*a^7*b^12*c^
3*d^9*f^4 - 560*a^8*b^11*c^2*d^10*f^4 + 400*a^8*b^11*c^4*d^8*f^4 - 240*a^9*b^10*c^3*d^9*f^4 - 1008*a^10*b^9*c^
2*d^10*f^4 + 48*a^10*b^9*c^4*d^8*f^4 + 240*a^11*b^8*c^3*d^9*f^4 - 784*a^12*b^7*c^2*d^10*f^4 - 48*a^12*b^7*c^4*
d^8*f^4 + 208*a^13*b^6*c^3*d^9*f^4 - 304*a^14*b^5*c^2*d^10*f^4 - 16*a^14*b^5*c^4*d^8*f^4 + 48*a^15*b^4*c^3*d^9
*f^4 - 48*a^16*b^3*c^2*d^10*f^4 - 64*a*b^18*c*d^11*f^4 - 80*a*b^18*c^3*d^9*f^4 - 304*a^3*b^16*c*d^11*f^4 - 464
*a^5*b^14*c*d^11*f^4 + 16*a^7*b^12*c*d^11*f^4 + 880*a^9*b^10*c*d^11*f^4 + 1136*a^11*b^8*c*d^11*f^4 + 656*a^13*
b^6*c*d^11*f^4 + 176*a^15*b^4*c*d^11*f^4 + 16*a^17*b^2*c*d^11*f^4))/((b^9*(8*a^2*c^3*f^2 + 6*a^2*c*d^2*f^2) +
b^3*(2*a^8*c^3*f^2 + 24*a^8*c*d^2*f^2) + b^7*(12*a^4*c^3*f^2 + 24*a^4*c*d^2*f^2) + b^5*(8*a^6*c^3*f^2 + 36*a^6
*c*d^2*f^2) - b^2*(8*a^9*d^3*f^2 + 6*a^9*c^2*d*f^2) - b^8*(2*a^3*d^3*f^2 + 24*a^3*c^2*d*f^2) - b^4*(12*a^7*d^3
*f^2 + 24*a^7*c^2*d*f^2) - b^6*(8*a^5*d^3*f^2 + 36*a^5*c^2*d*f^2) - 2*a^11*d^3*f^2 + 2*b^11*c^3*f^2 - 6*a*b^10
*c^2*d*f^2 + 6*a^10*b*c*d^2*f^2)*(a^10*d^2*f^4 + b^10*c^2*f^4 + 4*a^2*b^8*c^2*f^4 + 6*a^4*b^6*c^2*f^4 + 4*a^6*
b^4*c^2*f^4 + a^8*b^2*c^2*f^4 + a^2*b^8*d^2*f^4 + 4*a^4*b^6*d^2*f^4 + 6*a^6*b^4*d^2*f^4 + 4*a^8*b^2*d^2*f^4 -
2*a*b^9*c*d*f^4 - 2*a^9*b*c*d*f^4 - 8*a^3*b^7*c*d*f^4 - 12*a^5*b^5*c*d*f^4 - 8*a^7*b^3*c*d*f^4)))*(-(b^7*d^2 +
16*a^2*b^5*c^2 + 10*a^2*b^5*d^2 + 25*a^4*b^3*d^2 - 8*a*b^6*c*d - 40*a^3*b^4*c*d)*(a^11*d^3*f^2 - b^11*c^3*f^2
- 4*a^2*b^9*c^3*f^2 - 6*a^4*b^7*c^3*f^2 - 4*a^6*b^5*c^3*f^2 - a^8*b^3*c^3*f^2 + a^3*b^8*d^3*f^2 + 4*a^5*b^6*d
^3*f^2 + 6*a^7*b^4*d^3*f^2 + 4*a^9*b^2*d^3*f^2 + 3*a*b^10*c^2*d*f^2 - 3*a^10*b*c*d^2*f^2 - 3*a^2*b^9*c*d^2*f^2
+ 12*a^3*b^8*c^2*d*f^2 - 12*a^4*b^7*c*d^2*f^2 + 18*a^5*b^6*c^2*d*f^2 - 18*a^6*b^5*c*d^2*f^2 + 12*a^7*b^4*c^2*
d*f^2 - 12*a^8*b^3*c*d^2*f^2 + 3*a^9*b^2*c^2*d*f^2))^(1/2))/(b^9*(8*a^2*c^3*f^2 + 6*a^2*c*d^2*f^2) + b^3*(2*a^
8*c^3*f^2 + 24*a^8*c*d^2*f^2) + b^7*(12*a^4*c^3*f^2 + 24*a^4*c*d^2*f^2) + b^5*(8*a^6*c^3*f^2 + 36*a^6*c*d^2*f^
2) - b^2*(8*a^9*d^3*f^2 + 6*a^9*c^2*d*f^2) - b^8*(2*a^3*d^3*f^2 + 24*a^3*c^2*d*f^2) - b^4*(12*a^7*d^3*f^2 + 24
*a^7*c^2*d*f^2) - b^6*(8*a^5*d^3*f^2 + 36*a^5*c^2*d*f^2) - 2*a^11*d^3*f^2 + 2*b^11*c^3*f^2 - 6*a*b^10*c^2*d*f^
2 + 6*a^10*b*c*d^2*f^2))*(-(b^7*d^2 + 16*a^2*b^5*c^2 + 10*a^2*b^5*d^2 + 25*a^4*b^3*d^2 - 8*a*b^6*c*d - 40*a^3*
b^4*c*d)*(a^11*d^3*f^2 - b^11*c^3*f^2 - 4*a^2*b^9*c^3*f^2 - 6*a^4*b^7*c^3*f^2 - 4*a^6*b^5*c^3*f^2 - a^8*b^3*c^
3*f^2 + a^3*b^8*d^3*f^2 + 4*a^5*b^6*d^3*f^2 + 6*a^7*b^4*d^3*f^2 + 4*a^9*b^2*d^3*f^2 + 3*a*b^10*c^2*d*f^2 - 3*a
^10*b*c*d^2*f^2 - 3*a^2*b^9*c*d^2*f^2 + 12*a^3*b^8*c^2*d*f^2 - 12*a^4*b^7*c*d^2*f^2 + 18*a^5*b^6*c^2*d*f^2 - 1
8*a^6*b^5*c*d^2*f^2 + 12*a^7*b^4*c^2*d*f^2 - 12*a^8*b^3*c*d^2*f^2 + 3*a^9*b^2*c^2*d*f^2))^(1/2))/(b^9*(8*a^2*c
^3*f^2 + 6*a^2*c*d^2*f^2) + b^3*(2*a^8*c^3*f^2 + 24*a^8*c*d^2*f^2) + b^7*(12*a^4*c^3*f^2 + 24*a^4*c*d^2*f^2) +
b^5*(8*a^6*c^3*f^2 + 36*a^6*c*d^2*f^2) - b^2*(8*a^9*d^3*f^2 + 6*a^9*c^2*d*f^2) - b^8*(2*a^3*d^3*f^2 + 24*a^3*
c^2*d*f^2) - b^4*(12*a^7*d^3*f^2 + 24*a^7*c^2*d*f^2) - b^6*(8*a^5*d^3*f^2 + 36*a^5*c^2*d*f^2) - 2*a^11*d^3*f^2
+ 2*b^11*c^3*f^2 - 6*a*b^10*c^2*d*f^2 + 6*a^10*b*c*d^2*f^2))*(-(b^7*d^2 + 16*a^2*b^5*c^2 + 10*a^2*b^5*d^2 + 2
5*a^4*b^3*d^2 - 8*a*b^6*c*d - 40*a^3*b^4*c*d)*(a^11*d^3*f^2 - b^11*c^3*f^2 - 4*a^2*b^9*c^3*f^2 - 6*a^4*b^7*c^3
*f^2 - 4*a^6*b^5*c^3*f^2 - a^8*b^3*c^3*f^2 + a^3*b^8*d^3*f^2 + 4*a^5*b^6*d^3*f^2 + 6*a^7*b^4*d^3*f^2 + 4*a^9*b
^2*d^3*f^2 + 3*a*b^10*c^2*d*f^2 - 3*a^10*b*c*d^2*f^2 - 3*a^2*b^9*c*d^2*f^2 + 12*a^3*b^8*c^2*d*f^2 - 12*a^4*b^7
*c*d^2*f^2 + 18*a^5*b^6*c^2*d*f^2 - 18*a^6*b^5*c*d^2*f^2 + 12*a^7*b^4*c^2*d*f^2 - 12*a^8*b^3*c*d^2*f^2 + 3*a^9
*b^2*c^2*d*f^2))^(1/2))/(b^9*(8*a^2*c^3*f^2 + 6*a^2*c*d^2*f^2) + b^3*(2*a^8*c^3*f^2 + 24*a^8*c*d^2*f^2) + b^7*
(12*a^4*c^3*f^2 + 24*a^4*c*d^2*f^2) + b^5*(8*a^6*c^3*f^2 + 36*a^6*c*d^2*f^2) - b^2*(8*a^9*d^3*f^2 + 6*a^9*c^2*
d*f^2) - b^8*(2*a^3*d^3*f^2 + 24*a^3*c^2*d*f^2) - b^4*(12*a^7*d^3*f^2 + 24*a^7*c^2*d*f^2) - b^6*(8*a^5*d^3*f^2
+ 36*a^5*c^2*d*f^2) - 2*a^11*d^3*f^2 + 2*b^11*c^3*f^2 - 6*a*b^10*c^2*d*f^2 + 6*a^10*b*c*d^2*f^2) + (16*(c + d
*tan(e + f*x))^(1/2)*(b^11*d^10 + 7*a^2*b^9*d^10 + 11*a^4*b^7*d^10 - 27*a^6*b^5*d^10 - 2*b^11*c^2*d^8 - 24*a^3
*b^8*c*d^9 + 44*a^5*b^6*c*d^9 + 12*a^2*b^9*c^2*d^8 - 18*a^4*b^7*c^2*d^8 - 4*a*b^10*c*d^9))/(a^10*d^2*f^4 + b^1
0*c^2*f^4 + 4*a^2*b^8*c^2*f^4 + 6*a^4*b^6*c^2*f^4 + 4*a^6*b^4*c^2*f^4 + a^8*b^2*c^2*f^4 + a^2*b^8*d^2*f^4 + 4*
a^4*b^6*d^2*f^4 + 6*a^6*b^4*d^2*f^4 + 4*a^8*b^2*d^2*f^4 - 2*a*b^9*c*d*f^4 - 2*a^9*b*c*d*f^4 - 8*a^3*b^7*c*d*f^
4 - 12*a^5*b^5*c*d*f^4 - 8*a^7*b^3*c*d*f^4))*(-(b^7*d^2 + 16*a^2*b^5*c^2 + 10*a^2*b^5*d^2 + 25*a^4*b^3*d^2 - 8
*a*b^6*c*d - 40*a^3*b^4*c*d)*(a^11*d^3*f^2 - b^11*c^3*f^2 - 4*a^2*b^9*c^3*f^2 - 6*a^4*b^7*c^3*f^2 - 4*a^6*b^5*
c^3*f^2 - a^8*b^3*c^3*f^2 + a^3*b^8*d^3*f^2 + 4*a^5*b^6*d^3*f^2 + 6*a^7*b^4*d^3*f^2 + 4*a^9*b^2*d^3*f^2 + 3*a*
b^10*c^2*d*f^2 - 3*a^10*b*c*d^2*f^2 - 3*a^2*b^9*c*d^2*f^2 + 12*a^3*b^8*c^2*d*f^2 - 12*a^4*b^7*c*d^2*f^2 + 18*a
^5*b^6*c^2*d*f^2 - 18*a^6*b^5*c*d^2*f^2 + 12*a^7*b^4*c^2*d*f^2 - 12*a^8*b^3*c*d^2*f^2 + 3*a^9*b^2*c^2*d*f^2))^
(1/2))/(b^9*(8*a^2*c^3*f^2 + 6*a^2*c*d^2*f^2) + b^3*(2*a^8*c^3*f^2 + 24*a^8*c*d^2*f^2) + b^7*(12*a^4*c^3*f^2 +
24*a^4*c*d^2*f^2) + b^5*(8*a^6*c^3*f^2 + 36*a^6*c*d^2*f^2) - b^2*(8*a^9*d^3*f^2 + 6*a^9*c^2*d*f^2) - b^8*(2*a
^3*d^3*f^2 + 24*a^3*c^2*d*f^2) - b^4*(12*a^7*d^3*f^2 + 24*a^7*c^2*d*f^2) - b^6*(8*a^5*d^3*f^2 + 36*a^5*c^2*d*f
^2) - 2*a^11*d^3*f^2 + 2*b^11*c^3*f^2 - 6*a*b^10*c^2*d*f^2 + 6*a^10*b*c*d^2*f^2) - (32*(a*b^8*d^10 - b^9*c*d^9
+ 5*a^3*b^6*d^10 + 4*a*b^8*c^2*d^8 - 9*a^2*b^7*c*d^9))/(a^10*d^2*f^5 + b^10*c^2*f^5 + 4*a^2*b^8*c^2*f^5 + 6*a
^4*b^6*c^2*f^5 + 4*a^6*b^4*c^2*f^5 + a^8*b^2*c^2*f^5 + a^2*b^8*d^2*f^5 + 4*a^4*b^6*d^2*f^5 + 6*a^6*b^4*d^2*f^5
+ 4*a^8*b^2*d^2*f^5 - 2*a*b^9*c*d*f^5 - 2*a^9*b*c*d*f^5 - 8*a^3*b^7*c*d*f^5 - 12*a^5*b^5*c*d*f^5 - 8*a^7*b^3*
c*d*f^5) + (((((16*(2*b^13*d^11*f^2 - 24*a^2*b^11*d^11*f^2 - 196*a^4*b^9*d^11*f^2 - 120*a^6*b^7*d^11*f^2 + 50*
a^8*b^5*d^11*f^2 + 8*b^13*c^2*d^9*f^2 - 8*a^2*b^11*c^2*d^9*f^2 + 64*a^3*b^10*c^3*d^8*f^2 - 232*a^4*b^9*c^2*d^9
*f^2 + 96*a^5*b^8*c^3*d^8*f^2 - 216*a^6*b^7*c^2*d^9*f^2 - 32*a*b^12*c^3*d^8*f^2 + 208*a^3*b^10*c*d^10*f^2 + 28
8*a^5*b^8*c*d^10*f^2 + 80*a^7*b^6*c*d^10*f^2))/(a^10*d^2*f^5 + b^10*c^2*f^5 + 4*a^2*b^8*c^2*f^5 + 6*a^4*b^6*c^
2*f^5 + 4*a^6*b^4*c^2*f^5 + a^8*b^2*c^2*f^5 + a^2*b^8*d^2*f^5 + 4*a^4*b^6*d^2*f^5 + 6*a^6*b^4*d^2*f^5 + 4*a^8*
b^2*d^2*f^5 - 2*a*b^9*c*d*f^5 - 2*a^9*b*c*d*f^5 - 8*a^3*b^7*c*d*f^5 - 12*a^5*b^5*c*d*f^5 - 8*a^7*b^3*c*d*f^5)
- (((16*(c + d*tan(e + f*x))^(1/2)*(8*a*b^14*d^11*f^2 + 4*b^15*c*d^10*f^2 + 36*a^3*b^12*d^11*f^2 + 316*a^5*b^1
0*d^11*f^2 + 552*a^7*b^8*d^11*f^2 + 256*a^9*b^6*d^11*f^2 - 12*a^11*b^4*d^11*f^2 - 4*a^13*b^2*d^11*f^2 - 20*b^1
5*c^3*d^8*f^2 + 116*a^2*b^13*c^3*d^8*f^2 - 220*a^3*b^12*c^2*d^9*f^2 + 216*a^4*b^11*c^3*d^8*f^2 - 104*a^5*b^10*
c^2*d^9*f^2 + 8*a^6*b^9*c^3*d^8*f^2 + 232*a^7*b^8*c^2*d^9*f^2 - 68*a^8*b^7*c^3*d^8*f^2 + 156*a^9*b^6*c^2*d^9*f
^2 + 4*a^10*b^5*c^3*d^8*f^2 - 12*a^11*b^4*c^2*d^9*f^2 - 52*a*b^14*c^2*d^9*f^2 + 80*a^2*b^13*c*d^10*f^2 - 156*a
^4*b^11*c*d^10*f^2 - 640*a^6*b^9*c*d^10*f^2 - 500*a^8*b^7*c*d^10*f^2 - 80*a^10*b^5*c*d^10*f^2 + 12*a^12*b^3*c*
d^10*f^2))/(a^10*d^2*f^4 + b^10*c^2*f^4 + 4*a^2*b^8*c^2*f^4 + 6*a^4*b^6*c^2*f^4 + 4*a^6*b^4*c^2*f^4 + a^8*b^2*
c^2*f^4 + a^2*b^8*d^2*f^4 + 4*a^4*b^6*d^2*f^4 + 6*a^6*b^4*d^2*f^4 + 4*a^8*b^2*d^2*f^4 - 2*a*b^9*c*d*f^4 - 2*a^
9*b*c*d*f^4 - 8*a^3*b^7*c*d*f^4 - 12*a^5*b^5*c*d*f^4 - 8*a^7*b^3*c*d*f^4) + (((16*(16*a*b^16*d^12*f^4 - 16*b^1
7*c*d^11*f^4 + 136*a^3*b^14*d^12*f^4 + 432*a^5*b^12*d^12*f^4 + 680*a^7*b^10*d^12*f^4 + 560*a^9*b^8*d^12*f^4 +
216*a^11*b^6*d^12*f^4 + 16*a^13*b^4*d^12*f^4 - 8*a^15*b^2*d^12*f^4 - 8*b^17*c^3*d^9*f^4 - 128*a^2*b^15*c^3*d^9
*f^4 + 352*a^3*b^14*c^2*d^10*f^4 + 160*a^3*b^14*c^4*d^8*f^4 - 520*a^4*b^13*c^3*d^9*f^4 + 920*a^5*b^12*c^2*d^10
*f^4 + 320*a^5*b^12*c^4*d^8*f^4 - 960*a^6*b^11*c^3*d^9*f^4 + 1280*a^7*b^10*c^2*d^10*f^4 + 320*a^7*b^10*c^4*d^8
*f^4 - 920*a^8*b^9*c^3*d^9*f^4 + 1000*a^9*b^8*c^2*d^10*f^4 + 160*a^9*b^8*c^4*d^8*f^4 - 448*a^10*b^7*c^3*d^9*f^
4 + 416*a^11*b^6*c^2*d^10*f^4 + 32*a^11*b^6*c^4*d^8*f^4 - 88*a^12*b^5*c^3*d^9*f^4 + 72*a^13*b^4*c^2*d^10*f^4 +
56*a*b^16*c^2*d^10*f^4 + 32*a*b^16*c^4*d^8*f^4 - 184*a^2*b^15*c*d^11*f^4 - 688*a^4*b^13*c*d^11*f^4 - 1240*a^6
*b^11*c*d^11*f^4 - 1200*a^8*b^9*c*d^11*f^4 - 616*a^10*b^7*c*d^11*f^4 - 144*a^12*b^5*c*d^11*f^4 - 8*a^14*b^3*c*
d^11*f^4))/(a^10*d^2*f^5 + b^10*c^2*f^5 + 4*a^2*b^8*c^2*f^5 + 6*a^4*b^6*c^2*f^5 + 4*a^6*b^4*c^2*f^5 + a^8*b^2*
c^2*f^5 + a^2*b^8*d^2*f^5 + 4*a^4*b^6*d^2*f^5 + 6*a^6*b^4*d^2*f^5 + 4*a^8*b^2*d^2*f^5 - 2*a*b^9*c*d*f^5 - 2*a^
9*b*c*d*f^5 - 8*a^3*b^7*c*d*f^5 - 12*a^5*b^5*c*d*f^5 - 8*a^7*b^3*c*d*f^5) - (16*(-(b^7*d^2 + 16*a^2*b^5*c^2 +
10*a^2*b^5*d^2 + 25*a^4*b^3*d^2 - 8*a*b^6*c*d - 40*a^3*b^4*c*d)*(a^11*d^3*f^2 - b^11*c^3*f^2 - 4*a^2*b^9*c^3*f
^2 - 6*a^4*b^7*c^3*f^2 - 4*a^6*b^5*c^3*f^2 - a^8*b^3*c^3*f^2 + a^3*b^8*d^3*f^2 + 4*a^5*b^6*d^3*f^2 + 6*a^7*b^4
*d^3*f^2 + 4*a^9*b^2*d^3*f^2 + 3*a*b^10*c^2*d*f^2 - 3*a^10*b*c*d^2*f^2 - 3*a^2*b^9*c*d^2*f^2 + 12*a^3*b^8*c^2*
d*f^2 - 12*a^4*b^7*c*d^2*f^2 + 18*a^5*b^6*c^2*d*f^2 - 18*a^6*b^5*c*d^2*f^2 + 12*a^7*b^4*c^2*d*f^2 - 12*a^8*b^3
*c*d^2*f^2 + 3*a^9*b^2*c^2*d*f^2))^(1/2)*(c + d*tan(e + f*x))^(1/2)*(32*a^2*b^17*d^12*f^4 + 160*a^4*b^15*d^12*
f^4 + 288*a^6*b^13*d^12*f^4 + 160*a^8*b^11*d^12*f^4 - 160*a^10*b^9*d^12*f^4 - 288*a^12*b^7*d^12*f^4 - 160*a^14
*b^5*d^12*f^4 - 32*a^16*b^3*d^12*f^4 + 32*b^19*c^2*d^10*f^4 + 48*b^19*c^4*d^8*f^4 + 176*a^2*b^17*c^2*d^10*f^4
+ 272*a^2*b^17*c^4*d^8*f^4 - 432*a^3*b^16*c^3*d^9*f^4 + 336*a^4*b^15*c^2*d^10*f^4 + 624*a^4*b^15*c^4*d^8*f^4 -
912*a^5*b^14*c^3*d^9*f^4 + 112*a^6*b^13*c^2*d^10*f^4 + 720*a^6*b^13*c^4*d^8*f^4 - 880*a^7*b^12*c^3*d^9*f^4 -
560*a^8*b^11*c^2*d^10*f^4 + 400*a^8*b^11*c^4*d^8*f^4 - 240*a^9*b^10*c^3*d^9*f^4 - 1008*a^10*b^9*c^2*d^10*f^4 +
48*a^10*b^9*c^4*d^8*f^4 + 240*a^11*b^8*c^3*d^9*f^4 - 784*a^12*b^7*c^2*d^10*f^4 - 48*a^12*b^7*c^4*d^8*f^4 + 20
8*a^13*b^6*c^3*d^9*f^4 - 304*a^14*b^5*c^2*d^10*f^4 - 16*a^14*b^5*c^4*d^8*f^4 + 48*a^15*b^4*c^3*d^9*f^4 - 48*a^
16*b^3*c^2*d^10*f^4 - 64*a*b^18*c*d^11*f^4 - 80*a*b^18*c^3*d^9*f^4 - 304*a^3*b^16*c*d^11*f^4 - 464*a^5*b^14*c*
d^11*f^4 + 16*a^7*b^12*c*d^11*f^4 + 880*a^9*b^10*c*d^11*f^4 + 1136*a^11*b^8*c*d^11*f^4 + 656*a^13*b^6*c*d^11*f
^4 + 176*a^15*b^4*c*d^11*f^4 + 16*a^17*b^2*c*d^11*f^4))/((b^9*(8*a^2*c^3*f^2 + 6*a^2*c*d^2*f^2) + b^3*(2*a^8*c
^3*f^2 + 24*a^8*c*d^2*f^2) + b^7*(12*a^4*c^3*f^2 + 24*a^4*c*d^2*f^2) + b^5*(8*a^6*c^3*f^2 + 36*a^6*c*d^2*f^2)
- b^2*(8*a^9*d^3*f^2 + 6*a^9*c^2*d*f^2) - b^8*(2*a^3*d^3*f^2 + 24*a^3*c^2*d*f^2) - b^4*(12*a^7*d^3*f^2 + 24*a^
7*c^2*d*f^2) - b^6*(8*a^5*d^3*f^2 + 36*a^5*c^2*d*f^2) - 2*a^11*d^3*f^2 + 2*b^11*c^3*f^2 - 6*a*b^10*c^2*d*f^2 +
6*a^10*b*c*d^2*f^2)*(a^10*d^2*f^4 + b^10*c^2*f^4 + 4*a^2*b^8*c^2*f^4 + 6*a^4*b^6*c^2*f^4 + 4*a^6*b^4*c^2*f^4
+ a^8*b^2*c^2*f^4 + a^2*b^8*d^2*f^4 + 4*a^4*b^6*d^2*f^4 + 6*a^6*b^4*d^2*f^4 + 4*a^8*b^2*d^2*f^4 - 2*a*b^9*c*d*
f^4 - 2*a^9*b*c*d*f^4 - 8*a^3*b^7*c*d*f^4 - 12*a^5*b^5*c*d*f^4 - 8*a^7*b^3*c*d*f^4)))*(-(b^7*d^2 + 16*a^2*b^5*
c^2 + 10*a^2*b^5*d^2 + 25*a^4*b^3*d^2 - 8*a*b^6*c*d - 40*a^3*b^4*c*d)*(a^11*d^3*f^2 - b^11*c^3*f^2 - 4*a^2*b^9
*c^3*f^2 - 6*a^4*b^7*c^3*f^2 - 4*a^6*b^5*c^3*f^2 - a^8*b^3*c^3*f^2 + a^3*b^8*d^3*f^2 + 4*a^5*b^6*d^3*f^2 + 6*a
^7*b^4*d^3*f^2 + 4*a^9*b^2*d^3*f^2 + 3*a*b^10*c^2*d*f^2 - 3*a^10*b*c*d^2*f^2 - 3*a^2*b^9*c*d^2*f^2 + 12*a^3*b^
8*c^2*d*f^2 - 12*a^4*b^7*c*d^2*f^2 + 18*a^5*b^6*c^2*d*f^2 - 18*a^6*b^5*c*d^2*f^2 + 12*a^7*b^4*c^2*d*f^2 - 12*a
^8*b^3*c*d^2*f^2 + 3*a^9*b^2*c^2*d*f^2))^(1/2))/(b^9*(8*a^2*c^3*f^2 + 6*a^2*c*d^2*f^2) + b^3*(2*a^8*c^3*f^2 +
24*a^8*c*d^2*f^2) + b^7*(12*a^4*c^3*f^2 + 24*a^4*c*d^2*f^2) + b^5*(8*a^6*c^3*f^2 + 36*a^6*c*d^2*f^2) - b^2*(8*
a^9*d^3*f^2 + 6*a^9*c^2*d*f^2) - b^8*(2*a^3*d^3*f^2 + 24*a^3*c^2*d*f^2) - b^4*(12*a^7*d^3*f^2 + 24*a^7*c^2*d*f
^2) - b^6*(8*a^5*d^3*f^2 + 36*a^5*c^2*d*f^2) - 2*a^11*d^3*f^2 + 2*b^11*c^3*f^2 - 6*a*b^10*c^2*d*f^2 + 6*a^10*b
*c*d^2*f^2))*(-(b^7*d^2 + 16*a^2*b^5*c^2 + 10*a^2*b^5*d^2 + 25*a^4*b^3*d^2 - 8*a*b^6*c*d - 40*a^3*b^4*c*d)*(a^
11*d^3*f^2 - b^11*c^3*f^2 - 4*a^2*b^9*c^3*f^2 - 6*a^4*b^7*c^3*f^2 - 4*a^6*b^5*c^3*f^2 - a^8*b^3*c^3*f^2 + a^3*
b^8*d^3*f^2 + 4*a^5*b^6*d^3*f^2 + 6*a^7*b^4*d^3*f^2 + 4*a^9*b^2*d^3*f^2 + 3*a*b^10*c^2*d*f^2 - 3*a^10*b*c*d^2*
f^2 - 3*a^2*b^9*c*d^2*f^2 + 12*a^3*b^8*c^2*d*f^2 - 12*a^4*b^7*c*d^2*f^2 + 18*a^5*b^6*c^2*d*f^2 - 18*a^6*b^5*c*
d^2*f^2 + 12*a^7*b^4*c^2*d*f^2 - 12*a^8*b^3*c*d^2*f^2 + 3*a^9*b^2*c^2*d*f^2))^(1/2))/(b^9*(8*a^2*c^3*f^2 + 6*a
^2*c*d^2*f^2) + b^3*(2*a^8*c^3*f^2 + 24*a^8*c*d^2*f^2) + b^7*(12*a^4*c^3*f^2 + 24*a^4*c*d^2*f^2) + b^5*(8*a^6*
c^3*f^2 + 36*a^6*c*d^2*f^2) - b^2*(8*a^9*d^3*f^2 + 6*a^9*c^2*d*f^2) - b^8*(2*a^3*d^3*f^2 + 24*a^3*c^2*d*f^2) -
b^4*(12*a^7*d^3*f^2 + 24*a^7*c^2*d*f^2) - b^6*(8*a^5*d^3*f^2 + 36*a^5*c^2*d*f^2) - 2*a^11*d^3*f^2 + 2*b^11*c^
3*f^2 - 6*a*b^10*c^2*d*f^2 + 6*a^10*b*c*d^2*f^2))*(-(b^7*d^2 + 16*a^2*b^5*c^2 + 10*a^2*b^5*d^2 + 25*a^4*b^3*d^
2 - 8*a*b^6*c*d - 40*a^3*b^4*c*d)*(a^11*d^3*f^2 - b^11*c^3*f^2 - 4*a^2*b^9*c^3*f^2 - 6*a^4*b^7*c^3*f^2 - 4*a^6
*b^5*c^3*f^2 - a^8*b^3*c^3*f^2 + a^3*b^8*d^3*f^2 + 4*a^5*b^6*d^3*f^2 + 6*a^7*b^4*d^3*f^2 + 4*a^9*b^2*d^3*f^2 +
3*a*b^10*c^2*d*f^2 - 3*a^10*b*c*d^2*f^2 - 3*a^2*b^9*c*d^2*f^2 + 12*a^3*b^8*c^2*d*f^2 - 12*a^4*b^7*c*d^2*f^2 +
18*a^5*b^6*c^2*d*f^2 - 18*a^6*b^5*c*d^2*f^2 + 12*a^7*b^4*c^2*d*f^2 - 12*a^8*b^3*c*d^2*f^2 + 3*a^9*b^2*c^2*d*f
^2))^(1/2))/(b^9*(8*a^2*c^3*f^2 + 6*a^2*c*d^2*f^2) + b^3*(2*a^8*c^3*f^2 + 24*a^8*c*d^2*f^2) + b^7*(12*a^4*c^3*
f^2 + 24*a^4*c*d^2*f^2) + b^5*(8*a^6*c^3*f^2 + 36*a^6*c*d^2*f^2) - b^2*(8*a^9*d^3*f^2 + 6*a^9*c^2*d*f^2) - b^8
*(2*a^3*d^3*f^2 + 24*a^3*c^2*d*f^2) - b^4*(12*a^7*d^3*f^2 + 24*a^7*c^2*d*f^2) - b^6*(8*a^5*d^3*f^2 + 36*a^5*c^
2*d*f^2) - 2*a^11*d^3*f^2 + 2*b^11*c^3*f^2 - 6*a*b^10*c^2*d*f^2 + 6*a^10*b*c*d^2*f^2) - (16*(c + d*tan(e + f*x
))^(1/2)*(b^11*d^10 + 7*a^2*b^9*d^10 + 11*a^4*b^7*d^10 - 27*a^6*b^5*d^10 - 2*b^11*c^2*d^8 - 24*a^3*b^8*c*d^9 +
44*a^5*b^6*c*d^9 + 12*a^2*b^9*c^2*d^8 - 18*a^4*b^7*c^2*d^8 - 4*a*b^10*c*d^9))/(a^10*d^2*f^4 + b^10*c^2*f^4 +
4*a^2*b^8*c^2*f^4 + 6*a^4*b^6*c^2*f^4 + 4*a^6*b^4*c^2*f^4 + a^8*b^2*c^2*f^4 + a^2*b^8*d^2*f^4 + 4*a^4*b^6*d^2*
f^4 + 6*a^6*b^4*d^2*f^4 + 4*a^8*b^2*d^2*f^4 - 2*a*b^9*c*d*f^4 - 2*a^9*b*c*d*f^4 - 8*a^3*b^7*c*d*f^4 - 12*a^5*b
^5*c*d*f^4 - 8*a^7*b^3*c*d*f^4))*(-(b^7*d^2 + 16*a^2*b^5*c^2 + 10*a^2*b^5*d^2 + 25*a^4*b^3*d^2 - 8*a*b^6*c*d -
40*a^3*b^4*c*d)*(a^11*d^3*f^2 - b^11*c^3*f^2 - 4*a^2*b^9*c^3*f^2 - 6*a^4*b^7*c^3*f^2 - 4*a^6*b^5*c^3*f^2 - a^
8*b^3*c^3*f^2 + a^3*b^8*d^3*f^2 + 4*a^5*b^6*d^3*f^2 + 6*a^7*b^4*d^3*f^2 + 4*a^9*b^2*d^3*f^2 + 3*a*b^10*c^2*d*f
^2 - 3*a^10*b*c*d^2*f^2 - 3*a^2*b^9*c*d^2*f^2 + 12*a^3*b^8*c^2*d*f^2 - 12*a^4*b^7*c*d^2*f^2 + 18*a^5*b^6*c^2*d
*f^2 - 18*a^6*b^5*c*d^2*f^2 + 12*a^7*b^4*c^2*d*f^2 - 12*a^8*b^3*c*d^2*f^2 + 3*a^9*b^2*c^2*d*f^2))^(1/2))/(b^9*
(8*a^2*c^3*f^2 + 6*a^2*c*d^2*f^2) + b^3*(2*a^8*c^3*f^2 + 24*a^8*c*d^2*f^2) + b^7*(12*a^4*c^3*f^2 + 24*a^4*c*d^
2*f^2) + b^5*(8*a^6*c^3*f^2 + 36*a^6*c*d^2*f^2) - b^2*(8*a^9*d^3*f^2 + 6*a^9*c^2*d*f^2) - b^8*(2*a^3*d^3*f^2 +
24*a^3*c^2*d*f^2) - b^4*(12*a^7*d^3*f^2 + 24*a^7*c^2*d*f^2) - b^6*(8*a^5*d^3*f^2 + 36*a^5*c^2*d*f^2) - 2*a^11
*d^3*f^2 + 2*b^11*c^3*f^2 - 6*a*b^10*c^2*d*f^2 + 6*a^10*b*c*d^2*f^2)))*(-(b^7*d^2 + 16*a^2*b^5*c^2 + 10*a^2*b^
5*d^2 + 25*a^4*b^3*d^2 - 8*a*b^6*c*d - 40*a^3*b^4*c*d)*(a^11*d^3*f^2 - b^11*c^3*f^2 - 4*a^2*b^9*c^3*f^2 - 6*a^
4*b^7*c^3*f^2 - 4*a^6*b^5*c^3*f^2 - a^8*b^3*c^3*f^2 + a^3*b^8*d^3*f^2 + 4*a^5*b^6*d^3*f^2 + 6*a^7*b^4*d^3*f^2
+ 4*a^9*b^2*d^3*f^2 + 3*a*b^10*c^2*d*f^2 - 3*a^10*b*c*d^2*f^2 - 3*a^2*b^9*c*d^2*f^2 + 12*a^3*b^8*c^2*d*f^2 - 1
2*a^4*b^7*c*d^2*f^2 + 18*a^5*b^6*c^2*d*f^2 - 18*a^6*b^5*c*d^2*f^2 + 12*a^7*b^4*c^2*d*f^2 - 12*a^8*b^3*c*d^2*f^
2 + 3*a^9*b^2*c^2*d*f^2))^(1/2)*2i)/(b^9*(8*a^2*c^3*f^2 + 6*a^2*c*d^2*f^2) + b^3*(2*a^8*c^3*f^2 + 24*a^8*c*d^2
*f^2) + b^7*(12*a^4*c^3*f^2 + 24*a^4*c*d^2*f^2) + b^5*(8*a^6*c^3*f^2 + 36*a^6*c*d^2*f^2) - b^2*(8*a^9*d^3*f^2
+ 6*a^9*c^2*d*f^2) - b^8*(2*a^3*d^3*f^2 + 24*a^3*c^2*d*f^2) - b^4*(12*a^7*d^3*f^2 + 24*a^7*c^2*d*f^2) - b^6*(8
*a^5*d^3*f^2 + 36*a^5*c^2*d*f^2) - 2*a^11*d^3*f^2 + 2*b^11*c^3*f^2 - 6*a*b^10*c^2*d*f^2 + 6*a^10*b*c*d^2*f^2)
- (atan(((((16*(c + d*tan(e + f*x))^(1/2)*(b^11*d^10 + 7*a^2*b^9*d^10 + 11*a^4*b^7*d^10 - 27*a^6*b^5*d^10 - 2*
b^11*c^2*d^8 - 24*a^3*b^8*c*d^9 + 44*a^5*b^6*c*d^9 + 12*a^2*b^9*c^2*d^8 - 18*a^4*b^7*c^2*d^8 - 4*a*b^10*c*d^9)
)/(a^10*d^2*f^4 + b^10*c^2*f^4 + 4*a^2*b^8*c^2*f^4 + 6*a^4*b^6*c^2*f^4 + 4*a^6*b^4*c^2*f^4 + a^8*b^2*c^2*f^4 +
a^2*b^8*d^2*f^4 + 4*a^4*b^6*d^2*f^4 + 6*a^6*b^4*d^2*f^4 + 4*a^8*b^2*d^2*f^4 - 2*a*b^9*c*d*f^4 - 2*a^9*b*c*d*f
^4 - 8*a^3*b^7*c*d*f^4 - 12*a^5*b^5*c*d*f^4 - 8*a^7*b^3*c*d*f^4) - (((16*(2*b^13*d^11*f^2 - 24*a^2*b^11*d^11*f
^2 - 196*a^4*b^9*d^11*f^2 - 120*a^6*b^7*d^11*f^2 + 50*a^8*b^5*d^11*f^2 + 8*b^13*c^2*d^9*f^2 - 8*a^2*b^11*c^2*d
^9*f^2 + 64*a^3*b^10*c^3*d^8*f^2 - 232*a^4*b^9*c^2*d^9*f^2 + 96*a^5*b^8*c^3*d^8*f^2 - 216*a^6*b^7*c^2*d^9*f^2
- 32*a*b^12*c^3*d^8*f^2 + 208*a^3*b^10*c*d^10*f^2 + 288*a^5*b^8*c*d^10*f^2 + 80*a^7*b^6*c*d^10*f^2))/(a^10*d^2
*f^5 + b^10*c^2*f^5 + 4*a^2*b^8*c^2*f^5 + 6*a^4*b^6*c^2*f^5 + 4*a^6*b^4*c^2*f^5 + a^8*b^2*c^2*f^5 + a^2*b^8*d^
2*f^5 + 4*a^4*b^6*d^2*f^5 + 6*a^6*b^4*d^2*f^5 + 4*a^8*b^2*d^2*f^5 - 2*a*b^9*c*d*f^5 - 2*a^9*b*c*d*f^5 - 8*a^3*
b^7*c*d*f^5 - 12*a^5*b^5*c*d*f^5 - 8*a^7*b^3*c*d*f^5) - (((16*(c + d*tan(e + f*x))^(1/2)*(8*a*b^14*d^11*f^2 +
4*b^15*c*d^10*f^2 + 36*a^3*b^12*d^11*f^2 + 316*a^5*b^10*d^11*f^2 + 552*a^7*b^8*d^11*f^2 + 256*a^9*b^6*d^11*f^2
- 12*a^11*b^4*d^11*f^2 - 4*a^13*b^2*d^11*f^2 - 20*b^15*c^3*d^8*f^2 + 116*a^2*b^13*c^3*d^8*f^2 - 220*a^3*b^12*
c^2*d^9*f^2 + 216*a^4*b^11*c^3*d^8*f^2 - 104*a^5*b^10*c^2*d^9*f^2 + 8*a^6*b^9*c^3*d^8*f^2 + 232*a^7*b^8*c^2*d^
9*f^2 - 68*a^8*b^7*c^3*d^8*f^2 + 156*a^9*b^6*c^2*d^9*f^2 + 4*a^10*b^5*c^3*d^8*f^2 - 12*a^11*b^4*c^2*d^9*f^2 -
52*a*b^14*c^2*d^9*f^2 + 80*a^2*b^13*c*d^10*f^2 - 156*a^4*b^11*c*d^10*f^2 - 640*a^6*b^9*c*d^10*f^2 - 500*a^8*b^
7*c*d^10*f^2 - 80*a^10*b^5*c*d^10*f^2 + 12*a^12*b^3*c*d^10*f^2))/(a^10*d^2*f^4 + b^10*c^2*f^4 + 4*a^2*b^8*c^2*
f^4 + 6*a^4*b^6*c^2*f^4 + 4*a^6*b^4*c^2*f^4 + a^8*b^2*c^2*f^4 + a^2*b^8*d^2*f^4 + 4*a^4*b^6*d^2*f^4 + 6*a^6*b^
4*d^2*f^4 + 4*a^8*b^2*d^2*f^4 - 2*a*b^9*c*d*f^4 - 2*a^9*b*c*d*f^4 - 8*a^3*b^7*c*d*f^4 - 12*a^5*b^5*c*d*f^4 - 8
*a^7*b^3*c*d*f^4) + (((16*(16*a*b^16*d^12*f^4 - 16*b^17*c*d^11*f^4 + 136*a^3*b^14*d^12*f^4 + 432*a^5*b^12*d^12
*f^4 + 680*a^7*b^10*d^12*f^4 + 560*a^9*b^8*d^12*f^4 + 216*a^11*b^6*d^12*f^4 + 16*a^13*b^4*d^12*f^4 - 8*a^15*b^
2*d^12*f^4 - 8*b^17*c^3*d^9*f^4 - 128*a^2*b^15*c^3*d^9*f^4 + 352*a^3*b^14*c^2*d^10*f^4 + 160*a^3*b^14*c^4*d^8*
f^4 - 520*a^4*b^13*c^3*d^9*f^4 + 920*a^5*b^12*c^2*d^10*f^4 + 320*a^5*b^12*c^4*d^8*f^4 - 960*a^6*b^11*c^3*d^9*f
^4 + 1280*a^7*b^10*c^2*d^10*f^4 + 320*a^7*b^10*c^4*d^8*f^4 - 920*a^8*b^9*c^3*d^9*f^4 + 1000*a^9*b^8*c^2*d^10*f
^4 + 160*a^9*b^8*c^4*d^8*f^4 - 448*a^10*b^7*c^3*d^9*f^4 + 416*a^11*b^6*c^2*d^10*f^4 + 32*a^11*b^6*c^4*d^8*f^4
- 88*a^12*b^5*c^3*d^9*f^4 + 72*a^13*b^4*c^2*d^10*f^4 + 56*a*b^16*c^2*d^10*f^4 + 32*a*b^16*c^4*d^8*f^4 - 184*a^
2*b^15*c*d^11*f^4 - 688*a^4*b^13*c*d^11*f^4 - 1240*a^6*b^11*c*d^11*f^4 - 1200*a^8*b^9*c*d^11*f^4 - 616*a^10*b^
7*c*d^11*f^4 - 144*a^12*b^5*c*d^11*f^4 - 8*a^14*b^3*c*d^11*f^4))/(a^10*d^2*f^5 + b^10*c^2*f^5 + 4*a^2*b^8*c^2*
f^5 + 6*a^4*b^6*c^2*f^5 + 4*a^6*b^4*c^2*f^5 + a^8*b^2*c^2*f^5 + a^2*b^8*d^2*f^5 + 4*a^4*b^6*d^2*f^5 + 6*a^6*b^
4*d^2*f^5 + 4*a^8*b^2*d^2*f^5 - 2*a*b^9*c*d*f^5 - 2*a^9*b*c*d*f^5 - 8*a^3*b^7*c*d*f^5 - 12*a^5*b^5*c*d*f^5 - 8
*a^7*b^3*c*d*f^5) - (8*(1/(c - d*1i))^(1/2)*(c + d*tan(e + f*x))^(1/2)*(32*a^2*b^17*d^12*f^4 + 160*a^4*b^15*d^
12*f^4 + 288*a^6*b^13*d^12*f^4 + 160*a^8*b^11*d^12*f^4 - 160*a^10*b^9*d^12*f^4 - 288*a^12*b^7*d^12*f^4 - 160*a
^14*b^5*d^12*f^4 - 32*a^16*b^3*d^12*f^4 + 32*b^19*c^2*d^10*f^4 + 48*b^19*c^4*d^8*f^4 + 176*a^2*b^17*c^2*d^10*f
^4 + 272*a^2*b^17*c^4*d^8*f^4 - 432*a^3*b^16*c^3*d^9*f^4 + 336*a^4*b^15*c^2*d^10*f^4 + 624*a^4*b^15*c^4*d^8*f^
4 - 912*a^5*b^14*c^3*d^9*f^4 + 112*a^6*b^13*c^2*d^10*f^4 + 720*a^6*b^13*c^4*d^8*f^4 - 880*a^7*b^12*c^3*d^9*f^4
- 560*a^8*b^11*c^2*d^10*f^4 + 400*a^8*b^11*c^4*d^8*f^4 - 240*a^9*b^10*c^3*d^9*f^4 - 1008*a^10*b^9*c^2*d^10*f^
4 + 48*a^10*b^9*c^4*d^8*f^4 + 240*a^11*b^8*c^3*d^9*f^4 - 784*a^12*b^7*c^2*d^10*f^4 - 48*a^12*b^7*c^4*d^8*f^4 +
208*a^13*b^6*c^3*d^9*f^4 - 304*a^14*b^5*c^2*d^10*f^4 - 16*a^14*b^5*c^4*d^8*f^4 + 48*a^15*b^4*c^3*d^9*f^4 - 48
*a^16*b^3*c^2*d^10*f^4 - 64*a*b^18*c*d^11*f^4 - 80*a*b^18*c^3*d^9*f^4 - 304*a^3*b^16*c*d^11*f^4 - 464*a^5*b^14
*c*d^11*f^4 + 16*a^7*b^12*c*d^11*f^4 + 880*a^9*b^10*c*d^11*f^4 + 1136*a^11*b^8*c*d^11*f^4 + 656*a^13*b^6*c*d^1
1*f^4 + 176*a^15*b^4*c*d^11*f^4 + 16*a^17*b^2*c*d^11*f^4))/(f*(2*a*b + a^2*1i - b^2*1i)*(a^10*d^2*f^4 + b^10*c
^2*f^4 + 4*a^2*b^8*c^2*f^4 + 6*a^4*b^6*c^2*f^4 + 4*a^6*b^4*c^2*f^4 + a^8*b^2*c^2*f^4 + a^2*b^8*d^2*f^4 + 4*a^4
*b^6*d^2*f^4 + 6*a^6*b^4*d^2*f^4 + 4*a^8*b^2*d^2*f^4 - 2*a*b^9*c*d*f^4 - 2*a^9*b*c*d*f^4 - 8*a^3*b^7*c*d*f^4 -
12*a^5*b^5*c*d*f^4 - 8*a^7*b^3*c*d*f^4)))*(1/(c - d*1i))^(1/2))/(2*f*(2*a*b + a^2*1i - b^2*1i)))*(1/(c - d*1i
))^(1/2))/(2*f*(2*a*b + a^2*1i - b^2*1i)))*(1/(c - d*1i))^(1/2))/(2*f*(2*a*b + a^2*1i - b^2*1i)))*(1/(c - d*1i
))^(1/2)*1i)/(2*f*(2*a*b + a^2*1i - b^2*1i)) + (((16*(c + d*tan(e + f*x))^(1/2)*(b^11*d^10 + 7*a^2*b^9*d^10 +
11*a^4*b^7*d^10 - 27*a^6*b^5*d^10 - 2*b^11*c^2*d^8 - 24*a^3*b^8*c*d^9 + 44*a^5*b^6*c*d^9 + 12*a^2*b^9*c^2*d^8
- 18*a^4*b^7*c^2*d^8 - 4*a*b^10*c*d^9))/(a^10*d^2*f^4 + b^10*c^2*f^4 + 4*a^2*b^8*c^2*f^4 + 6*a^4*b^6*c^2*f^4 +
4*a^6*b^4*c^2*f^4 + a^8*b^2*c^2*f^4 + a^2*b^8*d^2*f^4 + 4*a^4*b^6*d^2*f^4 + 6*a^6*b^4*d^2*f^4 + 4*a^8*b^2*d^2
*f^4 - 2*a*b^9*c*d*f^4 - 2*a^9*b*c*d*f^4 - 8*a^3*b^7*c*d*f^4 - 12*a^5*b^5*c*d*f^4 - 8*a^7*b^3*c*d*f^4) + (((16
*(2*b^13*d^11*f^2 - 24*a^2*b^11*d^11*f^2 - 196*a^4*b^9*d^11*f^2 - 120*a^6*b^7*d^11*f^2 + 50*a^8*b^5*d^11*f^2 +
8*b^13*c^2*d^9*f^2 - 8*a^2*b^11*c^2*d^9*f^2 + 64*a^3*b^10*c^3*d^8*f^2 - 232*a^4*b^9*c^2*d^9*f^2 + 96*a^5*b^8*
c^3*d^8*f^2 - 216*a^6*b^7*c^2*d^9*f^2 - 32*a*b^12*c^3*d^8*f^2 + 208*a^3*b^10*c*d^10*f^2 + 288*a^5*b^8*c*d^10*f
^2 + 80*a^7*b^6*c*d^10*f^2))/(a^10*d^2*f^5 + b^10*c^2*f^5 + 4*a^2*b^8*c^2*f^5 + 6*a^4*b^6*c^2*f^5 + 4*a^6*b^4*
c^2*f^5 + a^8*b^2*c^2*f^5 + a^2*b^8*d^2*f^5 + 4*a^4*b^6*d^2*f^5 + 6*a^6*b^4*d^2*f^5 + 4*a^8*b^2*d^2*f^5 - 2*a*
b^9*c*d*f^5 - 2*a^9*b*c*d*f^5 - 8*a^3*b^7*c*d*f^5 - 12*a^5*b^5*c*d*f^5 - 8*a^7*b^3*c*d*f^5) + (((16*(c + d*tan
(e + f*x))^(1/2)*(8*a*b^14*d^11*f^2 + 4*b^15*c*d^10*f^2 + 36*a^3*b^12*d^11*f^2 + 316*a^5*b^10*d^11*f^2 + 552*a
^7*b^8*d^11*f^2 + 256*a^9*b^6*d^11*f^2 - 12*a^11*b^4*d^11*f^2 - 4*a^13*b^2*d^11*f^2 - 20*b^15*c^3*d^8*f^2 + 11
6*a^2*b^13*c^3*d^8*f^2 - 220*a^3*b^12*c^2*d^9*f^2 + 216*a^4*b^11*c^3*d^8*f^2 - 104*a^5*b^10*c^2*d^9*f^2 + 8*a^
6*b^9*c^3*d^8*f^2 + 232*a^7*b^8*c^2*d^9*f^2 - 68*a^8*b^7*c^3*d^8*f^2 + 156*a^9*b^6*c^2*d^9*f^2 + 4*a^10*b^5*c^
3*d^8*f^2 - 12*a^11*b^4*c^2*d^9*f^2 - 52*a*b^14*c^2*d^9*f^2 + 80*a^2*b^13*c*d^10*f^2 - 156*a^4*b^11*c*d^10*f^2
- 640*a^6*b^9*c*d^10*f^2 - 500*a^8*b^7*c*d^10*f^2 - 80*a^10*b^5*c*d^10*f^2 + 12*a^12*b^3*c*d^10*f^2))/(a^10*d
^2*f^4 + b^10*c^2*f^4 + 4*a^2*b^8*c^2*f^4 + 6*a^4*b^6*c^2*f^4 + 4*a^6*b^4*c^2*f^4 + a^8*b^2*c^2*f^4 + a^2*b^8*
d^2*f^4 + 4*a^4*b^6*d^2*f^4 + 6*a^6*b^4*d^2*f^4 + 4*a^8*b^2*d^2*f^4 - 2*a*b^9*c*d*f^4 - 2*a^9*b*c*d*f^4 - 8*a^
3*b^7*c*d*f^4 - 12*a^5*b^5*c*d*f^4 - 8*a^7*b^3*c*d*f^4) - (((16*(16*a*b^16*d^12*f^4 - 16*b^17*c*d^11*f^4 + 136
*a^3*b^14*d^12*f^4 + 432*a^5*b^12*d^12*f^4 + 680*a^7*b^10*d^12*f^4 + 560*a^9*b^8*d^12*f^4 + 216*a^11*b^6*d^12*
f^4 + 16*a^13*b^4*d^12*f^4 - 8*a^15*b^2*d^12*f^4 - 8*b^17*c^3*d^9*f^4 - 128*a^2*b^15*c^3*d^9*f^4 + 352*a^3*b^1
4*c^2*d^10*f^4 + 160*a^3*b^14*c^4*d^8*f^4 - 520*a^4*b^13*c^3*d^9*f^4 + 920*a^5*b^12*c^2*d^10*f^4 + 320*a^5*b^1
2*c^4*d^8*f^4 - 960*a^6*b^11*c^3*d^9*f^4 + 1280*a^7*b^10*c^2*d^10*f^4 + 320*a^7*b^10*c^4*d^8*f^4 - 920*a^8*b^9
*c^3*d^9*f^4 + 1000*a^9*b^8*c^2*d^10*f^4 + 160*a^9*b^8*c^4*d^8*f^4 - 448*a^10*b^7*c^3*d^9*f^4 + 416*a^11*b^6*c
^2*d^10*f^4 + 32*a^11*b^6*c^4*d^8*f^4 - 88*a^12*b^5*c^3*d^9*f^4 + 72*a^13*b^4*c^2*d^10*f^4 + 56*a*b^16*c^2*d^1
0*f^4 + 32*a*b^16*c^4*d^8*f^4 - 184*a^2*b^15*c*d^11*f^4 - 688*a^4*b^13*c*d^11*f^4 - 1240*a^6*b^11*c*d^11*f^4 -
1200*a^8*b^9*c*d^11*f^4 - 616*a^10*b^7*c*d^11*f^4 - 144*a^12*b^5*c*d^11*f^4 - 8*a^14*b^3*c*d^11*f^4))/(a^10*d
^2*f^5 + b^10*c^2*f^5 + 4*a^2*b^8*c^2*f^5 + 6*a^4*b^6*c^2*f^5 + 4*a^6*b^4*c^2*f^5 + a^8*b^2*c^2*f^5 + a^2*b^8*
d^2*f^5 + 4*a^4*b^6*d^2*f^5 + 6*a^6*b^4*d^2*f^5 + 4*a^8*b^2*d^2*f^5 - 2*a*b^9*c*d*f^5 - 2*a^9*b*c*d*f^5 - 8*a^
3*b^7*c*d*f^5 - 12*a^5*b^5*c*d*f^5 - 8*a^7*b^3*c*d*f^5) + (8*(1/(c - d*1i))^(1/2)*(c + d*tan(e + f*x))^(1/2)*(
32*a^2*b^17*d^12*f^4 + 160*a^4*b^15*d^12*f^4 + 288*a^6*b^13*d^12*f^4 + 160*a^8*b^11*d^12*f^4 - 160*a^10*b^9*d^
12*f^4 - 288*a^12*b^7*d^12*f^4 - 160*a^14*b^5*d^12*f^4 - 32*a^16*b^3*d^12*f^4 + 32*b^19*c^2*d^10*f^4 + 48*b^19
*c^4*d^8*f^4 + 176*a^2*b^17*c^2*d^10*f^4 + 272*a^2*b^17*c^4*d^8*f^4 - 432*a^3*b^16*c^3*d^9*f^4 + 336*a^4*b^15*
c^2*d^10*f^4 + 624*a^4*b^15*c^4*d^8*f^4 - 912*a^5*b^14*c^3*d^9*f^4 + 112*a^6*b^13*c^2*d^10*f^4 + 720*a^6*b^13*
c^4*d^8*f^4 - 880*a^7*b^12*c^3*d^9*f^4 - 560*a^8*b^11*c^2*d^10*f^4 + 400*a^8*b^11*c^4*d^8*f^4 - 240*a^9*b^10*c
^3*d^9*f^4 - 1008*a^10*b^9*c^2*d^10*f^4 + 48*a^10*b^9*c^4*d^8*f^4 + 240*a^11*b^8*c^3*d^9*f^4 - 784*a^12*b^7*c^
2*d^10*f^4 - 48*a^12*b^7*c^4*d^8*f^4 + 208*a^13*b^6*c^3*d^9*f^4 - 304*a^14*b^5*c^2*d^10*f^4 - 16*a^14*b^5*c^4*
d^8*f^4 + 48*a^15*b^4*c^3*d^9*f^4 - 48*a^16*b^3*c^2*d^10*f^4 - 64*a*b^18*c*d^11*f^4 - 80*a*b^18*c^3*d^9*f^4 -
304*a^3*b^16*c*d^11*f^4 - 464*a^5*b^14*c*d^11*f^4 + 16*a^7*b^12*c*d^11*f^4 + 880*a^9*b^10*c*d^11*f^4 + 1136*a^
11*b^8*c*d^11*f^4 + 656*a^13*b^6*c*d^11*f^4 + 176*a^15*b^4*c*d^11*f^4 + 16*a^17*b^2*c*d^11*f^4))/(f*(2*a*b + a
^2*1i - b^2*1i)*(a^10*d^2*f^4 + b^10*c^2*f^4 + 4*a^2*b^8*c^2*f^4 + 6*a^4*b^6*c^2*f^4 + 4*a^6*b^4*c^2*f^4 + a^8
*b^2*c^2*f^4 + a^2*b^8*d^2*f^4 + 4*a^4*b^6*d^2*f^4 + 6*a^6*b^4*d^2*f^4 + 4*a^8*b^2*d^2*f^4 - 2*a*b^9*c*d*f^4 -
2*a^9*b*c*d*f^4 - 8*a^3*b^7*c*d*f^4 - 12*a^5*b^5*c*d*f^4 - 8*a^7*b^3*c*d*f^4)))*(1/(c - d*1i))^(1/2))/(2*f*(2
*a*b + a^2*1i - b^2*1i)))*(1/(c - d*1i))^(1/2))/(2*f*(2*a*b + a^2*1i - b^2*1i)))*(1/(c - d*1i))^(1/2))/(2*f*(2
*a*b + a^2*1i - b^2*1i)))*(1/(c - d*1i))^(1/2)*1i)/(2*f*(2*a*b + a^2*1i - b^2*1i)))/((32*(a*b^8*d^10 - b^9*c*d
^9 + 5*a^3*b^6*d^10 + 4*a*b^8*c^2*d^8 - 9*a^2*b^7*c*d^9))/(a^10*d^2*f^5 + b^10*c^2*f^5 + 4*a^2*b^8*c^2*f^5 + 6
*a^4*b^6*c^2*f^5 + 4*a^6*b^4*c^2*f^5 + a^8*b^2*c^2*f^5 + a^2*b^8*d^2*f^5 + 4*a^4*b^6*d^2*f^5 + 6*a^6*b^4*d^2*f
^5 + 4*a^8*b^2*d^2*f^5 - 2*a*b^9*c*d*f^5 - 2*a^9*b*c*d*f^5 - 8*a^3*b^7*c*d*f^5 - 12*a^5*b^5*c*d*f^5 - 8*a^7*b^
3*c*d*f^5) + (((16*(c + d*tan(e + f*x))^(1/2)*(b^11*d^10 + 7*a^2*b^9*d^10 + 11*a^4*b^7*d^10 - 27*a^6*b^5*d^10
- 2*b^11*c^2*d^8 - 24*a^3*b^8*c*d^9 + 44*a^5*b^6*c*d^9 + 12*a^2*b^9*c^2*d^8 - 18*a^4*b^7*c^2*d^8 - 4*a*b^10*c*
d^9))/(a^10*d^2*f^4 + b^10*c^2*f^4 + 4*a^2*b^8*c^2*f^4 + 6*a^4*b^6*c^2*f^4 + 4*a^6*b^4*c^2*f^4 + a^8*b^2*c^2*f
^4 + a^2*b^8*d^2*f^4 + 4*a^4*b^6*d^2*f^4 + 6*a^6*b^4*d^2*f^4 + 4*a^8*b^2*d^2*f^4 - 2*a*b^9*c*d*f^4 - 2*a^9*b*c
*d*f^4 - 8*a^3*b^7*c*d*f^4 - 12*a^5*b^5*c*d*f^4 - 8*a^7*b^3*c*d*f^4) - (((16*(2*b^13*d^11*f^2 - 24*a^2*b^11*d^
11*f^2 - 196*a^4*b^9*d^11*f^2 - 120*a^6*b^7*d^11*f^2 + 50*a^8*b^5*d^11*f^2 + 8*b^13*c^2*d^9*f^2 - 8*a^2*b^11*c
^2*d^9*f^2 + 64*a^3*b^10*c^3*d^8*f^2 - 232*a^4*b^9*c^2*d^9*f^2 + 96*a^5*b^8*c^3*d^8*f^2 - 216*a^6*b^7*c^2*d^9*
f^2 - 32*a*b^12*c^3*d^8*f^2 + 208*a^3*b^10*c*d^10*f^2 + 288*a^5*b^8*c*d^10*f^2 + 80*a^7*b^6*c*d^10*f^2))/(a^10
*d^2*f^5 + b^10*c^2*f^5 + 4*a^2*b^8*c^2*f^5 + 6*a^4*b^6*c^2*f^5 + 4*a^6*b^4*c^2*f^5 + a^8*b^2*c^2*f^5 + a^2*b^
8*d^2*f^5 + 4*a^4*b^6*d^2*f^5 + 6*a^6*b^4*d^2*f^5 + 4*a^8*b^2*d^2*f^5 - 2*a*b^9*c*d*f^5 - 2*a^9*b*c*d*f^5 - 8*
a^3*b^7*c*d*f^5 - 12*a^5*b^5*c*d*f^5 - 8*a^7*b^3*c*d*f^5) - (((16*(c + d*tan(e + f*x))^(1/2)*(8*a*b^14*d^11*f^
2 + 4*b^15*c*d^10*f^2 + 36*a^3*b^12*d^11*f^2 + 316*a^5*b^10*d^11*f^2 + 552*a^7*b^8*d^11*f^2 + 256*a^9*b^6*d^11
*f^2 - 12*a^11*b^4*d^11*f^2 - 4*a^13*b^2*d^11*f^2 - 20*b^15*c^3*d^8*f^2 + 116*a^2*b^13*c^3*d^8*f^2 - 220*a^3*b
^12*c^2*d^9*f^2 + 216*a^4*b^11*c^3*d^8*f^2 - 104*a^5*b^10*c^2*d^9*f^2 + 8*a^6*b^9*c^3*d^8*f^2 + 232*a^7*b^8*c^
2*d^9*f^2 - 68*a^8*b^7*c^3*d^8*f^2 + 156*a^9*b^6*c^2*d^9*f^2 + 4*a^10*b^5*c^3*d^8*f^2 - 12*a^11*b^4*c^2*d^9*f^
2 - 52*a*b^14*c^2*d^9*f^2 + 80*a^2*b^13*c*d^10*f^2 - 156*a^4*b^11*c*d^10*f^2 - 640*a^6*b^9*c*d^10*f^2 - 500*a^
8*b^7*c*d^10*f^2 - 80*a^10*b^5*c*d^10*f^2 + 12*a^12*b^3*c*d^10*f^2))/(a^10*d^2*f^4 + b^10*c^2*f^4 + 4*a^2*b^8*
c^2*f^4 + 6*a^4*b^6*c^2*f^4 + 4*a^6*b^4*c^2*f^4 + a^8*b^2*c^2*f^4 + a^2*b^8*d^2*f^4 + 4*a^4*b^6*d^2*f^4 + 6*a^
6*b^4*d^2*f^4 + 4*a^8*b^2*d^2*f^4 - 2*a*b^9*c*d*f^4 - 2*a^9*b*c*d*f^4 - 8*a^3*b^7*c*d*f^4 - 12*a^5*b^5*c*d*f^4
- 8*a^7*b^3*c*d*f^4) + (((16*(16*a*b^16*d^12*f^4 - 16*b^17*c*d^11*f^4 + 136*a^3*b^14*d^12*f^4 + 432*a^5*b^12*
d^12*f^4 + 680*a^7*b^10*d^12*f^4 + 560*a^9*b^8*d^12*f^4 + 216*a^11*b^6*d^12*f^4 + 16*a^13*b^4*d^12*f^4 - 8*a^1
5*b^2*d^12*f^4 - 8*b^17*c^3*d^9*f^4 - 128*a^2*b^15*c^3*d^9*f^4 + 352*a^3*b^14*c^2*d^10*f^4 + 160*a^3*b^14*c^4*
d^8*f^4 - 520*a^4*b^13*c^3*d^9*f^4 + 920*a^5*b^12*c^2*d^10*f^4 + 320*a^5*b^12*c^4*d^8*f^4 - 960*a^6*b^11*c^3*d
^9*f^4 + 1280*a^7*b^10*c^2*d^10*f^4 + 320*a^7*b^10*c^4*d^8*f^4 - 920*a^8*b^9*c^3*d^9*f^4 + 1000*a^9*b^8*c^2*d^
10*f^4 + 160*a^9*b^8*c^4*d^8*f^4 - 448*a^10*b^7*c^3*d^9*f^4 + 416*a^11*b^6*c^2*d^10*f^4 + 32*a^11*b^6*c^4*d^8*
f^4 - 88*a^12*b^5*c^3*d^9*f^4 + 72*a^13*b^4*c^2*d^10*f^4 + 56*a*b^16*c^2*d^10*f^4 + 32*a*b^16*c^4*d^8*f^4 - 18
4*a^2*b^15*c*d^11*f^4 - 688*a^4*b^13*c*d^11*f^4 - 1240*a^6*b^11*c*d^11*f^4 - 1200*a^8*b^9*c*d^11*f^4 - 616*a^1
0*b^7*c*d^11*f^4 - 144*a^12*b^5*c*d^11*f^4 - 8*a^14*b^3*c*d^11*f^4))/(a^10*d^2*f^5 + b^10*c^2*f^5 + 4*a^2*b^8*
c^2*f^5 + 6*a^4*b^6*c^2*f^5 + 4*a^6*b^4*c^2*f^5 + a^8*b^2*c^2*f^5 + a^2*b^8*d^2*f^5 + 4*a^4*b^6*d^2*f^5 + 6*a^
6*b^4*d^2*f^5 + 4*a^8*b^2*d^2*f^5 - 2*a*b^9*c*d*f^5 - 2*a^9*b*c*d*f^5 - 8*a^3*b^7*c*d*f^5 - 12*a^5*b^5*c*d*f^5
- 8*a^7*b^3*c*d*f^5) - (8*(1/(c - d*1i))^(1/2)*(c + d*tan(e + f*x))^(1/2)*(32*a^2*b^17*d^12*f^4 + 160*a^4*b^1
5*d^12*f^4 + 288*a^6*b^13*d^12*f^4 + 160*a^8*b^11*d^12*f^4 - 160*a^10*b^9*d^12*f^4 - 288*a^12*b^7*d^12*f^4 - 1
60*a^14*b^5*d^12*f^4 - 32*a^16*b^3*d^12*f^4 + 32*b^19*c^2*d^10*f^4 + 48*b^19*c^4*d^8*f^4 + 176*a^2*b^17*c^2*d^
10*f^4 + 272*a^2*b^17*c^4*d^8*f^4 - 432*a^3*b^16*c^3*d^9*f^4 + 336*a^4*b^15*c^2*d^10*f^4 + 624*a^4*b^15*c^4*d^
8*f^4 - 912*a^5*b^14*c^3*d^9*f^4 + 112*a^6*b^13*c^2*d^10*f^4 + 720*a^6*b^13*c^4*d^8*f^4 - 880*a^7*b^12*c^3*d^9
*f^4 - 560*a^8*b^11*c^2*d^10*f^4 + 400*a^8*b^11*c^4*d^8*f^4 - 240*a^9*b^10*c^3*d^9*f^4 - 1008*a^10*b^9*c^2*d^1
0*f^4 + 48*a^10*b^9*c^4*d^8*f^4 + 240*a^11*b^8*c^3*d^9*f^4 - 784*a^12*b^7*c^2*d^10*f^4 - 48*a^12*b^7*c^4*d^8*f
^4 + 208*a^13*b^6*c^3*d^9*f^4 - 304*a^14*b^5*c^2*d^10*f^4 - 16*a^14*b^5*c^4*d^8*f^4 + 48*a^15*b^4*c^3*d^9*f^4
- 48*a^16*b^3*c^2*d^10*f^4 - 64*a*b^18*c*d^11*f^4 - 80*a*b^18*c^3*d^9*f^4 - 304*a^3*b^16*c*d^11*f^4 - 464*a^5*
b^14*c*d^11*f^4 + 16*a^7*b^12*c*d^11*f^4 + 880*a^9*b^10*c*d^11*f^4 + 1136*a^11*b^8*c*d^11*f^4 + 656*a^13*b^6*c
*d^11*f^4 + 176*a^15*b^4*c*d^11*f^4 + 16*a^17*b^2*c*d^11*f^4))/(f*(2*a*b + a^2*1i - b^2*1i)*(a^10*d^2*f^4 + b^
10*c^2*f^4 + 4*a^2*b^8*c^2*f^4 + 6*a^4*b^6*c^2*f^4 + 4*a^6*b^4*c^2*f^4 + a^8*b^2*c^2*f^4 + a^2*b^8*d^2*f^4 + 4
*a^4*b^6*d^2*f^4 + 6*a^6*b^4*d^2*f^4 + 4*a^8*b^2*d^2*f^4 - 2*a*b^9*c*d*f^4 - 2*a^9*b*c*d*f^4 - 8*a^3*b^7*c*d*f
^4 - 12*a^5*b^5*c*d*f^4 - 8*a^7*b^3*c*d*f^4)))*(1/(c - d*1i))^(1/2))/(2*f*(2*a*b + a^2*1i - b^2*1i)))*(1/(c -
d*1i))^(1/2))/(2*f*(2*a*b + a^2*1i - b^2*1i)))*(1/(c - d*1i))^(1/2))/(2*f*(2*a*b + a^2*1i - b^2*1i)))*(1/(c -
d*1i))^(1/2))/(2*f*(2*a*b + a^2*1i - b^2*1i)) - (((16*(c + d*tan(e + f*x))^(1/2)*(b^11*d^10 + 7*a^2*b^9*d^10 +
11*a^4*b^7*d^10 - 27*a^6*b^5*d^10 - 2*b^11*c^2*d^8 - 24*a^3*b^8*c*d^9 + 44*a^5*b^6*c*d^9 + 12*a^2*b^9*c^2*d^8
- 18*a^4*b^7*c^2*d^8 - 4*a*b^10*c*d^9))/(a^10*d^2*f^4 + b^10*c^2*f^4 + 4*a^2*b^8*c^2*f^4 + 6*a^4*b^6*c^2*f^4
+ 4*a^6*b^4*c^2*f^4 + a^8*b^2*c^2*f^4 + a^2*b^8*d^2*f^4 + 4*a^4*b^6*d^2*f^4 + 6*a^6*b^4*d^2*f^4 + 4*a^8*b^2*d^
2*f^4 - 2*a*b^9*c*d*f^4 - 2*a^9*b*c*d*f^4 - 8*a^3*b^7*c*d*f^4 - 12*a^5*b^5*c*d*f^4 - 8*a^7*b^3*c*d*f^4) + (((1
6*(2*b^13*d^11*f^2 - 24*a^2*b^11*d^11*f^2 - 196*a^4*b^9*d^11*f^2 - 120*a^6*b^7*d^11*f^2 + 50*a^8*b^5*d^11*f^2
+ 8*b^13*c^2*d^9*f^2 - 8*a^2*b^11*c^2*d^9*f^2 + 64*a^3*b^10*c^3*d^8*f^2 - 232*a^4*b^9*c^2*d^9*f^2 + 96*a^5*b^8
*c^3*d^8*f^2 - 216*a^6*b^7*c^2*d^9*f^2 - 32*a*b^12*c^3*d^8*f^2 + 208*a^3*b^10*c*d^10*f^2 + 288*a^5*b^8*c*d^10*
f^2 + 80*a^7*b^6*c*d^10*f^2))/(a^10*d^2*f^5 + b^10*c^2*f^5 + 4*a^2*b^8*c^2*f^5 + 6*a^4*b^6*c^2*f^5 + 4*a^6*b^4
*c^2*f^5 + a^8*b^2*c^2*f^5 + a^2*b^8*d^2*f^5 + 4*a^4*b^6*d^2*f^5 + 6*a^6*b^4*d^2*f^5 + 4*a^8*b^2*d^2*f^5 - 2*a
*b^9*c*d*f^5 - 2*a^9*b*c*d*f^5 - 8*a^3*b^7*c*d*f^5 - 12*a^5*b^5*c*d*f^5 - 8*a^7*b^3*c*d*f^5) + (((16*(c + d*ta
n(e + f*x))^(1/2)*(8*a*b^14*d^11*f^2 + 4*b^15*c*d^10*f^2 + 36*a^3*b^12*d^11*f^2 + 316*a^5*b^10*d^11*f^2 + 552*
a^7*b^8*d^11*f^2 + 256*a^9*b^6*d^11*f^2 - 12*a^11*b^4*d^11*f^2 - 4*a^13*b^2*d^11*f^2 - 20*b^15*c^3*d^8*f^2 + 1
16*a^2*b^13*c^3*d^8*f^2 - 220*a^3*b^12*c^2*d^9*f^2 + 216*a^4*b^11*c^3*d^8*f^2 - 104*a^5*b^10*c^2*d^9*f^2 + 8*a
^6*b^9*c^3*d^8*f^2 + 232*a^7*b^8*c^2*d^9*f^2 - 68*a^8*b^7*c^3*d^8*f^2 + 156*a^9*b^6*c^2*d^9*f^2 + 4*a^10*b^5*c
^3*d^8*f^2 - 12*a^11*b^4*c^2*d^9*f^2 - 52*a*b^14*c^2*d^9*f^2 + 80*a^2*b^13*c*d^10*f^2 - 156*a^4*b^11*c*d^10*f^
2 - 640*a^6*b^9*c*d^10*f^2 - 500*a^8*b^7*c*d^10*f^2 - 80*a^10*b^5*c*d^10*f^2 + 12*a^12*b^3*c*d^10*f^2))/(a^10*
d^2*f^4 + b^10*c^2*f^4 + 4*a^2*b^8*c^2*f^4 + 6*a^4*b^6*c^2*f^4 + 4*a^6*b^4*c^2*f^4 + a^8*b^2*c^2*f^4 + a^2*b^8
*d^2*f^4 + 4*a^4*b^6*d^2*f^4 + 6*a^6*b^4*d^2*f^4 + 4*a^8*b^2*d^2*f^4 - 2*a*b^9*c*d*f^4 - 2*a^9*b*c*d*f^4 - 8*a
^3*b^7*c*d*f^4 - 12*a^5*b^5*c*d*f^4 - 8*a^7*b^3*c*d*f^4) - (((16*(16*a*b^16*d^12*f^4 - 16*b^17*c*d^11*f^4 + 13
6*a^3*b^14*d^12*f^4 + 432*a^5*b^12*d^12*f^4 + 680*a^7*b^10*d^12*f^4 + 560*a^9*b^8*d^12*f^4 + 216*a^11*b^6*d^12
*f^4 + 16*a^13*b^4*d^12*f^4 - 8*a^15*b^2*d^12*f^4 - 8*b^17*c^3*d^9*f^4 - 128*a^2*b^15*c^3*d^9*f^4 + 352*a^3*b^
14*c^2*d^10*f^4 + 160*a^3*b^14*c^4*d^8*f^4 - 520*a^4*b^13*c^3*d^9*f^4 + 920*a^5*b^12*c^2*d^10*f^4 + 320*a^5*b^
12*c^4*d^8*f^4 - 960*a^6*b^11*c^3*d^9*f^4 + 1280*a^7*b^10*c^2*d^10*f^4 + 320*a^7*b^10*c^4*d^8*f^4 - 920*a^8*b^
9*c^3*d^9*f^4 + 1000*a^9*b^8*c^2*d^10*f^4 + 160*a^9*b^8*c^4*d^8*f^4 - 448*a^10*b^7*c^3*d^9*f^4 + 416*a^11*b^6*
c^2*d^10*f^4 + 32*a^11*b^6*c^4*d^8*f^4 - 88*a^12*b^5*c^3*d^9*f^4 + 72*a^13*b^4*c^2*d^10*f^4 + 56*a*b^16*c^2*d^
10*f^4 + 32*a*b^16*c^4*d^8*f^4 - 184*a^2*b^15*c*d^11*f^4 - 688*a^4*b^13*c*d^11*f^4 - 1240*a^6*b^11*c*d^11*f^4
- 1200*a^8*b^9*c*d^11*f^4 - 616*a^10*b^7*c*d^11*f^4 - 144*a^12*b^5*c*d^11*f^4 - 8*a^14*b^3*c*d^11*f^4))/(a^10*
d^2*f^5 + b^10*c^2*f^5 + 4*a^2*b^8*c^2*f^5 + 6*a^4*b^6*c^2*f^5 + 4*a^6*b^4*c^2*f^5 + a^8*b^2*c^2*f^5 + a^2*b^8
*d^2*f^5 + 4*a^4*b^6*d^2*f^5 + 6*a^6*b^4*d^2*f^5 + 4*a^8*b^2*d^2*f^5 - 2*a*b^9*c*d*f^5 - 2*a^9*b*c*d*f^5 - 8*a
^3*b^7*c*d*f^5 - 12*a^5*b^5*c*d*f^5 - 8*a^7*b^3*c*d*f^5) + (8*(1/(c - d*1i))^(1/2)*(c + d*tan(e + f*x))^(1/2)*
(32*a^2*b^17*d^12*f^4 + 160*a^4*b^15*d^12*f^4 + 288*a^6*b^13*d^12*f^4 + 160*a^8*b^11*d^12*f^4 - 160*a^10*b^9*d
^12*f^4 - 288*a^12*b^7*d^12*f^4 - 160*a^14*b^5*d^12*f^4 - 32*a^16*b^3*d^12*f^4 + 32*b^19*c^2*d^10*f^4 + 48*b^1
9*c^4*d^8*f^4 + 176*a^2*b^17*c^2*d^10*f^4 + 272*a^2*b^17*c^4*d^8*f^4 - 432*a^3*b^16*c^3*d^9*f^4 + 336*a^4*b^15
*c^2*d^10*f^4 + 624*a^4*b^15*c^4*d^8*f^4 - 912*a^5*b^14*c^3*d^9*f^4 + 112*a^6*b^13*c^2*d^10*f^4 + 720*a^6*b^13
*c^4*d^8*f^4 - 880*a^7*b^12*c^3*d^9*f^4 - 560*a^8*b^11*c^2*d^10*f^4 + 400*a^8*b^11*c^4*d^8*f^4 - 240*a^9*b^10*
c^3*d^9*f^4 - 1008*a^10*b^9*c^2*d^10*f^4 + 48*a^10*b^9*c^4*d^8*f^4 + 240*a^11*b^8*c^3*d^9*f^4 - 784*a^12*b^7*c
^2*d^10*f^4 - 48*a^12*b^7*c^4*d^8*f^4 + 208*a^13*b^6*c^3*d^9*f^4 - 304*a^14*b^5*c^2*d^10*f^4 - 16*a^14*b^5*c^4
*d^8*f^4 + 48*a^15*b^4*c^3*d^9*f^4 - 48*a^16*b^3*c^2*d^10*f^4 - 64*a*b^18*c*d^11*f^4 - 80*a*b^18*c^3*d^9*f^4 -
304*a^3*b^16*c*d^11*f^4 - 464*a^5*b^14*c*d^11*f^4 + 16*a^7*b^12*c*d^11*f^4 + 880*a^9*b^10*c*d^11*f^4 + 1136*a
^11*b^8*c*d^11*f^4 + 656*a^13*b^6*c*d^11*f^4 + 176*a^15*b^4*c*d^11*f^4 + 16*a^17*b^2*c*d^11*f^4))/(f*(2*a*b +
a^2*1i - b^2*1i)*(a^10*d^2*f^4 + b^10*c^2*f^4 + 4*a^2*b^8*c^2*f^4 + 6*a^4*b^6*c^2*f^4 + 4*a^6*b^4*c^2*f^4 + a^
8*b^2*c^2*f^4 + a^2*b^8*d^2*f^4 + 4*a^4*b^6*d^2*f^4 + 6*a^6*b^4*d^2*f^4 + 4*a^8*b^2*d^2*f^4 - 2*a*b^9*c*d*f^4
- 2*a^9*b*c*d*f^4 - 8*a^3*b^7*c*d*f^4 - 12*a^5*b^5*c*d*f^4 - 8*a^7*b^3*c*d*f^4)))*(1/(c - d*1i))^(1/2))/(2*f*(
2*a*b + a^2*1i - b^2*1i)))*(1/(c - d*1i))^(1/2))/(2*f*(2*a*b + a^2*1i - b^2*1i)))*(1/(c - d*1i))^(1/2))/(2*f*(
2*a*b + a^2*1i - b^2*1i)))*(1/(c - d*1i))^(1/2))/(2*f*(2*a*b + a^2*1i - b^2*1i))))*(1/(c - d*1i))^(1/2)*1i)/(f
*(2*a*b + a^2*1i - b^2*1i)) - (atan(((((16*(c + d*tan(e + f*x))^(1/2)*(b^11*d^10 + 7*a^2*b^9*d^10 + 11*a^4*b^7
*d^10 - 27*a^6*b^5*d^10 - 2*b^11*c^2*d^8 - 24*a^3*b^8*c*d^9 + 44*a^5*b^6*c*d^9 + 12*a^2*b^9*c^2*d^8 - 18*a^4*b
^7*c^2*d^8 - 4*a*b^10*c*d^9))/(a^10*d^2*f^4 + b^10*c^2*f^4 + 4*a^2*b^8*c^2*f^4 + 6*a^4*b^6*c^2*f^4 + 4*a^6*b^4
*c^2*f^4 + a^8*b^2*c^2*f^4 + a^2*b^8*d^2*f^4 + 4*a^4*b^6*d^2*f^4 + 6*a^6*b^4*d^2*f^4 + 4*a^8*b^2*d^2*f^4 - 2*a
*b^9*c*d*f^4 - 2*a^9*b*c*d*f^4 - 8*a^3*b^7*c*d*f^4 - 12*a^5*b^5*c*d*f^4 - 8*a^7*b^3*c*d*f^4) - (((16*(2*b^13*d
^11*f^2 - 24*a^2*b^11*d^11*f^2 - 196*a^4*b^9*d^11*f^2 - 120*a^6*b^7*d^11*f^2 + 50*a^8*b^5*d^11*f^2 + 8*b^13*c^
2*d^9*f^2 - 8*a^2*b^11*c^2*d^9*f^2 + 64*a^3*b^10*c^3*d^8*f^2 - 232*a^4*b^9*c^2*d^9*f^2 + 96*a^5*b^8*c^3*d^8*f^
2 - 216*a^6*b^7*c^2*d^9*f^2 - 32*a*b^12*c^3*d^8*f^2 + 208*a^3*b^10*c*d^10*f^2 + 288*a^5*b^8*c*d^10*f^2 + 80*a^
7*b^6*c*d^10*f^2))/(a^10*d^2*f^5 + b^10*c^2*f^5 + 4*a^2*b^8*c^2*f^5 + 6*a^4*b^6*c^2*f^5 + 4*a^6*b^4*c^2*f^5 +
a^8*b^2*c^2*f^5 + a^2*b^8*d^2*f^5 + 4*a^4*b^6*d^2*f^5 + 6*a^6*b^4*d^2*f^5 + 4*a^8*b^2*d^2*f^5 - 2*a*b^9*c*d*f^
5 - 2*a^9*b*c*d*f^5 - 8*a^3*b^7*c*d*f^5 - 12*a^5*b^5*c*d*f^5 - 8*a^7*b^3*c*d*f^5) - (((16*(c + d*tan(e + f*x))
^(1/2)*(8*a*b^14*d^11*f^2 + 4*b^15*c*d^10*f^2 + 36*a^3*b^12*d^11*f^2 + 316*a^5*b^10*d^11*f^2 + 552*a^7*b^8*d^1
1*f^2 + 256*a^9*b^6*d^11*f^2 - 12*a^11*b^4*d^11*f^2 - 4*a^13*b^2*d^11*f^2 - 20*b^15*c^3*d^8*f^2 + 116*a^2*b^13
*c^3*d^8*f^2 - 220*a^3*b^12*c^2*d^9*f^2 + 216*a^4*b^11*c^3*d^8*f^2 - 104*a^5*b^10*c^2*d^9*f^2 + 8*a^6*b^9*c^3*
d^8*f^2 + 232*a^7*b^8*c^2*d^9*f^2 - 68*a^8*b^7*c^3*d^8*f^2 + 156*a^9*b^6*c^2*d^9*f^2 + 4*a^10*b^5*c^3*d^8*f^2
- 12*a^11*b^4*c^2*d^9*f^2 - 52*a*b^14*c^2*d^9*f^2 + 80*a^2*b^13*c*d^10*f^2 - 156*a^4*b^11*c*d^10*f^2 - 640*a^6
*b^9*c*d^10*f^2 - 500*a^8*b^7*c*d^10*f^2 - 80*a^10*b^5*c*d^10*f^2 + 12*a^12*b^3*c*d^10*f^2))/(a^10*d^2*f^4 + b
^10*c^2*f^4 + 4*a^2*b^8*c^2*f^4 + 6*a^4*b^6*c^2*f^4 + 4*a^6*b^4*c^2*f^4 + a^8*b^2*c^2*f^4 + a^2*b^8*d^2*f^4 +
4*a^4*b^6*d^2*f^4 + 6*a^6*b^4*d^2*f^4 + 4*a^8*b^2*d^2*f^4 - 2*a*b^9*c*d*f^4 - 2*a^9*b*c*d*f^4 - 8*a^3*b^7*c*d*
f^4 - 12*a^5*b^5*c*d*f^4 - 8*a^7*b^3*c*d*f^4) + (((16*(16*a*b^16*d^12*f^4 - 16*b^17*c*d^11*f^4 + 136*a^3*b^14*
d^12*f^4 + 432*a^5*b^12*d^12*f^4 + 680*a^7*b^10*d^12*f^4 + 560*a^9*b^8*d^12*f^4 + 216*a^11*b^6*d^12*f^4 + 16*a
^13*b^4*d^12*f^4 - 8*a^15*b^2*d^12*f^4 - 8*b^17*c^3*d^9*f^4 - 128*a^2*b^15*c^3*d^9*f^4 + 352*a^3*b^14*c^2*d^10
*f^4 + 160*a^3*b^14*c^4*d^8*f^4 - 520*a^4*b^13*c^3*d^9*f^4 + 920*a^5*b^12*c^2*d^10*f^4 + 320*a^5*b^12*c^4*d^8*
f^4 - 960*a^6*b^11*c^3*d^9*f^4 + 1280*a^7*b^10*c^2*d^10*f^4 + 320*a^7*b^10*c^4*d^8*f^4 - 920*a^8*b^9*c^3*d^9*f
^4 + 1000*a^9*b^8*c^2*d^10*f^4 + 160*a^9*b^8*c^4*d^8*f^4 - 448*a^10*b^7*c^3*d^9*f^4 + 416*a^11*b^6*c^2*d^10*f^
4 + 32*a^11*b^6*c^4*d^8*f^4 - 88*a^12*b^5*c^3*d^9*f^4 + 72*a^13*b^4*c^2*d^10*f^4 + 56*a*b^16*c^2*d^10*f^4 + 32
*a*b^16*c^4*d^8*f^4 - 184*a^2*b^15*c*d^11*f^4 - 688*a^4*b^13*c*d^11*f^4 - 1240*a^6*b^11*c*d^11*f^4 - 1200*a^8*
b^9*c*d^11*f^4 - 616*a^10*b^7*c*d^11*f^4 - 144*a^12*b^5*c*d^11*f^4 - 8*a^14*b^3*c*d^11*f^4))/(a^10*d^2*f^5 + b
^10*c^2*f^5 + 4*a^2*b^8*c^2*f^5 + 6*a^4*b^6*c^2*f^5 + 4*a^6*b^4*c^2*f^5 + a^8*b^2*c^2*f^5 + a^2*b^8*d^2*f^5 +
4*a^4*b^6*d^2*f^5 + 6*a^6*b^4*d^2*f^5 + 4*a^8*b^2*d^2*f^5 - 2*a*b^9*c*d*f^5 - 2*a^9*b*c*d*f^5 - 8*a^3*b^7*c*d*
f^5 - 12*a^5*b^5*c*d*f^5 - 8*a^7*b^3*c*d*f^5) - ((1/(c + d*1i))^(1/2)*(c + d*tan(e + f*x))^(1/2)*(32*a^2*b^17*
d^12*f^4 + 160*a^4*b^15*d^12*f^4 + 288*a^6*b^13*d^12*f^4 + 160*a^8*b^11*d^12*f^4 - 160*a^10*b^9*d^12*f^4 - 288
*a^12*b^7*d^12*f^4 - 160*a^14*b^5*d^12*f^4 - 32*a^16*b^3*d^12*f^4 + 32*b^19*c^2*d^10*f^4 + 48*b^19*c^4*d^8*f^4
+ 176*a^2*b^17*c^2*d^10*f^4 + 272*a^2*b^17*c^4*d^8*f^4 - 432*a^3*b^16*c^3*d^9*f^4 + 336*a^4*b^15*c^2*d^10*f^4
+ 624*a^4*b^15*c^4*d^8*f^4 - 912*a^5*b^14*c^3*d^9*f^4 + 112*a^6*b^13*c^2*d^10*f^4 + 720*a^6*b^13*c^4*d^8*f^4
- 880*a^7*b^12*c^3*d^9*f^4 - 560*a^8*b^11*c^2*d^10*f^4 + 400*a^8*b^11*c^4*d^8*f^4 - 240*a^9*b^10*c^3*d^9*f^4 -
1008*a^10*b^9*c^2*d^10*f^4 + 48*a^10*b^9*c^4*d^8*f^4 + 240*a^11*b^8*c^3*d^9*f^4 - 784*a^12*b^7*c^2*d^10*f^4 -
48*a^12*b^7*c^4*d^8*f^4 + 208*a^13*b^6*c^3*d^9*f^4 - 304*a^14*b^5*c^2*d^10*f^4 - 16*a^14*b^5*c^4*d^8*f^4 + 48
*a^15*b^4*c^3*d^9*f^4 - 48*a^16*b^3*c^2*d^10*f^4 - 64*a*b^18*c*d^11*f^4 - 80*a*b^18*c^3*d^9*f^4 - 304*a^3*b^16
*c*d^11*f^4 - 464*a^5*b^14*c*d^11*f^4 + 16*a^7*b^12*c*d^11*f^4 + 880*a^9*b^10*c*d^11*f^4 + 1136*a^11*b^8*c*d^1
1*f^4 + 656*a^13*b^6*c*d^11*f^4 + 176*a^15*b^4*c*d^11*f^4 + 16*a^17*b^2*c*d^11*f^4)*8i)/(f*(a*b*2i + a^2 - b^2
)*(a^10*d^2*f^4 + b^10*c^2*f^4 + 4*a^2*b^8*c^2*f^4 + 6*a^4*b^6*c^2*f^4 + 4*a^6*b^4*c^2*f^4 + a^8*b^2*c^2*f^4 +
a^2*b^8*d^2*f^4 + 4*a^4*b^6*d^2*f^4 + 6*a^6*b^4*d^2*f^4 + 4*a^8*b^2*d^2*f^4 - 2*a*b^9*c*d*f^4 - 2*a^9*b*c*d*f
^4 - 8*a^3*b^7*c*d*f^4 - 12*a^5*b^5*c*d*f^4 - 8*a^7*b^3*c*d*f^4)))*(1/(c + d*1i))^(1/2)*1i)/(2*f*(a*b*2i + a^2
- b^2)))*(1/(c + d*1i))^(1/2)*1i)/(2*f*(a*b*2i + a^2 - b^2)))*(1/(c + d*1i))^(1/2)*1i)/(2*f*(a*b*2i + a^2 - b
^2)))*(1/(c + d*1i))^(1/2))/(2*f*(a*b*2i + a^2 - b^2)) + (((16*(c + d*tan(e + f*x))^(1/2)*(b^11*d^10 + 7*a^2*b
^9*d^10 + 11*a^4*b^7*d^10 - 27*a^6*b^5*d^10 - 2*b^11*c^2*d^8 - 24*a^3*b^8*c*d^9 + 44*a^5*b^6*c*d^9 + 12*a^2*b^
9*c^2*d^8 - 18*a^4*b^7*c^2*d^8 - 4*a*b^10*c*d^9))/(a^10*d^2*f^4 + b^10*c^2*f^4 + 4*a^2*b^8*c^2*f^4 + 6*a^4*b^6
*c^2*f^4 + 4*a^6*b^4*c^2*f^4 + a^8*b^2*c^2*f^4 + a^2*b^8*d^2*f^4 + 4*a^4*b^6*d^2*f^4 + 6*a^6*b^4*d^2*f^4 + 4*a
^8*b^2*d^2*f^4 - 2*a*b^9*c*d*f^4 - 2*a^9*b*c*d*f^4 - 8*a^3*b^7*c*d*f^4 - 12*a^5*b^5*c*d*f^4 - 8*a^7*b^3*c*d*f^
4) + (((16*(2*b^13*d^11*f^2 - 24*a^2*b^11*d^11*f^2 - 196*a^4*b^9*d^11*f^2 - 120*a^6*b^7*d^11*f^2 + 50*a^8*b^5*
d^11*f^2 + 8*b^13*c^2*d^9*f^2 - 8*a^2*b^11*c^2*d^9*f^2 + 64*a^3*b^10*c^3*d^8*f^2 - 232*a^4*b^9*c^2*d^9*f^2 + 9
6*a^5*b^8*c^3*d^8*f^2 - 216*a^6*b^7*c^2*d^9*f^2 - 32*a*b^12*c^3*d^8*f^2 + 208*a^3*b^10*c*d^10*f^2 + 288*a^5*b^
8*c*d^10*f^2 + 80*a^7*b^6*c*d^10*f^2))/(a^10*d^2*f^5 + b^10*c^2*f^5 + 4*a^2*b^8*c^2*f^5 + 6*a^4*b^6*c^2*f^5 +
4*a^6*b^4*c^2*f^5 + a^8*b^2*c^2*f^5 + a^2*b^8*d^2*f^5 + 4*a^4*b^6*d^2*f^5 + 6*a^6*b^4*d^2*f^5 + 4*a^8*b^2*d^2*
f^5 - 2*a*b^9*c*d*f^5 - 2*a^9*b*c*d*f^5 - 8*a^3*b^7*c*d*f^5 - 12*a^5*b^5*c*d*f^5 - 8*a^7*b^3*c*d*f^5) + (((16*
(c + d*tan(e + f*x))^(1/2)*(8*a*b^14*d^11*f^2 + 4*b^15*c*d^10*f^2 + 36*a^3*b^12*d^11*f^2 + 316*a^5*b^10*d^11*f
^2 + 552*a^7*b^8*d^11*f^2 + 256*a^9*b^6*d^11*f^2 - 12*a^11*b^4*d^11*f^2 - 4*a^13*b^2*d^11*f^2 - 20*b^15*c^3*d^
8*f^2 + 116*a^2*b^13*c^3*d^8*f^2 - 220*a^3*b^12*c^2*d^9*f^2 + 216*a^4*b^11*c^3*d^8*f^2 - 104*a^5*b^10*c^2*d^9*
f^2 + 8*a^6*b^9*c^3*d^8*f^2 + 232*a^7*b^8*c^2*d^9*f^2 - 68*a^8*b^7*c^3*d^8*f^2 + 156*a^9*b^6*c^2*d^9*f^2 + 4*a
^10*b^5*c^3*d^8*f^2 - 12*a^11*b^4*c^2*d^9*f^2 - 52*a*b^14*c^2*d^9*f^2 + 80*a^2*b^13*c*d^10*f^2 - 156*a^4*b^11*
c*d^10*f^2 - 640*a^6*b^9*c*d^10*f^2 - 500*a^8*b^7*c*d^10*f^2 - 80*a^10*b^5*c*d^10*f^2 + 12*a^12*b^3*c*d^10*f^2
))/(a^10*d^2*f^4 + b^10*c^2*f^4 + 4*a^2*b^8*c^2*f^4 + 6*a^4*b^6*c^2*f^4 + 4*a^6*b^4*c^2*f^4 + a^8*b^2*c^2*f^4
+ a^2*b^8*d^2*f^4 + 4*a^4*b^6*d^2*f^4 + 6*a^6*b^4*d^2*f^4 + 4*a^8*b^2*d^2*f^4 - 2*a*b^9*c*d*f^4 - 2*a^9*b*c*d*
f^4 - 8*a^3*b^7*c*d*f^4 - 12*a^5*b^5*c*d*f^4 - 8*a^7*b^3*c*d*f^4) - (((16*(16*a*b^16*d^12*f^4 - 16*b^17*c*d^11
*f^4 + 136*a^3*b^14*d^12*f^4 + 432*a^5*b^12*d^12*f^4 + 680*a^7*b^10*d^12*f^4 + 560*a^9*b^8*d^12*f^4 + 216*a^11
*b^6*d^12*f^4 + 16*a^13*b^4*d^12*f^4 - 8*a^15*b^2*d^12*f^4 - 8*b^17*c^3*d^9*f^4 - 128*a^2*b^15*c^3*d^9*f^4 + 3
52*a^3*b^14*c^2*d^10*f^4 + 160*a^3*b^14*c^4*d^8*f^4 - 520*a^4*b^13*c^3*d^9*f^4 + 920*a^5*b^12*c^2*d^10*f^4 + 3
20*a^5*b^12*c^4*d^8*f^4 - 960*a^6*b^11*c^3*d^9*f^4 + 1280*a^7*b^10*c^2*d^10*f^4 + 320*a^7*b^10*c^4*d^8*f^4 - 9
20*a^8*b^9*c^3*d^9*f^4 + 1000*a^9*b^8*c^2*d^10*f^4 + 160*a^9*b^8*c^4*d^8*f^4 - 448*a^10*b^7*c^3*d^9*f^4 + 416*
a^11*b^6*c^2*d^10*f^4 + 32*a^11*b^6*c^4*d^8*f^4 - 88*a^12*b^5*c^3*d^9*f^4 + 72*a^13*b^4*c^2*d^10*f^4 + 56*a*b^
16*c^2*d^10*f^4 + 32*a*b^16*c^4*d^8*f^4 - 184*a^2*b^15*c*d^11*f^4 - 688*a^4*b^13*c*d^11*f^4 - 1240*a^6*b^11*c*
d^11*f^4 - 1200*a^8*b^9*c*d^11*f^4 - 616*a^10*b^7*c*d^11*f^4 - 144*a^12*b^5*c*d^11*f^4 - 8*a^14*b^3*c*d^11*f^4
))/(a^10*d^2*f^5 + b^10*c^2*f^5 + 4*a^2*b^8*c^2*f^5 + 6*a^4*b^6*c^2*f^5 + 4*a^6*b^4*c^2*f^5 + a^8*b^2*c^2*f^5
+ a^2*b^8*d^2*f^5 + 4*a^4*b^6*d^2*f^5 + 6*a^6*b^4*d^2*f^5 + 4*a^8*b^2*d^2*f^5 - 2*a*b^9*c*d*f^5 - 2*a^9*b*c*d*
f^5 - 8*a^3*b^7*c*d*f^5 - 12*a^5*b^5*c*d*f^5 - 8*a^7*b^3*c*d*f^5) + ((1/(c + d*1i))^(1/2)*(c + d*tan(e + f*x))
^(1/2)*(32*a^2*b^17*d^12*f^4 + 160*a^4*b^15*d^12*f^4 + 288*a^6*b^13*d^12*f^4 + 160*a^8*b^11*d^12*f^4 - 160*a^1
0*b^9*d^12*f^4 - 288*a^12*b^7*d^12*f^4 - 160*a^14*b^5*d^12*f^4 - 32*a^16*b^3*d^12*f^4 + 32*b^19*c^2*d^10*f^4 +
48*b^19*c^4*d^8*f^4 + 176*a^2*b^17*c^2*d^10*f^4 + 272*a^2*b^17*c^4*d^8*f^4 - 432*a^3*b^16*c^3*d^9*f^4 + 336*a
^4*b^15*c^2*d^10*f^4 + 624*a^4*b^15*c^4*d^8*f^4 - 912*a^5*b^14*c^3*d^9*f^4 + 112*a^6*b^13*c^2*d^10*f^4 + 720*a
^6*b^13*c^4*d^8*f^4 - 880*a^7*b^12*c^3*d^9*f^4 - 560*a^8*b^11*c^2*d^10*f^4 + 400*a^8*b^11*c^4*d^8*f^4 - 240*a^
9*b^10*c^3*d^9*f^4 - 1008*a^10*b^9*c^2*d^10*f^4 + 48*a^10*b^9*c^4*d^8*f^4 + 240*a^11*b^8*c^3*d^9*f^4 - 784*a^1
2*b^7*c^2*d^10*f^4 - 48*a^12*b^7*c^4*d^8*f^4 + 208*a^13*b^6*c^3*d^9*f^4 - 304*a^14*b^5*c^2*d^10*f^4 - 16*a^14*
b^5*c^4*d^8*f^4 + 48*a^15*b^4*c^3*d^9*f^4 - 48*a^16*b^3*c^2*d^10*f^4 - 64*a*b^18*c*d^11*f^4 - 80*a*b^18*c^3*d^
9*f^4 - 304*a^3*b^16*c*d^11*f^4 - 464*a^5*b^14*c*d^11*f^4 + 16*a^7*b^12*c*d^11*f^4 + 880*a^9*b^10*c*d^11*f^4 +
1136*a^11*b^8*c*d^11*f^4 + 656*a^13*b^6*c*d^11*f^4 + 176*a^15*b^4*c*d^11*f^4 + 16*a^17*b^2*c*d^11*f^4)*8i)/(f
*(a*b*2i + a^2 - b^2)*(a^10*d^2*f^4 + b^10*c^2*f^4 + 4*a^2*b^8*c^2*f^4 + 6*a^4*b^6*c^2*f^4 + 4*a^6*b^4*c^2*f^4
+ a^8*b^2*c^2*f^4 + a^2*b^8*d^2*f^4 + 4*a^4*b^6*d^2*f^4 + 6*a^6*b^4*d^2*f^4 + 4*a^8*b^2*d^2*f^4 - 2*a*b^9*c*d
*f^4 - 2*a^9*b*c*d*f^4 - 8*a^3*b^7*c*d*f^4 - 12*a^5*b^5*c*d*f^4 - 8*a^7*b^3*c*d*f^4)))*(1/(c + d*1i))^(1/2)*1i
)/(2*f*(a*b*2i + a^2 - b^2)))*(1/(c + d*1i))^(1/2)*1i)/(2*f*(a*b*2i + a^2 - b^2)))*(1/(c + d*1i))^(1/2)*1i)/(2
*f*(a*b*2i + a^2 - b^2)))*(1/(c + d*1i))^(1/2))/(2*f*(a*b*2i + a^2 - b^2)))/((32*(a*b^8*d^10 - b^9*c*d^9 + 5*a
^3*b^6*d^10 + 4*a*b^8*c^2*d^8 - 9*a^2*b^7*c*d^9))/(a^10*d^2*f^5 + b^10*c^2*f^5 + 4*a^2*b^8*c^2*f^5 + 6*a^4*b^6
*c^2*f^5 + 4*a^6*b^4*c^2*f^5 + a^8*b^2*c^2*f^5 + a^2*b^8*d^2*f^5 + 4*a^4*b^6*d^2*f^5 + 6*a^6*b^4*d^2*f^5 + 4*a
^8*b^2*d^2*f^5 - 2*a*b^9*c*d*f^5 - 2*a^9*b*c*d*f^5 - 8*a^3*b^7*c*d*f^5 - 12*a^5*b^5*c*d*f^5 - 8*a^7*b^3*c*d*f^
5) + (((16*(c + d*tan(e + f*x))^(1/2)*(b^11*d^10 + 7*a^2*b^9*d^10 + 11*a^4*b^7*d^10 - 27*a^6*b^5*d^10 - 2*b^11
*c^2*d^8 - 24*a^3*b^8*c*d^9 + 44*a^5*b^6*c*d^9 + 12*a^2*b^9*c^2*d^8 - 18*a^4*b^7*c^2*d^8 - 4*a*b^10*c*d^9))/(a
^10*d^2*f^4 + b^10*c^2*f^4 + 4*a^2*b^8*c^2*f^4 + 6*a^4*b^6*c^2*f^4 + 4*a^6*b^4*c^2*f^4 + a^8*b^2*c^2*f^4 + a^2
*b^8*d^2*f^4 + 4*a^4*b^6*d^2*f^4 + 6*a^6*b^4*d^2*f^4 + 4*a^8*b^2*d^2*f^4 - 2*a*b^9*c*d*f^4 - 2*a^9*b*c*d*f^4 -
8*a^3*b^7*c*d*f^4 - 12*a^5*b^5*c*d*f^4 - 8*a^7*b^3*c*d*f^4) - (((16*(2*b^13*d^11*f^2 - 24*a^2*b^11*d^11*f^2 -
196*a^4*b^9*d^11*f^2 - 120*a^6*b^7*d^11*f^2 + 50*a^8*b^5*d^11*f^2 + 8*b^13*c^2*d^9*f^2 - 8*a^2*b^11*c^2*d^9*f
^2 + 64*a^3*b^10*c^3*d^8*f^2 - 232*a^4*b^9*c^2*d^9*f^2 + 96*a^5*b^8*c^3*d^8*f^2 - 216*a^6*b^7*c^2*d^9*f^2 - 32
*a*b^12*c^3*d^8*f^2 + 208*a^3*b^10*c*d^10*f^2 + 288*a^5*b^8*c*d^10*f^2 + 80*a^7*b^6*c*d^10*f^2))/(a^10*d^2*f^5
+ b^10*c^2*f^5 + 4*a^2*b^8*c^2*f^5 + 6*a^4*b^6*c^2*f^5 + 4*a^6*b^4*c^2*f^5 + a^8*b^2*c^2*f^5 + a^2*b^8*d^2*f^
5 + 4*a^4*b^6*d^2*f^5 + 6*a^6*b^4*d^2*f^5 + 4*a^8*b^2*d^2*f^5 - 2*a*b^9*c*d*f^5 - 2*a^9*b*c*d*f^5 - 8*a^3*b^7*
c*d*f^5 - 12*a^5*b^5*c*d*f^5 - 8*a^7*b^3*c*d*f^5) - (((16*(c + d*tan(e + f*x))^(1/2)*(8*a*b^14*d^11*f^2 + 4*b^
15*c*d^10*f^2 + 36*a^3*b^12*d^11*f^2 + 316*a^5*b^10*d^11*f^2 + 552*a^7*b^8*d^11*f^2 + 256*a^9*b^6*d^11*f^2 - 1
2*a^11*b^4*d^11*f^2 - 4*a^13*b^2*d^11*f^2 - 20*b^15*c^3*d^8*f^2 + 116*a^2*b^13*c^3*d^8*f^2 - 220*a^3*b^12*c^2*
d^9*f^2 + 216*a^4*b^11*c^3*d^8*f^2 - 104*a^5*b^10*c^2*d^9*f^2 + 8*a^6*b^9*c^3*d^8*f^2 + 232*a^7*b^8*c^2*d^9*f^
2 - 68*a^8*b^7*c^3*d^8*f^2 + 156*a^9*b^6*c^2*d^9*f^2 + 4*a^10*b^5*c^3*d^8*f^2 - 12*a^11*b^4*c^2*d^9*f^2 - 52*a
*b^14*c^2*d^9*f^2 + 80*a^2*b^13*c*d^10*f^2 - 156*a^4*b^11*c*d^10*f^2 - 640*a^6*b^9*c*d^10*f^2 - 500*a^8*b^7*c*
d^10*f^2 - 80*a^10*b^5*c*d^10*f^2 + 12*a^12*b^3*c*d^10*f^2))/(a^10*d^2*f^4 + b^10*c^2*f^4 + 4*a^2*b^8*c^2*f^4
+ 6*a^4*b^6*c^2*f^4 + 4*a^6*b^4*c^2*f^4 + a^8*b^2*c^2*f^4 + a^2*b^8*d^2*f^4 + 4*a^4*b^6*d^2*f^4 + 6*a^6*b^4*d^
2*f^4 + 4*a^8*b^2*d^2*f^4 - 2*a*b^9*c*d*f^4 - 2*a^9*b*c*d*f^4 - 8*a^3*b^7*c*d*f^4 - 12*a^5*b^5*c*d*f^4 - 8*a^7
*b^3*c*d*f^4) + (((16*(16*a*b^16*d^12*f^4 - 16*b^17*c*d^11*f^4 + 136*a^3*b^14*d^12*f^4 + 432*a^5*b^12*d^12*f^4
+ 680*a^7*b^10*d^12*f^4 + 560*a^9*b^8*d^12*f^4 + 216*a^11*b^6*d^12*f^4 + 16*a^13*b^4*d^12*f^4 - 8*a^15*b^2*d^
12*f^4 - 8*b^17*c^3*d^9*f^4 - 128*a^2*b^15*c^3*d^9*f^4 + 352*a^3*b^14*c^2*d^10*f^4 + 160*a^3*b^14*c^4*d^8*f^4
- 520*a^4*b^13*c^3*d^9*f^4 + 920*a^5*b^12*c^2*d^10*f^4 + 320*a^5*b^12*c^4*d^8*f^4 - 960*a^6*b^11*c^3*d^9*f^4 +
1280*a^7*b^10*c^2*d^10*f^4 + 320*a^7*b^10*c^4*d^8*f^4 - 920*a^8*b^9*c^3*d^9*f^4 + 1000*a^9*b^8*c^2*d^10*f^4 +
160*a^9*b^8*c^4*d^8*f^4 - 448*a^10*b^7*c^3*d^9*f^4 + 416*a^11*b^6*c^2*d^10*f^4 + 32*a^11*b^6*c^4*d^8*f^4 - 88
*a^12*b^5*c^3*d^9*f^4 + 72*a^13*b^4*c^2*d^10*f^4 + 56*a*b^16*c^2*d^10*f^4 + 32*a*b^16*c^4*d^8*f^4 - 184*a^2*b^
15*c*d^11*f^4 - 688*a^4*b^13*c*d^11*f^4 - 1240*a^6*b^11*c*d^11*f^4 - 1200*a^8*b^9*c*d^11*f^4 - 616*a^10*b^7*c*
d^11*f^4 - 144*a^12*b^5*c*d^11*f^4 - 8*a^14*b^3*c*d^11*f^4))/(a^10*d^2*f^5 + b^10*c^2*f^5 + 4*a^2*b^8*c^2*f^5
+ 6*a^4*b^6*c^2*f^5 + 4*a^6*b^4*c^2*f^5 + a^8*b^2*c^2*f^5 + a^2*b^8*d^2*f^5 + 4*a^4*b^6*d^2*f^5 + 6*a^6*b^4*d^
2*f^5 + 4*a^8*b^2*d^2*f^5 - 2*a*b^9*c*d*f^5 - 2*a^9*b*c*d*f^5 - 8*a^3*b^7*c*d*f^5 - 12*a^5*b^5*c*d*f^5 - 8*a^7
*b^3*c*d*f^5) - ((1/(c + d*1i))^(1/2)*(c + d*tan(e + f*x))^(1/2)*(32*a^2*b^17*d^12*f^4 + 160*a^4*b^15*d^12*f^4
+ 288*a^6*b^13*d^12*f^4 + 160*a^8*b^11*d^12*f^4 - 160*a^10*b^9*d^12*f^4 - 288*a^12*b^7*d^12*f^4 - 160*a^14*b^
5*d^12*f^4 - 32*a^16*b^3*d^12*f^4 + 32*b^19*c^2*d^10*f^4 + 48*b^19*c^4*d^8*f^4 + 176*a^2*b^17*c^2*d^10*f^4 + 2
72*a^2*b^17*c^4*d^8*f^4 - 432*a^3*b^16*c^3*d^9*f^4 + 336*a^4*b^15*c^2*d^10*f^4 + 624*a^4*b^15*c^4*d^8*f^4 - 91
2*a^5*b^14*c^3*d^9*f^4 + 112*a^6*b^13*c^2*d^10*f^4 + 720*a^6*b^13*c^4*d^8*f^4 - 880*a^7*b^12*c^3*d^9*f^4 - 560
*a^8*b^11*c^2*d^10*f^4 + 400*a^8*b^11*c^4*d^8*f^4 - 240*a^9*b^10*c^3*d^9*f^4 - 1008*a^10*b^9*c^2*d^10*f^4 + 48
*a^10*b^9*c^4*d^8*f^4 + 240*a^11*b^8*c^3*d^9*f^4 - 784*a^12*b^7*c^2*d^10*f^4 - 48*a^12*b^7*c^4*d^8*f^4 + 208*a
^13*b^6*c^3*d^9*f^4 - 304*a^14*b^5*c^2*d^10*f^4 - 16*a^14*b^5*c^4*d^8*f^4 + 48*a^15*b^4*c^3*d^9*f^4 - 48*a^16*
b^3*c^2*d^10*f^4 - 64*a*b^18*c*d^11*f^4 - 80*a*b^18*c^3*d^9*f^4 - 304*a^3*b^16*c*d^11*f^4 - 464*a^5*b^14*c*d^1
1*f^4 + 16*a^7*b^12*c*d^11*f^4 + 880*a^9*b^10*c*d^11*f^4 + 1136*a^11*b^8*c*d^11*f^4 + 656*a^13*b^6*c*d^11*f^4
+ 176*a^15*b^4*c*d^11*f^4 + 16*a^17*b^2*c*d^11*f^4)*8i)/(f*(a*b*2i + a^2 - b^2)*(a^10*d^2*f^4 + b^10*c^2*f^4 +
4*a^2*b^8*c^2*f^4 + 6*a^4*b^6*c^2*f^4 + 4*a^6*b^4*c^2*f^4 + a^8*b^2*c^2*f^4 + a^2*b^8*d^2*f^4 + 4*a^4*b^6*d^2
*f^4 + 6*a^6*b^4*d^2*f^4 + 4*a^8*b^2*d^2*f^4 - 2*a*b^9*c*d*f^4 - 2*a^9*b*c*d*f^4 - 8*a^3*b^7*c*d*f^4 - 12*a^5*
b^5*c*d*f^4 - 8*a^7*b^3*c*d*f^4)))*(1/(c + d*1i))^(1/2)*1i)/(2*f*(a*b*2i + a^2 - b^2)))*(1/(c + d*1i))^(1/2)*1
i)/(2*f*(a*b*2i + a^2 - b^2)))*(1/(c + d*1i))^(1/2)*1i)/(2*f*(a*b*2i + a^2 - b^2)))*(1/(c + d*1i))^(1/2)*1i)/(
2*f*(a*b*2i + a^2 - b^2)) - (((16*(c + d*tan(e + f*x))^(1/2)*(b^11*d^10 + 7*a^2*b^9*d^10 + 11*a^4*b^7*d^10 - 2
7*a^6*b^5*d^10 - 2*b^11*c^2*d^8 - 24*a^3*b^8*c*d^9 + 44*a^5*b^6*c*d^9 + 12*a^2*b^9*c^2*d^8 - 18*a^4*b^7*c^2*d^
8 - 4*a*b^10*c*d^9))/(a^10*d^2*f^4 + b^10*c^2*f^4 + 4*a^2*b^8*c^2*f^4 + 6*a^4*b^6*c^2*f^4 + 4*a^6*b^4*c^2*f^4
+ a^8*b^2*c^2*f^4 + a^2*b^8*d^2*f^4 + 4*a^4*b^6*d^2*f^4 + 6*a^6*b^4*d^2*f^4 + 4*a^8*b^2*d^2*f^4 - 2*a*b^9*c*d*
f^4 - 2*a^9*b*c*d*f^4 - 8*a^3*b^7*c*d*f^4 - 12*a^5*b^5*c*d*f^4 - 8*a^7*b^3*c*d*f^4) + (((16*(2*b^13*d^11*f^2 -
24*a^2*b^11*d^11*f^2 - 196*a^4*b^9*d^11*f^2 - 120*a^6*b^7*d^11*f^2 + 50*a^8*b^5*d^11*f^2 + 8*b^13*c^2*d^9*f^2
- 8*a^2*b^11*c^2*d^9*f^2 + 64*a^3*b^10*c^3*d^8*f^2 - 232*a^4*b^9*c^2*d^9*f^2 + 96*a^5*b^8*c^3*d^8*f^2 - 216*a
^6*b^7*c^2*d^9*f^2 - 32*a*b^12*c^3*d^8*f^2 + 208*a^3*b^10*c*d^10*f^2 + 288*a^5*b^8*c*d^10*f^2 + 80*a^7*b^6*c*d
^10*f^2))/(a^10*d^2*f^5 + b^10*c^2*f^5 + 4*a^2*b^8*c^2*f^5 + 6*a^4*b^6*c^2*f^5 + 4*a^6*b^4*c^2*f^5 + a^8*b^2*c
^2*f^5 + a^2*b^8*d^2*f^5 + 4*a^4*b^6*d^2*f^5 + 6*a^6*b^4*d^2*f^5 + 4*a^8*b^2*d^2*f^5 - 2*a*b^9*c*d*f^5 - 2*a^9
*b*c*d*f^5 - 8*a^3*b^7*c*d*f^5 - 12*a^5*b^5*c*d*f^5 - 8*a^7*b^3*c*d*f^5) + (((16*(c + d*tan(e + f*x))^(1/2)*(8
*a*b^14*d^11*f^2 + 4*b^15*c*d^10*f^2 + 36*a^3*b^12*d^11*f^2 + 316*a^5*b^10*d^11*f^2 + 552*a^7*b^8*d^11*f^2 + 2
56*a^9*b^6*d^11*f^2 - 12*a^11*b^4*d^11*f^2 - 4*a^13*b^2*d^11*f^2 - 20*b^15*c^3*d^8*f^2 + 116*a^2*b^13*c^3*d^8*
f^2 - 220*a^3*b^12*c^2*d^9*f^2 + 216*a^4*b^11*c^3*d^8*f^2 - 104*a^5*b^10*c^2*d^9*f^2 + 8*a^6*b^9*c^3*d^8*f^2 +
232*a^7*b^8*c^2*d^9*f^2 - 68*a^8*b^7*c^3*d^8*f^2 + 156*a^9*b^6*c^2*d^9*f^2 + 4*a^10*b^5*c^3*d^8*f^2 - 12*a^11
*b^4*c^2*d^9*f^2 - 52*a*b^14*c^2*d^9*f^2 + 80*a^2*b^13*c*d^10*f^2 - 156*a^4*b^11*c*d^10*f^2 - 640*a^6*b^9*c*d^
10*f^2 - 500*a^8*b^7*c*d^10*f^2 - 80*a^10*b^5*c*d^10*f^2 + 12*a^12*b^3*c*d^10*f^2))/(a^10*d^2*f^4 + b^10*c^2*f
^4 + 4*a^2*b^8*c^2*f^4 + 6*a^4*b^6*c^2*f^4 + 4*a^6*b^4*c^2*f^4 + a^8*b^2*c^2*f^4 + a^2*b^8*d^2*f^4 + 4*a^4*b^6
*d^2*f^4 + 6*a^6*b^4*d^2*f^4 + 4*a^8*b^2*d^2*f^4 - 2*a*b^9*c*d*f^4 - 2*a^9*b*c*d*f^4 - 8*a^3*b^7*c*d*f^4 - 12*
a^5*b^5*c*d*f^4 - 8*a^7*b^3*c*d*f^4) - (((16*(16*a*b^16*d^12*f^4 - 16*b^17*c*d^11*f^4 + 136*a^3*b^14*d^12*f^4
+ 432*a^5*b^12*d^12*f^4 + 680*a^7*b^10*d^12*f^4 + 560*a^9*b^8*d^12*f^4 + 216*a^11*b^6*d^12*f^4 + 16*a^13*b^4*d
^12*f^4 - 8*a^15*b^2*d^12*f^4 - 8*b^17*c^3*d^9*f^4 - 128*a^2*b^15*c^3*d^9*f^4 + 352*a^3*b^14*c^2*d^10*f^4 + 16
0*a^3*b^14*c^4*d^8*f^4 - 520*a^4*b^13*c^3*d^9*f^4 + 920*a^5*b^12*c^2*d^10*f^4 + 320*a^5*b^12*c^4*d^8*f^4 - 960
*a^6*b^11*c^3*d^9*f^4 + 1280*a^7*b^10*c^2*d^10*f^4 + 320*a^7*b^10*c^4*d^8*f^4 - 920*a^8*b^9*c^3*d^9*f^4 + 1000
*a^9*b^8*c^2*d^10*f^4 + 160*a^9*b^8*c^4*d^8*f^4 - 448*a^10*b^7*c^3*d^9*f^4 + 416*a^11*b^6*c^2*d^10*f^4 + 32*a^
11*b^6*c^4*d^8*f^4 - 88*a^12*b^5*c^3*d^9*f^4 + 72*a^13*b^4*c^2*d^10*f^4 + 56*a*b^16*c^2*d^10*f^4 + 32*a*b^16*c
^4*d^8*f^4 - 184*a^2*b^15*c*d^11*f^4 - 688*a^4*b^13*c*d^11*f^4 - 1240*a^6*b^11*c*d^11*f^4 - 1200*a^8*b^9*c*d^1
1*f^4 - 616*a^10*b^7*c*d^11*f^4 - 144*a^12*b^5*c*d^11*f^4 - 8*a^14*b^3*c*d^11*f^4))/(a^10*d^2*f^5 + b^10*c^2*f
^5 + 4*a^2*b^8*c^2*f^5 + 6*a^4*b^6*c^2*f^5 + 4*a^6*b^4*c^2*f^5 + a^8*b^2*c^2*f^5 + a^2*b^8*d^2*f^5 + 4*a^4*b^6
*d^2*f^5 + 6*a^6*b^4*d^2*f^5 + 4*a^8*b^2*d^2*f^5 - 2*a*b^9*c*d*f^5 - 2*a^9*b*c*d*f^5 - 8*a^3*b^7*c*d*f^5 - 12*
a^5*b^5*c*d*f^5 - 8*a^7*b^3*c*d*f^5) + ((1/(c + d*1i))^(1/2)*(c + d*tan(e + f*x))^(1/2)*(32*a^2*b^17*d^12*f^4
+ 160*a^4*b^15*d^12*f^4 + 288*a^6*b^13*d^12*f^4 + 160*a^8*b^11*d^12*f^4 - 160*a^10*b^9*d^12*f^4 - 288*a^12*b^7
*d^12*f^4 - 160*a^14*b^5*d^12*f^4 - 32*a^16*b^3*d^12*f^4 + 32*b^19*c^2*d^10*f^4 + 48*b^19*c^4*d^8*f^4 + 176*a^
2*b^17*c^2*d^10*f^4 + 272*a^2*b^17*c^4*d^8*f^4 - 432*a^3*b^16*c^3*d^9*f^4 + 336*a^4*b^15*c^2*d^10*f^4 + 624*a^
4*b^15*c^4*d^8*f^4 - 912*a^5*b^14*c^3*d^9*f^4 + 112*a^6*b^13*c^2*d^10*f^4 + 720*a^6*b^13*c^4*d^8*f^4 - 880*a^7
*b^12*c^3*d^9*f^4 - 560*a^8*b^11*c^2*d^10*f^4 + 400*a^8*b^11*c^4*d^8*f^4 - 240*a^9*b^10*c^3*d^9*f^4 - 1008*a^1
0*b^9*c^2*d^10*f^4 + 48*a^10*b^9*c^4*d^8*f^4 + 240*a^11*b^8*c^3*d^9*f^4 - 784*a^12*b^7*c^2*d^10*f^4 - 48*a^12*
b^7*c^4*d^8*f^4 + 208*a^13*b^6*c^3*d^9*f^4 - 304*a^14*b^5*c^2*d^10*f^4 - 16*a^14*b^5*c^4*d^8*f^4 + 48*a^15*b^4
*c^3*d^9*f^4 - 48*a^16*b^3*c^2*d^10*f^4 - 64*a*b^18*c*d^11*f^4 - 80*a*b^18*c^3*d^9*f^4 - 304*a^3*b^16*c*d^11*f
^4 - 464*a^5*b^14*c*d^11*f^4 + 16*a^7*b^12*c*d^11*f^4 + 880*a^9*b^10*c*d^11*f^4 + 1136*a^11*b^8*c*d^11*f^4 + 6
56*a^13*b^6*c*d^11*f^4 + 176*a^15*b^4*c*d^11*f^4 + 16*a^17*b^2*c*d^11*f^4)*8i)/(f*(a*b*2i + a^2 - b^2)*(a^10*d
^2*f^4 + b^10*c^2*f^4 + 4*a^2*b^8*c^2*f^4 + 6*a^4*b^6*c^2*f^4 + 4*a^6*b^4*c^2*f^4 + a^8*b^2*c^2*f^4 + a^2*b^8*
d^2*f^4 + 4*a^4*b^6*d^2*f^4 + 6*a^6*b^4*d^2*f^4 + 4*a^8*b^2*d^2*f^4 - 2*a*b^9*c*d*f^4 - 2*a^9*b*c*d*f^4 - 8*a^
3*b^7*c*d*f^4 - 12*a^5*b^5*c*d*f^4 - 8*a^7*b^3*c*d*f^4)))*(1/(c + d*1i))^(1/2)*1i)/(2*f*(a*b*2i + a^2 - b^2)))
*(1/(c + d*1i))^(1/2)*1i)/(2*f*(a*b*2i + a^2 - b^2)))*(1/(c + d*1i))^(1/2)*1i)/(2*f*(a*b*2i + a^2 - b^2)))*(1/
(c + d*1i))^(1/2)*1i)/(2*f*(a*b*2i + a^2 - b^2))))*(1/(c + d*1i))^(1/2))/(f*(a*b*2i + a^2 - b^2)) + (b^2*d*(c
+ d*tan(e + f*x))^(1/2))/((b*f*(c + d*tan(e + f*x)) + a*d*f - b*c*f)*(a^3*d - b^3*c - a^2*b*c + a*b^2*d))

________________________________________________________________________________________

sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (a + b \tan {\left (e + f x \right )}\right )^{2} \sqrt {c + d \tan {\left (e + f x \right )}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(c+d*tan(f*x+e))**(1/2)/(a+b*tan(f*x+e))**2,x)

[Out]

Integral(1/((a + b*tan(e + f*x))**2*sqrt(c + d*tan(e + f*x))), x)

________________________________________________________________________________________