\(\int \frac {(c+d \tan (e+f x))^{3/2}}{a+b \tan (e+f x)} \, dx\) [1238]

Optimal result

Integrand size = 27, antiderivative size = 170 \[ \int \frac {(c+d \tan (e+f x))^{3/2}}{a+b \tan (e+f x)} \, dx=\frac {(c-i d)^{3/2} \text {arctanh}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d}}\right )}{(i a+b) f}-\frac {(c+i d)^{3/2} \text {arctanh}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c+i d}}\right )}{(i a-b) f}-\frac {2 (b c-a d)^{3/2} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d \tan (e+f x)}}{\sqrt {b c-a d}}\right )}{\sqrt {b} \left (a^2+b^2\right ) f} \]

[Out]

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

Rubi [A] (verified)

Time = 0.62 (sec) , antiderivative size = 170, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {3654, 3620, 3618, 65, 214, 3715} \[ \int \frac {(c+d \tan (e+f x))^{3/2}}{a+b \tan (e+f x)} \, dx=-\frac {2 (b c-a d)^{3/2} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d \tan (e+f x)}}{\sqrt {b c-a d}}\right )}{\sqrt {b} f \left (a^2+b^2\right )}+\frac {(c-i d)^{3/2} \text {arctanh}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d}}\right )}{f (b+i a)}-\frac {(c+i d)^{3/2} \text {arctanh}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c+i d}}\right )}{f (-b+i a)} \]

[In]

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

[Out]

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

Rule 65

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 214

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

Rule 3618

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_) + (d_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Dist[c*(
d/f), Subst[Int[(a + (b/d)*x)^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 3620

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 3654

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(3/2)/((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Dist[1
/(c^2 + d^2), Int[Simp[a^2*c - b^2*c + 2*a*b*d + (2*a*b*c - a^2*d + b^2*d)*Tan[e + f*x], x]/Sqrt[a + b*Tan[e +
f*x]], x], x] + Dist[(b*c - a*d)^2/(c^2 + d^2), Int[(1 + Tan[e + f*x]^2)/(Sqrt[a + b*Tan[e + f*x]]*(c + d*Tan
[e + f*x])), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2
, 0]

Rule 3715

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]

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

Mathematica [A] (verified)

Time = 0.54 (sec) , antiderivative size = 168, normalized size of antiderivative = 0.99 \[ \int \frac {(c+d \tan (e+f x))^{3/2}}{a+b \tan (e+f x)} \, dx=\frac {\sqrt {b} (-i a+b) (c-i d)^{3/2} \text {arctanh}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d}}\right )+\sqrt {b} (i a+b) (c+i d)^{3/2} \text {arctanh}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c+i d}}\right )-2 (b c-a d)^{3/2} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d \tan (e+f x)}}{\sqrt {b c-a d}}\right )}{\sqrt {b} \left (a^2+b^2\right ) f} \]

[In]

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

[Out]

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

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1953\) vs. \(2(142)=284\).

Time = 0.89 (sec) , antiderivative size = 1954, normalized size of antiderivative = 11.49

method	result	size
derivativedivides	\(\text {Expression too large to display}\)	\(1954\)
default	\(\text {Expression too large to display}\)	\(1954\)

[In]

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

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 3795 vs. \(2 (136) = 272\).

Time = 3.59 (sec) , antiderivative size = 7610, normalized size of antiderivative = 44.76 \[ \int \frac {(c+d \tan (e+f x))^{3/2}}{a+b \tan (e+f x)} \, dx=\text {Too large to display} \]

[In]

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

[Out]

Too large to include

Sympy [F]

\[ \int \frac {(c+d \tan (e+f x))^{3/2}}{a+b \tan (e+f x)} \, dx=\int \frac {\left (c + d \tan {\left (e + f x \right )}\right )^{\frac {3}{2}}}{a + b \tan {\left (e + f x \right )}}\, dx \]

[In]

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

[Out]

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

Maxima [F(-2)]

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

[In]

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

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(a*d-b*c>0)', see `assume?` for
more detail

Giac [F(-1)]

Timed out. \[ \int \frac {(c+d \tan (e+f x))^{3/2}}{a+b \tan (e+f x)} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Mupad [B] (verification not implemented)

Time = 15.28 (sec) , antiderivative size = 19118, normalized size of antiderivative = 112.46 \[ \int \frac {(c+d \tan (e+f x))^{3/2}}{a+b \tan (e+f x)} \, dx=\text {Too large to display} \]

[In]

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

[Out]

(atan(-((((((32*(a*b^5*d^15*f^2 + 4*a^5*b*d^15*f^2 - b^6*c*d^14*f^2 - 15*a^3*b^3*d^15*f^2 + 23*b^6*c^3*d^12*f^
2 + 21*b^6*c^5*d^10*f^2 - 3*b^6*c^7*d^8*f^2 - 29*a^2*b^4*c^3*d^12*f^2 - 81*a^2*b^4*c^5*d^10*f^2 + a^2*b^4*c^7*
d^8*f^2 + 59*a^3*b^3*c^2*d^13*f^2 + 75*a^3*b^3*c^4*d^11*f^2 + a^3*b^3*c^6*d^9*f^2 - 32*a^4*b^2*c^3*d^12*f^2 -
2*a^4*b^2*c^5*d^10*f^2 - 61*a*b^5*c^2*d^13*f^2 - 25*a*b^5*c^4*d^11*f^2 + 37*a*b^5*c^6*d^9*f^2 + 53*a^2*b^4*c*d
^14*f^2 - 30*a^4*b^2*c*d^14*f^2 + 4*a^5*b*c^2*d^13*f^2))/f^5 + ((((-b*(a*d - b*c)^3)^(1/2)*((32*(4*a^2*b^6*d^1
2*f^4 + 8*a^4*b^4*d^12*f^4 + 4*a^6*b^2*d^12*f^4 + 12*b^8*c^2*d^10*f^4 + 12*b^8*c^4*d^8*f^4 + 28*a^2*b^6*c^2*d^
10*f^4 + 24*a^2*b^6*c^4*d^8*f^4 - 32*a^3*b^5*c^3*d^9*f^4 + 20*a^4*b^4*c^2*d^10*f^4 + 12*a^4*b^4*c^4*d^8*f^4 -
16*a^5*b^3*c^3*d^9*f^4 + 4*a^6*b^2*c^2*d^10*f^4 - 16*a*b^7*c*d^11*f^4 - 16*a*b^7*c^3*d^9*f^4 - 32*a^3*b^5*c*d^
11*f^4 - 16*a^5*b^3*c*d^11*f^4))/f^5 - (32*(-b*(a*d - b*c)^3)^(1/2)*(c + d*tan(e + f*x))^(1/2)*(16*b^9*d^10*f^
4 + 16*a^2*b^7*d^10*f^4 - 16*a^4*b^5*d^10*f^4 - 16*a^6*b^3*d^10*f^4 + 24*b^9*c^2*d^8*f^4 + 40*a^2*b^7*c^2*d^8*
f^4 + 8*a^4*b^5*c^2*d^8*f^4 - 8*a^6*b^3*c^2*d^8*f^4 + 8*a*b^8*c*d^9*f^4 + 24*a^3*b^6*c*d^9*f^4 + 24*a^5*b^4*c*
d^9*f^4 + 8*a^7*b^2*c*d^9*f^4))/(f^4*(b^3*f + a^2*b*f))))/(b^3*f + a^2*b*f) - (32*(c + d*tan(e + f*x))^(1/2)*(
22*b^7*c*d^12*f^2 - 14*a*b^6*d^13*f^2 + 4*a^3*b^4*d^13*f^2 - 14*a^5*b^2*d^13*f^2 + 28*b^7*c^3*d^10*f^2 - 18*b^
7*c^5*d^8*f^2 + 24*a^2*b^5*c^3*d^10*f^2 + 12*a^2*b^5*c^5*d^8*f^2 - 88*a^3*b^4*c^2*d^11*f^2 - 28*a^3*b^4*c^4*d^
9*f^2 + 60*a^4*b^3*c^3*d^10*f^2 - 2*a^4*b^3*c^5*d^8*f^2 - 44*a^5*b^2*c^2*d^11*f^2 + 2*a^5*b^2*c^4*d^9*f^2 + 8*
a^6*b*c*d^12*f^2 + 20*a*b^6*c^2*d^11*f^2 + 66*a*b^6*c^4*d^9*f^2 - 28*a^2*b^5*c*d^12*f^2 + 54*a^4*b^3*c*d^12*f^
2))/f^4)*(-b*(a*d - b*c)^3)^(1/2))/(b^3*f + a^2*b*f))*(-b*(a*d - b*c)^3)^(1/2))/(b^3*f + a^2*b*f) - (32*(c + d
*tan(e + f*x))^(1/2)*(b^5*d^16 + 2*a^4*b*d^16 + 4*b^5*c^2*d^14 + 8*b^5*c^4*d^12 - 8*b^5*c^6*d^10 + 3*b^5*c^8*d
^8 - 8*a*b^4*c^3*d^13 + 48*a*b^4*c^5*d^11 - 8*a*b^4*c^7*d^9 - 8*a^3*b^2*c*d^15 - 12*a^4*b*c^2*d^14 + 2*a^4*b*c
^4*d^12 + 12*a^2*b^3*c^2*d^14 - 72*a^2*b^3*c^4*d^12 + 12*a^2*b^3*c^6*d^10 + 48*a^3*b^2*c^3*d^13 - 8*a^3*b^2*c^
5*d^11))/f^4)*(-b*(a*d - b*c)^3)^(1/2)*1i)/(b^3*f + a^2*b*f) - (((((32*(a*b^5*d^15*f^2 + 4*a^5*b*d^15*f^2 - b^
6*c*d^14*f^2 - 15*a^3*b^3*d^15*f^2 + 23*b^6*c^3*d^12*f^2 + 21*b^6*c^5*d^10*f^2 - 3*b^6*c^7*d^8*f^2 - 29*a^2*b^
4*c^3*d^12*f^2 - 81*a^2*b^4*c^5*d^10*f^2 + a^2*b^4*c^7*d^8*f^2 + 59*a^3*b^3*c^2*d^13*f^2 + 75*a^3*b^3*c^4*d^11
*f^2 + a^3*b^3*c^6*d^9*f^2 - 32*a^4*b^2*c^3*d^12*f^2 - 2*a^4*b^2*c^5*d^10*f^2 - 61*a*b^5*c^2*d^13*f^2 - 25*a*b
^5*c^4*d^11*f^2 + 37*a*b^5*c^6*d^9*f^2 + 53*a^2*b^4*c*d^14*f^2 - 30*a^4*b^2*c*d^14*f^2 + 4*a^5*b*c^2*d^13*f^2)
)/f^5 + ((((-b*(a*d - b*c)^3)^(1/2)*((32*(4*a^2*b^6*d^12*f^4 + 8*a^4*b^4*d^12*f^4 + 4*a^6*b^2*d^12*f^4 + 12*b^
8*c^2*d^10*f^4 + 12*b^8*c^4*d^8*f^4 + 28*a^2*b^6*c^2*d^10*f^4 + 24*a^2*b^6*c^4*d^8*f^4 - 32*a^3*b^5*c^3*d^9*f^
4 + 20*a^4*b^4*c^2*d^10*f^4 + 12*a^4*b^4*c^4*d^8*f^4 - 16*a^5*b^3*c^3*d^9*f^4 + 4*a^6*b^2*c^2*d^10*f^4 - 16*a*
b^7*c*d^11*f^4 - 16*a*b^7*c^3*d^9*f^4 - 32*a^3*b^5*c*d^11*f^4 - 16*a^5*b^3*c*d^11*f^4))/f^5 + (32*(-b*(a*d - b
*c)^3)^(1/2)*(c + d*tan(e + f*x))^(1/2)*(16*b^9*d^10*f^4 + 16*a^2*b^7*d^10*f^4 - 16*a^4*b^5*d^10*f^4 - 16*a^6*
b^3*d^10*f^4 + 24*b^9*c^2*d^8*f^4 + 40*a^2*b^7*c^2*d^8*f^4 + 8*a^4*b^5*c^2*d^8*f^4 - 8*a^6*b^3*c^2*d^8*f^4 + 8
*a*b^8*c*d^9*f^4 + 24*a^3*b^6*c*d^9*f^4 + 24*a^5*b^4*c*d^9*f^4 + 8*a^7*b^2*c*d^9*f^4))/(f^4*(b^3*f + a^2*b*f))
))/(b^3*f + a^2*b*f) + (32*(c + d*tan(e + f*x))^(1/2)*(22*b^7*c*d^12*f^2 - 14*a*b^6*d^13*f^2 + 4*a^3*b^4*d^13*
f^2 - 14*a^5*b^2*d^13*f^2 + 28*b^7*c^3*d^10*f^2 - 18*b^7*c^5*d^8*f^2 + 24*a^2*b^5*c^3*d^10*f^2 + 12*a^2*b^5*c^
5*d^8*f^2 - 88*a^3*b^4*c^2*d^11*f^2 - 28*a^3*b^4*c^4*d^9*f^2 + 60*a^4*b^3*c^3*d^10*f^2 - 2*a^4*b^3*c^5*d^8*f^2
- 44*a^5*b^2*c^2*d^11*f^2 + 2*a^5*b^2*c^4*d^9*f^2 + 8*a^6*b*c*d^12*f^2 + 20*a*b^6*c^2*d^11*f^2 + 66*a*b^6*c^4
*d^9*f^2 - 28*a^2*b^5*c*d^12*f^2 + 54*a^4*b^3*c*d^12*f^2))/f^4)*(-b*(a*d - b*c)^3)^(1/2))/(b^3*f + a^2*b*f))*(
-b*(a*d - b*c)^3)^(1/2))/(b^3*f + a^2*b*f) + (32*(c + d*tan(e + f*x))^(1/2)*(b^5*d^16 + 2*a^4*b*d^16 + 4*b^5*c
^2*d^14 + 8*b^5*c^4*d^12 - 8*b^5*c^6*d^10 + 3*b^5*c^8*d^8 - 8*a*b^4*c^3*d^13 + 48*a*b^4*c^5*d^11 - 8*a*b^4*c^7
*d^9 - 8*a^3*b^2*c*d^15 - 12*a^4*b*c^2*d^14 + 2*a^4*b*c^4*d^12 + 12*a^2*b^3*c^2*d^14 - 72*a^2*b^3*c^4*d^12 + 1
2*a^2*b^3*c^6*d^10 + 48*a^3*b^2*c^3*d^13 - 8*a^3*b^2*c^5*d^11))/f^4)*(-b*(a*d - b*c)^3)^(1/2)*1i)/(b^3*f + a^2
*b*f))/((64*(a^2*b^2*d^18 + b^4*c^2*d^16 + 5*b^4*c^4*d^14 + 7*b^4*c^6*d^12 + 3*b^4*c^8*d^10 - 12*a*b^3*c^3*d^1
5 - 18*a*b^3*c^5*d^13 - 8*a*b^3*c^7*d^11 - 4*a^3*b*c^3*d^15 - 2*a^3*b*c^5*d^13 + 9*a^2*b^2*c^2*d^16 + 15*a^2*b
^2*c^4*d^14 + 7*a^2*b^2*c^6*d^12 - 2*a*b^3*c*d^17 - 2*a^3*b*c*d^17))/f^5 + (((((32*(a*b^5*d^15*f^2 + 4*a^5*b*d
^15*f^2 - b^6*c*d^14*f^2 - 15*a^3*b^3*d^15*f^2 + 23*b^6*c^3*d^12*f^2 + 21*b^6*c^5*d^10*f^2 - 3*b^6*c^7*d^8*f^2
- 29*a^2*b^4*c^3*d^12*f^2 - 81*a^2*b^4*c^5*d^10*f^2 + a^2*b^4*c^7*d^8*f^2 + 59*a^3*b^3*c^2*d^13*f^2 + 75*a^3*
b^3*c^4*d^11*f^2 + a^3*b^3*c^6*d^9*f^2 - 32*a^4*b^2*c^3*d^12*f^2 - 2*a^4*b^2*c^5*d^10*f^2 - 61*a*b^5*c^2*d^13*
f^2 - 25*a*b^5*c^4*d^11*f^2 + 37*a*b^5*c^6*d^9*f^2 + 53*a^2*b^4*c*d^14*f^2 - 30*a^4*b^2*c*d^14*f^2 + 4*a^5*b*c
^2*d^13*f^2))/f^5 + ((((-b*(a*d - b*c)^3)^(1/2)*((32*(4*a^2*b^6*d^12*f^4 + 8*a^4*b^4*d^12*f^4 + 4*a^6*b^2*d^12
*f^4 + 12*b^8*c^2*d^10*f^4 + 12*b^8*c^4*d^8*f^4 + 28*a^2*b^6*c^2*d^10*f^4 + 24*a^2*b^6*c^4*d^8*f^4 - 32*a^3*b^
5*c^3*d^9*f^4 + 20*a^4*b^4*c^2*d^10*f^4 + 12*a^4*b^4*c^4*d^8*f^4 - 16*a^5*b^3*c^3*d^9*f^4 + 4*a^6*b^2*c^2*d^10
*f^4 - 16*a*b^7*c*d^11*f^4 - 16*a*b^7*c^3*d^9*f^4 - 32*a^3*b^5*c*d^11*f^4 - 16*a^5*b^3*c*d^11*f^4))/f^5 - (32*
(-b*(a*d - b*c)^3)^(1/2)*(c + d*tan(e + f*x))^(1/2)*(16*b^9*d^10*f^4 + 16*a^2*b^7*d^10*f^4 - 16*a^4*b^5*d^10*f
^4 - 16*a^6*b^3*d^10*f^4 + 24*b^9*c^2*d^8*f^4 + 40*a^2*b^7*c^2*d^8*f^4 + 8*a^4*b^5*c^2*d^8*f^4 - 8*a^6*b^3*c^2
*d^8*f^4 + 8*a*b^8*c*d^9*f^4 + 24*a^3*b^6*c*d^9*f^4 + 24*a^5*b^4*c*d^9*f^4 + 8*a^7*b^2*c*d^9*f^4))/(f^4*(b^3*f
+ a^2*b*f))))/(b^3*f + a^2*b*f) - (32*(c + d*tan(e + f*x))^(1/2)*(22*b^7*c*d^12*f^2 - 14*a*b^6*d^13*f^2 + 4*a
^3*b^4*d^13*f^2 - 14*a^5*b^2*d^13*f^2 + 28*b^7*c^3*d^10*f^2 - 18*b^7*c^5*d^8*f^2 + 24*a^2*b^5*c^3*d^10*f^2 + 1
2*a^2*b^5*c^5*d^8*f^2 - 88*a^3*b^4*c^2*d^11*f^2 - 28*a^3*b^4*c^4*d^9*f^2 + 60*a^4*b^3*c^3*d^10*f^2 - 2*a^4*b^3
*c^5*d^8*f^2 - 44*a^5*b^2*c^2*d^11*f^2 + 2*a^5*b^2*c^4*d^9*f^2 + 8*a^6*b*c*d^12*f^2 + 20*a*b^6*c^2*d^11*f^2 +
66*a*b^6*c^4*d^9*f^2 - 28*a^2*b^5*c*d^12*f^2 + 54*a^4*b^3*c*d^12*f^2))/f^4)*(-b*(a*d - b*c)^3)^(1/2))/(b^3*f +
a^2*b*f))*(-b*(a*d - b*c)^3)^(1/2))/(b^3*f + a^2*b*f) - (32*(c + d*tan(e + f*x))^(1/2)*(b^5*d^16 + 2*a^4*b*d^
16 + 4*b^5*c^2*d^14 + 8*b^5*c^4*d^12 - 8*b^5*c^6*d^10 + 3*b^5*c^8*d^8 - 8*a*b^4*c^3*d^13 + 48*a*b^4*c^5*d^11 -
8*a*b^4*c^7*d^9 - 8*a^3*b^2*c*d^15 - 12*a^4*b*c^2*d^14 + 2*a^4*b*c^4*d^12 + 12*a^2*b^3*c^2*d^14 - 72*a^2*b^3*
c^4*d^12 + 12*a^2*b^3*c^6*d^10 + 48*a^3*b^2*c^3*d^13 - 8*a^3*b^2*c^5*d^11))/f^4)*(-b*(a*d - b*c)^3)^(1/2))/(b^
3*f + a^2*b*f) + (((((32*(a*b^5*d^15*f^2 + 4*a^5*b*d^15*f^2 - b^6*c*d^14*f^2 - 15*a^3*b^3*d^15*f^2 + 23*b^6*c^
3*d^12*f^2 + 21*b^6*c^5*d^10*f^2 - 3*b^6*c^7*d^8*f^2 - 29*a^2*b^4*c^3*d^12*f^2 - 81*a^2*b^4*c^5*d^10*f^2 + a^2
*b^4*c^7*d^8*f^2 + 59*a^3*b^3*c^2*d^13*f^2 + 75*a^3*b^3*c^4*d^11*f^2 + a^3*b^3*c^6*d^9*f^2 - 32*a^4*b^2*c^3*d^
12*f^2 - 2*a^4*b^2*c^5*d^10*f^2 - 61*a*b^5*c^2*d^13*f^2 - 25*a*b^5*c^4*d^11*f^2 + 37*a*b^5*c^6*d^9*f^2 + 53*a^
2*b^4*c*d^14*f^2 - 30*a^4*b^2*c*d^14*f^2 + 4*a^5*b*c^2*d^13*f^2))/f^5 + ((((-b*(a*d - b*c)^3)^(1/2)*((32*(4*a^
2*b^6*d^12*f^4 + 8*a^4*b^4*d^12*f^4 + 4*a^6*b^2*d^12*f^4 + 12*b^8*c^2*d^10*f^4 + 12*b^8*c^4*d^8*f^4 + 28*a^2*b
^6*c^2*d^10*f^4 + 24*a^2*b^6*c^4*d^8*f^4 - 32*a^3*b^5*c^3*d^9*f^4 + 20*a^4*b^4*c^2*d^10*f^4 + 12*a^4*b^4*c^4*d
^8*f^4 - 16*a^5*b^3*c^3*d^9*f^4 + 4*a^6*b^2*c^2*d^10*f^4 - 16*a*b^7*c*d^11*f^4 - 16*a*b^7*c^3*d^9*f^4 - 32*a^3
*b^5*c*d^11*f^4 - 16*a^5*b^3*c*d^11*f^4))/f^5 + (32*(-b*(a*d - b*c)^3)^(1/2)*(c + d*tan(e + f*x))^(1/2)*(16*b^
9*d^10*f^4 + 16*a^2*b^7*d^10*f^4 - 16*a^4*b^5*d^10*f^4 - 16*a^6*b^3*d^10*f^4 + 24*b^9*c^2*d^8*f^4 + 40*a^2*b^7
*c^2*d^8*f^4 + 8*a^4*b^5*c^2*d^8*f^4 - 8*a^6*b^3*c^2*d^8*f^4 + 8*a*b^8*c*d^9*f^4 + 24*a^3*b^6*c*d^9*f^4 + 24*a
^5*b^4*c*d^9*f^4 + 8*a^7*b^2*c*d^9*f^4))/(f^4*(b^3*f + a^2*b*f))))/(b^3*f + a^2*b*f) + (32*(c + d*tan(e + f*x)
)^(1/2)*(22*b^7*c*d^12*f^2 - 14*a*b^6*d^13*f^2 + 4*a^3*b^4*d^13*f^2 - 14*a^5*b^2*d^13*f^2 + 28*b^7*c^3*d^10*f^
2 - 18*b^7*c^5*d^8*f^2 + 24*a^2*b^5*c^3*d^10*f^2 + 12*a^2*b^5*c^5*d^8*f^2 - 88*a^3*b^4*c^2*d^11*f^2 - 28*a^3*b
^4*c^4*d^9*f^2 + 60*a^4*b^3*c^3*d^10*f^2 - 2*a^4*b^3*c^5*d^8*f^2 - 44*a^5*b^2*c^2*d^11*f^2 + 2*a^5*b^2*c^4*d^9
*f^2 + 8*a^6*b*c*d^12*f^2 + 20*a*b^6*c^2*d^11*f^2 + 66*a*b^6*c^4*d^9*f^2 - 28*a^2*b^5*c*d^12*f^2 + 54*a^4*b^3*
c*d^12*f^2))/f^4)*(-b*(a*d - b*c)^3)^(1/2))/(b^3*f + a^2*b*f))*(-b*(a*d - b*c)^3)^(1/2))/(b^3*f + a^2*b*f) + (
32*(c + d*tan(e + f*x))^(1/2)*(b^5*d^16 + 2*a^4*b*d^16 + 4*b^5*c^2*d^14 + 8*b^5*c^4*d^12 - 8*b^5*c^6*d^10 + 3*
b^5*c^8*d^8 - 8*a*b^4*c^3*d^13 + 48*a*b^4*c^5*d^11 - 8*a*b^4*c^7*d^9 - 8*a^3*b^2*c*d^15 - 12*a^4*b*c^2*d^14 +
2*a^4*b*c^4*d^12 + 12*a^2*b^3*c^2*d^14 - 72*a^2*b^3*c^4*d^12 + 12*a^2*b^3*c^6*d^10 + 48*a^3*b^2*c^3*d^13 - 8*a
^3*b^2*c^5*d^11))/f^4)*(-b*(a*d - b*c)^3)^(1/2))/(b^3*f + a^2*b*f)))*(-b*(a*d - b*c)^3)^(1/2)*2i)/(b^3*f + a^2
*b*f) - atan(((((32*(a*b^5*d^15*f^2 + 4*a^5*b*d^15*f^2 - b^6*c*d^14*f^2 - 15*a^3*b^3*d^15*f^2 + 23*b^6*c^3*d^1
2*f^2 + 21*b^6*c^5*d^10*f^2 - 3*b^6*c^7*d^8*f^2 - 29*a^2*b^4*c^3*d^12*f^2 - 81*a^2*b^4*c^5*d^10*f^2 + a^2*b^4*
c^7*d^8*f^2 + 59*a^3*b^3*c^2*d^13*f^2 + 75*a^3*b^3*c^4*d^11*f^2 + a^3*b^3*c^6*d^9*f^2 - 32*a^4*b^2*c^3*d^12*f^
2 - 2*a^4*b^2*c^5*d^10*f^2 - 61*a*b^5*c^2*d^13*f^2 - 25*a*b^5*c^4*d^11*f^2 + 37*a*b^5*c^6*d^9*f^2 + 53*a^2*b^4
*c*d^14*f^2 - 30*a^4*b^2*c*d^14*f^2 + 4*a^5*b*c^2*d^13*f^2))/f^5 + (((32*(4*a^2*b^6*d^12*f^4 + 8*a^4*b^4*d^12*
f^4 + 4*a^6*b^2*d^12*f^4 + 12*b^8*c^2*d^10*f^4 + 12*b^8*c^4*d^8*f^4 + 28*a^2*b^6*c^2*d^10*f^4 + 24*a^2*b^6*c^4
*d^8*f^4 - 32*a^3*b^5*c^3*d^9*f^4 + 20*a^4*b^4*c^2*d^10*f^4 + 12*a^4*b^4*c^4*d^8*f^4 - 16*a^5*b^3*c^3*d^9*f^4
+ 4*a^6*b^2*c^2*d^10*f^4 - 16*a*b^7*c*d^11*f^4 - 16*a*b^7*c^3*d^9*f^4 - 32*a^3*b^5*c*d^11*f^4 - 16*a^5*b^3*c*d
^11*f^4))/f^5 - (32*(c + d*tan(e + f*x))^(1/2)*((c*d^2*3i - 3*c^2*d - c^3*1i + d^3)/(4*(a^2*f^2*1i - b^2*f^2*1
i + 2*a*b*f^2)))^(1/2)*(16*b^9*d^10*f^4 + 16*a^2*b^7*d^10*f^4 - 16*a^4*b^5*d^10*f^4 - 16*a^6*b^3*d^10*f^4 + 24
*b^9*c^2*d^8*f^4 + 40*a^2*b^7*c^2*d^8*f^4 + 8*a^4*b^5*c^2*d^8*f^4 - 8*a^6*b^3*c^2*d^8*f^4 + 8*a*b^8*c*d^9*f^4
+ 24*a^3*b^6*c*d^9*f^4 + 24*a^5*b^4*c*d^9*f^4 + 8*a^7*b^2*c*d^9*f^4))/f^4)*((c*d^2*3i - 3*c^2*d - c^3*1i + d^3
)/(4*(a^2*f^2*1i - b^2*f^2*1i + 2*a*b*f^2)))^(1/2) - (32*(c + d*tan(e + f*x))^(1/2)*(22*b^7*c*d^12*f^2 - 14*a*
b^6*d^13*f^2 + 4*a^3*b^4*d^13*f^2 - 14*a^5*b^2*d^13*f^2 + 28*b^7*c^3*d^10*f^2 - 18*b^7*c^5*d^8*f^2 + 24*a^2*b^
5*c^3*d^10*f^2 + 12*a^2*b^5*c^5*d^8*f^2 - 88*a^3*b^4*c^2*d^11*f^2 - 28*a^3*b^4*c^4*d^9*f^2 + 60*a^4*b^3*c^3*d^
10*f^2 - 2*a^4*b^3*c^5*d^8*f^2 - 44*a^5*b^2*c^2*d^11*f^2 + 2*a^5*b^2*c^4*d^9*f^2 + 8*a^6*b*c*d^12*f^2 + 20*a*b
^6*c^2*d^11*f^2 + 66*a*b^6*c^4*d^9*f^2 - 28*a^2*b^5*c*d^12*f^2 + 54*a^4*b^3*c*d^12*f^2))/f^4)*((c*d^2*3i - 3*c
^2*d - c^3*1i + d^3)/(4*(a^2*f^2*1i - b^2*f^2*1i + 2*a*b*f^2)))^(1/2))*((c*d^2*3i - 3*c^2*d - c^3*1i + d^3)/(4
*(a^2*f^2*1i - b^2*f^2*1i + 2*a*b*f^2)))^(1/2) - (32*(c + d*tan(e + f*x))^(1/2)*(b^5*d^16 + 2*a^4*b*d^16 + 4*b
^5*c^2*d^14 + 8*b^5*c^4*d^12 - 8*b^5*c^6*d^10 + 3*b^5*c^8*d^8 - 8*a*b^4*c^3*d^13 + 48*a*b^4*c^5*d^11 - 8*a*b^4
*c^7*d^9 - 8*a^3*b^2*c*d^15 - 12*a^4*b*c^2*d^14 + 2*a^4*b*c^4*d^12 + 12*a^2*b^3*c^2*d^14 - 72*a^2*b^3*c^4*d^12
+ 12*a^2*b^3*c^6*d^10 + 48*a^3*b^2*c^3*d^13 - 8*a^3*b^2*c^5*d^11))/f^4)*((c*d^2*3i - 3*c^2*d - c^3*1i + d^3)/
(4*(a^2*f^2*1i - b^2*f^2*1i + 2*a*b*f^2)))^(1/2)*1i - (((32*(a*b^5*d^15*f^2 + 4*a^5*b*d^15*f^2 - b^6*c*d^14*f^
2 - 15*a^3*b^3*d^15*f^2 + 23*b^6*c^3*d^12*f^2 + 21*b^6*c^5*d^10*f^2 - 3*b^6*c^7*d^8*f^2 - 29*a^2*b^4*c^3*d^12*
f^2 - 81*a^2*b^4*c^5*d^10*f^2 + a^2*b^4*c^7*d^8*f^2 + 59*a^3*b^3*c^2*d^13*f^2 + 75*a^3*b^3*c^4*d^11*f^2 + a^3*
b^3*c^6*d^9*f^2 - 32*a^4*b^2*c^3*d^12*f^2 - 2*a^4*b^2*c^5*d^10*f^2 - 61*a*b^5*c^2*d^13*f^2 - 25*a*b^5*c^4*d^11
*f^2 + 37*a*b^5*c^6*d^9*f^2 + 53*a^2*b^4*c*d^14*f^2 - 30*a^4*b^2*c*d^14*f^2 + 4*a^5*b*c^2*d^13*f^2))/f^5 + (((
32*(4*a^2*b^6*d^12*f^4 + 8*a^4*b^4*d^12*f^4 + 4*a^6*b^2*d^12*f^4 + 12*b^8*c^2*d^10*f^4 + 12*b^8*c^4*d^8*f^4 +
28*a^2*b^6*c^2*d^10*f^4 + 24*a^2*b^6*c^4*d^8*f^4 - 32*a^3*b^5*c^3*d^9*f^4 + 20*a^4*b^4*c^2*d^10*f^4 + 12*a^4*b
^4*c^4*d^8*f^4 - 16*a^5*b^3*c^3*d^9*f^4 + 4*a^6*b^2*c^2*d^10*f^4 - 16*a*b^7*c*d^11*f^4 - 16*a*b^7*c^3*d^9*f^4
- 32*a^3*b^5*c*d^11*f^4 - 16*a^5*b^3*c*d^11*f^4))/f^5 + (32*(c + d*tan(e + f*x))^(1/2)*((c*d^2*3i - 3*c^2*d -
c^3*1i + d^3)/(4*(a^2*f^2*1i - b^2*f^2*1i + 2*a*b*f^2)))^(1/2)*(16*b^9*d^10*f^4 + 16*a^2*b^7*d^10*f^4 - 16*a^4
*b^5*d^10*f^4 - 16*a^6*b^3*d^10*f^4 + 24*b^9*c^2*d^8*f^4 + 40*a^2*b^7*c^2*d^8*f^4 + 8*a^4*b^5*c^2*d^8*f^4 - 8*
a^6*b^3*c^2*d^8*f^4 + 8*a*b^8*c*d^9*f^4 + 24*a^3*b^6*c*d^9*f^4 + 24*a^5*b^4*c*d^9*f^4 + 8*a^7*b^2*c*d^9*f^4))/
f^4)*((c*d^2*3i - 3*c^2*d - c^3*1i + d^3)/(4*(a^2*f^2*1i - b^2*f^2*1i + 2*a*b*f^2)))^(1/2) + (32*(c + d*tan(e
+ f*x))^(1/2)*(22*b^7*c*d^12*f^2 - 14*a*b^6*d^13*f^2 + 4*a^3*b^4*d^13*f^2 - 14*a^5*b^2*d^13*f^2 + 28*b^7*c^3*d
^10*f^2 - 18*b^7*c^5*d^8*f^2 + 24*a^2*b^5*c^3*d^10*f^2 + 12*a^2*b^5*c^5*d^8*f^2 - 88*a^3*b^4*c^2*d^11*f^2 - 28
*a^3*b^4*c^4*d^9*f^2 + 60*a^4*b^3*c^3*d^10*f^2 - 2*a^4*b^3*c^5*d^8*f^2 - 44*a^5*b^2*c^2*d^11*f^2 + 2*a^5*b^2*c
^4*d^9*f^2 + 8*a^6*b*c*d^12*f^2 + 20*a*b^6*c^2*d^11*f^2 + 66*a*b^6*c^4*d^9*f^2 - 28*a^2*b^5*c*d^12*f^2 + 54*a^
4*b^3*c*d^12*f^2))/f^4)*((c*d^2*3i - 3*c^2*d - c^3*1i + d^3)/(4*(a^2*f^2*1i - b^2*f^2*1i + 2*a*b*f^2)))^(1/2))
*((c*d^2*3i - 3*c^2*d - c^3*1i + d^3)/(4*(a^2*f^2*1i - b^2*f^2*1i + 2*a*b*f^2)))^(1/2) + (32*(c + d*tan(e + f*
x))^(1/2)*(b^5*d^16 + 2*a^4*b*d^16 + 4*b^5*c^2*d^14 + 8*b^5*c^4*d^12 - 8*b^5*c^6*d^10 + 3*b^5*c^8*d^8 - 8*a*b^
4*c^3*d^13 + 48*a*b^4*c^5*d^11 - 8*a*b^4*c^7*d^9 - 8*a^3*b^2*c*d^15 - 12*a^4*b*c^2*d^14 + 2*a^4*b*c^4*d^12 + 1
2*a^2*b^3*c^2*d^14 - 72*a^2*b^3*c^4*d^12 + 12*a^2*b^3*c^6*d^10 + 48*a^3*b^2*c^3*d^13 - 8*a^3*b^2*c^5*d^11))/f^
4)*((c*d^2*3i - 3*c^2*d - c^3*1i + d^3)/(4*(a^2*f^2*1i - b^2*f^2*1i + 2*a*b*f^2)))^(1/2)*1i)/((64*(a^2*b^2*d^1
8 + b^4*c^2*d^16 + 5*b^4*c^4*d^14 + 7*b^4*c^6*d^12 + 3*b^4*c^8*d^10 - 12*a*b^3*c^3*d^15 - 18*a*b^3*c^5*d^13 -
8*a*b^3*c^7*d^11 - 4*a^3*b*c^3*d^15 - 2*a^3*b*c^5*d^13 + 9*a^2*b^2*c^2*d^16 + 15*a^2*b^2*c^4*d^14 + 7*a^2*b^2*
c^6*d^12 - 2*a*b^3*c*d^17 - 2*a^3*b*c*d^17))/f^5 + (((32*(a*b^5*d^15*f^2 + 4*a^5*b*d^15*f^2 - b^6*c*d^14*f^2 -
15*a^3*b^3*d^15*f^2 + 23*b^6*c^3*d^12*f^2 + 21*b^6*c^5*d^10*f^2 - 3*b^6*c^7*d^8*f^2 - 29*a^2*b^4*c^3*d^12*f^2
- 81*a^2*b^4*c^5*d^10*f^2 + a^2*b^4*c^7*d^8*f^2 + 59*a^3*b^3*c^2*d^13*f^2 + 75*a^3*b^3*c^4*d^11*f^2 + a^3*b^3
*c^6*d^9*f^2 - 32*a^4*b^2*c^3*d^12*f^2 - 2*a^4*b^2*c^5*d^10*f^2 - 61*a*b^5*c^2*d^13*f^2 - 25*a*b^5*c^4*d^11*f^
2 + 37*a*b^5*c^6*d^9*f^2 + 53*a^2*b^4*c*d^14*f^2 - 30*a^4*b^2*c*d^14*f^2 + 4*a^5*b*c^2*d^13*f^2))/f^5 + (((32*
(4*a^2*b^6*d^12*f^4 + 8*a^4*b^4*d^12*f^4 + 4*a^6*b^2*d^12*f^4 + 12*b^8*c^2*d^10*f^4 + 12*b^8*c^4*d^8*f^4 + 28*
a^2*b^6*c^2*d^10*f^4 + 24*a^2*b^6*c^4*d^8*f^4 - 32*a^3*b^5*c^3*d^9*f^4 + 20*a^4*b^4*c^2*d^10*f^4 + 12*a^4*b^4*
c^4*d^8*f^4 - 16*a^5*b^3*c^3*d^9*f^4 + 4*a^6*b^2*c^2*d^10*f^4 - 16*a*b^7*c*d^11*f^4 - 16*a*b^7*c^3*d^9*f^4 - 3
2*a^3*b^5*c*d^11*f^4 - 16*a^5*b^3*c*d^11*f^4))/f^5 - (32*(c + d*tan(e + f*x))^(1/2)*((c*d^2*3i - 3*c^2*d - c^3
*1i + d^3)/(4*(a^2*f^2*1i - b^2*f^2*1i + 2*a*b*f^2)))^(1/2)*(16*b^9*d^10*f^4 + 16*a^2*b^7*d^10*f^4 - 16*a^4*b^
5*d^10*f^4 - 16*a^6*b^3*d^10*f^4 + 24*b^9*c^2*d^8*f^4 + 40*a^2*b^7*c^2*d^8*f^4 + 8*a^4*b^5*c^2*d^8*f^4 - 8*a^6
*b^3*c^2*d^8*f^4 + 8*a*b^8*c*d^9*f^4 + 24*a^3*b^6*c*d^9*f^4 + 24*a^5*b^4*c*d^9*f^4 + 8*a^7*b^2*c*d^9*f^4))/f^4
)*((c*d^2*3i - 3*c^2*d - c^3*1i + d^3)/(4*(a^2*f^2*1i - b^2*f^2*1i + 2*a*b*f^2)))^(1/2) - (32*(c + d*tan(e + f
*x))^(1/2)*(22*b^7*c*d^12*f^2 - 14*a*b^6*d^13*f^2 + 4*a^3*b^4*d^13*f^2 - 14*a^5*b^2*d^13*f^2 + 28*b^7*c^3*d^10
*f^2 - 18*b^7*c^5*d^8*f^2 + 24*a^2*b^5*c^3*d^10*f^2 + 12*a^2*b^5*c^5*d^8*f^2 - 88*a^3*b^4*c^2*d^11*f^2 - 28*a^
3*b^4*c^4*d^9*f^2 + 60*a^4*b^3*c^3*d^10*f^2 - 2*a^4*b^3*c^5*d^8*f^2 - 44*a^5*b^2*c^2*d^11*f^2 + 2*a^5*b^2*c^4*
d^9*f^2 + 8*a^6*b*c*d^12*f^2 + 20*a*b^6*c^2*d^11*f^2 + 66*a*b^6*c^4*d^9*f^2 - 28*a^2*b^5*c*d^12*f^2 + 54*a^4*b
^3*c*d^12*f^2))/f^4)*((c*d^2*3i - 3*c^2*d - c^3*1i + d^3)/(4*(a^2*f^2*1i - b^2*f^2*1i + 2*a*b*f^2)))^(1/2))*((
c*d^2*3i - 3*c^2*d - c^3*1i + d^3)/(4*(a^2*f^2*1i - b^2*f^2*1i + 2*a*b*f^2)))^(1/2) - (32*(c + d*tan(e + f*x))
^(1/2)*(b^5*d^16 + 2*a^4*b*d^16 + 4*b^5*c^2*d^14 + 8*b^5*c^4*d^12 - 8*b^5*c^6*d^10 + 3*b^5*c^8*d^8 - 8*a*b^4*c
^3*d^13 + 48*a*b^4*c^5*d^11 - 8*a*b^4*c^7*d^9 - 8*a^3*b^2*c*d^15 - 12*a^4*b*c^2*d^14 + 2*a^4*b*c^4*d^12 + 12*a
^2*b^3*c^2*d^14 - 72*a^2*b^3*c^4*d^12 + 12*a^2*b^3*c^6*d^10 + 48*a^3*b^2*c^3*d^13 - 8*a^3*b^2*c^5*d^11))/f^4)*
((c*d^2*3i - 3*c^2*d - c^3*1i + d^3)/(4*(a^2*f^2*1i - b^2*f^2*1i + 2*a*b*f^2)))^(1/2) + (((32*(a*b^5*d^15*f^2
+ 4*a^5*b*d^15*f^2 - b^6*c*d^14*f^2 - 15*a^3*b^3*d^15*f^2 + 23*b^6*c^3*d^12*f^2 + 21*b^6*c^5*d^10*f^2 - 3*b^6*
c^7*d^8*f^2 - 29*a^2*b^4*c^3*d^12*f^2 - 81*a^2*b^4*c^5*d^10*f^2 + a^2*b^4*c^7*d^8*f^2 + 59*a^3*b^3*c^2*d^13*f^
2 + 75*a^3*b^3*c^4*d^11*f^2 + a^3*b^3*c^6*d^9*f^2 - 32*a^4*b^2*c^3*d^12*f^2 - 2*a^4*b^2*c^5*d^10*f^2 - 61*a*b^
5*c^2*d^13*f^2 - 25*a*b^5*c^4*d^11*f^2 + 37*a*b^5*c^6*d^9*f^2 + 53*a^2*b^4*c*d^14*f^2 - 30*a^4*b^2*c*d^14*f^2
+ 4*a^5*b*c^2*d^13*f^2))/f^5 + (((32*(4*a^2*b^6*d^12*f^4 + 8*a^4*b^4*d^12*f^4 + 4*a^6*b^2*d^12*f^4 + 12*b^8*c^
2*d^10*f^4 + 12*b^8*c^4*d^8*f^4 + 28*a^2*b^6*c^2*d^10*f^4 + 24*a^2*b^6*c^4*d^8*f^4 - 32*a^3*b^5*c^3*d^9*f^4 +
20*a^4*b^4*c^2*d^10*f^4 + 12*a^4*b^4*c^4*d^8*f^4 - 16*a^5*b^3*c^3*d^9*f^4 + 4*a^6*b^2*c^2*d^10*f^4 - 16*a*b^7*
c*d^11*f^4 - 16*a*b^7*c^3*d^9*f^4 - 32*a^3*b^5*c*d^11*f^4 - 16*a^5*b^3*c*d^11*f^4))/f^5 + (32*(c + d*tan(e + f
*x))^(1/2)*((c*d^2*3i - 3*c^2*d - c^3*1i + d^3)/(4*(a^2*f^2*1i - b^2*f^2*1i + 2*a*b*f^2)))^(1/2)*(16*b^9*d^10*
f^4 + 16*a^2*b^7*d^10*f^4 - 16*a^4*b^5*d^10*f^4 - 16*a^6*b^3*d^10*f^4 + 24*b^9*c^2*d^8*f^4 + 40*a^2*b^7*c^2*d^
8*f^4 + 8*a^4*b^5*c^2*d^8*f^4 - 8*a^6*b^3*c^2*d^8*f^4 + 8*a*b^8*c*d^9*f^4 + 24*a^3*b^6*c*d^9*f^4 + 24*a^5*b^4*
c*d^9*f^4 + 8*a^7*b^2*c*d^9*f^4))/f^4)*((c*d^2*3i - 3*c^2*d - c^3*1i + d^3)/(4*(a^2*f^2*1i - b^2*f^2*1i + 2*a*
b*f^2)))^(1/2) + (32*(c + d*tan(e + f*x))^(1/2)*(22*b^7*c*d^12*f^2 - 14*a*b^6*d^13*f^2 + 4*a^3*b^4*d^13*f^2 -
14*a^5*b^2*d^13*f^2 + 28*b^7*c^3*d^10*f^2 - 18*b^7*c^5*d^8*f^2 + 24*a^2*b^5*c^3*d^10*f^2 + 12*a^2*b^5*c^5*d^8*
f^2 - 88*a^3*b^4*c^2*d^11*f^2 - 28*a^3*b^4*c^4*d^9*f^2 + 60*a^4*b^3*c^3*d^10*f^2 - 2*a^4*b^3*c^5*d^8*f^2 - 44*
a^5*b^2*c^2*d^11*f^2 + 2*a^5*b^2*c^4*d^9*f^2 + 8*a^6*b*c*d^12*f^2 + 20*a*b^6*c^2*d^11*f^2 + 66*a*b^6*c^4*d^9*f
^2 - 28*a^2*b^5*c*d^12*f^2 + 54*a^4*b^3*c*d^12*f^2))/f^4)*((c*d^2*3i - 3*c^2*d - c^3*1i + d^3)/(4*(a^2*f^2*1i
- b^2*f^2*1i + 2*a*b*f^2)))^(1/2))*((c*d^2*3i - 3*c^2*d - c^3*1i + d^3)/(4*(a^2*f^2*1i - b^2*f^2*1i + 2*a*b*f^
2)))^(1/2) + (32*(c + d*tan(e + f*x))^(1/2)*(b^5*d^16 + 2*a^4*b*d^16 + 4*b^5*c^2*d^14 + 8*b^5*c^4*d^12 - 8*b^5
*c^6*d^10 + 3*b^5*c^8*d^8 - 8*a*b^4*c^3*d^13 + 48*a*b^4*c^5*d^11 - 8*a*b^4*c^7*d^9 - 8*a^3*b^2*c*d^15 - 12*a^4
*b*c^2*d^14 + 2*a^4*b*c^4*d^12 + 12*a^2*b^3*c^2*d^14 - 72*a^2*b^3*c^4*d^12 + 12*a^2*b^3*c^6*d^10 + 48*a^3*b^2*
c^3*d^13 - 8*a^3*b^2*c^5*d^11))/f^4)*((c*d^2*3i - 3*c^2*d - c^3*1i + d^3)/(4*(a^2*f^2*1i - b^2*f^2*1i + 2*a*b*
f^2)))^(1/2)))*((c*d^2*3i - 3*c^2*d - c^3*1i + d^3)/(4*(a^2*f^2*1i - b^2*f^2*1i + 2*a*b*f^2)))^(1/2)*2i - atan
(((((((32*(4*a^2*b^6*d^12*f^4 + 8*a^4*b^4*d^12*f^4 + 4*a^6*b^2*d^12*f^4 + 12*b^8*c^2*d^10*f^4 + 12*b^8*c^4*d^8
*f^4 + 28*a^2*b^6*c^2*d^10*f^4 + 24*a^2*b^6*c^4*d^8*f^4 - 32*a^3*b^5*c^3*d^9*f^4 + 20*a^4*b^4*c^2*d^10*f^4 + 1
2*a^4*b^4*c^4*d^8*f^4 - 16*a^5*b^3*c^3*d^9*f^4 + 4*a^6*b^2*c^2*d^10*f^4 - 16*a*b^7*c*d^11*f^4 - 16*a*b^7*c^3*d
^9*f^4 - 32*a^3*b^5*c*d^11*f^4 - 16*a^5*b^3*c*d^11*f^4))/f^5 - (32*(c + d*tan(e + f*x))^(1/2)*((3*c*d^2 - c^2*
d*3i - c^3 + d^3*1i)/(4*(a^2*f^2 - b^2*f^2 + a*b*f^2*2i)))^(1/2)*(16*b^9*d^10*f^4 + 16*a^2*b^7*d^10*f^4 - 16*a
^4*b^5*d^10*f^4 - 16*a^6*b^3*d^10*f^4 + 24*b^9*c^2*d^8*f^4 + 40*a^2*b^7*c^2*d^8*f^4 + 8*a^4*b^5*c^2*d^8*f^4 -
8*a^6*b^3*c^2*d^8*f^4 + 8*a*b^8*c*d^9*f^4 + 24*a^3*b^6*c*d^9*f^4 + 24*a^5*b^4*c*d^9*f^4 + 8*a^7*b^2*c*d^9*f^4)
)/f^4)*((3*c*d^2 - c^2*d*3i - c^3 + d^3*1i)/(4*(a^2*f^2 - b^2*f^2 + a*b*f^2*2i)))^(1/2) - (32*(c + d*tan(e + f
*x))^(1/2)*(22*b^7*c*d^12*f^2 - 14*a*b^6*d^13*f^2 + 4*a^3*b^4*d^13*f^2 - 14*a^5*b^2*d^13*f^2 + 28*b^7*c^3*d^10
*f^2 - 18*b^7*c^5*d^8*f^2 + 24*a^2*b^5*c^3*d^10*f^2 + 12*a^2*b^5*c^5*d^8*f^2 - 88*a^3*b^4*c^2*d^11*f^2 - 28*a^
3*b^4*c^4*d^9*f^2 + 60*a^4*b^3*c^3*d^10*f^2 - 2*a^4*b^3*c^5*d^8*f^2 - 44*a^5*b^2*c^2*d^11*f^2 + 2*a^5*b^2*c^4*
d^9*f^2 + 8*a^6*b*c*d^12*f^2 + 20*a*b^6*c^2*d^11*f^2 + 66*a*b^6*c^4*d^9*f^2 - 28*a^2*b^5*c*d^12*f^2 + 54*a^4*b
^3*c*d^12*f^2))/f^4)*((3*c*d^2 - c^2*d*3i - c^3 + d^3*1i)/(4*(a^2*f^2 - b^2*f^2 + a*b*f^2*2i)))^(1/2) + (32*(a
*b^5*d^15*f^2 + 4*a^5*b*d^15*f^2 - b^6*c*d^14*f^2 - 15*a^3*b^3*d^15*f^2 + 23*b^6*c^3*d^12*f^2 + 21*b^6*c^5*d^1
0*f^2 - 3*b^6*c^7*d^8*f^2 - 29*a^2*b^4*c^3*d^12*f^2 - 81*a^2*b^4*c^5*d^10*f^2 + a^2*b^4*c^7*d^8*f^2 + 59*a^3*b
^3*c^2*d^13*f^2 + 75*a^3*b^3*c^4*d^11*f^2 + a^3*b^3*c^6*d^9*f^2 - 32*a^4*b^2*c^3*d^12*f^2 - 2*a^4*b^2*c^5*d^10
*f^2 - 61*a*b^5*c^2*d^13*f^2 - 25*a*b^5*c^4*d^11*f^2 + 37*a*b^5*c^6*d^9*f^2 + 53*a^2*b^4*c*d^14*f^2 - 30*a^4*b
^2*c*d^14*f^2 + 4*a^5*b*c^2*d^13*f^2))/f^5)*((3*c*d^2 - c^2*d*3i - c^3 + d^3*1i)/(4*(a^2*f^2 - b^2*f^2 + a*b*f
^2*2i)))^(1/2) - (32*(c + d*tan(e + f*x))^(1/2)*(b^5*d^16 + 2*a^4*b*d^16 + 4*b^5*c^2*d^14 + 8*b^5*c^4*d^12 - 8
*b^5*c^6*d^10 + 3*b^5*c^8*d^8 - 8*a*b^4*c^3*d^13 + 48*a*b^4*c^5*d^11 - 8*a*b^4*c^7*d^9 - 8*a^3*b^2*c*d^15 - 12
*a^4*b*c^2*d^14 + 2*a^4*b*c^4*d^12 + 12*a^2*b^3*c^2*d^14 - 72*a^2*b^3*c^4*d^12 + 12*a^2*b^3*c^6*d^10 + 48*a^3*
b^2*c^3*d^13 - 8*a^3*b^2*c^5*d^11))/f^4)*((3*c*d^2 - c^2*d*3i - c^3 + d^3*1i)/(4*(a^2*f^2 - b^2*f^2 + a*b*f^2*
2i)))^(1/2)*1i - (((((32*(4*a^2*b^6*d^12*f^4 + 8*a^4*b^4*d^12*f^4 + 4*a^6*b^2*d^12*f^4 + 12*b^8*c^2*d^10*f^4 +
12*b^8*c^4*d^8*f^4 + 28*a^2*b^6*c^2*d^10*f^4 + 24*a^2*b^6*c^4*d^8*f^4 - 32*a^3*b^5*c^3*d^9*f^4 + 20*a^4*b^4*c
^2*d^10*f^4 + 12*a^4*b^4*c^4*d^8*f^4 - 16*a^5*b^3*c^3*d^9*f^4 + 4*a^6*b^2*c^2*d^10*f^4 - 16*a*b^7*c*d^11*f^4 -
16*a*b^7*c^3*d^9*f^4 - 32*a^3*b^5*c*d^11*f^4 - 16*a^5*b^3*c*d^11*f^4))/f^5 + (32*(c + d*tan(e + f*x))^(1/2)*(
(3*c*d^2 - c^2*d*3i - c^3 + d^3*1i)/(4*(a^2*f^2 - b^2*f^2 + a*b*f^2*2i)))^(1/2)*(16*b^9*d^10*f^4 + 16*a^2*b^7*
d^10*f^4 - 16*a^4*b^5*d^10*f^4 - 16*a^6*b^3*d^10*f^4 + 24*b^9*c^2*d^8*f^4 + 40*a^2*b^7*c^2*d^8*f^4 + 8*a^4*b^5
*c^2*d^8*f^4 - 8*a^6*b^3*c^2*d^8*f^4 + 8*a*b^8*c*d^9*f^4 + 24*a^3*b^6*c*d^9*f^4 + 24*a^5*b^4*c*d^9*f^4 + 8*a^7
*b^2*c*d^9*f^4))/f^4)*((3*c*d^2 - c^2*d*3i - c^3 + d^3*1i)/(4*(a^2*f^2 - b^2*f^2 + a*b*f^2*2i)))^(1/2) + (32*(
c + d*tan(e + f*x))^(1/2)*(22*b^7*c*d^12*f^2 - 14*a*b^6*d^13*f^2 + 4*a^3*b^4*d^13*f^2 - 14*a^5*b^2*d^13*f^2 +
28*b^7*c^3*d^10*f^2 - 18*b^7*c^5*d^8*f^2 + 24*a^2*b^5*c^3*d^10*f^2 + 12*a^2*b^5*c^5*d^8*f^2 - 88*a^3*b^4*c^2*d
^11*f^2 - 28*a^3*b^4*c^4*d^9*f^2 + 60*a^4*b^3*c^3*d^10*f^2 - 2*a^4*b^3*c^5*d^8*f^2 - 44*a^5*b^2*c^2*d^11*f^2 +
2*a^5*b^2*c^4*d^9*f^2 + 8*a^6*b*c*d^12*f^2 + 20*a*b^6*c^2*d^11*f^2 + 66*a*b^6*c^4*d^9*f^2 - 28*a^2*b^5*c*d^12
*f^2 + 54*a^4*b^3*c*d^12*f^2))/f^4)*((3*c*d^2 - c^2*d*3i - c^3 + d^3*1i)/(4*(a^2*f^2 - b^2*f^2 + a*b*f^2*2i)))
^(1/2) + (32*(a*b^5*d^15*f^2 + 4*a^5*b*d^15*f^2 - b^6*c*d^14*f^2 - 15*a^3*b^3*d^15*f^2 + 23*b^6*c^3*d^12*f^2 +
21*b^6*c^5*d^10*f^2 - 3*b^6*c^7*d^8*f^2 - 29*a^2*b^4*c^3*d^12*f^2 - 81*a^2*b^4*c^5*d^10*f^2 + a^2*b^4*c^7*d^8
*f^2 + 59*a^3*b^3*c^2*d^13*f^2 + 75*a^3*b^3*c^4*d^11*f^2 + a^3*b^3*c^6*d^9*f^2 - 32*a^4*b^2*c^3*d^12*f^2 - 2*a
^4*b^2*c^5*d^10*f^2 - 61*a*b^5*c^2*d^13*f^2 - 25*a*b^5*c^4*d^11*f^2 + 37*a*b^5*c^6*d^9*f^2 + 53*a^2*b^4*c*d^14
*f^2 - 30*a^4*b^2*c*d^14*f^2 + 4*a^5*b*c^2*d^13*f^2))/f^5)*((3*c*d^2 - c^2*d*3i - c^3 + d^3*1i)/(4*(a^2*f^2 -
b^2*f^2 + a*b*f^2*2i)))^(1/2) + (32*(c + d*tan(e + f*x))^(1/2)*(b^5*d^16 + 2*a^4*b*d^16 + 4*b^5*c^2*d^14 + 8*b
^5*c^4*d^12 - 8*b^5*c^6*d^10 + 3*b^5*c^8*d^8 - 8*a*b^4*c^3*d^13 + 48*a*b^4*c^5*d^11 - 8*a*b^4*c^7*d^9 - 8*a^3*
b^2*c*d^15 - 12*a^4*b*c^2*d^14 + 2*a^4*b*c^4*d^12 + 12*a^2*b^3*c^2*d^14 - 72*a^2*b^3*c^4*d^12 + 12*a^2*b^3*c^6
*d^10 + 48*a^3*b^2*c^3*d^13 - 8*a^3*b^2*c^5*d^11))/f^4)*((3*c*d^2 - c^2*d*3i - c^3 + d^3*1i)/(4*(a^2*f^2 - b^2
*f^2 + a*b*f^2*2i)))^(1/2)*1i)/((64*(a^2*b^2*d^18 + b^4*c^2*d^16 + 5*b^4*c^4*d^14 + 7*b^4*c^6*d^12 + 3*b^4*c^8
*d^10 - 12*a*b^3*c^3*d^15 - 18*a*b^3*c^5*d^13 - 8*a*b^3*c^7*d^11 - 4*a^3*b*c^3*d^15 - 2*a^3*b*c^5*d^13 + 9*a^2
*b^2*c^2*d^16 + 15*a^2*b^2*c^4*d^14 + 7*a^2*b^2*c^6*d^12 - 2*a*b^3*c*d^17 - 2*a^3*b*c*d^17))/f^5 + (((((32*(4*
a^2*b^6*d^12*f^4 + 8*a^4*b^4*d^12*f^4 + 4*a^6*b^2*d^12*f^4 + 12*b^8*c^2*d^10*f^4 + 12*b^8*c^4*d^8*f^4 + 28*a^2
*b^6*c^2*d^10*f^4 + 24*a^2*b^6*c^4*d^8*f^4 - 32*a^3*b^5*c^3*d^9*f^4 + 20*a^4*b^4*c^2*d^10*f^4 + 12*a^4*b^4*c^4
*d^8*f^4 - 16*a^5*b^3*c^3*d^9*f^4 + 4*a^6*b^2*c^2*d^10*f^4 - 16*a*b^7*c*d^11*f^4 - 16*a*b^7*c^3*d^9*f^4 - 32*a
^3*b^5*c*d^11*f^4 - 16*a^5*b^3*c*d^11*f^4))/f^5 - (32*(c + d*tan(e + f*x))^(1/2)*((3*c*d^2 - c^2*d*3i - c^3 +
d^3*1i)/(4*(a^2*f^2 - b^2*f^2 + a*b*f^2*2i)))^(1/2)*(16*b^9*d^10*f^4 + 16*a^2*b^7*d^10*f^4 - 16*a^4*b^5*d^10*f
^4 - 16*a^6*b^3*d^10*f^4 + 24*b^9*c^2*d^8*f^4 + 40*a^2*b^7*c^2*d^8*f^4 + 8*a^4*b^5*c^2*d^8*f^4 - 8*a^6*b^3*c^2
*d^8*f^4 + 8*a*b^8*c*d^9*f^4 + 24*a^3*b^6*c*d^9*f^4 + 24*a^5*b^4*c*d^9*f^4 + 8*a^7*b^2*c*d^9*f^4))/f^4)*((3*c*
d^2 - c^2*d*3i - c^3 + d^3*1i)/(4*(a^2*f^2 - b^2*f^2 + a*b*f^2*2i)))^(1/2) - (32*(c + d*tan(e + f*x))^(1/2)*(2
2*b^7*c*d^12*f^2 - 14*a*b^6*d^13*f^2 + 4*a^3*b^4*d^13*f^2 - 14*a^5*b^2*d^13*f^2 + 28*b^7*c^3*d^10*f^2 - 18*b^7
*c^5*d^8*f^2 + 24*a^2*b^5*c^3*d^10*f^2 + 12*a^2*b^5*c^5*d^8*f^2 - 88*a^3*b^4*c^2*d^11*f^2 - 28*a^3*b^4*c^4*d^9
*f^2 + 60*a^4*b^3*c^3*d^10*f^2 - 2*a^4*b^3*c^5*d^8*f^2 - 44*a^5*b^2*c^2*d^11*f^2 + 2*a^5*b^2*c^4*d^9*f^2 + 8*a
^6*b*c*d^12*f^2 + 20*a*b^6*c^2*d^11*f^2 + 66*a*b^6*c^4*d^9*f^2 - 28*a^2*b^5*c*d^12*f^2 + 54*a^4*b^3*c*d^12*f^2
))/f^4)*((3*c*d^2 - c^2*d*3i - c^3 + d^3*1i)/(4*(a^2*f^2 - b^2*f^2 + a*b*f^2*2i)))^(1/2) + (32*(a*b^5*d^15*f^2
+ 4*a^5*b*d^15*f^2 - b^6*c*d^14*f^2 - 15*a^3*b^3*d^15*f^2 + 23*b^6*c^3*d^12*f^2 + 21*b^6*c^5*d^10*f^2 - 3*b^6
*c^7*d^8*f^2 - 29*a^2*b^4*c^3*d^12*f^2 - 81*a^2*b^4*c^5*d^10*f^2 + a^2*b^4*c^7*d^8*f^2 + 59*a^3*b^3*c^2*d^13*f
^2 + 75*a^3*b^3*c^4*d^11*f^2 + a^3*b^3*c^6*d^9*f^2 - 32*a^4*b^2*c^3*d^12*f^2 - 2*a^4*b^2*c^5*d^10*f^2 - 61*a*b
^5*c^2*d^13*f^2 - 25*a*b^5*c^4*d^11*f^2 + 37*a*b^5*c^6*d^9*f^2 + 53*a^2*b^4*c*d^14*f^2 - 30*a^4*b^2*c*d^14*f^2
+ 4*a^5*b*c^2*d^13*f^2))/f^5)*((3*c*d^2 - c^2*d*3i - c^3 + d^3*1i)/(4*(a^2*f^2 - b^2*f^2 + a*b*f^2*2i)))^(1/2
) - (32*(c + d*tan(e + f*x))^(1/2)*(b^5*d^16 + 2*a^4*b*d^16 + 4*b^5*c^2*d^14 + 8*b^5*c^4*d^12 - 8*b^5*c^6*d^10
+ 3*b^5*c^8*d^8 - 8*a*b^4*c^3*d^13 + 48*a*b^4*c^5*d^11 - 8*a*b^4*c^7*d^9 - 8*a^3*b^2*c*d^15 - 12*a^4*b*c^2*d^
14 + 2*a^4*b*c^4*d^12 + 12*a^2*b^3*c^2*d^14 - 72*a^2*b^3*c^4*d^12 + 12*a^2*b^3*c^6*d^10 + 48*a^3*b^2*c^3*d^13
- 8*a^3*b^2*c^5*d^11))/f^4)*((3*c*d^2 - c^2*d*3i - c^3 + d^3*1i)/(4*(a^2*f^2 - b^2*f^2 + a*b*f^2*2i)))^(1/2) +
(((((32*(4*a^2*b^6*d^12*f^4 + 8*a^4*b^4*d^12*f^4 + 4*a^6*b^2*d^12*f^4 + 12*b^8*c^2*d^10*f^4 + 12*b^8*c^4*d^8*
f^4 + 28*a^2*b^6*c^2*d^10*f^4 + 24*a^2*b^6*c^4*d^8*f^4 - 32*a^3*b^5*c^3*d^9*f^4 + 20*a^4*b^4*c^2*d^10*f^4 + 12
*a^4*b^4*c^4*d^8*f^4 - 16*a^5*b^3*c^3*d^9*f^4 + 4*a^6*b^2*c^2*d^10*f^4 - 16*a*b^7*c*d^11*f^4 - 16*a*b^7*c^3*d^
9*f^4 - 32*a^3*b^5*c*d^11*f^4 - 16*a^5*b^3*c*d^11*f^4))/f^5 + (32*(c + d*tan(e + f*x))^(1/2)*((3*c*d^2 - c^2*d
*3i - c^3 + d^3*1i)/(4*(a^2*f^2 - b^2*f^2 + a*b*f^2*2i)))^(1/2)*(16*b^9*d^10*f^4 + 16*a^2*b^7*d^10*f^4 - 16*a^
4*b^5*d^10*f^4 - 16*a^6*b^3*d^10*f^4 + 24*b^9*c^2*d^8*f^4 + 40*a^2*b^7*c^2*d^8*f^4 + 8*a^4*b^5*c^2*d^8*f^4 - 8
*a^6*b^3*c^2*d^8*f^4 + 8*a*b^8*c*d^9*f^4 + 24*a^3*b^6*c*d^9*f^4 + 24*a^5*b^4*c*d^9*f^4 + 8*a^7*b^2*c*d^9*f^4))
/f^4)*((3*c*d^2 - c^2*d*3i - c^3 + d^3*1i)/(4*(a^2*f^2 - b^2*f^2 + a*b*f^2*2i)))^(1/2) + (32*(c + d*tan(e + f*
x))^(1/2)*(22*b^7*c*d^12*f^2 - 14*a*b^6*d^13*f^2 + 4*a^3*b^4*d^13*f^2 - 14*a^5*b^2*d^13*f^2 + 28*b^7*c^3*d^10*
f^2 - 18*b^7*c^5*d^8*f^2 + 24*a^2*b^5*c^3*d^10*f^2 + 12*a^2*b^5*c^5*d^8*f^2 - 88*a^3*b^4*c^2*d^11*f^2 - 28*a^3
*b^4*c^4*d^9*f^2 + 60*a^4*b^3*c^3*d^10*f^2 - 2*a^4*b^3*c^5*d^8*f^2 - 44*a^5*b^2*c^2*d^11*f^2 + 2*a^5*b^2*c^4*d
^9*f^2 + 8*a^6*b*c*d^12*f^2 + 20*a*b^6*c^2*d^11*f^2 + 66*a*b^6*c^4*d^9*f^2 - 28*a^2*b^5*c*d^12*f^2 + 54*a^4*b^
3*c*d^12*f^2))/f^4)*((3*c*d^2 - c^2*d*3i - c^3 + d^3*1i)/(4*(a^2*f^2 - b^2*f^2 + a*b*f^2*2i)))^(1/2) + (32*(a*
b^5*d^15*f^2 + 4*a^5*b*d^15*f^2 - b^6*c*d^14*f^2 - 15*a^3*b^3*d^15*f^2 + 23*b^6*c^3*d^12*f^2 + 21*b^6*c^5*d^10
*f^2 - 3*b^6*c^7*d^8*f^2 - 29*a^2*b^4*c^3*d^12*f^2 - 81*a^2*b^4*c^5*d^10*f^2 + a^2*b^4*c^7*d^8*f^2 + 59*a^3*b^
3*c^2*d^13*f^2 + 75*a^3*b^3*c^4*d^11*f^2 + a^3*b^3*c^6*d^9*f^2 - 32*a^4*b^2*c^3*d^12*f^2 - 2*a^4*b^2*c^5*d^10*
f^2 - 61*a*b^5*c^2*d^13*f^2 - 25*a*b^5*c^4*d^11*f^2 + 37*a*b^5*c^6*d^9*f^2 + 53*a^2*b^4*c*d^14*f^2 - 30*a^4*b^
2*c*d^14*f^2 + 4*a^5*b*c^2*d^13*f^2))/f^5)*((3*c*d^2 - c^2*d*3i - c^3 + d^3*1i)/(4*(a^2*f^2 - b^2*f^2 + a*b*f^
2*2i)))^(1/2) + (32*(c + d*tan(e + f*x))^(1/2)*(b^5*d^16 + 2*a^4*b*d^16 + 4*b^5*c^2*d^14 + 8*b^5*c^4*d^12 - 8*
b^5*c^6*d^10 + 3*b^5*c^8*d^8 - 8*a*b^4*c^3*d^13 + 48*a*b^4*c^5*d^11 - 8*a*b^4*c^7*d^9 - 8*a^3*b^2*c*d^15 - 12*
a^4*b*c^2*d^14 + 2*a^4*b*c^4*d^12 + 12*a^2*b^3*c^2*d^14 - 72*a^2*b^3*c^4*d^12 + 12*a^2*b^3*c^6*d^10 + 48*a^3*b
^2*c^3*d^13 - 8*a^3*b^2*c^5*d^11))/f^4)*((3*c*d^2 - c^2*d*3i - c^3 + d^3*1i)/(4*(a^2*f^2 - b^2*f^2 + a*b*f^2*2
i)))^(1/2)))*((3*c*d^2 - c^2*d*3i - c^3 + d^3*1i)/(4*(a^2*f^2 - b^2*f^2 + a*b*f^2*2i)))^(1/2)*2i