\(\int \frac {1}{(a+i a \tan (e+f x))^2 \sqrt {c+d \tan (e+f x)}} \, dx\) [1123]

Optimal result

Integrand size = 30, antiderivative size = 221 \[ \int \frac {1}{(a+i a \tan (e+f x))^2 \sqrt {c+d \tan (e+f x)}} \, dx=-\frac {i \text {arctanh}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d}}\right )}{4 a^2 \sqrt {c-i d} f}+\frac {\left (2 i c^2-6 c d-7 i d^2\right ) \text {arctanh}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c+i d}}\right )}{8 a^2 (c+i d)^{5/2} f}+\frac {(2 i c-5 d) \sqrt {c+d \tan (e+f x)}}{8 a^2 (c+i d)^2 f (1+i \tan (e+f x))}-\frac {\sqrt {c+d \tan (e+f x)}}{4 (i c-d) f (a+i a \tan (e+f x))^2} \]

[Out]

1/8*(2*I*c^2-6*c*d-7*I*d^2)*arctanh((c+d*tan(f*x+e))^(1/2)/(c+I*d)^(1/2))/a^2/(c+I*d)^(5/2)/f-1/4*I*arctanh((c
+d*tan(f*x+e))^(1/2)/(c-I*d)^(1/2))/a^2/f/(c-I*d)^(1/2)+1/8*(2*I*c-5*d)*(c+d*tan(f*x+e))^(1/2)/a^2/(c+I*d)^2/f
/(1+I*tan(f*x+e))-1/4*(c+d*tan(f*x+e))^(1/2)/(I*c-d)/f/(a+I*a*tan(f*x+e))^2

Rubi [A] (verified)

Time = 0.73 (sec) , antiderivative size = 221, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {3640, 3677, 3620, 3618, 65, 214} \[ \int \frac {1}{(a+i a \tan (e+f x))^2 \sqrt {c+d \tan (e+f x)}} \, dx=\frac {\left (2 i c^2-6 c d-7 i d^2\right ) \text {arctanh}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c+i d}}\right )}{8 a^2 f (c+i d)^{5/2}}-\frac {i \text {arctanh}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d}}\right )}{4 a^2 f \sqrt {c-i d}}+\frac {(-5 d+2 i c) \sqrt {c+d \tan (e+f x)}}{8 a^2 f (c+i d)^2 (1+i \tan (e+f x))}-\frac {\sqrt {c+d \tan (e+f x)}}{4 f (-d+i c) (a+i a \tan (e+f x))^2} \]

[In]

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

[Out]

((-1/4*I)*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c - I*d]])/(a^2*Sqrt[c - I*d]*f) + (((2*I)*c^2 - 6*c*d - (7*I)
*d^2)*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c + I*d]])/(8*a^2*(c + I*d)^(5/2)*f) + (((2*I)*c - 5*d)*Sqrt[c + d
*Tan[e + f*x]])/(8*a^2*(c + I*d)^2*f*(1 + I*Tan[e + f*x])) - Sqrt[c + d*Tan[e + f*x]]/(4*(I*c - d)*f*(a + I*a*
Tan[e + f*x])^2)

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 3640

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

Rule 3677

Int[((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*tan[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*tan[(e_
.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[(a*A + b*B)*(a + b*Tan[e + f*x])^m*((c + d*Tan[e + f*x])^(n + 1)/(2*
f*m*(b*c - a*d))), x] + Dist[1/(2*a*m*(b*c - a*d)), Int[(a + b*Tan[e + f*x])^(m + 1)*(c + d*Tan[e + f*x])^n*Si
mp[A*(b*c*m - a*d*(2*m + n + 1)) + B*(a*c*m - b*d*(n + 1)) + d*(A*b - a*B)*(m + n + 1)*Tan[e + f*x], x], x], x
] /; FreeQ[{a, b, c, d, e, f, A, B, n}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 + b^2, 0] && LtQ[m, 0] &&  !GtQ[n,
0]

Rubi steps \begin{align*} \text {integral}& = -\frac {\sqrt {c+d \tan (e+f x)}}{4 (i c-d) f (a+i a \tan (e+f x))^2}-\frac {\int \frac {-\frac {1}{2} a (4 i c-7 d)-\frac {3}{2} i a d \tan (e+f x)}{(a+i a \tan (e+f x)) \sqrt {c+d \tan (e+f x)}} \, dx}{4 a^2 (i c-d)} \\ & = \frac {(2 i c-5 d) \sqrt {c+d \tan (e+f x)}}{8 a^2 (c+i d)^2 f (1+i \tan (e+f x))}-\frac {\sqrt {c+d \tan (e+f x)}}{4 (i c-d) f (a+i a \tan (e+f x))^2}-\frac {\int \frac {-\frac {1}{2} a^2 \left (4 c^2+10 i c d-9 d^2\right )-\frac {1}{2} a^2 (2 c+5 i d) d \tan (e+f x)}{\sqrt {c+d \tan (e+f x)}} \, dx}{8 a^4 (c+i d)^2} \\ & = \frac {(2 i c-5 d) \sqrt {c+d \tan (e+f x)}}{8 a^2 (c+i d)^2 f (1+i \tan (e+f x))}-\frac {\sqrt {c+d \tan (e+f x)}}{4 (i c-d) f (a+i a \tan (e+f x))^2}+\frac {\int \frac {1+i \tan (e+f x)}{\sqrt {c+d \tan (e+f x)}} \, dx}{8 a^2}+\frac {\left (2 c^2+6 i c d-7 d^2\right ) \int \frac {1-i \tan (e+f x)}{\sqrt {c+d \tan (e+f x)}} \, dx}{16 a^2 (c+i d)^2} \\ & = \frac {(2 i c-5 d) \sqrt {c+d \tan (e+f x)}}{8 a^2 (c+i d)^2 f (1+i \tan (e+f x))}-\frac {\sqrt {c+d \tan (e+f x)}}{4 (i c-d) f (a+i a \tan (e+f x))^2}+\frac {i \text {Subst}\left (\int \frac {1}{(-1+x) \sqrt {c-i d x}} \, dx,x,i \tan (e+f x)\right )}{8 a^2 f}-\frac {\left (i \left (2 c^2+6 i c d-7 d^2\right )\right ) \text {Subst}\left (\int \frac {1}{(-1+x) \sqrt {c+i d x}} \, dx,x,-i \tan (e+f x)\right )}{16 a^2 (c+i d)^2 f} \\ & = \frac {(2 i c-5 d) \sqrt {c+d \tan (e+f x)}}{8 a^2 (c+i d)^2 f (1+i \tan (e+f x))}-\frac {\sqrt {c+d \tan (e+f x)}}{4 (i c-d) f (a+i a \tan (e+f x))^2}-\frac {\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 )}{4 a^2 d f}-\frac {\left (2 c^2+6 i c d-7 d^2\right ) \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 )}{8 a^2 (c+i d)^2 d f} \\ & = -\frac {i \text {arctanh}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d}}\right )}{4 a^2 \sqrt {c-i d} f}-\frac {\left (6 c d-i \left (2 c^2-7 d^2\right )\right ) \text {arctanh}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c+i d}}\right )}{8 a^2 (c+i d)^{5/2} f}+\frac {(2 i c-5 d) \sqrt {c+d \tan (e+f x)}}{8 a^2 (c+i d)^2 f (1+i \tan (e+f x))}-\frac {\sqrt {c+d \tan (e+f x)}}{4 (i c-d) f (a+i a \tan (e+f x))^2} \\ \end{align*}

Mathematica [A] (verified)

Time = 1.76 (sec) , antiderivative size = 203, normalized size of antiderivative = 0.92 \[ \int \frac {1}{(a+i a \tan (e+f x))^2 \sqrt {c+d \tan (e+f x)}} \, dx=-\frac {\frac {2 i (c+i d)^2 \text {arctanh}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d}}\right )}{\sqrt {c-i d}}+\frac {\left (-2 i c^2+6 c d+7 i d^2\right ) \text {arctanh}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c+i d}}\right )}{\sqrt {c+i d}}+\frac {2 i (c+i d) \sqrt {c+d \tan (e+f x)}}{(-i+\tan (e+f x))^2}-\frac {(2 c+5 i d) \sqrt {c+d \tan (e+f x)}}{-i+\tan (e+f x)}}{8 a^2 (c+i d)^2 f} \]

[In]

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

[Out]

-1/8*(((2*I)*(c + I*d)^2*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c - I*d]])/Sqrt[c - I*d] + (((-2*I)*c^2 + 6*c*d
+ (7*I)*d^2)*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c + I*d]])/Sqrt[c + I*d] + ((2*I)*(c + I*d)*Sqrt[c + d*Tan
[e + f*x]])/(-I + Tan[e + f*x])^2 - ((2*c + (5*I)*d)*Sqrt[c + d*Tan[e + f*x]])/(-I + Tan[e + f*x]))/(a^2*(c +
I*d)^2*f)

Maple [A] (verified)

Time = 0.71 (sec) , antiderivative size = 232, normalized size of antiderivative = 1.05

method	result	size
derivativedivides	\(\frac {2 d^{3} \left (\frac {\frac {\frac {d \left (5 i d +2 c \right ) \left (c +d \tan \left (f x +e \right )\right )^{\frac {3}{2}}}{4 i c d +2 c^{2}-2 d^{2}}-\frac {d \left (9 i c d +2 c^{2}-7 d^{2}\right ) \sqrt {c +d \tan \left (f x +e \right )}}{2 \left (2 i c d +c^{2}-d^{2}\right )}}{\left (-d \tan \left (f x +e \right )+i d \right )^{2}}-\frac {\left (2 i c^{2}-7 i d^{2}-6 c d \right ) \arctan \left (\frac {\sqrt {c +d \tan \left (f x +e \right )}}{\sqrt {-i d -c}}\right )}{2 \left (2 i c d +c^{2}-d^{2}\right ) \sqrt {-i d -c}}}{8 d^{3}}+\frac {i \arctan \left (\frac {\sqrt {c +d \tan \left (f x +e \right )}}{\sqrt {i d -c}}\right )}{8 d^{3} \sqrt {i d -c}}\right )}{f \,a^{2}}\)	\(232\)
default	\(\frac {2 d^{3} \left (\frac {\frac {\frac {d \left (5 i d +2 c \right ) \left (c +d \tan \left (f x +e \right )\right )^{\frac {3}{2}}}{4 i c d +2 c^{2}-2 d^{2}}-\frac {d \left (9 i c d +2 c^{2}-7 d^{2}\right ) \sqrt {c +d \tan \left (f x +e \right )}}{2 \left (2 i c d +c^{2}-d^{2}\right )}}{\left (-d \tan \left (f x +e \right )+i d \right )^{2}}-\frac {\left (2 i c^{2}-7 i d^{2}-6 c d \right ) \arctan \left (\frac {\sqrt {c +d \tan \left (f x +e \right )}}{\sqrt {-i d -c}}\right )}{2 \left (2 i c d +c^{2}-d^{2}\right ) \sqrt {-i d -c}}}{8 d^{3}}+\frac {i \arctan \left (\frac {\sqrt {c +d \tan \left (f x +e \right )}}{\sqrt {i d -c}}\right )}{8 d^{3} \sqrt {i d -c}}\right )}{f \,a^{2}}\)	\(232\)

[In]

int(1/(c+d*tan(f*x+e))^(1/2)/(a+I*a*tan(f*x+e))^2,x,method=_RETURNVERBOSE)

[Out]

2/f/a^2*d^3*(1/8/d^3*((1/2*d*(5*I*d+2*c)/(2*I*c*d+c^2-d^2)*(c+d*tan(f*x+e))^(3/2)-1/2*d*(9*I*c*d+2*c^2-7*d^2)/
(2*I*c*d+c^2-d^2)*(c+d*tan(f*x+e))^(1/2))/(-d*tan(f*x+e)+I*d)^2-1/2*(2*I*c^2-6*c*d-7*I*d^2)/(2*I*c*d+c^2-d^2)/
(-I*d-c)^(1/2)*arctan((c+d*tan(f*x+e))^(1/2)/(-I*d-c)^(1/2)))+1/8*I/d^3/(I*d-c)^(1/2)*arctan((c+d*tan(f*x+e))^
(1/2)/(I*d-c)^(1/2)))

Fricas [B] (verification not implemented)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 1400 vs. \(2 (173) = 346\).

Time = 0.39 (sec) , antiderivative size = 1400, normalized size of antiderivative = 6.33 \[ \int \frac {1}{(a+i a \tan (e+f x))^2 \sqrt {c+d \tan (e+f x)}} \, dx=\text {Too large to display} \]

[In]

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

[Out]

-1/32*(8*(a^2*c^2 + 2*I*a^2*c*d - a^2*d^2)*f*sqrt(1/16*I/((-I*a^4*c - a^4*d)*f^2))*e^(4*I*f*x + 4*I*e)*log(-2*
(4*((I*a^2*c + a^2*d)*f*e^(2*I*f*x + 2*I*e) + (I*a^2*c + a^2*d)*f)*sqrt(((c - I*d)*e^(2*I*f*x + 2*I*e) + c + I
*d)/(e^(2*I*f*x + 2*I*e) + 1))*sqrt(1/16*I/((-I*a^4*c - a^4*d)*f^2)) - (c - I*d)*e^(2*I*f*x + 2*I*e) - c)*e^(-
2*I*f*x - 2*I*e)) - 8*(a^2*c^2 + 2*I*a^2*c*d - a^2*d^2)*f*sqrt(1/16*I/((-I*a^4*c - a^4*d)*f^2))*e^(4*I*f*x + 4
*I*e)*log(-2*(4*((-I*a^2*c - a^2*d)*f*e^(2*I*f*x + 2*I*e) + (-I*a^2*c - a^2*d)*f)*sqrt(((c - I*d)*e^(2*I*f*x +
2*I*e) + c + I*d)/(e^(2*I*f*x + 2*I*e) + 1))*sqrt(1/16*I/((-I*a^4*c - a^4*d)*f^2)) - (c - I*d)*e^(2*I*f*x + 2
*I*e) - c)*e^(-2*I*f*x - 2*I*e)) + (a^2*c^2 + 2*I*a^2*c*d - a^2*d^2)*f*sqrt(-(4*I*c^4 - 24*c^3*d - 64*I*c^2*d^
2 + 84*c*d^3 + 49*I*d^4)/((I*a^4*c^5 - 5*a^4*c^4*d - 10*I*a^4*c^3*d^2 + 10*a^4*c^2*d^3 + 5*I*a^4*c*d^4 - a^4*d
^5)*f^2))*e^(4*I*f*x + 4*I*e)*log(1/8*(2*c^3 + 8*I*c^2*d - 13*c*d^2 - 7*I*d^3 + ((I*a^2*c^3 - 3*a^2*c^2*d - 3*
I*a^2*c*d^2 + a^2*d^3)*f*e^(2*I*f*x + 2*I*e) + (I*a^2*c^3 - 3*a^2*c^2*d - 3*I*a^2*c*d^2 + a^2*d^3)*f)*sqrt(((c
- I*d)*e^(2*I*f*x + 2*I*e) + c + I*d)/(e^(2*I*f*x + 2*I*e) + 1))*sqrt(-(4*I*c^4 - 24*c^3*d - 64*I*c^2*d^2 + 8
4*c*d^3 + 49*I*d^4)/((I*a^4*c^5 - 5*a^4*c^4*d - 10*I*a^4*c^3*d^2 + 10*a^4*c^2*d^3 + 5*I*a^4*c*d^4 - a^4*d^5)*f
^2)) + (2*c^3 + 6*I*c^2*d - 7*c*d^2)*e^(2*I*f*x + 2*I*e))*e^(-2*I*f*x - 2*I*e)/((-I*a^2*c^3 + 3*a^2*c^2*d + 3*
I*a^2*c*d^2 - a^2*d^3)*f)) - (a^2*c^2 + 2*I*a^2*c*d - a^2*d^2)*f*sqrt(-(4*I*c^4 - 24*c^3*d - 64*I*c^2*d^2 + 84
*c*d^3 + 49*I*d^4)/((I*a^4*c^5 - 5*a^4*c^4*d - 10*I*a^4*c^3*d^2 + 10*a^4*c^2*d^3 + 5*I*a^4*c*d^4 - a^4*d^5)*f^
2))*e^(4*I*f*x + 4*I*e)*log(1/8*(2*c^3 + 8*I*c^2*d - 13*c*d^2 - 7*I*d^3 + ((-I*a^2*c^3 + 3*a^2*c^2*d + 3*I*a^2
*c*d^2 - a^2*d^3)*f*e^(2*I*f*x + 2*I*e) + (-I*a^2*c^3 + 3*a^2*c^2*d + 3*I*a^2*c*d^2 - a^2*d^3)*f)*sqrt(((c - I
*d)*e^(2*I*f*x + 2*I*e) + c + I*d)/(e^(2*I*f*x + 2*I*e) + 1))*sqrt(-(4*I*c^4 - 24*c^3*d - 64*I*c^2*d^2 + 84*c*
d^3 + 49*I*d^4)/((I*a^4*c^5 - 5*a^4*c^4*d - 10*I*a^4*c^3*d^2 + 10*a^4*c^2*d^3 + 5*I*a^4*c*d^4 - a^4*d^5)*f^2))
+ (2*c^3 + 6*I*c^2*d - 7*c*d^2)*e^(2*I*f*x + 2*I*e))*e^(-2*I*f*x - 2*I*e)/((-I*a^2*c^3 + 3*a^2*c^2*d + 3*I*a^
2*c*d^2 - a^2*d^3)*f)) - 2*(3*(I*c - 2*d)*e^(4*I*f*x + 4*I*e) - (-4*I*c + 7*d)*e^(2*I*f*x + 2*I*e) + I*c - d)*
sqrt(((c - I*d)*e^(2*I*f*x + 2*I*e) + c + I*d)/(e^(2*I*f*x + 2*I*e) + 1)))*e^(-4*I*f*x - 4*I*e)/((a^2*c^2 + 2*
I*a^2*c*d - a^2*d^2)*f)

Sympy [F]

\[ \int \frac {1}{(a+i a \tan (e+f x))^2 \sqrt {c+d \tan (e+f x)}} \, dx=- \frac {\int \frac {1}{\sqrt {c + d \tan {\left (e + f x \right )}} \tan ^{2}{\left (e + f x \right )} - 2 i \sqrt {c + d \tan {\left (e + f x \right )}} \tan {\left (e + f x \right )} - \sqrt {c + d \tan {\left (e + f x \right )}}}\, dx}{a^{2}} \]

[In]

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

[Out]

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

Maxima [F(-2)]

Exception generated. \[ \int \frac {1}{(a+i a \tan (e+f x))^2 \sqrt {c+d \tan (e+f x)}} \, dx=\text {Exception raised: RuntimeError} \]

[In]

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

[Out]

Exception raised: RuntimeError >> ECL says: expt: undefined: 0 to a negative exponent.

Giac [B] (verification not implemented)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 497 vs. \(2 (173) = 346\).

Time = 0.71 (sec) , antiderivative size = 497, normalized size of antiderivative = 2.25 \[ \int \frac {1}{(a+i a \tan (e+f x))^2 \sqrt {c+d \tan (e+f x)}} \, dx=\frac {1}{8} \, d^{3} {\left (\frac {16 \, {\left (2 \, c^{2} + 6 i \, c d - 7 \, d^{2}\right )} \arctan \left (\frac {2 \, {\left (\sqrt {d \tan \left (f x + e\right ) + c} c - \sqrt {c^{2} + d^{2}} \sqrt {d \tan \left (f x + e\right ) + c}\right )}}{c \sqrt {-2 \, c + 2 \, \sqrt {c^{2} + d^{2}}} + i \, \sqrt {-2 \, c + 2 \, \sqrt {c^{2} + d^{2}}} d - \sqrt {c^{2} + d^{2}} \sqrt {-2 \, c + 2 \, \sqrt {c^{2} + d^{2}}}}\right )}{-8 \, {\left (-i \, a^{2} c^{2} d^{3} f + 2 \, a^{2} c d^{4} f + i \, a^{2} d^{5} f\right )} \sqrt {-2 \, c + 2 \, \sqrt {c^{2} + d^{2}}} {\left (\frac {i \, d}{c - \sqrt {c^{2} + d^{2}}} + 1\right )}} + \frac {2 \, {\left (d \tan \left (f x + e\right ) + c\right )}^{\frac {3}{2}} c - 2 \, \sqrt {d \tan \left (f x + e\right ) + c} c^{2} + 5 i \, {\left (d \tan \left (f x + e\right ) + c\right )}^{\frac {3}{2}} d - 9 i \, \sqrt {d \tan \left (f x + e\right ) + c} c d + 7 \, \sqrt {d \tan \left (f x + e\right ) + c} d^{2}}{{\left (a^{2} c^{2} d^{2} f + 2 i \, a^{2} c d^{3} f - a^{2} d^{4} f\right )} {\left (d \tan \left (f x + e\right ) - i \, d\right )}^{2}} + \frac {4 i \, \arctan \left (\frac {2 \, {\left (\sqrt {d \tan \left (f x + e\right ) + c} c - \sqrt {c^{2} + d^{2}} \sqrt {d \tan \left (f x + e\right ) + c}\right )}}{c \sqrt {-2 \, c + 2 \, \sqrt {c^{2} + d^{2}}} - i \, \sqrt {-2 \, c + 2 \, \sqrt {c^{2} + d^{2}}} d - \sqrt {c^{2} + d^{2}} \sqrt {-2 \, c + 2 \, \sqrt {c^{2} + d^{2}}}}\right )}{a^{2} \sqrt {-2 \, c + 2 \, \sqrt {c^{2} + d^{2}}} d^{3} f {\left (-\frac {i \, d}{c - \sqrt {c^{2} + d^{2}}} + 1\right )}}\right )} \]

[In]

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

[Out]

1/8*d^3*(16*(2*c^2 + 6*I*c*d - 7*d^2)*arctan(2*(sqrt(d*tan(f*x + e) + c)*c - sqrt(c^2 + d^2)*sqrt(d*tan(f*x +
e) + c))/(c*sqrt(-2*c + 2*sqrt(c^2 + d^2)) + I*sqrt(-2*c + 2*sqrt(c^2 + d^2))*d - sqrt(c^2 + d^2)*sqrt(-2*c +
2*sqrt(c^2 + d^2))))/((8*I*a^2*c^2*d^3*f - 16*a^2*c*d^4*f - 8*I*a^2*d^5*f)*sqrt(-2*c + 2*sqrt(c^2 + d^2))*(I*d
/(c - sqrt(c^2 + d^2)) + 1)) + (2*(d*tan(f*x + e) + c)^(3/2)*c - 2*sqrt(d*tan(f*x + e) + c)*c^2 + 5*I*(d*tan(f
*x + e) + c)^(3/2)*d - 9*I*sqrt(d*tan(f*x + e) + c)*c*d + 7*sqrt(d*tan(f*x + e) + c)*d^2)/((a^2*c^2*d^2*f + 2*
I*a^2*c*d^3*f - a^2*d^4*f)*(d*tan(f*x + e) - I*d)^2) + 4*I*arctan(2*(sqrt(d*tan(f*x + e) + c)*c - sqrt(c^2 + d
^2)*sqrt(d*tan(f*x + e) + c))/(c*sqrt(-2*c + 2*sqrt(c^2 + d^2)) - I*sqrt(-2*c + 2*sqrt(c^2 + d^2))*d - sqrt(c^
2 + d^2)*sqrt(-2*c + 2*sqrt(c^2 + d^2))))/(a^2*sqrt(-2*c + 2*sqrt(c^2 + d^2))*d^3*f*(-I*d/(c - sqrt(c^2 + d^2)
) + 1)))

Mupad [B] (verification not implemented)

Time = 10.19 (sec) , antiderivative size = 32178, normalized size of antiderivative = 145.60 \[ \int \frac {1}{(a+i a \tan (e+f x))^2 \sqrt {c+d \tan (e+f x)}} \, dx=\text {Too large to display} \]

[In]

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

[Out]

log(a^2*d^10*f*35i - ((-(45*d^9 - c*d^8*15i + 60*c^2*d^7 - c^3*d^6*80i - 40*c^4*d^5 + c^5*d^4*8i + a^4*c^4*f^2
*(((165*c*d^12 + 70*c^3*d^10 + 73*c^5*d^8 + 32*c^7*d^6 + 8*c^9*d^4)/(a^4*c^8*f^2 + a^4*d^8*f^2 + 4*a^4*c^2*d^6
*f^2 + 6*a^4*c^4*d^4*f^2 + 4*a^4*c^6*d^2*f^2) + ((150*c^2*d^11 - 45*d^13 + 95*c^4*d^9 + 52*c^6*d^7 + 8*c^8*d^5
)*1i)/(a^4*c^8*f^2 + a^4*d^8*f^2 + 4*a^4*c^2*d^6*f^2 + 6*a^4*c^4*d^4*f^2 + 4*a^4*c^6*d^2*f^2))^2 - 4*(256*d^6
+ 256*c^2*d^4)*((((7*c*d^11)/4 + (19*c^3*d^9)/16 + (11*c^5*d^7)/16 + (c^7*d^5)/8)*1i)/(a^8*c^8*f^4 + a^8*d^8*f
^4 + 4*a^8*c^2*d^6*f^4 + 6*a^8*c^4*d^4*f^4 + 4*a^8*c^6*d^2*f^4) + ((49*d^12)/64 - (11*c^2*d^10)/32 + (5*c^4*d^
8)/64 + (c^6*d^6)/8 + (c^8*d^4)/16)/(a^8*c^8*f^4 + a^8*d^8*f^4 + 4*a^8*c^2*d^6*f^4 + 6*a^8*c^4*d^4*f^4 + 4*a^8
*c^6*d^2*f^4)))^(1/2)*1i + a^4*d^4*f^2*(((165*c*d^12 + 70*c^3*d^10 + 73*c^5*d^8 + 32*c^7*d^6 + 8*c^9*d^4)/(a^4
*c^8*f^2 + a^4*d^8*f^2 + 4*a^4*c^2*d^6*f^2 + 6*a^4*c^4*d^4*f^2 + 4*a^4*c^6*d^2*f^2) + ((150*c^2*d^11 - 45*d^13
+ 95*c^4*d^9 + 52*c^6*d^7 + 8*c^8*d^5)*1i)/(a^4*c^8*f^2 + a^4*d^8*f^2 + 4*a^4*c^2*d^6*f^2 + 6*a^4*c^4*d^4*f^2
+ 4*a^4*c^6*d^2*f^2))^2 - 4*(256*d^6 + 256*c^2*d^4)*((((7*c*d^11)/4 + (19*c^3*d^9)/16 + (11*c^5*d^7)/16 + (c^
7*d^5)/8)*1i)/(a^8*c^8*f^4 + a^8*d^8*f^4 + 4*a^8*c^2*d^6*f^4 + 6*a^8*c^4*d^4*f^4 + 4*a^8*c^6*d^2*f^4) + ((49*d
^12)/64 - (11*c^2*d^10)/32 + (5*c^4*d^8)/64 + (c^6*d^6)/8 + (c^8*d^4)/16)/(a^8*c^8*f^4 + a^8*d^8*f^4 + 4*a^8*c
^2*d^6*f^4 + 6*a^8*c^4*d^4*f^4 + 4*a^8*c^6*d^2*f^4)))^(1/2)*1i - a^4*c^2*d^2*f^2*(((165*c*d^12 + 70*c^3*d^10 +
73*c^5*d^8 + 32*c^7*d^6 + 8*c^9*d^4)/(a^4*c^8*f^2 + a^4*d^8*f^2 + 4*a^4*c^2*d^6*f^2 + 6*a^4*c^4*d^4*f^2 + 4*a
^4*c^6*d^2*f^2) + ((150*c^2*d^11 - 45*d^13 + 95*c^4*d^9 + 52*c^6*d^7 + 8*c^8*d^5)*1i)/(a^4*c^8*f^2 + a^4*d^8*f
^2 + 4*a^4*c^2*d^6*f^2 + 6*a^4*c^4*d^4*f^2 + 4*a^4*c^6*d^2*f^2))^2 - 4*(256*d^6 + 256*c^2*d^4)*((((7*c*d^11)/4
+ (19*c^3*d^9)/16 + (11*c^5*d^7)/16 + (c^7*d^5)/8)*1i)/(a^8*c^8*f^4 + a^8*d^8*f^4 + 4*a^8*c^2*d^6*f^4 + 6*a^8
*c^4*d^4*f^4 + 4*a^8*c^6*d^2*f^4) + ((49*d^12)/64 - (11*c^2*d^10)/32 + (5*c^4*d^8)/64 + (c^6*d^6)/8 + (c^8*d^4
)/16)/(a^8*c^8*f^4 + a^8*d^8*f^4 + 4*a^8*c^2*d^6*f^4 + 6*a^8*c^4*d^4*f^4 + 4*a^8*c^6*d^2*f^4)))^(1/2)*6i + 4*a
^4*c*d^3*f^2*(((165*c*d^12 + 70*c^3*d^10 + 73*c^5*d^8 + 32*c^7*d^6 + 8*c^9*d^4)/(a^4*c^8*f^2 + a^4*d^8*f^2 + 4
*a^4*c^2*d^6*f^2 + 6*a^4*c^4*d^4*f^2 + 4*a^4*c^6*d^2*f^2) + ((150*c^2*d^11 - 45*d^13 + 95*c^4*d^9 + 52*c^6*d^7
+ 8*c^8*d^5)*1i)/(a^4*c^8*f^2 + a^4*d^8*f^2 + 4*a^4*c^2*d^6*f^2 + 6*a^4*c^4*d^4*f^2 + 4*a^4*c^6*d^2*f^2))^2 -
4*(256*d^6 + 256*c^2*d^4)*((((7*c*d^11)/4 + (19*c^3*d^9)/16 + (11*c^5*d^7)/16 + (c^7*d^5)/8)*1i)/(a^8*c^8*f^4
+ a^8*d^8*f^4 + 4*a^8*c^2*d^6*f^4 + 6*a^8*c^4*d^4*f^4 + 4*a^8*c^6*d^2*f^4) + ((49*d^12)/64 - (11*c^2*d^10)/32
+ (5*c^4*d^8)/64 + (c^6*d^6)/8 + (c^8*d^4)/16)/(a^8*c^8*f^4 + a^8*d^8*f^4 + 4*a^8*c^2*d^6*f^4 + 6*a^8*c^4*d^4
*f^4 + 4*a^8*c^6*d^2*f^4)))^(1/2) - 4*a^4*c^3*d*f^2*(((165*c*d^12 + 70*c^3*d^10 + 73*c^5*d^8 + 32*c^7*d^6 + 8*
c^9*d^4)/(a^4*c^8*f^2 + a^4*d^8*f^2 + 4*a^4*c^2*d^6*f^2 + 6*a^4*c^4*d^4*f^2 + 4*a^4*c^6*d^2*f^2) + ((150*c^2*d
^11 - 45*d^13 + 95*c^4*d^9 + 52*c^6*d^7 + 8*c^8*d^5)*1i)/(a^4*c^8*f^2 + a^4*d^8*f^2 + 4*a^4*c^2*d^6*f^2 + 6*a^
4*c^4*d^4*f^2 + 4*a^4*c^6*d^2*f^2))^2 - 4*(256*d^6 + 256*c^2*d^4)*((((7*c*d^11)/4 + (19*c^3*d^9)/16 + (11*c^5*
d^7)/16 + (c^7*d^5)/8)*1i)/(a^8*c^8*f^4 + a^8*d^8*f^4 + 4*a^8*c^2*d^6*f^4 + 6*a^8*c^4*d^4*f^4 + 4*a^8*c^6*d^2*
f^4) + ((49*d^12)/64 - (11*c^2*d^10)/32 + (5*c^4*d^8)/64 + (c^6*d^6)/8 + (c^8*d^4)/16)/(a^8*c^8*f^4 + a^8*d^8*
f^4 + 4*a^8*c^2*d^6*f^4 + 6*a^8*c^4*d^4*f^4 + 4*a^8*c^6*d^2*f^4)))^(1/2))/(512*(d^6 + c^2*d^4)*(a^4*c^4*f^2*1i
+ a^4*d^4*f^2*1i + 4*a^4*c*d^3*f^2 - 4*a^4*c^3*d*f^2 - a^4*c^2*d^2*f^2*6i)))^(1/2)*(1152*a^6*d^11*f^3 + a^6*c
*d^10*f^3*1664i + 8*(c + d*tan(e + f*x))^(1/2)*(-(45*d^9 - c*d^8*15i + 60*c^2*d^7 - c^3*d^6*80i - 40*c^4*d^5 +
c^5*d^4*8i + a^4*c^4*f^2*(((165*c*d^12 + 70*c^3*d^10 + 73*c^5*d^8 + 32*c^7*d^6 + 8*c^9*d^4)/(a^4*c^8*f^2 + a^
4*d^8*f^2 + 4*a^4*c^2*d^6*f^2 + 6*a^4*c^4*d^4*f^2 + 4*a^4*c^6*d^2*f^2) + ((150*c^2*d^11 - 45*d^13 + 95*c^4*d^9
+ 52*c^6*d^7 + 8*c^8*d^5)*1i)/(a^4*c^8*f^2 + a^4*d^8*f^2 + 4*a^4*c^2*d^6*f^2 + 6*a^4*c^4*d^4*f^2 + 4*a^4*c^6*
d^2*f^2))^2 - 4*(256*d^6 + 256*c^2*d^4)*((((7*c*d^11)/4 + (19*c^3*d^9)/16 + (11*c^5*d^7)/16 + (c^7*d^5)/8)*1i)
/(a^8*c^8*f^4 + a^8*d^8*f^4 + 4*a^8*c^2*d^6*f^4 + 6*a^8*c^4*d^4*f^4 + 4*a^8*c^6*d^2*f^4) + ((49*d^12)/64 - (11
*c^2*d^10)/32 + (5*c^4*d^8)/64 + (c^6*d^6)/8 + (c^8*d^4)/16)/(a^8*c^8*f^4 + a^8*d^8*f^4 + 4*a^8*c^2*d^6*f^4 +
6*a^8*c^4*d^4*f^4 + 4*a^8*c^6*d^2*f^4)))^(1/2)*1i + a^4*d^4*f^2*(((165*c*d^12 + 70*c^3*d^10 + 73*c^5*d^8 + 32*
c^7*d^6 + 8*c^9*d^4)/(a^4*c^8*f^2 + a^4*d^8*f^2 + 4*a^4*c^2*d^6*f^2 + 6*a^4*c^4*d^4*f^2 + 4*a^4*c^6*d^2*f^2) +
((150*c^2*d^11 - 45*d^13 + 95*c^4*d^9 + 52*c^6*d^7 + 8*c^8*d^5)*1i)/(a^4*c^8*f^2 + a^4*d^8*f^2 + 4*a^4*c^2*d^
6*f^2 + 6*a^4*c^4*d^4*f^2 + 4*a^4*c^6*d^2*f^2))^2 - 4*(256*d^6 + 256*c^2*d^4)*((((7*c*d^11)/4 + (19*c^3*d^9)/1
6 + (11*c^5*d^7)/16 + (c^7*d^5)/8)*1i)/(a^8*c^8*f^4 + a^8*d^8*f^4 + 4*a^8*c^2*d^6*f^4 + 6*a^8*c^4*d^4*f^4 + 4*
a^8*c^6*d^2*f^4) + ((49*d^12)/64 - (11*c^2*d^10)/32 + (5*c^4*d^8)/64 + (c^6*d^6)/8 + (c^8*d^4)/16)/(a^8*c^8*f^
4 + a^8*d^8*f^4 + 4*a^8*c^2*d^6*f^4 + 6*a^8*c^4*d^4*f^4 + 4*a^8*c^6*d^2*f^4)))^(1/2)*1i - a^4*c^2*d^2*f^2*(((1
65*c*d^12 + 70*c^3*d^10 + 73*c^5*d^8 + 32*c^7*d^6 + 8*c^9*d^4)/(a^4*c^8*f^2 + a^4*d^8*f^2 + 4*a^4*c^2*d^6*f^2
+ 6*a^4*c^4*d^4*f^2 + 4*a^4*c^6*d^2*f^2) + ((150*c^2*d^11 - 45*d^13 + 95*c^4*d^9 + 52*c^6*d^7 + 8*c^8*d^5)*1i)
/(a^4*c^8*f^2 + a^4*d^8*f^2 + 4*a^4*c^2*d^6*f^2 + 6*a^4*c^4*d^4*f^2 + 4*a^4*c^6*d^2*f^2))^2 - 4*(256*d^6 + 256
*c^2*d^4)*((((7*c*d^11)/4 + (19*c^3*d^9)/16 + (11*c^5*d^7)/16 + (c^7*d^5)/8)*1i)/(a^8*c^8*f^4 + a^8*d^8*f^4 +
4*a^8*c^2*d^6*f^4 + 6*a^8*c^4*d^4*f^4 + 4*a^8*c^6*d^2*f^4) + ((49*d^12)/64 - (11*c^2*d^10)/32 + (5*c^4*d^8)/64
+ (c^6*d^6)/8 + (c^8*d^4)/16)/(a^8*c^8*f^4 + a^8*d^8*f^4 + 4*a^8*c^2*d^6*f^4 + 6*a^8*c^4*d^4*f^4 + 4*a^8*c^6*
d^2*f^4)))^(1/2)*6i + 4*a^4*c*d^3*f^2*(((165*c*d^12 + 70*c^3*d^10 + 73*c^5*d^8 + 32*c^7*d^6 + 8*c^9*d^4)/(a^4*
c^8*f^2 + a^4*d^8*f^2 + 4*a^4*c^2*d^6*f^2 + 6*a^4*c^4*d^4*f^2 + 4*a^4*c^6*d^2*f^2) + ((150*c^2*d^11 - 45*d^13
+ 95*c^4*d^9 + 52*c^6*d^7 + 8*c^8*d^5)*1i)/(a^4*c^8*f^2 + a^4*d^8*f^2 + 4*a^4*c^2*d^6*f^2 + 6*a^4*c^4*d^4*f^2
+ 4*a^4*c^6*d^2*f^2))^2 - 4*(256*d^6 + 256*c^2*d^4)*((((7*c*d^11)/4 + (19*c^3*d^9)/16 + (11*c^5*d^7)/16 + (c^7
*d^5)/8)*1i)/(a^8*c^8*f^4 + a^8*d^8*f^4 + 4*a^8*c^2*d^6*f^4 + 6*a^8*c^4*d^4*f^4 + 4*a^8*c^6*d^2*f^4) + ((49*d^
12)/64 - (11*c^2*d^10)/32 + (5*c^4*d^8)/64 + (c^6*d^6)/8 + (c^8*d^4)/16)/(a^8*c^8*f^4 + a^8*d^8*f^4 + 4*a^8*c^
2*d^6*f^4 + 6*a^8*c^4*d^4*f^4 + 4*a^8*c^6*d^2*f^4)))^(1/2) - 4*a^4*c^3*d*f^2*(((165*c*d^12 + 70*c^3*d^10 + 73*
c^5*d^8 + 32*c^7*d^6 + 8*c^9*d^4)/(a^4*c^8*f^2 + a^4*d^8*f^2 + 4*a^4*c^2*d^6*f^2 + 6*a^4*c^4*d^4*f^2 + 4*a^4*c
^6*d^2*f^2) + ((150*c^2*d^11 - 45*d^13 + 95*c^4*d^9 + 52*c^6*d^7 + 8*c^8*d^5)*1i)/(a^4*c^8*f^2 + a^4*d^8*f^2 +
4*a^4*c^2*d^6*f^2 + 6*a^4*c^4*d^4*f^2 + 4*a^4*c^6*d^2*f^2))^2 - 4*(256*d^6 + 256*c^2*d^4)*((((7*c*d^11)/4 + (
19*c^3*d^9)/16 + (11*c^5*d^7)/16 + (c^7*d^5)/8)*1i)/(a^8*c^8*f^4 + a^8*d^8*f^4 + 4*a^8*c^2*d^6*f^4 + 6*a^8*c^4
*d^4*f^4 + 4*a^8*c^6*d^2*f^4) + ((49*d^12)/64 - (11*c^2*d^10)/32 + (5*c^4*d^8)/64 + (c^6*d^6)/8 + (c^8*d^4)/16
)/(a^8*c^8*f^4 + a^8*d^8*f^4 + 4*a^8*c^2*d^6*f^4 + 6*a^8*c^4*d^4*f^4 + 4*a^8*c^6*d^2*f^4)))^(1/2))/(512*(d^6 +
c^2*d^4)*(a^4*c^4*f^2*1i + a^4*d^4*f^2*1i + 4*a^4*c*d^3*f^2 - 4*a^4*c^3*d*f^2 - a^4*c^2*d^2*f^2*6i)))^(1/2)*(
512*a^4*c*d^6*f^2 - a^4*c^2*d^5*f^2*2048i - 3072*a^4*c^3*d^4*f^2 + a^4*c^4*d^3*f^2*2048i + 512*a^4*c^5*d^2*f^2
)*(a^4*c^4*f^2 + a^4*d^4*f^2 + a^4*c*d^3*f^2*4i - a^4*c^3*d*f^2*4i - 6*a^4*c^2*d^2*f^2) + 2176*a^6*c^2*d^9*f^3
+ a^6*c^3*d^8*f^3*3456i + 1152*a^6*c^4*d^7*f^3 + a^6*c^5*d^6*f^3*1920i + 384*a^6*c^6*d^5*f^3 + a^6*c^7*d^4*f^
3*128i + 256*a^6*c^8*d^3*f^3) + 8*(c + d*tan(e + f*x))^(1/2)*(a^4*c^4*f^2 + a^4*d^4*f^2 + a^4*c*d^3*f^2*4i - a
^4*c^3*d*f^2*4i - 6*a^4*c^2*d^2*f^2)*(53*d^6 - c*d^5*100i - 88*c^2*d^4 + c^3*d^3*40i + 8*c^4*d^2))*(-(45*d^9 -
c*d^8*15i + 60*c^2*d^7 - c^3*d^6*80i - 40*c^4*d^5 + c^5*d^4*8i + a^4*c^4*f^2*(((165*c*d^12 + 70*c^3*d^10 + 73
*c^5*d^8 + 32*c^7*d^6 + 8*c^9*d^4)/(a^4*c^8*f^2 + a^4*d^8*f^2 + 4*a^4*c^2*d^6*f^2 + 6*a^4*c^4*d^4*f^2 + 4*a^4*
c^6*d^2*f^2) + ((150*c^2*d^11 - 45*d^13 + 95*c^4*d^9 + 52*c^6*d^7 + 8*c^8*d^5)*1i)/(a^4*c^8*f^2 + a^4*d^8*f^2
+ 4*a^4*c^2*d^6*f^2 + 6*a^4*c^4*d^4*f^2 + 4*a^4*c^6*d^2*f^2))^2 - 4*(256*d^6 + 256*c^2*d^4)*((((7*c*d^11)/4 +
(19*c^3*d^9)/16 + (11*c^5*d^7)/16 + (c^7*d^5)/8)*1i)/(a^8*c^8*f^4 + a^8*d^8*f^4 + 4*a^8*c^2*d^6*f^4 + 6*a^8*c^
4*d^4*f^4 + 4*a^8*c^6*d^2*f^4) + ((49*d^12)/64 - (11*c^2*d^10)/32 + (5*c^4*d^8)/64 + (c^6*d^6)/8 + (c^8*d^4)/1
6)/(a^8*c^8*f^4 + a^8*d^8*f^4 + 4*a^8*c^2*d^6*f^4 + 6*a^8*c^4*d^4*f^4 + 4*a^8*c^6*d^2*f^4)))^(1/2)*1i + a^4*d^
4*f^2*(((165*c*d^12 + 70*c^3*d^10 + 73*c^5*d^8 + 32*c^7*d^6 + 8*c^9*d^4)/(a^4*c^8*f^2 + a^4*d^8*f^2 + 4*a^4*c^
2*d^6*f^2 + 6*a^4*c^4*d^4*f^2 + 4*a^4*c^6*d^2*f^2) + ((150*c^2*d^11 - 45*d^13 + 95*c^4*d^9 + 52*c^6*d^7 + 8*c^
8*d^5)*1i)/(a^4*c^8*f^2 + a^4*d^8*f^2 + 4*a^4*c^2*d^6*f^2 + 6*a^4*c^4*d^4*f^2 + 4*a^4*c^6*d^2*f^2))^2 - 4*(256
*d^6 + 256*c^2*d^4)*((((7*c*d^11)/4 + (19*c^3*d^9)/16 + (11*c^5*d^7)/16 + (c^7*d^5)/8)*1i)/(a^8*c^8*f^4 + a^8*
d^8*f^4 + 4*a^8*c^2*d^6*f^4 + 6*a^8*c^4*d^4*f^4 + 4*a^8*c^6*d^2*f^4) + ((49*d^12)/64 - (11*c^2*d^10)/32 + (5*c
^4*d^8)/64 + (c^6*d^6)/8 + (c^8*d^4)/16)/(a^8*c^8*f^4 + a^8*d^8*f^4 + 4*a^8*c^2*d^6*f^4 + 6*a^8*c^4*d^4*f^4 +
4*a^8*c^6*d^2*f^4)))^(1/2)*1i - a^4*c^2*d^2*f^2*(((165*c*d^12 + 70*c^3*d^10 + 73*c^5*d^8 + 32*c^7*d^6 + 8*c^9*
d^4)/(a^4*c^8*f^2 + a^4*d^8*f^2 + 4*a^4*c^2*d^6*f^2 + 6*a^4*c^4*d^4*f^2 + 4*a^4*c^6*d^2*f^2) + ((150*c^2*d^11
- 45*d^13 + 95*c^4*d^9 + 52*c^6*d^7 + 8*c^8*d^5)*1i)/(a^4*c^8*f^2 + a^4*d^8*f^2 + 4*a^4*c^2*d^6*f^2 + 6*a^4*c^
4*d^4*f^2 + 4*a^4*c^6*d^2*f^2))^2 - 4*(256*d^6 + 256*c^2*d^4)*((((7*c*d^11)/4 + (19*c^3*d^9)/16 + (11*c^5*d^7)
/16 + (c^7*d^5)/8)*1i)/(a^8*c^8*f^4 + a^8*d^8*f^4 + 4*a^8*c^2*d^6*f^4 + 6*a^8*c^4*d^4*f^4 + 4*a^8*c^6*d^2*f^4)
+ ((49*d^12)/64 - (11*c^2*d^10)/32 + (5*c^4*d^8)/64 + (c^6*d^6)/8 + (c^8*d^4)/16)/(a^8*c^8*f^4 + a^8*d^8*f^4
+ 4*a^8*c^2*d^6*f^4 + 6*a^8*c^4*d^4*f^4 + 4*a^8*c^6*d^2*f^4)))^(1/2)*6i + 4*a^4*c*d^3*f^2*(((165*c*d^12 + 70*c
^3*d^10 + 73*c^5*d^8 + 32*c^7*d^6 + 8*c^9*d^4)/(a^4*c^8*f^2 + a^4*d^8*f^2 + 4*a^4*c^2*d^6*f^2 + 6*a^4*c^4*d^4*
f^2 + 4*a^4*c^6*d^2*f^2) + ((150*c^2*d^11 - 45*d^13 + 95*c^4*d^9 + 52*c^6*d^7 + 8*c^8*d^5)*1i)/(a^4*c^8*f^2 +
a^4*d^8*f^2 + 4*a^4*c^2*d^6*f^2 + 6*a^4*c^4*d^4*f^2 + 4*a^4*c^6*d^2*f^2))^2 - 4*(256*d^6 + 256*c^2*d^4)*((((7*
c*d^11)/4 + (19*c^3*d^9)/16 + (11*c^5*d^7)/16 + (c^7*d^5)/8)*1i)/(a^8*c^8*f^4 + a^8*d^8*f^4 + 4*a^8*c^2*d^6*f^
4 + 6*a^8*c^4*d^4*f^4 + 4*a^8*c^6*d^2*f^4) + ((49*d^12)/64 - (11*c^2*d^10)/32 + (5*c^4*d^8)/64 + (c^6*d^6)/8 +
(c^8*d^4)/16)/(a^8*c^8*f^4 + a^8*d^8*f^4 + 4*a^8*c^2*d^6*f^4 + 6*a^8*c^4*d^4*f^4 + 4*a^8*c^6*d^2*f^4)))^(1/2)
- 4*a^4*c^3*d*f^2*(((165*c*d^12 + 70*c^3*d^10 + 73*c^5*d^8 + 32*c^7*d^6 + 8*c^9*d^4)/(a^4*c^8*f^2 + a^4*d^8*f
^2 + 4*a^4*c^2*d^6*f^2 + 6*a^4*c^4*d^4*f^2 + 4*a^4*c^6*d^2*f^2) + ((150*c^2*d^11 - 45*d^13 + 95*c^4*d^9 + 52*c
^6*d^7 + 8*c^8*d^5)*1i)/(a^4*c^8*f^2 + a^4*d^8*f^2 + 4*a^4*c^2*d^6*f^2 + 6*a^4*c^4*d^4*f^2 + 4*a^4*c^6*d^2*f^2
))^2 - 4*(256*d^6 + 256*c^2*d^4)*((((7*c*d^11)/4 + (19*c^3*d^9)/16 + (11*c^5*d^7)/16 + (c^7*d^5)/8)*1i)/(a^8*c
^8*f^4 + a^8*d^8*f^4 + 4*a^8*c^2*d^6*f^4 + 6*a^8*c^4*d^4*f^4 + 4*a^8*c^6*d^2*f^4) + ((49*d^12)/64 - (11*c^2*d^
10)/32 + (5*c^4*d^8)/64 + (c^6*d^6)/8 + (c^8*d^4)/16)/(a^8*c^8*f^4 + a^8*d^8*f^4 + 4*a^8*c^2*d^6*f^4 + 6*a^8*c
^4*d^4*f^4 + 4*a^8*c^6*d^2*f^4)))^(1/2))/(512*(d^6 + c^2*d^4)*(a^4*c^4*f^2*1i + a^4*d^4*f^2*1i + 4*a^4*c*d^3*f
^2 - 4*a^4*c^3*d*f^2 - a^4*c^2*d^2*f^2*6i)))^(1/2) - a^2*c^2*d^8*f*56i - 40*a^2*c^3*d^7*f - a^2*c^4*d^6*f*25i
- 20*a^2*c^5*d^5*f - a^2*c^6*d^4*f*6i - 4*a^2*c^7*d^3*f - 96*a^2*c*d^9*f)*(-(45*d^9 - c*d^8*15i + 60*c^2*d^7 -
c^3*d^6*80i - 40*c^4*d^5 + c^5*d^4*8i + a^4*c^4*f^2*(((165*c*d^12 + 70*c^3*d^10 + 73*c^5*d^8 + 32*c^7*d^6 + 8
*c^9*d^4)/(a^4*c^8*f^2 + a^4*d^8*f^2 + 4*a^4*c^2*d^6*f^2 + 6*a^4*c^4*d^4*f^2 + 4*a^4*c^6*d^2*f^2) + ((150*c^2*
d^11 - 45*d^13 + 95*c^4*d^9 + 52*c^6*d^7 + 8*c^8*d^5)*1i)/(a^4*c^8*f^2 + a^4*d^8*f^2 + 4*a^4*c^2*d^6*f^2 + 6*a
^4*c^4*d^4*f^2 + 4*a^4*c^6*d^2*f^2))^2 - 4*(256*d^6 + 256*c^2*d^4)*((((7*c*d^11)/4 + (19*c^3*d^9)/16 + (11*c^5
*d^7)/16 + (c^7*d^5)/8)*1i)/(a^8*c^8*f^4 + a^8*d^8*f^4 + 4*a^8*c^2*d^6*f^4 + 6*a^8*c^4*d^4*f^4 + 4*a^8*c^6*d^2
*f^4) + ((49*d^12)/64 - (11*c^2*d^10)/32 + (5*c^4*d^8)/64 + (c^6*d^6)/8 + (c^8*d^4)/16)/(a^8*c^8*f^4 + a^8*d^8
*f^4 + 4*a^8*c^2*d^6*f^4 + 6*a^8*c^4*d^4*f^4 + 4*a^8*c^6*d^2*f^4)))^(1/2)*1i + a^4*d^4*f^2*(((165*c*d^12 + 70*
c^3*d^10 + 73*c^5*d^8 + 32*c^7*d^6 + 8*c^9*d^4)/(a^4*c^8*f^2 + a^4*d^8*f^2 + 4*a^4*c^2*d^6*f^2 + 6*a^4*c^4*d^4
*f^2 + 4*a^4*c^6*d^2*f^2) + ((150*c^2*d^11 - 45*d^13 + 95*c^4*d^9 + 52*c^6*d^7 + 8*c^8*d^5)*1i)/(a^4*c^8*f^2 +
a^4*d^8*f^2 + 4*a^4*c^2*d^6*f^2 + 6*a^4*c^4*d^4*f^2 + 4*a^4*c^6*d^2*f^2))^2 - 4*(256*d^6 + 256*c^2*d^4)*((((7
*c*d^11)/4 + (19*c^3*d^9)/16 + (11*c^5*d^7)/16 + (c^7*d^5)/8)*1i)/(a^8*c^8*f^4 + a^8*d^8*f^4 + 4*a^8*c^2*d^6*f
^4 + 6*a^8*c^4*d^4*f^4 + 4*a^8*c^6*d^2*f^4) + ((49*d^12)/64 - (11*c^2*d^10)/32 + (5*c^4*d^8)/64 + (c^6*d^6)/8
+ (c^8*d^4)/16)/(a^8*c^8*f^4 + a^8*d^8*f^4 + 4*a^8*c^2*d^6*f^4 + 6*a^8*c^4*d^4*f^4 + 4*a^8*c^6*d^2*f^4)))^(1/2
)*1i - a^4*c^2*d^2*f^2*(((165*c*d^12 + 70*c^3*d^10 + 73*c^5*d^8 + 32*c^7*d^6 + 8*c^9*d^4)/(a^4*c^8*f^2 + a^4*d
^8*f^2 + 4*a^4*c^2*d^6*f^2 + 6*a^4*c^4*d^4*f^2 + 4*a^4*c^6*d^2*f^2) + ((150*c^2*d^11 - 45*d^13 + 95*c^4*d^9 +
52*c^6*d^7 + 8*c^8*d^5)*1i)/(a^4*c^8*f^2 + a^4*d^8*f^2 + 4*a^4*c^2*d^6*f^2 + 6*a^4*c^4*d^4*f^2 + 4*a^4*c^6*d^2
*f^2))^2 - 4*(256*d^6 + 256*c^2*d^4)*((((7*c*d^11)/4 + (19*c^3*d^9)/16 + (11*c^5*d^7)/16 + (c^7*d^5)/8)*1i)/(a
^8*c^8*f^4 + a^8*d^8*f^4 + 4*a^8*c^2*d^6*f^4 + 6*a^8*c^4*d^4*f^4 + 4*a^8*c^6*d^2*f^4) + ((49*d^12)/64 - (11*c^
2*d^10)/32 + (5*c^4*d^8)/64 + (c^6*d^6)/8 + (c^8*d^4)/16)/(a^8*c^8*f^4 + a^8*d^8*f^4 + 4*a^8*c^2*d^6*f^4 + 6*a
^8*c^4*d^4*f^4 + 4*a^8*c^6*d^2*f^4)))^(1/2)*6i + 4*a^4*c*d^3*f^2*(((165*c*d^12 + 70*c^3*d^10 + 73*c^5*d^8 + 32
*c^7*d^6 + 8*c^9*d^4)/(a^4*c^8*f^2 + a^4*d^8*f^2 + 4*a^4*c^2*d^6*f^2 + 6*a^4*c^4*d^4*f^2 + 4*a^4*c^6*d^2*f^2)
+ ((150*c^2*d^11 - 45*d^13 + 95*c^4*d^9 + 52*c^6*d^7 + 8*c^8*d^5)*1i)/(a^4*c^8*f^2 + a^4*d^8*f^2 + 4*a^4*c^2*d
^6*f^2 + 6*a^4*c^4*d^4*f^2 + 4*a^4*c^6*d^2*f^2))^2 - 4*(256*d^6 + 256*c^2*d^4)*((((7*c*d^11)/4 + (19*c^3*d^9)/
16 + (11*c^5*d^7)/16 + (c^7*d^5)/8)*1i)/(a^8*c^8*f^4 + a^8*d^8*f^4 + 4*a^8*c^2*d^6*f^4 + 6*a^8*c^4*d^4*f^4 + 4
*a^8*c^6*d^2*f^4) + ((49*d^12)/64 - (11*c^2*d^10)/32 + (5*c^4*d^8)/64 + (c^6*d^6)/8 + (c^8*d^4)/16)/(a^8*c^8*f
^4 + a^8*d^8*f^4 + 4*a^8*c^2*d^6*f^4 + 6*a^8*c^4*d^4*f^4 + 4*a^8*c^6*d^2*f^4)))^(1/2) - 4*a^4*c^3*d*f^2*(((165
*c*d^12 + 70*c^3*d^10 + 73*c^5*d^8 + 32*c^7*d^6 + 8*c^9*d^4)/(a^4*c^8*f^2 + a^4*d^8*f^2 + 4*a^4*c^2*d^6*f^2 +
6*a^4*c^4*d^4*f^2 + 4*a^4*c^6*d^2*f^2) + ((150*c^2*d^11 - 45*d^13 + 95*c^4*d^9 + 52*c^6*d^7 + 8*c^8*d^5)*1i)/(
a^4*c^8*f^2 + a^4*d^8*f^2 + 4*a^4*c^2*d^6*f^2 + 6*a^4*c^4*d^4*f^2 + 4*a^4*c^6*d^2*f^2))^2 - 4*(256*d^6 + 256*c
^2*d^4)*((((7*c*d^11)/4 + (19*c^3*d^9)/16 + (11*c^5*d^7)/16 + (c^7*d^5)/8)*1i)/(a^8*c^8*f^4 + a^8*d^8*f^4 + 4*
a^8*c^2*d^6*f^4 + 6*a^8*c^4*d^4*f^4 + 4*a^8*c^6*d^2*f^4) + ((49*d^12)/64 - (11*c^2*d^10)/32 + (5*c^4*d^8)/64 +
(c^6*d^6)/8 + (c^8*d^4)/16)/(a^8*c^8*f^4 + a^8*d^8*f^4 + 4*a^8*c^2*d^6*f^4 + 6*a^8*c^4*d^4*f^4 + 4*a^8*c^6*d^
2*f^4)))^(1/2))/(512*(d^6 + c^2*d^4)*(a^4*c^4*f^2*1i + a^4*d^4*f^2*1i + 4*a^4*c*d^3*f^2 - 4*a^4*c^3*d*f^2 - a^
4*c^2*d^2*f^2*6i)))^(1/2) - log(a^2*d^10*f*35i - ((-(165*c*d^12 - d^13*45i + c^2*d^11*150i + 70*c^3*d^10 + c^4
*d^9*95i + 73*c^5*d^8 + c^6*d^7*52i + 32*c^7*d^6 + c^8*d^5*8i + 8*c^9*d^4 + a^4*c^8*f^2*((37157*c^2*d^24 - 280
9*d^26 - c*d^25*16642i + c^3*d^23*33024i + 2990*c^4*d^22 + c^5*d^21*21076i + 322*c^6*d^20 + c^7*d^19*15752i -
5029*c^8*d^18 + c^9*d^17*3870i - 1255*c^10*d^16 + c^11*d^15*1000i - 400*c^12*d^14)/(a^8*c^16*f^4 + a^8*d^16*f^
4 + 8*a^8*c^2*d^14*f^4 + 28*a^8*c^4*d^12*f^4 + 56*a^8*c^6*d^10*f^4 + 70*a^8*c^8*d^8*f^4 + 56*a^8*c^10*d^6*f^4
+ 28*a^8*c^12*d^4*f^4 + 8*a^8*c^14*d^2*f^4))^(1/2) + a^4*d^8*f^2*((37157*c^2*d^24 - 2809*d^26 - c*d^25*16642i
+ c^3*d^23*33024i + 2990*c^4*d^22 + c^5*d^21*21076i + 322*c^6*d^20 + c^7*d^19*15752i - 5029*c^8*d^18 + c^9*d^1
7*3870i - 1255*c^10*d^16 + c^11*d^15*1000i - 400*c^12*d^14)/(a^8*c^16*f^4 + a^8*d^16*f^4 + 8*a^8*c^2*d^14*f^4
+ 28*a^8*c^4*d^12*f^4 + 56*a^8*c^6*d^10*f^4 + 70*a^8*c^8*d^8*f^4 + 56*a^8*c^10*d^6*f^4 + 28*a^8*c^12*d^4*f^4 +
8*a^8*c^14*d^2*f^4))^(1/2) + 4*a^4*c^2*d^6*f^2*((37157*c^2*d^24 - 2809*d^26 - c*d^25*16642i + c^3*d^23*33024i
+ 2990*c^4*d^22 + c^5*d^21*21076i + 322*c^6*d^20 + c^7*d^19*15752i - 5029*c^8*d^18 + c^9*d^17*3870i - 1255*c^
10*d^16 + c^11*d^15*1000i - 400*c^12*d^14)/(a^8*c^16*f^4 + a^8*d^16*f^4 + 8*a^8*c^2*d^14*f^4 + 28*a^8*c^4*d^12
*f^4 + 56*a^8*c^6*d^10*f^4 + 70*a^8*c^8*d^8*f^4 + 56*a^8*c^10*d^6*f^4 + 28*a^8*c^12*d^4*f^4 + 8*a^8*c^14*d^2*f
^4))^(1/2) + 6*a^4*c^4*d^4*f^2*((37157*c^2*d^24 - 2809*d^26 - c*d^25*16642i + c^3*d^23*33024i + 2990*c^4*d^22
+ c^5*d^21*21076i + 322*c^6*d^20 + c^7*d^19*15752i - 5029*c^8*d^18 + c^9*d^17*3870i - 1255*c^10*d^16 + c^11*d^
15*1000i - 400*c^12*d^14)/(a^8*c^16*f^4 + a^8*d^16*f^4 + 8*a^8*c^2*d^14*f^4 + 28*a^8*c^4*d^12*f^4 + 56*a^8*c^6
*d^10*f^4 + 70*a^8*c^8*d^8*f^4 + 56*a^8*c^10*d^6*f^4 + 28*a^8*c^12*d^4*f^4 + 8*a^8*c^14*d^2*f^4))^(1/2) + 4*a^
4*c^6*d^2*f^2*((37157*c^2*d^24 - 2809*d^26 - c*d^25*16642i + c^3*d^23*33024i + 2990*c^4*d^22 + c^5*d^21*21076i
+ 322*c^6*d^20 + c^7*d^19*15752i - 5029*c^8*d^18 + c^9*d^17*3870i - 1255*c^10*d^16 + c^11*d^15*1000i - 400*c^
12*d^14)/(a^8*c^16*f^4 + a^8*d^16*f^4 + 8*a^8*c^2*d^14*f^4 + 28*a^8*c^4*d^12*f^4 + 56*a^8*c^6*d^10*f^4 + 70*a^
8*c^8*d^8*f^4 + 56*a^8*c^10*d^6*f^4 + 28*a^8*c^12*d^4*f^4 + 8*a^8*c^14*d^2*f^4))^(1/2))/(512*a^4*d^14*f^2 + 25
60*a^4*c^2*d^12*f^2 + 5120*a^4*c^4*d^10*f^2 + 5120*a^4*c^6*d^8*f^2 + 2560*a^4*c^8*d^6*f^2 + 512*a^4*c^10*d^4*f
^2))^(1/2)*(1152*a^6*d^11*f^3 + a^6*c*d^10*f^3*1664i - 8*(-(165*c*d^12 - d^13*45i + c^2*d^11*150i + 70*c^3*d^1
0 + c^4*d^9*95i + 73*c^5*d^8 + c^6*d^7*52i + 32*c^7*d^6 + c^8*d^5*8i + 8*c^9*d^4 + a^4*c^8*f^2*((37157*c^2*d^2
4 - 2809*d^26 - c*d^25*16642i + c^3*d^23*33024i + 2990*c^4*d^22 + c^5*d^21*21076i + 322*c^6*d^20 + c^7*d^19*15
752i - 5029*c^8*d^18 + c^9*d^17*3870i - 1255*c^10*d^16 + c^11*d^15*1000i - 400*c^12*d^14)/(a^8*c^16*f^4 + a^8*
d^16*f^4 + 8*a^8*c^2*d^14*f^4 + 28*a^8*c^4*d^12*f^4 + 56*a^8*c^6*d^10*f^4 + 70*a^8*c^8*d^8*f^4 + 56*a^8*c^10*d
^6*f^4 + 28*a^8*c^12*d^4*f^4 + 8*a^8*c^14*d^2*f^4))^(1/2) + a^4*d^8*f^2*((37157*c^2*d^24 - 2809*d^26 - c*d^25*
16642i + c^3*d^23*33024i + 2990*c^4*d^22 + c^5*d^21*21076i + 322*c^6*d^20 + c^7*d^19*15752i - 5029*c^8*d^18 +
c^9*d^17*3870i - 1255*c^10*d^16 + c^11*d^15*1000i - 400*c^12*d^14)/(a^8*c^16*f^4 + a^8*d^16*f^4 + 8*a^8*c^2*d^
14*f^4 + 28*a^8*c^4*d^12*f^4 + 56*a^8*c^6*d^10*f^4 + 70*a^8*c^8*d^8*f^4 + 56*a^8*c^10*d^6*f^4 + 28*a^8*c^12*d^
4*f^4 + 8*a^8*c^14*d^2*f^4))^(1/2) + 4*a^4*c^2*d^6*f^2*((37157*c^2*d^24 - 2809*d^26 - c*d^25*16642i + c^3*d^23
*33024i + 2990*c^4*d^22 + c^5*d^21*21076i + 322*c^6*d^20 + c^7*d^19*15752i - 5029*c^8*d^18 + c^9*d^17*3870i -
1255*c^10*d^16 + c^11*d^15*1000i - 400*c^12*d^14)/(a^8*c^16*f^4 + a^8*d^16*f^4 + 8*a^8*c^2*d^14*f^4 + 28*a^8*c
^4*d^12*f^4 + 56*a^8*c^6*d^10*f^4 + 70*a^8*c^8*d^8*f^4 + 56*a^8*c^10*d^6*f^4 + 28*a^8*c^12*d^4*f^4 + 8*a^8*c^1
4*d^2*f^4))^(1/2) + 6*a^4*c^4*d^4*f^2*((37157*c^2*d^24 - 2809*d^26 - c*d^25*16642i + c^3*d^23*33024i + 2990*c^
4*d^22 + c^5*d^21*21076i + 322*c^6*d^20 + c^7*d^19*15752i - 5029*c^8*d^18 + c^9*d^17*3870i - 1255*c^10*d^16 +
c^11*d^15*1000i - 400*c^12*d^14)/(a^8*c^16*f^4 + a^8*d^16*f^4 + 8*a^8*c^2*d^14*f^4 + 28*a^8*c^4*d^12*f^4 + 56*
a^8*c^6*d^10*f^4 + 70*a^8*c^8*d^8*f^4 + 56*a^8*c^10*d^6*f^4 + 28*a^8*c^12*d^4*f^4 + 8*a^8*c^14*d^2*f^4))^(1/2)
+ 4*a^4*c^6*d^2*f^2*((37157*c^2*d^24 - 2809*d^26 - c*d^25*16642i + c^3*d^23*33024i + 2990*c^4*d^22 + c^5*d^21
*21076i + 322*c^6*d^20 + c^7*d^19*15752i - 5029*c^8*d^18 + c^9*d^17*3870i - 1255*c^10*d^16 + c^11*d^15*1000i -
400*c^12*d^14)/(a^8*c^16*f^4 + a^8*d^16*f^4 + 8*a^8*c^2*d^14*f^4 + 28*a^8*c^4*d^12*f^4 + 56*a^8*c^6*d^10*f^4
+ 70*a^8*c^8*d^8*f^4 + 56*a^8*c^10*d^6*f^4 + 28*a^8*c^12*d^4*f^4 + 8*a^8*c^14*d^2*f^4))^(1/2))/(512*a^4*d^14*f
^2 + 2560*a^4*c^2*d^12*f^2 + 5120*a^4*c^4*d^10*f^2 + 5120*a^4*c^6*d^8*f^2 + 2560*a^4*c^8*d^6*f^2 + 512*a^4*c^1
0*d^4*f^2))^(1/2)*(c + d*tan(e + f*x))^(1/2)*(512*a^4*c*d^6*f^2 - a^4*c^2*d^5*f^2*2048i - 3072*a^4*c^3*d^4*f^2
+ a^4*c^4*d^3*f^2*2048i + 512*a^4*c^5*d^2*f^2)*(a^4*c^4*f^2 + a^4*d^4*f^2 + a^4*c*d^3*f^2*4i - a^4*c^3*d*f^2*
4i - 6*a^4*c^2*d^2*f^2) + 2176*a^6*c^2*d^9*f^3 + a^6*c^3*d^8*f^3*3456i + 1152*a^6*c^4*d^7*f^3 + a^6*c^5*d^6*f^
3*1920i + 384*a^6*c^6*d^5*f^3 + a^6*c^7*d^4*f^3*128i + 256*a^6*c^8*d^3*f^3) - 8*(c + d*tan(e + f*x))^(1/2)*(a^
4*c^4*f^2 + a^4*d^4*f^2 + a^4*c*d^3*f^2*4i - a^4*c^3*d*f^2*4i - 6*a^4*c^2*d^2*f^2)*(53*d^6 - c*d^5*100i - 88*c
^2*d^4 + c^3*d^3*40i + 8*c^4*d^2))*(-(165*c*d^12 - d^13*45i + c^2*d^11*150i + 70*c^3*d^10 + c^4*d^9*95i + 73*c
^5*d^8 + c^6*d^7*52i + 32*c^7*d^6 + c^8*d^5*8i + 8*c^9*d^4 + a^4*c^8*f^2*((37157*c^2*d^24 - 2809*d^26 - c*d^25
*16642i + c^3*d^23*33024i + 2990*c^4*d^22 + c^5*d^21*21076i + 322*c^6*d^20 + c^7*d^19*15752i - 5029*c^8*d^18 +
c^9*d^17*3870i - 1255*c^10*d^16 + c^11*d^15*1000i - 400*c^12*d^14)/(a^8*c^16*f^4 + a^8*d^16*f^4 + 8*a^8*c^2*d
^14*f^4 + 28*a^8*c^4*d^12*f^4 + 56*a^8*c^6*d^10*f^4 + 70*a^8*c^8*d^8*f^4 + 56*a^8*c^10*d^6*f^4 + 28*a^8*c^12*d
^4*f^4 + 8*a^8*c^14*d^2*f^4))^(1/2) + a^4*d^8*f^2*((37157*c^2*d^24 - 2809*d^26 - c*d^25*16642i + c^3*d^23*3302
4i + 2990*c^4*d^22 + c^5*d^21*21076i + 322*c^6*d^20 + c^7*d^19*15752i - 5029*c^8*d^18 + c^9*d^17*3870i - 1255*
c^10*d^16 + c^11*d^15*1000i - 400*c^12*d^14)/(a^8*c^16*f^4 + a^8*d^16*f^4 + 8*a^8*c^2*d^14*f^4 + 28*a^8*c^4*d^
12*f^4 + 56*a^8*c^6*d^10*f^4 + 70*a^8*c^8*d^8*f^4 + 56*a^8*c^10*d^6*f^4 + 28*a^8*c^12*d^4*f^4 + 8*a^8*c^14*d^2
*f^4))^(1/2) + 4*a^4*c^2*d^6*f^2*((37157*c^2*d^24 - 2809*d^26 - c*d^25*16642i + c^3*d^23*33024i + 2990*c^4*d^2
2 + c^5*d^21*21076i + 322*c^6*d^20 + c^7*d^19*15752i - 5029*c^8*d^18 + c^9*d^17*3870i - 1255*c^10*d^16 + c^11*
d^15*1000i - 400*c^12*d^14)/(a^8*c^16*f^4 + a^8*d^16*f^4 + 8*a^8*c^2*d^14*f^4 + 28*a^8*c^4*d^12*f^4 + 56*a^8*c
^6*d^10*f^4 + 70*a^8*c^8*d^8*f^4 + 56*a^8*c^10*d^6*f^4 + 28*a^8*c^12*d^4*f^4 + 8*a^8*c^14*d^2*f^4))^(1/2) + 6*
a^4*c^4*d^4*f^2*((37157*c^2*d^24 - 2809*d^26 - c*d^25*16642i + c^3*d^23*33024i + 2990*c^4*d^22 + c^5*d^21*2107
6i + 322*c^6*d^20 + c^7*d^19*15752i - 5029*c^8*d^18 + c^9*d^17*3870i - 1255*c^10*d^16 + c^11*d^15*1000i - 400*
c^12*d^14)/(a^8*c^16*f^4 + a^8*d^16*f^4 + 8*a^8*c^2*d^14*f^4 + 28*a^8*c^4*d^12*f^4 + 56*a^8*c^6*d^10*f^4 + 70*
a^8*c^8*d^8*f^4 + 56*a^8*c^10*d^6*f^4 + 28*a^8*c^12*d^4*f^4 + 8*a^8*c^14*d^2*f^4))^(1/2) + 4*a^4*c^6*d^2*f^2*(
(37157*c^2*d^24 - 2809*d^26 - c*d^25*16642i + c^3*d^23*33024i + 2990*c^4*d^22 + c^5*d^21*21076i + 322*c^6*d^20
+ c^7*d^19*15752i - 5029*c^8*d^18 + c^9*d^17*3870i - 1255*c^10*d^16 + c^11*d^15*1000i - 400*c^12*d^14)/(a^8*c
^16*f^4 + a^8*d^16*f^4 + 8*a^8*c^2*d^14*f^4 + 28*a^8*c^4*d^12*f^4 + 56*a^8*c^6*d^10*f^4 + 70*a^8*c^8*d^8*f^4 +
56*a^8*c^10*d^6*f^4 + 28*a^8*c^12*d^4*f^4 + 8*a^8*c^14*d^2*f^4))^(1/2))/(512*a^4*d^14*f^2 + 2560*a^4*c^2*d^12
*f^2 + 5120*a^4*c^4*d^10*f^2 + 5120*a^4*c^6*d^8*f^2 + 2560*a^4*c^8*d^6*f^2 + 512*a^4*c^10*d^4*f^2))^(1/2) - a^
2*c^2*d^8*f*56i - 40*a^2*c^3*d^7*f - a^2*c^4*d^6*f*25i - 20*a^2*c^5*d^5*f - a^2*c^6*d^4*f*6i - 4*a^2*c^7*d^3*f
- 96*a^2*c*d^9*f)*(-(165*c*d^12 - d^13*45i + c^2*d^11*150i + 70*c^3*d^10 + c^4*d^9*95i + 73*c^5*d^8 + c^6*d^7
*52i + 32*c^7*d^6 + c^8*d^5*8i + 8*c^9*d^4 + a^4*c^8*f^2*((37157*c^2*d^24 - 2809*d^26 - c*d^25*16642i + c^3*d^
23*33024i + 2990*c^4*d^22 + c^5*d^21*21076i + 322*c^6*d^20 + c^7*d^19*15752i - 5029*c^8*d^18 + c^9*d^17*3870i
- 1255*c^10*d^16 + c^11*d^15*1000i - 400*c^12*d^14)/(a^8*c^16*f^4 + a^8*d^16*f^4 + 8*a^8*c^2*d^14*f^4 + 28*a^8
*c^4*d^12*f^4 + 56*a^8*c^6*d^10*f^4 + 70*a^8*c^8*d^8*f^4 + 56*a^8*c^10*d^6*f^4 + 28*a^8*c^12*d^4*f^4 + 8*a^8*c
^14*d^2*f^4))^(1/2) + a^4*d^8*f^2*((37157*c^2*d^24 - 2809*d^26 - c*d^25*16642i + c^3*d^23*33024i + 2990*c^4*d^
22 + c^5*d^21*21076i + 322*c^6*d^20 + c^7*d^19*15752i - 5029*c^8*d^18 + c^9*d^17*3870i - 1255*c^10*d^16 + c^11
*d^15*1000i - 400*c^12*d^14)/(a^8*c^16*f^4 + a^8*d^16*f^4 + 8*a^8*c^2*d^14*f^4 + 28*a^8*c^4*d^12*f^4 + 56*a^8*
c^6*d^10*f^4 + 70*a^8*c^8*d^8*f^4 + 56*a^8*c^10*d^6*f^4 + 28*a^8*c^12*d^4*f^4 + 8*a^8*c^14*d^2*f^4))^(1/2) + 4
*a^4*c^2*d^6*f^2*((37157*c^2*d^24 - 2809*d^26 - c*d^25*16642i + c^3*d^23*33024i + 2990*c^4*d^22 + c^5*d^21*210
76i + 322*c^6*d^20 + c^7*d^19*15752i - 5029*c^8*d^18 + c^9*d^17*3870i - 1255*c^10*d^16 + c^11*d^15*1000i - 400
*c^12*d^14)/(a^8*c^16*f^4 + a^8*d^16*f^4 + 8*a^8*c^2*d^14*f^4 + 28*a^8*c^4*d^12*f^4 + 56*a^8*c^6*d^10*f^4 + 70
*a^8*c^8*d^8*f^4 + 56*a^8*c^10*d^6*f^4 + 28*a^8*c^12*d^4*f^4 + 8*a^8*c^14*d^2*f^4))^(1/2) + 6*a^4*c^4*d^4*f^2*
((37157*c^2*d^24 - 2809*d^26 - c*d^25*16642i + c^3*d^23*33024i + 2990*c^4*d^22 + c^5*d^21*21076i + 322*c^6*d^2
0 + c^7*d^19*15752i - 5029*c^8*d^18 + c^9*d^17*3870i - 1255*c^10*d^16 + c^11*d^15*1000i - 400*c^12*d^14)/(a^8*
c^16*f^4 + a^8*d^16*f^4 + 8*a^8*c^2*d^14*f^4 + 28*a^8*c^4*d^12*f^4 + 56*a^8*c^6*d^10*f^4 + 70*a^8*c^8*d^8*f^4
+ 56*a^8*c^10*d^6*f^4 + 28*a^8*c^12*d^4*f^4 + 8*a^8*c^14*d^2*f^4))^(1/2) + 4*a^4*c^6*d^2*f^2*((37157*c^2*d^24
- 2809*d^26 - c*d^25*16642i + c^3*d^23*33024i + 2990*c^4*d^22 + c^5*d^21*21076i + 322*c^6*d^20 + c^7*d^19*1575
2i - 5029*c^8*d^18 + c^9*d^17*3870i - 1255*c^10*d^16 + c^11*d^15*1000i - 400*c^12*d^14)/(a^8*c^16*f^4 + a^8*d^
16*f^4 + 8*a^8*c^2*d^14*f^4 + 28*a^8*c^4*d^12*f^4 + 56*a^8*c^6*d^10*f^4 + 70*a^8*c^8*d^8*f^4 + 56*a^8*c^10*d^6
*f^4 + 28*a^8*c^12*d^4*f^4 + 8*a^8*c^14*d^2*f^4))^(1/2))/(512*a^4*d^14*f^2 + 2560*a^4*c^2*d^12*f^2 + 5120*a^4*
c^4*d^10*f^2 + 5120*a^4*c^6*d^8*f^2 + 2560*a^4*c^8*d^6*f^2 + 512*a^4*c^10*d^4*f^2))^(1/2) - log(a^2*d^10*f*35i
- ((-(165*c*d^12 - d^13*45i + c^2*d^11*150i + 70*c^3*d^10 + c^4*d^9*95i + 73*c^5*d^8 + c^6*d^7*52i + 32*c^7*d
^6 + c^8*d^5*8i + 8*c^9*d^4 - a^4*c^8*f^2*((37157*c^2*d^24 - 2809*d^26 - c*d^25*16642i + c^3*d^23*33024i + 299
0*c^4*d^22 + c^5*d^21*21076i + 322*c^6*d^20 + c^7*d^19*15752i - 5029*c^8*d^18 + c^9*d^17*3870i - 1255*c^10*d^1
6 + c^11*d^15*1000i - 400*c^12*d^14)/(a^8*c^16*f^4 + a^8*d^16*f^4 + 8*a^8*c^2*d^14*f^4 + 28*a^8*c^4*d^12*f^4 +
56*a^8*c^6*d^10*f^4 + 70*a^8*c^8*d^8*f^4 + 56*a^8*c^10*d^6*f^4 + 28*a^8*c^12*d^4*f^4 + 8*a^8*c^14*d^2*f^4))^(
1/2) - a^4*d^8*f^2*((37157*c^2*d^24 - 2809*d^26 - c*d^25*16642i + c^3*d^23*33024i + 2990*c^4*d^22 + c^5*d^21*2
1076i + 322*c^6*d^20 + c^7*d^19*15752i - 5029*c^8*d^18 + c^9*d^17*3870i - 1255*c^10*d^16 + c^11*d^15*1000i - 4
00*c^12*d^14)/(a^8*c^16*f^4 + a^8*d^16*f^4 + 8*a^8*c^2*d^14*f^4 + 28*a^8*c^4*d^12*f^4 + 56*a^8*c^6*d^10*f^4 +
70*a^8*c^8*d^8*f^4 + 56*a^8*c^10*d^6*f^4 + 28*a^8*c^12*d^4*f^4 + 8*a^8*c^14*d^2*f^4))^(1/2) - 4*a^4*c^2*d^6*f^
2*((37157*c^2*d^24 - 2809*d^26 - c*d^25*16642i + c^3*d^23*33024i + 2990*c^4*d^22 + c^5*d^21*21076i + 322*c^6*d
^20 + c^7*d^19*15752i - 5029*c^8*d^18 + c^9*d^17*3870i - 1255*c^10*d^16 + c^11*d^15*1000i - 400*c^12*d^14)/(a^
8*c^16*f^4 + a^8*d^16*f^4 + 8*a^8*c^2*d^14*f^4 + 28*a^8*c^4*d^12*f^4 + 56*a^8*c^6*d^10*f^4 + 70*a^8*c^8*d^8*f^
4 + 56*a^8*c^10*d^6*f^4 + 28*a^8*c^12*d^4*f^4 + 8*a^8*c^14*d^2*f^4))^(1/2) - 6*a^4*c^4*d^4*f^2*((37157*c^2*d^2
4 - 2809*d^26 - c*d^25*16642i + c^3*d^23*33024i + 2990*c^4*d^22 + c^5*d^21*21076i + 322*c^6*d^20 + c^7*d^19*15
752i - 5029*c^8*d^18 + c^9*d^17*3870i - 1255*c^10*d^16 + c^11*d^15*1000i - 400*c^12*d^14)/(a^8*c^16*f^4 + a^8*
d^16*f^4 + 8*a^8*c^2*d^14*f^4 + 28*a^8*c^4*d^12*f^4 + 56*a^8*c^6*d^10*f^4 + 70*a^8*c^8*d^8*f^4 + 56*a^8*c^10*d
^6*f^4 + 28*a^8*c^12*d^4*f^4 + 8*a^8*c^14*d^2*f^4))^(1/2) - 4*a^4*c^6*d^2*f^2*((37157*c^2*d^24 - 2809*d^26 - c
*d^25*16642i + c^3*d^23*33024i + 2990*c^4*d^22 + c^5*d^21*21076i + 322*c^6*d^20 + c^7*d^19*15752i - 5029*c^8*d
^18 + c^9*d^17*3870i - 1255*c^10*d^16 + c^11*d^15*1000i - 400*c^12*d^14)/(a^8*c^16*f^4 + a^8*d^16*f^4 + 8*a^8*
c^2*d^14*f^4 + 28*a^8*c^4*d^12*f^4 + 56*a^8*c^6*d^10*f^4 + 70*a^8*c^8*d^8*f^4 + 56*a^8*c^10*d^6*f^4 + 28*a^8*c
^12*d^4*f^4 + 8*a^8*c^14*d^2*f^4))^(1/2))/(512*a^4*d^14*f^2 + 2560*a^4*c^2*d^12*f^2 + 5120*a^4*c^4*d^10*f^2 +
5120*a^4*c^6*d^8*f^2 + 2560*a^4*c^8*d^6*f^2 + 512*a^4*c^10*d^4*f^2))^(1/2)*(1152*a^6*d^11*f^3 + a^6*c*d^10*f^3
*1664i - 8*(-(165*c*d^12 - d^13*45i + c^2*d^11*150i + 70*c^3*d^10 + c^4*d^9*95i + 73*c^5*d^8 + c^6*d^7*52i + 3
2*c^7*d^6 + c^8*d^5*8i + 8*c^9*d^4 - a^4*c^8*f^2*((37157*c^2*d^24 - 2809*d^26 - c*d^25*16642i + c^3*d^23*33024
i + 2990*c^4*d^22 + c^5*d^21*21076i + 322*c^6*d^20 + c^7*d^19*15752i - 5029*c^8*d^18 + c^9*d^17*3870i - 1255*c
^10*d^16 + c^11*d^15*1000i - 400*c^12*d^14)/(a^8*c^16*f^4 + a^8*d^16*f^4 + 8*a^8*c^2*d^14*f^4 + 28*a^8*c^4*d^1
2*f^4 + 56*a^8*c^6*d^10*f^4 + 70*a^8*c^8*d^8*f^4 + 56*a^8*c^10*d^6*f^4 + 28*a^8*c^12*d^4*f^4 + 8*a^8*c^14*d^2*
f^4))^(1/2) - a^4*d^8*f^2*((37157*c^2*d^24 - 2809*d^26 - c*d^25*16642i + c^3*d^23*33024i + 2990*c^4*d^22 + c^5
*d^21*21076i + 322*c^6*d^20 + c^7*d^19*15752i - 5029*c^8*d^18 + c^9*d^17*3870i - 1255*c^10*d^16 + c^11*d^15*10
00i - 400*c^12*d^14)/(a^8*c^16*f^4 + a^8*d^16*f^4 + 8*a^8*c^2*d^14*f^4 + 28*a^8*c^4*d^12*f^4 + 56*a^8*c^6*d^10
*f^4 + 70*a^8*c^8*d^8*f^4 + 56*a^8*c^10*d^6*f^4 + 28*a^8*c^12*d^4*f^4 + 8*a^8*c^14*d^2*f^4))^(1/2) - 4*a^4*c^2
*d^6*f^2*((37157*c^2*d^24 - 2809*d^26 - c*d^25*16642i + c^3*d^23*33024i + 2990*c^4*d^22 + c^5*d^21*21076i + 32
2*c^6*d^20 + c^7*d^19*15752i - 5029*c^8*d^18 + c^9*d^17*3870i - 1255*c^10*d^16 + c^11*d^15*1000i - 400*c^12*d^
14)/(a^8*c^16*f^4 + a^8*d^16*f^4 + 8*a^8*c^2*d^14*f^4 + 28*a^8*c^4*d^12*f^4 + 56*a^8*c^6*d^10*f^4 + 70*a^8*c^8
*d^8*f^4 + 56*a^8*c^10*d^6*f^4 + 28*a^8*c^12*d^4*f^4 + 8*a^8*c^14*d^2*f^4))^(1/2) - 6*a^4*c^4*d^4*f^2*((37157*
c^2*d^24 - 2809*d^26 - c*d^25*16642i + c^3*d^23*33024i + 2990*c^4*d^22 + c^5*d^21*21076i + 322*c^6*d^20 + c^7*
d^19*15752i - 5029*c^8*d^18 + c^9*d^17*3870i - 1255*c^10*d^16 + c^11*d^15*1000i - 400*c^12*d^14)/(a^8*c^16*f^4
+ a^8*d^16*f^4 + 8*a^8*c^2*d^14*f^4 + 28*a^8*c^4*d^12*f^4 + 56*a^8*c^6*d^10*f^4 + 70*a^8*c^8*d^8*f^4 + 56*a^8
*c^10*d^6*f^4 + 28*a^8*c^12*d^4*f^4 + 8*a^8*c^14*d^2*f^4))^(1/2) - 4*a^4*c^6*d^2*f^2*((37157*c^2*d^24 - 2809*d
^26 - c*d^25*16642i + c^3*d^23*33024i + 2990*c^4*d^22 + c^5*d^21*21076i + 322*c^6*d^20 + c^7*d^19*15752i - 502
9*c^8*d^18 + c^9*d^17*3870i - 1255*c^10*d^16 + c^11*d^15*1000i - 400*c^12*d^14)/(a^8*c^16*f^4 + a^8*d^16*f^4 +
8*a^8*c^2*d^14*f^4 + 28*a^8*c^4*d^12*f^4 + 56*a^8*c^6*d^10*f^4 + 70*a^8*c^8*d^8*f^4 + 56*a^8*c^10*d^6*f^4 + 2
8*a^8*c^12*d^4*f^4 + 8*a^8*c^14*d^2*f^4))^(1/2))/(512*a^4*d^14*f^2 + 2560*a^4*c^2*d^12*f^2 + 5120*a^4*c^4*d^10
*f^2 + 5120*a^4*c^6*d^8*f^2 + 2560*a^4*c^8*d^6*f^2 + 512*a^4*c^10*d^4*f^2))^(1/2)*(c + d*tan(e + f*x))^(1/2)*(
512*a^4*c*d^6*f^2 - a^4*c^2*d^5*f^2*2048i - 3072*a^4*c^3*d^4*f^2 + a^4*c^4*d^3*f^2*2048i + 512*a^4*c^5*d^2*f^2
)*(a^4*c^4*f^2 + a^4*d^4*f^2 + a^4*c*d^3*f^2*4i - a^4*c^3*d*f^2*4i - 6*a^4*c^2*d^2*f^2) + 2176*a^6*c^2*d^9*f^3
+ a^6*c^3*d^8*f^3*3456i + 1152*a^6*c^4*d^7*f^3 + a^6*c^5*d^6*f^3*1920i + 384*a^6*c^6*d^5*f^3 + a^6*c^7*d^4*f^
3*128i + 256*a^6*c^8*d^3*f^3) - 8*(c + d*tan(e + f*x))^(1/2)*(a^4*c^4*f^2 + a^4*d^4*f^2 + a^4*c*d^3*f^2*4i - a
^4*c^3*d*f^2*4i - 6*a^4*c^2*d^2*f^2)*(53*d^6 - c*d^5*100i - 88*c^2*d^4 + c^3*d^3*40i + 8*c^4*d^2))*(-(165*c*d^
12 - d^13*45i + c^2*d^11*150i + 70*c^3*d^10 + c^4*d^9*95i + 73*c^5*d^8 + c^6*d^7*52i + 32*c^7*d^6 + c^8*d^5*8i
+ 8*c^9*d^4 - a^4*c^8*f^2*((37157*c^2*d^24 - 2809*d^26 - c*d^25*16642i + c^3*d^23*33024i + 2990*c^4*d^22 + c^
5*d^21*21076i + 322*c^6*d^20 + c^7*d^19*15752i - 5029*c^8*d^18 + c^9*d^17*3870i - 1255*c^10*d^16 + c^11*d^15*1
000i - 400*c^12*d^14)/(a^8*c^16*f^4 + a^8*d^16*f^4 + 8*a^8*c^2*d^14*f^4 + 28*a^8*c^4*d^12*f^4 + 56*a^8*c^6*d^1
0*f^4 + 70*a^8*c^8*d^8*f^4 + 56*a^8*c^10*d^6*f^4 + 28*a^8*c^12*d^4*f^4 + 8*a^8*c^14*d^2*f^4))^(1/2) - a^4*d^8*
f^2*((37157*c^2*d^24 - 2809*d^26 - c*d^25*16642i + c^3*d^23*33024i + 2990*c^4*d^22 + c^5*d^21*21076i + 322*c^6
*d^20 + c^7*d^19*15752i - 5029*c^8*d^18 + c^9*d^17*3870i - 1255*c^10*d^16 + c^11*d^15*1000i - 400*c^12*d^14)/(
a^8*c^16*f^4 + a^8*d^16*f^4 + 8*a^8*c^2*d^14*f^4 + 28*a^8*c^4*d^12*f^4 + 56*a^8*c^6*d^10*f^4 + 70*a^8*c^8*d^8*
f^4 + 56*a^8*c^10*d^6*f^4 + 28*a^8*c^12*d^4*f^4 + 8*a^8*c^14*d^2*f^4))^(1/2) - 4*a^4*c^2*d^6*f^2*((37157*c^2*d
^24 - 2809*d^26 - c*d^25*16642i + c^3*d^23*33024i + 2990*c^4*d^22 + c^5*d^21*21076i + 322*c^6*d^20 + c^7*d^19*
15752i - 5029*c^8*d^18 + c^9*d^17*3870i - 1255*c^10*d^16 + c^11*d^15*1000i - 400*c^12*d^14)/(a^8*c^16*f^4 + a^
8*d^16*f^4 + 8*a^8*c^2*d^14*f^4 + 28*a^8*c^4*d^12*f^4 + 56*a^8*c^6*d^10*f^4 + 70*a^8*c^8*d^8*f^4 + 56*a^8*c^10
*d^6*f^4 + 28*a^8*c^12*d^4*f^4 + 8*a^8*c^14*d^2*f^4))^(1/2) - 6*a^4*c^4*d^4*f^2*((37157*c^2*d^24 - 2809*d^26 -
c*d^25*16642i + c^3*d^23*33024i + 2990*c^4*d^22 + c^5*d^21*21076i + 322*c^6*d^20 + c^7*d^19*15752i - 5029*c^8
*d^18 + c^9*d^17*3870i - 1255*c^10*d^16 + c^11*d^15*1000i - 400*c^12*d^14)/(a^8*c^16*f^4 + a^8*d^16*f^4 + 8*a^
8*c^2*d^14*f^4 + 28*a^8*c^4*d^12*f^4 + 56*a^8*c^6*d^10*f^4 + 70*a^8*c^8*d^8*f^4 + 56*a^8*c^10*d^6*f^4 + 28*a^8
*c^12*d^4*f^4 + 8*a^8*c^14*d^2*f^4))^(1/2) - 4*a^4*c^6*d^2*f^2*((37157*c^2*d^24 - 2809*d^26 - c*d^25*16642i +
c^3*d^23*33024i + 2990*c^4*d^22 + c^5*d^21*21076i + 322*c^6*d^20 + c^7*d^19*15752i - 5029*c^8*d^18 + c^9*d^17*
3870i - 1255*c^10*d^16 + c^11*d^15*1000i - 400*c^12*d^14)/(a^8*c^16*f^4 + a^8*d^16*f^4 + 8*a^8*c^2*d^14*f^4 +
28*a^8*c^4*d^12*f^4 + 56*a^8*c^6*d^10*f^4 + 70*a^8*c^8*d^8*f^4 + 56*a^8*c^10*d^6*f^4 + 28*a^8*c^12*d^4*f^4 + 8
*a^8*c^14*d^2*f^4))^(1/2))/(512*a^4*d^14*f^2 + 2560*a^4*c^2*d^12*f^2 + 5120*a^4*c^4*d^10*f^2 + 5120*a^4*c^6*d^
8*f^2 + 2560*a^4*c^8*d^6*f^2 + 512*a^4*c^10*d^4*f^2))^(1/2) - a^2*c^2*d^8*f*56i - 40*a^2*c^3*d^7*f - a^2*c^4*d
^6*f*25i - 20*a^2*c^5*d^5*f - a^2*c^6*d^4*f*6i - 4*a^2*c^7*d^3*f - 96*a^2*c*d^9*f)*(-(165*c*d^12 - d^13*45i +
c^2*d^11*150i + 70*c^3*d^10 + c^4*d^9*95i + 73*c^5*d^8 + c^6*d^7*52i + 32*c^7*d^6 + c^8*d^5*8i + 8*c^9*d^4 - a
^4*c^8*f^2*((37157*c^2*d^24 - 2809*d^26 - c*d^25*16642i + c^3*d^23*33024i + 2990*c^4*d^22 + c^5*d^21*21076i +
322*c^6*d^20 + c^7*d^19*15752i - 5029*c^8*d^18 + c^9*d^17*3870i - 1255*c^10*d^16 + c^11*d^15*1000i - 400*c^12*
d^14)/(a^8*c^16*f^4 + a^8*d^16*f^4 + 8*a^8*c^2*d^14*f^4 + 28*a^8*c^4*d^12*f^4 + 56*a^8*c^6*d^10*f^4 + 70*a^8*c
^8*d^8*f^4 + 56*a^8*c^10*d^6*f^4 + 28*a^8*c^12*d^4*f^4 + 8*a^8*c^14*d^2*f^4))^(1/2) - a^4*d^8*f^2*((37157*c^2*
d^24 - 2809*d^26 - c*d^25*16642i + c^3*d^23*33024i + 2990*c^4*d^22 + c^5*d^21*21076i + 322*c^6*d^20 + c^7*d^19
*15752i - 5029*c^8*d^18 + c^9*d^17*3870i - 1255*c^10*d^16 + c^11*d^15*1000i - 400*c^12*d^14)/(a^8*c^16*f^4 + a
^8*d^16*f^4 + 8*a^8*c^2*d^14*f^4 + 28*a^8*c^4*d^12*f^4 + 56*a^8*c^6*d^10*f^4 + 70*a^8*c^8*d^8*f^4 + 56*a^8*c^1
0*d^6*f^4 + 28*a^8*c^12*d^4*f^4 + 8*a^8*c^14*d^2*f^4))^(1/2) - 4*a^4*c^2*d^6*f^2*((37157*c^2*d^24 - 2809*d^26
- c*d^25*16642i + c^3*d^23*33024i + 2990*c^4*d^22 + c^5*d^21*21076i + 322*c^6*d^20 + c^7*d^19*15752i - 5029*c^
8*d^18 + c^9*d^17*3870i - 1255*c^10*d^16 + c^11*d^15*1000i - 400*c^12*d^14)/(a^8*c^16*f^4 + a^8*d^16*f^4 + 8*a
^8*c^2*d^14*f^4 + 28*a^8*c^4*d^12*f^4 + 56*a^8*c^6*d^10*f^4 + 70*a^8*c^8*d^8*f^4 + 56*a^8*c^10*d^6*f^4 + 28*a^
8*c^12*d^4*f^4 + 8*a^8*c^14*d^2*f^4))^(1/2) - 6*a^4*c^4*d^4*f^2*((37157*c^2*d^24 - 2809*d^26 - c*d^25*16642i +
c^3*d^23*33024i + 2990*c^4*d^22 + c^5*d^21*21076i + 322*c^6*d^20 + c^7*d^19*15752i - 5029*c^8*d^18 + c^9*d^17
*3870i - 1255*c^10*d^16 + c^11*d^15*1000i - 400*c^12*d^14)/(a^8*c^16*f^4 + a^8*d^16*f^4 + 8*a^8*c^2*d^14*f^4 +
28*a^8*c^4*d^12*f^4 + 56*a^8*c^6*d^10*f^4 + 70*a^8*c^8*d^8*f^4 + 56*a^8*c^10*d^6*f^4 + 28*a^8*c^12*d^4*f^4 +
8*a^8*c^14*d^2*f^4))^(1/2) - 4*a^4*c^6*d^2*f^2*((37157*c^2*d^24 - 2809*d^26 - c*d^25*16642i + c^3*d^23*33024i
+ 2990*c^4*d^22 + c^5*d^21*21076i + 322*c^6*d^20 + c^7*d^19*15752i - 5029*c^8*d^18 + c^9*d^17*3870i - 1255*c^1
0*d^16 + c^11*d^15*1000i - 400*c^12*d^14)/(a^8*c^16*f^4 + a^8*d^16*f^4 + 8*a^8*c^2*d^14*f^4 + 28*a^8*c^4*d^12*
f^4 + 56*a^8*c^6*d^10*f^4 + 70*a^8*c^8*d^8*f^4 + 56*a^8*c^10*d^6*f^4 + 28*a^8*c^12*d^4*f^4 + 8*a^8*c^14*d^2*f^
4))^(1/2))/(512*a^4*d^14*f^2 + 2560*a^4*c^2*d^12*f^2 + 5120*a^4*c^4*d^10*f^2 + 5120*a^4*c^6*d^8*f^2 + 2560*a^4
*c^8*d^6*f^2 + 512*a^4*c^10*d^4*f^2))^(1/2) - (((c*d*2i - 7*d^2)*(c + d*tan(e + f*x))^(1/2))/(8*a^2*f*(c*1i -
d)) - (d*(2*c + d*5i)*(c + d*tan(e + f*x))^(3/2))/(8*a^2*f*(c*d*2i + c^2 - d^2)))/(c*d*2i - (2*c + d*2i)*(c +
d*tan(e + f*x)) + (c + d*tan(e + f*x))^2 + c^2 - d^2) + log(a^2*d^10*f*35i - (((c*d^8*15i - 45*d^9 - 60*c^2*d^
7 + c^3*d^6*80i + 40*c^4*d^5 - c^5*d^4*8i + a^4*c^4*f^2*(((165*c*d^12 + 70*c^3*d^10 + 73*c^5*d^8 + 32*c^7*d^6
+ 8*c^9*d^4)/(a^4*c^8*f^2 + a^4*d^8*f^2 + 4*a^4*c^2*d^6*f^2 + 6*a^4*c^4*d^4*f^2 + 4*a^4*c^6*d^2*f^2) + ((150*c
^2*d^11 - 45*d^13 + 95*c^4*d^9 + 52*c^6*d^7 + 8*c^8*d^5)*1i)/(a^4*c^8*f^2 + a^4*d^8*f^2 + 4*a^4*c^2*d^6*f^2 +
6*a^4*c^4*d^4*f^2 + 4*a^4*c^6*d^2*f^2))^2 - 4*(256*d^6 + 256*c^2*d^4)*((((7*c*d^11)/4 + (19*c^3*d^9)/16 + (11*
c^5*d^7)/16 + (c^7*d^5)/8)*1i)/(a^8*c^8*f^4 + a^8*d^8*f^4 + 4*a^8*c^2*d^6*f^4 + 6*a^8*c^4*d^4*f^4 + 4*a^8*c^6*
d^2*f^4) + ((49*d^12)/64 - (11*c^2*d^10)/32 + (5*c^4*d^8)/64 + (c^6*d^6)/8 + (c^8*d^4)/16)/(a^8*c^8*f^4 + a^8*
d^8*f^4 + 4*a^8*c^2*d^6*f^4 + 6*a^8*c^4*d^4*f^4 + 4*a^8*c^6*d^2*f^4)))^(1/2)*1i + a^4*d^4*f^2*(((165*c*d^12 +
70*c^3*d^10 + 73*c^5*d^8 + 32*c^7*d^6 + 8*c^9*d^4)/(a^4*c^8*f^2 + a^4*d^8*f^2 + 4*a^4*c^2*d^6*f^2 + 6*a^4*c^4*
d^4*f^2 + 4*a^4*c^6*d^2*f^2) + ((150*c^2*d^11 - 45*d^13 + 95*c^4*d^9 + 52*c^6*d^7 + 8*c^8*d^5)*1i)/(a^4*c^8*f^
2 + a^4*d^8*f^2 + 4*a^4*c^2*d^6*f^2 + 6*a^4*c^4*d^4*f^2 + 4*a^4*c^6*d^2*f^2))^2 - 4*(256*d^6 + 256*c^2*d^4)*((
((7*c*d^11)/4 + (19*c^3*d^9)/16 + (11*c^5*d^7)/16 + (c^7*d^5)/8)*1i)/(a^8*c^8*f^4 + a^8*d^8*f^4 + 4*a^8*c^2*d^
6*f^4 + 6*a^8*c^4*d^4*f^4 + 4*a^8*c^6*d^2*f^4) + ((49*d^12)/64 - (11*c^2*d^10)/32 + (5*c^4*d^8)/64 + (c^6*d^6)
/8 + (c^8*d^4)/16)/(a^8*c^8*f^4 + a^8*d^8*f^4 + 4*a^8*c^2*d^6*f^4 + 6*a^8*c^4*d^4*f^4 + 4*a^8*c^6*d^2*f^4)))^(
1/2)*1i - a^4*c^2*d^2*f^2*(((165*c*d^12 + 70*c^3*d^10 + 73*c^5*d^8 + 32*c^7*d^6 + 8*c^9*d^4)/(a^4*c^8*f^2 + a^
4*d^8*f^2 + 4*a^4*c^2*d^6*f^2 + 6*a^4*c^4*d^4*f^2 + 4*a^4*c^6*d^2*f^2) + ((150*c^2*d^11 - 45*d^13 + 95*c^4*d^9
+ 52*c^6*d^7 + 8*c^8*d^5)*1i)/(a^4*c^8*f^2 + a^4*d^8*f^2 + 4*a^4*c^2*d^6*f^2 + 6*a^4*c^4*d^4*f^2 + 4*a^4*c^6*
d^2*f^2))^2 - 4*(256*d^6 + 256*c^2*d^4)*((((7*c*d^11)/4 + (19*c^3*d^9)/16 + (11*c^5*d^7)/16 + (c^7*d^5)/8)*1i)
/(a^8*c^8*f^4 + a^8*d^8*f^4 + 4*a^8*c^2*d^6*f^4 + 6*a^8*c^4*d^4*f^4 + 4*a^8*c^6*d^2*f^4) + ((49*d^12)/64 - (11
*c^2*d^10)/32 + (5*c^4*d^8)/64 + (c^6*d^6)/8 + (c^8*d^4)/16)/(a^8*c^8*f^4 + a^8*d^8*f^4 + 4*a^8*c^2*d^6*f^4 +
6*a^8*c^4*d^4*f^4 + 4*a^8*c^6*d^2*f^4)))^(1/2)*6i + 4*a^4*c*d^3*f^2*(((165*c*d^12 + 70*c^3*d^10 + 73*c^5*d^8 +
32*c^7*d^6 + 8*c^9*d^4)/(a^4*c^8*f^2 + a^4*d^8*f^2 + 4*a^4*c^2*d^6*f^2 + 6*a^4*c^4*d^4*f^2 + 4*a^4*c^6*d^2*f^
2) + ((150*c^2*d^11 - 45*d^13 + 95*c^4*d^9 + 52*c^6*d^7 + 8*c^8*d^5)*1i)/(a^4*c^8*f^2 + a^4*d^8*f^2 + 4*a^4*c^
2*d^6*f^2 + 6*a^4*c^4*d^4*f^2 + 4*a^4*c^6*d^2*f^2))^2 - 4*(256*d^6 + 256*c^2*d^4)*((((7*c*d^11)/4 + (19*c^3*d^
9)/16 + (11*c^5*d^7)/16 + (c^7*d^5)/8)*1i)/(a^8*c^8*f^4 + a^8*d^8*f^4 + 4*a^8*c^2*d^6*f^4 + 6*a^8*c^4*d^4*f^4
+ 4*a^8*c^6*d^2*f^4) + ((49*d^12)/64 - (11*c^2*d^10)/32 + (5*c^4*d^8)/64 + (c^6*d^6)/8 + (c^8*d^4)/16)/(a^8*c^
8*f^4 + a^8*d^8*f^4 + 4*a^8*c^2*d^6*f^4 + 6*a^8*c^4*d^4*f^4 + 4*a^8*c^6*d^2*f^4)))^(1/2) - 4*a^4*c^3*d*f^2*(((
165*c*d^12 + 70*c^3*d^10 + 73*c^5*d^8 + 32*c^7*d^6 + 8*c^9*d^4)/(a^4*c^8*f^2 + a^4*d^8*f^2 + 4*a^4*c^2*d^6*f^2
+ 6*a^4*c^4*d^4*f^2 + 4*a^4*c^6*d^2*f^2) + ((150*c^2*d^11 - 45*d^13 + 95*c^4*d^9 + 52*c^6*d^7 + 8*c^8*d^5)*1i
)/(a^4*c^8*f^2 + a^4*d^8*f^2 + 4*a^4*c^2*d^6*f^2 + 6*a^4*c^4*d^4*f^2 + 4*a^4*c^6*d^2*f^2))^2 - 4*(256*d^6 + 25
6*c^2*d^4)*((((7*c*d^11)/4 + (19*c^3*d^9)/16 + (11*c^5*d^7)/16 + (c^7*d^5)/8)*1i)/(a^8*c^8*f^4 + a^8*d^8*f^4 +
4*a^8*c^2*d^6*f^4 + 6*a^8*c^4*d^4*f^4 + 4*a^8*c^6*d^2*f^4) + ((49*d^12)/64 - (11*c^2*d^10)/32 + (5*c^4*d^8)/6
4 + (c^6*d^6)/8 + (c^8*d^4)/16)/(a^8*c^8*f^4 + a^8*d^8*f^4 + 4*a^8*c^2*d^6*f^4 + 6*a^8*c^4*d^4*f^4 + 4*a^8*c^6
*d^2*f^4)))^(1/2))/(512*(d^6 + c^2*d^4)*(a^4*c^4*f^2*1i + a^4*d^4*f^2*1i + 4*a^4*c*d^3*f^2 - 4*a^4*c^3*d*f^2 -
a^4*c^2*d^2*f^2*6i)))^(1/2)*(1152*a^6*d^11*f^3 + a^6*c*d^10*f^3*1664i + 8*(c + d*tan(e + f*x))^(1/2)*((c*d^8*
15i - 45*d^9 - 60*c^2*d^7 + c^3*d^6*80i + 40*c^4*d^5 - c^5*d^4*8i + a^4*c^4*f^2*(((165*c*d^12 + 70*c^3*d^10 +
73*c^5*d^8 + 32*c^7*d^6 + 8*c^9*d^4)/(a^4*c^8*f^2 + a^4*d^8*f^2 + 4*a^4*c^2*d^6*f^2 + 6*a^4*c^4*d^4*f^2 + 4*a^
4*c^6*d^2*f^2) + ((150*c^2*d^11 - 45*d^13 + 95*c^4*d^9 + 52*c^6*d^7 + 8*c^8*d^5)*1i)/(a^4*c^8*f^2 + a^4*d^8*f^
2 + 4*a^4*c^2*d^6*f^2 + 6*a^4*c^4*d^4*f^2 + 4*a^4*c^6*d^2*f^2))^2 - 4*(256*d^6 + 256*c^2*d^4)*((((7*c*d^11)/4
+ (19*c^3*d^9)/16 + (11*c^5*d^7)/16 + (c^7*d^5)/8)*1i)/(a^8*c^8*f^4 + a^8*d^8*f^4 + 4*a^8*c^2*d^6*f^4 + 6*a^8*
c^4*d^4*f^4 + 4*a^8*c^6*d^2*f^4) + ((49*d^12)/64 - (11*c^2*d^10)/32 + (5*c^4*d^8)/64 + (c^6*d^6)/8 + (c^8*d^4)
/16)/(a^8*c^8*f^4 + a^8*d^8*f^4 + 4*a^8*c^2*d^6*f^4 + 6*a^8*c^4*d^4*f^4 + 4*a^8*c^6*d^2*f^4)))^(1/2)*1i + a^4*
d^4*f^2*(((165*c*d^12 + 70*c^3*d^10 + 73*c^5*d^8 + 32*c^7*d^6 + 8*c^9*d^4)/(a^4*c^8*f^2 + a^4*d^8*f^2 + 4*a^4*
c^2*d^6*f^2 + 6*a^4*c^4*d^4*f^2 + 4*a^4*c^6*d^2*f^2) + ((150*c^2*d^11 - 45*d^13 + 95*c^4*d^9 + 52*c^6*d^7 + 8*
c^8*d^5)*1i)/(a^4*c^8*f^2 + a^4*d^8*f^2 + 4*a^4*c^2*d^6*f^2 + 6*a^4*c^4*d^4*f^2 + 4*a^4*c^6*d^2*f^2))^2 - 4*(2
56*d^6 + 256*c^2*d^4)*((((7*c*d^11)/4 + (19*c^3*d^9)/16 + (11*c^5*d^7)/16 + (c^7*d^5)/8)*1i)/(a^8*c^8*f^4 + a^
8*d^8*f^4 + 4*a^8*c^2*d^6*f^4 + 6*a^8*c^4*d^4*f^4 + 4*a^8*c^6*d^2*f^4) + ((49*d^12)/64 - (11*c^2*d^10)/32 + (5
*c^4*d^8)/64 + (c^6*d^6)/8 + (c^8*d^4)/16)/(a^8*c^8*f^4 + a^8*d^8*f^4 + 4*a^8*c^2*d^6*f^4 + 6*a^8*c^4*d^4*f^4
+ 4*a^8*c^6*d^2*f^4)))^(1/2)*1i - a^4*c^2*d^2*f^2*(((165*c*d^12 + 70*c^3*d^10 + 73*c^5*d^8 + 32*c^7*d^6 + 8*c^
9*d^4)/(a^4*c^8*f^2 + a^4*d^8*f^2 + 4*a^4*c^2*d^6*f^2 + 6*a^4*c^4*d^4*f^2 + 4*a^4*c^6*d^2*f^2) + ((150*c^2*d^1
1 - 45*d^13 + 95*c^4*d^9 + 52*c^6*d^7 + 8*c^8*d^5)*1i)/(a^4*c^8*f^2 + a^4*d^8*f^2 + 4*a^4*c^2*d^6*f^2 + 6*a^4*
c^4*d^4*f^2 + 4*a^4*c^6*d^2*f^2))^2 - 4*(256*d^6 + 256*c^2*d^4)*((((7*c*d^11)/4 + (19*c^3*d^9)/16 + (11*c^5*d^
7)/16 + (c^7*d^5)/8)*1i)/(a^8*c^8*f^4 + a^8*d^8*f^4 + 4*a^8*c^2*d^6*f^4 + 6*a^8*c^4*d^4*f^4 + 4*a^8*c^6*d^2*f^
4) + ((49*d^12)/64 - (11*c^2*d^10)/32 + (5*c^4*d^8)/64 + (c^6*d^6)/8 + (c^8*d^4)/16)/(a^8*c^8*f^4 + a^8*d^8*f^
4 + 4*a^8*c^2*d^6*f^4 + 6*a^8*c^4*d^4*f^4 + 4*a^8*c^6*d^2*f^4)))^(1/2)*6i + 4*a^4*c*d^3*f^2*(((165*c*d^12 + 70
*c^3*d^10 + 73*c^5*d^8 + 32*c^7*d^6 + 8*c^9*d^4)/(a^4*c^8*f^2 + a^4*d^8*f^2 + 4*a^4*c^2*d^6*f^2 + 6*a^4*c^4*d^
4*f^2 + 4*a^4*c^6*d^2*f^2) + ((150*c^2*d^11 - 45*d^13 + 95*c^4*d^9 + 52*c^6*d^7 + 8*c^8*d^5)*1i)/(a^4*c^8*f^2
+ a^4*d^8*f^2 + 4*a^4*c^2*d^6*f^2 + 6*a^4*c^4*d^4*f^2 + 4*a^4*c^6*d^2*f^2))^2 - 4*(256*d^6 + 256*c^2*d^4)*((((
7*c*d^11)/4 + (19*c^3*d^9)/16 + (11*c^5*d^7)/16 + (c^7*d^5)/8)*1i)/(a^8*c^8*f^4 + a^8*d^8*f^4 + 4*a^8*c^2*d^6*
f^4 + 6*a^8*c^4*d^4*f^4 + 4*a^8*c^6*d^2*f^4) + ((49*d^12)/64 - (11*c^2*d^10)/32 + (5*c^4*d^8)/64 + (c^6*d^6)/8
+ (c^8*d^4)/16)/(a^8*c^8*f^4 + a^8*d^8*f^4 + 4*a^8*c^2*d^6*f^4 + 6*a^8*c^4*d^4*f^4 + 4*a^8*c^6*d^2*f^4)))^(1/
2) - 4*a^4*c^3*d*f^2*(((165*c*d^12 + 70*c^3*d^10 + 73*c^5*d^8 + 32*c^7*d^6 + 8*c^9*d^4)/(a^4*c^8*f^2 + a^4*d^8
*f^2 + 4*a^4*c^2*d^6*f^2 + 6*a^4*c^4*d^4*f^2 + 4*a^4*c^6*d^2*f^2) + ((150*c^2*d^11 - 45*d^13 + 95*c^4*d^9 + 52
*c^6*d^7 + 8*c^8*d^5)*1i)/(a^4*c^8*f^2 + a^4*d^8*f^2 + 4*a^4*c^2*d^6*f^2 + 6*a^4*c^4*d^4*f^2 + 4*a^4*c^6*d^2*f
^2))^2 - 4*(256*d^6 + 256*c^2*d^4)*((((7*c*d^11)/4 + (19*c^3*d^9)/16 + (11*c^5*d^7)/16 + (c^7*d^5)/8)*1i)/(a^8
*c^8*f^4 + a^8*d^8*f^4 + 4*a^8*c^2*d^6*f^4 + 6*a^8*c^4*d^4*f^4 + 4*a^8*c^6*d^2*f^4) + ((49*d^12)/64 - (11*c^2*
d^10)/32 + (5*c^4*d^8)/64 + (c^6*d^6)/8 + (c^8*d^4)/16)/(a^8*c^8*f^4 + a^8*d^8*f^4 + 4*a^8*c^2*d^6*f^4 + 6*a^8
*c^4*d^4*f^4 + 4*a^8*c^6*d^2*f^4)))^(1/2))/(512*(d^6 + c^2*d^4)*(a^4*c^4*f^2*1i + a^4*d^4*f^2*1i + 4*a^4*c*d^3
*f^2 - 4*a^4*c^3*d*f^2 - a^4*c^2*d^2*f^2*6i)))^(1/2)*(512*a^4*c*d^6*f^2 - a^4*c^2*d^5*f^2*2048i - 3072*a^4*c^3
*d^4*f^2 + a^4*c^4*d^3*f^2*2048i + 512*a^4*c^5*d^2*f^2)*(a^4*c^4*f^2 + a^4*d^4*f^2 + a^4*c*d^3*f^2*4i - a^4*c^
3*d*f^2*4i - 6*a^4*c^2*d^2*f^2) + 2176*a^6*c^2*d^9*f^3 + a^6*c^3*d^8*f^3*3456i + 1152*a^6*c^4*d^7*f^3 + a^6*c^
5*d^6*f^3*1920i + 384*a^6*c^6*d^5*f^3 + a^6*c^7*d^4*f^3*128i + 256*a^6*c^8*d^3*f^3) + 8*(c + d*tan(e + f*x))^(
1/2)*(a^4*c^4*f^2 + a^4*d^4*f^2 + a^4*c*d^3*f^2*4i - a^4*c^3*d*f^2*4i - 6*a^4*c^2*d^2*f^2)*(53*d^6 - c*d^5*100
i - 88*c^2*d^4 + c^3*d^3*40i + 8*c^4*d^2))*((c*d^8*15i - 45*d^9 - 60*c^2*d^7 + c^3*d^6*80i + 40*c^4*d^5 - c^5*
d^4*8i + a^4*c^4*f^2*(((165*c*d^12 + 70*c^3*d^10 + 73*c^5*d^8 + 32*c^7*d^6 + 8*c^9*d^4)/(a^4*c^8*f^2 + a^4*d^8
*f^2 + 4*a^4*c^2*d^6*f^2 + 6*a^4*c^4*d^4*f^2 + 4*a^4*c^6*d^2*f^2) + ((150*c^2*d^11 - 45*d^13 + 95*c^4*d^9 + 52
*c^6*d^7 + 8*c^8*d^5)*1i)/(a^4*c^8*f^2 + a^4*d^8*f^2 + 4*a^4*c^2*d^6*f^2 + 6*a^4*c^4*d^4*f^2 + 4*a^4*c^6*d^2*f
^2))^2 - 4*(256*d^6 + 256*c^2*d^4)*((((7*c*d^11)/4 + (19*c^3*d^9)/16 + (11*c^5*d^7)/16 + (c^7*d^5)/8)*1i)/(a^8
*c^8*f^4 + a^8*d^8*f^4 + 4*a^8*c^2*d^6*f^4 + 6*a^8*c^4*d^4*f^4 + 4*a^8*c^6*d^2*f^4) + ((49*d^12)/64 - (11*c^2*
d^10)/32 + (5*c^4*d^8)/64 + (c^6*d^6)/8 + (c^8*d^4)/16)/(a^8*c^8*f^4 + a^8*d^8*f^4 + 4*a^8*c^2*d^6*f^4 + 6*a^8
*c^4*d^4*f^4 + 4*a^8*c^6*d^2*f^4)))^(1/2)*1i + a^4*d^4*f^2*(((165*c*d^12 + 70*c^3*d^10 + 73*c^5*d^8 + 32*c^7*d
^6 + 8*c^9*d^4)/(a^4*c^8*f^2 + a^4*d^8*f^2 + 4*a^4*c^2*d^6*f^2 + 6*a^4*c^4*d^4*f^2 + 4*a^4*c^6*d^2*f^2) + ((15
0*c^2*d^11 - 45*d^13 + 95*c^4*d^9 + 52*c^6*d^7 + 8*c^8*d^5)*1i)/(a^4*c^8*f^2 + a^4*d^8*f^2 + 4*a^4*c^2*d^6*f^2
+ 6*a^4*c^4*d^4*f^2 + 4*a^4*c^6*d^2*f^2))^2 - 4*(256*d^6 + 256*c^2*d^4)*((((7*c*d^11)/4 + (19*c^3*d^9)/16 + (
11*c^5*d^7)/16 + (c^7*d^5)/8)*1i)/(a^8*c^8*f^4 + a^8*d^8*f^4 + 4*a^8*c^2*d^6*f^4 + 6*a^8*c^4*d^4*f^4 + 4*a^8*c
^6*d^2*f^4) + ((49*d^12)/64 - (11*c^2*d^10)/32 + (5*c^4*d^8)/64 + (c^6*d^6)/8 + (c^8*d^4)/16)/(a^8*c^8*f^4 + a
^8*d^8*f^4 + 4*a^8*c^2*d^6*f^4 + 6*a^8*c^4*d^4*f^4 + 4*a^8*c^6*d^2*f^4)))^(1/2)*1i - a^4*c^2*d^2*f^2*(((165*c*
d^12 + 70*c^3*d^10 + 73*c^5*d^8 + 32*c^7*d^6 + 8*c^9*d^4)/(a^4*c^8*f^2 + a^4*d^8*f^2 + 4*a^4*c^2*d^6*f^2 + 6*a
^4*c^4*d^4*f^2 + 4*a^4*c^6*d^2*f^2) + ((150*c^2*d^11 - 45*d^13 + 95*c^4*d^9 + 52*c^6*d^7 + 8*c^8*d^5)*1i)/(a^4
*c^8*f^2 + a^4*d^8*f^2 + 4*a^4*c^2*d^6*f^2 + 6*a^4*c^4*d^4*f^2 + 4*a^4*c^6*d^2*f^2))^2 - 4*(256*d^6 + 256*c^2*
d^4)*((((7*c*d^11)/4 + (19*c^3*d^9)/16 + (11*c^5*d^7)/16 + (c^7*d^5)/8)*1i)/(a^8*c^8*f^4 + a^8*d^8*f^4 + 4*a^8
*c^2*d^6*f^4 + 6*a^8*c^4*d^4*f^4 + 4*a^8*c^6*d^2*f^4) + ((49*d^12)/64 - (11*c^2*d^10)/32 + (5*c^4*d^8)/64 + (c
^6*d^6)/8 + (c^8*d^4)/16)/(a^8*c^8*f^4 + a^8*d^8*f^4 + 4*a^8*c^2*d^6*f^4 + 6*a^8*c^4*d^4*f^4 + 4*a^8*c^6*d^2*f
^4)))^(1/2)*6i + 4*a^4*c*d^3*f^2*(((165*c*d^12 + 70*c^3*d^10 + 73*c^5*d^8 + 32*c^7*d^6 + 8*c^9*d^4)/(a^4*c^8*f
^2 + a^4*d^8*f^2 + 4*a^4*c^2*d^6*f^2 + 6*a^4*c^4*d^4*f^2 + 4*a^4*c^6*d^2*f^2) + ((150*c^2*d^11 - 45*d^13 + 95*
c^4*d^9 + 52*c^6*d^7 + 8*c^8*d^5)*1i)/(a^4*c^8*f^2 + a^4*d^8*f^2 + 4*a^4*c^2*d^6*f^2 + 6*a^4*c^4*d^4*f^2 + 4*a
^4*c^6*d^2*f^2))^2 - 4*(256*d^6 + 256*c^2*d^4)*((((7*c*d^11)/4 + (19*c^3*d^9)/16 + (11*c^5*d^7)/16 + (c^7*d^5)
/8)*1i)/(a^8*c^8*f^4 + a^8*d^8*f^4 + 4*a^8*c^2*d^6*f^4 + 6*a^8*c^4*d^4*f^4 + 4*a^8*c^6*d^2*f^4) + ((49*d^12)/6
4 - (11*c^2*d^10)/32 + (5*c^4*d^8)/64 + (c^6*d^6)/8 + (c^8*d^4)/16)/(a^8*c^8*f^4 + a^8*d^8*f^4 + 4*a^8*c^2*d^6
*f^4 + 6*a^8*c^4*d^4*f^4 + 4*a^8*c^6*d^2*f^4)))^(1/2) - 4*a^4*c^3*d*f^2*(((165*c*d^12 + 70*c^3*d^10 + 73*c^5*d
^8 + 32*c^7*d^6 + 8*c^9*d^4)/(a^4*c^8*f^2 + a^4*d^8*f^2 + 4*a^4*c^2*d^6*f^2 + 6*a^4*c^4*d^4*f^2 + 4*a^4*c^6*d^
2*f^2) + ((150*c^2*d^11 - 45*d^13 + 95*c^4*d^9 + 52*c^6*d^7 + 8*c^8*d^5)*1i)/(a^4*c^8*f^2 + a^4*d^8*f^2 + 4*a^
4*c^2*d^6*f^2 + 6*a^4*c^4*d^4*f^2 + 4*a^4*c^6*d^2*f^2))^2 - 4*(256*d^6 + 256*c^2*d^4)*((((7*c*d^11)/4 + (19*c^
3*d^9)/16 + (11*c^5*d^7)/16 + (c^7*d^5)/8)*1i)/(a^8*c^8*f^4 + a^8*d^8*f^4 + 4*a^8*c^2*d^6*f^4 + 6*a^8*c^4*d^4*
f^4 + 4*a^8*c^6*d^2*f^4) + ((49*d^12)/64 - (11*c^2*d^10)/32 + (5*c^4*d^8)/64 + (c^6*d^6)/8 + (c^8*d^4)/16)/(a^
8*c^8*f^4 + a^8*d^8*f^4 + 4*a^8*c^2*d^6*f^4 + 6*a^8*c^4*d^4*f^4 + 4*a^8*c^6*d^2*f^4)))^(1/2))/(512*(d^6 + c^2*
d^4)*(a^4*c^4*f^2*1i + a^4*d^4*f^2*1i + 4*a^4*c*d^3*f^2 - 4*a^4*c^3*d*f^2 - a^4*c^2*d^2*f^2*6i)))^(1/2) - a^2*
c^2*d^8*f*56i - 40*a^2*c^3*d^7*f - a^2*c^4*d^6*f*25i - 20*a^2*c^5*d^5*f - a^2*c^6*d^4*f*6i - 4*a^2*c^7*d^3*f -
96*a^2*c*d^9*f)*((c*d^8*15i - 45*d^9 - 60*c^2*d^7 + c^3*d^6*80i + 40*c^4*d^5 - c^5*d^4*8i + a^4*c^4*f^2*(((16
5*c*d^12 + 70*c^3*d^10 + 73*c^5*d^8 + 32*c^7*d^6 + 8*c^9*d^4)/(a^4*c^8*f^2 + a^4*d^8*f^2 + 4*a^4*c^2*d^6*f^2 +
6*a^4*c^4*d^4*f^2 + 4*a^4*c^6*d^2*f^2) + ((150*c^2*d^11 - 45*d^13 + 95*c^4*d^9 + 52*c^6*d^7 + 8*c^8*d^5)*1i)/
(a^4*c^8*f^2 + a^4*d^8*f^2 + 4*a^4*c^2*d^6*f^2 + 6*a^4*c^4*d^4*f^2 + 4*a^4*c^6*d^2*f^2))^2 - 4*(256*d^6 + 256*
c^2*d^4)*((((7*c*d^11)/4 + (19*c^3*d^9)/16 + (11*c^5*d^7)/16 + (c^7*d^5)/8)*1i)/(a^8*c^8*f^4 + a^8*d^8*f^4 + 4
*a^8*c^2*d^6*f^4 + 6*a^8*c^4*d^4*f^4 + 4*a^8*c^6*d^2*f^4) + ((49*d^12)/64 - (11*c^2*d^10)/32 + (5*c^4*d^8)/64
+ (c^6*d^6)/8 + (c^8*d^4)/16)/(a^8*c^8*f^4 + a^8*d^8*f^4 + 4*a^8*c^2*d^6*f^4 + 6*a^8*c^4*d^4*f^4 + 4*a^8*c^6*d
^2*f^4)))^(1/2)*1i + a^4*d^4*f^2*(((165*c*d^12 + 70*c^3*d^10 + 73*c^5*d^8 + 32*c^7*d^6 + 8*c^9*d^4)/(a^4*c^8*f
^2 + a^4*d^8*f^2 + 4*a^4*c^2*d^6*f^2 + 6*a^4*c^4*d^4*f^2 + 4*a^4*c^6*d^2*f^2) + ((150*c^2*d^11 - 45*d^13 + 95*
c^4*d^9 + 52*c^6*d^7 + 8*c^8*d^5)*1i)/(a^4*c^8*f^2 + a^4*d^8*f^2 + 4*a^4*c^2*d^6*f^2 + 6*a^4*c^4*d^4*f^2 + 4*a
^4*c^6*d^2*f^2))^2 - 4*(256*d^6 + 256*c^2*d^4)*((((7*c*d^11)/4 + (19*c^3*d^9)/16 + (11*c^5*d^7)/16 + (c^7*d^5)
/8)*1i)/(a^8*c^8*f^4 + a^8*d^8*f^4 + 4*a^8*c^2*d^6*f^4 + 6*a^8*c^4*d^4*f^4 + 4*a^8*c^6*d^2*f^4) + ((49*d^12)/6
4 - (11*c^2*d^10)/32 + (5*c^4*d^8)/64 + (c^6*d^6)/8 + (c^8*d^4)/16)/(a^8*c^8*f^4 + a^8*d^8*f^4 + 4*a^8*c^2*d^6
*f^4 + 6*a^8*c^4*d^4*f^4 + 4*a^8*c^6*d^2*f^4)))^(1/2)*1i - a^4*c^2*d^2*f^2*(((165*c*d^12 + 70*c^3*d^10 + 73*c^
5*d^8 + 32*c^7*d^6 + 8*c^9*d^4)/(a^4*c^8*f^2 + a^4*d^8*f^2 + 4*a^4*c^2*d^6*f^2 + 6*a^4*c^4*d^4*f^2 + 4*a^4*c^6
*d^2*f^2) + ((150*c^2*d^11 - 45*d^13 + 95*c^4*d^9 + 52*c^6*d^7 + 8*c^8*d^5)*1i)/(a^4*c^8*f^2 + a^4*d^8*f^2 + 4
*a^4*c^2*d^6*f^2 + 6*a^4*c^4*d^4*f^2 + 4*a^4*c^6*d^2*f^2))^2 - 4*(256*d^6 + 256*c^2*d^4)*((((7*c*d^11)/4 + (19
*c^3*d^9)/16 + (11*c^5*d^7)/16 + (c^7*d^5)/8)*1i)/(a^8*c^8*f^4 + a^8*d^8*f^4 + 4*a^8*c^2*d^6*f^4 + 6*a^8*c^4*d
^4*f^4 + 4*a^8*c^6*d^2*f^4) + ((49*d^12)/64 - (11*c^2*d^10)/32 + (5*c^4*d^8)/64 + (c^6*d^6)/8 + (c^8*d^4)/16)/
(a^8*c^8*f^4 + a^8*d^8*f^4 + 4*a^8*c^2*d^6*f^4 + 6*a^8*c^4*d^4*f^4 + 4*a^8*c^6*d^2*f^4)))^(1/2)*6i + 4*a^4*c*d
^3*f^2*(((165*c*d^12 + 70*c^3*d^10 + 73*c^5*d^8 + 32*c^7*d^6 + 8*c^9*d^4)/(a^4*c^8*f^2 + a^4*d^8*f^2 + 4*a^4*c
^2*d^6*f^2 + 6*a^4*c^4*d^4*f^2 + 4*a^4*c^6*d^2*f^2) + ((150*c^2*d^11 - 45*d^13 + 95*c^4*d^9 + 52*c^6*d^7 + 8*c
^8*d^5)*1i)/(a^4*c^8*f^2 + a^4*d^8*f^2 + 4*a^4*c^2*d^6*f^2 + 6*a^4*c^4*d^4*f^2 + 4*a^4*c^6*d^2*f^2))^2 - 4*(25
6*d^6 + 256*c^2*d^4)*((((7*c*d^11)/4 + (19*c^3*d^9)/16 + (11*c^5*d^7)/16 + (c^7*d^5)/8)*1i)/(a^8*c^8*f^4 + a^8
*d^8*f^4 + 4*a^8*c^2*d^6*f^4 + 6*a^8*c^4*d^4*f^4 + 4*a^8*c^6*d^2*f^4) + ((49*d^12)/64 - (11*c^2*d^10)/32 + (5*
c^4*d^8)/64 + (c^6*d^6)/8 + (c^8*d^4)/16)/(a^8*c^8*f^4 + a^8*d^8*f^4 + 4*a^8*c^2*d^6*f^4 + 6*a^8*c^4*d^4*f^4 +
4*a^8*c^6*d^2*f^4)))^(1/2) - 4*a^4*c^3*d*f^2*(((165*c*d^12 + 70*c^3*d^10 + 73*c^5*d^8 + 32*c^7*d^6 + 8*c^9*d^
4)/(a^4*c^8*f^2 + a^4*d^8*f^2 + 4*a^4*c^2*d^6*f^2 + 6*a^4*c^4*d^4*f^2 + 4*a^4*c^6*d^2*f^2) + ((150*c^2*d^11 -
45*d^13 + 95*c^4*d^9 + 52*c^6*d^7 + 8*c^8*d^5)*1i)/(a^4*c^8*f^2 + a^4*d^8*f^2 + 4*a^4*c^2*d^6*f^2 + 6*a^4*c^4*
d^4*f^2 + 4*a^4*c^6*d^2*f^2))^2 - 4*(256*d^6 + 256*c^2*d^4)*((((7*c*d^11)/4 + (19*c^3*d^9)/16 + (11*c^5*d^7)/1
6 + (c^7*d^5)/8)*1i)/(a^8*c^8*f^4 + a^8*d^8*f^4 + 4*a^8*c^2*d^6*f^4 + 6*a^8*c^4*d^4*f^4 + 4*a^8*c^6*d^2*f^4) +
((49*d^12)/64 - (11*c^2*d^10)/32 + (5*c^4*d^8)/64 + (c^6*d^6)/8 + (c^8*d^4)/16)/(a^8*c^8*f^4 + a^8*d^8*f^4 +
4*a^8*c^2*d^6*f^4 + 6*a^8*c^4*d^4*f^4 + 4*a^8*c^6*d^2*f^4)))^(1/2))/(512*(d^6 + c^2*d^4)*(a^4*c^4*f^2*1i + a^4
*d^4*f^2*1i + 4*a^4*c*d^3*f^2 - 4*a^4*c^3*d*f^2 - a^4*c^2*d^2*f^2*6i)))^(1/2)