\(\int \cot (c+d x) (a+b \tan (c+d x))^{3/2} (A+B \tan (c+d x)) \, dx\) [328]

Optimal result

Integrand size = 31, antiderivative size = 152 \[ \int \cot (c+d x) (a+b \tan (c+d x))^{3/2} (A+B \tan (c+d x)) \, dx=-\frac {2 a^{3/2} A \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a}}\right )}{d}+\frac {(a-i b)^{3/2} (A-i B) \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a-i b}}\right )}{d}+\frac {(a+i b)^{3/2} (A+i B) \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a+i b}}\right )}{d}+\frac {2 b B \sqrt {a+b \tan (c+d x)}}{d} \]

[Out]

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

Rubi [A] (verified)

Time = 0.73 (sec) , antiderivative size = 152, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.226, Rules used = {3688, 3734, 3620, 3618, 65, 214, 3715} \[ \int \cot (c+d x) (a+b \tan (c+d x))^{3/2} (A+B \tan (c+d x)) \, dx=-\frac {2 a^{3/2} A \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a}}\right )}{d}+\frac {(a-i b)^{3/2} (A-i B) \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a-i b}}\right )}{d}+\frac {(a+i b)^{3/2} (A+i B) \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a+i b}}\right )}{d}+\frac {2 b B \sqrt {a+b \tan (c+d x)}}{d} \]

[In]

Int[Cot[c + d*x]*(a + b*Tan[c + d*x])^(3/2)*(A + B*Tan[c + d*x]),x]

[Out]

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

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 3688

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

Rule 3734

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

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

Mathematica [A] (verified)

Time = 0.41 (sec) , antiderivative size = 144, normalized size of antiderivative = 0.95 \[ \int \cot (c+d x) (a+b \tan (c+d x))^{3/2} (A+B \tan (c+d x)) \, dx=\frac {-2 a^{3/2} A \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a}}\right )+(a-i b)^{3/2} (A-i B) \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a-i b}}\right )+(a+i b)^{3/2} (A+i B) \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a+i b}}\right )+2 b B \sqrt {a+b \tan (c+d x)}}{d} \]

[In]

Integrate[Cot[c + d*x]*(a + b*Tan[c + d*x])^(3/2)*(A + B*Tan[c + d*x]),x]

[Out]

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

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1653\) vs. \(2(126)=252\).

Time = 0.24 (sec) , antiderivative size = 1654, normalized size of antiderivative = 10.88

method	result	size
derivativedivides	\(\text {Expression too large to display}\)	\(1654\)
default	\(\text {Expression too large to display}\)	\(1654\)

[In]

int(cot(d*x+c)*(a+b*tan(d*x+c))^(3/2)*(A+B*tan(d*x+c)),x,method=_RETURNVERBOSE)

[Out]

2*b*B*(a+b*tan(d*x+c))^(1/2)/d-1/d*b^2/(2*(a^2+b^2)^(1/2)-2*a)^(1/2)*arctan(((2*(a^2+b^2)^(1/2)+2*a)^(1/2)-2*(
a+b*tan(d*x+c))^(1/2))/(2*(a^2+b^2)^(1/2)-2*a)^(1/2))*A+1/d*b^2/(2*(a^2+b^2)^(1/2)-2*a)^(1/2)*arctan((2*(a+b*t
an(d*x+c))^(1/2)+(2*(a^2+b^2)^(1/2)+2*a)^(1/2))/(2*(a^2+b^2)^(1/2)-2*a)^(1/2))*A+1/4/d*b*ln((a+b*tan(d*x+c))^(
1/2)*(2*(a^2+b^2)^(1/2)+2*a)^(1/2)-b*tan(d*x+c)-a-(a^2+b^2)^(1/2))*B*(2*(a^2+b^2)^(1/2)+2*a)^(1/2)-1/4/d*b*ln(
b*tan(d*x+c)+a+(a+b*tan(d*x+c))^(1/2)*(2*(a^2+b^2)^(1/2)+2*a)^(1/2)+(a^2+b^2)^(1/2))*B*(2*(a^2+b^2)^(1/2)+2*a)
^(1/2)+1/4/d/b*ln((a+b*tan(d*x+c))^(1/2)*(2*(a^2+b^2)^(1/2)+2*a)^(1/2)-b*tan(d*x+c)-a-(a^2+b^2)^(1/2))*B*(2*(a
^2+b^2)^(1/2)+2*a)^(1/2)*(a^2+b^2)^(1/2)*a-1/d/(2*(a^2+b^2)^(1/2)-2*a)^(1/2)*arctan(((2*(a^2+b^2)^(1/2)+2*a)^(
1/2)-2*(a+b*tan(d*x+c))^(1/2))/(2*(a^2+b^2)^(1/2)-2*a)^(1/2))*A*(a^2+b^2)^(1/2)*a+1/d/(2*(a^2+b^2)^(1/2)-2*a)^
(1/2)*arctan((2*(a+b*tan(d*x+c))^(1/2)+(2*(a^2+b^2)^(1/2)+2*a)^(1/2))/(2*(a^2+b^2)^(1/2)-2*a)^(1/2))*A*(a^2+b^
2)^(1/2)*a-1/4/d/b*ln(b*tan(d*x+c)+a+(a+b*tan(d*x+c))^(1/2)*(2*(a^2+b^2)^(1/2)+2*a)^(1/2)+(a^2+b^2)^(1/2))*B*(
2*(a^2+b^2)^(1/2)+2*a)^(1/2)*(a^2+b^2)^(1/2)*a-1/4/d*ln(b*tan(d*x+c)+a+(a+b*tan(d*x+c))^(1/2)*(2*(a^2+b^2)^(1/
2)+2*a)^(1/2)+(a^2+b^2)^(1/2))*A*(2*(a^2+b^2)^(1/2)+2*a)^(1/2)*(a^2+b^2)^(1/2)+1/2/d*ln(b*tan(d*x+c)+a+(a+b*ta
n(d*x+c))^(1/2)*(2*(a^2+b^2)^(1/2)+2*a)^(1/2)+(a^2+b^2)^(1/2))*A*(2*(a^2+b^2)^(1/2)+2*a)^(1/2)*a+1/4/d/b*ln(b*
tan(d*x+c)+a+(a+b*tan(d*x+c))^(1/2)*(2*(a^2+b^2)^(1/2)+2*a)^(1/2)+(a^2+b^2)^(1/2))*B*(2*(a^2+b^2)^(1/2)+2*a)^(
1/2)*a^2-1/d*b/(2*(a^2+b^2)^(1/2)-2*a)^(1/2)*arctan((2*(a+b*tan(d*x+c))^(1/2)+(2*(a^2+b^2)^(1/2)+2*a)^(1/2))/(
2*(a^2+b^2)^(1/2)-2*a)^(1/2))*B*(a^2+b^2)^(1/2)-1/d/(2*(a^2+b^2)^(1/2)-2*a)^(1/2)*arctan((2*(a+b*tan(d*x+c))^(
1/2)+(2*(a^2+b^2)^(1/2)+2*a)^(1/2))/(2*(a^2+b^2)^(1/2)-2*a)^(1/2))*A*a^2+2/d*b/(2*(a^2+b^2)^(1/2)-2*a)^(1/2)*a
rctan((2*(a+b*tan(d*x+c))^(1/2)+(2*(a^2+b^2)^(1/2)+2*a)^(1/2))/(2*(a^2+b^2)^(1/2)-2*a)^(1/2))*B*a+1/4/d*ln((a+
b*tan(d*x+c))^(1/2)*(2*(a^2+b^2)^(1/2)+2*a)^(1/2)-b*tan(d*x+c)-a-(a^2+b^2)^(1/2))*A*(2*(a^2+b^2)^(1/2)+2*a)^(1
/2)*(a^2+b^2)^(1/2)-1/2/d*ln((a+b*tan(d*x+c))^(1/2)*(2*(a^2+b^2)^(1/2)+2*a)^(1/2)-b*tan(d*x+c)-a-(a^2+b^2)^(1/
2))*A*(2*(a^2+b^2)^(1/2)+2*a)^(1/2)*a-1/4/d/b*ln((a+b*tan(d*x+c))^(1/2)*(2*(a^2+b^2)^(1/2)+2*a)^(1/2)-b*tan(d*
x+c)-a-(a^2+b^2)^(1/2))*B*(2*(a^2+b^2)^(1/2)+2*a)^(1/2)*a^2+1/d*b/(2*(a^2+b^2)^(1/2)-2*a)^(1/2)*arctan(((2*(a^
2+b^2)^(1/2)+2*a)^(1/2)-2*(a+b*tan(d*x+c))^(1/2))/(2*(a^2+b^2)^(1/2)-2*a)^(1/2))*B*(a^2+b^2)^(1/2)+1/d/(2*(a^2
+b^2)^(1/2)-2*a)^(1/2)*arctan(((2*(a^2+b^2)^(1/2)+2*a)^(1/2)-2*(a+b*tan(d*x+c))^(1/2))/(2*(a^2+b^2)^(1/2)-2*a)
^(1/2))*A*a^2-2/d*b/(2*(a^2+b^2)^(1/2)-2*a)^(1/2)*arctan(((2*(a^2+b^2)^(1/2)+2*a)^(1/2)-2*(a+b*tan(d*x+c))^(1/
2))/(2*(a^2+b^2)^(1/2)-2*a)^(1/2))*B*a-2*a^(3/2)*A*arctanh((a+b*tan(d*x+c))^(1/2)/a^(1/2))/d

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 3075 vs. \(2 (120) = 240\).

Time = 2.67 (sec) , antiderivative size = 6165, normalized size of antiderivative = 40.56 \[ \int \cot (c+d x) (a+b \tan (c+d x))^{3/2} (A+B \tan (c+d x)) \, dx=\text {Too large to display} \]

[In]

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

[Out]

Too large to include

Sympy [F]

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

[In]

integrate(cot(d*x+c)*(a+b*tan(d*x+c))**(3/2)*(A+B*tan(d*x+c)),x)

[Out]

Integral((A + B*tan(c + d*x))*(a + b*tan(c + d*x))**(3/2)*cot(c + d*x), x)

Maxima [F]

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

[In]

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

[Out]

integrate((B*tan(d*x + c) + A)*(b*tan(d*x + c) + a)^(3/2)*cot(d*x + c), x)

Giac [F(-1)]

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

[In]

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

[Out]

Timed out

Mupad [B] (verification not implemented)

Time = 11.80 (sec) , antiderivative size = 20255, normalized size of antiderivative = 133.26 \[ \int \cot (c+d x) (a+b \tan (c+d x))^{3/2} (A+B \tan (c+d x)) \, dx=\text {Too large to display} \]

[In]

int(cot(c + d*x)*(A + B*tan(c + d*x))*(a + b*tan(c + d*x))^(3/2),x)

[Out]

(2*B*b*(a + b*tan(c + d*x))^(1/2))/d - atan(((((((32*(4*B*a*b^11*d^4 + 12*A*a^2*b^10*d^4 + 12*A*a^4*b^8*d^4 +
4*B*a^3*b^9*d^4))/d^5 - (32*(16*b^10*d^4 + 24*a^2*b^8*d^4)*(a + b*tan(c + d*x))^(1/2)*(-(((8*A^2*a^3*d^2 - 8*B
^2*a^3*d^2 + 16*A*B*b^3*d^2 - 24*A^2*a*b^2*d^2 + 24*B^2*a*b^2*d^2 - 48*A*B*a^2*b*d^2)^2/64 - d^4*(A^4*a^6 + A^
4*b^6 + B^4*a^6 + B^4*b^6 + 2*A^2*B^2*a^6 + 2*A^2*B^2*b^6 + 3*A^4*a^2*b^4 + 3*A^4*a^4*b^2 + 3*B^4*a^2*b^4 + 3*
B^4*a^4*b^2 + 6*A^2*B^2*a^2*b^4 + 6*A^2*B^2*a^4*b^2))^(1/2) - A^2*a^3*d^2 + B^2*a^3*d^2 - 2*A*B*b^3*d^2 + 3*A^
2*a*b^2*d^2 - 3*B^2*a*b^2*d^2 + 6*A*B*a^2*b*d^2)/(4*d^4))^(1/2))/d^4)*(-(((8*A^2*a^3*d^2 - 8*B^2*a^3*d^2 + 16*
A*B*b^3*d^2 - 24*A^2*a*b^2*d^2 + 24*B^2*a*b^2*d^2 - 48*A*B*a^2*b*d^2)^2/64 - d^4*(A^4*a^6 + A^4*b^6 + B^4*a^6
+ B^4*b^6 + 2*A^2*B^2*a^6 + 2*A^2*B^2*b^6 + 3*A^4*a^2*b^4 + 3*A^4*a^4*b^2 + 3*B^4*a^2*b^4 + 3*B^4*a^4*b^2 + 6*
A^2*B^2*a^2*b^4 + 6*A^2*B^2*a^4*b^2))^(1/2) - A^2*a^3*d^2 + B^2*a^3*d^2 - 2*A*B*b^3*d^2 + 3*A^2*a*b^2*d^2 - 3*
B^2*a*b^2*d^2 + 6*A*B*a^2*b*d^2)/(4*d^4))^(1/2) - (32*(a + b*tan(c + d*x))^(1/2)*(28*A^2*a^3*b^10*d^2 - 18*A^2
*a^5*b^8*d^2 - 28*B^2*a^3*b^10*d^2 + 10*B^2*a^5*b^8*d^2 - 16*A*B*b^13*d^2 + 22*A^2*a*b^12*d^2 - 22*B^2*a*b^12*
d^2 + 16*A*B*a^2*b^11*d^2 + 64*A*B*a^4*b^9*d^2))/d^4)*(-(((8*A^2*a^3*d^2 - 8*B^2*a^3*d^2 + 16*A*B*b^3*d^2 - 24
*A^2*a*b^2*d^2 + 24*B^2*a*b^2*d^2 - 48*A*B*a^2*b*d^2)^2/64 - d^4*(A^4*a^6 + A^4*b^6 + B^4*a^6 + B^4*b^6 + 2*A^
2*B^2*a^6 + 2*A^2*B^2*b^6 + 3*A^4*a^2*b^4 + 3*A^4*a^4*b^2 + 3*B^4*a^2*b^4 + 3*B^4*a^4*b^2 + 6*A^2*B^2*a^2*b^4
+ 6*A^2*B^2*a^4*b^2))^(1/2) - A^2*a^3*d^2 + B^2*a^3*d^2 - 2*A*B*b^3*d^2 + 3*A^2*a*b^2*d^2 - 3*B^2*a*b^2*d^2 +
6*A*B*a^2*b*d^2)/(4*d^4))^(1/2) - (32*(3*A^3*a^7*b^8*d^2 - 21*A^3*a^5*b^10*d^2 - 23*A^3*a^3*b^12*d^2 + 2*B^3*a
^2*b^13*d^2 + 4*B^3*a^4*b^11*d^2 + 2*B^3*a^6*b^9*d^2 + A^3*a*b^14*d^2 + A*B^2*a*b^14*d^2 + 25*A*B^2*a^3*b^12*d
^2 + 15*A*B^2*a^5*b^10*d^2 - 9*A*B^2*a^7*b^8*d^2 + 18*A^2*B*a^2*b^13*d^2 - 24*A^2*B*a^4*b^11*d^2 - 42*A^2*B*a^
6*b^9*d^2))/d^5)*(-(((8*A^2*a^3*d^2 - 8*B^2*a^3*d^2 + 16*A*B*b^3*d^2 - 24*A^2*a*b^2*d^2 + 24*B^2*a*b^2*d^2 - 4
8*A*B*a^2*b*d^2)^2/64 - d^4*(A^4*a^6 + A^4*b^6 + B^4*a^6 + B^4*b^6 + 2*A^2*B^2*a^6 + 2*A^2*B^2*b^6 + 3*A^4*a^2
*b^4 + 3*A^4*a^4*b^2 + 3*B^4*a^2*b^4 + 3*B^4*a^4*b^2 + 6*A^2*B^2*a^2*b^4 + 6*A^2*B^2*a^4*b^2))^(1/2) - A^2*a^3
*d^2 + B^2*a^3*d^2 - 2*A*B*b^3*d^2 + 3*A^2*a*b^2*d^2 - 3*B^2*a*b^2*d^2 + 6*A*B*a^2*b*d^2)/(4*d^4))^(1/2) - (32
*(a + b*tan(c + d*x))^(1/2)*(A^4*b^16 + B^4*b^16 + 2*A^2*B^2*b^16 + 4*A^4*a^2*b^14 + 8*A^4*a^4*b^12 - 8*A^4*a^
6*b^10 + 3*A^4*a^8*b^8 + 4*B^4*a^2*b^14 + 6*B^4*a^4*b^12 + 4*B^4*a^6*b^10 + B^4*a^8*b^8 + 8*A^2*B^2*a^2*b^14 +
10*A^2*B^2*a^4*b^12 + 20*A^2*B^2*a^6*b^10 + 16*A^3*B*a^5*b^11 - 16*A^3*B*a^7*b^9))/d^4)*(-(((8*A^2*a^3*d^2 -
8*B^2*a^3*d^2 + 16*A*B*b^3*d^2 - 24*A^2*a*b^2*d^2 + 24*B^2*a*b^2*d^2 - 48*A*B*a^2*b*d^2)^2/64 - d^4*(A^4*a^6 +
A^4*b^6 + B^4*a^6 + B^4*b^6 + 2*A^2*B^2*a^6 + 2*A^2*B^2*b^6 + 3*A^4*a^2*b^4 + 3*A^4*a^4*b^2 + 3*B^4*a^2*b^4 +
3*B^4*a^4*b^2 + 6*A^2*B^2*a^2*b^4 + 6*A^2*B^2*a^4*b^2))^(1/2) - A^2*a^3*d^2 + B^2*a^3*d^2 - 2*A*B*b^3*d^2 + 3
*A^2*a*b^2*d^2 - 3*B^2*a*b^2*d^2 + 6*A*B*a^2*b*d^2)/(4*d^4))^(1/2)*1i - (((((32*(4*B*a*b^11*d^4 + 12*A*a^2*b^1
0*d^4 + 12*A*a^4*b^8*d^4 + 4*B*a^3*b^9*d^4))/d^5 + (32*(16*b^10*d^4 + 24*a^2*b^8*d^4)*(a + b*tan(c + d*x))^(1/
2)*(-(((8*A^2*a^3*d^2 - 8*B^2*a^3*d^2 + 16*A*B*b^3*d^2 - 24*A^2*a*b^2*d^2 + 24*B^2*a*b^2*d^2 - 48*A*B*a^2*b*d^
2)^2/64 - d^4*(A^4*a^6 + A^4*b^6 + B^4*a^6 + B^4*b^6 + 2*A^2*B^2*a^6 + 2*A^2*B^2*b^6 + 3*A^4*a^2*b^4 + 3*A^4*a
^4*b^2 + 3*B^4*a^2*b^4 + 3*B^4*a^4*b^2 + 6*A^2*B^2*a^2*b^4 + 6*A^2*B^2*a^4*b^2))^(1/2) - A^2*a^3*d^2 + B^2*a^3
*d^2 - 2*A*B*b^3*d^2 + 3*A^2*a*b^2*d^2 - 3*B^2*a*b^2*d^2 + 6*A*B*a^2*b*d^2)/(4*d^4))^(1/2))/d^4)*(-(((8*A^2*a^
3*d^2 - 8*B^2*a^3*d^2 + 16*A*B*b^3*d^2 - 24*A^2*a*b^2*d^2 + 24*B^2*a*b^2*d^2 - 48*A*B*a^2*b*d^2)^2/64 - d^4*(A
^4*a^6 + A^4*b^6 + B^4*a^6 + B^4*b^6 + 2*A^2*B^2*a^6 + 2*A^2*B^2*b^6 + 3*A^4*a^2*b^4 + 3*A^4*a^4*b^2 + 3*B^4*a
^2*b^4 + 3*B^4*a^4*b^2 + 6*A^2*B^2*a^2*b^4 + 6*A^2*B^2*a^4*b^2))^(1/2) - A^2*a^3*d^2 + B^2*a^3*d^2 - 2*A*B*b^3
*d^2 + 3*A^2*a*b^2*d^2 - 3*B^2*a*b^2*d^2 + 6*A*B*a^2*b*d^2)/(4*d^4))^(1/2) + (32*(a + b*tan(c + d*x))^(1/2)*(2
8*A^2*a^3*b^10*d^2 - 18*A^2*a^5*b^8*d^2 - 28*B^2*a^3*b^10*d^2 + 10*B^2*a^5*b^8*d^2 - 16*A*B*b^13*d^2 + 22*A^2*
a*b^12*d^2 - 22*B^2*a*b^12*d^2 + 16*A*B*a^2*b^11*d^2 + 64*A*B*a^4*b^9*d^2))/d^4)*(-(((8*A^2*a^3*d^2 - 8*B^2*a^
3*d^2 + 16*A*B*b^3*d^2 - 24*A^2*a*b^2*d^2 + 24*B^2*a*b^2*d^2 - 48*A*B*a^2*b*d^2)^2/64 - d^4*(A^4*a^6 + A^4*b^6
+ B^4*a^6 + B^4*b^6 + 2*A^2*B^2*a^6 + 2*A^2*B^2*b^6 + 3*A^4*a^2*b^4 + 3*A^4*a^4*b^2 + 3*B^4*a^2*b^4 + 3*B^4*a
^4*b^2 + 6*A^2*B^2*a^2*b^4 + 6*A^2*B^2*a^4*b^2))^(1/2) - A^2*a^3*d^2 + B^2*a^3*d^2 - 2*A*B*b^3*d^2 + 3*A^2*a*b
^2*d^2 - 3*B^2*a*b^2*d^2 + 6*A*B*a^2*b*d^2)/(4*d^4))^(1/2) - (32*(3*A^3*a^7*b^8*d^2 - 21*A^3*a^5*b^10*d^2 - 23
*A^3*a^3*b^12*d^2 + 2*B^3*a^2*b^13*d^2 + 4*B^3*a^4*b^11*d^2 + 2*B^3*a^6*b^9*d^2 + A^3*a*b^14*d^2 + A*B^2*a*b^1
4*d^2 + 25*A*B^2*a^3*b^12*d^2 + 15*A*B^2*a^5*b^10*d^2 - 9*A*B^2*a^7*b^8*d^2 + 18*A^2*B*a^2*b^13*d^2 - 24*A^2*B
*a^4*b^11*d^2 - 42*A^2*B*a^6*b^9*d^2))/d^5)*(-(((8*A^2*a^3*d^2 - 8*B^2*a^3*d^2 + 16*A*B*b^3*d^2 - 24*A^2*a*b^2
*d^2 + 24*B^2*a*b^2*d^2 - 48*A*B*a^2*b*d^2)^2/64 - d^4*(A^4*a^6 + A^4*b^6 + B^4*a^6 + B^4*b^6 + 2*A^2*B^2*a^6
+ 2*A^2*B^2*b^6 + 3*A^4*a^2*b^4 + 3*A^4*a^4*b^2 + 3*B^4*a^2*b^4 + 3*B^4*a^4*b^2 + 6*A^2*B^2*a^2*b^4 + 6*A^2*B^
2*a^4*b^2))^(1/2) - A^2*a^3*d^2 + B^2*a^3*d^2 - 2*A*B*b^3*d^2 + 3*A^2*a*b^2*d^2 - 3*B^2*a*b^2*d^2 + 6*A*B*a^2*
b*d^2)/(4*d^4))^(1/2) + (32*(a + b*tan(c + d*x))^(1/2)*(A^4*b^16 + B^4*b^16 + 2*A^2*B^2*b^16 + 4*A^4*a^2*b^14
+ 8*A^4*a^4*b^12 - 8*A^4*a^6*b^10 + 3*A^4*a^8*b^8 + 4*B^4*a^2*b^14 + 6*B^4*a^4*b^12 + 4*B^4*a^6*b^10 + B^4*a^8
*b^8 + 8*A^2*B^2*a^2*b^14 + 10*A^2*B^2*a^4*b^12 + 20*A^2*B^2*a^6*b^10 + 16*A^3*B*a^5*b^11 - 16*A^3*B*a^7*b^9))
/d^4)*(-(((8*A^2*a^3*d^2 - 8*B^2*a^3*d^2 + 16*A*B*b^3*d^2 - 24*A^2*a*b^2*d^2 + 24*B^2*a*b^2*d^2 - 48*A*B*a^2*b
*d^2)^2/64 - d^4*(A^4*a^6 + A^4*b^6 + B^4*a^6 + B^4*b^6 + 2*A^2*B^2*a^6 + 2*A^2*B^2*b^6 + 3*A^4*a^2*b^4 + 3*A^
4*a^4*b^2 + 3*B^4*a^2*b^4 + 3*B^4*a^4*b^2 + 6*A^2*B^2*a^2*b^4 + 6*A^2*B^2*a^4*b^2))^(1/2) - A^2*a^3*d^2 + B^2*
a^3*d^2 - 2*A*B*b^3*d^2 + 3*A^2*a*b^2*d^2 - 3*B^2*a*b^2*d^2 + 6*A*B*a^2*b*d^2)/(4*d^4))^(1/2)*1i)/((((((32*(4*
B*a*b^11*d^4 + 12*A*a^2*b^10*d^4 + 12*A*a^4*b^8*d^4 + 4*B*a^3*b^9*d^4))/d^5 - (32*(16*b^10*d^4 + 24*a^2*b^8*d^
4)*(a + b*tan(c + d*x))^(1/2)*(-(((8*A^2*a^3*d^2 - 8*B^2*a^3*d^2 + 16*A*B*b^3*d^2 - 24*A^2*a*b^2*d^2 + 24*B^2*
a*b^2*d^2 - 48*A*B*a^2*b*d^2)^2/64 - d^4*(A^4*a^6 + A^4*b^6 + B^4*a^6 + B^4*b^6 + 2*A^2*B^2*a^6 + 2*A^2*B^2*b^
6 + 3*A^4*a^2*b^4 + 3*A^4*a^4*b^2 + 3*B^4*a^2*b^4 + 3*B^4*a^4*b^2 + 6*A^2*B^2*a^2*b^4 + 6*A^2*B^2*a^4*b^2))^(1
/2) - A^2*a^3*d^2 + B^2*a^3*d^2 - 2*A*B*b^3*d^2 + 3*A^2*a*b^2*d^2 - 3*B^2*a*b^2*d^2 + 6*A*B*a^2*b*d^2)/(4*d^4)
)^(1/2))/d^4)*(-(((8*A^2*a^3*d^2 - 8*B^2*a^3*d^2 + 16*A*B*b^3*d^2 - 24*A^2*a*b^2*d^2 + 24*B^2*a*b^2*d^2 - 48*A
*B*a^2*b*d^2)^2/64 - d^4*(A^4*a^6 + A^4*b^6 + B^4*a^6 + B^4*b^6 + 2*A^2*B^2*a^6 + 2*A^2*B^2*b^6 + 3*A^4*a^2*b^
4 + 3*A^4*a^4*b^2 + 3*B^4*a^2*b^4 + 3*B^4*a^4*b^2 + 6*A^2*B^2*a^2*b^4 + 6*A^2*B^2*a^4*b^2))^(1/2) - A^2*a^3*d^
2 + B^2*a^3*d^2 - 2*A*B*b^3*d^2 + 3*A^2*a*b^2*d^2 - 3*B^2*a*b^2*d^2 + 6*A*B*a^2*b*d^2)/(4*d^4))^(1/2) - (32*(a
+ b*tan(c + d*x))^(1/2)*(28*A^2*a^3*b^10*d^2 - 18*A^2*a^5*b^8*d^2 - 28*B^2*a^3*b^10*d^2 + 10*B^2*a^5*b^8*d^2
- 16*A*B*b^13*d^2 + 22*A^2*a*b^12*d^2 - 22*B^2*a*b^12*d^2 + 16*A*B*a^2*b^11*d^2 + 64*A*B*a^4*b^9*d^2))/d^4)*(-
(((8*A^2*a^3*d^2 - 8*B^2*a^3*d^2 + 16*A*B*b^3*d^2 - 24*A^2*a*b^2*d^2 + 24*B^2*a*b^2*d^2 - 48*A*B*a^2*b*d^2)^2/
64 - d^4*(A^4*a^6 + A^4*b^6 + B^4*a^6 + B^4*b^6 + 2*A^2*B^2*a^6 + 2*A^2*B^2*b^6 + 3*A^4*a^2*b^4 + 3*A^4*a^4*b^
2 + 3*B^4*a^2*b^4 + 3*B^4*a^4*b^2 + 6*A^2*B^2*a^2*b^4 + 6*A^2*B^2*a^4*b^2))^(1/2) - A^2*a^3*d^2 + B^2*a^3*d^2
- 2*A*B*b^3*d^2 + 3*A^2*a*b^2*d^2 - 3*B^2*a*b^2*d^2 + 6*A*B*a^2*b*d^2)/(4*d^4))^(1/2) - (32*(3*A^3*a^7*b^8*d^2
- 21*A^3*a^5*b^10*d^2 - 23*A^3*a^3*b^12*d^2 + 2*B^3*a^2*b^13*d^2 + 4*B^3*a^4*b^11*d^2 + 2*B^3*a^6*b^9*d^2 + A
^3*a*b^14*d^2 + A*B^2*a*b^14*d^2 + 25*A*B^2*a^3*b^12*d^2 + 15*A*B^2*a^5*b^10*d^2 - 9*A*B^2*a^7*b^8*d^2 + 18*A^
2*B*a^2*b^13*d^2 - 24*A^2*B*a^4*b^11*d^2 - 42*A^2*B*a^6*b^9*d^2))/d^5)*(-(((8*A^2*a^3*d^2 - 8*B^2*a^3*d^2 + 16
*A*B*b^3*d^2 - 24*A^2*a*b^2*d^2 + 24*B^2*a*b^2*d^2 - 48*A*B*a^2*b*d^2)^2/64 - d^4*(A^4*a^6 + A^4*b^6 + B^4*a^6
+ B^4*b^6 + 2*A^2*B^2*a^6 + 2*A^2*B^2*b^6 + 3*A^4*a^2*b^4 + 3*A^4*a^4*b^2 + 3*B^4*a^2*b^4 + 3*B^4*a^4*b^2 + 6
*A^2*B^2*a^2*b^4 + 6*A^2*B^2*a^4*b^2))^(1/2) - A^2*a^3*d^2 + B^2*a^3*d^2 - 2*A*B*b^3*d^2 + 3*A^2*a*b^2*d^2 - 3
*B^2*a*b^2*d^2 + 6*A*B*a^2*b*d^2)/(4*d^4))^(1/2) - (32*(a + b*tan(c + d*x))^(1/2)*(A^4*b^16 + B^4*b^16 + 2*A^2
*B^2*b^16 + 4*A^4*a^2*b^14 + 8*A^4*a^4*b^12 - 8*A^4*a^6*b^10 + 3*A^4*a^8*b^8 + 4*B^4*a^2*b^14 + 6*B^4*a^4*b^12
+ 4*B^4*a^6*b^10 + B^4*a^8*b^8 + 8*A^2*B^2*a^2*b^14 + 10*A^2*B^2*a^4*b^12 + 20*A^2*B^2*a^6*b^10 + 16*A^3*B*a^
5*b^11 - 16*A^3*B*a^7*b^9))/d^4)*(-(((8*A^2*a^3*d^2 - 8*B^2*a^3*d^2 + 16*A*B*b^3*d^2 - 24*A^2*a*b^2*d^2 + 24*B
^2*a*b^2*d^2 - 48*A*B*a^2*b*d^2)^2/64 - d^4*(A^4*a^6 + A^4*b^6 + B^4*a^6 + B^4*b^6 + 2*A^2*B^2*a^6 + 2*A^2*B^2
*b^6 + 3*A^4*a^2*b^4 + 3*A^4*a^4*b^2 + 3*B^4*a^2*b^4 + 3*B^4*a^4*b^2 + 6*A^2*B^2*a^2*b^4 + 6*A^2*B^2*a^4*b^2))
^(1/2) - A^2*a^3*d^2 + B^2*a^3*d^2 - 2*A*B*b^3*d^2 + 3*A^2*a*b^2*d^2 - 3*B^2*a*b^2*d^2 + 6*A*B*a^2*b*d^2)/(4*d
^4))^(1/2) + (((((32*(4*B*a*b^11*d^4 + 12*A*a^2*b^10*d^4 + 12*A*a^4*b^8*d^4 + 4*B*a^3*b^9*d^4))/d^5 + (32*(16*
b^10*d^4 + 24*a^2*b^8*d^4)*(a + b*tan(c + d*x))^(1/2)*(-(((8*A^2*a^3*d^2 - 8*B^2*a^3*d^2 + 16*A*B*b^3*d^2 - 24
*A^2*a*b^2*d^2 + 24*B^2*a*b^2*d^2 - 48*A*B*a^2*b*d^2)^2/64 - d^4*(A^4*a^6 + A^4*b^6 + B^4*a^6 + B^4*b^6 + 2*A^
2*B^2*a^6 + 2*A^2*B^2*b^6 + 3*A^4*a^2*b^4 + 3*A^4*a^4*b^2 + 3*B^4*a^2*b^4 + 3*B^4*a^4*b^2 + 6*A^2*B^2*a^2*b^4
+ 6*A^2*B^2*a^4*b^2))^(1/2) - A^2*a^3*d^2 + B^2*a^3*d^2 - 2*A*B*b^3*d^2 + 3*A^2*a*b^2*d^2 - 3*B^2*a*b^2*d^2 +
6*A*B*a^2*b*d^2)/(4*d^4))^(1/2))/d^4)*(-(((8*A^2*a^3*d^2 - 8*B^2*a^3*d^2 + 16*A*B*b^3*d^2 - 24*A^2*a*b^2*d^2 +
24*B^2*a*b^2*d^2 - 48*A*B*a^2*b*d^2)^2/64 - d^4*(A^4*a^6 + A^4*b^6 + B^4*a^6 + B^4*b^6 + 2*A^2*B^2*a^6 + 2*A^
2*B^2*b^6 + 3*A^4*a^2*b^4 + 3*A^4*a^4*b^2 + 3*B^4*a^2*b^4 + 3*B^4*a^4*b^2 + 6*A^2*B^2*a^2*b^4 + 6*A^2*B^2*a^4*
b^2))^(1/2) - A^2*a^3*d^2 + B^2*a^3*d^2 - 2*A*B*b^3*d^2 + 3*A^2*a*b^2*d^2 - 3*B^2*a*b^2*d^2 + 6*A*B*a^2*b*d^2)
/(4*d^4))^(1/2) + (32*(a + b*tan(c + d*x))^(1/2)*(28*A^2*a^3*b^10*d^2 - 18*A^2*a^5*b^8*d^2 - 28*B^2*a^3*b^10*d
^2 + 10*B^2*a^5*b^8*d^2 - 16*A*B*b^13*d^2 + 22*A^2*a*b^12*d^2 - 22*B^2*a*b^12*d^2 + 16*A*B*a^2*b^11*d^2 + 64*A
*B*a^4*b^9*d^2))/d^4)*(-(((8*A^2*a^3*d^2 - 8*B^2*a^3*d^2 + 16*A*B*b^3*d^2 - 24*A^2*a*b^2*d^2 + 24*B^2*a*b^2*d^
2 - 48*A*B*a^2*b*d^2)^2/64 - d^4*(A^4*a^6 + A^4*b^6 + B^4*a^6 + B^4*b^6 + 2*A^2*B^2*a^6 + 2*A^2*B^2*b^6 + 3*A^
4*a^2*b^4 + 3*A^4*a^4*b^2 + 3*B^4*a^2*b^4 + 3*B^4*a^4*b^2 + 6*A^2*B^2*a^2*b^4 + 6*A^2*B^2*a^4*b^2))^(1/2) - A^
2*a^3*d^2 + B^2*a^3*d^2 - 2*A*B*b^3*d^2 + 3*A^2*a*b^2*d^2 - 3*B^2*a*b^2*d^2 + 6*A*B*a^2*b*d^2)/(4*d^4))^(1/2)
- (32*(3*A^3*a^7*b^8*d^2 - 21*A^3*a^5*b^10*d^2 - 23*A^3*a^3*b^12*d^2 + 2*B^3*a^2*b^13*d^2 + 4*B^3*a^4*b^11*d^2
+ 2*B^3*a^6*b^9*d^2 + A^3*a*b^14*d^2 + A*B^2*a*b^14*d^2 + 25*A*B^2*a^3*b^12*d^2 + 15*A*B^2*a^5*b^10*d^2 - 9*A
*B^2*a^7*b^8*d^2 + 18*A^2*B*a^2*b^13*d^2 - 24*A^2*B*a^4*b^11*d^2 - 42*A^2*B*a^6*b^9*d^2))/d^5)*(-(((8*A^2*a^3*
d^2 - 8*B^2*a^3*d^2 + 16*A*B*b^3*d^2 - 24*A^2*a*b^2*d^2 + 24*B^2*a*b^2*d^2 - 48*A*B*a^2*b*d^2)^2/64 - d^4*(A^4
*a^6 + A^4*b^6 + B^4*a^6 + B^4*b^6 + 2*A^2*B^2*a^6 + 2*A^2*B^2*b^6 + 3*A^4*a^2*b^4 + 3*A^4*a^4*b^2 + 3*B^4*a^2
*b^4 + 3*B^4*a^4*b^2 + 6*A^2*B^2*a^2*b^4 + 6*A^2*B^2*a^4*b^2))^(1/2) - A^2*a^3*d^2 + B^2*a^3*d^2 - 2*A*B*b^3*d
^2 + 3*A^2*a*b^2*d^2 - 3*B^2*a*b^2*d^2 + 6*A*B*a^2*b*d^2)/(4*d^4))^(1/2) + (32*(a + b*tan(c + d*x))^(1/2)*(A^4
*b^16 + B^4*b^16 + 2*A^2*B^2*b^16 + 4*A^4*a^2*b^14 + 8*A^4*a^4*b^12 - 8*A^4*a^6*b^10 + 3*A^4*a^8*b^8 + 4*B^4*a
^2*b^14 + 6*B^4*a^4*b^12 + 4*B^4*a^6*b^10 + B^4*a^8*b^8 + 8*A^2*B^2*a^2*b^14 + 10*A^2*B^2*a^4*b^12 + 20*A^2*B^
2*a^6*b^10 + 16*A^3*B*a^5*b^11 - 16*A^3*B*a^7*b^9))/d^4)*(-(((8*A^2*a^3*d^2 - 8*B^2*a^3*d^2 + 16*A*B*b^3*d^2 -
24*A^2*a*b^2*d^2 + 24*B^2*a*b^2*d^2 - 48*A*B*a^2*b*d^2)^2/64 - d^4*(A^4*a^6 + A^4*b^6 + B^4*a^6 + B^4*b^6 + 2
*A^2*B^2*a^6 + 2*A^2*B^2*b^6 + 3*A^4*a^2*b^4 + 3*A^4*a^4*b^2 + 3*B^4*a^2*b^4 + 3*B^4*a^4*b^2 + 6*A^2*B^2*a^2*b
^4 + 6*A^2*B^2*a^4*b^2))^(1/2) - A^2*a^3*d^2 + B^2*a^3*d^2 - 2*A*B*b^3*d^2 + 3*A^2*a*b^2*d^2 - 3*B^2*a*b^2*d^2
+ 6*A*B*a^2*b*d^2)/(4*d^4))^(1/2) + (64*(A^5*a^2*b^16 + 5*A^5*a^4*b^14 + 7*A^5*a^6*b^12 + 3*A^5*a^8*b^10 + 2*
A^2*B^3*a^5*b^13 + 4*A^2*B^3*a^7*b^11 + 2*A^2*B^3*a^9*b^9 + 2*A^3*B^2*a^2*b^16 + 9*A^3*B^2*a^4*b^14 + 13*A^3*B
^2*a^6*b^12 + 7*A^3*B^2*a^8*b^10 + A^3*B^2*a^10*b^8 + A*B^4*a^2*b^16 + 4*A*B^4*a^4*b^14 + 6*A*B^4*a^6*b^12 + 4
*A*B^4*a^8*b^10 + A*B^4*a^10*b^8 + 2*A^4*B*a^5*b^13 + 4*A^4*B*a^7*b^11 + 2*A^4*B*a^9*b^9))/d^5))*(-(((8*A^2*a^
3*d^2 - 8*B^2*a^3*d^2 + 16*A*B*b^3*d^2 - 24*A^2*a*b^2*d^2 + 24*B^2*a*b^2*d^2 - 48*A*B*a^2*b*d^2)^2/64 - d^4*(A
^4*a^6 + A^4*b^6 + B^4*a^6 + B^4*b^6 + 2*A^2*B^2*a^6 + 2*A^2*B^2*b^6 + 3*A^4*a^2*b^4 + 3*A^4*a^4*b^2 + 3*B^4*a
^2*b^4 + 3*B^4*a^4*b^2 + 6*A^2*B^2*a^2*b^4 + 6*A^2*B^2*a^4*b^2))^(1/2) - A^2*a^3*d^2 + B^2*a^3*d^2 - 2*A*B*b^3
*d^2 + 3*A^2*a*b^2*d^2 - 3*B^2*a*b^2*d^2 + 6*A*B*a^2*b*d^2)/(4*d^4))^(1/2)*2i - atan(((((((32*(4*B*a*b^11*d^4
+ 12*A*a^2*b^10*d^4 + 12*A*a^4*b^8*d^4 + 4*B*a^3*b^9*d^4))/d^5 - (32*(16*b^10*d^4 + 24*a^2*b^8*d^4)*(a + b*tan
(c + d*x))^(1/2)*((((8*A^2*a^3*d^2 - 8*B^2*a^3*d^2 + 16*A*B*b^3*d^2 - 24*A^2*a*b^2*d^2 + 24*B^2*a*b^2*d^2 - 48
*A*B*a^2*b*d^2)^2/64 - d^4*(A^4*a^6 + A^4*b^6 + B^4*a^6 + B^4*b^6 + 2*A^2*B^2*a^6 + 2*A^2*B^2*b^6 + 3*A^4*a^2*
b^4 + 3*A^4*a^4*b^2 + 3*B^4*a^2*b^4 + 3*B^4*a^4*b^2 + 6*A^2*B^2*a^2*b^4 + 6*A^2*B^2*a^4*b^2))^(1/2) + A^2*a^3*
d^2 - B^2*a^3*d^2 + 2*A*B*b^3*d^2 - 3*A^2*a*b^2*d^2 + 3*B^2*a*b^2*d^2 - 6*A*B*a^2*b*d^2)/(4*d^4))^(1/2))/d^4)*
((((8*A^2*a^3*d^2 - 8*B^2*a^3*d^2 + 16*A*B*b^3*d^2 - 24*A^2*a*b^2*d^2 + 24*B^2*a*b^2*d^2 - 48*A*B*a^2*b*d^2)^2
/64 - d^4*(A^4*a^6 + A^4*b^6 + B^4*a^6 + B^4*b^6 + 2*A^2*B^2*a^6 + 2*A^2*B^2*b^6 + 3*A^4*a^2*b^4 + 3*A^4*a^4*b
^2 + 3*B^4*a^2*b^4 + 3*B^4*a^4*b^2 + 6*A^2*B^2*a^2*b^4 + 6*A^2*B^2*a^4*b^2))^(1/2) + A^2*a^3*d^2 - B^2*a^3*d^2
+ 2*A*B*b^3*d^2 - 3*A^2*a*b^2*d^2 + 3*B^2*a*b^2*d^2 - 6*A*B*a^2*b*d^2)/(4*d^4))^(1/2) - (32*(a + b*tan(c + d*
x))^(1/2)*(28*A^2*a^3*b^10*d^2 - 18*A^2*a^5*b^8*d^2 - 28*B^2*a^3*b^10*d^2 + 10*B^2*a^5*b^8*d^2 - 16*A*B*b^13*d
^2 + 22*A^2*a*b^12*d^2 - 22*B^2*a*b^12*d^2 + 16*A*B*a^2*b^11*d^2 + 64*A*B*a^4*b^9*d^2))/d^4)*((((8*A^2*a^3*d^2
- 8*B^2*a^3*d^2 + 16*A*B*b^3*d^2 - 24*A^2*a*b^2*d^2 + 24*B^2*a*b^2*d^2 - 48*A*B*a^2*b*d^2)^2/64 - d^4*(A^4*a^
6 + A^4*b^6 + B^4*a^6 + B^4*b^6 + 2*A^2*B^2*a^6 + 2*A^2*B^2*b^6 + 3*A^4*a^2*b^4 + 3*A^4*a^4*b^2 + 3*B^4*a^2*b^
4 + 3*B^4*a^4*b^2 + 6*A^2*B^2*a^2*b^4 + 6*A^2*B^2*a^4*b^2))^(1/2) + A^2*a^3*d^2 - B^2*a^3*d^2 + 2*A*B*b^3*d^2
- 3*A^2*a*b^2*d^2 + 3*B^2*a*b^2*d^2 - 6*A*B*a^2*b*d^2)/(4*d^4))^(1/2) - (32*(3*A^3*a^7*b^8*d^2 - 21*A^3*a^5*b^
10*d^2 - 23*A^3*a^3*b^12*d^2 + 2*B^3*a^2*b^13*d^2 + 4*B^3*a^4*b^11*d^2 + 2*B^3*a^6*b^9*d^2 + A^3*a*b^14*d^2 +
A*B^2*a*b^14*d^2 + 25*A*B^2*a^3*b^12*d^2 + 15*A*B^2*a^5*b^10*d^2 - 9*A*B^2*a^7*b^8*d^2 + 18*A^2*B*a^2*b^13*d^2
- 24*A^2*B*a^4*b^11*d^2 - 42*A^2*B*a^6*b^9*d^2))/d^5)*((((8*A^2*a^3*d^2 - 8*B^2*a^3*d^2 + 16*A*B*b^3*d^2 - 24
*A^2*a*b^2*d^2 + 24*B^2*a*b^2*d^2 - 48*A*B*a^2*b*d^2)^2/64 - d^4*(A^4*a^6 + A^4*b^6 + B^4*a^6 + B^4*b^6 + 2*A^
2*B^2*a^6 + 2*A^2*B^2*b^6 + 3*A^4*a^2*b^4 + 3*A^4*a^4*b^2 + 3*B^4*a^2*b^4 + 3*B^4*a^4*b^2 + 6*A^2*B^2*a^2*b^4
+ 6*A^2*B^2*a^4*b^2))^(1/2) + A^2*a^3*d^2 - B^2*a^3*d^2 + 2*A*B*b^3*d^2 - 3*A^2*a*b^2*d^2 + 3*B^2*a*b^2*d^2 -
6*A*B*a^2*b*d^2)/(4*d^4))^(1/2) - (32*(a + b*tan(c + d*x))^(1/2)*(A^4*b^16 + B^4*b^16 + 2*A^2*B^2*b^16 + 4*A^4
*a^2*b^14 + 8*A^4*a^4*b^12 - 8*A^4*a^6*b^10 + 3*A^4*a^8*b^8 + 4*B^4*a^2*b^14 + 6*B^4*a^4*b^12 + 4*B^4*a^6*b^10
+ B^4*a^8*b^8 + 8*A^2*B^2*a^2*b^14 + 10*A^2*B^2*a^4*b^12 + 20*A^2*B^2*a^6*b^10 + 16*A^3*B*a^5*b^11 - 16*A^3*B
*a^7*b^9))/d^4)*((((8*A^2*a^3*d^2 - 8*B^2*a^3*d^2 + 16*A*B*b^3*d^2 - 24*A^2*a*b^2*d^2 + 24*B^2*a*b^2*d^2 - 48*
A*B*a^2*b*d^2)^2/64 - d^4*(A^4*a^6 + A^4*b^6 + B^4*a^6 + B^4*b^6 + 2*A^2*B^2*a^6 + 2*A^2*B^2*b^6 + 3*A^4*a^2*b
^4 + 3*A^4*a^4*b^2 + 3*B^4*a^2*b^4 + 3*B^4*a^4*b^2 + 6*A^2*B^2*a^2*b^4 + 6*A^2*B^2*a^4*b^2))^(1/2) + A^2*a^3*d
^2 - B^2*a^3*d^2 + 2*A*B*b^3*d^2 - 3*A^2*a*b^2*d^2 + 3*B^2*a*b^2*d^2 - 6*A*B*a^2*b*d^2)/(4*d^4))^(1/2)*1i - ((
(((32*(4*B*a*b^11*d^4 + 12*A*a^2*b^10*d^4 + 12*A*a^4*b^8*d^4 + 4*B*a^3*b^9*d^4))/d^5 + (32*(16*b^10*d^4 + 24*a
^2*b^8*d^4)*(a + b*tan(c + d*x))^(1/2)*((((8*A^2*a^3*d^2 - 8*B^2*a^3*d^2 + 16*A*B*b^3*d^2 - 24*A^2*a*b^2*d^2 +
24*B^2*a*b^2*d^2 - 48*A*B*a^2*b*d^2)^2/64 - d^4*(A^4*a^6 + A^4*b^6 + B^4*a^6 + B^4*b^6 + 2*A^2*B^2*a^6 + 2*A^
2*B^2*b^6 + 3*A^4*a^2*b^4 + 3*A^4*a^4*b^2 + 3*B^4*a^2*b^4 + 3*B^4*a^4*b^2 + 6*A^2*B^2*a^2*b^4 + 6*A^2*B^2*a^4*
b^2))^(1/2) + A^2*a^3*d^2 - B^2*a^3*d^2 + 2*A*B*b^3*d^2 - 3*A^2*a*b^2*d^2 + 3*B^2*a*b^2*d^2 - 6*A*B*a^2*b*d^2)
/(4*d^4))^(1/2))/d^4)*((((8*A^2*a^3*d^2 - 8*B^2*a^3*d^2 + 16*A*B*b^3*d^2 - 24*A^2*a*b^2*d^2 + 24*B^2*a*b^2*d^2
- 48*A*B*a^2*b*d^2)^2/64 - d^4*(A^4*a^6 + A^4*b^6 + B^4*a^6 + B^4*b^6 + 2*A^2*B^2*a^6 + 2*A^2*B^2*b^6 + 3*A^4
*a^2*b^4 + 3*A^4*a^4*b^2 + 3*B^4*a^2*b^4 + 3*B^4*a^4*b^2 + 6*A^2*B^2*a^2*b^4 + 6*A^2*B^2*a^4*b^2))^(1/2) + A^2
*a^3*d^2 - B^2*a^3*d^2 + 2*A*B*b^3*d^2 - 3*A^2*a*b^2*d^2 + 3*B^2*a*b^2*d^2 - 6*A*B*a^2*b*d^2)/(4*d^4))^(1/2) +
(32*(a + b*tan(c + d*x))^(1/2)*(28*A^2*a^3*b^10*d^2 - 18*A^2*a^5*b^8*d^2 - 28*B^2*a^3*b^10*d^2 + 10*B^2*a^5*b
^8*d^2 - 16*A*B*b^13*d^2 + 22*A^2*a*b^12*d^2 - 22*B^2*a*b^12*d^2 + 16*A*B*a^2*b^11*d^2 + 64*A*B*a^4*b^9*d^2))/
d^4)*((((8*A^2*a^3*d^2 - 8*B^2*a^3*d^2 + 16*A*B*b^3*d^2 - 24*A^2*a*b^2*d^2 + 24*B^2*a*b^2*d^2 - 48*A*B*a^2*b*d
^2)^2/64 - d^4*(A^4*a^6 + A^4*b^6 + B^4*a^6 + B^4*b^6 + 2*A^2*B^2*a^6 + 2*A^2*B^2*b^6 + 3*A^4*a^2*b^4 + 3*A^4*
a^4*b^2 + 3*B^4*a^2*b^4 + 3*B^4*a^4*b^2 + 6*A^2*B^2*a^2*b^4 + 6*A^2*B^2*a^4*b^2))^(1/2) + A^2*a^3*d^2 - B^2*a^
3*d^2 + 2*A*B*b^3*d^2 - 3*A^2*a*b^2*d^2 + 3*B^2*a*b^2*d^2 - 6*A*B*a^2*b*d^2)/(4*d^4))^(1/2) - (32*(3*A^3*a^7*b
^8*d^2 - 21*A^3*a^5*b^10*d^2 - 23*A^3*a^3*b^12*d^2 + 2*B^3*a^2*b^13*d^2 + 4*B^3*a^4*b^11*d^2 + 2*B^3*a^6*b^9*d
^2 + A^3*a*b^14*d^2 + A*B^2*a*b^14*d^2 + 25*A*B^2*a^3*b^12*d^2 + 15*A*B^2*a^5*b^10*d^2 - 9*A*B^2*a^7*b^8*d^2 +
18*A^2*B*a^2*b^13*d^2 - 24*A^2*B*a^4*b^11*d^2 - 42*A^2*B*a^6*b^9*d^2))/d^5)*((((8*A^2*a^3*d^2 - 8*B^2*a^3*d^2
+ 16*A*B*b^3*d^2 - 24*A^2*a*b^2*d^2 + 24*B^2*a*b^2*d^2 - 48*A*B*a^2*b*d^2)^2/64 - d^4*(A^4*a^6 + A^4*b^6 + B^
4*a^6 + B^4*b^6 + 2*A^2*B^2*a^6 + 2*A^2*B^2*b^6 + 3*A^4*a^2*b^4 + 3*A^4*a^4*b^2 + 3*B^4*a^2*b^4 + 3*B^4*a^4*b^
2 + 6*A^2*B^2*a^2*b^4 + 6*A^2*B^2*a^4*b^2))^(1/2) + A^2*a^3*d^2 - B^2*a^3*d^2 + 2*A*B*b^3*d^2 - 3*A^2*a*b^2*d^
2 + 3*B^2*a*b^2*d^2 - 6*A*B*a^2*b*d^2)/(4*d^4))^(1/2) + (32*(a + b*tan(c + d*x))^(1/2)*(A^4*b^16 + B^4*b^16 +
2*A^2*B^2*b^16 + 4*A^4*a^2*b^14 + 8*A^4*a^4*b^12 - 8*A^4*a^6*b^10 + 3*A^4*a^8*b^8 + 4*B^4*a^2*b^14 + 6*B^4*a^4
*b^12 + 4*B^4*a^6*b^10 + B^4*a^8*b^8 + 8*A^2*B^2*a^2*b^14 + 10*A^2*B^2*a^4*b^12 + 20*A^2*B^2*a^6*b^10 + 16*A^3
*B*a^5*b^11 - 16*A^3*B*a^7*b^9))/d^4)*((((8*A^2*a^3*d^2 - 8*B^2*a^3*d^2 + 16*A*B*b^3*d^2 - 24*A^2*a*b^2*d^2 +
24*B^2*a*b^2*d^2 - 48*A*B*a^2*b*d^2)^2/64 - d^4*(A^4*a^6 + A^4*b^6 + B^4*a^6 + B^4*b^6 + 2*A^2*B^2*a^6 + 2*A^2
*B^2*b^6 + 3*A^4*a^2*b^4 + 3*A^4*a^4*b^2 + 3*B^4*a^2*b^4 + 3*B^4*a^4*b^2 + 6*A^2*B^2*a^2*b^4 + 6*A^2*B^2*a^4*b
^2))^(1/2) + A^2*a^3*d^2 - B^2*a^3*d^2 + 2*A*B*b^3*d^2 - 3*A^2*a*b^2*d^2 + 3*B^2*a*b^2*d^2 - 6*A*B*a^2*b*d^2)/
(4*d^4))^(1/2)*1i)/((((((32*(4*B*a*b^11*d^4 + 12*A*a^2*b^10*d^4 + 12*A*a^4*b^8*d^4 + 4*B*a^3*b^9*d^4))/d^5 - (
32*(16*b^10*d^4 + 24*a^2*b^8*d^4)*(a + b*tan(c + d*x))^(1/2)*((((8*A^2*a^3*d^2 - 8*B^2*a^3*d^2 + 16*A*B*b^3*d^
2 - 24*A^2*a*b^2*d^2 + 24*B^2*a*b^2*d^2 - 48*A*B*a^2*b*d^2)^2/64 - d^4*(A^4*a^6 + A^4*b^6 + B^4*a^6 + B^4*b^6
+ 2*A^2*B^2*a^6 + 2*A^2*B^2*b^6 + 3*A^4*a^2*b^4 + 3*A^4*a^4*b^2 + 3*B^4*a^2*b^4 + 3*B^4*a^4*b^2 + 6*A^2*B^2*a^
2*b^4 + 6*A^2*B^2*a^4*b^2))^(1/2) + A^2*a^3*d^2 - B^2*a^3*d^2 + 2*A*B*b^3*d^2 - 3*A^2*a*b^2*d^2 + 3*B^2*a*b^2*
d^2 - 6*A*B*a^2*b*d^2)/(4*d^4))^(1/2))/d^4)*((((8*A^2*a^3*d^2 - 8*B^2*a^3*d^2 + 16*A*B*b^3*d^2 - 24*A^2*a*b^2*
d^2 + 24*B^2*a*b^2*d^2 - 48*A*B*a^2*b*d^2)^2/64 - d^4*(A^4*a^6 + A^4*b^6 + B^4*a^6 + B^4*b^6 + 2*A^2*B^2*a^6 +
2*A^2*B^2*b^6 + 3*A^4*a^2*b^4 + 3*A^4*a^4*b^2 + 3*B^4*a^2*b^4 + 3*B^4*a^4*b^2 + 6*A^2*B^2*a^2*b^4 + 6*A^2*B^2
*a^4*b^2))^(1/2) + A^2*a^3*d^2 - B^2*a^3*d^2 + 2*A*B*b^3*d^2 - 3*A^2*a*b^2*d^2 + 3*B^2*a*b^2*d^2 - 6*A*B*a^2*b
*d^2)/(4*d^4))^(1/2) - (32*(a + b*tan(c + d*x))^(1/2)*(28*A^2*a^3*b^10*d^2 - 18*A^2*a^5*b^8*d^2 - 28*B^2*a^3*b
^10*d^2 + 10*B^2*a^5*b^8*d^2 - 16*A*B*b^13*d^2 + 22*A^2*a*b^12*d^2 - 22*B^2*a*b^12*d^2 + 16*A*B*a^2*b^11*d^2 +
64*A*B*a^4*b^9*d^2))/d^4)*((((8*A^2*a^3*d^2 - 8*B^2*a^3*d^2 + 16*A*B*b^3*d^2 - 24*A^2*a*b^2*d^2 + 24*B^2*a*b^
2*d^2 - 48*A*B*a^2*b*d^2)^2/64 - d^4*(A^4*a^6 + A^4*b^6 + B^4*a^6 + B^4*b^6 + 2*A^2*B^2*a^6 + 2*A^2*B^2*b^6 +
3*A^4*a^2*b^4 + 3*A^4*a^4*b^2 + 3*B^4*a^2*b^4 + 3*B^4*a^4*b^2 + 6*A^2*B^2*a^2*b^4 + 6*A^2*B^2*a^4*b^2))^(1/2)
+ A^2*a^3*d^2 - B^2*a^3*d^2 + 2*A*B*b^3*d^2 - 3*A^2*a*b^2*d^2 + 3*B^2*a*b^2*d^2 - 6*A*B*a^2*b*d^2)/(4*d^4))^(1
/2) - (32*(3*A^3*a^7*b^8*d^2 - 21*A^3*a^5*b^10*d^2 - 23*A^3*a^3*b^12*d^2 + 2*B^3*a^2*b^13*d^2 + 4*B^3*a^4*b^11
*d^2 + 2*B^3*a^6*b^9*d^2 + A^3*a*b^14*d^2 + A*B^2*a*b^14*d^2 + 25*A*B^2*a^3*b^12*d^2 + 15*A*B^2*a^5*b^10*d^2 -
9*A*B^2*a^7*b^8*d^2 + 18*A^2*B*a^2*b^13*d^2 - 24*A^2*B*a^4*b^11*d^2 - 42*A^2*B*a^6*b^9*d^2))/d^5)*((((8*A^2*a
^3*d^2 - 8*B^2*a^3*d^2 + 16*A*B*b^3*d^2 - 24*A^2*a*b^2*d^2 + 24*B^2*a*b^2*d^2 - 48*A*B*a^2*b*d^2)^2/64 - d^4*(
A^4*a^6 + A^4*b^6 + B^4*a^6 + B^4*b^6 + 2*A^2*B^2*a^6 + 2*A^2*B^2*b^6 + 3*A^4*a^2*b^4 + 3*A^4*a^4*b^2 + 3*B^4*
a^2*b^4 + 3*B^4*a^4*b^2 + 6*A^2*B^2*a^2*b^4 + 6*A^2*B^2*a^4*b^2))^(1/2) + A^2*a^3*d^2 - B^2*a^3*d^2 + 2*A*B*b^
3*d^2 - 3*A^2*a*b^2*d^2 + 3*B^2*a*b^2*d^2 - 6*A*B*a^2*b*d^2)/(4*d^4))^(1/2) - (32*(a + b*tan(c + d*x))^(1/2)*(
A^4*b^16 + B^4*b^16 + 2*A^2*B^2*b^16 + 4*A^4*a^2*b^14 + 8*A^4*a^4*b^12 - 8*A^4*a^6*b^10 + 3*A^4*a^8*b^8 + 4*B^
4*a^2*b^14 + 6*B^4*a^4*b^12 + 4*B^4*a^6*b^10 + B^4*a^8*b^8 + 8*A^2*B^2*a^2*b^14 + 10*A^2*B^2*a^4*b^12 + 20*A^2
*B^2*a^6*b^10 + 16*A^3*B*a^5*b^11 - 16*A^3*B*a^7*b^9))/d^4)*((((8*A^2*a^3*d^2 - 8*B^2*a^3*d^2 + 16*A*B*b^3*d^2
- 24*A^2*a*b^2*d^2 + 24*B^2*a*b^2*d^2 - 48*A*B*a^2*b*d^2)^2/64 - d^4*(A^4*a^6 + A^4*b^6 + B^4*a^6 + B^4*b^6 +
2*A^2*B^2*a^6 + 2*A^2*B^2*b^6 + 3*A^4*a^2*b^4 + 3*A^4*a^4*b^2 + 3*B^4*a^2*b^4 + 3*B^4*a^4*b^2 + 6*A^2*B^2*a^2
*b^4 + 6*A^2*B^2*a^4*b^2))^(1/2) + A^2*a^3*d^2 - B^2*a^3*d^2 + 2*A*B*b^3*d^2 - 3*A^2*a*b^2*d^2 + 3*B^2*a*b^2*d
^2 - 6*A*B*a^2*b*d^2)/(4*d^4))^(1/2) + (((((32*(4*B*a*b^11*d^4 + 12*A*a^2*b^10*d^4 + 12*A*a^4*b^8*d^4 + 4*B*a^
3*b^9*d^4))/d^5 + (32*(16*b^10*d^4 + 24*a^2*b^8*d^4)*(a + b*tan(c + d*x))^(1/2)*((((8*A^2*a^3*d^2 - 8*B^2*a^3*
d^2 + 16*A*B*b^3*d^2 - 24*A^2*a*b^2*d^2 + 24*B^2*a*b^2*d^2 - 48*A*B*a^2*b*d^2)^2/64 - d^4*(A^4*a^6 + A^4*b^6 +
B^4*a^6 + B^4*b^6 + 2*A^2*B^2*a^6 + 2*A^2*B^2*b^6 + 3*A^4*a^2*b^4 + 3*A^4*a^4*b^2 + 3*B^4*a^2*b^4 + 3*B^4*a^4
*b^2 + 6*A^2*B^2*a^2*b^4 + 6*A^2*B^2*a^4*b^2))^(1/2) + A^2*a^3*d^2 - B^2*a^3*d^2 + 2*A*B*b^3*d^2 - 3*A^2*a*b^2
*d^2 + 3*B^2*a*b^2*d^2 - 6*A*B*a^2*b*d^2)/(4*d^4))^(1/2))/d^4)*((((8*A^2*a^3*d^2 - 8*B^2*a^3*d^2 + 16*A*B*b^3*
d^2 - 24*A^2*a*b^2*d^2 + 24*B^2*a*b^2*d^2 - 48*A*B*a^2*b*d^2)^2/64 - d^4*(A^4*a^6 + A^4*b^6 + B^4*a^6 + B^4*b^
6 + 2*A^2*B^2*a^6 + 2*A^2*B^2*b^6 + 3*A^4*a^2*b^4 + 3*A^4*a^4*b^2 + 3*B^4*a^2*b^4 + 3*B^4*a^4*b^2 + 6*A^2*B^2*
a^2*b^4 + 6*A^2*B^2*a^4*b^2))^(1/2) + A^2*a^3*d^2 - B^2*a^3*d^2 + 2*A*B*b^3*d^2 - 3*A^2*a*b^2*d^2 + 3*B^2*a*b^
2*d^2 - 6*A*B*a^2*b*d^2)/(4*d^4))^(1/2) + (32*(a + b*tan(c + d*x))^(1/2)*(28*A^2*a^3*b^10*d^2 - 18*A^2*a^5*b^8
*d^2 - 28*B^2*a^3*b^10*d^2 + 10*B^2*a^5*b^8*d^2 - 16*A*B*b^13*d^2 + 22*A^2*a*b^12*d^2 - 22*B^2*a*b^12*d^2 + 16
*A*B*a^2*b^11*d^2 + 64*A*B*a^4*b^9*d^2))/d^4)*((((8*A^2*a^3*d^2 - 8*B^2*a^3*d^2 + 16*A*B*b^3*d^2 - 24*A^2*a*b^
2*d^2 + 24*B^2*a*b^2*d^2 - 48*A*B*a^2*b*d^2)^2/64 - d^4*(A^4*a^6 + A^4*b^6 + B^4*a^6 + B^4*b^6 + 2*A^2*B^2*a^6
+ 2*A^2*B^2*b^6 + 3*A^4*a^2*b^4 + 3*A^4*a^4*b^2 + 3*B^4*a^2*b^4 + 3*B^4*a^4*b^2 + 6*A^2*B^2*a^2*b^4 + 6*A^2*B
^2*a^4*b^2))^(1/2) + A^2*a^3*d^2 - B^2*a^3*d^2 + 2*A*B*b^3*d^2 - 3*A^2*a*b^2*d^2 + 3*B^2*a*b^2*d^2 - 6*A*B*a^2
*b*d^2)/(4*d^4))^(1/2) - (32*(3*A^3*a^7*b^8*d^2 - 21*A^3*a^5*b^10*d^2 - 23*A^3*a^3*b^12*d^2 + 2*B^3*a^2*b^13*d
^2 + 4*B^3*a^4*b^11*d^2 + 2*B^3*a^6*b^9*d^2 + A^3*a*b^14*d^2 + A*B^2*a*b^14*d^2 + 25*A*B^2*a^3*b^12*d^2 + 15*A
*B^2*a^5*b^10*d^2 - 9*A*B^2*a^7*b^8*d^2 + 18*A^2*B*a^2*b^13*d^2 - 24*A^2*B*a^4*b^11*d^2 - 42*A^2*B*a^6*b^9*d^2
))/d^5)*((((8*A^2*a^3*d^2 - 8*B^2*a^3*d^2 + 16*A*B*b^3*d^2 - 24*A^2*a*b^2*d^2 + 24*B^2*a*b^2*d^2 - 48*A*B*a^2*
b*d^2)^2/64 - d^4*(A^4*a^6 + A^4*b^6 + B^4*a^6 + B^4*b^6 + 2*A^2*B^2*a^6 + 2*A^2*B^2*b^6 + 3*A^4*a^2*b^4 + 3*A
^4*a^4*b^2 + 3*B^4*a^2*b^4 + 3*B^4*a^4*b^2 + 6*A^2*B^2*a^2*b^4 + 6*A^2*B^2*a^4*b^2))^(1/2) + A^2*a^3*d^2 - B^2
*a^3*d^2 + 2*A*B*b^3*d^2 - 3*A^2*a*b^2*d^2 + 3*B^2*a*b^2*d^2 - 6*A*B*a^2*b*d^2)/(4*d^4))^(1/2) + (32*(a + b*ta
n(c + d*x))^(1/2)*(A^4*b^16 + B^4*b^16 + 2*A^2*B^2*b^16 + 4*A^4*a^2*b^14 + 8*A^4*a^4*b^12 - 8*A^4*a^6*b^10 + 3
*A^4*a^8*b^8 + 4*B^4*a^2*b^14 + 6*B^4*a^4*b^12 + 4*B^4*a^6*b^10 + B^4*a^8*b^8 + 8*A^2*B^2*a^2*b^14 + 10*A^2*B^
2*a^4*b^12 + 20*A^2*B^2*a^6*b^10 + 16*A^3*B*a^5*b^11 - 16*A^3*B*a^7*b^9))/d^4)*((((8*A^2*a^3*d^2 - 8*B^2*a^3*d
^2 + 16*A*B*b^3*d^2 - 24*A^2*a*b^2*d^2 + 24*B^2*a*b^2*d^2 - 48*A*B*a^2*b*d^2)^2/64 - d^4*(A^4*a^6 + A^4*b^6 +
B^4*a^6 + B^4*b^6 + 2*A^2*B^2*a^6 + 2*A^2*B^2*b^6 + 3*A^4*a^2*b^4 + 3*A^4*a^4*b^2 + 3*B^4*a^2*b^4 + 3*B^4*a^4*
b^2 + 6*A^2*B^2*a^2*b^4 + 6*A^2*B^2*a^4*b^2))^(1/2) + A^2*a^3*d^2 - B^2*a^3*d^2 + 2*A*B*b^3*d^2 - 3*A^2*a*b^2*
d^2 + 3*B^2*a*b^2*d^2 - 6*A*B*a^2*b*d^2)/(4*d^4))^(1/2) + (64*(A^5*a^2*b^16 + 5*A^5*a^4*b^14 + 7*A^5*a^6*b^12
+ 3*A^5*a^8*b^10 + 2*A^2*B^3*a^5*b^13 + 4*A^2*B^3*a^7*b^11 + 2*A^2*B^3*a^9*b^9 + 2*A^3*B^2*a^2*b^16 + 9*A^3*B^
2*a^4*b^14 + 13*A^3*B^2*a^6*b^12 + 7*A^3*B^2*a^8*b^10 + A^3*B^2*a^10*b^8 + A*B^4*a^2*b^16 + 4*A*B^4*a^4*b^14 +
6*A*B^4*a^6*b^12 + 4*A*B^4*a^8*b^10 + A*B^4*a^10*b^8 + 2*A^4*B*a^5*b^13 + 4*A^4*B*a^7*b^11 + 2*A^4*B*a^9*b^9)
)/d^5))*((((8*A^2*a^3*d^2 - 8*B^2*a^3*d^2 + 16*A*B*b^3*d^2 - 24*A^2*a*b^2*d^2 + 24*B^2*a*b^2*d^2 - 48*A*B*a^2*
b*d^2)^2/64 - d^4*(A^4*a^6 + A^4*b^6 + B^4*a^6 + B^4*b^6 + 2*A^2*B^2*a^6 + 2*A^2*B^2*b^6 + 3*A^4*a^2*b^4 + 3*A
^4*a^4*b^2 + 3*B^4*a^2*b^4 + 3*B^4*a^4*b^2 + 6*A^2*B^2*a^2*b^4 + 6*A^2*B^2*a^4*b^2))^(1/2) + A^2*a^3*d^2 - B^2
*a^3*d^2 + 2*A*B*b^3*d^2 - 3*A^2*a*b^2*d^2 + 3*B^2*a*b^2*d^2 - 6*A*B*a^2*b*d^2)/(4*d^4))^(1/2)*2i + (A*atan(((
A*((32*(a + b*tan(c + d*x))^(1/2)*(A^4*b^16 + B^4*b^16 + 2*A^2*B^2*b^16 + 4*A^4*a^2*b^14 + 8*A^4*a^4*b^12 - 8*
A^4*a^6*b^10 + 3*A^4*a^8*b^8 + 4*B^4*a^2*b^14 + 6*B^4*a^4*b^12 + 4*B^4*a^6*b^10 + B^4*a^8*b^8 + 8*A^2*B^2*a^2*
b^14 + 10*A^2*B^2*a^4*b^12 + 20*A^2*B^2*a^6*b^10 + 16*A^3*B*a^5*b^11 - 16*A^3*B*a^7*b^9))/d^4 + (A*((32*(3*A^3
*a^7*b^8*d^2 - 21*A^3*a^5*b^10*d^2 - 23*A^3*a^3*b^12*d^2 + 2*B^3*a^2*b^13*d^2 + 4*B^3*a^4*b^11*d^2 + 2*B^3*a^6
*b^9*d^2 + A^3*a*b^14*d^2 + A*B^2*a*b^14*d^2 + 25*A*B^2*a^3*b^12*d^2 + 15*A*B^2*a^5*b^10*d^2 - 9*A*B^2*a^7*b^8
*d^2 + 18*A^2*B*a^2*b^13*d^2 - 24*A^2*B*a^4*b^11*d^2 - 42*A^2*B*a^6*b^9*d^2))/d^5 + (A*((32*(a + b*tan(c + d*x
))^(1/2)*(28*A^2*a^3*b^10*d^2 - 18*A^2*a^5*b^8*d^2 - 28*B^2*a^3*b^10*d^2 + 10*B^2*a^5*b^8*d^2 - 16*A*B*b^13*d^
2 + 22*A^2*a*b^12*d^2 - 22*B^2*a*b^12*d^2 + 16*A*B*a^2*b^11*d^2 + 64*A*B*a^4*b^9*d^2))/d^4 - (A*(a^3)^(1/2)*((
32*(4*B*a*b^11*d^4 + 12*A*a^2*b^10*d^4 + 12*A*a^4*b^8*d^4 + 4*B*a^3*b^9*d^4))/d^5 - (32*A*(a^3)^(1/2)*(16*b^10
*d^4 + 24*a^2*b^8*d^4)*(a + b*tan(c + d*x))^(1/2))/d^5))/d)*(a^3)^(1/2))/d)*(a^3)^(1/2))/d)*(a^3)^(1/2)*1i)/d
+ (A*((32*(a + b*tan(c + d*x))^(1/2)*(A^4*b^16 + B^4*b^16 + 2*A^2*B^2*b^16 + 4*A^4*a^2*b^14 + 8*A^4*a^4*b^12 -
8*A^4*a^6*b^10 + 3*A^4*a^8*b^8 + 4*B^4*a^2*b^14 + 6*B^4*a^4*b^12 + 4*B^4*a^6*b^10 + B^4*a^8*b^8 + 8*A^2*B^2*a
^2*b^14 + 10*A^2*B^2*a^4*b^12 + 20*A^2*B^2*a^6*b^10 + 16*A^3*B*a^5*b^11 - 16*A^3*B*a^7*b^9))/d^4 - (A*((32*(3*
A^3*a^7*b^8*d^2 - 21*A^3*a^5*b^10*d^2 - 23*A^3*a^3*b^12*d^2 + 2*B^3*a^2*b^13*d^2 + 4*B^3*a^4*b^11*d^2 + 2*B^3*
a^6*b^9*d^2 + A^3*a*b^14*d^2 + A*B^2*a*b^14*d^2 + 25*A*B^2*a^3*b^12*d^2 + 15*A*B^2*a^5*b^10*d^2 - 9*A*B^2*a^7*
b^8*d^2 + 18*A^2*B*a^2*b^13*d^2 - 24*A^2*B*a^4*b^11*d^2 - 42*A^2*B*a^6*b^9*d^2))/d^5 - (A*((32*(a + b*tan(c +
d*x))^(1/2)*(28*A^2*a^3*b^10*d^2 - 18*A^2*a^5*b^8*d^2 - 28*B^2*a^3*b^10*d^2 + 10*B^2*a^5*b^8*d^2 - 16*A*B*b^13
*d^2 + 22*A^2*a*b^12*d^2 - 22*B^2*a*b^12*d^2 + 16*A*B*a^2*b^11*d^2 + 64*A*B*a^4*b^9*d^2))/d^4 + (A*(a^3)^(1/2)
*((32*(4*B*a*b^11*d^4 + 12*A*a^2*b^10*d^4 + 12*A*a^4*b^8*d^4 + 4*B*a^3*b^9*d^4))/d^5 + (32*A*(a^3)^(1/2)*(16*b
^10*d^4 + 24*a^2*b^8*d^4)*(a + b*tan(c + d*x))^(1/2))/d^5))/d)*(a^3)^(1/2))/d)*(a^3)^(1/2))/d)*(a^3)^(1/2)*1i)
/d)/((64*(A^5*a^2*b^16 + 5*A^5*a^4*b^14 + 7*A^5*a^6*b^12 + 3*A^5*a^8*b^10 + 2*A^2*B^3*a^5*b^13 + 4*A^2*B^3*a^7
*b^11 + 2*A^2*B^3*a^9*b^9 + 2*A^3*B^2*a^2*b^16 + 9*A^3*B^2*a^4*b^14 + 13*A^3*B^2*a^6*b^12 + 7*A^3*B^2*a^8*b^10
+ A^3*B^2*a^10*b^8 + A*B^4*a^2*b^16 + 4*A*B^4*a^4*b^14 + 6*A*B^4*a^6*b^12 + 4*A*B^4*a^8*b^10 + A*B^4*a^10*b^8
+ 2*A^4*B*a^5*b^13 + 4*A^4*B*a^7*b^11 + 2*A^4*B*a^9*b^9))/d^5 - (A*((32*(a + b*tan(c + d*x))^(1/2)*(A^4*b^16
+ B^4*b^16 + 2*A^2*B^2*b^16 + 4*A^4*a^2*b^14 + 8*A^4*a^4*b^12 - 8*A^4*a^6*b^10 + 3*A^4*a^8*b^8 + 4*B^4*a^2*b^1
4 + 6*B^4*a^4*b^12 + 4*B^4*a^6*b^10 + B^4*a^8*b^8 + 8*A^2*B^2*a^2*b^14 + 10*A^2*B^2*a^4*b^12 + 20*A^2*B^2*a^6*
b^10 + 16*A^3*B*a^5*b^11 - 16*A^3*B*a^7*b^9))/d^4 + (A*((32*(3*A^3*a^7*b^8*d^2 - 21*A^3*a^5*b^10*d^2 - 23*A^3*
a^3*b^12*d^2 + 2*B^3*a^2*b^13*d^2 + 4*B^3*a^4*b^11*d^2 + 2*B^3*a^6*b^9*d^2 + A^3*a*b^14*d^2 + A*B^2*a*b^14*d^2
+ 25*A*B^2*a^3*b^12*d^2 + 15*A*B^2*a^5*b^10*d^2 - 9*A*B^2*a^7*b^8*d^2 + 18*A^2*B*a^2*b^13*d^2 - 24*A^2*B*a^4*
b^11*d^2 - 42*A^2*B*a^6*b^9*d^2))/d^5 + (A*((32*(a + b*tan(c + d*x))^(1/2)*(28*A^2*a^3*b^10*d^2 - 18*A^2*a^5*b
^8*d^2 - 28*B^2*a^3*b^10*d^2 + 10*B^2*a^5*b^8*d^2 - 16*A*B*b^13*d^2 + 22*A^2*a*b^12*d^2 - 22*B^2*a*b^12*d^2 +
16*A*B*a^2*b^11*d^2 + 64*A*B*a^4*b^9*d^2))/d^4 - (A*(a^3)^(1/2)*((32*(4*B*a*b^11*d^4 + 12*A*a^2*b^10*d^4 + 12*
A*a^4*b^8*d^4 + 4*B*a^3*b^9*d^4))/d^5 - (32*A*(a^3)^(1/2)*(16*b^10*d^4 + 24*a^2*b^8*d^4)*(a + b*tan(c + d*x))^
(1/2))/d^5))/d)*(a^3)^(1/2))/d)*(a^3)^(1/2))/d)*(a^3)^(1/2))/d + (A*((32*(a + b*tan(c + d*x))^(1/2)*(A^4*b^16
+ B^4*b^16 + 2*A^2*B^2*b^16 + 4*A^4*a^2*b^14 + 8*A^4*a^4*b^12 - 8*A^4*a^6*b^10 + 3*A^4*a^8*b^8 + 4*B^4*a^2*b^1
4 + 6*B^4*a^4*b^12 + 4*B^4*a^6*b^10 + B^4*a^8*b^8 + 8*A^2*B^2*a^2*b^14 + 10*A^2*B^2*a^4*b^12 + 20*A^2*B^2*a^6*
b^10 + 16*A^3*B*a^5*b^11 - 16*A^3*B*a^7*b^9))/d^4 - (A*((32*(3*A^3*a^7*b^8*d^2 - 21*A^3*a^5*b^10*d^2 - 23*A^3*
a^3*b^12*d^2 + 2*B^3*a^2*b^13*d^2 + 4*B^3*a^4*b^11*d^2 + 2*B^3*a^6*b^9*d^2 + A^3*a*b^14*d^2 + A*B^2*a*b^14*d^2
+ 25*A*B^2*a^3*b^12*d^2 + 15*A*B^2*a^5*b^10*d^2 - 9*A*B^2*a^7*b^8*d^2 + 18*A^2*B*a^2*b^13*d^2 - 24*A^2*B*a^4*
b^11*d^2 - 42*A^2*B*a^6*b^9*d^2))/d^5 - (A*((32*(a + b*tan(c + d*x))^(1/2)*(28*A^2*a^3*b^10*d^2 - 18*A^2*a^5*b
^8*d^2 - 28*B^2*a^3*b^10*d^2 + 10*B^2*a^5*b^8*d^2 - 16*A*B*b^13*d^2 + 22*A^2*a*b^12*d^2 - 22*B^2*a*b^12*d^2 +
16*A*B*a^2*b^11*d^2 + 64*A*B*a^4*b^9*d^2))/d^4 + (A*(a^3)^(1/2)*((32*(4*B*a*b^11*d^4 + 12*A*a^2*b^10*d^4 + 12*
A*a^4*b^8*d^4 + 4*B*a^3*b^9*d^4))/d^5 + (32*A*(a^3)^(1/2)*(16*b^10*d^4 + 24*a^2*b^8*d^4)*(a + b*tan(c + d*x))^
(1/2))/d^5))/d)*(a^3)^(1/2))/d)*(a^3)^(1/2))/d)*(a^3)^(1/2))/d))*(a^3)^(1/2)*2i)/d