\(\int \cot ^4(c+d x) (a+b \tan (c+d x))^{3/2} (A+B \tan (c+d x)) \, dx\) [331]

Optimal result

Integrand size = 33, antiderivative size = 278 \[ \int \cot ^4(c+d x) (a+b \tan (c+d x))^{3/2} (A+B \tan (c+d x)) \, dx=\frac {\left (24 a^2 A b+A b^3+16 a^3 B-6 a b^2 B\right ) \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a}}\right )}{8 a^{3/2} d}-\frac {(a-i b)^{3/2} (i A+B) \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a-i b}}\right )}{d}+\frac {(a+i b)^{3/2} (i A-B) \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a+i b}}\right )}{d}+\frac {\left (8 a^2 A-A b^2-10 a b B\right ) \cot (c+d x) \sqrt {a+b \tan (c+d x)}}{8 a d}-\frac {(7 A b+6 a B) \cot ^2(c+d x) \sqrt {a+b \tan (c+d x)}}{12 d}-\frac {a A \cot ^3(c+d x) \sqrt {a+b \tan (c+d x)}}{3 d} \]

[Out]

1/8*(24*A*a^2*b+A*b^3+16*B*a^3-6*B*a*b^2)*arctanh((a+b*tan(d*x+c))^(1/2)/a^(1/2))/a^(3/2)/d-(a-I*b)^(3/2)*(I*A
+B)*arctanh((a+b*tan(d*x+c))^(1/2)/(a-I*b)^(1/2))/d+(a+I*b)^(3/2)*(I*A-B)*arctanh((a+b*tan(d*x+c))^(1/2)/(a+I*
b)^(1/2))/d+1/8*(8*A*a^2-A*b^2-10*B*a*b)*cot(d*x+c)*(a+b*tan(d*x+c))^(1/2)/a/d-1/12*(7*A*b+6*B*a)*cot(d*x+c)^2
*(a+b*tan(d*x+c))^(1/2)/d-1/3*a*A*cot(d*x+c)^3*(a+b*tan(d*x+c))^(1/2)/d

Rubi [A] (verified)

Time = 1.59 (sec) , antiderivative size = 278, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.242, Rules used = {3686, 3730, 3734, 3620, 3618, 65, 214, 3715} \[ \int \cot ^4(c+d x) (a+b \tan (c+d x))^{3/2} (A+B \tan (c+d x)) \, dx=\frac {\left (8 a^2 A-10 a b B-A b^2\right ) \cot (c+d x) \sqrt {a+b \tan (c+d x)}}{8 a d}+\frac {\left (16 a^3 B+24 a^2 A b-6 a b^2 B+A b^3\right ) \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a}}\right )}{8 a^{3/2} d}-\frac {(a-i b)^{3/2} (B+i A) \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a-i b}}\right )}{d}+\frac {(a+i b)^{3/2} (-B+i A) \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a+i b}}\right )}{d}-\frac {(6 a B+7 A b) \cot ^2(c+d x) \sqrt {a+b \tan (c+d x)}}{12 d}-\frac {a A \cot ^3(c+d x) \sqrt {a+b \tan (c+d x)}}{3 d} \]

[In]

Int[Cot[c + d*x]^4*(a + b*Tan[c + d*x])^(3/2)*(A + B*Tan[c + d*x]),x]

[Out]

((24*a^2*A*b + A*b^3 + 16*a^3*B - 6*a*b^2*B)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a]])/(8*a^(3/2)*d) - ((a -
I*b)^(3/2)*(I*A + B)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a - I*b]])/d + ((a + I*b)^(3/2)*(I*A - B)*ArcTanh[S
qrt[a + b*Tan[c + d*x]]/Sqrt[a + I*b]])/d + ((8*a^2*A - A*b^2 - 10*a*b*B)*Cot[c + d*x]*Sqrt[a + b*Tan[c + d*x]
])/(8*a*d) - ((7*A*b + 6*a*B)*Cot[c + d*x]^2*Sqrt[a + b*Tan[c + d*x]])/(12*d) - (a*A*Cot[c + d*x]^3*Sqrt[a + b
*Tan[c + d*x]])/(3*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 3686

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*c - a*d)*(B*c - A*d)*(a + b*Tan[e + f*x])^(m - 1)*((c + d*Tan[e
+ f*x])^(n + 1)/(d*f*(n + 1)*(c^2 + d^2))), x] - Dist[1/(d*(n + 1)*(c^2 + d^2)), Int[(a + b*Tan[e + f*x])^(m -
2)*(c + d*Tan[e + f*x])^(n + 1)*Simp[a*A*d*(b*d*(m - 1) - a*c*(n + 1)) + (b*B*c - (A*b + a*B)*d)*(b*c*(m - 1)
+ a*d*(n + 1)) - d*((a*A - b*B)*(b*c - a*d) + (A*b + a*B)*(a*c + b*d))*(n + 1)*Tan[e + f*x] - b*(d*(A*b*c + a
*B*c - a*A*d)*(m + n) - b*B*(c^2*(m - 1) - d^2*(n + 1)))*Tan[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f
, A, B}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && GtQ[m, 1] && LtQ[n, -1] && (Inte
gerQ[m] || IntegersQ[2*m, 2*n])

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 3730

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

Rule 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 {a A \cot ^3(c+d x) \sqrt {a+b \tan (c+d x)}}{3 d}+\frac {1}{3} \int \frac {\cot ^3(c+d x) \left (\frac {1}{2} a (7 A b+6 a B)-3 \left (a^2 A-A b^2-2 a b B\right ) \tan (c+d x)-\frac {1}{2} b (5 a A-6 b B) \tan ^2(c+d x)\right )}{\sqrt {a+b \tan (c+d x)}} \, dx \\ & = -\frac {(7 A b+6 a B) \cot ^2(c+d x) \sqrt {a+b \tan (c+d x)}}{12 d}-\frac {a A \cot ^3(c+d x) \sqrt {a+b \tan (c+d x)}}{3 d}-\frac {\int \frac {\cot ^2(c+d x) \left (\frac {3}{4} a \left (8 a^2 A-A b^2-10 a b B\right )+6 a \left (2 a A b+a^2 B-b^2 B\right ) \tan (c+d x)+\frac {3}{4} a b (7 A b+6 a B) \tan ^2(c+d x)\right )}{\sqrt {a+b \tan (c+d x)}} \, dx}{6 a} \\ & = \frac {\left (8 a^2 A-A b^2-10 a b B\right ) \cot (c+d x) \sqrt {a+b \tan (c+d x)}}{8 a d}-\frac {(7 A b+6 a B) \cot ^2(c+d x) \sqrt {a+b \tan (c+d x)}}{12 d}-\frac {a A \cot ^3(c+d x) \sqrt {a+b \tan (c+d x)}}{3 d}+\frac {\int \frac {\cot (c+d x) \left (-\frac {3}{8} a \left (24 a^2 A b+A b^3+16 a^3 B-6 a b^2 B\right )+6 a^2 \left (a^2 A-A b^2-2 a b B\right ) \tan (c+d x)+\frac {3}{8} a b \left (8 a^2 A-A b^2-10 a b B\right ) \tan ^2(c+d x)\right )}{\sqrt {a+b \tan (c+d x)}} \, dx}{6 a^2} \\ & = \frac {\left (8 a^2 A-A b^2-10 a b B\right ) \cot (c+d x) \sqrt {a+b \tan (c+d x)}}{8 a d}-\frac {(7 A b+6 a B) \cot ^2(c+d x) \sqrt {a+b \tan (c+d x)}}{12 d}-\frac {a A \cot ^3(c+d x) \sqrt {a+b \tan (c+d x)}}{3 d}+\frac {\int \frac {6 a^2 \left (a^2 A-A b^2-2 a b B\right )+6 a^2 \left (2 a A b+a^2 B-b^2 B\right ) \tan (c+d x)}{\sqrt {a+b \tan (c+d x)}} \, dx}{6 a^2}-\frac {\left (24 a^2 A b+A b^3+16 a^3 B-6 a b^2 B\right ) \int \frac {\cot (c+d x) \left (1+\tan ^2(c+d x)\right )}{\sqrt {a+b \tan (c+d x)}} \, dx}{16 a} \\ & = \frac {\left (8 a^2 A-A b^2-10 a b B\right ) \cot (c+d x) \sqrt {a+b \tan (c+d x)}}{8 a d}-\frac {(7 A b+6 a B) \cot ^2(c+d x) \sqrt {a+b \tan (c+d x)}}{12 d}-\frac {a A \cot ^3(c+d x) \sqrt {a+b \tan (c+d x)}}{3 d}+\frac {1}{2} \left ((a-i b)^2 (A-i 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 (A+i B)\right ) \int \frac {1-i \tan (c+d x)}{\sqrt {a+b \tan (c+d x)}} \, dx-\frac {\left (24 a^2 A b+A b^3+16 a^3 B-6 a b^2 B\right ) \text {Subst}\left (\int \frac {1}{x \sqrt {a+b x}} \, dx,x,\tan (c+d x)\right )}{16 a d} \\ & = \frac {\left (8 a^2 A-A b^2-10 a b B\right ) \cot (c+d x) \sqrt {a+b \tan (c+d x)}}{8 a d}-\frac {(7 A b+6 a B) \cot ^2(c+d x) \sqrt {a+b \tan (c+d x)}}{12 d}-\frac {a A \cot ^3(c+d x) \sqrt {a+b \tan (c+d x)}}{3 d}-\frac {\left ((a+i b)^2 (i A-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 (i A+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 (24 a^2 A b+A b^3+16 a^3 B-6 a b^2 B\right ) \text {Subst}\left (\int \frac {1}{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b \tan (c+d x)}\right )}{8 a b d} \\ & = \frac {\left (24 a^2 A b+A b^3+16 a^3 B-6 a b^2 B\right ) \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a}}\right )}{8 a^{3/2} d}+\frac {\left (8 a^2 A-A b^2-10 a b B\right ) \cot (c+d x) \sqrt {a+b \tan (c+d x)}}{8 a d}-\frac {(7 A b+6 a B) \cot ^2(c+d x) \sqrt {a+b \tan (c+d x)}}{12 d}-\frac {a A \cot ^3(c+d x) \sqrt {a+b \tan (c+d x)}}{3 d}-\frac {\left ((a-i b)^2 (A-i 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 (A+i 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 (24 a^2 A b+A b^3+16 a^3 B-6 a b^2 B\right ) \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a}}\right )}{8 a^{3/2} d}-\frac {(a-i b)^{3/2} (i A+B) \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a-i b}}\right )}{d}+\frac {(a+i b)^{3/2} (i A-B) \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a+i b}}\right )}{d}+\frac {\left (8 a^2 A-A b^2-10 a b B\right ) \cot (c+d x) \sqrt {a+b \tan (c+d x)}}{8 a d}-\frac {(7 A b+6 a B) \cot ^2(c+d x) \sqrt {a+b \tan (c+d x)}}{12 d}-\frac {a A \cot ^3(c+d x) \sqrt {a+b \tan (c+d x)}}{3 d} \\ \end{align*}

Mathematica [A] (verified)

Time = 5.97 (sec) , antiderivative size = 241, normalized size of antiderivative = 0.87 \[ \int \cot ^4(c+d x) (a+b \tan (c+d x))^{3/2} (A+B \tan (c+d x)) \, dx=\frac {3 \left (24 a^2 A b+A b^3+16 a^3 B-6 a b^2 B\right ) \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a}}\right )+\sqrt {a} \left (-24 i a (a-i b)^{3/2} (A-i B) \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a-i b}}\right )+24 i a (a+i b)^{3/2} (A+i B) \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a+i b}}\right )-\cot (c+d x) \left (-24 a^2 A+3 A b^2+30 a b B+2 a (7 A b+6 a B) \cot (c+d x)+8 a^2 A \cot ^2(c+d x)\right ) \sqrt {a+b \tan (c+d x)}\right )}{24 a^{3/2} d} \]

[In]

Integrate[Cot[c + d*x]^4*(a + b*Tan[c + d*x])^(3/2)*(A + B*Tan[c + d*x]),x]

[Out]

(3*(24*a^2*A*b + A*b^3 + 16*a^3*B - 6*a*b^2*B)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a]] + Sqrt[a]*((-24*I)*a*
(a - I*b)^(3/2)*(A - I*B)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a - I*b]] + (24*I)*a*(a + I*b)^(3/2)*(A + I*B)
*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a + I*b]] - Cot[c + d*x]*(-24*a^2*A + 3*A*b^2 + 30*a*b*B + 2*a*(7*A*b +
6*a*B)*Cot[c + d*x] + 8*a^2*A*Cot[c + d*x]^2)*Sqrt[a + b*Tan[c + d*x]]))/(24*a^(3/2)*d)

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1984\) vs. \(2(240)=480\).

Time = 0.22 (sec) , antiderivative size = 1985, normalized size of antiderivative = 7.14

method	result	size
derivativedivides	\(\text {Expression too large to display}\)	\(1985\)
default	\(\text {Expression too large to display}\)	\(1985\)

[In]

int(cot(d*x+c)^4*(a+b*tan(d*x+c))^(3/2)*(A+B*tan(d*x+c)),x,method=_RETURNVERBOSE)

[Out]

2/d*a^(3/2)*arctanh((a+b*tan(d*x+c))^(1/2)/a^(1/2))*B-1/3/d/tan(d*x+c)^3*A*(a+b*tan(d*x+c))^(3/2)+3/d*b*A*a^(1
/2)*arctanh((a+b*tan(d*x+c))^(1/2)/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))*A*(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))*A*(a^2+b^2)^(1/2)-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))*A*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))*B*(a^2+b^2)^(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))*A*(a^2+b^2)^(1/2)*(2*(a^2+b^2)^(1/2)+2*a)^(1/2)*a-1/4/d/b*ln(b*ta
n(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*(a^2+b^2)^(1/2)*(2*(a^2+b^2
)^(1/2)+2*a)^(1/2)*a-5/4/d/b/tan(d*x+c)^3*(a+b*tan(d*x+c))^(5/2)*B-1/8/d/tan(d*x+c)^3/a*(a+b*tan(d*x+c))^(5/2)
*A+1/8/d/tan(d*x+c)^3*(a+b*tan(d*x+c))^(1/2)*A*a+1/8/d*b^3/a^(3/2)*arctanh((a+b*tan(d*x+c))^(1/2)/a^(1/2))*A-3
/4/d*b^2/a^(1/2)*arctanh((a+b*tan(d*x+c))^(1/2)/a^(1/2))*B+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))*B*(a^2+b^2)^(1/2)*(2*(a^2+b^2)^(1/2)+2*a)^(1/2)-1/2/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))*B*(2*(a^2+b^2)^(1/2)+2*a)^(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))*B*a^2-1/d*b^2/(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-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))*A*(2*(a^2+b^2)^(1/2)+2*a)^(1/2)-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))*B*(a^2+b^2)^(1/2)*(2*(a^2+b^2)^(1/2)+2*a)^(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))*B*(2*(a^2+b^2)^(1/2)+2*a)^(
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))*B*b^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))*A*(2*(a^2+b^2)^(1/2)+2*a)^(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))*B*a^2-1/d/(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^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))*A*(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))*A*(a^2+b^2)^(1/2)+2/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))*A*a+1/d/b^2/tan(d*x+c)^3
*(a+b*tan(d*x+c))^(1/2)*A*a^3-3/4/d/b/tan(d*x+c)^3*(a+b*tan(d*x+c))^(1/2)*B*a^2+1/d/b^2/tan(d*x+c)^3*a*(a+b*ta
n(d*x+c))^(5/2)*A-2/d/b^2/tan(d*x+c)^3*A*(a+b*tan(d*x+c))^(3/2)*a^2+2/d/b/tan(d*x+c)^3*B*(a+b*tan(d*x+c))^(3/2
)*a

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 3218 vs. \(2 (234) = 468\).

Time = 15.33 (sec) , antiderivative size = 6452, normalized size of antiderivative = 23.21 \[ \int \cot ^4(c+d x) (a+b \tan (c+d x))^{3/2} (A+B \tan (c+d x)) \, dx=\text {Too large to display} \]

[In]

integrate(cot(d*x+c)^4*(a+b*tan(d*x+c))^(3/2)*(A+B*tan(d*x+c)),x, algorithm="fricas")

[Out]

Too large to include

Sympy [F]

\[ \int \cot ^4(c+d x) (a+b \tan (c+d x))^{3/2} (A+B \tan (c+d x)) \, dx=\int \left (A + B \tan {\left (c + d x \right )}\right ) \left (a + b \tan {\left (c + d x \right )}\right )^{\frac {3}{2}} \cot ^{4}{\left (c + d x \right )}\, dx \]

[In]

integrate(cot(d*x+c)**4*(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)**4, x)

Maxima [F(-1)]

Timed out. \[ \int \cot ^4(c+d x) (a+b \tan (c+d x))^{3/2} (A+B \tan (c+d x)) \, dx=\text {Timed out} \]

[In]

integrate(cot(d*x+c)^4*(a+b*tan(d*x+c))^(3/2)*(A+B*tan(d*x+c)),x, algorithm="maxima")

[Out]

Timed out

Giac [F(-1)]

Timed out. \[ \int \cot ^4(c+d x) (a+b \tan (c+d x))^{3/2} (A+B \tan (c+d x)) \, dx=\text {Timed out} \]

[In]

integrate(cot(d*x+c)^4*(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 = 10.79 (sec) , antiderivative size = 25789, normalized size of antiderivative = 92.77 \[ \int \cot ^4(c+d x) (a+b \tan (c+d x))^{3/2} (A+B \tan (c+d x)) \, dx=\text {Too large to display} \]

[In]

int(cot(c + d*x)^4*(A + B*tan(c + d*x))*(a + b*tan(c + d*x))^(3/2),x)

[Out]

atan(((((((256*A*a*b^13*d^4 + 5376*A*a^3*b^11*d^4 + 5120*A*a^5*b^9*d^4 - 1536*B*a^2*b^12*d^4 + 1536*B*a^4*b^10
*d^4 + 3072*B*a^6*b^8*d^4)/(8*a^2*d^5) - ((2048*a^2*b^10*d^4 + 3072*a^4*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))/(4*a^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) + ((a + b*tan(c + d*x))^(1/2)*(3
008*A^2*a^3*b^12*d^2 + 5888*A^2*a^5*b^10*d^2 - 1280*A^2*a^7*b^8*d^2 - 2672*B^2*a^3*b^12*d^2 - 4352*B^2*a^5*b^1
0*d^2 + 2304*B^2*a^7*b^8*d^2 + 4*A^2*a*b^14*d^2 - 2096*A*B*a^2*b^13*d^2 + 1024*A*B*a^4*b^11*d^2 + 11264*A*B*a^
6*b^9*d^2))/(4*a^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) + (4*A^3*b^17*d^2 - 188*A^3*a^2*b^15*d^2 - 6464*A^3*a^4*b^13*d^2 - 2688*A^3*a^6*b^11*d^2 + 3584*A^3*a^8*b^9
*d^2 - 2416*B^3*a^3*b^14*d^2 + 3728*B^3*a^5*b^12*d^2 + 5376*B^3*a^7*b^10*d^2 - 768*B^3*a^9*b^8*d^2 + 212*A^2*B
*a*b^16*d^2 - 1056*A*B^2*a^2*b^15*d^2 + 15456*A*B^2*a^4*b^13*d^2 + 5760*A*B^2*a^6*b^11*d^2 - 10752*A*B^2*a^8*b
^9*d^2 + 6804*A^2*B*a^3*b^14*d^2 - 13376*A^2*B*a^5*b^12*d^2 - 17664*A^2*B*a^7*b^10*d^2 + 2304*A^2*B*a^9*b^8*d^
2)/(8*a^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) - ((a
+ b*tan(c + d*x))^(1/2)*(A^2*B^2*b^18 - A^4*b^18 + 86*A^4*a^2*b^16 + 223*A^4*a^4*b^14 + 4176*A^4*a^6*b^12 - 64
*A^4*a^8*b^10 + 128*A^4*a^10*b^8 + 164*B^4*a^2*b^16 + 104*B^4*a^4*b^14 + 2212*B^4*a^6*b^12 - 1216*B^4*a^8*b^10
+ 384*B^4*a^10*b^8 + 358*A^2*B^2*a^2*b^16 + 3673*A^2*B^2*a^4*b^14 - 11508*A^2*B^2*a^6*b^12 + 9472*A^2*B^2*a^8
*b^10 - 12*A*B^3*a*b^17 + 4*A^3*B*a*b^17 - 472*A*B^3*a^3*b^15 + 4116*A*B^3*a^5*b^13 - 8448*A*B^3*a^7*b^11 + 28
16*A*B^3*a^9*b^9 - 192*A^3*B*a^3*b^15 - 6516*A^3*B*a^5*b^13 + 9472*A^3*B*a^7*b^11 - 768*A^3*B*a^9*b^9))/(4*a^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)*1i - (((((256*A
*a*b^13*d^4 + 5376*A*a^3*b^11*d^4 + 5120*A*a^5*b^9*d^4 - 1536*B*a^2*b^12*d^4 + 1536*B*a^4*b^10*d^4 + 3072*B*a^
6*b^8*d^4)/(8*a^2*d^5) + ((2048*a^2*b^10*d^4 + 3072*a^4*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))/(4*a^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) - ((a + b*tan(c + d*x))^(1/2)*(3008*A^2*a^3*b^12
*d^2 + 5888*A^2*a^5*b^10*d^2 - 1280*A^2*a^7*b^8*d^2 - 2672*B^2*a^3*b^12*d^2 - 4352*B^2*a^5*b^10*d^2 + 2304*B^2
*a^7*b^8*d^2 + 4*A^2*a*b^14*d^2 - 2096*A*B*a^2*b^13*d^2 + 1024*A*B*a^4*b^11*d^2 + 11264*A*B*a^6*b^9*d^2))/(4*a
^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) + (4*A^3*b^17
*d^2 - 188*A^3*a^2*b^15*d^2 - 6464*A^3*a^4*b^13*d^2 - 2688*A^3*a^6*b^11*d^2 + 3584*A^3*a^8*b^9*d^2 - 2416*B^3*
a^3*b^14*d^2 + 3728*B^3*a^5*b^12*d^2 + 5376*B^3*a^7*b^10*d^2 - 768*B^3*a^9*b^8*d^2 + 212*A^2*B*a*b^16*d^2 - 10
56*A*B^2*a^2*b^15*d^2 + 15456*A*B^2*a^4*b^13*d^2 + 5760*A*B^2*a^6*b^11*d^2 - 10752*A*B^2*a^8*b^9*d^2 + 6804*A^
2*B*a^3*b^14*d^2 - 13376*A^2*B*a^5*b^12*d^2 - 17664*A^2*B*a^7*b^10*d^2 + 2304*A^2*B*a^9*b^8*d^2)/(8*a^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) + ((a + b*tan(c + d*x)
)^(1/2)*(A^2*B^2*b^18 - A^4*b^18 + 86*A^4*a^2*b^16 + 223*A^4*a^4*b^14 + 4176*A^4*a^6*b^12 - 64*A^4*a^8*b^10 +
128*A^4*a^10*b^8 + 164*B^4*a^2*b^16 + 104*B^4*a^4*b^14 + 2212*B^4*a^6*b^12 - 1216*B^4*a^8*b^10 + 384*B^4*a^10*
b^8 + 358*A^2*B^2*a^2*b^16 + 3673*A^2*B^2*a^4*b^14 - 11508*A^2*B^2*a^6*b^12 + 9472*A^2*B^2*a^8*b^10 - 12*A*B^3
*a*b^17 + 4*A^3*B*a*b^17 - 472*A*B^3*a^3*b^15 + 4116*A*B^3*a^5*b^13 - 8448*A*B^3*a^7*b^11 + 2816*A*B^3*a^9*b^9
- 192*A^3*B*a^3*b^15 - 6516*A^3*B*a^5*b^13 + 9472*A^3*B*a^7*b^11 - 768*A^3*B*a^9*b^9))/(4*a^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)*1i)/((((((256*A*a*b^13*d^4 + 53
76*A*a^3*b^11*d^4 + 5120*A*a^5*b^9*d^4 - 1536*B*a^2*b^12*d^4 + 1536*B*a^4*b^10*d^4 + 3072*B*a^6*b^8*d^4)/(8*a^
2*d^5) - ((2048*a^2*b^10*d^4 + 3072*a^4*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))/(4*a^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) + ((a + b*tan(c + d*x))^(1/2)*(3008*A^2*a^3*b^12*d^2 + 5888*A^2*
a^5*b^10*d^2 - 1280*A^2*a^7*b^8*d^2 - 2672*B^2*a^3*b^12*d^2 - 4352*B^2*a^5*b^10*d^2 + 2304*B^2*a^7*b^8*d^2 + 4
*A^2*a*b^14*d^2 - 2096*A*B*a^2*b^13*d^2 + 1024*A*B*a^4*b^11*d^2 + 11264*A*B*a^6*b^9*d^2))/(4*a^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) + (4*A^3*b^17*d^2 - 188*A^3*a
^2*b^15*d^2 - 6464*A^3*a^4*b^13*d^2 - 2688*A^3*a^6*b^11*d^2 + 3584*A^3*a^8*b^9*d^2 - 2416*B^3*a^3*b^14*d^2 + 3
728*B^3*a^5*b^12*d^2 + 5376*B^3*a^7*b^10*d^2 - 768*B^3*a^9*b^8*d^2 + 212*A^2*B*a*b^16*d^2 - 1056*A*B^2*a^2*b^1
5*d^2 + 15456*A*B^2*a^4*b^13*d^2 + 5760*A*B^2*a^6*b^11*d^2 - 10752*A*B^2*a^8*b^9*d^2 + 6804*A^2*B*a^3*b^14*d^2
- 13376*A^2*B*a^5*b^12*d^2 - 17664*A^2*B*a^7*b^10*d^2 + 2304*A^2*B*a^9*b^8*d^2)/(8*a^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) - ((a + b*tan(c + d*x))^(1/2)*(A^2*B^2
*b^18 - A^4*b^18 + 86*A^4*a^2*b^16 + 223*A^4*a^4*b^14 + 4176*A^4*a^6*b^12 - 64*A^4*a^8*b^10 + 128*A^4*a^10*b^8
+ 164*B^4*a^2*b^16 + 104*B^4*a^4*b^14 + 2212*B^4*a^6*b^12 - 1216*B^4*a^8*b^10 + 384*B^4*a^10*b^8 + 358*A^2*B^
2*a^2*b^16 + 3673*A^2*B^2*a^4*b^14 - 11508*A^2*B^2*a^6*b^12 + 9472*A^2*B^2*a^8*b^10 - 12*A*B^3*a*b^17 + 4*A^3*
B*a*b^17 - 472*A*B^3*a^3*b^15 + 4116*A*B^3*a^5*b^13 - 8448*A*B^3*a^7*b^11 + 2816*A*B^3*a^9*b^9 - 192*A^3*B*a^3
*b^15 - 6516*A^3*B*a^5*b^13 + 9472*A^3*B*a^7*b^11 - 768*A^3*B*a^9*b^9))/(4*a^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) + (((((256*A*a*b^13*d^4 + 5376*A*a^3*b^11*d^4 +
5120*A*a^5*b^9*d^4 - 1536*B*a^2*b^12*d^4 + 1536*B*a^4*b^10*d^4 + 3072*B*a^6*b^8*d^4)/(8*a^2*d^5) + ((2048*a^2
*b^10*d^4 + 3072*a^4*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))/(4*a^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) - ((a + b*tan(c + d*x))^(1/2)*(3008*A^2*a^3*b^12*d^2 + 5888*A^2*a^5*b^10*d^2 - 1280
*A^2*a^7*b^8*d^2 - 2672*B^2*a^3*b^12*d^2 - 4352*B^2*a^5*b^10*d^2 + 2304*B^2*a^7*b^8*d^2 + 4*A^2*a*b^14*d^2 - 2
096*A*B*a^2*b^13*d^2 + 1024*A*B*a^4*b^11*d^2 + 11264*A*B*a^6*b^9*d^2))/(4*a^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) + (4*A^3*b^17*d^2 - 188*A^3*a^2*b^15*d^2 - 6464*
A^3*a^4*b^13*d^2 - 2688*A^3*a^6*b^11*d^2 + 3584*A^3*a^8*b^9*d^2 - 2416*B^3*a^3*b^14*d^2 + 3728*B^3*a^5*b^12*d^
2 + 5376*B^3*a^7*b^10*d^2 - 768*B^3*a^9*b^8*d^2 + 212*A^2*B*a*b^16*d^2 - 1056*A*B^2*a^2*b^15*d^2 + 15456*A*B^2
*a^4*b^13*d^2 + 5760*A*B^2*a^6*b^11*d^2 - 10752*A*B^2*a^8*b^9*d^2 + 6804*A^2*B*a^3*b^14*d^2 - 13376*A^2*B*a^5*
b^12*d^2 - 17664*A^2*B*a^7*b^10*d^2 + 2304*A^2*B*a^9*b^8*d^2)/(8*a^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) + ((a + b*tan(c + d*x))^(1/2)*(A^2*B^2*b^18 - A^4*b^18 +
86*A^4*a^2*b^16 + 223*A^4*a^4*b^14 + 4176*A^4*a^6*b^12 - 64*A^4*a^8*b^10 + 128*A^4*a^10*b^8 + 164*B^4*a^2*b^16
+ 104*B^4*a^4*b^14 + 2212*B^4*a^6*b^12 - 1216*B^4*a^8*b^10 + 384*B^4*a^10*b^8 + 358*A^2*B^2*a^2*b^16 + 3673*A
^2*B^2*a^4*b^14 - 11508*A^2*B^2*a^6*b^12 + 9472*A^2*B^2*a^8*b^10 - 12*A*B^3*a*b^17 + 4*A^3*B*a*b^17 - 472*A*B^
3*a^3*b^15 + 4116*A*B^3*a^5*b^13 - 8448*A*B^3*a^7*b^11 + 2816*A*B^3*a^9*b^9 - 192*A^3*B*a^3*b^15 - 6516*A^3*B*
a^5*b^13 + 9472*A^3*B*a^7*b^11 - 768*A^3*B*a^9*b^9))/(4*a^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) + (A^4*B*b^20 + 14*A^5*a*b^19 + A^2*B^3*b^20 + 348*A^5*a^3*b^17 +
286*A^5*a^5*b^15 - 32*A^5*a^7*b^13 + 400*A^5*a^9*b^11 + 384*A^5*a^11*b^9 - 60*B^5*a^2*b^18 - 284*B^5*a^4*b^16
+ 476*B^5*a^6*b^14 + 1564*B^5*a^8*b^12 + 864*B^5*a^10*b^10 - 83*A^2*B^3*a^2*b^18 + 771*A^2*B^3*a^4*b^16 + 963*
A^2*B^3*a^6*b^14 - 468*A^2*B^3*a^8*b^12 - 320*A^2*B^3*a^10*b^10 + 256*A^2*B^3*a^12*b^8 + 456*A^3*B^2*a^3*b^17
+ 2682*A^3*B^2*a^5*b^15 + 3556*A^3*B^2*a^7*b^13 + 800*A^3*B^2*a^9*b^11 - 512*A^3*B^2*a^11*b^9 + 4*A*B^4*a*b^19
+ 108*A*B^4*a^3*b^17 + 2396*A*B^4*a^5*b^15 + 3588*A*B^4*a^7*b^13 + 400*A*B^4*a^9*b^11 - 896*A*B^4*a^11*b^9 +
18*A^3*B^2*a*b^19 - 23*A^4*B*a^2*b^18 + 1055*A^4*B*a^4*b^16 + 487*A^4*B*a^6*b^14 - 2032*A^4*B*a^8*b^12 - 1184*
A^4*B*a^10*b^10 + 256*A^4*B*a^12*b^8)/(4*a^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)*2i - ((a + b*tan(c + d*x))^(3/2)*((A*b^3)/3 + 2*A*a^2*b - 2*B*a*b^2) - (a + b*ta
n(c + d*x))^(1/2)*((A*a*b^3)/8 - (3*B*a^2*b^2)/4 + A*a^3*b) + ((a + b*tan(c + d*x))^(5/2)*(A*b^3 - 8*A*a^2*b +
10*B*a*b^2))/(8*a))/(d*(a + b*tan(c + d*x))^3 - a^3*d - 3*a*d*(a + b*tan(c + d*x))^2 + 3*a^2*d*(a + b*tan(c +
d*x))) + atan(((((((256*A*a*b^13*d^4 + 5376*A*a^3*b^11*d^4 + 5120*A*a^5*b^9*d^4 - 1536*B*a^2*b^12*d^4 + 1536*
B*a^4*b^10*d^4 + 3072*B*a^6*b^8*d^4)/(8*a^2*d^5) - ((2048*a^2*b^10*d^4 + 3072*a^4*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))/(4*a^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) + ((a + b*tan(c + d*x))^
(1/2)*(3008*A^2*a^3*b^12*d^2 + 5888*A^2*a^5*b^10*d^2 - 1280*A^2*a^7*b^8*d^2 - 2672*B^2*a^3*b^12*d^2 - 4352*B^2
*a^5*b^10*d^2 + 2304*B^2*a^7*b^8*d^2 + 4*A^2*a*b^14*d^2 - 2096*A*B*a^2*b^13*d^2 + 1024*A*B*a^4*b^11*d^2 + 1126
4*A*B*a^6*b^9*d^2))/(4*a^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) + (4*A^3*b^17*d^2 - 188*A^3*a^2*b^15*d^2 - 6464*A^3*a^4*b^13*d^2 - 2688*A^3*a^6*b^11*d^2 + 3584*A^3*
a^8*b^9*d^2 - 2416*B^3*a^3*b^14*d^2 + 3728*B^3*a^5*b^12*d^2 + 5376*B^3*a^7*b^10*d^2 - 768*B^3*a^9*b^8*d^2 + 21
2*A^2*B*a*b^16*d^2 - 1056*A*B^2*a^2*b^15*d^2 + 15456*A*B^2*a^4*b^13*d^2 + 5760*A*B^2*a^6*b^11*d^2 - 10752*A*B^
2*a^8*b^9*d^2 + 6804*A^2*B*a^3*b^14*d^2 - 13376*A^2*B*a^5*b^12*d^2 - 17664*A^2*B*a^7*b^10*d^2 + 2304*A^2*B*a^9
*b^8*d^2)/(8*a^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)
- ((a + b*tan(c + d*x))^(1/2)*(A^2*B^2*b^18 - A^4*b^18 + 86*A^4*a^2*b^16 + 223*A^4*a^4*b^14 + 4176*A^4*a^6*b^1
2 - 64*A^4*a^8*b^10 + 128*A^4*a^10*b^8 + 164*B^4*a^2*b^16 + 104*B^4*a^4*b^14 + 2212*B^4*a^6*b^12 - 1216*B^4*a^
8*b^10 + 384*B^4*a^10*b^8 + 358*A^2*B^2*a^2*b^16 + 3673*A^2*B^2*a^4*b^14 - 11508*A^2*B^2*a^6*b^12 + 9472*A^2*B
^2*a^8*b^10 - 12*A*B^3*a*b^17 + 4*A^3*B*a*b^17 - 472*A*B^3*a^3*b^15 + 4116*A*B^3*a^5*b^13 - 8448*A*B^3*a^7*b^1
1 + 2816*A*B^3*a^9*b^9 - 192*A^3*B*a^3*b^15 - 6516*A^3*B*a^5*b^13 + 9472*A^3*B*a^7*b^11 - 768*A^3*B*a^9*b^9))/
(4*a^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)*1i - (((((
256*A*a*b^13*d^4 + 5376*A*a^3*b^11*d^4 + 5120*A*a^5*b^9*d^4 - 1536*B*a^2*b^12*d^4 + 1536*B*a^4*b^10*d^4 + 3072
*B*a^6*b^8*d^4)/(8*a^2*d^5) + ((2048*a^2*b^10*d^4 + 3072*a^4*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))/(4*a^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) - ((a + b*tan(c + d*x))^(1/2)*(3008*A^2*a^3*b
^12*d^2 + 5888*A^2*a^5*b^10*d^2 - 1280*A^2*a^7*b^8*d^2 - 2672*B^2*a^3*b^12*d^2 - 4352*B^2*a^5*b^10*d^2 + 2304*
B^2*a^7*b^8*d^2 + 4*A^2*a*b^14*d^2 - 2096*A*B*a^2*b^13*d^2 + 1024*A*B*a^4*b^11*d^2 + 11264*A*B*a^6*b^9*d^2))/(
4*a^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) + (4*A^3*b^
17*d^2 - 188*A^3*a^2*b^15*d^2 - 6464*A^3*a^4*b^13*d^2 - 2688*A^3*a^6*b^11*d^2 + 3584*A^3*a^8*b^9*d^2 - 2416*B^
3*a^3*b^14*d^2 + 3728*B^3*a^5*b^12*d^2 + 5376*B^3*a^7*b^10*d^2 - 768*B^3*a^9*b^8*d^2 + 212*A^2*B*a*b^16*d^2 -
1056*A*B^2*a^2*b^15*d^2 + 15456*A*B^2*a^4*b^13*d^2 + 5760*A*B^2*a^6*b^11*d^2 - 10752*A*B^2*a^8*b^9*d^2 + 6804*
A^2*B*a^3*b^14*d^2 - 13376*A^2*B*a^5*b^12*d^2 - 17664*A^2*B*a^7*b^10*d^2 + 2304*A^2*B*a^9*b^8*d^2)/(8*a^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) + ((a + b*tan(c + d*x
))^(1/2)*(A^2*B^2*b^18 - A^4*b^18 + 86*A^4*a^2*b^16 + 223*A^4*a^4*b^14 + 4176*A^4*a^6*b^12 - 64*A^4*a^8*b^10 +
128*A^4*a^10*b^8 + 164*B^4*a^2*b^16 + 104*B^4*a^4*b^14 + 2212*B^4*a^6*b^12 - 1216*B^4*a^8*b^10 + 384*B^4*a^10
*b^8 + 358*A^2*B^2*a^2*b^16 + 3673*A^2*B^2*a^4*b^14 - 11508*A^2*B^2*a^6*b^12 + 9472*A^2*B^2*a^8*b^10 - 12*A*B^
3*a*b^17 + 4*A^3*B*a*b^17 - 472*A*B^3*a^3*b^15 + 4116*A*B^3*a^5*b^13 - 8448*A*B^3*a^7*b^11 + 2816*A*B^3*a^9*b^
9 - 192*A^3*B*a^3*b^15 - 6516*A^3*B*a^5*b^13 + 9472*A^3*B*a^7*b^11 - 768*A^3*B*a^9*b^9))/(4*a^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)*1i)/((((((256*A*a*b^13*d^4 + 53
76*A*a^3*b^11*d^4 + 5120*A*a^5*b^9*d^4 - 1536*B*a^2*b^12*d^4 + 1536*B*a^4*b^10*d^4 + 3072*B*a^6*b^8*d^4)/(8*a^
2*d^5) - ((2048*a^2*b^10*d^4 + 3072*a^4*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))/(4*a^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) + ((a + b*tan(c + d*x))^(1/2)*(3008*A^2*a^3*b^12*d^2 + 5888*A^2*a^
5*b^10*d^2 - 1280*A^2*a^7*b^8*d^2 - 2672*B^2*a^3*b^12*d^2 - 4352*B^2*a^5*b^10*d^2 + 2304*B^2*a^7*b^8*d^2 + 4*A
^2*a*b^14*d^2 - 2096*A*B*a^2*b^13*d^2 + 1024*A*B*a^4*b^11*d^2 + 11264*A*B*a^6*b^9*d^2))/(4*a^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) + (4*A^3*b^17*d^2 - 188*A^3*a^2*
b^15*d^2 - 6464*A^3*a^4*b^13*d^2 - 2688*A^3*a^6*b^11*d^2 + 3584*A^3*a^8*b^9*d^2 - 2416*B^3*a^3*b^14*d^2 + 3728
*B^3*a^5*b^12*d^2 + 5376*B^3*a^7*b^10*d^2 - 768*B^3*a^9*b^8*d^2 + 212*A^2*B*a*b^16*d^2 - 1056*A*B^2*a^2*b^15*d
^2 + 15456*A*B^2*a^4*b^13*d^2 + 5760*A*B^2*a^6*b^11*d^2 - 10752*A*B^2*a^8*b^9*d^2 + 6804*A^2*B*a^3*b^14*d^2 -
13376*A^2*B*a^5*b^12*d^2 - 17664*A^2*B*a^7*b^10*d^2 + 2304*A^2*B*a^9*b^8*d^2)/(8*a^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) - ((a + b*tan(c + d*x))^(1/2)*(A^2*B^2*b^1
8 - A^4*b^18 + 86*A^4*a^2*b^16 + 223*A^4*a^4*b^14 + 4176*A^4*a^6*b^12 - 64*A^4*a^8*b^10 + 128*A^4*a^10*b^8 + 1
64*B^4*a^2*b^16 + 104*B^4*a^4*b^14 + 2212*B^4*a^6*b^12 - 1216*B^4*a^8*b^10 + 384*B^4*a^10*b^8 + 358*A^2*B^2*a^
2*b^16 + 3673*A^2*B^2*a^4*b^14 - 11508*A^2*B^2*a^6*b^12 + 9472*A^2*B^2*a^8*b^10 - 12*A*B^3*a*b^17 + 4*A^3*B*a*
b^17 - 472*A*B^3*a^3*b^15 + 4116*A*B^3*a^5*b^13 - 8448*A*B^3*a^7*b^11 + 2816*A*B^3*a^9*b^9 - 192*A^3*B*a^3*b^1
5 - 6516*A^3*B*a^5*b^13 + 9472*A^3*B*a^7*b^11 - 768*A^3*B*a^9*b^9))/(4*a^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) + (((((256*A*a*b^13*d^4 + 5376*A*a^3*b^11*d^4 + 5120
*A*a^5*b^9*d^4 - 1536*B*a^2*b^12*d^4 + 1536*B*a^4*b^10*d^4 + 3072*B*a^6*b^8*d^4)/(8*a^2*d^5) + ((2048*a^2*b^10
*d^4 + 3072*a^4*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))/(4*a^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) - ((a + b*tan(c + d*x))^(1/2)*(3008*A^2*a^3*b^12*d^2 + 5888*A^2*a^5*b^10*d^2 - 1280*A^2*a^
7*b^8*d^2 - 2672*B^2*a^3*b^12*d^2 - 4352*B^2*a^5*b^10*d^2 + 2304*B^2*a^7*b^8*d^2 + 4*A^2*a*b^14*d^2 - 2096*A*B
*a^2*b^13*d^2 + 1024*A*B*a^4*b^11*d^2 + 11264*A*B*a^6*b^9*d^2))/(4*a^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) + (4*A^3*b^17*d^2 - 188*A^3*a^2*b^15*d^2 - 6464*A^3*a^4*
b^13*d^2 - 2688*A^3*a^6*b^11*d^2 + 3584*A^3*a^8*b^9*d^2 - 2416*B^3*a^3*b^14*d^2 + 3728*B^3*a^5*b^12*d^2 + 5376
*B^3*a^7*b^10*d^2 - 768*B^3*a^9*b^8*d^2 + 212*A^2*B*a*b^16*d^2 - 1056*A*B^2*a^2*b^15*d^2 + 15456*A*B^2*a^4*b^1
3*d^2 + 5760*A*B^2*a^6*b^11*d^2 - 10752*A*B^2*a^8*b^9*d^2 + 6804*A^2*B*a^3*b^14*d^2 - 13376*A^2*B*a^5*b^12*d^2
- 17664*A^2*B*a^7*b^10*d^2 + 2304*A^2*B*a^9*b^8*d^2)/(8*a^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) + ((a + b*tan(c + d*x))^(1/2)*(A^2*B^2*b^18 - A^4*b^18 + 86*A^4*a^
2*b^16 + 223*A^4*a^4*b^14 + 4176*A^4*a^6*b^12 - 64*A^4*a^8*b^10 + 128*A^4*a^10*b^8 + 164*B^4*a^2*b^16 + 104*B^
4*a^4*b^14 + 2212*B^4*a^6*b^12 - 1216*B^4*a^8*b^10 + 384*B^4*a^10*b^8 + 358*A^2*B^2*a^2*b^16 + 3673*A^2*B^2*a^
4*b^14 - 11508*A^2*B^2*a^6*b^12 + 9472*A^2*B^2*a^8*b^10 - 12*A*B^3*a*b^17 + 4*A^3*B*a*b^17 - 472*A*B^3*a^3*b^1
5 + 4116*A*B^3*a^5*b^13 - 8448*A*B^3*a^7*b^11 + 2816*A*B^3*a^9*b^9 - 192*A^3*B*a^3*b^15 - 6516*A^3*B*a^5*b^13
+ 9472*A^3*B*a^7*b^11 - 768*A^3*B*a^9*b^9))/(4*a^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) + (A^4*B*b^20 + 14*A^5*a*b^19 + A^2*B^3*b^20 + 348*A^5*a^3*b^17 + 286*A^5*a^
5*b^15 - 32*A^5*a^7*b^13 + 400*A^5*a^9*b^11 + 384*A^5*a^11*b^9 - 60*B^5*a^2*b^18 - 284*B^5*a^4*b^16 + 476*B^5*
a^6*b^14 + 1564*B^5*a^8*b^12 + 864*B^5*a^10*b^10 - 83*A^2*B^3*a^2*b^18 + 771*A^2*B^3*a^4*b^16 + 963*A^2*B^3*a^
6*b^14 - 468*A^2*B^3*a^8*b^12 - 320*A^2*B^3*a^10*b^10 + 256*A^2*B^3*a^12*b^8 + 456*A^3*B^2*a^3*b^17 + 2682*A^3
*B^2*a^5*b^15 + 3556*A^3*B^2*a^7*b^13 + 800*A^3*B^2*a^9*b^11 - 512*A^3*B^2*a^11*b^9 + 4*A*B^4*a*b^19 + 108*A*B
^4*a^3*b^17 + 2396*A*B^4*a^5*b^15 + 3588*A*B^4*a^7*b^13 + 400*A*B^4*a^9*b^11 - 896*A*B^4*a^11*b^9 + 18*A^3*B^2
*a*b^19 - 23*A^4*B*a^2*b^18 + 1055*A^4*B*a^4*b^16 + 487*A^4*B*a^6*b^14 - 2032*A^4*B*a^8*b^12 - 1184*A^4*B*a^10
*b^10 + 256*A^4*B*a^12*b^8)/(4*a^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)*2i + (atan((((((((A^3*b^17*d^2)/2 - (47*A^3*a^2*b^15*d^2)/2 - 808*A^3*a^4*b^13*d^2 - 336*A^
3*a^6*b^11*d^2 + 448*A^3*a^8*b^9*d^2 - 302*B^3*a^3*b^14*d^2 + 466*B^3*a^5*b^12*d^2 + 672*B^3*a^7*b^10*d^2 - 96
*B^3*a^9*b^8*d^2 + (53*A^2*B*a*b^16*d^2)/2 - 132*A*B^2*a^2*b^15*d^2 + 1932*A*B^2*a^4*b^13*d^2 + 720*A*B^2*a^6*
b^11*d^2 - 1344*A*B^2*a^8*b^9*d^2 + (1701*A^2*B*a^3*b^14*d^2)/2 - 1672*A^2*B*a^5*b^12*d^2 - 2208*A^2*B*a^7*b^1
0*d^2 + 288*A^2*B*a^9*b^8*d^2)/(16*a^2*d^5) + (((((32*A*a*b^13*d^4 + 672*A*a^3*b^11*d^4 + 640*A*a^5*b^9*d^4 -
192*B*a^2*b^12*d^4 + 192*B*a^4*b^10*d^4 + 384*B*a^6*b^8*d^4)/(16*a^2*d^5) - ((2048*a^2*b^10*d^4 + 3072*a^4*b^8
*d^4)*(a + b*tan(c + d*x))^(1/2)*(256*B^2*a^9 + A^2*a^3*b^6 + 48*A^2*a^5*b^4 + 576*A^2*a^7*b^2 + 36*B^2*a^5*b^
4 - 192*B^2*a^7*b^2 + 768*A*B*a^8*b - 12*A*B*a^4*b^5 - 256*A*B*a^6*b^3)^(1/2))/(1024*a^5*d^5))*(256*B^2*a^9 +
A^2*a^3*b^6 + 48*A^2*a^5*b^4 + 576*A^2*a^7*b^2 + 36*B^2*a^5*b^4 - 192*B^2*a^7*b^2 + 768*A*B*a^8*b - 12*A*B*a^4
*b^5 - 256*A*B*a^6*b^3)^(1/2))/(16*a^3*d) + ((a + b*tan(c + d*x))^(1/2)*(3008*A^2*a^3*b^12*d^2 + 5888*A^2*a^5*
b^10*d^2 - 1280*A^2*a^7*b^8*d^2 - 2672*B^2*a^3*b^12*d^2 - 4352*B^2*a^5*b^10*d^2 + 2304*B^2*a^7*b^8*d^2 + 4*A^2
*a*b^14*d^2 - 2096*A*B*a^2*b^13*d^2 + 1024*A*B*a^4*b^11*d^2 + 11264*A*B*a^6*b^9*d^2))/(64*a^2*d^4))*(256*B^2*a
^9 + A^2*a^3*b^6 + 48*A^2*a^5*b^4 + 576*A^2*a^7*b^2 + 36*B^2*a^5*b^4 - 192*B^2*a^7*b^2 + 768*A*B*a^8*b - 12*A*
B*a^4*b^5 - 256*A*B*a^6*b^3)^(1/2))/(16*a^3*d))*(256*B^2*a^9 + A^2*a^3*b^6 + 48*A^2*a^5*b^4 + 576*A^2*a^7*b^2
+ 36*B^2*a^5*b^4 - 192*B^2*a^7*b^2 + 768*A*B*a^8*b - 12*A*B*a^4*b^5 - 256*A*B*a^6*b^3)^(1/2))/(16*a^3*d) - ((a
+ b*tan(c + d*x))^(1/2)*(A^2*B^2*b^18 - A^4*b^18 + 86*A^4*a^2*b^16 + 223*A^4*a^4*b^14 + 4176*A^4*a^6*b^12 - 6
4*A^4*a^8*b^10 + 128*A^4*a^10*b^8 + 164*B^4*a^2*b^16 + 104*B^4*a^4*b^14 + 2212*B^4*a^6*b^12 - 1216*B^4*a^8*b^1
0 + 384*B^4*a^10*b^8 + 358*A^2*B^2*a^2*b^16 + 3673*A^2*B^2*a^4*b^14 - 11508*A^2*B^2*a^6*b^12 + 9472*A^2*B^2*a^
8*b^10 - 12*A*B^3*a*b^17 + 4*A^3*B*a*b^17 - 472*A*B^3*a^3*b^15 + 4116*A*B^3*a^5*b^13 - 8448*A*B^3*a^7*b^11 + 2
816*A*B^3*a^9*b^9 - 192*A^3*B*a^3*b^15 - 6516*A^3*B*a^5*b^13 + 9472*A^3*B*a^7*b^11 - 768*A^3*B*a^9*b^9))/(64*a
^2*d^4))*(256*B^2*a^9 + A^2*a^3*b^6 + 48*A^2*a^5*b^4 + 576*A^2*a^7*b^2 + 36*B^2*a^5*b^4 - 192*B^2*a^7*b^2 + 76
8*A*B*a^8*b - 12*A*B*a^4*b^5 - 256*A*B*a^6*b^3)^(1/2)*1i)/(a^3*d) - ((((((A^3*b^17*d^2)/2 - (47*A^3*a^2*b^15*d
^2)/2 - 808*A^3*a^4*b^13*d^2 - 336*A^3*a^6*b^11*d^2 + 448*A^3*a^8*b^9*d^2 - 302*B^3*a^3*b^14*d^2 + 466*B^3*a^5
*b^12*d^2 + 672*B^3*a^7*b^10*d^2 - 96*B^3*a^9*b^8*d^2 + (53*A^2*B*a*b^16*d^2)/2 - 132*A*B^2*a^2*b^15*d^2 + 193
2*A*B^2*a^4*b^13*d^2 + 720*A*B^2*a^6*b^11*d^2 - 1344*A*B^2*a^8*b^9*d^2 + (1701*A^2*B*a^3*b^14*d^2)/2 - 1672*A^
2*B*a^5*b^12*d^2 - 2208*A^2*B*a^7*b^10*d^2 + 288*A^2*B*a^9*b^8*d^2)/(16*a^2*d^5) + (((((32*A*a*b^13*d^4 + 672*
A*a^3*b^11*d^4 + 640*A*a^5*b^9*d^4 - 192*B*a^2*b^12*d^4 + 192*B*a^4*b^10*d^4 + 384*B*a^6*b^8*d^4)/(16*a^2*d^5)
+ ((2048*a^2*b^10*d^4 + 3072*a^4*b^8*d^4)*(a + b*tan(c + d*x))^(1/2)*(256*B^2*a^9 + A^2*a^3*b^6 + 48*A^2*a^5*
b^4 + 576*A^2*a^7*b^2 + 36*B^2*a^5*b^4 - 192*B^2*a^7*b^2 + 768*A*B*a^8*b - 12*A*B*a^4*b^5 - 256*A*B*a^6*b^3)^(
1/2))/(1024*a^5*d^5))*(256*B^2*a^9 + A^2*a^3*b^6 + 48*A^2*a^5*b^4 + 576*A^2*a^7*b^2 + 36*B^2*a^5*b^4 - 192*B^2
*a^7*b^2 + 768*A*B*a^8*b - 12*A*B*a^4*b^5 - 256*A*B*a^6*b^3)^(1/2))/(16*a^3*d) - ((a + b*tan(c + d*x))^(1/2)*(
3008*A^2*a^3*b^12*d^2 + 5888*A^2*a^5*b^10*d^2 - 1280*A^2*a^7*b^8*d^2 - 2672*B^2*a^3*b^12*d^2 - 4352*B^2*a^5*b^
10*d^2 + 2304*B^2*a^7*b^8*d^2 + 4*A^2*a*b^14*d^2 - 2096*A*B*a^2*b^13*d^2 + 1024*A*B*a^4*b^11*d^2 + 11264*A*B*a
^6*b^9*d^2))/(64*a^2*d^4))*(256*B^2*a^9 + A^2*a^3*b^6 + 48*A^2*a^5*b^4 + 576*A^2*a^7*b^2 + 36*B^2*a^5*b^4 - 19
2*B^2*a^7*b^2 + 768*A*B*a^8*b - 12*A*B*a^4*b^5 - 256*A*B*a^6*b^3)^(1/2))/(16*a^3*d))*(256*B^2*a^9 + A^2*a^3*b^
6 + 48*A^2*a^5*b^4 + 576*A^2*a^7*b^2 + 36*B^2*a^5*b^4 - 192*B^2*a^7*b^2 + 768*A*B*a^8*b - 12*A*B*a^4*b^5 - 256
*A*B*a^6*b^3)^(1/2))/(16*a^3*d) + ((a + b*tan(c + d*x))^(1/2)*(A^2*B^2*b^18 - A^4*b^18 + 86*A^4*a^2*b^16 + 223
*A^4*a^4*b^14 + 4176*A^4*a^6*b^12 - 64*A^4*a^8*b^10 + 128*A^4*a^10*b^8 + 164*B^4*a^2*b^16 + 104*B^4*a^4*b^14 +
2212*B^4*a^6*b^12 - 1216*B^4*a^8*b^10 + 384*B^4*a^10*b^8 + 358*A^2*B^2*a^2*b^16 + 3673*A^2*B^2*a^4*b^14 - 115
08*A^2*B^2*a^6*b^12 + 9472*A^2*B^2*a^8*b^10 - 12*A*B^3*a*b^17 + 4*A^3*B*a*b^17 - 472*A*B^3*a^3*b^15 + 4116*A*B
^3*a^5*b^13 - 8448*A*B^3*a^7*b^11 + 2816*A*B^3*a^9*b^9 - 192*A^3*B*a^3*b^15 - 6516*A^3*B*a^5*b^13 + 9472*A^3*B
*a^7*b^11 - 768*A^3*B*a^9*b^9))/(64*a^2*d^4))*(256*B^2*a^9 + A^2*a^3*b^6 + 48*A^2*a^5*b^4 + 576*A^2*a^7*b^2 +
36*B^2*a^5*b^4 - 192*B^2*a^7*b^2 + 768*A*B*a^8*b - 12*A*B*a^4*b^5 - 256*A*B*a^6*b^3)^(1/2)*1i)/(a^3*d))/(((A^4
*B*b^20)/4 + (7*A^5*a*b^19)/2 + (A^2*B^3*b^20)/4 + 87*A^5*a^3*b^17 + (143*A^5*a^5*b^15)/2 - 8*A^5*a^7*b^13 + 1
00*A^5*a^9*b^11 + 96*A^5*a^11*b^9 - 15*B^5*a^2*b^18 - 71*B^5*a^4*b^16 + 119*B^5*a^6*b^14 + 391*B^5*a^8*b^12 +
216*B^5*a^10*b^10 - (83*A^2*B^3*a^2*b^18)/4 + (771*A^2*B^3*a^4*b^16)/4 + (963*A^2*B^3*a^6*b^14)/4 - 117*A^2*B^
3*a^8*b^12 - 80*A^2*B^3*a^10*b^10 + 64*A^2*B^3*a^12*b^8 + 114*A^3*B^2*a^3*b^17 + (1341*A^3*B^2*a^5*b^15)/2 + 8
89*A^3*B^2*a^7*b^13 + 200*A^3*B^2*a^9*b^11 - 128*A^3*B^2*a^11*b^9 + A*B^4*a*b^19 + 27*A*B^4*a^3*b^17 + 599*A*B
^4*a^5*b^15 + 897*A*B^4*a^7*b^13 + 100*A*B^4*a^9*b^11 - 224*A*B^4*a^11*b^9 + (9*A^3*B^2*a*b^19)/2 - (23*A^4*B*
a^2*b^18)/4 + (1055*A^4*B*a^4*b^16)/4 + (487*A^4*B*a^6*b^14)/4 - 508*A^4*B*a^8*b^12 - 296*A^4*B*a^10*b^10 + 64
*A^4*B*a^12*b^8)/(a^2*d^5) + ((((((A^3*b^17*d^2)/2 - (47*A^3*a^2*b^15*d^2)/2 - 808*A^3*a^4*b^13*d^2 - 336*A^3*
a^6*b^11*d^2 + 448*A^3*a^8*b^9*d^2 - 302*B^3*a^3*b^14*d^2 + 466*B^3*a^5*b^12*d^2 + 672*B^3*a^7*b^10*d^2 - 96*B
^3*a^9*b^8*d^2 + (53*A^2*B*a*b^16*d^2)/2 - 132*A*B^2*a^2*b^15*d^2 + 1932*A*B^2*a^4*b^13*d^2 + 720*A*B^2*a^6*b^
11*d^2 - 1344*A*B^2*a^8*b^9*d^2 + (1701*A^2*B*a^3*b^14*d^2)/2 - 1672*A^2*B*a^5*b^12*d^2 - 2208*A^2*B*a^7*b^10*
d^2 + 288*A^2*B*a^9*b^8*d^2)/(16*a^2*d^5) + (((((32*A*a*b^13*d^4 + 672*A*a^3*b^11*d^4 + 640*A*a^5*b^9*d^4 - 19
2*B*a^2*b^12*d^4 + 192*B*a^4*b^10*d^4 + 384*B*a^6*b^8*d^4)/(16*a^2*d^5) - ((2048*a^2*b^10*d^4 + 3072*a^4*b^8*d
^4)*(a + b*tan(c + d*x))^(1/2)*(256*B^2*a^9 + A^2*a^3*b^6 + 48*A^2*a^5*b^4 + 576*A^2*a^7*b^2 + 36*B^2*a^5*b^4
- 192*B^2*a^7*b^2 + 768*A*B*a^8*b - 12*A*B*a^4*b^5 - 256*A*B*a^6*b^3)^(1/2))/(1024*a^5*d^5))*(256*B^2*a^9 + A^
2*a^3*b^6 + 48*A^2*a^5*b^4 + 576*A^2*a^7*b^2 + 36*B^2*a^5*b^4 - 192*B^2*a^7*b^2 + 768*A*B*a^8*b - 12*A*B*a^4*b
^5 - 256*A*B*a^6*b^3)^(1/2))/(16*a^3*d) + ((a + b*tan(c + d*x))^(1/2)*(3008*A^2*a^3*b^12*d^2 + 5888*A^2*a^5*b^
10*d^2 - 1280*A^2*a^7*b^8*d^2 - 2672*B^2*a^3*b^12*d^2 - 4352*B^2*a^5*b^10*d^2 + 2304*B^2*a^7*b^8*d^2 + 4*A^2*a
*b^14*d^2 - 2096*A*B*a^2*b^13*d^2 + 1024*A*B*a^4*b^11*d^2 + 11264*A*B*a^6*b^9*d^2))/(64*a^2*d^4))*(256*B^2*a^9
+ A^2*a^3*b^6 + 48*A^2*a^5*b^4 + 576*A^2*a^7*b^2 + 36*B^2*a^5*b^4 - 192*B^2*a^7*b^2 + 768*A*B*a^8*b - 12*A*B*
a^4*b^5 - 256*A*B*a^6*b^3)^(1/2))/(16*a^3*d))*(256*B^2*a^9 + A^2*a^3*b^6 + 48*A^2*a^5*b^4 + 576*A^2*a^7*b^2 +
36*B^2*a^5*b^4 - 192*B^2*a^7*b^2 + 768*A*B*a^8*b - 12*A*B*a^4*b^5 - 256*A*B*a^6*b^3)^(1/2))/(16*a^3*d) - ((a +
b*tan(c + d*x))^(1/2)*(A^2*B^2*b^18 - A^4*b^18 + 86*A^4*a^2*b^16 + 223*A^4*a^4*b^14 + 4176*A^4*a^6*b^12 - 64*
A^4*a^8*b^10 + 128*A^4*a^10*b^8 + 164*B^4*a^2*b^16 + 104*B^4*a^4*b^14 + 2212*B^4*a^6*b^12 - 1216*B^4*a^8*b^10
+ 384*B^4*a^10*b^8 + 358*A^2*B^2*a^2*b^16 + 3673*A^2*B^2*a^4*b^14 - 11508*A^2*B^2*a^6*b^12 + 9472*A^2*B^2*a^8*
b^10 - 12*A*B^3*a*b^17 + 4*A^3*B*a*b^17 - 472*A*B^3*a^3*b^15 + 4116*A*B^3*a^5*b^13 - 8448*A*B^3*a^7*b^11 + 281
6*A*B^3*a^9*b^9 - 192*A^3*B*a^3*b^15 - 6516*A^3*B*a^5*b^13 + 9472*A^3*B*a^7*b^11 - 768*A^3*B*a^9*b^9))/(64*a^2
*d^4))*(256*B^2*a^9 + A^2*a^3*b^6 + 48*A^2*a^5*b^4 + 576*A^2*a^7*b^2 + 36*B^2*a^5*b^4 - 192*B^2*a^7*b^2 + 768*
A*B*a^8*b - 12*A*B*a^4*b^5 - 256*A*B*a^6*b^3)^(1/2))/(a^3*d) + ((((((A^3*b^17*d^2)/2 - (47*A^3*a^2*b^15*d^2)/2
- 808*A^3*a^4*b^13*d^2 - 336*A^3*a^6*b^11*d^2 + 448*A^3*a^8*b^9*d^2 - 302*B^3*a^3*b^14*d^2 + 466*B^3*a^5*b^12
*d^2 + 672*B^3*a^7*b^10*d^2 - 96*B^3*a^9*b^8*d^2 + (53*A^2*B*a*b^16*d^2)/2 - 132*A*B^2*a^2*b^15*d^2 + 1932*A*B
^2*a^4*b^13*d^2 + 720*A*B^2*a^6*b^11*d^2 - 1344*A*B^2*a^8*b^9*d^2 + (1701*A^2*B*a^3*b^14*d^2)/2 - 1672*A^2*B*a
^5*b^12*d^2 - 2208*A^2*B*a^7*b^10*d^2 + 288*A^2*B*a^9*b^8*d^2)/(16*a^2*d^5) + (((((32*A*a*b^13*d^4 + 672*A*a^3
*b^11*d^4 + 640*A*a^5*b^9*d^4 - 192*B*a^2*b^12*d^4 + 192*B*a^4*b^10*d^4 + 384*B*a^6*b^8*d^4)/(16*a^2*d^5) + ((
2048*a^2*b^10*d^4 + 3072*a^4*b^8*d^4)*(a + b*tan(c + d*x))^(1/2)*(256*B^2*a^9 + A^2*a^3*b^6 + 48*A^2*a^5*b^4 +
576*A^2*a^7*b^2 + 36*B^2*a^5*b^4 - 192*B^2*a^7*b^2 + 768*A*B*a^8*b - 12*A*B*a^4*b^5 - 256*A*B*a^6*b^3)^(1/2))
/(1024*a^5*d^5))*(256*B^2*a^9 + A^2*a^3*b^6 + 48*A^2*a^5*b^4 + 576*A^2*a^7*b^2 + 36*B^2*a^5*b^4 - 192*B^2*a^7*
b^2 + 768*A*B*a^8*b - 12*A*B*a^4*b^5 - 256*A*B*a^6*b^3)^(1/2))/(16*a^3*d) - ((a + b*tan(c + d*x))^(1/2)*(3008*
A^2*a^3*b^12*d^2 + 5888*A^2*a^5*b^10*d^2 - 1280*A^2*a^7*b^8*d^2 - 2672*B^2*a^3*b^12*d^2 - 4352*B^2*a^5*b^10*d^
2 + 2304*B^2*a^7*b^8*d^2 + 4*A^2*a*b^14*d^2 - 2096*A*B*a^2*b^13*d^2 + 1024*A*B*a^4*b^11*d^2 + 11264*A*B*a^6*b^
9*d^2))/(64*a^2*d^4))*(256*B^2*a^9 + A^2*a^3*b^6 + 48*A^2*a^5*b^4 + 576*A^2*a^7*b^2 + 36*B^2*a^5*b^4 - 192*B^2
*a^7*b^2 + 768*A*B*a^8*b - 12*A*B*a^4*b^5 - 256*A*B*a^6*b^3)^(1/2))/(16*a^3*d))*(256*B^2*a^9 + A^2*a^3*b^6 + 4
8*A^2*a^5*b^4 + 576*A^2*a^7*b^2 + 36*B^2*a^5*b^4 - 192*B^2*a^7*b^2 + 768*A*B*a^8*b - 12*A*B*a^4*b^5 - 256*A*B*
a^6*b^3)^(1/2))/(16*a^3*d) + ((a + b*tan(c + d*x))^(1/2)*(A^2*B^2*b^18 - A^4*b^18 + 86*A^4*a^2*b^16 + 223*A^4*
a^4*b^14 + 4176*A^4*a^6*b^12 - 64*A^4*a^8*b^10 + 128*A^4*a^10*b^8 + 164*B^4*a^2*b^16 + 104*B^4*a^4*b^14 + 2212
*B^4*a^6*b^12 - 1216*B^4*a^8*b^10 + 384*B^4*a^10*b^8 + 358*A^2*B^2*a^2*b^16 + 3673*A^2*B^2*a^4*b^14 - 11508*A^
2*B^2*a^6*b^12 + 9472*A^2*B^2*a^8*b^10 - 12*A*B^3*a*b^17 + 4*A^3*B*a*b^17 - 472*A*B^3*a^3*b^15 + 4116*A*B^3*a^
5*b^13 - 8448*A*B^3*a^7*b^11 + 2816*A*B^3*a^9*b^9 - 192*A^3*B*a^3*b^15 - 6516*A^3*B*a^5*b^13 + 9472*A^3*B*a^7*
b^11 - 768*A^3*B*a^9*b^9))/(64*a^2*d^4))*(256*B^2*a^9 + A^2*a^3*b^6 + 48*A^2*a^5*b^4 + 576*A^2*a^7*b^2 + 36*B^
2*a^5*b^4 - 192*B^2*a^7*b^2 + 768*A*B*a^8*b - 12*A*B*a^4*b^5 - 256*A*B*a^6*b^3)^(1/2))/(a^3*d)))*(256*B^2*a^9
+ A^2*a^3*b^6 + 48*A^2*a^5*b^4 + 576*A^2*a^7*b^2 + 36*B^2*a^5*b^4 - 192*B^2*a^7*b^2 + 768*A*B*a^8*b - 12*A*B*a
^4*b^5 - 256*A*B*a^6*b^3)^(1/2)*1i)/(8*a^3*d)