\(\int \cot ^2(c+d x) (a+b \tan (c+d x))^{5/2} (A+B \tan (c+d x)) \, dx\) [336]

Optimal result

Integrand size = 33, antiderivative size = 196 \[ \int \cot ^2(c+d x) (a+b \tan (c+d x))^{5/2} (A+B \tan (c+d x)) \, dx=-\frac {a^{3/2} (5 A b+2 a B) \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a}}\right )}{d}+\frac {(a-i b)^{5/2} (i A+B) \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a-i b}}\right )}{d}-\frac {(a+i b)^{5/2} (i A-B) \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a+i b}}\right )}{d}+\frac {b (a A+2 b B) \sqrt {a+b \tan (c+d x)}}{d}-\frac {a A \cot (c+d x) (a+b \tan (c+d x))^{3/2}}{d} \]

[Out]

-a^(3/2)*(5*A*b+2*B*a)*arctanh((a+b*tan(d*x+c))^(1/2)/a^(1/2))/d+(a-I*b)^(5/2)*(I*A+B)*arctanh((a+b*tan(d*x+c)
)^(1/2)/(a-I*b)^(1/2))/d-(a+I*b)^(5/2)*(I*A-B)*arctanh((a+b*tan(d*x+c))^(1/2)/(a+I*b)^(1/2))/d+b*(A*a+2*B*b)*(
a+b*tan(d*x+c))^(1/2)/d-a*A*cot(d*x+c)*(a+b*tan(d*x+c))^(3/2)/d

Rubi [A] (verified)

Time = 1.03 (sec) , antiderivative size = 196, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.242, Rules used = {3686, 3728, 3734, 3620, 3618, 65, 214, 3715} \[ \int \cot ^2(c+d x) (a+b \tan (c+d x))^{5/2} (A+B \tan (c+d x)) \, dx=-\frac {a^{3/2} (2 a B+5 A b) \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a}}\right )}{d}+\frac {(a-i b)^{5/2} (B+i A) \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a-i b}}\right )}{d}-\frac {(a+i b)^{5/2} (-B+i A) \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a+i b}}\right )}{d}+\frac {b (a A+2 b B) \sqrt {a+b \tan (c+d x)}}{d}-\frac {a A \cot (c+d x) (a+b \tan (c+d x))^{3/2}}{d} \]

[In]

Int[Cot[c + d*x]^2*(a + b*Tan[c + d*x])^(5/2)*(A + B*Tan[c + d*x]),x]

[Out]

-((a^(3/2)*(5*A*b + 2*a*B)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a]])/d) + ((a - I*b)^(5/2)*(I*A + B)*ArcTanh[
Sqrt[a + b*Tan[c + d*x]]/Sqrt[a - I*b]])/d - ((a + I*b)^(5/2)*(I*A - B)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[
a + I*b]])/d + (b*(a*A + 2*b*B)*Sqrt[a + b*Tan[c + d*x]])/d - (a*A*Cot[c + d*x]*(a + b*Tan[c + d*x])^(3/2))/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 3728

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_.)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_)*((A_.) + (B_.)*
tan[(e_.) + (f_.)*(x_)] + (C_.)*tan[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[C*(a + b*Tan[e + f*x])^m*((c + d
*Tan[e + f*x])^(n + 1)/(d*f*(m + n + 1))), x] + Dist[1/(d*(m + n + 1)), Int[(a + b*Tan[e + f*x])^(m - 1)*(c +
d*Tan[e + f*x])^n*Simp[a*A*d*(m + n + 1) - C*(b*c*m + a*d*(n + 1)) + d*(A*b + a*B - b*C)*(m + n + 1)*Tan[e + f
*x] - (C*m*(b*c - a*d) - b*B*d*(m + n + 1))*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] && GtQ[m, 0] &&  !(IGtQ[n, 0] && ( !Intege
rQ[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 (c+d x) (a+b \tan (c+d x))^{3/2}}{d}+\int \cot (c+d x) \sqrt {a+b \tan (c+d x)} \left (\frac {1}{2} a (5 A b+2 a B)-\left (a^2 A-A b^2-2 a b B\right ) \tan (c+d x)+\frac {1}{2} b (a A+2 b B) \tan ^2(c+d x)\right ) \, dx \\ & = \frac {b (a A+2 b B) \sqrt {a+b \tan (c+d x)}}{d}-\frac {a A \cot (c+d x) (a+b \tan (c+d x))^{3/2}}{d}+2 \int \frac {\cot (c+d x) \left (\frac {1}{4} a^2 (5 A b+2 a B)-\frac {1}{2} \left (a^3 A-3 a A b^2-3 a^2 b B+b^3 B\right ) \tan (c+d x)-\frac {1}{4} b \left (a^2 A-2 A b^2-6 a b B\right ) \tan ^2(c+d x)\right )}{\sqrt {a+b \tan (c+d x)}} \, dx \\ & = \frac {b (a A+2 b B) \sqrt {a+b \tan (c+d x)}}{d}-\frac {a A \cot (c+d x) (a+b \tan (c+d x))^{3/2}}{d}+2 \int \frac {\frac {1}{2} \left (-a^3 A+3 a A b^2+3 a^2 b B-b^3 B\right )-\frac {1}{2} \left (3 a^2 A b-A b^3+a^3 B-3 a b^2 B\right ) \tan (c+d x)}{\sqrt {a+b \tan (c+d x)}} \, dx+\frac {1}{2} \left (a^2 (5 A b+2 a B)\right ) \int \frac {\cot (c+d x) \left (1+\tan ^2(c+d x)\right )}{\sqrt {a+b \tan (c+d x)}} \, dx \\ & = \frac {b (a A+2 b B) \sqrt {a+b \tan (c+d x)}}{d}-\frac {a A \cot (c+d x) (a+b \tan (c+d x))^{3/2}}{d}-\frac {1}{2} \left ((a-i b)^3 (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)^3 (A+i B)\right ) \int \frac {1-i \tan (c+d x)}{\sqrt {a+b \tan (c+d x)}} \, dx+\frac {\left (a^2 (5 A b+2 a B)\right ) \text {Subst}\left (\int \frac {1}{x \sqrt {a+b x}} \, dx,x,\tan (c+d x)\right )}{2 d} \\ & = \frac {b (a A+2 b B) \sqrt {a+b \tan (c+d x)}}{d}-\frac {a A \cot (c+d x) (a+b \tan (c+d x))^{3/2}}{d}+\frac {\left (i (a+i b)^3 (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)^3 (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^2 (5 A b+2 a 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 )}{b d} \\ & = -\frac {a^{3/2} (5 A b+2 a B) \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a}}\right )}{d}+\frac {b (a A+2 b B) \sqrt {a+b \tan (c+d x)}}{d}-\frac {a A \cot (c+d x) (a+b \tan (c+d x))^{3/2}}{d}+\frac {\left ((a-i b)^3 (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)^3 (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 {a^{3/2} (5 A b+2 a B) \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a}}\right )}{d}+\frac {(a-i b)^{5/2} (i A+B) \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a-i b}}\right )}{d}-\frac {(a+i b)^{5/2} (i A-B) \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a+i b}}\right )}{d}+\frac {b (a A+2 b B) \sqrt {a+b \tan (c+d x)}}{d}-\frac {a A \cot (c+d x) (a+b \tan (c+d x))^{3/2}}{d} \\ \end{align*}

Mathematica [B] (verified)

Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(400\) vs. \(2(196)=392\).

Time = 1.13 (sec) , antiderivative size = 400, normalized size of antiderivative = 2.04 \[ \int \cot ^2(c+d x) (a+b \tan (c+d x))^{5/2} (A+B \tan (c+d x)) \, dx=\frac {2 b B \cot (c+d x) (a+b \tan (c+d x))^{3/2}}{d}+2 \left (-\frac {b (A b+4 a B) \cot (c+d x) \sqrt {a+b \tan (c+d x)}}{d}-2 \left (-\frac {-\frac {a^{5/2} (5 A b+2 a B) \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a}}\right )}{4 d}+\frac {i \sqrt {a-i b} \left (\frac {1}{4} i a \left (3 a^2 A b-A b^3+a^3 B-3 a b^2 B\right )-\frac {1}{4} a \left (a^3 A-3 a A b^2-3 a^2 b B+b^3 B\right )\right ) \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a-i b}}\right )}{(-a+i b) d}-\frac {i \sqrt {a+i b} \left (-\frac {1}{4} i a \left (3 a^2 A b-A b^3+a^3 B-3 a b^2 B\right )-\frac {1}{4} a \left (a^3 A-3 a A b^2-3 a^2 b B+b^3 B\right )\right ) \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a+i b}}\right )}{(-a-i b) d}}{a}+\frac {\left (a^2 A-2 A b^2-6 a b B\right ) \cot (c+d x) \sqrt {a+b \tan (c+d x)}}{4 d}\right )\right ) \]

[In]

Integrate[Cot[c + d*x]^2*(a + b*Tan[c + d*x])^(5/2)*(A + B*Tan[c + d*x]),x]

[Out]

(2*b*B*Cot[c + d*x]*(a + b*Tan[c + d*x])^(3/2))/d + 2*(-((b*(A*b + 4*a*B)*Cot[c + d*x]*Sqrt[a + b*Tan[c + d*x]
])/d) - 2*(-((-1/4*(a^(5/2)*(5*A*b + 2*a*B)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a]])/d + (I*Sqrt[a - I*b]*((
I/4)*a*(3*a^2*A*b - A*b^3 + a^3*B - 3*a*b^2*B) - (a*(a^3*A - 3*a*A*b^2 - 3*a^2*b*B + b^3*B))/4)*ArcTanh[Sqrt[a
+ b*Tan[c + d*x]]/Sqrt[a - I*b]])/((-a + I*b)*d) - (I*Sqrt[a + I*b]*((-1/4*I)*a*(3*a^2*A*b - A*b^3 + a^3*B -
3*a*b^2*B) - (a*(a^3*A - 3*a*A*b^2 - 3*a^2*b*B + b^3*B))/4)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a + I*b]])/(
(-a - I*b)*d))/a) + ((a^2*A - 2*A*b^2 - 6*a*b*B)*Cot[c + d*x]*Sqrt[a + b*Tan[c + d*x]])/(4*d)))

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(2389\) vs. \(2(168)=336\).

Time = 0.24 (sec) , antiderivative size = 2390, normalized size of antiderivative = 12.19

method	result	size
derivativedivides	\(\text {Expression too large to display}\)	\(2390\)
default	\(\text {Expression too large to display}\)	\(2390\)

[In]

int(cot(d*x+c)^2*(a+b*tan(d*x+c))^(5/2)*(A+B*tan(d*x+c)),x,method=_RETURNVERBOSE)

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 4900 vs. \(2 (162) = 324\).

Time = 13.31 (sec) , antiderivative size = 9815, normalized size of antiderivative = 50.08 \[ \int \cot ^2(c+d x) (a+b \tan (c+d x))^{5/2} (A+B \tan (c+d x)) \, dx=\text {Too large to display} \]

[In]

integrate(cot(d*x+c)^2*(a+b*tan(d*x+c))^(5/2)*(A+B*tan(d*x+c)),x, algorithm="fricas")

[Out]

Too large to include

Sympy [F(-1)]

Timed out. \[ \int \cot ^2(c+d x) (a+b \tan (c+d x))^{5/2} (A+B \tan (c+d x)) \, dx=\text {Timed out} \]

[In]

integrate(cot(d*x+c)**2*(a+b*tan(d*x+c))**(5/2)*(A+B*tan(d*x+c)),x)

[Out]

Timed out

Maxima [F]

\[ \int \cot ^2(c+d x) (a+b \tan (c+d x))^{5/2} (A+B \tan (c+d x)) \, dx=\int { {\left (B \tan \left (d x + c\right ) + A\right )} {\left (b \tan \left (d x + c\right ) + a\right )}^{\frac {5}{2}} \cot \left (d x + c\right )^{2} \,d x } \]

[In]

integrate(cot(d*x+c)^2*(a+b*tan(d*x+c))^(5/2)*(A+B*tan(d*x+c)),x, algorithm="maxima")

[Out]

integrate((B*tan(d*x + c) + A)*(b*tan(d*x + c) + a)^(5/2)*cot(d*x + c)^2, x)

Giac [F(-1)]

Timed out. \[ \int \cot ^2(c+d x) (a+b \tan (c+d x))^{5/2} (A+B \tan (c+d x)) \, dx=\text {Timed out} \]

[In]

integrate(cot(d*x+c)^2*(a+b*tan(d*x+c))^(5/2)*(A+B*tan(d*x+c)),x, algorithm="giac")

[Out]

Timed out

Mupad [B] (verification not implemented)

Time = 11.72 (sec) , antiderivative size = 31186, normalized size of antiderivative = 159.11 \[ \int \cot ^2(c+d x) (a+b \tan (c+d x))^{5/2} (A+B \tan (c+d x)) \, dx=\text {Too large to display} \]

[In]

int(cot(c + d*x)^2*(A + B*tan(c + d*x))*(a + b*tan(c + d*x))^(5/2),x)

[Out]

(2*B*b^2*(a + b*tan(c + d*x))^(1/2))/d - atan(((((8*(176*A^3*a^5*b^13*d^2 - 400*A^3*a^3*b^15*d^2 + 488*A^3*a^7
*b^11*d^2 - 92*A^3*a^9*b^9*d^2 + 12*B^3*a^2*b^16*d^2 + 192*B^3*a^4*b^14*d^2 - 120*B^3*a^6*b^12*d^2 - 288*B^3*a
^8*b^10*d^2 + 12*B^3*a^10*b^8*d^2 + 4*A^3*a*b^17*d^2 + 4*A*B^2*a*b^17*d^2 + 464*A*B^2*a^3*b^15*d^2 - 920*A*B^2
*a^5*b^13*d^2 - 1104*A*B^2*a^7*b^11*d^2 + 276*A*B^2*a^9*b^9*d^2 + 172*A^2*B*a^2*b^16*d^2 - 1468*A^2*B*a^4*b^14
*d^2 - 776*A^2*B*a^6*b^12*d^2 + 828*A^2*B*a^8*b^10*d^2 - 36*A^2*B*a^10*b^8*d^2))/d^5 - (((8*(16*B*a*b^12*d^4 +
128*A*a^2*b^11*d^4 + 128*A*a^4*b^9*d^4 + 64*B*a^3*b^10*d^4 + 48*B*a^5*b^8*d^4))/d^5 - (16*(32*b^10*d^4 + 48*a
^2*b^8*d^4)*(a + b*tan(c + d*x))^(1/2)*((((8*B^2*a^5*d^2 - 8*A^2*a^5*d^2 + 80*A^2*a^3*b^2*d^2 - 80*B^2*a^3*b^2
*d^2 + 16*A*B*b^5*d^2 - 40*A^2*a*b^4*d^2 + 40*B^2*a*b^4*d^2 - 160*A*B*a^2*b^3*d^2 + 80*A*B*a^4*b*d^2)^2/64 - d
^4*(A^4*a^10 + A^4*b^10 + B^4*a^10 + B^4*b^10 + 2*A^2*B^2*a^10 + 2*A^2*B^2*b^10 + 5*A^4*a^2*b^8 + 10*A^4*a^4*b
^6 + 10*A^4*a^6*b^4 + 5*A^4*a^8*b^2 + 5*B^4*a^2*b^8 + 10*B^4*a^4*b^6 + 10*B^4*a^6*b^4 + 5*B^4*a^8*b^2 + 10*A^2
*B^2*a^2*b^8 + 20*A^2*B^2*a^4*b^6 + 20*A^2*B^2*a^6*b^4 + 10*A^2*B^2*a^8*b^2))^(1/2) - A^2*a^5*d^2 + B^2*a^5*d^
2 + 10*A^2*a^3*b^2*d^2 - 10*B^2*a^3*b^2*d^2 + 2*A*B*b^5*d^2 - 5*A^2*a*b^4*d^2 + 5*B^2*a*b^4*d^2 - 20*A*B*a^2*b
^3*d^2 + 10*A*B*a^4*b*d^2)/(4*d^4))^(1/2))/d^4)*((((8*B^2*a^5*d^2 - 8*A^2*a^5*d^2 + 80*A^2*a^3*b^2*d^2 - 80*B^
2*a^3*b^2*d^2 + 16*A*B*b^5*d^2 - 40*A^2*a*b^4*d^2 + 40*B^2*a*b^4*d^2 - 160*A*B*a^2*b^3*d^2 + 80*A*B*a^4*b*d^2)
^2/64 - d^4*(A^4*a^10 + A^4*b^10 + B^4*a^10 + B^4*b^10 + 2*A^2*B^2*a^10 + 2*A^2*B^2*b^10 + 5*A^4*a^2*b^8 + 10*
A^4*a^4*b^6 + 10*A^4*a^6*b^4 + 5*A^4*a^8*b^2 + 5*B^4*a^2*b^8 + 10*B^4*a^4*b^6 + 10*B^4*a^6*b^4 + 5*B^4*a^8*b^2
+ 10*A^2*B^2*a^2*b^8 + 20*A^2*B^2*a^4*b^6 + 20*A^2*B^2*a^6*b^4 + 10*A^2*B^2*a^8*b^2))^(1/2) - A^2*a^5*d^2 + B
^2*a^5*d^2 + 10*A^2*a^3*b^2*d^2 - 10*B^2*a^3*b^2*d^2 + 2*A*B*b^5*d^2 - 5*A^2*a*b^4*d^2 + 5*B^2*a*b^4*d^2 - 20*
A*B*a^2*b^3*d^2 + 10*A*B*a^4*b*d^2)/(4*d^4))^(1/2) + (16*(a + b*tan(c + d*x))^(1/2)*(20*A^2*a^3*b^12*d^2 + 304
*A^2*a^5*b^10*d^2 - 20*A^2*a^7*b^8*d^2 - 20*B^2*a^3*b^12*d^2 - 204*B^2*a^5*b^10*d^2 + 36*B^2*a^7*b^8*d^2 + 32*
A*B*b^15*d^2 - 76*A^2*a*b^14*d^2 + 76*B^2*a*b^14*d^2 - 240*A*B*a^2*b^13*d^2 - 320*A*B*a^4*b^11*d^2 + 288*A*B*a
^6*b^9*d^2))/d^4)*((((8*B^2*a^5*d^2 - 8*A^2*a^5*d^2 + 80*A^2*a^3*b^2*d^2 - 80*B^2*a^3*b^2*d^2 + 16*A*B*b^5*d^2
- 40*A^2*a*b^4*d^2 + 40*B^2*a*b^4*d^2 - 160*A*B*a^2*b^3*d^2 + 80*A*B*a^4*b*d^2)^2/64 - d^4*(A^4*a^10 + A^4*b^
10 + B^4*a^10 + B^4*b^10 + 2*A^2*B^2*a^10 + 2*A^2*B^2*b^10 + 5*A^4*a^2*b^8 + 10*A^4*a^4*b^6 + 10*A^4*a^6*b^4 +
5*A^4*a^8*b^2 + 5*B^4*a^2*b^8 + 10*B^4*a^4*b^6 + 10*B^4*a^6*b^4 + 5*B^4*a^8*b^2 + 10*A^2*B^2*a^2*b^8 + 20*A^2
*B^2*a^4*b^6 + 20*A^2*B^2*a^6*b^4 + 10*A^2*B^2*a^8*b^2))^(1/2) - A^2*a^5*d^2 + B^2*a^5*d^2 + 10*A^2*a^3*b^2*d^
2 - 10*B^2*a^3*b^2*d^2 + 2*A*B*b^5*d^2 - 5*A^2*a*b^4*d^2 + 5*B^2*a*b^4*d^2 - 20*A*B*a^2*b^3*d^2 + 10*A*B*a^4*b
*d^2)/(4*d^4))^(1/2))*((((8*B^2*a^5*d^2 - 8*A^2*a^5*d^2 + 80*A^2*a^3*b^2*d^2 - 80*B^2*a^3*b^2*d^2 + 16*A*B*b^5
*d^2 - 40*A^2*a*b^4*d^2 + 40*B^2*a*b^4*d^2 - 160*A*B*a^2*b^3*d^2 + 80*A*B*a^4*b*d^2)^2/64 - d^4*(A^4*a^10 + A^
4*b^10 + B^4*a^10 + B^4*b^10 + 2*A^2*B^2*a^10 + 2*A^2*B^2*b^10 + 5*A^4*a^2*b^8 + 10*A^4*a^4*b^6 + 10*A^4*a^6*b
^4 + 5*A^4*a^8*b^2 + 5*B^4*a^2*b^8 + 10*B^4*a^4*b^6 + 10*B^4*a^6*b^4 + 5*B^4*a^8*b^2 + 10*A^2*B^2*a^2*b^8 + 20
*A^2*B^2*a^4*b^6 + 20*A^2*B^2*a^6*b^4 + 10*A^2*B^2*a^8*b^2))^(1/2) - A^2*a^5*d^2 + B^2*a^5*d^2 + 10*A^2*a^3*b^
2*d^2 - 10*B^2*a^3*b^2*d^2 + 2*A*B*b^5*d^2 - 5*A^2*a*b^4*d^2 + 5*B^2*a*b^4*d^2 - 20*A*B*a^2*b^3*d^2 + 10*A*B*a
^4*b*d^2)/(4*d^4))^(1/2) + (16*(a + b*tan(c + d*x))^(1/2)*(2*A^4*b^20 + 2*B^4*b^20 + 4*A^2*B^2*b^20 + 12*A^4*a
^2*b^18 + 55*A^4*a^4*b^16 - 335*A^4*a^6*b^14 + 405*A^4*a^8*b^12 - 13*A^4*a^10*b^10 + 2*A^4*a^12*b^8 + 12*B^4*a
^2*b^18 + 30*B^4*a^4*b^16 + 36*B^4*a^6*b^14 + 90*B^4*a^8*b^12 - 48*B^4*a^10*b^10 + 6*B^4*a^12*b^8 + 24*A^2*B^2
*a^2*b^18 + 35*A^2*B^2*a^4*b^16 + 699*A^2*B^2*a^6*b^14 - 1175*A^2*B^2*a^8*b^12 + 349*A^2*B^2*a^10*b^10 - 20*A*
B^3*a^5*b^15 + 348*A*B^3*a^7*b^13 - 460*A*B^3*a^9*b^11 + 68*A*B^3*a^11*b^9 + 320*A^3*B*a^5*b^15 - 1300*A^3*B*a
^7*b^13 + 600*A^3*B*a^9*b^11 - 20*A^3*B*a^11*b^9))/d^4)*((((8*B^2*a^5*d^2 - 8*A^2*a^5*d^2 + 80*A^2*a^3*b^2*d^2
- 80*B^2*a^3*b^2*d^2 + 16*A*B*b^5*d^2 - 40*A^2*a*b^4*d^2 + 40*B^2*a*b^4*d^2 - 160*A*B*a^2*b^3*d^2 + 80*A*B*a^
4*b*d^2)^2/64 - d^4*(A^4*a^10 + A^4*b^10 + B^4*a^10 + B^4*b^10 + 2*A^2*B^2*a^10 + 2*A^2*B^2*b^10 + 5*A^4*a^2*b
^8 + 10*A^4*a^4*b^6 + 10*A^4*a^6*b^4 + 5*A^4*a^8*b^2 + 5*B^4*a^2*b^8 + 10*B^4*a^4*b^6 + 10*B^4*a^6*b^4 + 5*B^4
*a^8*b^2 + 10*A^2*B^2*a^2*b^8 + 20*A^2*B^2*a^4*b^6 + 20*A^2*B^2*a^6*b^4 + 10*A^2*B^2*a^8*b^2))^(1/2) - A^2*a^5
*d^2 + B^2*a^5*d^2 + 10*A^2*a^3*b^2*d^2 - 10*B^2*a^3*b^2*d^2 + 2*A*B*b^5*d^2 - 5*A^2*a*b^4*d^2 + 5*B^2*a*b^4*d
^2 - 20*A*B*a^2*b^3*d^2 + 10*A*B*a^4*b*d^2)/(4*d^4))^(1/2)*1i - (((8*(176*A^3*a^5*b^13*d^2 - 400*A^3*a^3*b^15*
d^2 + 488*A^3*a^7*b^11*d^2 - 92*A^3*a^9*b^9*d^2 + 12*B^3*a^2*b^16*d^2 + 192*B^3*a^4*b^14*d^2 - 120*B^3*a^6*b^1
2*d^2 - 288*B^3*a^8*b^10*d^2 + 12*B^3*a^10*b^8*d^2 + 4*A^3*a*b^17*d^2 + 4*A*B^2*a*b^17*d^2 + 464*A*B^2*a^3*b^1
5*d^2 - 920*A*B^2*a^5*b^13*d^2 - 1104*A*B^2*a^7*b^11*d^2 + 276*A*B^2*a^9*b^9*d^2 + 172*A^2*B*a^2*b^16*d^2 - 14
68*A^2*B*a^4*b^14*d^2 - 776*A^2*B*a^6*b^12*d^2 + 828*A^2*B*a^8*b^10*d^2 - 36*A^2*B*a^10*b^8*d^2))/d^5 - (((8*(
16*B*a*b^12*d^4 + 128*A*a^2*b^11*d^4 + 128*A*a^4*b^9*d^4 + 64*B*a^3*b^10*d^4 + 48*B*a^5*b^8*d^4))/d^5 + (16*(3
2*b^10*d^4 + 48*a^2*b^8*d^4)*(a + b*tan(c + d*x))^(1/2)*((((8*B^2*a^5*d^2 - 8*A^2*a^5*d^2 + 80*A^2*a^3*b^2*d^2
- 80*B^2*a^3*b^2*d^2 + 16*A*B*b^5*d^2 - 40*A^2*a*b^4*d^2 + 40*B^2*a*b^4*d^2 - 160*A*B*a^2*b^3*d^2 + 80*A*B*a^
4*b*d^2)^2/64 - d^4*(A^4*a^10 + A^4*b^10 + B^4*a^10 + B^4*b^10 + 2*A^2*B^2*a^10 + 2*A^2*B^2*b^10 + 5*A^4*a^2*b
^8 + 10*A^4*a^4*b^6 + 10*A^4*a^6*b^4 + 5*A^4*a^8*b^2 + 5*B^4*a^2*b^8 + 10*B^4*a^4*b^6 + 10*B^4*a^6*b^4 + 5*B^4
*a^8*b^2 + 10*A^2*B^2*a^2*b^8 + 20*A^2*B^2*a^4*b^6 + 20*A^2*B^2*a^6*b^4 + 10*A^2*B^2*a^8*b^2))^(1/2) - A^2*a^5
*d^2 + B^2*a^5*d^2 + 10*A^2*a^3*b^2*d^2 - 10*B^2*a^3*b^2*d^2 + 2*A*B*b^5*d^2 - 5*A^2*a*b^4*d^2 + 5*B^2*a*b^4*d
^2 - 20*A*B*a^2*b^3*d^2 + 10*A*B*a^4*b*d^2)/(4*d^4))^(1/2))/d^4)*((((8*B^2*a^5*d^2 - 8*A^2*a^5*d^2 + 80*A^2*a^
3*b^2*d^2 - 80*B^2*a^3*b^2*d^2 + 16*A*B*b^5*d^2 - 40*A^2*a*b^4*d^2 + 40*B^2*a*b^4*d^2 - 160*A*B*a^2*b^3*d^2 +
80*A*B*a^4*b*d^2)^2/64 - d^4*(A^4*a^10 + A^4*b^10 + B^4*a^10 + B^4*b^10 + 2*A^2*B^2*a^10 + 2*A^2*B^2*b^10 + 5*
A^4*a^2*b^8 + 10*A^4*a^4*b^6 + 10*A^4*a^6*b^4 + 5*A^4*a^8*b^2 + 5*B^4*a^2*b^8 + 10*B^4*a^4*b^6 + 10*B^4*a^6*b^
4 + 5*B^4*a^8*b^2 + 10*A^2*B^2*a^2*b^8 + 20*A^2*B^2*a^4*b^6 + 20*A^2*B^2*a^6*b^4 + 10*A^2*B^2*a^8*b^2))^(1/2)
- A^2*a^5*d^2 + B^2*a^5*d^2 + 10*A^2*a^3*b^2*d^2 - 10*B^2*a^3*b^2*d^2 + 2*A*B*b^5*d^2 - 5*A^2*a*b^4*d^2 + 5*B^
2*a*b^4*d^2 - 20*A*B*a^2*b^3*d^2 + 10*A*B*a^4*b*d^2)/(4*d^4))^(1/2) - (16*(a + b*tan(c + d*x))^(1/2)*(20*A^2*a
^3*b^12*d^2 + 304*A^2*a^5*b^10*d^2 - 20*A^2*a^7*b^8*d^2 - 20*B^2*a^3*b^12*d^2 - 204*B^2*a^5*b^10*d^2 + 36*B^2*
a^7*b^8*d^2 + 32*A*B*b^15*d^2 - 76*A^2*a*b^14*d^2 + 76*B^2*a*b^14*d^2 - 240*A*B*a^2*b^13*d^2 - 320*A*B*a^4*b^1
1*d^2 + 288*A*B*a^6*b^9*d^2))/d^4)*((((8*B^2*a^5*d^2 - 8*A^2*a^5*d^2 + 80*A^2*a^3*b^2*d^2 - 80*B^2*a^3*b^2*d^2
+ 16*A*B*b^5*d^2 - 40*A^2*a*b^4*d^2 + 40*B^2*a*b^4*d^2 - 160*A*B*a^2*b^3*d^2 + 80*A*B*a^4*b*d^2)^2/64 - d^4*(
A^4*a^10 + A^4*b^10 + B^4*a^10 + B^4*b^10 + 2*A^2*B^2*a^10 + 2*A^2*B^2*b^10 + 5*A^4*a^2*b^8 + 10*A^4*a^4*b^6 +
10*A^4*a^6*b^4 + 5*A^4*a^8*b^2 + 5*B^4*a^2*b^8 + 10*B^4*a^4*b^6 + 10*B^4*a^6*b^4 + 5*B^4*a^8*b^2 + 10*A^2*B^2
*a^2*b^8 + 20*A^2*B^2*a^4*b^6 + 20*A^2*B^2*a^6*b^4 + 10*A^2*B^2*a^8*b^2))^(1/2) - A^2*a^5*d^2 + B^2*a^5*d^2 +
10*A^2*a^3*b^2*d^2 - 10*B^2*a^3*b^2*d^2 + 2*A*B*b^5*d^2 - 5*A^2*a*b^4*d^2 + 5*B^2*a*b^4*d^2 - 20*A*B*a^2*b^3*d
^2 + 10*A*B*a^4*b*d^2)/(4*d^4))^(1/2))*((((8*B^2*a^5*d^2 - 8*A^2*a^5*d^2 + 80*A^2*a^3*b^2*d^2 - 80*B^2*a^3*b^2
*d^2 + 16*A*B*b^5*d^2 - 40*A^2*a*b^4*d^2 + 40*B^2*a*b^4*d^2 - 160*A*B*a^2*b^3*d^2 + 80*A*B*a^4*b*d^2)^2/64 - d
^4*(A^4*a^10 + A^4*b^10 + B^4*a^10 + B^4*b^10 + 2*A^2*B^2*a^10 + 2*A^2*B^2*b^10 + 5*A^4*a^2*b^8 + 10*A^4*a^4*b
^6 + 10*A^4*a^6*b^4 + 5*A^4*a^8*b^2 + 5*B^4*a^2*b^8 + 10*B^4*a^4*b^6 + 10*B^4*a^6*b^4 + 5*B^4*a^8*b^2 + 10*A^2
*B^2*a^2*b^8 + 20*A^2*B^2*a^4*b^6 + 20*A^2*B^2*a^6*b^4 + 10*A^2*B^2*a^8*b^2))^(1/2) - A^2*a^5*d^2 + B^2*a^5*d^
2 + 10*A^2*a^3*b^2*d^2 - 10*B^2*a^3*b^2*d^2 + 2*A*B*b^5*d^2 - 5*A^2*a*b^4*d^2 + 5*B^2*a*b^4*d^2 - 20*A*B*a^2*b
^3*d^2 + 10*A*B*a^4*b*d^2)/(4*d^4))^(1/2) - (16*(a + b*tan(c + d*x))^(1/2)*(2*A^4*b^20 + 2*B^4*b^20 + 4*A^2*B^
2*b^20 + 12*A^4*a^2*b^18 + 55*A^4*a^4*b^16 - 335*A^4*a^6*b^14 + 405*A^4*a^8*b^12 - 13*A^4*a^10*b^10 + 2*A^4*a^
12*b^8 + 12*B^4*a^2*b^18 + 30*B^4*a^4*b^16 + 36*B^4*a^6*b^14 + 90*B^4*a^8*b^12 - 48*B^4*a^10*b^10 + 6*B^4*a^12
*b^8 + 24*A^2*B^2*a^2*b^18 + 35*A^2*B^2*a^4*b^16 + 699*A^2*B^2*a^6*b^14 - 1175*A^2*B^2*a^8*b^12 + 349*A^2*B^2*
a^10*b^10 - 20*A*B^3*a^5*b^15 + 348*A*B^3*a^7*b^13 - 460*A*B^3*a^9*b^11 + 68*A*B^3*a^11*b^9 + 320*A^3*B*a^5*b^
15 - 1300*A^3*B*a^7*b^13 + 600*A^3*B*a^9*b^11 - 20*A^3*B*a^11*b^9))/d^4)*((((8*B^2*a^5*d^2 - 8*A^2*a^5*d^2 + 8
0*A^2*a^3*b^2*d^2 - 80*B^2*a^3*b^2*d^2 + 16*A*B*b^5*d^2 - 40*A^2*a*b^4*d^2 + 40*B^2*a*b^4*d^2 - 160*A*B*a^2*b^
3*d^2 + 80*A*B*a^4*b*d^2)^2/64 - d^4*(A^4*a^10 + A^4*b^10 + B^4*a^10 + B^4*b^10 + 2*A^2*B^2*a^10 + 2*A^2*B^2*b
^10 + 5*A^4*a^2*b^8 + 10*A^4*a^4*b^6 + 10*A^4*a^6*b^4 + 5*A^4*a^8*b^2 + 5*B^4*a^2*b^8 + 10*B^4*a^4*b^6 + 10*B^
4*a^6*b^4 + 5*B^4*a^8*b^2 + 10*A^2*B^2*a^2*b^8 + 20*A^2*B^2*a^4*b^6 + 20*A^2*B^2*a^6*b^4 + 10*A^2*B^2*a^8*b^2)
)^(1/2) - A^2*a^5*d^2 + B^2*a^5*d^2 + 10*A^2*a^3*b^2*d^2 - 10*B^2*a^3*b^2*d^2 + 2*A*B*b^5*d^2 - 5*A^2*a*b^4*d^
2 + 5*B^2*a*b^4*d^2 - 20*A*B*a^2*b^3*d^2 + 10*A*B*a^4*b*d^2)/(4*d^4))^(1/2)*1i)/((((8*(176*A^3*a^5*b^13*d^2 -
400*A^3*a^3*b^15*d^2 + 488*A^3*a^7*b^11*d^2 - 92*A^3*a^9*b^9*d^2 + 12*B^3*a^2*b^16*d^2 + 192*B^3*a^4*b^14*d^2
- 120*B^3*a^6*b^12*d^2 - 288*B^3*a^8*b^10*d^2 + 12*B^3*a^10*b^8*d^2 + 4*A^3*a*b^17*d^2 + 4*A*B^2*a*b^17*d^2 +
464*A*B^2*a^3*b^15*d^2 - 920*A*B^2*a^5*b^13*d^2 - 1104*A*B^2*a^7*b^11*d^2 + 276*A*B^2*a^9*b^9*d^2 + 172*A^2*B*
a^2*b^16*d^2 - 1468*A^2*B*a^4*b^14*d^2 - 776*A^2*B*a^6*b^12*d^2 + 828*A^2*B*a^8*b^10*d^2 - 36*A^2*B*a^10*b^8*d
^2))/d^5 - (((8*(16*B*a*b^12*d^4 + 128*A*a^2*b^11*d^4 + 128*A*a^4*b^9*d^4 + 64*B*a^3*b^10*d^4 + 48*B*a^5*b^8*d
^4))/d^5 - (16*(32*b^10*d^4 + 48*a^2*b^8*d^4)*(a + b*tan(c + d*x))^(1/2)*((((8*B^2*a^5*d^2 - 8*A^2*a^5*d^2 + 8
0*A^2*a^3*b^2*d^2 - 80*B^2*a^3*b^2*d^2 + 16*A*B*b^5*d^2 - 40*A^2*a*b^4*d^2 + 40*B^2*a*b^4*d^2 - 160*A*B*a^2*b^
3*d^2 + 80*A*B*a^4*b*d^2)^2/64 - d^4*(A^4*a^10 + A^4*b^10 + B^4*a^10 + B^4*b^10 + 2*A^2*B^2*a^10 + 2*A^2*B^2*b
^10 + 5*A^4*a^2*b^8 + 10*A^4*a^4*b^6 + 10*A^4*a^6*b^4 + 5*A^4*a^8*b^2 + 5*B^4*a^2*b^8 + 10*B^4*a^4*b^6 + 10*B^
4*a^6*b^4 + 5*B^4*a^8*b^2 + 10*A^2*B^2*a^2*b^8 + 20*A^2*B^2*a^4*b^6 + 20*A^2*B^2*a^6*b^4 + 10*A^2*B^2*a^8*b^2)
)^(1/2) - A^2*a^5*d^2 + B^2*a^5*d^2 + 10*A^2*a^3*b^2*d^2 - 10*B^2*a^3*b^2*d^2 + 2*A*B*b^5*d^2 - 5*A^2*a*b^4*d^
2 + 5*B^2*a*b^4*d^2 - 20*A*B*a^2*b^3*d^2 + 10*A*B*a^4*b*d^2)/(4*d^4))^(1/2))/d^4)*((((8*B^2*a^5*d^2 - 8*A^2*a^
5*d^2 + 80*A^2*a^3*b^2*d^2 - 80*B^2*a^3*b^2*d^2 + 16*A*B*b^5*d^2 - 40*A^2*a*b^4*d^2 + 40*B^2*a*b^4*d^2 - 160*A
*B*a^2*b^3*d^2 + 80*A*B*a^4*b*d^2)^2/64 - d^4*(A^4*a^10 + A^4*b^10 + B^4*a^10 + B^4*b^10 + 2*A^2*B^2*a^10 + 2*
A^2*B^2*b^10 + 5*A^4*a^2*b^8 + 10*A^4*a^4*b^6 + 10*A^4*a^6*b^4 + 5*A^4*a^8*b^2 + 5*B^4*a^2*b^8 + 10*B^4*a^4*b^
6 + 10*B^4*a^6*b^4 + 5*B^4*a^8*b^2 + 10*A^2*B^2*a^2*b^8 + 20*A^2*B^2*a^4*b^6 + 20*A^2*B^2*a^6*b^4 + 10*A^2*B^2
*a^8*b^2))^(1/2) - A^2*a^5*d^2 + B^2*a^5*d^2 + 10*A^2*a^3*b^2*d^2 - 10*B^2*a^3*b^2*d^2 + 2*A*B*b^5*d^2 - 5*A^2
*a*b^4*d^2 + 5*B^2*a*b^4*d^2 - 20*A*B*a^2*b^3*d^2 + 10*A*B*a^4*b*d^2)/(4*d^4))^(1/2) + (16*(a + b*tan(c + d*x)
)^(1/2)*(20*A^2*a^3*b^12*d^2 + 304*A^2*a^5*b^10*d^2 - 20*A^2*a^7*b^8*d^2 - 20*B^2*a^3*b^12*d^2 - 204*B^2*a^5*b
^10*d^2 + 36*B^2*a^7*b^8*d^2 + 32*A*B*b^15*d^2 - 76*A^2*a*b^14*d^2 + 76*B^2*a*b^14*d^2 - 240*A*B*a^2*b^13*d^2
- 320*A*B*a^4*b^11*d^2 + 288*A*B*a^6*b^9*d^2))/d^4)*((((8*B^2*a^5*d^2 - 8*A^2*a^5*d^2 + 80*A^2*a^3*b^2*d^2 - 8
0*B^2*a^3*b^2*d^2 + 16*A*B*b^5*d^2 - 40*A^2*a*b^4*d^2 + 40*B^2*a*b^4*d^2 - 160*A*B*a^2*b^3*d^2 + 80*A*B*a^4*b*
d^2)^2/64 - d^4*(A^4*a^10 + A^4*b^10 + B^4*a^10 + B^4*b^10 + 2*A^2*B^2*a^10 + 2*A^2*B^2*b^10 + 5*A^4*a^2*b^8 +
10*A^4*a^4*b^6 + 10*A^4*a^6*b^4 + 5*A^4*a^8*b^2 + 5*B^4*a^2*b^8 + 10*B^4*a^4*b^6 + 10*B^4*a^6*b^4 + 5*B^4*a^8
*b^2 + 10*A^2*B^2*a^2*b^8 + 20*A^2*B^2*a^4*b^6 + 20*A^2*B^2*a^6*b^4 + 10*A^2*B^2*a^8*b^2))^(1/2) - A^2*a^5*d^2
+ B^2*a^5*d^2 + 10*A^2*a^3*b^2*d^2 - 10*B^2*a^3*b^2*d^2 + 2*A*B*b^5*d^2 - 5*A^2*a*b^4*d^2 + 5*B^2*a*b^4*d^2 -
20*A*B*a^2*b^3*d^2 + 10*A*B*a^4*b*d^2)/(4*d^4))^(1/2))*((((8*B^2*a^5*d^2 - 8*A^2*a^5*d^2 + 80*A^2*a^3*b^2*d^2
- 80*B^2*a^3*b^2*d^2 + 16*A*B*b^5*d^2 - 40*A^2*a*b^4*d^2 + 40*B^2*a*b^4*d^2 - 160*A*B*a^2*b^3*d^2 + 80*A*B*a^
4*b*d^2)^2/64 - d^4*(A^4*a^10 + A^4*b^10 + B^4*a^10 + B^4*b^10 + 2*A^2*B^2*a^10 + 2*A^2*B^2*b^10 + 5*A^4*a^2*b
^8 + 10*A^4*a^4*b^6 + 10*A^4*a^6*b^4 + 5*A^4*a^8*b^2 + 5*B^4*a^2*b^8 + 10*B^4*a^4*b^6 + 10*B^4*a^6*b^4 + 5*B^4
*a^8*b^2 + 10*A^2*B^2*a^2*b^8 + 20*A^2*B^2*a^4*b^6 + 20*A^2*B^2*a^6*b^4 + 10*A^2*B^2*a^8*b^2))^(1/2) - A^2*a^5
*d^2 + B^2*a^5*d^2 + 10*A^2*a^3*b^2*d^2 - 10*B^2*a^3*b^2*d^2 + 2*A*B*b^5*d^2 - 5*A^2*a*b^4*d^2 + 5*B^2*a*b^4*d
^2 - 20*A*B*a^2*b^3*d^2 + 10*A*B*a^4*b*d^2)/(4*d^4))^(1/2) + (16*(a + b*tan(c + d*x))^(1/2)*(2*A^4*b^20 + 2*B^
4*b^20 + 4*A^2*B^2*b^20 + 12*A^4*a^2*b^18 + 55*A^4*a^4*b^16 - 335*A^4*a^6*b^14 + 405*A^4*a^8*b^12 - 13*A^4*a^1
0*b^10 + 2*A^4*a^12*b^8 + 12*B^4*a^2*b^18 + 30*B^4*a^4*b^16 + 36*B^4*a^6*b^14 + 90*B^4*a^8*b^12 - 48*B^4*a^10*
b^10 + 6*B^4*a^12*b^8 + 24*A^2*B^2*a^2*b^18 + 35*A^2*B^2*a^4*b^16 + 699*A^2*B^2*a^6*b^14 - 1175*A^2*B^2*a^8*b^
12 + 349*A^2*B^2*a^10*b^10 - 20*A*B^3*a^5*b^15 + 348*A*B^3*a^7*b^13 - 460*A*B^3*a^9*b^11 + 68*A*B^3*a^11*b^9 +
320*A^3*B*a^5*b^15 - 1300*A^3*B*a^7*b^13 + 600*A^3*B*a^9*b^11 - 20*A^3*B*a^11*b^9))/d^4)*((((8*B^2*a^5*d^2 -
8*A^2*a^5*d^2 + 80*A^2*a^3*b^2*d^2 - 80*B^2*a^3*b^2*d^2 + 16*A*B*b^5*d^2 - 40*A^2*a*b^4*d^2 + 40*B^2*a*b^4*d^2
- 160*A*B*a^2*b^3*d^2 + 80*A*B*a^4*b*d^2)^2/64 - d^4*(A^4*a^10 + A^4*b^10 + B^4*a^10 + B^4*b^10 + 2*A^2*B^2*a
^10 + 2*A^2*B^2*b^10 + 5*A^4*a^2*b^8 + 10*A^4*a^4*b^6 + 10*A^4*a^6*b^4 + 5*A^4*a^8*b^2 + 5*B^4*a^2*b^8 + 10*B^
4*a^4*b^6 + 10*B^4*a^6*b^4 + 5*B^4*a^8*b^2 + 10*A^2*B^2*a^2*b^8 + 20*A^2*B^2*a^4*b^6 + 20*A^2*B^2*a^6*b^4 + 10
*A^2*B^2*a^8*b^2))^(1/2) - A^2*a^5*d^2 + B^2*a^5*d^2 + 10*A^2*a^3*b^2*d^2 - 10*B^2*a^3*b^2*d^2 + 2*A*B*b^5*d^2
- 5*A^2*a*b^4*d^2 + 5*B^2*a*b^4*d^2 - 20*A*B*a^2*b^3*d^2 + 10*A*B*a^4*b*d^2)/(4*d^4))^(1/2) + (((8*(176*A^3*a
^5*b^13*d^2 - 400*A^3*a^3*b^15*d^2 + 488*A^3*a^7*b^11*d^2 - 92*A^3*a^9*b^9*d^2 + 12*B^3*a^2*b^16*d^2 + 192*B^3
*a^4*b^14*d^2 - 120*B^3*a^6*b^12*d^2 - 288*B^3*a^8*b^10*d^2 + 12*B^3*a^10*b^8*d^2 + 4*A^3*a*b^17*d^2 + 4*A*B^2
*a*b^17*d^2 + 464*A*B^2*a^3*b^15*d^2 - 920*A*B^2*a^5*b^13*d^2 - 1104*A*B^2*a^7*b^11*d^2 + 276*A*B^2*a^9*b^9*d^
2 + 172*A^2*B*a^2*b^16*d^2 - 1468*A^2*B*a^4*b^14*d^2 - 776*A^2*B*a^6*b^12*d^2 + 828*A^2*B*a^8*b^10*d^2 - 36*A^
2*B*a^10*b^8*d^2))/d^5 - (((8*(16*B*a*b^12*d^4 + 128*A*a^2*b^11*d^4 + 128*A*a^4*b^9*d^4 + 64*B*a^3*b^10*d^4 +
48*B*a^5*b^8*d^4))/d^5 + (16*(32*b^10*d^4 + 48*a^2*b^8*d^4)*(a + b*tan(c + d*x))^(1/2)*((((8*B^2*a^5*d^2 - 8*A
^2*a^5*d^2 + 80*A^2*a^3*b^2*d^2 - 80*B^2*a^3*b^2*d^2 + 16*A*B*b^5*d^2 - 40*A^2*a*b^4*d^2 + 40*B^2*a*b^4*d^2 -
160*A*B*a^2*b^3*d^2 + 80*A*B*a^4*b*d^2)^2/64 - d^4*(A^4*a^10 + A^4*b^10 + B^4*a^10 + B^4*b^10 + 2*A^2*B^2*a^10
+ 2*A^2*B^2*b^10 + 5*A^4*a^2*b^8 + 10*A^4*a^4*b^6 + 10*A^4*a^6*b^4 + 5*A^4*a^8*b^2 + 5*B^4*a^2*b^8 + 10*B^4*a
^4*b^6 + 10*B^4*a^6*b^4 + 5*B^4*a^8*b^2 + 10*A^2*B^2*a^2*b^8 + 20*A^2*B^2*a^4*b^6 + 20*A^2*B^2*a^6*b^4 + 10*A^
2*B^2*a^8*b^2))^(1/2) - A^2*a^5*d^2 + B^2*a^5*d^2 + 10*A^2*a^3*b^2*d^2 - 10*B^2*a^3*b^2*d^2 + 2*A*B*b^5*d^2 -
5*A^2*a*b^4*d^2 + 5*B^2*a*b^4*d^2 - 20*A*B*a^2*b^3*d^2 + 10*A*B*a^4*b*d^2)/(4*d^4))^(1/2))/d^4)*((((8*B^2*a^5*
d^2 - 8*A^2*a^5*d^2 + 80*A^2*a^3*b^2*d^2 - 80*B^2*a^3*b^2*d^2 + 16*A*B*b^5*d^2 - 40*A^2*a*b^4*d^2 + 40*B^2*a*b
^4*d^2 - 160*A*B*a^2*b^3*d^2 + 80*A*B*a^4*b*d^2)^2/64 - d^4*(A^4*a^10 + A^4*b^10 + B^4*a^10 + B^4*b^10 + 2*A^2
*B^2*a^10 + 2*A^2*B^2*b^10 + 5*A^4*a^2*b^8 + 10*A^4*a^4*b^6 + 10*A^4*a^6*b^4 + 5*A^4*a^8*b^2 + 5*B^4*a^2*b^8 +
10*B^4*a^4*b^6 + 10*B^4*a^6*b^4 + 5*B^4*a^8*b^2 + 10*A^2*B^2*a^2*b^8 + 20*A^2*B^2*a^4*b^6 + 20*A^2*B^2*a^6*b^
4 + 10*A^2*B^2*a^8*b^2))^(1/2) - A^2*a^5*d^2 + B^2*a^5*d^2 + 10*A^2*a^3*b^2*d^2 - 10*B^2*a^3*b^2*d^2 + 2*A*B*b
^5*d^2 - 5*A^2*a*b^4*d^2 + 5*B^2*a*b^4*d^2 - 20*A*B*a^2*b^3*d^2 + 10*A*B*a^4*b*d^2)/(4*d^4))^(1/2) - (16*(a +
b*tan(c + d*x))^(1/2)*(20*A^2*a^3*b^12*d^2 + 304*A^2*a^5*b^10*d^2 - 20*A^2*a^7*b^8*d^2 - 20*B^2*a^3*b^12*d^2 -
204*B^2*a^5*b^10*d^2 + 36*B^2*a^7*b^8*d^2 + 32*A*B*b^15*d^2 - 76*A^2*a*b^14*d^2 + 76*B^2*a*b^14*d^2 - 240*A*B
*a^2*b^13*d^2 - 320*A*B*a^4*b^11*d^2 + 288*A*B*a^6*b^9*d^2))/d^4)*((((8*B^2*a^5*d^2 - 8*A^2*a^5*d^2 + 80*A^2*a
^3*b^2*d^2 - 80*B^2*a^3*b^2*d^2 + 16*A*B*b^5*d^2 - 40*A^2*a*b^4*d^2 + 40*B^2*a*b^4*d^2 - 160*A*B*a^2*b^3*d^2 +
80*A*B*a^4*b*d^2)^2/64 - d^4*(A^4*a^10 + A^4*b^10 + B^4*a^10 + B^4*b^10 + 2*A^2*B^2*a^10 + 2*A^2*B^2*b^10 + 5
*A^4*a^2*b^8 + 10*A^4*a^4*b^6 + 10*A^4*a^6*b^4 + 5*A^4*a^8*b^2 + 5*B^4*a^2*b^8 + 10*B^4*a^4*b^6 + 10*B^4*a^6*b
^4 + 5*B^4*a^8*b^2 + 10*A^2*B^2*a^2*b^8 + 20*A^2*B^2*a^4*b^6 + 20*A^2*B^2*a^6*b^4 + 10*A^2*B^2*a^8*b^2))^(1/2)
- A^2*a^5*d^2 + B^2*a^5*d^2 + 10*A^2*a^3*b^2*d^2 - 10*B^2*a^3*b^2*d^2 + 2*A*B*b^5*d^2 - 5*A^2*a*b^4*d^2 + 5*B
^2*a*b^4*d^2 - 20*A*B*a^2*b^3*d^2 + 10*A*B*a^4*b*d^2)/(4*d^4))^(1/2))*((((8*B^2*a^5*d^2 - 8*A^2*a^5*d^2 + 80*A
^2*a^3*b^2*d^2 - 80*B^2*a^3*b^2*d^2 + 16*A*B*b^5*d^2 - 40*A^2*a*b^4*d^2 + 40*B^2*a*b^4*d^2 - 160*A*B*a^2*b^3*d
^2 + 80*A*B*a^4*b*d^2)^2/64 - d^4*(A^4*a^10 + A^4*b^10 + B^4*a^10 + B^4*b^10 + 2*A^2*B^2*a^10 + 2*A^2*B^2*b^10
+ 5*A^4*a^2*b^8 + 10*A^4*a^4*b^6 + 10*A^4*a^6*b^4 + 5*A^4*a^8*b^2 + 5*B^4*a^2*b^8 + 10*B^4*a^4*b^6 + 10*B^4*a
^6*b^4 + 5*B^4*a^8*b^2 + 10*A^2*B^2*a^2*b^8 + 20*A^2*B^2*a^4*b^6 + 20*A^2*B^2*a^6*b^4 + 10*A^2*B^2*a^8*b^2))^(
1/2) - A^2*a^5*d^2 + B^2*a^5*d^2 + 10*A^2*a^3*b^2*d^2 - 10*B^2*a^3*b^2*d^2 + 2*A*B*b^5*d^2 - 5*A^2*a*b^4*d^2 +
5*B^2*a*b^4*d^2 - 20*A*B*a^2*b^3*d^2 + 10*A*B*a^4*b*d^2)/(4*d^4))^(1/2) - (16*(a + b*tan(c + d*x))^(1/2)*(2*A
^4*b^20 + 2*B^4*b^20 + 4*A^2*B^2*b^20 + 12*A^4*a^2*b^18 + 55*A^4*a^4*b^16 - 335*A^4*a^6*b^14 + 405*A^4*a^8*b^1
2 - 13*A^4*a^10*b^10 + 2*A^4*a^12*b^8 + 12*B^4*a^2*b^18 + 30*B^4*a^4*b^16 + 36*B^4*a^6*b^14 + 90*B^4*a^8*b^12
- 48*B^4*a^10*b^10 + 6*B^4*a^12*b^8 + 24*A^2*B^2*a^2*b^18 + 35*A^2*B^2*a^4*b^16 + 699*A^2*B^2*a^6*b^14 - 1175*
A^2*B^2*a^8*b^12 + 349*A^2*B^2*a^10*b^10 - 20*A*B^3*a^5*b^15 + 348*A*B^3*a^7*b^13 - 460*A*B^3*a^9*b^11 + 68*A*
B^3*a^11*b^9 + 320*A^3*B*a^5*b^15 - 1300*A^3*B*a^7*b^13 + 600*A^3*B*a^9*b^11 - 20*A^3*B*a^11*b^9))/d^4)*((((8*
B^2*a^5*d^2 - 8*A^2*a^5*d^2 + 80*A^2*a^3*b^2*d^2 - 80*B^2*a^3*b^2*d^2 + 16*A*B*b^5*d^2 - 40*A^2*a*b^4*d^2 + 40
*B^2*a*b^4*d^2 - 160*A*B*a^2*b^3*d^2 + 80*A*B*a^4*b*d^2)^2/64 - d^4*(A^4*a^10 + A^4*b^10 + B^4*a^10 + B^4*b^10
+ 2*A^2*B^2*a^10 + 2*A^2*B^2*b^10 + 5*A^4*a^2*b^8 + 10*A^4*a^4*b^6 + 10*A^4*a^6*b^4 + 5*A^4*a^8*b^2 + 5*B^4*a
^2*b^8 + 10*B^4*a^4*b^6 + 10*B^4*a^6*b^4 + 5*B^4*a^8*b^2 + 10*A^2*B^2*a^2*b^8 + 20*A^2*B^2*a^4*b^6 + 20*A^2*B^
2*a^6*b^4 + 10*A^2*B^2*a^8*b^2))^(1/2) - A^2*a^5*d^2 + B^2*a^5*d^2 + 10*A^2*a^3*b^2*d^2 - 10*B^2*a^3*b^2*d^2 +
2*A*B*b^5*d^2 - 5*A^2*a*b^4*d^2 + 5*B^2*a*b^4*d^2 - 20*A*B*a^2*b^3*d^2 + 10*A*B*a^4*b*d^2)/(4*d^4))^(1/2) - (
16*(10*A^5*a^2*b^21 + 85*A^5*a^4*b^19 + 150*A^5*a^6*b^17 + 50*A^5*a^8*b^15 - 50*A^5*a^10*b^13 - 15*A^5*a^12*b^
11 + 10*A^5*a^14*b^9 + 4*B^5*a^3*b^20 + 24*B^5*a^5*b^18 + 72*B^5*a^7*b^16 + 112*B^5*a^9*b^14 + 84*B^5*a^11*b^1
2 + 24*B^5*a^13*b^10 + 8*A^2*B^3*a^3*b^20 + 143*A^2*B^3*a^5*b^18 + 332*A^2*B^3*a^7*b^16 + 222*A^2*B^3*a^9*b^14
- 16*A^2*B^3*a^11*b^12 - 37*A^2*B^3*a^13*b^10 + 4*A^2*B^3*a^15*b^8 + 20*A^3*B^2*a^2*b^21 + 145*A^3*B^2*a^4*b^
19 + 364*A^3*B^2*a^6*b^17 + 410*A^3*B^2*a^8*b^15 + 196*A^3*B^2*a^10*b^13 + 13*A^3*B^2*a^12*b^11 - 12*A^3*B^2*a
^14*b^9 + 10*A*B^4*a^2*b^21 + 60*A*B^4*a^4*b^19 + 214*A*B^4*a^6*b^17 + 360*A*B^4*a^8*b^15 + 246*A*B^4*a^10*b^1
3 + 28*A*B^4*a^12*b^11 - 22*A*B^4*a^14*b^9 + 4*A^4*B*a^3*b^20 + 119*A^4*B*a^5*b^18 + 260*A^4*B*a^7*b^16 + 110*
A^4*B*a^9*b^14 - 100*A^4*B*a^11*b^12 - 61*A^4*B*a^13*b^10 + 4*A^4*B*a^15*b^8))/d^5))*((((8*B^2*a^5*d^2 - 8*A^2
*a^5*d^2 + 80*A^2*a^3*b^2*d^2 - 80*B^2*a^3*b^2*d^2 + 16*A*B*b^5*d^2 - 40*A^2*a*b^4*d^2 + 40*B^2*a*b^4*d^2 - 16
0*A*B*a^2*b^3*d^2 + 80*A*B*a^4*b*d^2)^2/64 - d^4*(A^4*a^10 + A^4*b^10 + B^4*a^10 + B^4*b^10 + 2*A^2*B^2*a^10 +
2*A^2*B^2*b^10 + 5*A^4*a^2*b^8 + 10*A^4*a^4*b^6 + 10*A^4*a^6*b^4 + 5*A^4*a^8*b^2 + 5*B^4*a^2*b^8 + 10*B^4*a^4
*b^6 + 10*B^4*a^6*b^4 + 5*B^4*a^8*b^2 + 10*A^2*B^2*a^2*b^8 + 20*A^2*B^2*a^4*b^6 + 20*A^2*B^2*a^6*b^4 + 10*A^2*
B^2*a^8*b^2))^(1/2) - A^2*a^5*d^2 + B^2*a^5*d^2 + 10*A^2*a^3*b^2*d^2 - 10*B^2*a^3*b^2*d^2 + 2*A*B*b^5*d^2 - 5*
A^2*a*b^4*d^2 + 5*B^2*a*b^4*d^2 - 20*A*B*a^2*b^3*d^2 + 10*A*B*a^4*b*d^2)/(4*d^4))^(1/2)*2i - atan(((((8*(176*A
^3*a^5*b^13*d^2 - 400*A^3*a^3*b^15*d^2 + 488*A^3*a^7*b^11*d^2 - 92*A^3*a^9*b^9*d^2 + 12*B^3*a^2*b^16*d^2 + 192
*B^3*a^4*b^14*d^2 - 120*B^3*a^6*b^12*d^2 - 288*B^3*a^8*b^10*d^2 + 12*B^3*a^10*b^8*d^2 + 4*A^3*a*b^17*d^2 + 4*A
*B^2*a*b^17*d^2 + 464*A*B^2*a^3*b^15*d^2 - 920*A*B^2*a^5*b^13*d^2 - 1104*A*B^2*a^7*b^11*d^2 + 276*A*B^2*a^9*b^
9*d^2 + 172*A^2*B*a^2*b^16*d^2 - 1468*A^2*B*a^4*b^14*d^2 - 776*A^2*B*a^6*b^12*d^2 + 828*A^2*B*a^8*b^10*d^2 - 3
6*A^2*B*a^10*b^8*d^2))/d^5 - (((8*(16*B*a*b^12*d^4 + 128*A*a^2*b^11*d^4 + 128*A*a^4*b^9*d^4 + 64*B*a^3*b^10*d^
4 + 48*B*a^5*b^8*d^4))/d^5 - (16*(32*b^10*d^4 + 48*a^2*b^8*d^4)*(a + b*tan(c + d*x))^(1/2)*(-(((8*B^2*a^5*d^2
- 8*A^2*a^5*d^2 + 80*A^2*a^3*b^2*d^2 - 80*B^2*a^3*b^2*d^2 + 16*A*B*b^5*d^2 - 40*A^2*a*b^4*d^2 + 40*B^2*a*b^4*d
^2 - 160*A*B*a^2*b^3*d^2 + 80*A*B*a^4*b*d^2)^2/64 - d^4*(A^4*a^10 + A^4*b^10 + B^4*a^10 + B^4*b^10 + 2*A^2*B^2
*a^10 + 2*A^2*B^2*b^10 + 5*A^4*a^2*b^8 + 10*A^4*a^4*b^6 + 10*A^4*a^6*b^4 + 5*A^4*a^8*b^2 + 5*B^4*a^2*b^8 + 10*
B^4*a^4*b^6 + 10*B^4*a^6*b^4 + 5*B^4*a^8*b^2 + 10*A^2*B^2*a^2*b^8 + 20*A^2*B^2*a^4*b^6 + 20*A^2*B^2*a^6*b^4 +
10*A^2*B^2*a^8*b^2))^(1/2) + A^2*a^5*d^2 - B^2*a^5*d^2 - 10*A^2*a^3*b^2*d^2 + 10*B^2*a^3*b^2*d^2 - 2*A*B*b^5*d
^2 + 5*A^2*a*b^4*d^2 - 5*B^2*a*b^4*d^2 + 20*A*B*a^2*b^3*d^2 - 10*A*B*a^4*b*d^2)/(4*d^4))^(1/2))/d^4)*(-(((8*B^
2*a^5*d^2 - 8*A^2*a^5*d^2 + 80*A^2*a^3*b^2*d^2 - 80*B^2*a^3*b^2*d^2 + 16*A*B*b^5*d^2 - 40*A^2*a*b^4*d^2 + 40*B
^2*a*b^4*d^2 - 160*A*B*a^2*b^3*d^2 + 80*A*B*a^4*b*d^2)^2/64 - d^4*(A^4*a^10 + A^4*b^10 + B^4*a^10 + B^4*b^10 +
2*A^2*B^2*a^10 + 2*A^2*B^2*b^10 + 5*A^4*a^2*b^8 + 10*A^4*a^4*b^6 + 10*A^4*a^6*b^4 + 5*A^4*a^8*b^2 + 5*B^4*a^2
*b^8 + 10*B^4*a^4*b^6 + 10*B^4*a^6*b^4 + 5*B^4*a^8*b^2 + 10*A^2*B^2*a^2*b^8 + 20*A^2*B^2*a^4*b^6 + 20*A^2*B^2*
a^6*b^4 + 10*A^2*B^2*a^8*b^2))^(1/2) + A^2*a^5*d^2 - B^2*a^5*d^2 - 10*A^2*a^3*b^2*d^2 + 10*B^2*a^3*b^2*d^2 - 2
*A*B*b^5*d^2 + 5*A^2*a*b^4*d^2 - 5*B^2*a*b^4*d^2 + 20*A*B*a^2*b^3*d^2 - 10*A*B*a^4*b*d^2)/(4*d^4))^(1/2) + (16
*(a + b*tan(c + d*x))^(1/2)*(20*A^2*a^3*b^12*d^2 + 304*A^2*a^5*b^10*d^2 - 20*A^2*a^7*b^8*d^2 - 20*B^2*a^3*b^12
*d^2 - 204*B^2*a^5*b^10*d^2 + 36*B^2*a^7*b^8*d^2 + 32*A*B*b^15*d^2 - 76*A^2*a*b^14*d^2 + 76*B^2*a*b^14*d^2 - 2
40*A*B*a^2*b^13*d^2 - 320*A*B*a^4*b^11*d^2 + 288*A*B*a^6*b^9*d^2))/d^4)*(-(((8*B^2*a^5*d^2 - 8*A^2*a^5*d^2 + 8
0*A^2*a^3*b^2*d^2 - 80*B^2*a^3*b^2*d^2 + 16*A*B*b^5*d^2 - 40*A^2*a*b^4*d^2 + 40*B^2*a*b^4*d^2 - 160*A*B*a^2*b^
3*d^2 + 80*A*B*a^4*b*d^2)^2/64 - d^4*(A^4*a^10 + A^4*b^10 + B^4*a^10 + B^4*b^10 + 2*A^2*B^2*a^10 + 2*A^2*B^2*b
^10 + 5*A^4*a^2*b^8 + 10*A^4*a^4*b^6 + 10*A^4*a^6*b^4 + 5*A^4*a^8*b^2 + 5*B^4*a^2*b^8 + 10*B^4*a^4*b^6 + 10*B^
4*a^6*b^4 + 5*B^4*a^8*b^2 + 10*A^2*B^2*a^2*b^8 + 20*A^2*B^2*a^4*b^6 + 20*A^2*B^2*a^6*b^4 + 10*A^2*B^2*a^8*b^2)
)^(1/2) + A^2*a^5*d^2 - B^2*a^5*d^2 - 10*A^2*a^3*b^2*d^2 + 10*B^2*a^3*b^2*d^2 - 2*A*B*b^5*d^2 + 5*A^2*a*b^4*d^
2 - 5*B^2*a*b^4*d^2 + 20*A*B*a^2*b^3*d^2 - 10*A*B*a^4*b*d^2)/(4*d^4))^(1/2))*(-(((8*B^2*a^5*d^2 - 8*A^2*a^5*d^
2 + 80*A^2*a^3*b^2*d^2 - 80*B^2*a^3*b^2*d^2 + 16*A*B*b^5*d^2 - 40*A^2*a*b^4*d^2 + 40*B^2*a*b^4*d^2 - 160*A*B*a
^2*b^3*d^2 + 80*A*B*a^4*b*d^2)^2/64 - d^4*(A^4*a^10 + A^4*b^10 + B^4*a^10 + B^4*b^10 + 2*A^2*B^2*a^10 + 2*A^2*
B^2*b^10 + 5*A^4*a^2*b^8 + 10*A^4*a^4*b^6 + 10*A^4*a^6*b^4 + 5*A^4*a^8*b^2 + 5*B^4*a^2*b^8 + 10*B^4*a^4*b^6 +
10*B^4*a^6*b^4 + 5*B^4*a^8*b^2 + 10*A^2*B^2*a^2*b^8 + 20*A^2*B^2*a^4*b^6 + 20*A^2*B^2*a^6*b^4 + 10*A^2*B^2*a^8
*b^2))^(1/2) + A^2*a^5*d^2 - B^2*a^5*d^2 - 10*A^2*a^3*b^2*d^2 + 10*B^2*a^3*b^2*d^2 - 2*A*B*b^5*d^2 + 5*A^2*a*b
^4*d^2 - 5*B^2*a*b^4*d^2 + 20*A*B*a^2*b^3*d^2 - 10*A*B*a^4*b*d^2)/(4*d^4))^(1/2) + (16*(a + b*tan(c + d*x))^(1
/2)*(2*A^4*b^20 + 2*B^4*b^20 + 4*A^2*B^2*b^20 + 12*A^4*a^2*b^18 + 55*A^4*a^4*b^16 - 335*A^4*a^6*b^14 + 405*A^4
*a^8*b^12 - 13*A^4*a^10*b^10 + 2*A^4*a^12*b^8 + 12*B^4*a^2*b^18 + 30*B^4*a^4*b^16 + 36*B^4*a^6*b^14 + 90*B^4*a
^8*b^12 - 48*B^4*a^10*b^10 + 6*B^4*a^12*b^8 + 24*A^2*B^2*a^2*b^18 + 35*A^2*B^2*a^4*b^16 + 699*A^2*B^2*a^6*b^14
- 1175*A^2*B^2*a^8*b^12 + 349*A^2*B^2*a^10*b^10 - 20*A*B^3*a^5*b^15 + 348*A*B^3*a^7*b^13 - 460*A*B^3*a^9*b^11
+ 68*A*B^3*a^11*b^9 + 320*A^3*B*a^5*b^15 - 1300*A^3*B*a^7*b^13 + 600*A^3*B*a^9*b^11 - 20*A^3*B*a^11*b^9))/d^4
)*(-(((8*B^2*a^5*d^2 - 8*A^2*a^5*d^2 + 80*A^2*a^3*b^2*d^2 - 80*B^2*a^3*b^2*d^2 + 16*A*B*b^5*d^2 - 40*A^2*a*b^4
*d^2 + 40*B^2*a*b^4*d^2 - 160*A*B*a^2*b^3*d^2 + 80*A*B*a^4*b*d^2)^2/64 - d^4*(A^4*a^10 + A^4*b^10 + B^4*a^10 +
B^4*b^10 + 2*A^2*B^2*a^10 + 2*A^2*B^2*b^10 + 5*A^4*a^2*b^8 + 10*A^4*a^4*b^6 + 10*A^4*a^6*b^4 + 5*A^4*a^8*b^2
+ 5*B^4*a^2*b^8 + 10*B^4*a^4*b^6 + 10*B^4*a^6*b^4 + 5*B^4*a^8*b^2 + 10*A^2*B^2*a^2*b^8 + 20*A^2*B^2*a^4*b^6 +
20*A^2*B^2*a^6*b^4 + 10*A^2*B^2*a^8*b^2))^(1/2) + A^2*a^5*d^2 - B^2*a^5*d^2 - 10*A^2*a^3*b^2*d^2 + 10*B^2*a^3*
b^2*d^2 - 2*A*B*b^5*d^2 + 5*A^2*a*b^4*d^2 - 5*B^2*a*b^4*d^2 + 20*A*B*a^2*b^3*d^2 - 10*A*B*a^4*b*d^2)/(4*d^4))^
(1/2)*1i - (((8*(176*A^3*a^5*b^13*d^2 - 400*A^3*a^3*b^15*d^2 + 488*A^3*a^7*b^11*d^2 - 92*A^3*a^9*b^9*d^2 + 12*
B^3*a^2*b^16*d^2 + 192*B^3*a^4*b^14*d^2 - 120*B^3*a^6*b^12*d^2 - 288*B^3*a^8*b^10*d^2 + 12*B^3*a^10*b^8*d^2 +
4*A^3*a*b^17*d^2 + 4*A*B^2*a*b^17*d^2 + 464*A*B^2*a^3*b^15*d^2 - 920*A*B^2*a^5*b^13*d^2 - 1104*A*B^2*a^7*b^11*
d^2 + 276*A*B^2*a^9*b^9*d^2 + 172*A^2*B*a^2*b^16*d^2 - 1468*A^2*B*a^4*b^14*d^2 - 776*A^2*B*a^6*b^12*d^2 + 828*
A^2*B*a^8*b^10*d^2 - 36*A^2*B*a^10*b^8*d^2))/d^5 - (((8*(16*B*a*b^12*d^4 + 128*A*a^2*b^11*d^4 + 128*A*a^4*b^9*
d^4 + 64*B*a^3*b^10*d^4 + 48*B*a^5*b^8*d^4))/d^5 + (16*(32*b^10*d^4 + 48*a^2*b^8*d^4)*(a + b*tan(c + d*x))^(1/
2)*(-(((8*B^2*a^5*d^2 - 8*A^2*a^5*d^2 + 80*A^2*a^3*b^2*d^2 - 80*B^2*a^3*b^2*d^2 + 16*A*B*b^5*d^2 - 40*A^2*a*b^
4*d^2 + 40*B^2*a*b^4*d^2 - 160*A*B*a^2*b^3*d^2 + 80*A*B*a^4*b*d^2)^2/64 - d^4*(A^4*a^10 + A^4*b^10 + B^4*a^10
+ B^4*b^10 + 2*A^2*B^2*a^10 + 2*A^2*B^2*b^10 + 5*A^4*a^2*b^8 + 10*A^4*a^4*b^6 + 10*A^4*a^6*b^4 + 5*A^4*a^8*b^2
+ 5*B^4*a^2*b^8 + 10*B^4*a^4*b^6 + 10*B^4*a^6*b^4 + 5*B^4*a^8*b^2 + 10*A^2*B^2*a^2*b^8 + 20*A^2*B^2*a^4*b^6 +
20*A^2*B^2*a^6*b^4 + 10*A^2*B^2*a^8*b^2))^(1/2) + A^2*a^5*d^2 - B^2*a^5*d^2 - 10*A^2*a^3*b^2*d^2 + 10*B^2*a^3
*b^2*d^2 - 2*A*B*b^5*d^2 + 5*A^2*a*b^4*d^2 - 5*B^2*a*b^4*d^2 + 20*A*B*a^2*b^3*d^2 - 10*A*B*a^4*b*d^2)/(4*d^4))
^(1/2))/d^4)*(-(((8*B^2*a^5*d^2 - 8*A^2*a^5*d^2 + 80*A^2*a^3*b^2*d^2 - 80*B^2*a^3*b^2*d^2 + 16*A*B*b^5*d^2 - 4
0*A^2*a*b^4*d^2 + 40*B^2*a*b^4*d^2 - 160*A*B*a^2*b^3*d^2 + 80*A*B*a^4*b*d^2)^2/64 - d^4*(A^4*a^10 + A^4*b^10 +
B^4*a^10 + B^4*b^10 + 2*A^2*B^2*a^10 + 2*A^2*B^2*b^10 + 5*A^4*a^2*b^8 + 10*A^4*a^4*b^6 + 10*A^4*a^6*b^4 + 5*A
^4*a^8*b^2 + 5*B^4*a^2*b^8 + 10*B^4*a^4*b^6 + 10*B^4*a^6*b^4 + 5*B^4*a^8*b^2 + 10*A^2*B^2*a^2*b^8 + 20*A^2*B^2
*a^4*b^6 + 20*A^2*B^2*a^6*b^4 + 10*A^2*B^2*a^8*b^2))^(1/2) + A^2*a^5*d^2 - B^2*a^5*d^2 - 10*A^2*a^3*b^2*d^2 +
10*B^2*a^3*b^2*d^2 - 2*A*B*b^5*d^2 + 5*A^2*a*b^4*d^2 - 5*B^2*a*b^4*d^2 + 20*A*B*a^2*b^3*d^2 - 10*A*B*a^4*b*d^2
)/(4*d^4))^(1/2) - (16*(a + b*tan(c + d*x))^(1/2)*(20*A^2*a^3*b^12*d^2 + 304*A^2*a^5*b^10*d^2 - 20*A^2*a^7*b^8
*d^2 - 20*B^2*a^3*b^12*d^2 - 204*B^2*a^5*b^10*d^2 + 36*B^2*a^7*b^8*d^2 + 32*A*B*b^15*d^2 - 76*A^2*a*b^14*d^2 +
76*B^2*a*b^14*d^2 - 240*A*B*a^2*b^13*d^2 - 320*A*B*a^4*b^11*d^2 + 288*A*B*a^6*b^9*d^2))/d^4)*(-(((8*B^2*a^5*d
^2 - 8*A^2*a^5*d^2 + 80*A^2*a^3*b^2*d^2 - 80*B^2*a^3*b^2*d^2 + 16*A*B*b^5*d^2 - 40*A^2*a*b^4*d^2 + 40*B^2*a*b^
4*d^2 - 160*A*B*a^2*b^3*d^2 + 80*A*B*a^4*b*d^2)^2/64 - d^4*(A^4*a^10 + A^4*b^10 + B^4*a^10 + B^4*b^10 + 2*A^2*
B^2*a^10 + 2*A^2*B^2*b^10 + 5*A^4*a^2*b^8 + 10*A^4*a^4*b^6 + 10*A^4*a^6*b^4 + 5*A^4*a^8*b^2 + 5*B^4*a^2*b^8 +
10*B^4*a^4*b^6 + 10*B^4*a^6*b^4 + 5*B^4*a^8*b^2 + 10*A^2*B^2*a^2*b^8 + 20*A^2*B^2*a^4*b^6 + 20*A^2*B^2*a^6*b^4
+ 10*A^2*B^2*a^8*b^2))^(1/2) + A^2*a^5*d^2 - B^2*a^5*d^2 - 10*A^2*a^3*b^2*d^2 + 10*B^2*a^3*b^2*d^2 - 2*A*B*b^
5*d^2 + 5*A^2*a*b^4*d^2 - 5*B^2*a*b^4*d^2 + 20*A*B*a^2*b^3*d^2 - 10*A*B*a^4*b*d^2)/(4*d^4))^(1/2))*(-(((8*B^2*
a^5*d^2 - 8*A^2*a^5*d^2 + 80*A^2*a^3*b^2*d^2 - 80*B^2*a^3*b^2*d^2 + 16*A*B*b^5*d^2 - 40*A^2*a*b^4*d^2 + 40*B^2
*a*b^4*d^2 - 160*A*B*a^2*b^3*d^2 + 80*A*B*a^4*b*d^2)^2/64 - d^4*(A^4*a^10 + A^4*b^10 + B^4*a^10 + B^4*b^10 + 2
*A^2*B^2*a^10 + 2*A^2*B^2*b^10 + 5*A^4*a^2*b^8 + 10*A^4*a^4*b^6 + 10*A^4*a^6*b^4 + 5*A^4*a^8*b^2 + 5*B^4*a^2*b
^8 + 10*B^4*a^4*b^6 + 10*B^4*a^6*b^4 + 5*B^4*a^8*b^2 + 10*A^2*B^2*a^2*b^8 + 20*A^2*B^2*a^4*b^6 + 20*A^2*B^2*a^
6*b^4 + 10*A^2*B^2*a^8*b^2))^(1/2) + A^2*a^5*d^2 - B^2*a^5*d^2 - 10*A^2*a^3*b^2*d^2 + 10*B^2*a^3*b^2*d^2 - 2*A
*B*b^5*d^2 + 5*A^2*a*b^4*d^2 - 5*B^2*a*b^4*d^2 + 20*A*B*a^2*b^3*d^2 - 10*A*B*a^4*b*d^2)/(4*d^4))^(1/2) - (16*(
a + b*tan(c + d*x))^(1/2)*(2*A^4*b^20 + 2*B^4*b^20 + 4*A^2*B^2*b^20 + 12*A^4*a^2*b^18 + 55*A^4*a^4*b^16 - 335*
A^4*a^6*b^14 + 405*A^4*a^8*b^12 - 13*A^4*a^10*b^10 + 2*A^4*a^12*b^8 + 12*B^4*a^2*b^18 + 30*B^4*a^4*b^16 + 36*B
^4*a^6*b^14 + 90*B^4*a^8*b^12 - 48*B^4*a^10*b^10 + 6*B^4*a^12*b^8 + 24*A^2*B^2*a^2*b^18 + 35*A^2*B^2*a^4*b^16
+ 699*A^2*B^2*a^6*b^14 - 1175*A^2*B^2*a^8*b^12 + 349*A^2*B^2*a^10*b^10 - 20*A*B^3*a^5*b^15 + 348*A*B^3*a^7*b^1
3 - 460*A*B^3*a^9*b^11 + 68*A*B^3*a^11*b^9 + 320*A^3*B*a^5*b^15 - 1300*A^3*B*a^7*b^13 + 600*A^3*B*a^9*b^11 - 2
0*A^3*B*a^11*b^9))/d^4)*(-(((8*B^2*a^5*d^2 - 8*A^2*a^5*d^2 + 80*A^2*a^3*b^2*d^2 - 80*B^2*a^3*b^2*d^2 + 16*A*B*
b^5*d^2 - 40*A^2*a*b^4*d^2 + 40*B^2*a*b^4*d^2 - 160*A*B*a^2*b^3*d^2 + 80*A*B*a^4*b*d^2)^2/64 - d^4*(A^4*a^10 +
A^4*b^10 + B^4*a^10 + B^4*b^10 + 2*A^2*B^2*a^10 + 2*A^2*B^2*b^10 + 5*A^4*a^2*b^8 + 10*A^4*a^4*b^6 + 10*A^4*a^
6*b^4 + 5*A^4*a^8*b^2 + 5*B^4*a^2*b^8 + 10*B^4*a^4*b^6 + 10*B^4*a^6*b^4 + 5*B^4*a^8*b^2 + 10*A^2*B^2*a^2*b^8 +
20*A^2*B^2*a^4*b^6 + 20*A^2*B^2*a^6*b^4 + 10*A^2*B^2*a^8*b^2))^(1/2) + A^2*a^5*d^2 - B^2*a^5*d^2 - 10*A^2*a^3
*b^2*d^2 + 10*B^2*a^3*b^2*d^2 - 2*A*B*b^5*d^2 + 5*A^2*a*b^4*d^2 - 5*B^2*a*b^4*d^2 + 20*A*B*a^2*b^3*d^2 - 10*A*
B*a^4*b*d^2)/(4*d^4))^(1/2)*1i)/((((8*(176*A^3*a^5*b^13*d^2 - 400*A^3*a^3*b^15*d^2 + 488*A^3*a^7*b^11*d^2 - 92
*A^3*a^9*b^9*d^2 + 12*B^3*a^2*b^16*d^2 + 192*B^3*a^4*b^14*d^2 - 120*B^3*a^6*b^12*d^2 - 288*B^3*a^8*b^10*d^2 +
12*B^3*a^10*b^8*d^2 + 4*A^3*a*b^17*d^2 + 4*A*B^2*a*b^17*d^2 + 464*A*B^2*a^3*b^15*d^2 - 920*A*B^2*a^5*b^13*d^2
- 1104*A*B^2*a^7*b^11*d^2 + 276*A*B^2*a^9*b^9*d^2 + 172*A^2*B*a^2*b^16*d^2 - 1468*A^2*B*a^4*b^14*d^2 - 776*A^2
*B*a^6*b^12*d^2 + 828*A^2*B*a^8*b^10*d^2 - 36*A^2*B*a^10*b^8*d^2))/d^5 - (((8*(16*B*a*b^12*d^4 + 128*A*a^2*b^1
1*d^4 + 128*A*a^4*b^9*d^4 + 64*B*a^3*b^10*d^4 + 48*B*a^5*b^8*d^4))/d^5 - (16*(32*b^10*d^4 + 48*a^2*b^8*d^4)*(a
+ b*tan(c + d*x))^(1/2)*(-(((8*B^2*a^5*d^2 - 8*A^2*a^5*d^2 + 80*A^2*a^3*b^2*d^2 - 80*B^2*a^3*b^2*d^2 + 16*A*B
*b^5*d^2 - 40*A^2*a*b^4*d^2 + 40*B^2*a*b^4*d^2 - 160*A*B*a^2*b^3*d^2 + 80*A*B*a^4*b*d^2)^2/64 - d^4*(A^4*a^10
+ A^4*b^10 + B^4*a^10 + B^4*b^10 + 2*A^2*B^2*a^10 + 2*A^2*B^2*b^10 + 5*A^4*a^2*b^8 + 10*A^4*a^4*b^6 + 10*A^4*a
^6*b^4 + 5*A^4*a^8*b^2 + 5*B^4*a^2*b^8 + 10*B^4*a^4*b^6 + 10*B^4*a^6*b^4 + 5*B^4*a^8*b^2 + 10*A^2*B^2*a^2*b^8
+ 20*A^2*B^2*a^4*b^6 + 20*A^2*B^2*a^6*b^4 + 10*A^2*B^2*a^8*b^2))^(1/2) + A^2*a^5*d^2 - B^2*a^5*d^2 - 10*A^2*a^
3*b^2*d^2 + 10*B^2*a^3*b^2*d^2 - 2*A*B*b^5*d^2 + 5*A^2*a*b^4*d^2 - 5*B^2*a*b^4*d^2 + 20*A*B*a^2*b^3*d^2 - 10*A
*B*a^4*b*d^2)/(4*d^4))^(1/2))/d^4)*(-(((8*B^2*a^5*d^2 - 8*A^2*a^5*d^2 + 80*A^2*a^3*b^2*d^2 - 80*B^2*a^3*b^2*d^
2 + 16*A*B*b^5*d^2 - 40*A^2*a*b^4*d^2 + 40*B^2*a*b^4*d^2 - 160*A*B*a^2*b^3*d^2 + 80*A*B*a^4*b*d^2)^2/64 - d^4*
(A^4*a^10 + A^4*b^10 + B^4*a^10 + B^4*b^10 + 2*A^2*B^2*a^10 + 2*A^2*B^2*b^10 + 5*A^4*a^2*b^8 + 10*A^4*a^4*b^6
+ 10*A^4*a^6*b^4 + 5*A^4*a^8*b^2 + 5*B^4*a^2*b^8 + 10*B^4*a^4*b^6 + 10*B^4*a^6*b^4 + 5*B^4*a^8*b^2 + 10*A^2*B^
2*a^2*b^8 + 20*A^2*B^2*a^4*b^6 + 20*A^2*B^2*a^6*b^4 + 10*A^2*B^2*a^8*b^2))^(1/2) + A^2*a^5*d^2 - B^2*a^5*d^2 -
10*A^2*a^3*b^2*d^2 + 10*B^2*a^3*b^2*d^2 - 2*A*B*b^5*d^2 + 5*A^2*a*b^4*d^2 - 5*B^2*a*b^4*d^2 + 20*A*B*a^2*b^3*
d^2 - 10*A*B*a^4*b*d^2)/(4*d^4))^(1/2) + (16*(a + b*tan(c + d*x))^(1/2)*(20*A^2*a^3*b^12*d^2 + 304*A^2*a^5*b^1
0*d^2 - 20*A^2*a^7*b^8*d^2 - 20*B^2*a^3*b^12*d^2 - 204*B^2*a^5*b^10*d^2 + 36*B^2*a^7*b^8*d^2 + 32*A*B*b^15*d^2
- 76*A^2*a*b^14*d^2 + 76*B^2*a*b^14*d^2 - 240*A*B*a^2*b^13*d^2 - 320*A*B*a^4*b^11*d^2 + 288*A*B*a^6*b^9*d^2))
/d^4)*(-(((8*B^2*a^5*d^2 - 8*A^2*a^5*d^2 + 80*A^2*a^3*b^2*d^2 - 80*B^2*a^3*b^2*d^2 + 16*A*B*b^5*d^2 - 40*A^2*a
*b^4*d^2 + 40*B^2*a*b^4*d^2 - 160*A*B*a^2*b^3*d^2 + 80*A*B*a^4*b*d^2)^2/64 - d^4*(A^4*a^10 + A^4*b^10 + B^4*a^
10 + B^4*b^10 + 2*A^2*B^2*a^10 + 2*A^2*B^2*b^10 + 5*A^4*a^2*b^8 + 10*A^4*a^4*b^6 + 10*A^4*a^6*b^4 + 5*A^4*a^8*
b^2 + 5*B^4*a^2*b^8 + 10*B^4*a^4*b^6 + 10*B^4*a^6*b^4 + 5*B^4*a^8*b^2 + 10*A^2*B^2*a^2*b^8 + 20*A^2*B^2*a^4*b^
6 + 20*A^2*B^2*a^6*b^4 + 10*A^2*B^2*a^8*b^2))^(1/2) + A^2*a^5*d^2 - B^2*a^5*d^2 - 10*A^2*a^3*b^2*d^2 + 10*B^2*
a^3*b^2*d^2 - 2*A*B*b^5*d^2 + 5*A^2*a*b^4*d^2 - 5*B^2*a*b^4*d^2 + 20*A*B*a^2*b^3*d^2 - 10*A*B*a^4*b*d^2)/(4*d^
4))^(1/2))*(-(((8*B^2*a^5*d^2 - 8*A^2*a^5*d^2 + 80*A^2*a^3*b^2*d^2 - 80*B^2*a^3*b^2*d^2 + 16*A*B*b^5*d^2 - 40*
A^2*a*b^4*d^2 + 40*B^2*a*b^4*d^2 - 160*A*B*a^2*b^3*d^2 + 80*A*B*a^4*b*d^2)^2/64 - d^4*(A^4*a^10 + A^4*b^10 + B
^4*a^10 + B^4*b^10 + 2*A^2*B^2*a^10 + 2*A^2*B^2*b^10 + 5*A^4*a^2*b^8 + 10*A^4*a^4*b^6 + 10*A^4*a^6*b^4 + 5*A^4
*a^8*b^2 + 5*B^4*a^2*b^8 + 10*B^4*a^4*b^6 + 10*B^4*a^6*b^4 + 5*B^4*a^8*b^2 + 10*A^2*B^2*a^2*b^8 + 20*A^2*B^2*a
^4*b^6 + 20*A^2*B^2*a^6*b^4 + 10*A^2*B^2*a^8*b^2))^(1/2) + A^2*a^5*d^2 - B^2*a^5*d^2 - 10*A^2*a^3*b^2*d^2 + 10
*B^2*a^3*b^2*d^2 - 2*A*B*b^5*d^2 + 5*A^2*a*b^4*d^2 - 5*B^2*a*b^4*d^2 + 20*A*B*a^2*b^3*d^2 - 10*A*B*a^4*b*d^2)/
(4*d^4))^(1/2) + (16*(a + b*tan(c + d*x))^(1/2)*(2*A^4*b^20 + 2*B^4*b^20 + 4*A^2*B^2*b^20 + 12*A^4*a^2*b^18 +
55*A^4*a^4*b^16 - 335*A^4*a^6*b^14 + 405*A^4*a^8*b^12 - 13*A^4*a^10*b^10 + 2*A^4*a^12*b^8 + 12*B^4*a^2*b^18 +
30*B^4*a^4*b^16 + 36*B^4*a^6*b^14 + 90*B^4*a^8*b^12 - 48*B^4*a^10*b^10 + 6*B^4*a^12*b^8 + 24*A^2*B^2*a^2*b^18
+ 35*A^2*B^2*a^4*b^16 + 699*A^2*B^2*a^6*b^14 - 1175*A^2*B^2*a^8*b^12 + 349*A^2*B^2*a^10*b^10 - 20*A*B^3*a^5*b^
15 + 348*A*B^3*a^7*b^13 - 460*A*B^3*a^9*b^11 + 68*A*B^3*a^11*b^9 + 320*A^3*B*a^5*b^15 - 1300*A^3*B*a^7*b^13 +
600*A^3*B*a^9*b^11 - 20*A^3*B*a^11*b^9))/d^4)*(-(((8*B^2*a^5*d^2 - 8*A^2*a^5*d^2 + 80*A^2*a^3*b^2*d^2 - 80*B^2
*a^3*b^2*d^2 + 16*A*B*b^5*d^2 - 40*A^2*a*b^4*d^2 + 40*B^2*a*b^4*d^2 - 160*A*B*a^2*b^3*d^2 + 80*A*B*a^4*b*d^2)^
2/64 - d^4*(A^4*a^10 + A^4*b^10 + B^4*a^10 + B^4*b^10 + 2*A^2*B^2*a^10 + 2*A^2*B^2*b^10 + 5*A^4*a^2*b^8 + 10*A
^4*a^4*b^6 + 10*A^4*a^6*b^4 + 5*A^4*a^8*b^2 + 5*B^4*a^2*b^8 + 10*B^4*a^4*b^6 + 10*B^4*a^6*b^4 + 5*B^4*a^8*b^2
+ 10*A^2*B^2*a^2*b^8 + 20*A^2*B^2*a^4*b^6 + 20*A^2*B^2*a^6*b^4 + 10*A^2*B^2*a^8*b^2))^(1/2) + A^2*a^5*d^2 - B^
2*a^5*d^2 - 10*A^2*a^3*b^2*d^2 + 10*B^2*a^3*b^2*d^2 - 2*A*B*b^5*d^2 + 5*A^2*a*b^4*d^2 - 5*B^2*a*b^4*d^2 + 20*A
*B*a^2*b^3*d^2 - 10*A*B*a^4*b*d^2)/(4*d^4))^(1/2) + (((8*(176*A^3*a^5*b^13*d^2 - 400*A^3*a^3*b^15*d^2 + 488*A^
3*a^7*b^11*d^2 - 92*A^3*a^9*b^9*d^2 + 12*B^3*a^2*b^16*d^2 + 192*B^3*a^4*b^14*d^2 - 120*B^3*a^6*b^12*d^2 - 288*
B^3*a^8*b^10*d^2 + 12*B^3*a^10*b^8*d^2 + 4*A^3*a*b^17*d^2 + 4*A*B^2*a*b^17*d^2 + 464*A*B^2*a^3*b^15*d^2 - 920*
A*B^2*a^5*b^13*d^2 - 1104*A*B^2*a^7*b^11*d^2 + 276*A*B^2*a^9*b^9*d^2 + 172*A^2*B*a^2*b^16*d^2 - 1468*A^2*B*a^4
*b^14*d^2 - 776*A^2*B*a^6*b^12*d^2 + 828*A^2*B*a^8*b^10*d^2 - 36*A^2*B*a^10*b^8*d^2))/d^5 - (((8*(16*B*a*b^12*
d^4 + 128*A*a^2*b^11*d^4 + 128*A*a^4*b^9*d^4 + 64*B*a^3*b^10*d^4 + 48*B*a^5*b^8*d^4))/d^5 + (16*(32*b^10*d^4 +
48*a^2*b^8*d^4)*(a + b*tan(c + d*x))^(1/2)*(-(((8*B^2*a^5*d^2 - 8*A^2*a^5*d^2 + 80*A^2*a^3*b^2*d^2 - 80*B^2*a
^3*b^2*d^2 + 16*A*B*b^5*d^2 - 40*A^2*a*b^4*d^2 + 40*B^2*a*b^4*d^2 - 160*A*B*a^2*b^3*d^2 + 80*A*B*a^4*b*d^2)^2/
64 - d^4*(A^4*a^10 + A^4*b^10 + B^4*a^10 + B^4*b^10 + 2*A^2*B^2*a^10 + 2*A^2*B^2*b^10 + 5*A^4*a^2*b^8 + 10*A^4
*a^4*b^6 + 10*A^4*a^6*b^4 + 5*A^4*a^8*b^2 + 5*B^4*a^2*b^8 + 10*B^4*a^4*b^6 + 10*B^4*a^6*b^4 + 5*B^4*a^8*b^2 +
10*A^2*B^2*a^2*b^8 + 20*A^2*B^2*a^4*b^6 + 20*A^2*B^2*a^6*b^4 + 10*A^2*B^2*a^8*b^2))^(1/2) + A^2*a^5*d^2 - B^2*
a^5*d^2 - 10*A^2*a^3*b^2*d^2 + 10*B^2*a^3*b^2*d^2 - 2*A*B*b^5*d^2 + 5*A^2*a*b^4*d^2 - 5*B^2*a*b^4*d^2 + 20*A*B
*a^2*b^3*d^2 - 10*A*B*a^4*b*d^2)/(4*d^4))^(1/2))/d^4)*(-(((8*B^2*a^5*d^2 - 8*A^2*a^5*d^2 + 80*A^2*a^3*b^2*d^2
- 80*B^2*a^3*b^2*d^2 + 16*A*B*b^5*d^2 - 40*A^2*a*b^4*d^2 + 40*B^2*a*b^4*d^2 - 160*A*B*a^2*b^3*d^2 + 80*A*B*a^4
*b*d^2)^2/64 - d^4*(A^4*a^10 + A^4*b^10 + B^4*a^10 + B^4*b^10 + 2*A^2*B^2*a^10 + 2*A^2*B^2*b^10 + 5*A^4*a^2*b^
8 + 10*A^4*a^4*b^6 + 10*A^4*a^6*b^4 + 5*A^4*a^8*b^2 + 5*B^4*a^2*b^8 + 10*B^4*a^4*b^6 + 10*B^4*a^6*b^4 + 5*B^4*
a^8*b^2 + 10*A^2*B^2*a^2*b^8 + 20*A^2*B^2*a^4*b^6 + 20*A^2*B^2*a^6*b^4 + 10*A^2*B^2*a^8*b^2))^(1/2) + A^2*a^5*
d^2 - B^2*a^5*d^2 - 10*A^2*a^3*b^2*d^2 + 10*B^2*a^3*b^2*d^2 - 2*A*B*b^5*d^2 + 5*A^2*a*b^4*d^2 - 5*B^2*a*b^4*d^
2 + 20*A*B*a^2*b^3*d^2 - 10*A*B*a^4*b*d^2)/(4*d^4))^(1/2) - (16*(a + b*tan(c + d*x))^(1/2)*(20*A^2*a^3*b^12*d^
2 + 304*A^2*a^5*b^10*d^2 - 20*A^2*a^7*b^8*d^2 - 20*B^2*a^3*b^12*d^2 - 204*B^2*a^5*b^10*d^2 + 36*B^2*a^7*b^8*d^
2 + 32*A*B*b^15*d^2 - 76*A^2*a*b^14*d^2 + 76*B^2*a*b^14*d^2 - 240*A*B*a^2*b^13*d^2 - 320*A*B*a^4*b^11*d^2 + 28
8*A*B*a^6*b^9*d^2))/d^4)*(-(((8*B^2*a^5*d^2 - 8*A^2*a^5*d^2 + 80*A^2*a^3*b^2*d^2 - 80*B^2*a^3*b^2*d^2 + 16*A*B
*b^5*d^2 - 40*A^2*a*b^4*d^2 + 40*B^2*a*b^4*d^2 - 160*A*B*a^2*b^3*d^2 + 80*A*B*a^4*b*d^2)^2/64 - d^4*(A^4*a^10
+ A^4*b^10 + B^4*a^10 + B^4*b^10 + 2*A^2*B^2*a^10 + 2*A^2*B^2*b^10 + 5*A^4*a^2*b^8 + 10*A^4*a^4*b^6 + 10*A^4*a
^6*b^4 + 5*A^4*a^8*b^2 + 5*B^4*a^2*b^8 + 10*B^4*a^4*b^6 + 10*B^4*a^6*b^4 + 5*B^4*a^8*b^2 + 10*A^2*B^2*a^2*b^8
+ 20*A^2*B^2*a^4*b^6 + 20*A^2*B^2*a^6*b^4 + 10*A^2*B^2*a^8*b^2))^(1/2) + A^2*a^5*d^2 - B^2*a^5*d^2 - 10*A^2*a^
3*b^2*d^2 + 10*B^2*a^3*b^2*d^2 - 2*A*B*b^5*d^2 + 5*A^2*a*b^4*d^2 - 5*B^2*a*b^4*d^2 + 20*A*B*a^2*b^3*d^2 - 10*A
*B*a^4*b*d^2)/(4*d^4))^(1/2))*(-(((8*B^2*a^5*d^2 - 8*A^2*a^5*d^2 + 80*A^2*a^3*b^2*d^2 - 80*B^2*a^3*b^2*d^2 + 1
6*A*B*b^5*d^2 - 40*A^2*a*b^4*d^2 + 40*B^2*a*b^4*d^2 - 160*A*B*a^2*b^3*d^2 + 80*A*B*a^4*b*d^2)^2/64 - d^4*(A^4*
a^10 + A^4*b^10 + B^4*a^10 + B^4*b^10 + 2*A^2*B^2*a^10 + 2*A^2*B^2*b^10 + 5*A^4*a^2*b^8 + 10*A^4*a^4*b^6 + 10*
A^4*a^6*b^4 + 5*A^4*a^8*b^2 + 5*B^4*a^2*b^8 + 10*B^4*a^4*b^6 + 10*B^4*a^6*b^4 + 5*B^4*a^8*b^2 + 10*A^2*B^2*a^2
*b^8 + 20*A^2*B^2*a^4*b^6 + 20*A^2*B^2*a^6*b^4 + 10*A^2*B^2*a^8*b^2))^(1/2) + A^2*a^5*d^2 - B^2*a^5*d^2 - 10*A
^2*a^3*b^2*d^2 + 10*B^2*a^3*b^2*d^2 - 2*A*B*b^5*d^2 + 5*A^2*a*b^4*d^2 - 5*B^2*a*b^4*d^2 + 20*A*B*a^2*b^3*d^2 -
10*A*B*a^4*b*d^2)/(4*d^4))^(1/2) - (16*(a + b*tan(c + d*x))^(1/2)*(2*A^4*b^20 + 2*B^4*b^20 + 4*A^2*B^2*b^20 +
12*A^4*a^2*b^18 + 55*A^4*a^4*b^16 - 335*A^4*a^6*b^14 + 405*A^4*a^8*b^12 - 13*A^4*a^10*b^10 + 2*A^4*a^12*b^8 +
12*B^4*a^2*b^18 + 30*B^4*a^4*b^16 + 36*B^4*a^6*b^14 + 90*B^4*a^8*b^12 - 48*B^4*a^10*b^10 + 6*B^4*a^12*b^8 + 2
4*A^2*B^2*a^2*b^18 + 35*A^2*B^2*a^4*b^16 + 699*A^2*B^2*a^6*b^14 - 1175*A^2*B^2*a^8*b^12 + 349*A^2*B^2*a^10*b^1
0 - 20*A*B^3*a^5*b^15 + 348*A*B^3*a^7*b^13 - 460*A*B^3*a^9*b^11 + 68*A*B^3*a^11*b^9 + 320*A^3*B*a^5*b^15 - 130
0*A^3*B*a^7*b^13 + 600*A^3*B*a^9*b^11 - 20*A^3*B*a^11*b^9))/d^4)*(-(((8*B^2*a^5*d^2 - 8*A^2*a^5*d^2 + 80*A^2*a
^3*b^2*d^2 - 80*B^2*a^3*b^2*d^2 + 16*A*B*b^5*d^2 - 40*A^2*a*b^4*d^2 + 40*B^2*a*b^4*d^2 - 160*A*B*a^2*b^3*d^2 +
80*A*B*a^4*b*d^2)^2/64 - d^4*(A^4*a^10 + A^4*b^10 + B^4*a^10 + B^4*b^10 + 2*A^2*B^2*a^10 + 2*A^2*B^2*b^10 + 5
*A^4*a^2*b^8 + 10*A^4*a^4*b^6 + 10*A^4*a^6*b^4 + 5*A^4*a^8*b^2 + 5*B^4*a^2*b^8 + 10*B^4*a^4*b^6 + 10*B^4*a^6*b
^4 + 5*B^4*a^8*b^2 + 10*A^2*B^2*a^2*b^8 + 20*A^2*B^2*a^4*b^6 + 20*A^2*B^2*a^6*b^4 + 10*A^2*B^2*a^8*b^2))^(1/2)
+ A^2*a^5*d^2 - B^2*a^5*d^2 - 10*A^2*a^3*b^2*d^2 + 10*B^2*a^3*b^2*d^2 - 2*A*B*b^5*d^2 + 5*A^2*a*b^4*d^2 - 5*B
^2*a*b^4*d^2 + 20*A*B*a^2*b^3*d^2 - 10*A*B*a^4*b*d^2)/(4*d^4))^(1/2) - (16*(10*A^5*a^2*b^21 + 85*A^5*a^4*b^19
+ 150*A^5*a^6*b^17 + 50*A^5*a^8*b^15 - 50*A^5*a^10*b^13 - 15*A^5*a^12*b^11 + 10*A^5*a^14*b^9 + 4*B^5*a^3*b^20
+ 24*B^5*a^5*b^18 + 72*B^5*a^7*b^16 + 112*B^5*a^9*b^14 + 84*B^5*a^11*b^12 + 24*B^5*a^13*b^10 + 8*A^2*B^3*a^3*b
^20 + 143*A^2*B^3*a^5*b^18 + 332*A^2*B^3*a^7*b^16 + 222*A^2*B^3*a^9*b^14 - 16*A^2*B^3*a^11*b^12 - 37*A^2*B^3*a
^13*b^10 + 4*A^2*B^3*a^15*b^8 + 20*A^3*B^2*a^2*b^21 + 145*A^3*B^2*a^4*b^19 + 364*A^3*B^2*a^6*b^17 + 410*A^3*B^
2*a^8*b^15 + 196*A^3*B^2*a^10*b^13 + 13*A^3*B^2*a^12*b^11 - 12*A^3*B^2*a^14*b^9 + 10*A*B^4*a^2*b^21 + 60*A*B^4
*a^4*b^19 + 214*A*B^4*a^6*b^17 + 360*A*B^4*a^8*b^15 + 246*A*B^4*a^10*b^13 + 28*A*B^4*a^12*b^11 - 22*A*B^4*a^14
*b^9 + 4*A^4*B*a^3*b^20 + 119*A^4*B*a^5*b^18 + 260*A^4*B*a^7*b^16 + 110*A^4*B*a^9*b^14 - 100*A^4*B*a^11*b^12 -
61*A^4*B*a^13*b^10 + 4*A^4*B*a^15*b^8))/d^5))*(-(((8*B^2*a^5*d^2 - 8*A^2*a^5*d^2 + 80*A^2*a^3*b^2*d^2 - 80*B^
2*a^3*b^2*d^2 + 16*A*B*b^5*d^2 - 40*A^2*a*b^4*d^2 + 40*B^2*a*b^4*d^2 - 160*A*B*a^2*b^3*d^2 + 80*A*B*a^4*b*d^2)
^2/64 - d^4*(A^4*a^10 + A^4*b^10 + B^4*a^10 + B^4*b^10 + 2*A^2*B^2*a^10 + 2*A^2*B^2*b^10 + 5*A^4*a^2*b^8 + 10*
A^4*a^4*b^6 + 10*A^4*a^6*b^4 + 5*A^4*a^8*b^2 + 5*B^4*a^2*b^8 + 10*B^4*a^4*b^6 + 10*B^4*a^6*b^4 + 5*B^4*a^8*b^2
+ 10*A^2*B^2*a^2*b^8 + 20*A^2*B^2*a^4*b^6 + 20*A^2*B^2*a^6*b^4 + 10*A^2*B^2*a^8*b^2))^(1/2) + A^2*a^5*d^2 - B
^2*a^5*d^2 - 10*A^2*a^3*b^2*d^2 + 10*B^2*a^3*b^2*d^2 - 2*A*B*b^5*d^2 + 5*A^2*a*b^4*d^2 - 5*B^2*a*b^4*d^2 + 20*
A*B*a^2*b^3*d^2 - 10*A*B*a^4*b*d^2)/(4*d^4))^(1/2)*2i + (atan((((5*A*b + 2*B*a)*((16*(a + b*tan(c + d*x))^(1/2
)*(2*A^4*b^20 + 2*B^4*b^20 + 4*A^2*B^2*b^20 + 12*A^4*a^2*b^18 + 55*A^4*a^4*b^16 - 335*A^4*a^6*b^14 + 405*A^4*a
^8*b^12 - 13*A^4*a^10*b^10 + 2*A^4*a^12*b^8 + 12*B^4*a^2*b^18 + 30*B^4*a^4*b^16 + 36*B^4*a^6*b^14 + 90*B^4*a^8
*b^12 - 48*B^4*a^10*b^10 + 6*B^4*a^12*b^8 + 24*A^2*B^2*a^2*b^18 + 35*A^2*B^2*a^4*b^16 + 699*A^2*B^2*a^6*b^14 -
1175*A^2*B^2*a^8*b^12 + 349*A^2*B^2*a^10*b^10 - 20*A*B^3*a^5*b^15 + 348*A*B^3*a^7*b^13 - 460*A*B^3*a^9*b^11 +
68*A*B^3*a^11*b^9 + 320*A^3*B*a^5*b^15 - 1300*A^3*B*a^7*b^13 + 600*A^3*B*a^9*b^11 - 20*A^3*B*a^11*b^9))/d^4 +
((5*A*b + 2*B*a)*(a^3)^(1/2)*((8*(176*A^3*a^5*b^13*d^2 - 400*A^3*a^3*b^15*d^2 + 488*A^3*a^7*b^11*d^2 - 92*A^3
*a^9*b^9*d^2 + 12*B^3*a^2*b^16*d^2 + 192*B^3*a^4*b^14*d^2 - 120*B^3*a^6*b^12*d^2 - 288*B^3*a^8*b^10*d^2 + 12*B
^3*a^10*b^8*d^2 + 4*A^3*a*b^17*d^2 + 4*A*B^2*a*b^17*d^2 + 464*A*B^2*a^3*b^15*d^2 - 920*A*B^2*a^5*b^13*d^2 - 11
04*A*B^2*a^7*b^11*d^2 + 276*A*B^2*a^9*b^9*d^2 + 172*A^2*B*a^2*b^16*d^2 - 1468*A^2*B*a^4*b^14*d^2 - 776*A^2*B*a
^6*b^12*d^2 + 828*A^2*B*a^8*b^10*d^2 - 36*A^2*B*a^10*b^8*d^2))/d^5 - (((16*(a + b*tan(c + d*x))^(1/2)*(20*A^2*
a^3*b^12*d^2 + 304*A^2*a^5*b^10*d^2 - 20*A^2*a^7*b^8*d^2 - 20*B^2*a^3*b^12*d^2 - 204*B^2*a^5*b^10*d^2 + 36*B^2
*a^7*b^8*d^2 + 32*A*B*b^15*d^2 - 76*A^2*a*b^14*d^2 + 76*B^2*a*b^14*d^2 - 240*A*B*a^2*b^13*d^2 - 320*A*B*a^4*b^
11*d^2 + 288*A*B*a^6*b^9*d^2))/d^4 + ((5*A*b + 2*B*a)*(a^3)^(1/2)*((8*(16*B*a*b^12*d^4 + 128*A*a^2*b^11*d^4 +
128*A*a^4*b^9*d^4 + 64*B*a^3*b^10*d^4 + 48*B*a^5*b^8*d^4))/d^5 - (8*(5*A*b + 2*B*a)*(a^3)^(1/2)*(32*b^10*d^4 +
48*a^2*b^8*d^4)*(a + b*tan(c + d*x))^(1/2))/d^5))/(2*d))*(5*A*b + 2*B*a)*(a^3)^(1/2))/(2*d)))/(2*d))*(a^3)^(1
/2)*1i)/(2*d) + ((5*A*b + 2*B*a)*((16*(a + b*tan(c + d*x))^(1/2)*(2*A^4*b^20 + 2*B^4*b^20 + 4*A^2*B^2*b^20 + 1
2*A^4*a^2*b^18 + 55*A^4*a^4*b^16 - 335*A^4*a^6*b^14 + 405*A^4*a^8*b^12 - 13*A^4*a^10*b^10 + 2*A^4*a^12*b^8 + 1
2*B^4*a^2*b^18 + 30*B^4*a^4*b^16 + 36*B^4*a^6*b^14 + 90*B^4*a^8*b^12 - 48*B^4*a^10*b^10 + 6*B^4*a^12*b^8 + 24*
A^2*B^2*a^2*b^18 + 35*A^2*B^2*a^4*b^16 + 699*A^2*B^2*a^6*b^14 - 1175*A^2*B^2*a^8*b^12 + 349*A^2*B^2*a^10*b^10
- 20*A*B^3*a^5*b^15 + 348*A*B^3*a^7*b^13 - 460*A*B^3*a^9*b^11 + 68*A*B^3*a^11*b^9 + 320*A^3*B*a^5*b^15 - 1300*
A^3*B*a^7*b^13 + 600*A^3*B*a^9*b^11 - 20*A^3*B*a^11*b^9))/d^4 - ((5*A*b + 2*B*a)*(a^3)^(1/2)*((8*(176*A^3*a^5*
b^13*d^2 - 400*A^3*a^3*b^15*d^2 + 488*A^3*a^7*b^11*d^2 - 92*A^3*a^9*b^9*d^2 + 12*B^3*a^2*b^16*d^2 + 192*B^3*a^
4*b^14*d^2 - 120*B^3*a^6*b^12*d^2 - 288*B^3*a^8*b^10*d^2 + 12*B^3*a^10*b^8*d^2 + 4*A^3*a*b^17*d^2 + 4*A*B^2*a*
b^17*d^2 + 464*A*B^2*a^3*b^15*d^2 - 920*A*B^2*a^5*b^13*d^2 - 1104*A*B^2*a^7*b^11*d^2 + 276*A*B^2*a^9*b^9*d^2 +
172*A^2*B*a^2*b^16*d^2 - 1468*A^2*B*a^4*b^14*d^2 - 776*A^2*B*a^6*b^12*d^2 + 828*A^2*B*a^8*b^10*d^2 - 36*A^2*B
*a^10*b^8*d^2))/d^5 + (((16*(a + b*tan(c + d*x))^(1/2)*(20*A^2*a^3*b^12*d^2 + 304*A^2*a^5*b^10*d^2 - 20*A^2*a^
7*b^8*d^2 - 20*B^2*a^3*b^12*d^2 - 204*B^2*a^5*b^10*d^2 + 36*B^2*a^7*b^8*d^2 + 32*A*B*b^15*d^2 - 76*A^2*a*b^14*
d^2 + 76*B^2*a*b^14*d^2 - 240*A*B*a^2*b^13*d^2 - 320*A*B*a^4*b^11*d^2 + 288*A*B*a^6*b^9*d^2))/d^4 - ((5*A*b +
2*B*a)*(a^3)^(1/2)*((8*(16*B*a*b^12*d^4 + 128*A*a^2*b^11*d^4 + 128*A*a^4*b^9*d^4 + 64*B*a^3*b^10*d^4 + 48*B*a^
5*b^8*d^4))/d^5 + (8*(5*A*b + 2*B*a)*(a^3)^(1/2)*(32*b^10*d^4 + 48*a^2*b^8*d^4)*(a + b*tan(c + d*x))^(1/2))/d^
5))/(2*d))*(5*A*b + 2*B*a)*(a^3)^(1/2))/(2*d)))/(2*d))*(a^3)^(1/2)*1i)/(2*d))/((16*(10*A^5*a^2*b^21 + 85*A^5*a
^4*b^19 + 150*A^5*a^6*b^17 + 50*A^5*a^8*b^15 - 50*A^5*a^10*b^13 - 15*A^5*a^12*b^11 + 10*A^5*a^14*b^9 + 4*B^5*a
^3*b^20 + 24*B^5*a^5*b^18 + 72*B^5*a^7*b^16 + 112*B^5*a^9*b^14 + 84*B^5*a^11*b^12 + 24*B^5*a^13*b^10 + 8*A^2*B
^3*a^3*b^20 + 143*A^2*B^3*a^5*b^18 + 332*A^2*B^3*a^7*b^16 + 222*A^2*B^3*a^9*b^14 - 16*A^2*B^3*a^11*b^12 - 37*A
^2*B^3*a^13*b^10 + 4*A^2*B^3*a^15*b^8 + 20*A^3*B^2*a^2*b^21 + 145*A^3*B^2*a^4*b^19 + 364*A^3*B^2*a^6*b^17 + 41
0*A^3*B^2*a^8*b^15 + 196*A^3*B^2*a^10*b^13 + 13*A^3*B^2*a^12*b^11 - 12*A^3*B^2*a^14*b^9 + 10*A*B^4*a^2*b^21 +
60*A*B^4*a^4*b^19 + 214*A*B^4*a^6*b^17 + 360*A*B^4*a^8*b^15 + 246*A*B^4*a^10*b^13 + 28*A*B^4*a^12*b^11 - 22*A*
B^4*a^14*b^9 + 4*A^4*B*a^3*b^20 + 119*A^4*B*a^5*b^18 + 260*A^4*B*a^7*b^16 + 110*A^4*B*a^9*b^14 - 100*A^4*B*a^1
1*b^12 - 61*A^4*B*a^13*b^10 + 4*A^4*B*a^15*b^8))/d^5 - ((5*A*b + 2*B*a)*((16*(a + b*tan(c + d*x))^(1/2)*(2*A^4
*b^20 + 2*B^4*b^20 + 4*A^2*B^2*b^20 + 12*A^4*a^2*b^18 + 55*A^4*a^4*b^16 - 335*A^4*a^6*b^14 + 405*A^4*a^8*b^12
- 13*A^4*a^10*b^10 + 2*A^4*a^12*b^8 + 12*B^4*a^2*b^18 + 30*B^4*a^4*b^16 + 36*B^4*a^6*b^14 + 90*B^4*a^8*b^12 -
48*B^4*a^10*b^10 + 6*B^4*a^12*b^8 + 24*A^2*B^2*a^2*b^18 + 35*A^2*B^2*a^4*b^16 + 699*A^2*B^2*a^6*b^14 - 1175*A^
2*B^2*a^8*b^12 + 349*A^2*B^2*a^10*b^10 - 20*A*B^3*a^5*b^15 + 348*A*B^3*a^7*b^13 - 460*A*B^3*a^9*b^11 + 68*A*B^
3*a^11*b^9 + 320*A^3*B*a^5*b^15 - 1300*A^3*B*a^7*b^13 + 600*A^3*B*a^9*b^11 - 20*A^3*B*a^11*b^9))/d^4 + ((5*A*b
+ 2*B*a)*(a^3)^(1/2)*((8*(176*A^3*a^5*b^13*d^2 - 400*A^3*a^3*b^15*d^2 + 488*A^3*a^7*b^11*d^2 - 92*A^3*a^9*b^9
*d^2 + 12*B^3*a^2*b^16*d^2 + 192*B^3*a^4*b^14*d^2 - 120*B^3*a^6*b^12*d^2 - 288*B^3*a^8*b^10*d^2 + 12*B^3*a^10*
b^8*d^2 + 4*A^3*a*b^17*d^2 + 4*A*B^2*a*b^17*d^2 + 464*A*B^2*a^3*b^15*d^2 - 920*A*B^2*a^5*b^13*d^2 - 1104*A*B^2
*a^7*b^11*d^2 + 276*A*B^2*a^9*b^9*d^2 + 172*A^2*B*a^2*b^16*d^2 - 1468*A^2*B*a^4*b^14*d^2 - 776*A^2*B*a^6*b^12*
d^2 + 828*A^2*B*a^8*b^10*d^2 - 36*A^2*B*a^10*b^8*d^2))/d^5 - (((16*(a + b*tan(c + d*x))^(1/2)*(20*A^2*a^3*b^12
*d^2 + 304*A^2*a^5*b^10*d^2 - 20*A^2*a^7*b^8*d^2 - 20*B^2*a^3*b^12*d^2 - 204*B^2*a^5*b^10*d^2 + 36*B^2*a^7*b^8
*d^2 + 32*A*B*b^15*d^2 - 76*A^2*a*b^14*d^2 + 76*B^2*a*b^14*d^2 - 240*A*B*a^2*b^13*d^2 - 320*A*B*a^4*b^11*d^2 +
288*A*B*a^6*b^9*d^2))/d^4 + ((5*A*b + 2*B*a)*(a^3)^(1/2)*((8*(16*B*a*b^12*d^4 + 128*A*a^2*b^11*d^4 + 128*A*a^
4*b^9*d^4 + 64*B*a^3*b^10*d^4 + 48*B*a^5*b^8*d^4))/d^5 - (8*(5*A*b + 2*B*a)*(a^3)^(1/2)*(32*b^10*d^4 + 48*a^2*
b^8*d^4)*(a + b*tan(c + d*x))^(1/2))/d^5))/(2*d))*(5*A*b + 2*B*a)*(a^3)^(1/2))/(2*d)))/(2*d))*(a^3)^(1/2))/(2*
d) + ((5*A*b + 2*B*a)*((16*(a + b*tan(c + d*x))^(1/2)*(2*A^4*b^20 + 2*B^4*b^20 + 4*A^2*B^2*b^20 + 12*A^4*a^2*b
^18 + 55*A^4*a^4*b^16 - 335*A^4*a^6*b^14 + 405*A^4*a^8*b^12 - 13*A^4*a^10*b^10 + 2*A^4*a^12*b^8 + 12*B^4*a^2*b
^18 + 30*B^4*a^4*b^16 + 36*B^4*a^6*b^14 + 90*B^4*a^8*b^12 - 48*B^4*a^10*b^10 + 6*B^4*a^12*b^8 + 24*A^2*B^2*a^2
*b^18 + 35*A^2*B^2*a^4*b^16 + 699*A^2*B^2*a^6*b^14 - 1175*A^2*B^2*a^8*b^12 + 349*A^2*B^2*a^10*b^10 - 20*A*B^3*
a^5*b^15 + 348*A*B^3*a^7*b^13 - 460*A*B^3*a^9*b^11 + 68*A*B^3*a^11*b^9 + 320*A^3*B*a^5*b^15 - 1300*A^3*B*a^7*b
^13 + 600*A^3*B*a^9*b^11 - 20*A^3*B*a^11*b^9))/d^4 - ((5*A*b + 2*B*a)*(a^3)^(1/2)*((8*(176*A^3*a^5*b^13*d^2 -
400*A^3*a^3*b^15*d^2 + 488*A^3*a^7*b^11*d^2 - 92*A^3*a^9*b^9*d^2 + 12*B^3*a^2*b^16*d^2 + 192*B^3*a^4*b^14*d^2
- 120*B^3*a^6*b^12*d^2 - 288*B^3*a^8*b^10*d^2 + 12*B^3*a^10*b^8*d^2 + 4*A^3*a*b^17*d^2 + 4*A*B^2*a*b^17*d^2 +
464*A*B^2*a^3*b^15*d^2 - 920*A*B^2*a^5*b^13*d^2 - 1104*A*B^2*a^7*b^11*d^2 + 276*A*B^2*a^9*b^9*d^2 + 172*A^2*B*
a^2*b^16*d^2 - 1468*A^2*B*a^4*b^14*d^2 - 776*A^2*B*a^6*b^12*d^2 + 828*A^2*B*a^8*b^10*d^2 - 36*A^2*B*a^10*b^8*d
^2))/d^5 + (((16*(a + b*tan(c + d*x))^(1/2)*(20*A^2*a^3*b^12*d^2 + 304*A^2*a^5*b^10*d^2 - 20*A^2*a^7*b^8*d^2 -
20*B^2*a^3*b^12*d^2 - 204*B^2*a^5*b^10*d^2 + 36*B^2*a^7*b^8*d^2 + 32*A*B*b^15*d^2 - 76*A^2*a*b^14*d^2 + 76*B^
2*a*b^14*d^2 - 240*A*B*a^2*b^13*d^2 - 320*A*B*a^4*b^11*d^2 + 288*A*B*a^6*b^9*d^2))/d^4 - ((5*A*b + 2*B*a)*(a^3
)^(1/2)*((8*(16*B*a*b^12*d^4 + 128*A*a^2*b^11*d^4 + 128*A*a^4*b^9*d^4 + 64*B*a^3*b^10*d^4 + 48*B*a^5*b^8*d^4))
/d^5 + (8*(5*A*b + 2*B*a)*(a^3)^(1/2)*(32*b^10*d^4 + 48*a^2*b^8*d^4)*(a + b*tan(c + d*x))^(1/2))/d^5))/(2*d))*
(5*A*b + 2*B*a)*(a^3)^(1/2))/(2*d)))/(2*d))*(a^3)^(1/2))/(2*d)))*(5*A*b + 2*B*a)*(a^3)^(1/2)*1i)/d + (A*a^2*b*
(a + b*tan(c + d*x))^(1/2))/(a*d - d*(a + b*tan(c + d*x)))