\(\int \cot ^2(c+d x) (a+b \tan (c+d x))^{3/2} (A+B \tan (c+d x)) \, dx\) [329]

Optimal result

Integrand size = 33, antiderivative size = 169 \[ \int \cot ^2(c+d x) (a+b \tan (c+d x))^{3/2} (A+B \tan (c+d x)) \, dx=-\frac {\sqrt {a} (3 A b+2 a B) \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a}}\right )}{d}+\frac {(a-i b)^{3/2} (i A+B) \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a-i b}}\right )}{d}-\frac {(a+i b)^{3/2} (i A-B) \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a+i b}}\right )}{d}-\frac {a A \cot (c+d x) \sqrt {a+b \tan (c+d x)}}{d} \]

[Out]

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

Rubi [A] (verified)

Time = 0.75 (sec) , antiderivative size = 169, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.212, Rules used = {3686, 3734, 3620, 3618, 65, 214, 3715} \[ \int \cot ^2(c+d x) (a+b \tan (c+d x))^{3/2} (A+B \tan (c+d x)) \, dx=\frac {(a-i b)^{3/2} (B+i A) \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a-i b}}\right )}{d}-\frac {\sqrt {a} (2 a B+3 A b) \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a}}\right )}{d}-\frac {(a+i b)^{3/2} (-B+i A) \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a+i b}}\right )}{d}-\frac {a A \cot (c+d x) \sqrt {a+b \tan (c+d x)}}{d} \]

[In]

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

[Out]

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

Rule 65

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 214

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

Rule 3618

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

Rule 3620

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

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

Mathematica [A] (verified)

Time = 0.64 (sec) , antiderivative size = 282, normalized size of antiderivative = 1.67 \[ \int \cot ^2(c+d x) (a+b \tan (c+d x))^{3/2} (A+B \tan (c+d x)) \, dx=\frac {-\sqrt {a} (3 A b+2 a B) \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a}}\right )+(a-i b)^{3/2} (i A+B) \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a-i b}}\right )-i a A \sqrt {a+i b} \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a+i b}}\right )+A \sqrt {a+i b} b \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a+i b}}\right )+a \sqrt {a+i b} B \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a+i b}}\right )+i \sqrt {a+i b} b B \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a+i b}}\right )-a A \cot (c+d x) \sqrt {a+b \tan (c+d x)}}{d} \]

[In]

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

[Out]

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

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1688\) vs. \(2(143)=286\).

Time = 0.24 (sec) , antiderivative size = 1689, normalized size of antiderivative = 9.99

method	result	size
derivativedivides	\(\text {Expression too large to display}\)	\(1689\)
default	\(\text {Expression too large to display}\)	\(1689\)

[In]

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

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 3116 vs. \(2 (137) = 274\).

Time = 3.34 (sec) , antiderivative size = 6248, normalized size of antiderivative = 36.97 \[ \int \cot ^2(c+d x) (a+b \tan (c+d x))^{3/2} (A+B \tan (c+d x)) \, dx=\text {Too large to display} \]

[In]

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

[Out]

Too large to include

Sympy [F]

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

[In]

integrate(cot(d*x+c)**2*(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)**2, x)

Maxima [F]

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

[In]

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

[Out]

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

Giac [F(-1)]

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

[In]

integrate(cot(d*x+c)^2*(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 = 9.84 (sec) , antiderivative size = 21319, normalized size of antiderivative = 126.15 \[ \int \cot ^2(c+d x) (a+b \tan (c+d x))^{3/2} (A+B \tan (c+d x)) \, dx=\text {Too large to display} \]

[In]

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

[Out]

(a^(1/2)*atan(((a^(1/2)*((16*(a + b*tan(c + d*x))^(1/2)*(2*A^4*b^16 + 2*B^4*b^16 + 4*A^2*B^2*b^16 - A^4*a^2*b^
14 + 66*A^4*a^4*b^12 - A^4*a^6*b^10 + 2*A^4*a^8*b^8 + 8*B^4*a^2*b^14 + 16*B^4*a^4*b^12 - 16*B^4*a^6*b^10 + 6*B
^4*a^8*b^8 + 25*A^2*B^2*a^2*b^14 - 130*A^2*B^2*a^4*b^12 + 145*A^2*B^2*a^6*b^10 + 12*A*B^3*a^3*b^13 - 104*A*B^3
*a^5*b^11 + 44*A*B^3*a^7*b^9 - 84*A^3*B*a^3*b^13 + 144*A^3*B*a^5*b^11 - 12*A^3*B*a^7*b^9))/d^4 + (a^(1/2)*(3*A
*b + 2*B*a)*((8*(100*A^3*a^2*b^13*d^2 + 44*A^3*a^4*b^11*d^2 - 56*A^3*a^6*b^9*d^2 - 92*B^3*a^3*b^12*d^2 - 84*B^
3*a^5*b^10*d^2 + 12*B^3*a^7*b^8*d^2 + 4*B^3*a*b^14*d^2 - 92*A^2*B*a*b^14*d^2 - 216*A*B^2*a^2*b^13*d^2 - 48*A*B
^2*a^4*b^11*d^2 + 168*A*B^2*a^6*b^9*d^2 + 208*A^2*B*a^3*b^12*d^2 + 264*A^2*B*a^5*b^10*d^2 - 36*A^2*B*a^7*b^8*d
^2))/d^5 - (a^(1/2)*(3*A*b + 2*B*a)*((16*(a + b*tan(c + d*x))^(1/2)*(92*A^2*a^3*b^10*d^2 - 20*A^2*a^5*b^8*d^2
- 56*B^2*a^3*b^10*d^2 + 36*B^2*a^5*b^8*d^2 - 32*A*B*b^13*d^2 + 44*A^2*a*b^12*d^2 - 44*B^2*a*b^12*d^2 + 32*A*B*
a^2*b^11*d^2 + 176*A*B*a^4*b^9*d^2))/d^4 + (a^(1/2)*(3*A*b + 2*B*a)*((8*(80*A*a*b^11*d^4 + 80*A*a^3*b^9*d^4 +
48*B*a^2*b^10*d^4 + 48*B*a^4*b^8*d^4))/d^5 - (8*a^(1/2)*(3*A*b + 2*B*a)*(32*b^10*d^4 + 48*a^2*b^8*d^4)*(a + b*
tan(c + d*x))^(1/2))/d^5))/(2*d)))/(2*d)))/(2*d))*(3*A*b + 2*B*a)*1i)/(2*d) + (a^(1/2)*((16*(a + b*tan(c + d*x
))^(1/2)*(2*A^4*b^16 + 2*B^4*b^16 + 4*A^2*B^2*b^16 - A^4*a^2*b^14 + 66*A^4*a^4*b^12 - A^4*a^6*b^10 + 2*A^4*a^8
*b^8 + 8*B^4*a^2*b^14 + 16*B^4*a^4*b^12 - 16*B^4*a^6*b^10 + 6*B^4*a^8*b^8 + 25*A^2*B^2*a^2*b^14 - 130*A^2*B^2*
a^4*b^12 + 145*A^2*B^2*a^6*b^10 + 12*A*B^3*a^3*b^13 - 104*A*B^3*a^5*b^11 + 44*A*B^3*a^7*b^9 - 84*A^3*B*a^3*b^1
3 + 144*A^3*B*a^5*b^11 - 12*A^3*B*a^7*b^9))/d^4 - (a^(1/2)*(3*A*b + 2*B*a)*((8*(100*A^3*a^2*b^13*d^2 + 44*A^3*
a^4*b^11*d^2 - 56*A^3*a^6*b^9*d^2 - 92*B^3*a^3*b^12*d^2 - 84*B^3*a^5*b^10*d^2 + 12*B^3*a^7*b^8*d^2 + 4*B^3*a*b
^14*d^2 - 92*A^2*B*a*b^14*d^2 - 216*A*B^2*a^2*b^13*d^2 - 48*A*B^2*a^4*b^11*d^2 + 168*A*B^2*a^6*b^9*d^2 + 208*A
^2*B*a^3*b^12*d^2 + 264*A^2*B*a^5*b^10*d^2 - 36*A^2*B*a^7*b^8*d^2))/d^5 + (a^(1/2)*(3*A*b + 2*B*a)*((16*(a + b
*tan(c + d*x))^(1/2)*(92*A^2*a^3*b^10*d^2 - 20*A^2*a^5*b^8*d^2 - 56*B^2*a^3*b^10*d^2 + 36*B^2*a^5*b^8*d^2 - 32
*A*B*b^13*d^2 + 44*A^2*a*b^12*d^2 - 44*B^2*a*b^12*d^2 + 32*A*B*a^2*b^11*d^2 + 176*A*B*a^4*b^9*d^2))/d^4 - (a^(
1/2)*(3*A*b + 2*B*a)*((8*(80*A*a*b^11*d^4 + 80*A*a^3*b^9*d^4 + 48*B*a^2*b^10*d^4 + 48*B*a^4*b^8*d^4))/d^5 + (8
*a^(1/2)*(3*A*b + 2*B*a)*(32*b^10*d^4 + 48*a^2*b^8*d^4)*(a + b*tan(c + d*x))^(1/2))/d^5))/(2*d)))/(2*d)))/(2*d
))*(3*A*b + 2*B*a)*1i)/(2*d))/((16*(6*A^5*a*b^17 + 6*A^5*a^3*b^15 + 6*A^5*a^7*b^11 + 6*A^5*a^9*b^9 + 4*B^5*a^2
*b^16 + 20*B^5*a^4*b^14 + 28*B^5*a^6*b^12 + 12*B^5*a^8*b^10 + 17*A^2*B^3*a^2*b^16 + 21*A^2*B^3*a^4*b^14 - 5*A^
2*B^3*a^6*b^12 - 5*A^2*B^3*a^8*b^10 + 4*A^2*B^3*a^10*b^8 + 42*A^3*B^2*a^3*b^15 + 40*A^3*B^2*a^5*b^13 + 2*A^3*B
^2*a^7*b^11 - 8*A^3*B^2*a^9*b^9 + 6*A*B^4*a*b^17 + 36*A*B^4*a^3*b^15 + 40*A*B^4*a^5*b^13 - 4*A*B^4*a^7*b^11 -
14*A*B^4*a^9*b^9 + 12*A^3*B^2*a*b^17 + 13*A^4*B*a^2*b^16 + A^4*B*a^4*b^14 - 33*A^4*B*a^6*b^12 - 17*A^4*B*a^8*b
^10 + 4*A^4*B*a^10*b^8))/d^5 - (a^(1/2)*((16*(a + b*tan(c + d*x))^(1/2)*(2*A^4*b^16 + 2*B^4*b^16 + 4*A^2*B^2*b
^16 - A^4*a^2*b^14 + 66*A^4*a^4*b^12 - A^4*a^6*b^10 + 2*A^4*a^8*b^8 + 8*B^4*a^2*b^14 + 16*B^4*a^4*b^12 - 16*B^
4*a^6*b^10 + 6*B^4*a^8*b^8 + 25*A^2*B^2*a^2*b^14 - 130*A^2*B^2*a^4*b^12 + 145*A^2*B^2*a^6*b^10 + 12*A*B^3*a^3*
b^13 - 104*A*B^3*a^5*b^11 + 44*A*B^3*a^7*b^9 - 84*A^3*B*a^3*b^13 + 144*A^3*B*a^5*b^11 - 12*A^3*B*a^7*b^9))/d^4
+ (a^(1/2)*(3*A*b + 2*B*a)*((8*(100*A^3*a^2*b^13*d^2 + 44*A^3*a^4*b^11*d^2 - 56*A^3*a^6*b^9*d^2 - 92*B^3*a^3*
b^12*d^2 - 84*B^3*a^5*b^10*d^2 + 12*B^3*a^7*b^8*d^2 + 4*B^3*a*b^14*d^2 - 92*A^2*B*a*b^14*d^2 - 216*A*B^2*a^2*b
^13*d^2 - 48*A*B^2*a^4*b^11*d^2 + 168*A*B^2*a^6*b^9*d^2 + 208*A^2*B*a^3*b^12*d^2 + 264*A^2*B*a^5*b^10*d^2 - 36
*A^2*B*a^7*b^8*d^2))/d^5 - (a^(1/2)*(3*A*b + 2*B*a)*((16*(a + b*tan(c + d*x))^(1/2)*(92*A^2*a^3*b^10*d^2 - 20*
A^2*a^5*b^8*d^2 - 56*B^2*a^3*b^10*d^2 + 36*B^2*a^5*b^8*d^2 - 32*A*B*b^13*d^2 + 44*A^2*a*b^12*d^2 - 44*B^2*a*b^
12*d^2 + 32*A*B*a^2*b^11*d^2 + 176*A*B*a^4*b^9*d^2))/d^4 + (a^(1/2)*(3*A*b + 2*B*a)*((8*(80*A*a*b^11*d^4 + 80*
A*a^3*b^9*d^4 + 48*B*a^2*b^10*d^4 + 48*B*a^4*b^8*d^4))/d^5 - (8*a^(1/2)*(3*A*b + 2*B*a)*(32*b^10*d^4 + 48*a^2*
b^8*d^4)*(a + b*tan(c + d*x))^(1/2))/d^5))/(2*d)))/(2*d)))/(2*d))*(3*A*b + 2*B*a))/(2*d) + (a^(1/2)*((16*(a +
b*tan(c + d*x))^(1/2)*(2*A^4*b^16 + 2*B^4*b^16 + 4*A^2*B^2*b^16 - A^4*a^2*b^14 + 66*A^4*a^4*b^12 - A^4*a^6*b^1
0 + 2*A^4*a^8*b^8 + 8*B^4*a^2*b^14 + 16*B^4*a^4*b^12 - 16*B^4*a^6*b^10 + 6*B^4*a^8*b^8 + 25*A^2*B^2*a^2*b^14 -
130*A^2*B^2*a^4*b^12 + 145*A^2*B^2*a^6*b^10 + 12*A*B^3*a^3*b^13 - 104*A*B^3*a^5*b^11 + 44*A*B^3*a^7*b^9 - 84*
A^3*B*a^3*b^13 + 144*A^3*B*a^5*b^11 - 12*A^3*B*a^7*b^9))/d^4 - (a^(1/2)*(3*A*b + 2*B*a)*((8*(100*A^3*a^2*b^13*
d^2 + 44*A^3*a^4*b^11*d^2 - 56*A^3*a^6*b^9*d^2 - 92*B^3*a^3*b^12*d^2 - 84*B^3*a^5*b^10*d^2 + 12*B^3*a^7*b^8*d^
2 + 4*B^3*a*b^14*d^2 - 92*A^2*B*a*b^14*d^2 - 216*A*B^2*a^2*b^13*d^2 - 48*A*B^2*a^4*b^11*d^2 + 168*A*B^2*a^6*b^
9*d^2 + 208*A^2*B*a^3*b^12*d^2 + 264*A^2*B*a^5*b^10*d^2 - 36*A^2*B*a^7*b^8*d^2))/d^5 + (a^(1/2)*(3*A*b + 2*B*a
)*((16*(a + b*tan(c + d*x))^(1/2)*(92*A^2*a^3*b^10*d^2 - 20*A^2*a^5*b^8*d^2 - 56*B^2*a^3*b^10*d^2 + 36*B^2*a^5
*b^8*d^2 - 32*A*B*b^13*d^2 + 44*A^2*a*b^12*d^2 - 44*B^2*a*b^12*d^2 + 32*A*B*a^2*b^11*d^2 + 176*A*B*a^4*b^9*d^2
))/d^4 - (a^(1/2)*(3*A*b + 2*B*a)*((8*(80*A*a*b^11*d^4 + 80*A*a^3*b^9*d^4 + 48*B*a^2*b^10*d^4 + 48*B*a^4*b^8*d
^4))/d^5 + (8*a^(1/2)*(3*A*b + 2*B*a)*(32*b^10*d^4 + 48*a^2*b^8*d^4)*(a + b*tan(c + d*x))^(1/2))/d^5))/(2*d)))
/(2*d)))/(2*d))*(3*A*b + 2*B*a))/(2*d)))*(3*A*b + 2*B*a)*1i)/d - atan(((((((8*(80*A*a*b^11*d^4 + 80*A*a^3*b^9*
d^4 + 48*B*a^2*b^10*d^4 + 48*B*a^4*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*A^2*a^3*d^2 - 8*B^2*a^3*d^2 + 16*A*B*b^3*d^2 - 24*A^2*a*b^2*d^2 + 24*B^2*a*b^2*d^2 - 48*A*B*a^2*b*d^2
)^2/64 - d^4*(A^4*a^6 + A^4*b^6 + B^4*a^6 + B^4*b^6 + 2*A^2*B^2*a^6 + 2*A^2*B^2*b^6 + 3*A^4*a^2*b^4 + 3*A^4*a^
4*b^2 + 3*B^4*a^2*b^4 + 3*B^4*a^4*b^2 + 6*A^2*B^2*a^2*b^4 + 6*A^2*B^2*a^4*b^2))^(1/2) - A^2*a^3*d^2 + B^2*a^3*
d^2 - 2*A*B*b^3*d^2 + 3*A^2*a*b^2*d^2 - 3*B^2*a*b^2*d^2 + 6*A*B*a^2*b*d^2)/(4*d^4))^(1/2))/d^4)*((((8*A^2*a^3*
d^2 - 8*B^2*a^3*d^2 + 16*A*B*b^3*d^2 - 24*A^2*a*b^2*d^2 + 24*B^2*a*b^2*d^2 - 48*A*B*a^2*b*d^2)^2/64 - d^4*(A^4
*a^6 + A^4*b^6 + B^4*a^6 + B^4*b^6 + 2*A^2*B^2*a^6 + 2*A^2*B^2*b^6 + 3*A^4*a^2*b^4 + 3*A^4*a^4*b^2 + 3*B^4*a^2
*b^4 + 3*B^4*a^4*b^2 + 6*A^2*B^2*a^2*b^4 + 6*A^2*B^2*a^4*b^2))^(1/2) - A^2*a^3*d^2 + B^2*a^3*d^2 - 2*A*B*b^3*d
^2 + 3*A^2*a*b^2*d^2 - 3*B^2*a*b^2*d^2 + 6*A*B*a^2*b*d^2)/(4*d^4))^(1/2) + (16*(a + b*tan(c + d*x))^(1/2)*(92*
A^2*a^3*b^10*d^2 - 20*A^2*a^5*b^8*d^2 - 56*B^2*a^3*b^10*d^2 + 36*B^2*a^5*b^8*d^2 - 32*A*B*b^13*d^2 + 44*A^2*a*
b^12*d^2 - 44*B^2*a*b^12*d^2 + 32*A*B*a^2*b^11*d^2 + 176*A*B*a^4*b^9*d^2))/d^4)*((((8*A^2*a^3*d^2 - 8*B^2*a^3*
d^2 + 16*A*B*b^3*d^2 - 24*A^2*a*b^2*d^2 + 24*B^2*a*b^2*d^2 - 48*A*B*a^2*b*d^2)^2/64 - d^4*(A^4*a^6 + A^4*b^6 +
B^4*a^6 + B^4*b^6 + 2*A^2*B^2*a^6 + 2*A^2*B^2*b^6 + 3*A^4*a^2*b^4 + 3*A^4*a^4*b^2 + 3*B^4*a^2*b^4 + 3*B^4*a^4
*b^2 + 6*A^2*B^2*a^2*b^4 + 6*A^2*B^2*a^4*b^2))^(1/2) - A^2*a^3*d^2 + B^2*a^3*d^2 - 2*A*B*b^3*d^2 + 3*A^2*a*b^2
*d^2 - 3*B^2*a*b^2*d^2 + 6*A*B*a^2*b*d^2)/(4*d^4))^(1/2) - (8*(100*A^3*a^2*b^13*d^2 + 44*A^3*a^4*b^11*d^2 - 56
*A^3*a^6*b^9*d^2 - 92*B^3*a^3*b^12*d^2 - 84*B^3*a^5*b^10*d^2 + 12*B^3*a^7*b^8*d^2 + 4*B^3*a*b^14*d^2 - 92*A^2*
B*a*b^14*d^2 - 216*A*B^2*a^2*b^13*d^2 - 48*A*B^2*a^4*b^11*d^2 + 168*A*B^2*a^6*b^9*d^2 + 208*A^2*B*a^3*b^12*d^2
+ 264*A^2*B*a^5*b^10*d^2 - 36*A^2*B*a^7*b^8*d^2))/d^5)*((((8*A^2*a^3*d^2 - 8*B^2*a^3*d^2 + 16*A*B*b^3*d^2 - 2
4*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) - (16*(a + b*tan(c + d*x))^(1/2)*(2*A^4*b^16 + 2*B^4*b^16 + 4*A^2*B^2*b^16 -
A^4*a^2*b^14 + 66*A^4*a^4*b^12 - A^4*a^6*b^10 + 2*A^4*a^8*b^8 + 8*B^4*a^2*b^14 + 16*B^4*a^4*b^12 - 16*B^4*a^6*
b^10 + 6*B^4*a^8*b^8 + 25*A^2*B^2*a^2*b^14 - 130*A^2*B^2*a^4*b^12 + 145*A^2*B^2*a^6*b^10 + 12*A*B^3*a^3*b^13 -
104*A*B^3*a^5*b^11 + 44*A*B^3*a^7*b^9 - 84*A^3*B*a^3*b^13 + 144*A^3*B*a^5*b^11 - 12*A^3*B*a^7*b^9))/d^4)*((((
8*A^2*a^3*d^2 - 8*B^2*a^3*d^2 + 16*A*B*b^3*d^2 - 24*A^2*a*b^2*d^2 + 24*B^2*a*b^2*d^2 - 48*A*B*a^2*b*d^2)^2/64
- d^4*(A^4*a^6 + A^4*b^6 + B^4*a^6 + B^4*b^6 + 2*A^2*B^2*a^6 + 2*A^2*B^2*b^6 + 3*A^4*a^2*b^4 + 3*A^4*a^4*b^2 +
3*B^4*a^2*b^4 + 3*B^4*a^4*b^2 + 6*A^2*B^2*a^2*b^4 + 6*A^2*B^2*a^4*b^2))^(1/2) - A^2*a^3*d^2 + B^2*a^3*d^2 - 2
*A*B*b^3*d^2 + 3*A^2*a*b^2*d^2 - 3*B^2*a*b^2*d^2 + 6*A*B*a^2*b*d^2)/(4*d^4))^(1/2)*1i - (((((8*(80*A*a*b^11*d^
4 + 80*A*a^3*b^9*d^4 + 48*B*a^2*b^10*d^4 + 48*B*a^4*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*A^2*a^3*d^2 - 8*B^2*a^3*d^2 + 16*A*B*b^3*d^2 - 24*A^2*a*b^2*d^2 + 24*B^2*a*b^2*d^2 -
48*A*B*a^2*b*d^2)^2/64 - d^4*(A^4*a^6 + A^4*b^6 + B^4*a^6 + B^4*b^6 + 2*A^2*B^2*a^6 + 2*A^2*B^2*b^6 + 3*A^4*a
^2*b^4 + 3*A^4*a^4*b^2 + 3*B^4*a^2*b^4 + 3*B^4*a^4*b^2 + 6*A^2*B^2*a^2*b^4 + 6*A^2*B^2*a^4*b^2))^(1/2) - A^2*a
^3*d^2 + B^2*a^3*d^2 - 2*A*B*b^3*d^2 + 3*A^2*a*b^2*d^2 - 3*B^2*a*b^2*d^2 + 6*A*B*a^2*b*d^2)/(4*d^4))^(1/2))/d^
4)*((((8*A^2*a^3*d^2 - 8*B^2*a^3*d^2 + 16*A*B*b^3*d^2 - 24*A^2*a*b^2*d^2 + 24*B^2*a*b^2*d^2 - 48*A*B*a^2*b*d^2
)^2/64 - d^4*(A^4*a^6 + A^4*b^6 + B^4*a^6 + B^4*b^6 + 2*A^2*B^2*a^6 + 2*A^2*B^2*b^6 + 3*A^4*a^2*b^4 + 3*A^4*a^
4*b^2 + 3*B^4*a^2*b^4 + 3*B^4*a^4*b^2 + 6*A^2*B^2*a^2*b^4 + 6*A^2*B^2*a^4*b^2))^(1/2) - A^2*a^3*d^2 + B^2*a^3*
d^2 - 2*A*B*b^3*d^2 + 3*A^2*a*b^2*d^2 - 3*B^2*a*b^2*d^2 + 6*A*B*a^2*b*d^2)/(4*d^4))^(1/2) - (16*(a + b*tan(c +
d*x))^(1/2)*(92*A^2*a^3*b^10*d^2 - 20*A^2*a^5*b^8*d^2 - 56*B^2*a^3*b^10*d^2 + 36*B^2*a^5*b^8*d^2 - 32*A*B*b^1
3*d^2 + 44*A^2*a*b^12*d^2 - 44*B^2*a*b^12*d^2 + 32*A*B*a^2*b^11*d^2 + 176*A*B*a^4*b^9*d^2))/d^4)*((((8*A^2*a^3
*d^2 - 8*B^2*a^3*d^2 + 16*A*B*b^3*d^2 - 24*A^2*a*b^2*d^2 + 24*B^2*a*b^2*d^2 - 48*A*B*a^2*b*d^2)^2/64 - d^4*(A^
4*a^6 + A^4*b^6 + B^4*a^6 + B^4*b^6 + 2*A^2*B^2*a^6 + 2*A^2*B^2*b^6 + 3*A^4*a^2*b^4 + 3*A^4*a^4*b^2 + 3*B^4*a^
2*b^4 + 3*B^4*a^4*b^2 + 6*A^2*B^2*a^2*b^4 + 6*A^2*B^2*a^4*b^2))^(1/2) - A^2*a^3*d^2 + B^2*a^3*d^2 - 2*A*B*b^3*
d^2 + 3*A^2*a*b^2*d^2 - 3*B^2*a*b^2*d^2 + 6*A*B*a^2*b*d^2)/(4*d^4))^(1/2) - (8*(100*A^3*a^2*b^13*d^2 + 44*A^3*
a^4*b^11*d^2 - 56*A^3*a^6*b^9*d^2 - 92*B^3*a^3*b^12*d^2 - 84*B^3*a^5*b^10*d^2 + 12*B^3*a^7*b^8*d^2 + 4*B^3*a*b
^14*d^2 - 92*A^2*B*a*b^14*d^2 - 216*A*B^2*a^2*b^13*d^2 - 48*A*B^2*a^4*b^11*d^2 + 168*A*B^2*a^6*b^9*d^2 + 208*A
^2*B*a^3*b^12*d^2 + 264*A^2*B*a^5*b^10*d^2 - 36*A^2*B*a^7*b^8*d^2))/d^5)*((((8*A^2*a^3*d^2 - 8*B^2*a^3*d^2 + 1
6*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) + (16*(a + b*tan(c + d*x))^(1/2)*(2*A^4*b^16 + 2*B^4*b^16 +
4*A^2*B^2*b^16 - A^4*a^2*b^14 + 66*A^4*a^4*b^12 - A^4*a^6*b^10 + 2*A^4*a^8*b^8 + 8*B^4*a^2*b^14 + 16*B^4*a^4*b
^12 - 16*B^4*a^6*b^10 + 6*B^4*a^8*b^8 + 25*A^2*B^2*a^2*b^14 - 130*A^2*B^2*a^4*b^12 + 145*A^2*B^2*a^6*b^10 + 12
*A*B^3*a^3*b^13 - 104*A*B^3*a^5*b^11 + 44*A*B^3*a^7*b^9 - 84*A^3*B*a^3*b^13 + 144*A^3*B*a^5*b^11 - 12*A^3*B*a^
7*b^9))/d^4)*((((8*A^2*a^3*d^2 - 8*B^2*a^3*d^2 + 16*A*B*b^3*d^2 - 24*A^2*a*b^2*d^2 + 24*B^2*a*b^2*d^2 - 48*A*B
*a^2*b*d^2)^2/64 - d^4*(A^4*a^6 + A^4*b^6 + B^4*a^6 + B^4*b^6 + 2*A^2*B^2*a^6 + 2*A^2*B^2*b^6 + 3*A^4*a^2*b^4
+ 3*A^4*a^4*b^2 + 3*B^4*a^2*b^4 + 3*B^4*a^4*b^2 + 6*A^2*B^2*a^2*b^4 + 6*A^2*B^2*a^4*b^2))^(1/2) - A^2*a^3*d^2
+ B^2*a^3*d^2 - 2*A*B*b^3*d^2 + 3*A^2*a*b^2*d^2 - 3*B^2*a*b^2*d^2 + 6*A*B*a^2*b*d^2)/(4*d^4))^(1/2)*1i)/((16*(
6*A^5*a*b^17 + 6*A^5*a^3*b^15 + 6*A^5*a^7*b^11 + 6*A^5*a^9*b^9 + 4*B^5*a^2*b^16 + 20*B^5*a^4*b^14 + 28*B^5*a^6
*b^12 + 12*B^5*a^8*b^10 + 17*A^2*B^3*a^2*b^16 + 21*A^2*B^3*a^4*b^14 - 5*A^2*B^3*a^6*b^12 - 5*A^2*B^3*a^8*b^10
+ 4*A^2*B^3*a^10*b^8 + 42*A^3*B^2*a^3*b^15 + 40*A^3*B^2*a^5*b^13 + 2*A^3*B^2*a^7*b^11 - 8*A^3*B^2*a^9*b^9 + 6*
A*B^4*a*b^17 + 36*A*B^4*a^3*b^15 + 40*A*B^4*a^5*b^13 - 4*A*B^4*a^7*b^11 - 14*A*B^4*a^9*b^9 + 12*A^3*B^2*a*b^17
+ 13*A^4*B*a^2*b^16 + A^4*B*a^4*b^14 - 33*A^4*B*a^6*b^12 - 17*A^4*B*a^8*b^10 + 4*A^4*B*a^10*b^8))/d^5 + (((((
8*(80*A*a*b^11*d^4 + 80*A*a^3*b^9*d^4 + 48*B*a^2*b^10*d^4 + 48*B*a^4*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*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 + 2
4*B^2*a*b^2*d^2 - 48*A*B*a^2*b*d^2)^2/64 - d^4*(A^4*a^6 + A^4*b^6 + B^4*a^6 + B^4*b^6 + 2*A^2*B^2*a^6 + 2*A^2*
B^2*b^6 + 3*A^4*a^2*b^4 + 3*A^4*a^4*b^2 + 3*B^4*a^2*b^4 + 3*B^4*a^4*b^2 + 6*A^2*B^2*a^2*b^4 + 6*A^2*B^2*a^4*b^
2))^(1/2) - A^2*a^3*d^2 + B^2*a^3*d^2 - 2*A*B*b^3*d^2 + 3*A^2*a*b^2*d^2 - 3*B^2*a*b^2*d^2 + 6*A*B*a^2*b*d^2)/(
4*d^4))^(1/2))/d^4)*((((8*A^2*a^3*d^2 - 8*B^2*a^3*d^2 + 16*A*B*b^3*d^2 - 24*A^2*a*b^2*d^2 + 24*B^2*a*b^2*d^2 -
48*A*B*a^2*b*d^2)^2/64 - d^4*(A^4*a^6 + A^4*b^6 + B^4*a^6 + B^4*b^6 + 2*A^2*B^2*a^6 + 2*A^2*B^2*b^6 + 3*A^4*a
^2*b^4 + 3*A^4*a^4*b^2 + 3*B^4*a^2*b^4 + 3*B^4*a^4*b^2 + 6*A^2*B^2*a^2*b^4 + 6*A^2*B^2*a^4*b^2))^(1/2) - A^2*a
^3*d^2 + B^2*a^3*d^2 - 2*A*B*b^3*d^2 + 3*A^2*a*b^2*d^2 - 3*B^2*a*b^2*d^2 + 6*A*B*a^2*b*d^2)/(4*d^4))^(1/2) + (
16*(a + b*tan(c + d*x))^(1/2)*(92*A^2*a^3*b^10*d^2 - 20*A^2*a^5*b^8*d^2 - 56*B^2*a^3*b^10*d^2 + 36*B^2*a^5*b^8
*d^2 - 32*A*B*b^13*d^2 + 44*A^2*a*b^12*d^2 - 44*B^2*a*b^12*d^2 + 32*A*B*a^2*b^11*d^2 + 176*A*B*a^4*b^9*d^2))/d
^4)*((((8*A^2*a^3*d^2 - 8*B^2*a^3*d^2 + 16*A*B*b^3*d^2 - 24*A^2*a*b^2*d^2 + 24*B^2*a*b^2*d^2 - 48*A*B*a^2*b*d^
2)^2/64 - d^4*(A^4*a^6 + A^4*b^6 + B^4*a^6 + B^4*b^6 + 2*A^2*B^2*a^6 + 2*A^2*B^2*b^6 + 3*A^4*a^2*b^4 + 3*A^4*a
^4*b^2 + 3*B^4*a^2*b^4 + 3*B^4*a^4*b^2 + 6*A^2*B^2*a^2*b^4 + 6*A^2*B^2*a^4*b^2))^(1/2) - A^2*a^3*d^2 + B^2*a^3
*d^2 - 2*A*B*b^3*d^2 + 3*A^2*a*b^2*d^2 - 3*B^2*a*b^2*d^2 + 6*A*B*a^2*b*d^2)/(4*d^4))^(1/2) - (8*(100*A^3*a^2*b
^13*d^2 + 44*A^3*a^4*b^11*d^2 - 56*A^3*a^6*b^9*d^2 - 92*B^3*a^3*b^12*d^2 - 84*B^3*a^5*b^10*d^2 + 12*B^3*a^7*b^
8*d^2 + 4*B^3*a*b^14*d^2 - 92*A^2*B*a*b^14*d^2 - 216*A*B^2*a^2*b^13*d^2 - 48*A*B^2*a^4*b^11*d^2 + 168*A*B^2*a^
6*b^9*d^2 + 208*A^2*B*a^3*b^12*d^2 + 264*A^2*B*a^5*b^10*d^2 - 36*A^2*B*a^7*b^8*d^2))/d^5)*((((8*A^2*a^3*d^2 -
8*B^2*a^3*d^2 + 16*A*B*b^3*d^2 - 24*A^2*a*b^2*d^2 + 24*B^2*a*b^2*d^2 - 48*A*B*a^2*b*d^2)^2/64 - d^4*(A^4*a^6 +
A^4*b^6 + B^4*a^6 + B^4*b^6 + 2*A^2*B^2*a^6 + 2*A^2*B^2*b^6 + 3*A^4*a^2*b^4 + 3*A^4*a^4*b^2 + 3*B^4*a^2*b^4 +
3*B^4*a^4*b^2 + 6*A^2*B^2*a^2*b^4 + 6*A^2*B^2*a^4*b^2))^(1/2) - A^2*a^3*d^2 + B^2*a^3*d^2 - 2*A*B*b^3*d^2 + 3
*A^2*a*b^2*d^2 - 3*B^2*a*b^2*d^2 + 6*A*B*a^2*b*d^2)/(4*d^4))^(1/2) - (16*(a + b*tan(c + d*x))^(1/2)*(2*A^4*b^1
6 + 2*B^4*b^16 + 4*A^2*B^2*b^16 - A^4*a^2*b^14 + 66*A^4*a^4*b^12 - A^4*a^6*b^10 + 2*A^4*a^8*b^8 + 8*B^4*a^2*b^
14 + 16*B^4*a^4*b^12 - 16*B^4*a^6*b^10 + 6*B^4*a^8*b^8 + 25*A^2*B^2*a^2*b^14 - 130*A^2*B^2*a^4*b^12 + 145*A^2*
B^2*a^6*b^10 + 12*A*B^3*a^3*b^13 - 104*A*B^3*a^5*b^11 + 44*A*B^3*a^7*b^9 - 84*A^3*B*a^3*b^13 + 144*A^3*B*a^5*b
^11 - 12*A^3*B*a^7*b^9))/d^4)*((((8*A^2*a^3*d^2 - 8*B^2*a^3*d^2 + 16*A*B*b^3*d^2 - 24*A^2*a*b^2*d^2 + 24*B^2*a
*b^2*d^2 - 48*A*B*a^2*b*d^2)^2/64 - d^4*(A^4*a^6 + A^4*b^6 + B^4*a^6 + B^4*b^6 + 2*A^2*B^2*a^6 + 2*A^2*B^2*b^6
+ 3*A^4*a^2*b^4 + 3*A^4*a^4*b^2 + 3*B^4*a^2*b^4 + 3*B^4*a^4*b^2 + 6*A^2*B^2*a^2*b^4 + 6*A^2*B^2*a^4*b^2))^(1/
2) - A^2*a^3*d^2 + B^2*a^3*d^2 - 2*A*B*b^3*d^2 + 3*A^2*a*b^2*d^2 - 3*B^2*a*b^2*d^2 + 6*A*B*a^2*b*d^2)/(4*d^4))
^(1/2) + (((((8*(80*A*a*b^11*d^4 + 80*A*a^3*b^9*d^4 + 48*B*a^2*b^10*d^4 + 48*B*a^4*b^8*d^4))/d^5 + (16*(32*b^1
0*d^4 + 48*a^2*b^8*d^4)*(a + b*tan(c + d*x))^(1/2)*((((8*A^2*a^3*d^2 - 8*B^2*a^3*d^2 + 16*A*B*b^3*d^2 - 24*A^2
*a*b^2*d^2 + 24*B^2*a*b^2*d^2 - 48*A*B*a^2*b*d^2)^2/64 - d^4*(A^4*a^6 + A^4*b^6 + B^4*a^6 + B^4*b^6 + 2*A^2*B^
2*a^6 + 2*A^2*B^2*b^6 + 3*A^4*a^2*b^4 + 3*A^4*a^4*b^2 + 3*B^4*a^2*b^4 + 3*B^4*a^4*b^2 + 6*A^2*B^2*a^2*b^4 + 6*
A^2*B^2*a^4*b^2))^(1/2) - A^2*a^3*d^2 + B^2*a^3*d^2 - 2*A*B*b^3*d^2 + 3*A^2*a*b^2*d^2 - 3*B^2*a*b^2*d^2 + 6*A*
B*a^2*b*d^2)/(4*d^4))^(1/2))/d^4)*((((8*A^2*a^3*d^2 - 8*B^2*a^3*d^2 + 16*A*B*b^3*d^2 - 24*A^2*a*b^2*d^2 + 24*B
^2*a*b^2*d^2 - 48*A*B*a^2*b*d^2)^2/64 - d^4*(A^4*a^6 + A^4*b^6 + B^4*a^6 + B^4*b^6 + 2*A^2*B^2*a^6 + 2*A^2*B^2
*b^6 + 3*A^4*a^2*b^4 + 3*A^4*a^4*b^2 + 3*B^4*a^2*b^4 + 3*B^4*a^4*b^2 + 6*A^2*B^2*a^2*b^4 + 6*A^2*B^2*a^4*b^2))
^(1/2) - A^2*a^3*d^2 + B^2*a^3*d^2 - 2*A*B*b^3*d^2 + 3*A^2*a*b^2*d^2 - 3*B^2*a*b^2*d^2 + 6*A*B*a^2*b*d^2)/(4*d
^4))^(1/2) - (16*(a + b*tan(c + d*x))^(1/2)*(92*A^2*a^3*b^10*d^2 - 20*A^2*a^5*b^8*d^2 - 56*B^2*a^3*b^10*d^2 +
36*B^2*a^5*b^8*d^2 - 32*A*B*b^13*d^2 + 44*A^2*a*b^12*d^2 - 44*B^2*a*b^12*d^2 + 32*A*B*a^2*b^11*d^2 + 176*A*B*a
^4*b^9*d^2))/d^4)*((((8*A^2*a^3*d^2 - 8*B^2*a^3*d^2 + 16*A*B*b^3*d^2 - 24*A^2*a*b^2*d^2 + 24*B^2*a*b^2*d^2 - 4
8*A*B*a^2*b*d^2)^2/64 - d^4*(A^4*a^6 + A^4*b^6 + B^4*a^6 + B^4*b^6 + 2*A^2*B^2*a^6 + 2*A^2*B^2*b^6 + 3*A^4*a^2
*b^4 + 3*A^4*a^4*b^2 + 3*B^4*a^2*b^4 + 3*B^4*a^4*b^2 + 6*A^2*B^2*a^2*b^4 + 6*A^2*B^2*a^4*b^2))^(1/2) - A^2*a^3
*d^2 + B^2*a^3*d^2 - 2*A*B*b^3*d^2 + 3*A^2*a*b^2*d^2 - 3*B^2*a*b^2*d^2 + 6*A*B*a^2*b*d^2)/(4*d^4))^(1/2) - (8*
(100*A^3*a^2*b^13*d^2 + 44*A^3*a^4*b^11*d^2 - 56*A^3*a^6*b^9*d^2 - 92*B^3*a^3*b^12*d^2 - 84*B^3*a^5*b^10*d^2 +
12*B^3*a^7*b^8*d^2 + 4*B^3*a*b^14*d^2 - 92*A^2*B*a*b^14*d^2 - 216*A*B^2*a^2*b^13*d^2 - 48*A*B^2*a^4*b^11*d^2
+ 168*A*B^2*a^6*b^9*d^2 + 208*A^2*B*a^3*b^12*d^2 + 264*A^2*B*a^5*b^10*d^2 - 36*A^2*B*a^7*b^8*d^2))/d^5)*((((8*
A^2*a^3*d^2 - 8*B^2*a^3*d^2 + 16*A*B*b^3*d^2 - 24*A^2*a*b^2*d^2 + 24*B^2*a*b^2*d^2 - 48*A*B*a^2*b*d^2)^2/64 -
d^4*(A^4*a^6 + A^4*b^6 + B^4*a^6 + B^4*b^6 + 2*A^2*B^2*a^6 + 2*A^2*B^2*b^6 + 3*A^4*a^2*b^4 + 3*A^4*a^4*b^2 + 3
*B^4*a^2*b^4 + 3*B^4*a^4*b^2 + 6*A^2*B^2*a^2*b^4 + 6*A^2*B^2*a^4*b^2))^(1/2) - A^2*a^3*d^2 + B^2*a^3*d^2 - 2*A
*B*b^3*d^2 + 3*A^2*a*b^2*d^2 - 3*B^2*a*b^2*d^2 + 6*A*B*a^2*b*d^2)/(4*d^4))^(1/2) + (16*(a + b*tan(c + d*x))^(1
/2)*(2*A^4*b^16 + 2*B^4*b^16 + 4*A^2*B^2*b^16 - A^4*a^2*b^14 + 66*A^4*a^4*b^12 - A^4*a^6*b^10 + 2*A^4*a^8*b^8
+ 8*B^4*a^2*b^14 + 16*B^4*a^4*b^12 - 16*B^4*a^6*b^10 + 6*B^4*a^8*b^8 + 25*A^2*B^2*a^2*b^14 - 130*A^2*B^2*a^4*b
^12 + 145*A^2*B^2*a^6*b^10 + 12*A*B^3*a^3*b^13 - 104*A*B^3*a^5*b^11 + 44*A*B^3*a^7*b^9 - 84*A^3*B*a^3*b^13 + 1
44*A^3*B*a^5*b^11 - 12*A^3*B*a^7*b^9))/d^4)*((((8*A^2*a^3*d^2 - 8*B^2*a^3*d^2 + 16*A*B*b^3*d^2 - 24*A^2*a*b^2*
d^2 + 24*B^2*a*b^2*d^2 - 48*A*B*a^2*b*d^2)^2/64 - d^4*(A^4*a^6 + A^4*b^6 + B^4*a^6 + B^4*b^6 + 2*A^2*B^2*a^6 +
2*A^2*B^2*b^6 + 3*A^4*a^2*b^4 + 3*A^4*a^4*b^2 + 3*B^4*a^2*b^4 + 3*B^4*a^4*b^2 + 6*A^2*B^2*a^2*b^4 + 6*A^2*B^2
*a^4*b^2))^(1/2) - A^2*a^3*d^2 + B^2*a^3*d^2 - 2*A*B*b^3*d^2 + 3*A^2*a*b^2*d^2 - 3*B^2*a*b^2*d^2 + 6*A*B*a^2*b
*d^2)/(4*d^4))^(1/2)))*((((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(((((((8*(80*A*a*b^11*d^4 + 80*A*a^3*b^9*d^4 + 48*B*a^2*b^10*d^4 + 48*B*a^4*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*A^2*a^3*d^2 - 8*B^2*a^3*d^2 + 16*A*B*b^3*d^2 - 24*
A^2*a*b^2*d^2 + 24*B^2*a*b^2*d^2 - 48*A*B*a^2*b*d^2)^2/64 - d^4*(A^4*a^6 + A^4*b^6 + B^4*a^6 + B^4*b^6 + 2*A^2
*B^2*a^6 + 2*A^2*B^2*b^6 + 3*A^4*a^2*b^4 + 3*A^4*a^4*b^2 + 3*B^4*a^2*b^4 + 3*B^4*a^4*b^2 + 6*A^2*B^2*a^2*b^4 +
6*A^2*B^2*a^4*b^2))^(1/2) + A^2*a^3*d^2 - B^2*a^3*d^2 + 2*A*B*b^3*d^2 - 3*A^2*a*b^2*d^2 + 3*B^2*a*b^2*d^2 - 6
*A*B*a^2*b*d^2)/(4*d^4))^(1/2))/d^4)*(-(((8*A^2*a^3*d^2 - 8*B^2*a^3*d^2 + 16*A*B*b^3*d^2 - 24*A^2*a*b^2*d^2 +
24*B^2*a*b^2*d^2 - 48*A*B*a^2*b*d^2)^2/64 - d^4*(A^4*a^6 + A^4*b^6 + B^4*a^6 + B^4*b^6 + 2*A^2*B^2*a^6 + 2*A^2
*B^2*b^6 + 3*A^4*a^2*b^4 + 3*A^4*a^4*b^2 + 3*B^4*a^2*b^4 + 3*B^4*a^4*b^2 + 6*A^2*B^2*a^2*b^4 + 6*A^2*B^2*a^4*b
^2))^(1/2) + A^2*a^3*d^2 - B^2*a^3*d^2 + 2*A*B*b^3*d^2 - 3*A^2*a*b^2*d^2 + 3*B^2*a*b^2*d^2 - 6*A*B*a^2*b*d^2)/
(4*d^4))^(1/2) + (16*(a + b*tan(c + d*x))^(1/2)*(92*A^2*a^3*b^10*d^2 - 20*A^2*a^5*b^8*d^2 - 56*B^2*a^3*b^10*d^
2 + 36*B^2*a^5*b^8*d^2 - 32*A*B*b^13*d^2 + 44*A^2*a*b^12*d^2 - 44*B^2*a*b^12*d^2 + 32*A*B*a^2*b^11*d^2 + 176*A
*B*a^4*b^9*d^2))/d^4)*(-(((8*A^2*a^3*d^2 - 8*B^2*a^3*d^2 + 16*A*B*b^3*d^2 - 24*A^2*a*b^2*d^2 + 24*B^2*a*b^2*d^
2 - 48*A*B*a^2*b*d^2)^2/64 - d^4*(A^4*a^6 + A^4*b^6 + B^4*a^6 + B^4*b^6 + 2*A^2*B^2*a^6 + 2*A^2*B^2*b^6 + 3*A^
4*a^2*b^4 + 3*A^4*a^4*b^2 + 3*B^4*a^2*b^4 + 3*B^4*a^4*b^2 + 6*A^2*B^2*a^2*b^4 + 6*A^2*B^2*a^4*b^2))^(1/2) + A^
2*a^3*d^2 - B^2*a^3*d^2 + 2*A*B*b^3*d^2 - 3*A^2*a*b^2*d^2 + 3*B^2*a*b^2*d^2 - 6*A*B*a^2*b*d^2)/(4*d^4))^(1/2)
- (8*(100*A^3*a^2*b^13*d^2 + 44*A^3*a^4*b^11*d^2 - 56*A^3*a^6*b^9*d^2 - 92*B^3*a^3*b^12*d^2 - 84*B^3*a^5*b^10*
d^2 + 12*B^3*a^7*b^8*d^2 + 4*B^3*a*b^14*d^2 - 92*A^2*B*a*b^14*d^2 - 216*A*B^2*a^2*b^13*d^2 - 48*A*B^2*a^4*b^11
*d^2 + 168*A*B^2*a^6*b^9*d^2 + 208*A^2*B*a^3*b^12*d^2 + 264*A^2*B*a^5*b^10*d^2 - 36*A^2*B*a^7*b^8*d^2))/d^5)*(
-(((8*A^2*a^3*d^2 - 8*B^2*a^3*d^2 + 16*A*B*b^3*d^2 - 24*A^2*a*b^2*d^2 + 24*B^2*a*b^2*d^2 - 48*A*B*a^2*b*d^2)^2
/64 - d^4*(A^4*a^6 + A^4*b^6 + B^4*a^6 + B^4*b^6 + 2*A^2*B^2*a^6 + 2*A^2*B^2*b^6 + 3*A^4*a^2*b^4 + 3*A^4*a^4*b
^2 + 3*B^4*a^2*b^4 + 3*B^4*a^4*b^2 + 6*A^2*B^2*a^2*b^4 + 6*A^2*B^2*a^4*b^2))^(1/2) + A^2*a^3*d^2 - B^2*a^3*d^2
+ 2*A*B*b^3*d^2 - 3*A^2*a*b^2*d^2 + 3*B^2*a*b^2*d^2 - 6*A*B*a^2*b*d^2)/(4*d^4))^(1/2) - (16*(a + b*tan(c + d*
x))^(1/2)*(2*A^4*b^16 + 2*B^4*b^16 + 4*A^2*B^2*b^16 - A^4*a^2*b^14 + 66*A^4*a^4*b^12 - A^4*a^6*b^10 + 2*A^4*a^
8*b^8 + 8*B^4*a^2*b^14 + 16*B^4*a^4*b^12 - 16*B^4*a^6*b^10 + 6*B^4*a^8*b^8 + 25*A^2*B^2*a^2*b^14 - 130*A^2*B^2
*a^4*b^12 + 145*A^2*B^2*a^6*b^10 + 12*A*B^3*a^3*b^13 - 104*A*B^3*a^5*b^11 + 44*A*B^3*a^7*b^9 - 84*A^3*B*a^3*b^
13 + 144*A^3*B*a^5*b^11 - 12*A^3*B*a^7*b^9))/d^4)*(-(((8*A^2*a^3*d^2 - 8*B^2*a^3*d^2 + 16*A*B*b^3*d^2 - 24*A^2
*a*b^2*d^2 + 24*B^2*a*b^2*d^2 - 48*A*B*a^2*b*d^2)^2/64 - d^4*(A^4*a^6 + A^4*b^6 + B^4*a^6 + B^4*b^6 + 2*A^2*B^
2*a^6 + 2*A^2*B^2*b^6 + 3*A^4*a^2*b^4 + 3*A^4*a^4*b^2 + 3*B^4*a^2*b^4 + 3*B^4*a^4*b^2 + 6*A^2*B^2*a^2*b^4 + 6*
A^2*B^2*a^4*b^2))^(1/2) + A^2*a^3*d^2 - B^2*a^3*d^2 + 2*A*B*b^3*d^2 - 3*A^2*a*b^2*d^2 + 3*B^2*a*b^2*d^2 - 6*A*
B*a^2*b*d^2)/(4*d^4))^(1/2)*1i - (((((8*(80*A*a*b^11*d^4 + 80*A*a^3*b^9*d^4 + 48*B*a^2*b^10*d^4 + 48*B*a^4*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*A^2*a^3*d^2 - 8*B^2*a^3*d^2
+ 16*A*B*b^3*d^2 - 24*A^2*a*b^2*d^2 + 24*B^2*a*b^2*d^2 - 48*A*B*a^2*b*d^2)^2/64 - d^4*(A^4*a^6 + A^4*b^6 + B^4
*a^6 + B^4*b^6 + 2*A^2*B^2*a^6 + 2*A^2*B^2*b^6 + 3*A^4*a^2*b^4 + 3*A^4*a^4*b^2 + 3*B^4*a^2*b^4 + 3*B^4*a^4*b^2
+ 6*A^2*B^2*a^2*b^4 + 6*A^2*B^2*a^4*b^2))^(1/2) + A^2*a^3*d^2 - B^2*a^3*d^2 + 2*A*B*b^3*d^2 - 3*A^2*a*b^2*d^2
+ 3*B^2*a*b^2*d^2 - 6*A*B*a^2*b*d^2)/(4*d^4))^(1/2))/d^4)*(-(((8*A^2*a^3*d^2 - 8*B^2*a^3*d^2 + 16*A*B*b^3*d^2
- 24*A^2*a*b^2*d^2 + 24*B^2*a*b^2*d^2 - 48*A*B*a^2*b*d^2)^2/64 - d^4*(A^4*a^6 + A^4*b^6 + B^4*a^6 + B^4*b^6 +
2*A^2*B^2*a^6 + 2*A^2*B^2*b^6 + 3*A^4*a^2*b^4 + 3*A^4*a^4*b^2 + 3*B^4*a^2*b^4 + 3*B^4*a^4*b^2 + 6*A^2*B^2*a^2
*b^4 + 6*A^2*B^2*a^4*b^2))^(1/2) + A^2*a^3*d^2 - B^2*a^3*d^2 + 2*A*B*b^3*d^2 - 3*A^2*a*b^2*d^2 + 3*B^2*a*b^2*d
^2 - 6*A*B*a^2*b*d^2)/(4*d^4))^(1/2) - (16*(a + b*tan(c + d*x))^(1/2)*(92*A^2*a^3*b^10*d^2 - 20*A^2*a^5*b^8*d^
2 - 56*B^2*a^3*b^10*d^2 + 36*B^2*a^5*b^8*d^2 - 32*A*B*b^13*d^2 + 44*A^2*a*b^12*d^2 - 44*B^2*a*b^12*d^2 + 32*A*
B*a^2*b^11*d^2 + 176*A*B*a^4*b^9*d^2))/d^4)*(-(((8*A^2*a^3*d^2 - 8*B^2*a^3*d^2 + 16*A*B*b^3*d^2 - 24*A^2*a*b^2
*d^2 + 24*B^2*a*b^2*d^2 - 48*A*B*a^2*b*d^2)^2/64 - d^4*(A^4*a^6 + A^4*b^6 + B^4*a^6 + B^4*b^6 + 2*A^2*B^2*a^6
+ 2*A^2*B^2*b^6 + 3*A^4*a^2*b^4 + 3*A^4*a^4*b^2 + 3*B^4*a^2*b^4 + 3*B^4*a^4*b^2 + 6*A^2*B^2*a^2*b^4 + 6*A^2*B^
2*a^4*b^2))^(1/2) + A^2*a^3*d^2 - B^2*a^3*d^2 + 2*A*B*b^3*d^2 - 3*A^2*a*b^2*d^2 + 3*B^2*a*b^2*d^2 - 6*A*B*a^2*
b*d^2)/(4*d^4))^(1/2) - (8*(100*A^3*a^2*b^13*d^2 + 44*A^3*a^4*b^11*d^2 - 56*A^3*a^6*b^9*d^2 - 92*B^3*a^3*b^12*
d^2 - 84*B^3*a^5*b^10*d^2 + 12*B^3*a^7*b^8*d^2 + 4*B^3*a*b^14*d^2 - 92*A^2*B*a*b^14*d^2 - 216*A*B^2*a^2*b^13*d
^2 - 48*A*B^2*a^4*b^11*d^2 + 168*A*B^2*a^6*b^9*d^2 + 208*A^2*B*a^3*b^12*d^2 + 264*A^2*B*a^5*b^10*d^2 - 36*A^2*
B*a^7*b^8*d^2))/d^5)*(-(((8*A^2*a^3*d^2 - 8*B^2*a^3*d^2 + 16*A*B*b^3*d^2 - 24*A^2*a*b^2*d^2 + 24*B^2*a*b^2*d^2
- 48*A*B*a^2*b*d^2)^2/64 - d^4*(A^4*a^6 + A^4*b^6 + B^4*a^6 + B^4*b^6 + 2*A^2*B^2*a^6 + 2*A^2*B^2*b^6 + 3*A^4
*a^2*b^4 + 3*A^4*a^4*b^2 + 3*B^4*a^2*b^4 + 3*B^4*a^4*b^2 + 6*A^2*B^2*a^2*b^4 + 6*A^2*B^2*a^4*b^2))^(1/2) + A^2
*a^3*d^2 - B^2*a^3*d^2 + 2*A*B*b^3*d^2 - 3*A^2*a*b^2*d^2 + 3*B^2*a*b^2*d^2 - 6*A*B*a^2*b*d^2)/(4*d^4))^(1/2) +
(16*(a + b*tan(c + d*x))^(1/2)*(2*A^4*b^16 + 2*B^4*b^16 + 4*A^2*B^2*b^16 - A^4*a^2*b^14 + 66*A^4*a^4*b^12 - A
^4*a^6*b^10 + 2*A^4*a^8*b^8 + 8*B^4*a^2*b^14 + 16*B^4*a^4*b^12 - 16*B^4*a^6*b^10 + 6*B^4*a^8*b^8 + 25*A^2*B^2*
a^2*b^14 - 130*A^2*B^2*a^4*b^12 + 145*A^2*B^2*a^6*b^10 + 12*A*B^3*a^3*b^13 - 104*A*B^3*a^5*b^11 + 44*A*B^3*a^7
*b^9 - 84*A^3*B*a^3*b^13 + 144*A^3*B*a^5*b^11 - 12*A^3*B*a^7*b^9))/d^4)*(-(((8*A^2*a^3*d^2 - 8*B^2*a^3*d^2 + 1
6*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)/((16*(6*A^5*a*b^17 + 6*A^5*a^3*b^15 + 6*A^5*a^7*b^11 + 6
*A^5*a^9*b^9 + 4*B^5*a^2*b^16 + 20*B^5*a^4*b^14 + 28*B^5*a^6*b^12 + 12*B^5*a^8*b^10 + 17*A^2*B^3*a^2*b^16 + 21
*A^2*B^3*a^4*b^14 - 5*A^2*B^3*a^6*b^12 - 5*A^2*B^3*a^8*b^10 + 4*A^2*B^3*a^10*b^8 + 42*A^3*B^2*a^3*b^15 + 40*A^
3*B^2*a^5*b^13 + 2*A^3*B^2*a^7*b^11 - 8*A^3*B^2*a^9*b^9 + 6*A*B^4*a*b^17 + 36*A*B^4*a^3*b^15 + 40*A*B^4*a^5*b^
13 - 4*A*B^4*a^7*b^11 - 14*A*B^4*a^9*b^9 + 12*A^3*B^2*a*b^17 + 13*A^4*B*a^2*b^16 + A^4*B*a^4*b^14 - 33*A^4*B*a
^6*b^12 - 17*A^4*B*a^8*b^10 + 4*A^4*B*a^10*b^8))/d^5 + (((((8*(80*A*a*b^11*d^4 + 80*A*a^3*b^9*d^4 + 48*B*a^2*b
^10*d^4 + 48*B*a^4*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*A^2*a^
3*d^2 - 8*B^2*a^3*d^2 + 16*A*B*b^3*d^2 - 24*A^2*a*b^2*d^2 + 24*B^2*a*b^2*d^2 - 48*A*B*a^2*b*d^2)^2/64 - d^4*(A
^4*a^6 + A^4*b^6 + B^4*a^6 + B^4*b^6 + 2*A^2*B^2*a^6 + 2*A^2*B^2*b^6 + 3*A^4*a^2*b^4 + 3*A^4*a^4*b^2 + 3*B^4*a
^2*b^4 + 3*B^4*a^4*b^2 + 6*A^2*B^2*a^2*b^4 + 6*A^2*B^2*a^4*b^2))^(1/2) + A^2*a^3*d^2 - B^2*a^3*d^2 + 2*A*B*b^3
*d^2 - 3*A^2*a*b^2*d^2 + 3*B^2*a*b^2*d^2 - 6*A*B*a^2*b*d^2)/(4*d^4))^(1/2))/d^4)*(-(((8*A^2*a^3*d^2 - 8*B^2*a^
3*d^2 + 16*A*B*b^3*d^2 - 24*A^2*a*b^2*d^2 + 24*B^2*a*b^2*d^2 - 48*A*B*a^2*b*d^2)^2/64 - d^4*(A^4*a^6 + A^4*b^6
+ B^4*a^6 + B^4*b^6 + 2*A^2*B^2*a^6 + 2*A^2*B^2*b^6 + 3*A^4*a^2*b^4 + 3*A^4*a^4*b^2 + 3*B^4*a^2*b^4 + 3*B^4*a
^4*b^2 + 6*A^2*B^2*a^2*b^4 + 6*A^2*B^2*a^4*b^2))^(1/2) + A^2*a^3*d^2 - B^2*a^3*d^2 + 2*A*B*b^3*d^2 - 3*A^2*a*b
^2*d^2 + 3*B^2*a*b^2*d^2 - 6*A*B*a^2*b*d^2)/(4*d^4))^(1/2) + (16*(a + b*tan(c + d*x))^(1/2)*(92*A^2*a^3*b^10*d
^2 - 20*A^2*a^5*b^8*d^2 - 56*B^2*a^3*b^10*d^2 + 36*B^2*a^5*b^8*d^2 - 32*A*B*b^13*d^2 + 44*A^2*a*b^12*d^2 - 44*
B^2*a*b^12*d^2 + 32*A*B*a^2*b^11*d^2 + 176*A*B*a^4*b^9*d^2))/d^4)*(-(((8*A^2*a^3*d^2 - 8*B^2*a^3*d^2 + 16*A*B*
b^3*d^2 - 24*A^2*a*b^2*d^2 + 24*B^2*a*b^2*d^2 - 48*A*B*a^2*b*d^2)^2/64 - d^4*(A^4*a^6 + A^4*b^6 + B^4*a^6 + B^
4*b^6 + 2*A^2*B^2*a^6 + 2*A^2*B^2*b^6 + 3*A^4*a^2*b^4 + 3*A^4*a^4*b^2 + 3*B^4*a^2*b^4 + 3*B^4*a^4*b^2 + 6*A^2*
B^2*a^2*b^4 + 6*A^2*B^2*a^4*b^2))^(1/2) + A^2*a^3*d^2 - B^2*a^3*d^2 + 2*A*B*b^3*d^2 - 3*A^2*a*b^2*d^2 + 3*B^2*
a*b^2*d^2 - 6*A*B*a^2*b*d^2)/(4*d^4))^(1/2) - (8*(100*A^3*a^2*b^13*d^2 + 44*A^3*a^4*b^11*d^2 - 56*A^3*a^6*b^9*
d^2 - 92*B^3*a^3*b^12*d^2 - 84*B^3*a^5*b^10*d^2 + 12*B^3*a^7*b^8*d^2 + 4*B^3*a*b^14*d^2 - 92*A^2*B*a*b^14*d^2
- 216*A*B^2*a^2*b^13*d^2 - 48*A*B^2*a^4*b^11*d^2 + 168*A*B^2*a^6*b^9*d^2 + 208*A^2*B*a^3*b^12*d^2 + 264*A^2*B*
a^5*b^10*d^2 - 36*A^2*B*a^7*b^8*d^2))/d^5)*(-(((8*A^2*a^3*d^2 - 8*B^2*a^3*d^2 + 16*A*B*b^3*d^2 - 24*A^2*a*b^2*
d^2 + 24*B^2*a*b^2*d^2 - 48*A*B*a^2*b*d^2)^2/64 - d^4*(A^4*a^6 + A^4*b^6 + B^4*a^6 + B^4*b^6 + 2*A^2*B^2*a^6 +
2*A^2*B^2*b^6 + 3*A^4*a^2*b^4 + 3*A^4*a^4*b^2 + 3*B^4*a^2*b^4 + 3*B^4*a^4*b^2 + 6*A^2*B^2*a^2*b^4 + 6*A^2*B^2
*a^4*b^2))^(1/2) + A^2*a^3*d^2 - B^2*a^3*d^2 + 2*A*B*b^3*d^2 - 3*A^2*a*b^2*d^2 + 3*B^2*a*b^2*d^2 - 6*A*B*a^2*b
*d^2)/(4*d^4))^(1/2) - (16*(a + b*tan(c + d*x))^(1/2)*(2*A^4*b^16 + 2*B^4*b^16 + 4*A^2*B^2*b^16 - A^4*a^2*b^14
+ 66*A^4*a^4*b^12 - A^4*a^6*b^10 + 2*A^4*a^8*b^8 + 8*B^4*a^2*b^14 + 16*B^4*a^4*b^12 - 16*B^4*a^6*b^10 + 6*B^4
*a^8*b^8 + 25*A^2*B^2*a^2*b^14 - 130*A^2*B^2*a^4*b^12 + 145*A^2*B^2*a^6*b^10 + 12*A*B^3*a^3*b^13 - 104*A*B^3*a
^5*b^11 + 44*A*B^3*a^7*b^9 - 84*A^3*B*a^3*b^13 + 144*A^3*B*a^5*b^11 - 12*A^3*B*a^7*b^9))/d^4)*(-(((8*A^2*a^3*d
^2 - 8*B^2*a^3*d^2 + 16*A*B*b^3*d^2 - 24*A^2*a*b^2*d^2 + 24*B^2*a*b^2*d^2 - 48*A*B*a^2*b*d^2)^2/64 - d^4*(A^4*
a^6 + A^4*b^6 + B^4*a^6 + B^4*b^6 + 2*A^2*B^2*a^6 + 2*A^2*B^2*b^6 + 3*A^4*a^2*b^4 + 3*A^4*a^4*b^2 + 3*B^4*a^2*
b^4 + 3*B^4*a^4*b^2 + 6*A^2*B^2*a^2*b^4 + 6*A^2*B^2*a^4*b^2))^(1/2) + A^2*a^3*d^2 - B^2*a^3*d^2 + 2*A*B*b^3*d^
2 - 3*A^2*a*b^2*d^2 + 3*B^2*a*b^2*d^2 - 6*A*B*a^2*b*d^2)/(4*d^4))^(1/2) + (((((8*(80*A*a*b^11*d^4 + 80*A*a^3*b
^9*d^4 + 48*B*a^2*b^10*d^4 + 48*B*a^4*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*A^2*a^3*d^2 - 8*B^2*a^3*d^2 + 16*A*B*b^3*d^2 - 24*A^2*a*b^2*d^2 + 24*B^2*a*b^2*d^2 - 48*A*B*a^2*b
*d^2)^2/64 - d^4*(A^4*a^6 + A^4*b^6 + B^4*a^6 + B^4*b^6 + 2*A^2*B^2*a^6 + 2*A^2*B^2*b^6 + 3*A^4*a^2*b^4 + 3*A^
4*a^4*b^2 + 3*B^4*a^2*b^4 + 3*B^4*a^4*b^2 + 6*A^2*B^2*a^2*b^4 + 6*A^2*B^2*a^4*b^2))^(1/2) + A^2*a^3*d^2 - B^2*
a^3*d^2 + 2*A*B*b^3*d^2 - 3*A^2*a*b^2*d^2 + 3*B^2*a*b^2*d^2 - 6*A*B*a^2*b*d^2)/(4*d^4))^(1/2))/d^4)*(-(((8*A^2
*a^3*d^2 - 8*B^2*a^3*d^2 + 16*A*B*b^3*d^2 - 24*A^2*a*b^2*d^2 + 24*B^2*a*b^2*d^2 - 48*A*B*a^2*b*d^2)^2/64 - d^4
*(A^4*a^6 + A^4*b^6 + B^4*a^6 + B^4*b^6 + 2*A^2*B^2*a^6 + 2*A^2*B^2*b^6 + 3*A^4*a^2*b^4 + 3*A^4*a^4*b^2 + 3*B^
4*a^2*b^4 + 3*B^4*a^4*b^2 + 6*A^2*B^2*a^2*b^4 + 6*A^2*B^2*a^4*b^2))^(1/2) + A^2*a^3*d^2 - B^2*a^3*d^2 + 2*A*B*
b^3*d^2 - 3*A^2*a*b^2*d^2 + 3*B^2*a*b^2*d^2 - 6*A*B*a^2*b*d^2)/(4*d^4))^(1/2) - (16*(a + b*tan(c + d*x))^(1/2)
*(92*A^2*a^3*b^10*d^2 - 20*A^2*a^5*b^8*d^2 - 56*B^2*a^3*b^10*d^2 + 36*B^2*a^5*b^8*d^2 - 32*A*B*b^13*d^2 + 44*A
^2*a*b^12*d^2 - 44*B^2*a*b^12*d^2 + 32*A*B*a^2*b^11*d^2 + 176*A*B*a^4*b^9*d^2))/d^4)*(-(((8*A^2*a^3*d^2 - 8*B^
2*a^3*d^2 + 16*A*B*b^3*d^2 - 24*A^2*a*b^2*d^2 + 24*B^2*a*b^2*d^2 - 48*A*B*a^2*b*d^2)^2/64 - d^4*(A^4*a^6 + A^4
*b^6 + B^4*a^6 + B^4*b^6 + 2*A^2*B^2*a^6 + 2*A^2*B^2*b^6 + 3*A^4*a^2*b^4 + 3*A^4*a^4*b^2 + 3*B^4*a^2*b^4 + 3*B
^4*a^4*b^2 + 6*A^2*B^2*a^2*b^4 + 6*A^2*B^2*a^4*b^2))^(1/2) + A^2*a^3*d^2 - B^2*a^3*d^2 + 2*A*B*b^3*d^2 - 3*A^2
*a*b^2*d^2 + 3*B^2*a*b^2*d^2 - 6*A*B*a^2*b*d^2)/(4*d^4))^(1/2) - (8*(100*A^3*a^2*b^13*d^2 + 44*A^3*a^4*b^11*d^
2 - 56*A^3*a^6*b^9*d^2 - 92*B^3*a^3*b^12*d^2 - 84*B^3*a^5*b^10*d^2 + 12*B^3*a^7*b^8*d^2 + 4*B^3*a*b^14*d^2 - 9
2*A^2*B*a*b^14*d^2 - 216*A*B^2*a^2*b^13*d^2 - 48*A*B^2*a^4*b^11*d^2 + 168*A*B^2*a^6*b^9*d^2 + 208*A^2*B*a^3*b^
12*d^2 + 264*A^2*B*a^5*b^10*d^2 - 36*A^2*B*a^7*b^8*d^2))/d^5)*(-(((8*A^2*a^3*d^2 - 8*B^2*a^3*d^2 + 16*A*B*b^3*
d^2 - 24*A^2*a*b^2*d^2 + 24*B^2*a*b^2*d^2 - 48*A*B*a^2*b*d^2)^2/64 - d^4*(A^4*a^6 + A^4*b^6 + B^4*a^6 + B^4*b^
6 + 2*A^2*B^2*a^6 + 2*A^2*B^2*b^6 + 3*A^4*a^2*b^4 + 3*A^4*a^4*b^2 + 3*B^4*a^2*b^4 + 3*B^4*a^4*b^2 + 6*A^2*B^2*
a^2*b^4 + 6*A^2*B^2*a^4*b^2))^(1/2) + A^2*a^3*d^2 - B^2*a^3*d^2 + 2*A*B*b^3*d^2 - 3*A^2*a*b^2*d^2 + 3*B^2*a*b^
2*d^2 - 6*A*B*a^2*b*d^2)/(4*d^4))^(1/2) + (16*(a + b*tan(c + d*x))^(1/2)*(2*A^4*b^16 + 2*B^4*b^16 + 4*A^2*B^2*
b^16 - A^4*a^2*b^14 + 66*A^4*a^4*b^12 - A^4*a^6*b^10 + 2*A^4*a^8*b^8 + 8*B^4*a^2*b^14 + 16*B^4*a^4*b^12 - 16*B
^4*a^6*b^10 + 6*B^4*a^8*b^8 + 25*A^2*B^2*a^2*b^14 - 130*A^2*B^2*a^4*b^12 + 145*A^2*B^2*a^6*b^10 + 12*A*B^3*a^3
*b^13 - 104*A*B^3*a^5*b^11 + 44*A*B^3*a^7*b^9 - 84*A^3*B*a^3*b^13 + 144*A^3*B*a^5*b^11 - 12*A^3*B*a^7*b^9))/d^
4)*(-(((8*A^2*a^3*d^2 - 8*B^2*a^3*d^2 + 16*A*B*b^3*d^2 - 24*A^2*a*b^2*d^2 + 24*B^2*a*b^2*d^2 - 48*A*B*a^2*b*d^
2)^2/64 - d^4*(A^4*a^6 + A^4*b^6 + B^4*a^6 + B^4*b^6 + 2*A^2*B^2*a^6 + 2*A^2*B^2*b^6 + 3*A^4*a^2*b^4 + 3*A^4*a
^4*b^2 + 3*B^4*a^2*b^4 + 3*B^4*a^4*b^2 + 6*A^2*B^2*a^2*b^4 + 6*A^2*B^2*a^4*b^2))^(1/2) + A^2*a^3*d^2 - B^2*a^3
*d^2 + 2*A*B*b^3*d^2 - 3*A^2*a*b^2*d^2 + 3*B^2*a*b^2*d^2 - 6*A*B*a^2*b*d^2)/(4*d^4))^(1/2)))*(-(((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*a*b*(a + b*tan(c + d*x))^(1/2))
/(a*d - d*(a + b*tan(c + d*x)))