\(\int \cot (c+d x) (a+b \tan (c+d x))^{5/2} (A+B \tan (c+d x)) \, dx\) [335]
Optimal result
Integrand size = 31, antiderivative size = 182 \[
\int \cot (c+d x) (a+b \tan (c+d x))^{5/2} (A+B \tan (c+d x)) \, dx=-\frac {2 a^{5/2} A \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a}}\right )}{d}+\frac {(a-i b)^{5/2} (A-i B) \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a-i b}}\right )}{d}+\frac {(a+i b)^{5/2} (A+i B) \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a+i b}}\right )}{d}+\frac {2 b (A b+2 a B) \sqrt {a+b \tan (c+d x)}}{d}+\frac {2 b B (a+b \tan (c+d x))^{3/2}}{3 d}
\]
[Out]
-2*a^(5/2)*A*arctanh((a+b*tan(d*x+c))^(1/2)/a^(1/2))/d+(a-I*b)^(5/2)*(A-I*B)*arctanh((a+b*tan(d*x+c))^(1/2)/(a
-I*b)^(1/2))/d+(a+I*b)^(5/2)*(A+I*B)*arctanh((a+b*tan(d*x+c))^(1/2)/(a+I*b)^(1/2))/d+2*b*(A*b+2*B*a)*(a+b*tan(
d*x+c))^(1/2)/d+2/3*b*B*(a+b*tan(d*x+c))^(3/2)/d
Rubi [A] (verified)
Time = 1.01 (sec) , antiderivative size = 182, normalized size of antiderivative = 1.00, number of
steps used = 13, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.258, Rules used = {3688, 3728, 3734, 3620, 3618,
65, 214, 3715} \[
\int \cot (c+d x) (a+b \tan (c+d x))^{5/2} (A+B \tan (c+d x)) \, dx=-\frac {2 a^{5/2} A \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a}}\right )}{d}+\frac {(a-i b)^{5/2} (A-i B) \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a-i b}}\right )}{d}+\frac {(a+i b)^{5/2} (A+i B) \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a+i b}}\right )}{d}+\frac {2 b (2 a B+A b) \sqrt {a+b \tan (c+d x)}}{d}+\frac {2 b B (a+b \tan (c+d x))^{3/2}}{3 d}
\]
[In]
Int[Cot[c + d*x]*(a + b*Tan[c + d*x])^(5/2)*(A + B*Tan[c + d*x]),x]
[Out]
(-2*a^(5/2)*A*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a]])/d + ((a - I*b)^(5/2)*(A - I*B)*ArcTanh[Sqrt[a + b*Tan
[c + d*x]]/Sqrt[a - I*b]])/d + ((a + I*b)^(5/2)*(A + I*B)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a + I*b]])/d +
(2*b*(A*b + 2*a*B)*Sqrt[a + b*Tan[c + d*x]])/d + (2*b*B*(a + b*Tan[c + d*x])^(3/2))/(3*d)
Rule 65
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]
Rule 214
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x/Rt[-a/b, 2]], x] /; FreeQ[{a, b},
x] && NegQ[a/b]
Rule 3618
Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_) + (d_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Dist[c*(
d/f), Subst[Int[(a + (b/d)*x)^m/(d^2 + c*x), x], x, d*Tan[e + f*x]], x] /; FreeQ[{a, b, c, d, e, f, m}, x] &&
NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && EqQ[c^2 + d^2, 0]
Rule 3620
Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Dist[(c
+ I*d)/2, Int[(a + b*Tan[e + f*x])^m*(1 - I*Tan[e + f*x]), x], x] + Dist[(c - I*d)/2, Int[(a + b*Tan[e + f*x]
)^m*(1 + I*Tan[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0]
&& NeQ[c^2 + d^2, 0] && !IntegerQ[m]
Rule 3688
Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*tan[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*tan[(e
_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[b*B*(a + b*Tan[e + f*x])^(m - 1)*((c + d*Tan[e + f*x])^(n + 1)/(d*f
*(m + n))), x] + Dist[1/(d*(m + n)), Int[(a + b*Tan[e + f*x])^(m - 2)*(c + d*Tan[e + f*x])^n*Simp[a^2*A*d*(m +
n) - b*B*(b*c*(m - 1) + a*d*(n + 1)) + d*(m + n)*(2*a*A*b + B*(a^2 - b^2))*Tan[e + f*x] - (b*B*(b*c - a*d)*(m
- 1) - b*(A*b + a*B)*d*(m + n))*Tan[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, n}, x] && NeQ[b*
c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && GtQ[m, 1] && (IntegerQ[m] || IntegersQ[2*m, 2*n]) &&
!(IGtQ[n, 1] && ( !IntegerQ[m] || (EqQ[c, 0] && NeQ[a, 0])))
Rule 3715
Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_.)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_.)*((A_) + (C_.)*
tan[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Dist[A/f, Subst[Int[(a + b*x)^m*(c + d*x)^n, x], x, Tan[e + f*x]], x]
/; FreeQ[{a, b, c, d, e, f, A, C, m, n}, x] && EqQ[A, C]
Rule 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 {2 b B (a+b \tan (c+d x))^{3/2}}{3 d}+\frac {2}{3} \int \cot (c+d x) \sqrt {a+b \tan (c+d x)} \left (\frac {3 a^2 A}{2}+\frac {3}{2} \left (2 a A b+a^2 B-b^2 B\right ) \tan (c+d x)+\frac {3}{2} b (A b+2 a B) \tan ^2(c+d x)\right ) \, dx \\ & = \frac {2 b (A b+2 a B) \sqrt {a+b \tan (c+d x)}}{d}+\frac {2 b B (a+b \tan (c+d x))^{3/2}}{3 d}+\frac {4}{3} \int \frac {\cot (c+d x) \left (\frac {3 a^3 A}{4}+\frac {3}{4} \left (3 a^2 A b-A b^3+a^3 B-3 a b^2 B\right ) \tan (c+d x)+\frac {3}{4} b \left (3 a A b+3 a^2 B-b^2 B\right ) \tan ^2(c+d x)\right )}{\sqrt {a+b \tan (c+d x)}} \, dx \\ & = \frac {2 b (A b+2 a B) \sqrt {a+b \tan (c+d x)}}{d}+\frac {2 b B (a+b \tan (c+d x))^{3/2}}{3 d}+\frac {4}{3} \int \frac {\frac {3}{4} \left (3 a^2 A b-A b^3+a^3 B-3 a b^2 B\right )-\frac {3}{4} \left (a^3 A-3 a A b^2-3 a^2 b B+b^3 B\right ) \tan (c+d x)}{\sqrt {a+b \tan (c+d x)}} \, dx+\left (a^3 A\right ) \int \frac {\cot (c+d x) \left (1+\tan ^2(c+d x)\right )}{\sqrt {a+b \tan (c+d x)}} \, dx \\ & = \frac {2 b (A b+2 a B) \sqrt {a+b \tan (c+d x)}}{d}+\frac {2 b B (a+b \tan (c+d x))^{3/2}}{3 d}+\frac {1}{2} \left ((a-i b)^3 (i A+B)\right ) \int \frac {1+i \tan (c+d x)}{\sqrt {a+b \tan (c+d x)}} \, dx+\frac {1}{3} \left (2 \left (\frac {3}{4} \left (3 a^2 A b-A b^3+a^3 B-3 a b^2 B\right )-\frac {3}{4} i \left (a^3 A-3 a A b^2-3 a^2 b B+b^3 B\right )\right )\right ) \int \frac {1-i \tan (c+d x)}{\sqrt {a+b \tan (c+d x)}} \, dx+\frac {\left (a^3 A\right ) \text {Subst}\left (\int \frac {1}{x \sqrt {a+b x}} \, dx,x,\tan (c+d x)\right )}{d} \\ & = \frac {2 b (A b+2 a B) \sqrt {a+b \tan (c+d x)}}{d}+\frac {2 b B (a+b \tan (c+d x))^{3/2}}{3 d}+\frac {\left (2 a^3 A\right ) \text {Subst}\left (\int \frac {1}{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b \tan (c+d x)}\right )}{b d}-\frac {\left ((a-i b)^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 (A+i B)\right ) \text {Subst}\left (\int \frac {1}{(-1+x) \sqrt {a+i b x}} \, dx,x,-i \tan (c+d x)\right )}{2 d} \\ & = -\frac {2 a^{5/2} A \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a}}\right )}{d}+\frac {2 b (A b+2 a B) \sqrt {a+b \tan (c+d x)}}{d}+\frac {2 b B (a+b \tan (c+d x))^{3/2}}{3 d}+\frac {\left (i (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 (i A+B)\right ) \text {Subst}\left (\int \frac {1}{-1-\frac {i a}{b}+\frac {i x^2}{b}} \, dx,x,\sqrt {a+b \tan (c+d x)}\right )}{b d} \\ & = -\frac {2 a^{5/2} A \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a}}\right )}{d}+\frac {(a-i b)^{5/2} (A-i B) \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a-i b}}\right )}{d}+\frac {(a+i b)^{5/2} (A+i B) \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a+i b}}\right )}{d}+\frac {2 b (A b+2 a B) \sqrt {a+b \tan (c+d x)}}{d}+\frac {2 b B (a+b \tan (c+d x))^{3/2}}{3 d} \\
\end{align*}
Mathematica [A] (verified)
Time = 1.26 (sec) , antiderivative size = 177, normalized size of antiderivative = 0.97
\[
\int \cot (c+d x) (a+b \tan (c+d x))^{5/2} (A+B \tan (c+d x)) \, dx=\frac {2 \left (-3 a^{5/2} A \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a}}\right )+\frac {3}{2} (a-i b)^{5/2} (A-i B) \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a-i b}}\right )+\frac {3}{2} (a+i b)^{5/2} (A+i B) \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a+i b}}\right )+3 b (A b+2 a B) \sqrt {a+b \tan (c+d x)}+b B (a+b \tan (c+d x))^{3/2}\right )}{3 d}
\]
[In]
Integrate[Cot[c + d*x]*(a + b*Tan[c + d*x])^(5/2)*(A + B*Tan[c + d*x]),x]
[Out]
(2*(-3*a^(5/2)*A*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a]] + (3*(a - I*b)^(5/2)*(A - I*B)*ArcTanh[Sqrt[a + b*T
an[c + d*x]]/Sqrt[a - I*b]])/2 + (3*(a + I*b)^(5/2)*(A + I*B)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a + I*b]])
/2 + 3*b*(A*b + 2*a*B)*Sqrt[a + b*Tan[c + d*x]] + b*B*(a + b*Tan[c + d*x])^(3/2)))/(3*d)
Maple [B] (verified)
Leaf count of result is larger than twice the leaf count of optimal. \(2372\) vs. \(2(152)=304\).
Time = 0.25 (sec) , antiderivative size = 2373, normalized size of antiderivative =
13.04
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| method | result | size |
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| derivativedivides |
\(\text {Expression too large to display}\) |
\(2373\) |
| default |
\(\text {Expression too large to display}\) |
\(2373\) |
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[In]
int(cot(d*x+c)*(a+b*tan(d*x+c))^(5/2)*(A+B*tan(d*x+c)),x,method=_RETURNVERBOSE)
[Out]
2/d*b^2*(a+b*tan(d*x+c))^(1/2)*A-2*a^(5/2)*A*arctanh((a+b*tan(d*x+c))^(1/2)/a^(1/2))/d+2/3*b*B*(a+b*tan(d*x+c)
)^(3/2)/d-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))*B*(a^2+b^2)^(1/2)*a+1/4/d/b*ln((a+b*tan(d*x+c))^(1/2)*(2*(a^2+b^2)^(1/2)+2*a)^(
1/2)-b*tan(d*x+c)-a-(a^2+b^2)^(1/2))*B*(2*(a^2+b^2)^(1/2)+2*a)^(1/2)*(a^2+b^2)^(1/2)*a^2-1/4/d/b*ln(b*tan(d*x+
c)+a+(a+b*tan(d*x+c))^(1/2)*(2*(a^2+b^2)^(1/2)+2*a)^(1/2)+(a^2+b^2)^(1/2))*B*(2*(a^2+b^2)^(1/2)+2*a)^(1/2)*(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))*B*(a^2+b^2)^(1/2)*a-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))*A*a-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))*B*a^
2-1/4/d/b*ln((a+b*tan(d*x+c))^(1/2)*(2*(a^2+b^2)^(1/2)+2*a)^(1/2)-b*tan(d*x+c)-a-(a^2+b^2)^(1/2))*B*(2*(a^2+b^
2)^(1/2)+2*a)^(1/2)*a^3+1/4/d/b*ln(b*tan(d*x+c)+a+(a+b*tan(d*x+c))^(1/2)*(2*(a^2+b^2)^(1/2)+2*a)^(1/2)+(a^2+b^
2)^(1/2))*B*(2*(a^2+b^2)^(1/2)+2*a)^(1/2)*a^3-1/4/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))*A*(2*(a^2+b^2)^(1/2)+2*a)^(1/2)+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))*A*(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))*A*a^3
-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))*A*(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+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^3-1/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))*B+1/d*b^3/(2*(a^2+b^2)^(1/2)-2*a)^(1/2)*a
rctan(((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+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))*A*(2*(a^2+b^2)^(1/2)+2*a)^
(1/2)+4/d*b*(a+b*tan(d*x+c))^(1/2)*B*a-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))*B*(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))*A*(a^2+b^2)^(1/2)+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))*A*a+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))*B*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*ta
n(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-1/d/(2*(a^2+b^2)^(1/2)-2*a)^(1/2
)*arctan(((2*(a^2+b^2)^(1/2)+2*a)^(1/2)-2*(a+b*tan(d*x+c))^(1/2))/(2*(a^2+b^2)^(1/2)-2*a)^(1/2))*A*(a^2+b^2)^(
1/2)*a^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))*A*(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))*A*(a^2+b^2)^(1/2)*a^2-1/d*b^2/(2*(a^2+b^2)^(1/2)
-2*a)^(1/2)*arctan((2*(a+b*tan(d*x+c))^(1/2)+(2*(a^2+b^2)^(1/2)+2*a)^(1/2))/(2*(a^2+b^2)^(1/2)-2*a)^(1/2))*A*(
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))
*B*(2*(a^2+b^2)^(1/2)+2*a)^(1/2)*(a^2+b^2)^(1/2)+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))*B*(2*(a^2+b^2)^(1/2)+2*a)^(1/2)*a-1/4/d*b*ln((a+b*tan(d*x+c))^(1/2)*(2*(a^2
+b^2)^(1/2)+2*a)^(1/2)-b*tan(d*x+c)-a-(a^2+b^2)^(1/2))*B*(2*(a^2+b^2)^(1/2)+2*a)^(1/2)*(a^2+b^2)^(1/2)
Fricas [B] (verification not implemented)
Leaf count of result is larger than twice the leaf count of optimal. 4872 vs. \(2 (146) = 292\).
Time = 10.91 (sec) , antiderivative size = 9757, normalized size of antiderivative = 53.61
\[
\int \cot (c+d x) (a+b \tan (c+d x))^{5/2} (A+B \tan (c+d x)) \, dx=\text {Too large to display}
\]
[In]
integrate(cot(d*x+c)*(a+b*tan(d*x+c))^(5/2)*(A+B*tan(d*x+c)),x, algorithm="fricas")
[Out]
Too large to include
Sympy [F]
\[
\int \cot (c+d x) (a+b \tan (c+d x))^{5/2} (A+B \tan (c+d x)) \, dx=\int \left (A + B \tan {\left (c + d x \right )}\right ) \left (a + b \tan {\left (c + d x \right )}\right )^{\frac {5}{2}} \cot {\left (c + d x \right )}\, dx
\]
[In]
integrate(cot(d*x+c)*(a+b*tan(d*x+c))**(5/2)*(A+B*tan(d*x+c)),x)
[Out]
Integral((A + B*tan(c + d*x))*(a + b*tan(c + d*x))**(5/2)*cot(c + d*x), x)
Maxima [F]
\[
\int \cot (c+d x) (a+b \tan (c+d x))^{5/2} (A+B \tan (c+d x)) \, dx=\int { {\left (B \tan \left (d x + c\right ) + A\right )} {\left (b \tan \left (d x + c\right ) + a\right )}^{\frac {5}{2}} \cot \left (d x + c\right ) \,d x }
\]
[In]
integrate(cot(d*x+c)*(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), x)
Giac [F(-1)]
Timed out. \[
\int \cot (c+d x) (a+b \tan (c+d x))^{5/2} (A+B \tan (c+d x)) \, dx=\text {Timed out}
\]
[In]
integrate(cot(d*x+c)*(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 = 14.54 (sec) , antiderivative size = 29441, normalized size of antiderivative = 161.76
\[
\int \cot (c+d x) (a+b \tan (c+d x))^{5/2} (A+B \tan (c+d x)) \, dx=\text {Too large to display}
\]
[In]
int(cot(c + d*x)*(A + B*tan(c + d*x))*(a + b*tan(c + d*x))^(5/2),x)
[Out]
((2*A*b^2 - 2*B*a*b)/d + (6*B*a*b)/d)*(a + b*tan(c + d*x))^(1/2) - atan(((((32*(3*A^3*a^2*b^16*d^2 + 48*A^3*a^
4*b^14*d^2 - 30*A^3*a^6*b^12*d^2 - 72*A^3*a^8*b^10*d^2 + 3*A^3*a^10*b^8*d^2 + 6*B^3*a^5*b^13*d^2 + 8*B^3*a^7*b
^11*d^2 + 3*B^3*a^9*b^9*d^2 - B^3*a*b^17*d^2 - A^2*B*a*b^17*d^2 + 3*A*B^2*a^2*b^16*d^2 - 32*A*B^2*a^4*b^14*d^2
+ 46*A*B^2*a^6*b^12*d^2 + 72*A*B^2*a^8*b^10*d^2 - 9*A*B^2*a^10*b^8*d^2 - 16*A^2*B*a^3*b^15*d^2 + 150*A^2*B*a^
5*b^13*d^2 + 96*A^2*B*a^7*b^11*d^2 - 69*A^2*B*a^9*b^9*d^2))/d^5 - (((32*(4*A*a*b^12*d^4 + 16*A*a^3*b^10*d^4 +
12*A*a^5*b^8*d^4 + 8*B*a^2*b^11*d^4 + 8*B*a^4*b^9*d^4))/d^5 - (32*(16*b^10*d^4 + 24*a^2*b^8*d^4)*(a + b*tan(c
+ d*x))^(1/2)*((((8*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))/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 + 2
0*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) - (32*(a + b*tan(c + d*x))^(1/2)*(10*A^2*a^3*b^12*d^2 + 102*A^2*a^5*b^10*d^2 - 18*A^
2*a^7*b^8*d^2 - 10*B^2*a^3*b^12*d^2 - 102*B^2*a^5*b^10*d^2 + 10*B^2*a^7*b^8*d^2 + 16*A*B*b^15*d^2 - 38*A^2*a*b
^14*d^2 + 38*B^2*a*b^14*d^2 - 120*A*B*a^2*b^13*d^2 - 160*A*B*a^4*b^11*d^2 + 104*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)
+ (32*(a + b*tan(c + d*x))^(1/2)*(A^4*b^20 + B^4*b^20 + 2*A^2*B^2*b^20 + 6*A^4*a^2*b^18 + 15*A^4*a^4*b^16 + 18
*A^4*a^6*b^14 + 45*A^4*a^8*b^12 - 24*A^4*a^10*b^10 + 3*A^4*a^12*b^8 + 6*B^4*a^2*b^18 + 15*B^4*a^4*b^16 + 20*B^
4*a^6*b^14 + 15*B^4*a^8*b^12 + 6*B^4*a^10*b^10 + B^4*a^12*b^8 + 12*A^2*B^2*a^2*b^18 + 30*A^2*B^2*a^4*b^16 + 42
*A^2*B^2*a^6*b^14 + 42*A^2*B^2*a^10*b^10 - 24*A^3*B*a^7*b^13 + 80*A^3*B*a^9*b^11 - 24*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 - (((32*(3*A^3*a^2*b^16*d^2 + 48*A^3*a^4*b^14*d^2 - 30*A^3*a^6*b^12*d^2 - 72*A^3*a^8*b^10*d^2 + 3*A^3*a^10
*b^8*d^2 + 6*B^3*a^5*b^13*d^2 + 8*B^3*a^7*b^11*d^2 + 3*B^3*a^9*b^9*d^2 - B^3*a*b^17*d^2 - A^2*B*a*b^17*d^2 + 3
*A*B^2*a^2*b^16*d^2 - 32*A*B^2*a^4*b^14*d^2 + 46*A*B^2*a^6*b^12*d^2 + 72*A*B^2*a^8*b^10*d^2 - 9*A*B^2*a^10*b^8
*d^2 - 16*A^2*B*a^3*b^15*d^2 + 150*A^2*B*a^5*b^13*d^2 + 96*A^2*B*a^7*b^11*d^2 - 69*A^2*B*a^9*b^9*d^2))/d^5 - (
((32*(4*A*a*b^12*d^4 + 16*A*a^3*b^10*d^4 + 12*A*a^5*b^8*d^4 + 8*B*a^2*b^11*d^4 + 8*B*a^4*b^9*d^4))/d^5 + (32*(
16*b^10*d^4 + 24*a^2*b^8*d^4)*(a + b*tan(c + d*x))^(1/2)*((((8*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) + (32*(a + b*tan(c + d*x))^(1/2)*(10*A^2*
a^3*b^12*d^2 + 102*A^2*a^5*b^10*d^2 - 18*A^2*a^7*b^8*d^2 - 10*B^2*a^3*b^12*d^2 - 102*B^2*a^5*b^10*d^2 + 10*B^2
*a^7*b^8*d^2 + 16*A*B*b^15*d^2 - 38*A^2*a*b^14*d^2 + 38*B^2*a*b^14*d^2 - 120*A*B*a^2*b^13*d^2 - 160*A*B*a^4*b^
11*d^2 + 104*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) - (32*(a + b*tan(c + d*x))^(1/2)*(A^4*b^20 + B^4*b^20 + 2*A^2*B^2*b
^20 + 6*A^4*a^2*b^18 + 15*A^4*a^4*b^16 + 18*A^4*a^6*b^14 + 45*A^4*a^8*b^12 - 24*A^4*a^10*b^10 + 3*A^4*a^12*b^8
+ 6*B^4*a^2*b^18 + 15*B^4*a^4*b^16 + 20*B^4*a^6*b^14 + 15*B^4*a^8*b^12 + 6*B^4*a^10*b^10 + B^4*a^12*b^8 + 12*
A^2*B^2*a^2*b^18 + 30*A^2*B^2*a^4*b^16 + 42*A^2*B^2*a^6*b^14 + 42*A^2*B^2*a^10*b^10 - 24*A^3*B*a^7*b^13 + 80*A
^3*B*a^9*b^11 - 24*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)/((((32*(3*A^3*a^2*b^16*d^2 + 48*A^3*a^4*b^14*d^2 - 30*A^3*a^6*
b^12*d^2 - 72*A^3*a^8*b^10*d^2 + 3*A^3*a^10*b^8*d^2 + 6*B^3*a^5*b^13*d^2 + 8*B^3*a^7*b^11*d^2 + 3*B^3*a^9*b^9*
d^2 - B^3*a*b^17*d^2 - A^2*B*a*b^17*d^2 + 3*A*B^2*a^2*b^16*d^2 - 32*A*B^2*a^4*b^14*d^2 + 46*A*B^2*a^6*b^12*d^2
+ 72*A*B^2*a^8*b^10*d^2 - 9*A*B^2*a^10*b^8*d^2 - 16*A^2*B*a^3*b^15*d^2 + 150*A^2*B*a^5*b^13*d^2 + 96*A^2*B*a^
7*b^11*d^2 - 69*A^2*B*a^9*b^9*d^2))/d^5 - (((32*(4*A*a*b^12*d^4 + 16*A*a^3*b^10*d^4 + 12*A*a^5*b^8*d^4 + 8*B*a
^2*b^11*d^4 + 8*B*a^4*b^9*d^4))/d^5 - (32*(16*b^10*d^4 + 24*a^2*b^8*d^4)*(a + b*tan(c + d*x))^(1/2)*((((8*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
) - (32*(a + b*tan(c + d*x))^(1/2)*(10*A^2*a^3*b^12*d^2 + 102*A^2*a^5*b^10*d^2 - 18*A^2*a^7*b^8*d^2 - 10*B^2*a
^3*b^12*d^2 - 102*B^2*a^5*b^10*d^2 + 10*B^2*a^7*b^8*d^2 + 16*A*B*b^15*d^2 - 38*A^2*a*b^14*d^2 + 38*B^2*a*b^14*
d^2 - 120*A*B*a^2*b^13*d^2 - 160*A*B*a^4*b^11*d^2 + 104*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) + (32*(a + b*tan(c + d*x
))^(1/2)*(A^4*b^20 + B^4*b^20 + 2*A^2*B^2*b^20 + 6*A^4*a^2*b^18 + 15*A^4*a^4*b^16 + 18*A^4*a^6*b^14 + 45*A^4*a
^8*b^12 - 24*A^4*a^10*b^10 + 3*A^4*a^12*b^8 + 6*B^4*a^2*b^18 + 15*B^4*a^4*b^16 + 20*B^4*a^6*b^14 + 15*B^4*a^8*
b^12 + 6*B^4*a^10*b^10 + B^4*a^12*b^8 + 12*A^2*B^2*a^2*b^18 + 30*A^2*B^2*a^4*b^16 + 42*A^2*B^2*a^6*b^14 + 42*A
^2*B^2*a^10*b^10 - 24*A^3*B*a^7*b^13 + 80*A^3*B*a^9*b^11 - 24*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) - (64*(A^5*a^3*b^20 + 6
*A^5*a^5*b^18 + 18*A^5*a^7*b^16 + 28*A^5*a^9*b^14 + 21*A^5*a^11*b^12 + 6*A^5*a^13*b^10 - A^2*B^3*a^6*b^17 + 6*
A^2*B^3*a^10*b^13 + 8*A^2*B^3*a^12*b^11 + 3*A^2*B^3*a^14*b^9 + 2*A^3*B^2*a^3*b^20 + 12*A^3*B^2*a^5*b^18 + 33*A
^3*B^2*a^7*b^16 + 48*A^3*B^2*a^9*b^14 + 36*A^3*B^2*a^11*b^12 + 12*A^3*B^2*a^13*b^10 + A^3*B^2*a^15*b^8 + A*B^4
*a^3*b^20 + 6*A*B^4*a^5*b^18 + 15*A*B^4*a^7*b^16 + 20*A*B^4*a^9*b^14 + 15*A*B^4*a^11*b^12 + 6*A*B^4*a^13*b^10
+ A*B^4*a^15*b^8 - A^4*B*a^6*b^17 + 6*A^4*B*a^10*b^13 + 8*A^4*B*a^12*b^11 + 3*A^4*B*a^14*b^9))/d^5 + (((32*(3*
A^3*a^2*b^16*d^2 + 48*A^3*a^4*b^14*d^2 - 30*A^3*a^6*b^12*d^2 - 72*A^3*a^8*b^10*d^2 + 3*A^3*a^10*b^8*d^2 + 6*B^
3*a^5*b^13*d^2 + 8*B^3*a^7*b^11*d^2 + 3*B^3*a^9*b^9*d^2 - B^3*a*b^17*d^2 - A^2*B*a*b^17*d^2 + 3*A*B^2*a^2*b^16
*d^2 - 32*A*B^2*a^4*b^14*d^2 + 46*A*B^2*a^6*b^12*d^2 + 72*A*B^2*a^8*b^10*d^2 - 9*A*B^2*a^10*b^8*d^2 - 16*A^2*B
*a^3*b^15*d^2 + 150*A^2*B*a^5*b^13*d^2 + 96*A^2*B*a^7*b^11*d^2 - 69*A^2*B*a^9*b^9*d^2))/d^5 - (((32*(4*A*a*b^1
2*d^4 + 16*A*a^3*b^10*d^4 + 12*A*a^5*b^8*d^4 + 8*B*a^2*b^11*d^4 + 8*B*a^4*b^9*d^4))/d^5 + (32*(16*b^10*d^4 + 2
4*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) + (32*(a + b*tan(c + d*x))^(1/2)*(10*A^2*a^3*b^12*d^2 +
102*A^2*a^5*b^10*d^2 - 18*A^2*a^7*b^8*d^2 - 10*B^2*a^3*b^12*d^2 - 102*B^2*a^5*b^10*d^2 + 10*B^2*a^7*b^8*d^2 +
16*A*B*b^15*d^2 - 38*A^2*a*b^14*d^2 + 38*B^2*a*b^14*d^2 - 120*A*B*a^2*b^13*d^2 - 160*A*B*a^4*b^11*d^2 + 104*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) - (32*(a + b*tan(c + d*x))^(1/2)*(A^4*b^20 + B^4*b^20 + 2*A^2*B^2*b^20 + 6*A^4*a^2
*b^18 + 15*A^4*a^4*b^16 + 18*A^4*a^6*b^14 + 45*A^4*a^8*b^12 - 24*A^4*a^10*b^10 + 3*A^4*a^12*b^8 + 6*B^4*a^2*b^
18 + 15*B^4*a^4*b^16 + 20*B^4*a^6*b^14 + 15*B^4*a^8*b^12 + 6*B^4*a^10*b^10 + B^4*a^12*b^8 + 12*A^2*B^2*a^2*b^1
8 + 30*A^2*B^2*a^4*b^16 + 42*A^2*B^2*a^6*b^14 + 42*A^2*B^2*a^10*b^10 - 24*A^3*B*a^7*b^13 + 80*A^3*B*a^9*b^11 -
24*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*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)*2i - atan(((((32*(3*A^3*a^2*b^16*d^2 + 48*A^3*a^4*b^14*d^2 - 30*A^3*a^6*b^12
*d^2 - 72*A^3*a^8*b^10*d^2 + 3*A^3*a^10*b^8*d^2 + 6*B^3*a^5*b^13*d^2 + 8*B^3*a^7*b^11*d^2 + 3*B^3*a^9*b^9*d^2
- B^3*a*b^17*d^2 - A^2*B*a*b^17*d^2 + 3*A*B^2*a^2*b^16*d^2 - 32*A*B^2*a^4*b^14*d^2 + 46*A*B^2*a^6*b^12*d^2 + 7
2*A*B^2*a^8*b^10*d^2 - 9*A*B^2*a^10*b^8*d^2 - 16*A^2*B*a^3*b^15*d^2 + 150*A^2*B*a^5*b^13*d^2 + 96*A^2*B*a^7*b^
11*d^2 - 69*A^2*B*a^9*b^9*d^2))/d^5 - (((32*(4*A*a*b^12*d^4 + 16*A*a^3*b^10*d^4 + 12*A*a^5*b^8*d^4 + 8*B*a^2*b
^11*d^4 + 8*B*a^4*b^9*d^4))/d^5 - (32*(16*b^10*d^4 + 24*a^2*b^8*d^4)*(a + b*tan(c + d*x))^(1/2)*(-(((8*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)
- (32*(a + b*tan(c + d*x))^(1/2)*(10*A^2*a^3*b^12*d^2 + 102*A^2*a^5*b^10*d^2 - 18*A^2*a^7*b^8*d^2 - 10*B^2*a^3
*b^12*d^2 - 102*B^2*a^5*b^10*d^2 + 10*B^2*a^7*b^8*d^2 + 16*A*B*b^15*d^2 - 38*A^2*a*b^14*d^2 + 38*B^2*a*b^14*d^
2 - 120*A*B*a^2*b^13*d^2 - 160*A*B*a^4*b^11*d^2 + 104*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) + (32*(a + b*tan(c + d*x
))^(1/2)*(A^4*b^20 + B^4*b^20 + 2*A^2*B^2*b^20 + 6*A^4*a^2*b^18 + 15*A^4*a^4*b^16 + 18*A^4*a^6*b^14 + 45*A^4*a
^8*b^12 - 24*A^4*a^10*b^10 + 3*A^4*a^12*b^8 + 6*B^4*a^2*b^18 + 15*B^4*a^4*b^16 + 20*B^4*a^6*b^14 + 15*B^4*a^8*
b^12 + 6*B^4*a^10*b^10 + B^4*a^12*b^8 + 12*A^2*B^2*a^2*b^18 + 30*A^2*B^2*a^4*b^16 + 42*A^2*B^2*a^6*b^14 + 42*A
^2*B^2*a^10*b^10 - 24*A^3*B*a^7*b^13 + 80*A^3*B*a^9*b^11 - 24*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 - 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)*1i - (((32*(3*A^3*a^2*
b^16*d^2 + 48*A^3*a^4*b^14*d^2 - 30*A^3*a^6*b^12*d^2 - 72*A^3*a^8*b^10*d^2 + 3*A^3*a^10*b^8*d^2 + 6*B^3*a^5*b^
13*d^2 + 8*B^3*a^7*b^11*d^2 + 3*B^3*a^9*b^9*d^2 - B^3*a*b^17*d^2 - A^2*B*a*b^17*d^2 + 3*A*B^2*a^2*b^16*d^2 - 3
2*A*B^2*a^4*b^14*d^2 + 46*A*B^2*a^6*b^12*d^2 + 72*A*B^2*a^8*b^10*d^2 - 9*A*B^2*a^10*b^8*d^2 - 16*A^2*B*a^3*b^1
5*d^2 + 150*A^2*B*a^5*b^13*d^2 + 96*A^2*B*a^7*b^11*d^2 - 69*A^2*B*a^9*b^9*d^2))/d^5 - (((32*(4*A*a*b^12*d^4 +
16*A*a^3*b^10*d^4 + 12*A*a^5*b^8*d^4 + 8*B*a^2*b^11*d^4 + 8*B*a^4*b^9*d^4))/d^5 + (32*(16*b^10*d^4 + 24*a^2*b^
8*d^4)*(a + b*tan(c + d*x))^(1/2)*(-(((8*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) + (32*(a + b*tan(c + d*x))^(1/2)*(10*A^2*a^3*b^12*d^2 + 102*A^
2*a^5*b^10*d^2 - 18*A^2*a^7*b^8*d^2 - 10*B^2*a^3*b^12*d^2 - 102*B^2*a^5*b^10*d^2 + 10*B^2*a^7*b^8*d^2 + 16*A*B
*b^15*d^2 - 38*A^2*a*b^14*d^2 + 38*B^2*a*b^14*d^2 - 120*A*B*a^2*b^13*d^2 - 160*A*B*a^4*b^11*d^2 + 104*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) - (32*(a + b*tan(c + d*x))^(1/2)*(A^4*b^20 + B^4*b^20 + 2*A^2*B^2*b^20 + 6*A^4*a^2*b^1
8 + 15*A^4*a^4*b^16 + 18*A^4*a^6*b^14 + 45*A^4*a^8*b^12 - 24*A^4*a^10*b^10 + 3*A^4*a^12*b^8 + 6*B^4*a^2*b^18 +
15*B^4*a^4*b^16 + 20*B^4*a^6*b^14 + 15*B^4*a^8*b^12 + 6*B^4*a^10*b^10 + B^4*a^12*b^8 + 12*A^2*B^2*a^2*b^18 +
30*A^2*B^2*a^4*b^16 + 42*A^2*B^2*a^6*b^14 + 42*A^2*B^2*a^10*b^10 - 24*A^3*B*a^7*b^13 + 80*A^3*B*a^9*b^11 - 24*
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 + 2
0*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)/((((32*(3*A^3*a^2*b^16*d^2 + 48*A^3*a^4*b^14*d^2 - 30*A^3*a^6*b^12*d^2 - 72*A^3*
a^8*b^10*d^2 + 3*A^3*a^10*b^8*d^2 + 6*B^3*a^5*b^13*d^2 + 8*B^3*a^7*b^11*d^2 + 3*B^3*a^9*b^9*d^2 - B^3*a*b^17*d
^2 - A^2*B*a*b^17*d^2 + 3*A*B^2*a^2*b^16*d^2 - 32*A*B^2*a^4*b^14*d^2 + 46*A*B^2*a^6*b^12*d^2 + 72*A*B^2*a^8*b^
10*d^2 - 9*A*B^2*a^10*b^8*d^2 - 16*A^2*B*a^3*b^15*d^2 + 150*A^2*B*a^5*b^13*d^2 + 96*A^2*B*a^7*b^11*d^2 - 69*A^
2*B*a^9*b^9*d^2))/d^5 - (((32*(4*A*a*b^12*d^4 + 16*A*a^3*b^10*d^4 + 12*A*a^5*b^8*d^4 + 8*B*a^2*b^11*d^4 + 8*B*
a^4*b^9*d^4))/d^5 - (32*(16*b^10*d^4 + 24*a^2*b^8*d^4)*(a + b*tan(c + d*x))^(1/2)*(-(((8*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) - (32*(a + b*t
an(c + d*x))^(1/2)*(10*A^2*a^3*b^12*d^2 + 102*A^2*a^5*b^10*d^2 - 18*A^2*a^7*b^8*d^2 - 10*B^2*a^3*b^12*d^2 - 10
2*B^2*a^5*b^10*d^2 + 10*B^2*a^7*b^8*d^2 + 16*A*B*b^15*d^2 - 38*A^2*a*b^14*d^2 + 38*B^2*a*b^14*d^2 - 120*A*B*a^
2*b^13*d^2 - 160*A*B*a^4*b^11*d^2 + 104*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 + 8
0*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) + (32*(a + b*tan(c + d*x))^(1/2)*(A^4*
b^20 + B^4*b^20 + 2*A^2*B^2*b^20 + 6*A^4*a^2*b^18 + 15*A^4*a^4*b^16 + 18*A^4*a^6*b^14 + 45*A^4*a^8*b^12 - 24*A
^4*a^10*b^10 + 3*A^4*a^12*b^8 + 6*B^4*a^2*b^18 + 15*B^4*a^4*b^16 + 20*B^4*a^6*b^14 + 15*B^4*a^8*b^12 + 6*B^4*a
^10*b^10 + B^4*a^12*b^8 + 12*A^2*B^2*a^2*b^18 + 30*A^2*B^2*a^4*b^16 + 42*A^2*B^2*a^6*b^14 + 42*A^2*B^2*a^10*b^
10 - 24*A^3*B*a^7*b^13 + 80*A^3*B*a^9*b^11 - 24*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^1
0 + 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) - (64*(A^5*a^3*b^20 + 6*A^5*a^5*b^18
+ 18*A^5*a^7*b^16 + 28*A^5*a^9*b^14 + 21*A^5*a^11*b^12 + 6*A^5*a^13*b^10 - A^2*B^3*a^6*b^17 + 6*A^2*B^3*a^10*
b^13 + 8*A^2*B^3*a^12*b^11 + 3*A^2*B^3*a^14*b^9 + 2*A^3*B^2*a^3*b^20 + 12*A^3*B^2*a^5*b^18 + 33*A^3*B^2*a^7*b^
16 + 48*A^3*B^2*a^9*b^14 + 36*A^3*B^2*a^11*b^12 + 12*A^3*B^2*a^13*b^10 + A^3*B^2*a^15*b^8 + A*B^4*a^3*b^20 + 6
*A*B^4*a^5*b^18 + 15*A*B^4*a^7*b^16 + 20*A*B^4*a^9*b^14 + 15*A*B^4*a^11*b^12 + 6*A*B^4*a^13*b^10 + A*B^4*a^15*
b^8 - A^4*B*a^6*b^17 + 6*A^4*B*a^10*b^13 + 8*A^4*B*a^12*b^11 + 3*A^4*B*a^14*b^9))/d^5 + (((32*(3*A^3*a^2*b^16*
d^2 + 48*A^3*a^4*b^14*d^2 - 30*A^3*a^6*b^12*d^2 - 72*A^3*a^8*b^10*d^2 + 3*A^3*a^10*b^8*d^2 + 6*B^3*a^5*b^13*d^
2 + 8*B^3*a^7*b^11*d^2 + 3*B^3*a^9*b^9*d^2 - B^3*a*b^17*d^2 - A^2*B*a*b^17*d^2 + 3*A*B^2*a^2*b^16*d^2 - 32*A*B
^2*a^4*b^14*d^2 + 46*A*B^2*a^6*b^12*d^2 + 72*A*B^2*a^8*b^10*d^2 - 9*A*B^2*a^10*b^8*d^2 - 16*A^2*B*a^3*b^15*d^2
+ 150*A^2*B*a^5*b^13*d^2 + 96*A^2*B*a^7*b^11*d^2 - 69*A^2*B*a^9*b^9*d^2))/d^5 - (((32*(4*A*a*b^12*d^4 + 16*A*
a^3*b^10*d^4 + 12*A*a^5*b^8*d^4 + 8*B*a^2*b^11*d^4 + 8*B*a^4*b^9*d^4))/d^5 + (32*(16*b^10*d^4 + 24*a^2*b^8*d^4
)*(a + b*tan(c + d*x))^(1/2)*(-(((8*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) + (32*(a + b*tan(c + d*x))^(1/2)*(10*A^2*a^3*b^12*d^2 + 102*A^2*a^5
*b^10*d^2 - 18*A^2*a^7*b^8*d^2 - 10*B^2*a^3*b^12*d^2 - 102*B^2*a^5*b^10*d^2 + 10*B^2*a^7*b^8*d^2 + 16*A*B*b^15
*d^2 - 38*A^2*a*b^14*d^2 + 38*B^2*a*b^14*d^2 - 120*A*B*a^2*b^13*d^2 - 160*A*B*a^4*b^11*d^2 + 104*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) - (32*(a + b*tan(c + d*x))^(1/2)*(A^4*b^20 + B^4*b^20 + 2*A^2*B^2*b^20 + 6*A^4*a^2*b^18 + 1
5*A^4*a^4*b^16 + 18*A^4*a^6*b^14 + 45*A^4*a^8*b^12 - 24*A^4*a^10*b^10 + 3*A^4*a^12*b^8 + 6*B^4*a^2*b^18 + 15*B
^4*a^4*b^16 + 20*B^4*a^6*b^14 + 15*B^4*a^8*b^12 + 6*B^4*a^10*b^10 + B^4*a^12*b^8 + 12*A^2*B^2*a^2*b^18 + 30*A^
2*B^2*a^4*b^16 + 42*A^2*B^2*a^6*b^14 + 42*A^2*B^2*a^10*b^10 - 24*A^3*B*a^7*b^13 + 80*A^3*B*a^9*b^11 - 24*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*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 + (2*B*b*(a + b*tan(c + d*x))^(3/2))/(3*d) + (A*atan(((A*((32*(a + b*tan(c + d*x
))^(1/2)*(A^4*b^20 + B^4*b^20 + 2*A^2*B^2*b^20 + 6*A^4*a^2*b^18 + 15*A^4*a^4*b^16 + 18*A^4*a^6*b^14 + 45*A^4*a
^8*b^12 - 24*A^4*a^10*b^10 + 3*A^4*a^12*b^8 + 6*B^4*a^2*b^18 + 15*B^4*a^4*b^16 + 20*B^4*a^6*b^14 + 15*B^4*a^8*
b^12 + 6*B^4*a^10*b^10 + B^4*a^12*b^8 + 12*A^2*B^2*a^2*b^18 + 30*A^2*B^2*a^4*b^16 + 42*A^2*B^2*a^6*b^14 + 42*A
^2*B^2*a^10*b^10 - 24*A^3*B*a^7*b^13 + 80*A^3*B*a^9*b^11 - 24*A^3*B*a^11*b^9))/d^4 + (A*((32*(3*A^3*a^2*b^16*d
^2 + 48*A^3*a^4*b^14*d^2 - 30*A^3*a^6*b^12*d^2 - 72*A^3*a^8*b^10*d^2 + 3*A^3*a^10*b^8*d^2 + 6*B^3*a^5*b^13*d^2
+ 8*B^3*a^7*b^11*d^2 + 3*B^3*a^9*b^9*d^2 - B^3*a*b^17*d^2 - A^2*B*a*b^17*d^2 + 3*A*B^2*a^2*b^16*d^2 - 32*A*B^
2*a^4*b^14*d^2 + 46*A*B^2*a^6*b^12*d^2 + 72*A*B^2*a^8*b^10*d^2 - 9*A*B^2*a^10*b^8*d^2 - 16*A^2*B*a^3*b^15*d^2
+ 150*A^2*B*a^5*b^13*d^2 + 96*A^2*B*a^7*b^11*d^2 - 69*A^2*B*a^9*b^9*d^2))/d^5 + (A*(a^5)^(1/2)*((32*(a + b*tan
(c + d*x))^(1/2)*(10*A^2*a^3*b^12*d^2 + 102*A^2*a^5*b^10*d^2 - 18*A^2*a^7*b^8*d^2 - 10*B^2*a^3*b^12*d^2 - 102*
B^2*a^5*b^10*d^2 + 10*B^2*a^7*b^8*d^2 + 16*A*B*b^15*d^2 - 38*A^2*a*b^14*d^2 + 38*B^2*a*b^14*d^2 - 120*A*B*a^2*
b^13*d^2 - 160*A*B*a^4*b^11*d^2 + 104*A*B*a^6*b^9*d^2))/d^4 - (A*(a^5)^(1/2)*((32*(4*A*a*b^12*d^4 + 16*A*a^3*b
^10*d^4 + 12*A*a^5*b^8*d^4 + 8*B*a^2*b^11*d^4 + 8*B*a^4*b^9*d^4))/d^5 - (32*A*(a^5)^(1/2)*(16*b^10*d^4 + 24*a^
2*b^8*d^4)*(a + b*tan(c + d*x))^(1/2))/d^5))/d))/d)*(a^5)^(1/2))/d)*(a^5)^(1/2)*1i)/d + (A*((32*(a + b*tan(c +
d*x))^(1/2)*(A^4*b^20 + B^4*b^20 + 2*A^2*B^2*b^20 + 6*A^4*a^2*b^18 + 15*A^4*a^4*b^16 + 18*A^4*a^6*b^14 + 45*A
^4*a^8*b^12 - 24*A^4*a^10*b^10 + 3*A^4*a^12*b^8 + 6*B^4*a^2*b^18 + 15*B^4*a^4*b^16 + 20*B^4*a^6*b^14 + 15*B^4*
a^8*b^12 + 6*B^4*a^10*b^10 + B^4*a^12*b^8 + 12*A^2*B^2*a^2*b^18 + 30*A^2*B^2*a^4*b^16 + 42*A^2*B^2*a^6*b^14 +
42*A^2*B^2*a^10*b^10 - 24*A^3*B*a^7*b^13 + 80*A^3*B*a^9*b^11 - 24*A^3*B*a^11*b^9))/d^4 - (A*((32*(3*A^3*a^2*b^
16*d^2 + 48*A^3*a^4*b^14*d^2 - 30*A^3*a^6*b^12*d^2 - 72*A^3*a^8*b^10*d^2 + 3*A^3*a^10*b^8*d^2 + 6*B^3*a^5*b^13
*d^2 + 8*B^3*a^7*b^11*d^2 + 3*B^3*a^9*b^9*d^2 - B^3*a*b^17*d^2 - A^2*B*a*b^17*d^2 + 3*A*B^2*a^2*b^16*d^2 - 32*
A*B^2*a^4*b^14*d^2 + 46*A*B^2*a^6*b^12*d^2 + 72*A*B^2*a^8*b^10*d^2 - 9*A*B^2*a^10*b^8*d^2 - 16*A^2*B*a^3*b^15*
d^2 + 150*A^2*B*a^5*b^13*d^2 + 96*A^2*B*a^7*b^11*d^2 - 69*A^2*B*a^9*b^9*d^2))/d^5 - (A*(a^5)^(1/2)*((32*(a + b
*tan(c + d*x))^(1/2)*(10*A^2*a^3*b^12*d^2 + 102*A^2*a^5*b^10*d^2 - 18*A^2*a^7*b^8*d^2 - 10*B^2*a^3*b^12*d^2 -
102*B^2*a^5*b^10*d^2 + 10*B^2*a^7*b^8*d^2 + 16*A*B*b^15*d^2 - 38*A^2*a*b^14*d^2 + 38*B^2*a*b^14*d^2 - 120*A*B*
a^2*b^13*d^2 - 160*A*B*a^4*b^11*d^2 + 104*A*B*a^6*b^9*d^2))/d^4 + (A*(a^5)^(1/2)*((32*(4*A*a*b^12*d^4 + 16*A*a
^3*b^10*d^4 + 12*A*a^5*b^8*d^4 + 8*B*a^2*b^11*d^4 + 8*B*a^4*b^9*d^4))/d^5 + (32*A*(a^5)^(1/2)*(16*b^10*d^4 + 2
4*a^2*b^8*d^4)*(a + b*tan(c + d*x))^(1/2))/d^5))/d))/d)*(a^5)^(1/2))/d)*(a^5)^(1/2)*1i)/d)/((64*(A^5*a^3*b^20
+ 6*A^5*a^5*b^18 + 18*A^5*a^7*b^16 + 28*A^5*a^9*b^14 + 21*A^5*a^11*b^12 + 6*A^5*a^13*b^10 - A^2*B^3*a^6*b^17 +
6*A^2*B^3*a^10*b^13 + 8*A^2*B^3*a^12*b^11 + 3*A^2*B^3*a^14*b^9 + 2*A^3*B^2*a^3*b^20 + 12*A^3*B^2*a^5*b^18 + 3
3*A^3*B^2*a^7*b^16 + 48*A^3*B^2*a^9*b^14 + 36*A^3*B^2*a^11*b^12 + 12*A^3*B^2*a^13*b^10 + A^3*B^2*a^15*b^8 + A*
B^4*a^3*b^20 + 6*A*B^4*a^5*b^18 + 15*A*B^4*a^7*b^16 + 20*A*B^4*a^9*b^14 + 15*A*B^4*a^11*b^12 + 6*A*B^4*a^13*b^
10 + A*B^4*a^15*b^8 - A^4*B*a^6*b^17 + 6*A^4*B*a^10*b^13 + 8*A^4*B*a^12*b^11 + 3*A^4*B*a^14*b^9))/d^5 - (A*((3
2*(a + b*tan(c + d*x))^(1/2)*(A^4*b^20 + B^4*b^20 + 2*A^2*B^2*b^20 + 6*A^4*a^2*b^18 + 15*A^4*a^4*b^16 + 18*A^4
*a^6*b^14 + 45*A^4*a^8*b^12 - 24*A^4*a^10*b^10 + 3*A^4*a^12*b^8 + 6*B^4*a^2*b^18 + 15*B^4*a^4*b^16 + 20*B^4*a^
6*b^14 + 15*B^4*a^8*b^12 + 6*B^4*a^10*b^10 + B^4*a^12*b^8 + 12*A^2*B^2*a^2*b^18 + 30*A^2*B^2*a^4*b^16 + 42*A^2
*B^2*a^6*b^14 + 42*A^2*B^2*a^10*b^10 - 24*A^3*B*a^7*b^13 + 80*A^3*B*a^9*b^11 - 24*A^3*B*a^11*b^9))/d^4 + (A*((
32*(3*A^3*a^2*b^16*d^2 + 48*A^3*a^4*b^14*d^2 - 30*A^3*a^6*b^12*d^2 - 72*A^3*a^8*b^10*d^2 + 3*A^3*a^10*b^8*d^2
+ 6*B^3*a^5*b^13*d^2 + 8*B^3*a^7*b^11*d^2 + 3*B^3*a^9*b^9*d^2 - B^3*a*b^17*d^2 - A^2*B*a*b^17*d^2 + 3*A*B^2*a^
2*b^16*d^2 - 32*A*B^2*a^4*b^14*d^2 + 46*A*B^2*a^6*b^12*d^2 + 72*A*B^2*a^8*b^10*d^2 - 9*A*B^2*a^10*b^8*d^2 - 16
*A^2*B*a^3*b^15*d^2 + 150*A^2*B*a^5*b^13*d^2 + 96*A^2*B*a^7*b^11*d^2 - 69*A^2*B*a^9*b^9*d^2))/d^5 + (A*(a^5)^(
1/2)*((32*(a + b*tan(c + d*x))^(1/2)*(10*A^2*a^3*b^12*d^2 + 102*A^2*a^5*b^10*d^2 - 18*A^2*a^7*b^8*d^2 - 10*B^2
*a^3*b^12*d^2 - 102*B^2*a^5*b^10*d^2 + 10*B^2*a^7*b^8*d^2 + 16*A*B*b^15*d^2 - 38*A^2*a*b^14*d^2 + 38*B^2*a*b^1
4*d^2 - 120*A*B*a^2*b^13*d^2 - 160*A*B*a^4*b^11*d^2 + 104*A*B*a^6*b^9*d^2))/d^4 - (A*(a^5)^(1/2)*((32*(4*A*a*b
^12*d^4 + 16*A*a^3*b^10*d^4 + 12*A*a^5*b^8*d^4 + 8*B*a^2*b^11*d^4 + 8*B*a^4*b^9*d^4))/d^5 - (32*A*(a^5)^(1/2)*
(16*b^10*d^4 + 24*a^2*b^8*d^4)*(a + b*tan(c + d*x))^(1/2))/d^5))/d))/d)*(a^5)^(1/2))/d)*(a^5)^(1/2))/d + (A*((
32*(a + b*tan(c + d*x))^(1/2)*(A^4*b^20 + B^4*b^20 + 2*A^2*B^2*b^20 + 6*A^4*a^2*b^18 + 15*A^4*a^4*b^16 + 18*A^
4*a^6*b^14 + 45*A^4*a^8*b^12 - 24*A^4*a^10*b^10 + 3*A^4*a^12*b^8 + 6*B^4*a^2*b^18 + 15*B^4*a^4*b^16 + 20*B^4*a
^6*b^14 + 15*B^4*a^8*b^12 + 6*B^4*a^10*b^10 + B^4*a^12*b^8 + 12*A^2*B^2*a^2*b^18 + 30*A^2*B^2*a^4*b^16 + 42*A^
2*B^2*a^6*b^14 + 42*A^2*B^2*a^10*b^10 - 24*A^3*B*a^7*b^13 + 80*A^3*B*a^9*b^11 - 24*A^3*B*a^11*b^9))/d^4 - (A*(
(32*(3*A^3*a^2*b^16*d^2 + 48*A^3*a^4*b^14*d^2 - 30*A^3*a^6*b^12*d^2 - 72*A^3*a^8*b^10*d^2 + 3*A^3*a^10*b^8*d^2
+ 6*B^3*a^5*b^13*d^2 + 8*B^3*a^7*b^11*d^2 + 3*B^3*a^9*b^9*d^2 - B^3*a*b^17*d^2 - A^2*B*a*b^17*d^2 + 3*A*B^2*a
^2*b^16*d^2 - 32*A*B^2*a^4*b^14*d^2 + 46*A*B^2*a^6*b^12*d^2 + 72*A*B^2*a^8*b^10*d^2 - 9*A*B^2*a^10*b^8*d^2 - 1
6*A^2*B*a^3*b^15*d^2 + 150*A^2*B*a^5*b^13*d^2 + 96*A^2*B*a^7*b^11*d^2 - 69*A^2*B*a^9*b^9*d^2))/d^5 - (A*(a^5)^
(1/2)*((32*(a + b*tan(c + d*x))^(1/2)*(10*A^2*a^3*b^12*d^2 + 102*A^2*a^5*b^10*d^2 - 18*A^2*a^7*b^8*d^2 - 10*B^
2*a^3*b^12*d^2 - 102*B^2*a^5*b^10*d^2 + 10*B^2*a^7*b^8*d^2 + 16*A*B*b^15*d^2 - 38*A^2*a*b^14*d^2 + 38*B^2*a*b^
14*d^2 - 120*A*B*a^2*b^13*d^2 - 160*A*B*a^4*b^11*d^2 + 104*A*B*a^6*b^9*d^2))/d^4 + (A*(a^5)^(1/2)*((32*(4*A*a*
b^12*d^4 + 16*A*a^3*b^10*d^4 + 12*A*a^5*b^8*d^4 + 8*B*a^2*b^11*d^4 + 8*B*a^4*b^9*d^4))/d^5 + (32*A*(a^5)^(1/2)
*(16*b^10*d^4 + 24*a^2*b^8*d^4)*(a + b*tan(c + d*x))^(1/2))/d^5))/d))/d)*(a^5)^(1/2))/d)*(a^5)^(1/2))/d))*(a^5
)^(1/2)*2i)/d