\(\int \frac {\sqrt {\tan (c+d x)} (a B+b B \tan (c+d x))}{a+b \tan (c+d x)} \, dx\) [418]

Optimal result

Integrand size = 36, antiderivative size = 138 \[ \int \frac {\sqrt {\tan (c+d x)} (a B+b B \tan (c+d x))}{a+b \tan (c+d x)} \, dx=-\frac {B \arctan \left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} d}+\frac {B \arctan \left (1+\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} d}+\frac {B \log \left (1-\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right )}{2 \sqrt {2} d}-\frac {B \log \left (1+\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right )}{2 \sqrt {2} d} \]

[Out]

1/2*B*arctan(-1+2^(1/2)*tan(d*x+c)^(1/2))/d*2^(1/2)+1/2*B*arctan(1+2^(1/2)*tan(d*x+c)^(1/2))/d*2^(1/2)+1/4*B*l
n(1-2^(1/2)*tan(d*x+c)^(1/2)+tan(d*x+c))/d*2^(1/2)-1/4*B*ln(1+2^(1/2)*tan(d*x+c)^(1/2)+tan(d*x+c))/d*2^(1/2)

Rubi [A] (verified)

Time = 0.11 (sec) , antiderivative size = 138, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {21, 3557, 335, 303, 1176, 631, 210, 1179, 642} \[ \int \frac {\sqrt {\tan (c+d x)} (a B+b B \tan (c+d x))}{a+b \tan (c+d x)} \, dx=-\frac {B \arctan \left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} d}+\frac {B \arctan \left (\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{\sqrt {2} d}+\frac {B \log \left (\tan (c+d x)-\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{2 \sqrt {2} d}-\frac {B \log \left (\tan (c+d x)+\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{2 \sqrt {2} d} \]

[In]

Int[(Sqrt[Tan[c + d*x]]*(a*B + b*B*Tan[c + d*x]))/(a + b*Tan[c + d*x]),x]

[Out]

-((B*ArcTan[1 - Sqrt[2]*Sqrt[Tan[c + d*x]]])/(Sqrt[2]*d)) + (B*ArcTan[1 + Sqrt[2]*Sqrt[Tan[c + d*x]]])/(Sqrt[2
]*d) + (B*Log[1 - Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]])/(2*Sqrt[2]*d) - (B*Log[1 + Sqrt[2]*Sqrt[Tan[c +
d*x]] + Tan[c + d*x]])/(2*Sqrt[2]*d)

Rule 21

Int[(u_.)*((a_) + (b_.)*(v_))^(m_.)*((c_) + (d_.)*(v_))^(n_.), x_Symbol] :> Dist[(b/d)^m, Int[u*(c + d*v)^(m +
n), x], x] /; FreeQ[{a, b, c, d, n}, x] && EqQ[b*c - a*d, 0] && IntegerQ[m] && ( !IntegerQ[n] || SimplerQ[c +
d*x, a + b*x])

Rule 210

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^(-1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])
], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rule 303

Int[(x_)^2/((a_) + (b_.)*(x_)^4), x_Symbol] :> With[{r = Numerator[Rt[a/b, 2]], s = Denominator[Rt[a/b, 2]]},
Dist[1/(2*s), Int[(r + s*x^2)/(a + b*x^4), x], x] - Dist[1/(2*s), Int[(r - s*x^2)/(a + b*x^4), x], x]] /; Free
Q[{a, b}, x] && (GtQ[a/b, 0] || (PosQ[a/b] && AtomQ[SplitProduct[SumBaseQ, a]] && AtomQ[SplitProduct[SumBaseQ,
b]]))

Rule 335

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = Denominator[m]}, Dist[k/c, Subst[I
nt[x^(k*(m + 1) - 1)*(a + b*(x^(k*n)/c^n))^p, x], x, (c*x)^(1/k)], x]] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0]
&& FractionQ[m] && IntBinomialQ[a, b, c, n, m, p, x]

Rule 631

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*Simplify[a*(c/b^2)]}, Dist[-2/b, Sub
st[Int[1/(q - x^2), x], x, 1 + 2*c*(x/b)], x] /; RationalQ[q] && (EqQ[q^2, 1] ||  !RationalQ[b^2 - 4*a*c])] /;
FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 642

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[d*(Log[RemoveContent[a + b*x +
c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rule 1176

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[2*(d/e), 2]}, Dist[e/(2*c), Int[1/S
imp[d/e + q*x + x^2, x], x], x] + Dist[e/(2*c), Int[1/Simp[d/e - q*x + x^2, x], x], x]] /; FreeQ[{a, c, d, e},
x] && EqQ[c*d^2 - a*e^2, 0] && PosQ[d*e]

Rule 1179

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[-2*(d/e), 2]}, Dist[e/(2*c*q), Int[
(q - 2*x)/Simp[d/e + q*x - x^2, x], x], x] + Dist[e/(2*c*q), Int[(q + 2*x)/Simp[d/e - q*x - x^2, x], x], x]] /
; FreeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && NegQ[d*e]

Rule 3557

Int[((b_.)*tan[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Dist[b/d, Subst[Int[x^n/(b^2 + x^2), x], x, b*Tan[c + d
*x]], x] /; FreeQ[{b, c, d, n}, x] &&  !IntegerQ[n]

Rubi steps \begin{align*} \text {integral}& = B \int \sqrt {\tan (c+d x)} \, dx \\ & = \frac {B \text {Subst}\left (\int \frac {\sqrt {x}}{1+x^2} \, dx,x,\tan (c+d x)\right )}{d} \\ & = \frac {(2 B) \text {Subst}\left (\int \frac {x^2}{1+x^4} \, dx,x,\sqrt {\tan (c+d x)}\right )}{d} \\ & = -\frac {B \text {Subst}\left (\int \frac {1-x^2}{1+x^4} \, dx,x,\sqrt {\tan (c+d x)}\right )}{d}+\frac {B \text {Subst}\left (\int \frac {1+x^2}{1+x^4} \, dx,x,\sqrt {\tan (c+d x)}\right )}{d} \\ & = \frac {B \text {Subst}\left (\int \frac {1}{1-\sqrt {2} x+x^2} \, dx,x,\sqrt {\tan (c+d x)}\right )}{2 d}+\frac {B \text {Subst}\left (\int \frac {1}{1+\sqrt {2} x+x^2} \, dx,x,\sqrt {\tan (c+d x)}\right )}{2 d}+\frac {B \text {Subst}\left (\int \frac {\sqrt {2}+2 x}{-1-\sqrt {2} x-x^2} \, dx,x,\sqrt {\tan (c+d x)}\right )}{2 \sqrt {2} d}+\frac {B \text {Subst}\left (\int \frac {\sqrt {2}-2 x}{-1+\sqrt {2} x-x^2} \, dx,x,\sqrt {\tan (c+d x)}\right )}{2 \sqrt {2} d} \\ & = \frac {B \log \left (1-\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right )}{2 \sqrt {2} d}-\frac {B \log \left (1+\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right )}{2 \sqrt {2} d}+\frac {B \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} d}-\frac {B \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} d} \\ & = -\frac {B \arctan \left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} d}+\frac {B \arctan \left (1+\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} d}+\frac {B \log \left (1-\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right )}{2 \sqrt {2} d}-\frac {B \log \left (1+\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right )}{2 \sqrt {2} d} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.06 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.43 \[ \int \frac {\sqrt {\tan (c+d x)} (a B+b B \tan (c+d x))}{a+b \tan (c+d x)} \, dx=\frac {B \left (\arctan \left (\sqrt [4]{-\tan ^2(c+d x)}\right )-\text {arctanh}\left (\sqrt [4]{-\tan ^2(c+d x)}\right )\right ) \sqrt [4]{-\tan (c+d x)}}{d \sqrt [4]{\tan (c+d x)}} \]

[In]

Integrate[(Sqrt[Tan[c + d*x]]*(a*B + b*B*Tan[c + d*x]))/(a + b*Tan[c + d*x]),x]

[Out]

(B*(ArcTan[(-Tan[c + d*x]^2)^(1/4)] - ArcTanh[(-Tan[c + d*x]^2)^(1/4)])*(-Tan[c + d*x])^(1/4))/(d*Tan[c + d*x]
^(1/4))

Maple [A] (verified)

Time = 0.06 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.65

method	result	size
derivativedivides	\(\frac {B \sqrt {2}\, \left (\ln \left (\frac {1-\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}{1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}\right )+2 \arctan \left (1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )+2 \arctan \left (-1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )\right )}{4 d}\)	\(90\)
default	\(\frac {B \sqrt {2}\, \left (\ln \left (\frac {1-\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}{1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}\right )+2 \arctan \left (1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )+2 \arctan \left (-1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )\right )}{4 d}\)	\(90\)

[In]

int(tan(d*x+c)^(1/2)*(B*a+b*B*tan(d*x+c))/(a+b*tan(d*x+c)),x,method=_RETURNVERBOSE)

[Out]

1/4/d*B*2^(1/2)*(ln((1-2^(1/2)*tan(d*x+c)^(1/2)+tan(d*x+c))/(1+2^(1/2)*tan(d*x+c)^(1/2)+tan(d*x+c)))+2*arctan(
1+2^(1/2)*tan(d*x+c)^(1/2))+2*arctan(-1+2^(1/2)*tan(d*x+c)^(1/2)))

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.25 (sec) , antiderivative size = 164, normalized size of antiderivative = 1.19 \[ \int \frac {\sqrt {\tan (c+d x)} (a B+b B \tan (c+d x))}{a+b \tan (c+d x)} \, dx=\frac {1}{2} \, \left (-\frac {B^{4}}{d^{4}}\right )^{\frac {1}{4}} \log \left (d^{3} \left (-\frac {B^{4}}{d^{4}}\right )^{\frac {3}{4}} + B^{3} \sqrt {\tan \left (d x + c\right )}\right ) - \frac {1}{2} i \, \left (-\frac {B^{4}}{d^{4}}\right )^{\frac {1}{4}} \log \left (i \, d^{3} \left (-\frac {B^{4}}{d^{4}}\right )^{\frac {3}{4}} + B^{3} \sqrt {\tan \left (d x + c\right )}\right ) + \frac {1}{2} i \, \left (-\frac {B^{4}}{d^{4}}\right )^{\frac {1}{4}} \log \left (-i \, d^{3} \left (-\frac {B^{4}}{d^{4}}\right )^{\frac {3}{4}} + B^{3} \sqrt {\tan \left (d x + c\right )}\right ) - \frac {1}{2} \, \left (-\frac {B^{4}}{d^{4}}\right )^{\frac {1}{4}} \log \left (-d^{3} \left (-\frac {B^{4}}{d^{4}}\right )^{\frac {3}{4}} + B^{3} \sqrt {\tan \left (d x + c\right )}\right ) \]

[In]

integrate(tan(d*x+c)^(1/2)*(B*a+b*B*tan(d*x+c))/(a+b*tan(d*x+c)),x, algorithm="fricas")

[Out]

1/2*(-B^4/d^4)^(1/4)*log(d^3*(-B^4/d^4)^(3/4) + B^3*sqrt(tan(d*x + c))) - 1/2*I*(-B^4/d^4)^(1/4)*log(I*d^3*(-B
^4/d^4)^(3/4) + B^3*sqrt(tan(d*x + c))) + 1/2*I*(-B^4/d^4)^(1/4)*log(-I*d^3*(-B^4/d^4)^(3/4) + B^3*sqrt(tan(d*
x + c))) - 1/2*(-B^4/d^4)^(1/4)*log(-d^3*(-B^4/d^4)^(3/4) + B^3*sqrt(tan(d*x + c)))

Sympy [F]

\[ \int \frac {\sqrt {\tan (c+d x)} (a B+b B \tan (c+d x))}{a+b \tan (c+d x)} \, dx=B \int \sqrt {\tan {\left (c + d x \right )}}\, dx \]

[In]

integrate(tan(d*x+c)**(1/2)*(B*a+b*B*tan(d*x+c))/(a+b*tan(d*x+c)),x)

[Out]

B*Integral(sqrt(tan(c + d*x)), x)

Maxima [A] (verification not implemented)

none

Time = 0.36 (sec) , antiderivative size = 109, normalized size of antiderivative = 0.79 \[ \int \frac {\sqrt {\tan (c+d x)} (a B+b B \tan (c+d x))}{a+b \tan (c+d x)} \, dx=\frac {{\left (2 \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + 2 \, \sqrt {\tan \left (d x + c\right )}\right )}\right ) + 2 \, \sqrt {2} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - 2 \, \sqrt {\tan \left (d x + c\right )}\right )}\right ) - \sqrt {2} \log \left (\sqrt {2} \sqrt {\tan \left (d x + c\right )} + \tan \left (d x + c\right ) + 1\right ) + \sqrt {2} \log \left (-\sqrt {2} \sqrt {\tan \left (d x + c\right )} + \tan \left (d x + c\right ) + 1\right )\right )} B}{4 \, d} \]

[In]

integrate(tan(d*x+c)^(1/2)*(B*a+b*B*tan(d*x+c))/(a+b*tan(d*x+c)),x, algorithm="maxima")

[Out]

1/4*(2*sqrt(2)*arctan(1/2*sqrt(2)*(sqrt(2) + 2*sqrt(tan(d*x + c)))) + 2*sqrt(2)*arctan(-1/2*sqrt(2)*(sqrt(2) -
2*sqrt(tan(d*x + c)))) - sqrt(2)*log(sqrt(2)*sqrt(tan(d*x + c)) + tan(d*x + c) + 1) + sqrt(2)*log(-sqrt(2)*sq
rt(tan(d*x + c)) + tan(d*x + c) + 1))*B/d

Giac [F(-1)]

Timed out. \[ \int \frac {\sqrt {\tan (c+d x)} (a B+b B \tan (c+d x))}{a+b \tan (c+d x)} \, dx=\text {Timed out} \]

[In]

integrate(tan(d*x+c)^(1/2)*(B*a+b*B*tan(d*x+c))/(a+b*tan(d*x+c)),x, algorithm="giac")

[Out]

Timed out

Mupad [B] (verification not implemented)

Time = 13.38 (sec) , antiderivative size = 15753, normalized size of antiderivative = 114.15 \[ \int \frac {\sqrt {\tan (c+d x)} (a B+b B \tan (c+d x))}{a+b \tan (c+d x)} \, dx=\text {Too large to display} \]

[In]

int((tan(c + d*x)^(1/2)*(B*a + B*b*tan(c + d*x)))/(a + b*tan(c + d*x)),x)

[Out]

atan(((((32*(13*B^3*a^5*b^4*d^2 + B^3*a^7*b^2*d^2))/d^5 + (((32*(12*B*a^2*b^7*d^4 + 24*B*a^4*b^5*d^4 + 12*B*a^
6*b^3*d^4))/d^5 - (32*tan(c + d*x)^(1/2)*(((64*B^4*a^6*b^2*d^4 - B^4*a^4*(16*a^4*d^4 + 16*b^4*d^4 + 32*a^2*b^2
*d^4))^(1/2) - 8*B^2*a^3*b*d^2)/(16*(a^4*d^4 + b^4*d^4 + 2*a^2*b^2*d^4)))^(1/2)*(16*b^9*d^4 + 16*a^2*b^7*d^4 -
16*a^4*b^5*d^4 - 16*a^6*b^3*d^4))/d^4)*(((64*B^4*a^6*b^2*d^4 - B^4*a^4*(16*a^4*d^4 + 16*b^4*d^4 + 32*a^2*b^2*
d^4))^(1/2) - 8*B^2*a^3*b*d^2)/(16*(a^4*d^4 + b^4*d^4 + 2*a^2*b^2*d^4)))^(1/2) + (32*tan(c + d*x)^(1/2)*(20*B^
2*a^5*b^4*d^2 - 14*B^2*a^3*b^6*d^2 + 2*B^2*a^7*b^2*d^2))/d^4)*(((64*B^4*a^6*b^2*d^4 - B^4*a^4*(16*a^4*d^4 + 16
*b^4*d^4 + 32*a^2*b^2*d^4))^(1/2) - 8*B^2*a^3*b*d^2)/(16*(a^4*d^4 + b^4*d^4 + 2*a^2*b^2*d^4)))^(1/2))*(((64*B^
4*a^6*b^2*d^4 - B^4*a^4*(16*a^4*d^4 + 16*b^4*d^4 + 32*a^2*b^2*d^4))^(1/2) - 8*B^2*a^3*b*d^2)/(16*(a^4*d^4 + b^
4*d^4 + 2*a^2*b^2*d^4)))^(1/2) - (32*tan(c + d*x)^(1/2)*(B^4*a^4*b^5 - 2*B^4*a^6*b^3))/d^4)*(((64*B^4*a^6*b^2*
d^4 - B^4*a^4*(16*a^4*d^4 + 16*b^4*d^4 + 32*a^2*b^2*d^4))^(1/2) - 8*B^2*a^3*b*d^2)/(16*(a^4*d^4 + b^4*d^4 + 2*
a^2*b^2*d^4)))^(1/2)*1i - (((32*(13*B^3*a^5*b^4*d^2 + B^3*a^7*b^2*d^2))/d^5 + (((32*(12*B*a^2*b^7*d^4 + 24*B*a
^4*b^5*d^4 + 12*B*a^6*b^3*d^4))/d^5 + (32*tan(c + d*x)^(1/2)*(((64*B^4*a^6*b^2*d^4 - B^4*a^4*(16*a^4*d^4 + 16*
b^4*d^4 + 32*a^2*b^2*d^4))^(1/2) - 8*B^2*a^3*b*d^2)/(16*(a^4*d^4 + b^4*d^4 + 2*a^2*b^2*d^4)))^(1/2)*(16*b^9*d^
4 + 16*a^2*b^7*d^4 - 16*a^4*b^5*d^4 - 16*a^6*b^3*d^4))/d^4)*(((64*B^4*a^6*b^2*d^4 - B^4*a^4*(16*a^4*d^4 + 16*b
^4*d^4 + 32*a^2*b^2*d^4))^(1/2) - 8*B^2*a^3*b*d^2)/(16*(a^4*d^4 + b^4*d^4 + 2*a^2*b^2*d^4)))^(1/2) - (32*tan(c
+ d*x)^(1/2)*(20*B^2*a^5*b^4*d^2 - 14*B^2*a^3*b^6*d^2 + 2*B^2*a^7*b^2*d^2))/d^4)*(((64*B^4*a^6*b^2*d^4 - B^4*
a^4*(16*a^4*d^4 + 16*b^4*d^4 + 32*a^2*b^2*d^4))^(1/2) - 8*B^2*a^3*b*d^2)/(16*(a^4*d^4 + b^4*d^4 + 2*a^2*b^2*d^
4)))^(1/2))*(((64*B^4*a^6*b^2*d^4 - B^4*a^4*(16*a^4*d^4 + 16*b^4*d^4 + 32*a^2*b^2*d^4))^(1/2) - 8*B^2*a^3*b*d^
2)/(16*(a^4*d^4 + b^4*d^4 + 2*a^2*b^2*d^4)))^(1/2) + (32*tan(c + d*x)^(1/2)*(B^4*a^4*b^5 - 2*B^4*a^6*b^3))/d^4
)*(((64*B^4*a^6*b^2*d^4 - B^4*a^4*(16*a^4*d^4 + 16*b^4*d^4 + 32*a^2*b^2*d^4))^(1/2) - 8*B^2*a^3*b*d^2)/(16*(a^
4*d^4 + b^4*d^4 + 2*a^2*b^2*d^4)))^(1/2)*1i)/((((32*(13*B^3*a^5*b^4*d^2 + B^3*a^7*b^2*d^2))/d^5 + (((32*(12*B*
a^2*b^7*d^4 + 24*B*a^4*b^5*d^4 + 12*B*a^6*b^3*d^4))/d^5 - (32*tan(c + d*x)^(1/2)*(((64*B^4*a^6*b^2*d^4 - B^4*a
^4*(16*a^4*d^4 + 16*b^4*d^4 + 32*a^2*b^2*d^4))^(1/2) - 8*B^2*a^3*b*d^2)/(16*(a^4*d^4 + b^4*d^4 + 2*a^2*b^2*d^4
)))^(1/2)*(16*b^9*d^4 + 16*a^2*b^7*d^4 - 16*a^4*b^5*d^4 - 16*a^6*b^3*d^4))/d^4)*(((64*B^4*a^6*b^2*d^4 - B^4*a^
4*(16*a^4*d^4 + 16*b^4*d^4 + 32*a^2*b^2*d^4))^(1/2) - 8*B^2*a^3*b*d^2)/(16*(a^4*d^4 + b^4*d^4 + 2*a^2*b^2*d^4)
))^(1/2) + (32*tan(c + d*x)^(1/2)*(20*B^2*a^5*b^4*d^2 - 14*B^2*a^3*b^6*d^2 + 2*B^2*a^7*b^2*d^2))/d^4)*(((64*B^
4*a^6*b^2*d^4 - B^4*a^4*(16*a^4*d^4 + 16*b^4*d^4 + 32*a^2*b^2*d^4))^(1/2) - 8*B^2*a^3*b*d^2)/(16*(a^4*d^4 + b^
4*d^4 + 2*a^2*b^2*d^4)))^(1/2))*(((64*B^4*a^6*b^2*d^4 - B^4*a^4*(16*a^4*d^4 + 16*b^4*d^4 + 32*a^2*b^2*d^4))^(1
/2) - 8*B^2*a^3*b*d^2)/(16*(a^4*d^4 + b^4*d^4 + 2*a^2*b^2*d^4)))^(1/2) - (32*tan(c + d*x)^(1/2)*(B^4*a^4*b^5 -
2*B^4*a^6*b^3))/d^4)*(((64*B^4*a^6*b^2*d^4 - B^4*a^4*(16*a^4*d^4 + 16*b^4*d^4 + 32*a^2*b^2*d^4))^(1/2) - 8*B^
2*a^3*b*d^2)/(16*(a^4*d^4 + b^4*d^4 + 2*a^2*b^2*d^4)))^(1/2) + (((32*(13*B^3*a^5*b^4*d^2 + B^3*a^7*b^2*d^2))/d
^5 + (((32*(12*B*a^2*b^7*d^4 + 24*B*a^4*b^5*d^4 + 12*B*a^6*b^3*d^4))/d^5 + (32*tan(c + d*x)^(1/2)*(((64*B^4*a^
6*b^2*d^4 - B^4*a^4*(16*a^4*d^4 + 16*b^4*d^4 + 32*a^2*b^2*d^4))^(1/2) - 8*B^2*a^3*b*d^2)/(16*(a^4*d^4 + b^4*d^
4 + 2*a^2*b^2*d^4)))^(1/2)*(16*b^9*d^4 + 16*a^2*b^7*d^4 - 16*a^4*b^5*d^4 - 16*a^6*b^3*d^4))/d^4)*(((64*B^4*a^6
*b^2*d^4 - B^4*a^4*(16*a^4*d^4 + 16*b^4*d^4 + 32*a^2*b^2*d^4))^(1/2) - 8*B^2*a^3*b*d^2)/(16*(a^4*d^4 + b^4*d^4
+ 2*a^2*b^2*d^4)))^(1/2) - (32*tan(c + d*x)^(1/2)*(20*B^2*a^5*b^4*d^2 - 14*B^2*a^3*b^6*d^2 + 2*B^2*a^7*b^2*d^
2))/d^4)*(((64*B^4*a^6*b^2*d^4 - B^4*a^4*(16*a^4*d^4 + 16*b^4*d^4 + 32*a^2*b^2*d^4))^(1/2) - 8*B^2*a^3*b*d^2)/
(16*(a^4*d^4 + b^4*d^4 + 2*a^2*b^2*d^4)))^(1/2))*(((64*B^4*a^6*b^2*d^4 - B^4*a^4*(16*a^4*d^4 + 16*b^4*d^4 + 32
*a^2*b^2*d^4))^(1/2) - 8*B^2*a^3*b*d^2)/(16*(a^4*d^4 + b^4*d^4 + 2*a^2*b^2*d^4)))^(1/2) + (32*tan(c + d*x)^(1/
2)*(B^4*a^4*b^5 - 2*B^4*a^6*b^3))/d^4)*(((64*B^4*a^6*b^2*d^4 - B^4*a^4*(16*a^4*d^4 + 16*b^4*d^4 + 32*a^2*b^2*d
^4))^(1/2) - 8*B^2*a^3*b*d^2)/(16*(a^4*d^4 + b^4*d^4 + 2*a^2*b^2*d^4)))^(1/2) + (64*B^5*a^6*b^3)/d^5))*(((64*B
^4*a^6*b^2*d^4 - B^4*a^4*(16*a^4*d^4 + 16*b^4*d^4 + 32*a^2*b^2*d^4))^(1/2) - 8*B^2*a^3*b*d^2)/(16*(a^4*d^4 + b
^4*d^4 + 2*a^2*b^2*d^4)))^(1/2)*2i + atan(((((32*(13*B^3*a^5*b^4*d^2 + B^3*a^7*b^2*d^2))/d^5 + (((32*(12*B*a^2
*b^7*d^4 + 24*B*a^4*b^5*d^4 + 12*B*a^6*b^3*d^4))/d^5 - (32*tan(c + d*x)^(1/2)*(-((64*B^4*a^6*b^2*d^4 - B^4*a^4
*(16*a^4*d^4 + 16*b^4*d^4 + 32*a^2*b^2*d^4))^(1/2) + 8*B^2*a^3*b*d^2)/(16*(a^4*d^4 + b^4*d^4 + 2*a^2*b^2*d^4))
)^(1/2)*(16*b^9*d^4 + 16*a^2*b^7*d^4 - 16*a^4*b^5*d^4 - 16*a^6*b^3*d^4))/d^4)*(-((64*B^4*a^6*b^2*d^4 - B^4*a^4
*(16*a^4*d^4 + 16*b^4*d^4 + 32*a^2*b^2*d^4))^(1/2) + 8*B^2*a^3*b*d^2)/(16*(a^4*d^4 + b^4*d^4 + 2*a^2*b^2*d^4))
)^(1/2) + (32*tan(c + d*x)^(1/2)*(20*B^2*a^5*b^4*d^2 - 14*B^2*a^3*b^6*d^2 + 2*B^2*a^7*b^2*d^2))/d^4)*(-((64*B^
4*a^6*b^2*d^4 - B^4*a^4*(16*a^4*d^4 + 16*b^4*d^4 + 32*a^2*b^2*d^4))^(1/2) + 8*B^2*a^3*b*d^2)/(16*(a^4*d^4 + b^
4*d^4 + 2*a^2*b^2*d^4)))^(1/2))*(-((64*B^4*a^6*b^2*d^4 - B^4*a^4*(16*a^4*d^4 + 16*b^4*d^4 + 32*a^2*b^2*d^4))^(
1/2) + 8*B^2*a^3*b*d^2)/(16*(a^4*d^4 + b^4*d^4 + 2*a^2*b^2*d^4)))^(1/2) - (32*tan(c + d*x)^(1/2)*(B^4*a^4*b^5
- 2*B^4*a^6*b^3))/d^4)*(-((64*B^4*a^6*b^2*d^4 - B^4*a^4*(16*a^4*d^4 + 16*b^4*d^4 + 32*a^2*b^2*d^4))^(1/2) + 8*
B^2*a^3*b*d^2)/(16*(a^4*d^4 + b^4*d^4 + 2*a^2*b^2*d^4)))^(1/2)*1i - (((32*(13*B^3*a^5*b^4*d^2 + B^3*a^7*b^2*d^
2))/d^5 + (((32*(12*B*a^2*b^7*d^4 + 24*B*a^4*b^5*d^4 + 12*B*a^6*b^3*d^4))/d^5 + (32*tan(c + d*x)^(1/2)*(-((64*
B^4*a^6*b^2*d^4 - B^4*a^4*(16*a^4*d^4 + 16*b^4*d^4 + 32*a^2*b^2*d^4))^(1/2) + 8*B^2*a^3*b*d^2)/(16*(a^4*d^4 +
b^4*d^4 + 2*a^2*b^2*d^4)))^(1/2)*(16*b^9*d^4 + 16*a^2*b^7*d^4 - 16*a^4*b^5*d^4 - 16*a^6*b^3*d^4))/d^4)*(-((64*
B^4*a^6*b^2*d^4 - B^4*a^4*(16*a^4*d^4 + 16*b^4*d^4 + 32*a^2*b^2*d^4))^(1/2) + 8*B^2*a^3*b*d^2)/(16*(a^4*d^4 +
b^4*d^4 + 2*a^2*b^2*d^4)))^(1/2) - (32*tan(c + d*x)^(1/2)*(20*B^2*a^5*b^4*d^2 - 14*B^2*a^3*b^6*d^2 + 2*B^2*a^7
*b^2*d^2))/d^4)*(-((64*B^4*a^6*b^2*d^4 - B^4*a^4*(16*a^4*d^4 + 16*b^4*d^4 + 32*a^2*b^2*d^4))^(1/2) + 8*B^2*a^3
*b*d^2)/(16*(a^4*d^4 + b^4*d^4 + 2*a^2*b^2*d^4)))^(1/2))*(-((64*B^4*a^6*b^2*d^4 - B^4*a^4*(16*a^4*d^4 + 16*b^4
*d^4 + 32*a^2*b^2*d^4))^(1/2) + 8*B^2*a^3*b*d^2)/(16*(a^4*d^4 + b^4*d^4 + 2*a^2*b^2*d^4)))^(1/2) + (32*tan(c +
d*x)^(1/2)*(B^4*a^4*b^5 - 2*B^4*a^6*b^3))/d^4)*(-((64*B^4*a^6*b^2*d^4 - B^4*a^4*(16*a^4*d^4 + 16*b^4*d^4 + 32
*a^2*b^2*d^4))^(1/2) + 8*B^2*a^3*b*d^2)/(16*(a^4*d^4 + b^4*d^4 + 2*a^2*b^2*d^4)))^(1/2)*1i)/((((32*(13*B^3*a^5
*b^4*d^2 + B^3*a^7*b^2*d^2))/d^5 + (((32*(12*B*a^2*b^7*d^4 + 24*B*a^4*b^5*d^4 + 12*B*a^6*b^3*d^4))/d^5 - (32*t
an(c + d*x)^(1/2)*(-((64*B^4*a^6*b^2*d^4 - B^4*a^4*(16*a^4*d^4 + 16*b^4*d^4 + 32*a^2*b^2*d^4))^(1/2) + 8*B^2*a
^3*b*d^2)/(16*(a^4*d^4 + b^4*d^4 + 2*a^2*b^2*d^4)))^(1/2)*(16*b^9*d^4 + 16*a^2*b^7*d^4 - 16*a^4*b^5*d^4 - 16*a
^6*b^3*d^4))/d^4)*(-((64*B^4*a^6*b^2*d^4 - B^4*a^4*(16*a^4*d^4 + 16*b^4*d^4 + 32*a^2*b^2*d^4))^(1/2) + 8*B^2*a
^3*b*d^2)/(16*(a^4*d^4 + b^4*d^4 + 2*a^2*b^2*d^4)))^(1/2) + (32*tan(c + d*x)^(1/2)*(20*B^2*a^5*b^4*d^2 - 14*B^
2*a^3*b^6*d^2 + 2*B^2*a^7*b^2*d^2))/d^4)*(-((64*B^4*a^6*b^2*d^4 - B^4*a^4*(16*a^4*d^4 + 16*b^4*d^4 + 32*a^2*b^
2*d^4))^(1/2) + 8*B^2*a^3*b*d^2)/(16*(a^4*d^4 + b^4*d^4 + 2*a^2*b^2*d^4)))^(1/2))*(-((64*B^4*a^6*b^2*d^4 - B^4
*a^4*(16*a^4*d^4 + 16*b^4*d^4 + 32*a^2*b^2*d^4))^(1/2) + 8*B^2*a^3*b*d^2)/(16*(a^4*d^4 + b^4*d^4 + 2*a^2*b^2*d
^4)))^(1/2) - (32*tan(c + d*x)^(1/2)*(B^4*a^4*b^5 - 2*B^4*a^6*b^3))/d^4)*(-((64*B^4*a^6*b^2*d^4 - B^4*a^4*(16*
a^4*d^4 + 16*b^4*d^4 + 32*a^2*b^2*d^4))^(1/2) + 8*B^2*a^3*b*d^2)/(16*(a^4*d^4 + b^4*d^4 + 2*a^2*b^2*d^4)))^(1/
2) + (((32*(13*B^3*a^5*b^4*d^2 + B^3*a^7*b^2*d^2))/d^5 + (((32*(12*B*a^2*b^7*d^4 + 24*B*a^4*b^5*d^4 + 12*B*a^6
*b^3*d^4))/d^5 + (32*tan(c + d*x)^(1/2)*(-((64*B^4*a^6*b^2*d^4 - B^4*a^4*(16*a^4*d^4 + 16*b^4*d^4 + 32*a^2*b^2
*d^4))^(1/2) + 8*B^2*a^3*b*d^2)/(16*(a^4*d^4 + b^4*d^4 + 2*a^2*b^2*d^4)))^(1/2)*(16*b^9*d^4 + 16*a^2*b^7*d^4 -
16*a^4*b^5*d^4 - 16*a^6*b^3*d^4))/d^4)*(-((64*B^4*a^6*b^2*d^4 - B^4*a^4*(16*a^4*d^4 + 16*b^4*d^4 + 32*a^2*b^2
*d^4))^(1/2) + 8*B^2*a^3*b*d^2)/(16*(a^4*d^4 + b^4*d^4 + 2*a^2*b^2*d^4)))^(1/2) - (32*tan(c + d*x)^(1/2)*(20*B
^2*a^5*b^4*d^2 - 14*B^2*a^3*b^6*d^2 + 2*B^2*a^7*b^2*d^2))/d^4)*(-((64*B^4*a^6*b^2*d^4 - B^4*a^4*(16*a^4*d^4 +
16*b^4*d^4 + 32*a^2*b^2*d^4))^(1/2) + 8*B^2*a^3*b*d^2)/(16*(a^4*d^4 + b^4*d^4 + 2*a^2*b^2*d^4)))^(1/2))*(-((64
*B^4*a^6*b^2*d^4 - B^4*a^4*(16*a^4*d^4 + 16*b^4*d^4 + 32*a^2*b^2*d^4))^(1/2) + 8*B^2*a^3*b*d^2)/(16*(a^4*d^4 +
b^4*d^4 + 2*a^2*b^2*d^4)))^(1/2) + (32*tan(c + d*x)^(1/2)*(B^4*a^4*b^5 - 2*B^4*a^6*b^3))/d^4)*(-((64*B^4*a^6*
b^2*d^4 - B^4*a^4*(16*a^4*d^4 + 16*b^4*d^4 + 32*a^2*b^2*d^4))^(1/2) + 8*B^2*a^3*b*d^2)/(16*(a^4*d^4 + b^4*d^4
+ 2*a^2*b^2*d^4)))^(1/2) + (64*B^5*a^6*b^3)/d^5))*(-((64*B^4*a^6*b^2*d^4 - B^4*a^4*(16*a^4*d^4 + 16*b^4*d^4 +
32*a^2*b^2*d^4))^(1/2) + 8*B^2*a^3*b*d^2)/(16*(a^4*d^4 + b^4*d^4 + 2*a^2*b^2*d^4)))^(1/2)*2i - atan(((((((32*(
4*B*a^2*b^7*d^4 + 8*B*a^4*b^5*d^4 + 4*B*a^6*b^3*d^4))/d^5 - (32*tan(c + d*x)^(1/2)*(-((64*B^4*a^2*b^6*d^4 - B^
4*b^4*(16*a^4*d^4 + 16*b^4*d^4 + 32*a^2*b^2*d^4))^(1/2) - 8*B^2*a*b^3*d^2)/(16*(a^4*d^4 + b^4*d^4 + 2*a^2*b^2*
d^4)))^(1/2)*(16*b^9*d^4 + 16*a^2*b^7*d^4 - 16*a^4*b^5*d^4 - 16*a^6*b^3*d^4))/d^4)*(-((64*B^4*a^2*b^6*d^4 - B^
4*b^4*(16*a^4*d^4 + 16*b^4*d^4 + 32*a^2*b^2*d^4))^(1/2) - 8*B^2*a*b^3*d^2)/(16*(a^4*d^4 + b^4*d^4 + 2*a^2*b^2*
d^4)))^(1/2) + (32*tan(c + d*x)^(1/2)*(14*B^2*a^5*b^4*d^2 - 4*B^2*a^3*b^6*d^2 + 14*B^2*a*b^8*d^2))/d^4)*(-((64
*B^4*a^2*b^6*d^4 - B^4*b^4*(16*a^4*d^4 + 16*b^4*d^4 + 32*a^2*b^2*d^4))^(1/2) - 8*B^2*a*b^3*d^2)/(16*(a^4*d^4 +
b^4*d^4 + 2*a^2*b^2*d^4)))^(1/2) + (32*(4*B^3*a^5*b^4*d^2 - 15*B^3*a^3*b^6*d^2 + B^3*a*b^8*d^2))/d^5)*(-((64*
B^4*a^2*b^6*d^4 - B^4*b^4*(16*a^4*d^4 + 16*b^4*d^4 + 32*a^2*b^2*d^4))^(1/2) - 8*B^2*a*b^3*d^2)/(16*(a^4*d^4 +
b^4*d^4 + 2*a^2*b^2*d^4)))^(1/2) - (32*tan(c + d*x)^(1/2)*(B^4*b^9 + 2*B^4*a^4*b^5))/d^4)*(-((64*B^4*a^2*b^6*d
^4 - B^4*b^4*(16*a^4*d^4 + 16*b^4*d^4 + 32*a^2*b^2*d^4))^(1/2) - 8*B^2*a*b^3*d^2)/(16*(a^4*d^4 + b^4*d^4 + 2*a
^2*b^2*d^4)))^(1/2)*1i - (((((32*(4*B*a^2*b^7*d^4 + 8*B*a^4*b^5*d^4 + 4*B*a^6*b^3*d^4))/d^5 + (32*tan(c + d*x)
^(1/2)*(-((64*B^4*a^2*b^6*d^4 - B^4*b^4*(16*a^4*d^4 + 16*b^4*d^4 + 32*a^2*b^2*d^4))^(1/2) - 8*B^2*a*b^3*d^2)/(
16*(a^4*d^4 + b^4*d^4 + 2*a^2*b^2*d^4)))^(1/2)*(16*b^9*d^4 + 16*a^2*b^7*d^4 - 16*a^4*b^5*d^4 - 16*a^6*b^3*d^4)
)/d^4)*(-((64*B^4*a^2*b^6*d^4 - B^4*b^4*(16*a^4*d^4 + 16*b^4*d^4 + 32*a^2*b^2*d^4))^(1/2) - 8*B^2*a*b^3*d^2)/(
16*(a^4*d^4 + b^4*d^4 + 2*a^2*b^2*d^4)))^(1/2) - (32*tan(c + d*x)^(1/2)*(14*B^2*a^5*b^4*d^2 - 4*B^2*a^3*b^6*d^
2 + 14*B^2*a*b^8*d^2))/d^4)*(-((64*B^4*a^2*b^6*d^4 - B^4*b^4*(16*a^4*d^4 + 16*b^4*d^4 + 32*a^2*b^2*d^4))^(1/2)
- 8*B^2*a*b^3*d^2)/(16*(a^4*d^4 + b^4*d^4 + 2*a^2*b^2*d^4)))^(1/2) + (32*(4*B^3*a^5*b^4*d^2 - 15*B^3*a^3*b^6*
d^2 + B^3*a*b^8*d^2))/d^5)*(-((64*B^4*a^2*b^6*d^4 - B^4*b^4*(16*a^4*d^4 + 16*b^4*d^4 + 32*a^2*b^2*d^4))^(1/2)
- 8*B^2*a*b^3*d^2)/(16*(a^4*d^4 + b^4*d^4 + 2*a^2*b^2*d^4)))^(1/2) + (32*tan(c + d*x)^(1/2)*(B^4*b^9 + 2*B^4*a
^4*b^5))/d^4)*(-((64*B^4*a^2*b^6*d^4 - B^4*b^4*(16*a^4*d^4 + 16*b^4*d^4 + 32*a^2*b^2*d^4))^(1/2) - 8*B^2*a*b^3
*d^2)/(16*(a^4*d^4 + b^4*d^4 + 2*a^2*b^2*d^4)))^(1/2)*1i)/((((((32*(4*B*a^2*b^7*d^4 + 8*B*a^4*b^5*d^4 + 4*B*a^
6*b^3*d^4))/d^5 - (32*tan(c + d*x)^(1/2)*(-((64*B^4*a^2*b^6*d^4 - B^4*b^4*(16*a^4*d^4 + 16*b^4*d^4 + 32*a^2*b^
2*d^4))^(1/2) - 8*B^2*a*b^3*d^2)/(16*(a^4*d^4 + b^4*d^4 + 2*a^2*b^2*d^4)))^(1/2)*(16*b^9*d^4 + 16*a^2*b^7*d^4
- 16*a^4*b^5*d^4 - 16*a^6*b^3*d^4))/d^4)*(-((64*B^4*a^2*b^6*d^4 - B^4*b^4*(16*a^4*d^4 + 16*b^4*d^4 + 32*a^2*b^
2*d^4))^(1/2) - 8*B^2*a*b^3*d^2)/(16*(a^4*d^4 + b^4*d^4 + 2*a^2*b^2*d^4)))^(1/2) + (32*tan(c + d*x)^(1/2)*(14*
B^2*a^5*b^4*d^2 - 4*B^2*a^3*b^6*d^2 + 14*B^2*a*b^8*d^2))/d^4)*(-((64*B^4*a^2*b^6*d^4 - B^4*b^4*(16*a^4*d^4 + 1
6*b^4*d^4 + 32*a^2*b^2*d^4))^(1/2) - 8*B^2*a*b^3*d^2)/(16*(a^4*d^4 + b^4*d^4 + 2*a^2*b^2*d^4)))^(1/2) + (32*(4
*B^3*a^5*b^4*d^2 - 15*B^3*a^3*b^6*d^2 + B^3*a*b^8*d^2))/d^5)*(-((64*B^4*a^2*b^6*d^4 - B^4*b^4*(16*a^4*d^4 + 16
*b^4*d^4 + 32*a^2*b^2*d^4))^(1/2) - 8*B^2*a*b^3*d^2)/(16*(a^4*d^4 + b^4*d^4 + 2*a^2*b^2*d^4)))^(1/2) - (32*tan
(c + d*x)^(1/2)*(B^4*b^9 + 2*B^4*a^4*b^5))/d^4)*(-((64*B^4*a^2*b^6*d^4 - B^4*b^4*(16*a^4*d^4 + 16*b^4*d^4 + 32
*a^2*b^2*d^4))^(1/2) - 8*B^2*a*b^3*d^2)/(16*(a^4*d^4 + b^4*d^4 + 2*a^2*b^2*d^4)))^(1/2) + (((((32*(4*B*a^2*b^7
*d^4 + 8*B*a^4*b^5*d^4 + 4*B*a^6*b^3*d^4))/d^5 + (32*tan(c + d*x)^(1/2)*(-((64*B^4*a^2*b^6*d^4 - B^4*b^4*(16*a
^4*d^4 + 16*b^4*d^4 + 32*a^2*b^2*d^4))^(1/2) - 8*B^2*a*b^3*d^2)/(16*(a^4*d^4 + b^4*d^4 + 2*a^2*b^2*d^4)))^(1/2
)*(16*b^9*d^4 + 16*a^2*b^7*d^4 - 16*a^4*b^5*d^4 - 16*a^6*b^3*d^4))/d^4)*(-((64*B^4*a^2*b^6*d^4 - B^4*b^4*(16*a
^4*d^4 + 16*b^4*d^4 + 32*a^2*b^2*d^4))^(1/2) - 8*B^2*a*b^3*d^2)/(16*(a^4*d^4 + b^4*d^4 + 2*a^2*b^2*d^4)))^(1/2
) - (32*tan(c + d*x)^(1/2)*(14*B^2*a^5*b^4*d^2 - 4*B^2*a^3*b^6*d^2 + 14*B^2*a*b^8*d^2))/d^4)*(-((64*B^4*a^2*b^
6*d^4 - B^4*b^4*(16*a^4*d^4 + 16*b^4*d^4 + 32*a^2*b^2*d^4))^(1/2) - 8*B^2*a*b^3*d^2)/(16*(a^4*d^4 + b^4*d^4 +
2*a^2*b^2*d^4)))^(1/2) + (32*(4*B^3*a^5*b^4*d^2 - 15*B^3*a^3*b^6*d^2 + B^3*a*b^8*d^2))/d^5)*(-((64*B^4*a^2*b^6
*d^4 - B^4*b^4*(16*a^4*d^4 + 16*b^4*d^4 + 32*a^2*b^2*d^4))^(1/2) - 8*B^2*a*b^3*d^2)/(16*(a^4*d^4 + b^4*d^4 + 2
*a^2*b^2*d^4)))^(1/2) + (32*tan(c + d*x)^(1/2)*(B^4*b^9 + 2*B^4*a^4*b^5))/d^4)*(-((64*B^4*a^2*b^6*d^4 - B^4*b^
4*(16*a^4*d^4 + 16*b^4*d^4 + 32*a^2*b^2*d^4))^(1/2) - 8*B^2*a*b^3*d^2)/(16*(a^4*d^4 + b^4*d^4 + 2*a^2*b^2*d^4)
))^(1/2) + (64*B^5*a^2*b^7)/d^5))*(-((64*B^4*a^2*b^6*d^4 - B^4*b^4*(16*a^4*d^4 + 16*b^4*d^4 + 32*a^2*b^2*d^4))
^(1/2) - 8*B^2*a*b^3*d^2)/(16*(a^4*d^4 + b^4*d^4 + 2*a^2*b^2*d^4)))^(1/2)*2i - atan(((((((32*(4*B*a^2*b^7*d^4
+ 8*B*a^4*b^5*d^4 + 4*B*a^6*b^3*d^4))/d^5 - (32*tan(c + d*x)^(1/2)*(((64*B^4*a^2*b^6*d^4 - B^4*b^4*(16*a^4*d^4
+ 16*b^4*d^4 + 32*a^2*b^2*d^4))^(1/2) + 8*B^2*a*b^3*d^2)/(16*(a^4*d^4 + b^4*d^4 + 2*a^2*b^2*d^4)))^(1/2)*(16*
b^9*d^4 + 16*a^2*b^7*d^4 - 16*a^4*b^5*d^4 - 16*a^6*b^3*d^4))/d^4)*(((64*B^4*a^2*b^6*d^4 - B^4*b^4*(16*a^4*d^4
+ 16*b^4*d^4 + 32*a^2*b^2*d^4))^(1/2) + 8*B^2*a*b^3*d^2)/(16*(a^4*d^4 + b^4*d^4 + 2*a^2*b^2*d^4)))^(1/2) + (32
*tan(c + d*x)^(1/2)*(14*B^2*a^5*b^4*d^2 - 4*B^2*a^3*b^6*d^2 + 14*B^2*a*b^8*d^2))/d^4)*(((64*B^4*a^2*b^6*d^4 -
B^4*b^4*(16*a^4*d^4 + 16*b^4*d^4 + 32*a^2*b^2*d^4))^(1/2) + 8*B^2*a*b^3*d^2)/(16*(a^4*d^4 + b^4*d^4 + 2*a^2*b^
2*d^4)))^(1/2) + (32*(4*B^3*a^5*b^4*d^2 - 15*B^3*a^3*b^6*d^2 + B^3*a*b^8*d^2))/d^5)*(((64*B^4*a^2*b^6*d^4 - B^
4*b^4*(16*a^4*d^4 + 16*b^4*d^4 + 32*a^2*b^2*d^4))^(1/2) + 8*B^2*a*b^3*d^2)/(16*(a^4*d^4 + b^4*d^4 + 2*a^2*b^2*
d^4)))^(1/2) - (32*tan(c + d*x)^(1/2)*(B^4*b^9 + 2*B^4*a^4*b^5))/d^4)*(((64*B^4*a^2*b^6*d^4 - B^4*b^4*(16*a^4*
d^4 + 16*b^4*d^4 + 32*a^2*b^2*d^4))^(1/2) + 8*B^2*a*b^3*d^2)/(16*(a^4*d^4 + b^4*d^4 + 2*a^2*b^2*d^4)))^(1/2)*1
i - (((((32*(4*B*a^2*b^7*d^4 + 8*B*a^4*b^5*d^4 + 4*B*a^6*b^3*d^4))/d^5 + (32*tan(c + d*x)^(1/2)*(((64*B^4*a^2*
b^6*d^4 - B^4*b^4*(16*a^4*d^4 + 16*b^4*d^4 + 32*a^2*b^2*d^4))^(1/2) + 8*B^2*a*b^3*d^2)/(16*(a^4*d^4 + b^4*d^4
+ 2*a^2*b^2*d^4)))^(1/2)*(16*b^9*d^4 + 16*a^2*b^7*d^4 - 16*a^4*b^5*d^4 - 16*a^6*b^3*d^4))/d^4)*(((64*B^4*a^2*b
^6*d^4 - B^4*b^4*(16*a^4*d^4 + 16*b^4*d^4 + 32*a^2*b^2*d^4))^(1/2) + 8*B^2*a*b^3*d^2)/(16*(a^4*d^4 + b^4*d^4 +
2*a^2*b^2*d^4)))^(1/2) - (32*tan(c + d*x)^(1/2)*(14*B^2*a^5*b^4*d^2 - 4*B^2*a^3*b^6*d^2 + 14*B^2*a*b^8*d^2))/
d^4)*(((64*B^4*a^2*b^6*d^4 - B^4*b^4*(16*a^4*d^4 + 16*b^4*d^4 + 32*a^2*b^2*d^4))^(1/2) + 8*B^2*a*b^3*d^2)/(16*
(a^4*d^4 + b^4*d^4 + 2*a^2*b^2*d^4)))^(1/2) + (32*(4*B^3*a^5*b^4*d^2 - 15*B^3*a^3*b^6*d^2 + B^3*a*b^8*d^2))/d^
5)*(((64*B^4*a^2*b^6*d^4 - B^4*b^4*(16*a^4*d^4 + 16*b^4*d^4 + 32*a^2*b^2*d^4))^(1/2) + 8*B^2*a*b^3*d^2)/(16*(a
^4*d^4 + b^4*d^4 + 2*a^2*b^2*d^4)))^(1/2) + (32*tan(c + d*x)^(1/2)*(B^4*b^9 + 2*B^4*a^4*b^5))/d^4)*(((64*B^4*a
^2*b^6*d^4 - B^4*b^4*(16*a^4*d^4 + 16*b^4*d^4 + 32*a^2*b^2*d^4))^(1/2) + 8*B^2*a*b^3*d^2)/(16*(a^4*d^4 + b^4*d
^4 + 2*a^2*b^2*d^4)))^(1/2)*1i)/((((((32*(4*B*a^2*b^7*d^4 + 8*B*a^4*b^5*d^4 + 4*B*a^6*b^3*d^4))/d^5 - (32*tan(
c + d*x)^(1/2)*(((64*B^4*a^2*b^6*d^4 - B^4*b^4*(16*a^4*d^4 + 16*b^4*d^4 + 32*a^2*b^2*d^4))^(1/2) + 8*B^2*a*b^3
*d^2)/(16*(a^4*d^4 + b^4*d^4 + 2*a^2*b^2*d^4)))^(1/2)*(16*b^9*d^4 + 16*a^2*b^7*d^4 - 16*a^4*b^5*d^4 - 16*a^6*b
^3*d^4))/d^4)*(((64*B^4*a^2*b^6*d^4 - B^4*b^4*(16*a^4*d^4 + 16*b^4*d^4 + 32*a^2*b^2*d^4))^(1/2) + 8*B^2*a*b^3*
d^2)/(16*(a^4*d^4 + b^4*d^4 + 2*a^2*b^2*d^4)))^(1/2) + (32*tan(c + d*x)^(1/2)*(14*B^2*a^5*b^4*d^2 - 4*B^2*a^3*
b^6*d^2 + 14*B^2*a*b^8*d^2))/d^4)*(((64*B^4*a^2*b^6*d^4 - B^4*b^4*(16*a^4*d^4 + 16*b^4*d^4 + 32*a^2*b^2*d^4))^
(1/2) + 8*B^2*a*b^3*d^2)/(16*(a^4*d^4 + b^4*d^4 + 2*a^2*b^2*d^4)))^(1/2) + (32*(4*B^3*a^5*b^4*d^2 - 15*B^3*a^3
*b^6*d^2 + B^3*a*b^8*d^2))/d^5)*(((64*B^4*a^2*b^6*d^4 - B^4*b^4*(16*a^4*d^4 + 16*b^4*d^4 + 32*a^2*b^2*d^4))^(1
/2) + 8*B^2*a*b^3*d^2)/(16*(a^4*d^4 + b^4*d^4 + 2*a^2*b^2*d^4)))^(1/2) - (32*tan(c + d*x)^(1/2)*(B^4*b^9 + 2*B
^4*a^4*b^5))/d^4)*(((64*B^4*a^2*b^6*d^4 - B^4*b^4*(16*a^4*d^4 + 16*b^4*d^4 + 32*a^2*b^2*d^4))^(1/2) + 8*B^2*a*
b^3*d^2)/(16*(a^4*d^4 + b^4*d^4 + 2*a^2*b^2*d^4)))^(1/2) + (((((32*(4*B*a^2*b^7*d^4 + 8*B*a^4*b^5*d^4 + 4*B*a^
6*b^3*d^4))/d^5 + (32*tan(c + d*x)^(1/2)*(((64*B^4*a^2*b^6*d^4 - B^4*b^4*(16*a^4*d^4 + 16*b^4*d^4 + 32*a^2*b^2
*d^4))^(1/2) + 8*B^2*a*b^3*d^2)/(16*(a^4*d^4 + b^4*d^4 + 2*a^2*b^2*d^4)))^(1/2)*(16*b^9*d^4 + 16*a^2*b^7*d^4 -
16*a^4*b^5*d^4 - 16*a^6*b^3*d^4))/d^4)*(((64*B^4*a^2*b^6*d^4 - B^4*b^4*(16*a^4*d^4 + 16*b^4*d^4 + 32*a^2*b^2*
d^4))^(1/2) + 8*B^2*a*b^3*d^2)/(16*(a^4*d^4 + b^4*d^4 + 2*a^2*b^2*d^4)))^(1/2) - (32*tan(c + d*x)^(1/2)*(14*B^
2*a^5*b^4*d^2 - 4*B^2*a^3*b^6*d^2 + 14*B^2*a*b^8*d^2))/d^4)*(((64*B^4*a^2*b^6*d^4 - B^4*b^4*(16*a^4*d^4 + 16*b
^4*d^4 + 32*a^2*b^2*d^4))^(1/2) + 8*B^2*a*b^3*d^2)/(16*(a^4*d^4 + b^4*d^4 + 2*a^2*b^2*d^4)))^(1/2) + (32*(4*B^
3*a^5*b^4*d^2 - 15*B^3*a^3*b^6*d^2 + B^3*a*b^8*d^2))/d^5)*(((64*B^4*a^2*b^6*d^4 - B^4*b^4*(16*a^4*d^4 + 16*b^4
*d^4 + 32*a^2*b^2*d^4))^(1/2) + 8*B^2*a*b^3*d^2)/(16*(a^4*d^4 + b^4*d^4 + 2*a^2*b^2*d^4)))^(1/2) + (32*tan(c +
d*x)^(1/2)*(B^4*b^9 + 2*B^4*a^4*b^5))/d^4)*(((64*B^4*a^2*b^6*d^4 - B^4*b^4*(16*a^4*d^4 + 16*b^4*d^4 + 32*a^2*
b^2*d^4))^(1/2) + 8*B^2*a*b^3*d^2)/(16*(a^4*d^4 + b^4*d^4 + 2*a^2*b^2*d^4)))^(1/2) + (64*B^5*a^2*b^7)/d^5))*((
(64*B^4*a^2*b^6*d^4 - B^4*b^4*(16*a^4*d^4 + 16*b^4*d^4 + 32*a^2*b^2*d^4))^(1/2) + 8*B^2*a*b^3*d^2)/(16*(a^4*d^
4 + b^4*d^4 + 2*a^2*b^2*d^4)))^(1/2)*2i - (B*a^3*b*atan(((B*a^3*b*((32*tan(c + d*x)^(1/2)*(B^4*a^4*b^5 - 2*B^4
*a^6*b^3))/d^4 - (B*a^3*b*((32*(13*B^3*a^5*b^4*d^2 + B^3*a^7*b^2*d^2))/d^5 + (B*a^3*b*((32*tan(c + d*x)^(1/2)*
(20*B^2*a^5*b^4*d^2 - 14*B^2*a^3*b^6*d^2 + 2*B^2*a^7*b^2*d^2))/d^4 + (B*a^3*b*((32*(12*B*a^2*b^7*d^4 + 24*B*a^
4*b^5*d^4 + 12*B*a^6*b^3*d^4))/d^5 - (32*B*a^3*b*tan(c + d*x)^(1/2)*(16*b^9*d^4 + 16*a^2*b^7*d^4 - 16*a^4*b^5*
d^4 - 16*a^6*b^3*d^4))/(d^4*(- a^7*b*d^2 - a^3*b^5*d^2 - 2*a^5*b^3*d^2)^(1/2))))/(- a^7*b*d^2 - a^3*b^5*d^2 -
2*a^5*b^3*d^2)^(1/2)))/(- a^7*b*d^2 - a^3*b^5*d^2 - 2*a^5*b^3*d^2)^(1/2)))/(- a^7*b*d^2 - a^3*b^5*d^2 - 2*a^5*
b^3*d^2)^(1/2))*1i)/(- a^7*b*d^2 - a^3*b^5*d^2 - 2*a^5*b^3*d^2)^(1/2) + (B*a^3*b*((32*tan(c + d*x)^(1/2)*(B^4*
a^4*b^5 - 2*B^4*a^6*b^3))/d^4 + (B*a^3*b*((32*(13*B^3*a^5*b^4*d^2 + B^3*a^7*b^2*d^2))/d^5 - (B*a^3*b*((32*tan(
c + d*x)^(1/2)*(20*B^2*a^5*b^4*d^2 - 14*B^2*a^3*b^6*d^2 + 2*B^2*a^7*b^2*d^2))/d^4 - (B*a^3*b*((32*(12*B*a^2*b^
7*d^4 + 24*B*a^4*b^5*d^4 + 12*B*a^6*b^3*d^4))/d^5 + (32*B*a^3*b*tan(c + d*x)^(1/2)*(16*b^9*d^4 + 16*a^2*b^7*d^
4 - 16*a^4*b^5*d^4 - 16*a^6*b^3*d^4))/(d^4*(- a^7*b*d^2 - a^3*b^5*d^2 - 2*a^5*b^3*d^2)^(1/2))))/(- a^7*b*d^2 -
a^3*b^5*d^2 - 2*a^5*b^3*d^2)^(1/2)))/(- a^7*b*d^2 - a^3*b^5*d^2 - 2*a^5*b^3*d^2)^(1/2)))/(- a^7*b*d^2 - a^3*b
^5*d^2 - 2*a^5*b^3*d^2)^(1/2))*1i)/(- a^7*b*d^2 - a^3*b^5*d^2 - 2*a^5*b^3*d^2)^(1/2))/((64*B^5*a^6*b^3)/d^5 -
(B*a^3*b*((32*tan(c + d*x)^(1/2)*(B^4*a^4*b^5 - 2*B^4*a^6*b^3))/d^4 - (B*a^3*b*((32*(13*B^3*a^5*b^4*d^2 + B^3*
a^7*b^2*d^2))/d^5 + (B*a^3*b*((32*tan(c + d*x)^(1/2)*(20*B^2*a^5*b^4*d^2 - 14*B^2*a^3*b^6*d^2 + 2*B^2*a^7*b^2*
d^2))/d^4 + (B*a^3*b*((32*(12*B*a^2*b^7*d^4 + 24*B*a^4*b^5*d^4 + 12*B*a^6*b^3*d^4))/d^5 - (32*B*a^3*b*tan(c +
d*x)^(1/2)*(16*b^9*d^4 + 16*a^2*b^7*d^4 - 16*a^4*b^5*d^4 - 16*a^6*b^3*d^4))/(d^4*(- a^7*b*d^2 - a^3*b^5*d^2 -
2*a^5*b^3*d^2)^(1/2))))/(- a^7*b*d^2 - a^3*b^5*d^2 - 2*a^5*b^3*d^2)^(1/2)))/(- a^7*b*d^2 - a^3*b^5*d^2 - 2*a^5
*b^3*d^2)^(1/2)))/(- a^7*b*d^2 - a^3*b^5*d^2 - 2*a^5*b^3*d^2)^(1/2)))/(- a^7*b*d^2 - a^3*b^5*d^2 - 2*a^5*b^3*d
^2)^(1/2) + (B*a^3*b*((32*tan(c + d*x)^(1/2)*(B^4*a^4*b^5 - 2*B^4*a^6*b^3))/d^4 + (B*a^3*b*((32*(13*B^3*a^5*b^
4*d^2 + B^3*a^7*b^2*d^2))/d^5 - (B*a^3*b*((32*tan(c + d*x)^(1/2)*(20*B^2*a^5*b^4*d^2 - 14*B^2*a^3*b^6*d^2 + 2*
B^2*a^7*b^2*d^2))/d^4 - (B*a^3*b*((32*(12*B*a^2*b^7*d^4 + 24*B*a^4*b^5*d^4 + 12*B*a^6*b^3*d^4))/d^5 + (32*B*a^
3*b*tan(c + d*x)^(1/2)*(16*b^9*d^4 + 16*a^2*b^7*d^4 - 16*a^4*b^5*d^4 - 16*a^6*b^3*d^4))/(d^4*(- a^7*b*d^2 - a^
3*b^5*d^2 - 2*a^5*b^3*d^2)^(1/2))))/(- a^7*b*d^2 - a^3*b^5*d^2 - 2*a^5*b^3*d^2)^(1/2)))/(- a^7*b*d^2 - a^3*b^5
*d^2 - 2*a^5*b^3*d^2)^(1/2)))/(- a^7*b*d^2 - a^3*b^5*d^2 - 2*a^5*b^3*d^2)^(1/2)))/(- a^7*b*d^2 - a^3*b^5*d^2 -
2*a^5*b^3*d^2)^(1/2)))*2i)/(- a^7*b*d^2 - a^3*b^5*d^2 - 2*a^5*b^3*d^2)^(1/2) + (B*a^3*b*atan(((B*a^3*b*((32*t
an(c + d*x)^(1/2)*(B^4*b^9 + 2*B^4*a^4*b^5))/d^4 - (B*a^3*b*((32*(4*B^3*a^5*b^4*d^2 - 15*B^3*a^3*b^6*d^2 + B^3
*a*b^8*d^2))/d^5 + (B*a^3*b*((32*tan(c + d*x)^(1/2)*(14*B^2*a^5*b^4*d^2 - 4*B^2*a^3*b^6*d^2 + 14*B^2*a*b^8*d^2
))/d^4 + (B*a^3*b*((32*(4*B*a^2*b^7*d^4 + 8*B*a^4*b^5*d^4 + 4*B*a^6*b^3*d^4))/d^5 - (32*B*a^3*b*tan(c + d*x)^(
1/2)*(16*b^9*d^4 + 16*a^2*b^7*d^4 - 16*a^4*b^5*d^4 - 16*a^6*b^3*d^4))/(d^4*(- a^7*b*d^2 - a^3*b^5*d^2 - 2*a^5*
b^3*d^2)^(1/2))))/(- a^7*b*d^2 - a^3*b^5*d^2 - 2*a^5*b^3*d^2)^(1/2)))/(- a^7*b*d^2 - a^3*b^5*d^2 - 2*a^5*b^3*d
^2)^(1/2)))/(- a^7*b*d^2 - a^3*b^5*d^2 - 2*a^5*b^3*d^2)^(1/2))*1i)/(- a^7*b*d^2 - a^3*b^5*d^2 - 2*a^5*b^3*d^2)
^(1/2) + (B*a^3*b*((32*tan(c + d*x)^(1/2)*(B^4*b^9 + 2*B^4*a^4*b^5))/d^4 + (B*a^3*b*((32*(4*B^3*a^5*b^4*d^2 -
15*B^3*a^3*b^6*d^2 + B^3*a*b^8*d^2))/d^5 - (B*a^3*b*((32*tan(c + d*x)^(1/2)*(14*B^2*a^5*b^4*d^2 - 4*B^2*a^3*b^
6*d^2 + 14*B^2*a*b^8*d^2))/d^4 - (B*a^3*b*((32*(4*B*a^2*b^7*d^4 + 8*B*a^4*b^5*d^4 + 4*B*a^6*b^3*d^4))/d^5 + (3
2*B*a^3*b*tan(c + d*x)^(1/2)*(16*b^9*d^4 + 16*a^2*b^7*d^4 - 16*a^4*b^5*d^4 - 16*a^6*b^3*d^4))/(d^4*(- a^7*b*d^
2 - a^3*b^5*d^2 - 2*a^5*b^3*d^2)^(1/2))))/(- a^7*b*d^2 - a^3*b^5*d^2 - 2*a^5*b^3*d^2)^(1/2)))/(- a^7*b*d^2 - a
^3*b^5*d^2 - 2*a^5*b^3*d^2)^(1/2)))/(- a^7*b*d^2 - a^3*b^5*d^2 - 2*a^5*b^3*d^2)^(1/2))*1i)/(- a^7*b*d^2 - a^3*
b^5*d^2 - 2*a^5*b^3*d^2)^(1/2))/((64*B^5*a^2*b^7)/d^5 - (B*a^3*b*((32*tan(c + d*x)^(1/2)*(B^4*b^9 + 2*B^4*a^4*
b^5))/d^4 - (B*a^3*b*((32*(4*B^3*a^5*b^4*d^2 - 15*B^3*a^3*b^6*d^2 + B^3*a*b^8*d^2))/d^5 + (B*a^3*b*((32*tan(c
+ d*x)^(1/2)*(14*B^2*a^5*b^4*d^2 - 4*B^2*a^3*b^6*d^2 + 14*B^2*a*b^8*d^2))/d^4 + (B*a^3*b*((32*(4*B*a^2*b^7*d^4
+ 8*B*a^4*b^5*d^4 + 4*B*a^6*b^3*d^4))/d^5 - (32*B*a^3*b*tan(c + d*x)^(1/2)*(16*b^9*d^4 + 16*a^2*b^7*d^4 - 16*
a^4*b^5*d^4 - 16*a^6*b^3*d^4))/(d^4*(- a^7*b*d^2 - a^3*b^5*d^2 - 2*a^5*b^3*d^2)^(1/2))))/(- a^7*b*d^2 - a^3*b^
5*d^2 - 2*a^5*b^3*d^2)^(1/2)))/(- a^7*b*d^2 - a^3*b^5*d^2 - 2*a^5*b^3*d^2)^(1/2)))/(- a^7*b*d^2 - a^3*b^5*d^2
- 2*a^5*b^3*d^2)^(1/2)))/(- a^7*b*d^2 - a^3*b^5*d^2 - 2*a^5*b^3*d^2)^(1/2) + (B*a^3*b*((32*tan(c + d*x)^(1/2)*
(B^4*b^9 + 2*B^4*a^4*b^5))/d^4 + (B*a^3*b*((32*(4*B^3*a^5*b^4*d^2 - 15*B^3*a^3*b^6*d^2 + B^3*a*b^8*d^2))/d^5 -
(B*a^3*b*((32*tan(c + d*x)^(1/2)*(14*B^2*a^5*b^4*d^2 - 4*B^2*a^3*b^6*d^2 + 14*B^2*a*b^8*d^2))/d^4 - (B*a^3*b*
((32*(4*B*a^2*b^7*d^4 + 8*B*a^4*b^5*d^4 + 4*B*a^6*b^3*d^4))/d^5 + (32*B*a^3*b*tan(c + d*x)^(1/2)*(16*b^9*d^4 +
16*a^2*b^7*d^4 - 16*a^4*b^5*d^4 - 16*a^6*b^3*d^4))/(d^4*(- a^7*b*d^2 - a^3*b^5*d^2 - 2*a^5*b^3*d^2)^(1/2))))/
(- a^7*b*d^2 - a^3*b^5*d^2 - 2*a^5*b^3*d^2)^(1/2)))/(- a^7*b*d^2 - a^3*b^5*d^2 - 2*a^5*b^3*d^2)^(1/2)))/(- a^7
*b*d^2 - a^3*b^5*d^2 - 2*a^5*b^3*d^2)^(1/2)))/(- a^7*b*d^2 - a^3*b^5*d^2 - 2*a^5*b^3*d^2)^(1/2)))*2i)/(- a^7*b
*d^2 - a^3*b^5*d^2 - 2*a^5*b^3*d^2)^(1/2)