\(\int \frac {\tan ^{\frac {5}{2}}(c+d x) (a B+b B \tan (c+d x))}{(a+b \tan (c+d x))^2} \, dx\) [422]

Optimal result

Integrand size = 36, antiderivative size = 256 \[ \int \frac {\tan ^{\frac {5}{2}}(c+d x) (a B+b B \tan (c+d x))}{(a+b \tan (c+d x))^2} \, dx=\frac {(a+b) B \arctan \left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} \left (a^2+b^2\right ) d}-\frac {(a+b) B \arctan \left (1+\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} \left (a^2+b^2\right ) d}-\frac {2 a^{5/2} B \arctan \left (\frac {\sqrt {b} \sqrt {\tan (c+d x)}}{\sqrt {a}}\right )}{b^{3/2} \left (a^2+b^2\right ) d}-\frac {(a-b) B \log \left (1-\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right )}{2 \sqrt {2} \left (a^2+b^2\right ) d}+\frac {(a-b) B \log \left (1+\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right )}{2 \sqrt {2} \left (a^2+b^2\right ) d}+\frac {2 B \sqrt {\tan (c+d x)}}{b d} \]

[Out]

-2*a^(5/2)*B*arctan(b^(1/2)*tan(d*x+c)^(1/2)/a^(1/2))/b^(3/2)/(a^2+b^2)/d-1/2*(a+b)*B*arctan(-1+2^(1/2)*tan(d*
x+c)^(1/2))/(a^2+b^2)/d*2^(1/2)-1/2*(a+b)*B*arctan(1+2^(1/2)*tan(d*x+c)^(1/2))/(a^2+b^2)/d*2^(1/2)-1/4*(a-b)*B
*ln(1-2^(1/2)*tan(d*x+c)^(1/2)+tan(d*x+c))/(a^2+b^2)/d*2^(1/2)+1/4*(a-b)*B*ln(1+2^(1/2)*tan(d*x+c)^(1/2)+tan(d
*x+c))/(a^2+b^2)/d*2^(1/2)+2*B*tan(d*x+c)^(1/2)/b/d

Rubi [A] (verified)

Time = 0.55 (sec) , antiderivative size = 256, normalized size of antiderivative = 1.00, number of steps used = 16, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.361, Rules used = {21, 3647, 3734, 3615, 1182, 1176, 631, 210, 1179, 642, 3715, 65, 211} \[ \int \frac {\tan ^{\frac {5}{2}}(c+d x) (a B+b B \tan (c+d x))}{(a+b \tan (c+d x))^2} \, dx=\frac {B (a+b) \arctan \left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} d \left (a^2+b^2\right )}-\frac {B (a+b) \arctan \left (\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{\sqrt {2} d \left (a^2+b^2\right )}-\frac {B (a-b) \log \left (\tan (c+d x)-\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{2 \sqrt {2} d \left (a^2+b^2\right )}+\frac {B (a-b) \log \left (\tan (c+d x)+\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{2 \sqrt {2} d \left (a^2+b^2\right )}-\frac {2 a^{5/2} B \arctan \left (\frac {\sqrt {b} \sqrt {\tan (c+d x)}}{\sqrt {a}}\right )}{b^{3/2} d \left (a^2+b^2\right )}+\frac {2 B \sqrt {\tan (c+d x)}}{b d} \]

[In]

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

[Out]

((a + b)*B*ArcTan[1 - Sqrt[2]*Sqrt[Tan[c + d*x]]])/(Sqrt[2]*(a^2 + b^2)*d) - ((a + b)*B*ArcTan[1 + Sqrt[2]*Sqr
t[Tan[c + d*x]]])/(Sqrt[2]*(a^2 + b^2)*d) - (2*a^(5/2)*B*ArcTan[(Sqrt[b]*Sqrt[Tan[c + d*x]])/Sqrt[a]])/(b^(3/2
)*(a^2 + b^2)*d) - ((a - b)*B*Log[1 - Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]])/(2*Sqrt[2]*(a^2 + b^2)*d) +
((a - b)*B*Log[1 + Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]])/(2*Sqrt[2]*(a^2 + b^2)*d) + (2*B*Sqrt[Tan[c + d
*x]])/(b*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 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 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 211

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

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 1182

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

Rule 3615

Int[((c_) + (d_.)*tan[(e_.) + (f_.)*(x_)])/Sqrt[(b_.)*tan[(e_.) + (f_.)*(x_)]], x_Symbol] :> Dist[2/f, Subst[I
nt[(b*c + d*x^2)/(b^2 + x^4), x], x, Sqrt[b*Tan[e + f*x]]], x] /; FreeQ[{b, c, d, e, f}, x] && NeQ[c^2 - d^2,
0] && NeQ[c^2 + d^2, 0]

Rule 3647

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Si
mp[b^2*(a + b*Tan[e + f*x])^(m - 2)*((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 - 3)*(c + d*Tan[e + f*x])^n*Simp[a^3*d*(m + n - 1) - b^2*(b*c*(m - 2) + a*d*(
1 + n)) + b*d*(m + n - 1)*(3*a^2 - b^2)*Tan[e + f*x] - b^2*(b*c*(m - 2) - a*d*(3*m + 2*n - 4))*Tan[e + f*x]^2,
x], x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0]
&& IntegerQ[2*m] && GtQ[m, 2] && (GeQ[n, -1] || IntegerQ[m]) &&  !(IGtQ[n, 2] && ( !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 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}& = B \int \frac {\tan ^{\frac {5}{2}}(c+d x)}{a+b \tan (c+d x)} \, dx \\ & = \frac {2 B \sqrt {\tan (c+d x)}}{b d}+\frac {(2 B) \int \frac {-\frac {a}{2}-\frac {1}{2} b \tan (c+d x)-\frac {1}{2} a \tan ^2(c+d x)}{\sqrt {\tan (c+d x)} (a+b \tan (c+d x))} \, dx}{b} \\ & = \frac {2 B \sqrt {\tan (c+d x)}}{b d}+\frac {(2 B) \int \frac {-\frac {b^2}{2}-\frac {1}{2} a b \tan (c+d x)}{\sqrt {\tan (c+d x)}} \, dx}{b \left (a^2+b^2\right )}-\frac {\left (a^3 B\right ) \int \frac {1+\tan ^2(c+d x)}{\sqrt {\tan (c+d x)} (a+b \tan (c+d x))} \, dx}{b \left (a^2+b^2\right )} \\ & = \frac {2 B \sqrt {\tan (c+d x)}}{b d}+\frac {(4 B) \text {Subst}\left (\int \frac {-\frac {b^2}{2}-\frac {1}{2} a b x^2}{1+x^4} \, dx,x,\sqrt {\tan (c+d x)}\right )}{b \left (a^2+b^2\right ) d}-\frac {\left (a^3 B\right ) \text {Subst}\left (\int \frac {1}{\sqrt {x} (a+b x)} \, dx,x,\tan (c+d x)\right )}{b \left (a^2+b^2\right ) d} \\ & = \frac {2 B \sqrt {\tan (c+d x)}}{b d}+\frac {((a-b) B) \text {Subst}\left (\int \frac {1-x^2}{1+x^4} \, dx,x,\sqrt {\tan (c+d x)}\right )}{\left (a^2+b^2\right ) d}-\frac {\left (2 a^3 B\right ) \text {Subst}\left (\int \frac {1}{a+b x^2} \, dx,x,\sqrt {\tan (c+d x)}\right )}{b \left (a^2+b^2\right ) d}-\frac {((a+b) B) \text {Subst}\left (\int \frac {1+x^2}{1+x^4} \, dx,x,\sqrt {\tan (c+d x)}\right )}{\left (a^2+b^2\right ) d} \\ & = -\frac {2 a^{5/2} B \arctan \left (\frac {\sqrt {b} \sqrt {\tan (c+d x)}}{\sqrt {a}}\right )}{b^{3/2} \left (a^2+b^2\right ) d}+\frac {2 B \sqrt {\tan (c+d x)}}{b d}-\frac {((a-b) 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} \left (a^2+b^2\right ) d}-\frac {((a-b) 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} \left (a^2+b^2\right ) d}-\frac {((a+b) B) \text {Subst}\left (\int \frac {1}{1-\sqrt {2} x+x^2} \, dx,x,\sqrt {\tan (c+d x)}\right )}{2 \left (a^2+b^2\right ) d}-\frac {((a+b) B) \text {Subst}\left (\int \frac {1}{1+\sqrt {2} x+x^2} \, dx,x,\sqrt {\tan (c+d x)}\right )}{2 \left (a^2+b^2\right ) d} \\ & = -\frac {2 a^{5/2} B \arctan \left (\frac {\sqrt {b} \sqrt {\tan (c+d x)}}{\sqrt {a}}\right )}{b^{3/2} \left (a^2+b^2\right ) d}-\frac {(a-b) B \log \left (1-\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right )}{2 \sqrt {2} \left (a^2+b^2\right ) d}+\frac {(a-b) B \log \left (1+\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right )}{2 \sqrt {2} \left (a^2+b^2\right ) d}+\frac {2 B \sqrt {\tan (c+d x)}}{b d}-\frac {((a+b) B) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} \left (a^2+b^2\right ) d}+\frac {((a+b) B) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} \left (a^2+b^2\right ) d} \\ & = \frac {(a+b) B \arctan \left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} \left (a^2+b^2\right ) d}-\frac {(a+b) B \arctan \left (1+\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} \left (a^2+b^2\right ) d}-\frac {2 a^{5/2} B \arctan \left (\frac {\sqrt {b} \sqrt {\tan (c+d x)}}{\sqrt {a}}\right )}{b^{3/2} \left (a^2+b^2\right ) d}-\frac {(a-b) B \log \left (1-\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right )}{2 \sqrt {2} \left (a^2+b^2\right ) d}+\frac {(a-b) B \log \left (1+\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right )}{2 \sqrt {2} \left (a^2+b^2\right ) d}+\frac {2 B \sqrt {\tan (c+d x)}}{b d} \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 0.17 (sec) , antiderivative size = 156, normalized size of antiderivative = 0.61 \[ \int \frac {\tan ^{\frac {5}{2}}(c+d x) (a B+b B \tan (c+d x))}{(a+b \tan (c+d x))^2} \, dx=\frac {B \left (\sqrt [4]{-1} b^{3/2} (-i a+b) \arctan \left ((-1)^{3/4} \sqrt {\tan (c+d x)}\right )-2 a^{5/2} \arctan \left (\frac {\sqrt {b} \sqrt {\tan (c+d x)}}{\sqrt {a}}\right )+\sqrt [4]{-1} b^{3/2} (i a+b) \text {arctanh}\left ((-1)^{3/4} \sqrt {\tan (c+d x)}\right )+2 a^2 \sqrt {b} \sqrt {\tan (c+d x)}+2 b^{5/2} \sqrt {\tan (c+d x)}\right )}{b^{3/2} \left (a^2+b^2\right ) d} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 0.05 (sec) , antiderivative size = 242, normalized size of antiderivative = 0.95

method	result	size
derivativedivides	\(\frac {B \left (\frac {2 \left (\sqrt {\tan }\left (d x +c \right )\right )}{b}-\frac {2 a^{3} \arctan \left (\frac {b \left (\sqrt {\tan }\left (d x +c \right )\right )}{\sqrt {a b}}\right )}{b \left (a^{2}+b^{2}\right ) \sqrt {a b}}+\frac {-\frac {b \sqrt {2}\, \left (\ln \left (\frac {1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}{1-\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}\right )+2 \arctan \left (1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )+2 \arctan \left (-1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )\right )}{4}-\frac {a \sqrt {2}\, \left (\ln \left (\frac {1-\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}{1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}\right )+2 \arctan \left (1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )+2 \arctan \left (-1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )\right )}{4}}{a^{2}+b^{2}}\right )}{d}\)	\(242\)
default	\(\frac {B \left (\frac {2 \left (\sqrt {\tan }\left (d x +c \right )\right )}{b}-\frac {2 a^{3} \arctan \left (\frac {b \left (\sqrt {\tan }\left (d x +c \right )\right )}{\sqrt {a b}}\right )}{b \left (a^{2}+b^{2}\right ) \sqrt {a b}}+\frac {-\frac {b \sqrt {2}\, \left (\ln \left (\frac {1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}{1-\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}\right )+2 \arctan \left (1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )+2 \arctan \left (-1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )\right )}{4}-\frac {a \sqrt {2}\, \left (\ln \left (\frac {1-\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}{1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}\right )+2 \arctan \left (1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )+2 \arctan \left (-1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )\right )}{4}}{a^{2}+b^{2}}\right )}{d}\)	\(242\)

[In]

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

[Out]

1/d*B*(2*tan(d*x+c)^(1/2)/b-2/b*a^3/(a^2+b^2)/(a*b)^(1/2)*arctan(b*tan(d*x+c)^(1/2)/(a*b)^(1/2))+2/(a^2+b^2)*(
-1/8*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)))-1/8*a*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 [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1606 vs. \(2 (216) = 432\).

Time = 0.34 (sec) , antiderivative size = 3238, normalized size of antiderivative = 12.65 \[ \int \frac {\tan ^{\frac {5}{2}}(c+d x) (a B+b B \tan (c+d x))}{(a+b \tan (c+d x))^2} \, dx=\text {Too large to display} \]

[In]

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

[Out]

[1/2*(2*B*a^2*sqrt(-a/b)*log(-(2*b*sqrt(-a/b)*sqrt(tan(d*x + c)) - b*tan(d*x + c) + a)/(b*tan(d*x + c) + a)) -
(a^2*b + b^3)*d*sqrt(-(2*B^2*a*b + (a^4 + 2*a^2*b^2 + b^4)*d^2*sqrt(-(B^4*a^4 - 2*B^4*a^2*b^2 + B^4*b^4)/((a^
8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8)*d^4)))/((a^4 + 2*a^2*b^2 + b^4)*d^2))*log(((a^5 + 2*a^3*b^2 + a*b
^4)*d^3*sqrt(-(B^4*a^4 - 2*B^4*a^2*b^2 + B^4*b^4)/((a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8)*d^4)) - (B^
2*a^2*b - B^2*b^3)*d)*sqrt(-(2*B^2*a*b + (a^4 + 2*a^2*b^2 + b^4)*d^2*sqrt(-(B^4*a^4 - 2*B^4*a^2*b^2 + B^4*b^4)
/((a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8)*d^4)))/((a^4 + 2*a^2*b^2 + b^4)*d^2)) - (B^3*a^2 - B^3*b^2)*
sqrt(tan(d*x + c))) + (a^2*b + b^3)*d*sqrt(-(2*B^2*a*b + (a^4 + 2*a^2*b^2 + b^4)*d^2*sqrt(-(B^4*a^4 - 2*B^4*a^
2*b^2 + B^4*b^4)/((a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8)*d^4)))/((a^4 + 2*a^2*b^2 + b^4)*d^2))*log(-(
(a^5 + 2*a^3*b^2 + a*b^4)*d^3*sqrt(-(B^4*a^4 - 2*B^4*a^2*b^2 + B^4*b^4)/((a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*
b^6 + b^8)*d^4)) - (B^2*a^2*b - B^2*b^3)*d)*sqrt(-(2*B^2*a*b + (a^4 + 2*a^2*b^2 + b^4)*d^2*sqrt(-(B^4*a^4 - 2*
B^4*a^2*b^2 + B^4*b^4)/((a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8)*d^4)))/((a^4 + 2*a^2*b^2 + b^4)*d^2))
- (B^3*a^2 - B^3*b^2)*sqrt(tan(d*x + c))) + (a^2*b + b^3)*d*sqrt(-(2*B^2*a*b - (a^4 + 2*a^2*b^2 + b^4)*d^2*sqr
t(-(B^4*a^4 - 2*B^4*a^2*b^2 + B^4*b^4)/((a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8)*d^4)))/((a^4 + 2*a^2*b
^2 + b^4)*d^2))*log(((a^5 + 2*a^3*b^2 + a*b^4)*d^3*sqrt(-(B^4*a^4 - 2*B^4*a^2*b^2 + B^4*b^4)/((a^8 + 4*a^6*b^2
+ 6*a^4*b^4 + 4*a^2*b^6 + b^8)*d^4)) + (B^2*a^2*b - B^2*b^3)*d)*sqrt(-(2*B^2*a*b - (a^4 + 2*a^2*b^2 + b^4)*d^
2*sqrt(-(B^4*a^4 - 2*B^4*a^2*b^2 + B^4*b^4)/((a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8)*d^4)))/((a^4 + 2*
a^2*b^2 + b^4)*d^2)) - (B^3*a^2 - B^3*b^2)*sqrt(tan(d*x + c))) - (a^2*b + b^3)*d*sqrt(-(2*B^2*a*b - (a^4 + 2*a
^2*b^2 + b^4)*d^2*sqrt(-(B^4*a^4 - 2*B^4*a^2*b^2 + B^4*b^4)/((a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8)*d
^4)))/((a^4 + 2*a^2*b^2 + b^4)*d^2))*log(-((a^5 + 2*a^3*b^2 + a*b^4)*d^3*sqrt(-(B^4*a^4 - 2*B^4*a^2*b^2 + B^4*
b^4)/((a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8)*d^4)) + (B^2*a^2*b - B^2*b^3)*d)*sqrt(-(2*B^2*a*b - (a^4
+ 2*a^2*b^2 + b^4)*d^2*sqrt(-(B^4*a^4 - 2*B^4*a^2*b^2 + B^4*b^4)/((a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 +
b^8)*d^4)))/((a^4 + 2*a^2*b^2 + b^4)*d^2)) - (B^3*a^2 - B^3*b^2)*sqrt(tan(d*x + c))) + 4*(B*a^2 + B*b^2)*sqrt(
tan(d*x + c)))/((a^2*b + b^3)*d), -1/2*(4*B*a^2*sqrt(a/b)*arctan(b*sqrt(a/b)*sqrt(tan(d*x + c))/a) + (a^2*b +
b^3)*d*sqrt(-(2*B^2*a*b + (a^4 + 2*a^2*b^2 + b^4)*d^2*sqrt(-(B^4*a^4 - 2*B^4*a^2*b^2 + B^4*b^4)/((a^8 + 4*a^6*
b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8)*d^4)))/((a^4 + 2*a^2*b^2 + b^4)*d^2))*log(((a^5 + 2*a^3*b^2 + a*b^4)*d^3*sq
rt(-(B^4*a^4 - 2*B^4*a^2*b^2 + B^4*b^4)/((a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8)*d^4)) - (B^2*a^2*b -
B^2*b^3)*d)*sqrt(-(2*B^2*a*b + (a^4 + 2*a^2*b^2 + b^4)*d^2*sqrt(-(B^4*a^4 - 2*B^4*a^2*b^2 + B^4*b^4)/((a^8 + 4
*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8)*d^4)))/((a^4 + 2*a^2*b^2 + b^4)*d^2)) - (B^3*a^2 - B^3*b^2)*sqrt(tan(d
*x + c))) - (a^2*b + b^3)*d*sqrt(-(2*B^2*a*b + (a^4 + 2*a^2*b^2 + b^4)*d^2*sqrt(-(B^4*a^4 - 2*B^4*a^2*b^2 + B^
4*b^4)/((a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8)*d^4)))/((a^4 + 2*a^2*b^2 + b^4)*d^2))*log(-((a^5 + 2*a
^3*b^2 + a*b^4)*d^3*sqrt(-(B^4*a^4 - 2*B^4*a^2*b^2 + B^4*b^4)/((a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8)
*d^4)) - (B^2*a^2*b - B^2*b^3)*d)*sqrt(-(2*B^2*a*b + (a^4 + 2*a^2*b^2 + b^4)*d^2*sqrt(-(B^4*a^4 - 2*B^4*a^2*b^
2 + B^4*b^4)/((a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8)*d^4)))/((a^4 + 2*a^2*b^2 + b^4)*d^2)) - (B^3*a^2
- B^3*b^2)*sqrt(tan(d*x + c))) - (a^2*b + b^3)*d*sqrt(-(2*B^2*a*b - (a^4 + 2*a^2*b^2 + b^4)*d^2*sqrt(-(B^4*a^
4 - 2*B^4*a^2*b^2 + B^4*b^4)/((a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8)*d^4)))/((a^4 + 2*a^2*b^2 + b^4)*
d^2))*log(((a^5 + 2*a^3*b^2 + a*b^4)*d^3*sqrt(-(B^4*a^4 - 2*B^4*a^2*b^2 + B^4*b^4)/((a^8 + 4*a^6*b^2 + 6*a^4*b
^4 + 4*a^2*b^6 + b^8)*d^4)) + (B^2*a^2*b - B^2*b^3)*d)*sqrt(-(2*B^2*a*b - (a^4 + 2*a^2*b^2 + b^4)*d^2*sqrt(-(B
^4*a^4 - 2*B^4*a^2*b^2 + B^4*b^4)/((a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8)*d^4)))/((a^4 + 2*a^2*b^2 +
b^4)*d^2)) - (B^3*a^2 - B^3*b^2)*sqrt(tan(d*x + c))) + (a^2*b + b^3)*d*sqrt(-(2*B^2*a*b - (a^4 + 2*a^2*b^2 + b
^4)*d^2*sqrt(-(B^4*a^4 - 2*B^4*a^2*b^2 + B^4*b^4)/((a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8)*d^4)))/((a^
4 + 2*a^2*b^2 + b^4)*d^2))*log(-((a^5 + 2*a^3*b^2 + a*b^4)*d^3*sqrt(-(B^4*a^4 - 2*B^4*a^2*b^2 + B^4*b^4)/((a^8
+ 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8)*d^4)) + (B^2*a^2*b - B^2*b^3)*d)*sqrt(-(2*B^2*a*b - (a^4 + 2*a^2*b
^2 + b^4)*d^2*sqrt(-(B^4*a^4 - 2*B^4*a^2*b^2 + B^4*b^4)/((a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8)*d^4))
)/((a^4 + 2*a^2*b^2 + b^4)*d^2)) - (B^3*a^2 - B^3*b^2)*sqrt(tan(d*x + c))) - 4*(B*a^2 + B*b^2)*sqrt(tan(d*x +
c)))/((a^2*b + b^3)*d)]

Sympy [F]

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

[In]

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

[Out]

B*Integral(tan(c + d*x)**(5/2)/(a + b*tan(c + d*x)), x)

Maxima [A] (verification not implemented)

none

Time = 0.34 (sec) , antiderivative size = 188, normalized size of antiderivative = 0.73 \[ \int \frac {\tan ^{\frac {5}{2}}(c+d x) (a B+b B \tan (c+d x))}{(a+b \tan (c+d x))^2} \, dx=-\frac {\frac {8 \, B a^{3} \arctan \left (\frac {b \sqrt {\tan \left (d x + c\right )}}{\sqrt {a b}}\right )}{{\left (a^{2} b + b^{3}\right )} \sqrt {a b}} + \frac {{\left (2 \, \sqrt {2} {\left (a + b\right )} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + 2 \, \sqrt {\tan \left (d x + c\right )}\right )}\right ) + 2 \, \sqrt {2} {\left (a + b\right )} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - 2 \, \sqrt {\tan \left (d x + c\right )}\right )}\right ) - \sqrt {2} {\left (a - b\right )} \log \left (\sqrt {2} \sqrt {\tan \left (d x + c\right )} + \tan \left (d x + c\right ) + 1\right ) + \sqrt {2} {\left (a - b\right )} \log \left (-\sqrt {2} \sqrt {\tan \left (d x + c\right )} + \tan \left (d x + c\right ) + 1\right )\right )} B}{a^{2} + b^{2}} - \frac {8 \, B \sqrt {\tan \left (d x + c\right )}}{b}}{4 \, d} \]

[In]

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

[Out]

-1/4*(8*B*a^3*arctan(b*sqrt(tan(d*x + c))/sqrt(a*b))/((a^2*b + b^3)*sqrt(a*b)) + (2*sqrt(2)*(a + b)*arctan(1/2
*sqrt(2)*(sqrt(2) + 2*sqrt(tan(d*x + c)))) + 2*sqrt(2)*(a + b)*arctan(-1/2*sqrt(2)*(sqrt(2) - 2*sqrt(tan(d*x +
c)))) - sqrt(2)*(a - b)*log(sqrt(2)*sqrt(tan(d*x + c)) + tan(d*x + c) + 1) + sqrt(2)*(a - b)*log(-sqrt(2)*sqr
t(tan(d*x + c)) + tan(d*x + c) + 1))*B/(a^2 + b^2) - 8*B*sqrt(tan(d*x + c))/b)/d

Giac [F(-1)]

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

[In]

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

[Out]

Timed out

Mupad [B] (verification not implemented)

Time = 42.44 (sec) , antiderivative size = 18514, normalized size of antiderivative = 72.32 \[ \int \frac {\tan ^{\frac {5}{2}}(c+d x) (a B+b B \tan (c+d x))}{(a+b \tan (c+d x))^2} \, dx=\text {Too large to display} \]

[In]

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

[Out]

(log(((((((((128*b^3*tan(c + d*x)^(1/2)*(a^2 - b^2)*(a^2 + b^2)^2*((4*(-B^4*a^4*d^4*(a^4 + b^4 - 6*a^2*b^2)^2)
^(1/2) + 16*B^2*a^3*b^3*d^2 - 16*B^2*a^5*b*d^2)/(d^4*(a^2 + b^2)^4))^(1/2) + (768*B*a^3*b^3*(a^2 + b^2))/d)*((
4*(-B^4*a^4*d^4*(a^4 + b^4 - 6*a^2*b^2)^2)^(1/2) + 16*B^2*a^3*b^3*d^2 - 16*B^2*a^5*b*d^2)/(d^4*(a^2 + b^2)^4))
^(1/2))/4 + (64*B^2*a^3*tan(c + d*x)^(1/2)*(2*a^8 + 15*b^8 - 17*a^2*b^6 + 51*a^4*b^4 + 21*a^6*b^2))/(d^2*(a^2
+ b^2)^2))*((4*(-B^4*a^4*d^4*(a^4 + b^4 - 6*a^2*b^2)^2)^(1/2) + 16*B^2*a^3*b^3*d^2 - 16*B^2*a^5*b*d^2)/(d^4*(a
^2 + b^2)^4))^(1/2))/4 + (32*B^3*a^4*(4*a^8 + b^8 - 77*a^2*b^6 + 47*a^4*b^4 + 33*a^6*b^2))/(d^3*(a^2 + b^2)^3)
)*((4*(-B^4*a^4*d^4*(a^4 + b^4 - 6*a^2*b^2)^2)^(1/2) + 16*B^2*a^3*b^3*d^2 - 16*B^2*a^5*b*d^2)/(d^4*(a^2 + b^2)
^4))^(1/2))/4 + (16*B^4*a^4*tan(c + d*x)^(1/2)*(a^10 - 2*b^10 - 4*a^2*b^8 - 27*a^4*b^6 + 15*a^6*b^4 + 9*a^8*b^
2))/(b*d^4*(a^2 + b^2)^4))*((4*(-B^4*a^4*d^4*(a^4 + b^4 - 6*a^2*b^2)^2)^(1/2) + 16*B^2*a^3*b^3*d^2 - 16*B^2*a^
5*b*d^2)/(d^4*(a^2 + b^2)^4))^(1/2))/4 + (8*B^5*a^7*(a^6 + 10*b^6 + 27*a^2*b^4 + 10*a^4*b^2))/(b*d^5*(a^2 + b^
2)^4))*(((192*B^4*a^6*b^6*d^4 - 16*B^4*a^4*b^8*d^4 - 16*B^4*a^12*d^4 - 608*B^4*a^8*b^4*d^4 + 192*B^4*a^10*b^2*
d^4)^(1/2) + 16*B^2*a^3*b^3*d^2 - 16*B^2*a^5*b*d^2)/(a^8*d^4 + b^8*d^4 + 4*a^2*b^6*d^4 + 6*a^4*b^4*d^4 + 4*a^6
*b^2*d^4))^(1/2))/4 + (log(((((((((128*b^3*tan(c + d*x)^(1/2)*(a^2 - b^2)*(a^2 + b^2)^2*(-(4*(-B^4*a^4*d^4*(a^
4 + b^4 - 6*a^2*b^2)^2)^(1/2) - 16*B^2*a^3*b^3*d^2 + 16*B^2*a^5*b*d^2)/(d^4*(a^2 + b^2)^4))^(1/2) + (768*B*a^3
*b^3*(a^2 + b^2))/d)*(-(4*(-B^4*a^4*d^4*(a^4 + b^4 - 6*a^2*b^2)^2)^(1/2) - 16*B^2*a^3*b^3*d^2 + 16*B^2*a^5*b*d
^2)/(d^4*(a^2 + b^2)^4))^(1/2))/4 + (64*B^2*a^3*tan(c + d*x)^(1/2)*(2*a^8 + 15*b^8 - 17*a^2*b^6 + 51*a^4*b^4 +
21*a^6*b^2))/(d^2*(a^2 + b^2)^2))*(-(4*(-B^4*a^4*d^4*(a^4 + b^4 - 6*a^2*b^2)^2)^(1/2) - 16*B^2*a^3*b^3*d^2 +
16*B^2*a^5*b*d^2)/(d^4*(a^2 + b^2)^4))^(1/2))/4 + (32*B^3*a^4*(4*a^8 + b^8 - 77*a^2*b^6 + 47*a^4*b^4 + 33*a^6*
b^2))/(d^3*(a^2 + b^2)^3))*(-(4*(-B^4*a^4*d^4*(a^4 + b^4 - 6*a^2*b^2)^2)^(1/2) - 16*B^2*a^3*b^3*d^2 + 16*B^2*a
^5*b*d^2)/(d^4*(a^2 + b^2)^4))^(1/2))/4 + (16*B^4*a^4*tan(c + d*x)^(1/2)*(a^10 - 2*b^10 - 4*a^2*b^8 - 27*a^4*b
^6 + 15*a^6*b^4 + 9*a^8*b^2))/(b*d^4*(a^2 + b^2)^4))*(-(4*(-B^4*a^4*d^4*(a^4 + b^4 - 6*a^2*b^2)^2)^(1/2) - 16*
B^2*a^3*b^3*d^2 + 16*B^2*a^5*b*d^2)/(d^4*(a^2 + b^2)^4))^(1/2))/4 + (8*B^5*a^7*(a^6 + 10*b^6 + 27*a^2*b^4 + 10
*a^4*b^2))/(b*d^5*(a^2 + b^2)^4))*(-((192*B^4*a^6*b^6*d^4 - 16*B^4*a^4*b^8*d^4 - 16*B^4*a^12*d^4 - 608*B^4*a^8
*b^4*d^4 + 192*B^4*a^10*b^2*d^4)^(1/2) - 16*B^2*a^3*b^3*d^2 + 16*B^2*a^5*b*d^2)/(a^8*d^4 + b^8*d^4 + 4*a^2*b^6
*d^4 + 6*a^4*b^4*d^4 + 4*a^6*b^2*d^4))^(1/2))/4 - log((8*B^5*a^7*(a^6 + 10*b^6 + 27*a^2*b^4 + 10*a^4*b^2))/(b*
d^5*(a^2 + b^2)^4) - ((((((((128*b^3*tan(c + d*x)^(1/2)*(a^2 - b^2)*(a^2 + b^2)^2*((4*(-B^4*a^4*d^4*(a^4 + b^4
- 6*a^2*b^2)^2)^(1/2) + 16*B^2*a^3*b^3*d^2 - 16*B^2*a^5*b*d^2)/(d^4*(a^2 + b^2)^4))^(1/2) - (768*B*a^3*b^3*(a
^2 + b^2))/d)*((4*(-B^4*a^4*d^4*(a^4 + b^4 - 6*a^2*b^2)^2)^(1/2) + 16*B^2*a^3*b^3*d^2 - 16*B^2*a^5*b*d^2)/(d^4
*(a^2 + b^2)^4))^(1/2))/4 + (64*B^2*a^3*tan(c + d*x)^(1/2)*(2*a^8 + 15*b^8 - 17*a^2*b^6 + 51*a^4*b^4 + 21*a^6*
b^2))/(d^2*(a^2 + b^2)^2))*((4*(-B^4*a^4*d^4*(a^4 + b^4 - 6*a^2*b^2)^2)^(1/2) + 16*B^2*a^3*b^3*d^2 - 16*B^2*a^
5*b*d^2)/(d^4*(a^2 + b^2)^4))^(1/2))/4 - (32*B^3*a^4*(4*a^8 + b^8 - 77*a^2*b^6 + 47*a^4*b^4 + 33*a^6*b^2))/(d^
3*(a^2 + b^2)^3))*((4*(-B^4*a^4*d^4*(a^4 + b^4 - 6*a^2*b^2)^2)^(1/2) + 16*B^2*a^3*b^3*d^2 - 16*B^2*a^5*b*d^2)/
(d^4*(a^2 + b^2)^4))^(1/2))/4 + (16*B^4*a^4*tan(c + d*x)^(1/2)*(a^10 - 2*b^10 - 4*a^2*b^8 - 27*a^4*b^6 + 15*a^
6*b^4 + 9*a^8*b^2))/(b*d^4*(a^2 + b^2)^4))*((4*(-B^4*a^4*d^4*(a^4 + b^4 - 6*a^2*b^2)^2)^(1/2) + 16*B^2*a^3*b^3
*d^2 - 16*B^2*a^5*b*d^2)/(d^4*(a^2 + b^2)^4))^(1/2))/4)*(((192*B^4*a^6*b^6*d^4 - 16*B^4*a^4*b^8*d^4 - 16*B^4*a
^12*d^4 - 608*B^4*a^8*b^4*d^4 + 192*B^4*a^10*b^2*d^4)^(1/2) + 16*B^2*a^3*b^3*d^2 - 16*B^2*a^5*b*d^2)/(16*a^8*d
^4 + 16*b^8*d^4 + 64*a^2*b^6*d^4 + 96*a^4*b^4*d^4 + 64*a^6*b^2*d^4))^(1/2) - log((8*B^5*a^7*(a^6 + 10*b^6 + 27
*a^2*b^4 + 10*a^4*b^2))/(b*d^5*(a^2 + b^2)^4) - ((((((((128*b^3*tan(c + d*x)^(1/2)*(a^2 - b^2)*(a^2 + b^2)^2*(
-(4*(-B^4*a^4*d^4*(a^4 + b^4 - 6*a^2*b^2)^2)^(1/2) - 16*B^2*a^3*b^3*d^2 + 16*B^2*a^5*b*d^2)/(d^4*(a^2 + b^2)^4
))^(1/2) - (768*B*a^3*b^3*(a^2 + b^2))/d)*(-(4*(-B^4*a^4*d^4*(a^4 + b^4 - 6*a^2*b^2)^2)^(1/2) - 16*B^2*a^3*b^3
*d^2 + 16*B^2*a^5*b*d^2)/(d^4*(a^2 + b^2)^4))^(1/2))/4 + (64*B^2*a^3*tan(c + d*x)^(1/2)*(2*a^8 + 15*b^8 - 17*a
^2*b^6 + 51*a^4*b^4 + 21*a^6*b^2))/(d^2*(a^2 + b^2)^2))*(-(4*(-B^4*a^4*d^4*(a^4 + b^4 - 6*a^2*b^2)^2)^(1/2) -
16*B^2*a^3*b^3*d^2 + 16*B^2*a^5*b*d^2)/(d^4*(a^2 + b^2)^4))^(1/2))/4 - (32*B^3*a^4*(4*a^8 + b^8 - 77*a^2*b^6 +
47*a^4*b^4 + 33*a^6*b^2))/(d^3*(a^2 + b^2)^3))*(-(4*(-B^4*a^4*d^4*(a^4 + b^4 - 6*a^2*b^2)^2)^(1/2) - 16*B^2*a
^3*b^3*d^2 + 16*B^2*a^5*b*d^2)/(d^4*(a^2 + b^2)^4))^(1/2))/4 + (16*B^4*a^4*tan(c + d*x)^(1/2)*(a^10 - 2*b^10 -
4*a^2*b^8 - 27*a^4*b^6 + 15*a^6*b^4 + 9*a^8*b^2))/(b*d^4*(a^2 + b^2)^4))*(-(4*(-B^4*a^4*d^4*(a^4 + b^4 - 6*a^
2*b^2)^2)^(1/2) - 16*B^2*a^3*b^3*d^2 + 16*B^2*a^5*b*d^2)/(d^4*(a^2 + b^2)^4))^(1/2))/4)*(-((192*B^4*a^6*b^6*d^
4 - 16*B^4*a^4*b^8*d^4 - 16*B^4*a^12*d^4 - 608*B^4*a^8*b^4*d^4 + 192*B^4*a^10*b^2*d^4)^(1/2) - 16*B^2*a^3*b^3*
d^2 + 16*B^2*a^5*b*d^2)/(16*a^8*d^4 + 16*b^8*d^4 + 64*a^2*b^6*d^4 + 96*a^4*b^4*d^4 + 64*a^6*b^2*d^4))^(1/2) +
(log(((((((((128*b^3*tan(c + d*x)^(1/2)*(a^2 - b^2)*(a^2 + b^2)^2*((4*(-B^4*b^4*d^4*(a^4 + b^4 - 6*a^2*b^2)^2)
^(1/2) + 16*B^2*a^3*b^3*d^2 - 16*B^2*a*b^5*d^2)/(d^4*(a^2 + b^2)^4))^(1/2) - (128*B*a*b^3*(7*a^4 + b^4 + 8*a^2
*b^2))/d)*((4*(-B^4*b^4*d^4*(a^4 + b^4 - 6*a^2*b^2)^2)^(1/2) + 16*B^2*a^3*b^3*d^2 - 16*B^2*a*b^5*d^2)/(d^4*(a^
2 + b^2)^4))^(1/2))/4 + (64*B^2*a*tan(c + d*x)^(1/2)*(18*a^10 - 15*b^10 + 17*a^2*b^8 - a^4*b^6 + 97*a^6*b^4 +
84*a^8*b^2))/(d^2*(a^2 + b^2)^2))*((4*(-B^4*b^4*d^4*(a^4 + b^4 - 6*a^2*b^2)^2)^(1/2) + 16*B^2*a^3*b^3*d^2 - 16
*B^2*a*b^5*d^2)/(d^4*(a^2 + b^2)^4))^(1/2))/4 + (32*B^3*a^4*(127*a^2*b^6 - 112*b^8 - 9*a^8 + 173*a^4*b^4 + 21*
a^6*b^2))/(d^3*(a^2 + b^2)^3))*((4*(-B^4*b^4*d^4*(a^4 + b^4 - 6*a^2*b^2)^2)^(1/2) + 16*B^2*a^3*b^3*d^2 - 16*B^
2*a*b^5*d^2)/(d^4*(a^2 + b^2)^4))^(1/2))/4 - (16*B^4*b*tan(c + d*x)^(1/2)*(9*a^12 + 2*b^12 + 4*a^2*b^10 + 2*a^
4*b^8 - 49*a^6*b^6 + 7*a^8*b^4 + 33*a^10*b^2))/(d^4*(a^2 + b^2)^4))*((4*(-B^4*b^4*d^4*(a^4 + b^4 - 6*a^2*b^2)^
2)^(1/2) + 16*B^2*a^3*b^3*d^2 - 16*B^2*a*b^5*d^2)/(d^4*(a^2 + b^2)^4))^(1/2))/4 - (16*B^5*a^3*b^7*(3*a^2 + 7*b
^2))/(d^5*(a^2 + b^2)^4))*(((192*B^4*a^2*b^10*d^4 - 16*B^4*b^12*d^4 - 608*B^4*a^4*b^8*d^4 + 192*B^4*a^6*b^6*d^
4 - 16*B^4*a^8*b^4*d^4)^(1/2) + 16*B^2*a^3*b^3*d^2 - 16*B^2*a*b^5*d^2)/(a^8*d^4 + b^8*d^4 + 4*a^2*b^6*d^4 + 6*
a^4*b^4*d^4 + 4*a^6*b^2*d^4))^(1/2))/4 + (log(((((((((128*b^3*tan(c + d*x)^(1/2)*(a^2 - b^2)*(a^2 + b^2)^2*(-(
4*(-B^4*b^4*d^4*(a^4 + b^4 - 6*a^2*b^2)^2)^(1/2) - 16*B^2*a^3*b^3*d^2 + 16*B^2*a*b^5*d^2)/(d^4*(a^2 + b^2)^4))
^(1/2) - (128*B*a*b^3*(7*a^4 + b^4 + 8*a^2*b^2))/d)*(-(4*(-B^4*b^4*d^4*(a^4 + b^4 - 6*a^2*b^2)^2)^(1/2) - 16*B
^2*a^3*b^3*d^2 + 16*B^2*a*b^5*d^2)/(d^4*(a^2 + b^2)^4))^(1/2))/4 + (64*B^2*a*tan(c + d*x)^(1/2)*(18*a^10 - 15*
b^10 + 17*a^2*b^8 - a^4*b^6 + 97*a^6*b^4 + 84*a^8*b^2))/(d^2*(a^2 + b^2)^2))*(-(4*(-B^4*b^4*d^4*(a^4 + b^4 - 6
*a^2*b^2)^2)^(1/2) - 16*B^2*a^3*b^3*d^2 + 16*B^2*a*b^5*d^2)/(d^4*(a^2 + b^2)^4))^(1/2))/4 + (32*B^3*a^4*(127*a
^2*b^6 - 112*b^8 - 9*a^8 + 173*a^4*b^4 + 21*a^6*b^2))/(d^3*(a^2 + b^2)^3))*(-(4*(-B^4*b^4*d^4*(a^4 + b^4 - 6*a
^2*b^2)^2)^(1/2) - 16*B^2*a^3*b^3*d^2 + 16*B^2*a*b^5*d^2)/(d^4*(a^2 + b^2)^4))^(1/2))/4 - (16*B^4*b*tan(c + d*
x)^(1/2)*(9*a^12 + 2*b^12 + 4*a^2*b^10 + 2*a^4*b^8 - 49*a^6*b^6 + 7*a^8*b^4 + 33*a^10*b^2))/(d^4*(a^2 + b^2)^4
))*(-(4*(-B^4*b^4*d^4*(a^4 + b^4 - 6*a^2*b^2)^2)^(1/2) - 16*B^2*a^3*b^3*d^2 + 16*B^2*a*b^5*d^2)/(d^4*(a^2 + b^
2)^4))^(1/2))/4 - (16*B^5*a^3*b^7*(3*a^2 + 7*b^2))/(d^5*(a^2 + b^2)^4))*(-((192*B^4*a^2*b^10*d^4 - 16*B^4*b^12
*d^4 - 608*B^4*a^4*b^8*d^4 + 192*B^4*a^6*b^6*d^4 - 16*B^4*a^8*b^4*d^4)^(1/2) - 16*B^2*a^3*b^3*d^2 + 16*B^2*a*b
^5*d^2)/(a^8*d^4 + b^8*d^4 + 4*a^2*b^6*d^4 + 6*a^4*b^4*d^4 + 4*a^6*b^2*d^4))^(1/2))/4 - log(- ((((((((128*b^3*
tan(c + d*x)^(1/2)*(a^2 - b^2)*(a^2 + b^2)^2*((4*(-B^4*b^4*d^4*(a^4 + b^4 - 6*a^2*b^2)^2)^(1/2) + 16*B^2*a^3*b
^3*d^2 - 16*B^2*a*b^5*d^2)/(d^4*(a^2 + b^2)^4))^(1/2) + (128*B*a*b^3*(7*a^4 + b^4 + 8*a^2*b^2))/d)*((4*(-B^4*b
^4*d^4*(a^4 + b^4 - 6*a^2*b^2)^2)^(1/2) + 16*B^2*a^3*b^3*d^2 - 16*B^2*a*b^5*d^2)/(d^4*(a^2 + b^2)^4))^(1/2))/4
+ (64*B^2*a*tan(c + d*x)^(1/2)*(18*a^10 - 15*b^10 + 17*a^2*b^8 - a^4*b^6 + 97*a^6*b^4 + 84*a^8*b^2))/(d^2*(a^
2 + b^2)^2))*((4*(-B^4*b^4*d^4*(a^4 + b^4 - 6*a^2*b^2)^2)^(1/2) + 16*B^2*a^3*b^3*d^2 - 16*B^2*a*b^5*d^2)/(d^4*
(a^2 + b^2)^4))^(1/2))/4 - (32*B^3*a^4*(127*a^2*b^6 - 112*b^8 - 9*a^8 + 173*a^4*b^4 + 21*a^6*b^2))/(d^3*(a^2 +
b^2)^3))*((4*(-B^4*b^4*d^4*(a^4 + b^4 - 6*a^2*b^2)^2)^(1/2) + 16*B^2*a^3*b^3*d^2 - 16*B^2*a*b^5*d^2)/(d^4*(a^
2 + b^2)^4))^(1/2))/4 - (16*B^4*b*tan(c + d*x)^(1/2)*(9*a^12 + 2*b^12 + 4*a^2*b^10 + 2*a^4*b^8 - 49*a^6*b^6 +
7*a^8*b^4 + 33*a^10*b^2))/(d^4*(a^2 + b^2)^4))*((4*(-B^4*b^4*d^4*(a^4 + b^4 - 6*a^2*b^2)^2)^(1/2) + 16*B^2*a^3
*b^3*d^2 - 16*B^2*a*b^5*d^2)/(d^4*(a^2 + b^2)^4))^(1/2))/4 - (16*B^5*a^3*b^7*(3*a^2 + 7*b^2))/(d^5*(a^2 + b^2)
^4))*(((192*B^4*a^2*b^10*d^4 - 16*B^4*b^12*d^4 - 608*B^4*a^4*b^8*d^4 + 192*B^4*a^6*b^6*d^4 - 16*B^4*a^8*b^4*d^
4)^(1/2) + 16*B^2*a^3*b^3*d^2 - 16*B^2*a*b^5*d^2)/(16*a^8*d^4 + 16*b^8*d^4 + 64*a^2*b^6*d^4 + 96*a^4*b^4*d^4 +
64*a^6*b^2*d^4))^(1/2) - log(- ((((((((128*b^3*tan(c + d*x)^(1/2)*(a^2 - b^2)*(a^2 + b^2)^2*(-(4*(-B^4*b^4*d^
4*(a^4 + b^4 - 6*a^2*b^2)^2)^(1/2) - 16*B^2*a^3*b^3*d^2 + 16*B^2*a*b^5*d^2)/(d^4*(a^2 + b^2)^4))^(1/2) + (128*
B*a*b^3*(7*a^4 + b^4 + 8*a^2*b^2))/d)*(-(4*(-B^4*b^4*d^4*(a^4 + b^4 - 6*a^2*b^2)^2)^(1/2) - 16*B^2*a^3*b^3*d^2
+ 16*B^2*a*b^5*d^2)/(d^4*(a^2 + b^2)^4))^(1/2))/4 + (64*B^2*a*tan(c + d*x)^(1/2)*(18*a^10 - 15*b^10 + 17*a^2*
b^8 - a^4*b^6 + 97*a^6*b^4 + 84*a^8*b^2))/(d^2*(a^2 + b^2)^2))*(-(4*(-B^4*b^4*d^4*(a^4 + b^4 - 6*a^2*b^2)^2)^(
1/2) - 16*B^2*a^3*b^3*d^2 + 16*B^2*a*b^5*d^2)/(d^4*(a^2 + b^2)^4))^(1/2))/4 - (32*B^3*a^4*(127*a^2*b^6 - 112*b
^8 - 9*a^8 + 173*a^4*b^4 + 21*a^6*b^2))/(d^3*(a^2 + b^2)^3))*(-(4*(-B^4*b^4*d^4*(a^4 + b^4 - 6*a^2*b^2)^2)^(1/
2) - 16*B^2*a^3*b^3*d^2 + 16*B^2*a*b^5*d^2)/(d^4*(a^2 + b^2)^4))^(1/2))/4 - (16*B^4*b*tan(c + d*x)^(1/2)*(9*a^
12 + 2*b^12 + 4*a^2*b^10 + 2*a^4*b^8 - 49*a^6*b^6 + 7*a^8*b^4 + 33*a^10*b^2))/(d^4*(a^2 + b^2)^4))*(-(4*(-B^4*
b^4*d^4*(a^4 + b^4 - 6*a^2*b^2)^2)^(1/2) - 16*B^2*a^3*b^3*d^2 + 16*B^2*a*b^5*d^2)/(d^4*(a^2 + b^2)^4))^(1/2))/
4 - (16*B^5*a^3*b^7*(3*a^2 + 7*b^2))/(d^5*(a^2 + b^2)^4))*(-((192*B^4*a^2*b^10*d^4 - 16*B^4*b^12*d^4 - 608*B^4
*a^4*b^8*d^4 + 192*B^4*a^6*b^6*d^4 - 16*B^4*a^8*b^4*d^4)^(1/2) - 16*B^2*a^3*b^3*d^2 + 16*B^2*a*b^5*d^2)/(16*a^
8*d^4 + 16*b^8*d^4 + 64*a^2*b^6*d^4 + 96*a^4*b^4*d^4 + 64*a^6*b^2*d^4))^(1/2) - (atan(((((16*tan(c + d*x)^(1/2
)*(2*B^4*b^13 + 9*B^4*a^12*b + 4*B^4*a^2*b^11 + 2*B^4*a^4*b^9 - 49*B^4*a^6*b^7 + 7*B^4*a^8*b^5 + 33*B^4*a^10*b
^3))/(a^8*d^4 + b^8*d^4 + 4*a^2*b^6*d^4 + 6*a^4*b^4*d^4 + 4*a^6*b^2*d^4) + (((16*(30*B^3*a^6*b^8*d^2 - 224*B^3
*a^4*b^10*d^2 - 18*B^3*a^14*d^2 + 600*B^3*a^8*b^6*d^2 + 388*B^3*a^10*b^4*d^2 + 24*B^3*a^12*b^2*d^2))/(a^8*d^5
+ b^8*d^5 + 4*a^2*b^6*d^5 + 6*a^4*b^4*d^5 + 4*a^6*b^2*d^5) - (((16*tan(c + d*x)^(1/2)*(72*B^2*a^15*d^2 - 52*B^
2*a^3*b^12*d^2 + 72*B^2*a^5*b^10*d^2 + 448*B^2*a^7*b^8*d^2 + 1108*B^2*a^9*b^6*d^2 + 1132*B^2*a^11*b^4*d^2 + 48
0*B^2*a^13*b^2*d^2 - 60*B^2*a*b^14*d^2))/(a^8*d^4 + b^8*d^4 + 4*a^2*b^6*d^4 + 6*a^4*b^4*d^4 + 4*a^6*b^2*d^4) +
(((16*(8*B*a*b^15*d^4 + 96*B*a^3*b^13*d^4 + 360*B*a^5*b^11*d^4 + 640*B*a^7*b^9*d^4 + 600*B*a^9*b^7*d^4 + 288*
B*a^11*b^5*d^4 + 56*B*a^13*b^3*d^4))/(a^8*d^5 + b^8*d^5 + 4*a^2*b^6*d^5 + 6*a^4*b^4*d^5 + 4*a^6*b^2*d^5) - (4*
tan(c + d*x)^(1/2)*(-4*(9*B^2*a^9 + 49*B^2*a^5*b^4 + 42*B^2*a^7*b^2)*(b^11*d^2 + 4*a^2*b^9*d^2 + 6*a^4*b^7*d^2
+ 4*a^6*b^5*d^2 + a^8*b^3*d^2))^(1/2)*(32*b^17*d^4 + 160*a^2*b^15*d^4 + 288*a^4*b^13*d^4 + 160*a^6*b^11*d^4 -
160*a^8*b^9*d^4 - 288*a^10*b^7*d^4 - 160*a^12*b^5*d^4 - 32*a^14*b^3*d^4))/((a^8*d^4 + b^8*d^4 + 4*a^2*b^6*d^4
+ 6*a^4*b^4*d^4 + 4*a^6*b^2*d^4)*(b^11*d^2 + 4*a^2*b^9*d^2 + 6*a^4*b^7*d^2 + 4*a^6*b^5*d^2 + a^8*b^3*d^2)))*(
-4*(9*B^2*a^9 + 49*B^2*a^5*b^4 + 42*B^2*a^7*b^2)*(b^11*d^2 + 4*a^2*b^9*d^2 + 6*a^4*b^7*d^2 + 4*a^6*b^5*d^2 + a
^8*b^3*d^2))^(1/2))/(4*(b^11*d^2 + 4*a^2*b^9*d^2 + 6*a^4*b^7*d^2 + 4*a^6*b^5*d^2 + a^8*b^3*d^2)))*(-4*(9*B^2*a
^9 + 49*B^2*a^5*b^4 + 42*B^2*a^7*b^2)*(b^11*d^2 + 4*a^2*b^9*d^2 + 6*a^4*b^7*d^2 + 4*a^6*b^5*d^2 + a^8*b^3*d^2)
)^(1/2))/(4*(b^11*d^2 + 4*a^2*b^9*d^2 + 6*a^4*b^7*d^2 + 4*a^6*b^5*d^2 + a^8*b^3*d^2)))*(-4*(9*B^2*a^9 + 49*B^2
*a^5*b^4 + 42*B^2*a^7*b^2)*(b^11*d^2 + 4*a^2*b^9*d^2 + 6*a^4*b^7*d^2 + 4*a^6*b^5*d^2 + a^8*b^3*d^2))^(1/2))/(4
*(b^11*d^2 + 4*a^2*b^9*d^2 + 6*a^4*b^7*d^2 + 4*a^6*b^5*d^2 + a^8*b^3*d^2)))*(-4*(9*B^2*a^9 + 49*B^2*a^5*b^4 +
42*B^2*a^7*b^2)*(b^11*d^2 + 4*a^2*b^9*d^2 + 6*a^4*b^7*d^2 + 4*a^6*b^5*d^2 + a^8*b^3*d^2))^(1/2)*1i)/(4*(b^11*d
^2 + 4*a^2*b^9*d^2 + 6*a^4*b^7*d^2 + 4*a^6*b^5*d^2 + a^8*b^3*d^2)) + (((16*tan(c + d*x)^(1/2)*(2*B^4*b^13 + 9*
B^4*a^12*b + 4*B^4*a^2*b^11 + 2*B^4*a^4*b^9 - 49*B^4*a^6*b^7 + 7*B^4*a^8*b^5 + 33*B^4*a^10*b^3))/(a^8*d^4 + b^
8*d^4 + 4*a^2*b^6*d^4 + 6*a^4*b^4*d^4 + 4*a^6*b^2*d^4) - (((16*(30*B^3*a^6*b^8*d^2 - 224*B^3*a^4*b^10*d^2 - 18
*B^3*a^14*d^2 + 600*B^3*a^8*b^6*d^2 + 388*B^3*a^10*b^4*d^2 + 24*B^3*a^12*b^2*d^2))/(a^8*d^5 + b^8*d^5 + 4*a^2*
b^6*d^5 + 6*a^4*b^4*d^5 + 4*a^6*b^2*d^5) + (((16*tan(c + d*x)^(1/2)*(72*B^2*a^15*d^2 - 52*B^2*a^3*b^12*d^2 + 7
2*B^2*a^5*b^10*d^2 + 448*B^2*a^7*b^8*d^2 + 1108*B^2*a^9*b^6*d^2 + 1132*B^2*a^11*b^4*d^2 + 480*B^2*a^13*b^2*d^2
- 60*B^2*a*b^14*d^2))/(a^8*d^4 + b^8*d^4 + 4*a^2*b^6*d^4 + 6*a^4*b^4*d^4 + 4*a^6*b^2*d^4) - (((16*(8*B*a*b^15
*d^4 + 96*B*a^3*b^13*d^4 + 360*B*a^5*b^11*d^4 + 640*B*a^7*b^9*d^4 + 600*B*a^9*b^7*d^4 + 288*B*a^11*b^5*d^4 + 5
6*B*a^13*b^3*d^4))/(a^8*d^5 + b^8*d^5 + 4*a^2*b^6*d^5 + 6*a^4*b^4*d^5 + 4*a^6*b^2*d^5) + (4*tan(c + d*x)^(1/2)
*(-4*(9*B^2*a^9 + 49*B^2*a^5*b^4 + 42*B^2*a^7*b^2)*(b^11*d^2 + 4*a^2*b^9*d^2 + 6*a^4*b^7*d^2 + 4*a^6*b^5*d^2 +
a^8*b^3*d^2))^(1/2)*(32*b^17*d^4 + 160*a^2*b^15*d^4 + 288*a^4*b^13*d^4 + 160*a^6*b^11*d^4 - 160*a^8*b^9*d^4 -
288*a^10*b^7*d^4 - 160*a^12*b^5*d^4 - 32*a^14*b^3*d^4))/((a^8*d^4 + b^8*d^4 + 4*a^2*b^6*d^4 + 6*a^4*b^4*d^4 +
4*a^6*b^2*d^4)*(b^11*d^2 + 4*a^2*b^9*d^2 + 6*a^4*b^7*d^2 + 4*a^6*b^5*d^2 + a^8*b^3*d^2)))*(-4*(9*B^2*a^9 + 49
*B^2*a^5*b^4 + 42*B^2*a^7*b^2)*(b^11*d^2 + 4*a^2*b^9*d^2 + 6*a^4*b^7*d^2 + 4*a^6*b^5*d^2 + a^8*b^3*d^2))^(1/2)
)/(4*(b^11*d^2 + 4*a^2*b^9*d^2 + 6*a^4*b^7*d^2 + 4*a^6*b^5*d^2 + a^8*b^3*d^2)))*(-4*(9*B^2*a^9 + 49*B^2*a^5*b^
4 + 42*B^2*a^7*b^2)*(b^11*d^2 + 4*a^2*b^9*d^2 + 6*a^4*b^7*d^2 + 4*a^6*b^5*d^2 + a^8*b^3*d^2))^(1/2))/(4*(b^11*
d^2 + 4*a^2*b^9*d^2 + 6*a^4*b^7*d^2 + 4*a^6*b^5*d^2 + a^8*b^3*d^2)))*(-4*(9*B^2*a^9 + 49*B^2*a^5*b^4 + 42*B^2*
a^7*b^2)*(b^11*d^2 + 4*a^2*b^9*d^2 + 6*a^4*b^7*d^2 + 4*a^6*b^5*d^2 + a^8*b^3*d^2))^(1/2))/(4*(b^11*d^2 + 4*a^2
*b^9*d^2 + 6*a^4*b^7*d^2 + 4*a^6*b^5*d^2 + a^8*b^3*d^2)))*(-4*(9*B^2*a^9 + 49*B^2*a^5*b^4 + 42*B^2*a^7*b^2)*(b
^11*d^2 + 4*a^2*b^9*d^2 + 6*a^4*b^7*d^2 + 4*a^6*b^5*d^2 + a^8*b^3*d^2))^(1/2)*1i)/(4*(b^11*d^2 + 4*a^2*b^9*d^2
+ 6*a^4*b^7*d^2 + 4*a^6*b^5*d^2 + a^8*b^3*d^2)))/((32*(7*B^5*a^3*b^9 + 3*B^5*a^5*b^7))/(a^8*d^5 + b^8*d^5 + 4
*a^2*b^6*d^5 + 6*a^4*b^4*d^5 + 4*a^6*b^2*d^5) - (((16*tan(c + d*x)^(1/2)*(2*B^4*b^13 + 9*B^4*a^12*b + 4*B^4*a^
2*b^11 + 2*B^4*a^4*b^9 - 49*B^4*a^6*b^7 + 7*B^4*a^8*b^5 + 33*B^4*a^10*b^3))/(a^8*d^4 + b^8*d^4 + 4*a^2*b^6*d^4
+ 6*a^4*b^4*d^4 + 4*a^6*b^2*d^4) + (((16*(30*B^3*a^6*b^8*d^2 - 224*B^3*a^4*b^10*d^2 - 18*B^3*a^14*d^2 + 600*B
^3*a^8*b^6*d^2 + 388*B^3*a^10*b^4*d^2 + 24*B^3*a^12*b^2*d^2))/(a^8*d^5 + b^8*d^5 + 4*a^2*b^6*d^5 + 6*a^4*b^4*d
^5 + 4*a^6*b^2*d^5) - (((16*tan(c + d*x)^(1/2)*(72*B^2*a^15*d^2 - 52*B^2*a^3*b^12*d^2 + 72*B^2*a^5*b^10*d^2 +
448*B^2*a^7*b^8*d^2 + 1108*B^2*a^9*b^6*d^2 + 1132*B^2*a^11*b^4*d^2 + 480*B^2*a^13*b^2*d^2 - 60*B^2*a*b^14*d^2)
)/(a^8*d^4 + b^8*d^4 + 4*a^2*b^6*d^4 + 6*a^4*b^4*d^4 + 4*a^6*b^2*d^4) + (((16*(8*B*a*b^15*d^4 + 96*B*a^3*b^13*
d^4 + 360*B*a^5*b^11*d^4 + 640*B*a^7*b^9*d^4 + 600*B*a^9*b^7*d^4 + 288*B*a^11*b^5*d^4 + 56*B*a^13*b^3*d^4))/(a
^8*d^5 + b^8*d^5 + 4*a^2*b^6*d^5 + 6*a^4*b^4*d^5 + 4*a^6*b^2*d^5) - (4*tan(c + d*x)^(1/2)*(-4*(9*B^2*a^9 + 49*
B^2*a^5*b^4 + 42*B^2*a^7*b^2)*(b^11*d^2 + 4*a^2*b^9*d^2 + 6*a^4*b^7*d^2 + 4*a^6*b^5*d^2 + a^8*b^3*d^2))^(1/2)*
(32*b^17*d^4 + 160*a^2*b^15*d^4 + 288*a^4*b^13*d^4 + 160*a^6*b^11*d^4 - 160*a^8*b^9*d^4 - 288*a^10*b^7*d^4 - 1
60*a^12*b^5*d^4 - 32*a^14*b^3*d^4))/((a^8*d^4 + b^8*d^4 + 4*a^2*b^6*d^4 + 6*a^4*b^4*d^4 + 4*a^6*b^2*d^4)*(b^11
*d^2 + 4*a^2*b^9*d^2 + 6*a^4*b^7*d^2 + 4*a^6*b^5*d^2 + a^8*b^3*d^2)))*(-4*(9*B^2*a^9 + 49*B^2*a^5*b^4 + 42*B^2
*a^7*b^2)*(b^11*d^2 + 4*a^2*b^9*d^2 + 6*a^4*b^7*d^2 + 4*a^6*b^5*d^2 + a^8*b^3*d^2))^(1/2))/(4*(b^11*d^2 + 4*a^
2*b^9*d^2 + 6*a^4*b^7*d^2 + 4*a^6*b^5*d^2 + a^8*b^3*d^2)))*(-4*(9*B^2*a^9 + 49*B^2*a^5*b^4 + 42*B^2*a^7*b^2)*(
b^11*d^2 + 4*a^2*b^9*d^2 + 6*a^4*b^7*d^2 + 4*a^6*b^5*d^2 + a^8*b^3*d^2))^(1/2))/(4*(b^11*d^2 + 4*a^2*b^9*d^2 +
6*a^4*b^7*d^2 + 4*a^6*b^5*d^2 + a^8*b^3*d^2)))*(-4*(9*B^2*a^9 + 49*B^2*a^5*b^4 + 42*B^2*a^7*b^2)*(b^11*d^2 +
4*a^2*b^9*d^2 + 6*a^4*b^7*d^2 + 4*a^6*b^5*d^2 + a^8*b^3*d^2))^(1/2))/(4*(b^11*d^2 + 4*a^2*b^9*d^2 + 6*a^4*b^7*
d^2 + 4*a^6*b^5*d^2 + a^8*b^3*d^2)))*(-4*(9*B^2*a^9 + 49*B^2*a^5*b^4 + 42*B^2*a^7*b^2)*(b^11*d^2 + 4*a^2*b^9*d
^2 + 6*a^4*b^7*d^2 + 4*a^6*b^5*d^2 + a^8*b^3*d^2))^(1/2))/(4*(b^11*d^2 + 4*a^2*b^9*d^2 + 6*a^4*b^7*d^2 + 4*a^6
*b^5*d^2 + a^8*b^3*d^2)) + (((16*tan(c + d*x)^(1/2)*(2*B^4*b^13 + 9*B^4*a^12*b + 4*B^4*a^2*b^11 + 2*B^4*a^4*b^
9 - 49*B^4*a^6*b^7 + 7*B^4*a^8*b^5 + 33*B^4*a^10*b^3))/(a^8*d^4 + b^8*d^4 + 4*a^2*b^6*d^4 + 6*a^4*b^4*d^4 + 4*
a^6*b^2*d^4) - (((16*(30*B^3*a^6*b^8*d^2 - 224*B^3*a^4*b^10*d^2 - 18*B^3*a^14*d^2 + 600*B^3*a^8*b^6*d^2 + 388*
B^3*a^10*b^4*d^2 + 24*B^3*a^12*b^2*d^2))/(a^8*d^5 + b^8*d^5 + 4*a^2*b^6*d^5 + 6*a^4*b^4*d^5 + 4*a^6*b^2*d^5) +
(((16*tan(c + d*x)^(1/2)*(72*B^2*a^15*d^2 - 52*B^2*a^3*b^12*d^2 + 72*B^2*a^5*b^10*d^2 + 448*B^2*a^7*b^8*d^2 +
1108*B^2*a^9*b^6*d^2 + 1132*B^2*a^11*b^4*d^2 + 480*B^2*a^13*b^2*d^2 - 60*B^2*a*b^14*d^2))/(a^8*d^4 + b^8*d^4
+ 4*a^2*b^6*d^4 + 6*a^4*b^4*d^4 + 4*a^6*b^2*d^4) - (((16*(8*B*a*b^15*d^4 + 96*B*a^3*b^13*d^4 + 360*B*a^5*b^11*
d^4 + 640*B*a^7*b^9*d^4 + 600*B*a^9*b^7*d^4 + 288*B*a^11*b^5*d^4 + 56*B*a^13*b^3*d^4))/(a^8*d^5 + b^8*d^5 + 4*
a^2*b^6*d^5 + 6*a^4*b^4*d^5 + 4*a^6*b^2*d^5) + (4*tan(c + d*x)^(1/2)*(-4*(9*B^2*a^9 + 49*B^2*a^5*b^4 + 42*B^2*
a^7*b^2)*(b^11*d^2 + 4*a^2*b^9*d^2 + 6*a^4*b^7*d^2 + 4*a^6*b^5*d^2 + a^8*b^3*d^2))^(1/2)*(32*b^17*d^4 + 160*a^
2*b^15*d^4 + 288*a^4*b^13*d^4 + 160*a^6*b^11*d^4 - 160*a^8*b^9*d^4 - 288*a^10*b^7*d^4 - 160*a^12*b^5*d^4 - 32*
a^14*b^3*d^4))/((a^8*d^4 + b^8*d^4 + 4*a^2*b^6*d^4 + 6*a^4*b^4*d^4 + 4*a^6*b^2*d^4)*(b^11*d^2 + 4*a^2*b^9*d^2
+ 6*a^4*b^7*d^2 + 4*a^6*b^5*d^2 + a^8*b^3*d^2)))*(-4*(9*B^2*a^9 + 49*B^2*a^5*b^4 + 42*B^2*a^7*b^2)*(b^11*d^2 +
4*a^2*b^9*d^2 + 6*a^4*b^7*d^2 + 4*a^6*b^5*d^2 + a^8*b^3*d^2))^(1/2))/(4*(b^11*d^2 + 4*a^2*b^9*d^2 + 6*a^4*b^7
*d^2 + 4*a^6*b^5*d^2 + a^8*b^3*d^2)))*(-4*(9*B^2*a^9 + 49*B^2*a^5*b^4 + 42*B^2*a^7*b^2)*(b^11*d^2 + 4*a^2*b^9*
d^2 + 6*a^4*b^7*d^2 + 4*a^6*b^5*d^2 + a^8*b^3*d^2))^(1/2))/(4*(b^11*d^2 + 4*a^2*b^9*d^2 + 6*a^4*b^7*d^2 + 4*a^
6*b^5*d^2 + a^8*b^3*d^2)))*(-4*(9*B^2*a^9 + 49*B^2*a^5*b^4 + 42*B^2*a^7*b^2)*(b^11*d^2 + 4*a^2*b^9*d^2 + 6*a^4
*b^7*d^2 + 4*a^6*b^5*d^2 + a^8*b^3*d^2))^(1/2))/(4*(b^11*d^2 + 4*a^2*b^9*d^2 + 6*a^4*b^7*d^2 + 4*a^6*b^5*d^2 +
a^8*b^3*d^2)))*(-4*(9*B^2*a^9 + 49*B^2*a^5*b^4 + 42*B^2*a^7*b^2)*(b^11*d^2 + 4*a^2*b^9*d^2 + 6*a^4*b^7*d^2 +
4*a^6*b^5*d^2 + a^8*b^3*d^2))^(1/2))/(4*(b^11*d^2 + 4*a^2*b^9*d^2 + 6*a^4*b^7*d^2 + 4*a^6*b^5*d^2 + a^8*b^3*d^
2))))*(-4*(9*B^2*a^9 + 49*B^2*a^5*b^4 + 42*B^2*a^7*b^2)*(b^11*d^2 + 4*a^2*b^9*d^2 + 6*a^4*b^7*d^2 + 4*a^6*b^5*
d^2 + a^8*b^3*d^2))^(1/2)*1i)/(2*(b^11*d^2 + 4*a^2*b^9*d^2 + 6*a^4*b^7*d^2 + 4*a^6*b^5*d^2 + a^8*b^3*d^2)) - (
atan(((((((8*(4*B^3*a^4*b^11*d^2 - 304*B^3*a^6*b^9*d^2 - 120*B^3*a^8*b^7*d^2 + 320*B^3*a^10*b^5*d^2 + 148*B^3*
a^12*b^3*d^2 + 16*B^3*a^14*b*d^2))/(b^9*d^5 + a^8*b*d^5 + 4*a^2*b^7*d^5 + 6*a^4*b^5*d^5 + 4*a^6*b^3*d^5) + (((
((8*(96*B*a^3*b^14*d^4 + 480*B*a^5*b^12*d^4 + 960*B*a^7*b^10*d^4 + 960*B*a^9*b^8*d^4 + 480*B*a^11*b^6*d^4 + 96
*B*a^13*b^4*d^4))/(b^9*d^5 + a^8*b*d^5 + 4*a^2*b^7*d^5 + 6*a^4*b^5*d^5 + 4*a^6*b^3*d^5) - (4*tan(c + d*x)^(1/2
)*(-4*(B^2*a^9 + 25*B^2*a^5*b^4 + 10*B^2*a^7*b^2)*(b^11*d^2 + 4*a^2*b^9*d^2 + 6*a^4*b^7*d^2 + 4*a^6*b^5*d^2 +
a^8*b^3*d^2))^(1/2)*(32*b^18*d^4 + 160*a^2*b^16*d^4 + 288*a^4*b^14*d^4 + 160*a^6*b^12*d^4 - 160*a^8*b^10*d^4 -
288*a^10*b^8*d^4 - 160*a^12*b^6*d^4 - 32*a^14*b^4*d^4))/((b^9*d^4 + a^8*b*d^4 + 4*a^2*b^7*d^4 + 6*a^4*b^5*d^4
+ 4*a^6*b^3*d^4)*(b^11*d^2 + 4*a^2*b^9*d^2 + 6*a^4*b^7*d^2 + 4*a^6*b^5*d^2 + a^8*b^3*d^2)))*(-4*(B^2*a^9 + 25
*B^2*a^5*b^4 + 10*B^2*a^7*b^2)*(b^11*d^2 + 4*a^2*b^9*d^2 + 6*a^4*b^7*d^2 + 4*a^6*b^5*d^2 + a^8*b^3*d^2))^(1/2)
)/(4*(b^11*d^2 + 4*a^2*b^9*d^2 + 6*a^4*b^7*d^2 + 4*a^6*b^5*d^2 + a^8*b^3*d^2)) + (16*tan(c + d*x)^(1/2)*(60*B^
2*a^3*b^13*d^2 + 52*B^2*a^5*b^11*d^2 + 128*B^2*a^7*b^9*d^2 + 424*B^2*a^9*b^7*d^2 + 380*B^2*a^11*b^5*d^2 + 100*
B^2*a^13*b^3*d^2 + 8*B^2*a^15*b*d^2))/(b^9*d^4 + a^8*b*d^4 + 4*a^2*b^7*d^4 + 6*a^4*b^5*d^4 + 4*a^6*b^3*d^4))*(
-4*(B^2*a^9 + 25*B^2*a^5*b^4 + 10*B^2*a^7*b^2)*(b^11*d^2 + 4*a^2*b^9*d^2 + 6*a^4*b^7*d^2 + 4*a^6*b^5*d^2 + a^8
*b^3*d^2))^(1/2))/(4*(b^11*d^2 + 4*a^2*b^9*d^2 + 6*a^4*b^7*d^2 + 4*a^6*b^5*d^2 + a^8*b^3*d^2)))*(-4*(B^2*a^9 +
25*B^2*a^5*b^4 + 10*B^2*a^7*b^2)*(b^11*d^2 + 4*a^2*b^9*d^2 + 6*a^4*b^7*d^2 + 4*a^6*b^5*d^2 + a^8*b^3*d^2))^(1
/2))/(4*(b^11*d^2 + 4*a^2*b^9*d^2 + 6*a^4*b^7*d^2 + 4*a^6*b^5*d^2 + a^8*b^3*d^2)) + (16*tan(c + d*x)^(1/2)*(B^
4*a^14 - 2*B^4*a^4*b^10 - 4*B^4*a^6*b^8 - 27*B^4*a^8*b^6 + 15*B^4*a^10*b^4 + 9*B^4*a^12*b^2))/(b^9*d^4 + a^8*b
*d^4 + 4*a^2*b^7*d^4 + 6*a^4*b^5*d^4 + 4*a^6*b^3*d^4))*(-4*(B^2*a^9 + 25*B^2*a^5*b^4 + 10*B^2*a^7*b^2)*(b^11*d
^2 + 4*a^2*b^9*d^2 + 6*a^4*b^7*d^2 + 4*a^6*b^5*d^2 + a^8*b^3*d^2))^(1/2)*1i)/(4*(b^11*d^2 + 4*a^2*b^9*d^2 + 6*
a^4*b^7*d^2 + 4*a^6*b^5*d^2 + a^8*b^3*d^2)) - (((((8*(4*B^3*a^4*b^11*d^2 - 304*B^3*a^6*b^9*d^2 - 120*B^3*a^8*b
^7*d^2 + 320*B^3*a^10*b^5*d^2 + 148*B^3*a^12*b^3*d^2 + 16*B^3*a^14*b*d^2))/(b^9*d^5 + a^8*b*d^5 + 4*a^2*b^7*d^
5 + 6*a^4*b^5*d^5 + 4*a^6*b^3*d^5) + (((((8*(96*B*a^3*b^14*d^4 + 480*B*a^5*b^12*d^4 + 960*B*a^7*b^10*d^4 + 960
*B*a^9*b^8*d^4 + 480*B*a^11*b^6*d^4 + 96*B*a^13*b^4*d^4))/(b^9*d^5 + a^8*b*d^5 + 4*a^2*b^7*d^5 + 6*a^4*b^5*d^5
+ 4*a^6*b^3*d^5) + (4*tan(c + d*x)^(1/2)*(-4*(B^2*a^9 + 25*B^2*a^5*b^4 + 10*B^2*a^7*b^2)*(b^11*d^2 + 4*a^2*b^
9*d^2 + 6*a^4*b^7*d^2 + 4*a^6*b^5*d^2 + a^8*b^3*d^2))^(1/2)*(32*b^18*d^4 + 160*a^2*b^16*d^4 + 288*a^4*b^14*d^4
+ 160*a^6*b^12*d^4 - 160*a^8*b^10*d^4 - 288*a^10*b^8*d^4 - 160*a^12*b^6*d^4 - 32*a^14*b^4*d^4))/((b^9*d^4 + a
^8*b*d^4 + 4*a^2*b^7*d^4 + 6*a^4*b^5*d^4 + 4*a^6*b^3*d^4)*(b^11*d^2 + 4*a^2*b^9*d^2 + 6*a^4*b^7*d^2 + 4*a^6*b^
5*d^2 + a^8*b^3*d^2)))*(-4*(B^2*a^9 + 25*B^2*a^5*b^4 + 10*B^2*a^7*b^2)*(b^11*d^2 + 4*a^2*b^9*d^2 + 6*a^4*b^7*d
^2 + 4*a^6*b^5*d^2 + a^8*b^3*d^2))^(1/2))/(4*(b^11*d^2 + 4*a^2*b^9*d^2 + 6*a^4*b^7*d^2 + 4*a^6*b^5*d^2 + a^8*b
^3*d^2)) - (16*tan(c + d*x)^(1/2)*(60*B^2*a^3*b^13*d^2 + 52*B^2*a^5*b^11*d^2 + 128*B^2*a^7*b^9*d^2 + 424*B^2*a
^9*b^7*d^2 + 380*B^2*a^11*b^5*d^2 + 100*B^2*a^13*b^3*d^2 + 8*B^2*a^15*b*d^2))/(b^9*d^4 + a^8*b*d^4 + 4*a^2*b^7
*d^4 + 6*a^4*b^5*d^4 + 4*a^6*b^3*d^4))*(-4*(B^2*a^9 + 25*B^2*a^5*b^4 + 10*B^2*a^7*b^2)*(b^11*d^2 + 4*a^2*b^9*d
^2 + 6*a^4*b^7*d^2 + 4*a^6*b^5*d^2 + a^8*b^3*d^2))^(1/2))/(4*(b^11*d^2 + 4*a^2*b^9*d^2 + 6*a^4*b^7*d^2 + 4*a^6
*b^5*d^2 + a^8*b^3*d^2)))*(-4*(B^2*a^9 + 25*B^2*a^5*b^4 + 10*B^2*a^7*b^2)*(b^11*d^2 + 4*a^2*b^9*d^2 + 6*a^4*b^
7*d^2 + 4*a^6*b^5*d^2 + a^8*b^3*d^2))^(1/2))/(4*(b^11*d^2 + 4*a^2*b^9*d^2 + 6*a^4*b^7*d^2 + 4*a^6*b^5*d^2 + a^
8*b^3*d^2)) - (16*tan(c + d*x)^(1/2)*(B^4*a^14 - 2*B^4*a^4*b^10 - 4*B^4*a^6*b^8 - 27*B^4*a^8*b^6 + 15*B^4*a^10
*b^4 + 9*B^4*a^12*b^2))/(b^9*d^4 + a^8*b*d^4 + 4*a^2*b^7*d^4 + 6*a^4*b^5*d^4 + 4*a^6*b^3*d^4))*(-4*(B^2*a^9 +
25*B^2*a^5*b^4 + 10*B^2*a^7*b^2)*(b^11*d^2 + 4*a^2*b^9*d^2 + 6*a^4*b^7*d^2 + 4*a^6*b^5*d^2 + a^8*b^3*d^2))^(1/
2)*1i)/(4*(b^11*d^2 + 4*a^2*b^9*d^2 + 6*a^4*b^7*d^2 + 4*a^6*b^5*d^2 + a^8*b^3*d^2)))/((16*(B^5*a^13 + 10*B^5*a
^7*b^6 + 27*B^5*a^9*b^4 + 10*B^5*a^11*b^2))/(b^9*d^5 + a^8*b*d^5 + 4*a^2*b^7*d^5 + 6*a^4*b^5*d^5 + 4*a^6*b^3*d
^5) + (((((8*(4*B^3*a^4*b^11*d^2 - 304*B^3*a^6*b^9*d^2 - 120*B^3*a^8*b^7*d^2 + 320*B^3*a^10*b^5*d^2 + 148*B^3*
a^12*b^3*d^2 + 16*B^3*a^14*b*d^2))/(b^9*d^5 + a^8*b*d^5 + 4*a^2*b^7*d^5 + 6*a^4*b^5*d^5 + 4*a^6*b^3*d^5) + (((
((8*(96*B*a^3*b^14*d^4 + 480*B*a^5*b^12*d^4 + 960*B*a^7*b^10*d^4 + 960*B*a^9*b^8*d^4 + 480*B*a^11*b^6*d^4 + 96
*B*a^13*b^4*d^4))/(b^9*d^5 + a^8*b*d^5 + 4*a^2*b^7*d^5 + 6*a^4*b^5*d^5 + 4*a^6*b^3*d^5) - (4*tan(c + d*x)^(1/2
)*(-4*(B^2*a^9 + 25*B^2*a^5*b^4 + 10*B^2*a^7*b^2)*(b^11*d^2 + 4*a^2*b^9*d^2 + 6*a^4*b^7*d^2 + 4*a^6*b^5*d^2 +
a^8*b^3*d^2))^(1/2)*(32*b^18*d^4 + 160*a^2*b^16*d^4 + 288*a^4*b^14*d^4 + 160*a^6*b^12*d^4 - 160*a^8*b^10*d^4 -
288*a^10*b^8*d^4 - 160*a^12*b^6*d^4 - 32*a^14*b^4*d^4))/((b^9*d^4 + a^8*b*d^4 + 4*a^2*b^7*d^4 + 6*a^4*b^5*d^4
+ 4*a^6*b^3*d^4)*(b^11*d^2 + 4*a^2*b^9*d^2 + 6*a^4*b^7*d^2 + 4*a^6*b^5*d^2 + a^8*b^3*d^2)))*(-4*(B^2*a^9 + 25
*B^2*a^5*b^4 + 10*B^2*a^7*b^2)*(b^11*d^2 + 4*a^2*b^9*d^2 + 6*a^4*b^7*d^2 + 4*a^6*b^5*d^2 + a^8*b^3*d^2))^(1/2)
)/(4*(b^11*d^2 + 4*a^2*b^9*d^2 + 6*a^4*b^7*d^2 + 4*a^6*b^5*d^2 + a^8*b^3*d^2)) + (16*tan(c + d*x)^(1/2)*(60*B^
2*a^3*b^13*d^2 + 52*B^2*a^5*b^11*d^2 + 128*B^2*a^7*b^9*d^2 + 424*B^2*a^9*b^7*d^2 + 380*B^2*a^11*b^5*d^2 + 100*
B^2*a^13*b^3*d^2 + 8*B^2*a^15*b*d^2))/(b^9*d^4 + a^8*b*d^4 + 4*a^2*b^7*d^4 + 6*a^4*b^5*d^4 + 4*a^6*b^3*d^4))*(
-4*(B^2*a^9 + 25*B^2*a^5*b^4 + 10*B^2*a^7*b^2)*(b^11*d^2 + 4*a^2*b^9*d^2 + 6*a^4*b^7*d^2 + 4*a^6*b^5*d^2 + a^8
*b^3*d^2))^(1/2))/(4*(b^11*d^2 + 4*a^2*b^9*d^2 + 6*a^4*b^7*d^2 + 4*a^6*b^5*d^2 + a^8*b^3*d^2)))*(-4*(B^2*a^9 +
25*B^2*a^5*b^4 + 10*B^2*a^7*b^2)*(b^11*d^2 + 4*a^2*b^9*d^2 + 6*a^4*b^7*d^2 + 4*a^6*b^5*d^2 + a^8*b^3*d^2))^(1
/2))/(4*(b^11*d^2 + 4*a^2*b^9*d^2 + 6*a^4*b^7*d^2 + 4*a^6*b^5*d^2 + a^8*b^3*d^2)) + (16*tan(c + d*x)^(1/2)*(B^
4*a^14 - 2*B^4*a^4*b^10 - 4*B^4*a^6*b^8 - 27*B^4*a^8*b^6 + 15*B^4*a^10*b^4 + 9*B^4*a^12*b^2))/(b^9*d^4 + a^8*b
*d^4 + 4*a^2*b^7*d^4 + 6*a^4*b^5*d^4 + 4*a^6*b^3*d^4))*(-4*(B^2*a^9 + 25*B^2*a^5*b^4 + 10*B^2*a^7*b^2)*(b^11*d
^2 + 4*a^2*b^9*d^2 + 6*a^4*b^7*d^2 + 4*a^6*b^5*d^2 + a^8*b^3*d^2))^(1/2))/(4*(b^11*d^2 + 4*a^2*b^9*d^2 + 6*a^4
*b^7*d^2 + 4*a^6*b^5*d^2 + a^8*b^3*d^2)) + (((((8*(4*B^3*a^4*b^11*d^2 - 304*B^3*a^6*b^9*d^2 - 120*B^3*a^8*b^7*
d^2 + 320*B^3*a^10*b^5*d^2 + 148*B^3*a^12*b^3*d^2 + 16*B^3*a^14*b*d^2))/(b^9*d^5 + a^8*b*d^5 + 4*a^2*b^7*d^5 +
6*a^4*b^5*d^5 + 4*a^6*b^3*d^5) + (((((8*(96*B*a^3*b^14*d^4 + 480*B*a^5*b^12*d^4 + 960*B*a^7*b^10*d^4 + 960*B*
a^9*b^8*d^4 + 480*B*a^11*b^6*d^4 + 96*B*a^13*b^4*d^4))/(b^9*d^5 + a^8*b*d^5 + 4*a^2*b^7*d^5 + 6*a^4*b^5*d^5 +
4*a^6*b^3*d^5) + (4*tan(c + d*x)^(1/2)*(-4*(B^2*a^9 + 25*B^2*a^5*b^4 + 10*B^2*a^7*b^2)*(b^11*d^2 + 4*a^2*b^9*d
^2 + 6*a^4*b^7*d^2 + 4*a^6*b^5*d^2 + a^8*b^3*d^2))^(1/2)*(32*b^18*d^4 + 160*a^2*b^16*d^4 + 288*a^4*b^14*d^4 +
160*a^6*b^12*d^4 - 160*a^8*b^10*d^4 - 288*a^10*b^8*d^4 - 160*a^12*b^6*d^4 - 32*a^14*b^4*d^4))/((b^9*d^4 + a^8*
b*d^4 + 4*a^2*b^7*d^4 + 6*a^4*b^5*d^4 + 4*a^6*b^3*d^4)*(b^11*d^2 + 4*a^2*b^9*d^2 + 6*a^4*b^7*d^2 + 4*a^6*b^5*d
^2 + a^8*b^3*d^2)))*(-4*(B^2*a^9 + 25*B^2*a^5*b^4 + 10*B^2*a^7*b^2)*(b^11*d^2 + 4*a^2*b^9*d^2 + 6*a^4*b^7*d^2
+ 4*a^6*b^5*d^2 + a^8*b^3*d^2))^(1/2))/(4*(b^11*d^2 + 4*a^2*b^9*d^2 + 6*a^4*b^7*d^2 + 4*a^6*b^5*d^2 + a^8*b^3*
d^2)) - (16*tan(c + d*x)^(1/2)*(60*B^2*a^3*b^13*d^2 + 52*B^2*a^5*b^11*d^2 + 128*B^2*a^7*b^9*d^2 + 424*B^2*a^9*
b^7*d^2 + 380*B^2*a^11*b^5*d^2 + 100*B^2*a^13*b^3*d^2 + 8*B^2*a^15*b*d^2))/(b^9*d^4 + a^8*b*d^4 + 4*a^2*b^7*d^
4 + 6*a^4*b^5*d^4 + 4*a^6*b^3*d^4))*(-4*(B^2*a^9 + 25*B^2*a^5*b^4 + 10*B^2*a^7*b^2)*(b^11*d^2 + 4*a^2*b^9*d^2
+ 6*a^4*b^7*d^2 + 4*a^6*b^5*d^2 + a^8*b^3*d^2))^(1/2))/(4*(b^11*d^2 + 4*a^2*b^9*d^2 + 6*a^4*b^7*d^2 + 4*a^6*b^
5*d^2 + a^8*b^3*d^2)))*(-4*(B^2*a^9 + 25*B^2*a^5*b^4 + 10*B^2*a^7*b^2)*(b^11*d^2 + 4*a^2*b^9*d^2 + 6*a^4*b^7*d
^2 + 4*a^6*b^5*d^2 + a^8*b^3*d^2))^(1/2))/(4*(b^11*d^2 + 4*a^2*b^9*d^2 + 6*a^4*b^7*d^2 + 4*a^6*b^5*d^2 + a^8*b
^3*d^2)) - (16*tan(c + d*x)^(1/2)*(B^4*a^14 - 2*B^4*a^4*b^10 - 4*B^4*a^6*b^8 - 27*B^4*a^8*b^6 + 15*B^4*a^10*b^
4 + 9*B^4*a^12*b^2))/(b^9*d^4 + a^8*b*d^4 + 4*a^2*b^7*d^4 + 6*a^4*b^5*d^4 + 4*a^6*b^3*d^4))*(-4*(B^2*a^9 + 25*
B^2*a^5*b^4 + 10*B^2*a^7*b^2)*(b^11*d^2 + 4*a^2*b^9*d^2 + 6*a^4*b^7*d^2 + 4*a^6*b^5*d^2 + a^8*b^3*d^2))^(1/2))
/(4*(b^11*d^2 + 4*a^2*b^9*d^2 + 6*a^4*b^7*d^2 + 4*a^6*b^5*d^2 + a^8*b^3*d^2))))*(-4*(B^2*a^9 + 25*B^2*a^5*b^4
+ 10*B^2*a^7*b^2)*(b^11*d^2 + 4*a^2*b^9*d^2 + 6*a^4*b^7*d^2 + 4*a^6*b^5*d^2 + a^8*b^3*d^2))^(1/2)*1i)/(2*(b^11
*d^2 + 4*a^2*b^9*d^2 + 6*a^4*b^7*d^2 + 4*a^6*b^5*d^2 + a^8*b^3*d^2)) + (2*B*tan(c + d*x)^(1/2))/(b*d) + (B*a^3
*tan(c + d*x)^(1/2))/((a*b*d + b^2*d*tan(c + d*x))*(a^2 + b^2)) - (B*a^3*tan(c + d*x)^(1/2))/(b*(a*d + b*d*tan
(c + d*x))*(a^2 + b^2))