\(\int \frac {\sqrt {\tan (c+d x)} (a B+b B \tan (c+d x))}{(a+b \tan (c+d x))^2} \, dx\) [424]

Optimal result

Integrand size = 36, antiderivative size = 237 \[ \int \frac {\sqrt {\tan (c+d x)} (a B+b B \tan (c+d x))}{(a+b \tan (c+d x))^2} \, dx=-\frac {(a+b) B \arctan \left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} \left (a^2+b^2\right ) d}+\frac {(a+b) B \arctan \left (1+\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} \left (a^2+b^2\right ) d}-\frac {2 \sqrt {a} \sqrt {b} B \arctan \left (\frac {\sqrt {b} \sqrt {\tan (c+d x)}}{\sqrt {a}}\right )}{\left (a^2+b^2\right ) d}+\frac {(a-b) B \log \left (1-\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right )}{2 \sqrt {2} \left (a^2+b^2\right ) d}-\frac {(a-b) B \log \left (1+\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right )}{2 \sqrt {2} \left (a^2+b^2\right ) d} \]

[Out]

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*arctan(b^(1/2)*tan(d*x+c)^(1/2)/a^(1/2))*a^
(1/2)*b^(1/2)/(a^2+b^2)/d

Rubi [A] (verified)

Time = 0.44 (sec) , antiderivative size = 237, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {21, 3653, 3615, 1182, 1176, 631, 210, 1179, 642, 3715, 65, 211} \[ \int \frac {\sqrt {\tan (c+d x)} (a B+b B \tan (c+d x))}{(a+b \tan (c+d x))^2} \, dx=-\frac {B (a+b) \arctan \left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} d \left (a^2+b^2\right )}+\frac {B (a+b) \arctan \left (\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{\sqrt {2} d \left (a^2+b^2\right )}-\frac {2 \sqrt {a} \sqrt {b} B \arctan \left (\frac {\sqrt {b} \sqrt {\tan (c+d x)}}{\sqrt {a}}\right )}{d \left (a^2+b^2\right )}+\frac {B (a-b) \log \left (\tan (c+d x)-\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{2 \sqrt {2} d \left (a^2+b^2\right )}-\frac {B (a-b) \log \left (\tan (c+d x)+\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{2 \sqrt {2} d \left (a^2+b^2\right )} \]

[In]

Int[(Sqrt[Tan[c + d*x]]*(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]*
Sqrt[Tan[c + d*x]]])/(Sqrt[2]*(a^2 + b^2)*d) - (2*Sqrt[a]*Sqrt[b]*B*ArcTan[(Sqrt[b]*Sqrt[Tan[c + d*x]])/Sqrt[a
]])/((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)

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 3653

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

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

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.

Time = 0.10 (sec) , antiderivative size = 205, normalized size of antiderivative = 0.86 \[ \int \frac {\sqrt {\tan (c+d x)} (a B+b B \tan (c+d x))}{(a+b \tan (c+d x))^2} \, dx=\frac {B \left (-6 \sqrt {2} b \arctan \left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )+6 \sqrt {2} b \arctan \left (1+\sqrt {2} \sqrt {\tan (c+d x)}\right )-24 \sqrt {a} \sqrt {b} \arctan \left (\frac {\sqrt {b} \sqrt {\tan (c+d x)}}{\sqrt {a}}\right )-3 \sqrt {2} b \log \left (1-\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right )+3 \sqrt {2} b \log \left (1+\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right )+8 a \operatorname {Hypergeometric2F1}\left (\frac {3}{4},1,\frac {7}{4},-\tan ^2(c+d x)\right ) \tan ^{\frac {3}{2}}(c+d x)\right )}{12 \left (a^2+b^2\right ) d} \]

[In]

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

[Out]

(B*(-6*Sqrt[2]*b*ArcTan[1 - Sqrt[2]*Sqrt[Tan[c + d*x]]] + 6*Sqrt[2]*b*ArcTan[1 + Sqrt[2]*Sqrt[Tan[c + d*x]]] -
24*Sqrt[a]*Sqrt[b]*ArcTan[(Sqrt[b]*Sqrt[Tan[c + d*x]])/Sqrt[a]] - 3*Sqrt[2]*b*Log[1 - Sqrt[2]*Sqrt[Tan[c + d*
x]] + Tan[c + d*x]] + 3*Sqrt[2]*b*Log[1 + Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]] + 8*a*Hypergeometric2F1[3
/4, 1, 7/4, -Tan[c + d*x]^2]*Tan[c + d*x]^(3/2)))/(12*(a^2 + b^2)*d)

Maple [A] (verified)

Time = 0.05 (sec) , antiderivative size = 225, normalized size of antiderivative = 0.95

method	result	size
derivativedivides	\(\frac {B \left (-\frac {2 a b \arctan \left (\frac {b \left (\sqrt {\tan }\left (d x +c \right )\right )}{\sqrt {a b}}\right )}{\left (a^{2}+b^{2}\right ) \sqrt {a b}}+\frac {\frac {b \sqrt {2}\, \left (\ln \left (\frac {1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}{1-\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}\right )+2 \arctan \left (1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )+2 \arctan \left (-1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )\right )}{4}+\frac {a \sqrt {2}\, \left (\ln \left (\frac {1-\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}{1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}\right )+2 \arctan \left (1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )+2 \arctan \left (-1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )\right )}{4}}{a^{2}+b^{2}}\right )}{d}\)	\(225\)
default	\(\frac {B \left (-\frac {2 a b \arctan \left (\frac {b \left (\sqrt {\tan }\left (d x +c \right )\right )}{\sqrt {a b}}\right )}{\left (a^{2}+b^{2}\right ) \sqrt {a b}}+\frac {\frac {b \sqrt {2}\, \left (\ln \left (\frac {1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}{1-\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}\right )+2 \arctan \left (1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )+2 \arctan \left (-1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )\right )}{4}+\frac {a \sqrt {2}\, \left (\ln \left (\frac {1-\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}{1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}\right )+2 \arctan \left (1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )+2 \arctan \left (-1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )\right )}{4}}{a^{2}+b^{2}}\right )}{d}\)	\(225\)

[In]

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

[Out]

1/d*B*(-2*a*b/(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. 1567 vs. \(2 (199) = 398\).

Time = 0.32 (sec) , antiderivative size = 3160, normalized size of antiderivative = 13.33 \[ \int \frac {\sqrt {\tan (c+d x)} (a B+b B \tan (c+d x))}{(a+b \tan (c+d x))^2} \, dx=\text {Too large to display} \]

[In]

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

[Out]

[1/2*((a^2 + b^2)*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^2)*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^2)*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^2)*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))) + 2*sqrt(-a*b)*B*log((b*tan(
d*x + c) - a - 2*sqrt(-a*b)*sqrt(tan(d*x + c)))/(b*tan(d*x + c) + a)))/((a^2 + b^2)*d), 1/2*((a^2 + b^2)*d*sqr
t(-(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^2)*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^2)*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^2)*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*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)))/((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*sqrt(a*b)*B*arctan(sqrt(a*b)/(b*sqrt(tan(d*x +
c)))))/((a^2 + b^2)*d)]

Sympy [F]

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

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.33 (sec) , antiderivative size = 172, normalized size of antiderivative = 0.73 \[ \int \frac {\sqrt {\tan (c+d x)} (a B+b B \tan (c+d x))}{(a+b \tan (c+d x))^2} \, dx=-\frac {\frac {8 \, B a b \arctan \left (\frac {b \sqrt {\tan \left (d x + c\right )}}{\sqrt {a b}}\right )}{{\left (a^{2} + b^{2}\right )} \sqrt {a b}} - \frac {{\left (2 \, \sqrt {2} {\left (a + b\right )} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + 2 \, \sqrt {\tan \left (d x + c\right )}\right )}\right ) + 2 \, \sqrt {2} {\left (a + b\right )} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - 2 \, \sqrt {\tan \left (d x + c\right )}\right )}\right ) - \sqrt {2} {\left (a - b\right )} \log \left (\sqrt {2} \sqrt {\tan \left (d x + c\right )} + \tan \left (d x + c\right ) + 1\right ) + \sqrt {2} {\left (a - b\right )} \log \left (-\sqrt {2} \sqrt {\tan \left (d x + c\right )} + \tan \left (d x + c\right ) + 1\right )\right )} B}{a^{2} + b^{2}}}{4 \, d} \]

[In]

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

[Out]

-1/4*(8*B*a*b*arctan(b*sqrt(tan(d*x + c))/sqrt(a*b))/((a^2 + b^2)*sqrt(a*b)) - (2*sqrt(2)*(a + b)*arctan(1/2*s
qrt(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)*sqrt(
tan(d*x + c)) + tan(d*x + c) + 1))*B/(a^2 + b^2))/d

Giac [F(-1)]

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

[In]

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

[Out]

Timed out

Mupad [B] (verification not implemented)

Time = 37.53 (sec) , antiderivative size = 17323, normalized size of antiderivative = 73.09 \[ \int \frac {\sqrt {\tan (c+d x)} (a B+b B \tan (c+d x))}{(a+b \tan (c+d x))^2} \, dx=\text {Too large to display} \]

[In]

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

[Out]

(log(- (((((((((256*B*a*b^3*(2*a^4 - b^4 + a^2*b^2))/d - 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))*((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*b^2*tan(c + d*x)^(1/2)*(a^6 + 17*b^6 - 29*a^2*b^4 + 19*a^4*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*b^2*(a^6 + 13*b^6 - 45*a^2*b^4 + 39*a^4*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*b^3*tan(c + d*x)^(1/2)*(9*a^6 - 3*b^6 + 3*a^2*b^4 - 17*a^4*b^2))/(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^5*b^3*(9*a^4 - b^4))/(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(- (((((((((256*B*a*b^3*(2*a^
4 - b^4 + a^2*b^2))/d - 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))*(-(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*b^2*tan(c + d*x)^(1/2)*(a^6 + 17*b^6 - 29*a^2*b^4 + 19*a^4*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*b^2*(a^6 + 13*b^6 - 45*a^2*b^4 + 39*a^4*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*b^3*tan(
c + d*x)^(1/2)*(9*a^6 - 3*b^6 + 3*a^2*b^4 - 17*a^4*b^2))/(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^5*b^3*(9
*a^4 - b^4))/(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(- (((((((((256*B*a*b^3*(2*a^4 - b^4 + a^2*b^2))/d + 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))*((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*b^2*tan(c + d*x)^(1/2)*(a
^6 + 17*b^6 - 29*a^2*b^4 + 19*a^4*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*b^2*(a^6 + 13*b^6 - 45*
a^2*b^4 + 39*a^4*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*b^3*tan(c + d*x)^(1/2)*(9*a^6 - 3*b^6 +
3*a^2*b^4 - 17*a^4*b^2))/(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^5*b^3*(9*a^4 - b^4))/(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)/(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(- (((((((((256*B*a*b^3*(2*a^4 - b^4 + a^2*b^2))/d + 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))*(-(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*b^2*tan(c + d*x)^(1/2)*(a^6 + 17*b^6 - 29*a^2*b^4 +
19*a^4*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*b^2*(a^6 + 13*b^6 - 45*a^2*b^4 + 39*a^4*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*b^3*tan(c + d*x)^(1/2)*(9*a^6 - 3*b^6 + 3*a^2*b^4 - 17*a^4*b^2))
/(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^5*b^3*(9*a^4 - b^4))/(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)/(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((16*B^5*a*b^9*(a^2 - 3*b^2))/(d^5*(a^2 + b^2)^4) - (((((((((128*B*a*b^3*(5*b^4 - a^4 + 4*a^2*b^2))/d - 12
8*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))*((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*b^4*tan(c + d*x)^(1/2)*(a
^6 - 15*b^6 + 35*a^2*b^4 - 13*a^4*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^2*b^4*(a^6 - 39*b^6 + 43*
a^2*b^4 - 13*a^4*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^5*tan(c + d*x)^(1/2)*(a^8 + 2*b^8 - 5*a^2*
b^6 + 17*a^4*b^4 - 7*a^6*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)*(((192*B^4*a^2*b^10*d^4 - 16*B^4*b^12*d^4 - 6
08*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((16*B^5*a*b^9*(a^2 - 3*b^
2))/(d^5*(a^2 + b^2)^4) - (((((((((128*B*a*b^3*(5*b^4 - a^4 + 4*a^2*b^2))/d - 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))*(-(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*b^4*tan(c + d*x)^(1/2)*(a^6 - 15*b^6 + 35*a^2*b^4 - 1
3*a^4*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^2*b^4*(a^6 - 39*b^6 + 43*a^2*b^4 - 13*a^4*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^5*tan(c + d*x)^(1/2)*(a^8 + 2*b^8 - 5*a^2*b^6 + 17*a^4*b^4 - 7*a^6*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)*(-((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((16*B^5*a*b^9*(a^2 - 3*b^2))/(d^5*(a^2 + b^2)^4) -
(((((((((128*B*a*b^3*(5*b^4 - a^4 + 4*a^2*b^2))/d + 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))*((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*b^4*tan(c + d*x)^(1/2)*(a^6 - 15*b^6 + 35*a^2*b^4 - 13*a^4*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^2*b^4*(a^6 - 39*b^6 + 43*a^2*b^4 - 13*a^4*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^5*tan(c + d*x)^(1/2)*(a^8 + 2*b^8 - 5*a^2*b^6 + 17*a^4*b^4 - 7*a^6*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)*(((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((16*B^5*a*b^9*(a^2 - 3*b^2))/(d^5*(a^2 + b^2)^4) - (((((((((128*B*a*b^3*(5*
b^4 - a^4 + 4*a^2*b^2))/d + 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))*(-(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*b^4*tan(c + d*x)^(1/2)*(a^6 - 15*b^6 + 35*a^2*b^4 - 13*a^4*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 - (3
2*B^3*a^2*b^4*(a^6 - 39*b^6 + 43*a^2*b^4 - 13*a^4*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^5*tan(c
+ d*x)^(1/2)*(a^8 + 2*b^8 - 5*a^2*b^6 + 17*a^4*b^4 - 7*a^6*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)*(-((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)*(3*B^4*a^4*b^9 - 3*B^4*a^6*b^7 + 17*B^4*a^8*b^5 - 9*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) - (((8*(52*B^3*a^4*b^10*d^2 - 128*B^3*a^6
*b^8*d^2 - 24*B^3*a^8*b^6*d^2 + 160*B^3*a^10*b^4*d^2 + 4*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) + (((((8*(320*B*a^7*b^9*d^4 - 96*B*a^3*b^13*d^4 - 32*B*a*b^15*d^4 + 480*B*a^
9*b^7*d^4 + 288*B*a^11*b^5*d^4 + 64*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*(B^2*a*b^5 + 9*B^2*a^5*b - 6*B^2*a^3*b^3)*(a^8*d^2 + b^8*d^2 + 4*a^2*b^
6*d^2 + 6*a^4*b^4*d^2 + 4*a^6*b^2*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^1
1*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^2 + b^8*d^2 + 4*a^2*
b^6*d^2 + 6*a^4*b^4*d^2 + 4*a^6*b^2*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)))
*(-4*(B^2*a*b^5 + 9*B^2*a^5*b - 6*B^2*a^3*b^3)*(a^8*d^2 + b^8*d^2 + 4*a^2*b^6*d^2 + 6*a^4*b^4*d^2 + 4*a^6*b^2*
d^2))^(1/2))/(4*(a^8*d^2 + b^8*d^2 + 4*a^2*b^6*d^2 + 6*a^4*b^4*d^2 + 4*a^6*b^2*d^2)) + (16*tan(c + d*x)^(1/2)*
(68*B^2*a^3*b^12*d^2 + 20*B^2*a^5*b^10*d^2 - 88*B^2*a^7*b^8*d^2 + 40*B^2*a^9*b^6*d^2 + 84*B^2*a^11*b^4*d^2 + 4
*B^2*a^13*b^2*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))*(-4*(B^2*a*b^5 + 9*B^
2*a^5*b - 6*B^2*a^3*b^3)*(a^8*d^2 + b^8*d^2 + 4*a^2*b^6*d^2 + 6*a^4*b^4*d^2 + 4*a^6*b^2*d^2))^(1/2))/(4*(a^8*d
^2 + b^8*d^2 + 4*a^2*b^6*d^2 + 6*a^4*b^4*d^2 + 4*a^6*b^2*d^2)))*(-4*(B^2*a*b^5 + 9*B^2*a^5*b - 6*B^2*a^3*b^3)*
(a^8*d^2 + b^8*d^2 + 4*a^2*b^6*d^2 + 6*a^4*b^4*d^2 + 4*a^6*b^2*d^2))^(1/2))/(4*(a^8*d^2 + b^8*d^2 + 4*a^2*b^6*
d^2 + 6*a^4*b^4*d^2 + 4*a^6*b^2*d^2)))*(-4*(B^2*a*b^5 + 9*B^2*a^5*b - 6*B^2*a^3*b^3)*(a^8*d^2 + b^8*d^2 + 4*a^
2*b^6*d^2 + 6*a^4*b^4*d^2 + 4*a^6*b^2*d^2))^(1/2)*1i)/(4*(a^8*d^2 + b^8*d^2 + 4*a^2*b^6*d^2 + 6*a^4*b^4*d^2 +
4*a^6*b^2*d^2)) + (((16*tan(c + d*x)^(1/2)*(3*B^4*a^4*b^9 - 3*B^4*a^6*b^7 + 17*B^4*a^8*b^5 - 9*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) + (((8*(52*B^3*a^4*b^10*d^2 - 128*B^3*a^6*
b^8*d^2 - 24*B^3*a^8*b^6*d^2 + 160*B^3*a^10*b^4*d^2 + 4*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) + (((((8*(320*B*a^7*b^9*d^4 - 96*B*a^3*b^13*d^4 - 32*B*a*b^15*d^4 + 480*B*a^9
*b^7*d^4 + 288*B*a^11*b^5*d^4 + 64*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*(B^2*a*b^5 + 9*B^2*a^5*b - 6*B^2*a^3*b^3)*(a^8*d^2 + b^8*d^2 + 4*a^2*b^6
*d^2 + 6*a^4*b^4*d^2 + 4*a^6*b^2*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^2 + b^8*d^2 + 4*a^2*b
^6*d^2 + 6*a^4*b^4*d^2 + 4*a^6*b^2*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)))*
(-4*(B^2*a*b^5 + 9*B^2*a^5*b - 6*B^2*a^3*b^3)*(a^8*d^2 + b^8*d^2 + 4*a^2*b^6*d^2 + 6*a^4*b^4*d^2 + 4*a^6*b^2*d
^2))^(1/2))/(4*(a^8*d^2 + b^8*d^2 + 4*a^2*b^6*d^2 + 6*a^4*b^4*d^2 + 4*a^6*b^2*d^2)) - (16*tan(c + d*x)^(1/2)*(
68*B^2*a^3*b^12*d^2 + 20*B^2*a^5*b^10*d^2 - 88*B^2*a^7*b^8*d^2 + 40*B^2*a^9*b^6*d^2 + 84*B^2*a^11*b^4*d^2 + 4*
B^2*a^13*b^2*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))*(-4*(B^2*a*b^5 + 9*B^2
*a^5*b - 6*B^2*a^3*b^3)*(a^8*d^2 + b^8*d^2 + 4*a^2*b^6*d^2 + 6*a^4*b^4*d^2 + 4*a^6*b^2*d^2))^(1/2))/(4*(a^8*d^
2 + b^8*d^2 + 4*a^2*b^6*d^2 + 6*a^4*b^4*d^2 + 4*a^6*b^2*d^2)))*(-4*(B^2*a*b^5 + 9*B^2*a^5*b - 6*B^2*a^3*b^3)*(
a^8*d^2 + b^8*d^2 + 4*a^2*b^6*d^2 + 6*a^4*b^4*d^2 + 4*a^6*b^2*d^2))^(1/2))/(4*(a^8*d^2 + b^8*d^2 + 4*a^2*b^6*d
^2 + 6*a^4*b^4*d^2 + 4*a^6*b^2*d^2)))*(-4*(B^2*a*b^5 + 9*B^2*a^5*b - 6*B^2*a^3*b^3)*(a^8*d^2 + b^8*d^2 + 4*a^2
*b^6*d^2 + 6*a^4*b^4*d^2 + 4*a^6*b^2*d^2))^(1/2)*1i)/(4*(a^8*d^2 + b^8*d^2 + 4*a^2*b^6*d^2 + 6*a^4*b^4*d^2 + 4
*a^6*b^2*d^2)))/((16*(B^5*a^5*b^7 - 9*B^5*a^9*b^3))/(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)*(3*B^4*a^4*b^9 - 3*B^4*a^6*b^7 + 17*B^4*a^8*b^5 - 9*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) - (((8*(52*B^3*a^4*b^10*d^2 - 128*B^3*a^6*b^8*d^
2 - 24*B^3*a^8*b^6*d^2 + 160*B^3*a^10*b^4*d^2 + 4*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) + (((((8*(320*B*a^7*b^9*d^4 - 96*B*a^3*b^13*d^4 - 32*B*a*b^15*d^4 + 480*B*a^9*b^7*d
^4 + 288*B*a^11*b^5*d^4 + 64*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*(B^2*a*b^5 + 9*B^2*a^5*b - 6*B^2*a^3*b^3)*(a^8*d^2 + b^8*d^2 + 4*a^2*b^6*d^2 +
6*a^4*b^4*d^2 + 4*a^6*b^2*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^2 + b^8*d^2 + 4*a^2*b^6*d^2
+ 6*a^4*b^4*d^2 + 4*a^6*b^2*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)))*(-4*(B
^2*a*b^5 + 9*B^2*a^5*b - 6*B^2*a^3*b^3)*(a^8*d^2 + b^8*d^2 + 4*a^2*b^6*d^2 + 6*a^4*b^4*d^2 + 4*a^6*b^2*d^2))^(
1/2))/(4*(a^8*d^2 + b^8*d^2 + 4*a^2*b^6*d^2 + 6*a^4*b^4*d^2 + 4*a^6*b^2*d^2)) + (16*tan(c + d*x)^(1/2)*(68*B^2
*a^3*b^12*d^2 + 20*B^2*a^5*b^10*d^2 - 88*B^2*a^7*b^8*d^2 + 40*B^2*a^9*b^6*d^2 + 84*B^2*a^11*b^4*d^2 + 4*B^2*a^
13*b^2*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))*(-4*(B^2*a*b^5 + 9*B^2*a^5*b
- 6*B^2*a^3*b^3)*(a^8*d^2 + b^8*d^2 + 4*a^2*b^6*d^2 + 6*a^4*b^4*d^2 + 4*a^6*b^2*d^2))^(1/2))/(4*(a^8*d^2 + b^
8*d^2 + 4*a^2*b^6*d^2 + 6*a^4*b^4*d^2 + 4*a^6*b^2*d^2)))*(-4*(B^2*a*b^5 + 9*B^2*a^5*b - 6*B^2*a^3*b^3)*(a^8*d^
2 + b^8*d^2 + 4*a^2*b^6*d^2 + 6*a^4*b^4*d^2 + 4*a^6*b^2*d^2))^(1/2))/(4*(a^8*d^2 + b^8*d^2 + 4*a^2*b^6*d^2 + 6
*a^4*b^4*d^2 + 4*a^6*b^2*d^2)))*(-4*(B^2*a*b^5 + 9*B^2*a^5*b - 6*B^2*a^3*b^3)*(a^8*d^2 + b^8*d^2 + 4*a^2*b^6*d
^2 + 6*a^4*b^4*d^2 + 4*a^6*b^2*d^2))^(1/2))/(4*(a^8*d^2 + b^8*d^2 + 4*a^2*b^6*d^2 + 6*a^4*b^4*d^2 + 4*a^6*b^2*
d^2)) - (((16*tan(c + d*x)^(1/2)*(3*B^4*a^4*b^9 - 3*B^4*a^6*b^7 + 17*B^4*a^8*b^5 - 9*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) + (((8*(52*B^3*a^4*b^10*d^2 - 128*B^3*a^6*b^8*d^2 -
24*B^3*a^8*b^6*d^2 + 160*B^3*a^10*b^4*d^2 + 4*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) + (((((8*(320*B*a^7*b^9*d^4 - 96*B*a^3*b^13*d^4 - 32*B*a*b^15*d^4 + 480*B*a^9*b^7*d^4 +
288*B*a^11*b^5*d^4 + 64*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*(B^2*a*b^5 + 9*B^2*a^5*b - 6*B^2*a^3*b^3)*(a^8*d^2 + b^8*d^2 + 4*a^2*b^6*d^2 + 6*a
^4*b^4*d^2 + 4*a^6*b^2*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^2 + b^8*d^2 + 4*a^2*b^6*d^2 + 6
*a^4*b^4*d^2 + 4*a^6*b^2*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)))*(-4*(B^2*a
*b^5 + 9*B^2*a^5*b - 6*B^2*a^3*b^3)*(a^8*d^2 + b^8*d^2 + 4*a^2*b^6*d^2 + 6*a^4*b^4*d^2 + 4*a^6*b^2*d^2))^(1/2)
)/(4*(a^8*d^2 + b^8*d^2 + 4*a^2*b^6*d^2 + 6*a^4*b^4*d^2 + 4*a^6*b^2*d^2)) - (16*tan(c + d*x)^(1/2)*(68*B^2*a^3
*b^12*d^2 + 20*B^2*a^5*b^10*d^2 - 88*B^2*a^7*b^8*d^2 + 40*B^2*a^9*b^6*d^2 + 84*B^2*a^11*b^4*d^2 + 4*B^2*a^13*b
^2*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))*(-4*(B^2*a*b^5 + 9*B^2*a^5*b - 6
*B^2*a^3*b^3)*(a^8*d^2 + b^8*d^2 + 4*a^2*b^6*d^2 + 6*a^4*b^4*d^2 + 4*a^6*b^2*d^2))^(1/2))/(4*(a^8*d^2 + b^8*d^
2 + 4*a^2*b^6*d^2 + 6*a^4*b^4*d^2 + 4*a^6*b^2*d^2)))*(-4*(B^2*a*b^5 + 9*B^2*a^5*b - 6*B^2*a^3*b^3)*(a^8*d^2 +
b^8*d^2 + 4*a^2*b^6*d^2 + 6*a^4*b^4*d^2 + 4*a^6*b^2*d^2))^(1/2))/(4*(a^8*d^2 + b^8*d^2 + 4*a^2*b^6*d^2 + 6*a^4
*b^4*d^2 + 4*a^6*b^2*d^2)))*(-4*(B^2*a*b^5 + 9*B^2*a^5*b - 6*B^2*a^3*b^3)*(a^8*d^2 + b^8*d^2 + 4*a^2*b^6*d^2 +
6*a^4*b^4*d^2 + 4*a^6*b^2*d^2))^(1/2))/(4*(a^8*d^2 + b^8*d^2 + 4*a^2*b^6*d^2 + 6*a^4*b^4*d^2 + 4*a^6*b^2*d^2)
)))*(-4*(B^2*a*b^5 + 9*B^2*a^5*b - 6*B^2*a^3*b^3)*(a^8*d^2 + b^8*d^2 + 4*a^2*b^6*d^2 + 6*a^4*b^4*d^2 + 4*a^6*b
^2*d^2))^(1/2)*1i)/(2*(a^8*d^2 + b^8*d^2 + 4*a^2*b^6*d^2 + 6*a^4*b^4*d^2 + 4*a^6*b^2*d^2)) + (atan(((((((16*(8
*B^3*a^4*b^10*d^2 - 78*B^3*a^2*b^12*d^2 + 60*B^3*a^6*b^8*d^2 - 24*B^3*a^8*b^6*d^2 + 2*B^3*a^10*b^4*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*(40*B*a*b^15*d^4 + 192*B*a^3*b^13*d^4
+ 360*B*a^5*b^11*d^4 + 320*B*a^7*b^9*d^4 + 120*B*a^9*b^7*d^4 - 8*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*b^5 + B^2*a^5*b - 6*B^2*a^3*b^3)
*(a^8*d^2 + b^8*d^2 + 4*a^2*b^6*d^2 + 6*a^4*b^4*d^2 + 4*a^6*b^2*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^2 + b^8*d^2 + 4*a^2*b^6*d^2 + 6*a^4*b^4*d^2 + 4*a^6*b^2*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)))*(-4*(9*B^2*a*b^5 + B^2*a^5*b - 6*B^2*a^3*b^3)*(a^8*d^2 + b^8*d^2 + 4*a^2*b^6*d^
2 + 6*a^4*b^4*d^2 + 4*a^6*b^2*d^2))^(1/2))/(4*(a^8*d^2 + b^8*d^2 + 4*a^2*b^6*d^2 + 6*a^4*b^4*d^2 + 4*a^6*b^2*d
^2)) + (16*tan(c + d*x)^(1/2)*(20*B^2*a^3*b^12*d^2 + 168*B^2*a^5*b^10*d^2 + 40*B^2*a^7*b^8*d^2 - 44*B^2*a^9*b^
6*d^2 + 4*B^2*a^11*b^4*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))*(-4*(9*B^2*a*b^5 + B^2*a^5*b - 6*B^2*a^3*b^3)*(a^8*d^2 + b^8*d^2 + 4*a^2*b^6*d^2 + 6*a^4*b^4*d^2 + 4*a
^6*b^2*d^2))^(1/2))/(4*(a^8*d^2 + b^8*d^2 + 4*a^2*b^6*d^2 + 6*a^4*b^4*d^2 + 4*a^6*b^2*d^2)))*(-4*(9*B^2*a*b^5
+ B^2*a^5*b - 6*B^2*a^3*b^3)*(a^8*d^2 + b^8*d^2 + 4*a^2*b^6*d^2 + 6*a^4*b^4*d^2 + 4*a^6*b^2*d^2))^(1/2))/(4*(a
^8*d^2 + b^8*d^2 + 4*a^2*b^6*d^2 + 6*a^4*b^4*d^2 + 4*a^6*b^2*d^2)) + (16*tan(c + d*x)^(1/2)*(2*B^4*b^13 - 5*B^
4*a^2*b^11 + 17*B^4*a^4*b^9 - 7*B^4*a^6*b^7 + B^4*a^8*b^5))/(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))*(-4*(9*B^2*a*b^5 + B^2*a^5*b - 6*B^2*a^3*b^3)*(a^8*d^2 + b^8*d^2 + 4*a^2*b^6*d^2 + 6*a^4*b^
4*d^2 + 4*a^6*b^2*d^2))^(1/2)*1i)/(4*(a^8*d^2 + b^8*d^2 + 4*a^2*b^6*d^2 + 6*a^4*b^4*d^2 + 4*a^6*b^2*d^2)) - ((
(((16*(8*B^3*a^4*b^10*d^2 - 78*B^3*a^2*b^12*d^2 + 60*B^3*a^6*b^8*d^2 - 24*B^3*a^8*b^6*d^2 + 2*B^3*a^10*b^4*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*(40*B*a*b^15*d^4 + 192*B*a^3*
b^13*d^4 + 360*B*a^5*b^11*d^4 + 320*B*a^7*b^9*d^4 + 120*B*a^9*b^7*d^4 - 8*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*b^5 + B^2*a^5*b - 6*B^2*
a^3*b^3)*(a^8*d^2 + b^8*d^2 + 4*a^2*b^6*d^2 + 6*a^4*b^4*d^2 + 4*a^6*b^2*d^2))^(1/2)*(32*b^17*d^4 + 160*a^2*b^1
5*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^2 + b^8*d^2 + 4*a^2*b^6*d^2 + 6*a^4*b^4*d^2 + 4*a^6*b^2*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)))*(-4*(9*B^2*a*b^5 + B^2*a^5*b - 6*B^2*a^3*b^3)*(a^8*d^2 + b^8*d^2 + 4*a^
2*b^6*d^2 + 6*a^4*b^4*d^2 + 4*a^6*b^2*d^2))^(1/2))/(4*(a^8*d^2 + b^8*d^2 + 4*a^2*b^6*d^2 + 6*a^4*b^4*d^2 + 4*a
^6*b^2*d^2)) - (16*tan(c + d*x)^(1/2)*(20*B^2*a^3*b^12*d^2 + 168*B^2*a^5*b^10*d^2 + 40*B^2*a^7*b^8*d^2 - 44*B^
2*a^9*b^6*d^2 + 4*B^2*a^11*b^4*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))*(-4*(9*B^2*a*b^5 + B^2*a^5*b - 6*B^2*a^3*b^3)*(a^8*d^2 + b^8*d^2 + 4*a^2*b^6*d^2 + 6*a^4*b^4*d
^2 + 4*a^6*b^2*d^2))^(1/2))/(4*(a^8*d^2 + b^8*d^2 + 4*a^2*b^6*d^2 + 6*a^4*b^4*d^2 + 4*a^6*b^2*d^2)))*(-4*(9*B^
2*a*b^5 + B^2*a^5*b - 6*B^2*a^3*b^3)*(a^8*d^2 + b^8*d^2 + 4*a^2*b^6*d^2 + 6*a^4*b^4*d^2 + 4*a^6*b^2*d^2))^(1/2
))/(4*(a^8*d^2 + b^8*d^2 + 4*a^2*b^6*d^2 + 6*a^4*b^4*d^2 + 4*a^6*b^2*d^2)) - (16*tan(c + d*x)^(1/2)*(2*B^4*b^1
3 - 5*B^4*a^2*b^11 + 17*B^4*a^4*b^9 - 7*B^4*a^6*b^7 + B^4*a^8*b^5))/(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))*(-4*(9*B^2*a*b^5 + B^2*a^5*b - 6*B^2*a^3*b^3)*(a^8*d^2 + b^8*d^2 + 4*a^2*b^6*d^2 +
6*a^4*b^4*d^2 + 4*a^6*b^2*d^2))^(1/2)*1i)/(4*(a^8*d^2 + b^8*d^2 + 4*a^2*b^6*d^2 + 6*a^4*b^4*d^2 + 4*a^6*b^2*d^
2)))/((((((16*(8*B^3*a^4*b^10*d^2 - 78*B^3*a^2*b^12*d^2 + 60*B^3*a^6*b^8*d^2 - 24*B^3*a^8*b^6*d^2 + 2*B^3*a^10
*b^4*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*(40*B*a*b^15*d^4 + 19
2*B*a^3*b^13*d^4 + 360*B*a^5*b^11*d^4 + 320*B*a^7*b^9*d^4 + 120*B*a^9*b^7*d^4 - 8*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*b^5 + B^2*a^5*b
- 6*B^2*a^3*b^3)*(a^8*d^2 + b^8*d^2 + 4*a^2*b^6*d^2 + 6*a^4*b^4*d^2 + 4*a^6*b^2*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^2 + b^8*d^2 + 4*a^2*b^6*d^2 + 6*a^4*b^4*d^2 + 4*a^6*b^2*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)))*(-4*(9*B^2*a*b^5 + B^2*a^5*b - 6*B^2*a^3*b^3)*(a^8*d^2 + b^8*d^
2 + 4*a^2*b^6*d^2 + 6*a^4*b^4*d^2 + 4*a^6*b^2*d^2))^(1/2))/(4*(a^8*d^2 + b^8*d^2 + 4*a^2*b^6*d^2 + 6*a^4*b^4*d
^2 + 4*a^6*b^2*d^2)) + (16*tan(c + d*x)^(1/2)*(20*B^2*a^3*b^12*d^2 + 168*B^2*a^5*b^10*d^2 + 40*B^2*a^7*b^8*d^2
- 44*B^2*a^9*b^6*d^2 + 4*B^2*a^11*b^4*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))*(-4*(9*B^2*a*b^5 + B^2*a^5*b - 6*B^2*a^3*b^3)*(a^8*d^2 + b^8*d^2 + 4*a^2*b^6*d^2 + 6*a
^4*b^4*d^2 + 4*a^6*b^2*d^2))^(1/2))/(4*(a^8*d^2 + b^8*d^2 + 4*a^2*b^6*d^2 + 6*a^4*b^4*d^2 + 4*a^6*b^2*d^2)))*(
-4*(9*B^2*a*b^5 + B^2*a^5*b - 6*B^2*a^3*b^3)*(a^8*d^2 + b^8*d^2 + 4*a^2*b^6*d^2 + 6*a^4*b^4*d^2 + 4*a^6*b^2*d^
2))^(1/2))/(4*(a^8*d^2 + b^8*d^2 + 4*a^2*b^6*d^2 + 6*a^4*b^4*d^2 + 4*a^6*b^2*d^2)) + (16*tan(c + d*x)^(1/2)*(2
*B^4*b^13 - 5*B^4*a^2*b^11 + 17*B^4*a^4*b^9 - 7*B^4*a^6*b^7 + B^4*a^8*b^5))/(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))*(-4*(9*B^2*a*b^5 + B^2*a^5*b - 6*B^2*a^3*b^3)*(a^8*d^2 + b^8*d^2 + 4*a^2*b^
6*d^2 + 6*a^4*b^4*d^2 + 4*a^6*b^2*d^2))^(1/2))/(4*(a^8*d^2 + b^8*d^2 + 4*a^2*b^6*d^2 + 6*a^4*b^4*d^2 + 4*a^6*b
^2*d^2)) - (32*(3*B^5*a*b^11 - B^5*a^3*b^9))/(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*(8*B^3*a^4*b^10*d^2 - 78*B^3*a^2*b^12*d^2 + 60*B^3*a^6*b^8*d^2 - 24*B^3*a^8*b^6*d^2 + 2*B^3*a^10*
b^4*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*(40*B*a*b^15*d^4 + 192
*B*a^3*b^13*d^4 + 360*B*a^5*b^11*d^4 + 320*B*a^7*b^9*d^4 + 120*B*a^9*b^7*d^4 - 8*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*b^5 + B^2*a^5*b -
6*B^2*a^3*b^3)*(a^8*d^2 + b^8*d^2 + 4*a^2*b^6*d^2 + 6*a^4*b^4*d^2 + 4*a^6*b^2*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 - 3
2*a^14*b^3*d^4))/((a^8*d^2 + b^8*d^2 + 4*a^2*b^6*d^2 + 6*a^4*b^4*d^2 + 4*a^6*b^2*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)))*(-4*(9*B^2*a*b^5 + B^2*a^5*b - 6*B^2*a^3*b^3)*(a^8*d^2 + b^8*d^2
+ 4*a^2*b^6*d^2 + 6*a^4*b^4*d^2 + 4*a^6*b^2*d^2))^(1/2))/(4*(a^8*d^2 + b^8*d^2 + 4*a^2*b^6*d^2 + 6*a^4*b^4*d^
2 + 4*a^6*b^2*d^2)) - (16*tan(c + d*x)^(1/2)*(20*B^2*a^3*b^12*d^2 + 168*B^2*a^5*b^10*d^2 + 40*B^2*a^7*b^8*d^2
- 44*B^2*a^9*b^6*d^2 + 4*B^2*a^11*b^4*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))*(-4*(9*B^2*a*b^5 + B^2*a^5*b - 6*B^2*a^3*b^3)*(a^8*d^2 + b^8*d^2 + 4*a^2*b^6*d^2 + 6*a^
4*b^4*d^2 + 4*a^6*b^2*d^2))^(1/2))/(4*(a^8*d^2 + b^8*d^2 + 4*a^2*b^6*d^2 + 6*a^4*b^4*d^2 + 4*a^6*b^2*d^2)))*(-
4*(9*B^2*a*b^5 + B^2*a^5*b - 6*B^2*a^3*b^3)*(a^8*d^2 + b^8*d^2 + 4*a^2*b^6*d^2 + 6*a^4*b^4*d^2 + 4*a^6*b^2*d^2
))^(1/2))/(4*(a^8*d^2 + b^8*d^2 + 4*a^2*b^6*d^2 + 6*a^4*b^4*d^2 + 4*a^6*b^2*d^2)) - (16*tan(c + d*x)^(1/2)*(2*
B^4*b^13 - 5*B^4*a^2*b^11 + 17*B^4*a^4*b^9 - 7*B^4*a^6*b^7 + B^4*a^8*b^5))/(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))*(-4*(9*B^2*a*b^5 + B^2*a^5*b - 6*B^2*a^3*b^3)*(a^8*d^2 + b^8*d^2 + 4*a^2*b^6
*d^2 + 6*a^4*b^4*d^2 + 4*a^6*b^2*d^2))^(1/2))/(4*(a^8*d^2 + b^8*d^2 + 4*a^2*b^6*d^2 + 6*a^4*b^4*d^2 + 4*a^6*b^
2*d^2))))*(-4*(9*B^2*a*b^5 + B^2*a^5*b - 6*B^2*a^3*b^3)*(a^8*d^2 + b^8*d^2 + 4*a^2*b^6*d^2 + 6*a^4*b^4*d^2 + 4
*a^6*b^2*d^2))^(1/2)*1i)/(2*(a^8*d^2 + b^8*d^2 + 4*a^2*b^6*d^2 + 6*a^4*b^4*d^2 + 4*a^6*b^2*d^2))