\(\int \frac {A+B \tan (c+d x)}{\tan ^{\frac {3}{2}}(c+d x) (a+b \tan (c+d x))^2} \, dx\) [408]

Optimal result

Integrand size = 33, antiderivative size = 439 \[ \int \frac {A+B \tan (c+d x)}{\tan ^{\frac {3}{2}}(c+d x) (a+b \tan (c+d x))^2} \, dx=\frac {\left (a^2 (A-B)-b^2 (A-B)+2 a b (A+B)\right ) \arctan \left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} \left (a^2+b^2\right )^2 d}-\frac {\left (a^2 (A-B)-b^2 (A-B)+2 a b (A+B)\right ) \arctan \left (1+\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} \left (a^2+b^2\right )^2 d}-\frac {b^{3/2} \left (7 a^2 A b+3 A b^3-5 a^3 B-a b^2 B\right ) \arctan \left (\frac {\sqrt {b} \sqrt {\tan (c+d x)}}{\sqrt {a}}\right )}{a^{5/2} \left (a^2+b^2\right )^2 d}+\frac {\left (2 a b (A-B)-a^2 (A+B)+b^2 (A+B)\right ) \log \left (1-\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right )}{2 \sqrt {2} \left (a^2+b^2\right )^2 d}-\frac {\left (2 a b (A-B)-a^2 (A+B)+b^2 (A+B)\right ) \log \left (1+\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right )}{2 \sqrt {2} \left (a^2+b^2\right )^2 d}-\frac {2 a^2 A+3 A b^2-a b B}{a^2 \left (a^2+b^2\right ) d \sqrt {\tan (c+d x)}}+\frac {b (A b-a B)}{a \left (a^2+b^2\right ) d \sqrt {\tan (c+d x)} (a+b \tan (c+d x))} \]

[Out]

-b^(3/2)*(7*A*a^2*b+3*A*b^3-5*B*a^3-B*a*b^2)*arctan(b^(1/2)*tan(d*x+c)^(1/2)/a^(1/2))/a^(5/2)/(a^2+b^2)^2/d-1/
2*(a^2*(A-B)-b^2*(A-B)+2*a*b*(A+B))*arctan(-1+2^(1/2)*tan(d*x+c)^(1/2))/(a^2+b^2)^2/d*2^(1/2)-1/2*(a^2*(A-B)-b
^2*(A-B)+2*a*b*(A+B))*arctan(1+2^(1/2)*tan(d*x+c)^(1/2))/(a^2+b^2)^2/d*2^(1/2)+1/4*(2*a*b*(A-B)-a^2*(A+B)+b^2*
(A+B))*ln(1-2^(1/2)*tan(d*x+c)^(1/2)+tan(d*x+c))/(a^2+b^2)^2/d*2^(1/2)-1/4*(2*a*b*(A-B)-a^2*(A+B)+b^2*(A+B))*l
n(1+2^(1/2)*tan(d*x+c)^(1/2)+tan(d*x+c))/(a^2+b^2)^2/d*2^(1/2)+(-2*A*a^2-3*A*b^2+B*a*b)/a^2/(a^2+b^2)/d/tan(d*
x+c)^(1/2)+b*(A*b-B*a)/a/(a^2+b^2)/d/tan(d*x+c)^(1/2)/(a+b*tan(d*x+c))

Rubi [A] (verified)

Time = 1.56 (sec) , antiderivative size = 439, normalized size of antiderivative = 1.00, number of steps used = 16, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.394, Rules used = {3690, 3730, 3734, 3615, 1182, 1176, 631, 210, 1179, 642, 3715, 65, 211} \[ \int \frac {A+B \tan (c+d x)}{\tan ^{\frac {3}{2}}(c+d x) (a+b \tan (c+d x))^2} \, dx=\frac {\left (a^2 (A-B)+2 a b (A+B)-b^2 (A-B)\right ) \arctan \left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} d \left (a^2+b^2\right )^2}-\frac {\left (a^2 (A-B)+2 a b (A+B)-b^2 (A-B)\right ) \arctan \left (\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{\sqrt {2} d \left (a^2+b^2\right )^2}+\frac {b (A b-a B)}{a d \left (a^2+b^2\right ) \sqrt {\tan (c+d x)} (a+b \tan (c+d x))}-\frac {2 a^2 A-a b B+3 A b^2}{a^2 d \left (a^2+b^2\right ) \sqrt {\tan (c+d x)}}+\frac {\left (-\left (a^2 (A+B)\right )+2 a b (A-B)+b^2 (A+B)\right ) \log \left (\tan (c+d x)-\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{2 \sqrt {2} d \left (a^2+b^2\right )^2}-\frac {\left (-\left (a^2 (A+B)\right )+2 a b (A-B)+b^2 (A+B)\right ) \log \left (\tan (c+d x)+\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{2 \sqrt {2} d \left (a^2+b^2\right )^2}-\frac {b^{3/2} \left (-5 a^3 B+7 a^2 A b-a b^2 B+3 A b^3\right ) \arctan \left (\frac {\sqrt {b} \sqrt {\tan (c+d x)}}{\sqrt {a}}\right )}{a^{5/2} d \left (a^2+b^2\right )^2} \]

[In]

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

[Out]

((a^2*(A - B) - b^2*(A - B) + 2*a*b*(A + B))*ArcTan[1 - Sqrt[2]*Sqrt[Tan[c + d*x]]])/(Sqrt[2]*(a^2 + b^2)^2*d)
- ((a^2*(A - B) - b^2*(A - B) + 2*a*b*(A + B))*ArcTan[1 + Sqrt[2]*Sqrt[Tan[c + d*x]]])/(Sqrt[2]*(a^2 + b^2)^2
*d) - (b^(3/2)*(7*a^2*A*b + 3*A*b^3 - 5*a^3*B - a*b^2*B)*ArcTan[(Sqrt[b]*Sqrt[Tan[c + d*x]])/Sqrt[a]])/(a^(5/2
)*(a^2 + b^2)^2*d) + ((2*a*b*(A - B) - a^2*(A + B) + b^2*(A + B))*Log[1 - Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c +
d*x]])/(2*Sqrt[2]*(a^2 + b^2)^2*d) - ((2*a*b*(A - B) - a^2*(A + B) + b^2*(A + B))*Log[1 + Sqrt[2]*Sqrt[Tan[c
+ d*x]] + Tan[c + d*x]])/(2*Sqrt[2]*(a^2 + b^2)^2*d) - (2*a^2*A + 3*A*b^2 - a*b*B)/(a^2*(a^2 + b^2)*d*Sqrt[Tan
[c + d*x]]) + (b*(A*b - a*B))/(a*(a^2 + b^2)*d*Sqrt[Tan[c + d*x]]*(a + b*Tan[c + d*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 3690

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*tan[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*tan[(e
_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[b*(A*b - a*B)*(a + b*Tan[e + f*x])^(m + 1)*((c + d*Tan[e + f*x])^(n
+ 1)/(f*(m + 1)*(b*c - a*d)*(a^2 + b^2))), x] + Dist[1/((m + 1)*(b*c - a*d)*(a^2 + b^2)), Int[(a + b*Tan[e +
f*x])^(m + 1)*(c + d*Tan[e + f*x])^n*Simp[b*B*(b*c*(m + 1) + a*d*(n + 1)) + A*(a*(b*c - a*d)*(m + 1) - b^2*d*(
m + n + 2)) - (A*b - a*B)*(b*c - a*d)*(m + 1)*Tan[e + f*x] - b*d*(A*b - a*B)*(m + n + 2)*Tan[e + f*x]^2, x], x
], x] /; FreeQ[{a, b, c, d, e, f, A, B, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0]
&& LtQ[m, -1] && (IntegerQ[m] || IntegersQ[2*m, 2*n]) &&  !(ILtQ[n, -1] && ( !IntegerQ[m] || (EqQ[c, 0] && NeQ
[a, 0])))

Rule 3715

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

Rule 3730

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_)*((A_.) + (B_.)*t
an[(e_.) + (f_.)*(x_)] + (C_.)*tan[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(A*b^2 - a*(b*B - a*C))*(a + b*Ta
n[e + f*x])^(m + 1)*((c + d*Tan[e + f*x])^(n + 1)/(f*(m + 1)*(b*c - a*d)*(a^2 + b^2))), x] + Dist[1/((m + 1)*(
b*c - a*d)*(a^2 + b^2)), Int[(a + b*Tan[e + f*x])^(m + 1)*(c + d*Tan[e + f*x])^n*Simp[A*(a*(b*c - a*d)*(m + 1)
- b^2*d*(m + n + 2)) + (b*B - a*C)*(b*c*(m + 1) + a*d*(n + 1)) - (m + 1)*(b*c - a*d)*(A*b - a*B - b*C)*Tan[e
+ f*x] - d*(A*b^2 - a*(b*B - a*C))*(m + n + 2)*Tan[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C,
n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && LtQ[m, -1] &&  !(ILtQ[n, -1] && ( !I
ntegerQ[m] || (EqQ[c, 0] && NeQ[a, 0])))

Rule 3734

Int[(((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_)*((A_.) + (B_.)*tan[(e_.) + (f_.)*(x_)] + (C_.)*tan[(e_.) + (
f_.)*(x_)]^2))/((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Dist[1/(a^2 + b^2), Int[(c + d*Tan[e + f*
x])^n*Simp[b*B + a*(A - C) + (a*B - b*(A - C))*Tan[e + f*x], x], x], x] + Dist[(A*b^2 - a*b*B + a^2*C)/(a^2 +
b^2), Int[(c + d*Tan[e + f*x])^n*((1 + Tan[e + f*x]^2)/(a + b*Tan[e + f*x])), x], x] /; FreeQ[{a, b, c, d, e,
f, A, B, C, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] &&  !GtQ[n, 0] &&  !LeQ[n, -
1]

Rubi steps \begin{align*} \text {integral}& = \frac {b (A b-a B)}{a \left (a^2+b^2\right ) d \sqrt {\tan (c+d x)} (a+b \tan (c+d x))}+\frac {\int \frac {\frac {1}{2} \left (2 a^2 A+3 A b^2-a b B\right )-a (A b-a B) \tan (c+d x)+\frac {3}{2} b (A b-a B) \tan ^2(c+d x)}{\tan ^{\frac {3}{2}}(c+d x) (a+b \tan (c+d x))} \, dx}{a \left (a^2+b^2\right )} \\ & = -\frac {2 a^2 A+3 A b^2-a b B}{a^2 \left (a^2+b^2\right ) d \sqrt {\tan (c+d x)}}+\frac {b (A b-a B)}{a \left (a^2+b^2\right ) d \sqrt {\tan (c+d x)} (a+b \tan (c+d x))}-\frac {2 \int \frac {\frac {1}{4} \left (4 a^2 A b+3 A b^3-2 a^3 B-a b^2 B\right )+\frac {1}{2} a^2 (a A+b B) \tan (c+d x)+\frac {1}{4} b \left (2 a^2 A+3 A b^2-a b B\right ) \tan ^2(c+d x)}{\sqrt {\tan (c+d x)} (a+b \tan (c+d x))} \, dx}{a^2 \left (a^2+b^2\right )} \\ & = -\frac {2 a^2 A+3 A b^2-a b B}{a^2 \left (a^2+b^2\right ) d \sqrt {\tan (c+d x)}}+\frac {b (A b-a B)}{a \left (a^2+b^2\right ) d \sqrt {\tan (c+d x)} (a+b \tan (c+d x))}-\frac {2 \int \frac {\frac {1}{2} a^2 \left (2 a A b-a^2 B+b^2 B\right )+\frac {1}{2} a^2 \left (a^2 A-A b^2+2 a b B\right ) \tan (c+d x)}{\sqrt {\tan (c+d x)}} \, dx}{a^2 \left (a^2+b^2\right )^2}-\frac {\left (b^2 \left (7 a^2 A b+3 A b^3-5 a^3 B-a b^2 B\right )\right ) \int \frac {1+\tan ^2(c+d x)}{\sqrt {\tan (c+d x)} (a+b \tan (c+d x))} \, dx}{2 a^2 \left (a^2+b^2\right )^2} \\ & = -\frac {2 a^2 A+3 A b^2-a b B}{a^2 \left (a^2+b^2\right ) d \sqrt {\tan (c+d x)}}+\frac {b (A b-a B)}{a \left (a^2+b^2\right ) d \sqrt {\tan (c+d x)} (a+b \tan (c+d x))}-\frac {4 \text {Subst}\left (\int \frac {\frac {1}{2} a^2 \left (2 a A b-a^2 B+b^2 B\right )+\frac {1}{2} a^2 \left (a^2 A-A b^2+2 a b B\right ) x^2}{1+x^4} \, dx,x,\sqrt {\tan (c+d x)}\right )}{a^2 \left (a^2+b^2\right )^2 d}-\frac {\left (b^2 \left (7 a^2 A b+3 A b^3-5 a^3 B-a b^2 B\right )\right ) \text {Subst}\left (\int \frac {1}{\sqrt {x} (a+b x)} \, dx,x,\tan (c+d x)\right )}{2 a^2 \left (a^2+b^2\right )^2 d} \\ & = -\frac {2 a^2 A+3 A b^2-a b B}{a^2 \left (a^2+b^2\right ) d \sqrt {\tan (c+d x)}}+\frac {b (A b-a B)}{a \left (a^2+b^2\right ) d \sqrt {\tan (c+d x)} (a+b \tan (c+d x))}-\frac {\left (b^2 \left (7 a^2 A b+3 A b^3-5 a^3 B-a b^2 B\right )\right ) \text {Subst}\left (\int \frac {1}{a+b x^2} \, dx,x,\sqrt {\tan (c+d x)}\right )}{a^2 \left (a^2+b^2\right )^2 d}-\frac {\left (a^2 (A-B)-b^2 (A-B)+2 a b (A+B)\right ) \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 )^2 d}-\frac {\left (2 a b (A-B)-a^2 (A+B)+b^2 (A+B)\right ) \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 )^2 d} \\ & = -\frac {b^{3/2} \left (7 a^2 A b+3 A b^3-5 a^3 B-a b^2 B\right ) \arctan \left (\frac {\sqrt {b} \sqrt {\tan (c+d x)}}{\sqrt {a}}\right )}{a^{5/2} \left (a^2+b^2\right )^2 d}-\frac {2 a^2 A+3 A b^2-a b B}{a^2 \left (a^2+b^2\right ) d \sqrt {\tan (c+d x)}}+\frac {b (A b-a B)}{a \left (a^2+b^2\right ) d \sqrt {\tan (c+d x)} (a+b \tan (c+d x))}-\frac {\left (a^2 (A-B)-b^2 (A-B)+2 a b (A+B)\right ) \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 )^2 d}-\frac {\left (a^2 (A-B)-b^2 (A-B)+2 a b (A+B)\right ) \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 )^2 d}+\frac {\left (2 a b (A-B)-a^2 (A+B)+b^2 (A+B)\right ) \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 )^2 d}+\frac {\left (2 a b (A-B)-a^2 (A+B)+b^2 (A+B)\right ) \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 )^2 d} \\ & = -\frac {b^{3/2} \left (7 a^2 A b+3 A b^3-5 a^3 B-a b^2 B\right ) \arctan \left (\frac {\sqrt {b} \sqrt {\tan (c+d x)}}{\sqrt {a}}\right )}{a^{5/2} \left (a^2+b^2\right )^2 d}+\frac {\left (2 a b (A-B)-a^2 (A+B)+b^2 (A+B)\right ) \log \left (1-\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right )}{2 \sqrt {2} \left (a^2+b^2\right )^2 d}-\frac {\left (2 a b (A-B)-a^2 (A+B)+b^2 (A+B)\right ) \log \left (1+\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right )}{2 \sqrt {2} \left (a^2+b^2\right )^2 d}-\frac {2 a^2 A+3 A b^2-a b B}{a^2 \left (a^2+b^2\right ) d \sqrt {\tan (c+d x)}}+\frac {b (A b-a B)}{a \left (a^2+b^2\right ) d \sqrt {\tan (c+d x)} (a+b \tan (c+d x))}-\frac {\left (a^2 (A-B)-b^2 (A-B)+2 a b (A+B)\right ) \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 )^2 d}+\frac {\left (a^2 (A-B)-b^2 (A-B)+2 a b (A+B)\right ) \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 )^2 d} \\ & = \frac {\left (a^2 (A-B)-b^2 (A-B)+2 a b (A+B)\right ) \arctan \left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} \left (a^2+b^2\right )^2 d}-\frac {\left (a^2 (A-B)-b^2 (A-B)+2 a b (A+B)\right ) \arctan \left (1+\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} \left (a^2+b^2\right )^2 d}-\frac {b^{3/2} \left (7 a^2 A b+3 A b^3-5 a^3 B-a b^2 B\right ) \arctan \left (\frac {\sqrt {b} \sqrt {\tan (c+d x)}}{\sqrt {a}}\right )}{a^{5/2} \left (a^2+b^2\right )^2 d}+\frac {\left (2 a b (A-B)-a^2 (A+B)+b^2 (A+B)\right ) \log \left (1-\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right )}{2 \sqrt {2} \left (a^2+b^2\right )^2 d}-\frac {\left (2 a b (A-B)-a^2 (A+B)+b^2 (A+B)\right ) \log \left (1+\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right )}{2 \sqrt {2} \left (a^2+b^2\right )^2 d}-\frac {2 a^2 A+3 A b^2-a b B}{a^2 \left (a^2+b^2\right ) d \sqrt {\tan (c+d x)}}+\frac {b (A b-a B)}{a \left (a^2+b^2\right ) d \sqrt {\tan (c+d x)} (a+b \tan (c+d x))} \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 2.51 (sec) , antiderivative size = 239, normalized size of antiderivative = 0.54 \[ \int \frac {A+B \tan (c+d x)}{\tan ^{\frac {3}{2}}(c+d x) (a+b \tan (c+d x))^2} \, dx=\frac {\frac {b^{3/2} \left (-7 a^2 A b-3 A b^3+5 a^3 B+a b^2 B\right ) \arctan \left (\frac {\sqrt {b} \sqrt {\tan (c+d x)}}{\sqrt {a}}\right )}{a^{3/2} \left (a^2+b^2\right )}+\frac {\sqrt [4]{-1} a \left (-i (a+i b)^2 (A-i B) \arctan \left ((-1)^{3/4} \sqrt {\tan (c+d x)}\right )+i (a-i b)^2 (A+i B) \text {arctanh}\left ((-1)^{3/4} \sqrt {\tan (c+d x)}\right )\right )}{a^2+b^2}+\frac {-2 a^2 A-3 A b^2+a b B}{a \sqrt {\tan (c+d x)}}+\frac {b (A b-a B)}{\sqrt {\tan (c+d x)} (a+b \tan (c+d x))}}{a \left (a^2+b^2\right ) d} \]

[In]

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

[Out]

((b^(3/2)*(-7*a^2*A*b - 3*A*b^3 + 5*a^3*B + a*b^2*B)*ArcTan[(Sqrt[b]*Sqrt[Tan[c + d*x]])/Sqrt[a]])/(a^(3/2)*(a
^2 + b^2)) + ((-1)^(1/4)*a*((-I)*(a + I*b)^2*(A - I*B)*ArcTan[(-1)^(3/4)*Sqrt[Tan[c + d*x]]] + I*(a - I*b)^2*(
A + I*B)*ArcTanh[(-1)^(3/4)*Sqrt[Tan[c + d*x]]]))/(a^2 + b^2) + (-2*a^2*A - 3*A*b^2 + a*b*B)/(a*Sqrt[Tan[c + d
*x]]) + (b*(A*b - a*B))/(Sqrt[Tan[c + d*x]]*(a + b*Tan[c + d*x])))/(a*(a^2 + b^2)*d)

Maple [A] (verified)

Time = 0.05 (sec) , antiderivative size = 352, normalized size of antiderivative = 0.80

method	result	size
derivativedivides	\(\frac {-\frac {2 b^{2} \left (\frac {\left (\frac {1}{2} A \,a^{2} b +\frac {1}{2} A \,b^{3}-\frac {1}{2} B \,a^{3}-\frac {1}{2} B a \,b^{2}\right ) \left (\sqrt {\tan }\left (d x +c \right )\right )}{a +b \tan \left (d x +c \right )}+\frac {\left (7 A \,a^{2} b +3 A \,b^{3}-5 B \,a^{3}-B a \,b^{2}\right ) \arctan \left (\frac {b \left (\sqrt {\tan }\left (d x +c \right )\right )}{\sqrt {a b}}\right )}{2 \sqrt {a b}}\right )}{a^{2} \left (a^{2}+b^{2}\right )^{2}}-\frac {2 A}{a^{2} \sqrt {\tan \left (d x +c \right )}}+\frac {\frac {\left (-2 A a b +B \,a^{2}-B \,b^{2}\right ) \sqrt {2}\, \left (\ln \left (\frac {1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}{1-\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}\right )+2 \arctan \left (1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )+2 \arctan \left (-1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )\right )}{4}+\frac {\left (-A \,a^{2}+A \,b^{2}-2 B a b \right ) \sqrt {2}\, \left (\ln \left (\frac {1-\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}{1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}\right )+2 \arctan \left (1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )+2 \arctan \left (-1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )\right )}{4}}{\left (a^{2}+b^{2}\right )^{2}}}{d}\)	\(352\)
default	\(\frac {-\frac {2 b^{2} \left (\frac {\left (\frac {1}{2} A \,a^{2} b +\frac {1}{2} A \,b^{3}-\frac {1}{2} B \,a^{3}-\frac {1}{2} B a \,b^{2}\right ) \left (\sqrt {\tan }\left (d x +c \right )\right )}{a +b \tan \left (d x +c \right )}+\frac {\left (7 A \,a^{2} b +3 A \,b^{3}-5 B \,a^{3}-B a \,b^{2}\right ) \arctan \left (\frac {b \left (\sqrt {\tan }\left (d x +c \right )\right )}{\sqrt {a b}}\right )}{2 \sqrt {a b}}\right )}{a^{2} \left (a^{2}+b^{2}\right )^{2}}-\frac {2 A}{a^{2} \sqrt {\tan \left (d x +c \right )}}+\frac {\frac {\left (-2 A a b +B \,a^{2}-B \,b^{2}\right ) \sqrt {2}\, \left (\ln \left (\frac {1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}{1-\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}\right )+2 \arctan \left (1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )+2 \arctan \left (-1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )\right )}{4}+\frac {\left (-A \,a^{2}+A \,b^{2}-2 B a b \right ) \sqrt {2}\, \left (\ln \left (\frac {1-\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}{1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}\right )+2 \arctan \left (1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )+2 \arctan \left (-1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )\right )}{4}}{\left (a^{2}+b^{2}\right )^{2}}}{d}\)	\(352\)

[In]

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

[Out]

1/d*(-2*b^2/a^2/(a^2+b^2)^2*((1/2*A*a^2*b+1/2*A*b^3-1/2*B*a^3-1/2*B*a*b^2)*tan(d*x+c)^(1/2)/(a+b*tan(d*x+c))+1
/2*(7*A*a^2*b+3*A*b^3-5*B*a^3-B*a*b^2)/(a*b)^(1/2)*arctan(b*tan(d*x+c)^(1/2)/(a*b)^(1/2)))-2/a^2*A/tan(d*x+c)^
(1/2)+2/(a^2+b^2)^2*(1/8*(-2*A*a*b+B*a^2-B*b^2)*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*a^2+A*b^2-2*B*a*b)*2^(1/2)*(ln((1-2^(1/2)*tan(d*x+c)^(1/2)+tan(d*x+c))/(1+2^(1/2)*tan(d*x+c)^(1/2)+tan(d*
x+c)))+2*arctan(1+2^(1/2)*tan(d*x+c)^(1/2))+2*arctan(-1+2^(1/2)*tan(d*x+c)^(1/2)))))

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 5991 vs. \(2 (396) = 792\).

Time = 39.86 (sec) , antiderivative size = 12008, normalized size of antiderivative = 27.35 \[ \int \frac {A+B \tan (c+d x)}{\tan ^{\frac {3}{2}}(c+d x) (a+b \tan (c+d x))^2} \, dx=\text {Too large to display} \]

[In]

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

[Out]

Too large to include

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [A] (verification not implemented)

none

Time = 0.37 (sec) , antiderivative size = 396, normalized size of antiderivative = 0.90 \[ \int \frac {A+B \tan (c+d x)}{\tan ^{\frac {3}{2}}(c+d x) (a+b \tan (c+d x))^2} \, dx=\frac {\frac {4 \, {\left (5 \, B a^{3} b^{2} - 7 \, A a^{2} b^{3} + B a b^{4} - 3 \, A b^{5}\right )} \arctan \left (\frac {b \sqrt {\tan \left (d x + c\right )}}{\sqrt {a b}}\right )}{{\left (a^{6} + 2 \, a^{4} b^{2} + a^{2} b^{4}\right )} \sqrt {a b}} - \frac {2 \, \sqrt {2} {\left ({\left (A - B\right )} a^{2} + 2 \, {\left (A + B\right )} a b - {\left (A - B\right )} b^{2}\right )} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + 2 \, \sqrt {\tan \left (d x + c\right )}\right )}\right ) + 2 \, \sqrt {2} {\left ({\left (A - B\right )} a^{2} + 2 \, {\left (A + B\right )} a b - {\left (A - B\right )} b^{2}\right )} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - 2 \, \sqrt {\tan \left (d x + c\right )}\right )}\right ) - \sqrt {2} {\left ({\left (A + B\right )} a^{2} - 2 \, {\left (A - B\right )} a b - {\left (A + B\right )} b^{2}\right )} \log \left (\sqrt {2} \sqrt {\tan \left (d x + c\right )} + \tan \left (d x + c\right ) + 1\right ) + \sqrt {2} {\left ({\left (A + B\right )} a^{2} - 2 \, {\left (A - B\right )} a b - {\left (A + B\right )} b^{2}\right )} \log \left (-\sqrt {2} \sqrt {\tan \left (d x + c\right )} + \tan \left (d x + c\right ) + 1\right )}{a^{4} + 2 \, a^{2} b^{2} + b^{4}} - \frac {4 \, {\left (2 \, A a^{3} + 2 \, A a b^{2} + {\left (2 \, A a^{2} b - B a b^{2} + 3 \, A b^{3}\right )} \tan \left (d x + c\right )\right )}}{{\left (a^{4} b + a^{2} b^{3}\right )} \tan \left (d x + c\right )^{\frac {3}{2}} + {\left (a^{5} + a^{3} b^{2}\right )} \sqrt {\tan \left (d x + c\right )}}}{4 \, d} \]

[In]

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

[Out]

1/4*(4*(5*B*a^3*b^2 - 7*A*a^2*b^3 + B*a*b^4 - 3*A*b^5)*arctan(b*sqrt(tan(d*x + c))/sqrt(a*b))/((a^6 + 2*a^4*b^
2 + a^2*b^4)*sqrt(a*b)) - (2*sqrt(2)*((A - B)*a^2 + 2*(A + B)*a*b - (A - B)*b^2)*arctan(1/2*sqrt(2)*(sqrt(2) +
2*sqrt(tan(d*x + c)))) + 2*sqrt(2)*((A - B)*a^2 + 2*(A + B)*a*b - (A - B)*b^2)*arctan(-1/2*sqrt(2)*(sqrt(2) -
2*sqrt(tan(d*x + c)))) - sqrt(2)*((A + B)*a^2 - 2*(A - B)*a*b - (A + B)*b^2)*log(sqrt(2)*sqrt(tan(d*x + c)) +
tan(d*x + c) + 1) + sqrt(2)*((A + B)*a^2 - 2*(A - B)*a*b - (A + B)*b^2)*log(-sqrt(2)*sqrt(tan(d*x + c)) + tan
(d*x + c) + 1))/(a^4 + 2*a^2*b^2 + b^4) - 4*(2*A*a^3 + 2*A*a*b^2 + (2*A*a^2*b - B*a*b^2 + 3*A*b^3)*tan(d*x + c
))/((a^4*b + a^2*b^3)*tan(d*x + c)^(3/2) + (a^5 + a^3*b^2)*sqrt(tan(d*x + c))))/d

Giac [F(-1)]

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

[In]

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

[Out]

Timed out

Mupad [B] (verification not implemented)

Time = 29.80 (sec) , antiderivative size = 22667, normalized size of antiderivative = 51.63 \[ \int \frac {A+B \tan (c+d x)}{\tan ^{\frac {3}{2}}(c+d x) (a+b \tan (c+d x))^2} \, dx=\text {Too large to display} \]

[In]

int((A + B*tan(c + d*x))/(tan(c + d*x)^(3/2)*(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*d^4*(a^4 + b^4 - 6*a^2*b^2)^2)^(1/
2) - 16*B^2*a*b^3*d^2 + 16*B^2*a^3*b*d^2)/(d^4*(a^2 + b^2)^4))^(1/2) + (128*B*b^2*(2*b^6 - a^6 + 9*a^2*b^4 + 6
*a^4*b^2))/(a*d))*((4*(-B^4*d^4*(a^4 + b^4 - 6*a^2*b^2)^2)^(1/2) - 16*B^2*a*b^3*d^2 + 16*B^2*a^3*b*d^2)/(d^4*(
a^2 + b^2)^4))^(1/2))/4 + (64*B^2*b^2*tan(c + d*x)^(1/2)*(2*b^8 - a^8 + 5*a^2*b^6 + 67*a^4*b^4 - a^6*b^2))/(a*
d^2*(a^2 + b^2)^2))*((4*(-B^4*d^4*(a^4 + b^4 - 6*a^2*b^2)^2)^(1/2) - 16*B^2*a*b^3*d^2 + 16*B^2*a^3*b*d^2)/(d^4
*(a^2 + b^2)^4))^(1/2))/4 - (32*B^3*b^5*(25*a^6 + b^6 - 13*a^2*b^4 - 85*a^4*b^2))/(a^2*d^3*(a^2 + b^2)^3))*((4
*(-B^4*d^4*(a^4 + b^4 - 6*a^2*b^2)^2)^(1/2) - 16*B^2*a*b^3*d^2 + 16*B^2*a^3*b*d^2)/(d^4*(a^2 + b^2)^4))^(1/2))
/4 + (16*B^4*b^5*tan(c + d*x)^(1/2)*(b^6 - 27*a^6 + 7*a^2*b^4 + 11*a^4*b^2))/(a^2*d^4*(a^2 + b^2)^4))*((4*(-B^
4*d^4*(a^4 + b^4 - 6*a^2*b^2)^2)^(1/2) - 16*B^2*a*b^3*d^2 + 16*B^2*a^3*b*d^2)/(d^4*(a^2 + b^2)^4))^(1/2))/4 +
(16*B^5*b^6*(5*a^2 + b^2))/(a*d^5*(a^2 + b^2)^4))*(((192*B^4*a^2*b^6*d^4 - 16*B^4*b^8*d^4 - 16*B^4*a^8*d^4 - 6
08*B^4*a^4*b^4*d^4 + 192*B^4*a^6*b^2*d^4)^(1/2) - 16*B^2*a*b^3*d^2 + 16*B^2*a^3*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*d^4*(a^4 + b^4 - 6*a^2*b^2)^2)^(1/2) + 16*B^2*a*b^3*d^2 - 16*B^2*a^3*b*d^2)/(d^4*(a^2
+ b^2)^4))^(1/2) + (128*B*b^2*(2*b^6 - a^6 + 9*a^2*b^4 + 6*a^4*b^2))/(a*d))*(-(4*(-B^4*d^4*(a^4 + b^4 - 6*a^2*
b^2)^2)^(1/2) + 16*B^2*a*b^3*d^2 - 16*B^2*a^3*b*d^2)/(d^4*(a^2 + b^2)^4))^(1/2))/4 + (64*B^2*b^2*tan(c + d*x)^
(1/2)*(2*b^8 - a^8 + 5*a^2*b^6 + 67*a^4*b^4 - a^6*b^2))/(a*d^2*(a^2 + b^2)^2))*(-(4*(-B^4*d^4*(a^4 + b^4 - 6*a
^2*b^2)^2)^(1/2) + 16*B^2*a*b^3*d^2 - 16*B^2*a^3*b*d^2)/(d^4*(a^2 + b^2)^4))^(1/2))/4 - (32*B^3*b^5*(25*a^6 +
b^6 - 13*a^2*b^4 - 85*a^4*b^2))/(a^2*d^3*(a^2 + b^2)^3))*(-(4*(-B^4*d^4*(a^4 + b^4 - 6*a^2*b^2)^2)^(1/2) + 16*
B^2*a*b^3*d^2 - 16*B^2*a^3*b*d^2)/(d^4*(a^2 + b^2)^4))^(1/2))/4 + (16*B^4*b^5*tan(c + d*x)^(1/2)*(b^6 - 27*a^6
+ 7*a^2*b^4 + 11*a^4*b^2))/(a^2*d^4*(a^2 + b^2)^4))*(-(4*(-B^4*d^4*(a^4 + b^4 - 6*a^2*b^2)^2)^(1/2) + 16*B^2*
a*b^3*d^2 - 16*B^2*a^3*b*d^2)/(d^4*(a^2 + b^2)^4))^(1/2))/4 + (16*B^5*b^6*(5*a^2 + b^2))/(a*d^5*(a^2 + b^2)^4)
)*(-((192*B^4*a^2*b^6*d^4 - 16*B^4*b^8*d^4 - 16*B^4*a^8*d^4 - 608*B^4*a^4*b^4*d^4 + 192*B^4*a^6*b^2*d^4)^(1/2)
+ 16*B^2*a*b^3*d^2 - 16*B^2*a^3*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((16*B^5*b^6*(5*a^2 + b^2))/(a*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*d^4*(a^4 + b^4 - 6*a^2*b^2)^2)^(1/2) - 16*B^2*a*b^3*d^2 + 16*B^2*a^3*b*d^2)/(d^4*(a^
2 + b^2)^4))^(1/2) - (128*B*b^2*(2*b^6 - a^6 + 9*a^2*b^4 + 6*a^4*b^2))/(a*d))*((4*(-B^4*d^4*(a^4 + b^4 - 6*a^2
*b^2)^2)^(1/2) - 16*B^2*a*b^3*d^2 + 16*B^2*a^3*b*d^2)/(d^4*(a^2 + b^2)^4))^(1/2))/4 + (64*B^2*b^2*tan(c + d*x)
^(1/2)*(2*b^8 - a^8 + 5*a^2*b^6 + 67*a^4*b^4 - a^6*b^2))/(a*d^2*(a^2 + b^2)^2))*((4*(-B^4*d^4*(a^4 + b^4 - 6*a
^2*b^2)^2)^(1/2) - 16*B^2*a*b^3*d^2 + 16*B^2*a^3*b*d^2)/(d^4*(a^2 + b^2)^4))^(1/2))/4 + (32*B^3*b^5*(25*a^6 +
b^6 - 13*a^2*b^4 - 85*a^4*b^2))/(a^2*d^3*(a^2 + b^2)^3))*((4*(-B^4*d^4*(a^4 + b^4 - 6*a^2*b^2)^2)^(1/2) - 16*B
^2*a*b^3*d^2 + 16*B^2*a^3*b*d^2)/(d^4*(a^2 + b^2)^4))^(1/2))/4 + (16*B^4*b^5*tan(c + d*x)^(1/2)*(b^6 - 27*a^6
+ 7*a^2*b^4 + 11*a^4*b^2))/(a^2*d^4*(a^2 + b^2)^4))*((4*(-B^4*d^4*(a^4 + b^4 - 6*a^2*b^2)^2)^(1/2) - 16*B^2*a*
b^3*d^2 + 16*B^2*a^3*b*d^2)/(d^4*(a^2 + b^2)^4))^(1/2))/4)*(((192*B^4*a^2*b^6*d^4 - 16*B^4*b^8*d^4 - 16*B^4*a^
8*d^4 - 608*B^4*a^4*b^4*d^4 + 192*B^4*a^6*b^2*d^4)^(1/2) - 16*B^2*a*b^3*d^2 + 16*B^2*a^3*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*b^6*(5*a^2 + b^2))/(a*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*d^4*(a^4 + b^4 - 6*a^
2*b^2)^2)^(1/2) + 16*B^2*a*b^3*d^2 - 16*B^2*a^3*b*d^2)/(d^4*(a^2 + b^2)^4))^(1/2) - (128*B*b^2*(2*b^6 - a^6 +
9*a^2*b^4 + 6*a^4*b^2))/(a*d))*(-(4*(-B^4*d^4*(a^4 + b^4 - 6*a^2*b^2)^2)^(1/2) + 16*B^2*a*b^3*d^2 - 16*B^2*a^3
*b*d^2)/(d^4*(a^2 + b^2)^4))^(1/2))/4 + (64*B^2*b^2*tan(c + d*x)^(1/2)*(2*b^8 - a^8 + 5*a^2*b^6 + 67*a^4*b^4 -
a^6*b^2))/(a*d^2*(a^2 + b^2)^2))*(-(4*(-B^4*d^4*(a^4 + b^4 - 6*a^2*b^2)^2)^(1/2) + 16*B^2*a*b^3*d^2 - 16*B^2*
a^3*b*d^2)/(d^4*(a^2 + b^2)^4))^(1/2))/4 + (32*B^3*b^5*(25*a^6 + b^6 - 13*a^2*b^4 - 85*a^4*b^2))/(a^2*d^3*(a^2
+ b^2)^3))*(-(4*(-B^4*d^4*(a^4 + b^4 - 6*a^2*b^2)^2)^(1/2) + 16*B^2*a*b^3*d^2 - 16*B^2*a^3*b*d^2)/(d^4*(a^2 +
b^2)^4))^(1/2))/4 + (16*B^4*b^5*tan(c + d*x)^(1/2)*(b^6 - 27*a^6 + 7*a^2*b^4 + 11*a^4*b^2))/(a^2*d^4*(a^2 + b
^2)^4))*(-(4*(-B^4*d^4*(a^4 + b^4 - 6*a^2*b^2)^2)^(1/2) + 16*B^2*a*b^3*d^2 - 16*B^2*a^3*b*d^2)/(d^4*(a^2 + b^2
)^4))^(1/2))/4)*(-((192*B^4*a^2*b^6*d^4 - 16*B^4*b^8*d^4 - 16*B^4*a^8*d^4 - 608*B^4*a^4*b^4*d^4 + 192*B^4*a^6*
b^2*d^4)^(1/2) + 16*B^2*a*b^3*d^2 - 16*B^2*a^3*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(72*A^5*a^14*b^21*d^4 - ((((192*A^4*a^2*b^6*d^4 - 16*A^4*b^8*d^4 - 16*A^4*a^
8*d^4 - 608*A^4*a^4*b^4*d^4 + 192*A^4*a^6*b^2*d^4)^(1/2) + 16*A^2*a*b^3*d^2 - 16*A^2*a^3*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)*(tan(c + d*x)^(1/2)*(144*A^4*a^14*b^23*d^5 + 1248
*A^4*a^16*b^21*d^5 + 4224*A^4*a^18*b^19*d^5 + 6720*A^4*a^20*b^17*d^5 + 3872*A^4*a^22*b^15*d^5 - 2816*A^4*a^24*
b^13*d^5 - 5632*A^4*a^26*b^11*d^5 - 3136*A^4*a^28*b^9*d^5 - 560*A^4*a^30*b^7*d^5 + 32*A^4*a^32*b^5*d^5) - ((((
192*A^4*a^2*b^6*d^4 - 16*A^4*b^8*d^4 - 16*A^4*a^8*d^4 - 608*A^4*a^4*b^4*d^4 + 192*A^4*a^6*b^2*d^4)^(1/2) + 16*
A^2*a*b^3*d^2 - 16*A^2*a^3*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)*(
((tan(c + d*x)^(1/2)*(1152*A^2*a^15*b^26*d^7 + 13440*A^2*a^17*b^24*d^7 + 69056*A^2*a^19*b^22*d^7 + 202752*A^2*
a^21*b^20*d^7 + 372800*A^2*a^23*b^18*d^7 + 443136*A^2*a^25*b^16*d^7 + 337792*A^2*a^27*b^14*d^7 + 156160*A^2*a^
29*b^12*d^7 + 37632*A^2*a^31*b^10*d^7 + 3200*A^2*a^33*b^8*d^7 + 704*A^2*a^35*b^6*d^7 + 512*A^2*a^37*b^4*d^7 +
64*A^2*a^39*b^2*d^7) - ((((192*A^4*a^2*b^6*d^4 - 16*A^4*b^8*d^4 - 16*A^4*a^8*d^4 - 608*A^4*a^4*b^4*d^4 + 192*A
^4*a^6*b^2*d^4)^(1/2) + 16*A^2*a*b^3*d^2 - 16*A^2*a^3*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)*((tan(c + d*x)^(1/2)*(((192*A^4*a^2*b^6*d^4 - 16*A^4*b^8*d^4 - 16*A^4*a^8*d^4 - 608*
A^4*a^4*b^4*d^4 + 192*A^4*a^6*b^2*d^4)^(1/2) + 16*A^2*a*b^3*d^2 - 16*A^2*a^3*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)*(512*a^18*b^27*d^9 + 5120*a^20*b^25*d^9 + 22528*a^22*b^23*d^9
+ 56320*a^24*b^21*d^9 + 84480*a^26*b^19*d^9 + 67584*a^28*b^17*d^9 - 67584*a^32*b^13*d^9 - 84480*a^34*b^11*d^9
- 56320*a^36*b^9*d^9 - 22528*a^38*b^7*d^9 - 5120*a^40*b^5*d^9 - 512*a^42*b^3*d^9))/4 + 768*A*a^16*b^27*d^8 +
8704*A*a^18*b^25*d^8 + 44288*A*a^20*b^23*d^8 + 133120*A*a^22*b^21*d^8 + 261120*A*a^24*b^19*d^8 + 347136*A*a^26
*b^17*d^8 + 311808*A*a^28*b^15*d^8 + 178176*A*a^30*b^13*d^8 + 49920*A*a^32*b^11*d^8 - 7680*A*a^34*b^9*d^8 - 12
032*A*a^36*b^7*d^8 - 4096*A*a^38*b^5*d^8 - 512*A*a^40*b^3*d^8))/4)*(((192*A^4*a^2*b^6*d^4 - 16*A^4*b^8*d^4 - 1
6*A^4*a^8*d^4 - 608*A^4*a^4*b^4*d^4 + 192*A^4*a^6*b^2*d^4)^(1/2) + 16*A^2*a*b^3*d^2 - 16*A^2*a^3*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 - 1152*A^3*a^15*b^24*d^6 - 8448*A^3*a^
17*b^22*d^6 - 23776*A^3*a^19*b^20*d^6 - 29664*A^3*a^21*b^18*d^6 - 6528*A^3*a^23*b^16*d^6 + 26496*A^3*a^25*b^14
*d^6 + 33984*A^3*a^27*b^12*d^6 + 18624*A^3*a^29*b^10*d^6 + 5376*A^3*a^31*b^8*d^6 + 1152*A^3*a^33*b^6*d^6 + 288
*A^3*a^35*b^4*d^6 + 32*A^3*a^37*b^2*d^6))/4))/4 + 648*A^5*a^16*b^19*d^4 + 2440*A^5*a^18*b^17*d^4 + 5000*A^5*a^
20*b^15*d^4 + 6040*A^5*a^22*b^13*d^4 + 4312*A^5*a^24*b^11*d^4 + 1688*A^5*a^26*b^9*d^4 + 280*A^5*a^28*b^7*d^4)*
(((192*A^4*a^2*b^6*d^4 - 16*A^4*b^8*d^4 - 16*A^4*a^8*d^4 - 608*A^4*a^4*b^4*d^4 + 192*A^4*a^6*b^2*d^4)^(1/2) +
16*A^2*a*b^3*d^2 - 16*A^2*a^3*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(72*A^5*a^14*b^21*d^4 - ((-((192*A^4*a^2*b^6*d^4 - 16*A^4*b^8*d^4 - 16*A^4*a^8*d^4 - 608*A^4*a^4*b^
4*d^4 + 192*A^4*a^6*b^2*d^4)^(1/2) - 16*A^2*a*b^3*d^2 + 16*A^2*a^3*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)*(tan(c + d*x)^(1/2)*(144*A^4*a^14*b^23*d^5 + 1248*A^4*a^16*b^21*d^5 + 4
224*A^4*a^18*b^19*d^5 + 6720*A^4*a^20*b^17*d^5 + 3872*A^4*a^22*b^15*d^5 - 2816*A^4*a^24*b^13*d^5 - 5632*A^4*a^
26*b^11*d^5 - 3136*A^4*a^28*b^9*d^5 - 560*A^4*a^30*b^7*d^5 + 32*A^4*a^32*b^5*d^5) - ((-((192*A^4*a^2*b^6*d^4 -
16*A^4*b^8*d^4 - 16*A^4*a^8*d^4 - 608*A^4*a^4*b^4*d^4 + 192*A^4*a^6*b^2*d^4)^(1/2) - 16*A^2*a*b^3*d^2 + 16*A^
2*a^3*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)*(((tan(c + d*x)^(1/2)*
(1152*A^2*a^15*b^26*d^7 + 13440*A^2*a^17*b^24*d^7 + 69056*A^2*a^19*b^22*d^7 + 202752*A^2*a^21*b^20*d^7 + 37280
0*A^2*a^23*b^18*d^7 + 443136*A^2*a^25*b^16*d^7 + 337792*A^2*a^27*b^14*d^7 + 156160*A^2*a^29*b^12*d^7 + 37632*A
^2*a^31*b^10*d^7 + 3200*A^2*a^33*b^8*d^7 + 704*A^2*a^35*b^6*d^7 + 512*A^2*a^37*b^4*d^7 + 64*A^2*a^39*b^2*d^7)
- ((-((192*A^4*a^2*b^6*d^4 - 16*A^4*b^8*d^4 - 16*A^4*a^8*d^4 - 608*A^4*a^4*b^4*d^4 + 192*A^4*a^6*b^2*d^4)^(1/2
) - 16*A^2*a*b^3*d^2 + 16*A^2*a^3*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)*((tan(c + d*x)^(1/2)*(-((192*A^4*a^2*b^6*d^4 - 16*A^4*b^8*d^4 - 16*A^4*a^8*d^4 - 608*A^4*a^4*b^4*d^4 + 1
92*A^4*a^6*b^2*d^4)^(1/2) - 16*A^2*a*b^3*d^2 + 16*A^2*a^3*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)*(512*a^18*b^27*d^9 + 5120*a^20*b^25*d^9 + 22528*a^22*b^23*d^9 + 56320*a^24*b^21*
d^9 + 84480*a^26*b^19*d^9 + 67584*a^28*b^17*d^9 - 67584*a^32*b^13*d^9 - 84480*a^34*b^11*d^9 - 56320*a^36*b^9*d
^9 - 22528*a^38*b^7*d^9 - 5120*a^40*b^5*d^9 - 512*a^42*b^3*d^9))/4 + 768*A*a^16*b^27*d^8 + 8704*A*a^18*b^25*d^
8 + 44288*A*a^20*b^23*d^8 + 133120*A*a^22*b^21*d^8 + 261120*A*a^24*b^19*d^8 + 347136*A*a^26*b^17*d^8 + 311808*
A*a^28*b^15*d^8 + 178176*A*a^30*b^13*d^8 + 49920*A*a^32*b^11*d^8 - 7680*A*a^34*b^9*d^8 - 12032*A*a^36*b^7*d^8
- 4096*A*a^38*b^5*d^8 - 512*A*a^40*b^3*d^8))/4)*(-((192*A^4*a^2*b^6*d^4 - 16*A^4*b^8*d^4 - 16*A^4*a^8*d^4 - 60
8*A^4*a^4*b^4*d^4 + 192*A^4*a^6*b^2*d^4)^(1/2) - 16*A^2*a*b^3*d^2 + 16*A^2*a^3*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 - 1152*A^3*a^15*b^24*d^6 - 8448*A^3*a^17*b^22*d^6 - 2377
6*A^3*a^19*b^20*d^6 - 29664*A^3*a^21*b^18*d^6 - 6528*A^3*a^23*b^16*d^6 + 26496*A^3*a^25*b^14*d^6 + 33984*A^3*a
^27*b^12*d^6 + 18624*A^3*a^29*b^10*d^6 + 5376*A^3*a^31*b^8*d^6 + 1152*A^3*a^33*b^6*d^6 + 288*A^3*a^35*b^4*d^6
+ 32*A^3*a^37*b^2*d^6))/4))/4 + 648*A^5*a^16*b^19*d^4 + 2440*A^5*a^18*b^17*d^4 + 5000*A^5*a^20*b^15*d^4 + 6040
*A^5*a^22*b^13*d^4 + 4312*A^5*a^24*b^11*d^4 + 1688*A^5*a^26*b^9*d^4 + 280*A^5*a^28*b^7*d^4)*(-((192*A^4*a^2*b^
6*d^4 - 16*A^4*b^8*d^4 - 16*A^4*a^8*d^4 - 608*A^4*a^4*b^4*d^4 + 192*A^4*a^6*b^2*d^4)^(1/2) - 16*A^2*a*b^3*d^2
+ 16*A^2*a^3*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((((192
*A^4*a^2*b^6*d^4 - 16*A^4*b^8*d^4 - 16*A^4*a^8*d^4 - 608*A^4*a^4*b^4*d^4 + 192*A^4*a^6*b^2*d^4)^(1/2) + 16*A^2
*a*b^3*d^2 - 16*A^2*a^3*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)*(tan(c + d*x)^(1/2)*(144*A^4*a^14*b^23*d^5 + 1248*A^4*a^16*b^21*d^5 + 4224*A^4*a^18*b^19*d^5 + 6720*A^4*a
^20*b^17*d^5 + 3872*A^4*a^22*b^15*d^5 - 2816*A^4*a^24*b^13*d^5 - 5632*A^4*a^26*b^11*d^5 - 3136*A^4*a^28*b^9*d^
5 - 560*A^4*a^30*b^7*d^5 + 32*A^4*a^32*b^5*d^5) + (((192*A^4*a^2*b^6*d^4 - 16*A^4*b^8*d^4 - 16*A^4*a^8*d^4 - 6
08*A^4*a^4*b^4*d^4 + 192*A^4*a^6*b^2*d^4)^(1/2) + 16*A^2*a*b^3*d^2 - 16*A^2*a^3*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)*(26496*A^3*a^25*b^14*d^6 - 1152*A^3*a^15*b^24*d^6
- 8448*A^3*a^17*b^22*d^6 - 23776*A^3*a^19*b^20*d^6 - 29664*A^3*a^21*b^18*d^6 - 6528*A^3*a^23*b^16*d^6 - (tan(
c + d*x)^(1/2)*(1152*A^2*a^15*b^26*d^7 + 13440*A^2*a^17*b^24*d^7 + 69056*A^2*a^19*b^22*d^7 + 202752*A^2*a^21*b
^20*d^7 + 372800*A^2*a^23*b^18*d^7 + 443136*A^2*a^25*b^16*d^7 + 337792*A^2*a^27*b^14*d^7 + 156160*A^2*a^29*b^1
2*d^7 + 37632*A^2*a^31*b^10*d^7 + 3200*A^2*a^33*b^8*d^7 + 704*A^2*a^35*b^6*d^7 + 512*A^2*a^37*b^4*d^7 + 64*A^2
*a^39*b^2*d^7) + (((192*A^4*a^2*b^6*d^4 - 16*A^4*b^8*d^4 - 16*A^4*a^8*d^4 - 608*A^4*a^4*b^4*d^4 + 192*A^4*a^6*
b^2*d^4)^(1/2) + 16*A^2*a*b^3*d^2 - 16*A^2*a^3*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)*(768*A*a^16*b^27*d^8 - tan(c + d*x)^(1/2)*(((192*A^4*a^2*b^6*d^4 - 16*A^4*b^8*d^4
- 16*A^4*a^8*d^4 - 608*A^4*a^4*b^4*d^4 + 192*A^4*a^6*b^2*d^4)^(1/2) + 16*A^2*a*b^3*d^2 - 16*A^2*a^3*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)*(512*a^18*b^27*d^9 + 5120*a^2
0*b^25*d^9 + 22528*a^22*b^23*d^9 + 56320*a^24*b^21*d^9 + 84480*a^26*b^19*d^9 + 67584*a^28*b^17*d^9 - 67584*a^3
2*b^13*d^9 - 84480*a^34*b^11*d^9 - 56320*a^36*b^9*d^9 - 22528*a^38*b^7*d^9 - 5120*a^40*b^5*d^9 - 512*a^42*b^3*
d^9) + 8704*A*a^18*b^25*d^8 + 44288*A*a^20*b^23*d^8 + 133120*A*a^22*b^21*d^8 + 261120*A*a^24*b^19*d^8 + 347136
*A*a^26*b^17*d^8 + 311808*A*a^28*b^15*d^8 + 178176*A*a^30*b^13*d^8 + 49920*A*a^32*b^11*d^8 - 7680*A*a^34*b^9*d
^8 - 12032*A*a^36*b^7*d^8 - 4096*A*a^38*b^5*d^8 - 512*A*a^40*b^3*d^8))*(((192*A^4*a^2*b^6*d^4 - 16*A^4*b^8*d^4
- 16*A^4*a^8*d^4 - 608*A^4*a^4*b^4*d^4 + 192*A^4*a^6*b^2*d^4)^(1/2) + 16*A^2*a*b^3*d^2 - 16*A^2*a^3*b*d^2)/(1
6*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) + 33984*A^3*a^27*b^12*d^6 +
18624*A^3*a^29*b^10*d^6 + 5376*A^3*a^31*b^8*d^6 + 1152*A^3*a^33*b^6*d^6 + 288*A^3*a^35*b^4*d^6 + 32*A^3*a^37*b
^2*d^6)) + 72*A^5*a^14*b^21*d^4 + 648*A^5*a^16*b^19*d^4 + 2440*A^5*a^18*b^17*d^4 + 5000*A^5*a^20*b^15*d^4 + 60
40*A^5*a^22*b^13*d^4 + 4312*A^5*a^24*b^11*d^4 + 1688*A^5*a^26*b^9*d^4 + 280*A^5*a^28*b^7*d^4)*(((192*A^4*a^2*b
^6*d^4 - 16*A^4*b^8*d^4 - 16*A^4*a^8*d^4 - 608*A^4*a^4*b^4*d^4 + 192*A^4*a^6*b^2*d^4)^(1/2) + 16*A^2*a*b^3*d^2
- 16*A^2*a^3*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
((-((192*A^4*a^2*b^6*d^4 - 16*A^4*b^8*d^4 - 16*A^4*a^8*d^4 - 608*A^4*a^4*b^4*d^4 + 192*A^4*a^6*b^2*d^4)^(1/2)
- 16*A^2*a*b^3*d^2 + 16*A^2*a^3*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)*(tan(c + d*x)^(1/2)*(144*A^4*a^14*b^23*d^5 + 1248*A^4*a^16*b^21*d^5 + 4224*A^4*a^18*b^19*d^5 + 67
20*A^4*a^20*b^17*d^5 + 3872*A^4*a^22*b^15*d^5 - 2816*A^4*a^24*b^13*d^5 - 5632*A^4*a^26*b^11*d^5 - 3136*A^4*a^2
8*b^9*d^5 - 560*A^4*a^30*b^7*d^5 + 32*A^4*a^32*b^5*d^5) + (-((192*A^4*a^2*b^6*d^4 - 16*A^4*b^8*d^4 - 16*A^4*a^
8*d^4 - 608*A^4*a^4*b^4*d^4 + 192*A^4*a^6*b^2*d^4)^(1/2) - 16*A^2*a*b^3*d^2 + 16*A^2*a^3*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)*(26496*A^3*a^25*b^14*d^6 - 1152*A^3*a^15
*b^24*d^6 - 8448*A^3*a^17*b^22*d^6 - 23776*A^3*a^19*b^20*d^6 - 29664*A^3*a^21*b^18*d^6 - 6528*A^3*a^23*b^16*d^
6 - (tan(c + d*x)^(1/2)*(1152*A^2*a^15*b^26*d^7 + 13440*A^2*a^17*b^24*d^7 + 69056*A^2*a^19*b^22*d^7 + 202752*A
^2*a^21*b^20*d^7 + 372800*A^2*a^23*b^18*d^7 + 443136*A^2*a^25*b^16*d^7 + 337792*A^2*a^27*b^14*d^7 + 156160*A^2
*a^29*b^12*d^7 + 37632*A^2*a^31*b^10*d^7 + 3200*A^2*a^33*b^8*d^7 + 704*A^2*a^35*b^6*d^7 + 512*A^2*a^37*b^4*d^7
+ 64*A^2*a^39*b^2*d^7) + (-((192*A^4*a^2*b^6*d^4 - 16*A^4*b^8*d^4 - 16*A^4*a^8*d^4 - 608*A^4*a^4*b^4*d^4 + 19
2*A^4*a^6*b^2*d^4)^(1/2) - 16*A^2*a*b^3*d^2 + 16*A^2*a^3*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)*(768*A*a^16*b^27*d^8 - tan(c + d*x)^(1/2)*(-((192*A^4*a^2*b^6*d^4 - 16*A
^4*b^8*d^4 - 16*A^4*a^8*d^4 - 608*A^4*a^4*b^4*d^4 + 192*A^4*a^6*b^2*d^4)^(1/2) - 16*A^2*a*b^3*d^2 + 16*A^2*a^3
*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)*(512*a^18*b^27*d^9
+ 5120*a^20*b^25*d^9 + 22528*a^22*b^23*d^9 + 56320*a^24*b^21*d^9 + 84480*a^26*b^19*d^9 + 67584*a^28*b^17*d^9
- 67584*a^32*b^13*d^9 - 84480*a^34*b^11*d^9 - 56320*a^36*b^9*d^9 - 22528*a^38*b^7*d^9 - 5120*a^40*b^5*d^9 - 51
2*a^42*b^3*d^9) + 8704*A*a^18*b^25*d^8 + 44288*A*a^20*b^23*d^8 + 133120*A*a^22*b^21*d^8 + 261120*A*a^24*b^19*d
^8 + 347136*A*a^26*b^17*d^8 + 311808*A*a^28*b^15*d^8 + 178176*A*a^30*b^13*d^8 + 49920*A*a^32*b^11*d^8 - 7680*A
*a^34*b^9*d^8 - 12032*A*a^36*b^7*d^8 - 4096*A*a^38*b^5*d^8 - 512*A*a^40*b^3*d^8))*(-((192*A^4*a^2*b^6*d^4 - 16
*A^4*b^8*d^4 - 16*A^4*a^8*d^4 - 608*A^4*a^4*b^4*d^4 + 192*A^4*a^6*b^2*d^4)^(1/2) - 16*A^2*a*b^3*d^2 + 16*A^2*a
^3*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) + 33984*A^3*a^27
*b^12*d^6 + 18624*A^3*a^29*b^10*d^6 + 5376*A^3*a^31*b^8*d^6 + 1152*A^3*a^33*b^6*d^6 + 288*A^3*a^35*b^4*d^6 + 3
2*A^3*a^37*b^2*d^6)) + 72*A^5*a^14*b^21*d^4 + 648*A^5*a^16*b^19*d^4 + 2440*A^5*a^18*b^17*d^4 + 5000*A^5*a^20*b
^15*d^4 + 6040*A^5*a^22*b^13*d^4 + 4312*A^5*a^24*b^11*d^4 + 1688*A^5*a^26*b^9*d^4 + 280*A^5*a^28*b^7*d^4)*(-((
192*A^4*a^2*b^6*d^4 - 16*A^4*b^8*d^4 - 16*A^4*a^8*d^4 - 608*A^4*a^4*b^4*d^4 + 192*A^4*a^6*b^2*d^4)^(1/2) - 16*
A^2*a*b^3*d^2 + 16*A^2*a^3*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) - ((2*A)/a + (A*tan(c + d*x)*(2*a^2*b + 3*b^3))/(a^2*(a^2 + b^2)))/(a*d*tan(c + d*x)^(1/2) + b*d*tan(c
+ d*x)^(3/2)) + (atan(((((tan(c + d*x)^(1/2)*(144*A^4*a^14*b^23*d^5 + 1248*A^4*a^16*b^21*d^5 + 4224*A^4*a^18*
b^19*d^5 + 6720*A^4*a^20*b^17*d^5 + 3872*A^4*a^22*b^15*d^5 - 2816*A^4*a^24*b^13*d^5 - 5632*A^4*a^26*b^11*d^5 -
3136*A^4*a^28*b^9*d^5 - 560*A^4*a^30*b^7*d^5 + 32*A^4*a^32*b^5*d^5))/4 + ((-4*(9*A^2*b^9 + 42*A^2*a^2*b^7 + 4
9*A^2*a^4*b^5)*(a^13*d^2 + a^5*b^8*d^2 + 4*a^7*b^6*d^2 + 6*a^9*b^4*d^2 + 4*a^11*b^2*d^2))^(1/2)*(6624*A^3*a^25
*b^14*d^6 - 288*A^3*a^15*b^24*d^6 - 2112*A^3*a^17*b^22*d^6 - 5944*A^3*a^19*b^20*d^6 - 7416*A^3*a^21*b^18*d^6 -
1632*A^3*a^23*b^16*d^6 - (((tan(c + d*x)^(1/2)*(1152*A^2*a^15*b^26*d^7 + 13440*A^2*a^17*b^24*d^7 + 69056*A^2*
a^19*b^22*d^7 + 202752*A^2*a^21*b^20*d^7 + 372800*A^2*a^23*b^18*d^7 + 443136*A^2*a^25*b^16*d^7 + 337792*A^2*a^
27*b^14*d^7 + 156160*A^2*a^29*b^12*d^7 + 37632*A^2*a^31*b^10*d^7 + 3200*A^2*a^33*b^8*d^7 + 704*A^2*a^35*b^6*d^
7 + 512*A^2*a^37*b^4*d^7 + 64*A^2*a^39*b^2*d^7))/4 + ((-4*(9*A^2*b^9 + 42*A^2*a^2*b^7 + 49*A^2*a^4*b^5)*(a^13*
d^2 + a^5*b^8*d^2 + 4*a^7*b^6*d^2 + 6*a^9*b^4*d^2 + 4*a^11*b^2*d^2))^(1/2)*(192*A*a^16*b^27*d^8 - (tan(c + d*x
)^(1/2)*(-4*(9*A^2*b^9 + 42*A^2*a^2*b^7 + 49*A^2*a^4*b^5)*(a^13*d^2 + a^5*b^8*d^2 + 4*a^7*b^6*d^2 + 6*a^9*b^4*
d^2 + 4*a^11*b^2*d^2))^(1/2)*(512*a^18*b^27*d^9 + 5120*a^20*b^25*d^9 + 22528*a^22*b^23*d^9 + 56320*a^24*b^21*d
^9 + 84480*a^26*b^19*d^9 + 67584*a^28*b^17*d^9 - 67584*a^32*b^13*d^9 - 84480*a^34*b^11*d^9 - 56320*a^36*b^9*d^
9 - 22528*a^38*b^7*d^9 - 5120*a^40*b^5*d^9 - 512*a^42*b^3*d^9))/(16*(a^13*d^2 + a^5*b^8*d^2 + 4*a^7*b^6*d^2 +
6*a^9*b^4*d^2 + 4*a^11*b^2*d^2)) + 2176*A*a^18*b^25*d^8 + 11072*A*a^20*b^23*d^8 + 33280*A*a^22*b^21*d^8 + 6528
0*A*a^24*b^19*d^8 + 86784*A*a^26*b^17*d^8 + 77952*A*a^28*b^15*d^8 + 44544*A*a^30*b^13*d^8 + 12480*A*a^32*b^11*
d^8 - 1920*A*a^34*b^9*d^8 - 3008*A*a^36*b^7*d^8 - 1024*A*a^38*b^5*d^8 - 128*A*a^40*b^3*d^8))/(4*(a^13*d^2 + a^
5*b^8*d^2 + 4*a^7*b^6*d^2 + 6*a^9*b^4*d^2 + 4*a^11*b^2*d^2)))*(-4*(9*A^2*b^9 + 42*A^2*a^2*b^7 + 49*A^2*a^4*b^5
)*(a^13*d^2 + a^5*b^8*d^2 + 4*a^7*b^6*d^2 + 6*a^9*b^4*d^2 + 4*a^11*b^2*d^2))^(1/2))/(4*(a^13*d^2 + a^5*b^8*d^2
+ 4*a^7*b^6*d^2 + 6*a^9*b^4*d^2 + 4*a^11*b^2*d^2)) + 8496*A^3*a^27*b^12*d^6 + 4656*A^3*a^29*b^10*d^6 + 1344*A
^3*a^31*b^8*d^6 + 288*A^3*a^33*b^6*d^6 + 72*A^3*a^35*b^4*d^6 + 8*A^3*a^37*b^2*d^6))/(4*(a^13*d^2 + a^5*b^8*d^2
+ 4*a^7*b^6*d^2 + 6*a^9*b^4*d^2 + 4*a^11*b^2*d^2)))*(-4*(9*A^2*b^9 + 42*A^2*a^2*b^7 + 49*A^2*a^4*b^5)*(a^13*d
^2 + a^5*b^8*d^2 + 4*a^7*b^6*d^2 + 6*a^9*b^4*d^2 + 4*a^11*b^2*d^2))^(1/2)*1i)/(a^13*d^2 + a^5*b^8*d^2 + 4*a^7*
b^6*d^2 + 6*a^9*b^4*d^2 + 4*a^11*b^2*d^2) + (((tan(c + d*x)^(1/2)*(144*A^4*a^14*b^23*d^5 + 1248*A^4*a^16*b^21*
d^5 + 4224*A^4*a^18*b^19*d^5 + 6720*A^4*a^20*b^17*d^5 + 3872*A^4*a^22*b^15*d^5 - 2816*A^4*a^24*b^13*d^5 - 5632
*A^4*a^26*b^11*d^5 - 3136*A^4*a^28*b^9*d^5 - 560*A^4*a^30*b^7*d^5 + 32*A^4*a^32*b^5*d^5))/4 - ((-4*(9*A^2*b^9
+ 42*A^2*a^2*b^7 + 49*A^2*a^4*b^5)*(a^13*d^2 + a^5*b^8*d^2 + 4*a^7*b^6*d^2 + 6*a^9*b^4*d^2 + 4*a^11*b^2*d^2))^
(1/2)*((((tan(c + d*x)^(1/2)*(1152*A^2*a^15*b^26*d^7 + 13440*A^2*a^17*b^24*d^7 + 69056*A^2*a^19*b^22*d^7 + 202
752*A^2*a^21*b^20*d^7 + 372800*A^2*a^23*b^18*d^7 + 443136*A^2*a^25*b^16*d^7 + 337792*A^2*a^27*b^14*d^7 + 15616
0*A^2*a^29*b^12*d^7 + 37632*A^2*a^31*b^10*d^7 + 3200*A^2*a^33*b^8*d^7 + 704*A^2*a^35*b^6*d^7 + 512*A^2*a^37*b^
4*d^7 + 64*A^2*a^39*b^2*d^7))/4 - ((-4*(9*A^2*b^9 + 42*A^2*a^2*b^7 + 49*A^2*a^4*b^5)*(a^13*d^2 + a^5*b^8*d^2 +
4*a^7*b^6*d^2 + 6*a^9*b^4*d^2 + 4*a^11*b^2*d^2))^(1/2)*((tan(c + d*x)^(1/2)*(-4*(9*A^2*b^9 + 42*A^2*a^2*b^7 +
49*A^2*a^4*b^5)*(a^13*d^2 + a^5*b^8*d^2 + 4*a^7*b^6*d^2 + 6*a^9*b^4*d^2 + 4*a^11*b^2*d^2))^(1/2)*(512*a^18*b^
27*d^9 + 5120*a^20*b^25*d^9 + 22528*a^22*b^23*d^9 + 56320*a^24*b^21*d^9 + 84480*a^26*b^19*d^9 + 67584*a^28*b^1
7*d^9 - 67584*a^32*b^13*d^9 - 84480*a^34*b^11*d^9 - 56320*a^36*b^9*d^9 - 22528*a^38*b^7*d^9 - 5120*a^40*b^5*d^
9 - 512*a^42*b^3*d^9))/(16*(a^13*d^2 + a^5*b^8*d^2 + 4*a^7*b^6*d^2 + 6*a^9*b^4*d^2 + 4*a^11*b^2*d^2)) + 192*A*
a^16*b^27*d^8 + 2176*A*a^18*b^25*d^8 + 11072*A*a^20*b^23*d^8 + 33280*A*a^22*b^21*d^8 + 65280*A*a^24*b^19*d^8 +
86784*A*a^26*b^17*d^8 + 77952*A*a^28*b^15*d^8 + 44544*A*a^30*b^13*d^8 + 12480*A*a^32*b^11*d^8 - 1920*A*a^34*b
^9*d^8 - 3008*A*a^36*b^7*d^8 - 1024*A*a^38*b^5*d^8 - 128*A*a^40*b^3*d^8))/(4*(a^13*d^2 + a^5*b^8*d^2 + 4*a^7*b
^6*d^2 + 6*a^9*b^4*d^2 + 4*a^11*b^2*d^2)))*(-4*(9*A^2*b^9 + 42*A^2*a^2*b^7 + 49*A^2*a^4*b^5)*(a^13*d^2 + a^5*b
^8*d^2 + 4*a^7*b^6*d^2 + 6*a^9*b^4*d^2 + 4*a^11*b^2*d^2))^(1/2))/(4*(a^13*d^2 + a^5*b^8*d^2 + 4*a^7*b^6*d^2 +
6*a^9*b^4*d^2 + 4*a^11*b^2*d^2)) - 288*A^3*a^15*b^24*d^6 - 2112*A^3*a^17*b^22*d^6 - 5944*A^3*a^19*b^20*d^6 - 7
416*A^3*a^21*b^18*d^6 - 1632*A^3*a^23*b^16*d^6 + 6624*A^3*a^25*b^14*d^6 + 8496*A^3*a^27*b^12*d^6 + 4656*A^3*a^
29*b^10*d^6 + 1344*A^3*a^31*b^8*d^6 + 288*A^3*a^33*b^6*d^6 + 72*A^3*a^35*b^4*d^6 + 8*A^3*a^37*b^2*d^6))/(4*(a^
13*d^2 + a^5*b^8*d^2 + 4*a^7*b^6*d^2 + 6*a^9*b^4*d^2 + 4*a^11*b^2*d^2)))*(-4*(9*A^2*b^9 + 42*A^2*a^2*b^7 + 49*
A^2*a^4*b^5)*(a^13*d^2 + a^5*b^8*d^2 + 4*a^7*b^6*d^2 + 6*a^9*b^4*d^2 + 4*a^11*b^2*d^2))^(1/2)*1i)/(a^13*d^2 +
a^5*b^8*d^2 + 4*a^7*b^6*d^2 + 6*a^9*b^4*d^2 + 4*a^11*b^2*d^2))/((((tan(c + d*x)^(1/2)*(144*A^4*a^14*b^23*d^5 +
1248*A^4*a^16*b^21*d^5 + 4224*A^4*a^18*b^19*d^5 + 6720*A^4*a^20*b^17*d^5 + 3872*A^4*a^22*b^15*d^5 - 2816*A^4*
a^24*b^13*d^5 - 5632*A^4*a^26*b^11*d^5 - 3136*A^4*a^28*b^9*d^5 - 560*A^4*a^30*b^7*d^5 + 32*A^4*a^32*b^5*d^5))/
4 + ((-4*(9*A^2*b^9 + 42*A^2*a^2*b^7 + 49*A^2*a^4*b^5)*(a^13*d^2 + a^5*b^8*d^2 + 4*a^7*b^6*d^2 + 6*a^9*b^4*d^2
+ 4*a^11*b^2*d^2))^(1/2)*(6624*A^3*a^25*b^14*d^6 - 288*A^3*a^15*b^24*d^6 - 2112*A^3*a^17*b^22*d^6 - 5944*A^3*
a^19*b^20*d^6 - 7416*A^3*a^21*b^18*d^6 - 1632*A^3*a^23*b^16*d^6 - (((tan(c + d*x)^(1/2)*(1152*A^2*a^15*b^26*d^
7 + 13440*A^2*a^17*b^24*d^7 + 69056*A^2*a^19*b^22*d^7 + 202752*A^2*a^21*b^20*d^7 + 372800*A^2*a^23*b^18*d^7 +
443136*A^2*a^25*b^16*d^7 + 337792*A^2*a^27*b^14*d^7 + 156160*A^2*a^29*b^12*d^7 + 37632*A^2*a^31*b^10*d^7 + 320
0*A^2*a^33*b^8*d^7 + 704*A^2*a^35*b^6*d^7 + 512*A^2*a^37*b^4*d^7 + 64*A^2*a^39*b^2*d^7))/4 + ((-4*(9*A^2*b^9 +
42*A^2*a^2*b^7 + 49*A^2*a^4*b^5)*(a^13*d^2 + a^5*b^8*d^2 + 4*a^7*b^6*d^2 + 6*a^9*b^4*d^2 + 4*a^11*b^2*d^2))^(
1/2)*(192*A*a^16*b^27*d^8 - (tan(c + d*x)^(1/2)*(-4*(9*A^2*b^9 + 42*A^2*a^2*b^7 + 49*A^2*a^4*b^5)*(a^13*d^2 +
a^5*b^8*d^2 + 4*a^7*b^6*d^2 + 6*a^9*b^4*d^2 + 4*a^11*b^2*d^2))^(1/2)*(512*a^18*b^27*d^9 + 5120*a^20*b^25*d^9 +
22528*a^22*b^23*d^9 + 56320*a^24*b^21*d^9 + 84480*a^26*b^19*d^9 + 67584*a^28*b^17*d^9 - 67584*a^32*b^13*d^9 -
84480*a^34*b^11*d^9 - 56320*a^36*b^9*d^9 - 22528*a^38*b^7*d^9 - 5120*a^40*b^5*d^9 - 512*a^42*b^3*d^9))/(16*(a
^13*d^2 + a^5*b^8*d^2 + 4*a^7*b^6*d^2 + 6*a^9*b^4*d^2 + 4*a^11*b^2*d^2)) + 2176*A*a^18*b^25*d^8 + 11072*A*a^20
*b^23*d^8 + 33280*A*a^22*b^21*d^8 + 65280*A*a^24*b^19*d^8 + 86784*A*a^26*b^17*d^8 + 77952*A*a^28*b^15*d^8 + 44
544*A*a^30*b^13*d^8 + 12480*A*a^32*b^11*d^8 - 1920*A*a^34*b^9*d^8 - 3008*A*a^36*b^7*d^8 - 1024*A*a^38*b^5*d^8
- 128*A*a^40*b^3*d^8))/(4*(a^13*d^2 + a^5*b^8*d^2 + 4*a^7*b^6*d^2 + 6*a^9*b^4*d^2 + 4*a^11*b^2*d^2)))*(-4*(9*A
^2*b^9 + 42*A^2*a^2*b^7 + 49*A^2*a^4*b^5)*(a^13*d^2 + a^5*b^8*d^2 + 4*a^7*b^6*d^2 + 6*a^9*b^4*d^2 + 4*a^11*b^2
*d^2))^(1/2))/(4*(a^13*d^2 + a^5*b^8*d^2 + 4*a^7*b^6*d^2 + 6*a^9*b^4*d^2 + 4*a^11*b^2*d^2)) + 8496*A^3*a^27*b^
12*d^6 + 4656*A^3*a^29*b^10*d^6 + 1344*A^3*a^31*b^8*d^6 + 288*A^3*a^33*b^6*d^6 + 72*A^3*a^35*b^4*d^6 + 8*A^3*a
^37*b^2*d^6))/(4*(a^13*d^2 + a^5*b^8*d^2 + 4*a^7*b^6*d^2 + 6*a^9*b^4*d^2 + 4*a^11*b^2*d^2)))*(-4*(9*A^2*b^9 +
42*A^2*a^2*b^7 + 49*A^2*a^4*b^5)*(a^13*d^2 + a^5*b^8*d^2 + 4*a^7*b^6*d^2 + 6*a^9*b^4*d^2 + 4*a^11*b^2*d^2))^(1
/2))/(a^13*d^2 + a^5*b^8*d^2 + 4*a^7*b^6*d^2 + 6*a^9*b^4*d^2 + 4*a^11*b^2*d^2) - (((tan(c + d*x)^(1/2)*(144*A^
4*a^14*b^23*d^5 + 1248*A^4*a^16*b^21*d^5 + 4224*A^4*a^18*b^19*d^5 + 6720*A^4*a^20*b^17*d^5 + 3872*A^4*a^22*b^1
5*d^5 - 2816*A^4*a^24*b^13*d^5 - 5632*A^4*a^26*b^11*d^5 - 3136*A^4*a^28*b^9*d^5 - 560*A^4*a^30*b^7*d^5 + 32*A^
4*a^32*b^5*d^5))/4 - ((-4*(9*A^2*b^9 + 42*A^2*a^2*b^7 + 49*A^2*a^4*b^5)*(a^13*d^2 + a^5*b^8*d^2 + 4*a^7*b^6*d^
2 + 6*a^9*b^4*d^2 + 4*a^11*b^2*d^2))^(1/2)*((((tan(c + d*x)^(1/2)*(1152*A^2*a^15*b^26*d^7 + 13440*A^2*a^17*b^2
4*d^7 + 69056*A^2*a^19*b^22*d^7 + 202752*A^2*a^21*b^20*d^7 + 372800*A^2*a^23*b^18*d^7 + 443136*A^2*a^25*b^16*d
^7 + 337792*A^2*a^27*b^14*d^7 + 156160*A^2*a^29*b^12*d^7 + 37632*A^2*a^31*b^10*d^7 + 3200*A^2*a^33*b^8*d^7 + 7
04*A^2*a^35*b^6*d^7 + 512*A^2*a^37*b^4*d^7 + 64*A^2*a^39*b^2*d^7))/4 - ((-4*(9*A^2*b^9 + 42*A^2*a^2*b^7 + 49*A
^2*a^4*b^5)*(a^13*d^2 + a^5*b^8*d^2 + 4*a^7*b^6*d^2 + 6*a^9*b^4*d^2 + 4*a^11*b^2*d^2))^(1/2)*((tan(c + d*x)^(1
/2)*(-4*(9*A^2*b^9 + 42*A^2*a^2*b^7 + 49*A^2*a^4*b^5)*(a^13*d^2 + a^5*b^8*d^2 + 4*a^7*b^6*d^2 + 6*a^9*b^4*d^2
+ 4*a^11*b^2*d^2))^(1/2)*(512*a^18*b^27*d^9 + 5120*a^20*b^25*d^9 + 22528*a^22*b^23*d^9 + 56320*a^24*b^21*d^9 +
84480*a^26*b^19*d^9 + 67584*a^28*b^17*d^9 - 67584*a^32*b^13*d^9 - 84480*a^34*b^11*d^9 - 56320*a^36*b^9*d^9 -
22528*a^38*b^7*d^9 - 5120*a^40*b^5*d^9 - 512*a^42*b^3*d^9))/(16*(a^13*d^2 + a^5*b^8*d^2 + 4*a^7*b^6*d^2 + 6*a^
9*b^4*d^2 + 4*a^11*b^2*d^2)) + 192*A*a^16*b^27*d^8 + 2176*A*a^18*b^25*d^8 + 11072*A*a^20*b^23*d^8 + 33280*A*a^
22*b^21*d^8 + 65280*A*a^24*b^19*d^8 + 86784*A*a^26*b^17*d^8 + 77952*A*a^28*b^15*d^8 + 44544*A*a^30*b^13*d^8 +
12480*A*a^32*b^11*d^8 - 1920*A*a^34*b^9*d^8 - 3008*A*a^36*b^7*d^8 - 1024*A*a^38*b^5*d^8 - 128*A*a^40*b^3*d^8))
/(4*(a^13*d^2 + a^5*b^8*d^2 + 4*a^7*b^6*d^2 + 6*a^9*b^4*d^2 + 4*a^11*b^2*d^2)))*(-4*(9*A^2*b^9 + 42*A^2*a^2*b^
7 + 49*A^2*a^4*b^5)*(a^13*d^2 + a^5*b^8*d^2 + 4*a^7*b^6*d^2 + 6*a^9*b^4*d^2 + 4*a^11*b^2*d^2))^(1/2))/(4*(a^13
*d^2 + a^5*b^8*d^2 + 4*a^7*b^6*d^2 + 6*a^9*b^4*d^2 + 4*a^11*b^2*d^2)) - 288*A^3*a^15*b^24*d^6 - 2112*A^3*a^17*
b^22*d^6 - 5944*A^3*a^19*b^20*d^6 - 7416*A^3*a^21*b^18*d^6 - 1632*A^3*a^23*b^16*d^6 + 6624*A^3*a^25*b^14*d^6 +
8496*A^3*a^27*b^12*d^6 + 4656*A^3*a^29*b^10*d^6 + 1344*A^3*a^31*b^8*d^6 + 288*A^3*a^33*b^6*d^6 + 72*A^3*a^35*
b^4*d^6 + 8*A^3*a^37*b^2*d^6))/(4*(a^13*d^2 + a^5*b^8*d^2 + 4*a^7*b^6*d^2 + 6*a^9*b^4*d^2 + 4*a^11*b^2*d^2)))*
(-4*(9*A^2*b^9 + 42*A^2*a^2*b^7 + 49*A^2*a^4*b^5)*(a^13*d^2 + a^5*b^8*d^2 + 4*a^7*b^6*d^2 + 6*a^9*b^4*d^2 + 4*
a^11*b^2*d^2))^(1/2))/(a^13*d^2 + a^5*b^8*d^2 + 4*a^7*b^6*d^2 + 6*a^9*b^4*d^2 + 4*a^11*b^2*d^2) + 144*A^5*a^14
*b^21*d^4 + 1296*A^5*a^16*b^19*d^4 + 4880*A^5*a^18*b^17*d^4 + 10000*A^5*a^20*b^15*d^4 + 12080*A^5*a^22*b^13*d^
4 + 8624*A^5*a^24*b^11*d^4 + 3376*A^5*a^26*b^9*d^4 + 560*A^5*a^28*b^7*d^4))*(-4*(9*A^2*b^9 + 42*A^2*a^2*b^7 +
49*A^2*a^4*b^5)*(a^13*d^2 + a^5*b^8*d^2 + 4*a^7*b^6*d^2 + 6*a^9*b^4*d^2 + 4*a^11*b^2*d^2))^(1/2)*1i)/(2*(a^13*
d^2 + a^5*b^8*d^2 + 4*a^7*b^6*d^2 + 6*a^9*b^4*d^2 + 4*a^11*b^2*d^2)) - (atan(((((16*tan(c + d*x)^(1/2)*(B^4*b^
11 + 7*B^4*a^2*b^9 + 11*B^4*a^4*b^7 - 27*B^4*a^6*b^5))/(a^10*d^4 + a^2*b^8*d^4 + 4*a^4*b^6*d^4 + 6*a^6*b^4*d^4
+ 4*a^8*b^2*d^4) + (((16*(24*B^3*a^2*b^11*d^2 - 2*B^3*b^13*d^2 + 196*B^3*a^4*b^9*d^2 + 120*B^3*a^6*b^7*d^2 -
50*B^3*a^8*b^5*d^2))/(a^10*d^5 + a^2*b^8*d^5 + 4*a^4*b^6*d^5 + 6*a^6*b^4*d^5 + 4*a^8*b^2*d^5) + (((16*tan(c +
d*x)^(1/2)*(36*B^2*a^3*b^12*d^2 + 316*B^2*a^5*b^10*d^2 + 552*B^2*a^7*b^8*d^2 + 256*B^2*a^9*b^6*d^2 - 12*B^2*a^
11*b^4*d^2 - 4*B^2*a^13*b^2*d^2 + 8*B^2*a*b^14*d^2))/(a^10*d^4 + a^2*b^8*d^4 + 4*a^4*b^6*d^4 + 6*a^6*b^4*d^4 +
4*a^8*b^2*d^4) + (((16*(16*B*a*b^16*d^4 + 136*B*a^3*b^14*d^4 + 432*B*a^5*b^12*d^4 + 680*B*a^7*b^10*d^4 + 560*
B*a^9*b^8*d^4 + 216*B*a^11*b^6*d^4 + 16*B*a^13*b^4*d^4 - 8*B*a^15*b^2*d^4))/(a^10*d^5 + a^2*b^8*d^5 + 4*a^4*b^
6*d^5 + 6*a^6*b^4*d^5 + 4*a^8*b^2*d^5) - (4*tan(c + d*x)^(1/2)*(-4*(B^2*b^7 + 10*B^2*a^2*b^5 + 25*B^2*a^4*b^3)
*(a^11*d^2 + a^3*b^8*d^2 + 4*a^5*b^6*d^2 + 6*a^7*b^4*d^2 + 4*a^9*b^2*d^2))^(1/2)*(32*a^2*b^17*d^4 + 160*a^4*b^
15*d^4 + 288*a^6*b^13*d^4 + 160*a^8*b^11*d^4 - 160*a^10*b^9*d^4 - 288*a^12*b^7*d^4 - 160*a^14*b^5*d^4 - 32*a^1
6*b^3*d^4))/((a^11*d^2 + a^3*b^8*d^2 + 4*a^5*b^6*d^2 + 6*a^7*b^4*d^2 + 4*a^9*b^2*d^2)*(a^10*d^4 + a^2*b^8*d^4
+ 4*a^4*b^6*d^4 + 6*a^6*b^4*d^4 + 4*a^8*b^2*d^4)))*(-4*(B^2*b^7 + 10*B^2*a^2*b^5 + 25*B^2*a^4*b^3)*(a^11*d^2 +
a^3*b^8*d^2 + 4*a^5*b^6*d^2 + 6*a^7*b^4*d^2 + 4*a^9*b^2*d^2))^(1/2))/(4*(a^11*d^2 + a^3*b^8*d^2 + 4*a^5*b^6*d
^2 + 6*a^7*b^4*d^2 + 4*a^9*b^2*d^2)))*(-4*(B^2*b^7 + 10*B^2*a^2*b^5 + 25*B^2*a^4*b^3)*(a^11*d^2 + a^3*b^8*d^2
+ 4*a^5*b^6*d^2 + 6*a^7*b^4*d^2 + 4*a^9*b^2*d^2))^(1/2))/(4*(a^11*d^2 + a^3*b^8*d^2 + 4*a^5*b^6*d^2 + 6*a^7*b^
4*d^2 + 4*a^9*b^2*d^2)))*(-4*(B^2*b^7 + 10*B^2*a^2*b^5 + 25*B^2*a^4*b^3)*(a^11*d^2 + a^3*b^8*d^2 + 4*a^5*b^6*d
^2 + 6*a^7*b^4*d^2 + 4*a^9*b^2*d^2))^(1/2))/(4*(a^11*d^2 + a^3*b^8*d^2 + 4*a^5*b^6*d^2 + 6*a^7*b^4*d^2 + 4*a^9
*b^2*d^2)))*(-4*(B^2*b^7 + 10*B^2*a^2*b^5 + 25*B^2*a^4*b^3)*(a^11*d^2 + a^3*b^8*d^2 + 4*a^5*b^6*d^2 + 6*a^7*b^
4*d^2 + 4*a^9*b^2*d^2))^(1/2)*1i)/(4*(a^11*d^2 + a^3*b^8*d^2 + 4*a^5*b^6*d^2 + 6*a^7*b^4*d^2 + 4*a^9*b^2*d^2))
+ (((16*tan(c + d*x)^(1/2)*(B^4*b^11 + 7*B^4*a^2*b^9 + 11*B^4*a^4*b^7 - 27*B^4*a^6*b^5))/(a^10*d^4 + a^2*b^8*
d^4 + 4*a^4*b^6*d^4 + 6*a^6*b^4*d^4 + 4*a^8*b^2*d^4) - (((16*(24*B^3*a^2*b^11*d^2 - 2*B^3*b^13*d^2 + 196*B^3*a
^4*b^9*d^2 + 120*B^3*a^6*b^7*d^2 - 50*B^3*a^8*b^5*d^2))/(a^10*d^5 + a^2*b^8*d^5 + 4*a^4*b^6*d^5 + 6*a^6*b^4*d^
5 + 4*a^8*b^2*d^5) - (((16*tan(c + d*x)^(1/2)*(36*B^2*a^3*b^12*d^2 + 316*B^2*a^5*b^10*d^2 + 552*B^2*a^7*b^8*d^
2 + 256*B^2*a^9*b^6*d^2 - 12*B^2*a^11*b^4*d^2 - 4*B^2*a^13*b^2*d^2 + 8*B^2*a*b^14*d^2))/(a^10*d^4 + a^2*b^8*d^
4 + 4*a^4*b^6*d^4 + 6*a^6*b^4*d^4 + 4*a^8*b^2*d^4) - (((16*(16*B*a*b^16*d^4 + 136*B*a^3*b^14*d^4 + 432*B*a^5*b
^12*d^4 + 680*B*a^7*b^10*d^4 + 560*B*a^9*b^8*d^4 + 216*B*a^11*b^6*d^4 + 16*B*a^13*b^4*d^4 - 8*B*a^15*b^2*d^4))
/(a^10*d^5 + a^2*b^8*d^5 + 4*a^4*b^6*d^5 + 6*a^6*b^4*d^5 + 4*a^8*b^2*d^5) + (4*tan(c + d*x)^(1/2)*(-4*(B^2*b^7
+ 10*B^2*a^2*b^5 + 25*B^2*a^4*b^3)*(a^11*d^2 + a^3*b^8*d^2 + 4*a^5*b^6*d^2 + 6*a^7*b^4*d^2 + 4*a^9*b^2*d^2))^
(1/2)*(32*a^2*b^17*d^4 + 160*a^4*b^15*d^4 + 288*a^6*b^13*d^4 + 160*a^8*b^11*d^4 - 160*a^10*b^9*d^4 - 288*a^12*
b^7*d^4 - 160*a^14*b^5*d^4 - 32*a^16*b^3*d^4))/((a^11*d^2 + a^3*b^8*d^2 + 4*a^5*b^6*d^2 + 6*a^7*b^4*d^2 + 4*a^
9*b^2*d^2)*(a^10*d^4 + a^2*b^8*d^4 + 4*a^4*b^6*d^4 + 6*a^6*b^4*d^4 + 4*a^8*b^2*d^4)))*(-4*(B^2*b^7 + 10*B^2*a^
2*b^5 + 25*B^2*a^4*b^3)*(a^11*d^2 + a^3*b^8*d^2 + 4*a^5*b^6*d^2 + 6*a^7*b^4*d^2 + 4*a^9*b^2*d^2))^(1/2))/(4*(a
^11*d^2 + a^3*b^8*d^2 + 4*a^5*b^6*d^2 + 6*a^7*b^4*d^2 + 4*a^9*b^2*d^2)))*(-4*(B^2*b^7 + 10*B^2*a^2*b^5 + 25*B^
2*a^4*b^3)*(a^11*d^2 + a^3*b^8*d^2 + 4*a^5*b^6*d^2 + 6*a^7*b^4*d^2 + 4*a^9*b^2*d^2))^(1/2))/(4*(a^11*d^2 + a^3
*b^8*d^2 + 4*a^5*b^6*d^2 + 6*a^7*b^4*d^2 + 4*a^9*b^2*d^2)))*(-4*(B^2*b^7 + 10*B^2*a^2*b^5 + 25*B^2*a^4*b^3)*(a
^11*d^2 + a^3*b^8*d^2 + 4*a^5*b^6*d^2 + 6*a^7*b^4*d^2 + 4*a^9*b^2*d^2))^(1/2))/(4*(a^11*d^2 + a^3*b^8*d^2 + 4*
a^5*b^6*d^2 + 6*a^7*b^4*d^2 + 4*a^9*b^2*d^2)))*(-4*(B^2*b^7 + 10*B^2*a^2*b^5 + 25*B^2*a^4*b^3)*(a^11*d^2 + a^3
*b^8*d^2 + 4*a^5*b^6*d^2 + 6*a^7*b^4*d^2 + 4*a^9*b^2*d^2))^(1/2)*1i)/(4*(a^11*d^2 + a^3*b^8*d^2 + 4*a^5*b^6*d^
2 + 6*a^7*b^4*d^2 + 4*a^9*b^2*d^2)))/((32*(B^5*a*b^8 + 5*B^5*a^3*b^6))/(a^10*d^5 + a^2*b^8*d^5 + 4*a^4*b^6*d^5
+ 6*a^6*b^4*d^5 + 4*a^8*b^2*d^5) + (((16*tan(c + d*x)^(1/2)*(B^4*b^11 + 7*B^4*a^2*b^9 + 11*B^4*a^4*b^7 - 27*B
^4*a^6*b^5))/(a^10*d^4 + a^2*b^8*d^4 + 4*a^4*b^6*d^4 + 6*a^6*b^4*d^4 + 4*a^8*b^2*d^4) + (((16*(24*B^3*a^2*b^11
*d^2 - 2*B^3*b^13*d^2 + 196*B^3*a^4*b^9*d^2 + 120*B^3*a^6*b^7*d^2 - 50*B^3*a^8*b^5*d^2))/(a^10*d^5 + a^2*b^8*d
^5 + 4*a^4*b^6*d^5 + 6*a^6*b^4*d^5 + 4*a^8*b^2*d^5) + (((16*tan(c + d*x)^(1/2)*(36*B^2*a^3*b^12*d^2 + 316*B^2*
a^5*b^10*d^2 + 552*B^2*a^7*b^8*d^2 + 256*B^2*a^9*b^6*d^2 - 12*B^2*a^11*b^4*d^2 - 4*B^2*a^13*b^2*d^2 + 8*B^2*a*
b^14*d^2))/(a^10*d^4 + a^2*b^8*d^4 + 4*a^4*b^6*d^4 + 6*a^6*b^4*d^4 + 4*a^8*b^2*d^4) + (((16*(16*B*a*b^16*d^4 +
136*B*a^3*b^14*d^4 + 432*B*a^5*b^12*d^4 + 680*B*a^7*b^10*d^4 + 560*B*a^9*b^8*d^4 + 216*B*a^11*b^6*d^4 + 16*B*
a^13*b^4*d^4 - 8*B*a^15*b^2*d^4))/(a^10*d^5 + a^2*b^8*d^5 + 4*a^4*b^6*d^5 + 6*a^6*b^4*d^5 + 4*a^8*b^2*d^5) - (
4*tan(c + d*x)^(1/2)*(-4*(B^2*b^7 + 10*B^2*a^2*b^5 + 25*B^2*a^4*b^3)*(a^11*d^2 + a^3*b^8*d^2 + 4*a^5*b^6*d^2 +
6*a^7*b^4*d^2 + 4*a^9*b^2*d^2))^(1/2)*(32*a^2*b^17*d^4 + 160*a^4*b^15*d^4 + 288*a^6*b^13*d^4 + 160*a^8*b^11*d
^4 - 160*a^10*b^9*d^4 - 288*a^12*b^7*d^4 - 160*a^14*b^5*d^4 - 32*a^16*b^3*d^4))/((a^11*d^2 + a^3*b^8*d^2 + 4*a
^5*b^6*d^2 + 6*a^7*b^4*d^2 + 4*a^9*b^2*d^2)*(a^10*d^4 + a^2*b^8*d^4 + 4*a^4*b^6*d^4 + 6*a^6*b^4*d^4 + 4*a^8*b^
2*d^4)))*(-4*(B^2*b^7 + 10*B^2*a^2*b^5 + 25*B^2*a^4*b^3)*(a^11*d^2 + a^3*b^8*d^2 + 4*a^5*b^6*d^2 + 6*a^7*b^4*d
^2 + 4*a^9*b^2*d^2))^(1/2))/(4*(a^11*d^2 + a^3*b^8*d^2 + 4*a^5*b^6*d^2 + 6*a^7*b^4*d^2 + 4*a^9*b^2*d^2)))*(-4*
(B^2*b^7 + 10*B^2*a^2*b^5 + 25*B^2*a^4*b^3)*(a^11*d^2 + a^3*b^8*d^2 + 4*a^5*b^6*d^2 + 6*a^7*b^4*d^2 + 4*a^9*b^
2*d^2))^(1/2))/(4*(a^11*d^2 + a^3*b^8*d^2 + 4*a^5*b^6*d^2 + 6*a^7*b^4*d^2 + 4*a^9*b^2*d^2)))*(-4*(B^2*b^7 + 10
*B^2*a^2*b^5 + 25*B^2*a^4*b^3)*(a^11*d^2 + a^3*b^8*d^2 + 4*a^5*b^6*d^2 + 6*a^7*b^4*d^2 + 4*a^9*b^2*d^2))^(1/2)
)/(4*(a^11*d^2 + a^3*b^8*d^2 + 4*a^5*b^6*d^2 + 6*a^7*b^4*d^2 + 4*a^9*b^2*d^2)))*(-4*(B^2*b^7 + 10*B^2*a^2*b^5
+ 25*B^2*a^4*b^3)*(a^11*d^2 + a^3*b^8*d^2 + 4*a^5*b^6*d^2 + 6*a^7*b^4*d^2 + 4*a^9*b^2*d^2))^(1/2))/(4*(a^11*d^
2 + a^3*b^8*d^2 + 4*a^5*b^6*d^2 + 6*a^7*b^4*d^2 + 4*a^9*b^2*d^2)) - (((16*tan(c + d*x)^(1/2)*(B^4*b^11 + 7*B^4
*a^2*b^9 + 11*B^4*a^4*b^7 - 27*B^4*a^6*b^5))/(a^10*d^4 + a^2*b^8*d^4 + 4*a^4*b^6*d^4 + 6*a^6*b^4*d^4 + 4*a^8*b
^2*d^4) - (((16*(24*B^3*a^2*b^11*d^2 - 2*B^3*b^13*d^2 + 196*B^3*a^4*b^9*d^2 + 120*B^3*a^6*b^7*d^2 - 50*B^3*a^8
*b^5*d^2))/(a^10*d^5 + a^2*b^8*d^5 + 4*a^4*b^6*d^5 + 6*a^6*b^4*d^5 + 4*a^8*b^2*d^5) - (((16*tan(c + d*x)^(1/2)
*(36*B^2*a^3*b^12*d^2 + 316*B^2*a^5*b^10*d^2 + 552*B^2*a^7*b^8*d^2 + 256*B^2*a^9*b^6*d^2 - 12*B^2*a^11*b^4*d^2
- 4*B^2*a^13*b^2*d^2 + 8*B^2*a*b^14*d^2))/(a^10*d^4 + a^2*b^8*d^4 + 4*a^4*b^6*d^4 + 6*a^6*b^4*d^4 + 4*a^8*b^2
*d^4) - (((16*(16*B*a*b^16*d^4 + 136*B*a^3*b^14*d^4 + 432*B*a^5*b^12*d^4 + 680*B*a^7*b^10*d^4 + 560*B*a^9*b^8*
d^4 + 216*B*a^11*b^6*d^4 + 16*B*a^13*b^4*d^4 - 8*B*a^15*b^2*d^4))/(a^10*d^5 + a^2*b^8*d^5 + 4*a^4*b^6*d^5 + 6*
a^6*b^4*d^5 + 4*a^8*b^2*d^5) + (4*tan(c + d*x)^(1/2)*(-4*(B^2*b^7 + 10*B^2*a^2*b^5 + 25*B^2*a^4*b^3)*(a^11*d^2
+ a^3*b^8*d^2 + 4*a^5*b^6*d^2 + 6*a^7*b^4*d^2 + 4*a^9*b^2*d^2))^(1/2)*(32*a^2*b^17*d^4 + 160*a^4*b^15*d^4 + 2
88*a^6*b^13*d^4 + 160*a^8*b^11*d^4 - 160*a^10*b^9*d^4 - 288*a^12*b^7*d^4 - 160*a^14*b^5*d^4 - 32*a^16*b^3*d^4)
)/((a^11*d^2 + a^3*b^8*d^2 + 4*a^5*b^6*d^2 + 6*a^7*b^4*d^2 + 4*a^9*b^2*d^2)*(a^10*d^4 + a^2*b^8*d^4 + 4*a^4*b^
6*d^4 + 6*a^6*b^4*d^4 + 4*a^8*b^2*d^4)))*(-4*(B^2*b^7 + 10*B^2*a^2*b^5 + 25*B^2*a^4*b^3)*(a^11*d^2 + a^3*b^8*d
^2 + 4*a^5*b^6*d^2 + 6*a^7*b^4*d^2 + 4*a^9*b^2*d^2))^(1/2))/(4*(a^11*d^2 + a^3*b^8*d^2 + 4*a^5*b^6*d^2 + 6*a^7
*b^4*d^2 + 4*a^9*b^2*d^2)))*(-4*(B^2*b^7 + 10*B^2*a^2*b^5 + 25*B^2*a^4*b^3)*(a^11*d^2 + a^3*b^8*d^2 + 4*a^5*b^
6*d^2 + 6*a^7*b^4*d^2 + 4*a^9*b^2*d^2))^(1/2))/(4*(a^11*d^2 + a^3*b^8*d^2 + 4*a^5*b^6*d^2 + 6*a^7*b^4*d^2 + 4*
a^9*b^2*d^2)))*(-4*(B^2*b^7 + 10*B^2*a^2*b^5 + 25*B^2*a^4*b^3)*(a^11*d^2 + a^3*b^8*d^2 + 4*a^5*b^6*d^2 + 6*a^7
*b^4*d^2 + 4*a^9*b^2*d^2))^(1/2))/(4*(a^11*d^2 + a^3*b^8*d^2 + 4*a^5*b^6*d^2 + 6*a^7*b^4*d^2 + 4*a^9*b^2*d^2))
)*(-4*(B^2*b^7 + 10*B^2*a^2*b^5 + 25*B^2*a^4*b^3)*(a^11*d^2 + a^3*b^8*d^2 + 4*a^5*b^6*d^2 + 6*a^7*b^4*d^2 + 4*
a^9*b^2*d^2))^(1/2))/(4*(a^11*d^2 + a^3*b^8*d^2 + 4*a^5*b^6*d^2 + 6*a^7*b^4*d^2 + 4*a^9*b^2*d^2))))*(-4*(B^2*b
^7 + 10*B^2*a^2*b^5 + 25*B^2*a^4*b^3)*(a^11*d^2 + a^3*b^8*d^2 + 4*a^5*b^6*d^2 + 6*a^7*b^4*d^2 + 4*a^9*b^2*d^2)
)^(1/2)*1i)/(2*(a^11*d^2 + a^3*b^8*d^2 + 4*a^5*b^6*d^2 + 6*a^7*b^4*d^2 + 4*a^9*b^2*d^2)) + (B*b^2*tan(c + d*x)
^(1/2))/(a*(a*d + b*d*tan(c + d*x))*(a^2 + b^2))