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

Optimal. Leaf size=601 \[ \frac {b (A b-a B)}{2 a d \left (a^2+b^2\right ) \sqrt {\tan (c+d x)} (a+b \tan (c+d x))^2}+\frac {\left (a^3 (A-B)+3 a^2 b (A+B)-3 a b^2 (A-B)-b^3 (A+B)\right ) \tan ^{-1}\left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} d \left (a^2+b^2\right )^3}-\frac {\left (a^3 (A-B)+3 a^2 b (A+B)-3 a b^2 (A-B)-b^3 (A+B)\right ) \tan ^{-1}\left (\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{\sqrt {2} d \left (a^2+b^2\right )^3}+\frac {b \left (-9 a^3 B+13 a^2 A b-a b^2 B+5 A b^3\right )}{4 a^2 d \left (a^2+b^2\right )^2 \sqrt {\tan (c+d x)} (a+b \tan (c+d x))}+\frac {\left (-\left (a^3 (A+B)\right )+3 a^2 b (A-B)+3 a b^2 (A+B)-b^3 (A-B)\right ) \log \left (\tan (c+d x)-\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{2 \sqrt {2} d \left (a^2+b^2\right )^3}-\frac {\left (-\left (a^3 (A+B)\right )+3 a^2 b (A-B)+3 a b^2 (A+B)-b^3 (A-B)\right ) \log \left (\tan (c+d x)+\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{2 \sqrt {2} d \left (a^2+b^2\right )^3}-\frac {8 a^4 A-11 a^3 b B+31 a^2 A b^2-3 a b^3 B+15 A b^4}{4 a^3 d \left (a^2+b^2\right )^2 \sqrt {\tan (c+d x)}}-\frac {b^{3/2} \left (-35 a^5 B+63 a^4 A b-6 a^3 b^2 B+46 a^2 A b^3-3 a b^4 B+15 A b^5\right ) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {\tan (c+d x)}}{\sqrt {a}}\right )}{4 a^{7/2} d \left (a^2+b^2\right )^3} \]

[Out]

-1/4*b^(3/2)*(63*A*a^4*b+46*A*a^2*b^3+15*A*b^5-35*B*a^5-6*B*a^3*b^2-3*B*a*b^4)*arctan(b^(1/2)*tan(d*x+c)^(1/2)
/a^(1/2))/a^(7/2)/(a^2+b^2)^3/d-1/2*(a^3*(A-B)-3*a*b^2*(A-B)+3*a^2*b*(A+B)-b^3*(A+B))*arctan(-1+2^(1/2)*tan(d*
x+c)^(1/2))/(a^2+b^2)^3/d*2^(1/2)-1/2*(a^3*(A-B)-3*a*b^2*(A-B)+3*a^2*b*(A+B)-b^3*(A+B))*arctan(1+2^(1/2)*tan(d
*x+c)^(1/2))/(a^2+b^2)^3/d*2^(1/2)+1/4*(3*a^2*b*(A-B)-b^3*(A-B)-a^3*(A+B)+3*a*b^2*(A+B))*ln(1-2^(1/2)*tan(d*x+
c)^(1/2)+tan(d*x+c))/(a^2+b^2)^3/d*2^(1/2)-1/4*(3*a^2*b*(A-B)-b^3*(A-B)-a^3*(A+B)+3*a*b^2*(A+B))*ln(1+2^(1/2)*
tan(d*x+c)^(1/2)+tan(d*x+c))/(a^2+b^2)^3/d*2^(1/2)+1/4*(-8*A*a^4-31*A*a^2*b^2-15*A*b^4+11*B*a^3*b+3*B*a*b^3)/a
^3/(a^2+b^2)^2/d/tan(d*x+c)^(1/2)+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))^2+1/4*b*(13*
A*a^2*b+5*A*b^3-9*B*a^3-B*a*b^2)/a^2/(a^2+b^2)^2/d/tan(d*x+c)^(1/2)/(a+b*tan(d*x+c))

________________________________________________________________________________________

Rubi [A]  time = 1.69, antiderivative size = 601, normalized size of antiderivative = 1.00, number of steps used = 17, number of rules used = 13, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.394, Rules used = {3609, 3649, 3653, 3534, 1168, 1162, 617, 204, 1165, 628, 3634, 63, 205} \[ -\frac {b^{3/2} \left (46 a^2 A b^3+63 a^4 A b-6 a^3 b^2 B-35 a^5 B-3 a b^4 B+15 A b^5\right ) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {\tan (c+d x)}}{\sqrt {a}}\right )}{4 a^{7/2} d \left (a^2+b^2\right )^3}+\frac {\left (3 a^2 b (A+B)+a^3 (A-B)-3 a b^2 (A-B)-b^3 (A+B)\right ) \tan ^{-1}\left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} d \left (a^2+b^2\right )^3}-\frac {\left (3 a^2 b (A+B)+a^3 (A-B)-3 a b^2 (A-B)-b^3 (A+B)\right ) \tan ^{-1}\left (\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{\sqrt {2} d \left (a^2+b^2\right )^3}+\frac {b \left (13 a^2 A b-9 a^3 B-a b^2 B+5 A b^3\right )}{4 a^2 d \left (a^2+b^2\right )^2 \sqrt {\tan (c+d x)} (a+b \tan (c+d x))}+\frac {b (A b-a B)}{2 a d \left (a^2+b^2\right ) \sqrt {\tan (c+d x)} (a+b \tan (c+d x))^2}-\frac {31 a^2 A b^2+8 a^4 A-11 a^3 b B-3 a b^3 B+15 A b^4}{4 a^3 d \left (a^2+b^2\right )^2 \sqrt {\tan (c+d x)}}+\frac {\left (3 a^2 b (A-B)+a^3 (-(A+B))+3 a b^2 (A+B)-b^3 (A-B)\right ) \log \left (\tan (c+d x)-\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{2 \sqrt {2} d \left (a^2+b^2\right )^3}-\frac {\left (3 a^2 b (A-B)+a^3 (-(A+B))+3 a b^2 (A+B)-b^3 (A-B)\right ) \log \left (\tan (c+d x)+\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{2 \sqrt {2} d \left (a^2+b^2\right )^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((a^3*(A - B) - 3*a*b^2*(A - B) + 3*a^2*b*(A + B) - b^3*(A + B))*ArcTan[1 - Sqrt[2]*Sqrt[Tan[c + d*x]]])/(Sqrt
[2]*(a^2 + b^2)^3*d) - ((a^3*(A - B) - 3*a*b^2*(A - B) + 3*a^2*b*(A + B) - b^3*(A + B))*ArcTan[1 + Sqrt[2]*Sqr
t[Tan[c + d*x]]])/(Sqrt[2]*(a^2 + b^2)^3*d) - (b^(3/2)*(63*a^4*A*b + 46*a^2*A*b^3 + 15*A*b^5 - 35*a^5*B - 6*a^
3*b^2*B - 3*a*b^4*B)*ArcTan[(Sqrt[b]*Sqrt[Tan[c + d*x]])/Sqrt[a]])/(4*a^(7/2)*(a^2 + b^2)^3*d) + ((3*a^2*b*(A
- B) - b^3*(A - B) - a^3*(A + B) + 3*a*b^2*(A + B))*Log[1 - Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]])/(2*Sqr
t[2]*(a^2 + b^2)^3*d) - ((3*a^2*b*(A - B) - b^3*(A - B) - a^3*(A + B) + 3*a*b^2*(A + B))*Log[1 + Sqrt[2]*Sqrt[
Tan[c + d*x]] + Tan[c + d*x]])/(2*Sqrt[2]*(a^2 + b^2)^3*d) - (8*a^4*A + 31*a^2*A*b^2 + 15*A*b^4 - 11*a^3*b*B -
3*a*b^3*B)/(4*a^3*(a^2 + b^2)^2*d*Sqrt[Tan[c + d*x]]) + (b*(A*b - a*B))/(2*a*(a^2 + b^2)*d*Sqrt[Tan[c + d*x]]
*(a + b*Tan[c + d*x])^2) + (b*(13*a^2*A*b + 5*A*b^3 - 9*a^3*B - a*b^2*B))/(4*a^2*(a^2 + b^2)^2*d*Sqrt[Tan[c +
d*x]]*(a + b*Tan[c + d*x]))

Rule 63

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 204

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

Rule 205

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

Rule 617

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 628

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 1162

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 1165

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 1168

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 3534

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 3609

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 3634

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 3649

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*T
an[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 3653

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*} \int \frac {A+B \tan (c+d x)}{\tan ^{\frac {3}{2}}(c+d x) (a+b \tan (c+d x))^3} \, dx &=\frac {b (A b-a B)}{2 a \left (a^2+b^2\right ) d \sqrt {\tan (c+d x)} (a+b \tan (c+d x))^2}+\frac {\int \frac {\frac {1}{2} \left (4 a^2 A+5 A b^2-a b B\right )-2 a (A b-a B) \tan (c+d x)+\frac {5}{2} b (A b-a B) \tan ^2(c+d x)}{\tan ^{\frac {3}{2}}(c+d x) (a+b \tan (c+d x))^2} \, dx}{2 a \left (a^2+b^2\right )}\\ &=\frac {b (A b-a B)}{2 a \left (a^2+b^2\right ) d \sqrt {\tan (c+d x)} (a+b \tan (c+d x))^2}+\frac {b \left (13 a^2 A b+5 A b^3-9 a^3 B-a b^2 B\right )}{4 a^2 \left (a^2+b^2\right )^2 d \sqrt {\tan (c+d x)} (a+b \tan (c+d x))}+\frac {\int \frac {\frac {1}{4} \left (8 a^4 A+31 a^2 A b^2+15 A b^4-11 a^3 b B-3 a b^3 B\right )-2 a^2 \left (2 a A b-a^2 B+b^2 B\right ) \tan (c+d x)+\frac {3}{4} b \left (13 a^2 A b+5 A b^3-9 a^3 B-a b^2 B\right ) \tan ^2(c+d x)}{\tan ^{\frac {3}{2}}(c+d x) (a+b \tan (c+d x))} \, dx}{2 a^2 \left (a^2+b^2\right )^2}\\ &=-\frac {8 a^4 A+31 a^2 A b^2+15 A b^4-11 a^3 b B-3 a b^3 B}{4 a^3 \left (a^2+b^2\right )^2 d \sqrt {\tan (c+d x)}}+\frac {b (A b-a B)}{2 a \left (a^2+b^2\right ) d \sqrt {\tan (c+d x)} (a+b \tan (c+d x))^2}+\frac {b \left (13 a^2 A b+5 A b^3-9 a^3 B-a b^2 B\right )}{4 a^2 \left (a^2+b^2\right )^2 d \sqrt {\tan (c+d x)} (a+b \tan (c+d x))}-\frac {\int \frac {\frac {1}{8} \left (24 a^4 A b+31 a^2 A b^3+15 A b^5-8 a^5 B-3 a^3 b^2 B-3 a b^4 B\right )+a^3 \left (a^2 A-A b^2+2 a b B\right ) \tan (c+d x)+\frac {1}{8} b \left (8 a^4 A+31 a^2 A b^2+15 A b^4-11 a^3 b B-3 a b^3 B\right ) \tan ^2(c+d x)}{\sqrt {\tan (c+d x)} (a+b \tan (c+d x))} \, dx}{a^3 \left (a^2+b^2\right )^2}\\ &=-\frac {8 a^4 A+31 a^2 A b^2+15 A b^4-11 a^3 b B-3 a b^3 B}{4 a^3 \left (a^2+b^2\right )^2 d \sqrt {\tan (c+d x)}}+\frac {b (A b-a B)}{2 a \left (a^2+b^2\right ) d \sqrt {\tan (c+d x)} (a+b \tan (c+d x))^2}+\frac {b \left (13 a^2 A b+5 A b^3-9 a^3 B-a b^2 B\right )}{4 a^2 \left (a^2+b^2\right )^2 d \sqrt {\tan (c+d x)} (a+b \tan (c+d x))}-\frac {\int \frac {a^3 \left (3 a^2 A b-A b^3-a^3 B+3 a b^2 B\right )+a^3 \left (a^3 A-3 a A b^2+3 a^2 b B-b^3 B\right ) \tan (c+d x)}{\sqrt {\tan (c+d x)}} \, dx}{a^3 \left (a^2+b^2\right )^3}-\frac {\left (b^2 \left (63 a^4 A b+46 a^2 A b^3+15 A b^5-35 a^5 B-6 a^3 b^2 B-3 a b^4 B\right )\right ) \int \frac {1+\tan ^2(c+d x)}{\sqrt {\tan (c+d x)} (a+b \tan (c+d x))} \, dx}{8 a^3 \left (a^2+b^2\right )^3}\\ &=-\frac {8 a^4 A+31 a^2 A b^2+15 A b^4-11 a^3 b B-3 a b^3 B}{4 a^3 \left (a^2+b^2\right )^2 d \sqrt {\tan (c+d x)}}+\frac {b (A b-a B)}{2 a \left (a^2+b^2\right ) d \sqrt {\tan (c+d x)} (a+b \tan (c+d x))^2}+\frac {b \left (13 a^2 A b+5 A b^3-9 a^3 B-a b^2 B\right )}{4 a^2 \left (a^2+b^2\right )^2 d \sqrt {\tan (c+d x)} (a+b \tan (c+d x))}-\frac {2 \operatorname {Subst}\left (\int \frac {a^3 \left (3 a^2 A b-A b^3-a^3 B+3 a b^2 B\right )+a^3 \left (a^3 A-3 a A b^2+3 a^2 b B-b^3 B\right ) x^2}{1+x^4} \, dx,x,\sqrt {\tan (c+d x)}\right )}{a^3 \left (a^2+b^2\right )^3 d}-\frac {\left (b^2 \left (63 a^4 A b+46 a^2 A b^3+15 A b^5-35 a^5 B-6 a^3 b^2 B-3 a b^4 B\right )\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {x} (a+b x)} \, dx,x,\tan (c+d x)\right )}{8 a^3 \left (a^2+b^2\right )^3 d}\\ &=-\frac {8 a^4 A+31 a^2 A b^2+15 A b^4-11 a^3 b B-3 a b^3 B}{4 a^3 \left (a^2+b^2\right )^2 d \sqrt {\tan (c+d x)}}+\frac {b (A b-a B)}{2 a \left (a^2+b^2\right ) d \sqrt {\tan (c+d x)} (a+b \tan (c+d x))^2}+\frac {b \left (13 a^2 A b+5 A b^3-9 a^3 B-a b^2 B\right )}{4 a^2 \left (a^2+b^2\right )^2 d \sqrt {\tan (c+d x)} (a+b \tan (c+d x))}-\frac {\left (b^2 \left (63 a^4 A b+46 a^2 A b^3+15 A b^5-35 a^5 B-6 a^3 b^2 B-3 a b^4 B\right )\right ) \operatorname {Subst}\left (\int \frac {1}{a+b x^2} \, dx,x,\sqrt {\tan (c+d x)}\right )}{4 a^3 \left (a^2+b^2\right )^3 d}-\frac {\left (3 a^2 b (A-B)-b^3 (A-B)-a^3 (A+B)+3 a b^2 (A+B)\right ) \operatorname {Subst}\left (\int \frac {1-x^2}{1+x^4} \, dx,x,\sqrt {\tan (c+d x)}\right )}{\left (a^2+b^2\right )^3 d}-\frac {\left (a^3 (A-B)-3 a b^2 (A-B)+3 a^2 b (A+B)-b^3 (A+B)\right ) \operatorname {Subst}\left (\int \frac {1+x^2}{1+x^4} \, dx,x,\sqrt {\tan (c+d x)}\right )}{\left (a^2+b^2\right )^3 d}\\ &=-\frac {b^{3/2} \left (63 a^4 A b+46 a^2 A b^3+15 A b^5-35 a^5 B-6 a^3 b^2 B-3 a b^4 B\right ) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {\tan (c+d x)}}{\sqrt {a}}\right )}{4 a^{7/2} \left (a^2+b^2\right )^3 d}-\frac {8 a^4 A+31 a^2 A b^2+15 A b^4-11 a^3 b B-3 a b^3 B}{4 a^3 \left (a^2+b^2\right )^2 d \sqrt {\tan (c+d x)}}+\frac {b (A b-a B)}{2 a \left (a^2+b^2\right ) d \sqrt {\tan (c+d x)} (a+b \tan (c+d x))^2}+\frac {b \left (13 a^2 A b+5 A b^3-9 a^3 B-a b^2 B\right )}{4 a^2 \left (a^2+b^2\right )^2 d \sqrt {\tan (c+d x)} (a+b \tan (c+d x))}+\frac {\left (3 a^2 b (A-B)-b^3 (A-B)-a^3 (A+B)+3 a b^2 (A+B)\right ) \operatorname {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 )^3 d}+\frac {\left (3 a^2 b (A-B)-b^3 (A-B)-a^3 (A+B)+3 a b^2 (A+B)\right ) \operatorname {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 )^3 d}-\frac {\left (a^3 (A-B)-3 a b^2 (A-B)+3 a^2 b (A+B)-b^3 (A+B)\right ) \operatorname {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 )^3 d}-\frac {\left (a^3 (A-B)-3 a b^2 (A-B)+3 a^2 b (A+B)-b^3 (A+B)\right ) \operatorname {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 )^3 d}\\ &=-\frac {b^{3/2} \left (63 a^4 A b+46 a^2 A b^3+15 A b^5-35 a^5 B-6 a^3 b^2 B-3 a b^4 B\right ) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {\tan (c+d x)}}{\sqrt {a}}\right )}{4 a^{7/2} \left (a^2+b^2\right )^3 d}+\frac {\left (3 a^2 b (A-B)-b^3 (A-B)-a^3 (A+B)+3 a 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 )^3 d}-\frac {\left (3 a^2 b (A-B)-b^3 (A-B)-a^3 (A+B)+3 a 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 )^3 d}-\frac {8 a^4 A+31 a^2 A b^2+15 A b^4-11 a^3 b B-3 a b^3 B}{4 a^3 \left (a^2+b^2\right )^2 d \sqrt {\tan (c+d x)}}+\frac {b (A b-a B)}{2 a \left (a^2+b^2\right ) d \sqrt {\tan (c+d x)} (a+b \tan (c+d x))^2}+\frac {b \left (13 a^2 A b+5 A b^3-9 a^3 B-a b^2 B\right )}{4 a^2 \left (a^2+b^2\right )^2 d \sqrt {\tan (c+d x)} (a+b \tan (c+d x))}-\frac {\left (a^3 (A-B)-3 a b^2 (A-B)+3 a^2 b (A+B)-b^3 (A+B)\right ) \operatorname {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 )^3 d}+\frac {\left (a^3 (A-B)-3 a b^2 (A-B)+3 a^2 b (A+B)-b^3 (A+B)\right ) \operatorname {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 )^3 d}\\ &=\frac {\left (a^3 (A-B)-3 a b^2 (A-B)+3 a^2 b (A+B)-b^3 (A+B)\right ) \tan ^{-1}\left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} \left (a^2+b^2\right )^3 d}-\frac {\left (a^3 (A-B)-3 a b^2 (A-B)+3 a^2 b (A+B)-b^3 (A+B)\right ) \tan ^{-1}\left (1+\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} \left (a^2+b^2\right )^3 d}-\frac {b^{3/2} \left (63 a^4 A b+46 a^2 A b^3+15 A b^5-35 a^5 B-6 a^3 b^2 B-3 a b^4 B\right ) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {\tan (c+d x)}}{\sqrt {a}}\right )}{4 a^{7/2} \left (a^2+b^2\right )^3 d}+\frac {\left (3 a^2 b (A-B)-b^3 (A-B)-a^3 (A+B)+3 a 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 )^3 d}-\frac {\left (3 a^2 b (A-B)-b^3 (A-B)-a^3 (A+B)+3 a 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 )^3 d}-\frac {8 a^4 A+31 a^2 A b^2+15 A b^4-11 a^3 b B-3 a b^3 B}{4 a^3 \left (a^2+b^2\right )^2 d \sqrt {\tan (c+d x)}}+\frac {b (A b-a B)}{2 a \left (a^2+b^2\right ) d \sqrt {\tan (c+d x)} (a+b \tan (c+d x))^2}+\frac {b \left (13 a^2 A b+5 A b^3-9 a^3 B-a b^2 B\right )}{4 a^2 \left (a^2+b^2\right )^2 d \sqrt {\tan (c+d x)} (a+b \tan (c+d x))}\\ \end {align*}

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Mathematica [C]  time = 6.30, size = 585, normalized size = 0.97 \[ \frac {b (A b-a B)}{2 a d \left (a^2+b^2\right ) \sqrt {\tan (c+d x)} (a+b \tan (c+d x))^2}+\frac {\frac {\frac {1}{2} b^2 \left (4 a^2 A-a b B+5 A b^2\right )+\frac {9}{2} a^2 b (A b-a B)}{a d \left (a^2+b^2\right ) \sqrt {\tan (c+d x)} (a+b \tan (c+d x))}+\frac {-\frac {8 a^4 A-11 a^3 b B+31 a^2 A b^2-3 a b^3 B+15 A b^4}{2 a d \sqrt {\tan (c+d x)}}-\frac {2 \left (\frac {2 \left (a^4 (-b) \left (a^2 A+2 a b B-A b^2\right )+\frac {1}{8} a^2 b \left (8 a^4 A-11 a^3 b B+31 a^2 A b^2-3 a b^3 B+15 A b^4\right )+\frac {1}{8} b^2 \left (-8 a^5 B+24 a^4 A b-3 a^3 b^2 B+31 a^2 A b^3-3 a b^4 B+15 A b^5\right )\right ) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {\tan (c+d x)}}{\sqrt {a}}\right )}{\sqrt {a} \sqrt {b} d \left (a^2+b^2\right )}+\frac {-\frac {\sqrt [4]{-1} \left (a^3 \left (a^3 (-B)+3 a^2 A b+3 a b^2 B-A b^3\right )-i a^3 \left (a^3 A+3 a^2 b B-3 a A b^2-b^3 B\right )\right ) \tan ^{-1}\left ((-1)^{3/4} \sqrt {\tan (c+d x)}\right )}{d}-\frac {\sqrt [4]{-1} \left (a^3 \left (a^3 (-B)+3 a^2 A b+3 a b^2 B-A b^3\right )+i a^3 \left (a^3 A+3 a^2 b B-3 a A b^2-b^3 B\right )\right ) \tanh ^{-1}\left ((-1)^{3/4} \sqrt {\tan (c+d x)}\right )}{d}}{a^2+b^2}\right )}{a}}{a \left (a^2+b^2\right )}}{2 a \left (a^2+b^2\right )} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(b*(A*b - a*B))/(2*a*(a^2 + b^2)*d*Sqrt[Tan[c + d*x]]*(a + b*Tan[c + d*x])^2) + (((-2*((2*(-(a^4*b*(a^2*A - A*
b^2 + 2*a*b*B)) + (a^2*b*(8*a^4*A + 31*a^2*A*b^2 + 15*A*b^4 - 11*a^3*b*B - 3*a*b^3*B))/8 + (b^2*(24*a^4*A*b +
31*a^2*A*b^3 + 15*A*b^5 - 8*a^5*B - 3*a^3*b^2*B - 3*a*b^4*B))/8)*ArcTan[(Sqrt[b]*Sqrt[Tan[c + d*x]])/Sqrt[a]])
/(Sqrt[a]*Sqrt[b]*(a^2 + b^2)*d) + (-(((-1)^(1/4)*(a^3*(3*a^2*A*b - A*b^3 - a^3*B + 3*a*b^2*B) - I*a^3*(a^3*A
- 3*a*A*b^2 + 3*a^2*b*B - b^3*B))*ArcTan[(-1)^(3/4)*Sqrt[Tan[c + d*x]]])/d) - ((-1)^(1/4)*(a^3*(3*a^2*A*b - A*
b^3 - a^3*B + 3*a*b^2*B) + I*a^3*(a^3*A - 3*a*A*b^2 + 3*a^2*b*B - b^3*B))*ArcTanh[(-1)^(3/4)*Sqrt[Tan[c + d*x]
]])/d)/(a^2 + b^2)))/a - (8*a^4*A + 31*a^2*A*b^2 + 15*A*b^4 - 11*a^3*b*B - 3*a*b^3*B)/(2*a*d*Sqrt[Tan[c + d*x]
]))/(a*(a^2 + b^2)) + ((9*a^2*b*(A*b - a*B))/2 + (b^2*(4*a^2*A + 5*A*b^2 - a*b*B))/2)/(a*(a^2 + b^2)*d*Sqrt[Ta
n[c + d*x]]*(a + b*Tan[c + d*x])))/(2*a*(a^2 + b^2))

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fricas [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {B \tan \left (d x + c\right ) + A}{{\left (b \tan \left (d x + c\right ) + a\right )}^{3} \tan \left (d x + c\right )^{\frac {3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [B]  time = 0.42, size = 1864, normalized size = 3.10 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-3/4/d/(a^2+b^2)^3*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)
))*a*b^2+3/4/d/(a^2+b^2)^3*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)+ta
n(d*x+c)))*a*b^2+13/4/d*a^3/(a^2+b^2)^3/(a+b*tan(d*x+c))^2*b^2*tan(d*x+c)^(1/2)*B-17/4/d*a^2/(a^2+b^2)^3/(a+b*
tan(d*x+c))^2*b^3*tan(d*x+c)^(1/2)*A+35/4/d*a^2/(a^2+b^2)^3*b^2/(a*b)^(1/2)*arctan(tan(d*x+c)^(1/2)*b/(a*b)^(1
/2))*B-63/4/d*a/(a^2+b^2)^3*b^3/(a*b)^(1/2)*arctan(tan(d*x+c)^(1/2)*b/(a*b)^(1/2))*A-2/d*A/a^3/tan(d*x+c)^(1/2
)-3/2/d/(a^2+b^2)^3*B*2^(1/2)*arctan(-1+2^(1/2)*tan(d*x+c)^(1/2))*a^2*b-3/4/d/(a^2+b^2)^3*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)))*a^2*b-3/2/d/(a^2+b^2)^3*B*2^(1/2)*ar
ctan(1+2^(1/2)*tan(d*x+c)^(1/2))*a*b^2+5/4/d*b^6/(a^2+b^2)^3/a/(a+b*tan(d*x+c))^2*B*tan(d*x+c)^(1/2)+3/4/d*b^6
/(a^2+b^2)^3/a^2/(a*b)^(1/2)*arctan(tan(d*x+c)^(1/2)*b/(a*b)^(1/2))*B-3/2/d/(a^2+b^2)^3*B*2^(1/2)*arctan(-1+2^
(1/2)*tan(d*x+c)^(1/2))*a*b^2+3/4/d*b^7/(a^2+b^2)^3/a^2/(a+b*tan(d*x+c))^2*tan(d*x+c)^(3/2)*B-7/4/d*b^8/(a^2+b
^2)^3/a^3/(a+b*tan(d*x+c))^2*tan(d*x+c)^(3/2)*A-11/2/d/(a^2+b^2)^3/(a+b*tan(d*x+c))^2*b^6/a*tan(d*x+c)^(3/2)*A
-15/4/d*b^7/(a^2+b^2)^3/a^3/(a*b)^(1/2)*arctan(tan(d*x+c)^(1/2)*b/(a*b)^(1/2))*A-9/4/d*b^7/(a^2+b^2)^3/a^2/(a+
b*tan(d*x+c))^2*A*tan(d*x+c)^(1/2)+3/2/d/(a^2+b^2)^3*A*2^(1/2)*arctan(1+2^(1/2)*tan(d*x+c)^(1/2))*a*b^2-3/2/d/
(a^2+b^2)^3*A*2^(1/2)*arctan(-1+2^(1/2)*tan(d*x+c)^(1/2))*a^2*b-3/2/d/(a^2+b^2)^3*B*2^(1/2)*arctan(1+2^(1/2)*t
an(d*x+c)^(1/2))*a^2*b+3/2/d/(a^2+b^2)^3*A*2^(1/2)*arctan(-1+2^(1/2)*tan(d*x+c)^(1/2))*a*b^2-3/4/d/(a^2+b^2)^3
*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)))*a^2*b-15/4/d*a/
(a^2+b^2)^3/(a+b*tan(d*x+c))^2*b^4*tan(d*x+c)^(3/2)*A+11/4/d*a^2/(a^2+b^2)^3/(a+b*tan(d*x+c))^2*b^3*tan(d*x+c)
^(3/2)*B-23/2/d/(a^2+b^2)^3/a/(a*b)^(1/2)*arctan(tan(d*x+c)^(1/2)*b/(a*b)^(1/2))*A*b^5+9/2/d/(a^2+b^2)^3/(a+b*
tan(d*x+c))^2*tan(d*x+c)^(1/2)*a*b^4*B-3/2/d/(a^2+b^2)^3*A*2^(1/2)*arctan(1+2^(1/2)*tan(d*x+c)^(1/2))*a^2*b+1/
2/d/(a^2+b^2)^3*A*2^(1/2)*arctan(1+2^(1/2)*tan(d*x+c)^(1/2))*b^3-1/2/d/(a^2+b^2)^3*A*2^(1/2)*arctan(1+2^(1/2)*
tan(d*x+c)^(1/2))*a^3+1/4/d/(a^2+b^2)^3*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)))*b^3+1/4/d/(a^2+b^2)^3*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)))*a^3+1/4/d/(a^2+b^2)^3*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)))*b^3-1/4/d/(a^2+b^2)^3*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)))*a^3+1/2/d/(a^2+b^2)^3*B*2^(1/2)*arctan(1+2^(1/2)*tan(d*x+c)^(1/2))*a
^3+1/2/d/(a^2+b^2)^3*A*2^(1/2)*arctan(-1+2^(1/2)*tan(d*x+c)^(1/2))*b^3+1/2/d/(a^2+b^2)^3*B*2^(1/2)*arctan(1+2^
(1/2)*tan(d*x+c)^(1/2))*b^3+1/2/d/(a^2+b^2)^3*B*2^(1/2)*arctan(-1+2^(1/2)*tan(d*x+c)^(1/2))*b^3+7/2/d/(a^2+b^2
)^3/(a+b*tan(d*x+c))^2*b^5*tan(d*x+c)^(3/2)*B-13/2/d/(a^2+b^2)^3/(a+b*tan(d*x+c))^2*tan(d*x+c)^(1/2)*A*b^5+3/2
/d/(a^2+b^2)^3/(a*b)^(1/2)*arctan(tan(d*x+c)^(1/2)*b/(a*b)^(1/2))*b^4*B-1/2/d/(a^2+b^2)^3*A*2^(1/2)*arctan(-1+
2^(1/2)*tan(d*x+c)^(1/2))*a^3+1/2/d/(a^2+b^2)^3*B*2^(1/2)*arctan(-1+2^(1/2)*tan(d*x+c)^(1/2))*a^3

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maxima [A]  time = 0.48, size = 607, normalized size = 1.01 \[ \frac {\frac {{\left (35 \, B a^{5} b^{2} - 63 \, A a^{4} b^{3} + 6 \, B a^{3} b^{4} - 46 \, A a^{2} b^{5} + 3 \, B a b^{6} - 15 \, A b^{7}\right )} \arctan \left (\frac {b \sqrt {\tan \left (d x + c\right )}}{\sqrt {a b}}\right )}{{\left (a^{9} + 3 \, a^{7} b^{2} + 3 \, a^{5} b^{4} + a^{3} b^{6}\right )} \sqrt {a b}} - \frac {8 \, A a^{6} + 16 \, A a^{4} b^{2} + 8 \, A a^{2} b^{4} + {\left (8 \, A a^{4} b^{2} - 11 \, B a^{3} b^{3} + 31 \, A a^{2} b^{4} - 3 \, B a b^{5} + 15 \, A b^{6}\right )} \tan \left (d x + c\right )^{2} + {\left (16 \, A a^{5} b - 13 \, B a^{4} b^{2} + 49 \, A a^{3} b^{3} - 5 \, B a^{2} b^{4} + 25 \, A a b^{5}\right )} \tan \left (d x + c\right )}{{\left (a^{7} b^{2} + 2 \, a^{5} b^{4} + a^{3} b^{6}\right )} \tan \left (d x + c\right )^{\frac {5}{2}} + 2 \, {\left (a^{8} b + 2 \, a^{6} b^{3} + a^{4} b^{5}\right )} \tan \left (d x + c\right )^{\frac {3}{2}} + {\left (a^{9} + 2 \, a^{7} b^{2} + a^{5} b^{4}\right )} \sqrt {\tan \left (d x + c\right )}} - \frac {2 \, \sqrt {2} {\left ({\left (A - B\right )} a^{3} + 3 \, {\left (A + B\right )} a^{2} b - 3 \, {\left (A - B\right )} a b^{2} - {\left (A + B\right )} b^{3}\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^{3} + 3 \, {\left (A + B\right )} a^{2} b - 3 \, {\left (A - B\right )} a b^{2} - {\left (A + B\right )} b^{3}\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^{3} - 3 \, {\left (A - B\right )} a^{2} b - 3 \, {\left (A + B\right )} a b^{2} + {\left (A - B\right )} b^{3}\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^{3} - 3 \, {\left (A - B\right )} a^{2} b - 3 \, {\left (A + B\right )} a b^{2} + {\left (A - B\right )} b^{3}\right )} \log \left (-\sqrt {2} \sqrt {\tan \left (d x + c\right )} + \tan \left (d x + c\right ) + 1\right )}{a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}}}{4 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*((35*B*a^5*b^2 - 63*A*a^4*b^3 + 6*B*a^3*b^4 - 46*A*a^2*b^5 + 3*B*a*b^6 - 15*A*b^7)*arctan(b*sqrt(tan(d*x +
c))/sqrt(a*b))/((a^9 + 3*a^7*b^2 + 3*a^5*b^4 + a^3*b^6)*sqrt(a*b)) - (8*A*a^6 + 16*A*a^4*b^2 + 8*A*a^2*b^4 +
(8*A*a^4*b^2 - 11*B*a^3*b^3 + 31*A*a^2*b^4 - 3*B*a*b^5 + 15*A*b^6)*tan(d*x + c)^2 + (16*A*a^5*b - 13*B*a^4*b^2
+ 49*A*a^3*b^3 - 5*B*a^2*b^4 + 25*A*a*b^5)*tan(d*x + c))/((a^7*b^2 + 2*a^5*b^4 + a^3*b^6)*tan(d*x + c)^(5/2)
+ 2*(a^8*b + 2*a^6*b^3 + a^4*b^5)*tan(d*x + c)^(3/2) + (a^9 + 2*a^7*b^2 + a^5*b^4)*sqrt(tan(d*x + c))) - (2*sq
rt(2)*((A - B)*a^3 + 3*(A + B)*a^2*b - 3*(A - B)*a*b^2 - (A + B)*b^3)*arctan(1/2*sqrt(2)*(sqrt(2) + 2*sqrt(tan
(d*x + c)))) + 2*sqrt(2)*((A - B)*a^3 + 3*(A + B)*a^2*b - 3*(A - B)*a*b^2 - (A + B)*b^3)*arctan(-1/2*sqrt(2)*(
sqrt(2) - 2*sqrt(tan(d*x + c)))) - sqrt(2)*((A + B)*a^3 - 3*(A - B)*a^2*b - 3*(A + B)*a*b^2 + (A - B)*b^3)*log
(sqrt(2)*sqrt(tan(d*x + c)) + tan(d*x + c) + 1) + sqrt(2)*((A + B)*a^3 - 3*(A - B)*a^2*b - 3*(A + B)*a*b^2 + (
A - B)*b^3)*log(-sqrt(2)*sqrt(tan(d*x + c)) + tan(d*x + c) + 1))/(a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6))/d

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mupad [B]  time = 48.48, size = 35300, normalized size = 58.74 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((B*tan(c + d*x)^(1/2)*(5*b^4 + 13*a^2*b^2))/(4*a*(a^4 + b^4 + 2*a^2*b^2)) + (B*b*tan(c + d*x)^(3/2)*(3*b^4 +
11*a^2*b^2))/(4*a^2*(a^4 + b^4 + 2*a^2*b^2)))/(a^2*d + b^2*d*tan(c + d*x)^2 + 2*a*b*d*tan(c + d*x)) - ((2*A)/a
+ (A*tan(c + d*x)^2*(15*b^6 + 31*a^2*b^4 + 8*a^4*b^2))/(4*a^3*(a^4 + b^4 + 2*a^2*b^2)) + (A*tan(c + d*x)*(16*
a^4*b + 25*b^5 + 49*a^2*b^3))/(4*a^2*(a^4 + b^4 + 2*a^2*b^2)))/(a^2*d*tan(c + d*x)^(1/2) + b^2*d*tan(c + d*x)^
(5/2) + 2*a*b*d*tan(c + d*x)^(3/2)) + (log(29491200*A^5*a^22*b^35*d^4 - ((tan(c + d*x)^(1/2)*(7610564608*A^4*a
^27*b^33*d^5 - 597688320*A^4*a^23*b^37*d^5 - 1671430144*A^4*a^25*b^35*d^5 - 58982400*A^4*a^21*b^39*d^5 + 85774
565376*A^4*a^29*b^31*d^5 + 385487994880*A^4*a^31*b^29*d^5 + 1104303620096*A^4*a^33*b^27*d^5 + 2240523796480*A^
4*a^35*b^25*d^5 + 3345249468416*A^4*a^37*b^23*d^5 + 3717287903232*A^4*a^39*b^21*d^5 + 3053967114240*A^4*a^41*b
^19*d^5 + 1807474491392*A^4*a^43*b^17*d^5 + 726513221632*A^4*a^45*b^15*d^5 + 170768990208*A^4*a^47*b^13*d^5 +
10492051456*A^4*a^49*b^11*d^5 - 4917821440*A^4*a^51*b^9*d^5 - 923009024*A^4*a^53*b^7*d^5 + 8388608*A^4*a^55*b^
5*d^5) + ((((480*A^4*a^2*b^10*d^4 - 16*A^4*b^12*d^4 - 16*A^4*a^12*d^4 - 4080*A^4*a^4*b^8*d^4 + 7232*A^4*a^6*b^
6*d^4 - 4080*A^4*a^8*b^4*d^4 + 480*A^4*a^10*b^2*d^4)^(1/2) + 80*A^2*a^3*b^3*d^2 - 24*A^2*a*b^5*d^2 - 24*A^2*a^
5*b*d^2)/(a^12*d^4 + b^12*d^4 + 6*a^2*b^10*d^4 + 15*a^4*b^8*d^4 + 20*a^6*b^6*d^4 + 15*a^8*b^4*d^4 + 6*a^10*b^2
*d^4))^(1/2)*(((((((480*A^4*a^2*b^10*d^4 - 16*A^4*b^12*d^4 - 16*A^4*a^12*d^4 - 4080*A^4*a^4*b^8*d^4 + 7232*A^4
*a^6*b^6*d^4 - 4080*A^4*a^8*b^4*d^4 + 480*A^4*a^10*b^2*d^4)^(1/2) + 80*A^2*a^3*b^3*d^2 - 24*A^2*a*b^5*d^2 - 24
*A^2*a^5*b*d^2)/(a^12*d^4 + b^12*d^4 + 6*a^2*b^10*d^4 + 15*a^4*b^8*d^4 + 20*a^6*b^6*d^4 + 15*a^8*b^4*d^4 + 6*a
^10*b^2*d^4))^(1/2)*((tan(c + d*x)^(1/2)*(((480*A^4*a^2*b^10*d^4 - 16*A^4*b^12*d^4 - 16*A^4*a^12*d^4 - 4080*A^
4*a^4*b^8*d^4 + 7232*A^4*a^6*b^6*d^4 - 4080*A^4*a^8*b^4*d^4 + 480*A^4*a^10*b^2*d^4)^(1/2) + 80*A^2*a^3*b^3*d^2
- 24*A^2*a*b^5*d^2 - 24*A^2*a^5*b*d^2)/(a^12*d^4 + b^12*d^4 + 6*a^2*b^10*d^4 + 15*a^4*b^8*d^4 + 20*a^6*b^6*d^
4 + 15*a^8*b^4*d^4 + 6*a^10*b^2*d^4))^(1/2)*(134217728*a^27*b^45*d^9 + 2550136832*a^29*b^43*d^9 + 22817013760*
a^31*b^41*d^9 + 127506841600*a^33*b^39*d^9 + 497276682240*a^35*b^37*d^9 + 1430626762752*a^37*b^35*d^9 + 312136
7482368*a^39*b^33*d^9 + 5202279137280*a^41*b^31*d^9 + 6502848921600*a^43*b^29*d^9 + 5635802398720*a^45*b^27*d^
9 + 2254320959488*a^47*b^25*d^9 - 2254320959488*a^49*b^23*d^9 - 5635802398720*a^51*b^21*d^9 - 6502848921600*a^
53*b^19*d^9 - 5202279137280*a^55*b^17*d^9 - 3121367482368*a^57*b^15*d^9 - 1430626762752*a^59*b^13*d^9 - 497276
682240*a^61*b^11*d^9 - 127506841600*a^63*b^9*d^9 - 22817013760*a^65*b^7*d^9 - 2550136832*a^67*b^5*d^9 - 134217
728*a^69*b^3*d^9))/4 + 251658240*A*a^24*b^45*d^8 + 5049942016*A*a^26*b^43*d^8 + 48368713728*A*a^28*b^41*d^8 +
293819383808*A*a^30*b^39*d^8 + 1268458192896*A*a^32*b^37*d^8 + 4132731617280*A*a^34*b^35*d^8 + 10531192700928*
A*a^36*b^33*d^8 + 21462823993344*A*a^38*b^31*d^8 + 35469618315264*A*a^40*b^29*d^8 + 47896904859648*A*a^42*b^27
*d^8 + 52983958077440*A*a^44*b^25*d^8 + 47896904859648*A*a^46*b^23*d^8 + 35090285461504*A*a^48*b^21*d^8 + 2048
7396655104*A*a^50*b^19*d^8 + 9230622916608*A*a^52*b^17*d^8 + 2994733056000*A*a^54*b^15*d^8 + 565576728576*A*a^
56*b^13*d^8 - 18572378112*A*a^58*b^11*d^8 - 50281316352*A*a^60*b^9*d^8 - 16089350144*A*a^62*b^7*d^8 - 25165824
00*A*a^64*b^5*d^8 - 167772160*A*a^66*b^3*d^8))/4 - tan(c + d*x)^(1/2)*(471859200*A^2*a^22*b^44*d^7 + 950009856
0*A^2*a^24*b^42*d^7 + 91857354752*A^2*a^26*b^40*d^7 + 564502986752*A^2*a^28*b^38*d^7 + 2464648527872*A^2*a^30*
b^36*d^7 + 8104469069824*A^2*a^32*b^34*d^7 + 20769933361152*A^2*a^34*b^32*d^7 + 42351565209600*A^2*a^36*b^30*d
^7 + 69534945902592*A^2*a^38*b^28*d^7 + 92434029608960*A^2*a^40*b^26*d^7 + 99508717355008*A^2*a^42*b^24*d^7 +
86342935511040*A^2*a^44*b^22*d^7 + 59767095558144*A^2*a^46*b^20*d^7 + 32432589897728*A^2*a^48*b^18*d^7 + 13411
815522304*A^2*a^50*b^16*d^7 + 4030457708544*A^2*a^52*b^14*d^7 + 805425905664*A^2*a^54*b^12*d^7 + 86608183296*A
^2*a^56*b^10*d^7 + 1612709888*A^2*a^58*b^8*d^7 + 16777216*A^2*a^60*b^6*d^7 + 167772160*A^2*a^62*b^4*d^7 + 1677
7216*A^2*a^64*b^2*d^7))*(((480*A^4*a^2*b^10*d^4 - 16*A^4*b^12*d^4 - 16*A^4*a^12*d^4 - 4080*A^4*a^4*b^8*d^4 + 7
232*A^4*a^6*b^6*d^4 - 4080*A^4*a^8*b^4*d^4 + 480*A^4*a^10*b^2*d^4)^(1/2) + 80*A^2*a^3*b^3*d^2 - 24*A^2*a*b^5*d
^2 - 24*A^2*a^5*b*d^2)/(a^12*d^4 + b^12*d^4 + 6*a^2*b^10*d^4 + 15*a^4*b^8*d^4 + 20*a^6*b^6*d^4 + 15*a^8*b^4*d^
4 + 6*a^10*b^2*d^4))^(1/2))/4 - 117964800*A^3*a^21*b^42*d^6 - 841482240*A^3*a^23*b^40*d^6 + 3829399552*A^3*a^2
5*b^38*d^6 + 78068580352*A^3*a^27*b^36*d^6 + 497438162944*A^3*a^29*b^34*d^6 + 1899895980032*A^3*a^31*b^32*d^6
+ 4972695519232*A^3*a^33*b^30*d^6 + 9371195015168*A^3*a^35*b^28*d^6 + 12890720436224*A^3*a^37*b^26*d^6 + 12726
089809920*A^3*a^39*b^24*d^6 + 8366961197056*A^3*a^41*b^22*d^6 + 2597662490624*A^3*a^43*b^20*d^6 - 117183610880
0*A^3*a^45*b^18*d^6 - 1986881650688*A^3*a^47*b^16*d^6 - 1237583921152*A^3*a^49*b^14*d^6 - 449507753984*A^3*a^5
1*b^12*d^6 - 97476149248*A^3*a^53*b^10*d^6 - 11931222016*A^3*a^55*b^8*d^6 - 1006632960*A^3*a^57*b^6*d^6 - 1342
17728*A^3*a^59*b^4*d^6 - 8388608*A^3*a^61*b^2*d^6))/4)*(((480*A^4*a^2*b^10*d^4 - 16*A^4*b^12*d^4 - 16*A^4*a^12
*d^4 - 4080*A^4*a^4*b^8*d^4 + 7232*A^4*a^6*b^6*d^4 - 4080*A^4*a^8*b^4*d^4 + 480*A^4*a^10*b^2*d^4)^(1/2) + 80*A
^2*a^3*b^3*d^2 - 24*A^2*a*b^5*d^2 - 24*A^2*a^5*b*d^2)/(a^12*d^4 + b^12*d^4 + 6*a^2*b^10*d^4 + 15*a^4*b^8*d^4 +
20*a^6*b^6*d^4 + 15*a^8*b^4*d^4 + 6*a^10*b^2*d^4))^(1/2))/4 + 460062720*A^5*a^24*b^33*d^4 + 3439722496*A^5*a^
26*b^31*d^4 + 16227237888*A^5*a^28*b^29*d^4 + 53669396480*A^5*a^30*b^27*d^4 + 131031367680*A^5*a^32*b^25*d^4 +
242529730560*A^5*a^34*b^23*d^4 + 344454070272*A^5*a^36*b^21*d^4 + 375993532416*A^5*a^38*b^19*d^4 + 3130431897
60*A^5*a^40*b^17*d^4 + 195253370880*A^5*a^42*b^15*d^4 + 88318935040*A^5*a^44*b^13*d^4 + 27352498176*A^5*a^46*b
^11*d^4 + 5187043328*A^5*a^48*b^9*d^4 + 454164480*A^5*a^50*b^7*d^4)*(((480*A^4*a^2*b^10*d^4 - 16*A^4*b^12*d^4
- 16*A^4*a^12*d^4 - 4080*A^4*a^4*b^8*d^4 + 7232*A^4*a^6*b^6*d^4 - 4080*A^4*a^8*b^4*d^4 + 480*A^4*a^10*b^2*d^4)
^(1/2) + 80*A^2*a^3*b^3*d^2 - 24*A^2*a*b^5*d^2 - 24*A^2*a^5*b*d^2)/(a^12*d^4 + b^12*d^4 + 6*a^2*b^10*d^4 + 15*
a^4*b^8*d^4 + 20*a^6*b^6*d^4 + 15*a^8*b^4*d^4 + 6*a^10*b^2*d^4))^(1/2))/4 + (log(29491200*A^5*a^22*b^35*d^4 -
((tan(c + d*x)^(1/2)*(7610564608*A^4*a^27*b^33*d^5 - 597688320*A^4*a^23*b^37*d^5 - 1671430144*A^4*a^25*b^35*d^
5 - 58982400*A^4*a^21*b^39*d^5 + 85774565376*A^4*a^29*b^31*d^5 + 385487994880*A^4*a^31*b^29*d^5 + 110430362009
6*A^4*a^33*b^27*d^5 + 2240523796480*A^4*a^35*b^25*d^5 + 3345249468416*A^4*a^37*b^23*d^5 + 3717287903232*A^4*a^
39*b^21*d^5 + 3053967114240*A^4*a^41*b^19*d^5 + 1807474491392*A^4*a^43*b^17*d^5 + 726513221632*A^4*a^45*b^15*d
^5 + 170768990208*A^4*a^47*b^13*d^5 + 10492051456*A^4*a^49*b^11*d^5 - 4917821440*A^4*a^51*b^9*d^5 - 923009024*
A^4*a^53*b^7*d^5 + 8388608*A^4*a^55*b^5*d^5) + ((-((480*A^4*a^2*b^10*d^4 - 16*A^4*b^12*d^4 - 16*A^4*a^12*d^4 -
4080*A^4*a^4*b^8*d^4 + 7232*A^4*a^6*b^6*d^4 - 4080*A^4*a^8*b^4*d^4 + 480*A^4*a^10*b^2*d^4)^(1/2) - 80*A^2*a^3
*b^3*d^2 + 24*A^2*a*b^5*d^2 + 24*A^2*a^5*b*d^2)/(a^12*d^4 + b^12*d^4 + 6*a^2*b^10*d^4 + 15*a^4*b^8*d^4 + 20*a^
6*b^6*d^4 + 15*a^8*b^4*d^4 + 6*a^10*b^2*d^4))^(1/2)*(((((-((480*A^4*a^2*b^10*d^4 - 16*A^4*b^12*d^4 - 16*A^4*a^
12*d^4 - 4080*A^4*a^4*b^8*d^4 + 7232*A^4*a^6*b^6*d^4 - 4080*A^4*a^8*b^4*d^4 + 480*A^4*a^10*b^2*d^4)^(1/2) - 80
*A^2*a^3*b^3*d^2 + 24*A^2*a*b^5*d^2 + 24*A^2*a^5*b*d^2)/(a^12*d^4 + b^12*d^4 + 6*a^2*b^10*d^4 + 15*a^4*b^8*d^4
+ 20*a^6*b^6*d^4 + 15*a^8*b^4*d^4 + 6*a^10*b^2*d^4))^(1/2)*((tan(c + d*x)^(1/2)*(-((480*A^4*a^2*b^10*d^4 - 16
*A^4*b^12*d^4 - 16*A^4*a^12*d^4 - 4080*A^4*a^4*b^8*d^4 + 7232*A^4*a^6*b^6*d^4 - 4080*A^4*a^8*b^4*d^4 + 480*A^4
*a^10*b^2*d^4)^(1/2) - 80*A^2*a^3*b^3*d^2 + 24*A^2*a*b^5*d^2 + 24*A^2*a^5*b*d^2)/(a^12*d^4 + b^12*d^4 + 6*a^2*
b^10*d^4 + 15*a^4*b^8*d^4 + 20*a^6*b^6*d^4 + 15*a^8*b^4*d^4 + 6*a^10*b^2*d^4))^(1/2)*(134217728*a^27*b^45*d^9
+ 2550136832*a^29*b^43*d^9 + 22817013760*a^31*b^41*d^9 + 127506841600*a^33*b^39*d^9 + 497276682240*a^35*b^37*d
^9 + 1430626762752*a^37*b^35*d^9 + 3121367482368*a^39*b^33*d^9 + 5202279137280*a^41*b^31*d^9 + 6502848921600*a
^43*b^29*d^9 + 5635802398720*a^45*b^27*d^9 + 2254320959488*a^47*b^25*d^9 - 2254320959488*a^49*b^23*d^9 - 56358
02398720*a^51*b^21*d^9 - 6502848921600*a^53*b^19*d^9 - 5202279137280*a^55*b^17*d^9 - 3121367482368*a^57*b^15*d
^9 - 1430626762752*a^59*b^13*d^9 - 497276682240*a^61*b^11*d^9 - 127506841600*a^63*b^9*d^9 - 22817013760*a^65*b
^7*d^9 - 2550136832*a^67*b^5*d^9 - 134217728*a^69*b^3*d^9))/4 + 251658240*A*a^24*b^45*d^8 + 5049942016*A*a^26*
b^43*d^8 + 48368713728*A*a^28*b^41*d^8 + 293819383808*A*a^30*b^39*d^8 + 1268458192896*A*a^32*b^37*d^8 + 413273
1617280*A*a^34*b^35*d^8 + 10531192700928*A*a^36*b^33*d^8 + 21462823993344*A*a^38*b^31*d^8 + 35469618315264*A*a
^40*b^29*d^8 + 47896904859648*A*a^42*b^27*d^8 + 52983958077440*A*a^44*b^25*d^8 + 47896904859648*A*a^46*b^23*d^
8 + 35090285461504*A*a^48*b^21*d^8 + 20487396655104*A*a^50*b^19*d^8 + 9230622916608*A*a^52*b^17*d^8 + 29947330
56000*A*a^54*b^15*d^8 + 565576728576*A*a^56*b^13*d^8 - 18572378112*A*a^58*b^11*d^8 - 50281316352*A*a^60*b^9*d^
8 - 16089350144*A*a^62*b^7*d^8 - 2516582400*A*a^64*b^5*d^8 - 167772160*A*a^66*b^3*d^8))/4 - tan(c + d*x)^(1/2)
*(471859200*A^2*a^22*b^44*d^7 + 9500098560*A^2*a^24*b^42*d^7 + 91857354752*A^2*a^26*b^40*d^7 + 564502986752*A^
2*a^28*b^38*d^7 + 2464648527872*A^2*a^30*b^36*d^7 + 8104469069824*A^2*a^32*b^34*d^7 + 20769933361152*A^2*a^34*
b^32*d^7 + 42351565209600*A^2*a^36*b^30*d^7 + 69534945902592*A^2*a^38*b^28*d^7 + 92434029608960*A^2*a^40*b^26*
d^7 + 99508717355008*A^2*a^42*b^24*d^7 + 86342935511040*A^2*a^44*b^22*d^7 + 59767095558144*A^2*a^46*b^20*d^7 +
32432589897728*A^2*a^48*b^18*d^7 + 13411815522304*A^2*a^50*b^16*d^7 + 4030457708544*A^2*a^52*b^14*d^7 + 80542
5905664*A^2*a^54*b^12*d^7 + 86608183296*A^2*a^56*b^10*d^7 + 1612709888*A^2*a^58*b^8*d^7 + 16777216*A^2*a^60*b^
6*d^7 + 167772160*A^2*a^62*b^4*d^7 + 16777216*A^2*a^64*b^2*d^7))*(-((480*A^4*a^2*b^10*d^4 - 16*A^4*b^12*d^4 -
16*A^4*a^12*d^4 - 4080*A^4*a^4*b^8*d^4 + 7232*A^4*a^6*b^6*d^4 - 4080*A^4*a^8*b^4*d^4 + 480*A^4*a^10*b^2*d^4)^(
1/2) - 80*A^2*a^3*b^3*d^2 + 24*A^2*a*b^5*d^2 + 24*A^2*a^5*b*d^2)/(a^12*d^4 + b^12*d^4 + 6*a^2*b^10*d^4 + 15*a^
4*b^8*d^4 + 20*a^6*b^6*d^4 + 15*a^8*b^4*d^4 + 6*a^10*b^2*d^4))^(1/2))/4 - 117964800*A^3*a^21*b^42*d^6 - 841482
240*A^3*a^23*b^40*d^6 + 3829399552*A^3*a^25*b^38*d^6 + 78068580352*A^3*a^27*b^36*d^6 + 497438162944*A^3*a^29*b
^34*d^6 + 1899895980032*A^3*a^31*b^32*d^6 + 4972695519232*A^3*a^33*b^30*d^6 + 9371195015168*A^3*a^35*b^28*d^6
+ 12890720436224*A^3*a^37*b^26*d^6 + 12726089809920*A^3*a^39*b^24*d^6 + 8366961197056*A^3*a^41*b^22*d^6 + 2597
662490624*A^3*a^43*b^20*d^6 - 1171836108800*A^3*a^45*b^18*d^6 - 1986881650688*A^3*a^47*b^16*d^6 - 123758392115
2*A^3*a^49*b^14*d^6 - 449507753984*A^3*a^51*b^12*d^6 - 97476149248*A^3*a^53*b^10*d^6 - 11931222016*A^3*a^55*b^
8*d^6 - 1006632960*A^3*a^57*b^6*d^6 - 134217728*A^3*a^59*b^4*d^6 - 8388608*A^3*a^61*b^2*d^6))/4)*(-((480*A^4*a
^2*b^10*d^4 - 16*A^4*b^12*d^4 - 16*A^4*a^12*d^4 - 4080*A^4*a^4*b^8*d^4 + 7232*A^4*a^6*b^6*d^4 - 4080*A^4*a^8*b
^4*d^4 + 480*A^4*a^10*b^2*d^4)^(1/2) - 80*A^2*a^3*b^3*d^2 + 24*A^2*a*b^5*d^2 + 24*A^2*a^5*b*d^2)/(a^12*d^4 + b
^12*d^4 + 6*a^2*b^10*d^4 + 15*a^4*b^8*d^4 + 20*a^6*b^6*d^4 + 15*a^8*b^4*d^4 + 6*a^10*b^2*d^4))^(1/2))/4 + 4600
62720*A^5*a^24*b^33*d^4 + 3439722496*A^5*a^26*b^31*d^4 + 16227237888*A^5*a^28*b^29*d^4 + 53669396480*A^5*a^30*
b^27*d^4 + 131031367680*A^5*a^32*b^25*d^4 + 242529730560*A^5*a^34*b^23*d^4 + 344454070272*A^5*a^36*b^21*d^4 +
375993532416*A^5*a^38*b^19*d^4 + 313043189760*A^5*a^40*b^17*d^4 + 195253370880*A^5*a^42*b^15*d^4 + 88318935040
*A^5*a^44*b^13*d^4 + 27352498176*A^5*a^46*b^11*d^4 + 5187043328*A^5*a^48*b^9*d^4 + 454164480*A^5*a^50*b^7*d^4)
*(-((480*A^4*a^2*b^10*d^4 - 16*A^4*b^12*d^4 - 16*A^4*a^12*d^4 - 4080*A^4*a^4*b^8*d^4 + 7232*A^4*a^6*b^6*d^4 -
4080*A^4*a^8*b^4*d^4 + 480*A^4*a^10*b^2*d^4)^(1/2) - 80*A^2*a^3*b^3*d^2 + 24*A^2*a*b^5*d^2 + 24*A^2*a^5*b*d^2)
/(a^12*d^4 + b^12*d^4 + 6*a^2*b^10*d^4 + 15*a^4*b^8*d^4 + 20*a^6*b^6*d^4 + 15*a^8*b^4*d^4 + 6*a^10*b^2*d^4))^(
1/2))/4 - log((tan(c + d*x)^(1/2)*(7610564608*A^4*a^27*b^33*d^5 - 597688320*A^4*a^23*b^37*d^5 - 1671430144*A^4
*a^25*b^35*d^5 - 58982400*A^4*a^21*b^39*d^5 + 85774565376*A^4*a^29*b^31*d^5 + 385487994880*A^4*a^31*b^29*d^5 +
1104303620096*A^4*a^33*b^27*d^5 + 2240523796480*A^4*a^35*b^25*d^5 + 3345249468416*A^4*a^37*b^23*d^5 + 3717287
903232*A^4*a^39*b^21*d^5 + 3053967114240*A^4*a^41*b^19*d^5 + 1807474491392*A^4*a^43*b^17*d^5 + 726513221632*A^
4*a^45*b^15*d^5 + 170768990208*A^4*a^47*b^13*d^5 + 10492051456*A^4*a^49*b^11*d^5 - 4917821440*A^4*a^51*b^9*d^5
- 923009024*A^4*a^53*b^7*d^5 + 8388608*A^4*a^55*b^5*d^5) - (((480*A^4*a^2*b^10*d^4 - 16*A^4*b^12*d^4 - 16*A^4
*a^12*d^4 - 4080*A^4*a^4*b^8*d^4 + 7232*A^4*a^6*b^6*d^4 - 4080*A^4*a^8*b^4*d^4 + 480*A^4*a^10*b^2*d^4)^(1/2) +
80*A^2*a^3*b^3*d^2 - 24*A^2*a*b^5*d^2 - 24*A^2*a^5*b*d^2)/(16*a^12*d^4 + 16*b^12*d^4 + 96*a^2*b^10*d^4 + 240*
a^4*b^8*d^4 + 320*a^6*b^6*d^4 + 240*a^8*b^4*d^4 + 96*a^10*b^2*d^4))^(1/2)*(((((480*A^4*a^2*b^10*d^4 - 16*A^4*b
^12*d^4 - 16*A^4*a^12*d^4 - 4080*A^4*a^4*b^8*d^4 + 7232*A^4*a^6*b^6*d^4 - 4080*A^4*a^8*b^4*d^4 + 480*A^4*a^10*
b^2*d^4)^(1/2) + 80*A^2*a^3*b^3*d^2 - 24*A^2*a*b^5*d^2 - 24*A^2*a^5*b*d^2)/(16*a^12*d^4 + 16*b^12*d^4 + 96*a^2
*b^10*d^4 + 240*a^4*b^8*d^4 + 320*a^6*b^6*d^4 + 240*a^8*b^4*d^4 + 96*a^10*b^2*d^4))^(1/2)*(251658240*A*a^24*b^
45*d^8 - tan(c + d*x)^(1/2)*(((480*A^4*a^2*b^10*d^4 - 16*A^4*b^12*d^4 - 16*A^4*a^12*d^4 - 4080*A^4*a^4*b^8*d^4
+ 7232*A^4*a^6*b^6*d^4 - 4080*A^4*a^8*b^4*d^4 + 480*A^4*a^10*b^2*d^4)^(1/2) + 80*A^2*a^3*b^3*d^2 - 24*A^2*a*b
^5*d^2 - 24*A^2*a^5*b*d^2)/(16*a^12*d^4 + 16*b^12*d^4 + 96*a^2*b^10*d^4 + 240*a^4*b^8*d^4 + 320*a^6*b^6*d^4 +
240*a^8*b^4*d^4 + 96*a^10*b^2*d^4))^(1/2)*(134217728*a^27*b^45*d^9 + 2550136832*a^29*b^43*d^9 + 22817013760*a^
31*b^41*d^9 + 127506841600*a^33*b^39*d^9 + 497276682240*a^35*b^37*d^9 + 1430626762752*a^37*b^35*d^9 + 31213674
82368*a^39*b^33*d^9 + 5202279137280*a^41*b^31*d^9 + 6502848921600*a^43*b^29*d^9 + 5635802398720*a^45*b^27*d^9
+ 2254320959488*a^47*b^25*d^9 - 2254320959488*a^49*b^23*d^9 - 5635802398720*a^51*b^21*d^9 - 6502848921600*a^53
*b^19*d^9 - 5202279137280*a^55*b^17*d^9 - 3121367482368*a^57*b^15*d^9 - 1430626762752*a^59*b^13*d^9 - 49727668
2240*a^61*b^11*d^9 - 127506841600*a^63*b^9*d^9 - 22817013760*a^65*b^7*d^9 - 2550136832*a^67*b^5*d^9 - 13421772
8*a^69*b^3*d^9) + 5049942016*A*a^26*b^43*d^8 + 48368713728*A*a^28*b^41*d^8 + 293819383808*A*a^30*b^39*d^8 + 12
68458192896*A*a^32*b^37*d^8 + 4132731617280*A*a^34*b^35*d^8 + 10531192700928*A*a^36*b^33*d^8 + 21462823993344*
A*a^38*b^31*d^8 + 35469618315264*A*a^40*b^29*d^8 + 47896904859648*A*a^42*b^27*d^8 + 52983958077440*A*a^44*b^25
*d^8 + 47896904859648*A*a^46*b^23*d^8 + 35090285461504*A*a^48*b^21*d^8 + 20487396655104*A*a^50*b^19*d^8 + 9230
622916608*A*a^52*b^17*d^8 + 2994733056000*A*a^54*b^15*d^8 + 565576728576*A*a^56*b^13*d^8 - 18572378112*A*a^58*
b^11*d^8 - 50281316352*A*a^60*b^9*d^8 - 16089350144*A*a^62*b^7*d^8 - 2516582400*A*a^64*b^5*d^8 - 167772160*A*a
^66*b^3*d^8) + tan(c + d*x)^(1/2)*(471859200*A^2*a^22*b^44*d^7 + 9500098560*A^2*a^24*b^42*d^7 + 91857354752*A^
2*a^26*b^40*d^7 + 564502986752*A^2*a^28*b^38*d^7 + 2464648527872*A^2*a^30*b^36*d^7 + 8104469069824*A^2*a^32*b^
34*d^7 + 20769933361152*A^2*a^34*b^32*d^7 + 42351565209600*A^2*a^36*b^30*d^7 + 69534945902592*A^2*a^38*b^28*d^
7 + 92434029608960*A^2*a^40*b^26*d^7 + 99508717355008*A^2*a^42*b^24*d^7 + 86342935511040*A^2*a^44*b^22*d^7 + 5
9767095558144*A^2*a^46*b^20*d^7 + 32432589897728*A^2*a^48*b^18*d^7 + 13411815522304*A^2*a^50*b^16*d^7 + 403045
7708544*A^2*a^52*b^14*d^7 + 805425905664*A^2*a^54*b^12*d^7 + 86608183296*A^2*a^56*b^10*d^7 + 1612709888*A^2*a^
58*b^8*d^7 + 16777216*A^2*a^60*b^6*d^7 + 167772160*A^2*a^62*b^4*d^7 + 16777216*A^2*a^64*b^2*d^7))*(((480*A^4*a
^2*b^10*d^4 - 16*A^4*b^12*d^4 - 16*A^4*a^12*d^4 - 4080*A^4*a^4*b^8*d^4 + 7232*A^4*a^6*b^6*d^4 - 4080*A^4*a^8*b
^4*d^4 + 480*A^4*a^10*b^2*d^4)^(1/2) + 80*A^2*a^3*b^3*d^2 - 24*A^2*a*b^5*d^2 - 24*A^2*a^5*b*d^2)/(16*a^12*d^4
+ 16*b^12*d^4 + 96*a^2*b^10*d^4 + 240*a^4*b^8*d^4 + 320*a^6*b^6*d^4 + 240*a^8*b^4*d^4 + 96*a^10*b^2*d^4))^(1/2
) - 117964800*A^3*a^21*b^42*d^6 - 841482240*A^3*a^23*b^40*d^6 + 3829399552*A^3*a^25*b^38*d^6 + 78068580352*A^3
*a^27*b^36*d^6 + 497438162944*A^3*a^29*b^34*d^6 + 1899895980032*A^3*a^31*b^32*d^6 + 4972695519232*A^3*a^33*b^3
0*d^6 + 9371195015168*A^3*a^35*b^28*d^6 + 12890720436224*A^3*a^37*b^26*d^6 + 12726089809920*A^3*a^39*b^24*d^6
+ 8366961197056*A^3*a^41*b^22*d^6 + 2597662490624*A^3*a^43*b^20*d^6 - 1171836108800*A^3*a^45*b^18*d^6 - 198688
1650688*A^3*a^47*b^16*d^6 - 1237583921152*A^3*a^49*b^14*d^6 - 449507753984*A^3*a^51*b^12*d^6 - 97476149248*A^3
*a^53*b^10*d^6 - 11931222016*A^3*a^55*b^8*d^6 - 1006632960*A^3*a^57*b^6*d^6 - 134217728*A^3*a^59*b^4*d^6 - 838
8608*A^3*a^61*b^2*d^6))*(((480*A^4*a^2*b^10*d^4 - 16*A^4*b^12*d^4 - 16*A^4*a^12*d^4 - 4080*A^4*a^4*b^8*d^4 + 7
232*A^4*a^6*b^6*d^4 - 4080*A^4*a^8*b^4*d^4 + 480*A^4*a^10*b^2*d^4)^(1/2) + 80*A^2*a^3*b^3*d^2 - 24*A^2*a*b^5*d
^2 - 24*A^2*a^5*b*d^2)/(16*a^12*d^4 + 16*b^12*d^4 + 96*a^2*b^10*d^4 + 240*a^4*b^8*d^4 + 320*a^6*b^6*d^4 + 240*
a^8*b^4*d^4 + 96*a^10*b^2*d^4))^(1/2) + 29491200*A^5*a^22*b^35*d^4 + 460062720*A^5*a^24*b^33*d^4 + 3439722496*
A^5*a^26*b^31*d^4 + 16227237888*A^5*a^28*b^29*d^4 + 53669396480*A^5*a^30*b^27*d^4 + 131031367680*A^5*a^32*b^25
*d^4 + 242529730560*A^5*a^34*b^23*d^4 + 344454070272*A^5*a^36*b^21*d^4 + 375993532416*A^5*a^38*b^19*d^4 + 3130
43189760*A^5*a^40*b^17*d^4 + 195253370880*A^5*a^42*b^15*d^4 + 88318935040*A^5*a^44*b^13*d^4 + 27352498176*A^5*
a^46*b^11*d^4 + 5187043328*A^5*a^48*b^9*d^4 + 454164480*A^5*a^50*b^7*d^4)*(((480*A^4*a^2*b^10*d^4 - 16*A^4*b^1
2*d^4 - 16*A^4*a^12*d^4 - 4080*A^4*a^4*b^8*d^4 + 7232*A^4*a^6*b^6*d^4 - 4080*A^4*a^8*b^4*d^4 + 480*A^4*a^10*b^
2*d^4)^(1/2) + 80*A^2*a^3*b^3*d^2 - 24*A^2*a*b^5*d^2 - 24*A^2*a^5*b*d^2)/(16*a^12*d^4 + 16*b^12*d^4 + 96*a^2*b
^10*d^4 + 240*a^4*b^8*d^4 + 320*a^6*b^6*d^4 + 240*a^8*b^4*d^4 + 96*a^10*b^2*d^4))^(1/2) - log((tan(c + d*x)^(1
/2)*(7610564608*A^4*a^27*b^33*d^5 - 597688320*A^4*a^23*b^37*d^5 - 1671430144*A^4*a^25*b^35*d^5 - 58982400*A^4*
a^21*b^39*d^5 + 85774565376*A^4*a^29*b^31*d^5 + 385487994880*A^4*a^31*b^29*d^5 + 1104303620096*A^4*a^33*b^27*d
^5 + 2240523796480*A^4*a^35*b^25*d^5 + 3345249468416*A^4*a^37*b^23*d^5 + 3717287903232*A^4*a^39*b^21*d^5 + 305
3967114240*A^4*a^41*b^19*d^5 + 1807474491392*A^4*a^43*b^17*d^5 + 726513221632*A^4*a^45*b^15*d^5 + 170768990208
*A^4*a^47*b^13*d^5 + 10492051456*A^4*a^49*b^11*d^5 - 4917821440*A^4*a^51*b^9*d^5 - 923009024*A^4*a^53*b^7*d^5
+ 8388608*A^4*a^55*b^5*d^5) - (-((480*A^4*a^2*b^10*d^4 - 16*A^4*b^12*d^4 - 16*A^4*a^12*d^4 - 4080*A^4*a^4*b^8*
d^4 + 7232*A^4*a^6*b^6*d^4 - 4080*A^4*a^8*b^4*d^4 + 480*A^4*a^10*b^2*d^4)^(1/2) - 80*A^2*a^3*b^3*d^2 + 24*A^2*
a*b^5*d^2 + 24*A^2*a^5*b*d^2)/(16*a^12*d^4 + 16*b^12*d^4 + 96*a^2*b^10*d^4 + 240*a^4*b^8*d^4 + 320*a^6*b^6*d^4
+ 240*a^8*b^4*d^4 + 96*a^10*b^2*d^4))^(1/2)*(((-((480*A^4*a^2*b^10*d^4 - 16*A^4*b^12*d^4 - 16*A^4*a^12*d^4 -
4080*A^4*a^4*b^8*d^4 + 7232*A^4*a^6*b^6*d^4 - 4080*A^4*a^8*b^4*d^4 + 480*A^4*a^10*b^2*d^4)^(1/2) - 80*A^2*a^3*
b^3*d^2 + 24*A^2*a*b^5*d^2 + 24*A^2*a^5*b*d^2)/(16*a^12*d^4 + 16*b^12*d^4 + 96*a^2*b^10*d^4 + 240*a^4*b^8*d^4
+ 320*a^6*b^6*d^4 + 240*a^8*b^4*d^4 + 96*a^10*b^2*d^4))^(1/2)*(251658240*A*a^24*b^45*d^8 - tan(c + d*x)^(1/2)*
(-((480*A^4*a^2*b^10*d^4 - 16*A^4*b^12*d^4 - 16*A^4*a^12*d^4 - 4080*A^4*a^4*b^8*d^4 + 7232*A^4*a^6*b^6*d^4 - 4
080*A^4*a^8*b^4*d^4 + 480*A^4*a^10*b^2*d^4)^(1/2) - 80*A^2*a^3*b^3*d^2 + 24*A^2*a*b^5*d^2 + 24*A^2*a^5*b*d^2)/
(16*a^12*d^4 + 16*b^12*d^4 + 96*a^2*b^10*d^4 + 240*a^4*b^8*d^4 + 320*a^6*b^6*d^4 + 240*a^8*b^4*d^4 + 96*a^10*b
^2*d^4))^(1/2)*(134217728*a^27*b^45*d^9 + 2550136832*a^29*b^43*d^9 + 22817013760*a^31*b^41*d^9 + 127506841600*
a^33*b^39*d^9 + 497276682240*a^35*b^37*d^9 + 1430626762752*a^37*b^35*d^9 + 3121367482368*a^39*b^33*d^9 + 52022
79137280*a^41*b^31*d^9 + 6502848921600*a^43*b^29*d^9 + 5635802398720*a^45*b^27*d^9 + 2254320959488*a^47*b^25*d
^9 - 2254320959488*a^49*b^23*d^9 - 5635802398720*a^51*b^21*d^9 - 6502848921600*a^53*b^19*d^9 - 5202279137280*a
^55*b^17*d^9 - 3121367482368*a^57*b^15*d^9 - 1430626762752*a^59*b^13*d^9 - 497276682240*a^61*b^11*d^9 - 127506
841600*a^63*b^9*d^9 - 22817013760*a^65*b^7*d^9 - 2550136832*a^67*b^5*d^9 - 134217728*a^69*b^3*d^9) + 504994201
6*A*a^26*b^43*d^8 + 48368713728*A*a^28*b^41*d^8 + 293819383808*A*a^30*b^39*d^8 + 1268458192896*A*a^32*b^37*d^8
+ 4132731617280*A*a^34*b^35*d^8 + 10531192700928*A*a^36*b^33*d^8 + 21462823993344*A*a^38*b^31*d^8 + 354696183
15264*A*a^40*b^29*d^8 + 47896904859648*A*a^42*b^27*d^8 + 52983958077440*A*a^44*b^25*d^8 + 47896904859648*A*a^4
6*b^23*d^8 + 35090285461504*A*a^48*b^21*d^8 + 20487396655104*A*a^50*b^19*d^8 + 9230622916608*A*a^52*b^17*d^8 +
2994733056000*A*a^54*b^15*d^8 + 565576728576*A*a^56*b^13*d^8 - 18572378112*A*a^58*b^11*d^8 - 50281316352*A*a^
60*b^9*d^8 - 16089350144*A*a^62*b^7*d^8 - 2516582400*A*a^64*b^5*d^8 - 167772160*A*a^66*b^3*d^8) + tan(c + d*x)
^(1/2)*(471859200*A^2*a^22*b^44*d^7 + 9500098560*A^2*a^24*b^42*d^7 + 91857354752*A^2*a^26*b^40*d^7 + 564502986
752*A^2*a^28*b^38*d^7 + 2464648527872*A^2*a^30*b^36*d^7 + 8104469069824*A^2*a^32*b^34*d^7 + 20769933361152*A^2
*a^34*b^32*d^7 + 42351565209600*A^2*a^36*b^30*d^7 + 69534945902592*A^2*a^38*b^28*d^7 + 92434029608960*A^2*a^40
*b^26*d^7 + 99508717355008*A^2*a^42*b^24*d^7 + 86342935511040*A^2*a^44*b^22*d^7 + 59767095558144*A^2*a^46*b^20
*d^7 + 32432589897728*A^2*a^48*b^18*d^7 + 13411815522304*A^2*a^50*b^16*d^7 + 4030457708544*A^2*a^52*b^14*d^7 +
805425905664*A^2*a^54*b^12*d^7 + 86608183296*A^2*a^56*b^10*d^7 + 1612709888*A^2*a^58*b^8*d^7 + 16777216*A^2*a
^60*b^6*d^7 + 167772160*A^2*a^62*b^4*d^7 + 16777216*A^2*a^64*b^2*d^7))*(-((480*A^4*a^2*b^10*d^4 - 16*A^4*b^12*
d^4 - 16*A^4*a^12*d^4 - 4080*A^4*a^4*b^8*d^4 + 7232*A^4*a^6*b^6*d^4 - 4080*A^4*a^8*b^4*d^4 + 480*A^4*a^10*b^2*
d^4)^(1/2) - 80*A^2*a^3*b^3*d^2 + 24*A^2*a*b^5*d^2 + 24*A^2*a^5*b*d^2)/(16*a^12*d^4 + 16*b^12*d^4 + 96*a^2*b^1
0*d^4 + 240*a^4*b^8*d^4 + 320*a^6*b^6*d^4 + 240*a^8*b^4*d^4 + 96*a^10*b^2*d^4))^(1/2) - 117964800*A^3*a^21*b^4
2*d^6 - 841482240*A^3*a^23*b^40*d^6 + 3829399552*A^3*a^25*b^38*d^6 + 78068580352*A^3*a^27*b^36*d^6 + 497438162
944*A^3*a^29*b^34*d^6 + 1899895980032*A^3*a^31*b^32*d^6 + 4972695519232*A^3*a^33*b^30*d^6 + 9371195015168*A^3*
a^35*b^28*d^6 + 12890720436224*A^3*a^37*b^26*d^6 + 12726089809920*A^3*a^39*b^24*d^6 + 8366961197056*A^3*a^41*b
^22*d^6 + 2597662490624*A^3*a^43*b^20*d^6 - 1171836108800*A^3*a^45*b^18*d^6 - 1986881650688*A^3*a^47*b^16*d^6
- 1237583921152*A^3*a^49*b^14*d^6 - 449507753984*A^3*a^51*b^12*d^6 - 97476149248*A^3*a^53*b^10*d^6 - 119312220
16*A^3*a^55*b^8*d^6 - 1006632960*A^3*a^57*b^6*d^6 - 134217728*A^3*a^59*b^4*d^6 - 8388608*A^3*a^61*b^2*d^6))*(-
((480*A^4*a^2*b^10*d^4 - 16*A^4*b^12*d^4 - 16*A^4*a^12*d^4 - 4080*A^4*a^4*b^8*d^4 + 7232*A^4*a^6*b^6*d^4 - 408
0*A^4*a^8*b^4*d^4 + 480*A^4*a^10*b^2*d^4)^(1/2) - 80*A^2*a^3*b^3*d^2 + 24*A^2*a*b^5*d^2 + 24*A^2*a^5*b*d^2)/(1
6*a^12*d^4 + 16*b^12*d^4 + 96*a^2*b^10*d^4 + 240*a^4*b^8*d^4 + 320*a^6*b^6*d^4 + 240*a^8*b^4*d^4 + 96*a^10*b^2
*d^4))^(1/2) + 29491200*A^5*a^22*b^35*d^4 + 460062720*A^5*a^24*b^33*d^4 + 3439722496*A^5*a^26*b^31*d^4 + 16227
237888*A^5*a^28*b^29*d^4 + 53669396480*A^5*a^30*b^27*d^4 + 131031367680*A^5*a^32*b^25*d^4 + 242529730560*A^5*a
^34*b^23*d^4 + 344454070272*A^5*a^36*b^21*d^4 + 375993532416*A^5*a^38*b^19*d^4 + 313043189760*A^5*a^40*b^17*d^
4 + 195253370880*A^5*a^42*b^15*d^4 + 88318935040*A^5*a^44*b^13*d^4 + 27352498176*A^5*a^46*b^11*d^4 + 518704332
8*A^5*a^48*b^9*d^4 + 454164480*A^5*a^50*b^7*d^4)*(-((480*A^4*a^2*b^10*d^4 - 16*A^4*b^12*d^4 - 16*A^4*a^12*d^4
- 4080*A^4*a^4*b^8*d^4 + 7232*A^4*a^6*b^6*d^4 - 4080*A^4*a^8*b^4*d^4 + 480*A^4*a^10*b^2*d^4)^(1/2) - 80*A^2*a^
3*b^3*d^2 + 24*A^2*a*b^5*d^2 + 24*A^2*a^5*b*d^2)/(16*a^12*d^4 + 16*b^12*d^4 + 96*a^2*b^10*d^4 + 240*a^4*b^8*d^
4 + 320*a^6*b^6*d^4 + 240*a^8*b^4*d^4 + 96*a^10*b^2*d^4))^(1/2) + (log((((((((((64*B*b^2*(3*b^6 - 2*a^6 + 3*a^
2*b^4 + 22*a^4*b^2))/(a^2*d) + 128*b^3*tan(c + d*x)^(1/2)*(a^2 - b^2)*(a^2 + b^2)^2*((4*(-B^4*d^4*(a^6 - b^6 +
15*a^2*b^4 - 15*a^4*b^2)^2)^(1/2) - 80*B^2*a^3*b^3*d^2 + 24*B^2*a*b^5*d^2 + 24*B^2*a^5*b*d^2)/(d^4*(a^2 + b^2
)^6))^(1/2))*((4*(-B^4*d^4*(a^6 - b^6 + 15*a^2*b^4 - 15*a^4*b^2)^2)^(1/2) - 80*B^2*a^3*b^3*d^2 + 24*B^2*a*b^5*
d^2 + 24*B^2*a^5*b*d^2)/(d^4*(a^2 + b^2)^6))^(1/2))/4 + (8*B^2*b^2*tan(c + d*x)^(1/2)*(9*b^12 - 8*a^12 + 36*a^
2*b^10 + 430*a^4*b^8 - 188*a^6*b^6 + 1497*a^8*b^4 + 32*a^10*b^2))/(a^3*d^2*(a^2 + b^2)^4))*((4*(-B^4*d^4*(a^6
- b^6 + 15*a^2*b^4 - 15*a^4*b^2)^2)^(1/2) - 80*B^2*a^3*b^3*d^2 + 24*B^2*a*b^5*d^2 + 24*B^2*a^5*b*d^2)/(d^4*(a^
2 + b^2)^6))^(1/2))/4 - (2*B^3*b^3*(45*b^12 - 16*a^12 + 333*a^2*b^10 + 146*a^4*b^8 + 1178*a^6*b^6 - 9791*a^8*b
^4 + 1161*a^10*b^2))/(a^3*d^3*(a^2 + b^2)^6))*((4*(-B^4*d^4*(a^6 - b^6 + 15*a^2*b^4 - 15*a^4*b^2)^2)^(1/2) - 8
0*B^2*a^3*b^3*d^2 + 24*B^2*a*b^5*d^2 + 24*B^2*a^5*b*d^2)/(d^4*(a^2 + b^2)^6))^(1/2))/4 + (B^4*b^5*tan(c + d*x)
^(1/2)*(18*a^2*b^10 - 9*b^12 - 1257*a^12 - 71*a^4*b^8 + 892*a^6*b^6 + 857*a^8*b^4 + 6802*a^10*b^2))/(a^4*d^4*(
a^2 + b^2)^8))*((4*(-B^4*d^4*(a^6 - b^6 + 15*a^2*b^4 - 15*a^4*b^2)^2)^(1/2) - 80*B^2*a^3*b^3*d^2 + 24*B^2*a*b^
5*d^2 + 24*B^2*a^5*b*d^2)/(d^4*(a^2 + b^2)^6))^(1/2))/4 + (B^5*b^6*(1505*a^8 + 9*b^8 + 60*a^2*b^6 + 318*a^4*b^
4 + 748*a^6*b^2))/(2*a^4*d^5*(a^2 + b^2)^8))*(((480*B^4*a^2*b^10*d^4 - 16*B^4*b^12*d^4 - 16*B^4*a^12*d^4 - 408
0*B^4*a^4*b^8*d^4 + 7232*B^4*a^6*b^6*d^4 - 4080*B^4*a^8*b^4*d^4 + 480*B^4*a^10*b^2*d^4)^(1/2) - 80*B^2*a^3*b^3
*d^2 + 24*B^2*a*b^5*d^2 + 24*B^2*a^5*b*d^2)/(a^12*d^4 + b^12*d^4 + 6*a^2*b^10*d^4 + 15*a^4*b^8*d^4 + 20*a^6*b^
6*d^4 + 15*a^8*b^4*d^4 + 6*a^10*b^2*d^4))^(1/2))/4 + (log((((((((((64*B*b^2*(3*b^6 - 2*a^6 + 3*a^2*b^4 + 22*a^
4*b^2))/(a^2*d) + 128*b^3*tan(c + d*x)^(1/2)*(a^2 - b^2)*(a^2 + b^2)^2*(-(4*(-B^4*d^4*(a^6 - b^6 + 15*a^2*b^4
- 15*a^4*b^2)^2)^(1/2) + 80*B^2*a^3*b^3*d^2 - 24*B^2*a*b^5*d^2 - 24*B^2*a^5*b*d^2)/(d^4*(a^2 + b^2)^6))^(1/2))
*(-(4*(-B^4*d^4*(a^6 - b^6 + 15*a^2*b^4 - 15*a^4*b^2)^2)^(1/2) + 80*B^2*a^3*b^3*d^2 - 24*B^2*a*b^5*d^2 - 24*B^
2*a^5*b*d^2)/(d^4*(a^2 + b^2)^6))^(1/2))/4 + (8*B^2*b^2*tan(c + d*x)^(1/2)*(9*b^12 - 8*a^12 + 36*a^2*b^10 + 43
0*a^4*b^8 - 188*a^6*b^6 + 1497*a^8*b^4 + 32*a^10*b^2))/(a^3*d^2*(a^2 + b^2)^4))*(-(4*(-B^4*d^4*(a^6 - b^6 + 15
*a^2*b^4 - 15*a^4*b^2)^2)^(1/2) + 80*B^2*a^3*b^3*d^2 - 24*B^2*a*b^5*d^2 - 24*B^2*a^5*b*d^2)/(d^4*(a^2 + b^2)^6
))^(1/2))/4 - (2*B^3*b^3*(45*b^12 - 16*a^12 + 333*a^2*b^10 + 146*a^4*b^8 + 1178*a^6*b^6 - 9791*a^8*b^4 + 1161*
a^10*b^2))/(a^3*d^3*(a^2 + b^2)^6))*(-(4*(-B^4*d^4*(a^6 - b^6 + 15*a^2*b^4 - 15*a^4*b^2)^2)^(1/2) + 80*B^2*a^3
*b^3*d^2 - 24*B^2*a*b^5*d^2 - 24*B^2*a^5*b*d^2)/(d^4*(a^2 + b^2)^6))^(1/2))/4 + (B^4*b^5*tan(c + d*x)^(1/2)*(1
8*a^2*b^10 - 9*b^12 - 1257*a^12 - 71*a^4*b^8 + 892*a^6*b^6 + 857*a^8*b^4 + 6802*a^10*b^2))/(a^4*d^4*(a^2 + b^2
)^8))*(-(4*(-B^4*d^4*(a^6 - b^6 + 15*a^2*b^4 - 15*a^4*b^2)^2)^(1/2) + 80*B^2*a^3*b^3*d^2 - 24*B^2*a*b^5*d^2 -
24*B^2*a^5*b*d^2)/(d^4*(a^2 + b^2)^6))^(1/2))/4 + (B^5*b^6*(1505*a^8 + 9*b^8 + 60*a^2*b^6 + 318*a^4*b^4 + 748*
a^6*b^2))/(2*a^4*d^5*(a^2 + b^2)^8))*(-((480*B^4*a^2*b^10*d^4 - 16*B^4*b^12*d^4 - 16*B^4*a^12*d^4 - 4080*B^4*a
^4*b^8*d^4 + 7232*B^4*a^6*b^6*d^4 - 4080*B^4*a^8*b^4*d^4 + 480*B^4*a^10*b^2*d^4)^(1/2) + 80*B^2*a^3*b^3*d^2 -
24*B^2*a*b^5*d^2 - 24*B^2*a^5*b*d^2)/(a^12*d^4 + b^12*d^4 + 6*a^2*b^10*d^4 + 15*a^4*b^8*d^4 + 20*a^6*b^6*d^4 +
15*a^8*b^4*d^4 + 6*a^10*b^2*d^4))^(1/2))/4 - log((((((((((64*B*b^2*(3*b^6 - 2*a^6 + 3*a^2*b^4 + 22*a^4*b^2))/
(a^2*d) - 128*b^3*tan(c + d*x)^(1/2)*(a^2 - b^2)*(a^2 + b^2)^2*((4*(-B^4*d^4*(a^6 - b^6 + 15*a^2*b^4 - 15*a^4*
b^2)^2)^(1/2) - 80*B^2*a^3*b^3*d^2 + 24*B^2*a*b^5*d^2 + 24*B^2*a^5*b*d^2)/(d^4*(a^2 + b^2)^6))^(1/2))*((4*(-B^
4*d^4*(a^6 - b^6 + 15*a^2*b^4 - 15*a^4*b^2)^2)^(1/2) - 80*B^2*a^3*b^3*d^2 + 24*B^2*a*b^5*d^2 + 24*B^2*a^5*b*d^
2)/(d^4*(a^2 + b^2)^6))^(1/2))/4 - (8*B^2*b^2*tan(c + d*x)^(1/2)*(9*b^12 - 8*a^12 + 36*a^2*b^10 + 430*a^4*b^8
- 188*a^6*b^6 + 1497*a^8*b^4 + 32*a^10*b^2))/(a^3*d^2*(a^2 + b^2)^4))*((4*(-B^4*d^4*(a^6 - b^6 + 15*a^2*b^4 -
15*a^4*b^2)^2)^(1/2) - 80*B^2*a^3*b^3*d^2 + 24*B^2*a*b^5*d^2 + 24*B^2*a^5*b*d^2)/(d^4*(a^2 + b^2)^6))^(1/2))/4
- (2*B^3*b^3*(45*b^12 - 16*a^12 + 333*a^2*b^10 + 146*a^4*b^8 + 1178*a^6*b^6 - 9791*a^8*b^4 + 1161*a^10*b^2))/
(a^3*d^3*(a^2 + b^2)^6))*((4*(-B^4*d^4*(a^6 - b^6 + 15*a^2*b^4 - 15*a^4*b^2)^2)^(1/2) - 80*B^2*a^3*b^3*d^2 + 2
4*B^2*a*b^5*d^2 + 24*B^2*a^5*b*d^2)/(d^4*(a^2 + b^2)^6))^(1/2))/4 - (B^4*b^5*tan(c + d*x)^(1/2)*(18*a^2*b^10 -
9*b^12 - 1257*a^12 - 71*a^4*b^8 + 892*a^6*b^6 + 857*a^8*b^4 + 6802*a^10*b^2))/(a^4*d^4*(a^2 + b^2)^8))*((4*(-
B^4*d^4*(a^6 - b^6 + 15*a^2*b^4 - 15*a^4*b^2)^2)^(1/2) - 80*B^2*a^3*b^3*d^2 + 24*B^2*a*b^5*d^2 + 24*B^2*a^5*b*
d^2)/(d^4*(a^2 + b^2)^6))^(1/2))/4 + (B^5*b^6*(1505*a^8 + 9*b^8 + 60*a^2*b^6 + 318*a^4*b^4 + 748*a^6*b^2))/(2*
a^4*d^5*(a^2 + b^2)^8))*(((480*B^4*a^2*b^10*d^4 - 16*B^4*b^12*d^4 - 16*B^4*a^12*d^4 - 4080*B^4*a^4*b^8*d^4 + 7
232*B^4*a^6*b^6*d^4 - 4080*B^4*a^8*b^4*d^4 + 480*B^4*a^10*b^2*d^4)^(1/2) - 80*B^2*a^3*b^3*d^2 + 24*B^2*a*b^5*d
^2 + 24*B^2*a^5*b*d^2)/(16*a^12*d^4 + 16*b^12*d^4 + 96*a^2*b^10*d^4 + 240*a^4*b^8*d^4 + 320*a^6*b^6*d^4 + 240*
a^8*b^4*d^4 + 96*a^10*b^2*d^4))^(1/2) - log((((((((((64*B*b^2*(3*b^6 - 2*a^6 + 3*a^2*b^4 + 22*a^4*b^2))/(a^2*d
) - 128*b^3*tan(c + d*x)^(1/2)*(a^2 - b^2)*(a^2 + b^2)^2*(-(4*(-B^4*d^4*(a^6 - b^6 + 15*a^2*b^4 - 15*a^4*b^2)^
2)^(1/2) + 80*B^2*a^3*b^3*d^2 - 24*B^2*a*b^5*d^2 - 24*B^2*a^5*b*d^2)/(d^4*(a^2 + b^2)^6))^(1/2))*(-(4*(-B^4*d^
4*(a^6 - b^6 + 15*a^2*b^4 - 15*a^4*b^2)^2)^(1/2) + 80*B^2*a^3*b^3*d^2 - 24*B^2*a*b^5*d^2 - 24*B^2*a^5*b*d^2)/(
d^4*(a^2 + b^2)^6))^(1/2))/4 - (8*B^2*b^2*tan(c + d*x)^(1/2)*(9*b^12 - 8*a^12 + 36*a^2*b^10 + 430*a^4*b^8 - 18
8*a^6*b^6 + 1497*a^8*b^4 + 32*a^10*b^2))/(a^3*d^2*(a^2 + b^2)^4))*(-(4*(-B^4*d^4*(a^6 - b^6 + 15*a^2*b^4 - 15*
a^4*b^2)^2)^(1/2) + 80*B^2*a^3*b^3*d^2 - 24*B^2*a*b^5*d^2 - 24*B^2*a^5*b*d^2)/(d^4*(a^2 + b^2)^6))^(1/2))/4 -
(2*B^3*b^3*(45*b^12 - 16*a^12 + 333*a^2*b^10 + 146*a^4*b^8 + 1178*a^6*b^6 - 9791*a^8*b^4 + 1161*a^10*b^2))/(a^
3*d^3*(a^2 + b^2)^6))*(-(4*(-B^4*d^4*(a^6 - b^6 + 15*a^2*b^4 - 15*a^4*b^2)^2)^(1/2) + 80*B^2*a^3*b^3*d^2 - 24*
B^2*a*b^5*d^2 - 24*B^2*a^5*b*d^2)/(d^4*(a^2 + b^2)^6))^(1/2))/4 - (B^4*b^5*tan(c + d*x)^(1/2)*(18*a^2*b^10 - 9
*b^12 - 1257*a^12 - 71*a^4*b^8 + 892*a^6*b^6 + 857*a^8*b^4 + 6802*a^10*b^2))/(a^4*d^4*(a^2 + b^2)^8))*(-(4*(-B
^4*d^4*(a^6 - b^6 + 15*a^2*b^4 - 15*a^4*b^2)^2)^(1/2) + 80*B^2*a^3*b^3*d^2 - 24*B^2*a*b^5*d^2 - 24*B^2*a^5*b*d
^2)/(d^4*(a^2 + b^2)^6))^(1/2))/4 + (B^5*b^6*(1505*a^8 + 9*b^8 + 60*a^2*b^6 + 318*a^4*b^4 + 748*a^6*b^2))/(2*a
^4*d^5*(a^2 + b^2)^8))*(-((480*B^4*a^2*b^10*d^4 - 16*B^4*b^12*d^4 - 16*B^4*a^12*d^4 - 4080*B^4*a^4*b^8*d^4 + 7
232*B^4*a^6*b^6*d^4 - 4080*B^4*a^8*b^4*d^4 + 480*B^4*a^10*b^2*d^4)^(1/2) + 80*B^2*a^3*b^3*d^2 - 24*B^2*a*b^5*d
^2 - 24*B^2*a^5*b*d^2)/(16*a^12*d^4 + 16*b^12*d^4 + 96*a^2*b^10*d^4 + 240*a^4*b^8*d^4 + 320*a^6*b^6*d^4 + 240*
a^8*b^4*d^4 + 96*a^10*b^2*d^4))^(1/2) - (atan(((((((846*B^3*a^3*b^17*d^2 + 1714*B^3*a^5*b^15*d^2 + 3606*B^3*a^
7*b^13*d^2 - 14578*B^3*a^9*b^11*d^2 - 34486*B^3*a^11*b^9*d^2 - 14970*B^3*a^13*b^7*d^2 + 2258*B^3*a^15*b^5*d^2
- 32*B^3*a^17*b^3*d^2 + 90*B^3*a*b^19*d^2)/(64*(a^20*d^5 + a^4*b^16*d^5 + 8*a^6*b^14*d^5 + 28*a^8*b^12*d^5 + 5
6*a^10*b^10*d^5 + 70*a^12*b^8*d^5 + 56*a^14*b^6*d^5 + 28*a^16*b^4*d^5 + 8*a^18*b^2*d^5)) - (((((192*B*a^2*b^24
*d^4 + 1728*B*a^4*b^22*d^4 + 8320*B*a^6*b^20*d^4 + 27264*B*a^8*b^18*d^4 + 62592*B*a^10*b^16*d^4 + 99456*B*a^12
*b^14*d^4 + 107520*B*a^14*b^12*d^4 + 76800*B*a^16*b^10*d^4 + 33984*B*a^18*b^8*d^4 + 7872*B*a^20*b^6*d^4 + 384*
B*a^22*b^4*d^4 - 128*B*a^24*b^2*d^4)/(64*(a^20*d^5 + a^4*b^16*d^5 + 8*a^6*b^14*d^5 + 28*a^8*b^12*d^5 + 56*a^10
*b^10*d^5 + 70*a^12*b^8*d^5 + 56*a^14*b^6*d^5 + 28*a^16*b^4*d^5 + 8*a^18*b^2*d^5)) - (tan(c + d*x)^(1/2)*(-64*
(9*B^2*b^11 + 36*B^2*a^2*b^9 + 246*B^2*a^4*b^7 + 420*B^2*a^6*b^5 + 1225*B^2*a^8*b^3)*(a^17*d^2 + a^5*b^12*d^2
+ 6*a^7*b^10*d^2 + 15*a^9*b^8*d^2 + 20*a^11*b^6*d^2 + 15*a^13*b^4*d^2 + 6*a^15*b^2*d^2))^(1/2)*(512*a^4*b^25*d
^4 + 4608*a^6*b^23*d^4 + 17920*a^8*b^21*d^4 + 38400*a^10*b^19*d^4 + 46080*a^12*b^17*d^4 + 21504*a^14*b^15*d^4
- 21504*a^16*b^13*d^4 - 46080*a^18*b^11*d^4 - 38400*a^20*b^9*d^4 - 17920*a^22*b^7*d^4 - 4608*a^24*b^5*d^4 - 51
2*a^26*b^3*d^4))/(4096*(a^17*d^2 + a^5*b^12*d^2 + 6*a^7*b^10*d^2 + 15*a^9*b^8*d^2 + 20*a^11*b^6*d^2 + 15*a^13*
b^4*d^2 + 6*a^15*b^2*d^2)*(a^20*d^4 + a^4*b^16*d^4 + 8*a^6*b^14*d^4 + 28*a^8*b^12*d^4 + 56*a^10*b^10*d^4 + 70*
a^12*b^8*d^4 + 56*a^14*b^6*d^4 + 28*a^16*b^4*d^4 + 8*a^18*b^2*d^4)))*(-64*(9*B^2*b^11 + 36*B^2*a^2*b^9 + 246*B
^2*a^4*b^7 + 420*B^2*a^6*b^5 + 1225*B^2*a^8*b^3)*(a^17*d^2 + a^5*b^12*d^2 + 6*a^7*b^10*d^2 + 15*a^9*b^8*d^2 +
20*a^11*b^6*d^2 + 15*a^13*b^4*d^2 + 6*a^15*b^2*d^2))^(1/2))/(64*(a^17*d^2 + a^5*b^12*d^2 + 6*a^7*b^10*d^2 + 15
*a^9*b^8*d^2 + 20*a^11*b^6*d^2 + 15*a^13*b^4*d^2 + 6*a^15*b^2*d^2)) + (tan(c + d*x)^(1/2)*(576*B^2*a^3*b^20*d^
2 + 5024*B^2*a^5*b^18*d^2 + 14272*B^2*a^7*b^16*d^2 + 27824*B^2*a^9*b^14*d^2 + 53184*B^2*a^11*b^12*d^2 + 70240*
B^2*a^13*b^10*d^2 + 47680*B^2*a^15*b^8*d^2 + 12616*B^2*a^17*b^6*d^2 - 64*B^2*a^21*b^2*d^2 + 72*B^2*a*b^22*d^2)
)/(64*(a^20*d^4 + a^4*b^16*d^4 + 8*a^6*b^14*d^4 + 28*a^8*b^12*d^4 + 56*a^10*b^10*d^4 + 70*a^12*b^8*d^4 + 56*a^
14*b^6*d^4 + 28*a^16*b^4*d^4 + 8*a^18*b^2*d^4)))*(-64*(9*B^2*b^11 + 36*B^2*a^2*b^9 + 246*B^2*a^4*b^7 + 420*B^2
*a^6*b^5 + 1225*B^2*a^8*b^3)*(a^17*d^2 + a^5*b^12*d^2 + 6*a^7*b^10*d^2 + 15*a^9*b^8*d^2 + 20*a^11*b^6*d^2 + 15
*a^13*b^4*d^2 + 6*a^15*b^2*d^2))^(1/2))/(64*(a^17*d^2 + a^5*b^12*d^2 + 6*a^7*b^10*d^2 + 15*a^9*b^8*d^2 + 20*a^
11*b^6*d^2 + 15*a^13*b^4*d^2 + 6*a^15*b^2*d^2)))*(-64*(9*B^2*b^11 + 36*B^2*a^2*b^9 + 246*B^2*a^4*b^7 + 420*B^2
*a^6*b^5 + 1225*B^2*a^8*b^3)*(a^17*d^2 + a^5*b^12*d^2 + 6*a^7*b^10*d^2 + 15*a^9*b^8*d^2 + 20*a^11*b^6*d^2 + 15
*a^13*b^4*d^2 + 6*a^15*b^2*d^2))^(1/2))/(64*(a^17*d^2 + a^5*b^12*d^2 + 6*a^7*b^10*d^2 + 15*a^9*b^8*d^2 + 20*a^
11*b^6*d^2 + 15*a^13*b^4*d^2 + 6*a^15*b^2*d^2)) - (tan(c + d*x)^(1/2)*(18*B^4*a^2*b^15 - 9*B^4*b^17 - 71*B^4*a
^4*b^13 + 892*B^4*a^6*b^11 + 857*B^4*a^8*b^9 + 6802*B^4*a^10*b^7 - 1257*B^4*a^12*b^5))/(64*(a^20*d^4 + a^4*b^1
6*d^4 + 8*a^6*b^14*d^4 + 28*a^8*b^12*d^4 + 56*a^10*b^10*d^4 + 70*a^12*b^8*d^4 + 56*a^14*b^6*d^4 + 28*a^16*b^4*
d^4 + 8*a^18*b^2*d^4)))*(-64*(9*B^2*b^11 + 36*B^2*a^2*b^9 + 246*B^2*a^4*b^7 + 420*B^2*a^6*b^5 + 1225*B^2*a^8*b
^3)*(a^17*d^2 + a^5*b^12*d^2 + 6*a^7*b^10*d^2 + 15*a^9*b^8*d^2 + 20*a^11*b^6*d^2 + 15*a^13*b^4*d^2 + 6*a^15*b^
2*d^2))^(1/2)*1i)/(a^17*d^2 + a^5*b^12*d^2 + 6*a^7*b^10*d^2 + 15*a^9*b^8*d^2 + 20*a^11*b^6*d^2 + 15*a^13*b^4*d
^2 + 6*a^15*b^2*d^2) - (((((846*B^3*a^3*b^17*d^2 + 1714*B^3*a^5*b^15*d^2 + 3606*B^3*a^7*b^13*d^2 - 14578*B^3*a
^9*b^11*d^2 - 34486*B^3*a^11*b^9*d^2 - 14970*B^3*a^13*b^7*d^2 + 2258*B^3*a^15*b^5*d^2 - 32*B^3*a^17*b^3*d^2 +
90*B^3*a*b^19*d^2)/(64*(a^20*d^5 + a^4*b^16*d^5 + 8*a^6*b^14*d^5 + 28*a^8*b^12*d^5 + 56*a^10*b^10*d^5 + 70*a^1
2*b^8*d^5 + 56*a^14*b^6*d^5 + 28*a^16*b^4*d^5 + 8*a^18*b^2*d^5)) - (((((192*B*a^2*b^24*d^4 + 1728*B*a^4*b^22*d
^4 + 8320*B*a^6*b^20*d^4 + 27264*B*a^8*b^18*d^4 + 62592*B*a^10*b^16*d^4 + 99456*B*a^12*b^14*d^4 + 107520*B*a^1
4*b^12*d^4 + 76800*B*a^16*b^10*d^4 + 33984*B*a^18*b^8*d^4 + 7872*B*a^20*b^6*d^4 + 384*B*a^22*b^4*d^4 - 128*B*a
^24*b^2*d^4)/(64*(a^20*d^5 + a^4*b^16*d^5 + 8*a^6*b^14*d^5 + 28*a^8*b^12*d^5 + 56*a^10*b^10*d^5 + 70*a^12*b^8*
d^5 + 56*a^14*b^6*d^5 + 28*a^16*b^4*d^5 + 8*a^18*b^2*d^5)) + (tan(c + d*x)^(1/2)*(-64*(9*B^2*b^11 + 36*B^2*a^2
*b^9 + 246*B^2*a^4*b^7 + 420*B^2*a^6*b^5 + 1225*B^2*a^8*b^3)*(a^17*d^2 + a^5*b^12*d^2 + 6*a^7*b^10*d^2 + 15*a^
9*b^8*d^2 + 20*a^11*b^6*d^2 + 15*a^13*b^4*d^2 + 6*a^15*b^2*d^2))^(1/2)*(512*a^4*b^25*d^4 + 4608*a^6*b^23*d^4 +
17920*a^8*b^21*d^4 + 38400*a^10*b^19*d^4 + 46080*a^12*b^17*d^4 + 21504*a^14*b^15*d^4 - 21504*a^16*b^13*d^4 -
46080*a^18*b^11*d^4 - 38400*a^20*b^9*d^4 - 17920*a^22*b^7*d^4 - 4608*a^24*b^5*d^4 - 512*a^26*b^3*d^4))/(4096*(
a^17*d^2 + a^5*b^12*d^2 + 6*a^7*b^10*d^2 + 15*a^9*b^8*d^2 + 20*a^11*b^6*d^2 + 15*a^13*b^4*d^2 + 6*a^15*b^2*d^2
)*(a^20*d^4 + a^4*b^16*d^4 + 8*a^6*b^14*d^4 + 28*a^8*b^12*d^4 + 56*a^10*b^10*d^4 + 70*a^12*b^8*d^4 + 56*a^14*b
^6*d^4 + 28*a^16*b^4*d^4 + 8*a^18*b^2*d^4)))*(-64*(9*B^2*b^11 + 36*B^2*a^2*b^9 + 246*B^2*a^4*b^7 + 420*B^2*a^6
*b^5 + 1225*B^2*a^8*b^3)*(a^17*d^2 + a^5*b^12*d^2 + 6*a^7*b^10*d^2 + 15*a^9*b^8*d^2 + 20*a^11*b^6*d^2 + 15*a^1
3*b^4*d^2 + 6*a^15*b^2*d^2))^(1/2))/(64*(a^17*d^2 + a^5*b^12*d^2 + 6*a^7*b^10*d^2 + 15*a^9*b^8*d^2 + 20*a^11*b
^6*d^2 + 15*a^13*b^4*d^2 + 6*a^15*b^2*d^2)) - (tan(c + d*x)^(1/2)*(576*B^2*a^3*b^20*d^2 + 5024*B^2*a^5*b^18*d^
2 + 14272*B^2*a^7*b^16*d^2 + 27824*B^2*a^9*b^14*d^2 + 53184*B^2*a^11*b^12*d^2 + 70240*B^2*a^13*b^10*d^2 + 4768
0*B^2*a^15*b^8*d^2 + 12616*B^2*a^17*b^6*d^2 - 64*B^2*a^21*b^2*d^2 + 72*B^2*a*b^22*d^2))/(64*(a^20*d^4 + a^4*b^
16*d^4 + 8*a^6*b^14*d^4 + 28*a^8*b^12*d^4 + 56*a^10*b^10*d^4 + 70*a^12*b^8*d^4 + 56*a^14*b^6*d^4 + 28*a^16*b^4
*d^4 + 8*a^18*b^2*d^4)))*(-64*(9*B^2*b^11 + 36*B^2*a^2*b^9 + 246*B^2*a^4*b^7 + 420*B^2*a^6*b^5 + 1225*B^2*a^8*
b^3)*(a^17*d^2 + a^5*b^12*d^2 + 6*a^7*b^10*d^2 + 15*a^9*b^8*d^2 + 20*a^11*b^6*d^2 + 15*a^13*b^4*d^2 + 6*a^15*b
^2*d^2))^(1/2))/(64*(a^17*d^2 + a^5*b^12*d^2 + 6*a^7*b^10*d^2 + 15*a^9*b^8*d^2 + 20*a^11*b^6*d^2 + 15*a^13*b^4
*d^2 + 6*a^15*b^2*d^2)))*(-64*(9*B^2*b^11 + 36*B^2*a^2*b^9 + 246*B^2*a^4*b^7 + 420*B^2*a^6*b^5 + 1225*B^2*a^8*
b^3)*(a^17*d^2 + a^5*b^12*d^2 + 6*a^7*b^10*d^2 + 15*a^9*b^8*d^2 + 20*a^11*b^6*d^2 + 15*a^13*b^4*d^2 + 6*a^15*b
^2*d^2))^(1/2))/(64*(a^17*d^2 + a^5*b^12*d^2 + 6*a^7*b^10*d^2 + 15*a^9*b^8*d^2 + 20*a^11*b^6*d^2 + 15*a^13*b^4
*d^2 + 6*a^15*b^2*d^2)) + (tan(c + d*x)^(1/2)*(18*B^4*a^2*b^15 - 9*B^4*b^17 - 71*B^4*a^4*b^13 + 892*B^4*a^6*b^
11 + 857*B^4*a^8*b^9 + 6802*B^4*a^10*b^7 - 1257*B^4*a^12*b^5))/(64*(a^20*d^4 + a^4*b^16*d^4 + 8*a^6*b^14*d^4 +
28*a^8*b^12*d^4 + 56*a^10*b^10*d^4 + 70*a^12*b^8*d^4 + 56*a^14*b^6*d^4 + 28*a^16*b^4*d^4 + 8*a^18*b^2*d^4)))*
(-64*(9*B^2*b^11 + 36*B^2*a^2*b^9 + 246*B^2*a^4*b^7 + 420*B^2*a^6*b^5 + 1225*B^2*a^8*b^3)*(a^17*d^2 + a^5*b^12
*d^2 + 6*a^7*b^10*d^2 + 15*a^9*b^8*d^2 + 20*a^11*b^6*d^2 + 15*a^13*b^4*d^2 + 6*a^15*b^2*d^2))^(1/2)*1i)/(a^17*
d^2 + a^5*b^12*d^2 + 6*a^7*b^10*d^2 + 15*a^9*b^8*d^2 + 20*a^11*b^6*d^2 + 15*a^13*b^4*d^2 + 6*a^15*b^2*d^2))/((
((((846*B^3*a^3*b^17*d^2 + 1714*B^3*a^5*b^15*d^2 + 3606*B^3*a^7*b^13*d^2 - 14578*B^3*a^9*b^11*d^2 - 34486*B^3*
a^11*b^9*d^2 - 14970*B^3*a^13*b^7*d^2 + 2258*B^3*a^15*b^5*d^2 - 32*B^3*a^17*b^3*d^2 + 90*B^3*a*b^19*d^2)/(64*(
a^20*d^5 + a^4*b^16*d^5 + 8*a^6*b^14*d^5 + 28*a^8*b^12*d^5 + 56*a^10*b^10*d^5 + 70*a^12*b^8*d^5 + 56*a^14*b^6*
d^5 + 28*a^16*b^4*d^5 + 8*a^18*b^2*d^5)) - (((((192*B*a^2*b^24*d^4 + 1728*B*a^4*b^22*d^4 + 8320*B*a^6*b^20*d^4
+ 27264*B*a^8*b^18*d^4 + 62592*B*a^10*b^16*d^4 + 99456*B*a^12*b^14*d^4 + 107520*B*a^14*b^12*d^4 + 76800*B*a^1
6*b^10*d^4 + 33984*B*a^18*b^8*d^4 + 7872*B*a^20*b^6*d^4 + 384*B*a^22*b^4*d^4 - 128*B*a^24*b^2*d^4)/(64*(a^20*d
^5 + a^4*b^16*d^5 + 8*a^6*b^14*d^5 + 28*a^8*b^12*d^5 + 56*a^10*b^10*d^5 + 70*a^12*b^8*d^5 + 56*a^14*b^6*d^5 +
28*a^16*b^4*d^5 + 8*a^18*b^2*d^5)) - (tan(c + d*x)^(1/2)*(-64*(9*B^2*b^11 + 36*B^2*a^2*b^9 + 246*B^2*a^4*b^7 +
420*B^2*a^6*b^5 + 1225*B^2*a^8*b^3)*(a^17*d^2 + a^5*b^12*d^2 + 6*a^7*b^10*d^2 + 15*a^9*b^8*d^2 + 20*a^11*b^6*
d^2 + 15*a^13*b^4*d^2 + 6*a^15*b^2*d^2))^(1/2)*(512*a^4*b^25*d^4 + 4608*a^6*b^23*d^4 + 17920*a^8*b^21*d^4 + 38
400*a^10*b^19*d^4 + 46080*a^12*b^17*d^4 + 21504*a^14*b^15*d^4 - 21504*a^16*b^13*d^4 - 46080*a^18*b^11*d^4 - 38
400*a^20*b^9*d^4 - 17920*a^22*b^7*d^4 - 4608*a^24*b^5*d^4 - 512*a^26*b^3*d^4))/(4096*(a^17*d^2 + a^5*b^12*d^2
+ 6*a^7*b^10*d^2 + 15*a^9*b^8*d^2 + 20*a^11*b^6*d^2 + 15*a^13*b^4*d^2 + 6*a^15*b^2*d^2)*(a^20*d^4 + a^4*b^16*d
^4 + 8*a^6*b^14*d^4 + 28*a^8*b^12*d^4 + 56*a^10*b^10*d^4 + 70*a^12*b^8*d^4 + 56*a^14*b^6*d^4 + 28*a^16*b^4*d^4
+ 8*a^18*b^2*d^4)))*(-64*(9*B^2*b^11 + 36*B^2*a^2*b^9 + 246*B^2*a^4*b^7 + 420*B^2*a^6*b^5 + 1225*B^2*a^8*b^3)
*(a^17*d^2 + a^5*b^12*d^2 + 6*a^7*b^10*d^2 + 15*a^9*b^8*d^2 + 20*a^11*b^6*d^2 + 15*a^13*b^4*d^2 + 6*a^15*b^2*d
^2))^(1/2))/(64*(a^17*d^2 + a^5*b^12*d^2 + 6*a^7*b^10*d^2 + 15*a^9*b^8*d^2 + 20*a^11*b^6*d^2 + 15*a^13*b^4*d^2
+ 6*a^15*b^2*d^2)) + (tan(c + d*x)^(1/2)*(576*B^2*a^3*b^20*d^2 + 5024*B^2*a^5*b^18*d^2 + 14272*B^2*a^7*b^16*d
^2 + 27824*B^2*a^9*b^14*d^2 + 53184*B^2*a^11*b^12*d^2 + 70240*B^2*a^13*b^10*d^2 + 47680*B^2*a^15*b^8*d^2 + 126
16*B^2*a^17*b^6*d^2 - 64*B^2*a^21*b^2*d^2 + 72*B^2*a*b^22*d^2))/(64*(a^20*d^4 + a^4*b^16*d^4 + 8*a^6*b^14*d^4
+ 28*a^8*b^12*d^4 + 56*a^10*b^10*d^4 + 70*a^12*b^8*d^4 + 56*a^14*b^6*d^4 + 28*a^16*b^4*d^4 + 8*a^18*b^2*d^4)))
*(-64*(9*B^2*b^11 + 36*B^2*a^2*b^9 + 246*B^2*a^4*b^7 + 420*B^2*a^6*b^5 + 1225*B^2*a^8*b^3)*(a^17*d^2 + a^5*b^1
2*d^2 + 6*a^7*b^10*d^2 + 15*a^9*b^8*d^2 + 20*a^11*b^6*d^2 + 15*a^13*b^4*d^2 + 6*a^15*b^2*d^2))^(1/2))/(64*(a^1
7*d^2 + a^5*b^12*d^2 + 6*a^7*b^10*d^2 + 15*a^9*b^8*d^2 + 20*a^11*b^6*d^2 + 15*a^13*b^4*d^2 + 6*a^15*b^2*d^2)))
*(-64*(9*B^2*b^11 + 36*B^2*a^2*b^9 + 246*B^2*a^4*b^7 + 420*B^2*a^6*b^5 + 1225*B^2*a^8*b^3)*(a^17*d^2 + a^5*b^1
2*d^2 + 6*a^7*b^10*d^2 + 15*a^9*b^8*d^2 + 20*a^11*b^6*d^2 + 15*a^13*b^4*d^2 + 6*a^15*b^2*d^2))^(1/2))/(64*(a^1
7*d^2 + a^5*b^12*d^2 + 6*a^7*b^10*d^2 + 15*a^9*b^8*d^2 + 20*a^11*b^6*d^2 + 15*a^13*b^4*d^2 + 6*a^15*b^2*d^2))
- (tan(c + d*x)^(1/2)*(18*B^4*a^2*b^15 - 9*B^4*b^17 - 71*B^4*a^4*b^13 + 892*B^4*a^6*b^11 + 857*B^4*a^8*b^9 + 6
802*B^4*a^10*b^7 - 1257*B^4*a^12*b^5))/(64*(a^20*d^4 + a^4*b^16*d^4 + 8*a^6*b^14*d^4 + 28*a^8*b^12*d^4 + 56*a^
10*b^10*d^4 + 70*a^12*b^8*d^4 + 56*a^14*b^6*d^4 + 28*a^16*b^4*d^4 + 8*a^18*b^2*d^4)))*(-64*(9*B^2*b^11 + 36*B^
2*a^2*b^9 + 246*B^2*a^4*b^7 + 420*B^2*a^6*b^5 + 1225*B^2*a^8*b^3)*(a^17*d^2 + a^5*b^12*d^2 + 6*a^7*b^10*d^2 +
15*a^9*b^8*d^2 + 20*a^11*b^6*d^2 + 15*a^13*b^4*d^2 + 6*a^15*b^2*d^2))^(1/2))/(a^17*d^2 + a^5*b^12*d^2 + 6*a^7*
b^10*d^2 + 15*a^9*b^8*d^2 + 20*a^11*b^6*d^2 + 15*a^13*b^4*d^2 + 6*a^15*b^2*d^2) - (9*B^5*b^14 + 60*B^5*a^2*b^1
2 + 318*B^5*a^4*b^10 + 748*B^5*a^6*b^8 + 1505*B^5*a^8*b^6)/(a^20*d^5 + a^4*b^16*d^5 + 8*a^6*b^14*d^5 + 28*a^8*
b^12*d^5 + 56*a^10*b^10*d^5 + 70*a^12*b^8*d^5 + 56*a^14*b^6*d^5 + 28*a^16*b^4*d^5 + 8*a^18*b^2*d^5) + (((((846
*B^3*a^3*b^17*d^2 + 1714*B^3*a^5*b^15*d^2 + 3606*B^3*a^7*b^13*d^2 - 14578*B^3*a^9*b^11*d^2 - 34486*B^3*a^11*b^
9*d^2 - 14970*B^3*a^13*b^7*d^2 + 2258*B^3*a^15*b^5*d^2 - 32*B^3*a^17*b^3*d^2 + 90*B^3*a*b^19*d^2)/(64*(a^20*d^
5 + a^4*b^16*d^5 + 8*a^6*b^14*d^5 + 28*a^8*b^12*d^5 + 56*a^10*b^10*d^5 + 70*a^12*b^8*d^5 + 56*a^14*b^6*d^5 + 2
8*a^16*b^4*d^5 + 8*a^18*b^2*d^5)) - (((((192*B*a^2*b^24*d^4 + 1728*B*a^4*b^22*d^4 + 8320*B*a^6*b^20*d^4 + 2726
4*B*a^8*b^18*d^4 + 62592*B*a^10*b^16*d^4 + 99456*B*a^12*b^14*d^4 + 107520*B*a^14*b^12*d^4 + 76800*B*a^16*b^10*
d^4 + 33984*B*a^18*b^8*d^4 + 7872*B*a^20*b^6*d^4 + 384*B*a^22*b^4*d^4 - 128*B*a^24*b^2*d^4)/(64*(a^20*d^5 + a^
4*b^16*d^5 + 8*a^6*b^14*d^5 + 28*a^8*b^12*d^5 + 56*a^10*b^10*d^5 + 70*a^12*b^8*d^5 + 56*a^14*b^6*d^5 + 28*a^16
*b^4*d^5 + 8*a^18*b^2*d^5)) + (tan(c + d*x)^(1/2)*(-64*(9*B^2*b^11 + 36*B^2*a^2*b^9 + 246*B^2*a^4*b^7 + 420*B^
2*a^6*b^5 + 1225*B^2*a^8*b^3)*(a^17*d^2 + a^5*b^12*d^2 + 6*a^7*b^10*d^2 + 15*a^9*b^8*d^2 + 20*a^11*b^6*d^2 + 1
5*a^13*b^4*d^2 + 6*a^15*b^2*d^2))^(1/2)*(512*a^4*b^25*d^4 + 4608*a^6*b^23*d^4 + 17920*a^8*b^21*d^4 + 38400*a^1
0*b^19*d^4 + 46080*a^12*b^17*d^4 + 21504*a^14*b^15*d^4 - 21504*a^16*b^13*d^4 - 46080*a^18*b^11*d^4 - 38400*a^2
0*b^9*d^4 - 17920*a^22*b^7*d^4 - 4608*a^24*b^5*d^4 - 512*a^26*b^3*d^4))/(4096*(a^17*d^2 + a^5*b^12*d^2 + 6*a^7
*b^10*d^2 + 15*a^9*b^8*d^2 + 20*a^11*b^6*d^2 + 15*a^13*b^4*d^2 + 6*a^15*b^2*d^2)*(a^20*d^4 + a^4*b^16*d^4 + 8*
a^6*b^14*d^4 + 28*a^8*b^12*d^4 + 56*a^10*b^10*d^4 + 70*a^12*b^8*d^4 + 56*a^14*b^6*d^4 + 28*a^16*b^4*d^4 + 8*a^
18*b^2*d^4)))*(-64*(9*B^2*b^11 + 36*B^2*a^2*b^9 + 246*B^2*a^4*b^7 + 420*B^2*a^6*b^5 + 1225*B^2*a^8*b^3)*(a^17*
d^2 + a^5*b^12*d^2 + 6*a^7*b^10*d^2 + 15*a^9*b^8*d^2 + 20*a^11*b^6*d^2 + 15*a^13*b^4*d^2 + 6*a^15*b^2*d^2))^(1
/2))/(64*(a^17*d^2 + a^5*b^12*d^2 + 6*a^7*b^10*d^2 + 15*a^9*b^8*d^2 + 20*a^11*b^6*d^2 + 15*a^13*b^4*d^2 + 6*a^
15*b^2*d^2)) - (tan(c + d*x)^(1/2)*(576*B^2*a^3*b^20*d^2 + 5024*B^2*a^5*b^18*d^2 + 14272*B^2*a^7*b^16*d^2 + 27
824*B^2*a^9*b^14*d^2 + 53184*B^2*a^11*b^12*d^2 + 70240*B^2*a^13*b^10*d^2 + 47680*B^2*a^15*b^8*d^2 + 12616*B^2*
a^17*b^6*d^2 - 64*B^2*a^21*b^2*d^2 + 72*B^2*a*b^22*d^2))/(64*(a^20*d^4 + a^4*b^16*d^4 + 8*a^6*b^14*d^4 + 28*a^
8*b^12*d^4 + 56*a^10*b^10*d^4 + 70*a^12*b^8*d^4 + 56*a^14*b^6*d^4 + 28*a^16*b^4*d^4 + 8*a^18*b^2*d^4)))*(-64*(
9*B^2*b^11 + 36*B^2*a^2*b^9 + 246*B^2*a^4*b^7 + 420*B^2*a^6*b^5 + 1225*B^2*a^8*b^3)*(a^17*d^2 + a^5*b^12*d^2 +
6*a^7*b^10*d^2 + 15*a^9*b^8*d^2 + 20*a^11*b^6*d^2 + 15*a^13*b^4*d^2 + 6*a^15*b^2*d^2))^(1/2))/(64*(a^17*d^2 +
a^5*b^12*d^2 + 6*a^7*b^10*d^2 + 15*a^9*b^8*d^2 + 20*a^11*b^6*d^2 + 15*a^13*b^4*d^2 + 6*a^15*b^2*d^2)))*(-64*(
9*B^2*b^11 + 36*B^2*a^2*b^9 + 246*B^2*a^4*b^7 + 420*B^2*a^6*b^5 + 1225*B^2*a^8*b^3)*(a^17*d^2 + a^5*b^12*d^2 +
6*a^7*b^10*d^2 + 15*a^9*b^8*d^2 + 20*a^11*b^6*d^2 + 15*a^13*b^4*d^2 + 6*a^15*b^2*d^2))^(1/2))/(64*(a^17*d^2 +
a^5*b^12*d^2 + 6*a^7*b^10*d^2 + 15*a^9*b^8*d^2 + 20*a^11*b^6*d^2 + 15*a^13*b^4*d^2 + 6*a^15*b^2*d^2)) + (tan(
c + d*x)^(1/2)*(18*B^4*a^2*b^15 - 9*B^4*b^17 - 71*B^4*a^4*b^13 + 892*B^4*a^6*b^11 + 857*B^4*a^8*b^9 + 6802*B^4
*a^10*b^7 - 1257*B^4*a^12*b^5))/(64*(a^20*d^4 + a^4*b^16*d^4 + 8*a^6*b^14*d^4 + 28*a^8*b^12*d^4 + 56*a^10*b^10
*d^4 + 70*a^12*b^8*d^4 + 56*a^14*b^6*d^4 + 28*a^16*b^4*d^4 + 8*a^18*b^2*d^4)))*(-64*(9*B^2*b^11 + 36*B^2*a^2*b
^9 + 246*B^2*a^4*b^7 + 420*B^2*a^6*b^5 + 1225*B^2*a^8*b^3)*(a^17*d^2 + a^5*b^12*d^2 + 6*a^7*b^10*d^2 + 15*a^9*
b^8*d^2 + 20*a^11*b^6*d^2 + 15*a^13*b^4*d^2 + 6*a^15*b^2*d^2))^(1/2))/(a^17*d^2 + a^5*b^12*d^2 + 6*a^7*b^10*d^
2 + 15*a^9*b^8*d^2 + 20*a^11*b^6*d^2 + 15*a^13*b^4*d^2 + 6*a^15*b^2*d^2)))*(-64*(9*B^2*b^11 + 36*B^2*a^2*b^9 +
246*B^2*a^4*b^7 + 420*B^2*a^6*b^5 + 1225*B^2*a^8*b^3)*(a^17*d^2 + a^5*b^12*d^2 + 6*a^7*b^10*d^2 + 15*a^9*b^8*
d^2 + 20*a^11*b^6*d^2 + 15*a^13*b^4*d^2 + 6*a^15*b^2*d^2))^(1/2)*1i)/(32*(a^17*d^2 + a^5*b^12*d^2 + 6*a^7*b^10
*d^2 + 15*a^9*b^8*d^2 + 20*a^11*b^6*d^2 + 15*a^13*b^4*d^2 + 6*a^15*b^2*d^2)) + (atan(((((tan(c + d*x)^(1/2)*(7
610564608*A^4*a^27*b^33*d^5 - 597688320*A^4*a^23*b^37*d^5 - 1671430144*A^4*a^25*b^35*d^5 - 58982400*A^4*a^21*b
^39*d^5 + 85774565376*A^4*a^29*b^31*d^5 + 385487994880*A^4*a^31*b^29*d^5 + 1104303620096*A^4*a^33*b^27*d^5 + 2
240523796480*A^4*a^35*b^25*d^5 + 3345249468416*A^4*a^37*b^23*d^5 + 3717287903232*A^4*a^39*b^21*d^5 + 305396711
4240*A^4*a^41*b^19*d^5 + 1807474491392*A^4*a^43*b^17*d^5 + 726513221632*A^4*a^45*b^15*d^5 + 170768990208*A^4*a
^47*b^13*d^5 + 10492051456*A^4*a^49*b^11*d^5 - 4917821440*A^4*a^51*b^9*d^5 - 923009024*A^4*a^53*b^7*d^5 + 8388
608*A^4*a^55*b^5*d^5))/64 - ((-64*(225*A^2*b^13 + 1380*A^2*a^2*b^11 + 4006*A^2*a^4*b^9 + 5796*A^2*a^6*b^7 + 39
69*A^2*a^8*b^5)*(a^19*d^2 + a^7*b^12*d^2 + 6*a^9*b^10*d^2 + 15*a^11*b^8*d^2 + 20*a^13*b^6*d^2 + 15*a^15*b^4*d^
2 + 6*a^17*b^2*d^2))^(1/2)*((((tan(c + d*x)^(1/2)*(471859200*A^2*a^22*b^44*d^7 + 9500098560*A^2*a^24*b^42*d^7
+ 91857354752*A^2*a^26*b^40*d^7 + 564502986752*A^2*a^28*b^38*d^7 + 2464648527872*A^2*a^30*b^36*d^7 + 810446906
9824*A^2*a^32*b^34*d^7 + 20769933361152*A^2*a^34*b^32*d^7 + 42351565209600*A^2*a^36*b^30*d^7 + 69534945902592*
A^2*a^38*b^28*d^7 + 92434029608960*A^2*a^40*b^26*d^7 + 99508717355008*A^2*a^42*b^24*d^7 + 86342935511040*A^2*a
^44*b^22*d^7 + 59767095558144*A^2*a^46*b^20*d^7 + 32432589897728*A^2*a^48*b^18*d^7 + 13411815522304*A^2*a^50*b
^16*d^7 + 4030457708544*A^2*a^52*b^14*d^7 + 805425905664*A^2*a^54*b^12*d^7 + 86608183296*A^2*a^56*b^10*d^7 + 1
612709888*A^2*a^58*b^8*d^7 + 16777216*A^2*a^60*b^6*d^7 + 167772160*A^2*a^62*b^4*d^7 + 16777216*A^2*a^64*b^2*d^
7))/64 + ((-64*(225*A^2*b^13 + 1380*A^2*a^2*b^11 + 4006*A^2*a^4*b^9 + 5796*A^2*a^6*b^7 + 3969*A^2*a^8*b^5)*(a^
19*d^2 + a^7*b^12*d^2 + 6*a^9*b^10*d^2 + 15*a^11*b^8*d^2 + 20*a^13*b^6*d^2 + 15*a^15*b^4*d^2 + 6*a^17*b^2*d^2)
)^(1/2)*(3932160*A*a^24*b^45*d^8 - (tan(c + d*x)^(1/2)*(-64*(225*A^2*b^13 + 1380*A^2*a^2*b^11 + 4006*A^2*a^4*b
^9 + 5796*A^2*a^6*b^7 + 3969*A^2*a^8*b^5)*(a^19*d^2 + a^7*b^12*d^2 + 6*a^9*b^10*d^2 + 15*a^11*b^8*d^2 + 20*a^1
3*b^6*d^2 + 15*a^15*b^4*d^2 + 6*a^17*b^2*d^2))^(1/2)*(134217728*a^27*b^45*d^9 + 2550136832*a^29*b^43*d^9 + 228
17013760*a^31*b^41*d^9 + 127506841600*a^33*b^39*d^9 + 497276682240*a^35*b^37*d^9 + 1430626762752*a^37*b^35*d^9
+ 3121367482368*a^39*b^33*d^9 + 5202279137280*a^41*b^31*d^9 + 6502848921600*a^43*b^29*d^9 + 5635802398720*a^4
5*b^27*d^9 + 2254320959488*a^47*b^25*d^9 - 2254320959488*a^49*b^23*d^9 - 5635802398720*a^51*b^21*d^9 - 6502848
921600*a^53*b^19*d^9 - 5202279137280*a^55*b^17*d^9 - 3121367482368*a^57*b^15*d^9 - 1430626762752*a^59*b^13*d^9
- 497276682240*a^61*b^11*d^9 - 127506841600*a^63*b^9*d^9 - 22817013760*a^65*b^7*d^9 - 2550136832*a^67*b^5*d^9
- 134217728*a^69*b^3*d^9))/(4096*(a^19*d^2 + a^7*b^12*d^2 + 6*a^9*b^10*d^2 + 15*a^11*b^8*d^2 + 20*a^13*b^6*d^
2 + 15*a^15*b^4*d^2 + 6*a^17*b^2*d^2)) + 78905344*A*a^26*b^43*d^8 + 755761152*A*a^28*b^41*d^8 + 4590927872*A*a
^30*b^39*d^8 + 19819659264*A*a^32*b^37*d^8 + 64573931520*A*a^34*b^35*d^8 + 164549885952*A*a^36*b^33*d^8 + 3353
56624896*A*a^38*b^31*d^8 + 554212786176*A*a^40*b^29*d^8 + 748389138432*A*a^42*b^27*d^8 + 827874344960*A*a^44*b
^25*d^8 + 748389138432*A*a^46*b^23*d^8 + 548285710336*A*a^48*b^21*d^8 + 320115572736*A*a^50*b^19*d^8 + 1442284
83072*A*a^52*b^17*d^8 + 46792704000*A*a^54*b^15*d^8 + 8837136384*A*a^56*b^13*d^8 - 290193408*A*a^58*b^11*d^8 -
785645568*A*a^60*b^9*d^8 - 251396096*A*a^62*b^7*d^8 - 39321600*A*a^64*b^5*d^8 - 2621440*A*a^66*b^3*d^8))/(64*
(a^19*d^2 + a^7*b^12*d^2 + 6*a^9*b^10*d^2 + 15*a^11*b^8*d^2 + 20*a^13*b^6*d^2 + 15*a^15*b^4*d^2 + 6*a^17*b^2*d
^2)))*(-64*(225*A^2*b^13 + 1380*A^2*a^2*b^11 + 4006*A^2*a^4*b^9 + 5796*A^2*a^6*b^7 + 3969*A^2*a^8*b^5)*(a^19*d
^2 + a^7*b^12*d^2 + 6*a^9*b^10*d^2 + 15*a^11*b^8*d^2 + 20*a^13*b^6*d^2 + 15*a^15*b^4*d^2 + 6*a^17*b^2*d^2))^(1
/2))/(64*(a^19*d^2 + a^7*b^12*d^2 + 6*a^9*b^10*d^2 + 15*a^11*b^8*d^2 + 20*a^13*b^6*d^2 + 15*a^15*b^4*d^2 + 6*a
^17*b^2*d^2)) - 1843200*A^3*a^21*b^42*d^6 - 13148160*A^3*a^23*b^40*d^6 + 59834368*A^3*a^25*b^38*d^6 + 12198215
68*A^3*a^27*b^36*d^6 + 7772471296*A^3*a^29*b^34*d^6 + 29685874688*A^3*a^31*b^32*d^6 + 77698367488*A^3*a^33*b^3
0*d^6 + 146424922112*A^3*a^35*b^28*d^6 + 201417506816*A^3*a^37*b^26*d^6 + 198845153280*A^3*a^39*b^24*d^6 + 130
733768704*A^3*a^41*b^22*d^6 + 40588476416*A^3*a^43*b^20*d^6 - 18309939200*A^3*a^45*b^18*d^6 - 31045025792*A^3*
a^47*b^16*d^6 - 19337248768*A^3*a^49*b^14*d^6 - 7023558656*A^3*a^51*b^12*d^6 - 1523064832*A^3*a^53*b^10*d^6 -
186425344*A^3*a^55*b^8*d^6 - 15728640*A^3*a^57*b^6*d^6 - 2097152*A^3*a^59*b^4*d^6 - 131072*A^3*a^61*b^2*d^6))/
(64*(a^19*d^2 + a^7*b^12*d^2 + 6*a^9*b^10*d^2 + 15*a^11*b^8*d^2 + 20*a^13*b^6*d^2 + 15*a^15*b^4*d^2 + 6*a^17*b
^2*d^2)))*(-64*(225*A^2*b^13 + 1380*A^2*a^2*b^11 + 4006*A^2*a^4*b^9 + 5796*A^2*a^6*b^7 + 3969*A^2*a^8*b^5)*(a^
19*d^2 + a^7*b^12*d^2 + 6*a^9*b^10*d^2 + 15*a^11*b^8*d^2 + 20*a^13*b^6*d^2 + 15*a^15*b^4*d^2 + 6*a^17*b^2*d^2)
)^(1/2)*1i)/(a^19*d^2 + a^7*b^12*d^2 + 6*a^9*b^10*d^2 + 15*a^11*b^8*d^2 + 20*a^13*b^6*d^2 + 15*a^15*b^4*d^2 +
6*a^17*b^2*d^2) + (((tan(c + d*x)^(1/2)*(7610564608*A^4*a^27*b^33*d^5 - 597688320*A^4*a^23*b^37*d^5 - 16714301
44*A^4*a^25*b^35*d^5 - 58982400*A^4*a^21*b^39*d^5 + 85774565376*A^4*a^29*b^31*d^5 + 385487994880*A^4*a^31*b^29
*d^5 + 1104303620096*A^4*a^33*b^27*d^5 + 2240523796480*A^4*a^35*b^25*d^5 + 3345249468416*A^4*a^37*b^23*d^5 + 3
717287903232*A^4*a^39*b^21*d^5 + 3053967114240*A^4*a^41*b^19*d^5 + 1807474491392*A^4*a^43*b^17*d^5 + 726513221
632*A^4*a^45*b^15*d^5 + 170768990208*A^4*a^47*b^13*d^5 + 10492051456*A^4*a^49*b^11*d^5 - 4917821440*A^4*a^51*b
^9*d^5 - 923009024*A^4*a^53*b^7*d^5 + 8388608*A^4*a^55*b^5*d^5))/64 - ((-64*(225*A^2*b^13 + 1380*A^2*a^2*b^11
+ 4006*A^2*a^4*b^9 + 5796*A^2*a^6*b^7 + 3969*A^2*a^8*b^5)*(a^19*d^2 + a^7*b^12*d^2 + 6*a^9*b^10*d^2 + 15*a^11*
b^8*d^2 + 20*a^13*b^6*d^2 + 15*a^15*b^4*d^2 + 6*a^17*b^2*d^2))^(1/2)*((((tan(c + d*x)^(1/2)*(471859200*A^2*a^2
2*b^44*d^7 + 9500098560*A^2*a^24*b^42*d^7 + 91857354752*A^2*a^26*b^40*d^7 + 564502986752*A^2*a^28*b^38*d^7 + 2
464648527872*A^2*a^30*b^36*d^7 + 8104469069824*A^2*a^32*b^34*d^7 + 20769933361152*A^2*a^34*b^32*d^7 + 42351565
209600*A^2*a^36*b^30*d^7 + 69534945902592*A^2*a^38*b^28*d^7 + 92434029608960*A^2*a^40*b^26*d^7 + 9950871735500
8*A^2*a^42*b^24*d^7 + 86342935511040*A^2*a^44*b^22*d^7 + 59767095558144*A^2*a^46*b^20*d^7 + 32432589897728*A^2
*a^48*b^18*d^7 + 13411815522304*A^2*a^50*b^16*d^7 + 4030457708544*A^2*a^52*b^14*d^7 + 805425905664*A^2*a^54*b^
12*d^7 + 86608183296*A^2*a^56*b^10*d^7 + 1612709888*A^2*a^58*b^8*d^7 + 16777216*A^2*a^60*b^6*d^7 + 167772160*A
^2*a^62*b^4*d^7 + 16777216*A^2*a^64*b^2*d^7))/64 - ((-64*(225*A^2*b^13 + 1380*A^2*a^2*b^11 + 4006*A^2*a^4*b^9
+ 5796*A^2*a^6*b^7 + 3969*A^2*a^8*b^5)*(a^19*d^2 + a^7*b^12*d^2 + 6*a^9*b^10*d^2 + 15*a^11*b^8*d^2 + 20*a^13*b
^6*d^2 + 15*a^15*b^4*d^2 + 6*a^17*b^2*d^2))^(1/2)*((tan(c + d*x)^(1/2)*(-64*(225*A^2*b^13 + 1380*A^2*a^2*b^11
+ 4006*A^2*a^4*b^9 + 5796*A^2*a^6*b^7 + 3969*A^2*a^8*b^5)*(a^19*d^2 + a^7*b^12*d^2 + 6*a^9*b^10*d^2 + 15*a^11*
b^8*d^2 + 20*a^13*b^6*d^2 + 15*a^15*b^4*d^2 + 6*a^17*b^2*d^2))^(1/2)*(134217728*a^27*b^45*d^9 + 2550136832*a^2
9*b^43*d^9 + 22817013760*a^31*b^41*d^9 + 127506841600*a^33*b^39*d^9 + 497276682240*a^35*b^37*d^9 + 14306267627
52*a^37*b^35*d^9 + 3121367482368*a^39*b^33*d^9 + 5202279137280*a^41*b^31*d^9 + 6502848921600*a^43*b^29*d^9 + 5
635802398720*a^45*b^27*d^9 + 2254320959488*a^47*b^25*d^9 - 2254320959488*a^49*b^23*d^9 - 5635802398720*a^51*b^
21*d^9 - 6502848921600*a^53*b^19*d^9 - 5202279137280*a^55*b^17*d^9 - 3121367482368*a^57*b^15*d^9 - 14306267627
52*a^59*b^13*d^9 - 497276682240*a^61*b^11*d^9 - 127506841600*a^63*b^9*d^9 - 22817013760*a^65*b^7*d^9 - 2550136
832*a^67*b^5*d^9 - 134217728*a^69*b^3*d^9))/(4096*(a^19*d^2 + a^7*b^12*d^2 + 6*a^9*b^10*d^2 + 15*a^11*b^8*d^2
+ 20*a^13*b^6*d^2 + 15*a^15*b^4*d^2 + 6*a^17*b^2*d^2)) + 3932160*A*a^24*b^45*d^8 + 78905344*A*a^26*b^43*d^8 +
755761152*A*a^28*b^41*d^8 + 4590927872*A*a^30*b^39*d^8 + 19819659264*A*a^32*b^37*d^8 + 64573931520*A*a^34*b^35
*d^8 + 164549885952*A*a^36*b^33*d^8 + 335356624896*A*a^38*b^31*d^8 + 554212786176*A*a^40*b^29*d^8 + 7483891384
32*A*a^42*b^27*d^8 + 827874344960*A*a^44*b^25*d^8 + 748389138432*A*a^46*b^23*d^8 + 548285710336*A*a^48*b^21*d^
8 + 320115572736*A*a^50*b^19*d^8 + 144228483072*A*a^52*b^17*d^8 + 46792704000*A*a^54*b^15*d^8 + 8837136384*A*a
^56*b^13*d^8 - 290193408*A*a^58*b^11*d^8 - 785645568*A*a^60*b^9*d^8 - 251396096*A*a^62*b^7*d^8 - 39321600*A*a^
64*b^5*d^8 - 2621440*A*a^66*b^3*d^8))/(64*(a^19*d^2 + a^7*b^12*d^2 + 6*a^9*b^10*d^2 + 15*a^11*b^8*d^2 + 20*a^1
3*b^6*d^2 + 15*a^15*b^4*d^2 + 6*a^17*b^2*d^2)))*(-64*(225*A^2*b^13 + 1380*A^2*a^2*b^11 + 4006*A^2*a^4*b^9 + 57
96*A^2*a^6*b^7 + 3969*A^2*a^8*b^5)*(a^19*d^2 + a^7*b^12*d^2 + 6*a^9*b^10*d^2 + 15*a^11*b^8*d^2 + 20*a^13*b^6*d
^2 + 15*a^15*b^4*d^2 + 6*a^17*b^2*d^2))^(1/2))/(64*(a^19*d^2 + a^7*b^12*d^2 + 6*a^9*b^10*d^2 + 15*a^11*b^8*d^2
+ 20*a^13*b^6*d^2 + 15*a^15*b^4*d^2 + 6*a^17*b^2*d^2)) + 1843200*A^3*a^21*b^42*d^6 + 13148160*A^3*a^23*b^40*d
^6 - 59834368*A^3*a^25*b^38*d^6 - 1219821568*A^3*a^27*b^36*d^6 - 7772471296*A^3*a^29*b^34*d^6 - 29685874688*A^
3*a^31*b^32*d^6 - 77698367488*A^3*a^33*b^30*d^6 - 146424922112*A^3*a^35*b^28*d^6 - 201417506816*A^3*a^37*b^26*
d^6 - 198845153280*A^3*a^39*b^24*d^6 - 130733768704*A^3*a^41*b^22*d^6 - 40588476416*A^3*a^43*b^20*d^6 + 183099
39200*A^3*a^45*b^18*d^6 + 31045025792*A^3*a^47*b^16*d^6 + 19337248768*A^3*a^49*b^14*d^6 + 7023558656*A^3*a^51*
b^12*d^6 + 1523064832*A^3*a^53*b^10*d^6 + 186425344*A^3*a^55*b^8*d^6 + 15728640*A^3*a^57*b^6*d^6 + 2097152*A^3
*a^59*b^4*d^6 + 131072*A^3*a^61*b^2*d^6))/(64*(a^19*d^2 + a^7*b^12*d^2 + 6*a^9*b^10*d^2 + 15*a^11*b^8*d^2 + 20
*a^13*b^6*d^2 + 15*a^15*b^4*d^2 + 6*a^17*b^2*d^2)))*(-64*(225*A^2*b^13 + 1380*A^2*a^2*b^11 + 4006*A^2*a^4*b^9
+ 5796*A^2*a^6*b^7 + 3969*A^2*a^8*b^5)*(a^19*d^2 + a^7*b^12*d^2 + 6*a^9*b^10*d^2 + 15*a^11*b^8*d^2 + 20*a^13*b
^6*d^2 + 15*a^15*b^4*d^2 + 6*a^17*b^2*d^2))^(1/2)*1i)/(a^19*d^2 + a^7*b^12*d^2 + 6*a^9*b^10*d^2 + 15*a^11*b^8*
d^2 + 20*a^13*b^6*d^2 + 15*a^15*b^4*d^2 + 6*a^17*b^2*d^2))/((((tan(c + d*x)^(1/2)*(7610564608*A^4*a^27*b^33*d^
5 - 597688320*A^4*a^23*b^37*d^5 - 1671430144*A^4*a^25*b^35*d^5 - 58982400*A^4*a^21*b^39*d^5 + 85774565376*A^4*
a^29*b^31*d^5 + 385487994880*A^4*a^31*b^29*d^5 + 1104303620096*A^4*a^33*b^27*d^5 + 2240523796480*A^4*a^35*b^25
*d^5 + 3345249468416*A^4*a^37*b^23*d^5 + 3717287903232*A^4*a^39*b^21*d^5 + 3053967114240*A^4*a^41*b^19*d^5 + 1
807474491392*A^4*a^43*b^17*d^5 + 726513221632*A^4*a^45*b^15*d^5 + 170768990208*A^4*a^47*b^13*d^5 + 10492051456
*A^4*a^49*b^11*d^5 - 4917821440*A^4*a^51*b^9*d^5 - 923009024*A^4*a^53*b^7*d^5 + 8388608*A^4*a^55*b^5*d^5))/64
- ((-64*(225*A^2*b^13 + 1380*A^2*a^2*b^11 + 4006*A^2*a^4*b^9 + 5796*A^2*a^6*b^7 + 3969*A^2*a^8*b^5)*(a^19*d^2
+ a^7*b^12*d^2 + 6*a^9*b^10*d^2 + 15*a^11*b^8*d^2 + 20*a^13*b^6*d^2 + 15*a^15*b^4*d^2 + 6*a^17*b^2*d^2))^(1/2)
*((((tan(c + d*x)^(1/2)*(471859200*A^2*a^22*b^44*d^7 + 9500098560*A^2*a^24*b^42*d^7 + 91857354752*A^2*a^26*b^4
0*d^7 + 564502986752*A^2*a^28*b^38*d^7 + 2464648527872*A^2*a^30*b^36*d^7 + 8104469069824*A^2*a^32*b^34*d^7 + 2
0769933361152*A^2*a^34*b^32*d^7 + 42351565209600*A^2*a^36*b^30*d^7 + 69534945902592*A^2*a^38*b^28*d^7 + 924340
29608960*A^2*a^40*b^26*d^7 + 99508717355008*A^2*a^42*b^24*d^7 + 86342935511040*A^2*a^44*b^22*d^7 + 59767095558
144*A^2*a^46*b^20*d^7 + 32432589897728*A^2*a^48*b^18*d^7 + 13411815522304*A^2*a^50*b^16*d^7 + 4030457708544*A^
2*a^52*b^14*d^7 + 805425905664*A^2*a^54*b^12*d^7 + 86608183296*A^2*a^56*b^10*d^7 + 1612709888*A^2*a^58*b^8*d^7
+ 16777216*A^2*a^60*b^6*d^7 + 167772160*A^2*a^62*b^4*d^7 + 16777216*A^2*a^64*b^2*d^7))/64 + ((-64*(225*A^2*b^
13 + 1380*A^2*a^2*b^11 + 4006*A^2*a^4*b^9 + 5796*A^2*a^6*b^7 + 3969*A^2*a^8*b^5)*(a^19*d^2 + a^7*b^12*d^2 + 6*
a^9*b^10*d^2 + 15*a^11*b^8*d^2 + 20*a^13*b^6*d^2 + 15*a^15*b^4*d^2 + 6*a^17*b^2*d^2))^(1/2)*(3932160*A*a^24*b^
45*d^8 - (tan(c + d*x)^(1/2)*(-64*(225*A^2*b^13 + 1380*A^2*a^2*b^11 + 4006*A^2*a^4*b^9 + 5796*A^2*a^6*b^7 + 39
69*A^2*a^8*b^5)*(a^19*d^2 + a^7*b^12*d^2 + 6*a^9*b^10*d^2 + 15*a^11*b^8*d^2 + 20*a^13*b^6*d^2 + 15*a^15*b^4*d^
2 + 6*a^17*b^2*d^2))^(1/2)*(134217728*a^27*b^45*d^9 + 2550136832*a^29*b^43*d^9 + 22817013760*a^31*b^41*d^9 + 1
27506841600*a^33*b^39*d^9 + 497276682240*a^35*b^37*d^9 + 1430626762752*a^37*b^35*d^9 + 3121367482368*a^39*b^33
*d^9 + 5202279137280*a^41*b^31*d^9 + 6502848921600*a^43*b^29*d^9 + 5635802398720*a^45*b^27*d^9 + 2254320959488
*a^47*b^25*d^9 - 2254320959488*a^49*b^23*d^9 - 5635802398720*a^51*b^21*d^9 - 6502848921600*a^53*b^19*d^9 - 520
2279137280*a^55*b^17*d^9 - 3121367482368*a^57*b^15*d^9 - 1430626762752*a^59*b^13*d^9 - 497276682240*a^61*b^11*
d^9 - 127506841600*a^63*b^9*d^9 - 22817013760*a^65*b^7*d^9 - 2550136832*a^67*b^5*d^9 - 134217728*a^69*b^3*d^9)
)/(4096*(a^19*d^2 + a^7*b^12*d^2 + 6*a^9*b^10*d^2 + 15*a^11*b^8*d^2 + 20*a^13*b^6*d^2 + 15*a^15*b^4*d^2 + 6*a^
17*b^2*d^2)) + 78905344*A*a^26*b^43*d^8 + 755761152*A*a^28*b^41*d^8 + 4590927872*A*a^30*b^39*d^8 + 19819659264
*A*a^32*b^37*d^8 + 64573931520*A*a^34*b^35*d^8 + 164549885952*A*a^36*b^33*d^8 + 335356624896*A*a^38*b^31*d^8 +
554212786176*A*a^40*b^29*d^8 + 748389138432*A*a^42*b^27*d^8 + 827874344960*A*a^44*b^25*d^8 + 748389138432*A*a
^46*b^23*d^8 + 548285710336*A*a^48*b^21*d^8 + 320115572736*A*a^50*b^19*d^8 + 144228483072*A*a^52*b^17*d^8 + 46
792704000*A*a^54*b^15*d^8 + 8837136384*A*a^56*b^13*d^8 - 290193408*A*a^58*b^11*d^8 - 785645568*A*a^60*b^9*d^8
- 251396096*A*a^62*b^7*d^8 - 39321600*A*a^64*b^5*d^8 - 2621440*A*a^66*b^3*d^8))/(64*(a^19*d^2 + a^7*b^12*d^2 +
6*a^9*b^10*d^2 + 15*a^11*b^8*d^2 + 20*a^13*b^6*d^2 + 15*a^15*b^4*d^2 + 6*a^17*b^2*d^2)))*(-64*(225*A^2*b^13 +
1380*A^2*a^2*b^11 + 4006*A^2*a^4*b^9 + 5796*A^2*a^6*b^7 + 3969*A^2*a^8*b^5)*(a^19*d^2 + a^7*b^12*d^2 + 6*a^9*
b^10*d^2 + 15*a^11*b^8*d^2 + 20*a^13*b^6*d^2 + 15*a^15*b^4*d^2 + 6*a^17*b^2*d^2))^(1/2))/(64*(a^19*d^2 + a^7*b
^12*d^2 + 6*a^9*b^10*d^2 + 15*a^11*b^8*d^2 + 20*a^13*b^6*d^2 + 15*a^15*b^4*d^2 + 6*a^17*b^2*d^2)) - 1843200*A^
3*a^21*b^42*d^6 - 13148160*A^3*a^23*b^40*d^6 + 59834368*A^3*a^25*b^38*d^6 + 1219821568*A^3*a^27*b^36*d^6 + 777
2471296*A^3*a^29*b^34*d^6 + 29685874688*A^3*a^31*b^32*d^6 + 77698367488*A^3*a^33*b^30*d^6 + 146424922112*A^3*a
^35*b^28*d^6 + 201417506816*A^3*a^37*b^26*d^6 + 198845153280*A^3*a^39*b^24*d^6 + 130733768704*A^3*a^41*b^22*d^
6 + 40588476416*A^3*a^43*b^20*d^6 - 18309939200*A^3*a^45*b^18*d^6 - 31045025792*A^3*a^47*b^16*d^6 - 1933724876
8*A^3*a^49*b^14*d^6 - 7023558656*A^3*a^51*b^12*d^6 - 1523064832*A^3*a^53*b^10*d^6 - 186425344*A^3*a^55*b^8*d^6
- 15728640*A^3*a^57*b^6*d^6 - 2097152*A^3*a^59*b^4*d^6 - 131072*A^3*a^61*b^2*d^6))/(64*(a^19*d^2 + a^7*b^12*d
^2 + 6*a^9*b^10*d^2 + 15*a^11*b^8*d^2 + 20*a^13*b^6*d^2 + 15*a^15*b^4*d^2 + 6*a^17*b^2*d^2)))*(-64*(225*A^2*b^
13 + 1380*A^2*a^2*b^11 + 4006*A^2*a^4*b^9 + 5796*A^2*a^6*b^7 + 3969*A^2*a^8*b^5)*(a^19*d^2 + a^7*b^12*d^2 + 6*
a^9*b^10*d^2 + 15*a^11*b^8*d^2 + 20*a^13*b^6*d^2 + 15*a^15*b^4*d^2 + 6*a^17*b^2*d^2))^(1/2))/(a^19*d^2 + a^7*b
^12*d^2 + 6*a^9*b^10*d^2 + 15*a^11*b^8*d^2 + 20*a^13*b^6*d^2 + 15*a^15*b^4*d^2 + 6*a^17*b^2*d^2) - (((tan(c +
d*x)^(1/2)*(7610564608*A^4*a^27*b^33*d^5 - 597688320*A^4*a^23*b^37*d^5 - 1671430144*A^4*a^25*b^35*d^5 - 589824
00*A^4*a^21*b^39*d^5 + 85774565376*A^4*a^29*b^31*d^5 + 385487994880*A^4*a^31*b^29*d^5 + 1104303620096*A^4*a^33
*b^27*d^5 + 2240523796480*A^4*a^35*b^25*d^5 + 3345249468416*A^4*a^37*b^23*d^5 + 3717287903232*A^4*a^39*b^21*d^
5 + 3053967114240*A^4*a^41*b^19*d^5 + 1807474491392*A^4*a^43*b^17*d^5 + 726513221632*A^4*a^45*b^15*d^5 + 17076
8990208*A^4*a^47*b^13*d^5 + 10492051456*A^4*a^49*b^11*d^5 - 4917821440*A^4*a^51*b^9*d^5 - 923009024*A^4*a^53*b
^7*d^5 + 8388608*A^4*a^55*b^5*d^5))/64 - ((-64*(225*A^2*b^13 + 1380*A^2*a^2*b^11 + 4006*A^2*a^4*b^9 + 5796*A^2
*a^6*b^7 + 3969*A^2*a^8*b^5)*(a^19*d^2 + a^7*b^12*d^2 + 6*a^9*b^10*d^2 + 15*a^11*b^8*d^2 + 20*a^13*b^6*d^2 + 1
5*a^15*b^4*d^2 + 6*a^17*b^2*d^2))^(1/2)*((((tan(c + d*x)^(1/2)*(471859200*A^2*a^22*b^44*d^7 + 9500098560*A^2*a
^24*b^42*d^7 + 91857354752*A^2*a^26*b^40*d^7 + 564502986752*A^2*a^28*b^38*d^7 + 2464648527872*A^2*a^30*b^36*d^
7 + 8104469069824*A^2*a^32*b^34*d^7 + 20769933361152*A^2*a^34*b^32*d^7 + 42351565209600*A^2*a^36*b^30*d^7 + 69
534945902592*A^2*a^38*b^28*d^7 + 92434029608960*A^2*a^40*b^26*d^7 + 99508717355008*A^2*a^42*b^24*d^7 + 8634293
5511040*A^2*a^44*b^22*d^7 + 59767095558144*A^2*a^46*b^20*d^7 + 32432589897728*A^2*a^48*b^18*d^7 + 134118155223
04*A^2*a^50*b^16*d^7 + 4030457708544*A^2*a^52*b^14*d^7 + 805425905664*A^2*a^54*b^12*d^7 + 86608183296*A^2*a^56
*b^10*d^7 + 1612709888*A^2*a^58*b^8*d^7 + 16777216*A^2*a^60*b^6*d^7 + 167772160*A^2*a^62*b^4*d^7 + 16777216*A^
2*a^64*b^2*d^7))/64 - ((-64*(225*A^2*b^13 + 1380*A^2*a^2*b^11 + 4006*A^2*a^4*b^9 + 5796*A^2*a^6*b^7 + 3969*A^2
*a^8*b^5)*(a^19*d^2 + a^7*b^12*d^2 + 6*a^9*b^10*d^2 + 15*a^11*b^8*d^2 + 20*a^13*b^6*d^2 + 15*a^15*b^4*d^2 + 6*
a^17*b^2*d^2))^(1/2)*((tan(c + d*x)^(1/2)*(-64*(225*A^2*b^13 + 1380*A^2*a^2*b^11 + 4006*A^2*a^4*b^9 + 5796*A^2
*a^6*b^7 + 3969*A^2*a^8*b^5)*(a^19*d^2 + a^7*b^12*d^2 + 6*a^9*b^10*d^2 + 15*a^11*b^8*d^2 + 20*a^13*b^6*d^2 + 1
5*a^15*b^4*d^2 + 6*a^17*b^2*d^2))^(1/2)*(134217728*a^27*b^45*d^9 + 2550136832*a^29*b^43*d^9 + 22817013760*a^31
*b^41*d^9 + 127506841600*a^33*b^39*d^9 + 497276682240*a^35*b^37*d^9 + 1430626762752*a^37*b^35*d^9 + 3121367482
368*a^39*b^33*d^9 + 5202279137280*a^41*b^31*d^9 + 6502848921600*a^43*b^29*d^9 + 5635802398720*a^45*b^27*d^9 +
2254320959488*a^47*b^25*d^9 - 2254320959488*a^49*b^23*d^9 - 5635802398720*a^51*b^21*d^9 - 6502848921600*a^53*b
^19*d^9 - 5202279137280*a^55*b^17*d^9 - 3121367482368*a^57*b^15*d^9 - 1430626762752*a^59*b^13*d^9 - 4972766822
40*a^61*b^11*d^9 - 127506841600*a^63*b^9*d^9 - 22817013760*a^65*b^7*d^9 - 2550136832*a^67*b^5*d^9 - 134217728*
a^69*b^3*d^9))/(4096*(a^19*d^2 + a^7*b^12*d^2 + 6*a^9*b^10*d^2 + 15*a^11*b^8*d^2 + 20*a^13*b^6*d^2 + 15*a^15*b
^4*d^2 + 6*a^17*b^2*d^2)) + 3932160*A*a^24*b^45*d^8 + 78905344*A*a^26*b^43*d^8 + 755761152*A*a^28*b^41*d^8 + 4
590927872*A*a^30*b^39*d^8 + 19819659264*A*a^32*b^37*d^8 + 64573931520*A*a^34*b^35*d^8 + 164549885952*A*a^36*b^
33*d^8 + 335356624896*A*a^38*b^31*d^8 + 554212786176*A*a^40*b^29*d^8 + 748389138432*A*a^42*b^27*d^8 + 82787434
4960*A*a^44*b^25*d^8 + 748389138432*A*a^46*b^23*d^8 + 548285710336*A*a^48*b^21*d^8 + 320115572736*A*a^50*b^19*
d^8 + 144228483072*A*a^52*b^17*d^8 + 46792704000*A*a^54*b^15*d^8 + 8837136384*A*a^56*b^13*d^8 - 290193408*A*a^
58*b^11*d^8 - 785645568*A*a^60*b^9*d^8 - 251396096*A*a^62*b^7*d^8 - 39321600*A*a^64*b^5*d^8 - 2621440*A*a^66*b
^3*d^8))/(64*(a^19*d^2 + a^7*b^12*d^2 + 6*a^9*b^10*d^2 + 15*a^11*b^8*d^2 + 20*a^13*b^6*d^2 + 15*a^15*b^4*d^2 +
6*a^17*b^2*d^2)))*(-64*(225*A^2*b^13 + 1380*A^2*a^2*b^11 + 4006*A^2*a^4*b^9 + 5796*A^2*a^6*b^7 + 3969*A^2*a^8
*b^5)*(a^19*d^2 + a^7*b^12*d^2 + 6*a^9*b^10*d^2 + 15*a^11*b^8*d^2 + 20*a^13*b^6*d^2 + 15*a^15*b^4*d^2 + 6*a^17
*b^2*d^2))^(1/2))/(64*(a^19*d^2 + a^7*b^12*d^2 + 6*a^9*b^10*d^2 + 15*a^11*b^8*d^2 + 20*a^13*b^6*d^2 + 15*a^15*
b^4*d^2 + 6*a^17*b^2*d^2)) + 1843200*A^3*a^21*b^42*d^6 + 13148160*A^3*a^23*b^40*d^6 - 59834368*A^3*a^25*b^38*d
^6 - 1219821568*A^3*a^27*b^36*d^6 - 7772471296*A^3*a^29*b^34*d^6 - 29685874688*A^3*a^31*b^32*d^6 - 77698367488
*A^3*a^33*b^30*d^6 - 146424922112*A^3*a^35*b^28*d^6 - 201417506816*A^3*a^37*b^26*d^6 - 198845153280*A^3*a^39*b
^24*d^6 - 130733768704*A^3*a^41*b^22*d^6 - 40588476416*A^3*a^43*b^20*d^6 + 18309939200*A^3*a^45*b^18*d^6 + 310
45025792*A^3*a^47*b^16*d^6 + 19337248768*A^3*a^49*b^14*d^6 + 7023558656*A^3*a^51*b^12*d^6 + 1523064832*A^3*a^5
3*b^10*d^6 + 186425344*A^3*a^55*b^8*d^6 + 15728640*A^3*a^57*b^6*d^6 + 2097152*A^3*a^59*b^4*d^6 + 131072*A^3*a^
61*b^2*d^6))/(64*(a^19*d^2 + a^7*b^12*d^2 + 6*a^9*b^10*d^2 + 15*a^11*b^8*d^2 + 20*a^13*b^6*d^2 + 15*a^15*b^4*d
^2 + 6*a^17*b^2*d^2)))*(-64*(225*A^2*b^13 + 1380*A^2*a^2*b^11 + 4006*A^2*a^4*b^9 + 5796*A^2*a^6*b^7 + 3969*A^2
*a^8*b^5)*(a^19*d^2 + a^7*b^12*d^2 + 6*a^9*b^10*d^2 + 15*a^11*b^8*d^2 + 20*a^13*b^6*d^2 + 15*a^15*b^4*d^2 + 6*
a^17*b^2*d^2))^(1/2))/(a^19*d^2 + a^7*b^12*d^2 + 6*a^9*b^10*d^2 + 15*a^11*b^8*d^2 + 20*a^13*b^6*d^2 + 15*a^15*
b^4*d^2 + 6*a^17*b^2*d^2) + 58982400*A^5*a^22*b^35*d^4 + 920125440*A^5*a^24*b^33*d^4 + 6879444992*A^5*a^26*b^3
1*d^4 + 32454475776*A^5*a^28*b^29*d^4 + 107338792960*A^5*a^30*b^27*d^4 + 262062735360*A^5*a^32*b^25*d^4 + 4850
59461120*A^5*a^34*b^23*d^4 + 688908140544*A^5*a^36*b^21*d^4 + 751987064832*A^5*a^38*b^19*d^4 + 626086379520*A^
5*a^40*b^17*d^4 + 390506741760*A^5*a^42*b^15*d^4 + 176637870080*A^5*a^44*b^13*d^4 + 54704996352*A^5*a^46*b^11*
d^4 + 10374086656*A^5*a^48*b^9*d^4 + 908328960*A^5*a^50*b^7*d^4))*(-64*(225*A^2*b^13 + 1380*A^2*a^2*b^11 + 400
6*A^2*a^4*b^9 + 5796*A^2*a^6*b^7 + 3969*A^2*a^8*b^5)*(a^19*d^2 + a^7*b^12*d^2 + 6*a^9*b^10*d^2 + 15*a^11*b^8*d
^2 + 20*a^13*b^6*d^2 + 15*a^15*b^4*d^2 + 6*a^17*b^2*d^2))^(1/2)*1i)/(32*(a^19*d^2 + a^7*b^12*d^2 + 6*a^9*b^10*
d^2 + 15*a^11*b^8*d^2 + 20*a^13*b^6*d^2 + 15*a^15*b^4*d^2 + 6*a^17*b^2*d^2))

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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