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

Optimal. Leaf size=531 \[ -\frac {(A b-a B) \sqrt {\tan (c+d x)}}{2 d \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}-\frac {\left (a^3 (A-B)+3 a^2 b (A+B)-3 a b^2 (A-B)-b^3 (A+B)\right ) \tan ^{-1}\left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} d \left (a^2+b^2\right )^3}+\frac {\left (a^3 (A-B)+3 a^2 b (A+B)-3 a b^2 (A-B)-b^3 (A+B)\right ) \tan ^{-1}\left (\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{\sqrt {2} d \left (a^2+b^2\right )^3}-\frac {\left (-3 a^3 B+7 a^2 A b+5 a b^2 B-A b^3\right ) \sqrt {\tan (c+d x)}}{4 a d \left (a^2+b^2\right )^2 (a+b \tan (c+d x))}-\frac {\left (-\left (a^3 (A+B)\right )+3 a^2 b (A-B)+3 a b^2 (A+B)-b^3 (A-B)\right ) \log \left (\tan (c+d x)-\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{2 \sqrt {2} d \left (a^2+b^2\right )^3}+\frac {\left (-\left (a^3 (A+B)\right )+3 a^2 b (A-B)+3 a b^2 (A+B)-b^3 (A-B)\right ) \log \left (\tan (c+d x)+\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{2 \sqrt {2} d \left (a^2+b^2\right )^3}-\frac {\left (-3 a^5 B+15 a^4 A b+26 a^3 b^2 B-18 a^2 A b^3-3 a b^4 B-A b^5\right ) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {\tan (c+d x)}}{\sqrt {a}}\right )}{4 a^{3/2} \sqrt {b} d \left (a^2+b^2\right )^3} \]

[Out]

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*(15*A*a^4*b-18*A*a^2*b^3-A*b^5-3*B*a^5+26*B*a^3*b^2-3*B*a*b^4)*arctan(b^(1/2)*tan(d*x+c
)^(1/2)/a^(1/2))/a^(3/2)/(a^2+b^2)^3/d/b^(1/2)-1/2*(A*b-B*a)*tan(d*x+c)^(1/2)/(a^2+b^2)/d/(a+b*tan(d*x+c))^2-1
/4*(7*A*a^2*b-A*b^3-3*B*a^3+5*B*a*b^2)*tan(d*x+c)^(1/2)/a/(a^2+b^2)^2/d/(a+b*tan(d*x+c))

________________________________________________________________________________________

Rubi [A]  time = 1.31, antiderivative size = 531, normalized size of antiderivative = 1.00, number of steps used = 16, number of rules used = 13, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.394, Rules used = {3608, 3649, 3653, 3534, 1168, 1162, 617, 204, 1165, 628, 3634, 63, 205} \[ -\frac {\left (3 a^2 b (A+B)+a^3 (A-B)-3 a b^2 (A-B)-b^3 (A+B)\right ) \tan ^{-1}\left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} d \left (a^2+b^2\right )^3}+\frac {\left (3 a^2 b (A+B)+a^3 (A-B)-3 a b^2 (A-B)-b^3 (A+B)\right ) \tan ^{-1}\left (\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{\sqrt {2} d \left (a^2+b^2\right )^3}-\frac {\left (-18 a^2 A b^3+15 a^4 A b+26 a^3 b^2 B-3 a^5 B-3 a b^4 B-A b^5\right ) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {\tan (c+d x)}}{\sqrt {a}}\right )}{4 a^{3/2} \sqrt {b} d \left (a^2+b^2\right )^3}-\frac {(A b-a B) \sqrt {\tan (c+d x)}}{2 d \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}-\frac {\left (7 a^2 A b-3 a^3 B+5 a b^2 B-A b^3\right ) \sqrt {\tan (c+d x)}}{4 a d \left (a^2+b^2\right )^2 (a+b \tan (c+d x))}-\frac {\left (3 a^2 b (A-B)+a^3 (-(A+B))+3 a b^2 (A+B)-b^3 (A-B)\right ) \log \left (\tan (c+d x)-\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{2 \sqrt {2} d \left (a^2+b^2\right )^3}+\frac {\left (3 a^2 b (A-B)+a^3 (-(A+B))+3 a b^2 (A+B)-b^3 (A-B)\right ) \log \left (\tan (c+d x)+\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{2 \sqrt {2} d \left (a^2+b^2\right )^3} \]

Antiderivative was successfully verified.

[In]

Int[(Sqrt[Tan[c + d*x]]*(A + B*Tan[c + d*x]))/(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]]])/(Sq
rt[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]*
Sqrt[Tan[c + d*x]]])/(Sqrt[2]*(a^2 + b^2)^3*d) - ((15*a^4*A*b - 18*a^2*A*b^3 - A*b^5 - 3*a^5*B + 26*a^3*b^2*B
- 3*a*b^4*B)*ArcTan[(Sqrt[b]*Sqrt[Tan[c + d*x]])/Sqrt[a]])/(4*a^(3/2)*Sqrt[b]*(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) - ((A*b - a*B)*Sqrt[Tan[c + d*x]])/(2*(a^2 + b^2)*d
*(a + b*Tan[c + d*x])^2) - ((7*a^2*A*b - A*b^3 - 3*a^3*B + 5*a*b^2*B)*Sqrt[Tan[c + d*x]])/(4*a*(a^2 + b^2)^2*d
*(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 3608

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

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 {\sqrt {\tan (c+d x)} (A+B \tan (c+d x))}{(a+b \tan (c+d x))^3} \, dx &=-\frac {(A b-a B) \sqrt {\tan (c+d x)}}{2 \left (a^2+b^2\right ) d (a+b \tan (c+d x))^2}-\frac {\int \frac {-\frac {1}{2} b (A b-a B)-2 b (a A+b B) \tan (c+d x)+\frac {3}{2} b (A b-a B) \tan ^2(c+d x)}{\sqrt {\tan (c+d x)} (a+b \tan (c+d x))^2} \, dx}{2 b \left (a^2+b^2\right )}\\ &=-\frac {(A b-a B) \sqrt {\tan (c+d x)}}{2 \left (a^2+b^2\right ) d (a+b \tan (c+d x))^2}-\frac {\left (7 a^2 A b-A b^3-3 a^3 B+5 a b^2 B\right ) \sqrt {\tan (c+d x)}}{4 a \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))}-\frac {\int \frac {-\frac {1}{4} b \left (9 a^2 A b+A b^3-5 a^3 B+3 a b^2 B\right )-2 a b \left (a^2 A-A b^2+2 a b B\right ) \tan (c+d x)+\frac {1}{4} b \left (7 a^2 A b-A b^3-3 a^3 B+5 a b^2 B\right ) \tan ^2(c+d x)}{\sqrt {\tan (c+d x)} (a+b \tan (c+d x))} \, dx}{2 a b \left (a^2+b^2\right )^2}\\ &=-\frac {(A b-a B) \sqrt {\tan (c+d x)}}{2 \left (a^2+b^2\right ) d (a+b \tan (c+d x))^2}-\frac {\left (7 a^2 A b-A b^3-3 a^3 B+5 a b^2 B\right ) \sqrt {\tan (c+d x)}}{4 a \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))}-\frac {\int \frac {-2 a b \left (3 a^2 A b-A b^3-a^3 B+3 a b^2 B\right )-2 a b \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}{2 a b \left (a^2+b^2\right )^3}-\frac {\left (15 a^4 A b-18 a^2 A b^3-A b^5-3 a^5 B+26 a^3 b^2 B-3 a b^4 B\right ) \int \frac {1+\tan ^2(c+d x)}{\sqrt {\tan (c+d x)} (a+b \tan (c+d x))} \, dx}{8 a \left (a^2+b^2\right )^3}\\ &=-\frac {(A b-a B) \sqrt {\tan (c+d x)}}{2 \left (a^2+b^2\right ) d (a+b \tan (c+d x))^2}-\frac {\left (7 a^2 A b-A b^3-3 a^3 B+5 a b^2 B\right ) \sqrt {\tan (c+d x)}}{4 a \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))}-\frac {\operatorname {Subst}\left (\int \frac {-2 a b \left (3 a^2 A b-A b^3-a^3 B+3 a b^2 B\right )-2 a b \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 b \left (a^2+b^2\right )^3 d}-\frac {\left (15 a^4 A b-18 a^2 A b^3-A b^5-3 a^5 B+26 a^3 b^2 B-3 a b^4 B\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {x} (a+b x)} \, dx,x,\tan (c+d x)\right )}{8 a \left (a^2+b^2\right )^3 d}\\ &=-\frac {(A b-a B) \sqrt {\tan (c+d x)}}{2 \left (a^2+b^2\right ) d (a+b \tan (c+d x))^2}-\frac {\left (7 a^2 A b-A b^3-3 a^3 B+5 a b^2 B\right ) \sqrt {\tan (c+d x)}}{4 a \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))}-\frac {\left (15 a^4 A b-18 a^2 A b^3-A b^5-3 a^5 B+26 a^3 b^2 B-3 a b^4 B\right ) \operatorname {Subst}\left (\int \frac {1}{a+b x^2} \, dx,x,\sqrt {\tan (c+d x)}\right )}{4 a \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 {\left (15 a^4 A b-18 a^2 A b^3-A b^5-3 a^5 B+26 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^{3/2} \sqrt {b} \left (a^2+b^2\right )^3 d}-\frac {(A b-a B) \sqrt {\tan (c+d x)}}{2 \left (a^2+b^2\right ) d (a+b \tan (c+d x))^2}-\frac {\left (7 a^2 A b-A b^3-3 a^3 B+5 a b^2 B\right ) \sqrt {\tan (c+d x)}}{4 a \left (a^2+b^2\right )^2 d (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 {\left (15 a^4 A b-18 a^2 A b^3-A b^5-3 a^5 B+26 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^{3/2} \sqrt {b} \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 {(A b-a B) \sqrt {\tan (c+d x)}}{2 \left (a^2+b^2\right ) d (a+b \tan (c+d x))^2}-\frac {\left (7 a^2 A b-A b^3-3 a^3 B+5 a b^2 B\right ) \sqrt {\tan (c+d x)}}{4 a \left (a^2+b^2\right )^2 d (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 {\left (15 a^4 A b-18 a^2 A b^3-A b^5-3 a^5 B+26 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^{3/2} \sqrt {b} \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 {(A b-a B) \sqrt {\tan (c+d x)}}{2 \left (a^2+b^2\right ) d (a+b \tan (c+d x))^2}-\frac {\left (7 a^2 A b-A b^3-3 a^3 B+5 a b^2 B\right ) \sqrt {\tan (c+d x)}}{4 a \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))}\\ \end {align*}

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Mathematica [C]  time = 6.28, size = 552, normalized size = 1.04 \[ \frac {b (A b-a B) \tan ^{\frac {3}{2}}(c+d x)}{2 a d \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}+\frac {-\frac {(A b-a B) \sqrt {\tan (c+d x)}}{d (a+b \tan (c+d x))}-\frac {2 \left (\frac {\left (-a \left (-\frac {3}{4} a^2 b (A b-a B)-a b^2 (a A+b B)\right )-\frac {1}{4} a b^3 (A b-a B)\right ) \sqrt {\tan (c+d x)}}{a d \left (a^2+b^2\right ) (a+b \tan (c+d x))}+\frac {\frac {2 \left (a^3 b^2 \left (a^2 A+2 a b B-A b^2\right )+\frac {1}{8} a^3 b \left (-3 a^3 B+7 a^2 A b+5 a b^2 B-A b^3\right )-\frac {1}{8} a b^3 \left (-5 a^3 B+9 a^2 A b+3 a b^2 B+A b^3\right )\right ) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {\tan (c+d x)}}{\sqrt {a}}\right )}{\sqrt {a} \sqrt {b} d \left (a^2+b^2\right )}+\frac {-\frac {\sqrt [4]{-1} \left (-a^2 b \left (a^3 (-B)+3 a^2 A b+3 a b^2 B-A b^3\right )+i a^2 b \left (a^3 A+3 a^2 b B-3 a A b^2-b^3 B\right )\right ) \tan ^{-1}\left ((-1)^{3/4} \sqrt {\tan (c+d x)}\right )}{d}-\frac {\sqrt [4]{-1} \left (a^2 (-b) \left (a^3 (-B)+3 a^2 A b+3 a b^2 B-A b^3\right )-i a^2 b \left (a^3 A+3 a^2 b B-3 a A b^2-b^3 B\right )\right ) \tanh ^{-1}\left ((-1)^{3/4} \sqrt {\tan (c+d x)}\right )}{d}}{a^2+b^2}}{a \left (a^2+b^2\right )}\right )}{b}}{2 a \left (a^2+b^2\right )} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(b*(A*b - a*B)*Tan[c + d*x]^(3/2))/(2*a*(a^2 + b^2)*d*(a + b*Tan[c + d*x])^2) + (-(((A*b - a*B)*Sqrt[Tan[c + d
*x]])/(d*(a + b*Tan[c + d*x]))) - (2*(((2*(a^3*b^2*(a^2*A - A*b^2 + 2*a*b*B) - (a*b^3*(9*a^2*A*b + A*b^3 - 5*a
^3*B + 3*a*b^2*B))/8 + (a^3*b*(7*a^2*A*b - A*b^3 - 3*a^3*B + 5*a*b^2*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^2*b*(3*a^2*A*b - A*b^3 - a^3*B + 3*a*b^2*B))
+ I*a^2*b*(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^2*b*(3*a^2*A*b - A*b^3 - a^3*B + 3*a*b^2*B)) - I*a^2*b*(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*(a^2 + b^2)) + ((-1/4*(a*b^3*(A*b - a*B)) - a*((-3*a^2*b*(A*b
- a*B))/4 - a*b^2*(a*A + b*B)))*Sqrt[Tan[c + d*x]])/(a*(a^2 + b^2)*d*(a + b*Tan[c + d*x]))))/b)/(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(tan(d*x+c)^(1/2)*(A+B*tan(d*x+c))/(a+b*tan(d*x+c))^3,x, algorithm="fricas")

[Out]

Timed out

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giac [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(tan(d*x+c)^(1/2)*(A+B*tan(d*x+c))/(a+b*tan(d*x+c))^3,x, algorithm="giac")

[Out]

Timed out

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maple [B]  time = 0.45, size = 1835, normalized size = 3.46 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(tan(d*x+c)^(1/2)*(A+B*tan(d*x+c))/(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)+tan
(d*x+c)))*a*b^2+1/2/d*a^3/(a^2+b^2)^3/(a+b*tan(d*x+c))^2*b^2*tan(d*x+c)^(1/2)*B-9/4/d*a^4/(a^2+b^2)^3/(a+b*tan
(d*x+c))^2*b*tan(d*x+c)^(1/2)*A-5/2/d*a^2/(a^2+b^2)^3/(a+b*tan(d*x+c))^2*b^3*tan(d*x+c)^(1/2)*A-13/2/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-7/4/d*a^3/(a^2+b^2)^3/(a+b*tan(d*x+c))^2*b^2
*tan(d*x+c)^(3/2)*A-15/4/d*a^3/(a^2+b^2)^3*b/(a*b)^(1/2)*arctan(tan(d*x+c)^(1/2)*b/(a*b)^(1/2))*A+9/2/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+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)*arctan(1+2^(1/2)*tan(d*x+c)^(1/2))*a*b^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*b^2+1/4/d/(a^2+b^2)^3/(a+b*tan(d*x+c))^2*b^6/a*tan(d
*x+c)^(3/2)*A-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)*tan(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-3/2/d*a/(a^2+b^2)^3/(a+b*tan(d
*x+c))^2*b^4*tan(d*x+c)^(3/2)*A+3/4/d*a^4/(a^2+b^2)^3/(a+b*tan(d*x+c))^2*b*tan(d*x+c)^(3/2)*B-1/2/d*a^2/(a^2+b
^2)^3/(a+b*tan(d*x+c))^2*b^3*tan(d*x+c)^(3/2)*B+1/4/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^5-3/4/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)*arcta
n(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+5/4/d*a^5/(a^2+b^2)^3/(a+b*tan(d*x+c))^2*tan(d*x+c
)^(1/2)*B+3/4/d*a^4/(a^2+b^2)^3/(a*b)^(1/2)*arctan(tan(d*x+c)^(1/2)*b/(a*b)^(1/2))*B-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^3-5/4/d/(a^2+b^2)^3/(a+b*tan(d*x+c))^2*b^5*tan(d*x+c)^(3/2)*B-1
/4/d/(a^2+b^2)^3/(a+b*tan(d*x+c))^2*tan(d*x+c)^(1/2)*A*b^5+3/4/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 = 0.91, size = 537, normalized size = 1.01 \[ \frac {\frac {{\left (3 \, B a^{5} - 15 \, A a^{4} b - 26 \, B a^{3} b^{2} + 18 \, A a^{2} b^{3} + 3 \, B a b^{4} + A b^{5}\right )} \arctan \left (\frac {b \sqrt {\tan \left (d x + c\right )}}{\sqrt {a b}}\right )}{{\left (a^{7} + 3 \, a^{5} b^{2} + 3 \, a^{3} b^{4} + a b^{6}\right )} \sqrt {a b}} + \frac {2 \, \sqrt {2} {\left ({\left (A - B\right )} a^{3} + 3 \, {\left (A + B\right )} a^{2} b - 3 \, {\left (A - B\right )} a b^{2} - {\left (A + B\right )} b^{3}\right )} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + 2 \, \sqrt {\tan \left (d x + c\right )}\right )}\right ) + 2 \, \sqrt {2} {\left ({\left (A - B\right )} a^{3} + 3 \, {\left (A + B\right )} a^{2} b - 3 \, {\left (A - B\right )} a b^{2} - {\left (A + B\right )} b^{3}\right )} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - 2 \, \sqrt {\tan \left (d x + c\right )}\right )}\right ) - \sqrt {2} {\left ({\left (A + B\right )} a^{3} - 3 \, {\left (A - B\right )} a^{2} b - 3 \, {\left (A + B\right )} a b^{2} + {\left (A - B\right )} b^{3}\right )} \log \left (\sqrt {2} \sqrt {\tan \left (d x + c\right )} + \tan \left (d x + c\right ) + 1\right ) + \sqrt {2} {\left ({\left (A + B\right )} a^{3} - 3 \, {\left (A - B\right )} a^{2} b - 3 \, {\left (A + B\right )} a b^{2} + {\left (A - B\right )} b^{3}\right )} \log \left (-\sqrt {2} \sqrt {\tan \left (d x + c\right )} + \tan \left (d x + c\right ) + 1\right )}{a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}} + \frac {{\left (3 \, B a^{3} b - 7 \, A a^{2} b^{2} - 5 \, B a b^{3} + A b^{4}\right )} \tan \left (d x + c\right )^{\frac {3}{2}} + {\left (5 \, B a^{4} - 9 \, A a^{3} b - 3 \, B a^{2} b^{2} - A a b^{3}\right )} \sqrt {\tan \left (d x + c\right )}}{a^{7} + 2 \, a^{5} b^{2} + a^{3} b^{4} + {\left (a^{5} b^{2} + 2 \, a^{3} b^{4} + a b^{6}\right )} \tan \left (d x + c\right )^{2} + 2 \, {\left (a^{6} b + 2 \, a^{4} b^{3} + a^{2} b^{5}\right )} \tan \left (d x + c\right )}}{4 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tan(d*x+c)^(1/2)*(A+B*tan(d*x+c))/(a+b*tan(d*x+c))^3,x, algorithm="maxima")

[Out]

1/4*((3*B*a^5 - 15*A*a^4*b - 26*B*a^3*b^2 + 18*A*a^2*b^3 + 3*B*a*b^4 + A*b^5)*arctan(b*sqrt(tan(d*x + c))/sqrt
(a*b))/((a^7 + 3*a^5*b^2 + 3*a^3*b^4 + a*b^6)*sqrt(a*b)) + (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)))) + 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) + ((3*B*a^3*b - 7*A*a^2*b^2 - 5*B*a*b^3 + A*b^4)*tan(d*
x + c)^(3/2) + (5*B*a^4 - 9*A*a^3*b - 3*B*a^2*b^2 - A*a*b^3)*sqrt(tan(d*x + c)))/(a^7 + 2*a^5*b^2 + a^3*b^4 +
(a^5*b^2 + 2*a^3*b^4 + a*b^6)*tan(d*x + c)^2 + 2*(a^6*b + 2*a^4*b^3 + a^2*b^5)*tan(d*x + c)))/d

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mupad [B]  time = 52.40, size = 26133, normalized size = 49.21 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((tan(c + d*x)^(1/2)*(A + B*tan(c + d*x)))/(a + b*tan(c + d*x))^3,x)

[Out]

(log((((((((((64*A*b^3*(b^4 - 10*a^4 + 15*a^2*b^2))/(a*d) + 128*b^3*tan(c + d*x)^(1/2)*(a^2 - b^2)*(a^2 + b^2)
^2*((4*(-A^4*d^4*(a^6 - b^6 + 15*a^2*b^4 - 15*a^4*b^2)^2)^(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)/(d^4*(a^2 + b^2)^6))^(1/2))*((4*(-A^4*d^4*(a^6 - b^6 + 15*a^2*b^4 - 15*a^4*b^2)^2)^(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)/(d^4*(a^2 + b^2)^6))^(1/2))/4 + (8*A^2*b^2*tan(c + d*x)^(
1/2)*(8*a^10 + b^10 - 148*a^2*b^8 + 902*a^4*b^6 - 812*a^6*b^4 + 193*a^8*b^2))/(a*d^2*(a^2 + b^2)^4))*((4*(-A^4
*d^4*(a^6 - b^6 + 15*a^2*b^4 - 15*a^4*b^2)^2)^(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
)/(d^4*(a^2 + b^2)^6))^(1/2))/4 - (2*A^3*b^2*(16*a^12 + b^12 - 71*a^2*b^10 - 1382*a^4*b^8 + 5266*a^6*b^6 - 453
9*a^8*b^4 + 1189*a^10*b^2))/(a^2*d^3*(a^2 + b^2)^6))*((4*(-A^4*d^4*(a^6 - b^6 + 15*a^2*b^4 - 15*a^4*b^2)^2)^(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)/(d^4*(a^2 + b^2)^6))^(1/2))/4 - (A^4*b^3*tan(c
+ d*x)^(1/2)*(2*a^2*b^10 - b^12 - 225*a^12 + 49*a^4*b^8 + 2460*a^6*b^6 - 3631*a^8*b^4 + 1922*a^10*b^2))/(a^2*
d^4*(a^2 + b^2)^8))*((4*(-A^4*d^4*(a^6 - b^6 + 15*a^2*b^4 - 15*a^4*b^2)^2)^(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)/(d^4*(a^2 + b^2)^6))^(1/2))/4 + (A^5*b^3*(7*b^8 - 225*a^8 + 116*a^2*b^6 - 270*a
^4*b^4 + 420*a^6*b^2))/(2*a*d^5*(a^2 + b^2)^8))*(((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 - ((tan(c + d*x)^(1/2)*(A*b^3 + 9*A*a^2*b))/(4*(a^4 + b^
4 + 2*a^2*b^2)) - (tan(c + d*x)^(3/2)*(A*b^4 - 7*A*a^2*b^2))/(4*a*(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)) - ((tan(c + d*x)^(3/2)*(5*B*b^3 - 3*B*a^2*b))/(4*(a^4 + b^4 + 2*a^2*b^2))
- (a*tan(c + d*x)^(1/2)*(5*B*a^2 - 3*B*b^2))/(4*(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)) + (log((((((((((64*A*b^3*(b^4 - 10*a^4 + 15*a^2*b^2))/(a*d) + 128*b^3*tan(c + d*x)^(1/2)*(a^
2 - b^2)*(a^2 + b^2)^2*(-(4*(-A^4*d^4*(a^6 - b^6 + 15*a^2*b^4 - 15*a^4*b^2)^2)^(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)/(d^4*(a^2 + b^2)^6))^(1/2))*(-(4*(-A^4*d^4*(a^6 - b^6 + 15*a^2*b^4 - 15*a^4
*b^2)^2)^(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)/(d^4*(a^2 + b^2)^6))^(1/2))/4 + (8*
A^2*b^2*tan(c + d*x)^(1/2)*(8*a^10 + b^10 - 148*a^2*b^8 + 902*a^4*b^6 - 812*a^6*b^4 + 193*a^8*b^2))/(a*d^2*(a^
2 + b^2)^4))*(-(4*(-A^4*d^4*(a^6 - b^6 + 15*a^2*b^4 - 15*a^4*b^2)^2)^(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)/(d^4*(a^2 + b^2)^6))^(1/2))/4 - (2*A^3*b^2*(16*a^12 + b^12 - 71*a^2*b^10 - 1382*a^4*b
^8 + 5266*a^6*b^6 - 4539*a^8*b^4 + 1189*a^10*b^2))/(a^2*d^3*(a^2 + b^2)^6))*(-(4*(-A^4*d^4*(a^6 - b^6 + 15*a^2
*b^4 - 15*a^4*b^2)^2)^(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)/(d^4*(a^2 + b^2)^6))^(
1/2))/4 - (A^4*b^3*tan(c + d*x)^(1/2)*(2*a^2*b^10 - b^12 - 225*a^12 + 49*a^4*b^8 + 2460*a^6*b^6 - 3631*a^8*b^4
+ 1922*a^10*b^2))/(a^2*d^4*(a^2 + b^2)^8))*(-(4*(-A^4*d^4*(a^6 - b^6 + 15*a^2*b^4 - 15*a^4*b^2)^2)^(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)/(d^4*(a^2 + b^2)^6))^(1/2))/4 + (A^5*b^3*(7*b^8 - 225*
a^8 + 116*a^2*b^6 - 270*a^4*b^4 + 420*a^6*b^2))/(2*a*d^5*(a^2 + b^2)^8))*(-((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)/(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*A*b^3*(b^4 -
10*a^4 + 15*a^2*b^2))/(a*d) - 128*b^3*tan(c + d*x)^(1/2)*(a^2 - b^2)*(a^2 + b^2)^2*((4*(-A^4*d^4*(a^6 - b^6 +
15*a^2*b^4 - 15*a^4*b^2)^2)^(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)/(d^4*(a^2 + b^2
)^6))^(1/2))*((4*(-A^4*d^4*(a^6 - b^6 + 15*a^2*b^4 - 15*a^4*b^2)^2)^(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)/(d^4*(a^2 + b^2)^6))^(1/2))/4 - (8*A^2*b^2*tan(c + d*x)^(1/2)*(8*a^10 + b^10 - 148*a^2
*b^8 + 902*a^4*b^6 - 812*a^6*b^4 + 193*a^8*b^2))/(a*d^2*(a^2 + b^2)^4))*((4*(-A^4*d^4*(a^6 - b^6 + 15*a^2*b^4
- 15*a^4*b^2)^2)^(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)/(d^4*(a^2 + b^2)^6))^(1/2))
/4 - (2*A^3*b^2*(16*a^12 + b^12 - 71*a^2*b^10 - 1382*a^4*b^8 + 5266*a^6*b^6 - 4539*a^8*b^4 + 1189*a^10*b^2))/(
a^2*d^3*(a^2 + b^2)^6))*((4*(-A^4*d^4*(a^6 - b^6 + 15*a^2*b^4 - 15*a^4*b^2)^2)^(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)/(d^4*(a^2 + b^2)^6))^(1/2))/4 + (A^4*b^3*tan(c + d*x)^(1/2)*(2*a^2*b^10 - b
^12 - 225*a^12 + 49*a^4*b^8 + 2460*a^6*b^6 - 3631*a^8*b^4 + 1922*a^10*b^2))/(a^2*d^4*(a^2 + b^2)^8))*((4*(-A^4
*d^4*(a^6 - b^6 + 15*a^2*b^4 - 15*a^4*b^2)^2)^(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
)/(d^4*(a^2 + b^2)^6))^(1/2))/4 + (A^5*b^3*(7*b^8 - 225*a^8 + 116*a^2*b^6 - 270*a^4*b^4 + 420*a^6*b^2))/(2*a*d
^5*(a^2 + b^2)^8))*(((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*A*b^3*(b^4 - 10*a^4 + 15*a^2*b^2))/(a*d) - 128*b^3*tan(c +
d*x)^(1/2)*(a^2 - b^2)*(a^2 + b^2)^2*(-(4*(-A^4*d^4*(a^6 - b^6 + 15*a^2*b^4 - 15*a^4*b^2)^2)^(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)/(d^4*(a^2 + b^2)^6))^(1/2))*(-(4*(-A^4*d^4*(a^6 - b^6 + 15*a^
2*b^4 - 15*a^4*b^2)^2)^(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)/(d^4*(a^2 + b^2)^6))^
(1/2))/4 - (8*A^2*b^2*tan(c + d*x)^(1/2)*(8*a^10 + b^10 - 148*a^2*b^8 + 902*a^4*b^6 - 812*a^6*b^4 + 193*a^8*b^
2))/(a*d^2*(a^2 + b^2)^4))*(-(4*(-A^4*d^4*(a^6 - b^6 + 15*a^2*b^4 - 15*a^4*b^2)^2)^(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)/(d^4*(a^2 + b^2)^6))^(1/2))/4 - (2*A^3*b^2*(16*a^12 + b^12 - 71*a^2*b^1
0 - 1382*a^4*b^8 + 5266*a^6*b^6 - 4539*a^8*b^4 + 1189*a^10*b^2))/(a^2*d^3*(a^2 + b^2)^6))*(-(4*(-A^4*d^4*(a^6
- b^6 + 15*a^2*b^4 - 15*a^4*b^2)^2)^(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)/(d^4*(a^
2 + b^2)^6))^(1/2))/4 + (A^4*b^3*tan(c + d*x)^(1/2)*(2*a^2*b^10 - b^12 - 225*a^12 + 49*a^4*b^8 + 2460*a^6*b^6
- 3631*a^8*b^4 + 1922*a^10*b^2))/(a^2*d^4*(a^2 + b^2)^8))*(-(4*(-A^4*d^4*(a^6 - b^6 + 15*a^2*b^4 - 15*a^4*b^2)
^2)^(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)/(d^4*(a^2 + b^2)^6))^(1/2))/4 + (A^5*b^3
*(7*b^8 - 225*a^8 + 116*a^2*b^6 - 270*a^4*b^4 + 420*a^6*b^2))/(2*a*d^5*(a^2 + b^2)^8))*(-((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 + 4
80*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*(2*a^4 + 3*b^4 - 19*a^2*b^2))/d + 128*b^3*tan(c + d*x)^(1/2)*(a^2 - b^2)*(a^2 + b^2)^2*((4*(-
B^4*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*a*b^2*tan(c + d*x)^(1/2)*(a
^8 + 193*b^8 - 764*a^2*b^6 + 966*a^4*b^4 - 124*a^6*b^2))/(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*a*b*(9*a^10 - 259*b^10 + 2765*a^2*b^8 - 6782*a^4*b^6 + 2202*a^6*b^4 - 271*a^8*b^2))/(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*tan(c + d*x)^(1/2)*(9*a^12 + 41*b^12 - 82*a^2*b
^10 + 1831*a^4*b^8 - 4348*a^6*b^6 + 1671*a^8*b^4 - 210*a^10*b^2))/(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^2*(9*a^8 - 15*b^8 + 28*a^2*b^6 + 878*a^4*b^4 - 180*a^6*b^2))/(2*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 - 4
080*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*(2*a^4 + 3*b^4 - 19*a^2*b^2))/d + 128*b^3*tan(c + d*x)^(1/2)*(a^2 - b^2)*(a^2
+ b^2)^2*(-(4*(-B^4*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*a*b^2*tan(
c + d*x)^(1/2)*(a^8 + 193*b^8 - 764*a^2*b^6 + 966*a^4*b^4 - 124*a^6*b^2))/(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*a*b*(9*a^10 - 259*b^10 + 2765*a^2*b^8 - 6782*a^4*b^6 + 2202*a^6*b^4 - 271*
a^8*b^2))/(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*tan(c + d*x)^(1/2)*(9*a^12 +
41*b^12 - 82*a^2*b^10 + 1831*a^4*b^8 - 4348*a^6*b^6 + 1671*a^8*b^4 - 210*a^10*b^2))/(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^2*(9*a^8 - 15*b^8 + 28*a^2*b^6 + 878*a^4*b^4 - 180*a^6*b^2))/(2
*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 + 723
2*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*(2*a^4 + 3*b^4 - 19*a^2*b^2))/d - 128*b^3*tan(c + d*x)^(1/
2)*(a^2 - b^2)*(a^2 + b^2)^2*((4*(-B^4*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*a*b^2*tan(c + d*x)^(1/2)*(a^8 + 193*b^8 - 764*a^2*b^6 + 966*a^4*b^4 - 124*a^6*b^2))/(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*a*b*(9*a^10 - 259*b^10 + 2765*a^2*b^8 - 6782*a^4*b^6 + 22
02*a^6*b^4 - 271*a^8*b^2))/(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*tan(c + d*x)
^(1/2)*(9*a^12 + 41*b^12 - 82*a^2*b^10 + 1831*a^4*b^8 - 4348*a^6*b^6 + 1671*a^8*b^4 - 210*a^10*b^2))/(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^2*(9*a^8 - 15*b^8 + 28*a^2*b^6 + 878*a^4*b^4 - 1
80*a^6*b^2))/(2*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)/(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*(2*a^4 + 3*b^4 - 19*a^2*b^2))/d - 12
8*b^3*tan(c + d*x)^(1/2)*(a^2 - b^2)*(a^2 + b^2)^2*(-(4*(-B^4*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*a*b^2*tan(c + d*x)^(1/2)*(a^8 + 193*b^8 - 764*a^2*b^6 + 966*a^4*b^4 - 124*a^6*
b^2))/(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*a*b*(9*a^10 - 259*b^10 + 2765*a^
2*b^8 - 6782*a^4*b^6 + 2202*a^6*b^4 - 271*a^8*b^2))/(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*tan(c + d*x)^(1/2)*(9*a^12 + 41*b^12 - 82*a^2*b^10 + 1831*a^4*b^8 - 4348*a^6*b^6 + 1671*a^8*b^4
- 210*a^10*b^2))/(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^2*(9*a^8 - 15*b^8 + 2
8*a^2*b^6 + 878*a^4*b^4 - 180*a^6*b^2))/(2*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)/(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(((((tan(c + d*x)^(1/2)*(
2*A^4*a^2*b^13 - A^4*b^15 + 49*A^4*a^4*b^11 + 2460*A^4*a^6*b^9 - 3631*A^4*a^8*b^7 + 1922*A^4*a^10*b^5 - 225*A^
4*a^12*b^3))/(64*(a^18*d^4 + a^2*b^16*d^4 + 8*a^4*b^14*d^4 + 28*a^6*b^12*d^4 + 56*a^8*b^10*d^4 + 70*a^10*b^8*d
^4 + 56*a^12*b^6*d^4 + 28*a^14*b^4*d^4 + 8*a^16*b^2*d^4)) + (((2*A^3*b^18*d^2 - 138*A^3*a^2*b^16*d^2 - 3046*A^
3*a^4*b^14*d^2 + 4862*A^3*a^6*b^12*d^2 + 9222*A^3*a^8*b^10*d^2 - 5246*A^3*a^10*b^8*d^2 - 4290*A^3*a^12*b^6*d^2
+ 2442*A^3*a^14*b^4*d^2 + 32*A^3*a^16*b^2*d^2)/(64*(a^18*d^5 + a^2*b^16*d^5 + 8*a^4*b^14*d^5 + 28*a^6*b^12*d^
5 + 56*a^8*b^10*d^5 + 70*a^10*b^8*d^5 + 56*a^12*b^6*d^5 + 28*a^14*b^4*d^5 + 8*a^16*b^2*d^5)) - (((((64*A*a*b^2
3*d^4 + 1472*A*a^3*b^21*d^4 + 8832*A*a^5*b^19*d^4 + 25344*A*a^7*b^17*d^4 + 40320*A*a^9*b^15*d^4 + 34944*A*a^11
*b^13*d^4 + 10752*A*a^13*b^11*d^4 - 8448*A*a^15*b^9*d^4 - 10176*A*a^17*b^7*d^4 - 4160*A*a^19*b^5*d^4 - 640*A*a
^21*b^3*d^4)/(64*(a^18*d^5 + a^2*b^16*d^5 + 8*a^4*b^14*d^5 + 28*a^6*b^12*d^5 + 56*a^8*b^10*d^5 + 70*a^10*b^8*d
^5 + 56*a^12*b^6*d^5 + 28*a^14*b^4*d^5 + 8*a^16*b^2*d^5)) - (tan(c + d*x)^(1/2)*(-64*(A^2*b^9 + 225*A^2*a^8*b
+ 36*A^2*a^2*b^7 + 294*A^2*a^4*b^5 - 540*A^2*a^6*b^3)*(a^15*d^2 + a^3*b^12*d^2 + 6*a^5*b^10*d^2 + 15*a^7*b^8*d
^2 + 20*a^9*b^6*d^2 + 15*a^11*b^4*d^2 + 6*a^13*b^2*d^2))^(1/2)*(512*a^2*b^25*d^4 + 4608*a^4*b^23*d^4 + 17920*a
^6*b^21*d^4 + 38400*a^8*b^19*d^4 + 46080*a^10*b^17*d^4 + 21504*a^12*b^15*d^4 - 21504*a^14*b^13*d^4 - 46080*a^1
6*b^11*d^4 - 38400*a^18*b^9*d^4 - 17920*a^20*b^7*d^4 - 4608*a^22*b^5*d^4 - 512*a^24*b^3*d^4))/(4096*(a^15*d^2
+ a^3*b^12*d^2 + 6*a^5*b^10*d^2 + 15*a^7*b^8*d^2 + 20*a^9*b^6*d^2 + 15*a^11*b^4*d^2 + 6*a^13*b^2*d^2)*(a^18*d^
4 + a^2*b^16*d^4 + 8*a^4*b^14*d^4 + 28*a^6*b^12*d^4 + 56*a^8*b^10*d^4 + 70*a^10*b^8*d^4 + 56*a^12*b^6*d^4 + 28
*a^14*b^4*d^4 + 8*a^16*b^2*d^4)))*(-64*(A^2*b^9 + 225*A^2*a^8*b + 36*A^2*a^2*b^7 + 294*A^2*a^4*b^5 - 540*A^2*a
^6*b^3)*(a^15*d^2 + a^3*b^12*d^2 + 6*a^5*b^10*d^2 + 15*a^7*b^8*d^2 + 20*a^9*b^6*d^2 + 15*a^11*b^4*d^2 + 6*a^13
*b^2*d^2))^(1/2))/(64*(a^15*d^2 + a^3*b^12*d^2 + 6*a^5*b^10*d^2 + 15*a^7*b^8*d^2 + 20*a^9*b^6*d^2 + 15*a^11*b^
4*d^2 + 6*a^13*b^2*d^2)) + (tan(c + d*x)^(1/2)*(2528*A^2*a^5*b^16*d^2 - 1152*A^2*a^3*b^18*d^2 + 15296*A^2*a^7*
b^14*d^2 + 14128*A^2*a^9*b^12*d^2 - 5056*A^2*a^11*b^10*d^2 - 9248*A^2*a^13*b^8*d^2 + 64*A^2*a^15*b^6*d^2 + 180
0*A^2*a^17*b^4*d^2 + 64*A^2*a^19*b^2*d^2 + 8*A^2*a*b^20*d^2))/(64*(a^18*d^4 + a^2*b^16*d^4 + 8*a^4*b^14*d^4 +
28*a^6*b^12*d^4 + 56*a^8*b^10*d^4 + 70*a^10*b^8*d^4 + 56*a^12*b^6*d^4 + 28*a^14*b^4*d^4 + 8*a^16*b^2*d^4)))*(-
64*(A^2*b^9 + 225*A^2*a^8*b + 36*A^2*a^2*b^7 + 294*A^2*a^4*b^5 - 540*A^2*a^6*b^3)*(a^15*d^2 + a^3*b^12*d^2 + 6
*a^5*b^10*d^2 + 15*a^7*b^8*d^2 + 20*a^9*b^6*d^2 + 15*a^11*b^4*d^2 + 6*a^13*b^2*d^2))^(1/2))/(64*(a^15*d^2 + a^
3*b^12*d^2 + 6*a^5*b^10*d^2 + 15*a^7*b^8*d^2 + 20*a^9*b^6*d^2 + 15*a^11*b^4*d^2 + 6*a^13*b^2*d^2)))*(-64*(A^2*
b^9 + 225*A^2*a^8*b + 36*A^2*a^2*b^7 + 294*A^2*a^4*b^5 - 540*A^2*a^6*b^3)*(a^15*d^2 + a^3*b^12*d^2 + 6*a^5*b^1
0*d^2 + 15*a^7*b^8*d^2 + 20*a^9*b^6*d^2 + 15*a^11*b^4*d^2 + 6*a^13*b^2*d^2))^(1/2))/(64*(a^15*d^2 + a^3*b^12*d
^2 + 6*a^5*b^10*d^2 + 15*a^7*b^8*d^2 + 20*a^9*b^6*d^2 + 15*a^11*b^4*d^2 + 6*a^13*b^2*d^2)))*(-64*(A^2*b^9 + 22
5*A^2*a^8*b + 36*A^2*a^2*b^7 + 294*A^2*a^4*b^5 - 540*A^2*a^6*b^3)*(a^15*d^2 + a^3*b^12*d^2 + 6*a^5*b^10*d^2 +
15*a^7*b^8*d^2 + 20*a^9*b^6*d^2 + 15*a^11*b^4*d^2 + 6*a^13*b^2*d^2))^(1/2)*1i)/(a^15*d^2 + a^3*b^12*d^2 + 6*a^
5*b^10*d^2 + 15*a^7*b^8*d^2 + 20*a^9*b^6*d^2 + 15*a^11*b^4*d^2 + 6*a^13*b^2*d^2) + (((tan(c + d*x)^(1/2)*(2*A^
4*a^2*b^13 - A^4*b^15 + 49*A^4*a^4*b^11 + 2460*A^4*a^6*b^9 - 3631*A^4*a^8*b^7 + 1922*A^4*a^10*b^5 - 225*A^4*a^
12*b^3))/(64*(a^18*d^4 + a^2*b^16*d^4 + 8*a^4*b^14*d^4 + 28*a^6*b^12*d^4 + 56*a^8*b^10*d^4 + 70*a^10*b^8*d^4 +
56*a^12*b^6*d^4 + 28*a^14*b^4*d^4 + 8*a^16*b^2*d^4)) - (((2*A^3*b^18*d^2 - 138*A^3*a^2*b^16*d^2 - 3046*A^3*a^
4*b^14*d^2 + 4862*A^3*a^6*b^12*d^2 + 9222*A^3*a^8*b^10*d^2 - 5246*A^3*a^10*b^8*d^2 - 4290*A^3*a^12*b^6*d^2 + 2
442*A^3*a^14*b^4*d^2 + 32*A^3*a^16*b^2*d^2)/(64*(a^18*d^5 + a^2*b^16*d^5 + 8*a^4*b^14*d^5 + 28*a^6*b^12*d^5 +
56*a^8*b^10*d^5 + 70*a^10*b^8*d^5 + 56*a^12*b^6*d^5 + 28*a^14*b^4*d^5 + 8*a^16*b^2*d^5)) - (((((64*A*a*b^23*d^
4 + 1472*A*a^3*b^21*d^4 + 8832*A*a^5*b^19*d^4 + 25344*A*a^7*b^17*d^4 + 40320*A*a^9*b^15*d^4 + 34944*A*a^11*b^1
3*d^4 + 10752*A*a^13*b^11*d^4 - 8448*A*a^15*b^9*d^4 - 10176*A*a^17*b^7*d^4 - 4160*A*a^19*b^5*d^4 - 640*A*a^21*
b^3*d^4)/(64*(a^18*d^5 + a^2*b^16*d^5 + 8*a^4*b^14*d^5 + 28*a^6*b^12*d^5 + 56*a^8*b^10*d^5 + 70*a^10*b^8*d^5 +
56*a^12*b^6*d^5 + 28*a^14*b^4*d^5 + 8*a^16*b^2*d^5)) + (tan(c + d*x)^(1/2)*(-64*(A^2*b^9 + 225*A^2*a^8*b + 36
*A^2*a^2*b^7 + 294*A^2*a^4*b^5 - 540*A^2*a^6*b^3)*(a^15*d^2 + a^3*b^12*d^2 + 6*a^5*b^10*d^2 + 15*a^7*b^8*d^2 +
20*a^9*b^6*d^2 + 15*a^11*b^4*d^2 + 6*a^13*b^2*d^2))^(1/2)*(512*a^2*b^25*d^4 + 4608*a^4*b^23*d^4 + 17920*a^6*b
^21*d^4 + 38400*a^8*b^19*d^4 + 46080*a^10*b^17*d^4 + 21504*a^12*b^15*d^4 - 21504*a^14*b^13*d^4 - 46080*a^16*b^
11*d^4 - 38400*a^18*b^9*d^4 - 17920*a^20*b^7*d^4 - 4608*a^22*b^5*d^4 - 512*a^24*b^3*d^4))/(4096*(a^15*d^2 + a^
3*b^12*d^2 + 6*a^5*b^10*d^2 + 15*a^7*b^8*d^2 + 20*a^9*b^6*d^2 + 15*a^11*b^4*d^2 + 6*a^13*b^2*d^2)*(a^18*d^4 +
a^2*b^16*d^4 + 8*a^4*b^14*d^4 + 28*a^6*b^12*d^4 + 56*a^8*b^10*d^4 + 70*a^10*b^8*d^4 + 56*a^12*b^6*d^4 + 28*a^1
4*b^4*d^4 + 8*a^16*b^2*d^4)))*(-64*(A^2*b^9 + 225*A^2*a^8*b + 36*A^2*a^2*b^7 + 294*A^2*a^4*b^5 - 540*A^2*a^6*b
^3)*(a^15*d^2 + a^3*b^12*d^2 + 6*a^5*b^10*d^2 + 15*a^7*b^8*d^2 + 20*a^9*b^6*d^2 + 15*a^11*b^4*d^2 + 6*a^13*b^2
*d^2))^(1/2))/(64*(a^15*d^2 + a^3*b^12*d^2 + 6*a^5*b^10*d^2 + 15*a^7*b^8*d^2 + 20*a^9*b^6*d^2 + 15*a^11*b^4*d^
2 + 6*a^13*b^2*d^2)) - (tan(c + d*x)^(1/2)*(2528*A^2*a^5*b^16*d^2 - 1152*A^2*a^3*b^18*d^2 + 15296*A^2*a^7*b^14
*d^2 + 14128*A^2*a^9*b^12*d^2 - 5056*A^2*a^11*b^10*d^2 - 9248*A^2*a^13*b^8*d^2 + 64*A^2*a^15*b^6*d^2 + 1800*A^
2*a^17*b^4*d^2 + 64*A^2*a^19*b^2*d^2 + 8*A^2*a*b^20*d^2))/(64*(a^18*d^4 + a^2*b^16*d^4 + 8*a^4*b^14*d^4 + 28*a
^6*b^12*d^4 + 56*a^8*b^10*d^4 + 70*a^10*b^8*d^4 + 56*a^12*b^6*d^4 + 28*a^14*b^4*d^4 + 8*a^16*b^2*d^4)))*(-64*(
A^2*b^9 + 225*A^2*a^8*b + 36*A^2*a^2*b^7 + 294*A^2*a^4*b^5 - 540*A^2*a^6*b^3)*(a^15*d^2 + a^3*b^12*d^2 + 6*a^5
*b^10*d^2 + 15*a^7*b^8*d^2 + 20*a^9*b^6*d^2 + 15*a^11*b^4*d^2 + 6*a^13*b^2*d^2))^(1/2))/(64*(a^15*d^2 + a^3*b^
12*d^2 + 6*a^5*b^10*d^2 + 15*a^7*b^8*d^2 + 20*a^9*b^6*d^2 + 15*a^11*b^4*d^2 + 6*a^13*b^2*d^2)))*(-64*(A^2*b^9
+ 225*A^2*a^8*b + 36*A^2*a^2*b^7 + 294*A^2*a^4*b^5 - 540*A^2*a^6*b^3)*(a^15*d^2 + a^3*b^12*d^2 + 6*a^5*b^10*d^
2 + 15*a^7*b^8*d^2 + 20*a^9*b^6*d^2 + 15*a^11*b^4*d^2 + 6*a^13*b^2*d^2))^(1/2))/(64*(a^15*d^2 + a^3*b^12*d^2 +
6*a^5*b^10*d^2 + 15*a^7*b^8*d^2 + 20*a^9*b^6*d^2 + 15*a^11*b^4*d^2 + 6*a^13*b^2*d^2)))*(-64*(A^2*b^9 + 225*A^
2*a^8*b + 36*A^2*a^2*b^7 + 294*A^2*a^4*b^5 - 540*A^2*a^6*b^3)*(a^15*d^2 + a^3*b^12*d^2 + 6*a^5*b^10*d^2 + 15*a
^7*b^8*d^2 + 20*a^9*b^6*d^2 + 15*a^11*b^4*d^2 + 6*a^13*b^2*d^2))^(1/2)*1i)/(a^15*d^2 + a^3*b^12*d^2 + 6*a^5*b^
10*d^2 + 15*a^7*b^8*d^2 + 20*a^9*b^6*d^2 + 15*a^11*b^4*d^2 + 6*a^13*b^2*d^2))/((7*A^5*a*b^11 + 116*A^5*a^3*b^9
- 270*A^5*a^5*b^7 + 420*A^5*a^7*b^5 - 225*A^5*a^9*b^3)/(a^18*d^5 + a^2*b^16*d^5 + 8*a^4*b^14*d^5 + 28*a^6*b^1
2*d^5 + 56*a^8*b^10*d^5 + 70*a^10*b^8*d^5 + 56*a^12*b^6*d^5 + 28*a^14*b^4*d^5 + 8*a^16*b^2*d^5) - (((tan(c + d
*x)^(1/2)*(2*A^4*a^2*b^13 - A^4*b^15 + 49*A^4*a^4*b^11 + 2460*A^4*a^6*b^9 - 3631*A^4*a^8*b^7 + 1922*A^4*a^10*b
^5 - 225*A^4*a^12*b^3))/(64*(a^18*d^4 + a^2*b^16*d^4 + 8*a^4*b^14*d^4 + 28*a^6*b^12*d^4 + 56*a^8*b^10*d^4 + 70
*a^10*b^8*d^4 + 56*a^12*b^6*d^4 + 28*a^14*b^4*d^4 + 8*a^16*b^2*d^4)) + (((2*A^3*b^18*d^2 - 138*A^3*a^2*b^16*d^
2 - 3046*A^3*a^4*b^14*d^2 + 4862*A^3*a^6*b^12*d^2 + 9222*A^3*a^8*b^10*d^2 - 5246*A^3*a^10*b^8*d^2 - 4290*A^3*a
^12*b^6*d^2 + 2442*A^3*a^14*b^4*d^2 + 32*A^3*a^16*b^2*d^2)/(64*(a^18*d^5 + a^2*b^16*d^5 + 8*a^4*b^14*d^5 + 28*
a^6*b^12*d^5 + 56*a^8*b^10*d^5 + 70*a^10*b^8*d^5 + 56*a^12*b^6*d^5 + 28*a^14*b^4*d^5 + 8*a^16*b^2*d^5)) - ((((
(64*A*a*b^23*d^4 + 1472*A*a^3*b^21*d^4 + 8832*A*a^5*b^19*d^4 + 25344*A*a^7*b^17*d^4 + 40320*A*a^9*b^15*d^4 + 3
4944*A*a^11*b^13*d^4 + 10752*A*a^13*b^11*d^4 - 8448*A*a^15*b^9*d^4 - 10176*A*a^17*b^7*d^4 - 4160*A*a^19*b^5*d^
4 - 640*A*a^21*b^3*d^4)/(64*(a^18*d^5 + a^2*b^16*d^5 + 8*a^4*b^14*d^5 + 28*a^6*b^12*d^5 + 56*a^8*b^10*d^5 + 70
*a^10*b^8*d^5 + 56*a^12*b^6*d^5 + 28*a^14*b^4*d^5 + 8*a^16*b^2*d^5)) - (tan(c + d*x)^(1/2)*(-64*(A^2*b^9 + 225
*A^2*a^8*b + 36*A^2*a^2*b^7 + 294*A^2*a^4*b^5 - 540*A^2*a^6*b^3)*(a^15*d^2 + a^3*b^12*d^2 + 6*a^5*b^10*d^2 + 1
5*a^7*b^8*d^2 + 20*a^9*b^6*d^2 + 15*a^11*b^4*d^2 + 6*a^13*b^2*d^2))^(1/2)*(512*a^2*b^25*d^4 + 4608*a^4*b^23*d^
4 + 17920*a^6*b^21*d^4 + 38400*a^8*b^19*d^4 + 46080*a^10*b^17*d^4 + 21504*a^12*b^15*d^4 - 21504*a^14*b^13*d^4
- 46080*a^16*b^11*d^4 - 38400*a^18*b^9*d^4 - 17920*a^20*b^7*d^4 - 4608*a^22*b^5*d^4 - 512*a^24*b^3*d^4))/(4096
*(a^15*d^2 + a^3*b^12*d^2 + 6*a^5*b^10*d^2 + 15*a^7*b^8*d^2 + 20*a^9*b^6*d^2 + 15*a^11*b^4*d^2 + 6*a^13*b^2*d^
2)*(a^18*d^4 + a^2*b^16*d^4 + 8*a^4*b^14*d^4 + 28*a^6*b^12*d^4 + 56*a^8*b^10*d^4 + 70*a^10*b^8*d^4 + 56*a^12*b
^6*d^4 + 28*a^14*b^4*d^4 + 8*a^16*b^2*d^4)))*(-64*(A^2*b^9 + 225*A^2*a^8*b + 36*A^2*a^2*b^7 + 294*A^2*a^4*b^5
- 540*A^2*a^6*b^3)*(a^15*d^2 + a^3*b^12*d^2 + 6*a^5*b^10*d^2 + 15*a^7*b^8*d^2 + 20*a^9*b^6*d^2 + 15*a^11*b^4*d
^2 + 6*a^13*b^2*d^2))^(1/2))/(64*(a^15*d^2 + a^3*b^12*d^2 + 6*a^5*b^10*d^2 + 15*a^7*b^8*d^2 + 20*a^9*b^6*d^2 +
15*a^11*b^4*d^2 + 6*a^13*b^2*d^2)) + (tan(c + d*x)^(1/2)*(2528*A^2*a^5*b^16*d^2 - 1152*A^2*a^3*b^18*d^2 + 152
96*A^2*a^7*b^14*d^2 + 14128*A^2*a^9*b^12*d^2 - 5056*A^2*a^11*b^10*d^2 - 9248*A^2*a^13*b^8*d^2 + 64*A^2*a^15*b^
6*d^2 + 1800*A^2*a^17*b^4*d^2 + 64*A^2*a^19*b^2*d^2 + 8*A^2*a*b^20*d^2))/(64*(a^18*d^4 + a^2*b^16*d^4 + 8*a^4*
b^14*d^4 + 28*a^6*b^12*d^4 + 56*a^8*b^10*d^4 + 70*a^10*b^8*d^4 + 56*a^12*b^6*d^4 + 28*a^14*b^4*d^4 + 8*a^16*b^
2*d^4)))*(-64*(A^2*b^9 + 225*A^2*a^8*b + 36*A^2*a^2*b^7 + 294*A^2*a^4*b^5 - 540*A^2*a^6*b^3)*(a^15*d^2 + a^3*b
^12*d^2 + 6*a^5*b^10*d^2 + 15*a^7*b^8*d^2 + 20*a^9*b^6*d^2 + 15*a^11*b^4*d^2 + 6*a^13*b^2*d^2))^(1/2))/(64*(a^
15*d^2 + a^3*b^12*d^2 + 6*a^5*b^10*d^2 + 15*a^7*b^8*d^2 + 20*a^9*b^6*d^2 + 15*a^11*b^4*d^2 + 6*a^13*b^2*d^2)))
*(-64*(A^2*b^9 + 225*A^2*a^8*b + 36*A^2*a^2*b^7 + 294*A^2*a^4*b^5 - 540*A^2*a^6*b^3)*(a^15*d^2 + a^3*b^12*d^2
+ 6*a^5*b^10*d^2 + 15*a^7*b^8*d^2 + 20*a^9*b^6*d^2 + 15*a^11*b^4*d^2 + 6*a^13*b^2*d^2))^(1/2))/(64*(a^15*d^2 +
a^3*b^12*d^2 + 6*a^5*b^10*d^2 + 15*a^7*b^8*d^2 + 20*a^9*b^6*d^2 + 15*a^11*b^4*d^2 + 6*a^13*b^2*d^2)))*(-64*(A
^2*b^9 + 225*A^2*a^8*b + 36*A^2*a^2*b^7 + 294*A^2*a^4*b^5 - 540*A^2*a^6*b^3)*(a^15*d^2 + a^3*b^12*d^2 + 6*a^5*
b^10*d^2 + 15*a^7*b^8*d^2 + 20*a^9*b^6*d^2 + 15*a^11*b^4*d^2 + 6*a^13*b^2*d^2))^(1/2))/(a^15*d^2 + a^3*b^12*d^
2 + 6*a^5*b^10*d^2 + 15*a^7*b^8*d^2 + 20*a^9*b^6*d^2 + 15*a^11*b^4*d^2 + 6*a^13*b^2*d^2) + (((tan(c + d*x)^(1/
2)*(2*A^4*a^2*b^13 - A^4*b^15 + 49*A^4*a^4*b^11 + 2460*A^4*a^6*b^9 - 3631*A^4*a^8*b^7 + 1922*A^4*a^10*b^5 - 22
5*A^4*a^12*b^3))/(64*(a^18*d^4 + a^2*b^16*d^4 + 8*a^4*b^14*d^4 + 28*a^6*b^12*d^4 + 56*a^8*b^10*d^4 + 70*a^10*b
^8*d^4 + 56*a^12*b^6*d^4 + 28*a^14*b^4*d^4 + 8*a^16*b^2*d^4)) - (((2*A^3*b^18*d^2 - 138*A^3*a^2*b^16*d^2 - 304
6*A^3*a^4*b^14*d^2 + 4862*A^3*a^6*b^12*d^2 + 9222*A^3*a^8*b^10*d^2 - 5246*A^3*a^10*b^8*d^2 - 4290*A^3*a^12*b^6
*d^2 + 2442*A^3*a^14*b^4*d^2 + 32*A^3*a^16*b^2*d^2)/(64*(a^18*d^5 + a^2*b^16*d^5 + 8*a^4*b^14*d^5 + 28*a^6*b^1
2*d^5 + 56*a^8*b^10*d^5 + 70*a^10*b^8*d^5 + 56*a^12*b^6*d^5 + 28*a^14*b^4*d^5 + 8*a^16*b^2*d^5)) - (((((64*A*a
*b^23*d^4 + 1472*A*a^3*b^21*d^4 + 8832*A*a^5*b^19*d^4 + 25344*A*a^7*b^17*d^4 + 40320*A*a^9*b^15*d^4 + 34944*A*
a^11*b^13*d^4 + 10752*A*a^13*b^11*d^4 - 8448*A*a^15*b^9*d^4 - 10176*A*a^17*b^7*d^4 - 4160*A*a^19*b^5*d^4 - 640
*A*a^21*b^3*d^4)/(64*(a^18*d^5 + a^2*b^16*d^5 + 8*a^4*b^14*d^5 + 28*a^6*b^12*d^5 + 56*a^8*b^10*d^5 + 70*a^10*b
^8*d^5 + 56*a^12*b^6*d^5 + 28*a^14*b^4*d^5 + 8*a^16*b^2*d^5)) + (tan(c + d*x)^(1/2)*(-64*(A^2*b^9 + 225*A^2*a^
8*b + 36*A^2*a^2*b^7 + 294*A^2*a^4*b^5 - 540*A^2*a^6*b^3)*(a^15*d^2 + a^3*b^12*d^2 + 6*a^5*b^10*d^2 + 15*a^7*b
^8*d^2 + 20*a^9*b^6*d^2 + 15*a^11*b^4*d^2 + 6*a^13*b^2*d^2))^(1/2)*(512*a^2*b^25*d^4 + 4608*a^4*b^23*d^4 + 179
20*a^6*b^21*d^4 + 38400*a^8*b^19*d^4 + 46080*a^10*b^17*d^4 + 21504*a^12*b^15*d^4 - 21504*a^14*b^13*d^4 - 46080
*a^16*b^11*d^4 - 38400*a^18*b^9*d^4 - 17920*a^20*b^7*d^4 - 4608*a^22*b^5*d^4 - 512*a^24*b^3*d^4))/(4096*(a^15*
d^2 + a^3*b^12*d^2 + 6*a^5*b^10*d^2 + 15*a^7*b^8*d^2 + 20*a^9*b^6*d^2 + 15*a^11*b^4*d^2 + 6*a^13*b^2*d^2)*(a^1
8*d^4 + a^2*b^16*d^4 + 8*a^4*b^14*d^4 + 28*a^6*b^12*d^4 + 56*a^8*b^10*d^4 + 70*a^10*b^8*d^4 + 56*a^12*b^6*d^4
+ 28*a^14*b^4*d^4 + 8*a^16*b^2*d^4)))*(-64*(A^2*b^9 + 225*A^2*a^8*b + 36*A^2*a^2*b^7 + 294*A^2*a^4*b^5 - 540*A
^2*a^6*b^3)*(a^15*d^2 + a^3*b^12*d^2 + 6*a^5*b^10*d^2 + 15*a^7*b^8*d^2 + 20*a^9*b^6*d^2 + 15*a^11*b^4*d^2 + 6*
a^13*b^2*d^2))^(1/2))/(64*(a^15*d^2 + a^3*b^12*d^2 + 6*a^5*b^10*d^2 + 15*a^7*b^8*d^2 + 20*a^9*b^6*d^2 + 15*a^1
1*b^4*d^2 + 6*a^13*b^2*d^2)) - (tan(c + d*x)^(1/2)*(2528*A^2*a^5*b^16*d^2 - 1152*A^2*a^3*b^18*d^2 + 15296*A^2*
a^7*b^14*d^2 + 14128*A^2*a^9*b^12*d^2 - 5056*A^2*a^11*b^10*d^2 - 9248*A^2*a^13*b^8*d^2 + 64*A^2*a^15*b^6*d^2 +
1800*A^2*a^17*b^4*d^2 + 64*A^2*a^19*b^2*d^2 + 8*A^2*a*b^20*d^2))/(64*(a^18*d^4 + a^2*b^16*d^4 + 8*a^4*b^14*d^
4 + 28*a^6*b^12*d^4 + 56*a^8*b^10*d^4 + 70*a^10*b^8*d^4 + 56*a^12*b^6*d^4 + 28*a^14*b^4*d^4 + 8*a^16*b^2*d^4))
)*(-64*(A^2*b^9 + 225*A^2*a^8*b + 36*A^2*a^2*b^7 + 294*A^2*a^4*b^5 - 540*A^2*a^6*b^3)*(a^15*d^2 + a^3*b^12*d^2
+ 6*a^5*b^10*d^2 + 15*a^7*b^8*d^2 + 20*a^9*b^6*d^2 + 15*a^11*b^4*d^2 + 6*a^13*b^2*d^2))^(1/2))/(64*(a^15*d^2
+ a^3*b^12*d^2 + 6*a^5*b^10*d^2 + 15*a^7*b^8*d^2 + 20*a^9*b^6*d^2 + 15*a^11*b^4*d^2 + 6*a^13*b^2*d^2)))*(-64*(
A^2*b^9 + 225*A^2*a^8*b + 36*A^2*a^2*b^7 + 294*A^2*a^4*b^5 - 540*A^2*a^6*b^3)*(a^15*d^2 + a^3*b^12*d^2 + 6*a^5
*b^10*d^2 + 15*a^7*b^8*d^2 + 20*a^9*b^6*d^2 + 15*a^11*b^4*d^2 + 6*a^13*b^2*d^2))^(1/2))/(64*(a^15*d^2 + a^3*b^
12*d^2 + 6*a^5*b^10*d^2 + 15*a^7*b^8*d^2 + 20*a^9*b^6*d^2 + 15*a^11*b^4*d^2 + 6*a^13*b^2*d^2)))*(-64*(A^2*b^9
+ 225*A^2*a^8*b + 36*A^2*a^2*b^7 + 294*A^2*a^4*b^5 - 540*A^2*a^6*b^3)*(a^15*d^2 + a^3*b^12*d^2 + 6*a^5*b^10*d^
2 + 15*a^7*b^8*d^2 + 20*a^9*b^6*d^2 + 15*a^11*b^4*d^2 + 6*a^13*b^2*d^2))^(1/2))/(a^15*d^2 + a^3*b^12*d^2 + 6*a
^5*b^10*d^2 + 15*a^7*b^8*d^2 + 20*a^9*b^6*d^2 + 15*a^11*b^4*d^2 + 6*a^13*b^2*d^2)))*(-64*(A^2*b^9 + 225*A^2*a^
8*b + 36*A^2*a^2*b^7 + 294*A^2*a^4*b^5 - 540*A^2*a^6*b^3)*(a^15*d^2 + a^3*b^12*d^2 + 6*a^5*b^10*d^2 + 15*a^7*b
^8*d^2 + 20*a^9*b^6*d^2 + 15*a^11*b^4*d^2 + 6*a^13*b^2*d^2))^(1/2)*1i)/(32*(a^15*d^2 + a^3*b^12*d^2 + 6*a^5*b^
10*d^2 + 15*a^7*b^8*d^2 + 20*a^9*b^6*d^2 + 15*a^11*b^4*d^2 + 6*a^13*b^2*d^2)) + (atan(-((((tan(c + d*x)^(1/2)*
(41*B^4*b^13 + 9*B^4*a^12*b - 82*B^4*a^2*b^11 + 1831*B^4*a^4*b^9 - 4348*B^4*a^6*b^7 + 1671*B^4*a^8*b^5 - 210*B
^4*a^10*b^3))/(64*(a^16*d^4 + b^16*d^4 + 8*a^2*b^14*d^4 + 28*a^4*b^12*d^4 + 56*a^6*b^10*d^4 + 70*a^8*b^8*d^4 +
56*a^10*b^6*d^4 + 28*a^12*b^4*d^4 + 8*a^14*b^2*d^4)) - (((3022*B^3*a^5*b^11*d^2 - 4494*B^3*a^3*b^13*d^2 + 171
94*B^3*a^7*b^9*d^2 + 5298*B^3*a^9*b^7*d^2 - 3338*B^3*a^11*b^5*d^2 + 506*B^3*a^13*b^3*d^2 + 518*B^3*a*b^15*d^2
- 18*B^3*a^15*b*d^2)/(64*(a^16*d^5 + b^16*d^5 + 8*a^2*b^14*d^5 + 28*a^4*b^12*d^5 + 56*a^6*b^10*d^5 + 70*a^8*b^
8*d^5 + 56*a^10*b^6*d^5 + 28*a^12*b^4*d^5 + 8*a^14*b^2*d^5)) + (((tan(c + d*x)^(1/2)*(64*B^2*a^3*b^16*d^2 - 74
56*B^2*a^5*b^14*d^2 - 576*B^2*a^7*b^12*d^2 + 19504*B^2*a^9*b^10*d^2 + 18880*B^2*a^11*b^8*d^2 + 3808*B^2*a^13*b
^6*d^2 - 960*B^2*a^15*b^4*d^2 + 8*B^2*a^17*b^2*d^2 + 1544*B^2*a*b^18*d^2))/(64*(a^16*d^4 + b^16*d^4 + 8*a^2*b^
14*d^4 + 28*a^4*b^12*d^4 + 56*a^6*b^10*d^4 + 70*a^8*b^8*d^4 + 56*a^10*b^6*d^4 + 28*a^12*b^4*d^4 + 8*a^14*b^2*d
^4)) + (((4224*B*a^4*b^18*d^4 - 320*B*a^2*b^20*d^4 - 192*B*b^22*d^4 + 22272*B*a^6*b^16*d^4 + 51072*B*a^8*b^14*
d^4 + 67200*B*a^10*b^12*d^4 + 53760*B*a^12*b^10*d^4 + 25344*B*a^14*b^8*d^4 + 5952*B*a^16*b^6*d^4 + 192*B*a^18*
b^4*d^4 - 128*B*a^20*b^2*d^4)/(64*(a^16*d^5 + b^16*d^5 + 8*a^2*b^14*d^5 + 28*a^4*b^12*d^5 + 56*a^6*b^10*d^5 +
70*a^8*b^8*d^5 + 56*a^10*b^6*d^5 + 28*a^12*b^4*d^5 + 8*a^14*b^2*d^5)) - (tan(c + d*x)^(1/2)*(-64*(9*B^2*a^8 +
9*B^2*b^8 - 156*B^2*a^2*b^6 + 694*B^2*a^4*b^4 - 156*B^2*a^6*b^2)*(a*b^13*d^2 + a^13*b*d^2 + 6*a^3*b^11*d^2 + 1
5*a^5*b^9*d^2 + 20*a^7*b^7*d^2 + 15*a^9*b^5*d^2 + 6*a^11*b^3*d^2))^(1/2)*(512*b^25*d^4 + 4608*a^2*b^23*d^4 + 1
7920*a^4*b^21*d^4 + 38400*a^6*b^19*d^4 + 46080*a^8*b^17*d^4 + 21504*a^10*b^15*d^4 - 21504*a^12*b^13*d^4 - 4608
0*a^14*b^11*d^4 - 38400*a^16*b^9*d^4 - 17920*a^18*b^7*d^4 - 4608*a^20*b^5*d^4 - 512*a^22*b^3*d^4))/(4096*(a*b^
13*d^2 + a^13*b*d^2 + 6*a^3*b^11*d^2 + 15*a^5*b^9*d^2 + 20*a^7*b^7*d^2 + 15*a^9*b^5*d^2 + 6*a^11*b^3*d^2)*(a^1
6*d^4 + b^16*d^4 + 8*a^2*b^14*d^4 + 28*a^4*b^12*d^4 + 56*a^6*b^10*d^4 + 70*a^8*b^8*d^4 + 56*a^10*b^6*d^4 + 28*
a^12*b^4*d^4 + 8*a^14*b^2*d^4)))*(-64*(9*B^2*a^8 + 9*B^2*b^8 - 156*B^2*a^2*b^6 + 694*B^2*a^4*b^4 - 156*B^2*a^6
*b^2)*(a*b^13*d^2 + a^13*b*d^2 + 6*a^3*b^11*d^2 + 15*a^5*b^9*d^2 + 20*a^7*b^7*d^2 + 15*a^9*b^5*d^2 + 6*a^11*b^
3*d^2))^(1/2))/(64*(a*b^13*d^2 + a^13*b*d^2 + 6*a^3*b^11*d^2 + 15*a^5*b^9*d^2 + 20*a^7*b^7*d^2 + 15*a^9*b^5*d^
2 + 6*a^11*b^3*d^2)))*(-64*(9*B^2*a^8 + 9*B^2*b^8 - 156*B^2*a^2*b^6 + 694*B^2*a^4*b^4 - 156*B^2*a^6*b^2)*(a*b^
13*d^2 + a^13*b*d^2 + 6*a^3*b^11*d^2 + 15*a^5*b^9*d^2 + 20*a^7*b^7*d^2 + 15*a^9*b^5*d^2 + 6*a^11*b^3*d^2))^(1/
2))/(64*(a*b^13*d^2 + a^13*b*d^2 + 6*a^3*b^11*d^2 + 15*a^5*b^9*d^2 + 20*a^7*b^7*d^2 + 15*a^9*b^5*d^2 + 6*a^11*
b^3*d^2)))*(-64*(9*B^2*a^8 + 9*B^2*b^8 - 156*B^2*a^2*b^6 + 694*B^2*a^4*b^4 - 156*B^2*a^6*b^2)*(a*b^13*d^2 + a^
13*b*d^2 + 6*a^3*b^11*d^2 + 15*a^5*b^9*d^2 + 20*a^7*b^7*d^2 + 15*a^9*b^5*d^2 + 6*a^11*b^3*d^2))^(1/2))/(64*(a*
b^13*d^2 + a^13*b*d^2 + 6*a^3*b^11*d^2 + 15*a^5*b^9*d^2 + 20*a^7*b^7*d^2 + 15*a^9*b^5*d^2 + 6*a^11*b^3*d^2)))*
(-64*(9*B^2*a^8 + 9*B^2*b^8 - 156*B^2*a^2*b^6 + 694*B^2*a^4*b^4 - 156*B^2*a^6*b^2)*(a*b^13*d^2 + a^13*b*d^2 +
6*a^3*b^11*d^2 + 15*a^5*b^9*d^2 + 20*a^7*b^7*d^2 + 15*a^9*b^5*d^2 + 6*a^11*b^3*d^2))^(1/2)*1i)/(a*b^13*d^2 + a
^13*b*d^2 + 6*a^3*b^11*d^2 + 15*a^5*b^9*d^2 + 20*a^7*b^7*d^2 + 15*a^9*b^5*d^2 + 6*a^11*b^3*d^2) + (((tan(c + d
*x)^(1/2)*(41*B^4*b^13 + 9*B^4*a^12*b - 82*B^4*a^2*b^11 + 1831*B^4*a^4*b^9 - 4348*B^4*a^6*b^7 + 1671*B^4*a^8*b
^5 - 210*B^4*a^10*b^3))/(64*(a^16*d^4 + b^16*d^4 + 8*a^2*b^14*d^4 + 28*a^4*b^12*d^4 + 56*a^6*b^10*d^4 + 70*a^8
*b^8*d^4 + 56*a^10*b^6*d^4 + 28*a^12*b^4*d^4 + 8*a^14*b^2*d^4)) + (((3022*B^3*a^5*b^11*d^2 - 4494*B^3*a^3*b^13
*d^2 + 17194*B^3*a^7*b^9*d^2 + 5298*B^3*a^9*b^7*d^2 - 3338*B^3*a^11*b^5*d^2 + 506*B^3*a^13*b^3*d^2 + 518*B^3*a
*b^15*d^2 - 18*B^3*a^15*b*d^2)/(64*(a^16*d^5 + b^16*d^5 + 8*a^2*b^14*d^5 + 28*a^4*b^12*d^5 + 56*a^6*b^10*d^5 +
70*a^8*b^8*d^5 + 56*a^10*b^6*d^5 + 28*a^12*b^4*d^5 + 8*a^14*b^2*d^5)) - (((tan(c + d*x)^(1/2)*(64*B^2*a^3*b^1
6*d^2 - 7456*B^2*a^5*b^14*d^2 - 576*B^2*a^7*b^12*d^2 + 19504*B^2*a^9*b^10*d^2 + 18880*B^2*a^11*b^8*d^2 + 3808*
B^2*a^13*b^6*d^2 - 960*B^2*a^15*b^4*d^2 + 8*B^2*a^17*b^2*d^2 + 1544*B^2*a*b^18*d^2))/(64*(a^16*d^4 + b^16*d^4
+ 8*a^2*b^14*d^4 + 28*a^4*b^12*d^4 + 56*a^6*b^10*d^4 + 70*a^8*b^8*d^4 + 56*a^10*b^6*d^4 + 28*a^12*b^4*d^4 + 8*
a^14*b^2*d^4)) - (((4224*B*a^4*b^18*d^4 - 320*B*a^2*b^20*d^4 - 192*B*b^22*d^4 + 22272*B*a^6*b^16*d^4 + 51072*B
*a^8*b^14*d^4 + 67200*B*a^10*b^12*d^4 + 53760*B*a^12*b^10*d^4 + 25344*B*a^14*b^8*d^4 + 5952*B*a^16*b^6*d^4 + 1
92*B*a^18*b^4*d^4 - 128*B*a^20*b^2*d^4)/(64*(a^16*d^5 + b^16*d^5 + 8*a^2*b^14*d^5 + 28*a^4*b^12*d^5 + 56*a^6*b
^10*d^5 + 70*a^8*b^8*d^5 + 56*a^10*b^6*d^5 + 28*a^12*b^4*d^5 + 8*a^14*b^2*d^5)) + (tan(c + d*x)^(1/2)*(-64*(9*
B^2*a^8 + 9*B^2*b^8 - 156*B^2*a^2*b^6 + 694*B^2*a^4*b^4 - 156*B^2*a^6*b^2)*(a*b^13*d^2 + a^13*b*d^2 + 6*a^3*b^
11*d^2 + 15*a^5*b^9*d^2 + 20*a^7*b^7*d^2 + 15*a^9*b^5*d^2 + 6*a^11*b^3*d^2))^(1/2)*(512*b^25*d^4 + 4608*a^2*b^
23*d^4 + 17920*a^4*b^21*d^4 + 38400*a^6*b^19*d^4 + 46080*a^8*b^17*d^4 + 21504*a^10*b^15*d^4 - 21504*a^12*b^13*
d^4 - 46080*a^14*b^11*d^4 - 38400*a^16*b^9*d^4 - 17920*a^18*b^7*d^4 - 4608*a^20*b^5*d^4 - 512*a^22*b^3*d^4))/(
4096*(a*b^13*d^2 + a^13*b*d^2 + 6*a^3*b^11*d^2 + 15*a^5*b^9*d^2 + 20*a^7*b^7*d^2 + 15*a^9*b^5*d^2 + 6*a^11*b^3
*d^2)*(a^16*d^4 + b^16*d^4 + 8*a^2*b^14*d^4 + 28*a^4*b^12*d^4 + 56*a^6*b^10*d^4 + 70*a^8*b^8*d^4 + 56*a^10*b^6
*d^4 + 28*a^12*b^4*d^4 + 8*a^14*b^2*d^4)))*(-64*(9*B^2*a^8 + 9*B^2*b^8 - 156*B^2*a^2*b^6 + 694*B^2*a^4*b^4 - 1
56*B^2*a^6*b^2)*(a*b^13*d^2 + a^13*b*d^2 + 6*a^3*b^11*d^2 + 15*a^5*b^9*d^2 + 20*a^7*b^7*d^2 + 15*a^9*b^5*d^2 +
6*a^11*b^3*d^2))^(1/2))/(64*(a*b^13*d^2 + a^13*b*d^2 + 6*a^3*b^11*d^2 + 15*a^5*b^9*d^2 + 20*a^7*b^7*d^2 + 15*
a^9*b^5*d^2 + 6*a^11*b^3*d^2)))*(-64*(9*B^2*a^8 + 9*B^2*b^8 - 156*B^2*a^2*b^6 + 694*B^2*a^4*b^4 - 156*B^2*a^6*
b^2)*(a*b^13*d^2 + a^13*b*d^2 + 6*a^3*b^11*d^2 + 15*a^5*b^9*d^2 + 20*a^7*b^7*d^2 + 15*a^9*b^5*d^2 + 6*a^11*b^3
*d^2))^(1/2))/(64*(a*b^13*d^2 + a^13*b*d^2 + 6*a^3*b^11*d^2 + 15*a^5*b^9*d^2 + 20*a^7*b^7*d^2 + 15*a^9*b^5*d^2
+ 6*a^11*b^3*d^2)))*(-64*(9*B^2*a^8 + 9*B^2*b^8 - 156*B^2*a^2*b^6 + 694*B^2*a^4*b^4 - 156*B^2*a^6*b^2)*(a*b^1
3*d^2 + a^13*b*d^2 + 6*a^3*b^11*d^2 + 15*a^5*b^9*d^2 + 20*a^7*b^7*d^2 + 15*a^9*b^5*d^2 + 6*a^11*b^3*d^2))^(1/2
))/(64*(a*b^13*d^2 + a^13*b*d^2 + 6*a^3*b^11*d^2 + 15*a^5*b^9*d^2 + 20*a^7*b^7*d^2 + 15*a^9*b^5*d^2 + 6*a^11*b
^3*d^2)))*(-64*(9*B^2*a^8 + 9*B^2*b^8 - 156*B^2*a^2*b^6 + 694*B^2*a^4*b^4 - 156*B^2*a^6*b^2)*(a*b^13*d^2 + a^1
3*b*d^2 + 6*a^3*b^11*d^2 + 15*a^5*b^9*d^2 + 20*a^7*b^7*d^2 + 15*a^9*b^5*d^2 + 6*a^11*b^3*d^2))^(1/2)*1i)/(a*b^
13*d^2 + a^13*b*d^2 + 6*a^3*b^11*d^2 + 15*a^5*b^9*d^2 + 20*a^7*b^7*d^2 + 15*a^9*b^5*d^2 + 6*a^11*b^3*d^2))/((2
8*B^5*a^2*b^8 - 15*B^5*b^10 + 878*B^5*a^4*b^6 - 180*B^5*a^6*b^4 + 9*B^5*a^8*b^2)/(a^16*d^5 + b^16*d^5 + 8*a^2*
b^14*d^5 + 28*a^4*b^12*d^5 + 56*a^6*b^10*d^5 + 70*a^8*b^8*d^5 + 56*a^10*b^6*d^5 + 28*a^12*b^4*d^5 + 8*a^14*b^2
*d^5) - (((tan(c + d*x)^(1/2)*(41*B^4*b^13 + 9*B^4*a^12*b - 82*B^4*a^2*b^11 + 1831*B^4*a^4*b^9 - 4348*B^4*a^6*
b^7 + 1671*B^4*a^8*b^5 - 210*B^4*a^10*b^3))/(64*(a^16*d^4 + b^16*d^4 + 8*a^2*b^14*d^4 + 28*a^4*b^12*d^4 + 56*a
^6*b^10*d^4 + 70*a^8*b^8*d^4 + 56*a^10*b^6*d^4 + 28*a^12*b^4*d^4 + 8*a^14*b^2*d^4)) - (((3022*B^3*a^5*b^11*d^2
- 4494*B^3*a^3*b^13*d^2 + 17194*B^3*a^7*b^9*d^2 + 5298*B^3*a^9*b^7*d^2 - 3338*B^3*a^11*b^5*d^2 + 506*B^3*a^13
*b^3*d^2 + 518*B^3*a*b^15*d^2 - 18*B^3*a^15*b*d^2)/(64*(a^16*d^5 + b^16*d^5 + 8*a^2*b^14*d^5 + 28*a^4*b^12*d^5
+ 56*a^6*b^10*d^5 + 70*a^8*b^8*d^5 + 56*a^10*b^6*d^5 + 28*a^12*b^4*d^5 + 8*a^14*b^2*d^5)) + (((tan(c + d*x)^(
1/2)*(64*B^2*a^3*b^16*d^2 - 7456*B^2*a^5*b^14*d^2 - 576*B^2*a^7*b^12*d^2 + 19504*B^2*a^9*b^10*d^2 + 18880*B^2*
a^11*b^8*d^2 + 3808*B^2*a^13*b^6*d^2 - 960*B^2*a^15*b^4*d^2 + 8*B^2*a^17*b^2*d^2 + 1544*B^2*a*b^18*d^2))/(64*(
a^16*d^4 + b^16*d^4 + 8*a^2*b^14*d^4 + 28*a^4*b^12*d^4 + 56*a^6*b^10*d^4 + 70*a^8*b^8*d^4 + 56*a^10*b^6*d^4 +
28*a^12*b^4*d^4 + 8*a^14*b^2*d^4)) + (((4224*B*a^4*b^18*d^4 - 320*B*a^2*b^20*d^4 - 192*B*b^22*d^4 + 22272*B*a^
6*b^16*d^4 + 51072*B*a^8*b^14*d^4 + 67200*B*a^10*b^12*d^4 + 53760*B*a^12*b^10*d^4 + 25344*B*a^14*b^8*d^4 + 595
2*B*a^16*b^6*d^4 + 192*B*a^18*b^4*d^4 - 128*B*a^20*b^2*d^4)/(64*(a^16*d^5 + b^16*d^5 + 8*a^2*b^14*d^5 + 28*a^4
*b^12*d^5 + 56*a^6*b^10*d^5 + 70*a^8*b^8*d^5 + 56*a^10*b^6*d^5 + 28*a^12*b^4*d^5 + 8*a^14*b^2*d^5)) - (tan(c +
d*x)^(1/2)*(-64*(9*B^2*a^8 + 9*B^2*b^8 - 156*B^2*a^2*b^6 + 694*B^2*a^4*b^4 - 156*B^2*a^6*b^2)*(a*b^13*d^2 + a
^13*b*d^2 + 6*a^3*b^11*d^2 + 15*a^5*b^9*d^2 + 20*a^7*b^7*d^2 + 15*a^9*b^5*d^2 + 6*a^11*b^3*d^2))^(1/2)*(512*b^
25*d^4 + 4608*a^2*b^23*d^4 + 17920*a^4*b^21*d^4 + 38400*a^6*b^19*d^4 + 46080*a^8*b^17*d^4 + 21504*a^10*b^15*d^
4 - 21504*a^12*b^13*d^4 - 46080*a^14*b^11*d^4 - 38400*a^16*b^9*d^4 - 17920*a^18*b^7*d^4 - 4608*a^20*b^5*d^4 -
512*a^22*b^3*d^4))/(4096*(a*b^13*d^2 + a^13*b*d^2 + 6*a^3*b^11*d^2 + 15*a^5*b^9*d^2 + 20*a^7*b^7*d^2 + 15*a^9*
b^5*d^2 + 6*a^11*b^3*d^2)*(a^16*d^4 + b^16*d^4 + 8*a^2*b^14*d^4 + 28*a^4*b^12*d^4 + 56*a^6*b^10*d^4 + 70*a^8*b
^8*d^4 + 56*a^10*b^6*d^4 + 28*a^12*b^4*d^4 + 8*a^14*b^2*d^4)))*(-64*(9*B^2*a^8 + 9*B^2*b^8 - 156*B^2*a^2*b^6 +
694*B^2*a^4*b^4 - 156*B^2*a^6*b^2)*(a*b^13*d^2 + a^13*b*d^2 + 6*a^3*b^11*d^2 + 15*a^5*b^9*d^2 + 20*a^7*b^7*d^
2 + 15*a^9*b^5*d^2 + 6*a^11*b^3*d^2))^(1/2))/(64*(a*b^13*d^2 + a^13*b*d^2 + 6*a^3*b^11*d^2 + 15*a^5*b^9*d^2 +
20*a^7*b^7*d^2 + 15*a^9*b^5*d^2 + 6*a^11*b^3*d^2)))*(-64*(9*B^2*a^8 + 9*B^2*b^8 - 156*B^2*a^2*b^6 + 694*B^2*a^
4*b^4 - 156*B^2*a^6*b^2)*(a*b^13*d^2 + a^13*b*d^2 + 6*a^3*b^11*d^2 + 15*a^5*b^9*d^2 + 20*a^7*b^7*d^2 + 15*a^9*
b^5*d^2 + 6*a^11*b^3*d^2))^(1/2))/(64*(a*b^13*d^2 + a^13*b*d^2 + 6*a^3*b^11*d^2 + 15*a^5*b^9*d^2 + 20*a^7*b^7*
d^2 + 15*a^9*b^5*d^2 + 6*a^11*b^3*d^2)))*(-64*(9*B^2*a^8 + 9*B^2*b^8 - 156*B^2*a^2*b^6 + 694*B^2*a^4*b^4 - 156
*B^2*a^6*b^2)*(a*b^13*d^2 + a^13*b*d^2 + 6*a^3*b^11*d^2 + 15*a^5*b^9*d^2 + 20*a^7*b^7*d^2 + 15*a^9*b^5*d^2 + 6
*a^11*b^3*d^2))^(1/2))/(64*(a*b^13*d^2 + a^13*b*d^2 + 6*a^3*b^11*d^2 + 15*a^5*b^9*d^2 + 20*a^7*b^7*d^2 + 15*a^
9*b^5*d^2 + 6*a^11*b^3*d^2)))*(-64*(9*B^2*a^8 + 9*B^2*b^8 - 156*B^2*a^2*b^6 + 694*B^2*a^4*b^4 - 156*B^2*a^6*b^
2)*(a*b^13*d^2 + a^13*b*d^2 + 6*a^3*b^11*d^2 + 15*a^5*b^9*d^2 + 20*a^7*b^7*d^2 + 15*a^9*b^5*d^2 + 6*a^11*b^3*d
^2))^(1/2))/(a*b^13*d^2 + a^13*b*d^2 + 6*a^3*b^11*d^2 + 15*a^5*b^9*d^2 + 20*a^7*b^7*d^2 + 15*a^9*b^5*d^2 + 6*a
^11*b^3*d^2) + (((tan(c + d*x)^(1/2)*(41*B^4*b^13 + 9*B^4*a^12*b - 82*B^4*a^2*b^11 + 1831*B^4*a^4*b^9 - 4348*B
^4*a^6*b^7 + 1671*B^4*a^8*b^5 - 210*B^4*a^10*b^3))/(64*(a^16*d^4 + b^16*d^4 + 8*a^2*b^14*d^4 + 28*a^4*b^12*d^4
+ 56*a^6*b^10*d^4 + 70*a^8*b^8*d^4 + 56*a^10*b^6*d^4 + 28*a^12*b^4*d^4 + 8*a^14*b^2*d^4)) + (((3022*B^3*a^5*b
^11*d^2 - 4494*B^3*a^3*b^13*d^2 + 17194*B^3*a^7*b^9*d^2 + 5298*B^3*a^9*b^7*d^2 - 3338*B^3*a^11*b^5*d^2 + 506*B
^3*a^13*b^3*d^2 + 518*B^3*a*b^15*d^2 - 18*B^3*a^15*b*d^2)/(64*(a^16*d^5 + b^16*d^5 + 8*a^2*b^14*d^5 + 28*a^4*b
^12*d^5 + 56*a^6*b^10*d^5 + 70*a^8*b^8*d^5 + 56*a^10*b^6*d^5 + 28*a^12*b^4*d^5 + 8*a^14*b^2*d^5)) - (((tan(c +
d*x)^(1/2)*(64*B^2*a^3*b^16*d^2 - 7456*B^2*a^5*b^14*d^2 - 576*B^2*a^7*b^12*d^2 + 19504*B^2*a^9*b^10*d^2 + 188
80*B^2*a^11*b^8*d^2 + 3808*B^2*a^13*b^6*d^2 - 960*B^2*a^15*b^4*d^2 + 8*B^2*a^17*b^2*d^2 + 1544*B^2*a*b^18*d^2)
)/(64*(a^16*d^4 + b^16*d^4 + 8*a^2*b^14*d^4 + 28*a^4*b^12*d^4 + 56*a^6*b^10*d^4 + 70*a^8*b^8*d^4 + 56*a^10*b^6
*d^4 + 28*a^12*b^4*d^4 + 8*a^14*b^2*d^4)) - (((4224*B*a^4*b^18*d^4 - 320*B*a^2*b^20*d^4 - 192*B*b^22*d^4 + 222
72*B*a^6*b^16*d^4 + 51072*B*a^8*b^14*d^4 + 67200*B*a^10*b^12*d^4 + 53760*B*a^12*b^10*d^4 + 25344*B*a^14*b^8*d^
4 + 5952*B*a^16*b^6*d^4 + 192*B*a^18*b^4*d^4 - 128*B*a^20*b^2*d^4)/(64*(a^16*d^5 + b^16*d^5 + 8*a^2*b^14*d^5 +
28*a^4*b^12*d^5 + 56*a^6*b^10*d^5 + 70*a^8*b^8*d^5 + 56*a^10*b^6*d^5 + 28*a^12*b^4*d^5 + 8*a^14*b^2*d^5)) + (
tan(c + d*x)^(1/2)*(-64*(9*B^2*a^8 + 9*B^2*b^8 - 156*B^2*a^2*b^6 + 694*B^2*a^4*b^4 - 156*B^2*a^6*b^2)*(a*b^13*
d^2 + a^13*b*d^2 + 6*a^3*b^11*d^2 + 15*a^5*b^9*d^2 + 20*a^7*b^7*d^2 + 15*a^9*b^5*d^2 + 6*a^11*b^3*d^2))^(1/2)*
(512*b^25*d^4 + 4608*a^2*b^23*d^4 + 17920*a^4*b^21*d^4 + 38400*a^6*b^19*d^4 + 46080*a^8*b^17*d^4 + 21504*a^10*
b^15*d^4 - 21504*a^12*b^13*d^4 - 46080*a^14*b^11*d^4 - 38400*a^16*b^9*d^4 - 17920*a^18*b^7*d^4 - 4608*a^20*b^5
*d^4 - 512*a^22*b^3*d^4))/(4096*(a*b^13*d^2 + a^13*b*d^2 + 6*a^3*b^11*d^2 + 15*a^5*b^9*d^2 + 20*a^7*b^7*d^2 +
15*a^9*b^5*d^2 + 6*a^11*b^3*d^2)*(a^16*d^4 + b^16*d^4 + 8*a^2*b^14*d^4 + 28*a^4*b^12*d^4 + 56*a^6*b^10*d^4 + 7
0*a^8*b^8*d^4 + 56*a^10*b^6*d^4 + 28*a^12*b^4*d^4 + 8*a^14*b^2*d^4)))*(-64*(9*B^2*a^8 + 9*B^2*b^8 - 156*B^2*a^
2*b^6 + 694*B^2*a^4*b^4 - 156*B^2*a^6*b^2)*(a*b^13*d^2 + a^13*b*d^2 + 6*a^3*b^11*d^2 + 15*a^5*b^9*d^2 + 20*a^7
*b^7*d^2 + 15*a^9*b^5*d^2 + 6*a^11*b^3*d^2))^(1/2))/(64*(a*b^13*d^2 + a^13*b*d^2 + 6*a^3*b^11*d^2 + 15*a^5*b^9
*d^2 + 20*a^7*b^7*d^2 + 15*a^9*b^5*d^2 + 6*a^11*b^3*d^2)))*(-64*(9*B^2*a^8 + 9*B^2*b^8 - 156*B^2*a^2*b^6 + 694
*B^2*a^4*b^4 - 156*B^2*a^6*b^2)*(a*b^13*d^2 + a^13*b*d^2 + 6*a^3*b^11*d^2 + 15*a^5*b^9*d^2 + 20*a^7*b^7*d^2 +
15*a^9*b^5*d^2 + 6*a^11*b^3*d^2))^(1/2))/(64*(a*b^13*d^2 + a^13*b*d^2 + 6*a^3*b^11*d^2 + 15*a^5*b^9*d^2 + 20*a
^7*b^7*d^2 + 15*a^9*b^5*d^2 + 6*a^11*b^3*d^2)))*(-64*(9*B^2*a^8 + 9*B^2*b^8 - 156*B^2*a^2*b^6 + 694*B^2*a^4*b^
4 - 156*B^2*a^6*b^2)*(a*b^13*d^2 + a^13*b*d^2 + 6*a^3*b^11*d^2 + 15*a^5*b^9*d^2 + 20*a^7*b^7*d^2 + 15*a^9*b^5*
d^2 + 6*a^11*b^3*d^2))^(1/2))/(64*(a*b^13*d^2 + a^13*b*d^2 + 6*a^3*b^11*d^2 + 15*a^5*b^9*d^2 + 20*a^7*b^7*d^2
+ 15*a^9*b^5*d^2 + 6*a^11*b^3*d^2)))*(-64*(9*B^2*a^8 + 9*B^2*b^8 - 156*B^2*a^2*b^6 + 694*B^2*a^4*b^4 - 156*B^2
*a^6*b^2)*(a*b^13*d^2 + a^13*b*d^2 + 6*a^3*b^11*d^2 + 15*a^5*b^9*d^2 + 20*a^7*b^7*d^2 + 15*a^9*b^5*d^2 + 6*a^1
1*b^3*d^2))^(1/2))/(a*b^13*d^2 + a^13*b*d^2 + 6*a^3*b^11*d^2 + 15*a^5*b^9*d^2 + 20*a^7*b^7*d^2 + 15*a^9*b^5*d^
2 + 6*a^11*b^3*d^2)))*(-64*(9*B^2*a^8 + 9*B^2*b^8 - 156*B^2*a^2*b^6 + 694*B^2*a^4*b^4 - 156*B^2*a^6*b^2)*(a*b^
13*d^2 + a^13*b*d^2 + 6*a^3*b^11*d^2 + 15*a^5*b^9*d^2 + 20*a^7*b^7*d^2 + 15*a^9*b^5*d^2 + 6*a^11*b^3*d^2))^(1/
2)*1i)/(32*(a*b^13*d^2 + a^13*b*d^2 + 6*a^3*b^11*d^2 + 15*a^5*b^9*d^2 + 20*a^7*b^7*d^2 + 15*a^9*b^5*d^2 + 6*a^
11*b^3*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(tan(d*x+c)**(1/2)*(A+B*tan(d*x+c))/(a+b*tan(d*x+c))**3,x)

[Out]

Timed out

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