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

Optimal. Leaf size=534 \[ \frac {b (A b-a B) \sqrt {\tan (c+d x)}}{2 a d \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}+\frac {\left (-\left (a^3 (A+B)\right )+3 a^2 b (A-B)+3 a b^2 (A+B)-b^3 (A-B)\right ) \tan ^{-1}\left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} d \left (a^2+b^2\right )^3}-\frac {\left (-\left (a^3 (A+B)\right )+3 a^2 b (A-B)+3 a b^2 (A+B)-b^3 (A-B)\right ) \tan ^{-1}\left (\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{\sqrt {2} d \left (a^2+b^2\right )^3}+\frac {b \left (-7 a^3 B+11 a^2 A b+a b^2 B+3 A b^3\right ) \sqrt {\tan (c+d x)}}{4 a^2 d \left (a^2+b^2\right )^2 (a+b \tan (c+d x))}-\frac {\left (a^3 (A-B)+3 a^2 b (A+B)-3 a b^2 (A-B)-b^3 (A+B)\right ) \log \left (\tan (c+d x)-\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{2 \sqrt {2} d \left (a^2+b^2\right )^3}+\frac {\left (a^3 (A-B)+3 a^2 b (A+B)-3 a b^2 (A-B)-b^3 (A+B)\right ) \log \left (\tan (c+d x)+\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{2 \sqrt {2} d \left (a^2+b^2\right )^3}+\frac {\sqrt {b} \left (-15 a^5 B+35 a^4 A b+18 a^3 b^2 B+6 a^2 A b^3+a b^4 B+3 A b^5\right ) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {\tan (c+d x)}}{\sqrt {a}}\right )}{4 a^{5/2} d \left (a^2+b^2\right )^3} \]

[Out]

-1/2*(3*a^2*b*(A-B)-b^3*(A-B)-a^3*(A+B)+3*a*b^2*(A+B))*arctan(-1+2^(1/2)*tan(d*x+c)^(1/2))/(a^2+b^2)^3/d*2^(1/
2)-1/2*(3*a^2*b*(A-B)-b^3*(A-B)-a^3*(A+B)+3*a*b^2*(A+B))*arctan(1+2^(1/2)*tan(d*x+c)^(1/2))/(a^2+b^2)^3/d*2^(1
/2)-1/4*(a^3*(A-B)-3*a*b^2*(A-B)+3*a^2*b*(A+B)-b^3*(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*(a^3*(A-B)-3*a*b^2*(A-B)+3*a^2*b*(A+B)-b^3*(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*(35*A*a^4*b+6*A*a^2*b^3+3*A*b^5-15*B*a^5+18*B*a^3*b^2+B*a*b^4)*arctan(b^(1/2)*tan(d*x+
c)^(1/2)/a^(1/2))*b^(1/2)/a^(5/2)/(a^2+b^2)^3/d+1/2*b*(A*b-B*a)*tan(d*x+c)^(1/2)/a/(a^2+b^2)/d/(a+b*tan(d*x+c)
)^2+1/4*b*(11*A*a^2*b+3*A*b^3-7*B*a^3+B*a*b^2)*tan(d*x+c)^(1/2)/a^2/(a^2+b^2)^2/d/(a+b*tan(d*x+c))

________________________________________________________________________________________

Rubi [A]  time = 1.25, antiderivative size = 534, 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 = {3609, 3649, 3653, 3534, 1168, 1162, 617, 204, 1165, 628, 3634, 63, 205} \[ \frac {\left (3 a^2 b (A-B)+a^3 (-(A+B))+3 a b^2 (A+B)-b^3 (A-B)\right ) \tan ^{-1}\left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} d \left (a^2+b^2\right )^3}-\frac {\left (3 a^2 b (A-B)+a^3 (-(A+B))+3 a b^2 (A+B)-b^3 (A-B)\right ) \tan ^{-1}\left (\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{\sqrt {2} d \left (a^2+b^2\right )^3}+\frac {\sqrt {b} \left (6 a^2 A b^3+35 a^4 A b+18 a^3 b^2 B-15 a^5 B+a b^4 B+3 A b^5\right ) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {\tan (c+d x)}}{\sqrt {a}}\right )}{4 a^{5/2} d \left (a^2+b^2\right )^3}+\frac {b (A b-a B) \sqrt {\tan (c+d x)}}{2 a d \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}+\frac {b \left (11 a^2 A b-7 a^3 B+a b^2 B+3 A b^3\right ) \sqrt {\tan (c+d x)}}{4 a^2 d \left (a^2+b^2\right )^2 (a+b \tan (c+d x))}-\frac {\left (3 a^2 b (A+B)+a^3 (A-B)-3 a b^2 (A-B)-b^3 (A+B)\right ) \log \left (\tan (c+d x)-\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{2 \sqrt {2} d \left (a^2+b^2\right )^3}+\frac {\left (3 a^2 b (A+B)+a^3 (A-B)-3 a b^2 (A-B)-b^3 (A+B)\right ) \log \left (\tan (c+d x)+\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{2 \sqrt {2} d \left (a^2+b^2\right )^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((3*a^2*b*(A - B) - b^3*(A - B) - a^3*(A + B) + 3*a*b^2*(A + B))*ArcTan[1 - Sqrt[2]*Sqrt[Tan[c + d*x]]])/(Sqrt
[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))*ArcTan[1 + Sqrt[2]*Sqr
t[Tan[c + d*x]]])/(Sqrt[2]*(a^2 + b^2)^3*d) + (Sqrt[b]*(35*a^4*A*b + 6*a^2*A*b^3 + 3*A*b^5 - 15*a^5*B + 18*a^3
*b^2*B + a*b^4*B)*ArcTan[(Sqrt[b]*Sqrt[Tan[c + d*x]])/Sqrt[a]])/(4*a^(5/2)*(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))*Log[1 - Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]])/(2*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))*Log[1 + Sqrt[2]*Sqrt[Tan
[c + d*x]] + Tan[c + d*x]])/(2*Sqrt[2]*(a^2 + b^2)^3*d) + (b*(A*b - a*B)*Sqrt[Tan[c + d*x]])/(2*a*(a^2 + b^2)*
d*(a + b*Tan[c + d*x])^2) + (b*(11*a^2*A*b + 3*A*b^3 - 7*a^3*B + a*b^2*B)*Sqrt[Tan[c + d*x]])/(4*a^2*(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 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)}{\sqrt {\tan (c+d x)} (a+b \tan (c+d x))^3} \, dx &=\frac {b (A b-a B) \sqrt {\tan (c+d x)}}{2 a \left (a^2+b^2\right ) d (a+b \tan (c+d x))^2}+\frac {\int \frac {\frac {1}{2} \left (4 a^2 A+3 A b^2+a b B\right )-2 a (A b-a 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 a \left (a^2+b^2\right )}\\ &=\frac {b (A b-a B) \sqrt {\tan (c+d x)}}{2 a \left (a^2+b^2\right ) d (a+b \tan (c+d x))^2}+\frac {b \left (11 a^2 A b+3 A b^3-7 a^3 B+a b^2 B\right ) \sqrt {\tan (c+d x)}}{4 a^2 \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))}+\frac {\int \frac {\frac {1}{4} \left (8 a^4 A+3 a^2 A b^2+3 A b^4+9 a^3 b B+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 {1}{4} b \left (11 a^2 A b+3 A b^3-7 a^3 B+a b^2 B\right ) \tan ^2(c+d x)}{\sqrt {\tan (c+d x)} (a+b \tan (c+d x))} \, dx}{2 a^2 \left (a^2+b^2\right )^2}\\ &=\frac {b (A b-a B) \sqrt {\tan (c+d x)}}{2 a \left (a^2+b^2\right ) d (a+b \tan (c+d x))^2}+\frac {b \left (11 a^2 A b+3 A b^3-7 a^3 B+a b^2 B\right ) \sqrt {\tan (c+d x)}}{4 a^2 \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))}+\frac {\int \frac {2 a^2 \left (a^3 A-3 a A b^2+3 a^2 b B-b^3 B\right )-2 a^2 \left (3 a^2 A b-A b^3-a^3 B+3 a b^2 B\right ) \tan (c+d x)}{\sqrt {\tan (c+d x)}} \, dx}{2 a^2 \left (a^2+b^2\right )^3}+\frac {\left (b \left (35 a^4 A b+6 a^2 A b^3+3 A b^5-15 a^5 B+18 a^3 b^2 B+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^2 \left (a^2+b^2\right )^3}\\ &=\frac {b (A b-a B) \sqrt {\tan (c+d x)}}{2 a \left (a^2+b^2\right ) d (a+b \tan (c+d x))^2}+\frac {b \left (11 a^2 A b+3 A b^3-7 a^3 B+a b^2 B\right ) \sqrt {\tan (c+d x)}}{4 a^2 \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))}+\frac {\operatorname {Subst}\left (\int \frac {2 a^2 \left (a^3 A-3 a A b^2+3 a^2 b B-b^3 B\right )-2 a^2 \left (3 a^2 A b-A b^3-a^3 B+3 a b^2 B\right ) x^2}{1+x^4} \, dx,x,\sqrt {\tan (c+d x)}\right )}{a^2 \left (a^2+b^2\right )^3 d}+\frac {\left (b \left (35 a^4 A b+6 a^2 A b^3+3 A b^5-15 a^5 B+18 a^3 b^2 B+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^2 \left (a^2+b^2\right )^3 d}\\ &=\frac {b (A b-a B) \sqrt {\tan (c+d x)}}{2 a \left (a^2+b^2\right ) d (a+b \tan (c+d x))^2}+\frac {b \left (11 a^2 A b+3 A b^3-7 a^3 B+a b^2 B\right ) \sqrt {\tan (c+d x)}}{4 a^2 \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))}+\frac {\left (b \left (35 a^4 A b+6 a^2 A b^3+3 A b^5-15 a^5 B+18 a^3 b^2 B+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^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 {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 {\sqrt {b} \left (35 a^4 A b+6 a^2 A b^3+3 A b^5-15 a^5 B+18 a^3 b^2 B+a b^4 B\right ) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {\tan (c+d x)}}{\sqrt {a}}\right )}{4 a^{5/2} \left (a^2+b^2\right )^3 d}+\frac {b (A b-a B) \sqrt {\tan (c+d x)}}{2 a \left (a^2+b^2\right ) d (a+b \tan (c+d x))^2}+\frac {b \left (11 a^2 A b+3 A b^3-7 a^3 B+a b^2 B\right ) \sqrt {\tan (c+d x)}}{4 a^2 \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 {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 (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}{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 {\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 {\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 {\sqrt {b} \left (35 a^4 A b+6 a^2 A b^3+3 A b^5-15 a^5 B+18 a^3 b^2 B+a b^4 B\right ) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {\tan (c+d x)}}{\sqrt {a}}\right )}{4 a^{5/2} \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 ) \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 (a^3 (A-B)-3 a b^2 (A-B)+3 a^2 b (A+B)-b^3 (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 {b (A b-a B) \sqrt {\tan (c+d x)}}{2 a \left (a^2+b^2\right ) d (a+b \tan (c+d x))^2}+\frac {b \left (11 a^2 A b+3 A b^3-7 a^3 B+a b^2 B\right ) \sqrt {\tan (c+d x)}}{4 a^2 \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 {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 (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}{-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 (3 a^2 b (A-B)-b^3 (A-B)-a^3 (A+B)+3 a b^2 (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 (3 a^2 b (A-B)-b^3 (A-B)-a^3 (A+B)+3 a b^2 (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 {\sqrt {b} \left (35 a^4 A b+6 a^2 A b^3+3 A b^5-15 a^5 B+18 a^3 b^2 B+a b^4 B\right ) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {\tan (c+d x)}}{\sqrt {a}}\right )}{4 a^{5/2} \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 ) \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 (a^3 (A-B)-3 a b^2 (A-B)+3 a^2 b (A+B)-b^3 (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 {b (A b-a B) \sqrt {\tan (c+d x)}}{2 a \left (a^2+b^2\right ) d (a+b \tan (c+d x))^2}+\frac {b \left (11 a^2 A b+3 A b^3-7 a^3 B+a b^2 B\right ) \sqrt {\tan (c+d x)}}{4 a^2 \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))}\\ \end {align*}

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Mathematica [C]  time = 4.68, size = 288, normalized size = 0.54 \[ \frac {\frac {b \left (-7 a^3 B+11 a^2 A b+a b^2 B+3 A b^3\right ) \sqrt {\tan (c+d x)}}{a \left (a^2+b^2\right ) (a+b \tan (c+d x))}+\frac {2 \left (\frac {1}{2} \sqrt {b} \left (-15 a^5 B+35 a^4 A b+18 a^3 b^2 B+6 a^2 A b^3+a b^4 B+3 A b^5\right ) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {\tan (c+d x)}}{\sqrt {a}}\right )-2 \sqrt [4]{-1} a^{5/2} \left ((a+i b)^3 (A-i B) \tan ^{-1}\left ((-1)^{3/4} \sqrt {\tan (c+d x)}\right )+(a-i b)^3 (A+i B) \tanh ^{-1}\left ((-1)^{3/4} \sqrt {\tan (c+d x)}\right )\right )\right )}{a^{3/2} \left (a^2+b^2\right )^2}+\frac {2 b (A b-a B) \sqrt {\tan (c+d x)}}{(a+b \tan (c+d x))^2}}{4 a d \left (a^2+b^2\right )} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((2*((Sqrt[b]*(35*a^4*A*b + 6*a^2*A*b^3 + 3*A*b^5 - 15*a^5*B + 18*a^3*b^2*B + a*b^4*B)*ArcTan[(Sqrt[b]*Sqrt[Ta
n[c + d*x]])/Sqrt[a]])/2 - 2*(-1)^(1/4)*a^(5/2)*((a + I*b)^3*(A - I*B)*ArcTan[(-1)^(3/4)*Sqrt[Tan[c + d*x]]] +
(a - I*b)^3*(A + I*B)*ArcTanh[(-1)^(3/4)*Sqrt[Tan[c + d*x]]])))/(a^(3/2)*(a^2 + b^2)^2) + (2*b*(A*b - a*B)*Sq
rt[Tan[c + d*x]])/(a + b*Tan[c + d*x])^2 + (b*(11*a^2*A*b + 3*A*b^3 - 7*a^3*B + a*b^2*B)*Sqrt[Tan[c + d*x]])/(
a*(a^2 + b^2)*(a + b*Tan[c + d*x])))/(4*a*(a^2 + b^2)*d)

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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)^(1/2)/(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 \[ \int \frac {B \tan \left (d x + c\right ) + A}{{\left (b \tan \left (d x + c\right ) + a\right )}^{3} \sqrt {\tan \left (d x + c\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((A+B*tan(d*x+c))/tan(d*x+c)^(1/2)/(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*sqrt(tan(d*x + c))), x)

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maple [B]  time = 0.48, size = 1843, normalized size = 3.45 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4/d*b^6/(a^2+b^2)^3/(a+b*tan(d*x+c))^2/a*tan(d*x+c)^(3/2)*B+1/4/d*b^5/(a^2+b^2)^3/a/(a*b)^(1/2)*arctan(tan(d
*x+c)^(1/2)*b/(a*b)^(1/2))*B+5/4/d*b^6/(a^2+b^2)^3/(a+b*tan(d*x+c))^2/a*tan(d*x+c)^(1/2)*A+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))*A+13/4/d*a^3*b^2/(a^2+b^2)^3/(a+b*tan(d*x+c))^2*A*ta
n(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-3/4/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+9/2/d/(a^
2+b^2)^3*b^3/(a*b)^(1/2)*arctan(tan(d*x+c)^(1/2)*b/(a*b)^(1/2))*a*B+9/2/d/(a^2+b^2)^3/(a+b*tan(d*x+c))^2*a*b^4
*tan(d*x+c)^(1/2)*A-5/2/d/(a^2+b^2)^3/(a+b*tan(d*x+c))^2*a^2*b^3*tan(d*x+c)^(1/2)*B+3/4/d*b^7/(a^2+b^2)^3/(a+b
*tan(d*x+c))^2/a^2*tan(d*x+c)^(3/2)*A-7/4/d*a^3*b^2/(a^2+b^2)^3/(a+b*tan(d*x+c))^2*tan(d*x+c)^(3/2)*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*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^2-9/4/d*a^4*b/(a^2+b^2)^3/(a+b*tan(d*x+c))^
2*B*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*b^2-3/2/d/(a^2+b^2)^3/(
a+b*tan(d*x+c))^2*tan(d*x+c)^(3/2)*a*b^4*B-15/4/d*a^3*b/(a^2+b^2)^3/(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*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)*ar
ctan(-1+2^(1/2)*tan(d*x+c)^(1/2))*a^2*b-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+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+35/4/d*a^2*b^2/(a^2+b^2)^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*A*2^(1/2)*arctan(1+2^(1/2)*tan(d*x+c)^(1/2))*
a^2*b+11/4/d*a^2*b^3/(a^2+b^2)^3/(a+b*tan(d*x+c))^2*tan(d*x+c)^(3/2)*A+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*b^5/(a^2+
b^2)^3/(a+b*tan(d*x+c))^2*tan(d*x+c)^(1/2)*B+3/2/d/(a^2+b^2)^3*b^4/(a*b)^(1/2)*arctan(tan(d*x+c)^(1/2)*b/(a*b)
^(1/2))*A+7/2/d/(a^2+b^2)^3/(a+b*tan(d*x+c))^2*tan(d*x+c)^(3/2)*A*b^5+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+1/2/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+1/4/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))*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^3-1/4/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)))*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+1/2/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 = 1.04, size = 551, normalized size = 1.03 \[ -\frac {\frac {{\left (15 \, B a^{5} b - 35 \, A a^{4} b^{2} - 18 \, B a^{3} b^{3} - 6 \, A a^{2} b^{4} - B a b^{5} - 3 \, A b^{6}\right )} \arctan \left (\frac {b \sqrt {\tan \left (d x + c\right )}}{\sqrt {a b}}\right )}{{\left (a^{8} + 3 \, a^{6} b^{2} + 3 \, a^{4} b^{4} + a^{2} b^{6}\right )} \sqrt {a b}} - \frac {2 \, \sqrt {2} {\left ({\left (A + B\right )} a^{3} - 3 \, {\left (A - B\right )} a^{2} b - 3 \, {\left (A + B\right )} a b^{2} + {\left (A - B\right )} b^{3}\right )} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + 2 \, \sqrt {\tan \left (d x + c\right )}\right )}\right ) + 2 \, \sqrt {2} {\left ({\left (A + B\right )} a^{3} - 3 \, {\left (A - B\right )} a^{2} b - 3 \, {\left (A + B\right )} a b^{2} + {\left (A - B\right )} b^{3}\right )} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - 2 \, \sqrt {\tan \left (d x + c\right )}\right )}\right ) + \sqrt {2} {\left ({\left (A - B\right )} a^{3} + 3 \, {\left (A + B\right )} a^{2} b - 3 \, {\left (A - B\right )} a b^{2} - {\left (A + B\right )} b^{3}\right )} \log \left (\sqrt {2} \sqrt {\tan \left (d x + c\right )} + \tan \left (d x + c\right ) + 1\right ) - \sqrt {2} {\left ({\left (A - B\right )} a^{3} + 3 \, {\left (A + B\right )} a^{2} b - 3 \, {\left (A - B\right )} a b^{2} - {\left (A + B\right )} b^{3}\right )} \log \left (-\sqrt {2} \sqrt {\tan \left (d x + c\right )} + \tan \left (d x + c\right ) + 1\right )}{a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}} + \frac {{\left (7 \, B a^{3} b^{2} - 11 \, A a^{2} b^{3} - B a b^{4} - 3 \, A b^{5}\right )} \tan \left (d x + c\right )^{\frac {3}{2}} + {\left (9 \, B a^{4} b - 13 \, A a^{3} b^{2} + B a^{2} b^{3} - 5 \, A a b^{4}\right )} \sqrt {\tan \left (d x + c\right )}}{a^{8} + 2 \, a^{6} b^{2} + a^{4} b^{4} + {\left (a^{6} b^{2} + 2 \, a^{4} b^{4} + a^{2} b^{6}\right )} \tan \left (d x + c\right )^{2} + 2 \, {\left (a^{7} b + 2 \, a^{5} b^{3} + a^{3} b^{5}\right )} \tan \left (d x + c\right )}}{4 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*((15*B*a^5*b - 35*A*a^4*b^2 - 18*B*a^3*b^3 - 6*A*a^2*b^4 - B*a*b^5 - 3*A*b^6)*arctan(b*sqrt(tan(d*x + c))
/sqrt(a*b))/((a^8 + 3*a^6*b^2 + 3*a^4*b^4 + a^2*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) + ((7*B*a^3*b^2 - 11*A*a^2*b^3 - B*a*b^4 - 3*A*b
^5)*tan(d*x + c)^(3/2) + (9*B*a^4*b - 13*A*a^3*b^2 + B*a^2*b^3 - 5*A*a*b^4)*sqrt(tan(d*x + c)))/(a^8 + 2*a^6*b
^2 + a^4*b^4 + (a^6*b^2 + 2*a^4*b^4 + a^2*b^6)*tan(d*x + c)^2 + 2*(a^7*b + 2*a^5*b^3 + a^3*b^5)*tan(d*x + c)))
/d

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mupad [B]  time = 53.70, size = 26707, normalized size = 50.01 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((A*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)) + (A*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)) - ((tan(c
+ d*x)^(1/2)*(B*b^3 + 9*B*a^2*b))/(4*(a^4 + b^4 + 2*a^2*b^2)) - (tan(c + d*x)^(3/2)*(B*b^4 - 7*B*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)) + (log((((((((((64*A*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*(-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)*(9*b^1
2 - 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*(-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^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*(-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^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^1
0*b^2))/(a^4*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^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*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((((((((((64*A*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*(-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)*(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*(-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^3*(45*b^12 - 16*a^12 + 333*a^2*b^10 + 146*a^4*b^8 + 1178*a^6*b^6 - 97
91*a^8*b^4 + 1161*a^10*b^2))/(a^3*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^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*(-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 - 2
4*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^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*A^4*a^2*b^10*d^4 - 16*A^4*b^12*d^4 - 16*A^4*a^1
2*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^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*(-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)*(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*(-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^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*(-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^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*(-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^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*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^2*(3*b^6 - 2*a^6 + 3*a^2*b^4 + 2
2*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*(-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 - 2
4*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)*(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*(-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^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 + 1
161*a^10*b^2))/(a^3*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^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*(-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^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*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^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*(-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)*(8*a^10 + b^10 - 148*a^2*b^8 + 902*a^4*b^6 - 81
2*a^6*b^4 + 193*a^8*b^2))/(a*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^2*(16*a^1
2 + 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*(-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^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*(-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^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))*(((4
80*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^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*(-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)*(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*(-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^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*(-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
^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*(-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^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*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) - 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)/(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^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*(-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)*(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*(-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^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*(-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^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*(-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^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*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^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*(-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)*(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*(-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^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*(-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^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*(-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^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*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^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) + (atan(((((tan(c + d*x)
^(1/2)*(2*B^4*a^2*b^13 - B^4*b^15 + 49*B^4*a^4*b^11 + 2460*B^4*a^6*b^9 - 3631*B^4*a^8*b^7 + 1922*B^4*a^10*b^5
- 225*B^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*B^3*b^18*d^2 - 138*B^3*a^2*b^16*d^2 -
3046*B^3*a^4*b^14*d^2 + 4862*B^3*a^6*b^12*d^2 + 9222*B^3*a^8*b^10*d^2 - 5246*B^3*a^10*b^8*d^2 - 4290*B^3*a^12
*b^6*d^2 + 2442*B^3*a^14*b^4*d^2 + 32*B^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
*B*a*b^23*d^4 + 1472*B*a^3*b^21*d^4 + 8832*B*a^5*b^19*d^4 + 25344*B*a^7*b^17*d^4 + 40320*B*a^9*b^15*d^4 + 3494
4*B*a^11*b^13*d^4 + 10752*B*a^13*b^11*d^4 - 8448*B*a^15*b^9*d^4 - 10176*B*a^17*b^7*d^4 - 4160*B*a^19*b^5*d^4 -
640*B*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*(B^2*b^9 + 225*B^
2*a^8*b + 36*B^2*a^2*b^7 + 294*B^2*a^4*b^5 - 540*B^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 - 4
6080*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*(B^2*b^9 + 225*B^2*a^8*b + 36*B^2*a^2*b^7 + 294*B^2*a^4*b^5 - 5
40*B^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*B^2*a^5*b^16*d^2 - 1152*B^2*a^3*b^18*d^2 + 15296*
B^2*a^7*b^14*d^2 + 14128*B^2*a^9*b^12*d^2 - 5056*B^2*a^11*b^10*d^2 - 9248*B^2*a^13*b^8*d^2 + 64*B^2*a^15*b^6*d
^2 + 1800*B^2*a^17*b^4*d^2 + 64*B^2*a^19*b^2*d^2 + 8*B^2*a*b^20*d^2))/(64*(a^18*d^4 + a^2*b^16*d^4 + 8*a^4*b^1
4*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*(B^2*b^9 + 225*B^2*a^8*b + 36*B^2*a^2*b^7 + 294*B^2*a^4*b^5 - 540*B^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*(B^2*b^9 + 225*B^2*a^8*b + 36*B^2*a^2*b^7 + 294*B^2*a^4*b^5 - 540*B^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*(B^2*
b^9 + 225*B^2*a^8*b + 36*B^2*a^2*b^7 + 294*B^2*a^4*b^5 - 540*B^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)*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*B^4*a^2*b^13 - B^4*b^15 + 49*B^4*a^4*b^11 + 2460*B^4*a^6*b^9 - 3631*B^4*a^8*b^7 + 1922*B^4*a^10*b^5 - 22
5*B^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*B^3*b^18*d^2 - 138*B^3*a^2*b^16*d^2 - 304
6*B^3*a^4*b^14*d^2 + 4862*B^3*a^6*b^12*d^2 + 9222*B^3*a^8*b^10*d^2 - 5246*B^3*a^10*b^8*d^2 - 4290*B^3*a^12*b^6
*d^2 + 2442*B^3*a^14*b^4*d^2 + 32*B^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*B*a
*b^23*d^4 + 1472*B*a^3*b^21*d^4 + 8832*B*a^5*b^19*d^4 + 25344*B*a^7*b^17*d^4 + 40320*B*a^9*b^15*d^4 + 34944*B*
a^11*b^13*d^4 + 10752*B*a^13*b^11*d^4 - 8448*B*a^15*b^9*d^4 - 10176*B*a^17*b^7*d^4 - 4160*B*a^19*b^5*d^4 - 640
*B*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*(B^2*b^9 + 225*B^2*a^
8*b + 36*B^2*a^2*b^7 + 294*B^2*a^4*b^5 - 540*B^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*(B^2*b^9 + 225*B^2*a^8*b + 36*B^2*a^2*b^7 + 294*B^2*a^4*b^5 - 540*B
^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*B^2*a^5*b^16*d^2 - 1152*B^2*a^3*b^18*d^2 + 15296*B^2*
a^7*b^14*d^2 + 14128*B^2*a^9*b^12*d^2 - 5056*B^2*a^11*b^10*d^2 - 9248*B^2*a^13*b^8*d^2 + 64*B^2*a^15*b^6*d^2 +
1800*B^2*a^17*b^4*d^2 + 64*B^2*a^19*b^2*d^2 + 8*B^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*(B^2*b^9 + 225*B^2*a^8*b + 36*B^2*a^2*b^7 + 294*B^2*a^4*b^5 - 540*B^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*(
B^2*b^9 + 225*B^2*a^8*b + 36*B^2*a^2*b^7 + 294*B^2*a^4*b^5 - 540*B^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*(B^2*b^9
+ 225*B^2*a^8*b + 36*B^2*a^2*b^7 + 294*B^2*a^4*b^5 - 540*B^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*B^5*a*b^11 + 116*B^5
*a^3*b^9 - 270*B^5*a^5*b^7 + 420*B^5*a^7*b^5 - 225*B^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^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) - (((t
an(c + d*x)^(1/2)*(2*B^4*a^2*b^13 - B^4*b^15 + 49*B^4*a^4*b^11 + 2460*B^4*a^6*b^9 - 3631*B^4*a^8*b^7 + 1922*B^
4*a^10*b^5 - 225*B^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*B^3*b^18*d^2 - 138*B^3*a^2
*b^16*d^2 - 3046*B^3*a^4*b^14*d^2 + 4862*B^3*a^6*b^12*d^2 + 9222*B^3*a^8*b^10*d^2 - 5246*B^3*a^10*b^8*d^2 - 42
90*B^3*a^12*b^6*d^2 + 2442*B^3*a^14*b^4*d^2 + 32*B^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*B*a*b^23*d^4 + 1472*B*a^3*b^21*d^4 + 8832*B*a^5*b^19*d^4 + 25344*B*a^7*b^17*d^4 + 40320*B*a^9*b^15
*d^4 + 34944*B*a^11*b^13*d^4 + 10752*B*a^13*b^11*d^4 - 8448*B*a^15*b^9*d^4 - 10176*B*a^17*b^7*d^4 - 4160*B*a^1
9*b^5*d^4 - 640*B*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*(B^2*b
^9 + 225*B^2*a^8*b + 36*B^2*a^2*b^7 + 294*B^2*a^4*b^5 - 540*B^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^1
3*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 + 5
6*a^12*b^6*d^4 + 28*a^14*b^4*d^4 + 8*a^16*b^2*d^4)))*(-64*(B^2*b^9 + 225*B^2*a^8*b + 36*B^2*a^2*b^7 + 294*B^2*
a^4*b^5 - 540*B^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*B^2*a^5*b^16*d^2 - 1152*B^2*a^3*b^18*d
^2 + 15296*B^2*a^7*b^14*d^2 + 14128*B^2*a^9*b^12*d^2 - 5056*B^2*a^11*b^10*d^2 - 9248*B^2*a^13*b^8*d^2 + 64*B^2
*a^15*b^6*d^2 + 1800*B^2*a^17*b^4*d^2 + 64*B^2*a^19*b^2*d^2 + 8*B^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*(B^2*b^9 + 225*B^2*a^8*b + 36*B^2*a^2*b^7 + 294*B^2*a^4*b^5 - 540*B^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*(B^2*b^9 + 225*B^2*a^8*b + 36*B^2*a^2*b^7 + 294*B^2*a^4*b^5 - 540*B^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*(B^2*b^9 + 225*B^2*a^8*b + 36*B^2*a^2*b^7 + 294*B^2*a^4*b^5 - 540*B^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*B^4*a^2*b^13 - B^4*b^15 + 49*B^4*a^4*b^11 + 2460*B^4*a^6*b^9 - 3631*B^4*a^8*b^7 + 1922*B^4*a^10*
b^5 - 225*B^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 + 7
0*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*B^3*b^18*d^2 - 138*B^3*a^2*b^16*d
^2 - 3046*B^3*a^4*b^14*d^2 + 4862*B^3*a^6*b^12*d^2 + 9222*B^3*a^8*b^10*d^2 - 5246*B^3*a^10*b^8*d^2 - 4290*B^3*
a^12*b^6*d^2 + 2442*B^3*a^14*b^4*d^2 + 32*B^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*B*a*b^23*d^4 + 1472*B*a^3*b^21*d^4 + 8832*B*a^5*b^19*d^4 + 25344*B*a^7*b^17*d^4 + 40320*B*a^9*b^15*d^4 +
34944*B*a^11*b^13*d^4 + 10752*B*a^13*b^11*d^4 - 8448*B*a^15*b^9*d^4 - 10176*B*a^17*b^7*d^4 - 4160*B*a^19*b^5*d
^4 - 640*B*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 + 7
0*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*(B^2*b^9 + 22
5*B^2*a^8*b + 36*B^2*a^2*b^7 + 294*B^2*a^4*b^5 - 540*B^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))/(409
6*(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*(B^2*b^9 + 225*B^2*a^8*b + 36*B^2*a^2*b^7 + 294*B^2*a^4*b^5
- 540*B^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*B^2*a^5*b^16*d^2 - 1152*B^2*a^3*b^18*d^2 + 15
296*B^2*a^7*b^14*d^2 + 14128*B^2*a^9*b^12*d^2 - 5056*B^2*a^11*b^10*d^2 - 9248*B^2*a^13*b^8*d^2 + 64*B^2*a^15*b
^6*d^2 + 1800*B^2*a^17*b^4*d^2 + 64*B^2*a^19*b^2*d^2 + 8*B^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*(B^2*b^9 + 225*B^2*a^8*b + 36*B^2*a^2*b^7 + 294*B^2*a^4*b^5 - 540*B^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*(B^2*b^9 + 225*B^2*a^8*b + 36*B^2*a^2*b^7 + 294*B^2*a^4*b^5 - 540*B^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*(
B^2*b^9 + 225*B^2*a^8*b + 36*B^2*a^2*b^7 + 294*B^2*a^4*b^5 - 540*B^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*(B^2*b^9 + 22
5*B^2*a^8*b + 36*B^2*a^2*b^7 + 294*B^2*a^4*b^5 - 540*B^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)*(18*A^4*a^2*b^15 - 9*A^4*b^17 - 71*A^4*a^4*b^13 + 892*A^4*a^6*b^11 + 857*A^4*a^8*b^9 + 6802*A^4*a^10*b^
7 - 1257*A^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 + 7
0*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)) - (((846*A^3*a^3*b^17*d^2 + 1714*A^3*a^5
*b^15*d^2 + 3606*A^3*a^7*b^13*d^2 - 14578*A^3*a^9*b^11*d^2 - 34486*A^3*a^11*b^9*d^2 - 14970*A^3*a^13*b^7*d^2 +
2258*A^3*a^15*b^5*d^2 - 32*A^3*a^17*b^3*d^2 + 90*A^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*A*a^2*b^24*d^4 + 1728*A*a^4*b^22*d^4 + 8320*A*a^6*b^20*d^4 + 27264*A*a^8*b^18*d^4 + 62592*A*a^10*
b^16*d^4 + 99456*A*a^12*b^14*d^4 + 107520*A*a^14*b^12*d^4 + 76800*A*a^16*b^10*d^4 + 33984*A*a^18*b^8*d^4 + 787
2*A*a^20*b^6*d^4 + 384*A*a^22*b^4*d^4 - 128*A*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)) - (t
an(c + d*x)^(1/2)*(-64*(9*A^2*b^11 + 36*A^2*a^2*b^9 + 246*A^2*a^4*b^7 + 420*A^2*a^6*b^5 + 1225*A^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*A^2*b^11 +
36*A^2*a^2*b^9 + 246*A^2*a^4*b^7 + 420*A^2*a^6*b^5 + 1225*A^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*A^2*a^3*b^20*d^2 + 5024*A^2*a^5*b^18*d^2 + 14272*A^2*a^7*b^16*d^2 + 27824*A^2*a^9*b^14*d^2 + 53184*A^2
*a^11*b^12*d^2 + 70240*A^2*a^13*b^10*d^2 + 47680*A^2*a^15*b^8*d^2 + 12616*A^2*a^17*b^6*d^2 - 64*A^2*a^21*b^2*d
^2 + 72*A^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*A^2*b^11 + 36*A^2*a^2*b^9 + 24
6*A^2*a^4*b^7 + 420*A^2*a^6*b^5 + 1225*A^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*A^2*b^11 + 36*A^2*a^2*b^9 + 24
6*A^2*a^4*b^7 + 420*A^2*a^6*b^5 + 1225*A^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*A^2*b^11 + 36*A^2*a^2*b^9 + 24
6*A^2*a^4*b^7 + 420*A^2*a^6*b^5 + 1225*A^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) + (((tan(c + d*x)^(1/2)*(18*A^4*a^2*b^15
- 9*A^4*b^17 - 71*A^4*a^4*b^13 + 892*A^4*a^6*b^11 + 857*A^4*a^8*b^9 + 6802*A^4*a^10*b^7 - 1257*A^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)) + (((846*A^3*a^3*b^17*d^2 + 1714*A^3*a^5*b^15*d^2 + 3606*A^3*a^7
*b^13*d^2 - 14578*A^3*a^9*b^11*d^2 - 34486*A^3*a^11*b^9*d^2 - 14970*A^3*a^13*b^7*d^2 + 2258*A^3*a^15*b^5*d^2 -
32*A^3*a^17*b^3*d^2 + 90*A^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*A*a^2*b^24*
d^4 + 1728*A*a^4*b^22*d^4 + 8320*A*a^6*b^20*d^4 + 27264*A*a^8*b^18*d^4 + 62592*A*a^10*b^16*d^4 + 99456*A*a^12*
b^14*d^4 + 107520*A*a^14*b^12*d^4 + 76800*A*a^16*b^10*d^4 + 33984*A*a^18*b^8*d^4 + 7872*A*a^20*b^6*d^4 + 384*A
*a^22*b^4*d^4 - 128*A*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*A^2*b^11 + 36*A^2*a^2*b^9 + 246*A^2*a^4*b^7 + 420*A^2*a^6*b^5 + 1225*A^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*A^2*b^11 + 36*A^2*a^2*b^9 + 246*A^
2*a^4*b^7 + 420*A^2*a^6*b^5 + 1225*A^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 + 2
0*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*A^2*a^3*b^20*d^2
+ 5024*A^2*a^5*b^18*d^2 + 14272*A^2*a^7*b^16*d^2 + 27824*A^2*a^9*b^14*d^2 + 53184*A^2*a^11*b^12*d^2 + 70240*A
^2*a^13*b^10*d^2 + 47680*A^2*a^15*b^8*d^2 + 12616*A^2*a^17*b^6*d^2 - 64*A^2*a^21*b^2*d^2 + 72*A^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^1
4*b^6*d^4 + 28*a^16*b^4*d^4 + 8*a^18*b^2*d^4)))*(-64*(9*A^2*b^11 + 36*A^2*a^2*b^9 + 246*A^2*a^4*b^7 + 420*A^2*
a^6*b^5 + 1225*A^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^1
1*b^6*d^2 + 15*a^13*b^4*d^2 + 6*a^15*b^2*d^2)))*(-64*(9*A^2*b^11 + 36*A^2*a^2*b^9 + 246*A^2*a^4*b^7 + 420*A^2*
a^6*b^5 + 1225*A^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^1
1*b^6*d^2 + 15*a^13*b^4*d^2 + 6*a^15*b^2*d^2)))*(-64*(9*A^2*b^11 + 36*A^2*a^2*b^9 + 246*A^2*a^4*b^7 + 420*A^2*
a^6*b^5 + 1225*A^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))/((9*A^5*b^14 + 60*A^5*a^2*b^12 + 318*A^5*a^4*b^10 + 748*A^5*a^6*
b^8 + 1505*A^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) + (((tan(c + d*x)^(1/2)*(18*A^4*a^2*b^15 - 9*
A^4*b^17 - 71*A^4*a^4*b^13 + 892*A^4*a^6*b^11 + 857*A^4*a^8*b^9 + 6802*A^4*a^10*b^7 - 1257*A^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)) - (((846*A^3*a^3*b^17*d^2 + 1714*A^3*a^5*b^15*d^2 + 3606*A^3*a^7*b^1
3*d^2 - 14578*A^3*a^9*b^11*d^2 - 34486*A^3*a^11*b^9*d^2 - 14970*A^3*a^13*b^7*d^2 + 2258*A^3*a^15*b^5*d^2 - 32*
A^3*a^17*b^3*d^2 + 90*A^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^1
0*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*A*a^2*b^24*d^4
+ 1728*A*a^4*b^22*d^4 + 8320*A*a^6*b^20*d^4 + 27264*A*a^8*b^18*d^4 + 62592*A*a^10*b^16*d^4 + 99456*A*a^12*b^14
*d^4 + 107520*A*a^14*b^12*d^4 + 76800*A*a^16*b^10*d^4 + 33984*A*a^18*b^8*d^4 + 7872*A*a^20*b^6*d^4 + 384*A*a^2
2*b^4*d^4 - 128*A*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*A^
2*b^11 + 36*A^2*a^2*b^9 + 246*A^2*a^4*b^7 + 420*A^2*a^6*b^5 + 1225*A^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 - 215
04*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^2
6*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*A^2*b^11 + 36*A^2*a^2*b^9 + 246*A^2*a^
4*b^7 + 420*A^2*a^6*b^5 + 1225*A^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*A^2*a^3*b^20*d^2 + 5
024*A^2*a^5*b^18*d^2 + 14272*A^2*a^7*b^16*d^2 + 27824*A^2*a^9*b^14*d^2 + 53184*A^2*a^11*b^12*d^2 + 70240*A^2*a
^13*b^10*d^2 + 47680*A^2*a^15*b^8*d^2 + 12616*A^2*a^17*b^6*d^2 - 64*A^2*a^21*b^2*d^2 + 72*A^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*A^2*b^11 + 36*A^2*a^2*b^9 + 246*A^2*a^4*b^7 + 420*A^2*a^6*
b^5 + 1225*A^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*A^2*b^11 + 36*A^2*a^2*b^9 + 246*A^2*a^4*b^7 + 420*A^2*a^6*
b^5 + 1225*A^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*A^2*b^11 + 36*A^2*a^2*b^9 + 246*A^2*a^4*b^7 + 420*A^2*a^6*
b^5 + 1225*A^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) - (((tan(c + d*x)^(1/2)*(18*A^4*a^2*b^15 - 9*A^4*b^17 - 71*A^4*a^4*b^13
+ 892*A^4*a^6*b^11 + 857*A^4*a^8*b^9 + 6802*A^4*a^10*b^7 - 1257*A^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)) + (((846*A^3*a^3*b^17*d^2 + 1714*A^3*a^5*b^15*d^2 + 3606*A^3*a^7*b^13*d^2 - 14578*A^3*a^9*b^11*
d^2 - 34486*A^3*a^11*b^9*d^2 - 14970*A^3*a^13*b^7*d^2 + 2258*A^3*a^15*b^5*d^2 - 32*A^3*a^17*b^3*d^2 + 90*A^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*A*a^2*b^24*d^4 + 1728*A*a^4*b^22*d^4 + 832
0*A*a^6*b^20*d^4 + 27264*A*a^8*b^18*d^4 + 62592*A*a^10*b^16*d^4 + 99456*A*a^12*b^14*d^4 + 107520*A*a^14*b^12*d
^4 + 76800*A*a^16*b^10*d^4 + 33984*A*a^18*b^8*d^4 + 7872*A*a^20*b^6*d^4 + 384*A*a^22*b^4*d^4 - 128*A*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*A^2*b^11 + 36*A^2*a^2*b^9 + 2
46*A^2*a^4*b^7 + 420*A^2*a^6*b^5 + 1225*A^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*A^2*b^11 + 36*A^2*a^2*b^9 + 246*A^2*a^4*b^7 + 420*A^2*a^6*b^5 + 1
225*A^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*A^2*a^3*b^20*d^2 + 5024*A^2*a^5*b^18*d^2 + 1427
2*A^2*a^7*b^16*d^2 + 27824*A^2*a^9*b^14*d^2 + 53184*A^2*a^11*b^12*d^2 + 70240*A^2*a^13*b^10*d^2 + 47680*A^2*a^
15*b^8*d^2 + 12616*A^2*a^17*b^6*d^2 - 64*A^2*a^21*b^2*d^2 + 72*A^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*A^2*b^11 + 36*A^2*a^2*b^9 + 246*A^2*a^4*b^7 + 420*A^2*a^6*b^5 + 1225*A^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*A^2*b^11 + 36*A^2*a^2*b^9 + 246*A^2*a^4*b^7 + 420*A^2*a^6*b^5 + 1225*A^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*A^2*b^11 + 36*A^2*a^2*b^9 + 246*A^2*a^4*b^7 + 420*A^2*a^6*b^5 + 1225*A^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^1
5*b^2*d^2)))*(-64*(9*A^2*b^11 + 36*A^2*a^2*b^9 + 246*A^2*a^4*b^7 + 420*A^2*a^6*b^5 + 1225*A^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))

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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)**(1/2)/(a+b*tan(d*x+c))**3,x)

[Out]

Timed out

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