3.412 \(\int \frac {\tan ^{\frac {3}{2}}(c+d x) (A+B \tan (c+d x))}{(a+b \tan (c+d x))^3} \, dx\)
Optimal. Leaf size=533 \[ \frac {a (A b-a B) \sqrt {\tan (c+d x)}}{2 b d \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}-\frac {\left (-\left (a^3 (A+B)\right )+3 a^2 b (A-B)+3 a b^2 (A+B)-b^3 (A-B)\right ) \tan ^{-1}\left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} d \left (a^2+b^2\right )^3}+\frac {\left (-\left (a^3 (A+B)\right )+3 a^2 b (A-B)+3 a b^2 (A+B)-b^3 (A-B)\right ) \tan ^{-1}\left (\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{\sqrt {2} d \left (a^2+b^2\right )^3}+\frac {\left (a^3 B+3 a^2 A b+9 a b^2 B-5 A b^3\right ) \sqrt {\tan (c+d x)}}{4 b d \left (a^2+b^2\right )^2 (a+b \tan (c+d x))}+\frac {\left (a^3 (A-B)+3 a^2 b (A+B)-3 a b^2 (A-B)-b^3 (A+B)\right ) \log \left (\tan (c+d x)-\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{2 \sqrt {2} d \left (a^2+b^2\right )^3}-\frac {\left (a^3 (A-B)+3 a^2 b (A+B)-3 a b^2 (A-B)-b^3 (A+B)\right ) \log \left (\tan (c+d x)+\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{2 \sqrt {2} d \left (a^2+b^2\right )^3}+\frac {\left (a^5 B+3 a^4 A b+18 a^3 b^2 B-26 a^2 A b^3-15 a b^4 B+3 A b^5\right ) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {\tan (c+d x)}}{\sqrt {a}}\right )}{4 \sqrt {a} b^{3/2} d \left (a^2+b^2\right )^3} \]
[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*(3*A*a^4*b-26*A*a^2*b^3+3*A*b^5+B*a^5+18*B*a^3*b^2-15*B*a*b^4)*arctan(b^(1/2)*tan(d*x+c
)^(1/2)/a^(1/2))/b^(3/2)/(a^2+b^2)^3/d/a^(1/2)+1/2*a*(A*b-B*a)*tan(d*x+c)^(1/2)/b/(a^2+b^2)/d/(a+b*tan(d*x+c))
^2+1/4*(3*A*a^2*b-5*A*b^3+B*a^3+9*B*a*b^2)*tan(d*x+c)^(1/2)/b/(a^2+b^2)^2/d/(a+b*tan(d*x+c))
________________________________________________________________________________________
Rubi [A] time = 1.23, antiderivative size = 533, 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 = {3605, 3649, 3653, 3534, 1168, 1162, 617, 204, 1165, 628, 3634, 63, 205}
\[ -\frac {\left (3 a^2 b (A-B)+a^3 (-(A+B))+3 a b^2 (A+B)-b^3 (A-B)\right ) \tan ^{-1}\left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} d \left (a^2+b^2\right )^3}+\frac {\left (3 a^2 b (A-B)+a^3 (-(A+B))+3 a b^2 (A+B)-b^3 (A-B)\right ) \tan ^{-1}\left (\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{\sqrt {2} d \left (a^2+b^2\right )^3}+\frac {\left (-26 a^2 A b^3+3 a^4 A b+18 a^3 b^2 B+a^5 B-15 a b^4 B+3 A b^5\right ) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {\tan (c+d x)}}{\sqrt {a}}\right )}{4 \sqrt {a} b^{3/2} d \left (a^2+b^2\right )^3}+\frac {a (A b-a B) \sqrt {\tan (c+d x)}}{2 b d \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}+\frac {\left (3 a^2 A b+a^3 B+9 a b^2 B-5 A b^3\right ) \sqrt {\tan (c+d x)}}{4 b d \left (a^2+b^2\right )^2 (a+b \tan (c+d x))}+\frac {\left (3 a^2 b (A+B)+a^3 (A-B)-3 a b^2 (A-B)-b^3 (A+B)\right ) \log \left (\tan (c+d x)-\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{2 \sqrt {2} d \left (a^2+b^2\right )^3}-\frac {\left (3 a^2 b (A+B)+a^3 (A-B)-3 a b^2 (A-B)-b^3 (A+B)\right ) \log \left (\tan (c+d x)+\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{2 \sqrt {2} d \left (a^2+b^2\right )^3} \]
Antiderivative was successfully verified.
[In]
Int[(Tan[c + d*x]^(3/2)*(A + B*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]]])/(Sq
rt[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]*
Sqrt[Tan[c + d*x]]])/(Sqrt[2]*(a^2 + b^2)^3*d) + ((3*a^4*A*b - 26*a^2*A*b^3 + 3*A*b^5 + a^5*B + 18*a^3*b^2*B -
15*a*b^4*B)*ArcTan[(Sqrt[b]*Sqrt[Tan[c + d*x]])/Sqrt[a]])/(4*Sqrt[a]*b^(3/2)*(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*Sqr
t[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*(A*b - a*B)*Sqrt[Tan[c + d*x]])/(2*b*(a^2 + b^
2)*d*(a + b*Tan[c + d*x])^2) + ((3*a^2*A*b - 5*A*b^3 + a^3*B + 9*a*b^2*B)*Sqrt[Tan[c + d*x]])/(4*b*(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 3605
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*c - a*d)*(B*c - A*d)*(a + b*Tan[e + f*x])^(m - 1)*(c + d*Tan[e
+ f*x])^(n + 1))/(d*f*(n + 1)*(c^2 + d^2)), x] - Dist[1/(d*(n + 1)*(c^2 + d^2)), Int[(a + b*Tan[e + f*x])^(m -
2)*(c + d*Tan[e + f*x])^(n + 1)*Simp[a*A*d*(b*d*(m - 1) - a*c*(n + 1)) + (b*B*c - (A*b + a*B)*d)*(b*c*(m - 1)
+ a*d*(n + 1)) - d*((a*A - b*B)*(b*c - a*d) + (A*b + a*B)*(a*c + b*d))*(n + 1)*Tan[e + f*x] - b*(d*(A*b*c + a
*B*c - a*A*d)*(m + n) - b*B*(c^2*(m - 1) - d^2*(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] && GtQ[m, 1] && LtQ[n, -1] && (Inte
gerQ[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 {\tan ^{\frac {3}{2}}(c+d x) (A+B \tan (c+d x))}{(a+b \tan (c+d x))^3} \, dx &=\frac {a (A b-a B) \sqrt {\tan (c+d x)}}{2 b \left (a^2+b^2\right ) d (a+b \tan (c+d x))^2}+\frac {\int \frac {-\frac {1}{2} a (A b-a B)+2 b (A b-a B) \tan (c+d x)+\frac {1}{2} \left (3 a A b+a^2 B+4 b^2 B\right ) \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 (A b-a B) \sqrt {\tan (c+d x)}}{2 b \left (a^2+b^2\right ) d (a+b \tan (c+d x))^2}+\frac {\left (3 a^2 A b-5 A b^3+a^3 B+9 a b^2 B\right ) \sqrt {\tan (c+d x)}}{4 b \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))}+\frac {\int \frac {-\frac {1}{4} a \left (5 a^2 A b-3 A b^3-a^3 B+7 a b^2 B\right )+2 a b \left (2 a A b-a^2 B+b^2 B\right ) \tan (c+d x)+\frac {1}{4} a \left (3 a^2 A b-5 A b^3+a^3 B+9 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 (A b-a B) \sqrt {\tan (c+d x)}}{2 b \left (a^2+b^2\right ) d (a+b \tan (c+d x))^2}+\frac {\left (3 a^2 A b-5 A b^3+a^3 B+9 a b^2 B\right ) \sqrt {\tan (c+d x)}}{4 b \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))}+\frac {\int \frac {-2 a b \left (a^3 A-3 a A b^2+3 a^2 b B-b^3 B\right )+2 a b \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 b \left (a^2+b^2\right )^3}+\frac {\left (3 a^4 A b-26 a^2 A b^3+3 A b^5+a^5 B+18 a^3 b^2 B-15 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 b \left (a^2+b^2\right )^3}\\ &=\frac {a (A b-a B) \sqrt {\tan (c+d x)}}{2 b \left (a^2+b^2\right ) d (a+b \tan (c+d x))^2}+\frac {\left (3 a^2 A b-5 A b^3+a^3 B+9 a b^2 B\right ) \sqrt {\tan (c+d x)}}{4 b \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))}+\frac {\operatorname {Subst}\left (\int \frac {-2 a b \left (a^3 A-3 a A b^2+3 a^2 b B-b^3 B\right )+2 a b \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 b \left (a^2+b^2\right )^3 d}+\frac {\left (3 a^4 A b-26 a^2 A b^3+3 A b^5+a^5 B+18 a^3 b^2 B-15 a b^4 B\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {x} (a+b x)} \, dx,x,\tan (c+d x)\right )}{8 b \left (a^2+b^2\right )^3 d}\\ &=\frac {a (A b-a B) \sqrt {\tan (c+d x)}}{2 b \left (a^2+b^2\right ) d (a+b \tan (c+d x))^2}+\frac {\left (3 a^2 A b-5 A b^3+a^3 B+9 a b^2 B\right ) \sqrt {\tan (c+d x)}}{4 b \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))}+\frac {\left (3 a^4 A b-26 a^2 A b^3+3 A b^5+a^5 B+18 a^3 b^2 B-15 a b^4 B\right ) \operatorname {Subst}\left (\int \frac {1}{a+b x^2} \, dx,x,\sqrt {\tan (c+d x)}\right )}{4 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 ) \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 (3 a^4 A b-26 a^2 A b^3+3 A b^5+a^5 B+18 a^3 b^2 B-15 a b^4 B\right ) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {\tan (c+d x)}}{\sqrt {a}}\right )}{4 \sqrt {a} b^{3/2} \left (a^2+b^2\right )^3 d}+\frac {a (A b-a B) \sqrt {\tan (c+d x)}}{2 b \left (a^2+b^2\right ) d (a+b \tan (c+d x))^2}+\frac {\left (3 a^2 A b-5 A b^3+a^3 B+9 a b^2 B\right ) \sqrt {\tan (c+d x)}}{4 b \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 {\left (3 a^4 A b-26 a^2 A b^3+3 A b^5+a^5 B+18 a^3 b^2 B-15 a b^4 B\right ) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {\tan (c+d x)}}{\sqrt {a}}\right )}{4 \sqrt {a} b^{3/2} \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 {a (A b-a B) \sqrt {\tan (c+d x)}}{2 b \left (a^2+b^2\right ) d (a+b \tan (c+d x))^2}+\frac {\left (3 a^2 A b-5 A b^3+a^3 B+9 a b^2 B\right ) \sqrt {\tan (c+d x)}}{4 b \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 {\left (3 a^4 A b-26 a^2 A b^3+3 A b^5+a^5 B+18 a^3 b^2 B-15 a b^4 B\right ) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {\tan (c+d x)}}{\sqrt {a}}\right )}{4 \sqrt {a} b^{3/2} \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 {a (A b-a B) \sqrt {\tan (c+d x)}}{2 b \left (a^2+b^2\right ) d (a+b \tan (c+d x))^2}+\frac {\left (3 a^2 A b-5 A b^3+a^3 B+9 a b^2 B\right ) \sqrt {\tan (c+d x)}}{4 b \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))}\\ \end {align*}
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Mathematica [C] time = 6.08, size = 333, normalized size = 0.62 \[ \frac {\frac {\left (a^2 B+3 a A b+4 b^2 B\right ) \sqrt {\tan (c+d x)}}{a^2+b^2}-\frac {2 (a+b \tan (c+d x)) \left (-\frac {3}{4} a^{5/2} \sqrt {b} \left (a^2+b^2\right ) \left (a^3 B+3 a^2 A b+9 a b^2 B-5 A b^3\right ) \sqrt {\tan (c+d x)}+(a+b \tan (c+d x)) \left (-\frac {3}{4} a^2 \left (a^5 B+3 a^4 A b+18 a^3 b^2 B-26 a^2 A b^3-15 a b^4 B+3 A b^5\right ) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {\tan (c+d x)}}{\sqrt {a}}\right )-3 \sqrt [4]{-1} a^{5/2} b^{3/2} \left ((a+i b)^3 (A-i B) \tan ^{-1}\left ((-1)^{3/4} \sqrt {\tan (c+d x)}\right )+(a-i b)^3 (A+i B) \tanh ^{-1}\left ((-1)^{3/4} \sqrt {\tan (c+d x)}\right )\right )\right )\right )}{a^{5/2} \sqrt {b} \left (a^2+b^2\right )^3}-4 B \sqrt {\tan (c+d x)}}{6 b d (a+b \tan (c+d x))^2} \]
Antiderivative was successfully verified.
[In]
Integrate[(Tan[c + d*x]^(3/2)*(A + B*Tan[c + d*x]))/(a + b*Tan[c + d*x])^3,x]
[Out]
(-4*B*Sqrt[Tan[c + d*x]] + ((3*a*A*b + a^2*B + 4*b^2*B)*Sqrt[Tan[c + d*x]])/(a^2 + b^2) - (2*(a + b*Tan[c + d*
x])*((-3*a^(5/2)*Sqrt[b]*(a^2 + b^2)*(3*a^2*A*b - 5*A*b^3 + a^3*B + 9*a*b^2*B)*Sqrt[Tan[c + d*x]])/4 + ((-3*a^
2*(3*a^4*A*b - 26*a^2*A*b^3 + 3*A*b^5 + a^5*B + 18*a^3*b^2*B - 15*a*b^4*B)*ArcTan[(Sqrt[b]*Sqrt[Tan[c + d*x]])
/Sqrt[a]])/4 - 3*(-1)^(1/4)*a^(5/2)*b^(3/2)*((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 + b*Tan[c + d*x])))/(a^(5/2)*Sqrt[b]*(a^2 + b^2
)^3))/(6*b*d*(a + b*Tan[c + d*x])^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)^(3/2)*(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)^(3/2)*(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 = 0.43, size = 1835, normalized size = 3.44 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(tan(d*x+c)^(3/2)*(A+B*tan(d*x+c))/(a+b*tan(d*x+c))^3,x)
[Out]
1/2/d*a^3*b^2/(a^2+b^2)^3/(a+b*tan(d*x+c))^2*A*tan(d*x+c)^(1/2)-3/2/d/(a^2+b^2)^3*B*2^(1/2)*arctan(-1+2^(1/2)*
tan(d*x+c)^(1/2))*a^2*b-3/4/d/(a^2+b^2)^3*B*2^(1/2)*ln((1+2^(1/2)*tan(d*x+c)^(1/2)+tan(d*x+c))/(1-2^(1/2)*tan(
d*x+c)^(1/2)+tan(d*x+c)))*a^2*b+3/4/d/(a^2+b^2)^3*A*2^(1/2)*ln((1-2^(1/2)*tan(d*x+c)^(1/2)+tan(d*x+c))/(1+2^(1
/2)*tan(d*x+c)^(1/2)+tan(d*x+c)))*a^2*b-15/4/d/(a^2+b^2)^3*b^3/(a*b)^(1/2)*arctan(tan(d*x+c)^(1/2)*b/(a*b)^(1/
2))*a*B-3/4/d/(a^2+b^2)^3/(a+b*tan(d*x+c))^2*a*b^4*tan(d*x+c)^(1/2)*A+7/4/d/(a^2+b^2)^3/(a+b*tan(d*x+c))^2*a^2
*b^3*tan(d*x+c)^(1/2)*B+5/2/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+3/2/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+9/4/d/(a^2+b^2)^3/(a+b*tan(d*x+c)
)^2*tan(d*x+c)^(3/2)*a*b^4*B+1/4/d*a^5/b/(a^2+b^2)^3/(a*b)^(1/2)*arctan(tan(d*x+c)^(1/2)*b/(a*b)^(1/2))*B+9/2/
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)*arctan(-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^2-13/2/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-1/4/d*a^6/b/(a^2+b^2)^3/(a+b*tan(d*x+c))^2
*B*tan(d*x+c)^(1/2)+3/4/d*a^4*b/(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^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+3/4/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-5/4/d/(a^2+b^2)^3/(a+b*tan(d*x+c))^2*tan(d*x+c)^(3/2)*A*b^5+5/4/d*a^5/(a^2+b^2)^3/(a+
b*tan(d*x+c))^2*A*tan(d*x+c)^(1/2)+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))*A-
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+1/4/d*a^5/
(a^2+b^2)^3/(a+b*tan(d*x+c))^2*tan(d*x+c)^(3/2)*B
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maxima [A] time = 0.96, size = 538, normalized size = 1.01 \[ \frac {\frac {{\left (B a^{5} + 3 \, A a^{4} b + 18 \, B a^{3} b^{2} - 26 \, A a^{2} b^{3} - 15 \, B a b^{4} + 3 \, A b^{5}\right )} \arctan \left (\frac {b \sqrt {\tan \left (d x + c\right )}}{\sqrt {a b}}\right )}{{\left (a^{6} b + 3 \, a^{4} b^{3} + 3 \, a^{2} b^{5} + b^{7}\right )} \sqrt {a b}} - \frac {2 \, \sqrt {2} {\left ({\left (A + B\right )} a^{3} - 3 \, {\left (A - B\right )} a^{2} b - 3 \, {\left (A + B\right )} a b^{2} + {\left (A - B\right )} b^{3}\right )} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + 2 \, \sqrt {\tan \left (d x + c\right )}\right )}\right ) + 2 \, \sqrt {2} {\left ({\left (A + B\right )} a^{3} - 3 \, {\left (A - B\right )} a^{2} b - 3 \, {\left (A + B\right )} a b^{2} + {\left (A - B\right )} b^{3}\right )} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - 2 \, \sqrt {\tan \left (d x + c\right )}\right )}\right ) + \sqrt {2} {\left ({\left (A - B\right )} a^{3} + 3 \, {\left (A + B\right )} a^{2} b - 3 \, {\left (A - B\right )} a b^{2} - {\left (A + B\right )} b^{3}\right )} \log \left (\sqrt {2} \sqrt {\tan \left (d x + c\right )} + \tan \left (d x + c\right ) + 1\right ) - \sqrt {2} {\left ({\left (A - B\right )} a^{3} + 3 \, {\left (A + B\right )} a^{2} b - 3 \, {\left (A - B\right )} a b^{2} - {\left (A + B\right )} b^{3}\right )} \log \left (-\sqrt {2} \sqrt {\tan \left (d x + c\right )} + \tan \left (d x + c\right ) + 1\right )}{a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}} + \frac {{\left (B a^{3} b + 3 \, A a^{2} b^{2} + 9 \, B a b^{3} - 5 \, A b^{4}\right )} \tan \left (d x + c\right )^{\frac {3}{2}} - {\left (B a^{4} - 5 \, A a^{3} b - 7 \, B a^{2} b^{2} + 3 \, A a b^{3}\right )} \sqrt {\tan \left (d x + c\right )}}{a^{6} b + 2 \, a^{4} b^{3} + a^{2} b^{5} + {\left (a^{4} b^{3} + 2 \, a^{2} b^{5} + b^{7}\right )} \tan \left (d x + c\right )^{2} + 2 \, {\left (a^{5} b^{2} + 2 \, a^{3} b^{4} + a b^{6}\right )} \tan \left (d x + c\right )}}{4 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(tan(d*x+c)^(3/2)*(A+B*tan(d*x+c))/(a+b*tan(d*x+c))^3,x, algorithm="maxima")
[Out]
1/4*((B*a^5 + 3*A*a^4*b + 18*B*a^3*b^2 - 26*A*a^2*b^3 - 15*B*a*b^4 + 3*A*b^5)*arctan(b*sqrt(tan(d*x + c))/sqrt
(a*b))/((a^6*b + 3*a^4*b^3 + 3*a^2*b^5 + b^7)*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) + ((B*a^3*b + 3*A*a^2*b^2 + 9*B*a*b^3 - 5*A*b^4)*tan(d*
x + c)^(3/2) - (B*a^4 - 5*A*a^3*b - 7*B*a^2*b^2 + 3*A*a*b^3)*sqrt(tan(d*x + c)))/(a^6*b + 2*a^4*b^3 + a^2*b^5
+ (a^4*b^3 + 2*a^2*b^5 + b^7)*tan(d*x + c)^2 + 2*(a^5*b^2 + 2*a^3*b^4 + a*b^6)*tan(d*x + c)))/d
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mupad [B] time = 53.11, size = 25944, normalized size = 48.68 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((tan(c + d*x)^(3/2)*(A + B*tan(c + d*x)))/(a + b*tan(c + d*x))^3,x)
[Out]
(log((((((((((64*B*a*b^3*(11*a^2 - 13*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*tan(c + d*x)^(1/2)*(a^10 - 1
84*b^10 + 833*a^2*b^8 - 812*a^4*b^6 + 262*a^6*b^4 + 44*a^8*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^2*(5*a^10 - 1199*b^10 + 5017*a^2*b^8 - 5142*a^4*b^6 + 1106*a^6*b^4 + 181*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*tan(c + d*x)^(1/2)*(a^14 - 32*b^14 + 97*
a^2*b^12 - 2082*a^4*b^10 + 3631*a^6*b^8 - 2300*a^8*b^6 + 79*a^10*b^4 + 30*a^12*b^2))/(b*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*a*(a^10 - 120*b^10 + 249*a^2*b^8 - 388*a^4*b^6 + 302*a^6*b^4 + 3
6*a^8*b^2))/(2*b*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 - 2
4*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*a*b^3*(11*a^2 - 13*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*tan(c + d*x)^(1/2)*(a^10 - 184*b^10 + 833*a^2*b^8 - 812*a^4*b^6 + 262*a^6*b^4 + 44*a^8*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^2*(5*a^10 - 1199*b^10 + 5017*a^2*b
^8 - 5142*a^4*b^6 + 1106*a^6*b^4 + 181*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*tan(c + d*x)^(1/2)*(a^14 - 32*b^14 + 97*a^2*b^12 - 2082*a^4*b^10 + 3631*a^6*b^8 - 2300*a^8*b^6 + 79*a
^10*b^4 + 30*a^12*b^2))/(b*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*a*(a^10 - 120*
b^10 + 249*a^2*b^8 - 388*a^4*b^6 + 302*a^6*b^4 + 36*a^8*b^2))/(2*b*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 + 4
80*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((((((((((6
4*B*a*b^3*(11*a^2 - 13*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*tan(c + d*x)^(1/2)*(a^10 - 184*b^10 + 833*a
^2*b^8 - 812*a^4*b^6 + 262*a^6*b^4 + 44*a^8*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^2*(5*a^10 - 1199*b^10 + 5017*a^2*b^8 - 5142*a^4*b^6 + 1106*a^6*b^4 + 181*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*tan(c + d*x)^(1/2)*(a^14 - 32*b^14 + 97*a^2*b^12 - 2082
*a^4*b^10 + 3631*a^6*b^8 - 2300*a^8*b^6 + 79*a^10*b^4 + 30*a^12*b^2))/(b*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*a*(a^10 - 120*b^10 + 249*a^2*b^8 - 388*a^4*b^6 + 302*a^6*b^4 + 36*a^8*b^2))/(2*
b*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)/(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*a*b^3*(11*a^2 - 13*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 - 1
5*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*tan(c + d*x)^(1/2)*(a^10 - 184*b^10 + 833*a^2*b^8 - 812*a^4*b^6 + 262*a^6*b^4 + 44*a^8*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^2*(5*a^10 - 1199*b^10 + 5017*a^2*b^8 - 514
2*a^4*b^6 + 1106*a^6*b^4 + 181*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
*tan(c + d*x)^(1/2)*(a^14 - 32*b^14 + 97*a^2*b^12 - 2082*a^4*b^10 + 3631*a^6*b^8 - 2300*a^8*b^6 + 79*a^10*b^4
+ 30*a^12*b^2))/(b*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*a*(a^10 - 120*b^10 + 2
49*a^2*b^8 - 388*a^4*b^6 + 302*a^6*b^4 + 36*a^8*b^2))/(2*b*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) + ((tan(c + d*x)
^(3/2)*(B*a^3 + 9*B*a*b^2))/(4*(a^4 + b^4 + 2*a^2*b^2)) - (B*a^2*tan(c + d*x)^(1/2)*(a^2 - 7*b^2))/(4*b*(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*(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*(-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*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*(-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*a*b*(9*a^1
0 - 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*(-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*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*(-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^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*A^4*a^2*b^10*d^4 - 1
6*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*(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*(-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*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*(-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*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*(-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*tan(c + d*x)^(1/2)*(9*a^12 + 41*b^12 - 82*a^2*b^10 + 183
1*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*(-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^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
*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*(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*(-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*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*(-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*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*(-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*tan(c + d*x)^(1/2)*(9*a^12 + 41*b^12 - 8
2*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*(-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^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*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*(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*(-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*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*(-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*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*(-A^4*d^4*(a^6 - b^6 + 15*a^2*b^4 - 15*a^4*b^2)^2)^(1/2) + 8
0*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*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*(-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^2*(9*a^8 - 15*b^8 + 28*a^2*b^6 + 878*a^4*b^4 - 18
0*a^6*b^2))/(2*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) - ((tan(c + d*x)^(3/2)*(5*A*b^3 - 3*A*a^2*b))/(4*(a^4 + b^4
+ 2*a^2*b^2)) - (a*tan(c + d*x)^(1/2)*(5*A*a^2 - 3*A*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)) + (atan(-((((tan(c + d*x)^(1/2)*(41*A^4*b^13 + 9*A^4*a^12*b - 82*A^4*a^2*b^11
+ 1831*A^4*a^4*b^9 - 4348*A^4*a^6*b^7 + 1671*A^4*a^8*b^5 - 210*A^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*A^3*a^5*b^11*d^2 - 4494*A^3*a^3*b^13*d^2 + 17194*A^3*a^7*b^9*d^2 + 5298*A^3*a^9*b^7*d^2 - 33
38*A^3*a^11*b^5*d^2 + 506*A^3*a^13*b^3*d^2 + 518*A^3*a*b^15*d^2 - 18*A^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*A^2*a^3*b^16*d^2 - 7456*A^2*a^5*b^14*d^2 - 576*A^2*a^7*b^12*d^2 + 1
9504*A^2*a^9*b^10*d^2 + 18880*A^2*a^11*b^8*d^2 + 3808*A^2*a^13*b^6*d^2 - 960*A^2*a^15*b^4*d^2 + 8*A^2*a^17*b^2
*d^2 + 1544*A^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 + 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)) + (((4224*A*a^4*b^18*d^4 - 320*A*a^2*b^20
*d^4 - 192*A*b^22*d^4 + 22272*A*a^6*b^16*d^4 + 51072*A*a^8*b^14*d^4 + 67200*A*a^10*b^12*d^4 + 53760*A*a^12*b^1
0*d^4 + 25344*A*a^14*b^8*d^4 + 5952*A*a^16*b^6*d^4 + 192*A*a^18*b^4*d^4 - 128*A*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*A^2*a^8 + 9*A^2*b^8 - 156*A^2*a^2*b^6 + 694*A^2*a^4*b^4
- 156*A^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^1
2*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*A^2*a
^8 + 9*A^2*b^8 - 156*A^2*a^2*b^6 + 694*A^2*a^4*b^4 - 156*A^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*A^2*a^8 + 9*A^2*
b^8 - 156*A^2*a^2*b^6 + 694*A^2*a^4*b^4 - 156*A^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*A^2*a^8 + 9*A^2*b^8 - 156*A
^2*a^2*b^6 + 694*A^2*a^4*b^4 - 156*A^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 + 2
0*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*A^2*a^8 + 9*A^2*b^8 - 156*A^2*a^2*b^6
+ 694*A^2*a^4*b^4 - 156*A^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*A^4*b^13 + 9*A^4*a^12*b - 82*A^4
*a^2*b^11 + 1831*A^4*a^4*b^9 - 4348*A^4*a^6*b^7 + 1671*A^4*a^8*b^5 - 210*A^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*A^3*a^5*b^11*d^2 - 4494*A^3*a^3*b^13*d^2 + 17194*A^3*a^7*b^9*d^2 + 5298*A^3*a^9*b^
7*d^2 - 3338*A^3*a^11*b^5*d^2 + 506*A^3*a^13*b^3*d^2 + 518*A^3*a*b^15*d^2 - 18*A^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*A^2*a^3*b^16*d^2 - 7456*A^2*a^5*b^14*d^2 - 576*A^2*a^7*b^
12*d^2 + 19504*A^2*a^9*b^10*d^2 + 18880*A^2*a^11*b^8*d^2 + 3808*A^2*a^13*b^6*d^2 - 960*A^2*a^15*b^4*d^2 + 8*A^
2*a^17*b^2*d^2 + 1544*A^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*A*a^4*b^18*d^4 - 320*
A*a^2*b^20*d^4 - 192*A*b^22*d^4 + 22272*A*a^6*b^16*d^4 + 51072*A*a^8*b^14*d^4 + 67200*A*a^10*b^12*d^4 + 53760*
A*a^12*b^10*d^4 + 25344*A*a^14*b^8*d^4 + 5952*A*a^16*b^6*d^4 + 192*A*a^18*b^4*d^4 - 128*A*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 + 2
8*a^12*b^4*d^5 + 8*a^14*b^2*d^5)) + (tan(c + d*x)^(1/2)*(-64*(9*A^2*a^8 + 9*A^2*b^8 - 156*A^2*a^2*b^6 + 694*A^
2*a^4*b^4 - 156*A^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)))*(-6
4*(9*A^2*a^8 + 9*A^2*b^8 - 156*A^2*a^2*b^6 + 694*A^2*a^4*b^4 - 156*A^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^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)))*(-64*(9*A^2*a^
8 + 9*A^2*b^8 - 156*A^2*a^2*b^6 + 694*A^2*a^4*b^4 - 156*A^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*A^2*a^8 + 9*A^2*b
^8 - 156*A^2*a^2*b^6 + 694*A^2*a^4*b^4 - 156*A^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*A^2*a^8 + 9*A^2*b^8 - 156*A^
2*a^2*b^6 + 694*A^2*a^4*b^4 - 156*A^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))/((28*A^5*a^2*b^8 - 15*A^5*b^10 + 878*A^5*a^4*b^6
- 180*A^5*a^6*b^4 + 9*A^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*A^4*b^13 + 9
*A^4*a^12*b - 82*A^4*a^2*b^11 + 1831*A^4*a^4*b^9 - 4348*A^4*a^6*b^7 + 1671*A^4*a^8*b^5 - 210*A^4*a^10*b^3))/(6
4*(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*A^3*a^5*b^11*d^2 - 4494*A^3*a^3*b^13*d^2 + 17194*A^3*a^7*b^9*d
^2 + 5298*A^3*a^9*b^7*d^2 - 3338*A^3*a^11*b^5*d^2 + 506*A^3*a^13*b^3*d^2 + 518*A^3*a*b^15*d^2 - 18*A^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*A^2*a^3*b^16*d^2 - 7456*A^2*a^5*b^14*
d^2 - 576*A^2*a^7*b^12*d^2 + 19504*A^2*a^9*b^10*d^2 + 18880*A^2*a^11*b^8*d^2 + 3808*A^2*a^13*b^6*d^2 - 960*A^2
*a^15*b^4*d^2 + 8*A^2*a^17*b^2*d^2 + 1544*A^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*A
*a^4*b^18*d^4 - 320*A*a^2*b^20*d^4 - 192*A*b^22*d^4 + 22272*A*a^6*b^16*d^4 + 51072*A*a^8*b^14*d^4 + 67200*A*a^
10*b^12*d^4 + 53760*A*a^12*b^10*d^4 + 25344*A*a^14*b^8*d^4 + 5952*A*a^16*b^6*d^4 + 192*A*a^18*b^4*d^4 - 128*A*
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*A^2*a^8 + 9*A^2*b^8 - 156*
A^2*a^2*b^6 + 694*A^2*a^4*b^4 - 156*A^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*A^2*a^8 + 9*A^2*b^8 - 156*A^2*a^2*b^6 + 694*A^2*a^4*b^4 - 156*A^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*A^2*a^8 + 9*A^2*b^8 - 156*A^2*a^2*b^6 + 694*A^2*a^4*b^4 - 156*A^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*A^2*a^8 + 9*A^2*b^8 - 156*A^2*a^2*b^6 + 694*A^2*a^4*b^4 - 156*A^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*A^2*a^8
+ 9*A^2*b^8 - 156*A^2*a^2*b^6 + 694*A^2*a^4*b^4 - 156*A^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*A^4*b
^13 + 9*A^4*a^12*b - 82*A^4*a^2*b^11 + 1831*A^4*a^4*b^9 - 4348*A^4*a^6*b^7 + 1671*A^4*a^8*b^5 - 210*A^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*A^3*a^5*b^11*d^2 - 4494*A^3*a^3*b^13*d^2 + 17194*A^3*a^
7*b^9*d^2 + 5298*A^3*a^9*b^7*d^2 - 3338*A^3*a^11*b^5*d^2 + 506*A^3*a^13*b^3*d^2 + 518*A^3*a*b^15*d^2 - 18*A^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 + 5
6*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*A^2*a^3*b^16*d^2 - 7456*A^2*a^
5*b^14*d^2 - 576*A^2*a^7*b^12*d^2 + 19504*A^2*a^9*b^10*d^2 + 18880*A^2*a^11*b^8*d^2 + 3808*A^2*a^13*b^6*d^2 -
960*A^2*a^15*b^4*d^2 + 8*A^2*a^17*b^2*d^2 + 1544*A^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*A*a^4*b^18*d^4 - 320*A*a^2*b^20*d^4 - 192*A*b^22*d^4 + 22272*A*a^6*b^16*d^4 + 51072*A*a^8*b^14*d^4 + 672
00*A*a^10*b^12*d^4 + 53760*A*a^12*b^10*d^4 + 25344*A*a^14*b^8*d^4 + 5952*A*a^16*b^6*d^4 + 192*A*a^18*b^4*d^4 -
128*A*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*A^2*a^8 + 9*A^2*b^8
- 156*A^2*a^2*b^6 + 694*A^2*a^4*b^4 - 156*A^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*A^2*a^8 + 9*A^2*b^8 - 156*A^2*a^2*b^6 + 694*A^2*a^4*b^4 - 156*A^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^1
1*b^3*d^2)))*(-64*(9*A^2*a^8 + 9*A^2*b^8 - 156*A^2*a^2*b^6 + 694*A^2*a^4*b^4 - 156*A^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*A^2*a^8 + 9*A^2*b^8 - 156*A^2*a^2*b^6 + 694*A^2*a^4*b^4 - 156*A^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*A
^2*a^8 + 9*A^2*b^8 - 156*A^2*a^2*b^6 + 694*A^2*a^4*b^4 - 156*A^2*a^6*b^2)*(a*b^13*d^2 + a^13*b*d^2 + 6*a^3*b^1
1*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)))*(-64*(9*A^2*a^8 + 9*A^2*
b^8 - 156*A^2*a^2*b^6 + 694*A^2*a^4*b^4 - 156*A^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)) + (atan(-((((tan(c + d*x)^(1/2)*(
B^4*a^14 - 32*B^4*b^14 + 97*B^4*a^2*b^12 - 2082*B^4*a^4*b^10 + 3631*B^4*a^6*b^8 - 2300*B^4*a^8*b^6 + 79*B^4*a^
10*b^4 + 30*B^4*a^12*b^2))/(64*(b^17*d^4 + a^16*b*d^4 + 8*a^2*b^15*d^4 + 28*a^4*b^13*d^4 + 56*a^6*b^11*d^4 + 7
0*a^8*b^9*d^4 + 56*a^10*b^7*d^4 + 28*a^12*b^5*d^4 + 8*a^14*b^3*d^4)) - (((5238*B^3*a^4*b^13*d^2 - 2398*B^3*a^2
*b^15*d^2 + 7386*B^3*a^6*b^11*d^2 - 8322*B^3*a^8*b^9*d^2 - 5498*B^3*a^10*b^7*d^2 + 2946*B^3*a^12*b^5*d^2 + 382
*B^3*a^14*b^3*d^2 + 10*B^3*a^16*b*d^2)/(64*(b^17*d^5 + a^16*b*d^5 + 8*a^2*b^15*d^5 + 28*a^4*b^13*d^5 + 56*a^6*
b^11*d^5 + 70*a^8*b^9*d^5 + 56*a^10*b^7*d^5 + 28*a^12*b^5*d^5 + 8*a^14*b^3*d^5)) - (((((832*B*a*b^22*d^4 + 595
2*B*a^3*b^20*d^4 + 17664*B*a^5*b^18*d^4 + 26880*B*a^7*b^16*d^4 + 18816*B*a^9*b^14*d^4 - 2688*B*a^11*b^12*d^4 -
16128*B*a^13*b^10*d^4 - 13056*B*a^15*b^8*d^4 - 4800*B*a^17*b^6*d^4 - 704*B*a^19*b^4*d^4)/(64*(b^17*d^5 + a^16
*b*d^5 + 8*a^2*b^15*d^5 + 28*a^4*b^13*d^5 + 56*a^6*b^11*d^5 + 70*a^8*b^9*d^5 + 56*a^10*b^7*d^5 + 28*a^12*b^5*d
^5 + 8*a^14*b^3*d^5)) - (tan(c + d*x)^(1/2)*(-64*(B^2*a^9 + 225*B^2*a*b^8 - 540*B^2*a^3*b^6 + 294*B^2*a^5*b^4
+ 36*B^2*a^7*b^2)*(b^15*d^2 + 6*a^2*b^13*d^2 + 15*a^4*b^11*d^2 + 20*a^6*b^9*d^2 + 15*a^8*b^7*d^2 + 6*a^10*b^5*
d^2 + a^12*b^3*d^2))^(1/2)*(512*b^26*d^4 + 4608*a^2*b^24*d^4 + 17920*a^4*b^22*d^4 + 38400*a^6*b^20*d^4 + 46080
*a^8*b^18*d^4 + 21504*a^10*b^16*d^4 - 21504*a^12*b^14*d^4 - 46080*a^14*b^12*d^4 - 38400*a^16*b^10*d^4 - 17920*
a^18*b^8*d^4 - 4608*a^20*b^6*d^4 - 512*a^22*b^4*d^4))/(4096*(b^15*d^2 + 6*a^2*b^13*d^2 + 15*a^4*b^11*d^2 + 20*
a^6*b^9*d^2 + 15*a^8*b^7*d^2 + 6*a^10*b^5*d^2 + a^12*b^3*d^2)*(b^17*d^4 + a^16*b*d^4 + 8*a^2*b^15*d^4 + 28*a^4
*b^13*d^4 + 56*a^6*b^11*d^4 + 70*a^8*b^9*d^4 + 56*a^10*b^7*d^4 + 28*a^12*b^5*d^4 + 8*a^14*b^3*d^4)))*(-64*(B^2
*a^9 + 225*B^2*a*b^8 - 540*B^2*a^3*b^6 + 294*B^2*a^5*b^4 + 36*B^2*a^7*b^2)*(b^15*d^2 + 6*a^2*b^13*d^2 + 15*a^4
*b^11*d^2 + 20*a^6*b^9*d^2 + 15*a^8*b^7*d^2 + 6*a^10*b^5*d^2 + a^12*b^3*d^2))^(1/2))/(64*(b^15*d^2 + 6*a^2*b^1
3*d^2 + 15*a^4*b^11*d^2 + 20*a^6*b^9*d^2 + 15*a^8*b^7*d^2 + 6*a^10*b^5*d^2 + a^12*b^3*d^2)) + (tan(c + d*x)^(1
/2)*(776*B^2*a^3*b^17*d^2 + 11328*B^2*a^5*b^15*d^2 + 10208*B^2*a^7*b^13*d^2 - 5056*B^2*a^9*b^11*d^2 - 5328*B^2
*a^11*b^9*d^2 + 4032*B^2*a^13*b^7*d^2 + 3552*B^2*a^15*b^5*d^2 + 384*B^2*a^17*b^3*d^2 - 1472*B^2*a*b^19*d^2 + 8
*B^2*a^19*b*d^2))/(64*(b^17*d^4 + a^16*b*d^4 + 8*a^2*b^15*d^4 + 28*a^4*b^13*d^4 + 56*a^6*b^11*d^4 + 70*a^8*b^9
*d^4 + 56*a^10*b^7*d^4 + 28*a^12*b^5*d^4 + 8*a^14*b^3*d^4)))*(-64*(B^2*a^9 + 225*B^2*a*b^8 - 540*B^2*a^3*b^6 +
294*B^2*a^5*b^4 + 36*B^2*a^7*b^2)*(b^15*d^2 + 6*a^2*b^13*d^2 + 15*a^4*b^11*d^2 + 20*a^6*b^9*d^2 + 15*a^8*b^7*
d^2 + 6*a^10*b^5*d^2 + a^12*b^3*d^2))^(1/2))/(64*(b^15*d^2 + 6*a^2*b^13*d^2 + 15*a^4*b^11*d^2 + 20*a^6*b^9*d^2
+ 15*a^8*b^7*d^2 + 6*a^10*b^5*d^2 + a^12*b^3*d^2)))*(-64*(B^2*a^9 + 225*B^2*a*b^8 - 540*B^2*a^3*b^6 + 294*B^2
*a^5*b^4 + 36*B^2*a^7*b^2)*(b^15*d^2 + 6*a^2*b^13*d^2 + 15*a^4*b^11*d^2 + 20*a^6*b^9*d^2 + 15*a^8*b^7*d^2 + 6*
a^10*b^5*d^2 + a^12*b^3*d^2))^(1/2))/(64*(b^15*d^2 + 6*a^2*b^13*d^2 + 15*a^4*b^11*d^2 + 20*a^6*b^9*d^2 + 15*a^
8*b^7*d^2 + 6*a^10*b^5*d^2 + a^12*b^3*d^2)))*(-64*(B^2*a^9 + 225*B^2*a*b^8 - 540*B^2*a^3*b^6 + 294*B^2*a^5*b^4
+ 36*B^2*a^7*b^2)*(b^15*d^2 + 6*a^2*b^13*d^2 + 15*a^4*b^11*d^2 + 20*a^6*b^9*d^2 + 15*a^8*b^7*d^2 + 6*a^10*b^5
*d^2 + a^12*b^3*d^2))^(1/2)*1i)/(b^15*d^2 + 6*a^2*b^13*d^2 + 15*a^4*b^11*d^2 + 20*a^6*b^9*d^2 + 15*a^8*b^7*d^2
+ 6*a^10*b^5*d^2 + a^12*b^3*d^2) + (((tan(c + d*x)^(1/2)*(B^4*a^14 - 32*B^4*b^14 + 97*B^4*a^2*b^12 - 2082*B^4
*a^4*b^10 + 3631*B^4*a^6*b^8 - 2300*B^4*a^8*b^6 + 79*B^4*a^10*b^4 + 30*B^4*a^12*b^2))/(64*(b^17*d^4 + a^16*b*d
^4 + 8*a^2*b^15*d^4 + 28*a^4*b^13*d^4 + 56*a^6*b^11*d^4 + 70*a^8*b^9*d^4 + 56*a^10*b^7*d^4 + 28*a^12*b^5*d^4 +
8*a^14*b^3*d^4)) + (((5238*B^3*a^4*b^13*d^2 - 2398*B^3*a^2*b^15*d^2 + 7386*B^3*a^6*b^11*d^2 - 8322*B^3*a^8*b^
9*d^2 - 5498*B^3*a^10*b^7*d^2 + 2946*B^3*a^12*b^5*d^2 + 382*B^3*a^14*b^3*d^2 + 10*B^3*a^16*b*d^2)/(64*(b^17*d^
5 + a^16*b*d^5 + 8*a^2*b^15*d^5 + 28*a^4*b^13*d^5 + 56*a^6*b^11*d^5 + 70*a^8*b^9*d^5 + 56*a^10*b^7*d^5 + 28*a^
12*b^5*d^5 + 8*a^14*b^3*d^5)) - (((((832*B*a*b^22*d^4 + 5952*B*a^3*b^20*d^4 + 17664*B*a^5*b^18*d^4 + 26880*B*a
^7*b^16*d^4 + 18816*B*a^9*b^14*d^4 - 2688*B*a^11*b^12*d^4 - 16128*B*a^13*b^10*d^4 - 13056*B*a^15*b^8*d^4 - 480
0*B*a^17*b^6*d^4 - 704*B*a^19*b^4*d^4)/(64*(b^17*d^5 + a^16*b*d^5 + 8*a^2*b^15*d^5 + 28*a^4*b^13*d^5 + 56*a^6*
b^11*d^5 + 70*a^8*b^9*d^5 + 56*a^10*b^7*d^5 + 28*a^12*b^5*d^5 + 8*a^14*b^3*d^5)) + (tan(c + d*x)^(1/2)*(-64*(B
^2*a^9 + 225*B^2*a*b^8 - 540*B^2*a^3*b^6 + 294*B^2*a^5*b^4 + 36*B^2*a^7*b^2)*(b^15*d^2 + 6*a^2*b^13*d^2 + 15*a
^4*b^11*d^2 + 20*a^6*b^9*d^2 + 15*a^8*b^7*d^2 + 6*a^10*b^5*d^2 + a^12*b^3*d^2))^(1/2)*(512*b^26*d^4 + 4608*a^2
*b^24*d^4 + 17920*a^4*b^22*d^4 + 38400*a^6*b^20*d^4 + 46080*a^8*b^18*d^4 + 21504*a^10*b^16*d^4 - 21504*a^12*b^
14*d^4 - 46080*a^14*b^12*d^4 - 38400*a^16*b^10*d^4 - 17920*a^18*b^8*d^4 - 4608*a^20*b^6*d^4 - 512*a^22*b^4*d^4
))/(4096*(b^15*d^2 + 6*a^2*b^13*d^2 + 15*a^4*b^11*d^2 + 20*a^6*b^9*d^2 + 15*a^8*b^7*d^2 + 6*a^10*b^5*d^2 + a^1
2*b^3*d^2)*(b^17*d^4 + a^16*b*d^4 + 8*a^2*b^15*d^4 + 28*a^4*b^13*d^4 + 56*a^6*b^11*d^4 + 70*a^8*b^9*d^4 + 56*a
^10*b^7*d^4 + 28*a^12*b^5*d^4 + 8*a^14*b^3*d^4)))*(-64*(B^2*a^9 + 225*B^2*a*b^8 - 540*B^2*a^3*b^6 + 294*B^2*a^
5*b^4 + 36*B^2*a^7*b^2)*(b^15*d^2 + 6*a^2*b^13*d^2 + 15*a^4*b^11*d^2 + 20*a^6*b^9*d^2 + 15*a^8*b^7*d^2 + 6*a^1
0*b^5*d^2 + a^12*b^3*d^2))^(1/2))/(64*(b^15*d^2 + 6*a^2*b^13*d^2 + 15*a^4*b^11*d^2 + 20*a^6*b^9*d^2 + 15*a^8*b
^7*d^2 + 6*a^10*b^5*d^2 + a^12*b^3*d^2)) - (tan(c + d*x)^(1/2)*(776*B^2*a^3*b^17*d^2 + 11328*B^2*a^5*b^15*d^2
+ 10208*B^2*a^7*b^13*d^2 - 5056*B^2*a^9*b^11*d^2 - 5328*B^2*a^11*b^9*d^2 + 4032*B^2*a^13*b^7*d^2 + 3552*B^2*a^
15*b^5*d^2 + 384*B^2*a^17*b^3*d^2 - 1472*B^2*a*b^19*d^2 + 8*B^2*a^19*b*d^2))/(64*(b^17*d^4 + a^16*b*d^4 + 8*a^
2*b^15*d^4 + 28*a^4*b^13*d^4 + 56*a^6*b^11*d^4 + 70*a^8*b^9*d^4 + 56*a^10*b^7*d^4 + 28*a^12*b^5*d^4 + 8*a^14*b
^3*d^4)))*(-64*(B^2*a^9 + 225*B^2*a*b^8 - 540*B^2*a^3*b^6 + 294*B^2*a^5*b^4 + 36*B^2*a^7*b^2)*(b^15*d^2 + 6*a^
2*b^13*d^2 + 15*a^4*b^11*d^2 + 20*a^6*b^9*d^2 + 15*a^8*b^7*d^2 + 6*a^10*b^5*d^2 + a^12*b^3*d^2))^(1/2))/(64*(b
^15*d^2 + 6*a^2*b^13*d^2 + 15*a^4*b^11*d^2 + 20*a^6*b^9*d^2 + 15*a^8*b^7*d^2 + 6*a^10*b^5*d^2 + a^12*b^3*d^2))
)*(-64*(B^2*a^9 + 225*B^2*a*b^8 - 540*B^2*a^3*b^6 + 294*B^2*a^5*b^4 + 36*B^2*a^7*b^2)*(b^15*d^2 + 6*a^2*b^13*d
^2 + 15*a^4*b^11*d^2 + 20*a^6*b^9*d^2 + 15*a^8*b^7*d^2 + 6*a^10*b^5*d^2 + a^12*b^3*d^2))^(1/2))/(64*(b^15*d^2
+ 6*a^2*b^13*d^2 + 15*a^4*b^11*d^2 + 20*a^6*b^9*d^2 + 15*a^8*b^7*d^2 + 6*a^10*b^5*d^2 + a^12*b^3*d^2)))*(-64*(
B^2*a^9 + 225*B^2*a*b^8 - 540*B^2*a^3*b^6 + 294*B^2*a^5*b^4 + 36*B^2*a^7*b^2)*(b^15*d^2 + 6*a^2*b^13*d^2 + 15*
a^4*b^11*d^2 + 20*a^6*b^9*d^2 + 15*a^8*b^7*d^2 + 6*a^10*b^5*d^2 + a^12*b^3*d^2))^(1/2)*1i)/(b^15*d^2 + 6*a^2*b
^13*d^2 + 15*a^4*b^11*d^2 + 20*a^6*b^9*d^2 + 15*a^8*b^7*d^2 + 6*a^10*b^5*d^2 + a^12*b^3*d^2))/((B^5*a^11 - 120
*B^5*a*b^10 + 249*B^5*a^3*b^8 - 388*B^5*a^5*b^6 + 302*B^5*a^7*b^4 + 36*B^5*a^9*b^2)/(b^17*d^5 + a^16*b*d^5 + 8
*a^2*b^15*d^5 + 28*a^4*b^13*d^5 + 56*a^6*b^11*d^5 + 70*a^8*b^9*d^5 + 56*a^10*b^7*d^5 + 28*a^12*b^5*d^5 + 8*a^1
4*b^3*d^5) - (((tan(c + d*x)^(1/2)*(B^4*a^14 - 32*B^4*b^14 + 97*B^4*a^2*b^12 - 2082*B^4*a^4*b^10 + 3631*B^4*a^
6*b^8 - 2300*B^4*a^8*b^6 + 79*B^4*a^10*b^4 + 30*B^4*a^12*b^2))/(64*(b^17*d^4 + a^16*b*d^4 + 8*a^2*b^15*d^4 + 2
8*a^4*b^13*d^4 + 56*a^6*b^11*d^4 + 70*a^8*b^9*d^4 + 56*a^10*b^7*d^4 + 28*a^12*b^5*d^4 + 8*a^14*b^3*d^4)) - (((
5238*B^3*a^4*b^13*d^2 - 2398*B^3*a^2*b^15*d^2 + 7386*B^3*a^6*b^11*d^2 - 8322*B^3*a^8*b^9*d^2 - 5498*B^3*a^10*b
^7*d^2 + 2946*B^3*a^12*b^5*d^2 + 382*B^3*a^14*b^3*d^2 + 10*B^3*a^16*b*d^2)/(64*(b^17*d^5 + a^16*b*d^5 + 8*a^2*
b^15*d^5 + 28*a^4*b^13*d^5 + 56*a^6*b^11*d^5 + 70*a^8*b^9*d^5 + 56*a^10*b^7*d^5 + 28*a^12*b^5*d^5 + 8*a^14*b^3
*d^5)) - (((((832*B*a*b^22*d^4 + 5952*B*a^3*b^20*d^4 + 17664*B*a^5*b^18*d^4 + 26880*B*a^7*b^16*d^4 + 18816*B*a
^9*b^14*d^4 - 2688*B*a^11*b^12*d^4 - 16128*B*a^13*b^10*d^4 - 13056*B*a^15*b^8*d^4 - 4800*B*a^17*b^6*d^4 - 704*
B*a^19*b^4*d^4)/(64*(b^17*d^5 + a^16*b*d^5 + 8*a^2*b^15*d^5 + 28*a^4*b^13*d^5 + 56*a^6*b^11*d^5 + 70*a^8*b^9*d
^5 + 56*a^10*b^7*d^5 + 28*a^12*b^5*d^5 + 8*a^14*b^3*d^5)) - (tan(c + d*x)^(1/2)*(-64*(B^2*a^9 + 225*B^2*a*b^8
- 540*B^2*a^3*b^6 + 294*B^2*a^5*b^4 + 36*B^2*a^7*b^2)*(b^15*d^2 + 6*a^2*b^13*d^2 + 15*a^4*b^11*d^2 + 20*a^6*b^
9*d^2 + 15*a^8*b^7*d^2 + 6*a^10*b^5*d^2 + a^12*b^3*d^2))^(1/2)*(512*b^26*d^4 + 4608*a^2*b^24*d^4 + 17920*a^4*b
^22*d^4 + 38400*a^6*b^20*d^4 + 46080*a^8*b^18*d^4 + 21504*a^10*b^16*d^4 - 21504*a^12*b^14*d^4 - 46080*a^14*b^1
2*d^4 - 38400*a^16*b^10*d^4 - 17920*a^18*b^8*d^4 - 4608*a^20*b^6*d^4 - 512*a^22*b^4*d^4))/(4096*(b^15*d^2 + 6*
a^2*b^13*d^2 + 15*a^4*b^11*d^2 + 20*a^6*b^9*d^2 + 15*a^8*b^7*d^2 + 6*a^10*b^5*d^2 + a^12*b^3*d^2)*(b^17*d^4 +
a^16*b*d^4 + 8*a^2*b^15*d^4 + 28*a^4*b^13*d^4 + 56*a^6*b^11*d^4 + 70*a^8*b^9*d^4 + 56*a^10*b^7*d^4 + 28*a^12*b
^5*d^4 + 8*a^14*b^3*d^4)))*(-64*(B^2*a^9 + 225*B^2*a*b^8 - 540*B^2*a^3*b^6 + 294*B^2*a^5*b^4 + 36*B^2*a^7*b^2)
*(b^15*d^2 + 6*a^2*b^13*d^2 + 15*a^4*b^11*d^2 + 20*a^6*b^9*d^2 + 15*a^8*b^7*d^2 + 6*a^10*b^5*d^2 + a^12*b^3*d^
2))^(1/2))/(64*(b^15*d^2 + 6*a^2*b^13*d^2 + 15*a^4*b^11*d^2 + 20*a^6*b^9*d^2 + 15*a^8*b^7*d^2 + 6*a^10*b^5*d^2
+ a^12*b^3*d^2)) + (tan(c + d*x)^(1/2)*(776*B^2*a^3*b^17*d^2 + 11328*B^2*a^5*b^15*d^2 + 10208*B^2*a^7*b^13*d^
2 - 5056*B^2*a^9*b^11*d^2 - 5328*B^2*a^11*b^9*d^2 + 4032*B^2*a^13*b^7*d^2 + 3552*B^2*a^15*b^5*d^2 + 384*B^2*a^
17*b^3*d^2 - 1472*B^2*a*b^19*d^2 + 8*B^2*a^19*b*d^2))/(64*(b^17*d^4 + a^16*b*d^4 + 8*a^2*b^15*d^4 + 28*a^4*b^1
3*d^4 + 56*a^6*b^11*d^4 + 70*a^8*b^9*d^4 + 56*a^10*b^7*d^4 + 28*a^12*b^5*d^4 + 8*a^14*b^3*d^4)))*(-64*(B^2*a^9
+ 225*B^2*a*b^8 - 540*B^2*a^3*b^6 + 294*B^2*a^5*b^4 + 36*B^2*a^7*b^2)*(b^15*d^2 + 6*a^2*b^13*d^2 + 15*a^4*b^1
1*d^2 + 20*a^6*b^9*d^2 + 15*a^8*b^7*d^2 + 6*a^10*b^5*d^2 + a^12*b^3*d^2))^(1/2))/(64*(b^15*d^2 + 6*a^2*b^13*d^
2 + 15*a^4*b^11*d^2 + 20*a^6*b^9*d^2 + 15*a^8*b^7*d^2 + 6*a^10*b^5*d^2 + a^12*b^3*d^2)))*(-64*(B^2*a^9 + 225*B
^2*a*b^8 - 540*B^2*a^3*b^6 + 294*B^2*a^5*b^4 + 36*B^2*a^7*b^2)*(b^15*d^2 + 6*a^2*b^13*d^2 + 15*a^4*b^11*d^2 +
20*a^6*b^9*d^2 + 15*a^8*b^7*d^2 + 6*a^10*b^5*d^2 + a^12*b^3*d^2))^(1/2))/(64*(b^15*d^2 + 6*a^2*b^13*d^2 + 15*a
^4*b^11*d^2 + 20*a^6*b^9*d^2 + 15*a^8*b^7*d^2 + 6*a^10*b^5*d^2 + a^12*b^3*d^2)))*(-64*(B^2*a^9 + 225*B^2*a*b^8
- 540*B^2*a^3*b^6 + 294*B^2*a^5*b^4 + 36*B^2*a^7*b^2)*(b^15*d^2 + 6*a^2*b^13*d^2 + 15*a^4*b^11*d^2 + 20*a^6*b
^9*d^2 + 15*a^8*b^7*d^2 + 6*a^10*b^5*d^2 + a^12*b^3*d^2))^(1/2))/(b^15*d^2 + 6*a^2*b^13*d^2 + 15*a^4*b^11*d^2
+ 20*a^6*b^9*d^2 + 15*a^8*b^7*d^2 + 6*a^10*b^5*d^2 + a^12*b^3*d^2) + (((tan(c + d*x)^(1/2)*(B^4*a^14 - 32*B^4*
b^14 + 97*B^4*a^2*b^12 - 2082*B^4*a^4*b^10 + 3631*B^4*a^6*b^8 - 2300*B^4*a^8*b^6 + 79*B^4*a^10*b^4 + 30*B^4*a^
12*b^2))/(64*(b^17*d^4 + a^16*b*d^4 + 8*a^2*b^15*d^4 + 28*a^4*b^13*d^4 + 56*a^6*b^11*d^4 + 70*a^8*b^9*d^4 + 56
*a^10*b^7*d^4 + 28*a^12*b^5*d^4 + 8*a^14*b^3*d^4)) + (((5238*B^3*a^4*b^13*d^2 - 2398*B^3*a^2*b^15*d^2 + 7386*B
^3*a^6*b^11*d^2 - 8322*B^3*a^8*b^9*d^2 - 5498*B^3*a^10*b^7*d^2 + 2946*B^3*a^12*b^5*d^2 + 382*B^3*a^14*b^3*d^2
+ 10*B^3*a^16*b*d^2)/(64*(b^17*d^5 + a^16*b*d^5 + 8*a^2*b^15*d^5 + 28*a^4*b^13*d^5 + 56*a^6*b^11*d^5 + 70*a^8*
b^9*d^5 + 56*a^10*b^7*d^5 + 28*a^12*b^5*d^5 + 8*a^14*b^3*d^5)) - (((((832*B*a*b^22*d^4 + 5952*B*a^3*b^20*d^4 +
17664*B*a^5*b^18*d^4 + 26880*B*a^7*b^16*d^4 + 18816*B*a^9*b^14*d^4 - 2688*B*a^11*b^12*d^4 - 16128*B*a^13*b^10
*d^4 - 13056*B*a^15*b^8*d^4 - 4800*B*a^17*b^6*d^4 - 704*B*a^19*b^4*d^4)/(64*(b^17*d^5 + a^16*b*d^5 + 8*a^2*b^1
5*d^5 + 28*a^4*b^13*d^5 + 56*a^6*b^11*d^5 + 70*a^8*b^9*d^5 + 56*a^10*b^7*d^5 + 28*a^12*b^5*d^5 + 8*a^14*b^3*d^
5)) + (tan(c + d*x)^(1/2)*(-64*(B^2*a^9 + 225*B^2*a*b^8 - 540*B^2*a^3*b^6 + 294*B^2*a^5*b^4 + 36*B^2*a^7*b^2)*
(b^15*d^2 + 6*a^2*b^13*d^2 + 15*a^4*b^11*d^2 + 20*a^6*b^9*d^2 + 15*a^8*b^7*d^2 + 6*a^10*b^5*d^2 + a^12*b^3*d^2
))^(1/2)*(512*b^26*d^4 + 4608*a^2*b^24*d^4 + 17920*a^4*b^22*d^4 + 38400*a^6*b^20*d^4 + 46080*a^8*b^18*d^4 + 21
504*a^10*b^16*d^4 - 21504*a^12*b^14*d^4 - 46080*a^14*b^12*d^4 - 38400*a^16*b^10*d^4 - 17920*a^18*b^8*d^4 - 460
8*a^20*b^6*d^4 - 512*a^22*b^4*d^4))/(4096*(b^15*d^2 + 6*a^2*b^13*d^2 + 15*a^4*b^11*d^2 + 20*a^6*b^9*d^2 + 15*a
^8*b^7*d^2 + 6*a^10*b^5*d^2 + a^12*b^3*d^2)*(b^17*d^4 + a^16*b*d^4 + 8*a^2*b^15*d^4 + 28*a^4*b^13*d^4 + 56*a^6
*b^11*d^4 + 70*a^8*b^9*d^4 + 56*a^10*b^7*d^4 + 28*a^12*b^5*d^4 + 8*a^14*b^3*d^4)))*(-64*(B^2*a^9 + 225*B^2*a*b
^8 - 540*B^2*a^3*b^6 + 294*B^2*a^5*b^4 + 36*B^2*a^7*b^2)*(b^15*d^2 + 6*a^2*b^13*d^2 + 15*a^4*b^11*d^2 + 20*a^6
*b^9*d^2 + 15*a^8*b^7*d^2 + 6*a^10*b^5*d^2 + a^12*b^3*d^2))^(1/2))/(64*(b^15*d^2 + 6*a^2*b^13*d^2 + 15*a^4*b^1
1*d^2 + 20*a^6*b^9*d^2 + 15*a^8*b^7*d^2 + 6*a^10*b^5*d^2 + a^12*b^3*d^2)) - (tan(c + d*x)^(1/2)*(776*B^2*a^3*b
^17*d^2 + 11328*B^2*a^5*b^15*d^2 + 10208*B^2*a^7*b^13*d^2 - 5056*B^2*a^9*b^11*d^2 - 5328*B^2*a^11*b^9*d^2 + 40
32*B^2*a^13*b^7*d^2 + 3552*B^2*a^15*b^5*d^2 + 384*B^2*a^17*b^3*d^2 - 1472*B^2*a*b^19*d^2 + 8*B^2*a^19*b*d^2))/
(64*(b^17*d^4 + a^16*b*d^4 + 8*a^2*b^15*d^4 + 28*a^4*b^13*d^4 + 56*a^6*b^11*d^4 + 70*a^8*b^9*d^4 + 56*a^10*b^7
*d^4 + 28*a^12*b^5*d^4 + 8*a^14*b^3*d^4)))*(-64*(B^2*a^9 + 225*B^2*a*b^8 - 540*B^2*a^3*b^6 + 294*B^2*a^5*b^4 +
36*B^2*a^7*b^2)*(b^15*d^2 + 6*a^2*b^13*d^2 + 15*a^4*b^11*d^2 + 20*a^6*b^9*d^2 + 15*a^8*b^7*d^2 + 6*a^10*b^5*d
^2 + a^12*b^3*d^2))^(1/2))/(64*(b^15*d^2 + 6*a^2*b^13*d^2 + 15*a^4*b^11*d^2 + 20*a^6*b^9*d^2 + 15*a^8*b^7*d^2
+ 6*a^10*b^5*d^2 + a^12*b^3*d^2)))*(-64*(B^2*a^9 + 225*B^2*a*b^8 - 540*B^2*a^3*b^6 + 294*B^2*a^5*b^4 + 36*B^2*
a^7*b^2)*(b^15*d^2 + 6*a^2*b^13*d^2 + 15*a^4*b^11*d^2 + 20*a^6*b^9*d^2 + 15*a^8*b^7*d^2 + 6*a^10*b^5*d^2 + a^1
2*b^3*d^2))^(1/2))/(64*(b^15*d^2 + 6*a^2*b^13*d^2 + 15*a^4*b^11*d^2 + 20*a^6*b^9*d^2 + 15*a^8*b^7*d^2 + 6*a^10
*b^5*d^2 + a^12*b^3*d^2)))*(-64*(B^2*a^9 + 225*B^2*a*b^8 - 540*B^2*a^3*b^6 + 294*B^2*a^5*b^4 + 36*B^2*a^7*b^2)
*(b^15*d^2 + 6*a^2*b^13*d^2 + 15*a^4*b^11*d^2 + 20*a^6*b^9*d^2 + 15*a^8*b^7*d^2 + 6*a^10*b^5*d^2 + a^12*b^3*d^
2))^(1/2))/(b^15*d^2 + 6*a^2*b^13*d^2 + 15*a^4*b^11*d^2 + 20*a^6*b^9*d^2 + 15*a^8*b^7*d^2 + 6*a^10*b^5*d^2 + a
^12*b^3*d^2)))*(-64*(B^2*a^9 + 225*B^2*a*b^8 - 540*B^2*a^3*b^6 + 294*B^2*a^5*b^4 + 36*B^2*a^7*b^2)*(b^15*d^2 +
6*a^2*b^13*d^2 + 15*a^4*b^11*d^2 + 20*a^6*b^9*d^2 + 15*a^8*b^7*d^2 + 6*a^10*b^5*d^2 + a^12*b^3*d^2))^(1/2)*1i
)/(32*(b^15*d^2 + 6*a^2*b^13*d^2 + 15*a^4*b^11*d^2 + 20*a^6*b^9*d^2 + 15*a^8*b^7*d^2 + 6*a^10*b^5*d^2 + a^12*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)**(3/2)*(A+B*tan(d*x+c))/(a+b*tan(d*x+c))**3,x)
[Out]
Timed out
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