3.409 \(\int \frac {A+B \tan (c+d x)}{\tan ^{\frac {5}{2}}(c+d x) (a+b \tan (c+d x))^2} \, dx\)

Optimal. Leaf size=493 \[ -\frac {\left (-\left (a^2 (A+B)\right )+2 a b (A-B)+b^2 (A+B)\right ) \tan ^{-1}\left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} d \left (a^2+b^2\right )^2}+\frac {\left (-\left (a^2 (A+B)\right )+2 a b (A-B)+b^2 (A+B)\right ) \tan ^{-1}\left (\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{\sqrt {2} d \left (a^2+b^2\right )^2}+\frac {b (A b-a B)}{a d \left (a^2+b^2\right ) \tan ^{\frac {3}{2}}(c+d x) (a+b \tan (c+d x))}-\frac {2 a^2 A-3 a b B+5 A b^2}{3 a^2 d \left (a^2+b^2\right ) \tan ^{\frac {3}{2}}(c+d x)}+\frac {\left (a^2 (A-B)+2 a b (A+B)-b^2 (A-B)\right ) \log \left (\tan (c+d x)-\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{2 \sqrt {2} d \left (a^2+b^2\right )^2}-\frac {\left (a^2 (A-B)+2 a b (A+B)-b^2 (A-B)\right ) \log \left (\tan (c+d x)+\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{2 \sqrt {2} d \left (a^2+b^2\right )^2}+\frac {-2 a^3 B+4 a^2 A b-3 a b^2 B+5 A b^3}{a^3 d \left (a^2+b^2\right ) \sqrt {\tan (c+d x)}}+\frac {b^{5/2} \left (-7 a^3 B+9 a^2 A b-3 a b^2 B+5 A b^3\right ) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {\tan (c+d x)}}{\sqrt {a}}\right )}{a^{7/2} d \left (a^2+b^2\right )^2} \]

[Out]

b^(5/2)*(9*A*a^2*b+5*A*b^3-7*B*a^3-3*B*a*b^2)*arctan(b^(1/2)*tan(d*x+c)^(1/2)/a^(1/2))/a^(7/2)/(a^2+b^2)^2/d+1
/2*(2*a*b*(A-B)-a^2*(A+B)+b^2*(A+B))*arctan(-1+2^(1/2)*tan(d*x+c)^(1/2))/(a^2+b^2)^2/d*2^(1/2)+1/2*(2*a*b*(A-B
)-a^2*(A+B)+b^2*(A+B))*arctan(1+2^(1/2)*tan(d*x+c)^(1/2))/(a^2+b^2)^2/d*2^(1/2)+1/4*(a^2*(A-B)-b^2*(A-B)+2*a*b
*(A+B))*ln(1-2^(1/2)*tan(d*x+c)^(1/2)+tan(d*x+c))/(a^2+b^2)^2/d*2^(1/2)-1/4*(a^2*(A-B)-b^2*(A-B)+2*a*b*(A+B))*
ln(1+2^(1/2)*tan(d*x+c)^(1/2)+tan(d*x+c))/(a^2+b^2)^2/d*2^(1/2)+(4*A*a^2*b+5*A*b^3-2*B*a^3-3*B*a*b^2)/a^3/(a^2
+b^2)/d/tan(d*x+c)^(1/2)+1/3*(-2*A*a^2-5*A*b^2+3*B*a*b)/a^2/(a^2+b^2)/d/tan(d*x+c)^(3/2)+b*(A*b-B*a)/a/(a^2+b^
2)/d/tan(d*x+c)^(3/2)/(a+b*tan(d*x+c))

________________________________________________________________________________________

Rubi [A]  time = 1.53, antiderivative size = 493, normalized size of antiderivative = 1.00, number of steps used = 17, number of rules used = 13, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.394, Rules used = {3609, 3649, 3653, 3534, 1168, 1162, 617, 204, 1165, 628, 3634, 63, 205} \[ \frac {b^{5/2} \left (9 a^2 A b-7 a^3 B-3 a b^2 B+5 A b^3\right ) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {\tan (c+d x)}}{\sqrt {a}}\right )}{a^{7/2} d \left (a^2+b^2\right )^2}+\frac {b (A b-a B)}{a d \left (a^2+b^2\right ) \tan ^{\frac {3}{2}}(c+d x) (a+b \tan (c+d x))}-\frac {\left (a^2 (-(A+B))+2 a b (A-B)+b^2 (A+B)\right ) \tan ^{-1}\left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} d \left (a^2+b^2\right )^2}+\frac {\left (a^2 (-(A+B))+2 a b (A-B)+b^2 (A+B)\right ) \tan ^{-1}\left (\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{\sqrt {2} d \left (a^2+b^2\right )^2}-\frac {2 a^2 A-3 a b B+5 A b^2}{3 a^2 d \left (a^2+b^2\right ) \tan ^{\frac {3}{2}}(c+d x)}+\frac {4 a^2 A b-2 a^3 B-3 a b^2 B+5 A b^3}{a^3 d \left (a^2+b^2\right ) \sqrt {\tan (c+d x)}}+\frac {\left (a^2 (A-B)+2 a b (A+B)-b^2 (A-B)\right ) \log \left (\tan (c+d x)-\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{2 \sqrt {2} d \left (a^2+b^2\right )^2}-\frac {\left (a^2 (A-B)+2 a b (A+B)-b^2 (A-B)\right ) \log \left (\tan (c+d x)+\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{2 \sqrt {2} d \left (a^2+b^2\right )^2} \]

Antiderivative was successfully verified.

[In]

Int[(A + B*Tan[c + d*x])/(Tan[c + d*x]^(5/2)*(a + b*Tan[c + d*x])^2),x]

[Out]

-(((2*a*b*(A - B) - a^2*(A + B) + b^2*(A + B))*ArcTan[1 - Sqrt[2]*Sqrt[Tan[c + d*x]]])/(Sqrt[2]*(a^2 + b^2)^2*
d)) + ((2*a*b*(A - B) - a^2*(A + B) + b^2*(A + B))*ArcTan[1 + Sqrt[2]*Sqrt[Tan[c + d*x]]])/(Sqrt[2]*(a^2 + b^2
)^2*d) + (b^(5/2)*(9*a^2*A*b + 5*A*b^3 - 7*a^3*B - 3*a*b^2*B)*ArcTan[(Sqrt[b]*Sqrt[Tan[c + d*x]])/Sqrt[a]])/(a
^(7/2)*(a^2 + b^2)^2*d) + ((a^2*(A - B) - b^2*(A - B) + 2*a*b*(A + B))*Log[1 - Sqrt[2]*Sqrt[Tan[c + d*x]] + Ta
n[c + d*x]])/(2*Sqrt[2]*(a^2 + b^2)^2*d) - ((a^2*(A - B) - b^2*(A - B) + 2*a*b*(A + B))*Log[1 + Sqrt[2]*Sqrt[T
an[c + d*x]] + Tan[c + d*x]])/(2*Sqrt[2]*(a^2 + b^2)^2*d) - (2*a^2*A + 5*A*b^2 - 3*a*b*B)/(3*a^2*(a^2 + b^2)*d
*Tan[c + d*x]^(3/2)) + (4*a^2*A*b + 5*A*b^3 - 2*a^3*B - 3*a*b^2*B)/(a^3*(a^2 + b^2)*d*Sqrt[Tan[c + d*x]]) + (b
*(A*b - a*B))/(a*(a^2 + b^2)*d*Tan[c + d*x]^(3/2)*(a + b*Tan[c + d*x]))

Rule 63

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - (a*d)/b + (d*x^p)/b)^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 204

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

Rule 205

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

Rule 617

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*Simplify[(a*c)/b^2]}, Dist[-2/b, Sub
st[Int[1/(q - x^2), x], x, 1 + (2*c*x)/b], x] /; RationalQ[q] && (EqQ[q^2, 1] ||  !RationalQ[b^2 - 4*a*c])] /;
FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 628

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(d*Log[RemoveContent[a + b*x +
c*x^2, x]])/b, x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rule 1162

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[(2*d)/e, 2]}, Dist[e/(2*c), Int[1/S
imp[d/e + q*x + x^2, x], x], x] + Dist[e/(2*c), Int[1/Simp[d/e - q*x + x^2, x], x], x]] /; FreeQ[{a, c, d, e},
x] && EqQ[c*d^2 - a*e^2, 0] && PosQ[d*e]

Rule 1165

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[(-2*d)/e, 2]}, Dist[e/(2*c*q), Int[
(q - 2*x)/Simp[d/e + q*x - x^2, x], x], x] + Dist[e/(2*c*q), Int[(q + 2*x)/Simp[d/e - q*x - x^2, x], x], x]] /
; FreeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && NegQ[d*e]

Rule 1168

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[a*c, 2]}, Dist[(d*q + a*e)/(2*a*c),
Int[(q + c*x^2)/(a + c*x^4), x], x] + Dist[(d*q - a*e)/(2*a*c), Int[(q - c*x^2)/(a + c*x^4), x], x]] /; FreeQ
[{a, c, d, e}, x] && NeQ[c*d^2 + a*e^2, 0] && NeQ[c*d^2 - a*e^2, 0] && NegQ[-(a*c)]

Rule 3534

Int[((c_) + (d_.)*tan[(e_.) + (f_.)*(x_)])/Sqrt[(b_.)*tan[(e_.) + (f_.)*(x_)]], x_Symbol] :> Dist[2/f, Subst[I
nt[(b*c + d*x^2)/(b^2 + x^4), x], x, Sqrt[b*Tan[e + f*x]]], x] /; FreeQ[{b, c, d, e, f}, x] && NeQ[c^2 - d^2,
0] && NeQ[c^2 + d^2, 0]

Rule 3609

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

Rule 3634

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

Rule 3649

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

Rule 3653

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

Rubi steps

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

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Mathematica [C]  time = 4.07, size = 287, normalized size = 0.58 \[ \frac {\frac {-2 a^2 A+3 a b B-5 A b^2}{a \tan ^{\frac {3}{2}}(c+d x)}+\frac {3 \left (-2 a^3 B+4 a^2 A b-3 a b^2 B+5 A b^3\right )}{a^2 \sqrt {\tan (c+d x)}}+\frac {3 \left (\sqrt [4]{-1} a^{7/2} (a+i b)^2 (A-i B) \tan ^{-1}\left ((-1)^{3/4} \sqrt {\tan (c+d x)}\right )+\sqrt [4]{-1} a^{7/2} (a-i b)^2 (A+i B) \tanh ^{-1}\left ((-1)^{3/4} \sqrt {\tan (c+d x)}\right )+b^{5/2} \left (-7 a^3 B+9 a^2 A b-3 a b^2 B+5 A b^3\right ) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {\tan (c+d x)}}{\sqrt {a}}\right )\right )}{a^{5/2} \left (a^2+b^2\right )}+\frac {3 b (A b-a B)}{\tan ^{\frac {3}{2}}(c+d x) (a+b \tan (c+d x))}}{3 a d \left (a^2+b^2\right )} \]

Antiderivative was successfully verified.

[In]

Integrate[(A + B*Tan[c + d*x])/(Tan[c + d*x]^(5/2)*(a + b*Tan[c + d*x])^2),x]

[Out]

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

[Out]

Timed out

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {B \tan \left (d x + c\right ) + A}{{\left (b \tan \left (d x + c\right ) + a\right )}^{2} \tan \left (d x + c\right )^{\frac {5}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((A+B*tan(d*x+c))/tan(d*x+c)^(5/2)/(a+b*tan(d*x+c))^2,x, algorithm="giac")

[Out]

integrate((B*tan(d*x + c) + A)/((b*tan(d*x + c) + a)^2*tan(d*x + c)^(5/2)), x)

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maple [B]  time = 0.37, size = 1198, normalized size = 2.43 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((A+B*tan(d*x+c))/tan(d*x+c)^(5/2)/(a+b*tan(d*x+c))^2,x)

[Out]

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

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maxima [A]  time = 0.89, size = 449, normalized size = 0.91 \[ -\frac {\frac {12 \, {\left (7 \, B a^{3} b^{3} - 9 \, A a^{2} b^{4} + 3 \, B a b^{5} - 5 \, A b^{6}\right )} \arctan \left (\frac {b \sqrt {\tan \left (d x + c\right )}}{\sqrt {a b}}\right )}{{\left (a^{7} + 2 \, a^{5} b^{2} + a^{3} b^{4}\right )} \sqrt {a b}} + \frac {4 \, {\left (2 \, A a^{4} + 2 \, A a^{2} b^{2} + 3 \, {\left (2 \, B a^{3} b - 4 \, A a^{2} b^{2} + 3 \, B a b^{3} - 5 \, A b^{4}\right )} \tan \left (d x + c\right )^{2} + 2 \, {\left (3 \, B a^{4} - 5 \, A a^{3} b + 3 \, B a^{2} b^{2} - 5 \, A a b^{3}\right )} \tan \left (d x + c\right )\right )}}{{\left (a^{5} b + a^{3} b^{3}\right )} \tan \left (d x + c\right )^{\frac {5}{2}} + {\left (a^{6} + a^{4} b^{2}\right )} \tan \left (d x + c\right )^{\frac {3}{2}}} + \frac {3 \, {\left (2 \, \sqrt {2} {\left ({\left (A + B\right )} a^{2} - 2 \, {\left (A - B\right )} a b - {\left (A + B\right )} b^{2}\right )} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + 2 \, \sqrt {\tan \left (d x + c\right )}\right )}\right ) + 2 \, \sqrt {2} {\left ({\left (A + B\right )} a^{2} - 2 \, {\left (A - B\right )} a b - {\left (A + B\right )} b^{2}\right )} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - 2 \, \sqrt {\tan \left (d x + c\right )}\right )}\right ) + \sqrt {2} {\left ({\left (A - B\right )} a^{2} + 2 \, {\left (A + B\right )} a b - {\left (A - B\right )} b^{2}\right )} \log \left (\sqrt {2} \sqrt {\tan \left (d x + c\right )} + \tan \left (d x + c\right ) + 1\right ) - \sqrt {2} {\left ({\left (A - B\right )} a^{2} + 2 \, {\left (A + B\right )} a b - {\left (A - B\right )} b^{2}\right )} \log \left (-\sqrt {2} \sqrt {\tan \left (d x + c\right )} + \tan \left (d x + c\right ) + 1\right )\right )}}{a^{4} + 2 \, a^{2} b^{2} + b^{4}}}{12 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((A+B*tan(d*x+c))/tan(d*x+c)^(5/2)/(a+b*tan(d*x+c))^2,x, algorithm="maxima")

[Out]

-1/12*(12*(7*B*a^3*b^3 - 9*A*a^2*b^4 + 3*B*a*b^5 - 5*A*b^6)*arctan(b*sqrt(tan(d*x + c))/sqrt(a*b))/((a^7 + 2*a
^5*b^2 + a^3*b^4)*sqrt(a*b)) + 4*(2*A*a^4 + 2*A*a^2*b^2 + 3*(2*B*a^3*b - 4*A*a^2*b^2 + 3*B*a*b^3 - 5*A*b^4)*ta
n(d*x + c)^2 + 2*(3*B*a^4 - 5*A*a^3*b + 3*B*a^2*b^2 - 5*A*a*b^3)*tan(d*x + c))/((a^5*b + a^3*b^3)*tan(d*x + c)
^(5/2) + (a^6 + a^4*b^2)*tan(d*x + c)^(3/2)) + 3*(2*sqrt(2)*((A + B)*a^2 - 2*(A - B)*a*b - (A + B)*b^2)*arctan
(1/2*sqrt(2)*(sqrt(2) + 2*sqrt(tan(d*x + c)))) + 2*sqrt(2)*((A + B)*a^2 - 2*(A - B)*a*b - (A + B)*b^2)*arctan(
-1/2*sqrt(2)*(sqrt(2) - 2*sqrt(tan(d*x + c)))) + sqrt(2)*((A - B)*a^2 + 2*(A + B)*a*b - (A - B)*b^2)*log(sqrt(
2)*sqrt(tan(d*x + c)) + tan(d*x + c) + 1) - sqrt(2)*((A - B)*a^2 + 2*(A + B)*a*b - (A - B)*b^2)*log(-sqrt(2)*s
qrt(tan(d*x + c)) + tan(d*x + c) + 1))/(a^4 + 2*a^2*b^2 + b^4))/d

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mupad [B]  time = 22.90, size = 24620, normalized size = 49.94 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((A + B*tan(c + d*x))/(tan(c + d*x)^(5/2)*(a + b*tan(c + d*x))^2),x)

[Out]

(log(80*A^5*a^24*b^20*d^4 - ((((192*A^4*a^2*b^6*d^4 - 16*A^4*b^8*d^4 - 16*A^4*a^8*d^4 - 608*A^4*a^4*b^4*d^4 +
192*A^4*a^6*b^2*d^4)^(1/2) - 16*A^2*a*b^3*d^2 + 16*A^2*a^3*b*d^2)/(a^8*d^4 + b^8*d^4 + 4*a^2*b^6*d^4 + 6*a^4*b
^4*d^4 + 4*a^6*b^2*d^4))^(1/2)*(tan(c + d*x)^(1/2)*(9472*A^4*a^31*b^15*d^5 - 3040*A^4*a^23*b^23*d^5 - 9056*A^4
*a^25*b^21*d^5 - 12352*A^4*a^27*b^19*d^5 - 4256*A^4*a^29*b^17*d^5 - 400*A^4*a^21*b^25*d^5 + 13760*A^4*a^33*b^1
3*d^5 + 7744*A^4*a^35*b^11*d^5 + 1968*A^4*a^37*b^9*d^5 + 224*A^4*a^39*b^7*d^5 + 32*A^4*a^41*b^5*d^5) + ((((192
*A^4*a^2*b^6*d^4 - 16*A^4*b^8*d^4 - 16*A^4*a^8*d^4 - 608*A^4*a^4*b^4*d^4 + 192*A^4*a^6*b^2*d^4)^(1/2) - 16*A^2
*a*b^3*d^2 + 16*A^2*a^3*b*d^2)/(a^8*d^4 + b^8*d^4 + 4*a^2*b^6*d^4 + 6*a^4*b^4*d^4 + 4*a^6*b^2*d^4))^(1/2)*(129
28*A^3*a^25*b^23*d^6 - 800*A^3*a^21*b^27*d^6 - 2080*A^3*a^23*b^25*d^6 - ((tan(c + d*x)^(1/2)*(3200*A^2*a^22*b^
28*d^7 + 33920*A^2*a^24*b^26*d^7 + 158208*A^2*a^26*b^24*d^7 + 425536*A^2*a^28*b^22*d^7 + 727296*A^2*a^30*b^20*
d^7 + 820672*A^2*a^32*b^18*d^7 + 615936*A^2*a^34*b^16*d^7 + 304256*A^2*a^36*b^14*d^7 + 98432*A^2*a^38*b^12*d^7
+ 22016*A^2*a^40*b^10*d^7 + 3072*A^2*a^42*b^8*d^7 - 704*A^2*a^44*b^6*d^7 - 512*A^2*a^46*b^4*d^7 - 64*A^2*a^48
*b^2*d^7) + ((((192*A^4*a^2*b^6*d^4 - 16*A^4*b^8*d^4 - 16*A^4*a^8*d^4 - 608*A^4*a^4*b^4*d^4 + 192*A^4*a^6*b^2*
d^4)^(1/2) - 16*A^2*a*b^3*d^2 + 16*A^2*a^3*b*d^2)/(a^8*d^4 + b^8*d^4 + 4*a^2*b^6*d^4 + 6*a^4*b^4*d^4 + 4*a^6*b
^2*d^4))^(1/2)*(1280*A*a^24*b^28*d^8 - (tan(c + d*x)^(1/2)*(((192*A^4*a^2*b^6*d^4 - 16*A^4*b^8*d^4 - 16*A^4*a^
8*d^4 - 608*A^4*a^4*b^4*d^4 + 192*A^4*a^6*b^2*d^4)^(1/2) - 16*A^2*a*b^3*d^2 + 16*A^2*a^3*b*d^2)/(a^8*d^4 + b^8
*d^4 + 4*a^2*b^6*d^4 + 6*a^4*b^4*d^4 + 4*a^6*b^2*d^4))^(1/2)*(512*a^27*b^27*d^9 + 5120*a^29*b^25*d^9 + 22528*a
^31*b^23*d^9 + 56320*a^33*b^21*d^9 + 84480*a^35*b^19*d^9 + 67584*a^37*b^17*d^9 - 67584*a^41*b^13*d^9 - 84480*a
^43*b^11*d^9 - 56320*a^45*b^9*d^9 - 22528*a^47*b^7*d^9 - 5120*a^49*b^5*d^9 - 512*a^51*b^3*d^9))/4 + 13824*A*a^
26*b^26*d^8 + 66944*A*a^28*b^24*d^8 + 190848*A*a^30*b^22*d^8 + 352640*A*a^32*b^20*d^8 + 435840*A*a^34*b^18*d^8
+ 354048*A*a^36*b^16*d^8 + 169728*A*a^38*b^14*d^8 + 24576*A*a^40*b^12*d^8 - 21760*A*a^42*b^10*d^8 - 13440*A*a
^44*b^8*d^8 - 2176*A*a^46*b^6*d^8 + 384*A*a^48*b^4*d^8 + 128*A*a^50*b^2*d^8))/4)*(((192*A^4*a^2*b^6*d^4 - 16*A
^4*b^8*d^4 - 16*A^4*a^8*d^4 - 608*A^4*a^4*b^4*d^4 + 192*A^4*a^6*b^2*d^4)^(1/2) - 16*A^2*a*b^3*d^2 + 16*A^2*a^3
*b*d^2)/(a^8*d^4 + b^8*d^4 + 4*a^2*b^6*d^4 + 6*a^4*b^4*d^4 + 4*a^6*b^2*d^4))^(1/2))/4 + 78464*A^3*a^27*b^21*d^
6 + 183616*A^3*a^29*b^19*d^6 + 238400*A^3*a^31*b^17*d^6 + 184960*A^3*a^33*b^15*d^6 + 84608*A^3*a^35*b^13*d^6 +
20704*A^3*a^37*b^11*d^6 + 2016*A^3*a^39*b^9*d^6))/4))/4 + 544*A^5*a^26*b^18*d^4 + 1520*A^5*a^28*b^16*d^4 + 22
40*A^5*a^30*b^14*d^4 + 1840*A^5*a^32*b^12*d^4 + 800*A^5*a^34*b^10*d^4 + 144*A^5*a^36*b^8*d^4)*(((192*A^4*a^2*b
^6*d^4 - 16*A^4*b^8*d^4 - 16*A^4*a^8*d^4 - 608*A^4*a^4*b^4*d^4 + 192*A^4*a^6*b^2*d^4)^(1/2) - 16*A^2*a*b^3*d^2
+ 16*A^2*a^3*b*d^2)/(a^8*d^4 + b^8*d^4 + 4*a^2*b^6*d^4 + 6*a^4*b^4*d^4 + 4*a^6*b^2*d^4))^(1/2))/4 + (log(80*A
^5*a^24*b^20*d^4 - ((-((192*A^4*a^2*b^6*d^4 - 16*A^4*b^8*d^4 - 16*A^4*a^8*d^4 - 608*A^4*a^4*b^4*d^4 + 192*A^4*
a^6*b^2*d^4)^(1/2) + 16*A^2*a*b^3*d^2 - 16*A^2*a^3*b*d^2)/(a^8*d^4 + b^8*d^4 + 4*a^2*b^6*d^4 + 6*a^4*b^4*d^4 +
4*a^6*b^2*d^4))^(1/2)*(tan(c + d*x)^(1/2)*(9472*A^4*a^31*b^15*d^5 - 3040*A^4*a^23*b^23*d^5 - 9056*A^4*a^25*b^
21*d^5 - 12352*A^4*a^27*b^19*d^5 - 4256*A^4*a^29*b^17*d^5 - 400*A^4*a^21*b^25*d^5 + 13760*A^4*a^33*b^13*d^5 +
7744*A^4*a^35*b^11*d^5 + 1968*A^4*a^37*b^9*d^5 + 224*A^4*a^39*b^7*d^5 + 32*A^4*a^41*b^5*d^5) + ((-((192*A^4*a^
2*b^6*d^4 - 16*A^4*b^8*d^4 - 16*A^4*a^8*d^4 - 608*A^4*a^4*b^4*d^4 + 192*A^4*a^6*b^2*d^4)^(1/2) + 16*A^2*a*b^3*
d^2 - 16*A^2*a^3*b*d^2)/(a^8*d^4 + b^8*d^4 + 4*a^2*b^6*d^4 + 6*a^4*b^4*d^4 + 4*a^6*b^2*d^4))^(1/2)*(12928*A^3*
a^25*b^23*d^6 - 800*A^3*a^21*b^27*d^6 - 2080*A^3*a^23*b^25*d^6 - ((tan(c + d*x)^(1/2)*(3200*A^2*a^22*b^28*d^7
+ 33920*A^2*a^24*b^26*d^7 + 158208*A^2*a^26*b^24*d^7 + 425536*A^2*a^28*b^22*d^7 + 727296*A^2*a^30*b^20*d^7 + 8
20672*A^2*a^32*b^18*d^7 + 615936*A^2*a^34*b^16*d^7 + 304256*A^2*a^36*b^14*d^7 + 98432*A^2*a^38*b^12*d^7 + 2201
6*A^2*a^40*b^10*d^7 + 3072*A^2*a^42*b^8*d^7 - 704*A^2*a^44*b^6*d^7 - 512*A^2*a^46*b^4*d^7 - 64*A^2*a^48*b^2*d^
7) + ((-((192*A^4*a^2*b^6*d^4 - 16*A^4*b^8*d^4 - 16*A^4*a^8*d^4 - 608*A^4*a^4*b^4*d^4 + 192*A^4*a^6*b^2*d^4)^(
1/2) + 16*A^2*a*b^3*d^2 - 16*A^2*a^3*b*d^2)/(a^8*d^4 + b^8*d^4 + 4*a^2*b^6*d^4 + 6*a^4*b^4*d^4 + 4*a^6*b^2*d^4
))^(1/2)*(1280*A*a^24*b^28*d^8 - (tan(c + d*x)^(1/2)*(-((192*A^4*a^2*b^6*d^4 - 16*A^4*b^8*d^4 - 16*A^4*a^8*d^4
- 608*A^4*a^4*b^4*d^4 + 192*A^4*a^6*b^2*d^4)^(1/2) + 16*A^2*a*b^3*d^2 - 16*A^2*a^3*b*d^2)/(a^8*d^4 + b^8*d^4
+ 4*a^2*b^6*d^4 + 6*a^4*b^4*d^4 + 4*a^6*b^2*d^4))^(1/2)*(512*a^27*b^27*d^9 + 5120*a^29*b^25*d^9 + 22528*a^31*b
^23*d^9 + 56320*a^33*b^21*d^9 + 84480*a^35*b^19*d^9 + 67584*a^37*b^17*d^9 - 67584*a^41*b^13*d^9 - 84480*a^43*b
^11*d^9 - 56320*a^45*b^9*d^9 - 22528*a^47*b^7*d^9 - 5120*a^49*b^5*d^9 - 512*a^51*b^3*d^9))/4 + 13824*A*a^26*b^
26*d^8 + 66944*A*a^28*b^24*d^8 + 190848*A*a^30*b^22*d^8 + 352640*A*a^32*b^20*d^8 + 435840*A*a^34*b^18*d^8 + 35
4048*A*a^36*b^16*d^8 + 169728*A*a^38*b^14*d^8 + 24576*A*a^40*b^12*d^8 - 21760*A*a^42*b^10*d^8 - 13440*A*a^44*b
^8*d^8 - 2176*A*a^46*b^6*d^8 + 384*A*a^48*b^4*d^8 + 128*A*a^50*b^2*d^8))/4)*(-((192*A^4*a^2*b^6*d^4 - 16*A^4*b
^8*d^4 - 16*A^4*a^8*d^4 - 608*A^4*a^4*b^4*d^4 + 192*A^4*a^6*b^2*d^4)^(1/2) + 16*A^2*a*b^3*d^2 - 16*A^2*a^3*b*d
^2)/(a^8*d^4 + b^8*d^4 + 4*a^2*b^6*d^4 + 6*a^4*b^4*d^4 + 4*a^6*b^2*d^4))^(1/2))/4 + 78464*A^3*a^27*b^21*d^6 +
183616*A^3*a^29*b^19*d^6 + 238400*A^3*a^31*b^17*d^6 + 184960*A^3*a^33*b^15*d^6 + 84608*A^3*a^35*b^13*d^6 + 207
04*A^3*a^37*b^11*d^6 + 2016*A^3*a^39*b^9*d^6))/4))/4 + 544*A^5*a^26*b^18*d^4 + 1520*A^5*a^28*b^16*d^4 + 2240*A
^5*a^30*b^14*d^4 + 1840*A^5*a^32*b^12*d^4 + 800*A^5*a^34*b^10*d^4 + 144*A^5*a^36*b^8*d^4)*(-((192*A^4*a^2*b^6*
d^4 - 16*A^4*b^8*d^4 - 16*A^4*a^8*d^4 - 608*A^4*a^4*b^4*d^4 + 192*A^4*a^6*b^2*d^4)^(1/2) + 16*A^2*a*b^3*d^2 -
16*A^2*a^3*b*d^2)/(a^8*d^4 + b^8*d^4 + 4*a^2*b^6*d^4 + 6*a^4*b^4*d^4 + 4*a^6*b^2*d^4))^(1/2))/4 - log((((192*A
^4*a^2*b^6*d^4 - 16*A^4*b^8*d^4 - 16*A^4*a^8*d^4 - 608*A^4*a^4*b^4*d^4 + 192*A^4*a^6*b^2*d^4)^(1/2) - 16*A^2*a
*b^3*d^2 + 16*A^2*a^3*b*d^2)/(16*a^8*d^4 + 16*b^8*d^4 + 64*a^2*b^6*d^4 + 96*a^4*b^4*d^4 + 64*a^6*b^2*d^4))^(1/
2)*(tan(c + d*x)^(1/2)*(9472*A^4*a^31*b^15*d^5 - 3040*A^4*a^23*b^23*d^5 - 9056*A^4*a^25*b^21*d^5 - 12352*A^4*a
^27*b^19*d^5 - 4256*A^4*a^29*b^17*d^5 - 400*A^4*a^21*b^25*d^5 + 13760*A^4*a^33*b^13*d^5 + 7744*A^4*a^35*b^11*d
^5 + 1968*A^4*a^37*b^9*d^5 + 224*A^4*a^39*b^7*d^5 + 32*A^4*a^41*b^5*d^5) - (((192*A^4*a^2*b^6*d^4 - 16*A^4*b^8
*d^4 - 16*A^4*a^8*d^4 - 608*A^4*a^4*b^4*d^4 + 192*A^4*a^6*b^2*d^4)^(1/2) - 16*A^2*a*b^3*d^2 + 16*A^2*a^3*b*d^2
)/(16*a^8*d^4 + 16*b^8*d^4 + 64*a^2*b^6*d^4 + 96*a^4*b^4*d^4 + 64*a^6*b^2*d^4))^(1/2)*((tan(c + d*x)^(1/2)*(32
00*A^2*a^22*b^28*d^7 + 33920*A^2*a^24*b^26*d^7 + 158208*A^2*a^26*b^24*d^7 + 425536*A^2*a^28*b^22*d^7 + 727296*
A^2*a^30*b^20*d^7 + 820672*A^2*a^32*b^18*d^7 + 615936*A^2*a^34*b^16*d^7 + 304256*A^2*a^36*b^14*d^7 + 98432*A^2
*a^38*b^12*d^7 + 22016*A^2*a^40*b^10*d^7 + 3072*A^2*a^42*b^8*d^7 - 704*A^2*a^44*b^6*d^7 - 512*A^2*a^46*b^4*d^7
- 64*A^2*a^48*b^2*d^7) - (((192*A^4*a^2*b^6*d^4 - 16*A^4*b^8*d^4 - 16*A^4*a^8*d^4 - 608*A^4*a^4*b^4*d^4 + 192
*A^4*a^6*b^2*d^4)^(1/2) - 16*A^2*a*b^3*d^2 + 16*A^2*a^3*b*d^2)/(16*a^8*d^4 + 16*b^8*d^4 + 64*a^2*b^6*d^4 + 96*
a^4*b^4*d^4 + 64*a^6*b^2*d^4))^(1/2)*(tan(c + d*x)^(1/2)*(((192*A^4*a^2*b^6*d^4 - 16*A^4*b^8*d^4 - 16*A^4*a^8*
d^4 - 608*A^4*a^4*b^4*d^4 + 192*A^4*a^6*b^2*d^4)^(1/2) - 16*A^2*a*b^3*d^2 + 16*A^2*a^3*b*d^2)/(16*a^8*d^4 + 16
*b^8*d^4 + 64*a^2*b^6*d^4 + 96*a^4*b^4*d^4 + 64*a^6*b^2*d^4))^(1/2)*(512*a^27*b^27*d^9 + 5120*a^29*b^25*d^9 +
22528*a^31*b^23*d^9 + 56320*a^33*b^21*d^9 + 84480*a^35*b^19*d^9 + 67584*a^37*b^17*d^9 - 67584*a^41*b^13*d^9 -
84480*a^43*b^11*d^9 - 56320*a^45*b^9*d^9 - 22528*a^47*b^7*d^9 - 5120*a^49*b^5*d^9 - 512*a^51*b^3*d^9) + 1280*A
*a^24*b^28*d^8 + 13824*A*a^26*b^26*d^8 + 66944*A*a^28*b^24*d^8 + 190848*A*a^30*b^22*d^8 + 352640*A*a^32*b^20*d
^8 + 435840*A*a^34*b^18*d^8 + 354048*A*a^36*b^16*d^8 + 169728*A*a^38*b^14*d^8 + 24576*A*a^40*b^12*d^8 - 21760*
A*a^42*b^10*d^8 - 13440*A*a^44*b^8*d^8 - 2176*A*a^46*b^6*d^8 + 384*A*a^48*b^4*d^8 + 128*A*a^50*b^2*d^8))*(((19
2*A^4*a^2*b^6*d^4 - 16*A^4*b^8*d^4 - 16*A^4*a^8*d^4 - 608*A^4*a^4*b^4*d^4 + 192*A^4*a^6*b^2*d^4)^(1/2) - 16*A^
2*a*b^3*d^2 + 16*A^2*a^3*b*d^2)/(16*a^8*d^4 + 16*b^8*d^4 + 64*a^2*b^6*d^4 + 96*a^4*b^4*d^4 + 64*a^6*b^2*d^4))^
(1/2) - 800*A^3*a^21*b^27*d^6 - 2080*A^3*a^23*b^25*d^6 + 12928*A^3*a^25*b^23*d^6 + 78464*A^3*a^27*b^21*d^6 + 1
83616*A^3*a^29*b^19*d^6 + 238400*A^3*a^31*b^17*d^6 + 184960*A^3*a^33*b^15*d^6 + 84608*A^3*a^35*b^13*d^6 + 2070
4*A^3*a^37*b^11*d^6 + 2016*A^3*a^39*b^9*d^6)) + 80*A^5*a^24*b^20*d^4 + 544*A^5*a^26*b^18*d^4 + 1520*A^5*a^28*b
^16*d^4 + 2240*A^5*a^30*b^14*d^4 + 1840*A^5*a^32*b^12*d^4 + 800*A^5*a^34*b^10*d^4 + 144*A^5*a^36*b^8*d^4)*(((1
92*A^4*a^2*b^6*d^4 - 16*A^4*b^8*d^4 - 16*A^4*a^8*d^4 - 608*A^4*a^4*b^4*d^4 + 192*A^4*a^6*b^2*d^4)^(1/2) - 16*A
^2*a*b^3*d^2 + 16*A^2*a^3*b*d^2)/(16*a^8*d^4 + 16*b^8*d^4 + 64*a^2*b^6*d^4 + 96*a^4*b^4*d^4 + 64*a^6*b^2*d^4))
^(1/2) - log((-((192*A^4*a^2*b^6*d^4 - 16*A^4*b^8*d^4 - 16*A^4*a^8*d^4 - 608*A^4*a^4*b^4*d^4 + 192*A^4*a^6*b^2
*d^4)^(1/2) + 16*A^2*a*b^3*d^2 - 16*A^2*a^3*b*d^2)/(16*a^8*d^4 + 16*b^8*d^4 + 64*a^2*b^6*d^4 + 96*a^4*b^4*d^4
+ 64*a^6*b^2*d^4))^(1/2)*(tan(c + d*x)^(1/2)*(9472*A^4*a^31*b^15*d^5 - 3040*A^4*a^23*b^23*d^5 - 9056*A^4*a^25*
b^21*d^5 - 12352*A^4*a^27*b^19*d^5 - 4256*A^4*a^29*b^17*d^5 - 400*A^4*a^21*b^25*d^5 + 13760*A^4*a^33*b^13*d^5
+ 7744*A^4*a^35*b^11*d^5 + 1968*A^4*a^37*b^9*d^5 + 224*A^4*a^39*b^7*d^5 + 32*A^4*a^41*b^5*d^5) - (-((192*A^4*a
^2*b^6*d^4 - 16*A^4*b^8*d^4 - 16*A^4*a^8*d^4 - 608*A^4*a^4*b^4*d^4 + 192*A^4*a^6*b^2*d^4)^(1/2) + 16*A^2*a*b^3
*d^2 - 16*A^2*a^3*b*d^2)/(16*a^8*d^4 + 16*b^8*d^4 + 64*a^2*b^6*d^4 + 96*a^4*b^4*d^4 + 64*a^6*b^2*d^4))^(1/2)*(
(tan(c + d*x)^(1/2)*(3200*A^2*a^22*b^28*d^7 + 33920*A^2*a^24*b^26*d^7 + 158208*A^2*a^26*b^24*d^7 + 425536*A^2*
a^28*b^22*d^7 + 727296*A^2*a^30*b^20*d^7 + 820672*A^2*a^32*b^18*d^7 + 615936*A^2*a^34*b^16*d^7 + 304256*A^2*a^
36*b^14*d^7 + 98432*A^2*a^38*b^12*d^7 + 22016*A^2*a^40*b^10*d^7 + 3072*A^2*a^42*b^8*d^7 - 704*A^2*a^44*b^6*d^7
- 512*A^2*a^46*b^4*d^7 - 64*A^2*a^48*b^2*d^7) - (-((192*A^4*a^2*b^6*d^4 - 16*A^4*b^8*d^4 - 16*A^4*a^8*d^4 - 6
08*A^4*a^4*b^4*d^4 + 192*A^4*a^6*b^2*d^4)^(1/2) + 16*A^2*a*b^3*d^2 - 16*A^2*a^3*b*d^2)/(16*a^8*d^4 + 16*b^8*d^
4 + 64*a^2*b^6*d^4 + 96*a^4*b^4*d^4 + 64*a^6*b^2*d^4))^(1/2)*(tan(c + d*x)^(1/2)*(-((192*A^4*a^2*b^6*d^4 - 16*
A^4*b^8*d^4 - 16*A^4*a^8*d^4 - 608*A^4*a^4*b^4*d^4 + 192*A^4*a^6*b^2*d^4)^(1/2) + 16*A^2*a*b^3*d^2 - 16*A^2*a^
3*b*d^2)/(16*a^8*d^4 + 16*b^8*d^4 + 64*a^2*b^6*d^4 + 96*a^4*b^4*d^4 + 64*a^6*b^2*d^4))^(1/2)*(512*a^27*b^27*d^
9 + 5120*a^29*b^25*d^9 + 22528*a^31*b^23*d^9 + 56320*a^33*b^21*d^9 + 84480*a^35*b^19*d^9 + 67584*a^37*b^17*d^9
- 67584*a^41*b^13*d^9 - 84480*a^43*b^11*d^9 - 56320*a^45*b^9*d^9 - 22528*a^47*b^7*d^9 - 5120*a^49*b^5*d^9 - 5
12*a^51*b^3*d^9) + 1280*A*a^24*b^28*d^8 + 13824*A*a^26*b^26*d^8 + 66944*A*a^28*b^24*d^8 + 190848*A*a^30*b^22*d
^8 + 352640*A*a^32*b^20*d^8 + 435840*A*a^34*b^18*d^8 + 354048*A*a^36*b^16*d^8 + 169728*A*a^38*b^14*d^8 + 24576
*A*a^40*b^12*d^8 - 21760*A*a^42*b^10*d^8 - 13440*A*a^44*b^8*d^8 - 2176*A*a^46*b^6*d^8 + 384*A*a^48*b^4*d^8 + 1
28*A*a^50*b^2*d^8))*(-((192*A^4*a^2*b^6*d^4 - 16*A^4*b^8*d^4 - 16*A^4*a^8*d^4 - 608*A^4*a^4*b^4*d^4 + 192*A^4*
a^6*b^2*d^4)^(1/2) + 16*A^2*a*b^3*d^2 - 16*A^2*a^3*b*d^2)/(16*a^8*d^4 + 16*b^8*d^4 + 64*a^2*b^6*d^4 + 96*a^4*b
^4*d^4 + 64*a^6*b^2*d^4))^(1/2) - 800*A^3*a^21*b^27*d^6 - 2080*A^3*a^23*b^25*d^6 + 12928*A^3*a^25*b^23*d^6 + 7
8464*A^3*a^27*b^21*d^6 + 183616*A^3*a^29*b^19*d^6 + 238400*A^3*a^31*b^17*d^6 + 184960*A^3*a^33*b^15*d^6 + 8460
8*A^3*a^35*b^13*d^6 + 20704*A^3*a^37*b^11*d^6 + 2016*A^3*a^39*b^9*d^6)) + 80*A^5*a^24*b^20*d^4 + 544*A^5*a^26*
b^18*d^4 + 1520*A^5*a^28*b^16*d^4 + 2240*A^5*a^30*b^14*d^4 + 1840*A^5*a^32*b^12*d^4 + 800*A^5*a^34*b^10*d^4 +
144*A^5*a^36*b^8*d^4)*(-((192*A^4*a^2*b^6*d^4 - 16*A^4*b^8*d^4 - 16*A^4*a^8*d^4 - 608*A^4*a^4*b^4*d^4 + 192*A^
4*a^6*b^2*d^4)^(1/2) + 16*A^2*a*b^3*d^2 - 16*A^2*a^3*b*d^2)/(16*a^8*d^4 + 16*b^8*d^4 + 64*a^2*b^6*d^4 + 96*a^4
*b^4*d^4 + 64*a^6*b^2*d^4))^(1/2) + ((10*A*b*tan(c + d*x))/(3*a^2) - (2*A)/(3*a) + (A*tan(c + d*x)^2*(5*b^4 +
4*a^2*b^2))/(a^3*(a^2 + b^2)))/(a*d*tan(c + d*x)^(3/2) + b*d*tan(c + d*x)^(5/2)) - ((2*B)/a + (B*tan(c + d*x)*
(2*a^2*b + 3*b^3))/(a^2*(a^2 + b^2)))/(a*d*tan(c + d*x)^(1/2) + b*d*tan(c + d*x)^(3/2)) + (log(72*B^5*a^14*b^2
1*d^4 - ((tan(c + d*x)^(1/2)*(144*B^4*a^14*b^23*d^5 + 1248*B^4*a^16*b^21*d^5 + 4224*B^4*a^18*b^19*d^5 + 6720*B
^4*a^20*b^17*d^5 + 3872*B^4*a^22*b^15*d^5 - 2816*B^4*a^24*b^13*d^5 - 5632*B^4*a^26*b^11*d^5 - 3136*B^4*a^28*b^
9*d^5 - 560*B^4*a^30*b^7*d^5 + 32*B^4*a^32*b^5*d^5) - ((((192*B^4*a^2*b^6*d^4 - 16*B^4*b^8*d^4 - 16*B^4*a^8*d^
4 - 608*B^4*a^4*b^4*d^4 + 192*B^4*a^6*b^2*d^4)^(1/2) + 16*B^2*a*b^3*d^2 - 16*B^2*a^3*b*d^2)/(a^8*d^4 + b^8*d^4
+ 4*a^2*b^6*d^4 + 6*a^4*b^4*d^4 + 4*a^6*b^2*d^4))^(1/2)*(((tan(c + d*x)^(1/2)*(1152*B^2*a^15*b^26*d^7 + 13440
*B^2*a^17*b^24*d^7 + 69056*B^2*a^19*b^22*d^7 + 202752*B^2*a^21*b^20*d^7 + 372800*B^2*a^23*b^18*d^7 + 443136*B^
2*a^25*b^16*d^7 + 337792*B^2*a^27*b^14*d^7 + 156160*B^2*a^29*b^12*d^7 + 37632*B^2*a^31*b^10*d^7 + 3200*B^2*a^3
3*b^8*d^7 + 704*B^2*a^35*b^6*d^7 + 512*B^2*a^37*b^4*d^7 + 64*B^2*a^39*b^2*d^7) - ((((192*B^4*a^2*b^6*d^4 - 16*
B^4*b^8*d^4 - 16*B^4*a^8*d^4 - 608*B^4*a^4*b^4*d^4 + 192*B^4*a^6*b^2*d^4)^(1/2) + 16*B^2*a*b^3*d^2 - 16*B^2*a^
3*b*d^2)/(a^8*d^4 + b^8*d^4 + 4*a^2*b^6*d^4 + 6*a^4*b^4*d^4 + 4*a^6*b^2*d^4))^(1/2)*((tan(c + d*x)^(1/2)*(((19
2*B^4*a^2*b^6*d^4 - 16*B^4*b^8*d^4 - 16*B^4*a^8*d^4 - 608*B^4*a^4*b^4*d^4 + 192*B^4*a^6*b^2*d^4)^(1/2) + 16*B^
2*a*b^3*d^2 - 16*B^2*a^3*b*d^2)/(a^8*d^4 + b^8*d^4 + 4*a^2*b^6*d^4 + 6*a^4*b^4*d^4 + 4*a^6*b^2*d^4))^(1/2)*(51
2*a^18*b^27*d^9 + 5120*a^20*b^25*d^9 + 22528*a^22*b^23*d^9 + 56320*a^24*b^21*d^9 + 84480*a^26*b^19*d^9 + 67584
*a^28*b^17*d^9 - 67584*a^32*b^13*d^9 - 84480*a^34*b^11*d^9 - 56320*a^36*b^9*d^9 - 22528*a^38*b^7*d^9 - 5120*a^
40*b^5*d^9 - 512*a^42*b^3*d^9))/4 + 768*B*a^16*b^27*d^8 + 8704*B*a^18*b^25*d^8 + 44288*B*a^20*b^23*d^8 + 13312
0*B*a^22*b^21*d^8 + 261120*B*a^24*b^19*d^8 + 347136*B*a^26*b^17*d^8 + 311808*B*a^28*b^15*d^8 + 178176*B*a^30*b
^13*d^8 + 49920*B*a^32*b^11*d^8 - 7680*B*a^34*b^9*d^8 - 12032*B*a^36*b^7*d^8 - 4096*B*a^38*b^5*d^8 - 512*B*a^4
0*b^3*d^8))/4)*(((192*B^4*a^2*b^6*d^4 - 16*B^4*b^8*d^4 - 16*B^4*a^8*d^4 - 608*B^4*a^4*b^4*d^4 + 192*B^4*a^6*b^
2*d^4)^(1/2) + 16*B^2*a*b^3*d^2 - 16*B^2*a^3*b*d^2)/(a^8*d^4 + b^8*d^4 + 4*a^2*b^6*d^4 + 6*a^4*b^4*d^4 + 4*a^6
*b^2*d^4))^(1/2))/4 - 1152*B^3*a^15*b^24*d^6 - 8448*B^3*a^17*b^22*d^6 - 23776*B^3*a^19*b^20*d^6 - 29664*B^3*a^
21*b^18*d^6 - 6528*B^3*a^23*b^16*d^6 + 26496*B^3*a^25*b^14*d^6 + 33984*B^3*a^27*b^12*d^6 + 18624*B^3*a^29*b^10
*d^6 + 5376*B^3*a^31*b^8*d^6 + 1152*B^3*a^33*b^6*d^6 + 288*B^3*a^35*b^4*d^6 + 32*B^3*a^37*b^2*d^6))/4)*(((192*
B^4*a^2*b^6*d^4 - 16*B^4*b^8*d^4 - 16*B^4*a^8*d^4 - 608*B^4*a^4*b^4*d^4 + 192*B^4*a^6*b^2*d^4)^(1/2) + 16*B^2*
a*b^3*d^2 - 16*B^2*a^3*b*d^2)/(a^8*d^4 + b^8*d^4 + 4*a^2*b^6*d^4 + 6*a^4*b^4*d^4 + 4*a^6*b^2*d^4))^(1/2))/4 +
648*B^5*a^16*b^19*d^4 + 2440*B^5*a^18*b^17*d^4 + 5000*B^5*a^20*b^15*d^4 + 6040*B^5*a^22*b^13*d^4 + 4312*B^5*a^
24*b^11*d^4 + 1688*B^5*a^26*b^9*d^4 + 280*B^5*a^28*b^7*d^4)*(((192*B^4*a^2*b^6*d^4 - 16*B^4*b^8*d^4 - 16*B^4*a
^8*d^4 - 608*B^4*a^4*b^4*d^4 + 192*B^4*a^6*b^2*d^4)^(1/2) + 16*B^2*a*b^3*d^2 - 16*B^2*a^3*b*d^2)/(a^8*d^4 + b^
8*d^4 + 4*a^2*b^6*d^4 + 6*a^4*b^4*d^4 + 4*a^6*b^2*d^4))^(1/2))/4 + (log(72*B^5*a^14*b^21*d^4 - ((tan(c + d*x)^
(1/2)*(144*B^4*a^14*b^23*d^5 + 1248*B^4*a^16*b^21*d^5 + 4224*B^4*a^18*b^19*d^5 + 6720*B^4*a^20*b^17*d^5 + 3872
*B^4*a^22*b^15*d^5 - 2816*B^4*a^24*b^13*d^5 - 5632*B^4*a^26*b^11*d^5 - 3136*B^4*a^28*b^9*d^5 - 560*B^4*a^30*b^
7*d^5 + 32*B^4*a^32*b^5*d^5) - ((-((192*B^4*a^2*b^6*d^4 - 16*B^4*b^8*d^4 - 16*B^4*a^8*d^4 - 608*B^4*a^4*b^4*d^
4 + 192*B^4*a^6*b^2*d^4)^(1/2) - 16*B^2*a*b^3*d^2 + 16*B^2*a^3*b*d^2)/(a^8*d^4 + b^8*d^4 + 4*a^2*b^6*d^4 + 6*a
^4*b^4*d^4 + 4*a^6*b^2*d^4))^(1/2)*(((tan(c + d*x)^(1/2)*(1152*B^2*a^15*b^26*d^7 + 13440*B^2*a^17*b^24*d^7 + 6
9056*B^2*a^19*b^22*d^7 + 202752*B^2*a^21*b^20*d^7 + 372800*B^2*a^23*b^18*d^7 + 443136*B^2*a^25*b^16*d^7 + 3377
92*B^2*a^27*b^14*d^7 + 156160*B^2*a^29*b^12*d^7 + 37632*B^2*a^31*b^10*d^7 + 3200*B^2*a^33*b^8*d^7 + 704*B^2*a^
35*b^6*d^7 + 512*B^2*a^37*b^4*d^7 + 64*B^2*a^39*b^2*d^7) - ((-((192*B^4*a^2*b^6*d^4 - 16*B^4*b^8*d^4 - 16*B^4*
a^8*d^4 - 608*B^4*a^4*b^4*d^4 + 192*B^4*a^6*b^2*d^4)^(1/2) - 16*B^2*a*b^3*d^2 + 16*B^2*a^3*b*d^2)/(a^8*d^4 + b
^8*d^4 + 4*a^2*b^6*d^4 + 6*a^4*b^4*d^4 + 4*a^6*b^2*d^4))^(1/2)*((tan(c + d*x)^(1/2)*(-((192*B^4*a^2*b^6*d^4 -
16*B^4*b^8*d^4 - 16*B^4*a^8*d^4 - 608*B^4*a^4*b^4*d^4 + 192*B^4*a^6*b^2*d^4)^(1/2) - 16*B^2*a*b^3*d^2 + 16*B^2
*a^3*b*d^2)/(a^8*d^4 + b^8*d^4 + 4*a^2*b^6*d^4 + 6*a^4*b^4*d^4 + 4*a^6*b^2*d^4))^(1/2)*(512*a^18*b^27*d^9 + 51
20*a^20*b^25*d^9 + 22528*a^22*b^23*d^9 + 56320*a^24*b^21*d^9 + 84480*a^26*b^19*d^9 + 67584*a^28*b^17*d^9 - 675
84*a^32*b^13*d^9 - 84480*a^34*b^11*d^9 - 56320*a^36*b^9*d^9 - 22528*a^38*b^7*d^9 - 5120*a^40*b^5*d^9 - 512*a^4
2*b^3*d^9))/4 + 768*B*a^16*b^27*d^8 + 8704*B*a^18*b^25*d^8 + 44288*B*a^20*b^23*d^8 + 133120*B*a^22*b^21*d^8 +
261120*B*a^24*b^19*d^8 + 347136*B*a^26*b^17*d^8 + 311808*B*a^28*b^15*d^8 + 178176*B*a^30*b^13*d^8 + 49920*B*a^
32*b^11*d^8 - 7680*B*a^34*b^9*d^8 - 12032*B*a^36*b^7*d^8 - 4096*B*a^38*b^5*d^8 - 512*B*a^40*b^3*d^8))/4)*(-((1
92*B^4*a^2*b^6*d^4 - 16*B^4*b^8*d^4 - 16*B^4*a^8*d^4 - 608*B^4*a^4*b^4*d^4 + 192*B^4*a^6*b^2*d^4)^(1/2) - 16*B
^2*a*b^3*d^2 + 16*B^2*a^3*b*d^2)/(a^8*d^4 + b^8*d^4 + 4*a^2*b^6*d^4 + 6*a^4*b^4*d^4 + 4*a^6*b^2*d^4))^(1/2))/4
- 1152*B^3*a^15*b^24*d^6 - 8448*B^3*a^17*b^22*d^6 - 23776*B^3*a^19*b^20*d^6 - 29664*B^3*a^21*b^18*d^6 - 6528*
B^3*a^23*b^16*d^6 + 26496*B^3*a^25*b^14*d^6 + 33984*B^3*a^27*b^12*d^6 + 18624*B^3*a^29*b^10*d^6 + 5376*B^3*a^3
1*b^8*d^6 + 1152*B^3*a^33*b^6*d^6 + 288*B^3*a^35*b^4*d^6 + 32*B^3*a^37*b^2*d^6))/4)*(-((192*B^4*a^2*b^6*d^4 -
16*B^4*b^8*d^4 - 16*B^4*a^8*d^4 - 608*B^4*a^4*b^4*d^4 + 192*B^4*a^6*b^2*d^4)^(1/2) - 16*B^2*a*b^3*d^2 + 16*B^2
*a^3*b*d^2)/(a^8*d^4 + b^8*d^4 + 4*a^2*b^6*d^4 + 6*a^4*b^4*d^4 + 4*a^6*b^2*d^4))^(1/2))/4 + 648*B^5*a^16*b^19*
d^4 + 2440*B^5*a^18*b^17*d^4 + 5000*B^5*a^20*b^15*d^4 + 6040*B^5*a^22*b^13*d^4 + 4312*B^5*a^24*b^11*d^4 + 1688
*B^5*a^26*b^9*d^4 + 280*B^5*a^28*b^7*d^4)*(-((192*B^4*a^2*b^6*d^4 - 16*B^4*b^8*d^4 - 16*B^4*a^8*d^4 - 608*B^4*
a^4*b^4*d^4 + 192*B^4*a^6*b^2*d^4)^(1/2) - 16*B^2*a*b^3*d^2 + 16*B^2*a^3*b*d^2)/(a^8*d^4 + b^8*d^4 + 4*a^2*b^6
*d^4 + 6*a^4*b^4*d^4 + 4*a^6*b^2*d^4))^(1/2))/4 - log((tan(c + d*x)^(1/2)*(144*B^4*a^14*b^23*d^5 + 1248*B^4*a^
16*b^21*d^5 + 4224*B^4*a^18*b^19*d^5 + 6720*B^4*a^20*b^17*d^5 + 3872*B^4*a^22*b^15*d^5 - 2816*B^4*a^24*b^13*d^
5 - 5632*B^4*a^26*b^11*d^5 - 3136*B^4*a^28*b^9*d^5 - 560*B^4*a^30*b^7*d^5 + 32*B^4*a^32*b^5*d^5) + (((192*B^4*
a^2*b^6*d^4 - 16*B^4*b^8*d^4 - 16*B^4*a^8*d^4 - 608*B^4*a^4*b^4*d^4 + 192*B^4*a^6*b^2*d^4)^(1/2) + 16*B^2*a*b^
3*d^2 - 16*B^2*a^3*b*d^2)/(16*a^8*d^4 + 16*b^8*d^4 + 64*a^2*b^6*d^4 + 96*a^4*b^4*d^4 + 64*a^6*b^2*d^4))^(1/2)*
(26496*B^3*a^25*b^14*d^6 - 1152*B^3*a^15*b^24*d^6 - 8448*B^3*a^17*b^22*d^6 - 23776*B^3*a^19*b^20*d^6 - 29664*B
^3*a^21*b^18*d^6 - 6528*B^3*a^23*b^16*d^6 - (tan(c + d*x)^(1/2)*(1152*B^2*a^15*b^26*d^7 + 13440*B^2*a^17*b^24*
d^7 + 69056*B^2*a^19*b^22*d^7 + 202752*B^2*a^21*b^20*d^7 + 372800*B^2*a^23*b^18*d^7 + 443136*B^2*a^25*b^16*d^7
+ 337792*B^2*a^27*b^14*d^7 + 156160*B^2*a^29*b^12*d^7 + 37632*B^2*a^31*b^10*d^7 + 3200*B^2*a^33*b^8*d^7 + 704
*B^2*a^35*b^6*d^7 + 512*B^2*a^37*b^4*d^7 + 64*B^2*a^39*b^2*d^7) + (((192*B^4*a^2*b^6*d^4 - 16*B^4*b^8*d^4 - 16
*B^4*a^8*d^4 - 608*B^4*a^4*b^4*d^4 + 192*B^4*a^6*b^2*d^4)^(1/2) + 16*B^2*a*b^3*d^2 - 16*B^2*a^3*b*d^2)/(16*a^8
*d^4 + 16*b^8*d^4 + 64*a^2*b^6*d^4 + 96*a^4*b^4*d^4 + 64*a^6*b^2*d^4))^(1/2)*(768*B*a^16*b^27*d^8 - tan(c + d*
x)^(1/2)*(((192*B^4*a^2*b^6*d^4 - 16*B^4*b^8*d^4 - 16*B^4*a^8*d^4 - 608*B^4*a^4*b^4*d^4 + 192*B^4*a^6*b^2*d^4)
^(1/2) + 16*B^2*a*b^3*d^2 - 16*B^2*a^3*b*d^2)/(16*a^8*d^4 + 16*b^8*d^4 + 64*a^2*b^6*d^4 + 96*a^4*b^4*d^4 + 64*
a^6*b^2*d^4))^(1/2)*(512*a^18*b^27*d^9 + 5120*a^20*b^25*d^9 + 22528*a^22*b^23*d^9 + 56320*a^24*b^21*d^9 + 8448
0*a^26*b^19*d^9 + 67584*a^28*b^17*d^9 - 67584*a^32*b^13*d^9 - 84480*a^34*b^11*d^9 - 56320*a^36*b^9*d^9 - 22528
*a^38*b^7*d^9 - 5120*a^40*b^5*d^9 - 512*a^42*b^3*d^9) + 8704*B*a^18*b^25*d^8 + 44288*B*a^20*b^23*d^8 + 133120*
B*a^22*b^21*d^8 + 261120*B*a^24*b^19*d^8 + 347136*B*a^26*b^17*d^8 + 311808*B*a^28*b^15*d^8 + 178176*B*a^30*b^1
3*d^8 + 49920*B*a^32*b^11*d^8 - 7680*B*a^34*b^9*d^8 - 12032*B*a^36*b^7*d^8 - 4096*B*a^38*b^5*d^8 - 512*B*a^40*
b^3*d^8))*(((192*B^4*a^2*b^6*d^4 - 16*B^4*b^8*d^4 - 16*B^4*a^8*d^4 - 608*B^4*a^4*b^4*d^4 + 192*B^4*a^6*b^2*d^4
)^(1/2) + 16*B^2*a*b^3*d^2 - 16*B^2*a^3*b*d^2)/(16*a^8*d^4 + 16*b^8*d^4 + 64*a^2*b^6*d^4 + 96*a^4*b^4*d^4 + 64
*a^6*b^2*d^4))^(1/2) + 33984*B^3*a^27*b^12*d^6 + 18624*B^3*a^29*b^10*d^6 + 5376*B^3*a^31*b^8*d^6 + 1152*B^3*a^
33*b^6*d^6 + 288*B^3*a^35*b^4*d^6 + 32*B^3*a^37*b^2*d^6))*(((192*B^4*a^2*b^6*d^4 - 16*B^4*b^8*d^4 - 16*B^4*a^8
*d^4 - 608*B^4*a^4*b^4*d^4 + 192*B^4*a^6*b^2*d^4)^(1/2) + 16*B^2*a*b^3*d^2 - 16*B^2*a^3*b*d^2)/(16*a^8*d^4 + 1
6*b^8*d^4 + 64*a^2*b^6*d^4 + 96*a^4*b^4*d^4 + 64*a^6*b^2*d^4))^(1/2) + 72*B^5*a^14*b^21*d^4 + 648*B^5*a^16*b^1
9*d^4 + 2440*B^5*a^18*b^17*d^4 + 5000*B^5*a^20*b^15*d^4 + 6040*B^5*a^22*b^13*d^4 + 4312*B^5*a^24*b^11*d^4 + 16
88*B^5*a^26*b^9*d^4 + 280*B^5*a^28*b^7*d^4)*(((192*B^4*a^2*b^6*d^4 - 16*B^4*b^8*d^4 - 16*B^4*a^8*d^4 - 608*B^4
*a^4*b^4*d^4 + 192*B^4*a^6*b^2*d^4)^(1/2) + 16*B^2*a*b^3*d^2 - 16*B^2*a^3*b*d^2)/(16*a^8*d^4 + 16*b^8*d^4 + 64
*a^2*b^6*d^4 + 96*a^4*b^4*d^4 + 64*a^6*b^2*d^4))^(1/2) - log((tan(c + d*x)^(1/2)*(144*B^4*a^14*b^23*d^5 + 1248
*B^4*a^16*b^21*d^5 + 4224*B^4*a^18*b^19*d^5 + 6720*B^4*a^20*b^17*d^5 + 3872*B^4*a^22*b^15*d^5 - 2816*B^4*a^24*
b^13*d^5 - 5632*B^4*a^26*b^11*d^5 - 3136*B^4*a^28*b^9*d^5 - 560*B^4*a^30*b^7*d^5 + 32*B^4*a^32*b^5*d^5) + (-((
192*B^4*a^2*b^6*d^4 - 16*B^4*b^8*d^4 - 16*B^4*a^8*d^4 - 608*B^4*a^4*b^4*d^4 + 192*B^4*a^6*b^2*d^4)^(1/2) - 16*
B^2*a*b^3*d^2 + 16*B^2*a^3*b*d^2)/(16*a^8*d^4 + 16*b^8*d^4 + 64*a^2*b^6*d^4 + 96*a^4*b^4*d^4 + 64*a^6*b^2*d^4)
)^(1/2)*(26496*B^3*a^25*b^14*d^6 - 1152*B^3*a^15*b^24*d^6 - 8448*B^3*a^17*b^22*d^6 - 23776*B^3*a^19*b^20*d^6 -
29664*B^3*a^21*b^18*d^6 - 6528*B^3*a^23*b^16*d^6 - (tan(c + d*x)^(1/2)*(1152*B^2*a^15*b^26*d^7 + 13440*B^2*a^
17*b^24*d^7 + 69056*B^2*a^19*b^22*d^7 + 202752*B^2*a^21*b^20*d^7 + 372800*B^2*a^23*b^18*d^7 + 443136*B^2*a^25*
b^16*d^7 + 337792*B^2*a^27*b^14*d^7 + 156160*B^2*a^29*b^12*d^7 + 37632*B^2*a^31*b^10*d^7 + 3200*B^2*a^33*b^8*d
^7 + 704*B^2*a^35*b^6*d^7 + 512*B^2*a^37*b^4*d^7 + 64*B^2*a^39*b^2*d^7) + (-((192*B^4*a^2*b^6*d^4 - 16*B^4*b^8
*d^4 - 16*B^4*a^8*d^4 - 608*B^4*a^4*b^4*d^4 + 192*B^4*a^6*b^2*d^4)^(1/2) - 16*B^2*a*b^3*d^2 + 16*B^2*a^3*b*d^2
)/(16*a^8*d^4 + 16*b^8*d^4 + 64*a^2*b^6*d^4 + 96*a^4*b^4*d^4 + 64*a^6*b^2*d^4))^(1/2)*(768*B*a^16*b^27*d^8 - t
an(c + d*x)^(1/2)*(-((192*B^4*a^2*b^6*d^4 - 16*B^4*b^8*d^4 - 16*B^4*a^8*d^4 - 608*B^4*a^4*b^4*d^4 + 192*B^4*a^
6*b^2*d^4)^(1/2) - 16*B^2*a*b^3*d^2 + 16*B^2*a^3*b*d^2)/(16*a^8*d^4 + 16*b^8*d^4 + 64*a^2*b^6*d^4 + 96*a^4*b^4
*d^4 + 64*a^6*b^2*d^4))^(1/2)*(512*a^18*b^27*d^9 + 5120*a^20*b^25*d^9 + 22528*a^22*b^23*d^9 + 56320*a^24*b^21*
d^9 + 84480*a^26*b^19*d^9 + 67584*a^28*b^17*d^9 - 67584*a^32*b^13*d^9 - 84480*a^34*b^11*d^9 - 56320*a^36*b^9*d
^9 - 22528*a^38*b^7*d^9 - 5120*a^40*b^5*d^9 - 512*a^42*b^3*d^9) + 8704*B*a^18*b^25*d^8 + 44288*B*a^20*b^23*d^8
+ 133120*B*a^22*b^21*d^8 + 261120*B*a^24*b^19*d^8 + 347136*B*a^26*b^17*d^8 + 311808*B*a^28*b^15*d^8 + 178176*
B*a^30*b^13*d^8 + 49920*B*a^32*b^11*d^8 - 7680*B*a^34*b^9*d^8 - 12032*B*a^36*b^7*d^8 - 4096*B*a^38*b^5*d^8 - 5
12*B*a^40*b^3*d^8))*(-((192*B^4*a^2*b^6*d^4 - 16*B^4*b^8*d^4 - 16*B^4*a^8*d^4 - 608*B^4*a^4*b^4*d^4 + 192*B^4*
a^6*b^2*d^4)^(1/2) - 16*B^2*a*b^3*d^2 + 16*B^2*a^3*b*d^2)/(16*a^8*d^4 + 16*b^8*d^4 + 64*a^2*b^6*d^4 + 96*a^4*b
^4*d^4 + 64*a^6*b^2*d^4))^(1/2) + 33984*B^3*a^27*b^12*d^6 + 18624*B^3*a^29*b^10*d^6 + 5376*B^3*a^31*b^8*d^6 +
1152*B^3*a^33*b^6*d^6 + 288*B^3*a^35*b^4*d^6 + 32*B^3*a^37*b^2*d^6))*(-((192*B^4*a^2*b^6*d^4 - 16*B^4*b^8*d^4
- 16*B^4*a^8*d^4 - 608*B^4*a^4*b^4*d^4 + 192*B^4*a^6*b^2*d^4)^(1/2) - 16*B^2*a*b^3*d^2 + 16*B^2*a^3*b*d^2)/(16
*a^8*d^4 + 16*b^8*d^4 + 64*a^2*b^6*d^4 + 96*a^4*b^4*d^4 + 64*a^6*b^2*d^4))^(1/2) + 72*B^5*a^14*b^21*d^4 + 648*
B^5*a^16*b^19*d^4 + 2440*B^5*a^18*b^17*d^4 + 5000*B^5*a^20*b^15*d^4 + 6040*B^5*a^22*b^13*d^4 + 4312*B^5*a^24*b
^11*d^4 + 1688*B^5*a^26*b^9*d^4 + 280*B^5*a^28*b^7*d^4)*(-((192*B^4*a^2*b^6*d^4 - 16*B^4*b^8*d^4 - 16*B^4*a^8*
d^4 - 608*B^4*a^4*b^4*d^4 + 192*B^4*a^6*b^2*d^4)^(1/2) - 16*B^2*a*b^3*d^2 + 16*B^2*a^3*b*d^2)/(16*a^8*d^4 + 16
*b^8*d^4 + 64*a^2*b^6*d^4 + 96*a^4*b^4*d^4 + 64*a^6*b^2*d^4))^(1/2) + (atan(((((tan(c + d*x)^(1/2)*(144*B^4*a^
14*b^23*d^5 + 1248*B^4*a^16*b^21*d^5 + 4224*B^4*a^18*b^19*d^5 + 6720*B^4*a^20*b^17*d^5 + 3872*B^4*a^22*b^15*d^
5 - 2816*B^4*a^24*b^13*d^5 - 5632*B^4*a^26*b^11*d^5 - 3136*B^4*a^28*b^9*d^5 - 560*B^4*a^30*b^7*d^5 + 32*B^4*a^
32*b^5*d^5))/4 + ((-4*(9*B^2*b^9 + 42*B^2*a^2*b^7 + 49*B^2*a^4*b^5)*(a^13*d^2 + a^5*b^8*d^2 + 4*a^7*b^6*d^2 +
6*a^9*b^4*d^2 + 4*a^11*b^2*d^2))^(1/2)*(6624*B^3*a^25*b^14*d^6 - 288*B^3*a^15*b^24*d^6 - 2112*B^3*a^17*b^22*d^
6 - 5944*B^3*a^19*b^20*d^6 - 7416*B^3*a^21*b^18*d^6 - 1632*B^3*a^23*b^16*d^6 - (((tan(c + d*x)^(1/2)*(1152*B^2
*a^15*b^26*d^7 + 13440*B^2*a^17*b^24*d^7 + 69056*B^2*a^19*b^22*d^7 + 202752*B^2*a^21*b^20*d^7 + 372800*B^2*a^2
3*b^18*d^7 + 443136*B^2*a^25*b^16*d^7 + 337792*B^2*a^27*b^14*d^7 + 156160*B^2*a^29*b^12*d^7 + 37632*B^2*a^31*b
^10*d^7 + 3200*B^2*a^33*b^8*d^7 + 704*B^2*a^35*b^6*d^7 + 512*B^2*a^37*b^4*d^7 + 64*B^2*a^39*b^2*d^7))/4 + ((-4
*(9*B^2*b^9 + 42*B^2*a^2*b^7 + 49*B^2*a^4*b^5)*(a^13*d^2 + a^5*b^8*d^2 + 4*a^7*b^6*d^2 + 6*a^9*b^4*d^2 + 4*a^1
1*b^2*d^2))^(1/2)*(192*B*a^16*b^27*d^8 - (tan(c + d*x)^(1/2)*(-4*(9*B^2*b^9 + 42*B^2*a^2*b^7 + 49*B^2*a^4*b^5)
*(a^13*d^2 + a^5*b^8*d^2 + 4*a^7*b^6*d^2 + 6*a^9*b^4*d^2 + 4*a^11*b^2*d^2))^(1/2)*(512*a^18*b^27*d^9 + 5120*a^
20*b^25*d^9 + 22528*a^22*b^23*d^9 + 56320*a^24*b^21*d^9 + 84480*a^26*b^19*d^9 + 67584*a^28*b^17*d^9 - 67584*a^
32*b^13*d^9 - 84480*a^34*b^11*d^9 - 56320*a^36*b^9*d^9 - 22528*a^38*b^7*d^9 - 5120*a^40*b^5*d^9 - 512*a^42*b^3
*d^9))/(16*(a^13*d^2 + a^5*b^8*d^2 + 4*a^7*b^6*d^2 + 6*a^9*b^4*d^2 + 4*a^11*b^2*d^2)) + 2176*B*a^18*b^25*d^8 +
11072*B*a^20*b^23*d^8 + 33280*B*a^22*b^21*d^8 + 65280*B*a^24*b^19*d^8 + 86784*B*a^26*b^17*d^8 + 77952*B*a^28*
b^15*d^8 + 44544*B*a^30*b^13*d^8 + 12480*B*a^32*b^11*d^8 - 1920*B*a^34*b^9*d^8 - 3008*B*a^36*b^7*d^8 - 1024*B*
a^38*b^5*d^8 - 128*B*a^40*b^3*d^8))/(4*(a^13*d^2 + a^5*b^8*d^2 + 4*a^7*b^6*d^2 + 6*a^9*b^4*d^2 + 4*a^11*b^2*d^
2)))*(-4*(9*B^2*b^9 + 42*B^2*a^2*b^7 + 49*B^2*a^4*b^5)*(a^13*d^2 + a^5*b^8*d^2 + 4*a^7*b^6*d^2 + 6*a^9*b^4*d^2
+ 4*a^11*b^2*d^2))^(1/2))/(4*(a^13*d^2 + a^5*b^8*d^2 + 4*a^7*b^6*d^2 + 6*a^9*b^4*d^2 + 4*a^11*b^2*d^2)) + 849
6*B^3*a^27*b^12*d^6 + 4656*B^3*a^29*b^10*d^6 + 1344*B^3*a^31*b^8*d^6 + 288*B^3*a^33*b^6*d^6 + 72*B^3*a^35*b^4*
d^6 + 8*B^3*a^37*b^2*d^6))/(4*(a^13*d^2 + a^5*b^8*d^2 + 4*a^7*b^6*d^2 + 6*a^9*b^4*d^2 + 4*a^11*b^2*d^2)))*(-4*
(9*B^2*b^9 + 42*B^2*a^2*b^7 + 49*B^2*a^4*b^5)*(a^13*d^2 + a^5*b^8*d^2 + 4*a^7*b^6*d^2 + 6*a^9*b^4*d^2 + 4*a^11
*b^2*d^2))^(1/2)*1i)/(a^13*d^2 + a^5*b^8*d^2 + 4*a^7*b^6*d^2 + 6*a^9*b^4*d^2 + 4*a^11*b^2*d^2) + (((tan(c + d*
x)^(1/2)*(144*B^4*a^14*b^23*d^5 + 1248*B^4*a^16*b^21*d^5 + 4224*B^4*a^18*b^19*d^5 + 6720*B^4*a^20*b^17*d^5 + 3
872*B^4*a^22*b^15*d^5 - 2816*B^4*a^24*b^13*d^5 - 5632*B^4*a^26*b^11*d^5 - 3136*B^4*a^28*b^9*d^5 - 560*B^4*a^30
*b^7*d^5 + 32*B^4*a^32*b^5*d^5))/4 - ((-4*(9*B^2*b^9 + 42*B^2*a^2*b^7 + 49*B^2*a^4*b^5)*(a^13*d^2 + a^5*b^8*d^
2 + 4*a^7*b^6*d^2 + 6*a^9*b^4*d^2 + 4*a^11*b^2*d^2))^(1/2)*((((tan(c + d*x)^(1/2)*(1152*B^2*a^15*b^26*d^7 + 13
440*B^2*a^17*b^24*d^7 + 69056*B^2*a^19*b^22*d^7 + 202752*B^2*a^21*b^20*d^7 + 372800*B^2*a^23*b^18*d^7 + 443136
*B^2*a^25*b^16*d^7 + 337792*B^2*a^27*b^14*d^7 + 156160*B^2*a^29*b^12*d^7 + 37632*B^2*a^31*b^10*d^7 + 3200*B^2*
a^33*b^8*d^7 + 704*B^2*a^35*b^6*d^7 + 512*B^2*a^37*b^4*d^7 + 64*B^2*a^39*b^2*d^7))/4 - ((-4*(9*B^2*b^9 + 42*B^
2*a^2*b^7 + 49*B^2*a^4*b^5)*(a^13*d^2 + a^5*b^8*d^2 + 4*a^7*b^6*d^2 + 6*a^9*b^4*d^2 + 4*a^11*b^2*d^2))^(1/2)*(
(tan(c + d*x)^(1/2)*(-4*(9*B^2*b^9 + 42*B^2*a^2*b^7 + 49*B^2*a^4*b^5)*(a^13*d^2 + a^5*b^8*d^2 + 4*a^7*b^6*d^2
+ 6*a^9*b^4*d^2 + 4*a^11*b^2*d^2))^(1/2)*(512*a^18*b^27*d^9 + 5120*a^20*b^25*d^9 + 22528*a^22*b^23*d^9 + 56320
*a^24*b^21*d^9 + 84480*a^26*b^19*d^9 + 67584*a^28*b^17*d^9 - 67584*a^32*b^13*d^9 - 84480*a^34*b^11*d^9 - 56320
*a^36*b^9*d^9 - 22528*a^38*b^7*d^9 - 5120*a^40*b^5*d^9 - 512*a^42*b^3*d^9))/(16*(a^13*d^2 + a^5*b^8*d^2 + 4*a^
7*b^6*d^2 + 6*a^9*b^4*d^2 + 4*a^11*b^2*d^2)) + 192*B*a^16*b^27*d^8 + 2176*B*a^18*b^25*d^8 + 11072*B*a^20*b^23*
d^8 + 33280*B*a^22*b^21*d^8 + 65280*B*a^24*b^19*d^8 + 86784*B*a^26*b^17*d^8 + 77952*B*a^28*b^15*d^8 + 44544*B*
a^30*b^13*d^8 + 12480*B*a^32*b^11*d^8 - 1920*B*a^34*b^9*d^8 - 3008*B*a^36*b^7*d^8 - 1024*B*a^38*b^5*d^8 - 128*
B*a^40*b^3*d^8))/(4*(a^13*d^2 + a^5*b^8*d^2 + 4*a^7*b^6*d^2 + 6*a^9*b^4*d^2 + 4*a^11*b^2*d^2)))*(-4*(9*B^2*b^9
+ 42*B^2*a^2*b^7 + 49*B^2*a^4*b^5)*(a^13*d^2 + a^5*b^8*d^2 + 4*a^7*b^6*d^2 + 6*a^9*b^4*d^2 + 4*a^11*b^2*d^2))
^(1/2))/(4*(a^13*d^2 + a^5*b^8*d^2 + 4*a^7*b^6*d^2 + 6*a^9*b^4*d^2 + 4*a^11*b^2*d^2)) - 288*B^3*a^15*b^24*d^6
- 2112*B^3*a^17*b^22*d^6 - 5944*B^3*a^19*b^20*d^6 - 7416*B^3*a^21*b^18*d^6 - 1632*B^3*a^23*b^16*d^6 + 6624*B^3
*a^25*b^14*d^6 + 8496*B^3*a^27*b^12*d^6 + 4656*B^3*a^29*b^10*d^6 + 1344*B^3*a^31*b^8*d^6 + 288*B^3*a^33*b^6*d^
6 + 72*B^3*a^35*b^4*d^6 + 8*B^3*a^37*b^2*d^6))/(4*(a^13*d^2 + a^5*b^8*d^2 + 4*a^7*b^6*d^2 + 6*a^9*b^4*d^2 + 4*
a^11*b^2*d^2)))*(-4*(9*B^2*b^9 + 42*B^2*a^2*b^7 + 49*B^2*a^4*b^5)*(a^13*d^2 + a^5*b^8*d^2 + 4*a^7*b^6*d^2 + 6*
a^9*b^4*d^2 + 4*a^11*b^2*d^2))^(1/2)*1i)/(a^13*d^2 + a^5*b^8*d^2 + 4*a^7*b^6*d^2 + 6*a^9*b^4*d^2 + 4*a^11*b^2*
d^2))/((((tan(c + d*x)^(1/2)*(144*B^4*a^14*b^23*d^5 + 1248*B^4*a^16*b^21*d^5 + 4224*B^4*a^18*b^19*d^5 + 6720*B
^4*a^20*b^17*d^5 + 3872*B^4*a^22*b^15*d^5 - 2816*B^4*a^24*b^13*d^5 - 5632*B^4*a^26*b^11*d^5 - 3136*B^4*a^28*b^
9*d^5 - 560*B^4*a^30*b^7*d^5 + 32*B^4*a^32*b^5*d^5))/4 + ((-4*(9*B^2*b^9 + 42*B^2*a^2*b^7 + 49*B^2*a^4*b^5)*(a
^13*d^2 + a^5*b^8*d^2 + 4*a^7*b^6*d^2 + 6*a^9*b^4*d^2 + 4*a^11*b^2*d^2))^(1/2)*(6624*B^3*a^25*b^14*d^6 - 288*B
^3*a^15*b^24*d^6 - 2112*B^3*a^17*b^22*d^6 - 5944*B^3*a^19*b^20*d^6 - 7416*B^3*a^21*b^18*d^6 - 1632*B^3*a^23*b^
16*d^6 - (((tan(c + d*x)^(1/2)*(1152*B^2*a^15*b^26*d^7 + 13440*B^2*a^17*b^24*d^7 + 69056*B^2*a^19*b^22*d^7 + 2
02752*B^2*a^21*b^20*d^7 + 372800*B^2*a^23*b^18*d^7 + 443136*B^2*a^25*b^16*d^7 + 337792*B^2*a^27*b^14*d^7 + 156
160*B^2*a^29*b^12*d^7 + 37632*B^2*a^31*b^10*d^7 + 3200*B^2*a^33*b^8*d^7 + 704*B^2*a^35*b^6*d^7 + 512*B^2*a^37*
b^4*d^7 + 64*B^2*a^39*b^2*d^7))/4 + ((-4*(9*B^2*b^9 + 42*B^2*a^2*b^7 + 49*B^2*a^4*b^5)*(a^13*d^2 + a^5*b^8*d^2
+ 4*a^7*b^6*d^2 + 6*a^9*b^4*d^2 + 4*a^11*b^2*d^2))^(1/2)*(192*B*a^16*b^27*d^8 - (tan(c + d*x)^(1/2)*(-4*(9*B^
2*b^9 + 42*B^2*a^2*b^7 + 49*B^2*a^4*b^5)*(a^13*d^2 + a^5*b^8*d^2 + 4*a^7*b^6*d^2 + 6*a^9*b^4*d^2 + 4*a^11*b^2*
d^2))^(1/2)*(512*a^18*b^27*d^9 + 5120*a^20*b^25*d^9 + 22528*a^22*b^23*d^9 + 56320*a^24*b^21*d^9 + 84480*a^26*b
^19*d^9 + 67584*a^28*b^17*d^9 - 67584*a^32*b^13*d^9 - 84480*a^34*b^11*d^9 - 56320*a^36*b^9*d^9 - 22528*a^38*b^
7*d^9 - 5120*a^40*b^5*d^9 - 512*a^42*b^3*d^9))/(16*(a^13*d^2 + a^5*b^8*d^2 + 4*a^7*b^6*d^2 + 6*a^9*b^4*d^2 + 4
*a^11*b^2*d^2)) + 2176*B*a^18*b^25*d^8 + 11072*B*a^20*b^23*d^8 + 33280*B*a^22*b^21*d^8 + 65280*B*a^24*b^19*d^8
+ 86784*B*a^26*b^17*d^8 + 77952*B*a^28*b^15*d^8 + 44544*B*a^30*b^13*d^8 + 12480*B*a^32*b^11*d^8 - 1920*B*a^34
*b^9*d^8 - 3008*B*a^36*b^7*d^8 - 1024*B*a^38*b^5*d^8 - 128*B*a^40*b^3*d^8))/(4*(a^13*d^2 + a^5*b^8*d^2 + 4*a^7
*b^6*d^2 + 6*a^9*b^4*d^2 + 4*a^11*b^2*d^2)))*(-4*(9*B^2*b^9 + 42*B^2*a^2*b^7 + 49*B^2*a^4*b^5)*(a^13*d^2 + a^5
*b^8*d^2 + 4*a^7*b^6*d^2 + 6*a^9*b^4*d^2 + 4*a^11*b^2*d^2))^(1/2))/(4*(a^13*d^2 + a^5*b^8*d^2 + 4*a^7*b^6*d^2
+ 6*a^9*b^4*d^2 + 4*a^11*b^2*d^2)) + 8496*B^3*a^27*b^12*d^6 + 4656*B^3*a^29*b^10*d^6 + 1344*B^3*a^31*b^8*d^6 +
288*B^3*a^33*b^6*d^6 + 72*B^3*a^35*b^4*d^6 + 8*B^3*a^37*b^2*d^6))/(4*(a^13*d^2 + a^5*b^8*d^2 + 4*a^7*b^6*d^2
+ 6*a^9*b^4*d^2 + 4*a^11*b^2*d^2)))*(-4*(9*B^2*b^9 + 42*B^2*a^2*b^7 + 49*B^2*a^4*b^5)*(a^13*d^2 + a^5*b^8*d^2
+ 4*a^7*b^6*d^2 + 6*a^9*b^4*d^2 + 4*a^11*b^2*d^2))^(1/2))/(a^13*d^2 + a^5*b^8*d^2 + 4*a^7*b^6*d^2 + 6*a^9*b^4*
d^2 + 4*a^11*b^2*d^2) - (((tan(c + d*x)^(1/2)*(144*B^4*a^14*b^23*d^5 + 1248*B^4*a^16*b^21*d^5 + 4224*B^4*a^18*
b^19*d^5 + 6720*B^4*a^20*b^17*d^5 + 3872*B^4*a^22*b^15*d^5 - 2816*B^4*a^24*b^13*d^5 - 5632*B^4*a^26*b^11*d^5 -
3136*B^4*a^28*b^9*d^5 - 560*B^4*a^30*b^7*d^5 + 32*B^4*a^32*b^5*d^5))/4 - ((-4*(9*B^2*b^9 + 42*B^2*a^2*b^7 + 4
9*B^2*a^4*b^5)*(a^13*d^2 + a^5*b^8*d^2 + 4*a^7*b^6*d^2 + 6*a^9*b^4*d^2 + 4*a^11*b^2*d^2))^(1/2)*((((tan(c + d*
x)^(1/2)*(1152*B^2*a^15*b^26*d^7 + 13440*B^2*a^17*b^24*d^7 + 69056*B^2*a^19*b^22*d^7 + 202752*B^2*a^21*b^20*d^
7 + 372800*B^2*a^23*b^18*d^7 + 443136*B^2*a^25*b^16*d^7 + 337792*B^2*a^27*b^14*d^7 + 156160*B^2*a^29*b^12*d^7
+ 37632*B^2*a^31*b^10*d^7 + 3200*B^2*a^33*b^8*d^7 + 704*B^2*a^35*b^6*d^7 + 512*B^2*a^37*b^4*d^7 + 64*B^2*a^39*
b^2*d^7))/4 - ((-4*(9*B^2*b^9 + 42*B^2*a^2*b^7 + 49*B^2*a^4*b^5)*(a^13*d^2 + a^5*b^8*d^2 + 4*a^7*b^6*d^2 + 6*a
^9*b^4*d^2 + 4*a^11*b^2*d^2))^(1/2)*((tan(c + d*x)^(1/2)*(-4*(9*B^2*b^9 + 42*B^2*a^2*b^7 + 49*B^2*a^4*b^5)*(a^
13*d^2 + a^5*b^8*d^2 + 4*a^7*b^6*d^2 + 6*a^9*b^4*d^2 + 4*a^11*b^2*d^2))^(1/2)*(512*a^18*b^27*d^9 + 5120*a^20*b
^25*d^9 + 22528*a^22*b^23*d^9 + 56320*a^24*b^21*d^9 + 84480*a^26*b^19*d^9 + 67584*a^28*b^17*d^9 - 67584*a^32*b
^13*d^9 - 84480*a^34*b^11*d^9 - 56320*a^36*b^9*d^9 - 22528*a^38*b^7*d^9 - 5120*a^40*b^5*d^9 - 512*a^42*b^3*d^9
))/(16*(a^13*d^2 + a^5*b^8*d^2 + 4*a^7*b^6*d^2 + 6*a^9*b^4*d^2 + 4*a^11*b^2*d^2)) + 192*B*a^16*b^27*d^8 + 2176
*B*a^18*b^25*d^8 + 11072*B*a^20*b^23*d^8 + 33280*B*a^22*b^21*d^8 + 65280*B*a^24*b^19*d^8 + 86784*B*a^26*b^17*d
^8 + 77952*B*a^28*b^15*d^8 + 44544*B*a^30*b^13*d^8 + 12480*B*a^32*b^11*d^8 - 1920*B*a^34*b^9*d^8 - 3008*B*a^36
*b^7*d^8 - 1024*B*a^38*b^5*d^8 - 128*B*a^40*b^3*d^8))/(4*(a^13*d^2 + a^5*b^8*d^2 + 4*a^7*b^6*d^2 + 6*a^9*b^4*d
^2 + 4*a^11*b^2*d^2)))*(-4*(9*B^2*b^9 + 42*B^2*a^2*b^7 + 49*B^2*a^4*b^5)*(a^13*d^2 + a^5*b^8*d^2 + 4*a^7*b^6*d
^2 + 6*a^9*b^4*d^2 + 4*a^11*b^2*d^2))^(1/2))/(4*(a^13*d^2 + a^5*b^8*d^2 + 4*a^7*b^6*d^2 + 6*a^9*b^4*d^2 + 4*a^
11*b^2*d^2)) - 288*B^3*a^15*b^24*d^6 - 2112*B^3*a^17*b^22*d^6 - 5944*B^3*a^19*b^20*d^6 - 7416*B^3*a^21*b^18*d^
6 - 1632*B^3*a^23*b^16*d^6 + 6624*B^3*a^25*b^14*d^6 + 8496*B^3*a^27*b^12*d^6 + 4656*B^3*a^29*b^10*d^6 + 1344*B
^3*a^31*b^8*d^6 + 288*B^3*a^33*b^6*d^6 + 72*B^3*a^35*b^4*d^6 + 8*B^3*a^37*b^2*d^6))/(4*(a^13*d^2 + a^5*b^8*d^2
+ 4*a^7*b^6*d^2 + 6*a^9*b^4*d^2 + 4*a^11*b^2*d^2)))*(-4*(9*B^2*b^9 + 42*B^2*a^2*b^7 + 49*B^2*a^4*b^5)*(a^13*d
^2 + a^5*b^8*d^2 + 4*a^7*b^6*d^2 + 6*a^9*b^4*d^2 + 4*a^11*b^2*d^2))^(1/2))/(a^13*d^2 + a^5*b^8*d^2 + 4*a^7*b^6
*d^2 + 6*a^9*b^4*d^2 + 4*a^11*b^2*d^2) + 144*B^5*a^14*b^21*d^4 + 1296*B^5*a^16*b^19*d^4 + 4880*B^5*a^18*b^17*d
^4 + 10000*B^5*a^20*b^15*d^4 + 12080*B^5*a^22*b^13*d^4 + 8624*B^5*a^24*b^11*d^4 + 3376*B^5*a^26*b^9*d^4 + 560*
B^5*a^28*b^7*d^4))*(-4*(9*B^2*b^9 + 42*B^2*a^2*b^7 + 49*B^2*a^4*b^5)*(a^13*d^2 + a^5*b^8*d^2 + 4*a^7*b^6*d^2 +
6*a^9*b^4*d^2 + 4*a^11*b^2*d^2))^(1/2)*1i)/(2*(a^13*d^2 + a^5*b^8*d^2 + 4*a^7*b^6*d^2 + 6*a^9*b^4*d^2 + 4*a^1
1*b^2*d^2)) - (atan(((25*b^11 + 90*a^2*b^9 + 81*a^4*b^7)^2*(a^2*b*tan(c + d*x)^(1/2)*(-(25*A^2*b^11 + 90*A^2*a
^2*b^9 + 81*A^2*a^4*b^7)*(a^15*d^2 + a^7*b^8*d^2 + 4*a^9*b^6*d^2 + 6*a^11*b^4*d^2 + 4*a^13*b^2*d^2))^(5/2)*2i
- b^3*tan(c + d*x)^(1/2)*(-(25*A^2*b^11 + 90*A^2*a^2*b^9 + 81*A^2*a^4*b^7)*(a^15*d^2 + a^7*b^8*d^2 + 4*a^9*b^6
*d^2 + 6*a^11*b^4*d^2 + 4*a^13*b^2*d^2))^(5/2)*2i - A^2*a^29*d^2*tan(c + d*x)^(1/2)*(-(25*A^2*b^11 + 90*A^2*a^
2*b^9 + 81*A^2*a^4*b^7)*(a^15*d^2 + a^7*b^8*d^2 + 4*a^9*b^6*d^2 + 6*a^11*b^4*d^2 + 4*a^13*b^2*d^2))^(3/2)*1i +
A^2*a^9*b^20*d^2*tan(c + d*x)^(1/2)*(-(25*A^2*b^11 + 90*A^2*a^2*b^9 + 81*A^2*a^4*b^7)*(a^15*d^2 + a^7*b^8*d^2
+ 4*a^9*b^6*d^2 + 6*a^11*b^4*d^2 + 4*a^13*b^2*d^2))^(3/2)*50i + A^2*a^11*b^18*d^2*tan(c + d*x)^(1/2)*(-(25*A^
2*b^11 + 90*A^2*a^2*b^9 + 81*A^2*a^4*b^7)*(a^15*d^2 + a^7*b^8*d^2 + 4*a^9*b^6*d^2 + 6*a^11*b^4*d^2 + 4*a^13*b^
2*d^2))^(3/2)*380i + A^2*a^13*b^16*d^2*tan(c + d*x)^(1/2)*(-(25*A^2*b^11 + 90*A^2*a^2*b^9 + 81*A^2*a^4*b^7)*(a
^15*d^2 + a^7*b^8*d^2 + 4*a^9*b^6*d^2 + 6*a^11*b^4*d^2 + 4*a^13*b^2*d^2))^(3/2)*1182i + A^2*a^15*b^14*d^2*tan(
c + d*x)^(1/2)*(-(25*A^2*b^11 + 90*A^2*a^2*b^9 + 81*A^2*a^4*b^7)*(a^15*d^2 + a^7*b^8*d^2 + 4*a^9*b^6*d^2 + 6*a
^11*b^4*d^2 + 4*a^13*b^2*d^2))^(3/2)*1913i + A^2*a^17*b^12*d^2*tan(c + d*x)^(1/2)*(-(25*A^2*b^11 + 90*A^2*a^2*
b^9 + 81*A^2*a^4*b^7)*(a^15*d^2 + a^7*b^8*d^2 + 4*a^9*b^6*d^2 + 6*a^11*b^4*d^2 + 4*a^13*b^2*d^2))^(3/2)*1699i
+ A^2*a^19*b^10*d^2*tan(c + d*x)^(1/2)*(-(25*A^2*b^11 + 90*A^2*a^2*b^9 + 81*A^2*a^4*b^7)*(a^15*d^2 + a^7*b^8*d
^2 + 4*a^9*b^6*d^2 + 6*a^11*b^4*d^2 + 4*a^13*b^2*d^2))^(3/2)*805i + A^2*a^21*b^8*d^2*tan(c + d*x)^(1/2)*(-(25*
A^2*b^11 + 90*A^2*a^2*b^9 + 81*A^2*a^4*b^7)*(a^15*d^2 + a^7*b^8*d^2 + 4*a^9*b^6*d^2 + 6*a^11*b^4*d^2 + 4*a^13*
b^2*d^2))^(3/2)*199i + A^2*a^23*b^6*d^2*tan(c + d*x)^(1/2)*(-(25*A^2*b^11 + 90*A^2*a^2*b^9 + 81*A^2*a^4*b^7)*(
a^15*d^2 + a^7*b^8*d^2 + 4*a^9*b^6*d^2 + 6*a^11*b^4*d^2 + 4*a^13*b^2*d^2))^(3/2)*43i + A^2*a^25*b^4*d^2*tan(c
+ d*x)^(1/2)*(-(25*A^2*b^11 + 90*A^2*a^2*b^9 + 81*A^2*a^4*b^7)*(a^15*d^2 + a^7*b^8*d^2 + 4*a^9*b^6*d^2 + 6*a^1
1*b^4*d^2 + 4*a^13*b^2*d^2))^(3/2)*7i - A^2*a^27*b^2*d^2*tan(c + d*x)^(1/2)*(-(25*A^2*b^11 + 90*A^2*a^2*b^9 +
81*A^2*a^4*b^7)*(a^15*d^2 + a^7*b^8*d^2 + 4*a^9*b^6*d^2 + 6*a^11*b^4*d^2 + 4*a^13*b^2*d^2))^(3/2)*5i + A^4*a^2
2*b^33*d^4*tan(c + d*x)^(1/2)*(-(25*A^2*b^11 + 90*A^2*a^2*b^9 + 81*A^2*a^4*b^7)*(a^15*d^2 + a^7*b^8*d^2 + 4*a^
9*b^6*d^2 + 6*a^11*b^4*d^2 + 4*a^13*b^2*d^2))^(1/2)*25i + A^4*a^24*b^31*d^4*tan(c + d*x)^(1/2)*(-(25*A^2*b^11
+ 90*A^2*a^2*b^9 + 81*A^2*a^4*b^7)*(a^15*d^2 + a^7*b^8*d^2 + 4*a^9*b^6*d^2 + 6*a^11*b^4*d^2 + 4*a^13*b^2*d^2))
^(1/2)*315i + A^4*a^26*b^29*d^4*tan(c + d*x)^(1/2)*(-(25*A^2*b^11 + 90*A^2*a^2*b^9 + 81*A^2*a^4*b^7)*(a^15*d^2
+ a^7*b^8*d^2 + 4*a^9*b^6*d^2 + 6*a^11*b^4*d^2 + 4*a^13*b^2*d^2))^(1/2)*1766i + A^4*a^28*b^27*d^4*tan(c + d*x
)^(1/2)*(-(25*A^2*b^11 + 90*A^2*a^2*b^9 + 81*A^2*a^4*b^7)*(a^15*d^2 + a^7*b^8*d^2 + 4*a^9*b^6*d^2 + 6*a^11*b^4
*d^2 + 4*a^13*b^2*d^2))^(1/2)*5752i + A^4*a^30*b^25*d^4*tan(c + d*x)^(1/2)*(-(25*A^2*b^11 + 90*A^2*a^2*b^9 + 8
1*A^2*a^4*b^7)*(a^15*d^2 + a^7*b^8*d^2 + 4*a^9*b^6*d^2 + 6*a^11*b^4*d^2 + 4*a^13*b^2*d^2))^(1/2)*11811i + A^4*
a^32*b^23*d^4*tan(c + d*x)^(1/2)*(-(25*A^2*b^11 + 90*A^2*a^2*b^9 + 81*A^2*a^4*b^7)*(a^15*d^2 + a^7*b^8*d^2 + 4
*a^9*b^6*d^2 + 6*a^11*b^4*d^2 + 4*a^13*b^2*d^2))^(1/2)*15093i + A^4*a^34*b^21*d^4*tan(c + d*x)^(1/2)*(-(25*A^2
*b^11 + 90*A^2*a^2*b^9 + 81*A^2*a^4*b^7)*(a^15*d^2 + a^7*b^8*d^2 + 4*a^9*b^6*d^2 + 6*a^11*b^4*d^2 + 4*a^13*b^2
*d^2))^(1/2)*9580i - A^4*a^36*b^19*d^4*tan(c + d*x)^(1/2)*(-(25*A^2*b^11 + 90*A^2*a^2*b^9 + 81*A^2*a^4*b^7)*(a
^15*d^2 + a^7*b^8*d^2 + 4*a^9*b^6*d^2 + 6*a^11*b^4*d^2 + 4*a^13*b^2*d^2))^(1/2)*3618i - A^4*a^38*b^17*d^4*tan(
c + d*x)^(1/2)*(-(25*A^2*b^11 + 90*A^2*a^2*b^9 + 81*A^2*a^4*b^7)*(a^15*d^2 + a^7*b^8*d^2 + 4*a^9*b^6*d^2 + 6*a
^11*b^4*d^2 + 4*a^13*b^2*d^2))^(1/2)*14961i - A^4*a^40*b^15*d^4*tan(c + d*x)^(1/2)*(-(25*A^2*b^11 + 90*A^2*a^2
*b^9 + 81*A^2*a^4*b^7)*(a^15*d^2 + a^7*b^8*d^2 + 4*a^9*b^6*d^2 + 6*a^11*b^4*d^2 + 4*a^13*b^2*d^2))^(1/2)*16763
i - A^4*a^42*b^13*d^4*tan(c + d*x)^(1/2)*(-(25*A^2*b^11 + 90*A^2*a^2*b^9 + 81*A^2*a^4*b^7)*(a^15*d^2 + a^7*b^8
*d^2 + 4*a^9*b^6*d^2 + 6*a^11*b^4*d^2 + 4*a^13*b^2*d^2))^(1/2)*11034i - A^4*a^44*b^11*d^4*tan(c + d*x)^(1/2)*(
-(25*A^2*b^11 + 90*A^2*a^2*b^9 + 81*A^2*a^4*b^7)*(a^15*d^2 + a^7*b^8*d^2 + 4*a^9*b^6*d^2 + 6*a^11*b^4*d^2 + 4*
a^13*b^2*d^2))^(1/2)*4660i - A^4*a^46*b^9*d^4*tan(c + d*x)^(1/2)*(-(25*A^2*b^11 + 90*A^2*a^2*b^9 + 81*A^2*a^4*
b^7)*(a^15*d^2 + a^7*b^8*d^2 + 4*a^9*b^6*d^2 + 6*a^11*b^4*d^2 + 4*a^13*b^2*d^2))^(1/2)*1259i - A^4*a^48*b^7*d^
4*tan(c + d*x)^(1/2)*(-(25*A^2*b^11 + 90*A^2*a^2*b^9 + 81*A^2*a^4*b^7)*(a^15*d^2 + a^7*b^8*d^2 + 4*a^9*b^6*d^2
+ 6*a^11*b^4*d^2 + 4*a^13*b^2*d^2))^(1/2)*213i - A^4*a^50*b^5*d^4*tan(c + d*x)^(1/2)*(-(25*A^2*b^11 + 90*A^2*
a^2*b^9 + 81*A^2*a^4*b^7)*(a^15*d^2 + a^7*b^8*d^2 + 4*a^9*b^6*d^2 + 6*a^11*b^4*d^2 + 4*a^13*b^2*d^2))^(1/2)*24
i - A^4*a^52*b^3*d^4*tan(c + d*x)^(1/2)*(-(25*A^2*b^11 + 90*A^2*a^2*b^9 + 81*A^2*a^4*b^7)*(a^15*d^2 + a^7*b^8*
d^2 + 4*a^9*b^6*d^2 + 6*a^11*b^4*d^2 + 4*a^13*b^2*d^2))^(1/2)*2i))/(A*(25*A^2*b^11 + 90*A^2*a^2*b^9 + 81*A^2*a
^4*b^7)^2*(6250*a^18*b^50*d^5 + 118750*a^20*b^48*d^5 + 1046250*a^22*b^46*d^5 + 5670750*a^24*b^44*d^5 + 2114605
0*a^26*b^42*d^5 + 57459798*a^28*b^40*d^5 + 117509090*a^30*b^38*d^5 + 184103296*a^32*b^36*d^5 + 222697292*a^34*
b^34*d^5 + 207986852*a^36*b^32*d^5 + 149083724*a^38*b^30*d^5 + 81434252*a^40*b^28*d^5 + 34268140*a^42*b^26*d^5
+ 12144964*a^44*b^24*d^5 + 4501164*a^46*b^22*d^5 + 1864304*a^48*b^20*d^5 + 668074*a^50*b^18*d^5 + 170462*a^52
*b^16*d^5 + 33322*a^54*b^14*d^5 + 7462*a^56*b^12*d^5 + 1778*a^58*b^10*d^5 + 262*a^60*b^8*d^5 + 18*a^62*b^6*d^5
)))*(-4*(25*A^2*b^11 + 90*A^2*a^2*b^9 + 81*A^2*a^4*b^7)*(a^15*d^2 + a^7*b^8*d^2 + 4*a^9*b^6*d^2 + 6*a^11*b^4*d
^2 + 4*a^13*b^2*d^2))^(1/2)*1i)/(2*(a^15*d^2 + a^7*b^8*d^2 + 4*a^9*b^6*d^2 + 6*a^11*b^4*d^2 + 4*a^13*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)**(5/2)/(a+b*tan(d*x+c))**2,x)

[Out]

Timed out

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