\(\int \frac {\tan ^{\frac {7}{2}}(c+d x)}{(a+b \tan (c+d x))^3} \, dx\) [601]

Optimal result

Integrand size = 23, antiderivative size = 396 \[ \int \frac {\tan ^{\frac {7}{2}}(c+d x)}{(a+b \tan (c+d x))^3} \, dx=-\frac {(a+b) \left (a^2-4 a b+b^2\right ) \arctan \left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} \left (a^2+b^2\right )^3 d}+\frac {(a+b) \left (a^2-4 a b+b^2\right ) \arctan \left (1+\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} \left (a^2+b^2\right )^3 d}+\frac {a^{3/2} \left (3 a^4+6 a^2 b^2+35 b^4\right ) \arctan \left (\frac {\sqrt {b} \sqrt {\tan (c+d x)}}{\sqrt {a}}\right )}{4 b^{5/2} \left (a^2+b^2\right )^3 d}-\frac {(a-b) \left (a^2+4 a b+b^2\right ) \log \left (1-\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right )}{2 \sqrt {2} \left (a^2+b^2\right )^3 d}+\frac {(a-b) \left (a^2+4 a b+b^2\right ) \log \left (1+\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right )}{2 \sqrt {2} \left (a^2+b^2\right )^3 d}-\frac {a^2 \tan ^{\frac {3}{2}}(c+d x)}{2 b \left (a^2+b^2\right ) d (a+b \tan (c+d x))^2}-\frac {a^2 \left (3 a^2+11 b^2\right ) \sqrt {\tan (c+d x)}}{4 b^2 \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))} \]

[Out]

1/4*a^(3/2)*(3*a^4+6*a^2*b^2+35*b^4)*arctan(b^(1/2)*tan(d*x+c)^(1/2)/a^(1/2))/b^(5/2)/(a^2+b^2)^3/d+1/2*(a+b)*
(a^2-4*a*b+b^2)*arctan(-1+2^(1/2)*tan(d*x+c)^(1/2))/(a^2+b^2)^3/d*2^(1/2)+1/2*(a+b)*(a^2-4*a*b+b^2)*arctan(1+2
^(1/2)*tan(d*x+c)^(1/2))/(a^2+b^2)^3/d*2^(1/2)-1/4*(a-b)*(a^2+4*a*b+b^2)*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-b)*(a^2+4*a*b+b^2)*ln(1+2^(1/2)*tan(d*x+c)^(1/2)+tan(d*x+c))/(a^2+b^2)^3/d*2
^(1/2)-1/2*a^2*tan(d*x+c)^(3/2)/b/(a^2+b^2)/d/(a+b*tan(d*x+c))^2-1/4*a^2*(3*a^2+11*b^2)*tan(d*x+c)^(1/2)/b^2/(
a^2+b^2)^2/d/(a+b*tan(d*x+c))

Rubi [A] (verified)

Time = 0.98 (sec) , antiderivative size = 396, normalized size of antiderivative = 1.00, number of steps used = 16, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.565, Rules used = {3646, 3726, 3734, 3615, 1182, 1176, 631, 210, 1179, 642, 3715, 65, 211} \[ \int \frac {\tan ^{\frac {7}{2}}(c+d x)}{(a+b \tan (c+d x))^3} \, dx=-\frac {(a+b) \left (a^2-4 a b+b^2\right ) \arctan \left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} d \left (a^2+b^2\right )^3}+\frac {(a+b) \left (a^2-4 a b+b^2\right ) \arctan \left (\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{\sqrt {2} d \left (a^2+b^2\right )^3}-\frac {a^2 \tan ^{\frac {3}{2}}(c+d x)}{2 b d \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}-\frac {a^2 \left (3 a^2+11 b^2\right ) \sqrt {\tan (c+d x)}}{4 b^2 d \left (a^2+b^2\right )^2 (a+b \tan (c+d x))}-\frac {(a-b) \left (a^2+4 a b+b^2\right ) \log \left (\tan (c+d x)-\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{2 \sqrt {2} d \left (a^2+b^2\right )^3}+\frac {(a-b) \left (a^2+4 a b+b^2\right ) \log \left (\tan (c+d x)+\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{2 \sqrt {2} d \left (a^2+b^2\right )^3}+\frac {a^{3/2} \left (3 a^4+6 a^2 b^2+35 b^4\right ) \arctan \left (\frac {\sqrt {b} \sqrt {\tan (c+d x)}}{\sqrt {a}}\right )}{4 b^{5/2} d \left (a^2+b^2\right )^3} \]

[In]

Int[Tan[c + d*x]^(7/2)/(a + b*Tan[c + d*x])^3,x]

[Out]

-(((a + b)*(a^2 - 4*a*b + b^2)*ArcTan[1 - Sqrt[2]*Sqrt[Tan[c + d*x]]])/(Sqrt[2]*(a^2 + b^2)^3*d)) + ((a + b)*(
a^2 - 4*a*b + b^2)*ArcTan[1 + Sqrt[2]*Sqrt[Tan[c + d*x]]])/(Sqrt[2]*(a^2 + b^2)^3*d) + (a^(3/2)*(3*a^4 + 6*a^2
*b^2 + 35*b^4)*ArcTan[(Sqrt[b]*Sqrt[Tan[c + d*x]])/Sqrt[a]])/(4*b^(5/2)*(a^2 + b^2)^3*d) - ((a - b)*(a^2 + 4*a
*b + b^2)*Log[1 - Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]])/(2*Sqrt[2]*(a^2 + b^2)^3*d) + ((a - b)*(a^2 + 4*
a*b + b^2)*Log[1 + Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]])/(2*Sqrt[2]*(a^2 + b^2)^3*d) - (a^2*Tan[c + d*x]
^(3/2))/(2*b*(a^2 + b^2)*d*(a + b*Tan[c + d*x])^2) - (a^2*(3*a^2 + 11*b^2)*Sqrt[Tan[c + d*x]])/(4*b^2*(a^2 + b
^2)^2*d*(a + b*Tan[c + d*x]))

Rule 65

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 210

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

Rule 211

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

Rule 631

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 642

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 1176

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 1179

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 1182

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 3615

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 3646

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Si
mp[(b*c - a*d)^2*(a + b*Tan[e + f*x])^(m - 2)*((c + d*Tan[e + f*x])^(n + 1)/(d*f*(n + 1)*(c^2 + d^2))), x] - D
ist[1/(d*(n + 1)*(c^2 + d^2)), Int[(a + b*Tan[e + f*x])^(m - 3)*(c + d*Tan[e + f*x])^(n + 1)*Simp[a^2*d*(b*d*(
m - 2) - a*c*(n + 1)) + b*(b*c - 2*a*d)*(b*c*(m - 2) + a*d*(n + 1)) - d*(n + 1)*(3*a^2*b*c - b^3*c - a^3*d + 3
*a*b^2*d)*Tan[e + f*x] - b*(a*d*(2*b*c - a*d)*(m + n - 1) - b^2*(c^2*(m - 2) - d^2*(n + 1)))*Tan[e + f*x]^2, x
], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && Gt
Q[m, 2] && LtQ[n, -1] && IntegerQ[2*m]

Rule 3715

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 3726

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*d^2 + c*(c*C - B*d))*(a + b*Ta
n[e + f*x])^m*((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)), I
nt[(a + b*Tan[e + f*x])^(m - 1)*(c + d*Tan[e + f*x])^(n + 1)*Simp[A*d*(b*d*m - a*c*(n + 1)) + (c*C - B*d)*(b*c
*m + a*d*(n + 1)) - d*(n + 1)*((A - C)*(b*c - a*d) + B*(a*c + b*d))*Tan[e + f*x] - b*(d*(B*c - A*d)*(m + n + 1
) - C*(c^2*m - d^2*(n + 1)))*Tan[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C}, x] && NeQ[b*c -
a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && GtQ[m, 0] && LtQ[n, -1]

Rule 3734

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*} \text {integral}& = -\frac {a^2 \tan ^{\frac {3}{2}}(c+d x)}{2 b \left (a^2+b^2\right ) d (a+b \tan (c+d x))^2}+\frac {\int \frac {\sqrt {\tan (c+d x)} \left (\frac {3 a^2}{2}-2 a b \tan (c+d x)+\frac {1}{2} \left (3 a^2+4 b^2\right ) \tan ^2(c+d x)\right )}{(a+b \tan (c+d x))^2} \, dx}{2 b \left (a^2+b^2\right )} \\ & = -\frac {a^2 \tan ^{\frac {3}{2}}(c+d x)}{2 b \left (a^2+b^2\right ) d (a+b \tan (c+d x))^2}-\frac {a^2 \left (3 a^2+11 b^2\right ) \sqrt {\tan (c+d x)}}{4 b^2 \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))}+\frac {\int \frac {\frac {1}{4} a^2 \left (3 a^2+11 b^2\right )-4 a b^3 \tan (c+d x)+\frac {1}{4} \left (3 a^4+3 a^2 b^2+8 b^4\right ) \tan ^2(c+d x)}{\sqrt {\tan (c+d x)} (a+b \tan (c+d x))} \, dx}{2 b^2 \left (a^2+b^2\right )^2} \\ & = -\frac {a^2 \tan ^{\frac {3}{2}}(c+d x)}{2 b \left (a^2+b^2\right ) d (a+b \tan (c+d x))^2}-\frac {a^2 \left (3 a^2+11 b^2\right ) \sqrt {\tan (c+d x)}}{4 b^2 \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))}+\frac {\int \frac {2 a b^2 \left (a^2-3 b^2\right )-2 b^3 \left (3 a^2-b^2\right ) \tan (c+d x)}{\sqrt {\tan (c+d x)}} \, dx}{2 b^2 \left (a^2+b^2\right )^3}+\frac {\left (a^2 \left (3 a^4+6 a^2 b^2+35 b^4\right )\right ) \int \frac {1+\tan ^2(c+d x)}{\sqrt {\tan (c+d x)} (a+b \tan (c+d x))} \, dx}{8 b^2 \left (a^2+b^2\right )^3} \\ & = -\frac {a^2 \tan ^{\frac {3}{2}}(c+d x)}{2 b \left (a^2+b^2\right ) d (a+b \tan (c+d x))^2}-\frac {a^2 \left (3 a^2+11 b^2\right ) \sqrt {\tan (c+d x)}}{4 b^2 \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))}+\frac {\text {Subst}\left (\int \frac {2 a b^2 \left (a^2-3 b^2\right )-2 b^3 \left (3 a^2-b^2\right ) x^2}{1+x^4} \, dx,x,\sqrt {\tan (c+d x)}\right )}{b^2 \left (a^2+b^2\right )^3 d}+\frac {\left (a^2 \left (3 a^4+6 a^2 b^2+35 b^4\right )\right ) \text {Subst}\left (\int \frac {1}{\sqrt {x} (a+b x)} \, dx,x,\tan (c+d x)\right )}{8 b^2 \left (a^2+b^2\right )^3 d} \\ & = -\frac {a^2 \tan ^{\frac {3}{2}}(c+d x)}{2 b \left (a^2+b^2\right ) d (a+b \tan (c+d x))^2}-\frac {a^2 \left (3 a^2+11 b^2\right ) \sqrt {\tan (c+d x)}}{4 b^2 \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))}+\frac {\left ((a+b) \left (a^2-4 a b+b^2\right )\right ) \text {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-b) \left (a^2+4 a b+b^2\right )\right ) \text {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^2 \left (3 a^4+6 a^2 b^2+35 b^4\right )\right ) \text {Subst}\left (\int \frac {1}{a+b x^2} \, dx,x,\sqrt {\tan (c+d x)}\right )}{4 b^2 \left (a^2+b^2\right )^3 d} \\ & = \frac {a^{3/2} \left (3 a^4+6 a^2 b^2+35 b^4\right ) \arctan \left (\frac {\sqrt {b} \sqrt {\tan (c+d x)}}{\sqrt {a}}\right )}{4 b^{5/2} \left (a^2+b^2\right )^3 d}-\frac {a^2 \tan ^{\frac {3}{2}}(c+d x)}{2 b \left (a^2+b^2\right ) d (a+b \tan (c+d x))^2}-\frac {a^2 \left (3 a^2+11 b^2\right ) \sqrt {\tan (c+d x)}}{4 b^2 \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))}+\frac {\left ((a+b) \left (a^2-4 a b+b^2\right )\right ) \text {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+b) \left (a^2-4 a b+b^2\right )\right ) \text {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-b) \left (a^2+4 a b+b^2\right )\right ) \text {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-b) \left (a^2+4 a b+b^2\right )\right ) \text {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 {a^{3/2} \left (3 a^4+6 a^2 b^2+35 b^4\right ) \arctan \left (\frac {\sqrt {b} \sqrt {\tan (c+d x)}}{\sqrt {a}}\right )}{4 b^{5/2} \left (a^2+b^2\right )^3 d}-\frac {(a-b) \left (a^2+4 a b+b^2\right ) \log \left (1-\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right )}{2 \sqrt {2} \left (a^2+b^2\right )^3 d}+\frac {(a-b) \left (a^2+4 a b+b^2\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^2 \tan ^{\frac {3}{2}}(c+d x)}{2 b \left (a^2+b^2\right ) d (a+b \tan (c+d x))^2}-\frac {a^2 \left (3 a^2+11 b^2\right ) \sqrt {\tan (c+d x)}}{4 b^2 \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))}+\frac {\left ((a+b) \left (a^2-4 a b+b^2\right )\right ) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} \left (a^2+b^2\right )^3 d}-\frac {\left ((a+b) \left (a^2-4 a b+b^2\right )\right ) \text {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 {(a+b) \left (a^2-4 a b+b^2\right ) \arctan \left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} \left (a^2+b^2\right )^3 d}+\frac {(a+b) \left (a^2-4 a b+b^2\right ) \arctan \left (1+\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} \left (a^2+b^2\right )^3 d}+\frac {a^{3/2} \left (3 a^4+6 a^2 b^2+35 b^4\right ) \arctan \left (\frac {\sqrt {b} \sqrt {\tan (c+d x)}}{\sqrt {a}}\right )}{4 b^{5/2} \left (a^2+b^2\right )^3 d}-\frac {(a-b) \left (a^2+4 a b+b^2\right ) \log \left (1-\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right )}{2 \sqrt {2} \left (a^2+b^2\right )^3 d}+\frac {(a-b) \left (a^2+4 a b+b^2\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^2 \tan ^{\frac {3}{2}}(c+d x)}{2 b \left (a^2+b^2\right ) d (a+b \tan (c+d x))^2}-\frac {a^2 \left (3 a^2+11 b^2\right ) \sqrt {\tan (c+d x)}}{4 b^2 \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))} \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 5.27 (sec) , antiderivative size = 373, normalized size of antiderivative = 0.94 \[ \int \frac {\tan ^{\frac {7}{2}}(c+d x)}{(a+b \tan (c+d x))^3} \, dx=\frac {2 b^{9/2} \tan ^{\frac {9}{2}}(c+d x)-2 b^{7/2} \tan ^{\frac {7}{2}}(c+d x) (a+b \tan (c+d x))-\frac {a (a+b \tan (c+d x)) \left (2 \sqrt {b} \left (a^2+b^2\right )^2 \left (3 a^2+4 b^2\right ) \sqrt {\tan (c+d x)}-\sqrt {b} \left (a^2+b^2\right ) \left (3 a^4+3 a^2 b^2+8 b^4\right ) \sqrt {\tan (c+d x)}+2 a b^{3/2} \left (a^2+b^2\right )^2 \tan ^{\frac {3}{2}}(c+d x)-2 b^{5/2} \left (a^2+b^2\right )^2 \tan ^{\frac {5}{2}}(c+d x)-\left (-4 \sqrt [4]{-1} (a+i b)^3 b^{5/2} \arctan \left ((-1)^{3/4} \sqrt {\tan (c+d x)}\right )+a^{3/2} \left (3 a^4+6 a^2 b^2+35 b^4\right ) \arctan \left (\frac {\sqrt {b} \sqrt {\tan (c+d x)}}{\sqrt {a}}\right )-4 \sqrt [4]{-1} (a-i b)^3 b^{5/2} \text {arctanh}\left ((-1)^{3/4} \sqrt {\tan (c+d x)}\right )\right ) (a+b \tan (c+d x))\right )}{\left (a^2+b^2\right )^2}}{4 a b^{5/2} \left (a^2+b^2\right ) d (a+b \tan (c+d x))^2} \]

[In]

Integrate[Tan[c + d*x]^(7/2)/(a + b*Tan[c + d*x])^3,x]

[Out]

(2*b^(9/2)*Tan[c + d*x]^(9/2) - 2*b^(7/2)*Tan[c + d*x]^(7/2)*(a + b*Tan[c + d*x]) - (a*(a + b*Tan[c + d*x])*(2
*Sqrt[b]*(a^2 + b^2)^2*(3*a^2 + 4*b^2)*Sqrt[Tan[c + d*x]] - Sqrt[b]*(a^2 + b^2)*(3*a^4 + 3*a^2*b^2 + 8*b^4)*Sq
rt[Tan[c + d*x]] + 2*a*b^(3/2)*(a^2 + b^2)^2*Tan[c + d*x]^(3/2) - 2*b^(5/2)*(a^2 + b^2)^2*Tan[c + d*x]^(5/2) -
(-4*(-1)^(1/4)*(a + I*b)^3*b^(5/2)*ArcTan[(-1)^(3/4)*Sqrt[Tan[c + d*x]]] + a^(3/2)*(3*a^4 + 6*a^2*b^2 + 35*b^
4)*ArcTan[(Sqrt[b]*Sqrt[Tan[c + d*x]])/Sqrt[a]] - 4*(-1)^(1/4)*(a - I*b)^3*b^(5/2)*ArcTanh[(-1)^(3/4)*Sqrt[Tan
[c + d*x]]])*(a + b*Tan[c + d*x])))/(a^2 + b^2)^2)/(4*a*b^(5/2)*(a^2 + b^2)*d*(a + b*Tan[c + d*x])^2)

Maple [A] (verified)

Time = 0.49 (sec) , antiderivative size = 347, normalized size of antiderivative = 0.88

method	result	size
derivativedivides	\(\frac {\frac {\frac {\left (a^{3}-3 a \,b^{2}\right ) \sqrt {2}\, \left (\ln \left (\frac {1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}{1-\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}\right )+2 \arctan \left (1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )+2 \arctan \left (-1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )\right )}{4}+\frac {\left (-3 a^{2} b +b^{3}\right ) \sqrt {2}\, \left (\ln \left (\frac {1-\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}{1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}\right )+2 \arctan \left (1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )+2 \arctan \left (-1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )\right )}{4}}{\left (a^{2}+b^{2}\right )^{3}}+\frac {2 a^{2} \left (\frac {-\frac {\left (5 a^{4}+18 a^{2} b^{2}+13 b^{4}\right ) \left (\tan ^{\frac {3}{2}}\left (d x +c \right )\right )}{8 b}-\frac {a \left (3 a^{4}+14 a^{2} b^{2}+11 b^{4}\right ) \left (\sqrt {\tan }\left (d x +c \right )\right )}{8 b^{2}}}{\left (a +b \tan \left (d x +c \right )\right )^{2}}+\frac {\left (3 a^{4}+6 a^{2} b^{2}+35 b^{4}\right ) \arctan \left (\frac {b \left (\sqrt {\tan }\left (d x +c \right )\right )}{\sqrt {a b}}\right )}{8 b^{2} \sqrt {a b}}\right )}{\left (a^{2}+b^{2}\right )^{3}}}{d}\)	\(347\)
default	\(\frac {\frac {\frac {\left (a^{3}-3 a \,b^{2}\right ) \sqrt {2}\, \left (\ln \left (\frac {1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}{1-\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}\right )+2 \arctan \left (1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )+2 \arctan \left (-1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )\right )}{4}+\frac {\left (-3 a^{2} b +b^{3}\right ) \sqrt {2}\, \left (\ln \left (\frac {1-\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}{1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}\right )+2 \arctan \left (1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )+2 \arctan \left (-1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )\right )}{4}}{\left (a^{2}+b^{2}\right )^{3}}+\frac {2 a^{2} \left (\frac {-\frac {\left (5 a^{4}+18 a^{2} b^{2}+13 b^{4}\right ) \left (\tan ^{\frac {3}{2}}\left (d x +c \right )\right )}{8 b}-\frac {a \left (3 a^{4}+14 a^{2} b^{2}+11 b^{4}\right ) \left (\sqrt {\tan }\left (d x +c \right )\right )}{8 b^{2}}}{\left (a +b \tan \left (d x +c \right )\right )^{2}}+\frac {\left (3 a^{4}+6 a^{2} b^{2}+35 b^{4}\right ) \arctan \left (\frac {b \left (\sqrt {\tan }\left (d x +c \right )\right )}{\sqrt {a b}}\right )}{8 b^{2} \sqrt {a b}}\right )}{\left (a^{2}+b^{2}\right )^{3}}}{d}\)	\(347\)

[In]

int(tan(d*x+c)^(7/2)/(a+b*tan(d*x+c))^3,x,method=_RETURNVERBOSE)

[Out]

1/d*(2/(a^2+b^2)^3*(1/8*(a^3-3*a*b^2)*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)))+2*arctan(1+2^(1/2)*tan(d*x+c)^(1/2))+2*arctan(-1+2^(1/2)*tan(d*x+c)^(1/2)))+1/8*(-3*a^2*b
+b^3)*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)))+2*arctan(1+
2^(1/2)*tan(d*x+c)^(1/2))+2*arctan(-1+2^(1/2)*tan(d*x+c)^(1/2))))+2*a^2/(a^2+b^2)^3*((-1/8*(5*a^4+18*a^2*b^2+1
3*b^4)/b*tan(d*x+c)^(3/2)-1/8*a*(3*a^4+14*a^2*b^2+11*b^4)/b^2*tan(d*x+c)^(1/2))/(a+b*tan(d*x+c))^2+1/8*(3*a^4+
6*a^2*b^2+35*b^4)/b^2/(a*b)^(1/2)*arctan(b*tan(d*x+c)^(1/2)/(a*b)^(1/2))))

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 4126 vs. \(2 (348) = 696\).

Time = 1.17 (sec) , antiderivative size = 8279, normalized size of antiderivative = 20.91 \[ \int \frac {\tan ^{\frac {7}{2}}(c+d x)}{(a+b \tan (c+d x))^3} \, dx=\text {Too large to display} \]

[In]

integrate(tan(d*x+c)^(7/2)/(a+b*tan(d*x+c))^3,x, algorithm="fricas")

[Out]

Too large to include

Sympy [F]

\[ \int \frac {\tan ^{\frac {7}{2}}(c+d x)}{(a+b \tan (c+d x))^3} \, dx=\int \frac {\tan ^{\frac {7}{2}}{\left (c + d x \right )}}{\left (a + b \tan {\left (c + d x \right )}\right )^{3}}\, dx \]

[In]

integrate(tan(d*x+c)**(7/2)/(a+b*tan(d*x+c))**3,x)

[Out]

Integral(tan(c + d*x)**(7/2)/(a + b*tan(c + d*x))**3, x)

Maxima [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 420, normalized size of antiderivative = 1.06 \[ \int \frac {\tan ^{\frac {7}{2}}(c+d x)}{(a+b \tan (c+d x))^3} \, dx=\frac {\frac {{\left (3 \, a^{6} + 6 \, a^{4} b^{2} + 35 \, a^{2} b^{4}\right )} \arctan \left (\frac {b \sqrt {\tan \left (d x + c\right )}}{\sqrt {a b}}\right )}{{\left (a^{6} b^{2} + 3 \, a^{4} b^{4} + 3 \, a^{2} b^{6} + b^{8}\right )} \sqrt {a b}} + \frac {2 \, \sqrt {2} {\left (a^{3} - 3 \, a^{2} b - 3 \, a b^{2} + b^{3}\right )} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + 2 \, \sqrt {\tan \left (d x + c\right )}\right )}\right ) + 2 \, \sqrt {2} {\left (a^{3} - 3 \, a^{2} b - 3 \, a b^{2} + b^{3}\right )} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - 2 \, \sqrt {\tan \left (d x + c\right )}\right )}\right ) + \sqrt {2} {\left (a^{3} + 3 \, a^{2} b - 3 \, a b^{2} - b^{3}\right )} \log \left (\sqrt {2} \sqrt {\tan \left (d x + c\right )} + \tan \left (d x + c\right ) + 1\right ) - \sqrt {2} {\left (a^{3} + 3 \, a^{2} b - 3 \, a b^{2} - b^{3}\right )} \log \left (-\sqrt {2} \sqrt {\tan \left (d x + c\right )} + \tan \left (d x + c\right ) + 1\right )}{a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}} - \frac {{\left (5 \, a^{4} b + 13 \, a^{2} b^{3}\right )} \tan \left (d x + c\right )^{\frac {3}{2}} + {\left (3 \, a^{5} + 11 \, a^{3} b^{2}\right )} \sqrt {\tan \left (d x + c\right )}}{a^{6} b^{2} + 2 \, a^{4} b^{4} + a^{2} b^{6} + {\left (a^{4} b^{4} + 2 \, a^{2} b^{6} + b^{8}\right )} \tan \left (d x + c\right )^{2} + 2 \, {\left (a^{5} b^{3} + 2 \, a^{3} b^{5} + a b^{7}\right )} \tan \left (d x + c\right )}}{4 \, d} \]

[In]

integrate(tan(d*x+c)^(7/2)/(a+b*tan(d*x+c))^3,x, algorithm="maxima")

[Out]

1/4*((3*a^6 + 6*a^4*b^2 + 35*a^2*b^4)*arctan(b*sqrt(tan(d*x + c))/sqrt(a*b))/((a^6*b^2 + 3*a^4*b^4 + 3*a^2*b^6
+ b^8)*sqrt(a*b)) + (2*sqrt(2)*(a^3 - 3*a^2*b - 3*a*b^2 + b^3)*arctan(1/2*sqrt(2)*(sqrt(2) + 2*sqrt(tan(d*x +
c)))) + 2*sqrt(2)*(a^3 - 3*a^2*b - 3*a*b^2 + b^3)*arctan(-1/2*sqrt(2)*(sqrt(2) - 2*sqrt(tan(d*x + c)))) + sqr
t(2)*(a^3 + 3*a^2*b - 3*a*b^2 - b^3)*log(sqrt(2)*sqrt(tan(d*x + c)) + tan(d*x + c) + 1) - sqrt(2)*(a^3 + 3*a^2
*b - 3*a*b^2 - 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) -
((5*a^4*b + 13*a^2*b^3)*tan(d*x + c)^(3/2) + (3*a^5 + 11*a^3*b^2)*sqrt(tan(d*x + c)))/(a^6*b^2 + 2*a^4*b^4 +
a^2*b^6 + (a^4*b^4 + 2*a^2*b^6 + b^8)*tan(d*x + c)^2 + 2*(a^5*b^3 + 2*a^3*b^5 + a*b^7)*tan(d*x + c)))/d

Giac [F(-1)]

Timed out. \[ \int \frac {\tan ^{\frac {7}{2}}(c+d x)}{(a+b \tan (c+d x))^3} \, dx=\text {Timed out} \]

[In]

integrate(tan(d*x+c)^(7/2)/(a+b*tan(d*x+c))^3,x, algorithm="giac")

[Out]

Timed out

Mupad [B] (verification not implemented)

Time = 11.35 (sec) , antiderivative size = 17912, normalized size of antiderivative = 45.23 \[ \int \frac {\tan ^{\frac {7}{2}}(c+d x)}{(a+b \tan (c+d x))^3} \, dx=\text {Too large to display} \]

[In]

int(tan(c + d*x)^(7/2)/(a + b*tan(c + d*x))^3,x)

[Out]

atan((((1i/(4*(b^6*d^2 - a^6*d^2 + a*b^5*d^2*6i + a^5*b*d^2*6i - 15*a^2*b^4*d^2 - a^3*b^3*d^2*20i + 15*a^4*b^2
*d^2)))^(1/2)*((1i/(4*(b^6*d^2 - a^6*d^2 + a*b^5*d^2*6i + a^5*b*d^2*6i - 15*a^2*b^4*d^2 - a^3*b^3*d^2*20i + 15
*a^4*b^2*d^2)))^(1/2)*((1i/(4*(b^6*d^2 - a^6*d^2 + a*b^5*d^2*6i + a^5*b*d^2*6i - 15*a^2*b^4*d^2 - a^3*b^3*d^2*
20i + 15*a^4*b^2*d^2)))^(1/2)*((1600*a^2*b^23*d^4 + 12864*a^4*b^21*d^4 + 45312*a^6*b^19*d^4 + 91392*a^8*b^17*d
^4 + 115584*a^10*b^15*d^4 + 94080*a^12*b^13*d^4 + 48384*a^14*b^11*d^4 + 14592*a^16*b^9*d^4 + 2112*a^18*b^7*d^4
+ 64*a^20*b^5*d^4)/(b^19*d^5 + 8*a^2*b^17*d^5 + 28*a^4*b^15*d^5 + 56*a^6*b^13*d^5 + 70*a^8*b^11*d^5 + 56*a^10
*b^9*d^5 + 28*a^12*b^7*d^5 + 8*a^14*b^5*d^5 + a^16*b^3*d^5) + (tan(c + d*x)^(1/2)*(1i/(4*(b^6*d^2 - a^6*d^2 +
a*b^5*d^2*6i + a^5*b*d^2*6i - 15*a^2*b^4*d^2 - a^3*b^3*d^2*20i + 15*a^4*b^2*d^2)))^(1/2)*(512*b^28*d^4 + 4608*
a^2*b^26*d^4 + 17920*a^4*b^24*d^4 + 38400*a^6*b^22*d^4 + 46080*a^8*b^20*d^4 + 21504*a^10*b^18*d^4 - 21504*a^12
*b^16*d^4 - 46080*a^14*b^14*d^4 - 38400*a^16*b^12*d^4 - 17920*a^18*b^10*d^4 - 4608*a^20*b^8*d^4 - 512*a^22*b^6
*d^4))/(b^19*d^4 + 8*a^2*b^17*d^4 + 28*a^4*b^15*d^4 + 56*a^6*b^13*d^4 + 70*a^8*b^11*d^4 + 56*a^10*b^9*d^4 + 28
*a^12*b^7*d^4 + 8*a^14*b^5*d^4 + a^16*b^3*d^4)) - (tan(c + d*x)^(1/2)*(1472*a*b^21*d^2 + 72*a^21*b*d^2 + 1024*
a^3*b^19*d^2 + 1352*a^5*b^17*d^2 + 28224*a^7*b^15*d^2 + 70240*a^9*b^13*d^2 + 72640*a^11*b^11*d^2 + 39088*a^13*
b^9*d^2 + 13248*a^15*b^7*d^2 + 3488*a^17*b^5*d^2 + 576*a^19*b^3*d^2))/(b^19*d^4 + 8*a^2*b^17*d^4 + 28*a^4*b^15
*d^4 + 56*a^6*b^13*d^4 + 70*a^8*b^11*d^4 + 56*a^10*b^9*d^4 + 28*a^12*b^7*d^4 + 8*a^14*b^5*d^4 + a^16*b^3*d^4))
+ (32*a*b^18*d^2 - 18*a^19*d^2 - 6528*a^3*b^16*d^2 + 2758*a^5*b^14*d^2 + 26482*a^7*b^12*d^2 + 21582*a^9*b^10*
d^2 + 7594*a^11*b^8*d^2 + 3314*a^13*b^6*d^2 + 246*a^15*b^4*d^2 + 90*a^17*b^2*d^2)/(b^19*d^5 + 8*a^2*b^17*d^5 +
28*a^4*b^15*d^5 + 56*a^6*b^13*d^5 + 70*a^8*b^11*d^5 + 56*a^10*b^9*d^5 + 28*a^12*b^7*d^5 + 8*a^14*b^5*d^5 + a^
16*b^3*d^5)) + (tan(c + d*x)^(1/2)*(9*a^16 + 32*b^16 + 128*a^2*b^14 + 1417*a^4*b^12 - 6802*a^6*b^10 - 1017*a^8
*b^8 - 1020*a^10*b^6 + 39*a^12*b^4 - 18*a^14*b^2))/(b^19*d^4 + 8*a^2*b^17*d^4 + 28*a^4*b^15*d^4 + 56*a^6*b^13*
d^4 + 70*a^8*b^11*d^4 + 56*a^10*b^9*d^4 + 28*a^12*b^7*d^4 + 8*a^14*b^5*d^4 + a^16*b^3*d^4))*(1i/(4*(b^6*d^2 -
a^6*d^2 + a*b^5*d^2*6i + a^5*b*d^2*6i - 15*a^2*b^4*d^2 - a^3*b^3*d^2*20i + 15*a^4*b^2*d^2)))^(1/2)*1i - ((1i/(
4*(b^6*d^2 - a^6*d^2 + a*b^5*d^2*6i + a^5*b*d^2*6i - 15*a^2*b^4*d^2 - a^3*b^3*d^2*20i + 15*a^4*b^2*d^2)))^(1/2
)*((1i/(4*(b^6*d^2 - a^6*d^2 + a*b^5*d^2*6i + a^5*b*d^2*6i - 15*a^2*b^4*d^2 - a^3*b^3*d^2*20i + 15*a^4*b^2*d^2
)))^(1/2)*((1i/(4*(b^6*d^2 - a^6*d^2 + a*b^5*d^2*6i + a^5*b*d^2*6i - 15*a^2*b^4*d^2 - a^3*b^3*d^2*20i + 15*a^4
*b^2*d^2)))^(1/2)*((1600*a^2*b^23*d^4 + 12864*a^4*b^21*d^4 + 45312*a^6*b^19*d^4 + 91392*a^8*b^17*d^4 + 115584*
a^10*b^15*d^4 + 94080*a^12*b^13*d^4 + 48384*a^14*b^11*d^4 + 14592*a^16*b^9*d^4 + 2112*a^18*b^7*d^4 + 64*a^20*b
^5*d^4)/(b^19*d^5 + 8*a^2*b^17*d^5 + 28*a^4*b^15*d^5 + 56*a^6*b^13*d^5 + 70*a^8*b^11*d^5 + 56*a^10*b^9*d^5 + 2
8*a^12*b^7*d^5 + 8*a^14*b^5*d^5 + a^16*b^3*d^5) - (tan(c + d*x)^(1/2)*(1i/(4*(b^6*d^2 - a^6*d^2 + a*b^5*d^2*6i
+ a^5*b*d^2*6i - 15*a^2*b^4*d^2 - a^3*b^3*d^2*20i + 15*a^4*b^2*d^2)))^(1/2)*(512*b^28*d^4 + 4608*a^2*b^26*d^4
+ 17920*a^4*b^24*d^4 + 38400*a^6*b^22*d^4 + 46080*a^8*b^20*d^4 + 21504*a^10*b^18*d^4 - 21504*a^12*b^16*d^4 -
46080*a^14*b^14*d^4 - 38400*a^16*b^12*d^4 - 17920*a^18*b^10*d^4 - 4608*a^20*b^8*d^4 - 512*a^22*b^6*d^4))/(b^19
*d^4 + 8*a^2*b^17*d^4 + 28*a^4*b^15*d^4 + 56*a^6*b^13*d^4 + 70*a^8*b^11*d^4 + 56*a^10*b^9*d^4 + 28*a^12*b^7*d^
4 + 8*a^14*b^5*d^4 + a^16*b^3*d^4)) + (tan(c + d*x)^(1/2)*(1472*a*b^21*d^2 + 72*a^21*b*d^2 + 1024*a^3*b^19*d^2
+ 1352*a^5*b^17*d^2 + 28224*a^7*b^15*d^2 + 70240*a^9*b^13*d^2 + 72640*a^11*b^11*d^2 + 39088*a^13*b^9*d^2 + 13
248*a^15*b^7*d^2 + 3488*a^17*b^5*d^2 + 576*a^19*b^3*d^2))/(b^19*d^4 + 8*a^2*b^17*d^4 + 28*a^4*b^15*d^4 + 56*a^
6*b^13*d^4 + 70*a^8*b^11*d^4 + 56*a^10*b^9*d^4 + 28*a^12*b^7*d^4 + 8*a^14*b^5*d^4 + a^16*b^3*d^4)) + (32*a*b^1
8*d^2 - 18*a^19*d^2 - 6528*a^3*b^16*d^2 + 2758*a^5*b^14*d^2 + 26482*a^7*b^12*d^2 + 21582*a^9*b^10*d^2 + 7594*a
^11*b^8*d^2 + 3314*a^13*b^6*d^2 + 246*a^15*b^4*d^2 + 90*a^17*b^2*d^2)/(b^19*d^5 + 8*a^2*b^17*d^5 + 28*a^4*b^15
*d^5 + 56*a^6*b^13*d^5 + 70*a^8*b^11*d^5 + 56*a^10*b^9*d^5 + 28*a^12*b^7*d^5 + 8*a^14*b^5*d^5 + a^16*b^3*d^5))
- (tan(c + d*x)^(1/2)*(9*a^16 + 32*b^16 + 128*a^2*b^14 + 1417*a^4*b^12 - 6802*a^6*b^10 - 1017*a^8*b^8 - 1020*
a^10*b^6 + 39*a^12*b^4 - 18*a^14*b^2))/(b^19*d^4 + 8*a^2*b^17*d^4 + 28*a^4*b^15*d^4 + 56*a^6*b^13*d^4 + 70*a^8
*b^11*d^4 + 56*a^10*b^9*d^4 + 28*a^12*b^7*d^4 + 8*a^14*b^5*d^4 + a^16*b^3*d^4))*(1i/(4*(b^6*d^2 - a^6*d^2 + a*
b^5*d^2*6i + a^5*b*d^2*6i - 15*a^2*b^4*d^2 - a^3*b^3*d^2*20i + 15*a^4*b^2*d^2)))^(1/2)*1i)/(((1i/(4*(b^6*d^2 -
a^6*d^2 + a*b^5*d^2*6i + a^5*b*d^2*6i - 15*a^2*b^4*d^2 - a^3*b^3*d^2*20i + 15*a^4*b^2*d^2)))^(1/2)*((1i/(4*(b
^6*d^2 - a^6*d^2 + a*b^5*d^2*6i + a^5*b*d^2*6i - 15*a^2*b^4*d^2 - a^3*b^3*d^2*20i + 15*a^4*b^2*d^2)))^(1/2)*((
1i/(4*(b^6*d^2 - a^6*d^2 + a*b^5*d^2*6i + a^5*b*d^2*6i - 15*a^2*b^4*d^2 - a^3*b^3*d^2*20i + 15*a^4*b^2*d^2)))^
(1/2)*((1600*a^2*b^23*d^4 + 12864*a^4*b^21*d^4 + 45312*a^6*b^19*d^4 + 91392*a^8*b^17*d^4 + 115584*a^10*b^15*d^
4 + 94080*a^12*b^13*d^4 + 48384*a^14*b^11*d^4 + 14592*a^16*b^9*d^4 + 2112*a^18*b^7*d^4 + 64*a^20*b^5*d^4)/(b^1
9*d^5 + 8*a^2*b^17*d^5 + 28*a^4*b^15*d^5 + 56*a^6*b^13*d^5 + 70*a^8*b^11*d^5 + 56*a^10*b^9*d^5 + 28*a^12*b^7*d
^5 + 8*a^14*b^5*d^5 + a^16*b^3*d^5) + (tan(c + d*x)^(1/2)*(1i/(4*(b^6*d^2 - a^6*d^2 + a*b^5*d^2*6i + a^5*b*d^2
*6i - 15*a^2*b^4*d^2 - a^3*b^3*d^2*20i + 15*a^4*b^2*d^2)))^(1/2)*(512*b^28*d^4 + 4608*a^2*b^26*d^4 + 17920*a^4
*b^24*d^4 + 38400*a^6*b^22*d^4 + 46080*a^8*b^20*d^4 + 21504*a^10*b^18*d^4 - 21504*a^12*b^16*d^4 - 46080*a^14*b
^14*d^4 - 38400*a^16*b^12*d^4 - 17920*a^18*b^10*d^4 - 4608*a^20*b^8*d^4 - 512*a^22*b^6*d^4))/(b^19*d^4 + 8*a^2
*b^17*d^4 + 28*a^4*b^15*d^4 + 56*a^6*b^13*d^4 + 70*a^8*b^11*d^4 + 56*a^10*b^9*d^4 + 28*a^12*b^7*d^4 + 8*a^14*b
^5*d^4 + a^16*b^3*d^4)) - (tan(c + d*x)^(1/2)*(1472*a*b^21*d^2 + 72*a^21*b*d^2 + 1024*a^3*b^19*d^2 + 1352*a^5*
b^17*d^2 + 28224*a^7*b^15*d^2 + 70240*a^9*b^13*d^2 + 72640*a^11*b^11*d^2 + 39088*a^13*b^9*d^2 + 13248*a^15*b^7
*d^2 + 3488*a^17*b^5*d^2 + 576*a^19*b^3*d^2))/(b^19*d^4 + 8*a^2*b^17*d^4 + 28*a^4*b^15*d^4 + 56*a^6*b^13*d^4 +
70*a^8*b^11*d^4 + 56*a^10*b^9*d^4 + 28*a^12*b^7*d^4 + 8*a^14*b^5*d^4 + a^16*b^3*d^4)) + (32*a*b^18*d^2 - 18*a
^19*d^2 - 6528*a^3*b^16*d^2 + 2758*a^5*b^14*d^2 + 26482*a^7*b^12*d^2 + 21582*a^9*b^10*d^2 + 7594*a^11*b^8*d^2
+ 3314*a^13*b^6*d^2 + 246*a^15*b^4*d^2 + 90*a^17*b^2*d^2)/(b^19*d^5 + 8*a^2*b^17*d^5 + 28*a^4*b^15*d^5 + 56*a^
6*b^13*d^5 + 70*a^8*b^11*d^5 + 56*a^10*b^9*d^5 + 28*a^12*b^7*d^5 + 8*a^14*b^5*d^5 + a^16*b^3*d^5)) + (tan(c +
d*x)^(1/2)*(9*a^16 + 32*b^16 + 128*a^2*b^14 + 1417*a^4*b^12 - 6802*a^6*b^10 - 1017*a^8*b^8 - 1020*a^10*b^6 + 3
9*a^12*b^4 - 18*a^14*b^2))/(b^19*d^4 + 8*a^2*b^17*d^4 + 28*a^4*b^15*d^4 + 56*a^6*b^13*d^4 + 70*a^8*b^11*d^4 +
56*a^10*b^9*d^4 + 28*a^12*b^7*d^4 + 8*a^14*b^5*d^4 + a^16*b^3*d^4))*(1i/(4*(b^6*d^2 - a^6*d^2 + a*b^5*d^2*6i +
a^5*b*d^2*6i - 15*a^2*b^4*d^2 - a^3*b^3*d^2*20i + 15*a^4*b^2*d^2)))^(1/2) + ((1i/(4*(b^6*d^2 - a^6*d^2 + a*b^
5*d^2*6i + a^5*b*d^2*6i - 15*a^2*b^4*d^2 - a^3*b^3*d^2*20i + 15*a^4*b^2*d^2)))^(1/2)*((1i/(4*(b^6*d^2 - a^6*d^
2 + a*b^5*d^2*6i + a^5*b*d^2*6i - 15*a^2*b^4*d^2 - a^3*b^3*d^2*20i + 15*a^4*b^2*d^2)))^(1/2)*((1i/(4*(b^6*d^2
- a^6*d^2 + a*b^5*d^2*6i + a^5*b*d^2*6i - 15*a^2*b^4*d^2 - a^3*b^3*d^2*20i + 15*a^4*b^2*d^2)))^(1/2)*((1600*a^
2*b^23*d^4 + 12864*a^4*b^21*d^4 + 45312*a^6*b^19*d^4 + 91392*a^8*b^17*d^4 + 115584*a^10*b^15*d^4 + 94080*a^12*
b^13*d^4 + 48384*a^14*b^11*d^4 + 14592*a^16*b^9*d^4 + 2112*a^18*b^7*d^4 + 64*a^20*b^5*d^4)/(b^19*d^5 + 8*a^2*b
^17*d^5 + 28*a^4*b^15*d^5 + 56*a^6*b^13*d^5 + 70*a^8*b^11*d^5 + 56*a^10*b^9*d^5 + 28*a^12*b^7*d^5 + 8*a^14*b^5
*d^5 + a^16*b^3*d^5) - (tan(c + d*x)^(1/2)*(1i/(4*(b^6*d^2 - a^6*d^2 + a*b^5*d^2*6i + a^5*b*d^2*6i - 15*a^2*b^
4*d^2 - a^3*b^3*d^2*20i + 15*a^4*b^2*d^2)))^(1/2)*(512*b^28*d^4 + 4608*a^2*b^26*d^4 + 17920*a^4*b^24*d^4 + 384
00*a^6*b^22*d^4 + 46080*a^8*b^20*d^4 + 21504*a^10*b^18*d^4 - 21504*a^12*b^16*d^4 - 46080*a^14*b^14*d^4 - 38400
*a^16*b^12*d^4 - 17920*a^18*b^10*d^4 - 4608*a^20*b^8*d^4 - 512*a^22*b^6*d^4))/(b^19*d^4 + 8*a^2*b^17*d^4 + 28*
a^4*b^15*d^4 + 56*a^6*b^13*d^4 + 70*a^8*b^11*d^4 + 56*a^10*b^9*d^4 + 28*a^12*b^7*d^4 + 8*a^14*b^5*d^4 + a^16*b
^3*d^4)) + (tan(c + d*x)^(1/2)*(1472*a*b^21*d^2 + 72*a^21*b*d^2 + 1024*a^3*b^19*d^2 + 1352*a^5*b^17*d^2 + 2822
4*a^7*b^15*d^2 + 70240*a^9*b^13*d^2 + 72640*a^11*b^11*d^2 + 39088*a^13*b^9*d^2 + 13248*a^15*b^7*d^2 + 3488*a^1
7*b^5*d^2 + 576*a^19*b^3*d^2))/(b^19*d^4 + 8*a^2*b^17*d^4 + 28*a^4*b^15*d^4 + 56*a^6*b^13*d^4 + 70*a^8*b^11*d^
4 + 56*a^10*b^9*d^4 + 28*a^12*b^7*d^4 + 8*a^14*b^5*d^4 + a^16*b^3*d^4)) + (32*a*b^18*d^2 - 18*a^19*d^2 - 6528*
a^3*b^16*d^2 + 2758*a^5*b^14*d^2 + 26482*a^7*b^12*d^2 + 21582*a^9*b^10*d^2 + 7594*a^11*b^8*d^2 + 3314*a^13*b^6
*d^2 + 246*a^15*b^4*d^2 + 90*a^17*b^2*d^2)/(b^19*d^5 + 8*a^2*b^17*d^5 + 28*a^4*b^15*d^5 + 56*a^6*b^13*d^5 + 70
*a^8*b^11*d^5 + 56*a^10*b^9*d^5 + 28*a^12*b^7*d^5 + 8*a^14*b^5*d^5 + a^16*b^3*d^5)) - (tan(c + d*x)^(1/2)*(9*a
^16 + 32*b^16 + 128*a^2*b^14 + 1417*a^4*b^12 - 6802*a^6*b^10 - 1017*a^8*b^8 - 1020*a^10*b^6 + 39*a^12*b^4 - 18
*a^14*b^2))/(b^19*d^4 + 8*a^2*b^17*d^4 + 28*a^4*b^15*d^4 + 56*a^6*b^13*d^4 + 70*a^8*b^11*d^4 + 56*a^10*b^9*d^4
+ 28*a^12*b^7*d^4 + 8*a^14*b^5*d^4 + a^16*b^3*d^4))*(1i/(4*(b^6*d^2 - a^6*d^2 + a*b^5*d^2*6i + a^5*b*d^2*6i -
15*a^2*b^4*d^2 - a^3*b^3*d^2*20i + 15*a^4*b^2*d^2)))^(1/2) + (9*a^12*b + 280*a^2*b^11 + 1553*a^4*b^9 + 492*a^
6*b^7 + 270*a^8*b^5 + 36*a^10*b^3)/(b^19*d^5 + 8*a^2*b^17*d^5 + 28*a^4*b^15*d^5 + 56*a^6*b^13*d^5 + 70*a^8*b^1
1*d^5 + 56*a^10*b^9*d^5 + 28*a^12*b^7*d^5 + 8*a^14*b^5*d^5 + a^16*b^3*d^5)))*(1i/(4*(b^6*d^2 - a^6*d^2 + a*b^5
*d^2*6i + a^5*b*d^2*6i - 15*a^2*b^4*d^2 - a^3*b^3*d^2*20i + 15*a^4*b^2*d^2)))^(1/2)*2i - ((a^3*tan(c + d*x)^(1
/2)*(3*a^2 + 11*b^2))/(4*b^2*(a^4 + b^4 + 2*a^2*b^2)) + (a^2*tan(c + d*x)^(3/2)*(5*a^2 + 13*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)) - atan((((((((((1600*a^2*b^23*d^4 + 1
2864*a^4*b^21*d^4 + 45312*a^6*b^19*d^4 + 91392*a^8*b^17*d^4 + 115584*a^10*b^15*d^4 + 94080*a^12*b^13*d^4 + 483
84*a^14*b^11*d^4 + 14592*a^16*b^9*d^4 + 2112*a^18*b^7*d^4 + 64*a^20*b^5*d^4)/(2*(b^19*d^5 + 8*a^2*b^17*d^5 + 2
8*a^4*b^15*d^5 + 56*a^6*b^13*d^5 + 70*a^8*b^11*d^5 + 56*a^10*b^9*d^5 + 28*a^12*b^7*d^5 + 8*a^14*b^5*d^5 + a^16
*b^3*d^5)) - (tan(c + d*x)^(1/2)*(1/(b^6*d^2*1i - a^6*d^2*1i + 6*a*b^5*d^2 + 6*a^5*b*d^2 - a^2*b^4*d^2*15i - 2
0*a^3*b^3*d^2 + a^4*b^2*d^2*15i))^(1/2)*(512*b^28*d^4 + 4608*a^2*b^26*d^4 + 17920*a^4*b^24*d^4 + 38400*a^6*b^2
2*d^4 + 46080*a^8*b^20*d^4 + 21504*a^10*b^18*d^4 - 21504*a^12*b^16*d^4 - 46080*a^14*b^14*d^4 - 38400*a^16*b^12
*d^4 - 17920*a^18*b^10*d^4 - 4608*a^20*b^8*d^4 - 512*a^22*b^6*d^4))/(4*(b^19*d^4 + 8*a^2*b^17*d^4 + 28*a^4*b^1
5*d^4 + 56*a^6*b^13*d^4 + 70*a^8*b^11*d^4 + 56*a^10*b^9*d^4 + 28*a^12*b^7*d^4 + 8*a^14*b^5*d^4 + a^16*b^3*d^4)
))*(1/(b^6*d^2*1i - a^6*d^2*1i + 6*a*b^5*d^2 + 6*a^5*b*d^2 - a^2*b^4*d^2*15i - 20*a^3*b^3*d^2 + a^4*b^2*d^2*15
i))^(1/2))/2 + (tan(c + d*x)^(1/2)*(1472*a*b^21*d^2 + 72*a^21*b*d^2 + 1024*a^3*b^19*d^2 + 1352*a^5*b^17*d^2 +
28224*a^7*b^15*d^2 + 70240*a^9*b^13*d^2 + 72640*a^11*b^11*d^2 + 39088*a^13*b^9*d^2 + 13248*a^15*b^7*d^2 + 3488
*a^17*b^5*d^2 + 576*a^19*b^3*d^2))/(2*(b^19*d^4 + 8*a^2*b^17*d^4 + 28*a^4*b^15*d^4 + 56*a^6*b^13*d^4 + 70*a^8*
b^11*d^4 + 56*a^10*b^9*d^4 + 28*a^12*b^7*d^4 + 8*a^14*b^5*d^4 + a^16*b^3*d^4)))*(1/(b^6*d^2*1i - a^6*d^2*1i +
6*a*b^5*d^2 + 6*a^5*b*d^2 - a^2*b^4*d^2*15i - 20*a^3*b^3*d^2 + a^4*b^2*d^2*15i))^(1/2))/2 + (32*a*b^18*d^2 - 1
8*a^19*d^2 - 6528*a^3*b^16*d^2 + 2758*a^5*b^14*d^2 + 26482*a^7*b^12*d^2 + 21582*a^9*b^10*d^2 + 7594*a^11*b^8*d
^2 + 3314*a^13*b^6*d^2 + 246*a^15*b^4*d^2 + 90*a^17*b^2*d^2)/(2*(b^19*d^5 + 8*a^2*b^17*d^5 + 28*a^4*b^15*d^5 +
56*a^6*b^13*d^5 + 70*a^8*b^11*d^5 + 56*a^10*b^9*d^5 + 28*a^12*b^7*d^5 + 8*a^14*b^5*d^5 + a^16*b^3*d^5)))*(1/(
b^6*d^2*1i - a^6*d^2*1i + 6*a*b^5*d^2 + 6*a^5*b*d^2 - a^2*b^4*d^2*15i - 20*a^3*b^3*d^2 + a^4*b^2*d^2*15i))^(1/
2))/2 - (tan(c + d*x)^(1/2)*(9*a^16 + 32*b^16 + 128*a^2*b^14 + 1417*a^4*b^12 - 6802*a^6*b^10 - 1017*a^8*b^8 -
1020*a^10*b^6 + 39*a^12*b^4 - 18*a^14*b^2))/(2*(b^19*d^4 + 8*a^2*b^17*d^4 + 28*a^4*b^15*d^4 + 56*a^6*b^13*d^4
+ 70*a^8*b^11*d^4 + 56*a^10*b^9*d^4 + 28*a^12*b^7*d^4 + 8*a^14*b^5*d^4 + a^16*b^3*d^4)))*(1/(b^6*d^2*1i - a^6*
d^2*1i + 6*a*b^5*d^2 + 6*a^5*b*d^2 - a^2*b^4*d^2*15i - 20*a^3*b^3*d^2 + a^4*b^2*d^2*15i))^(1/2)*1i - ((((((((1
600*a^2*b^23*d^4 + 12864*a^4*b^21*d^4 + 45312*a^6*b^19*d^4 + 91392*a^8*b^17*d^4 + 115584*a^10*b^15*d^4 + 94080
*a^12*b^13*d^4 + 48384*a^14*b^11*d^4 + 14592*a^16*b^9*d^4 + 2112*a^18*b^7*d^4 + 64*a^20*b^5*d^4)/(2*(b^19*d^5
+ 8*a^2*b^17*d^5 + 28*a^4*b^15*d^5 + 56*a^6*b^13*d^5 + 70*a^8*b^11*d^5 + 56*a^10*b^9*d^5 + 28*a^12*b^7*d^5 + 8
*a^14*b^5*d^5 + a^16*b^3*d^5)) + (tan(c + d*x)^(1/2)*(1/(b^6*d^2*1i - a^6*d^2*1i + 6*a*b^5*d^2 + 6*a^5*b*d^2 -
a^2*b^4*d^2*15i - 20*a^3*b^3*d^2 + a^4*b^2*d^2*15i))^(1/2)*(512*b^28*d^4 + 4608*a^2*b^26*d^4 + 17920*a^4*b^24
*d^4 + 38400*a^6*b^22*d^4 + 46080*a^8*b^20*d^4 + 21504*a^10*b^18*d^4 - 21504*a^12*b^16*d^4 - 46080*a^14*b^14*d
^4 - 38400*a^16*b^12*d^4 - 17920*a^18*b^10*d^4 - 4608*a^20*b^8*d^4 - 512*a^22*b^6*d^4))/(4*(b^19*d^4 + 8*a^2*b
^17*d^4 + 28*a^4*b^15*d^4 + 56*a^6*b^13*d^4 + 70*a^8*b^11*d^4 + 56*a^10*b^9*d^4 + 28*a^12*b^7*d^4 + 8*a^14*b^5
*d^4 + a^16*b^3*d^4)))*(1/(b^6*d^2*1i - a^6*d^2*1i + 6*a*b^5*d^2 + 6*a^5*b*d^2 - a^2*b^4*d^2*15i - 20*a^3*b^3*
d^2 + a^4*b^2*d^2*15i))^(1/2))/2 - (tan(c + d*x)^(1/2)*(1472*a*b^21*d^2 + 72*a^21*b*d^2 + 1024*a^3*b^19*d^2 +
1352*a^5*b^17*d^2 + 28224*a^7*b^15*d^2 + 70240*a^9*b^13*d^2 + 72640*a^11*b^11*d^2 + 39088*a^13*b^9*d^2 + 13248
*a^15*b^7*d^2 + 3488*a^17*b^5*d^2 + 576*a^19*b^3*d^2))/(2*(b^19*d^4 + 8*a^2*b^17*d^4 + 28*a^4*b^15*d^4 + 56*a^
6*b^13*d^4 + 70*a^8*b^11*d^4 + 56*a^10*b^9*d^4 + 28*a^12*b^7*d^4 + 8*a^14*b^5*d^4 + a^16*b^3*d^4)))*(1/(b^6*d^
2*1i - a^6*d^2*1i + 6*a*b^5*d^2 + 6*a^5*b*d^2 - a^2*b^4*d^2*15i - 20*a^3*b^3*d^2 + a^4*b^2*d^2*15i))^(1/2))/2
+ (32*a*b^18*d^2 - 18*a^19*d^2 - 6528*a^3*b^16*d^2 + 2758*a^5*b^14*d^2 + 26482*a^7*b^12*d^2 + 21582*a^9*b^10*d
^2 + 7594*a^11*b^8*d^2 + 3314*a^13*b^6*d^2 + 246*a^15*b^4*d^2 + 90*a^17*b^2*d^2)/(2*(b^19*d^5 + 8*a^2*b^17*d^5
+ 28*a^4*b^15*d^5 + 56*a^6*b^13*d^5 + 70*a^8*b^11*d^5 + 56*a^10*b^9*d^5 + 28*a^12*b^7*d^5 + 8*a^14*b^5*d^5 +
a^16*b^3*d^5)))*(1/(b^6*d^2*1i - a^6*d^2*1i + 6*a*b^5*d^2 + 6*a^5*b*d^2 - a^2*b^4*d^2*15i - 20*a^3*b^3*d^2 + a
^4*b^2*d^2*15i))^(1/2))/2 + (tan(c + d*x)^(1/2)*(9*a^16 + 32*b^16 + 128*a^2*b^14 + 1417*a^4*b^12 - 6802*a^6*b^
10 - 1017*a^8*b^8 - 1020*a^10*b^6 + 39*a^12*b^4 - 18*a^14*b^2))/(2*(b^19*d^4 + 8*a^2*b^17*d^4 + 28*a^4*b^15*d^
4 + 56*a^6*b^13*d^4 + 70*a^8*b^11*d^4 + 56*a^10*b^9*d^4 + 28*a^12*b^7*d^4 + 8*a^14*b^5*d^4 + a^16*b^3*d^4)))*(
1/(b^6*d^2*1i - a^6*d^2*1i + 6*a*b^5*d^2 + 6*a^5*b*d^2 - a^2*b^4*d^2*15i - 20*a^3*b^3*d^2 + a^4*b^2*d^2*15i))^
(1/2)*1i)/(((((((((1600*a^2*b^23*d^4 + 12864*a^4*b^21*d^4 + 45312*a^6*b^19*d^4 + 91392*a^8*b^17*d^4 + 115584*a
^10*b^15*d^4 + 94080*a^12*b^13*d^4 + 48384*a^14*b^11*d^4 + 14592*a^16*b^9*d^4 + 2112*a^18*b^7*d^4 + 64*a^20*b^
5*d^4)/(2*(b^19*d^5 + 8*a^2*b^17*d^5 + 28*a^4*b^15*d^5 + 56*a^6*b^13*d^5 + 70*a^8*b^11*d^5 + 56*a^10*b^9*d^5 +
28*a^12*b^7*d^5 + 8*a^14*b^5*d^5 + a^16*b^3*d^5)) - (tan(c + d*x)^(1/2)*(1/(b^6*d^2*1i - a^6*d^2*1i + 6*a*b^5
*d^2 + 6*a^5*b*d^2 - a^2*b^4*d^2*15i - 20*a^3*b^3*d^2 + a^4*b^2*d^2*15i))^(1/2)*(512*b^28*d^4 + 4608*a^2*b^26*
d^4 + 17920*a^4*b^24*d^4 + 38400*a^6*b^22*d^4 + 46080*a^8*b^20*d^4 + 21504*a^10*b^18*d^4 - 21504*a^12*b^16*d^4
- 46080*a^14*b^14*d^4 - 38400*a^16*b^12*d^4 - 17920*a^18*b^10*d^4 - 4608*a^20*b^8*d^4 - 512*a^22*b^6*d^4))/(4
*(b^19*d^4 + 8*a^2*b^17*d^4 + 28*a^4*b^15*d^4 + 56*a^6*b^13*d^4 + 70*a^8*b^11*d^4 + 56*a^10*b^9*d^4 + 28*a^12*
b^7*d^4 + 8*a^14*b^5*d^4 + a^16*b^3*d^4)))*(1/(b^6*d^2*1i - a^6*d^2*1i + 6*a*b^5*d^2 + 6*a^5*b*d^2 - a^2*b^4*d
^2*15i - 20*a^3*b^3*d^2 + a^4*b^2*d^2*15i))^(1/2))/2 + (tan(c + d*x)^(1/2)*(1472*a*b^21*d^2 + 72*a^21*b*d^2 +
1024*a^3*b^19*d^2 + 1352*a^5*b^17*d^2 + 28224*a^7*b^15*d^2 + 70240*a^9*b^13*d^2 + 72640*a^11*b^11*d^2 + 39088*
a^13*b^9*d^2 + 13248*a^15*b^7*d^2 + 3488*a^17*b^5*d^2 + 576*a^19*b^3*d^2))/(2*(b^19*d^4 + 8*a^2*b^17*d^4 + 28*
a^4*b^15*d^4 + 56*a^6*b^13*d^4 + 70*a^8*b^11*d^4 + 56*a^10*b^9*d^4 + 28*a^12*b^7*d^4 + 8*a^14*b^5*d^4 + a^16*b
^3*d^4)))*(1/(b^6*d^2*1i - a^6*d^2*1i + 6*a*b^5*d^2 + 6*a^5*b*d^2 - a^2*b^4*d^2*15i - 20*a^3*b^3*d^2 + a^4*b^2
*d^2*15i))^(1/2))/2 + (32*a*b^18*d^2 - 18*a^19*d^2 - 6528*a^3*b^16*d^2 + 2758*a^5*b^14*d^2 + 26482*a^7*b^12*d^
2 + 21582*a^9*b^10*d^2 + 7594*a^11*b^8*d^2 + 3314*a^13*b^6*d^2 + 246*a^15*b^4*d^2 + 90*a^17*b^2*d^2)/(2*(b^19*
d^5 + 8*a^2*b^17*d^5 + 28*a^4*b^15*d^5 + 56*a^6*b^13*d^5 + 70*a^8*b^11*d^5 + 56*a^10*b^9*d^5 + 28*a^12*b^7*d^5
+ 8*a^14*b^5*d^5 + a^16*b^3*d^5)))*(1/(b^6*d^2*1i - a^6*d^2*1i + 6*a*b^5*d^2 + 6*a^5*b*d^2 - a^2*b^4*d^2*15i
- 20*a^3*b^3*d^2 + a^4*b^2*d^2*15i))^(1/2))/2 - (tan(c + d*x)^(1/2)*(9*a^16 + 32*b^16 + 128*a^2*b^14 + 1417*a^
4*b^12 - 6802*a^6*b^10 - 1017*a^8*b^8 - 1020*a^10*b^6 + 39*a^12*b^4 - 18*a^14*b^2))/(2*(b^19*d^4 + 8*a^2*b^17*
d^4 + 28*a^4*b^15*d^4 + 56*a^6*b^13*d^4 + 70*a^8*b^11*d^4 + 56*a^10*b^9*d^4 + 28*a^12*b^7*d^4 + 8*a^14*b^5*d^4
+ a^16*b^3*d^4)))*(1/(b^6*d^2*1i - a^6*d^2*1i + 6*a*b^5*d^2 + 6*a^5*b*d^2 - a^2*b^4*d^2*15i - 20*a^3*b^3*d^2
+ a^4*b^2*d^2*15i))^(1/2) + ((((((((1600*a^2*b^23*d^4 + 12864*a^4*b^21*d^4 + 45312*a^6*b^19*d^4 + 91392*a^8*b^
17*d^4 + 115584*a^10*b^15*d^4 + 94080*a^12*b^13*d^4 + 48384*a^14*b^11*d^4 + 14592*a^16*b^9*d^4 + 2112*a^18*b^7
*d^4 + 64*a^20*b^5*d^4)/(2*(b^19*d^5 + 8*a^2*b^17*d^5 + 28*a^4*b^15*d^5 + 56*a^6*b^13*d^5 + 70*a^8*b^11*d^5 +
56*a^10*b^9*d^5 + 28*a^12*b^7*d^5 + 8*a^14*b^5*d^5 + a^16*b^3*d^5)) + (tan(c + d*x)^(1/2)*(1/(b^6*d^2*1i - a^6
*d^2*1i + 6*a*b^5*d^2 + 6*a^5*b*d^2 - a^2*b^4*d^2*15i - 20*a^3*b^3*d^2 + a^4*b^2*d^2*15i))^(1/2)*(512*b^28*d^4
+ 4608*a^2*b^26*d^4 + 17920*a^4*b^24*d^4 + 38400*a^6*b^22*d^4 + 46080*a^8*b^20*d^4 + 21504*a^10*b^18*d^4 - 21
504*a^12*b^16*d^4 - 46080*a^14*b^14*d^4 - 38400*a^16*b^12*d^4 - 17920*a^18*b^10*d^4 - 4608*a^20*b^8*d^4 - 512*
a^22*b^6*d^4))/(4*(b^19*d^4 + 8*a^2*b^17*d^4 + 28*a^4*b^15*d^4 + 56*a^6*b^13*d^4 + 70*a^8*b^11*d^4 + 56*a^10*b
^9*d^4 + 28*a^12*b^7*d^4 + 8*a^14*b^5*d^4 + a^16*b^3*d^4)))*(1/(b^6*d^2*1i - a^6*d^2*1i + 6*a*b^5*d^2 + 6*a^5*
b*d^2 - a^2*b^4*d^2*15i - 20*a^3*b^3*d^2 + a^4*b^2*d^2*15i))^(1/2))/2 - (tan(c + d*x)^(1/2)*(1472*a*b^21*d^2 +
72*a^21*b*d^2 + 1024*a^3*b^19*d^2 + 1352*a^5*b^17*d^2 + 28224*a^7*b^15*d^2 + 70240*a^9*b^13*d^2 + 72640*a^11*
b^11*d^2 + 39088*a^13*b^9*d^2 + 13248*a^15*b^7*d^2 + 3488*a^17*b^5*d^2 + 576*a^19*b^3*d^2))/(2*(b^19*d^4 + 8*a
^2*b^17*d^4 + 28*a^4*b^15*d^4 + 56*a^6*b^13*d^4 + 70*a^8*b^11*d^4 + 56*a^10*b^9*d^4 + 28*a^12*b^7*d^4 + 8*a^14
*b^5*d^4 + a^16*b^3*d^4)))*(1/(b^6*d^2*1i - a^6*d^2*1i + 6*a*b^5*d^2 + 6*a^5*b*d^2 - a^2*b^4*d^2*15i - 20*a^3*
b^3*d^2 + a^4*b^2*d^2*15i))^(1/2))/2 + (32*a*b^18*d^2 - 18*a^19*d^2 - 6528*a^3*b^16*d^2 + 2758*a^5*b^14*d^2 +
26482*a^7*b^12*d^2 + 21582*a^9*b^10*d^2 + 7594*a^11*b^8*d^2 + 3314*a^13*b^6*d^2 + 246*a^15*b^4*d^2 + 90*a^17*b
^2*d^2)/(2*(b^19*d^5 + 8*a^2*b^17*d^5 + 28*a^4*b^15*d^5 + 56*a^6*b^13*d^5 + 70*a^8*b^11*d^5 + 56*a^10*b^9*d^5
+ 28*a^12*b^7*d^5 + 8*a^14*b^5*d^5 + a^16*b^3*d^5)))*(1/(b^6*d^2*1i - a^6*d^2*1i + 6*a*b^5*d^2 + 6*a^5*b*d^2 -
a^2*b^4*d^2*15i - 20*a^3*b^3*d^2 + a^4*b^2*d^2*15i))^(1/2))/2 + (tan(c + d*x)^(1/2)*(9*a^16 + 32*b^16 + 128*a
^2*b^14 + 1417*a^4*b^12 - 6802*a^6*b^10 - 1017*a^8*b^8 - 1020*a^10*b^6 + 39*a^12*b^4 - 18*a^14*b^2))/(2*(b^19*
d^4 + 8*a^2*b^17*d^4 + 28*a^4*b^15*d^4 + 56*a^6*b^13*d^4 + 70*a^8*b^11*d^4 + 56*a^10*b^9*d^4 + 28*a^12*b^7*d^4
+ 8*a^14*b^5*d^4 + a^16*b^3*d^4)))*(1/(b^6*d^2*1i - a^6*d^2*1i + 6*a*b^5*d^2 + 6*a^5*b*d^2 - a^2*b^4*d^2*15i
- 20*a^3*b^3*d^2 + a^4*b^2*d^2*15i))^(1/2) + (9*a^12*b + 280*a^2*b^11 + 1553*a^4*b^9 + 492*a^6*b^7 + 270*a^8*b
^5 + 36*a^10*b^3)/(b^19*d^5 + 8*a^2*b^17*d^5 + 28*a^4*b^15*d^5 + 56*a^6*b^13*d^5 + 70*a^8*b^11*d^5 + 56*a^10*b
^9*d^5 + 28*a^12*b^7*d^5 + 8*a^14*b^5*d^5 + a^16*b^3*d^5)))*(1/(b^6*d^2*1i - a^6*d^2*1i + 6*a*b^5*d^2 + 6*a^5*
b*d^2 - a^2*b^4*d^2*15i - 20*a^3*b^3*d^2 + a^4*b^2*d^2*15i))^(1/2)*1i + (atan(((((tan(c + d*x)^(1/2)*(9*a^16 +
32*b^16 + 128*a^2*b^14 + 1417*a^4*b^12 - 6802*a^6*b^10 - 1017*a^8*b^8 - 1020*a^10*b^6 + 39*a^12*b^4 - 18*a^14
*b^2))/(b^19*d^4 + 8*a^2*b^17*d^4 + 28*a^4*b^15*d^4 + 56*a^6*b^13*d^4 + 70*a^8*b^11*d^4 + 56*a^10*b^9*d^4 + 28
*a^12*b^7*d^4 + 8*a^14*b^5*d^4 + a^16*b^3*d^4) - (((32*a*b^18*d^2 - 18*a^19*d^2 - 6528*a^3*b^16*d^2 + 2758*a^5
*b^14*d^2 + 26482*a^7*b^12*d^2 + 21582*a^9*b^10*d^2 + 7594*a^11*b^8*d^2 + 3314*a^13*b^6*d^2 + 246*a^15*b^4*d^2
+ 90*a^17*b^2*d^2)/(b^19*d^5 + 8*a^2*b^17*d^5 + 28*a^4*b^15*d^5 + 56*a^6*b^13*d^5 + 70*a^8*b^11*d^5 + 56*a^10
*b^9*d^5 + 28*a^12*b^7*d^5 + 8*a^14*b^5*d^5 + a^16*b^3*d^5) + (((tan(c + d*x)^(1/2)*(1472*a*b^21*d^2 + 72*a^21
*b*d^2 + 1024*a^3*b^19*d^2 + 1352*a^5*b^17*d^2 + 28224*a^7*b^15*d^2 + 70240*a^9*b^13*d^2 + 72640*a^11*b^11*d^2
+ 39088*a^13*b^9*d^2 + 13248*a^15*b^7*d^2 + 3488*a^17*b^5*d^2 + 576*a^19*b^3*d^2))/(b^19*d^4 + 8*a^2*b^17*d^4
+ 28*a^4*b^15*d^4 + 56*a^6*b^13*d^4 + 70*a^8*b^11*d^4 + 56*a^10*b^9*d^4 + 28*a^12*b^7*d^4 + 8*a^14*b^5*d^4 +
a^16*b^3*d^4) + (((1600*a^2*b^23*d^4 + 12864*a^4*b^21*d^4 + 45312*a^6*b^19*d^4 + 91392*a^8*b^17*d^4 + 115584*a
^10*b^15*d^4 + 94080*a^12*b^13*d^4 + 48384*a^14*b^11*d^4 + 14592*a^16*b^9*d^4 + 2112*a^18*b^7*d^4 + 64*a^20*b^
5*d^4)/(b^19*d^5 + 8*a^2*b^17*d^5 + 28*a^4*b^15*d^5 + 56*a^6*b^13*d^5 + 70*a^8*b^11*d^5 + 56*a^10*b^9*d^5 + 28
*a^12*b^7*d^5 + 8*a^14*b^5*d^5 + a^16*b^3*d^5) - (tan(c + d*x)^(1/2)*(-a^3*b^5)^(1/2)*(3*a^4 + 35*b^4 + 6*a^2*
b^2)*(512*b^28*d^4 + 4608*a^2*b^26*d^4 + 17920*a^4*b^24*d^4 + 38400*a^6*b^22*d^4 + 46080*a^8*b^20*d^4 + 21504*
a^10*b^18*d^4 - 21504*a^12*b^16*d^4 - 46080*a^14*b^14*d^4 - 38400*a^16*b^12*d^4 - 17920*a^18*b^10*d^4 - 4608*a
^20*b^8*d^4 - 512*a^22*b^6*d^4))/(8*(b^11*d + 3*a^2*b^9*d + 3*a^4*b^7*d + a^6*b^5*d)*(b^19*d^4 + 8*a^2*b^17*d^
4 + 28*a^4*b^15*d^4 + 56*a^6*b^13*d^4 + 70*a^8*b^11*d^4 + 56*a^10*b^9*d^4 + 28*a^12*b^7*d^4 + 8*a^14*b^5*d^4 +
a^16*b^3*d^4)))*(-a^3*b^5)^(1/2)*(3*a^4 + 35*b^4 + 6*a^2*b^2))/(8*(b^11*d + 3*a^2*b^9*d + 3*a^4*b^7*d + a^6*b
^5*d)))*(-a^3*b^5)^(1/2)*(3*a^4 + 35*b^4 + 6*a^2*b^2))/(8*(b^11*d + 3*a^2*b^9*d + 3*a^4*b^7*d + a^6*b^5*d)))*(
-a^3*b^5)^(1/2)*(3*a^4 + 35*b^4 + 6*a^2*b^2))/(8*(b^11*d + 3*a^2*b^9*d + 3*a^4*b^7*d + a^6*b^5*d)))*(-a^3*b^5)
^(1/2)*(3*a^4 + 35*b^4 + 6*a^2*b^2)*1i)/(8*(b^11*d + 3*a^2*b^9*d + 3*a^4*b^7*d + a^6*b^5*d)) + (((tan(c + d*x)
^(1/2)*(9*a^16 + 32*b^16 + 128*a^2*b^14 + 1417*a^4*b^12 - 6802*a^6*b^10 - 1017*a^8*b^8 - 1020*a^10*b^6 + 39*a^
12*b^4 - 18*a^14*b^2))/(b^19*d^4 + 8*a^2*b^17*d^4 + 28*a^4*b^15*d^4 + 56*a^6*b^13*d^4 + 70*a^8*b^11*d^4 + 56*a
^10*b^9*d^4 + 28*a^12*b^7*d^4 + 8*a^14*b^5*d^4 + a^16*b^3*d^4) + (((32*a*b^18*d^2 - 18*a^19*d^2 - 6528*a^3*b^1
6*d^2 + 2758*a^5*b^14*d^2 + 26482*a^7*b^12*d^2 + 21582*a^9*b^10*d^2 + 7594*a^11*b^8*d^2 + 3314*a^13*b^6*d^2 +
246*a^15*b^4*d^2 + 90*a^17*b^2*d^2)/(b^19*d^5 + 8*a^2*b^17*d^5 + 28*a^4*b^15*d^5 + 56*a^6*b^13*d^5 + 70*a^8*b^
11*d^5 + 56*a^10*b^9*d^5 + 28*a^12*b^7*d^5 + 8*a^14*b^5*d^5 + a^16*b^3*d^5) - (((tan(c + d*x)^(1/2)*(1472*a*b^
21*d^2 + 72*a^21*b*d^2 + 1024*a^3*b^19*d^2 + 1352*a^5*b^17*d^2 + 28224*a^7*b^15*d^2 + 70240*a^9*b^13*d^2 + 726
40*a^11*b^11*d^2 + 39088*a^13*b^9*d^2 + 13248*a^15*b^7*d^2 + 3488*a^17*b^5*d^2 + 576*a^19*b^3*d^2))/(b^19*d^4
+ 8*a^2*b^17*d^4 + 28*a^4*b^15*d^4 + 56*a^6*b^13*d^4 + 70*a^8*b^11*d^4 + 56*a^10*b^9*d^4 + 28*a^12*b^7*d^4 + 8
*a^14*b^5*d^4 + a^16*b^3*d^4) - (((1600*a^2*b^23*d^4 + 12864*a^4*b^21*d^4 + 45312*a^6*b^19*d^4 + 91392*a^8*b^1
7*d^4 + 115584*a^10*b^15*d^4 + 94080*a^12*b^13*d^4 + 48384*a^14*b^11*d^4 + 14592*a^16*b^9*d^4 + 2112*a^18*b^7*
d^4 + 64*a^20*b^5*d^4)/(b^19*d^5 + 8*a^2*b^17*d^5 + 28*a^4*b^15*d^5 + 56*a^6*b^13*d^5 + 70*a^8*b^11*d^5 + 56*a
^10*b^9*d^5 + 28*a^12*b^7*d^5 + 8*a^14*b^5*d^5 + a^16*b^3*d^5) + (tan(c + d*x)^(1/2)*(-a^3*b^5)^(1/2)*(3*a^4 +
35*b^4 + 6*a^2*b^2)*(512*b^28*d^4 + 4608*a^2*b^26*d^4 + 17920*a^4*b^24*d^4 + 38400*a^6*b^22*d^4 + 46080*a^8*b
^20*d^4 + 21504*a^10*b^18*d^4 - 21504*a^12*b^16*d^4 - 46080*a^14*b^14*d^4 - 38400*a^16*b^12*d^4 - 17920*a^18*b
^10*d^4 - 4608*a^20*b^8*d^4 - 512*a^22*b^6*d^4))/(8*(b^11*d + 3*a^2*b^9*d + 3*a^4*b^7*d + a^6*b^5*d)*(b^19*d^4
+ 8*a^2*b^17*d^4 + 28*a^4*b^15*d^4 + 56*a^6*b^13*d^4 + 70*a^8*b^11*d^4 + 56*a^10*b^9*d^4 + 28*a^12*b^7*d^4 +
8*a^14*b^5*d^4 + a^16*b^3*d^4)))*(-a^3*b^5)^(1/2)*(3*a^4 + 35*b^4 + 6*a^2*b^2))/(8*(b^11*d + 3*a^2*b^9*d + 3*a
^4*b^7*d + a^6*b^5*d)))*(-a^3*b^5)^(1/2)*(3*a^4 + 35*b^4 + 6*a^2*b^2))/(8*(b^11*d + 3*a^2*b^9*d + 3*a^4*b^7*d
+ a^6*b^5*d)))*(-a^3*b^5)^(1/2)*(3*a^4 + 35*b^4 + 6*a^2*b^2))/(8*(b^11*d + 3*a^2*b^9*d + 3*a^4*b^7*d + a^6*b^5
*d)))*(-a^3*b^5)^(1/2)*(3*a^4 + 35*b^4 + 6*a^2*b^2)*1i)/(8*(b^11*d + 3*a^2*b^9*d + 3*a^4*b^7*d + a^6*b^5*d)))/
((9*a^12*b + 280*a^2*b^11 + 1553*a^4*b^9 + 492*a^6*b^7 + 270*a^8*b^5 + 36*a^10*b^3)/(b^19*d^5 + 8*a^2*b^17*d^5
+ 28*a^4*b^15*d^5 + 56*a^6*b^13*d^5 + 70*a^8*b^11*d^5 + 56*a^10*b^9*d^5 + 28*a^12*b^7*d^5 + 8*a^14*b^5*d^5 +
a^16*b^3*d^5) - (((tan(c + d*x)^(1/2)*(9*a^16 + 32*b^16 + 128*a^2*b^14 + 1417*a^4*b^12 - 6802*a^6*b^10 - 1017*
a^8*b^8 - 1020*a^10*b^6 + 39*a^12*b^4 - 18*a^14*b^2))/(b^19*d^4 + 8*a^2*b^17*d^4 + 28*a^4*b^15*d^4 + 56*a^6*b^
13*d^4 + 70*a^8*b^11*d^4 + 56*a^10*b^9*d^4 + 28*a^12*b^7*d^4 + 8*a^14*b^5*d^4 + a^16*b^3*d^4) - (((32*a*b^18*d
^2 - 18*a^19*d^2 - 6528*a^3*b^16*d^2 + 2758*a^5*b^14*d^2 + 26482*a^7*b^12*d^2 + 21582*a^9*b^10*d^2 + 7594*a^11
*b^8*d^2 + 3314*a^13*b^6*d^2 + 246*a^15*b^4*d^2 + 90*a^17*b^2*d^2)/(b^19*d^5 + 8*a^2*b^17*d^5 + 28*a^4*b^15*d^
5 + 56*a^6*b^13*d^5 + 70*a^8*b^11*d^5 + 56*a^10*b^9*d^5 + 28*a^12*b^7*d^5 + 8*a^14*b^5*d^5 + a^16*b^3*d^5) + (
((tan(c + d*x)^(1/2)*(1472*a*b^21*d^2 + 72*a^21*b*d^2 + 1024*a^3*b^19*d^2 + 1352*a^5*b^17*d^2 + 28224*a^7*b^15
*d^2 + 70240*a^9*b^13*d^2 + 72640*a^11*b^11*d^2 + 39088*a^13*b^9*d^2 + 13248*a^15*b^7*d^2 + 3488*a^17*b^5*d^2
+ 576*a^19*b^3*d^2))/(b^19*d^4 + 8*a^2*b^17*d^4 + 28*a^4*b^15*d^4 + 56*a^6*b^13*d^4 + 70*a^8*b^11*d^4 + 56*a^1
0*b^9*d^4 + 28*a^12*b^7*d^4 + 8*a^14*b^5*d^4 + a^16*b^3*d^4) + (((1600*a^2*b^23*d^4 + 12864*a^4*b^21*d^4 + 453
12*a^6*b^19*d^4 + 91392*a^8*b^17*d^4 + 115584*a^10*b^15*d^4 + 94080*a^12*b^13*d^4 + 48384*a^14*b^11*d^4 + 1459
2*a^16*b^9*d^4 + 2112*a^18*b^7*d^4 + 64*a^20*b^5*d^4)/(b^19*d^5 + 8*a^2*b^17*d^5 + 28*a^4*b^15*d^5 + 56*a^6*b^
13*d^5 + 70*a^8*b^11*d^5 + 56*a^10*b^9*d^5 + 28*a^12*b^7*d^5 + 8*a^14*b^5*d^5 + a^16*b^3*d^5) - (tan(c + d*x)^
(1/2)*(-a^3*b^5)^(1/2)*(3*a^4 + 35*b^4 + 6*a^2*b^2)*(512*b^28*d^4 + 4608*a^2*b^26*d^4 + 17920*a^4*b^24*d^4 + 3
8400*a^6*b^22*d^4 + 46080*a^8*b^20*d^4 + 21504*a^10*b^18*d^4 - 21504*a^12*b^16*d^4 - 46080*a^14*b^14*d^4 - 384
00*a^16*b^12*d^4 - 17920*a^18*b^10*d^4 - 4608*a^20*b^8*d^4 - 512*a^22*b^6*d^4))/(8*(b^11*d + 3*a^2*b^9*d + 3*a
^4*b^7*d + a^6*b^5*d)*(b^19*d^4 + 8*a^2*b^17*d^4 + 28*a^4*b^15*d^4 + 56*a^6*b^13*d^4 + 70*a^8*b^11*d^4 + 56*a^
10*b^9*d^4 + 28*a^12*b^7*d^4 + 8*a^14*b^5*d^4 + a^16*b^3*d^4)))*(-a^3*b^5)^(1/2)*(3*a^4 + 35*b^4 + 6*a^2*b^2))
/(8*(b^11*d + 3*a^2*b^9*d + 3*a^4*b^7*d + a^6*b^5*d)))*(-a^3*b^5)^(1/2)*(3*a^4 + 35*b^4 + 6*a^2*b^2))/(8*(b^11
*d + 3*a^2*b^9*d + 3*a^4*b^7*d + a^6*b^5*d)))*(-a^3*b^5)^(1/2)*(3*a^4 + 35*b^4 + 6*a^2*b^2))/(8*(b^11*d + 3*a^
2*b^9*d + 3*a^4*b^7*d + a^6*b^5*d)))*(-a^3*b^5)^(1/2)*(3*a^4 + 35*b^4 + 6*a^2*b^2))/(8*(b^11*d + 3*a^2*b^9*d +
3*a^4*b^7*d + a^6*b^5*d)) + (((tan(c + d*x)^(1/2)*(9*a^16 + 32*b^16 + 128*a^2*b^14 + 1417*a^4*b^12 - 6802*a^6
*b^10 - 1017*a^8*b^8 - 1020*a^10*b^6 + 39*a^12*b^4 - 18*a^14*b^2))/(b^19*d^4 + 8*a^2*b^17*d^4 + 28*a^4*b^15*d^
4 + 56*a^6*b^13*d^4 + 70*a^8*b^11*d^4 + 56*a^10*b^9*d^4 + 28*a^12*b^7*d^4 + 8*a^14*b^5*d^4 + a^16*b^3*d^4) + (
((32*a*b^18*d^2 - 18*a^19*d^2 - 6528*a^3*b^16*d^2 + 2758*a^5*b^14*d^2 + 26482*a^7*b^12*d^2 + 21582*a^9*b^10*d^
2 + 7594*a^11*b^8*d^2 + 3314*a^13*b^6*d^2 + 246*a^15*b^4*d^2 + 90*a^17*b^2*d^2)/(b^19*d^5 + 8*a^2*b^17*d^5 + 2
8*a^4*b^15*d^5 + 56*a^6*b^13*d^5 + 70*a^8*b^11*d^5 + 56*a^10*b^9*d^5 + 28*a^12*b^7*d^5 + 8*a^14*b^5*d^5 + a^16
*b^3*d^5) - (((tan(c + d*x)^(1/2)*(1472*a*b^21*d^2 + 72*a^21*b*d^2 + 1024*a^3*b^19*d^2 + 1352*a^5*b^17*d^2 + 2
8224*a^7*b^15*d^2 + 70240*a^9*b^13*d^2 + 72640*a^11*b^11*d^2 + 39088*a^13*b^9*d^2 + 13248*a^15*b^7*d^2 + 3488*
a^17*b^5*d^2 + 576*a^19*b^3*d^2))/(b^19*d^4 + 8*a^2*b^17*d^4 + 28*a^4*b^15*d^4 + 56*a^6*b^13*d^4 + 70*a^8*b^11
*d^4 + 56*a^10*b^9*d^4 + 28*a^12*b^7*d^4 + 8*a^14*b^5*d^4 + a^16*b^3*d^4) - (((1600*a^2*b^23*d^4 + 12864*a^4*b
^21*d^4 + 45312*a^6*b^19*d^4 + 91392*a^8*b^17*d^4 + 115584*a^10*b^15*d^4 + 94080*a^12*b^13*d^4 + 48384*a^14*b^
11*d^4 + 14592*a^16*b^9*d^4 + 2112*a^18*b^7*d^4 + 64*a^20*b^5*d^4)/(b^19*d^5 + 8*a^2*b^17*d^5 + 28*a^4*b^15*d^
5 + 56*a^6*b^13*d^5 + 70*a^8*b^11*d^5 + 56*a^10*b^9*d^5 + 28*a^12*b^7*d^5 + 8*a^14*b^5*d^5 + a^16*b^3*d^5) + (
tan(c + d*x)^(1/2)*(-a^3*b^5)^(1/2)*(3*a^4 + 35*b^4 + 6*a^2*b^2)*(512*b^28*d^4 + 4608*a^2*b^26*d^4 + 17920*a^4
*b^24*d^4 + 38400*a^6*b^22*d^4 + 46080*a^8*b^20*d^4 + 21504*a^10*b^18*d^4 - 21504*a^12*b^16*d^4 - 46080*a^14*b
^14*d^4 - 38400*a^16*b^12*d^4 - 17920*a^18*b^10*d^4 - 4608*a^20*b^8*d^4 - 512*a^22*b^6*d^4))/(8*(b^11*d + 3*a^
2*b^9*d + 3*a^4*b^7*d + a^6*b^5*d)*(b^19*d^4 + 8*a^2*b^17*d^4 + 28*a^4*b^15*d^4 + 56*a^6*b^13*d^4 + 70*a^8*b^1
1*d^4 + 56*a^10*b^9*d^4 + 28*a^12*b^7*d^4 + 8*a^14*b^5*d^4 + a^16*b^3*d^4)))*(-a^3*b^5)^(1/2)*(3*a^4 + 35*b^4
+ 6*a^2*b^2))/(8*(b^11*d + 3*a^2*b^9*d + 3*a^4*b^7*d + a^6*b^5*d)))*(-a^3*b^5)^(1/2)*(3*a^4 + 35*b^4 + 6*a^2*b
^2))/(8*(b^11*d + 3*a^2*b^9*d + 3*a^4*b^7*d + a^6*b^5*d)))*(-a^3*b^5)^(1/2)*(3*a^4 + 35*b^4 + 6*a^2*b^2))/(8*(
b^11*d + 3*a^2*b^9*d + 3*a^4*b^7*d + a^6*b^5*d)))*(-a^3*b^5)^(1/2)*(3*a^4 + 35*b^4 + 6*a^2*b^2))/(8*(b^11*d +
3*a^2*b^9*d + 3*a^4*b^7*d + a^6*b^5*d))))*(-a^3*b^5)^(1/2)*(3*a^4 + 35*b^4 + 6*a^2*b^2)*1i)/(4*(b^11*d + 3*a^2
*b^9*d + 3*a^4*b^7*d + a^6*b^5*d))