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3.607 \(\int \frac {1}{\tan ^{\frac {5}{2}}(c+d x) (a+b \tan (c+d x))^3} \, dx\)

Optimal. Leaf size=493 \[ \frac {(a+b) \left (a^2-4 a b+b^2\right ) \tan ^{-1}\left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} d \left (a^2+b^2\right )^3}-\frac {(a+b) \left (a^2-4 a b+b^2\right ) \tan ^{-1}\left (\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{\sqrt {2} d \left (a^2+b^2\right )^3}+\frac {b^2 \left (15 a^2+7 b^2\right )}{4 a^2 d \left (a^2+b^2\right )^2 \tan ^{\frac {3}{2}}(c+d x) (a+b \tan (c+d x))}+\frac {b^2}{2 a d \left (a^2+b^2\right ) \tan ^{\frac {3}{2}}(c+d x) (a+b \tan (c+d x))^2}+\frac {(a-b) \left (a^2+4 a b+b^2\right ) \log \left (\tan (c+d x)-\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{2 \sqrt {2} d \left (a^2+b^2\right )^3}-\frac {(a-b) \left (a^2+4 a b+b^2\right ) \log \left (\tan (c+d x)+\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{2 \sqrt {2} d \left (a^2+b^2\right )^3}+\frac {b \left (24 a^4+67 a^2 b^2+35 b^4\right )}{4 a^4 d \left (a^2+b^2\right )^2 \sqrt {\tan (c+d x)}}+\frac {b^{7/2} \left (99 a^4+102 a^2 b^2+35 b^4\right ) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {\tan (c+d x)}}{\sqrt {a}}\right )}{4 a^{9/2} d \left (a^2+b^2\right )^3}-\frac {8 a^4+67 a^2 b^2+35 b^4}{12 a^3 d \left (a^2+b^2\right )^2 \tan ^{\frac {3}{2}}(c+d x)} \]

[Out]

1/4*b^(7/2)*(99*a^4+102*a^2*b^2+35*b^4)*arctan(b^(1/2)*tan(d*x+c)^(1/2)/a^(1/2))/a^(9/2)/(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)+1/4*b*(24*a^4+67*a^2*b^2+35*b^4)/a^4/(a^2+b^2)^2/d/tan(d*x+c)^(1/2)+1/12*(-8*a^4-67*a^2*b^2-35*b^4)/
a^3/(a^2+b^2)^2/d/tan(d*x+c)^(3/2)+1/2*b^2/a/(a^2+b^2)/d/tan(d*x+c)^(3/2)/(a+b*tan(d*x+c))^2+1/4*b^2*(15*a^2+7
*b^2)/a^2/(a^2+b^2)^2/d/tan(d*x+c)^(3/2)/(a+b*tan(d*x+c))

________________________________________________________________________________________

Rubi [A]  time = 1.40, antiderivative size = 493, normalized size of antiderivative = 1.00, number of steps used = 18, number of rules used = 13, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.565, Rules used = {3569, 3649, 3653, 3534, 1168, 1162, 617, 204, 1165, 628, 3634, 63, 205} \[ \frac {b^{7/2} \left (102 a^2 b^2+99 a^4+35 b^4\right ) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {\tan (c+d x)}}{\sqrt {a}}\right )}{4 a^{9/2} d \left (a^2+b^2\right )^3}+\frac {b^2 \left (15 a^2+7 b^2\right )}{4 a^2 d \left (a^2+b^2\right )^2 \tan ^{\frac {3}{2}}(c+d x) (a+b \tan (c+d x))}+\frac {b^2}{2 a d \left (a^2+b^2\right ) \tan ^{\frac {3}{2}}(c+d x) (a+b \tan (c+d x))^2}+\frac {(a+b) \left (a^2-4 a b+b^2\right ) \tan ^{-1}\left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} d \left (a^2+b^2\right )^3}-\frac {(a+b) \left (a^2-4 a b+b^2\right ) \tan ^{-1}\left (\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{\sqrt {2} d \left (a^2+b^2\right )^3}-\frac {67 a^2 b^2+8 a^4+35 b^4}{12 a^3 d \left (a^2+b^2\right )^2 \tan ^{\frac {3}{2}}(c+d x)}+\frac {b \left (67 a^2 b^2+24 a^4+35 b^4\right )}{4 a^4 d \left (a^2+b^2\right )^2 \sqrt {\tan (c+d x)}}+\frac {(a-b) \left (a^2+4 a b+b^2\right ) \log \left (\tan (c+d x)-\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{2 \sqrt {2} d \left (a^2+b^2\right )^3}-\frac {(a-b) \left (a^2+4 a b+b^2\right ) \log \left (\tan (c+d x)+\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{2 \sqrt {2} d \left (a^2+b^2\right )^3} \]

Antiderivative was successfully verified.

[In]

Int[1/(Tan[c + d*x]^(5/2)*(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) + (b^(7/2)*(99*a^4 + 102*a^2
*b^2 + 35*b^4)*ArcTan[(Sqrt[b]*Sqrt[Tan[c + d*x]])/Sqrt[a]])/(4*a^(9/2)*(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) - (8*a^4 + 67*a^2*b
^2 + 35*b^4)/(12*a^3*(a^2 + b^2)^2*d*Tan[c + d*x]^(3/2)) + (b*(24*a^4 + 67*a^2*b^2 + 35*b^4))/(4*a^4*(a^2 + b^
2)^2*d*Sqrt[Tan[c + d*x]]) + b^2/(2*a*(a^2 + b^2)*d*Tan[c + d*x]^(3/2)*(a + b*Tan[c + d*x])^2) + (b^2*(15*a^2
+ 7*b^2))/(4*a^2*(a^2 + b^2)^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 3569

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Si
mp[(b^2*(a + b*Tan[e + f*x])^(m + 1)*(c + d*Tan[e + f*x])^(n + 1))/(f*(m + 1)*(a^2 + b^2)*(b*c - a*d)), x] + D
ist[1/((m + 1)*(a^2 + b^2)*(b*c - a*d)), Int[(a + b*Tan[e + f*x])^(m + 1)*(c + d*Tan[e + f*x])^n*Simp[a*(b*c -
 a*d)*(m + 1) - b^2*d*(m + n + 2) - b*(b*c - a*d)*(m + 1)*Tan[e + f*x] - b^2*d*(m + n + 2)*Tan[e + f*x]^2, x],
 x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && I
ntegerQ[2*m] && LtQ[m, -1] && (LtQ[n, 0] || IntegerQ[m]) &&  !(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 {1}{\tan ^{\frac {5}{2}}(c+d x) (a+b \tan (c+d x))^3} \, dx &=\frac {b^2}{2 a \left (a^2+b^2\right ) d \tan ^{\frac {3}{2}}(c+d x) (a+b \tan (c+d x))^2}+\frac {\int \frac {\frac {1}{2} \left (4 a^2+7 b^2\right )-2 a b \tan (c+d x)+\frac {7}{2} b^2 \tan ^2(c+d x)}{\tan ^{\frac {5}{2}}(c+d x) (a+b \tan (c+d x))^2} \, dx}{2 a \left (a^2+b^2\right )}\\ &=\frac {b^2}{2 a \left (a^2+b^2\right ) d \tan ^{\frac {3}{2}}(c+d x) (a+b \tan (c+d x))^2}+\frac {b^2 \left (15 a^2+7 b^2\right )}{4 a^2 \left (a^2+b^2\right )^2 d \tan ^{\frac {3}{2}}(c+d x) (a+b \tan (c+d x))}+\frac {\int \frac {\frac {1}{4} \left (8 a^4+67 a^2 b^2+35 b^4\right )-4 a^3 b \tan (c+d x)+\frac {5}{4} b^2 \left (15 a^2+7 b^2\right ) \tan ^2(c+d x)}{\tan ^{\frac {5}{2}}(c+d x) (a+b \tan (c+d x))} \, dx}{2 a^2 \left (a^2+b^2\right )^2}\\ &=-\frac {8 a^4+67 a^2 b^2+35 b^4}{12 a^3 \left (a^2+b^2\right )^2 d \tan ^{\frac {3}{2}}(c+d x)}+\frac {b^2}{2 a \left (a^2+b^2\right ) d \tan ^{\frac {3}{2}}(c+d x) (a+b \tan (c+d x))^2}+\frac {b^2 \left (15 a^2+7 b^2\right )}{4 a^2 \left (a^2+b^2\right )^2 d \tan ^{\frac {3}{2}}(c+d x) (a+b \tan (c+d x))}-\frac {\int \frac {\frac {3}{8} b \left (24 a^4+67 a^2 b^2+35 b^4\right )+3 a^3 \left (a^2-b^2\right ) \tan (c+d x)+\frac {3}{8} b \left (8 a^4+67 a^2 b^2+35 b^4\right ) \tan ^2(c+d x)}{\tan ^{\frac {3}{2}}(c+d x) (a+b \tan (c+d x))} \, dx}{3 a^3 \left (a^2+b^2\right )^2}\\ &=-\frac {8 a^4+67 a^2 b^2+35 b^4}{12 a^3 \left (a^2+b^2\right )^2 d \tan ^{\frac {3}{2}}(c+d x)}+\frac {b \left (24 a^4+67 a^2 b^2+35 b^4\right )}{4 a^4 \left (a^2+b^2\right )^2 d \sqrt {\tan (c+d x)}}+\frac {b^2}{2 a \left (a^2+b^2\right ) d \tan ^{\frac {3}{2}}(c+d x) (a+b \tan (c+d x))^2}+\frac {b^2 \left (15 a^2+7 b^2\right )}{4 a^2 \left (a^2+b^2\right )^2 d \tan ^{\frac {3}{2}}(c+d x) (a+b \tan (c+d x))}+\frac {2 \int \frac {-\frac {3}{16} \left (8 a^6-32 a^4 b^2-67 a^2 b^4-35 b^6\right )+3 a^5 b \tan (c+d x)+\frac {3}{16} b^2 \left (24 a^4+67 a^2 b^2+35 b^4\right ) \tan ^2(c+d x)}{\sqrt {\tan (c+d x)} (a+b \tan (c+d x))} \, dx}{3 a^4 \left (a^2+b^2\right )^2}\\ &=-\frac {8 a^4+67 a^2 b^2+35 b^4}{12 a^3 \left (a^2+b^2\right )^2 d \tan ^{\frac {3}{2}}(c+d x)}+\frac {b \left (24 a^4+67 a^2 b^2+35 b^4\right )}{4 a^4 \left (a^2+b^2\right )^2 d \sqrt {\tan (c+d x)}}+\frac {b^2}{2 a \left (a^2+b^2\right ) d \tan ^{\frac {3}{2}}(c+d x) (a+b \tan (c+d x))^2}+\frac {b^2 \left (15 a^2+7 b^2\right )}{4 a^2 \left (a^2+b^2\right )^2 d \tan ^{\frac {3}{2}}(c+d x) (a+b \tan (c+d x))}+\frac {2 \int \frac {-\frac {3}{2} a^5 \left (a^2-3 b^2\right )+\frac {3}{2} a^4 b \left (3 a^2-b^2\right ) \tan (c+d x)}{\sqrt {\tan (c+d x)}} \, dx}{3 a^4 \left (a^2+b^2\right )^3}+\frac {\left (b^4 \left (99 a^4+102 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 a^4 \left (a^2+b^2\right )^3}\\ &=-\frac {8 a^4+67 a^2 b^2+35 b^4}{12 a^3 \left (a^2+b^2\right )^2 d \tan ^{\frac {3}{2}}(c+d x)}+\frac {b \left (24 a^4+67 a^2 b^2+35 b^4\right )}{4 a^4 \left (a^2+b^2\right )^2 d \sqrt {\tan (c+d x)}}+\frac {b^2}{2 a \left (a^2+b^2\right ) d \tan ^{\frac {3}{2}}(c+d x) (a+b \tan (c+d x))^2}+\frac {b^2 \left (15 a^2+7 b^2\right )}{4 a^2 \left (a^2+b^2\right )^2 d \tan ^{\frac {3}{2}}(c+d x) (a+b \tan (c+d x))}+\frac {4 \operatorname {Subst}\left (\int \frac {-\frac {3}{2} a^5 \left (a^2-3 b^2\right )+\frac {3}{2} a^4 b \left (3 a^2-b^2\right ) x^2}{1+x^4} \, dx,x,\sqrt {\tan (c+d x)}\right )}{3 a^4 \left (a^2+b^2\right )^3 d}+\frac {\left (b^4 \left (99 a^4+102 a^2 b^2+35 b^4\right )\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {x} (a+b x)} \, dx,x,\tan (c+d x)\right )}{8 a^4 \left (a^2+b^2\right )^3 d}\\ &=-\frac {8 a^4+67 a^2 b^2+35 b^4}{12 a^3 \left (a^2+b^2\right )^2 d \tan ^{\frac {3}{2}}(c+d x)}+\frac {b \left (24 a^4+67 a^2 b^2+35 b^4\right )}{4 a^4 \left (a^2+b^2\right )^2 d \sqrt {\tan (c+d x)}}+\frac {b^2}{2 a \left (a^2+b^2\right ) d \tan ^{\frac {3}{2}}(c+d x) (a+b \tan (c+d x))^2}+\frac {b^2 \left (15 a^2+7 b^2\right )}{4 a^2 \left (a^2+b^2\right )^2 d \tan ^{\frac {3}{2}}(c+d x) (a+b \tan (c+d x))}-\frac {\left ((a+b) \left (a^2-4 a b+b^2\right )\right ) \operatorname {Subst}\left (\int \frac {1+x^2}{1+x^4} \, dx,x,\sqrt {\tan (c+d x)}\right )}{\left (a^2+b^2\right )^3 d}-\frac {\left ((a-b) \left (a^2+4 a b+b^2\right )\right ) \operatorname {Subst}\left (\int \frac {1-x^2}{1+x^4} \, dx,x,\sqrt {\tan (c+d x)}\right )}{\left (a^2+b^2\right )^3 d}+\frac {\left (b^4 \left (99 a^4+102 a^2 b^2+35 b^4\right )\right ) \operatorname {Subst}\left (\int \frac {1}{a+b x^2} \, dx,x,\sqrt {\tan (c+d x)}\right )}{4 a^4 \left (a^2+b^2\right )^3 d}\\ &=\frac {b^{7/2} \left (99 a^4+102 a^2 b^2+35 b^4\right ) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {\tan (c+d x)}}{\sqrt {a}}\right )}{4 a^{9/2} \left (a^2+b^2\right )^3 d}-\frac {8 a^4+67 a^2 b^2+35 b^4}{12 a^3 \left (a^2+b^2\right )^2 d \tan ^{\frac {3}{2}}(c+d x)}+\frac {b \left (24 a^4+67 a^2 b^2+35 b^4\right )}{4 a^4 \left (a^2+b^2\right )^2 d \sqrt {\tan (c+d x)}}+\frac {b^2}{2 a \left (a^2+b^2\right ) d \tan ^{\frac {3}{2}}(c+d x) (a+b \tan (c+d x))^2}+\frac {b^2 \left (15 a^2+7 b^2\right )}{4 a^2 \left (a^2+b^2\right )^2 d \tan ^{\frac {3}{2}}(c+d x) (a+b \tan (c+d x))}-\frac {\left ((a+b) \left (a^2-4 a b+b^2\right )\right ) \operatorname {Subst}\left (\int \frac {1}{1-\sqrt {2} x+x^2} \, dx,x,\sqrt {\tan (c+d x)}\right )}{2 \left (a^2+b^2\right )^3 d}-\frac {\left ((a+b) \left (a^2-4 a b+b^2\right )\right ) \operatorname {Subst}\left (\int \frac {1}{1+\sqrt {2} x+x^2} \, dx,x,\sqrt {\tan (c+d x)}\right )}{2 \left (a^2+b^2\right )^3 d}+\frac {\left ((a-b) \left (a^2+4 a b+b^2\right )\right ) \operatorname {Subst}\left (\int \frac {\sqrt {2}+2 x}{-1-\sqrt {2} x-x^2} \, dx,x,\sqrt {\tan (c+d x)}\right )}{2 \sqrt {2} \left (a^2+b^2\right )^3 d}+\frac {\left ((a-b) \left (a^2+4 a b+b^2\right )\right ) \operatorname {Subst}\left (\int \frac {\sqrt {2}-2 x}{-1+\sqrt {2} x-x^2} \, dx,x,\sqrt {\tan (c+d x)}\right )}{2 \sqrt {2} \left (a^2+b^2\right )^3 d}\\ &=\frac {b^{7/2} \left (99 a^4+102 a^2 b^2+35 b^4\right ) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {\tan (c+d x)}}{\sqrt {a}}\right )}{4 a^{9/2} \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 {8 a^4+67 a^2 b^2+35 b^4}{12 a^3 \left (a^2+b^2\right )^2 d \tan ^{\frac {3}{2}}(c+d x)}+\frac {b \left (24 a^4+67 a^2 b^2+35 b^4\right )}{4 a^4 \left (a^2+b^2\right )^2 d \sqrt {\tan (c+d x)}}+\frac {b^2}{2 a \left (a^2+b^2\right ) d \tan ^{\frac {3}{2}}(c+d x) (a+b \tan (c+d x))^2}+\frac {b^2 \left (15 a^2+7 b^2\right )}{4 a^2 \left (a^2+b^2\right )^2 d \tan ^{\frac {3}{2}}(c+d x) (a+b \tan (c+d x))}-\frac {\left ((a+b) \left (a^2-4 a b+b^2\right )\right ) \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} \left (a^2+b^2\right )^3 d}+\frac {\left ((a+b) \left (a^2-4 a b+b^2\right )\right ) \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} \left (a^2+b^2\right )^3 d}\\ &=\frac {(a+b) \left (a^2-4 a b+b^2\right ) \tan ^{-1}\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 ) \tan ^{-1}\left (1+\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} \left (a^2+b^2\right )^3 d}+\frac {b^{7/2} \left (99 a^4+102 a^2 b^2+35 b^4\right ) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {\tan (c+d x)}}{\sqrt {a}}\right )}{4 a^{9/2} \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 {8 a^4+67 a^2 b^2+35 b^4}{12 a^3 \left (a^2+b^2\right )^2 d \tan ^{\frac {3}{2}}(c+d x)}+\frac {b \left (24 a^4+67 a^2 b^2+35 b^4\right )}{4 a^4 \left (a^2+b^2\right )^2 d \sqrt {\tan (c+d x)}}+\frac {b^2}{2 a \left (a^2+b^2\right ) d \tan ^{\frac {3}{2}}(c+d x) (a+b \tan (c+d x))^2}+\frac {b^2 \left (15 a^2+7 b^2\right )}{4 a^2 \left (a^2+b^2\right )^2 d \tan ^{\frac {3}{2}}(c+d x) (a+b \tan (c+d x))}\\ \end {align*}

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Mathematica [C]  time = 6.20, size = 495, normalized size = 1.00 \[ \frac {b^2}{2 a d \left (a^2+b^2\right ) \tan ^{\frac {3}{2}}(c+d x) (a+b \tan (c+d x))^2}+\frac {\frac {\frac {11 a^2 b^2}{2}+\frac {1}{2} b^2 \left (4 a^2+7 b^2\right )}{a d \left (a^2+b^2\right ) \tan ^{\frac {3}{2}}(c+d x) (a+b \tan (c+d x))}+\frac {-\frac {8 a^4+67 a^2 b^2+35 b^4}{6 a d \tan ^{\frac {3}{2}}(c+d x)}-\frac {2 \left (-\frac {3 b \left (24 a^4+67 a^2 b^2+35 b^4\right )}{4 a d \sqrt {\tan (c+d x)}}-\frac {2 \left (\frac {2 \left (-3 a^6 b^2+\frac {3}{16} a^2 b^2 \left (24 a^4+67 a^2 b^2+35 b^4\right )-\frac {3}{16} b^2 \left (8 a^6-32 a^4 b^2-67 a^2 b^4-35 b^6\right )\right ) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {\tan (c+d x)}}{\sqrt {a}}\right )}{\sqrt {a} \sqrt {b} d \left (a^2+b^2\right )}+\frac {-\frac {\sqrt [4]{-1} \left (-\frac {3}{2} a^5 \left (a^2-3 b^2\right )-\frac {3}{2} i a^4 b \left (3 a^2-b^2\right )\right ) \tan ^{-1}\left ((-1)^{3/4} \sqrt {\tan (c+d x)}\right )}{d}-\frac {\sqrt [4]{-1} \left (-\frac {3}{2} a^5 \left (a^2-3 b^2\right )+\frac {3}{2} i a^4 b \left (3 a^2-b^2\right )\right ) \tanh ^{-1}\left ((-1)^{3/4} \sqrt {\tan (c+d x)}\right )}{d}}{a^2+b^2}\right )}{a}\right )}{3 a}}{a \left (a^2+b^2\right )}}{2 a \left (a^2+b^2\right )} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(Tan[c + d*x]^(5/2)*(a + b*Tan[c + d*x])^3),x]

[Out]

b^2/(2*a*(a^2 + b^2)*d*Tan[c + d*x]^(3/2)*(a + b*Tan[c + d*x])^2) + (((-2*((-2*((2*(-3*a^6*b^2 + (3*a^2*b^2*(2
4*a^4 + 67*a^2*b^2 + 35*b^4))/16 - (3*b^2*(8*a^6 - 32*a^4*b^2 - 67*a^2*b^4 - 35*b^6))/16)*ArcTan[(Sqrt[b]*Sqrt
[Tan[c + d*x]])/Sqrt[a]])/(Sqrt[a]*Sqrt[b]*(a^2 + b^2)*d) + (-(((-1)^(1/4)*((-3*a^5*(a^2 - 3*b^2))/2 - ((3*I)/
2)*a^4*b*(3*a^2 - b^2))*ArcTan[(-1)^(3/4)*Sqrt[Tan[c + d*x]]])/d) - ((-1)^(1/4)*((-3*a^5*(a^2 - 3*b^2))/2 + ((
3*I)/2)*a^4*b*(3*a^2 - b^2))*ArcTanh[(-1)^(3/4)*Sqrt[Tan[c + d*x]]])/d)/(a^2 + b^2)))/a - (3*b*(24*a^4 + 67*a^
2*b^2 + 35*b^4))/(4*a*d*Sqrt[Tan[c + d*x]])))/(3*a) - (8*a^4 + 67*a^2*b^2 + 35*b^4)/(6*a*d*Tan[c + d*x]^(3/2))
)/(a*(a^2 + b^2)) + ((11*a^2*b^2)/2 + (b^2*(4*a^2 + 7*b^2))/2)/(a*(a^2 + b^2)*d*Tan[c + d*x]^(3/2)*(a + b*Tan[
c + d*x])))/(2*a*(a^2 + b^2))

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fricas [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/tan(d*x+c)^(5/2)/(a+b*tan(d*x+c))^3,x, algorithm="fricas")

[Out]

Timed out

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (b \tan \left (d x + c\right ) + a\right )}^{3} \tan \left (d x + c\right )^{\frac {5}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/tan(d*x+c)^(5/2)/(a+b*tan(d*x+c))^3,x, algorithm="giac")

[Out]

integrate(1/((b*tan(d*x + c) + a)^3*tan(d*x + c)^(5/2)), x)

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maple [B]  time = 0.35, size = 936, normalized size = 1.90 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/tan(d*x+c)^(5/2)/(a+b*tan(d*x+c))^3,x)

[Out]

19/4/d/(a^2+b^2)^3/(a+b*tan(d*x+c))^2*tan(d*x+c)^(3/2)*b^5+15/2/d*b^7/(a^2+b^2)^3/(a+b*tan(d*x+c))^2/a^2*tan(d
*x+c)^(3/2)+11/4/d*b^9/a^4/(a^2+b^2)^3/(a+b*tan(d*x+c))^2*tan(d*x+c)^(3/2)+21/4/d/(a^2+b^2)^3/(a+b*tan(d*x+c))
^2*tan(d*x+c)^(1/2)*a*b^4+17/2/d*b^6/(a^2+b^2)^3/(a+b*tan(d*x+c))^2/a*tan(d*x+c)^(1/2)+13/4/d*b^8/a^3/(a^2+b^2
)^3/(a+b*tan(d*x+c))^2*tan(d*x+c)^(1/2)+99/4/d/(a^2+b^2)^3/(a*b)^(1/2)*arctan(tan(d*x+c)^(1/2)*b/(a*b)^(1/2))*
b^4+51/2/d*b^6/(a^2+b^2)^3/a^2/(a*b)^(1/2)*arctan(tan(d*x+c)^(1/2)*b/(a*b)^(1/2))+35/4/d*b^8/a^4/(a^2+b^2)^3/(
a*b)^(1/2)*arctan(tan(d*x+c)^(1/2)*b/(a*b)^(1/2))-2/3/d/a^3/tan(d*x+c)^(3/2)+6/d/a^4*b/tan(d*x+c)^(1/2)-1/2/d/
(a^2+b^2)^3*2^(1/2)*arctan(-1+2^(1/2)*tan(d*x+c)^(1/2))*a^3+3/2/d/(a^2+b^2)^3*2^(1/2)*arctan(-1+2^(1/2)*tan(d*
x+c)^(1/2))*a*b^2-1/4/d/(a^2+b^2)^3*2^(1/2)*ln((1+2^(1/2)*tan(d*x+c)^(1/2)+tan(d*x+c))/(1-2^(1/2)*tan(d*x+c)^(
1/2)+tan(d*x+c)))*a^3+3/4/d/(a^2+b^2)^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)))*a*b^2-1/2/d/(a^2+b^2)^3*2^(1/2)*arctan(1+2^(1/2)*tan(d*x+c)^(1/2))*a^3+3/2/d/(a^2+b^2)^3
*2^(1/2)*arctan(1+2^(1/2)*tan(d*x+c)^(1/2))*a*b^2+3/2/d/(a^2+b^2)^3*2^(1/2)*arctan(-1+2^(1/2)*tan(d*x+c)^(1/2)
)*a^2*b-1/2/d/(a^2+b^2)^3*2^(1/2)*arctan(-1+2^(1/2)*tan(d*x+c)^(1/2))*b^3+3/2/d/(a^2+b^2)^3*2^(1/2)*arctan(1+2
^(1/2)*tan(d*x+c)^(1/2))*a^2*b-1/2/d/(a^2+b^2)^3*2^(1/2)*arctan(1+2^(1/2)*tan(d*x+c)^(1/2))*b^3+3/4/d/(a^2+b^2
)^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)))*a^2*b-1/4/d/(a
^2+b^2)^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)))*b^3

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maxima [A]  time = 0.62, size = 502, normalized size = 1.02 \[ \frac {\frac {3 \, {\left (99 \, a^{4} b^{4} + 102 \, a^{2} b^{6} + 35 \, b^{8}\right )} \arctan \left (\frac {b \sqrt {\tan \left (d x + c\right )}}{\sqrt {a b}}\right )}{{\left (a^{10} + 3 \, a^{8} b^{2} + 3 \, a^{6} b^{4} + a^{4} b^{6}\right )} \sqrt {a b}} - \frac {8 \, a^{7} + 16 \, a^{5} b^{2} + 8 \, a^{3} b^{4} - 3 \, {\left (24 \, a^{4} b^{3} + 67 \, a^{2} b^{5} + 35 \, b^{7}\right )} \tan \left (d x + c\right )^{3} - {\left (136 \, a^{5} b^{2} + 335 \, a^{3} b^{4} + 175 \, a b^{6}\right )} \tan \left (d x + c\right )^{2} - 56 \, {\left (a^{6} b + 2 \, a^{4} b^{3} + a^{2} b^{5}\right )} \tan \left (d x + c\right )}{{\left (a^{8} b^{2} + 2 \, a^{6} b^{4} + a^{4} b^{6}\right )} \tan \left (d x + c\right )^{\frac {7}{2}} + 2 \, {\left (a^{9} b + 2 \, a^{7} b^{3} + a^{5} b^{5}\right )} \tan \left (d x + c\right )^{\frac {5}{2}} + {\left (a^{10} + 2 \, a^{8} b^{2} + a^{6} b^{4}\right )} \tan \left (d x + c\right )^{\frac {3}{2}}} - \frac {3 \, {\left (2 \, \sqrt {2} {\left (a^{3} - 3 \, a^{2} b - 3 \, a b^{2} + b^{3}\right )} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + 2 \, \sqrt {\tan \left (d x + c\right )}\right )}\right ) + 2 \, \sqrt {2} {\left (a^{3} - 3 \, a^{2} b - 3 \, a b^{2} + b^{3}\right )} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - 2 \, \sqrt {\tan \left (d x + c\right )}\right )}\right ) + \sqrt {2} {\left (a^{3} + 3 \, a^{2} b - 3 \, a b^{2} - b^{3}\right )} \log \left (\sqrt {2} \sqrt {\tan \left (d x + c\right )} + \tan \left (d x + c\right ) + 1\right ) - \sqrt {2} {\left (a^{3} + 3 \, a^{2} b - 3 \, a b^{2} - b^{3}\right )} \log \left (-\sqrt {2} \sqrt {\tan \left (d x + c\right )} + \tan \left (d x + c\right ) + 1\right )\right )}}{a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}}}{12 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/tan(d*x+c)^(5/2)/(a+b*tan(d*x+c))^3,x, algorithm="maxima")

[Out]

1/12*(3*(99*a^4*b^4 + 102*a^2*b^6 + 35*b^8)*arctan(b*sqrt(tan(d*x + c))/sqrt(a*b))/((a^10 + 3*a^8*b^2 + 3*a^6*
b^4 + a^4*b^6)*sqrt(a*b)) - (8*a^7 + 16*a^5*b^2 + 8*a^3*b^4 - 3*(24*a^4*b^3 + 67*a^2*b^5 + 35*b^7)*tan(d*x + c
)^3 - (136*a^5*b^2 + 335*a^3*b^4 + 175*a*b^6)*tan(d*x + c)^2 - 56*(a^6*b + 2*a^4*b^3 + a^2*b^5)*tan(d*x + c))/
((a^8*b^2 + 2*a^6*b^4 + a^4*b^6)*tan(d*x + c)^(7/2) + 2*(a^9*b + 2*a^7*b^3 + a^5*b^5)*tan(d*x + c)^(5/2) + (a^
10 + 2*a^8*b^2 + a^6*b^4)*tan(d*x + c)^(3/2)) - 3*(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)))) + 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) - 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))/d

________________________________________________________________________________________

mupad [B]  time = 18.88, size = 20088, normalized size = 40.75 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(tan(c + d*x)^(5/2)*(a + b*tan(c + d*x))^3),x)

[Out]

atan(((tan(c + d*x)^(1/2)*(47691333632*a^34*b^35*d^5 - 3156213760*a^30*b^39*d^5 - 7535067136*a^32*b^37*d^5 - 3
21126400*a^28*b^41*d^5 + 451224272896*a^36*b^33*d^5 + 1855390220288*a^38*b^31*d^5 + 4902111674368*a^40*b^29*d^
5 + 9182617010176*a^42*b^27*d^5 + 12661071282176*a^44*b^25*d^5 + 12996430528512*a^46*b^23*d^5 + 9861255397376*
a^48*b^21*d^5 + 5375636013056*a^50*b^19*d^5 + 1966525644800*a^52*b^17*d^5 + 396976193536*a^54*b^15*d^5 + 18517
85216*a^56*b^13*d^5 - 18624806912*a^58*b^11*d^5 - 3332636672*a^60*b^9*d^5 - 117440512*a^62*b^7*d^5 - 8388608*a
^64*b^5*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)*((tan(c + d*x)^(1/2)*(2569011200*a^29*b^46*d^7 + 50939822080*a^31*b^44*d^7 + 47976336588
8*a^33*b^42*d^7 + 2849006157824*a^35*b^40*d^7 + 11943926562816*a^37*b^38*d^7 + 37510046547968*a^39*b^36*d^7 +
91385554272256*a^41*b^34*d^7 + 176470173417472*a^43*b^32*d^7 + 273612095356928*a^45*b^30*d^7 + 342917730271232
*a^47*b^28*d^7 + 347997439262720*a^49*b^26*d^7 + 285130161651712*a^51*b^24*d^7 + 187202969534464*a^53*b^22*d^7
 + 97245760323584*a^55*b^20*d^7 + 39238359842816*a^57*b^18*d^7 + 12009902964736*a^59*b^16*d^7 + 2725695717376*
a^61*b^14*d^7 + 460706545664*a^63*b^12*d^7 + 62631444480*a^65*b^10*d^7 + 6710886400*a^67*b^8*d^7 - 16777216*a^
69*b^6*d^7 - 167772160*a^71*b^4*d^7 - 16777216*a^73*b^2*d^7) - (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)*(587202560*a^32*b^46*d^8 + 11693719552*a
^34*b^44*d^8 + 110612185088*a^36*b^42*d^8 + 660435107840*a^38*b^40*d^8 + 2789614813184*a^40*b^38*d^8 + 8853034
893312*a^42*b^36*d^8 + 21878664069120*a^44*b^34*d^8 + 43052282413056*a^46*b^32*d^8 + 68374033858560*a^48*b^30*
d^8 + 88257920499712*a^50*b^28*d^8 + 92716364988416*a^52*b^26*d^8 + 78893818052608*a^54*b^24*d^8 + 53688433377
280*a^56*b^22*d^8 + 28464224665600*a^58*b^20*d^8 + 11116449103872*a^60*b^18*d^8 + 2734619099136*a^62*b^16*d^8
+ 110377304064*a^64*b^14*d^8 - 221308256256*a^66*b^12*d^8 - 95546245120*a^68*b^10*d^8 - 18303942656*a^70*b^8*d
^8 - 973078528*a^72*b^6*d^8 + 234881024*a^74*b^4*d^8 + 33554432*a^76*b^2*d^8 + 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)*(1342
17728*a^36*b^45*d^9 + 2550136832*a^38*b^43*d^9 + 22817013760*a^40*b^41*d^9 + 127506841600*a^42*b^39*d^9 + 4972
76682240*a^44*b^37*d^9 + 1430626762752*a^46*b^35*d^9 + 3121367482368*a^48*b^33*d^9 + 5202279137280*a^50*b^31*d
^9 + 6502848921600*a^52*b^29*d^9 + 5635802398720*a^54*b^27*d^9 + 2254320959488*a^56*b^25*d^9 - 2254320959488*a
^58*b^23*d^9 - 5635802398720*a^60*b^21*d^9 - 6502848921600*a^62*b^19*d^9 - 5202279137280*a^64*b^17*d^9 - 31213
67482368*a^66*b^15*d^9 - 1430626762752*a^68*b^13*d^9 - 497276682240*a^70*b^11*d^9 - 127506841600*a^72*b^9*d^9
- 22817013760*a^74*b^7*d^9 - 2550136832*a^76*b^5*d^9 - 134217728*a^78*b^3*d^9)))*(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) - 3211264000*a^29*b^43
*d^6 - 47618457600*a^31*b^41*d^6 - 318746132480*a^33*b^39*d^6 - 1248354893824*a^35*b^37*d^6 - 3038318166016*a^
37*b^35*d^6 - 4139457183744*a^39*b^33*d^6 - 252148973568*a^41*b^31*d^6 + 12756182892544*a^43*b^29*d^6 + 325756
25101312*a^45*b^27*d^6 + 48725922676736*a^47*b^25*d^6 + 50814978621440*a^49*b^23*d^6 + 38616415862784*a^51*b^2
1*d^6 + 21485550829568*a^53*b^19*d^6 + 8584215658496*a^55*b^17*d^6 + 2355709870080*a^57*b^15*d^6 + 41134797619
2*a^59*b^13*d^6 + 42875748352*a^61*b^11*d^6 + 4541906944*a^63*b^9*d^6 + 1006632960*a^65*b^7*d^6 + 134217728*a^
67*b^5*d^6 + 8388608*a^69*b^3*d^6))*(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 + (tan(c + d*x)^(1/2)*(47691333632*a^34*b^35*d^5 - 3156213760*a^
30*b^39*d^5 - 7535067136*a^32*b^37*d^5 - 321126400*a^28*b^41*d^5 + 451224272896*a^36*b^33*d^5 + 1855390220288*
a^38*b^31*d^5 + 4902111674368*a^40*b^29*d^5 + 9182617010176*a^42*b^27*d^5 + 12661071282176*a^44*b^25*d^5 + 129
96430528512*a^46*b^23*d^5 + 9861255397376*a^48*b^21*d^5 + 5375636013056*a^50*b^19*d^5 + 1966525644800*a^52*b^1
7*d^5 + 396976193536*a^54*b^15*d^5 + 1851785216*a^56*b^13*d^5 - 18624806912*a^58*b^11*d^5 - 3332636672*a^60*b^
9*d^5 - 117440512*a^62*b^7*d^5 - 8388608*a^64*b^5*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)*(12756182892544*a^43*b^29*d^6 - 3211264000*a^2
9*b^43*d^6 - 47618457600*a^31*b^41*d^6 - 318746132480*a^33*b^39*d^6 - 1248354893824*a^35*b^37*d^6 - 3038318166
016*a^37*b^35*d^6 - 4139457183744*a^39*b^33*d^6 - 252148973568*a^41*b^31*d^6 - (tan(c + d*x)^(1/2)*(2569011200
*a^29*b^46*d^7 + 50939822080*a^31*b^44*d^7 + 479763365888*a^33*b^42*d^7 + 2849006157824*a^35*b^40*d^7 + 119439
26562816*a^37*b^38*d^7 + 37510046547968*a^39*b^36*d^7 + 91385554272256*a^41*b^34*d^7 + 176470173417472*a^43*b^
32*d^7 + 273612095356928*a^45*b^30*d^7 + 342917730271232*a^47*b^28*d^7 + 347997439262720*a^49*b^26*d^7 + 28513
0161651712*a^51*b^24*d^7 + 187202969534464*a^53*b^22*d^7 + 97245760323584*a^55*b^20*d^7 + 39238359842816*a^57*
b^18*d^7 + 12009902964736*a^59*b^16*d^7 + 2725695717376*a^61*b^14*d^7 + 460706545664*a^63*b^12*d^7 + 626314444
80*a^65*b^10*d^7 + 6710886400*a^67*b^8*d^7 - 16777216*a^69*b^6*d^7 - 167772160*a^71*b^4*d^7 - 16777216*a^73*b^
2*d^7) + (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)*(587202560*a^32*b^46*d^8 + 11693719552*a^34*b^44*d^8 + 110612185088*a^36*b^42*d^8 + 6604351078
40*a^38*b^40*d^8 + 2789614813184*a^40*b^38*d^8 + 8853034893312*a^42*b^36*d^8 + 21878664069120*a^44*b^34*d^8 +
43052282413056*a^46*b^32*d^8 + 68374033858560*a^48*b^30*d^8 + 88257920499712*a^50*b^28*d^8 + 92716364988416*a^
52*b^26*d^8 + 78893818052608*a^54*b^24*d^8 + 53688433377280*a^56*b^22*d^8 + 28464224665600*a^58*b^20*d^8 + 111
16449103872*a^60*b^18*d^8 + 2734619099136*a^62*b^16*d^8 + 110377304064*a^64*b^14*d^8 - 221308256256*a^66*b^12*
d^8 - 95546245120*a^68*b^10*d^8 - 18303942656*a^70*b^8*d^8 - 973078528*a^72*b^6*d^8 + 234881024*a^74*b^4*d^8 +
 33554432*a^76*b^2*d^8 - 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)*(134217728*a^36*b^45*d^9 + 2550136832*a^38*b^43*d^9 + 22817
013760*a^40*b^41*d^9 + 127506841600*a^42*b^39*d^9 + 497276682240*a^44*b^37*d^9 + 1430626762752*a^46*b^35*d^9 +
 3121367482368*a^48*b^33*d^9 + 5202279137280*a^50*b^31*d^9 + 6502848921600*a^52*b^29*d^9 + 5635802398720*a^54*
b^27*d^9 + 2254320959488*a^56*b^25*d^9 - 2254320959488*a^58*b^23*d^9 - 5635802398720*a^60*b^21*d^9 - 650284892
1600*a^62*b^19*d^9 - 5202279137280*a^64*b^17*d^9 - 3121367482368*a^66*b^15*d^9 - 1430626762752*a^68*b^13*d^9 -
 497276682240*a^70*b^11*d^9 - 127506841600*a^72*b^9*d^9 - 22817013760*a^74*b^7*d^9 - 2550136832*a^76*b^5*d^9 -
 134217728*a^78*b^3*d^9)))*(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) + 32575625101312*a^45*b^27*d^6 + 48725922676736*a^47*b^25*d^6 + 508149786214
40*a^49*b^23*d^6 + 38616415862784*a^51*b^21*d^6 + 21485550829568*a^53*b^19*d^6 + 8584215658496*a^55*b^17*d^6 +
 2355709870080*a^57*b^15*d^6 + 411347976192*a^59*b^13*d^6 + 42875748352*a^61*b^11*d^6 + 4541906944*a^63*b^9*d^
6 + 1006632960*a^65*b^7*d^6 + 134217728*a^67*b^5*d^6 + 8388608*a^69*b^3*d^6))*(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)/((tan(c + d*x)^(1/2)*
(47691333632*a^34*b^35*d^5 - 3156213760*a^30*b^39*d^5 - 7535067136*a^32*b^37*d^5 - 321126400*a^28*b^41*d^5 + 4
51224272896*a^36*b^33*d^5 + 1855390220288*a^38*b^31*d^5 + 4902111674368*a^40*b^29*d^5 + 9182617010176*a^42*b^2
7*d^5 + 12661071282176*a^44*b^25*d^5 + 12996430528512*a^46*b^23*d^5 + 9861255397376*a^48*b^21*d^5 + 5375636013
056*a^50*b^19*d^5 + 1966525644800*a^52*b^17*d^5 + 396976193536*a^54*b^15*d^5 + 1851785216*a^56*b^13*d^5 - 1862
4806912*a^58*b^11*d^5 - 3332636672*a^60*b^9*d^5 - 117440512*a^62*b^7*d^5 - 8388608*a^64*b^5*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)*((ta
n(c + d*x)^(1/2)*(2569011200*a^29*b^46*d^7 + 50939822080*a^31*b^44*d^7 + 479763365888*a^33*b^42*d^7 + 28490061
57824*a^35*b^40*d^7 + 11943926562816*a^37*b^38*d^7 + 37510046547968*a^39*b^36*d^7 + 91385554272256*a^41*b^34*d
^7 + 176470173417472*a^43*b^32*d^7 + 273612095356928*a^45*b^30*d^7 + 342917730271232*a^47*b^28*d^7 + 347997439
262720*a^49*b^26*d^7 + 285130161651712*a^51*b^24*d^7 + 187202969534464*a^53*b^22*d^7 + 97245760323584*a^55*b^2
0*d^7 + 39238359842816*a^57*b^18*d^7 + 12009902964736*a^59*b^16*d^7 + 2725695717376*a^61*b^14*d^7 + 4607065456
64*a^63*b^12*d^7 + 62631444480*a^65*b^10*d^7 + 6710886400*a^67*b^8*d^7 - 16777216*a^69*b^6*d^7 - 167772160*a^7
1*b^4*d^7 - 16777216*a^73*b^2*d^7) - (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)*(587202560*a^32*b^46*d^8 + 11693719552*a^34*b^44*d^8 + 11061218508
8*a^36*b^42*d^8 + 660435107840*a^38*b^40*d^8 + 2789614813184*a^40*b^38*d^8 + 8853034893312*a^42*b^36*d^8 + 218
78664069120*a^44*b^34*d^8 + 43052282413056*a^46*b^32*d^8 + 68374033858560*a^48*b^30*d^8 + 88257920499712*a^50*
b^28*d^8 + 92716364988416*a^52*b^26*d^8 + 78893818052608*a^54*b^24*d^8 + 53688433377280*a^56*b^22*d^8 + 284642
24665600*a^58*b^20*d^8 + 11116449103872*a^60*b^18*d^8 + 2734619099136*a^62*b^16*d^8 + 110377304064*a^64*b^14*d
^8 - 221308256256*a^66*b^12*d^8 - 95546245120*a^68*b^10*d^8 - 18303942656*a^70*b^8*d^8 - 973078528*a^72*b^6*d^
8 + 234881024*a^74*b^4*d^8 + 33554432*a^76*b^2*d^8 + 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)*(134217728*a^36*b^45*d^9 + 2550
136832*a^38*b^43*d^9 + 22817013760*a^40*b^41*d^9 + 127506841600*a^42*b^39*d^9 + 497276682240*a^44*b^37*d^9 + 1
430626762752*a^46*b^35*d^9 + 3121367482368*a^48*b^33*d^9 + 5202279137280*a^50*b^31*d^9 + 6502848921600*a^52*b^
29*d^9 + 5635802398720*a^54*b^27*d^9 + 2254320959488*a^56*b^25*d^9 - 2254320959488*a^58*b^23*d^9 - 56358023987
20*a^60*b^21*d^9 - 6502848921600*a^62*b^19*d^9 - 5202279137280*a^64*b^17*d^9 - 3121367482368*a^66*b^15*d^9 - 1
430626762752*a^68*b^13*d^9 - 497276682240*a^70*b^11*d^9 - 127506841600*a^72*b^9*d^9 - 22817013760*a^74*b^7*d^9
 - 2550136832*a^76*b^5*d^9 - 134217728*a^78*b^3*d^9)))*(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) - 3211264000*a^29*b^43*d^6 - 47618457600*a^31*b^
41*d^6 - 318746132480*a^33*b^39*d^6 - 1248354893824*a^35*b^37*d^6 - 3038318166016*a^37*b^35*d^6 - 413945718374
4*a^39*b^33*d^6 - 252148973568*a^41*b^31*d^6 + 12756182892544*a^43*b^29*d^6 + 32575625101312*a^45*b^27*d^6 + 4
8725922676736*a^47*b^25*d^6 + 50814978621440*a^49*b^23*d^6 + 38616415862784*a^51*b^21*d^6 + 21485550829568*a^5
3*b^19*d^6 + 8584215658496*a^55*b^17*d^6 + 2355709870080*a^57*b^15*d^6 + 411347976192*a^59*b^13*d^6 + 42875748
352*a^61*b^11*d^6 + 4541906944*a^63*b^9*d^6 + 1006632960*a^65*b^7*d^6 + 134217728*a^67*b^5*d^6 + 8388608*a^69*
b^3*d^6))*(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) - (tan(c + d*x)^(1/2)*(47691333632*a^34*b^35*d^5 - 3156213760*a^30*b^39*d^5 - 7535067136*a^32
*b^37*d^5 - 321126400*a^28*b^41*d^5 + 451224272896*a^36*b^33*d^5 + 1855390220288*a^38*b^31*d^5 + 4902111674368
*a^40*b^29*d^5 + 9182617010176*a^42*b^27*d^5 + 12661071282176*a^44*b^25*d^5 + 12996430528512*a^46*b^23*d^5 + 9
861255397376*a^48*b^21*d^5 + 5375636013056*a^50*b^19*d^5 + 1966525644800*a^52*b^17*d^5 + 396976193536*a^54*b^1
5*d^5 + 1851785216*a^56*b^13*d^5 - 18624806912*a^58*b^11*d^5 - 3332636672*a^60*b^9*d^5 - 117440512*a^62*b^7*d^
5 - 8388608*a^64*b^5*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)*(12756182892544*a^43*b^29*d^6 - 3211264000*a^29*b^43*d^6 - 47618457600*a^31
*b^41*d^6 - 318746132480*a^33*b^39*d^6 - 1248354893824*a^35*b^37*d^6 - 3038318166016*a^37*b^35*d^6 - 413945718
3744*a^39*b^33*d^6 - 252148973568*a^41*b^31*d^6 - (tan(c + d*x)^(1/2)*(2569011200*a^29*b^46*d^7 + 50939822080*
a^31*b^44*d^7 + 479763365888*a^33*b^42*d^7 + 2849006157824*a^35*b^40*d^7 + 11943926562816*a^37*b^38*d^7 + 3751
0046547968*a^39*b^36*d^7 + 91385554272256*a^41*b^34*d^7 + 176470173417472*a^43*b^32*d^7 + 273612095356928*a^45
*b^30*d^7 + 342917730271232*a^47*b^28*d^7 + 347997439262720*a^49*b^26*d^7 + 285130161651712*a^51*b^24*d^7 + 18
7202969534464*a^53*b^22*d^7 + 97245760323584*a^55*b^20*d^7 + 39238359842816*a^57*b^18*d^7 + 12009902964736*a^5
9*b^16*d^7 + 2725695717376*a^61*b^14*d^7 + 460706545664*a^63*b^12*d^7 + 62631444480*a^65*b^10*d^7 + 6710886400
*a^67*b^8*d^7 - 16777216*a^69*b^6*d^7 - 167772160*a^71*b^4*d^7 - 16777216*a^73*b^2*d^7) + (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)*(587202560*a^
32*b^46*d^8 + 11693719552*a^34*b^44*d^8 + 110612185088*a^36*b^42*d^8 + 660435107840*a^38*b^40*d^8 + 2789614813
184*a^40*b^38*d^8 + 8853034893312*a^42*b^36*d^8 + 21878664069120*a^44*b^34*d^8 + 43052282413056*a^46*b^32*d^8
+ 68374033858560*a^48*b^30*d^8 + 88257920499712*a^50*b^28*d^8 + 92716364988416*a^52*b^26*d^8 + 78893818052608*
a^54*b^24*d^8 + 53688433377280*a^56*b^22*d^8 + 28464224665600*a^58*b^20*d^8 + 11116449103872*a^60*b^18*d^8 + 2
734619099136*a^62*b^16*d^8 + 110377304064*a^64*b^14*d^8 - 221308256256*a^66*b^12*d^8 - 95546245120*a^68*b^10*d
^8 - 18303942656*a^70*b^8*d^8 - 973078528*a^72*b^6*d^8 + 234881024*a^74*b^4*d^8 + 33554432*a^76*b^2*d^8 - 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)*(134217728*a^36*b^45*d^9 + 2550136832*a^38*b^43*d^9 + 22817013760*a^40*b^41*d^9 + 127506
841600*a^42*b^39*d^9 + 497276682240*a^44*b^37*d^9 + 1430626762752*a^46*b^35*d^9 + 3121367482368*a^48*b^33*d^9
+ 5202279137280*a^50*b^31*d^9 + 6502848921600*a^52*b^29*d^9 + 5635802398720*a^54*b^27*d^9 + 2254320959488*a^56
*b^25*d^9 - 2254320959488*a^58*b^23*d^9 - 5635802398720*a^60*b^21*d^9 - 6502848921600*a^62*b^19*d^9 - 52022791
37280*a^64*b^17*d^9 - 3121367482368*a^66*b^15*d^9 - 1430626762752*a^68*b^13*d^9 - 497276682240*a^70*b^11*d^9 -
 127506841600*a^72*b^9*d^9 - 22817013760*a^74*b^7*d^9 - 2550136832*a^76*b^5*d^9 - 134217728*a^78*b^3*d^9)))*(1
i/(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) + 32575625101312*a^45*b^27*d^6 + 48725922676736*a^47*b^25*d^6 + 50814978621440*a^49*b^23*d^6 + 3861641586
2784*a^51*b^21*d^6 + 21485550829568*a^53*b^19*d^6 + 8584215658496*a^55*b^17*d^6 + 2355709870080*a^57*b^15*d^6
+ 411347976192*a^59*b^13*d^6 + 42875748352*a^61*b^11*d^6 + 4541906944*a^63*b^9*d^6 + 1006632960*a^65*b^7*d^6 +
 134217728*a^67*b^5*d^6 + 8388608*a^69*b^3*d^6))*(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) + 321126400*a^28*b^38*d^4 + 5082972160*a^30*b^36*d^4 +
 37638373376*a^32*b^34*d^4 + 172475023360*a^34*b^32*d^4 + 545435942912*a^36*b^30*d^4 + 1257037627392*a^38*b^28
*d^4 + 2173340221440*a^40*b^26*d^4 + 2858032300032*a^42*b^24*d^4 + 2865746411520*a^44*b^22*d^4 + 2173002317824
*a^46*b^20*d^4 + 1219756294144*a^48*b^18*d^4 + 486020218880*a^50*b^16*d^4 + 126274502656*a^52*b^14*d^4 + 17141
596160*a^54*b^12*d^4 + 71565312*a^56*b^10*d^4 - 207618048*a^58*b^8*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)*2i + ((14*b*tan(c + d*x))/(3*a
^2) - 2/(3*a) + (tan(c + d*x)^2*(175*b^6 + 335*a^2*b^4 + 136*a^4*b^2))/(12*a^3*(a^4 + b^4 + 2*a^2*b^2)) + (b*t
an(c + d*x)^3*(35*b^6 + 67*a^2*b^4 + 24*a^4*b^2))/(4*a^4*(a^4 + b^4 + 2*a^2*b^2)))/(a^2*d*tan(c + d*x)^(3/2) +
 b^2*d*tan(c + d*x)^(7/2) + 2*a*b*d*tan(c + d*x)^(5/2)) + atan((((tan(c + d*x)^(1/2)*(47691333632*a^34*b^35*d^
5 - 3156213760*a^30*b^39*d^5 - 7535067136*a^32*b^37*d^5 - 321126400*a^28*b^41*d^5 + 451224272896*a^36*b^33*d^5
 + 1855390220288*a^38*b^31*d^5 + 4902111674368*a^40*b^29*d^5 + 9182617010176*a^42*b^27*d^5 + 12661071282176*a^
44*b^25*d^5 + 12996430528512*a^46*b^23*d^5 + 9861255397376*a^48*b^21*d^5 + 5375636013056*a^50*b^19*d^5 + 19665
25644800*a^52*b^17*d^5 + 396976193536*a^54*b^15*d^5 + 1851785216*a^56*b^13*d^5 - 18624806912*a^58*b^11*d^5 - 3
332636672*a^60*b^9*d^5 - 117440512*a^62*b^7*d^5 - 8388608*a^64*b^5*d^5))/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)*(6378091446272*a^43*b^29*
d^6 - 1605632000*a^29*b^43*d^6 - 23809228800*a^31*b^41*d^6 - 159373066240*a^33*b^39*d^6 - 624177446912*a^35*b^
37*d^6 - 1519159083008*a^37*b^35*d^6 - 2069728591872*a^39*b^33*d^6 - 126074486784*a^41*b^31*d^6 - (((tan(c + d
*x)^(1/2)*(2569011200*a^29*b^46*d^7 + 50939822080*a^31*b^44*d^7 + 479763365888*a^33*b^42*d^7 + 2849006157824*a
^35*b^40*d^7 + 11943926562816*a^37*b^38*d^7 + 37510046547968*a^39*b^36*d^7 + 91385554272256*a^41*b^34*d^7 + 17
6470173417472*a^43*b^32*d^7 + 273612095356928*a^45*b^30*d^7 + 342917730271232*a^47*b^28*d^7 + 347997439262720*
a^49*b^26*d^7 + 285130161651712*a^51*b^24*d^7 + 187202969534464*a^53*b^22*d^7 + 97245760323584*a^55*b^20*d^7 +
 39238359842816*a^57*b^18*d^7 + 12009902964736*a^59*b^16*d^7 + 2725695717376*a^61*b^14*d^7 + 460706545664*a^63
*b^12*d^7 + 62631444480*a^65*b^10*d^7 + 6710886400*a^67*b^8*d^7 - 16777216*a^69*b^6*d^7 - 167772160*a^71*b^4*d
^7 - 16777216*a^73*b^2*d^7))/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)*(293601280*a^32*b^46*d^8 - (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)*(134217728*a^36
*b^45*d^9 + 2550136832*a^38*b^43*d^9 + 22817013760*a^40*b^41*d^9 + 127506841600*a^42*b^39*d^9 + 497276682240*a
^44*b^37*d^9 + 1430626762752*a^46*b^35*d^9 + 3121367482368*a^48*b^33*d^9 + 5202279137280*a^50*b^31*d^9 + 65028
48921600*a^52*b^29*d^9 + 5635802398720*a^54*b^27*d^9 + 2254320959488*a^56*b^25*d^9 - 2254320959488*a^58*b^23*d
^9 - 5635802398720*a^60*b^21*d^9 - 6502848921600*a^62*b^19*d^9 - 5202279137280*a^64*b^17*d^9 - 3121367482368*a
^66*b^15*d^9 - 1430626762752*a^68*b^13*d^9 - 497276682240*a^70*b^11*d^9 - 127506841600*a^72*b^9*d^9 - 22817013
760*a^74*b^7*d^9 - 2550136832*a^76*b^5*d^9 - 134217728*a^78*b^3*d^9))/4 + 5846859776*a^34*b^44*d^8 + 553060925
44*a^36*b^42*d^8 + 330217553920*a^38*b^40*d^8 + 1394807406592*a^40*b^38*d^8 + 4426517446656*a^42*b^36*d^8 + 10
939332034560*a^44*b^34*d^8 + 21526141206528*a^46*b^32*d^8 + 34187016929280*a^48*b^30*d^8 + 44128960249856*a^50
*b^28*d^8 + 46358182494208*a^52*b^26*d^8 + 39446909026304*a^54*b^24*d^8 + 26844216688640*a^56*b^22*d^8 + 14232
112332800*a^58*b^20*d^8 + 5558224551936*a^60*b^18*d^8 + 1367309549568*a^62*b^16*d^8 + 55188652032*a^64*b^14*d^
8 - 110654128128*a^66*b^12*d^8 - 47773122560*a^68*b^10*d^8 - 9151971328*a^70*b^8*d^8 - 486539264*a^72*b^6*d^8
+ 117440512*a^74*b^4*d^8 + 16777216*a^76*b^2*d^8))/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))/2 + 16287812550656*a^45*b^27*d^6 + 2436296133836
8*a^47*b^25*d^6 + 25407489310720*a^49*b^23*d^6 + 19308207931392*a^51*b^21*d^6 + 10742775414784*a^53*b^19*d^6 +
 4292107829248*a^55*b^17*d^6 + 1177854935040*a^57*b^15*d^6 + 205673988096*a^59*b^13*d^6 + 21437874176*a^61*b^1
1*d^6 + 2270953472*a^63*b^9*d^6 + 503316480*a^65*b^7*d^6 + 67108864*a^67*b^5*d^6 + 4194304*a^69*b^3*d^6))/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)*1i + ((tan(c + d*x)^(1/2)*(47691333632*a^34*b^35*d^5 - 3156213760*a^30*b^39*d^5 - 7535067136*a^32*b^37*d
^5 - 321126400*a^28*b^41*d^5 + 451224272896*a^36*b^33*d^5 + 1855390220288*a^38*b^31*d^5 + 4902111674368*a^40*b
^29*d^5 + 9182617010176*a^42*b^27*d^5 + 12661071282176*a^44*b^25*d^5 + 12996430528512*a^46*b^23*d^5 + 98612553
97376*a^48*b^21*d^5 + 5375636013056*a^50*b^19*d^5 + 1966525644800*a^52*b^17*d^5 + 396976193536*a^54*b^15*d^5 +
 1851785216*a^56*b^13*d^5 - 18624806912*a^58*b^11*d^5 - 3332636672*a^60*b^9*d^5 - 117440512*a^62*b^7*d^5 - 838
8608*a^64*b^5*d^5))/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)*((((tan(c + d*x)^(1/2)*(2569011200*a^29*b^46*d^7 + 50939822080*a^31*b^44*d^7 +
 479763365888*a^33*b^42*d^7 + 2849006157824*a^35*b^40*d^7 + 11943926562816*a^37*b^38*d^7 + 37510046547968*a^39
*b^36*d^7 + 91385554272256*a^41*b^34*d^7 + 176470173417472*a^43*b^32*d^7 + 273612095356928*a^45*b^30*d^7 + 342
917730271232*a^47*b^28*d^7 + 347997439262720*a^49*b^26*d^7 + 285130161651712*a^51*b^24*d^7 + 187202969534464*a
^53*b^22*d^7 + 97245760323584*a^55*b^20*d^7 + 39238359842816*a^57*b^18*d^7 + 12009902964736*a^59*b^16*d^7 + 27
25695717376*a^61*b^14*d^7 + 460706545664*a^63*b^12*d^7 + 62631444480*a^65*b^10*d^7 + 6710886400*a^67*b^8*d^7 -
 16777216*a^69*b^6*d^7 - 167772160*a^71*b^4*d^7 - 16777216*a^73*b^2*d^7))/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)*((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)*(134217728*a^36*b^45*d^9 + 2550136832*a^38*b^43*d^9 + 22817013760*a^40*b^41*d^9 + 127506841600*a^42*b^39*d
^9 + 497276682240*a^44*b^37*d^9 + 1430626762752*a^46*b^35*d^9 + 3121367482368*a^48*b^33*d^9 + 5202279137280*a^
50*b^31*d^9 + 6502848921600*a^52*b^29*d^9 + 5635802398720*a^54*b^27*d^9 + 2254320959488*a^56*b^25*d^9 - 225432
0959488*a^58*b^23*d^9 - 5635802398720*a^60*b^21*d^9 - 6502848921600*a^62*b^19*d^9 - 5202279137280*a^64*b^17*d^
9 - 3121367482368*a^66*b^15*d^9 - 1430626762752*a^68*b^13*d^9 - 497276682240*a^70*b^11*d^9 - 127506841600*a^72
*b^9*d^9 - 22817013760*a^74*b^7*d^9 - 2550136832*a^76*b^5*d^9 - 134217728*a^78*b^3*d^9))/4 + 293601280*a^32*b^
46*d^8 + 5846859776*a^34*b^44*d^8 + 55306092544*a^36*b^42*d^8 + 330217553920*a^38*b^40*d^8 + 1394807406592*a^4
0*b^38*d^8 + 4426517446656*a^42*b^36*d^8 + 10939332034560*a^44*b^34*d^8 + 21526141206528*a^46*b^32*d^8 + 34187
016929280*a^48*b^30*d^8 + 44128960249856*a^50*b^28*d^8 + 46358182494208*a^52*b^26*d^8 + 39446909026304*a^54*b^
24*d^8 + 26844216688640*a^56*b^22*d^8 + 14232112332800*a^58*b^20*d^8 + 5558224551936*a^60*b^18*d^8 + 136730954
9568*a^62*b^16*d^8 + 55188652032*a^64*b^14*d^8 - 110654128128*a^66*b^12*d^8 - 47773122560*a^68*b^10*d^8 - 9151
971328*a^70*b^8*d^8 - 486539264*a^72*b^6*d^8 + 117440512*a^74*b^4*d^8 + 16777216*a^76*b^2*d^8))/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))/2 -
 1605632000*a^29*b^43*d^6 - 23809228800*a^31*b^41*d^6 - 159373066240*a^33*b^39*d^6 - 624177446912*a^35*b^37*d^
6 - 1519159083008*a^37*b^35*d^6 - 2069728591872*a^39*b^33*d^6 - 126074486784*a^41*b^31*d^6 + 6378091446272*a^4
3*b^29*d^6 + 16287812550656*a^45*b^27*d^6 + 24362961338368*a^47*b^25*d^6 + 25407489310720*a^49*b^23*d^6 + 1930
8207931392*a^51*b^21*d^6 + 10742775414784*a^53*b^19*d^6 + 4292107829248*a^55*b^17*d^6 + 1177854935040*a^57*b^1
5*d^6 + 205673988096*a^59*b^13*d^6 + 21437874176*a^61*b^11*d^6 + 2270953472*a^63*b^9*d^6 + 503316480*a^65*b^7*
d^6 + 67108864*a^67*b^5*d^6 + 4194304*a^69*b^3*d^6))/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)*1i)/(((tan(c + d*x)^(1/2)*(47691333632*a^34*b^3
5*d^5 - 3156213760*a^30*b^39*d^5 - 7535067136*a^32*b^37*d^5 - 321126400*a^28*b^41*d^5 + 451224272896*a^36*b^33
*d^5 + 1855390220288*a^38*b^31*d^5 + 4902111674368*a^40*b^29*d^5 + 9182617010176*a^42*b^27*d^5 + 1266107128217
6*a^44*b^25*d^5 + 12996430528512*a^46*b^23*d^5 + 9861255397376*a^48*b^21*d^5 + 5375636013056*a^50*b^19*d^5 + 1
966525644800*a^52*b^17*d^5 + 396976193536*a^54*b^15*d^5 + 1851785216*a^56*b^13*d^5 - 18624806912*a^58*b^11*d^5
 - 3332636672*a^60*b^9*d^5 - 117440512*a^62*b^7*d^5 - 8388608*a^64*b^5*d^5))/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)*((((tan(c + d*x)^(1/2
)*(2569011200*a^29*b^46*d^7 + 50939822080*a^31*b^44*d^7 + 479763365888*a^33*b^42*d^7 + 2849006157824*a^35*b^40
*d^7 + 11943926562816*a^37*b^38*d^7 + 37510046547968*a^39*b^36*d^7 + 91385554272256*a^41*b^34*d^7 + 1764701734
17472*a^43*b^32*d^7 + 273612095356928*a^45*b^30*d^7 + 342917730271232*a^47*b^28*d^7 + 347997439262720*a^49*b^2
6*d^7 + 285130161651712*a^51*b^24*d^7 + 187202969534464*a^53*b^22*d^7 + 97245760323584*a^55*b^20*d^7 + 3923835
9842816*a^57*b^18*d^7 + 12009902964736*a^59*b^16*d^7 + 2725695717376*a^61*b^14*d^7 + 460706545664*a^63*b^12*d^
7 + 62631444480*a^65*b^10*d^7 + 6710886400*a^67*b^8*d^7 - 16777216*a^69*b^6*d^7 - 167772160*a^71*b^4*d^7 - 167
77216*a^73*b^2*d^7))/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)*((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)*(134217728*a^36*b^45*d^9 + 2550136832*a^38*b^43*d
^9 + 22817013760*a^40*b^41*d^9 + 127506841600*a^42*b^39*d^9 + 497276682240*a^44*b^37*d^9 + 1430626762752*a^46*
b^35*d^9 + 3121367482368*a^48*b^33*d^9 + 5202279137280*a^50*b^31*d^9 + 6502848921600*a^52*b^29*d^9 + 563580239
8720*a^54*b^27*d^9 + 2254320959488*a^56*b^25*d^9 - 2254320959488*a^58*b^23*d^9 - 5635802398720*a^60*b^21*d^9 -
 6502848921600*a^62*b^19*d^9 - 5202279137280*a^64*b^17*d^9 - 3121367482368*a^66*b^15*d^9 - 1430626762752*a^68*
b^13*d^9 - 497276682240*a^70*b^11*d^9 - 127506841600*a^72*b^9*d^9 - 22817013760*a^74*b^7*d^9 - 2550136832*a^76
*b^5*d^9 - 134217728*a^78*b^3*d^9))/4 + 293601280*a^32*b^46*d^8 + 5846859776*a^34*b^44*d^8 + 55306092544*a^36*
b^42*d^8 + 330217553920*a^38*b^40*d^8 + 1394807406592*a^40*b^38*d^8 + 4426517446656*a^42*b^36*d^8 + 1093933203
4560*a^44*b^34*d^8 + 21526141206528*a^46*b^32*d^8 + 34187016929280*a^48*b^30*d^8 + 44128960249856*a^50*b^28*d^
8 + 46358182494208*a^52*b^26*d^8 + 39446909026304*a^54*b^24*d^8 + 26844216688640*a^56*b^22*d^8 + 1423211233280
0*a^58*b^20*d^8 + 5558224551936*a^60*b^18*d^8 + 1367309549568*a^62*b^16*d^8 + 55188652032*a^64*b^14*d^8 - 1106
54128128*a^66*b^12*d^8 - 47773122560*a^68*b^10*d^8 - 9151971328*a^70*b^8*d^8 - 486539264*a^72*b^6*d^8 + 117440
512*a^74*b^4*d^8 + 16777216*a^76*b^2*d^8))/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))/2 - 1605632000*a^29*b^43*d^6 - 23809228800*a^31*b^41*d^6
 - 159373066240*a^33*b^39*d^6 - 624177446912*a^35*b^37*d^6 - 1519159083008*a^37*b^35*d^6 - 2069728591872*a^39*
b^33*d^6 - 126074486784*a^41*b^31*d^6 + 6378091446272*a^43*b^29*d^6 + 16287812550656*a^45*b^27*d^6 + 243629613
38368*a^47*b^25*d^6 + 25407489310720*a^49*b^23*d^6 + 19308207931392*a^51*b^21*d^6 + 10742775414784*a^53*b^19*d
^6 + 4292107829248*a^55*b^17*d^6 + 1177854935040*a^57*b^15*d^6 + 205673988096*a^59*b^13*d^6 + 21437874176*a^61
*b^11*d^6 + 2270953472*a^63*b^9*d^6 + 503316480*a^65*b^7*d^6 + 67108864*a^67*b^5*d^6 + 4194304*a^69*b^3*d^6))/
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*15
i))^(1/2) - ((tan(c + d*x)^(1/2)*(47691333632*a^34*b^35*d^5 - 3156213760*a^30*b^39*d^5 - 7535067136*a^32*b^37*
d^5 - 321126400*a^28*b^41*d^5 + 451224272896*a^36*b^33*d^5 + 1855390220288*a^38*b^31*d^5 + 4902111674368*a^40*
b^29*d^5 + 9182617010176*a^42*b^27*d^5 + 12661071282176*a^44*b^25*d^5 + 12996430528512*a^46*b^23*d^5 + 9861255
397376*a^48*b^21*d^5 + 5375636013056*a^50*b^19*d^5 + 1966525644800*a^52*b^17*d^5 + 396976193536*a^54*b^15*d^5
+ 1851785216*a^56*b^13*d^5 - 18624806912*a^58*b^11*d^5 - 3332636672*a^60*b^9*d^5 - 117440512*a^62*b^7*d^5 - 83
88608*a^64*b^5*d^5))/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)*(6378091446272*a^43*b^29*d^6 - 1605632000*a^29*b^43*d^6 - 23809228800*a^31*b^
41*d^6 - 159373066240*a^33*b^39*d^6 - 624177446912*a^35*b^37*d^6 - 1519159083008*a^37*b^35*d^6 - 2069728591872
*a^39*b^33*d^6 - 126074486784*a^41*b^31*d^6 - (((tan(c + d*x)^(1/2)*(2569011200*a^29*b^46*d^7 + 50939822080*a^
31*b^44*d^7 + 479763365888*a^33*b^42*d^7 + 2849006157824*a^35*b^40*d^7 + 11943926562816*a^37*b^38*d^7 + 375100
46547968*a^39*b^36*d^7 + 91385554272256*a^41*b^34*d^7 + 176470173417472*a^43*b^32*d^7 + 273612095356928*a^45*b
^30*d^7 + 342917730271232*a^47*b^28*d^7 + 347997439262720*a^49*b^26*d^7 + 285130161651712*a^51*b^24*d^7 + 1872
02969534464*a^53*b^22*d^7 + 97245760323584*a^55*b^20*d^7 + 39238359842816*a^57*b^18*d^7 + 12009902964736*a^59*
b^16*d^7 + 2725695717376*a^61*b^14*d^7 + 460706545664*a^63*b^12*d^7 + 62631444480*a^65*b^10*d^7 + 6710886400*a
^67*b^8*d^7 - 16777216*a^69*b^6*d^7 - 167772160*a^71*b^4*d^7 - 16777216*a^73*b^2*d^7))/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)*(293601280*
a^32*b^46*d^8 - (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)*(134217728*a^36*b^45*d^9 + 2550136832*a^38*b^43*d^9 + 22817013760*a
^40*b^41*d^9 + 127506841600*a^42*b^39*d^9 + 497276682240*a^44*b^37*d^9 + 1430626762752*a^46*b^35*d^9 + 3121367
482368*a^48*b^33*d^9 + 5202279137280*a^50*b^31*d^9 + 6502848921600*a^52*b^29*d^9 + 5635802398720*a^54*b^27*d^9
 + 2254320959488*a^56*b^25*d^9 - 2254320959488*a^58*b^23*d^9 - 5635802398720*a^60*b^21*d^9 - 6502848921600*a^6
2*b^19*d^9 - 5202279137280*a^64*b^17*d^9 - 3121367482368*a^66*b^15*d^9 - 1430626762752*a^68*b^13*d^9 - 4972766
82240*a^70*b^11*d^9 - 127506841600*a^72*b^9*d^9 - 22817013760*a^74*b^7*d^9 - 2550136832*a^76*b^5*d^9 - 1342177
28*a^78*b^3*d^9))/4 + 5846859776*a^34*b^44*d^8 + 55306092544*a^36*b^42*d^8 + 330217553920*a^38*b^40*d^8 + 1394
807406592*a^40*b^38*d^8 + 4426517446656*a^42*b^36*d^8 + 10939332034560*a^44*b^34*d^8 + 21526141206528*a^46*b^3
2*d^8 + 34187016929280*a^48*b^30*d^8 + 44128960249856*a^50*b^28*d^8 + 46358182494208*a^52*b^26*d^8 + 394469090
26304*a^54*b^24*d^8 + 26844216688640*a^56*b^22*d^8 + 14232112332800*a^58*b^20*d^8 + 5558224551936*a^60*b^18*d^
8 + 1367309549568*a^62*b^16*d^8 + 55188652032*a^64*b^14*d^8 - 110654128128*a^66*b^12*d^8 - 47773122560*a^68*b^
10*d^8 - 9151971328*a^70*b^8*d^8 - 486539264*a^72*b^6*d^8 + 117440512*a^74*b^4*d^8 + 16777216*a^76*b^2*d^8))/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))/2 + 16287812550656*a^45*b^27*d^6 + 24362961338368*a^47*b^25*d^6 + 25407489310720*a^49*b^23*d^6 + 193
08207931392*a^51*b^21*d^6 + 10742775414784*a^53*b^19*d^6 + 4292107829248*a^55*b^17*d^6 + 1177854935040*a^57*b^
15*d^6 + 205673988096*a^59*b^13*d^6 + 21437874176*a^61*b^11*d^6 + 2270953472*a^63*b^9*d^6 + 503316480*a^65*b^7
*d^6 + 67108864*a^67*b^5*d^6 + 4194304*a^69*b^3*d^6))/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) + 321126400*a^28*b^38*d^4 + 5082972160*a^30*b^
36*d^4 + 37638373376*a^32*b^34*d^4 + 172475023360*a^34*b^32*d^4 + 545435942912*a^36*b^30*d^4 + 1257037627392*a
^38*b^28*d^4 + 2173340221440*a^40*b^26*d^4 + 2858032300032*a^42*b^24*d^4 + 2865746411520*a^44*b^22*d^4 + 21730
02317824*a^46*b^20*d^4 + 1219756294144*a^48*b^18*d^4 + 486020218880*a^50*b^16*d^4 + 126274502656*a^52*b^14*d^4
 + 17141596160*a^54*b^12*d^4 + 71565312*a^56*b^10*d^4 - 207618048*a^58*b^8*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 + (atan((((tan(c +
d*x)^(1/2)*(47691333632*a^34*b^35*d^5 - 3156213760*a^30*b^39*d^5 - 7535067136*a^32*b^37*d^5 - 321126400*a^28*b
^41*d^5 + 451224272896*a^36*b^33*d^5 + 1855390220288*a^38*b^31*d^5 + 4902111674368*a^40*b^29*d^5 + 91826170101
76*a^42*b^27*d^5 + 12661071282176*a^44*b^25*d^5 + 12996430528512*a^46*b^23*d^5 + 9861255397376*a^48*b^21*d^5 +
 5375636013056*a^50*b^19*d^5 + 1966525644800*a^52*b^17*d^5 + 396976193536*a^54*b^15*d^5 + 1851785216*a^56*b^13
*d^5 - 18624806912*a^58*b^11*d^5 - 3332636672*a^60*b^9*d^5 - 117440512*a^62*b^7*d^5 - 8388608*a^64*b^5*d^5) -
((-a^9*b^7)^(1/2)*(99*a^4 + 35*b^4 + 102*a^2*b^2)*(12756182892544*a^43*b^29*d^6 - 47618457600*a^31*b^41*d^6 -
318746132480*a^33*b^39*d^6 - 1248354893824*a^35*b^37*d^6 - 3038318166016*a^37*b^35*d^6 - 4139457183744*a^39*b^
33*d^6 - 252148973568*a^41*b^31*d^6 - 3211264000*a^29*b^43*d^6 + 32575625101312*a^45*b^27*d^6 + 48725922676736
*a^47*b^25*d^6 + 50814978621440*a^49*b^23*d^6 + 38616415862784*a^51*b^21*d^6 + 21485550829568*a^53*b^19*d^6 +
8584215658496*a^55*b^17*d^6 + 2355709870080*a^57*b^15*d^6 + 411347976192*a^59*b^13*d^6 + 42875748352*a^61*b^11
*d^6 + 4541906944*a^63*b^9*d^6 + 1006632960*a^65*b^7*d^6 + 134217728*a^67*b^5*d^6 + 8388608*a^69*b^3*d^6 - ((t
an(c + d*x)^(1/2)*(2569011200*a^29*b^46*d^7 + 50939822080*a^31*b^44*d^7 + 479763365888*a^33*b^42*d^7 + 2849006
157824*a^35*b^40*d^7 + 11943926562816*a^37*b^38*d^7 + 37510046547968*a^39*b^36*d^7 + 91385554272256*a^41*b^34*
d^7 + 176470173417472*a^43*b^32*d^7 + 273612095356928*a^45*b^30*d^7 + 342917730271232*a^47*b^28*d^7 + 34799743
9262720*a^49*b^26*d^7 + 285130161651712*a^51*b^24*d^7 + 187202969534464*a^53*b^22*d^7 + 97245760323584*a^55*b^
20*d^7 + 39238359842816*a^57*b^18*d^7 + 12009902964736*a^59*b^16*d^7 + 2725695717376*a^61*b^14*d^7 + 460706545
664*a^63*b^12*d^7 + 62631444480*a^65*b^10*d^7 + 6710886400*a^67*b^8*d^7 - 16777216*a^69*b^6*d^7 - 167772160*a^
71*b^4*d^7 - 16777216*a^73*b^2*d^7) + ((-a^9*b^7)^(1/2)*(99*a^4 + 35*b^4 + 102*a^2*b^2)*(587202560*a^32*b^46*d
^8 + 11693719552*a^34*b^44*d^8 + 110612185088*a^36*b^42*d^8 + 660435107840*a^38*b^40*d^8 + 2789614813184*a^40*
b^38*d^8 + 8853034893312*a^42*b^36*d^8 + 21878664069120*a^44*b^34*d^8 + 43052282413056*a^46*b^32*d^8 + 6837403
3858560*a^48*b^30*d^8 + 88257920499712*a^50*b^28*d^8 + 92716364988416*a^52*b^26*d^8 + 78893818052608*a^54*b^24
*d^8 + 53688433377280*a^56*b^22*d^8 + 28464224665600*a^58*b^20*d^8 + 11116449103872*a^60*b^18*d^8 + 2734619099
136*a^62*b^16*d^8 + 110377304064*a^64*b^14*d^8 - 221308256256*a^66*b^12*d^8 - 95546245120*a^68*b^10*d^8 - 1830
3942656*a^70*b^8*d^8 - 973078528*a^72*b^6*d^8 + 234881024*a^74*b^4*d^8 + 33554432*a^76*b^2*d^8 - (tan(c + d*x)
^(1/2)*(-a^9*b^7)^(1/2)*(99*a^4 + 35*b^4 + 102*a^2*b^2)*(134217728*a^36*b^45*d^9 + 2550136832*a^38*b^43*d^9 +
22817013760*a^40*b^41*d^9 + 127506841600*a^42*b^39*d^9 + 497276682240*a^44*b^37*d^9 + 1430626762752*a^46*b^35*
d^9 + 3121367482368*a^48*b^33*d^9 + 5202279137280*a^50*b^31*d^9 + 6502848921600*a^52*b^29*d^9 + 5635802398720*
a^54*b^27*d^9 + 2254320959488*a^56*b^25*d^9 - 2254320959488*a^58*b^23*d^9 - 5635802398720*a^60*b^21*d^9 - 6502
848921600*a^62*b^19*d^9 - 5202279137280*a^64*b^17*d^9 - 3121367482368*a^66*b^15*d^9 - 1430626762752*a^68*b^13*
d^9 - 497276682240*a^70*b^11*d^9 - 127506841600*a^72*b^9*d^9 - 22817013760*a^74*b^7*d^9 - 2550136832*a^76*b^5*
d^9 - 134217728*a^78*b^3*d^9))/(8*(a^15*d + a^9*b^6*d + 3*a^11*b^4*d + 3*a^13*b^2*d))))/(8*(a^15*d + a^9*b^6*d
 + 3*a^11*b^4*d + 3*a^13*b^2*d)))*(-a^9*b^7)^(1/2)*(99*a^4 + 35*b^4 + 102*a^2*b^2))/(8*(a^15*d + a^9*b^6*d + 3
*a^11*b^4*d + 3*a^13*b^2*d))))/(8*(a^15*d + a^9*b^6*d + 3*a^11*b^4*d + 3*a^13*b^2*d)))*(-a^9*b^7)^(1/2)*(99*a^
4 + 35*b^4 + 102*a^2*b^2)*1i)/(8*(a^15*d + a^9*b^6*d + 3*a^11*b^4*d + 3*a^13*b^2*d)) + ((tan(c + d*x)^(1/2)*(4
7691333632*a^34*b^35*d^5 - 3156213760*a^30*b^39*d^5 - 7535067136*a^32*b^37*d^5 - 321126400*a^28*b^41*d^5 + 451
224272896*a^36*b^33*d^5 + 1855390220288*a^38*b^31*d^5 + 4902111674368*a^40*b^29*d^5 + 9182617010176*a^42*b^27*
d^5 + 12661071282176*a^44*b^25*d^5 + 12996430528512*a^46*b^23*d^5 + 9861255397376*a^48*b^21*d^5 + 537563601305
6*a^50*b^19*d^5 + 1966525644800*a^52*b^17*d^5 + 396976193536*a^54*b^15*d^5 + 1851785216*a^56*b^13*d^5 - 186248
06912*a^58*b^11*d^5 - 3332636672*a^60*b^9*d^5 - 117440512*a^62*b^7*d^5 - 8388608*a^64*b^5*d^5) + ((-a^9*b^7)^(
1/2)*(99*a^4 + 35*b^4 + 102*a^2*b^2)*(12756182892544*a^43*b^29*d^6 - 47618457600*a^31*b^41*d^6 - 318746132480*
a^33*b^39*d^6 - 1248354893824*a^35*b^37*d^6 - 3038318166016*a^37*b^35*d^6 - 4139457183744*a^39*b^33*d^6 - 2521
48973568*a^41*b^31*d^6 - 3211264000*a^29*b^43*d^6 + 32575625101312*a^45*b^27*d^6 + 48725922676736*a^47*b^25*d^
6 + 50814978621440*a^49*b^23*d^6 + 38616415862784*a^51*b^21*d^6 + 21485550829568*a^53*b^19*d^6 + 8584215658496
*a^55*b^17*d^6 + 2355709870080*a^57*b^15*d^6 + 411347976192*a^59*b^13*d^6 + 42875748352*a^61*b^11*d^6 + 454190
6944*a^63*b^9*d^6 + 1006632960*a^65*b^7*d^6 + 134217728*a^67*b^5*d^6 + 8388608*a^69*b^3*d^6 + ((tan(c + d*x)^(
1/2)*(2569011200*a^29*b^46*d^7 + 50939822080*a^31*b^44*d^7 + 479763365888*a^33*b^42*d^7 + 2849006157824*a^35*b
^40*d^7 + 11943926562816*a^37*b^38*d^7 + 37510046547968*a^39*b^36*d^7 + 91385554272256*a^41*b^34*d^7 + 1764701
73417472*a^43*b^32*d^7 + 273612095356928*a^45*b^30*d^7 + 342917730271232*a^47*b^28*d^7 + 347997439262720*a^49*
b^26*d^7 + 285130161651712*a^51*b^24*d^7 + 187202969534464*a^53*b^22*d^7 + 97245760323584*a^55*b^20*d^7 + 3923
8359842816*a^57*b^18*d^7 + 12009902964736*a^59*b^16*d^7 + 2725695717376*a^61*b^14*d^7 + 460706545664*a^63*b^12
*d^7 + 62631444480*a^65*b^10*d^7 + 6710886400*a^67*b^8*d^7 - 16777216*a^69*b^6*d^7 - 167772160*a^71*b^4*d^7 -
16777216*a^73*b^2*d^7) - ((-a^9*b^7)^(1/2)*(99*a^4 + 35*b^4 + 102*a^2*b^2)*(587202560*a^32*b^46*d^8 + 11693719
552*a^34*b^44*d^8 + 110612185088*a^36*b^42*d^8 + 660435107840*a^38*b^40*d^8 + 2789614813184*a^40*b^38*d^8 + 88
53034893312*a^42*b^36*d^8 + 21878664069120*a^44*b^34*d^8 + 43052282413056*a^46*b^32*d^8 + 68374033858560*a^48*
b^30*d^8 + 88257920499712*a^50*b^28*d^8 + 92716364988416*a^52*b^26*d^8 + 78893818052608*a^54*b^24*d^8 + 536884
33377280*a^56*b^22*d^8 + 28464224665600*a^58*b^20*d^8 + 11116449103872*a^60*b^18*d^8 + 2734619099136*a^62*b^16
*d^8 + 110377304064*a^64*b^14*d^8 - 221308256256*a^66*b^12*d^8 - 95546245120*a^68*b^10*d^8 - 18303942656*a^70*
b^8*d^8 - 973078528*a^72*b^6*d^8 + 234881024*a^74*b^4*d^8 + 33554432*a^76*b^2*d^8 + (tan(c + d*x)^(1/2)*(-a^9*
b^7)^(1/2)*(99*a^4 + 35*b^4 + 102*a^2*b^2)*(134217728*a^36*b^45*d^9 + 2550136832*a^38*b^43*d^9 + 22817013760*a
^40*b^41*d^9 + 127506841600*a^42*b^39*d^9 + 497276682240*a^44*b^37*d^9 + 1430626762752*a^46*b^35*d^9 + 3121367
482368*a^48*b^33*d^9 + 5202279137280*a^50*b^31*d^9 + 6502848921600*a^52*b^29*d^9 + 5635802398720*a^54*b^27*d^9
 + 2254320959488*a^56*b^25*d^9 - 2254320959488*a^58*b^23*d^9 - 5635802398720*a^60*b^21*d^9 - 6502848921600*a^6
2*b^19*d^9 - 5202279137280*a^64*b^17*d^9 - 3121367482368*a^66*b^15*d^9 - 1430626762752*a^68*b^13*d^9 - 4972766
82240*a^70*b^11*d^9 - 127506841600*a^72*b^9*d^9 - 22817013760*a^74*b^7*d^9 - 2550136832*a^76*b^5*d^9 - 1342177
28*a^78*b^3*d^9))/(8*(a^15*d + a^9*b^6*d + 3*a^11*b^4*d + 3*a^13*b^2*d))))/(8*(a^15*d + a^9*b^6*d + 3*a^11*b^4
*d + 3*a^13*b^2*d)))*(-a^9*b^7)^(1/2)*(99*a^4 + 35*b^4 + 102*a^2*b^2))/(8*(a^15*d + a^9*b^6*d + 3*a^11*b^4*d +
 3*a^13*b^2*d))))/(8*(a^15*d + a^9*b^6*d + 3*a^11*b^4*d + 3*a^13*b^2*d)))*(-a^9*b^7)^(1/2)*(99*a^4 + 35*b^4 +
102*a^2*b^2)*1i)/(8*(a^15*d + a^9*b^6*d + 3*a^11*b^4*d + 3*a^13*b^2*d)))/(321126400*a^28*b^38*d^4 + 5082972160
*a^30*b^36*d^4 + 37638373376*a^32*b^34*d^4 + 172475023360*a^34*b^32*d^4 + 545435942912*a^36*b^30*d^4 + 1257037
627392*a^38*b^28*d^4 + 2173340221440*a^40*b^26*d^4 + 2858032300032*a^42*b^24*d^4 + 2865746411520*a^44*b^22*d^4
 + 2173002317824*a^46*b^20*d^4 + 1219756294144*a^48*b^18*d^4 + 486020218880*a^50*b^16*d^4 + 126274502656*a^52*
b^14*d^4 + 17141596160*a^54*b^12*d^4 + 71565312*a^56*b^10*d^4 - 207618048*a^58*b^8*d^4 - ((tan(c + d*x)^(1/2)*
(47691333632*a^34*b^35*d^5 - 3156213760*a^30*b^39*d^5 - 7535067136*a^32*b^37*d^5 - 321126400*a^28*b^41*d^5 + 4
51224272896*a^36*b^33*d^5 + 1855390220288*a^38*b^31*d^5 + 4902111674368*a^40*b^29*d^5 + 9182617010176*a^42*b^2
7*d^5 + 12661071282176*a^44*b^25*d^5 + 12996430528512*a^46*b^23*d^5 + 9861255397376*a^48*b^21*d^5 + 5375636013
056*a^50*b^19*d^5 + 1966525644800*a^52*b^17*d^5 + 396976193536*a^54*b^15*d^5 + 1851785216*a^56*b^13*d^5 - 1862
4806912*a^58*b^11*d^5 - 3332636672*a^60*b^9*d^5 - 117440512*a^62*b^7*d^5 - 8388608*a^64*b^5*d^5) - ((-a^9*b^7)
^(1/2)*(99*a^4 + 35*b^4 + 102*a^2*b^2)*(12756182892544*a^43*b^29*d^6 - 47618457600*a^31*b^41*d^6 - 31874613248
0*a^33*b^39*d^6 - 1248354893824*a^35*b^37*d^6 - 3038318166016*a^37*b^35*d^6 - 4139457183744*a^39*b^33*d^6 - 25
2148973568*a^41*b^31*d^6 - 3211264000*a^29*b^43*d^6 + 32575625101312*a^45*b^27*d^6 + 48725922676736*a^47*b^25*
d^6 + 50814978621440*a^49*b^23*d^6 + 38616415862784*a^51*b^21*d^6 + 21485550829568*a^53*b^19*d^6 + 85842156584
96*a^55*b^17*d^6 + 2355709870080*a^57*b^15*d^6 + 411347976192*a^59*b^13*d^6 + 42875748352*a^61*b^11*d^6 + 4541
906944*a^63*b^9*d^6 + 1006632960*a^65*b^7*d^6 + 134217728*a^67*b^5*d^6 + 8388608*a^69*b^3*d^6 - ((tan(c + d*x)
^(1/2)*(2569011200*a^29*b^46*d^7 + 50939822080*a^31*b^44*d^7 + 479763365888*a^33*b^42*d^7 + 2849006157824*a^35
*b^40*d^7 + 11943926562816*a^37*b^38*d^7 + 37510046547968*a^39*b^36*d^7 + 91385554272256*a^41*b^34*d^7 + 17647
0173417472*a^43*b^32*d^7 + 273612095356928*a^45*b^30*d^7 + 342917730271232*a^47*b^28*d^7 + 347997439262720*a^4
9*b^26*d^7 + 285130161651712*a^51*b^24*d^7 + 187202969534464*a^53*b^22*d^7 + 97245760323584*a^55*b^20*d^7 + 39
238359842816*a^57*b^18*d^7 + 12009902964736*a^59*b^16*d^7 + 2725695717376*a^61*b^14*d^7 + 460706545664*a^63*b^
12*d^7 + 62631444480*a^65*b^10*d^7 + 6710886400*a^67*b^8*d^7 - 16777216*a^69*b^6*d^7 - 167772160*a^71*b^4*d^7
- 16777216*a^73*b^2*d^7) + ((-a^9*b^7)^(1/2)*(99*a^4 + 35*b^4 + 102*a^2*b^2)*(587202560*a^32*b^46*d^8 + 116937
19552*a^34*b^44*d^8 + 110612185088*a^36*b^42*d^8 + 660435107840*a^38*b^40*d^8 + 2789614813184*a^40*b^38*d^8 +
8853034893312*a^42*b^36*d^8 + 21878664069120*a^44*b^34*d^8 + 43052282413056*a^46*b^32*d^8 + 68374033858560*a^4
8*b^30*d^8 + 88257920499712*a^50*b^28*d^8 + 92716364988416*a^52*b^26*d^8 + 78893818052608*a^54*b^24*d^8 + 5368
8433377280*a^56*b^22*d^8 + 28464224665600*a^58*b^20*d^8 + 11116449103872*a^60*b^18*d^8 + 2734619099136*a^62*b^
16*d^8 + 110377304064*a^64*b^14*d^8 - 221308256256*a^66*b^12*d^8 - 95546245120*a^68*b^10*d^8 - 18303942656*a^7
0*b^8*d^8 - 973078528*a^72*b^6*d^8 + 234881024*a^74*b^4*d^8 + 33554432*a^76*b^2*d^8 - (tan(c + d*x)^(1/2)*(-a^
9*b^7)^(1/2)*(99*a^4 + 35*b^4 + 102*a^2*b^2)*(134217728*a^36*b^45*d^9 + 2550136832*a^38*b^43*d^9 + 22817013760
*a^40*b^41*d^9 + 127506841600*a^42*b^39*d^9 + 497276682240*a^44*b^37*d^9 + 1430626762752*a^46*b^35*d^9 + 31213
67482368*a^48*b^33*d^9 + 5202279137280*a^50*b^31*d^9 + 6502848921600*a^52*b^29*d^9 + 5635802398720*a^54*b^27*d
^9 + 2254320959488*a^56*b^25*d^9 - 2254320959488*a^58*b^23*d^9 - 5635802398720*a^60*b^21*d^9 - 6502848921600*a
^62*b^19*d^9 - 5202279137280*a^64*b^17*d^9 - 3121367482368*a^66*b^15*d^9 - 1430626762752*a^68*b^13*d^9 - 49727
6682240*a^70*b^11*d^9 - 127506841600*a^72*b^9*d^9 - 22817013760*a^74*b^7*d^9 - 2550136832*a^76*b^5*d^9 - 13421
7728*a^78*b^3*d^9))/(8*(a^15*d + a^9*b^6*d + 3*a^11*b^4*d + 3*a^13*b^2*d))))/(8*(a^15*d + a^9*b^6*d + 3*a^11*b
^4*d + 3*a^13*b^2*d)))*(-a^9*b^7)^(1/2)*(99*a^4 + 35*b^4 + 102*a^2*b^2))/(8*(a^15*d + a^9*b^6*d + 3*a^11*b^4*d
 + 3*a^13*b^2*d))))/(8*(a^15*d + a^9*b^6*d + 3*a^11*b^4*d + 3*a^13*b^2*d)))*(-a^9*b^7)^(1/2)*(99*a^4 + 35*b^4
+ 102*a^2*b^2))/(8*(a^15*d + a^9*b^6*d + 3*a^11*b^4*d + 3*a^13*b^2*d)) + ((tan(c + d*x)^(1/2)*(47691333632*a^3
4*b^35*d^5 - 3156213760*a^30*b^39*d^5 - 7535067136*a^32*b^37*d^5 - 321126400*a^28*b^41*d^5 + 451224272896*a^36
*b^33*d^5 + 1855390220288*a^38*b^31*d^5 + 4902111674368*a^40*b^29*d^5 + 9182617010176*a^42*b^27*d^5 + 12661071
282176*a^44*b^25*d^5 + 12996430528512*a^46*b^23*d^5 + 9861255397376*a^48*b^21*d^5 + 5375636013056*a^50*b^19*d^
5 + 1966525644800*a^52*b^17*d^5 + 396976193536*a^54*b^15*d^5 + 1851785216*a^56*b^13*d^5 - 18624806912*a^58*b^1
1*d^5 - 3332636672*a^60*b^9*d^5 - 117440512*a^62*b^7*d^5 - 8388608*a^64*b^5*d^5) + ((-a^9*b^7)^(1/2)*(99*a^4 +
 35*b^4 + 102*a^2*b^2)*(12756182892544*a^43*b^29*d^6 - 47618457600*a^31*b^41*d^6 - 318746132480*a^33*b^39*d^6
- 1248354893824*a^35*b^37*d^6 - 3038318166016*a^37*b^35*d^6 - 4139457183744*a^39*b^33*d^6 - 252148973568*a^41*
b^31*d^6 - 3211264000*a^29*b^43*d^6 + 32575625101312*a^45*b^27*d^6 + 48725922676736*a^47*b^25*d^6 + 5081497862
1440*a^49*b^23*d^6 + 38616415862784*a^51*b^21*d^6 + 21485550829568*a^53*b^19*d^6 + 8584215658496*a^55*b^17*d^6
 + 2355709870080*a^57*b^15*d^6 + 411347976192*a^59*b^13*d^6 + 42875748352*a^61*b^11*d^6 + 4541906944*a^63*b^9*
d^6 + 1006632960*a^65*b^7*d^6 + 134217728*a^67*b^5*d^6 + 8388608*a^69*b^3*d^6 + ((tan(c + d*x)^(1/2)*(25690112
00*a^29*b^46*d^7 + 50939822080*a^31*b^44*d^7 + 479763365888*a^33*b^42*d^7 + 2849006157824*a^35*b^40*d^7 + 1194
3926562816*a^37*b^38*d^7 + 37510046547968*a^39*b^36*d^7 + 91385554272256*a^41*b^34*d^7 + 176470173417472*a^43*
b^32*d^7 + 273612095356928*a^45*b^30*d^7 + 342917730271232*a^47*b^28*d^7 + 347997439262720*a^49*b^26*d^7 + 285
130161651712*a^51*b^24*d^7 + 187202969534464*a^53*b^22*d^7 + 97245760323584*a^55*b^20*d^7 + 39238359842816*a^5
7*b^18*d^7 + 12009902964736*a^59*b^16*d^7 + 2725695717376*a^61*b^14*d^7 + 460706545664*a^63*b^12*d^7 + 6263144
4480*a^65*b^10*d^7 + 6710886400*a^67*b^8*d^7 - 16777216*a^69*b^6*d^7 - 167772160*a^71*b^4*d^7 - 16777216*a^73*
b^2*d^7) - ((-a^9*b^7)^(1/2)*(99*a^4 + 35*b^4 + 102*a^2*b^2)*(587202560*a^32*b^46*d^8 + 11693719552*a^34*b^44*
d^8 + 110612185088*a^36*b^42*d^8 + 660435107840*a^38*b^40*d^8 + 2789614813184*a^40*b^38*d^8 + 8853034893312*a^
42*b^36*d^8 + 21878664069120*a^44*b^34*d^8 + 43052282413056*a^46*b^32*d^8 + 68374033858560*a^48*b^30*d^8 + 882
57920499712*a^50*b^28*d^8 + 92716364988416*a^52*b^26*d^8 + 78893818052608*a^54*b^24*d^8 + 53688433377280*a^56*
b^22*d^8 + 28464224665600*a^58*b^20*d^8 + 11116449103872*a^60*b^18*d^8 + 2734619099136*a^62*b^16*d^8 + 1103773
04064*a^64*b^14*d^8 - 221308256256*a^66*b^12*d^8 - 95546245120*a^68*b^10*d^8 - 18303942656*a^70*b^8*d^8 - 9730
78528*a^72*b^6*d^8 + 234881024*a^74*b^4*d^8 + 33554432*a^76*b^2*d^8 + (tan(c + d*x)^(1/2)*(-a^9*b^7)^(1/2)*(99
*a^4 + 35*b^4 + 102*a^2*b^2)*(134217728*a^36*b^45*d^9 + 2550136832*a^38*b^43*d^9 + 22817013760*a^40*b^41*d^9 +
 127506841600*a^42*b^39*d^9 + 497276682240*a^44*b^37*d^9 + 1430626762752*a^46*b^35*d^9 + 3121367482368*a^48*b^
33*d^9 + 5202279137280*a^50*b^31*d^9 + 6502848921600*a^52*b^29*d^9 + 5635802398720*a^54*b^27*d^9 + 22543209594
88*a^56*b^25*d^9 - 2254320959488*a^58*b^23*d^9 - 5635802398720*a^60*b^21*d^9 - 6502848921600*a^62*b^19*d^9 - 5
202279137280*a^64*b^17*d^9 - 3121367482368*a^66*b^15*d^9 - 1430626762752*a^68*b^13*d^9 - 497276682240*a^70*b^1
1*d^9 - 127506841600*a^72*b^9*d^9 - 22817013760*a^74*b^7*d^9 - 2550136832*a^76*b^5*d^9 - 134217728*a^78*b^3*d^
9))/(8*(a^15*d + a^9*b^6*d + 3*a^11*b^4*d + 3*a^13*b^2*d))))/(8*(a^15*d + a^9*b^6*d + 3*a^11*b^4*d + 3*a^13*b^
2*d)))*(-a^9*b^7)^(1/2)*(99*a^4 + 35*b^4 + 102*a^2*b^2))/(8*(a^15*d + a^9*b^6*d + 3*a^11*b^4*d + 3*a^13*b^2*d)
)))/(8*(a^15*d + a^9*b^6*d + 3*a^11*b^4*d + 3*a^13*b^2*d)))*(-a^9*b^7)^(1/2)*(99*a^4 + 35*b^4 + 102*a^2*b^2))/
(8*(a^15*d + a^9*b^6*d + 3*a^11*b^4*d + 3*a^13*b^2*d))))*(-a^9*b^7)^(1/2)*(99*a^4 + 35*b^4 + 102*a^2*b^2)*1i)/
(4*(a^15*d + a^9*b^6*d + 3*a^11*b^4*d + 3*a^13*b^2*d))

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (a + b \tan {\left (c + d x \right )}\right )^{3} \tan ^{\frac {5}{2}}{\left (c + d x \right )}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/tan(d*x+c)**(5/2)/(a+b*tan(d*x+c))**3,x)

[Out]

Integral(1/((a + b*tan(c + d*x))**3*tan(c + d*x)**(5/2)), x)

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