3.599 \(\int \frac {\tan ^{\frac {11}{2}}(c+d x)}{(a+b \tan (c+d x))^3} \, dx\)

Optimal. Leaf size=493 \[ \frac {(a+b) \left (a^2-4 a b+b^2\right ) \tan ^{-1}\left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} d \left (a^2+b^2\right )^3}-\frac {(a+b) \left (a^2-4 a b+b^2\right ) \tan ^{-1}\left (\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{\sqrt {2} d \left (a^2+b^2\right )^3}-\frac {a^2 \left (7 a^2+15 b^2\right ) \tan ^{\frac {5}{2}}(c+d x)}{4 b^2 d \left (a^2+b^2\right )^2 (a+b \tan (c+d x))}-\frac {a^2 \tan ^{\frac {7}{2}}(c+d x)}{2 b d \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}+\frac {(a-b) \left (a^2+4 a b+b^2\right ) \log \left (\tan (c+d x)-\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{2 \sqrt {2} d \left (a^2+b^2\right )^3}-\frac {(a-b) \left (a^2+4 a b+b^2\right ) \log \left (\tan (c+d x)+\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{2 \sqrt {2} d \left (a^2+b^2\right )^3}-\frac {a \left (35 a^4+67 a^2 b^2+24 b^4\right ) \sqrt {\tan (c+d x)}}{4 b^4 d \left (a^2+b^2\right )^2}+\frac {\left (35 a^4+67 a^2 b^2+8 b^4\right ) \tan ^{\frac {3}{2}}(c+d x)}{12 b^3 d \left (a^2+b^2\right )^2}+\frac {a^{7/2} \left (35 a^4+102 a^2 b^2+99 b^4\right ) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {\tan (c+d x)}}{\sqrt {a}}\right )}{4 b^{9/2} d \left (a^2+b^2\right )^3} \]

[Out]

1/4*a^(7/2)*(35*a^4+102*a^2*b^2+99*b^4)*arctan(b^(1/2)*tan(d*x+c)^(1/2)/a^(1/2))/b^(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*a*(35*a^4+67*a^2*b^2+24*b^4)*tan(d*x+c)^(1/2)/b^4/(a^2+b^2)^2/d+1/12*(35*a^4+67*a^2*b^2+8*b^4)*t
an(d*x+c)^(3/2)/b^3/(a^2+b^2)^2/d-1/2*a^2*tan(d*x+c)^(7/2)/b/(a^2+b^2)/d/(a+b*tan(d*x+c))^2-1/4*a^2*(7*a^2+15*
b^2)*tan(d*x+c)^(5/2)/b^2/(a^2+b^2)^2/d/(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 = 14, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.609, Rules used = {3565, 3645, 3647, 3653, 3534, 1168, 1162, 617, 204, 1165, 628, 3634, 63, 205} \[ \frac {a^{7/2} \left (102 a^2 b^2+35 a^4+99 b^4\right ) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {\tan (c+d x)}}{\sqrt {a}}\right )}{4 b^{9/2} d \left (a^2+b^2\right )^3}-\frac {a^2 \left (7 a^2+15 b^2\right ) \tan ^{\frac {5}{2}}(c+d x)}{4 b^2 d \left (a^2+b^2\right )^2 (a+b \tan (c+d x))}-\frac {a^2 \tan ^{\frac {7}{2}}(c+d x)}{2 b d \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}+\frac {\left (67 a^2 b^2+35 a^4+8 b^4\right ) \tan ^{\frac {3}{2}}(c+d x)}{12 b^3 d \left (a^2+b^2\right )^2}+\frac {(a+b) \left (a^2-4 a b+b^2\right ) \tan ^{-1}\left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} d \left (a^2+b^2\right )^3}-\frac {(a+b) \left (a^2-4 a b+b^2\right ) \tan ^{-1}\left (\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{\sqrt {2} d \left (a^2+b^2\right )^3}-\frac {a \left (67 a^2 b^2+35 a^4+24 b^4\right ) \sqrt {\tan (c+d x)}}{4 b^4 d \left (a^2+b^2\right )^2}+\frac {(a-b) \left (a^2+4 a b+b^2\right ) \log \left (\tan (c+d x)-\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{2 \sqrt {2} d \left (a^2+b^2\right )^3}-\frac {(a-b) \left (a^2+4 a b+b^2\right ) \log \left (\tan (c+d x)+\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{2 \sqrt {2} d \left (a^2+b^2\right )^3} \]

Antiderivative was successfully verified.

[In]

Int[Tan[c + d*x]^(11/2)/(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^(7/2)*(35*a^4 + 102*a^2
*b^2 + 99*b^4)*ArcTan[(Sqrt[b]*Sqrt[Tan[c + d*x]])/Sqrt[a]])/(4*b^(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) - (a*(35*a^4 + 67*a
^2*b^2 + 24*b^4)*Sqrt[Tan[c + d*x]])/(4*b^4*(a^2 + b^2)^2*d) + ((35*a^4 + 67*a^2*b^2 + 8*b^4)*Tan[c + d*x]^(3/
2))/(12*b^3*(a^2 + b^2)^2*d) - (a^2*Tan[c + d*x]^(7/2))/(2*b*(a^2 + b^2)*d*(a + b*Tan[c + d*x])^2) - (a^2*(7*a
^2 + 15*b^2)*Tan[c + d*x]^(5/2))/(4*b^2*(a^2 + b^2)^2*d*(a + b*Tan[c + d*x]))

Rule 63

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

Rule 204

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

Rule 205

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

Rule 617

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

Rule 628

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

Rule 1162

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

Rule 1165

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

Rule 1168

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

Rule 3534

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

Rule 3565

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 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 3645

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*T
an[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 3647

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

Rule 3653

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

Rubi steps

\begin {align*} \int \frac {\tan ^{\frac {11}{2}}(c+d x)}{(a+b \tan (c+d x))^3} \, dx &=-\frac {a^2 \tan ^{\frac {7}{2}}(c+d x)}{2 b \left (a^2+b^2\right ) d (a+b \tan (c+d x))^2}+\frac {\int \frac {\tan ^{\frac {5}{2}}(c+d x) \left (\frac {7 a^2}{2}-2 a b \tan (c+d x)+\frac {1}{2} \left (7 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 {7}{2}}(c+d x)}{2 b \left (a^2+b^2\right ) d (a+b \tan (c+d x))^2}-\frac {a^2 \left (7 a^2+15 b^2\right ) \tan ^{\frac {5}{2}}(c+d x)}{4 b^2 \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))}+\frac {\int \frac {\tan ^{\frac {3}{2}}(c+d x) \left (\frac {5}{4} a^2 \left (7 a^2+15 b^2\right )-4 a b^3 \tan (c+d x)+\frac {1}{4} \left (35 a^4+67 a^2 b^2+8 b^4\right ) \tan ^2(c+d x)\right )}{a+b \tan (c+d x)} \, dx}{2 b^2 \left (a^2+b^2\right )^2}\\ &=\frac {\left (35 a^4+67 a^2 b^2+8 b^4\right ) \tan ^{\frac {3}{2}}(c+d x)}{12 b^3 \left (a^2+b^2\right )^2 d}-\frac {a^2 \tan ^{\frac {7}{2}}(c+d x)}{2 b \left (a^2+b^2\right ) d (a+b \tan (c+d x))^2}-\frac {a^2 \left (7 a^2+15 b^2\right ) \tan ^{\frac {5}{2}}(c+d x)}{4 b^2 \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))}+\frac {\int \frac {\sqrt {\tan (c+d x)} \left (-\frac {3}{8} a \left (35 a^4+67 a^2 b^2+8 b^4\right )+3 b^3 \left (a^2-b^2\right ) \tan (c+d x)-\frac {3}{8} a \left (35 a^4+67 a^2 b^2+24 b^4\right ) \tan ^2(c+d x)\right )}{a+b \tan (c+d x)} \, dx}{3 b^3 \left (a^2+b^2\right )^2}\\ &=-\frac {a \left (35 a^4+67 a^2 b^2+24 b^4\right ) \sqrt {\tan (c+d x)}}{4 b^4 \left (a^2+b^2\right )^2 d}+\frac {\left (35 a^4+67 a^2 b^2+8 b^4\right ) \tan ^{\frac {3}{2}}(c+d x)}{12 b^3 \left (a^2+b^2\right )^2 d}-\frac {a^2 \tan ^{\frac {7}{2}}(c+d x)}{2 b \left (a^2+b^2\right ) d (a+b \tan (c+d x))^2}-\frac {a^2 \left (7 a^2+15 b^2\right ) \tan ^{\frac {5}{2}}(c+d x)}{4 b^2 \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))}+\frac {2 \int \frac {\frac {3}{16} a^2 \left (35 a^4+67 a^2 b^2+24 b^4\right )+3 a b^5 \tan (c+d x)+\frac {3}{16} \left (35 a^6+67 a^4 b^2+32 a^2 b^4-8 b^6\right ) \tan ^2(c+d x)}{\sqrt {\tan (c+d x)} (a+b \tan (c+d x))} \, dx}{3 b^4 \left (a^2+b^2\right )^2}\\ &=-\frac {a \left (35 a^4+67 a^2 b^2+24 b^4\right ) \sqrt {\tan (c+d x)}}{4 b^4 \left (a^2+b^2\right )^2 d}+\frac {\left (35 a^4+67 a^2 b^2+8 b^4\right ) \tan ^{\frac {3}{2}}(c+d x)}{12 b^3 \left (a^2+b^2\right )^2 d}-\frac {a^2 \tan ^{\frac {7}{2}}(c+d x)}{2 b \left (a^2+b^2\right ) d (a+b \tan (c+d x))^2}-\frac {a^2 \left (7 a^2+15 b^2\right ) \tan ^{\frac {5}{2}}(c+d x)}{4 b^2 \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))}+\frac {2 \int \frac {-\frac {3}{2} a b^4 \left (a^2-3 b^2\right )+\frac {3}{2} b^5 \left (3 a^2-b^2\right ) \tan (c+d x)}{\sqrt {\tan (c+d x)}} \, dx}{3 b^4 \left (a^2+b^2\right )^3}+\frac {\left (a^4 \left (35 a^4+102 a^2 b^2+99 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^4 \left (a^2+b^2\right )^3}\\ &=-\frac {a \left (35 a^4+67 a^2 b^2+24 b^4\right ) \sqrt {\tan (c+d x)}}{4 b^4 \left (a^2+b^2\right )^2 d}+\frac {\left (35 a^4+67 a^2 b^2+8 b^4\right ) \tan ^{\frac {3}{2}}(c+d x)}{12 b^3 \left (a^2+b^2\right )^2 d}-\frac {a^2 \tan ^{\frac {7}{2}}(c+d x)}{2 b \left (a^2+b^2\right ) d (a+b \tan (c+d x))^2}-\frac {a^2 \left (7 a^2+15 b^2\right ) \tan ^{\frac {5}{2}}(c+d x)}{4 b^2 \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))}+\frac {4 \operatorname {Subst}\left (\int \frac {-\frac {3}{2} a b^4 \left (a^2-3 b^2\right )+\frac {3}{2} b^5 \left (3 a^2-b^2\right ) x^2}{1+x^4} \, dx,x,\sqrt {\tan (c+d x)}\right )}{3 b^4 \left (a^2+b^2\right )^3 d}+\frac {\left (a^4 \left (35 a^4+102 a^2 b^2+99 b^4\right )\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {x} (a+b x)} \, dx,x,\tan (c+d x)\right )}{8 b^4 \left (a^2+b^2\right )^3 d}\\ &=-\frac {a \left (35 a^4+67 a^2 b^2+24 b^4\right ) \sqrt {\tan (c+d x)}}{4 b^4 \left (a^2+b^2\right )^2 d}+\frac {\left (35 a^4+67 a^2 b^2+8 b^4\right ) \tan ^{\frac {3}{2}}(c+d x)}{12 b^3 \left (a^2+b^2\right )^2 d}-\frac {a^2 \tan ^{\frac {7}{2}}(c+d x)}{2 b \left (a^2+b^2\right ) d (a+b \tan (c+d x))^2}-\frac {a^2 \left (7 a^2+15 b^2\right ) \tan ^{\frac {5}{2}}(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 ) \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 (a^4 \left (35 a^4+102 a^2 b^2+99 b^4\right )\right ) \operatorname {Subst}\left (\int \frac {1}{a+b x^2} \, dx,x,\sqrt {\tan (c+d x)}\right )}{4 b^4 \left (a^2+b^2\right )^3 d}\\ &=\frac {a^{7/2} \left (35 a^4+102 a^2 b^2+99 b^4\right ) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {\tan (c+d x)}}{\sqrt {a}}\right )}{4 b^{9/2} \left (a^2+b^2\right )^3 d}-\frac {a \left (35 a^4+67 a^2 b^2+24 b^4\right ) \sqrt {\tan (c+d x)}}{4 b^4 \left (a^2+b^2\right )^2 d}+\frac {\left (35 a^4+67 a^2 b^2+8 b^4\right ) \tan ^{\frac {3}{2}}(c+d x)}{12 b^3 \left (a^2+b^2\right )^2 d}-\frac {a^2 \tan ^{\frac {7}{2}}(c+d x)}{2 b \left (a^2+b^2\right ) d (a+b \tan (c+d x))^2}-\frac {a^2 \left (7 a^2+15 b^2\right ) \tan ^{\frac {5}{2}}(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 ) \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 {a^{7/2} \left (35 a^4+102 a^2 b^2+99 b^4\right ) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {\tan (c+d x)}}{\sqrt {a}}\right )}{4 b^{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 {a \left (35 a^4+67 a^2 b^2+24 b^4\right ) \sqrt {\tan (c+d x)}}{4 b^4 \left (a^2+b^2\right )^2 d}+\frac {\left (35 a^4+67 a^2 b^2+8 b^4\right ) \tan ^{\frac {3}{2}}(c+d x)}{12 b^3 \left (a^2+b^2\right )^2 d}-\frac {a^2 \tan ^{\frac {7}{2}}(c+d x)}{2 b \left (a^2+b^2\right ) d (a+b \tan (c+d x))^2}-\frac {a^2 \left (7 a^2+15 b^2\right ) \tan ^{\frac {5}{2}}(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 ) \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 {a^{7/2} \left (35 a^4+102 a^2 b^2+99 b^4\right ) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {\tan (c+d x)}}{\sqrt {a}}\right )}{4 b^{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 {a \left (35 a^4+67 a^2 b^2+24 b^4\right ) \sqrt {\tan (c+d x)}}{4 b^4 \left (a^2+b^2\right )^2 d}+\frac {\left (35 a^4+67 a^2 b^2+8 b^4\right ) \tan ^{\frac {3}{2}}(c+d x)}{12 b^3 \left (a^2+b^2\right )^2 d}-\frac {a^2 \tan ^{\frac {7}{2}}(c+d x)}{2 b \left (a^2+b^2\right ) d (a+b \tan (c+d x))^2}-\frac {a^2 \left (7 a^2+15 b^2\right ) \tan ^{\frac {5}{2}}(c+d x)}{4 b^2 \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))}\\ \end {align*}

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Mathematica [C]  time = 6.40, size = 723, normalized size = 1.47 \[ \frac {b^2 \tan ^{\frac {13}{2}}(c+d x)}{2 a d \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}+\frac {-\frac {b \tan ^{\frac {11}{2}}(c+d x)}{d (a+b \tan (c+d x))}+\frac {2 \left (\frac {9 a b \tan ^{\frac {9}{2}}(c+d x)}{2 d (a+b \tan (c+d x))}+\frac {2 \left (-\frac {63 a^2 b \tan ^{\frac {7}{2}}(c+d x)}{4 d (a+b \tan (c+d x))}+\frac {2 \left (\frac {105 a b \left (7 a^2+4 b^2\right ) \tan ^{\frac {5}{2}}(c+d x)}{8 d (a+b \tan (c+d x))}+\frac {2 \left (-\frac {315 a^2 b \left (35 a^2+32 b^2\right ) \tan ^{\frac {3}{2}}(c+d x)}{16 d (a+b \tan (c+d x))}+\frac {2 \left (-\frac {945 a b \left (35 a^4+32 a^2 b^2-4 b^4\right ) \sqrt {\tan (c+d x)}}{32 d (a+b \tan (c+d x))}-\frac {2 \left (\frac {\left (-a \left (\frac {945}{128} a^5 b^2 \left (35 a^2+32 b^2\right )-\frac {945 a b^8}{32}\right )-\frac {945}{128} a^2 b^4 \left (35 a^4+32 a^2 b^2-4 b^4\right )\right ) \sqrt {\tan (c+d x)}}{a d \left (a^2+b^2\right ) (a+b \tan (c+d x))}+\frac {\frac {2 \left (\frac {945 a^4 b^8}{16}-\frac {945}{256} a^4 b^4 \left (35 a^4+67 a^2 b^2+24 b^4\right )-\frac {945}{256} a^4 b^2 \left (35 a^6+67 a^4 b^2+32 a^2 b^4-8 b^6\right )\right ) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {\tan (c+d x)}}{\sqrt {a}}\right )}{\sqrt {a} \sqrt {b} d \left (a^2+b^2\right )}+\frac {-\frac {\sqrt [4]{-1} \left (\frac {945}{32} a^3 b^6 \left (a^2-3 b^2\right )+\frac {945}{32} i a^2 b^7 \left (3 a^2-b^2\right )\right ) \tan ^{-1}\left ((-1)^{3/4} \sqrt {\tan (c+d x)}\right )}{d}-\frac {\sqrt [4]{-1} \left (\frac {945}{32} a^3 b^6 \left (a^2-3 b^2\right )-\frac {945}{32} i a^2 b^7 \left (3 a^2-b^2\right )\right ) \tanh ^{-1}\left ((-1)^{3/4} \sqrt {\tan (c+d x)}\right )}{d}}{a^2+b^2}}{a \left (a^2+b^2\right )}\right )}{b}\right )}{b}\right )}{3 b}\right )}{5 b}\right )}{7 b}\right )}{9 b}}{2 a \left (a^2+b^2\right )} \]

Antiderivative was successfully verified.

[In]

Integrate[Tan[c + d*x]^(11/2)/(a + b*Tan[c + d*x])^3,x]

[Out]

(b^2*Tan[c + d*x]^(13/2))/(2*a*(a^2 + b^2)*d*(a + b*Tan[c + d*x])^2) + (-((b*Tan[c + d*x]^(11/2))/(d*(a + b*Ta
n[c + d*x]))) + (2*((9*a*b*Tan[c + d*x]^(9/2))/(2*d*(a + b*Tan[c + d*x])) + (2*((-63*a^2*b*Tan[c + d*x]^(7/2))
/(4*d*(a + b*Tan[c + d*x])) + (2*((105*a*b*(7*a^2 + 4*b^2)*Tan[c + d*x]^(5/2))/(8*d*(a + b*Tan[c + d*x])) + (2
*((-315*a^2*b*(35*a^2 + 32*b^2)*Tan[c + d*x]^(3/2))/(16*d*(a + b*Tan[c + d*x])) + (2*((-945*a*b*(35*a^4 + 32*a
^2*b^2 - 4*b^4)*Sqrt[Tan[c + d*x]])/(32*d*(a + b*Tan[c + d*x])) - (2*(((2*((945*a^4*b^8)/16 - (945*a^4*b^4*(35
*a^4 + 67*a^2*b^2 + 24*b^4))/256 - (945*a^4*b^2*(35*a^6 + 67*a^4*b^2 + 32*a^2*b^4 - 8*b^6))/256)*ArcTan[(Sqrt[
b]*Sqrt[Tan[c + d*x]])/Sqrt[a]])/(Sqrt[a]*Sqrt[b]*(a^2 + b^2)*d) + (-(((-1)^(1/4)*((945*a^3*b^6*(a^2 - 3*b^2))
/32 + ((945*I)/32)*a^2*b^7*(3*a^2 - b^2))*ArcTan[(-1)^(3/4)*Sqrt[Tan[c + d*x]]])/d) - ((-1)^(1/4)*((945*a^3*b^
6*(a^2 - 3*b^2))/32 - ((945*I)/32)*a^2*b^7*(3*a^2 - b^2))*ArcTanh[(-1)^(3/4)*Sqrt[Tan[c + d*x]]])/d)/(a^2 + b^
2))/(a*(a^2 + b^2)) + (((-945*a^2*b^4*(35*a^4 + 32*a^2*b^2 - 4*b^4))/128 - a*((-945*a*b^8)/32 + (945*a^5*b^2*(
35*a^2 + 32*b^2))/128))*Sqrt[Tan[c + d*x]])/(a*(a^2 + b^2)*d*(a + b*Tan[c + d*x]))))/b))/b))/(3*b)))/(5*b)))/(
7*b)))/(9*b))/(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(tan(d*x+c)^(11/2)/(a+b*tan(d*x+c))^3,x, algorithm="fricas")

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tan(d*x+c)^(11/2)/(a+b*tan(d*x+c))^3,x, algorithm="giac")

[Out]

Timed out

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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(tan(d*x+c)^(11/2)/(a+b*tan(d*x+c))^3,x)

[Out]

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

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maxima [A]  time = 0.57, size = 449, normalized size = 0.91 \[ \frac {\frac {3 \, {\left (35 \, a^{8} + 102 \, a^{6} b^{2} + 99 \, a^{4} b^{4}\right )} \arctan \left (\frac {b \sqrt {\tan \left (d x + c\right )}}{\sqrt {a b}}\right )}{{\left (a^{6} b^{4} + 3 \, a^{4} b^{6} + 3 \, a^{2} b^{8} + b^{10}\right )} \sqrt {a b}} - \frac {3 \, {\left (2 \, \sqrt {2} {\left (a^{3} - 3 \, a^{2} b - 3 \, a b^{2} + b^{3}\right )} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + 2 \, \sqrt {\tan \left (d x + c\right )}\right )}\right ) + 2 \, \sqrt {2} {\left (a^{3} - 3 \, a^{2} b - 3 \, a b^{2} + b^{3}\right )} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - 2 \, \sqrt {\tan \left (d x + c\right )}\right )}\right ) + \sqrt {2} {\left (a^{3} + 3 \, a^{2} b - 3 \, a b^{2} - b^{3}\right )} \log \left (\sqrt {2} \sqrt {\tan \left (d x + c\right )} + \tan \left (d x + c\right ) + 1\right ) - \sqrt {2} {\left (a^{3} + 3 \, a^{2} b - 3 \, a b^{2} - b^{3}\right )} \log \left (-\sqrt {2} \sqrt {\tan \left (d x + c\right )} + \tan \left (d x + c\right ) + 1\right )\right )}}{a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}} - \frac {3 \, {\left ({\left (13 \, a^{6} b + 21 \, a^{4} b^{3}\right )} \tan \left (d x + c\right )^{\frac {3}{2}} + {\left (11 \, a^{7} + 19 \, a^{5} b^{2}\right )} \sqrt {\tan \left (d x + c\right )}\right )}}{a^{6} b^{4} + 2 \, a^{4} b^{6} + a^{2} b^{8} + {\left (a^{4} b^{6} + 2 \, a^{2} b^{8} + b^{10}\right )} \tan \left (d x + c\right )^{2} + 2 \, {\left (a^{5} b^{5} + 2 \, a^{3} b^{7} + a b^{9}\right )} \tan \left (d x + c\right )} + \frac {8 \, {\left (b \tan \left (d x + c\right )^{\frac {3}{2}} - 9 \, a \sqrt {\tan \left (d x + c\right )}\right )}}{b^{4}}}{12 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tan(d*x+c)^(11/2)/(a+b*tan(d*x+c))^3,x, algorithm="maxima")

[Out]

1/12*(3*(35*a^8 + 102*a^6*b^2 + 99*a^4*b^4)*arctan(b*sqrt(tan(d*x + c))/sqrt(a*b))/((a^6*b^4 + 3*a^4*b^6 + 3*a
^2*b^8 + b^10)*sqrt(a*b)) - 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) - 3*((13*a^6*b + 21*a^4*b^3)*tan(d*x + c)^(3/2) + (11*a^7 + 19*a^5*b^2)*sqrt(tan(d*x + c)))/(a^6*b^4 +
2*a^4*b^6 + a^2*b^8 + (a^4*b^6 + 2*a^2*b^8 + b^10)*tan(d*x + c)^2 + 2*(a^5*b^5 + 2*a^3*b^7 + a*b^9)*tan(d*x +
c)) + 8*(b*tan(d*x + c)^(3/2) - 9*a*sqrt(tan(d*x + c)))/b^4)/d

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mupad [B]  time = 20.45, size = 18832, normalized size = 38.20 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(tan(c + d*x)^(11/2)/(a + b*tan(c + d*x))^3,x)

[Out]

(2*tan(c + d*x)^(3/2))/(3*b^3*d) - atan((((((32*a*b^22*d^2 - 2450*a^23*d^2 + 192*a^3*b^20*d^2 + 19488*a^5*b^18
*d^2 - 24128*a^7*b^16*d^2 - 180858*a^9*b^14*d^2 - 146126*a^11*b^12*d^2 + 208974*a^13*b^10*d^2 + 452586*a^15*b^
8*d^2 + 330770*a^17*b^6*d^2 + 106102*a^19*b^4*d^2 + 7770*a^21*b^2*d^2)/(2*(b^23*d^5 + 8*a^2*b^21*d^5 + 28*a^4*
b^19*d^5 + 56*a^6*b^17*d^5 + 70*a^8*b^15*d^5 + 56*a^10*b^13*d^5 + 28*a^12*b^11*d^5 + 8*a^14*b^9*d^5 + a^16*b^7
*d^5)) - (((((640*a^2*b^27*d^4 + 9536*a^4*b^25*d^4 + 55488*a^6*b^23*d^4 + 177408*a^8*b^21*d^4 + 354816*a^10*b^
19*d^4 + 470400*a^12*b^17*d^4 + 422016*a^14*b^15*d^4 + 254208*a^16*b^13*d^4 + 98688*a^18*b^11*d^4 + 22336*a^20
*b^9*d^4 + 2240*a^22*b^7*d^4)/(2*(b^23*d^5 + 8*a^2*b^21*d^5 + 28*a^4*b^19*d^5 + 56*a^6*b^17*d^5 + 70*a^8*b^15*
d^5 + 56*a^10*b^13*d^5 + 28*a^12*b^11*d^5 + 8*a^14*b^9*d^5 + a^16*b^7*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^32*d^4 + 4608*a^2*b^30*d^4 + 17920*a^4*b^28*d^4 + 38400*a^6*b^26*d^4 + 46080*a^8*b^24*d^4 + 21504*a^10*b^22*
d^4 - 21504*a^12*b^20*d^4 - 46080*a^14*b^18*d^4 - 38400*a^16*b^16*d^4 - 17920*a^18*b^14*d^4 - 4608*a^20*b^12*d
^4 - 512*a^22*b^10*d^4))/(4*(b^23*d^4 + 8*a^2*b^21*d^4 + 28*a^4*b^19*d^4 + 56*a^6*b^17*d^4 + 70*a^8*b^15*d^4 +
56*a^10*b^13*d^4 + 28*a^12*b^11*d^4 + 8*a^14*b^9*d^4 + a^16*b^7*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^25*d^2 + 9800*a^25*b*d^2 + 1024*a^3*b^23*d^2 - 8448*a^5*b^21*d^2 - 14336*a^7*b^19*d^2 + 74440*a^9*b^17*d^2
+ 480320*a^11*b^15*d^2 + 1258208*a^13*b^13*d^2 + 1894848*a^15*b^11*d^2 + 1794928*a^17*b^9*d^2 + 1098176*a^19*
b^7*d^2 + 425952*a^21*b^5*d^2 + 96320*a^23*b^3*d^2))/(2*(b^23*d^4 + 8*a^2*b^21*d^4 + 28*a^4*b^19*d^4 + 56*a^6*
b^17*d^4 + 70*a^8*b^15*d^4 + 56*a^10*b^13*d^4 + 28*a^12*b^11*d^4 + 8*a^14*b^9*d^4 + a^16*b^7*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)
*(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)*(1225*a^20 + 32*b^20 + 128*a^2*b^18 + 192*a^4*b^16 + 128*a^6*b^14 + 9833*a^8*
b^12 - 38610*a^10*b^10 - 94041*a^12*b^8 - 76668*a^14*b^6 - 24281*a^16*b^4 - 210*a^18*b^2))/(2*(b^23*d^4 + 8*a^
2*b^21*d^4 + 28*a^4*b^19*d^4 + 56*a^6*b^17*d^4 + 70*a^8*b^15*d^4 + 56*a^10*b^13*d^4 + 28*a^12*b^11*d^4 + 8*a^1
4*b^9*d^4 + a^16*b^7*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 - ((((32*a*b^22*d^2 - 2450*a^23*d^2 + 192*a^3*b^20*d^2 + 19488*a^5*b^18*
d^2 - 24128*a^7*b^16*d^2 - 180858*a^9*b^14*d^2 - 146126*a^11*b^12*d^2 + 208974*a^13*b^10*d^2 + 452586*a^15*b^8
*d^2 + 330770*a^17*b^6*d^2 + 106102*a^19*b^4*d^2 + 7770*a^21*b^2*d^2)/(2*(b^23*d^5 + 8*a^2*b^21*d^5 + 28*a^4*b
^19*d^5 + 56*a^6*b^17*d^5 + 70*a^8*b^15*d^5 + 56*a^10*b^13*d^5 + 28*a^12*b^11*d^5 + 8*a^14*b^9*d^5 + a^16*b^7*
d^5)) - (((((640*a^2*b^27*d^4 + 9536*a^4*b^25*d^4 + 55488*a^6*b^23*d^4 + 177408*a^8*b^21*d^4 + 354816*a^10*b^1
9*d^4 + 470400*a^12*b^17*d^4 + 422016*a^14*b^15*d^4 + 254208*a^16*b^13*d^4 + 98688*a^18*b^11*d^4 + 22336*a^20*
b^9*d^4 + 2240*a^22*b^7*d^4)/(2*(b^23*d^5 + 8*a^2*b^21*d^5 + 28*a^4*b^19*d^5 + 56*a^6*b^17*d^5 + 70*a^8*b^15*d
^5 + 56*a^10*b^13*d^5 + 28*a^12*b^11*d^5 + 8*a^14*b^9*d^5 + a^16*b^7*d^5)) + (tan(c + d*x)^(1/2)*(1/(b^6*d^2*1
i - 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
^32*d^4 + 4608*a^2*b^30*d^4 + 17920*a^4*b^28*d^4 + 38400*a^6*b^26*d^4 + 46080*a^8*b^24*d^4 + 21504*a^10*b^22*d
^4 - 21504*a^12*b^20*d^4 - 46080*a^14*b^18*d^4 - 38400*a^16*b^16*d^4 - 17920*a^18*b^14*d^4 - 4608*a^20*b^12*d^
4 - 512*a^22*b^10*d^4))/(4*(b^23*d^4 + 8*a^2*b^21*d^4 + 28*a^4*b^19*d^4 + 56*a^6*b^17*d^4 + 70*a^8*b^15*d^4 +
56*a^10*b^13*d^4 + 28*a^12*b^11*d^4 + 8*a^14*b^9*d^4 + a^16*b^7*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^25*d^2 + 9800*a^25*b*d^2 + 1024*a^3*b^23*d^2 - 8448*a^5*b^21*d^2 - 14336*a^7*b^19*d^2 + 74440*a^9*b^17*d^2
+ 480320*a^11*b^15*d^2 + 1258208*a^13*b^13*d^2 + 1894848*a^15*b^11*d^2 + 1794928*a^17*b^9*d^2 + 1098176*a^19*b
^7*d^2 + 425952*a^21*b^5*d^2 + 96320*a^23*b^3*d^2))/(2*(b^23*d^4 + 8*a^2*b^21*d^4 + 28*a^4*b^19*d^4 + 56*a^6*b
^17*d^4 + 70*a^8*b^15*d^4 + 56*a^10*b^13*d^4 + 28*a^12*b^11*d^4 + 8*a^14*b^9*d^4 + a^16*b^7*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)*
(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)*(1225*a^20 + 32*b^20 + 128*a^2*b^18 + 192*a^4*b^16 + 128*a^6*b^14 + 9833*a^8*b
^12 - 38610*a^10*b^10 - 94041*a^12*b^8 - 76668*a^14*b^6 - 24281*a^16*b^4 - 210*a^18*b^2))/(2*(b^23*d^4 + 8*a^2
*b^21*d^4 + 28*a^4*b^19*d^4 + 56*a^6*b^17*d^4 + 70*a^8*b^15*d^4 + 56*a^10*b^13*d^4 + 28*a^12*b^11*d^4 + 8*a^14
*b^9*d^4 + a^16*b^7*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)/((1225*a^16*b - 792*a^4*b^13 - 1608*a^6*b^11 + 8705*a^8*b^9 + 19916*a^10
*b^7 + 17334*a^12*b^5 + 7140*a^14*b^3)/(b^23*d^5 + 8*a^2*b^21*d^5 + 28*a^4*b^19*d^5 + 56*a^6*b^17*d^5 + 70*a^8
*b^15*d^5 + 56*a^10*b^13*d^5 + 28*a^12*b^11*d^5 + 8*a^14*b^9*d^5 + a^16*b^7*d^5) + ((((32*a*b^22*d^2 - 2450*a^
23*d^2 + 192*a^3*b^20*d^2 + 19488*a^5*b^18*d^2 - 24128*a^7*b^16*d^2 - 180858*a^9*b^14*d^2 - 146126*a^11*b^12*d
^2 + 208974*a^13*b^10*d^2 + 452586*a^15*b^8*d^2 + 330770*a^17*b^6*d^2 + 106102*a^19*b^4*d^2 + 7770*a^21*b^2*d^
2)/(2*(b^23*d^5 + 8*a^2*b^21*d^5 + 28*a^4*b^19*d^5 + 56*a^6*b^17*d^5 + 70*a^8*b^15*d^5 + 56*a^10*b^13*d^5 + 28
*a^12*b^11*d^5 + 8*a^14*b^9*d^5 + a^16*b^7*d^5)) - (((((640*a^2*b^27*d^4 + 9536*a^4*b^25*d^4 + 55488*a^6*b^23*
d^4 + 177408*a^8*b^21*d^4 + 354816*a^10*b^19*d^4 + 470400*a^12*b^17*d^4 + 422016*a^14*b^15*d^4 + 254208*a^16*b
^13*d^4 + 98688*a^18*b^11*d^4 + 22336*a^20*b^9*d^4 + 2240*a^22*b^7*d^4)/(2*(b^23*d^5 + 8*a^2*b^21*d^5 + 28*a^4
*b^19*d^5 + 56*a^6*b^17*d^5 + 70*a^8*b^15*d^5 + 56*a^10*b^13*d^5 + 28*a^12*b^11*d^5 + 8*a^14*b^9*d^5 + a^16*b^
7*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^32*d^4 + 4608*a^2*b^30*d^4 + 17920*a^4*b^28*d^4 + 38400*a^6*b^26*d
^4 + 46080*a^8*b^24*d^4 + 21504*a^10*b^22*d^4 - 21504*a^12*b^20*d^4 - 46080*a^14*b^18*d^4 - 38400*a^16*b^16*d^
4 - 17920*a^18*b^14*d^4 - 4608*a^20*b^12*d^4 - 512*a^22*b^10*d^4))/(4*(b^23*d^4 + 8*a^2*b^21*d^4 + 28*a^4*b^19
*d^4 + 56*a^6*b^17*d^4 + 70*a^8*b^15*d^4 + 56*a^10*b^13*d^4 + 28*a^12*b^11*d^4 + 8*a^14*b^9*d^4 + a^16*b^7*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*1
5i))^(1/2))/2 + (tan(c + d*x)^(1/2)*(1472*a*b^25*d^2 + 9800*a^25*b*d^2 + 1024*a^3*b^23*d^2 - 8448*a^5*b^21*d^2
- 14336*a^7*b^19*d^2 + 74440*a^9*b^17*d^2 + 480320*a^11*b^15*d^2 + 1258208*a^13*b^13*d^2 + 1894848*a^15*b^11*
d^2 + 1794928*a^17*b^9*d^2 + 1098176*a^19*b^7*d^2 + 425952*a^21*b^5*d^2 + 96320*a^23*b^3*d^2))/(2*(b^23*d^4 +
8*a^2*b^21*d^4 + 28*a^4*b^19*d^4 + 56*a^6*b^17*d^4 + 70*a^8*b^15*d^4 + 56*a^10*b^13*d^4 + 28*a^12*b^11*d^4 + 8
*a^14*b^9*d^4 + a^16*b^7*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)*(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)*(1225*a^20 + 32*b^20 + 128*a^2*b^18
+ 192*a^4*b^16 + 128*a^6*b^14 + 9833*a^8*b^12 - 38610*a^10*b^10 - 94041*a^12*b^8 - 76668*a^14*b^6 - 24281*a^1
6*b^4 - 210*a^18*b^2))/(2*(b^23*d^4 + 8*a^2*b^21*d^4 + 28*a^4*b^19*d^4 + 56*a^6*b^17*d^4 + 70*a^8*b^15*d^4 + 5
6*a^10*b^13*d^4 + 28*a^12*b^11*d^4 + 8*a^14*b^9*d^4 + a^16*b^7*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) + ((((32*a*b^22*d^2 - 2450*a^23*d
^2 + 192*a^3*b^20*d^2 + 19488*a^5*b^18*d^2 - 24128*a^7*b^16*d^2 - 180858*a^9*b^14*d^2 - 146126*a^11*b^12*d^2 +
208974*a^13*b^10*d^2 + 452586*a^15*b^8*d^2 + 330770*a^17*b^6*d^2 + 106102*a^19*b^4*d^2 + 7770*a^21*b^2*d^2)/(
2*(b^23*d^5 + 8*a^2*b^21*d^5 + 28*a^4*b^19*d^5 + 56*a^6*b^17*d^5 + 70*a^8*b^15*d^5 + 56*a^10*b^13*d^5 + 28*a^1
2*b^11*d^5 + 8*a^14*b^9*d^5 + a^16*b^7*d^5)) - (((((640*a^2*b^27*d^4 + 9536*a^4*b^25*d^4 + 55488*a^6*b^23*d^4
+ 177408*a^8*b^21*d^4 + 354816*a^10*b^19*d^4 + 470400*a^12*b^17*d^4 + 422016*a^14*b^15*d^4 + 254208*a^16*b^13*
d^4 + 98688*a^18*b^11*d^4 + 22336*a^20*b^9*d^4 + 2240*a^22*b^7*d^4)/(2*(b^23*d^5 + 8*a^2*b^21*d^5 + 28*a^4*b^1
9*d^5 + 56*a^6*b^17*d^5 + 70*a^8*b^15*d^5 + 56*a^10*b^13*d^5 + 28*a^12*b^11*d^5 + 8*a^14*b^9*d^5 + a^16*b^7*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^32*d^4 + 4608*a^2*b^30*d^4 + 17920*a^4*b^28*d^4 + 38400*a^6*b^26*d^4 +
46080*a^8*b^24*d^4 + 21504*a^10*b^22*d^4 - 21504*a^12*b^20*d^4 - 46080*a^14*b^18*d^4 - 38400*a^16*b^16*d^4 -
17920*a^18*b^14*d^4 - 4608*a^20*b^12*d^4 - 512*a^22*b^10*d^4))/(4*(b^23*d^4 + 8*a^2*b^21*d^4 + 28*a^4*b^19*d^4
+ 56*a^6*b^17*d^4 + 70*a^8*b^15*d^4 + 56*a^10*b^13*d^4 + 28*a^12*b^11*d^4 + 8*a^14*b^9*d^4 + a^16*b^7*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^25*d^2 + 9800*a^25*b*d^2 + 1024*a^3*b^23*d^2 - 8448*a^5*b^21*d^2 - 1
4336*a^7*b^19*d^2 + 74440*a^9*b^17*d^2 + 480320*a^11*b^15*d^2 + 1258208*a^13*b^13*d^2 + 1894848*a^15*b^11*d^2
+ 1794928*a^17*b^9*d^2 + 1098176*a^19*b^7*d^2 + 425952*a^21*b^5*d^2 + 96320*a^23*b^3*d^2))/(2*(b^23*d^4 + 8*a^
2*b^21*d^4 + 28*a^4*b^19*d^4 + 56*a^6*b^17*d^4 + 70*a^8*b^15*d^4 + 56*a^10*b^13*d^4 + 28*a^12*b^11*d^4 + 8*a^1
4*b^9*d^4 + a^16*b^7*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)*(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*1
5i - 20*a^3*b^3*d^2 + a^4*b^2*d^2*15i))^(1/2))/2 - (tan(c + d*x)^(1/2)*(1225*a^20 + 32*b^20 + 128*a^2*b^18 + 1
92*a^4*b^16 + 128*a^6*b^14 + 9833*a^8*b^12 - 38610*a^10*b^10 - 94041*a^12*b^8 - 76668*a^14*b^6 - 24281*a^16*b^
4 - 210*a^18*b^2))/(2*(b^23*d^4 + 8*a^2*b^21*d^4 + 28*a^4*b^19*d^4 + 56*a^6*b^17*d^4 + 70*a^8*b^15*d^4 + 56*a^
10*b^13*d^4 + 28*a^12*b^11*d^4 + 8*a^14*b^9*d^4 + a^16*b^7*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)))*(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)^(3/2)*(1
3*a^6*b + 21*a^4*b^3))/(4*(a^4 + b^4 + 2*a^2*b^2)) + (a*tan(c + d*x)^(1/2)*(11*a^6 + 19*a^4*b^2))/(4*(a^4 + b^
4 + 2*a^2*b^2)))/(a^2*b^4*d + b^6*d*tan(c + d*x)^2 + 2*a*b^5*d*tan(c + d*x)) - atan(((((32*a*b^22*d^2 - 2450*a
^23*d^2 + 192*a^3*b^20*d^2 + 19488*a^5*b^18*d^2 - 24128*a^7*b^16*d^2 - 180858*a^9*b^14*d^2 - 146126*a^11*b^12*
d^2 + 208974*a^13*b^10*d^2 + 452586*a^15*b^8*d^2 + 330770*a^17*b^6*d^2 + 106102*a^19*b^4*d^2 + 7770*a^21*b^2*d
^2)/(b^23*d^5 + 8*a^2*b^21*d^5 + 28*a^4*b^19*d^5 + 56*a^6*b^17*d^5 + 70*a^8*b^15*d^5 + 56*a^10*b^13*d^5 + 28*a
^12*b^11*d^5 + 8*a^14*b^9*d^5 + a^16*b^7*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)*(((640*a^2*b^27*d^4 + 9536*a^4*b^25*d^4 + 55488*a^6*b^2
3*d^4 + 177408*a^8*b^21*d^4 + 354816*a^10*b^19*d^4 + 470400*a^12*b^17*d^4 + 422016*a^14*b^15*d^4 + 254208*a^16
*b^13*d^4 + 98688*a^18*b^11*d^4 + 22336*a^20*b^9*d^4 + 2240*a^22*b^7*d^4)/(b^23*d^5 + 8*a^2*b^21*d^5 + 28*a^4*
b^19*d^5 + 56*a^6*b^17*d^5 + 70*a^8*b^15*d^5 + 56*a^10*b^13*d^5 + 28*a^12*b^11*d^5 + 8*a^14*b^9*d^5 + a^16*b^7
*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^32*d^4 + 4608*a^2*b^30*d^4 + 17920*a^4*b^28*d^4 + 38400*a^6*b^26*d^4
+ 46080*a^8*b^24*d^4 + 21504*a^10*b^22*d^4 - 21504*a^12*b^20*d^4 - 46080*a^14*b^18*d^4 - 38400*a^16*b^16*d^4
- 17920*a^18*b^14*d^4 - 4608*a^20*b^12*d^4 - 512*a^22*b^10*d^4))/(b^23*d^4 + 8*a^2*b^21*d^4 + 28*a^4*b^19*d^4
+ 56*a^6*b^17*d^4 + 70*a^8*b^15*d^4 + 56*a^10*b^13*d^4 + 28*a^12*b^11*d^4 + 8*a^14*b^9*d^4 + a^16*b^7*d^4))*(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) + (tan(c + d*x)^(1/2)*(1472*a*b^25*d^2 + 9800*a^25*b*d^2 + 1024*a^3*b^23*d^2 - 8448*a^5*b^21*d^2 - 14336*
a^7*b^19*d^2 + 74440*a^9*b^17*d^2 + 480320*a^11*b^15*d^2 + 1258208*a^13*b^13*d^2 + 1894848*a^15*b^11*d^2 + 179
4928*a^17*b^9*d^2 + 1098176*a^19*b^7*d^2 + 425952*a^21*b^5*d^2 + 96320*a^23*b^3*d^2))/(b^23*d^4 + 8*a^2*b^21*d
^4 + 28*a^4*b^19*d^4 + 56*a^6*b^17*d^4 + 70*a^8*b^15*d^4 + 56*a^10*b^13*d^4 + 28*a^12*b^11*d^4 + 8*a^14*b^9*d^
4 + a^16*b^7*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) + (tan(c + d*x)^(1/2)*(1225*a^20 + 32*b^20 + 128*a^2*b^18 + 192*a^4*b^16 + 128*a^6*
b^14 + 9833*a^8*b^12 - 38610*a^10*b^10 - 94041*a^12*b^8 - 76668*a^14*b^6 - 24281*a^16*b^4 - 210*a^18*b^2))/(b^
23*d^4 + 8*a^2*b^21*d^4 + 28*a^4*b^19*d^4 + 56*a^6*b^17*d^4 + 70*a^8*b^15*d^4 + 56*a^10*b^13*d^4 + 28*a^12*b^1
1*d^4 + 8*a^14*b^9*d^4 + a^16*b^7*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 - (((32*a*b^22*d^2 - 2450*a^23*d^2 + 192*a^3*b^20*d^2 + 1948
8*a^5*b^18*d^2 - 24128*a^7*b^16*d^2 - 180858*a^9*b^14*d^2 - 146126*a^11*b^12*d^2 + 208974*a^13*b^10*d^2 + 4525
86*a^15*b^8*d^2 + 330770*a^17*b^6*d^2 + 106102*a^19*b^4*d^2 + 7770*a^21*b^2*d^2)/(b^23*d^5 + 8*a^2*b^21*d^5 +
28*a^4*b^19*d^5 + 56*a^6*b^17*d^5 + 70*a^8*b^15*d^5 + 56*a^10*b^13*d^5 + 28*a^12*b^11*d^5 + 8*a^14*b^9*d^5 + a
^16*b^7*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)*(((640*a^2*b^27*d^4 + 9536*a^4*b^25*d^4 + 55488*a^6*b^23*d^4 + 177408*a^8*b^21*d^4 + 354
816*a^10*b^19*d^4 + 470400*a^12*b^17*d^4 + 422016*a^14*b^15*d^4 + 254208*a^16*b^13*d^4 + 98688*a^18*b^11*d^4 +
22336*a^20*b^9*d^4 + 2240*a^22*b^7*d^4)/(b^23*d^5 + 8*a^2*b^21*d^5 + 28*a^4*b^19*d^5 + 56*a^6*b^17*d^5 + 70*a
^8*b^15*d^5 + 56*a^10*b^13*d^5 + 28*a^12*b^11*d^5 + 8*a^14*b^9*d^5 + a^16*b^7*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^32*d^4 + 4608*a^2*b^30*d^4 + 17920*a^4*b^28*d^4 + 38400*a^6*b^26*d^4 + 46080*a^8*b^24*d^4 + 21504*a^1
0*b^22*d^4 - 21504*a^12*b^20*d^4 - 46080*a^14*b^18*d^4 - 38400*a^16*b^16*d^4 - 17920*a^18*b^14*d^4 - 4608*a^20
*b^12*d^4 - 512*a^22*b^10*d^4))/(b^23*d^4 + 8*a^2*b^21*d^4 + 28*a^4*b^19*d^4 + 56*a^6*b^17*d^4 + 70*a^8*b^15*d
^4 + 56*a^10*b^13*d^4 + 28*a^12*b^11*d^4 + 8*a^14*b^9*d^4 + a^16*b^7*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) - (tan(c + d*x)^(1/2)*(1472*
a*b^25*d^2 + 9800*a^25*b*d^2 + 1024*a^3*b^23*d^2 - 8448*a^5*b^21*d^2 - 14336*a^7*b^19*d^2 + 74440*a^9*b^17*d^2
+ 480320*a^11*b^15*d^2 + 1258208*a^13*b^13*d^2 + 1894848*a^15*b^11*d^2 + 1794928*a^17*b^9*d^2 + 1098176*a^19*
b^7*d^2 + 425952*a^21*b^5*d^2 + 96320*a^23*b^3*d^2))/(b^23*d^4 + 8*a^2*b^21*d^4 + 28*a^4*b^19*d^4 + 56*a^6*b^1
7*d^4 + 70*a^8*b^15*d^4 + 56*a^10*b^13*d^4 + 28*a^12*b^11*d^4 + 8*a^14*b^9*d^4 + a^16*b^7*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) - (tan
(c + d*x)^(1/2)*(1225*a^20 + 32*b^20 + 128*a^2*b^18 + 192*a^4*b^16 + 128*a^6*b^14 + 9833*a^8*b^12 - 38610*a^10
*b^10 - 94041*a^12*b^8 - 76668*a^14*b^6 - 24281*a^16*b^4 - 210*a^18*b^2))/(b^23*d^4 + 8*a^2*b^21*d^4 + 28*a^4*
b^19*d^4 + 56*a^6*b^17*d^4 + 70*a^8*b^15*d^4 + 56*a^10*b^13*d^4 + 28*a^12*b^11*d^4 + 8*a^14*b^9*d^4 + a^16*b^7
*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)/((((32*a*b^22*d^2 - 2450*a^23*d^2 + 192*a^3*b^20*d^2 + 19488*a^5*b^18*d^2 - 24128*a^7*b^16*d
^2 - 180858*a^9*b^14*d^2 - 146126*a^11*b^12*d^2 + 208974*a^13*b^10*d^2 + 452586*a^15*b^8*d^2 + 330770*a^17*b^6
*d^2 + 106102*a^19*b^4*d^2 + 7770*a^21*b^2*d^2)/(b^23*d^5 + 8*a^2*b^21*d^5 + 28*a^4*b^19*d^5 + 56*a^6*b^17*d^5
+ 70*a^8*b^15*d^5 + 56*a^10*b^13*d^5 + 28*a^12*b^11*d^5 + 8*a^14*b^9*d^5 + a^16*b^7*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)*(((640*a^2*
b^27*d^4 + 9536*a^4*b^25*d^4 + 55488*a^6*b^23*d^4 + 177408*a^8*b^21*d^4 + 354816*a^10*b^19*d^4 + 470400*a^12*b
^17*d^4 + 422016*a^14*b^15*d^4 + 254208*a^16*b^13*d^4 + 98688*a^18*b^11*d^4 + 22336*a^20*b^9*d^4 + 2240*a^22*b
^7*d^4)/(b^23*d^5 + 8*a^2*b^21*d^5 + 28*a^4*b^19*d^5 + 56*a^6*b^17*d^5 + 70*a^8*b^15*d^5 + 56*a^10*b^13*d^5 +
28*a^12*b^11*d^5 + 8*a^14*b^9*d^5 + a^16*b^7*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^32*d^4 + 4608*a^2*b^30*d
^4 + 17920*a^4*b^28*d^4 + 38400*a^6*b^26*d^4 + 46080*a^8*b^24*d^4 + 21504*a^10*b^22*d^4 - 21504*a^12*b^20*d^4
- 46080*a^14*b^18*d^4 - 38400*a^16*b^16*d^4 - 17920*a^18*b^14*d^4 - 4608*a^20*b^12*d^4 - 512*a^22*b^10*d^4))/(
b^23*d^4 + 8*a^2*b^21*d^4 + 28*a^4*b^19*d^4 + 56*a^6*b^17*d^4 + 70*a^8*b^15*d^4 + 56*a^10*b^13*d^4 + 28*a^12*b
^11*d^4 + 8*a^14*b^9*d^4 + a^16*b^7*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) + (tan(c + d*x)^(1/2)*(1472*a*b^25*d^2 + 9800*a^25*b*d^2 + 10
24*a^3*b^23*d^2 - 8448*a^5*b^21*d^2 - 14336*a^7*b^19*d^2 + 74440*a^9*b^17*d^2 + 480320*a^11*b^15*d^2 + 1258208
*a^13*b^13*d^2 + 1894848*a^15*b^11*d^2 + 1794928*a^17*b^9*d^2 + 1098176*a^19*b^7*d^2 + 425952*a^21*b^5*d^2 + 9
6320*a^23*b^3*d^2))/(b^23*d^4 + 8*a^2*b^21*d^4 + 28*a^4*b^19*d^4 + 56*a^6*b^17*d^4 + 70*a^8*b^15*d^4 + 56*a^10
*b^13*d^4 + 28*a^12*b^11*d^4 + 8*a^14*b^9*d^4 + a^16*b^7*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) + (tan(c + d*x)^(1/2)*(1225*a^20 + 32*b
^20 + 128*a^2*b^18 + 192*a^4*b^16 + 128*a^6*b^14 + 9833*a^8*b^12 - 38610*a^10*b^10 - 94041*a^12*b^8 - 76668*a^
14*b^6 - 24281*a^16*b^4 - 210*a^18*b^2))/(b^23*d^4 + 8*a^2*b^21*d^4 + 28*a^4*b^19*d^4 + 56*a^6*b^17*d^4 + 70*a
^8*b^15*d^4 + 56*a^10*b^13*d^4 + 28*a^12*b^11*d^4 + 8*a^14*b^9*d^4 + a^16*b^7*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) + (((32*a*b^22*d^2
- 2450*a^23*d^2 + 192*a^3*b^20*d^2 + 19488*a^5*b^18*d^2 - 24128*a^7*b^16*d^2 - 180858*a^9*b^14*d^2 - 146126*a^
11*b^12*d^2 + 208974*a^13*b^10*d^2 + 452586*a^15*b^8*d^2 + 330770*a^17*b^6*d^2 + 106102*a^19*b^4*d^2 + 7770*a^
21*b^2*d^2)/(b^23*d^5 + 8*a^2*b^21*d^5 + 28*a^4*b^19*d^5 + 56*a^6*b^17*d^5 + 70*a^8*b^15*d^5 + 56*a^10*b^13*d^
5 + 28*a^12*b^11*d^5 + 8*a^14*b^9*d^5 + a^16*b^7*d^5) - (1i/(4*(b^6*d^2 - a^6*d^2 + a*b^5*d^2*6i + a^5*b*d^2*6
i - 15*a^2*b^4*d^2 - a^3*b^3*d^2*20i + 15*a^4*b^2*d^2)))^(1/2)*(((640*a^2*b^27*d^4 + 9536*a^4*b^25*d^4 + 55488
*a^6*b^23*d^4 + 177408*a^8*b^21*d^4 + 354816*a^10*b^19*d^4 + 470400*a^12*b^17*d^4 + 422016*a^14*b^15*d^4 + 254
208*a^16*b^13*d^4 + 98688*a^18*b^11*d^4 + 22336*a^20*b^9*d^4 + 2240*a^22*b^7*d^4)/(b^23*d^5 + 8*a^2*b^21*d^5 +
28*a^4*b^19*d^5 + 56*a^6*b^17*d^5 + 70*a^8*b^15*d^5 + 56*a^10*b^13*d^5 + 28*a^12*b^11*d^5 + 8*a^14*b^9*d^5 +
a^16*b^7*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^32*d^4 + 4608*a^2*b^30*d^4 + 17920*a^4*b^28*d^4 + 38400*a^6*
b^26*d^4 + 46080*a^8*b^24*d^4 + 21504*a^10*b^22*d^4 - 21504*a^12*b^20*d^4 - 46080*a^14*b^18*d^4 - 38400*a^16*b
^16*d^4 - 17920*a^18*b^14*d^4 - 4608*a^20*b^12*d^4 - 512*a^22*b^10*d^4))/(b^23*d^4 + 8*a^2*b^21*d^4 + 28*a^4*b
^19*d^4 + 56*a^6*b^17*d^4 + 70*a^8*b^15*d^4 + 56*a^10*b^13*d^4 + 28*a^12*b^11*d^4 + 8*a^14*b^9*d^4 + a^16*b^7*
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) - (tan(c + d*x)^(1/2)*(1472*a*b^25*d^2 + 9800*a^25*b*d^2 + 1024*a^3*b^23*d^2 - 8448*a^5*b^21*d^2
- 14336*a^7*b^19*d^2 + 74440*a^9*b^17*d^2 + 480320*a^11*b^15*d^2 + 1258208*a^13*b^13*d^2 + 1894848*a^15*b^11*d
^2 + 1794928*a^17*b^9*d^2 + 1098176*a^19*b^7*d^2 + 425952*a^21*b^5*d^2 + 96320*a^23*b^3*d^2))/(b^23*d^4 + 8*a^
2*b^21*d^4 + 28*a^4*b^19*d^4 + 56*a^6*b^17*d^4 + 70*a^8*b^15*d^4 + 56*a^10*b^13*d^4 + 28*a^12*b^11*d^4 + 8*a^1
4*b^9*d^4 + a^16*b^7*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) - (tan(c + d*x)^(1/2)*(1225*a^20 + 32*b^20 + 128*a^2*b^18 + 192*a^4*b^16 +
128*a^6*b^14 + 9833*a^8*b^12 - 38610*a^10*b^10 - 94041*a^12*b^8 - 76668*a^14*b^6 - 24281*a^16*b^4 - 210*a^18*b
^2))/(b^23*d^4 + 8*a^2*b^21*d^4 + 28*a^4*b^19*d^4 + 56*a^6*b^17*d^4 + 70*a^8*b^15*d^4 + 56*a^10*b^13*d^4 + 28*
a^12*b^11*d^4 + 8*a^14*b^9*d^4 + a^16*b^7*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) + (1225*a^16*b - 792*a^4*b^13 - 1608*a^6*b^11 + 8705*a^
8*b^9 + 19916*a^10*b^7 + 17334*a^12*b^5 + 7140*a^14*b^3)/(b^23*d^5 + 8*a^2*b^21*d^5 + 28*a^4*b^19*d^5 + 56*a^6
*b^17*d^5 + 70*a^8*b^15*d^5 + 56*a^10*b^13*d^5 + 28*a^12*b^11*d^5 + 8*a^14*b^9*d^5 + a^16*b^7*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
- (6*a*tan(c + d*x)^(1/2))/(b^4*d) - (atan(((((tan(c + d*x)^(1/2)*(1225*a^20 + 32*b^20 + 128*a^2*b^18 + 192*a
^4*b^16 + 128*a^6*b^14 + 9833*a^8*b^12 - 38610*a^10*b^10 - 94041*a^12*b^8 - 76668*a^14*b^6 - 24281*a^16*b^4 -
210*a^18*b^2))/(b^23*d^4 + 8*a^2*b^21*d^4 + 28*a^4*b^19*d^4 + 56*a^6*b^17*d^4 + 70*a^8*b^15*d^4 + 56*a^10*b^13
*d^4 + 28*a^12*b^11*d^4 + 8*a^14*b^9*d^4 + a^16*b^7*d^4) + (((32*a*b^22*d^2 - 2450*a^23*d^2 + 192*a^3*b^20*d^2
+ 19488*a^5*b^18*d^2 - 24128*a^7*b^16*d^2 - 180858*a^9*b^14*d^2 - 146126*a^11*b^12*d^2 + 208974*a^13*b^10*d^2
+ 452586*a^15*b^8*d^2 + 330770*a^17*b^6*d^2 + 106102*a^19*b^4*d^2 + 7770*a^21*b^2*d^2)/(b^23*d^5 + 8*a^2*b^21
*d^5 + 28*a^4*b^19*d^5 + 56*a^6*b^17*d^5 + 70*a^8*b^15*d^5 + 56*a^10*b^13*d^5 + 28*a^12*b^11*d^5 + 8*a^14*b^9*
d^5 + a^16*b^7*d^5) - (((tan(c + d*x)^(1/2)*(1472*a*b^25*d^2 + 9800*a^25*b*d^2 + 1024*a^3*b^23*d^2 - 8448*a^5*
b^21*d^2 - 14336*a^7*b^19*d^2 + 74440*a^9*b^17*d^2 + 480320*a^11*b^15*d^2 + 1258208*a^13*b^13*d^2 + 1894848*a^
15*b^11*d^2 + 1794928*a^17*b^9*d^2 + 1098176*a^19*b^7*d^2 + 425952*a^21*b^5*d^2 + 96320*a^23*b^3*d^2))/(b^23*d
^4 + 8*a^2*b^21*d^4 + 28*a^4*b^19*d^4 + 56*a^6*b^17*d^4 + 70*a^8*b^15*d^4 + 56*a^10*b^13*d^4 + 28*a^12*b^11*d^
4 + 8*a^14*b^9*d^4 + a^16*b^7*d^4) + (((640*a^2*b^27*d^4 + 9536*a^4*b^25*d^4 + 55488*a^6*b^23*d^4 + 177408*a^8
*b^21*d^4 + 354816*a^10*b^19*d^4 + 470400*a^12*b^17*d^4 + 422016*a^14*b^15*d^4 + 254208*a^16*b^13*d^4 + 98688*
a^18*b^11*d^4 + 22336*a^20*b^9*d^4 + 2240*a^22*b^7*d^4)/(b^23*d^5 + 8*a^2*b^21*d^5 + 28*a^4*b^19*d^5 + 56*a^6*
b^17*d^5 + 70*a^8*b^15*d^5 + 56*a^10*b^13*d^5 + 28*a^12*b^11*d^5 + 8*a^14*b^9*d^5 + a^16*b^7*d^5) - (tan(c + d
*x)^(1/2)*(-a^7*b^9)^(1/2)*(35*a^4 + 99*b^4 + 102*a^2*b^2)*(512*b^32*d^4 + 4608*a^2*b^30*d^4 + 17920*a^4*b^28*
d^4 + 38400*a^6*b^26*d^4 + 46080*a^8*b^24*d^4 + 21504*a^10*b^22*d^4 - 21504*a^12*b^20*d^4 - 46080*a^14*b^18*d^
4 - 38400*a^16*b^16*d^4 - 17920*a^18*b^14*d^4 - 4608*a^20*b^12*d^4 - 512*a^22*b^10*d^4))/(8*(b^15*d + 3*a^2*b^
13*d + 3*a^4*b^11*d + a^6*b^9*d)*(b^23*d^4 + 8*a^2*b^21*d^4 + 28*a^4*b^19*d^4 + 56*a^6*b^17*d^4 + 70*a^8*b^15*
d^4 + 56*a^10*b^13*d^4 + 28*a^12*b^11*d^4 + 8*a^14*b^9*d^4 + a^16*b^7*d^4)))*(-a^7*b^9)^(1/2)*(35*a^4 + 99*b^4
+ 102*a^2*b^2))/(8*(b^15*d + 3*a^2*b^13*d + 3*a^4*b^11*d + a^6*b^9*d)))*(-a^7*b^9)^(1/2)*(35*a^4 + 99*b^4 + 1
02*a^2*b^2))/(8*(b^15*d + 3*a^2*b^13*d + 3*a^4*b^11*d + a^6*b^9*d)))*(-a^7*b^9)^(1/2)*(35*a^4 + 99*b^4 + 102*a
^2*b^2))/(8*(b^15*d + 3*a^2*b^13*d + 3*a^4*b^11*d + a^6*b^9*d)))*(-a^7*b^9)^(1/2)*(35*a^4 + 99*b^4 + 102*a^2*b
^2)*1i)/(8*(b^15*d + 3*a^2*b^13*d + 3*a^4*b^11*d + a^6*b^9*d)) + (((tan(c + d*x)^(1/2)*(1225*a^20 + 32*b^20 +
128*a^2*b^18 + 192*a^4*b^16 + 128*a^6*b^14 + 9833*a^8*b^12 - 38610*a^10*b^10 - 94041*a^12*b^8 - 76668*a^14*b^6
- 24281*a^16*b^4 - 210*a^18*b^2))/(b^23*d^4 + 8*a^2*b^21*d^4 + 28*a^4*b^19*d^4 + 56*a^6*b^17*d^4 + 70*a^8*b^1
5*d^4 + 56*a^10*b^13*d^4 + 28*a^12*b^11*d^4 + 8*a^14*b^9*d^4 + a^16*b^7*d^4) - (((32*a*b^22*d^2 - 2450*a^23*d^
2 + 192*a^3*b^20*d^2 + 19488*a^5*b^18*d^2 - 24128*a^7*b^16*d^2 - 180858*a^9*b^14*d^2 - 146126*a^11*b^12*d^2 +
208974*a^13*b^10*d^2 + 452586*a^15*b^8*d^2 + 330770*a^17*b^6*d^2 + 106102*a^19*b^4*d^2 + 7770*a^21*b^2*d^2)/(b
^23*d^5 + 8*a^2*b^21*d^5 + 28*a^4*b^19*d^5 + 56*a^6*b^17*d^5 + 70*a^8*b^15*d^5 + 56*a^10*b^13*d^5 + 28*a^12*b^
11*d^5 + 8*a^14*b^9*d^5 + a^16*b^7*d^5) + (((tan(c + d*x)^(1/2)*(1472*a*b^25*d^2 + 9800*a^25*b*d^2 + 1024*a^3*
b^23*d^2 - 8448*a^5*b^21*d^2 - 14336*a^7*b^19*d^2 + 74440*a^9*b^17*d^2 + 480320*a^11*b^15*d^2 + 1258208*a^13*b
^13*d^2 + 1894848*a^15*b^11*d^2 + 1794928*a^17*b^9*d^2 + 1098176*a^19*b^7*d^2 + 425952*a^21*b^5*d^2 + 96320*a^
23*b^3*d^2))/(b^23*d^4 + 8*a^2*b^21*d^4 + 28*a^4*b^19*d^4 + 56*a^6*b^17*d^4 + 70*a^8*b^15*d^4 + 56*a^10*b^13*d
^4 + 28*a^12*b^11*d^4 + 8*a^14*b^9*d^4 + a^16*b^7*d^4) - (((640*a^2*b^27*d^4 + 9536*a^4*b^25*d^4 + 55488*a^6*b
^23*d^4 + 177408*a^8*b^21*d^4 + 354816*a^10*b^19*d^4 + 470400*a^12*b^17*d^4 + 422016*a^14*b^15*d^4 + 254208*a^
16*b^13*d^4 + 98688*a^18*b^11*d^4 + 22336*a^20*b^9*d^4 + 2240*a^22*b^7*d^4)/(b^23*d^5 + 8*a^2*b^21*d^5 + 28*a^
4*b^19*d^5 + 56*a^6*b^17*d^5 + 70*a^8*b^15*d^5 + 56*a^10*b^13*d^5 + 28*a^12*b^11*d^5 + 8*a^14*b^9*d^5 + a^16*b
^7*d^5) + (tan(c + d*x)^(1/2)*(-a^7*b^9)^(1/2)*(35*a^4 + 99*b^4 + 102*a^2*b^2)*(512*b^32*d^4 + 4608*a^2*b^30*d
^4 + 17920*a^4*b^28*d^4 + 38400*a^6*b^26*d^4 + 46080*a^8*b^24*d^4 + 21504*a^10*b^22*d^4 - 21504*a^12*b^20*d^4
- 46080*a^14*b^18*d^4 - 38400*a^16*b^16*d^4 - 17920*a^18*b^14*d^4 - 4608*a^20*b^12*d^4 - 512*a^22*b^10*d^4))/(
8*(b^15*d + 3*a^2*b^13*d + 3*a^4*b^11*d + a^6*b^9*d)*(b^23*d^4 + 8*a^2*b^21*d^4 + 28*a^4*b^19*d^4 + 56*a^6*b^1
7*d^4 + 70*a^8*b^15*d^4 + 56*a^10*b^13*d^4 + 28*a^12*b^11*d^4 + 8*a^14*b^9*d^4 + a^16*b^7*d^4)))*(-a^7*b^9)^(1
/2)*(35*a^4 + 99*b^4 + 102*a^2*b^2))/(8*(b^15*d + 3*a^2*b^13*d + 3*a^4*b^11*d + a^6*b^9*d)))*(-a^7*b^9)^(1/2)*
(35*a^4 + 99*b^4 + 102*a^2*b^2))/(8*(b^15*d + 3*a^2*b^13*d + 3*a^4*b^11*d + a^6*b^9*d)))*(-a^7*b^9)^(1/2)*(35*
a^4 + 99*b^4 + 102*a^2*b^2))/(8*(b^15*d + 3*a^2*b^13*d + 3*a^4*b^11*d + a^6*b^9*d)))*(-a^7*b^9)^(1/2)*(35*a^4
+ 99*b^4 + 102*a^2*b^2)*1i)/(8*(b^15*d + 3*a^2*b^13*d + 3*a^4*b^11*d + a^6*b^9*d)))/((1225*a^16*b - 792*a^4*b^
13 - 1608*a^6*b^11 + 8705*a^8*b^9 + 19916*a^10*b^7 + 17334*a^12*b^5 + 7140*a^14*b^3)/(b^23*d^5 + 8*a^2*b^21*d^
5 + 28*a^4*b^19*d^5 + 56*a^6*b^17*d^5 + 70*a^8*b^15*d^5 + 56*a^10*b^13*d^5 + 28*a^12*b^11*d^5 + 8*a^14*b^9*d^5
+ a^16*b^7*d^5) + (((tan(c + d*x)^(1/2)*(1225*a^20 + 32*b^20 + 128*a^2*b^18 + 192*a^4*b^16 + 128*a^6*b^14 + 9
833*a^8*b^12 - 38610*a^10*b^10 - 94041*a^12*b^8 - 76668*a^14*b^6 - 24281*a^16*b^4 - 210*a^18*b^2))/(b^23*d^4 +
8*a^2*b^21*d^4 + 28*a^4*b^19*d^4 + 56*a^6*b^17*d^4 + 70*a^8*b^15*d^4 + 56*a^10*b^13*d^4 + 28*a^12*b^11*d^4 +
8*a^14*b^9*d^4 + a^16*b^7*d^4) + (((32*a*b^22*d^2 - 2450*a^23*d^2 + 192*a^3*b^20*d^2 + 19488*a^5*b^18*d^2 - 24
128*a^7*b^16*d^2 - 180858*a^9*b^14*d^2 - 146126*a^11*b^12*d^2 + 208974*a^13*b^10*d^2 + 452586*a^15*b^8*d^2 + 3
30770*a^17*b^6*d^2 + 106102*a^19*b^4*d^2 + 7770*a^21*b^2*d^2)/(b^23*d^5 + 8*a^2*b^21*d^5 + 28*a^4*b^19*d^5 + 5
6*a^6*b^17*d^5 + 70*a^8*b^15*d^5 + 56*a^10*b^13*d^5 + 28*a^12*b^11*d^5 + 8*a^14*b^9*d^5 + a^16*b^7*d^5) - (((t
an(c + d*x)^(1/2)*(1472*a*b^25*d^2 + 9800*a^25*b*d^2 + 1024*a^3*b^23*d^2 - 8448*a^5*b^21*d^2 - 14336*a^7*b^19*
d^2 + 74440*a^9*b^17*d^2 + 480320*a^11*b^15*d^2 + 1258208*a^13*b^13*d^2 + 1894848*a^15*b^11*d^2 + 1794928*a^17
*b^9*d^2 + 1098176*a^19*b^7*d^2 + 425952*a^21*b^5*d^2 + 96320*a^23*b^3*d^2))/(b^23*d^4 + 8*a^2*b^21*d^4 + 28*a
^4*b^19*d^4 + 56*a^6*b^17*d^4 + 70*a^8*b^15*d^4 + 56*a^10*b^13*d^4 + 28*a^12*b^11*d^4 + 8*a^14*b^9*d^4 + a^16*
b^7*d^4) + (((640*a^2*b^27*d^4 + 9536*a^4*b^25*d^4 + 55488*a^6*b^23*d^4 + 177408*a^8*b^21*d^4 + 354816*a^10*b^
19*d^4 + 470400*a^12*b^17*d^4 + 422016*a^14*b^15*d^4 + 254208*a^16*b^13*d^4 + 98688*a^18*b^11*d^4 + 22336*a^20
*b^9*d^4 + 2240*a^22*b^7*d^4)/(b^23*d^5 + 8*a^2*b^21*d^5 + 28*a^4*b^19*d^5 + 56*a^6*b^17*d^5 + 70*a^8*b^15*d^5
+ 56*a^10*b^13*d^5 + 28*a^12*b^11*d^5 + 8*a^14*b^9*d^5 + a^16*b^7*d^5) - (tan(c + d*x)^(1/2)*(-a^7*b^9)^(1/2)
*(35*a^4 + 99*b^4 + 102*a^2*b^2)*(512*b^32*d^4 + 4608*a^2*b^30*d^4 + 17920*a^4*b^28*d^4 + 38400*a^6*b^26*d^4 +
46080*a^8*b^24*d^4 + 21504*a^10*b^22*d^4 - 21504*a^12*b^20*d^4 - 46080*a^14*b^18*d^4 - 38400*a^16*b^16*d^4 -
17920*a^18*b^14*d^4 - 4608*a^20*b^12*d^4 - 512*a^22*b^10*d^4))/(8*(b^15*d + 3*a^2*b^13*d + 3*a^4*b^11*d + a^6*
b^9*d)*(b^23*d^4 + 8*a^2*b^21*d^4 + 28*a^4*b^19*d^4 + 56*a^6*b^17*d^4 + 70*a^8*b^15*d^4 + 56*a^10*b^13*d^4 + 2
8*a^12*b^11*d^4 + 8*a^14*b^9*d^4 + a^16*b^7*d^4)))*(-a^7*b^9)^(1/2)*(35*a^4 + 99*b^4 + 102*a^2*b^2))/(8*(b^15*
d + 3*a^2*b^13*d + 3*a^4*b^11*d + a^6*b^9*d)))*(-a^7*b^9)^(1/2)*(35*a^4 + 99*b^4 + 102*a^2*b^2))/(8*(b^15*d +
3*a^2*b^13*d + 3*a^4*b^11*d + a^6*b^9*d)))*(-a^7*b^9)^(1/2)*(35*a^4 + 99*b^4 + 102*a^2*b^2))/(8*(b^15*d + 3*a^
2*b^13*d + 3*a^4*b^11*d + a^6*b^9*d)))*(-a^7*b^9)^(1/2)*(35*a^4 + 99*b^4 + 102*a^2*b^2))/(8*(b^15*d + 3*a^2*b^
13*d + 3*a^4*b^11*d + a^6*b^9*d)) - (((tan(c + d*x)^(1/2)*(1225*a^20 + 32*b^20 + 128*a^2*b^18 + 192*a^4*b^16 +
128*a^6*b^14 + 9833*a^8*b^12 - 38610*a^10*b^10 - 94041*a^12*b^8 - 76668*a^14*b^6 - 24281*a^16*b^4 - 210*a^18*
b^2))/(b^23*d^4 + 8*a^2*b^21*d^4 + 28*a^4*b^19*d^4 + 56*a^6*b^17*d^4 + 70*a^8*b^15*d^4 + 56*a^10*b^13*d^4 + 28
*a^12*b^11*d^4 + 8*a^14*b^9*d^4 + a^16*b^7*d^4) - (((32*a*b^22*d^2 - 2450*a^23*d^2 + 192*a^3*b^20*d^2 + 19488*
a^5*b^18*d^2 - 24128*a^7*b^16*d^2 - 180858*a^9*b^14*d^2 - 146126*a^11*b^12*d^2 + 208974*a^13*b^10*d^2 + 452586
*a^15*b^8*d^2 + 330770*a^17*b^6*d^2 + 106102*a^19*b^4*d^2 + 7770*a^21*b^2*d^2)/(b^23*d^5 + 8*a^2*b^21*d^5 + 28
*a^4*b^19*d^5 + 56*a^6*b^17*d^5 + 70*a^8*b^15*d^5 + 56*a^10*b^13*d^5 + 28*a^12*b^11*d^5 + 8*a^14*b^9*d^5 + a^1
6*b^7*d^5) + (((tan(c + d*x)^(1/2)*(1472*a*b^25*d^2 + 9800*a^25*b*d^2 + 1024*a^3*b^23*d^2 - 8448*a^5*b^21*d^2
- 14336*a^7*b^19*d^2 + 74440*a^9*b^17*d^2 + 480320*a^11*b^15*d^2 + 1258208*a^13*b^13*d^2 + 1894848*a^15*b^11*d
^2 + 1794928*a^17*b^9*d^2 + 1098176*a^19*b^7*d^2 + 425952*a^21*b^5*d^2 + 96320*a^23*b^3*d^2))/(b^23*d^4 + 8*a^
2*b^21*d^4 + 28*a^4*b^19*d^4 + 56*a^6*b^17*d^4 + 70*a^8*b^15*d^4 + 56*a^10*b^13*d^4 + 28*a^12*b^11*d^4 + 8*a^1
4*b^9*d^4 + a^16*b^7*d^4) - (((640*a^2*b^27*d^4 + 9536*a^4*b^25*d^4 + 55488*a^6*b^23*d^4 + 177408*a^8*b^21*d^4
+ 354816*a^10*b^19*d^4 + 470400*a^12*b^17*d^4 + 422016*a^14*b^15*d^4 + 254208*a^16*b^13*d^4 + 98688*a^18*b^11
*d^4 + 22336*a^20*b^9*d^4 + 2240*a^22*b^7*d^4)/(b^23*d^5 + 8*a^2*b^21*d^5 + 28*a^4*b^19*d^5 + 56*a^6*b^17*d^5
+ 70*a^8*b^15*d^5 + 56*a^10*b^13*d^5 + 28*a^12*b^11*d^5 + 8*a^14*b^9*d^5 + a^16*b^7*d^5) + (tan(c + d*x)^(1/2)
*(-a^7*b^9)^(1/2)*(35*a^4 + 99*b^4 + 102*a^2*b^2)*(512*b^32*d^4 + 4608*a^2*b^30*d^4 + 17920*a^4*b^28*d^4 + 384
00*a^6*b^26*d^4 + 46080*a^8*b^24*d^4 + 21504*a^10*b^22*d^4 - 21504*a^12*b^20*d^4 - 46080*a^14*b^18*d^4 - 38400
*a^16*b^16*d^4 - 17920*a^18*b^14*d^4 - 4608*a^20*b^12*d^4 - 512*a^22*b^10*d^4))/(8*(b^15*d + 3*a^2*b^13*d + 3*
a^4*b^11*d + a^6*b^9*d)*(b^23*d^4 + 8*a^2*b^21*d^4 + 28*a^4*b^19*d^4 + 56*a^6*b^17*d^4 + 70*a^8*b^15*d^4 + 56*
a^10*b^13*d^4 + 28*a^12*b^11*d^4 + 8*a^14*b^9*d^4 + a^16*b^7*d^4)))*(-a^7*b^9)^(1/2)*(35*a^4 + 99*b^4 + 102*a^
2*b^2))/(8*(b^15*d + 3*a^2*b^13*d + 3*a^4*b^11*d + a^6*b^9*d)))*(-a^7*b^9)^(1/2)*(35*a^4 + 99*b^4 + 102*a^2*b^
2))/(8*(b^15*d + 3*a^2*b^13*d + 3*a^4*b^11*d + a^6*b^9*d)))*(-a^7*b^9)^(1/2)*(35*a^4 + 99*b^4 + 102*a^2*b^2))/
(8*(b^15*d + 3*a^2*b^13*d + 3*a^4*b^11*d + a^6*b^9*d)))*(-a^7*b^9)^(1/2)*(35*a^4 + 99*b^4 + 102*a^2*b^2))/(8*(
b^15*d + 3*a^2*b^13*d + 3*a^4*b^11*d + a^6*b^9*d))))*(-a^7*b^9)^(1/2)*(35*a^4 + 99*b^4 + 102*a^2*b^2)*1i)/(4*(
b^15*d + 3*a^2*b^13*d + 3*a^4*b^11*d + a^6*b^9*d))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tan(d*x+c)**(11/2)/(a+b*tan(d*x+c))**3,x)

[Out]

Timed out

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