3.410 \(\int \frac {\tan ^{\frac {7}{2}}(c+d x) (A+B \tan (c+d x))}{(a+b \tan (c+d x))^3} \, dx\)

Optimal. Leaf size=600 \[ \frac {a (A b-a B) \tan ^{\frac {5}{2}}(c+d x)}{2 b d \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}+\frac {\left (-\left (a^3 (A+B)\right )+3 a^2 b (A-B)+3 a b^2 (A+B)-b^3 (A-B)\right ) \tan ^{-1}\left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} d \left (a^2+b^2\right )^3}-\frac {\left (-\left (a^3 (A+B)\right )+3 a^2 b (A-B)+3 a b^2 (A+B)-b^3 (A-B)\right ) \tan ^{-1}\left (\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{\sqrt {2} d \left (a^2+b^2\right )^3}+\frac {a \left (-5 a^3 B+a^2 A b-13 a b^2 B+9 A b^3\right ) \tan ^{\frac {3}{2}}(c+d x)}{4 b^2 d \left (a^2+b^2\right )^2 (a+b \tan (c+d x))}-\frac {\left (a^3 (A-B)+3 a^2 b (A+B)-3 a b^2 (A-B)-b^3 (A+B)\right ) \log \left (\tan (c+d x)-\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{2 \sqrt {2} d \left (a^2+b^2\right )^3}+\frac {\left (a^3 (A-B)+3 a^2 b (A+B)-3 a b^2 (A-B)-b^3 (A+B)\right ) \log \left (\tan (c+d x)+\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{2 \sqrt {2} d \left (a^2+b^2\right )^3}-\frac {\left (-15 a^4 B+3 a^3 A b-31 a^2 b^2 B+11 a A b^3-8 b^4 B\right ) \sqrt {\tan (c+d x)}}{4 b^3 d \left (a^2+b^2\right )^2}+\frac {a^{3/2} \left (-15 a^5 B+3 a^4 A b-46 a^3 b^2 B+6 a^2 A b^3-63 a b^4 B+35 A b^5\right ) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {\tan (c+d x)}}{\sqrt {a}}\right )}{4 b^{7/2} d \left (a^2+b^2\right )^3} \]

[Out]

1/4*a^(3/2)*(3*A*a^4*b+6*A*a^2*b^3+35*A*b^5-15*B*a^5-46*B*a^3*b^2-63*B*a*b^4)*arctan(b^(1/2)*tan(d*x+c)^(1/2)/
a^(1/2))/b^(7/2)/(a^2+b^2)^3/d-1/2*(3*a^2*b*(A-B)-b^3*(A-B)-a^3*(A+B)+3*a*b^2*(A+B))*arctan(-1+2^(1/2)*tan(d*x
+c)^(1/2))/(a^2+b^2)^3/d*2^(1/2)-1/2*(3*a^2*b*(A-B)-b^3*(A-B)-a^3*(A+B)+3*a*b^2*(A+B))*arctan(1+2^(1/2)*tan(d*
x+c)^(1/2))/(a^2+b^2)^3/d*2^(1/2)-1/4*(a^3*(A-B)-3*a*b^2*(A-B)+3*a^2*b*(A+B)-b^3*(A+B))*ln(1-2^(1/2)*tan(d*x+c
)^(1/2)+tan(d*x+c))/(a^2+b^2)^3/d*2^(1/2)+1/4*(a^3*(A-B)-3*a*b^2*(A-B)+3*a^2*b*(A+B)-b^3*(A+B))*ln(1+2^(1/2)*t
an(d*x+c)^(1/2)+tan(d*x+c))/(a^2+b^2)^3/d*2^(1/2)-1/4*(3*A*a^3*b+11*A*a*b^3-15*B*a^4-31*B*a^2*b^2-8*B*b^4)*tan
(d*x+c)^(1/2)/b^3/(a^2+b^2)^2/d+1/2*a*(A*b-B*a)*tan(d*x+c)^(5/2)/b/(a^2+b^2)/d/(a+b*tan(d*x+c))^2+1/4*a*(A*a^2
*b+9*A*b^3-5*B*a^3-13*B*a*b^2)*tan(d*x+c)^(3/2)/b^2/(a^2+b^2)^2/d/(a+b*tan(d*x+c))

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Rubi [A]  time = 1.72, antiderivative size = 600, normalized size of antiderivative = 1.00, number of steps used = 17, number of rules used = 14, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.424, Rules used = {3605, 3645, 3647, 3653, 3534, 1168, 1162, 617, 204, 1165, 628, 3634, 63, 205} \[ \frac {a (A b-a B) \tan ^{\frac {5}{2}}(c+d x)}{2 b d \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}+\frac {a \left (a^2 A b-5 a^3 B-13 a b^2 B+9 A b^3\right ) \tan ^{\frac {3}{2}}(c+d x)}{4 b^2 d \left (a^2+b^2\right )^2 (a+b \tan (c+d x))}+\frac {\left (3 a^2 b (A-B)+a^3 (-(A+B))+3 a b^2 (A+B)-b^3 (A-B)\right ) \tan ^{-1}\left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} d \left (a^2+b^2\right )^3}-\frac {\left (3 a^2 b (A-B)+a^3 (-(A+B))+3 a b^2 (A+B)-b^3 (A-B)\right ) \tan ^{-1}\left (\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{\sqrt {2} d \left (a^2+b^2\right )^3}+\frac {a^{3/2} \left (6 a^2 A b^3+3 a^4 A b-46 a^3 b^2 B-15 a^5 B-63 a b^4 B+35 A b^5\right ) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {\tan (c+d x)}}{\sqrt {a}}\right )}{4 b^{7/2} d \left (a^2+b^2\right )^3}-\frac {\left (3 a^3 A b-31 a^2 b^2 B-15 a^4 B+11 a A b^3-8 b^4 B\right ) \sqrt {\tan (c+d x)}}{4 b^3 d \left (a^2+b^2\right )^2}-\frac {\left (3 a^2 b (A+B)+a^3 (A-B)-3 a b^2 (A-B)-b^3 (A+B)\right ) \log \left (\tan (c+d x)-\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{2 \sqrt {2} d \left (a^2+b^2\right )^3}+\frac {\left (3 a^2 b (A+B)+a^3 (A-B)-3 a b^2 (A-B)-b^3 (A+B)\right ) \log \left (\tan (c+d x)+\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{2 \sqrt {2} d \left (a^2+b^2\right )^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((3*a^2*b*(A - B) - b^3*(A - B) - a^3*(A + B) + 3*a*b^2*(A + B))*ArcTan[1 - Sqrt[2]*Sqrt[Tan[c + d*x]]])/(Sqrt
[2]*(a^2 + b^2)^3*d) - ((3*a^2*b*(A - B) - b^3*(A - B) - a^3*(A + B) + 3*a*b^2*(A + B))*ArcTan[1 + Sqrt[2]*Sqr
t[Tan[c + d*x]]])/(Sqrt[2]*(a^2 + b^2)^3*d) + (a^(3/2)*(3*a^4*A*b + 6*a^2*A*b^3 + 35*A*b^5 - 15*a^5*B - 46*a^3
*b^2*B - 63*a*b^4*B)*ArcTan[(Sqrt[b]*Sqrt[Tan[c + d*x]])/Sqrt[a]])/(4*b^(7/2)*(a^2 + b^2)^3*d) - ((a^3*(A - B)
- 3*a*b^2*(A - B) + 3*a^2*b*(A + B) - b^3*(A + B))*Log[1 - Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]])/(2*Sqr
t[2]*(a^2 + b^2)^3*d) + ((a^3*(A - B) - 3*a*b^2*(A - B) + 3*a^2*b*(A + B) - b^3*(A + B))*Log[1 + Sqrt[2]*Sqrt[
Tan[c + d*x]] + Tan[c + d*x]])/(2*Sqrt[2]*(a^2 + b^2)^3*d) - ((3*a^3*A*b + 11*a*A*b^3 - 15*a^4*B - 31*a^2*b^2*
B - 8*b^4*B)*Sqrt[Tan[c + d*x]])/(4*b^3*(a^2 + b^2)^2*d) + (a*(A*b - a*B)*Tan[c + d*x]^(5/2))/(2*b*(a^2 + b^2)
*d*(a + b*Tan[c + d*x])^2) + (a*(a^2*A*b + 9*A*b^3 - 5*a^3*B - 13*a*b^2*B)*Tan[c + d*x]^(3/2))/(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 3605

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

Rule 3634

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

Rule 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 {7}{2}}(c+d x) (A+B \tan (c+d x))}{(a+b \tan (c+d x))^3} \, dx &=\frac {a (A b-a B) \tan ^{\frac {5}{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 {3}{2}}(c+d x) \left (-\frac {5}{2} a (A b-a B)+2 b (A b-a B) \tan (c+d x)-\frac {1}{2} \left (a A b-5 a^2 B-4 b^2 B\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 (A b-a B) \tan ^{\frac {5}{2}}(c+d x)}{2 b \left (a^2+b^2\right ) d (a+b \tan (c+d x))^2}+\frac {a \left (a^2 A b+9 A b^3-5 a^3 B-13 a b^2 B\right ) \tan ^{\frac {3}{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}{4} a \left (a^2 A b+9 A b^3-5 a^3 B-13 a b^2 B\right )-2 b^2 \left (a^2 A-A b^2+2 a b B\right ) \tan (c+d x)-\frac {1}{4} \left (3 a^3 A b+11 a A b^3-15 a^4 B-31 a^2 b^2 B-8 b^4 B\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 (3 a^3 A b+11 a A b^3-15 a^4 B-31 a^2 b^2 B-8 b^4 B\right ) \sqrt {\tan (c+d x)}}{4 b^3 \left (a^2+b^2\right )^2 d}+\frac {a (A b-a B) \tan ^{\frac {5}{2}}(c+d x)}{2 b \left (a^2+b^2\right ) d (a+b \tan (c+d x))^2}+\frac {a \left (a^2 A b+9 A b^3-5 a^3 B-13 a b^2 B\right ) \tan ^{\frac {3}{2}}(c+d x)}{4 b^2 \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))}+\frac {\int \frac {\frac {1}{8} a \left (3 a^3 A b+11 a A b^3-15 a^4 B-31 a^2 b^2 B-8 b^4 B\right )-b^3 \left (2 a A b-a^2 B+b^2 B\right ) \tan (c+d x)+\frac {1}{8} \left (3 a^4 A b+3 a^2 A b^3+8 A b^5-15 a^5 B-31 a^3 b^2 B-24 a b^4 B\right ) \tan ^2(c+d x)}{\sqrt {\tan (c+d x)} (a+b \tan (c+d x))} \, dx}{b^3 \left (a^2+b^2\right )^2}\\ &=-\frac {\left (3 a^3 A b+11 a A b^3-15 a^4 B-31 a^2 b^2 B-8 b^4 B\right ) \sqrt {\tan (c+d x)}}{4 b^3 \left (a^2+b^2\right )^2 d}+\frac {a (A b-a B) \tan ^{\frac {5}{2}}(c+d x)}{2 b \left (a^2+b^2\right ) d (a+b \tan (c+d x))^2}+\frac {a \left (a^2 A b+9 A b^3-5 a^3 B-13 a b^2 B\right ) \tan ^{\frac {3}{2}}(c+d x)}{4 b^2 \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))}+\frac {\int \frac {b^3 \left (a^3 A-3 a A b^2+3 a^2 b B-b^3 B\right )-b^3 \left (3 a^2 A b-A b^3-a^3 B+3 a b^2 B\right ) \tan (c+d x)}{\sqrt {\tan (c+d x)}} \, dx}{b^3 \left (a^2+b^2\right )^3}+\frac {\left (a^2 \left (3 a^4 A b+6 a^2 A b^3+35 A b^5-15 a^5 B-46 a^3 b^2 B-63 a b^4 B\right )\right ) \int \frac {1+\tan ^2(c+d x)}{\sqrt {\tan (c+d x)} (a+b \tan (c+d x))} \, dx}{8 b^3 \left (a^2+b^2\right )^3}\\ &=-\frac {\left (3 a^3 A b+11 a A b^3-15 a^4 B-31 a^2 b^2 B-8 b^4 B\right ) \sqrt {\tan (c+d x)}}{4 b^3 \left (a^2+b^2\right )^2 d}+\frac {a (A b-a B) \tan ^{\frac {5}{2}}(c+d x)}{2 b \left (a^2+b^2\right ) d (a+b \tan (c+d x))^2}+\frac {a \left (a^2 A b+9 A b^3-5 a^3 B-13 a b^2 B\right ) \tan ^{\frac {3}{2}}(c+d x)}{4 b^2 \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))}+\frac {2 \operatorname {Subst}\left (\int \frac {b^3 \left (a^3 A-3 a A b^2+3 a^2 b B-b^3 B\right )-b^3 \left (3 a^2 A b-A b^3-a^3 B+3 a b^2 B\right ) x^2}{1+x^4} \, dx,x,\sqrt {\tan (c+d x)}\right )}{b^3 \left (a^2+b^2\right )^3 d}+\frac {\left (a^2 \left (3 a^4 A b+6 a^2 A b^3+35 A b^5-15 a^5 B-46 a^3 b^2 B-63 a b^4 B\right )\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {x} (a+b x)} \, dx,x,\tan (c+d x)\right )}{8 b^3 \left (a^2+b^2\right )^3 d}\\ &=-\frac {\left (3 a^3 A b+11 a A b^3-15 a^4 B-31 a^2 b^2 B-8 b^4 B\right ) \sqrt {\tan (c+d x)}}{4 b^3 \left (a^2+b^2\right )^2 d}+\frac {a (A b-a B) \tan ^{\frac {5}{2}}(c+d x)}{2 b \left (a^2+b^2\right ) d (a+b \tan (c+d x))^2}+\frac {a \left (a^2 A b+9 A b^3-5 a^3 B-13 a b^2 B\right ) \tan ^{\frac {3}{2}}(c+d x)}{4 b^2 \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))}+\frac {\left (a^2 \left (3 a^4 A b+6 a^2 A b^3+35 A b^5-15 a^5 B-46 a^3 b^2 B-63 a b^4 B\right )\right ) \operatorname {Subst}\left (\int \frac {1}{a+b x^2} \, dx,x,\sqrt {\tan (c+d x)}\right )}{4 b^3 \left (a^2+b^2\right )^3 d}-\frac {\left (3 a^2 b (A-B)-b^3 (A-B)-a^3 (A+B)+3 a b^2 (A+B)\right ) \operatorname {Subst}\left (\int \frac {1+x^2}{1+x^4} \, dx,x,\sqrt {\tan (c+d x)}\right )}{\left (a^2+b^2\right )^3 d}+\frac {\left (a^3 (A-B)-3 a b^2 (A-B)+3 a^2 b (A+B)-b^3 (A+B)\right ) \operatorname {Subst}\left (\int \frac {1-x^2}{1+x^4} \, dx,x,\sqrt {\tan (c+d x)}\right )}{\left (a^2+b^2\right )^3 d}\\ &=\frac {a^{3/2} \left (3 a^4 A b+6 a^2 A b^3+35 A b^5-15 a^5 B-46 a^3 b^2 B-63 a b^4 B\right ) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {\tan (c+d x)}}{\sqrt {a}}\right )}{4 b^{7/2} \left (a^2+b^2\right )^3 d}-\frac {\left (3 a^3 A b+11 a A b^3-15 a^4 B-31 a^2 b^2 B-8 b^4 B\right ) \sqrt {\tan (c+d x)}}{4 b^3 \left (a^2+b^2\right )^2 d}+\frac {a (A b-a B) \tan ^{\frac {5}{2}}(c+d x)}{2 b \left (a^2+b^2\right ) d (a+b \tan (c+d x))^2}+\frac {a \left (a^2 A b+9 A b^3-5 a^3 B-13 a b^2 B\right ) \tan ^{\frac {3}{2}}(c+d x)}{4 b^2 \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))}-\frac {\left (3 a^2 b (A-B)-b^3 (A-B)-a^3 (A+B)+3 a b^2 (A+B)\right ) \operatorname {Subst}\left (\int \frac {1}{1-\sqrt {2} x+x^2} \, dx,x,\sqrt {\tan (c+d x)}\right )}{2 \left (a^2+b^2\right )^3 d}-\frac {\left (3 a^2 b (A-B)-b^3 (A-B)-a^3 (A+B)+3 a b^2 (A+B)\right ) \operatorname {Subst}\left (\int \frac {1}{1+\sqrt {2} x+x^2} \, dx,x,\sqrt {\tan (c+d x)}\right )}{2 \left (a^2+b^2\right )^3 d}-\frac {\left (a^3 (A-B)-3 a b^2 (A-B)+3 a^2 b (A+B)-b^3 (A+B)\right ) \operatorname {Subst}\left (\int \frac {\sqrt {2}+2 x}{-1-\sqrt {2} x-x^2} \, dx,x,\sqrt {\tan (c+d x)}\right )}{2 \sqrt {2} \left (a^2+b^2\right )^3 d}-\frac {\left (a^3 (A-B)-3 a b^2 (A-B)+3 a^2 b (A+B)-b^3 (A+B)\right ) \operatorname {Subst}\left (\int \frac {\sqrt {2}-2 x}{-1+\sqrt {2} x-x^2} \, dx,x,\sqrt {\tan (c+d x)}\right )}{2 \sqrt {2} \left (a^2+b^2\right )^3 d}\\ &=\frac {a^{3/2} \left (3 a^4 A b+6 a^2 A b^3+35 A b^5-15 a^5 B-46 a^3 b^2 B-63 a b^4 B\right ) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {\tan (c+d x)}}{\sqrt {a}}\right )}{4 b^{7/2} \left (a^2+b^2\right )^3 d}-\frac {\left (a^3 (A-B)-3 a b^2 (A-B)+3 a^2 b (A+B)-b^3 (A+B)\right ) \log \left (1-\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right )}{2 \sqrt {2} \left (a^2+b^2\right )^3 d}+\frac {\left (a^3 (A-B)-3 a b^2 (A-B)+3 a^2 b (A+B)-b^3 (A+B)\right ) \log \left (1+\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right )}{2 \sqrt {2} \left (a^2+b^2\right )^3 d}-\frac {\left (3 a^3 A b+11 a A b^3-15 a^4 B-31 a^2 b^2 B-8 b^4 B\right ) \sqrt {\tan (c+d x)}}{4 b^3 \left (a^2+b^2\right )^2 d}+\frac {a (A b-a B) \tan ^{\frac {5}{2}}(c+d x)}{2 b \left (a^2+b^2\right ) d (a+b \tan (c+d x))^2}+\frac {a \left (a^2 A b+9 A b^3-5 a^3 B-13 a b^2 B\right ) \tan ^{\frac {3}{2}}(c+d x)}{4 b^2 \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))}-\frac {\left (3 a^2 b (A-B)-b^3 (A-B)-a^3 (A+B)+3 a b^2 (A+B)\right ) \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} \left (a^2+b^2\right )^3 d}+\frac {\left (3 a^2 b (A-B)-b^3 (A-B)-a^3 (A+B)+3 a b^2 (A+B)\right ) \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} \left (a^2+b^2\right )^3 d}\\ &=\frac {\left (3 a^2 b (A-B)-b^3 (A-B)-a^3 (A+B)+3 a b^2 (A+B)\right ) \tan ^{-1}\left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} \left (a^2+b^2\right )^3 d}-\frac {\left (3 a^2 b (A-B)-b^3 (A-B)-a^3 (A+B)+3 a b^2 (A+B)\right ) \tan ^{-1}\left (1+\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} \left (a^2+b^2\right )^3 d}+\frac {a^{3/2} \left (3 a^4 A b+6 a^2 A b^3+35 A b^5-15 a^5 B-46 a^3 b^2 B-63 a b^4 B\right ) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {\tan (c+d x)}}{\sqrt {a}}\right )}{4 b^{7/2} \left (a^2+b^2\right )^3 d}-\frac {\left (a^3 (A-B)-3 a b^2 (A-B)+3 a^2 b (A+B)-b^3 (A+B)\right ) \log \left (1-\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right )}{2 \sqrt {2} \left (a^2+b^2\right )^3 d}+\frac {\left (a^3 (A-B)-3 a b^2 (A-B)+3 a^2 b (A+B)-b^3 (A+B)\right ) \log \left (1+\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right )}{2 \sqrt {2} \left (a^2+b^2\right )^3 d}-\frac {\left (3 a^3 A b+11 a A b^3-15 a^4 B-31 a^2 b^2 B-8 b^4 B\right ) \sqrt {\tan (c+d x)}}{4 b^3 \left (a^2+b^2\right )^2 d}+\frac {a (A b-a B) \tan ^{\frac {5}{2}}(c+d x)}{2 b \left (a^2+b^2\right ) d (a+b \tan (c+d x))^2}+\frac {a \left (a^2 A b+9 A b^3-5 a^3 B-13 a b^2 B\right ) \tan ^{\frac {3}{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 [B]  time = 6.35, size = 1563, normalized size = 2.60 \[ \text {result too large to display} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(((-1)^(1/4)*a^3*A*ArcTan[(-1)^(3/4)*Sqrt[Tan[c + d*x]]])/((a^2 + b^2)^3*d)) - (3*(-1)^(3/4)*a^2*A*b*ArcTan[(
-1)^(3/4)*Sqrt[Tan[c + d*x]]])/((a^2 + b^2)^3*d) + (3*(-1)^(1/4)*a*A*b^2*ArcTan[(-1)^(3/4)*Sqrt[Tan[c + d*x]]]
)/((a^2 + b^2)^3*d) + ((-1)^(3/4)*A*b^3*ArcTan[(-1)^(3/4)*Sqrt[Tan[c + d*x]]])/((a^2 + b^2)^3*d) + ((-1)^(3/4)
*a^3*B*ArcTan[(-1)^(3/4)*Sqrt[Tan[c + d*x]]])/((a^2 + b^2)^3*d) - (3*(-1)^(1/4)*a^2*b*B*ArcTan[(-1)^(3/4)*Sqrt
[Tan[c + d*x]]])/((a^2 + b^2)^3*d) - (3*(-1)^(3/4)*a*b^2*B*ArcTan[(-1)^(3/4)*Sqrt[Tan[c + d*x]]])/((a^2 + b^2)
^3*d) + ((-1)^(1/4)*b^3*B*ArcTan[(-1)^(3/4)*Sqrt[Tan[c + d*x]]])/((a^2 + b^2)^3*d) + (3*a^(11/2)*A*ArcTan[(Sqr
t[b]*Sqrt[Tan[c + d*x]])/Sqrt[a]])/(4*b^(5/2)*(a^2 + b^2)^3*d) + (3*a^(7/2)*A*ArcTan[(Sqrt[b]*Sqrt[Tan[c + d*x
]])/Sqrt[a]])/(2*Sqrt[b]*(a^2 + b^2)^3*d) + (35*a^(3/2)*A*b^(3/2)*ArcTan[(Sqrt[b]*Sqrt[Tan[c + d*x]])/Sqrt[a]]
)/(4*(a^2 + b^2)^3*d) - (15*a^(13/2)*B*ArcTan[(Sqrt[b]*Sqrt[Tan[c + d*x]])/Sqrt[a]])/(4*b^(7/2)*(a^2 + b^2)^3*
d) - (23*a^(9/2)*B*ArcTan[(Sqrt[b]*Sqrt[Tan[c + d*x]])/Sqrt[a]])/(2*b^(3/2)*(a^2 + b^2)^3*d) - (63*a^(5/2)*Sqr
t[b]*B*ArcTan[(Sqrt[b]*Sqrt[Tan[c + d*x]])/Sqrt[a]])/(4*(a^2 + b^2)^3*d) - ((-1)^(1/4)*a^3*A*ArcTanh[(-1)^(3/4
)*Sqrt[Tan[c + d*x]]])/((a^2 + b^2)^3*d) + (3*(-1)^(3/4)*a^2*A*b*ArcTanh[(-1)^(3/4)*Sqrt[Tan[c + d*x]]])/((a^2
+ b^2)^3*d) + (3*(-1)^(1/4)*a*A*b^2*ArcTanh[(-1)^(3/4)*Sqrt[Tan[c + d*x]]])/((a^2 + b^2)^3*d) - ((-1)^(3/4)*A
*b^3*ArcTanh[(-1)^(3/4)*Sqrt[Tan[c + d*x]]])/((a^2 + b^2)^3*d) - ((-1)^(3/4)*a^3*B*ArcTanh[(-1)^(3/4)*Sqrt[Tan
[c + d*x]]])/((a^2 + b^2)^3*d) - (3*(-1)^(1/4)*a^2*b*B*ArcTanh[(-1)^(3/4)*Sqrt[Tan[c + d*x]]])/((a^2 + b^2)^3*
d) + (3*(-1)^(3/4)*a*b^2*B*ArcTanh[(-1)^(3/4)*Sqrt[Tan[c + d*x]]])/((a^2 + b^2)^3*d) + ((-1)^(1/4)*b^3*B*ArcTa
nh[(-1)^(3/4)*Sqrt[Tan[c + d*x]]])/((a^2 + b^2)^3*d) - (2*a*A*Sqrt[Tan[c + d*x]])/(b^2*d*(a + b*Tan[c + d*x])^
2) + (a^3*A*Sqrt[Tan[c + d*x]])/(2*b^2*(a^2 + b^2)*d*(a + b*Tan[c + d*x])^2) + (10*a^2*B*Sqrt[Tan[c + d*x]])/(
b^3*d*(a + b*Tan[c + d*x])^2) + (2*B*Sqrt[Tan[c + d*x]])/(3*b*d*(a + b*Tan[c + d*x])^2) - (5*a^4*B*Sqrt[Tan[c
+ d*x]])/(2*b^3*(a^2 + b^2)*d*(a + b*Tan[c + d*x])^2) - (8*a^2*B*Sqrt[Tan[c + d*x]])/(3*b*(a^2 + b^2)*d*(a + b
*Tan[c + d*x])^2) - (2*b*B*Sqrt[Tan[c + d*x]])/(3*(a^2 + b^2)*d*(a + b*Tan[c + d*x])^2) - (2*A*Tan[c + d*x]^(3
/2))/(b*d*(a + b*Tan[c + d*x])^2) + (10*a*B*Tan[c + d*x]^(3/2))/(b^2*d*(a + b*Tan[c + d*x])^2) + (2*B*Tan[c +
d*x]^(5/2))/(b*d*(a + b*Tan[c + d*x])^2) + (3*a^2*A*Sqrt[Tan[c + d*x]])/(4*(a^2 + b^2)^2*d*(a + b*Tan[c + d*x]
)) + (3*a^4*A*Sqrt[Tan[c + d*x]])/(4*b^2*(a^2 + b^2)^2*d*(a + b*Tan[c + d*x])) + (2*A*b^2*Sqrt[Tan[c + d*x]])/
((a^2 + b^2)^2*d*(a + b*Tan[c + d*x])) - (15*a^5*B*Sqrt[Tan[c + d*x]])/(4*b^3*(a^2 + b^2)^2*d*(a + b*Tan[c + d
*x])) - (31*a^3*B*Sqrt[Tan[c + d*x]])/(4*b*(a^2 + b^2)^2*d*(a + b*Tan[c + d*x])) - (6*a*b*B*Sqrt[Tan[c + d*x]]
)/((a^2 + b^2)^2*d*(a + b*Tan[c + d*x]))

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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maple [B]  time = 0.46, size = 1864, normalized size = 3.11 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-3/4/d*a^7/b^2/(a^2+b^2)^3/(a+b*tan(d*x+c))^2*A*tan(d*x+c)^(1/2)-11/4/d*a^3*b^2/(a^2+b^2)^3/(a+b*tan(d*x+c))^2
*A*tan(d*x+c)^(1/2)+3/2/d/(a^2+b^2)^3*B*2^(1/2)*arctan(-1+2^(1/2)*tan(d*x+c)^(1/2))*a^2*b+3/4/d/(a^2+b^2)^3*B*
2^(1/2)*ln((1+2^(1/2)*tan(d*x+c)^(1/2)+tan(d*x+c))/(1-2^(1/2)*tan(d*x+c)^(1/2)+tan(d*x+c)))*a^2*b-3/4/d/(a^2+b
^2)^3*A*2^(1/2)*ln((1-2^(1/2)*tan(d*x+c)^(1/2)+tan(d*x+c))/(1+2^(1/2)*tan(d*x+c)^(1/2)+tan(d*x+c)))*a^2*b+9/4/
d*a^7/b^2/(a^2+b^2)^3/(a+b*tan(d*x+c))^2*tan(d*x+c)^(3/2)*B+17/4/d*a^3*b^2/(a^2+b^2)^3/(a+b*tan(d*x+c))^2*tan(
d*x+c)^(3/2)*B-3/4/d/(a^2+b^2)^3*B*2^(1/2)*ln((1-2^(1/2)*tan(d*x+c)^(1/2)+tan(d*x+c))/(1+2^(1/2)*tan(d*x+c)^(1
/2)+tan(d*x+c)))*a*b^2-3/2/d/(a^2+b^2)^3*B*2^(1/2)*arctan(1+2^(1/2)*tan(d*x+c)^(1/2))*a*b^2+15/4/d*a^4*b/(a^2+
b^2)^3/(a+b*tan(d*x+c))^2*B*tan(d*x+c)^(1/2)+3/4/d*a^6/b^2/(a^2+b^2)^3/(a*b)^(1/2)*arctan(tan(d*x+c)^(1/2)*b/(
a*b)^(1/2))*A-3/2/d/(a^2+b^2)^3*B*2^(1/2)*arctan(-1+2^(1/2)*tan(d*x+c)^(1/2))*a*b^2-23/2/d*a^5/b/(a^2+b^2)^3/(
a*b)^(1/2)*arctan(tan(d*x+c)^(1/2)*b/(a*b)^(1/2))*B-63/4/d*a^3*b/(a^2+b^2)^3/(a*b)^(1/2)*arctan(tan(d*x+c)^(1/
2)*b/(a*b)^(1/2))*B-3/2/d/(a^2+b^2)^3*A*2^(1/2)*arctan(1+2^(1/2)*tan(d*x+c)^(1/2))*a*b^2-3/2/d/(a^2+b^2)^3*A*2
^(1/2)*arctan(-1+2^(1/2)*tan(d*x+c)^(1/2))*a^2*b-3/4/d/(a^2+b^2)^3*A*2^(1/2)*ln((1+2^(1/2)*tan(d*x+c)^(1/2)+ta
n(d*x+c))/(1-2^(1/2)*tan(d*x+c)^(1/2)+tan(d*x+c)))*a*b^2+3/2/d/(a^2+b^2)^3*B*2^(1/2)*arctan(1+2^(1/2)*tan(d*x+
c)^(1/2))*a^2*b-3/2/d/(a^2+b^2)^3*A*2^(1/2)*arctan(-1+2^(1/2)*tan(d*x+c)^(1/2))*a*b^2+35/4/d*a^2*b^2/(a^2+b^2)
^3/(a*b)^(1/2)*arctan(tan(d*x+c)^(1/2)*b/(a*b)^(1/2))*A-15/4/d*a^7/b^3/(a^2+b^2)^3/(a*b)^(1/2)*arctan(tan(d*x+
c)^(1/2)*b/(a*b)^(1/2))*B-3/2/d/(a^2+b^2)^3*A*2^(1/2)*arctan(1+2^(1/2)*tan(d*x+c)^(1/2))*a^2*b+7/4/d*a^8/b^3/(
a^2+b^2)^3/(a+b*tan(d*x+c))^2*B*tan(d*x+c)^(1/2)+11/2/d*a^6/b/(a^2+b^2)^3/(a+b*tan(d*x+c))^2*B*tan(d*x+c)^(1/2
)-5/4/d*a^6/b/(a^2+b^2)^3/(a+b*tan(d*x+c))^2*tan(d*x+c)^(3/2)*A-9/2/d*a^4*b/(a^2+b^2)^3/(a+b*tan(d*x+c))^2*tan
(d*x+c)^(3/2)*A-13/4/d*a^2*b^3/(a^2+b^2)^3/(a+b*tan(d*x+c))^2*tan(d*x+c)^(3/2)*A+1/2/d/(a^2+b^2)^3*A*2^(1/2)*a
rctan(1+2^(1/2)*tan(d*x+c)^(1/2))*b^3+1/2/d/(a^2+b^2)^3*A*2^(1/2)*arctan(1+2^(1/2)*tan(d*x+c)^(1/2))*a^3-7/2/d
*a^5/(a^2+b^2)^3/(a+b*tan(d*x+c))^2*A*tan(d*x+c)^(1/2)+3/2/d*a^4/(a^2+b^2)^3/(a*b)^(1/2)*arctan(tan(d*x+c)^(1/
2)*b/(a*b)^(1/2))*A+1/4/d/(a^2+b^2)^3*B*2^(1/2)*ln((1-2^(1/2)*tan(d*x+c)^(1/2)+tan(d*x+c))/(1+2^(1/2)*tan(d*x+
c)^(1/2)+tan(d*x+c)))*a^3+1/2/d/(a^2+b^2)^3*B*2^(1/2)*arctan(1+2^(1/2)*tan(d*x+c)^(1/2))*a^3+1/2/d/(a^2+b^2)^3
*A*2^(1/2)*arctan(-1+2^(1/2)*tan(d*x+c)^(1/2))*b^3+1/4/d/(a^2+b^2)^3*A*2^(1/2)*ln((1+2^(1/2)*tan(d*x+c)^(1/2)+
tan(d*x+c))/(1-2^(1/2)*tan(d*x+c)^(1/2)+tan(d*x+c)))*a^3-1/2/d/(a^2+b^2)^3*B*2^(1/2)*arctan(1+2^(1/2)*tan(d*x+
c)^(1/2))*b^3-1/2/d/(a^2+b^2)^3*B*2^(1/2)*arctan(-1+2^(1/2)*tan(d*x+c)^(1/2))*b^3-1/4/d/(a^2+b^2)^3*B*2^(1/2)*
ln((1+2^(1/2)*tan(d*x+c)^(1/2)+tan(d*x+c))/(1-2^(1/2)*tan(d*x+c)^(1/2)+tan(d*x+c)))*b^3+1/4/d/(a^2+b^2)^3*A*2^
(1/2)*ln((1-2^(1/2)*tan(d*x+c)^(1/2)+tan(d*x+c))/(1+2^(1/2)*tan(d*x+c)^(1/2)+tan(d*x+c)))*b^3+2/d*B/b^3*tan(d*
x+c)^(1/2)+1/2/d/(a^2+b^2)^3*A*2^(1/2)*arctan(-1+2^(1/2)*tan(d*x+c)^(1/2))*a^3+1/2/d/(a^2+b^2)^3*B*2^(1/2)*arc
tan(-1+2^(1/2)*tan(d*x+c)^(1/2))*a^3+13/2/d*a^5/(a^2+b^2)^3/(a+b*tan(d*x+c))^2*tan(d*x+c)^(3/2)*B

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maxima [A]  time = 1.06, size = 571, normalized size = 0.95 \[ -\frac {\frac {{\left (15 \, B a^{7} - 3 \, A a^{6} b + 46 \, B a^{5} b^{2} - 6 \, A a^{4} b^{3} + 63 \, B a^{3} b^{4} - 35 \, A a^{2} b^{5}\right )} \arctan \left (\frac {b \sqrt {\tan \left (d x + c\right )}}{\sqrt {a b}}\right )}{{\left (a^{6} b^{3} + 3 \, a^{4} b^{5} + 3 \, a^{2} b^{7} + b^{9}\right )} \sqrt {a b}} - \frac {2 \, \sqrt {2} {\left ({\left (A + B\right )} a^{3} - 3 \, {\left (A - B\right )} a^{2} b - 3 \, {\left (A + B\right )} a b^{2} + {\left (A - B\right )} b^{3}\right )} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + 2 \, \sqrt {\tan \left (d x + c\right )}\right )}\right ) + 2 \, \sqrt {2} {\left ({\left (A + B\right )} a^{3} - 3 \, {\left (A - B\right )} a^{2} b - 3 \, {\left (A + B\right )} a b^{2} + {\left (A - B\right )} b^{3}\right )} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - 2 \, \sqrt {\tan \left (d x + c\right )}\right )}\right ) + \sqrt {2} {\left ({\left (A - B\right )} a^{3} + 3 \, {\left (A + B\right )} a^{2} b - 3 \, {\left (A - B\right )} a b^{2} - {\left (A + B\right )} b^{3}\right )} \log \left (\sqrt {2} \sqrt {\tan \left (d x + c\right )} + \tan \left (d x + c\right ) + 1\right ) - \sqrt {2} {\left ({\left (A - B\right )} a^{3} + 3 \, {\left (A + B\right )} a^{2} b - 3 \, {\left (A - B\right )} a b^{2} - {\left (A + B\right )} b^{3}\right )} \log \left (-\sqrt {2} \sqrt {\tan \left (d x + c\right )} + \tan \left (d x + c\right ) + 1\right )}{a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}} - \frac {{\left (9 \, B a^{5} b - 5 \, A a^{4} b^{2} + 17 \, B a^{3} b^{3} - 13 \, A a^{2} b^{4}\right )} \tan \left (d x + c\right )^{\frac {3}{2}} + {\left (7 \, B a^{6} - 3 \, A a^{5} b + 15 \, B a^{4} b^{2} - 11 \, A a^{3} b^{3}\right )} \sqrt {\tan \left (d x + c\right )}}{a^{6} b^{3} + 2 \, a^{4} b^{5} + a^{2} b^{7} + {\left (a^{4} b^{5} + 2 \, a^{2} b^{7} + b^{9}\right )} \tan \left (d x + c\right )^{2} + 2 \, {\left (a^{5} b^{4} + 2 \, a^{3} b^{6} + a b^{8}\right )} \tan \left (d x + c\right )} - \frac {8 \, B \sqrt {\tan \left (d x + c\right )}}{b^{3}}}{4 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*((15*B*a^7 - 3*A*a^6*b + 46*B*a^5*b^2 - 6*A*a^4*b^3 + 63*B*a^3*b^4 - 35*A*a^2*b^5)*arctan(b*sqrt(tan(d*x
+ c))/sqrt(a*b))/((a^6*b^3 + 3*a^4*b^5 + 3*a^2*b^7 + b^9)*sqrt(a*b)) - (2*sqrt(2)*((A + B)*a^3 - 3*(A - B)*a^2
*b - 3*(A + B)*a*b^2 + (A - B)*b^3)*arctan(1/2*sqrt(2)*(sqrt(2) + 2*sqrt(tan(d*x + c)))) + 2*sqrt(2)*((A + B)*
a^3 - 3*(A - B)*a^2*b - 3*(A + B)*a*b^2 + (A - B)*b^3)*arctan(-1/2*sqrt(2)*(sqrt(2) - 2*sqrt(tan(d*x + c)))) +
sqrt(2)*((A - B)*a^3 + 3*(A + B)*a^2*b - 3*(A - B)*a*b^2 - (A + B)*b^3)*log(sqrt(2)*sqrt(tan(d*x + c)) + tan(
d*x + c) + 1) - sqrt(2)*((A - B)*a^3 + 3*(A + B)*a^2*b - 3*(A - B)*a*b^2 - (A + B)*b^3)*log(-sqrt(2)*sqrt(tan(
d*x + c)) + tan(d*x + c) + 1))/(a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6) - ((9*B*a^5*b - 5*A*a^4*b^2 + 17*B*a^3*b^3
- 13*A*a^2*b^4)*tan(d*x + c)^(3/2) + (7*B*a^6 - 3*A*a^5*b + 15*B*a^4*b^2 - 11*A*a^3*b^3)*sqrt(tan(d*x + c)))/(
a^6*b^3 + 2*a^4*b^5 + a^2*b^7 + (a^4*b^5 + 2*a^2*b^7 + b^9)*tan(d*x + c)^2 + 2*(a^5*b^4 + 2*a^3*b^6 + a*b^8)*t
an(d*x + c)) - 8*B*sqrt(tan(d*x + c))/b^3)/d

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mupad [B]  time = 60.84, size = 27429, normalized size = 45.72 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(log(((((((((128*b^3*tan(c + d*x)^(1/2)*(a^2 - b^2)*(a^2 + b^2)^2*((4*(-B^4*d^4*(a^6 - b^6 + 15*a^2*b^4 - 15*a
^4*b^2)^2)^(1/2) + 80*B^2*a^3*b^3*d^2 - 24*B^2*a*b^5*d^2 - 24*B^2*a^5*b*d^2)/(d^4*(a^2 + b^2)^6))^(1/2) - (64*
B*a*b*(15*a^4 + 2*b^4 + 41*a^2*b^2))/d)*((4*(-B^4*d^4*(a^6 - b^6 + 15*a^2*b^4 - 15*a^4*b^2)^2)^(1/2) + 80*B^2*
a^3*b^3*d^2 - 24*B^2*a*b^5*d^2 - 24*B^2*a^5*b*d^2)/(d^4*(a^2 + b^2)^6))^(1/2))/4 + (8*B^2*a*tan(c + d*x)^(1/2)
*(225*a^14 - 184*b^14 + 608*a^2*b^12 - 272*a^4*b^10 + 3937*a^6*b^8 + 5804*a^8*b^6 + 4006*a^10*b^4 + 1380*a^12*
b^2))/(b^4*d^2*(a^2 + b^2)^4))*((4*(-B^4*d^4*(a^6 - b^6 + 15*a^2*b^4 - 15*a^4*b^2)^2)^(1/2) + 80*B^2*a^3*b^3*d
^2 - 24*B^2*a*b^5*d^2 - 24*B^2*a^5*b*d^2)/(d^4*(a^2 + b^2)^6))^(1/2))/4 - (2*B^3*a^2*(1125*a^14 + 16*b^14 + 61
12*a^2*b^12 - 17727*a^4*b^10 - 23239*a^6*b^8 - 11174*a^8*b^6 + 2930*a^10*b^4 + 3525*a^12*b^2))/(b^4*d^3*(a^2 +
b^2)^6))*((4*(-B^4*d^4*(a^6 - b^6 + 15*a^2*b^4 - 15*a^4*b^2)^2)^(1/2) + 80*B^2*a^3*b^3*d^2 - 24*B^2*a*b^5*d^2
- 24*B^2*a^5*b*d^2)/(d^4*(a^2 + b^2)^6))^(1/2))/4 - (B^4*tan(c + d*x)^(1/2)*(32*b^18 - 225*a^18 + 128*a^2*b^1
6 + 192*a^4*b^14 - 3841*a^6*b^12 + 18050*a^8*b^10 + 26801*a^10*b^8 + 16860*a^12*b^6 + 4049*a^14*b^4 - 30*a^16*
b^2))/(b^5*d^4*(a^2 + b^2)^8))*((4*(-B^4*d^4*(a^6 - b^6 + 15*a^2*b^4 - 15*a^4*b^2)^2)^(1/2) + 80*B^2*a^3*b^3*d
^2 - 24*B^2*a*b^5*d^2 - 24*B^2*a^5*b*d^2)/(d^4*(a^2 + b^2)^6))^(1/2))/4 - (B^5*a^3*(225*a^12 + 504*b^12 + 872*
a^2*b^10 + 4457*a^4*b^8 + 5916*a^6*b^6 + 4006*a^8*b^4 + 1380*a^10*b^2))/(2*b^5*d^5*(a^2 + b^2)^8))*(((480*B^4*
a^2*b^10*d^4 - 16*B^4*b^12*d^4 - 16*B^4*a^12*d^4 - 4080*B^4*a^4*b^8*d^4 + 7232*B^4*a^6*b^6*d^4 - 4080*B^4*a^8*
b^4*d^4 + 480*B^4*a^10*b^2*d^4)^(1/2) + 80*B^2*a^3*b^3*d^2 - 24*B^2*a*b^5*d^2 - 24*B^2*a^5*b*d^2)/(a^12*d^4 +
b^12*d^4 + 6*a^2*b^10*d^4 + 15*a^4*b^8*d^4 + 20*a^6*b^6*d^4 + 15*a^8*b^4*d^4 + 6*a^10*b^2*d^4))^(1/2))/4 + (lo
g(((((((((128*b^3*tan(c + d*x)^(1/2)*(a^2 - b^2)*(a^2 + b^2)^2*(-(4*(-B^4*d^4*(a^6 - b^6 + 15*a^2*b^4 - 15*a^4
*b^2)^2)^(1/2) - 80*B^2*a^3*b^3*d^2 + 24*B^2*a*b^5*d^2 + 24*B^2*a^5*b*d^2)/(d^4*(a^2 + b^2)^6))^(1/2) - (64*B*
a*b*(15*a^4 + 2*b^4 + 41*a^2*b^2))/d)*(-(4*(-B^4*d^4*(a^6 - b^6 + 15*a^2*b^4 - 15*a^4*b^2)^2)^(1/2) - 80*B^2*a
^3*b^3*d^2 + 24*B^2*a*b^5*d^2 + 24*B^2*a^5*b*d^2)/(d^4*(a^2 + b^2)^6))^(1/2))/4 + (8*B^2*a*tan(c + d*x)^(1/2)*
(225*a^14 - 184*b^14 + 608*a^2*b^12 - 272*a^4*b^10 + 3937*a^6*b^8 + 5804*a^8*b^6 + 4006*a^10*b^4 + 1380*a^12*b
^2))/(b^4*d^2*(a^2 + b^2)^4))*(-(4*(-B^4*d^4*(a^6 - b^6 + 15*a^2*b^4 - 15*a^4*b^2)^2)^(1/2) - 80*B^2*a^3*b^3*d
^2 + 24*B^2*a*b^5*d^2 + 24*B^2*a^5*b*d^2)/(d^4*(a^2 + b^2)^6))^(1/2))/4 - (2*B^3*a^2*(1125*a^14 + 16*b^14 + 61
12*a^2*b^12 - 17727*a^4*b^10 - 23239*a^6*b^8 - 11174*a^8*b^6 + 2930*a^10*b^4 + 3525*a^12*b^2))/(b^4*d^3*(a^2 +
b^2)^6))*(-(4*(-B^4*d^4*(a^6 - b^6 + 15*a^2*b^4 - 15*a^4*b^2)^2)^(1/2) - 80*B^2*a^3*b^3*d^2 + 24*B^2*a*b^5*d^
2 + 24*B^2*a^5*b*d^2)/(d^4*(a^2 + b^2)^6))^(1/2))/4 - (B^4*tan(c + d*x)^(1/2)*(32*b^18 - 225*a^18 + 128*a^2*b^
16 + 192*a^4*b^14 - 3841*a^6*b^12 + 18050*a^8*b^10 + 26801*a^10*b^8 + 16860*a^12*b^6 + 4049*a^14*b^4 - 30*a^16
*b^2))/(b^5*d^4*(a^2 + b^2)^8))*(-(4*(-B^4*d^4*(a^6 - b^6 + 15*a^2*b^4 - 15*a^4*b^2)^2)^(1/2) - 80*B^2*a^3*b^3
*d^2 + 24*B^2*a*b^5*d^2 + 24*B^2*a^5*b*d^2)/(d^4*(a^2 + b^2)^6))^(1/2))/4 - (B^5*a^3*(225*a^12 + 504*b^12 + 87
2*a^2*b^10 + 4457*a^4*b^8 + 5916*a^6*b^6 + 4006*a^8*b^4 + 1380*a^10*b^2))/(2*b^5*d^5*(a^2 + b^2)^8))*(-((480*B
^4*a^2*b^10*d^4 - 16*B^4*b^12*d^4 - 16*B^4*a^12*d^4 - 4080*B^4*a^4*b^8*d^4 + 7232*B^4*a^6*b^6*d^4 - 4080*B^4*a
^8*b^4*d^4 + 480*B^4*a^10*b^2*d^4)^(1/2) - 80*B^2*a^3*b^3*d^2 + 24*B^2*a*b^5*d^2 + 24*B^2*a^5*b*d^2)/(a^12*d^4
+ b^12*d^4 + 6*a^2*b^10*d^4 + 15*a^4*b^8*d^4 + 20*a^6*b^6*d^4 + 15*a^8*b^4*d^4 + 6*a^10*b^2*d^4))^(1/2))/4 -
log(- ((((((((128*b^3*tan(c + d*x)^(1/2)*(a^2 - b^2)*(a^2 + b^2)^2*((4*(-B^4*d^4*(a^6 - b^6 + 15*a^2*b^4 - 15*
a^4*b^2)^2)^(1/2) + 80*B^2*a^3*b^3*d^2 - 24*B^2*a*b^5*d^2 - 24*B^2*a^5*b*d^2)/(d^4*(a^2 + b^2)^6))^(1/2) + (64
*B*a*b*(15*a^4 + 2*b^4 + 41*a^2*b^2))/d)*((4*(-B^4*d^4*(a^6 - b^6 + 15*a^2*b^4 - 15*a^4*b^2)^2)^(1/2) + 80*B^2
*a^3*b^3*d^2 - 24*B^2*a*b^5*d^2 - 24*B^2*a^5*b*d^2)/(d^4*(a^2 + b^2)^6))^(1/2))/4 + (8*B^2*a*tan(c + d*x)^(1/2
)*(225*a^14 - 184*b^14 + 608*a^2*b^12 - 272*a^4*b^10 + 3937*a^6*b^8 + 5804*a^8*b^6 + 4006*a^10*b^4 + 1380*a^12
*b^2))/(b^4*d^2*(a^2 + b^2)^4))*((4*(-B^4*d^4*(a^6 - b^6 + 15*a^2*b^4 - 15*a^4*b^2)^2)^(1/2) + 80*B^2*a^3*b^3*
d^2 - 24*B^2*a*b^5*d^2 - 24*B^2*a^5*b*d^2)/(d^4*(a^2 + b^2)^6))^(1/2))/4 + (2*B^3*a^2*(1125*a^14 + 16*b^14 + 6
112*a^2*b^12 - 17727*a^4*b^10 - 23239*a^6*b^8 - 11174*a^8*b^6 + 2930*a^10*b^4 + 3525*a^12*b^2))/(b^4*d^3*(a^2
+ b^2)^6))*((4*(-B^4*d^4*(a^6 - b^6 + 15*a^2*b^4 - 15*a^4*b^2)^2)^(1/2) + 80*B^2*a^3*b^3*d^2 - 24*B^2*a*b^5*d^
2 - 24*B^2*a^5*b*d^2)/(d^4*(a^2 + b^2)^6))^(1/2))/4 - (B^4*tan(c + d*x)^(1/2)*(32*b^18 - 225*a^18 + 128*a^2*b^
16 + 192*a^4*b^14 - 3841*a^6*b^12 + 18050*a^8*b^10 + 26801*a^10*b^8 + 16860*a^12*b^6 + 4049*a^14*b^4 - 30*a^16
*b^2))/(b^5*d^4*(a^2 + b^2)^8))*((4*(-B^4*d^4*(a^6 - b^6 + 15*a^2*b^4 - 15*a^4*b^2)^2)^(1/2) + 80*B^2*a^3*b^3*
d^2 - 24*B^2*a*b^5*d^2 - 24*B^2*a^5*b*d^2)/(d^4*(a^2 + b^2)^6))^(1/2))/4 - (B^5*a^3*(225*a^12 + 504*b^12 + 872
*a^2*b^10 + 4457*a^4*b^8 + 5916*a^6*b^6 + 4006*a^8*b^4 + 1380*a^10*b^2))/(2*b^5*d^5*(a^2 + b^2)^8))*(((480*B^4
*a^2*b^10*d^4 - 16*B^4*b^12*d^4 - 16*B^4*a^12*d^4 - 4080*B^4*a^4*b^8*d^4 + 7232*B^4*a^6*b^6*d^4 - 4080*B^4*a^8
*b^4*d^4 + 480*B^4*a^10*b^2*d^4)^(1/2) + 80*B^2*a^3*b^3*d^2 - 24*B^2*a*b^5*d^2 - 24*B^2*a^5*b*d^2)/(16*a^12*d^
4 + 16*b^12*d^4 + 96*a^2*b^10*d^4 + 240*a^4*b^8*d^4 + 320*a^6*b^6*d^4 + 240*a^8*b^4*d^4 + 96*a^10*b^2*d^4))^(1
/2) - log(- ((((((((128*b^3*tan(c + d*x)^(1/2)*(a^2 - b^2)*(a^2 + b^2)^2*(-(4*(-B^4*d^4*(a^6 - b^6 + 15*a^2*b^
4 - 15*a^4*b^2)^2)^(1/2) - 80*B^2*a^3*b^3*d^2 + 24*B^2*a*b^5*d^2 + 24*B^2*a^5*b*d^2)/(d^4*(a^2 + b^2)^6))^(1/2
) + (64*B*a*b*(15*a^4 + 2*b^4 + 41*a^2*b^2))/d)*(-(4*(-B^4*d^4*(a^6 - b^6 + 15*a^2*b^4 - 15*a^4*b^2)^2)^(1/2)
- 80*B^2*a^3*b^3*d^2 + 24*B^2*a*b^5*d^2 + 24*B^2*a^5*b*d^2)/(d^4*(a^2 + b^2)^6))^(1/2))/4 + (8*B^2*a*tan(c + d
*x)^(1/2)*(225*a^14 - 184*b^14 + 608*a^2*b^12 - 272*a^4*b^10 + 3937*a^6*b^8 + 5804*a^8*b^6 + 4006*a^10*b^4 + 1
380*a^12*b^2))/(b^4*d^2*(a^2 + b^2)^4))*(-(4*(-B^4*d^4*(a^6 - b^6 + 15*a^2*b^4 - 15*a^4*b^2)^2)^(1/2) - 80*B^2
*a^3*b^3*d^2 + 24*B^2*a*b^5*d^2 + 24*B^2*a^5*b*d^2)/(d^4*(a^2 + b^2)^6))^(1/2))/4 + (2*B^3*a^2*(1125*a^14 + 16
*b^14 + 6112*a^2*b^12 - 17727*a^4*b^10 - 23239*a^6*b^8 - 11174*a^8*b^6 + 2930*a^10*b^4 + 3525*a^12*b^2))/(b^4*
d^3*(a^2 + b^2)^6))*(-(4*(-B^4*d^4*(a^6 - b^6 + 15*a^2*b^4 - 15*a^4*b^2)^2)^(1/2) - 80*B^2*a^3*b^3*d^2 + 24*B^
2*a*b^5*d^2 + 24*B^2*a^5*b*d^2)/(d^4*(a^2 + b^2)^6))^(1/2))/4 - (B^4*tan(c + d*x)^(1/2)*(32*b^18 - 225*a^18 +
128*a^2*b^16 + 192*a^4*b^14 - 3841*a^6*b^12 + 18050*a^8*b^10 + 26801*a^10*b^8 + 16860*a^12*b^6 + 4049*a^14*b^4
- 30*a^16*b^2))/(b^5*d^4*(a^2 + b^2)^8))*(-(4*(-B^4*d^4*(a^6 - b^6 + 15*a^2*b^4 - 15*a^4*b^2)^2)^(1/2) - 80*B
^2*a^3*b^3*d^2 + 24*B^2*a*b^5*d^2 + 24*B^2*a^5*b*d^2)/(d^4*(a^2 + b^2)^6))^(1/2))/4 - (B^5*a^3*(225*a^12 + 504
*b^12 + 872*a^2*b^10 + 4457*a^4*b^8 + 5916*a^6*b^6 + 4006*a^8*b^4 + 1380*a^10*b^2))/(2*b^5*d^5*(a^2 + b^2)^8))
*(-((480*B^4*a^2*b^10*d^4 - 16*B^4*b^12*d^4 - 16*B^4*a^12*d^4 - 4080*B^4*a^4*b^8*d^4 + 7232*B^4*a^6*b^6*d^4 -
4080*B^4*a^8*b^4*d^4 + 480*B^4*a^10*b^2*d^4)^(1/2) - 80*B^2*a^3*b^3*d^2 + 24*B^2*a*b^5*d^2 + 24*B^2*a^5*b*d^2)
/(16*a^12*d^4 + 16*b^12*d^4 + 96*a^2*b^10*d^4 + 240*a^4*b^8*d^4 + 320*a^6*b^6*d^4 + 240*a^8*b^4*d^4 + 96*a^10*
b^2*d^4))^(1/2) + (log((((((((((64*A*a^2*b^2*(a^2 + 25*b^2))/d + 128*b^3*tan(c + d*x)^(1/2)*(a^2 - b^2)*(a^2 +
b^2)^2*((4*(-A^4*d^4*(a^6 - b^6 + 15*a^2*b^4 - 15*a^4*b^2)^2)^(1/2) - 80*A^2*a^3*b^3*d^2 + 24*A^2*a*b^5*d^2 +
24*A^2*a^5*b*d^2)/(d^4*(a^2 + b^2)^6))^(1/2))*((4*(-A^4*d^4*(a^6 - b^6 + 15*a^2*b^4 - 15*a^4*b^2)^2)^(1/2) -
80*A^2*a^3*b^3*d^2 + 24*A^2*a*b^5*d^2 + 24*A^2*a^5*b*d^2)/(d^4*(a^2 + b^2)^6))^(1/2))/4 + (8*A^2*a*tan(c + d*x
)^(1/2)*(9*a^12 + 184*b^12 - 608*a^2*b^10 + 1497*a^4*b^8 + 452*a^6*b^6 + 238*a^8*b^4 + 36*a^10*b^2))/(b^2*d^2*
(a^2 + b^2)^4))*((4*(-A^4*d^4*(a^6 - b^6 + 15*a^2*b^4 - 15*a^4*b^2)^2)^(1/2) - 80*A^2*a^3*b^3*d^2 + 24*A^2*a*b
^5*d^2 + 24*A^2*a^5*b*d^2)/(d^4*(a^2 + b^2)^6))^(1/2))/4 + (2*A^3*a*(16*b^14 - 9*a^14 - 3296*a^2*b^12 + 7955*a
^4*b^10 + 627*a^6*b^8 + 1582*a^8*b^6 + 6*a^10*b^4 + 63*a^12*b^2))/(b^3*d^3*(a^2 + b^2)^6))*((4*(-A^4*d^4*(a^6
- b^6 + 15*a^2*b^4 - 15*a^4*b^2)^2)^(1/2) - 80*A^2*a^3*b^3*d^2 + 24*A^2*a*b^5*d^2 + 24*A^2*a^5*b*d^2)/(d^4*(a^
2 + b^2)^6))^(1/2))/4 - (A^4*tan(c + d*x)^(1/2)*(9*a^16 + 32*b^16 + 128*a^2*b^14 + 1417*a^4*b^12 - 6802*a^6*b^
10 - 1017*a^8*b^8 - 1020*a^10*b^6 + 39*a^12*b^4 - 18*a^14*b^2))/(b^3*d^4*(a^2 + b^2)^8))*((4*(-A^4*d^4*(a^6 -
b^6 + 15*a^2*b^4 - 15*a^4*b^2)^2)^(1/2) - 80*A^2*a^3*b^3*d^2 + 24*A^2*a*b^5*d^2 + 24*A^2*a^5*b*d^2)/(d^4*(a^2
+ b^2)^6))^(1/2))/4 + (A^5*a^2*(9*a^10 + 280*b^10 + 1553*a^2*b^8 + 492*a^4*b^6 + 270*a^6*b^4 + 36*a^8*b^2))/(2
*b^2*d^5*(a^2 + b^2)^8))*(((480*A^4*a^2*b^10*d^4 - 16*A^4*b^12*d^4 - 16*A^4*a^12*d^4 - 4080*A^4*a^4*b^8*d^4 +
7232*A^4*a^6*b^6*d^4 - 4080*A^4*a^8*b^4*d^4 + 480*A^4*a^10*b^2*d^4)^(1/2) - 80*A^2*a^3*b^3*d^2 + 24*A^2*a*b^5*
d^2 + 24*A^2*a^5*b*d^2)/(a^12*d^4 + b^12*d^4 + 6*a^2*b^10*d^4 + 15*a^4*b^8*d^4 + 20*a^6*b^6*d^4 + 15*a^8*b^4*d
^4 + 6*a^10*b^2*d^4))^(1/2))/4 + (log((((((((((64*A*a^2*b^2*(a^2 + 25*b^2))/d + 128*b^3*tan(c + d*x)^(1/2)*(a^
2 - b^2)*(a^2 + b^2)^2*(-(4*(-A^4*d^4*(a^6 - b^6 + 15*a^2*b^4 - 15*a^4*b^2)^2)^(1/2) + 80*A^2*a^3*b^3*d^2 - 24
*A^2*a*b^5*d^2 - 24*A^2*a^5*b*d^2)/(d^4*(a^2 + b^2)^6))^(1/2))*(-(4*(-A^4*d^4*(a^6 - b^6 + 15*a^2*b^4 - 15*a^4
*b^2)^2)^(1/2) + 80*A^2*a^3*b^3*d^2 - 24*A^2*a*b^5*d^2 - 24*A^2*a^5*b*d^2)/(d^4*(a^2 + b^2)^6))^(1/2))/4 + (8*
A^2*a*tan(c + d*x)^(1/2)*(9*a^12 + 184*b^12 - 608*a^2*b^10 + 1497*a^4*b^8 + 452*a^6*b^6 + 238*a^8*b^4 + 36*a^1
0*b^2))/(b^2*d^2*(a^2 + b^2)^4))*(-(4*(-A^4*d^4*(a^6 - b^6 + 15*a^2*b^4 - 15*a^4*b^2)^2)^(1/2) + 80*A^2*a^3*b^
3*d^2 - 24*A^2*a*b^5*d^2 - 24*A^2*a^5*b*d^2)/(d^4*(a^2 + b^2)^6))^(1/2))/4 + (2*A^3*a*(16*b^14 - 9*a^14 - 3296
*a^2*b^12 + 7955*a^4*b^10 + 627*a^6*b^8 + 1582*a^8*b^6 + 6*a^10*b^4 + 63*a^12*b^2))/(b^3*d^3*(a^2 + b^2)^6))*(
-(4*(-A^4*d^4*(a^6 - b^6 + 15*a^2*b^4 - 15*a^4*b^2)^2)^(1/2) + 80*A^2*a^3*b^3*d^2 - 24*A^2*a*b^5*d^2 - 24*A^2*
a^5*b*d^2)/(d^4*(a^2 + b^2)^6))^(1/2))/4 - (A^4*tan(c + d*x)^(1/2)*(9*a^16 + 32*b^16 + 128*a^2*b^14 + 1417*a^4
*b^12 - 6802*a^6*b^10 - 1017*a^8*b^8 - 1020*a^10*b^6 + 39*a^12*b^4 - 18*a^14*b^2))/(b^3*d^4*(a^2 + b^2)^8))*(-
(4*(-A^4*d^4*(a^6 - b^6 + 15*a^2*b^4 - 15*a^4*b^2)^2)^(1/2) + 80*A^2*a^3*b^3*d^2 - 24*A^2*a*b^5*d^2 - 24*A^2*a
^5*b*d^2)/(d^4*(a^2 + b^2)^6))^(1/2))/4 + (A^5*a^2*(9*a^10 + 280*b^10 + 1553*a^2*b^8 + 492*a^4*b^6 + 270*a^6*b
^4 + 36*a^8*b^2))/(2*b^2*d^5*(a^2 + b^2)^8))*(-((480*A^4*a^2*b^10*d^4 - 16*A^4*b^12*d^4 - 16*A^4*a^12*d^4 - 40
80*A^4*a^4*b^8*d^4 + 7232*A^4*a^6*b^6*d^4 - 4080*A^4*a^8*b^4*d^4 + 480*A^4*a^10*b^2*d^4)^(1/2) + 80*A^2*a^3*b^
3*d^2 - 24*A^2*a*b^5*d^2 - 24*A^2*a^5*b*d^2)/(a^12*d^4 + b^12*d^4 + 6*a^2*b^10*d^4 + 15*a^4*b^8*d^4 + 20*a^6*b
^6*d^4 + 15*a^8*b^4*d^4 + 6*a^10*b^2*d^4))^(1/2))/4 - log((((((((((64*A*a^2*b^2*(a^2 + 25*b^2))/d - 128*b^3*ta
n(c + d*x)^(1/2)*(a^2 - b^2)*(a^2 + b^2)^2*((4*(-A^4*d^4*(a^6 - b^6 + 15*a^2*b^4 - 15*a^4*b^2)^2)^(1/2) - 80*A
^2*a^3*b^3*d^2 + 24*A^2*a*b^5*d^2 + 24*A^2*a^5*b*d^2)/(d^4*(a^2 + b^2)^6))^(1/2))*((4*(-A^4*d^4*(a^6 - b^6 + 1
5*a^2*b^4 - 15*a^4*b^2)^2)^(1/2) - 80*A^2*a^3*b^3*d^2 + 24*A^2*a*b^5*d^2 + 24*A^2*a^5*b*d^2)/(d^4*(a^2 + b^2)^
6))^(1/2))/4 - (8*A^2*a*tan(c + d*x)^(1/2)*(9*a^12 + 184*b^12 - 608*a^2*b^10 + 1497*a^4*b^8 + 452*a^6*b^6 + 23
8*a^8*b^4 + 36*a^10*b^2))/(b^2*d^2*(a^2 + b^2)^4))*((4*(-A^4*d^4*(a^6 - b^6 + 15*a^2*b^4 - 15*a^4*b^2)^2)^(1/2
) - 80*A^2*a^3*b^3*d^2 + 24*A^2*a*b^5*d^2 + 24*A^2*a^5*b*d^2)/(d^4*(a^2 + b^2)^6))^(1/2))/4 + (2*A^3*a*(16*b^1
4 - 9*a^14 - 3296*a^2*b^12 + 7955*a^4*b^10 + 627*a^6*b^8 + 1582*a^8*b^6 + 6*a^10*b^4 + 63*a^12*b^2))/(b^3*d^3*
(a^2 + b^2)^6))*((4*(-A^4*d^4*(a^6 - b^6 + 15*a^2*b^4 - 15*a^4*b^2)^2)^(1/2) - 80*A^2*a^3*b^3*d^2 + 24*A^2*a*b
^5*d^2 + 24*A^2*a^5*b*d^2)/(d^4*(a^2 + b^2)^6))^(1/2))/4 + (A^4*tan(c + d*x)^(1/2)*(9*a^16 + 32*b^16 + 128*a^2
*b^14 + 1417*a^4*b^12 - 6802*a^6*b^10 - 1017*a^8*b^8 - 1020*a^10*b^6 + 39*a^12*b^4 - 18*a^14*b^2))/(b^3*d^4*(a
^2 + b^2)^8))*((4*(-A^4*d^4*(a^6 - b^6 + 15*a^2*b^4 - 15*a^4*b^2)^2)^(1/2) - 80*A^2*a^3*b^3*d^2 + 24*A^2*a*b^5
*d^2 + 24*A^2*a^5*b*d^2)/(d^4*(a^2 + b^2)^6))^(1/2))/4 + (A^5*a^2*(9*a^10 + 280*b^10 + 1553*a^2*b^8 + 492*a^4*
b^6 + 270*a^6*b^4 + 36*a^8*b^2))/(2*b^2*d^5*(a^2 + b^2)^8))*(((480*A^4*a^2*b^10*d^4 - 16*A^4*b^12*d^4 - 16*A^4
*a^12*d^4 - 4080*A^4*a^4*b^8*d^4 + 7232*A^4*a^6*b^6*d^4 - 4080*A^4*a^8*b^4*d^4 + 480*A^4*a^10*b^2*d^4)^(1/2) -
80*A^2*a^3*b^3*d^2 + 24*A^2*a*b^5*d^2 + 24*A^2*a^5*b*d^2)/(16*a^12*d^4 + 16*b^12*d^4 + 96*a^2*b^10*d^4 + 240*
a^4*b^8*d^4 + 320*a^6*b^6*d^4 + 240*a^8*b^4*d^4 + 96*a^10*b^2*d^4))^(1/2) - log((((((((((64*A*a^2*b^2*(a^2 + 2
5*b^2))/d - 128*b^3*tan(c + d*x)^(1/2)*(a^2 - b^2)*(a^2 + b^2)^2*(-(4*(-A^4*d^4*(a^6 - b^6 + 15*a^2*b^4 - 15*a
^4*b^2)^2)^(1/2) + 80*A^2*a^3*b^3*d^2 - 24*A^2*a*b^5*d^2 - 24*A^2*a^5*b*d^2)/(d^4*(a^2 + b^2)^6))^(1/2))*(-(4*
(-A^4*d^4*(a^6 - b^6 + 15*a^2*b^4 - 15*a^4*b^2)^2)^(1/2) + 80*A^2*a^3*b^3*d^2 - 24*A^2*a*b^5*d^2 - 24*A^2*a^5*
b*d^2)/(d^4*(a^2 + b^2)^6))^(1/2))/4 - (8*A^2*a*tan(c + d*x)^(1/2)*(9*a^12 + 184*b^12 - 608*a^2*b^10 + 1497*a^
4*b^8 + 452*a^6*b^6 + 238*a^8*b^4 + 36*a^10*b^2))/(b^2*d^2*(a^2 + b^2)^4))*(-(4*(-A^4*d^4*(a^6 - b^6 + 15*a^2*
b^4 - 15*a^4*b^2)^2)^(1/2) + 80*A^2*a^3*b^3*d^2 - 24*A^2*a*b^5*d^2 - 24*A^2*a^5*b*d^2)/(d^4*(a^2 + b^2)^6))^(1
/2))/4 + (2*A^3*a*(16*b^14 - 9*a^14 - 3296*a^2*b^12 + 7955*a^4*b^10 + 627*a^6*b^8 + 1582*a^8*b^6 + 6*a^10*b^4
+ 63*a^12*b^2))/(b^3*d^3*(a^2 + b^2)^6))*(-(4*(-A^4*d^4*(a^6 - b^6 + 15*a^2*b^4 - 15*a^4*b^2)^2)^(1/2) + 80*A^
2*a^3*b^3*d^2 - 24*A^2*a*b^5*d^2 - 24*A^2*a^5*b*d^2)/(d^4*(a^2 + b^2)^6))^(1/2))/4 + (A^4*tan(c + d*x)^(1/2)*(
9*a^16 + 32*b^16 + 128*a^2*b^14 + 1417*a^4*b^12 - 6802*a^6*b^10 - 1017*a^8*b^8 - 1020*a^10*b^6 + 39*a^12*b^4 -
18*a^14*b^2))/(b^3*d^4*(a^2 + b^2)^8))*(-(4*(-A^4*d^4*(a^6 - b^6 + 15*a^2*b^4 - 15*a^4*b^2)^2)^(1/2) + 80*A^2
*a^3*b^3*d^2 - 24*A^2*a*b^5*d^2 - 24*A^2*a^5*b*d^2)/(d^4*(a^2 + b^2)^6))^(1/2))/4 + (A^5*a^2*(9*a^10 + 280*b^1
0 + 1553*a^2*b^8 + 492*a^4*b^6 + 270*a^6*b^4 + 36*a^8*b^2))/(2*b^2*d^5*(a^2 + b^2)^8))*(-((480*A^4*a^2*b^10*d^
4 - 16*A^4*b^12*d^4 - 16*A^4*a^12*d^4 - 4080*A^4*a^4*b^8*d^4 + 7232*A^4*a^6*b^6*d^4 - 4080*A^4*a^8*b^4*d^4 + 4
80*A^4*a^10*b^2*d^4)^(1/2) + 80*A^2*a^3*b^3*d^2 - 24*A^2*a*b^5*d^2 - 24*A^2*a^5*b*d^2)/(16*a^12*d^4 + 16*b^12*
d^4 + 96*a^2*b^10*d^4 + 240*a^4*b^8*d^4 + 320*a^6*b^6*d^4 + 240*a^8*b^4*d^4 + 96*a^10*b^2*d^4))^(1/2) + ((tan(
c + d*x)^(3/2)*(17*B*a^3*b^3 + 9*B*a^5*b))/(4*(a^4 + b^4 + 2*a^2*b^2)) + (a*tan(c + d*x)^(1/2)*(7*B*a^5 + 15*B
*a^3*b^2))/(4*(a^4 + b^4 + 2*a^2*b^2)))/(a^2*b^3*d + b^5*d*tan(c + d*x)^2 + 2*a*b^4*d*tan(c + d*x)) - ((A*a^3*
tan(c + d*x)^(1/2)*(3*a^2 + 11*b^2))/(4*b^2*(a^4 + b^4 + 2*a^2*b^2)) + (A*a^2*tan(c + d*x)^(3/2)*(5*a^2 + 13*b
^2))/(4*b*(a^4 + b^4 + 2*a^2*b^2)))/(a^2*d + b^2*d*tan(c + d*x)^2 + 2*a*b*d*tan(c + d*x)) + (atan(((((((32*B^3
*a^2*b^19*d^2 + 12288*B^3*a^4*b^17*d^2 - 10974*B^3*a^6*b^15*d^2 - 105162*B^3*a^8*b^13*d^2 - 150758*B^3*a^10*b^
11*d^2 - 85314*B^3*a^12*b^9*d^2 - 3578*B^3*a^14*b^7*d^2 + 22210*B^3*a^16*b^5*d^2 + 11550*B^3*a^18*b^3*d^2 + 22
50*B^3*a^20*b*d^2)/(64*(b^21*d^5 + 8*a^2*b^19*d^5 + 28*a^4*b^17*d^5 + 56*a^6*b^15*d^5 + 70*a^8*b^13*d^5 + 56*a
^10*b^11*d^5 + 28*a^12*b^9*d^5 + 8*a^14*b^7*d^5 + a^16*b^5*d^5)) + (((((128*B*a*b^26*d^4 + 3648*B*a^3*b^24*d^4
+ 25536*B*a^5*b^22*d^4 + 88320*B*a^7*b^20*d^4 + 182784*B*a^9*b^18*d^4 + 244608*B*a^11*b^16*d^4 + 217728*B*a^1
3*b^14*d^4 + 128256*B*a^15*b^12*d^4 + 48000*B*a^17*b^10*d^4 + 10304*B*a^19*b^8*d^4 + 960*B*a^21*b^6*d^4)/(64*(
b^21*d^5 + 8*a^2*b^19*d^5 + 28*a^4*b^17*d^5 + 56*a^6*b^15*d^5 + 70*a^8*b^13*d^5 + 56*a^10*b^11*d^5 + 28*a^12*b
^9*d^5 + 8*a^14*b^7*d^5 + a^16*b^5*d^5)) - (tan(c + d*x)^(1/2)*(-64*(225*B^2*a^13 + 3969*B^2*a^5*b^8 + 5796*B^
2*a^7*b^6 + 4006*B^2*a^9*b^4 + 1380*B^2*a^11*b^2)*(b^19*d^2 + 6*a^2*b^17*d^2 + 15*a^4*b^15*d^2 + 20*a^6*b^13*d
^2 + 15*a^8*b^11*d^2 + 6*a^10*b^9*d^2 + a^12*b^7*d^2))^(1/2)*(512*b^30*d^4 + 4608*a^2*b^28*d^4 + 17920*a^4*b^2
6*d^4 + 38400*a^6*b^24*d^4 + 46080*a^8*b^22*d^4 + 21504*a^10*b^20*d^4 - 21504*a^12*b^18*d^4 - 46080*a^14*b^16*
d^4 - 38400*a^16*b^14*d^4 - 17920*a^18*b^12*d^4 - 4608*a^20*b^10*d^4 - 512*a^22*b^8*d^4))/(4096*(b^19*d^2 + 6*
a^2*b^17*d^2 + 15*a^4*b^15*d^2 + 20*a^6*b^13*d^2 + 15*a^8*b^11*d^2 + 6*a^10*b^9*d^2 + a^12*b^7*d^2)*(b^21*d^4
+ 8*a^2*b^19*d^4 + 28*a^4*b^17*d^4 + 56*a^6*b^15*d^4 + 70*a^8*b^13*d^4 + 56*a^10*b^11*d^4 + 28*a^12*b^9*d^4 +
8*a^14*b^7*d^4 + a^16*b^5*d^4)))*(-64*(225*B^2*a^13 + 3969*B^2*a^5*b^8 + 5796*B^2*a^7*b^6 + 4006*B^2*a^9*b^4 +
1380*B^2*a^11*b^2)*(b^19*d^2 + 6*a^2*b^17*d^2 + 15*a^4*b^15*d^2 + 20*a^6*b^13*d^2 + 15*a^8*b^11*d^2 + 6*a^10*
b^9*d^2 + a^12*b^7*d^2))^(1/2))/(64*(b^19*d^2 + 6*a^2*b^17*d^2 + 15*a^4*b^15*d^2 + 20*a^6*b^13*d^2 + 15*a^8*b^
11*d^2 + 6*a^10*b^9*d^2 + a^12*b^7*d^2)) + (tan(c + d*x)^(1/2)*(8448*B^2*a^5*b^19*d^2 - 1024*B^2*a^3*b^21*d^2
+ 46088*B^2*a^7*b^17*d^2 + 177344*B^2*a^9*b^15*d^2 + 402912*B^2*a^11*b^13*d^2 + 541632*B^2*a^13*b^11*d^2 + 455
472*B^2*a^15*b^9*d^2 + 248064*B^2*a^17*b^7*d^2 + 87008*B^2*a^19*b^5*d^2 + 18240*B^2*a^21*b^3*d^2 - 1472*B^2*a*
b^23*d^2 + 1800*B^2*a^23*b*d^2))/(64*(b^21*d^4 + 8*a^2*b^19*d^4 + 28*a^4*b^17*d^4 + 56*a^6*b^15*d^4 + 70*a^8*b
^13*d^4 + 56*a^10*b^11*d^4 + 28*a^12*b^9*d^4 + 8*a^14*b^7*d^4 + a^16*b^5*d^4)))*(-64*(225*B^2*a^13 + 3969*B^2*
a^5*b^8 + 5796*B^2*a^7*b^6 + 4006*B^2*a^9*b^4 + 1380*B^2*a^11*b^2)*(b^19*d^2 + 6*a^2*b^17*d^2 + 15*a^4*b^15*d^
2 + 20*a^6*b^13*d^2 + 15*a^8*b^11*d^2 + 6*a^10*b^9*d^2 + a^12*b^7*d^2))^(1/2))/(64*(b^19*d^2 + 6*a^2*b^17*d^2
+ 15*a^4*b^15*d^2 + 20*a^6*b^13*d^2 + 15*a^8*b^11*d^2 + 6*a^10*b^9*d^2 + a^12*b^7*d^2)))*(-64*(225*B^2*a^13 +
3969*B^2*a^5*b^8 + 5796*B^2*a^7*b^6 + 4006*B^2*a^9*b^4 + 1380*B^2*a^11*b^2)*(b^19*d^2 + 6*a^2*b^17*d^2 + 15*a^
4*b^15*d^2 + 20*a^6*b^13*d^2 + 15*a^8*b^11*d^2 + 6*a^10*b^9*d^2 + a^12*b^7*d^2))^(1/2))/(64*(b^19*d^2 + 6*a^2*
b^17*d^2 + 15*a^4*b^15*d^2 + 20*a^6*b^13*d^2 + 15*a^8*b^11*d^2 + 6*a^10*b^9*d^2 + a^12*b^7*d^2)) - (tan(c + d*
x)^(1/2)*(32*B^4*b^18 - 225*B^4*a^18 + 128*B^4*a^2*b^16 + 192*B^4*a^4*b^14 - 3841*B^4*a^6*b^12 + 18050*B^4*a^8
*b^10 + 26801*B^4*a^10*b^8 + 16860*B^4*a^12*b^6 + 4049*B^4*a^14*b^4 - 30*B^4*a^16*b^2))/(64*(b^21*d^4 + 8*a^2*
b^19*d^4 + 28*a^4*b^17*d^4 + 56*a^6*b^15*d^4 + 70*a^8*b^13*d^4 + 56*a^10*b^11*d^4 + 28*a^12*b^9*d^4 + 8*a^14*b
^7*d^4 + a^16*b^5*d^4)))*(-64*(225*B^2*a^13 + 3969*B^2*a^5*b^8 + 5796*B^2*a^7*b^6 + 4006*B^2*a^9*b^4 + 1380*B^
2*a^11*b^2)*(b^19*d^2 + 6*a^2*b^17*d^2 + 15*a^4*b^15*d^2 + 20*a^6*b^13*d^2 + 15*a^8*b^11*d^2 + 6*a^10*b^9*d^2
+ a^12*b^7*d^2))^(1/2)*1i)/(b^19*d^2 + 6*a^2*b^17*d^2 + 15*a^4*b^15*d^2 + 20*a^6*b^13*d^2 + 15*a^8*b^11*d^2 +
6*a^10*b^9*d^2 + a^12*b^7*d^2) - (((((32*B^3*a^2*b^19*d^2 + 12288*B^3*a^4*b^17*d^2 - 10974*B^3*a^6*b^15*d^2 -
105162*B^3*a^8*b^13*d^2 - 150758*B^3*a^10*b^11*d^2 - 85314*B^3*a^12*b^9*d^2 - 3578*B^3*a^14*b^7*d^2 + 22210*B^
3*a^16*b^5*d^2 + 11550*B^3*a^18*b^3*d^2 + 2250*B^3*a^20*b*d^2)/(64*(b^21*d^5 + 8*a^2*b^19*d^5 + 28*a^4*b^17*d^
5 + 56*a^6*b^15*d^5 + 70*a^8*b^13*d^5 + 56*a^10*b^11*d^5 + 28*a^12*b^9*d^5 + 8*a^14*b^7*d^5 + a^16*b^5*d^5)) +
(((((128*B*a*b^26*d^4 + 3648*B*a^3*b^24*d^4 + 25536*B*a^5*b^22*d^4 + 88320*B*a^7*b^20*d^4 + 182784*B*a^9*b^18
*d^4 + 244608*B*a^11*b^16*d^4 + 217728*B*a^13*b^14*d^4 + 128256*B*a^15*b^12*d^4 + 48000*B*a^17*b^10*d^4 + 1030
4*B*a^19*b^8*d^4 + 960*B*a^21*b^6*d^4)/(64*(b^21*d^5 + 8*a^2*b^19*d^5 + 28*a^4*b^17*d^5 + 56*a^6*b^15*d^5 + 70
*a^8*b^13*d^5 + 56*a^10*b^11*d^5 + 28*a^12*b^9*d^5 + 8*a^14*b^7*d^5 + a^16*b^5*d^5)) + (tan(c + d*x)^(1/2)*(-6
4*(225*B^2*a^13 + 3969*B^2*a^5*b^8 + 5796*B^2*a^7*b^6 + 4006*B^2*a^9*b^4 + 1380*B^2*a^11*b^2)*(b^19*d^2 + 6*a^
2*b^17*d^2 + 15*a^4*b^15*d^2 + 20*a^6*b^13*d^2 + 15*a^8*b^11*d^2 + 6*a^10*b^9*d^2 + a^12*b^7*d^2))^(1/2)*(512*
b^30*d^4 + 4608*a^2*b^28*d^4 + 17920*a^4*b^26*d^4 + 38400*a^6*b^24*d^4 + 46080*a^8*b^22*d^4 + 21504*a^10*b^20*
d^4 - 21504*a^12*b^18*d^4 - 46080*a^14*b^16*d^4 - 38400*a^16*b^14*d^4 - 17920*a^18*b^12*d^4 - 4608*a^20*b^10*d
^4 - 512*a^22*b^8*d^4))/(4096*(b^19*d^2 + 6*a^2*b^17*d^2 + 15*a^4*b^15*d^2 + 20*a^6*b^13*d^2 + 15*a^8*b^11*d^2
+ 6*a^10*b^9*d^2 + a^12*b^7*d^2)*(b^21*d^4 + 8*a^2*b^19*d^4 + 28*a^4*b^17*d^4 + 56*a^6*b^15*d^4 + 70*a^8*b^13
*d^4 + 56*a^10*b^11*d^4 + 28*a^12*b^9*d^4 + 8*a^14*b^7*d^4 + a^16*b^5*d^4)))*(-64*(225*B^2*a^13 + 3969*B^2*a^5
*b^8 + 5796*B^2*a^7*b^6 + 4006*B^2*a^9*b^4 + 1380*B^2*a^11*b^2)*(b^19*d^2 + 6*a^2*b^17*d^2 + 15*a^4*b^15*d^2 +
20*a^6*b^13*d^2 + 15*a^8*b^11*d^2 + 6*a^10*b^9*d^2 + a^12*b^7*d^2))^(1/2))/(64*(b^19*d^2 + 6*a^2*b^17*d^2 + 1
5*a^4*b^15*d^2 + 20*a^6*b^13*d^2 + 15*a^8*b^11*d^2 + 6*a^10*b^9*d^2 + a^12*b^7*d^2)) - (tan(c + d*x)^(1/2)*(84
48*B^2*a^5*b^19*d^2 - 1024*B^2*a^3*b^21*d^2 + 46088*B^2*a^7*b^17*d^2 + 177344*B^2*a^9*b^15*d^2 + 402912*B^2*a^
11*b^13*d^2 + 541632*B^2*a^13*b^11*d^2 + 455472*B^2*a^15*b^9*d^2 + 248064*B^2*a^17*b^7*d^2 + 87008*B^2*a^19*b^
5*d^2 + 18240*B^2*a^21*b^3*d^2 - 1472*B^2*a*b^23*d^2 + 1800*B^2*a^23*b*d^2))/(64*(b^21*d^4 + 8*a^2*b^19*d^4 +
28*a^4*b^17*d^4 + 56*a^6*b^15*d^4 + 70*a^8*b^13*d^4 + 56*a^10*b^11*d^4 + 28*a^12*b^9*d^4 + 8*a^14*b^7*d^4 + a^
16*b^5*d^4)))*(-64*(225*B^2*a^13 + 3969*B^2*a^5*b^8 + 5796*B^2*a^7*b^6 + 4006*B^2*a^9*b^4 + 1380*B^2*a^11*b^2)
*(b^19*d^2 + 6*a^2*b^17*d^2 + 15*a^4*b^15*d^2 + 20*a^6*b^13*d^2 + 15*a^8*b^11*d^2 + 6*a^10*b^9*d^2 + a^12*b^7*
d^2))^(1/2))/(64*(b^19*d^2 + 6*a^2*b^17*d^2 + 15*a^4*b^15*d^2 + 20*a^6*b^13*d^2 + 15*a^8*b^11*d^2 + 6*a^10*b^9
*d^2 + a^12*b^7*d^2)))*(-64*(225*B^2*a^13 + 3969*B^2*a^5*b^8 + 5796*B^2*a^7*b^6 + 4006*B^2*a^9*b^4 + 1380*B^2*
a^11*b^2)*(b^19*d^2 + 6*a^2*b^17*d^2 + 15*a^4*b^15*d^2 + 20*a^6*b^13*d^2 + 15*a^8*b^11*d^2 + 6*a^10*b^9*d^2 +
a^12*b^7*d^2))^(1/2))/(64*(b^19*d^2 + 6*a^2*b^17*d^2 + 15*a^4*b^15*d^2 + 20*a^6*b^13*d^2 + 15*a^8*b^11*d^2 + 6
*a^10*b^9*d^2 + a^12*b^7*d^2)) + (tan(c + d*x)^(1/2)*(32*B^4*b^18 - 225*B^4*a^18 + 128*B^4*a^2*b^16 + 192*B^4*
a^4*b^14 - 3841*B^4*a^6*b^12 + 18050*B^4*a^8*b^10 + 26801*B^4*a^10*b^8 + 16860*B^4*a^12*b^6 + 4049*B^4*a^14*b^
4 - 30*B^4*a^16*b^2))/(64*(b^21*d^4 + 8*a^2*b^19*d^4 + 28*a^4*b^17*d^4 + 56*a^6*b^15*d^4 + 70*a^8*b^13*d^4 + 5
6*a^10*b^11*d^4 + 28*a^12*b^9*d^4 + 8*a^14*b^7*d^4 + a^16*b^5*d^4)))*(-64*(225*B^2*a^13 + 3969*B^2*a^5*b^8 + 5
796*B^2*a^7*b^6 + 4006*B^2*a^9*b^4 + 1380*B^2*a^11*b^2)*(b^19*d^2 + 6*a^2*b^17*d^2 + 15*a^4*b^15*d^2 + 20*a^6*
b^13*d^2 + 15*a^8*b^11*d^2 + 6*a^10*b^9*d^2 + a^12*b^7*d^2))^(1/2)*1i)/(b^19*d^2 + 6*a^2*b^17*d^2 + 15*a^4*b^1
5*d^2 + 20*a^6*b^13*d^2 + 15*a^8*b^11*d^2 + 6*a^10*b^9*d^2 + a^12*b^7*d^2))/((225*B^5*a^15 + 504*B^5*a^3*b^12
+ 872*B^5*a^5*b^10 + 4457*B^5*a^7*b^8 + 5916*B^5*a^9*b^6 + 4006*B^5*a^11*b^4 + 1380*B^5*a^13*b^2)/(b^21*d^5 +
8*a^2*b^19*d^5 + 28*a^4*b^17*d^5 + 56*a^6*b^15*d^5 + 70*a^8*b^13*d^5 + 56*a^10*b^11*d^5 + 28*a^12*b^9*d^5 + 8*
a^14*b^7*d^5 + a^16*b^5*d^5) + (((((32*B^3*a^2*b^19*d^2 + 12288*B^3*a^4*b^17*d^2 - 10974*B^3*a^6*b^15*d^2 - 10
5162*B^3*a^8*b^13*d^2 - 150758*B^3*a^10*b^11*d^2 - 85314*B^3*a^12*b^9*d^2 - 3578*B^3*a^14*b^7*d^2 + 22210*B^3*
a^16*b^5*d^2 + 11550*B^3*a^18*b^3*d^2 + 2250*B^3*a^20*b*d^2)/(64*(b^21*d^5 + 8*a^2*b^19*d^5 + 28*a^4*b^17*d^5
+ 56*a^6*b^15*d^5 + 70*a^8*b^13*d^5 + 56*a^10*b^11*d^5 + 28*a^12*b^9*d^5 + 8*a^14*b^7*d^5 + a^16*b^5*d^5)) + (
((((128*B*a*b^26*d^4 + 3648*B*a^3*b^24*d^4 + 25536*B*a^5*b^22*d^4 + 88320*B*a^7*b^20*d^4 + 182784*B*a^9*b^18*d
^4 + 244608*B*a^11*b^16*d^4 + 217728*B*a^13*b^14*d^4 + 128256*B*a^15*b^12*d^4 + 48000*B*a^17*b^10*d^4 + 10304*
B*a^19*b^8*d^4 + 960*B*a^21*b^6*d^4)/(64*(b^21*d^5 + 8*a^2*b^19*d^5 + 28*a^4*b^17*d^5 + 56*a^6*b^15*d^5 + 70*a
^8*b^13*d^5 + 56*a^10*b^11*d^5 + 28*a^12*b^9*d^5 + 8*a^14*b^7*d^5 + a^16*b^5*d^5)) - (tan(c + d*x)^(1/2)*(-64*
(225*B^2*a^13 + 3969*B^2*a^5*b^8 + 5796*B^2*a^7*b^6 + 4006*B^2*a^9*b^4 + 1380*B^2*a^11*b^2)*(b^19*d^2 + 6*a^2*
b^17*d^2 + 15*a^4*b^15*d^2 + 20*a^6*b^13*d^2 + 15*a^8*b^11*d^2 + 6*a^10*b^9*d^2 + a^12*b^7*d^2))^(1/2)*(512*b^
30*d^4 + 4608*a^2*b^28*d^4 + 17920*a^4*b^26*d^4 + 38400*a^6*b^24*d^4 + 46080*a^8*b^22*d^4 + 21504*a^10*b^20*d^
4 - 21504*a^12*b^18*d^4 - 46080*a^14*b^16*d^4 - 38400*a^16*b^14*d^4 - 17920*a^18*b^12*d^4 - 4608*a^20*b^10*d^4
- 512*a^22*b^8*d^4))/(4096*(b^19*d^2 + 6*a^2*b^17*d^2 + 15*a^4*b^15*d^2 + 20*a^6*b^13*d^2 + 15*a^8*b^11*d^2 +
6*a^10*b^9*d^2 + a^12*b^7*d^2)*(b^21*d^4 + 8*a^2*b^19*d^4 + 28*a^4*b^17*d^4 + 56*a^6*b^15*d^4 + 70*a^8*b^13*d
^4 + 56*a^10*b^11*d^4 + 28*a^12*b^9*d^4 + 8*a^14*b^7*d^4 + a^16*b^5*d^4)))*(-64*(225*B^2*a^13 + 3969*B^2*a^5*b
^8 + 5796*B^2*a^7*b^6 + 4006*B^2*a^9*b^4 + 1380*B^2*a^11*b^2)*(b^19*d^2 + 6*a^2*b^17*d^2 + 15*a^4*b^15*d^2 + 2
0*a^6*b^13*d^2 + 15*a^8*b^11*d^2 + 6*a^10*b^9*d^2 + a^12*b^7*d^2))^(1/2))/(64*(b^19*d^2 + 6*a^2*b^17*d^2 + 15*
a^4*b^15*d^2 + 20*a^6*b^13*d^2 + 15*a^8*b^11*d^2 + 6*a^10*b^9*d^2 + a^12*b^7*d^2)) + (tan(c + d*x)^(1/2)*(8448
*B^2*a^5*b^19*d^2 - 1024*B^2*a^3*b^21*d^2 + 46088*B^2*a^7*b^17*d^2 + 177344*B^2*a^9*b^15*d^2 + 402912*B^2*a^11
*b^13*d^2 + 541632*B^2*a^13*b^11*d^2 + 455472*B^2*a^15*b^9*d^2 + 248064*B^2*a^17*b^7*d^2 + 87008*B^2*a^19*b^5*
d^2 + 18240*B^2*a^21*b^3*d^2 - 1472*B^2*a*b^23*d^2 + 1800*B^2*a^23*b*d^2))/(64*(b^21*d^4 + 8*a^2*b^19*d^4 + 28
*a^4*b^17*d^4 + 56*a^6*b^15*d^4 + 70*a^8*b^13*d^4 + 56*a^10*b^11*d^4 + 28*a^12*b^9*d^4 + 8*a^14*b^7*d^4 + a^16
*b^5*d^4)))*(-64*(225*B^2*a^13 + 3969*B^2*a^5*b^8 + 5796*B^2*a^7*b^6 + 4006*B^2*a^9*b^4 + 1380*B^2*a^11*b^2)*(
b^19*d^2 + 6*a^2*b^17*d^2 + 15*a^4*b^15*d^2 + 20*a^6*b^13*d^2 + 15*a^8*b^11*d^2 + 6*a^10*b^9*d^2 + a^12*b^7*d^
2))^(1/2))/(64*(b^19*d^2 + 6*a^2*b^17*d^2 + 15*a^4*b^15*d^2 + 20*a^6*b^13*d^2 + 15*a^8*b^11*d^2 + 6*a^10*b^9*d
^2 + a^12*b^7*d^2)))*(-64*(225*B^2*a^13 + 3969*B^2*a^5*b^8 + 5796*B^2*a^7*b^6 + 4006*B^2*a^9*b^4 + 1380*B^2*a^
11*b^2)*(b^19*d^2 + 6*a^2*b^17*d^2 + 15*a^4*b^15*d^2 + 20*a^6*b^13*d^2 + 15*a^8*b^11*d^2 + 6*a^10*b^9*d^2 + a^
12*b^7*d^2))^(1/2))/(64*(b^19*d^2 + 6*a^2*b^17*d^2 + 15*a^4*b^15*d^2 + 20*a^6*b^13*d^2 + 15*a^8*b^11*d^2 + 6*a
^10*b^9*d^2 + a^12*b^7*d^2)) - (tan(c + d*x)^(1/2)*(32*B^4*b^18 - 225*B^4*a^18 + 128*B^4*a^2*b^16 + 192*B^4*a^
4*b^14 - 3841*B^4*a^6*b^12 + 18050*B^4*a^8*b^10 + 26801*B^4*a^10*b^8 + 16860*B^4*a^12*b^6 + 4049*B^4*a^14*b^4
- 30*B^4*a^16*b^2))/(64*(b^21*d^4 + 8*a^2*b^19*d^4 + 28*a^4*b^17*d^4 + 56*a^6*b^15*d^4 + 70*a^8*b^13*d^4 + 56*
a^10*b^11*d^4 + 28*a^12*b^9*d^4 + 8*a^14*b^7*d^4 + a^16*b^5*d^4)))*(-64*(225*B^2*a^13 + 3969*B^2*a^5*b^8 + 579
6*B^2*a^7*b^6 + 4006*B^2*a^9*b^4 + 1380*B^2*a^11*b^2)*(b^19*d^2 + 6*a^2*b^17*d^2 + 15*a^4*b^15*d^2 + 20*a^6*b^
13*d^2 + 15*a^8*b^11*d^2 + 6*a^10*b^9*d^2 + a^12*b^7*d^2))^(1/2))/(b^19*d^2 + 6*a^2*b^17*d^2 + 15*a^4*b^15*d^2
+ 20*a^6*b^13*d^2 + 15*a^8*b^11*d^2 + 6*a^10*b^9*d^2 + a^12*b^7*d^2) + (((((32*B^3*a^2*b^19*d^2 + 12288*B^3*a
^4*b^17*d^2 - 10974*B^3*a^6*b^15*d^2 - 105162*B^3*a^8*b^13*d^2 - 150758*B^3*a^10*b^11*d^2 - 85314*B^3*a^12*b^9
*d^2 - 3578*B^3*a^14*b^7*d^2 + 22210*B^3*a^16*b^5*d^2 + 11550*B^3*a^18*b^3*d^2 + 2250*B^3*a^20*b*d^2)/(64*(b^2
1*d^5 + 8*a^2*b^19*d^5 + 28*a^4*b^17*d^5 + 56*a^6*b^15*d^5 + 70*a^8*b^13*d^5 + 56*a^10*b^11*d^5 + 28*a^12*b^9*
d^5 + 8*a^14*b^7*d^5 + a^16*b^5*d^5)) + (((((128*B*a*b^26*d^4 + 3648*B*a^3*b^24*d^4 + 25536*B*a^5*b^22*d^4 + 8
8320*B*a^7*b^20*d^4 + 182784*B*a^9*b^18*d^4 + 244608*B*a^11*b^16*d^4 + 217728*B*a^13*b^14*d^4 + 128256*B*a^15*
b^12*d^4 + 48000*B*a^17*b^10*d^4 + 10304*B*a^19*b^8*d^4 + 960*B*a^21*b^6*d^4)/(64*(b^21*d^5 + 8*a^2*b^19*d^5 +
28*a^4*b^17*d^5 + 56*a^6*b^15*d^5 + 70*a^8*b^13*d^5 + 56*a^10*b^11*d^5 + 28*a^12*b^9*d^5 + 8*a^14*b^7*d^5 + a
^16*b^5*d^5)) + (tan(c + d*x)^(1/2)*(-64*(225*B^2*a^13 + 3969*B^2*a^5*b^8 + 5796*B^2*a^7*b^6 + 4006*B^2*a^9*b^
4 + 1380*B^2*a^11*b^2)*(b^19*d^2 + 6*a^2*b^17*d^2 + 15*a^4*b^15*d^2 + 20*a^6*b^13*d^2 + 15*a^8*b^11*d^2 + 6*a^
10*b^9*d^2 + a^12*b^7*d^2))^(1/2)*(512*b^30*d^4 + 4608*a^2*b^28*d^4 + 17920*a^4*b^26*d^4 + 38400*a^6*b^24*d^4
+ 46080*a^8*b^22*d^4 + 21504*a^10*b^20*d^4 - 21504*a^12*b^18*d^4 - 46080*a^14*b^16*d^4 - 38400*a^16*b^14*d^4 -
17920*a^18*b^12*d^4 - 4608*a^20*b^10*d^4 - 512*a^22*b^8*d^4))/(4096*(b^19*d^2 + 6*a^2*b^17*d^2 + 15*a^4*b^15*
d^2 + 20*a^6*b^13*d^2 + 15*a^8*b^11*d^2 + 6*a^10*b^9*d^2 + a^12*b^7*d^2)*(b^21*d^4 + 8*a^2*b^19*d^4 + 28*a^4*b
^17*d^4 + 56*a^6*b^15*d^4 + 70*a^8*b^13*d^4 + 56*a^10*b^11*d^4 + 28*a^12*b^9*d^4 + 8*a^14*b^7*d^4 + a^16*b^5*d
^4)))*(-64*(225*B^2*a^13 + 3969*B^2*a^5*b^8 + 5796*B^2*a^7*b^6 + 4006*B^2*a^9*b^4 + 1380*B^2*a^11*b^2)*(b^19*d
^2 + 6*a^2*b^17*d^2 + 15*a^4*b^15*d^2 + 20*a^6*b^13*d^2 + 15*a^8*b^11*d^2 + 6*a^10*b^9*d^2 + a^12*b^7*d^2))^(1
/2))/(64*(b^19*d^2 + 6*a^2*b^17*d^2 + 15*a^4*b^15*d^2 + 20*a^6*b^13*d^2 + 15*a^8*b^11*d^2 + 6*a^10*b^9*d^2 + a
^12*b^7*d^2)) - (tan(c + d*x)^(1/2)*(8448*B^2*a^5*b^19*d^2 - 1024*B^2*a^3*b^21*d^2 + 46088*B^2*a^7*b^17*d^2 +
177344*B^2*a^9*b^15*d^2 + 402912*B^2*a^11*b^13*d^2 + 541632*B^2*a^13*b^11*d^2 + 455472*B^2*a^15*b^9*d^2 + 2480
64*B^2*a^17*b^7*d^2 + 87008*B^2*a^19*b^5*d^2 + 18240*B^2*a^21*b^3*d^2 - 1472*B^2*a*b^23*d^2 + 1800*B^2*a^23*b*
d^2))/(64*(b^21*d^4 + 8*a^2*b^19*d^4 + 28*a^4*b^17*d^4 + 56*a^6*b^15*d^4 + 70*a^8*b^13*d^4 + 56*a^10*b^11*d^4
+ 28*a^12*b^9*d^4 + 8*a^14*b^7*d^4 + a^16*b^5*d^4)))*(-64*(225*B^2*a^13 + 3969*B^2*a^5*b^8 + 5796*B^2*a^7*b^6
+ 4006*B^2*a^9*b^4 + 1380*B^2*a^11*b^2)*(b^19*d^2 + 6*a^2*b^17*d^2 + 15*a^4*b^15*d^2 + 20*a^6*b^13*d^2 + 15*a^
8*b^11*d^2 + 6*a^10*b^9*d^2 + a^12*b^7*d^2))^(1/2))/(64*(b^19*d^2 + 6*a^2*b^17*d^2 + 15*a^4*b^15*d^2 + 20*a^6*
b^13*d^2 + 15*a^8*b^11*d^2 + 6*a^10*b^9*d^2 + a^12*b^7*d^2)))*(-64*(225*B^2*a^13 + 3969*B^2*a^5*b^8 + 5796*B^2
*a^7*b^6 + 4006*B^2*a^9*b^4 + 1380*B^2*a^11*b^2)*(b^19*d^2 + 6*a^2*b^17*d^2 + 15*a^4*b^15*d^2 + 20*a^6*b^13*d^
2 + 15*a^8*b^11*d^2 + 6*a^10*b^9*d^2 + a^12*b^7*d^2))^(1/2))/(64*(b^19*d^2 + 6*a^2*b^17*d^2 + 15*a^4*b^15*d^2
+ 20*a^6*b^13*d^2 + 15*a^8*b^11*d^2 + 6*a^10*b^9*d^2 + a^12*b^7*d^2)) + (tan(c + d*x)^(1/2)*(32*B^4*b^18 - 225
*B^4*a^18 + 128*B^4*a^2*b^16 + 192*B^4*a^4*b^14 - 3841*B^4*a^6*b^12 + 18050*B^4*a^8*b^10 + 26801*B^4*a^10*b^8
+ 16860*B^4*a^12*b^6 + 4049*B^4*a^14*b^4 - 30*B^4*a^16*b^2))/(64*(b^21*d^4 + 8*a^2*b^19*d^4 + 28*a^4*b^17*d^4
+ 56*a^6*b^15*d^4 + 70*a^8*b^13*d^4 + 56*a^10*b^11*d^4 + 28*a^12*b^9*d^4 + 8*a^14*b^7*d^4 + a^16*b^5*d^4)))*(-
64*(225*B^2*a^13 + 3969*B^2*a^5*b^8 + 5796*B^2*a^7*b^6 + 4006*B^2*a^9*b^4 + 1380*B^2*a^11*b^2)*(b^19*d^2 + 6*a
^2*b^17*d^2 + 15*a^4*b^15*d^2 + 20*a^6*b^13*d^2 + 15*a^8*b^11*d^2 + 6*a^10*b^9*d^2 + a^12*b^7*d^2))^(1/2))/(b^
19*d^2 + 6*a^2*b^17*d^2 + 15*a^4*b^15*d^2 + 20*a^6*b^13*d^2 + 15*a^8*b^11*d^2 + 6*a^10*b^9*d^2 + a^12*b^7*d^2)
))*(-64*(225*B^2*a^13 + 3969*B^2*a^5*b^8 + 5796*B^2*a^7*b^6 + 4006*B^2*a^9*b^4 + 1380*B^2*a^11*b^2)*(b^19*d^2
+ 6*a^2*b^17*d^2 + 15*a^4*b^15*d^2 + 20*a^6*b^13*d^2 + 15*a^8*b^11*d^2 + 6*a^10*b^9*d^2 + a^12*b^7*d^2))^(1/2)
*1i)/(32*(b^19*d^2 + 6*a^2*b^17*d^2 + 15*a^4*b^15*d^2 + 20*a^6*b^13*d^2 + 15*a^8*b^11*d^2 + 6*a^10*b^9*d^2 + a
^12*b^7*d^2)) + (atan(((((tan(c + d*x)^(1/2)*(9*A^4*a^16 + 32*A^4*b^16 + 128*A^4*a^2*b^14 + 1417*A^4*a^4*b^12
- 6802*A^4*a^6*b^10 - 1017*A^4*a^8*b^8 - 1020*A^4*a^10*b^6 + 39*A^4*a^12*b^4 - 18*A^4*a^14*b^2))/(64*(b^19*d^4
+ 8*a^2*b^17*d^4 + 28*a^4*b^15*d^4 + 56*a^6*b^13*d^4 + 70*a^8*b^11*d^4 + 56*a^10*b^9*d^4 + 28*a^12*b^7*d^4 +
8*a^14*b^5*d^4 + a^16*b^3*d^4)) - (((2758*A^3*a^5*b^14*d^2 - 6528*A^3*a^3*b^16*d^2 - 18*A^3*a^19*d^2 + 26482*A
^3*a^7*b^12*d^2 + 21582*A^3*a^9*b^10*d^2 + 7594*A^3*a^11*b^8*d^2 + 3314*A^3*a^13*b^6*d^2 + 246*A^3*a^15*b^4*d^
2 + 90*A^3*a^17*b^2*d^2 + 32*A^3*a*b^18*d^2)/(64*(b^19*d^5 + 8*a^2*b^17*d^5 + 28*a^4*b^15*d^5 + 56*a^6*b^13*d^
5 + 70*a^8*b^11*d^5 + 56*a^10*b^9*d^5 + 28*a^12*b^7*d^5 + 8*a^14*b^5*d^5 + a^16*b^3*d^5)) + (((tan(c + d*x)^(1
/2)*(1024*A^2*a^3*b^19*d^2 + 1352*A^2*a^5*b^17*d^2 + 28224*A^2*a^7*b^15*d^2 + 70240*A^2*a^9*b^13*d^2 + 72640*A
^2*a^11*b^11*d^2 + 39088*A^2*a^13*b^9*d^2 + 13248*A^2*a^15*b^7*d^2 + 3488*A^2*a^17*b^5*d^2 + 576*A^2*a^19*b^3*
d^2 + 1472*A^2*a*b^21*d^2 + 72*A^2*a^21*b*d^2))/(64*(b^19*d^4 + 8*a^2*b^17*d^4 + 28*a^4*b^15*d^4 + 56*a^6*b^13
*d^4 + 70*a^8*b^11*d^4 + 56*a^10*b^9*d^4 + 28*a^12*b^7*d^4 + 8*a^14*b^5*d^4 + a^16*b^3*d^4)) + (((1600*A*a^2*b
^23*d^4 + 12864*A*a^4*b^21*d^4 + 45312*A*a^6*b^19*d^4 + 91392*A*a^8*b^17*d^4 + 115584*A*a^10*b^15*d^4 + 94080*
A*a^12*b^13*d^4 + 48384*A*a^14*b^11*d^4 + 14592*A*a^16*b^9*d^4 + 2112*A*a^18*b^7*d^4 + 64*A*a^20*b^5*d^4)/(64*
(b^19*d^5 + 8*a^2*b^17*d^5 + 28*a^4*b^15*d^5 + 56*a^6*b^13*d^5 + 70*a^8*b^11*d^5 + 56*a^10*b^9*d^5 + 28*a^12*b
^7*d^5 + 8*a^14*b^5*d^5 + a^16*b^3*d^5)) - (tan(c + d*x)^(1/2)*(-64*(9*A^2*a^11 + 1225*A^2*a^3*b^8 + 420*A^2*a
^5*b^6 + 246*A^2*a^7*b^4 + 36*A^2*a^9*b^2)*(b^17*d^2 + 6*a^2*b^15*d^2 + 15*a^4*b^13*d^2 + 20*a^6*b^11*d^2 + 15
*a^8*b^9*d^2 + 6*a^10*b^7*d^2 + a^12*b^5*d^2))^(1/2)*(512*b^28*d^4 + 4608*a^2*b^26*d^4 + 17920*a^4*b^24*d^4 +
38400*a^6*b^22*d^4 + 46080*a^8*b^20*d^4 + 21504*a^10*b^18*d^4 - 21504*a^12*b^16*d^4 - 46080*a^14*b^14*d^4 - 38
400*a^16*b^12*d^4 - 17920*a^18*b^10*d^4 - 4608*a^20*b^8*d^4 - 512*a^22*b^6*d^4))/(4096*(b^17*d^2 + 6*a^2*b^15*
d^2 + 15*a^4*b^13*d^2 + 20*a^6*b^11*d^2 + 15*a^8*b^9*d^2 + 6*a^10*b^7*d^2 + a^12*b^5*d^2)*(b^19*d^4 + 8*a^2*b^
17*d^4 + 28*a^4*b^15*d^4 + 56*a^6*b^13*d^4 + 70*a^8*b^11*d^4 + 56*a^10*b^9*d^4 + 28*a^12*b^7*d^4 + 8*a^14*b^5*
d^4 + a^16*b^3*d^4)))*(-64*(9*A^2*a^11 + 1225*A^2*a^3*b^8 + 420*A^2*a^5*b^6 + 246*A^2*a^7*b^4 + 36*A^2*a^9*b^2
)*(b^17*d^2 + 6*a^2*b^15*d^2 + 15*a^4*b^13*d^2 + 20*a^6*b^11*d^2 + 15*a^8*b^9*d^2 + 6*a^10*b^7*d^2 + a^12*b^5*
d^2))^(1/2))/(64*(b^17*d^2 + 6*a^2*b^15*d^2 + 15*a^4*b^13*d^2 + 20*a^6*b^11*d^2 + 15*a^8*b^9*d^2 + 6*a^10*b^7*
d^2 + a^12*b^5*d^2)))*(-64*(9*A^2*a^11 + 1225*A^2*a^3*b^8 + 420*A^2*a^5*b^6 + 246*A^2*a^7*b^4 + 36*A^2*a^9*b^2
)*(b^17*d^2 + 6*a^2*b^15*d^2 + 15*a^4*b^13*d^2 + 20*a^6*b^11*d^2 + 15*a^8*b^9*d^2 + 6*a^10*b^7*d^2 + a^12*b^5*
d^2))^(1/2))/(64*(b^17*d^2 + 6*a^2*b^15*d^2 + 15*a^4*b^13*d^2 + 20*a^6*b^11*d^2 + 15*a^8*b^9*d^2 + 6*a^10*b^7*
d^2 + a^12*b^5*d^2)))*(-64*(9*A^2*a^11 + 1225*A^2*a^3*b^8 + 420*A^2*a^5*b^6 + 246*A^2*a^7*b^4 + 36*A^2*a^9*b^2
)*(b^17*d^2 + 6*a^2*b^15*d^2 + 15*a^4*b^13*d^2 + 20*a^6*b^11*d^2 + 15*a^8*b^9*d^2 + 6*a^10*b^7*d^2 + a^12*b^5*
d^2))^(1/2))/(64*(b^17*d^2 + 6*a^2*b^15*d^2 + 15*a^4*b^13*d^2 + 20*a^6*b^11*d^2 + 15*a^8*b^9*d^2 + 6*a^10*b^7*
d^2 + a^12*b^5*d^2)))*(-64*(9*A^2*a^11 + 1225*A^2*a^3*b^8 + 420*A^2*a^5*b^6 + 246*A^2*a^7*b^4 + 36*A^2*a^9*b^2
)*(b^17*d^2 + 6*a^2*b^15*d^2 + 15*a^4*b^13*d^2 + 20*a^6*b^11*d^2 + 15*a^8*b^9*d^2 + 6*a^10*b^7*d^2 + a^12*b^5*
d^2))^(1/2)*1i)/(b^17*d^2 + 6*a^2*b^15*d^2 + 15*a^4*b^13*d^2 + 20*a^6*b^11*d^2 + 15*a^8*b^9*d^2 + 6*a^10*b^7*d
^2 + a^12*b^5*d^2) + (((tan(c + d*x)^(1/2)*(9*A^4*a^16 + 32*A^4*b^16 + 128*A^4*a^2*b^14 + 1417*A^4*a^4*b^12 -
6802*A^4*a^6*b^10 - 1017*A^4*a^8*b^8 - 1020*A^4*a^10*b^6 + 39*A^4*a^12*b^4 - 18*A^4*a^14*b^2))/(64*(b^19*d^4 +
8*a^2*b^17*d^4 + 28*a^4*b^15*d^4 + 56*a^6*b^13*d^4 + 70*a^8*b^11*d^4 + 56*a^10*b^9*d^4 + 28*a^12*b^7*d^4 + 8*
a^14*b^5*d^4 + a^16*b^3*d^4)) + (((2758*A^3*a^5*b^14*d^2 - 6528*A^3*a^3*b^16*d^2 - 18*A^3*a^19*d^2 + 26482*A^3
*a^7*b^12*d^2 + 21582*A^3*a^9*b^10*d^2 + 7594*A^3*a^11*b^8*d^2 + 3314*A^3*a^13*b^6*d^2 + 246*A^3*a^15*b^4*d^2
+ 90*A^3*a^17*b^2*d^2 + 32*A^3*a*b^18*d^2)/(64*(b^19*d^5 + 8*a^2*b^17*d^5 + 28*a^4*b^15*d^5 + 56*a^6*b^13*d^5
+ 70*a^8*b^11*d^5 + 56*a^10*b^9*d^5 + 28*a^12*b^7*d^5 + 8*a^14*b^5*d^5 + a^16*b^3*d^5)) - (((tan(c + d*x)^(1/2
)*(1024*A^2*a^3*b^19*d^2 + 1352*A^2*a^5*b^17*d^2 + 28224*A^2*a^7*b^15*d^2 + 70240*A^2*a^9*b^13*d^2 + 72640*A^2
*a^11*b^11*d^2 + 39088*A^2*a^13*b^9*d^2 + 13248*A^2*a^15*b^7*d^2 + 3488*A^2*a^17*b^5*d^2 + 576*A^2*a^19*b^3*d^
2 + 1472*A^2*a*b^21*d^2 + 72*A^2*a^21*b*d^2))/(64*(b^19*d^4 + 8*a^2*b^17*d^4 + 28*a^4*b^15*d^4 + 56*a^6*b^13*d
^4 + 70*a^8*b^11*d^4 + 56*a^10*b^9*d^4 + 28*a^12*b^7*d^4 + 8*a^14*b^5*d^4 + a^16*b^3*d^4)) - (((1600*A*a^2*b^2
3*d^4 + 12864*A*a^4*b^21*d^4 + 45312*A*a^6*b^19*d^4 + 91392*A*a^8*b^17*d^4 + 115584*A*a^10*b^15*d^4 + 94080*A*
a^12*b^13*d^4 + 48384*A*a^14*b^11*d^4 + 14592*A*a^16*b^9*d^4 + 2112*A*a^18*b^7*d^4 + 64*A*a^20*b^5*d^4)/(64*(b
^19*d^5 + 8*a^2*b^17*d^5 + 28*a^4*b^15*d^5 + 56*a^6*b^13*d^5 + 70*a^8*b^11*d^5 + 56*a^10*b^9*d^5 + 28*a^12*b^7
*d^5 + 8*a^14*b^5*d^5 + a^16*b^3*d^5)) + (tan(c + d*x)^(1/2)*(-64*(9*A^2*a^11 + 1225*A^2*a^3*b^8 + 420*A^2*a^5
*b^6 + 246*A^2*a^7*b^4 + 36*A^2*a^9*b^2)*(b^17*d^2 + 6*a^2*b^15*d^2 + 15*a^4*b^13*d^2 + 20*a^6*b^11*d^2 + 15*a
^8*b^9*d^2 + 6*a^10*b^7*d^2 + a^12*b^5*d^2))^(1/2)*(512*b^28*d^4 + 4608*a^2*b^26*d^4 + 17920*a^4*b^24*d^4 + 38
400*a^6*b^22*d^4 + 46080*a^8*b^20*d^4 + 21504*a^10*b^18*d^4 - 21504*a^12*b^16*d^4 - 46080*a^14*b^14*d^4 - 3840
0*a^16*b^12*d^4 - 17920*a^18*b^10*d^4 - 4608*a^20*b^8*d^4 - 512*a^22*b^6*d^4))/(4096*(b^17*d^2 + 6*a^2*b^15*d^
2 + 15*a^4*b^13*d^2 + 20*a^6*b^11*d^2 + 15*a^8*b^9*d^2 + 6*a^10*b^7*d^2 + a^12*b^5*d^2)*(b^19*d^4 + 8*a^2*b^17
*d^4 + 28*a^4*b^15*d^4 + 56*a^6*b^13*d^4 + 70*a^8*b^11*d^4 + 56*a^10*b^9*d^4 + 28*a^12*b^7*d^4 + 8*a^14*b^5*d^
4 + a^16*b^3*d^4)))*(-64*(9*A^2*a^11 + 1225*A^2*a^3*b^8 + 420*A^2*a^5*b^6 + 246*A^2*a^7*b^4 + 36*A^2*a^9*b^2)*
(b^17*d^2 + 6*a^2*b^15*d^2 + 15*a^4*b^13*d^2 + 20*a^6*b^11*d^2 + 15*a^8*b^9*d^2 + 6*a^10*b^7*d^2 + a^12*b^5*d^
2))^(1/2))/(64*(b^17*d^2 + 6*a^2*b^15*d^2 + 15*a^4*b^13*d^2 + 20*a^6*b^11*d^2 + 15*a^8*b^9*d^2 + 6*a^10*b^7*d^
2 + a^12*b^5*d^2)))*(-64*(9*A^2*a^11 + 1225*A^2*a^3*b^8 + 420*A^2*a^5*b^6 + 246*A^2*a^7*b^4 + 36*A^2*a^9*b^2)*
(b^17*d^2 + 6*a^2*b^15*d^2 + 15*a^4*b^13*d^2 + 20*a^6*b^11*d^2 + 15*a^8*b^9*d^2 + 6*a^10*b^7*d^2 + a^12*b^5*d^
2))^(1/2))/(64*(b^17*d^2 + 6*a^2*b^15*d^2 + 15*a^4*b^13*d^2 + 20*a^6*b^11*d^2 + 15*a^8*b^9*d^2 + 6*a^10*b^7*d^
2 + a^12*b^5*d^2)))*(-64*(9*A^2*a^11 + 1225*A^2*a^3*b^8 + 420*A^2*a^5*b^6 + 246*A^2*a^7*b^4 + 36*A^2*a^9*b^2)*
(b^17*d^2 + 6*a^2*b^15*d^2 + 15*a^4*b^13*d^2 + 20*a^6*b^11*d^2 + 15*a^8*b^9*d^2 + 6*a^10*b^7*d^2 + a^12*b^5*d^
2))^(1/2))/(64*(b^17*d^2 + 6*a^2*b^15*d^2 + 15*a^4*b^13*d^2 + 20*a^6*b^11*d^2 + 15*a^8*b^9*d^2 + 6*a^10*b^7*d^
2 + a^12*b^5*d^2)))*(-64*(9*A^2*a^11 + 1225*A^2*a^3*b^8 + 420*A^2*a^5*b^6 + 246*A^2*a^7*b^4 + 36*A^2*a^9*b^2)*
(b^17*d^2 + 6*a^2*b^15*d^2 + 15*a^4*b^13*d^2 + 20*a^6*b^11*d^2 + 15*a^8*b^9*d^2 + 6*a^10*b^7*d^2 + a^12*b^5*d^
2))^(1/2)*1i)/(b^17*d^2 + 6*a^2*b^15*d^2 + 15*a^4*b^13*d^2 + 20*a^6*b^11*d^2 + 15*a^8*b^9*d^2 + 6*a^10*b^7*d^2
+ a^12*b^5*d^2))/((9*A^5*a^12*b + 280*A^5*a^2*b^11 + 1553*A^5*a^4*b^9 + 492*A^5*a^6*b^7 + 270*A^5*a^8*b^5 + 3
6*A^5*a^10*b^3)/(b^19*d^5 + 8*a^2*b^17*d^5 + 28*a^4*b^15*d^5 + 56*a^6*b^13*d^5 + 70*a^8*b^11*d^5 + 56*a^10*b^9
*d^5 + 28*a^12*b^7*d^5 + 8*a^14*b^5*d^5 + a^16*b^3*d^5) - (((tan(c + d*x)^(1/2)*(9*A^4*a^16 + 32*A^4*b^16 + 12
8*A^4*a^2*b^14 + 1417*A^4*a^4*b^12 - 6802*A^4*a^6*b^10 - 1017*A^4*a^8*b^8 - 1020*A^4*a^10*b^6 + 39*A^4*a^12*b^
4 - 18*A^4*a^14*b^2))/(64*(b^19*d^4 + 8*a^2*b^17*d^4 + 28*a^4*b^15*d^4 + 56*a^6*b^13*d^4 + 70*a^8*b^11*d^4 + 5
6*a^10*b^9*d^4 + 28*a^12*b^7*d^4 + 8*a^14*b^5*d^4 + a^16*b^3*d^4)) - (((2758*A^3*a^5*b^14*d^2 - 6528*A^3*a^3*b
^16*d^2 - 18*A^3*a^19*d^2 + 26482*A^3*a^7*b^12*d^2 + 21582*A^3*a^9*b^10*d^2 + 7594*A^3*a^11*b^8*d^2 + 3314*A^3
*a^13*b^6*d^2 + 246*A^3*a^15*b^4*d^2 + 90*A^3*a^17*b^2*d^2 + 32*A^3*a*b^18*d^2)/(64*(b^19*d^5 + 8*a^2*b^17*d^5
+ 28*a^4*b^15*d^5 + 56*a^6*b^13*d^5 + 70*a^8*b^11*d^5 + 56*a^10*b^9*d^5 + 28*a^12*b^7*d^5 + 8*a^14*b^5*d^5 +
a^16*b^3*d^5)) + (((tan(c + d*x)^(1/2)*(1024*A^2*a^3*b^19*d^2 + 1352*A^2*a^5*b^17*d^2 + 28224*A^2*a^7*b^15*d^2
+ 70240*A^2*a^9*b^13*d^2 + 72640*A^2*a^11*b^11*d^2 + 39088*A^2*a^13*b^9*d^2 + 13248*A^2*a^15*b^7*d^2 + 3488*A
^2*a^17*b^5*d^2 + 576*A^2*a^19*b^3*d^2 + 1472*A^2*a*b^21*d^2 + 72*A^2*a^21*b*d^2))/(64*(b^19*d^4 + 8*a^2*b^17*
d^4 + 28*a^4*b^15*d^4 + 56*a^6*b^13*d^4 + 70*a^8*b^11*d^4 + 56*a^10*b^9*d^4 + 28*a^12*b^7*d^4 + 8*a^14*b^5*d^4
+ a^16*b^3*d^4)) + (((1600*A*a^2*b^23*d^4 + 12864*A*a^4*b^21*d^4 + 45312*A*a^6*b^19*d^4 + 91392*A*a^8*b^17*d^
4 + 115584*A*a^10*b^15*d^4 + 94080*A*a^12*b^13*d^4 + 48384*A*a^14*b^11*d^4 + 14592*A*a^16*b^9*d^4 + 2112*A*a^1
8*b^7*d^4 + 64*A*a^20*b^5*d^4)/(64*(b^19*d^5 + 8*a^2*b^17*d^5 + 28*a^4*b^15*d^5 + 56*a^6*b^13*d^5 + 70*a^8*b^1
1*d^5 + 56*a^10*b^9*d^5 + 28*a^12*b^7*d^5 + 8*a^14*b^5*d^5 + a^16*b^3*d^5)) - (tan(c + d*x)^(1/2)*(-64*(9*A^2*
a^11 + 1225*A^2*a^3*b^8 + 420*A^2*a^5*b^6 + 246*A^2*a^7*b^4 + 36*A^2*a^9*b^2)*(b^17*d^2 + 6*a^2*b^15*d^2 + 15*
a^4*b^13*d^2 + 20*a^6*b^11*d^2 + 15*a^8*b^9*d^2 + 6*a^10*b^7*d^2 + a^12*b^5*d^2))^(1/2)*(512*b^28*d^4 + 4608*a
^2*b^26*d^4 + 17920*a^4*b^24*d^4 + 38400*a^6*b^22*d^4 + 46080*a^8*b^20*d^4 + 21504*a^10*b^18*d^4 - 21504*a^12*
b^16*d^4 - 46080*a^14*b^14*d^4 - 38400*a^16*b^12*d^4 - 17920*a^18*b^10*d^4 - 4608*a^20*b^8*d^4 - 512*a^22*b^6*
d^4))/(4096*(b^17*d^2 + 6*a^2*b^15*d^2 + 15*a^4*b^13*d^2 + 20*a^6*b^11*d^2 + 15*a^8*b^9*d^2 + 6*a^10*b^7*d^2 +
a^12*b^5*d^2)*(b^19*d^4 + 8*a^2*b^17*d^4 + 28*a^4*b^15*d^4 + 56*a^6*b^13*d^4 + 70*a^8*b^11*d^4 + 56*a^10*b^9*
d^4 + 28*a^12*b^7*d^4 + 8*a^14*b^5*d^4 + a^16*b^3*d^4)))*(-64*(9*A^2*a^11 + 1225*A^2*a^3*b^8 + 420*A^2*a^5*b^6
+ 246*A^2*a^7*b^4 + 36*A^2*a^9*b^2)*(b^17*d^2 + 6*a^2*b^15*d^2 + 15*a^4*b^13*d^2 + 20*a^6*b^11*d^2 + 15*a^8*b
^9*d^2 + 6*a^10*b^7*d^2 + a^12*b^5*d^2))^(1/2))/(64*(b^17*d^2 + 6*a^2*b^15*d^2 + 15*a^4*b^13*d^2 + 20*a^6*b^11
*d^2 + 15*a^8*b^9*d^2 + 6*a^10*b^7*d^2 + a^12*b^5*d^2)))*(-64*(9*A^2*a^11 + 1225*A^2*a^3*b^8 + 420*A^2*a^5*b^6
+ 246*A^2*a^7*b^4 + 36*A^2*a^9*b^2)*(b^17*d^2 + 6*a^2*b^15*d^2 + 15*a^4*b^13*d^2 + 20*a^6*b^11*d^2 + 15*a^8*b
^9*d^2 + 6*a^10*b^7*d^2 + a^12*b^5*d^2))^(1/2))/(64*(b^17*d^2 + 6*a^2*b^15*d^2 + 15*a^4*b^13*d^2 + 20*a^6*b^11
*d^2 + 15*a^8*b^9*d^2 + 6*a^10*b^7*d^2 + a^12*b^5*d^2)))*(-64*(9*A^2*a^11 + 1225*A^2*a^3*b^8 + 420*A^2*a^5*b^6
+ 246*A^2*a^7*b^4 + 36*A^2*a^9*b^2)*(b^17*d^2 + 6*a^2*b^15*d^2 + 15*a^4*b^13*d^2 + 20*a^6*b^11*d^2 + 15*a^8*b
^9*d^2 + 6*a^10*b^7*d^2 + a^12*b^5*d^2))^(1/2))/(64*(b^17*d^2 + 6*a^2*b^15*d^2 + 15*a^4*b^13*d^2 + 20*a^6*b^11
*d^2 + 15*a^8*b^9*d^2 + 6*a^10*b^7*d^2 + a^12*b^5*d^2)))*(-64*(9*A^2*a^11 + 1225*A^2*a^3*b^8 + 420*A^2*a^5*b^6
+ 246*A^2*a^7*b^4 + 36*A^2*a^9*b^2)*(b^17*d^2 + 6*a^2*b^15*d^2 + 15*a^4*b^13*d^2 + 20*a^6*b^11*d^2 + 15*a^8*b
^9*d^2 + 6*a^10*b^7*d^2 + a^12*b^5*d^2))^(1/2))/(b^17*d^2 + 6*a^2*b^15*d^2 + 15*a^4*b^13*d^2 + 20*a^6*b^11*d^2
+ 15*a^8*b^9*d^2 + 6*a^10*b^7*d^2 + a^12*b^5*d^2) + (((tan(c + d*x)^(1/2)*(9*A^4*a^16 + 32*A^4*b^16 + 128*A^4
*a^2*b^14 + 1417*A^4*a^4*b^12 - 6802*A^4*a^6*b^10 - 1017*A^4*a^8*b^8 - 1020*A^4*a^10*b^6 + 39*A^4*a^12*b^4 - 1
8*A^4*a^14*b^2))/(64*(b^19*d^4 + 8*a^2*b^17*d^4 + 28*a^4*b^15*d^4 + 56*a^6*b^13*d^4 + 70*a^8*b^11*d^4 + 56*a^1
0*b^9*d^4 + 28*a^12*b^7*d^4 + 8*a^14*b^5*d^4 + a^16*b^3*d^4)) + (((2758*A^3*a^5*b^14*d^2 - 6528*A^3*a^3*b^16*d
^2 - 18*A^3*a^19*d^2 + 26482*A^3*a^7*b^12*d^2 + 21582*A^3*a^9*b^10*d^2 + 7594*A^3*a^11*b^8*d^2 + 3314*A^3*a^13
*b^6*d^2 + 246*A^3*a^15*b^4*d^2 + 90*A^3*a^17*b^2*d^2 + 32*A^3*a*b^18*d^2)/(64*(b^19*d^5 + 8*a^2*b^17*d^5 + 28
*a^4*b^15*d^5 + 56*a^6*b^13*d^5 + 70*a^8*b^11*d^5 + 56*a^10*b^9*d^5 + 28*a^12*b^7*d^5 + 8*a^14*b^5*d^5 + a^16*
b^3*d^5)) - (((tan(c + d*x)^(1/2)*(1024*A^2*a^3*b^19*d^2 + 1352*A^2*a^5*b^17*d^2 + 28224*A^2*a^7*b^15*d^2 + 70
240*A^2*a^9*b^13*d^2 + 72640*A^2*a^11*b^11*d^2 + 39088*A^2*a^13*b^9*d^2 + 13248*A^2*a^15*b^7*d^2 + 3488*A^2*a^
17*b^5*d^2 + 576*A^2*a^19*b^3*d^2 + 1472*A^2*a*b^21*d^2 + 72*A^2*a^21*b*d^2))/(64*(b^19*d^4 + 8*a^2*b^17*d^4 +
28*a^4*b^15*d^4 + 56*a^6*b^13*d^4 + 70*a^8*b^11*d^4 + 56*a^10*b^9*d^4 + 28*a^12*b^7*d^4 + 8*a^14*b^5*d^4 + a^
16*b^3*d^4)) - (((1600*A*a^2*b^23*d^4 + 12864*A*a^4*b^21*d^4 + 45312*A*a^6*b^19*d^4 + 91392*A*a^8*b^17*d^4 + 1
15584*A*a^10*b^15*d^4 + 94080*A*a^12*b^13*d^4 + 48384*A*a^14*b^11*d^4 + 14592*A*a^16*b^9*d^4 + 2112*A*a^18*b^7
*d^4 + 64*A*a^20*b^5*d^4)/(64*(b^19*d^5 + 8*a^2*b^17*d^5 + 28*a^4*b^15*d^5 + 56*a^6*b^13*d^5 + 70*a^8*b^11*d^5
+ 56*a^10*b^9*d^5 + 28*a^12*b^7*d^5 + 8*a^14*b^5*d^5 + a^16*b^3*d^5)) + (tan(c + d*x)^(1/2)*(-64*(9*A^2*a^11
+ 1225*A^2*a^3*b^8 + 420*A^2*a^5*b^6 + 246*A^2*a^7*b^4 + 36*A^2*a^9*b^2)*(b^17*d^2 + 6*a^2*b^15*d^2 + 15*a^4*b
^13*d^2 + 20*a^6*b^11*d^2 + 15*a^8*b^9*d^2 + 6*a^10*b^7*d^2 + a^12*b^5*d^2))^(1/2)*(512*b^28*d^4 + 4608*a^2*b^
26*d^4 + 17920*a^4*b^24*d^4 + 38400*a^6*b^22*d^4 + 46080*a^8*b^20*d^4 + 21504*a^10*b^18*d^4 - 21504*a^12*b^16*
d^4 - 46080*a^14*b^14*d^4 - 38400*a^16*b^12*d^4 - 17920*a^18*b^10*d^4 - 4608*a^20*b^8*d^4 - 512*a^22*b^6*d^4))
/(4096*(b^17*d^2 + 6*a^2*b^15*d^2 + 15*a^4*b^13*d^2 + 20*a^6*b^11*d^2 + 15*a^8*b^9*d^2 + 6*a^10*b^7*d^2 + a^12
*b^5*d^2)*(b^19*d^4 + 8*a^2*b^17*d^4 + 28*a^4*b^15*d^4 + 56*a^6*b^13*d^4 + 70*a^8*b^11*d^4 + 56*a^10*b^9*d^4 +
28*a^12*b^7*d^4 + 8*a^14*b^5*d^4 + a^16*b^3*d^4)))*(-64*(9*A^2*a^11 + 1225*A^2*a^3*b^8 + 420*A^2*a^5*b^6 + 24
6*A^2*a^7*b^4 + 36*A^2*a^9*b^2)*(b^17*d^2 + 6*a^2*b^15*d^2 + 15*a^4*b^13*d^2 + 20*a^6*b^11*d^2 + 15*a^8*b^9*d^
2 + 6*a^10*b^7*d^2 + a^12*b^5*d^2))^(1/2))/(64*(b^17*d^2 + 6*a^2*b^15*d^2 + 15*a^4*b^13*d^2 + 20*a^6*b^11*d^2
+ 15*a^8*b^9*d^2 + 6*a^10*b^7*d^2 + a^12*b^5*d^2)))*(-64*(9*A^2*a^11 + 1225*A^2*a^3*b^8 + 420*A^2*a^5*b^6 + 24
6*A^2*a^7*b^4 + 36*A^2*a^9*b^2)*(b^17*d^2 + 6*a^2*b^15*d^2 + 15*a^4*b^13*d^2 + 20*a^6*b^11*d^2 + 15*a^8*b^9*d^
2 + 6*a^10*b^7*d^2 + a^12*b^5*d^2))^(1/2))/(64*(b^17*d^2 + 6*a^2*b^15*d^2 + 15*a^4*b^13*d^2 + 20*a^6*b^11*d^2
+ 15*a^8*b^9*d^2 + 6*a^10*b^7*d^2 + a^12*b^5*d^2)))*(-64*(9*A^2*a^11 + 1225*A^2*a^3*b^8 + 420*A^2*a^5*b^6 + 24
6*A^2*a^7*b^4 + 36*A^2*a^9*b^2)*(b^17*d^2 + 6*a^2*b^15*d^2 + 15*a^4*b^13*d^2 + 20*a^6*b^11*d^2 + 15*a^8*b^9*d^
2 + 6*a^10*b^7*d^2 + a^12*b^5*d^2))^(1/2))/(64*(b^17*d^2 + 6*a^2*b^15*d^2 + 15*a^4*b^13*d^2 + 20*a^6*b^11*d^2
+ 15*a^8*b^9*d^2 + 6*a^10*b^7*d^2 + a^12*b^5*d^2)))*(-64*(9*A^2*a^11 + 1225*A^2*a^3*b^8 + 420*A^2*a^5*b^6 + 24
6*A^2*a^7*b^4 + 36*A^2*a^9*b^2)*(b^17*d^2 + 6*a^2*b^15*d^2 + 15*a^4*b^13*d^2 + 20*a^6*b^11*d^2 + 15*a^8*b^9*d^
2 + 6*a^10*b^7*d^2 + a^12*b^5*d^2))^(1/2))/(b^17*d^2 + 6*a^2*b^15*d^2 + 15*a^4*b^13*d^2 + 20*a^6*b^11*d^2 + 15
*a^8*b^9*d^2 + 6*a^10*b^7*d^2 + a^12*b^5*d^2)))*(-64*(9*A^2*a^11 + 1225*A^2*a^3*b^8 + 420*A^2*a^5*b^6 + 246*A^
2*a^7*b^4 + 36*A^2*a^9*b^2)*(b^17*d^2 + 6*a^2*b^15*d^2 + 15*a^4*b^13*d^2 + 20*a^6*b^11*d^2 + 15*a^8*b^9*d^2 +
6*a^10*b^7*d^2 + a^12*b^5*d^2))^(1/2)*1i)/(32*(b^17*d^2 + 6*a^2*b^15*d^2 + 15*a^4*b^13*d^2 + 20*a^6*b^11*d^2 +
15*a^8*b^9*d^2 + 6*a^10*b^7*d^2 + a^12*b^5*d^2)) + (2*B*tan(c + d*x)^(1/2))/(b^3*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)**(7/2)*(A+B*tan(d*x+c))/(a+b*tan(d*x+c))**3,x)

[Out]

Timed out

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