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

Optimal. Leaf size=436 \[ \frac {\left (a^2 (A-B)+2 a b (A+B)-b^2 (A-B)\right ) \tan ^{-1}\left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} d \left (a^2+b^2\right )^2}-\frac {\left (a^2 (A-B)+2 a b (A+B)-b^2 (A-B)\right ) \tan ^{-1}\left (\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{\sqrt {2} d \left (a^2+b^2\right )^2}+\frac {a (A b-a B) \tan ^{\frac {3}{2}}(c+d x)}{b d \left (a^2+b^2\right ) (a+b \tan (c+d x))}-\frac {\left (-3 a^2 B+a A b-2 b^2 B\right ) \sqrt {\tan (c+d x)}}{b^2 d \left (a^2+b^2\right )}+\frac {\left (-\left (a^2 (A+B)\right )+2 a b (A-B)+b^2 (A+B)\right ) \log \left (\tan (c+d x)-\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{2 \sqrt {2} d \left (a^2+b^2\right )^2}-\frac {\left (-\left (a^2 (A+B)\right )+2 a b (A-B)+b^2 (A+B)\right ) \log \left (\tan (c+d x)+\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{2 \sqrt {2} d \left (a^2+b^2\right )^2}+\frac {a^{3/2} \left (-3 a^3 B+a^2 A b-7 a b^2 B+5 A b^3\right ) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {\tan (c+d x)}}{\sqrt {a}}\right )}{b^{5/2} d \left (a^2+b^2\right )^2} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 1.16, antiderivative size = 436, normalized size of antiderivative = 1.00, number of steps used = 16, number of rules used = 13, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.394, Rules used = {3605, 3647, 3653, 3534, 1168, 1162, 617, 204, 1165, 628, 3634, 63, 205} \[ \frac {a^{3/2} \left (a^2 A b-3 a^3 B-7 a b^2 B+5 A b^3\right ) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {\tan (c+d x)}}{\sqrt {a}}\right )}{b^{5/2} d \left (a^2+b^2\right )^2}+\frac {a (A b-a B) \tan ^{\frac {3}{2}}(c+d x)}{b d \left (a^2+b^2\right ) (a+b \tan (c+d x))}+\frac {\left (a^2 (A-B)+2 a b (A+B)-b^2 (A-B)\right ) \tan ^{-1}\left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} d \left (a^2+b^2\right )^2}-\frac {\left (a^2 (A-B)+2 a b (A+B)-b^2 (A-B)\right ) \tan ^{-1}\left (\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{\sqrt {2} d \left (a^2+b^2\right )^2}-\frac {\left (-3 a^2 B+a A b-2 b^2 B\right ) \sqrt {\tan (c+d x)}}{b^2 d \left (a^2+b^2\right )}+\frac {\left (a^2 (-(A+B))+2 a b (A-B)+b^2 (A+B)\right ) \log \left (\tan (c+d x)-\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{2 \sqrt {2} d \left (a^2+b^2\right )^2}-\frac {\left (a^2 (-(A+B))+2 a b (A-B)+b^2 (A+B)\right ) \log \left (\tan (c+d x)+\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{2 \sqrt {2} d \left (a^2+b^2\right )^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Mathematica [C]  time = 2.40, size = 275, normalized size = 0.63 \[ \frac {2 \left (\frac {\left (-3 a^3 B+a^2 A b-4 a b^2 B+2 A b^3\right ) \sqrt {\tan (c+d x)}}{2 \left (a^2+b^2\right )}-\frac {(a+b \tan (c+d x)) \left (a^{3/2} \left (3 a^3 B-a^2 A b+7 a b^2 B-5 A b^3\right ) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {\tan (c+d x)}}{\sqrt {a}}\right )+\sqrt [4]{-1} b^{5/2} (a+i b)^2 (B+i A) \tan ^{-1}\left ((-1)^{3/4} \sqrt {\tan (c+d x)}\right )+(-1)^{3/4} b^{5/2} (b+i a)^2 (A+i B) \tanh ^{-1}\left ((-1)^{3/4} \sqrt {\tan (c+d x)}\right )\right )}{2 \sqrt {b} \left (a^2+b^2\right )^2}+(3 a B-A b) \sqrt {\tan (c+d x)}+b B \tan ^{\frac {3}{2}}(c+d x)\right )}{b^2 d (a+b \tan (c+d x))} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

[Out]

Timed out

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maple [B]  time = 0.44, size = 1160, normalized size = 2.66 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.95, size = 377, normalized size = 0.86 \[ -\frac {\frac {4 \, {\left (3 \, B a^{5} - A a^{4} b + 7 \, B a^{3} b^{2} - 5 \, A a^{2} b^{3}\right )} \arctan \left (\frac {b \sqrt {\tan \left (d x + c\right )}}{\sqrt {a b}}\right )}{{\left (a^{4} b^{2} + 2 \, a^{2} b^{4} + b^{6}\right )} \sqrt {a b}} - \frac {4 \, {\left (B a^{3} - A a^{2} b\right )} \sqrt {\tan \left (d x + c\right )}}{a^{3} b^{2} + a b^{4} + {\left (a^{2} b^{3} + b^{5}\right )} \tan \left (d x + c\right )} + \frac {2 \, \sqrt {2} {\left ({\left (A - B\right )} a^{2} + 2 \, {\left (A + B\right )} a b - {\left (A - B\right )} b^{2}\right )} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + 2 \, \sqrt {\tan \left (d x + c\right )}\right )}\right ) + 2 \, \sqrt {2} {\left ({\left (A - B\right )} a^{2} + 2 \, {\left (A + B\right )} a b - {\left (A - B\right )} b^{2}\right )} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - 2 \, \sqrt {\tan \left (d x + c\right )}\right )}\right ) - \sqrt {2} {\left ({\left (A + B\right )} a^{2} - 2 \, {\left (A - B\right )} a b - {\left (A + B\right )} b^{2}\right )} \log \left (\sqrt {2} \sqrt {\tan \left (d x + c\right )} + \tan \left (d x + c\right ) + 1\right ) + \sqrt {2} {\left ({\left (A + B\right )} a^{2} - 2 \, {\left (A - B\right )} a b - {\left (A + B\right )} b^{2}\right )} \log \left (-\sqrt {2} \sqrt {\tan \left (d x + c\right )} + \tan \left (d x + c\right ) + 1\right )}{a^{4} + 2 \, a^{2} b^{2} + b^{4}} - \frac {8 \, B \sqrt {\tan \left (d x + c\right )}}{b^{2}}}{4 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*(4*(3*B*a^5 - A*a^4*b + 7*B*a^3*b^2 - 5*A*a^2*b^3)*arctan(b*sqrt(tan(d*x + c))/sqrt(a*b))/((a^4*b^2 + 2*a
^2*b^4 + b^6)*sqrt(a*b)) - 4*(B*a^3 - A*a^2*b)*sqrt(tan(d*x + c))/(a^3*b^2 + a*b^4 + (a^2*b^3 + b^5)*tan(d*x +
c)) + (2*sqrt(2)*((A - B)*a^2 + 2*(A + B)*a*b - (A - B)*b^2)*arctan(1/2*sqrt(2)*(sqrt(2) + 2*sqrt(tan(d*x + c
)))) + 2*sqrt(2)*((A - B)*a^2 + 2*(A + B)*a*b - (A - B)*b^2)*arctan(-1/2*sqrt(2)*(sqrt(2) - 2*sqrt(tan(d*x + c
)))) - sqrt(2)*((A + B)*a^2 - 2*(A - B)*a*b - (A + B)*b^2)*log(sqrt(2)*sqrt(tan(d*x + c)) + tan(d*x + c) + 1)
+ sqrt(2)*((A + B)*a^2 - 2*(A - B)*a*b - (A + B)*b^2)*log(-sqrt(2)*sqrt(tan(d*x + c)) + tan(d*x + c) + 1))/(a^
4 + 2*a^2*b^2 + b^4) - 8*B*sqrt(tan(d*x + c))/b^2)/d

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mupad [B]  time = 36.19, size = 18313, normalized size = 42.00 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((tan(c + d*x)^(5/2)*(A + B*tan(c + d*x)))/(a + b*tan(c + d*x))^2,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^4 + b^4 - 6*a^2*b^2)^2)^(1/
2) - 16*B^2*a*b^3*d^2 + 16*B^2*a^3*b*d^2)/(d^4*(a^2 + b^2)^4))^(1/2) - (128*B*a*b^2*(7*a^4 + b^4 + 8*a^2*b^2))
/d)*((4*(-B^4*d^4*(a^4 + b^4 - 6*a^2*b^2)^2)^(1/2) - 16*B^2*a*b^3*d^2 + 16*B^2*a^3*b*d^2)/(d^4*(a^2 + b^2)^4))
^(1/2))/4 + (64*B^2*a*tan(c + d*x)^(1/2)*(18*a^10 - 15*b^10 + 17*a^2*b^8 - a^4*b^6 + 97*a^6*b^4 + 84*a^8*b^2))
/(b^2*d^2*(a^2 + b^2)^2))*((4*(-B^4*d^4*(a^4 + b^4 - 6*a^2*b^2)^2)^(1/2) - 16*B^2*a*b^3*d^2 + 16*B^2*a^3*b*d^2
)/(d^4*(a^2 + b^2)^4))^(1/2))/4 + (32*B^3*a^4*(127*a^2*b^6 - 112*b^8 - 9*a^8 + 173*a^4*b^4 + 21*a^6*b^2))/(b^3
*d^3*(a^2 + b^2)^3))*((4*(-B^4*d^4*(a^4 + b^4 - 6*a^2*b^2)^2)^(1/2) - 16*B^2*a*b^3*d^2 + 16*B^2*a^3*b*d^2)/(d^
4*(a^2 + b^2)^4))^(1/2))/4 - (16*B^4*tan(c + d*x)^(1/2)*(9*a^12 + 2*b^12 + 4*a^2*b^10 + 2*a^4*b^8 - 49*a^6*b^6
+ 7*a^8*b^4 + 33*a^10*b^2))/(b^3*d^4*(a^2 + b^2)^4))*((4*(-B^4*d^4*(a^4 + b^4 - 6*a^2*b^2)^2)^(1/2) - 16*B^2*
a*b^3*d^2 + 16*B^2*a^3*b*d^2)/(d^4*(a^2 + b^2)^4))^(1/2))/4 - (16*B^5*a^3*b^2*(3*a^2 + 7*b^2))/(d^5*(a^2 + b^2
)^4))*(((192*B^4*a^2*b^6*d^4 - 16*B^4*b^8*d^4 - 16*B^4*a^8*d^4 - 608*B^4*a^4*b^4*d^4 + 192*B^4*a^6*b^2*d^4)^(1
/2) - 16*B^2*a*b^3*d^2 + 16*B^2*a^3*b*d^2)/(a^8*d^4 + b^8*d^4 + 4*a^2*b^6*d^4 + 6*a^4*b^4*d^4 + 4*a^6*b^2*d^4)
)^(1/2))/4 + (log(((((((((128*b^3*tan(c + d*x)^(1/2)*(a^2 - b^2)*(a^2 + b^2)^2*(-(4*(-B^4*d^4*(a^4 + b^4 - 6*a
^2*b^2)^2)^(1/2) + 16*B^2*a*b^3*d^2 - 16*B^2*a^3*b*d^2)/(d^4*(a^2 + b^2)^4))^(1/2) - (128*B*a*b^2*(7*a^4 + b^4
+ 8*a^2*b^2))/d)*(-(4*(-B^4*d^4*(a^4 + b^4 - 6*a^2*b^2)^2)^(1/2) + 16*B^2*a*b^3*d^2 - 16*B^2*a^3*b*d^2)/(d^4*
(a^2 + b^2)^4))^(1/2))/4 + (64*B^2*a*tan(c + d*x)^(1/2)*(18*a^10 - 15*b^10 + 17*a^2*b^8 - a^4*b^6 + 97*a^6*b^4
+ 84*a^8*b^2))/(b^2*d^2*(a^2 + b^2)^2))*(-(4*(-B^4*d^4*(a^4 + b^4 - 6*a^2*b^2)^2)^(1/2) + 16*B^2*a*b^3*d^2 -
16*B^2*a^3*b*d^2)/(d^4*(a^2 + b^2)^4))^(1/2))/4 + (32*B^3*a^4*(127*a^2*b^6 - 112*b^8 - 9*a^8 + 173*a^4*b^4 + 2
1*a^6*b^2))/(b^3*d^3*(a^2 + b^2)^3))*(-(4*(-B^4*d^4*(a^4 + b^4 - 6*a^2*b^2)^2)^(1/2) + 16*B^2*a*b^3*d^2 - 16*B
^2*a^3*b*d^2)/(d^4*(a^2 + b^2)^4))^(1/2))/4 - (16*B^4*tan(c + d*x)^(1/2)*(9*a^12 + 2*b^12 + 4*a^2*b^10 + 2*a^4
*b^8 - 49*a^6*b^6 + 7*a^8*b^4 + 33*a^10*b^2))/(b^3*d^4*(a^2 + b^2)^4))*(-(4*(-B^4*d^4*(a^4 + b^4 - 6*a^2*b^2)^
2)^(1/2) + 16*B^2*a*b^3*d^2 - 16*B^2*a^3*b*d^2)/(d^4*(a^2 + b^2)^4))^(1/2))/4 - (16*B^5*a^3*b^2*(3*a^2 + 7*b^2
))/(d^5*(a^2 + b^2)^4))*(-((192*B^4*a^2*b^6*d^4 - 16*B^4*b^8*d^4 - 16*B^4*a^8*d^4 - 608*B^4*a^4*b^4*d^4 + 192*
B^4*a^6*b^2*d^4)^(1/2) + 16*B^2*a*b^3*d^2 - 16*B^2*a^3*b*d^2)/(a^8*d^4 + b^8*d^4 + 4*a^2*b^6*d^4 + 6*a^4*b^4*d
^4 + 4*a^6*b^2*d^4))^(1/2))/4 - log(- ((((((((128*b^3*tan(c + d*x)^(1/2)*(a^2 - b^2)*(a^2 + b^2)^2*((4*(-B^4*d
^4*(a^4 + b^4 - 6*a^2*b^2)^2)^(1/2) - 16*B^2*a*b^3*d^2 + 16*B^2*a^3*b*d^2)/(d^4*(a^2 + b^2)^4))^(1/2) + (128*B
*a*b^2*(7*a^4 + b^4 + 8*a^2*b^2))/d)*((4*(-B^4*d^4*(a^4 + b^4 - 6*a^2*b^2)^2)^(1/2) - 16*B^2*a*b^3*d^2 + 16*B^
2*a^3*b*d^2)/(d^4*(a^2 + b^2)^4))^(1/2))/4 + (64*B^2*a*tan(c + d*x)^(1/2)*(18*a^10 - 15*b^10 + 17*a^2*b^8 - a^
4*b^6 + 97*a^6*b^4 + 84*a^8*b^2))/(b^2*d^2*(a^2 + b^2)^2))*((4*(-B^4*d^4*(a^4 + b^4 - 6*a^2*b^2)^2)^(1/2) - 16
*B^2*a*b^3*d^2 + 16*B^2*a^3*b*d^2)/(d^4*(a^2 + b^2)^4))^(1/2))/4 - (32*B^3*a^4*(127*a^2*b^6 - 112*b^8 - 9*a^8
+ 173*a^4*b^4 + 21*a^6*b^2))/(b^3*d^3*(a^2 + b^2)^3))*((4*(-B^4*d^4*(a^4 + b^4 - 6*a^2*b^2)^2)^(1/2) - 16*B^2*
a*b^3*d^2 + 16*B^2*a^3*b*d^2)/(d^4*(a^2 + b^2)^4))^(1/2))/4 - (16*B^4*tan(c + d*x)^(1/2)*(9*a^12 + 2*b^12 + 4*
a^2*b^10 + 2*a^4*b^8 - 49*a^6*b^6 + 7*a^8*b^4 + 33*a^10*b^2))/(b^3*d^4*(a^2 + b^2)^4))*((4*(-B^4*d^4*(a^4 + b^
4 - 6*a^2*b^2)^2)^(1/2) - 16*B^2*a*b^3*d^2 + 16*B^2*a^3*b*d^2)/(d^4*(a^2 + b^2)^4))^(1/2))/4 - (16*B^5*a^3*b^2
*(3*a^2 + 7*b^2))/(d^5*(a^2 + b^2)^4))*(((192*B^4*a^2*b^6*d^4 - 16*B^4*b^8*d^4 - 16*B^4*a^8*d^4 - 608*B^4*a^4*
b^4*d^4 + 192*B^4*a^6*b^2*d^4)^(1/2) - 16*B^2*a*b^3*d^2 + 16*B^2*a^3*b*d^2)/(16*a^8*d^4 + 16*b^8*d^4 + 64*a^2*
b^6*d^4 + 96*a^4*b^4*d^4 + 64*a^6*b^2*d^4))^(1/2) - log(- ((((((((128*b^3*tan(c + d*x)^(1/2)*(a^2 - b^2)*(a^2
+ b^2)^2*(-(4*(-B^4*d^4*(a^4 + b^4 - 6*a^2*b^2)^2)^(1/2) + 16*B^2*a*b^3*d^2 - 16*B^2*a^3*b*d^2)/(d^4*(a^2 + b^
2)^4))^(1/2) + (128*B*a*b^2*(7*a^4 + b^4 + 8*a^2*b^2))/d)*(-(4*(-B^4*d^4*(a^4 + b^4 - 6*a^2*b^2)^2)^(1/2) + 16
*B^2*a*b^3*d^2 - 16*B^2*a^3*b*d^2)/(d^4*(a^2 + b^2)^4))^(1/2))/4 + (64*B^2*a*tan(c + d*x)^(1/2)*(18*a^10 - 15*
b^10 + 17*a^2*b^8 - a^4*b^6 + 97*a^6*b^4 + 84*a^8*b^2))/(b^2*d^2*(a^2 + b^2)^2))*(-(4*(-B^4*d^4*(a^4 + b^4 - 6
*a^2*b^2)^2)^(1/2) + 16*B^2*a*b^3*d^2 - 16*B^2*a^3*b*d^2)/(d^4*(a^2 + b^2)^4))^(1/2))/4 - (32*B^3*a^4*(127*a^2
*b^6 - 112*b^8 - 9*a^8 + 173*a^4*b^4 + 21*a^6*b^2))/(b^3*d^3*(a^2 + b^2)^3))*(-(4*(-B^4*d^4*(a^4 + b^4 - 6*a^2
*b^2)^2)^(1/2) + 16*B^2*a*b^3*d^2 - 16*B^2*a^3*b*d^2)/(d^4*(a^2 + b^2)^4))^(1/2))/4 - (16*B^4*tan(c + d*x)^(1/
2)*(9*a^12 + 2*b^12 + 4*a^2*b^10 + 2*a^4*b^8 - 49*a^6*b^6 + 7*a^8*b^4 + 33*a^10*b^2))/(b^3*d^4*(a^2 + b^2)^4))
*(-(4*(-B^4*d^4*(a^4 + b^4 - 6*a^2*b^2)^2)^(1/2) + 16*B^2*a*b^3*d^2 - 16*B^2*a^3*b*d^2)/(d^4*(a^2 + b^2)^4))^(
1/2))/4 - (16*B^5*a^3*b^2*(3*a^2 + 7*b^2))/(d^5*(a^2 + b^2)^4))*(-((192*B^4*a^2*b^6*d^4 - 16*B^4*b^8*d^4 - 16*
B^4*a^8*d^4 - 608*B^4*a^4*b^4*d^4 + 192*B^4*a^6*b^2*d^4)^(1/2) + 16*B^2*a*b^3*d^2 - 16*B^2*a^3*b*d^2)/(16*a^8*
d^4 + 16*b^8*d^4 + 64*a^2*b^6*d^4 + 96*a^4*b^4*d^4 + 64*a^6*b^2*d^4))^(1/2) + (log(((((((((128*b^3*tan(c + d*x
)^(1/2)*(a^2 - b^2)*(a^2 + b^2)^2*((4*(-A^4*d^4*(a^4 + b^4 - 6*a^2*b^2)^2)^(1/2) + 16*A^2*a*b^3*d^2 - 16*A^2*a
^3*b*d^2)/(d^4*(a^2 + b^2)^4))^(1/2) + (768*A*a^2*b^3*(a^2 + b^2))/d)*((4*(-A^4*d^4*(a^4 + b^4 - 6*a^2*b^2)^2)
^(1/2) + 16*A^2*a*b^3*d^2 - 16*A^2*a^3*b*d^2)/(d^4*(a^2 + b^2)^4))^(1/2))/4 + (64*A^2*a*tan(c + d*x)^(1/2)*(2*
a^8 + 15*b^8 - 17*a^2*b^6 + 51*a^4*b^4 + 21*a^6*b^2))/(d^2*(a^2 + b^2)^2))*((4*(-A^4*d^4*(a^4 + b^4 - 6*a^2*b^
2)^2)^(1/2) + 16*A^2*a*b^3*d^2 - 16*A^2*a^3*b*d^2)/(d^4*(a^2 + b^2)^4))^(1/2))/4 + (32*A^3*a*(4*a^8 + b^8 - 77
*a^2*b^6 + 47*a^4*b^4 + 33*a^6*b^2))/(d^3*(a^2 + b^2)^3))*((4*(-A^4*d^4*(a^4 + b^4 - 6*a^2*b^2)^2)^(1/2) + 16*
A^2*a*b^3*d^2 - 16*A^2*a^3*b*d^2)/(d^4*(a^2 + b^2)^4))^(1/2))/4 + (16*A^4*tan(c + d*x)^(1/2)*(a^10 - 2*b^10 -
4*a^2*b^8 - 27*a^4*b^6 + 15*a^6*b^4 + 9*a^8*b^2))/(b*d^4*(a^2 + b^2)^4))*((4*(-A^4*d^4*(a^4 + b^4 - 6*a^2*b^2)
^2)^(1/2) + 16*A^2*a*b^3*d^2 - 16*A^2*a^3*b*d^2)/(d^4*(a^2 + b^2)^4))^(1/2))/4 + (8*A^5*a^2*(a^6 + 10*b^6 + 27
*a^2*b^4 + 10*a^4*b^2))/(b*d^5*(a^2 + b^2)^4))*(((192*A^4*a^2*b^6*d^4 - 16*A^4*b^8*d^4 - 16*A^4*a^8*d^4 - 608*
A^4*a^4*b^4*d^4 + 192*A^4*a^6*b^2*d^4)^(1/2) + 16*A^2*a*b^3*d^2 - 16*A^2*a^3*b*d^2)/(a^8*d^4 + b^8*d^4 + 4*a^2
*b^6*d^4 + 6*a^4*b^4*d^4 + 4*a^6*b^2*d^4))^(1/2))/4 + (log(((((((((128*b^3*tan(c + d*x)^(1/2)*(a^2 - b^2)*(a^2
+ b^2)^2*(-(4*(-A^4*d^4*(a^4 + b^4 - 6*a^2*b^2)^2)^(1/2) - 16*A^2*a*b^3*d^2 + 16*A^2*a^3*b*d^2)/(d^4*(a^2 + b
^2)^4))^(1/2) + (768*A*a^2*b^3*(a^2 + b^2))/d)*(-(4*(-A^4*d^4*(a^4 + b^4 - 6*a^2*b^2)^2)^(1/2) - 16*A^2*a*b^3*
d^2 + 16*A^2*a^3*b*d^2)/(d^4*(a^2 + b^2)^4))^(1/2))/4 + (64*A^2*a*tan(c + d*x)^(1/2)*(2*a^8 + 15*b^8 - 17*a^2*
b^6 + 51*a^4*b^4 + 21*a^6*b^2))/(d^2*(a^2 + b^2)^2))*(-(4*(-A^4*d^4*(a^4 + b^4 - 6*a^2*b^2)^2)^(1/2) - 16*A^2*
a*b^3*d^2 + 16*A^2*a^3*b*d^2)/(d^4*(a^2 + b^2)^4))^(1/2))/4 + (32*A^3*a*(4*a^8 + b^8 - 77*a^2*b^6 + 47*a^4*b^4
+ 33*a^6*b^2))/(d^3*(a^2 + b^2)^3))*(-(4*(-A^4*d^4*(a^4 + b^4 - 6*a^2*b^2)^2)^(1/2) - 16*A^2*a*b^3*d^2 + 16*A
^2*a^3*b*d^2)/(d^4*(a^2 + b^2)^4))^(1/2))/4 + (16*A^4*tan(c + d*x)^(1/2)*(a^10 - 2*b^10 - 4*a^2*b^8 - 27*a^4*b
^6 + 15*a^6*b^4 + 9*a^8*b^2))/(b*d^4*(a^2 + b^2)^4))*(-(4*(-A^4*d^4*(a^4 + b^4 - 6*a^2*b^2)^2)^(1/2) - 16*A^2*
a*b^3*d^2 + 16*A^2*a^3*b*d^2)/(d^4*(a^2 + b^2)^4))^(1/2))/4 + (8*A^5*a^2*(a^6 + 10*b^6 + 27*a^2*b^4 + 10*a^4*b
^2))/(b*d^5*(a^2 + b^2)^4))*(-((192*A^4*a^2*b^6*d^4 - 16*A^4*b^8*d^4 - 16*A^4*a^8*d^4 - 608*A^4*a^4*b^4*d^4 +
192*A^4*a^6*b^2*d^4)^(1/2) - 16*A^2*a*b^3*d^2 + 16*A^2*a^3*b*d^2)/(a^8*d^4 + b^8*d^4 + 4*a^2*b^6*d^4 + 6*a^4*b
^4*d^4 + 4*a^6*b^2*d^4))^(1/2))/4 - log((8*A^5*a^2*(a^6 + 10*b^6 + 27*a^2*b^4 + 10*a^4*b^2))/(b*d^5*(a^2 + b^2
)^4) - ((((((((128*b^3*tan(c + d*x)^(1/2)*(a^2 - b^2)*(a^2 + b^2)^2*((4*(-A^4*d^4*(a^4 + b^4 - 6*a^2*b^2)^2)^(
1/2) + 16*A^2*a*b^3*d^2 - 16*A^2*a^3*b*d^2)/(d^4*(a^2 + b^2)^4))^(1/2) - (768*A*a^2*b^3*(a^2 + b^2))/d)*((4*(-
A^4*d^4*(a^4 + b^4 - 6*a^2*b^2)^2)^(1/2) + 16*A^2*a*b^3*d^2 - 16*A^2*a^3*b*d^2)/(d^4*(a^2 + b^2)^4))^(1/2))/4
+ (64*A^2*a*tan(c + d*x)^(1/2)*(2*a^8 + 15*b^8 - 17*a^2*b^6 + 51*a^4*b^4 + 21*a^6*b^2))/(d^2*(a^2 + b^2)^2))*(
(4*(-A^4*d^4*(a^4 + b^4 - 6*a^2*b^2)^2)^(1/2) + 16*A^2*a*b^3*d^2 - 16*A^2*a^3*b*d^2)/(d^4*(a^2 + b^2)^4))^(1/2
))/4 - (32*A^3*a*(4*a^8 + b^8 - 77*a^2*b^6 + 47*a^4*b^4 + 33*a^6*b^2))/(d^3*(a^2 + b^2)^3))*((4*(-A^4*d^4*(a^4
+ b^4 - 6*a^2*b^2)^2)^(1/2) + 16*A^2*a*b^3*d^2 - 16*A^2*a^3*b*d^2)/(d^4*(a^2 + b^2)^4))^(1/2))/4 + (16*A^4*ta
n(c + d*x)^(1/2)*(a^10 - 2*b^10 - 4*a^2*b^8 - 27*a^4*b^6 + 15*a^6*b^4 + 9*a^8*b^2))/(b*d^4*(a^2 + b^2)^4))*((4
*(-A^4*d^4*(a^4 + b^4 - 6*a^2*b^2)^2)^(1/2) + 16*A^2*a*b^3*d^2 - 16*A^2*a^3*b*d^2)/(d^4*(a^2 + b^2)^4))^(1/2))
/4)*(((192*A^4*a^2*b^6*d^4 - 16*A^4*b^8*d^4 - 16*A^4*a^8*d^4 - 608*A^4*a^4*b^4*d^4 + 192*A^4*a^6*b^2*d^4)^(1/2
) + 16*A^2*a*b^3*d^2 - 16*A^2*a^3*b*d^2)/(16*a^8*d^4 + 16*b^8*d^4 + 64*a^2*b^6*d^4 + 96*a^4*b^4*d^4 + 64*a^6*b
^2*d^4))^(1/2) - log((8*A^5*a^2*(a^6 + 10*b^6 + 27*a^2*b^4 + 10*a^4*b^2))/(b*d^5*(a^2 + b^2)^4) - ((((((((128*
b^3*tan(c + d*x)^(1/2)*(a^2 - b^2)*(a^2 + b^2)^2*(-(4*(-A^4*d^4*(a^4 + b^4 - 6*a^2*b^2)^2)^(1/2) - 16*A^2*a*b^
3*d^2 + 16*A^2*a^3*b*d^2)/(d^4*(a^2 + b^2)^4))^(1/2) - (768*A*a^2*b^3*(a^2 + b^2))/d)*(-(4*(-A^4*d^4*(a^4 + b^
4 - 6*a^2*b^2)^2)^(1/2) - 16*A^2*a*b^3*d^2 + 16*A^2*a^3*b*d^2)/(d^4*(a^2 + b^2)^4))^(1/2))/4 + (64*A^2*a*tan(c
+ d*x)^(1/2)*(2*a^8 + 15*b^8 - 17*a^2*b^6 + 51*a^4*b^4 + 21*a^6*b^2))/(d^2*(a^2 + b^2)^2))*(-(4*(-A^4*d^4*(a^
4 + b^4 - 6*a^2*b^2)^2)^(1/2) - 16*A^2*a*b^3*d^2 + 16*A^2*a^3*b*d^2)/(d^4*(a^2 + b^2)^4))^(1/2))/4 - (32*A^3*a
*(4*a^8 + b^8 - 77*a^2*b^6 + 47*a^4*b^4 + 33*a^6*b^2))/(d^3*(a^2 + b^2)^3))*(-(4*(-A^4*d^4*(a^4 + b^4 - 6*a^2*
b^2)^2)^(1/2) - 16*A^2*a*b^3*d^2 + 16*A^2*a^3*b*d^2)/(d^4*(a^2 + b^2)^4))^(1/2))/4 + (16*A^4*tan(c + d*x)^(1/2
)*(a^10 - 2*b^10 - 4*a^2*b^8 - 27*a^4*b^6 + 15*a^6*b^4 + 9*a^8*b^2))/(b*d^4*(a^2 + b^2)^4))*(-(4*(-A^4*d^4*(a^
4 + b^4 - 6*a^2*b^2)^2)^(1/2) - 16*A^2*a*b^3*d^2 + 16*A^2*a^3*b*d^2)/(d^4*(a^2 + b^2)^4))^(1/2))/4)*(-((192*A^
4*a^2*b^6*d^4 - 16*A^4*b^8*d^4 - 16*A^4*a^8*d^4 - 608*A^4*a^4*b^4*d^4 + 192*A^4*a^6*b^2*d^4)^(1/2) - 16*A^2*a*
b^3*d^2 + 16*A^2*a^3*b*d^2)/(16*a^8*d^4 + 16*b^8*d^4 + 64*a^2*b^6*d^4 + 96*a^4*b^4*d^4 + 64*a^6*b^2*d^4))^(1/2
) - (atan(((((((8*(320*A^3*a^7*b^5*d^2 - 120*A^3*a^5*b^7*d^2 - 304*A^3*a^3*b^9*d^2 + 148*A^3*a^9*b^3*d^2 + 4*A
^3*a*b^11*d^2 + 16*A^3*a^11*b*d^2))/(b^9*d^5 + a^8*b*d^5 + 4*a^2*b^7*d^5 + 6*a^4*b^5*d^5 + 4*a^6*b^3*d^5) + ((
(((8*(96*A*a^2*b^14*d^4 + 480*A*a^4*b^12*d^4 + 960*A*a^6*b^10*d^4 + 960*A*a^8*b^8*d^4 + 480*A*a^10*b^6*d^4 + 9
6*A*a^12*b^4*d^4))/(b^9*d^5 + a^8*b*d^5 + 4*a^2*b^7*d^5 + 6*a^4*b^5*d^5 + 4*a^6*b^3*d^5) - (4*tan(c + d*x)^(1/
2)*(-4*(A^2*a^7 + 25*A^2*a^3*b^4 + 10*A^2*a^5*b^2)*(b^11*d^2 + 4*a^2*b^9*d^2 + 6*a^4*b^7*d^2 + 4*a^6*b^5*d^2 +
a^8*b^3*d^2))^(1/2)*(32*b^18*d^4 + 160*a^2*b^16*d^4 + 288*a^4*b^14*d^4 + 160*a^6*b^12*d^4 - 160*a^8*b^10*d^4
- 288*a^10*b^8*d^4 - 160*a^12*b^6*d^4 - 32*a^14*b^4*d^4))/((b^9*d^4 + a^8*b*d^4 + 4*a^2*b^7*d^4 + 6*a^4*b^5*d^
4 + 4*a^6*b^3*d^4)*(b^11*d^2 + 4*a^2*b^9*d^2 + 6*a^4*b^7*d^2 + 4*a^6*b^5*d^2 + a^8*b^3*d^2)))*(-4*(A^2*a^7 + 2
5*A^2*a^3*b^4 + 10*A^2*a^5*b^2)*(b^11*d^2 + 4*a^2*b^9*d^2 + 6*a^4*b^7*d^2 + 4*a^6*b^5*d^2 + a^8*b^3*d^2))^(1/2
))/(4*(b^11*d^2 + 4*a^2*b^9*d^2 + 6*a^4*b^7*d^2 + 4*a^6*b^5*d^2 + a^8*b^3*d^2)) + (16*tan(c + d*x)^(1/2)*(52*A
^2*a^3*b^11*d^2 + 128*A^2*a^5*b^9*d^2 + 424*A^2*a^7*b^7*d^2 + 380*A^2*a^9*b^5*d^2 + 100*A^2*a^11*b^3*d^2 + 60*
A^2*a*b^13*d^2 + 8*A^2*a^13*b*d^2))/(b^9*d^4 + a^8*b*d^4 + 4*a^2*b^7*d^4 + 6*a^4*b^5*d^4 + 4*a^6*b^3*d^4))*(-4
*(A^2*a^7 + 25*A^2*a^3*b^4 + 10*A^2*a^5*b^2)*(b^11*d^2 + 4*a^2*b^9*d^2 + 6*a^4*b^7*d^2 + 4*a^6*b^5*d^2 + a^8*b
^3*d^2))^(1/2))/(4*(b^11*d^2 + 4*a^2*b^9*d^2 + 6*a^4*b^7*d^2 + 4*a^6*b^5*d^2 + a^8*b^3*d^2)))*(-4*(A^2*a^7 + 2
5*A^2*a^3*b^4 + 10*A^2*a^5*b^2)*(b^11*d^2 + 4*a^2*b^9*d^2 + 6*a^4*b^7*d^2 + 4*a^6*b^5*d^2 + a^8*b^3*d^2))^(1/2
))/(4*(b^11*d^2 + 4*a^2*b^9*d^2 + 6*a^4*b^7*d^2 + 4*a^6*b^5*d^2 + a^8*b^3*d^2)) + (16*tan(c + d*x)^(1/2)*(A^4*
a^10 - 2*A^4*b^10 - 4*A^4*a^2*b^8 - 27*A^4*a^4*b^6 + 15*A^4*a^6*b^4 + 9*A^4*a^8*b^2))/(b^9*d^4 + a^8*b*d^4 + 4
*a^2*b^7*d^4 + 6*a^4*b^5*d^4 + 4*a^6*b^3*d^4))*(-4*(A^2*a^7 + 25*A^2*a^3*b^4 + 10*A^2*a^5*b^2)*(b^11*d^2 + 4*a
^2*b^9*d^2 + 6*a^4*b^7*d^2 + 4*a^6*b^5*d^2 + a^8*b^3*d^2))^(1/2)*1i)/(4*(b^11*d^2 + 4*a^2*b^9*d^2 + 6*a^4*b^7*
d^2 + 4*a^6*b^5*d^2 + a^8*b^3*d^2)) - (((((8*(320*A^3*a^7*b^5*d^2 - 120*A^3*a^5*b^7*d^2 - 304*A^3*a^3*b^9*d^2
+ 148*A^3*a^9*b^3*d^2 + 4*A^3*a*b^11*d^2 + 16*A^3*a^11*b*d^2))/(b^9*d^5 + a^8*b*d^5 + 4*a^2*b^7*d^5 + 6*a^4*b^
5*d^5 + 4*a^6*b^3*d^5) + (((((8*(96*A*a^2*b^14*d^4 + 480*A*a^4*b^12*d^4 + 960*A*a^6*b^10*d^4 + 960*A*a^8*b^8*d
^4 + 480*A*a^10*b^6*d^4 + 96*A*a^12*b^4*d^4))/(b^9*d^5 + a^8*b*d^5 + 4*a^2*b^7*d^5 + 6*a^4*b^5*d^5 + 4*a^6*b^3
*d^5) + (4*tan(c + d*x)^(1/2)*(-4*(A^2*a^7 + 25*A^2*a^3*b^4 + 10*A^2*a^5*b^2)*(b^11*d^2 + 4*a^2*b^9*d^2 + 6*a^
4*b^7*d^2 + 4*a^6*b^5*d^2 + a^8*b^3*d^2))^(1/2)*(32*b^18*d^4 + 160*a^2*b^16*d^4 + 288*a^4*b^14*d^4 + 160*a^6*b
^12*d^4 - 160*a^8*b^10*d^4 - 288*a^10*b^8*d^4 - 160*a^12*b^6*d^4 - 32*a^14*b^4*d^4))/((b^9*d^4 + a^8*b*d^4 + 4
*a^2*b^7*d^4 + 6*a^4*b^5*d^4 + 4*a^6*b^3*d^4)*(b^11*d^2 + 4*a^2*b^9*d^2 + 6*a^4*b^7*d^2 + 4*a^6*b^5*d^2 + a^8*
b^3*d^2)))*(-4*(A^2*a^7 + 25*A^2*a^3*b^4 + 10*A^2*a^5*b^2)*(b^11*d^2 + 4*a^2*b^9*d^2 + 6*a^4*b^7*d^2 + 4*a^6*b
^5*d^2 + a^8*b^3*d^2))^(1/2))/(4*(b^11*d^2 + 4*a^2*b^9*d^2 + 6*a^4*b^7*d^2 + 4*a^6*b^5*d^2 + a^8*b^3*d^2)) - (
16*tan(c + d*x)^(1/2)*(52*A^2*a^3*b^11*d^2 + 128*A^2*a^5*b^9*d^2 + 424*A^2*a^7*b^7*d^2 + 380*A^2*a^9*b^5*d^2 +
100*A^2*a^11*b^3*d^2 + 60*A^2*a*b^13*d^2 + 8*A^2*a^13*b*d^2))/(b^9*d^4 + a^8*b*d^4 + 4*a^2*b^7*d^4 + 6*a^4*b^
5*d^4 + 4*a^6*b^3*d^4))*(-4*(A^2*a^7 + 25*A^2*a^3*b^4 + 10*A^2*a^5*b^2)*(b^11*d^2 + 4*a^2*b^9*d^2 + 6*a^4*b^7*
d^2 + 4*a^6*b^5*d^2 + a^8*b^3*d^2))^(1/2))/(4*(b^11*d^2 + 4*a^2*b^9*d^2 + 6*a^4*b^7*d^2 + 4*a^6*b^5*d^2 + a^8*
b^3*d^2)))*(-4*(A^2*a^7 + 25*A^2*a^3*b^4 + 10*A^2*a^5*b^2)*(b^11*d^2 + 4*a^2*b^9*d^2 + 6*a^4*b^7*d^2 + 4*a^6*b
^5*d^2 + a^8*b^3*d^2))^(1/2))/(4*(b^11*d^2 + 4*a^2*b^9*d^2 + 6*a^4*b^7*d^2 + 4*a^6*b^5*d^2 + a^8*b^3*d^2)) - (
16*tan(c + d*x)^(1/2)*(A^4*a^10 - 2*A^4*b^10 - 4*A^4*a^2*b^8 - 27*A^4*a^4*b^6 + 15*A^4*a^6*b^4 + 9*A^4*a^8*b^2
))/(b^9*d^4 + a^8*b*d^4 + 4*a^2*b^7*d^4 + 6*a^4*b^5*d^4 + 4*a^6*b^3*d^4))*(-4*(A^2*a^7 + 25*A^2*a^3*b^4 + 10*A
^2*a^5*b^2)*(b^11*d^2 + 4*a^2*b^9*d^2 + 6*a^4*b^7*d^2 + 4*a^6*b^5*d^2 + a^8*b^3*d^2))^(1/2)*1i)/(4*(b^11*d^2 +
4*a^2*b^9*d^2 + 6*a^4*b^7*d^2 + 4*a^6*b^5*d^2 + a^8*b^3*d^2)))/((16*(A^5*a^8 + 10*A^5*a^2*b^6 + 27*A^5*a^4*b^
4 + 10*A^5*a^6*b^2))/(b^9*d^5 + a^8*b*d^5 + 4*a^2*b^7*d^5 + 6*a^4*b^5*d^5 + 4*a^6*b^3*d^5) + (((((8*(320*A^3*a
^7*b^5*d^2 - 120*A^3*a^5*b^7*d^2 - 304*A^3*a^3*b^9*d^2 + 148*A^3*a^9*b^3*d^2 + 4*A^3*a*b^11*d^2 + 16*A^3*a^11*
b*d^2))/(b^9*d^5 + a^8*b*d^5 + 4*a^2*b^7*d^5 + 6*a^4*b^5*d^5 + 4*a^6*b^3*d^5) + (((((8*(96*A*a^2*b^14*d^4 + 48
0*A*a^4*b^12*d^4 + 960*A*a^6*b^10*d^4 + 960*A*a^8*b^8*d^4 + 480*A*a^10*b^6*d^4 + 96*A*a^12*b^4*d^4))/(b^9*d^5
+ a^8*b*d^5 + 4*a^2*b^7*d^5 + 6*a^4*b^5*d^5 + 4*a^6*b^3*d^5) - (4*tan(c + d*x)^(1/2)*(-4*(A^2*a^7 + 25*A^2*a^3
*b^4 + 10*A^2*a^5*b^2)*(b^11*d^2 + 4*a^2*b^9*d^2 + 6*a^4*b^7*d^2 + 4*a^6*b^5*d^2 + a^8*b^3*d^2))^(1/2)*(32*b^1
8*d^4 + 160*a^2*b^16*d^4 + 288*a^4*b^14*d^4 + 160*a^6*b^12*d^4 - 160*a^8*b^10*d^4 - 288*a^10*b^8*d^4 - 160*a^1
2*b^6*d^4 - 32*a^14*b^4*d^4))/((b^9*d^4 + a^8*b*d^4 + 4*a^2*b^7*d^4 + 6*a^4*b^5*d^4 + 4*a^6*b^3*d^4)*(b^11*d^2
+ 4*a^2*b^9*d^2 + 6*a^4*b^7*d^2 + 4*a^6*b^5*d^2 + a^8*b^3*d^2)))*(-4*(A^2*a^7 + 25*A^2*a^3*b^4 + 10*A^2*a^5*b
^2)*(b^11*d^2 + 4*a^2*b^9*d^2 + 6*a^4*b^7*d^2 + 4*a^6*b^5*d^2 + a^8*b^3*d^2))^(1/2))/(4*(b^11*d^2 + 4*a^2*b^9*
d^2 + 6*a^4*b^7*d^2 + 4*a^6*b^5*d^2 + a^8*b^3*d^2)) + (16*tan(c + d*x)^(1/2)*(52*A^2*a^3*b^11*d^2 + 128*A^2*a^
5*b^9*d^2 + 424*A^2*a^7*b^7*d^2 + 380*A^2*a^9*b^5*d^2 + 100*A^2*a^11*b^3*d^2 + 60*A^2*a*b^13*d^2 + 8*A^2*a^13*
b*d^2))/(b^9*d^4 + a^8*b*d^4 + 4*a^2*b^7*d^4 + 6*a^4*b^5*d^4 + 4*a^6*b^3*d^4))*(-4*(A^2*a^7 + 25*A^2*a^3*b^4 +
10*A^2*a^5*b^2)*(b^11*d^2 + 4*a^2*b^9*d^2 + 6*a^4*b^7*d^2 + 4*a^6*b^5*d^2 + a^8*b^3*d^2))^(1/2))/(4*(b^11*d^2
+ 4*a^2*b^9*d^2 + 6*a^4*b^7*d^2 + 4*a^6*b^5*d^2 + a^8*b^3*d^2)))*(-4*(A^2*a^7 + 25*A^2*a^3*b^4 + 10*A^2*a^5*b
^2)*(b^11*d^2 + 4*a^2*b^9*d^2 + 6*a^4*b^7*d^2 + 4*a^6*b^5*d^2 + a^8*b^3*d^2))^(1/2))/(4*(b^11*d^2 + 4*a^2*b^9*
d^2 + 6*a^4*b^7*d^2 + 4*a^6*b^5*d^2 + a^8*b^3*d^2)) + (16*tan(c + d*x)^(1/2)*(A^4*a^10 - 2*A^4*b^10 - 4*A^4*a^
2*b^8 - 27*A^4*a^4*b^6 + 15*A^4*a^6*b^4 + 9*A^4*a^8*b^2))/(b^9*d^4 + a^8*b*d^4 + 4*a^2*b^7*d^4 + 6*a^4*b^5*d^4
+ 4*a^6*b^3*d^4))*(-4*(A^2*a^7 + 25*A^2*a^3*b^4 + 10*A^2*a^5*b^2)*(b^11*d^2 + 4*a^2*b^9*d^2 + 6*a^4*b^7*d^2 +
4*a^6*b^5*d^2 + a^8*b^3*d^2))^(1/2))/(4*(b^11*d^2 + 4*a^2*b^9*d^2 + 6*a^4*b^7*d^2 + 4*a^6*b^5*d^2 + a^8*b^3*d
^2)) + (((((8*(320*A^3*a^7*b^5*d^2 - 120*A^3*a^5*b^7*d^2 - 304*A^3*a^3*b^9*d^2 + 148*A^3*a^9*b^3*d^2 + 4*A^3*a
*b^11*d^2 + 16*A^3*a^11*b*d^2))/(b^9*d^5 + a^8*b*d^5 + 4*a^2*b^7*d^5 + 6*a^4*b^5*d^5 + 4*a^6*b^3*d^5) + (((((8
*(96*A*a^2*b^14*d^4 + 480*A*a^4*b^12*d^4 + 960*A*a^6*b^10*d^4 + 960*A*a^8*b^8*d^4 + 480*A*a^10*b^6*d^4 + 96*A*
a^12*b^4*d^4))/(b^9*d^5 + a^8*b*d^5 + 4*a^2*b^7*d^5 + 6*a^4*b^5*d^5 + 4*a^6*b^3*d^5) + (4*tan(c + d*x)^(1/2)*(
-4*(A^2*a^7 + 25*A^2*a^3*b^4 + 10*A^2*a^5*b^2)*(b^11*d^2 + 4*a^2*b^9*d^2 + 6*a^4*b^7*d^2 + 4*a^6*b^5*d^2 + a^8
*b^3*d^2))^(1/2)*(32*b^18*d^4 + 160*a^2*b^16*d^4 + 288*a^4*b^14*d^4 + 160*a^6*b^12*d^4 - 160*a^8*b^10*d^4 - 28
8*a^10*b^8*d^4 - 160*a^12*b^6*d^4 - 32*a^14*b^4*d^4))/((b^9*d^4 + a^8*b*d^4 + 4*a^2*b^7*d^4 + 6*a^4*b^5*d^4 +
4*a^6*b^3*d^4)*(b^11*d^2 + 4*a^2*b^9*d^2 + 6*a^4*b^7*d^2 + 4*a^6*b^5*d^2 + a^8*b^3*d^2)))*(-4*(A^2*a^7 + 25*A^
2*a^3*b^4 + 10*A^2*a^5*b^2)*(b^11*d^2 + 4*a^2*b^9*d^2 + 6*a^4*b^7*d^2 + 4*a^6*b^5*d^2 + a^8*b^3*d^2))^(1/2))/(
4*(b^11*d^2 + 4*a^2*b^9*d^2 + 6*a^4*b^7*d^2 + 4*a^6*b^5*d^2 + a^8*b^3*d^2)) - (16*tan(c + d*x)^(1/2)*(52*A^2*a
^3*b^11*d^2 + 128*A^2*a^5*b^9*d^2 + 424*A^2*a^7*b^7*d^2 + 380*A^2*a^9*b^5*d^2 + 100*A^2*a^11*b^3*d^2 + 60*A^2*
a*b^13*d^2 + 8*A^2*a^13*b*d^2))/(b^9*d^4 + a^8*b*d^4 + 4*a^2*b^7*d^4 + 6*a^4*b^5*d^4 + 4*a^6*b^3*d^4))*(-4*(A^
2*a^7 + 25*A^2*a^3*b^4 + 10*A^2*a^5*b^2)*(b^11*d^2 + 4*a^2*b^9*d^2 + 6*a^4*b^7*d^2 + 4*a^6*b^5*d^2 + a^8*b^3*d
^2))^(1/2))/(4*(b^11*d^2 + 4*a^2*b^9*d^2 + 6*a^4*b^7*d^2 + 4*a^6*b^5*d^2 + a^8*b^3*d^2)))*(-4*(A^2*a^7 + 25*A^
2*a^3*b^4 + 10*A^2*a^5*b^2)*(b^11*d^2 + 4*a^2*b^9*d^2 + 6*a^4*b^7*d^2 + 4*a^6*b^5*d^2 + a^8*b^3*d^2))^(1/2))/(
4*(b^11*d^2 + 4*a^2*b^9*d^2 + 6*a^4*b^7*d^2 + 4*a^6*b^5*d^2 + a^8*b^3*d^2)) - (16*tan(c + d*x)^(1/2)*(A^4*a^10
- 2*A^4*b^10 - 4*A^4*a^2*b^8 - 27*A^4*a^4*b^6 + 15*A^4*a^6*b^4 + 9*A^4*a^8*b^2))/(b^9*d^4 + a^8*b*d^4 + 4*a^2
*b^7*d^4 + 6*a^4*b^5*d^4 + 4*a^6*b^3*d^4))*(-4*(A^2*a^7 + 25*A^2*a^3*b^4 + 10*A^2*a^5*b^2)*(b^11*d^2 + 4*a^2*b
^9*d^2 + 6*a^4*b^7*d^2 + 4*a^6*b^5*d^2 + a^8*b^3*d^2))^(1/2))/(4*(b^11*d^2 + 4*a^2*b^9*d^2 + 6*a^4*b^7*d^2 + 4
*a^6*b^5*d^2 + a^8*b^3*d^2))))*(-4*(A^2*a^7 + 25*A^2*a^3*b^4 + 10*A^2*a^5*b^2)*(b^11*d^2 + 4*a^2*b^9*d^2 + 6*a
^4*b^7*d^2 + 4*a^6*b^5*d^2 + a^8*b^3*d^2))^(1/2)*1i)/(2*(b^11*d^2 + 4*a^2*b^9*d^2 + 6*a^4*b^7*d^2 + 4*a^6*b^5*
d^2 + a^8*b^3*d^2)) - (atan(-((((((16*(30*B^3*a^6*b^8*d^2 - 224*B^3*a^4*b^10*d^2 - 18*B^3*a^14*d^2 + 600*B^3*a
^8*b^6*d^2 + 388*B^3*a^10*b^4*d^2 + 24*B^3*a^12*b^2*d^2))/(b^11*d^5 + 4*a^2*b^9*d^5 + 6*a^4*b^7*d^5 + 4*a^6*b^
5*d^5 + a^8*b^3*d^5) - (((16*tan(c + d*x)^(1/2)*(72*B^2*a^5*b^11*d^2 - 52*B^2*a^3*b^13*d^2 + 448*B^2*a^7*b^9*d
^2 + 1108*B^2*a^9*b^7*d^2 + 1132*B^2*a^11*b^5*d^2 + 480*B^2*a^13*b^3*d^2 - 60*B^2*a*b^15*d^2 + 72*B^2*a^15*b*d
^2))/(b^11*d^4 + 4*a^2*b^9*d^4 + 6*a^4*b^7*d^4 + 4*a^6*b^5*d^4 + a^8*b^3*d^4) + (((16*(8*B*a*b^17*d^4 + 96*B*a
^3*b^15*d^4 + 360*B*a^5*b^13*d^4 + 640*B*a^7*b^11*d^4 + 600*B*a^9*b^9*d^4 + 288*B*a^11*b^7*d^4 + 56*B*a^13*b^5
*d^4))/(b^11*d^5 + 4*a^2*b^9*d^5 + 6*a^4*b^7*d^5 + 4*a^6*b^5*d^5 + a^8*b^3*d^5) - (4*tan(c + d*x)^(1/2)*(-4*(9
*B^2*a^9 + 49*B^2*a^5*b^4 + 42*B^2*a^7*b^2)*(b^13*d^2 + 4*a^2*b^11*d^2 + 6*a^4*b^9*d^2 + 4*a^6*b^7*d^2 + a^8*b
^5*d^2))^(1/2)*(32*b^20*d^4 + 160*a^2*b^18*d^4 + 288*a^4*b^16*d^4 + 160*a^6*b^14*d^4 - 160*a^8*b^12*d^4 - 288*
a^10*b^10*d^4 - 160*a^12*b^8*d^4 - 32*a^14*b^6*d^4))/((b^11*d^4 + 4*a^2*b^9*d^4 + 6*a^4*b^7*d^4 + 4*a^6*b^5*d^
4 + a^8*b^3*d^4)*(b^13*d^2 + 4*a^2*b^11*d^2 + 6*a^4*b^9*d^2 + 4*a^6*b^7*d^2 + a^8*b^5*d^2)))*(-4*(9*B^2*a^9 +
49*B^2*a^5*b^4 + 42*B^2*a^7*b^2)*(b^13*d^2 + 4*a^2*b^11*d^2 + 6*a^4*b^9*d^2 + 4*a^6*b^7*d^2 + a^8*b^5*d^2))^(1
/2))/(4*(b^13*d^2 + 4*a^2*b^11*d^2 + 6*a^4*b^9*d^2 + 4*a^6*b^7*d^2 + a^8*b^5*d^2)))*(-4*(9*B^2*a^9 + 49*B^2*a^
5*b^4 + 42*B^2*a^7*b^2)*(b^13*d^2 + 4*a^2*b^11*d^2 + 6*a^4*b^9*d^2 + 4*a^6*b^7*d^2 + a^8*b^5*d^2))^(1/2))/(4*(
b^13*d^2 + 4*a^2*b^11*d^2 + 6*a^4*b^9*d^2 + 4*a^6*b^7*d^2 + a^8*b^5*d^2)))*(-4*(9*B^2*a^9 + 49*B^2*a^5*b^4 + 4
2*B^2*a^7*b^2)*(b^13*d^2 + 4*a^2*b^11*d^2 + 6*a^4*b^9*d^2 + 4*a^6*b^7*d^2 + a^8*b^5*d^2))^(1/2))/(4*(b^13*d^2
+ 4*a^2*b^11*d^2 + 6*a^4*b^9*d^2 + 4*a^6*b^7*d^2 + a^8*b^5*d^2)) + (16*tan(c + d*x)^(1/2)*(9*B^4*a^12 + 2*B^4*
b^12 + 4*B^4*a^2*b^10 + 2*B^4*a^4*b^8 - 49*B^4*a^6*b^6 + 7*B^4*a^8*b^4 + 33*B^4*a^10*b^2))/(b^11*d^4 + 4*a^2*b
^9*d^4 + 6*a^4*b^7*d^4 + 4*a^6*b^5*d^4 + a^8*b^3*d^4))*(-4*(9*B^2*a^9 + 49*B^2*a^5*b^4 + 42*B^2*a^7*b^2)*(b^13
*d^2 + 4*a^2*b^11*d^2 + 6*a^4*b^9*d^2 + 4*a^6*b^7*d^2 + a^8*b^5*d^2))^(1/2)*1i)/(4*(b^13*d^2 + 4*a^2*b^11*d^2
+ 6*a^4*b^9*d^2 + 4*a^6*b^7*d^2 + a^8*b^5*d^2)) - (((((16*(30*B^3*a^6*b^8*d^2 - 224*B^3*a^4*b^10*d^2 - 18*B^3*
a^14*d^2 + 600*B^3*a^8*b^6*d^2 + 388*B^3*a^10*b^4*d^2 + 24*B^3*a^12*b^2*d^2))/(b^11*d^5 + 4*a^2*b^9*d^5 + 6*a^
4*b^7*d^5 + 4*a^6*b^5*d^5 + a^8*b^3*d^5) + (((16*tan(c + d*x)^(1/2)*(72*B^2*a^5*b^11*d^2 - 52*B^2*a^3*b^13*d^2
+ 448*B^2*a^7*b^9*d^2 + 1108*B^2*a^9*b^7*d^2 + 1132*B^2*a^11*b^5*d^2 + 480*B^2*a^13*b^3*d^2 - 60*B^2*a*b^15*d
^2 + 72*B^2*a^15*b*d^2))/(b^11*d^4 + 4*a^2*b^9*d^4 + 6*a^4*b^7*d^4 + 4*a^6*b^5*d^4 + a^8*b^3*d^4) - (((16*(8*B
*a*b^17*d^4 + 96*B*a^3*b^15*d^4 + 360*B*a^5*b^13*d^4 + 640*B*a^7*b^11*d^4 + 600*B*a^9*b^9*d^4 + 288*B*a^11*b^7
*d^4 + 56*B*a^13*b^5*d^4))/(b^11*d^5 + 4*a^2*b^9*d^5 + 6*a^4*b^7*d^5 + 4*a^6*b^5*d^5 + a^8*b^3*d^5) + (4*tan(c
+ d*x)^(1/2)*(-4*(9*B^2*a^9 + 49*B^2*a^5*b^4 + 42*B^2*a^7*b^2)*(b^13*d^2 + 4*a^2*b^11*d^2 + 6*a^4*b^9*d^2 + 4
*a^6*b^7*d^2 + a^8*b^5*d^2))^(1/2)*(32*b^20*d^4 + 160*a^2*b^18*d^4 + 288*a^4*b^16*d^4 + 160*a^6*b^14*d^4 - 160
*a^8*b^12*d^4 - 288*a^10*b^10*d^4 - 160*a^12*b^8*d^4 - 32*a^14*b^6*d^4))/((b^11*d^4 + 4*a^2*b^9*d^4 + 6*a^4*b^
7*d^4 + 4*a^6*b^5*d^4 + a^8*b^3*d^4)*(b^13*d^2 + 4*a^2*b^11*d^2 + 6*a^4*b^9*d^2 + 4*a^6*b^7*d^2 + a^8*b^5*d^2)
))*(-4*(9*B^2*a^9 + 49*B^2*a^5*b^4 + 42*B^2*a^7*b^2)*(b^13*d^2 + 4*a^2*b^11*d^2 + 6*a^4*b^9*d^2 + 4*a^6*b^7*d^
2 + a^8*b^5*d^2))^(1/2))/(4*(b^13*d^2 + 4*a^2*b^11*d^2 + 6*a^4*b^9*d^2 + 4*a^6*b^7*d^2 + a^8*b^5*d^2)))*(-4*(9
*B^2*a^9 + 49*B^2*a^5*b^4 + 42*B^2*a^7*b^2)*(b^13*d^2 + 4*a^2*b^11*d^2 + 6*a^4*b^9*d^2 + 4*a^6*b^7*d^2 + a^8*b
^5*d^2))^(1/2))/(4*(b^13*d^2 + 4*a^2*b^11*d^2 + 6*a^4*b^9*d^2 + 4*a^6*b^7*d^2 + a^8*b^5*d^2)))*(-4*(9*B^2*a^9
+ 49*B^2*a^5*b^4 + 42*B^2*a^7*b^2)*(b^13*d^2 + 4*a^2*b^11*d^2 + 6*a^4*b^9*d^2 + 4*a^6*b^7*d^2 + a^8*b^5*d^2))^
(1/2))/(4*(b^13*d^2 + 4*a^2*b^11*d^2 + 6*a^4*b^9*d^2 + 4*a^6*b^7*d^2 + a^8*b^5*d^2)) - (16*tan(c + d*x)^(1/2)*
(9*B^4*a^12 + 2*B^4*b^12 + 4*B^4*a^2*b^10 + 2*B^4*a^4*b^8 - 49*B^4*a^6*b^6 + 7*B^4*a^8*b^4 + 33*B^4*a^10*b^2))
/(b^11*d^4 + 4*a^2*b^9*d^4 + 6*a^4*b^7*d^4 + 4*a^6*b^5*d^4 + a^8*b^3*d^4))*(-4*(9*B^2*a^9 + 49*B^2*a^5*b^4 + 4
2*B^2*a^7*b^2)*(b^13*d^2 + 4*a^2*b^11*d^2 + 6*a^4*b^9*d^2 + 4*a^6*b^7*d^2 + a^8*b^5*d^2))^(1/2)*1i)/(4*(b^13*d
^2 + 4*a^2*b^11*d^2 + 6*a^4*b^9*d^2 + 4*a^6*b^7*d^2 + a^8*b^5*d^2)))/((((((16*(30*B^3*a^6*b^8*d^2 - 224*B^3*a^
4*b^10*d^2 - 18*B^3*a^14*d^2 + 600*B^3*a^8*b^6*d^2 + 388*B^3*a^10*b^4*d^2 + 24*B^3*a^12*b^2*d^2))/(b^11*d^5 +
4*a^2*b^9*d^5 + 6*a^4*b^7*d^5 + 4*a^6*b^5*d^5 + a^8*b^3*d^5) - (((16*tan(c + d*x)^(1/2)*(72*B^2*a^5*b^11*d^2 -
52*B^2*a^3*b^13*d^2 + 448*B^2*a^7*b^9*d^2 + 1108*B^2*a^9*b^7*d^2 + 1132*B^2*a^11*b^5*d^2 + 480*B^2*a^13*b^3*d
^2 - 60*B^2*a*b^15*d^2 + 72*B^2*a^15*b*d^2))/(b^11*d^4 + 4*a^2*b^9*d^4 + 6*a^4*b^7*d^4 + 4*a^6*b^5*d^4 + a^8*b
^3*d^4) + (((16*(8*B*a*b^17*d^4 + 96*B*a^3*b^15*d^4 + 360*B*a^5*b^13*d^4 + 640*B*a^7*b^11*d^4 + 600*B*a^9*b^9*
d^4 + 288*B*a^11*b^7*d^4 + 56*B*a^13*b^5*d^4))/(b^11*d^5 + 4*a^2*b^9*d^5 + 6*a^4*b^7*d^5 + 4*a^6*b^5*d^5 + a^8
*b^3*d^5) - (4*tan(c + d*x)^(1/2)*(-4*(9*B^2*a^9 + 49*B^2*a^5*b^4 + 42*B^2*a^7*b^2)*(b^13*d^2 + 4*a^2*b^11*d^2
+ 6*a^4*b^9*d^2 + 4*a^6*b^7*d^2 + a^8*b^5*d^2))^(1/2)*(32*b^20*d^4 + 160*a^2*b^18*d^4 + 288*a^4*b^16*d^4 + 16
0*a^6*b^14*d^4 - 160*a^8*b^12*d^4 - 288*a^10*b^10*d^4 - 160*a^12*b^8*d^4 - 32*a^14*b^6*d^4))/((b^11*d^4 + 4*a^
2*b^9*d^4 + 6*a^4*b^7*d^4 + 4*a^6*b^5*d^4 + a^8*b^3*d^4)*(b^13*d^2 + 4*a^2*b^11*d^2 + 6*a^4*b^9*d^2 + 4*a^6*b^
7*d^2 + a^8*b^5*d^2)))*(-4*(9*B^2*a^9 + 49*B^2*a^5*b^4 + 42*B^2*a^7*b^2)*(b^13*d^2 + 4*a^2*b^11*d^2 + 6*a^4*b^
9*d^2 + 4*a^6*b^7*d^2 + a^8*b^5*d^2))^(1/2))/(4*(b^13*d^2 + 4*a^2*b^11*d^2 + 6*a^4*b^9*d^2 + 4*a^6*b^7*d^2 + a
^8*b^5*d^2)))*(-4*(9*B^2*a^9 + 49*B^2*a^5*b^4 + 42*B^2*a^7*b^2)*(b^13*d^2 + 4*a^2*b^11*d^2 + 6*a^4*b^9*d^2 + 4
*a^6*b^7*d^2 + a^8*b^5*d^2))^(1/2))/(4*(b^13*d^2 + 4*a^2*b^11*d^2 + 6*a^4*b^9*d^2 + 4*a^6*b^7*d^2 + a^8*b^5*d^
2)))*(-4*(9*B^2*a^9 + 49*B^2*a^5*b^4 + 42*B^2*a^7*b^2)*(b^13*d^2 + 4*a^2*b^11*d^2 + 6*a^4*b^9*d^2 + 4*a^6*b^7*
d^2 + a^8*b^5*d^2))^(1/2))/(4*(b^13*d^2 + 4*a^2*b^11*d^2 + 6*a^4*b^9*d^2 + 4*a^6*b^7*d^2 + a^8*b^5*d^2)) + (16
*tan(c + d*x)^(1/2)*(9*B^4*a^12 + 2*B^4*b^12 + 4*B^4*a^2*b^10 + 2*B^4*a^4*b^8 - 49*B^4*a^6*b^6 + 7*B^4*a^8*b^4
+ 33*B^4*a^10*b^2))/(b^11*d^4 + 4*a^2*b^9*d^4 + 6*a^4*b^7*d^4 + 4*a^6*b^5*d^4 + a^8*b^3*d^4))*(-4*(9*B^2*a^9
+ 49*B^2*a^5*b^4 + 42*B^2*a^7*b^2)*(b^13*d^2 + 4*a^2*b^11*d^2 + 6*a^4*b^9*d^2 + 4*a^6*b^7*d^2 + a^8*b^5*d^2))^
(1/2))/(4*(b^13*d^2 + 4*a^2*b^11*d^2 + 6*a^4*b^9*d^2 + 4*a^6*b^7*d^2 + a^8*b^5*d^2)) - (32*(7*B^5*a^3*b^7 + 3*
B^5*a^5*b^5))/(b^11*d^5 + 4*a^2*b^9*d^5 + 6*a^4*b^7*d^5 + 4*a^6*b^5*d^5 + a^8*b^3*d^5) + (((((16*(30*B^3*a^6*b
^8*d^2 - 224*B^3*a^4*b^10*d^2 - 18*B^3*a^14*d^2 + 600*B^3*a^8*b^6*d^2 + 388*B^3*a^10*b^4*d^2 + 24*B^3*a^12*b^2
*d^2))/(b^11*d^5 + 4*a^2*b^9*d^5 + 6*a^4*b^7*d^5 + 4*a^6*b^5*d^5 + a^8*b^3*d^5) + (((16*tan(c + d*x)^(1/2)*(72
*B^2*a^5*b^11*d^2 - 52*B^2*a^3*b^13*d^2 + 448*B^2*a^7*b^9*d^2 + 1108*B^2*a^9*b^7*d^2 + 1132*B^2*a^11*b^5*d^2 +
480*B^2*a^13*b^3*d^2 - 60*B^2*a*b^15*d^2 + 72*B^2*a^15*b*d^2))/(b^11*d^4 + 4*a^2*b^9*d^4 + 6*a^4*b^7*d^4 + 4*
a^6*b^5*d^4 + a^8*b^3*d^4) - (((16*(8*B*a*b^17*d^4 + 96*B*a^3*b^15*d^4 + 360*B*a^5*b^13*d^4 + 640*B*a^7*b^11*d
^4 + 600*B*a^9*b^9*d^4 + 288*B*a^11*b^7*d^4 + 56*B*a^13*b^5*d^4))/(b^11*d^5 + 4*a^2*b^9*d^5 + 6*a^4*b^7*d^5 +
4*a^6*b^5*d^5 + a^8*b^3*d^5) + (4*tan(c + d*x)^(1/2)*(-4*(9*B^2*a^9 + 49*B^2*a^5*b^4 + 42*B^2*a^7*b^2)*(b^13*d
^2 + 4*a^2*b^11*d^2 + 6*a^4*b^9*d^2 + 4*a^6*b^7*d^2 + a^8*b^5*d^2))^(1/2)*(32*b^20*d^4 + 160*a^2*b^18*d^4 + 28
8*a^4*b^16*d^4 + 160*a^6*b^14*d^4 - 160*a^8*b^12*d^4 - 288*a^10*b^10*d^4 - 160*a^12*b^8*d^4 - 32*a^14*b^6*d^4)
)/((b^11*d^4 + 4*a^2*b^9*d^4 + 6*a^4*b^7*d^4 + 4*a^6*b^5*d^4 + a^8*b^3*d^4)*(b^13*d^2 + 4*a^2*b^11*d^2 + 6*a^4
*b^9*d^2 + 4*a^6*b^7*d^2 + a^8*b^5*d^2)))*(-4*(9*B^2*a^9 + 49*B^2*a^5*b^4 + 42*B^2*a^7*b^2)*(b^13*d^2 + 4*a^2*
b^11*d^2 + 6*a^4*b^9*d^2 + 4*a^6*b^7*d^2 + a^8*b^5*d^2))^(1/2))/(4*(b^13*d^2 + 4*a^2*b^11*d^2 + 6*a^4*b^9*d^2
+ 4*a^6*b^7*d^2 + a^8*b^5*d^2)))*(-4*(9*B^2*a^9 + 49*B^2*a^5*b^4 + 42*B^2*a^7*b^2)*(b^13*d^2 + 4*a^2*b^11*d^2
+ 6*a^4*b^9*d^2 + 4*a^6*b^7*d^2 + a^8*b^5*d^2))^(1/2))/(4*(b^13*d^2 + 4*a^2*b^11*d^2 + 6*a^4*b^9*d^2 + 4*a^6*b
^7*d^2 + a^8*b^5*d^2)))*(-4*(9*B^2*a^9 + 49*B^2*a^5*b^4 + 42*B^2*a^7*b^2)*(b^13*d^2 + 4*a^2*b^11*d^2 + 6*a^4*b
^9*d^2 + 4*a^6*b^7*d^2 + a^8*b^5*d^2))^(1/2))/(4*(b^13*d^2 + 4*a^2*b^11*d^2 + 6*a^4*b^9*d^2 + 4*a^6*b^7*d^2 +
a^8*b^5*d^2)) - (16*tan(c + d*x)^(1/2)*(9*B^4*a^12 + 2*B^4*b^12 + 4*B^4*a^2*b^10 + 2*B^4*a^4*b^8 - 49*B^4*a^6*
b^6 + 7*B^4*a^8*b^4 + 33*B^4*a^10*b^2))/(b^11*d^4 + 4*a^2*b^9*d^4 + 6*a^4*b^7*d^4 + 4*a^6*b^5*d^4 + a^8*b^3*d^
4))*(-4*(9*B^2*a^9 + 49*B^2*a^5*b^4 + 42*B^2*a^7*b^2)*(b^13*d^2 + 4*a^2*b^11*d^2 + 6*a^4*b^9*d^2 + 4*a^6*b^7*d
^2 + a^8*b^5*d^2))^(1/2))/(4*(b^13*d^2 + 4*a^2*b^11*d^2 + 6*a^4*b^9*d^2 + 4*a^6*b^7*d^2 + a^8*b^5*d^2))))*(-4*
(9*B^2*a^9 + 49*B^2*a^5*b^4 + 42*B^2*a^7*b^2)*(b^13*d^2 + 4*a^2*b^11*d^2 + 6*a^4*b^9*d^2 + 4*a^6*b^7*d^2 + a^8
*b^5*d^2))^(1/2)*1i)/(2*(b^13*d^2 + 4*a^2*b^11*d^2 + 6*a^4*b^9*d^2 + 4*a^6*b^7*d^2 + a^8*b^5*d^2)) + (2*B*tan(
c + d*x)^(1/2))/(b^2*d) + (B*a^3*tan(c + d*x)^(1/2))/((a*b^2*d + b^3*d*tan(c + d*x))*(a^2 + b^2)) - (A*a^2*tan
(c + d*x)^(1/2))/(b*(a*d + b*d*tan(c + d*x))*(a^2 + b^2))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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