3.83 \(\int \frac {(e \cot (c+d x))^{5/2}}{(a+b \cot (c+d x))^3} \, dx\)

Optimal. Leaf size=470 \[ \frac {e^{5/2} (a+b) \left (a^2-4 a b+b^2\right ) \log \left (\sqrt {e} \cot (c+d x)-\sqrt {2} \sqrt {e \cot (c+d x)}+\sqrt {e}\right )}{2 \sqrt {2} d \left (a^2+b^2\right )^3}-\frac {e^{5/2} (a+b) \left (a^2-4 a b+b^2\right ) \log \left (\sqrt {e} \cot (c+d x)+\sqrt {2} \sqrt {e \cot (c+d x)}+\sqrt {e}\right )}{2 \sqrt {2} d \left (a^2+b^2\right )^3}-\frac {e^{5/2} (a-b) \left (a^2+4 a b+b^2\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} d \left (a^2+b^2\right )^3}+\frac {e^{5/2} (a-b) \left (a^2+4 a b+b^2\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}+1\right )}{\sqrt {2} d \left (a^2+b^2\right )^3}-\frac {a e^2 \left (a^2+9 b^2\right ) \sqrt {e \cot (c+d x)}}{4 b d \left (a^2+b^2\right )^2 (a+b \cot (c+d x))}+\frac {a^2 e^2 \sqrt {e \cot (c+d x)}}{2 b d \left (a^2+b^2\right ) (a+b \cot (c+d x))^2}-\frac {\sqrt {a} e^{5/2} \left (a^4+18 a^2 b^2-15 b^4\right ) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {e \cot (c+d x)}}{\sqrt {a} \sqrt {e}}\right )}{4 b^{3/2} d \left (a^2+b^2\right )^3} \]

[Out]

-1/2*(a-b)*(a^2+4*a*b+b^2)*e^(5/2)*arctan(1-2^(1/2)*(e*cot(d*x+c))^(1/2)/e^(1/2))/(a^2+b^2)^3/d*2^(1/2)+1/2*(a
-b)*(a^2+4*a*b+b^2)*e^(5/2)*arctan(1+2^(1/2)*(e*cot(d*x+c))^(1/2)/e^(1/2))/(a^2+b^2)^3/d*2^(1/2)+1/4*(a+b)*(a^
2-4*a*b+b^2)*e^(5/2)*ln(e^(1/2)+cot(d*x+c)*e^(1/2)-2^(1/2)*(e*cot(d*x+c))^(1/2))/(a^2+b^2)^3/d*2^(1/2)-1/4*(a+
b)*(a^2-4*a*b+b^2)*e^(5/2)*ln(e^(1/2)+cot(d*x+c)*e^(1/2)+2^(1/2)*(e*cot(d*x+c))^(1/2))/(a^2+b^2)^3/d*2^(1/2)-1
/4*(a^4+18*a^2*b^2-15*b^4)*e^(5/2)*arctan(b^(1/2)*(e*cot(d*x+c))^(1/2)/a^(1/2)/e^(1/2))*a^(1/2)/b^(3/2)/(a^2+b
^2)^3/d+1/2*a^2*e^2*(e*cot(d*x+c))^(1/2)/b/(a^2+b^2)/d/(a+b*cot(d*x+c))^2-1/4*a*(a^2+9*b^2)*e^2*(e*cot(d*x+c))
^(1/2)/b/(a^2+b^2)^2/d/(a+b*cot(d*x+c))

________________________________________________________________________________________

Rubi [A]  time = 1.29, antiderivative size = 470, normalized size of antiderivative = 1.00, number of steps used = 16, number of rules used = 13, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.520, Rules used = {3565, 3649, 3653, 3534, 1168, 1162, 617, 204, 1165, 628, 3634, 63, 205} \[ -\frac {a e^2 \left (a^2+9 b^2\right ) \sqrt {e \cot (c+d x)}}{4 b d \left (a^2+b^2\right )^2 (a+b \cot (c+d x))}+\frac {a^2 e^2 \sqrt {e \cot (c+d x)}}{2 b d \left (a^2+b^2\right ) (a+b \cot (c+d x))^2}+\frac {e^{5/2} (a+b) \left (a^2-4 a b+b^2\right ) \log \left (\sqrt {e} \cot (c+d x)-\sqrt {2} \sqrt {e \cot (c+d x)}+\sqrt {e}\right )}{2 \sqrt {2} d \left (a^2+b^2\right )^3}-\frac {e^{5/2} (a+b) \left (a^2-4 a b+b^2\right ) \log \left (\sqrt {e} \cot (c+d x)+\sqrt {2} \sqrt {e \cot (c+d x)}+\sqrt {e}\right )}{2 \sqrt {2} d \left (a^2+b^2\right )^3}-\frac {\sqrt {a} e^{5/2} \left (18 a^2 b^2+a^4-15 b^4\right ) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {e \cot (c+d x)}}{\sqrt {a} \sqrt {e}}\right )}{4 b^{3/2} d \left (a^2+b^2\right )^3}-\frac {e^{5/2} (a-b) \left (a^2+4 a b+b^2\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} d \left (a^2+b^2\right )^3}+\frac {e^{5/2} (a-b) \left (a^2+4 a b+b^2\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}+1\right )}{\sqrt {2} d \left (a^2+b^2\right )^3} \]

Antiderivative was successfully verified.

[In]

Int[(e*Cot[c + d*x])^(5/2)/(a + b*Cot[c + d*x])^3,x]

[Out]

-(Sqrt[a]*(a^4 + 18*a^2*b^2 - 15*b^4)*e^(5/2)*ArcTan[(Sqrt[b]*Sqrt[e*Cot[c + d*x]])/(Sqrt[a]*Sqrt[e])])/(4*b^(
3/2)*(a^2 + b^2)^3*d) - ((a - b)*(a^2 + 4*a*b + b^2)*e^(5/2)*ArcTan[1 - (Sqrt[2]*Sqrt[e*Cot[c + d*x]])/Sqrt[e]
])/(Sqrt[2]*(a^2 + b^2)^3*d) + ((a - b)*(a^2 + 4*a*b + b^2)*e^(5/2)*ArcTan[1 + (Sqrt[2]*Sqrt[e*Cot[c + d*x]])/
Sqrt[e]])/(Sqrt[2]*(a^2 + b^2)^3*d) + (a^2*e^2*Sqrt[e*Cot[c + d*x]])/(2*b*(a^2 + b^2)*d*(a + b*Cot[c + d*x])^2
) - (a*(a^2 + 9*b^2)*e^2*Sqrt[e*Cot[c + d*x]])/(4*b*(a^2 + b^2)^2*d*(a + b*Cot[c + d*x])) + ((a + b)*(a^2 - 4*
a*b + b^2)*e^(5/2)*Log[Sqrt[e] + Sqrt[e]*Cot[c + d*x] - Sqrt[2]*Sqrt[e*Cot[c + d*x]]])/(2*Sqrt[2]*(a^2 + b^2)^
3*d) - ((a + b)*(a^2 - 4*a*b + b^2)*e^(5/2)*Log[Sqrt[e] + Sqrt[e]*Cot[c + d*x] + Sqrt[2]*Sqrt[e*Cot[c + d*x]]]
)/(2*Sqrt[2]*(a^2 + b^2)^3*d)

Rule 63

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

Rule 204

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

Rule 205

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

Rule 617

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

Rule 628

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

Rule 1162

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

Rule 1165

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

Rule 1168

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

Rule 3534

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

Rule 3565

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

Rule 3634

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

Rule 3649

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_)*((A_.) + (B_.)*t
an[(e_.) + (f_.)*(x_)] + (C_.)*tan[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[((A*b^2 - a*(b*B - a*C))*(a + b*T
an[e + f*x])^(m + 1)*(c + d*Tan[e + f*x])^(n + 1))/(f*(m + 1)*(b*c - a*d)*(a^2 + b^2)), x] + Dist[1/((m + 1)*(
b*c - a*d)*(a^2 + b^2)), Int[(a + b*Tan[e + f*x])^(m + 1)*(c + d*Tan[e + f*x])^n*Simp[A*(a*(b*c - a*d)*(m + 1)
- b^2*d*(m + n + 2)) + (b*B - a*C)*(b*c*(m + 1) + a*d*(n + 1)) - (m + 1)*(b*c - a*d)*(A*b - a*B - b*C)*Tan[e
+ f*x] - d*(A*b^2 - a*(b*B - a*C))*(m + n + 2)*Tan[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C,
n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && LtQ[m, -1] &&  !(ILtQ[n, -1] && ( !I
ntegerQ[m] || (EqQ[c, 0] && NeQ[a, 0])))

Rule 3653

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

Rubi steps

\begin {align*} \int \frac {(e \cot (c+d x))^{5/2}}{(a+b \cot (c+d x))^3} \, dx &=\frac {a^2 e^2 \sqrt {e \cot (c+d x)}}{2 b \left (a^2+b^2\right ) d (a+b \cot (c+d x))^2}-\frac {\int \frac {-\frac {1}{2} a^2 e^3+2 a b e^3 \cot (c+d x)-\frac {1}{2} \left (a^2+4 b^2\right ) e^3 \cot ^2(c+d x)}{\sqrt {e \cot (c+d x)} (a+b \cot (c+d x))^2} \, dx}{2 b \left (a^2+b^2\right )}\\ &=\frac {a^2 e^2 \sqrt {e \cot (c+d x)}}{2 b \left (a^2+b^2\right ) d (a+b \cot (c+d x))^2}-\frac {a \left (a^2+9 b^2\right ) e^2 \sqrt {e \cot (c+d x)}}{4 b \left (a^2+b^2\right )^2 d (a+b \cot (c+d x))}+\frac {\int \frac {\frac {1}{4} a^2 \left (a^2-7 b^2\right ) e^4-2 a b \left (a^2-b^2\right ) e^4 \cot (c+d x)+\frac {1}{4} a^2 \left (a^2+9 b^2\right ) e^4 \cot ^2(c+d x)}{\sqrt {e \cot (c+d x)} (a+b \cot (c+d x))} \, dx}{2 a b \left (a^2+b^2\right )^2 e}\\ &=\frac {a^2 e^2 \sqrt {e \cot (c+d x)}}{2 b \left (a^2+b^2\right ) d (a+b \cot (c+d x))^2}-\frac {a \left (a^2+9 b^2\right ) e^2 \sqrt {e \cot (c+d x)}}{4 b \left (a^2+b^2\right )^2 d (a+b \cot (c+d x))}+\frac {\int \frac {-2 a b^2 \left (3 a^2-b^2\right ) e^4-2 a^2 b \left (a^2-3 b^2\right ) e^4 \cot (c+d x)}{\sqrt {e \cot (c+d x)}} \, dx}{2 a b \left (a^2+b^2\right )^3 e}+\frac {\left (a \left (a^4+18 a^2 b^2-15 b^4\right ) e^3\right ) \int \frac {1+\cot ^2(c+d x)}{\sqrt {e \cot (c+d x)} (a+b \cot (c+d x))} \, dx}{8 b \left (a^2+b^2\right )^3}\\ &=\frac {a^2 e^2 \sqrt {e \cot (c+d x)}}{2 b \left (a^2+b^2\right ) d (a+b \cot (c+d x))^2}-\frac {a \left (a^2+9 b^2\right ) e^2 \sqrt {e \cot (c+d x)}}{4 b \left (a^2+b^2\right )^2 d (a+b \cot (c+d x))}+\frac {\operatorname {Subst}\left (\int \frac {2 a b^2 \left (3 a^2-b^2\right ) e^5+2 a^2 b \left (a^2-3 b^2\right ) e^4 x^2}{e^2+x^4} \, dx,x,\sqrt {e \cot (c+d x)}\right )}{a b \left (a^2+b^2\right )^3 d e}+\frac {\left (a \left (a^4+18 a^2 b^2-15 b^4\right ) e^3\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {-e x} (a-b x)} \, dx,x,-\cot (c+d x)\right )}{8 b \left (a^2+b^2\right )^3 d}\\ &=\frac {a^2 e^2 \sqrt {e \cot (c+d x)}}{2 b \left (a^2+b^2\right ) d (a+b \cot (c+d x))^2}-\frac {a \left (a^2+9 b^2\right ) e^2 \sqrt {e \cot (c+d x)}}{4 b \left (a^2+b^2\right )^2 d (a+b \cot (c+d x))}-\frac {\left (a \left (a^4+18 a^2 b^2-15 b^4\right ) e^2\right ) \operatorname {Subst}\left (\int \frac {1}{a+\frac {b x^2}{e}} \, dx,x,\sqrt {e \cot (c+d x)}\right )}{4 b \left (a^2+b^2\right )^3 d}-\frac {\left ((a+b) \left (a^2-4 a b+b^2\right ) e^3\right ) \operatorname {Subst}\left (\int \frac {e-x^2}{e^2+x^4} \, dx,x,\sqrt {e \cot (c+d x)}\right )}{\left (a^2+b^2\right )^3 d}+\frac {\left ((a-b) \left (a^2+4 a b+b^2\right ) e^3\right ) \operatorname {Subst}\left (\int \frac {e+x^2}{e^2+x^4} \, dx,x,\sqrt {e \cot (c+d x)}\right )}{\left (a^2+b^2\right )^3 d}\\ &=-\frac {\sqrt {a} \left (a^4+18 a^2 b^2-15 b^4\right ) e^{5/2} \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {e \cot (c+d x)}}{\sqrt {a} \sqrt {e}}\right )}{4 b^{3/2} \left (a^2+b^2\right )^3 d}+\frac {a^2 e^2 \sqrt {e \cot (c+d x)}}{2 b \left (a^2+b^2\right ) d (a+b \cot (c+d x))^2}-\frac {a \left (a^2+9 b^2\right ) e^2 \sqrt {e \cot (c+d x)}}{4 b \left (a^2+b^2\right )^2 d (a+b \cot (c+d x))}+\frac {\left ((a+b) \left (a^2-4 a b+b^2\right ) e^{5/2}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {2} \sqrt {e}+2 x}{-e-\sqrt {2} \sqrt {e} x-x^2} \, dx,x,\sqrt {e \cot (c+d x)}\right )}{2 \sqrt {2} \left (a^2+b^2\right )^3 d}+\frac {\left ((a+b) \left (a^2-4 a b+b^2\right ) e^{5/2}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {2} \sqrt {e}-2 x}{-e+\sqrt {2} \sqrt {e} x-x^2} \, dx,x,\sqrt {e \cot (c+d x)}\right )}{2 \sqrt {2} \left (a^2+b^2\right )^3 d}+\frac {\left ((a-b) \left (a^2+4 a b+b^2\right ) e^3\right ) \operatorname {Subst}\left (\int \frac {1}{e-\sqrt {2} \sqrt {e} x+x^2} \, dx,x,\sqrt {e \cot (c+d x)}\right )}{2 \left (a^2+b^2\right )^3 d}+\frac {\left ((a-b) \left (a^2+4 a b+b^2\right ) e^3\right ) \operatorname {Subst}\left (\int \frac {1}{e+\sqrt {2} \sqrt {e} x+x^2} \, dx,x,\sqrt {e \cot (c+d x)}\right )}{2 \left (a^2+b^2\right )^3 d}\\ &=-\frac {\sqrt {a} \left (a^4+18 a^2 b^2-15 b^4\right ) e^{5/2} \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {e \cot (c+d x)}}{\sqrt {a} \sqrt {e}}\right )}{4 b^{3/2} \left (a^2+b^2\right )^3 d}+\frac {a^2 e^2 \sqrt {e \cot (c+d x)}}{2 b \left (a^2+b^2\right ) d (a+b \cot (c+d x))^2}-\frac {a \left (a^2+9 b^2\right ) e^2 \sqrt {e \cot (c+d x)}}{4 b \left (a^2+b^2\right )^2 d (a+b \cot (c+d x))}+\frac {(a+b) \left (a^2-4 a b+b^2\right ) e^{5/2} \log \left (\sqrt {e}+\sqrt {e} \cot (c+d x)-\sqrt {2} \sqrt {e \cot (c+d x)}\right )}{2 \sqrt {2} \left (a^2+b^2\right )^3 d}-\frac {(a+b) \left (a^2-4 a b+b^2\right ) e^{5/2} \log \left (\sqrt {e}+\sqrt {e} \cot (c+d x)+\sqrt {2} \sqrt {e \cot (c+d x)}\right )}{2 \sqrt {2} \left (a^2+b^2\right )^3 d}+\frac {\left ((a-b) \left (a^2+4 a b+b^2\right ) e^{5/2}\right ) \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} \left (a^2+b^2\right )^3 d}-\frac {\left ((a-b) \left (a^2+4 a b+b^2\right ) e^{5/2}\right ) \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} \left (a^2+b^2\right )^3 d}\\ &=-\frac {\sqrt {a} \left (a^4+18 a^2 b^2-15 b^4\right ) e^{5/2} \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {e \cot (c+d x)}}{\sqrt {a} \sqrt {e}}\right )}{4 b^{3/2} \left (a^2+b^2\right )^3 d}-\frac {(a-b) \left (a^2+4 a b+b^2\right ) e^{5/2} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} \left (a^2+b^2\right )^3 d}+\frac {(a-b) \left (a^2+4 a b+b^2\right ) e^{5/2} \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} \left (a^2+b^2\right )^3 d}+\frac {a^2 e^2 \sqrt {e \cot (c+d x)}}{2 b \left (a^2+b^2\right ) d (a+b \cot (c+d x))^2}-\frac {a \left (a^2+9 b^2\right ) e^2 \sqrt {e \cot (c+d x)}}{4 b \left (a^2+b^2\right )^2 d (a+b \cot (c+d x))}+\frac {(a+b) \left (a^2-4 a b+b^2\right ) e^{5/2} \log \left (\sqrt {e}+\sqrt {e} \cot (c+d x)-\sqrt {2} \sqrt {e \cot (c+d x)}\right )}{2 \sqrt {2} \left (a^2+b^2\right )^3 d}-\frac {(a+b) \left (a^2-4 a b+b^2\right ) e^{5/2} \log \left (\sqrt {e}+\sqrt {e} \cot (c+d x)+\sqrt {2} \sqrt {e \cot (c+d x)}\right )}{2 \sqrt {2} \left (a^2+b^2\right )^3 d}\\ \end {align*}

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Mathematica [C]  time = 6.20, size = 488, normalized size = 1.04 \[ -\frac {(e \cot (c+d x))^{5/2} \left (\frac {4 b^2 \cot ^{\frac {7}{2}}(c+d x) \, _2F_1\left (2,\frac {7}{2};\frac {9}{2};-\frac {b \cot (c+d x)}{a}\right )}{7 a \left (a^2+b^2\right )^2}+\frac {2 a \left (a^2-3 b^2\right ) \left (\cot ^{\frac {3}{2}}(c+d x)-\cot ^{\frac {3}{2}}(c+d x) \, _2F_1\left (\frac {3}{4},1;\frac {7}{4};-\cot ^2(c+d x)\right )\right )}{3 \left (a^2+b^2\right )^3}+\frac {2 b \left (3 a^2-b^2\right ) \cot ^{\frac {5}{2}}(c+d x)}{5 \left (a^2+b^2\right )^3}+\frac {b \left (3 a^2-b^2\right ) \left (-8 \cot ^{\frac {5}{2}}(c+d x)+40 \sqrt {\cot (c+d x)}+\frac {5}{2} \left (2 \sqrt {2} \log \left (\cot (c+d x)-\sqrt {2} \sqrt {\cot (c+d x)}+1\right )-2 \sqrt {2} \log \left (\cot (c+d x)+\sqrt {2} \sqrt {\cot (c+d x)}+1\right )+4 \left (\sqrt {2} \tan ^{-1}\left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )-\sqrt {2} \tan ^{-1}\left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )\right )\right )\right )}{20 \left (a^2+b^2\right )^3}-\frac {2 a \left (3 a^2-b^2\right ) \left (\cot ^{\frac {3}{2}}(c+d x)-3 a \left (\frac {\sqrt {\cot (c+d x)}}{b}-\frac {\sqrt {a} \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {\cot (c+d x)}}{\sqrt {a}}\right )}{b^{3/2}}\right )\right )}{3 \left (a^2+b^2\right )^3}+\frac {2 b^2 \cot ^{\frac {7}{2}}(c+d x) \, _2F_1\left (3,\frac {7}{2};\frac {9}{2};-\frac {b \cot (c+d x)}{a}\right )}{7 a^3 \left (a^2+b^2\right )}\right )}{d \cot ^{\frac {5}{2}}(c+d x)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(((e*Cot[c + d*x])^(5/2)*((2*b*(3*a^2 - b^2)*Cot[c + d*x]^(5/2))/(5*(a^2 + b^2)^3) - (2*a*(3*a^2 - b^2)*(-3*a
*(-((Sqrt[a]*ArcTan[(Sqrt[b]*Sqrt[Cot[c + d*x]])/Sqrt[a]])/b^(3/2)) + Sqrt[Cot[c + d*x]]/b) + Cot[c + d*x]^(3/
2)))/(3*(a^2 + b^2)^3) + (2*a*(a^2 - 3*b^2)*(Cot[c + d*x]^(3/2) - Cot[c + d*x]^(3/2)*Hypergeometric2F1[3/4, 1,
7/4, -Cot[c + d*x]^2]))/(3*(a^2 + b^2)^3) + (4*b^2*Cot[c + d*x]^(7/2)*Hypergeometric2F1[2, 7/2, 9/2, -((b*Cot
[c + d*x])/a)])/(7*a*(a^2 + b^2)^2) + (2*b^2*Cot[c + d*x]^(7/2)*Hypergeometric2F1[3, 7/2, 9/2, -((b*Cot[c + d*
x])/a)])/(7*a^3*(a^2 + b^2)) + (b*(3*a^2 - b^2)*(40*Sqrt[Cot[c + d*x]] - 8*Cot[c + d*x]^(5/2) + (5*(4*(Sqrt[2]
*ArcTan[1 - Sqrt[2]*Sqrt[Cot[c + d*x]]] - Sqrt[2]*ArcTan[1 + Sqrt[2]*Sqrt[Cot[c + d*x]]]) + 2*Sqrt[2]*Log[1 -
Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]] - 2*Sqrt[2]*Log[1 + Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]]))/2)
)/(20*(a^2 + b^2)^3)))/(d*Cot[c + d*x]^(5/2)))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [B]  time = 1.00, size = 1229, normalized size = 2.61 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4/d*e^3*a^5/(a^2+b^2)^3/(e*cot(d*x+c)*b+a*e)^2*(e*cot(d*x+c))^(3/2)-5/2/d*e^3*a^3/(a^2+b^2)^3/(e*cot(d*x+c)
*b+a*e)^2*(e*cot(d*x+c))^(3/2)*b^2-9/4/d*e^3*a/(a^2+b^2)^3/(e*cot(d*x+c)*b+a*e)^2*(e*cot(d*x+c))^(3/2)*b^4+1/4
/d*e^4*a^6/(a^2+b^2)^3/(e*cot(d*x+c)*b+a*e)^2/b*(e*cot(d*x+c))^(1/2)-3/2/d*e^4*a^4/(a^2+b^2)^3/(e*cot(d*x+c)*b
+a*e)^2*b*(e*cot(d*x+c))^(1/2)-7/4/d*e^4*a^2/(a^2+b^2)^3/(e*cot(d*x+c)*b+a*e)^2*b^3*(e*cot(d*x+c))^(1/2)-1/4/d
*e^3*a^5/(a^2+b^2)^3/b/(a*e*b)^(1/2)*arctan((e*cot(d*x+c))^(1/2)*b/(a*e*b)^(1/2))-9/2/d*e^3*a^3/(a^2+b^2)^3*b/
(a*e*b)^(1/2)*arctan((e*cot(d*x+c))^(1/2)*b/(a*e*b)^(1/2))+15/4/d*e^3*a/(a^2+b^2)^3*b^3/(a*e*b)^(1/2)*arctan((
e*cot(d*x+c))^(1/2)*b/(a*e*b)^(1/2))+3/2/d*e^2/(a^2+b^2)^3*(e^2)^(1/4)*2^(1/2)*arctan(2^(1/2)/(e^2)^(1/4)*(e*c
ot(d*x+c))^(1/2)+1)*a^2*b-1/2/d*e^2/(a^2+b^2)^3*(e^2)^(1/4)*2^(1/2)*arctan(2^(1/2)/(e^2)^(1/4)*(e*cot(d*x+c))^
(1/2)+1)*b^3-3/2/d*e^2/(a^2+b^2)^3*(e^2)^(1/4)*2^(1/2)*arctan(-2^(1/2)/(e^2)^(1/4)*(e*cot(d*x+c))^(1/2)+1)*a^2
*b+1/2/d*e^2/(a^2+b^2)^3*(e^2)^(1/4)*2^(1/2)*arctan(-2^(1/2)/(e^2)^(1/4)*(e*cot(d*x+c))^(1/2)+1)*b^3+3/4/d*e^2
/(a^2+b^2)^3*(e^2)^(1/4)*2^(1/2)*ln((e*cot(d*x+c)+(e^2)^(1/4)*(e*cot(d*x+c))^(1/2)*2^(1/2)+(e^2)^(1/2))/(e*cot
(d*x+c)-(e^2)^(1/4)*(e*cot(d*x+c))^(1/2)*2^(1/2)+(e^2)^(1/2)))*a^2*b-1/4/d*e^2/(a^2+b^2)^3*(e^2)^(1/4)*2^(1/2)
*ln((e*cot(d*x+c)+(e^2)^(1/4)*(e*cot(d*x+c))^(1/2)*2^(1/2)+(e^2)^(1/2))/(e*cot(d*x+c)-(e^2)^(1/4)*(e*cot(d*x+c
))^(1/2)*2^(1/2)+(e^2)^(1/2)))*b^3+1/2/d*e^3/(a^2+b^2)^3*2^(1/2)/(e^2)^(1/4)*arctan(2^(1/2)/(e^2)^(1/4)*(e*cot
(d*x+c))^(1/2)+1)*a^3-3/2/d*e^3/(a^2+b^2)^3*2^(1/2)/(e^2)^(1/4)*arctan(2^(1/2)/(e^2)^(1/4)*(e*cot(d*x+c))^(1/2
)+1)*a*b^2-1/2/d*e^3/(a^2+b^2)^3*2^(1/2)/(e^2)^(1/4)*arctan(-2^(1/2)/(e^2)^(1/4)*(e*cot(d*x+c))^(1/2)+1)*a^3+3
/2/d*e^3/(a^2+b^2)^3*2^(1/2)/(e^2)^(1/4)*arctan(-2^(1/2)/(e^2)^(1/4)*(e*cot(d*x+c))^(1/2)+1)*a*b^2+1/4/d*e^3/(
a^2+b^2)^3*2^(1/2)/(e^2)^(1/4)*ln((e*cot(d*x+c)-(e^2)^(1/4)*(e*cot(d*x+c))^(1/2)*2^(1/2)+(e^2)^(1/2))/(e*cot(d
*x+c)+(e^2)^(1/4)*(e*cot(d*x+c))^(1/2)*2^(1/2)+(e^2)^(1/2)))*a^3-3/4/d*e^3/(a^2+b^2)^3*2^(1/2)/(e^2)^(1/4)*ln(
(e*cot(d*x+c)-(e^2)^(1/4)*(e*cot(d*x+c))^(1/2)*2^(1/2)+(e^2)^(1/2))/(e*cot(d*x+c)+(e^2)^(1/4)*(e*cot(d*x+c))^(
1/2)*2^(1/2)+(e^2)^(1/2)))*a*b^2

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maxima [A]  time = 0.56, size = 505, normalized size = 1.07 \[ -\frac {{\left (\frac {{\left (a^{5} + 18 \, a^{3} b^{2} - 15 \, a b^{4}\right )} e^{2} \arctan \left (\frac {b \sqrt {\frac {e}{\tan \left (d x + c\right )}}}{\sqrt {a b e}}\right )}{{\left (a^{6} b + 3 \, a^{4} b^{3} + 3 \, a^{2} b^{5} + b^{7}\right )} \sqrt {a b e}} - \frac {{\left (\frac {2 \, \sqrt {2} {\left (a^{3} + 3 \, a^{2} b - 3 \, a b^{2} - b^{3}\right )} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \sqrt {e} + 2 \, \sqrt {\frac {e}{\tan \left (d x + c\right )}}\right )}}{2 \, \sqrt {e}}\right )}{\sqrt {e}} + \frac {2 \, \sqrt {2} {\left (a^{3} + 3 \, a^{2} b - 3 \, a b^{2} - b^{3}\right )} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \sqrt {e} - 2 \, \sqrt {\frac {e}{\tan \left (d x + c\right )}}\right )}}{2 \, \sqrt {e}}\right )}{\sqrt {e}} - \frac {\sqrt {2} {\left (a^{3} - 3 \, a^{2} b - 3 \, a b^{2} + b^{3}\right )} \log \left (\sqrt {2} \sqrt {e} \sqrt {\frac {e}{\tan \left (d x + c\right )}} + e + \frac {e}{\tan \left (d x + c\right )}\right )}{\sqrt {e}} + \frac {\sqrt {2} {\left (a^{3} - 3 \, a^{2} b - 3 \, a b^{2} + b^{3}\right )} \log \left (-\sqrt {2} \sqrt {e} \sqrt {\frac {e}{\tan \left (d x + c\right )}} + e + \frac {e}{\tan \left (d x + c\right )}\right )}{\sqrt {e}}\right )} e^{2}}{a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}} - \frac {{\left (a^{4} - 7 \, a^{2} b^{2}\right )} e^{3} \sqrt {\frac {e}{\tan \left (d x + c\right )}} - {\left (a^{3} b + 9 \, a b^{3}\right )} e^{2} \left (\frac {e}{\tan \left (d x + c\right )}\right )^{\frac {3}{2}}}{{\left (a^{6} b + 2 \, a^{4} b^{3} + a^{2} b^{5}\right )} e^{2} + \frac {2 \, {\left (a^{5} b^{2} + 2 \, a^{3} b^{4} + a b^{6}\right )} e^{2}}{\tan \left (d x + c\right )} + \frac {{\left (a^{4} b^{3} + 2 \, a^{2} b^{5} + b^{7}\right )} e^{2}}{\tan \left (d x + c\right )^{2}}}\right )} e}{4 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*((a^5 + 18*a^3*b^2 - 15*a*b^4)*e^2*arctan(b*sqrt(e/tan(d*x + c))/sqrt(a*b*e))/((a^6*b + 3*a^4*b^3 + 3*a^2
*b^5 + b^7)*sqrt(a*b*e)) - (2*sqrt(2)*(a^3 + 3*a^2*b - 3*a*b^2 - b^3)*arctan(1/2*sqrt(2)*(sqrt(2)*sqrt(e) + 2*
sqrt(e/tan(d*x + c)))/sqrt(e))/sqrt(e) + 2*sqrt(2)*(a^3 + 3*a^2*b - 3*a*b^2 - b^3)*arctan(-1/2*sqrt(2)*(sqrt(2
)*sqrt(e) - 2*sqrt(e/tan(d*x + c)))/sqrt(e))/sqrt(e) - sqrt(2)*(a^3 - 3*a^2*b - 3*a*b^2 + b^3)*log(sqrt(2)*sqr
t(e)*sqrt(e/tan(d*x + c)) + e + e/tan(d*x + c))/sqrt(e) + sqrt(2)*(a^3 - 3*a^2*b - 3*a*b^2 + b^3)*log(-sqrt(2)
*sqrt(e)*sqrt(e/tan(d*x + c)) + e + e/tan(d*x + c))/sqrt(e))*e^2/(a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6) - ((a^4 -
7*a^2*b^2)*e^3*sqrt(e/tan(d*x + c)) - (a^3*b + 9*a*b^3)*e^2*(e/tan(d*x + c))^(3/2))/((a^6*b + 2*a^4*b^3 + a^2
*b^5)*e^2 + 2*(a^5*b^2 + 2*a^3*b^4 + a*b^6)*e^2/tan(d*x + c) + (a^4*b^3 + 2*a^2*b^5 + b^7)*e^2/tan(d*x + c)^2)
)*e/d

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mupad [B]  time = 6.51, size = 19256, normalized size = 40.97 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

atan(((((10*a^16*b*d^2*e^18 - 2398*a^2*b^15*d^2*e^18 + 5238*a^4*b^13*d^2*e^18 + 7386*a^6*b^11*d^2*e^18 - 8322*
a^8*b^9*d^2*e^18 - 5498*a^10*b^7*d^2*e^18 + 2946*a^12*b^5*d^2*e^18 + 382*a^14*b^3*d^2*e^18)/(b^17*d^5 + a^16*b
*d^5 + 8*a^2*b^15*d^5 + 28*a^4*b^13*d^5 + 56*a^6*b^11*d^5 + 70*a^8*b^9*d^5 + 56*a^10*b^7*d^5 + 28*a^12*b^5*d^5
+ 8*a^14*b^3*d^5) - (((832*a*b^22*d^4*e^13 + 5952*a^3*b^20*d^4*e^13 + 17664*a^5*b^18*d^4*e^13 + 26880*a^7*b^1
6*d^4*e^13 + 18816*a^9*b^14*d^4*e^13 - 2688*a^11*b^12*d^4*e^13 - 16128*a^13*b^10*d^4*e^13 - 13056*a^15*b^8*d^4
*e^13 - 4800*a^17*b^6*d^4*e^13 - 704*a^19*b^4*d^4*e^13)/(b^17*d^5 + a^16*b*d^5 + 8*a^2*b^15*d^5 + 28*a^4*b^13*
d^5 + 56*a^6*b^11*d^5 + 70*a^8*b^9*d^5 + 56*a^10*b^7*d^5 + 28*a^12*b^5*d^5 + 8*a^14*b^3*d^5) + ((e*cot(c + d*x
))^(1/2)*(-(e^5*1i)/(4*(b^6*d^2 - a^6*d^2 + a*b^5*d^2*6i + a^5*b*d^2*6i - 15*a^2*b^4*d^2 - a^3*b^3*d^2*20i + 1
5*a^4*b^2*d^2)))^(1/2)*(512*b^26*d^4*e^10 + 4608*a^2*b^24*d^4*e^10 + 17920*a^4*b^22*d^4*e^10 + 38400*a^6*b^20*
d^4*e^10 + 46080*a^8*b^18*d^4*e^10 + 21504*a^10*b^16*d^4*e^10 - 21504*a^12*b^14*d^4*e^10 - 46080*a^14*b^12*d^4
*e^10 - 38400*a^16*b^10*d^4*e^10 - 17920*a^18*b^8*d^4*e^10 - 4608*a^20*b^6*d^4*e^10 - 512*a^22*b^4*d^4*e^10))/
(b^17*d^4 + a^16*b*d^4 + 8*a^2*b^15*d^4 + 28*a^4*b^13*d^4 + 56*a^6*b^11*d^4 + 70*a^8*b^9*d^4 + 56*a^10*b^7*d^4
+ 28*a^12*b^5*d^4 + 8*a^14*b^3*d^4))*(-(e^5*1i)/(4*(b^6*d^2 - a^6*d^2 + a*b^5*d^2*6i + a^5*b*d^2*6i - 15*a^2*
b^4*d^2 - a^3*b^3*d^2*20i + 15*a^4*b^2*d^2)))^(1/2) - ((e*cot(c + d*x))^(1/2)*(8*a^19*b*d^2*e^15 - 1472*a*b^19
*d^2*e^15 + 776*a^3*b^17*d^2*e^15 + 11328*a^5*b^15*d^2*e^15 + 10208*a^7*b^13*d^2*e^15 - 5056*a^9*b^11*d^2*e^15
- 5328*a^11*b^9*d^2*e^15 + 4032*a^13*b^7*d^2*e^15 + 3552*a^15*b^5*d^2*e^15 + 384*a^17*b^3*d^2*e^15))/(b^17*d^
4 + a^16*b*d^4 + 8*a^2*b^15*d^4 + 28*a^4*b^13*d^4 + 56*a^6*b^11*d^4 + 70*a^8*b^9*d^4 + 56*a^10*b^7*d^4 + 28*a^
12*b^5*d^4 + 8*a^14*b^3*d^4))*(-(e^5*1i)/(4*(b^6*d^2 - a^6*d^2 + a*b^5*d^2*6i + a^5*b*d^2*6i - 15*a^2*b^4*d^2
- a^3*b^3*d^2*20i + 15*a^4*b^2*d^2)))^(1/2))*(-(e^5*1i)/(4*(b^6*d^2 - a^6*d^2 + a*b^5*d^2*6i + a^5*b*d^2*6i -
15*a^2*b^4*d^2 - a^3*b^3*d^2*20i + 15*a^4*b^2*d^2)))^(1/2) + ((e*cot(c + d*x))^(1/2)*(a^14*e^20 - 32*b^14*e^20
+ 97*a^2*b^12*e^20 - 2082*a^4*b^10*e^20 + 3631*a^6*b^8*e^20 - 2300*a^8*b^6*e^20 + 79*a^10*b^4*e^20 + 30*a^12*
b^2*e^20))/(b^17*d^4 + a^16*b*d^4 + 8*a^2*b^15*d^4 + 28*a^4*b^13*d^4 + 56*a^6*b^11*d^4 + 70*a^8*b^9*d^4 + 56*a
^10*b^7*d^4 + 28*a^12*b^5*d^4 + 8*a^14*b^3*d^4))*(-(e^5*1i)/(4*(b^6*d^2 - a^6*d^2 + a*b^5*d^2*6i + a^5*b*d^2*6
i - 15*a^2*b^4*d^2 - a^3*b^3*d^2*20i + 15*a^4*b^2*d^2)))^(1/2)*1i - (((10*a^16*b*d^2*e^18 - 2398*a^2*b^15*d^2*
e^18 + 5238*a^4*b^13*d^2*e^18 + 7386*a^6*b^11*d^2*e^18 - 8322*a^8*b^9*d^2*e^18 - 5498*a^10*b^7*d^2*e^18 + 2946
*a^12*b^5*d^2*e^18 + 382*a^14*b^3*d^2*e^18)/(b^17*d^5 + a^16*b*d^5 + 8*a^2*b^15*d^5 + 28*a^4*b^13*d^5 + 56*a^6
*b^11*d^5 + 70*a^8*b^9*d^5 + 56*a^10*b^7*d^5 + 28*a^12*b^5*d^5 + 8*a^14*b^3*d^5) - (((832*a*b^22*d^4*e^13 + 59
52*a^3*b^20*d^4*e^13 + 17664*a^5*b^18*d^4*e^13 + 26880*a^7*b^16*d^4*e^13 + 18816*a^9*b^14*d^4*e^13 - 2688*a^11
*b^12*d^4*e^13 - 16128*a^13*b^10*d^4*e^13 - 13056*a^15*b^8*d^4*e^13 - 4800*a^17*b^6*d^4*e^13 - 704*a^19*b^4*d^
4*e^13)/(b^17*d^5 + a^16*b*d^5 + 8*a^2*b^15*d^5 + 28*a^4*b^13*d^5 + 56*a^6*b^11*d^5 + 70*a^8*b^9*d^5 + 56*a^10
*b^7*d^5 + 28*a^12*b^5*d^5 + 8*a^14*b^3*d^5) - ((e*cot(c + d*x))^(1/2)*(-(e^5*1i)/(4*(b^6*d^2 - a^6*d^2 + a*b^
5*d^2*6i + a^5*b*d^2*6i - 15*a^2*b^4*d^2 - a^3*b^3*d^2*20i + 15*a^4*b^2*d^2)))^(1/2)*(512*b^26*d^4*e^10 + 4608
*a^2*b^24*d^4*e^10 + 17920*a^4*b^22*d^4*e^10 + 38400*a^6*b^20*d^4*e^10 + 46080*a^8*b^18*d^4*e^10 + 21504*a^10*
b^16*d^4*e^10 - 21504*a^12*b^14*d^4*e^10 - 46080*a^14*b^12*d^4*e^10 - 38400*a^16*b^10*d^4*e^10 - 17920*a^18*b^
8*d^4*e^10 - 4608*a^20*b^6*d^4*e^10 - 512*a^22*b^4*d^4*e^10))/(b^17*d^4 + a^16*b*d^4 + 8*a^2*b^15*d^4 + 28*a^4
*b^13*d^4 + 56*a^6*b^11*d^4 + 70*a^8*b^9*d^4 + 56*a^10*b^7*d^4 + 28*a^12*b^5*d^4 + 8*a^14*b^3*d^4))*(-(e^5*1i)
/(4*(b^6*d^2 - a^6*d^2 + a*b^5*d^2*6i + a^5*b*d^2*6i - 15*a^2*b^4*d^2 - a^3*b^3*d^2*20i + 15*a^4*b^2*d^2)))^(1
/2) + ((e*cot(c + d*x))^(1/2)*(8*a^19*b*d^2*e^15 - 1472*a*b^19*d^2*e^15 + 776*a^3*b^17*d^2*e^15 + 11328*a^5*b^
15*d^2*e^15 + 10208*a^7*b^13*d^2*e^15 - 5056*a^9*b^11*d^2*e^15 - 5328*a^11*b^9*d^2*e^15 + 4032*a^13*b^7*d^2*e^
15 + 3552*a^15*b^5*d^2*e^15 + 384*a^17*b^3*d^2*e^15))/(b^17*d^4 + a^16*b*d^4 + 8*a^2*b^15*d^4 + 28*a^4*b^13*d^
4 + 56*a^6*b^11*d^4 + 70*a^8*b^9*d^4 + 56*a^10*b^7*d^4 + 28*a^12*b^5*d^4 + 8*a^14*b^3*d^4))*(-(e^5*1i)/(4*(b^6
*d^2 - a^6*d^2 + a*b^5*d^2*6i + a^5*b*d^2*6i - 15*a^2*b^4*d^2 - a^3*b^3*d^2*20i + 15*a^4*b^2*d^2)))^(1/2))*(-(
e^5*1i)/(4*(b^6*d^2 - a^6*d^2 + a*b^5*d^2*6i + a^5*b*d^2*6i - 15*a^2*b^4*d^2 - a^3*b^3*d^2*20i + 15*a^4*b^2*d^
2)))^(1/2) - ((e*cot(c + d*x))^(1/2)*(a^14*e^20 - 32*b^14*e^20 + 97*a^2*b^12*e^20 - 2082*a^4*b^10*e^20 + 3631*
a^6*b^8*e^20 - 2300*a^8*b^6*e^20 + 79*a^10*b^4*e^20 + 30*a^12*b^2*e^20))/(b^17*d^4 + a^16*b*d^4 + 8*a^2*b^15*d
^4 + 28*a^4*b^13*d^4 + 56*a^6*b^11*d^4 + 70*a^8*b^9*d^4 + 56*a^10*b^7*d^4 + 28*a^12*b^5*d^4 + 8*a^14*b^3*d^4))
*(-(e^5*1i)/(4*(b^6*d^2 - a^6*d^2 + a*b^5*d^2*6i + a^5*b*d^2*6i - 15*a^2*b^4*d^2 - a^3*b^3*d^2*20i + 15*a^4*b^
2*d^2)))^(1/2)*1i)/((((10*a^16*b*d^2*e^18 - 2398*a^2*b^15*d^2*e^18 + 5238*a^4*b^13*d^2*e^18 + 7386*a^6*b^11*d^
2*e^18 - 8322*a^8*b^9*d^2*e^18 - 5498*a^10*b^7*d^2*e^18 + 2946*a^12*b^5*d^2*e^18 + 382*a^14*b^3*d^2*e^18)/(b^1
7*d^5 + a^16*b*d^5 + 8*a^2*b^15*d^5 + 28*a^4*b^13*d^5 + 56*a^6*b^11*d^5 + 70*a^8*b^9*d^5 + 56*a^10*b^7*d^5 + 2
8*a^12*b^5*d^5 + 8*a^14*b^3*d^5) - (((832*a*b^22*d^4*e^13 + 5952*a^3*b^20*d^4*e^13 + 17664*a^5*b^18*d^4*e^13 +
26880*a^7*b^16*d^4*e^13 + 18816*a^9*b^14*d^4*e^13 - 2688*a^11*b^12*d^4*e^13 - 16128*a^13*b^10*d^4*e^13 - 1305
6*a^15*b^8*d^4*e^13 - 4800*a^17*b^6*d^4*e^13 - 704*a^19*b^4*d^4*e^13)/(b^17*d^5 + a^16*b*d^5 + 8*a^2*b^15*d^5
+ 28*a^4*b^13*d^5 + 56*a^6*b^11*d^5 + 70*a^8*b^9*d^5 + 56*a^10*b^7*d^5 + 28*a^12*b^5*d^5 + 8*a^14*b^3*d^5) + (
(e*cot(c + d*x))^(1/2)*(-(e^5*1i)/(4*(b^6*d^2 - a^6*d^2 + a*b^5*d^2*6i + a^5*b*d^2*6i - 15*a^2*b^4*d^2 - a^3*b
^3*d^2*20i + 15*a^4*b^2*d^2)))^(1/2)*(512*b^26*d^4*e^10 + 4608*a^2*b^24*d^4*e^10 + 17920*a^4*b^22*d^4*e^10 + 3
8400*a^6*b^20*d^4*e^10 + 46080*a^8*b^18*d^4*e^10 + 21504*a^10*b^16*d^4*e^10 - 21504*a^12*b^14*d^4*e^10 - 46080
*a^14*b^12*d^4*e^10 - 38400*a^16*b^10*d^4*e^10 - 17920*a^18*b^8*d^4*e^10 - 4608*a^20*b^6*d^4*e^10 - 512*a^22*b
^4*d^4*e^10))/(b^17*d^4 + a^16*b*d^4 + 8*a^2*b^15*d^4 + 28*a^4*b^13*d^4 + 56*a^6*b^11*d^4 + 70*a^8*b^9*d^4 + 5
6*a^10*b^7*d^4 + 28*a^12*b^5*d^4 + 8*a^14*b^3*d^4))*(-(e^5*1i)/(4*(b^6*d^2 - a^6*d^2 + a*b^5*d^2*6i + a^5*b*d^
2*6i - 15*a^2*b^4*d^2 - a^3*b^3*d^2*20i + 15*a^4*b^2*d^2)))^(1/2) - ((e*cot(c + d*x))^(1/2)*(8*a^19*b*d^2*e^15
- 1472*a*b^19*d^2*e^15 + 776*a^3*b^17*d^2*e^15 + 11328*a^5*b^15*d^2*e^15 + 10208*a^7*b^13*d^2*e^15 - 5056*a^9
*b^11*d^2*e^15 - 5328*a^11*b^9*d^2*e^15 + 4032*a^13*b^7*d^2*e^15 + 3552*a^15*b^5*d^2*e^15 + 384*a^17*b^3*d^2*e
^15))/(b^17*d^4 + a^16*b*d^4 + 8*a^2*b^15*d^4 + 28*a^4*b^13*d^4 + 56*a^6*b^11*d^4 + 70*a^8*b^9*d^4 + 56*a^10*b
^7*d^4 + 28*a^12*b^5*d^4 + 8*a^14*b^3*d^4))*(-(e^5*1i)/(4*(b^6*d^2 - a^6*d^2 + a*b^5*d^2*6i + a^5*b*d^2*6i - 1
5*a^2*b^4*d^2 - a^3*b^3*d^2*20i + 15*a^4*b^2*d^2)))^(1/2))*(-(e^5*1i)/(4*(b^6*d^2 - a^6*d^2 + a*b^5*d^2*6i + a
^5*b*d^2*6i - 15*a^2*b^4*d^2 - a^3*b^3*d^2*20i + 15*a^4*b^2*d^2)))^(1/2) + ((e*cot(c + d*x))^(1/2)*(a^14*e^20
- 32*b^14*e^20 + 97*a^2*b^12*e^20 - 2082*a^4*b^10*e^20 + 3631*a^6*b^8*e^20 - 2300*a^8*b^6*e^20 + 79*a^10*b^4*e
^20 + 30*a^12*b^2*e^20))/(b^17*d^4 + a^16*b*d^4 + 8*a^2*b^15*d^4 + 28*a^4*b^13*d^4 + 56*a^6*b^11*d^4 + 70*a^8*
b^9*d^4 + 56*a^10*b^7*d^4 + 28*a^12*b^5*d^4 + 8*a^14*b^3*d^4))*(-(e^5*1i)/(4*(b^6*d^2 - a^6*d^2 + a*b^5*d^2*6i
+ a^5*b*d^2*6i - 15*a^2*b^4*d^2 - a^3*b^3*d^2*20i + 15*a^4*b^2*d^2)))^(1/2) + (((10*a^16*b*d^2*e^18 - 2398*a^
2*b^15*d^2*e^18 + 5238*a^4*b^13*d^2*e^18 + 7386*a^6*b^11*d^2*e^18 - 8322*a^8*b^9*d^2*e^18 - 5498*a^10*b^7*d^2*
e^18 + 2946*a^12*b^5*d^2*e^18 + 382*a^14*b^3*d^2*e^18)/(b^17*d^5 + a^16*b*d^5 + 8*a^2*b^15*d^5 + 28*a^4*b^13*d
^5 + 56*a^6*b^11*d^5 + 70*a^8*b^9*d^5 + 56*a^10*b^7*d^5 + 28*a^12*b^5*d^5 + 8*a^14*b^3*d^5) - (((832*a*b^22*d^
4*e^13 + 5952*a^3*b^20*d^4*e^13 + 17664*a^5*b^18*d^4*e^13 + 26880*a^7*b^16*d^4*e^13 + 18816*a^9*b^14*d^4*e^13
- 2688*a^11*b^12*d^4*e^13 - 16128*a^13*b^10*d^4*e^13 - 13056*a^15*b^8*d^4*e^13 - 4800*a^17*b^6*d^4*e^13 - 704*
a^19*b^4*d^4*e^13)/(b^17*d^5 + a^16*b*d^5 + 8*a^2*b^15*d^5 + 28*a^4*b^13*d^5 + 56*a^6*b^11*d^5 + 70*a^8*b^9*d^
5 + 56*a^10*b^7*d^5 + 28*a^12*b^5*d^5 + 8*a^14*b^3*d^5) - ((e*cot(c + d*x))^(1/2)*(-(e^5*1i)/(4*(b^6*d^2 - a^6
*d^2 + a*b^5*d^2*6i + a^5*b*d^2*6i - 15*a^2*b^4*d^2 - a^3*b^3*d^2*20i + 15*a^4*b^2*d^2)))^(1/2)*(512*b^26*d^4*
e^10 + 4608*a^2*b^24*d^4*e^10 + 17920*a^4*b^22*d^4*e^10 + 38400*a^6*b^20*d^4*e^10 + 46080*a^8*b^18*d^4*e^10 +
21504*a^10*b^16*d^4*e^10 - 21504*a^12*b^14*d^4*e^10 - 46080*a^14*b^12*d^4*e^10 - 38400*a^16*b^10*d^4*e^10 - 17
920*a^18*b^8*d^4*e^10 - 4608*a^20*b^6*d^4*e^10 - 512*a^22*b^4*d^4*e^10))/(b^17*d^4 + a^16*b*d^4 + 8*a^2*b^15*d
^4 + 28*a^4*b^13*d^4 + 56*a^6*b^11*d^4 + 70*a^8*b^9*d^4 + 56*a^10*b^7*d^4 + 28*a^12*b^5*d^4 + 8*a^14*b^3*d^4))
*(-(e^5*1i)/(4*(b^6*d^2 - a^6*d^2 + a*b^5*d^2*6i + a^5*b*d^2*6i - 15*a^2*b^4*d^2 - a^3*b^3*d^2*20i + 15*a^4*b^
2*d^2)))^(1/2) + ((e*cot(c + d*x))^(1/2)*(8*a^19*b*d^2*e^15 - 1472*a*b^19*d^2*e^15 + 776*a^3*b^17*d^2*e^15 + 1
1328*a^5*b^15*d^2*e^15 + 10208*a^7*b^13*d^2*e^15 - 5056*a^9*b^11*d^2*e^15 - 5328*a^11*b^9*d^2*e^15 + 4032*a^13
*b^7*d^2*e^15 + 3552*a^15*b^5*d^2*e^15 + 384*a^17*b^3*d^2*e^15))/(b^17*d^4 + a^16*b*d^4 + 8*a^2*b^15*d^4 + 28*
a^4*b^13*d^4 + 56*a^6*b^11*d^4 + 70*a^8*b^9*d^4 + 56*a^10*b^7*d^4 + 28*a^12*b^5*d^4 + 8*a^14*b^3*d^4))*(-(e^5*
1i)/(4*(b^6*d^2 - a^6*d^2 + a*b^5*d^2*6i + a^5*b*d^2*6i - 15*a^2*b^4*d^2 - a^3*b^3*d^2*20i + 15*a^4*b^2*d^2)))
^(1/2))*(-(e^5*1i)/(4*(b^6*d^2 - a^6*d^2 + a*b^5*d^2*6i + a^5*b*d^2*6i - 15*a^2*b^4*d^2 - a^3*b^3*d^2*20i + 15
*a^4*b^2*d^2)))^(1/2) - ((e*cot(c + d*x))^(1/2)*(a^14*e^20 - 32*b^14*e^20 + 97*a^2*b^12*e^20 - 2082*a^4*b^10*e
^20 + 3631*a^6*b^8*e^20 - 2300*a^8*b^6*e^20 + 79*a^10*b^4*e^20 + 30*a^12*b^2*e^20))/(b^17*d^4 + a^16*b*d^4 + 8
*a^2*b^15*d^4 + 28*a^4*b^13*d^4 + 56*a^6*b^11*d^4 + 70*a^8*b^9*d^4 + 56*a^10*b^7*d^4 + 28*a^12*b^5*d^4 + 8*a^1
4*b^3*d^4))*(-(e^5*1i)/(4*(b^6*d^2 - a^6*d^2 + a*b^5*d^2*6i + a^5*b*d^2*6i - 15*a^2*b^4*d^2 - a^3*b^3*d^2*20i
+ 15*a^4*b^2*d^2)))^(1/2) + (a^11*e^23 - 120*a*b^10*e^23 + 249*a^3*b^8*e^23 - 388*a^5*b^6*e^23 + 302*a^7*b^4*e
^23 + 36*a^9*b^2*e^23)/(b^17*d^5 + a^16*b*d^5 + 8*a^2*b^15*d^5 + 28*a^4*b^13*d^5 + 56*a^6*b^11*d^5 + 70*a^8*b^
9*d^5 + 56*a^10*b^7*d^5 + 28*a^12*b^5*d^5 + 8*a^14*b^3*d^5)))*(-(e^5*1i)/(4*(b^6*d^2 - a^6*d^2 + a*b^5*d^2*6i
+ a^5*b*d^2*6i - 15*a^2*b^4*d^2 - a^3*b^3*d^2*20i + 15*a^4*b^2*d^2)))^(1/2)*2i - ((e^3*(e*cot(c + d*x))^(3/2)*
(9*a*b^2 + a^3))/(4*(a^4 + b^4 + 2*a^2*b^2)) - ((e*cot(c + d*x))^(1/2)*(a^4*e^4 - 7*a^2*b^2*e^4))/(4*b*(a^4 +
b^4 + 2*a^2*b^2)))/(a^2*d*e^2 + b^2*d*e^2*cot(c + d*x)^2 + 2*a*b*d*e^2*cot(c + d*x)) + atan(((((10*a^16*b*d^2*
e^18 - 2398*a^2*b^15*d^2*e^18 + 5238*a^4*b^13*d^2*e^18 + 7386*a^6*b^11*d^2*e^18 - 8322*a^8*b^9*d^2*e^18 - 5498
*a^10*b^7*d^2*e^18 + 2946*a^12*b^5*d^2*e^18 + 382*a^14*b^3*d^2*e^18)/(b^17*d^5 + a^16*b*d^5 + 8*a^2*b^15*d^5 +
28*a^4*b^13*d^5 + 56*a^6*b^11*d^5 + 70*a^8*b^9*d^5 + 56*a^10*b^7*d^5 + 28*a^12*b^5*d^5 + 8*a^14*b^3*d^5) - ((
(832*a*b^22*d^4*e^13 + 5952*a^3*b^20*d^4*e^13 + 17664*a^5*b^18*d^4*e^13 + 26880*a^7*b^16*d^4*e^13 + 18816*a^9*
b^14*d^4*e^13 - 2688*a^11*b^12*d^4*e^13 - 16128*a^13*b^10*d^4*e^13 - 13056*a^15*b^8*d^4*e^13 - 4800*a^17*b^6*d
^4*e^13 - 704*a^19*b^4*d^4*e^13)/(b^17*d^5 + a^16*b*d^5 + 8*a^2*b^15*d^5 + 28*a^4*b^13*d^5 + 56*a^6*b^11*d^5 +
70*a^8*b^9*d^5 + 56*a^10*b^7*d^5 + 28*a^12*b^5*d^5 + 8*a^14*b^3*d^5) + ((e*cot(c + d*x))^(1/2)*(-e^5/(4*(b^6*
d^2*1i - a^6*d^2*1i + 6*a*b^5*d^2 + 6*a^5*b*d^2 - a^2*b^4*d^2*15i - 20*a^3*b^3*d^2 + a^4*b^2*d^2*15i)))^(1/2)*
(512*b^26*d^4*e^10 + 4608*a^2*b^24*d^4*e^10 + 17920*a^4*b^22*d^4*e^10 + 38400*a^6*b^20*d^4*e^10 + 46080*a^8*b^
18*d^4*e^10 + 21504*a^10*b^16*d^4*e^10 - 21504*a^12*b^14*d^4*e^10 - 46080*a^14*b^12*d^4*e^10 - 38400*a^16*b^10
*d^4*e^10 - 17920*a^18*b^8*d^4*e^10 - 4608*a^20*b^6*d^4*e^10 - 512*a^22*b^4*d^4*e^10))/(b^17*d^4 + a^16*b*d^4
+ 8*a^2*b^15*d^4 + 28*a^4*b^13*d^4 + 56*a^6*b^11*d^4 + 70*a^8*b^9*d^4 + 56*a^10*b^7*d^4 + 28*a^12*b^5*d^4 + 8*
a^14*b^3*d^4))*(-e^5/(4*(b^6*d^2*1i - a^6*d^2*1i + 6*a*b^5*d^2 + 6*a^5*b*d^2 - a^2*b^4*d^2*15i - 20*a^3*b^3*d^
2 + a^4*b^2*d^2*15i)))^(1/2) - ((e*cot(c + d*x))^(1/2)*(8*a^19*b*d^2*e^15 - 1472*a*b^19*d^2*e^15 + 776*a^3*b^1
7*d^2*e^15 + 11328*a^5*b^15*d^2*e^15 + 10208*a^7*b^13*d^2*e^15 - 5056*a^9*b^11*d^2*e^15 - 5328*a^11*b^9*d^2*e^
15 + 4032*a^13*b^7*d^2*e^15 + 3552*a^15*b^5*d^2*e^15 + 384*a^17*b^3*d^2*e^15))/(b^17*d^4 + a^16*b*d^4 + 8*a^2*
b^15*d^4 + 28*a^4*b^13*d^4 + 56*a^6*b^11*d^4 + 70*a^8*b^9*d^4 + 56*a^10*b^7*d^4 + 28*a^12*b^5*d^4 + 8*a^14*b^3
*d^4))*(-e^5/(4*(b^6*d^2*1i - a^6*d^2*1i + 6*a*b^5*d^2 + 6*a^5*b*d^2 - a^2*b^4*d^2*15i - 20*a^3*b^3*d^2 + a^4*
b^2*d^2*15i)))^(1/2))*(-e^5/(4*(b^6*d^2*1i - a^6*d^2*1i + 6*a*b^5*d^2 + 6*a^5*b*d^2 - a^2*b^4*d^2*15i - 20*a^3
*b^3*d^2 + a^4*b^2*d^2*15i)))^(1/2) + ((e*cot(c + d*x))^(1/2)*(a^14*e^20 - 32*b^14*e^20 + 97*a^2*b^12*e^20 - 2
082*a^4*b^10*e^20 + 3631*a^6*b^8*e^20 - 2300*a^8*b^6*e^20 + 79*a^10*b^4*e^20 + 30*a^12*b^2*e^20))/(b^17*d^4 +
a^16*b*d^4 + 8*a^2*b^15*d^4 + 28*a^4*b^13*d^4 + 56*a^6*b^11*d^4 + 70*a^8*b^9*d^4 + 56*a^10*b^7*d^4 + 28*a^12*b
^5*d^4 + 8*a^14*b^3*d^4))*(-e^5/(4*(b^6*d^2*1i - a^6*d^2*1i + 6*a*b^5*d^2 + 6*a^5*b*d^2 - a^2*b^4*d^2*15i - 20
*a^3*b^3*d^2 + a^4*b^2*d^2*15i)))^(1/2)*1i - (((10*a^16*b*d^2*e^18 - 2398*a^2*b^15*d^2*e^18 + 5238*a^4*b^13*d^
2*e^18 + 7386*a^6*b^11*d^2*e^18 - 8322*a^8*b^9*d^2*e^18 - 5498*a^10*b^7*d^2*e^18 + 2946*a^12*b^5*d^2*e^18 + 38
2*a^14*b^3*d^2*e^18)/(b^17*d^5 + a^16*b*d^5 + 8*a^2*b^15*d^5 + 28*a^4*b^13*d^5 + 56*a^6*b^11*d^5 + 70*a^8*b^9*
d^5 + 56*a^10*b^7*d^5 + 28*a^12*b^5*d^5 + 8*a^14*b^3*d^5) - (((832*a*b^22*d^4*e^13 + 5952*a^3*b^20*d^4*e^13 +
17664*a^5*b^18*d^4*e^13 + 26880*a^7*b^16*d^4*e^13 + 18816*a^9*b^14*d^4*e^13 - 2688*a^11*b^12*d^4*e^13 - 16128*
a^13*b^10*d^4*e^13 - 13056*a^15*b^8*d^4*e^13 - 4800*a^17*b^6*d^4*e^13 - 704*a^19*b^4*d^4*e^13)/(b^17*d^5 + a^1
6*b*d^5 + 8*a^2*b^15*d^5 + 28*a^4*b^13*d^5 + 56*a^6*b^11*d^5 + 70*a^8*b^9*d^5 + 56*a^10*b^7*d^5 + 28*a^12*b^5*
d^5 + 8*a^14*b^3*d^5) - ((e*cot(c + d*x))^(1/2)*(-e^5/(4*(b^6*d^2*1i - a^6*d^2*1i + 6*a*b^5*d^2 + 6*a^5*b*d^2
- a^2*b^4*d^2*15i - 20*a^3*b^3*d^2 + a^4*b^2*d^2*15i)))^(1/2)*(512*b^26*d^4*e^10 + 4608*a^2*b^24*d^4*e^10 + 17
920*a^4*b^22*d^4*e^10 + 38400*a^6*b^20*d^4*e^10 + 46080*a^8*b^18*d^4*e^10 + 21504*a^10*b^16*d^4*e^10 - 21504*a
^12*b^14*d^4*e^10 - 46080*a^14*b^12*d^4*e^10 - 38400*a^16*b^10*d^4*e^10 - 17920*a^18*b^8*d^4*e^10 - 4608*a^20*
b^6*d^4*e^10 - 512*a^22*b^4*d^4*e^10))/(b^17*d^4 + a^16*b*d^4 + 8*a^2*b^15*d^4 + 28*a^4*b^13*d^4 + 56*a^6*b^11
*d^4 + 70*a^8*b^9*d^4 + 56*a^10*b^7*d^4 + 28*a^12*b^5*d^4 + 8*a^14*b^3*d^4))*(-e^5/(4*(b^6*d^2*1i - a^6*d^2*1i
+ 6*a*b^5*d^2 + 6*a^5*b*d^2 - a^2*b^4*d^2*15i - 20*a^3*b^3*d^2 + a^4*b^2*d^2*15i)))^(1/2) + ((e*cot(c + d*x))
^(1/2)*(8*a^19*b*d^2*e^15 - 1472*a*b^19*d^2*e^15 + 776*a^3*b^17*d^2*e^15 + 11328*a^5*b^15*d^2*e^15 + 10208*a^7
*b^13*d^2*e^15 - 5056*a^9*b^11*d^2*e^15 - 5328*a^11*b^9*d^2*e^15 + 4032*a^13*b^7*d^2*e^15 + 3552*a^15*b^5*d^2*
e^15 + 384*a^17*b^3*d^2*e^15))/(b^17*d^4 + a^16*b*d^4 + 8*a^2*b^15*d^4 + 28*a^4*b^13*d^4 + 56*a^6*b^11*d^4 + 7
0*a^8*b^9*d^4 + 56*a^10*b^7*d^4 + 28*a^12*b^5*d^4 + 8*a^14*b^3*d^4))*(-e^5/(4*(b^6*d^2*1i - a^6*d^2*1i + 6*a*b
^5*d^2 + 6*a^5*b*d^2 - a^2*b^4*d^2*15i - 20*a^3*b^3*d^2 + a^4*b^2*d^2*15i)))^(1/2))*(-e^5/(4*(b^6*d^2*1i - a^6
*d^2*1i + 6*a*b^5*d^2 + 6*a^5*b*d^2 - a^2*b^4*d^2*15i - 20*a^3*b^3*d^2 + a^4*b^2*d^2*15i)))^(1/2) - ((e*cot(c
+ d*x))^(1/2)*(a^14*e^20 - 32*b^14*e^20 + 97*a^2*b^12*e^20 - 2082*a^4*b^10*e^20 + 3631*a^6*b^8*e^20 - 2300*a^8
*b^6*e^20 + 79*a^10*b^4*e^20 + 30*a^12*b^2*e^20))/(b^17*d^4 + a^16*b*d^4 + 8*a^2*b^15*d^4 + 28*a^4*b^13*d^4 +
56*a^6*b^11*d^4 + 70*a^8*b^9*d^4 + 56*a^10*b^7*d^4 + 28*a^12*b^5*d^4 + 8*a^14*b^3*d^4))*(-e^5/(4*(b^6*d^2*1i -
a^6*d^2*1i + 6*a*b^5*d^2 + 6*a^5*b*d^2 - a^2*b^4*d^2*15i - 20*a^3*b^3*d^2 + a^4*b^2*d^2*15i)))^(1/2)*1i)/((((
10*a^16*b*d^2*e^18 - 2398*a^2*b^15*d^2*e^18 + 5238*a^4*b^13*d^2*e^18 + 7386*a^6*b^11*d^2*e^18 - 8322*a^8*b^9*d
^2*e^18 - 5498*a^10*b^7*d^2*e^18 + 2946*a^12*b^5*d^2*e^18 + 382*a^14*b^3*d^2*e^18)/(b^17*d^5 + a^16*b*d^5 + 8*
a^2*b^15*d^5 + 28*a^4*b^13*d^5 + 56*a^6*b^11*d^5 + 70*a^8*b^9*d^5 + 56*a^10*b^7*d^5 + 28*a^12*b^5*d^5 + 8*a^14
*b^3*d^5) - (((832*a*b^22*d^4*e^13 + 5952*a^3*b^20*d^4*e^13 + 17664*a^5*b^18*d^4*e^13 + 26880*a^7*b^16*d^4*e^1
3 + 18816*a^9*b^14*d^4*e^13 - 2688*a^11*b^12*d^4*e^13 - 16128*a^13*b^10*d^4*e^13 - 13056*a^15*b^8*d^4*e^13 - 4
800*a^17*b^6*d^4*e^13 - 704*a^19*b^4*d^4*e^13)/(b^17*d^5 + a^16*b*d^5 + 8*a^2*b^15*d^5 + 28*a^4*b^13*d^5 + 56*
a^6*b^11*d^5 + 70*a^8*b^9*d^5 + 56*a^10*b^7*d^5 + 28*a^12*b^5*d^5 + 8*a^14*b^3*d^5) + ((e*cot(c + d*x))^(1/2)*
(-e^5/(4*(b^6*d^2*1i - a^6*d^2*1i + 6*a*b^5*d^2 + 6*a^5*b*d^2 - a^2*b^4*d^2*15i - 20*a^3*b^3*d^2 + a^4*b^2*d^2
*15i)))^(1/2)*(512*b^26*d^4*e^10 + 4608*a^2*b^24*d^4*e^10 + 17920*a^4*b^22*d^4*e^10 + 38400*a^6*b^20*d^4*e^10
+ 46080*a^8*b^18*d^4*e^10 + 21504*a^10*b^16*d^4*e^10 - 21504*a^12*b^14*d^4*e^10 - 46080*a^14*b^12*d^4*e^10 - 3
8400*a^16*b^10*d^4*e^10 - 17920*a^18*b^8*d^4*e^10 - 4608*a^20*b^6*d^4*e^10 - 512*a^22*b^4*d^4*e^10))/(b^17*d^4
+ a^16*b*d^4 + 8*a^2*b^15*d^4 + 28*a^4*b^13*d^4 + 56*a^6*b^11*d^4 + 70*a^8*b^9*d^4 + 56*a^10*b^7*d^4 + 28*a^1
2*b^5*d^4 + 8*a^14*b^3*d^4))*(-e^5/(4*(b^6*d^2*1i - a^6*d^2*1i + 6*a*b^5*d^2 + 6*a^5*b*d^2 - a^2*b^4*d^2*15i -
20*a^3*b^3*d^2 + a^4*b^2*d^2*15i)))^(1/2) - ((e*cot(c + d*x))^(1/2)*(8*a^19*b*d^2*e^15 - 1472*a*b^19*d^2*e^15
+ 776*a^3*b^17*d^2*e^15 + 11328*a^5*b^15*d^2*e^15 + 10208*a^7*b^13*d^2*e^15 - 5056*a^9*b^11*d^2*e^15 - 5328*a
^11*b^9*d^2*e^15 + 4032*a^13*b^7*d^2*e^15 + 3552*a^15*b^5*d^2*e^15 + 384*a^17*b^3*d^2*e^15))/(b^17*d^4 + a^16*
b*d^4 + 8*a^2*b^15*d^4 + 28*a^4*b^13*d^4 + 56*a^6*b^11*d^4 + 70*a^8*b^9*d^4 + 56*a^10*b^7*d^4 + 28*a^12*b^5*d^
4 + 8*a^14*b^3*d^4))*(-e^5/(4*(b^6*d^2*1i - a^6*d^2*1i + 6*a*b^5*d^2 + 6*a^5*b*d^2 - a^2*b^4*d^2*15i - 20*a^3*
b^3*d^2 + a^4*b^2*d^2*15i)))^(1/2))*(-e^5/(4*(b^6*d^2*1i - a^6*d^2*1i + 6*a*b^5*d^2 + 6*a^5*b*d^2 - a^2*b^4*d^
2*15i - 20*a^3*b^3*d^2 + a^4*b^2*d^2*15i)))^(1/2) + ((e*cot(c + d*x))^(1/2)*(a^14*e^20 - 32*b^14*e^20 + 97*a^2
*b^12*e^20 - 2082*a^4*b^10*e^20 + 3631*a^6*b^8*e^20 - 2300*a^8*b^6*e^20 + 79*a^10*b^4*e^20 + 30*a^12*b^2*e^20)
)/(b^17*d^4 + a^16*b*d^4 + 8*a^2*b^15*d^4 + 28*a^4*b^13*d^4 + 56*a^6*b^11*d^4 + 70*a^8*b^9*d^4 + 56*a^10*b^7*d
^4 + 28*a^12*b^5*d^4 + 8*a^14*b^3*d^4))*(-e^5/(4*(b^6*d^2*1i - a^6*d^2*1i + 6*a*b^5*d^2 + 6*a^5*b*d^2 - a^2*b^
4*d^2*15i - 20*a^3*b^3*d^2 + a^4*b^2*d^2*15i)))^(1/2) + (((10*a^16*b*d^2*e^18 - 2398*a^2*b^15*d^2*e^18 + 5238*
a^4*b^13*d^2*e^18 + 7386*a^6*b^11*d^2*e^18 - 8322*a^8*b^9*d^2*e^18 - 5498*a^10*b^7*d^2*e^18 + 2946*a^12*b^5*d^
2*e^18 + 382*a^14*b^3*d^2*e^18)/(b^17*d^5 + a^16*b*d^5 + 8*a^2*b^15*d^5 + 28*a^4*b^13*d^5 + 56*a^6*b^11*d^5 +
70*a^8*b^9*d^5 + 56*a^10*b^7*d^5 + 28*a^12*b^5*d^5 + 8*a^14*b^3*d^5) - (((832*a*b^22*d^4*e^13 + 5952*a^3*b^20*
d^4*e^13 + 17664*a^5*b^18*d^4*e^13 + 26880*a^7*b^16*d^4*e^13 + 18816*a^9*b^14*d^4*e^13 - 2688*a^11*b^12*d^4*e^
13 - 16128*a^13*b^10*d^4*e^13 - 13056*a^15*b^8*d^4*e^13 - 4800*a^17*b^6*d^4*e^13 - 704*a^19*b^4*d^4*e^13)/(b^1
7*d^5 + a^16*b*d^5 + 8*a^2*b^15*d^5 + 28*a^4*b^13*d^5 + 56*a^6*b^11*d^5 + 70*a^8*b^9*d^5 + 56*a^10*b^7*d^5 + 2
8*a^12*b^5*d^5 + 8*a^14*b^3*d^5) - ((e*cot(c + d*x))^(1/2)*(-e^5/(4*(b^6*d^2*1i - a^6*d^2*1i + 6*a*b^5*d^2 + 6
*a^5*b*d^2 - a^2*b^4*d^2*15i - 20*a^3*b^3*d^2 + a^4*b^2*d^2*15i)))^(1/2)*(512*b^26*d^4*e^10 + 4608*a^2*b^24*d^
4*e^10 + 17920*a^4*b^22*d^4*e^10 + 38400*a^6*b^20*d^4*e^10 + 46080*a^8*b^18*d^4*e^10 + 21504*a^10*b^16*d^4*e^1
0 - 21504*a^12*b^14*d^4*e^10 - 46080*a^14*b^12*d^4*e^10 - 38400*a^16*b^10*d^4*e^10 - 17920*a^18*b^8*d^4*e^10 -
4608*a^20*b^6*d^4*e^10 - 512*a^22*b^4*d^4*e^10))/(b^17*d^4 + a^16*b*d^4 + 8*a^2*b^15*d^4 + 28*a^4*b^13*d^4 +
56*a^6*b^11*d^4 + 70*a^8*b^9*d^4 + 56*a^10*b^7*d^4 + 28*a^12*b^5*d^4 + 8*a^14*b^3*d^4))*(-e^5/(4*(b^6*d^2*1i -
a^6*d^2*1i + 6*a*b^5*d^2 + 6*a^5*b*d^2 - a^2*b^4*d^2*15i - 20*a^3*b^3*d^2 + a^4*b^2*d^2*15i)))^(1/2) + ((e*co
t(c + d*x))^(1/2)*(8*a^19*b*d^2*e^15 - 1472*a*b^19*d^2*e^15 + 776*a^3*b^17*d^2*e^15 + 11328*a^5*b^15*d^2*e^15
+ 10208*a^7*b^13*d^2*e^15 - 5056*a^9*b^11*d^2*e^15 - 5328*a^11*b^9*d^2*e^15 + 4032*a^13*b^7*d^2*e^15 + 3552*a^
15*b^5*d^2*e^15 + 384*a^17*b^3*d^2*e^15))/(b^17*d^4 + a^16*b*d^4 + 8*a^2*b^15*d^4 + 28*a^4*b^13*d^4 + 56*a^6*b
^11*d^4 + 70*a^8*b^9*d^4 + 56*a^10*b^7*d^4 + 28*a^12*b^5*d^4 + 8*a^14*b^3*d^4))*(-e^5/(4*(b^6*d^2*1i - a^6*d^2
*1i + 6*a*b^5*d^2 + 6*a^5*b*d^2 - a^2*b^4*d^2*15i - 20*a^3*b^3*d^2 + a^4*b^2*d^2*15i)))^(1/2))*(-e^5/(4*(b^6*d
^2*1i - a^6*d^2*1i + 6*a*b^5*d^2 + 6*a^5*b*d^2 - a^2*b^4*d^2*15i - 20*a^3*b^3*d^2 + a^4*b^2*d^2*15i)))^(1/2) -
((e*cot(c + d*x))^(1/2)*(a^14*e^20 - 32*b^14*e^20 + 97*a^2*b^12*e^20 - 2082*a^4*b^10*e^20 + 3631*a^6*b^8*e^20
- 2300*a^8*b^6*e^20 + 79*a^10*b^4*e^20 + 30*a^12*b^2*e^20))/(b^17*d^4 + a^16*b*d^4 + 8*a^2*b^15*d^4 + 28*a^4*
b^13*d^4 + 56*a^6*b^11*d^4 + 70*a^8*b^9*d^4 + 56*a^10*b^7*d^4 + 28*a^12*b^5*d^4 + 8*a^14*b^3*d^4))*(-e^5/(4*(b
^6*d^2*1i - a^6*d^2*1i + 6*a*b^5*d^2 + 6*a^5*b*d^2 - a^2*b^4*d^2*15i - 20*a^3*b^3*d^2 + a^4*b^2*d^2*15i)))^(1/
2) + (a^11*e^23 - 120*a*b^10*e^23 + 249*a^3*b^8*e^23 - 388*a^5*b^6*e^23 + 302*a^7*b^4*e^23 + 36*a^9*b^2*e^23)/
(b^17*d^5 + a^16*b*d^5 + 8*a^2*b^15*d^5 + 28*a^4*b^13*d^5 + 56*a^6*b^11*d^5 + 70*a^8*b^9*d^5 + 56*a^10*b^7*d^5
+ 28*a^12*b^5*d^5 + 8*a^14*b^3*d^5)))*(-e^5/(4*(b^6*d^2*1i - a^6*d^2*1i + 6*a*b^5*d^2 + 6*a^5*b*d^2 - a^2*b^4
*d^2*15i - 20*a^3*b^3*d^2 + a^4*b^2*d^2*15i)))^(1/2)*2i + (atan((((((e*cot(c + d*x))^(1/2)*(a^14*e^20 - 32*b^1
4*e^20 + 97*a^2*b^12*e^20 - 2082*a^4*b^10*e^20 + 3631*a^6*b^8*e^20 - 2300*a^8*b^6*e^20 + 79*a^10*b^4*e^20 + 30
*a^12*b^2*e^20))/(b^17*d^4 + a^16*b*d^4 + 8*a^2*b^15*d^4 + 28*a^4*b^13*d^4 + 56*a^6*b^11*d^4 + 70*a^8*b^9*d^4
+ 56*a^10*b^7*d^4 + 28*a^12*b^5*d^4 + 8*a^14*b^3*d^4) - (((10*a^16*b*d^2*e^18 - 2398*a^2*b^15*d^2*e^18 + 5238*
a^4*b^13*d^2*e^18 + 7386*a^6*b^11*d^2*e^18 - 8322*a^8*b^9*d^2*e^18 - 5498*a^10*b^7*d^2*e^18 + 2946*a^12*b^5*d^
2*e^18 + 382*a^14*b^3*d^2*e^18)/(b^17*d^5 + a^16*b*d^5 + 8*a^2*b^15*d^5 + 28*a^4*b^13*d^5 + 56*a^6*b^11*d^5 +
70*a^8*b^9*d^5 + 56*a^10*b^7*d^5 + 28*a^12*b^5*d^5 + 8*a^14*b^3*d^5) - ((((e*cot(c + d*x))^(1/2)*(8*a^19*b*d^2
*e^15 - 1472*a*b^19*d^2*e^15 + 776*a^3*b^17*d^2*e^15 + 11328*a^5*b^15*d^2*e^15 + 10208*a^7*b^13*d^2*e^15 - 505
6*a^9*b^11*d^2*e^15 - 5328*a^11*b^9*d^2*e^15 + 4032*a^13*b^7*d^2*e^15 + 3552*a^15*b^5*d^2*e^15 + 384*a^17*b^3*
d^2*e^15))/(b^17*d^4 + a^16*b*d^4 + 8*a^2*b^15*d^4 + 28*a^4*b^13*d^4 + 56*a^6*b^11*d^4 + 70*a^8*b^9*d^4 + 56*a
^10*b^7*d^4 + 28*a^12*b^5*d^4 + 8*a^14*b^3*d^4) + (((832*a*b^22*d^4*e^13 + 5952*a^3*b^20*d^4*e^13 + 17664*a^5*
b^18*d^4*e^13 + 26880*a^7*b^16*d^4*e^13 + 18816*a^9*b^14*d^4*e^13 - 2688*a^11*b^12*d^4*e^13 - 16128*a^13*b^10*
d^4*e^13 - 13056*a^15*b^8*d^4*e^13 - 4800*a^17*b^6*d^4*e^13 - 704*a^19*b^4*d^4*e^13)/(b^17*d^5 + a^16*b*d^5 +
8*a^2*b^15*d^5 + 28*a^4*b^13*d^5 + 56*a^6*b^11*d^5 + 70*a^8*b^9*d^5 + 56*a^10*b^7*d^5 + 28*a^12*b^5*d^5 + 8*a^
14*b^3*d^5) - ((e*cot(c + d*x))^(1/2)*(a^4 - 15*b^4 + 18*a^2*b^2)*(-a*b^3*e^5)^(1/2)*(512*b^26*d^4*e^10 + 4608
*a^2*b^24*d^4*e^10 + 17920*a^4*b^22*d^4*e^10 + 38400*a^6*b^20*d^4*e^10 + 46080*a^8*b^18*d^4*e^10 + 21504*a^10*
b^16*d^4*e^10 - 21504*a^12*b^14*d^4*e^10 - 46080*a^14*b^12*d^4*e^10 - 38400*a^16*b^10*d^4*e^10 - 17920*a^18*b^
8*d^4*e^10 - 4608*a^20*b^6*d^4*e^10 - 512*a^22*b^4*d^4*e^10))/(8*(b^9*d + 3*a^2*b^7*d + 3*a^4*b^5*d + a^6*b^3*
d)*(b^17*d^4 + a^16*b*d^4 + 8*a^2*b^15*d^4 + 28*a^4*b^13*d^4 + 56*a^6*b^11*d^4 + 70*a^8*b^9*d^4 + 56*a^10*b^7*
d^4 + 28*a^12*b^5*d^4 + 8*a^14*b^3*d^4)))*(a^4 - 15*b^4 + 18*a^2*b^2)*(-a*b^3*e^5)^(1/2))/(8*(b^9*d + 3*a^2*b^
7*d + 3*a^4*b^5*d + a^6*b^3*d)))*(a^4 - 15*b^4 + 18*a^2*b^2)*(-a*b^3*e^5)^(1/2))/(8*(b^9*d + 3*a^2*b^7*d + 3*a
^4*b^5*d + a^6*b^3*d)))*(a^4 - 15*b^4 + 18*a^2*b^2)*(-a*b^3*e^5)^(1/2))/(8*(b^9*d + 3*a^2*b^7*d + 3*a^4*b^5*d
+ a^6*b^3*d)))*(a^4 - 15*b^4 + 18*a^2*b^2)*(-a*b^3*e^5)^(1/2)*1i)/(8*(b^9*d + 3*a^2*b^7*d + 3*a^4*b^5*d + a^6*
b^3*d)) + ((((e*cot(c + d*x))^(1/2)*(a^14*e^20 - 32*b^14*e^20 + 97*a^2*b^12*e^20 - 2082*a^4*b^10*e^20 + 3631*a
^6*b^8*e^20 - 2300*a^8*b^6*e^20 + 79*a^10*b^4*e^20 + 30*a^12*b^2*e^20))/(b^17*d^4 + a^16*b*d^4 + 8*a^2*b^15*d^
4 + 28*a^4*b^13*d^4 + 56*a^6*b^11*d^4 + 70*a^8*b^9*d^4 + 56*a^10*b^7*d^4 + 28*a^12*b^5*d^4 + 8*a^14*b^3*d^4) +
(((10*a^16*b*d^2*e^18 - 2398*a^2*b^15*d^2*e^18 + 5238*a^4*b^13*d^2*e^18 + 7386*a^6*b^11*d^2*e^18 - 8322*a^8*b
^9*d^2*e^18 - 5498*a^10*b^7*d^2*e^18 + 2946*a^12*b^5*d^2*e^18 + 382*a^14*b^3*d^2*e^18)/(b^17*d^5 + a^16*b*d^5
+ 8*a^2*b^15*d^5 + 28*a^4*b^13*d^5 + 56*a^6*b^11*d^5 + 70*a^8*b^9*d^5 + 56*a^10*b^7*d^5 + 28*a^12*b^5*d^5 + 8*
a^14*b^3*d^5) + ((((e*cot(c + d*x))^(1/2)*(8*a^19*b*d^2*e^15 - 1472*a*b^19*d^2*e^15 + 776*a^3*b^17*d^2*e^15 +
11328*a^5*b^15*d^2*e^15 + 10208*a^7*b^13*d^2*e^15 - 5056*a^9*b^11*d^2*e^15 - 5328*a^11*b^9*d^2*e^15 + 4032*a^1
3*b^7*d^2*e^15 + 3552*a^15*b^5*d^2*e^15 + 384*a^17*b^3*d^2*e^15))/(b^17*d^4 + a^16*b*d^4 + 8*a^2*b^15*d^4 + 28
*a^4*b^13*d^4 + 56*a^6*b^11*d^4 + 70*a^8*b^9*d^4 + 56*a^10*b^7*d^4 + 28*a^12*b^5*d^4 + 8*a^14*b^3*d^4) - (((83
2*a*b^22*d^4*e^13 + 5952*a^3*b^20*d^4*e^13 + 17664*a^5*b^18*d^4*e^13 + 26880*a^7*b^16*d^4*e^13 + 18816*a^9*b^1
4*d^4*e^13 - 2688*a^11*b^12*d^4*e^13 - 16128*a^13*b^10*d^4*e^13 - 13056*a^15*b^8*d^4*e^13 - 4800*a^17*b^6*d^4*
e^13 - 704*a^19*b^4*d^4*e^13)/(b^17*d^5 + a^16*b*d^5 + 8*a^2*b^15*d^5 + 28*a^4*b^13*d^5 + 56*a^6*b^11*d^5 + 70
*a^8*b^9*d^5 + 56*a^10*b^7*d^5 + 28*a^12*b^5*d^5 + 8*a^14*b^3*d^5) + ((e*cot(c + d*x))^(1/2)*(a^4 - 15*b^4 + 1
8*a^2*b^2)*(-a*b^3*e^5)^(1/2)*(512*b^26*d^4*e^10 + 4608*a^2*b^24*d^4*e^10 + 17920*a^4*b^22*d^4*e^10 + 38400*a^
6*b^20*d^4*e^10 + 46080*a^8*b^18*d^4*e^10 + 21504*a^10*b^16*d^4*e^10 - 21504*a^12*b^14*d^4*e^10 - 46080*a^14*b
^12*d^4*e^10 - 38400*a^16*b^10*d^4*e^10 - 17920*a^18*b^8*d^4*e^10 - 4608*a^20*b^6*d^4*e^10 - 512*a^22*b^4*d^4*
e^10))/(8*(b^9*d + 3*a^2*b^7*d + 3*a^4*b^5*d + a^6*b^3*d)*(b^17*d^4 + a^16*b*d^4 + 8*a^2*b^15*d^4 + 28*a^4*b^1
3*d^4 + 56*a^6*b^11*d^4 + 70*a^8*b^9*d^4 + 56*a^10*b^7*d^4 + 28*a^12*b^5*d^4 + 8*a^14*b^3*d^4)))*(a^4 - 15*b^4
+ 18*a^2*b^2)*(-a*b^3*e^5)^(1/2))/(8*(b^9*d + 3*a^2*b^7*d + 3*a^4*b^5*d + a^6*b^3*d)))*(a^4 - 15*b^4 + 18*a^2
*b^2)*(-a*b^3*e^5)^(1/2))/(8*(b^9*d + 3*a^2*b^7*d + 3*a^4*b^5*d + a^6*b^3*d)))*(a^4 - 15*b^4 + 18*a^2*b^2)*(-a
*b^3*e^5)^(1/2))/(8*(b^9*d + 3*a^2*b^7*d + 3*a^4*b^5*d + a^6*b^3*d)))*(a^4 - 15*b^4 + 18*a^2*b^2)*(-a*b^3*e^5)
^(1/2)*1i)/(8*(b^9*d + 3*a^2*b^7*d + 3*a^4*b^5*d + a^6*b^3*d)))/((a^11*e^23 - 120*a*b^10*e^23 + 249*a^3*b^8*e^
23 - 388*a^5*b^6*e^23 + 302*a^7*b^4*e^23 + 36*a^9*b^2*e^23)/(b^17*d^5 + a^16*b*d^5 + 8*a^2*b^15*d^5 + 28*a^4*b
^13*d^5 + 56*a^6*b^11*d^5 + 70*a^8*b^9*d^5 + 56*a^10*b^7*d^5 + 28*a^12*b^5*d^5 + 8*a^14*b^3*d^5) - ((((e*cot(c
+ d*x))^(1/2)*(a^14*e^20 - 32*b^14*e^20 + 97*a^2*b^12*e^20 - 2082*a^4*b^10*e^20 + 3631*a^6*b^8*e^20 - 2300*a^
8*b^6*e^20 + 79*a^10*b^4*e^20 + 30*a^12*b^2*e^20))/(b^17*d^4 + a^16*b*d^4 + 8*a^2*b^15*d^4 + 28*a^4*b^13*d^4 +
56*a^6*b^11*d^4 + 70*a^8*b^9*d^4 + 56*a^10*b^7*d^4 + 28*a^12*b^5*d^4 + 8*a^14*b^3*d^4) - (((10*a^16*b*d^2*e^1
8 - 2398*a^2*b^15*d^2*e^18 + 5238*a^4*b^13*d^2*e^18 + 7386*a^6*b^11*d^2*e^18 - 8322*a^8*b^9*d^2*e^18 - 5498*a^
10*b^7*d^2*e^18 + 2946*a^12*b^5*d^2*e^18 + 382*a^14*b^3*d^2*e^18)/(b^17*d^5 + a^16*b*d^5 + 8*a^2*b^15*d^5 + 28
*a^4*b^13*d^5 + 56*a^6*b^11*d^5 + 70*a^8*b^9*d^5 + 56*a^10*b^7*d^5 + 28*a^12*b^5*d^5 + 8*a^14*b^3*d^5) - ((((e
*cot(c + d*x))^(1/2)*(8*a^19*b*d^2*e^15 - 1472*a*b^19*d^2*e^15 + 776*a^3*b^17*d^2*e^15 + 11328*a^5*b^15*d^2*e^
15 + 10208*a^7*b^13*d^2*e^15 - 5056*a^9*b^11*d^2*e^15 - 5328*a^11*b^9*d^2*e^15 + 4032*a^13*b^7*d^2*e^15 + 3552
*a^15*b^5*d^2*e^15 + 384*a^17*b^3*d^2*e^15))/(b^17*d^4 + a^16*b*d^4 + 8*a^2*b^15*d^4 + 28*a^4*b^13*d^4 + 56*a^
6*b^11*d^4 + 70*a^8*b^9*d^4 + 56*a^10*b^7*d^4 + 28*a^12*b^5*d^4 + 8*a^14*b^3*d^4) + (((832*a*b^22*d^4*e^13 + 5
952*a^3*b^20*d^4*e^13 + 17664*a^5*b^18*d^4*e^13 + 26880*a^7*b^16*d^4*e^13 + 18816*a^9*b^14*d^4*e^13 - 2688*a^1
1*b^12*d^4*e^13 - 16128*a^13*b^10*d^4*e^13 - 13056*a^15*b^8*d^4*e^13 - 4800*a^17*b^6*d^4*e^13 - 704*a^19*b^4*d
^4*e^13)/(b^17*d^5 + a^16*b*d^5 + 8*a^2*b^15*d^5 + 28*a^4*b^13*d^5 + 56*a^6*b^11*d^5 + 70*a^8*b^9*d^5 + 56*a^1
0*b^7*d^5 + 28*a^12*b^5*d^5 + 8*a^14*b^3*d^5) - ((e*cot(c + d*x))^(1/2)*(a^4 - 15*b^4 + 18*a^2*b^2)*(-a*b^3*e^
5)^(1/2)*(512*b^26*d^4*e^10 + 4608*a^2*b^24*d^4*e^10 + 17920*a^4*b^22*d^4*e^10 + 38400*a^6*b^20*d^4*e^10 + 460
80*a^8*b^18*d^4*e^10 + 21504*a^10*b^16*d^4*e^10 - 21504*a^12*b^14*d^4*e^10 - 46080*a^14*b^12*d^4*e^10 - 38400*
a^16*b^10*d^4*e^10 - 17920*a^18*b^8*d^4*e^10 - 4608*a^20*b^6*d^4*e^10 - 512*a^22*b^4*d^4*e^10))/(8*(b^9*d + 3*
a^2*b^7*d + 3*a^4*b^5*d + a^6*b^3*d)*(b^17*d^4 + a^16*b*d^4 + 8*a^2*b^15*d^4 + 28*a^4*b^13*d^4 + 56*a^6*b^11*d
^4 + 70*a^8*b^9*d^4 + 56*a^10*b^7*d^4 + 28*a^12*b^5*d^4 + 8*a^14*b^3*d^4)))*(a^4 - 15*b^4 + 18*a^2*b^2)*(-a*b^
3*e^5)^(1/2))/(8*(b^9*d + 3*a^2*b^7*d + 3*a^4*b^5*d + a^6*b^3*d)))*(a^4 - 15*b^4 + 18*a^2*b^2)*(-a*b^3*e^5)^(1
/2))/(8*(b^9*d + 3*a^2*b^7*d + 3*a^4*b^5*d + a^6*b^3*d)))*(a^4 - 15*b^4 + 18*a^2*b^2)*(-a*b^3*e^5)^(1/2))/(8*(
b^9*d + 3*a^2*b^7*d + 3*a^4*b^5*d + a^6*b^3*d)))*(a^4 - 15*b^4 + 18*a^2*b^2)*(-a*b^3*e^5)^(1/2))/(8*(b^9*d + 3
*a^2*b^7*d + 3*a^4*b^5*d + a^6*b^3*d)) + ((((e*cot(c + d*x))^(1/2)*(a^14*e^20 - 32*b^14*e^20 + 97*a^2*b^12*e^2
0 - 2082*a^4*b^10*e^20 + 3631*a^6*b^8*e^20 - 2300*a^8*b^6*e^20 + 79*a^10*b^4*e^20 + 30*a^12*b^2*e^20))/(b^17*d
^4 + a^16*b*d^4 + 8*a^2*b^15*d^4 + 28*a^4*b^13*d^4 + 56*a^6*b^11*d^4 + 70*a^8*b^9*d^4 + 56*a^10*b^7*d^4 + 28*a
^12*b^5*d^4 + 8*a^14*b^3*d^4) + (((10*a^16*b*d^2*e^18 - 2398*a^2*b^15*d^2*e^18 + 5238*a^4*b^13*d^2*e^18 + 7386
*a^6*b^11*d^2*e^18 - 8322*a^8*b^9*d^2*e^18 - 5498*a^10*b^7*d^2*e^18 + 2946*a^12*b^5*d^2*e^18 + 382*a^14*b^3*d^
2*e^18)/(b^17*d^5 + a^16*b*d^5 + 8*a^2*b^15*d^5 + 28*a^4*b^13*d^5 + 56*a^6*b^11*d^5 + 70*a^8*b^9*d^5 + 56*a^10
*b^7*d^5 + 28*a^12*b^5*d^5 + 8*a^14*b^3*d^5) + ((((e*cot(c + d*x))^(1/2)*(8*a^19*b*d^2*e^15 - 1472*a*b^19*d^2*
e^15 + 776*a^3*b^17*d^2*e^15 + 11328*a^5*b^15*d^2*e^15 + 10208*a^7*b^13*d^2*e^15 - 5056*a^9*b^11*d^2*e^15 - 53
28*a^11*b^9*d^2*e^15 + 4032*a^13*b^7*d^2*e^15 + 3552*a^15*b^5*d^2*e^15 + 384*a^17*b^3*d^2*e^15))/(b^17*d^4 + a
^16*b*d^4 + 8*a^2*b^15*d^4 + 28*a^4*b^13*d^4 + 56*a^6*b^11*d^4 + 70*a^8*b^9*d^4 + 56*a^10*b^7*d^4 + 28*a^12*b^
5*d^4 + 8*a^14*b^3*d^4) - (((832*a*b^22*d^4*e^13 + 5952*a^3*b^20*d^4*e^13 + 17664*a^5*b^18*d^4*e^13 + 26880*a^
7*b^16*d^4*e^13 + 18816*a^9*b^14*d^4*e^13 - 2688*a^11*b^12*d^4*e^13 - 16128*a^13*b^10*d^4*e^13 - 13056*a^15*b^
8*d^4*e^13 - 4800*a^17*b^6*d^4*e^13 - 704*a^19*b^4*d^4*e^13)/(b^17*d^5 + a^16*b*d^5 + 8*a^2*b^15*d^5 + 28*a^4*
b^13*d^5 + 56*a^6*b^11*d^5 + 70*a^8*b^9*d^5 + 56*a^10*b^7*d^5 + 28*a^12*b^5*d^5 + 8*a^14*b^3*d^5) + ((e*cot(c
+ d*x))^(1/2)*(a^4 - 15*b^4 + 18*a^2*b^2)*(-a*b^3*e^5)^(1/2)*(512*b^26*d^4*e^10 + 4608*a^2*b^24*d^4*e^10 + 179
20*a^4*b^22*d^4*e^10 + 38400*a^6*b^20*d^4*e^10 + 46080*a^8*b^18*d^4*e^10 + 21504*a^10*b^16*d^4*e^10 - 21504*a^
12*b^14*d^4*e^10 - 46080*a^14*b^12*d^4*e^10 - 38400*a^16*b^10*d^4*e^10 - 17920*a^18*b^8*d^4*e^10 - 4608*a^20*b
^6*d^4*e^10 - 512*a^22*b^4*d^4*e^10))/(8*(b^9*d + 3*a^2*b^7*d + 3*a^4*b^5*d + a^6*b^3*d)*(b^17*d^4 + a^16*b*d^
4 + 8*a^2*b^15*d^4 + 28*a^4*b^13*d^4 + 56*a^6*b^11*d^4 + 70*a^8*b^9*d^4 + 56*a^10*b^7*d^4 + 28*a^12*b^5*d^4 +
8*a^14*b^3*d^4)))*(a^4 - 15*b^4 + 18*a^2*b^2)*(-a*b^3*e^5)^(1/2))/(8*(b^9*d + 3*a^2*b^7*d + 3*a^4*b^5*d + a^6*
b^3*d)))*(a^4 - 15*b^4 + 18*a^2*b^2)*(-a*b^3*e^5)^(1/2))/(8*(b^9*d + 3*a^2*b^7*d + 3*a^4*b^5*d + a^6*b^3*d)))*
(a^4 - 15*b^4 + 18*a^2*b^2)*(-a*b^3*e^5)^(1/2))/(8*(b^9*d + 3*a^2*b^7*d + 3*a^4*b^5*d + a^6*b^3*d)))*(a^4 - 15
*b^4 + 18*a^2*b^2)*(-a*b^3*e^5)^(1/2))/(8*(b^9*d + 3*a^2*b^7*d + 3*a^4*b^5*d + a^6*b^3*d))))*(a^4 - 15*b^4 + 1
8*a^2*b^2)*(-a*b^3*e^5)^(1/2)*1i)/(4*(b^9*d + 3*a^2*b^7*d + 3*a^4*b^5*d + a^6*b^3*d))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*cot(d*x+c))**(5/2)/(a+b*cot(d*x+c))**3,x)

[Out]

Integral((e*cot(c + d*x))**(5/2)/(a + b*cot(c + d*x))**3, x)

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