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

Optimal. Leaf size=461 \[ -\frac {e^{3/2} (a-b) \left (a^2+4 a b+b^2\right ) \log \left (\sqrt {e} \cot (c+d x)-\sqrt {2} \sqrt {e \cot (c+d x)}+\sqrt {e}\right )}{2 \sqrt {2} d \left (a^2+b^2\right )^3}+\frac {e^{3/2} (a-b) \left (a^2+4 a b+b^2\right ) \log \left (\sqrt {e} \cot (c+d x)+\sqrt {2} \sqrt {e \cot (c+d x)}+\sqrt {e}\right )}{2 \sqrt {2} d \left (a^2+b^2\right )^3}-\frac {e^{3/2} (a+b) \left (a^2-4 a b+b^2\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} d \left (a^2+b^2\right )^3}+\frac {e^{3/2} (a+b) \left (a^2-4 a b+b^2\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}+1\right )}{\sqrt {2} d \left (a^2+b^2\right )^3}-\frac {e \left (3 a^2-5 b^2\right ) \sqrt {e \cot (c+d x)}}{4 d \left (a^2+b^2\right )^2 (a+b \cot (c+d x))}-\frac {a e \sqrt {e \cot (c+d x)}}{2 d \left (a^2+b^2\right ) (a+b \cot (c+d x))^2}-\frac {e^{3/2} \left (3 a^4-26 a^2 b^2+3 b^4\right ) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {e \cot (c+d x)}}{\sqrt {a} \sqrt {e}}\right )}{4 \sqrt {a} \sqrt {b} d \left (a^2+b^2\right )^3} \]

[Out]

-1/2*(a+b)*(a^2-4*a*b+b^2)*e^(3/2)*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^(3/2)*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^(3/2)*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^(3/2)*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*(3*a^4-26*a^2*b^2+3*b^4)*e^(3/2)*arctan(b^(1/2)*(e*cot(d*x+c))^(1/2)/a^(1/2)/e^(1/2))/(a^2+b^2)^3/d/a^(1/2)
/b^(1/2)-1/2*a*e*(e*cot(d*x+c))^(1/2)/(a^2+b^2)/d/(a+b*cot(d*x+c))^2-1/4*(3*a^2-5*b^2)*e*(e*cot(d*x+c))^(1/2)/
(a^2+b^2)^2/d/(a+b*cot(d*x+c))

________________________________________________________________________________________

Rubi [A]  time = 1.23, antiderivative size = 461, 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 = {3567, 3649, 3653, 3534, 1168, 1162, 617, 204, 1165, 628, 3634, 63, 205} \[ -\frac {e^{3/2} (a-b) \left (a^2+4 a b+b^2\right ) \log \left (\sqrt {e} \cot (c+d x)-\sqrt {2} \sqrt {e \cot (c+d x)}+\sqrt {e}\right )}{2 \sqrt {2} d \left (a^2+b^2\right )^3}+\frac {e^{3/2} (a-b) \left (a^2+4 a b+b^2\right ) \log \left (\sqrt {e} \cot (c+d x)+\sqrt {2} \sqrt {e \cot (c+d x)}+\sqrt {e}\right )}{2 \sqrt {2} d \left (a^2+b^2\right )^3}-\frac {e^{3/2} \left (-26 a^2 b^2+3 a^4+3 b^4\right ) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {e \cot (c+d x)}}{\sqrt {a} \sqrt {e}}\right )}{4 \sqrt {a} \sqrt {b} d \left (a^2+b^2\right )^3}-\frac {e^{3/2} (a+b) \left (a^2-4 a b+b^2\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} d \left (a^2+b^2\right )^3}+\frac {e^{3/2} (a+b) \left (a^2-4 a b+b^2\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}+1\right )}{\sqrt {2} d \left (a^2+b^2\right )^3}-\frac {e \left (3 a^2-5 b^2\right ) \sqrt {e \cot (c+d x)}}{4 d \left (a^2+b^2\right )^2 (a+b \cot (c+d x))}-\frac {a e \sqrt {e \cot (c+d x)}}{2 d \left (a^2+b^2\right ) (a+b \cot (c+d x))^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((3*a^4 - 26*a^2*b^2 + 3*b^4)*e^(3/2)*ArcTan[(Sqrt[b]*Sqrt[e*Cot[c + d*x]])/(Sqrt[a]*Sqrt[e])])/(4*Sqrt[a]*Sq
rt[b]*(a^2 + b^2)^3*d) - ((a + b)*(a^2 - 4*a*b + b^2)*e^(3/2)*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^(3/2)*ArcTan[1 + (Sqrt[2]*Sqrt[e*Cot[c + d*x]])
/Sqrt[e]])/(Sqrt[2]*(a^2 + b^2)^3*d) - (a*e*Sqrt[e*Cot[c + d*x]])/(2*(a^2 + b^2)*d*(a + b*Cot[c + d*x])^2) - (
(3*a^2 - 5*b^2)*e*Sqrt[e*Cot[c + d*x]])/(4*(a^2 + b^2)^2*d*(a + b*Cot[c + d*x])) - ((a - b)*(a^2 + 4*a*b + b^2
)*e^(3/2)*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^(3/2)*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 3567

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Si
mp[((b*c - a*d)*(a + b*Tan[e + f*x])^(m + 1)*(c + d*Tan[e + f*x])^(n - 1))/(f*(m + 1)*(a^2 + b^2)), x] + Dist[
1/((m + 1)*(a^2 + b^2)), Int[(a + b*Tan[e + f*x])^(m + 1)*(c + d*Tan[e + f*x])^(n - 2)*Simp[a*c^2*(m + 1) + a*
d^2*(n - 1) + b*c*d*(m - n + 2) - (b*c^2 - 2*a*c*d - b*d^2)*(m + 1)*Tan[e + f*x] - d*(b*c - a*d)*(m + n)*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] && LtQ[m, -1] && LtQ[1, n, 2] && 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))^{3/2}}{(a+b \cot (c+d x))^3} \, dx &=-\frac {a e \sqrt {e \cot (c+d x)}}{2 \left (a^2+b^2\right ) d (a+b \cot (c+d x))^2}-\frac {\int \frac {\frac {a e^2}{2}-2 b e^2 \cot (c+d x)-\frac {3}{2} a e^2 \cot ^2(c+d x)}{\sqrt {e \cot (c+d x)} (a+b \cot (c+d x))^2} \, dx}{2 \left (a^2+b^2\right )}\\ &=-\frac {a e \sqrt {e \cot (c+d x)}}{2 \left (a^2+b^2\right ) d (a+b \cot (c+d x))^2}-\frac {\left (3 a^2-5 b^2\right ) e \sqrt {e \cot (c+d x)}}{4 \left (a^2+b^2\right )^2 d (a+b \cot (c+d x))}+\frac {\int \frac {-\frac {1}{4} a \left (5 a^2-3 b^2\right ) e^3+4 a^2 b e^3 \cot (c+d x)+\frac {1}{4} a \left (3 a^2-5 b^2\right ) e^3 \cot ^2(c+d x)}{\sqrt {e \cot (c+d x)} (a+b \cot (c+d x))} \, dx}{2 a \left (a^2+b^2\right )^2 e}\\ &=-\frac {a e \sqrt {e \cot (c+d x)}}{2 \left (a^2+b^2\right ) d (a+b \cot (c+d x))^2}-\frac {\left (3 a^2-5 b^2\right ) e \sqrt {e \cot (c+d x)}}{4 \left (a^2+b^2\right )^2 d (a+b \cot (c+d x))}+\frac {\int \frac {-2 a^2 \left (a^2-3 b^2\right ) e^3+2 a b \left (3 a^2-b^2\right ) e^3 \cot (c+d x)}{\sqrt {e \cot (c+d x)}} \, dx}{2 a \left (a^2+b^2\right )^3 e}+\frac {\left (\left (3 a^4-26 a^2 b^2+3 b^4\right ) e^2\right ) \int \frac {1+\cot ^2(c+d x)}{\sqrt {e \cot (c+d x)} (a+b \cot (c+d x))} \, dx}{8 \left (a^2+b^2\right )^3}\\ &=-\frac {a e \sqrt {e \cot (c+d x)}}{2 \left (a^2+b^2\right ) d (a+b \cot (c+d x))^2}-\frac {\left (3 a^2-5 b^2\right ) e \sqrt {e \cot (c+d x)}}{4 \left (a^2+b^2\right )^2 d (a+b \cot (c+d x))}+\frac {\operatorname {Subst}\left (\int \frac {2 a^2 \left (a^2-3 b^2\right ) e^4-2 a b \left (3 a^2-b^2\right ) e^3 x^2}{e^2+x^4} \, dx,x,\sqrt {e \cot (c+d x)}\right )}{a \left (a^2+b^2\right )^3 d e}+\frac {\left (\left (3 a^4-26 a^2 b^2+3 b^4\right ) e^2\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {-e x} (a-b x)} \, dx,x,-\cot (c+d x)\right )}{8 \left (a^2+b^2\right )^3 d}\\ &=-\frac {a e \sqrt {e \cot (c+d x)}}{2 \left (a^2+b^2\right ) d (a+b \cot (c+d x))^2}-\frac {\left (3 a^2-5 b^2\right ) e \sqrt {e \cot (c+d x)}}{4 \left (a^2+b^2\right )^2 d (a+b \cot (c+d x))}-\frac {\left (\left (3 a^4-26 a^2 b^2+3 b^4\right ) e\right ) \operatorname {Subst}\left (\int \frac {1}{a+\frac {b x^2}{e}} \, dx,x,\sqrt {e \cot (c+d x)}\right )}{4 \left (a^2+b^2\right )^3 d}+\frac {\left ((a+b) \left (a^2-4 a b+b^2\right ) e^2\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^2\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 (3 a^4-26 a^2 b^2+3 b^4\right ) e^{3/2} \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {e \cot (c+d x)}}{\sqrt {a} \sqrt {e}}\right )}{4 \sqrt {a} \sqrt {b} \left (a^2+b^2\right )^3 d}-\frac {a e \sqrt {e \cot (c+d x)}}{2 \left (a^2+b^2\right ) d (a+b \cot (c+d x))^2}-\frac {\left (3 a^2-5 b^2\right ) e \sqrt {e \cot (c+d x)}}{4 \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^{3/2}\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^{3/2}\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^2\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^2\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 (3 a^4-26 a^2 b^2+3 b^4\right ) e^{3/2} \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {e \cot (c+d x)}}{\sqrt {a} \sqrt {e}}\right )}{4 \sqrt {a} \sqrt {b} \left (a^2+b^2\right )^3 d}-\frac {a e \sqrt {e \cot (c+d x)}}{2 \left (a^2+b^2\right ) d (a+b \cot (c+d x))^2}-\frac {\left (3 a^2-5 b^2\right ) e \sqrt {e \cot (c+d x)}}{4 \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^{3/2} \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^{3/2} \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^{3/2}\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^{3/2}\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 (3 a^4-26 a^2 b^2+3 b^4\right ) e^{3/2} \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {e \cot (c+d x)}}{\sqrt {a} \sqrt {e}}\right )}{4 \sqrt {a} \sqrt {b} \left (a^2+b^2\right )^3 d}-\frac {(a+b) \left (a^2-4 a b+b^2\right ) e^{3/2} \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^{3/2} \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 e \sqrt {e \cot (c+d x)}}{2 \left (a^2+b^2\right ) d (a+b \cot (c+d x))^2}-\frac {\left (3 a^2-5 b^2\right ) e \sqrt {e \cot (c+d x)}}{4 \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^{3/2} \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^{3/2} \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 = 6.16, size = 518, normalized size = 1.12 \[ -\frac {(e \cot (c+d x))^{3/2} \left (\frac {4 b^2 \cot ^{\frac {5}{2}}(c+d x) \, _2F_1\left (2,\frac {5}{2};\frac {7}{2};-\frac {b \cot (c+d x)}{a}\right )}{5 a \left (a^2+b^2\right )^2}-\frac {2 b \left (3 a^2-b^2\right ) \left (\cot ^{\frac {3}{2}}(c+d x)-\cot ^{\frac {3}{2}}(c+d x) \, _2F_1\left (\frac {3}{4},1;\frac {7}{4};-\cot ^2(c+d x)\right )\right )}{3 \left (a^2+b^2\right )^3}+\frac {2 b \left (3 a^2-b^2\right ) \cot ^{\frac {3}{2}}(c+d x)}{3 \left (a^2+b^2\right )^3}-\frac {\frac {2 b^2 \cot ^2(c+d x)}{(a+b \cot (c+d x))^2}+\frac {3 b \cot (c+d x)}{a+b \cot (c+d x)}-\frac {3 \sqrt {b} \sqrt {\cot (c+d x)} \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {\cot (c+d x)}}{\sqrt {a}}\right )}{\sqrt {a}}}{4 b \left (a^2+b^2\right ) \sqrt {\cot (c+d x)}}-\frac {2 a \left (3 a^2-b^2\right ) \left (\sqrt {\cot (c+d x)}-\frac {\sqrt {a} \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {\cot (c+d x)}}{\sqrt {a}}\right )}{\sqrt {b}}\right )}{\left (a^2+b^2\right )^3}+\frac {a \left (a^2-3 b^2\right ) \left (8 \sqrt {\cot (c+d x)}+\sqrt {2} \log \left (\cot (c+d x)-\sqrt {2} \sqrt {\cot (c+d x)}+1\right )-\sqrt {2} \log \left (\cot (c+d x)+\sqrt {2} \sqrt {\cot (c+d x)}+1\right )+2 \left (\sqrt {2} \tan ^{-1}\left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )-\sqrt {2} \tan ^{-1}\left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )\right )\right )}{4 \left (a^2+b^2\right )^3}\right )}{d \cot ^{\frac {3}{2}}(c+d x)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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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))^(3/2)/(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 \[ \int \frac {\left (e \cot \left (d x + c\right )\right )^{\frac {3}{2}}}{{\left (b \cot \left (d x + c\right ) + a\right )}^{3}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [B]  time = 0.87, size = 1212, normalized size = 2.63 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-3/4/d*e^2/(a^2+b^2)^3/(e*cot(d*x+c)*b+a*e)^2*(e*cot(d*x+c))^(3/2)*b*a^4+1/2/d*e^2/(a^2+b^2)^3/(e*cot(d*x+c)*b
+a*e)^2*(e*cot(d*x+c))^(3/2)*a^2*b^3+5/4/d*e^2/(a^2+b^2)^3/(e*cot(d*x+c)*b+a*e)^2*(e*cot(d*x+c))^(3/2)*b^5-5/4
/d*e^3/(a^2+b^2)^3/(e*cot(d*x+c)*b+a*e)^2*(e*cot(d*x+c))^(1/2)*a^5-1/2/d*e^3/(a^2+b^2)^3/(e*cot(d*x+c)*b+a*e)^
2*(e*cot(d*x+c))^(1/2)*a^3*b^2+3/4/d*e^3/(a^2+b^2)^3/(e*cot(d*x+c)*b+a*e)^2*(e*cot(d*x+c))^(1/2)*a*b^4-3/4/d*e
^2/(a^2+b^2)^3/(a*e*b)^(1/2)*arctan((e*cot(d*x+c))^(1/2)*b/(a*e*b)^(1/2))*a^4+13/2/d*e^2/(a^2+b^2)^3/(a*e*b)^(
1/2)*arctan((e*cot(d*x+c))^(1/2)*b/(a*e*b)^(1/2))*a^2*b^2-3/4/d*e^2/(a^2+b^2)^3/(a*e*b)^(1/2)*arctan((e*cot(d*
x+c))^(1/2)*b/(a*e*b)^(1/2))*b^4-1/2/d*e/(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^3+3/2/d*e/(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*b^2+1/4/d*e/(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^3-3/4/d*e/(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*b^2+1/2/d*e/(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^3-3/2/d*e/(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*b^2+3/2/d*e^2/(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^2*b-1/2/d*e^2/(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)*b^3-3/2
/d*e^2/(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^2*b+1/2/d*e^2/(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)*b^3-3/4/d*e^2/(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^2*b+1/4/d*e^2/(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)))*b^3

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maxima [A]  time = 0.78, size = 492, normalized size = 1.07 \[ -\frac {{\left (\frac {{\left (3 \, a^{4} - 26 \, a^{2} b^{2} + 3 \, b^{4}\right )} e \arctan \left (\frac {b \sqrt {\frac {e}{\tan \left (d x + c\right )}}}{\sqrt {a b e}}\right )}{{\left (a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}\right )} \sqrt {a b e}} - \frac {{\left (\frac {2 \, \sqrt {2} {\left (a^{3} - 3 \, a^{2} b - 3 \, a b^{2} + b^{3}\right )} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \sqrt {e} + 2 \, \sqrt {\frac {e}{\tan \left (d x + c\right )}}\right )}}{2 \, \sqrt {e}}\right )}{\sqrt {e}} + \frac {2 \, \sqrt {2} {\left (a^{3} - 3 \, a^{2} b - 3 \, a b^{2} + b^{3}\right )} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \sqrt {e} - 2 \, \sqrt {\frac {e}{\tan \left (d x + c\right )}}\right )}}{2 \, \sqrt {e}}\right )}{\sqrt {e}} + \frac {\sqrt {2} {\left (a^{3} + 3 \, a^{2} b - 3 \, a b^{2} - b^{3}\right )} \log \left (\sqrt {2} \sqrt {e} \sqrt {\frac {e}{\tan \left (d x + c\right )}} + e + \frac {e}{\tan \left (d x + c\right )}\right )}{\sqrt {e}} - \frac {\sqrt {2} {\left (a^{3} + 3 \, a^{2} b - 3 \, a b^{2} - b^{3}\right )} \log \left (-\sqrt {2} \sqrt {e} \sqrt {\frac {e}{\tan \left (d x + c\right )}} + e + \frac {e}{\tan \left (d x + c\right )}\right )}{\sqrt {e}}\right )} e}{a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}} + \frac {{\left (5 \, a^{3} - 3 \, a b^{2}\right )} e^{2} \sqrt {\frac {e}{\tan \left (d x + c\right )}} + {\left (3 \, a^{2} b - 5 \, b^{3}\right )} e \left (\frac {e}{\tan \left (d x + c\right )}\right )^{\frac {3}{2}}}{{\left (a^{6} + 2 \, a^{4} b^{2} + a^{2} b^{4}\right )} e^{2} + \frac {2 \, {\left (a^{5} b + 2 \, a^{3} b^{3} + a b^{5}\right )} e^{2}}{\tan \left (d x + c\right )} + \frac {{\left (a^{4} b^{2} + 2 \, a^{2} b^{4} + b^{6}\right )} e^{2}}{\tan \left (d x + c\right )^{2}}}\right )} e}{4 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*((3*a^4 - 26*a^2*b^2 + 3*b^4)*e*arctan(b*sqrt(e/tan(d*x + c))/sqrt(a*b*e))/((a^6 + 3*a^4*b^2 + 3*a^2*b^4
+ b^6)*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)*sqr
t(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)*sqrt(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/(a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6) + ((5*a^3 - 3*a*
b^2)*e^2*sqrt(e/tan(d*x + c)) + (3*a^2*b - 5*b^3)*e*(e/tan(d*x + c))^(3/2))/((a^6 + 2*a^4*b^2 + a^2*b^4)*e^2 +
2*(a^5*b + 2*a^3*b^3 + a*b^5)*e^2/tan(d*x + c) + (a^4*b^2 + 2*a^2*b^4 + b^6)*e^2/tan(d*x + c)^2))*e/d

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mupad [B]  time = 6.21, size = 19000, normalized size = 41.21 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

atan(((((518*a*b^15*d^2*e^15 - 18*a^15*b*d^2*e^15 - 4494*a^3*b^13*d^2*e^15 + 3022*a^5*b^11*d^2*e^15 + 17194*a^
7*b^9*d^2*e^15 + 5298*a^9*b^7*d^2*e^15 - 3338*a^11*b^5*d^2*e^15 + 506*a^13*b^3*d^2*e^15)/(a^16*d^5 + b^16*d^5
+ 8*a^2*b^14*d^5 + 28*a^4*b^12*d^5 + 56*a^6*b^10*d^5 + 70*a^8*b^8*d^5 + 56*a^10*b^6*d^5 + 28*a^12*b^4*d^5 + 8*
a^14*b^2*d^5) + (((4224*a^4*b^18*d^4*e^12 - 320*a^2*b^20*d^4*e^12 - 192*b^22*d^4*e^12 + 22272*a^6*b^16*d^4*e^1
2 + 51072*a^8*b^14*d^4*e^12 + 67200*a^10*b^12*d^4*e^12 + 53760*a^12*b^10*d^4*e^12 + 25344*a^14*b^8*d^4*e^12 +
5952*a^16*b^6*d^4*e^12 + 192*a^18*b^4*d^4*e^12 - 128*a^20*b^2*d^4*e^12)/(a^16*d^5 + b^16*d^5 + 8*a^2*b^14*d^5
+ 28*a^4*b^12*d^5 + 56*a^6*b^10*d^5 + 70*a^8*b^8*d^5 + 56*a^10*b^6*d^5 + 28*a^12*b^4*d^5 + 8*a^14*b^2*d^5) + (
(e*cot(c + d*x))^(1/2)*((e^3*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^25*d^4*e^10 + 4608*a^2*b^23*d^4*e^10 + 17920*a^4*b^21*d^4*e^10 + 38
400*a^6*b^19*d^4*e^10 + 46080*a^8*b^17*d^4*e^10 + 21504*a^10*b^15*d^4*e^10 - 21504*a^12*b^13*d^4*e^10 - 46080*
a^14*b^11*d^4*e^10 - 38400*a^16*b^9*d^4*e^10 - 17920*a^18*b^7*d^4*e^10 - 4608*a^20*b^5*d^4*e^10 - 512*a^22*b^3
*d^4*e^10))/(a^16*d^4 + b^16*d^4 + 8*a^2*b^14*d^4 + 28*a^4*b^12*d^4 + 56*a^6*b^10*d^4 + 70*a^8*b^8*d^4 + 56*a^
10*b^6*d^4 + 28*a^12*b^4*d^4 + 8*a^14*b^2*d^4))*((e^3*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)*(1544*a*b^18*d^2*e^13 +
64*a^3*b^16*d^2*e^13 - 7456*a^5*b^14*d^2*e^13 - 576*a^7*b^12*d^2*e^13 + 19504*a^9*b^10*d^2*e^13 + 18880*a^11*
b^8*d^2*e^13 + 3808*a^13*b^6*d^2*e^13 - 960*a^15*b^4*d^2*e^13 + 8*a^17*b^2*d^2*e^13))/(a^16*d^4 + b^16*d^4 + 8
*a^2*b^14*d^4 + 28*a^4*b^12*d^4 + 56*a^6*b^10*d^4 + 70*a^8*b^8*d^4 + 56*a^10*b^6*d^4 + 28*a^12*b^4*d^4 + 8*a^1
4*b^2*d^4))*((e^3*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^3*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)*(41*b^13*e^16 + 9*a^12*b*e^16 - 82*a^2*b^11*e
^16 + 1831*a^4*b^9*e^16 - 4348*a^6*b^7*e^16 + 1671*a^8*b^5*e^16 - 210*a^10*b^3*e^16))/(a^16*d^4 + b^16*d^4 + 8
*a^2*b^14*d^4 + 28*a^4*b^12*d^4 + 56*a^6*b^10*d^4 + 70*a^8*b^8*d^4 + 56*a^10*b^6*d^4 + 28*a^12*b^4*d^4 + 8*a^1
4*b^2*d^4))*((e^3*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 - (((518*a*b^15*d^2*e^15 - 18*a^15*b*d^2*e^15 - 4494*a^3*b^13*d^2*e^15 + 3022*a^5*
b^11*d^2*e^15 + 17194*a^7*b^9*d^2*e^15 + 5298*a^9*b^7*d^2*e^15 - 3338*a^11*b^5*d^2*e^15 + 506*a^13*b^3*d^2*e^1
5)/(a^16*d^5 + b^16*d^5 + 8*a^2*b^14*d^5 + 28*a^4*b^12*d^5 + 56*a^6*b^10*d^5 + 70*a^8*b^8*d^5 + 56*a^10*b^6*d^
5 + 28*a^12*b^4*d^5 + 8*a^14*b^2*d^5) + (((4224*a^4*b^18*d^4*e^12 - 320*a^2*b^20*d^4*e^12 - 192*b^22*d^4*e^12
+ 22272*a^6*b^16*d^4*e^12 + 51072*a^8*b^14*d^4*e^12 + 67200*a^10*b^12*d^4*e^12 + 53760*a^12*b^10*d^4*e^12 + 25
344*a^14*b^8*d^4*e^12 + 5952*a^16*b^6*d^4*e^12 + 192*a^18*b^4*d^4*e^12 - 128*a^20*b^2*d^4*e^12)/(a^16*d^5 + b^
16*d^5 + 8*a^2*b^14*d^5 + 28*a^4*b^12*d^5 + 56*a^6*b^10*d^5 + 70*a^8*b^8*d^5 + 56*a^10*b^6*d^5 + 28*a^12*b^4*d
^5 + 8*a^14*b^2*d^5) - ((e*cot(c + d*x))^(1/2)*((e^3*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^25*d^4*e^10 + 4608*a^2*b^23*d^4*e^10 + 1792
0*a^4*b^21*d^4*e^10 + 38400*a^6*b^19*d^4*e^10 + 46080*a^8*b^17*d^4*e^10 + 21504*a^10*b^15*d^4*e^10 - 21504*a^1
2*b^13*d^4*e^10 - 46080*a^14*b^11*d^4*e^10 - 38400*a^16*b^9*d^4*e^10 - 17920*a^18*b^7*d^4*e^10 - 4608*a^20*b^5
*d^4*e^10 - 512*a^22*b^3*d^4*e^10))/(a^16*d^4 + b^16*d^4 + 8*a^2*b^14*d^4 + 28*a^4*b^12*d^4 + 56*a^6*b^10*d^4
+ 70*a^8*b^8*d^4 + 56*a^10*b^6*d^4 + 28*a^12*b^4*d^4 + 8*a^14*b^2*d^4))*((e^3*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)
*(1544*a*b^18*d^2*e^13 + 64*a^3*b^16*d^2*e^13 - 7456*a^5*b^14*d^2*e^13 - 576*a^7*b^12*d^2*e^13 + 19504*a^9*b^1
0*d^2*e^13 + 18880*a^11*b^8*d^2*e^13 + 3808*a^13*b^6*d^2*e^13 - 960*a^15*b^4*d^2*e^13 + 8*a^17*b^2*d^2*e^13))/
(a^16*d^4 + b^16*d^4 + 8*a^2*b^14*d^4 + 28*a^4*b^12*d^4 + 56*a^6*b^10*d^4 + 70*a^8*b^8*d^4 + 56*a^10*b^6*d^4 +
28*a^12*b^4*d^4 + 8*a^14*b^2*d^4))*((e^3*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^3*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) - ((e*cot(c + d*x))^(1/2)*(41*b^13*e^16 + 9*a^1
2*b*e^16 - 82*a^2*b^11*e^16 + 1831*a^4*b^9*e^16 - 4348*a^6*b^7*e^16 + 1671*a^8*b^5*e^16 - 210*a^10*b^3*e^16))/
(a^16*d^4 + b^16*d^4 + 8*a^2*b^14*d^4 + 28*a^4*b^12*d^4 + 56*a^6*b^10*d^4 + 70*a^8*b^8*d^4 + 56*a^10*b^6*d^4 +
28*a^12*b^4*d^4 + 8*a^14*b^2*d^4))*((e^3*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)/((((518*a*b^15*d^2*e^15 - 18*a^15*b*d^2*e^15 - 4494*a^3*b
^13*d^2*e^15 + 3022*a^5*b^11*d^2*e^15 + 17194*a^7*b^9*d^2*e^15 + 5298*a^9*b^7*d^2*e^15 - 3338*a^11*b^5*d^2*e^1
5 + 506*a^13*b^3*d^2*e^15)/(a^16*d^5 + b^16*d^5 + 8*a^2*b^14*d^5 + 28*a^4*b^12*d^5 + 56*a^6*b^10*d^5 + 70*a^8*
b^8*d^5 + 56*a^10*b^6*d^5 + 28*a^12*b^4*d^5 + 8*a^14*b^2*d^5) + (((4224*a^4*b^18*d^4*e^12 - 320*a^2*b^20*d^4*e
^12 - 192*b^22*d^4*e^12 + 22272*a^6*b^16*d^4*e^12 + 51072*a^8*b^14*d^4*e^12 + 67200*a^10*b^12*d^4*e^12 + 53760
*a^12*b^10*d^4*e^12 + 25344*a^14*b^8*d^4*e^12 + 5952*a^16*b^6*d^4*e^12 + 192*a^18*b^4*d^4*e^12 - 128*a^20*b^2*
d^4*e^12)/(a^16*d^5 + b^16*d^5 + 8*a^2*b^14*d^5 + 28*a^4*b^12*d^5 + 56*a^6*b^10*d^5 + 70*a^8*b^8*d^5 + 56*a^10
*b^6*d^5 + 28*a^12*b^4*d^5 + 8*a^14*b^2*d^5) + ((e*cot(c + d*x))^(1/2)*((e^3*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^25*d^4*e^10 + 4608*
a^2*b^23*d^4*e^10 + 17920*a^4*b^21*d^4*e^10 + 38400*a^6*b^19*d^4*e^10 + 46080*a^8*b^17*d^4*e^10 + 21504*a^10*b
^15*d^4*e^10 - 21504*a^12*b^13*d^4*e^10 - 46080*a^14*b^11*d^4*e^10 - 38400*a^16*b^9*d^4*e^10 - 17920*a^18*b^7*
d^4*e^10 - 4608*a^20*b^5*d^4*e^10 - 512*a^22*b^3*d^4*e^10))/(a^16*d^4 + b^16*d^4 + 8*a^2*b^14*d^4 + 28*a^4*b^1
2*d^4 + 56*a^6*b^10*d^4 + 70*a^8*b^8*d^4 + 56*a^10*b^6*d^4 + 28*a^12*b^4*d^4 + 8*a^14*b^2*d^4))*((e^3*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)*(1544*a*b^18*d^2*e^13 + 64*a^3*b^16*d^2*e^13 - 7456*a^5*b^14*d^2*e^13 - 576*a^7*b^12*
d^2*e^13 + 19504*a^9*b^10*d^2*e^13 + 18880*a^11*b^8*d^2*e^13 + 3808*a^13*b^6*d^2*e^13 - 960*a^15*b^4*d^2*e^13
+ 8*a^17*b^2*d^2*e^13))/(a^16*d^4 + b^16*d^4 + 8*a^2*b^14*d^4 + 28*a^4*b^12*d^4 + 56*a^6*b^10*d^4 + 70*a^8*b^8
*d^4 + 56*a^10*b^6*d^4 + 28*a^12*b^4*d^4 + 8*a^14*b^2*d^4))*((e^3*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^3*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)*(41*b^13*e^16 + 9*a^12*b*e^16 - 82*a^2*b^11*e^16 + 1831*a^4*b^9*e^16 - 4348*a^6*b^7*e^16 + 1671*a^8*b^5*e^1
6 - 210*a^10*b^3*e^16))/(a^16*d^4 + b^16*d^4 + 8*a^2*b^14*d^4 + 28*a^4*b^12*d^4 + 56*a^6*b^10*d^4 + 70*a^8*b^8
*d^4 + 56*a^10*b^6*d^4 + 28*a^12*b^4*d^4 + 8*a^14*b^2*d^4))*((e^3*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) + (((518*a*b^15*d^2*e^15 - 18*a^15*b*
d^2*e^15 - 4494*a^3*b^13*d^2*e^15 + 3022*a^5*b^11*d^2*e^15 + 17194*a^7*b^9*d^2*e^15 + 5298*a^9*b^7*d^2*e^15 -
3338*a^11*b^5*d^2*e^15 + 506*a^13*b^3*d^2*e^15)/(a^16*d^5 + b^16*d^5 + 8*a^2*b^14*d^5 + 28*a^4*b^12*d^5 + 56*a
^6*b^10*d^5 + 70*a^8*b^8*d^5 + 56*a^10*b^6*d^5 + 28*a^12*b^4*d^5 + 8*a^14*b^2*d^5) + (((4224*a^4*b^18*d^4*e^12
- 320*a^2*b^20*d^4*e^12 - 192*b^22*d^4*e^12 + 22272*a^6*b^16*d^4*e^12 + 51072*a^8*b^14*d^4*e^12 + 67200*a^10*
b^12*d^4*e^12 + 53760*a^12*b^10*d^4*e^12 + 25344*a^14*b^8*d^4*e^12 + 5952*a^16*b^6*d^4*e^12 + 192*a^18*b^4*d^4
*e^12 - 128*a^20*b^2*d^4*e^12)/(a^16*d^5 + b^16*d^5 + 8*a^2*b^14*d^5 + 28*a^4*b^12*d^5 + 56*a^6*b^10*d^5 + 70*
a^8*b^8*d^5 + 56*a^10*b^6*d^5 + 28*a^12*b^4*d^5 + 8*a^14*b^2*d^5) - ((e*cot(c + d*x))^(1/2)*((e^3*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^25*d^4*e^10 + 4608*a^2*b^23*d^4*e^10 + 17920*a^4*b^21*d^4*e^10 + 38400*a^6*b^19*d^4*e^10 + 46080*a^8*b^17*d^
4*e^10 + 21504*a^10*b^15*d^4*e^10 - 21504*a^12*b^13*d^4*e^10 - 46080*a^14*b^11*d^4*e^10 - 38400*a^16*b^9*d^4*e
^10 - 17920*a^18*b^7*d^4*e^10 - 4608*a^20*b^5*d^4*e^10 - 512*a^22*b^3*d^4*e^10))/(a^16*d^4 + b^16*d^4 + 8*a^2*
b^14*d^4 + 28*a^4*b^12*d^4 + 56*a^6*b^10*d^4 + 70*a^8*b^8*d^4 + 56*a^10*b^6*d^4 + 28*a^12*b^4*d^4 + 8*a^14*b^2
*d^4))*((e^3*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)*(1544*a*b^18*d^2*e^13 + 64*a^3*b^16*d^2*e^13 - 7456*a^5*b^14*d^2
*e^13 - 576*a^7*b^12*d^2*e^13 + 19504*a^9*b^10*d^2*e^13 + 18880*a^11*b^8*d^2*e^13 + 3808*a^13*b^6*d^2*e^13 - 9
60*a^15*b^4*d^2*e^13 + 8*a^17*b^2*d^2*e^13))/(a^16*d^4 + b^16*d^4 + 8*a^2*b^14*d^4 + 28*a^4*b^12*d^4 + 56*a^6*
b^10*d^4 + 70*a^8*b^8*d^4 + 56*a^10*b^6*d^4 + 28*a^12*b^4*d^4 + 8*a^14*b^2*d^4))*((e^3*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^3*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)*(41*b^13*e^16 + 9*a^12*b*e^16 - 82*a^2*b^11*e^16 + 1831*a^4*b^9*e^16 - 4348*a^6*b^7*e^
16 + 1671*a^8*b^5*e^16 - 210*a^10*b^3*e^16))/(a^16*d^4 + b^16*d^4 + 8*a^2*b^14*d^4 + 28*a^4*b^12*d^4 + 56*a^6*
b^10*d^4 + 70*a^8*b^8*d^4 + 56*a^10*b^6*d^4 + 28*a^12*b^4*d^4 + 8*a^14*b^2*d^4))*((e^3*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) + (28*a^2*b^8*e^
18 - 15*b^10*e^18 + 878*a^4*b^6*e^18 - 180*a^6*b^4*e^18 + 9*a^8*b^2*e^18)/(a^16*d^5 + b^16*d^5 + 8*a^2*b^14*d^
5 + 28*a^4*b^12*d^5 + 56*a^6*b^10*d^5 + 70*a^8*b^8*d^5 + 56*a^10*b^6*d^5 + 28*a^12*b^4*d^5 + 8*a^14*b^2*d^5)))
*((e^3*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 + atan(((((518*a*b^15*d^2*e^15 - 18*a^15*b*d^2*e^15 - 4494*a^3*b^13*d^2*e^15 + 3022*a^5*b^11*
d^2*e^15 + 17194*a^7*b^9*d^2*e^15 + 5298*a^9*b^7*d^2*e^15 - 3338*a^11*b^5*d^2*e^15 + 506*a^13*b^3*d^2*e^15)/(a
^16*d^5 + b^16*d^5 + 8*a^2*b^14*d^5 + 28*a^4*b^12*d^5 + 56*a^6*b^10*d^5 + 70*a^8*b^8*d^5 + 56*a^10*b^6*d^5 + 2
8*a^12*b^4*d^5 + 8*a^14*b^2*d^5) + (((4224*a^4*b^18*d^4*e^12 - 320*a^2*b^20*d^4*e^12 - 192*b^22*d^4*e^12 + 222
72*a^6*b^16*d^4*e^12 + 51072*a^8*b^14*d^4*e^12 + 67200*a^10*b^12*d^4*e^12 + 53760*a^12*b^10*d^4*e^12 + 25344*a
^14*b^8*d^4*e^12 + 5952*a^16*b^6*d^4*e^12 + 192*a^18*b^4*d^4*e^12 - 128*a^20*b^2*d^4*e^12)/(a^16*d^5 + b^16*d^
5 + 8*a^2*b^14*d^5 + 28*a^4*b^12*d^5 + 56*a^6*b^10*d^5 + 70*a^8*b^8*d^5 + 56*a^10*b^6*d^5 + 28*a^12*b^4*d^5 +
8*a^14*b^2*d^5) + ((e*cot(c + d*x))^(1/2)*(e^3/(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^25*d^4*e^10 + 4608*a^2*b^23*d^4*e^10 + 17920*a^4
*b^21*d^4*e^10 + 38400*a^6*b^19*d^4*e^10 + 46080*a^8*b^17*d^4*e^10 + 21504*a^10*b^15*d^4*e^10 - 21504*a^12*b^1
3*d^4*e^10 - 46080*a^14*b^11*d^4*e^10 - 38400*a^16*b^9*d^4*e^10 - 17920*a^18*b^7*d^4*e^10 - 4608*a^20*b^5*d^4*
e^10 - 512*a^22*b^3*d^4*e^10))/(a^16*d^4 + b^16*d^4 + 8*a^2*b^14*d^4 + 28*a^4*b^12*d^4 + 56*a^6*b^10*d^4 + 70*
a^8*b^8*d^4 + 56*a^10*b^6*d^4 + 28*a^12*b^4*d^4 + 8*a^14*b^2*d^4))*(e^3/(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)*(154
4*a*b^18*d^2*e^13 + 64*a^3*b^16*d^2*e^13 - 7456*a^5*b^14*d^2*e^13 - 576*a^7*b^12*d^2*e^13 + 19504*a^9*b^10*d^2
*e^13 + 18880*a^11*b^8*d^2*e^13 + 3808*a^13*b^6*d^2*e^13 - 960*a^15*b^4*d^2*e^13 + 8*a^17*b^2*d^2*e^13))/(a^16
*d^4 + b^16*d^4 + 8*a^2*b^14*d^4 + 28*a^4*b^12*d^4 + 56*a^6*b^10*d^4 + 70*a^8*b^8*d^4 + 56*a^10*b^6*d^4 + 28*a
^12*b^4*d^4 + 8*a^14*b^2*d^4))*(e^3/(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^3/(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)*(41*b^13*e^16 + 9*a^12*b*e
^16 - 82*a^2*b^11*e^16 + 1831*a^4*b^9*e^16 - 4348*a^6*b^7*e^16 + 1671*a^8*b^5*e^16 - 210*a^10*b^3*e^16))/(a^16
*d^4 + b^16*d^4 + 8*a^2*b^14*d^4 + 28*a^4*b^12*d^4 + 56*a^6*b^10*d^4 + 70*a^8*b^8*d^4 + 56*a^10*b^6*d^4 + 28*a
^12*b^4*d^4 + 8*a^14*b^2*d^4))*(e^3/(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 - (((518*a*b^15*d^2*e^15 - 18*a^15*b*d^2*e^15 - 4494*a^3*b^13*d
^2*e^15 + 3022*a^5*b^11*d^2*e^15 + 17194*a^7*b^9*d^2*e^15 + 5298*a^9*b^7*d^2*e^15 - 3338*a^11*b^5*d^2*e^15 + 5
06*a^13*b^3*d^2*e^15)/(a^16*d^5 + b^16*d^5 + 8*a^2*b^14*d^5 + 28*a^4*b^12*d^5 + 56*a^6*b^10*d^5 + 70*a^8*b^8*d
^5 + 56*a^10*b^6*d^5 + 28*a^12*b^4*d^5 + 8*a^14*b^2*d^5) + (((4224*a^4*b^18*d^4*e^12 - 320*a^2*b^20*d^4*e^12 -
192*b^22*d^4*e^12 + 22272*a^6*b^16*d^4*e^12 + 51072*a^8*b^14*d^4*e^12 + 67200*a^10*b^12*d^4*e^12 + 53760*a^12
*b^10*d^4*e^12 + 25344*a^14*b^8*d^4*e^12 + 5952*a^16*b^6*d^4*e^12 + 192*a^18*b^4*d^4*e^12 - 128*a^20*b^2*d^4*e
^12)/(a^16*d^5 + b^16*d^5 + 8*a^2*b^14*d^5 + 28*a^4*b^12*d^5 + 56*a^6*b^10*d^5 + 70*a^8*b^8*d^5 + 56*a^10*b^6*
d^5 + 28*a^12*b^4*d^5 + 8*a^14*b^2*d^5) - ((e*cot(c + d*x))^(1/2)*(e^3/(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^25*d^4*e^10 + 4608*a^2*b
^23*d^4*e^10 + 17920*a^4*b^21*d^4*e^10 + 38400*a^6*b^19*d^4*e^10 + 46080*a^8*b^17*d^4*e^10 + 21504*a^10*b^15*d
^4*e^10 - 21504*a^12*b^13*d^4*e^10 - 46080*a^14*b^11*d^4*e^10 - 38400*a^16*b^9*d^4*e^10 - 17920*a^18*b^7*d^4*e
^10 - 4608*a^20*b^5*d^4*e^10 - 512*a^22*b^3*d^4*e^10))/(a^16*d^4 + b^16*d^4 + 8*a^2*b^14*d^4 + 28*a^4*b^12*d^4
+ 56*a^6*b^10*d^4 + 70*a^8*b^8*d^4 + 56*a^10*b^6*d^4 + 28*a^12*b^4*d^4 + 8*a^14*b^2*d^4))*(e^3/(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)*(1544*a*b^18*d^2*e^13 + 64*a^3*b^16*d^2*e^13 - 7456*a^5*b^14*d^2*e^13 - 576*a^7*b^12*d^2*e
^13 + 19504*a^9*b^10*d^2*e^13 + 18880*a^11*b^8*d^2*e^13 + 3808*a^13*b^6*d^2*e^13 - 960*a^15*b^4*d^2*e^13 + 8*a
^17*b^2*d^2*e^13))/(a^16*d^4 + b^16*d^4 + 8*a^2*b^14*d^4 + 28*a^4*b^12*d^4 + 56*a^6*b^10*d^4 + 70*a^8*b^8*d^4
+ 56*a^10*b^6*d^4 + 28*a^12*b^4*d^4 + 8*a^14*b^2*d^4))*(e^3/(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^3/(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)*(4
1*b^13*e^16 + 9*a^12*b*e^16 - 82*a^2*b^11*e^16 + 1831*a^4*b^9*e^16 - 4348*a^6*b^7*e^16 + 1671*a^8*b^5*e^16 - 2
10*a^10*b^3*e^16))/(a^16*d^4 + b^16*d^4 + 8*a^2*b^14*d^4 + 28*a^4*b^12*d^4 + 56*a^6*b^10*d^4 + 70*a^8*b^8*d^4
+ 56*a^10*b^6*d^4 + 28*a^12*b^4*d^4 + 8*a^14*b^2*d^4))*(e^3/(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)/((((518*a*b^15*d^2*e^15 - 18*a^15*b*d^
2*e^15 - 4494*a^3*b^13*d^2*e^15 + 3022*a^5*b^11*d^2*e^15 + 17194*a^7*b^9*d^2*e^15 + 5298*a^9*b^7*d^2*e^15 - 33
38*a^11*b^5*d^2*e^15 + 506*a^13*b^3*d^2*e^15)/(a^16*d^5 + b^16*d^5 + 8*a^2*b^14*d^5 + 28*a^4*b^12*d^5 + 56*a^6
*b^10*d^5 + 70*a^8*b^8*d^5 + 56*a^10*b^6*d^5 + 28*a^12*b^4*d^5 + 8*a^14*b^2*d^5) + (((4224*a^4*b^18*d^4*e^12 -
320*a^2*b^20*d^4*e^12 - 192*b^22*d^4*e^12 + 22272*a^6*b^16*d^4*e^12 + 51072*a^8*b^14*d^4*e^12 + 67200*a^10*b^
12*d^4*e^12 + 53760*a^12*b^10*d^4*e^12 + 25344*a^14*b^8*d^4*e^12 + 5952*a^16*b^6*d^4*e^12 + 192*a^18*b^4*d^4*e
^12 - 128*a^20*b^2*d^4*e^12)/(a^16*d^5 + b^16*d^5 + 8*a^2*b^14*d^5 + 28*a^4*b^12*d^5 + 56*a^6*b^10*d^5 + 70*a^
8*b^8*d^5 + 56*a^10*b^6*d^5 + 28*a^12*b^4*d^5 + 8*a^14*b^2*d^5) + ((e*cot(c + d*x))^(1/2)*(e^3/(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^
25*d^4*e^10 + 4608*a^2*b^23*d^4*e^10 + 17920*a^4*b^21*d^4*e^10 + 38400*a^6*b^19*d^4*e^10 + 46080*a^8*b^17*d^4*
e^10 + 21504*a^10*b^15*d^4*e^10 - 21504*a^12*b^13*d^4*e^10 - 46080*a^14*b^11*d^4*e^10 - 38400*a^16*b^9*d^4*e^1
0 - 17920*a^18*b^7*d^4*e^10 - 4608*a^20*b^5*d^4*e^10 - 512*a^22*b^3*d^4*e^10))/(a^16*d^4 + b^16*d^4 + 8*a^2*b^
14*d^4 + 28*a^4*b^12*d^4 + 56*a^6*b^10*d^4 + 70*a^8*b^8*d^4 + 56*a^10*b^6*d^4 + 28*a^12*b^4*d^4 + 8*a^14*b^2*d
^4))*(e^3/(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)*(1544*a*b^18*d^2*e^13 + 64*a^3*b^16*d^2*e^13 - 7456*a^5*b^14*d^2*e
^13 - 576*a^7*b^12*d^2*e^13 + 19504*a^9*b^10*d^2*e^13 + 18880*a^11*b^8*d^2*e^13 + 3808*a^13*b^6*d^2*e^13 - 960
*a^15*b^4*d^2*e^13 + 8*a^17*b^2*d^2*e^13))/(a^16*d^4 + b^16*d^4 + 8*a^2*b^14*d^4 + 28*a^4*b^12*d^4 + 56*a^6*b^
10*d^4 + 70*a^8*b^8*d^4 + 56*a^10*b^6*d^4 + 28*a^12*b^4*d^4 + 8*a^14*b^2*d^4))*(e^3/(4*(b^6*d^2*1i - a^6*d^2*1
i + 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^3/(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)*(41*b^13*e^16 + 9*a^12*b*e^16 - 82*a^2*b^11*e^16 + 1831*a^4*b^9*e^16 - 4348*a^6*b^7*e^16
+ 1671*a^8*b^5*e^16 - 210*a^10*b^3*e^16))/(a^16*d^4 + b^16*d^4 + 8*a^2*b^14*d^4 + 28*a^4*b^12*d^4 + 56*a^6*b^
10*d^4 + 70*a^8*b^8*d^4 + 56*a^10*b^6*d^4 + 28*a^12*b^4*d^4 + 8*a^14*b^2*d^4))*(e^3/(4*(b^6*d^2*1i - a^6*d^2*1
i + 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) + (((518*a*b^15*d^
2*e^15 - 18*a^15*b*d^2*e^15 - 4494*a^3*b^13*d^2*e^15 + 3022*a^5*b^11*d^2*e^15 + 17194*a^7*b^9*d^2*e^15 + 5298*
a^9*b^7*d^2*e^15 - 3338*a^11*b^5*d^2*e^15 + 506*a^13*b^3*d^2*e^15)/(a^16*d^5 + b^16*d^5 + 8*a^2*b^14*d^5 + 28*
a^4*b^12*d^5 + 56*a^6*b^10*d^5 + 70*a^8*b^8*d^5 + 56*a^10*b^6*d^5 + 28*a^12*b^4*d^5 + 8*a^14*b^2*d^5) + (((422
4*a^4*b^18*d^4*e^12 - 320*a^2*b^20*d^4*e^12 - 192*b^22*d^4*e^12 + 22272*a^6*b^16*d^4*e^12 + 51072*a^8*b^14*d^4
*e^12 + 67200*a^10*b^12*d^4*e^12 + 53760*a^12*b^10*d^4*e^12 + 25344*a^14*b^8*d^4*e^12 + 5952*a^16*b^6*d^4*e^12
+ 192*a^18*b^4*d^4*e^12 - 128*a^20*b^2*d^4*e^12)/(a^16*d^5 + b^16*d^5 + 8*a^2*b^14*d^5 + 28*a^4*b^12*d^5 + 56
*a^6*b^10*d^5 + 70*a^8*b^8*d^5 + 56*a^10*b^6*d^5 + 28*a^12*b^4*d^5 + 8*a^14*b^2*d^5) - ((e*cot(c + d*x))^(1/2)
*(e^3/(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^25*d^4*e^10 + 4608*a^2*b^23*d^4*e^10 + 17920*a^4*b^21*d^4*e^10 + 38400*a^6*b^19*d^4*e^10
+ 46080*a^8*b^17*d^4*e^10 + 21504*a^10*b^15*d^4*e^10 - 21504*a^12*b^13*d^4*e^10 - 46080*a^14*b^11*d^4*e^10 - 3
8400*a^16*b^9*d^4*e^10 - 17920*a^18*b^7*d^4*e^10 - 4608*a^20*b^5*d^4*e^10 - 512*a^22*b^3*d^4*e^10))/(a^16*d^4
+ b^16*d^4 + 8*a^2*b^14*d^4 + 28*a^4*b^12*d^4 + 56*a^6*b^10*d^4 + 70*a^8*b^8*d^4 + 56*a^10*b^6*d^4 + 28*a^12*b
^4*d^4 + 8*a^14*b^2*d^4))*(e^3/(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)*(1544*a*b^18*d^2*e^13 + 64*a^3*b^16*d^2*e^13
- 7456*a^5*b^14*d^2*e^13 - 576*a^7*b^12*d^2*e^13 + 19504*a^9*b^10*d^2*e^13 + 18880*a^11*b^8*d^2*e^13 + 3808*a^
13*b^6*d^2*e^13 - 960*a^15*b^4*d^2*e^13 + 8*a^17*b^2*d^2*e^13))/(a^16*d^4 + b^16*d^4 + 8*a^2*b^14*d^4 + 28*a^4
*b^12*d^4 + 56*a^6*b^10*d^4 + 70*a^8*b^8*d^4 + 56*a^10*b^6*d^4 + 28*a^12*b^4*d^4 + 8*a^14*b^2*d^4))*(e^3/(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^3/(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)*(41*b^13*e^16 + 9*a^12*b*e^16 - 82*a^2*b^11*e^16 + 1831*a^4*b^9*e^1
6 - 4348*a^6*b^7*e^16 + 1671*a^8*b^5*e^16 - 210*a^10*b^3*e^16))/(a^16*d^4 + b^16*d^4 + 8*a^2*b^14*d^4 + 28*a^4
*b^12*d^4 + 56*a^6*b^10*d^4 + 70*a^8*b^8*d^4 + 56*a^10*b^6*d^4 + 28*a^12*b^4*d^4 + 8*a^14*b^2*d^4))*(e^3/(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) + (28*a^2*b^8*e^18 - 15*b^10*e^18 + 878*a^4*b^6*e^18 - 180*a^6*b^4*e^18 + 9*a^8*b^2*e^18)/(a^16*d^5 + b^16*
d^5 + 8*a^2*b^14*d^5 + 28*a^4*b^12*d^5 + 56*a^6*b^10*d^5 + 70*a^8*b^8*d^5 + 56*a^10*b^6*d^5 + 28*a^12*b^4*d^5
+ 8*a^14*b^2*d^5)))*(e^3/(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 - (((e*cot(c + d*x))^(1/2)*(5*a^3*e^3 - 3*a*b^2*e^3))/(4*(a^4 + b^4 + 2*a^
2*b^2)) + (b*e^2*(e*cot(c + d*x))^(3/2)*(3*a^2 - 5*b^2))/(4*(a^4 + b^4 + 2*a^2*b^2)))/(a^2*d*e^2 + b^2*d*e^2*c
ot(c + d*x)^2 + 2*a*b*d*e^2*cot(c + d*x)) + (atan((((((e*cot(c + d*x))^(1/2)*(41*b^13*e^16 + 9*a^12*b*e^16 - 8
2*a^2*b^11*e^16 + 1831*a^4*b^9*e^16 - 4348*a^6*b^7*e^16 + 1671*a^8*b^5*e^16 - 210*a^10*b^3*e^16))/(a^16*d^4 +
b^16*d^4 + 8*a^2*b^14*d^4 + 28*a^4*b^12*d^4 + 56*a^6*b^10*d^4 + 70*a^8*b^8*d^4 + 56*a^10*b^6*d^4 + 28*a^12*b^4
*d^4 + 8*a^14*b^2*d^4) - (((518*a*b^15*d^2*e^15 - 18*a^15*b*d^2*e^15 - 4494*a^3*b^13*d^2*e^15 + 3022*a^5*b^11*
d^2*e^15 + 17194*a^7*b^9*d^2*e^15 + 5298*a^9*b^7*d^2*e^15 - 3338*a^11*b^5*d^2*e^15 + 506*a^13*b^3*d^2*e^15)/(a
^16*d^5 + b^16*d^5 + 8*a^2*b^14*d^5 + 28*a^4*b^12*d^5 + 56*a^6*b^10*d^5 + 70*a^8*b^8*d^5 + 56*a^10*b^6*d^5 + 2
8*a^12*b^4*d^5 + 8*a^14*b^2*d^5) + ((((e*cot(c + d*x))^(1/2)*(1544*a*b^18*d^2*e^13 + 64*a^3*b^16*d^2*e^13 - 74
56*a^5*b^14*d^2*e^13 - 576*a^7*b^12*d^2*e^13 + 19504*a^9*b^10*d^2*e^13 + 18880*a^11*b^8*d^2*e^13 + 3808*a^13*b
^6*d^2*e^13 - 960*a^15*b^4*d^2*e^13 + 8*a^17*b^2*d^2*e^13))/(a^16*d^4 + b^16*d^4 + 8*a^2*b^14*d^4 + 28*a^4*b^1
2*d^4 + 56*a^6*b^10*d^4 + 70*a^8*b^8*d^4 + 56*a^10*b^6*d^4 + 28*a^12*b^4*d^4 + 8*a^14*b^2*d^4) + (((4224*a^4*b
^18*d^4*e^12 - 320*a^2*b^20*d^4*e^12 - 192*b^22*d^4*e^12 + 22272*a^6*b^16*d^4*e^12 + 51072*a^8*b^14*d^4*e^12 +
67200*a^10*b^12*d^4*e^12 + 53760*a^12*b^10*d^4*e^12 + 25344*a^14*b^8*d^4*e^12 + 5952*a^16*b^6*d^4*e^12 + 192*
a^18*b^4*d^4*e^12 - 128*a^20*b^2*d^4*e^12)/(a^16*d^5 + b^16*d^5 + 8*a^2*b^14*d^5 + 28*a^4*b^12*d^5 + 56*a^6*b^
10*d^5 + 70*a^8*b^8*d^5 + 56*a^10*b^6*d^5 + 28*a^12*b^4*d^5 + 8*a^14*b^2*d^5) - ((e*cot(c + d*x))^(1/2)*(3*a^4
+ 3*b^4 - 26*a^2*b^2)*(-a*b*e^3)^(1/2)*(512*b^25*d^4*e^10 + 4608*a^2*b^23*d^4*e^10 + 17920*a^4*b^21*d^4*e^10
+ 38400*a^6*b^19*d^4*e^10 + 46080*a^8*b^17*d^4*e^10 + 21504*a^10*b^15*d^4*e^10 - 21504*a^12*b^13*d^4*e^10 - 46
080*a^14*b^11*d^4*e^10 - 38400*a^16*b^9*d^4*e^10 - 17920*a^18*b^7*d^4*e^10 - 4608*a^20*b^5*d^4*e^10 - 512*a^22
*b^3*d^4*e^10))/(8*(3*a^3*b^5*d + 3*a^5*b^3*d + a*b^7*d + a^7*b*d)*(a^16*d^4 + b^16*d^4 + 8*a^2*b^14*d^4 + 28*
a^4*b^12*d^4 + 56*a^6*b^10*d^4 + 70*a^8*b^8*d^4 + 56*a^10*b^6*d^4 + 28*a^12*b^4*d^4 + 8*a^14*b^2*d^4)))*(3*a^4
+ 3*b^4 - 26*a^2*b^2)*(-a*b*e^3)^(1/2))/(8*(3*a^3*b^5*d + 3*a^5*b^3*d + a*b^7*d + a^7*b*d)))*(3*a^4 + 3*b^4 -
26*a^2*b^2)*(-a*b*e^3)^(1/2))/(8*(3*a^3*b^5*d + 3*a^5*b^3*d + a*b^7*d + a^7*b*d)))*(3*a^4 + 3*b^4 - 26*a^2*b^
2)*(-a*b*e^3)^(1/2))/(8*(3*a^3*b^5*d + 3*a^5*b^3*d + a*b^7*d + a^7*b*d)))*(3*a^4 + 3*b^4 - 26*a^2*b^2)*(-a*b*e
^3)^(1/2)*1i)/(8*(3*a^3*b^5*d + 3*a^5*b^3*d + a*b^7*d + a^7*b*d)) + ((((e*cot(c + d*x))^(1/2)*(41*b^13*e^16 +
9*a^12*b*e^16 - 82*a^2*b^11*e^16 + 1831*a^4*b^9*e^16 - 4348*a^6*b^7*e^16 + 1671*a^8*b^5*e^16 - 210*a^10*b^3*e^
16))/(a^16*d^4 + b^16*d^4 + 8*a^2*b^14*d^4 + 28*a^4*b^12*d^4 + 56*a^6*b^10*d^4 + 70*a^8*b^8*d^4 + 56*a^10*b^6*
d^4 + 28*a^12*b^4*d^4 + 8*a^14*b^2*d^4) + (((518*a*b^15*d^2*e^15 - 18*a^15*b*d^2*e^15 - 4494*a^3*b^13*d^2*e^15
+ 3022*a^5*b^11*d^2*e^15 + 17194*a^7*b^9*d^2*e^15 + 5298*a^9*b^7*d^2*e^15 - 3338*a^11*b^5*d^2*e^15 + 506*a^13
*b^3*d^2*e^15)/(a^16*d^5 + b^16*d^5 + 8*a^2*b^14*d^5 + 28*a^4*b^12*d^5 + 56*a^6*b^10*d^5 + 70*a^8*b^8*d^5 + 56
*a^10*b^6*d^5 + 28*a^12*b^4*d^5 + 8*a^14*b^2*d^5) - ((((e*cot(c + d*x))^(1/2)*(1544*a*b^18*d^2*e^13 + 64*a^3*b
^16*d^2*e^13 - 7456*a^5*b^14*d^2*e^13 - 576*a^7*b^12*d^2*e^13 + 19504*a^9*b^10*d^2*e^13 + 18880*a^11*b^8*d^2*e
^13 + 3808*a^13*b^6*d^2*e^13 - 960*a^15*b^4*d^2*e^13 + 8*a^17*b^2*d^2*e^13))/(a^16*d^4 + b^16*d^4 + 8*a^2*b^14
*d^4 + 28*a^4*b^12*d^4 + 56*a^6*b^10*d^4 + 70*a^8*b^8*d^4 + 56*a^10*b^6*d^4 + 28*a^12*b^4*d^4 + 8*a^14*b^2*d^4
) - (((4224*a^4*b^18*d^4*e^12 - 320*a^2*b^20*d^4*e^12 - 192*b^22*d^4*e^12 + 22272*a^6*b^16*d^4*e^12 + 51072*a^
8*b^14*d^4*e^12 + 67200*a^10*b^12*d^4*e^12 + 53760*a^12*b^10*d^4*e^12 + 25344*a^14*b^8*d^4*e^12 + 5952*a^16*b^
6*d^4*e^12 + 192*a^18*b^4*d^4*e^12 - 128*a^20*b^2*d^4*e^12)/(a^16*d^5 + b^16*d^5 + 8*a^2*b^14*d^5 + 28*a^4*b^1
2*d^5 + 56*a^6*b^10*d^5 + 70*a^8*b^8*d^5 + 56*a^10*b^6*d^5 + 28*a^12*b^4*d^5 + 8*a^14*b^2*d^5) + ((e*cot(c + d
*x))^(1/2)*(3*a^4 + 3*b^4 - 26*a^2*b^2)*(-a*b*e^3)^(1/2)*(512*b^25*d^4*e^10 + 4608*a^2*b^23*d^4*e^10 + 17920*a
^4*b^21*d^4*e^10 + 38400*a^6*b^19*d^4*e^10 + 46080*a^8*b^17*d^4*e^10 + 21504*a^10*b^15*d^4*e^10 - 21504*a^12*b
^13*d^4*e^10 - 46080*a^14*b^11*d^4*e^10 - 38400*a^16*b^9*d^4*e^10 - 17920*a^18*b^7*d^4*e^10 - 4608*a^20*b^5*d^
4*e^10 - 512*a^22*b^3*d^4*e^10))/(8*(3*a^3*b^5*d + 3*a^5*b^3*d + a*b^7*d + a^7*b*d)*(a^16*d^4 + b^16*d^4 + 8*a
^2*b^14*d^4 + 28*a^4*b^12*d^4 + 56*a^6*b^10*d^4 + 70*a^8*b^8*d^4 + 56*a^10*b^6*d^4 + 28*a^12*b^4*d^4 + 8*a^14*
b^2*d^4)))*(3*a^4 + 3*b^4 - 26*a^2*b^2)*(-a*b*e^3)^(1/2))/(8*(3*a^3*b^5*d + 3*a^5*b^3*d + a*b^7*d + a^7*b*d)))
*(3*a^4 + 3*b^4 - 26*a^2*b^2)*(-a*b*e^3)^(1/2))/(8*(3*a^3*b^5*d + 3*a^5*b^3*d + a*b^7*d + a^7*b*d)))*(3*a^4 +
3*b^4 - 26*a^2*b^2)*(-a*b*e^3)^(1/2))/(8*(3*a^3*b^5*d + 3*a^5*b^3*d + a*b^7*d + a^7*b*d)))*(3*a^4 + 3*b^4 - 26
*a^2*b^2)*(-a*b*e^3)^(1/2)*1i)/(8*(3*a^3*b^5*d + 3*a^5*b^3*d + a*b^7*d + a^7*b*d)))/((28*a^2*b^8*e^18 - 15*b^1
0*e^18 + 878*a^4*b^6*e^18 - 180*a^6*b^4*e^18 + 9*a^8*b^2*e^18)/(a^16*d^5 + b^16*d^5 + 8*a^2*b^14*d^5 + 28*a^4*
b^12*d^5 + 56*a^6*b^10*d^5 + 70*a^8*b^8*d^5 + 56*a^10*b^6*d^5 + 28*a^12*b^4*d^5 + 8*a^14*b^2*d^5) - ((((e*cot(
c + d*x))^(1/2)*(41*b^13*e^16 + 9*a^12*b*e^16 - 82*a^2*b^11*e^16 + 1831*a^4*b^9*e^16 - 4348*a^6*b^7*e^16 + 167
1*a^8*b^5*e^16 - 210*a^10*b^3*e^16))/(a^16*d^4 + b^16*d^4 + 8*a^2*b^14*d^4 + 28*a^4*b^12*d^4 + 56*a^6*b^10*d^4
+ 70*a^8*b^8*d^4 + 56*a^10*b^6*d^4 + 28*a^12*b^4*d^4 + 8*a^14*b^2*d^4) - (((518*a*b^15*d^2*e^15 - 18*a^15*b*d
^2*e^15 - 4494*a^3*b^13*d^2*e^15 + 3022*a^5*b^11*d^2*e^15 + 17194*a^7*b^9*d^2*e^15 + 5298*a^9*b^7*d^2*e^15 - 3
338*a^11*b^5*d^2*e^15 + 506*a^13*b^3*d^2*e^15)/(a^16*d^5 + b^16*d^5 + 8*a^2*b^14*d^5 + 28*a^4*b^12*d^5 + 56*a^
6*b^10*d^5 + 70*a^8*b^8*d^5 + 56*a^10*b^6*d^5 + 28*a^12*b^4*d^5 + 8*a^14*b^2*d^5) + ((((e*cot(c + d*x))^(1/2)*
(1544*a*b^18*d^2*e^13 + 64*a^3*b^16*d^2*e^13 - 7456*a^5*b^14*d^2*e^13 - 576*a^7*b^12*d^2*e^13 + 19504*a^9*b^10
*d^2*e^13 + 18880*a^11*b^8*d^2*e^13 + 3808*a^13*b^6*d^2*e^13 - 960*a^15*b^4*d^2*e^13 + 8*a^17*b^2*d^2*e^13))/(
a^16*d^4 + b^16*d^4 + 8*a^2*b^14*d^4 + 28*a^4*b^12*d^4 + 56*a^6*b^10*d^4 + 70*a^8*b^8*d^4 + 56*a^10*b^6*d^4 +
28*a^12*b^4*d^4 + 8*a^14*b^2*d^4) + (((4224*a^4*b^18*d^4*e^12 - 320*a^2*b^20*d^4*e^12 - 192*b^22*d^4*e^12 + 22
272*a^6*b^16*d^4*e^12 + 51072*a^8*b^14*d^4*e^12 + 67200*a^10*b^12*d^4*e^12 + 53760*a^12*b^10*d^4*e^12 + 25344*
a^14*b^8*d^4*e^12 + 5952*a^16*b^6*d^4*e^12 + 192*a^18*b^4*d^4*e^12 - 128*a^20*b^2*d^4*e^12)/(a^16*d^5 + b^16*d
^5 + 8*a^2*b^14*d^5 + 28*a^4*b^12*d^5 + 56*a^6*b^10*d^5 + 70*a^8*b^8*d^5 + 56*a^10*b^6*d^5 + 28*a^12*b^4*d^5 +
8*a^14*b^2*d^5) - ((e*cot(c + d*x))^(1/2)*(3*a^4 + 3*b^4 - 26*a^2*b^2)*(-a*b*e^3)^(1/2)*(512*b^25*d^4*e^10 +
4608*a^2*b^23*d^4*e^10 + 17920*a^4*b^21*d^4*e^10 + 38400*a^6*b^19*d^4*e^10 + 46080*a^8*b^17*d^4*e^10 + 21504*a
^10*b^15*d^4*e^10 - 21504*a^12*b^13*d^4*e^10 - 46080*a^14*b^11*d^4*e^10 - 38400*a^16*b^9*d^4*e^10 - 17920*a^18
*b^7*d^4*e^10 - 4608*a^20*b^5*d^4*e^10 - 512*a^22*b^3*d^4*e^10))/(8*(3*a^3*b^5*d + 3*a^5*b^3*d + a*b^7*d + a^7
*b*d)*(a^16*d^4 + b^16*d^4 + 8*a^2*b^14*d^4 + 28*a^4*b^12*d^4 + 56*a^6*b^10*d^4 + 70*a^8*b^8*d^4 + 56*a^10*b^6
*d^4 + 28*a^12*b^4*d^4 + 8*a^14*b^2*d^4)))*(3*a^4 + 3*b^4 - 26*a^2*b^2)*(-a*b*e^3)^(1/2))/(8*(3*a^3*b^5*d + 3*
a^5*b^3*d + a*b^7*d + a^7*b*d)))*(3*a^4 + 3*b^4 - 26*a^2*b^2)*(-a*b*e^3)^(1/2))/(8*(3*a^3*b^5*d + 3*a^5*b^3*d
+ a*b^7*d + a^7*b*d)))*(3*a^4 + 3*b^4 - 26*a^2*b^2)*(-a*b*e^3)^(1/2))/(8*(3*a^3*b^5*d + 3*a^5*b^3*d + a*b^7*d
+ a^7*b*d)))*(3*a^4 + 3*b^4 - 26*a^2*b^2)*(-a*b*e^3)^(1/2))/(8*(3*a^3*b^5*d + 3*a^5*b^3*d + a*b^7*d + a^7*b*d)
) + ((((e*cot(c + d*x))^(1/2)*(41*b^13*e^16 + 9*a^12*b*e^16 - 82*a^2*b^11*e^16 + 1831*a^4*b^9*e^16 - 4348*a^6*
b^7*e^16 + 1671*a^8*b^5*e^16 - 210*a^10*b^3*e^16))/(a^16*d^4 + b^16*d^4 + 8*a^2*b^14*d^4 + 28*a^4*b^12*d^4 + 5
6*a^6*b^10*d^4 + 70*a^8*b^8*d^4 + 56*a^10*b^6*d^4 + 28*a^12*b^4*d^4 + 8*a^14*b^2*d^4) + (((518*a*b^15*d^2*e^15
- 18*a^15*b*d^2*e^15 - 4494*a^3*b^13*d^2*e^15 + 3022*a^5*b^11*d^2*e^15 + 17194*a^7*b^9*d^2*e^15 + 5298*a^9*b^
7*d^2*e^15 - 3338*a^11*b^5*d^2*e^15 + 506*a^13*b^3*d^2*e^15)/(a^16*d^5 + b^16*d^5 + 8*a^2*b^14*d^5 + 28*a^4*b^
12*d^5 + 56*a^6*b^10*d^5 + 70*a^8*b^8*d^5 + 56*a^10*b^6*d^5 + 28*a^12*b^4*d^5 + 8*a^14*b^2*d^5) - ((((e*cot(c
+ d*x))^(1/2)*(1544*a*b^18*d^2*e^13 + 64*a^3*b^16*d^2*e^13 - 7456*a^5*b^14*d^2*e^13 - 576*a^7*b^12*d^2*e^13 +
19504*a^9*b^10*d^2*e^13 + 18880*a^11*b^8*d^2*e^13 + 3808*a^13*b^6*d^2*e^13 - 960*a^15*b^4*d^2*e^13 + 8*a^17*b^
2*d^2*e^13))/(a^16*d^4 + b^16*d^4 + 8*a^2*b^14*d^4 + 28*a^4*b^12*d^4 + 56*a^6*b^10*d^4 + 70*a^8*b^8*d^4 + 56*a
^10*b^6*d^4 + 28*a^12*b^4*d^4 + 8*a^14*b^2*d^4) - (((4224*a^4*b^18*d^4*e^12 - 320*a^2*b^20*d^4*e^12 - 192*b^22
*d^4*e^12 + 22272*a^6*b^16*d^4*e^12 + 51072*a^8*b^14*d^4*e^12 + 67200*a^10*b^12*d^4*e^12 + 53760*a^12*b^10*d^4
*e^12 + 25344*a^14*b^8*d^4*e^12 + 5952*a^16*b^6*d^4*e^12 + 192*a^18*b^4*d^4*e^12 - 128*a^20*b^2*d^4*e^12)/(a^1
6*d^5 + b^16*d^5 + 8*a^2*b^14*d^5 + 28*a^4*b^12*d^5 + 56*a^6*b^10*d^5 + 70*a^8*b^8*d^5 + 56*a^10*b^6*d^5 + 28*
a^12*b^4*d^5 + 8*a^14*b^2*d^5) + ((e*cot(c + d*x))^(1/2)*(3*a^4 + 3*b^4 - 26*a^2*b^2)*(-a*b*e^3)^(1/2)*(512*b^
25*d^4*e^10 + 4608*a^2*b^23*d^4*e^10 + 17920*a^4*b^21*d^4*e^10 + 38400*a^6*b^19*d^4*e^10 + 46080*a^8*b^17*d^4*
e^10 + 21504*a^10*b^15*d^4*e^10 - 21504*a^12*b^13*d^4*e^10 - 46080*a^14*b^11*d^4*e^10 - 38400*a^16*b^9*d^4*e^1
0 - 17920*a^18*b^7*d^4*e^10 - 4608*a^20*b^5*d^4*e^10 - 512*a^22*b^3*d^4*e^10))/(8*(3*a^3*b^5*d + 3*a^5*b^3*d +
a*b^7*d + a^7*b*d)*(a^16*d^4 + b^16*d^4 + 8*a^2*b^14*d^4 + 28*a^4*b^12*d^4 + 56*a^6*b^10*d^4 + 70*a^8*b^8*d^4
+ 56*a^10*b^6*d^4 + 28*a^12*b^4*d^4 + 8*a^14*b^2*d^4)))*(3*a^4 + 3*b^4 - 26*a^2*b^2)*(-a*b*e^3)^(1/2))/(8*(3*
a^3*b^5*d + 3*a^5*b^3*d + a*b^7*d + a^7*b*d)))*(3*a^4 + 3*b^4 - 26*a^2*b^2)*(-a*b*e^3)^(1/2))/(8*(3*a^3*b^5*d
+ 3*a^5*b^3*d + a*b^7*d + a^7*b*d)))*(3*a^4 + 3*b^4 - 26*a^2*b^2)*(-a*b*e^3)^(1/2))/(8*(3*a^3*b^5*d + 3*a^5*b^
3*d + a*b^7*d + a^7*b*d)))*(3*a^4 + 3*b^4 - 26*a^2*b^2)*(-a*b*e^3)^(1/2))/(8*(3*a^3*b^5*d + 3*a^5*b^3*d + a*b^
7*d + a^7*b*d))))*(3*a^4 + 3*b^4 - 26*a^2*b^2)*(-a*b*e^3)^(1/2)*1i)/(4*(3*a^3*b^5*d + 3*a^5*b^3*d + a*b^7*d +
a^7*b*d))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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