3.85 \(\int \frac {\sqrt {e \cot (c+d x)}}{(a+b \cot (c+d x))^3} \, dx\)

Optimal. Leaf size=463 \[ \frac {b \left (7 a^2-b^2\right ) \sqrt {e \cot (c+d x)}}{4 a d \left (a^2+b^2\right )^2 (a+b \cot (c+d x))}+\frac {b \sqrt {e \cot (c+d x)}}{2 d \left (a^2+b^2\right ) (a+b \cot (c+d x))^2}-\frac {\sqrt {e} (a+b) \left (a^2-4 a b+b^2\right ) \log \left (\sqrt {e} \cot (c+d x)-\sqrt {2} \sqrt {e \cot (c+d x)}+\sqrt {e}\right )}{2 \sqrt {2} d \left (a^2+b^2\right )^3}+\frac {\sqrt {e} (a+b) \left (a^2-4 a b+b^2\right ) \log \left (\sqrt {e} \cot (c+d x)+\sqrt {2} \sqrt {e \cot (c+d x)}+\sqrt {e}\right )}{2 \sqrt {2} d \left (a^2+b^2\right )^3}+\frac {\sqrt {e} (a-b) \left (a^2+4 a b+b^2\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} d \left (a^2+b^2\right )^3}-\frac {\sqrt {e} (a-b) \left (a^2+4 a b+b^2\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}+1\right )}{\sqrt {2} d \left (a^2+b^2\right )^3}+\frac {\sqrt {b} \sqrt {e} \left (15 a^4-18 a^2 b^2-b^4\right ) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {e \cot (c+d x)}}{\sqrt {a} \sqrt {e}}\right )}{4 a^{3/2} d \left (a^2+b^2\right )^3} \]

[Out]

1/2*(a-b)*(a^2+4*a*b+b^2)*arctan(1-2^(1/2)*(e*cot(d*x+c))^(1/2)/e^(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)*arctan(1+2^(1/2)*(e*cot(d*x+c))^(1/2)/e^(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)*ln(e^(1/2)+cot(d*x+c)*e^(1/2)-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)*ln(e^(1/2)+cot(d*x+c)*e^(1/2)+2^(1/2)*(e*cot(d*x+c))^(1/2))*e^(1/2)/(a^2+b^2)^3/d*2^(1/2)+1/
4*(15*a^4-18*a^2*b^2-b^4)*arctan(b^(1/2)*(e*cot(d*x+c))^(1/2)/a^(1/2)/e^(1/2))*b^(1/2)*e^(1/2)/a^(3/2)/(a^2+b^
2)^3/d+1/2*b*(e*cot(d*x+c))^(1/2)/(a^2+b^2)/d/(a+b*cot(d*x+c))^2+1/4*b*(7*a^2-b^2)*(e*cot(d*x+c))^(1/2)/a/(a^2
+b^2)^2/d/(a+b*cot(d*x+c))

________________________________________________________________________________________

Rubi [A]  time = 1.15, antiderivative size = 463, 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 = {3568, 3649, 3653, 3534, 1168, 1162, 617, 204, 1165, 628, 3634, 63, 205} \[ \frac {b \left (7 a^2-b^2\right ) \sqrt {e \cot (c+d x)}}{4 a d \left (a^2+b^2\right )^2 (a+b \cot (c+d x))}+\frac {b \sqrt {e \cot (c+d x)}}{2 d \left (a^2+b^2\right ) (a+b \cot (c+d x))^2}-\frac {\sqrt {e} (a+b) \left (a^2-4 a b+b^2\right ) \log \left (\sqrt {e} \cot (c+d x)-\sqrt {2} \sqrt {e \cot (c+d x)}+\sqrt {e}\right )}{2 \sqrt {2} d \left (a^2+b^2\right )^3}+\frac {\sqrt {e} (a+b) \left (a^2-4 a b+b^2\right ) \log \left (\sqrt {e} \cot (c+d x)+\sqrt {2} \sqrt {e \cot (c+d x)}+\sqrt {e}\right )}{2 \sqrt {2} d \left (a^2+b^2\right )^3}+\frac {\sqrt {b} \sqrt {e} \left (-18 a^2 b^2+15 a^4-b^4\right ) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {e \cot (c+d x)}}{\sqrt {a} \sqrt {e}}\right )}{4 a^{3/2} d \left (a^2+b^2\right )^3}+\frac {\sqrt {e} (a-b) \left (a^2+4 a b+b^2\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} d \left (a^2+b^2\right )^3}-\frac {\sqrt {e} (a-b) \left (a^2+4 a b+b^2\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}+1\right )}{\sqrt {2} d \left (a^2+b^2\right )^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[b]*(15*a^4 - 18*a^2*b^2 - b^4)*Sqrt[e]*ArcTan[(Sqrt[b]*Sqrt[e*Cot[c + d*x]])/(Sqrt[a]*Sqrt[e])])/(4*a^(3
/2)*(a^2 + b^2)^3*d) + ((a - b)*(a^2 + 4*a*b + b^2)*Sqrt[e]*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)*Sqrt[e]*ArcTan[1 + (Sqrt[2]*Sqrt[e*Cot[c + d*x]])/S
qrt[e]])/(Sqrt[2]*(a^2 + b^2)^3*d) + (b*Sqrt[e*Cot[c + d*x]])/(2*(a^2 + b^2)*d*(a + b*Cot[c + d*x])^2) + (b*(7
*a^2 - b^2)*Sqrt[e*Cot[c + d*x]])/(4*a*(a^2 + b^2)^2*d*(a + b*Cot[c + d*x])) - ((a + b)*(a^2 - 4*a*b + b^2)*Sq
rt[e]*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)*Sqrt[e]*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 3568

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Si
mp[(b*(a + b*Tan[e + f*x])^(m + 1)*(c + d*Tan[e + f*x])^n)/(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 - 1)*Simp[a*c*(m + 1) - b*d*n - (b*c - a*d)*
(m + 1)*Tan[e + f*x] - b*d*(m + 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] && LtQ[m, -1] && GtQ[n, 0] && 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 {\sqrt {e \cot (c+d x)}}{(a+b \cot (c+d x))^3} \, dx &=\frac {b \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 {b e}{2}-2 a e \cot (c+d x)+\frac {3}{2} b e \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 {b \sqrt {e \cot (c+d x)}}{2 \left (a^2+b^2\right ) d (a+b \cot (c+d x))^2}+\frac {b \left (7 a^2-b^2\right ) \sqrt {e \cot (c+d x)}}{4 a \left (a^2+b^2\right )^2 d (a+b \cot (c+d x))}+\frac {\int \frac {\frac {1}{4} b \left (9 a^2+b^2\right ) e^2+2 a \left (a^2-b^2\right ) e^2 \cot (c+d x)-\frac {1}{4} b \left (7 a^2-b^2\right ) e^2 \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 {b \sqrt {e \cot (c+d x)}}{2 \left (a^2+b^2\right ) d (a+b \cot (c+d x))^2}+\frac {b \left (7 a^2-b^2\right ) \sqrt {e \cot (c+d x)}}{4 a \left (a^2+b^2\right )^2 d (a+b \cot (c+d x))}+\frac {\int \frac {2 a b \left (3 a^2-b^2\right ) e^2+2 a^2 \left (a^2-3 b^2\right ) e^2 \cot (c+d x)}{\sqrt {e \cot (c+d x)}} \, dx}{2 a \left (a^2+b^2\right )^3 e}-\frac {\left (b \left (15 a^4-18 a^2 b^2-b^4\right ) e\right ) \int \frac {1+\cot ^2(c+d x)}{\sqrt {e \cot (c+d x)} (a+b \cot (c+d x))} \, dx}{8 a \left (a^2+b^2\right )^3}\\ &=\frac {b \sqrt {e \cot (c+d x)}}{2 \left (a^2+b^2\right ) d (a+b \cot (c+d x))^2}+\frac {b \left (7 a^2-b^2\right ) \sqrt {e \cot (c+d x)}}{4 a \left (a^2+b^2\right )^2 d (a+b \cot (c+d x))}+\frac {\operatorname {Subst}\left (\int \frac {-2 a b \left (3 a^2-b^2\right ) e^3-2 a^2 \left (a^2-3 b^2\right ) e^2 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 (b \left (15 a^4-18 a^2 b^2-b^4\right ) e\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {-e x} (a-b x)} \, dx,x,-\cot (c+d x)\right )}{8 a \left (a^2+b^2\right )^3 d}\\ &=\frac {b \sqrt {e \cot (c+d x)}}{2 \left (a^2+b^2\right ) d (a+b \cot (c+d x))^2}+\frac {b \left (7 a^2-b^2\right ) \sqrt {e \cot (c+d x)}}{4 a \left (a^2+b^2\right )^2 d (a+b \cot (c+d x))}+\frac {\left (b \left (15 a^4-18 a^2 b^2-b^4\right )\right ) \operatorname {Subst}\left (\int \frac {1}{a+\frac {b x^2}{e}} \, dx,x,\sqrt {e \cot (c+d x)}\right )}{4 a \left (a^2+b^2\right )^3 d}+\frac {\left ((a+b) \left (a^2-4 a b+b^2\right ) e\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\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 {b} \left (15 a^4-18 a^2 b^2-b^4\right ) \sqrt {e} \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {e \cot (c+d x)}}{\sqrt {a} \sqrt {e}}\right )}{4 a^{3/2} \left (a^2+b^2\right )^3 d}+\frac {b \sqrt {e \cot (c+d x)}}{2 \left (a^2+b^2\right ) d (a+b \cot (c+d x))^2}+\frac {b \left (7 a^2-b^2\right ) \sqrt {e \cot (c+d x)}}{4 a \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 ) \sqrt {e}\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 ) \sqrt {e}\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\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\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 {b} \left (15 a^4-18 a^2 b^2-b^4\right ) \sqrt {e} \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {e \cot (c+d x)}}{\sqrt {a} \sqrt {e}}\right )}{4 a^{3/2} \left (a^2+b^2\right )^3 d}+\frac {b \sqrt {e \cot (c+d x)}}{2 \left (a^2+b^2\right ) d (a+b \cot (c+d x))^2}+\frac {b \left (7 a^2-b^2\right ) \sqrt {e \cot (c+d x)}}{4 a \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 ) \sqrt {e} \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 ) \sqrt {e} \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 ) \sqrt {e}\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 ) \sqrt {e}\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 {b} \left (15 a^4-18 a^2 b^2-b^4\right ) \sqrt {e} \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {e \cot (c+d x)}}{\sqrt {a} \sqrt {e}}\right )}{4 a^{3/2} \left (a^2+b^2\right )^3 d}+\frac {(a-b) \left (a^2+4 a b+b^2\right ) \sqrt {e} \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 ) \sqrt {e} \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 {b \sqrt {e \cot (c+d x)}}{2 \left (a^2+b^2\right ) d (a+b \cot (c+d x))^2}+\frac {b \left (7 a^2-b^2\right ) \sqrt {e \cot (c+d x)}}{4 a \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 ) \sqrt {e} \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 ) \sqrt {e} \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 = 6.19, size = 483, normalized size = 1.04 \[ -\frac {\sqrt {e \cot (c+d x)} \left (\frac {2 a \left (a^2-3 b^2\right ) \cot ^{\frac {3}{2}}(c+d x) \, _2F_1\left (\frac {3}{4},1;\frac {7}{4};-\cot ^2(c+d x)\right )}{3 \left (a^2+b^2\right )^3}+\frac {2 b \left (3 a^2-b^2\right ) \sqrt {\cot (c+d x)}}{\left (a^2+b^2\right )^3}-\frac {2 \sqrt {a} \sqrt {b} \left (3 a^2-b^2\right ) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {\cot (c+d x)}}{\sqrt {a}}\right )}{\left (a^2+b^2\right )^3}-\frac {2 \sqrt {a} \sqrt {b} \left (\sqrt {a} \sqrt {b} \sqrt {\cot (c+d x)}-a \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {\cot (c+d x)}}{\sqrt {a}}\right )-b \cot (c+d x) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {\cot (c+d x)}}{\sqrt {a}}\right )\right )}{\left (a^2+b^2\right )^2 (a+b \cot (c+d x))}-\frac {b \left (3 a^2-b^2\right ) \left (8 \sqrt {\cot (c+d x)}+\sqrt {2} \log \left (\cot (c+d x)-\sqrt {2} \sqrt {\cot (c+d x)}+1\right )-\sqrt {2} \log \left (\cot (c+d x)+\sqrt {2} \sqrt {\cot (c+d x)}+1\right )+2 \left (\sqrt {2} \tan ^{-1}\left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )-\sqrt {2} \tan ^{-1}\left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )\right )\right )}{4 \left (a^2+b^2\right )^3}+\frac {2 b^2 \cot ^{\frac {3}{2}}(c+d x) \, _2F_1\left (\frac {3}{2},3;\frac {5}{2};-\frac {b \cot (c+d x)}{a}\right )}{3 a^3 \left (a^2+b^2\right )}\right )}{d \sqrt {\cot (c+d x)}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((Sqrt[e*Cot[c + d*x]]*((-2*Sqrt[a]*Sqrt[b]*(3*a^2 - b^2)*ArcTan[(Sqrt[b]*Sqrt[Cot[c + d*x]])/Sqrt[a]])/(a^2
+ b^2)^3 + (2*b*(3*a^2 - b^2)*Sqrt[Cot[c + d*x]])/(a^2 + b^2)^3 - (2*Sqrt[a]*Sqrt[b]*(-(a*ArcTan[(Sqrt[b]*Sqrt
[Cot[c + d*x]])/Sqrt[a]]) + Sqrt[a]*Sqrt[b]*Sqrt[Cot[c + d*x]] - b*ArcTan[(Sqrt[b]*Sqrt[Cot[c + d*x]])/Sqrt[a]
]*Cot[c + d*x]))/((a^2 + b^2)^2*(a + b*Cot[c + d*x])) + (2*a*(a^2 - 3*b^2)*Cot[c + d*x]^(3/2)*Hypergeometric2F
1[3/4, 1, 7/4, -Cot[c + d*x]^2])/(3*(a^2 + b^2)^3) + (2*b^2*Cot[c + d*x]^(3/2)*Hypergeometric2F1[3/2, 3, 5/2,
-((b*Cot[c + d*x])/a)])/(3*a^3*(a^2 + b^2)) - (b*(3*a^2 - 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[Co
t[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*Sqrt[Cot[c + d*x]]))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*cot(d*x+c))^(1/2)/(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 \[ \int \frac {\sqrt {e \cot \left (d x + c\right )}}{{\left (b \cot \left (d x + c\right ) + a\right )}^{3}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*cot(d*x+c))^(1/2)/(a+b*cot(d*x+c))^3,x, algorithm="giac")

[Out]

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

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maple [B]  time = 0.97, size = 1187, normalized size = 2.56 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*cot(d*x+c))^(1/2)/(a+b*cot(d*x+c))^3,x)

[Out]

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

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maxima [A]  time = 0.84, size = 496, normalized size = 1.07 \[ \frac {e {\left (\frac {{\left (15 \, a^{4} b - 18 \, a^{2} b^{3} - b^{5}\right )} \arctan \left (\frac {b \sqrt {\frac {e}{\tan \left (d x + c\right )}}}{\sqrt {a b e}}\right )}{{\left (a^{7} + 3 \, a^{5} b^{2} + 3 \, a^{3} b^{4} + a b^{6}\right )} \sqrt {a b e}} + \frac {{\left (9 \, a^{3} b + a b^{3}\right )} e \sqrt {\frac {e}{\tan \left (d x + c\right )}} + {\left (7 \, a^{2} b^{2} - b^{4}\right )} \left (\frac {e}{\tan \left (d x + c\right )}\right )^{\frac {3}{2}}}{{\left (a^{7} + 2 \, a^{5} b^{2} + a^{3} b^{4}\right )} e^{2} + \frac {2 \, {\left (a^{6} b + 2 \, a^{4} b^{3} + a^{2} b^{5}\right )} e^{2}}{\tan \left (d x + c\right )} + \frac {{\left (a^{5} b^{2} + 2 \, a^{3} b^{4} + a b^{6}\right )} e^{2}}{\tan \left (d x + c\right )^{2}}} - \frac {\frac {2 \, \sqrt {2} {\left (a^{3} + 3 \, a^{2} b - 3 \, a b^{2} - b^{3}\right )} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \sqrt {e} + 2 \, \sqrt {\frac {e}{\tan \left (d x + c\right )}}\right )}}{2 \, \sqrt {e}}\right )}{\sqrt {e}} + \frac {2 \, \sqrt {2} {\left (a^{3} + 3 \, a^{2} b - 3 \, a b^{2} - b^{3}\right )} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \sqrt {e} - 2 \, \sqrt {\frac {e}{\tan \left (d x + c\right )}}\right )}}{2 \, \sqrt {e}}\right )}{\sqrt {e}} - \frac {\sqrt {2} {\left (a^{3} - 3 \, a^{2} b - 3 \, a b^{2} + b^{3}\right )} \log \left (\sqrt {2} \sqrt {e} \sqrt {\frac {e}{\tan \left (d x + c\right )}} + e + \frac {e}{\tan \left (d x + c\right )}\right )}{\sqrt {e}} + \frac {\sqrt {2} {\left (a^{3} - 3 \, a^{2} b - 3 \, a b^{2} + b^{3}\right )} \log \left (-\sqrt {2} \sqrt {e} \sqrt {\frac {e}{\tan \left (d x + c\right )}} + e + \frac {e}{\tan \left (d x + c\right )}\right )}{\sqrt {e}}}{a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}}\right )}}{4 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*cot(d*x+c))^(1/2)/(a+b*cot(d*x+c))^3,x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 6.13, size = 19534, normalized size = 42.19 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*cot(c + d*x))^(1/2)/(a + b*cot(c + d*x))^3,x)

[Out]

(((e*cot(c + d*x))^(1/2)*(b^3*e^2 + 9*a^2*b*e^2))/(4*(a^4 + b^4 + 2*a^2*b^2)) + (b^2*e*(e*cot(c + d*x))^(3/2)*
(7*a^2 - b^2))/(4*a*(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(((((((64*a*b^23*d^4*e^11 + 1472*a^3*b^21*d^4*e^11 + 8832*a^5*b^19*d^4*e^11 + 25344*a^7*b^17*d^4*e^11
+ 40320*a^9*b^15*d^4*e^11 + 34944*a^11*b^13*d^4*e^11 + 10752*a^13*b^11*d^4*e^11 - 8448*a^15*b^9*d^4*e^11 - 101
76*a^17*b^7*d^4*e^11 - 4160*a^19*b^5*d^4*e^11 - 640*a^21*b^3*d^4*e^11)/(a^18*d^5 + a^2*b^16*d^5 + 8*a^4*b^14*d
^5 + 28*a^6*b^12*d^5 + 56*a^8*b^10*d^5 + 70*a^10*b^8*d^5 + 56*a^12*b^6*d^5 + 28*a^14*b^4*d^5 + 8*a^16*b^2*d^5)
+ ((e*cot(c + d*x))^(1/2)*(-e/(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*a^2*b^25*d^4*e^10 + 4608*a^4*b^23*d^4*e^10 + 17920*a^6*b^21*d^4*e^
10 + 38400*a^8*b^19*d^4*e^10 + 46080*a^10*b^17*d^4*e^10 + 21504*a^12*b^15*d^4*e^10 - 21504*a^14*b^13*d^4*e^10
- 46080*a^16*b^11*d^4*e^10 - 38400*a^18*b^9*d^4*e^10 - 17920*a^20*b^7*d^4*e^10 - 4608*a^22*b^5*d^4*e^10 - 512*
a^24*b^3*d^4*e^10))/(a^18*d^4 + a^2*b^16*d^4 + 8*a^4*b^14*d^4 + 28*a^6*b^12*d^4 + 56*a^8*b^10*d^4 + 70*a^10*b^
8*d^4 + 56*a^12*b^6*d^4 + 28*a^14*b^4*d^4 + 8*a^16*b^2*d^4))*(-e/(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*b^20*d
^2*e^11 - 1152*a^3*b^18*d^2*e^11 + 2528*a^5*b^16*d^2*e^11 + 15296*a^7*b^14*d^2*e^11 + 14128*a^9*b^12*d^2*e^11
- 5056*a^11*b^10*d^2*e^11 - 9248*a^13*b^8*d^2*e^11 + 64*a^15*b^6*d^2*e^11 + 1800*a^17*b^4*d^2*e^11 + 64*a^19*b
^2*d^2*e^11))/(a^18*d^4 + a^2*b^16*d^4 + 8*a^4*b^14*d^4 + 28*a^6*b^12*d^4 + 56*a^8*b^10*d^4 + 70*a^10*b^8*d^4
+ 56*a^12*b^6*d^4 + 28*a^14*b^4*d^4 + 8*a^16*b^2*d^4))*(-e/(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) - (2*b^18*d^2*e^12 - 138*a^2*b^16*d^2*e^12
- 3046*a^4*b^14*d^2*e^12 + 4862*a^6*b^12*d^2*e^12 + 9222*a^8*b^10*d^2*e^12 - 5246*a^10*b^8*d^2*e^12 - 4290*a^1
2*b^6*d^2*e^12 + 2442*a^14*b^4*d^2*e^12 + 32*a^16*b^2*d^2*e^12)/(a^18*d^5 + a^2*b^16*d^5 + 8*a^4*b^14*d^5 + 28
*a^6*b^12*d^5 + 56*a^8*b^10*d^5 + 70*a^10*b^8*d^5 + 56*a^12*b^6*d^5 + 28*a^14*b^4*d^5 + 8*a^16*b^2*d^5))*(-e/(
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)*(2*a^2*b^13*e^12 - b^15*e^12 + 49*a^4*b^11*e^12 + 2460*a^6*b^9*e^12 - 3631*a^
8*b^7*e^12 + 1922*a^10*b^5*e^12 - 225*a^12*b^3*e^12))/(a^18*d^4 + a^2*b^16*d^4 + 8*a^4*b^14*d^4 + 28*a^6*b^12*
d^4 + 56*a^8*b^10*d^4 + 70*a^10*b^8*d^4 + 56*a^12*b^6*d^4 + 28*a^14*b^4*d^4 + 8*a^16*b^2*d^4))*(-e/(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
- (((((64*a*b^23*d^4*e^11 + 1472*a^3*b^21*d^4*e^11 + 8832*a^5*b^19*d^4*e^11 + 25344*a^7*b^17*d^4*e^11 + 40320*
a^9*b^15*d^4*e^11 + 34944*a^11*b^13*d^4*e^11 + 10752*a^13*b^11*d^4*e^11 - 8448*a^15*b^9*d^4*e^11 - 10176*a^17*
b^7*d^4*e^11 - 4160*a^19*b^5*d^4*e^11 - 640*a^21*b^3*d^4*e^11)/(a^18*d^5 + a^2*b^16*d^5 + 8*a^4*b^14*d^5 + 28*
a^6*b^12*d^5 + 56*a^8*b^10*d^5 + 70*a^10*b^8*d^5 + 56*a^12*b^6*d^5 + 28*a^14*b^4*d^5 + 8*a^16*b^2*d^5) - ((e*c
ot(c + d*x))^(1/2)*(-e/(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*a^2*b^25*d^4*e^10 + 4608*a^4*b^23*d^4*e^10 + 17920*a^6*b^21*d^4*e^10 + 384
00*a^8*b^19*d^4*e^10 + 46080*a^10*b^17*d^4*e^10 + 21504*a^12*b^15*d^4*e^10 - 21504*a^14*b^13*d^4*e^10 - 46080*
a^16*b^11*d^4*e^10 - 38400*a^18*b^9*d^4*e^10 - 17920*a^20*b^7*d^4*e^10 - 4608*a^22*b^5*d^4*e^10 - 512*a^24*b^3
*d^4*e^10))/(a^18*d^4 + a^2*b^16*d^4 + 8*a^4*b^14*d^4 + 28*a^6*b^12*d^4 + 56*a^8*b^10*d^4 + 70*a^10*b^8*d^4 +
56*a^12*b^6*d^4 + 28*a^14*b^4*d^4 + 8*a^16*b^2*d^4))*(-e/(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*b^20*d^2*e^11
- 1152*a^3*b^18*d^2*e^11 + 2528*a^5*b^16*d^2*e^11 + 15296*a^7*b^14*d^2*e^11 + 14128*a^9*b^12*d^2*e^11 - 5056*a
^11*b^10*d^2*e^11 - 9248*a^13*b^8*d^2*e^11 + 64*a^15*b^6*d^2*e^11 + 1800*a^17*b^4*d^2*e^11 + 64*a^19*b^2*d^2*e
^11))/(a^18*d^4 + a^2*b^16*d^4 + 8*a^4*b^14*d^4 + 28*a^6*b^12*d^4 + 56*a^8*b^10*d^4 + 70*a^10*b^8*d^4 + 56*a^1
2*b^6*d^4 + 28*a^14*b^4*d^4 + 8*a^16*b^2*d^4))*(-e/(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) - (2*b^18*d^2*e^12 - 138*a^2*b^16*d^2*e^12 - 3046*a
^4*b^14*d^2*e^12 + 4862*a^6*b^12*d^2*e^12 + 9222*a^8*b^10*d^2*e^12 - 5246*a^10*b^8*d^2*e^12 - 4290*a^12*b^6*d^
2*e^12 + 2442*a^14*b^4*d^2*e^12 + 32*a^16*b^2*d^2*e^12)/(a^18*d^5 + a^2*b^16*d^5 + 8*a^4*b^14*d^5 + 28*a^6*b^1
2*d^5 + 56*a^8*b^10*d^5 + 70*a^10*b^8*d^5 + 56*a^12*b^6*d^5 + 28*a^14*b^4*d^5 + 8*a^16*b^2*d^5))*(-e/(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)*(2*a^2*b^13*e^12 - b^15*e^12 + 49*a^4*b^11*e^12 + 2460*a^6*b^9*e^12 - 3631*a^8*b^7*e^
12 + 1922*a^10*b^5*e^12 - 225*a^12*b^3*e^12))/(a^18*d^4 + a^2*b^16*d^4 + 8*a^4*b^14*d^4 + 28*a^6*b^12*d^4 + 56
*a^8*b^10*d^4 + 70*a^10*b^8*d^4 + 56*a^12*b^6*d^4 + 28*a^14*b^4*d^4 + 8*a^16*b^2*d^4))*(-e/(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)/((7*a*b
^11*e^13 + 116*a^3*b^9*e^13 - 270*a^5*b^7*e^13 + 420*a^7*b^5*e^13 - 225*a^9*b^3*e^13)/(a^18*d^5 + a^2*b^16*d^5
+ 8*a^4*b^14*d^5 + 28*a^6*b^12*d^5 + 56*a^8*b^10*d^5 + 70*a^10*b^8*d^5 + 56*a^12*b^6*d^5 + 28*a^14*b^4*d^5 +
8*a^16*b^2*d^5) + (((((64*a*b^23*d^4*e^11 + 1472*a^3*b^21*d^4*e^11 + 8832*a^5*b^19*d^4*e^11 + 25344*a^7*b^17*d
^4*e^11 + 40320*a^9*b^15*d^4*e^11 + 34944*a^11*b^13*d^4*e^11 + 10752*a^13*b^11*d^4*e^11 - 8448*a^15*b^9*d^4*e^
11 - 10176*a^17*b^7*d^4*e^11 - 4160*a^19*b^5*d^4*e^11 - 640*a^21*b^3*d^4*e^11)/(a^18*d^5 + a^2*b^16*d^5 + 8*a^
4*b^14*d^5 + 28*a^6*b^12*d^5 + 56*a^8*b^10*d^5 + 70*a^10*b^8*d^5 + 56*a^12*b^6*d^5 + 28*a^14*b^4*d^5 + 8*a^16*
b^2*d^5) + ((e*cot(c + d*x))^(1/2)*(-e/(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*1
5i - 20*a^3*b^3*d^2 + a^4*b^2*d^2*15i)))^(1/2)*(512*a^2*b^25*d^4*e^10 + 4608*a^4*b^23*d^4*e^10 + 17920*a^6*b^2
1*d^4*e^10 + 38400*a^8*b^19*d^4*e^10 + 46080*a^10*b^17*d^4*e^10 + 21504*a^12*b^15*d^4*e^10 - 21504*a^14*b^13*d
^4*e^10 - 46080*a^16*b^11*d^4*e^10 - 38400*a^18*b^9*d^4*e^10 - 17920*a^20*b^7*d^4*e^10 - 4608*a^22*b^5*d^4*e^1
0 - 512*a^24*b^3*d^4*e^10))/(a^18*d^4 + a^2*b^16*d^4 + 8*a^4*b^14*d^4 + 28*a^6*b^12*d^4 + 56*a^8*b^10*d^4 + 70
*a^10*b^8*d^4 + 56*a^12*b^6*d^4 + 28*a^14*b^4*d^4 + 8*a^16*b^2*d^4))*(-e/(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*b^20*d^2*e^11 - 1152*a^3*b^18*d^2*e^11 + 2528*a^5*b^16*d^2*e^11 + 15296*a^7*b^14*d^2*e^11 + 14128*a^9*b^12*d
^2*e^11 - 5056*a^11*b^10*d^2*e^11 - 9248*a^13*b^8*d^2*e^11 + 64*a^15*b^6*d^2*e^11 + 1800*a^17*b^4*d^2*e^11 + 6
4*a^19*b^2*d^2*e^11))/(a^18*d^4 + a^2*b^16*d^4 + 8*a^4*b^14*d^4 + 28*a^6*b^12*d^4 + 56*a^8*b^10*d^4 + 70*a^10*
b^8*d^4 + 56*a^12*b^6*d^4 + 28*a^14*b^4*d^4 + 8*a^16*b^2*d^4))*(-e/(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) - (2*b^18*d^2*e^12 - 138*a^2*b^16*d
^2*e^12 - 3046*a^4*b^14*d^2*e^12 + 4862*a^6*b^12*d^2*e^12 + 9222*a^8*b^10*d^2*e^12 - 5246*a^10*b^8*d^2*e^12 -
4290*a^12*b^6*d^2*e^12 + 2442*a^14*b^4*d^2*e^12 + 32*a^16*b^2*d^2*e^12)/(a^18*d^5 + a^2*b^16*d^5 + 8*a^4*b^14*
d^5 + 28*a^6*b^12*d^5 + 56*a^8*b^10*d^5 + 70*a^10*b^8*d^5 + 56*a^12*b^6*d^5 + 28*a^14*b^4*d^5 + 8*a^16*b^2*d^5
))*(-e/(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)*(2*a^2*b^13*e^12 - b^15*e^12 + 49*a^4*b^11*e^12 + 2460*a^6*b^9*e^12 -
3631*a^8*b^7*e^12 + 1922*a^10*b^5*e^12 - 225*a^12*b^3*e^12))/(a^18*d^4 + a^2*b^16*d^4 + 8*a^4*b^14*d^4 + 28*a
^6*b^12*d^4 + 56*a^8*b^10*d^4 + 70*a^10*b^8*d^4 + 56*a^12*b^6*d^4 + 28*a^14*b^4*d^4 + 8*a^16*b^2*d^4))*(-e/(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) + (((((64*a*b^23*d^4*e^11 + 1472*a^3*b^21*d^4*e^11 + 8832*a^5*b^19*d^4*e^11 + 25344*a^7*b^17*d^4*e^11 + 4
0320*a^9*b^15*d^4*e^11 + 34944*a^11*b^13*d^4*e^11 + 10752*a^13*b^11*d^4*e^11 - 8448*a^15*b^9*d^4*e^11 - 10176*
a^17*b^7*d^4*e^11 - 4160*a^19*b^5*d^4*e^11 - 640*a^21*b^3*d^4*e^11)/(a^18*d^5 + a^2*b^16*d^5 + 8*a^4*b^14*d^5
+ 28*a^6*b^12*d^5 + 56*a^8*b^10*d^5 + 70*a^10*b^8*d^5 + 56*a^12*b^6*d^5 + 28*a^14*b^4*d^5 + 8*a^16*b^2*d^5) -
((e*cot(c + d*x))^(1/2)*(-e/(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*a^2*b^25*d^4*e^10 + 4608*a^4*b^23*d^4*e^10 + 17920*a^6*b^21*d^4*e^10
+ 38400*a^8*b^19*d^4*e^10 + 46080*a^10*b^17*d^4*e^10 + 21504*a^12*b^15*d^4*e^10 - 21504*a^14*b^13*d^4*e^10 - 4
6080*a^16*b^11*d^4*e^10 - 38400*a^18*b^9*d^4*e^10 - 17920*a^20*b^7*d^4*e^10 - 4608*a^22*b^5*d^4*e^10 - 512*a^2
4*b^3*d^4*e^10))/(a^18*d^4 + a^2*b^16*d^4 + 8*a^4*b^14*d^4 + 28*a^6*b^12*d^4 + 56*a^8*b^10*d^4 + 70*a^10*b^8*d
^4 + 56*a^12*b^6*d^4 + 28*a^14*b^4*d^4 + 8*a^16*b^2*d^4))*(-e/(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*b^20*d^2*
e^11 - 1152*a^3*b^18*d^2*e^11 + 2528*a^5*b^16*d^2*e^11 + 15296*a^7*b^14*d^2*e^11 + 14128*a^9*b^12*d^2*e^11 - 5
056*a^11*b^10*d^2*e^11 - 9248*a^13*b^8*d^2*e^11 + 64*a^15*b^6*d^2*e^11 + 1800*a^17*b^4*d^2*e^11 + 64*a^19*b^2*
d^2*e^11))/(a^18*d^4 + a^2*b^16*d^4 + 8*a^4*b^14*d^4 + 28*a^6*b^12*d^4 + 56*a^8*b^10*d^4 + 70*a^10*b^8*d^4 + 5
6*a^12*b^6*d^4 + 28*a^14*b^4*d^4 + 8*a^16*b^2*d^4))*(-e/(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) - (2*b^18*d^2*e^12 - 138*a^2*b^16*d^2*e^12 - 3
046*a^4*b^14*d^2*e^12 + 4862*a^6*b^12*d^2*e^12 + 9222*a^8*b^10*d^2*e^12 - 5246*a^10*b^8*d^2*e^12 - 4290*a^12*b
^6*d^2*e^12 + 2442*a^14*b^4*d^2*e^12 + 32*a^16*b^2*d^2*e^12)/(a^18*d^5 + a^2*b^16*d^5 + 8*a^4*b^14*d^5 + 28*a^
6*b^12*d^5 + 56*a^8*b^10*d^5 + 70*a^10*b^8*d^5 + 56*a^12*b^6*d^5 + 28*a^14*b^4*d^5 + 8*a^16*b^2*d^5))*(-e/(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)*(2*a^2*b^13*e^12 - b^15*e^12 + 49*a^4*b^11*e^12 + 2460*a^6*b^9*e^12 - 3631*a^8*b
^7*e^12 + 1922*a^10*b^5*e^12 - 225*a^12*b^3*e^12))/(a^18*d^4 + a^2*b^16*d^4 + 8*a^4*b^14*d^4 + 28*a^6*b^12*d^4
+ 56*a^8*b^10*d^4 + 70*a^10*b^8*d^4 + 56*a^12*b^6*d^4 + 28*a^14*b^4*d^4 + 8*a^16*b^2*d^4))*(-e/(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/
(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(((((((64*a*b^23*d^4*e^11 + 1472*a^3*b^21*d^4*e^11 + 8832*a^5*b^19*d^4*e^11 + 25344*a^7*b^17*
d^4*e^11 + 40320*a^9*b^15*d^4*e^11 + 34944*a^11*b^13*d^4*e^11 + 10752*a^13*b^11*d^4*e^11 - 8448*a^15*b^9*d^4*e
^11 - 10176*a^17*b^7*d^4*e^11 - 4160*a^19*b^5*d^4*e^11 - 640*a^21*b^3*d^4*e^11)/(a^18*d^5 + a^2*b^16*d^5 + 8*a
^4*b^14*d^5 + 28*a^6*b^12*d^5 + 56*a^8*b^10*d^5 + 70*a^10*b^8*d^5 + 56*a^12*b^6*d^5 + 28*a^14*b^4*d^5 + 8*a^16
*b^2*d^5) + ((e*cot(c + d*x))^(1/2)*(-(e*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*a^2*b^25*d^4*e^10 + 4608*a^4*b^23*d^4*e^10 + 17920*a^6*b^
21*d^4*e^10 + 38400*a^8*b^19*d^4*e^10 + 46080*a^10*b^17*d^4*e^10 + 21504*a^12*b^15*d^4*e^10 - 21504*a^14*b^13*
d^4*e^10 - 46080*a^16*b^11*d^4*e^10 - 38400*a^18*b^9*d^4*e^10 - 17920*a^20*b^7*d^4*e^10 - 4608*a^22*b^5*d^4*e^
10 - 512*a^24*b^3*d^4*e^10))/(a^18*d^4 + a^2*b^16*d^4 + 8*a^4*b^14*d^4 + 28*a^6*b^12*d^4 + 56*a^8*b^10*d^4 + 7
0*a^10*b^8*d^4 + 56*a^12*b^6*d^4 + 28*a^14*b^4*d^4 + 8*a^16*b^2*d^4))*(-(e*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*b^20*d^2*e^11 - 1152*a^3*b^18*d^2*e^11 + 2528*a^5*b^16*d^2*e^11 + 15296*a^7*b^14*d^2*e^11 + 14128*a^9*b^12*
d^2*e^11 - 5056*a^11*b^10*d^2*e^11 - 9248*a^13*b^8*d^2*e^11 + 64*a^15*b^6*d^2*e^11 + 1800*a^17*b^4*d^2*e^11 +
64*a^19*b^2*d^2*e^11))/(a^18*d^4 + a^2*b^16*d^4 + 8*a^4*b^14*d^4 + 28*a^6*b^12*d^4 + 56*a^8*b^10*d^4 + 70*a^10
*b^8*d^4 + 56*a^12*b^6*d^4 + 28*a^14*b^4*d^4 + 8*a^16*b^2*d^4))*(-(e*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) - (2*b^18*d^2*e^12 - 138*a^2*b^16*
d^2*e^12 - 3046*a^4*b^14*d^2*e^12 + 4862*a^6*b^12*d^2*e^12 + 9222*a^8*b^10*d^2*e^12 - 5246*a^10*b^8*d^2*e^12 -
4290*a^12*b^6*d^2*e^12 + 2442*a^14*b^4*d^2*e^12 + 32*a^16*b^2*d^2*e^12)/(a^18*d^5 + a^2*b^16*d^5 + 8*a^4*b^14
*d^5 + 28*a^6*b^12*d^5 + 56*a^8*b^10*d^5 + 70*a^10*b^8*d^5 + 56*a^12*b^6*d^5 + 28*a^14*b^4*d^5 + 8*a^16*b^2*d^
5))*(-(e*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)*(2*a^2*b^13*e^12 - b^15*e^12 + 49*a^4*b^11*e^12 + 2460*a^6*b^9*e^12
- 3631*a^8*b^7*e^12 + 1922*a^10*b^5*e^12 - 225*a^12*b^3*e^12))/(a^18*d^4 + a^2*b^16*d^4 + 8*a^4*b^14*d^4 + 28*
a^6*b^12*d^4 + 56*a^8*b^10*d^4 + 70*a^10*b^8*d^4 + 56*a^12*b^6*d^4 + 28*a^14*b^4*d^4 + 8*a^16*b^2*d^4))*(-(e*1
i)/(4*(b^6*d^2 - a^6*d^2 + a*b^5*d^2*6i + a^5*b*d^2*6i - 15*a^2*b^4*d^2 - a^3*b^3*d^2*20i + 15*a^4*b^2*d^2)))^
(1/2)*1i - (((((64*a*b^23*d^4*e^11 + 1472*a^3*b^21*d^4*e^11 + 8832*a^5*b^19*d^4*e^11 + 25344*a^7*b^17*d^4*e^11
+ 40320*a^9*b^15*d^4*e^11 + 34944*a^11*b^13*d^4*e^11 + 10752*a^13*b^11*d^4*e^11 - 8448*a^15*b^9*d^4*e^11 - 10
176*a^17*b^7*d^4*e^11 - 4160*a^19*b^5*d^4*e^11 - 640*a^21*b^3*d^4*e^11)/(a^18*d^5 + a^2*b^16*d^5 + 8*a^4*b^14*
d^5 + 28*a^6*b^12*d^5 + 56*a^8*b^10*d^5 + 70*a^10*b^8*d^5 + 56*a^12*b^6*d^5 + 28*a^14*b^4*d^5 + 8*a^16*b^2*d^5
) - ((e*cot(c + d*x))^(1/2)*(-(e*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*a^2*b^25*d^4*e^10 + 4608*a^4*b^23*d^4*e^10 + 17920*a^6*b^21*d^4*e
^10 + 38400*a^8*b^19*d^4*e^10 + 46080*a^10*b^17*d^4*e^10 + 21504*a^12*b^15*d^4*e^10 - 21504*a^14*b^13*d^4*e^10
- 46080*a^16*b^11*d^4*e^10 - 38400*a^18*b^9*d^4*e^10 - 17920*a^20*b^7*d^4*e^10 - 4608*a^22*b^5*d^4*e^10 - 512
*a^24*b^3*d^4*e^10))/(a^18*d^4 + a^2*b^16*d^4 + 8*a^4*b^14*d^4 + 28*a^6*b^12*d^4 + 56*a^8*b^10*d^4 + 70*a^10*b
^8*d^4 + 56*a^12*b^6*d^4 + 28*a^14*b^4*d^4 + 8*a^16*b^2*d^4))*(-(e*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*b^20*
d^2*e^11 - 1152*a^3*b^18*d^2*e^11 + 2528*a^5*b^16*d^2*e^11 + 15296*a^7*b^14*d^2*e^11 + 14128*a^9*b^12*d^2*e^11
- 5056*a^11*b^10*d^2*e^11 - 9248*a^13*b^8*d^2*e^11 + 64*a^15*b^6*d^2*e^11 + 1800*a^17*b^4*d^2*e^11 + 64*a^19*
b^2*d^2*e^11))/(a^18*d^4 + a^2*b^16*d^4 + 8*a^4*b^14*d^4 + 28*a^6*b^12*d^4 + 56*a^8*b^10*d^4 + 70*a^10*b^8*d^4
+ 56*a^12*b^6*d^4 + 28*a^14*b^4*d^4 + 8*a^16*b^2*d^4))*(-(e*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) - (2*b^18*d^2*e^12 - 138*a^2*b^16*d^2*e^12
- 3046*a^4*b^14*d^2*e^12 + 4862*a^6*b^12*d^2*e^12 + 9222*a^8*b^10*d^2*e^12 - 5246*a^10*b^8*d^2*e^12 - 4290*a^
12*b^6*d^2*e^12 + 2442*a^14*b^4*d^2*e^12 + 32*a^16*b^2*d^2*e^12)/(a^18*d^5 + a^2*b^16*d^5 + 8*a^4*b^14*d^5 + 2
8*a^6*b^12*d^5 + 56*a^8*b^10*d^5 + 70*a^10*b^8*d^5 + 56*a^12*b^6*d^5 + 28*a^14*b^4*d^5 + 8*a^16*b^2*d^5))*(-(e
*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)*(2*a^2*b^13*e^12 - b^15*e^12 + 49*a^4*b^11*e^12 + 2460*a^6*b^9*e^12 - 3631*a
^8*b^7*e^12 + 1922*a^10*b^5*e^12 - 225*a^12*b^3*e^12))/(a^18*d^4 + a^2*b^16*d^4 + 8*a^4*b^14*d^4 + 28*a^6*b^12
*d^4 + 56*a^8*b^10*d^4 + 70*a^10*b^8*d^4 + 56*a^12*b^6*d^4 + 28*a^14*b^4*d^4 + 8*a^16*b^2*d^4))*(-(e*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
)/((7*a*b^11*e^13 + 116*a^3*b^9*e^13 - 270*a^5*b^7*e^13 + 420*a^7*b^5*e^13 - 225*a^9*b^3*e^13)/(a^18*d^5 + a^2
*b^16*d^5 + 8*a^4*b^14*d^5 + 28*a^6*b^12*d^5 + 56*a^8*b^10*d^5 + 70*a^10*b^8*d^5 + 56*a^12*b^6*d^5 + 28*a^14*b
^4*d^5 + 8*a^16*b^2*d^5) + (((((64*a*b^23*d^4*e^11 + 1472*a^3*b^21*d^4*e^11 + 8832*a^5*b^19*d^4*e^11 + 25344*a
^7*b^17*d^4*e^11 + 40320*a^9*b^15*d^4*e^11 + 34944*a^11*b^13*d^4*e^11 + 10752*a^13*b^11*d^4*e^11 - 8448*a^15*b
^9*d^4*e^11 - 10176*a^17*b^7*d^4*e^11 - 4160*a^19*b^5*d^4*e^11 - 640*a^21*b^3*d^4*e^11)/(a^18*d^5 + a^2*b^16*d
^5 + 8*a^4*b^14*d^5 + 28*a^6*b^12*d^5 + 56*a^8*b^10*d^5 + 70*a^10*b^8*d^5 + 56*a^12*b^6*d^5 + 28*a^14*b^4*d^5
+ 8*a^16*b^2*d^5) + ((e*cot(c + d*x))^(1/2)*(-(e*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*a^2*b^25*d^4*e^10 + 4608*a^4*b^23*d^4*e^10 + 1792
0*a^6*b^21*d^4*e^10 + 38400*a^8*b^19*d^4*e^10 + 46080*a^10*b^17*d^4*e^10 + 21504*a^12*b^15*d^4*e^10 - 21504*a^
14*b^13*d^4*e^10 - 46080*a^16*b^11*d^4*e^10 - 38400*a^18*b^9*d^4*e^10 - 17920*a^20*b^7*d^4*e^10 - 4608*a^22*b^
5*d^4*e^10 - 512*a^24*b^3*d^4*e^10))/(a^18*d^4 + a^2*b^16*d^4 + 8*a^4*b^14*d^4 + 28*a^6*b^12*d^4 + 56*a^8*b^10
*d^4 + 70*a^10*b^8*d^4 + 56*a^12*b^6*d^4 + 28*a^14*b^4*d^4 + 8*a^16*b^2*d^4))*(-(e*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*b^20*d^2*e^11 - 1152*a^3*b^18*d^2*e^11 + 2528*a^5*b^16*d^2*e^11 + 15296*a^7*b^14*d^2*e^11 + 14128*a
^9*b^12*d^2*e^11 - 5056*a^11*b^10*d^2*e^11 - 9248*a^13*b^8*d^2*e^11 + 64*a^15*b^6*d^2*e^11 + 1800*a^17*b^4*d^2
*e^11 + 64*a^19*b^2*d^2*e^11))/(a^18*d^4 + a^2*b^16*d^4 + 8*a^4*b^14*d^4 + 28*a^6*b^12*d^4 + 56*a^8*b^10*d^4 +
70*a^10*b^8*d^4 + 56*a^12*b^6*d^4 + 28*a^14*b^4*d^4 + 8*a^16*b^2*d^4))*(-(e*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) - (2*b^18*d^2*e^12 - 138*a
^2*b^16*d^2*e^12 - 3046*a^4*b^14*d^2*e^12 + 4862*a^6*b^12*d^2*e^12 + 9222*a^8*b^10*d^2*e^12 - 5246*a^10*b^8*d^
2*e^12 - 4290*a^12*b^6*d^2*e^12 + 2442*a^14*b^4*d^2*e^12 + 32*a^16*b^2*d^2*e^12)/(a^18*d^5 + a^2*b^16*d^5 + 8*
a^4*b^14*d^5 + 28*a^6*b^12*d^5 + 56*a^8*b^10*d^5 + 70*a^10*b^8*d^5 + 56*a^12*b^6*d^5 + 28*a^14*b^4*d^5 + 8*a^1
6*b^2*d^5))*(-(e*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)*(2*a^2*b^13*e^12 - b^15*e^12 + 49*a^4*b^11*e^12 + 2460*a^6*b
^9*e^12 - 3631*a^8*b^7*e^12 + 1922*a^10*b^5*e^12 - 225*a^12*b^3*e^12))/(a^18*d^4 + a^2*b^16*d^4 + 8*a^4*b^14*d
^4 + 28*a^6*b^12*d^4 + 56*a^8*b^10*d^4 + 70*a^10*b^8*d^4 + 56*a^12*b^6*d^4 + 28*a^14*b^4*d^4 + 8*a^16*b^2*d^4)
)*(-(e*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) + (((((64*a*b^23*d^4*e^11 + 1472*a^3*b^21*d^4*e^11 + 8832*a^5*b^19*d^4*e^11 + 25344*a^7*b^17*d^4
*e^11 + 40320*a^9*b^15*d^4*e^11 + 34944*a^11*b^13*d^4*e^11 + 10752*a^13*b^11*d^4*e^11 - 8448*a^15*b^9*d^4*e^11
- 10176*a^17*b^7*d^4*e^11 - 4160*a^19*b^5*d^4*e^11 - 640*a^21*b^3*d^4*e^11)/(a^18*d^5 + a^2*b^16*d^5 + 8*a^4*
b^14*d^5 + 28*a^6*b^12*d^5 + 56*a^8*b^10*d^5 + 70*a^10*b^8*d^5 + 56*a^12*b^6*d^5 + 28*a^14*b^4*d^5 + 8*a^16*b^
2*d^5) - ((e*cot(c + d*x))^(1/2)*(-(e*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*a^2*b^25*d^4*e^10 + 4608*a^4*b^23*d^4*e^10 + 17920*a^6*b^21*
d^4*e^10 + 38400*a^8*b^19*d^4*e^10 + 46080*a^10*b^17*d^4*e^10 + 21504*a^12*b^15*d^4*e^10 - 21504*a^14*b^13*d^4
*e^10 - 46080*a^16*b^11*d^4*e^10 - 38400*a^18*b^9*d^4*e^10 - 17920*a^20*b^7*d^4*e^10 - 4608*a^22*b^5*d^4*e^10
- 512*a^24*b^3*d^4*e^10))/(a^18*d^4 + a^2*b^16*d^4 + 8*a^4*b^14*d^4 + 28*a^6*b^12*d^4 + 56*a^8*b^10*d^4 + 70*a
^10*b^8*d^4 + 56*a^12*b^6*d^4 + 28*a^14*b^4*d^4 + 8*a^16*b^2*d^4))*(-(e*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*
b^20*d^2*e^11 - 1152*a^3*b^18*d^2*e^11 + 2528*a^5*b^16*d^2*e^11 + 15296*a^7*b^14*d^2*e^11 + 14128*a^9*b^12*d^2
*e^11 - 5056*a^11*b^10*d^2*e^11 - 9248*a^13*b^8*d^2*e^11 + 64*a^15*b^6*d^2*e^11 + 1800*a^17*b^4*d^2*e^11 + 64*
a^19*b^2*d^2*e^11))/(a^18*d^4 + a^2*b^16*d^4 + 8*a^4*b^14*d^4 + 28*a^6*b^12*d^4 + 56*a^8*b^10*d^4 + 70*a^10*b^
8*d^4 + 56*a^12*b^6*d^4 + 28*a^14*b^4*d^4 + 8*a^16*b^2*d^4))*(-(e*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) - (2*b^18*d^2*e^12 - 138*a^2*b^16*d^2
*e^12 - 3046*a^4*b^14*d^2*e^12 + 4862*a^6*b^12*d^2*e^12 + 9222*a^8*b^10*d^2*e^12 - 5246*a^10*b^8*d^2*e^12 - 42
90*a^12*b^6*d^2*e^12 + 2442*a^14*b^4*d^2*e^12 + 32*a^16*b^2*d^2*e^12)/(a^18*d^5 + a^2*b^16*d^5 + 8*a^4*b^14*d^
5 + 28*a^6*b^12*d^5 + 56*a^8*b^10*d^5 + 70*a^10*b^8*d^5 + 56*a^12*b^6*d^5 + 28*a^14*b^4*d^5 + 8*a^16*b^2*d^5))
*(-(e*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)*(2*a^2*b^13*e^12 - b^15*e^12 + 49*a^4*b^11*e^12 + 2460*a^6*b^9*e^12 - 3
631*a^8*b^7*e^12 + 1922*a^10*b^5*e^12 - 225*a^12*b^3*e^12))/(a^18*d^4 + a^2*b^16*d^4 + 8*a^4*b^14*d^4 + 28*a^6
*b^12*d^4 + 56*a^8*b^10*d^4 + 70*a^10*b^8*d^4 + 56*a^12*b^6*d^4 + 28*a^14*b^4*d^4 + 8*a^16*b^2*d^4))*(-(e*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*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((((((e*cot(c + d*x))^(1/2)*(2*a^2*b^13*e^12 - b^15*e^12 + 49*a^4*b^11*e^12 + 2460*
a^6*b^9*e^12 - 3631*a^8*b^7*e^12 + 1922*a^10*b^5*e^12 - 225*a^12*b^3*e^12))/(a^18*d^4 + a^2*b^16*d^4 + 8*a^4*b
^14*d^4 + 28*a^6*b^12*d^4 + 56*a^8*b^10*d^4 + 70*a^10*b^8*d^4 + 56*a^12*b^6*d^4 + 28*a^14*b^4*d^4 + 8*a^16*b^2
*d^4) + (((2*b^18*d^2*e^12 - 138*a^2*b^16*d^2*e^12 - 3046*a^4*b^14*d^2*e^12 + 4862*a^6*b^12*d^2*e^12 + 9222*a^
8*b^10*d^2*e^12 - 5246*a^10*b^8*d^2*e^12 - 4290*a^12*b^6*d^2*e^12 + 2442*a^14*b^4*d^2*e^12 + 32*a^16*b^2*d^2*e
^12)/(a^18*d^5 + a^2*b^16*d^5 + 8*a^4*b^14*d^5 + 28*a^6*b^12*d^5 + 56*a^8*b^10*d^5 + 70*a^10*b^8*d^5 + 56*a^12
*b^6*d^5 + 28*a^14*b^4*d^5 + 8*a^16*b^2*d^5) - ((((e*cot(c + d*x))^(1/2)*(8*a*b^20*d^2*e^11 - 1152*a^3*b^18*d^
2*e^11 + 2528*a^5*b^16*d^2*e^11 + 15296*a^7*b^14*d^2*e^11 + 14128*a^9*b^12*d^2*e^11 - 5056*a^11*b^10*d^2*e^11
- 9248*a^13*b^8*d^2*e^11 + 64*a^15*b^6*d^2*e^11 + 1800*a^17*b^4*d^2*e^11 + 64*a^19*b^2*d^2*e^11))/(a^18*d^4 +
a^2*b^16*d^4 + 8*a^4*b^14*d^4 + 28*a^6*b^12*d^4 + 56*a^8*b^10*d^4 + 70*a^10*b^8*d^4 + 56*a^12*b^6*d^4 + 28*a^1
4*b^4*d^4 + 8*a^16*b^2*d^4) + (((64*a*b^23*d^4*e^11 + 1472*a^3*b^21*d^4*e^11 + 8832*a^5*b^19*d^4*e^11 + 25344*
a^7*b^17*d^4*e^11 + 40320*a^9*b^15*d^4*e^11 + 34944*a^11*b^13*d^4*e^11 + 10752*a^13*b^11*d^4*e^11 - 8448*a^15*
b^9*d^4*e^11 - 10176*a^17*b^7*d^4*e^11 - 4160*a^19*b^5*d^4*e^11 - 640*a^21*b^3*d^4*e^11)/(a^18*d^5 + a^2*b^16*
d^5 + 8*a^4*b^14*d^5 + 28*a^6*b^12*d^5 + 56*a^8*b^10*d^5 + 70*a^10*b^8*d^5 + 56*a^12*b^6*d^5 + 28*a^14*b^4*d^5
+ 8*a^16*b^2*d^5) - ((e*cot(c + d*x))^(1/2)*(b^4 - 15*a^4 + 18*a^2*b^2)*(-a^3*b*e)^(1/2)*(512*a^2*b^25*d^4*e^
10 + 4608*a^4*b^23*d^4*e^10 + 17920*a^6*b^21*d^4*e^10 + 38400*a^8*b^19*d^4*e^10 + 46080*a^10*b^17*d^4*e^10 + 2
1504*a^12*b^15*d^4*e^10 - 21504*a^14*b^13*d^4*e^10 - 46080*a^16*b^11*d^4*e^10 - 38400*a^18*b^9*d^4*e^10 - 1792
0*a^20*b^7*d^4*e^10 - 4608*a^22*b^5*d^4*e^10 - 512*a^24*b^3*d^4*e^10))/(8*(a^9*d + a^3*b^6*d + 3*a^5*b^4*d + 3
*a^7*b^2*d)*(a^18*d^4 + a^2*b^16*d^4 + 8*a^4*b^14*d^4 + 28*a^6*b^12*d^4 + 56*a^8*b^10*d^4 + 70*a^10*b^8*d^4 +
56*a^12*b^6*d^4 + 28*a^14*b^4*d^4 + 8*a^16*b^2*d^4)))*(b^4 - 15*a^4 + 18*a^2*b^2)*(-a^3*b*e)^(1/2))/(8*(a^9*d
+ a^3*b^6*d + 3*a^5*b^4*d + 3*a^7*b^2*d)))*(b^4 - 15*a^4 + 18*a^2*b^2)*(-a^3*b*e)^(1/2))/(8*(a^9*d + a^3*b^6*d
+ 3*a^5*b^4*d + 3*a^7*b^2*d)))*(b^4 - 15*a^4 + 18*a^2*b^2)*(-a^3*b*e)^(1/2))/(8*(a^9*d + a^3*b^6*d + 3*a^5*b^
4*d + 3*a^7*b^2*d)))*(b^4 - 15*a^4 + 18*a^2*b^2)*(-a^3*b*e)^(1/2)*1i)/(8*(a^9*d + a^3*b^6*d + 3*a^5*b^4*d + 3*
a^7*b^2*d)) + ((((e*cot(c + d*x))^(1/2)*(2*a^2*b^13*e^12 - b^15*e^12 + 49*a^4*b^11*e^12 + 2460*a^6*b^9*e^12 -
3631*a^8*b^7*e^12 + 1922*a^10*b^5*e^12 - 225*a^12*b^3*e^12))/(a^18*d^4 + a^2*b^16*d^4 + 8*a^4*b^14*d^4 + 28*a^
6*b^12*d^4 + 56*a^8*b^10*d^4 + 70*a^10*b^8*d^4 + 56*a^12*b^6*d^4 + 28*a^14*b^4*d^4 + 8*a^16*b^2*d^4) - (((2*b^
18*d^2*e^12 - 138*a^2*b^16*d^2*e^12 - 3046*a^4*b^14*d^2*e^12 + 4862*a^6*b^12*d^2*e^12 + 9222*a^8*b^10*d^2*e^12
- 5246*a^10*b^8*d^2*e^12 - 4290*a^12*b^6*d^2*e^12 + 2442*a^14*b^4*d^2*e^12 + 32*a^16*b^2*d^2*e^12)/(a^18*d^5
+ a^2*b^16*d^5 + 8*a^4*b^14*d^5 + 28*a^6*b^12*d^5 + 56*a^8*b^10*d^5 + 70*a^10*b^8*d^5 + 56*a^12*b^6*d^5 + 28*a
^14*b^4*d^5 + 8*a^16*b^2*d^5) + ((((e*cot(c + d*x))^(1/2)*(8*a*b^20*d^2*e^11 - 1152*a^3*b^18*d^2*e^11 + 2528*a
^5*b^16*d^2*e^11 + 15296*a^7*b^14*d^2*e^11 + 14128*a^9*b^12*d^2*e^11 - 5056*a^11*b^10*d^2*e^11 - 9248*a^13*b^8
*d^2*e^11 + 64*a^15*b^6*d^2*e^11 + 1800*a^17*b^4*d^2*e^11 + 64*a^19*b^2*d^2*e^11))/(a^18*d^4 + a^2*b^16*d^4 +
8*a^4*b^14*d^4 + 28*a^6*b^12*d^4 + 56*a^8*b^10*d^4 + 70*a^10*b^8*d^4 + 56*a^12*b^6*d^4 + 28*a^14*b^4*d^4 + 8*a
^16*b^2*d^4) - (((64*a*b^23*d^4*e^11 + 1472*a^3*b^21*d^4*e^11 + 8832*a^5*b^19*d^4*e^11 + 25344*a^7*b^17*d^4*e^
11 + 40320*a^9*b^15*d^4*e^11 + 34944*a^11*b^13*d^4*e^11 + 10752*a^13*b^11*d^4*e^11 - 8448*a^15*b^9*d^4*e^11 -
10176*a^17*b^7*d^4*e^11 - 4160*a^19*b^5*d^4*e^11 - 640*a^21*b^3*d^4*e^11)/(a^18*d^5 + a^2*b^16*d^5 + 8*a^4*b^1
4*d^5 + 28*a^6*b^12*d^5 + 56*a^8*b^10*d^5 + 70*a^10*b^8*d^5 + 56*a^12*b^6*d^5 + 28*a^14*b^4*d^5 + 8*a^16*b^2*d
^5) + ((e*cot(c + d*x))^(1/2)*(b^4 - 15*a^4 + 18*a^2*b^2)*(-a^3*b*e)^(1/2)*(512*a^2*b^25*d^4*e^10 + 4608*a^4*b
^23*d^4*e^10 + 17920*a^6*b^21*d^4*e^10 + 38400*a^8*b^19*d^4*e^10 + 46080*a^10*b^17*d^4*e^10 + 21504*a^12*b^15*
d^4*e^10 - 21504*a^14*b^13*d^4*e^10 - 46080*a^16*b^11*d^4*e^10 - 38400*a^18*b^9*d^4*e^10 - 17920*a^20*b^7*d^4*
e^10 - 4608*a^22*b^5*d^4*e^10 - 512*a^24*b^3*d^4*e^10))/(8*(a^9*d + a^3*b^6*d + 3*a^5*b^4*d + 3*a^7*b^2*d)*(a^
18*d^4 + a^2*b^16*d^4 + 8*a^4*b^14*d^4 + 28*a^6*b^12*d^4 + 56*a^8*b^10*d^4 + 70*a^10*b^8*d^4 + 56*a^12*b^6*d^4
+ 28*a^14*b^4*d^4 + 8*a^16*b^2*d^4)))*(b^4 - 15*a^4 + 18*a^2*b^2)*(-a^3*b*e)^(1/2))/(8*(a^9*d + a^3*b^6*d + 3
*a^5*b^4*d + 3*a^7*b^2*d)))*(b^4 - 15*a^4 + 18*a^2*b^2)*(-a^3*b*e)^(1/2))/(8*(a^9*d + a^3*b^6*d + 3*a^5*b^4*d
+ 3*a^7*b^2*d)))*(b^4 - 15*a^4 + 18*a^2*b^2)*(-a^3*b*e)^(1/2))/(8*(a^9*d + a^3*b^6*d + 3*a^5*b^4*d + 3*a^7*b^2
*d)))*(b^4 - 15*a^4 + 18*a^2*b^2)*(-a^3*b*e)^(1/2)*1i)/(8*(a^9*d + a^3*b^6*d + 3*a^5*b^4*d + 3*a^7*b^2*d)))/((
7*a*b^11*e^13 + 116*a^3*b^9*e^13 - 270*a^5*b^7*e^13 + 420*a^7*b^5*e^13 - 225*a^9*b^3*e^13)/(a^18*d^5 + a^2*b^1
6*d^5 + 8*a^4*b^14*d^5 + 28*a^6*b^12*d^5 + 56*a^8*b^10*d^5 + 70*a^10*b^8*d^5 + 56*a^12*b^6*d^5 + 28*a^14*b^4*d
^5 + 8*a^16*b^2*d^5) - ((((e*cot(c + d*x))^(1/2)*(2*a^2*b^13*e^12 - b^15*e^12 + 49*a^4*b^11*e^12 + 2460*a^6*b^
9*e^12 - 3631*a^8*b^7*e^12 + 1922*a^10*b^5*e^12 - 225*a^12*b^3*e^12))/(a^18*d^4 + a^2*b^16*d^4 + 8*a^4*b^14*d^
4 + 28*a^6*b^12*d^4 + 56*a^8*b^10*d^4 + 70*a^10*b^8*d^4 + 56*a^12*b^6*d^4 + 28*a^14*b^4*d^4 + 8*a^16*b^2*d^4)
+ (((2*b^18*d^2*e^12 - 138*a^2*b^16*d^2*e^12 - 3046*a^4*b^14*d^2*e^12 + 4862*a^6*b^12*d^2*e^12 + 9222*a^8*b^10
*d^2*e^12 - 5246*a^10*b^8*d^2*e^12 - 4290*a^12*b^6*d^2*e^12 + 2442*a^14*b^4*d^2*e^12 + 32*a^16*b^2*d^2*e^12)/(
a^18*d^5 + a^2*b^16*d^5 + 8*a^4*b^14*d^5 + 28*a^6*b^12*d^5 + 56*a^8*b^10*d^5 + 70*a^10*b^8*d^5 + 56*a^12*b^6*d
^5 + 28*a^14*b^4*d^5 + 8*a^16*b^2*d^5) - ((((e*cot(c + d*x))^(1/2)*(8*a*b^20*d^2*e^11 - 1152*a^3*b^18*d^2*e^11
+ 2528*a^5*b^16*d^2*e^11 + 15296*a^7*b^14*d^2*e^11 + 14128*a^9*b^12*d^2*e^11 - 5056*a^11*b^10*d^2*e^11 - 9248
*a^13*b^8*d^2*e^11 + 64*a^15*b^6*d^2*e^11 + 1800*a^17*b^4*d^2*e^11 + 64*a^19*b^2*d^2*e^11))/(a^18*d^4 + a^2*b^
16*d^4 + 8*a^4*b^14*d^4 + 28*a^6*b^12*d^4 + 56*a^8*b^10*d^4 + 70*a^10*b^8*d^4 + 56*a^12*b^6*d^4 + 28*a^14*b^4*
d^4 + 8*a^16*b^2*d^4) + (((64*a*b^23*d^4*e^11 + 1472*a^3*b^21*d^4*e^11 + 8832*a^5*b^19*d^4*e^11 + 25344*a^7*b^
17*d^4*e^11 + 40320*a^9*b^15*d^4*e^11 + 34944*a^11*b^13*d^4*e^11 + 10752*a^13*b^11*d^4*e^11 - 8448*a^15*b^9*d^
4*e^11 - 10176*a^17*b^7*d^4*e^11 - 4160*a^19*b^5*d^4*e^11 - 640*a^21*b^3*d^4*e^11)/(a^18*d^5 + a^2*b^16*d^5 +
8*a^4*b^14*d^5 + 28*a^6*b^12*d^5 + 56*a^8*b^10*d^5 + 70*a^10*b^8*d^5 + 56*a^12*b^6*d^5 + 28*a^14*b^4*d^5 + 8*a
^16*b^2*d^5) - ((e*cot(c + d*x))^(1/2)*(b^4 - 15*a^4 + 18*a^2*b^2)*(-a^3*b*e)^(1/2)*(512*a^2*b^25*d^4*e^10 + 4
608*a^4*b^23*d^4*e^10 + 17920*a^6*b^21*d^4*e^10 + 38400*a^8*b^19*d^4*e^10 + 46080*a^10*b^17*d^4*e^10 + 21504*a
^12*b^15*d^4*e^10 - 21504*a^14*b^13*d^4*e^10 - 46080*a^16*b^11*d^4*e^10 - 38400*a^18*b^9*d^4*e^10 - 17920*a^20
*b^7*d^4*e^10 - 4608*a^22*b^5*d^4*e^10 - 512*a^24*b^3*d^4*e^10))/(8*(a^9*d + a^3*b^6*d + 3*a^5*b^4*d + 3*a^7*b
^2*d)*(a^18*d^4 + a^2*b^16*d^4 + 8*a^4*b^14*d^4 + 28*a^6*b^12*d^4 + 56*a^8*b^10*d^4 + 70*a^10*b^8*d^4 + 56*a^1
2*b^6*d^4 + 28*a^14*b^4*d^4 + 8*a^16*b^2*d^4)))*(b^4 - 15*a^4 + 18*a^2*b^2)*(-a^3*b*e)^(1/2))/(8*(a^9*d + a^3*
b^6*d + 3*a^5*b^4*d + 3*a^7*b^2*d)))*(b^4 - 15*a^4 + 18*a^2*b^2)*(-a^3*b*e)^(1/2))/(8*(a^9*d + a^3*b^6*d + 3*a
^5*b^4*d + 3*a^7*b^2*d)))*(b^4 - 15*a^4 + 18*a^2*b^2)*(-a^3*b*e)^(1/2))/(8*(a^9*d + a^3*b^6*d + 3*a^5*b^4*d +
3*a^7*b^2*d)))*(b^4 - 15*a^4 + 18*a^2*b^2)*(-a^3*b*e)^(1/2))/(8*(a^9*d + a^3*b^6*d + 3*a^5*b^4*d + 3*a^7*b^2*d
)) + ((((e*cot(c + d*x))^(1/2)*(2*a^2*b^13*e^12 - b^15*e^12 + 49*a^4*b^11*e^12 + 2460*a^6*b^9*e^12 - 3631*a^8*
b^7*e^12 + 1922*a^10*b^5*e^12 - 225*a^12*b^3*e^12))/(a^18*d^4 + a^2*b^16*d^4 + 8*a^4*b^14*d^4 + 28*a^6*b^12*d^
4 + 56*a^8*b^10*d^4 + 70*a^10*b^8*d^4 + 56*a^12*b^6*d^4 + 28*a^14*b^4*d^4 + 8*a^16*b^2*d^4) - (((2*b^18*d^2*e^
12 - 138*a^2*b^16*d^2*e^12 - 3046*a^4*b^14*d^2*e^12 + 4862*a^6*b^12*d^2*e^12 + 9222*a^8*b^10*d^2*e^12 - 5246*a
^10*b^8*d^2*e^12 - 4290*a^12*b^6*d^2*e^12 + 2442*a^14*b^4*d^2*e^12 + 32*a^16*b^2*d^2*e^12)/(a^18*d^5 + a^2*b^1
6*d^5 + 8*a^4*b^14*d^5 + 28*a^6*b^12*d^5 + 56*a^8*b^10*d^5 + 70*a^10*b^8*d^5 + 56*a^12*b^6*d^5 + 28*a^14*b^4*d
^5 + 8*a^16*b^2*d^5) + ((((e*cot(c + d*x))^(1/2)*(8*a*b^20*d^2*e^11 - 1152*a^3*b^18*d^2*e^11 + 2528*a^5*b^16*d
^2*e^11 + 15296*a^7*b^14*d^2*e^11 + 14128*a^9*b^12*d^2*e^11 - 5056*a^11*b^10*d^2*e^11 - 9248*a^13*b^8*d^2*e^11
+ 64*a^15*b^6*d^2*e^11 + 1800*a^17*b^4*d^2*e^11 + 64*a^19*b^2*d^2*e^11))/(a^18*d^4 + a^2*b^16*d^4 + 8*a^4*b^1
4*d^4 + 28*a^6*b^12*d^4 + 56*a^8*b^10*d^4 + 70*a^10*b^8*d^4 + 56*a^12*b^6*d^4 + 28*a^14*b^4*d^4 + 8*a^16*b^2*d
^4) - (((64*a*b^23*d^4*e^11 + 1472*a^3*b^21*d^4*e^11 + 8832*a^5*b^19*d^4*e^11 + 25344*a^7*b^17*d^4*e^11 + 4032
0*a^9*b^15*d^4*e^11 + 34944*a^11*b^13*d^4*e^11 + 10752*a^13*b^11*d^4*e^11 - 8448*a^15*b^9*d^4*e^11 - 10176*a^1
7*b^7*d^4*e^11 - 4160*a^19*b^5*d^4*e^11 - 640*a^21*b^3*d^4*e^11)/(a^18*d^5 + a^2*b^16*d^5 + 8*a^4*b^14*d^5 + 2
8*a^6*b^12*d^5 + 56*a^8*b^10*d^5 + 70*a^10*b^8*d^5 + 56*a^12*b^6*d^5 + 28*a^14*b^4*d^5 + 8*a^16*b^2*d^5) + ((e
*cot(c + d*x))^(1/2)*(b^4 - 15*a^4 + 18*a^2*b^2)*(-a^3*b*e)^(1/2)*(512*a^2*b^25*d^4*e^10 + 4608*a^4*b^23*d^4*e
^10 + 17920*a^6*b^21*d^4*e^10 + 38400*a^8*b^19*d^4*e^10 + 46080*a^10*b^17*d^4*e^10 + 21504*a^12*b^15*d^4*e^10
- 21504*a^14*b^13*d^4*e^10 - 46080*a^16*b^11*d^4*e^10 - 38400*a^18*b^9*d^4*e^10 - 17920*a^20*b^7*d^4*e^10 - 46
08*a^22*b^5*d^4*e^10 - 512*a^24*b^3*d^4*e^10))/(8*(a^9*d + a^3*b^6*d + 3*a^5*b^4*d + 3*a^7*b^2*d)*(a^18*d^4 +
a^2*b^16*d^4 + 8*a^4*b^14*d^4 + 28*a^6*b^12*d^4 + 56*a^8*b^10*d^4 + 70*a^10*b^8*d^4 + 56*a^12*b^6*d^4 + 28*a^1
4*b^4*d^4 + 8*a^16*b^2*d^4)))*(b^4 - 15*a^4 + 18*a^2*b^2)*(-a^3*b*e)^(1/2))/(8*(a^9*d + a^3*b^6*d + 3*a^5*b^4*
d + 3*a^7*b^2*d)))*(b^4 - 15*a^4 + 18*a^2*b^2)*(-a^3*b*e)^(1/2))/(8*(a^9*d + a^3*b^6*d + 3*a^5*b^4*d + 3*a^7*b
^2*d)))*(b^4 - 15*a^4 + 18*a^2*b^2)*(-a^3*b*e)^(1/2))/(8*(a^9*d + a^3*b^6*d + 3*a^5*b^4*d + 3*a^7*b^2*d)))*(b^
4 - 15*a^4 + 18*a^2*b^2)*(-a^3*b*e)^(1/2))/(8*(a^9*d + a^3*b^6*d + 3*a^5*b^4*d + 3*a^7*b^2*d))))*(b^4 - 15*a^4
+ 18*a^2*b^2)*(-a^3*b*e)^(1/2)*1i)/(4*(a^9*d + a^3*b^6*d + 3*a^5*b^4*d + 3*a^7*b^2*d))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*cot(d*x+c))**(1/2)/(a+b*cot(d*x+c))**3,x)

[Out]

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

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