3.86 \(\int \frac {1}{\sqrt {e \cot (c+d x)} (a+b \cot (c+d x))^3} \, dx\)

Optimal. Leaf size=476 \[ -\frac {b^2 \left (11 a^2+3 b^2\right ) \sqrt {e \cot (c+d x)}}{4 a^2 d e \left (a^2+b^2\right )^2 (a+b \cot (c+d x))}-\frac {b^2 \sqrt {e \cot (c+d x)}}{2 a d e \left (a^2+b^2\right ) (a+b \cot (c+d x))^2}+\frac {(a-b) \left (a^2+4 a b+b^2\right ) \log \left (\sqrt {e} \cot (c+d x)-\sqrt {2} \sqrt {e \cot (c+d x)}+\sqrt {e}\right )}{2 \sqrt {2} d \sqrt {e} \left (a^2+b^2\right )^3}-\frac {(a-b) \left (a^2+4 a b+b^2\right ) \log \left (\sqrt {e} \cot (c+d x)+\sqrt {2} \sqrt {e \cot (c+d x)}+\sqrt {e}\right )}{2 \sqrt {2} d \sqrt {e} \left (a^2+b^2\right )^3}+\frac {(a+b) \left (a^2-4 a b+b^2\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} d \sqrt {e} \left (a^2+b^2\right )^3}-\frac {(a+b) \left (a^2-4 a b+b^2\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}+1\right )}{\sqrt {2} d \sqrt {e} \left (a^2+b^2\right )^3}-\frac {b^{3/2} \left (35 a^4+6 a^2 b^2+3 b^4\right ) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {e \cot (c+d x)}}{\sqrt {a} \sqrt {e}}\right )}{4 a^{5/2} d \sqrt {e} \left (a^2+b^2\right )^3} \]

[Out]

-1/4*b^(3/2)*(35*a^4+6*a^2*b^2+3*b^4)*arctan(b^(1/2)*(e*cot(d*x+c))^(1/2)/a^(1/2)/e^(1/2))/a^(5/2)/(a^2+b^2)^3
/d/e^(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))/(a^2+b^2)^3/d*2^(1/2)/e^(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))/(a^2+b^2)^3/d*2^(1/2)/e^(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))/(a^2+b^2)^3/d*2^(1/2)/e^(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))/(a^2+b^2)^3/d*2^(1/2)
/e^(1/2)-1/2*b^2*(e*cot(d*x+c))^(1/2)/a/(a^2+b^2)/d/e/(a+b*cot(d*x+c))^2-1/4*b^2*(11*a^2+3*b^2)*(e*cot(d*x+c))
^(1/2)/a^2/(a^2+b^2)^2/d/e/(a+b*cot(d*x+c))

________________________________________________________________________________________

Rubi [A]  time = 1.24, antiderivative size = 476, 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 = {3569, 3649, 3653, 3534, 1168, 1162, 617, 204, 1165, 628, 3634, 63, 205} \[ -\frac {b^2 \left (11 a^2+3 b^2\right ) \sqrt {e \cot (c+d x)}}{4 a^2 d e \left (a^2+b^2\right )^2 (a+b \cot (c+d x))}-\frac {b^2 \sqrt {e \cot (c+d x)}}{2 a d e \left (a^2+b^2\right ) (a+b \cot (c+d x))^2}+\frac {(a-b) \left (a^2+4 a b+b^2\right ) \log \left (\sqrt {e} \cot (c+d x)-\sqrt {2} \sqrt {e \cot (c+d x)}+\sqrt {e}\right )}{2 \sqrt {2} d \sqrt {e} \left (a^2+b^2\right )^3}-\frac {(a-b) \left (a^2+4 a b+b^2\right ) \log \left (\sqrt {e} \cot (c+d x)+\sqrt {2} \sqrt {e \cot (c+d x)}+\sqrt {e}\right )}{2 \sqrt {2} d \sqrt {e} \left (a^2+b^2\right )^3}-\frac {b^{3/2} \left (6 a^2 b^2+35 a^4+3 b^4\right ) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {e \cot (c+d x)}}{\sqrt {a} \sqrt {e}}\right )}{4 a^{5/2} d \sqrt {e} \left (a^2+b^2\right )^3}+\frac {(a+b) \left (a^2-4 a b+b^2\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} d \sqrt {e} \left (a^2+b^2\right )^3}-\frac {(a+b) \left (a^2-4 a b+b^2\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}+1\right )}{\sqrt {2} d \sqrt {e} \left (a^2+b^2\right )^3} \]

Antiderivative was successfully verified.

[In]

Int[1/(Sqrt[e*Cot[c + d*x]]*(a + b*Cot[c + d*x])^3),x]

[Out]

-(b^(3/2)*(35*a^4 + 6*a^2*b^2 + 3*b^4)*ArcTan[(Sqrt[b]*Sqrt[e*Cot[c + d*x]])/(Sqrt[a]*Sqrt[e])])/(4*a^(5/2)*(a
^2 + b^2)^3*d*Sqrt[e]) + ((a + b)*(a^2 - 4*a*b + b^2)*ArcTan[1 - (Sqrt[2]*Sqrt[e*Cot[c + d*x]])/Sqrt[e]])/(Sqr
t[2]*(a^2 + b^2)^3*d*Sqrt[e]) - ((a + b)*(a^2 - 4*a*b + b^2)*ArcTan[1 + (Sqrt[2]*Sqrt[e*Cot[c + d*x]])/Sqrt[e]
])/(Sqrt[2]*(a^2 + b^2)^3*d*Sqrt[e]) - (b^2*Sqrt[e*Cot[c + d*x]])/(2*a*(a^2 + b^2)*d*e*(a + b*Cot[c + d*x])^2)
- (b^2*(11*a^2 + 3*b^2)*Sqrt[e*Cot[c + d*x]])/(4*a^2*(a^2 + b^2)^2*d*e*(a + b*Cot[c + d*x])) + ((a - b)*(a^2
+ 4*a*b + b^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*
Sqrt[e]) - ((a - b)*(a^2 + 4*a*b + b^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*Sqrt[e])

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 3569

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

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 {1}{\sqrt {e \cot (c+d x)} (a+b \cot (c+d x))^3} \, dx &=-\frac {b^2 \sqrt {e \cot (c+d x)}}{2 a \left (a^2+b^2\right ) d e (a+b \cot (c+d x))^2}-\frac {\int \frac {-\frac {1}{2} \left (4 a^2+3 b^2\right ) e+2 a b e \cot (c+d x)-\frac {3}{2} b^2 e \cot ^2(c+d x)}{\sqrt {e \cot (c+d x)} (a+b \cot (c+d x))^2} \, dx}{2 a \left (a^2+b^2\right ) e}\\ &=-\frac {b^2 \sqrt {e \cot (c+d x)}}{2 a \left (a^2+b^2\right ) d e (a+b \cot (c+d x))^2}-\frac {b^2 \left (11 a^2+3 b^2\right ) \sqrt {e \cot (c+d x)}}{4 a^2 \left (a^2+b^2\right )^2 d e (a+b \cot (c+d x))}+\frac {\int \frac {\frac {1}{4} \left (8 a^4+3 a^2 b^2+3 b^4\right ) e^2-4 a^3 b e^2 \cot (c+d x)+\frac {1}{4} b^2 \left (11 a^2+3 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^2 \left (a^2+b^2\right )^2 e^2}\\ &=-\frac {b^2 \sqrt {e \cot (c+d x)}}{2 a \left (a^2+b^2\right ) d e (a+b \cot (c+d x))^2}-\frac {b^2 \left (11 a^2+3 b^2\right ) \sqrt {e \cot (c+d x)}}{4 a^2 \left (a^2+b^2\right )^2 d e (a+b \cot (c+d x))}+\frac {\left (b^2 \left (35 a^4+6 a^2 b^2+3 b^4\right )\right ) \int \frac {1+\cot ^2(c+d x)}{\sqrt {e \cot (c+d x)} (a+b \cot (c+d x))} \, dx}{8 a^2 \left (a^2+b^2\right )^3}+\frac {\int \frac {2 a^3 \left (a^2-3 b^2\right ) e^2-2 a^2 b \left (3 a^2-b^2\right ) e^2 \cot (c+d x)}{\sqrt {e \cot (c+d x)}} \, dx}{2 a^2 \left (a^2+b^2\right )^3 e^2}\\ &=-\frac {b^2 \sqrt {e \cot (c+d x)}}{2 a \left (a^2+b^2\right ) d e (a+b \cot (c+d x))^2}-\frac {b^2 \left (11 a^2+3 b^2\right ) \sqrt {e \cot (c+d x)}}{4 a^2 \left (a^2+b^2\right )^2 d e (a+b \cot (c+d x))}+\frac {\left (b^2 \left (35 a^4+6 a^2 b^2+3 b^4\right )\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {-e x} (a-b x)} \, dx,x,-\cot (c+d x)\right )}{8 a^2 \left (a^2+b^2\right )^3 d}+\frac {\operatorname {Subst}\left (\int \frac {-2 a^3 \left (a^2-3 b^2\right ) e^3+2 a^2 b \left (3 a^2-b^2\right ) e^2 x^2}{e^2+x^4} \, dx,x,\sqrt {e \cot (c+d x)}\right )}{a^2 \left (a^2+b^2\right )^3 d e^2}\\ &=-\frac {b^2 \sqrt {e \cot (c+d x)}}{2 a \left (a^2+b^2\right ) d e (a+b \cot (c+d x))^2}-\frac {b^2 \left (11 a^2+3 b^2\right ) \sqrt {e \cot (c+d x)}}{4 a^2 \left (a^2+b^2\right )^2 d e (a+b \cot (c+d x))}-\frac {\left ((a+b) \left (a^2-4 a b+b^2\right )\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 )\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 (b^2 \left (35 a^4+6 a^2 b^2+3 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^2 \left (a^2+b^2\right )^3 d e}\\ &=-\frac {b^{3/2} \left (35 a^4+6 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 a^{5/2} \left (a^2+b^2\right )^3 d \sqrt {e}}-\frac {b^2 \sqrt {e \cot (c+d x)}}{2 a \left (a^2+b^2\right ) d e (a+b \cot (c+d x))^2}-\frac {b^2 \left (11 a^2+3 b^2\right ) \sqrt {e \cot (c+d x)}}{4 a^2 \left (a^2+b^2\right )^2 d e (a+b \cot (c+d x))}-\frac {\left ((a+b) \left (a^2-4 a b+b^2\right )\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 )\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 )\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 \sqrt {e}}+\frac {\left ((a-b) \left (a^2+4 a b+b^2\right )\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 \sqrt {e}}\\ &=-\frac {b^{3/2} \left (35 a^4+6 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 a^{5/2} \left (a^2+b^2\right )^3 d \sqrt {e}}-\frac {b^2 \sqrt {e \cot (c+d x)}}{2 a \left (a^2+b^2\right ) d e (a+b \cot (c+d x))^2}-\frac {b^2 \left (11 a^2+3 b^2\right ) \sqrt {e \cot (c+d x)}}{4 a^2 \left (a^2+b^2\right )^2 d e (a+b \cot (c+d x))}+\frac {(a-b) \left (a^2+4 a b+b^2\right ) \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 \sqrt {e}}-\frac {(a-b) \left (a^2+4 a b+b^2\right ) \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 \sqrt {e}}-\frac {\left ((a+b) \left (a^2-4 a b+b^2\right )\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 \sqrt {e}}+\frac {\left ((a+b) \left (a^2-4 a b+b^2\right )\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 \sqrt {e}}\\ &=-\frac {b^{3/2} \left (35 a^4+6 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 a^{5/2} \left (a^2+b^2\right )^3 d \sqrt {e}}+\frac {(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} \left (a^2+b^2\right )^3 d \sqrt {e}}-\frac {(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} \left (a^2+b^2\right )^3 d \sqrt {e}}-\frac {b^2 \sqrt {e \cot (c+d x)}}{2 a \left (a^2+b^2\right ) d e (a+b \cot (c+d x))^2}-\frac {b^2 \left (11 a^2+3 b^2\right ) \sqrt {e \cot (c+d x)}}{4 a^2 \left (a^2+b^2\right )^2 d e (a+b \cot (c+d x))}+\frac {(a-b) \left (a^2+4 a b+b^2\right ) \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 \sqrt {e}}-\frac {(a-b) \left (a^2+4 a b+b^2\right ) \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 \sqrt {e}}\\ \end {align*}

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Mathematica [C]  time = 6.13, size = 411, normalized size = 0.86 \[ -\frac {\sqrt {\cot (c+d x)} \left (-\frac {2 b \left (3 a^2-b^2\right ) \cot ^{\frac {3}{2}}(c+d x) \, _2F_1\left (\frac {3}{4},1;\frac {7}{4};-\cot ^2(c+d x)\right )}{3 \left (a^2+b^2\right )^3}+\frac {2 b^2 \sqrt {\cot (c+d x)} \left (\frac {a}{a+b \cot (c+d x)}+\frac {\sqrt {a} \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {\cot (c+d x)}}{\sqrt {a}}\right )}{\sqrt {b} \sqrt {\cot (c+d x)}}\right )}{a \left (a^2+b^2\right )^2}-\frac {a \left (a^2-3 b^2\right ) \left (2 \sqrt {2} \log \left (\cot (c+d x)-\sqrt {2} \sqrt {\cot (c+d x)}+1\right )-2 \sqrt {2} \log \left (\cot (c+d x)+\sqrt {2} \sqrt {\cot (c+d x)}+1\right )+4 \left (\sqrt {2} \tan ^{-1}\left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )-\sqrt {2} \tan ^{-1}\left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )\right )\right )}{8 \left (a^2+b^2\right )^3}+\frac {2 b^{3/2} \left (3 a^2-b^2\right ) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {\cot (c+d x)}}{\sqrt {a}}\right )}{\sqrt {a} \left (a^2+b^2\right )^3}+\frac {2 b^2 \sqrt {\cot (c+d x)} \, _2F_1\left (\frac {1}{2},3;\frac {3}{2};-\frac {b \cot (c+d x)}{a}\right )}{a^3 \left (a^2+b^2\right )}\right )}{d \sqrt {e \cot (c+d x)}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(Sqrt[e*Cot[c + d*x]]*(a + b*Cot[c + d*x])^3),x]

[Out]

-((Sqrt[Cot[c + d*x]]*((2*b^(3/2)*(3*a^2 - b^2)*ArcTan[(Sqrt[b]*Sqrt[Cot[c + d*x]])/Sqrt[a]])/(Sqrt[a]*(a^2 +
b^2)^3) + (2*b^2*Sqrt[Cot[c + d*x]]*((Sqrt[a]*ArcTan[(Sqrt[b]*Sqrt[Cot[c + d*x]])/Sqrt[a]])/(Sqrt[b]*Sqrt[Cot[
c + d*x]]) + a/(a + b*Cot[c + d*x])))/(a*(a^2 + b^2)^2) + (2*b^2*Sqrt[Cot[c + d*x]]*Hypergeometric2F1[1/2, 3,
3/2, -((b*Cot[c + d*x])/a)])/(a^3*(a^2 + b^2)) - (2*b*(3*a^2 - b^2)*Cot[c + d*x]^(3/2)*Hypergeometric2F1[3/4,
1, 7/4, -Cot[c + d*x]^2])/(3*(a^2 + b^2)^3) - (a*(a^2 - 3*b^2)*(4*(Sqrt[2]*ArcTan[1 - Sqrt[2]*Sqrt[Cot[c + d*x
]]] - Sqrt[2]*ArcTan[1 + Sqrt[2]*Sqrt[Cot[c + d*x]]]) + 2*Sqrt[2]*Log[1 - Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c +
d*x]] - 2*Sqrt[2]*Log[1 + Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]]))/(8*(a^2 + b^2)^3)))/(d*Sqrt[e*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(1/(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 \[ \int \frac {1}{{\left (b \cot \left (d x + c\right ) + a\right )}^{3} \sqrt {e \cot \left (d x + c\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [B]  time = 0.84, size = 1190, normalized size = 2.50 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-11/4/d*b^3/(a^2+b^2)^3/(e*cot(d*x+c)*b+a*e)^2*a^2*(e*cot(d*x+c))^(3/2)-7/2/d*b^5/(a^2+b^2)^3/(e*cot(d*x+c)*b+
a*e)^2*(e*cot(d*x+c))^(3/2)-3/4/d*b^7/(a^2+b^2)^3/(e*cot(d*x+c)*b+a*e)^2/a^2*(e*cot(d*x+c))^(3/2)-13/4/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))^(1/2)-9/2/d*e*b^4/(a^2+b^2)^3/(e*cot(d*x+c)*b+a*e)^2*a*
(e*cot(d*x+c))^(1/2)-5/4/d*e*b^6/(a^2+b^2)^3/(e*cot(d*x+c)*b+a*e)^2/a*(e*cot(d*x+c))^(1/2)-35/4/d*b^2/(a^2+b^2
)^3*a^2/(a*e*b)^(1/2)*arctan((e*cot(d*x+c))^(1/2)*b/(a*e*b)^(1/2))-3/2/d*b^4/(a^2+b^2)^3/(a*e*b)^(1/2)*arctan(
(e*cot(d*x+c))^(1/2)*b/(a*e*b)^(1/2))-3/4/d*b^6/(a^2+b^2)^3/a^2/(a*e*b)^(1/2)*arctan((e*cot(d*x+c))^(1/2)*b/(a
*e*b)^(1/2))+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^2-1/2/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)*l
n((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^2-3
/2/d/(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/(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/(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/(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/(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/(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 = 0.48, size = 510, normalized size = 1.07 \[ -\frac {e {\left (\frac {{\left (13 \, a^{3} b^{2} + 5 \, a b^{4}\right )} e \sqrt {\frac {e}{\tan \left (d x + c\right )}} + {\left (11 \, a^{2} b^{3} + 3 \, b^{5}\right )} \left (\frac {e}{\tan \left (d x + c\right )}\right )^{\frac {3}{2}}}{{\left (a^{8} + 2 \, a^{6} b^{2} + a^{4} b^{4}\right )} e^{3} + \frac {2 \, {\left (a^{7} b + 2 \, a^{5} b^{3} + a^{3} b^{5}\right )} e^{3}}{\tan \left (d x + c\right )} + \frac {{\left (a^{6} b^{2} + 2 \, a^{4} b^{4} + a^{2} b^{6}\right )} e^{3}}{\tan \left (d x + c\right )^{2}}} + \frac {{\left (35 \, a^{4} b^{2} + 6 \, a^{2} b^{4} + 3 \, b^{6}\right )} \arctan \left (\frac {b \sqrt {\frac {e}{\tan \left (d x + c\right )}}}{\sqrt {a b e}}\right )}{{\left (a^{8} + 3 \, a^{6} b^{2} + 3 \, a^{4} b^{4} + a^{2} b^{6}\right )} \sqrt {a b e} e} + \frac {\frac {2 \, \sqrt {2} {\left (a^{3} - 3 \, a^{2} b - 3 \, a b^{2} + b^{3}\right )} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \sqrt {e} + 2 \, \sqrt {\frac {e}{\tan \left (d x + c\right )}}\right )}}{2 \, \sqrt {e}}\right )}{\sqrt {e}} + \frac {2 \, \sqrt {2} {\left (a^{3} - 3 \, a^{2} b - 3 \, a b^{2} + b^{3}\right )} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \sqrt {e} - 2 \, \sqrt {\frac {e}{\tan \left (d x + c\right )}}\right )}}{2 \, \sqrt {e}}\right )}{\sqrt {e}} + \frac {\sqrt {2} {\left (a^{3} + 3 \, a^{2} b - 3 \, a b^{2} - b^{3}\right )} \log \left (\sqrt {2} \sqrt {e} \sqrt {\frac {e}{\tan \left (d x + c\right )}} + e + \frac {e}{\tan \left (d x + c\right )}\right )}{\sqrt {e}} - \frac {\sqrt {2} {\left (a^{3} + 3 \, a^{2} b - 3 \, a b^{2} - b^{3}\right )} \log \left (-\sqrt {2} \sqrt {e} \sqrt {\frac {e}{\tan \left (d x + c\right )}} + e + \frac {e}{\tan \left (d x + c\right )}\right )}{\sqrt {e}}}{{\left (a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}\right )} e}\right )}}{4 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*e*(((13*a^3*b^2 + 5*a*b^4)*e*sqrt(e/tan(d*x + c)) + (11*a^2*b^3 + 3*b^5)*(e/tan(d*x + c))^(3/2))/((a^8 +
2*a^6*b^2 + a^4*b^4)*e^3 + 2*(a^7*b + 2*a^5*b^3 + a^3*b^5)*e^3/tan(d*x + c) + (a^6*b^2 + 2*a^4*b^4 + a^2*b^6)*
e^3/tan(d*x + c)^2) + (35*a^4*b^2 + 6*a^2*b^4 + 3*b^6)*arctan(b*sqrt(e/tan(d*x + c))/sqrt(a*b*e))/((a^8 + 3*a^
6*b^2 + 3*a^4*b^4 + a^2*b^6)*sqrt(a*b*e)*e) + (2*sqrt(2)*(a^3 - 3*a^2*b - 3*a*b^2 + b^3)*arctan(1/2*sqrt(2)*(s
qrt(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)*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))/((a^6 + 3*a^4*b^2 + 3*a^2*b^4
+ b^6)*e))/d

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mupad [B]  time = 6.79, size = 20155, normalized size = 42.34 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

atan(((1i/(4*(b^6*d^2*e - a^6*d^2*e - 15*a^2*b^4*d^2*e - a^3*b^3*d^2*e*20i + 15*a^4*b^2*d^2*e + a*b^5*d^2*e*6i
+ a^5*b*d^2*e*6i)))^(1/2)*((1i/(4*(b^6*d^2*e - a^6*d^2*e - 15*a^2*b^4*d^2*e - a^3*b^3*d^2*e*20i + 15*a^4*b^2*
d^2*e + a*b^5*d^2*e*6i + a^5*b*d^2*e*6i)))^(1/2)*((1i/(4*(b^6*d^2*e - a^6*d^2*e - 15*a^2*b^4*d^2*e - a^3*b^3*d
^2*e*20i + 15*a^4*b^2*d^2*e + a*b^5*d^2*e*6i + a^5*b*d^2*e*6i)))^(1/2)*((1i/(4*(b^6*d^2*e - a^6*d^2*e - 15*a^2
*b^4*d^2*e - a^3*b^3*d^2*e*20i + 15*a^4*b^2*d^2*e + a*b^5*d^2*e*6i + a^5*b*d^2*e*6i)))^(1/2)*((192*a^2*b^24*d^
4*e^10 + 1728*a^4*b^22*d^4*e^10 + 8320*a^6*b^20*d^4*e^10 + 27264*a^8*b^18*d^4*e^10 + 62592*a^10*b^16*d^4*e^10
+ 99456*a^12*b^14*d^4*e^10 + 107520*a^14*b^12*d^4*e^10 + 76800*a^16*b^10*d^4*e^10 + 33984*a^18*b^8*d^4*e^10 +
7872*a^20*b^6*d^4*e^10 + 384*a^22*b^4*d^4*e^10 - 128*a^24*b^2*d^4*e^10)/(a^20*d^5 + a^4*b^16*d^5 + 8*a^6*b^14*
d^5 + 28*a^8*b^12*d^5 + 56*a^10*b^10*d^5 + 70*a^12*b^8*d^5 + 56*a^14*b^6*d^5 + 28*a^16*b^4*d^5 + 8*a^18*b^2*d^
5) - ((1i/(4*(b^6*d^2*e - a^6*d^2*e - 15*a^2*b^4*d^2*e - a^3*b^3*d^2*e*20i + 15*a^4*b^2*d^2*e + a*b^5*d^2*e*6i
+ a^5*b*d^2*e*6i)))^(1/2)*(e*cot(c + d*x))^(1/2)*(512*a^4*b^25*d^4*e^10 + 4608*a^6*b^23*d^4*e^10 + 17920*a^8*
b^21*d^4*e^10 + 38400*a^10*b^19*d^4*e^10 + 46080*a^12*b^17*d^4*e^10 + 21504*a^14*b^15*d^4*e^10 - 21504*a^16*b^
13*d^4*e^10 - 46080*a^18*b^11*d^4*e^10 - 38400*a^20*b^9*d^4*e^10 - 17920*a^22*b^7*d^4*e^10 - 4608*a^24*b^5*d^4
*e^10 - 512*a^26*b^3*d^4*e^10))/(a^20*d^4 + a^4*b^16*d^4 + 8*a^6*b^14*d^4 + 28*a^8*b^12*d^4 + 56*a^10*b^10*d^4
+ 70*a^12*b^8*d^4 + 56*a^14*b^6*d^4 + 28*a^16*b^4*d^4 + 8*a^18*b^2*d^4)) + ((e*cot(c + d*x))^(1/2)*(72*a*b^22
*d^2*e^9 + 576*a^3*b^20*d^2*e^9 + 5024*a^5*b^18*d^2*e^9 + 14272*a^7*b^16*d^2*e^9 + 27824*a^9*b^14*d^2*e^9 + 53
184*a^11*b^12*d^2*e^9 + 70240*a^13*b^10*d^2*e^9 + 47680*a^15*b^8*d^2*e^9 + 12616*a^17*b^6*d^2*e^9 - 64*a^21*b^
2*d^2*e^9))/(a^20*d^4 + a^4*b^16*d^4 + 8*a^6*b^14*d^4 + 28*a^8*b^12*d^4 + 56*a^10*b^10*d^4 + 70*a^12*b^8*d^4 +
56*a^14*b^6*d^4 + 28*a^16*b^4*d^4 + 8*a^18*b^2*d^4)) - (90*a*b^19*d^2*e^9 + 846*a^3*b^17*d^2*e^9 + 1714*a^5*b
^15*d^2*e^9 + 3606*a^7*b^13*d^2*e^9 - 14578*a^9*b^11*d^2*e^9 - 34486*a^11*b^9*d^2*e^9 - 14970*a^13*b^7*d^2*e^9
+ 2258*a^15*b^5*d^2*e^9 - 32*a^17*b^3*d^2*e^9)/(a^20*d^5 + a^4*b^16*d^5 + 8*a^6*b^14*d^5 + 28*a^8*b^12*d^5 +
56*a^10*b^10*d^5 + 70*a^12*b^8*d^5 + 56*a^14*b^6*d^5 + 28*a^16*b^4*d^5 + 8*a^18*b^2*d^5)) + ((e*cot(c + d*x))^
(1/2)*(18*a^2*b^15*e^8 - 9*b^17*e^8 - 71*a^4*b^13*e^8 + 892*a^6*b^11*e^8 + 857*a^8*b^9*e^8 + 6802*a^10*b^7*e^8
- 1257*a^12*b^5*e^8))/(a^20*d^4 + a^4*b^16*d^4 + 8*a^6*b^14*d^4 + 28*a^8*b^12*d^4 + 56*a^10*b^10*d^4 + 70*a^1
2*b^8*d^4 + 56*a^14*b^6*d^4 + 28*a^16*b^4*d^4 + 8*a^18*b^2*d^4))*1i - (1i/(4*(b^6*d^2*e - a^6*d^2*e - 15*a^2*b
^4*d^2*e - a^3*b^3*d^2*e*20i + 15*a^4*b^2*d^2*e + a*b^5*d^2*e*6i + a^5*b*d^2*e*6i)))^(1/2)*((1i/(4*(b^6*d^2*e
- a^6*d^2*e - 15*a^2*b^4*d^2*e - a^3*b^3*d^2*e*20i + 15*a^4*b^2*d^2*e + a*b^5*d^2*e*6i + a^5*b*d^2*e*6i)))^(1/
2)*((1i/(4*(b^6*d^2*e - a^6*d^2*e - 15*a^2*b^4*d^2*e - a^3*b^3*d^2*e*20i + 15*a^4*b^2*d^2*e + a*b^5*d^2*e*6i +
a^5*b*d^2*e*6i)))^(1/2)*((1i/(4*(b^6*d^2*e - a^6*d^2*e - 15*a^2*b^4*d^2*e - a^3*b^3*d^2*e*20i + 15*a^4*b^2*d^
2*e + a*b^5*d^2*e*6i + a^5*b*d^2*e*6i)))^(1/2)*((192*a^2*b^24*d^4*e^10 + 1728*a^4*b^22*d^4*e^10 + 8320*a^6*b^2
0*d^4*e^10 + 27264*a^8*b^18*d^4*e^10 + 62592*a^10*b^16*d^4*e^10 + 99456*a^12*b^14*d^4*e^10 + 107520*a^14*b^12*
d^4*e^10 + 76800*a^16*b^10*d^4*e^10 + 33984*a^18*b^8*d^4*e^10 + 7872*a^20*b^6*d^4*e^10 + 384*a^22*b^4*d^4*e^10
- 128*a^24*b^2*d^4*e^10)/(a^20*d^5 + a^4*b^16*d^5 + 8*a^6*b^14*d^5 + 28*a^8*b^12*d^5 + 56*a^10*b^10*d^5 + 70*
a^12*b^8*d^5 + 56*a^14*b^6*d^5 + 28*a^16*b^4*d^5 + 8*a^18*b^2*d^5) + ((1i/(4*(b^6*d^2*e - a^6*d^2*e - 15*a^2*b
^4*d^2*e - a^3*b^3*d^2*e*20i + 15*a^4*b^2*d^2*e + a*b^5*d^2*e*6i + a^5*b*d^2*e*6i)))^(1/2)*(e*cot(c + d*x))^(1
/2)*(512*a^4*b^25*d^4*e^10 + 4608*a^6*b^23*d^4*e^10 + 17920*a^8*b^21*d^4*e^10 + 38400*a^10*b^19*d^4*e^10 + 460
80*a^12*b^17*d^4*e^10 + 21504*a^14*b^15*d^4*e^10 - 21504*a^16*b^13*d^4*e^10 - 46080*a^18*b^11*d^4*e^10 - 38400
*a^20*b^9*d^4*e^10 - 17920*a^22*b^7*d^4*e^10 - 4608*a^24*b^5*d^4*e^10 - 512*a^26*b^3*d^4*e^10))/(a^20*d^4 + a^
4*b^16*d^4 + 8*a^6*b^14*d^4 + 28*a^8*b^12*d^4 + 56*a^10*b^10*d^4 + 70*a^12*b^8*d^4 + 56*a^14*b^6*d^4 + 28*a^16
*b^4*d^4 + 8*a^18*b^2*d^4)) - ((e*cot(c + d*x))^(1/2)*(72*a*b^22*d^2*e^9 + 576*a^3*b^20*d^2*e^9 + 5024*a^5*b^1
8*d^2*e^9 + 14272*a^7*b^16*d^2*e^9 + 27824*a^9*b^14*d^2*e^9 + 53184*a^11*b^12*d^2*e^9 + 70240*a^13*b^10*d^2*e^
9 + 47680*a^15*b^8*d^2*e^9 + 12616*a^17*b^6*d^2*e^9 - 64*a^21*b^2*d^2*e^9))/(a^20*d^4 + a^4*b^16*d^4 + 8*a^6*b
^14*d^4 + 28*a^8*b^12*d^4 + 56*a^10*b^10*d^4 + 70*a^12*b^8*d^4 + 56*a^14*b^6*d^4 + 28*a^16*b^4*d^4 + 8*a^18*b^
2*d^4)) - (90*a*b^19*d^2*e^9 + 846*a^3*b^17*d^2*e^9 + 1714*a^5*b^15*d^2*e^9 + 3606*a^7*b^13*d^2*e^9 - 14578*a^
9*b^11*d^2*e^9 - 34486*a^11*b^9*d^2*e^9 - 14970*a^13*b^7*d^2*e^9 + 2258*a^15*b^5*d^2*e^9 - 32*a^17*b^3*d^2*e^9
)/(a^20*d^5 + a^4*b^16*d^5 + 8*a^6*b^14*d^5 + 28*a^8*b^12*d^5 + 56*a^10*b^10*d^5 + 70*a^12*b^8*d^5 + 56*a^14*b
^6*d^5 + 28*a^16*b^4*d^5 + 8*a^18*b^2*d^5)) - ((e*cot(c + d*x))^(1/2)*(18*a^2*b^15*e^8 - 9*b^17*e^8 - 71*a^4*b
^13*e^8 + 892*a^6*b^11*e^8 + 857*a^8*b^9*e^8 + 6802*a^10*b^7*e^8 - 1257*a^12*b^5*e^8))/(a^20*d^4 + a^4*b^16*d^
4 + 8*a^6*b^14*d^4 + 28*a^8*b^12*d^4 + 56*a^10*b^10*d^4 + 70*a^12*b^8*d^4 + 56*a^14*b^6*d^4 + 28*a^16*b^4*d^4
+ 8*a^18*b^2*d^4))*1i)/((9*b^14*e^8 + 60*a^2*b^12*e^8 + 318*a^4*b^10*e^8 + 748*a^6*b^8*e^8 + 1505*a^8*b^6*e^8)
/(a^20*d^5 + a^4*b^16*d^5 + 8*a^6*b^14*d^5 + 28*a^8*b^12*d^5 + 56*a^10*b^10*d^5 + 70*a^12*b^8*d^5 + 56*a^14*b^
6*d^5 + 28*a^16*b^4*d^5 + 8*a^18*b^2*d^5) + (1i/(4*(b^6*d^2*e - a^6*d^2*e - 15*a^2*b^4*d^2*e - a^3*b^3*d^2*e*2
0i + 15*a^4*b^2*d^2*e + a*b^5*d^2*e*6i + a^5*b*d^2*e*6i)))^(1/2)*((1i/(4*(b^6*d^2*e - a^6*d^2*e - 15*a^2*b^4*d
^2*e - a^3*b^3*d^2*e*20i + 15*a^4*b^2*d^2*e + a*b^5*d^2*e*6i + a^5*b*d^2*e*6i)))^(1/2)*((1i/(4*(b^6*d^2*e - a^
6*d^2*e - 15*a^2*b^4*d^2*e - a^3*b^3*d^2*e*20i + 15*a^4*b^2*d^2*e + a*b^5*d^2*e*6i + a^5*b*d^2*e*6i)))^(1/2)*(
(1i/(4*(b^6*d^2*e - a^6*d^2*e - 15*a^2*b^4*d^2*e - a^3*b^3*d^2*e*20i + 15*a^4*b^2*d^2*e + a*b^5*d^2*e*6i + a^5
*b*d^2*e*6i)))^(1/2)*((192*a^2*b^24*d^4*e^10 + 1728*a^4*b^22*d^4*e^10 + 8320*a^6*b^20*d^4*e^10 + 27264*a^8*b^1
8*d^4*e^10 + 62592*a^10*b^16*d^4*e^10 + 99456*a^12*b^14*d^4*e^10 + 107520*a^14*b^12*d^4*e^10 + 76800*a^16*b^10
*d^4*e^10 + 33984*a^18*b^8*d^4*e^10 + 7872*a^20*b^6*d^4*e^10 + 384*a^22*b^4*d^4*e^10 - 128*a^24*b^2*d^4*e^10)/
(a^20*d^5 + a^4*b^16*d^5 + 8*a^6*b^14*d^5 + 28*a^8*b^12*d^5 + 56*a^10*b^10*d^5 + 70*a^12*b^8*d^5 + 56*a^14*b^6
*d^5 + 28*a^16*b^4*d^5 + 8*a^18*b^2*d^5) - ((1i/(4*(b^6*d^2*e - a^6*d^2*e - 15*a^2*b^4*d^2*e - a^3*b^3*d^2*e*2
0i + 15*a^4*b^2*d^2*e + a*b^5*d^2*e*6i + a^5*b*d^2*e*6i)))^(1/2)*(e*cot(c + d*x))^(1/2)*(512*a^4*b^25*d^4*e^10
+ 4608*a^6*b^23*d^4*e^10 + 17920*a^8*b^21*d^4*e^10 + 38400*a^10*b^19*d^4*e^10 + 46080*a^12*b^17*d^4*e^10 + 21
504*a^14*b^15*d^4*e^10 - 21504*a^16*b^13*d^4*e^10 - 46080*a^18*b^11*d^4*e^10 - 38400*a^20*b^9*d^4*e^10 - 17920
*a^22*b^7*d^4*e^10 - 4608*a^24*b^5*d^4*e^10 - 512*a^26*b^3*d^4*e^10))/(a^20*d^4 + a^4*b^16*d^4 + 8*a^6*b^14*d^
4 + 28*a^8*b^12*d^4 + 56*a^10*b^10*d^4 + 70*a^12*b^8*d^4 + 56*a^14*b^6*d^4 + 28*a^16*b^4*d^4 + 8*a^18*b^2*d^4)
) + ((e*cot(c + d*x))^(1/2)*(72*a*b^22*d^2*e^9 + 576*a^3*b^20*d^2*e^9 + 5024*a^5*b^18*d^2*e^9 + 14272*a^7*b^16
*d^2*e^9 + 27824*a^9*b^14*d^2*e^9 + 53184*a^11*b^12*d^2*e^9 + 70240*a^13*b^10*d^2*e^9 + 47680*a^15*b^8*d^2*e^9
+ 12616*a^17*b^6*d^2*e^9 - 64*a^21*b^2*d^2*e^9))/(a^20*d^4 + a^4*b^16*d^4 + 8*a^6*b^14*d^4 + 28*a^8*b^12*d^4
+ 56*a^10*b^10*d^4 + 70*a^12*b^8*d^4 + 56*a^14*b^6*d^4 + 28*a^16*b^4*d^4 + 8*a^18*b^2*d^4)) - (90*a*b^19*d^2*e
^9 + 846*a^3*b^17*d^2*e^9 + 1714*a^5*b^15*d^2*e^9 + 3606*a^7*b^13*d^2*e^9 - 14578*a^9*b^11*d^2*e^9 - 34486*a^1
1*b^9*d^2*e^9 - 14970*a^13*b^7*d^2*e^9 + 2258*a^15*b^5*d^2*e^9 - 32*a^17*b^3*d^2*e^9)/(a^20*d^5 + a^4*b^16*d^5
+ 8*a^6*b^14*d^5 + 28*a^8*b^12*d^5 + 56*a^10*b^10*d^5 + 70*a^12*b^8*d^5 + 56*a^14*b^6*d^5 + 28*a^16*b^4*d^5 +
8*a^18*b^2*d^5)) + ((e*cot(c + d*x))^(1/2)*(18*a^2*b^15*e^8 - 9*b^17*e^8 - 71*a^4*b^13*e^8 + 892*a^6*b^11*e^8
+ 857*a^8*b^9*e^8 + 6802*a^10*b^7*e^8 - 1257*a^12*b^5*e^8))/(a^20*d^4 + a^4*b^16*d^4 + 8*a^6*b^14*d^4 + 28*a^
8*b^12*d^4 + 56*a^10*b^10*d^4 + 70*a^12*b^8*d^4 + 56*a^14*b^6*d^4 + 28*a^16*b^4*d^4 + 8*a^18*b^2*d^4)) + (1i/(
4*(b^6*d^2*e - a^6*d^2*e - 15*a^2*b^4*d^2*e - a^3*b^3*d^2*e*20i + 15*a^4*b^2*d^2*e + a*b^5*d^2*e*6i + a^5*b*d^
2*e*6i)))^(1/2)*((1i/(4*(b^6*d^2*e - a^6*d^2*e - 15*a^2*b^4*d^2*e - a^3*b^3*d^2*e*20i + 15*a^4*b^2*d^2*e + a*b
^5*d^2*e*6i + a^5*b*d^2*e*6i)))^(1/2)*((1i/(4*(b^6*d^2*e - a^6*d^2*e - 15*a^2*b^4*d^2*e - a^3*b^3*d^2*e*20i +
15*a^4*b^2*d^2*e + a*b^5*d^2*e*6i + a^5*b*d^2*e*6i)))^(1/2)*((1i/(4*(b^6*d^2*e - a^6*d^2*e - 15*a^2*b^4*d^2*e
- a^3*b^3*d^2*e*20i + 15*a^4*b^2*d^2*e + a*b^5*d^2*e*6i + a^5*b*d^2*e*6i)))^(1/2)*((192*a^2*b^24*d^4*e^10 + 17
28*a^4*b^22*d^4*e^10 + 8320*a^6*b^20*d^4*e^10 + 27264*a^8*b^18*d^4*e^10 + 62592*a^10*b^16*d^4*e^10 + 99456*a^1
2*b^14*d^4*e^10 + 107520*a^14*b^12*d^4*e^10 + 76800*a^16*b^10*d^4*e^10 + 33984*a^18*b^8*d^4*e^10 + 7872*a^20*b
^6*d^4*e^10 + 384*a^22*b^4*d^4*e^10 - 128*a^24*b^2*d^4*e^10)/(a^20*d^5 + a^4*b^16*d^5 + 8*a^6*b^14*d^5 + 28*a^
8*b^12*d^5 + 56*a^10*b^10*d^5 + 70*a^12*b^8*d^5 + 56*a^14*b^6*d^5 + 28*a^16*b^4*d^5 + 8*a^18*b^2*d^5) + ((1i/(
4*(b^6*d^2*e - a^6*d^2*e - 15*a^2*b^4*d^2*e - a^3*b^3*d^2*e*20i + 15*a^4*b^2*d^2*e + a*b^5*d^2*e*6i + a^5*b*d^
2*e*6i)))^(1/2)*(e*cot(c + d*x))^(1/2)*(512*a^4*b^25*d^4*e^10 + 4608*a^6*b^23*d^4*e^10 + 17920*a^8*b^21*d^4*e^
10 + 38400*a^10*b^19*d^4*e^10 + 46080*a^12*b^17*d^4*e^10 + 21504*a^14*b^15*d^4*e^10 - 21504*a^16*b^13*d^4*e^10
- 46080*a^18*b^11*d^4*e^10 - 38400*a^20*b^9*d^4*e^10 - 17920*a^22*b^7*d^4*e^10 - 4608*a^24*b^5*d^4*e^10 - 512
*a^26*b^3*d^4*e^10))/(a^20*d^4 + a^4*b^16*d^4 + 8*a^6*b^14*d^4 + 28*a^8*b^12*d^4 + 56*a^10*b^10*d^4 + 70*a^12*
b^8*d^4 + 56*a^14*b^6*d^4 + 28*a^16*b^4*d^4 + 8*a^18*b^2*d^4)) - ((e*cot(c + d*x))^(1/2)*(72*a*b^22*d^2*e^9 +
576*a^3*b^20*d^2*e^9 + 5024*a^5*b^18*d^2*e^9 + 14272*a^7*b^16*d^2*e^9 + 27824*a^9*b^14*d^2*e^9 + 53184*a^11*b^
12*d^2*e^9 + 70240*a^13*b^10*d^2*e^9 + 47680*a^15*b^8*d^2*e^9 + 12616*a^17*b^6*d^2*e^9 - 64*a^21*b^2*d^2*e^9))
/(a^20*d^4 + a^4*b^16*d^4 + 8*a^6*b^14*d^4 + 28*a^8*b^12*d^4 + 56*a^10*b^10*d^4 + 70*a^12*b^8*d^4 + 56*a^14*b^
6*d^4 + 28*a^16*b^4*d^4 + 8*a^18*b^2*d^4)) - (90*a*b^19*d^2*e^9 + 846*a^3*b^17*d^2*e^9 + 1714*a^5*b^15*d^2*e^9
+ 3606*a^7*b^13*d^2*e^9 - 14578*a^9*b^11*d^2*e^9 - 34486*a^11*b^9*d^2*e^9 - 14970*a^13*b^7*d^2*e^9 + 2258*a^1
5*b^5*d^2*e^9 - 32*a^17*b^3*d^2*e^9)/(a^20*d^5 + a^4*b^16*d^5 + 8*a^6*b^14*d^5 + 28*a^8*b^12*d^5 + 56*a^10*b^1
0*d^5 + 70*a^12*b^8*d^5 + 56*a^14*b^6*d^5 + 28*a^16*b^4*d^5 + 8*a^18*b^2*d^5)) - ((e*cot(c + d*x))^(1/2)*(18*a
^2*b^15*e^8 - 9*b^17*e^8 - 71*a^4*b^13*e^8 + 892*a^6*b^11*e^8 + 857*a^8*b^9*e^8 + 6802*a^10*b^7*e^8 - 1257*a^1
2*b^5*e^8))/(a^20*d^4 + a^4*b^16*d^4 + 8*a^6*b^14*d^4 + 28*a^8*b^12*d^4 + 56*a^10*b^10*d^4 + 70*a^12*b^8*d^4 +
56*a^14*b^6*d^4 + 28*a^16*b^4*d^4 + 8*a^18*b^2*d^4))))*(1i/(4*(b^6*d^2*e - a^6*d^2*e - 15*a^2*b^4*d^2*e - a^3
*b^3*d^2*e*20i + 15*a^4*b^2*d^2*e + a*b^5*d^2*e*6i + a^5*b*d^2*e*6i)))^(1/2)*2i - ((b^3*(e*cot(c + d*x))^(3/2)
*(11*a^2 + 3*b^2))/(4*a^2*(a^4 + b^4 + 2*a^2*b^2)) + (b^2*e*(e*cot(c + d*x))^(1/2)*(13*a^2 + 5*b^2))/(4*a*(a^2
+ b^2)^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(((((((((1/(b^6*d^2*e*1i -
a^6*d^2*e*1i - a^2*b^4*d^2*e*15i - 20*a^3*b^3*d^2*e + a^4*b^2*d^2*e*15i + 6*a*b^5*d^2*e + 6*a^5*b*d^2*e))^(1/
2)*((192*a^2*b^24*d^4*e^10 + 1728*a^4*b^22*d^4*e^10 + 8320*a^6*b^20*d^4*e^10 + 27264*a^8*b^18*d^4*e^10 + 62592
*a^10*b^16*d^4*e^10 + 99456*a^12*b^14*d^4*e^10 + 107520*a^14*b^12*d^4*e^10 + 76800*a^16*b^10*d^4*e^10 + 33984*
a^18*b^8*d^4*e^10 + 7872*a^20*b^6*d^4*e^10 + 384*a^22*b^4*d^4*e^10 - 128*a^24*b^2*d^4*e^10)/(2*(a^20*d^5 + a^4
*b^16*d^5 + 8*a^6*b^14*d^5 + 28*a^8*b^12*d^5 + 56*a^10*b^10*d^5 + 70*a^12*b^8*d^5 + 56*a^14*b^6*d^5 + 28*a^16*
b^4*d^5 + 8*a^18*b^2*d^5)) - ((e*cot(c + d*x))^(1/2)*(1/(b^6*d^2*e*1i - a^6*d^2*e*1i - a^2*b^4*d^2*e*15i - 20*
a^3*b^3*d^2*e + a^4*b^2*d^2*e*15i + 6*a*b^5*d^2*e + 6*a^5*b*d^2*e))^(1/2)*(512*a^4*b^25*d^4*e^10 + 4608*a^6*b^
23*d^4*e^10 + 17920*a^8*b^21*d^4*e^10 + 38400*a^10*b^19*d^4*e^10 + 46080*a^12*b^17*d^4*e^10 + 21504*a^14*b^15*
d^4*e^10 - 21504*a^16*b^13*d^4*e^10 - 46080*a^18*b^11*d^4*e^10 - 38400*a^20*b^9*d^4*e^10 - 17920*a^22*b^7*d^4*
e^10 - 4608*a^24*b^5*d^4*e^10 - 512*a^26*b^3*d^4*e^10))/(4*(a^20*d^4 + a^4*b^16*d^4 + 8*a^6*b^14*d^4 + 28*a^8*
b^12*d^4 + 56*a^10*b^10*d^4 + 70*a^12*b^8*d^4 + 56*a^14*b^6*d^4 + 28*a^16*b^4*d^4 + 8*a^18*b^2*d^4))))/2 + ((e
*cot(c + d*x))^(1/2)*(72*a*b^22*d^2*e^9 + 576*a^3*b^20*d^2*e^9 + 5024*a^5*b^18*d^2*e^9 + 14272*a^7*b^16*d^2*e^
9 + 27824*a^9*b^14*d^2*e^9 + 53184*a^11*b^12*d^2*e^9 + 70240*a^13*b^10*d^2*e^9 + 47680*a^15*b^8*d^2*e^9 + 1261
6*a^17*b^6*d^2*e^9 - 64*a^21*b^2*d^2*e^9))/(2*(a^20*d^4 + a^4*b^16*d^4 + 8*a^6*b^14*d^4 + 28*a^8*b^12*d^4 + 56
*a^10*b^10*d^4 + 70*a^12*b^8*d^4 + 56*a^14*b^6*d^4 + 28*a^16*b^4*d^4 + 8*a^18*b^2*d^4)))*(1/(b^6*d^2*e*1i - a^
6*d^2*e*1i - a^2*b^4*d^2*e*15i - 20*a^3*b^3*d^2*e + a^4*b^2*d^2*e*15i + 6*a*b^5*d^2*e + 6*a^5*b*d^2*e))^(1/2))
/2 - (90*a*b^19*d^2*e^9 + 846*a^3*b^17*d^2*e^9 + 1714*a^5*b^15*d^2*e^9 + 3606*a^7*b^13*d^2*e^9 - 14578*a^9*b^1
1*d^2*e^9 - 34486*a^11*b^9*d^2*e^9 - 14970*a^13*b^7*d^2*e^9 + 2258*a^15*b^5*d^2*e^9 - 32*a^17*b^3*d^2*e^9)/(2*
(a^20*d^5 + a^4*b^16*d^5 + 8*a^6*b^14*d^5 + 28*a^8*b^12*d^5 + 56*a^10*b^10*d^5 + 70*a^12*b^8*d^5 + 56*a^14*b^6
*d^5 + 28*a^16*b^4*d^5 + 8*a^18*b^2*d^5)))*(1/(b^6*d^2*e*1i - a^6*d^2*e*1i - a^2*b^4*d^2*e*15i - 20*a^3*b^3*d^
2*e + a^4*b^2*d^2*e*15i + 6*a*b^5*d^2*e + 6*a^5*b*d^2*e))^(1/2))/2 + ((e*cot(c + d*x))^(1/2)*(18*a^2*b^15*e^8
- 9*b^17*e^8 - 71*a^4*b^13*e^8 + 892*a^6*b^11*e^8 + 857*a^8*b^9*e^8 + 6802*a^10*b^7*e^8 - 1257*a^12*b^5*e^8))/
(2*(a^20*d^4 + a^4*b^16*d^4 + 8*a^6*b^14*d^4 + 28*a^8*b^12*d^4 + 56*a^10*b^10*d^4 + 70*a^12*b^8*d^4 + 56*a^14*
b^6*d^4 + 28*a^16*b^4*d^4 + 8*a^18*b^2*d^4)))*(1/(b^6*d^2*e*1i - a^6*d^2*e*1i - a^2*b^4*d^2*e*15i - 20*a^3*b^3
*d^2*e + a^4*b^2*d^2*e*15i + 6*a*b^5*d^2*e + 6*a^5*b*d^2*e))^(1/2)*1i - (((((((1/(b^6*d^2*e*1i - a^6*d^2*e*1i
- a^2*b^4*d^2*e*15i - 20*a^3*b^3*d^2*e + a^4*b^2*d^2*e*15i + 6*a*b^5*d^2*e + 6*a^5*b*d^2*e))^(1/2)*((192*a^2*b
^24*d^4*e^10 + 1728*a^4*b^22*d^4*e^10 + 8320*a^6*b^20*d^4*e^10 + 27264*a^8*b^18*d^4*e^10 + 62592*a^10*b^16*d^4
*e^10 + 99456*a^12*b^14*d^4*e^10 + 107520*a^14*b^12*d^4*e^10 + 76800*a^16*b^10*d^4*e^10 + 33984*a^18*b^8*d^4*e
^10 + 7872*a^20*b^6*d^4*e^10 + 384*a^22*b^4*d^4*e^10 - 128*a^24*b^2*d^4*e^10)/(2*(a^20*d^5 + a^4*b^16*d^5 + 8*
a^6*b^14*d^5 + 28*a^8*b^12*d^5 + 56*a^10*b^10*d^5 + 70*a^12*b^8*d^5 + 56*a^14*b^6*d^5 + 28*a^16*b^4*d^5 + 8*a^
18*b^2*d^5)) + ((e*cot(c + d*x))^(1/2)*(1/(b^6*d^2*e*1i - a^6*d^2*e*1i - a^2*b^4*d^2*e*15i - 20*a^3*b^3*d^2*e
+ a^4*b^2*d^2*e*15i + 6*a*b^5*d^2*e + 6*a^5*b*d^2*e))^(1/2)*(512*a^4*b^25*d^4*e^10 + 4608*a^6*b^23*d^4*e^10 +
17920*a^8*b^21*d^4*e^10 + 38400*a^10*b^19*d^4*e^10 + 46080*a^12*b^17*d^4*e^10 + 21504*a^14*b^15*d^4*e^10 - 215
04*a^16*b^13*d^4*e^10 - 46080*a^18*b^11*d^4*e^10 - 38400*a^20*b^9*d^4*e^10 - 17920*a^22*b^7*d^4*e^10 - 4608*a^
24*b^5*d^4*e^10 - 512*a^26*b^3*d^4*e^10))/(4*(a^20*d^4 + a^4*b^16*d^4 + 8*a^6*b^14*d^4 + 28*a^8*b^12*d^4 + 56*
a^10*b^10*d^4 + 70*a^12*b^8*d^4 + 56*a^14*b^6*d^4 + 28*a^16*b^4*d^4 + 8*a^18*b^2*d^4))))/2 - ((e*cot(c + d*x))
^(1/2)*(72*a*b^22*d^2*e^9 + 576*a^3*b^20*d^2*e^9 + 5024*a^5*b^18*d^2*e^9 + 14272*a^7*b^16*d^2*e^9 + 27824*a^9*
b^14*d^2*e^9 + 53184*a^11*b^12*d^2*e^9 + 70240*a^13*b^10*d^2*e^9 + 47680*a^15*b^8*d^2*e^9 + 12616*a^17*b^6*d^2
*e^9 - 64*a^21*b^2*d^2*e^9))/(2*(a^20*d^4 + a^4*b^16*d^4 + 8*a^6*b^14*d^4 + 28*a^8*b^12*d^4 + 56*a^10*b^10*d^4
+ 70*a^12*b^8*d^4 + 56*a^14*b^6*d^4 + 28*a^16*b^4*d^4 + 8*a^18*b^2*d^4)))*(1/(b^6*d^2*e*1i - a^6*d^2*e*1i - a
^2*b^4*d^2*e*15i - 20*a^3*b^3*d^2*e + a^4*b^2*d^2*e*15i + 6*a*b^5*d^2*e + 6*a^5*b*d^2*e))^(1/2))/2 - (90*a*b^1
9*d^2*e^9 + 846*a^3*b^17*d^2*e^9 + 1714*a^5*b^15*d^2*e^9 + 3606*a^7*b^13*d^2*e^9 - 14578*a^9*b^11*d^2*e^9 - 34
486*a^11*b^9*d^2*e^9 - 14970*a^13*b^7*d^2*e^9 + 2258*a^15*b^5*d^2*e^9 - 32*a^17*b^3*d^2*e^9)/(2*(a^20*d^5 + a^
4*b^16*d^5 + 8*a^6*b^14*d^5 + 28*a^8*b^12*d^5 + 56*a^10*b^10*d^5 + 70*a^12*b^8*d^5 + 56*a^14*b^6*d^5 + 28*a^16
*b^4*d^5 + 8*a^18*b^2*d^5)))*(1/(b^6*d^2*e*1i - a^6*d^2*e*1i - a^2*b^4*d^2*e*15i - 20*a^3*b^3*d^2*e + a^4*b^2*
d^2*e*15i + 6*a*b^5*d^2*e + 6*a^5*b*d^2*e))^(1/2))/2 - ((e*cot(c + d*x))^(1/2)*(18*a^2*b^15*e^8 - 9*b^17*e^8 -
71*a^4*b^13*e^8 + 892*a^6*b^11*e^8 + 857*a^8*b^9*e^8 + 6802*a^10*b^7*e^8 - 1257*a^12*b^5*e^8))/(2*(a^20*d^4 +
a^4*b^16*d^4 + 8*a^6*b^14*d^4 + 28*a^8*b^12*d^4 + 56*a^10*b^10*d^4 + 70*a^12*b^8*d^4 + 56*a^14*b^6*d^4 + 28*a
^16*b^4*d^4 + 8*a^18*b^2*d^4)))*(1/(b^6*d^2*e*1i - a^6*d^2*e*1i - a^2*b^4*d^2*e*15i - 20*a^3*b^3*d^2*e + a^4*b
^2*d^2*e*15i + 6*a*b^5*d^2*e + 6*a^5*b*d^2*e))^(1/2)*1i)/((9*b^14*e^8 + 60*a^2*b^12*e^8 + 318*a^4*b^10*e^8 + 7
48*a^6*b^8*e^8 + 1505*a^8*b^6*e^8)/(a^20*d^5 + a^4*b^16*d^5 + 8*a^6*b^14*d^5 + 28*a^8*b^12*d^5 + 56*a^10*b^10*
d^5 + 70*a^12*b^8*d^5 + 56*a^14*b^6*d^5 + 28*a^16*b^4*d^5 + 8*a^18*b^2*d^5) + (((((((1/(b^6*d^2*e*1i - a^6*d^2
*e*1i - a^2*b^4*d^2*e*15i - 20*a^3*b^3*d^2*e + a^4*b^2*d^2*e*15i + 6*a*b^5*d^2*e + 6*a^5*b*d^2*e))^(1/2)*((192
*a^2*b^24*d^4*e^10 + 1728*a^4*b^22*d^4*e^10 + 8320*a^6*b^20*d^4*e^10 + 27264*a^8*b^18*d^4*e^10 + 62592*a^10*b^
16*d^4*e^10 + 99456*a^12*b^14*d^4*e^10 + 107520*a^14*b^12*d^4*e^10 + 76800*a^16*b^10*d^4*e^10 + 33984*a^18*b^8
*d^4*e^10 + 7872*a^20*b^6*d^4*e^10 + 384*a^22*b^4*d^4*e^10 - 128*a^24*b^2*d^4*e^10)/(2*(a^20*d^5 + a^4*b^16*d^
5 + 8*a^6*b^14*d^5 + 28*a^8*b^12*d^5 + 56*a^10*b^10*d^5 + 70*a^12*b^8*d^5 + 56*a^14*b^6*d^5 + 28*a^16*b^4*d^5
+ 8*a^18*b^2*d^5)) - ((e*cot(c + d*x))^(1/2)*(1/(b^6*d^2*e*1i - a^6*d^2*e*1i - a^2*b^4*d^2*e*15i - 20*a^3*b^3*
d^2*e + a^4*b^2*d^2*e*15i + 6*a*b^5*d^2*e + 6*a^5*b*d^2*e))^(1/2)*(512*a^4*b^25*d^4*e^10 + 4608*a^6*b^23*d^4*e
^10 + 17920*a^8*b^21*d^4*e^10 + 38400*a^10*b^19*d^4*e^10 + 46080*a^12*b^17*d^4*e^10 + 21504*a^14*b^15*d^4*e^10
- 21504*a^16*b^13*d^4*e^10 - 46080*a^18*b^11*d^4*e^10 - 38400*a^20*b^9*d^4*e^10 - 17920*a^22*b^7*d^4*e^10 - 4
608*a^24*b^5*d^4*e^10 - 512*a^26*b^3*d^4*e^10))/(4*(a^20*d^4 + a^4*b^16*d^4 + 8*a^6*b^14*d^4 + 28*a^8*b^12*d^4
+ 56*a^10*b^10*d^4 + 70*a^12*b^8*d^4 + 56*a^14*b^6*d^4 + 28*a^16*b^4*d^4 + 8*a^18*b^2*d^4))))/2 + ((e*cot(c +
d*x))^(1/2)*(72*a*b^22*d^2*e^9 + 576*a^3*b^20*d^2*e^9 + 5024*a^5*b^18*d^2*e^9 + 14272*a^7*b^16*d^2*e^9 + 2782
4*a^9*b^14*d^2*e^9 + 53184*a^11*b^12*d^2*e^9 + 70240*a^13*b^10*d^2*e^9 + 47680*a^15*b^8*d^2*e^9 + 12616*a^17*b
^6*d^2*e^9 - 64*a^21*b^2*d^2*e^9))/(2*(a^20*d^4 + a^4*b^16*d^4 + 8*a^6*b^14*d^4 + 28*a^8*b^12*d^4 + 56*a^10*b^
10*d^4 + 70*a^12*b^8*d^4 + 56*a^14*b^6*d^4 + 28*a^16*b^4*d^4 + 8*a^18*b^2*d^4)))*(1/(b^6*d^2*e*1i - a^6*d^2*e*
1i - a^2*b^4*d^2*e*15i - 20*a^3*b^3*d^2*e + a^4*b^2*d^2*e*15i + 6*a*b^5*d^2*e + 6*a^5*b*d^2*e))^(1/2))/2 - (90
*a*b^19*d^2*e^9 + 846*a^3*b^17*d^2*e^9 + 1714*a^5*b^15*d^2*e^9 + 3606*a^7*b^13*d^2*e^9 - 14578*a^9*b^11*d^2*e^
9 - 34486*a^11*b^9*d^2*e^9 - 14970*a^13*b^7*d^2*e^9 + 2258*a^15*b^5*d^2*e^9 - 32*a^17*b^3*d^2*e^9)/(2*(a^20*d^
5 + a^4*b^16*d^5 + 8*a^6*b^14*d^5 + 28*a^8*b^12*d^5 + 56*a^10*b^10*d^5 + 70*a^12*b^8*d^5 + 56*a^14*b^6*d^5 + 2
8*a^16*b^4*d^5 + 8*a^18*b^2*d^5)))*(1/(b^6*d^2*e*1i - a^6*d^2*e*1i - a^2*b^4*d^2*e*15i - 20*a^3*b^3*d^2*e + a^
4*b^2*d^2*e*15i + 6*a*b^5*d^2*e + 6*a^5*b*d^2*e))^(1/2))/2 + ((e*cot(c + d*x))^(1/2)*(18*a^2*b^15*e^8 - 9*b^17
*e^8 - 71*a^4*b^13*e^8 + 892*a^6*b^11*e^8 + 857*a^8*b^9*e^8 + 6802*a^10*b^7*e^8 - 1257*a^12*b^5*e^8))/(2*(a^20
*d^4 + a^4*b^16*d^4 + 8*a^6*b^14*d^4 + 28*a^8*b^12*d^4 + 56*a^10*b^10*d^4 + 70*a^12*b^8*d^4 + 56*a^14*b^6*d^4
+ 28*a^16*b^4*d^4 + 8*a^18*b^2*d^4)))*(1/(b^6*d^2*e*1i - a^6*d^2*e*1i - a^2*b^4*d^2*e*15i - 20*a^3*b^3*d^2*e +
a^4*b^2*d^2*e*15i + 6*a*b^5*d^2*e + 6*a^5*b*d^2*e))^(1/2) + (((((((1/(b^6*d^2*e*1i - a^6*d^2*e*1i - a^2*b^4*d
^2*e*15i - 20*a^3*b^3*d^2*e + a^4*b^2*d^2*e*15i + 6*a*b^5*d^2*e + 6*a^5*b*d^2*e))^(1/2)*((192*a^2*b^24*d^4*e^1
0 + 1728*a^4*b^22*d^4*e^10 + 8320*a^6*b^20*d^4*e^10 + 27264*a^8*b^18*d^4*e^10 + 62592*a^10*b^16*d^4*e^10 + 994
56*a^12*b^14*d^4*e^10 + 107520*a^14*b^12*d^4*e^10 + 76800*a^16*b^10*d^4*e^10 + 33984*a^18*b^8*d^4*e^10 + 7872*
a^20*b^6*d^4*e^10 + 384*a^22*b^4*d^4*e^10 - 128*a^24*b^2*d^4*e^10)/(2*(a^20*d^5 + a^4*b^16*d^5 + 8*a^6*b^14*d^
5 + 28*a^8*b^12*d^5 + 56*a^10*b^10*d^5 + 70*a^12*b^8*d^5 + 56*a^14*b^6*d^5 + 28*a^16*b^4*d^5 + 8*a^18*b^2*d^5)
) + ((e*cot(c + d*x))^(1/2)*(1/(b^6*d^2*e*1i - a^6*d^2*e*1i - a^2*b^4*d^2*e*15i - 20*a^3*b^3*d^2*e + a^4*b^2*d
^2*e*15i + 6*a*b^5*d^2*e + 6*a^5*b*d^2*e))^(1/2)*(512*a^4*b^25*d^4*e^10 + 4608*a^6*b^23*d^4*e^10 + 17920*a^8*b
^21*d^4*e^10 + 38400*a^10*b^19*d^4*e^10 + 46080*a^12*b^17*d^4*e^10 + 21504*a^14*b^15*d^4*e^10 - 21504*a^16*b^1
3*d^4*e^10 - 46080*a^18*b^11*d^4*e^10 - 38400*a^20*b^9*d^4*e^10 - 17920*a^22*b^7*d^4*e^10 - 4608*a^24*b^5*d^4*
e^10 - 512*a^26*b^3*d^4*e^10))/(4*(a^20*d^4 + a^4*b^16*d^4 + 8*a^6*b^14*d^4 + 28*a^8*b^12*d^4 + 56*a^10*b^10*d
^4 + 70*a^12*b^8*d^4 + 56*a^14*b^6*d^4 + 28*a^16*b^4*d^4 + 8*a^18*b^2*d^4))))/2 - ((e*cot(c + d*x))^(1/2)*(72*
a*b^22*d^2*e^9 + 576*a^3*b^20*d^2*e^9 + 5024*a^5*b^18*d^2*e^9 + 14272*a^7*b^16*d^2*e^9 + 27824*a^9*b^14*d^2*e^
9 + 53184*a^11*b^12*d^2*e^9 + 70240*a^13*b^10*d^2*e^9 + 47680*a^15*b^8*d^2*e^9 + 12616*a^17*b^6*d^2*e^9 - 64*a
^21*b^2*d^2*e^9))/(2*(a^20*d^4 + a^4*b^16*d^4 + 8*a^6*b^14*d^4 + 28*a^8*b^12*d^4 + 56*a^10*b^10*d^4 + 70*a^12*
b^8*d^4 + 56*a^14*b^6*d^4 + 28*a^16*b^4*d^4 + 8*a^18*b^2*d^4)))*(1/(b^6*d^2*e*1i - a^6*d^2*e*1i - a^2*b^4*d^2*
e*15i - 20*a^3*b^3*d^2*e + a^4*b^2*d^2*e*15i + 6*a*b^5*d^2*e + 6*a^5*b*d^2*e))^(1/2))/2 - (90*a*b^19*d^2*e^9 +
846*a^3*b^17*d^2*e^9 + 1714*a^5*b^15*d^2*e^9 + 3606*a^7*b^13*d^2*e^9 - 14578*a^9*b^11*d^2*e^9 - 34486*a^11*b^
9*d^2*e^9 - 14970*a^13*b^7*d^2*e^9 + 2258*a^15*b^5*d^2*e^9 - 32*a^17*b^3*d^2*e^9)/(2*(a^20*d^5 + a^4*b^16*d^5
+ 8*a^6*b^14*d^5 + 28*a^8*b^12*d^5 + 56*a^10*b^10*d^5 + 70*a^12*b^8*d^5 + 56*a^14*b^6*d^5 + 28*a^16*b^4*d^5 +
8*a^18*b^2*d^5)))*(1/(b^6*d^2*e*1i - a^6*d^2*e*1i - a^2*b^4*d^2*e*15i - 20*a^3*b^3*d^2*e + a^4*b^2*d^2*e*15i +
6*a*b^5*d^2*e + 6*a^5*b*d^2*e))^(1/2))/2 - ((e*cot(c + d*x))^(1/2)*(18*a^2*b^15*e^8 - 9*b^17*e^8 - 71*a^4*b^1
3*e^8 + 892*a^6*b^11*e^8 + 857*a^8*b^9*e^8 + 6802*a^10*b^7*e^8 - 1257*a^12*b^5*e^8))/(2*(a^20*d^4 + a^4*b^16*d
^4 + 8*a^6*b^14*d^4 + 28*a^8*b^12*d^4 + 56*a^10*b^10*d^4 + 70*a^12*b^8*d^4 + 56*a^14*b^6*d^4 + 28*a^16*b^4*d^4
+ 8*a^18*b^2*d^4)))*(1/(b^6*d^2*e*1i - a^6*d^2*e*1i - a^2*b^4*d^2*e*15i - 20*a^3*b^3*d^2*e + a^4*b^2*d^2*e*15
i + 6*a*b^5*d^2*e + 6*a^5*b*d^2*e))^(1/2)))*(1/(b^6*d^2*e*1i - a^6*d^2*e*1i - a^2*b^4*d^2*e*15i - 20*a^3*b^3*d
^2*e + a^4*b^2*d^2*e*15i + 6*a*b^5*d^2*e + 6*a^5*b*d^2*e))^(1/2)*1i + (atan((((((e*cot(c + d*x))^(1/2)*(18*a^2
*b^15*e^8 - 9*b^17*e^8 - 71*a^4*b^13*e^8 + 892*a^6*b^11*e^8 + 857*a^8*b^9*e^8 + 6802*a^10*b^7*e^8 - 1257*a^12*
b^5*e^8))/(a^20*d^4 + a^4*b^16*d^4 + 8*a^6*b^14*d^4 + 28*a^8*b^12*d^4 + 56*a^10*b^10*d^4 + 70*a^12*b^8*d^4 + 5
6*a^14*b^6*d^4 + 28*a^16*b^4*d^4 + 8*a^18*b^2*d^4) - (((90*a*b^19*d^2*e^9 + 846*a^3*b^17*d^2*e^9 + 1714*a^5*b^
15*d^2*e^9 + 3606*a^7*b^13*d^2*e^9 - 14578*a^9*b^11*d^2*e^9 - 34486*a^11*b^9*d^2*e^9 - 14970*a^13*b^7*d^2*e^9
+ 2258*a^15*b^5*d^2*e^9 - 32*a^17*b^3*d^2*e^9)/(a^20*d^5 + a^4*b^16*d^5 + 8*a^6*b^14*d^5 + 28*a^8*b^12*d^5 + 5
6*a^10*b^10*d^5 + 70*a^12*b^8*d^5 + 56*a^14*b^6*d^5 + 28*a^16*b^4*d^5 + 8*a^18*b^2*d^5) - ((((e*cot(c + d*x))^
(1/2)*(72*a*b^22*d^2*e^9 + 576*a^3*b^20*d^2*e^9 + 5024*a^5*b^18*d^2*e^9 + 14272*a^7*b^16*d^2*e^9 + 27824*a^9*b
^14*d^2*e^9 + 53184*a^11*b^12*d^2*e^9 + 70240*a^13*b^10*d^2*e^9 + 47680*a^15*b^8*d^2*e^9 + 12616*a^17*b^6*d^2*
e^9 - 64*a^21*b^2*d^2*e^9))/(a^20*d^4 + a^4*b^16*d^4 + 8*a^6*b^14*d^4 + 28*a^8*b^12*d^4 + 56*a^10*b^10*d^4 + 7
0*a^12*b^8*d^4 + 56*a^14*b^6*d^4 + 28*a^16*b^4*d^4 + 8*a^18*b^2*d^4) + (((192*a^2*b^24*d^4*e^10 + 1728*a^4*b^2
2*d^4*e^10 + 8320*a^6*b^20*d^4*e^10 + 27264*a^8*b^18*d^4*e^10 + 62592*a^10*b^16*d^4*e^10 + 99456*a^12*b^14*d^4
*e^10 + 107520*a^14*b^12*d^4*e^10 + 76800*a^16*b^10*d^4*e^10 + 33984*a^18*b^8*d^4*e^10 + 7872*a^20*b^6*d^4*e^1
0 + 384*a^22*b^4*d^4*e^10 - 128*a^24*b^2*d^4*e^10)/(a^20*d^5 + a^4*b^16*d^5 + 8*a^6*b^14*d^5 + 28*a^8*b^12*d^5
+ 56*a^10*b^10*d^5 + 70*a^12*b^8*d^5 + 56*a^14*b^6*d^5 + 28*a^16*b^4*d^5 + 8*a^18*b^2*d^5) - ((e*cot(c + d*x)
)^(1/2)*(-a^5*b^3*e)^(1/2)*(35*a^4 + 3*b^4 + 6*a^2*b^2)*(512*a^4*b^25*d^4*e^10 + 4608*a^6*b^23*d^4*e^10 + 1792
0*a^8*b^21*d^4*e^10 + 38400*a^10*b^19*d^4*e^10 + 46080*a^12*b^17*d^4*e^10 + 21504*a^14*b^15*d^4*e^10 - 21504*a
^16*b^13*d^4*e^10 - 46080*a^18*b^11*d^4*e^10 - 38400*a^20*b^9*d^4*e^10 - 17920*a^22*b^7*d^4*e^10 - 4608*a^24*b
^5*d^4*e^10 - 512*a^26*b^3*d^4*e^10))/(8*(a^11*d*e + a^5*b^6*d*e + 3*a^7*b^4*d*e + 3*a^9*b^2*d*e)*(a^20*d^4 +
a^4*b^16*d^4 + 8*a^6*b^14*d^4 + 28*a^8*b^12*d^4 + 56*a^10*b^10*d^4 + 70*a^12*b^8*d^4 + 56*a^14*b^6*d^4 + 28*a^
16*b^4*d^4 + 8*a^18*b^2*d^4)))*(-a^5*b^3*e)^(1/2)*(35*a^4 + 3*b^4 + 6*a^2*b^2))/(8*(a^11*d*e + a^5*b^6*d*e + 3
*a^7*b^4*d*e + 3*a^9*b^2*d*e)))*(-a^5*b^3*e)^(1/2)*(35*a^4 + 3*b^4 + 6*a^2*b^2))/(8*(a^11*d*e + a^5*b^6*d*e +
3*a^7*b^4*d*e + 3*a^9*b^2*d*e)))*(-a^5*b^3*e)^(1/2)*(35*a^4 + 3*b^4 + 6*a^2*b^2))/(8*(a^11*d*e + a^5*b^6*d*e +
3*a^7*b^4*d*e + 3*a^9*b^2*d*e)))*(-a^5*b^3*e)^(1/2)*(35*a^4 + 3*b^4 + 6*a^2*b^2)*1i)/(8*(a^11*d*e + a^5*b^6*d
*e + 3*a^7*b^4*d*e + 3*a^9*b^2*d*e)) + ((((e*cot(c + d*x))^(1/2)*(18*a^2*b^15*e^8 - 9*b^17*e^8 - 71*a^4*b^13*e
^8 + 892*a^6*b^11*e^8 + 857*a^8*b^9*e^8 + 6802*a^10*b^7*e^8 - 1257*a^12*b^5*e^8))/(a^20*d^4 + a^4*b^16*d^4 + 8
*a^6*b^14*d^4 + 28*a^8*b^12*d^4 + 56*a^10*b^10*d^4 + 70*a^12*b^8*d^4 + 56*a^14*b^6*d^4 + 28*a^16*b^4*d^4 + 8*a
^18*b^2*d^4) + (((90*a*b^19*d^2*e^9 + 846*a^3*b^17*d^2*e^9 + 1714*a^5*b^15*d^2*e^9 + 3606*a^7*b^13*d^2*e^9 - 1
4578*a^9*b^11*d^2*e^9 - 34486*a^11*b^9*d^2*e^9 - 14970*a^13*b^7*d^2*e^9 + 2258*a^15*b^5*d^2*e^9 - 32*a^17*b^3*
d^2*e^9)/(a^20*d^5 + a^4*b^16*d^5 + 8*a^6*b^14*d^5 + 28*a^8*b^12*d^5 + 56*a^10*b^10*d^5 + 70*a^12*b^8*d^5 + 56
*a^14*b^6*d^5 + 28*a^16*b^4*d^5 + 8*a^18*b^2*d^5) + ((((e*cot(c + d*x))^(1/2)*(72*a*b^22*d^2*e^9 + 576*a^3*b^2
0*d^2*e^9 + 5024*a^5*b^18*d^2*e^9 + 14272*a^7*b^16*d^2*e^9 + 27824*a^9*b^14*d^2*e^9 + 53184*a^11*b^12*d^2*e^9
+ 70240*a^13*b^10*d^2*e^9 + 47680*a^15*b^8*d^2*e^9 + 12616*a^17*b^6*d^2*e^9 - 64*a^21*b^2*d^2*e^9))/(a^20*d^4
+ a^4*b^16*d^4 + 8*a^6*b^14*d^4 + 28*a^8*b^12*d^4 + 56*a^10*b^10*d^4 + 70*a^12*b^8*d^4 + 56*a^14*b^6*d^4 + 28*
a^16*b^4*d^4 + 8*a^18*b^2*d^4) - (((192*a^2*b^24*d^4*e^10 + 1728*a^4*b^22*d^4*e^10 + 8320*a^6*b^20*d^4*e^10 +
27264*a^8*b^18*d^4*e^10 + 62592*a^10*b^16*d^4*e^10 + 99456*a^12*b^14*d^4*e^10 + 107520*a^14*b^12*d^4*e^10 + 76
800*a^16*b^10*d^4*e^10 + 33984*a^18*b^8*d^4*e^10 + 7872*a^20*b^6*d^4*e^10 + 384*a^22*b^4*d^4*e^10 - 128*a^24*b
^2*d^4*e^10)/(a^20*d^5 + a^4*b^16*d^5 + 8*a^6*b^14*d^5 + 28*a^8*b^12*d^5 + 56*a^10*b^10*d^5 + 70*a^12*b^8*d^5
+ 56*a^14*b^6*d^5 + 28*a^16*b^4*d^5 + 8*a^18*b^2*d^5) + ((e*cot(c + d*x))^(1/2)*(-a^5*b^3*e)^(1/2)*(35*a^4 + 3
*b^4 + 6*a^2*b^2)*(512*a^4*b^25*d^4*e^10 + 4608*a^6*b^23*d^4*e^10 + 17920*a^8*b^21*d^4*e^10 + 38400*a^10*b^19*
d^4*e^10 + 46080*a^12*b^17*d^4*e^10 + 21504*a^14*b^15*d^4*e^10 - 21504*a^16*b^13*d^4*e^10 - 46080*a^18*b^11*d^
4*e^10 - 38400*a^20*b^9*d^4*e^10 - 17920*a^22*b^7*d^4*e^10 - 4608*a^24*b^5*d^4*e^10 - 512*a^26*b^3*d^4*e^10))/
(8*(a^11*d*e + a^5*b^6*d*e + 3*a^7*b^4*d*e + 3*a^9*b^2*d*e)*(a^20*d^4 + a^4*b^16*d^4 + 8*a^6*b^14*d^4 + 28*a^8
*b^12*d^4 + 56*a^10*b^10*d^4 + 70*a^12*b^8*d^4 + 56*a^14*b^6*d^4 + 28*a^16*b^4*d^4 + 8*a^18*b^2*d^4)))*(-a^5*b
^3*e)^(1/2)*(35*a^4 + 3*b^4 + 6*a^2*b^2))/(8*(a^11*d*e + a^5*b^6*d*e + 3*a^7*b^4*d*e + 3*a^9*b^2*d*e)))*(-a^5*
b^3*e)^(1/2)*(35*a^4 + 3*b^4 + 6*a^2*b^2))/(8*(a^11*d*e + a^5*b^6*d*e + 3*a^7*b^4*d*e + 3*a^9*b^2*d*e)))*(-a^5
*b^3*e)^(1/2)*(35*a^4 + 3*b^4 + 6*a^2*b^2))/(8*(a^11*d*e + a^5*b^6*d*e + 3*a^7*b^4*d*e + 3*a^9*b^2*d*e)))*(-a^
5*b^3*e)^(1/2)*(35*a^4 + 3*b^4 + 6*a^2*b^2)*1i)/(8*(a^11*d*e + a^5*b^6*d*e + 3*a^7*b^4*d*e + 3*a^9*b^2*d*e)))/
((9*b^14*e^8 + 60*a^2*b^12*e^8 + 318*a^4*b^10*e^8 + 748*a^6*b^8*e^8 + 1505*a^8*b^6*e^8)/(a^20*d^5 + a^4*b^16*d
^5 + 8*a^6*b^14*d^5 + 28*a^8*b^12*d^5 + 56*a^10*b^10*d^5 + 70*a^12*b^8*d^5 + 56*a^14*b^6*d^5 + 28*a^16*b^4*d^5
+ 8*a^18*b^2*d^5) + ((((e*cot(c + d*x))^(1/2)*(18*a^2*b^15*e^8 - 9*b^17*e^8 - 71*a^4*b^13*e^8 + 892*a^6*b^11*
e^8 + 857*a^8*b^9*e^8 + 6802*a^10*b^7*e^8 - 1257*a^12*b^5*e^8))/(a^20*d^4 + a^4*b^16*d^4 + 8*a^6*b^14*d^4 + 28
*a^8*b^12*d^4 + 56*a^10*b^10*d^4 + 70*a^12*b^8*d^4 + 56*a^14*b^6*d^4 + 28*a^16*b^4*d^4 + 8*a^18*b^2*d^4) - (((
90*a*b^19*d^2*e^9 + 846*a^3*b^17*d^2*e^9 + 1714*a^5*b^15*d^2*e^9 + 3606*a^7*b^13*d^2*e^9 - 14578*a^9*b^11*d^2*
e^9 - 34486*a^11*b^9*d^2*e^9 - 14970*a^13*b^7*d^2*e^9 + 2258*a^15*b^5*d^2*e^9 - 32*a^17*b^3*d^2*e^9)/(a^20*d^5
+ a^4*b^16*d^5 + 8*a^6*b^14*d^5 + 28*a^8*b^12*d^5 + 56*a^10*b^10*d^5 + 70*a^12*b^8*d^5 + 56*a^14*b^6*d^5 + 28
*a^16*b^4*d^5 + 8*a^18*b^2*d^5) - ((((e*cot(c + d*x))^(1/2)*(72*a*b^22*d^2*e^9 + 576*a^3*b^20*d^2*e^9 + 5024*a
^5*b^18*d^2*e^9 + 14272*a^7*b^16*d^2*e^9 + 27824*a^9*b^14*d^2*e^9 + 53184*a^11*b^12*d^2*e^9 + 70240*a^13*b^10*
d^2*e^9 + 47680*a^15*b^8*d^2*e^9 + 12616*a^17*b^6*d^2*e^9 - 64*a^21*b^2*d^2*e^9))/(a^20*d^4 + a^4*b^16*d^4 + 8
*a^6*b^14*d^4 + 28*a^8*b^12*d^4 + 56*a^10*b^10*d^4 + 70*a^12*b^8*d^4 + 56*a^14*b^6*d^4 + 28*a^16*b^4*d^4 + 8*a
^18*b^2*d^4) + (((192*a^2*b^24*d^4*e^10 + 1728*a^4*b^22*d^4*e^10 + 8320*a^6*b^20*d^4*e^10 + 27264*a^8*b^18*d^4
*e^10 + 62592*a^10*b^16*d^4*e^10 + 99456*a^12*b^14*d^4*e^10 + 107520*a^14*b^12*d^4*e^10 + 76800*a^16*b^10*d^4*
e^10 + 33984*a^18*b^8*d^4*e^10 + 7872*a^20*b^6*d^4*e^10 + 384*a^22*b^4*d^4*e^10 - 128*a^24*b^2*d^4*e^10)/(a^20
*d^5 + a^4*b^16*d^5 + 8*a^6*b^14*d^5 + 28*a^8*b^12*d^5 + 56*a^10*b^10*d^5 + 70*a^12*b^8*d^5 + 56*a^14*b^6*d^5
+ 28*a^16*b^4*d^5 + 8*a^18*b^2*d^5) - ((e*cot(c + d*x))^(1/2)*(-a^5*b^3*e)^(1/2)*(35*a^4 + 3*b^4 + 6*a^2*b^2)*
(512*a^4*b^25*d^4*e^10 + 4608*a^6*b^23*d^4*e^10 + 17920*a^8*b^21*d^4*e^10 + 38400*a^10*b^19*d^4*e^10 + 46080*a
^12*b^17*d^4*e^10 + 21504*a^14*b^15*d^4*e^10 - 21504*a^16*b^13*d^4*e^10 - 46080*a^18*b^11*d^4*e^10 - 38400*a^2
0*b^9*d^4*e^10 - 17920*a^22*b^7*d^4*e^10 - 4608*a^24*b^5*d^4*e^10 - 512*a^26*b^3*d^4*e^10))/(8*(a^11*d*e + a^5
*b^6*d*e + 3*a^7*b^4*d*e + 3*a^9*b^2*d*e)*(a^20*d^4 + a^4*b^16*d^4 + 8*a^6*b^14*d^4 + 28*a^8*b^12*d^4 + 56*a^1
0*b^10*d^4 + 70*a^12*b^8*d^4 + 56*a^14*b^6*d^4 + 28*a^16*b^4*d^4 + 8*a^18*b^2*d^4)))*(-a^5*b^3*e)^(1/2)*(35*a^
4 + 3*b^4 + 6*a^2*b^2))/(8*(a^11*d*e + a^5*b^6*d*e + 3*a^7*b^4*d*e + 3*a^9*b^2*d*e)))*(-a^5*b^3*e)^(1/2)*(35*a
^4 + 3*b^4 + 6*a^2*b^2))/(8*(a^11*d*e + a^5*b^6*d*e + 3*a^7*b^4*d*e + 3*a^9*b^2*d*e)))*(-a^5*b^3*e)^(1/2)*(35*
a^4 + 3*b^4 + 6*a^2*b^2))/(8*(a^11*d*e + a^5*b^6*d*e + 3*a^7*b^4*d*e + 3*a^9*b^2*d*e)))*(-a^5*b^3*e)^(1/2)*(35
*a^4 + 3*b^4 + 6*a^2*b^2))/(8*(a^11*d*e + a^5*b^6*d*e + 3*a^7*b^4*d*e + 3*a^9*b^2*d*e)) - ((((e*cot(c + d*x))^
(1/2)*(18*a^2*b^15*e^8 - 9*b^17*e^8 - 71*a^4*b^13*e^8 + 892*a^6*b^11*e^8 + 857*a^8*b^9*e^8 + 6802*a^10*b^7*e^8
- 1257*a^12*b^5*e^8))/(a^20*d^4 + a^4*b^16*d^4 + 8*a^6*b^14*d^4 + 28*a^8*b^12*d^4 + 56*a^10*b^10*d^4 + 70*a^1
2*b^8*d^4 + 56*a^14*b^6*d^4 + 28*a^16*b^4*d^4 + 8*a^18*b^2*d^4) + (((90*a*b^19*d^2*e^9 + 846*a^3*b^17*d^2*e^9
+ 1714*a^5*b^15*d^2*e^9 + 3606*a^7*b^13*d^2*e^9 - 14578*a^9*b^11*d^2*e^9 - 34486*a^11*b^9*d^2*e^9 - 14970*a^13
*b^7*d^2*e^9 + 2258*a^15*b^5*d^2*e^9 - 32*a^17*b^3*d^2*e^9)/(a^20*d^5 + a^4*b^16*d^5 + 8*a^6*b^14*d^5 + 28*a^8
*b^12*d^5 + 56*a^10*b^10*d^5 + 70*a^12*b^8*d^5 + 56*a^14*b^6*d^5 + 28*a^16*b^4*d^5 + 8*a^18*b^2*d^5) + ((((e*c
ot(c + d*x))^(1/2)*(72*a*b^22*d^2*e^9 + 576*a^3*b^20*d^2*e^9 + 5024*a^5*b^18*d^2*e^9 + 14272*a^7*b^16*d^2*e^9
+ 27824*a^9*b^14*d^2*e^9 + 53184*a^11*b^12*d^2*e^9 + 70240*a^13*b^10*d^2*e^9 + 47680*a^15*b^8*d^2*e^9 + 12616*
a^17*b^6*d^2*e^9 - 64*a^21*b^2*d^2*e^9))/(a^20*d^4 + a^4*b^16*d^4 + 8*a^6*b^14*d^4 + 28*a^8*b^12*d^4 + 56*a^10
*b^10*d^4 + 70*a^12*b^8*d^4 + 56*a^14*b^6*d^4 + 28*a^16*b^4*d^4 + 8*a^18*b^2*d^4) - (((192*a^2*b^24*d^4*e^10 +
1728*a^4*b^22*d^4*e^10 + 8320*a^6*b^20*d^4*e^10 + 27264*a^8*b^18*d^4*e^10 + 62592*a^10*b^16*d^4*e^10 + 99456*
a^12*b^14*d^4*e^10 + 107520*a^14*b^12*d^4*e^10 + 76800*a^16*b^10*d^4*e^10 + 33984*a^18*b^8*d^4*e^10 + 7872*a^2
0*b^6*d^4*e^10 + 384*a^22*b^4*d^4*e^10 - 128*a^24*b^2*d^4*e^10)/(a^20*d^5 + a^4*b^16*d^5 + 8*a^6*b^14*d^5 + 28
*a^8*b^12*d^5 + 56*a^10*b^10*d^5 + 70*a^12*b^8*d^5 + 56*a^14*b^6*d^5 + 28*a^16*b^4*d^5 + 8*a^18*b^2*d^5) + ((e
*cot(c + d*x))^(1/2)*(-a^5*b^3*e)^(1/2)*(35*a^4 + 3*b^4 + 6*a^2*b^2)*(512*a^4*b^25*d^4*e^10 + 4608*a^6*b^23*d^
4*e^10 + 17920*a^8*b^21*d^4*e^10 + 38400*a^10*b^19*d^4*e^10 + 46080*a^12*b^17*d^4*e^10 + 21504*a^14*b^15*d^4*e
^10 - 21504*a^16*b^13*d^4*e^10 - 46080*a^18*b^11*d^4*e^10 - 38400*a^20*b^9*d^4*e^10 - 17920*a^22*b^7*d^4*e^10
- 4608*a^24*b^5*d^4*e^10 - 512*a^26*b^3*d^4*e^10))/(8*(a^11*d*e + a^5*b^6*d*e + 3*a^7*b^4*d*e + 3*a^9*b^2*d*e)
*(a^20*d^4 + a^4*b^16*d^4 + 8*a^6*b^14*d^4 + 28*a^8*b^12*d^4 + 56*a^10*b^10*d^4 + 70*a^12*b^8*d^4 + 56*a^14*b^
6*d^4 + 28*a^16*b^4*d^4 + 8*a^18*b^2*d^4)))*(-a^5*b^3*e)^(1/2)*(35*a^4 + 3*b^4 + 6*a^2*b^2))/(8*(a^11*d*e + a^
5*b^6*d*e + 3*a^7*b^4*d*e + 3*a^9*b^2*d*e)))*(-a^5*b^3*e)^(1/2)*(35*a^4 + 3*b^4 + 6*a^2*b^2))/(8*(a^11*d*e + a
^5*b^6*d*e + 3*a^7*b^4*d*e + 3*a^9*b^2*d*e)))*(-a^5*b^3*e)^(1/2)*(35*a^4 + 3*b^4 + 6*a^2*b^2))/(8*(a^11*d*e +
a^5*b^6*d*e + 3*a^7*b^4*d*e + 3*a^9*b^2*d*e)))*(-a^5*b^3*e)^(1/2)*(35*a^4 + 3*b^4 + 6*a^2*b^2))/(8*(a^11*d*e +
a^5*b^6*d*e + 3*a^7*b^4*d*e + 3*a^9*b^2*d*e))))*(-a^5*b^3*e)^(1/2)*(35*a^4 + 3*b^4 + 6*a^2*b^2)*1i)/(4*(a^11*
d*e + a^5*b^6*d*e + 3*a^7*b^4*d*e + 3*a^9*b^2*d*e))

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {e \cot {\left (c + d x \right )}} \left (a + b \cot {\left (c + d x \right )}\right )^{3}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(1/(sqrt(e*cot(c + d*x))*(a + b*cot(c + d*x))**3), x)

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