3.1.87 \(\int \frac {1}{(e \cot (c+d x))^{3/2} (a+b \cot (c+d x))^3} \, dx\) [87]

Optimal. Leaf size=529 \[ \frac {b^{5/2} \left (63 a^4+46 a^2 b^2+15 b^4\right ) \text {ArcTan}\left (\frac {\sqrt {b} \sqrt {e \cot (c+d x)}}{\sqrt {a} \sqrt {e}}\right )}{4 a^{7/2} \left (a^2+b^2\right )^3 d e^{3/2}}-\frac {(a-b) \left (a^2+4 a b+b^2\right ) \text {ArcTan}\left (1-\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} \left (a^2+b^2\right )^3 d e^{3/2}}+\frac {(a-b) \left (a^2+4 a b+b^2\right ) \text {ArcTan}\left (1+\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} \left (a^2+b^2\right )^3 d e^{3/2}}+\frac {8 a^4+31 a^2 b^2+15 b^4}{4 a^3 \left (a^2+b^2\right )^2 d e \sqrt {e \cot (c+d x)}}-\frac {b^2}{2 a \left (a^2+b^2\right ) d e \sqrt {e \cot (c+d x)} (a+b \cot (c+d x))^2}-\frac {b^2 \left (13 a^2+5 b^2\right )}{4 a^2 \left (a^2+b^2\right )^2 d e \sqrt {e \cot (c+d x)} (a+b \cot (c+d x))}+\frac {(a+b) \left (a^2-4 a b+b^2\right ) \log \left (\sqrt {e}+\sqrt {e} \cot (c+d x)-\sqrt {2} \sqrt {e \cot (c+d x)}\right )}{2 \sqrt {2} \left (a^2+b^2\right )^3 d e^{3/2}}-\frac {(a+b) \left (a^2-4 a b+b^2\right ) \log \left (\sqrt {e}+\sqrt {e} \cot (c+d x)+\sqrt {2} \sqrt {e \cot (c+d x)}\right )}{2 \sqrt {2} \left (a^2+b^2\right )^3 d e^{3/2}} \]

[Out]

1/4*b^(5/2)*(63*a^4+46*a^2*b^2+15*b^4)*arctan(b^(1/2)*(e*cot(d*x+c))^(1/2)/a^(1/2)/e^(1/2))/a^(7/2)/(a^2+b^2)^
3/d/e^(3/2)-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/e^(3/2)*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))/(a^2+b^2)^3/d/e^(3/2)*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))/(a^2+b^2)^3/d/e^(3/2)*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))/(a^2+b^2)^3/d/e^(3/2
)*2^(1/2)+1/4*(8*a^4+31*a^2*b^2+15*b^4)/a^3/(a^2+b^2)^2/d/e/(e*cot(d*x+c))^(1/2)-1/2*b^2/a/(a^2+b^2)/d/e/(a+b*
cot(d*x+c))^2/(e*cot(d*x+c))^(1/2)-1/4*b^2*(13*a^2+5*b^2)/a^2/(a^2+b^2)^2/d/e/(a+b*cot(d*x+c))/(e*cot(d*x+c))^
(1/2)

________________________________________________________________________________________

Rubi [A]
time = 1.13, antiderivative size = 529, normalized size of antiderivative = 1.00, number of steps used = 17, number of rules used = 13, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.520, Rules used = {3650, 3730, 3734, 3615, 1182, 1176, 631, 210, 1179, 642, 3715, 65, 211} \begin {gather*} -\frac {(a-b) \left (a^2+4 a b+b^2\right ) \text {ArcTan}\left (1-\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} d e^{3/2} \left (a^2+b^2\right )^3}+\frac {(a-b) \left (a^2+4 a b+b^2\right ) \text {ArcTan}\left (\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}+1\right )}{\sqrt {2} d e^{3/2} \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 e^{3/2} \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 e^{3/2} \left (a^2+b^2\right )^3}-\frac {b^2 \left (13 a^2+5 b^2\right )}{4 a^2 d e \left (a^2+b^2\right )^2 \sqrt {e \cot (c+d x)} (a+b \cot (c+d x))}-\frac {b^2}{2 a d e \left (a^2+b^2\right ) \sqrt {e \cot (c+d x)} (a+b \cot (c+d x))^2}+\frac {b^{5/2} \left (63 a^4+46 a^2 b^2+15 b^4\right ) \text {ArcTan}\left (\frac {\sqrt {b} \sqrt {e \cot (c+d x)}}{\sqrt {a} \sqrt {e}}\right )}{4 a^{7/2} d e^{3/2} \left (a^2+b^2\right )^3}+\frac {8 a^4+31 a^2 b^2+15 b^4}{4 a^3 d e \left (a^2+b^2\right )^2 \sqrt {e \cot (c+d x)}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/((e*Cot[c + d*x])^(3/2)*(a + b*Cot[c + d*x])^3),x]

[Out]

(b^(5/2)*(63*a^4 + 46*a^2*b^2 + 15*b^4)*ArcTan[(Sqrt[b]*Sqrt[e*Cot[c + d*x]])/(Sqrt[a]*Sqrt[e])])/(4*a^(7/2)*(
a^2 + b^2)^3*d*e^(3/2)) - ((a - b)*(a^2 + 4*a*b + b^2)*ArcTan[1 - (Sqrt[2]*Sqrt[e*Cot[c + d*x]])/Sqrt[e]])/(Sq
rt[2]*(a^2 + b^2)^3*d*e^(3/2)) + ((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*e^(3/2)) + (8*a^4 + 31*a^2*b^2 + 15*b^4)/(4*a^3*(a^2 + b^2)^2*d*e*Sqrt[e*Cot[c +
d*x]]) - b^2/(2*a*(a^2 + b^2)*d*e*Sqrt[e*Cot[c + d*x]]*(a + b*Cot[c + d*x])^2) - (b^2*(13*a^2 + 5*b^2))/(4*a^2
*(a^2 + b^2)^2*d*e*Sqrt[e*Cot[c + d*x]]*(a + b*Cot[c + d*x])) + ((a + b)*(a^2 - 4*a*b + b^2)*Log[Sqrt[e] + Sqr
t[e]*Cot[c + d*x] - Sqrt[2]*Sqrt[e*Cot[c + d*x]]])/(2*Sqrt[2]*(a^2 + b^2)^3*d*e^(3/2)) - ((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*e^(3/2)
)

Rule 65

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 210

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

Rule 211

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

Rule 631

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 642

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 1176

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 1179

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 1182

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 3615

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 3650

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 3715

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 3730

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*Ta
n[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 3734

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}{(e \cot (c+d x))^{3/2} (a+b \cot (c+d x))^3} \, dx &=-\frac {b^2}{2 a \left (a^2+b^2\right ) d e \sqrt {e \cot (c+d x)} (a+b \cot (c+d x))^2}-\frac {\int \frac {-\frac {1}{2} \left (4 a^2+5 b^2\right ) e+2 a b e \cot (c+d x)-\frac {5}{2} b^2 e \cot ^2(c+d x)}{(e \cot (c+d x))^{3/2} (a+b \cot (c+d x))^2} \, dx}{2 a \left (a^2+b^2\right ) e}\\ &=-\frac {b^2}{2 a \left (a^2+b^2\right ) d e \sqrt {e \cot (c+d x)} (a+b \cot (c+d x))^2}-\frac {b^2 \left (13 a^2+5 b^2\right )}{4 a^2 \left (a^2+b^2\right )^2 d e \sqrt {e \cot (c+d x)} (a+b \cot (c+d x))}+\frac {\int \frac {\frac {1}{4} \left (8 a^4+31 a^2 b^2+15 b^4\right ) e^2-4 a^3 b e^2 \cot (c+d x)+\frac {3}{4} b^2 \left (13 a^2+5 b^2\right ) e^2 \cot ^2(c+d x)}{(e \cot (c+d x))^{3/2} (a+b \cot (c+d x))} \, dx}{2 a^2 \left (a^2+b^2\right )^2 e^2}\\ &=\frac {8 a^4+31 a^2 b^2+15 b^4}{4 a^3 \left (a^2+b^2\right )^2 d e \sqrt {e \cot (c+d x)}}-\frac {b^2}{2 a \left (a^2+b^2\right ) d e \sqrt {e \cot (c+d x)} (a+b \cot (c+d x))^2}-\frac {b^2 \left (13 a^2+5 b^2\right )}{4 a^2 \left (a^2+b^2\right )^2 d e \sqrt {e \cot (c+d x)} (a+b \cot (c+d x))}+\frac {\int \frac {-\frac {1}{8} b \left (24 a^4+31 a^2 b^2+15 b^4\right ) e^4-a^3 \left (a^2-b^2\right ) e^4 \cot (c+d x)-\frac {1}{8} b \left (8 a^4+31 a^2 b^2+15 b^4\right ) e^4 \cot ^2(c+d x)}{\sqrt {e \cot (c+d x)} (a+b \cot (c+d x))} \, dx}{a^3 \left (a^2+b^2\right )^2 e^5}\\ &=\frac {8 a^4+31 a^2 b^2+15 b^4}{4 a^3 \left (a^2+b^2\right )^2 d e \sqrt {e \cot (c+d x)}}-\frac {b^2}{2 a \left (a^2+b^2\right ) d e \sqrt {e \cot (c+d x)} (a+b \cot (c+d x))^2}-\frac {b^2 \left (13 a^2+5 b^2\right )}{4 a^2 \left (a^2+b^2\right )^2 d e \sqrt {e \cot (c+d x)} (a+b \cot (c+d x))}+\frac {\int \frac {-a^3 b \left (3 a^2-b^2\right ) e^4-a^4 \left (a^2-3 b^2\right ) e^4 \cot (c+d x)}{\sqrt {e \cot (c+d x)}} \, dx}{a^3 \left (a^2+b^2\right )^3 e^5}-\frac {\left (b^3 \left (63 a^4+46 a^2 b^2+15 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^3 \left (a^2+b^2\right )^3 e}\\ &=\frac {8 a^4+31 a^2 b^2+15 b^4}{4 a^3 \left (a^2+b^2\right )^2 d e \sqrt {e \cot (c+d x)}}-\frac {b^2}{2 a \left (a^2+b^2\right ) d e \sqrt {e \cot (c+d x)} (a+b \cot (c+d x))^2}-\frac {b^2 \left (13 a^2+5 b^2\right )}{4 a^2 \left (a^2+b^2\right )^2 d e \sqrt {e \cot (c+d x)} (a+b \cot (c+d x))}+\frac {2 \text {Subst}\left (\int \frac {a^3 b \left (3 a^2-b^2\right ) e^5+a^4 \left (a^2-3 b^2\right ) e^4 x^2}{e^2+x^4} \, dx,x,\sqrt {e \cot (c+d x)}\right )}{a^3 \left (a^2+b^2\right )^3 d e^5}-\frac {\left (b^3 \left (63 a^4+46 a^2 b^2+15 b^4\right )\right ) \text {Subst}\left (\int \frac {1}{\sqrt {-e x} (a-b x)} \, dx,x,-\cot (c+d x)\right )}{8 a^3 \left (a^2+b^2\right )^3 d e}\\ &=\frac {8 a^4+31 a^2 b^2+15 b^4}{4 a^3 \left (a^2+b^2\right )^2 d e \sqrt {e \cot (c+d x)}}-\frac {b^2}{2 a \left (a^2+b^2\right ) d e \sqrt {e \cot (c+d x)} (a+b \cot (c+d x))^2}-\frac {b^2 \left (13 a^2+5 b^2\right )}{4 a^2 \left (a^2+b^2\right )^2 d e \sqrt {e \cot (c+d x)} (a+b \cot (c+d x))}+\frac {\left (b^3 \left (63 a^4+46 a^2 b^2+15 b^4\right )\right ) \text {Subst}\left (\int \frac {1}{a+\frac {b x^2}{e}} \, dx,x,\sqrt {e \cot (c+d x)}\right )}{4 a^3 \left (a^2+b^2\right )^3 d e^2}-\frac {\left ((a+b) \left (a^2-4 a b+b^2\right )\right ) \text {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 e}+\frac {\left ((a-b) \left (a^2+4 a b+b^2\right )\right ) \text {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 e}\\ &=\frac {b^{5/2} \left (63 a^4+46 a^2 b^2+15 b^4\right ) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {e \cot (c+d x)}}{\sqrt {a} \sqrt {e}}\right )}{4 a^{7/2} \left (a^2+b^2\right )^3 d e^{3/2}}+\frac {8 a^4+31 a^2 b^2+15 b^4}{4 a^3 \left (a^2+b^2\right )^2 d e \sqrt {e \cot (c+d x)}}-\frac {b^2}{2 a \left (a^2+b^2\right ) d e \sqrt {e \cot (c+d x)} (a+b \cot (c+d x))^2}-\frac {b^2 \left (13 a^2+5 b^2\right )}{4 a^2 \left (a^2+b^2\right )^2 d e \sqrt {e \cot (c+d x)} (a+b \cot (c+d x))}+\frac {\left ((a+b) \left (a^2-4 a b+b^2\right )\right ) \text {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 e^{3/2}}+\frac {\left ((a+b) \left (a^2-4 a b+b^2\right )\right ) \text {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 e^{3/2}}+\frac {\left ((a-b) \left (a^2+4 a b+b^2\right )\right ) \text {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 e}+\frac {\left ((a-b) \left (a^2+4 a b+b^2\right )\right ) \text {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 e}\\ &=\frac {b^{5/2} \left (63 a^4+46 a^2 b^2+15 b^4\right ) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {e \cot (c+d x)}}{\sqrt {a} \sqrt {e}}\right )}{4 a^{7/2} \left (a^2+b^2\right )^3 d e^{3/2}}+\frac {8 a^4+31 a^2 b^2+15 b^4}{4 a^3 \left (a^2+b^2\right )^2 d e \sqrt {e \cot (c+d x)}}-\frac {b^2}{2 a \left (a^2+b^2\right ) d e \sqrt {e \cot (c+d x)} (a+b \cot (c+d x))^2}-\frac {b^2 \left (13 a^2+5 b^2\right )}{4 a^2 \left (a^2+b^2\right )^2 d e \sqrt {e \cot (c+d x)} (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 e^{3/2}}-\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 e^{3/2}}+\frac {\left ((a-b) \left (a^2+4 a b+b^2\right )\right ) \text {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 e^{3/2}}-\frac {\left ((a-b) \left (a^2+4 a b+b^2\right )\right ) \text {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 e^{3/2}}\\ &=\frac {b^{5/2} \left (63 a^4+46 a^2 b^2+15 b^4\right ) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {e \cot (c+d x)}}{\sqrt {a} \sqrt {e}}\right )}{4 a^{7/2} \left (a^2+b^2\right )^3 d e^{3/2}}-\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 e^{3/2}}+\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 e^{3/2}}+\frac {8 a^4+31 a^2 b^2+15 b^4}{4 a^3 \left (a^2+b^2\right )^2 d e \sqrt {e \cot (c+d x)}}-\frac {b^2}{2 a \left (a^2+b^2\right ) d e \sqrt {e \cot (c+d x)} (a+b \cot (c+d x))^2}-\frac {b^2 \left (13 a^2+5 b^2\right )}{4 a^2 \left (a^2+b^2\right )^2 d e \sqrt {e \cot (c+d x)} (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 e^{3/2}}-\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 e^{3/2}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
time = 1.90, size = 303, normalized size = 0.57 \begin {gather*} -\frac {-8 a^2 b^2 \left (3 a^2-b^2\right ) \, _2F_1\left (-\frac {1}{2},1;\frac {1}{2};-\frac {b \cot (c+d x)}{a}\right )-16 a^2 b^2 \left (a^2+b^2\right ) \, _2F_1\left (-\frac {1}{2},2;\frac {1}{2};-\frac {b \cot (c+d x)}{a}\right )-8 b^2 \left (a^2+b^2\right )^2 \, _2F_1\left (-\frac {1}{2},3;\frac {1}{2};-\frac {b \cot (c+d x)}{a}\right )-8 a^4 \left (a^2-3 b^2\right ) \, _2F_1\left (-\frac {1}{4},1;\frac {3}{4};-\cot ^2(c+d x)\right )+\sqrt {2} a^3 b \left (3 a^2-b^2\right ) \sqrt {\cot (c+d x)} \left (2 \text {ArcTan}\left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )-2 \text {ArcTan}\left (1+\sqrt {2} \sqrt {\cot (c+d x)}\right )+\log \left (1-\sqrt {2} \sqrt {\cot (c+d x)}+\cot (c+d x)\right )-\log \left (1+\sqrt {2} \sqrt {\cot (c+d x)}+\cot (c+d x)\right )\right )}{4 a^3 \left (a^2+b^2\right )^3 d e \sqrt {e \cot (c+d x)}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/((e*Cot[c + d*x])^(3/2)*(a + b*Cot[c + d*x])^3),x]

[Out]

-1/4*(-8*a^2*b^2*(3*a^2 - b^2)*Hypergeometric2F1[-1/2, 1, 1/2, -((b*Cot[c + d*x])/a)] - 16*a^2*b^2*(a^2 + b^2)
*Hypergeometric2F1[-1/2, 2, 1/2, -((b*Cot[c + d*x])/a)] - 8*b^2*(a^2 + b^2)^2*Hypergeometric2F1[-1/2, 3, 1/2,
-((b*Cot[c + d*x])/a)] - 8*a^4*(a^2 - 3*b^2)*Hypergeometric2F1[-1/4, 1, 3/4, -Cot[c + d*x]^2] + Sqrt[2]*a^3*b*
(3*a^2 - b^2)*Sqrt[Cot[c + d*x]]*(2*ArcTan[1 - Sqrt[2]*Sqrt[Cot[c + d*x]]] - 2*ArcTan[1 + Sqrt[2]*Sqrt[Cot[c +
d*x]]] + Log[1 - Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]] - Log[1 + Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*
x]]))/(a^3*(a^2 + b^2)^3*d*e*Sqrt[e*Cot[c + d*x]])

________________________________________________________________________________________

Maple [A]
time = 0.65, size = 480, normalized size = 0.91

method	result	size
derivativedivides	\(-\frac {2 e^{4} \left (-\frac {b^{3} \left (\frac {\left (\frac {15}{8} a^{4} b +\frac {11}{4} a^{2} b^{3}+\frac {7}{8} b^{5}\right ) \left (e \cot \left (d x +c \right )\right )^{\frac {3}{2}}+\frac {a e \left (17 a^{4}+26 a^{2} b^{2}+9 b^{4}\right ) \sqrt {e \cot \left (d x +c \right )}}{8}}{\left (e \cot \left (d x +c \right ) b +a e \right )^{2}}+\frac {\left (63 a^{4}+46 a^{2} b^{2}+15 b^{4}\right ) \arctan \left (\frac {b \sqrt {e \cot \left (d x +c \right )}}{\sqrt {a e b}}\right )}{8 \sqrt {a e b}}\right )}{a^{3} e^{5} \left (a^{2}+b^{2}\right )^{3}}+\frac {\frac {\left (-3 a^{2} b e +b^{3} e \right ) \left (e^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 e^{2}}+\frac {\left (-a^{3}+3 a \,b^{2}\right ) \sqrt {2}\, \left (\ln \left (\frac {e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 \left (e^{2}\right )^{\frac {1}{4}}}}{\left (a^{2}+b^{2}\right )^{3} e^{5}}-\frac {1}{a^{3} e^{5} \sqrt {e \cot \left (d x +c \right )}}\right )}{d}\)	\(480\)
default	\(-\frac {2 e^{4} \left (-\frac {b^{3} \left (\frac {\left (\frac {15}{8} a^{4} b +\frac {11}{4} a^{2} b^{3}+\frac {7}{8} b^{5}\right ) \left (e \cot \left (d x +c \right )\right )^{\frac {3}{2}}+\frac {a e \left (17 a^{4}+26 a^{2} b^{2}+9 b^{4}\right ) \sqrt {e \cot \left (d x +c \right )}}{8}}{\left (e \cot \left (d x +c \right ) b +a e \right )^{2}}+\frac {\left (63 a^{4}+46 a^{2} b^{2}+15 b^{4}\right ) \arctan \left (\frac {b \sqrt {e \cot \left (d x +c \right )}}{\sqrt {a e b}}\right )}{8 \sqrt {a e b}}\right )}{a^{3} e^{5} \left (a^{2}+b^{2}\right )^{3}}+\frac {\frac {\left (-3 a^{2} b e +b^{3} e \right ) \left (e^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 e^{2}}+\frac {\left (-a^{3}+3 a \,b^{2}\right ) \sqrt {2}\, \left (\ln \left (\frac {e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 \left (e^{2}\right )^{\frac {1}{4}}}}{\left (a^{2}+b^{2}\right )^{3} e^{5}}-\frac {1}{a^{3} e^{5} \sqrt {e \cot \left (d x +c \right )}}\right )}{d}\)	\(480\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/d*e^4*(-b^3/a^3/e^5/(a^2+b^2)^3*(((15/8*a^4*b+11/4*a^2*b^3+7/8*b^5)*(e*cot(d*x+c))^(3/2)+1/8*a*e*(17*a^4+26
*a^2*b^2+9*b^4)*(e*cot(d*x+c))^(1/2))/(e*cot(d*x+c)*b+a*e)^2+1/8*(63*a^4+46*a^2*b^2+15*b^4)/(a*e*b)^(1/2)*arct
an(b*(e*cot(d*x+c))^(1/2)/(a*e*b)^(1/2)))+1/(a^2+b^2)^3/e^5*(1/8*(-3*a^2*b*e+b^3*e)*(e^2)^(1/4)/e^2*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)))+2*arctan(2^(1/2)/(e^2)^(1/4)*(e*cot(d*x+c))^(1/2)+1)-2*arctan(-2^(1/2)/(e^2)^(1/4
)*(e*cot(d*x+c))^(1/2)+1))+1/8*(-a^3+3*a*b^2)/(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)))+2*arctan(2^(1
/2)/(e^2)^(1/4)*(e*cot(d*x+c))^(1/2)+1)-2*arctan(-2^(1/2)/(e^2)^(1/4)*(e*cot(d*x+c))^(1/2)+1)))-1/a^3/e^5/(e*c
ot(d*x+c))^(1/2))

________________________________________________________________________________________

Maxima [A]
time = 0.53, size = 472, normalized size = 0.89 \begin {gather*} \frac {{\left (\frac {{\left (63 \, a^{4} b^{3} + 46 \, a^{2} b^{5} + 15 \, b^{7}\right )} \arctan \left (\frac {b}{\sqrt {a b} \sqrt {\tan \left (d x + c\right )}}\right )}{{\left (a^{9} + 3 \, a^{7} b^{2} + 3 \, a^{5} b^{4} + a^{3} b^{6}\right )} \sqrt {a b}} + \frac {2 \, \sqrt {2} {\left (a^{3} + 3 \, a^{2} b - 3 \, a b^{2} - b^{3}\right )} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + \frac {2}{\sqrt {\tan \left (d x + c\right )}}\right )}\right ) + 2 \, \sqrt {2} {\left (a^{3} + 3 \, a^{2} b - 3 \, a b^{2} - b^{3}\right )} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - \frac {2}{\sqrt {\tan \left (d x + c\right )}}\right )}\right ) - \sqrt {2} {\left (a^{3} - 3 \, a^{2} b - 3 \, a b^{2} + b^{3}\right )} \log \left (\frac {\sqrt {2}}{\sqrt {\tan \left (d x + c\right )}} + \frac {1}{\tan \left (d x + c\right )} + 1\right ) + \sqrt {2} {\left (a^{3} - 3 \, a^{2} b - 3 \, a b^{2} + b^{3}\right )} \log \left (-\frac {\sqrt {2}}{\sqrt {\tan \left (d x + c\right )}} + \frac {1}{\tan \left (d x + c\right )} + 1\right )}{a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}} + \frac {8 \, a^{6} + 16 \, a^{4} b^{2} + 8 \, a^{2} b^{4} + \frac {16 \, a^{5} b + 49 \, a^{3} b^{3} + 25 \, a b^{5}}{\tan \left (d x + c\right )} + \frac {8 \, a^{4} b^{2} + 31 \, a^{2} b^{4} + 15 \, b^{6}}{\tan \left (d x + c\right )^{2}}}{\frac {a^{9} + 2 \, a^{7} b^{2} + a^{5} b^{4}}{\sqrt {\tan \left (d x + c\right )}} + \frac {2 \, {\left (a^{8} b + 2 \, a^{6} b^{3} + a^{4} b^{5}\right )}}{\tan \left (d x + c\right )^{\frac {3}{2}}} + \frac {a^{7} b^{2} + 2 \, a^{5} b^{4} + a^{3} b^{6}}{\tan \left (d x + c\right )^{\frac {5}{2}}}}\right )} e^{\left (-\frac {3}{2}\right )}}{4 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*((63*a^4*b^3 + 46*a^2*b^5 + 15*b^7)*arctan(b/(sqrt(a*b)*sqrt(tan(d*x + c))))/((a^9 + 3*a^7*b^2 + 3*a^5*b^4
+ a^3*b^6)*sqrt(a*b)) + (2*sqrt(2)*(a^3 + 3*a^2*b - 3*a*b^2 - b^3)*arctan(1/2*sqrt(2)*(sqrt(2) + 2/sqrt(tan(d
*x + c)))) + 2*sqrt(2)*(a^3 + 3*a^2*b - 3*a*b^2 - b^3)*arctan(-1/2*sqrt(2)*(sqrt(2) - 2/sqrt(tan(d*x + c)))) -
sqrt(2)*(a^3 - 3*a^2*b - 3*a*b^2 + b^3)*log(sqrt(2)/sqrt(tan(d*x + c)) + 1/tan(d*x + c) + 1) + sqrt(2)*(a^3 -
3*a^2*b - 3*a*b^2 + b^3)*log(-sqrt(2)/sqrt(tan(d*x + c)) + 1/tan(d*x + c) + 1))/(a^6 + 3*a^4*b^2 + 3*a^2*b^4
+ b^6) + (8*a^6 + 16*a^4*b^2 + 8*a^2*b^4 + (16*a^5*b + 49*a^3*b^3 + 25*a*b^5)/tan(d*x + c) + (8*a^4*b^2 + 31*a
^2*b^4 + 15*b^6)/tan(d*x + c)^2)/((a^9 + 2*a^7*b^2 + a^5*b^4)/sqrt(tan(d*x + c)) + 2*(a^8*b + 2*a^6*b^3 + a^4*
b^5)/tan(d*x + c)^(3/2) + (a^7*b^2 + 2*a^5*b^4 + a^3*b^6)/tan(d*x + c)^(5/2)))*e^(-3/2)/d

________________________________________________________________________________________

Fricas [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

________________________________________________________________________________________

Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\left (e \cot {\left (c + d x \right )}\right )^{\frac {3}{2}} \left (a + b \cot {\left (c + d x \right )}\right )^{3}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Mupad [B]
time = 10.00, size = 2500, normalized size = 4.73 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((2*e)/a + (e*cot(c + d*x)*(16*a^4*b + 25*b^5 + 49*a^2*b^3))/(4*a^2*(a^4 + b^4 + 2*a^2*b^2)) + (b^2*e^2*cot(c
+ d*x)^2*(8*a^4 + 15*b^4 + 31*a^2*b^2))/(4*a^3*(a^4*e + b^4*e + 2*a^2*b^2*e)))/(b^2*d*(e*cot(c + d*x))^(5/2) +
a^2*d*e^2*(e*cot(c + d*x))^(1/2) + 2*a*b*d*e*(e*cot(c + d*x))^(3/2)) + atan((((-1i/(4*(b^6*d^2*e^3 - a^6*d^2*
e^3 + a*b^5*d^2*e^3*6i + a^5*b*d^2*e^3*6i - 15*a^2*b^4*d^2*e^3 - a^3*b^3*d^2*e^3*20i + 15*a^4*b^2*d^2*e^3)))^(
1/2)*(((e*cot(c + d*x))^(1/2)*(471859200*a^22*b^44*d^7*e^16 + 9500098560*a^24*b^42*d^7*e^16 + 91857354752*a^26
*b^40*d^7*e^16 + 564502986752*a^28*b^38*d^7*e^16 + 2464648527872*a^30*b^36*d^7*e^16 + 8104469069824*a^32*b^34*
d^7*e^16 + 20769933361152*a^34*b^32*d^7*e^16 + 42351565209600*a^36*b^30*d^7*e^16 + 69534945902592*a^38*b^28*d^
7*e^16 + 92434029608960*a^40*b^26*d^7*e^16 + 99508717355008*a^42*b^24*d^7*e^16 + 86342935511040*a^44*b^22*d^7*
e^16 + 59767095558144*a^46*b^20*d^7*e^16 + 32432589897728*a^48*b^18*d^7*e^16 + 13411815522304*a^50*b^16*d^7*e^
16 + 4030457708544*a^52*b^14*d^7*e^16 + 805425905664*a^54*b^12*d^7*e^16 + 86608183296*a^56*b^10*d^7*e^16 + 161
2709888*a^58*b^8*d^7*e^16 + 16777216*a^60*b^6*d^7*e^16 + 167772160*a^62*b^4*d^7*e^16 + 16777216*a^64*b^2*d^7*e
^16) + (-1i/(4*(b^6*d^2*e^3 - a^6*d^2*e^3 + a*b^5*d^2*e^3*6i + a^5*b*d^2*e^3*6i - 15*a^2*b^4*d^2*e^3 - a^3*b^3
*d^2*e^3*20i + 15*a^4*b^2*d^2*e^3)))^(1/2)*(251658240*a^24*b^45*d^8*e^18 - (e*cot(c + d*x))^(1/2)*(-1i/(4*(b^6
*d^2*e^3 - a^6*d^2*e^3 + a*b^5*d^2*e^3*6i + a^5*b*d^2*e^3*6i - 15*a^2*b^4*d^2*e^3 - a^3*b^3*d^2*e^3*20i + 15*a
^4*b^2*d^2*e^3)))^(1/2)*(134217728*a^27*b^45*d^9*e^19 + 2550136832*a^29*b^43*d^9*e^19 + 22817013760*a^31*b^41*
d^9*e^19 + 127506841600*a^33*b^39*d^9*e^19 + 497276682240*a^35*b^37*d^9*e^19 + 1430626762752*a^37*b^35*d^9*e^1
9 + 3121367482368*a^39*b^33*d^9*e^19 + 5202279137280*a^41*b^31*d^9*e^19 + 6502848921600*a^43*b^29*d^9*e^19 + 5
635802398720*a^45*b^27*d^9*e^19 + 2254320959488*a^47*b^25*d^9*e^19 - 2254320959488*a^49*b^23*d^9*e^19 - 563580
2398720*a^51*b^21*d^9*e^19 - 6502848921600*a^53*b^19*d^9*e^19 - 5202279137280*a^55*b^17*d^9*e^19 - 31213674823
68*a^57*b^15*d^9*e^19 - 1430626762752*a^59*b^13*d^9*e^19 - 497276682240*a^61*b^11*d^9*e^19 - 127506841600*a^63
*b^9*d^9*e^19 - 22817013760*a^65*b^7*d^9*e^19 - 2550136832*a^67*b^5*d^9*e^19 - 134217728*a^69*b^3*d^9*e^19) +
5049942016*a^26*b^43*d^8*e^18 + 48368713728*a^28*b^41*d^8*e^18 + 293819383808*a^30*b^39*d^8*e^18 + 12684581928
96*a^32*b^37*d^8*e^18 + 4132731617280*a^34*b^35*d^8*e^18 + 10531192700928*a^36*b^33*d^8*e^18 + 21462823993344*
a^38*b^31*d^8*e^18 + 35469618315264*a^40*b^29*d^8*e^18 + 47896904859648*a^42*b^27*d^8*e^18 + 52983958077440*a^
44*b^25*d^8*e^18 + 47896904859648*a^46*b^23*d^8*e^18 + 35090285461504*a^48*b^21*d^8*e^18 + 20487396655104*a^50
*b^19*d^8*e^18 + 9230622916608*a^52*b^17*d^8*e^18 + 2994733056000*a^54*b^15*d^8*e^18 + 565576728576*a^56*b^13*
d^8*e^18 - 18572378112*a^58*b^11*d^8*e^18 - 50281316352*a^60*b^9*d^8*e^18 - 16089350144*a^62*b^7*d^8*e^18 - 25
16582400*a^64*b^5*d^8*e^18 - 167772160*a^66*b^3*d^8*e^18))*(-1i/(4*(b^6*d^2*e^3 - a^6*d^2*e^3 + a*b^5*d^2*e^3*
6i + a^5*b*d^2*e^3*6i - 15*a^2*b^4*d^2*e^3 - a^3*b^3*d^2*e^3*20i + 15*a^4*b^2*d^2*e^3)))^(1/2) - 117964800*a^2
1*b^42*d^6*e^15 - 841482240*a^23*b^40*d^6*e^15 + 3829399552*a^25*b^38*d^6*e^15 + 78068580352*a^27*b^36*d^6*e^1
5 + 497438162944*a^29*b^34*d^6*e^15 + 1899895980032*a^31*b^32*d^6*e^15 + 4972695519232*a^33*b^30*d^6*e^15 + 93
71195015168*a^35*b^28*d^6*e^15 + 12890720436224*a^37*b^26*d^6*e^15 + 12726089809920*a^39*b^24*d^6*e^15 + 83669
61197056*a^41*b^22*d^6*e^15 + 2597662490624*a^43*b^20*d^6*e^15 - 1171836108800*a^45*b^18*d^6*e^15 - 1986881650
688*a^47*b^16*d^6*e^15 - 1237583921152*a^49*b^14*d^6*e^15 - 449507753984*a^51*b^12*d^6*e^15 - 97476149248*a^53
*b^10*d^6*e^15 - 11931222016*a^55*b^8*d^6*e^15 - 1006632960*a^57*b^6*d^6*e^15 - 134217728*a^59*b^4*d^6*e^15 -
8388608*a^61*b^2*d^6*e^15) - (e*cot(c + d*x))^(1/2)*(7610564608*a^27*b^33*d^5*e^13 - 597688320*a^23*b^37*d^5*e
^13 - 1671430144*a^25*b^35*d^5*e^13 - 58982400*a^21*b^39*d^5*e^13 + 85774565376*a^29*b^31*d^5*e^13 + 385487994
880*a^31*b^29*d^5*e^13 + 1104303620096*a^33*b^27*d^5*e^13 + 2240523796480*a^35*b^25*d^5*e^13 + 3345249468416*a
^37*b^23*d^5*e^13 + 3717287903232*a^39*b^21*d^5*e^13 + 3053967114240*a^41*b^19*d^5*e^13 + 1807474491392*a^43*b
^17*d^5*e^13 + 726513221632*a^45*b^15*d^5*e^13 + 170768990208*a^47*b^13*d^5*e^13 + 10492051456*a^49*b^11*d^5*e
^13 - 4917821440*a^51*b^9*d^5*e^13 - 923009024*a^53*b^7*d^5*e^13 + 8388608*a^55*b^5*d^5*e^13))*(-1i/(4*(b^6*d^
2*e^3 - a^6*d^2*e^3 + a*b^5*d^2*e^3*6i + a^5*b*d^2*e^3*6i - 15*a^2*b^4*d^2*e^3 - a^3*b^3*d^2*e^3*20i + 15*a^4*
b^2*d^2*e^3)))^(1/2)*1i + ((-1i/(4*(b^6*d^2*e^3 - a^6*d^2*e^3 + a*b^5*d^2*e^3*6i + a^5*b*d^2*e^3*6i - 15*a^2*b
^4*d^2*e^3 - a^3*b^3*d^2*e^3*20i + 15*a^4*b^2*d^2*e^3)))^(1/2)*(((e*cot(c + d*x))^(1/2)*(471859200*a^22*b^44*d
^7*e^16 + 9500098560*a^24*b^42*d^7*e^16 + 91857354752*a^26*b^40*d^7*e^16 + 564502986752*a^28*b^38*d^7*e^16 + 2
464648527872*a^30*b^36*d^7*e^16 + 8104469069824...

________________________________________________________________________________________