3.1.80 \(\int \frac {1}{(e \cot (c+d x))^{3/2} (a+b \cot (c+d x))^2} \, dx\) [80]

Optimal. Leaf size=437 \[ \frac {b^{5/2} \left (7 a^2+3 b^2\right ) \text {ArcTan}\left (\frac {\sqrt {b} \sqrt {e \cot (c+d x)}}{\sqrt {a} \sqrt {e}}\right )}{a^{5/2} \left (a^2+b^2\right )^2 d e^{3/2}}-\frac {\left (a^2+2 a b-b^2\right ) \text {ArcTan}\left (1-\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} \left (a^2+b^2\right )^2 d e^{3/2}}+\frac {\left (a^2+2 a b-b^2\right ) \text {ArcTan}\left (1+\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} \left (a^2+b^2\right )^2 d e^{3/2}}+\frac {2 a^2+3 b^2}{a^2 \left (a^2+b^2\right ) d e \sqrt {e \cot (c+d x)}}-\frac {b^2}{a \left (a^2+b^2\right ) d e \sqrt {e \cot (c+d x)} (a+b \cot (c+d x))}+\frac {\left (a^2-2 a b-b^2\right ) \log \left (\sqrt {e}+\sqrt {e} \cot (c+d x)-\sqrt {2} \sqrt {e \cot (c+d x)}\right )}{2 \sqrt {2} \left (a^2+b^2\right )^2 d e^{3/2}}-\frac {\left (a^2-2 a b-b^2\right ) \log \left (\sqrt {e}+\sqrt {e} \cot (c+d x)+\sqrt {2} \sqrt {e \cot (c+d x)}\right )}{2 \sqrt {2} \left (a^2+b^2\right )^2 d e^{3/2}} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 0.74, antiderivative size = 437, 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 = {3650, 3730, 3734, 3615, 1182, 1176, 631, 210, 1179, 642, 3715, 65, 211} \begin {gather*} -\frac {\left (a^2+2 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 )^2}+\frac {\left (a^2+2 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 )^2}+\frac {\left (a^2-2 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 )^2}-\frac {\left (a^2-2 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 )^2}-\frac {b^2}{a d e \left (a^2+b^2\right ) \sqrt {e \cot (c+d x)} (a+b \cot (c+d x))}+\frac {2 a^2+3 b^2}{a^2 d e \left (a^2+b^2\right ) \sqrt {e \cot (c+d x)}}+\frac {b^{5/2} \left (7 a^2+3 b^2\right ) \text {ArcTan}\left (\frac {\sqrt {b} \sqrt {e \cot (c+d x)}}{\sqrt {a} \sqrt {e}}\right )}{a^{5/2} d e^{3/2} \left (a^2+b^2\right )^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(b^(5/2)*(7*a^2 + 3*b^2)*ArcTan[(Sqrt[b]*Sqrt[e*Cot[c + d*x]])/(Sqrt[a]*Sqrt[e])])/(a^(5/2)*(a^2 + b^2)^2*d*e^
(3/2)) - ((a^2 + 2*a*b - b^2)*ArcTan[1 - (Sqrt[2]*Sqrt[e*Cot[c + d*x]])/Sqrt[e]])/(Sqrt[2]*(a^2 + b^2)^2*d*e^(
3/2)) + ((a^2 + 2*a*b - b^2)*ArcTan[1 + (Sqrt[2]*Sqrt[e*Cot[c + d*x]])/Sqrt[e]])/(Sqrt[2]*(a^2 + b^2)^2*d*e^(3
/2)) + (2*a^2 + 3*b^2)/(a^2*(a^2 + b^2)*d*e*Sqrt[e*Cot[c + d*x]]) - b^2/(a*(a^2 + b^2)*d*e*Sqrt[e*Cot[c + d*x]
]*(a + b*Cot[c + d*x])) + ((a^2 - 2*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)^2*d*e^(3/2)) - ((a^2 - 2*a*b - b^2)*Log[Sqrt[e] + Sqrt[e]*Cot[c + d*x] + Sqrt[2]*S
qrt[e*Cot[c + d*x]]])/(2*Sqrt[2]*(a^2 + b^2)^2*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))^2} \, dx &=-\frac {b^2}{a \left (a^2+b^2\right ) d e \sqrt {e \cot (c+d x)} (a+b \cot (c+d x))}-\frac {\int \frac {-\frac {1}{2} \left (2 a^2+3 b^2\right ) e+a b e \cot (c+d x)-\frac {3}{2} b^2 e \cot ^2(c+d x)}{(e \cot (c+d x))^{3/2} (a+b \cot (c+d x))} \, dx}{a \left (a^2+b^2\right ) e}\\ &=\frac {2 a^2+3 b^2}{a^2 \left (a^2+b^2\right ) d e \sqrt {e \cot (c+d x)}}-\frac {b^2}{a \left (a^2+b^2\right ) d e \sqrt {e \cot (c+d x)} (a+b \cot (c+d x))}-\frac {2 \int \frac {\frac {1}{4} b \left (4 a^2+3 b^2\right ) e^3+\frac {1}{2} a^3 e^3 \cot (c+d x)+\frac {1}{4} b \left (2 a^2+3 b^2\right ) e^3 \cot ^2(c+d x)}{\sqrt {e \cot (c+d x)} (a+b \cot (c+d x))} \, dx}{a^2 \left (a^2+b^2\right ) e^4}\\ &=\frac {2 a^2+3 b^2}{a^2 \left (a^2+b^2\right ) d e \sqrt {e \cot (c+d x)}}-\frac {b^2}{a \left (a^2+b^2\right ) d e \sqrt {e \cot (c+d x)} (a+b \cot (c+d x))}-\frac {2 \int \frac {a^3 b e^3+\frac {1}{2} a^2 \left (a^2-b^2\right ) e^3 \cot (c+d x)}{\sqrt {e \cot (c+d x)}} \, dx}{a^2 \left (a^2+b^2\right )^2 e^4}-\frac {\left (b^3 \left (7 a^2+3 b^2\right )\right ) \int \frac {1+\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}\\ &=\frac {2 a^2+3 b^2}{a^2 \left (a^2+b^2\right ) d e \sqrt {e \cot (c+d x)}}-\frac {b^2}{a \left (a^2+b^2\right ) d e \sqrt {e \cot (c+d x)} (a+b \cot (c+d x))}-\frac {4 \text {Subst}\left (\int \frac {-a^3 b e^4-\frac {1}{2} a^2 \left (a^2-b^2\right ) e^3 x^2}{e^2+x^4} \, dx,x,\sqrt {e \cot (c+d x)}\right )}{a^2 \left (a^2+b^2\right )^2 d e^4}-\frac {\left (b^3 \left (7 a^2+3 b^2\right )\right ) \text {Subst}\left (\int \frac {1}{\sqrt {-e x} (a-b x)} \, dx,x,-\cot (c+d x)\right )}{2 a^2 \left (a^2+b^2\right )^2 d e}\\ &=\frac {2 a^2+3 b^2}{a^2 \left (a^2+b^2\right ) d e \sqrt {e \cot (c+d x)}}-\frac {b^2}{a \left (a^2+b^2\right ) d e \sqrt {e \cot (c+d x)} (a+b \cot (c+d x))}+\frac {\left (b^3 \left (7 a^2+3 b^2\right )\right ) \text {Subst}\left (\int \frac {1}{a+\frac {b x^2}{e}} \, dx,x,\sqrt {e \cot (c+d x)}\right )}{a^2 \left (a^2+b^2\right )^2 d e^2}-\frac {\left (a^2-2 a b-b^2\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 )^2 d e}+\frac {\left (a^2+2 a b-b^2\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 )^2 d e}\\ &=\frac {b^{5/2} \left (7 a^2+3 b^2\right ) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {e \cot (c+d x)}}{\sqrt {a} \sqrt {e}}\right )}{a^{5/2} \left (a^2+b^2\right )^2 d e^{3/2}}+\frac {2 a^2+3 b^2}{a^2 \left (a^2+b^2\right ) d e \sqrt {e \cot (c+d x)}}-\frac {b^2}{a \left (a^2+b^2\right ) d e \sqrt {e \cot (c+d x)} (a+b \cot (c+d x))}+\frac {\left (a^2-2 a b-b^2\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 )^2 d e^{3/2}}+\frac {\left (a^2-2 a b-b^2\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 )^2 d e^{3/2}}+\frac {\left (a^2+2 a b-b^2\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 )^2 d e}+\frac {\left (a^2+2 a b-b^2\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 )^2 d e}\\ &=\frac {b^{5/2} \left (7 a^2+3 b^2\right ) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {e \cot (c+d x)}}{\sqrt {a} \sqrt {e}}\right )}{a^{5/2} \left (a^2+b^2\right )^2 d e^{3/2}}+\frac {2 a^2+3 b^2}{a^2 \left (a^2+b^2\right ) d e \sqrt {e \cot (c+d x)}}-\frac {b^2}{a \left (a^2+b^2\right ) d e \sqrt {e \cot (c+d x)} (a+b \cot (c+d x))}+\frac {\left (a^2-2 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 )^2 d e^{3/2}}-\frac {\left (a^2-2 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 )^2 d e^{3/2}}+\frac {\left (a^2+2 a b-b^2\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 )^2 d e^{3/2}}-\frac {\left (a^2+2 a b-b^2\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 )^2 d e^{3/2}}\\ &=\frac {b^{5/2} \left (7 a^2+3 b^2\right ) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {e \cot (c+d x)}}{\sqrt {a} \sqrt {e}}\right )}{a^{5/2} \left (a^2+b^2\right )^2 d e^{3/2}}-\frac {\left (a^2+2 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 )^2 d e^{3/2}}+\frac {\left (a^2+2 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 )^2 d e^{3/2}}+\frac {2 a^2+3 b^2}{a^2 \left (a^2+b^2\right ) d e \sqrt {e \cot (c+d x)}}-\frac {b^2}{a \left (a^2+b^2\right ) d e \sqrt {e \cot (c+d x)} (a+b \cot (c+d x))}+\frac {\left (a^2-2 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 )^2 d e^{3/2}}-\frac {\left (a^2-2 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 )^2 d e^{3/2}}\\ \end {align*}

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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
time = 0.62, size = 244, normalized size = 0.56 \begin {gather*} \frac {8 a^2 b^2 \, _2F_1\left (-\frac {1}{2},1;\frac {1}{2};-\frac {b \cot (c+d x)}{a}\right )+4 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 )+a^2 \left (4 \left (a^2-b^2\right ) \, _2F_1\left (-\frac {1}{4},1;\frac {3}{4};-\cot ^2(c+d x)\right )+\sqrt {2} a b \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 )\right )}{2 a^2 \left (a^2+b^2\right )^2 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])^2),x]

[Out]

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

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Maple [A]
time = 0.63, size = 414, normalized size = 0.95

method	result	size
derivativedivides	\(-\frac {2 e^{3} \left (-\frac {b^{3} \left (\frac {\left (\frac {a^{2}}{2}+\frac {b^{2}}{2}\right ) \sqrt {e \cot \left (d x +c \right )}}{e \cot \left (d x +c \right ) b +a e}+\frac {\left (7 a^{2}+3 b^{2}\right ) \arctan \left (\frac {b \sqrt {e \cot \left (d x +c \right )}}{\sqrt {a e b}}\right )}{2 \sqrt {a e b}}\right )}{a^{2} e^{4} \left (a^{2}+b^{2}\right )^{2}}+\frac {-\frac {a b \left (e^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{4 e}+\frac {\left (-a^{2}+b^{2}\right ) \sqrt {2}\, \left (\ln \left (\frac {e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 \left (e^{2}\right )^{\frac {1}{4}}}}{\left (a^{2}+b^{2}\right )^{2} e^{4}}-\frac {1}{a^{2} e^{4} \sqrt {e \cot \left (d x +c \right )}}\right )}{d}\)	\(414\)
default	\(-\frac {2 e^{3} \left (-\frac {b^{3} \left (\frac {\left (\frac {a^{2}}{2}+\frac {b^{2}}{2}\right ) \sqrt {e \cot \left (d x +c \right )}}{e \cot \left (d x +c \right ) b +a e}+\frac {\left (7 a^{2}+3 b^{2}\right ) \arctan \left (\frac {b \sqrt {e \cot \left (d x +c \right )}}{\sqrt {a e b}}\right )}{2 \sqrt {a e b}}\right )}{a^{2} e^{4} \left (a^{2}+b^{2}\right )^{2}}+\frac {-\frac {a b \left (e^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{4 e}+\frac {\left (-a^{2}+b^{2}\right ) \sqrt {2}\, \left (\ln \left (\frac {e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 \left (e^{2}\right )^{\frac {1}{4}}}}{\left (a^{2}+b^{2}\right )^{2} e^{4}}-\frac {1}{a^{2} e^{4} \sqrt {e \cot \left (d x +c \right )}}\right )}{d}\)	\(414\)

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))^2,x,method=_RETURNVERBOSE)

[Out]

-2/d*e^3*(-b^3/a^2/e^4/(a^2+b^2)^2*((1/2*a^2+1/2*b^2)*(e*cot(d*x+c))^(1/2)/(e*cot(d*x+c)*b+a*e)+1/2*(7*a^2+3*b
^2)/(a*e*b)^(1/2)*arctan(b*(e*cot(d*x+c))^(1/2)/(a*e*b)^(1/2)))+1/(a^2+b^2)^2/e^4*(-1/4*a/e*b*(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))+1/8*(-a^2+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^2/e^4/(e
*cot(d*x+c))^(1/2))

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Maxima [A]
time = 0.51, size = 323, normalized size = 0.74 \begin {gather*} \frac {{\left (\frac {4 \, {\left (7 \, a^{2} b^{3} + 3 \, b^{5}\right )} \arctan \left (\frac {b}{\sqrt {a b} \sqrt {\tan \left (d x + c\right )}}\right )}{{\left (a^{6} + 2 \, a^{4} b^{2} + a^{2} b^{4}\right )} \sqrt {a b}} + \frac {2 \, \sqrt {2} {\left (a^{2} + 2 \, a b - b^{2}\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^{2} + 2 \, a b - b^{2}\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^{2} - 2 \, a b - b^{2}\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^{2} - 2 \, a b - b^{2}\right )} \log \left (-\frac {\sqrt {2}}{\sqrt {\tan \left (d x + c\right )}} + \frac {1}{\tan \left (d x + c\right )} + 1\right )}{a^{4} + 2 \, a^{2} b^{2} + b^{4}} + \frac {4 \, {\left (2 \, a^{3} + 2 \, a b^{2} + \frac {2 \, a^{2} b + 3 \, b^{3}}{\tan \left (d x + c\right )}\right )}}{\frac {a^{5} + a^{3} b^{2}}{\sqrt {\tan \left (d x + c\right )}} + \frac {a^{4} b + a^{2} b^{3}}{\tan \left (d x + c\right )^{\frac {3}{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))^2,x, algorithm="maxima")

[Out]

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

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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))^2,x, algorithm="fricas")

[Out]

Timed out

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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 )^{2}}\, 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))**2,x)

[Out]

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

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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))^2,x, algorithm="giac")

[Out]

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

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Mupad [B]
time = 4.34, size = 2500, normalized size = 5.72 \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))^2),x)

[Out]

(2/a + (b*cot(c + d*x)*(2*a^2 + 3*b^2))/(a^2*(a^2 + b^2)))/(b*d*(e*cot(c + d*x))^(3/2) + a*d*e*(e*cot(c + d*x)
)^(1/2)) - atan((((e*cot(c + d*x))^(1/2)*(144*a^14*b^23*d^5*e^13 + 1248*a^16*b^21*d^5*e^13 + 4224*a^18*b^19*d^
5*e^13 + 6720*a^20*b^17*d^5*e^13 + 3872*a^22*b^15*d^5*e^13 - 2816*a^24*b^13*d^5*e^13 - 5632*a^26*b^11*d^5*e^13
- 3136*a^28*b^9*d^5*e^13 - 560*a^30*b^7*d^5*e^13 + 32*a^32*b^5*d^5*e^13) + (1i/(4*(a^4*d^2*e^3 + b^4*d^2*e^3
+ a*b^3*d^2*e^3*4i - a^3*b*d^2*e^3*4i - 6*a^2*b^2*d^2*e^3)))^(1/2)*(26496*a^25*b^14*d^6*e^15 - 1152*a^15*b^24*
d^6*e^15 - 8448*a^17*b^22*d^6*e^15 - 23776*a^19*b^20*d^6*e^15 - 29664*a^21*b^18*d^6*e^15 - 6528*a^23*b^16*d^6*
e^15 - ((e*cot(c + d*x))^(1/2)*(1152*a^15*b^26*d^7*e^16 + 13440*a^17*b^24*d^7*e^16 + 69056*a^19*b^22*d^7*e^16
+ 202752*a^21*b^20*d^7*e^16 + 372800*a^23*b^18*d^7*e^16 + 443136*a^25*b^16*d^7*e^16 + 337792*a^27*b^14*d^7*e^1
6 + 156160*a^29*b^12*d^7*e^16 + 37632*a^31*b^10*d^7*e^16 + 3200*a^33*b^8*d^7*e^16 + 704*a^35*b^6*d^7*e^16 + 51
2*a^37*b^4*d^7*e^16 + 64*a^39*b^2*d^7*e^16) + (1i/(4*(a^4*d^2*e^3 + b^4*d^2*e^3 + a*b^3*d^2*e^3*4i - a^3*b*d^2
*e^3*4i - 6*a^2*b^2*d^2*e^3)))^(1/2)*(768*a^16*b^27*d^8*e^18 - (e*cot(c + d*x))^(1/2)*(1i/(4*(a^4*d^2*e^3 + b^
4*d^2*e^3 + a*b^3*d^2*e^3*4i - a^3*b*d^2*e^3*4i - 6*a^2*b^2*d^2*e^3)))^(1/2)*(512*a^18*b^27*d^9*e^19 + 5120*a^
20*b^25*d^9*e^19 + 22528*a^22*b^23*d^9*e^19 + 56320*a^24*b^21*d^9*e^19 + 84480*a^26*b^19*d^9*e^19 + 67584*a^28
*b^17*d^9*e^19 - 67584*a^32*b^13*d^9*e^19 - 84480*a^34*b^11*d^9*e^19 - 56320*a^36*b^9*d^9*e^19 - 22528*a^38*b^
7*d^9*e^19 - 5120*a^40*b^5*d^9*e^19 - 512*a^42*b^3*d^9*e^19) + 8704*a^18*b^25*d^8*e^18 + 44288*a^20*b^23*d^8*e
^18 + 133120*a^22*b^21*d^8*e^18 + 261120*a^24*b^19*d^8*e^18 + 347136*a^26*b^17*d^8*e^18 + 311808*a^28*b^15*d^8
*e^18 + 178176*a^30*b^13*d^8*e^18 + 49920*a^32*b^11*d^8*e^18 - 7680*a^34*b^9*d^8*e^18 - 12032*a^36*b^7*d^8*e^1
8 - 4096*a^38*b^5*d^8*e^18 - 512*a^40*b^3*d^8*e^18))*(1i/(4*(a^4*d^2*e^3 + b^4*d^2*e^3 + a*b^3*d^2*e^3*4i - a^
3*b*d^2*e^3*4i - 6*a^2*b^2*d^2*e^3)))^(1/2) + 33984*a^27*b^12*d^6*e^15 + 18624*a^29*b^10*d^6*e^15 + 5376*a^31*
b^8*d^6*e^15 + 1152*a^33*b^6*d^6*e^15 + 288*a^35*b^4*d^6*e^15 + 32*a^37*b^2*d^6*e^15))*(1i/(4*(a^4*d^2*e^3 + b
^4*d^2*e^3 + a*b^3*d^2*e^3*4i - a^3*b*d^2*e^3*4i - 6*a^2*b^2*d^2*e^3)))^(1/2)*1i + ((e*cot(c + d*x))^(1/2)*(14
4*a^14*b^23*d^5*e^13 + 1248*a^16*b^21*d^5*e^13 + 4224*a^18*b^19*d^5*e^13 + 6720*a^20*b^17*d^5*e^13 + 3872*a^22
*b^15*d^5*e^13 - 2816*a^24*b^13*d^5*e^13 - 5632*a^26*b^11*d^5*e^13 - 3136*a^28*b^9*d^5*e^13 - 560*a^30*b^7*d^5
*e^13 + 32*a^32*b^5*d^5*e^13) - (1i/(4*(a^4*d^2*e^3 + b^4*d^2*e^3 + a*b^3*d^2*e^3*4i - a^3*b*d^2*e^3*4i - 6*a^
2*b^2*d^2*e^3)))^(1/2)*(((e*cot(c + d*x))^(1/2)*(1152*a^15*b^26*d^7*e^16 + 13440*a^17*b^24*d^7*e^16 + 69056*a^
19*b^22*d^7*e^16 + 202752*a^21*b^20*d^7*e^16 + 372800*a^23*b^18*d^7*e^16 + 443136*a^25*b^16*d^7*e^16 + 337792*
a^27*b^14*d^7*e^16 + 156160*a^29*b^12*d^7*e^16 + 37632*a^31*b^10*d^7*e^16 + 3200*a^33*b^8*d^7*e^16 + 704*a^35*
b^6*d^7*e^16 + 512*a^37*b^4*d^7*e^16 + 64*a^39*b^2*d^7*e^16) - (1i/(4*(a^4*d^2*e^3 + b^4*d^2*e^3 + a*b^3*d^2*e
^3*4i - a^3*b*d^2*e^3*4i - 6*a^2*b^2*d^2*e^3)))^(1/2)*((e*cot(c + d*x))^(1/2)*(1i/(4*(a^4*d^2*e^3 + b^4*d^2*e^
3 + a*b^3*d^2*e^3*4i - a^3*b*d^2*e^3*4i - 6*a^2*b^2*d^2*e^3)))^(1/2)*(512*a^18*b^27*d^9*e^19 + 5120*a^20*b^25*
d^9*e^19 + 22528*a^22*b^23*d^9*e^19 + 56320*a^24*b^21*d^9*e^19 + 84480*a^26*b^19*d^9*e^19 + 67584*a^28*b^17*d^
9*e^19 - 67584*a^32*b^13*d^9*e^19 - 84480*a^34*b^11*d^9*e^19 - 56320*a^36*b^9*d^9*e^19 - 22528*a^38*b^7*d^9*e^
19 - 5120*a^40*b^5*d^9*e^19 - 512*a^42*b^3*d^9*e^19) + 768*a^16*b^27*d^8*e^18 + 8704*a^18*b^25*d^8*e^18 + 4428
8*a^20*b^23*d^8*e^18 + 133120*a^22*b^21*d^8*e^18 + 261120*a^24*b^19*d^8*e^18 + 347136*a^26*b^17*d^8*e^18 + 311
808*a^28*b^15*d^8*e^18 + 178176*a^30*b^13*d^8*e^18 + 49920*a^32*b^11*d^8*e^18 - 7680*a^34*b^9*d^8*e^18 - 12032
*a^36*b^7*d^8*e^18 - 4096*a^38*b^5*d^8*e^18 - 512*a^40*b^3*d^8*e^18))*(1i/(4*(a^4*d^2*e^3 + b^4*d^2*e^3 + a*b^
3*d^2*e^3*4i - a^3*b*d^2*e^3*4i - 6*a^2*b^2*d^2*e^3)))^(1/2) - 1152*a^15*b^24*d^6*e^15 - 8448*a^17*b^22*d^6*e^
15 - 23776*a^19*b^20*d^6*e^15 - 29664*a^21*b^18*d^6*e^15 - 6528*a^23*b^16*d^6*e^15 + 26496*a^25*b^14*d^6*e^15
+ 33984*a^27*b^12*d^6*e^15 + 18624*a^29*b^10*d^6*e^15 + 5376*a^31*b^8*d^6*e^15 + 1152*a^33*b^6*d^6*e^15 + 288*
a^35*b^4*d^6*e^15 + 32*a^37*b^2*d^6*e^15))*(1i/(4*(a^4*d^2*e^3 + b^4*d^2*e^3 + a*b^3*d^2*e^3*4i - a^3*b*d^2*e^
3*4i - 6*a^2*b^2*d^2*e^3)))^(1/2)*1i)/(((e*cot(c + d*x))^(1/2)*(144*a^14*b^23*d^5*e^13 + 1248*a^16*b^21*d^5*e^
13 + 4224*a^18*b^19*d^5*e^13 + 6720*a^20*b^17*d^5*e^13 + 3872*a^22*b^15*d^5*e^13 - 2816*a^24*b^13*d^5*e^13 - 5
632*a^26*b^11*d^5*e^13 - 3136*a^28*b^9*d^5*e^13 - 560*a^30*b^7*d^5*e^13 + 32*a^32*b^5*d^5*e^13) + (1i/(4*(a^4*
d^2*e^3 + b^4*d^2*e^3 + a*b^3*d^2*e^3*4i - a^3*b*d^2*e^3*4i - 6*a^2*b^2*d^2*e^3)))^(1/2)*(26496*a^25*b^14*d^6*
e^15 - 1152*a^15*b^24*d^6*e^15 - 8448*a^17*b^22*d^6*e^15 - 23776*a^19*b^20*d^6*e^15 - 29664*a^21*b^18*d^6*e^15
- 6528*a^23*b^16*d^6*e^15 - ((e*cot(c + d*x))^...

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