3.2.10 \(\int \frac {a+b \csc ^{-1}(c x)}{x^2 (d+e x^2)^2} \, dx\) [110]

Optimal. Leaf size=806 \[ -\frac {b c \sqrt {1-\frac {1}{c^2 x^2}}}{d^2}-\frac {a}{d^2 x}-\frac {b \csc ^{-1}(c x)}{d^2 x}+\frac {e \left (a+b \csc ^{-1}(c x)\right )}{4 d^2 \left (\sqrt {-d} \sqrt {e}-\frac {d}{x}\right )}-\frac {e \left (a+b \csc ^{-1}(c x)\right )}{4 d^2 \left (\sqrt {-d} \sqrt {e}+\frac {d}{x}\right )}-\frac {b e \tanh ^{-1}\left (\frac {c^2 d-\frac {\sqrt {-d} \sqrt {e}}{x}}{c \sqrt {d} \sqrt {c^2 d+e} \sqrt {1-\frac {1}{c^2 x^2}}}\right )}{4 d^{5/2} \sqrt {c^2 d+e}}-\frac {b e \tanh ^{-1}\left (\frac {c^2 d+\frac {\sqrt {-d} \sqrt {e}}{x}}{c \sqrt {d} \sqrt {c^2 d+e} \sqrt {1-\frac {1}{c^2 x^2}}}\right )}{4 d^{5/2} \sqrt {c^2 d+e}}+\frac {3 \sqrt {e} \left (a+b \csc ^{-1}(c x)\right ) \log \left (1-\frac {i c \sqrt {-d} e^{i \csc ^{-1}(c x)}}{\sqrt {e}-\sqrt {c^2 d+e}}\right )}{4 (-d)^{5/2}}-\frac {3 \sqrt {e} \left (a+b \csc ^{-1}(c x)\right ) \log \left (1+\frac {i c \sqrt {-d} e^{i \csc ^{-1}(c x)}}{\sqrt {e}-\sqrt {c^2 d+e}}\right )}{4 (-d)^{5/2}}+\frac {3 \sqrt {e} \left (a+b \csc ^{-1}(c x)\right ) \log \left (1-\frac {i c \sqrt {-d} e^{i \csc ^{-1}(c x)}}{\sqrt {e}+\sqrt {c^2 d+e}}\right )}{4 (-d)^{5/2}}-\frac {3 \sqrt {e} \left (a+b \csc ^{-1}(c x)\right ) \log \left (1+\frac {i c \sqrt {-d} e^{i \csc ^{-1}(c x)}}{\sqrt {e}+\sqrt {c^2 d+e}}\right )}{4 (-d)^{5/2}}+\frac {3 i b \sqrt {e} \text {PolyLog}\left (2,-\frac {i c \sqrt {-d} e^{i \csc ^{-1}(c x)}}{\sqrt {e}-\sqrt {c^2 d+e}}\right )}{4 (-d)^{5/2}}-\frac {3 i b \sqrt {e} \text {PolyLog}\left (2,\frac {i c \sqrt {-d} e^{i \csc ^{-1}(c x)}}{\sqrt {e}-\sqrt {c^2 d+e}}\right )}{4 (-d)^{5/2}}+\frac {3 i b \sqrt {e} \text {PolyLog}\left (2,-\frac {i c \sqrt {-d} e^{i \csc ^{-1}(c x)}}{\sqrt {e}+\sqrt {c^2 d+e}}\right )}{4 (-d)^{5/2}}-\frac {3 i b \sqrt {e} \text {PolyLog}\left (2,\frac {i c \sqrt {-d} e^{i \csc ^{-1}(c x)}}{\sqrt {e}+\sqrt {c^2 d+e}}\right )}{4 (-d)^{5/2}} \]

[Out]

-a/d^2/x-b*arccsc(c*x)/d^2/x+3/4*(a+b*arccsc(c*x))*ln(1-I*c*(I/c/x+(1-1/c^2/x^2)^(1/2))*(-d)^(1/2)/(e^(1/2)-(c
^2*d+e)^(1/2)))*e^(1/2)/(-d)^(5/2)-3/4*(a+b*arccsc(c*x))*ln(1+I*c*(I/c/x+(1-1/c^2/x^2)^(1/2))*(-d)^(1/2)/(e^(1
/2)-(c^2*d+e)^(1/2)))*e^(1/2)/(-d)^(5/2)+3/4*(a+b*arccsc(c*x))*ln(1-I*c*(I/c/x+(1-1/c^2/x^2)^(1/2))*(-d)^(1/2)
/(e^(1/2)+(c^2*d+e)^(1/2)))*e^(1/2)/(-d)^(5/2)-3/4*(a+b*arccsc(c*x))*ln(1+I*c*(I/c/x+(1-1/c^2/x^2)^(1/2))*(-d)
^(1/2)/(e^(1/2)+(c^2*d+e)^(1/2)))*e^(1/2)/(-d)^(5/2)+3/4*I*b*polylog(2,-I*c*(I/c/x+(1-1/c^2/x^2)^(1/2))*(-d)^(
1/2)/(e^(1/2)-(c^2*d+e)^(1/2)))*e^(1/2)/(-d)^(5/2)-3/4*I*b*polylog(2,I*c*(I/c/x+(1-1/c^2/x^2)^(1/2))*(-d)^(1/2
)/(e^(1/2)-(c^2*d+e)^(1/2)))*e^(1/2)/(-d)^(5/2)+3/4*I*b*polylog(2,-I*c*(I/c/x+(1-1/c^2/x^2)^(1/2))*(-d)^(1/2)/
(e^(1/2)+(c^2*d+e)^(1/2)))*e^(1/2)/(-d)^(5/2)-3/4*I*b*polylog(2,I*c*(I/c/x+(1-1/c^2/x^2)^(1/2))*(-d)^(1/2)/(e^
(1/2)+(c^2*d+e)^(1/2)))*e^(1/2)/(-d)^(5/2)+1/4*e*(a+b*arccsc(c*x))/d^2/(-d/x+(-d)^(1/2)*e^(1/2))-1/4*e*(a+b*ar
ccsc(c*x))/d^2/(d/x+(-d)^(1/2)*e^(1/2))-1/4*b*e*arctanh((c^2*d-(-d)^(1/2)*e^(1/2)/x)/c/d^(1/2)/(c^2*d+e)^(1/2)
/(1-1/c^2/x^2)^(1/2))/d^(5/2)/(c^2*d+e)^(1/2)-1/4*b*e*arctanh((c^2*d+(-d)^(1/2)*e^(1/2)/x)/c/d^(1/2)/(c^2*d+e)
^(1/2)/(1-1/c^2/x^2)^(1/2))/d^(5/2)/(c^2*d+e)^(1/2)-b*c*(1-1/c^2/x^2)^(1/2)/d^2

________________________________________________________________________________________

Rubi [A]
time = 2.21, antiderivative size = 806, normalized size of antiderivative = 1.00, number of steps used = 50, number of rules used = 13, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.619, Rules used = {5349, 4817, 4715, 267, 4757, 4827, 739, 212, 4825, 4615, 2221, 2317, 2438} \begin {gather*} -\frac {a}{d^2 x}-\frac {b \csc ^{-1}(c x)}{d^2 x}+\frac {e \left (a+b \csc ^{-1}(c x)\right )}{4 d^2 \left (\sqrt {-d} \sqrt {e}-\frac {d}{x}\right )}-\frac {e \left (a+b \csc ^{-1}(c x)\right )}{4 d^2 \left (\frac {d}{x}+\sqrt {-d} \sqrt {e}\right )}-\frac {b e \tanh ^{-1}\left (\frac {c^2 d-\frac {\sqrt {-d} \sqrt {e}}{x}}{c \sqrt {d} \sqrt {d c^2+e} \sqrt {1-\frac {1}{c^2 x^2}}}\right )}{4 d^{5/2} \sqrt {d c^2+e}}-\frac {b e \tanh ^{-1}\left (\frac {d c^2+\frac {\sqrt {-d} \sqrt {e}}{x}}{c \sqrt {d} \sqrt {d c^2+e} \sqrt {1-\frac {1}{c^2 x^2}}}\right )}{4 d^{5/2} \sqrt {d c^2+e}}+\frac {3 \sqrt {e} \left (a+b \csc ^{-1}(c x)\right ) \log \left (1-\frac {i c \sqrt {-d} e^{i \csc ^{-1}(c x)}}{\sqrt {e}-\sqrt {d c^2+e}}\right )}{4 (-d)^{5/2}}-\frac {3 \sqrt {e} \left (a+b \csc ^{-1}(c x)\right ) \log \left (\frac {i \sqrt {-d} e^{i \csc ^{-1}(c x)} c}{\sqrt {e}-\sqrt {d c^2+e}}+1\right )}{4 (-d)^{5/2}}+\frac {3 \sqrt {e} \left (a+b \csc ^{-1}(c x)\right ) \log \left (1-\frac {i c \sqrt {-d} e^{i \csc ^{-1}(c x)}}{\sqrt {e}+\sqrt {d c^2+e}}\right )}{4 (-d)^{5/2}}-\frac {3 \sqrt {e} \left (a+b \csc ^{-1}(c x)\right ) \log \left (\frac {i \sqrt {-d} e^{i \csc ^{-1}(c x)} c}{\sqrt {e}+\sqrt {d c^2+e}}+1\right )}{4 (-d)^{5/2}}+\frac {3 i b \sqrt {e} \text {Li}_2\left (-\frac {i c \sqrt {-d} e^{i \csc ^{-1}(c x)}}{\sqrt {e}-\sqrt {d c^2+e}}\right )}{4 (-d)^{5/2}}-\frac {3 i b \sqrt {e} \text {Li}_2\left (\frac {i c \sqrt {-d} e^{i \csc ^{-1}(c x)}}{\sqrt {e}-\sqrt {d c^2+e}}\right )}{4 (-d)^{5/2}}+\frac {3 i b \sqrt {e} \text {Li}_2\left (-\frac {i c \sqrt {-d} e^{i \csc ^{-1}(c x)}}{\sqrt {e}+\sqrt {d c^2+e}}\right )}{4 (-d)^{5/2}}-\frac {3 i b \sqrt {e} \text {Li}_2\left (\frac {i c \sqrt {-d} e^{i \csc ^{-1}(c x)}}{\sqrt {e}+\sqrt {d c^2+e}}\right )}{4 (-d)^{5/2}}-\frac {b c \sqrt {1-\frac {1}{c^2 x^2}}}{d^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-((b*c*Sqrt[1 - 1/(c^2*x^2)])/d^2) - a/(d^2*x) - (b*ArcCsc[c*x])/(d^2*x) + (e*(a + b*ArcCsc[c*x]))/(4*d^2*(Sqr
t[-d]*Sqrt[e] - d/x)) - (e*(a + b*ArcCsc[c*x]))/(4*d^2*(Sqrt[-d]*Sqrt[e] + d/x)) - (b*e*ArcTanh[(c^2*d - (Sqrt
[-d]*Sqrt[e])/x)/(c*Sqrt[d]*Sqrt[c^2*d + e]*Sqrt[1 - 1/(c^2*x^2)])])/(4*d^(5/2)*Sqrt[c^2*d + e]) - (b*e*ArcTan
h[(c^2*d + (Sqrt[-d]*Sqrt[e])/x)/(c*Sqrt[d]*Sqrt[c^2*d + e]*Sqrt[1 - 1/(c^2*x^2)])])/(4*d^(5/2)*Sqrt[c^2*d + e
]) + (3*Sqrt[e]*(a + b*ArcCsc[c*x])*Log[1 - (I*c*Sqrt[-d]*E^(I*ArcCsc[c*x]))/(Sqrt[e] - Sqrt[c^2*d + e])])/(4*
(-d)^(5/2)) - (3*Sqrt[e]*(a + b*ArcCsc[c*x])*Log[1 + (I*c*Sqrt[-d]*E^(I*ArcCsc[c*x]))/(Sqrt[e] - Sqrt[c^2*d +
e])])/(4*(-d)^(5/2)) + (3*Sqrt[e]*(a + b*ArcCsc[c*x])*Log[1 - (I*c*Sqrt[-d]*E^(I*ArcCsc[c*x]))/(Sqrt[e] + Sqrt
[c^2*d + e])])/(4*(-d)^(5/2)) - (3*Sqrt[e]*(a + b*ArcCsc[c*x])*Log[1 + (I*c*Sqrt[-d]*E^(I*ArcCsc[c*x]))/(Sqrt[
e] + Sqrt[c^2*d + e])])/(4*(-d)^(5/2)) + (((3*I)/4)*b*Sqrt[e]*PolyLog[2, ((-I)*c*Sqrt[-d]*E^(I*ArcCsc[c*x]))/(
Sqrt[e] - Sqrt[c^2*d + e])])/(-d)^(5/2) - (((3*I)/4)*b*Sqrt[e]*PolyLog[2, (I*c*Sqrt[-d]*E^(I*ArcCsc[c*x]))/(Sq
rt[e] - Sqrt[c^2*d + e])])/(-d)^(5/2) + (((3*I)/4)*b*Sqrt[e]*PolyLog[2, ((-I)*c*Sqrt[-d]*E^(I*ArcCsc[c*x]))/(S
qrt[e] + Sqrt[c^2*d + e])])/(-d)^(5/2) - (((3*I)/4)*b*Sqrt[e]*PolyLog[2, (I*c*Sqrt[-d]*E^(I*ArcCsc[c*x]))/(Sqr
t[e] + Sqrt[c^2*d + e])])/(-d)^(5/2)

Rule 212

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

Rule 267

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(a + b*x^n)^(p + 1)/(b*n*(p + 1)), x] /; FreeQ
[{a, b, m, n, p}, x] && EqQ[m, n - 1] && NeQ[p, -1]

Rule 739

Int[1/(((d_) + (e_.)*(x_))*Sqrt[(a_) + (c_.)*(x_)^2]), x_Symbol] :> -Subst[Int[1/(c*d^2 + a*e^2 - x^2), x], x,
(a*e - c*d*x)/Sqrt[a + c*x^2]] /; FreeQ[{a, c, d, e}, x]

Rule 2221

Int[(((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.))/((a_) + (b_.)*((F_)^((g_.)*((e_.) +
(f_.)*(x_))))^(n_.)), x_Symbol] :> Simp[((c + d*x)^m/(b*f*g*n*Log[F]))*Log[1 + b*((F^(g*(e + f*x)))^n/a)], x]
- Dist[d*(m/(b*f*g*n*Log[F])), Int[(c + d*x)^(m - 1)*Log[1 + b*((F^(g*(e + f*x)))^n/a)], x], x] /; FreeQ[{F,
a, b, c, d, e, f, g, n}, x] && IGtQ[m, 0]

Rule 2317

Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] :> Dist[1/(d*e*n*Log[F]), Subst[Int
[Log[a + b*x]/x, x], x, (F^(e*(c + d*x)))^n], x] /; FreeQ[{F, a, b, c, d, e, n}, x] && GtQ[a, 0]

Rule 2438

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> Simp[-PolyLog[2, (-c)*e*x^n]/n, x] /; FreeQ[{c, d,
e, n}, x] && EqQ[c*d, 1]

Rule 4615

Int[(Cos[(c_.) + (d_.)*(x_)]*((e_.) + (f_.)*(x_))^(m_.))/((a_) + (b_.)*Sin[(c_.) + (d_.)*(x_)]), x_Symbol] :>
Simp[(-I)*((e + f*x)^(m + 1)/(b*f*(m + 1))), x] + (Int[(e + f*x)^m*(E^(I*(c + d*x))/(a - Rt[a^2 - b^2, 2] - I*
b*E^(I*(c + d*x)))), x] + Int[(e + f*x)^m*(E^(I*(c + d*x))/(a + Rt[a^2 - b^2, 2] - I*b*E^(I*(c + d*x)))), x])
/; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && PosQ[a^2 - b^2]

Rule 4715

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.), x_Symbol] :> Simp[x*(a + b*ArcSin[c*x])^n, x] - Dist[b*c*n, Int[
x*((a + b*ArcSin[c*x])^(n - 1)/Sqrt[1 - c^2*x^2]), x], x] /; FreeQ[{a, b, c}, x] && GtQ[n, 0]

Rule 4757

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(a
+ b*ArcSin[c*x])^n, (d + e*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, n}, x] && NeQ[c^2*d + e, 0] && IntegerQ[p]
&& (GtQ[p, 0] || IGtQ[n, 0])

Rule 4817

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> Int[
ExpandIntegrand[(a + b*ArcSin[c*x])^n, (f*x)^m*(d + e*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[c^
2*d + e, 0] && IGtQ[n, 0] && IntegerQ[p] && IntegerQ[m]

Rule 4825

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)/((d_) + (e_.)*(x_)), x_Symbol] :> Subst[Int[(a + b*x)^n*(Cos[x]/(
c*d + e*Sin[x])), x], x, ArcSin[c*x]] /; FreeQ[{a, b, c, d, e}, x] && IGtQ[n, 0]

Rule 4827

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((d_) + (e_.)*(x_))^(m_.), x_Symbol] :> Simp[(d + e*x)^(m + 1)*((
a + b*ArcSin[c*x])^n/(e*(m + 1))), x] - Dist[b*c*(n/(e*(m + 1))), Int[(d + e*x)^(m + 1)*((a + b*ArcSin[c*x])^(
n - 1)/Sqrt[1 - c^2*x^2]), x], x] /; FreeQ[{a, b, c, d, e, m}, x] && IGtQ[n, 0] && NeQ[m, -1]

Rule 5349

Int[((a_.) + ArcCsc[(c_.)*(x_)]*(b_.))^(n_.)*(x_)^(m_.)*((d_.) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> -Subst[Int[
(e + d*x^2)^p*((a + b*ArcSin[x/c])^n/x^(m + 2*(p + 1))), x], x, 1/x] /; FreeQ[{a, b, c, d, e, n}, x] && IGtQ[n
, 0] && IntegerQ[m] && IntegerQ[p]

Rubi steps

\begin {align*} \int \frac {a+b \csc ^{-1}(c x)}{x^2 \left (d+e x^2\right )^2} \, dx &=-\text {Subst}\left (\int \frac {x^4 \left (a+b \sin ^{-1}\left (\frac {x}{c}\right )\right )}{\left (e+d x^2\right )^2} \, dx,x,\frac {1}{x}\right )\\ &=-\text {Subst}\left (\int \left (\frac {a+b \sin ^{-1}\left (\frac {x}{c}\right )}{d^2}+\frac {e^2 \left (a+b \sin ^{-1}\left (\frac {x}{c}\right )\right )}{d^2 \left (e+d x^2\right )^2}-\frac {2 e \left (a+b \sin ^{-1}\left (\frac {x}{c}\right )\right )}{d^2 \left (e+d x^2\right )}\right ) \, dx,x,\frac {1}{x}\right )\\ &=-\frac {\text {Subst}\left (\int \left (a+b \sin ^{-1}\left (\frac {x}{c}\right )\right ) \, dx,x,\frac {1}{x}\right )}{d^2}+\frac {(2 e) \text {Subst}\left (\int \frac {a+b \sin ^{-1}\left (\frac {x}{c}\right )}{e+d x^2} \, dx,x,\frac {1}{x}\right )}{d^2}-\frac {e^2 \text {Subst}\left (\int \frac {a+b \sin ^{-1}\left (\frac {x}{c}\right )}{\left (e+d x^2\right )^2} \, dx,x,\frac {1}{x}\right )}{d^2}\\ &=-\frac {a}{d^2 x}-\frac {b \text {Subst}\left (\int \sin ^{-1}\left (\frac {x}{c}\right ) \, dx,x,\frac {1}{x}\right )}{d^2}+\frac {(2 e) \text {Subst}\left (\int \left (\frac {a+b \sin ^{-1}\left (\frac {x}{c}\right )}{2 \sqrt {e} \left (\sqrt {e}-\sqrt {-d} x\right )}+\frac {a+b \sin ^{-1}\left (\frac {x}{c}\right )}{2 \sqrt {e} \left (\sqrt {e}+\sqrt {-d} x\right )}\right ) \, dx,x,\frac {1}{x}\right )}{d^2}-\frac {e^2 \text {Subst}\left (\int \left (-\frac {d \left (a+b \sin ^{-1}\left (\frac {x}{c}\right )\right )}{4 e \left (\sqrt {-d} \sqrt {e}-d x\right )^2}-\frac {d \left (a+b \sin ^{-1}\left (\frac {x}{c}\right )\right )}{4 e \left (\sqrt {-d} \sqrt {e}+d x\right )^2}-\frac {d \left (a+b \sin ^{-1}\left (\frac {x}{c}\right )\right )}{2 e \left (-d e-d^2 x^2\right )}\right ) \, dx,x,\frac {1}{x}\right )}{d^2}\\ &=-\frac {a}{d^2 x}-\frac {b \csc ^{-1}(c x)}{d^2 x}+\frac {b \text {Subst}\left (\int \frac {x}{\sqrt {1-\frac {x^2}{c^2}}} \, dx,x,\frac {1}{x}\right )}{c d^2}+\frac {\sqrt {e} \text {Subst}\left (\int \frac {a+b \sin ^{-1}\left (\frac {x}{c}\right )}{\sqrt {e}-\sqrt {-d} x} \, dx,x,\frac {1}{x}\right )}{d^2}+\frac {\sqrt {e} \text {Subst}\left (\int \frac {a+b \sin ^{-1}\left (\frac {x}{c}\right )}{\sqrt {e}+\sqrt {-d} x} \, dx,x,\frac {1}{x}\right )}{d^2}+\frac {e \text {Subst}\left (\int \frac {a+b \sin ^{-1}\left (\frac {x}{c}\right )}{\left (\sqrt {-d} \sqrt {e}-d x\right )^2} \, dx,x,\frac {1}{x}\right )}{4 d}+\frac {e \text {Subst}\left (\int \frac {a+b \sin ^{-1}\left (\frac {x}{c}\right )}{\left (\sqrt {-d} \sqrt {e}+d x\right )^2} \, dx,x,\frac {1}{x}\right )}{4 d}+\frac {e \text {Subst}\left (\int \frac {a+b \sin ^{-1}\left (\frac {x}{c}\right )}{-d e-d^2 x^2} \, dx,x,\frac {1}{x}\right )}{2 d}\\ &=-\frac {b c \sqrt {1-\frac {1}{c^2 x^2}}}{d^2}-\frac {a}{d^2 x}-\frac {b \csc ^{-1}(c x)}{d^2 x}+\frac {e \left (a+b \csc ^{-1}(c x)\right )}{4 d^2 \left (\sqrt {-d} \sqrt {e}-\frac {d}{x}\right )}-\frac {e \left (a+b \csc ^{-1}(c x)\right )}{4 d^2 \left (\sqrt {-d} \sqrt {e}+\frac {d}{x}\right )}+\frac {\sqrt {e} \text {Subst}\left (\int \frac {(a+b x) \cos (x)}{\frac {\sqrt {e}}{c}-\sqrt {-d} \sin (x)} \, dx,x,\csc ^{-1}(c x)\right )}{d^2}+\frac {\sqrt {e} \text {Subst}\left (\int \frac {(a+b x) \cos (x)}{\frac {\sqrt {e}}{c}+\sqrt {-d} \sin (x)} \, dx,x,\csc ^{-1}(c x)\right )}{d^2}-\frac {(b e) \text {Subst}\left (\int \frac {1}{\left (\sqrt {-d} \sqrt {e}-d x\right ) \sqrt {1-\frac {x^2}{c^2}}} \, dx,x,\frac {1}{x}\right )}{4 c d^2}+\frac {(b e) \text {Subst}\left (\int \frac {1}{\left (\sqrt {-d} \sqrt {e}+d x\right ) \sqrt {1-\frac {x^2}{c^2}}} \, dx,x,\frac {1}{x}\right )}{4 c d^2}+\frac {e \text {Subst}\left (\int \left (-\frac {a+b \sin ^{-1}\left (\frac {x}{c}\right )}{2 d \sqrt {e} \left (\sqrt {e}-\sqrt {-d} x\right )}-\frac {a+b \sin ^{-1}\left (\frac {x}{c}\right )}{2 d \sqrt {e} \left (\sqrt {e}+\sqrt {-d} x\right )}\right ) \, dx,x,\frac {1}{x}\right )}{2 d}\\ &=-\frac {b c \sqrt {1-\frac {1}{c^2 x^2}}}{d^2}-\frac {a}{d^2 x}-\frac {b \csc ^{-1}(c x)}{d^2 x}+\frac {e \left (a+b \csc ^{-1}(c x)\right )}{4 d^2 \left (\sqrt {-d} \sqrt {e}-\frac {d}{x}\right )}-\frac {e \left (a+b \csc ^{-1}(c x)\right )}{4 d^2 \left (\sqrt {-d} \sqrt {e}+\frac {d}{x}\right )}-\frac {\sqrt {e} \text {Subst}\left (\int \frac {a+b \sin ^{-1}\left (\frac {x}{c}\right )}{\sqrt {e}-\sqrt {-d} x} \, dx,x,\frac {1}{x}\right )}{4 d^2}-\frac {\sqrt {e} \text {Subst}\left (\int \frac {a+b \sin ^{-1}\left (\frac {x}{c}\right )}{\sqrt {e}+\sqrt {-d} x} \, dx,x,\frac {1}{x}\right )}{4 d^2}+\frac {\sqrt {e} \text {Subst}\left (\int \frac {e^{i x} (a+b x)}{\frac {\sqrt {e}}{c}-\frac {\sqrt {c^2 d+e}}{c}-i \sqrt {-d} e^{i x}} \, dx,x,\csc ^{-1}(c x)\right )}{d^2}+\frac {\sqrt {e} \text {Subst}\left (\int \frac {e^{i x} (a+b x)}{\frac {\sqrt {e}}{c}+\frac {\sqrt {c^2 d+e}}{c}-i \sqrt {-d} e^{i x}} \, dx,x,\csc ^{-1}(c x)\right )}{d^2}+\frac {\sqrt {e} \text {Subst}\left (\int \frac {e^{i x} (a+b x)}{\frac {\sqrt {e}}{c}-\frac {\sqrt {c^2 d+e}}{c}+i \sqrt {-d} e^{i x}} \, dx,x,\csc ^{-1}(c x)\right )}{d^2}+\frac {\sqrt {e} \text {Subst}\left (\int \frac {e^{i x} (a+b x)}{\frac {\sqrt {e}}{c}+\frac {\sqrt {c^2 d+e}}{c}+i \sqrt {-d} e^{i x}} \, dx,x,\csc ^{-1}(c x)\right )}{d^2}+\frac {(b e) \text {Subst}\left (\int \frac {1}{d^2+\frac {d e}{c^2}-x^2} \, dx,x,\frac {-d+\frac {\sqrt {-d} \sqrt {e}}{c^2 x}}{\sqrt {1-\frac {1}{c^2 x^2}}}\right )}{4 c d^2}-\frac {(b e) \text {Subst}\left (\int \frac {1}{d^2+\frac {d e}{c^2}-x^2} \, dx,x,\frac {d+\frac {\sqrt {-d} \sqrt {e}}{c^2 x}}{\sqrt {1-\frac {1}{c^2 x^2}}}\right )}{4 c d^2}\\ &=-\frac {b c \sqrt {1-\frac {1}{c^2 x^2}}}{d^2}-\frac {a}{d^2 x}-\frac {b \csc ^{-1}(c x)}{d^2 x}+\frac {e \left (a+b \csc ^{-1}(c x)\right )}{4 d^2 \left (\sqrt {-d} \sqrt {e}-\frac {d}{x}\right )}-\frac {e \left (a+b \csc ^{-1}(c x)\right )}{4 d^2 \left (\sqrt {-d} \sqrt {e}+\frac {d}{x}\right )}-\frac {b e \tanh ^{-1}\left (\frac {c^2 d-\frac {\sqrt {-d} \sqrt {e}}{x}}{c \sqrt {d} \sqrt {c^2 d+e} \sqrt {1-\frac {1}{c^2 x^2}}}\right )}{4 d^{5/2} \sqrt {c^2 d+e}}-\frac {b e \tanh ^{-1}\left (\frac {c^2 d+\frac {\sqrt {-d} \sqrt {e}}{x}}{c \sqrt {d} \sqrt {c^2 d+e} \sqrt {1-\frac {1}{c^2 x^2}}}\right )}{4 d^{5/2} \sqrt {c^2 d+e}}+\frac {\sqrt {e} \left (a+b \csc ^{-1}(c x)\right ) \log \left (1-\frac {i c \sqrt {-d} e^{i \csc ^{-1}(c x)}}{\sqrt {e}-\sqrt {c^2 d+e}}\right )}{(-d)^{5/2}}-\frac {\sqrt {e} \left (a+b \csc ^{-1}(c x)\right ) \log \left (1+\frac {i c \sqrt {-d} e^{i \csc ^{-1}(c x)}}{\sqrt {e}-\sqrt {c^2 d+e}}\right )}{(-d)^{5/2}}+\frac {\sqrt {e} \left (a+b \csc ^{-1}(c x)\right ) \log \left (1-\frac {i c \sqrt {-d} e^{i \csc ^{-1}(c x)}}{\sqrt {e}+\sqrt {c^2 d+e}}\right )}{(-d)^{5/2}}-\frac {\sqrt {e} \left (a+b \csc ^{-1}(c x)\right ) \log \left (1+\frac {i c \sqrt {-d} e^{i \csc ^{-1}(c x)}}{\sqrt {e}+\sqrt {c^2 d+e}}\right )}{(-d)^{5/2}}-\frac {\left (b \sqrt {e}\right ) \text {Subst}\left (\int \log \left (1-\frac {i \sqrt {-d} e^{i x}}{\frac {\sqrt {e}}{c}-\frac {\sqrt {c^2 d+e}}{c}}\right ) \, dx,x,\csc ^{-1}(c x)\right )}{(-d)^{5/2}}+\frac {\left (b \sqrt {e}\right ) \text {Subst}\left (\int \log \left (1+\frac {i \sqrt {-d} e^{i x}}{\frac {\sqrt {e}}{c}-\frac {\sqrt {c^2 d+e}}{c}}\right ) \, dx,x,\csc ^{-1}(c x)\right )}{(-d)^{5/2}}-\frac {\left (b \sqrt {e}\right ) \text {Subst}\left (\int \log \left (1-\frac {i \sqrt {-d} e^{i x}}{\frac {\sqrt {e}}{c}+\frac {\sqrt {c^2 d+e}}{c}}\right ) \, dx,x,\csc ^{-1}(c x)\right )}{(-d)^{5/2}}+\frac {\left (b \sqrt {e}\right ) \text {Subst}\left (\int \log \left (1+\frac {i \sqrt {-d} e^{i x}}{\frac {\sqrt {e}}{c}+\frac {\sqrt {c^2 d+e}}{c}}\right ) \, dx,x,\csc ^{-1}(c x)\right )}{(-d)^{5/2}}-\frac {\sqrt {e} \text {Subst}\left (\int \frac {(a+b x) \cos (x)}{\frac {\sqrt {e}}{c}-\sqrt {-d} \sin (x)} \, dx,x,\csc ^{-1}(c x)\right )}{4 d^2}-\frac {\sqrt {e} \text {Subst}\left (\int \frac {(a+b x) \cos (x)}{\frac {\sqrt {e}}{c}+\sqrt {-d} \sin (x)} \, dx,x,\csc ^{-1}(c x)\right )}{4 d^2}\\ &=-\frac {b c \sqrt {1-\frac {1}{c^2 x^2}}}{d^2}-\frac {a}{d^2 x}-\frac {b \csc ^{-1}(c x)}{d^2 x}+\frac {e \left (a+b \csc ^{-1}(c x)\right )}{4 d^2 \left (\sqrt {-d} \sqrt {e}-\frac {d}{x}\right )}-\frac {e \left (a+b \csc ^{-1}(c x)\right )}{4 d^2 \left (\sqrt {-d} \sqrt {e}+\frac {d}{x}\right )}-\frac {b e \tanh ^{-1}\left (\frac {c^2 d-\frac {\sqrt {-d} \sqrt {e}}{x}}{c \sqrt {d} \sqrt {c^2 d+e} \sqrt {1-\frac {1}{c^2 x^2}}}\right )}{4 d^{5/2} \sqrt {c^2 d+e}}-\frac {b e \tanh ^{-1}\left (\frac {c^2 d+\frac {\sqrt {-d} \sqrt {e}}{x}}{c \sqrt {d} \sqrt {c^2 d+e} \sqrt {1-\frac {1}{c^2 x^2}}}\right )}{4 d^{5/2} \sqrt {c^2 d+e}}+\frac {\sqrt {e} \left (a+b \csc ^{-1}(c x)\right ) \log \left (1-\frac {i c \sqrt {-d} e^{i \csc ^{-1}(c x)}}{\sqrt {e}-\sqrt {c^2 d+e}}\right )}{(-d)^{5/2}}-\frac {\sqrt {e} \left (a+b \csc ^{-1}(c x)\right ) \log \left (1+\frac {i c \sqrt {-d} e^{i \csc ^{-1}(c x)}}{\sqrt {e}-\sqrt {c^2 d+e}}\right )}{(-d)^{5/2}}+\frac {\sqrt {e} \left (a+b \csc ^{-1}(c x)\right ) \log \left (1-\frac {i c \sqrt {-d} e^{i \csc ^{-1}(c x)}}{\sqrt {e}+\sqrt {c^2 d+e}}\right )}{(-d)^{5/2}}-\frac {\sqrt {e} \left (a+b \csc ^{-1}(c x)\right ) \log \left (1+\frac {i c \sqrt {-d} e^{i \csc ^{-1}(c x)}}{\sqrt {e}+\sqrt {c^2 d+e}}\right )}{(-d)^{5/2}}+\frac {\left (i b \sqrt {e}\right ) \text {Subst}\left (\int \frac {\log \left (1-\frac {i \sqrt {-d} x}{\frac {\sqrt {e}}{c}-\frac {\sqrt {c^2 d+e}}{c}}\right )}{x} \, dx,x,e^{i \csc ^{-1}(c x)}\right )}{(-d)^{5/2}}-\frac {\left (i b \sqrt {e}\right ) \text {Subst}\left (\int \frac {\log \left (1+\frac {i \sqrt {-d} x}{\frac {\sqrt {e}}{c}-\frac {\sqrt {c^2 d+e}}{c}}\right )}{x} \, dx,x,e^{i \csc ^{-1}(c x)}\right )}{(-d)^{5/2}}+\frac {\left (i b \sqrt {e}\right ) \text {Subst}\left (\int \frac {\log \left (1-\frac {i \sqrt {-d} x}{\frac {\sqrt {e}}{c}+\frac {\sqrt {c^2 d+e}}{c}}\right )}{x} \, dx,x,e^{i \csc ^{-1}(c x)}\right )}{(-d)^{5/2}}-\frac {\left (i b \sqrt {e}\right ) \text {Subst}\left (\int \frac {\log \left (1+\frac {i \sqrt {-d} x}{\frac {\sqrt {e}}{c}+\frac {\sqrt {c^2 d+e}}{c}}\right )}{x} \, dx,x,e^{i \csc ^{-1}(c x)}\right )}{(-d)^{5/2}}-\frac {\sqrt {e} \text {Subst}\left (\int \frac {e^{i x} (a+b x)}{\frac {\sqrt {e}}{c}-\frac {\sqrt {c^2 d+e}}{c}-i \sqrt {-d} e^{i x}} \, dx,x,\csc ^{-1}(c x)\right )}{4 d^2}-\frac {\sqrt {e} \text {Subst}\left (\int \frac {e^{i x} (a+b x)}{\frac {\sqrt {e}}{c}+\frac {\sqrt {c^2 d+e}}{c}-i \sqrt {-d} e^{i x}} \, dx,x,\csc ^{-1}(c x)\right )}{4 d^2}-\frac {\sqrt {e} \text {Subst}\left (\int \frac {e^{i x} (a+b x)}{\frac {\sqrt {e}}{c}-\frac {\sqrt {c^2 d+e}}{c}+i \sqrt {-d} e^{i x}} \, dx,x,\csc ^{-1}(c x)\right )}{4 d^2}-\frac {\sqrt {e} \text {Subst}\left (\int \frac {e^{i x} (a+b x)}{\frac {\sqrt {e}}{c}+\frac {\sqrt {c^2 d+e}}{c}+i \sqrt {-d} e^{i x}} \, dx,x,\csc ^{-1}(c x)\right )}{4 d^2}\\ &=-\frac {b c \sqrt {1-\frac {1}{c^2 x^2}}}{d^2}-\frac {a}{d^2 x}-\frac {b \csc ^{-1}(c x)}{d^2 x}+\frac {e \left (a+b \csc ^{-1}(c x)\right )}{4 d^2 \left (\sqrt {-d} \sqrt {e}-\frac {d}{x}\right )}-\frac {e \left (a+b \csc ^{-1}(c x)\right )}{4 d^2 \left (\sqrt {-d} \sqrt {e}+\frac {d}{x}\right )}-\frac {b e \tanh ^{-1}\left (\frac {c^2 d-\frac {\sqrt {-d} \sqrt {e}}{x}}{c \sqrt {d} \sqrt {c^2 d+e} \sqrt {1-\frac {1}{c^2 x^2}}}\right )}{4 d^{5/2} \sqrt {c^2 d+e}}-\frac {b e \tanh ^{-1}\left (\frac {c^2 d+\frac {\sqrt {-d} \sqrt {e}}{x}}{c \sqrt {d} \sqrt {c^2 d+e} \sqrt {1-\frac {1}{c^2 x^2}}}\right )}{4 d^{5/2} \sqrt {c^2 d+e}}+\frac {3 \sqrt {e} \left (a+b \csc ^{-1}(c x)\right ) \log \left (1-\frac {i c \sqrt {-d} e^{i \csc ^{-1}(c x)}}{\sqrt {e}-\sqrt {c^2 d+e}}\right )}{4 (-d)^{5/2}}-\frac {3 \sqrt {e} \left (a+b \csc ^{-1}(c x)\right ) \log \left (1+\frac {i c \sqrt {-d} e^{i \csc ^{-1}(c x)}}{\sqrt {e}-\sqrt {c^2 d+e}}\right )}{4 (-d)^{5/2}}+\frac {3 \sqrt {e} \left (a+b \csc ^{-1}(c x)\right ) \log \left (1-\frac {i c \sqrt {-d} e^{i \csc ^{-1}(c x)}}{\sqrt {e}+\sqrt {c^2 d+e}}\right )}{4 (-d)^{5/2}}-\frac {3 \sqrt {e} \left (a+b \csc ^{-1}(c x)\right ) \log \left (1+\frac {i c \sqrt {-d} e^{i \csc ^{-1}(c x)}}{\sqrt {e}+\sqrt {c^2 d+e}}\right )}{4 (-d)^{5/2}}+\frac {i b \sqrt {e} \text {Li}_2\left (-\frac {i c \sqrt {-d} e^{i \csc ^{-1}(c x)}}{\sqrt {e}-\sqrt {c^2 d+e}}\right )}{(-d)^{5/2}}-\frac {i b \sqrt {e} \text {Li}_2\left (\frac {i c \sqrt {-d} e^{i \csc ^{-1}(c x)}}{\sqrt {e}-\sqrt {c^2 d+e}}\right )}{(-d)^{5/2}}+\frac {i b \sqrt {e} \text {Li}_2\left (-\frac {i c \sqrt {-d} e^{i \csc ^{-1}(c x)}}{\sqrt {e}+\sqrt {c^2 d+e}}\right )}{(-d)^{5/2}}-\frac {i b \sqrt {e} \text {Li}_2\left (\frac {i c \sqrt {-d} e^{i \csc ^{-1}(c x)}}{\sqrt {e}+\sqrt {c^2 d+e}}\right )}{(-d)^{5/2}}+\frac {\left (b \sqrt {e}\right ) \text {Subst}\left (\int \log \left (1-\frac {i \sqrt {-d} e^{i x}}{\frac {\sqrt {e}}{c}-\frac {\sqrt {c^2 d+e}}{c}}\right ) \, dx,x,\csc ^{-1}(c x)\right )}{4 (-d)^{5/2}}-\frac {\left (b \sqrt {e}\right ) \text {Subst}\left (\int \log \left (1+\frac {i \sqrt {-d} e^{i x}}{\frac {\sqrt {e}}{c}-\frac {\sqrt {c^2 d+e}}{c}}\right ) \, dx,x,\csc ^{-1}(c x)\right )}{4 (-d)^{5/2}}+\frac {\left (b \sqrt {e}\right ) \text {Subst}\left (\int \log \left (1-\frac {i \sqrt {-d} e^{i x}}{\frac {\sqrt {e}}{c}+\frac {\sqrt {c^2 d+e}}{c}}\right ) \, dx,x,\csc ^{-1}(c x)\right )}{4 (-d)^{5/2}}-\frac {\left (b \sqrt {e}\right ) \text {Subst}\left (\int \log \left (1+\frac {i \sqrt {-d} e^{i x}}{\frac {\sqrt {e}}{c}+\frac {\sqrt {c^2 d+e}}{c}}\right ) \, dx,x,\csc ^{-1}(c x)\right )}{4 (-d)^{5/2}}\\ &=-\frac {b c \sqrt {1-\frac {1}{c^2 x^2}}}{d^2}-\frac {a}{d^2 x}-\frac {b \csc ^{-1}(c x)}{d^2 x}+\frac {e \left (a+b \csc ^{-1}(c x)\right )}{4 d^2 \left (\sqrt {-d} \sqrt {e}-\frac {d}{x}\right )}-\frac {e \left (a+b \csc ^{-1}(c x)\right )}{4 d^2 \left (\sqrt {-d} \sqrt {e}+\frac {d}{x}\right )}-\frac {b e \tanh ^{-1}\left (\frac {c^2 d-\frac {\sqrt {-d} \sqrt {e}}{x}}{c \sqrt {d} \sqrt {c^2 d+e} \sqrt {1-\frac {1}{c^2 x^2}}}\right )}{4 d^{5/2} \sqrt {c^2 d+e}}-\frac {b e \tanh ^{-1}\left (\frac {c^2 d+\frac {\sqrt {-d} \sqrt {e}}{x}}{c \sqrt {d} \sqrt {c^2 d+e} \sqrt {1-\frac {1}{c^2 x^2}}}\right )}{4 d^{5/2} \sqrt {c^2 d+e}}+\frac {3 \sqrt {e} \left (a+b \csc ^{-1}(c x)\right ) \log \left (1-\frac {i c \sqrt {-d} e^{i \csc ^{-1}(c x)}}{\sqrt {e}-\sqrt {c^2 d+e}}\right )}{4 (-d)^{5/2}}-\frac {3 \sqrt {e} \left (a+b \csc ^{-1}(c x)\right ) \log \left (1+\frac {i c \sqrt {-d} e^{i \csc ^{-1}(c x)}}{\sqrt {e}-\sqrt {c^2 d+e}}\right )}{4 (-d)^{5/2}}+\frac {3 \sqrt {e} \left (a+b \csc ^{-1}(c x)\right ) \log \left (1-\frac {i c \sqrt {-d} e^{i \csc ^{-1}(c x)}}{\sqrt {e}+\sqrt {c^2 d+e}}\right )}{4 (-d)^{5/2}}-\frac {3 \sqrt {e} \left (a+b \csc ^{-1}(c x)\right ) \log \left (1+\frac {i c \sqrt {-d} e^{i \csc ^{-1}(c x)}}{\sqrt {e}+\sqrt {c^2 d+e}}\right )}{4 (-d)^{5/2}}+\frac {i b \sqrt {e} \text {Li}_2\left (-\frac {i c \sqrt {-d} e^{i \csc ^{-1}(c x)}}{\sqrt {e}-\sqrt {c^2 d+e}}\right )}{(-d)^{5/2}}-\frac {i b \sqrt {e} \text {Li}_2\left (\frac {i c \sqrt {-d} e^{i \csc ^{-1}(c x)}}{\sqrt {e}-\sqrt {c^2 d+e}}\right )}{(-d)^{5/2}}+\frac {i b \sqrt {e} \text {Li}_2\left (-\frac {i c \sqrt {-d} e^{i \csc ^{-1}(c x)}}{\sqrt {e}+\sqrt {c^2 d+e}}\right )}{(-d)^{5/2}}-\frac {i b \sqrt {e} \text {Li}_2\left (\frac {i c \sqrt {-d} e^{i \csc ^{-1}(c x)}}{\sqrt {e}+\sqrt {c^2 d+e}}\right )}{(-d)^{5/2}}-\frac {\left (i b \sqrt {e}\right ) \text {Subst}\left (\int \frac {\log \left (1-\frac {i \sqrt {-d} x}{\frac {\sqrt {e}}{c}-\frac {\sqrt {c^2 d+e}}{c}}\right )}{x} \, dx,x,e^{i \csc ^{-1}(c x)}\right )}{4 (-d)^{5/2}}+\frac {\left (i b \sqrt {e}\right ) \text {Subst}\left (\int \frac {\log \left (1+\frac {i \sqrt {-d} x}{\frac {\sqrt {e}}{c}-\frac {\sqrt {c^2 d+e}}{c}}\right )}{x} \, dx,x,e^{i \csc ^{-1}(c x)}\right )}{4 (-d)^{5/2}}-\frac {\left (i b \sqrt {e}\right ) \text {Subst}\left (\int \frac {\log \left (1-\frac {i \sqrt {-d} x}{\frac {\sqrt {e}}{c}+\frac {\sqrt {c^2 d+e}}{c}}\right )}{x} \, dx,x,e^{i \csc ^{-1}(c x)}\right )}{4 (-d)^{5/2}}+\frac {\left (i b \sqrt {e}\right ) \text {Subst}\left (\int \frac {\log \left (1+\frac {i \sqrt {-d} x}{\frac {\sqrt {e}}{c}+\frac {\sqrt {c^2 d+e}}{c}}\right )}{x} \, dx,x,e^{i \csc ^{-1}(c x)}\right )}{4 (-d)^{5/2}}\\ &=-\frac {b c \sqrt {1-\frac {1}{c^2 x^2}}}{d^2}-\frac {a}{d^2 x}-\frac {b \csc ^{-1}(c x)}{d^2 x}+\frac {e \left (a+b \csc ^{-1}(c x)\right )}{4 d^2 \left (\sqrt {-d} \sqrt {e}-\frac {d}{x}\right )}-\frac {e \left (a+b \csc ^{-1}(c x)\right )}{4 d^2 \left (\sqrt {-d} \sqrt {e}+\frac {d}{x}\right )}-\frac {b e \tanh ^{-1}\left (\frac {c^2 d-\frac {\sqrt {-d} \sqrt {e}}{x}}{c \sqrt {d} \sqrt {c^2 d+e} \sqrt {1-\frac {1}{c^2 x^2}}}\right )}{4 d^{5/2} \sqrt {c^2 d+e}}-\frac {b e \tanh ^{-1}\left (\frac {c^2 d+\frac {\sqrt {-d} \sqrt {e}}{x}}{c \sqrt {d} \sqrt {c^2 d+e} \sqrt {1-\frac {1}{c^2 x^2}}}\right )}{4 d^{5/2} \sqrt {c^2 d+e}}+\frac {3 \sqrt {e} \left (a+b \csc ^{-1}(c x)\right ) \log \left (1-\frac {i c \sqrt {-d} e^{i \csc ^{-1}(c x)}}{\sqrt {e}-\sqrt {c^2 d+e}}\right )}{4 (-d)^{5/2}}-\frac {3 \sqrt {e} \left (a+b \csc ^{-1}(c x)\right ) \log \left (1+\frac {i c \sqrt {-d} e^{i \csc ^{-1}(c x)}}{\sqrt {e}-\sqrt {c^2 d+e}}\right )}{4 (-d)^{5/2}}+\frac {3 \sqrt {e} \left (a+b \csc ^{-1}(c x)\right ) \log \left (1-\frac {i c \sqrt {-d} e^{i \csc ^{-1}(c x)}}{\sqrt {e}+\sqrt {c^2 d+e}}\right )}{4 (-d)^{5/2}}-\frac {3 \sqrt {e} \left (a+b \csc ^{-1}(c x)\right ) \log \left (1+\frac {i c \sqrt {-d} e^{i \csc ^{-1}(c x)}}{\sqrt {e}+\sqrt {c^2 d+e}}\right )}{4 (-d)^{5/2}}+\frac {3 i b \sqrt {e} \text {Li}_2\left (-\frac {i c \sqrt {-d} e^{i \csc ^{-1}(c x)}}{\sqrt {e}-\sqrt {c^2 d+e}}\right )}{4 (-d)^{5/2}}-\frac {3 i b \sqrt {e} \text {Li}_2\left (\frac {i c \sqrt {-d} e^{i \csc ^{-1}(c x)}}{\sqrt {e}-\sqrt {c^2 d+e}}\right )}{4 (-d)^{5/2}}+\frac {3 i b \sqrt {e} \text {Li}_2\left (-\frac {i c \sqrt {-d} e^{i \csc ^{-1}(c x)}}{\sqrt {e}+\sqrt {c^2 d+e}}\right )}{4 (-d)^{5/2}}-\frac {3 i b \sqrt {e} \text {Li}_2\left (\frac {i c \sqrt {-d} e^{i \csc ^{-1}(c x)}}{\sqrt {e}+\sqrt {c^2 d+e}}\right )}{4 (-d)^{5/2}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 1.76, size = 1525, normalized size = 1.89 \begin {gather*} \frac {-\frac {8 a \sqrt {d}}{x}-\frac {4 a \sqrt {d} e x}{d+e x^2}-12 a \sqrt {e} \text {ArcTan}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )+b \left (-8 c \sqrt {d} \sqrt {1-\frac {1}{c^2 x^2}}-\frac {8 \sqrt {d} \csc ^{-1}(c x)}{x}-\frac {2 \sqrt {d} e \csc ^{-1}(c x)}{-i \sqrt {d} \sqrt {e}+e x}-\frac {2 \sqrt {d} e \csc ^{-1}(c x)}{i \sqrt {d} \sqrt {e}+e x}-24 \sqrt {e} \text {ArcSin}\left (\frac {\sqrt {1-\frac {i \sqrt {e}}{c \sqrt {d}}}}{\sqrt {2}}\right ) \text {ArcTan}\left (\frac {\left (-i c \sqrt {d}+\sqrt {e}\right ) \cot \left (\frac {1}{4} \left (\pi +2 \csc ^{-1}(c x)\right )\right )}{\sqrt {c^2 d+e}}\right )+24 \sqrt {e} \text {ArcSin}\left (\frac {\sqrt {1+\frac {i \sqrt {e}}{c \sqrt {d}}}}{\sqrt {2}}\right ) \text {ArcTan}\left (\frac {\left (i c \sqrt {d}+\sqrt {e}\right ) \cot \left (\frac {1}{4} \left (\pi +2 \csc ^{-1}(c x)\right )\right )}{\sqrt {c^2 d+e}}\right )+3 i \sqrt {e} \pi \log \left (1+\frac {\left (\sqrt {e}-\sqrt {c^2 d+e}\right ) e^{-i \csc ^{-1}(c x)}}{c \sqrt {d}}\right )-6 i \sqrt {e} \csc ^{-1}(c x) \log \left (1+\frac {\left (\sqrt {e}-\sqrt {c^2 d+e}\right ) e^{-i \csc ^{-1}(c x)}}{c \sqrt {d}}\right )+12 i \sqrt {e} \text {ArcSin}\left (\frac {\sqrt {1-\frac {i \sqrt {e}}{c \sqrt {d}}}}{\sqrt {2}}\right ) \log \left (1+\frac {\left (\sqrt {e}-\sqrt {c^2 d+e}\right ) e^{-i \csc ^{-1}(c x)}}{c \sqrt {d}}\right )-3 i \sqrt {e} \pi \log \left (1+\frac {\left (-\sqrt {e}+\sqrt {c^2 d+e}\right ) e^{-i \csc ^{-1}(c x)}}{c \sqrt {d}}\right )+6 i \sqrt {e} \csc ^{-1}(c x) \log \left (1+\frac {\left (-\sqrt {e}+\sqrt {c^2 d+e}\right ) e^{-i \csc ^{-1}(c x)}}{c \sqrt {d}}\right )-12 i \sqrt {e} \text {ArcSin}\left (\frac {\sqrt {1+\frac {i \sqrt {e}}{c \sqrt {d}}}}{\sqrt {2}}\right ) \log \left (1+\frac {\left (-\sqrt {e}+\sqrt {c^2 d+e}\right ) e^{-i \csc ^{-1}(c x)}}{c \sqrt {d}}\right )-3 i \sqrt {e} \pi \log \left (1-\frac {\left (\sqrt {e}+\sqrt {c^2 d+e}\right ) e^{-i \csc ^{-1}(c x)}}{c \sqrt {d}}\right )+6 i \sqrt {e} \csc ^{-1}(c x) \log \left (1-\frac {\left (\sqrt {e}+\sqrt {c^2 d+e}\right ) e^{-i \csc ^{-1}(c x)}}{c \sqrt {d}}\right )+12 i \sqrt {e} \text {ArcSin}\left (\frac {\sqrt {1+\frac {i \sqrt {e}}{c \sqrt {d}}}}{\sqrt {2}}\right ) \log \left (1-\frac {\left (\sqrt {e}+\sqrt {c^2 d+e}\right ) e^{-i \csc ^{-1}(c x)}}{c \sqrt {d}}\right )+3 i \sqrt {e} \pi \log \left (1+\frac {\left (\sqrt {e}+\sqrt {c^2 d+e}\right ) e^{-i \csc ^{-1}(c x)}}{c \sqrt {d}}\right )-6 i \sqrt {e} \csc ^{-1}(c x) \log \left (1+\frac {\left (\sqrt {e}+\sqrt {c^2 d+e}\right ) e^{-i \csc ^{-1}(c x)}}{c \sqrt {d}}\right )-12 i \sqrt {e} \text {ArcSin}\left (\frac {\sqrt {1-\frac {i \sqrt {e}}{c \sqrt {d}}}}{\sqrt {2}}\right ) \log \left (1+\frac {\left (\sqrt {e}+\sqrt {c^2 d+e}\right ) e^{-i \csc ^{-1}(c x)}}{c \sqrt {d}}\right )+3 i \sqrt {e} \pi \log \left (\sqrt {e}-\frac {i \sqrt {d}}{x}\right )-3 i \sqrt {e} \pi \log \left (\sqrt {e}+\frac {i \sqrt {d}}{x}\right )-\frac {2 i e \log \left (\frac {2 \sqrt {d} \sqrt {e} \left (\sqrt {e}+c \left (-i c \sqrt {d}-\sqrt {-c^2 d-e} \sqrt {1-\frac {1}{c^2 x^2}}\right ) x\right )}{\sqrt {-c^2 d-e} \left (\sqrt {d}+i \sqrt {e} x\right )}\right )}{\sqrt {-c^2 d-e}}+\frac {2 i e \log \left (\frac {2 \sqrt {d} \sqrt {e} \left (-\sqrt {e}+c \left (-i c \sqrt {d}+\sqrt {-c^2 d-e} \sqrt {1-\frac {1}{c^2 x^2}}\right ) x\right )}{\sqrt {-c^2 d-e} \left (\sqrt {d}-i \sqrt {e} x\right )}\right )}{\sqrt {-c^2 d-e}}-6 \sqrt {e} \text {PolyLog}\left (2,\frac {\left (\sqrt {e}-\sqrt {c^2 d+e}\right ) e^{-i \csc ^{-1}(c x)}}{c \sqrt {d}}\right )+6 \sqrt {e} \text {PolyLog}\left (2,\frac {\left (-\sqrt {e}+\sqrt {c^2 d+e}\right ) e^{-i \csc ^{-1}(c x)}}{c \sqrt {d}}\right )+6 \sqrt {e} \text {PolyLog}\left (2,-\frac {\left (\sqrt {e}+\sqrt {c^2 d+e}\right ) e^{-i \csc ^{-1}(c x)}}{c \sqrt {d}}\right )-6 \sqrt {e} \text {PolyLog}\left (2,\frac {\left (\sqrt {e}+\sqrt {c^2 d+e}\right ) e^{-i \csc ^{-1}(c x)}}{c \sqrt {d}}\right )\right )}{8 d^{5/2}} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

((-8*a*Sqrt[d])/x - (4*a*Sqrt[d]*e*x)/(d + e*x^2) - 12*a*Sqrt[e]*ArcTan[(Sqrt[e]*x)/Sqrt[d]] + b*(-8*c*Sqrt[d]
*Sqrt[1 - 1/(c^2*x^2)] - (8*Sqrt[d]*ArcCsc[c*x])/x - (2*Sqrt[d]*e*ArcCsc[c*x])/((-I)*Sqrt[d]*Sqrt[e] + e*x) -
(2*Sqrt[d]*e*ArcCsc[c*x])/(I*Sqrt[d]*Sqrt[e] + e*x) - 24*Sqrt[e]*ArcSin[Sqrt[1 - (I*Sqrt[e])/(c*Sqrt[d])]/Sqrt
[2]]*ArcTan[(((-I)*c*Sqrt[d] + Sqrt[e])*Cot[(Pi + 2*ArcCsc[c*x])/4])/Sqrt[c^2*d + e]] + 24*Sqrt[e]*ArcSin[Sqrt
[1 + (I*Sqrt[e])/(c*Sqrt[d])]/Sqrt[2]]*ArcTan[((I*c*Sqrt[d] + Sqrt[e])*Cot[(Pi + 2*ArcCsc[c*x])/4])/Sqrt[c^2*d
+ e]] + (3*I)*Sqrt[e]*Pi*Log[1 + (Sqrt[e] - Sqrt[c^2*d + e])/(c*Sqrt[d]*E^(I*ArcCsc[c*x]))] - (6*I)*Sqrt[e]*A
rcCsc[c*x]*Log[1 + (Sqrt[e] - Sqrt[c^2*d + e])/(c*Sqrt[d]*E^(I*ArcCsc[c*x]))] + (12*I)*Sqrt[e]*ArcSin[Sqrt[1 -
(I*Sqrt[e])/(c*Sqrt[d])]/Sqrt[2]]*Log[1 + (Sqrt[e] - Sqrt[c^2*d + e])/(c*Sqrt[d]*E^(I*ArcCsc[c*x]))] - (3*I)*
Sqrt[e]*Pi*Log[1 + (-Sqrt[e] + Sqrt[c^2*d + e])/(c*Sqrt[d]*E^(I*ArcCsc[c*x]))] + (6*I)*Sqrt[e]*ArcCsc[c*x]*Log
[1 + (-Sqrt[e] + Sqrt[c^2*d + e])/(c*Sqrt[d]*E^(I*ArcCsc[c*x]))] - (12*I)*Sqrt[e]*ArcSin[Sqrt[1 + (I*Sqrt[e])/
(c*Sqrt[d])]/Sqrt[2]]*Log[1 + (-Sqrt[e] + Sqrt[c^2*d + e])/(c*Sqrt[d]*E^(I*ArcCsc[c*x]))] - (3*I)*Sqrt[e]*Pi*L
og[1 - (Sqrt[e] + Sqrt[c^2*d + e])/(c*Sqrt[d]*E^(I*ArcCsc[c*x]))] + (6*I)*Sqrt[e]*ArcCsc[c*x]*Log[1 - (Sqrt[e]
+ Sqrt[c^2*d + e])/(c*Sqrt[d]*E^(I*ArcCsc[c*x]))] + (12*I)*Sqrt[e]*ArcSin[Sqrt[1 + (I*Sqrt[e])/(c*Sqrt[d])]/S
qrt[2]]*Log[1 - (Sqrt[e] + Sqrt[c^2*d + e])/(c*Sqrt[d]*E^(I*ArcCsc[c*x]))] + (3*I)*Sqrt[e]*Pi*Log[1 + (Sqrt[e]
+ Sqrt[c^2*d + e])/(c*Sqrt[d]*E^(I*ArcCsc[c*x]))] - (6*I)*Sqrt[e]*ArcCsc[c*x]*Log[1 + (Sqrt[e] + Sqrt[c^2*d +
e])/(c*Sqrt[d]*E^(I*ArcCsc[c*x]))] - (12*I)*Sqrt[e]*ArcSin[Sqrt[1 - (I*Sqrt[e])/(c*Sqrt[d])]/Sqrt[2]]*Log[1 +
(Sqrt[e] + Sqrt[c^2*d + e])/(c*Sqrt[d]*E^(I*ArcCsc[c*x]))] + (3*I)*Sqrt[e]*Pi*Log[Sqrt[e] - (I*Sqrt[d])/x] -
(3*I)*Sqrt[e]*Pi*Log[Sqrt[e] + (I*Sqrt[d])/x] - ((2*I)*e*Log[(2*Sqrt[d]*Sqrt[e]*(Sqrt[e] + c*((-I)*c*Sqrt[d] -
Sqrt[-(c^2*d) - e]*Sqrt[1 - 1/(c^2*x^2)])*x))/(Sqrt[-(c^2*d) - e]*(Sqrt[d] + I*Sqrt[e]*x))])/Sqrt[-(c^2*d) -
e] + ((2*I)*e*Log[(2*Sqrt[d]*Sqrt[e]*(-Sqrt[e] + c*((-I)*c*Sqrt[d] + Sqrt[-(c^2*d) - e]*Sqrt[1 - 1/(c^2*x^2)])
*x))/(Sqrt[-(c^2*d) - e]*(Sqrt[d] - I*Sqrt[e]*x))])/Sqrt[-(c^2*d) - e] - 6*Sqrt[e]*PolyLog[2, (Sqrt[e] - Sqrt[
c^2*d + e])/(c*Sqrt[d]*E^(I*ArcCsc[c*x]))] + 6*Sqrt[e]*PolyLog[2, (-Sqrt[e] + Sqrt[c^2*d + e])/(c*Sqrt[d]*E^(I
*ArcCsc[c*x]))] + 6*Sqrt[e]*PolyLog[2, -((Sqrt[e] + Sqrt[c^2*d + e])/(c*Sqrt[d]*E^(I*ArcCsc[c*x])))] - 6*Sqrt[
e]*PolyLog[2, (Sqrt[e] + Sqrt[c^2*d + e])/(c*Sqrt[d]*E^(I*ArcCsc[c*x]))]))/(8*d^(5/2))

________________________________________________________________________________________

Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 17.08, size = 1784, normalized size = 2.21 \[\text {Expression too large to display}\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*a*e/d^2*x*c^2/(c^2*e*x^2+c^2*d)-3/2*a*e/d^2/(d*e)^(1/2)*arctan(e*x/(d*e)^(1/2))-a/d^2/x-b/c^4*(-(c^2*d-2*
(e*(c^2*d+e))^(1/2)+2*e)*d)^(1/2)*e*arctan((I/c/x+(1-1/c^2/x^2)^(1/2))*c*d/((-c^2*d+2*(e*(c^2*d+e))^(1/2)-2*e)
*d)^(1/2))/d^5*(e*(c^2*d+e))^(1/2)+b/c^2*(-(c^2*d-2*(e*(c^2*d+e))^(1/2)+2*e)*d)^(1/2)*e^2*arctan((I/c/x+(1-1/c
^2/x^2)^(1/2))*c*d/((-c^2*d+2*(e*(c^2*d+e))^(1/2)-2*e)*d)^(1/2))/d^4/(c^2*d+e)+b/c^4*(-(c^2*d-2*(e*(c^2*d+e))^
(1/2)+2*e)*d)^(1/2)*e^3*arctan((I/c/x+(1-1/c^2/x^2)^(1/2))*c*d/((-c^2*d+2*(e*(c^2*d+e))^(1/2)-2*e)*d)^(1/2))/d
^5/(c^2*d+e)-1/2*b/c^2*((c^2*d+2*(e*(c^2*d+e))^(1/2)+2*e)*d)^(1/2)*e*arctanh((I/c/x+(1-1/c^2/x^2)^(1/2))*c*d/(
(c^2*d+2*(e*(c^2*d+e))^(1/2)+2*e)*d)^(1/2))/d^4-b/c^4*((c^2*d+2*(e*(c^2*d+e))^(1/2)+2*e)*d)^(1/2)*e^2*arctanh(
(I/c/x+(1-1/c^2/x^2)^(1/2))*c*d/((c^2*d+2*(e*(c^2*d+e))^(1/2)+2*e)*d)^(1/2))/d^5-1/2*b/c^2*(-(c^2*d-2*(e*(c^2*
d+e))^(1/2)+2*e)*d)^(1/2)*e*arctan((I/c/x+(1-1/c^2/x^2)^(1/2))*c*d/((-c^2*d+2*(e*(c^2*d+e))^(1/2)-2*e)*d)^(1/2
))/d^4-b/c^4*(-(c^2*d-2*(e*(c^2*d+e))^(1/2)+2*e)*d)^(1/2)*e^2*arctan((I/c/x+(1-1/c^2/x^2)^(1/2))*c*d/((-c^2*d+
2*(e*(c^2*d+e))^(1/2)-2*e)*d)^(1/2))/d^5-1/2*b*x*c^2*arccsc(c*x)*e/(c^2*e*x^2+c^2*d)/d^2+b/c^4*((c^2*d+2*(e*(c
^2*d+e))^(1/2)+2*e)*d)^(1/2)*e*arctanh((I/c/x+(1-1/c^2/x^2)^(1/2))*c*d/((c^2*d+2*(e*(c^2*d+e))^(1/2)+2*e)*d)^(
1/2))/d^5*(e*(c^2*d+e))^(1/2)+b/c^2*((c^2*d+2*(e*(c^2*d+e))^(1/2)+2*e)*d)^(1/2)*e^2*arctanh((I/c/x+(1-1/c^2/x^
2)^(1/2))*c*d/((c^2*d+2*(e*(c^2*d+e))^(1/2)+2*e)*d)^(1/2))/d^4/(c^2*d+e)+b/c^4*((c^2*d+2*(e*(c^2*d+e))^(1/2)+2
*e)*d)^(1/2)*e^3*arctanh((I/c/x+(1-1/c^2/x^2)^(1/2))*c*d/((c^2*d+2*(e*(c^2*d+e))^(1/2)+2*e)*d)^(1/2))/d^5/(c^2
*d+e)-1/2*b/c^2*((c^2*d+2*(e*(c^2*d+e))^(1/2)+2*e)*d)^(1/2)*e*arctanh((I/c/x+(1-1/c^2/x^2)^(1/2))*c*d/((c^2*d+
2*(e*(c^2*d+e))^(1/2)+2*e)*d)^(1/2))/d^4/(c^2*d+e)*(e*(c^2*d+e))^(1/2)-b/c^4*((c^2*d+2*(e*(c^2*d+e))^(1/2)+2*e
)*d)^(1/2)*e^2*arctanh((I/c/x+(1-1/c^2/x^2)^(1/2))*c*d/((c^2*d+2*(e*(c^2*d+e))^(1/2)+2*e)*d)^(1/2))/d^5/(c^2*d
+e)*(e*(c^2*d+e))^(1/2)+1/2*b/c^2*(-(c^2*d-2*(e*(c^2*d+e))^(1/2)+2*e)*d)^(1/2)*e*arctan((I/c/x+(1-1/c^2/x^2)^(
1/2))*c*d/((-c^2*d+2*(e*(c^2*d+e))^(1/2)-2*e)*d)^(1/2))/d^4/(c^2*d+e)*(e*(c^2*d+e))^(1/2)+b/c^4*(-(c^2*d-2*(e*
(c^2*d+e))^(1/2)+2*e)*d)^(1/2)*e^2*arctan((I/c/x+(1-1/c^2/x^2)^(1/2))*c*d/((-c^2*d+2*(e*(c^2*d+e))^(1/2)-2*e)*
d)^(1/2))/d^5/(c^2*d+e)*(e*(c^2*d+e))^(1/2)+3/4*c*b*e/d^2*sum(1/_R1/(_R1^2*c^2*d-c^2*d-2*e)*(I*arccsc(c*x)*ln(
(_R1-I/c/x-(1-1/c^2/x^2)^(1/2))/_R1)+dilog((_R1-I/c/x-(1-1/c^2/x^2)^(1/2))/_R1)),_R1=RootOf(c^2*d*_Z^4+(-2*c^2
*d-4*e)*_Z^2+c^2*d))+3/4*c*b*e/d^2*sum(_R1/(_R1^2*c^2*d-c^2*d-2*e)*(I*arccsc(c*x)*ln((_R1-I/c/x-(1-1/c^2/x^2)^
(1/2))/_R1)+dilog((_R1-I/c/x-(1-1/c^2/x^2)^(1/2))/_R1)),_R1=RootOf(c^2*d*_Z^4+(-2*c^2*d-4*e)*_Z^2+c^2*d))-b*ar
ccsc(c*x)/d^2/x-c*b/d^2*((c^2*x^2-1)/c^2/x^2)^(1/2)

________________________________________________________________________________________

Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*a*((3*x^2*e + 2*d)/(d^2*x^3*e + d^3*x) + 3*arctan(x*e^(1/2)/sqrt(d))*e^(1/2)/d^(5/2)) + b*integrate(arcta
n2(1, sqrt(c*x + 1)*sqrt(c*x - 1))/(x^6*e^2 + 2*d*x^4*e + d^2*x^2), x)

________________________________________________________________________________________

Fricas [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((a+b*arccsc(c*x))/x^2/(e*x^2+d)^2,x, algorithm="fricas")

[Out]

integral((b*arccsc(c*x) + a)/(x^6*e^2 + 2*d*x^4*e + d^2*x^2), x)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: RuntimeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: RuntimeError >> An error occurred running a Giac command:INPUT:sage2OUTPUT:Warning, integrat
ion of abs or sign assumes constant sign by intervals (correct if the argument is real):Check [abs(sageVARx)]s
ym2poly/r2sym(

________________________________________________________________________________________

Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {a+b\,\mathrm {asin}\left (\frac {1}{c\,x}\right )}{x^2\,{\left (e\,x^2+d\right )}^2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________