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

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

[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)

________________________________________________________________________________________

Rubi [A]  time = 2.36407, antiderivative size = 806, normalized size of antiderivative = 1., 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 = {5241, 4733, 4619, 261, 4667, 4743, 725, 206, 4741, 4519, 2190, 2279, 2391} \[ -\frac{a}{d^2 x}-\frac{b \csc ^{-1}(c x)}{d^2 x}+\frac{e \left (a+b \csc ^{-1}(c x)\right )}{4 d^2 \left (\sqrt{-d} \sqrt{e}-\frac{d}{x}\right )}-\frac{e \left (a+b \csc ^{-1}(c x)\right )}{4 d^2 \left (\frac{d}{x}+\sqrt{-d} \sqrt{e}\right )}-\frac{b e \tanh ^{-1}\left (\frac{c^2 d-\frac{\sqrt{-d} \sqrt{e}}{x}}{c \sqrt{d} \sqrt{d c^2+e} \sqrt{1-\frac{1}{c^2 x^2}}}\right )}{4 d^{5/2} \sqrt{d c^2+e}}-\frac{b e \tanh ^{-1}\left (\frac{d c^2+\frac{\sqrt{-d} \sqrt{e}}{x}}{c \sqrt{d} \sqrt{d c^2+e} \sqrt{1-\frac{1}{c^2 x^2}}}\right )}{4 d^{5/2} \sqrt{d c^2+e}}+\frac{3 \sqrt{e} \left (a+b \csc ^{-1}(c x)\right ) \log \left (1-\frac{i c \sqrt{-d} e^{i \csc ^{-1}(c x)}}{\sqrt{e}-\sqrt{d c^2+e}}\right )}{4 (-d)^{5/2}}-\frac{3 \sqrt{e} \left (a+b \csc ^{-1}(c x)\right ) \log \left (\frac{i \sqrt{-d} e^{i \csc ^{-1}(c x)} c}{\sqrt{e}-\sqrt{d c^2+e}}+1\right )}{4 (-d)^{5/2}}+\frac{3 \sqrt{e} \left (a+b \csc ^{-1}(c x)\right ) \log \left (1-\frac{i c \sqrt{-d} e^{i \csc ^{-1}(c x)}}{\sqrt{e}+\sqrt{d c^2+e}}\right )}{4 (-d)^{5/2}}-\frac{3 \sqrt{e} \left (a+b \csc ^{-1}(c x)\right ) \log \left (\frac{i \sqrt{-d} e^{i \csc ^{-1}(c x)} c}{\sqrt{e}+\sqrt{d c^2+e}}+1\right )}{4 (-d)^{5/2}}+\frac{3 i b \sqrt{e} \text{PolyLog}\left (2,-\frac{i c \sqrt{-d} e^{i \csc ^{-1}(c x)}}{\sqrt{e}-\sqrt{d c^2+e}}\right )}{4 (-d)^{5/2}}-\frac{3 i b \sqrt{e} \text{PolyLog}\left (2,\frac{i c \sqrt{-d} e^{i \csc ^{-1}(c x)}}{\sqrt{e}-\sqrt{d c^2+e}}\right )}{4 (-d)^{5/2}}+\frac{3 i b \sqrt{e} \text{PolyLog}\left (2,-\frac{i c \sqrt{-d} e^{i \csc ^{-1}(c x)}}{\sqrt{e}+\sqrt{d c^2+e}}\right )}{4 (-d)^{5/2}}-\frac{3 i b \sqrt{e} \text{PolyLog}\left (2,\frac{i c \sqrt{-d} e^{i \csc ^{-1}(c x)}}{\sqrt{e}+\sqrt{d c^2+e}}\right )}{4 (-d)^{5/2}}-\frac{b c \sqrt{1-\frac{1}{c^2 x^2}}}{d^2} \]

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 5241

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]

Rule 4733

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 4619

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 261

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 4667

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 4743

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 725

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 206

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

Rule 4741

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 4519

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 2190

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*Log[1 + (b*(F^(g*(e + f*x)))^n)/a])/(b*f*g*n*Log[F]), 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 2279

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 2391

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]

Rubi steps

\begin{align*} \int \frac{a+b \csc ^{-1}(c x)}{x^2 \left (d+e x^2\right )^2} \, dx &=-\operatorname{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 )\\ &=-\operatorname{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{\operatorname{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) \operatorname{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 \operatorname{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 \operatorname{Subst}\left (\int \sin ^{-1}\left (\frac{x}{c}\right ) \, dx,x,\frac{1}{x}\right )}{d^2}+\frac{(2 e) \operatorname{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 \operatorname{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 \operatorname{Subst}\left (\int \frac{x}{\sqrt{1-\frac{x^2}{c^2}}} \, dx,x,\frac{1}{x}\right )}{c d^2}+\frac{\sqrt{e} \operatorname{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} \operatorname{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 \operatorname{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 \operatorname{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 \operatorname{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} \operatorname{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} \operatorname{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) \operatorname{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) \operatorname{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 \operatorname{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} \operatorname{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} \operatorname{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} \operatorname{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} \operatorname{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} \operatorname{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} \operatorname{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) \operatorname{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) \operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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} \operatorname{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} \operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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} \operatorname{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} \operatorname{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} \operatorname{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} \operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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 = 2.46789, size = 1525, normalized size = 1.89 \[ \text{result too large to display} \]

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))

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Maple [C]  time = 10.007, size = 1784, normalized size = 2.2 \begin{align*} \text{result too large to display} \end{align*}

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/d^2*e*c^2*x/(c^2*e*x^2+c^2*d)-3/2*a/d^2*e/(d*e)^(1/2)*arctan(e*x/(d*e)^(1/2))-a/d^2/x-b*arccsc(c*x)/d^2
/x-1/2*b/c^2*(-(c^2*d-2*(e*(c^2*d+e))^(1/2)+2*e)*d)^(1/2)*arctan(d*c*(I/c/x+(1-1/c^2/x^2)^(1/2))/((-c^2*d+2*(e
*(c^2*d+e))^(1/2)-2*e)*d)^(1/2))/d^4*e-b/c^4*(-(c^2*d-2*(e*(c^2*d+e))^(1/2)+2*e)*d)^(1/2)*e^2*arctan(d*c*(I/c/
x+(1-1/c^2/x^2)^(1/2))/((-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)*arctanh(d*c*(I/c/x+(1-1/c^2/x^2)^(1/2))/((c^2*d+2*(e*(c^2*d+e))^(1/2)+2*e)*d)^(1/2))/d^4*e-b/
c^4*((c^2*d+2*(e*(c^2*d+e))^(1/2)+2*e)*d)^(1/2)*e^2*arctanh(d*c*(I/c/x+(1-1/c^2/x^2)^(1/2))/((c^2*d+2*(e*(c^2*
d+e))^(1/2)+2*e)*d)^(1/2))/d^5+3/4*c*b/d^2*e*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/d^2*e*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)+d
ilog((_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))+1/2*b/c^2*(-(c^2*
d-2*(e*(c^2*d+e))^(1/2)+2*e)*d)^(1/2)*arctan(d*c*(I/c/x+(1-1/c^2/x^2)^(1/2))/((-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)*e+b/c^4*(-(c^2*d-2*(e*(c^2*d+e))^(1/2)+2*e)*d)^(1/2)*e^2*arctan
(d*c*(I/c/x+(1-1/c^2/x^2)^(1/2))/((-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)*arctanh(d*c*(I/c/x+(1-1/c^2/x^2)^(1/2))/((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)*e-b/c^4*((c^2*d+2*(e*(c^2*d+e))^(1/2)+2*e)*d
)^(1/2)*e^2*arctanh(d*c*(I/c/x+(1-1/c^2/x^2)^(1/2))/((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)-b/c^4*(-(c^2*d-2*(e*(c^2*d+e))^(1/2)+2*e)*d)^(1/2)*e*arctan(d*c*(I/c/x+(1-1/c^2/x^2)^(1/2
))/((-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)*arctan(d*c*(I/c/x+(1-1/c^2/x^2)^(1/2))/((-c^2*d+2*(e*(c^2*d+e))^(1/2)-2*e)*d)^(1/2))/d^4/(c^2*d+
e)*e^2+b/c^4*(-(c^2*d-2*(e*(c^2*d+e))^(1/2)+2*e)*d)^(1/2)*e^3*arctan(d*c*(I/c/x+(1-1/c^2/x^2)^(1/2))/((-c^2*d+
2*(e*(c^2*d+e))^(1/2)-2*e)*d)^(1/2))/d^5/(c^2*d+e)-c*b/d^2*((c^2*x^2-1)/c^2/x^2)^(1/2)-1/2*b*c^2*x*e*arccsc(c*
x)/(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(d*c*(I/c/x+(1-1/c^2/x^2)^
(1/2))/((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)*arctanh(d*c*(I/c/x+(1-1/c^2/x^2)^(1/2))/((c^2*d+2*(e*(c^2*d+e))^(1/2)+2*e)*d)^(1/2))/d^4/(c^2*
d+e)*e^2+b/c^4*((c^2*d+2*(e*(c^2*d+e))^(1/2)+2*e)*d)^(1/2)*e^3*arctanh(d*c*(I/c/x+(1-1/c^2/x^2)^(1/2))/((c^2*d
+2*(e*(c^2*d+e))^(1/2)+2*e)*d)^(1/2))/d^5/(c^2*d+e)

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

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]

Exception raised: ValueError

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{b \operatorname{arccsc}\left (c x\right ) + a}{e^{2} x^{6} + 2 \, d e x^{4} + d^{2} x^{2}}, x\right ) \end{align*}

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)/(e^2*x^6 + 2*d*e*x^4 + d^2*x^2), x)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

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]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{b \operatorname{arccsc}\left (c x\right ) + a}{{\left (e x^{2} + d\right )}^{2} x^{2}}\,{d x} \end{align*}

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]

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