3.104 \(\int \frac{x^3 (a+b \csc ^{-1}(c x))}{(d+e x^2)^2} \, dx\)

Optimal. Leaf size=590 \[ -\frac{i b \text{PolyLog}\left (2,-\frac{i c \sqrt{-d} e^{i \csc ^{-1}(c x)}}{\sqrt{e}-\sqrt{c^2 d+e}}\right )}{2 e^2}-\frac{i b \text{PolyLog}\left (2,\frac{i c \sqrt{-d} e^{i \csc ^{-1}(c x)}}{\sqrt{e}-\sqrt{c^2 d+e}}\right )}{2 e^2}-\frac{i b \text{PolyLog}\left (2,-\frac{i c \sqrt{-d} e^{i \csc ^{-1}(c x)}}{\sqrt{c^2 d+e}+\sqrt{e}}\right )}{2 e^2}-\frac{i b \text{PolyLog}\left (2,\frac{i c \sqrt{-d} e^{i \csc ^{-1}(c x)}}{\sqrt{c^2 d+e}+\sqrt{e}}\right )}{2 e^2}+\frac{i b \text{PolyLog}\left (2,e^{2 i \csc ^{-1}(c x)}\right )}{2 e^2}+\frac{\left (a+b \csc ^{-1}(c x)\right ) \log \left (1-\frac{i c \sqrt{-d} e^{i \csc ^{-1}(c x)}}{\sqrt{e}-\sqrt{c^2 d+e}}\right )}{2 e^2}+\frac{\left (a+b \csc ^{-1}(c x)\right ) \log \left (1+\frac{i c \sqrt{-d} e^{i \csc ^{-1}(c x)}}{\sqrt{e}-\sqrt{c^2 d+e}}\right )}{2 e^2}+\frac{\left (a+b \csc ^{-1}(c x)\right ) \log \left (1-\frac{i c \sqrt{-d} e^{i \csc ^{-1}(c x)}}{\sqrt{c^2 d+e}+\sqrt{e}}\right )}{2 e^2}+\frac{\left (a+b \csc ^{-1}(c x)\right ) \log \left (1+\frac{i c \sqrt{-d} e^{i \csc ^{-1}(c x)}}{\sqrt{c^2 d+e}+\sqrt{e}}\right )}{2 e^2}-\frac{a+b \csc ^{-1}(c x)}{2 e \left (\frac{d}{x^2}+e\right )}-\frac{\log \left (1-e^{2 i \csc ^{-1}(c x)}\right ) \left (a+b \csc ^{-1}(c x)\right )}{e^2}+\frac{b \tan ^{-1}\left (\frac{\sqrt{c^2 d+e}}{c \sqrt{e} x \sqrt{1-\frac{1}{c^2 x^2}}}\right )}{2 e^{3/2} \sqrt{c^2 d+e}} \]

[Out]

-(a + b*ArcCsc[c*x])/(2*e*(e + d/x^2)) + (b*ArcTan[Sqrt[c^2*d + e]/(c*Sqrt[e]*Sqrt[1 - 1/(c^2*x^2)]*x)])/(2*e^
(3/2)*Sqrt[c^2*d + e]) + ((a + b*ArcCsc[c*x])*Log[1 - (I*c*Sqrt[-d]*E^(I*ArcCsc[c*x]))/(Sqrt[e] - Sqrt[c^2*d +
e])])/(2*e^2) + ((a + b*ArcCsc[c*x])*Log[1 + (I*c*Sqrt[-d]*E^(I*ArcCsc[c*x]))/(Sqrt[e] - Sqrt[c^2*d + e])])/(
2*e^2) + ((a + b*ArcCsc[c*x])*Log[1 - (I*c*Sqrt[-d]*E^(I*ArcCsc[c*x]))/(Sqrt[e] + Sqrt[c^2*d + e])])/(2*e^2) +
((a + b*ArcCsc[c*x])*Log[1 + (I*c*Sqrt[-d]*E^(I*ArcCsc[c*x]))/(Sqrt[e] + Sqrt[c^2*d + e])])/(2*e^2) - ((a + b
*ArcCsc[c*x])*Log[1 - E^((2*I)*ArcCsc[c*x])])/e^2 - ((I/2)*b*PolyLog[2, ((-I)*c*Sqrt[-d]*E^(I*ArcCsc[c*x]))/(S
qrt[e] - Sqrt[c^2*d + e])])/e^2 - ((I/2)*b*PolyLog[2, (I*c*Sqrt[-d]*E^(I*ArcCsc[c*x]))/(Sqrt[e] - Sqrt[c^2*d +
e])])/e^2 - ((I/2)*b*PolyLog[2, ((-I)*c*Sqrt[-d]*E^(I*ArcCsc[c*x]))/(Sqrt[e] + Sqrt[c^2*d + e])])/e^2 - ((I/2
)*b*PolyLog[2, (I*c*Sqrt[-d]*E^(I*ArcCsc[c*x]))/(Sqrt[e] + Sqrt[c^2*d + e])])/e^2 + ((I/2)*b*PolyLog[2, E^((2*
I)*ArcCsc[c*x])])/e^2

________________________________________________________________________________________

Rubi [A]  time = 1.23845, antiderivative size = 590, normalized size of antiderivative = 1., number of steps used = 29, number of rules used = 12, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.571, Rules used = {5241, 4733, 4625, 3717, 2190, 2279, 2391, 4729, 377, 205, 4741, 4519} \[ -\frac{i b \text{PolyLog}\left (2,-\frac{i c \sqrt{-d} e^{i \csc ^{-1}(c x)}}{\sqrt{e}-\sqrt{c^2 d+e}}\right )}{2 e^2}-\frac{i b \text{PolyLog}\left (2,\frac{i c \sqrt{-d} e^{i \csc ^{-1}(c x)}}{\sqrt{e}-\sqrt{c^2 d+e}}\right )}{2 e^2}-\frac{i b \text{PolyLog}\left (2,-\frac{i c \sqrt{-d} e^{i \csc ^{-1}(c x)}}{\sqrt{c^2 d+e}+\sqrt{e}}\right )}{2 e^2}-\frac{i b \text{PolyLog}\left (2,\frac{i c \sqrt{-d} e^{i \csc ^{-1}(c x)}}{\sqrt{c^2 d+e}+\sqrt{e}}\right )}{2 e^2}+\frac{i b \text{PolyLog}\left (2,e^{2 i \csc ^{-1}(c x)}\right )}{2 e^2}+\frac{\left (a+b \csc ^{-1}(c x)\right ) \log \left (1-\frac{i c \sqrt{-d} e^{i \csc ^{-1}(c x)}}{\sqrt{e}-\sqrt{c^2 d+e}}\right )}{2 e^2}+\frac{\left (a+b \csc ^{-1}(c x)\right ) \log \left (1+\frac{i c \sqrt{-d} e^{i \csc ^{-1}(c x)}}{\sqrt{e}-\sqrt{c^2 d+e}}\right )}{2 e^2}+\frac{\left (a+b \csc ^{-1}(c x)\right ) \log \left (1-\frac{i c \sqrt{-d} e^{i \csc ^{-1}(c x)}}{\sqrt{c^2 d+e}+\sqrt{e}}\right )}{2 e^2}+\frac{\left (a+b \csc ^{-1}(c x)\right ) \log \left (1+\frac{i c \sqrt{-d} e^{i \csc ^{-1}(c x)}}{\sqrt{c^2 d+e}+\sqrt{e}}\right )}{2 e^2}-\frac{a+b \csc ^{-1}(c x)}{2 e \left (\frac{d}{x^2}+e\right )}-\frac{\log \left (1-e^{2 i \csc ^{-1}(c x)}\right ) \left (a+b \csc ^{-1}(c x)\right )}{e^2}+\frac{b \tan ^{-1}\left (\frac{\sqrt{c^2 d+e}}{c \sqrt{e} x \sqrt{1-\frac{1}{c^2 x^2}}}\right )}{2 e^{3/2} \sqrt{c^2 d+e}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(a + b*ArcCsc[c*x])/(2*e*(e + d/x^2)) + (b*ArcTan[Sqrt[c^2*d + e]/(c*Sqrt[e]*Sqrt[1 - 1/(c^2*x^2)]*x)])/(2*e^
(3/2)*Sqrt[c^2*d + e]) + ((a + b*ArcCsc[c*x])*Log[1 - (I*c*Sqrt[-d]*E^(I*ArcCsc[c*x]))/(Sqrt[e] - Sqrt[c^2*d +
e])])/(2*e^2) + ((a + b*ArcCsc[c*x])*Log[1 + (I*c*Sqrt[-d]*E^(I*ArcCsc[c*x]))/(Sqrt[e] - Sqrt[c^2*d + e])])/(
2*e^2) + ((a + b*ArcCsc[c*x])*Log[1 - (I*c*Sqrt[-d]*E^(I*ArcCsc[c*x]))/(Sqrt[e] + Sqrt[c^2*d + e])])/(2*e^2) +
((a + b*ArcCsc[c*x])*Log[1 + (I*c*Sqrt[-d]*E^(I*ArcCsc[c*x]))/(Sqrt[e] + Sqrt[c^2*d + e])])/(2*e^2) - ((a + b
*ArcCsc[c*x])*Log[1 - E^((2*I)*ArcCsc[c*x])])/e^2 - ((I/2)*b*PolyLog[2, ((-I)*c*Sqrt[-d]*E^(I*ArcCsc[c*x]))/(S
qrt[e] - Sqrt[c^2*d + e])])/e^2 - ((I/2)*b*PolyLog[2, (I*c*Sqrt[-d]*E^(I*ArcCsc[c*x]))/(Sqrt[e] - Sqrt[c^2*d +
e])])/e^2 - ((I/2)*b*PolyLog[2, ((-I)*c*Sqrt[-d]*E^(I*ArcCsc[c*x]))/(Sqrt[e] + Sqrt[c^2*d + e])])/e^2 - ((I/2
)*b*PolyLog[2, (I*c*Sqrt[-d]*E^(I*ArcCsc[c*x]))/(Sqrt[e] + Sqrt[c^2*d + e])])/e^2 + ((I/2)*b*PolyLog[2, E^((2*
I)*ArcCsc[c*x])])/e^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 4625

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

Rule 3717

Int[((c_.) + (d_.)*(x_))^(m_.)*tan[(e_.) + Pi*(k_.) + (f_.)*(x_)], x_Symbol] :> Simp[(I*(c + d*x)^(m + 1))/(d*
(m + 1)), x] - Dist[2*I, Int[((c + d*x)^m*E^(2*I*k*Pi)*E^(2*I*(e + f*x)))/(1 + E^(2*I*k*Pi)*E^(2*I*(e + f*x)))
, x], x] /; FreeQ[{c, d, e, f}, x] && IntegerQ[4*k] && IGtQ[m, 0]

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]

Rule 4729

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

Rule 377

Int[((a_) + (b_.)*(x_)^(n_))^(p_)/((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Subst[Int[1/(c - (b*c - a*d)*x^n), x]
, x, x/(a + b*x^n)^(1/n)] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && EqQ[n*p + 1, 0] && IntegerQ[n]

Rule 205

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

Rule 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]

Rubi steps

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

Mathematica [B]  time = 2.04494, size = 1442, normalized size = 2.44 \[ \text{result too large to display} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(I*b*Pi^2 + (4*a*d)/(d + e*x^2) - (4*I)*b*Pi*ArcCsc[c*x] + (2*b*Sqrt[d]*ArcCsc[c*x])/(Sqrt[d] - I*Sqrt[e]*x) +
(2*b*Sqrt[d]*ArcCsc[c*x])/(Sqrt[d] + I*Sqrt[e]*x) + (8*I)*b*ArcCsc[c*x]^2 - 4*b*ArcSin[1/(c*x)] - (16*I)*b*Ar
cSin[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]] - (16*I)*b*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]] - 2*b*Pi*Log[1 + (Sqrt[e] - Sqrt[c^2*d + e])/(c*Sqrt[d]*E^(I*A
rcCsc[c*x]))] + 4*b*ArcCsc[c*x]*Log[1 + (Sqrt[e] - Sqrt[c^2*d + e])/(c*Sqrt[d]*E^(I*ArcCsc[c*x]))] - 8*b*ArcSi
n[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]))
] - 2*b*Pi*Log[1 + (-Sqrt[e] + Sqrt[c^2*d + e])/(c*Sqrt[d]*E^(I*ArcCsc[c*x]))] + 4*b*ArcCsc[c*x]*Log[1 + (-Sqr
t[e] + Sqrt[c^2*d + e])/(c*Sqrt[d]*E^(I*ArcCsc[c*x]))] - 8*b*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]))] - 2*b*Pi*Log[1 - (Sqrt[e] + Sqrt[c^2*d +
e])/(c*Sqrt[d]*E^(I*ArcCsc[c*x]))] + 4*b*ArcCsc[c*x]*Log[1 - (Sqrt[e] + Sqrt[c^2*d + e])/(c*Sqrt[d]*E^(I*ArcCs
c[c*x]))] + 8*b*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]))] - 2*b*Pi*Log[1 + (Sqrt[e] + Sqrt[c^2*d + e])/(c*Sqrt[d]*E^(I*ArcCsc[c*x]))] + 4*b*ArcCs
c[c*x]*Log[1 + (Sqrt[e] + Sqrt[c^2*d + e])/(c*Sqrt[d]*E^(I*ArcCsc[c*x]))] + 8*b*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]))] - 8*b*ArcCsc[c*x]*Log[1
- E^((2*I)*ArcCsc[c*x])] + 2*b*Pi*Log[Sqrt[e] - (I*Sqrt[d])/x] + 2*b*Pi*Log[Sqrt[e] + (I*Sqrt[d])/x] + (2*b*S
qrt[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))/(Sq
rt[-(c^2*d) - e]*(Sqrt[d] + I*Sqrt[e]*x))])/Sqrt[-(c^2*d) - e] + (2*b*Sqrt[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] + 4*a*Log[d + e*x^2] + (4*I)*b*PolyLog[2, (Sqrt[e] - Sqrt[c^2*d + e])/(c*Sqrt[d]*E^(I
*ArcCsc[c*x]))] + (4*I)*b*PolyLog[2, (-Sqrt[e] + Sqrt[c^2*d + e])/(c*Sqrt[d]*E^(I*ArcCsc[c*x]))] + (4*I)*b*Pol
yLog[2, -((Sqrt[e] + Sqrt[c^2*d + e])/(c*Sqrt[d]*E^(I*ArcCsc[c*x])))] + (4*I)*b*PolyLog[2, (Sqrt[e] + Sqrt[c^2
*d + e])/(c*Sqrt[d]*E^(I*ArcCsc[c*x]))] + (4*I)*b*PolyLog[2, E^((2*I)*ArcCsc[c*x])])/(8*e^2)

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Maple [C]  time = 0.536, size = 541, normalized size = 0.9 \begin{align*}{\frac{{c}^{2}ad}{2\,{e}^{2} \left ({c}^{2}e{x}^{2}+{c}^{2}d \right ) }}+{\frac{a\ln \left ({c}^{2}e{x}^{2}+{c}^{2}d \right ) }{2\,{e}^{2}}}-{\frac{{c}^{2}b{x}^{2}{\rm arccsc} \left (cx\right )}{2\,e \left ({c}^{2}e{x}^{2}+{c}^{2}d \right ) }}-{\frac{{\frac{i}{2}}b}{ \left ({c}^{2}d+e \right ){e}^{2}}\sqrt{e \left ({c}^{2}d+e \right ) }{\it Artanh} \left ({\frac{1}{4} \left ( 2\,{c}^{2}d \left ({\frac{i}{cx}}+\sqrt{1-{\frac{1}{{c}^{2}{x}^{2}}}} \right ) ^{2}-2\,{c}^{2}d-4\,e \right ){\frac{1}{\sqrt{{c}^{2}ed+{e}^{2}}}}} \right ) }-{\frac{{\frac{i}{4}}b}{{e}^{2}}\sum _{{\it \_R1}={\it RootOf} \left ({c}^{2}d{{\it \_Z}}^{4}+ \left ( -2\,{c}^{2}d-4\,e \right ){{\it \_Z}}^{2}+{c}^{2}d \right ) }{\frac{{{\it \_R1}}^{2}{c}^{2}d-{c}^{2}d-4\,e}{{{\it \_R1}}^{2}{c}^{2}d-{c}^{2}d-2\,e} \left ( i{\rm arccsc} \left (cx\right )\ln \left ({\frac{1}{{\it \_R1}} \left ({\it \_R1}-{\frac{i}{cx}}-\sqrt{1-{\frac{1}{{c}^{2}{x}^{2}}}} \right ) } \right ) +{\it dilog} \left ({\frac{1}{{\it \_R1}} \left ({\it \_R1}-{\frac{i}{cx}}-\sqrt{1-{\frac{1}{{c}^{2}{x}^{2}}}} \right ) } \right ) \right ) }}-{\frac{ib}{{e}^{2}}{\it dilog} \left ({\frac{i}{cx}}+\sqrt{1-{\frac{1}{{c}^{2}{x}^{2}}}} \right ) }-{\frac{b{\rm arccsc} \left (cx\right )}{{e}^{2}}\ln \left ( 1+{\frac{i}{cx}}+\sqrt{1-{\frac{1}{{c}^{2}{x}^{2}}}} \right ) }+{\frac{ib}{{e}^{2}}{\it dilog} \left ( 1+{\frac{i}{cx}}+\sqrt{1-{\frac{1}{{c}^{2}{x}^{2}}}} \right ) }-{\frac{{\frac{i}{4}}{c}^{2}bd}{{e}^{2}}\sum _{{\it \_R1}={\it RootOf} \left ({c}^{2}d{{\it \_Z}}^{4}+ \left ( -2\,{c}^{2}d-4\,e \right ){{\it \_Z}}^{2}+{c}^{2}d \right ) }{\frac{{{\it \_R1}}^{2}-1}{{{\it \_R1}}^{2}{c}^{2}d-{c}^{2}d-2\,e} \left ( i{\rm arccsc} \left (cx\right )\ln \left ({\frac{1}{{\it \_R1}} \left ({\it \_R1}-{\frac{i}{cx}}-\sqrt{1-{\frac{1}{{c}^{2}{x}^{2}}}} \right ) } \right ) +{\it dilog} \left ({\frac{1}{{\it \_R1}} \left ({\it \_R1}-{\frac{i}{cx}}-\sqrt{1-{\frac{1}{{c}^{2}{x}^{2}}}} \right ) } \right ) \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*c^2*a/e^2*d/(c^2*e*x^2+c^2*d)+1/2*a/e^2*ln(c^2*e*x^2+c^2*d)-1/2*c^2*b*x^2*arccsc(c*x)/e/(c^2*e*x^2+c^2*d)-
1/2*I*b*(e*(c^2*d+e))^(1/2)/(c^2*d+e)/e^2*arctanh(1/4*(2*c^2*d*(I/c/x+(1-1/c^2/x^2)^(1/2))^2-2*c^2*d-4*e)/(c^2
*d*e+e^2)^(1/2))-1/4*I*b/e^2*sum((_R1^2*c^2*d-c^2*d-4*e)/(_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))-I*b/e^2*dilog(I/c/x+(1-1/c^2/x^2)^(1/2))-b/e^2*arccsc(c*x)*ln(1+I/c/x+(1-1/c^2/x^2)^(1/2))+I*b/e^2*d
ilog(1+I/c/x+(1-1/c^2/x^2)^(1/2))-1/4*I*c^2*b/e^2*d*sum((_R1^2-1)/(_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))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{1}{2} \, a{\left (\frac{d}{e^{3} x^{2} + d e^{2}} + \frac{\log \left (e x^{2} + d\right )}{e^{2}}\right )} + b \int \frac{x^{3} \arctan \left (1, \sqrt{c x + 1} \sqrt{c x - 1}\right )}{e^{2} x^{4} + 2 \, d e x^{2} + d^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*a*(d/(e^3*x^2 + d*e^2) + log(e*x^2 + d)/e^2) + b*integrate(x^3*arctan2(1, sqrt(c*x + 1)*sqrt(c*x - 1))/(e^
2*x^4 + 2*d*e*x^2 + d^2), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{b x^{3} \operatorname{arccsc}\left (c x\right ) + a x^{3}}{e^{2} x^{4} + 2 \, d e x^{2} + d^{2}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral((b*x^3*arccsc(c*x) + a*x^3)/(e^2*x^4 + 2*d*e*x^2 + d^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(x**3*(a+b*acsc(c*x))/(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{{\left (b \operatorname{arccsc}\left (c x\right ) + a\right )} x^{3}}{{\left (e x^{2} + d\right )}^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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