3.34 \(\int x \csc ^{-1}(a+b x)^3 \, dx\)

Optimal. Leaf size=264 \[ \frac{6 i a \csc ^{-1}(a+b x) \text{PolyLog}\left (2,-e^{i \csc ^{-1}(a+b x)}\right )}{b^2}-\frac{6 i a \csc ^{-1}(a+b x) \text{PolyLog}\left (2,e^{i \csc ^{-1}(a+b x)}\right )}{b^2}+\frac{3 i \text{PolyLog}\left (2,e^{2 i \csc ^{-1}(a+b x)}\right )}{2 b^2}-\frac{6 a \text{PolyLog}\left (3,-e^{i \csc ^{-1}(a+b x)}\right )}{b^2}+\frac{6 a \text{PolyLog}\left (3,e^{i \csc ^{-1}(a+b x)}\right )}{b^2}-\frac{a^2 \csc ^{-1}(a+b x)^3}{2 b^2}+\frac{3 (a+b x) \sqrt{1-\frac{1}{(a+b x)^2}} \csc ^{-1}(a+b x)^2}{2 b^2}+\frac{3 i \csc ^{-1}(a+b x)^2}{2 b^2}-\frac{3 \csc ^{-1}(a+b x) \log \left (1-e^{2 i \csc ^{-1}(a+b x)}\right )}{b^2}-\frac{6 a \csc ^{-1}(a+b x)^2 \tanh ^{-1}\left (e^{i \csc ^{-1}(a+b x)}\right )}{b^2}+\frac{1}{2} x^2 \csc ^{-1}(a+b x)^3 \]

[Out]

(((3*I)/2)*ArcCsc[a + b*x]^2)/b^2 + (3*(a + b*x)*Sqrt[1 - (a + b*x)^(-2)]*ArcCsc[a + b*x]^2)/(2*b^2) - (a^2*Ar
cCsc[a + b*x]^3)/(2*b^2) + (x^2*ArcCsc[a + b*x]^3)/2 - (6*a*ArcCsc[a + b*x]^2*ArcTanh[E^(I*ArcCsc[a + b*x])])/
b^2 - (3*ArcCsc[a + b*x]*Log[1 - E^((2*I)*ArcCsc[a + b*x])])/b^2 + ((6*I)*a*ArcCsc[a + b*x]*PolyLog[2, -E^(I*A
rcCsc[a + b*x])])/b^2 - ((6*I)*a*ArcCsc[a + b*x]*PolyLog[2, E^(I*ArcCsc[a + b*x])])/b^2 + (((3*I)/2)*PolyLog[2
, E^((2*I)*ArcCsc[a + b*x])])/b^2 - (6*a*PolyLog[3, -E^(I*ArcCsc[a + b*x])])/b^2 + (6*a*PolyLog[3, E^(I*ArcCsc
[a + b*x])])/b^2

________________________________________________________________________________________

Rubi [A]  time = 0.255632, antiderivative size = 264, normalized size of antiderivative = 1., number of steps used = 16, number of rules used = 12, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 1.2, Rules used = {5259, 4427, 4190, 4183, 2531, 2282, 6589, 4184, 3717, 2190, 2279, 2391} \[ \frac{6 i a \csc ^{-1}(a+b x) \text{PolyLog}\left (2,-e^{i \csc ^{-1}(a+b x)}\right )}{b^2}-\frac{6 i a \csc ^{-1}(a+b x) \text{PolyLog}\left (2,e^{i \csc ^{-1}(a+b x)}\right )}{b^2}+\frac{3 i \text{PolyLog}\left (2,e^{2 i \csc ^{-1}(a+b x)}\right )}{2 b^2}-\frac{6 a \text{PolyLog}\left (3,-e^{i \csc ^{-1}(a+b x)}\right )}{b^2}+\frac{6 a \text{PolyLog}\left (3,e^{i \csc ^{-1}(a+b x)}\right )}{b^2}-\frac{a^2 \csc ^{-1}(a+b x)^3}{2 b^2}+\frac{3 (a+b x) \sqrt{1-\frac{1}{(a+b x)^2}} \csc ^{-1}(a+b x)^2}{2 b^2}+\frac{3 i \csc ^{-1}(a+b x)^2}{2 b^2}-\frac{3 \csc ^{-1}(a+b x) \log \left (1-e^{2 i \csc ^{-1}(a+b x)}\right )}{b^2}-\frac{6 a \csc ^{-1}(a+b x)^2 \tanh ^{-1}\left (e^{i \csc ^{-1}(a+b x)}\right )}{b^2}+\frac{1}{2} x^2 \csc ^{-1}(a+b x)^3 \]

Antiderivative was successfully verified.

[In]

Int[x*ArcCsc[a + b*x]^3,x]

[Out]

(((3*I)/2)*ArcCsc[a + b*x]^2)/b^2 + (3*(a + b*x)*Sqrt[1 - (a + b*x)^(-2)]*ArcCsc[a + b*x]^2)/(2*b^2) - (a^2*Ar
cCsc[a + b*x]^3)/(2*b^2) + (x^2*ArcCsc[a + b*x]^3)/2 - (6*a*ArcCsc[a + b*x]^2*ArcTanh[E^(I*ArcCsc[a + b*x])])/
b^2 - (3*ArcCsc[a + b*x]*Log[1 - E^((2*I)*ArcCsc[a + b*x])])/b^2 + ((6*I)*a*ArcCsc[a + b*x]*PolyLog[2, -E^(I*A
rcCsc[a + b*x])])/b^2 - ((6*I)*a*ArcCsc[a + b*x]*PolyLog[2, E^(I*ArcCsc[a + b*x])])/b^2 + (((3*I)/2)*PolyLog[2
, E^((2*I)*ArcCsc[a + b*x])])/b^2 - (6*a*PolyLog[3, -E^(I*ArcCsc[a + b*x])])/b^2 + (6*a*PolyLog[3, E^(I*ArcCsc
[a + b*x])])/b^2

Rule 5259

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

Rule 4427

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

Rule 4190

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

Rule 4183

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

Rule 2531

Int[Log[1 + (e_.)*((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.)]*((f_.) + (g_.)*(x_))^(m_.), x_Symbol] :> -Simp[((
f + g*x)^m*PolyLog[2, -(e*(F^(c*(a + b*x)))^n)])/(b*c*n*Log[F]), x] + Dist[(g*m)/(b*c*n*Log[F]), Int[(f + g*x)
^(m - 1)*PolyLog[2, -(e*(F^(c*(a + b*x)))^n)], x], x] /; FreeQ[{F, a, b, c, e, f, g, n}, x] && GtQ[m, 0]

Rule 2282

Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Dist[v/D[v, x], Subst[Int[FunctionOfExponentialFu
nction[u, x]/x, x], x, v], x]] /; FunctionOfExponentialQ[u, x] &&  !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; F
reeQ[{a, m, n}, x] && IntegerQ[m*n]] &&  !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x))*(F_)[v_] /; FreeQ[{a, b, c}, x
] && InverseFunctionQ[F[x]]]

Rule 6589

Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_Symbol] :> Simp[PolyLog[n + 1, c*(a
+ b*x)^p]/(e*p), x] /; FreeQ[{a, b, c, d, e, n, p}, x] && EqQ[b*d, a*e]

Rule 4184

Int[csc[(e_.) + (f_.)*(x_)]^2*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> -Simp[((c + d*x)^m*Cot[e + f*x])/f, x]
+ Dist[(d*m)/f, Int[(c + d*x)^(m - 1)*Cot[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && GtQ[m, 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]

Rubi steps

\begin{align*} \int x \csc ^{-1}(a+b x)^3 \, dx &=-\frac{\operatorname{Subst}\left (\int x^3 \cot (x) \csc (x) (-a+\csc (x)) \, dx,x,\csc ^{-1}(a+b x)\right )}{b^2}\\ &=\frac{1}{2} x^2 \csc ^{-1}(a+b x)^3-\frac{3 \operatorname{Subst}\left (\int x^2 (-a+\csc (x))^2 \, dx,x,\csc ^{-1}(a+b x)\right )}{2 b^2}\\ &=\frac{1}{2} x^2 \csc ^{-1}(a+b x)^3-\frac{3 \operatorname{Subst}\left (\int \left (a^2 x^2-2 a x^2 \csc (x)+x^2 \csc ^2(x)\right ) \, dx,x,\csc ^{-1}(a+b x)\right )}{2 b^2}\\ &=-\frac{a^2 \csc ^{-1}(a+b x)^3}{2 b^2}+\frac{1}{2} x^2 \csc ^{-1}(a+b x)^3-\frac{3 \operatorname{Subst}\left (\int x^2 \csc ^2(x) \, dx,x,\csc ^{-1}(a+b x)\right )}{2 b^2}+\frac{(3 a) \operatorname{Subst}\left (\int x^2 \csc (x) \, dx,x,\csc ^{-1}(a+b x)\right )}{b^2}\\ &=\frac{3 (a+b x) \sqrt{1-\frac{1}{(a+b x)^2}} \csc ^{-1}(a+b x)^2}{2 b^2}-\frac{a^2 \csc ^{-1}(a+b x)^3}{2 b^2}+\frac{1}{2} x^2 \csc ^{-1}(a+b x)^3-\frac{6 a \csc ^{-1}(a+b x)^2 \tanh ^{-1}\left (e^{i \csc ^{-1}(a+b x)}\right )}{b^2}-\frac{3 \operatorname{Subst}\left (\int x \cot (x) \, dx,x,\csc ^{-1}(a+b x)\right )}{b^2}-\frac{(6 a) \operatorname{Subst}\left (\int x \log \left (1-e^{i x}\right ) \, dx,x,\csc ^{-1}(a+b x)\right )}{b^2}+\frac{(6 a) \operatorname{Subst}\left (\int x \log \left (1+e^{i x}\right ) \, dx,x,\csc ^{-1}(a+b x)\right )}{b^2}\\ &=\frac{3 i \csc ^{-1}(a+b x)^2}{2 b^2}+\frac{3 (a+b x) \sqrt{1-\frac{1}{(a+b x)^2}} \csc ^{-1}(a+b x)^2}{2 b^2}-\frac{a^2 \csc ^{-1}(a+b x)^3}{2 b^2}+\frac{1}{2} x^2 \csc ^{-1}(a+b x)^3-\frac{6 a \csc ^{-1}(a+b x)^2 \tanh ^{-1}\left (e^{i \csc ^{-1}(a+b x)}\right )}{b^2}+\frac{6 i a \csc ^{-1}(a+b x) \text{Li}_2\left (-e^{i \csc ^{-1}(a+b x)}\right )}{b^2}-\frac{6 i a \csc ^{-1}(a+b x) \text{Li}_2\left (e^{i \csc ^{-1}(a+b x)}\right )}{b^2}+\frac{(6 i) \operatorname{Subst}\left (\int \frac{e^{2 i x} x}{1-e^{2 i x}} \, dx,x,\csc ^{-1}(a+b x)\right )}{b^2}-\frac{(6 i a) \operatorname{Subst}\left (\int \text{Li}_2\left (-e^{i x}\right ) \, dx,x,\csc ^{-1}(a+b x)\right )}{b^2}+\frac{(6 i a) \operatorname{Subst}\left (\int \text{Li}_2\left (e^{i x}\right ) \, dx,x,\csc ^{-1}(a+b x)\right )}{b^2}\\ &=\frac{3 i \csc ^{-1}(a+b x)^2}{2 b^2}+\frac{3 (a+b x) \sqrt{1-\frac{1}{(a+b x)^2}} \csc ^{-1}(a+b x)^2}{2 b^2}-\frac{a^2 \csc ^{-1}(a+b x)^3}{2 b^2}+\frac{1}{2} x^2 \csc ^{-1}(a+b x)^3-\frac{6 a \csc ^{-1}(a+b x)^2 \tanh ^{-1}\left (e^{i \csc ^{-1}(a+b x)}\right )}{b^2}-\frac{3 \csc ^{-1}(a+b x) \log \left (1-e^{2 i \csc ^{-1}(a+b x)}\right )}{b^2}+\frac{6 i a \csc ^{-1}(a+b x) \text{Li}_2\left (-e^{i \csc ^{-1}(a+b x)}\right )}{b^2}-\frac{6 i a \csc ^{-1}(a+b x) \text{Li}_2\left (e^{i \csc ^{-1}(a+b x)}\right )}{b^2}+\frac{3 \operatorname{Subst}\left (\int \log \left (1-e^{2 i x}\right ) \, dx,x,\csc ^{-1}(a+b x)\right )}{b^2}-\frac{(6 a) \operatorname{Subst}\left (\int \frac{\text{Li}_2(-x)}{x} \, dx,x,e^{i \csc ^{-1}(a+b x)}\right )}{b^2}+\frac{(6 a) \operatorname{Subst}\left (\int \frac{\text{Li}_2(x)}{x} \, dx,x,e^{i \csc ^{-1}(a+b x)}\right )}{b^2}\\ &=\frac{3 i \csc ^{-1}(a+b x)^2}{2 b^2}+\frac{3 (a+b x) \sqrt{1-\frac{1}{(a+b x)^2}} \csc ^{-1}(a+b x)^2}{2 b^2}-\frac{a^2 \csc ^{-1}(a+b x)^3}{2 b^2}+\frac{1}{2} x^2 \csc ^{-1}(a+b x)^3-\frac{6 a \csc ^{-1}(a+b x)^2 \tanh ^{-1}\left (e^{i \csc ^{-1}(a+b x)}\right )}{b^2}-\frac{3 \csc ^{-1}(a+b x) \log \left (1-e^{2 i \csc ^{-1}(a+b x)}\right )}{b^2}+\frac{6 i a \csc ^{-1}(a+b x) \text{Li}_2\left (-e^{i \csc ^{-1}(a+b x)}\right )}{b^2}-\frac{6 i a \csc ^{-1}(a+b x) \text{Li}_2\left (e^{i \csc ^{-1}(a+b x)}\right )}{b^2}-\frac{6 a \text{Li}_3\left (-e^{i \csc ^{-1}(a+b x)}\right )}{b^2}+\frac{6 a \text{Li}_3\left (e^{i \csc ^{-1}(a+b x)}\right )}{b^2}-\frac{(3 i) \operatorname{Subst}\left (\int \frac{\log (1-x)}{x} \, dx,x,e^{2 i \csc ^{-1}(a+b x)}\right )}{2 b^2}\\ &=\frac{3 i \csc ^{-1}(a+b x)^2}{2 b^2}+\frac{3 (a+b x) \sqrt{1-\frac{1}{(a+b x)^2}} \csc ^{-1}(a+b x)^2}{2 b^2}-\frac{a^2 \csc ^{-1}(a+b x)^3}{2 b^2}+\frac{1}{2} x^2 \csc ^{-1}(a+b x)^3-\frac{6 a \csc ^{-1}(a+b x)^2 \tanh ^{-1}\left (e^{i \csc ^{-1}(a+b x)}\right )}{b^2}-\frac{3 \csc ^{-1}(a+b x) \log \left (1-e^{2 i \csc ^{-1}(a+b x)}\right )}{b^2}+\frac{6 i a \csc ^{-1}(a+b x) \text{Li}_2\left (-e^{i \csc ^{-1}(a+b x)}\right )}{b^2}-\frac{6 i a \csc ^{-1}(a+b x) \text{Li}_2\left (e^{i \csc ^{-1}(a+b x)}\right )}{b^2}+\frac{3 i \text{Li}_2\left (e^{2 i \csc ^{-1}(a+b x)}\right )}{2 b^2}-\frac{6 a \text{Li}_3\left (-e^{i \csc ^{-1}(a+b x)}\right )}{b^2}+\frac{6 a \text{Li}_3\left (e^{i \csc ^{-1}(a+b x)}\right )}{b^2}\\ \end{align*}

Mathematica [A]  time = 0.805551, size = 314, normalized size = 1.19 \[ \frac{12 i a \csc ^{-1}(a+b x) \text{PolyLog}\left (2,-e^{i \csc ^{-1}(a+b x)}\right )-12 i a \csc ^{-1}(a+b x) \text{PolyLog}\left (2,e^{i \csc ^{-1}(a+b x)}\right )+3 i \text{PolyLog}\left (2,e^{2 i \csc ^{-1}(a+b x)}\right )-12 a \text{PolyLog}\left (3,-e^{i \csc ^{-1}(a+b x)}\right )+12 a \text{PolyLog}\left (3,e^{i \csc ^{-1}(a+b x)}\right )+3 a \sqrt{\frac{a^2+2 a b x+b^2 x^2-1}{(a+b x)^2}} \csc ^{-1}(a+b x)^2+3 b x \sqrt{\frac{a^2+2 a b x+b^2 x^2-1}{(a+b x)^2}} \csc ^{-1}(a+b x)^2-a^2 \csc ^{-1}(a+b x)^3+b^2 x^2 \csc ^{-1}(a+b x)^3+3 i \csc ^{-1}(a+b x)^2+6 a \csc ^{-1}(a+b x)^2 \log \left (1-e^{i \csc ^{-1}(a+b x)}\right )-6 a \csc ^{-1}(a+b x)^2 \log \left (1+e^{i \csc ^{-1}(a+b x)}\right )-6 \csc ^{-1}(a+b x) \log \left (1-e^{2 i \csc ^{-1}(a+b x)}\right )}{2 b^2} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[x*ArcCsc[a + b*x]^3,x]

[Out]

((3*I)*ArcCsc[a + b*x]^2 + 3*a*Sqrt[(-1 + a^2 + 2*a*b*x + b^2*x^2)/(a + b*x)^2]*ArcCsc[a + b*x]^2 + 3*b*x*Sqrt
[(-1 + a^2 + 2*a*b*x + b^2*x^2)/(a + b*x)^2]*ArcCsc[a + b*x]^2 - a^2*ArcCsc[a + b*x]^3 + b^2*x^2*ArcCsc[a + b*
x]^3 + 6*a*ArcCsc[a + b*x]^2*Log[1 - E^(I*ArcCsc[a + b*x])] - 6*a*ArcCsc[a + b*x]^2*Log[1 + E^(I*ArcCsc[a + b*
x])] - 6*ArcCsc[a + b*x]*Log[1 - E^((2*I)*ArcCsc[a + b*x])] + (12*I)*a*ArcCsc[a + b*x]*PolyLog[2, -E^(I*ArcCsc
[a + b*x])] - (12*I)*a*ArcCsc[a + b*x]*PolyLog[2, E^(I*ArcCsc[a + b*x])] + (3*I)*PolyLog[2, E^((2*I)*ArcCsc[a
+ b*x])] - 12*a*PolyLog[3, -E^(I*ArcCsc[a + b*x])] + 12*a*PolyLog[3, E^(I*ArcCsc[a + b*x])])/(2*b^2)

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Maple [A]  time = 0.425, size = 481, normalized size = 1.8 \begin{align*}{\frac{{x}^{2} \left ({\rm arccsc} \left (bx+a\right ) \right ) ^{3}}{2}}-{\frac{{a}^{2} \left ({\rm arccsc} \left (bx+a\right ) \right ) ^{3}}{2\,{b}^{2}}}+{\frac{3\,x \left ({\rm arccsc} \left (bx+a\right ) \right ) ^{2}}{2\,b}\sqrt{{\frac{-1+ \left ( bx+a \right ) ^{2}}{ \left ( bx+a \right ) ^{2}}}}}+{\frac{3\, \left ({\rm arccsc} \left (bx+a\right ) \right ) ^{2}a}{2\,{b}^{2}}\sqrt{{\frac{-1+ \left ( bx+a \right ) ^{2}}{ \left ( bx+a \right ) ^{2}}}}}+{\frac{{\frac{3\,i}{2}} \left ({\rm arccsc} \left (bx+a\right ) \right ) ^{2}}{{b}^{2}}}+3\,{\frac{ \left ({\rm arccsc} \left (bx+a\right ) \right ) ^{2}a}{{b}^{2}}\ln \left ( 1-{\frac{i}{bx+a}}-\sqrt{1- \left ( bx+a \right ) ^{-2}} \right ) }-3\,{\frac{ \left ({\rm arccsc} \left (bx+a\right ) \right ) ^{2}a}{{b}^{2}}\ln \left ( 1+{\frac{i}{bx+a}}+\sqrt{1- \left ( bx+a \right ) ^{-2}} \right ) }-3\,{\frac{{\rm arccsc} \left (bx+a\right )}{{b}^{2}}\ln \left ( 1-{\frac{i}{bx+a}}-\sqrt{1- \left ( bx+a \right ) ^{-2}} \right ) }+6\,{\frac{a}{{b}^{2}}{\it polylog} \left ( 3,{\frac{i}{bx+a}}+\sqrt{1- \left ( bx+a \right ) ^{-2}} \right ) }-3\,{\frac{{\rm arccsc} \left (bx+a\right )}{{b}^{2}}\ln \left ( 1+{\frac{i}{bx+a}}+\sqrt{1- \left ( bx+a \right ) ^{-2}} \right ) }-6\,{\frac{a}{{b}^{2}}{\it polylog} \left ( 3,{\frac{-i}{bx+a}}-\sqrt{1- \left ( bx+a \right ) ^{-2}} \right ) }-{\frac{6\,ia{\rm arccsc} \left (bx+a\right )}{{b}^{2}}{\it polylog} \left ( 2,{\frac{i}{bx+a}}+\sqrt{1- \left ( bx+a \right ) ^{-2}} \right ) }+{\frac{6\,ia{\rm arccsc} \left (bx+a\right )}{{b}^{2}}{\it polylog} \left ( 2,{\frac{-i}{bx+a}}-\sqrt{1- \left ( bx+a \right ) ^{-2}} \right ) }+{\frac{3\,i}{{b}^{2}}{\it polylog} \left ( 2,{\frac{i}{bx+a}}+\sqrt{1- \left ( bx+a \right ) ^{-2}} \right ) }+{\frac{3\,i}{{b}^{2}}{\it polylog} \left ( 2,{\frac{-i}{bx+a}}-\sqrt{1- \left ( bx+a \right ) ^{-2}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*arccsc(b*x+a)^3,x)

[Out]

1/2*x^2*arccsc(b*x+a)^3-1/2*a^2*arccsc(b*x+a)^3/b^2+3/2/b*arccsc(b*x+a)^2*((-1+(b*x+a)^2)/(b*x+a)^2)^(1/2)*x+3
/2/b^2*arccsc(b*x+a)^2*((-1+(b*x+a)^2)/(b*x+a)^2)^(1/2)*a+3/2*I*arccsc(b*x+a)^2/b^2+3/b^2*a*arccsc(b*x+a)^2*ln
(1-I/(b*x+a)-(1-1/(b*x+a)^2)^(1/2))-3/b^2*a*arccsc(b*x+a)^2*ln(1+I/(b*x+a)+(1-1/(b*x+a)^2)^(1/2))-3/b^2*arccsc
(b*x+a)*ln(1-I/(b*x+a)-(1-1/(b*x+a)^2)^(1/2))+6*a*polylog(3,I/(b*x+a)+(1-1/(b*x+a)^2)^(1/2))/b^2-3/b^2*arccsc(
b*x+a)*ln(1+I/(b*x+a)+(1-1/(b*x+a)^2)^(1/2))-6*a*polylog(3,-I/(b*x+a)-(1-1/(b*x+a)^2)^(1/2))/b^2-6*I*a*arccsc(
b*x+a)*polylog(2,I/(b*x+a)+(1-1/(b*x+a)^2)^(1/2))/b^2+6*I*a*arccsc(b*x+a)*polylog(2,-I/(b*x+a)-(1-1/(b*x+a)^2)
^(1/2))/b^2+3*I/b^2*polylog(2,I/(b*x+a)+(1-1/(b*x+a)^2)^(1/2))+3*I/b^2*polylog(2,-I/(b*x+a)-(1-1/(b*x+a)^2)^(1
/2))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{1}{2} \, x^{2} \arctan \left (1, \sqrt{b x + a + 1} \sqrt{b x + a - 1}\right )^{3} - \frac{3}{8} \, x^{2} \arctan \left (1, \sqrt{b x + a + 1} \sqrt{b x + a - 1}\right ) \log \left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )^{2} - \int \frac{3 \,{\left (8 \,{\left (b^{3} x^{4} \arctan \left (1, \sqrt{b x + a + 1} \sqrt{b x + a - 1}\right ) + 3 \, a b^{2} x^{3} \arctan \left (1, \sqrt{b x + a + 1} \sqrt{b x + a - 1}\right ) +{\left (3 \, a^{2} \arctan \left (1, \sqrt{b x + a + 1} \sqrt{b x + a - 1}\right ) - \arctan \left (1, \sqrt{b x + a + 1} \sqrt{b x + a - 1}\right )\right )} b x^{2} +{\left (a^{3} \arctan \left (1, \sqrt{b x + a + 1} \sqrt{b x + a - 1}\right ) - a \arctan \left (1, \sqrt{b x + a + 1} \sqrt{b x + a - 1}\right )\right )} x\right )} \log \left (b x + a\right )^{2} -{\left (4 \, b x^{2} \arctan \left (1, \sqrt{b x + a + 1} \sqrt{b x + a - 1}\right )^{2} - b x^{2} \log \left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )^{2}\right )} \sqrt{b x + a + 1} \sqrt{b x + a - 1} - 4 \,{\left (b^{3} x^{4} \arctan \left (1, \sqrt{b x + a + 1} \sqrt{b x + a - 1}\right ) + 2 \, a b^{2} x^{3} \arctan \left (1, \sqrt{b x + a + 1} \sqrt{b x + a - 1}\right ) +{\left (a^{2} \arctan \left (1, \sqrt{b x + a + 1} \sqrt{b x + a - 1}\right ) - \arctan \left (1, \sqrt{b x + a + 1} \sqrt{b x + a - 1}\right )\right )} b x^{2} + 2 \,{\left (b^{3} x^{4} \arctan \left (1, \sqrt{b x + a + 1} \sqrt{b x + a - 1}\right ) + 3 \, a b^{2} x^{3} \arctan \left (1, \sqrt{b x + a + 1} \sqrt{b x + a - 1}\right ) +{\left (3 \, a^{2} \arctan \left (1, \sqrt{b x + a + 1} \sqrt{b x + a - 1}\right ) - \arctan \left (1, \sqrt{b x + a + 1} \sqrt{b x + a - 1}\right )\right )} b x^{2} +{\left (a^{3} \arctan \left (1, \sqrt{b x + a + 1} \sqrt{b x + a - 1}\right ) - a \arctan \left (1, \sqrt{b x + a + 1} \sqrt{b x + a - 1}\right )\right )} x\right )} \log \left (b x + a\right )\right )} \log \left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )\right )}}{8 \,{\left (b^{3} x^{3} + 3 \, a b^{2} x^{2} + a^{3} +{\left (3 \, a^{2} - 1\right )} b x - a\right )}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*arccsc(b*x+a)^3,x, algorithm="maxima")

[Out]

1/2*x^2*arctan2(1, sqrt(b*x + a + 1)*sqrt(b*x + a - 1))^3 - 3/8*x^2*arctan2(1, sqrt(b*x + a + 1)*sqrt(b*x + a
- 1))*log(b^2*x^2 + 2*a*b*x + a^2)^2 - integrate(3/8*(8*(b^3*x^4*arctan2(1, sqrt(b*x + a + 1)*sqrt(b*x + a - 1
)) + 3*a*b^2*x^3*arctan2(1, sqrt(b*x + a + 1)*sqrt(b*x + a - 1)) + (3*a^2*arctan2(1, sqrt(b*x + a + 1)*sqrt(b*
x + a - 1)) - arctan2(1, sqrt(b*x + a + 1)*sqrt(b*x + a - 1)))*b*x^2 + (a^3*arctan2(1, sqrt(b*x + a + 1)*sqrt(
b*x + a - 1)) - a*arctan2(1, sqrt(b*x + a + 1)*sqrt(b*x + a - 1)))*x)*log(b*x + a)^2 - (4*b*x^2*arctan2(1, sqr
t(b*x + a + 1)*sqrt(b*x + a - 1))^2 - b*x^2*log(b^2*x^2 + 2*a*b*x + a^2)^2)*sqrt(b*x + a + 1)*sqrt(b*x + a - 1
) - 4*(b^3*x^4*arctan2(1, sqrt(b*x + a + 1)*sqrt(b*x + a - 1)) + 2*a*b^2*x^3*arctan2(1, sqrt(b*x + a + 1)*sqrt
(b*x + a - 1)) + (a^2*arctan2(1, sqrt(b*x + a + 1)*sqrt(b*x + a - 1)) - arctan2(1, sqrt(b*x + a + 1)*sqrt(b*x
+ a - 1)))*b*x^2 + 2*(b^3*x^4*arctan2(1, sqrt(b*x + a + 1)*sqrt(b*x + a - 1)) + 3*a*b^2*x^3*arctan2(1, sqrt(b*
x + a + 1)*sqrt(b*x + a - 1)) + (3*a^2*arctan2(1, sqrt(b*x + a + 1)*sqrt(b*x + a - 1)) - arctan2(1, sqrt(b*x +
a + 1)*sqrt(b*x + a - 1)))*b*x^2 + (a^3*arctan2(1, sqrt(b*x + a + 1)*sqrt(b*x + a - 1)) - a*arctan2(1, sqrt(b
*x + a + 1)*sqrt(b*x + a - 1)))*x)*log(b*x + a))*log(b^2*x^2 + 2*a*b*x + a^2))/(b^3*x^3 + 3*a*b^2*x^2 + a^3 +
(3*a^2 - 1)*b*x - a), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*arccsc(b*x+a)^3,x, algorithm="fricas")

[Out]

integral(x*arccsc(b*x + a)^3, x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int x \operatorname{acsc}^{3}{\left (a + b x \right )}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*acsc(b*x+a)**3,x)

[Out]

Integral(x*acsc(a + b*x)**3, x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int x \operatorname{arccsc}\left (b x + a\right )^{3}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*arccsc(b*x+a)^3,x, algorithm="giac")

[Out]

integrate(x*arccsc(b*x + a)^3, x)