3.1.33 \(\int x^2 \csc ^{-1}(a+b x)^3 \, dx\) [33]

Optimal. Leaf size=464 \[ \frac {(a+b x) \csc ^{-1}(a+b x)}{b^3}-\frac {3 i a \csc ^{-1}(a+b x)^2}{b^3}-\frac {3 a (a+b x) \sqrt {1-\frac {1}{(a+b x)^2}} \csc ^{-1}(a+b x)^2}{b^3}+\frac {(a+b x)^2 \sqrt {1-\frac {1}{(a+b x)^2}} \csc ^{-1}(a+b x)^2}{2 b^3}+\frac {a^3 \csc ^{-1}(a+b x)^3}{3 b^3}+\frac {1}{3} x^3 \csc ^{-1}(a+b x)^3+\frac {\csc ^{-1}(a+b x)^2 \tanh ^{-1}\left (e^{i \csc ^{-1}(a+b x)}\right )}{b^3}+\frac {6 a^2 \csc ^{-1}(a+b x)^2 \tanh ^{-1}\left (e^{i \csc ^{-1}(a+b x)}\right )}{b^3}+\frac {\tanh ^{-1}\left (\sqrt {1-\frac {1}{(a+b x)^2}}\right )}{b^3}+\frac {6 a \csc ^{-1}(a+b x) \log \left (1-e^{2 i \csc ^{-1}(a+b x)}\right )}{b^3}-\frac {i \csc ^{-1}(a+b x) \text {PolyLog}\left (2,-e^{i \csc ^{-1}(a+b x)}\right )}{b^3}-\frac {6 i a^2 \csc ^{-1}(a+b x) \text {PolyLog}\left (2,-e^{i \csc ^{-1}(a+b x)}\right )}{b^3}+\frac {i \csc ^{-1}(a+b x) \text {PolyLog}\left (2,e^{i \csc ^{-1}(a+b x)}\right )}{b^3}+\frac {6 i a^2 \csc ^{-1}(a+b x) \text {PolyLog}\left (2,e^{i \csc ^{-1}(a+b x)}\right )}{b^3}-\frac {3 i a \text {PolyLog}\left (2,e^{2 i \csc ^{-1}(a+b x)}\right )}{b^3}+\frac {\text {PolyLog}\left (3,-e^{i \csc ^{-1}(a+b x)}\right )}{b^3}+\frac {6 a^2 \text {PolyLog}\left (3,-e^{i \csc ^{-1}(a+b x)}\right )}{b^3}-\frac {\text {PolyLog}\left (3,e^{i \csc ^{-1}(a+b x)}\right )}{b^3}-\frac {6 a^2 \text {PolyLog}\left (3,e^{i \csc ^{-1}(a+b x)}\right )}{b^3} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 0.28, antiderivative size = 464, normalized size of antiderivative = 1.00, number of steps used = 25, number of rules used = 14, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.167, Rules used = {5367, 4512, 4275, 4268, 2611, 2320, 6724, 4269, 3798, 2221, 2317, 2438, 4271, 3855} \begin {gather*} \frac {a^3 \csc ^{-1}(a+b x)^3}{3 b^3}-\frac {6 i a^2 \csc ^{-1}(a+b x) \text {Li}_2\left (-e^{i \csc ^{-1}(a+b x)}\right )}{b^3}+\frac {6 i a^2 \csc ^{-1}(a+b x) \text {Li}_2\left (e^{i \csc ^{-1}(a+b x)}\right )}{b^3}+\frac {6 a^2 \text {Li}_3\left (-e^{i \csc ^{-1}(a+b x)}\right )}{b^3}-\frac {6 a^2 \text {Li}_3\left (e^{i \csc ^{-1}(a+b x)}\right )}{b^3}+\frac {6 a^2 \csc ^{-1}(a+b x)^2 \tanh ^{-1}\left (e^{i \csc ^{-1}(a+b x)}\right )}{b^3}-\frac {i \csc ^{-1}(a+b x) \text {Li}_2\left (-e^{i \csc ^{-1}(a+b x)}\right )}{b^3}+\frac {i \csc ^{-1}(a+b x) \text {Li}_2\left (e^{i \csc ^{-1}(a+b x)}\right )}{b^3}-\frac {3 i a \text {Li}_2\left (e^{2 i \csc ^{-1}(a+b x)}\right )}{b^3}+\frac {\text {Li}_3\left (-e^{i \csc ^{-1}(a+b x)}\right )}{b^3}-\frac {\text {Li}_3\left (e^{i \csc ^{-1}(a+b x)}\right )}{b^3}+\frac {(a+b x)^2 \sqrt {1-\frac {1}{(a+b x)^2}} \csc ^{-1}(a+b x)^2}{2 b^3}-\frac {3 a (a+b x) \sqrt {1-\frac {1}{(a+b x)^2}} \csc ^{-1}(a+b x)^2}{b^3}-\frac {3 i a \csc ^{-1}(a+b x)^2}{b^3}+\frac {(a+b x) \csc ^{-1}(a+b x)}{b^3}+\frac {\tanh ^{-1}\left (\sqrt {1-\frac {1}{(a+b x)^2}}\right )}{b^3}+\frac {6 a \csc ^{-1}(a+b x) \log \left (1-e^{2 i \csc ^{-1}(a+b x)}\right )}{b^3}+\frac {\csc ^{-1}(a+b x)^2 \tanh ^{-1}\left (e^{i \csc ^{-1}(a+b x)}\right )}{b^3}+\frac {1}{3} x^3 \csc ^{-1}(a+b x)^3 \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^2*ArcCsc[a + b*x]^3,x]

[Out]

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

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 2320

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 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 2611

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 3798

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 3855

Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]

Rule 4268

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 4269

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 4271

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

Rule 4275

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 4512

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 5367

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 6724

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]

Rubi steps

\begin {align*} \int x^2 \csc ^{-1}(a+b x)^3 \, dx &=-\frac {\text {Subst}\left (\int x^3 \cot (x) \csc (x) (-a+\csc (x))^2 \, dx,x,\csc ^{-1}(a+b x)\right )}{b^3}\\ &=\frac {1}{3} x^3 \csc ^{-1}(a+b x)^3-\frac {\text {Subst}\left (\int x^2 (-a+\csc (x))^3 \, dx,x,\csc ^{-1}(a+b x)\right )}{b^3}\\ &=\frac {1}{3} x^3 \csc ^{-1}(a+b x)^3-\frac {\text {Subst}\left (\int \left (-a^3 x^2+3 a^2 x^2 \csc (x)-3 a x^2 \csc ^2(x)+x^2 \csc ^3(x)\right ) \, dx,x,\csc ^{-1}(a+b x)\right )}{b^3}\\ &=\frac {a^3 \csc ^{-1}(a+b x)^3}{3 b^3}+\frac {1}{3} x^3 \csc ^{-1}(a+b x)^3-\frac {\text {Subst}\left (\int x^2 \csc ^3(x) \, dx,x,\csc ^{-1}(a+b x)\right )}{b^3}+\frac {(3 a) \text {Subst}\left (\int x^2 \csc ^2(x) \, dx,x,\csc ^{-1}(a+b x)\right )}{b^3}-\frac {\left (3 a^2\right ) \text {Subst}\left (\int x^2 \csc (x) \, dx,x,\csc ^{-1}(a+b x)\right )}{b^3}\\ &=\frac {(a+b x) \csc ^{-1}(a+b x)}{b^3}-\frac {3 a (a+b x) \sqrt {1-\frac {1}{(a+b x)^2}} \csc ^{-1}(a+b x)^2}{b^3}+\frac {(a+b x)^2 \sqrt {1-\frac {1}{(a+b x)^2}} \csc ^{-1}(a+b x)^2}{2 b^3}+\frac {a^3 \csc ^{-1}(a+b x)^3}{3 b^3}+\frac {1}{3} x^3 \csc ^{-1}(a+b x)^3+\frac {6 a^2 \csc ^{-1}(a+b x)^2 \tanh ^{-1}\left (e^{i \csc ^{-1}(a+b x)}\right )}{b^3}-\frac {\text {Subst}\left (\int x^2 \csc (x) \, dx,x,\csc ^{-1}(a+b x)\right )}{2 b^3}-\frac {\text {Subst}\left (\int \csc (x) \, dx,x,\csc ^{-1}(a+b x)\right )}{b^3}+\frac {(6 a) \text {Subst}\left (\int x \cot (x) \, dx,x,\csc ^{-1}(a+b x)\right )}{b^3}+\frac {\left (6 a^2\right ) \text {Subst}\left (\int x \log \left (1-e^{i x}\right ) \, dx,x,\csc ^{-1}(a+b x)\right )}{b^3}-\frac {\left (6 a^2\right ) \text {Subst}\left (\int x \log \left (1+e^{i x}\right ) \, dx,x,\csc ^{-1}(a+b x)\right )}{b^3}\\ &=\frac {(a+b x) \csc ^{-1}(a+b x)}{b^3}-\frac {3 i a \csc ^{-1}(a+b x)^2}{b^3}-\frac {3 a (a+b x) \sqrt {1-\frac {1}{(a+b x)^2}} \csc ^{-1}(a+b x)^2}{b^3}+\frac {(a+b x)^2 \sqrt {1-\frac {1}{(a+b x)^2}} \csc ^{-1}(a+b x)^2}{2 b^3}+\frac {a^3 \csc ^{-1}(a+b x)^3}{3 b^3}+\frac {1}{3} x^3 \csc ^{-1}(a+b x)^3+\frac {\csc ^{-1}(a+b x)^2 \tanh ^{-1}\left (e^{i \csc ^{-1}(a+b x)}\right )}{b^3}+\frac {6 a^2 \csc ^{-1}(a+b x)^2 \tanh ^{-1}\left (e^{i \csc ^{-1}(a+b x)}\right )}{b^3}+\frac {\tanh ^{-1}\left (\sqrt {1-\frac {1}{(a+b x)^2}}\right )}{b^3}-\frac {6 i a^2 \csc ^{-1}(a+b x) \text {Li}_2\left (-e^{i \csc ^{-1}(a+b x)}\right )}{b^3}+\frac {6 i a^2 \csc ^{-1}(a+b x) \text {Li}_2\left (e^{i \csc ^{-1}(a+b x)}\right )}{b^3}+\frac {\text {Subst}\left (\int x \log \left (1-e^{i x}\right ) \, dx,x,\csc ^{-1}(a+b x)\right )}{b^3}-\frac {\text {Subst}\left (\int x \log \left (1+e^{i x}\right ) \, dx,x,\csc ^{-1}(a+b x)\right )}{b^3}-\frac {(12 i a) \text {Subst}\left (\int \frac {e^{2 i x} x}{1-e^{2 i x}} \, dx,x,\csc ^{-1}(a+b x)\right )}{b^3}+\frac {\left (6 i a^2\right ) \text {Subst}\left (\int \text {Li}_2\left (-e^{i x}\right ) \, dx,x,\csc ^{-1}(a+b x)\right )}{b^3}-\frac {\left (6 i a^2\right ) \text {Subst}\left (\int \text {Li}_2\left (e^{i x}\right ) \, dx,x,\csc ^{-1}(a+b x)\right )}{b^3}\\ &=\frac {(a+b x) \csc ^{-1}(a+b x)}{b^3}-\frac {3 i a \csc ^{-1}(a+b x)^2}{b^3}-\frac {3 a (a+b x) \sqrt {1-\frac {1}{(a+b x)^2}} \csc ^{-1}(a+b x)^2}{b^3}+\frac {(a+b x)^2 \sqrt {1-\frac {1}{(a+b x)^2}} \csc ^{-1}(a+b x)^2}{2 b^3}+\frac {a^3 \csc ^{-1}(a+b x)^3}{3 b^3}+\frac {1}{3} x^3 \csc ^{-1}(a+b x)^3+\frac {\csc ^{-1}(a+b x)^2 \tanh ^{-1}\left (e^{i \csc ^{-1}(a+b x)}\right )}{b^3}+\frac {6 a^2 \csc ^{-1}(a+b x)^2 \tanh ^{-1}\left (e^{i \csc ^{-1}(a+b x)}\right )}{b^3}+\frac {\tanh ^{-1}\left (\sqrt {1-\frac {1}{(a+b x)^2}}\right )}{b^3}+\frac {6 a \csc ^{-1}(a+b x) \log \left (1-e^{2 i \csc ^{-1}(a+b x)}\right )}{b^3}-\frac {i \csc ^{-1}(a+b x) \text {Li}_2\left (-e^{i \csc ^{-1}(a+b x)}\right )}{b^3}-\frac {6 i a^2 \csc ^{-1}(a+b x) \text {Li}_2\left (-e^{i \csc ^{-1}(a+b x)}\right )}{b^3}+\frac {i \csc ^{-1}(a+b x) \text {Li}_2\left (e^{i \csc ^{-1}(a+b x)}\right )}{b^3}+\frac {6 i a^2 \csc ^{-1}(a+b x) \text {Li}_2\left (e^{i \csc ^{-1}(a+b x)}\right )}{b^3}+\frac {i \text {Subst}\left (\int \text {Li}_2\left (-e^{i x}\right ) \, dx,x,\csc ^{-1}(a+b x)\right )}{b^3}-\frac {i \text {Subst}\left (\int \text {Li}_2\left (e^{i x}\right ) \, dx,x,\csc ^{-1}(a+b x)\right )}{b^3}-\frac {(6 a) \text {Subst}\left (\int \log \left (1-e^{2 i x}\right ) \, dx,x,\csc ^{-1}(a+b x)\right )}{b^3}+\frac {\left (6 a^2\right ) \text {Subst}\left (\int \frac {\text {Li}_2(-x)}{x} \, dx,x,e^{i \csc ^{-1}(a+b x)}\right )}{b^3}-\frac {\left (6 a^2\right ) \text {Subst}\left (\int \frac {\text {Li}_2(x)}{x} \, dx,x,e^{i \csc ^{-1}(a+b x)}\right )}{b^3}\\ &=\frac {(a+b x) \csc ^{-1}(a+b x)}{b^3}-\frac {3 i a \csc ^{-1}(a+b x)^2}{b^3}-\frac {3 a (a+b x) \sqrt {1-\frac {1}{(a+b x)^2}} \csc ^{-1}(a+b x)^2}{b^3}+\frac {(a+b x)^2 \sqrt {1-\frac {1}{(a+b x)^2}} \csc ^{-1}(a+b x)^2}{2 b^3}+\frac {a^3 \csc ^{-1}(a+b x)^3}{3 b^3}+\frac {1}{3} x^3 \csc ^{-1}(a+b x)^3+\frac {\csc ^{-1}(a+b x)^2 \tanh ^{-1}\left (e^{i \csc ^{-1}(a+b x)}\right )}{b^3}+\frac {6 a^2 \csc ^{-1}(a+b x)^2 \tanh ^{-1}\left (e^{i \csc ^{-1}(a+b x)}\right )}{b^3}+\frac {\tanh ^{-1}\left (\sqrt {1-\frac {1}{(a+b x)^2}}\right )}{b^3}+\frac {6 a \csc ^{-1}(a+b x) \log \left (1-e^{2 i \csc ^{-1}(a+b x)}\right )}{b^3}-\frac {i \csc ^{-1}(a+b x) \text {Li}_2\left (-e^{i \csc ^{-1}(a+b x)}\right )}{b^3}-\frac {6 i a^2 \csc ^{-1}(a+b x) \text {Li}_2\left (-e^{i \csc ^{-1}(a+b x)}\right )}{b^3}+\frac {i \csc ^{-1}(a+b x) \text {Li}_2\left (e^{i \csc ^{-1}(a+b x)}\right )}{b^3}+\frac {6 i a^2 \csc ^{-1}(a+b x) \text {Li}_2\left (e^{i \csc ^{-1}(a+b x)}\right )}{b^3}+\frac {6 a^2 \text {Li}_3\left (-e^{i \csc ^{-1}(a+b x)}\right )}{b^3}-\frac {6 a^2 \text {Li}_3\left (e^{i \csc ^{-1}(a+b x)}\right )}{b^3}+\frac {\text {Subst}\left (\int \frac {\text {Li}_2(-x)}{x} \, dx,x,e^{i \csc ^{-1}(a+b x)}\right )}{b^3}-\frac {\text {Subst}\left (\int \frac {\text {Li}_2(x)}{x} \, dx,x,e^{i \csc ^{-1}(a+b x)}\right )}{b^3}+\frac {(3 i a) \text {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{2 i \csc ^{-1}(a+b x)}\right )}{b^3}\\ &=\frac {(a+b x) \csc ^{-1}(a+b x)}{b^3}-\frac {3 i a \csc ^{-1}(a+b x)^2}{b^3}-\frac {3 a (a+b x) \sqrt {1-\frac {1}{(a+b x)^2}} \csc ^{-1}(a+b x)^2}{b^3}+\frac {(a+b x)^2 \sqrt {1-\frac {1}{(a+b x)^2}} \csc ^{-1}(a+b x)^2}{2 b^3}+\frac {a^3 \csc ^{-1}(a+b x)^3}{3 b^3}+\frac {1}{3} x^3 \csc ^{-1}(a+b x)^3+\frac {\csc ^{-1}(a+b x)^2 \tanh ^{-1}\left (e^{i \csc ^{-1}(a+b x)}\right )}{b^3}+\frac {6 a^2 \csc ^{-1}(a+b x)^2 \tanh ^{-1}\left (e^{i \csc ^{-1}(a+b x)}\right )}{b^3}+\frac {\tanh ^{-1}\left (\sqrt {1-\frac {1}{(a+b x)^2}}\right )}{b^3}+\frac {6 a \csc ^{-1}(a+b x) \log \left (1-e^{2 i \csc ^{-1}(a+b x)}\right )}{b^3}-\frac {i \csc ^{-1}(a+b x) \text {Li}_2\left (-e^{i \csc ^{-1}(a+b x)}\right )}{b^3}-\frac {6 i a^2 \csc ^{-1}(a+b x) \text {Li}_2\left (-e^{i \csc ^{-1}(a+b x)}\right )}{b^3}+\frac {i \csc ^{-1}(a+b x) \text {Li}_2\left (e^{i \csc ^{-1}(a+b x)}\right )}{b^3}+\frac {6 i a^2 \csc ^{-1}(a+b x) \text {Li}_2\left (e^{i \csc ^{-1}(a+b x)}\right )}{b^3}-\frac {3 i a \text {Li}_2\left (e^{2 i \csc ^{-1}(a+b x)}\right )}{b^3}+\frac {\text {Li}_3\left (-e^{i \csc ^{-1}(a+b x)}\right )}{b^3}+\frac {6 a^2 \text {Li}_3\left (-e^{i \csc ^{-1}(a+b x)}\right )}{b^3}-\frac {\text {Li}_3\left (e^{i \csc ^{-1}(a+b x)}\right )}{b^3}-\frac {6 a^2 \text {Li}_3\left (e^{i \csc ^{-1}(a+b x)}\right )}{b^3}\\ \end {align*}

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Mathematica [A]
time = 5.99, size = 656, normalized size = 1.41 \begin {gather*} -\frac {72 i a \csc ^{-1}(a+b x)^2-12 \csc ^{-1}(a+b x) \cot \left (\frac {1}{2} \csc ^{-1}(a+b x)\right )+36 a \csc ^{-1}(a+b x)^2 \cot \left (\frac {1}{2} \csc ^{-1}(a+b x)\right )-2 \csc ^{-1}(a+b x)^3 \cot \left (\frac {1}{2} \csc ^{-1}(a+b x)\right )-12 a^2 \csc ^{-1}(a+b x)^3 \cot \left (\frac {1}{2} \csc ^{-1}(a+b x)\right )-3 \csc ^{-1}(a+b x)^2 \csc ^2\left (\frac {1}{2} \csc ^{-1}(a+b x)\right )+6 a \csc ^{-1}(a+b x)^3 \csc ^2\left (\frac {1}{2} \csc ^{-1}(a+b x)\right )-\frac {\csc ^{-1}(a+b x)^3 \csc ^4\left (\frac {1}{2} \csc ^{-1}(a+b x)\right )}{2 (a+b x)}+12 \csc ^{-1}(a+b x)^2 \log \left (1-e^{i \csc ^{-1}(a+b x)}\right )+72 a^2 \csc ^{-1}(a+b x)^2 \log \left (1-e^{i \csc ^{-1}(a+b x)}\right )-12 \csc ^{-1}(a+b x)^2 \log \left (1+e^{i \csc ^{-1}(a+b x)}\right )-72 a^2 \csc ^{-1}(a+b x)^2 \log \left (1+e^{i \csc ^{-1}(a+b x)}\right )-144 a \csc ^{-1}(a+b x) \log \left (1-e^{2 i \csc ^{-1}(a+b x)}\right )+24 \log \left (\tan \left (\frac {1}{2} \csc ^{-1}(a+b x)\right )\right )+24 i \left (1+6 a^2\right ) \csc ^{-1}(a+b x) \text {PolyLog}\left (2,-e^{i \csc ^{-1}(a+b x)}\right )-24 i \left (1+6 a^2\right ) \csc ^{-1}(a+b x) \text {PolyLog}\left (2,e^{i \csc ^{-1}(a+b x)}\right )+72 i a \text {PolyLog}\left (2,e^{2 i \csc ^{-1}(a+b x)}\right )-24 \text {PolyLog}\left (3,-e^{i \csc ^{-1}(a+b x)}\right )-144 a^2 \text {PolyLog}\left (3,-e^{i \csc ^{-1}(a+b x)}\right )+24 \text {PolyLog}\left (3,e^{i \csc ^{-1}(a+b x)}\right )+144 a^2 \text {PolyLog}\left (3,e^{i \csc ^{-1}(a+b x)}\right )+3 \csc ^{-1}(a+b x)^2 \sec ^2\left (\frac {1}{2} \csc ^{-1}(a+b x)\right )+6 a \csc ^{-1}(a+b x)^3 \sec ^2\left (\frac {1}{2} \csc ^{-1}(a+b x)\right )-8 (a+b x)^3 \csc ^{-1}(a+b x)^3 \sin ^4\left (\frac {1}{2} \csc ^{-1}(a+b x)\right )-12 \csc ^{-1}(a+b x) \tan \left (\frac {1}{2} \csc ^{-1}(a+b x)\right )-36 a \csc ^{-1}(a+b x)^2 \tan \left (\frac {1}{2} \csc ^{-1}(a+b x)\right )-2 \csc ^{-1}(a+b x)^3 \tan \left (\frac {1}{2} \csc ^{-1}(a+b x)\right )-12 a^2 \csc ^{-1}(a+b x)^3 \tan \left (\frac {1}{2} \csc ^{-1}(a+b x)\right )}{24 b^3} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

Integrate[x^2*ArcCsc[a + b*x]^3,x]

[Out]

-1/24*((72*I)*a*ArcCsc[a + b*x]^2 - 12*ArcCsc[a + b*x]*Cot[ArcCsc[a + b*x]/2] + 36*a*ArcCsc[a + b*x]^2*Cot[Arc
Csc[a + b*x]/2] - 2*ArcCsc[a + b*x]^3*Cot[ArcCsc[a + b*x]/2] - 12*a^2*ArcCsc[a + b*x]^3*Cot[ArcCsc[a + b*x]/2]
- 3*ArcCsc[a + b*x]^2*Csc[ArcCsc[a + b*x]/2]^2 + 6*a*ArcCsc[a + b*x]^3*Csc[ArcCsc[a + b*x]/2]^2 - (ArcCsc[a +
b*x]^3*Csc[ArcCsc[a + b*x]/2]^4)/(2*(a + b*x)) + 12*ArcCsc[a + b*x]^2*Log[1 - E^(I*ArcCsc[a + b*x])] + 72*a^2
*ArcCsc[a + b*x]^2*Log[1 - E^(I*ArcCsc[a + b*x])] - 12*ArcCsc[a + b*x]^2*Log[1 + E^(I*ArcCsc[a + b*x])] - 72*a
^2*ArcCsc[a + b*x]^2*Log[1 + E^(I*ArcCsc[a + b*x])] - 144*a*ArcCsc[a + b*x]*Log[1 - E^((2*I)*ArcCsc[a + b*x])]
+ 24*Log[Tan[ArcCsc[a + b*x]/2]] + (24*I)*(1 + 6*a^2)*ArcCsc[a + b*x]*PolyLog[2, -E^(I*ArcCsc[a + b*x])] - (2
4*I)*(1 + 6*a^2)*ArcCsc[a + b*x]*PolyLog[2, E^(I*ArcCsc[a + b*x])] + (72*I)*a*PolyLog[2, E^((2*I)*ArcCsc[a + b
*x])] - 24*PolyLog[3, -E^(I*ArcCsc[a + b*x])] - 144*a^2*PolyLog[3, -E^(I*ArcCsc[a + b*x])] + 24*PolyLog[3, E^(
I*ArcCsc[a + b*x])] + 144*a^2*PolyLog[3, E^(I*ArcCsc[a + b*x])] + 3*ArcCsc[a + b*x]^2*Sec[ArcCsc[a + b*x]/2]^2
+ 6*a*ArcCsc[a + b*x]^3*Sec[ArcCsc[a + b*x]/2]^2 - 8*(a + b*x)^3*ArcCsc[a + b*x]^3*Sin[ArcCsc[a + b*x]/2]^4 -
12*ArcCsc[a + b*x]*Tan[ArcCsc[a + b*x]/2] - 36*a*ArcCsc[a + b*x]^2*Tan[ArcCsc[a + b*x]/2] - 2*ArcCsc[a + b*x]
^3*Tan[ArcCsc[a + b*x]/2] - 12*a^2*ArcCsc[a + b*x]^3*Tan[ArcCsc[a + b*x]/2])/b^3

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Maple [A]
time = 9.85, size = 749, normalized size = 1.61 Too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*arccsc(b*x+a)^3,x,method=_RETURNVERBOSE)

[Out]

1/b^3*(1/6*arccsc(b*x+a)*(6*arccsc(b*x+a)^2*a^2*(b*x+a)-6*arccsc(b*x+a)^2*a*(b*x+a)^2+2*arccsc(b*x+a)^2*(b*x+a
)^3-18*arccsc(b*x+a)*(((b*x+a)^2-1)/(b*x+a)^2)^(1/2)*a*(b*x+a)+3*arccsc(b*x+a)*(((b*x+a)^2-1)/(b*x+a)^2)^(1/2)
*(b*x+a)^2+18*I*arccsc(b*x+a)*a+6*b*x+6*a)-1/2*arccsc(b*x+a)^2*ln(1-I/(b*x+a)-(1-1/(b*x+a)^2)^(1/2))-6*I*a*pol
ylog(2,-I/(b*x+a)-(1-1/(b*x+a)^2)^(1/2))-polylog(3,I/(b*x+a)+(1-1/(b*x+a)^2)^(1/2))+1/2*arccsc(b*x+a)^2*ln(1+I
/(b*x+a)+(1-1/(b*x+a)^2)^(1/2))-I*arccsc(b*x+a)*polylog(2,-I/(b*x+a)-(1-1/(b*x+a)^2)^(1/2))+polylog(3,-I/(b*x+
a)-(1-1/(b*x+a)^2)^(1/2))+2*arctanh(I/(b*x+a)+(1-1/(b*x+a)^2)^(1/2))-6*I*a*polylog(2,I/(b*x+a)+(1-1/(b*x+a)^2)
^(1/2))+6*a*arccsc(b*x+a)*ln(1-I/(b*x+a)-(1-1/(b*x+a)^2)^(1/2))+6*I*a^2*arccsc(b*x+a)*polylog(2,I/(b*x+a)+(1-1
/(b*x+a)^2)^(1/2))+6*a*arccsc(b*x+a)*ln(1+I/(b*x+a)+(1-1/(b*x+a)^2)^(1/2))+I*arccsc(b*x+a)*polylog(2,I/(b*x+a)
+(1-1/(b*x+a)^2)^(1/2))-3*a^2*arccsc(b*x+a)^2*ln(1-I/(b*x+a)-(1-1/(b*x+a)^2)^(1/2))-6*I*arccsc(b*x+a)^2*a-6*a^
2*polylog(3,I/(b*x+a)+(1-1/(b*x+a)^2)^(1/2))+3*a^2*arccsc(b*x+a)^2*ln(1+I/(b*x+a)+(1-1/(b*x+a)^2)^(1/2))-6*I*a
^2*arccsc(b*x+a)*polylog(2,-I/(b*x+a)-(1-1/(b*x+a)^2)^(1/2))+6*a^2*polylog(3,-I/(b*x+a)-(1-1/(b*x+a)^2)^(1/2))
)

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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(x^2*arccsc(b*x+a)^3,x, algorithm="maxima")

[Out]

1/3*x^3*arctan2(1, sqrt(b*x + a + 1)*sqrt(b*x + a - 1))^3 - 1/4*x^3*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(1/4*(12*(b^3*x^5*arctan2(1, sqrt(b*x + a + 1)*sqrt(b*x + a -
1)) + 3*a*b^2*x^4*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^3 + (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^2)*log(b*x + a)^2 - (4*b*x^3*arctan2(1,
sqrt(b*x + a + 1)*sqrt(b*x + a - 1))^2 - b*x^3*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^5*arctan2(1, sqrt(b*x + a + 1)*sqrt(b*x + a - 1)) + 2*a*b^2*x^4*arctan2(1, sqrt(b*x + a + 1)*s
qrt(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^3 + 3*(b^3*x^5*arctan2(1, sqrt(b*x + a + 1)*sqrt(b*x + a - 1)) + 3*a*b^2*x^4*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^3 + (a^3*arctan2(1, sqrt(b*x + a + 1)*sqrt(b*x + a - 1)) - a*arctan2(1, sqr
t(b*x + a + 1)*sqrt(b*x + a - 1)))*x^2)*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 = 0.00, 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(x^2*arccsc(b*x+a)^3,x, algorithm="fricas")

[Out]

integral(x^2*arccsc(b*x + a)^3, x)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int x^{2} \operatorname {acsc}^{3}{\left (a + b x \right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**2*acsc(b*x+a)**3,x)

[Out]

Integral(x**2*acsc(a + b*x)**3, x)

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(x^2*arccsc(b*x + a)^3, x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int x^2\,{\mathrm {asin}\left (\frac {1}{a+b\,x}\right )}^3 \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*asin(1/(a + b*x))^3,x)

[Out]

int(x^2*asin(1/(a + b*x))^3, x)

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