\(\int x^2 (a+b \csc ^{-1}(c x))^3 \, dx\) [25]

Optimal result

Integrand size = 14, antiderivative size = 220 \[ \int x^2 \left (a+b \csc ^{-1}(c x)\right )^3 \, dx=\frac {b^2 x \left (a+b \csc ^{-1}(c x)\right )}{c^2}+\frac {b \sqrt {1-\frac {1}{c^2 x^2}} x^2 \left (a+b \csc ^{-1}(c x)\right )^2}{2 c}+\frac {1}{3} x^3 \left (a+b \csc ^{-1}(c x)\right )^3+\frac {b \left (a+b \csc ^{-1}(c x)\right )^2 \text {arctanh}\left (e^{i \csc ^{-1}(c x)}\right )}{c^3}+\frac {b^3 \text {arctanh}\left (\sqrt {1-\frac {1}{c^2 x^2}}\right )}{c^3}-\frac {i b^2 \left (a+b \csc ^{-1}(c x)\right ) \operatorname {PolyLog}\left (2,-e^{i \csc ^{-1}(c x)}\right )}{c^3}+\frac {i b^2 \left (a+b \csc ^{-1}(c x)\right ) \operatorname {PolyLog}\left (2,e^{i \csc ^{-1}(c x)}\right )}{c^3}+\frac {b^3 \operatorname {PolyLog}\left (3,-e^{i \csc ^{-1}(c x)}\right )}{c^3}-\frac {b^3 \operatorname {PolyLog}\left (3,e^{i \csc ^{-1}(c x)}\right )}{c^3} \]

[Out]

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

Rubi [A] (verified)

Time = 0.16 (sec) , antiderivative size = 220, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.571, Rules used = {5331, 4495, 4271, 3855, 4268, 2611, 2320, 6724} \[ \int x^2 \left (a+b \csc ^{-1}(c x)\right )^3 \, dx=\frac {b \text {arctanh}\left (e^{i \csc ^{-1}(c x)}\right ) \left (a+b \csc ^{-1}(c x)\right )^2}{c^3}-\frac {i b^2 \operatorname {PolyLog}\left (2,-e^{i \csc ^{-1}(c x)}\right ) \left (a+b \csc ^{-1}(c x)\right )}{c^3}+\frac {i b^2 \operatorname {PolyLog}\left (2,e^{i \csc ^{-1}(c x)}\right ) \left (a+b \csc ^{-1}(c x)\right )}{c^3}+\frac {b^2 x \left (a+b \csc ^{-1}(c x)\right )}{c^2}+\frac {b x^2 \sqrt {1-\frac {1}{c^2 x^2}} \left (a+b \csc ^{-1}(c x)\right )^2}{2 c}+\frac {1}{3} x^3 \left (a+b \csc ^{-1}(c x)\right )^3+\frac {b^3 \text {arctanh}\left (\sqrt {1-\frac {1}{c^2 x^2}}\right )}{c^3}+\frac {b^3 \operatorname {PolyLog}\left (3,-e^{i \csc ^{-1}(c x)}\right )}{c^3}-\frac {b^3 \operatorname {PolyLog}\left (3,e^{i \csc ^{-1}(c x)}\right )}{c^3} \]

[In]

Int[x^2*(a + b*ArcCsc[c*x])^3,x]

[Out]

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

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 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 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 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 4495

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

Rule 5331

Int[((a_.) + ArcCsc[(c_.)*(x_)]*(b_.))^(n_)*(x_)^(m_.), x_Symbol] :> Dist[-(c^(m + 1))^(-1), Subst[Int[(a + b*
x)^n*Csc[x]^(m + 1)*Cot[x], x], x, ArcCsc[c*x]], x] /; FreeQ[{a, b, c}, x] && IntegerQ[n] && IntegerQ[m] && (G
tQ[n, 0] || LtQ[m, -1])

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*} \text {integral}& = -\frac {\text {Subst}\left (\int (a+b x)^3 \cot (x) \csc ^3(x) \, dx,x,\csc ^{-1}(c x)\right )}{c^3} \\ & = \frac {1}{3} x^3 \left (a+b \csc ^{-1}(c x)\right )^3-\frac {b \text {Subst}\left (\int (a+b x)^2 \csc ^3(x) \, dx,x,\csc ^{-1}(c x)\right )}{c^3} \\ & = \frac {b^2 x \left (a+b \csc ^{-1}(c x)\right )}{c^2}+\frac {b \sqrt {1-\frac {1}{c^2 x^2}} x^2 \left (a+b \csc ^{-1}(c x)\right )^2}{2 c}+\frac {1}{3} x^3 \left (a+b \csc ^{-1}(c x)\right )^3-\frac {b \text {Subst}\left (\int (a+b x)^2 \csc (x) \, dx,x,\csc ^{-1}(c x)\right )}{2 c^3}-\frac {b^3 \text {Subst}\left (\int \csc (x) \, dx,x,\csc ^{-1}(c x)\right )}{c^3} \\ & = \frac {b^2 x \left (a+b \csc ^{-1}(c x)\right )}{c^2}+\frac {b \sqrt {1-\frac {1}{c^2 x^2}} x^2 \left (a+b \csc ^{-1}(c x)\right )^2}{2 c}+\frac {1}{3} x^3 \left (a+b \csc ^{-1}(c x)\right )^3+\frac {b \left (a+b \csc ^{-1}(c x)\right )^2 \text {arctanh}\left (e^{i \csc ^{-1}(c x)}\right )}{c^3}+\frac {b^3 \text {arctanh}\left (\sqrt {1-\frac {1}{c^2 x^2}}\right )}{c^3}+\frac {b^2 \text {Subst}\left (\int (a+b x) \log \left (1-e^{i x}\right ) \, dx,x,\csc ^{-1}(c x)\right )}{c^3}-\frac {b^2 \text {Subst}\left (\int (a+b x) \log \left (1+e^{i x}\right ) \, dx,x,\csc ^{-1}(c x)\right )}{c^3} \\ & = \frac {b^2 x \left (a+b \csc ^{-1}(c x)\right )}{c^2}+\frac {b \sqrt {1-\frac {1}{c^2 x^2}} x^2 \left (a+b \csc ^{-1}(c x)\right )^2}{2 c}+\frac {1}{3} x^3 \left (a+b \csc ^{-1}(c x)\right )^3+\frac {b \left (a+b \csc ^{-1}(c x)\right )^2 \text {arctanh}\left (e^{i \csc ^{-1}(c x)}\right )}{c^3}+\frac {b^3 \text {arctanh}\left (\sqrt {1-\frac {1}{c^2 x^2}}\right )}{c^3}-\frac {i b^2 \left (a+b \csc ^{-1}(c x)\right ) \operatorname {PolyLog}\left (2,-e^{i \csc ^{-1}(c x)}\right )}{c^3}+\frac {i b^2 \left (a+b \csc ^{-1}(c x)\right ) \operatorname {PolyLog}\left (2,e^{i \csc ^{-1}(c x)}\right )}{c^3}+\frac {\left (i b^3\right ) \text {Subst}\left (\int \operatorname {PolyLog}\left (2,-e^{i x}\right ) \, dx,x,\csc ^{-1}(c x)\right )}{c^3}-\frac {\left (i b^3\right ) \text {Subst}\left (\int \operatorname {PolyLog}\left (2,e^{i x}\right ) \, dx,x,\csc ^{-1}(c x)\right )}{c^3} \\ & = \frac {b^2 x \left (a+b \csc ^{-1}(c x)\right )}{c^2}+\frac {b \sqrt {1-\frac {1}{c^2 x^2}} x^2 \left (a+b \csc ^{-1}(c x)\right )^2}{2 c}+\frac {1}{3} x^3 \left (a+b \csc ^{-1}(c x)\right )^3+\frac {b \left (a+b \csc ^{-1}(c x)\right )^2 \text {arctanh}\left (e^{i \csc ^{-1}(c x)}\right )}{c^3}+\frac {b^3 \text {arctanh}\left (\sqrt {1-\frac {1}{c^2 x^2}}\right )}{c^3}-\frac {i b^2 \left (a+b \csc ^{-1}(c x)\right ) \operatorname {PolyLog}\left (2,-e^{i \csc ^{-1}(c x)}\right )}{c^3}+\frac {i b^2 \left (a+b \csc ^{-1}(c x)\right ) \operatorname {PolyLog}\left (2,e^{i \csc ^{-1}(c x)}\right )}{c^3}+\frac {b^3 \text {Subst}\left (\int \frac {\operatorname {PolyLog}(2,-x)}{x} \, dx,x,e^{i \csc ^{-1}(c x)}\right )}{c^3}-\frac {b^3 \text {Subst}\left (\int \frac {\operatorname {PolyLog}(2,x)}{x} \, dx,x,e^{i \csc ^{-1}(c x)}\right )}{c^3} \\ & = \frac {b^2 x \left (a+b \csc ^{-1}(c x)\right )}{c^2}+\frac {b \sqrt {1-\frac {1}{c^2 x^2}} x^2 \left (a+b \csc ^{-1}(c x)\right )^2}{2 c}+\frac {1}{3} x^3 \left (a+b \csc ^{-1}(c x)\right )^3+\frac {b \left (a+b \csc ^{-1}(c x)\right )^2 \text {arctanh}\left (e^{i \csc ^{-1}(c x)}\right )}{c^3}+\frac {b^3 \text {arctanh}\left (\sqrt {1-\frac {1}{c^2 x^2}}\right )}{c^3}-\frac {i b^2 \left (a+b \csc ^{-1}(c x)\right ) \operatorname {PolyLog}\left (2,-e^{i \csc ^{-1}(c x)}\right )}{c^3}+\frac {i b^2 \left (a+b \csc ^{-1}(c x)\right ) \operatorname {PolyLog}\left (2,e^{i \csc ^{-1}(c x)}\right )}{c^3}+\frac {b^3 \operatorname {PolyLog}\left (3,-e^{i \csc ^{-1}(c x)}\right )}{c^3}-\frac {b^3 \operatorname {PolyLog}\left (3,e^{i \csc ^{-1}(c x)}\right )}{c^3} \\ \end{align*}

Mathematica [B] (warning: unable to verify)

Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(580\) vs. \(2(220)=440\).

Time = 7.39 (sec) , antiderivative size = 580, normalized size of antiderivative = 2.64 \[ \int x^2 \left (a+b \csc ^{-1}(c x)\right )^3 \, dx=\frac {a^3 x^3}{3}+\frac {a^2 b x^2 \sqrt {\frac {-1+c^2 x^2}{c^2 x^2}}}{2 c}+a^2 b x^3 \csc ^{-1}(c x)+\frac {a^2 b \log \left (x \left (1+\sqrt {\frac {-1+c^2 x^2}{c^2 x^2}}\right )\right )}{2 c^3}+\frac {a b^2 \left (-8 i \operatorname {PolyLog}\left (2,-e^{i \csc ^{-1}(c x)}\right )+2 c^3 x^3 \left (2+4 \csc ^{-1}(c x)^2-2 \cos \left (2 \csc ^{-1}(c x)\right )-\frac {3 \csc ^{-1}(c x) \log \left (1-e^{i \csc ^{-1}(c x)}\right )}{c x}+\frac {3 \csc ^{-1}(c x) \log \left (1+e^{i \csc ^{-1}(c x)}\right )}{c x}+\frac {4 i \operatorname {PolyLog}\left (2,e^{i \csc ^{-1}(c x)}\right )}{c^3 x^3}+2 \csc ^{-1}(c x) \sin \left (2 \csc ^{-1}(c x)\right )+\csc ^{-1}(c x) \log \left (1-e^{i \csc ^{-1}(c x)}\right ) \sin \left (3 \csc ^{-1}(c x)\right )-\csc ^{-1}(c x) \log \left (1+e^{i \csc ^{-1}(c x)}\right ) \sin \left (3 \csc ^{-1}(c x)\right )\right )\right )}{8 c^3}+\frac {b^3 \left (24 \csc ^{-1}(c x) \cot \left (\frac {1}{2} \csc ^{-1}(c x)\right )+4 \csc ^{-1}(c x)^3 \cot \left (\frac {1}{2} \csc ^{-1}(c x)\right )+6 \csc ^{-1}(c x)^2 \csc ^2\left (\frac {1}{2} \csc ^{-1}(c x)\right )+\frac {\csc ^{-1}(c x)^3 \csc ^4\left (\frac {1}{2} \csc ^{-1}(c x)\right )}{c x}-24 \csc ^{-1}(c x)^2 \log \left (1-e^{i \csc ^{-1}(c x)}\right )+24 \csc ^{-1}(c x)^2 \log \left (1+e^{i \csc ^{-1}(c x)}\right )-48 \log \left (\tan \left (\frac {1}{2} \csc ^{-1}(c x)\right )\right )-48 i \csc ^{-1}(c x) \operatorname {PolyLog}\left (2,-e^{i \csc ^{-1}(c x)}\right )+48 i \csc ^{-1}(c x) \operatorname {PolyLog}\left (2,e^{i \csc ^{-1}(c x)}\right )+48 \operatorname {PolyLog}\left (3,-e^{i \csc ^{-1}(c x)}\right )-48 \operatorname {PolyLog}\left (3,e^{i \csc ^{-1}(c x)}\right )-6 \csc ^{-1}(c x)^2 \sec ^2\left (\frac {1}{2} \csc ^{-1}(c x)\right )+16 c^3 x^3 \csc ^{-1}(c x)^3 \sin ^4\left (\frac {1}{2} \csc ^{-1}(c x)\right )+24 \csc ^{-1}(c x) \tan \left (\frac {1}{2} \csc ^{-1}(c x)\right )+4 \csc ^{-1}(c x)^3 \tan \left (\frac {1}{2} \csc ^{-1}(c x)\right )\right )}{48 c^3} \]

[In]

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

[Out]

(a^3*x^3)/3 + (a^2*b*x^2*Sqrt[(-1 + c^2*x^2)/(c^2*x^2)])/(2*c) + a^2*b*x^3*ArcCsc[c*x] + (a^2*b*Log[x*(1 + Sqr
t[(-1 + c^2*x^2)/(c^2*x^2)])])/(2*c^3) + (a*b^2*((-8*I)*PolyLog[2, -E^(I*ArcCsc[c*x])] + 2*c^3*x^3*(2 + 4*ArcC
sc[c*x]^2 - 2*Cos[2*ArcCsc[c*x]] - (3*ArcCsc[c*x]*Log[1 - E^(I*ArcCsc[c*x])])/(c*x) + (3*ArcCsc[c*x]*Log[1 + E
^(I*ArcCsc[c*x])])/(c*x) + ((4*I)*PolyLog[2, E^(I*ArcCsc[c*x])])/(c^3*x^3) + 2*ArcCsc[c*x]*Sin[2*ArcCsc[c*x]]
+ ArcCsc[c*x]*Log[1 - E^(I*ArcCsc[c*x])]*Sin[3*ArcCsc[c*x]] - ArcCsc[c*x]*Log[1 + E^(I*ArcCsc[c*x])]*Sin[3*Arc
Csc[c*x]])))/(8*c^3) + (b^3*(24*ArcCsc[c*x]*Cot[ArcCsc[c*x]/2] + 4*ArcCsc[c*x]^3*Cot[ArcCsc[c*x]/2] + 6*ArcCsc
[c*x]^2*Csc[ArcCsc[c*x]/2]^2 + (ArcCsc[c*x]^3*Csc[ArcCsc[c*x]/2]^4)/(c*x) - 24*ArcCsc[c*x]^2*Log[1 - E^(I*ArcC
sc[c*x])] + 24*ArcCsc[c*x]^2*Log[1 + E^(I*ArcCsc[c*x])] - 48*Log[Tan[ArcCsc[c*x]/2]] - (48*I)*ArcCsc[c*x]*Poly
Log[2, -E^(I*ArcCsc[c*x])] + (48*I)*ArcCsc[c*x]*PolyLog[2, E^(I*ArcCsc[c*x])] + 48*PolyLog[3, -E^(I*ArcCsc[c*x
])] - 48*PolyLog[3, E^(I*ArcCsc[c*x])] - 6*ArcCsc[c*x]^2*Sec[ArcCsc[c*x]/2]^2 + 16*c^3*x^3*ArcCsc[c*x]^3*Sin[A
rcCsc[c*x]/2]^4 + 24*ArcCsc[c*x]*Tan[ArcCsc[c*x]/2] + 4*ArcCsc[c*x]^3*Tan[ArcCsc[c*x]/2]))/(48*c^3)

Maple [A] (verified)

Time = 1.90 (sec) , antiderivative size = 535, normalized size of antiderivative = 2.43

method	result	size
derivativedivides	\(\frac {\frac {c^{3} x^{3} a^{3}}{3}+b^{3} \left (\frac {\operatorname {arccsc}\left (c x \right ) \left (2 c^{2} x^{2} \operatorname {arccsc}\left (c x \right )^{2}+3 \,\operatorname {arccsc}\left (c x \right ) c x \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}+6\right ) c x}{6}-\frac {\operatorname {arccsc}\left (c x \right )^{2} \ln \left (1-\frac {i}{c x}-\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )}{2}+i \operatorname {arccsc}\left (c x \right ) \operatorname {polylog}\left (2, \frac {i}{c x}+\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )-\operatorname {polylog}\left (3, \frac {i}{c x}+\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )+\frac {\operatorname {arccsc}\left (c x \right )^{2} \ln \left (1+\frac {i}{c x}+\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )}{2}-i \operatorname {arccsc}\left (c x \right ) \operatorname {polylog}\left (2, -\frac {i}{c x}-\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )+\operatorname {polylog}\left (3, -\frac {i}{c x}-\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )+2 \,\operatorname {arctanh}\left (\frac {i}{c x}+\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )\right )+3 a \,b^{2} \left (\frac {\left (c^{2} x^{2} \operatorname {arccsc}\left (c x \right )^{2}+\operatorname {arccsc}\left (c x \right ) c x \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}+1\right ) c x}{3}-\frac {\operatorname {arccsc}\left (c x \right ) \ln \left (1-\frac {i}{c x}-\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )}{3}+\frac {i \operatorname {polylog}\left (2, \frac {i}{c x}+\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )}{3}+\frac {\operatorname {arccsc}\left (c x \right ) \ln \left (1+\frac {i}{c x}+\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )}{3}-\frac {i \operatorname {polylog}\left (2, -\frac {i}{c x}-\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )}{3}\right )+3 a^{2} b \left (\frac {c^{3} x^{3} \operatorname {arccsc}\left (c x \right )}{3}+\frac {\sqrt {c^{2} x^{2}-1}\, \left (c x \sqrt {c^{2} x^{2}-1}+\ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )\right )}{6 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, c x}\right )}{c^{3}}\)	\(535\)
default	\(\frac {\frac {c^{3} x^{3} a^{3}}{3}+b^{3} \left (\frac {\operatorname {arccsc}\left (c x \right ) \left (2 c^{2} x^{2} \operatorname {arccsc}\left (c x \right )^{2}+3 \,\operatorname {arccsc}\left (c x \right ) c x \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}+6\right ) c x}{6}-\frac {\operatorname {arccsc}\left (c x \right )^{2} \ln \left (1-\frac {i}{c x}-\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )}{2}+i \operatorname {arccsc}\left (c x \right ) \operatorname {polylog}\left (2, \frac {i}{c x}+\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )-\operatorname {polylog}\left (3, \frac {i}{c x}+\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )+\frac {\operatorname {arccsc}\left (c x \right )^{2} \ln \left (1+\frac {i}{c x}+\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )}{2}-i \operatorname {arccsc}\left (c x \right ) \operatorname {polylog}\left (2, -\frac {i}{c x}-\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )+\operatorname {polylog}\left (3, -\frac {i}{c x}-\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )+2 \,\operatorname {arctanh}\left (\frac {i}{c x}+\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )\right )+3 a \,b^{2} \left (\frac {\left (c^{2} x^{2} \operatorname {arccsc}\left (c x \right )^{2}+\operatorname {arccsc}\left (c x \right ) c x \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}+1\right ) c x}{3}-\frac {\operatorname {arccsc}\left (c x \right ) \ln \left (1-\frac {i}{c x}-\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )}{3}+\frac {i \operatorname {polylog}\left (2, \frac {i}{c x}+\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )}{3}+\frac {\operatorname {arccsc}\left (c x \right ) \ln \left (1+\frac {i}{c x}+\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )}{3}-\frac {i \operatorname {polylog}\left (2, -\frac {i}{c x}-\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )}{3}\right )+3 a^{2} b \left (\frac {c^{3} x^{3} \operatorname {arccsc}\left (c x \right )}{3}+\frac {\sqrt {c^{2} x^{2}-1}\, \left (c x \sqrt {c^{2} x^{2}-1}+\ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )\right )}{6 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, c x}\right )}{c^{3}}\)	\(535\)
parts	\(\frac {a^{3} x^{3}}{3}+\frac {b^{3} \left (\frac {\operatorname {arccsc}\left (c x \right ) \left (2 c^{2} x^{2} \operatorname {arccsc}\left (c x \right )^{2}+3 \,\operatorname {arccsc}\left (c x \right ) c x \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}+6\right ) c x}{6}-\frac {\operatorname {arccsc}\left (c x \right )^{2} \ln \left (1-\frac {i}{c x}-\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )}{2}+i \operatorname {arccsc}\left (c x \right ) \operatorname {polylog}\left (2, \frac {i}{c x}+\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )-\operatorname {polylog}\left (3, \frac {i}{c x}+\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )+\frac {\operatorname {arccsc}\left (c x \right )^{2} \ln \left (1+\frac {i}{c x}+\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )}{2}-i \operatorname {arccsc}\left (c x \right ) \operatorname {polylog}\left (2, -\frac {i}{c x}-\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )+\operatorname {polylog}\left (3, -\frac {i}{c x}-\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )+2 \,\operatorname {arctanh}\left (\frac {i}{c x}+\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )\right )}{c^{3}}+\frac {3 a \,b^{2} \left (\frac {\left (c^{2} x^{2} \operatorname {arccsc}\left (c x \right )^{2}+\operatorname {arccsc}\left (c x \right ) c x \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}+1\right ) c x}{3}-\frac {\operatorname {arccsc}\left (c x \right ) \ln \left (1-\frac {i}{c x}-\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )}{3}+\frac {i \operatorname {polylog}\left (2, \frac {i}{c x}+\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )}{3}+\frac {\operatorname {arccsc}\left (c x \right ) \ln \left (1+\frac {i}{c x}+\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )}{3}-\frac {i \operatorname {polylog}\left (2, -\frac {i}{c x}-\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )}{3}\right )}{c^{3}}+\frac {3 a^{2} b \left (\frac {c^{3} x^{3} \operatorname {arccsc}\left (c x \right )}{3}+\frac {\sqrt {c^{2} x^{2}-1}\, \left (c x \sqrt {c^{2} x^{2}-1}+\ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )\right )}{6 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, c x}\right )}{c^{3}}\)	\(537\)

[In]

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

[Out]

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

Fricas [F]

\[ \int x^2 \left (a+b \csc ^{-1}(c x)\right )^3 \, dx=\int { {\left (b \operatorname {arccsc}\left (c x\right ) + a\right )}^{3} x^{2} \,d x } \]

[In]

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

[Out]

integral(b^3*x^2*arccsc(c*x)^3 + 3*a*b^2*x^2*arccsc(c*x)^2 + 3*a^2*b*x^2*arccsc(c*x) + a^3*x^2, x)

Sympy [F]

\[ \int x^2 \left (a+b \csc ^{-1}(c x)\right )^3 \, dx=\int x^{2} \left (a + b \operatorname {acsc}{\left (c x \right )}\right )^{3}\, dx \]

[In]

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

[Out]

Integral(x**2*(a + b*acsc(c*x))**3, x)

Maxima [F]

\[ \int x^2 \left (a+b \csc ^{-1}(c x)\right )^3 \, dx=\int { {\left (b \operatorname {arccsc}\left (c x\right ) + a\right )}^{3} x^{2} \,d x } \]

[In]

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

[Out]

1/3*b^3*x^3*arctan2(1, sqrt(c*x + 1)*sqrt(c*x - 1))^3 - 1/4*b^3*x^3*arctan2(1, sqrt(c*x + 1)*sqrt(c*x - 1))*lo
g(c^2*x^2)^2 - 1/2*a*b^2*c^2*(2*(c^2*x^3 + 3*x)/c^4 - 3*log(c*x + 1)/c^5 + 3*log(c*x - 1)/c^5)*log(c)^2 - 12*b
^3*c^2*integrate(1/4*x^4*arctan(1/(sqrt(c*x + 1)*sqrt(c*x - 1)))/(c^2*x^2 - 1), x)*log(c)^2 + 12*b^3*c^2*integ
rate(1/4*x^4*arctan(1/(sqrt(c*x + 1)*sqrt(c*x - 1)))*log(c^2*x^2)/(c^2*x^2 - 1), x)*log(c) - 24*b^3*c^2*integr
ate(1/4*x^4*arctan(1/(sqrt(c*x + 1)*sqrt(c*x - 1)))*log(x)/(c^2*x^2 - 1), x)*log(c) + 12*a*b^2*c^2*integrate(1
/4*x^4*log(c^2*x^2)/(c^2*x^2 - 1), x)*log(c) - 24*a*b^2*c^2*integrate(1/4*x^4*log(x)/(c^2*x^2 - 1), x)*log(c)
+ 1/3*a^3*x^3 + 12*b^3*c^2*integrate(1/4*x^4*arctan(1/(sqrt(c*x + 1)*sqrt(c*x - 1)))*log(c^2*x^2)*log(x)/(c^2*
x^2 - 1), x) - 12*b^3*c^2*integrate(1/4*x^4*arctan(1/(sqrt(c*x + 1)*sqrt(c*x - 1)))*log(x)^2/(c^2*x^2 - 1), x)
+ 12*a*b^2*c^2*integrate(1/4*x^4*arctan(1/(sqrt(c*x + 1)*sqrt(c*x - 1)))^2/(c^2*x^2 - 1), x) + 4*b^3*c^2*inte
grate(1/4*x^4*arctan(1/(sqrt(c*x + 1)*sqrt(c*x - 1)))*log(c^2*x^2)/(c^2*x^2 - 1), x) - 3*a*b^2*c^2*integrate(1
/4*x^4*log(c^2*x^2)^2/(c^2*x^2 - 1), x) + 12*a*b^2*c^2*integrate(1/4*x^4*log(c^2*x^2)*log(x)/(c^2*x^2 - 1), x)
- 12*a*b^2*c^2*integrate(1/4*x^4*log(x)^2/(c^2*x^2 - 1), x) + 3/2*a*b^2*(2*x/c^2 - log(c*x + 1)/c^3 + log(c*x
- 1)/c^3)*log(c)^2 + 12*b^3*integrate(1/4*x^2*arctan(1/(sqrt(c*x + 1)*sqrt(c*x - 1)))/(c^2*x^2 - 1), x)*log(c
)^2 - 12*b^3*integrate(1/4*x^2*arctan(1/(sqrt(c*x + 1)*sqrt(c*x - 1)))*log(c^2*x^2)/(c^2*x^2 - 1), x)*log(c) +
24*b^3*integrate(1/4*x^2*arctan(1/(sqrt(c*x + 1)*sqrt(c*x - 1)))*log(x)/(c^2*x^2 - 1), x)*log(c) - 12*a*b^2*i
ntegrate(1/4*x^2*log(c^2*x^2)/(c^2*x^2 - 1), x)*log(c) + 24*a*b^2*integrate(1/4*x^2*log(x)/(c^2*x^2 - 1), x)*l
og(c) + 1/4*(4*x^3*arccsc(c*x) + (2*sqrt(-1/(c^2*x^2) + 1)/(c^2*(1/(c^2*x^2) - 1) + c^2) + log(sqrt(-1/(c^2*x^
2) + 1) + 1)/c^2 - log(sqrt(-1/(c^2*x^2) + 1) - 1)/c^2)/c)*a^2*b + 4*b^3*integrate(1/4*sqrt(c*x + 1)*sqrt(c*x
- 1)*x^2*arctan(1/(sqrt(c*x + 1)*sqrt(c*x - 1)))^2/(c^2*x^2 - 1), x) - b^3*integrate(1/4*sqrt(c*x + 1)*sqrt(c*
x - 1)*x^2*log(c^2*x^2)^2/(c^2*x^2 - 1), x) - 12*b^3*integrate(1/4*x^2*arctan(1/(sqrt(c*x + 1)*sqrt(c*x - 1)))
*log(c^2*x^2)*log(x)/(c^2*x^2 - 1), x) + 12*b^3*integrate(1/4*x^2*arctan(1/(sqrt(c*x + 1)*sqrt(c*x - 1)))*log(
x)^2/(c^2*x^2 - 1), x) - 12*a*b^2*integrate(1/4*x^2*arctan(1/(sqrt(c*x + 1)*sqrt(c*x - 1)))^2/(c^2*x^2 - 1), x
) - 4*b^3*integrate(1/4*x^2*arctan(1/(sqrt(c*x + 1)*sqrt(c*x - 1)))*log(c^2*x^2)/(c^2*x^2 - 1), x) + 3*a*b^2*i
ntegrate(1/4*x^2*log(c^2*x^2)^2/(c^2*x^2 - 1), x) - 12*a*b^2*integrate(1/4*x^2*log(c^2*x^2)*log(x)/(c^2*x^2 -
1), x) + 12*a*b^2*integrate(1/4*x^2*log(x)^2/(c^2*x^2 - 1), x)

Giac [F]

\[ \int x^2 \left (a+b \csc ^{-1}(c x)\right )^3 \, dx=\int { {\left (b \operatorname {arccsc}\left (c x\right ) + a\right )}^{3} x^{2} \,d x } \]

[In]

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

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int x^2 \left (a+b \csc ^{-1}(c x)\right )^3 \, dx=\int x^2\,{\left (a+b\,\mathrm {asin}\left (\frac {1}{c\,x}\right )\right )}^3 \,d x \]

[In]

int(x^2*(a + b*asin(1/(c*x)))^3,x)

[Out]

int(x^2*(a + b*asin(1/(c*x)))^3, x)