\(\int \frac {(a+b \csc ^{-1}(c x))^3}{x} \, dx\) [28]

Optimal result

Integrand size = 14, antiderivative size = 124 \[ \int \frac {\left (a+b \csc ^{-1}(c x)\right )^3}{x} \, dx=\frac {i \left (a+b \csc ^{-1}(c x)\right )^4}{4 b}-\left (a+b \csc ^{-1}(c x)\right )^3 \log \left (1-e^{2 i \csc ^{-1}(c x)}\right )+\frac {3}{2} i b \left (a+b \csc ^{-1}(c x)\right )^2 \operatorname {PolyLog}\left (2,e^{2 i \csc ^{-1}(c x)}\right )-\frac {3}{2} b^2 \left (a+b \csc ^{-1}(c x)\right ) \operatorname {PolyLog}\left (3,e^{2 i \csc ^{-1}(c x)}\right )-\frac {3}{4} i b^3 \operatorname {PolyLog}\left (4,e^{2 i \csc ^{-1}(c x)}\right ) \]

[Out]

1/4*I*(a+b*arccsc(c*x))^4/b-(a+b*arccsc(c*x))^3*ln(1-(I/c/x+(1-1/c^2/x^2)^(1/2))^2)+3/2*I*b*(a+b*arccsc(c*x))^
2*polylog(2,(I/c/x+(1-1/c^2/x^2)^(1/2))^2)-3/2*b^2*(a+b*arccsc(c*x))*polylog(3,(I/c/x+(1-1/c^2/x^2)^(1/2))^2)-
3/4*I*b^3*polylog(4,(I/c/x+(1-1/c^2/x^2)^(1/2))^2)

Rubi [A] (verified)

Time = 0.11 (sec) , antiderivative size = 124, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {5331, 3798, 2221, 2611, 6744, 2320, 6724} \[ \int \frac {\left (a+b \csc ^{-1}(c x)\right )^3}{x} \, dx=-\frac {3}{2} b^2 \operatorname {PolyLog}\left (3,e^{2 i \csc ^{-1}(c x)}\right ) \left (a+b \csc ^{-1}(c x)\right )+\frac {3}{2} i b \operatorname {PolyLog}\left (2,e^{2 i \csc ^{-1}(c x)}\right ) \left (a+b \csc ^{-1}(c x)\right )^2+\frac {i \left (a+b \csc ^{-1}(c x)\right )^4}{4 b}-\log \left (1-e^{2 i \csc ^{-1}(c x)}\right ) \left (a+b \csc ^{-1}(c x)\right )^3-\frac {3}{4} i b^3 \operatorname {PolyLog}\left (4,e^{2 i \csc ^{-1}(c x)}\right ) \]

[In]

Int[(a + b*ArcCsc[c*x])^3/x,x]

[Out]

((I/4)*(a + b*ArcCsc[c*x])^4)/b - (a + b*ArcCsc[c*x])^3*Log[1 - E^((2*I)*ArcCsc[c*x])] + ((3*I)/2)*b*(a + b*Ar
cCsc[c*x])^2*PolyLog[2, E^((2*I)*ArcCsc[c*x])] - (3*b^2*(a + b*ArcCsc[c*x])*PolyLog[3, E^((2*I)*ArcCsc[c*x])])
/2 - ((3*I)/4)*b^3*PolyLog[4, E^((2*I)*ArcCsc[c*x])]

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 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 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 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]

Rule 6744

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

Rubi steps \begin{align*} \text {integral}& = -\text {Subst}\left (\int (a+b x)^3 \cot (x) \, dx,x,\csc ^{-1}(c x)\right ) \\ & = \frac {i \left (a+b \csc ^{-1}(c x)\right )^4}{4 b}+2 i \text {Subst}\left (\int \frac {e^{2 i x} (a+b x)^3}{1-e^{2 i x}} \, dx,x,\csc ^{-1}(c x)\right ) \\ & = \frac {i \left (a+b \csc ^{-1}(c x)\right )^4}{4 b}-\left (a+b \csc ^{-1}(c x)\right )^3 \log \left (1-e^{2 i \csc ^{-1}(c x)}\right )+(3 b) \text {Subst}\left (\int (a+b x)^2 \log \left (1-e^{2 i x}\right ) \, dx,x,\csc ^{-1}(c x)\right ) \\ & = \frac {i \left (a+b \csc ^{-1}(c x)\right )^4}{4 b}-\left (a+b \csc ^{-1}(c x)\right )^3 \log \left (1-e^{2 i \csc ^{-1}(c x)}\right )+\frac {3}{2} i b \left (a+b \csc ^{-1}(c x)\right )^2 \operatorname {PolyLog}\left (2,e^{2 i \csc ^{-1}(c x)}\right )-\left (3 i b^2\right ) \text {Subst}\left (\int (a+b x) \operatorname {PolyLog}\left (2,e^{2 i x}\right ) \, dx,x,\csc ^{-1}(c x)\right ) \\ & = \frac {i \left (a+b \csc ^{-1}(c x)\right )^4}{4 b}-\left (a+b \csc ^{-1}(c x)\right )^3 \log \left (1-e^{2 i \csc ^{-1}(c x)}\right )+\frac {3}{2} i b \left (a+b \csc ^{-1}(c x)\right )^2 \operatorname {PolyLog}\left (2,e^{2 i \csc ^{-1}(c x)}\right )-\frac {3}{2} b^2 \left (a+b \csc ^{-1}(c x)\right ) \operatorname {PolyLog}\left (3,e^{2 i \csc ^{-1}(c x)}\right )+\frac {1}{2} \left (3 b^3\right ) \text {Subst}\left (\int \operatorname {PolyLog}\left (3,e^{2 i x}\right ) \, dx,x,\csc ^{-1}(c x)\right ) \\ & = \frac {i \left (a+b \csc ^{-1}(c x)\right )^4}{4 b}-\left (a+b \csc ^{-1}(c x)\right )^3 \log \left (1-e^{2 i \csc ^{-1}(c x)}\right )+\frac {3}{2} i b \left (a+b \csc ^{-1}(c x)\right )^2 \operatorname {PolyLog}\left (2,e^{2 i \csc ^{-1}(c x)}\right )-\frac {3}{2} b^2 \left (a+b \csc ^{-1}(c x)\right ) \operatorname {PolyLog}\left (3,e^{2 i \csc ^{-1}(c x)}\right )-\frac {1}{4} \left (3 i b^3\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(3,x)}{x} \, dx,x,e^{2 i \csc ^{-1}(c x)}\right ) \\ & = \frac {i \left (a+b \csc ^{-1}(c x)\right )^4}{4 b}-\left (a+b \csc ^{-1}(c x)\right )^3 \log \left (1-e^{2 i \csc ^{-1}(c x)}\right )+\frac {3}{2} i b \left (a+b \csc ^{-1}(c x)\right )^2 \operatorname {PolyLog}\left (2,e^{2 i \csc ^{-1}(c x)}\right )-\frac {3}{2} b^2 \left (a+b \csc ^{-1}(c x)\right ) \operatorname {PolyLog}\left (3,e^{2 i \csc ^{-1}(c x)}\right )-\frac {3}{4} i b^3 \operatorname {PolyLog}\left (4,e^{2 i \csc ^{-1}(c x)}\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.15 (sec) , antiderivative size = 242, normalized size of antiderivative = 1.95 \[ \int \frac {\left (a+b \csc ^{-1}(c x)\right )^3}{x} \, dx=a^3 \log (c x)+\frac {3}{2} i a^2 b \left (\csc ^{-1}(c x) \left (\csc ^{-1}(c x)+2 i \log \left (1-e^{2 i \csc ^{-1}(c x)}\right )\right )+\operatorname {PolyLog}\left (2,e^{2 i \csc ^{-1}(c x)}\right )\right )+\frac {1}{8} i a b^2 \left (\pi ^3-8 \csc ^{-1}(c x)^3+24 i \csc ^{-1}(c x)^2 \log \left (1-e^{-2 i \csc ^{-1}(c x)}\right )-24 \csc ^{-1}(c x) \operatorname {PolyLog}\left (2,e^{-2 i \csc ^{-1}(c x)}\right )+12 i \operatorname {PolyLog}\left (3,e^{-2 i \csc ^{-1}(c x)}\right )\right )+\frac {1}{64} i b^3 \left (\pi ^4-16 \csc ^{-1}(c x)^4+64 i \csc ^{-1}(c x)^3 \log \left (1-e^{-2 i \csc ^{-1}(c x)}\right )-96 \csc ^{-1}(c x)^2 \operatorname {PolyLog}\left (2,e^{-2 i \csc ^{-1}(c x)}\right )+96 i \csc ^{-1}(c x) \operatorname {PolyLog}\left (3,e^{-2 i \csc ^{-1}(c x)}\right )+48 \operatorname {PolyLog}\left (4,e^{-2 i \csc ^{-1}(c x)}\right )\right ) \]

[In]

Integrate[(a + b*ArcCsc[c*x])^3/x,x]

[Out]

a^3*Log[c*x] + ((3*I)/2)*a^2*b*(ArcCsc[c*x]*(ArcCsc[c*x] + (2*I)*Log[1 - E^((2*I)*ArcCsc[c*x])]) + PolyLog[2,
E^((2*I)*ArcCsc[c*x])]) + (I/8)*a*b^2*(Pi^3 - 8*ArcCsc[c*x]^3 + (24*I)*ArcCsc[c*x]^2*Log[1 - E^((-2*I)*ArcCsc[
c*x])] - 24*ArcCsc[c*x]*PolyLog[2, E^((-2*I)*ArcCsc[c*x])] + (12*I)*PolyLog[3, E^((-2*I)*ArcCsc[c*x])]) + (I/6
4)*b^3*(Pi^4 - 16*ArcCsc[c*x]^4 + (64*I)*ArcCsc[c*x]^3*Log[1 - E^((-2*I)*ArcCsc[c*x])] - 96*ArcCsc[c*x]^2*Poly
Log[2, E^((-2*I)*ArcCsc[c*x])] + (96*I)*ArcCsc[c*x]*PolyLog[3, E^((-2*I)*ArcCsc[c*x])] + 48*PolyLog[4, E^((-2*
I)*ArcCsc[c*x])])

Maple [B] (verified)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 607 vs. \(2 (169 ) = 338\).

Time = 1.20 (sec) , antiderivative size = 608, normalized size of antiderivative = 4.90

method	result	size
parts	\(a^{3} \ln \left (x \right )+b^{3} \left (\frac {i \operatorname {arccsc}\left (c x \right )^{4}}{4}-\operatorname {arccsc}\left (c x \right )^{3} \ln \left (1-\frac {i}{c x}-\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )+3 i \operatorname {arccsc}\left (c x \right )^{2} \operatorname {polylog}\left (2, \frac {i}{c x}+\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )-6 \,\operatorname {arccsc}\left (c x \right ) \operatorname {polylog}\left (3, \frac {i}{c x}+\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )-6 i \operatorname {polylog}\left (4, \frac {i}{c x}+\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )-\operatorname {arccsc}\left (c x \right )^{3} \ln \left (1+\frac {i}{c x}+\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )+3 i \operatorname {arccsc}\left (c x \right )^{2} \operatorname {polylog}\left (2, -\frac {i}{c x}-\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )-6 \,\operatorname {arccsc}\left (c x \right ) \operatorname {polylog}\left (3, -\frac {i}{c x}-\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )-6 i \operatorname {polylog}\left (4, -\frac {i}{c x}-\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )\right )+3 a \,b^{2} \left (\frac {i \operatorname {arccsc}\left (c x \right )^{3}}{3}-\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 )-2 \operatorname {polylog}\left (3, \frac {i}{c x}+\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )-\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 )-2 \operatorname {polylog}\left (3, -\frac {i}{c x}-\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )\right )+3 a^{2} b \left (\frac {i \operatorname {arccsc}\left (c x \right )^{2}}{2}-\operatorname {arccsc}\left (c x \right ) \ln \left (1-\frac {i}{c x}-\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )+i \operatorname {polylog}\left (2, \frac {i}{c x}+\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )-\operatorname {arccsc}\left (c x \right ) \ln \left (1+\frac {i}{c x}+\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )+i \operatorname {polylog}\left (2, -\frac {i}{c x}-\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )\right )\)	\(608\)
derivativedivides	\(a^{3} \ln \left (c x \right )+b^{3} \left (\frac {i \operatorname {arccsc}\left (c x \right )^{4}}{4}-\operatorname {arccsc}\left (c x \right )^{3} \ln \left (1-\frac {i}{c x}-\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )+3 i \operatorname {arccsc}\left (c x \right )^{2} \operatorname {polylog}\left (2, \frac {i}{c x}+\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )-6 \,\operatorname {arccsc}\left (c x \right ) \operatorname {polylog}\left (3, \frac {i}{c x}+\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )-6 i \operatorname {polylog}\left (4, \frac {i}{c x}+\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )-\operatorname {arccsc}\left (c x \right )^{3} \ln \left (1+\frac {i}{c x}+\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )+3 i \operatorname {arccsc}\left (c x \right )^{2} \operatorname {polylog}\left (2, -\frac {i}{c x}-\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )-6 \,\operatorname {arccsc}\left (c x \right ) \operatorname {polylog}\left (3, -\frac {i}{c x}-\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )-6 i \operatorname {polylog}\left (4, -\frac {i}{c x}-\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )\right )+3 a \,b^{2} \left (\frac {i \operatorname {arccsc}\left (c x \right )^{3}}{3}-\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 )-2 \operatorname {polylog}\left (3, \frac {i}{c x}+\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )-\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 )-2 \operatorname {polylog}\left (3, -\frac {i}{c x}-\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )\right )+3 a^{2} b \left (\frac {i \operatorname {arccsc}\left (c x \right )^{2}}{2}-\operatorname {arccsc}\left (c x \right ) \ln \left (1-\frac {i}{c x}-\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )+i \operatorname {polylog}\left (2, \frac {i}{c x}+\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )-\operatorname {arccsc}\left (c x \right ) \ln \left (1+\frac {i}{c x}+\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )+i \operatorname {polylog}\left (2, -\frac {i}{c x}-\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )\right )\)	\(610\)
default	\(a^{3} \ln \left (c x \right )+b^{3} \left (\frac {i \operatorname {arccsc}\left (c x \right )^{4}}{4}-\operatorname {arccsc}\left (c x \right )^{3} \ln \left (1-\frac {i}{c x}-\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )+3 i \operatorname {arccsc}\left (c x \right )^{2} \operatorname {polylog}\left (2, \frac {i}{c x}+\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )-6 \,\operatorname {arccsc}\left (c x \right ) \operatorname {polylog}\left (3, \frac {i}{c x}+\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )-6 i \operatorname {polylog}\left (4, \frac {i}{c x}+\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )-\operatorname {arccsc}\left (c x \right )^{3} \ln \left (1+\frac {i}{c x}+\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )+3 i \operatorname {arccsc}\left (c x \right )^{2} \operatorname {polylog}\left (2, -\frac {i}{c x}-\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )-6 \,\operatorname {arccsc}\left (c x \right ) \operatorname {polylog}\left (3, -\frac {i}{c x}-\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )-6 i \operatorname {polylog}\left (4, -\frac {i}{c x}-\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )\right )+3 a \,b^{2} \left (\frac {i \operatorname {arccsc}\left (c x \right )^{3}}{3}-\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 )-2 \operatorname {polylog}\left (3, \frac {i}{c x}+\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )-\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 )-2 \operatorname {polylog}\left (3, -\frac {i}{c x}-\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )\right )+3 a^{2} b \left (\frac {i \operatorname {arccsc}\left (c x \right )^{2}}{2}-\operatorname {arccsc}\left (c x \right ) \ln \left (1-\frac {i}{c x}-\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )+i \operatorname {polylog}\left (2, \frac {i}{c x}+\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )-\operatorname {arccsc}\left (c x \right ) \ln \left (1+\frac {i}{c x}+\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )+i \operatorname {polylog}\left (2, -\frac {i}{c x}-\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )\right )\)	\(610\)

[In]

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

[Out]

a^3*ln(x)+b^3*(1/4*I*arccsc(c*x)^4-arccsc(c*x)^3*ln(1-I/c/x-(1-1/c^2/x^2)^(1/2))+3*I*arccsc(c*x)^2*polylog(2,I
/c/x+(1-1/c^2/x^2)^(1/2))-6*arccsc(c*x)*polylog(3,I/c/x+(1-1/c^2/x^2)^(1/2))-6*I*polylog(4,I/c/x+(1-1/c^2/x^2)
^(1/2))-arccsc(c*x)^3*ln(1+I/c/x+(1-1/c^2/x^2)^(1/2))+3*I*arccsc(c*x)^2*polylog(2,-I/c/x-(1-1/c^2/x^2)^(1/2))-
6*arccsc(c*x)*polylog(3,-I/c/x-(1-1/c^2/x^2)^(1/2))-6*I*polylog(4,-I/c/x-(1-1/c^2/x^2)^(1/2)))+3*a*b^2*(1/3*I*
arccsc(c*x)^3-arccsc(c*x)^2*ln(1-I/c/x-(1-1/c^2/x^2)^(1/2))+2*I*arccsc(c*x)*polylog(2,I/c/x+(1-1/c^2/x^2)^(1/2
))-2*polylog(3,I/c/x+(1-1/c^2/x^2)^(1/2))-arccsc(c*x)^2*ln(1+I/c/x+(1-1/c^2/x^2)^(1/2))+2*I*arccsc(c*x)*polylo
g(2,-I/c/x-(1-1/c^2/x^2)^(1/2))-2*polylog(3,-I/c/x-(1-1/c^2/x^2)^(1/2)))+3*a^2*b*(1/2*I*arccsc(c*x)^2-arccsc(c
*x)*ln(1-I/c/x-(1-1/c^2/x^2)^(1/2))+I*polylog(2,I/c/x+(1-1/c^2/x^2)^(1/2))-arccsc(c*x)*ln(1+I/c/x+(1-1/c^2/x^2
)^(1/2))+I*polylog(2,-I/c/x-(1-1/c^2/x^2)^(1/2)))

Fricas [F]

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

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

Integral((a + b*acsc(c*x))**3/x, x)

Maxima [F]

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

[In]

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

[Out]

-3/2*a*b^2*c^2*(log(c*x + 1)/c^2 + log(c*x - 1)/c^2)*log(c)^2 - 12*b^3*c^2*integrate(1/4*x^2*arctan(1/(sqrt(c*
x + 1)*sqrt(c*x - 1)))/(c^2*x^3 - x), x)*log(c)^2 + 12*b^3*c^2*integrate(1/4*x^2*arctan(1/(sqrt(c*x + 1)*sqrt(
c*x - 1)))*log(c^2*x^2)/(c^2*x^3 - x), x)*log(c) - 24*b^3*c^2*integrate(1/4*x^2*arctan(1/(sqrt(c*x + 1)*sqrt(c
*x - 1)))*log(x)/(c^2*x^3 - x), x)*log(c) + 12*a*b^2*c^2*integrate(1/4*x^2*log(c^2*x^2)/(c^2*x^3 - x), x)*log(
c) - 24*a*b^2*c^2*integrate(1/4*x^2*log(x)/(c^2*x^3 - x), x)*log(c) + b^3*arctan2(1, sqrt(c*x + 1)*sqrt(c*x -
1))^3*log(x) - 3/4*b^3*arctan2(1, sqrt(c*x + 1)*sqrt(c*x - 1))*log(c^2*x^2)^2*log(x) + 24*b^3*c^2*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^3 - x), x) - 12*b^3*c^2*integrate(1/4
*x^2*arctan(1/(sqrt(c*x + 1)*sqrt(c*x - 1)))*log(x)^2/(c^2*x^3 - x), x) + 12*a*b^2*c^2*integrate(1/4*x^2*arcta
n(1/(sqrt(c*x + 1)*sqrt(c*x - 1)))^2/(c^2*x^3 - x), x) - 3*a*b^2*c^2*integrate(1/4*x^2*log(c^2*x^2)^2/(c^2*x^3
- x), x) + 12*a*b^2*c^2*integrate(1/4*x^2*log(c^2*x^2)*log(x)/(c^2*x^3 - x), x) - 12*a*b^2*c^2*integrate(1/4*
x^2*log(x)^2/(c^2*x^3 - x), x) + 12*a^2*b*c^2*integrate(1/4*x^2*arctan(1/(sqrt(c*x + 1)*sqrt(c*x - 1)))/(c^2*x
^3 - x), x) + 3/2*a*b^2*(log(c*x + 1) + log(c*x - 1) - 2*log(x))*log(c)^2 + 12*b^3*integrate(1/4*arctan(1/(sqr
t(c*x + 1)*sqrt(c*x - 1)))/(c^2*x^3 - x), x)*log(c)^2 - 12*b^3*integrate(1/4*arctan(1/(sqrt(c*x + 1)*sqrt(c*x
- 1)))*log(c^2*x^2)/(c^2*x^3 - x), x)*log(c) + 24*b^3*integrate(1/4*arctan(1/(sqrt(c*x + 1)*sqrt(c*x - 1)))*lo
g(x)/(c^2*x^3 - x), x)*log(c) - 12*a*b^2*integrate(1/4*log(c^2*x^2)/(c^2*x^3 - x), x)*log(c) + 24*a*b^2*integr
ate(1/4*log(x)/(c^2*x^3 - x), x)*log(c) + 12*b^3*integrate(1/4*sqrt(c*x + 1)*sqrt(c*x - 1)*arctan(1/(sqrt(c*x
+ 1)*sqrt(c*x - 1)))^2*log(x)/(c^2*x^3 - x), x) - 3*b^3*integrate(1/4*sqrt(c*x + 1)*sqrt(c*x - 1)*log(c^2*x^2)
^2*log(x)/(c^2*x^3 - x), x) - 24*b^3*integrate(1/4*arctan(1/(sqrt(c*x + 1)*sqrt(c*x - 1)))*log(c^2*x^2)*log(x)
/(c^2*x^3 - x), x) + 12*b^3*integrate(1/4*arctan(1/(sqrt(c*x + 1)*sqrt(c*x - 1)))*log(x)^2/(c^2*x^3 - x), x) -
12*a*b^2*integrate(1/4*arctan(1/(sqrt(c*x + 1)*sqrt(c*x - 1)))^2/(c^2*x^3 - x), x) + 3*a*b^2*integrate(1/4*lo
g(c^2*x^2)^2/(c^2*x^3 - x), x) - 12*a*b^2*integrate(1/4*log(c^2*x^2)*log(x)/(c^2*x^3 - x), x) + 12*a*b^2*integ
rate(1/4*log(x)^2/(c^2*x^3 - x), x) - 12*a^2*b*integrate(1/4*arctan(1/(sqrt(c*x + 1)*sqrt(c*x - 1)))/(c^2*x^3
- x), x) + a^3*log(x)

Giac [F]

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

[In]

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

[Out]

integrate((b*arccsc(c*x) + a)^3/x, x)

Mupad [F(-1)]

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

[In]

int((a + b*asin(1/(c*x)))^3/x,x)

[Out]

int((a + b*asin(1/(c*x)))^3/x, x)