Integrand size = 14, antiderivative size = 124 \[ \int \frac {\left (a+b \csc ^{-1}(c x)\right )^3}{x} \, dx=\frac {i \left (a+b \csc ^{-1}(c x)\right )^4}{4 b}-\left (a+b \csc ^{-1}(c x)\right )^3 \log \left (1-e^{2 i \csc ^{-1}(c x)}\right )+\frac {3}{2} i b \left (a+b \csc ^{-1}(c x)\right )^2 \operatorname {PolyLog}\left (2,e^{2 i \csc ^{-1}(c x)}\right )-\frac {3}{2} b^2 \left (a+b \csc ^{-1}(c x)\right ) \operatorname {PolyLog}\left (3,e^{2 i \csc ^{-1}(c x)}\right )-\frac {3}{4} i b^3 \operatorname {PolyLog}\left (4,e^{2 i \csc ^{-1}(c x)}\right ) \]
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)
Time = 0.15 (sec) , antiderivative size = 242, normalized size of antiderivative = 1.95 \[ \int \frac {\left (a+b \csc ^{-1}(c x)\right )^3}{x} \, dx=a^3 \log (c x)+\frac {3}{2} i a^2 b \left (\csc ^{-1}(c x) \left (\csc ^{-1}(c x)+2 i \log \left (1-e^{2 i \csc ^{-1}(c x)}\right )\right )+\operatorname {PolyLog}\left (2,e^{2 i \csc ^{-1}(c x)}\right )\right )+\frac {1}{8} i a b^2 \left (\pi ^3-8 \csc ^{-1}(c x)^3+24 i \csc ^{-1}(c x)^2 \log \left (1-e^{-2 i \csc ^{-1}(c x)}\right )-24 \csc ^{-1}(c x) \operatorname {PolyLog}\left (2,e^{-2 i \csc ^{-1}(c x)}\right )+12 i \operatorname {PolyLog}\left (3,e^{-2 i \csc ^{-1}(c x)}\right )\right )+\frac {1}{64} i b^3 \left (\pi ^4-16 \csc ^{-1}(c x)^4+64 i \csc ^{-1}(c x)^3 \log \left (1-e^{-2 i \csc ^{-1}(c x)}\right )-96 \csc ^{-1}(c x)^2 \operatorname {PolyLog}\left (2,e^{-2 i \csc ^{-1}(c x)}\right )+96 i \csc ^{-1}(c x) \operatorname {PolyLog}\left (3,e^{-2 i \csc ^{-1}(c x)}\right )+48 \operatorname {PolyLog}\left (4,e^{-2 i \csc ^{-1}(c x)}\right )\right ) \]
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/64)*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*PolyLog[ 2, E^((-2*I)*ArcCsc[c*x])] + (96*I)*ArcCsc[c*x]*PolyLog[3, E^((-2*I)*ArcCs c[c*x])] + 48*PolyLog[4, E^((-2*I)*ArcCsc[c*x])])
Time = 0.67 (sec) , antiderivative size = 141, normalized size of antiderivative = 1.14, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.714, Rules used = {5746, 3042, 25, 4200, 25, 2620, 3011, 7163, 2720, 7143}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {\left (a+b \csc ^{-1}(c x)\right )^3}{x} \, dx\) |
| \(\Big \downarrow \) 5746 |
| \(\displaystyle -\int c \sqrt {1-\frac {1}{c^2 x^2}} x \left (a+b \csc ^{-1}(c x)\right )^3d\csc ^{-1}(c x)\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\int -\left (a+b \csc ^{-1}(c x)\right )^3 \tan \left (\csc ^{-1}(c x)+\frac {\pi }{2}\right )d\csc ^{-1}(c x)\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \int \tan \left (\csc ^{-1}(c x)+\frac {\pi }{2}\right ) \left (a+b \csc ^{-1}(c x)\right )^3d\csc ^{-1}(c x)\) |
| \(\Big \downarrow \) 4200 |
| \(\displaystyle \frac {i \left (a+b \csc ^{-1}(c x)\right )^4}{4 b}-2 i \int -\frac {e^{2 i \csc ^{-1}(c x)} \left (a+b \csc ^{-1}(c x)\right )^3}{1-e^{2 i \csc ^{-1}(c x)}}d\csc ^{-1}(c x)\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle 2 i \int \frac {e^{2 i \csc ^{-1}(c x)} \left (a+b \csc ^{-1}(c x)\right )^3}{1-e^{2 i \csc ^{-1}(c x)}}d\csc ^{-1}(c x)+\frac {i \left (a+b \csc ^{-1}(c x)\right )^4}{4 b}\) |
| \(\Big \downarrow \) 2620 |
| \(\displaystyle 2 i \left (\frac {1}{2} i \log \left (1-e^{2 i \csc ^{-1}(c x)}\right ) \left (a+b \csc ^{-1}(c x)\right )^3-\frac {3}{2} i b \int \left (a+b \csc ^{-1}(c x)\right )^2 \log \left (1-e^{2 i \csc ^{-1}(c x)}\right )d\csc ^{-1}(c x)\right )+\frac {i \left (a+b \csc ^{-1}(c x)\right )^4}{4 b}\) |
| \(\Big \downarrow \) 3011 |
| \(\displaystyle 2 i \left (\frac {1}{2} i \log \left (1-e^{2 i \csc ^{-1}(c x)}\right ) \left (a+b \csc ^{-1}(c x)\right )^3-\frac {3}{2} i b \left (\frac {1}{2} i \operatorname {PolyLog}\left (2,e^{2 i \csc ^{-1}(c x)}\right ) \left (a+b \csc ^{-1}(c x)\right )^2-i b \int \left (a+b \csc ^{-1}(c x)\right ) \operatorname {PolyLog}\left (2,e^{2 i \csc ^{-1}(c x)}\right )d\csc ^{-1}(c x)\right )\right )+\frac {i \left (a+b \csc ^{-1}(c x)\right )^4}{4 b}\) |
| \(\Big \downarrow \) 7163 |
| \(\displaystyle 2 i \left (\frac {1}{2} i \log \left (1-e^{2 i \csc ^{-1}(c x)}\right ) \left (a+b \csc ^{-1}(c x)\right )^3-\frac {3}{2} i b \left (\frac {1}{2} i \operatorname {PolyLog}\left (2,e^{2 i \csc ^{-1}(c x)}\right ) \left (a+b \csc ^{-1}(c x)\right )^2-i b \left (\frac {1}{2} i b \int \operatorname {PolyLog}\left (3,e^{2 i \csc ^{-1}(c x)}\right )d\csc ^{-1}(c x)-\frac {1}{2} i \operatorname {PolyLog}\left (3,e^{2 i \csc ^{-1}(c x)}\right ) \left (a+b \csc ^{-1}(c x)\right )\right )\right )\right )+\frac {i \left (a+b \csc ^{-1}(c x)\right )^4}{4 b}\) |
| \(\Big \downarrow \) 2720 |
| \(\displaystyle 2 i \left (\frac {1}{2} i \log \left (1-e^{2 i \csc ^{-1}(c x)}\right ) \left (a+b \csc ^{-1}(c x)\right )^3-\frac {3}{2} i b \left (\frac {1}{2} i \operatorname {PolyLog}\left (2,e^{2 i \csc ^{-1}(c x)}\right ) \left (a+b \csc ^{-1}(c x)\right )^2-i b \left (\frac {1}{4} b \int e^{-2 i \csc ^{-1}(c x)} \operatorname {PolyLog}\left (3,e^{2 i \csc ^{-1}(c x)}\right )de^{2 i \csc ^{-1}(c x)}-\frac {1}{2} i \operatorname {PolyLog}\left (3,e^{2 i \csc ^{-1}(c x)}\right ) \left (a+b \csc ^{-1}(c x)\right )\right )\right )\right )+\frac {i \left (a+b \csc ^{-1}(c x)\right )^4}{4 b}\) |
| \(\Big \downarrow \) 7143 |
| \(\displaystyle 2 i \left (\frac {1}{2} i \log \left (1-e^{2 i \csc ^{-1}(c x)}\right ) \left (a+b \csc ^{-1}(c x)\right )^3-\frac {3}{2} i b \left (\frac {1}{2} i \operatorname {PolyLog}\left (2,e^{2 i \csc ^{-1}(c x)}\right ) \left (a+b \csc ^{-1}(c x)\right )^2-i b \left (\frac {1}{4} b \operatorname {PolyLog}\left (4,e^{2 i \csc ^{-1}(c x)}\right )-\frac {1}{2} i \operatorname {PolyLog}\left (3,e^{2 i \csc ^{-1}(c x)}\right ) \left (a+b \csc ^{-1}(c x)\right )\right )\right )\right )+\frac {i \left (a+b \csc ^{-1}(c x)\right )^4}{4 b}\) |
((I/4)*(a + b*ArcCsc[c*x])^4)/b + (2*I)*((I/2)*(a + b*ArcCsc[c*x])^3*Log[1 - E^((2*I)*ArcCsc[c*x])] - ((3*I)/2)*b*((I/2)*(a + b*ArcCsc[c*x])^2*PolyL og[2, E^((2*I)*ArcCsc[c*x])] - I*b*((-1/2*I)*(a + b*ArcCsc[c*x])*PolyLog[3 , E^((2*I)*ArcCsc[c*x])] + (b*PolyLog[4, E^((2*I)*ArcCsc[c*x])])/4)))
3.1.28.3.1 Defintions of rubi rules used
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] - Si mp[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]
Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Simp[v/D[v, x] Subst[Int[FunctionOfExponentialFunction[u, x]/x, x], x, v], x]] /; Funct ionOfExponentialQ[u, x] && !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; FreeQ [{a, m, n}, x] && IntegerQ[m*n]] && !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x)) *(F_)[v_] /; FreeQ[{a, b, c}, x] && InverseFunctionQ[F[x]]]
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] + Simp[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]
Int[((c_.) + (d_.)*(x_))^(m_.)*tan[(e_.) + Pi*(k_.) + (f_.)*(x_)], x_Symbol ] :> Simp[I*((c + d*x)^(m + 1)/(d*(m + 1))), x] - Simp[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]
Int[((a_.) + ArcCsc[(c_.)*(x_)]*(b_.))^(n_)*(x_)^(m_.), x_Symbol] :> Simp[- (c^(m + 1))^(-1) Subst[Int[(a + b*x)^n*Csc[x]^(m + 1)*Cot[x], x], x, ArcC sc[c*x]], x] /; FreeQ[{a, b, c}, x] && IntegerQ[n] && IntegerQ[m] && (GtQ[n , 0] || LtQ[m, -1])
Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_S ymbol] :> 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]
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] - Simp[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]
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 )\) | \(610\) |
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*poly log(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*arcc sc(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)*po lylog(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)*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)))+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*polylo g(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)))
\[ \int \frac {\left (a+b \csc ^{-1}(c x)\right )^3}{x} \, dx=\int { \frac {{\left (b \operatorname {arccsc}\left (c x\right ) + a\right )}^{3}}{x} \,d x } \]
\[ \int \frac {\left (a+b \csc ^{-1}(c x)\right )^3}{x} \, dx=\int \frac {\left (a + b \operatorname {acsc}{\left (c x \right )}\right )^{3}}{x}\, dx \]
\[ \int \frac {\left (a+b \csc ^{-1}(c x)\right )^3}{x} \, dx=\int { \frac {{\left (b \operatorname {arccsc}\left (c x\right ) + a\right )}^{3}}{x} \,d x } \]
-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*arct an2(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*arctan(1/(sqr t(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/(sqrt(c*x + 1)*s qrt(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) + ...
\[ \int \frac {\left (a+b \csc ^{-1}(c x)\right )^3}{x} \, dx=\int { \frac {{\left (b \operatorname {arccsc}\left (c x\right ) + a\right )}^{3}}{x} \,d x } \]
Timed out. \[ \int \frac {\left (a+b \csc ^{-1}(c x)\right )^3}{x} \, dx=\int \frac {{\left (a+b\,\mathrm {asin}\left (\frac {1}{c\,x}\right )\right )}^3}{x} \,d x \]