Integrand size = 10, antiderivative size = 144 \[ \int \left (a+b \csc ^{-1}(c x)\right )^3 \, dx=x \left (a+b \csc ^{-1}(c x)\right )^3+\frac {6 b \left (a+b \csc ^{-1}(c x)\right )^2 \text {arctanh}\left (e^{i \csc ^{-1}(c x)}\right )}{c}-\frac {6 i b^2 \left (a+b \csc ^{-1}(c x)\right ) \operatorname {PolyLog}\left (2,-e^{i \csc ^{-1}(c x)}\right )}{c}+\frac {6 i b^2 \left (a+b \csc ^{-1}(c x)\right ) \operatorname {PolyLog}\left (2,e^{i \csc ^{-1}(c x)}\right )}{c}+\frac {6 b^3 \operatorname {PolyLog}\left (3,-e^{i \csc ^{-1}(c x)}\right )}{c}-\frac {6 b^3 \operatorname {PolyLog}\left (3,e^{i \csc ^{-1}(c x)}\right )}{c} \]
x*(a+b*arccsc(c*x))^3+6*b*(a+b*arccsc(c*x))^2*arctanh(I/c/x+(1-1/c^2/x^2)^ (1/2))/c-6*I*b^2*(a+b*arccsc(c*x))*polylog(2,-I/c/x-(1-1/c^2/x^2)^(1/2))/c +6*I*b^2*(a+b*arccsc(c*x))*polylog(2,I/c/x+(1-1/c^2/x^2)^(1/2))/c+6*b^3*po lylog(3,-I/c/x-(1-1/c^2/x^2)^(1/2))/c-6*b^3*polylog(3,I/c/x+(1-1/c^2/x^2)^ (1/2))/c
Time = 0.28 (sec) , antiderivative size = 265, normalized size of antiderivative = 1.84 \[ \int \left (a+b \csc ^{-1}(c x)\right )^3 \, dx=\frac {a^3 c x+3 a^2 b c x \csc ^{-1}(c x)+3 a b^2 c x \csc ^{-1}(c x)^2+b^3 c x \csc ^{-1}(c x)^3-6 a b^2 \csc ^{-1}(c x) \log \left (1-e^{i \csc ^{-1}(c x)}\right )-3 b^3 \csc ^{-1}(c x)^2 \log \left (1-e^{i \csc ^{-1}(c x)}\right )+6 a b^2 \csc ^{-1}(c x) \log \left (1+e^{i \csc ^{-1}(c x)}\right )+3 b^3 \csc ^{-1}(c x)^2 \log \left (1+e^{i \csc ^{-1}(c x)}\right )+3 a^2 b \log \left (c \left (1+\sqrt {1-\frac {1}{c^2 x^2}}\right ) x\right )-6 i b^2 \left (a+b \csc ^{-1}(c x)\right ) \operatorname {PolyLog}\left (2,-e^{i \csc ^{-1}(c x)}\right )+6 i b^2 \left (a+b \csc ^{-1}(c x)\right ) \operatorname {PolyLog}\left (2,e^{i \csc ^{-1}(c x)}\right )+6 b^3 \operatorname {PolyLog}\left (3,-e^{i \csc ^{-1}(c x)}\right )-6 b^3 \operatorname {PolyLog}\left (3,e^{i \csc ^{-1}(c x)}\right )}{c} \]
(a^3*c*x + 3*a^2*b*c*x*ArcCsc[c*x] + 3*a*b^2*c*x*ArcCsc[c*x]^2 + b^3*c*x*A rcCsc[c*x]^3 - 6*a*b^2*ArcCsc[c*x]*Log[1 - E^(I*ArcCsc[c*x])] - 3*b^3*ArcC sc[c*x]^2*Log[1 - E^(I*ArcCsc[c*x])] + 6*a*b^2*ArcCsc[c*x]*Log[1 + E^(I*Ar cCsc[c*x])] + 3*b^3*ArcCsc[c*x]^2*Log[1 + E^(I*ArcCsc[c*x])] + 3*a^2*b*Log [c*(1 + Sqrt[1 - 1/(c^2*x^2)])*x] - (6*I)*b^2*(a + b*ArcCsc[c*x])*PolyLog[ 2, -E^(I*ArcCsc[c*x])] + (6*I)*b^2*(a + b*ArcCsc[c*x])*PolyLog[2, E^(I*Arc Csc[c*x])] + 6*b^3*PolyLog[3, -E^(I*ArcCsc[c*x])] - 6*b^3*PolyLog[3, E^(I* ArcCsc[c*x])])/c
Time = 0.56 (sec) , antiderivative size = 137, normalized size of antiderivative = 0.95, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.700, Rules used = {5740, 4910, 3042, 4671, 3011, 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.
| \(\displaystyle \int \left (a+b \csc ^{-1}(c x)\right )^3 \, dx\) |
| \(\Big \downarrow \) 5740 |
| \(\displaystyle -\frac {\int c^2 \sqrt {1-\frac {1}{c^2 x^2}} x^2 \left (a+b \csc ^{-1}(c x)\right )^3d\csc ^{-1}(c x)}{c}\) |
| \(\Big \downarrow \) 4910 |
| \(\displaystyle -\frac {3 b \int c x \left (a+b \csc ^{-1}(c x)\right )^2d\csc ^{-1}(c x)-c x \left (a+b \csc ^{-1}(c x)\right )^3}{c}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {3 b \int \left (a+b \csc ^{-1}(c x)\right )^2 \csc \left (\csc ^{-1}(c x)\right )d\csc ^{-1}(c x)-c x \left (a+b \csc ^{-1}(c x)\right )^3}{c}\) |
| \(\Big \downarrow \) 4671 |
| \(\displaystyle -\frac {-c x \left (a+b \csc ^{-1}(c x)\right )^3+3 b \left (-2 b \int \left (a+b \csc ^{-1}(c x)\right ) \log \left (1-e^{i \csc ^{-1}(c x)}\right )d\csc ^{-1}(c x)+2 b \int \left (a+b \csc ^{-1}(c x)\right ) \log \left (1+e^{i \csc ^{-1}(c x)}\right )d\csc ^{-1}(c x)-2 \text {arctanh}\left (e^{i \csc ^{-1}(c x)}\right ) \left (a+b \csc ^{-1}(c x)\right )^2\right )}{c}\) |
| \(\Big \downarrow \) 3011 |
| \(\displaystyle -\frac {-c x \left (a+b \csc ^{-1}(c x)\right )^3+3 b \left (2 b \left (i \operatorname {PolyLog}\left (2,-e^{i \csc ^{-1}(c x)}\right ) \left (a+b \csc ^{-1}(c x)\right )-i b \int \operatorname {PolyLog}\left (2,-e^{i \csc ^{-1}(c x)}\right )d\csc ^{-1}(c x)\right )-2 b \left (i \operatorname {PolyLog}\left (2,e^{i \csc ^{-1}(c x)}\right ) \left (a+b \csc ^{-1}(c x)\right )-i b \int \operatorname {PolyLog}\left (2,e^{i \csc ^{-1}(c x)}\right )d\csc ^{-1}(c x)\right )-2 \text {arctanh}\left (e^{i \csc ^{-1}(c x)}\right ) \left (a+b \csc ^{-1}(c x)\right )^2\right )}{c}\) |
| \(\Big \downarrow \) 2720 |
| \(\displaystyle -\frac {-c x \left (a+b \csc ^{-1}(c x)\right )^3+3 b \left (2 b \left (i \operatorname {PolyLog}\left (2,-e^{i \csc ^{-1}(c x)}\right ) \left (a+b \csc ^{-1}(c x)\right )-b \int e^{-i \csc ^{-1}(c x)} \operatorname {PolyLog}\left (2,-e^{i \csc ^{-1}(c x)}\right )de^{i \csc ^{-1}(c x)}\right )-2 b \left (i \operatorname {PolyLog}\left (2,e^{i \csc ^{-1}(c x)}\right ) \left (a+b \csc ^{-1}(c x)\right )-b \int e^{-i \csc ^{-1}(c x)} \operatorname {PolyLog}\left (2,e^{i \csc ^{-1}(c x)}\right )de^{i \csc ^{-1}(c x)}\right )-2 \text {arctanh}\left (e^{i \csc ^{-1}(c x)}\right ) \left (a+b \csc ^{-1}(c x)\right )^2\right )}{c}\) |
| \(\Big \downarrow \) 7143 |
| \(\displaystyle -\frac {-c x \left (a+b \csc ^{-1}(c x)\right )^3+3 b \left (-2 \text {arctanh}\left (e^{i \csc ^{-1}(c x)}\right ) \left (a+b \csc ^{-1}(c x)\right )^2+2 b \left (i \operatorname {PolyLog}\left (2,-e^{i \csc ^{-1}(c x)}\right ) \left (a+b \csc ^{-1}(c x)\right )-b \operatorname {PolyLog}\left (3,-e^{i \csc ^{-1}(c x)}\right )\right )-2 b \left (i \operatorname {PolyLog}\left (2,e^{i \csc ^{-1}(c x)}\right ) \left (a+b \csc ^{-1}(c x)\right )-b \operatorname {PolyLog}\left (3,e^{i \csc ^{-1}(c x)}\right )\right )\right )}{c}\) |
-((-(c*x*(a + b*ArcCsc[c*x])^3) + 3*b*(-2*(a + b*ArcCsc[c*x])^2*ArcTanh[E^ (I*ArcCsc[c*x])] + 2*b*(I*(a + b*ArcCsc[c*x])*PolyLog[2, -E^(I*ArcCsc[c*x] )] - b*PolyLog[3, -E^(I*ArcCsc[c*x])]) - 2*b*(I*(a + b*ArcCsc[c*x])*PolyLo g[2, E^(I*ArcCsc[c*x])] - b*PolyLog[3, E^(I*ArcCsc[c*x])])))/c)
3.1.27.3.1 Defintions of rubi rules used
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[csc[(e_.) + (f_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[- 2*(c + d*x)^m*(ArcTanh[E^(I*(e + f*x))]/f), x] + (-Simp[d*(m/f) Int[(c + d*x)^(m - 1)*Log[1 - E^(I*(e + f*x))], x], x] + Simp[d*(m/f) Int[(c + d*x )^(m - 1)*Log[1 + E^(I*(e + f*x))], x], x]) /; FreeQ[{c, d, e, f}, x] && IG tQ[m, 0]
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 ] + Simp[d*(m/(b*n)) Int[(c + d*x)^(m - 1)*Csc[a + b*x]^n, x], x] /; Free Q[{a, b, c, d, n}, x] && EqQ[p, 1] && GtQ[m, 0]
Int[((a_.) + ArcCsc[(c_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Simp[-c^(-1) Su bst[Int[(a + b*x)^n*Csc[x]*Cot[x], x], x, ArcCsc[c*x]], x] /; FreeQ[{a, b, c, n}, x] && IGtQ[n, 0]
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]
Time = 1.16 (sec) , antiderivative size = 378, normalized size of antiderivative = 2.62
| method | result | size |
| derivativedivides | \(\frac {c x \,a^{3}+b^{3} \left (\operatorname {arccsc}\left (c x \right )^{3} c x -3 \operatorname {arccsc}\left (c x \right )^{2} \ln \left (1-\frac {i}{c x}-\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )+6 i \operatorname {arccsc}\left (c x \right ) \operatorname {polylog}\left (2, \frac {i}{c x}+\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )-6 \operatorname {polylog}\left (3, \frac {i}{c x}+\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )+3 \operatorname {arccsc}\left (c x \right )^{2} \ln \left (1+\frac {i}{c x}+\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )-6 i \operatorname {arccsc}\left (c x \right ) \operatorname {polylog}\left (2, -\frac {i}{c x}-\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )+6 \operatorname {polylog}\left (3, -\frac {i}{c x}-\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )\right )+3 a \,b^{2} \left (\operatorname {arccsc}\left (c x \right )^{2} c x -2 \,\operatorname {arccsc}\left (c x \right ) \ln \left (1-\frac {i}{c x}-\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )+2 \,\operatorname {arccsc}\left (c x \right ) \ln \left (1+\frac {i}{c x}+\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )-2 i \operatorname {dilog}\left (1+\frac {i}{c x}+\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )+2 i \operatorname {dilog}\left (1-\frac {i}{c x}-\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )\right )+3 a^{2} b \left (\operatorname {arccsc}\left (c x \right ) c x +\ln \left (c x +c x \sqrt {1-\frac {1}{c^{2} x^{2}}}\right )\right )}{c}\) | \(378\) |
| default | \(\frac {c x \,a^{3}+b^{3} \left (\operatorname {arccsc}\left (c x \right )^{3} c x -3 \operatorname {arccsc}\left (c x \right )^{2} \ln \left (1-\frac {i}{c x}-\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )+6 i \operatorname {arccsc}\left (c x \right ) \operatorname {polylog}\left (2, \frac {i}{c x}+\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )-6 \operatorname {polylog}\left (3, \frac {i}{c x}+\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )+3 \operatorname {arccsc}\left (c x \right )^{2} \ln \left (1+\frac {i}{c x}+\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )-6 i \operatorname {arccsc}\left (c x \right ) \operatorname {polylog}\left (2, -\frac {i}{c x}-\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )+6 \operatorname {polylog}\left (3, -\frac {i}{c x}-\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )\right )+3 a \,b^{2} \left (\operatorname {arccsc}\left (c x \right )^{2} c x -2 \,\operatorname {arccsc}\left (c x \right ) \ln \left (1-\frac {i}{c x}-\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )+2 \,\operatorname {arccsc}\left (c x \right ) \ln \left (1+\frac {i}{c x}+\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )-2 i \operatorname {dilog}\left (1+\frac {i}{c x}+\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )+2 i \operatorname {dilog}\left (1-\frac {i}{c x}-\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )\right )+3 a^{2} b \left (\operatorname {arccsc}\left (c x \right ) c x +\ln \left (c x +c x \sqrt {1-\frac {1}{c^{2} x^{2}}}\right )\right )}{c}\) | \(378\) |
| parts | \(a^{3} x +\frac {b^{3} \left (\operatorname {arccsc}\left (c x \right )^{3} c x -3 \operatorname {arccsc}\left (c x \right )^{2} \ln \left (1-\frac {i}{c x}-\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )+6 i \operatorname {arccsc}\left (c x \right ) \operatorname {polylog}\left (2, \frac {i}{c x}+\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )-6 \operatorname {polylog}\left (3, \frac {i}{c x}+\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )+3 \operatorname {arccsc}\left (c x \right )^{2} \ln \left (1+\frac {i}{c x}+\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )-6 i \operatorname {arccsc}\left (c x \right ) \operatorname {polylog}\left (2, -\frac {i}{c x}-\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )+6 \operatorname {polylog}\left (3, -\frac {i}{c x}-\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )\right )}{c}+\frac {3 a \,b^{2} \left (\operatorname {arccsc}\left (c x \right )^{2} c x -2 \,\operatorname {arccsc}\left (c x \right ) \ln \left (1-\frac {i}{c x}-\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )+2 \,\operatorname {arccsc}\left (c x \right ) \ln \left (1+\frac {i}{c x}+\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )-2 i \operatorname {dilog}\left (1+\frac {i}{c x}+\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )+2 i \operatorname {dilog}\left (1-\frac {i}{c x}-\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )\right )}{c}+3 a^{2} b x \,\operatorname {arccsc}\left (c x \right )+\frac {3 a^{2} b \ln \left (c x +c x \sqrt {1-\frac {1}{c^{2} x^{2}}}\right )}{c}\) | \(385\) |
1/c*(c*x*a^3+b^3*(arccsc(c*x)^3*c*x-3*arccsc(c*x)^2*ln(1-I/c/x-(1-1/c^2/x^ 2)^(1/2))+6*I*arccsc(c*x)*polylog(2,I/c/x+(1-1/c^2/x^2)^(1/2))-6*polylog(3 ,I/c/x+(1-1/c^2/x^2)^(1/2))+3*arccsc(c*x)^2*ln(1+I/c/x+(1-1/c^2/x^2)^(1/2) )-6*I*arccsc(c*x)*polylog(2,-I/c/x-(1-1/c^2/x^2)^(1/2))+6*polylog(3,-I/c/x -(1-1/c^2/x^2)^(1/2)))+3*a*b^2*(arccsc(c*x)^2*c*x-2*arccsc(c*x)*ln(1-I/c/x -(1-1/c^2/x^2)^(1/2))+2*arccsc(c*x)*ln(1+I/c/x+(1-1/c^2/x^2)^(1/2))-2*I*di log(1+I/c/x+(1-1/c^2/x^2)^(1/2))+2*I*dilog(1-I/c/x-(1-1/c^2/x^2)^(1/2)))+3 *a^2*b*(arccsc(c*x)*c*x+ln(c*x+c*x*(1-1/c^2/x^2)^(1/2))))
\[ \int \left (a+b \csc ^{-1}(c x)\right )^3 \, dx=\int { {\left (b \operatorname {arccsc}\left (c x\right ) + a\right )}^{3} \,d x } \]
\[ \int \left (a+b \csc ^{-1}(c x)\right )^3 \, dx=\int \left (a + b \operatorname {acsc}{\left (c x \right )}\right )^{3}\, dx \]
\[ \int \left (a+b \csc ^{-1}(c x)\right )^3 \, dx=\int { {\left (b \operatorname {arccsc}\left (c x\right ) + a\right )}^{3} \,d x } \]
-3/2*a*b^2*c^2*(2*x/c^2 - log(c*x + 1)/c^3 + log(c*x - 1)/c^3)*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^2 - 1), x)*log(c)^2 + b^3*x*arctan2(1, sqrt(c*x + 1)*sqrt(c*x - 1))^3 - 3/4*b^3*x*arctan2(1, sqrt(c*x + 1)*sqrt(c*x - 1))*log(c^2*x^2)^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^2 - 1), 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^2 - 1), x)*log(c) + 12*a*b^2*c^2*inte grate(1/4*x^2*log(c^2*x^2)/(c^2*x^2 - 1), x)*log(c) - 24*a*b^2*c^2*integra te(1/4*x^2*log(x)/(c^2*x^2 - 1), x)*log(c) + 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)*log(x)/(c^2*x^2 - 1), 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^2 - 1), 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^2 - 1), x) + 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^2 - 1), x) - 3*a*b^2*c^2*integrate(1/4*x^2*log(c^2*x^2)^2/(c^2*x^2 - 1), x) + 12*a*b ^2*c^2*integrate(1/4*x^2*log(c^2*x^2)*log(x)/(c^2*x^2 - 1), x) - 12*a*b^2* c^2*integrate(1/4*x^2*log(x)^2/(c^2*x^2 - 1), x) - 3/2*a*b^2*(log(c*x + 1) /c - log(c*x - 1)/c)*log(c)^2 + 12*b^3*integrate(1/4*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*arcta n(1/(sqrt(c*x + 1)*sqrt(c*x - 1)))*log(c^2*x^2)/(c^2*x^2 - 1), x)*log(c...
\[ \int \left (a+b \csc ^{-1}(c x)\right )^3 \, dx=\int { {\left (b \operatorname {arccsc}\left (c x\right ) + a\right )}^{3} \,d x } \]
Timed out. \[ \int \left (a+b \csc ^{-1}(c x)\right )^3 \, dx=\int {\left (a+b\,\mathrm {asin}\left (\frac {1}{c\,x}\right )\right )}^3 \,d x \]