Integrand size = 10, antiderivative size = 71 \[ \int \frac {\arcsin (a x)^2}{x} \, dx=-\frac {1}{3} i \arcsin (a x)^3+\arcsin (a x)^2 \log \left (1-e^{2 i \arcsin (a x)}\right )-i \arcsin (a x) \operatorname {PolyLog}\left (2,e^{2 i \arcsin (a x)}\right )+\frac {1}{2} \operatorname {PolyLog}\left (3,e^{2 i \arcsin (a x)}\right ) \]
-1/3*I*arcsin(a*x)^3+arcsin(a*x)^2*ln(1-(I*a*x+(-a^2*x^2+1)^(1/2))^2)-I*ar csin(a*x)*polylog(2,(I*a*x+(-a^2*x^2+1)^(1/2))^2)+1/2*polylog(3,(I*a*x+(-a ^2*x^2+1)^(1/2))^2)
Time = 0.09 (sec) , antiderivative size = 71, normalized size of antiderivative = 1.00 \[ \int \frac {\arcsin (a x)^2}{x} \, dx=\frac {1}{3} i \arcsin (a x)^3+\arcsin (a x)^2 \log \left (1-e^{-2 i \arcsin (a x)}\right )+i \arcsin (a x) \operatorname {PolyLog}\left (2,e^{-2 i \arcsin (a x)}\right )+\frac {1}{2} \operatorname {PolyLog}\left (3,e^{-2 i \arcsin (a x)}\right ) \]
(I/3)*ArcSin[a*x]^3 + ArcSin[a*x]^2*Log[1 - E^((-2*I)*ArcSin[a*x])] + I*Ar cSin[a*x]*PolyLog[2, E^((-2*I)*ArcSin[a*x])] + PolyLog[3, E^((-2*I)*ArcSin [a*x])]/2
Time = 0.41 (sec) , antiderivative size = 88, normalized size of antiderivative = 1.24, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.900, Rules used = {5136, 3042, 25, 4200, 25, 2620, 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.
\(\displaystyle \int \frac {\arcsin (a x)^2}{x} \, dx\) |
\(\Big \downarrow \) 5136 |
\(\displaystyle \int \frac {\sqrt {1-a^2 x^2} \arcsin (a x)^2}{a x}d\arcsin (a x)\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int -\arcsin (a x)^2 \tan \left (\arcsin (a x)+\frac {\pi }{2}\right )d\arcsin (a x)\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\int \arcsin (a x)^2 \tan \left (\arcsin (a x)+\frac {\pi }{2}\right )d\arcsin (a x)\) |
\(\Big \downarrow \) 4200 |
\(\displaystyle 2 i \int -\frac {e^{2 i \arcsin (a x)} \arcsin (a x)^2}{1-e^{2 i \arcsin (a x)}}d\arcsin (a x)-\frac {1}{3} i \arcsin (a x)^3\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -2 i \int \frac {e^{2 i \arcsin (a x)} \arcsin (a x)^2}{1-e^{2 i \arcsin (a x)}}d\arcsin (a x)-\frac {1}{3} i \arcsin (a x)^3\) |
\(\Big \downarrow \) 2620 |
\(\displaystyle -2 i \left (\frac {1}{2} i \arcsin (a x)^2 \log \left (1-e^{2 i \arcsin (a x)}\right )-i \int \arcsin (a x) \log \left (1-e^{2 i \arcsin (a x)}\right )d\arcsin (a x)\right )-\frac {1}{3} i \arcsin (a x)^3\) |
\(\Big \downarrow \) 3011 |
\(\displaystyle -2 i \left (\frac {1}{2} i \arcsin (a x)^2 \log \left (1-e^{2 i \arcsin (a x)}\right )-i \left (\frac {1}{2} i \arcsin (a x) \operatorname {PolyLog}\left (2,e^{2 i \arcsin (a x)}\right )-\frac {1}{2} i \int \operatorname {PolyLog}\left (2,e^{2 i \arcsin (a x)}\right )d\arcsin (a x)\right )\right )-\frac {1}{3} i \arcsin (a x)^3\) |
\(\Big \downarrow \) 2720 |
\(\displaystyle -2 i \left (\frac {1}{2} i \arcsin (a x)^2 \log \left (1-e^{2 i \arcsin (a x)}\right )-i \left (\frac {1}{2} i \arcsin (a x) \operatorname {PolyLog}\left (2,e^{2 i \arcsin (a x)}\right )-\frac {1}{4} \int e^{-2 i \arcsin (a x)} \operatorname {PolyLog}\left (2,e^{2 i \arcsin (a x)}\right )de^{2 i \arcsin (a x)}\right )\right )-\frac {1}{3} i \arcsin (a x)^3\) |
\(\Big \downarrow \) 7143 |
\(\displaystyle -2 i \left (\frac {1}{2} i \arcsin (a x)^2 \log \left (1-e^{2 i \arcsin (a x)}\right )-i \left (\frac {1}{2} i \arcsin (a x) \operatorname {PolyLog}\left (2,e^{2 i \arcsin (a x)}\right )-\frac {1}{4} \operatorname {PolyLog}\left (3,e^{2 i \arcsin (a x)}\right )\right )\right )-\frac {1}{3} i \arcsin (a x)^3\) |
(-1/3*I)*ArcSin[a*x]^3 - (2*I)*((I/2)*ArcSin[a*x]^2*Log[1 - E^((2*I)*ArcSi n[a*x])] - I*((I/2)*ArcSin[a*x]*PolyLog[2, E^((2*I)*ArcSin[a*x])] - PolyLo g[3, E^((2*I)*ArcSin[a*x])]/4))
3.1.17.3.1 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_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)/(x_), x_Symbol] :> Subst[Int[( a + b*x)^n*Cot[x], x], x, ArcSin[c*x]] /; FreeQ[{a, b, c}, 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 = 0.04 (sec) , antiderivative size = 169, normalized size of antiderivative = 2.38
method | result | size |
derivativedivides | \(-\frac {i \arcsin \left (a x \right )^{3}}{3}+\arcsin \left (a x \right )^{2} \ln \left (1-i a x -\sqrt {-a^{2} x^{2}+1}\right )-2 i \arcsin \left (a x \right ) \operatorname {polylog}\left (2, i a x +\sqrt {-a^{2} x^{2}+1}\right )+2 \operatorname {polylog}\left (3, i a x +\sqrt {-a^{2} x^{2}+1}\right )+\arcsin \left (a x \right )^{2} \ln \left (1+i a x +\sqrt {-a^{2} x^{2}+1}\right )-2 i \arcsin \left (a x \right ) \operatorname {polylog}\left (2, -i a x -\sqrt {-a^{2} x^{2}+1}\right )+2 \operatorname {polylog}\left (3, -i a x -\sqrt {-a^{2} x^{2}+1}\right )\) | \(169\) |
default | \(-\frac {i \arcsin \left (a x \right )^{3}}{3}+\arcsin \left (a x \right )^{2} \ln \left (1-i a x -\sqrt {-a^{2} x^{2}+1}\right )-2 i \arcsin \left (a x \right ) \operatorname {polylog}\left (2, i a x +\sqrt {-a^{2} x^{2}+1}\right )+2 \operatorname {polylog}\left (3, i a x +\sqrt {-a^{2} x^{2}+1}\right )+\arcsin \left (a x \right )^{2} \ln \left (1+i a x +\sqrt {-a^{2} x^{2}+1}\right )-2 i \arcsin \left (a x \right ) \operatorname {polylog}\left (2, -i a x -\sqrt {-a^{2} x^{2}+1}\right )+2 \operatorname {polylog}\left (3, -i a x -\sqrt {-a^{2} x^{2}+1}\right )\) | \(169\) |
-1/3*I*arcsin(a*x)^3+arcsin(a*x)^2*ln(1-I*a*x-(-a^2*x^2+1)^(1/2))-2*I*arcs in(a*x)*polylog(2,I*a*x+(-a^2*x^2+1)^(1/2))+2*polylog(3,I*a*x+(-a^2*x^2+1) ^(1/2))+arcsin(a*x)^2*ln(1+I*a*x+(-a^2*x^2+1)^(1/2))-2*I*arcsin(a*x)*polyl og(2,-I*a*x-(-a^2*x^2+1)^(1/2))+2*polylog(3,-I*a*x-(-a^2*x^2+1)^(1/2))
\[ \int \frac {\arcsin (a x)^2}{x} \, dx=\int { \frac {\arcsin \left (a x\right )^{2}}{x} \,d x } \]
\[ \int \frac {\arcsin (a x)^2}{x} \, dx=\int \frac {\operatorname {asin}^{2}{\left (a x \right )}}{x}\, dx \]
\[ \int \frac {\arcsin (a x)^2}{x} \, dx=\int { \frac {\arcsin \left (a x\right )^{2}}{x} \,d x } \]
\[ \int \frac {\arcsin (a x)^2}{x} \, dx=\int { \frac {\arcsin \left (a x\right )^{2}}{x} \,d x } \]
Timed out. \[ \int \frac {\arcsin (a x)^2}{x} \, dx=\int \frac {{\mathrm {asin}\left (a\,x\right )}^2}{x} \,d x \]