Integrand size = 10, antiderivative size = 108 \[ \int \frac {\arcsin (a x)^3}{x^2} \, dx=-\frac {\arcsin (a x)^3}{x}-6 a \arcsin (a x)^2 \text {arctanh}\left (e^{i \arcsin (a x)}\right )+6 i a \arcsin (a x) \operatorname {PolyLog}\left (2,-e^{i \arcsin (a x)}\right )-6 i a \arcsin (a x) \operatorname {PolyLog}\left (2,e^{i \arcsin (a x)}\right )-6 a \operatorname {PolyLog}\left (3,-e^{i \arcsin (a x)}\right )+6 a \operatorname {PolyLog}\left (3,e^{i \arcsin (a x)}\right ) \]
-arcsin(a*x)^3/x-6*a*arcsin(a*x)^2*arctanh(I*a*x+(-a^2*x^2+1)^(1/2))+6*I*a *arcsin(a*x)*polylog(2,-I*a*x-(-a^2*x^2+1)^(1/2))-6*I*a*arcsin(a*x)*polylo g(2,I*a*x+(-a^2*x^2+1)^(1/2))-6*a*polylog(3,-I*a*x-(-a^2*x^2+1)^(1/2))+6*a *polylog(3,I*a*x+(-a^2*x^2+1)^(1/2))
Time = 0.12 (sec) , antiderivative size = 133, normalized size of antiderivative = 1.23 \[ \int \frac {\arcsin (a x)^3}{x^2} \, dx=a \left (-\frac {\arcsin (a x)^3}{a x}+3 \arcsin (a x)^2 \log \left (1-e^{i \arcsin (a x)}\right )-3 \arcsin (a x)^2 \log \left (1+e^{i \arcsin (a x)}\right )+6 i \arcsin (a x) \operatorname {PolyLog}\left (2,-e^{i \arcsin (a x)}\right )-6 i \arcsin (a x) \operatorname {PolyLog}\left (2,e^{i \arcsin (a x)}\right )-6 \operatorname {PolyLog}\left (3,-e^{i \arcsin (a x)}\right )+6 \operatorname {PolyLog}\left (3,e^{i \arcsin (a x)}\right )\right ) \]
a*(-(ArcSin[a*x]^3/(a*x)) + 3*ArcSin[a*x]^2*Log[1 - E^(I*ArcSin[a*x])] - 3 *ArcSin[a*x]^2*Log[1 + E^(I*ArcSin[a*x])] + (6*I)*ArcSin[a*x]*PolyLog[2, - E^(I*ArcSin[a*x])] - (6*I)*ArcSin[a*x]*PolyLog[2, E^(I*ArcSin[a*x])] - 6*P olyLog[3, -E^(I*ArcSin[a*x])] + 6*PolyLog[3, E^(I*ArcSin[a*x])])
Time = 0.53 (sec) , antiderivative size = 113, normalized size of antiderivative = 1.05, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.700, Rules used = {5138, 5218, 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.
\(\displaystyle \int \frac {\arcsin (a x)^3}{x^2} \, dx\) |
\(\Big \downarrow \) 5138 |
\(\displaystyle 3 a \int \frac {\arcsin (a x)^2}{x \sqrt {1-a^2 x^2}}dx-\frac {\arcsin (a x)^3}{x}\) |
\(\Big \downarrow \) 5218 |
\(\displaystyle 3 a \int \frac {\arcsin (a x)^2}{a x}d\arcsin (a x)-\frac {\arcsin (a x)^3}{x}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle 3 a \int \arcsin (a x)^2 \csc (\arcsin (a x))d\arcsin (a x)-\frac {\arcsin (a x)^3}{x}\) |
\(\Big \downarrow \) 4671 |
\(\displaystyle -\frac {\arcsin (a x)^3}{x}+3 a \left (-2 \int \arcsin (a x) \log \left (1-e^{i \arcsin (a x)}\right )d\arcsin (a x)+2 \int \arcsin (a x) \log \left (1+e^{i \arcsin (a x)}\right )d\arcsin (a x)-2 \arcsin (a x)^2 \text {arctanh}\left (e^{i \arcsin (a x)}\right )\right )\) |
\(\Big \downarrow \) 3011 |
\(\displaystyle -\frac {\arcsin (a x)^3}{x}+3 a \left (2 \left (i \arcsin (a x) \operatorname {PolyLog}\left (2,-e^{i \arcsin (a x)}\right )-i \int \operatorname {PolyLog}\left (2,-e^{i \arcsin (a x)}\right )d\arcsin (a x)\right )-2 \left (i \arcsin (a x) \operatorname {PolyLog}\left (2,e^{i \arcsin (a x)}\right )-i \int \operatorname {PolyLog}\left (2,e^{i \arcsin (a x)}\right )d\arcsin (a x)\right )-2 \arcsin (a x)^2 \text {arctanh}\left (e^{i \arcsin (a x)}\right )\right )\) |
\(\Big \downarrow \) 2720 |
\(\displaystyle -\frac {\arcsin (a x)^3}{x}+3 a \left (2 \left (i \arcsin (a x) \operatorname {PolyLog}\left (2,-e^{i \arcsin (a x)}\right )-\int e^{-i \arcsin (a x)} \operatorname {PolyLog}\left (2,-e^{i \arcsin (a x)}\right )de^{i \arcsin (a x)}\right )-2 \left (i \arcsin (a x) \operatorname {PolyLog}\left (2,e^{i \arcsin (a x)}\right )-\int e^{-i \arcsin (a x)} \operatorname {PolyLog}\left (2,e^{i \arcsin (a x)}\right )de^{i \arcsin (a x)}\right )-2 \arcsin (a x)^2 \text {arctanh}\left (e^{i \arcsin (a x)}\right )\right )\) |
\(\Big \downarrow \) 7143 |
\(\displaystyle -\frac {\arcsin (a x)^3}{x}+3 a \left (-2 \arcsin (a x)^2 \text {arctanh}\left (e^{i \arcsin (a x)}\right )+2 \left (i \arcsin (a x) \operatorname {PolyLog}\left (2,-e^{i \arcsin (a x)}\right )-\operatorname {PolyLog}\left (3,-e^{i \arcsin (a x)}\right )\right )-2 \left (i \arcsin (a x) \operatorname {PolyLog}\left (2,e^{i \arcsin (a x)}\right )-\operatorname {PolyLog}\left (3,e^{i \arcsin (a x)}\right )\right )\right )\) |
-(ArcSin[a*x]^3/x) + 3*a*(-2*ArcSin[a*x]^2*ArcTanh[E^(I*ArcSin[a*x])] + 2* (I*ArcSin[a*x]*PolyLog[2, -E^(I*ArcSin[a*x])] - PolyLog[3, -E^(I*ArcSin[a* x])]) - 2*(I*ArcSin[a*x]*PolyLog[2, E^(I*ArcSin[a*x])] - PolyLog[3, E^(I*A rcSin[a*x])]))
3.1.28.3.1 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[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[(d*x)^(m + 1)*((a + b*ArcSin[c*x])^n/(d*(m + 1))), x] - Simp[b*c*(n /(d*(m + 1))) Int[(d*x)^(m + 1)*((a + b*ArcSin[c*x])^(n - 1)/Sqrt[1 - c^2 *x^2]), x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] && NeQ[m, -1]
Int[(((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*(x_)^(m_))/Sqrt[(d_) + (e_.)* (x_)^2], x_Symbol] :> Simp[(1/c^(m + 1))*Simp[Sqrt[1 - c^2*x^2]/Sqrt[d + e* x^2]] Subst[Int[(a + b*x)^n*Sin[x]^m, x], x, ArcSin[c*x]], x] /; FreeQ[{a , b, c, d, e}, x] && EqQ[c^2*d + e, 0] && IGtQ[n, 0] && IntegerQ[m]
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 = 0.05 (sec) , antiderivative size = 178, normalized size of antiderivative = 1.65
method | result | size |
derivativedivides | \(a \left (-\frac {\arcsin \left (a x \right )^{3}}{a x}+3 \arcsin \left (a x \right )^{2} \ln \left (1-i a x -\sqrt {-a^{2} x^{2}+1}\right )-6 i \arcsin \left (a x \right ) \operatorname {polylog}\left (2, i a x +\sqrt {-a^{2} x^{2}+1}\right )+6 \operatorname {polylog}\left (3, i a x +\sqrt {-a^{2} x^{2}+1}\right )-3 \arcsin \left (a x \right )^{2} \ln \left (1+i a x +\sqrt {-a^{2} x^{2}+1}\right )+6 i \arcsin \left (a x \right ) \operatorname {polylog}\left (2, -i a x -\sqrt {-a^{2} x^{2}+1}\right )-6 \operatorname {polylog}\left (3, -i a x -\sqrt {-a^{2} x^{2}+1}\right )\right )\) | \(178\) |
default | \(a \left (-\frac {\arcsin \left (a x \right )^{3}}{a x}+3 \arcsin \left (a x \right )^{2} \ln \left (1-i a x -\sqrt {-a^{2} x^{2}+1}\right )-6 i \arcsin \left (a x \right ) \operatorname {polylog}\left (2, i a x +\sqrt {-a^{2} x^{2}+1}\right )+6 \operatorname {polylog}\left (3, i a x +\sqrt {-a^{2} x^{2}+1}\right )-3 \arcsin \left (a x \right )^{2} \ln \left (1+i a x +\sqrt {-a^{2} x^{2}+1}\right )+6 i \arcsin \left (a x \right ) \operatorname {polylog}\left (2, -i a x -\sqrt {-a^{2} x^{2}+1}\right )-6 \operatorname {polylog}\left (3, -i a x -\sqrt {-a^{2} x^{2}+1}\right )\right )\) | \(178\) |
a*(-arcsin(a*x)^3/a/x+3*arcsin(a*x)^2*ln(1-I*a*x-(-a^2*x^2+1)^(1/2))-6*I*a rcsin(a*x)*polylog(2,I*a*x+(-a^2*x^2+1)^(1/2))+6*polylog(3,I*a*x+(-a^2*x^2 +1)^(1/2))-3*arcsin(a*x)^2*ln(1+I*a*x+(-a^2*x^2+1)^(1/2))+6*I*arcsin(a*x)* polylog(2,-I*a*x-(-a^2*x^2+1)^(1/2))-6*polylog(3,-I*a*x-(-a^2*x^2+1)^(1/2) ))
\[ \int \frac {\arcsin (a x)^3}{x^2} \, dx=\int { \frac {\arcsin \left (a x\right )^{3}}{x^{2}} \,d x } \]
\[ \int \frac {\arcsin (a x)^3}{x^2} \, dx=\int \frac {\operatorname {asin}^{3}{\left (a x \right )}}{x^{2}}\, dx \]
\[ \int \frac {\arcsin (a x)^3}{x^2} \, dx=\int { \frac {\arcsin \left (a x\right )^{3}}{x^{2}} \,d x } \]
-(arctan2(a*x, sqrt(a*x + 1)*sqrt(-a*x + 1))^3 + 3*a*x*integrate(sqrt(a*x + 1)*sqrt(-a*x + 1)*arctan2(a*x, sqrt(a*x + 1)*sqrt(-a*x + 1))^2/(a^2*x^3 - x), x))/x
\[ \int \frac {\arcsin (a x)^3}{x^2} \, dx=\int { \frac {\arcsin \left (a x\right )^{3}}{x^{2}} \,d x } \]
Timed out. \[ \int \frac {\arcsin (a x)^3}{x^2} \, dx=\int \frac {{\mathrm {asin}\left (a\,x\right )}^3}{x^2} \,d x \]