Integrand size = 10, antiderivative size = 102 \[ \int \frac {\arcsin (a x)^3}{x^3} \, dx=-\frac {3}{2} i a^2 \arcsin (a x)^2-\frac {3 a \sqrt {1-a^2 x^2} \arcsin (a x)^2}{2 x}-\frac {\arcsin (a x)^3}{2 x^2}+3 a^2 \arcsin (a x) \log \left (1-e^{2 i \arcsin (a x)}\right )-\frac {3}{2} i a^2 \operatorname {PolyLog}\left (2,e^{2 i \arcsin (a x)}\right ) \]
-3/2*I*a^2*arcsin(a*x)^2-1/2*arcsin(a*x)^3/x^2+3*a^2*arcsin(a*x)*ln(1-(I*a *x+(-a^2*x^2+1)^(1/2))^2)-3/2*I*a^2*polylog(2,(I*a*x+(-a^2*x^2+1)^(1/2))^2 )-3/2*a*arcsin(a*x)^2*(-a^2*x^2+1)^(1/2)/x
Time = 0.28 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.90 \[ \int \frac {\arcsin (a x)^3}{x^3} \, dx=-\frac {\arcsin (a x) \left (3 a x \left (i a x+\sqrt {1-a^2 x^2}\right ) \arcsin (a x)+\arcsin (a x)^2-6 a^2 x^2 \log \left (1-e^{2 i \arcsin (a x)}\right )\right )}{2 x^2}-\frac {3}{2} i a^2 \operatorname {PolyLog}\left (2,e^{2 i \arcsin (a x)}\right ) \]
-1/2*(ArcSin[a*x]*(3*a*x*(I*a*x + Sqrt[1 - a^2*x^2])*ArcSin[a*x] + ArcSin[ a*x]^2 - 6*a^2*x^2*Log[1 - E^((2*I)*ArcSin[a*x])]))/x^2 - ((3*I)/2)*a^2*Po lyLog[2, E^((2*I)*ArcSin[a*x])]
Time = 0.55 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.05, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.000, Rules used = {5138, 5186, 5136, 3042, 25, 4200, 25, 2620, 2715, 2838}
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.
\(\displaystyle \int \frac {\arcsin (a x)^3}{x^3} \, dx\) |
\(\Big \downarrow \) 5138 |
\(\displaystyle \frac {3}{2} a \int \frac {\arcsin (a x)^2}{x^2 \sqrt {1-a^2 x^2}}dx-\frac {\arcsin (a x)^3}{2 x^2}\) |
\(\Big \downarrow \) 5186 |
\(\displaystyle \frac {3}{2} a \left (2 a \int \frac {\arcsin (a x)}{x}dx-\frac {\sqrt {1-a^2 x^2} \arcsin (a x)^2}{x}\right )-\frac {\arcsin (a x)^3}{2 x^2}\) |
\(\Big \downarrow \) 5136 |
\(\displaystyle \frac {3}{2} a \left (2 a \int \frac {\sqrt {1-a^2 x^2} \arcsin (a x)}{a x}d\arcsin (a x)-\frac {\sqrt {1-a^2 x^2} \arcsin (a x)^2}{x}\right )-\frac {\arcsin (a x)^3}{2 x^2}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {3}{2} a \left (2 a \int -\arcsin (a x) \tan \left (\arcsin (a x)+\frac {\pi }{2}\right )d\arcsin (a x)-\frac {\sqrt {1-a^2 x^2} \arcsin (a x)^2}{x}\right )-\frac {\arcsin (a x)^3}{2 x^2}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {3}{2} a \left (-2 a \int \arcsin (a x) \tan \left (\arcsin (a x)+\frac {\pi }{2}\right )d\arcsin (a x)-\frac {\sqrt {1-a^2 x^2} \arcsin (a x)^2}{x}\right )-\frac {\arcsin (a x)^3}{2 x^2}\) |
\(\Big \downarrow \) 4200 |
\(\displaystyle -\frac {\arcsin (a x)^3}{2 x^2}+\frac {3}{2} a \left (-\frac {\sqrt {1-a^2 x^2} \arcsin (a x)^2}{x}+2 a \left (2 i \int -\frac {e^{2 i \arcsin (a x)} \arcsin (a x)}{1-e^{2 i \arcsin (a x)}}d\arcsin (a x)-\frac {1}{2} i \arcsin (a x)^2\right )\right )\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\frac {\arcsin (a x)^3}{2 x^2}+\frac {3}{2} a \left (-\frac {\sqrt {1-a^2 x^2} \arcsin (a x)^2}{x}+2 a \left (-2 i \int \frac {e^{2 i \arcsin (a x)} \arcsin (a x)}{1-e^{2 i \arcsin (a x)}}d\arcsin (a x)-\frac {1}{2} i \arcsin (a x)^2\right )\right )\) |
\(\Big \downarrow \) 2620 |
\(\displaystyle -\frac {\arcsin (a x)^3}{2 x^2}+\frac {3}{2} a \left (-\frac {\sqrt {1-a^2 x^2} \arcsin (a x)^2}{x}+2 a \left (-2 i \left (\frac {1}{2} i \arcsin (a x) \log \left (1-e^{2 i \arcsin (a x)}\right )-\frac {1}{2} i \int \log \left (1-e^{2 i \arcsin (a x)}\right )d\arcsin (a x)\right )-\frac {1}{2} i \arcsin (a x)^2\right )\right )\) |
\(\Big \downarrow \) 2715 |
\(\displaystyle -\frac {\arcsin (a x)^3}{2 x^2}+\frac {3}{2} a \left (-\frac {\sqrt {1-a^2 x^2} \arcsin (a x)^2}{x}+2 a \left (-2 i \left (\frac {1}{2} i \arcsin (a x) \log \left (1-e^{2 i \arcsin (a x)}\right )-\frac {1}{4} \int e^{-2 i \arcsin (a x)} \log \left (1-e^{2 i \arcsin (a x)}\right )de^{2 i \arcsin (a x)}\right )-\frac {1}{2} i \arcsin (a x)^2\right )\right )\) |
\(\Big \downarrow \) 2838 |
\(\displaystyle -\frac {\arcsin (a x)^3}{2 x^2}+\frac {3}{2} a \left (-\frac {\sqrt {1-a^2 x^2} \arcsin (a x)^2}{x}+2 a \left (-2 i \left (\frac {1}{4} \operatorname {PolyLog}\left (2,e^{2 i \arcsin (a x)}\right )+\frac {1}{2} i \arcsin (a x) \log \left (1-e^{2 i \arcsin (a x)}\right )\right )-\frac {1}{2} i \arcsin (a x)^2\right )\right )\) |
-1/2*ArcSin[a*x]^3/x^2 + (3*a*(-((Sqrt[1 - a^2*x^2]*ArcSin[a*x]^2)/x) + 2* a*((-1/2*I)*ArcSin[a*x]^2 - (2*I)*((I/2)*ArcSin[a*x]*Log[1 - E^((2*I)*ArcS in[a*x])] + PolyLog[2, E^((2*I)*ArcSin[a*x])]/4))))/2
3.1.29.3.1 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[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] :> Simp[1/(d*e*n*Log[F]) Subst[Int[Log[a + b*x]/x, x], x, (F^(e*(c + d*x) ))^n], x] /; FreeQ[{F, a, b, c, d, e, n}, x] && GtQ[a, 0]
Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> Simp[-PolyLog[2 , (-c)*e*x^n]/n, x] /; FreeQ[{c, d, e, n}, x] && EqQ[c*d, 1]
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[((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_.)*((f_.)*(x_))^(m_)*((d_) + (e_. )*(x_)^2)^(p_), x_Symbol] :> Simp[(f*x)^(m + 1)*(d + e*x^2)^(p + 1)*((a + b *ArcSin[c*x])^n/(d*f*(m + 1))), x] - Simp[b*c*(n/(f*(m + 1)))*Simp[(d + e*x ^2)^p/(1 - c^2*x^2)^p] Int[(f*x)^(m + 1)*(1 - c^2*x^2)^(p + 1/2)*(a + b*A rcSin[c*x])^(n - 1), x], x] /; FreeQ[{a, b, c, d, e, f, m, p}, x] && EqQ[c^ 2*d + e, 0] && GtQ[n, 0] && EqQ[m + 2*p + 3, 0] && NeQ[m, -1]
Time = 0.06 (sec) , antiderivative size = 161, normalized size of antiderivative = 1.58
method | result | size |
derivativedivides | \(a^{2} \left (-\frac {\arcsin \left (a x \right )^{2} \left (-3 i a^{2} x^{2}+3 a x \sqrt {-a^{2} x^{2}+1}+\arcsin \left (a x \right )\right )}{2 a^{2} x^{2}}+3 \arcsin \left (a x \right ) \ln \left (1-i a x -\sqrt {-a^{2} x^{2}+1}\right )+3 \arcsin \left (a x \right ) \ln \left (1+i a x +\sqrt {-a^{2} x^{2}+1}\right )-3 i \arcsin \left (a x \right )^{2}-3 i \operatorname {polylog}\left (2, i a x +\sqrt {-a^{2} x^{2}+1}\right )-3 i \operatorname {polylog}\left (2, -i a x -\sqrt {-a^{2} x^{2}+1}\right )\right )\) | \(161\) |
default | \(a^{2} \left (-\frac {\arcsin \left (a x \right )^{2} \left (-3 i a^{2} x^{2}+3 a x \sqrt {-a^{2} x^{2}+1}+\arcsin \left (a x \right )\right )}{2 a^{2} x^{2}}+3 \arcsin \left (a x \right ) \ln \left (1-i a x -\sqrt {-a^{2} x^{2}+1}\right )+3 \arcsin \left (a x \right ) \ln \left (1+i a x +\sqrt {-a^{2} x^{2}+1}\right )-3 i \arcsin \left (a x \right )^{2}-3 i \operatorname {polylog}\left (2, i a x +\sqrt {-a^{2} x^{2}+1}\right )-3 i \operatorname {polylog}\left (2, -i a x -\sqrt {-a^{2} x^{2}+1}\right )\right )\) | \(161\) |
a^2*(-1/2*arcsin(a*x)^2*(-3*I*a^2*x^2+3*a*x*(-a^2*x^2+1)^(1/2)+arcsin(a*x) )/a^2/x^2+3*arcsin(a*x)*ln(1-I*a*x-(-a^2*x^2+1)^(1/2))+3*arcsin(a*x)*ln(1+ I*a*x+(-a^2*x^2+1)^(1/2))-3*I*arcsin(a*x)^2-3*I*polylog(2,I*a*x+(-a^2*x^2+ 1)^(1/2))-3*I*polylog(2,-I*a*x-(-a^2*x^2+1)^(1/2)))
\[ \int \frac {\arcsin (a x)^3}{x^3} \, dx=\int { \frac {\arcsin \left (a x\right )^{3}}{x^{3}} \,d x } \]
\[ \int \frac {\arcsin (a x)^3}{x^3} \, dx=\int \frac {\operatorname {asin}^{3}{\left (a x \right )}}{x^{3}}\, dx \]
\[ \int \frac {\arcsin (a x)^3}{x^3} \, dx=\int { \frac {\arcsin \left (a x\right )^{3}}{x^{3}} \,d x } \]
-1/2*(6*a*x^2*integrate(1/2*sqrt(a*x + 1)*sqrt(-a*x + 1)*arctan2(a*x, sqrt (a*x + 1)*sqrt(-a*x + 1))^2/(a^2*x^4 - x^2), x) + arctan2(a*x, sqrt(a*x + 1)*sqrt(-a*x + 1))^3)/x^2
\[ \int \frac {\arcsin (a x)^3}{x^3} \, dx=\int { \frac {\arcsin \left (a x\right )^{3}}{x^{3}} \,d x } \]
Timed out. \[ \int \frac {\arcsin (a x)^3}{x^3} \, dx=\int \frac {{\mathrm {asin}\left (a\,x\right )}^3}{x^3} \,d x \]