\(\int \frac {\arcsin (a x)^3}{x^2 \sqrt {1-a^2 x^2}} \, dx\) [288]

Optimal result

Integrand size = 24, antiderivative size = 99 \[ \int \frac {\arcsin (a x)^3}{x^2 \sqrt {1-a^2 x^2}} \, dx=-i a \arcsin (a x)^3-\frac {\sqrt {1-a^2 x^2} \arcsin (a x)^3}{x}+3 a \arcsin (a x)^2 \log \left (1-e^{2 i \arcsin (a x)}\right )-3 i a \arcsin (a x) \operatorname {PolyLog}\left (2,e^{2 i \arcsin (a x)}\right )+\frac {3}{2} a \operatorname {PolyLog}\left (3,e^{2 i \arcsin (a x)}\right ) \] Output:

-I*a*arcsin(a*x)^3-(-a^2*x^2+1)^(1/2)*arcsin(a*x)^3/x+3*a*arcsin(a*x)^2*ln 
(1-(I*a*x+(-a^2*x^2+1)^(1/2))^2)-3*I*a*arcsin(a*x)*polylog(2,(I*a*x+(-a^2* 
x^2+1)^(1/2))^2)+3/2*a*polylog(3,(I*a*x+(-a^2*x^2+1)^(1/2))^2)
 

Mathematica [A] (verified)

Time = 0.18 (sec) , antiderivative size = 108, normalized size of antiderivative = 1.09 \[ \int \frac {\arcsin (a x)^3}{x^2 \sqrt {1-a^2 x^2}} \, dx=\frac {1}{8} a \left (-i \pi ^3+8 i \arcsin (a x)^3-\frac {8 \sqrt {1-a^2 x^2} \arcsin (a x)^3}{a x}+24 \arcsin (a x)^2 \log \left (1-e^{-2 i \arcsin (a x)}\right )+24 i \arcsin (a x) \operatorname {PolyLog}\left (2,e^{-2 i \arcsin (a x)}\right )+12 \operatorname {PolyLog}\left (3,e^{-2 i \arcsin (a x)}\right )\right ) \] Input:

Integrate[ArcSin[a*x]^3/(x^2*Sqrt[1 - a^2*x^2]),x]
 

Output:

(a*((-I)*Pi^3 + (8*I)*ArcSin[a*x]^3 - (8*Sqrt[1 - a^2*x^2]*ArcSin[a*x]^3)/ 
(a*x) + 24*ArcSin[a*x]^2*Log[1 - E^((-2*I)*ArcSin[a*x])] + (24*I)*ArcSin[a 
*x]*PolyLog[2, E^((-2*I)*ArcSin[a*x])] + 12*PolyLog[3, E^((-2*I)*ArcSin[a* 
x])]))/8
 

Rubi [A] (verified)

Time = 0.68 (sec) , antiderivative size = 117, normalized size of antiderivative = 1.18, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {5186, 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.

\(\displaystyle \int \frac {\arcsin (a x)^3}{x^2 \sqrt {1-a^2 x^2}} \, dx\)

\(\Big \downarrow \) 5186

\(\displaystyle 3 a \int \frac {\arcsin (a x)^2}{x}dx-\frac {\sqrt {1-a^2 x^2} \arcsin (a x)^3}{x}\)

\(\Big \downarrow \) 5136

\(\displaystyle 3 a \int \frac {\sqrt {1-a^2 x^2} \arcsin (a x)^2}{a x}d\arcsin (a x)-\frac {\sqrt {1-a^2 x^2} \arcsin (a x)^3}{x}\)

\(\Big \downarrow \) 3042

\(\displaystyle 3 a \int -\arcsin (a x)^2 \tan \left (\arcsin (a x)+\frac {\pi }{2}\right )d\arcsin (a x)-\frac {\sqrt {1-a^2 x^2} \arcsin (a x)^3}{x}\)

\(\Big \downarrow \) 25

\(\displaystyle -3 a \int \arcsin (a x)^2 \tan \left (\arcsin (a x)+\frac {\pi }{2}\right )d\arcsin (a x)-\frac {\sqrt {1-a^2 x^2} \arcsin (a x)^3}{x}\)

\(\Big \downarrow \) 4200

\(\displaystyle -\frac {\sqrt {1-a^2 x^2} \arcsin (a x)^3}{x}+3 a \left (2 i \int -\frac {e^{2 i \arcsin (a x)} \arcsin (a x)^2}{1-e^{2 i \arcsin (a x)}}d\arcsin (a x)-\frac {1}{3} i \arcsin (a x)^3\right )\)

\(\Big \downarrow \) 25

\(\displaystyle -\frac {\sqrt {1-a^2 x^2} \arcsin (a x)^3}{x}+3 a \left (-2 i \int \frac {e^{2 i \arcsin (a x)} \arcsin (a x)^2}{1-e^{2 i \arcsin (a x)}}d\arcsin (a x)-\frac {1}{3} i \arcsin (a x)^3\right )\)

\(\Big \downarrow \) 2620

\(\displaystyle -\frac {\sqrt {1-a^2 x^2} \arcsin (a x)^3}{x}+3 a \left (-2 i \left (\frac {1}{2} i \arcsin (a x)^2 \log \left (1-e^{2 i \arcsin (a x)}\right )-i \int \arcsin (a x) \log \left (1-e^{2 i \arcsin (a x)}\right )d\arcsin (a x)\right )-\frac {1}{3} i \arcsin (a x)^3\right )\)

\(\Big \downarrow \) 3011

\(\displaystyle -\frac {\sqrt {1-a^2 x^2} \arcsin (a x)^3}{x}+3 a \left (-2 i \left (\frac {1}{2} i \arcsin (a x)^2 \log \left (1-e^{2 i \arcsin (a x)}\right )-i \left (\frac {1}{2} i \arcsin (a x) \operatorname {PolyLog}\left (2,e^{2 i \arcsin (a x)}\right )-\frac {1}{2} i \int \operatorname {PolyLog}\left (2,e^{2 i \arcsin (a x)}\right )d\arcsin (a x)\right )\right )-\frac {1}{3} i \arcsin (a x)^3\right )\)

\(\Big \downarrow \) 2720

\(\displaystyle -\frac {\sqrt {1-a^2 x^2} \arcsin (a x)^3}{x}+3 a \left (-2 i \left (\frac {1}{2} i \arcsin (a x)^2 \log \left (1-e^{2 i \arcsin (a x)}\right )-i \left (\frac {1}{2} i \arcsin (a x) \operatorname {PolyLog}\left (2,e^{2 i \arcsin (a x)}\right )-\frac {1}{4} \int e^{-2 i \arcsin (a x)} \operatorname {PolyLog}\left (2,e^{2 i \arcsin (a x)}\right )de^{2 i \arcsin (a x)}\right )\right )-\frac {1}{3} i \arcsin (a x)^3\right )\)

\(\Big \downarrow \) 7143

\(\displaystyle -\frac {\sqrt {1-a^2 x^2} \arcsin (a x)^3}{x}+3 a \left (-2 i \left (\frac {1}{2} i \arcsin (a x)^2 \log \left (1-e^{2 i \arcsin (a x)}\right )-i \left (\frac {1}{2} i \arcsin (a x) \operatorname {PolyLog}\left (2,e^{2 i \arcsin (a x)}\right )-\frac {1}{4} \operatorname {PolyLog}\left (3,e^{2 i \arcsin (a x)}\right )\right )\right )-\frac {1}{3} i \arcsin (a x)^3\right )\)

Input:

Int[ArcSin[a*x]^3/(x^2*Sqrt[1 - a^2*x^2]),x]
 

Output:

-((Sqrt[1 - a^2*x^2]*ArcSin[a*x]^3)/x) + 3*a*((-1/3*I)*ArcSin[a*x]^3 - (2* 
I)*((I/2)*ArcSin[a*x]^2*Log[1 - E^((2*I)*ArcSin[a*x])] - I*((I/2)*ArcSin[a 
*x]*PolyLog[2, E^((2*I)*ArcSin[a*x])] - PolyLog[3, E^((2*I)*ArcSin[a*x])]/ 
4)))
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 2620

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]
 

rule 2720

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]]]
 

rule 3011

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]
 

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 4200

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]
 

rule 5136

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]
 

rule 5186

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]
 

rule 7143

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]
 

Maple [A] (verified)

Time = 0.50 (sec) , antiderivative size = 205, normalized size of antiderivative = 2.07

method	result	size
default	\(\frac {\left (i a x -\sqrt {-a^{2} x^{2}+1}\right ) \arcsin \left (a x \right )^{3}}{x}-a \left (2 i \arcsin \left (a x \right )^{3}-3 \arcsin \left (a x \right )^{2} \ln \left (1+i a x +\sqrt {-a^{2} x^{2}+1}\right )-3 \arcsin \left (a x \right )^{2} \ln \left (1-i a x -\sqrt {-a^{2} x^{2}+1}\right )+6 i \arcsin \left (a x \right ) \operatorname {polylog}\left (2, -i a x -\sqrt {-a^{2} x^{2}+1}\right )+6 i \arcsin \left (a x \right ) \operatorname {polylog}\left (2, i a x +\sqrt {-a^{2} x^{2}+1}\right )-6 \operatorname {polylog}\left (3, -i a x -\sqrt {-a^{2} x^{2}+1}\right )-6 \operatorname {polylog}\left (3, i a x +\sqrt {-a^{2} x^{2}+1}\right )\right )\)	\(205\)

Input:

int(arcsin(a*x)^3/x^2/(-a^2*x^2+1)^(1/2),x,method=_RETURNVERBOSE)
 

Output:

(I*a*x-(-a^2*x^2+1)^(1/2))*arcsin(a*x)^3/x-a*(2*I*arcsin(a*x)^3-3*arcsin(a 
*x)^2*ln(1+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*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))-6*polylog(3,I*a*x+(-a^2*x^2+1)^(1/2)))
                                                                                    
                                                                                    
 

Fricas [F]

\[ \int \frac {\arcsin (a x)^3}{x^2 \sqrt {1-a^2 x^2}} \, dx=\int { \frac {\arcsin \left (a x\right )^{3}}{\sqrt {-a^{2} x^{2} + 1} x^{2}} \,d x } \] Input:

integrate(arcsin(a*x)^3/x^2/(-a^2*x^2+1)^(1/2),x, algorithm="fricas")
 

Output:

integral(-sqrt(-a^2*x^2 + 1)*arcsin(a*x)^3/(a^2*x^4 - x^2), x)
 

Sympy [F]

\[ \int \frac {\arcsin (a x)^3}{x^2 \sqrt {1-a^2 x^2}} \, dx=\int \frac {\operatorname {asin}^{3}{\left (a x \right )}}{x^{2} \sqrt {- \left (a x - 1\right ) \left (a x + 1\right )}}\, dx \] Input:

integrate(asin(a*x)**3/x**2/(-a**2*x**2+1)**(1/2),x)
 

Output:

Integral(asin(a*x)**3/(x**2*sqrt(-(a*x - 1)*(a*x + 1))), x)
 

Maxima [F]

\[ \int \frac {\arcsin (a x)^3}{x^2 \sqrt {1-a^2 x^2}} \, dx=\int { \frac {\arcsin \left (a x\right )^{3}}{\sqrt {-a^{2} x^{2} + 1} x^{2}} \,d x } \] Input:

integrate(arcsin(a*x)^3/x^2/(-a^2*x^2+1)^(1/2),x, algorithm="maxima")
 

Output:

(3*a^3*x*integrate(x*arctan2(a*x, sqrt(a*x + 1)*sqrt(-a*x + 1))^2, x) - sq 
rt(a*x + 1)*sqrt(-a*x + 1)*arctan2(a*x, sqrt(a*x + 1)*sqrt(-a*x + 1))^3)/x
 

Giac [F]

\[ \int \frac {\arcsin (a x)^3}{x^2 \sqrt {1-a^2 x^2}} \, dx=\int { \frac {\arcsin \left (a x\right )^{3}}{\sqrt {-a^{2} x^{2} + 1} x^{2}} \,d x } \] Input:

integrate(arcsin(a*x)^3/x^2/(-a^2*x^2+1)^(1/2),x, algorithm="giac")
 

Output:

integrate(arcsin(a*x)^3/(sqrt(-a^2*x^2 + 1)*x^2), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\arcsin (a x)^3}{x^2 \sqrt {1-a^2 x^2}} \, dx=\int \frac {{\mathrm {asin}\left (a\,x\right )}^3}{x^2\,\sqrt {1-a^2\,x^2}} \,d x \] Input:

int(asin(a*x)^3/(x^2*(1 - a^2*x^2)^(1/2)),x)
 

Output:

int(asin(a*x)^3/(x^2*(1 - a^2*x^2)^(1/2)), x)
 

Reduce [F]

\[ \int \frac {\arcsin (a x)^3}{x^2 \sqrt {1-a^2 x^2}} \, dx=\int \frac {\mathit {asin} \left (a x \right )^{3}}{\sqrt {-a^{2} x^{2}+1}\, x^{2}}d x \] Input:

int(asin(a*x)^3/x^2/(-a^2*x^2+1)^(1/2),x)
 

Output:

int(asin(a*x)**3/(sqrt( - a**2*x**2 + 1)*x**2),x)