3.1.30 \(\int \frac {\arcsin (a x)^3}{x^4} \, dx\) [30]

3.1.30.1 Optimal result

Integrand size = 10, antiderivative size = 179 \[ \int \frac {\arcsin (a x)^3}{x^4} \, dx=-\frac {a^2 \arcsin (a x)}{x}-\frac {a \sqrt {1-a^2 x^2} \arcsin (a x)^2}{2 x^2}-\frac {\arcsin (a x)^3}{3 x^3}-a^3 \arcsin (a x)^2 \text {arctanh}\left (e^{i \arcsin (a x)}\right )-a^3 \text {arctanh}\left (\sqrt {1-a^2 x^2}\right )+i a^3 \arcsin (a x) \operatorname {PolyLog}\left (2,-e^{i \arcsin (a x)}\right )-i a^3 \arcsin (a x) \operatorname {PolyLog}\left (2,e^{i \arcsin (a x)}\right )-a^3 \operatorname {PolyLog}\left (3,-e^{i \arcsin (a x)}\right )+a^3 \operatorname {PolyLog}\left (3,e^{i \arcsin (a x)}\right ) \]

output

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

3.1.30.2 Mathematica [A] (verified)

Time = 2.30 (sec) , antiderivative size = 284, normalized size of antiderivative = 1.59 \[ \int \frac {\arcsin (a x)^3}{x^4} \, dx=\frac {1}{48} a^3 \left (-24 \arcsin (a x) \cot \left (\frac {1}{2} \arcsin (a x)\right )-4 \arcsin (a x)^3 \cot \left (\frac {1}{2} \arcsin (a x)\right )-6 \arcsin (a x)^2 \csc ^2\left (\frac {1}{2} \arcsin (a x)\right )-a x \arcsin (a x)^3 \csc ^4\left (\frac {1}{2} \arcsin (a x)\right )+24 \arcsin (a x)^2 \log \left (1-e^{i \arcsin (a x)}\right )-24 \arcsin (a x)^2 \log \left (1+e^{i \arcsin (a x)}\right )+48 \log \left (\tan \left (\frac {1}{2} \arcsin (a x)\right )\right )+48 i \arcsin (a x) \operatorname {PolyLog}\left (2,-e^{i \arcsin (a x)}\right )-48 i \arcsin (a x) \operatorname {PolyLog}\left (2,e^{i \arcsin (a x)}\right )-48 \operatorname {PolyLog}\left (3,-e^{i \arcsin (a x)}\right )+48 \operatorname {PolyLog}\left (3,e^{i \arcsin (a x)}\right )+6 \arcsin (a x)^2 \sec ^2\left (\frac {1}{2} \arcsin (a x)\right )-\frac {16 \arcsin (a x)^3 \sin ^4\left (\frac {1}{2} \arcsin (a x)\right )}{a^3 x^3}-24 \arcsin (a x) \tan \left (\frac {1}{2} \arcsin (a x)\right )-4 \arcsin (a x)^3 \tan \left (\frac {1}{2} \arcsin (a x)\right )\right ) \]

input

Integrate[ArcSin[a*x]^3/x^4,x]
 

output

(a^3*(-24*ArcSin[a*x]*Cot[ArcSin[a*x]/2] - 4*ArcSin[a*x]^3*Cot[ArcSin[a*x] 
/2] - 6*ArcSin[a*x]^2*Csc[ArcSin[a*x]/2]^2 - a*x*ArcSin[a*x]^3*Csc[ArcSin[ 
a*x]/2]^4 + 24*ArcSin[a*x]^2*Log[1 - E^(I*ArcSin[a*x])] - 24*ArcSin[a*x]^2 
*Log[1 + E^(I*ArcSin[a*x])] + 48*Log[Tan[ArcSin[a*x]/2]] + (48*I)*ArcSin[a 
*x]*PolyLog[2, -E^(I*ArcSin[a*x])] - (48*I)*ArcSin[a*x]*PolyLog[2, E^(I*Ar 
cSin[a*x])] - 48*PolyLog[3, -E^(I*ArcSin[a*x])] + 48*PolyLog[3, E^(I*ArcSi 
n[a*x])] + 6*ArcSin[a*x]^2*Sec[ArcSin[a*x]/2]^2 - (16*ArcSin[a*x]^3*Sin[Ar 
cSin[a*x]/2]^4)/(a^3*x^3) - 24*ArcSin[a*x]*Tan[ArcSin[a*x]/2] - 4*ArcSin[a 
*x]^3*Tan[ArcSin[a*x]/2]))/48
 

3.1.30.3 Rubi [A] (verified)

Time = 0.96 (sec) , antiderivative size = 179, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.200, Rules used = {5138, 5204, 5138, 243, 73, 221, 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.

\(\displaystyle \int \frac {\arcsin (a x)^3}{x^4} \, dx\)

\(\Big \downarrow \) 5138

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

\(\Big \downarrow \) 5204

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

\(\Big \downarrow \) 5138

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

\(\Big \downarrow \) 243

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

\(\Big \downarrow \) 73

\(\displaystyle a \left (\frac {1}{2} a^2 \int \frac {\arcsin (a x)^2}{x \sqrt {1-a^2 x^2}}dx+a \left (-\frac {\int \frac {1}{\frac {1}{a^2}-\frac {x^4}{a^2}}d\sqrt {1-a^2 x^2}}{a}-\frac {\arcsin (a x)}{x}\right )-\frac {\sqrt {1-a^2 x^2} \arcsin (a x)^2}{2 x^2}\right )-\frac {\arcsin (a x)^3}{3 x^3}\)

\(\Big \downarrow \) 221

\(\displaystyle a \left (\frac {1}{2} a^2 \int \frac {\arcsin (a x)^2}{x \sqrt {1-a^2 x^2}}dx+a \left (-a \text {arctanh}\left (\sqrt {1-a^2 x^2}\right )-\frac {\arcsin (a x)}{x}\right )-\frac {\sqrt {1-a^2 x^2} \arcsin (a x)^2}{2 x^2}\right )-\frac {\arcsin (a x)^3}{3 x^3}\)

\(\Big \downarrow \) 5218

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4671

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

\(\Big \downarrow \) 3011

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

\(\Big \downarrow \) 2720

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

\(\Big \downarrow \) 7143

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

input

Int[ArcSin[a*x]^3/x^4,x]
 

output

-1/3*ArcSin[a*x]^3/x^3 + a*(-1/2*(Sqrt[1 - a^2*x^2]*ArcSin[a*x]^2)/x^2 + a 
*(-(ArcSin[a*x]/x) - a*ArcTanh[Sqrt[1 - a^2*x^2]]) + (a^2*(-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*ArcSin[a*x])])))/2)
 

3.1.30.3.1 Defintions of rubi rules used

rule 73

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ 
{p = Denominator[m]}, Simp[p/b   Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + 
 d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt 
Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL 
inearQ[a, b, c, d, m, n, x]
 

rule 221

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x 
/Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
 

rule 243

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[1/2   Subst[In 
t[x^((m - 1)/2)*(a + b*x)^p, x], x, x^2], x] /; FreeQ[{a, b, m, p}, x] && I 
ntegerQ[(m - 1)/2]
 

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 4671

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]
 

rule 5138

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]
 

rule 5204

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[c^2*((m + 2*p + 3)/(f^2*(m + 1)) 
)   Int[(f*x)^(m + 2)*(d + e*x^2)^p*(a + b*ArcSin[c*x])^n, x], 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*ArcSin[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, 
 c, d, e, f, p}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && ILtQ[m, -1]
 

rule 5218

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]
 

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]
 

3.1.30.4 Maple [A] (verified)

Time = 0.11 (sec) , antiderivative size = 234, normalized size of antiderivative = 1.31

method	result	size
derivativedivides	\(a^{3} \left (-\frac {\arcsin \left (a x \right ) \left (3 \arcsin \left (a x \right ) \sqrt {-a^{2} x^{2}+1}\, a x +2 \arcsin \left (a x \right )^{2}+6 a^{2} x^{2}\right )}{6 a^{3} x^{3}}+\frac {\arcsin \left (a x \right )^{2} \ln \left (1-i a x -\sqrt {-a^{2} x^{2}+1}\right )}{2}-i \arcsin \left (a x \right ) \operatorname {polylog}\left (2, i a x +\sqrt {-a^{2} x^{2}+1}\right )+\operatorname {polylog}\left (3, i a x +\sqrt {-a^{2} x^{2}+1}\right )-\frac {\arcsin \left (a x \right )^{2} \ln \left (1+i a x +\sqrt {-a^{2} x^{2}+1}\right )}{2}+i \arcsin \left (a x \right ) \operatorname {polylog}\left (2, -i a x -\sqrt {-a^{2} x^{2}+1}\right )-\operatorname {polylog}\left (3, -i a x -\sqrt {-a^{2} x^{2}+1}\right )-2 \,\operatorname {arctanh}\left (i a x +\sqrt {-a^{2} x^{2}+1}\right )\right )\)	\(234\)
default	\(a^{3} \left (-\frac {\arcsin \left (a x \right ) \left (3 \arcsin \left (a x \right ) \sqrt {-a^{2} x^{2}+1}\, a x +2 \arcsin \left (a x \right )^{2}+6 a^{2} x^{2}\right )}{6 a^{3} x^{3}}+\frac {\arcsin \left (a x \right )^{2} \ln \left (1-i a x -\sqrt {-a^{2} x^{2}+1}\right )}{2}-i \arcsin \left (a x \right ) \operatorname {polylog}\left (2, i a x +\sqrt {-a^{2} x^{2}+1}\right )+\operatorname {polylog}\left (3, i a x +\sqrt {-a^{2} x^{2}+1}\right )-\frac {\arcsin \left (a x \right )^{2} \ln \left (1+i a x +\sqrt {-a^{2} x^{2}+1}\right )}{2}+i \arcsin \left (a x \right ) \operatorname {polylog}\left (2, -i a x -\sqrt {-a^{2} x^{2}+1}\right )-\operatorname {polylog}\left (3, -i a x -\sqrt {-a^{2} x^{2}+1}\right )-2 \,\operatorname {arctanh}\left (i a x +\sqrt {-a^{2} x^{2}+1}\right )\right )\)	\(234\)

input

int(arcsin(a*x)^3/x^4,x,method=_RETURNVERBOSE)
 

output

a^3*(-1/6/a^3/x^3*arcsin(a*x)*(3*arcsin(a*x)*(-a^2*x^2+1)^(1/2)*a*x+2*arcs 
in(a*x)^2+6*a^2*x^2)+1/2*arcsin(a*x)^2*ln(1-I*a*x-(-a^2*x^2+1)^(1/2))-I*ar 
csin(a*x)*polylog(2,I*a*x+(-a^2*x^2+1)^(1/2))+polylog(3,I*a*x+(-a^2*x^2+1) 
^(1/2))-1/2*arcsin(a*x)^2*ln(1+I*a*x+(-a^2*x^2+1)^(1/2))+I*arcsin(a*x)*pol 
ylog(2,-I*a*x-(-a^2*x^2+1)^(1/2))-polylog(3,-I*a*x-(-a^2*x^2+1)^(1/2))-2*a 
rctanh(I*a*x+(-a^2*x^2+1)^(1/2)))
 

3.1.30.5 Fricas [F]

\[ \int \frac {\arcsin (a x)^3}{x^4} \, dx=\int { \frac {\arcsin \left (a x\right )^{3}}{x^{4}} \,d x } \]

input

integrate(arcsin(a*x)^3/x^4,x, algorithm="fricas")
 

output

integral(arcsin(a*x)^3/x^4, x)
 

3.1.30.6 Sympy [F]

\[ \int \frac {\arcsin (a x)^3}{x^4} \, dx=\int \frac {\operatorname {asin}^{3}{\left (a x \right )}}{x^{4}}\, dx \]

input

integrate(asin(a*x)**3/x**4,x)
 

output

Integral(asin(a*x)**3/x**4, x)
 

3.1.30.7 Maxima [F]

\[ \int \frac {\arcsin (a x)^3}{x^4} \, dx=\int { \frac {\arcsin \left (a x\right )^{3}}{x^{4}} \,d x } \]

input

integrate(arcsin(a*x)^3/x^4,x, algorithm="maxima")
 

output

-1/3*(3*a*x^3*integrate(sqrt(a*x + 1)*sqrt(-a*x + 1)*arctan2(a*x, sqrt(a*x 
 + 1)*sqrt(-a*x + 1))^2/(a^2*x^5 - x^3), x) + arctan2(a*x, sqrt(a*x + 1)*s 
qrt(-a*x + 1))^3)/x^3
 

3.1.30.8 Giac [F]

\[ \int \frac {\arcsin (a x)^3}{x^4} \, dx=\int { \frac {\arcsin \left (a x\right )^{3}}{x^{4}} \,d x } \]

input

integrate(arcsin(a*x)^3/x^4,x, algorithm="giac")
 

output

integrate(arcsin(a*x)^3/x^4, x)
 

3.1.30.9 Mupad [F(-1)]

Timed out. \[ \int \frac {\arcsin (a x)^3}{x^4} \, dx=\int \frac {{\mathrm {asin}\left (a\,x\right )}^3}{x^4} \,d x \]

input

int(asin(a*x)^3/x^4,x)
 

output

int(asin(a*x)^3/x^4, x)