3.1.23 \(\int x^3 \arcsin (a x)^3 \, dx\) [23]

3.1.23.1 Optimal result

Integrand size = 10, antiderivative size = 167 \[ \int x^3 \arcsin (a x)^3 \, dx=-\frac {45 x \sqrt {1-a^2 x^2}}{256 a^3}-\frac {3 x^3 \sqrt {1-a^2 x^2}}{128 a}+\frac {45 \arcsin (a x)}{256 a^4}-\frac {9 x^2 \arcsin (a x)}{32 a^2}-\frac {3}{32} x^4 \arcsin (a x)+\frac {9 x \sqrt {1-a^2 x^2} \arcsin (a x)^2}{32 a^3}+\frac {3 x^3 \sqrt {1-a^2 x^2} \arcsin (a x)^2}{16 a}-\frac {3 \arcsin (a x)^3}{32 a^4}+\frac {1}{4} x^4 \arcsin (a x)^3 \]

output

45/256*arcsin(a*x)/a^4-9/32*x^2*arcsin(a*x)/a^2-3/32*x^4*arcsin(a*x)-3/32* 
arcsin(a*x)^3/a^4+1/4*x^4*arcsin(a*x)^3-45/256*x*(-a^2*x^2+1)^(1/2)/a^3-3/ 
128*x^3*(-a^2*x^2+1)^(1/2)/a+9/32*x*arcsin(a*x)^2*(-a^2*x^2+1)^(1/2)/a^3+3 
/16*x^3*arcsin(a*x)^2*(-a^2*x^2+1)^(1/2)/a
 

3.1.23.2 Mathematica [A] (verified)

Time = 0.04 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.67 \[ \int x^3 \arcsin (a x)^3 \, dx=\frac {-3 a x \sqrt {1-a^2 x^2} \left (15+2 a^2 x^2\right )-3 \left (-15+24 a^2 x^2+8 a^4 x^4\right ) \arcsin (a x)+24 a x \sqrt {1-a^2 x^2} \left (3+2 a^2 x^2\right ) \arcsin (a x)^2+8 \left (-3+8 a^4 x^4\right ) \arcsin (a x)^3}{256 a^4} \]

input

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

output

(-3*a*x*Sqrt[1 - a^2*x^2]*(15 + 2*a^2*x^2) - 3*(-15 + 24*a^2*x^2 + 8*a^4*x 
^4)*ArcSin[a*x] + 24*a*x*Sqrt[1 - a^2*x^2]*(3 + 2*a^2*x^2)*ArcSin[a*x]^2 + 
 8*(-3 + 8*a^4*x^4)*ArcSin[a*x]^3)/(256*a^4)
 

3.1.23.3 Rubi [A] (verified)

Time = 0.95 (sec) , antiderivative size = 244, normalized size of antiderivative = 1.46, number of steps used = 11, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.100, Rules used = {5138, 5210, 5138, 262, 262, 223, 5210, 5138, 262, 223, 5152}

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.

\(\displaystyle \int x^3 \arcsin (a x)^3 \, dx\)

\(\Big \downarrow \) 5138

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

\(\Big \downarrow \) 5210

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

\(\Big \downarrow \) 5138

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

\(\Big \downarrow \) 262

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

\(\Big \downarrow \) 262

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

\(\Big \downarrow \) 223

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

\(\Big \downarrow \) 5210

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

\(\Big \downarrow \) 5138

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

\(\Big \downarrow \) 262

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

\(\Big \downarrow \) 223

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

\(\Big \downarrow \) 5152

\(\displaystyle \frac {1}{4} x^4 \arcsin (a x)^3-\frac {3}{4} a \left (-\frac {x^3 \sqrt {1-a^2 x^2} \arcsin (a x)^2}{4 a^2}+\frac {3 \left (\frac {\arcsin (a x)^3}{6 a^3}-\frac {x \sqrt {1-a^2 x^2} \arcsin (a x)^2}{2 a^2}+\frac {\frac {1}{2} x^2 \arcsin (a x)-\frac {1}{2} a \left (\frac {\arcsin (a x)}{2 a^3}-\frac {x \sqrt {1-a^2 x^2}}{2 a^2}\right )}{a}\right )}{4 a^2}+\frac {\frac {1}{4} x^4 \arcsin (a x)-\frac {1}{4} a \left (\frac {3 \left (\frac {\arcsin (a x)}{2 a^3}-\frac {x \sqrt {1-a^2 x^2}}{2 a^2}\right )}{4 a^2}-\frac {x^3 \sqrt {1-a^2 x^2}}{4 a^2}\right )}{2 a}\right )\)

input

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

output

(x^4*ArcSin[a*x]^3)/4 - (3*a*(-1/4*(x^3*Sqrt[1 - a^2*x^2]*ArcSin[a*x]^2)/a 
^2 + ((x^4*ArcSin[a*x])/4 - (a*(-1/4*(x^3*Sqrt[1 - a^2*x^2])/a^2 + (3*(-1/ 
2*(x*Sqrt[1 - a^2*x^2])/a^2 + ArcSin[a*x]/(2*a^3)))/(4*a^2)))/4)/(2*a) + ( 
3*(-1/2*(x*Sqrt[1 - a^2*x^2]*ArcSin[a*x]^2)/a^2 + ArcSin[a*x]^3/(6*a^3) + 
((x^2*ArcSin[a*x])/2 - (a*(-1/2*(x*Sqrt[1 - a^2*x^2])/a^2 + ArcSin[a*x]/(2 
*a^3)))/2)/a))/(4*a^2)))/4
 

3.1.23.3.1 Defintions of rubi rules used

rule 223

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSin[Rt[-b, 2]*(x/Sqrt 
[a])]/Rt[-b, 2], x] /; FreeQ[{a, b}, x] && GtQ[a, 0] && NegQ[b]
 

rule 262

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[c*(c*x) 
^(m - 1)*((a + b*x^2)^(p + 1)/(b*(m + 2*p + 1))), x] - Simp[a*c^2*((m - 1)/ 
(b*(m + 2*p + 1)))   Int[(c*x)^(m - 2)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b 
, c, p}, x] && GtQ[m, 2 - 1] && NeQ[m + 2*p + 1, 0] && IntBinomialQ[a, b, c 
, 2, m, p, x]
 

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 5152

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)/Sqrt[(d_) + (e_.)*(x_)^2], x_S 
ymbol] :> Simp[(1/(b*c*(n + 1)))*Simp[Sqrt[1 - c^2*x^2]/Sqrt[d + e*x^2]]*(a 
 + b*ArcSin[c*x])^(n + 1), x] /; FreeQ[{a, b, c, d, e, n}, x] && EqQ[c^2*d 
+ e, 0] && NeQ[n, -1]
 

rule 5210

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*((d_) + (e_. 
)*(x_)^2)^(p_), x_Symbol] :> Simp[f*(f*x)^(m - 1)*(d + e*x^2)^(p + 1)*((a + 
 b*ArcSin[c*x])^n/(e*(m + 2*p + 1))), x] + (Simp[f^2*((m - 1)/(c^2*(m + 2*p 
 + 1)))   Int[(f*x)^(m - 2)*(d + e*x^2)^p*(a + b*ArcSin[c*x])^n, x], x] + S 
imp[b*f*(n/(c*(m + 2*p + 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] && IGtQ[m 
, 1] && NeQ[m + 2*p + 1, 0]
 

3.1.23.4 Maple [A] (verified)

Time = 0.09 (sec) , antiderivative size = 154, normalized size of antiderivative = 0.92

method	result	size
derivativedivides	\(\frac {\frac {a^{4} x^{4} \arcsin \left (a x \right )^{3}}{4}-\frac {3 \arcsin \left (a x \right )^{2} \left (-2 a^{3} x^{3} \sqrt {-a^{2} x^{2}+1}-3 a x \sqrt {-a^{2} x^{2}+1}+3 \arcsin \left (a x \right )\right )}{32}-\frac {3 a^{4} x^{4} \arcsin \left (a x \right )}{32}-\frac {3 a x \left (2 a^{2} x^{2}+3\right ) \sqrt {-a^{2} x^{2}+1}}{256}-\frac {27 \arcsin \left (a x \right )}{256}-\frac {9 \left (a^{2} x^{2}-1\right ) \arcsin \left (a x \right )}{32}-\frac {9 a x \sqrt {-a^{2} x^{2}+1}}{64}+\frac {3 \arcsin \left (a x \right )^{3}}{16}}{a^{4}}\)	\(154\)
default	\(\frac {\frac {a^{4} x^{4} \arcsin \left (a x \right )^{3}}{4}-\frac {3 \arcsin \left (a x \right )^{2} \left (-2 a^{3} x^{3} \sqrt {-a^{2} x^{2}+1}-3 a x \sqrt {-a^{2} x^{2}+1}+3 \arcsin \left (a x \right )\right )}{32}-\frac {3 a^{4} x^{4} \arcsin \left (a x \right )}{32}-\frac {3 a x \left (2 a^{2} x^{2}+3\right ) \sqrt {-a^{2} x^{2}+1}}{256}-\frac {27 \arcsin \left (a x \right )}{256}-\frac {9 \left (a^{2} x^{2}-1\right ) \arcsin \left (a x \right )}{32}-\frac {9 a x \sqrt {-a^{2} x^{2}+1}}{64}+\frac {3 \arcsin \left (a x \right )^{3}}{16}}{a^{4}}\)	\(154\)

input

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

output

1/a^4*(1/4*a^4*x^4*arcsin(a*x)^3-3/32*arcsin(a*x)^2*(-2*a^3*x^3*(-a^2*x^2+ 
1)^(1/2)-3*a*x*(-a^2*x^2+1)^(1/2)+3*arcsin(a*x))-3/32*a^4*x^4*arcsin(a*x)- 
3/256*a*x*(2*a^2*x^2+3)*(-a^2*x^2+1)^(1/2)-27/256*arcsin(a*x)-9/32*(a^2*x^ 
2-1)*arcsin(a*x)-9/64*a*x*(-a^2*x^2+1)^(1/2)+3/16*arcsin(a*x)^3)
 

3.1.23.5 Fricas [A] (verification not implemented)

Time = 0.27 (sec) , antiderivative size = 96, normalized size of antiderivative = 0.57 \[ \int x^3 \arcsin (a x)^3 \, dx=\frac {8 \, {\left (8 \, a^{4} x^{4} - 3\right )} \arcsin \left (a x\right )^{3} - 3 \, {\left (8 \, a^{4} x^{4} + 24 \, a^{2} x^{2} - 15\right )} \arcsin \left (a x\right ) - 3 \, {\left (2 \, a^{3} x^{3} - 8 \, {\left (2 \, a^{3} x^{3} + 3 \, a x\right )} \arcsin \left (a x\right )^{2} + 15 \, a x\right )} \sqrt {-a^{2} x^{2} + 1}}{256 \, a^{4}} \]

input

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

output

1/256*(8*(8*a^4*x^4 - 3)*arcsin(a*x)^3 - 3*(8*a^4*x^4 + 24*a^2*x^2 - 15)*a 
rcsin(a*x) - 3*(2*a^3*x^3 - 8*(2*a^3*x^3 + 3*a*x)*arcsin(a*x)^2 + 15*a*x)* 
sqrt(-a^2*x^2 + 1))/a^4
 

3.1.23.6 Sympy [A] (verification not implemented)

Time = 0.42 (sec) , antiderivative size = 160, normalized size of antiderivative = 0.96 \[ \int x^3 \arcsin (a x)^3 \, dx=\begin {cases} \frac {x^{4} \operatorname {asin}^{3}{\left (a x \right )}}{4} - \frac {3 x^{4} \operatorname {asin}{\left (a x \right )}}{32} + \frac {3 x^{3} \sqrt {- a^{2} x^{2} + 1} \operatorname {asin}^{2}{\left (a x \right )}}{16 a} - \frac {3 x^{3} \sqrt {- a^{2} x^{2} + 1}}{128 a} - \frac {9 x^{2} \operatorname {asin}{\left (a x \right )}}{32 a^{2}} + \frac {9 x \sqrt {- a^{2} x^{2} + 1} \operatorname {asin}^{2}{\left (a x \right )}}{32 a^{3}} - \frac {45 x \sqrt {- a^{2} x^{2} + 1}}{256 a^{3}} - \frac {3 \operatorname {asin}^{3}{\left (a x \right )}}{32 a^{4}} + \frac {45 \operatorname {asin}{\left (a x \right )}}{256 a^{4}} & \text {for}\: a \neq 0 \\0 & \text {otherwise} \end {cases} \]

input

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

output

Piecewise((x**4*asin(a*x)**3/4 - 3*x**4*asin(a*x)/32 + 3*x**3*sqrt(-a**2*x 
**2 + 1)*asin(a*x)**2/(16*a) - 3*x**3*sqrt(-a**2*x**2 + 1)/(128*a) - 9*x** 
2*asin(a*x)/(32*a**2) + 9*x*sqrt(-a**2*x**2 + 1)*asin(a*x)**2/(32*a**3) - 
45*x*sqrt(-a**2*x**2 + 1)/(256*a**3) - 3*asin(a*x)**3/(32*a**4) + 45*asin( 
a*x)/(256*a**4), Ne(a, 0)), (0, True))
 

3.1.23.7 Maxima [F]

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

input

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

output

1/4*x^4*arctan2(a*x, sqrt(a*x + 1)*sqrt(-a*x + 1))^3 + 3*a*integrate(1/4*s 
qrt(a*x + 1)*sqrt(-a*x + 1)*x^4*arctan2(a*x, sqrt(a*x + 1)*sqrt(-a*x + 1)) 
^2/(a^2*x^2 - 1), x)
 

3.1.23.8 Giac [A] (verification not implemented)

Time = 0.28 (sec) , antiderivative size = 185, normalized size of antiderivative = 1.11 \[ \int x^3 \arcsin (a x)^3 \, dx=-\frac {3 \, {\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}} x \arcsin \left (a x\right )^{2}}{16 \, a^{3}} + \frac {{\left (a^{2} x^{2} - 1\right )}^{2} \arcsin \left (a x\right )^{3}}{4 \, a^{4}} + \frac {15 \, \sqrt {-a^{2} x^{2} + 1} x \arcsin \left (a x\right )^{2}}{32 \, a^{3}} + \frac {{\left (a^{2} x^{2} - 1\right )} \arcsin \left (a x\right )^{3}}{2 \, a^{4}} + \frac {3 \, {\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}} x}{128 \, a^{3}} - \frac {3 \, {\left (a^{2} x^{2} - 1\right )}^{2} \arcsin \left (a x\right )}{32 \, a^{4}} + \frac {5 \, \arcsin \left (a x\right )^{3}}{32 \, a^{4}} - \frac {51 \, \sqrt {-a^{2} x^{2} + 1} x}{256 \, a^{3}} - \frac {15 \, {\left (a^{2} x^{2} - 1\right )} \arcsin \left (a x\right )}{32 \, a^{4}} - \frac {51 \, \arcsin \left (a x\right )}{256 \, a^{4}} \]

input

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

output

-3/16*(-a^2*x^2 + 1)^(3/2)*x*arcsin(a*x)^2/a^3 + 1/4*(a^2*x^2 - 1)^2*arcsi 
n(a*x)^3/a^4 + 15/32*sqrt(-a^2*x^2 + 1)*x*arcsin(a*x)^2/a^3 + 1/2*(a^2*x^2 
 - 1)*arcsin(a*x)^3/a^4 + 3/128*(-a^2*x^2 + 1)^(3/2)*x/a^3 - 3/32*(a^2*x^2 
 - 1)^2*arcsin(a*x)/a^4 + 5/32*arcsin(a*x)^3/a^4 - 51/256*sqrt(-a^2*x^2 + 
1)*x/a^3 - 15/32*(a^2*x^2 - 1)*arcsin(a*x)/a^4 - 51/256*arcsin(a*x)/a^4
 

3.1.23.9 Mupad [F(-1)]

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

input

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

output

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