\(\int x^3 \arcsin (a x)^4 \, dx\) [31]

Optimal result

Integrand size = 10, antiderivative size = 198 \[ \int x^3 \arcsin (a x)^4 \, dx=\frac {45 x^2}{128 a^2}+\frac {3 x^4}{128}-\frac {45 x \sqrt {1-a^2 x^2} \arcsin (a x)}{64 a^3}-\frac {3 x^3 \sqrt {1-a^2 x^2} \arcsin (a x)}{32 a}+\frac {45 \arcsin (a x)^2}{128 a^4}-\frac {9 x^2 \arcsin (a x)^2}{16 a^2}-\frac {3}{16} x^4 \arcsin (a x)^2+\frac {3 x \sqrt {1-a^2 x^2} \arcsin (a x)^3}{8 a^3}+\frac {x^3 \sqrt {1-a^2 x^2} \arcsin (a x)^3}{4 a}-\frac {3 \arcsin (a x)^4}{32 a^4}+\frac {1}{4} x^4 \arcsin (a x)^4 \] Output:

45/128*x^2/a^2+3/128*x^4-45/64*x*(-a^2*x^2+1)^(1/2)*arcsin(a*x)/a^3-3/32*x 
^3*(-a^2*x^2+1)^(1/2)*arcsin(a*x)/a+45/128*arcsin(a*x)^2/a^4-9/16*x^2*arcs 
in(a*x)^2/a^2-3/16*x^4*arcsin(a*x)^2+3/8*x*(-a^2*x^2+1)^(1/2)*arcsin(a*x)^ 
3/a^3+1/4*x^3*(-a^2*x^2+1)^(1/2)*arcsin(a*x)^3/a-3/32*arcsin(a*x)^4/a^4+1/ 
4*x^4*arcsin(a*x)^4
 

Mathematica [A] (verified)

Time = 0.04 (sec) , antiderivative size = 135, normalized size of antiderivative = 0.68 \[ \int x^3 \arcsin (a x)^4 \, dx=\frac {3 a^2 x^2 \left (15+a^2 x^2\right )-6 a x \sqrt {1-a^2 x^2} \left (15+2 a^2 x^2\right ) \arcsin (a x)-3 \left (-15+24 a^2 x^2+8 a^4 x^4\right ) \arcsin (a x)^2+16 a x \sqrt {1-a^2 x^2} \left (3+2 a^2 x^2\right ) \arcsin (a x)^3+4 \left (-3+8 a^4 x^4\right ) \arcsin (a x)^4}{128 a^4} \] Input:

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

Output:

(3*a^2*x^2*(15 + a^2*x^2) - 6*a*x*Sqrt[1 - a^2*x^2]*(15 + 2*a^2*x^2)*ArcSi 
n[a*x] - 3*(-15 + 24*a^2*x^2 + 8*a^4*x^4)*ArcSin[a*x]^2 + 16*a*x*Sqrt[1 - 
a^2*x^2]*(3 + 2*a^2*x^2)*ArcSin[a*x]^3 + 4*(-3 + 8*a^4*x^4)*ArcSin[a*x]^4) 
/(128*a^4)
 

Rubi [A] (verified)

Time = 1.68 (sec) , antiderivative size = 293, normalized size of antiderivative = 1.48, number of steps used = 10, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.000, Rules used = {5138, 5210, 5138, 5210, 15, 5138, 5152, 5210, 15, 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.

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

\(\Big \downarrow \) 5138

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

\(\Big \downarrow \) 5210

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

\(\Big \downarrow \) 5138

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

\(\Big \downarrow \) 5210

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

\(\Big \downarrow \) 15

\(\displaystyle \frac {1}{4} x^4 \arcsin (a x)^4-a \left (\frac {3 \left (\frac {\int \frac {\arcsin (a x)^3}{\sqrt {1-a^2 x^2}}dx}{2 a^2}+\frac {3 \int x \arcsin (a x)^2dx}{2 a}-\frac {x \sqrt {1-a^2 x^2} \arcsin (a x)^3}{2 a^2}\right )}{4 a^2}+\frac {3 \left (\frac {1}{4} x^4 \arcsin (a x)^2-\frac {1}{2} a \left (\frac {3 \int \frac {x^2 \arcsin (a x)}{\sqrt {1-a^2 x^2}}dx}{4 a^2}-\frac {x^3 \sqrt {1-a^2 x^2} \arcsin (a x)}{4 a^2}+\frac {x^4}{16 a}\right )\right )}{4 a}-\frac {x^3 \sqrt {1-a^2 x^2} \arcsin (a x)^3}{4 a^2}\right )\)

\(\Big \downarrow \) 5138

\(\displaystyle \frac {1}{4} x^4 \arcsin (a x)^4-a \left (\frac {3 \left (\frac {3 \left (\frac {1}{2} x^2 \arcsin (a x)^2-a \int \frac {x^2 \arcsin (a x)}{\sqrt {1-a^2 x^2}}dx\right )}{2 a}+\frac {\int \frac {\arcsin (a x)^3}{\sqrt {1-a^2 x^2}}dx}{2 a^2}-\frac {x \sqrt {1-a^2 x^2} \arcsin (a x)^3}{2 a^2}\right )}{4 a^2}+\frac {3 \left (\frac {1}{4} x^4 \arcsin (a x)^2-\frac {1}{2} a \left (\frac {3 \int \frac {x^2 \arcsin (a x)}{\sqrt {1-a^2 x^2}}dx}{4 a^2}-\frac {x^3 \sqrt {1-a^2 x^2} \arcsin (a x)}{4 a^2}+\frac {x^4}{16 a}\right )\right )}{4 a}-\frac {x^3 \sqrt {1-a^2 x^2} \arcsin (a x)^3}{4 a^2}\right )\)

\(\Big \downarrow \) 5152

\(\displaystyle \frac {1}{4} x^4 \arcsin (a x)^4-a \left (\frac {3 \left (\frac {1}{4} x^4 \arcsin (a x)^2-\frac {1}{2} a \left (\frac {3 \int \frac {x^2 \arcsin (a x)}{\sqrt {1-a^2 x^2}}dx}{4 a^2}-\frac {x^3 \sqrt {1-a^2 x^2} \arcsin (a x)}{4 a^2}+\frac {x^4}{16 a}\right )\right )}{4 a}+\frac {3 \left (\frac {3 \left (\frac {1}{2} x^2 \arcsin (a x)^2-a \int \frac {x^2 \arcsin (a x)}{\sqrt {1-a^2 x^2}}dx\right )}{2 a}+\frac {\arcsin (a x)^4}{8 a^3}-\frac {x \sqrt {1-a^2 x^2} \arcsin (a x)^3}{2 a^2}\right )}{4 a^2}-\frac {x^3 \sqrt {1-a^2 x^2} \arcsin (a x)^3}{4 a^2}\right )\)

\(\Big \downarrow \) 5210

\(\displaystyle \frac {1}{4} x^4 \arcsin (a x)^4-a \left (\frac {3 \left (\frac {1}{4} x^4 \arcsin (a x)^2-\frac {1}{2} a \left (\frac {3 \left (\frac {\int \frac {\arcsin (a x)}{\sqrt {1-a^2 x^2}}dx}{2 a^2}+\frac {\int xdx}{2 a}-\frac {x \sqrt {1-a^2 x^2} \arcsin (a x)}{2 a^2}\right )}{4 a^2}-\frac {x^3 \sqrt {1-a^2 x^2} \arcsin (a x)}{4 a^2}+\frac {x^4}{16 a}\right )\right )}{4 a}+\frac {3 \left (\frac {3 \left (\frac {1}{2} x^2 \arcsin (a x)^2-a \left (\frac {\int \frac {\arcsin (a x)}{\sqrt {1-a^2 x^2}}dx}{2 a^2}+\frac {\int xdx}{2 a}-\frac {x \sqrt {1-a^2 x^2} \arcsin (a x)}{2 a^2}\right )\right )}{2 a}+\frac {\arcsin (a x)^4}{8 a^3}-\frac {x \sqrt {1-a^2 x^2} \arcsin (a x)^3}{2 a^2}\right )}{4 a^2}-\frac {x^3 \sqrt {1-a^2 x^2} \arcsin (a x)^3}{4 a^2}\right )\)

\(\Big \downarrow \) 15

\(\displaystyle \frac {1}{4} x^4 \arcsin (a x)^4-a \left (\frac {3 \left (\frac {1}{4} x^4 \arcsin (a x)^2-\frac {1}{2} a \left (\frac {3 \left (\frac {\int \frac {\arcsin (a x)}{\sqrt {1-a^2 x^2}}dx}{2 a^2}-\frac {x \sqrt {1-a^2 x^2} \arcsin (a x)}{2 a^2}+\frac {x^2}{4 a}\right )}{4 a^2}-\frac {x^3 \sqrt {1-a^2 x^2} \arcsin (a x)}{4 a^2}+\frac {x^4}{16 a}\right )\right )}{4 a}+\frac {3 \left (\frac {3 \left (\frac {1}{2} x^2 \arcsin (a x)^2-a \left (\frac {\int \frac {\arcsin (a x)}{\sqrt {1-a^2 x^2}}dx}{2 a^2}-\frac {x \sqrt {1-a^2 x^2} \arcsin (a x)}{2 a^2}+\frac {x^2}{4 a}\right )\right )}{2 a}+\frac {\arcsin (a x)^4}{8 a^3}-\frac {x \sqrt {1-a^2 x^2} \arcsin (a x)^3}{2 a^2}\right )}{4 a^2}-\frac {x^3 \sqrt {1-a^2 x^2} \arcsin (a x)^3}{4 a^2}\right )\)

\(\Big \downarrow \) 5152

\(\displaystyle \frac {1}{4} x^4 \arcsin (a x)^4-a \left (-\frac {x^3 \sqrt {1-a^2 x^2} \arcsin (a x)^3}{4 a^2}+\frac {3 \left (\frac {\arcsin (a x)^4}{8 a^3}-\frac {x \sqrt {1-a^2 x^2} \arcsin (a x)^3}{2 a^2}+\frac {3 \left (\frac {1}{2} x^2 \arcsin (a x)^2-a \left (\frac {\arcsin (a x)^2}{4 a^3}-\frac {x \sqrt {1-a^2 x^2} \arcsin (a x)}{2 a^2}+\frac {x^2}{4 a}\right )\right )}{2 a}\right )}{4 a^2}+\frac {3 \left (\frac {1}{4} x^4 \arcsin (a x)^2-\frac {1}{2} a \left (-\frac {x^3 \sqrt {1-a^2 x^2} \arcsin (a x)}{4 a^2}+\frac {3 \left (\frac {\arcsin (a x)^2}{4 a^3}-\frac {x \sqrt {1-a^2 x^2} \arcsin (a x)}{2 a^2}+\frac {x^2}{4 a}\right )}{4 a^2}+\frac {x^4}{16 a}\right )\right )}{4 a}\right )\)

Input:

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

Output:

(x^4*ArcSin[a*x]^4)/4 - a*(-1/4*(x^3*Sqrt[1 - a^2*x^2]*ArcSin[a*x]^3)/a^2 
+ (3*((x^4*ArcSin[a*x]^2)/4 - (a*(x^4/(16*a) - (x^3*Sqrt[1 - a^2*x^2]*ArcS 
in[a*x])/(4*a^2) + (3*(x^2/(4*a) - (x*Sqrt[1 - a^2*x^2]*ArcSin[a*x])/(2*a^ 
2) + ArcSin[a*x]^2/(4*a^3)))/(4*a^2)))/2))/(4*a) + (3*(-1/2*(x*Sqrt[1 - a^ 
2*x^2]*ArcSin[a*x]^3)/a^2 + ArcSin[a*x]^4/(8*a^3) + (3*((x^2*ArcSin[a*x]^2 
)/2 - a*(x^2/(4*a) - (x*Sqrt[1 - a^2*x^2]*ArcSin[a*x])/(2*a^2) + ArcSin[a* 
x]^2/(4*a^3))))/(2*a)))/(4*a^2))
 

Defintions of rubi rules used

rule 15

Int[(a_.)*(x_)^(m_.), x_Symbol] :> Simp[a*(x^(m + 1)/(m + 1)), x] /; FreeQ[ 
{a, m}, x] && NeQ[m, -1]
 

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]
 

Maple [A] (verified)

Time = 0.25 (sec) , antiderivative size = 215, normalized size of antiderivative = 1.09

method	result	size
derivativedivides	\(\frac {\frac {a^{4} x^{4} \arcsin \left (a x \right )^{4}}{4}-\frac {\arcsin \left (a x \right )^{3} \left (-2 \sqrt {-a^{2} x^{2}+1}\, a^{3} x^{3}-3 \sqrt {-a^{2} x^{2}+1}\, x a +3 \arcsin \left (a x \right )\right )}{8}-\frac {3 \arcsin \left (a x \right )^{2} a^{4} x^{4}}{16}+\frac {3 \arcsin \left (a x \right ) \left (-2 \sqrt {-a^{2} x^{2}+1}\, a^{3} x^{3}-3 \sqrt {-a^{2} x^{2}+1}\, x a +3 \arcsin \left (a x \right )\right )}{64}+\frac {27 \arcsin \left (a x \right )^{2}}{128}+\frac {3 \left (2 a^{2} x^{2}+3\right )^{2}}{512}-\frac {9 \arcsin \left (a x \right )^{2} \left (a^{2} x^{2}-1\right )}{16}-\frac {9 \arcsin \left (a x \right ) \left (\sqrt {-a^{2} x^{2}+1}\, x a +\arcsin \left (a x \right )\right )}{16}+\frac {9 a^{2} x^{2}}{32}+\frac {9 \arcsin \left (a x \right )^{4}}{32}}{a^{4}}\)	\(215\)
default	\(\frac {\frac {a^{4} x^{4} \arcsin \left (a x \right )^{4}}{4}-\frac {\arcsin \left (a x \right )^{3} \left (-2 \sqrt {-a^{2} x^{2}+1}\, a^{3} x^{3}-3 \sqrt {-a^{2} x^{2}+1}\, x a +3 \arcsin \left (a x \right )\right )}{8}-\frac {3 \arcsin \left (a x \right )^{2} a^{4} x^{4}}{16}+\frac {3 \arcsin \left (a x \right ) \left (-2 \sqrt {-a^{2} x^{2}+1}\, a^{3} x^{3}-3 \sqrt {-a^{2} x^{2}+1}\, x a +3 \arcsin \left (a x \right )\right )}{64}+\frac {27 \arcsin \left (a x \right )^{2}}{128}+\frac {3 \left (2 a^{2} x^{2}+3\right )^{2}}{512}-\frac {9 \arcsin \left (a x \right )^{2} \left (a^{2} x^{2}-1\right )}{16}-\frac {9 \arcsin \left (a x \right ) \left (\sqrt {-a^{2} x^{2}+1}\, x a +\arcsin \left (a x \right )\right )}{16}+\frac {9 a^{2} x^{2}}{32}+\frac {9 \arcsin \left (a x \right )^{4}}{32}}{a^{4}}\)	\(215\)
orering	\(\frac {\left (781 a^{6} x^{6}+1605 a^{4} x^{4}-9360 a^{2} x^{2}+7560\right ) \arcsin \left (a x \right )^{4}}{1024 a^{6} x^{2}}-\frac {\left (285 a^{6} x^{6}+1213 a^{4} x^{4}-6120 a^{2} x^{2}+4860\right ) \left (3 x^{2} \arcsin \left (a x \right )^{4}+\frac {4 x^{3} \arcsin \left (a x \right )^{3} a}{\sqrt {-a^{2} x^{2}+1}}\right )}{1024 a^{6} x^{4}}+\frac {\left (65 a^{6} x^{6}+433 a^{4} x^{4}-1830 a^{2} x^{2}+1380\right ) \left (6 x \arcsin \left (a x \right )^{4}+\frac {24 x^{2} \arcsin \left (a x \right )^{3} a}{\sqrt {-a^{2} x^{2}+1}}+\frac {12 x^{3} \arcsin \left (a x \right )^{2} a^{2}}{-a^{2} x^{2}+1}+\frac {4 x^{4} \arcsin \left (a x \right )^{3} a^{3}}{\left (-a^{2} x^{2}+1\right )^{\frac {3}{2}}}\right )}{1024 x^{3} a^{6}}-\frac {\left (a x -1\right ) \left (a x +1\right ) \left (5 a^{4} x^{4}+52 a^{2} x^{2}-105\right ) \left (6 \arcsin \left (a x \right )^{4}+\frac {72 x \arcsin \left (a x \right )^{3} a}{\sqrt {-a^{2} x^{2}+1}}+\frac {108 x^{2} \arcsin \left (a x \right )^{2} a^{2}}{-a^{2} x^{2}+1}+\frac {40 x^{3} \arcsin \left (a x \right )^{3} a^{3}}{\left (-a^{2} x^{2}+1\right )^{\frac {3}{2}}}+\frac {24 x^{3} \arcsin \left (a x \right ) a^{3}}{\left (-a^{2} x^{2}+1\right )^{\frac {3}{2}}}+\frac {36 x^{4} \arcsin \left (a x \right )^{2} a^{4}}{\left (-a^{2} x^{2}+1\right )^{2}}+\frac {12 x^{5} \arcsin \left (a x \right )^{3} a^{5}}{\left (-a^{2} x^{2}+1\right )^{\frac {5}{2}}}\right )}{512 x^{2} a^{6}}+\frac {\left (a^{2} x^{2}+15\right ) \left (a x +1\right )^{2} \left (a x -1\right )^{2} \left (\frac {96 \arcsin \left (a x \right )^{3} a}{\sqrt {-a^{2} x^{2}+1}}+\frac {432 x \arcsin \left (a x \right )^{2} a^{2}}{-a^{2} x^{2}+1}+\frac {192 x^{2} \arcsin \left (a x \right )^{3} a^{3}}{\left (-a^{2} x^{2}+1\right )^{\frac {3}{2}}}+\frac {288 x^{2} \arcsin \left (a x \right ) a^{3}}{\left (-a^{2} x^{2}+1\right )^{\frac {3}{2}}}+\frac {480 x^{3} \arcsin \left (a x \right )^{2} a^{4}}{\left (-a^{2} x^{2}+1\right )^{2}}+\frac {180 x^{4} \arcsin \left (a x \right )^{3} a^{5}}{\left (-a^{2} x^{2}+1\right )^{\frac {5}{2}}}+\frac {24 x^{3} a^{4}}{\left (-a^{2} x^{2}+1\right )^{2}}+\frac {144 x^{4} \arcsin \left (a x \right ) a^{5}}{\left (-a^{2} x^{2}+1\right )^{\frac {5}{2}}}+\frac {180 x^{5} \arcsin \left (a x \right )^{2} a^{6}}{\left (-a^{2} x^{2}+1\right )^{3}}+\frac {60 x^{6} \arcsin \left (a x \right )^{3} a^{7}}{\left (-a^{2} x^{2}+1\right )^{\frac {7}{2}}}\right )}{1024 x \,a^{6}}\)	\(702\)

Input:

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

Output:

1/a^4*(1/4*a^4*x^4*arcsin(a*x)^4-1/8*arcsin(a*x)^3*(-2*(-a^2*x^2+1)^(1/2)* 
a^3*x^3-3*(-a^2*x^2+1)^(1/2)*x*a+3*arcsin(a*x))-3/16*arcsin(a*x)^2*a^4*x^4 
+3/64*arcsin(a*x)*(-2*(-a^2*x^2+1)^(1/2)*a^3*x^3-3*(-a^2*x^2+1)^(1/2)*x*a+ 
3*arcsin(a*x))+27/128*arcsin(a*x)^2+3/512*(2*a^2*x^2+3)^2-9/16*arcsin(a*x) 
^2*(a^2*x^2-1)-9/16*arcsin(a*x)*((-a^2*x^2+1)^(1/2)*x*a+arcsin(a*x))+9/32* 
a^2*x^2+9/32*arcsin(a*x)^4)
 

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 121, normalized size of antiderivative = 0.61 \[ \int x^3 \arcsin (a x)^4 \, dx=\frac {3 \, a^{4} x^{4} + 4 \, {\left (8 \, a^{4} x^{4} - 3\right )} \arcsin \left (a x\right )^{4} + 45 \, a^{2} x^{2} - 3 \, {\left (8 \, a^{4} x^{4} + 24 \, a^{2} x^{2} - 15\right )} \arcsin \left (a x\right )^{2} + 2 \, \sqrt {-a^{2} x^{2} + 1} {\left (8 \, {\left (2 \, a^{3} x^{3} + 3 \, a x\right )} \arcsin \left (a x\right )^{3} - 3 \, {\left (2 \, a^{3} x^{3} + 15 \, a x\right )} \arcsin \left (a x\right )\right )}}{128 \, a^{4}} \] Input:

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

Output:

1/128*(3*a^4*x^4 + 4*(8*a^4*x^4 - 3)*arcsin(a*x)^4 + 45*a^2*x^2 - 3*(8*a^4 
*x^4 + 24*a^2*x^2 - 15)*arcsin(a*x)^2 + 2*sqrt(-a^2*x^2 + 1)*(8*(2*a^3*x^3 
 + 3*a*x)*arcsin(a*x)^3 - 3*(2*a^3*x^3 + 15*a*x)*arcsin(a*x)))/a^4
 

Sympy [A] (verification not implemented)

Time = 0.72 (sec) , antiderivative size = 190, normalized size of antiderivative = 0.96 \[ \int x^3 \arcsin (a x)^4 \, dx=\begin {cases} \frac {x^{4} \operatorname {asin}^{4}{\left (a x \right )}}{4} - \frac {3 x^{4} \operatorname {asin}^{2}{\left (a x \right )}}{16} + \frac {3 x^{4}}{128} + \frac {x^{3} \sqrt {- a^{2} x^{2} + 1} \operatorname {asin}^{3}{\left (a x \right )}}{4 a} - \frac {3 x^{3} \sqrt {- a^{2} x^{2} + 1} \operatorname {asin}{\left (a x \right )}}{32 a} - \frac {9 x^{2} \operatorname {asin}^{2}{\left (a x \right )}}{16 a^{2}} + \frac {45 x^{2}}{128 a^{2}} + \frac {3 x \sqrt {- a^{2} x^{2} + 1} \operatorname {asin}^{3}{\left (a x \right )}}{8 a^{3}} - \frac {45 x \sqrt {- a^{2} x^{2} + 1} \operatorname {asin}{\left (a x \right )}}{64 a^{3}} - \frac {3 \operatorname {asin}^{4}{\left (a x \right )}}{32 a^{4}} + \frac {45 \operatorname {asin}^{2}{\left (a x \right )}}{128 a^{4}} & \text {for}\: a \neq 0 \\0 & \text {otherwise} \end {cases} \] Input:

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

Output:

Piecewise((x**4*asin(a*x)**4/4 - 3*x**4*asin(a*x)**2/16 + 3*x**4/128 + x** 
3*sqrt(-a**2*x**2 + 1)*asin(a*x)**3/(4*a) - 3*x**3*sqrt(-a**2*x**2 + 1)*as 
in(a*x)/(32*a) - 9*x**2*asin(a*x)**2/(16*a**2) + 45*x**2/(128*a**2) + 3*x* 
sqrt(-a**2*x**2 + 1)*asin(a*x)**3/(8*a**3) - 45*x*sqrt(-a**2*x**2 + 1)*asi 
n(a*x)/(64*a**3) - 3*asin(a*x)**4/(32*a**4) + 45*asin(a*x)**2/(128*a**4), 
Ne(a, 0)), (0, True))
 

Maxima [F]

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

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

Output:

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

Giac [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 234, normalized size of antiderivative = 1.18 \[ \int x^3 \arcsin (a x)^4 \, dx=-\frac {{\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}} x \arcsin \left (a x\right )^{3}}{4 \, a^{3}} + \frac {{\left (a^{2} x^{2} - 1\right )}^{2} \arcsin \left (a x\right )^{4}}{4 \, a^{4}} + \frac {5 \, \sqrt {-a^{2} x^{2} + 1} x \arcsin \left (a x\right )^{3}}{8 \, a^{3}} + \frac {{\left (a^{2} x^{2} - 1\right )} \arcsin \left (a x\right )^{4}}{2 \, a^{4}} + \frac {3 \, {\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}} x \arcsin \left (a x\right )}{32 \, a^{3}} - \frac {3 \, {\left (a^{2} x^{2} - 1\right )}^{2} \arcsin \left (a x\right )^{2}}{16 \, a^{4}} + \frac {5 \, \arcsin \left (a x\right )^{4}}{32 \, a^{4}} - \frac {51 \, \sqrt {-a^{2} x^{2} + 1} x \arcsin \left (a x\right )}{64 \, a^{3}} - \frac {15 \, {\left (a^{2} x^{2} - 1\right )} \arcsin \left (a x\right )^{2}}{16 \, a^{4}} + \frac {3 \, {\left (a^{2} x^{2} - 1\right )}^{2}}{128 \, a^{4}} - \frac {51 \, \arcsin \left (a x\right )^{2}}{128 \, a^{4}} + \frac {51 \, {\left (a^{2} x^{2} - 1\right )}}{128 \, a^{4}} + \frac {195}{1024 \, a^{4}} \] Input:

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

Output:

-1/4*(-a^2*x^2 + 1)^(3/2)*x*arcsin(a*x)^3/a^3 + 1/4*(a^2*x^2 - 1)^2*arcsin 
(a*x)^4/a^4 + 5/8*sqrt(-a^2*x^2 + 1)*x*arcsin(a*x)^3/a^3 + 1/2*(a^2*x^2 - 
1)*arcsin(a*x)^4/a^4 + 3/32*(-a^2*x^2 + 1)^(3/2)*x*arcsin(a*x)/a^3 - 3/16* 
(a^2*x^2 - 1)^2*arcsin(a*x)^2/a^4 + 5/32*arcsin(a*x)^4/a^4 - 51/64*sqrt(-a 
^2*x^2 + 1)*x*arcsin(a*x)/a^3 - 15/16*(a^2*x^2 - 1)*arcsin(a*x)^2/a^4 + 3/ 
128*(a^2*x^2 - 1)^2/a^4 - 51/128*arcsin(a*x)^2/a^4 + 51/128*(a^2*x^2 - 1)/ 
a^4 + 195/1024/a^4
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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