\(\int \frac {x^4 \arccos (a x)^3}{\sqrt {1-a^2 x^2}} \, dx\) [305]

Optimal result

Integrand size = 24, antiderivative size = 191 \[ \int \frac {x^4 \arccos (a x)^3}{\sqrt {1-a^2 x^2}} \, dx=-\frac {45 x^2}{128 a^3}-\frac {3 x^4}{128 a}+\frac {45 x \sqrt {1-a^2 x^2} \arccos (a x)}{64 a^4}+\frac {3 x^3 \sqrt {1-a^2 x^2} \arccos (a x)}{32 a^2}-\frac {45 \arccos (a x)^2}{128 a^5}+\frac {9 x^2 \arccos (a x)^2}{16 a^3}+\frac {3 x^4 \arccos (a x)^2}{16 a}-\frac {3 x \sqrt {1-a^2 x^2} \arccos (a x)^3}{8 a^4}-\frac {x^3 \sqrt {1-a^2 x^2} \arccos (a x)^3}{4 a^2}+\frac {3 \arccos (a x)^4}{32 a^5} \] Output:

-45/128*x^2/a^3-3/128*x^4/a+45/64*x*(-a^2*x^2+1)^(1/2)*arccos(a*x)/a^4+3/3 
2*x^3*(-a^2*x^2+1)^(1/2)*arccos(a*x)/a^2-45/128*arccos(a*x)^2/a^5+9/16*x^2 
*arccos(a*x)^2/a^3+3/16*x^4*arccos(a*x)^2/a-3/8*x*(-a^2*x^2+1)^(1/2)*arcco 
s(a*x)^3/a^4-1/4*x^3*(-a^2*x^2+1)^(1/2)*arccos(a*x)^3/a^2+3/32*arccos(a*x) 
^4/a^5
 

Mathematica [A] (verified)

Time = 0.06 (sec) , antiderivative size = 125, normalized size of antiderivative = 0.65 \[ \int \frac {x^4 \arccos (a x)^3}{\sqrt {1-a^2 x^2}} \, dx=\frac {3 a^2 x^2 \left (15+a^2 x^2\right )+6 a x \sqrt {1-a^2 x^2} \left (15+2 a^2 x^2\right ) \arccos (a x)-3 \left (-15+24 a^2 x^2+8 a^4 x^4\right ) \arccos (a x)^2-16 a x \sqrt {1-a^2 x^2} \left (3+2 a^2 x^2\right ) \arccos (a x)^3-12 \arccos (a x)^4}{128 a^5} \] Input:

Integrate[(x^4*ArcCos[a*x]^3)/Sqrt[1 - a^2*x^2],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)*ArcCo 
s[a*x] - 3*(-15 + 24*a^2*x^2 + 8*a^4*x^4)*ArcCos[a*x]^2 - 16*a*x*Sqrt[1 - 
a^2*x^2]*(3 + 2*a^2*x^2)*ArcCos[a*x]^3 - 12*ArcCos[a*x]^4)/(128*a^5)
 

Rubi [A] (verified)

Time = 1.53 (sec) , antiderivative size = 275, normalized size of antiderivative = 1.44, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {5211, 5139, 5211, 15, 5139, 5153, 5211, 15, 5153}

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.

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

\(\Big \downarrow \) 5211

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

\(\Big \downarrow \) 5139

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

\(\Big \downarrow \) 5211

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

\(\Big \downarrow \) 15

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

\(\Big \downarrow \) 5139

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

\(\Big \downarrow \) 5153

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

\(\Big \downarrow \) 5211

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

\(\Big \downarrow \) 15

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

\(\Big \downarrow \) 5153

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

Input:

Int[(x^4*ArcCos[a*x]^3)/Sqrt[1 - a^2*x^2],x]
 

Output:

-1/4*(x^3*Sqrt[1 - a^2*x^2]*ArcCos[a*x]^3)/a^2 - (3*((x^4*ArcCos[a*x]^2)/4 
 + (a*(-1/16*x^4/a - (x^3*Sqrt[1 - a^2*x^2]*ArcCos[a*x])/(4*a^2) + (3*(-1/ 
4*x^2/a - (x*Sqrt[1 - a^2*x^2]*ArcCos[a*x])/(2*a^2) - ArcCos[a*x]^2/(4*a^3 
)))/(4*a^2)))/2))/(4*a) + (3*(-1/2*(x*Sqrt[1 - a^2*x^2]*ArcCos[a*x]^3)/a^2 
 - ArcCos[a*x]^4/(8*a^3) - (3*((x^2*ArcCos[a*x]^2)/2 + a*(-1/4*x^2/a - (x* 
Sqrt[1 - a^2*x^2]*ArcCos[a*x])/(2*a^2) - ArcCos[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 5139

Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_.)*((d_.)*(x_))^(m_.), x_Symbol] 
:> Simp[(d*x)^(m + 1)*((a + b*ArcCos[c*x])^n/(d*(m + 1))), x] + Simp[b*c*(n 
/(d*(m + 1)))   Int[(d*x)^(m + 1)*((a + b*ArcCos[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 5153

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

rule 5211

Int[((a_.) + ArcCos[(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*ArcCos[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*ArcCos[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*ArcCos[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 = 0.43 (sec) , antiderivative size = 160, normalized size of antiderivative = 0.84

method	result	size
default	\(-\frac {128 \arccos \left (a x \right )^{3} \sqrt {-a^{2} x^{2}+1}\, a^{3} x^{3}+96 a^{4} x^{4} \arccos \left (a x \right )^{2}-48 \sqrt {-a^{2} x^{2}+1}\, \arccos \left (a x \right ) a^{3} x^{3}-12 a^{4} x^{4}+192 \arccos \left (a x \right )^{3} \sqrt {-a^{2} x^{2}+1}\, a x +288 a^{2} x^{2} \arccos \left (a x \right )^{2}+48 \arccos \left (a x \right )^{4}-360 \arccos \left (a x \right ) \sqrt {-a^{2} x^{2}+1}\, a x -180 a^{2} x^{2}-180 \arccos \left (a x \right )^{2}+117}{512 a^{5}}\)	\(160\)

Input:

int(x^4*arccos(a*x)^3/(-a^2*x^2+1)^(1/2),x,method=_RETURNVERBOSE)
 

Output:

-1/512*(128*arccos(a*x)^3*(-a^2*x^2+1)^(1/2)*a^3*x^3+96*a^4*x^4*arccos(a*x 
)^2-48*(-a^2*x^2+1)^(1/2)*arccos(a*x)*a^3*x^3-12*a^4*x^4+192*arccos(a*x)^3 
*(-a^2*x^2+1)^(1/2)*a*x+288*a^2*x^2*arccos(a*x)^2+48*arccos(a*x)^4-360*arc 
cos(a*x)*(-a^2*x^2+1)^(1/2)*a*x-180*a^2*x^2-180*arccos(a*x)^2+117)/a^5
 

Fricas [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 111, normalized size of antiderivative = 0.58 \[ \int \frac {x^4 \arccos (a x)^3}{\sqrt {1-a^2 x^2}} \, dx=\frac {3 \, a^{4} x^{4} + 45 \, a^{2} x^{2} - 12 \, \arccos \left (a x\right )^{4} - 3 \, {\left (8 \, a^{4} x^{4} + 24 \, a^{2} x^{2} - 15\right )} \arccos \left (a x\right )^{2} - 2 \, \sqrt {-a^{2} x^{2} + 1} {\left (8 \, {\left (2 \, a^{3} x^{3} + 3 \, a x\right )} \arccos \left (a x\right )^{3} - 3 \, {\left (2 \, a^{3} x^{3} + 15 \, a x\right )} \arccos \left (a x\right )\right )}}{128 \, a^{5}} \] Input:

integrate(x^4*arccos(a*x)^3/(-a^2*x^2+1)^(1/2),x, algorithm="fricas")
 

Output:

1/128*(3*a^4*x^4 + 45*a^2*x^2 - 12*arccos(a*x)^4 - 3*(8*a^4*x^4 + 24*a^2*x 
^2 - 15)*arccos(a*x)^2 - 2*sqrt(-a^2*x^2 + 1)*(8*(2*a^3*x^3 + 3*a*x)*arcco 
s(a*x)^3 - 3*(2*a^3*x^3 + 15*a*x)*arccos(a*x)))/a^5
 

Sympy [A] (verification not implemented)

Time = 0.71 (sec) , antiderivative size = 192, normalized size of antiderivative = 1.01 \[ \int \frac {x^4 \arccos (a x)^3}{\sqrt {1-a^2 x^2}} \, dx=\begin {cases} - \frac {3 x^{4} \operatorname {acos}^{2}{\left (a x \right )}}{16 a} + \frac {3 x^{4}}{128 a} - \frac {x^{3} \sqrt {- a^{2} x^{2} + 1} \operatorname {acos}^{3}{\left (a x \right )}}{4 a^{2}} + \frac {3 x^{3} \sqrt {- a^{2} x^{2} + 1} \operatorname {acos}{\left (a x \right )}}{32 a^{2}} - \frac {9 x^{2} \operatorname {acos}^{2}{\left (a x \right )}}{16 a^{3}} + \frac {45 x^{2}}{128 a^{3}} - \frac {3 x \sqrt {- a^{2} x^{2} + 1} \operatorname {acos}^{3}{\left (a x \right )}}{8 a^{4}} + \frac {45 x \sqrt {- a^{2} x^{2} + 1} \operatorname {acos}{\left (a x \right )}}{64 a^{4}} - \frac {3 \operatorname {acos}^{4}{\left (a x \right )}}{32 a^{5}} + \frac {45 \operatorname {acos}^{2}{\left (a x \right )}}{128 a^{5}} & \text {for}\: a \neq 0 \\\frac {\pi ^{3} x^{5}}{40} & \text {otherwise} \end {cases} \] Input:

integrate(x**4*acos(a*x)**3/(-a**2*x**2+1)**(1/2),x)
 

Output:

Piecewise((-3*x**4*acos(a*x)**2/(16*a) + 3*x**4/(128*a) - x**3*sqrt(-a**2* 
x**2 + 1)*acos(a*x)**3/(4*a**2) + 3*x**3*sqrt(-a**2*x**2 + 1)*acos(a*x)/(3 
2*a**2) - 9*x**2*acos(a*x)**2/(16*a**3) + 45*x**2/(128*a**3) - 3*x*sqrt(-a 
**2*x**2 + 1)*acos(a*x)**3/(8*a**4) + 45*x*sqrt(-a**2*x**2 + 1)*acos(a*x)/ 
(64*a**4) - 3*acos(a*x)**4/(32*a**5) + 45*acos(a*x)**2/(128*a**5), Ne(a, 0 
)), (pi**3*x**5/40, True))
 

Maxima [F]

\[ \int \frac {x^4 \arccos (a x)^3}{\sqrt {1-a^2 x^2}} \, dx=\int { \frac {x^{4} \arccos \left (a x\right )^{3}}{\sqrt {-a^{2} x^{2} + 1}} \,d x } \] Input:

integrate(x^4*arccos(a*x)^3/(-a^2*x^2+1)^(1/2),x, algorithm="maxima")
 

Output:

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

Giac [A] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 168, normalized size of antiderivative = 0.88 \[ \int \frac {x^4 \arccos (a x)^3}{\sqrt {1-a^2 x^2}} \, dx=-\frac {3 \, x^{4} \arccos \left (a x\right )^{2}}{16 \, a} - \frac {\sqrt {-a^{2} x^{2} + 1} x^{3} \arccos \left (a x\right )^{3}}{4 \, a^{2}} + \frac {3 \, x^{4}}{128 \, a} + \frac {3 \, \sqrt {-a^{2} x^{2} + 1} x^{3} \arccos \left (a x\right )}{32 \, a^{2}} - \frac {9 \, x^{2} \arccos \left (a x\right )^{2}}{16 \, a^{3}} - \frac {3 \, \sqrt {-a^{2} x^{2} + 1} x \arccos \left (a x\right )^{3}}{8 \, a^{4}} + \frac {45 \, x^{2}}{128 \, a^{3}} - \frac {3 \, \arccos \left (a x\right )^{4}}{32 \, a^{5}} + \frac {45 \, \sqrt {-a^{2} x^{2} + 1} x \arccos \left (a x\right )}{64 \, a^{4}} + \frac {45 \, \arccos \left (a x\right )^{2}}{128 \, a^{5}} - \frac {189}{1024 \, a^{5}} \] Input:

integrate(x^4*arccos(a*x)^3/(-a^2*x^2+1)^(1/2),x, algorithm="giac")
 

Output:

-3/16*x^4*arccos(a*x)^2/a - 1/4*sqrt(-a^2*x^2 + 1)*x^3*arccos(a*x)^3/a^2 + 
 3/128*x^4/a + 3/32*sqrt(-a^2*x^2 + 1)*x^3*arccos(a*x)/a^2 - 9/16*x^2*arcc 
os(a*x)^2/a^3 - 3/8*sqrt(-a^2*x^2 + 1)*x*arccos(a*x)^3/a^4 + 45/128*x^2/a^ 
3 - 3/32*arccos(a*x)^4/a^5 + 45/64*sqrt(-a^2*x^2 + 1)*x*arccos(a*x)/a^4 + 
45/128*arccos(a*x)^2/a^5 - 189/1024/a^5
 

Mupad [F(-1)]

Timed out. \[ \int \frac {x^4 \arccos (a x)^3}{\sqrt {1-a^2 x^2}} \, dx=\int \frac {x^4\,{\mathrm {acos}\left (a\,x\right )}^3}{\sqrt {1-a^2\,x^2}} \,d x \] Input:

int((x^4*acos(a*x)^3)/(1 - a^2*x^2)^(1/2),x)
 

Output:

int((x^4*acos(a*x)^3)/(1 - a^2*x^2)^(1/2), x)
 

Reduce [F]

\[ \int \frac {x^4 \arccos (a x)^3}{\sqrt {1-a^2 x^2}} \, dx=\int \frac {\mathit {acos} \left (a x \right )^{3} x^{4}}{\sqrt {-a^{2} x^{2}+1}}d x \] Input:

int(x^4*acos(a*x)^3/(-a^2*x^2+1)^(1/2),x)
 

Output:

int((acos(a*x)**3*x**4)/sqrt( - a**2*x**2 + 1),x)