\(\int \frac {x^3 \arccos (a x)^3}{\sqrt {1-a^2 x^2}} \, dx\) [306]

Optimal result

Integrand size = 24, antiderivative size = 157 \[ \int \frac {x^3 \arccos (a x)^3}{\sqrt {1-a^2 x^2}} \, dx=-\frac {40 x}{9 a^3}-\frac {2 x^3}{27 a}+\frac {40 \sqrt {1-a^2 x^2} \arccos (a x)}{9 a^4}+\frac {2 x^2 \sqrt {1-a^2 x^2} \arccos (a x)}{9 a^2}+\frac {2 x \arccos (a x)^2}{a^3}+\frac {x^3 \arccos (a x)^2}{3 a}-\frac {2 \sqrt {1-a^2 x^2} \arccos (a x)^3}{3 a^4}-\frac {x^2 \sqrt {1-a^2 x^2} \arccos (a x)^3}{3 a^2} \] Output:

-40/9*x/a^3-2/27*x^3/a+40/9*(-a^2*x^2+1)^(1/2)*arccos(a*x)/a^4+2/9*x^2*(-a 
^2*x^2+1)^(1/2)*arccos(a*x)/a^2+2*x*arccos(a*x)^2/a^3+1/3*x^3*arccos(a*x)^ 
2/a-2/3*(-a^2*x^2+1)^(1/2)*arccos(a*x)^3/a^4-1/3*x^2*(-a^2*x^2+1)^(1/2)*ar 
ccos(a*x)^3/a^2
 

Mathematica [A] (verified)

Time = 0.05 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.64 \[ \int \frac {x^3 \arccos (a x)^3}{\sqrt {1-a^2 x^2}} \, dx=\frac {2 a x \left (60+a^2 x^2\right )+6 \sqrt {1-a^2 x^2} \left (20+a^2 x^2\right ) \arccos (a x)-9 a x \left (6+a^2 x^2\right ) \arccos (a x)^2-9 \sqrt {1-a^2 x^2} \left (2+a^2 x^2\right ) \arccos (a x)^3}{27 a^4} \] Input:

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

Output:

(2*a*x*(60 + a^2*x^2) + 6*Sqrt[1 - a^2*x^2]*(20 + a^2*x^2)*ArcCos[a*x] - 9 
*a*x*(6 + a^2*x^2)*ArcCos[a*x]^2 - 9*Sqrt[1 - a^2*x^2]*(2 + a^2*x^2)*ArcCo 
s[a*x]^3)/(27*a^4)
 

Rubi [A] (verified)

Time = 1.25 (sec) , antiderivative size = 211, normalized size of antiderivative = 1.34, number of steps used = 10, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {5211, 5139, 5183, 5131, 5183, 24, 5211, 15, 5183, 24}

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.

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

\(\Big \downarrow \) 5211

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

\(\Big \downarrow \) 5139

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

\(\Big \downarrow \) 5183

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

\(\Big \downarrow \) 5131

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

\(\Big \downarrow \) 5183

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

\(\Big \downarrow \) 24

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

\(\Big \downarrow \) 5211

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

\(\Big \downarrow \) 15

\(\displaystyle -\frac {\frac {2}{3} a \left (\frac {2 \int \frac {x \arccos (a x)}{\sqrt {1-a^2 x^2}}dx}{3 a^2}-\frac {x^2 \sqrt {1-a^2 x^2} \arccos (a x)}{3 a^2}-\frac {x^3}{9 a}\right )+\frac {1}{3} x^3 \arccos (a x)^2}{a}-\frac {x^2 \sqrt {1-a^2 x^2} \arccos (a x)^3}{3 a^2}+\frac {2 \left (-\frac {\sqrt {1-a^2 x^2} \arccos (a x)^3}{a^2}-\frac {3 \left (2 a \left (-\frac {\sqrt {1-a^2 x^2} \arccos (a x)}{a^2}-\frac {x}{a}\right )+x \arccos (a x)^2\right )}{a}\right )}{3 a^2}\)

\(\Big \downarrow \) 5183

\(\displaystyle -\frac {\frac {2}{3} a \left (\frac {2 \left (-\frac {\int 1dx}{a}-\frac {\sqrt {1-a^2 x^2} \arccos (a x)}{a^2}\right )}{3 a^2}-\frac {x^2 \sqrt {1-a^2 x^2} \arccos (a x)}{3 a^2}-\frac {x^3}{9 a}\right )+\frac {1}{3} x^3 \arccos (a x)^2}{a}-\frac {x^2 \sqrt {1-a^2 x^2} \arccos (a x)^3}{3 a^2}+\frac {2 \left (-\frac {\sqrt {1-a^2 x^2} \arccos (a x)^3}{a^2}-\frac {3 \left (2 a \left (-\frac {\sqrt {1-a^2 x^2} \arccos (a x)}{a^2}-\frac {x}{a}\right )+x \arccos (a x)^2\right )}{a}\right )}{3 a^2}\)

\(\Big \downarrow \) 24

\(\displaystyle -\frac {x^2 \sqrt {1-a^2 x^2} \arccos (a x)^3}{3 a^2}+\frac {2 \left (-\frac {\sqrt {1-a^2 x^2} \arccos (a x)^3}{a^2}-\frac {3 \left (2 a \left (-\frac {\sqrt {1-a^2 x^2} \arccos (a x)}{a^2}-\frac {x}{a}\right )+x \arccos (a x)^2\right )}{a}\right )}{3 a^2}-\frac {\frac {2}{3} a \left (-\frac {x^2 \sqrt {1-a^2 x^2} \arccos (a x)}{3 a^2}+\frac {2 \left (-\frac {\sqrt {1-a^2 x^2} \arccos (a x)}{a^2}-\frac {x}{a}\right )}{3 a^2}-\frac {x^3}{9 a}\right )+\frac {1}{3} x^3 \arccos (a x)^2}{a}\)

Input:

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

Output:

-1/3*(x^2*Sqrt[1 - a^2*x^2]*ArcCos[a*x]^3)/a^2 - ((x^3*ArcCos[a*x]^2)/3 + 
(2*a*(-1/9*x^3/a - (x^2*Sqrt[1 - a^2*x^2]*ArcCos[a*x])/(3*a^2) + (2*(-(x/a 
) - (Sqrt[1 - a^2*x^2]*ArcCos[a*x])/a^2))/(3*a^2)))/3)/a + (2*(-((Sqrt[1 - 
 a^2*x^2]*ArcCos[a*x]^3)/a^2) - (3*(x*ArcCos[a*x]^2 + 2*a*(-(x/a) - (Sqrt[ 
1 - a^2*x^2]*ArcCos[a*x])/a^2)))/a))/(3*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 24

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]
 

rule 5131

Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_.), x_Symbol] :> Simp[x*(a + b*Ar 
cCos[c*x])^n, x] + Simp[b*c*n   Int[x*((a + b*ArcCos[c*x])^(n - 1)/Sqrt[1 - 
 c^2*x^2]), x], x] /; FreeQ[{a, b, c}, x] && GtQ[n, 0]
 

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 5183

Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_.)*(x_)*((d_) + (e_.)*(x_)^2)^(p_ 
.), x_Symbol] :> Simp[(d + e*x^2)^(p + 1)*((a + b*ArcCos[c*x])^n/(2*e*(p + 
1))), x] - Simp[b*(n/(2*c*(p + 1)))*Simp[(d + e*x^2)^p/(1 - c^2*x^2)^p]   I 
nt[(1 - c^2*x^2)^(p + 1/2)*(a + b*ArcCos[c*x])^(n - 1), x], x] /; FreeQ[{a, 
 b, c, d, e, p}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && NeQ[p, -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 = 0.38 (sec) , antiderivative size = 180, normalized size of antiderivative = 1.15

method	result	size
default	\(-\frac {\sqrt {-a^{2} x^{2}+1}\, \left (9 a^{4} x^{4} \arccos \left (a x \right )^{3}+9 \arccos \left (a x \right )^{3} a^{2} x^{2}-9 \arccos \left (a x \right )^{2} \sqrt {-a^{2} x^{2}+1}\, a^{3} x^{3}-6 a^{4} x^{4} \arccos \left (a x \right )-114 a^{2} x^{2} \arccos \left (a x \right )+2 a^{3} x^{3} \sqrt {-a^{2} x^{2}+1}-18 \arccos \left (a x \right )^{3}-54 \arccos \left (a x \right )^{2} \sqrt {-a^{2} x^{2}+1}\, a x +120 \arccos \left (a x \right )+120 \sqrt {-a^{2} x^{2}+1}\, a x \right )}{27 a^{4} \left (a^{2} x^{2}-1\right )}\)	\(180\)
orering	\(\frac {5 \left (13 a^{6} x^{6}+144 a^{4} x^{4}-936 a^{2} x^{2}+864\right ) \arccos \left (a x \right )^{3}}{81 a^{6} x^{2} \sqrt {-a^{2} x^{2}+1}}-\frac {\left (25 a^{6} x^{6}+578 a^{4} x^{4}-2940 a^{2} x^{2}+2520\right ) \left (\frac {3 x^{2} \arccos \left (a x \right )^{3}}{\sqrt {-a^{2} x^{2}+1}}-\frac {3 x^{3} \arccos \left (a x \right )^{2} a}{-a^{2} x^{2}+1}+\frac {x^{4} \arccos \left (a x \right )^{3} a^{2}}{\left (-a^{2} x^{2}+1\right )^{\frac {3}{2}}}\right )}{81 a^{6} x^{4}}+\frac {2 \left (a x -1\right ) \left (a x +1\right ) \left (a^{4} x^{4}+38 a^{2} x^{2}-100\right ) \left (\frac {6 x \arccos \left (a x \right )^{3}}{\sqrt {-a^{2} x^{2}+1}}-\frac {18 x^{2} \arccos \left (a x \right )^{2} a}{-a^{2} x^{2}+1}+\frac {7 x^{3} \arccos \left (a x \right )^{3} a^{2}}{\left (-a^{2} x^{2}+1\right )^{\frac {3}{2}}}+\frac {6 x^{3} \arccos \left (a x \right ) a^{2}}{\left (-a^{2} x^{2}+1\right )^{\frac {3}{2}}}-\frac {9 x^{4} \arccos \left (a x \right )^{2} a^{3}}{\left (-a^{2} x^{2}+1\right )^{2}}+\frac {3 x^{5} \arccos \left (a x \right )^{3} a^{4}}{\left (-a^{2} x^{2}+1\right )^{\frac {5}{2}}}\right )}{27 x^{3} a^{6}}-\frac {\left (a^{2} x^{2}+60\right ) \left (a x -1\right )^{2} \left (a x +1\right )^{2} \left (\frac {6 \arccos \left (a x \right )^{3}}{\sqrt {-a^{2} x^{2}+1}}-\frac {54 x \arccos \left (a x \right )^{2} a}{-a^{2} x^{2}+1}+\frac {27 x^{2} \arccos \left (a x \right )^{3} a^{2}}{\left (-a^{2} x^{2}+1\right )^{\frac {3}{2}}}+\frac {54 x^{2} \arccos \left (a x \right ) a^{2}}{\left (-a^{2} x^{2}+1\right )^{\frac {3}{2}}}-\frac {93 x^{3} \arccos \left (a x \right )^{2} a^{3}}{\left (-a^{2} x^{2}+1\right )^{2}}+\frac {36 x^{4} \arccos \left (a x \right )^{3} a^{4}}{\left (-a^{2} x^{2}+1\right )^{\frac {5}{2}}}-\frac {6 x^{3} a^{3}}{\left (-a^{2} x^{2}+1\right )^{2}}+\frac {36 x^{4} \arccos \left (a x \right ) a^{4}}{\left (-a^{2} x^{2}+1\right )^{\frac {5}{2}}}-\frac {45 x^{5} \arccos \left (a x \right )^{2} a^{5}}{\left (-a^{2} x^{2}+1\right )^{3}}+\frac {15 x^{6} \arccos \left (a x \right )^{3} a^{6}}{\left (-a^{2} x^{2}+1\right )^{\frac {7}{2}}}\right )}{81 x^{2} a^{6}}\)	\(616\)

Input:

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

Output:

-1/27/a^4*(-a^2*x^2+1)^(1/2)/(a^2*x^2-1)*(9*a^4*x^4*arccos(a*x)^3+9*arccos 
(a*x)^3*a^2*x^2-9*arccos(a*x)^2*(-a^2*x^2+1)^(1/2)*a^3*x^3-6*a^4*x^4*arcco 
s(a*x)-114*a^2*x^2*arccos(a*x)+2*a^3*x^3*(-a^2*x^2+1)^(1/2)-18*arccos(a*x) 
^3-54*arccos(a*x)^2*(-a^2*x^2+1)^(1/2)*a*x+120*arccos(a*x)+120*(-a^2*x^2+1 
)^(1/2)*a*x)
 

Fricas [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.54 \[ \int \frac {x^3 \arccos (a x)^3}{\sqrt {1-a^2 x^2}} \, dx=\frac {2 \, a^{3} x^{3} - 9 \, {\left (a^{3} x^{3} + 6 \, a x\right )} \arccos \left (a x\right )^{2} + 120 \, a x - 3 \, \sqrt {-a^{2} x^{2} + 1} {\left (3 \, {\left (a^{2} x^{2} + 2\right )} \arccos \left (a x\right )^{3} - 2 \, {\left (a^{2} x^{2} + 20\right )} \arccos \left (a x\right )\right )}}{27 \, a^{4}} \] Input:

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

Output:

1/27*(2*a^3*x^3 - 9*(a^3*x^3 + 6*a*x)*arccos(a*x)^2 + 120*a*x - 3*sqrt(-a^ 
2*x^2 + 1)*(3*(a^2*x^2 + 2)*arccos(a*x)^3 - 2*(a^2*x^2 + 20)*arccos(a*x))) 
/a^4
 

Sympy [A] (verification not implemented)

Time = 0.52 (sec) , antiderivative size = 155, normalized size of antiderivative = 0.99 \[ \int \frac {x^3 \arccos (a x)^3}{\sqrt {1-a^2 x^2}} \, dx=\begin {cases} - \frac {x^{3} \operatorname {acos}^{2}{\left (a x \right )}}{3 a} + \frac {2 x^{3}}{27 a} - \frac {x^{2} \sqrt {- a^{2} x^{2} + 1} \operatorname {acos}^{3}{\left (a x \right )}}{3 a^{2}} + \frac {2 x^{2} \sqrt {- a^{2} x^{2} + 1} \operatorname {acos}{\left (a x \right )}}{9 a^{2}} - \frac {2 x \operatorname {acos}^{2}{\left (a x \right )}}{a^{3}} + \frac {40 x}{9 a^{3}} - \frac {2 \sqrt {- a^{2} x^{2} + 1} \operatorname {acos}^{3}{\left (a x \right )}}{3 a^{4}} + \frac {40 \sqrt {- a^{2} x^{2} + 1} \operatorname {acos}{\left (a x \right )}}{9 a^{4}} & \text {for}\: a \neq 0 \\\frac {\pi ^{3} x^{4}}{32} & \text {otherwise} \end {cases} \] Input:

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

Output:

Piecewise((-x**3*acos(a*x)**2/(3*a) + 2*x**3/(27*a) - x**2*sqrt(-a**2*x**2 
 + 1)*acos(a*x)**3/(3*a**2) + 2*x**2*sqrt(-a**2*x**2 + 1)*acos(a*x)/(9*a** 
2) - 2*x*acos(a*x)**2/a**3 + 40*x/(9*a**3) - 2*sqrt(-a**2*x**2 + 1)*acos(a 
*x)**3/(3*a**4) + 40*sqrt(-a**2*x**2 + 1)*acos(a*x)/(9*a**4), Ne(a, 0)), ( 
pi**3*x**4/32, True))
 

Maxima [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 130, normalized size of antiderivative = 0.83 \[ \int \frac {x^3 \arccos (a x)^3}{\sqrt {1-a^2 x^2}} \, dx=-\frac {1}{3} \, {\left (\frac {\sqrt {-a^{2} x^{2} + 1} x^{2}}{a^{2}} + \frac {2 \, \sqrt {-a^{2} x^{2} + 1}}{a^{4}}\right )} \arccos \left (a x\right )^{3} + \frac {2}{27} \, a {\left (\frac {3 \, {\left (\sqrt {-a^{2} x^{2} + 1} x^{2} + \frac {20 \, \sqrt {-a^{2} x^{2} + 1}}{a^{2}}\right )} \arccos \left (a x\right )}{a^{3}} + \frac {a^{2} x^{3} + 60 \, x}{a^{4}}\right )} - \frac {{\left (a^{2} x^{3} + 6 \, x\right )} \arccos \left (a x\right )^{2}}{3 \, a^{3}} \] Input:

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

Output:

-1/3*(sqrt(-a^2*x^2 + 1)*x^2/a^2 + 2*sqrt(-a^2*x^2 + 1)/a^4)*arccos(a*x)^3 
 + 2/27*a*(3*(sqrt(-a^2*x^2 + 1)*x^2 + 20*sqrt(-a^2*x^2 + 1)/a^2)*arccos(a 
*x)/a^3 + (a^2*x^3 + 60*x)/a^4) - 1/3*(a^2*x^3 + 6*x)*arccos(a*x)^2/a^3
 

Giac [F(-2)]

Exception generated. \[ \int \frac {x^3 \arccos (a x)^3}{\sqrt {1-a^2 x^2}} \, dx=\text {Exception raised: TypeError} \] Input:

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

Output:

Exception raised: TypeError >> an error occurred running a Giac command:IN 
PUT:sage2:=int(sage0,sageVARx):;OUTPUT:sym2poly/r2sym(const gen & e,const 
index_m & i,const vecteur & l) Error: Bad Argument Value
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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