\(\int x^2 \arccos (a x)^3 \, dx\) [24]

Optimal result

Integrand size = 10, antiderivative size = 136 \[ \int x^2 \arccos (a x)^3 \, dx=\frac {14 \sqrt {1-a^2 x^2}}{9 a^3}-\frac {2 \left (1-a^2 x^2\right )^{3/2}}{27 a^3}-\frac {4 x \arccos (a x)}{3 a^2}-\frac {2}{9} x^3 \arccos (a x)-\frac {2 \sqrt {1-a^2 x^2} \arccos (a x)^2}{3 a^3}-\frac {x^2 \sqrt {1-a^2 x^2} \arccos (a x)^2}{3 a}+\frac {1}{3} x^3 \arccos (a x)^3 \] Output:

14/9*(-a^2*x^2+1)^(1/2)/a^3-2/27*(-a^2*x^2+1)^(3/2)/a^3-4/3*x*arccos(a*x)/ 
a^2-2/9*x^3*arccos(a*x)-2/3*(-a^2*x^2+1)^(1/2)*arccos(a*x)^2/a^3-1/3*x^2*( 
-a^2*x^2+1)^(1/2)*arccos(a*x)^2/a+1/3*x^3*arccos(a*x)^3
 

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.70 \[ \int x^2 \arccos (a x)^3 \, dx=\frac {2 \sqrt {1-a^2 x^2} \left (20+a^2 x^2\right )-6 a x \left (6+a^2 x^2\right ) \arccos (a x)-9 \sqrt {1-a^2 x^2} \left (2+a^2 x^2\right ) \arccos (a x)^2+9 a^3 x^3 \arccos (a x)^3}{27 a^3} \] Input:

Integrate[x^2*ArcCos[a*x]^3,x]
 

Output:

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

Rubi [A] (verified)

Time = 0.88 (sec) , antiderivative size = 176, normalized size of antiderivative = 1.29, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.900, Rules used = {5139, 5211, 5139, 243, 53, 2009, 5183, 5131, 241}

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.

\(\displaystyle \int x^2 \arccos (a x)^3 \, dx\)

\(\Big \downarrow \) 5139

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

\(\Big \downarrow \) 5211

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

\(\Big \downarrow \) 5139

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

\(\Big \downarrow \) 243

\(\displaystyle a \left (\frac {2 \int \frac {x \arccos (a x)^2}{\sqrt {1-a^2 x^2}}dx}{3 a^2}-\frac {2 \left (\frac {1}{6} a \int \frac {x^2}{\sqrt {1-a^2 x^2}}dx^2+\frac {1}{3} x^3 \arccos (a x)\right )}{3 a}-\frac {x^2 \sqrt {1-a^2 x^2} \arccos (a x)^2}{3 a^2}\right )+\frac {1}{3} x^3 \arccos (a x)^3\)

\(\Big \downarrow \) 53

\(\displaystyle a \left (\frac {2 \int \frac {x \arccos (a x)^2}{\sqrt {1-a^2 x^2}}dx}{3 a^2}-\frac {2 \left (\frac {1}{6} a \int \left (\frac {1}{a^2 \sqrt {1-a^2 x^2}}-\frac {\sqrt {1-a^2 x^2}}{a^2}\right )dx^2+\frac {1}{3} x^3 \arccos (a x)\right )}{3 a}-\frac {x^2 \sqrt {1-a^2 x^2} \arccos (a x)^2}{3 a^2}\right )+\frac {1}{3} x^3 \arccos (a x)^3\)

\(\Big \downarrow \) 2009

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

\(\Big \downarrow \) 5183

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

\(\Big \downarrow \) 5131

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

\(\Big \downarrow \) 241

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

Input:

Int[x^2*ArcCos[a*x]^3,x]
 

Output:

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

Defintions of rubi rules used

rule 53

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int 
[ExpandIntegrand[(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, 
x] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0] && LeQ[7*m + 4*n + 4, 0]) 
|| LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])
 

rule 241

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

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 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

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 = 0.30 (sec) , antiderivative size = 106, normalized size of antiderivative = 0.78

method	result	size
derivativedivides	\(\frac {\frac {a^{3} x^{3} \arccos \left (a x \right )^{3}}{3}-\frac {\arccos \left (a x \right )^{2} \left (a^{2} x^{2}+2\right ) \sqrt {-a^{2} x^{2}+1}}{3}+\frac {4 \sqrt {-a^{2} x^{2}+1}}{3}-\frac {4 a x \arccos \left (a x \right )}{3}-\frac {2 a^{3} x^{3} \arccos \left (a x \right )}{9}+\frac {2 \left (a^{2} x^{2}+2\right ) \sqrt {-a^{2} x^{2}+1}}{27}}{a^{3}}\)	\(106\)
default	\(\frac {\frac {a^{3} x^{3} \arccos \left (a x \right )^{3}}{3}-\frac {\arccos \left (a x \right )^{2} \left (a^{2} x^{2}+2\right ) \sqrt {-a^{2} x^{2}+1}}{3}+\frac {4 \sqrt {-a^{2} x^{2}+1}}{3}-\frac {4 a x \arccos \left (a x \right )}{3}-\frac {2 a^{3} x^{3} \arccos \left (a x \right )}{9}+\frac {2 \left (a^{2} x^{2}+2\right ) \sqrt {-a^{2} x^{2}+1}}{27}}{a^{3}}\)	\(106\)
orering	\(\frac {5 \left (13 a^{6} x^{6}+40 a^{4} x^{4}-152 a^{2} x^{2}+96\right ) \arccos \left (a x \right )^{3}}{81 a^{6} x^{3}}-\frac {\left (25 a^{6} x^{6}+166 a^{4} x^{4}-572 a^{2} x^{2}+360\right ) \left (2 x \arccos \left (a x \right )^{3}-\frac {3 x^{2} \arccos \left (a x \right )^{2} a}{\sqrt {-a^{2} x^{2}+1}}\right )}{81 a^{6} x^{4}}+\frac {2 \left (a x -1\right ) \left (a x +1\right ) \left (a^{4} x^{4}+12 a^{2} x^{2}-20\right ) \left (2 \arccos \left (a x \right )^{3}-\frac {12 x \arccos \left (a x \right )^{2} a}{\sqrt {-a^{2} x^{2}+1}}+\frac {6 x^{2} \arccos \left (a x \right ) a^{2}}{-a^{2} x^{2}+1}-\frac {3 x^{3} \arccos \left (a x \right )^{2} a^{3}}{\left (-a^{2} x^{2}+1\right )^{\frac {3}{2}}}\right )}{27 x^{3} a^{6}}-\frac {\left (a^{2} x^{2}+20\right ) \left (a x -1\right )^{2} \left (a x +1\right )^{2} \left (-\frac {18 \arccos \left (a x \right )^{2} a}{\sqrt {-a^{2} x^{2}+1}}+\frac {36 x \arccos \left (a x \right ) a^{2}}{-a^{2} x^{2}+1}-\frac {21 x^{2} \arccos \left (a x \right )^{2} a^{3}}{\left (-a^{2} x^{2}+1\right )^{\frac {3}{2}}}-\frac {6 x^{2} a^{3}}{\left (-a^{2} x^{2}+1\right )^{\frac {3}{2}}}+\frac {18 x^{3} \arccos \left (a x \right ) a^{4}}{\left (-a^{2} x^{2}+1\right )^{2}}-\frac {9 x^{4} \arccos \left (a x \right )^{2} a^{5}}{\left (-a^{2} x^{2}+1\right )^{\frac {5}{2}}}\right )}{81 a^{6} x^{2}}\)	\(397\)

Input:

int(x^2*arccos(a*x)^3,x,method=_RETURNVERBOSE)
 

Output:

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

Fricas [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.57 \[ \int x^2 \arccos (a x)^3 \, dx=\frac {9 \, a^{3} x^{3} \arccos \left (a x\right )^{3} - 6 \, {\left (a^{3} x^{3} + 6 \, a x\right )} \arccos \left (a x\right ) + {\left (2 \, a^{2} x^{2} - 9 \, {\left (a^{2} x^{2} + 2\right )} \arccos \left (a x\right )^{2} + 40\right )} \sqrt {-a^{2} x^{2} + 1}}{27 \, a^{3}} \] Input:

integrate(x^2*arccos(a*x)^3,x, algorithm="fricas")
 

Output:

1/27*(9*a^3*x^3*arccos(a*x)^3 - 6*(a^3*x^3 + 6*a*x)*arccos(a*x) + (2*a^2*x 
^2 - 9*(a^2*x^2 + 2)*arccos(a*x)^2 + 40)*sqrt(-a^2*x^2 + 1))/a^3
 

Sympy [A] (verification not implemented)

Time = 0.32 (sec) , antiderivative size = 134, normalized size of antiderivative = 0.99 \[ \int x^2 \arccos (a x)^3 \, dx=\begin {cases} \frac {x^{3} \operatorname {acos}^{3}{\left (a x \right )}}{3} - \frac {2 x^{3} \operatorname {acos}{\left (a x \right )}}{9} - \frac {x^{2} \sqrt {- a^{2} x^{2} + 1} \operatorname {acos}^{2}{\left (a x \right )}}{3 a} + \frac {2 x^{2} \sqrt {- a^{2} x^{2} + 1}}{27 a} - \frac {4 x \operatorname {acos}{\left (a x \right )}}{3 a^{2}} - \frac {2 \sqrt {- a^{2} x^{2} + 1} \operatorname {acos}^{2}{\left (a x \right )}}{3 a^{3}} + \frac {40 \sqrt {- a^{2} x^{2} + 1}}{27 a^{3}} & \text {for}\: a \neq 0 \\\frac {\pi ^{3} x^{3}}{24} & \text {otherwise} \end {cases} \] Input:

integrate(x**2*acos(a*x)**3,x)
 

Output:

Piecewise((x**3*acos(a*x)**3/3 - 2*x**3*acos(a*x)/9 - x**2*sqrt(-a**2*x**2 
 + 1)*acos(a*x)**2/(3*a) + 2*x**2*sqrt(-a**2*x**2 + 1)/(27*a) - 4*x*acos(a 
*x)/(3*a**2) - 2*sqrt(-a**2*x**2 + 1)*acos(a*x)**2/(3*a**3) + 40*sqrt(-a** 
2*x**2 + 1)/(27*a**3), Ne(a, 0)), (pi**3*x**3/24, True))
 

Maxima [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 120, normalized size of antiderivative = 0.88 \[ \int x^2 \arccos (a x)^3 \, dx=\frac {1}{3} \, x^{3} \arccos \left (a x\right )^{3} - \frac {1}{3} \, a {\left (\frac {\sqrt {-a^{2} x^{2} + 1} x^{2}}{a^{2}} + \frac {2 \, \sqrt {-a^{2} x^{2} + 1}}{a^{4}}\right )} \arccos \left (a x\right )^{2} + \frac {2}{27} \, a {\left (\frac {\sqrt {-a^{2} x^{2} + 1} x^{2} + \frac {20 \, \sqrt {-a^{2} x^{2} + 1}}{a^{2}}}{a^{2}} - \frac {3 \, {\left (a^{2} x^{3} + 6 \, x\right )} \arccos \left (a x\right )}{a^{3}}\right )} \] Input:

integrate(x^2*arccos(a*x)^3,x, algorithm="maxima")
 

Output:

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

Giac [A] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 117, normalized size of antiderivative = 0.86 \[ \int x^2 \arccos (a x)^3 \, dx=\frac {1}{3} \, x^{3} \arccos \left (a x\right )^{3} - \frac {2}{9} \, x^{3} \arccos \left (a x\right ) - \frac {\sqrt {-a^{2} x^{2} + 1} x^{2} \arccos \left (a x\right )^{2}}{3 \, a} + \frac {2 \, \sqrt {-a^{2} x^{2} + 1} x^{2}}{27 \, a} - \frac {4 \, x \arccos \left (a x\right )}{3 \, a^{2}} - \frac {2 \, \sqrt {-a^{2} x^{2} + 1} \arccos \left (a x\right )^{2}}{3 \, a^{3}} + \frac {40 \, \sqrt {-a^{2} x^{2} + 1}}{27 \, a^{3}} \] Input:

integrate(x^2*arccos(a*x)^3,x, algorithm="giac")
 

Output:

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

Mupad [F(-1)]

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

int(x^2*acos(a*x)^3,x)
 

Output:

int(x^2*acos(a*x)^3, x)
 

Reduce [F]

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

int(x^2*acos(a*x)^3,x)
 

Output:

int(acos(a*x)**3*x**2,x)