Integrand size = 17, antiderivative size = 66 \[ \int x \sqrt {1-x^2} \arccos (x)^2 \, dx=\frac {4 \sqrt {1-x^2}}{9}+\frac {2}{27} \left (1-x^2\right )^{3/2}-\frac {2}{3} x \arccos (x)+\frac {2}{9} x^3 \arccos (x)-\frac {1}{3} \left (1-x^2\right )^{3/2} \arccos (x)^2 \] Output:
2/27*(-x^2+1)^(3/2)-2/3*x*arccos(x)+2/9*x^3*arccos(x)-1/3*(-x^2+1)^(3/2)*a rccos(x)^2+4/9*(-x^2+1)^(1/2)
Time = 0.03 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.76 \[ \int x \sqrt {1-x^2} \arccos (x)^2 \, dx=\frac {1}{27} \left (-2 \sqrt {1-x^2} \left (-7+x^2\right )+6 x \left (-3+x^2\right ) \arccos (x)-9 \left (1-x^2\right )^{3/2} \arccos (x)^2\right ) \] Input:
Integrate[x*Sqrt[1 - x^2]*ArcCos[x]^2,x]
Output:
(-2*Sqrt[1 - x^2]*(-7 + x^2) + 6*x*(-3 + x^2)*ArcCos[x] - 9*(1 - x^2)^(3/2 )*ArcCos[x]^2)/27
Time = 0.29 (sec) , antiderivative size = 71, normalized size of antiderivative = 1.08, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.353, Rules used = {5183, 5155, 27, 353, 53, 2009}
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.
\(\displaystyle \int x \sqrt {1-x^2} \arccos (x)^2 \, dx\) |
\(\Big \downarrow \) 5183 |
\(\displaystyle -\frac {2}{3} \int \left (1-x^2\right ) \arccos (x)dx-\frac {1}{3} \left (1-x^2\right )^{3/2} \arccos (x)^2\) |
\(\Big \downarrow \) 5155 |
\(\displaystyle -\frac {2}{3} \left (\int \frac {x \left (3-x^2\right )}{3 \sqrt {1-x^2}}dx-\frac {1}{3} x^3 \arccos (x)+x \arccos (x)\right )-\frac {1}{3} \left (1-x^2\right )^{3/2} \arccos (x)^2\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -\frac {2}{3} \left (\frac {1}{3} \int \frac {x \left (3-x^2\right )}{\sqrt {1-x^2}}dx-\frac {1}{3} x^3 \arccos (x)+x \arccos (x)\right )-\frac {1}{3} \left (1-x^2\right )^{3/2} \arccos (x)^2\) |
\(\Big \downarrow \) 353 |
\(\displaystyle -\frac {2}{3} \left (\frac {1}{6} \int \frac {3-x^2}{\sqrt {1-x^2}}dx^2-\frac {1}{3} x^3 \arccos (x)+x \arccos (x)\right )-\frac {1}{3} \left (1-x^2\right )^{3/2} \arccos (x)^2\) |
\(\Big \downarrow \) 53 |
\(\displaystyle -\frac {2}{3} \left (\frac {1}{6} \int \left (\sqrt {1-x^2}+\frac {2}{\sqrt {1-x^2}}\right )dx^2-\frac {1}{3} x^3 \arccos (x)+x \arccos (x)\right )-\frac {1}{3} \left (1-x^2\right )^{3/2} \arccos (x)^2\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {1}{3} \left (1-x^2\right )^{3/2} \arccos (x)^2-\frac {2}{3} \left (-\frac {1}{3} x^3 \arccos (x)+x \arccos (x)+\frac {1}{6} \left (-\frac {2}{3} \left (1-x^2\right )^{3/2}-4 \sqrt {1-x^2}\right )\right )\) |
Input:
Int[x*Sqrt[1 - x^2]*ArcCos[x]^2,x]
Output:
-1/3*((1 - x^2)^(3/2)*ArcCos[x]^2) - (2*((-4*Sqrt[1 - x^2] - (2*(1 - x^2)^ (3/2))/3)/6 + x*ArcCos[x] - (x^3*ArcCos[x])/3))/3
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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])
Int[(x_)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(q_.), x_Symbol] :> Simp[1/2 Subst[Int[(a + b*x)^p*(c + d*x)^q, x], x, x^2], x] /; FreeQ[ {a, b, c, d, p, q}, x] && NeQ[b*c - a*d, 0]
Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbo l] :> With[{u = IntHide[(d + e*x^2)^p, x]}, Simp[(a + b*ArcCos[c*x]) u, x ] + Simp[b*c Int[SimplifyIntegrand[u/Sqrt[1 - c^2*x^2], x], x], x]] /; Fr eeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && IGtQ[p, 0]
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]
Time = 0.46 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.89
method | result | size |
default | \(\frac {\left (x^{2}-1\right ) \sqrt {-x^{2}+1}\, \arccos \left (x \right )^{2}}{3}+\frac {2 \arccos \left (x \right ) \left (x^{2}-3\right ) x}{9}-\frac {2 \left (x^{2}-1\right ) \sqrt {-x^{2}+1}}{27}+\frac {4 \sqrt {-x^{2}+1}}{9}\) | \(59\) |
orering | \(\frac {\left (19 x^{6}-71 x^{4}+48 x^{2}-14\right ) \arccos \left (x \right )^{2} \sqrt {-x^{2}+1}}{27 x^{2} \left (x^{2}-1\right )}-\frac {2 \left (3 x^{4}-16 x^{2}+7\right ) \left (\arccos \left (x \right )^{2} \sqrt {-x^{2}+1}-2 x \arccos \left (x \right )-\frac {x^{2} \arccos \left (x \right )^{2}}{\sqrt {-x^{2}+1}}\right )}{27 x^{2}}+\frac {\left (x^{2}-7\right ) \left (-1+x \right ) \left (1+x \right ) \left (-4 \arccos \left (x \right )-\frac {3 \arccos \left (x \right )^{2} x}{\sqrt {-x^{2}+1}}+\frac {2 x}{\sqrt {-x^{2}+1}}+\frac {2 x^{2} \arccos \left (x \right )}{-x^{2}+1}-\frac {x^{3} \arccos \left (x \right )^{2}}{\left (-x^{2}+1\right )^{\frac {3}{2}}}\right )}{27 x}\) | \(182\) |
Input:
int(x*arccos(x)^2*(-x^2+1)^(1/2),x,method=_RETURNVERBOSE)
Output:
1/3*(x^2-1)*(-x^2+1)^(1/2)*arccos(x)^2+2/9*arccos(x)*(x^2-3)*x-2/27*(x^2-1 )*(-x^2+1)^(1/2)+4/9*(-x^2+1)^(1/2)
Time = 0.07 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.62 \[ \int x \sqrt {1-x^2} \arccos (x)^2 \, dx=\frac {2}{9} \, {\left (x^{3} - 3 \, x\right )} \arccos \left (x\right ) + \frac {1}{27} \, {\left (9 \, {\left (x^{2} - 1\right )} \arccos \left (x\right )^{2} - 2 \, x^{2} + 14\right )} \sqrt {-x^{2} + 1} \] Input:
integrate(x*arccos(x)^2*(-x^2+1)^(1/2),x, algorithm="fricas")
Output:
2/9*(x^3 - 3*x)*arccos(x) + 1/27*(9*(x^2 - 1)*arccos(x)^2 - 2*x^2 + 14)*sq rt(-x^2 + 1)
Time = 0.26 (sec) , antiderivative size = 78, normalized size of antiderivative = 1.18 \[ \int x \sqrt {1-x^2} \arccos (x)^2 \, dx=\frac {2 x^{3} \operatorname {acos}{\left (x \right )}}{9} + \frac {x^{2} \sqrt {1 - x^{2}} \operatorname {acos}^{2}{\left (x \right )}}{3} - \frac {2 x^{2} \sqrt {1 - x^{2}}}{27} - \frac {2 x \operatorname {acos}{\left (x \right )}}{3} - \frac {\sqrt {1 - x^{2}} \operatorname {acos}^{2}{\left (x \right )}}{3} + \frac {14 \sqrt {1 - x^{2}}}{27} \] Input:
integrate(x*acos(x)**2*(-x**2+1)**(1/2),x)
Output:
2*x**3*acos(x)/9 + x**2*sqrt(1 - x**2)*acos(x)**2/3 - 2*x**2*sqrt(1 - x**2 )/27 - 2*x*acos(x)/3 - sqrt(1 - x**2)*acos(x)**2/3 + 14*sqrt(1 - x**2)/27
Time = 0.10 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.79 \[ \int x \sqrt {1-x^2} \arccos (x)^2 \, dx=-\frac {1}{3} \, {\left (-x^{2} + 1\right )}^{\frac {3}{2}} \arccos \left (x\right )^{2} - \frac {2}{27} \, \sqrt {-x^{2} + 1} x^{2} + \frac {2}{9} \, {\left (x^{3} - 3 \, x\right )} \arccos \left (x\right ) + \frac {14}{27} \, \sqrt {-x^{2} + 1} \] Input:
integrate(x*arccos(x)^2*(-x^2+1)^(1/2),x, algorithm="maxima")
Output:
-1/3*(-x^2 + 1)^(3/2)*arccos(x)^2 - 2/27*sqrt(-x^2 + 1)*x^2 + 2/9*(x^3 - 3 *x)*arccos(x) + 14/27*sqrt(-x^2 + 1)
Time = 0.14 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.80 \[ \int x \sqrt {1-x^2} \arccos (x)^2 \, dx=\frac {2}{9} \, x^{3} \arccos \left (x\right ) - \frac {1}{3} \, {\left (-x^{2} + 1\right )}^{\frac {3}{2}} \arccos \left (x\right )^{2} - \frac {2}{27} \, \sqrt {-x^{2} + 1} x^{2} - \frac {2}{3} \, x \arccos \left (x\right ) + \frac {14}{27} \, \sqrt {-x^{2} + 1} \] Input:
integrate(x*arccos(x)^2*(-x^2+1)^(1/2),x, algorithm="giac")
Output:
2/9*x^3*arccos(x) - 1/3*(-x^2 + 1)^(3/2)*arccos(x)^2 - 2/27*sqrt(-x^2 + 1) *x^2 - 2/3*x*arccos(x) + 14/27*sqrt(-x^2 + 1)
Timed out. \[ \int x \sqrt {1-x^2} \arccos (x)^2 \, dx=\int x\,{\mathrm {acos}\left (x\right )}^2\,\sqrt {1-x^2} \,d x \] Input:
int(x*acos(x)^2*(1 - x^2)^(1/2),x)
Output:
int(x*acos(x)^2*(1 - x^2)^(1/2), x)
\[ \int x \sqrt {1-x^2} \arccos (x)^2 \, dx=\int \sqrt {-x^{2}+1}\, \mathit {acos} \left (x \right )^{2} x d x \] Input:
int(x*acos(x)^2*(-x^2+1)^(1/2),x)
Output:
int(sqrt( - x**2 + 1)*acos(x)**2*x,x)