Integrand size = 24, antiderivative size = 107 \[ \int \frac {x^2 \arccos (a x)^3}{\sqrt {1-a^2 x^2}} \, dx=-\frac {3 x^2}{8 a}+\frac {3 x \sqrt {1-a^2 x^2} \arccos (a x)}{4 a^2}-\frac {3 \arccos (a x)^2}{8 a^3}+\frac {3 x^2 \arccos (a x)^2}{4 a}-\frac {x \sqrt {1-a^2 x^2} \arccos (a x)^3}{2 a^2}+\frac {\arccos (a x)^4}{8 a^3} \] Output:
-3/8*x^2/a+3/4*x*(-a^2*x^2+1)^(1/2)*arccos(a*x)/a^2-3/8*arccos(a*x)^2/a^3+ 3/4*x^2*arccos(a*x)^2/a-1/2*x*(-a^2*x^2+1)^(1/2)*arccos(a*x)^3/a^2+1/8*arc cos(a*x)^4/a^3
Time = 0.04 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.81 \[ \int \frac {x^2 \arccos (a x)^3}{\sqrt {1-a^2 x^2}} \, dx=\frac {3 a^2 x^2+6 a x \sqrt {1-a^2 x^2} \arccos (a x)+\left (3-6 a^2 x^2\right ) \arccos (a x)^2-4 a x \sqrt {1-a^2 x^2} \arccos (a x)^3-\arccos (a x)^4}{8 a^3} \] Input:
Integrate[(x^2*ArcCos[a*x]^3)/Sqrt[1 - a^2*x^2],x]
Output:
(3*a^2*x^2 + 6*a*x*Sqrt[1 - a^2*x^2]*ArcCos[a*x] + (3 - 6*a^2*x^2)*ArcCos[ a*x]^2 - 4*a*x*Sqrt[1 - a^2*x^2]*ArcCos[a*x]^3 - ArcCos[a*x]^4)/(8*a^3)
Time = 0.72 (sec) , antiderivative size = 115, normalized size of antiderivative = 1.07, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {5211, 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.
\(\displaystyle \int \frac {x^2 \arccos (a x)^3}{\sqrt {1-a^2 x^2}} \, dx\) |
\(\Big \downarrow \) 5211 |
\(\displaystyle \frac {\int \frac {\arccos (a x)^3}{\sqrt {1-a^2 x^2}}dx}{2 a^2}-\frac {3 \int x \arccos (a x)^2dx}{2 a}-\frac {x \sqrt {1-a^2 x^2} \arccos (a x)^3}{2 a^2}\) |
\(\Big \downarrow \) 5139 |
\(\displaystyle -\frac {3 \left (a \int \frac {x^2 \arccos (a x)}{\sqrt {1-a^2 x^2}}dx+\frac {1}{2} x^2 \arccos (a x)^2\right )}{2 a}+\frac {\int \frac {\arccos (a x)^3}{\sqrt {1-a^2 x^2}}dx}{2 a^2}-\frac {x \sqrt {1-a^2 x^2} \arccos (a x)^3}{2 a^2}\) |
\(\Big \downarrow \) 5153 |
\(\displaystyle -\frac {3 \left (a \int \frac {x^2 \arccos (a x)}{\sqrt {1-a^2 x^2}}dx+\frac {1}{2} x^2 \arccos (a x)^2\right )}{2 a}-\frac {\arccos (a x)^4}{8 a^3}-\frac {x \sqrt {1-a^2 x^2} \arccos (a x)^3}{2 a^2}\) |
\(\Big \downarrow \) 5211 |
\(\displaystyle -\frac {3 \left (a \left (\frac {\int \frac {\arccos (a x)}{\sqrt {1-a^2 x^2}}dx}{2 a^2}-\frac {\int xdx}{2 a}-\frac {x \sqrt {1-a^2 x^2} \arccos (a x)}{2 a^2}\right )+\frac {1}{2} x^2 \arccos (a x)^2\right )}{2 a}-\frac {\arccos (a x)^4}{8 a^3}-\frac {x \sqrt {1-a^2 x^2} \arccos (a x)^3}{2 a^2}\) |
\(\Big \downarrow \) 15 |
\(\displaystyle -\frac {3 \left (a \left (\frac {\int \frac {\arccos (a x)}{\sqrt {1-a^2 x^2}}dx}{2 a^2}-\frac {x \sqrt {1-a^2 x^2} \arccos (a x)}{2 a^2}-\frac {x^2}{4 a}\right )+\frac {1}{2} x^2 \arccos (a x)^2\right )}{2 a}-\frac {\arccos (a x)^4}{8 a^3}-\frac {x \sqrt {1-a^2 x^2} \arccos (a x)^3}{2 a^2}\) |
\(\Big \downarrow \) 5153 |
\(\displaystyle -\frac {\arccos (a x)^4}{8 a^3}-\frac {x \sqrt {1-a^2 x^2} \arccos (a x)^3}{2 a^2}-\frac {3 \left (a \left (-\frac {\arccos (a x)^2}{4 a^3}-\frac {x \sqrt {1-a^2 x^2} \arccos (a x)}{2 a^2}-\frac {x^2}{4 a}\right )+\frac {1}{2} x^2 \arccos (a x)^2\right )}{2 a}\) |
Input:
Int[(x^2*ArcCos[a*x]^3)/Sqrt[1 - a^2*x^2],x]
Output:
-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)
Int[(a_.)*(x_)^(m_.), x_Symbol] :> Simp[a*(x^(m + 1)/(m + 1)), x] /; FreeQ[ {a, m}, x] && NeQ[m, -1]
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]
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]
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]
Time = 0.44 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.80
method | result | size |
default | \(-\frac {4 \arccos \left (a x \right )^{3} \sqrt {-a^{2} x^{2}+1}\, a x +6 a^{2} x^{2} \arccos \left (a x \right )^{2}+\arccos \left (a x \right )^{4}-6 \arccos \left (a x \right ) \sqrt {-a^{2} x^{2}+1}\, a x -3 a^{2} x^{2}-3 \arccos \left (a x \right )^{2}+3}{8 a^{3}}\) | \(86\) |
Input:
int(x^2*arccos(a*x)^3/(-a^2*x^2+1)^(1/2),x,method=_RETURNVERBOSE)
Output:
-1/8*(4*arccos(a*x)^3*(-a^2*x^2+1)^(1/2)*a*x+6*a^2*x^2*arccos(a*x)^2+arcco s(a*x)^4-6*arccos(a*x)*(-a^2*x^2+1)^(1/2)*a*x-3*a^2*x^2-3*arccos(a*x)^2+3) /a^3
Time = 0.10 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.68 \[ \int \frac {x^2 \arccos (a x)^3}{\sqrt {1-a^2 x^2}} \, dx=\frac {3 \, a^{2} x^{2} - \arccos \left (a x\right )^{4} - 3 \, {\left (2 \, a^{2} x^{2} - 1\right )} \arccos \left (a x\right )^{2} - 2 \, {\left (2 \, a x \arccos \left (a x\right )^{3} - 3 \, a x \arccos \left (a x\right )\right )} \sqrt {-a^{2} x^{2} + 1}}{8 \, a^{3}} \] Input:
integrate(x^2*arccos(a*x)^3/(-a^2*x^2+1)^(1/2),x, algorithm="fricas")
Output:
1/8*(3*a^2*x^2 - arccos(a*x)^4 - 3*(2*a^2*x^2 - 1)*arccos(a*x)^2 - 2*(2*a* x*arccos(a*x)^3 - 3*a*x*arccos(a*x))*sqrt(-a^2*x^2 + 1))/a^3
Time = 0.40 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.00 \[ \int \frac {x^2 \arccos (a x)^3}{\sqrt {1-a^2 x^2}} \, dx=\begin {cases} - \frac {3 x^{2} \operatorname {acos}^{2}{\left (a x \right )}}{4 a} + \frac {3 x^{2}}{8 a} - \frac {x \sqrt {- a^{2} x^{2} + 1} \operatorname {acos}^{3}{\left (a x \right )}}{2 a^{2}} + \frac {3 x \sqrt {- a^{2} x^{2} + 1} \operatorname {acos}{\left (a x \right )}}{4 a^{2}} - \frac {\operatorname {acos}^{4}{\left (a x \right )}}{8 a^{3}} + \frac {3 \operatorname {acos}^{2}{\left (a x \right )}}{8 a^{3}} & \text {for}\: a \neq 0 \\\frac {\pi ^{3} x^{3}}{24} & \text {otherwise} \end {cases} \] Input:
integrate(x**2*acos(a*x)**3/(-a**2*x**2+1)**(1/2),x)
Output:
Piecewise((-3*x**2*acos(a*x)**2/(4*a) + 3*x**2/(8*a) - x*sqrt(-a**2*x**2 + 1)*acos(a*x)**3/(2*a**2) + 3*x*sqrt(-a**2*x**2 + 1)*acos(a*x)/(4*a**2) - acos(a*x)**4/(8*a**3) + 3*acos(a*x)**2/(8*a**3), Ne(a, 0)), (pi**3*x**3/24 , True))
\[ \int \frac {x^2 \arccos (a x)^3}{\sqrt {1-a^2 x^2}} \, dx=\int { \frac {x^{2} \arccos \left (a x\right )^{3}}{\sqrt {-a^{2} x^{2} + 1}} \,d x } \] Input:
integrate(x^2*arccos(a*x)^3/(-a^2*x^2+1)^(1/2),x, algorithm="maxima")
Output:
integrate(x^2*arccos(a*x)^3/sqrt(-a^2*x^2 + 1), x)
Time = 0.17 (sec) , antiderivative size = 96, normalized size of antiderivative = 0.90 \[ \int \frac {x^2 \arccos (a x)^3}{\sqrt {1-a^2 x^2}} \, dx=-\frac {3 \, x^{2} \arccos \left (a x\right )^{2}}{4 \, a} - \frac {\sqrt {-a^{2} x^{2} + 1} x \arccos \left (a x\right )^{3}}{2 \, a^{2}} + \frac {3 \, x^{2}}{8 \, a} - \frac {\arccos \left (a x\right )^{4}}{8 \, a^{3}} + \frac {3 \, \sqrt {-a^{2} x^{2} + 1} x \arccos \left (a x\right )}{4 \, a^{2}} + \frac {3 \, \arccos \left (a x\right )^{2}}{8 \, a^{3}} - \frac {3}{16 \, a^{3}} \] Input:
integrate(x^2*arccos(a*x)^3/(-a^2*x^2+1)^(1/2),x, algorithm="giac")
Output:
-3/4*x^2*arccos(a*x)^2/a - 1/2*sqrt(-a^2*x^2 + 1)*x*arccos(a*x)^3/a^2 + 3/ 8*x^2/a - 1/8*arccos(a*x)^4/a^3 + 3/4*sqrt(-a^2*x^2 + 1)*x*arccos(a*x)/a^2 + 3/8*arccos(a*x)^2/a^3 - 3/16/a^3
Timed out. \[ \int \frac {x^2 \arccos (a x)^3}{\sqrt {1-a^2 x^2}} \, dx=\int \frac {x^2\,{\mathrm {acos}\left (a\,x\right )}^3}{\sqrt {1-a^2\,x^2}} \,d x \] Input:
int((x^2*acos(a*x)^3)/(1 - a^2*x^2)^(1/2),x)
Output:
int((x^2*acos(a*x)^3)/(1 - a^2*x^2)^(1/2), x)
\[ \int \frac {x^2 \arccos (a x)^3}{\sqrt {1-a^2 x^2}} \, dx=\int \frac {\mathit {acos} \left (a x \right )^{3} x^{2}}{\sqrt {-a^{2} x^{2}+1}}d x \] Input:
int(x^2*acos(a*x)^3/(-a^2*x^2+1)^(1/2),x)
Output:
int((acos(a*x)**3*x**2)/sqrt( - a**2*x**2 + 1),x)