Integrand size = 24, antiderivative size = 89 \[ \int \frac {x^2 \arccos (a x)^2}{\sqrt {1-a^2 x^2}} \, dx=\frac {x \sqrt {1-a^2 x^2}}{4 a^2}-\frac {\arccos (a x)}{4 a^3}+\frac {x^2 \arccos (a x)}{2 a}-\frac {x \sqrt {1-a^2 x^2} \arccos (a x)^2}{2 a^2}+\frac {\arccos (a x)^3}{6 a^3} \] Output:
1/4*x*(-a^2*x^2+1)^(1/2)/a^2-1/4*arccos(a*x)/a^3+1/2*x^2*arccos(a*x)/a-1/2 *x*(-a^2*x^2+1)^(1/2)*arccos(a*x)^2/a^2+1/6*arccos(a*x)^3/a^3
Time = 0.04 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.85 \[ \int \frac {x^2 \arccos (a x)^2}{\sqrt {1-a^2 x^2}} \, dx=-\frac {-3 a x \sqrt {1-a^2 x^2}+6 a^2 x^2 \arccos (a x)+6 a x \sqrt {1-a^2 x^2} \arccos (a x)^2+2 \arccos (a x)^3+3 \arcsin (a x)}{12 a^3} \] Input:
Integrate[(x^2*ArcCos[a*x]^2)/Sqrt[1 - a^2*x^2],x]
Output:
-1/12*(-3*a*x*Sqrt[1 - a^2*x^2] + 6*a^2*x^2*ArcCos[a*x] + 6*a*x*Sqrt[1 - a ^2*x^2]*ArcCos[a*x]^2 + 2*ArcCos[a*x]^3 + 3*ArcSin[a*x])/a^3
Time = 0.48 (sec) , antiderivative size = 98, normalized size of antiderivative = 1.10, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {5211, 5139, 262, 223, 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.
\(\displaystyle \int \frac {x^2 \arccos (a x)^2}{\sqrt {1-a^2 x^2}} \, dx\) |
\(\Big \downarrow \) 5211 |
\(\displaystyle \frac {\int \frac {\arccos (a x)^2}{\sqrt {1-a^2 x^2}}dx}{2 a^2}-\frac {\int x \arccos (a x)dx}{a}-\frac {x \sqrt {1-a^2 x^2} \arccos (a x)^2}{2 a^2}\) |
\(\Big \downarrow \) 5139 |
\(\displaystyle -\frac {\frac {1}{2} a \int \frac {x^2}{\sqrt {1-a^2 x^2}}dx+\frac {1}{2} x^2 \arccos (a x)}{a}+\frac {\int \frac {\arccos (a x)^2}{\sqrt {1-a^2 x^2}}dx}{2 a^2}-\frac {x \sqrt {1-a^2 x^2} \arccos (a x)^2}{2 a^2}\) |
\(\Big \downarrow \) 262 |
\(\displaystyle -\frac {\frac {1}{2} a \left (\frac {\int \frac {1}{\sqrt {1-a^2 x^2}}dx}{2 a^2}-\frac {x \sqrt {1-a^2 x^2}}{2 a^2}\right )+\frac {1}{2} x^2 \arccos (a x)}{a}+\frac {\int \frac {\arccos (a x)^2}{\sqrt {1-a^2 x^2}}dx}{2 a^2}-\frac {x \sqrt {1-a^2 x^2} \arccos (a x)^2}{2 a^2}\) |
\(\Big \downarrow \) 223 |
\(\displaystyle \frac {\int \frac {\arccos (a x)^2}{\sqrt {1-a^2 x^2}}dx}{2 a^2}-\frac {x \sqrt {1-a^2 x^2} \arccos (a x)^2}{2 a^2}-\frac {\frac {1}{2} a \left (\frac {\arcsin (a x)}{2 a^3}-\frac {x \sqrt {1-a^2 x^2}}{2 a^2}\right )+\frac {1}{2} x^2 \arccos (a x)}{a}\) |
\(\Big \downarrow \) 5153 |
\(\displaystyle -\frac {\arccos (a x)^3}{6 a^3}-\frac {x \sqrt {1-a^2 x^2} \arccos (a x)^2}{2 a^2}-\frac {\frac {1}{2} a \left (\frac {\arcsin (a x)}{2 a^3}-\frac {x \sqrt {1-a^2 x^2}}{2 a^2}\right )+\frac {1}{2} x^2 \arccos (a x)}{a}\) |
Input:
Int[(x^2*ArcCos[a*x]^2)/Sqrt[1 - a^2*x^2],x]
Output:
-1/2*(x*Sqrt[1 - a^2*x^2]*ArcCos[a*x]^2)/a^2 - ArcCos[a*x]^3/(6*a^3) - ((x ^2*ArcCos[a*x])/2 + (a*(-1/2*(x*Sqrt[1 - a^2*x^2])/a^2 + ArcSin[a*x]/(2*a^ 3)))/2)/a
Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSin[Rt[-b, 2]*(x/Sqrt [a])]/Rt[-b, 2], x] /; FreeQ[{a, b}, x] && GtQ[a, 0] && NegQ[b]
Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[c*(c*x) ^(m - 1)*((a + b*x^2)^(p + 1)/(b*(m + 2*p + 1))), x] - Simp[a*c^2*((m - 1)/ (b*(m + 2*p + 1))) Int[(c*x)^(m - 2)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b , c, p}, x] && GtQ[m, 2 - 1] && NeQ[m + 2*p + 1, 0] && IntBinomialQ[a, b, c , 2, m, p, x]
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 = 0.34 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.80
method | result | size |
default | \(-\frac {6 \arccos \left (a x \right )^{2} \sqrt {-a^{2} x^{2}+1}\, a x +6 a^{2} x^{2} \arccos \left (a x \right )+2 \arccos \left (a x \right )^{3}-3 \sqrt {-a^{2} x^{2}+1}\, a x -3 \arccos \left (a x \right )}{12 a^{3}}\) | \(71\) |
Input:
int(x^2*arccos(a*x)^2/(-a^2*x^2+1)^(1/2),x,method=_RETURNVERBOSE)
Output:
-1/12*(6*arccos(a*x)^2*(-a^2*x^2+1)^(1/2)*a*x+6*a^2*x^2*arccos(a*x)+2*arcc os(a*x)^3-3*(-a^2*x^2+1)^(1/2)*a*x-3*arccos(a*x))/a^3
Time = 0.10 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.66 \[ \int \frac {x^2 \arccos (a x)^2}{\sqrt {1-a^2 x^2}} \, dx=-\frac {2 \, \arccos \left (a x\right )^{3} + 3 \, {\left (2 \, a^{2} x^{2} - 1\right )} \arccos \left (a x\right ) + 3 \, \sqrt {-a^{2} x^{2} + 1} {\left (2 \, a x \arccos \left (a x\right )^{2} - a x\right )}}{12 \, a^{3}} \] Input:
integrate(x^2*arccos(a*x)^2/(-a^2*x^2+1)^(1/2),x, algorithm="fricas")
Output:
-1/12*(2*arccos(a*x)^3 + 3*(2*a^2*x^2 - 1)*arccos(a*x) + 3*sqrt(-a^2*x^2 + 1)*(2*a*x*arccos(a*x)^2 - a*x))/a^3
Time = 0.32 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.96 \[ \int \frac {x^2 \arccos (a x)^2}{\sqrt {1-a^2 x^2}} \, dx=\begin {cases} - \frac {x^{2} \operatorname {acos}{\left (a x \right )}}{2 a} - \frac {x \sqrt {- a^{2} x^{2} + 1} \operatorname {acos}^{2}{\left (a x \right )}}{2 a^{2}} + \frac {x \sqrt {- a^{2} x^{2} + 1}}{4 a^{2}} - \frac {\operatorname {acos}^{3}{\left (a x \right )}}{6 a^{3}} + \frac {\operatorname {acos}{\left (a x \right )}}{4 a^{3}} & \text {for}\: a \neq 0 \\\frac {\pi ^{2} x^{3}}{12} & \text {otherwise} \end {cases} \] Input:
integrate(x**2*acos(a*x)**2/(-a**2*x**2+1)**(1/2),x)
Output:
Piecewise((-x**2*acos(a*x)/(2*a) - x*sqrt(-a**2*x**2 + 1)*acos(a*x)**2/(2* a**2) + x*sqrt(-a**2*x**2 + 1)/(4*a**2) - acos(a*x)**3/(6*a**3) + acos(a*x )/(4*a**3), Ne(a, 0)), (pi**2*x**3/12, True))
\[ \int \frac {x^2 \arccos (a x)^2}{\sqrt {1-a^2 x^2}} \, dx=\int { \frac {x^{2} \arccos \left (a x\right )^{2}}{\sqrt {-a^{2} x^{2} + 1}} \,d x } \] Input:
integrate(x^2*arccos(a*x)^2/(-a^2*x^2+1)^(1/2),x, algorithm="maxima")
Output:
integrate(x^2*arccos(a*x)^2/sqrt(-a^2*x^2 + 1), x)
Time = 0.15 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.84 \[ \int \frac {x^2 \arccos (a x)^2}{\sqrt {1-a^2 x^2}} \, dx=-\frac {x^{2} \arccos \left (a x\right )}{2 \, a} - \frac {\sqrt {-a^{2} x^{2} + 1} x \arccos \left (a x\right )^{2}}{2 \, a^{2}} - \frac {\arccos \left (a x\right )^{3}}{6 \, a^{3}} + \frac {\sqrt {-a^{2} x^{2} + 1} x}{4 \, a^{2}} + \frac {\arccos \left (a x\right )}{4 \, a^{3}} \] Input:
integrate(x^2*arccos(a*x)^2/(-a^2*x^2+1)^(1/2),x, algorithm="giac")
Output:
-1/2*x^2*arccos(a*x)/a - 1/2*sqrt(-a^2*x^2 + 1)*x*arccos(a*x)^2/a^2 - 1/6* arccos(a*x)^3/a^3 + 1/4*sqrt(-a^2*x^2 + 1)*x/a^2 + 1/4*arccos(a*x)/a^3
Timed out. \[ \int \frac {x^2 \arccos (a x)^2}{\sqrt {1-a^2 x^2}} \, dx=\int \frac {x^2\,{\mathrm {acos}\left (a\,x\right )}^2}{\sqrt {1-a^2\,x^2}} \,d x \] Input:
int((x^2*acos(a*x)^2)/(1 - a^2*x^2)^(1/2),x)
Output:
int((x^2*acos(a*x)^2)/(1 - a^2*x^2)^(1/2), x)
\[ \int \frac {x^2 \arccos (a x)^2}{\sqrt {1-a^2 x^2}} \, dx=\int \frac {\mathit {acos} \left (a x \right )^{2} x^{2}}{\sqrt {-a^{2} x^{2}+1}}d x \] Input:
int(x^2*acos(a*x)^2/(-a^2*x^2+1)^(1/2),x)
Output:
int((acos(a*x)**2*x**2)/sqrt( - a**2*x**2 + 1),x)