Integrand size = 24, antiderivative size = 126 \[ \int \frac {x^3 \arcsin (a x)^2}{\sqrt {1-a^2 x^2}} \, dx=\frac {14 \sqrt {1-a^2 x^2}}{9 a^4}-\frac {2 \left (1-a^2 x^2\right )^{3/2}}{27 a^4}+\frac {4 x \arcsin (a x)}{3 a^3}+\frac {2 x^3 \arcsin (a x)}{9 a}-\frac {2 \sqrt {1-a^2 x^2} \arcsin (a x)^2}{3 a^4}-\frac {x^2 \sqrt {1-a^2 x^2} \arcsin (a x)^2}{3 a^2} \]
-2/27*(-a^2*x^2+1)^(3/2)/a^4+4/3*x*arcsin(a*x)/a^3+2/9*x^3*arcsin(a*x)/a+1 4/9*(-a^2*x^2+1)^(1/2)/a^4-2/3*arcsin(a*x)^2*(-a^2*x^2+1)^(1/2)/a^4-1/3*x^ 2*arcsin(a*x)^2*(-a^2*x^2+1)^(1/2)/a^2
Time = 0.04 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.64 \[ \int \frac {x^3 \arcsin (a x)^2}{\sqrt {1-a^2 x^2}} \, dx=\frac {2 \sqrt {1-a^2 x^2} \left (20+a^2 x^2\right )+6 a x \left (6+a^2 x^2\right ) \arcsin (a x)-9 \sqrt {1-a^2 x^2} \left (2+a^2 x^2\right ) \arcsin (a x)^2}{27 a^4} \]
(2*Sqrt[1 - a^2*x^2]*(20 + a^2*x^2) + 6*a*x*(6 + a^2*x^2)*ArcSin[a*x] - 9* Sqrt[1 - a^2*x^2]*(2 + a^2*x^2)*ArcSin[a*x]^2)/(27*a^4)
Time = 0.64 (sec) , antiderivative size = 159, normalized size of antiderivative = 1.26, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {5210, 5138, 243, 53, 2009, 5182, 5130, 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.
\(\displaystyle \int \frac {x^3 \arcsin (a x)^2}{\sqrt {1-a^2 x^2}} \, dx\) |
\(\Big \downarrow \) 5210 |
\(\displaystyle \frac {2 \int \frac {x \arcsin (a x)^2}{\sqrt {1-a^2 x^2}}dx}{3 a^2}+\frac {2 \int x^2 \arcsin (a x)dx}{3 a}-\frac {x^2 \sqrt {1-a^2 x^2} \arcsin (a x)^2}{3 a^2}\) |
\(\Big \downarrow \) 5138 |
\(\displaystyle \frac {2 \int \frac {x \arcsin (a x)^2}{\sqrt {1-a^2 x^2}}dx}{3 a^2}+\frac {2 \left (\frac {1}{3} x^3 \arcsin (a x)-\frac {1}{3} a \int \frac {x^3}{\sqrt {1-a^2 x^2}}dx\right )}{3 a}-\frac {x^2 \sqrt {1-a^2 x^2} \arcsin (a x)^2}{3 a^2}\) |
\(\Big \downarrow \) 243 |
\(\displaystyle \frac {2 \int \frac {x \arcsin (a x)^2}{\sqrt {1-a^2 x^2}}dx}{3 a^2}+\frac {2 \left (\frac {1}{3} x^3 \arcsin (a x)-\frac {1}{6} a \int \frac {x^2}{\sqrt {1-a^2 x^2}}dx^2\right )}{3 a}-\frac {x^2 \sqrt {1-a^2 x^2} \arcsin (a x)^2}{3 a^2}\) |
\(\Big \downarrow \) 53 |
\(\displaystyle \frac {2 \int \frac {x \arcsin (a x)^2}{\sqrt {1-a^2 x^2}}dx}{3 a^2}+\frac {2 \left (\frac {1}{3} x^3 \arcsin (a x)-\frac {1}{6} a \int \left (\frac {1}{a^2 \sqrt {1-a^2 x^2}}-\frac {\sqrt {1-a^2 x^2}}{a^2}\right )dx^2\right )}{3 a}-\frac {x^2 \sqrt {1-a^2 x^2} \arcsin (a x)^2}{3 a^2}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {2 \int \frac {x \arcsin (a x)^2}{\sqrt {1-a^2 x^2}}dx}{3 a^2}-\frac {x^2 \sqrt {1-a^2 x^2} \arcsin (a x)^2}{3 a^2}+\frac {2 \left (\frac {1}{3} x^3 \arcsin (a x)-\frac {1}{6} a \left (\frac {2 \left (1-a^2 x^2\right )^{3/2}}{3 a^4}-\frac {2 \sqrt {1-a^2 x^2}}{a^4}\right )\right )}{3 a}\) |
\(\Big \downarrow \) 5182 |
\(\displaystyle \frac {2 \left (\frac {2 \int \arcsin (a x)dx}{a}-\frac {\sqrt {1-a^2 x^2} \arcsin (a x)^2}{a^2}\right )}{3 a^2}-\frac {x^2 \sqrt {1-a^2 x^2} \arcsin (a x)^2}{3 a^2}+\frac {2 \left (\frac {1}{3} x^3 \arcsin (a x)-\frac {1}{6} a \left (\frac {2 \left (1-a^2 x^2\right )^{3/2}}{3 a^4}-\frac {2 \sqrt {1-a^2 x^2}}{a^4}\right )\right )}{3 a}\) |
\(\Big \downarrow \) 5130 |
\(\displaystyle \frac {2 \left (\frac {2 \left (x \arcsin (a x)-a \int \frac {x}{\sqrt {1-a^2 x^2}}dx\right )}{a}-\frac {\sqrt {1-a^2 x^2} \arcsin (a x)^2}{a^2}\right )}{3 a^2}-\frac {x^2 \sqrt {1-a^2 x^2} \arcsin (a x)^2}{3 a^2}+\frac {2 \left (\frac {1}{3} x^3 \arcsin (a x)-\frac {1}{6} a \left (\frac {2 \left (1-a^2 x^2\right )^{3/2}}{3 a^4}-\frac {2 \sqrt {1-a^2 x^2}}{a^4}\right )\right )}{3 a}\) |
\(\Big \downarrow \) 241 |
\(\displaystyle -\frac {x^2 \sqrt {1-a^2 x^2} \arcsin (a x)^2}{3 a^2}+\frac {2 \left (\frac {2 \left (\frac {\sqrt {1-a^2 x^2}}{a}+x \arcsin (a x)\right )}{a}-\frac {\sqrt {1-a^2 x^2} \arcsin (a x)^2}{a^2}\right )}{3 a^2}+\frac {2 \left (\frac {1}{3} x^3 \arcsin (a x)-\frac {1}{6} a \left (\frac {2 \left (1-a^2 x^2\right )^{3/2}}{3 a^4}-\frac {2 \sqrt {1-a^2 x^2}}{a^4}\right )\right )}{3 a}\) |
-1/3*(x^2*Sqrt[1 - a^2*x^2]*ArcSin[a*x]^2)/a^2 + (2*(-1/6*(a*((-2*Sqrt[1 - a^2*x^2])/a^4 + (2*(1 - a^2*x^2)^(3/2))/(3*a^4))) + (x^3*ArcSin[a*x])/3)) /(3*a) + (2*(-((Sqrt[1 - a^2*x^2]*ArcSin[a*x]^2)/a^2) + (2*(Sqrt[1 - a^2*x ^2]/a + x*ArcSin[a*x]))/a))/(3*a^2)
3.3.65.3.1 Defintions of rubi rules used
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_), x_Symbol] :> Simp[(a + b*x^2)^(p + 1)/ (2*b*(p + 1)), x] /; FreeQ[{a, b, p}, x] && NeQ[p, -1]
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]
Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.), x_Symbol] :> Simp[x*(a + b*Ar cSin[c*x])^n, x] - Simp[b*c*n Int[x*((a + b*ArcSin[c*x])^(n - 1)/Sqrt[1 - c^2*x^2]), x], x] /; FreeQ[{a, b, c}, x] && GtQ[n, 0]
Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[(d*x)^(m + 1)*((a + b*ArcSin[c*x])^n/(d*(m + 1))), x] - Simp[b*c*(n /(d*(m + 1))) Int[(d*x)^(m + 1)*((a + b*ArcSin[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_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*(x_)*((d_) + (e_.)*(x_)^2)^(p_ .), x_Symbol] :> Simp[(d + e*x^2)^(p + 1)*((a + b*ArcSin[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*ArcSin[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]
Int[((a_.) + ArcSin[(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*ArcSin[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*ArcSin[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*ArcSin[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 = 0.15 (sec) , antiderivative size = 127, normalized size of antiderivative = 1.01
method | result | size |
default | \(-\frac {\left (9 a^{4} x^{4} \arcsin \left (a x \right )^{2}+9 \arcsin \left (a x \right )^{2} a^{2} x^{2}+6 \arcsin \left (a x \right ) \sqrt {-a^{2} x^{2}+1}\, a^{3} x^{3}-2 a^{4} x^{4}-38 a^{2} x^{2}-18 \arcsin \left (a x \right )^{2}+36 \arcsin \left (a x \right ) \sqrt {-a^{2} x^{2}+1}\, a x +40\right ) \sqrt {-a^{2} x^{2}+1}}{27 a^{4} \left (a^{2} x^{2}-1\right )}\) | \(127\) |
-1/27/a^4*(9*a^4*x^4*arcsin(a*x)^2+9*arcsin(a*x)^2*a^2*x^2+6*arcsin(a*x)*( -a^2*x^2+1)^(1/2)*a^3*x^3-2*a^4*x^4-38*a^2*x^2-18*arcsin(a*x)^2+36*arcsin( a*x)*(-a^2*x^2+1)^(1/2)*a*x+40)*(-a^2*x^2+1)^(1/2)/(a^2*x^2-1)
Time = 0.27 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.51 \[ \int \frac {x^3 \arcsin (a x)^2}{\sqrt {1-a^2 x^2}} \, dx=\frac {6 \, {\left (a^{3} x^{3} + 6 \, a x\right )} \arcsin \left (a x\right ) + {\left (2 \, a^{2} x^{2} - 9 \, {\left (a^{2} x^{2} + 2\right )} \arcsin \left (a x\right )^{2} + 40\right )} \sqrt {-a^{2} x^{2} + 1}}{27 \, a^{4}} \]
1/27*(6*(a^3*x^3 + 6*a*x)*arcsin(a*x) + (2*a^2*x^2 - 9*(a^2*x^2 + 2)*arcsi n(a*x)^2 + 40)*sqrt(-a^2*x^2 + 1))/a^4
Time = 0.40 (sec) , antiderivative size = 121, normalized size of antiderivative = 0.96 \[ \int \frac {x^3 \arcsin (a x)^2}{\sqrt {1-a^2 x^2}} \, dx=\begin {cases} \frac {2 x^{3} \operatorname {asin}{\left (a x \right )}}{9 a} - \frac {x^{2} \sqrt {- a^{2} x^{2} + 1} \operatorname {asin}^{2}{\left (a x \right )}}{3 a^{2}} + \frac {2 x^{2} \sqrt {- a^{2} x^{2} + 1}}{27 a^{2}} + \frac {4 x \operatorname {asin}{\left (a x \right )}}{3 a^{3}} - \frac {2 \sqrt {- a^{2} x^{2} + 1} \operatorname {asin}^{2}{\left (a x \right )}}{3 a^{4}} + \frac {40 \sqrt {- a^{2} x^{2} + 1}}{27 a^{4}} & \text {for}\: a \neq 0 \\0 & \text {otherwise} \end {cases} \]
Piecewise((2*x**3*asin(a*x)/(9*a) - x**2*sqrt(-a**2*x**2 + 1)*asin(a*x)**2 /(3*a**2) + 2*x**2*sqrt(-a**2*x**2 + 1)/(27*a**2) + 4*x*asin(a*x)/(3*a**3) - 2*sqrt(-a**2*x**2 + 1)*asin(a*x)**2/(3*a**4) + 40*sqrt(-a**2*x**2 + 1)/ (27*a**4), Ne(a, 0)), (0, True))
Time = 0.29 (sec) , antiderivative size = 105, normalized size of antiderivative = 0.83 \[ \int \frac {x^3 \arcsin (a x)^2}{\sqrt {1-a^2 x^2}} \, dx=-\frac {1}{3} \, {\left (\frac {\sqrt {-a^{2} x^{2} + 1} x^{2}}{a^{2}} + \frac {2 \, \sqrt {-a^{2} x^{2} + 1}}{a^{4}}\right )} \arcsin \left (a x\right )^{2} + \frac {2 \, {\left (\sqrt {-a^{2} x^{2} + 1} x^{2} + \frac {20 \, \sqrt {-a^{2} x^{2} + 1}}{a^{2}}\right )}}{27 \, a^{2}} + \frac {2 \, {\left (a^{2} x^{3} + 6 \, x\right )} \arcsin \left (a x\right )}{9 \, a^{3}} \]
-1/3*(sqrt(-a^2*x^2 + 1)*x^2/a^2 + 2*sqrt(-a^2*x^2 + 1)/a^4)*arcsin(a*x)^2 + 2/27*(sqrt(-a^2*x^2 + 1)*x^2 + 20*sqrt(-a^2*x^2 + 1)/a^2)/a^2 + 2/9*(a^ 2*x^3 + 6*x)*arcsin(a*x)/a^3
Exception generated. \[ \int \frac {x^3 \arcsin (a x)^2}{\sqrt {1-a^2 x^2}} \, dx=\text {Exception raised: TypeError} \]
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
Timed out. \[ \int \frac {x^3 \arcsin (a x)^2}{\sqrt {1-a^2 x^2}} \, dx=\int \frac {x^3\,{\mathrm {asin}\left (a\,x\right )}^2}{\sqrt {1-a^2\,x^2}} \,d x \]