Integrand size = 24, antiderivative size = 157 \[ \int \frac {x^3 \arcsin (a x)^3}{\sqrt {1-a^2 x^2}} \, dx=-\frac {40 x}{9 a^3}-\frac {2 x^3}{27 a}+\frac {40 \sqrt {1-a^2 x^2} \arcsin (a x)}{9 a^4}+\frac {2 x^2 \sqrt {1-a^2 x^2} \arcsin (a x)}{9 a^2}+\frac {2 x \arcsin (a x)^2}{a^3}+\frac {x^3 \arcsin (a x)^2}{3 a}-\frac {2 \sqrt {1-a^2 x^2} \arcsin (a x)^3}{3 a^4}-\frac {x^2 \sqrt {1-a^2 x^2} \arcsin (a x)^3}{3 a^2} \]
-40/9*x/a^3-2/27*x^3/a+2*x*arcsin(a*x)^2/a^3+1/3*x^3*arcsin(a*x)^2/a+40/9* arcsin(a*x)*(-a^2*x^2+1)^(1/2)/a^4+2/9*x^2*arcsin(a*x)*(-a^2*x^2+1)^(1/2)/ a^2-2/3*arcsin(a*x)^3*(-a^2*x^2+1)^(1/2)/a^4-1/3*x^2*arcsin(a*x)^3*(-a^2*x ^2+1)^(1/2)/a^2
Time = 0.04 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.64 \[ \int \frac {x^3 \arcsin (a x)^3}{\sqrt {1-a^2 x^2}} \, dx=\frac {-2 a x \left (60+a^2 x^2\right )+6 \sqrt {1-a^2 x^2} \left (20+a^2 x^2\right ) \arcsin (a x)+9 a x \left (6+a^2 x^2\right ) \arcsin (a x)^2-9 \sqrt {1-a^2 x^2} \left (2+a^2 x^2\right ) \arcsin (a x)^3}{27 a^4} \]
(-2*a*x*(60 + a^2*x^2) + 6*Sqrt[1 - a^2*x^2]*(20 + a^2*x^2)*ArcSin[a*x] + 9*a*x*(6 + a^2*x^2)*ArcSin[a*x]^2 - 9*Sqrt[1 - a^2*x^2]*(2 + a^2*x^2)*ArcS in[a*x]^3)/(27*a^4)
Time = 1.11 (sec) , antiderivative size = 208, normalized size of antiderivative = 1.32, number of steps used = 10, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {5210, 5138, 5182, 5130, 5182, 24, 5210, 15, 5182, 24}
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.
| \(\displaystyle \int \frac {x^3 \arcsin (a x)^3}{\sqrt {1-a^2 x^2}} \, dx\) |
| \(\Big \downarrow \) 5210 |
| \(\displaystyle \frac {2 \int \frac {x \arcsin (a x)^3}{\sqrt {1-a^2 x^2}}dx}{3 a^2}+\frac {\int x^2 \arcsin (a x)^2dx}{a}-\frac {x^2 \sqrt {1-a^2 x^2} \arcsin (a x)^3}{3 a^2}\) |
| \(\Big \downarrow \) 5138 |
| \(\displaystyle \frac {2 \int \frac {x \arcsin (a x)^3}{\sqrt {1-a^2 x^2}}dx}{3 a^2}+\frac {\frac {1}{3} x^3 \arcsin (a x)^2-\frac {2}{3} a \int \frac {x^3 \arcsin (a x)}{\sqrt {1-a^2 x^2}}dx}{a}-\frac {x^2 \sqrt {1-a^2 x^2} \arcsin (a x)^3}{3 a^2}\) |
| \(\Big \downarrow \) 5182 |
| \(\displaystyle \frac {2 \left (\frac {3 \int \arcsin (a x)^2dx}{a}-\frac {\sqrt {1-a^2 x^2} \arcsin (a x)^3}{a^2}\right )}{3 a^2}+\frac {\frac {1}{3} x^3 \arcsin (a x)^2-\frac {2}{3} a \int \frac {x^3 \arcsin (a x)}{\sqrt {1-a^2 x^2}}dx}{a}-\frac {x^2 \sqrt {1-a^2 x^2} \arcsin (a x)^3}{3 a^2}\) |
| \(\Big \downarrow \) 5130 |
| \(\displaystyle \frac {2 \left (\frac {3 \left (x \arcsin (a x)^2-2 a \int \frac {x \arcsin (a x)}{\sqrt {1-a^2 x^2}}dx\right )}{a}-\frac {\sqrt {1-a^2 x^2} \arcsin (a x)^3}{a^2}\right )}{3 a^2}+\frac {\frac {1}{3} x^3 \arcsin (a x)^2-\frac {2}{3} a \int \frac {x^3 \arcsin (a x)}{\sqrt {1-a^2 x^2}}dx}{a}-\frac {x^2 \sqrt {1-a^2 x^2} \arcsin (a x)^3}{3 a^2}\) |
| \(\Big \downarrow \) 5182 |
| \(\displaystyle \frac {2 \left (\frac {3 \left (x \arcsin (a x)^2-2 a \left (\frac {\int 1dx}{a}-\frac {\sqrt {1-a^2 x^2} \arcsin (a x)}{a^2}\right )\right )}{a}-\frac {\sqrt {1-a^2 x^2} \arcsin (a x)^3}{a^2}\right )}{3 a^2}+\frac {\frac {1}{3} x^3 \arcsin (a x)^2-\frac {2}{3} a \int \frac {x^3 \arcsin (a x)}{\sqrt {1-a^2 x^2}}dx}{a}-\frac {x^2 \sqrt {1-a^2 x^2} \arcsin (a x)^3}{3 a^2}\) |
| \(\Big \downarrow \) 24 |
| \(\displaystyle \frac {\frac {1}{3} x^3 \arcsin (a x)^2-\frac {2}{3} a \int \frac {x^3 \arcsin (a x)}{\sqrt {1-a^2 x^2}}dx}{a}-\frac {x^2 \sqrt {1-a^2 x^2} \arcsin (a x)^3}{3 a^2}+\frac {2 \left (\frac {3 \left (x \arcsin (a x)^2-2 a \left (\frac {x}{a}-\frac {\sqrt {1-a^2 x^2} \arcsin (a x)}{a^2}\right )\right )}{a}-\frac {\sqrt {1-a^2 x^2} \arcsin (a x)^3}{a^2}\right )}{3 a^2}\) |
| \(\Big \downarrow \) 5210 |
| \(\displaystyle \frac {\frac {1}{3} x^3 \arcsin (a x)^2-\frac {2}{3} a \left (\frac {2 \int \frac {x \arcsin (a x)}{\sqrt {1-a^2 x^2}}dx}{3 a^2}+\frac {\int x^2dx}{3 a}-\frac {x^2 \sqrt {1-a^2 x^2} \arcsin (a x)}{3 a^2}\right )}{a}-\frac {x^2 \sqrt {1-a^2 x^2} \arcsin (a x)^3}{3 a^2}+\frac {2 \left (\frac {3 \left (x \arcsin (a x)^2-2 a \left (\frac {x}{a}-\frac {\sqrt {1-a^2 x^2} \arcsin (a x)}{a^2}\right )\right )}{a}-\frac {\sqrt {1-a^2 x^2} \arcsin (a x)^3}{a^2}\right )}{3 a^2}\) |
| \(\Big \downarrow \) 15 |
| \(\displaystyle \frac {\frac {1}{3} x^3 \arcsin (a x)^2-\frac {2}{3} a \left (\frac {2 \int \frac {x \arcsin (a x)}{\sqrt {1-a^2 x^2}}dx}{3 a^2}-\frac {x^2 \sqrt {1-a^2 x^2} \arcsin (a x)}{3 a^2}+\frac {x^3}{9 a}\right )}{a}-\frac {x^2 \sqrt {1-a^2 x^2} \arcsin (a x)^3}{3 a^2}+\frac {2 \left (\frac {3 \left (x \arcsin (a x)^2-2 a \left (\frac {x}{a}-\frac {\sqrt {1-a^2 x^2} \arcsin (a x)}{a^2}\right )\right )}{a}-\frac {\sqrt {1-a^2 x^2} \arcsin (a x)^3}{a^2}\right )}{3 a^2}\) |
| \(\Big \downarrow \) 5182 |
| \(\displaystyle \frac {\frac {1}{3} x^3 \arcsin (a x)^2-\frac {2}{3} a \left (\frac {2 \left (\frac {\int 1dx}{a}-\frac {\sqrt {1-a^2 x^2} \arcsin (a x)}{a^2}\right )}{3 a^2}-\frac {x^2 \sqrt {1-a^2 x^2} \arcsin (a x)}{3 a^2}+\frac {x^3}{9 a}\right )}{a}-\frac {x^2 \sqrt {1-a^2 x^2} \arcsin (a x)^3}{3 a^2}+\frac {2 \left (\frac {3 \left (x \arcsin (a x)^2-2 a \left (\frac {x}{a}-\frac {\sqrt {1-a^2 x^2} \arcsin (a x)}{a^2}\right )\right )}{a}-\frac {\sqrt {1-a^2 x^2} \arcsin (a x)^3}{a^2}\right )}{3 a^2}\) |
| \(\Big \downarrow \) 24 |
| \(\displaystyle -\frac {x^2 \sqrt {1-a^2 x^2} \arcsin (a x)^3}{3 a^2}+\frac {2 \left (\frac {3 \left (x \arcsin (a x)^2-2 a \left (\frac {x}{a}-\frac {\sqrt {1-a^2 x^2} \arcsin (a x)}{a^2}\right )\right )}{a}-\frac {\sqrt {1-a^2 x^2} \arcsin (a x)^3}{a^2}\right )}{3 a^2}+\frac {\frac {1}{3} x^3 \arcsin (a x)^2-\frac {2}{3} a \left (-\frac {x^2 \sqrt {1-a^2 x^2} \arcsin (a x)}{3 a^2}+\frac {2 \left (\frac {x}{a}-\frac {\sqrt {1-a^2 x^2} \arcsin (a x)}{a^2}\right )}{3 a^2}+\frac {x^3}{9 a}\right )}{a}\) |
-1/3*(x^2*Sqrt[1 - a^2*x^2]*ArcSin[a*x]^3)/a^2 + ((x^3*ArcSin[a*x]^2)/3 - (2*a*(x^3/(9*a) - (x^2*Sqrt[1 - a^2*x^2]*ArcSin[a*x])/(3*a^2) + (2*(x/a - (Sqrt[1 - a^2*x^2]*ArcSin[a*x])/a^2))/(3*a^2)))/3)/a + (2*(-((Sqrt[1 - a^2 *x^2]*ArcSin[a*x]^3)/a^2) + (3*(x*ArcSin[a*x]^2 - 2*a*(x/a - (Sqrt[1 - a^2 *x^2]*ArcSin[a*x])/a^2)))/a))/(3*a^2)
3.4.4.3.1 Defintions of rubi rules used
Int[(a_.)*(x_)^(m_.), x_Symbol] :> Simp[a*(x^(m + 1)/(m + 1)), x] /; FreeQ[ {a, m}, x] && NeQ[m, -1]
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 = 0.20 (sec) , antiderivative size = 180, normalized size of antiderivative = 1.15
| method | result | size |
| default | \(-\frac {\left (9 \arcsin \left (a x \right )^{3} a^{4} x^{4}+9 \arcsin \left (a x \right )^{3} a^{2} x^{2}+9 \arcsin \left (a x \right )^{2} \sqrt {-a^{2} x^{2}+1}\, a^{3} x^{3}-6 a^{4} x^{4} \arcsin \left (a x \right )-114 a^{2} x^{2} \arcsin \left (a x \right )-2 a^{3} x^{3} \sqrt {-a^{2} x^{2}+1}-18 \arcsin \left (a x \right )^{3}+54 \arcsin \left (a x \right )^{2} \sqrt {-a^{2} x^{2}+1}\, a x +120 \arcsin \left (a x \right )-120 a x \sqrt {-a^{2} x^{2}+1}\right ) \sqrt {-a^{2} x^{2}+1}}{27 a^{4} \left (a^{2} x^{2}-1\right )}\) | \(180\) |
-1/27/a^4*(9*arcsin(a*x)^3*a^4*x^4+9*arcsin(a*x)^3*a^2*x^2+9*arcsin(a*x)^2 *(-a^2*x^2+1)^(1/2)*a^3*x^3-6*a^4*x^4*arcsin(a*x)-114*a^2*x^2*arcsin(a*x)- 2*a^3*x^3*(-a^2*x^2+1)^(1/2)-18*arcsin(a*x)^3+54*arcsin(a*x)^2*(-a^2*x^2+1 )^(1/2)*a*x+120*arcsin(a*x)-120*a*x*(-a^2*x^2+1)^(1/2))*(-a^2*x^2+1)^(1/2) /(a^2*x^2-1)
Time = 0.25 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.54 \[ \int \frac {x^3 \arcsin (a x)^3}{\sqrt {1-a^2 x^2}} \, dx=-\frac {2 \, a^{3} x^{3} - 9 \, {\left (a^{3} x^{3} + 6 \, a x\right )} \arcsin \left (a x\right )^{2} + 120 \, a x + 3 \, \sqrt {-a^{2} x^{2} + 1} {\left (3 \, {\left (a^{2} x^{2} + 2\right )} \arcsin \left (a x\right )^{3} - 2 \, {\left (a^{2} x^{2} + 20\right )} \arcsin \left (a x\right )\right )}}{27 \, a^{4}} \]
-1/27*(2*a^3*x^3 - 9*(a^3*x^3 + 6*a*x)*arcsin(a*x)^2 + 120*a*x + 3*sqrt(-a ^2*x^2 + 1)*(3*(a^2*x^2 + 2)*arcsin(a*x)^3 - 2*(a^2*x^2 + 20)*arcsin(a*x)) )/a^4
Time = 0.55 (sec) , antiderivative size = 148, normalized size of antiderivative = 0.94 \[ \int \frac {x^3 \arcsin (a x)^3}{\sqrt {1-a^2 x^2}} \, dx=\begin {cases} \frac {x^{3} \operatorname {asin}^{2}{\left (a x \right )}}{3 a} - \frac {2 x^{3}}{27 a} - \frac {x^{2} \sqrt {- a^{2} x^{2} + 1} \operatorname {asin}^{3}{\left (a x \right )}}{3 a^{2}} + \frac {2 x^{2} \sqrt {- a^{2} x^{2} + 1} \operatorname {asin}{\left (a x \right )}}{9 a^{2}} + \frac {2 x \operatorname {asin}^{2}{\left (a x \right )}}{a^{3}} - \frac {40 x}{9 a^{3}} - \frac {2 \sqrt {- a^{2} x^{2} + 1} \operatorname {asin}^{3}{\left (a x \right )}}{3 a^{4}} + \frac {40 \sqrt {- a^{2} x^{2} + 1} \operatorname {asin}{\left (a x \right )}}{9 a^{4}} & \text {for}\: a \neq 0 \\0 & \text {otherwise} \end {cases} \]
Piecewise((x**3*asin(a*x)**2/(3*a) - 2*x**3/(27*a) - x**2*sqrt(-a**2*x**2 + 1)*asin(a*x)**3/(3*a**2) + 2*x**2*sqrt(-a**2*x**2 + 1)*asin(a*x)/(9*a**2 ) + 2*x*asin(a*x)**2/a**3 - 40*x/(9*a**3) - 2*sqrt(-a**2*x**2 + 1)*asin(a* x)**3/(3*a**4) + 40*sqrt(-a**2*x**2 + 1)*asin(a*x)/(9*a**4), Ne(a, 0)), (0 , True))
Time = 0.28 (sec) , antiderivative size = 131, normalized size of antiderivative = 0.83 \[ \int \frac {x^3 \arcsin (a x)^3}{\sqrt {1-a^2 x^2}} \, dx=-\frac {1}{3} \, {\left (\frac {\sqrt {-a^{2} x^{2} + 1} x^{2}}{a^{2}} + \frac {2 \, \sqrt {-a^{2} x^{2} + 1}}{a^{4}}\right )} \arcsin \left (a x\right )^{3} + \frac {2}{27} \, a {\left (\frac {3 \, {\left (\sqrt {-a^{2} x^{2} + 1} x^{2} + \frac {20 \, \sqrt {-a^{2} x^{2} + 1}}{a^{2}}\right )} \arcsin \left (a x\right )}{a^{3}} - \frac {a^{2} x^{3} + 60 \, x}{a^{4}}\right )} + \frac {{\left (a^{2} x^{3} + 6 \, x\right )} \arcsin \left (a x\right )^{2}}{3 \, a^{3}} \]
-1/3*(sqrt(-a^2*x^2 + 1)*x^2/a^2 + 2*sqrt(-a^2*x^2 + 1)/a^4)*arcsin(a*x)^3 + 2/27*a*(3*(sqrt(-a^2*x^2 + 1)*x^2 + 20*sqrt(-a^2*x^2 + 1)/a^2)*arcsin(a *x)/a^3 - (a^2*x^3 + 60*x)/a^4) + 1/3*(a^2*x^3 + 6*x)*arcsin(a*x)^2/a^3
Exception generated. \[ \int \frac {x^3 \arcsin (a x)^3}{\sqrt {1-a^2 x^2}} \, dx=\text {Exception raised: TypeError} \]
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. \[ \int \frac {x^3 \arcsin (a x)^3}{\sqrt {1-a^2 x^2}} \, dx=\int \frac {x^3\,{\mathrm {asin}\left (a\,x\right )}^3}{\sqrt {1-a^2\,x^2}} \,d x \]