Integrand size = 24, antiderivative size = 107 \[ \int \frac {x^2 \arcsin (a x)^3}{\sqrt {1-a^2 x^2}} \, dx=-\frac {3 x^2}{8 a}+\frac {3 x \sqrt {1-a^2 x^2} \arcsin (a x)}{4 a^2}-\frac {3 \arcsin (a x)^2}{8 a^3}+\frac {3 x^2 \arcsin (a x)^2}{4 a}-\frac {x \sqrt {1-a^2 x^2} \arcsin (a x)^3}{2 a^2}+\frac {\arcsin (a x)^4}{8 a^3} \]
-3/8*x^2/a-3/8*arcsin(a*x)^2/a^3+3/4*x^2*arcsin(a*x)^2/a+1/8*arcsin(a*x)^4 /a^3+3/4*x*arcsin(a*x)*(-a^2*x^2+1)^(1/2)/a^2-1/2*x*arcsin(a*x)^3*(-a^2*x^ 2+1)^(1/2)/a^2
Time = 0.02 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.79 \[ \int \frac {x^2 \arcsin (a x)^3}{\sqrt {1-a^2 x^2}} \, dx=\frac {-3 a^2 x^2+6 a x \sqrt {1-a^2 x^2} \arcsin (a x)+\left (-3+6 a^2 x^2\right ) \arcsin (a x)^2-4 a x \sqrt {1-a^2 x^2} \arcsin (a x)^3+\arcsin (a x)^4}{8 a^3} \]
(-3*a^2*x^2 + 6*a*x*Sqrt[1 - a^2*x^2]*ArcSin[a*x] + (-3 + 6*a^2*x^2)*ArcSi n[a*x]^2 - 4*a*x*Sqrt[1 - a^2*x^2]*ArcSin[a*x]^3 + ArcSin[a*x]^4)/(8*a^3)
Time = 0.69 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.08, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {5210, 5138, 5152, 5210, 15, 5152}
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.
\(\displaystyle \int \frac {x^2 \arcsin (a x)^3}{\sqrt {1-a^2 x^2}} \, dx\) |
\(\Big \downarrow \) 5210 |
\(\displaystyle \frac {\int \frac {\arcsin (a x)^3}{\sqrt {1-a^2 x^2}}dx}{2 a^2}+\frac {3 \int x \arcsin (a x)^2dx}{2 a}-\frac {x \sqrt {1-a^2 x^2} \arcsin (a x)^3}{2 a^2}\) |
\(\Big \downarrow \) 5138 |
\(\displaystyle \frac {3 \left (\frac {1}{2} x^2 \arcsin (a x)^2-a \int \frac {x^2 \arcsin (a x)}{\sqrt {1-a^2 x^2}}dx\right )}{2 a}+\frac {\int \frac {\arcsin (a x)^3}{\sqrt {1-a^2 x^2}}dx}{2 a^2}-\frac {x \sqrt {1-a^2 x^2} \arcsin (a x)^3}{2 a^2}\) |
\(\Big \downarrow \) 5152 |
\(\displaystyle \frac {3 \left (\frac {1}{2} x^2 \arcsin (a x)^2-a \int \frac {x^2 \arcsin (a x)}{\sqrt {1-a^2 x^2}}dx\right )}{2 a}+\frac {\arcsin (a x)^4}{8 a^3}-\frac {x \sqrt {1-a^2 x^2} \arcsin (a x)^3}{2 a^2}\) |
\(\Big \downarrow \) 5210 |
\(\displaystyle \frac {3 \left (\frac {1}{2} x^2 \arcsin (a x)^2-a \left (\frac {\int \frac {\arcsin (a x)}{\sqrt {1-a^2 x^2}}dx}{2 a^2}+\frac {\int xdx}{2 a}-\frac {x \sqrt {1-a^2 x^2} \arcsin (a x)}{2 a^2}\right )\right )}{2 a}+\frac {\arcsin (a x)^4}{8 a^3}-\frac {x \sqrt {1-a^2 x^2} \arcsin (a x)^3}{2 a^2}\) |
\(\Big \downarrow \) 15 |
\(\displaystyle \frac {3 \left (\frac {1}{2} x^2 \arcsin (a x)^2-a \left (\frac {\int \frac {\arcsin (a x)}{\sqrt {1-a^2 x^2}}dx}{2 a^2}-\frac {x \sqrt {1-a^2 x^2} \arcsin (a x)}{2 a^2}+\frac {x^2}{4 a}\right )\right )}{2 a}+\frac {\arcsin (a x)^4}{8 a^3}-\frac {x \sqrt {1-a^2 x^2} \arcsin (a x)^3}{2 a^2}\) |
\(\Big \downarrow \) 5152 |
\(\displaystyle \frac {\arcsin (a x)^4}{8 a^3}-\frac {x \sqrt {1-a^2 x^2} \arcsin (a x)^3}{2 a^2}+\frac {3 \left (\frac {1}{2} x^2 \arcsin (a x)^2-a \left (\frac {\arcsin (a x)^2}{4 a^3}-\frac {x \sqrt {1-a^2 x^2} \arcsin (a x)}{2 a^2}+\frac {x^2}{4 a}\right )\right )}{2 a}\) |
-1/2*(x*Sqrt[1 - a^2*x^2]*ArcSin[a*x]^3)/a^2 + ArcSin[a*x]^4/(8*a^3) + (3* ((x^2*ArcSin[a*x]^2)/2 - a*(x^2/(4*a) - (x*Sqrt[1 - a^2*x^2]*ArcSin[a*x])/ (2*a^2) + ArcSin[a*x]^2/(4*a^3))))/(2*a)
3.4.5.3.1 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_.)*((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_.)/Sqrt[(d_) + (e_.)*(x_)^2], x_S ymbol] :> Simp[(1/(b*c*(n + 1)))*Simp[Sqrt[1 - c^2*x^2]/Sqrt[d + e*x^2]]*(a + b*ArcSin[c*x])^(n + 1), x] /; FreeQ[{a, b, c, d, e, n}, x] && EqQ[c^2*d + e, 0] && NeQ[n, -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 = 0.12 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.79
method | result | size |
default | \(\frac {-4 \arcsin \left (a x \right )^{3} \sqrt {-a^{2} x^{2}+1}\, a x +6 \arcsin \left (a x \right )^{2} a^{2} x^{2}+\arcsin \left (a x \right )^{4}+6 \arcsin \left (a x \right ) \sqrt {-a^{2} x^{2}+1}\, a x -3 a^{2} x^{2}-3 \arcsin \left (a x \right )^{2}}{8 a^{3}}\) | \(85\) |
1/8*(-4*arcsin(a*x)^3*(-a^2*x^2+1)^(1/2)*a*x+6*arcsin(a*x)^2*a^2*x^2+arcsi n(a*x)^4+6*arcsin(a*x)*(-a^2*x^2+1)^(1/2)*a*x-3*a^2*x^2-3*arcsin(a*x)^2)/a ^3
Time = 0.25 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.68 \[ \int \frac {x^2 \arcsin (a x)^3}{\sqrt {1-a^2 x^2}} \, dx=-\frac {3 \, a^{2} x^{2} - \arcsin \left (a x\right )^{4} - 3 \, {\left (2 \, a^{2} x^{2} - 1\right )} \arcsin \left (a x\right )^{2} + 2 \, {\left (2 \, a x \arcsin \left (a x\right )^{3} - 3 \, a x \arcsin \left (a x\right )\right )} \sqrt {-a^{2} x^{2} + 1}}{8 \, a^{3}} \]
-1/8*(3*a^2*x^2 - arcsin(a*x)^4 - 3*(2*a^2*x^2 - 1)*arcsin(a*x)^2 + 2*(2*a *x*arcsin(a*x)^3 - 3*a*x*arcsin(a*x))*sqrt(-a^2*x^2 + 1))/a^3
Time = 0.40 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.93 \[ \int \frac {x^2 \arcsin (a x)^3}{\sqrt {1-a^2 x^2}} \, dx=\begin {cases} \frac {3 x^{2} \operatorname {asin}^{2}{\left (a x \right )}}{4 a} - \frac {3 x^{2}}{8 a} - \frac {x \sqrt {- a^{2} x^{2} + 1} \operatorname {asin}^{3}{\left (a x \right )}}{2 a^{2}} + \frac {3 x \sqrt {- a^{2} x^{2} + 1} \operatorname {asin}{\left (a x \right )}}{4 a^{2}} + \frac {\operatorname {asin}^{4}{\left (a x \right )}}{8 a^{3}} - \frac {3 \operatorname {asin}^{2}{\left (a x \right )}}{8 a^{3}} & \text {for}\: a \neq 0 \\0 & \text {otherwise} \end {cases} \]
Piecewise((3*x**2*asin(a*x)**2/(4*a) - 3*x**2/(8*a) - x*sqrt(-a**2*x**2 + 1)*asin(a*x)**3/(2*a**2) + 3*x*sqrt(-a**2*x**2 + 1)*asin(a*x)/(4*a**2) + a sin(a*x)**4/(8*a**3) - 3*asin(a*x)**2/(8*a**3), Ne(a, 0)), (0, True))
\[ \int \frac {x^2 \arcsin (a x)^3}{\sqrt {1-a^2 x^2}} \, dx=\int { \frac {x^{2} \arcsin \left (a x\right )^{3}}{\sqrt {-a^{2} x^{2} + 1}} \,d x } \]
Time = 0.33 (sec) , antiderivative size = 108, normalized size of antiderivative = 1.01 \[ \int \frac {x^2 \arcsin (a x)^3}{\sqrt {1-a^2 x^2}} \, dx=-\frac {\sqrt {-a^{2} x^{2} + 1} x \arcsin \left (a x\right )^{3}}{2 \, a^{2}} + \frac {\arcsin \left (a x\right )^{4}}{8 \, a^{3}} + \frac {3 \, \sqrt {-a^{2} x^{2} + 1} x \arcsin \left (a x\right )}{4 \, a^{2}} + \frac {3 \, {\left (a^{2} x^{2} - 1\right )} \arcsin \left (a x\right )^{2}}{4 \, a^{3}} + \frac {3 \, \arcsin \left (a x\right )^{2}}{8 \, a^{3}} - \frac {3 \, {\left (a^{2} x^{2} - 1\right )}}{8 \, a^{3}} - \frac {3}{16 \, a^{3}} \]
-1/2*sqrt(-a^2*x^2 + 1)*x*arcsin(a*x)^3/a^2 + 1/8*arcsin(a*x)^4/a^3 + 3/4* sqrt(-a^2*x^2 + 1)*x*arcsin(a*x)/a^2 + 3/4*(a^2*x^2 - 1)*arcsin(a*x)^2/a^3 + 3/8*arcsin(a*x)^2/a^3 - 3/8*(a^2*x^2 - 1)/a^3 - 3/16/a^3
Timed out. \[ \int \frac {x^2 \arcsin (a x)^3}{\sqrt {1-a^2 x^2}} \, dx=\int \frac {x^2\,{\mathrm {asin}\left (a\,x\right )}^3}{\sqrt {1-a^2\,x^2}} \,d x \]