Integrand size = 24, antiderivative size = 89 \[ \int \frac {x^2 \arcsin (a x)^2}{\sqrt {1-a^2 x^2}} \, dx=\frac {x \sqrt {1-a^2 x^2}}{4 a^2}-\frac {\arcsin (a x)}{4 a^3}+\frac {x^2 \arcsin (a x)}{2 a}-\frac {x \sqrt {1-a^2 x^2} \arcsin (a x)^2}{2 a^2}+\frac {\arcsin (a x)^3}{6 a^3} \]
-1/4*arcsin(a*x)/a^3+1/2*x^2*arcsin(a*x)/a+1/6*arcsin(a*x)^3/a^3+1/4*x*(-a ^2*x^2+1)^(1/2)/a^2-1/2*x*arcsin(a*x)^2*(-a^2*x^2+1)^(1/2)/a^2
Time = 0.03 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.82 \[ \int \frac {x^2 \arcsin (a x)^2}{\sqrt {1-a^2 x^2}} \, dx=\frac {3 a x \sqrt {1-a^2 x^2}+\left (-3+6 a^2 x^2\right ) \arcsin (a x)-6 a x \sqrt {1-a^2 x^2} \arcsin (a x)^2+2 \arcsin (a x)^3}{12 a^3} \]
(3*a*x*Sqrt[1 - a^2*x^2] + (-3 + 6*a^2*x^2)*ArcSin[a*x] - 6*a*x*Sqrt[1 - a ^2*x^2]*ArcSin[a*x]^2 + 2*ArcSin[a*x]^3)/(12*a^3)
Time = 0.42 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.09, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {5210, 5138, 262, 223, 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.
\(\displaystyle \int \frac {x^2 \arcsin (a x)^2}{\sqrt {1-a^2 x^2}} \, dx\) |
\(\Big \downarrow \) 5210 |
\(\displaystyle \frac {\int \frac {\arcsin (a x)^2}{\sqrt {1-a^2 x^2}}dx}{2 a^2}+\frac {\int x \arcsin (a x)dx}{a}-\frac {x \sqrt {1-a^2 x^2} \arcsin (a x)^2}{2 a^2}\) |
\(\Big \downarrow \) 5138 |
\(\displaystyle \frac {\frac {1}{2} x^2 \arcsin (a x)-\frac {1}{2} a \int \frac {x^2}{\sqrt {1-a^2 x^2}}dx}{a}+\frac {\int \frac {\arcsin (a x)^2}{\sqrt {1-a^2 x^2}}dx}{2 a^2}-\frac {x \sqrt {1-a^2 x^2} \arcsin (a x)^2}{2 a^2}\) |
\(\Big \downarrow \) 262 |
\(\displaystyle \frac {\frac {1}{2} x^2 \arcsin (a x)-\frac {1}{2} a \left (\frac {\int \frac {1}{\sqrt {1-a^2 x^2}}dx}{2 a^2}-\frac {x \sqrt {1-a^2 x^2}}{2 a^2}\right )}{a}+\frac {\int \frac {\arcsin (a x)^2}{\sqrt {1-a^2 x^2}}dx}{2 a^2}-\frac {x \sqrt {1-a^2 x^2} \arcsin (a x)^2}{2 a^2}\) |
\(\Big \downarrow \) 223 |
\(\displaystyle \frac {\int \frac {\arcsin (a x)^2}{\sqrt {1-a^2 x^2}}dx}{2 a^2}-\frac {x \sqrt {1-a^2 x^2} \arcsin (a x)^2}{2 a^2}+\frac {\frac {1}{2} x^2 \arcsin (a x)-\frac {1}{2} a \left (\frac {\arcsin (a x)}{2 a^3}-\frac {x \sqrt {1-a^2 x^2}}{2 a^2}\right )}{a}\) |
\(\Big \downarrow \) 5152 |
\(\displaystyle \frac {\arcsin (a x)^3}{6 a^3}-\frac {x \sqrt {1-a^2 x^2} \arcsin (a x)^2}{2 a^2}+\frac {\frac {1}{2} x^2 \arcsin (a x)-\frac {1}{2} a \left (\frac {\arcsin (a x)}{2 a^3}-\frac {x \sqrt {1-a^2 x^2}}{2 a^2}\right )}{a}\) |
-1/2*(x*Sqrt[1 - a^2*x^2]*ArcSin[a*x]^2)/a^2 + ArcSin[a*x]^3/(6*a^3) + ((x ^2*ArcSin[a*x])/2 - (a*(-1/2*(x*Sqrt[1 - a^2*x^2])/a^2 + ArcSin[a*x]/(2*a^ 3)))/2)/a
3.3.66.3.1 Defintions of rubi rules used
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_.) + 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 = 0.17 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.80
method | result | size |
default | \(\frac {-6 \arcsin \left (a x \right )^{2} \sqrt {-a^{2} x^{2}+1}\, a x +6 a^{2} x^{2} \arcsin \left (a x \right )+2 \arcsin \left (a x \right )^{3}+3 a x \sqrt {-a^{2} x^{2}+1}-3 \arcsin \left (a x \right )}{12 a^{3}}\) | \(71\) |
1/12*(-6*arcsin(a*x)^2*(-a^2*x^2+1)^(1/2)*a*x+6*a^2*x^2*arcsin(a*x)+2*arcs in(a*x)^3+3*a*x*(-a^2*x^2+1)^(1/2)-3*arcsin(a*x))/a^3
Time = 0.25 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.66 \[ \int \frac {x^2 \arcsin (a x)^2}{\sqrt {1-a^2 x^2}} \, dx=\frac {2 \, \arcsin \left (a x\right )^{3} + 3 \, {\left (2 \, a^{2} x^{2} - 1\right )} \arcsin \left (a x\right ) - 3 \, \sqrt {-a^{2} x^{2} + 1} {\left (2 \, a x \arcsin \left (a x\right )^{2} - a x\right )}}{12 \, a^{3}} \]
1/12*(2*arcsin(a*x)^3 + 3*(2*a^2*x^2 - 1)*arcsin(a*x) - 3*sqrt(-a^2*x^2 + 1)*(2*a*x*arcsin(a*x)^2 - a*x))/a^3
Time = 0.32 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.88 \[ \int \frac {x^2 \arcsin (a x)^2}{\sqrt {1-a^2 x^2}} \, dx=\begin {cases} \frac {x^{2} \operatorname {asin}{\left (a x \right )}}{2 a} - \frac {x \sqrt {- a^{2} x^{2} + 1} \operatorname {asin}^{2}{\left (a x \right )}}{2 a^{2}} + \frac {x \sqrt {- a^{2} x^{2} + 1}}{4 a^{2}} + \frac {\operatorname {asin}^{3}{\left (a x \right )}}{6 a^{3}} - \frac {\operatorname {asin}{\left (a x \right )}}{4 a^{3}} & \text {for}\: a \neq 0 \\0 & \text {otherwise} \end {cases} \]
Piecewise((x**2*asin(a*x)/(2*a) - x*sqrt(-a**2*x**2 + 1)*asin(a*x)**2/(2*a **2) + x*sqrt(-a**2*x**2 + 1)/(4*a**2) + asin(a*x)**3/(6*a**3) - asin(a*x) /(4*a**3), Ne(a, 0)), (0, True))
\[ \int \frac {x^2 \arcsin (a x)^2}{\sqrt {1-a^2 x^2}} \, dx=\int { \frac {x^{2} \arcsin \left (a x\right )^{2}}{\sqrt {-a^{2} x^{2} + 1}} \,d x } \]
Time = 0.33 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.91 \[ \int \frac {x^2 \arcsin (a x)^2}{\sqrt {1-a^2 x^2}} \, dx=-\frac {\sqrt {-a^{2} x^{2} + 1} x \arcsin \left (a x\right )^{2}}{2 \, a^{2}} + \frac {\arcsin \left (a x\right )^{3}}{6 \, a^{3}} + \frac {\sqrt {-a^{2} x^{2} + 1} x}{4 \, a^{2}} + \frac {{\left (a^{2} x^{2} - 1\right )} \arcsin \left (a x\right )}{2 \, a^{3}} + \frac {\arcsin \left (a x\right )}{4 \, a^{3}} \]
-1/2*sqrt(-a^2*x^2 + 1)*x*arcsin(a*x)^2/a^2 + 1/6*arcsin(a*x)^3/a^3 + 1/4* sqrt(-a^2*x^2 + 1)*x/a^2 + 1/2*(a^2*x^2 - 1)*arcsin(a*x)/a^3 + 1/4*arcsin( a*x)/a^3
Timed out. \[ \int \frac {x^2 \arcsin (a x)^2}{\sqrt {1-a^2 x^2}} \, dx=\int \frac {x^2\,{\mathrm {asin}\left (a\,x\right )}^2}{\sqrt {1-a^2\,x^2}} \,d x \]