Integrand size = 8, antiderivative size = 99 \[ \int x \arcsin (a x)^3 \, dx=-\frac {3 x \sqrt {1-a^2 x^2}}{8 a}+\frac {3 \arcsin (a x)}{8 a^2}-\frac {3}{4} x^2 \arcsin (a x)+\frac {3 x \sqrt {1-a^2 x^2} \arcsin (a x)^2}{4 a}-\frac {\arcsin (a x)^3}{4 a^2}+\frac {1}{2} x^2 \arcsin (a x)^3 \]
3/8*arcsin(a*x)/a^2-3/4*x^2*arcsin(a*x)-1/4*arcsin(a*x)^3/a^2+1/2*x^2*arcs in(a*x)^3-3/8*x*(-a^2*x^2+1)^(1/2)/a+3/4*x*arcsin(a*x)^2*(-a^2*x^2+1)^(1/2 )/a
Time = 0.02 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.83 \[ \int x \arcsin (a x)^3 \, dx=\frac {-3 a x \sqrt {1-a^2 x^2}+\left (3-6 a^2 x^2\right ) \arcsin (a x)+6 a x \sqrt {1-a^2 x^2} \arcsin (a x)^2+\left (-2+4 a^2 x^2\right ) \arcsin (a x)^3}{8 a^2} \]
(-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 + 4*a^2*x^2)*ArcSin[a*x]^3)/(8*a^2)
Time = 0.49 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.17, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.750, Rules used = {5138, 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.
\(\displaystyle \int x \arcsin (a x)^3 \, dx\) |
\(\Big \downarrow \) 5138 |
\(\displaystyle \frac {1}{2} x^2 \arcsin (a x)^3-\frac {3}{2} a \int \frac {x^2 \arcsin (a x)^2}{\sqrt {1-a^2 x^2}}dx\) |
\(\Big \downarrow \) 5210 |
\(\displaystyle \frac {1}{2} x^2 \arcsin (a x)^3-\frac {3}{2} a \left (\frac {\int \frac {\arcsin (a x)^2}{\sqrt {1-a^2 x^2}}dx}{2 a^2}+\frac {\int x \arcsin (a x)dx}{a}-\frac {x \sqrt {1-a^2 x^2} \arcsin (a x)^2}{2 a^2}\right )\) |
\(\Big \downarrow \) 5138 |
\(\displaystyle \frac {1}{2} x^2 \arcsin (a x)^3-\frac {3}{2} a \left (\frac {\frac {1}{2} x^2 \arcsin (a x)-\frac {1}{2} a \int \frac {x^2}{\sqrt {1-a^2 x^2}}dx}{a}+\frac {\int \frac {\arcsin (a x)^2}{\sqrt {1-a^2 x^2}}dx}{2 a^2}-\frac {x \sqrt {1-a^2 x^2} \arcsin (a x)^2}{2 a^2}\right )\) |
\(\Big \downarrow \) 262 |
\(\displaystyle \frac {1}{2} x^2 \arcsin (a x)^3-\frac {3}{2} a \left (\frac {\frac {1}{2} x^2 \arcsin (a x)-\frac {1}{2} a \left (\frac {\int \frac {1}{\sqrt {1-a^2 x^2}}dx}{2 a^2}-\frac {x \sqrt {1-a^2 x^2}}{2 a^2}\right )}{a}+\frac {\int \frac {\arcsin (a x)^2}{\sqrt {1-a^2 x^2}}dx}{2 a^2}-\frac {x \sqrt {1-a^2 x^2} \arcsin (a x)^2}{2 a^2}\right )\) |
\(\Big \downarrow \) 223 |
\(\displaystyle \frac {1}{2} x^2 \arcsin (a x)^3-\frac {3}{2} a \left (\frac {\int \frac {\arcsin (a x)^2}{\sqrt {1-a^2 x^2}}dx}{2 a^2}-\frac {x \sqrt {1-a^2 x^2} \arcsin (a x)^2}{2 a^2}+\frac {\frac {1}{2} x^2 \arcsin (a x)-\frac {1}{2} a \left (\frac {\arcsin (a x)}{2 a^3}-\frac {x \sqrt {1-a^2 x^2}}{2 a^2}\right )}{a}\right )\) |
\(\Big \downarrow \) 5152 |
\(\displaystyle \frac {1}{2} x^2 \arcsin (a x)^3-\frac {3}{2} a \left (\frac {\arcsin (a x)^3}{6 a^3}-\frac {x \sqrt {1-a^2 x^2} \arcsin (a x)^2}{2 a^2}+\frac {\frac {1}{2} x^2 \arcsin (a x)-\frac {1}{2} a \left (\frac {\arcsin (a x)}{2 a^3}-\frac {x \sqrt {1-a^2 x^2}}{2 a^2}\right )}{a}\right )\) |
(x^2*ArcSin[a*x]^3)/2 - (3*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))/2
3.1.25.3.1 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 = 0.05 (sec) , antiderivative size = 96, normalized size of antiderivative = 0.97
method | result | size |
derivativedivides | \(\frac {\frac {\arcsin \left (a x \right )^{3} \left (a^{2} x^{2}-1\right )}{2}+\frac {3 \arcsin \left (a x \right )^{2} \left (a x \sqrt {-a^{2} x^{2}+1}+\arcsin \left (a x \right )\right )}{4}-\frac {3 \left (a^{2} x^{2}-1\right ) \arcsin \left (a x \right )}{4}-\frac {3 a x \sqrt {-a^{2} x^{2}+1}}{8}-\frac {3 \arcsin \left (a x \right )}{8}-\frac {\arcsin \left (a x \right )^{3}}{2}}{a^{2}}\) | \(96\) |
default | \(\frac {\frac {\arcsin \left (a x \right )^{3} \left (a^{2} x^{2}-1\right )}{2}+\frac {3 \arcsin \left (a x \right )^{2} \left (a x \sqrt {-a^{2} x^{2}+1}+\arcsin \left (a x \right )\right )}{4}-\frac {3 \left (a^{2} x^{2}-1\right ) \arcsin \left (a x \right )}{4}-\frac {3 a x \sqrt {-a^{2} x^{2}+1}}{8}-\frac {3 \arcsin \left (a x \right )}{8}-\frac {\arcsin \left (a x \right )^{3}}{2}}{a^{2}}\) | \(96\) |
1/a^2*(1/2*arcsin(a*x)^3*(a^2*x^2-1)+3/4*arcsin(a*x)^2*(a*x*(-a^2*x^2+1)^( 1/2)+arcsin(a*x))-3/4*(a^2*x^2-1)*arcsin(a*x)-3/8*a*x*(-a^2*x^2+1)^(1/2)-3 /8*arcsin(a*x)-1/2*arcsin(a*x)^3)
Time = 0.27 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.70 \[ \int x \arcsin (a x)^3 \, dx=\frac {2 \, {\left (2 \, a^{2} x^{2} - 1\right )} \arcsin \left (a x\right )^{3} - 3 \, {\left (2 \, a^{2} x^{2} - 1\right )} \arcsin \left (a x\right ) + 3 \, \sqrt {-a^{2} x^{2} + 1} {\left (2 \, a x \arcsin \left (a x\right )^{2} - a x\right )}}{8 \, a^{2}} \]
1/8*(2*(2*a^2*x^2 - 1)*arcsin(a*x)^3 - 3*(2*a^2*x^2 - 1)*arcsin(a*x) + 3*s qrt(-a^2*x^2 + 1)*(2*a*x*arcsin(a*x)^2 - a*x))/a^2
Time = 0.23 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.93 \[ \int x \arcsin (a x)^3 \, dx=\begin {cases} \frac {x^{2} \operatorname {asin}^{3}{\left (a x \right )}}{2} - \frac {3 x^{2} \operatorname {asin}{\left (a x \right )}}{4} + \frac {3 x \sqrt {- a^{2} x^{2} + 1} \operatorname {asin}^{2}{\left (a x \right )}}{4 a} - \frac {3 x \sqrt {- a^{2} x^{2} + 1}}{8 a} - \frac {\operatorname {asin}^{3}{\left (a x \right )}}{4 a^{2}} + \frac {3 \operatorname {asin}{\left (a x \right )}}{8 a^{2}} & \text {for}\: a \neq 0 \\0 & \text {otherwise} \end {cases} \]
Piecewise((x**2*asin(a*x)**3/2 - 3*x**2*asin(a*x)/4 + 3*x*sqrt(-a**2*x**2 + 1)*asin(a*x)**2/(4*a) - 3*x*sqrt(-a**2*x**2 + 1)/(8*a) - asin(a*x)**3/(4 *a**2) + 3*asin(a*x)/(8*a**2), Ne(a, 0)), (0, True))
\[ \int x \arcsin (a x)^3 \, dx=\int { x \arcsin \left (a x\right )^{3} \,d x } \]
1/2*x^2*arctan2(a*x, sqrt(a*x + 1)*sqrt(-a*x + 1))^3 + 3*a*integrate(1/2*s qrt(a*x + 1)*sqrt(-a*x + 1)*x^2*arctan2(a*x, sqrt(a*x + 1)*sqrt(-a*x + 1)) ^2/(a^2*x^2 - 1), x)
Time = 0.29 (sec) , antiderivative size = 101, normalized size of antiderivative = 1.02 \[ \int x \arcsin (a x)^3 \, dx=\frac {3 \, \sqrt {-a^{2} x^{2} + 1} x \arcsin \left (a x\right )^{2}}{4 \, a} + \frac {{\left (a^{2} x^{2} - 1\right )} \arcsin \left (a x\right )^{3}}{2 \, a^{2}} + \frac {\arcsin \left (a x\right )^{3}}{4 \, a^{2}} - \frac {3 \, \sqrt {-a^{2} x^{2} + 1} x}{8 \, a} - \frac {3 \, {\left (a^{2} x^{2} - 1\right )} \arcsin \left (a x\right )}{4 \, a^{2}} - \frac {3 \, \arcsin \left (a x\right )}{8 \, a^{2}} \]
3/4*sqrt(-a^2*x^2 + 1)*x*arcsin(a*x)^2/a + 1/2*(a^2*x^2 - 1)*arcsin(a*x)^3 /a^2 + 1/4*arcsin(a*x)^3/a^2 - 3/8*sqrt(-a^2*x^2 + 1)*x/a - 3/4*(a^2*x^2 - 1)*arcsin(a*x)/a^2 - 3/8*arcsin(a*x)/a^2
Timed out. \[ \int x \arcsin (a x)^3 \, dx=\int x\,{\mathrm {asin}\left (a\,x\right )}^3 \,d x \]