3.24 \(\int x^2 \sin ^{-1}(a x)^3 \, dx\)

Optimal. Leaf size=136 \[ \frac{2 \left (1-a^2 x^2\right )^{3/2}}{27 a^3}-\frac{14 \sqrt{1-a^2 x^2}}{9 a^3}+\frac{x^2 \sqrt{1-a^2 x^2} \sin ^{-1}(a x)^2}{3 a}+\frac{2 \sqrt{1-a^2 x^2} \sin ^{-1}(a x)^2}{3 a^3}-\frac{4 x \sin ^{-1}(a x)}{3 a^2}+\frac{1}{3} x^3 \sin ^{-1}(a x)^3-\frac{2}{9} x^3 \sin ^{-1}(a x) \]

[Out]

(-14*Sqrt[1 - a^2*x^2])/(9*a^3) + (2*(1 - a^2*x^2)^(3/2))/(27*a^3) - (4*x*ArcSin[a*x])/(3*a^2) - (2*x^3*ArcSin
[a*x])/9 + (2*Sqrt[1 - a^2*x^2]*ArcSin[a*x]^2)/(3*a^3) + (x^2*Sqrt[1 - a^2*x^2]*ArcSin[a*x]^2)/(3*a) + (x^3*Ar
cSin[a*x]^3)/3

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Rubi [A]  time = 0.224747, antiderivative size = 136, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 7, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.7, Rules used = {4627, 4707, 4677, 4619, 261, 266, 43} \[ \frac{2 \left (1-a^2 x^2\right )^{3/2}}{27 a^3}-\frac{14 \sqrt{1-a^2 x^2}}{9 a^3}+\frac{x^2 \sqrt{1-a^2 x^2} \sin ^{-1}(a x)^2}{3 a}+\frac{2 \sqrt{1-a^2 x^2} \sin ^{-1}(a x)^2}{3 a^3}-\frac{4 x \sin ^{-1}(a x)}{3 a^2}+\frac{1}{3} x^3 \sin ^{-1}(a x)^3-\frac{2}{9} x^3 \sin ^{-1}(a x) \]

Antiderivative was successfully verified.

[In]

Int[x^2*ArcSin[a*x]^3,x]

[Out]

(-14*Sqrt[1 - a^2*x^2])/(9*a^3) + (2*(1 - a^2*x^2)^(3/2))/(27*a^3) - (4*x*ArcSin[a*x])/(3*a^2) - (2*x^3*ArcSin
[a*x])/9 + (2*Sqrt[1 - a^2*x^2]*ArcSin[a*x]^2)/(3*a^3) + (x^2*Sqrt[1 - a^2*x^2]*ArcSin[a*x]^2)/(3*a) + (x^3*Ar
cSin[a*x]^3)/3

Rule 4627

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[((d*x)^(m + 1)*(a + b*ArcSi
n[c*x])^n)/(d*(m + 1)), x] - Dist[(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]

Rule 4707

Int[(((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_))/Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Simp[
(f*(f*x)^(m - 1)*Sqrt[d + e*x^2]*(a + b*ArcSin[c*x])^n)/(e*m), x] + (Dist[(f^2*(m - 1))/(c^2*m), Int[((f*x)^(m
- 2)*(a + b*ArcSin[c*x])^n)/Sqrt[d + e*x^2], x], x] + Dist[(b*f*n*Sqrt[1 - c^2*x^2])/(c*m*Sqrt[d + e*x^2]), I
nt[(f*x)^(m - 1)*(a + b*ArcSin[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[c^2*d + e, 0] &&
GtQ[n, 0] && GtQ[m, 1] && IntegerQ[m]

Rule 4677

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] + Dist[(b*n*d^IntPart[p]*(d + e*x^2)^FracPart[p])/(2*c*(p + 1
)*(1 - c^2*x^2)^FracPart[p]), Int[(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]

Rule 4619

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.), x_Symbol] :> Simp[x*(a + b*ArcSin[c*x])^n, x] - Dist[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]

Rule 261

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(a + b*x^n)^(p + 1)/(b*n*(p + 1)), x] /; FreeQ
[{a, b, m, n, p}, x] && EqQ[m, n - 1] && NeQ[p, -1]

Rule 266

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rule 43

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] && NeQ[b*c - a*d, 0] && 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])

Rubi steps

\begin{align*} \int x^2 \sin ^{-1}(a x)^3 \, dx &=\frac{1}{3} x^3 \sin ^{-1}(a x)^3-a \int \frac{x^3 \sin ^{-1}(a x)^2}{\sqrt{1-a^2 x^2}} \, dx\\ &=\frac{x^2 \sqrt{1-a^2 x^2} \sin ^{-1}(a x)^2}{3 a}+\frac{1}{3} x^3 \sin ^{-1}(a x)^3-\frac{2}{3} \int x^2 \sin ^{-1}(a x) \, dx-\frac{2 \int \frac{x \sin ^{-1}(a x)^2}{\sqrt{1-a^2 x^2}} \, dx}{3 a}\\ &=-\frac{2}{9} x^3 \sin ^{-1}(a x)+\frac{2 \sqrt{1-a^2 x^2} \sin ^{-1}(a x)^2}{3 a^3}+\frac{x^2 \sqrt{1-a^2 x^2} \sin ^{-1}(a x)^2}{3 a}+\frac{1}{3} x^3 \sin ^{-1}(a x)^3-\frac{4 \int \sin ^{-1}(a x) \, dx}{3 a^2}+\frac{1}{9} (2 a) \int \frac{x^3}{\sqrt{1-a^2 x^2}} \, dx\\ &=-\frac{4 x \sin ^{-1}(a x)}{3 a^2}-\frac{2}{9} x^3 \sin ^{-1}(a x)+\frac{2 \sqrt{1-a^2 x^2} \sin ^{-1}(a x)^2}{3 a^3}+\frac{x^2 \sqrt{1-a^2 x^2} \sin ^{-1}(a x)^2}{3 a}+\frac{1}{3} x^3 \sin ^{-1}(a x)^3+\frac{4 \int \frac{x}{\sqrt{1-a^2 x^2}} \, dx}{3 a}+\frac{1}{9} a \operatorname{Subst}\left (\int \frac{x}{\sqrt{1-a^2 x}} \, dx,x,x^2\right )\\ &=-\frac{4 \sqrt{1-a^2 x^2}}{3 a^3}-\frac{4 x \sin ^{-1}(a x)}{3 a^2}-\frac{2}{9} x^3 \sin ^{-1}(a x)+\frac{2 \sqrt{1-a^2 x^2} \sin ^{-1}(a x)^2}{3 a^3}+\frac{x^2 \sqrt{1-a^2 x^2} \sin ^{-1}(a x)^2}{3 a}+\frac{1}{3} x^3 \sin ^{-1}(a x)^3+\frac{1}{9} a \operatorname{Subst}\left (\int \left (\frac{1}{a^2 \sqrt{1-a^2 x}}-\frac{\sqrt{1-a^2 x}}{a^2}\right ) \, dx,x,x^2\right )\\ &=-\frac{14 \sqrt{1-a^2 x^2}}{9 a^3}+\frac{2 \left (1-a^2 x^2\right )^{3/2}}{27 a^3}-\frac{4 x \sin ^{-1}(a x)}{3 a^2}-\frac{2}{9} x^3 \sin ^{-1}(a x)+\frac{2 \sqrt{1-a^2 x^2} \sin ^{-1}(a x)^2}{3 a^3}+\frac{x^2 \sqrt{1-a^2 x^2} \sin ^{-1}(a x)^2}{3 a}+\frac{1}{3} x^3 \sin ^{-1}(a x)^3\\ \end{align*}

Mathematica [A]  time = 0.0446312, size = 95, normalized size = 0.7 \[ \frac{-2 \sqrt{1-a^2 x^2} \left (a^2 x^2+20\right )+9 a^3 x^3 \sin ^{-1}(a x)^3+9 \sqrt{1-a^2 x^2} \left (a^2 x^2+2\right ) \sin ^{-1}(a x)^2-6 a x \left (a^2 x^2+6\right ) \sin ^{-1}(a x)}{27 a^3} \]

Antiderivative was successfully verified.

[In]

Integrate[x^2*ArcSin[a*x]^3,x]

[Out]

(-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)*Arc
Sin[a*x]^2 + 9*a^3*x^3*ArcSin[a*x]^3)/(27*a^3)

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Maple [A]  time = 0.045, size = 106, normalized size = 0.8 \begin{align*}{\frac{1}{{a}^{3}} \left ({\frac{{a}^{3}{x}^{3} \left ( \arcsin \left ( ax \right ) \right ) ^{3}}{3}}+{\frac{ \left ( \arcsin \left ( ax \right ) \right ) ^{2} \left ({a}^{2}{x}^{2}+2 \right ) }{3}\sqrt{-{a}^{2}{x}^{2}+1}}-{\frac{4}{3}\sqrt{-{a}^{2}{x}^{2}+1}}-{\frac{4\,ax\arcsin \left ( ax \right ) }{3}}-{\frac{2\,{a}^{3}{x}^{3}\arcsin \left ( ax \right ) }{9}}-{\frac{2\,{a}^{2}{x}^{2}+4}{27}\sqrt{-{a}^{2}{x}^{2}+1}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*arcsin(a*x)^3,x)

[Out]

1/a^3*(1/3*a^3*x^3*arcsin(a*x)^3+1/3*arcsin(a*x)^2*(a^2*x^2+2)*(-a^2*x^2+1)^(1/2)-4/3*(-a^2*x^2+1)^(1/2)-4/3*a
*x*arcsin(a*x)-2/9*a^3*x^3*arcsin(a*x)-2/27*(a^2*x^2+2)*(-a^2*x^2+1)^(1/2))

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Maxima [A]  time = 1.72559, size = 162, normalized size = 1.19 \begin{align*} \frac{1}{3} \, x^{3} \arcsin \left (a x\right )^{3} + \frac{1}{3} \, a{\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}{27} \, a{\left (\frac{\sqrt{-a^{2} x^{2} + 1} x^{2} + \frac{20 \, \sqrt{-a^{2} x^{2} + 1}}{a^{2}}}{a^{2}} + \frac{3 \,{\left (a^{2} x^{3} + 6 \, x\right )} \arcsin \left (a x\right )}{a^{3}}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*arcsin(a*x)^3,x, algorithm="maxima")

[Out]

1/3*x^3*arcsin(a*x)^3 + 1/3*a*(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*a*(
(sqrt(-a^2*x^2 + 1)*x^2 + 20*sqrt(-a^2*x^2 + 1)/a^2)/a^2 + 3*(a^2*x^3 + 6*x)*arcsin(a*x)/a^3)

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Fricas [A]  time = 2.16897, size = 189, normalized size = 1.39 \begin{align*} \frac{9 \, a^{3} x^{3} \arcsin \left (a x\right )^{3} - 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^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*arcsin(a*x)^3,x, algorithm="fricas")

[Out]

1/27*(9*a^3*x^3*arcsin(a*x)^3 - 6*(a^3*x^3 + 6*a*x)*arcsin(a*x) - (2*a^2*x^2 - 9*(a^2*x^2 + 2)*arcsin(a*x)^2 +
40)*sqrt(-a^2*x^2 + 1))/a^3

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Sympy [A]  time = 5.89492, size = 128, normalized size = 0.94 \begin{align*} \begin{cases} \frac{x^{3} \operatorname{asin}^{3}{\left (a x \right )}}{3} - \frac{2 x^{3} \operatorname{asin}{\left (a x \right )}}{9} + \frac{x^{2} \sqrt{- a^{2} x^{2} + 1} \operatorname{asin}^{2}{\left (a x \right )}}{3 a} - \frac{2 x^{2} \sqrt{- a^{2} x^{2} + 1}}{27 a} - \frac{4 x \operatorname{asin}{\left (a x \right )}}{3 a^{2}} + \frac{2 \sqrt{- a^{2} x^{2} + 1} \operatorname{asin}^{2}{\left (a x \right )}}{3 a^{3}} - \frac{40 \sqrt{- a^{2} x^{2} + 1}}{27 a^{3}} & \text{for}\: a \neq 0 \\0 & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**2*asin(a*x)**3,x)

[Out]

Piecewise((x**3*asin(a*x)**3/3 - 2*x**3*asin(a*x)/9 + x**2*sqrt(-a**2*x**2 + 1)*asin(a*x)**2/(3*a) - 2*x**2*sq
rt(-a**2*x**2 + 1)/(27*a) - 4*x*asin(a*x)/(3*a**2) + 2*sqrt(-a**2*x**2 + 1)*asin(a*x)**2/(3*a**3) - 40*sqrt(-a
**2*x**2 + 1)/(27*a**3), Ne(a, 0)), (0, True))

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Giac [A]  time = 1.29957, size = 192, normalized size = 1.41 \begin{align*} \frac{{\left (a^{2} x^{2} - 1\right )} x \arcsin \left (a x\right )^{3}}{3 \, a^{2}} + \frac{x \arcsin \left (a x\right )^{3}}{3 \, a^{2}} - \frac{2 \,{\left (a^{2} x^{2} - 1\right )} x \arcsin \left (a x\right )}{9 \, a^{2}} - \frac{{\left (-a^{2} x^{2} + 1\right )}^{\frac{3}{2}} \arcsin \left (a x\right )^{2}}{3 \, a^{3}} - \frac{14 \, x \arcsin \left (a x\right )}{9 \, a^{2}} + \frac{\sqrt{-a^{2} x^{2} + 1} \arcsin \left (a x\right )^{2}}{a^{3}} + \frac{2 \,{\left (-a^{2} x^{2} + 1\right )}^{\frac{3}{2}}}{27 \, a^{3}} - \frac{14 \, \sqrt{-a^{2} x^{2} + 1}}{9 \, a^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*arcsin(a*x)^3,x, algorithm="giac")

[Out]

1/3*(a^2*x^2 - 1)*x*arcsin(a*x)^3/a^2 + 1/3*x*arcsin(a*x)^3/a^2 - 2/9*(a^2*x^2 - 1)*x*arcsin(a*x)/a^2 - 1/3*(-
a^2*x^2 + 1)^(3/2)*arcsin(a*x)^2/a^3 - 14/9*x*arcsin(a*x)/a^2 + sqrt(-a^2*x^2 + 1)*arcsin(a*x)^2/a^3 + 2/27*(-
a^2*x^2 + 1)^(3/2)/a^3 - 14/9*sqrt(-a^2*x^2 + 1)/a^3