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

Optimal. Leaf size=72 \[ -\frac{x^2 \sqrt{1-a^2 x^2} \sin ^{-1}(a x)}{3 a^2}-\frac{2 \sqrt{1-a^2 x^2} \sin ^{-1}(a x)}{3 a^4}+\frac{2 x}{3 a^3}+\frac{x^3}{9 a} \]

[Out]

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

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Rubi [A]  time = 0.107182, antiderivative size = 72, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {4707, 4677, 8, 30} \[ -\frac{x^2 \sqrt{1-a^2 x^2} \sin ^{-1}(a x)}{3 a^2}-\frac{2 \sqrt{1-a^2 x^2} \sin ^{-1}(a x)}{3 a^4}+\frac{2 x}{3 a^3}+\frac{x^3}{9 a} \]

Antiderivative was successfully verified.

[In]

Int[(x^3*ArcSin[a*x])/Sqrt[1 - a^2*x^2],x]

[Out]

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

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 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rubi steps

\begin{align*} \int \frac{x^3 \sin ^{-1}(a x)}{\sqrt{1-a^2 x^2}} \, dx &=-\frac{x^2 \sqrt{1-a^2 x^2} \sin ^{-1}(a x)}{3 a^2}+\frac{2 \int \frac{x \sin ^{-1}(a x)}{\sqrt{1-a^2 x^2}} \, dx}{3 a^2}+\frac{\int x^2 \, dx}{3 a}\\ &=\frac{x^3}{9 a}-\frac{2 \sqrt{1-a^2 x^2} \sin ^{-1}(a x)}{3 a^4}-\frac{x^2 \sqrt{1-a^2 x^2} \sin ^{-1}(a x)}{3 a^2}+\frac{2 \int 1 \, dx}{3 a^3}\\ &=\frac{2 x}{3 a^3}+\frac{x^3}{9 a}-\frac{2 \sqrt{1-a^2 x^2} \sin ^{-1}(a x)}{3 a^4}-\frac{x^2 \sqrt{1-a^2 x^2} \sin ^{-1}(a x)}{3 a^2}\\ \end{align*}

Mathematica [A]  time = 0.0247226, size = 49, normalized size = 0.68 \[ \frac{a x \left (a^2 x^2+6\right )-3 \sqrt{1-a^2 x^2} \left (a^2 x^2+2\right ) \sin ^{-1}(a x)}{9 a^4} \]

Antiderivative was successfully verified.

[In]

Integrate[(x^3*ArcSin[a*x])/Sqrt[1 - a^2*x^2],x]

[Out]

(a*x*(6 + a^2*x^2) - 3*Sqrt[1 - a^2*x^2]*(2 + a^2*x^2)*ArcSin[a*x])/(9*a^4)

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Maple [A]  time = 0.046, size = 95, normalized size = 1.3 \begin{align*} -{\frac{1}{9\,{a}^{4} \left ({a}^{2}{x}^{2}-1 \right ) } \left ( 3\,{a}^{4}{x}^{4}\arcsin \left ( ax \right ) +3\,{a}^{2}{x}^{2}\arcsin \left ( ax \right ) +{a}^{3}{x}^{3}\sqrt{-{a}^{2}{x}^{2}+1}-6\,\arcsin \left ( ax \right ) +6\,ax\sqrt{-{a}^{2}{x}^{2}+1} \right ) \sqrt{-{a}^{2}{x}^{2}+1}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3*arcsin(a*x)/(-a^2*x^2+1)^(1/2),x)

[Out]

-1/9/a^4*(3*a^4*x^4*arcsin(a*x)+3*a^2*x^2*arcsin(a*x)+a^3*x^3*(-a^2*x^2+1)^(1/2)-6*arcsin(a*x)+6*a*x*(-a^2*x^2
+1)^(1/2))*(-a^2*x^2+1)^(1/2)/(a^2*x^2-1)

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Maxima [A]  time = 1.56754, size = 82, normalized size = 1.14 \begin{align*} \frac{1}{9} \, a{\left (\frac{x^{3}}{a^{2}} + \frac{6 \, x}{a^{4}}\right )} - \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 ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*arcsin(a*x)/(-a^2*x^2+1)^(1/2),x, algorithm="maxima")

[Out]

1/9*a*(x^3/a^2 + 6*x/a^4) - 1/3*(sqrt(-a^2*x^2 + 1)*x^2/a^2 + 2*sqrt(-a^2*x^2 + 1)/a^4)*arcsin(a*x)

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Fricas [A]  time = 2.07645, size = 103, normalized size = 1.43 \begin{align*} \frac{a^{3} x^{3} - 3 \,{\left (a^{2} x^{2} + 2\right )} \sqrt{-a^{2} x^{2} + 1} \arcsin \left (a x\right ) + 6 \, a x}{9 \, a^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*arcsin(a*x)/(-a^2*x^2+1)^(1/2),x, algorithm="fricas")

[Out]

1/9*(a^3*x^3 - 3*(a^2*x^2 + 2)*sqrt(-a^2*x^2 + 1)*arcsin(a*x) + 6*a*x)/a^4

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Sympy [A]  time = 1.57694, size = 65, normalized size = 0.9 \begin{align*} \begin{cases} \frac{x^{3}}{9 a} - \frac{x^{2} \sqrt{- a^{2} x^{2} + 1} \operatorname{asin}{\left (a x \right )}}{3 a^{2}} + \frac{2 x}{3 a^{3}} - \frac{2 \sqrt{- a^{2} x^{2} + 1} \operatorname{asin}{\left (a x \right )}}{3 a^{4}} & \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**3*asin(a*x)/(-a**2*x**2+1)**(1/2),x)

[Out]

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

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Giac [A]  time = 1.22243, size = 72, normalized size = 1. \begin{align*} \frac{a^{2} x^{3} + 6 \, x}{9 \, a^{3}} + \frac{{\left ({\left (-a^{2} x^{2} + 1\right )}^{\frac{3}{2}} - 3 \, \sqrt{-a^{2} x^{2} + 1}\right )} \arcsin \left (a x\right )}{3 \, a^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*arcsin(a*x)/(-a^2*x^2+1)^(1/2),x, algorithm="giac")

[Out]

1/9*(a^2*x^3 + 6*x)/a^3 + 1/3*((-a^2*x^2 + 1)^(3/2) - 3*sqrt(-a^2*x^2 + 1))*arcsin(a*x)/a^4