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

Optimal. Leaf size=157 \[ -\frac{x^2 \sqrt{1-a^2 x^2} \sin ^{-1}(a x)^3}{3 a^2}+\frac{2 x^2 \sqrt{1-a^2 x^2} \sin ^{-1}(a x)}{9 a^2}-\frac{2 \sqrt{1-a^2 x^2} \sin ^{-1}(a x)^3}{3 a^4}+\frac{40 \sqrt{1-a^2 x^2} \sin ^{-1}(a x)}{9 a^4}-\frac{40 x}{9 a^3}+\frac{2 x \sin ^{-1}(a x)^2}{a^3}-\frac{2 x^3}{27 a}+\frac{x^3 \sin ^{-1}(a x)^2}{3 a} \]

[Out]

(-40*x)/(9*a^3) - (2*x^3)/(27*a) + (40*Sqrt[1 - a^2*x^2]*ArcSin[a*x])/(9*a^4) + (2*x^2*Sqrt[1 - a^2*x^2]*ArcSi
n[a*x])/(9*a^2) + (2*x*ArcSin[a*x]^2)/a^3 + (x^3*ArcSin[a*x]^2)/(3*a) - (2*Sqrt[1 - a^2*x^2]*ArcSin[a*x]^3)/(3
*a^4) - (x^2*Sqrt[1 - a^2*x^2]*ArcSin[a*x]^3)/(3*a^2)

________________________________________________________________________________________

Rubi [A]  time = 0.320715, antiderivative size = 157, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 6, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {4707, 4677, 4619, 8, 4627, 30} \[ -\frac{x^2 \sqrt{1-a^2 x^2} \sin ^{-1}(a x)^3}{3 a^2}+\frac{2 x^2 \sqrt{1-a^2 x^2} \sin ^{-1}(a x)}{9 a^2}-\frac{2 \sqrt{1-a^2 x^2} \sin ^{-1}(a x)^3}{3 a^4}+\frac{40 \sqrt{1-a^2 x^2} \sin ^{-1}(a x)}{9 a^4}-\frac{40 x}{9 a^3}+\frac{2 x \sin ^{-1}(a x)^2}{a^3}-\frac{2 x^3}{27 a}+\frac{x^3 \sin ^{-1}(a x)^2}{3 a} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-40*x)/(9*a^3) - (2*x^3)/(27*a) + (40*Sqrt[1 - a^2*x^2]*ArcSin[a*x])/(9*a^4) + (2*x^2*Sqrt[1 - a^2*x^2]*ArcSi
n[a*x])/(9*a^2) + (2*x*ArcSin[a*x]^2)/a^3 + (x^3*ArcSin[a*x]^2)/(3*a) - (2*Sqrt[1 - a^2*x^2]*ArcSin[a*x]^3)/(3
*a^4) - (x^2*Sqrt[1 - a^2*x^2]*ArcSin[a*x]^3)/(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 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 8

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

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 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)^3}{\sqrt{1-a^2 x^2}} \, dx &=-\frac{x^2 \sqrt{1-a^2 x^2} \sin ^{-1}(a x)^3}{3 a^2}+\frac{2 \int \frac{x \sin ^{-1}(a x)^3}{\sqrt{1-a^2 x^2}} \, dx}{3 a^2}+\frac{\int x^2 \sin ^{-1}(a x)^2 \, dx}{a}\\ &=\frac{x^3 \sin ^{-1}(a x)^2}{3 a}-\frac{2 \sqrt{1-a^2 x^2} \sin ^{-1}(a x)^3}{3 a^4}-\frac{x^2 \sqrt{1-a^2 x^2} \sin ^{-1}(a x)^3}{3 a^2}-\frac{2}{3} \int \frac{x^3 \sin ^{-1}(a x)}{\sqrt{1-a^2 x^2}} \, dx+\frac{2 \int \sin ^{-1}(a x)^2 \, dx}{a^3}\\ &=\frac{2 x^2 \sqrt{1-a^2 x^2} \sin ^{-1}(a x)}{9 a^2}+\frac{2 x \sin ^{-1}(a x)^2}{a^3}+\frac{x^3 \sin ^{-1}(a x)^2}{3 a}-\frac{2 \sqrt{1-a^2 x^2} \sin ^{-1}(a x)^3}{3 a^4}-\frac{x^2 \sqrt{1-a^2 x^2} \sin ^{-1}(a x)^3}{3 a^2}-\frac{4 \int \frac{x \sin ^{-1}(a x)}{\sqrt{1-a^2 x^2}} \, dx}{9 a^2}-\frac{4 \int \frac{x \sin ^{-1}(a x)}{\sqrt{1-a^2 x^2}} \, dx}{a^2}-\frac{2 \int x^2 \, dx}{9 a}\\ &=-\frac{2 x^3}{27 a}+\frac{40 \sqrt{1-a^2 x^2} \sin ^{-1}(a x)}{9 a^4}+\frac{2 x^2 \sqrt{1-a^2 x^2} \sin ^{-1}(a x)}{9 a^2}+\frac{2 x \sin ^{-1}(a x)^2}{a^3}+\frac{x^3 \sin ^{-1}(a x)^2}{3 a}-\frac{2 \sqrt{1-a^2 x^2} \sin ^{-1}(a x)^3}{3 a^4}-\frac{x^2 \sqrt{1-a^2 x^2} \sin ^{-1}(a x)^3}{3 a^2}-\frac{4 \int 1 \, dx}{9 a^3}-\frac{4 \int 1 \, dx}{a^3}\\ &=-\frac{40 x}{9 a^3}-\frac{2 x^3}{27 a}+\frac{40 \sqrt{1-a^2 x^2} \sin ^{-1}(a x)}{9 a^4}+\frac{2 x^2 \sqrt{1-a^2 x^2} \sin ^{-1}(a x)}{9 a^2}+\frac{2 x \sin ^{-1}(a x)^2}{a^3}+\frac{x^3 \sin ^{-1}(a x)^2}{3 a}-\frac{2 \sqrt{1-a^2 x^2} \sin ^{-1}(a x)^3}{3 a^4}-\frac{x^2 \sqrt{1-a^2 x^2} \sin ^{-1}(a x)^3}{3 a^2}\\ \end{align*}

Mathematica [A]  time = 0.0518458, size = 100, normalized size = 0.64 \[ \frac{-2 a x \left (a^2 x^2+60\right )-9 \sqrt{1-a^2 x^2} \left (a^2 x^2+2\right ) \sin ^{-1}(a x)^3+9 a x \left (a^2 x^2+6\right ) \sin ^{-1}(a x)^2+6 \sqrt{1-a^2 x^2} \left (a^2 x^2+20\right ) \sin ^{-1}(a x)}{27 a^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*a*x*(60 + a^2*x^2) + 6*Sqrt[1 - a^2*x^2]*(20 + a^2*x^2)*ArcSin[a*x] + 9*a*x*(6 + a^2*x^2)*ArcSin[a*x]^2 -
9*Sqrt[1 - a^2*x^2]*(2 + a^2*x^2)*ArcSin[a*x]^3)/(27*a^4)

________________________________________________________________________________________

Maple [A]  time = 0.056, size = 180, normalized size = 1.2 \begin{align*} -{\frac{1}{27\,{a}^{4} \left ({a}^{2}{x}^{2}-1 \right ) } \left ( 9\,{a}^{4}{x}^{4} \left ( \arcsin \left ( ax \right ) \right ) ^{3}+9\, \left ( \arcsin \left ( ax \right ) \right ) ^{3}{x}^{2}{a}^{2}+9\, \left ( \arcsin \left ( ax \right ) \right ) ^{2}\sqrt{-{a}^{2}{x}^{2}+1}{x}^{3}{a}^{3}-6\,{a}^{4}{x}^{4}\arcsin \left ( ax \right ) -114\,{a}^{2}{x}^{2}\arcsin \left ( ax \right ) -2\,{a}^{3}{x}^{3}\sqrt{-{a}^{2}{x}^{2}+1}-18\, \left ( \arcsin \left ( ax \right ) \right ) ^{3}+54\, \left ( \arcsin \left ( ax \right ) \right ) ^{2}\sqrt{-{a}^{2}{x}^{2}+1}xa+120\,\arcsin \left ( ax \right ) -120\,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)^3/(-a^2*x^2+1)^(1/2),x)

[Out]

-1/27/a^4*(9*a^4*x^4*arcsin(a*x)^3+9*arcsin(a*x)^3*x^2*a^2+9*arcsin(a*x)^2*(-a^2*x^2+1)^(1/2)*x^3*a^3-6*a^4*x^
4*arcsin(a*x)-114*a^2*x^2*arcsin(a*x)-2*a^3*x^3*(-a^2*x^2+1)^(1/2)-18*arcsin(a*x)^3+54*arcsin(a*x)^2*(-a^2*x^2
+1)^(1/2)*x*a+120*arcsin(a*x)-120*a*x*(-a^2*x^2+1)^(1/2))*(-a^2*x^2+1)^(1/2)/(a^2*x^2-1)

________________________________________________________________________________________

Maxima [A]  time = 1.61041, size = 177, normalized size = 1.13 \begin{align*} -\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 )^{3} + \frac{2}{27} \, a{\left (\frac{3 \,{\left (\sqrt{-a^{2} x^{2} + 1} x^{2} + \frac{20 \, \sqrt{-a^{2} x^{2} + 1}}{a^{2}}\right )} \arcsin \left (a x\right )}{a^{3}} - \frac{a^{2} x^{3} + 60 \, x}{a^{4}}\right )} + \frac{{\left (a^{2} x^{3} + 6 \, x\right )} \arcsin \left (a x\right )^{2}}{3 \, a^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3*(sqrt(-a^2*x^2 + 1)*x^2/a^2 + 2*sqrt(-a^2*x^2 + 1)/a^4)*arcsin(a*x)^3 + 2/27*a*(3*(sqrt(-a^2*x^2 + 1)*x^2
+ 20*sqrt(-a^2*x^2 + 1)/a^2)*arcsin(a*x)/a^3 - (a^2*x^3 + 60*x)/a^4) + 1/3*(a^2*x^3 + 6*x)*arcsin(a*x)^2/a^3

________________________________________________________________________________________

Fricas [A]  time = 1.69654, size = 209, normalized size = 1.33 \begin{align*} -\frac{2 \, a^{3} x^{3} - 9 \,{\left (a^{3} x^{3} + 6 \, a x\right )} \arcsin \left (a x\right )^{2} + 120 \, a x + 3 \, \sqrt{-a^{2} x^{2} + 1}{\left (3 \,{\left (a^{2} x^{2} + 2\right )} \arcsin \left (a x\right )^{3} - 2 \,{\left (a^{2} x^{2} + 20\right )} \arcsin \left (a x\right )\right )}}{27 \, a^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/27*(2*a^3*x^3 - 9*(a^3*x^3 + 6*a*x)*arcsin(a*x)^2 + 120*a*x + 3*sqrt(-a^2*x^2 + 1)*(3*(a^2*x^2 + 2)*arcsin(
a*x)^3 - 2*(a^2*x^2 + 20)*arcsin(a*x)))/a^4

________________________________________________________________________________________

Sympy [A]  time = 4.78095, size = 148, normalized size = 0.94 \begin{align*} \begin{cases} \frac{x^{3} \operatorname{asin}^{2}{\left (a x \right )}}{3 a} - \frac{2 x^{3}}{27 a} - \frac{x^{2} \sqrt{- a^{2} x^{2} + 1} \operatorname{asin}^{3}{\left (a x \right )}}{3 a^{2}} + \frac{2 x^{2} \sqrt{- a^{2} x^{2} + 1} \operatorname{asin}{\left (a x \right )}}{9 a^{2}} + \frac{2 x \operatorname{asin}^{2}{\left (a x \right )}}{a^{3}} - \frac{40 x}{9 a^{3}} - \frac{2 \sqrt{- a^{2} x^{2} + 1} \operatorname{asin}^{3}{\left (a x \right )}}{3 a^{4}} + \frac{40 \sqrt{- a^{2} x^{2} + 1} \operatorname{asin}{\left (a x \right )}}{9 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)**3/(-a**2*x**2+1)**(1/2),x)

[Out]

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

________________________________________________________________________________________

Giac [A]  time = 1.47098, size = 174, normalized size = 1.11 \begin{align*} \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}}{3 \, a^{4}} + \frac{9 \,{\left (a^{2} x^{2} - 1\right )} x \arcsin \left (a x\right )^{2} + 63 \, x \arcsin \left (a x\right )^{2} - 2 \,{\left (a^{2} x^{2} - 1\right )} x - \frac{6 \,{\left (-a^{2} x^{2} + 1\right )}^{\frac{3}{2}} \arcsin \left (a x\right )}{a} - 122 \, x + \frac{126 \, \sqrt{-a^{2} x^{2} + 1} \arcsin \left (a x\right )}{a}}{27 \, a^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*((-a^2*x^2 + 1)^(3/2) - 3*sqrt(-a^2*x^2 + 1))*arcsin(a*x)^3/a^4 + 1/27*(9*(a^2*x^2 - 1)*x*arcsin(a*x)^2 +
63*x*arcsin(a*x)^2 - 2*(a^2*x^2 - 1)*x - 6*(-a^2*x^2 + 1)^(3/2)*arcsin(a*x)/a - 122*x + 126*sqrt(-a^2*x^2 + 1)
*arcsin(a*x)/a)/a^3