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

Optimal. Leaf size=67 \[ -\frac{\sqrt{1-a^2 x^2} \sin ^{-1}(a x)^3}{a^2}+\frac{6 \sqrt{1-a^2 x^2} \sin ^{-1}(a x)}{a^2}-\frac{6 x}{a}+\frac{3 x \sin ^{-1}(a x)^2}{a} \]

[Out]

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

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Rubi [A]  time = 0.105192, antiderivative size = 67, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136, Rules used = {4677, 4619, 8} \[ -\frac{\sqrt{1-a^2 x^2} \sin ^{-1}(a x)^3}{a^2}+\frac{6 \sqrt{1-a^2 x^2} \sin ^{-1}(a x)}{a^2}-\frac{6 x}{a}+\frac{3 x \sin ^{-1}(a x)^2}{a} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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]

Rubi steps

\begin{align*} \int \frac{x \sin ^{-1}(a x)^3}{\sqrt{1-a^2 x^2}} \, dx &=-\frac{\sqrt{1-a^2 x^2} \sin ^{-1}(a x)^3}{a^2}+\frac{3 \int \sin ^{-1}(a x)^2 \, dx}{a}\\ &=\frac{3 x \sin ^{-1}(a x)^2}{a}-\frac{\sqrt{1-a^2 x^2} \sin ^{-1}(a x)^3}{a^2}-6 \int \frac{x \sin ^{-1}(a x)}{\sqrt{1-a^2 x^2}} \, dx\\ &=\frac{6 \sqrt{1-a^2 x^2} \sin ^{-1}(a x)}{a^2}+\frac{3 x \sin ^{-1}(a x)^2}{a}-\frac{\sqrt{1-a^2 x^2} \sin ^{-1}(a x)^3}{a^2}-\frac{6 \int 1 \, dx}{a}\\ &=-\frac{6 x}{a}+\frac{6 \sqrt{1-a^2 x^2} \sin ^{-1}(a x)}{a^2}+\frac{3 x \sin ^{-1}(a x)^2}{a}-\frac{\sqrt{1-a^2 x^2} \sin ^{-1}(a x)^3}{a^2}\\ \end{align*}

Mathematica [A]  time = 0.0167272, size = 61, normalized size = 0.91 \[ \frac{-\sqrt{1-a^2 x^2} \sin ^{-1}(a x)^3+6 \sqrt{1-a^2 x^2} \sin ^{-1}(a x)-6 a x+3 a x \sin ^{-1}(a x)^2}{a^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.046, size = 107, normalized size = 1.6 \begin{align*} -{\frac{1}{{a}^{2} \left ({a}^{2}{x}^{2}-1 \right ) }\sqrt{-{a}^{2}{x}^{2}+1} \left ( \left ( \arcsin \left ( ax \right ) \right ) ^{3}{x}^{2}{a}^{2}- \left ( \arcsin \left ( ax \right ) \right ) ^{3}+3\, \left ( \arcsin \left ( ax \right ) \right ) ^{2}\sqrt{-{a}^{2}{x}^{2}+1}xa-6\,{a}^{2}{x}^{2}\arcsin \left ( ax \right ) +6\,\arcsin \left ( ax \right ) -6\,ax\sqrt{-{a}^{2}{x}^{2}+1} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.51214, size = 86, normalized size = 1.28 \begin{align*} \frac{3 \, x \arcsin \left (a x\right )^{2}}{a} - \frac{\sqrt{-a^{2} x^{2} + 1} \arcsin \left (a x\right )^{3}}{a^{2}} - \frac{6 \,{\left (x - \frac{\sqrt{-a^{2} x^{2} + 1} \arcsin \left (a x\right )}{a}\right )}}{a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.9285, size = 119, normalized size = 1.78 \begin{align*} \frac{3 \, a x \arcsin \left (a x\right )^{2} - 6 \, a x - \sqrt{-a^{2} x^{2} + 1}{\left (\arcsin \left (a x\right )^{3} - 6 \, \arcsin \left (a x\right )\right )}}{a^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 1.41598, size = 61, normalized size = 0.91 \begin{align*} \begin{cases} \frac{3 x \operatorname{asin}^{2}{\left (a x \right )}}{a} - \frac{6 x}{a} - \frac{\sqrt{- a^{2} x^{2} + 1} \operatorname{asin}^{3}{\left (a x \right )}}{a^{2}} + \frac{6 \sqrt{- a^{2} x^{2} + 1} \operatorname{asin}{\left (a x \right )}}{a^{2}} & \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*asin(a*x)**3/(-a**2*x**2+1)**(1/2),x)

[Out]

Piecewise((3*x*asin(a*x)**2/a - 6*x/a - sqrt(-a**2*x**2 + 1)*asin(a*x)**3/a**2 + 6*sqrt(-a**2*x**2 + 1)*asin(a
*x)/a**2, Ne(a, 0)), (0, True))

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Giac [A]  time = 1.41462, size = 84, normalized size = 1.25 \begin{align*} -\frac{\sqrt{-a^{2} x^{2} + 1} \arcsin \left (a x\right )^{3}}{a^{2}} + \frac{3 \,{\left (x \arcsin \left (a x\right )^{2} - 2 \, x + \frac{2 \, \sqrt{-a^{2} x^{2} + 1} \arcsin \left (a x\right )}{a}\right )}}{a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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