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

Optimal. Leaf size=55 \[ \frac{2 \sqrt{1-a^2 x^2}}{a^2}-\frac{\sqrt{1-a^2 x^2} \sin ^{-1}(a x)^2}{a^2}+\frac{2 x \sin ^{-1}(a x)}{a} \]

[Out]

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

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Rubi [A]  time = 0.0719107, antiderivative size = 55, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136, Rules used = {4677, 4619, 261} \[ \frac{2 \sqrt{1-a^2 x^2}}{a^2}-\frac{\sqrt{1-a^2 x^2} \sin ^{-1}(a x)^2}{a^2}+\frac{2 x \sin ^{-1}(a x)}{a} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rubi steps

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

Mathematica [A]  time = 0.0127663, size = 51, normalized size = 0.93 \[ \frac{2 \sqrt{1-a^2 x^2}-\sqrt{1-a^2 x^2} \sin ^{-1}(a x)^2+2 a x \sin ^{-1}(a x)}{a^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.044, size = 80, normalized size = 1.5 \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 ) ^{2}{x}^{2}{a}^{2}- \left ( \arcsin \left ( ax \right ) \right ) ^{2}+2\,\arcsin \left ( ax \right ) \sqrt{-{a}^{2}{x}^{2}+1}xa-2\,{a}^{2}{x}^{2}+2 \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

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Maxima [A]  time = 1.49674, size = 66, normalized size = 1.2 \begin{align*} -\frac{\sqrt{-a^{2} x^{2} + 1} \arcsin \left (a x\right )^{2}}{a^{2}} + \frac{2 \,{\left (a x \arcsin \left (a x\right ) + \sqrt{-a^{2} x^{2} + 1}\right )}}{a^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.6864, size = 89, normalized size = 1.62 \begin{align*} \frac{2 \, a x \arcsin \left (a x\right ) - \sqrt{-a^{2} x^{2} + 1}{\left (\arcsin \left (a x\right )^{2} - 2\right )}}{a^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0.87482, size = 49, normalized size = 0.89 \begin{align*} \begin{cases} \frac{2 x \operatorname{asin}{\left (a x \right )}}{a} - \frac{\sqrt{- a^{2} x^{2} + 1} \operatorname{asin}^{2}{\left (a x \right )}}{a^{2}} + \frac{2 \sqrt{- a^{2} x^{2} + 1}}{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)**2/(-a**2*x**2+1)**(1/2),x)

[Out]

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

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Giac [A]  time = 1.27632, size = 66, normalized size = 1.2 \begin{align*} -\frac{\sqrt{-a^{2} x^{2} + 1} \arcsin \left (a x\right )^{2}}{a^{2}} + \frac{2 \,{\left (a x \arcsin \left (a x\right ) + \sqrt{-a^{2} x^{2} + 1}\right )}}{a^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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