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

Optimal. Leaf size=29 \[ \frac{x}{a}-\frac{\sqrt{1-a^2 x^2} \sin ^{-1}(a x)}{a^2} \]

[Out]

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

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Rubi [A]  time = 0.0405081, antiderivative size = 29, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, Rules used = {4677, 8} \[ \frac{x}{a}-\frac{\sqrt{1-a^2 x^2} \sin ^{-1}(a x)}{a^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

x/a - (Sqrt[1 - a^2*x^2]*ArcSin[a*x])/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 8

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

Rubi steps

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

Mathematica [A]  time = 0.0083922, size = 29, normalized size = 1. \[ \frac{x}{a}-\frac{\sqrt{1-a^2 x^2} \sin ^{-1}(a x)}{a^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [B]  time = 0.038, size = 62, normalized size = 2.1 \begin{align*} -{\frac{1}{{a}^{2} \left ({a}^{2}{x}^{2}-1 \right ) }\sqrt{-{a}^{2}{x}^{2}+1} \left ({a}^{2}{x}^{2}\arcsin \left ( ax \right ) -\arcsin \left ( ax \right ) +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)/(-a^2*x^2+1)^(1/2),x)

[Out]

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

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Maxima [A]  time = 1.53852, size = 36, normalized size = 1.24 \begin{align*} \frac{x}{a} - \frac{\sqrt{-a^{2} x^{2} + 1} \arcsin \left (a x\right )}{a^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 2.13272, size = 59, normalized size = 2.03 \begin{align*} \frac{a x - \sqrt{-a^{2} x^{2} + 1} \arcsin \left (a x\right )}{a^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0.521907, size = 24, normalized size = 0.83 \begin{align*} \begin{cases} \frac{x}{a} - \frac{\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)/(-a**2*x**2+1)**(1/2),x)

[Out]

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

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Giac [A]  time = 1.40187, size = 36, normalized size = 1.24 \begin{align*} \frac{x}{a} - \frac{\sqrt{-a^{2} x^{2} + 1} \arcsin \left (a x\right )}{a^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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