### 3.273 $$\int x \sin ^{-1}(x) \, dx$$

Optimal. Leaf size=32 $\frac{1}{4} \sqrt{1-x^2} x+\frac{1}{2} x^2 \sin ^{-1}(x)-\frac{1}{4} \sin ^{-1}(x)$

[Out]

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

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Rubi [A]  time = 0.0101239, antiderivative size = 32, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 4, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.75, Rules used = {4627, 321, 216} $\frac{1}{4} \sqrt{1-x^2} x+\frac{1}{2} x^2 \sin ^{-1}(x)-\frac{1}{4} \sin ^{-1}(x)$

Antiderivative was successfully veriﬁed.

[In]

Int[x*ArcSin[x],x]

[Out]

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

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 321

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c^(n - 1)*(c*x)^(m - n + 1)*(a + b*x^n
)^(p + 1))/(b*(m + n*p + 1)), x] - Dist[(a*c^n*(m - n + 1))/(b*(m + n*p + 1)), Int[(c*x)^(m - n)*(a + b*x^n)^p
, x], x] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && GtQ[m, n - 1] && NeQ[m + n*p + 1, 0] && IntBinomialQ[a, b,
c, n, m, p, x]

Rule 216

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSin[(Rt[-b, 2]*x)/Sqrt[a]]/Rt[-b, 2], x] /; FreeQ[{a, b}
, x] && GtQ[a, 0] && NegQ[b]

Rubi steps

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

Mathematica [A]  time = 0.0091866, size = 28, normalized size = 0.88 $\frac{1}{4} \left (\sqrt{1-x^2} x+\left (2 x^2-1\right ) \sin ^{-1}(x)\right )$

Antiderivative was successfully veriﬁed.

[In]

Integrate[x*ArcSin[x],x]

[Out]

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

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Maple [A]  time = 0.003, size = 25, normalized size = 0.8 \begin{align*} -{\frac{\arcsin \left ( x \right ) }{4}}+{\frac{{x}^{2}\arcsin \left ( x \right ) }{2}}+{\frac{x}{4}\sqrt{-{x}^{2}+1}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(x*arcsin(x),x)

[Out]

-1/4*arcsin(x)+1/2*x^2*arcsin(x)+1/4*x*(-x^2+1)^(1/2)

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Maxima [A]  time = 1.4477, size = 32, normalized size = 1. \begin{align*} \frac{1}{2} \, x^{2} \arcsin \left (x\right ) + \frac{1}{4} \, \sqrt{-x^{2} + 1} x - \frac{1}{4} \, \arcsin \left (x\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*arcsin(x),x, algorithm="maxima")

[Out]

1/2*x^2*arcsin(x) + 1/4*sqrt(-x^2 + 1)*x - 1/4*arcsin(x)

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Fricas [A]  time = 1.98438, size = 68, normalized size = 2.12 \begin{align*} \frac{1}{4} \,{\left (2 \, x^{2} - 1\right )} \arcsin \left (x\right ) + \frac{1}{4} \, \sqrt{-x^{2} + 1} x \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*arcsin(x),x, algorithm="fricas")

[Out]

1/4*(2*x^2 - 1)*arcsin(x) + 1/4*sqrt(-x^2 + 1)*x

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Sympy [A]  time = 0.181605, size = 24, normalized size = 0.75 \begin{align*} \frac{x^{2} \operatorname{asin}{\left (x \right )}}{2} + \frac{x \sqrt{1 - x^{2}}}{4} - \frac{\operatorname{asin}{\left (x \right )}}{4} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*asin(x),x)

[Out]

x**2*asin(x)/2 + x*sqrt(1 - x**2)/4 - asin(x)/4

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Giac [A]  time = 1.07066, size = 35, normalized size = 1.09 \begin{align*} \frac{1}{2} \,{\left (x^{2} - 1\right )} \arcsin \left (x\right ) + \frac{1}{4} \, \sqrt{-x^{2} + 1} x + \frac{1}{4} \, \arcsin \left (x\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*arcsin(x),x, algorithm="giac")

[Out]

1/2*(x^2 - 1)*arcsin(x) + 1/4*sqrt(-x^2 + 1)*x + 1/4*arcsin(x)