### 3.27 $$\int \sin ^{-1}(x) \, dx$$

Optimal. Leaf size=16 $\sqrt{1-x^2}+x \sin ^{-1}(x)$

[Out]

Sqrt[1 - x^2] + x*ArcSin[x]

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Rubi [A]  time = 0.0038397, antiderivative size = 16, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 2, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 1., Rules used = {4619, 261} $\sqrt{1-x^2}+x \sin ^{-1}(x)$

Antiderivative was successfully veriﬁed.

[In]

Int[ArcSin[x],x]

[Out]

Sqrt[1 - x^2] + x*ArcSin[x]

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

Mathematica [A]  time = 0.0020695, size = 16, normalized size = 1. $\sqrt{1-x^2}+x \sin ^{-1}(x)$

Antiderivative was successfully veriﬁed.

[In]

Integrate[ArcSin[x],x]

[Out]

Sqrt[1 - x^2] + x*ArcSin[x]

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Maple [A]  time = 0., size = 15, normalized size = 0.9 \begin{align*} x\arcsin \left ( x \right ) +\sqrt{-{x}^{2}+1} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(arcsin(x),x)

[Out]

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

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Maxima [A]  time = 1.40238, size = 19, normalized size = 1.19 \begin{align*} x \arcsin \left (x\right ) + \sqrt{-x^{2} + 1} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x*arcsin(x) + sqrt(-x^2 + 1)

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Fricas [A]  time = 2.02459, size = 41, normalized size = 2.56 \begin{align*} x \arcsin \left (x\right ) + \sqrt{-x^{2} + 1} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x*arcsin(x) + sqrt(-x^2 + 1)

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Sympy [A]  time = 0.11681, size = 12, normalized size = 0.75 \begin{align*} x \operatorname{asin}{\left (x \right )} + \sqrt{1 - x^{2}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(asin(x),x)

[Out]

x*asin(x) + sqrt(1 - x**2)

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Giac [A]  time = 1.06511, size = 19, normalized size = 1.19 \begin{align*} x \arcsin \left (x\right ) + \sqrt{-x^{2} + 1} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x*arcsin(x) + sqrt(-x^2 + 1)