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

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

[Out]

x*ArcSin[x/Sqrt[1 - x^2]] + ArcTan[Sqrt[1 - 2*x^2]]

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Rubi [A]  time = 0.0241646, antiderivative size = 29, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286, Rules used = {4840, 444, 63, 203} \[ x \sin ^{-1}\left (\frac{x}{\sqrt{1-x^2}}\right )+\tan ^{-1}\left (\sqrt{1-2 x^2}\right ) \]

Antiderivative was successfully verified.

[In]

Int[ArcSin[x/Sqrt[1 - x^2]],x]

[Out]

x*ArcSin[x/Sqrt[1 - x^2]] + ArcTan[Sqrt[1 - 2*x^2]]

Rule 4840

Int[ArcSin[u_], x_Symbol] :> Simp[x*ArcSin[u], x] - Int[SimplifyIntegrand[(x*D[u, x])/Sqrt[1 - u^2], x], x] /;
InverseFunctionFreeQ[u, x] &&  !FunctionOfExponentialQ[u, x]

Rule 444

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> Dist[1/n, Subst[Int
[(a + b*x)^p*(c + d*x)^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p, q}, x] && NeQ[b*c - a*d, 0] && EqQ[m
- n + 1, 0]

Rule 63

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - (a*d)/b + (d*x^p)/b)^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 203

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

Rubi steps

\begin{align*} \int \sin ^{-1}\left (\frac{x}{\sqrt{1-x^2}}\right ) \, dx &=x \sin ^{-1}\left (\frac{x}{\sqrt{1-x^2}}\right )-\int \frac{x}{\sqrt{1-2 x^2} \left (1-x^2\right )} \, dx\\ &=x \sin ^{-1}\left (\frac{x}{\sqrt{1-x^2}}\right )-\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{\sqrt{1-2 x} (1-x)} \, dx,x,x^2\right )\\ &=x \sin ^{-1}\left (\frac{x}{\sqrt{1-x^2}}\right )+\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{\frac{1}{2}+\frac{x^2}{2}} \, dx,x,\sqrt{1-2 x^2}\right )\\ &=x \sin ^{-1}\left (\frac{x}{\sqrt{1-x^2}}\right )+\tan ^{-1}\left (\sqrt{1-2 x^2}\right )\\ \end{align*}

Mathematica [A]  time = 0.0111224, size = 29, normalized size = 1. \[ x \sin ^{-1}\left (\frac{x}{\sqrt{1-x^2}}\right )+\tan ^{-1}\left (\sqrt{1-2 x^2}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[ArcSin[x/Sqrt[1 - x^2]],x]

[Out]

x*ArcSin[x/Sqrt[1 - x^2]] + ArcTan[Sqrt[1 - 2*x^2]]

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Maple [B]  time = 0.072, size = 138, normalized size = 4.8 \begin{align*} x\arcsin \left ({x{\frac{1}{\sqrt{-{x}^{2}+1}}}} \right ) +{\frac{1}{ \left ( 2+\sqrt{2} \right ) \left ( -2+\sqrt{2} \right ) }\sqrt{{\frac{2\,{x}^{2}-1}{{x}^{2}-1}}}\sqrt{-{x}^{2}+1} \left ( \sqrt{-2\,{x}^{2}+1}-\arctan \left ({(1+2\,x){\frac{1}{\sqrt{-2\,{x}^{2}+1}}}} \right ) +\arctan \left ({(2\,x-1){\frac{1}{\sqrt{-2\,{x}^{2}+1}}}} \right ) \right ){\frac{1}{\sqrt{-2\,{x}^{2}+1}}}}+{\frac{1}{2}\sqrt{{\frac{2\,{x}^{2}-1}{{x}^{2}-1}}}\sqrt{-{x}^{2}+1}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \arcsin \left (\frac{x}{\sqrt{-x^{2} + 1}}\right )\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(arcsin(x/sqrt(-x^2 + 1)), x)

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Fricas [B]  time = 2.45746, size = 146, normalized size = 5.03 \begin{align*} -x \arcsin \left (\frac{\sqrt{-x^{2} + 1} x}{x^{2} - 1}\right ) + \arctan \left (\frac{x^{2} + \sqrt{-x^{2} + 1} \sqrt{\frac{2 \, x^{2} - 1}{x^{2} - 1}} - 1}{x^{2}}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(asin(x/(-x**2+1)**(1/2)),x)

[Out]

Timed out

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Giac [A]  time = 1.15469, size = 46, normalized size = 1.59 \begin{align*} x \arcsin \left (\frac{x}{\sqrt{-x^{2} + 1}}\right ) + \frac{\arctan \left (\sqrt{-2 \, x^{2} + 1}\right )}{\mathrm{sgn}\left (x^{2} - 1\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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