3.194 \(\int \frac{x}{\sqrt{1-x^4}} \, dx\)

Optimal. Leaf size=8 \[ \frac{1}{2} \sin ^{-1}\left (x^2\right ) \]

[Out]

ArcSin[x^2]/2

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Rubi [A]  time = 0.0027417, antiderivative size = 8, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {275, 216} \[ \frac{1}{2} \sin ^{-1}\left (x^2\right ) \]

Antiderivative was successfully verified.

[In]

Int[x/Sqrt[1 - x^4],x]

[Out]

ArcSin[x^2]/2

Rule 275

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = GCD[m + 1, n]}, Dist[1/k, Subst[Int[x^((m
+ 1)/k - 1)*(a + b*x^(n/k))^p, x], x, x^k], x] /; k != 1] /; FreeQ[{a, b, p}, x] && IGtQ[n, 0] && IntegerQ[m]

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 \frac{x}{\sqrt{1-x^4}} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{\sqrt{1-x^2}} \, dx,x,x^2\right )\\ &=\frac{1}{2} \sin ^{-1}\left (x^2\right )\\ \end{align*}

Mathematica [A]  time = 0.002432, size = 8, normalized size = 1. \[ \frac{1}{2} \sin ^{-1}\left (x^2\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[x/Sqrt[1 - x^4],x]

[Out]

ArcSin[x^2]/2

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Maple [A]  time = 0.008, size = 7, normalized size = 0.9 \begin{align*}{\frac{\arcsin \left ({x}^{2} \right ) }{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x/(-x^4+1)^(1/2),x)

[Out]

1/2*arcsin(x^2)

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Maxima [B]  time = 1.42137, size = 22, normalized size = 2.75 \begin{align*} -\frac{1}{2} \, \arctan \left (\frac{\sqrt{-x^{4} + 1}}{x^{2}}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/(-x^4+1)^(1/2),x, algorithm="maxima")

[Out]

-1/2*arctan(sqrt(-x^4 + 1)/x^2)

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Fricas [B]  time = 1.55369, size = 47, normalized size = 5.88 \begin{align*} -\arctan \left (\frac{\sqrt{-x^{4} + 1} - 1}{x^{2}}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/(-x^4+1)^(1/2),x, algorithm="fricas")

[Out]

-arctan((sqrt(-x^4 + 1) - 1)/x^2)

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Sympy [A]  time = 0.964442, size = 19, normalized size = 2.38 \begin{align*} \begin{cases} - \frac{i \operatorname{acosh}{\left (x^{2} \right )}}{2} & \text{for}\: \left |{x^{4}}\right | > 1 \\\frac{\operatorname{asin}{\left (x^{2} \right )}}{2} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/(-x**4+1)**(1/2),x)

[Out]

Piecewise((-I*acosh(x**2)/2, Abs(x**4) > 1), (asin(x**2)/2, True))

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Giac [A]  time = 1.11262, size = 8, normalized size = 1. \begin{align*} \frac{1}{2} \, \arcsin \left (x^{2}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/(-x^4+1)^(1/2),x, algorithm="giac")

[Out]

1/2*arcsin(x^2)