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

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

[Out]

1/2*arcsin(x^2)

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Rubi [A]  time = 0.00, antiderivative size = 8, normalized size of antiderivative = 1.00, 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 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]

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]

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*}

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Mathematica [A]  time = 0.00, size = 8, normalized size = 1.00 \[ \frac {1}{2} \sin ^{-1}\left (x^2\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[x/Sqrt[1 - x^4],x]

[Out]

ArcSin[x^2]/2

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fricas [B]  time = 0.40, size = 18, normalized size = 2.25 \[ -\arctan \left (\frac {\sqrt {-x^{4} + 1} - 1}{x^{2}}\right ) \]

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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giac [A]  time = 1.23, size = 6, normalized size = 0.75 \[ \frac {1}{2} \, \arcsin \left (x^{2}\right ) \]

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)

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maple [A]  time = 0.01, size = 7, normalized size = 0.88 \[ \frac {\arcsin \left (x^{2}\right )}{2} \]

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 = 0.96, size = 16, normalized size = 2.00 \[ -\frac {1}{2} \, \arctan \left (\frac {\sqrt {-x^{4} + 1}}{x^{2}}\right ) \]

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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mupad [B]  time = 0.31, size = 16, normalized size = 2.00 \[ \frac {\mathrm {atan}\left (\frac {x^2}{\sqrt {1-x^4}}\right )}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x/(1 - x^4)^(1/2),x)

[Out]

atan(x^2/(1 - x^4)^(1/2))/2

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sympy [A]  time = 0.92, size = 19, normalized size = 2.38 \[ \begin {cases} - \frac {i \operatorname {acosh}{\left (x^{2} \right )}}{2} & \text {for}\: \left |{x^{4}}\right | > 1 \\\frac {\operatorname {asin}{\left (x^{2} \right )}}{2} & \text {otherwise} \end {cases} \]

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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