3.923 \(\int \frac{x}{\sqrt{1+x^4}} \, dx\)

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

[Out]

ArcSinh[x^2]/2

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Rubi [A]  time = 0.0026005, antiderivative size = 8, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {275, 215} \[ \frac{1}{2} \sinh ^{-1}\left (x^2\right ) \]

Antiderivative was successfully verified.

[In]

Int[x/Sqrt[1 + x^4],x]

[Out]

ArcSinh[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 215

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSinh[(Rt[b, 2]*x)/Sqrt[a]]/Rt[b, 2], x] /; FreeQ[{a, b},
x] && GtQ[a, 0] && PosQ[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} \sinh ^{-1}\left (x^2\right )\\ \end{align*}

Mathematica [A]  time = 0.0018955, size = 8, normalized size = 1. \[ \frac{1}{2} \sinh ^{-1}\left (x^2\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[x/Sqrt[1 + x^4],x]

[Out]

ArcSinh[x^2]/2

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Maple [A]  time = 0.074, size = 7, normalized size = 0.9 \begin{align*}{\frac{{\it Arcsinh} \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*arcsinh(x^2)

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Maxima [B]  time = 0.983164, size = 45, normalized size = 5.62 \begin{align*} \frac{1}{4} \, \log \left (\frac{\sqrt{x^{4} + 1}}{x^{2}} + 1\right ) - \frac{1}{4} \, \log \left (\frac{\sqrt{x^{4} + 1}}{x^{2}} - 1\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]

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

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Fricas [B]  time = 1.48857, size = 43, normalized size = 5.38 \begin{align*} -\frac{1}{2} \, \log \left (-x^{2} + \sqrt{x^{4} + 1}\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]

-1/2*log(-x^2 + sqrt(x^4 + 1))

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Sympy [A]  time = 0.90055, size = 5, normalized size = 0.62 \begin{align*} \frac{\operatorname{asinh}{\left (x^{2} \right )}}{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

asinh(x**2)/2

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Giac [B]  time = 1.16653, size = 22, normalized size = 2.75 \begin{align*} -\frac{1}{2} \, \log \left (-x^{2} + \sqrt{x^{4} + 1}\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*log(-x^2 + sqrt(x^4 + 1))