3.974 \(\int \frac{x}{\sqrt{-4+x^4}} \, dx\)

Optimal. Leaf size=18 \[ \frac{1}{2} \tanh ^{-1}\left (\frac{x^2}{\sqrt{x^4-4}}\right ) \]

[Out]

ArcTanh[x^2/Sqrt[-4 + x^4]]/2

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Rubi [A]  time = 0.0070415, antiderivative size = 18, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273, Rules used = {275, 217, 206} \[ \frac{1}{2} \tanh ^{-1}\left (\frac{x^2}{\sqrt{x^4-4}}\right ) \]

Antiderivative was successfully verified.

[In]

Int[x/Sqrt[-4 + x^4],x]

[Out]

ArcTanh[x^2/Sqrt[-4 + x^4]]/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 217

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a,
b}, x] &&  !GtQ[a, 0]

Rule 206

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

Rubi steps

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

Mathematica [A]  time = 0.0027142, size = 18, normalized size = 1. \[ \frac{1}{2} \tanh ^{-1}\left (\frac{x^2}{\sqrt{x^4-4}}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[x/Sqrt[-4 + x^4],x]

[Out]

ArcTanh[x^2/Sqrt[-4 + x^4]]/2

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Maple [A]  time = 0.01, size = 15, normalized size = 0.8 \begin{align*}{\frac{1}{2}\ln \left ({x}^{2}+\sqrt{{x}^{4}-4} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*ln(x^2+(x^4-4)^(1/2))

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Maxima [B]  time = 0.956952, size = 45, normalized size = 2.5 \begin{align*} \frac{1}{4} \, \log \left (\frac{\sqrt{x^{4} - 4}}{x^{2}} + 1\right ) - \frac{1}{4} \, \log \left (\frac{\sqrt{x^{4} - 4}}{x^{2}} - 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.455, size = 43, normalized size = 2.39 \begin{align*} -\frac{1}{2} \, \log \left (-x^{2} + \sqrt{x^{4} - 4}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*log(-x^2 + sqrt(x^4 - 4))

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Sympy [A]  time = 1.20737, size = 24, normalized size = 1.33 \begin{align*} \begin{cases} \frac{\operatorname{acosh}{\left (\frac{x^{2}}{2} \right )}}{2} & \text{for}\: \frac{\left |{x^{4}}\right |}{4} > 1 \\- \frac{i \operatorname{asin}{\left (\frac{x^{2}}{2} \right )}}{2} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.20838, size = 22, normalized size = 1.22 \begin{align*} -\frac{1}{2} \, \log \left (x^{2} - \sqrt{x^{4} - 4}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*log(x^2 - sqrt(x^4 - 4))