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]

1/2*arctanh(x^2/(x^4-4)^(1/2))

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Rubi [A]  time = 0.01, antiderivative size = 18, normalized size of antiderivative = 1.00, 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 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])

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 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 {-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*}

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Mathematica [A]  time = 0.00, size = 18, normalized size = 1.00 \[ \frac {1}{2} \tanh ^{-1}\left (\frac {x^2}{\sqrt {x^4-4}}\right ) \]

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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fricas [A]  time = 1.14, size = 16, normalized size = 0.89 \[ -\frac {1}{2} \, \log \left (-x^{2} + \sqrt {x^{4} - 4}\right ) \]

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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giac [A]  time = 0.16, size = 16, normalized size = 0.89 \[ -\frac {1}{2} \, \log \left (x^{2} - \sqrt {x^{4} - 4}\right ) \]

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))

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maple [A]  time = 0.01, size = 15, normalized size = 0.83 \[ \frac {\ln \left (x^{2}+\sqrt {x^{4}-4}\right )}{2} \]

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 = 1.35, size = 33, normalized size = 1.83 \[ \frac {1}{4} \, \log \left (\frac {\sqrt {x^{4} - 4}}{x^{2}} + 1\right ) - \frac {1}{4} \, \log \left (\frac {\sqrt {x^{4} - 4}}{x^{2}} - 1\right ) \]

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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mupad [B]  time = 0.08, size = 14, normalized size = 0.78 \[ \frac {\ln \left (\sqrt {x^4-4}+x^2\right )}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x/(x^4 - 4)^(1/2),x)

[Out]

log((x^4 - 4)^(1/2) + x^2)/2

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sympy [A]  time = 1.96, size = 24, normalized size = 1.33 \[ \begin {cases} \frac {\operatorname {acosh}{\left (\frac {x^{2}}{2} \right )}}{2} & \text {for}\: \frac {\left |{x^{4}}\right |}{4} > 1 \\- \frac {i \operatorname {asin}{\left (\frac {x^{2}}{2} \right )}}{2} & \text {otherwise} \end {cases} \]

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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