3.10.74 \(\int \frac {x}{\sqrt {-4+x^4}} \, dx\) [974]

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

[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 = {281, 223, 212} \begin {gather*} \frac {1}{2} \tanh ^{-1}\left (\frac {x^2}{\sqrt {x^4-4}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x/Sqrt[-4 + x^4],x]

[Out]

ArcTanh[x^2/Sqrt[-4 + x^4]]/2

Rule 212

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

Rule 223

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 281

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} \text {Subst}\left (\int \frac {1}{\sqrt {-4+x^2}} \, dx,x,x^2\right )\\ &=\frac {1}{2} \text {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 = 0.09, size = 18, normalized size = 1.00 \begin {gather*} \frac {1}{2} \tanh ^{-1}\left (\frac {\sqrt {-4+x^4}}{x^2}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x/Sqrt[-4 + x^4],x]

[Out]

ArcTanh[Sqrt[-4 + x^4]/x^2]/2

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Maple [A]
time = 0.17, size = 15, normalized size = 0.83

method	result	size
default	\(\frac {\ln \left (x^{2}+\sqrt {x^{4}-4}\right )}{2}\)	\(15\)
trager	\(\frac {\ln \left (x^{2}+\sqrt {x^{4}-4}\right )}{2}\)	\(15\)
elliptic	\(\frac {\ln \left (x^{2}+\sqrt {x^{4}-4}\right )}{2}\)	\(15\)
meijerg	\(\frac {\sqrt {-\mathrm {signum}\left (-1+\frac {x^{4}}{4}\right )}\, \arcsin \left (\frac {x^{2}}{2}\right )}{2 \sqrt {\mathrm {signum}\left (-1+\frac {x^{4}}{4}\right )}}\)	\(31\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x/(x^4-4)^(1/2),x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 33 vs. \(2 (14) = 28\).
time = 0.29, size = 33, normalized size = 1.83 \begin {gather*} \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 {gather*}

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 = 0.36, size = 16, normalized size = 0.89 \begin {gather*} -\frac {1}{2} \, \log \left (-x^{2} + \sqrt {x^{4} - 4}\right ) \end {gather*}

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 [C] Result contains complex when optimal does not.
time = 0.41, size = 22, normalized size = 1.22 \begin {gather*} \begin {cases} \frac {\operatorname {acosh}{\left (\frac {x^{2}}{2} \right )}}{2} & \text {for}\: \left |{x^{4}}\right | > 4 \\- \frac {i \operatorname {asin}{\left (\frac {x^{2}}{2} \right )}}{2} & \text {otherwise} \end {cases} \end {gather*}

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), (-I*asin(x**2/2)/2, True))

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Giac [A]
time = 0.94, size = 16, normalized size = 0.89 \begin {gather*} -\frac {1}{2} \, \log \left (x^{2} - \sqrt {x^{4} - 4}\right ) \end {gather*}

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

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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