\(\int \frac {x}{\sqrt {4+x^4}} \, dx\) [975]

Optimal result

Integrand size = 11, antiderivative size = 12 \[ \int \frac {x}{\sqrt {4+x^4}} \, dx=\frac {1}{2} \text {arcsinh}\left (\frac {x^2}{2}\right ) \]

[Out]

1/2*arcsinh(1/2*x^2)

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {281, 221} \[ \int \frac {x}{\sqrt {4+x^4}} \, dx=\frac {1}{2} \text {arcsinh}\left (\frac {x^2}{2}\right ) \]

[In]

Int[x/Sqrt[4 + x^4],x]

[Out]

ArcSinh[x^2/2]/2

Rule 221

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]

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

Mathematica [A] (verified)

Time = 0.22 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.50 \[ \int \frac {x}{\sqrt {4+x^4}} \, dx=\frac {1}{2} \log \left (x^2+\sqrt {4+x^4}\right ) \]

[In]

Integrate[x/Sqrt[4 + x^4],x]

[Out]

Log[x^2 + Sqrt[4 + x^4]]/2

Maple [A] (verified)

Time = 4.12 (sec) , antiderivative size = 9, normalized size of antiderivative = 0.75

method	result	size
default	\(\frac {\operatorname {arcsinh}\left (\frac {x^{2}}{2}\right )}{2}\)	\(9\)
meijerg	\(\frac {\operatorname {arcsinh}\left (\frac {x^{2}}{2}\right )}{2}\)	\(9\)
elliptic	\(\frac {\operatorname {arcsinh}\left (\frac {x^{2}}{2}\right )}{2}\)	\(9\)
pseudoelliptic	\(\frac {\operatorname {arcsinh}\left (\frac {x^{2}}{2}\right )}{2}\)	\(9\)
trager	\(-\frac {\ln \left (x^{2}-\sqrt {x^{4}+4}\right )}{2}\)	\(17\)

[In]

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

[Out]

1/2*arcsinh(1/2*x^2)

Fricas [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.33 \[ \int \frac {x}{\sqrt {4+x^4}} \, dx=-\frac {1}{2} \, \log \left (-x^{2} + \sqrt {x^{4} + 4}\right ) \]

[In]

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

[Out]

-1/2*log(-x^2 + sqrt(x^4 + 4))

Sympy [A] (verification not implemented)

Time = 0.45 (sec) , antiderivative size = 7, normalized size of antiderivative = 0.58 \[ \int \frac {x}{\sqrt {4+x^4}} \, dx=\frac {\operatorname {asinh}{\left (\frac {x^{2}}{2} \right )}}{2} \]

[In]

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

[Out]

asinh(x**2/2)/2

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 33 vs. \(2 (8) = 16\).

Time = 0.22 (sec) , antiderivative size = 33, normalized size of antiderivative = 2.75 \[ \int \frac {x}{\sqrt {4+x^4}} \, dx=\frac {1}{4} \, \log \left (\frac {\sqrt {x^{4} + 4}}{x^{2}} + 1\right ) - \frac {1}{4} \, \log \left (\frac {\sqrt {x^{4} + 4}}{x^{2}} - 1\right ) \]

[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)

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.33 \[ \int \frac {x}{\sqrt {4+x^4}} \, dx=-\frac {1}{2} \, \log \left (-x^{2} + \sqrt {x^{4} + 4}\right ) \]

[In]

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

[Out]

-1/2*log(-x^2 + sqrt(x^4 + 4))

Mupad [B] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 8, normalized size of antiderivative = 0.67 \[ \int \frac {x}{\sqrt {4+x^4}} \, dx=\frac {\mathrm {asinh}\left (\frac {x^2}{2}\right )}{2} \]

[In]

int(x/(x^4 + 4)^(1/2),x)

[Out]

asinh(x^2/2)/2