3.10.23 \(\int \frac {x}{\sqrt {1+x^4}} \, dx\) [923]

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

[Out]

1/2*arcsinh(x^2)

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Rubi [A]
time = 0.00, antiderivative size = 8, normalized size of antiderivative = 1.00, 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 = {281, 221} \begin {gather*} \frac {1}{2} \sinh ^{-1}\left (x^2\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x/Sqrt[1 + x^4],x]

[Out]

ArcSinh[x^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*} \int \frac {x}{\sqrt {1+x^4}} \, dx &=\frac {1}{2} \text {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*}

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(18\) vs. \(2(8)=16\).
time = 0.07, size = 18, normalized size = 2.25 \begin {gather*} \frac {1}{2} \tanh ^{-1}\left (\frac {x^2}{\sqrt {1+x^4}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x/Sqrt[1 + x^4],x]

[Out]

ArcTanh[x^2/Sqrt[1 + x^4]]/2

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Maple [A]
time = 0.14, size = 7, normalized size = 0.88

method	result	size
default	\(\frac {\arcsinh \left (x^{2}\right )}{2}\)	\(7\)
meijerg	\(\frac {\arcsinh \left (x^{2}\right )}{2}\)	\(7\)
elliptic	\(\frac {\arcsinh \left (x^{2}\right )}{2}\)	\(7\)
trager	\(\frac {\ln \left (x^{2}+\sqrt {x^{4}+1}\right )}{2}\)	\(15\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*arcsinh(x^2)

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 33 vs. \(2 (6) = 12\).
time = 0.30, size = 33, normalized size = 4.12 \begin {gather*} \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 {gather*}

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] Leaf count of result is larger than twice the leaf count of optimal. 16 vs. \(2 (6) = 12\).
time = 0.36, size = 16, normalized size = 2.00 \begin {gather*} -\frac {1}{2} \, \log \left (-x^{2} + \sqrt {x^{4} + 1}\right ) \end {gather*}

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 = 0.40, size = 5, normalized size = 0.62 \begin {gather*} \frac {\operatorname {asinh}{\left (x^{2} \right )}}{2} \end {gather*}

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] Leaf count of result is larger than twice the leaf count of optimal. 16 vs. \(2 (6) = 12\).
time = 2.06, size = 16, normalized size = 2.00 \begin {gather*} -\frac {1}{2} \, \log \left (-x^{2} + \sqrt {x^{4} + 1}\right ) \end {gather*}

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))

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Mupad [B]
time = 1.02, size = 6, normalized size = 0.75 \begin {gather*} \frac {\mathrm {asinh}\left (x^2\right )}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x/(x^4 + 1)^(1/2),x)

[Out]

asinh(x^2)/2

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