3.80 \(\int \frac{x}{1+x^4} \, dx\)

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

[Out]

ArcTan[x^2]/2

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Rubi [A]  time = 0.0031274, antiderivative size = 8, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222, Rules used = {275, 203} \[ \frac{1}{2} \tan ^{-1}\left (x^2\right ) \]

Antiderivative was successfully verified.

[In]

Int[x/(1 + x^4),x]

[Out]

ArcTan[x^2]/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 203

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

Rubi steps

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

Mathematica [A]  time = 0.003168, size = 8, normalized size = 1. \[ \frac{1}{2} \tan ^{-1}\left (x^2\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[x/(1 + x^4),x]

[Out]

ArcTan[x^2]/2

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Maple [A]  time = 0.002, size = 7, normalized size = 0.9 \begin{align*}{\frac{\arctan \left ({x}^{2} \right ) }{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x/(x^4+1),x)

[Out]

1/2*arctan(x^2)

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Maxima [A]  time = 1.4093, size = 8, normalized size = 1. \begin{align*} \frac{1}{2} \, \arctan \left (x^{2}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*arctan(x^2)

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Fricas [A]  time = 2.18864, size = 23, normalized size = 2.88 \begin{align*} \frac{1}{2} \, \arctan \left (x^{2}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*arctan(x^2)

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Sympy [A]  time = 0.086513, size = 5, normalized size = 0.62 \begin{align*} \frac{\operatorname{atan}{\left (x^{2} \right )}}{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

atan(x**2)/2

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Giac [A]  time = 1.0647, size = 8, normalized size = 1. \begin{align*} \frac{1}{2} \, \arctan \left (x^{2}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*arctan(x^2)