3.1.80 \(\int \frac {x}{1+x^4} \, dx\) [80]

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

[Out]

1/2*arctan(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 = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {281, 209} \begin {gather*} \frac {1}{2} \tan ^{-1}\left (x^2\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x/(1 + x^4),x]

[Out]

ArcTan[x^2]/2

Rule 209

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*ArcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 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}{1+x^4} \, dx &=\frac {1}{2} \text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,x^2\right )\\ &=\frac {1}{2} \tan ^{-1}\left (x^2\right )\\ \end {align*}

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Mathematica [A]
time = 0.00, size = 8, normalized size = 1.00 \begin {gather*} \frac {1}{2} \tan ^{-1}\left (x^2\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x/(1 + x^4),x]

[Out]

ArcTan[x^2]/2

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Mathics [A]
time = 1.78, size = 6, normalized size = 0.75 \begin {gather*} \frac {\text {ArcTan}\left [x^2\right ]}{2} \end {gather*}

Antiderivative was successfully verified.

[In]

mathics('Integrate[x/(1 + x^4),x]')

[Out]

ArcTan[x ^ 2] / 2

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Maple [A]
time = 0.03, size = 7, normalized size = 0.88

method	result	size
default	\(\frac {\arctan \left (x^{2}\right )}{2}\)	\(7\)
meijerg	\(\frac {\arctan \left (x^{2}\right )}{2}\)	\(7\)
risch	\(\frac {\arctan \left (x^{2}\right )}{2}\)	\(7\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*arctan(x^2)

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Maxima [A]
time = 0.35, size = 6, normalized size = 0.75 \begin {gather*} \frac {1}{2} \, \arctan \left (x^{2}\right ) \end {gather*}

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 = 0.31, size = 6, normalized size = 0.75 \begin {gather*} \frac {1}{2} \, \arctan \left (x^{2}\right ) \end {gather*}

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 = 0.05, size = 5, normalized size = 0.62 \begin {gather*} \frac {\operatorname {atan}{\left (x^{2} \right )}}{2} \end {gather*}

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 = 0.00, size = 7, normalized size = 0.88 \begin {gather*} \frac {\arctan \left (x^{2}\right )}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*arctan(x^2)

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Mupad [B]
time = 0.06, size = 6, normalized size = 0.75 \begin {gather*} \frac {\mathrm {atan}\left (x^2\right )}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x/(x^4 + 1),x)

[Out]

atan(x^2)/2

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