3.1.69 \(\int \frac {x}{4+x^4} \, dx\) [69]

Optimal. Leaf size=12 \[ \frac {1}{4} \arctan \left (\frac {x^2}{2}\right ) \]

[Out]

1/4*arctan(1/2*x^2)

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Rubi [A]
time = 0.00, antiderivative size = 12, 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}{4} \arctan \left (\frac {x^2}{2}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x/(4 + x^4),x]

[Out]

ArcTan[x^2/2]/4

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 {gather*} \begin {aligned} \text {Integral} &=\frac {1}{2} \text {Subst}\left (\int \frac {1}{4+x^2} \, dx,x,x^2\right )\\ &=\frac {1}{4} \arctan \left (\frac {x^2}{2}\right )\\ \end {aligned} \end {gather*}

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Mathematica [A]
time = 0.00, size = 12, normalized size = 1.00 \begin {gather*} \frac {1}{4} \arctan \left (\frac {x^2}{2}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x/(4 + x^4),x]

[Out]

ArcTan[x^2/2]/4

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Maple [A]
time = 0.07, size = 9, normalized size = 0.75

method	result	size
default	\(\frac {\arctan \left (\frac {x^{2}}{2}\right )}{4}\)	\(9\)
meijerg	\(\frac {\arctan \left (\frac {x^{2}}{2}\right )}{4}\)	\(9\)
risch	\(\frac {\arctan \left (\frac {x^{2}}{2}\right )}{4}\)	\(9\)
parallelrisch	\(-\frac {i \ln \left (x -1-i\right )}{8}+\frac {i \ln \left (x -1+i\right )}{8}+\frac {i \ln \left (x +1-i\right )}{8}-\frac {i \ln \left (x +1+i\right )}{8}\)	\(38\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*arctan(1/2*x^2)

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Maxima [A]
time = 0.43, size = 8, normalized size = 0.67 \begin {gather*} \frac {1}{4} \, \arctan \left (\frac {1}{2} \, x^{2}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*arctan(1/2*x^2)

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Fricas [A]
time = 0.57, size = 8, normalized size = 0.67 \begin {gather*} \frac {1}{4} \, \arctan \left (\frac {1}{2} \, x^{2}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*arctan(1/2*x^2)

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Sympy [A]
time = 0.05, size = 7, normalized size = 0.58 \begin {gather*} \frac {\operatorname {atan}{\left (\frac {x^{2}}{2} \right )}}{4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

atan(x**2/2)/4

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Giac [A]
time = 0.41, size = 8, normalized size = 0.67 \begin {gather*} \frac {1}{4} \, \arctan \left (\frac {1}{2} \, x^{2}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*arctan(1/2*x^2)

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Mupad [B]
time = 0.09, size = 8, normalized size = 0.67 \begin {gather*} \frac {\mathrm {atan}\left (\frac {x^2}{2}\right )}{4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x/(x^4 + 4),x)

[Out]

atan(x^2/2)/4

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Chatgpt [A]
time = 1.00, size = 8, normalized size = 0.67 \begin {gather*} \frac {\arctan \left (\frac {x^{2}}{2}\right )}{4} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

1/4*arctan(1/2*x^2)

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