\(\int \frac {x}{a^4+x^4} \, dx\) [134]

Optimal result

Integrand size = 11, antiderivative size = 15 \[ \int \frac {x}{a^4+x^4} \, dx=\frac {\arctan \left (\frac {x^2}{a^2}\right )}{2 a^2} \]

[Out]

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

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 15, 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, 209} \[ \int \frac {x}{a^4+x^4} \, dx=\frac {\arctan \left (\frac {x^2}{a^2}\right )}{2 a^2} \]

[In]

Int[x/(a^4 + x^4),x]

[Out]

ArcTan[x^2/a^2]/(2*a^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*} \text {integral}& = \frac {1}{2} \text {Subst}\left (\int \frac {1}{a^4+x^2} \, dx,x,x^2\right ) \\ & = \frac {\arctan \left (\frac {x^2}{a^2}\right )}{2 a^2} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.00 \[ \int \frac {x}{a^4+x^4} \, dx=\frac {\arctan \left (\frac {x^2}{a^2}\right )}{2 a^2} \]

[In]

Integrate[x/(a^4 + x^4),x]

[Out]

ArcTan[x^2/a^2]/(2*a^2)

Maple [A] (verified)

Time = 0.24 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.93

method	result	size
default	\(\frac {\arctan \left (\frac {x^{2}}{a^{2}}\right )}{2 a^{2}}\)	\(14\)
risch	\(\frac {\arctan \left (\frac {x^{2}}{a^{2}}\right )}{2 a^{2}}\)	\(14\)
parallelrisch	\(-\frac {i \ln \left (-i a^{2}+x^{2}\right )-i \ln \left (i a^{2}+x^{2}\right )}{4 a^{2}}\)	\(35\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.87 \[ \int \frac {x}{a^4+x^4} \, dx=\frac {\arctan \left (\frac {x^{2}}{a^{2}}\right )}{2 \, a^{2}} \]

[In]

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

[Out]

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

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.07 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.93 \[ \int \frac {x}{a^4+x^4} \, dx=\frac {- \frac {i \log {\left (- i a^{2} + x^{2} \right )}}{4} + \frac {i \log {\left (i a^{2} + x^{2} \right )}}{4}}{a^{2}} \]

[In]

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

[Out]

(-I*log(-I*a**2 + x**2)/4 + I*log(I*a**2 + x**2)/4)/a**2

Maxima [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.87 \[ \int \frac {x}{a^4+x^4} \, dx=\frac {\arctan \left (\frac {x^{2}}{a^{2}}\right )}{2 \, a^{2}} \]

[In]

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

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.32 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.87 \[ \int \frac {x}{a^4+x^4} \, dx=\frac {\arctan \left (\frac {x^{2}}{a^{2}}\right )}{2 \, a^{2}} \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 0.26 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.87 \[ \int \frac {x}{a^4+x^4} \, dx=\frac {\mathrm {atan}\left (\frac {x^2}{a^2}\right )}{2\,a^2} \]

[In]

int(x/(a^4 + x^4),x)

[Out]

atan(x^2/a^2)/(2*a^2)