3.134 \(\int \frac{x}{a^4+x^4} \, dx\)

Optimal. Leaf size=15 \[ \frac{\tan ^{-1}\left (\frac{x^2}{a^2}\right )}{2 a^2} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.0165114, antiderivative size = 15, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182 \[ \frac{\tan ^{-1}\left (\frac{x^2}{a^2}\right )}{2 a^2} \]

Antiderivative was successfully verified.

[In]  Int[x/(a^4 + x^4),x]

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 1.59015, size = 12, normalized size = 0.8 \[ \frac{\operatorname{atan}{\left (\frac{x^{2}}{a^{2}} \right )}}{2 a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x/(a**4+x**4),x)

[Out]

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

_______________________________________________________________________________________

Mathematica [A]  time = 0.00483174, size = 15, normalized size = 1. \[ \frac{\tan ^{-1}\left (\frac{x^2}{a^2}\right )}{2 a^2} \]

Antiderivative was successfully verified.

[In]  Integrate[x/(a^4 + x^4),x]

[Out]

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

_______________________________________________________________________________________

Maple [A]  time = 0.006, size = 14, normalized size = 0.9 \[{\frac{1}{2\,{a}^{2}}\arctan \left ({\frac{{x}^{2}}{{a}^{2}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x/(a^4+x^4),x)

[Out]

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

_______________________________________________________________________________________

Maxima [A]  time = 1.48336, size = 18, normalized size = 1.2 \[ \frac{\arctan \left (\frac{x^{2}}{a^{2}}\right )}{2 \, a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Fricas [A]  time = 0.195442, size = 18, normalized size = 1.2 \[ \frac{\arctan \left (\frac{x^{2}}{a^{2}}\right )}{2 \, a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Sympy [A]  time = 0.154332, size = 29, normalized size = 1.93 \[ \frac{- \frac{i \log{\left (- i a^{2} + x^{2} \right )}}{4} + \frac{i \log{\left (i a^{2} + x^{2} \right )}}{4}}{a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[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

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.201055, size = 18, normalized size = 1.2 \[ \frac{\arctan \left (\frac{x^{2}}{a^{2}}\right )}{2 \, a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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