3.664 \(\int \frac{x}{x+\sqrt{x^6}} \, dx\)

Optimal. Leaf size=45 \[ \frac{\sqrt{x^6} \tan ^{-1}(x)}{2 x^3}-\frac{\sqrt{x^6} \tanh ^{-1}(x)}{2 x^3}+\frac{1}{2} \tan ^{-1}(x)+\frac{1}{2} \tanh ^{-1}(x) \]

[Out]

ArcTan[x]/2 + (Sqrt[x^6]*ArcTan[x])/(2*x^3) + ArcTanh[x]/2 - (Sqrt[x^6]*ArcTanh[
x])/(2*x^3)

_______________________________________________________________________________________

Rubi [A]  time = 0.216194, antiderivative size = 45, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 8, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.615 \[ \frac{\sqrt{x^6} \tan ^{-1}(x)}{2 x^3}-\frac{\sqrt{x^6} \tanh ^{-1}(x)}{2 x^3}+\frac{1}{2} \tan ^{-1}(x)+\frac{1}{2} \tanh ^{-1}(x) \]

Antiderivative was successfully verified.

[In]  Int[x/(x + Sqrt[x^6]),x]

[Out]

ArcTan[x]/2 + (Sqrt[x^6]*ArcTan[x])/(2*x^3) + ArcTanh[x]/2 - (Sqrt[x^6]*ArcTanh[
x])/(2*x^3)

_______________________________________________________________________________________

Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \frac{x}{2} + \frac{\sqrt{x^{6}}}{6} + \frac{\operatorname{atan}{\left (x \right )}}{2} + \int \frac{x - \sqrt{x^{6}}}{x}\, dx + \int \frac{- \frac{x}{4} + \frac{\sqrt{x^{6}}}{4}}{x - 1}\, dx + \int \frac{- \frac{x}{4} + \frac{\sqrt{x^{6}}}{4}}{x + 1}\, dx - \frac{\sqrt{x^{6}}}{2 x^{2}} + \frac{\sqrt{x^{6}} \operatorname{atan}{\left (x \right )}}{2 x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x/(x+(x**6)**(1/2)),x)

[Out]

-x/2 + sqrt(x**6)/6 + atan(x)/2 + Integral((x - sqrt(x**6))/x, x) + Integral((-x
/4 + sqrt(x**6)/4)/(x - 1), x) + Integral((-x/4 + sqrt(x**6)/4)/(x + 1), x) - sq
rt(x**6)/(2*x**2) + sqrt(x**6)*atan(x)/(2*x**3)

_______________________________________________________________________________________

Mathematica [A]  time = 0.0681589, size = 0, normalized size = 0. \[ \int \frac{x}{x+\sqrt{x^6}} \, dx \]

Verification is Not applicable to the result.

[In]  Integrate[x/(x + Sqrt[x^6]),x]

[Out]

Integrate[x/(x + Sqrt[x^6]), x]

_______________________________________________________________________________________

Maple [A]  time = 0.012, size = 27, normalized size = 0.6 \[{1\arctan \left ( \sqrt{{\frac{1}{{x}^{3}}\sqrt{{x}^{6}}}}x \right ){\frac{1}{\sqrt{{\frac{1}{{x}^{3}}\sqrt{{x}^{6}}}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x/(x+(x^6)^(1/2)),x)

[Out]

1/((x^6)^(1/2)/x^3)^(1/2)*arctan(((x^6)^(1/2)/x^3)^(1/2)*x)

_______________________________________________________________________________________

Maxima [A]  time = 0.833021, size = 3, normalized size = 0.07 \[ \arctan \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x/(x + sqrt(x^6)),x, algorithm="maxima")

[Out]

arctan(x)

_______________________________________________________________________________________

Fricas [A]  time = 0.26283, size = 3, normalized size = 0.07 \[ \arctan \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x/(x + sqrt(x^6)),x, algorithm="fricas")

[Out]

arctan(x)

_______________________________________________________________________________________

Sympy [A]  time = 0.180629, size = 2, normalized size = 0.04 \[ \operatorname{atan}{\left (x \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x/(x+(x**6)**(1/2)),x)

[Out]

atan(x)

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.264695, size = 16, normalized size = 0.36 \[ \frac{\arctan \left (x \sqrt{{\rm sign}\left (x\right )}\right )}{\sqrt{{\rm sign}\left (x\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x/(x + sqrt(x^6)),x, algorithm="giac")

[Out]

arctan(x*sqrt(sign(x)))/sqrt(sign(x))