Optimal. Leaf size=48 \[ \frac{x^2 \left (x^n\right )^{-2/n}}{\left (x^n\right )^{\frac{1}{n}}+1}+x^2 \left (x^n\right )^{-2/n} \log \left (\left (x^n\right )^{\frac{1}{n}}+1\right ) \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.0304985, antiderivative size = 48, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154 \[ \frac{x^2 \left (x^n\right )^{-2/n}}{\left (x^n\right )^{\frac{1}{n}}+1}+x^2 \left (x^n\right )^{-2/n} \log \left (\left (x^n\right )^{\frac{1}{n}}+1\right ) \]
Antiderivative was successfully verified.
[In] Int[x/(1 + (x^n)^n^(-1))^2,x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [A] time = 4.29697, size = 37, normalized size = 0.77 \[ x^{2} \left (x^{n}\right )^{- \frac{2}{n}} \log{\left (\left (x^{n}\right )^{\frac{1}{n}} + 1 \right )} + \frac{x^{2} \left (x^{n}\right )^{- \frac{2}{n}}}{\left (x^{n}\right )^{\frac{1}{n}} + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x/(1+(x**n)**(1/n))**2,x)
[Out]
_______________________________________________________________________________________
Mathematica [A] time = 2.12276, size = 0, normalized size = 0. \[ \int \frac{x}{\left (1+\left (x^n\right )^{\frac{1}{n}}\right )^2} \, dx \]
Verification is Not applicable to the result.
[In] Integrate[x/(1 + (x^n)^n^(-1))^2,x]
[Out]
_______________________________________________________________________________________
Maple [A] time = 0.058, size = 76, normalized size = 1.6 \[{\frac{{x}^{2}}{1+\sqrt [n]{{x}^{n}}}}-{{\rm e}^{{\frac{n\ln \left ( x \right ) -\ln \left ({x}^{n} \right ) }{n}}}}x+{{\rm e}^{2\,{\frac{n\ln \left ( x \right ) -\ln \left ({x}^{n} \right ) }{n}}}}\ln \left ( 1+{{\rm e}^{-{\frac{n\ln \left ( x \right ) -\ln \left ({x}^{n} \right ) }{n}}}}x \right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x/(1+(x^n)^(1/n))^2,x)
[Out]
_______________________________________________________________________________________
Maxima [A] time = 22.1823, size = 31, normalized size = 0.65 \[ -x + \frac{x^{2}}{{\left (x^{n}\right )}^{\left (\frac{1}{n}\right )} + 1} + \log \left (x + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x/((x^n)^(1/n) + 1)^2,x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [A] time = 0.222028, size = 22, normalized size = 0.46 \[ \frac{{\left (x + 1\right )} \log \left (x + 1\right ) + 1}{x + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x/((x^n)^(1/n) + 1)^2,x, algorithm="fricas")
[Out]
_______________________________________________________________________________________
Sympy [A] time = 0.135808, size = 19, normalized size = 0.4 \[ \log{\left (\left (x^{n}\right )^{\frac{1}{n}} + 1 \right )} + \frac{1}{\left (x^{n}\right )^{\frac{1}{n}} + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x/(1+(x**n)**(1/n))**2,x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [A] time = 0.224262, size = 15, normalized size = 0.31 \[ \frac{1}{x + 1} +{\rm ln}\left ({\left | x + 1 \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x/((x^n)^(1/n) + 1)^2,x, algorithm="giac")
[Out]