3.3014 \(\int \frac{x}{\left (1+\left (x^n\right )^{\frac{1}{n}}\right )^2} \, dx\)

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]

x^2/((x^n)^(2/n)*(1 + (x^n)^n^(-1))) + (x^2*Log[1 + (x^n)^n^(-1)])/(x^n)^(2/n)

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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]

x^2/((x^n)^(2/n)*(1 + (x^n)^n^(-1))) + (x^2*Log[1 + (x^n)^n^(-1)])/(x^n)^(2/n)

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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]

x**2*(x**n)**(-2/n)*log((x**n)**(1/n) + 1) + x**2*(x**n)**(-2/n)/((x**n)**(1/n)
+ 1)

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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]

Integrate[x/(1 + (x^n)^n^(-1))^2, x]

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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]

x^2/(1+(x^n)^(1/n))-exp((n*ln(x)-ln(x^n))/n)*x+exp(2*(n*ln(x)-ln(x^n))/n)*ln(1+e
xp(-(n*ln(x)-ln(x^n))/n)*x)

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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]

-x + x^2/((x^n)^(1/n) + 1) + log(x + 1)

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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]

((x + 1)*log(x + 1) + 1)/(x + 1)

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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]

log((x**n)**(1/n) + 1) + 1/((x**n)**(1/n) + 1)

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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]

1/(x + 1) + ln(abs(x + 1))