Optimal. Leaf size=48 \[ \frac{x^2 \left (x^n\right )^{-2/n}}{\left (x^n\right )^{\frac{1}{n}}+1}+x^2 \left (x^n\right )^{-2/n} \log \left (\left (x^n\right )^{\frac{1}{n}}+1\right ) \]
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Rubi [A] time = 0.0113031, 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, Rules used = {368, 43} \[ \frac{x^2 \left (x^n\right )^{-2/n}}{\left (x^n\right )^{\frac{1}{n}}+1}+x^2 \left (x^n\right )^{-2/n} \log \left (\left (x^n\right )^{\frac{1}{n}}+1\right ) \]
Antiderivative was successfully verified.
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Rule 368
Rule 43
Rubi steps
\begin{align*} \int \frac{x}{\left (1+\left (x^n\right )^{\frac{1}{n}}\right )^2} \, dx &=\left (x^2 \left (x^n\right )^{-2/n}\right ) \operatorname{Subst}\left (\int \frac{x}{(1+x)^2} \, dx,x,\left (x^n\right )^{\frac{1}{n}}\right )\\ &=\left (x^2 \left (x^n\right )^{-2/n}\right ) \operatorname{Subst}\left (\int \left (-\frac{1}{(1+x)^2}+\frac{1}{1+x}\right ) \, dx,x,\left (x^n\right )^{\frac{1}{n}}\right )\\ &=\frac{x^2 \left (x^n\right )^{-2/n}}{1+\left (x^n\right )^{\frac{1}{n}}}+x^2 \left (x^n\right )^{-2/n} \log \left (1+\left (x^n\right )^{\frac{1}{n}}\right )\\ \end{align*}
Mathematica [A] time = 0.0179509, size = 35, normalized size = 0.73 \[ x^2 \left (x^n\right )^{-2/n} \left (\frac{1}{\left (x^n\right )^{\frac{1}{n}}+1}+\log \left (\left (x^n\right )^{\frac{1}{n}}+1\right )\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.037, size = 75, normalized size = 1.6 \begin{align*}{\frac{{x}^{2}}{1+\sqrt [n]{{x}^{n}}}}-x{{\rm e}^{-{\frac{\ln \left ({x}^{n} \right ) -n\ln \left ( x \right ) }{n}}}}+\ln \left ( 1+{{\rm e}^{-{\frac{n\ln \left ( x \right ) -\ln \left ({x}^{n} \right ) }{n}}}}x \right ){{\rm e}^{-2\,{\frac{\ln \left ({x}^{n} \right ) -n\ln \left ( x \right ) }{n}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \frac{x^{2}}{{\left (x^{n}\right )}^{\left (\frac{1}{n}\right )} + 1} - \int \frac{x}{{\left (x^{n}\right )}^{\left (\frac{1}{n}\right )} + 1}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.44312, size = 46, normalized size = 0.96 \begin{align*} \frac{{\left (x + 1\right )} \log \left (x + 1\right ) + 1}{x + 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.086545, size = 19, normalized size = 0.4 \begin{align*} \log{\left (\left (x^{n}\right )^{\frac{1}{n}} + 1 \right )} + \frac{1}{\left (x^{n}\right )^{\frac{1}{n}} + 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.12246, size = 15, normalized size = 0.31 \begin{align*} \frac{1}{x + 1} + \log \left ({\left | x + 1 \right |}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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