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 = 0.01, antiderivative size = 48, normalized size of antiderivative = 1.00, 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} \begin {gather*} \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 ) \end {gather*}
Antiderivative was successfully verified.
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Rule 43
Rule 368
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*}
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Mathematica [A] time = 0.02, size = 35, normalized size = 0.73 \begin {gather*} 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 ) \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.08, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x}{\left (1+\left (x^n\right )^{\frac {1}{n}}\right )^2} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.82, size = 16, normalized size = 0.33 \begin {gather*} \frac {{\left (x + 1\right )} \log \left (x + 1\right ) + 1}{x + 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.17, size = 11, normalized size = 0.23 \begin {gather*} \frac {1}{x + 1} + \log \left ({\left | x + 1 \right |}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 63, normalized size = 1.31 \begin {gather*} x^{2} \left (\left (x^{n}\right )^{-\frac {1}{n}}\right )^{2} \ln \left (\left (x^{n}\right )^{\frac {1}{n}}+1\right )-x^{2} \left (x^{n}\right )^{-\frac {1}{n}}+\frac {x^{2}}{\left (x^{n}\right )^{\frac {1}{n}}+1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {x}{{\left ({\left (x^n\right )}^{1/n}+1\right )}^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.11, size = 19, normalized size = 0.40 \begin {gather*} \log {\left (\left (x^{n}\right )^{\frac {1}{n}} + 1 \right )} + \frac {1}{\left (x^{n}\right )^{\frac {1}{n}} + 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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