Optimal. Leaf size=61 \[ \frac{2^{-n} n (1-x)^{n+1} \, _2F_1\left (n,n+1;n+2;\frac{1-x}{2}\right )}{n+1}-\frac{1}{2} (1-x)^{n+1} (x+1)^{1-n} \]
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Rubi [A] time = 0.0126024, antiderivative size = 61, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {80, 69} \[ \frac{2^{-n} n (1-x)^{n+1} \, _2F_1\left (n,n+1;n+2;\frac{1-x}{2}\right )}{n+1}-\frac{1}{2} (1-x)^{n+1} (x+1)^{1-n} \]
Antiderivative was successfully verified.
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Rule 80
Rule 69
Rubi steps
\begin{align*} \int (1-x)^n x (1+x)^{-n} \, dx &=-\frac{1}{2} (1-x)^{1+n} (1+x)^{1-n}-n \int (1-x)^n (1+x)^{-n} \, dx\\ &=-\frac{1}{2} (1-x)^{1+n} (1+x)^{1-n}+\frac{2^{-n} n (1-x)^{1+n} \, _2F_1\left (n,1+n;2+n;\frac{1-x}{2}\right )}{1+n}\\ \end{align*}
Mathematica [A] time = 0.0280958, size = 56, normalized size = 0.92 \[ \frac{1}{2} (1-x)^{n+1} \left (\frac{2^{1-n} n \, _2F_1\left (n,n+1;n+2;\frac{1-x}{2}\right )}{n+1}-(x+1)^{1-n}\right ) \]
Antiderivative was successfully verified.
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Maple [F] time = 0.047, size = 0, normalized size = 0. \begin{align*} \int{\frac{x \left ( 1-x \right ) ^{n}}{ \left ( 1+x \right ) ^{n}}}\, dx \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*} \int \frac{x{\left (-x + 1\right )}^{n}}{{\left (x + 1\right )}^{n}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{x{\left (-x + 1\right )}^{n}}{{\left (x + 1\right )}^{n}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int x \left (1 - x\right )^{n} \left (x + 1\right )^{- n}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x{\left (-x + 1\right )}^{n}}{{\left (x + 1\right )}^{n}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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