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 = 0.01, antiderivative size = 61, normalized size of antiderivative = 1.00, 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 69
Rule 80
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*}
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Mathematica [A] time = 0.03, 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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fricas [F] time = 0.99, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {x {\left (-x + 1\right )}^{n}}{{\left (x + 1\right )}^{n}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x {\left (-x + 1\right )}^{n}}{{\left (x + 1\right )}^{n}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.14, size = 0, normalized size = 0.00 \[ \int x \left (-x +1\right )^{n} \left (x +1\right )^{-n}\, dx \]
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 \[ \int \frac {x {\left (-x + 1\right )}^{n}}{{\left (x + 1\right )}^{n}}\,{d x} \]
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 \[ \int \frac {x\,{\left (1-x\right )}^n}{{\left (x+1\right )}^n} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int x \left (1 - x\right )^{n} \left (x + 1\right )^{- n}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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