Optimal. Leaf size=38 \[ -\frac{2^{-n} (1-x)^{n+1} \, _2F_1\left (n,n+1;n+2;\frac{1-x}{2}\right )}{n+1} \]
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Rubi [A] time = 0.0061471, antiderivative size = 38, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067, Rules used = {69} \[ -\frac{2^{-n} (1-x)^{n+1} \, _2F_1\left (n,n+1;n+2;\frac{1-x}{2}\right )}{n+1} \]
Antiderivative was successfully verified.
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Rule 69
Rubi steps
\begin{align*} \int (1-x)^n (1+x)^{-n} \, dx &=-\frac{2^{-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 = 0.0052931, size = 38, normalized size = 1. \[ -\frac{2^{-n} (1-x)^{n+1} \, _2F_1\left (n,n+1;n+2;\frac{1-x}{2}\right )}{n+1} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.056, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \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{{\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{{\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 [C] time = 59.6141, size = 42, normalized size = 1.11 \begin{align*} \frac{2^{- n} \left (x - 1\right ) \left (x - 1\right )^{n} e^{i \pi n} \Gamma \left (n + 1\right ){{}_{2}F_{1}\left (\begin{matrix} n, n + 1 \\ n + 2 \end{matrix}\middle |{\frac{\left (x - 1\right ) e^{i \pi }}{2}} \right )}}{\Gamma \left (n + 2\right )} \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{{\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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