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 = 0.01, antiderivative size = 38, normalized size of antiderivative = 1.00, 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*}
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Mathematica [A] time = 0.01, size = 38, normalized size = 1.00 \[ -\frac {2^{-n} (1-x)^{n+1} \, _2F_1\left (n,n+1;n+2;\frac {1-x}{2}\right )}{n+1} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.93, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (-x + 1\right )}^{n}}{{\left (x + 1\right )}^{n}}, x\right ) \]
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 \[ \int \frac {{\left (-x + 1\right )}^{n}}{{\left (x + 1\right )}^{n}}\,{d x} \]
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 \[ \int \left (-x +1\right )^{n} \left (x +1\right )^{-n}\, dx \]
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 \[ \int \frac {{\left (-x + 1\right )}^{n}}{{\left (x + 1\right )}^{n}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.03 \[ \int \frac {{\left (1-x\right )}^n}{{\left (x+1\right )}^n} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 17.88, size = 42, normalized size = 1.11 \[ \frac {2^{- n} \left (x - 1\right ) \left (x - 1\right )^{n} e^{i \pi n} \Gamma \left (n + 1\right ) {{}_{2}F_{1}\left (\begin {matrix} n, n + 1 \\ n + 2 \end {matrix}\middle | {\frac {\left (x - 1\right ) e^{i \pi }}{2}} \right )}}{\Gamma \left (n + 2\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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