Optimal. Leaf size=68 \[ \frac{2^{-n} (1-x)^n \, _2F_1\left (n,n;n+1;\frac{1-x}{2}\right )}{n}-\frac{(1-x)^n (x+1)^{-n} \, _2F_1\left (1,n;n+1;\frac{1-x}{x+1}\right )}{n} \]
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Rubi [A] time = 0.021044, antiderivative size = 68, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {105, 69, 131} \[ \frac{2^{-n} (1-x)^n \, _2F_1\left (n,n;n+1;\frac{1-x}{2}\right )}{n}-\frac{(1-x)^n (x+1)^{-n} \, _2F_1\left (1,n;n+1;\frac{1-x}{x+1}\right )}{n} \]
Antiderivative was successfully verified.
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Rule 105
Rule 69
Rule 131
Rubi steps
\begin{align*} \int \frac{(1-x)^n (1+x)^{-n}}{x} \, dx &=-\int (1-x)^{-1+n} (1+x)^{-n} \, dx+\int \frac{(1-x)^{-1+n} (1+x)^{-n}}{x} \, dx\\ &=-\frac{(1-x)^n (1+x)^{-n} \, _2F_1\left (1,n;1+n;\frac{1-x}{1+x}\right )}{n}+\frac{2^{-n} (1-x)^n \, _2F_1\left (n,n;1+n;\frac{1-x}{2}\right )}{n}\\ \end{align*}
Mathematica [A] time = 0.0203599, size = 67, normalized size = 0.99 \[ \frac{2^{-n} (1-x)^n (x+1)^{-n} \left ((x+1)^n \, _2F_1\left (n,n;n+1;\frac{1-x}{2}\right )-2^n \, _2F_1\left (1,n;n+1;\frac{1-x}{x+1}\right )\right )}{n} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.049, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( 1-x \right ) ^{n}}{x \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} x}\,{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}, 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 \frac{\left (1 - x\right )^{n} \left (x + 1\right )^{- n}}{x}\, 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{{\left (-x + 1\right )}^{n}}{{\left (x + 1\right )}^{n} x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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