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 = 0.02, antiderivative size = 68, normalized size of antiderivative = 1.00, 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 69
Rule 105
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*}
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Mathematica [A] time = 0.02, 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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fricas [F] time = 0.93, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (-x + 1\right )}^{n}}{{\left (x + 1\right )}^{n} x}, x\right ) \]
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 \[ \int \frac {{\left (-x + 1\right )}^{n}}{{\left (x + 1\right )}^{n} x}\,{d x} \]
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 \[ \int \frac {\left (-x +1\right )^{n} \left (x +1\right )^{-n}}{x}\, dx \]
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 \[ \int \frac {{\left (-x + 1\right )}^{n}}{{\left (x + 1\right )}^{n} x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (1-x\right )}^n}{x\,{\left (x+1\right )}^n} \,d x \]
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 \[ \int \frac {\left (1 - x\right )^{n} \left (x + 1\right )^{- n}}{x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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