Optimal. Leaf size=86 \[ \frac {(1-a x)^{-n} (1+a x)^n \, _2F_1\left (1,-n;1-n;\frac {1-a x}{1+a x}\right )}{n}-\frac {2^n (1-a x)^{-n} \, _2F_1\left (-n,-n;1-n;\frac {1}{2} (1-a x)\right )}{n} \]
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Rubi [A]
time = 0.02, antiderivative size = 86, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {132, 71, 133}
\begin {gather*} \frac {(1-a x)^{-n} (a x+1)^n \, _2F_1\left (1,-n;1-n;\frac {1-a x}{a x+1}\right )}{n}-\frac {2^n (1-a x)^{-n} \, _2F_1\left (-n,-n;1-n;\frac {1}{2} (1-a x)\right )}{n} \end {gather*}
Antiderivative was successfully verified.
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Rule 71
Rule 132
Rule 133
Rubi steps
\begin {align*} \int \frac {(1-a x)^{-n} (1+a x)^n}{x} \, dx &=-\left (a \int (1-a x)^{-1-n} (1+a x)^n \, dx\right )+\int \frac {(1-a x)^{-1-n} (1+a x)^n}{x} \, dx\\ &=\frac {(1-a x)^{-n} (1+a x)^n \, _2F_1\left (1,-n;1-n;\frac {1-a x}{1+a x}\right )}{n}-\frac {2^n (1-a x)^{-n} \, _2F_1\left (-n,-n;1-n;\frac {1}{2} (1-a x)\right )}{n}\\ \end {align*}
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Mathematica [A]
time = 0.06, size = 74, normalized size = 0.86 \begin {gather*} \frac {(1-a x)^{-n} \left ((1+a x)^n \, _2F_1\left (1,-n;1-n;\frac {1-a x}{1+a x}\right )-2^n \, _2F_1\left (-n,-n;1-n;\frac {1}{2} (1-a x)\right )\right )}{n} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.02, size = 0, normalized size = 0.00 \[\int \frac {\left (a x +1\right )^{n} \left (-a x +1\right )^{-n}}{x}\, dx\]
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \frac {\left (- a x + 1\right )^{- n} \left (a x + 1\right )^{n}}{x}\, dx \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \frac {{\left (a\,x+1\right )}^n}{x\,{\left (1-a\,x\right )}^n} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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