Integrand size = 31, antiderivative size = 70 \[ \int \frac {\left (1-\frac {x}{a}\right )^{-n/2} \left (1+\frac {x}{a}\right )^{n/2}}{x^2} \, dx=-\frac {4 \left (1-\frac {x}{a}\right )^{1-\frac {n}{2}} \left (1+\frac {x}{a}\right )^{\frac {1}{2} (-2+n)} \operatorname {Hypergeometric2F1}\left (2,1-\frac {n}{2},2-\frac {n}{2},\frac {a-x}{a+x}\right )}{a (2-n)} \]
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Time = 0.01 (sec) , antiderivative size = 70, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.032, Rules used = {133} \[ \int \frac {\left (1-\frac {x}{a}\right )^{-n/2} \left (1+\frac {x}{a}\right )^{n/2}}{x^2} \, dx=-\frac {4 \left (1-\frac {x}{a}\right )^{1-\frac {n}{2}} \left (\frac {x}{a}+1\right )^{\frac {n-2}{2}} \operatorname {Hypergeometric2F1}\left (2,1-\frac {n}{2},2-\frac {n}{2},\frac {a-x}{a+x}\right )}{a (2-n)} \]
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Rule 133
Rubi steps \begin{align*} \text {integral}& = -\frac {4 \left (1-\frac {x}{a}\right )^{1-\frac {n}{2}} \left (1+\frac {x}{a}\right )^{\frac {1}{2} (-2+n)} \, _2F_1\left (2,1-\frac {n}{2};2-\frac {n}{2};\frac {a-x}{a+x}\right )}{a (2-n)} \\ \end{align*}
Time = 0.31 (sec) , antiderivative size = 70, normalized size of antiderivative = 1.00 \[ \int \frac {\left (1-\frac {x}{a}\right )^{-n/2} \left (1+\frac {x}{a}\right )^{n/2}}{x^2} \, dx=-\frac {4 \left (\frac {a+x}{a}\right )^{\frac {2+n}{2}} \left (1-\frac {x}{a}\right )^{-n/2} \operatorname {Hypergeometric2F1}\left (2,1+\frac {n}{2},2+\frac {n}{2},\frac {a+x}{a-x}\right )}{(2+n) (-a+x)} \]
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\[\int \frac {\left (1+\frac {x}{a}\right )^{\frac {n}{2}} \left (1-\frac {x}{a}\right )^{-\frac {n}{2}}}{x^{2}}d x\]
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\[ \int \frac {\left (1-\frac {x}{a}\right )^{-n/2} \left (1+\frac {x}{a}\right )^{n/2}}{x^2} \, dx=\int { \frac {{\left (\frac {x}{a} + 1\right )}^{\frac {1}{2} \, n}}{x^{2} {\left (-\frac {x}{a} + 1\right )}^{\frac {1}{2} \, n}} \,d x } \]
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\[ \int \frac {\left (1-\frac {x}{a}\right )^{-n/2} \left (1+\frac {x}{a}\right )^{n/2}}{x^2} \, dx=\int \frac {\left (1 - \frac {x}{a}\right )^{- \frac {n}{2}} \left (1 + \frac {x}{a}\right )^{\frac {n}{2}}}{x^{2}}\, dx \]
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\[ \int \frac {\left (1-\frac {x}{a}\right )^{-n/2} \left (1+\frac {x}{a}\right )^{n/2}}{x^2} \, dx=\int { \frac {{\left (\frac {x}{a} + 1\right )}^{\frac {1}{2} \, n}}{x^{2} {\left (-\frac {x}{a} + 1\right )}^{\frac {1}{2} \, n}} \,d x } \]
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\[ \int \frac {\left (1-\frac {x}{a}\right )^{-n/2} \left (1+\frac {x}{a}\right )^{n/2}}{x^2} \, dx=\int { \frac {{\left (\frac {x}{a} + 1\right )}^{\frac {1}{2} \, n}}{x^{2} {\left (-\frac {x}{a} + 1\right )}^{\frac {1}{2} \, n}} \,d x } \]
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Timed out. \[ \int \frac {\left (1-\frac {x}{a}\right )^{-n/2} \left (1+\frac {x}{a}\right )^{n/2}}{x^2} \, dx=\int \frac {{\left (\frac {x}{a}+1\right )}^{n/2}}{x^2\,{\left (1-\frac {x}{a}\right )}^{n/2}} \,d x \]
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