Optimal. Leaf size=70 \[ -\frac{4 \left (1-\frac{x}{a}\right )^{1-\frac{n}{2}} \left (\frac{x}{a}+1\right )^{\frac{n-2}{2}} \, _2F_1\left (2,1-\frac{n}{2};2-\frac{n}{2};\frac{a-x}{a+x}\right )}{a (2-n)} \]
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Rubi [A] time = 0.0169073, antiderivative size = 70, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.032, Rules used = {131} \[ -\frac{4 \left (1-\frac{x}{a}\right )^{1-\frac{n}{2}} \left (\frac{x}{a}+1\right )^{\frac{n-2}{2}} \, _2F_1\left (2,1-\frac{n}{2};2-\frac{n}{2};\frac{a-x}{a+x}\right )}{a (2-n)} \]
Antiderivative was successfully verified.
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Rule 131
Rubi steps
\begin{align*} \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)} \, _2F_1\left (2,1-\frac{n}{2};2-\frac{n}{2};\frac{a-x}{a+x}\right )}{a (2-n)}\\ \end{align*}
Mathematica [A] time = 0.0331712, size = 70, normalized size = 1. \[ -\frac{4 \left (\frac{a+x}{a}\right )^{\frac{n+2}{2}} \left (1-\frac{x}{a}\right )^{-n/2} \, _2F_1\left (2,\frac{n}{2}+1;\frac{n}{2}+2;\frac{a+x}{a-x}\right )}{(n+2) (x-a)} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.104, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{2}} \left ( 1+{\frac{x}{a}} \right ) ^{{\frac{n}{2}}} \left ( \left ( 1-{\frac{x}{a}} \right ) ^{{\frac{n}{2}}} \right ) ^{-1}}\, 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 (\frac{x}{a} + 1\right )}^{\frac{1}{2} \, n}}{x^{2}{\left (-\frac{x}{a} + 1\right )}^{\frac{1}{2} \, n}}\,{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 (\frac{a + x}{a}\right )^{\frac{1}{2} \, n}}{x^{2} \left (\frac{a - x}{a}\right )^{\frac{1}{2} \, n}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \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 (\frac{x}{a} + 1\right )}^{\frac{1}{2} \, n}}{x^{2}{\left (-\frac{x}{a} + 1\right )}^{\frac{1}{2} \, n}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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