Optimal. Leaf size=70 \[ -\frac {4 \left (1-\frac {x}{a}\right )^{1-\frac {n}{2}} \left (1+\frac {x}{a}\right )^{\frac {1}{2} (-2+n)} \, _2F_1\left (2,1-\frac {n}{2};2-\frac {n}{2};\frac {a-x}{a+x}\right )}{a (2-n)} \]
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Rubi [A]
time = 0.01, antiderivative size = 70, normalized size of antiderivative = 1.00, 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 = {133}
\begin {gather*} -\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)} \end {gather*}
Antiderivative was successfully verified.
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Rule 133
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*}
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Mathematica [A]
time = 0.10, size = 70, normalized size = 1.00 \begin {gather*} -\frac {4 \left (\frac {a+x}{a}\right )^{\frac {2+n}{2}} \left (1-\frac {x}{a}\right )^{-n/2} \, _2F_1\left (2,1+\frac {n}{2};2+\frac {n}{2};\frac {a+x}{a-x}\right )}{(2+n) (-a+x)} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.02, size = 0, normalized size = 0.00 \[\int \frac {\left (1+\frac {x}{a}\right )^{\frac {n}{2}} \left (1-\frac {x}{a}\right )^{-\frac {n}{2}}}{x^{2}}\, dx\]
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 (1 - \frac {x}{a}\right )^{- \frac {n}{2}} \left (1 + \frac {x}{a}\right )^{\frac {n}{2}}}{x^{2}}\, 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 (\frac {x}{a}+1\right )}^{n/2}}{x^2\,{\left (1-\frac {x}{a}\right )}^{n/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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