Optimal. Leaf size=63 \[ \frac{2 \left (x-\sqrt{a+x^2}\right )^{n+1} \, _2F_1\left (1,\frac{n+1}{2};\frac{n+3}{2};-\frac{\left (x-\sqrt{x^2+a}\right )^2}{a}\right )}{a (n+1)} \]
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Rubi [A] time = 0.0721823, antiderivative size = 63, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.087, Rules used = {2122, 364} \[ \frac{2 \left (x-\sqrt{a+x^2}\right )^{n+1} \, _2F_1\left (1,\frac{n+1}{2};\frac{n+3}{2};-\frac{\left (x-\sqrt{x^2+a}\right )^2}{a}\right )}{a (n+1)} \]
Antiderivative was successfully verified.
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Rule 2122
Rule 364
Rubi steps
\begin{align*} \int \frac{\left (x-\sqrt{a+x^2}\right )^n}{a+x^2} \, dx &=2 \operatorname{Subst}\left (\int \frac{x^n}{a+x^2} \, dx,x,x-\sqrt{a+x^2}\right )\\ &=\frac{2 \left (x-\sqrt{a+x^2}\right )^{1+n} \, _2F_1\left (1,\frac{1+n}{2};\frac{3+n}{2};-\frac{\left (x-\sqrt{a+x^2}\right )^2}{a}\right )}{a (1+n)}\\ \end{align*}
Mathematica [A] time = 0.0257508, size = 65, normalized size = 1.03 \[ \frac{2 \left (x-\sqrt{a+x^2}\right )^{n+1} \, _2F_1\left (1,\frac{n+1}{2};\frac{n+1}{2}+1;-\frac{\left (x-\sqrt{x^2+a}\right )^2}{a}\right )}{a (n+1)} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.021, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{2}+a} \left ( x-\sqrt{{x}^{2}+a} \right ) ^{n}}\, 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 (x - \sqrt{x^{2} + a}\right )}^{n}}{x^{2} + a}\,{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 (x - \sqrt{x^{2} + a}\right )}^{n}}{x^{2} + a}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (x - \sqrt{a + x^{2}}\right )^{n}}{a + x^{2}}\, dx \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 (x - \sqrt{x^{2} + a}\right )}^{n}}{x^{2} + a}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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