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 = 0.07, antiderivative size = 63, normalized size of antiderivative = 1.00, 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 364
Rule 2122
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*}
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Mathematica [A] time = 0.03, 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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fricas [F] time = 0.47, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (x - \sqrt {x^{2} + a}\right )}^{n}}{x^{2} + a}, x\right ) \]
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 \[ \int \frac {{\left (x - \sqrt {x^{2} + a}\right )}^{n}}{x^{2} + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.06, size = 0, normalized size = 0.00 \[ \int \frac {\left (x -\sqrt {x^{2}+a}\right )^{n}}{x^{2}+a}\, dx \]
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 \[ \int \frac {{\left (x - \sqrt {x^{2} + a}\right )}^{n}}{x^{2} + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {{\left (x-\sqrt {x^2+a}\right )}^n}{x^2+a} \,d x \]
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 \[ \int \frac {\left (x - \sqrt {a + x^{2}}\right )^{n}}{a + x^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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