Optimal. Leaf size=41 \[ \sqrt {-\frac {a-x}{a+x}} (a+x)-2 a \tanh ^{-1}\left (\sqrt {-\frac {a-x}{a+x}}\right ) \]
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Rubi [A] time = 0.02, antiderivative size = 41, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {1959, 288, 206} \[ \sqrt {-\frac {a-x}{a+x}} (a+x)-2 a \tanh ^{-1}\left (\sqrt {-\frac {a-x}{a+x}}\right ) \]
Antiderivative was successfully verified.
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Rule 206
Rule 288
Rule 1959
Rubi steps
\begin {align*} \int \sqrt {\frac {-a+x}{a+x}} \, dx &=(4 a) \operatorname {Subst}\left (\int \frac {x^2}{\left (1-x^2\right )^2} \, dx,x,\sqrt {\frac {-a+x}{a+x}}\right )\\ &=\sqrt {-\frac {a-x}{a+x}} (a+x)-(2 a) \operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\sqrt {\frac {-a+x}{a+x}}\right )\\ &=\sqrt {-\frac {a-x}{a+x}} (a+x)-2 a \tanh ^{-1}\left (\sqrt {-\frac {a-x}{a+x}}\right )\\ \end {align*}
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Mathematica [A] time = 0.07, size = 78, normalized size = 1.90 \[ \frac {\sqrt {\frac {x-a}{a+x}} \left (\sqrt {x-a} (a+x)-2 a^{3/2} \sqrt {\frac {a+x}{a}} \sinh ^{-1}\left (\frac {\sqrt {x-a}}{\sqrt {2} \sqrt {a}}\right )\right )}{\sqrt {x-a}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.40, size = 58, normalized size = 1.41 \[ -a \log \left (\sqrt {-\frac {a - x}{a + x}} + 1\right ) + a \log \left (\sqrt {-\frac {a - x}{a + x}} - 1\right ) + {\left (a + x\right )} \sqrt {-\frac {a - x}{a + x}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.31, size = 40, normalized size = 0.98 \[ a \log \left ({\left | -x + \sqrt {-a^{2} + x^{2}} \right |}\right ) \mathrm {sgn}\left (a + x\right ) + \sqrt {-a^{2} + x^{2}} \mathrm {sgn}\left (a + x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 60, normalized size = 1.46 \[ -\frac {\sqrt {\frac {-a +x}{a +x}}\, \left (a +x \right ) \left (a \ln \left (x +\sqrt {-a^{2}+x^{2}}\right )-\sqrt {-a^{2}+x^{2}}\right )}{\sqrt {\left (a +x \right ) \left (-a +x \right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.91, size = 70, normalized size = 1.71 \[ a {\left (\frac {2 \, \sqrt {-\frac {a - x}{a + x}}}{\frac {a - x}{a + x} + 1} - \log \left (\sqrt {-\frac {a - x}{a + x}} + 1\right ) + \log \left (\sqrt {-\frac {a - x}{a + x}} - 1\right )\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.05, size = 51, normalized size = 1.24 \[ \frac {2\,a\,\sqrt {-\frac {a-x}{a+x}}}{\frac {a-x}{a+x}+1}-2\,a\,\mathrm {atanh}\left (\sqrt {-\frac {a-x}{a+x}}\right ) \]
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 \[ \int \sqrt {\frac {- a + x}{a + x}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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