Optimal. Leaf size=52 \[ \frac {\sqrt {-\frac {x^2}{1-x^2}} \sqrt {x^2-1} \tan ^{-1}\left (\frac {\sqrt {x^2-1}}{\sqrt {2}}\right )}{\sqrt {2} x} \]
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Rubi [A] time = 0.10, antiderivative size = 52, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {6719, 444, 63, 203} \[ \frac {\sqrt {-\frac {x^2}{1-x^2}} \sqrt {x^2-1} \tan ^{-1}\left (\frac {\sqrt {x^2-1}}{\sqrt {2}}\right )}{\sqrt {2} x} \]
Antiderivative was successfully verified.
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Rule 63
Rule 203
Rule 444
Rule 6719
Rubi steps
\begin {align*} \int \frac {\sqrt {\frac {x^2}{-1+x^2}}}{1+x^2} \, dx &=\frac {\left (\sqrt {\frac {x^2}{-1+x^2}} \sqrt {-1+x^2}\right ) \int \frac {x}{\sqrt {-1+x^2} \left (1+x^2\right )} \, dx}{x}\\ &=\frac {\left (\sqrt {\frac {x^2}{-1+x^2}} \sqrt {-1+x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {-1+x} (1+x)} \, dx,x,x^2\right )}{2 x}\\ &=\frac {\left (\sqrt {\frac {x^2}{-1+x^2}} \sqrt {-1+x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{2+x^2} \, dx,x,\sqrt {-1+x^2}\right )}{x}\\ &=\frac {\sqrt {-\frac {x^2}{1-x^2}} \sqrt {-1+x^2} \tan ^{-1}\left (\frac {\sqrt {-1+x^2}}{\sqrt {2}}\right )}{\sqrt {2} x}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 49, normalized size = 0.94 \[ \frac {\sqrt {\frac {x^2}{x^2-1}} \sqrt {x^2-1} \tan ^{-1}\left (\frac {\sqrt {x^2-1}}{\sqrt {2}}\right )}{\sqrt {2} x} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.44, size = 32, normalized size = 0.62 \[ \frac {1}{2} \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (x^{2} - 1\right )} \sqrt {\frac {x^{2}}{x^{2} - 1}}}{2 \, x}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [C] time = 0.42, size = 40, normalized size = 0.77 \[ \frac {1}{2} \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} \sqrt {x^{2} - 1}\right ) \mathrm {sgn}\left (x^{2} - 1\right ) \mathrm {sgn}\relax (x) + \frac {1}{2} \, \sqrt {2} \arctan \left (\frac {1}{2} i \, \sqrt {2}\right ) \mathrm {sgn}\relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 42, normalized size = 0.81 \[ \frac {\sqrt {\frac {x^{2}}{x^{2}-1}}\, \sqrt {x^{2}-1}\, \sqrt {2}\, \arctan \left (\frac {\sqrt {x^{2}-1}\, \sqrt {2}}{2}\right )}{2 x} \]
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 \[ \int \frac {\sqrt {\frac {x^{2}}{x^{2} - 1}}}{x^{2} + 1}\,{d x} \]
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 \[ \int \frac {\sqrt {\frac {x^2}{x^2-1}}}{x^2+1} \,d x \]
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 \[ \int \frac {\sqrt {\frac {x^{2}}{x^{2} - 1}}}{x^{2} + 1}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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