Optimal. Leaf size=22 \[ \sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {\frac {1}{x}+1}}{\sqrt {2}}\right ) \]
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Rubi [A] time = 0.03, antiderivative size = 22, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {1446, 1469, 627, 63, 207} \[ \sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {\frac {1}{x}+1}}{\sqrt {2}}\right ) \]
Antiderivative was successfully verified.
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Rule 63
Rule 207
Rule 627
Rule 1446
Rule 1469
Rubi steps
\begin {align*} \int \frac {\sqrt {1+\frac {1}{x}}}{1-x^2} \, dx &=\int \frac {\sqrt {1+\frac {1}{x}}}{\left (-1+\frac {1}{x^2}\right ) x^2} \, dx\\ &=-\operatorname {Subst}\left (\int \frac {\sqrt {1+x}}{-1+x^2} \, dx,x,\frac {1}{x}\right )\\ &=-\operatorname {Subst}\left (\int \frac {1}{(-1+x) \sqrt {1+x}} \, dx,x,\frac {1}{x}\right )\\ &=-\left (2 \operatorname {Subst}\left (\int \frac {1}{-2+x^2} \, dx,x,\sqrt {1+\frac {1}{x}}\right )\right )\\ &=\sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {1+\frac {1}{x}}}{\sqrt {2}}\right )\\ \end {align*}
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Mathematica [A] time = 0.03, size = 22, normalized size = 1.00 \[ \sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {\frac {1}{x}+1}}{\sqrt {2}}\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.44, size = 33, normalized size = 1.50 \[ \frac {1}{2} \, \sqrt {2} \log \left (-\frac {2 \, \sqrt {2} x \sqrt {\frac {x + 1}{x}} + 3 \, x + 1}{x - 1}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.53, size = 73, normalized size = 3.32 \[ \frac {1}{2} \, \sqrt {2} \log \left (\frac {\sqrt {2} - 1}{\sqrt {2} + 1}\right ) \mathrm {sgn}\relax (x) - \frac {1}{2} \, \sqrt {2} \log \left (\frac {{\left | -2 \, x - 2 \, \sqrt {2} + 2 \, \sqrt {x^{2} + x} + 2 \right |}}{{\left | -2 \, x + 2 \, \sqrt {2} + 2 \, \sqrt {x^{2} + x} + 2 \right |}}\right ) \mathrm {sgn}\relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.02, size = 41, normalized size = 1.86 \[ \frac {\sqrt {\frac {x +1}{x}}\, \sqrt {2}\, x \arctanh \left (\frac {\left (3 x +1\right ) \sqrt {2}}{4 \sqrt {x^{2}+x}}\right )}{2 \sqrt {\left (x +1\right ) x}} \]
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 \[ -\int \frac {\sqrt {\frac {1}{x} + 1}}{x^{2} - 1}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.58, size = 17, normalized size = 0.77 \[ \sqrt {2}\,\mathrm {atanh}\left (\frac {\sqrt {2}\,\sqrt {\frac {1}{x}+1}}{2}\right ) \]
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 \[ - \int \frac {\sqrt {1 + \frac {1}{x}}}{x^{2} - 1}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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