Optimal. Leaf size=25 \[ \sqrt {1-x^2} \sqrt {\frac {1}{x^2-1}} \sin ^{-1}(x) \]
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Rubi [A] time = 0.02, antiderivative size = 33, normalized size of antiderivative = 1.32, number of steps used = 4, number of rules used = 4, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.235, Rules used = {6688, 6720, 217, 206} \[ \sqrt {\frac {1}{x^2-1}} \sqrt {x^2-1} \tanh ^{-1}\left (\frac {x}{\sqrt {x^2-1}}\right ) \]
Antiderivative was successfully verified.
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Rule 206
Rule 217
Rule 6688
Rule 6720
Rubi steps
\begin {align*} \int \sqrt {\frac {1+x^2}{-1+x^4}} \, dx &=\int \sqrt {\frac {1}{-1+x^2}} \, dx\\ &=\left (\sqrt {\frac {1}{-1+x^2}} \sqrt {-1+x^2}\right ) \int \frac {1}{\sqrt {-1+x^2}} \, dx\\ &=\left (\sqrt {\frac {1}{-1+x^2}} \sqrt {-1+x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\frac {x}{\sqrt {-1+x^2}}\right )\\ &=\sqrt {\frac {1}{-1+x^2}} \sqrt {-1+x^2} \tanh ^{-1}\left (\frac {x}{\sqrt {-1+x^2}}\right )\\ \end {align*}
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Mathematica [B] time = 0.00, size = 56, normalized size = 2.24 \[ \frac {1}{2} \sqrt {\frac {1}{x^2-1}} \sqrt {x^2-1} \left (\log \left (\frac {x}{\sqrt {x^2-1}}+1\right )-\log \left (1-\frac {x}{\sqrt {x^2-1}}\right )\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.41, size = 14, normalized size = 0.56 \[ -\log \left (-x + \sqrt {x^{2} - 1}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.32, size = 21, normalized size = 0.84 \[ -\log \left ({\left | -x + \sqrt {x^{2} - 1} \right |}\right ) \mathrm {sgn}\left (x^{2} - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 28, normalized size = 1.12 \[ \sqrt {\frac {1}{x^{2}-1}}\, \sqrt {x^{2}-1}\, \ln \left (x +\sqrt {x^{2}-1}\right ) \]
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 \[ \int \sqrt {\frac {x^{2} + 1}{x^{4} - 1}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.04 \[ \int \sqrt {\frac {x^2+1}{x^4-1}} \,d x \]
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 \[ \int \sqrt {\frac {x^{2} + 1}{x^{4} - 1}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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