Optimal. Leaf size=25 \[ \sqrt {1-x^2} \sqrt {\frac {1}{-1+x^2}} \sin ^{-1}(x) \]
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Rubi [A]
time = 0.01, antiderivative size = 25, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {6820, 1973,
222} \begin {gather*} \sqrt {1-x^2} \sqrt {\frac {1}{x^2-1}} \text {ArcSin}(x) \end {gather*}
Antiderivative was successfully verified.
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Rule 222
Rule 1973
Rule 6820
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 ) \text {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] Leaf count is larger than twice the leaf count of optimal. \(56\) vs. \(2(25)=50\).
time = 0.00, size = 56, normalized size = 2.24 \begin {gather*} \frac {1}{2} \sqrt {\frac {1}{-1+x^2}} \sqrt {-1+x^2} \left (-\log \left (1-\frac {x}{\sqrt {-1+x^2}}\right )+\log \left (1+\frac {x}{\sqrt {-1+x^2}}\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.08, size = 28, normalized size = 1.12
method | result | size |
default | \(\sqrt {\frac {1}{x^{2}-1}}\, \sqrt {x^{2}-1}\, \ln \left (x +\sqrt {x^{2}-1}\right )\) | \(28\) |
trager | \(-\ln \left (-\sqrt {\frac {1}{x^{2}-1}}\, x^{2}+\sqrt {\frac {1}{x^{2}-1}}+x \right )\) | \(29\) |
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.35, size = 14, normalized size = 0.56 \begin {gather*} -\log \left (-x + \sqrt {x^{2} - 1}\right ) \end {gather*}
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 \begin {gather*} \int \sqrt {\frac {x^{2} + 1}{x^{4} - 1}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 2.96, size = 21, normalized size = 0.84 \begin {gather*} -\log \left ({\left | -x + \sqrt {x^{2} - 1} \right |}\right ) \mathrm {sgn}\left (x^{2} - 1\right ) \end {gather*}
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 \begin {gather*} \int \sqrt {\frac {x^2+1}{x^4-1}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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