Optimal. Leaf size=47 \[ \frac {\left (1-x^2\right ) \sqrt {1-\frac {1}{\left (1-x^2\right )^2}} \tan ^{-1}\left (\sqrt {-2+x^2}\right )}{x \sqrt {-2+x^2}} \]
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Rubi [A]
time = 0.29, antiderivative size = 73, normalized size of antiderivative = 1.55, number of steps
used = 13, number of rules used = 10, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {6854, 6857,
2015, 1160, 21, 267, 455, 52, 65, 209} \begin {gather*} \frac {\left (1-x^2\right ) \sqrt {x^4-2 x^2} \sqrt {1-\frac {1}{\left (1-x^2\right )^2}} \text {ArcTan}\left (\sqrt {x^2-2}\right )}{x \sqrt {x^2-2} \sqrt {\left (x^2-1\right )^2-1}} \end {gather*}
Antiderivative was successfully verified.
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Rule 21
Rule 52
Rule 65
Rule 209
Rule 267
Rule 455
Rule 1160
Rule 2015
Rule 6854
Rule 6857
Rubi steps
\begin {align*} \int \frac {\sqrt {1-\frac {1}{\left (-1+x^2\right )^2}}}{2-x^2} \, dx &=\frac {\left (\left (-1+x^2\right ) \sqrt {1-\frac {1}{\left (-1+x^2\right )^2}}\right ) \int \frac {\sqrt {-1+\left (-1+x^2\right )^2}}{\left (2-x^2\right ) \left (-1+x^2\right )} \, dx}{\sqrt {-1+\left (-1+x^2\right )^2}}\\ &=\frac {\left (\left (-1+x^2\right ) \sqrt {1-\frac {1}{\left (-1+x^2\right )^2}}\right ) \int \left (\frac {\sqrt {-1+\left (-1+x^2\right )^2}}{2-x^2}+\frac {\sqrt {-1+\left (-1+x^2\right )^2}}{-1+x^2}\right ) \, dx}{\sqrt {-1+\left (-1+x^2\right )^2}}\\ &=\frac {\left (\left (-1+x^2\right ) \sqrt {1-\frac {1}{\left (-1+x^2\right )^2}}\right ) \int \frac {\sqrt {-1+\left (-1+x^2\right )^2}}{2-x^2} \, dx}{\sqrt {-1+\left (-1+x^2\right )^2}}+\frac {\left (\left (-1+x^2\right ) \sqrt {1-\frac {1}{\left (-1+x^2\right )^2}}\right ) \int \frac {\sqrt {-1+\left (-1+x^2\right )^2}}{-1+x^2} \, dx}{\sqrt {-1+\left (-1+x^2\right )^2}}\\ &=\frac {\left (\left (-1+x^2\right ) \sqrt {1-\frac {1}{\left (-1+x^2\right )^2}}\right ) \int \frac {\sqrt {-2 x^2+x^4}}{2-x^2} \, dx}{\sqrt {-1+\left (-1+x^2\right )^2}}+\frac {\left (\left (-1+x^2\right ) \sqrt {1-\frac {1}{\left (-1+x^2\right )^2}}\right ) \int \frac {\sqrt {-2 x^2+x^4}}{-1+x^2} \, dx}{\sqrt {-1+\left (-1+x^2\right )^2}}\\ &=\frac {\left (\left (-1+x^2\right ) \sqrt {-2 x^2+x^4} \sqrt {1-\frac {1}{\left (-1+x^2\right )^2}}\right ) \int \frac {x \sqrt {-2+x^2}}{2-x^2} \, dx}{x \sqrt {-2+x^2} \sqrt {-1+\left (-1+x^2\right )^2}}+\frac {\left (\left (-1+x^2\right ) \sqrt {-2 x^2+x^4} \sqrt {1-\frac {1}{\left (-1+x^2\right )^2}}\right ) \int \frac {x \sqrt {-2+x^2}}{-1+x^2} \, dx}{x \sqrt {-2+x^2} \sqrt {-1+\left (-1+x^2\right )^2}}\\ &=\frac {\left (\left (-1+x^2\right ) \sqrt {-2 x^2+x^4} \sqrt {1-\frac {1}{\left (-1+x^2\right )^2}}\right ) \text {Subst}\left (\int \frac {\sqrt {-2+x}}{-1+x} \, dx,x,x^2\right )}{2 x \sqrt {-2+x^2} \sqrt {-1+\left (-1+x^2\right )^2}}-\frac {\left (\left (-1+x^2\right ) \sqrt {-2 x^2+x^4} \sqrt {1-\frac {1}{\left (-1+x^2\right )^2}}\right ) \int \frac {x}{\sqrt {-2+x^2}} \, dx}{x \sqrt {-2+x^2} \sqrt {-1+\left (-1+x^2\right )^2}}\\ &=-\frac {\left (\left (-1+x^2\right ) \sqrt {-2 x^2+x^4} \sqrt {1-\frac {1}{\left (-1+x^2\right )^2}}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {-2+x} (-1+x)} \, dx,x,x^2\right )}{2 x \sqrt {-2+x^2} \sqrt {-1+\left (-1+x^2\right )^2}}\\ &=-\frac {\left (\left (-1+x^2\right ) \sqrt {-2 x^2+x^4} \sqrt {1-\frac {1}{\left (-1+x^2\right )^2}}\right ) \text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\sqrt {-2+x^2}\right )}{x \sqrt {-2+x^2} \sqrt {-1+\left (-1+x^2\right )^2}}\\ &=\frac {\left (1-x^2\right ) \sqrt {-2 x^2+x^4} \sqrt {1-\frac {1}{\left (1-x^2\right )^2}} \tan ^{-1}\left (\sqrt {-2+x^2}\right )}{x \sqrt {-2+x^2} \sqrt {-1+\left (-1+x^2\right )^2}}\\ \end {align*}
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Mathematica [A]
time = 5.52, size = 44, normalized size = 0.94 \begin {gather*} -\frac {\left (-1+x^2\right ) \sqrt {1-\frac {1}{\left (-1+x^2\right )^2}} \tan ^{-1}\left (\sqrt {-2+x^2}\right )}{x \sqrt {-2+x^2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.23, size = 63, normalized size = 1.34
method | result | size |
default | \(-\frac {\sqrt {\frac {x^{2} \left (x^{2}-2\right )}{\left (x^{2}-1\right )^{2}}}\, \left (x^{2}-1\right ) \left (\arctan \left (\frac {x -2}{\sqrt {x^{2}-2}}\right )-\arctan \left (\frac {x +2}{\sqrt {x^{2}-2}}\right )\right )}{2 x \sqrt {x^{2}-2}}\) | \(63\) |
trager | \(-\frac {\RootOf \left (\textit {\_Z}^{2}+1\right ) \ln \left (\frac {-\RootOf \left (\textit {\_Z}^{2}+1\right ) x^{3}+2 x^{2} \sqrt {-\frac {-x^{4}+2 x^{2}}{x^{4}-2 x^{2}+1}}+3 x \RootOf \left (\textit {\_Z}^{2}+1\right )-2 \sqrt {-\frac {-x^{4}+2 x^{2}}{x^{4}-2 x^{2}+1}}}{x \left (1+x \right ) \left (-1+x \right )}\right )}{2}\) | \(106\) |
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 = 0.37, size = 36, normalized size = 0.77 \begin {gather*} -\arctan \left (\frac {{\left (x^{2} - 1\right )} \sqrt {\frac {x^{4} - 2 \, x^{2}}{x^{4} - 2 \, x^{2} + 1}}}{x}\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 \frac {\sqrt {\frac {x^{4}}{x^{4} - 2 x^{2} + 1} - \frac {2 x^{2}}{x^{4} - 2 x^{2} + 1}}}{x^{2} - 2}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 5.51, size = 18, normalized size = 0.38 \begin {gather*} -\arctan \left (\sqrt {x^{2} - 2}\right ) \mathrm {sgn}\left (x^{3} - x\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 = -0.02 \begin {gather*} \int -\frac {\sqrt {1-\frac {1}{{\left (x^2-1\right )}^2}}}{x^2-2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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