Optimal. Leaf size=44 \[ \frac {3}{2} \sqrt {-1+\frac {1}{x^2}}-\frac {1}{2} \left (-1+\frac {1}{x^2}\right )^{3/2} x^2-\frac {3}{2} \tan ^{-1}\left (\sqrt {-1+\frac {1}{x^2}}\right ) \]
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Rubi [A]
time = 0.01, antiderivative size = 44, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 6, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {25, 272, 43, 52,
65, 209} \begin {gather*} -\frac {3}{2} \text {ArcTan}\left (\sqrt {\frac {1}{x^2}-1}\right )-\frac {1}{2} \left (\frac {1}{x^2}-1\right )^{3/2} x^2+\frac {3}{2} \sqrt {\frac {1}{x^2}-1} \end {gather*}
Antiderivative was successfully verified.
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Rule 25
Rule 43
Rule 52
Rule 65
Rule 209
Rule 272
Rubi steps
\begin {align*} \int \frac {\sqrt {-1+\frac {1}{x^2}} \left (-1+x^2\right )}{x} \, dx &=-\int \left (-1+\frac {1}{x^2}\right )^{3/2} x \, dx\\ &=\frac {1}{2} \text {Subst}\left (\int \frac {(-1+x)^{3/2}}{x^2} \, dx,x,\frac {1}{x^2}\right )\\ &=-\frac {1}{2} \left (-1+\frac {1}{x^2}\right )^{3/2} x^2+\frac {3}{4} \text {Subst}\left (\int \frac {\sqrt {-1+x}}{x} \, dx,x,\frac {1}{x^2}\right )\\ &=\frac {3}{2} \sqrt {-1+\frac {1}{x^2}}-\frac {1}{2} \left (-1+\frac {1}{x^2}\right )^{3/2} x^2-\frac {3}{4} \text {Subst}\left (\int \frac {1}{\sqrt {-1+x} x} \, dx,x,\frac {1}{x^2}\right )\\ &=\frac {3}{2} \sqrt {-1+\frac {1}{x^2}}-\frac {1}{2} \left (-1+\frac {1}{x^2}\right )^{3/2} x^2-\frac {3}{2} \text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\sqrt {-1+\frac {1}{x^2}}\right )\\ &=\frac {3}{2} \sqrt {-1+\frac {1}{x^2}}-\frac {1}{2} \left (-1+\frac {1}{x^2}\right )^{3/2} x^2-\frac {3}{2} \tan ^{-1}\left (\sqrt {-1+\frac {1}{x^2}}\right )\\ \end {align*}
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Mathematica [A]
time = 0.06, size = 46, normalized size = 1.05 \begin {gather*} \frac {1}{2} \sqrt {-1+\frac {1}{x^2}} \left (2+x^2-\frac {6 x \tanh ^{-1}\left (\frac {\sqrt {-1+x^2}}{-1+x}\right )}{\sqrt {-1+x^2}}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.11, size = 55, normalized size = 1.25
method | result | size |
default | \(\frac {\sqrt {-\frac {x^{2}-1}{x^{2}}}\, \left (2 \left (-x^{2}+1\right )^{\frac {3}{2}}+3 x^{2} \sqrt {-x^{2}+1}+3 \arcsin \left (x \right ) x \right )}{2 \sqrt {-x^{2}+1}}\) | \(55\) |
trager | \(2 \left (\frac {x^{2}}{4}+\frac {1}{2}\right ) \sqrt {-\frac {x^{2}-1}{x^{2}}}+\frac {3 \RootOf \left (\textit {\_Z}^{2}+1\right ) \ln \left (-\left (\RootOf \left (\textit {\_Z}^{2}+1\right )-\sqrt {-\frac {x^{2}-1}{x^{2}}}\right ) x \right )}{2}\) | \(56\) |
risch | \(\frac {\left (x^{4}+x^{2}-2\right ) \sqrt {-\frac {x^{2}-1}{x^{2}}}}{2 x^{2}-2}-\frac {3 \arcsin \left (x \right ) \sqrt {-\frac {x^{2}-1}{x^{2}}}\, x \sqrt {-x^{2}+1}}{2 \left (x^{2}-1\right )}\) | \(64\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.53, size = 30, normalized size = 0.68 \begin {gather*} \frac {1}{2} \, x^{2} \sqrt {\frac {1}{x^{2}} - 1} + \sqrt {\frac {1}{x^{2}} - 1} - \frac {3}{2} \, \arctan \left (\sqrt {\frac {1}{x^{2}} - 1}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.34, size = 43, normalized size = 0.98 \begin {gather*} \frac {1}{2} \, {\left (x^{2} + 2\right )} \sqrt {-\frac {x^{2} - 1}{x^{2}}} - 3 \, \arctan \left (\frac {x \sqrt {-\frac {x^{2} - 1}{x^{2}}} - 1}{x}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 22.77, size = 39, normalized size = 0.89 \begin {gather*} \frac {x^{2} \sqrt {-1 + \frac {1}{x^{2}}}}{2} + \sqrt {-1 + \frac {1}{x^{2}}} - \frac {3 \operatorname {atan}{\left (\sqrt {-1 + \frac {1}{x^{2}}} \right )}}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 2.26, size = 57, normalized size = 1.30 \begin {gather*} \frac {1}{2} \, \sqrt {-x^{2} + 1} x \mathrm {sgn}\left (x\right ) + \frac {3}{2} \, \arcsin \left (x\right ) \mathrm {sgn}\left (x\right ) - \frac {x \mathrm {sgn}\left (x\right )}{2 \, {\left (\sqrt {-x^{2} + 1} - 1\right )}} + \frac {{\left (\sqrt {-x^{2} + 1} - 1\right )} \mathrm {sgn}\left (x\right )}{2 \, x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 3.57, size = 30, normalized size = 0.68 \begin {gather*} \sqrt {\frac {1}{x^2}-1}-\frac {3\,\mathrm {atan}\left (\sqrt {\frac {1}{x^2}-1}\right )}{2}+\frac {x^2\,\sqrt {\frac {1}{x^2}-1}}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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