Optimal. Leaf size=12 \[ \tanh ^{-1}\left (\frac {x}{\sqrt {-1+x^2}}\right ) \]
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Rubi [A]
time = 0.00, antiderivative size = 12, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {223, 212}
\begin {gather*} \tanh ^{-1}\left (\frac {x}{\sqrt {x^2-1}}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 223
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {-1+x^2}} \, dx &=\text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\frac {x}{\sqrt {-1+x^2}}\right )\\ &=\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. \(38\) vs. \(2(12)=24\).
time = 0.00, size = 38, normalized size = 3.17 \begin {gather*} -\frac {1}{2} \log \left (1-\frac {x}{\sqrt {-1+x^2}}\right )+\frac {1}{2} \log \left (1+\frac {x}{\sqrt {-1+x^2}}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.05, size = 11, normalized size = 0.92
method | result | size |
default | \(\ln \left (x +\sqrt {x^{2}-1}\right )\) | \(11\) |
trager | \(\ln \left (x +\sqrt {x^{2}-1}\right )\) | \(11\) |
meijerg | \(\frac {\sqrt {-\mathrm {signum}\left (x^{2}-1\right )}\, \arcsin \left (x \right )}{\sqrt {\mathrm {signum}\left (x^{2}-1\right )}}\) | \(22\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 3.21, size = 14, normalized size = 1.17 \begin {gather*} \log \left (2 \, x + 2 \, \sqrt {x^{2} - 1}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.75, size = 14, normalized size = 1.17 \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 [A]
time = 0.04, size = 2, normalized size = 0.17 \begin {gather*} \operatorname {acosh}{\left (x \right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 26 vs.
\(2 (10) = 20\).
time = 0.50, size = 26, normalized size = 2.17 \begin {gather*} \frac {1}{2} \, \sqrt {x^{2} - 1} x + \frac {1}{2} \, \log \left ({\left | -x + \sqrt {x^{2} - 1} \right |}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.20, size = 10, normalized size = 0.83 \begin {gather*} \ln \left (x+\sqrt {x^2-1}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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