Optimal. Leaf size=22 \[ -\frac {\sqrt {-1+x} \sqrt {1+x}}{x}+\cosh ^{-1}(x) \]
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Rubi [A]
time = 0.00, antiderivative size = 22, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {99, 54}
\begin {gather*} \cosh ^{-1}(x)-\frac {\sqrt {x-1} \sqrt {x+1}}{x} \end {gather*}
Antiderivative was successfully verified.
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Rule 54
Rule 99
Rubi steps
\begin {align*} \int \frac {\sqrt {-1+x} \sqrt {1+x}}{x^2} \, dx &=-\frac {\sqrt {-1+x} \sqrt {1+x}}{x}+\int \frac {1}{\sqrt {-1+x} \sqrt {1+x}} \, dx\\ &=-\frac {\sqrt {-1+x} \sqrt {1+x}}{x}+\cosh ^{-1}(x)\\ \end {align*}
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Mathematica [A]
time = 0.05, size = 36, normalized size = 1.64 \begin {gather*} -\frac {\sqrt {-1+x} \sqrt {1+x}}{x}+2 \tanh ^{-1}\left (\sqrt {\frac {-1+x}{1+x}}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(43\) vs.
\(2(18)=36\).
time = 0.10, size = 44, normalized size = 2.00
method | result | size |
default | \(\frac {\sqrt {-1+x}\, \sqrt {1+x}\, \left (\ln \left (x +\sqrt {x^{2}-1}\right ) x -\sqrt {x^{2}-1}\right )}{x \sqrt {x^{2}-1}}\) | \(44\) |
risch | \(-\frac {\sqrt {-1+x}\, \sqrt {1+x}}{x}+\frac {\ln \left (x +\sqrt {x^{2}-1}\right ) \sqrt {\left (1+x \right ) \left (-1+x \right )}}{\sqrt {-1+x}\, \sqrt {1+x}}\) | \(47\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.55, size = 27, normalized size = 1.23 \begin {gather*} -\frac {\sqrt {x^{2} - 1}}{x} + \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 = 1.00, size = 36, normalized size = 1.64 \begin {gather*} -\frac {x \log \left (\sqrt {x + 1} \sqrt {x - 1} - x\right ) + \sqrt {x + 1} \sqrt {x - 1} + x}{x} \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 {x - 1} \sqrt {x + 1}}{x^{2}}\, dx \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. 40 vs.
\(2 (18) = 36\).
time = 1.90, size = 40, normalized size = 1.82 \begin {gather*} -\frac {8}{{\left (\sqrt {x + 1} - \sqrt {x - 1}\right )}^{4} + 4} - \frac {1}{2} \, \log \left ({\left (\sqrt {x + 1} - \sqrt {x - 1}\right )}^{4}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 2.00, size = 109, normalized size = 4.95 \begin {gather*} 4\,\mathrm {atanh}\left (\frac {\sqrt {x-1}-\mathrm {i}}{\sqrt {x+1}-1}\right )-\frac {\sqrt {x-1}-\mathrm {i}}{4\,\left (\sqrt {x+1}-1\right )}-\frac {\frac {5\,{\left (\sqrt {x-1}-\mathrm {i}\right )}^2}{4\,{\left (\sqrt {x+1}-1\right )}^2}+\frac {1}{4}}{\frac {{\left (\sqrt {x-1}-\mathrm {i}\right )}^3}{{\left (\sqrt {x+1}-1\right )}^3}+\frac {\sqrt {x-1}-\mathrm {i}}{\sqrt {x+1}-1}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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