Optimal. Leaf size=24 \[ \sqrt {\frac {1}{x+1}} \sqrt {x+1} \sin ^{-1}(x)+x \text {sech}^{-1}(x) \]
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Rubi [A] time = 0.01, antiderivative size = 24, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.750, Rules used = {5893, 6277, 216} \[ \sqrt {\frac {1}{x+1}} \sqrt {x+1} \sin ^{-1}(x)+x \text {sech}^{-1}(x) \]
Antiderivative was successfully verified.
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Rule 216
Rule 5893
Rule 6277
Rubi steps
\begin {align*} \int \cosh ^{-1}\left (\frac {1}{x}\right ) \, dx &=\int \text {sech}^{-1}(x) \, dx\\ &=x \text {sech}^{-1}(x)+\left (\sqrt {\frac {1}{1+x}} \sqrt {1+x}\right ) \int \frac {1}{\sqrt {1-x^2}} \, dx\\ &=x \text {sech}^{-1}(x)+\sqrt {\frac {1}{1+x}} \sqrt {1+x} \sin ^{-1}(x)\\ \end {align*}
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Mathematica [A] time = 0.05, size = 46, normalized size = 1.92 \[ x \cosh ^{-1}\left (\frac {1}{x}\right )-\frac {\sqrt {\frac {1}{x^2}-1} \tan ^{-1}\left (\sqrt {\frac {1}{x^2}-1}\right )}{\sqrt {\frac {1}{x}-1} \sqrt {\frac {1}{x}+1}} \]
Antiderivative was successfully verified.
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fricas [B] time = 1.02, size = 72, normalized size = 3.00 \[ {\left (x - 2\right )} \log \left (\frac {x \sqrt {-\frac {x^{2} - 1}{x^{2}}} + 1}{x}\right ) - 2 \, \arctan \left (\frac {x \sqrt {-\frac {x^{2} - 1}{x^{2}}} - 1}{x}\right ) - 2 \, \log \left (\frac {x \sqrt {-\frac {x^{2} - 1}{x^{2}}} - 1}{x}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.23, size = 22, normalized size = 0.92 \[ x \log \left (\sqrt {\frac {1}{x^{2}} - 1} + \frac {1}{x}\right ) + \frac {\arcsin \relax (x)}{\mathrm {sgn}\relax (x)} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 38, normalized size = 1.58 \[ \mathrm {arccosh}\left (\frac {1}{x}\right ) x +\frac {\sqrt {\frac {1}{x}-1}\, \sqrt {\frac {1}{x}+1}\, \arctan \left (\frac {1}{\sqrt {\frac {1}{x^{2}}-1}}\right )}{\sqrt {\frac {1}{x^{2}}-1}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.80, size = 17, normalized size = 0.71 \[ x \operatorname {arcosh}\left (\frac {1}{x}\right ) - \arctan \left (\sqrt {\frac {1}{x^{2}} - 1}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.37, size = 23, normalized size = 0.96 \[ \mathrm {atan}\left (\frac {1}{\sqrt {\frac {1}{x}-1}\,\sqrt {\frac {1}{x}+1}}\right )+x\,\mathrm {acosh}\left (\frac {1}{x}\right ) \]
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 \[ \int \operatorname {acosh}{\left (\frac {1}{x} \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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