Optimal. Leaf size=25 \[ \sqrt {x^2-1} \sec ^{-1}(x)-\frac {x \log (x)}{\sqrt {x^2}} \]
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Rubi [A] time = 0.03, antiderivative size = 25, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {5236, 29} \[ \sqrt {x^2-1} \sec ^{-1}(x)-\frac {x \log (x)}{\sqrt {x^2}} \]
Antiderivative was successfully verified.
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Rule 29
Rule 5236
Rubi steps
\begin {align*} \int \frac {x \sec ^{-1}(x)}{\sqrt {-1+x^2}} \, dx &=\sqrt {-1+x^2} \sec ^{-1}(x)-\frac {x \int \frac {1}{x} \, dx}{\sqrt {x^2}}\\ &=\sqrt {-1+x^2} \sec ^{-1}(x)-\frac {x \log (x)}{\sqrt {x^2}}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 35, normalized size = 1.40 \[ \frac {\left (x^2-1\right ) \sec ^{-1}(x)-\sqrt {1-\frac {1}{x^2}} x \log (x)}{\sqrt {x^2-1}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.44, size = 15, normalized size = 0.60 \[ \sqrt {x^{2} - 1} \operatorname {arcsec}\relax (x) - \log \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.09, size = 22, normalized size = 0.88 \[ \sqrt {x^{2} - 1} \arccos \left (\frac {1}{x}\right ) - \frac {\log \left ({\left | x \right |}\right )}{\mathrm {sgn}\relax (x)} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.51, size = 97, normalized size = 3.88 \[ -\frac {2 i \sqrt {\frac {x^{2}-1}{x^{2}}}\, x \,\mathrm {arcsec}\relax (x )}{\sqrt {x^{2}-1}}+\frac {\sqrt {\frac {x^{2}-1}{x^{2}}}\, x \ln \left (\left (\frac {1}{x}+i \sqrt {-\frac {1}{x^{2}}+1}\right )^{2}+1\right )}{\sqrt {x^{2}-1}}+\frac {\left (x^{2}+i \sqrt {\frac {x^{2}-1}{x^{2}}}\, x -1\right ) \mathrm {arcsec}\relax (x )}{\sqrt {x^{2}-1}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.47, size = 15, normalized size = 0.60 \[ \sqrt {x^{2} - 1} \operatorname {arcsec}\relax (x) - \log \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.04 \[ \int \frac {x\,\mathrm {acos}\left (\frac {1}{x}\right )}{\sqrt {x^2-1}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 19.71, size = 22, normalized size = 0.88 \[ \sqrt {x^{2} - 1} \operatorname {asec}{\relax (x )} - \begin {cases} - \log {\left (\frac {1}{x} \right )} & \text {for}\: x > -1 \wedge x < 1 \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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