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 = 0.0293864, antiderivative size = 25, normalized size of antiderivative = 1., 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 5236
Rule 29
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*}
Mathematica [A] time = 0.0418904, size = 35, normalized size = 1.4 \[ \frac{\left (x^2-1\right ) \sec ^{-1}(x)-\sqrt{1-\frac{1}{x^2}} x \log (x)}{\sqrt{x^2-1}} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.355, size = 97, normalized size = 3.9 \begin{align*}{-2\,ix{\rm arcsec} \left (x\right )\sqrt{{\frac{{x}^{2}-1}{{x}^{2}}}}{\frac{1}{\sqrt{{x}^{2}-1}}}}+{{\rm arcsec} \left (x\right ) \left ( i\sqrt{{\frac{{x}^{2}-1}{{x}^{2}}}}x+{x}^{2}-1 \right ){\frac{1}{\sqrt{{x}^{2}-1}}}}+{x\sqrt{{\frac{{x}^{2}-1}{{x}^{2}}}}\ln \left ( \left ({x}^{-1}+i\sqrt{1-{x}^{-2}} \right ) ^{2}+1 \right ){\frac{1}{\sqrt{{x}^{2}-1}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.97353, size = 20, normalized size = 0.8 \begin{align*} \sqrt{x^{2} - 1} \operatorname{arcsec}\left (x\right ) - \log \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.17265, size = 46, normalized size = 1.84 \begin{align*} \sqrt{x^{2} - 1} \operatorname{arcsec}\left (x\right ) - \log \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x \operatorname{asec}{\left (x \right )}}{\sqrt{\left (x - 1\right ) \left (x + 1\right )}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.11712, size = 30, normalized size = 1.2 \begin{align*} \sqrt{x^{2} - 1} \arccos \left (\frac{1}{x}\right ) - \frac{\log \left ({\left | x \right |}\right )}{\mathrm{sgn}\left (x\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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