Optimal. Leaf size=34 \[ \sqrt{x-1} \sqrt{x+1}-\tan ^{-1}\left (\sqrt{x-1} \sqrt{x+1}\right ) \]
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Rubi [A] time = 0.004916, antiderivative size = 34, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {101, 92, 203} \[ \sqrt{x-1} \sqrt{x+1}-\tan ^{-1}\left (\sqrt{x-1} \sqrt{x+1}\right ) \]
Antiderivative was successfully verified.
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Rule 101
Rule 92
Rule 203
Rubi steps
\begin{align*} \int \frac{\sqrt{-1+x} \sqrt{1+x}}{x} \, dx &=\sqrt{-1+x} \sqrt{1+x}-\int \frac{1}{\sqrt{-1+x} x \sqrt{1+x}} \, dx\\ &=\sqrt{-1+x} \sqrt{1+x}-\operatorname{Subst}\left (\int \frac{1}{1+x^2} \, dx,x,\sqrt{-1+x} \sqrt{1+x}\right )\\ &=\sqrt{-1+x} \sqrt{1+x}-\tan ^{-1}\left (\sqrt{-1+x} \sqrt{1+x}\right )\\ \end{align*}
Mathematica [B] time = 0.11536, size = 101, normalized size = 2.97 \[ \frac{\sqrt{1-x} \left (x^2-\sqrt{x^2-1} \tan ^{-1}\left (\sqrt{x^2-1}\right )-1\right )+2 (x-1) \sqrt{x+1} \sin ^{-1}\left (\frac{\sqrt{1-x}}{\sqrt{2}}\right )}{\sqrt{-(x-1)^2} \sqrt{x+1}}-2 \tanh ^{-1}\left (\sqrt{\frac{x-1}{x+1}}\right ) \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.006, size = 35, normalized size = 1. \begin{align*}{\sqrt{-1+x}\sqrt{1+x} \left ( \sqrt{{x}^{2}-1}+\arctan \left ({\frac{1}{\sqrt{{x}^{2}-1}}} \right ) \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 = 1.52845, size = 18, normalized size = 0.53 \begin{align*} \sqrt{x^{2} - 1} + \arcsin \left (\frac{1}{{\left | x \right |}}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.60815, size = 88, normalized size = 2.59 \begin{align*} \sqrt{x + 1} \sqrt{x - 1} - 2 \, \arctan \left (\sqrt{x + 1} \sqrt{x - 1} - 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{\sqrt{x - 1} \sqrt{x + 1}}{x}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 2.93792, size = 43, normalized size = 1.26 \begin{align*} \sqrt{x + 1} \sqrt{x - 1} + 2 \, \arctan \left (\frac{1}{2} \,{\left (\sqrt{x + 1} - \sqrt{x - 1}\right )}^{2}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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