Optimal. Leaf size=67 \[ -\frac{1}{6} \log \left (1-\sqrt{\frac{1}{x}+1}\right )+\frac{1}{2} \log \left (\sqrt{\frac{1}{x}+1}+1\right )-\frac{1}{3} \log \left (\sqrt{\frac{1}{x}+1}+2\right )+x \log \left (\sqrt{\frac{x+1}{x}}+2\right ) \]
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Rubi [A] time = 0.0742242, antiderivative size = 67, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.214, Rules used = {2548, 12, 2058} \[ -\frac{1}{6} \log \left (1-\sqrt{\frac{1}{x}+1}\right )+\frac{1}{2} \log \left (\sqrt{\frac{1}{x}+1}+1\right )-\frac{1}{3} \log \left (\sqrt{\frac{1}{x}+1}+2\right )+x \log \left (\sqrt{\frac{x+1}{x}}+2\right ) \]
Antiderivative was successfully verified.
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Rule 2548
Rule 12
Rule 2058
Rubi steps
\begin{align*} \int \log \left (2+\sqrt{\frac{1+x}{x}}\right ) \, dx &=x \log \left (2+\sqrt{\frac{1+x}{x}}\right )-\int \frac{1}{2 \left (-1-x-2 x \sqrt{\frac{1+x}{x}}\right )} \, dx\\ &=x \log \left (2+\sqrt{\frac{1+x}{x}}\right )-\frac{1}{2} \int \frac{1}{-1-x-2 x \sqrt{\frac{1+x}{x}}} \, dx\\ &=x \log \left (2+\sqrt{\frac{1+x}{x}}\right )+\operatorname{Subst}\left (\int \frac{1}{2+x-2 x^2-x^3} \, dx,x,\sqrt{\frac{1+x}{x}}\right )\\ &=x \log \left (2+\sqrt{\frac{1+x}{x}}\right )+\operatorname{Subst}\left (\int \left (-\frac{1}{6 (-1+x)}+\frac{1}{2 (1+x)}-\frac{1}{3 (2+x)}\right ) \, dx,x,\sqrt{\frac{1+x}{x}}\right )\\ &=-\frac{1}{6} \log \left (1-\sqrt{1+\frac{1}{x}}\right )+\frac{1}{2} \log \left (1+\sqrt{1+\frac{1}{x}}\right )-\frac{1}{3} \log \left (2+\sqrt{1+\frac{1}{x}}\right )+x \log \left (2+\sqrt{\frac{1+x}{x}}\right )\\ \end{align*}
Mathematica [A] time = 0.0305202, size = 53, normalized size = 0.79 \[ x \log \left (\sqrt{\frac{1}{x}+1}+2\right )+\frac{1}{3} \tanh ^{-1}\left (\frac{1}{3} \left (2 \sqrt{\frac{1}{x}+1}+1\right )\right )-\tanh ^{-1}\left (2 \sqrt{\frac{1}{x}+1}+3\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.046, size = 107, normalized size = 1.6 \begin{align*} x\ln \left ( 2+\sqrt{{\frac{1+x}{x}}} \right ) -{\frac{1}{18\,x} \left ( \sqrt{9}\ln \left ({\frac{1}{9\,x-3} \left ( 4\,\sqrt{9}\sqrt{{x}^{2}+x}+15\,x+3 \right ) } \right ) \sqrt{x \left ( 1+x \right ) }+3\,\sqrt{{\frac{1+x}{x}}}x\ln \left ( -3\,x+1 \right ) -6\,\ln \left ( 1/2+x+\sqrt{{x}^{2}+x} \right ) \sqrt{x \left ( 1+x \right ) } \right ){\frac{1}{\sqrt{{\frac{1+x}{x}}}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.01949, size = 90, normalized size = 1.34 \begin{align*} \frac{\log \left (\sqrt{\frac{x + 1}{x}} + 2\right )}{\frac{x + 1}{x} - 1} - \frac{1}{3} \, \log \left (\sqrt{\frac{x + 1}{x}} + 2\right ) + \frac{1}{2} \, \log \left (\sqrt{\frac{x + 1}{x}} + 1\right ) - \frac{1}{6} \, \log \left (\sqrt{\frac{x + 1}{x}} - 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.25612, size = 138, normalized size = 2.06 \begin{align*} \frac{1}{3} \,{\left (3 \, x - 1\right )} \log \left (\sqrt{\frac{x + 1}{x}} + 2\right ) + \frac{1}{2} \, \log \left (\sqrt{\frac{x + 1}{x}} + 1\right ) - \frac{1}{6} \, \log \left (\sqrt{\frac{x + 1}{x}} - 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 161.722, size = 53, normalized size = 0.79 \begin{align*} x \log{\left (\sqrt{\frac{x + 1}{x}} + 2 \right )} - \frac{\log{\left (\sqrt{1 + \frac{1}{x}} - 1 \right )}}{6} + \frac{\log{\left (\sqrt{1 + \frac{1}{x}} + 1 \right )}}{2} - \frac{\log{\left (\sqrt{1 + \frac{1}{x}} + 2 \right )}}{3} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.3405, size = 157, normalized size = 2.34 \begin{align*} x \log \left (\sqrt{\frac{x + 1}{x}} + 2\right ) - \frac{1}{6} \,{\left (\frac{\log \left ({\left | -2 \,{\left (x - \sqrt{x^{2} + x}\right )} \mathrm{sgn}\left (x\right ) + x - \sqrt{x^{2} + x} + 1 \right |}\right )}{\mathrm{sgn}\left (x\right )} - \frac{\log \left ({\left | -2 \,{\left (x - \sqrt{x^{2} + x}\right )} \mathrm{sgn}\left (x\right ) - x + \sqrt{x^{2} + x} - 1 \right |}\right )}{\mathrm{sgn}\left (x\right )} + 2 \, \log \left ({\left | -2 \, x + 2 \, \sqrt{x^{2} + x} - 1 \right |}\right )\right )} \mathrm{sgn}\left (x\right ) - \frac{1}{6} \, \log \left ({\left | 3 \, x - 1 \right |}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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