Optimal. Leaf size=69 \[ \frac {1}{2} \log \left (1-\sqrt {\frac {1}{x}+1}\right )-\frac {1}{3} \log \left (2-\sqrt {\frac {1}{x}+1}\right )-\frac {1}{6} \log \left (\sqrt {\frac {1}{x}+1}+1\right )+x \log \left (\sqrt {\frac {x+1}{x}}-2\right ) \]
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Rubi [A] time = 0.05, antiderivative size = 69, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 4, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {2548, 706, 31, 633} \[ \frac {1}{2} \log \left (1-\sqrt {\frac {1}{x}+1}\right )-\frac {1}{3} \log \left (2-\sqrt {\frac {1}{x}+1}\right )-\frac {1}{6} \log \left (\sqrt {\frac {1}{x}+1}+1\right )+x \log \left (\sqrt {\frac {x+1}{x}}-2\right ) \]
Antiderivative was successfully verified.
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Rule 31
Rule 633
Rule 706
Rule 2548
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 (-2+4 \sqrt {1+\frac {1}{x}}\right ) x} \, dx\\ &=x \log \left (-2+\sqrt {\frac {1+x}{x}}\right )-\operatorname {Subst}\left (\int \frac {1}{(-2+x) \left (-1+x^2\right )} \, dx,x,\sqrt {1+\frac {1}{x}}\right )\\ &=x \log \left (-2+\sqrt {\frac {1+x}{x}}\right )-\frac {1}{3} \operatorname {Subst}\left (\int \frac {1}{-2+x} \, dx,x,\sqrt {1+\frac {1}{x}}\right )-\frac {1}{3} \operatorname {Subst}\left (\int \frac {-2-x}{-1+x^2} \, dx,x,\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 )-\frac {1}{6} \operatorname {Subst}\left (\int \frac {1}{1+x} \, dx,x,\sqrt {1+\frac {1}{x}}\right )+\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{-1+x} \, dx,x,\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 )-\frac {1}{6} \log \left (1+\sqrt {1+\frac {1}{x}}\right )+x \log \left (-2+\sqrt {\frac {1+x}{x}}\right )\\ \end {align*}
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Mathematica [A] time = 0.03, size = 64, normalized size = 0.93 \[ \frac {1}{6} \left (\log \left (2-\sqrt {\frac {1}{x}+1}\right )+6 x \log \left (\sqrt {\frac {1}{x}+1}-2\right )-\log \left (\sqrt {\frac {1}{x}+1}+1\right )-6 \tanh ^{-1}\left (3-2 \sqrt {\frac {1}{x}+1}\right )\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.43, size = 48, normalized size = 0.70 \[ \frac {1}{3} \, {\left (3 \, x - 1\right )} \log \left (\sqrt {\frac {x + 1}{x}} - 2\right ) - \frac {1}{6} \, \log \left (\sqrt {\frac {x + 1}{x}} + 1\right ) + \frac {1}{2} \, \log \left (\sqrt {\frac {x + 1}{x}} - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.27, size = 88, normalized size = 1.28 \[ x \log \left (\sqrt {\frac {x + 1}{x}} - 2\right ) + \frac {\log \left ({\left | -x + \sqrt {x^{2} + x} + 1 \right |}\right )}{6 \, \mathrm {sgn}\relax (x)} + \frac {\log \left ({\left | -2 \, x + 2 \, \sqrt {x^{2} + x} - 1 \right |}\right )}{3 \, \mathrm {sgn}\relax (x)} - \frac {\log \left ({\left | -3 \, x + 3 \, \sqrt {x^{2} + x} - 1 \right |}\right )}{6 \, \mathrm {sgn}\relax (x)} - \frac {1}{6} \, \log \left ({\left | 3 \, x - 1 \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.14, size = 107, normalized size = 1.55 \[ x \ln \left (-2+\sqrt {\frac {x +1}{x}}\right )+\frac {-3 \sqrt {\frac {x +1}{x}}\, x \ln \left (-3 x +1\right )+\sqrt {9}\, \sqrt {\left (x +1\right ) x}\, \ln \left (\frac {15 x +4 \sqrt {9}\, \sqrt {x^{2}+x}+3}{9 x -3}\right )-6 \sqrt {\left (x +1\right ) x}\, \ln \left (x +\frac {1}{2}+\sqrt {x^{2}+x}\right )}{18 \sqrt {\frac {x +1}{x}}\, x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.61, size = 67, normalized size = 0.97 \[ \frac {\log \left (\sqrt {\frac {x + 1}{x}} - 2\right )}{\frac {x + 1}{x} - 1} - \frac {1}{6} \, \log \left (\sqrt {\frac {x + 1}{x}} + 1\right ) + \frac {1}{2} \, \log \left (\sqrt {\frac {x + 1}{x}} - 1\right ) - \frac {1}{3} \, \log \left (\sqrt {\frac {x + 1}{x}} - 2\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.32, size = 63, normalized size = 0.91 \[ \frac {\ln \left (5-5\,\sqrt {\frac {x+1}{x}}\right )}{2}-\frac {\ln \left (\frac {\sqrt {\frac {x+1}{x}}}{9}+\frac {1}{9}\right )}{6}-\frac {\ln \left (\frac {10}{9}-\frac {5\,\sqrt {\frac {x+1}{x}}}{9}\right )}{3}+x\,\ln \left (\sqrt {\frac {x+1}{x}}-2\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 41.30, size = 53, normalized size = 0.77 \[ x \log {\left (\sqrt {\frac {x + 1}{x}} - 2 \right )} - \frac {\log {\left (\sqrt {1 + \frac {1}{x}} - 2 \right )}}{3} + \frac {\log {\left (\sqrt {1 + \frac {1}{x}} - 1 \right )}}{2} - \frac {\log {\left (\sqrt {1 + \frac {1}{x}} + 1 \right )}}{6} \]
Verification of antiderivative is not currently implemented for this CAS.
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