Optimal. Leaf size=67 \[ -\frac {1}{6} \log \left (1-\sqrt {\frac {1}{x}+1}\right )+\frac {1}{2} \log \left (\sqrt {\frac {1}{x}+1}+1\right )-\frac {1}{3} \log \left (\sqrt {\frac {1}{x}+1}+2\right )+x \log \left (\sqrt {\frac {x+1}{x}}+2\right ) \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.07, antiderivative size = 67, normalized size of antiderivative = 1.00, 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.
[In]
[Out]
Rule 12
Rule 2058
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 (-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 = 0.03, 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.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.45, size = 48, normalized size = 0.72 \[ \frac {1}{3} \, {\left (3 \, x - 1\right )} \log \left (\sqrt {\frac {x + 1}{x}} + 2\right ) + \frac {1}{2} \, \log \left (\sqrt {\frac {x + 1}{x}} + 1\right ) - \frac {1}{6} \, \log \left (\sqrt {\frac {x + 1}{x}} - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.25, size = 88, normalized size = 1.31 \[ x \log \left (\sqrt {\frac {x + 1}{x}} + 2\right ) - \frac {\log \left ({\left | -x + \sqrt {x^{2} + x} + 1 \right |}\right )}{6 \, \mathrm {sgn}\relax (x)} - \frac {\log \left ({\left | -2 \, x + 2 \, \sqrt {x^{2} + x} - 1 \right |}\right )}{3 \, \mathrm {sgn}\relax (x)} + \frac {\log \left ({\left | -3 \, x + 3 \, \sqrt {x^{2} + x} - 1 \right |}\right )}{6 \, \mathrm {sgn}\relax (x)} - \frac {1}{6} \, \log \left ({\left | 3 \, x - 1 \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [B] time = 0.14, size = 108, normalized size = 1.61 \[ x \ln \left (2+\sqrt {\frac {x +1}{x}}\right )+\frac {-3 \sqrt {\frac {x +1}{x}}\, x \ln \left (-3 x +1\right )-\sqrt {9}\, \sqrt {\left (x +1\right ) x}\, \ln \left (\frac {15 x +4 \sqrt {9}\, \sqrt {x^{2}+x}+3}{9 x -3}\right )+6 \sqrt {\left (x +1\right ) x}\, \ln \left (x +\frac {1}{2}+\sqrt {x^{2}+x}\right )}{18 \sqrt {\frac {x +1}{x}}\, x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.56, size = 67, normalized size = 1.00 \[ \frac {\log \left (\sqrt {\frac {x + 1}{x}} + 2\right )}{\frac {x + 1}{x} - 1} - \frac {1}{3} \, \log \left (\sqrt {\frac {x + 1}{x}} + 2\right ) + \frac {1}{2} \, \log \left (\sqrt {\frac {x + 1}{x}} + 1\right ) - \frac {1}{6} \, \log \left (\sqrt {\frac {x + 1}{x}} - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.64, size = 63, normalized size = 0.94 \[ \frac {\ln \left (-5\,\sqrt {\frac {x+1}{x}}-5\right )}{2}-\frac {\ln \left (\frac {\sqrt {\frac {x+1}{x}}}{9}-\frac {1}{9}\right )}{6}-\frac {\ln \left (-\frac {5\,\sqrt {\frac {x+1}{x}}}{9}-\frac {10}{9}\right )}{3}+x\,\ln \left (\sqrt {\frac {x+1}{x}}+2\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 40.98, size = 53, normalized size = 0.79 \[ x \log {\left (\sqrt {\frac {x + 1}{x}} + 2 \right )} - \frac {\log {\left (\sqrt {1 + \frac {1}{x}} - 1 \right )}}{6} + \frac {\log {\left (\sqrt {1 + \frac {1}{x}} + 1 \right )}}{2} - \frac {\log {\left (\sqrt {1 + \frac {1}{x}} + 2 \right )}}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________