Integrand size = 13, antiderivative size = 34 \[ \int \frac {x \log (x)}{\sqrt {-1+x^2}} \, dx=-\sqrt {-1+x^2}+\arctan \left (\sqrt {-1+x^2}\right )+\sqrt {-1+x^2} \log (x) \]
[Out]
Time = 0.03 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.385, Rules used = {2376, 272, 52, 65, 209} \[ \int \frac {x \log (x)}{\sqrt {-1+x^2}} \, dx=\arctan \left (\sqrt {x^2-1}\right )-\sqrt {x^2-1}+\sqrt {x^2-1} \log (x) \]
[In]
[Out]
Rule 52
Rule 65
Rule 209
Rule 272
Rule 2376
Rubi steps \begin{align*} \text {integral}& = \sqrt {-1+x^2} \log (x)-\int \frac {\sqrt {-1+x^2}}{x} \, dx \\ & = \sqrt {-1+x^2} \log (x)-\frac {1}{2} \text {Subst}\left (\int \frac {\sqrt {-1+x}}{x} \, dx,x,x^2\right ) \\ & = -\sqrt {-1+x^2}+\sqrt {-1+x^2} \log (x)+\frac {1}{2} \text {Subst}\left (\int \frac {1}{\sqrt {-1+x} x} \, dx,x,x^2\right ) \\ & = -\sqrt {-1+x^2}+\sqrt {-1+x^2} \log (x)+\text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\sqrt {-1+x^2}\right ) \\ & = -\sqrt {-1+x^2}+\tan ^{-1}\left (\sqrt {-1+x^2}\right )+\sqrt {-1+x^2} \log (x) \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.79 \[ \int \frac {x \log (x)}{\sqrt {-1+x^2}} \, dx=-\arctan \left (\frac {1}{\sqrt {-1+x^2}}\right )+\sqrt {-1+x^2} (-1+\log (x)) \]
[In]
[Out]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.42 (sec) , antiderivative size = 119, normalized size of antiderivative = 3.50
method | result | size |
meijerg | \(-\frac {\sqrt {-\operatorname {signum}\left (x^{2}-1\right )}\, \left (2-2 \sqrt {-x^{2}+1}\right )}{4 \sqrt {\operatorname {signum}\left (x^{2}-1\right )}}+\frac {\sqrt {-\operatorname {signum}\left (x^{2}-1\right )}\, \ln \left (x \right ) \left (2-2 \sqrt {-x^{2}+1}\right )}{2 \sqrt {\operatorname {signum}\left (x^{2}-1\right )}}+\frac {\sqrt {-\operatorname {signum}\left (x^{2}-1\right )}\, \left (-16+16 \sqrt {-x^{2}+1}-32 \ln \left (\frac {1}{2}+\frac {\sqrt {-x^{2}+1}}{2}\right )\right )}{32 \sqrt {\operatorname {signum}\left (x^{2}-1\right )}}\) | \(119\) |
[In]
[Out]
none
Time = 0.28 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.79 \[ \int \frac {x \log (x)}{\sqrt {-1+x^2}} \, dx=\sqrt {x^{2} - 1} {\left (\log \left (x\right ) - 1\right )} + 2 \, \arctan \left (-x + \sqrt {x^{2} - 1}\right ) \]
[In]
[Out]
Time = 1.21 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.85 \[ \int \frac {x \log (x)}{\sqrt {-1+x^2}} \, dx=\sqrt {x^{2} - 1} \log {\left (x \right )} - \begin {cases} \sqrt {x^{2} - 1} - \operatorname {acos}{\left (\frac {1}{x} \right )} & \text {for}\: x > -1 \wedge x < 1 \end {cases} \]
[In]
[Out]
none
Time = 0.26 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.79 \[ \int \frac {x \log (x)}{\sqrt {-1+x^2}} \, dx=\sqrt {x^{2} - 1} \log \left (x\right ) - \sqrt {x^{2} - 1} - \arcsin \left (\frac {1}{{\left | x \right |}}\right ) \]
[In]
[Out]
none
Time = 0.35 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.82 \[ \int \frac {x \log (x)}{\sqrt {-1+x^2}} \, dx=\sqrt {x^{2} - 1} \log \left (x\right ) - \sqrt {x^{2} - 1} + \arctan \left (\sqrt {x^{2} - 1}\right ) \]
[In]
[Out]
Timed out. \[ \int \frac {x \log (x)}{\sqrt {-1+x^2}} \, dx=\int \frac {x\,\ln \left (x\right )}{\sqrt {x^2-1}} \,d x \]
[In]
[Out]