Integrand size = 85, antiderivative size = 21 \[ \int \frac {x \left (-\sqrt {-4+x^2}+x^2 \sqrt {-4+x^2}-4 \sqrt {-1+x^2}+x^2 \sqrt {-1+x^2}\right )}{\left (4-5 x^2+x^4\right ) \left (1+\sqrt {-4+x^2}+\sqrt {-1+x^2}\right )} \, dx=\log \left (1+\sqrt {-4+x^2}+\sqrt {-1+x^2}\right ) \]
[Out]
Time = 0.21 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.012, Rules used = {6816} \[ \int \frac {x \left (-\sqrt {-4+x^2}+x^2 \sqrt {-4+x^2}-4 \sqrt {-1+x^2}+x^2 \sqrt {-1+x^2}\right )}{\left (4-5 x^2+x^4\right ) \left (1+\sqrt {-4+x^2}+\sqrt {-1+x^2}\right )} \, dx=\log \left (\sqrt {x^2-4}+\sqrt {x^2-1}+1\right ) \]
[In]
[Out]
Rule 6816
Rubi steps \begin{align*} \text {integral}& = \log \left (1+\sqrt {-4+x^2}+\sqrt {-1+x^2}\right ) \\ \end{align*}
Time = 0.92 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.48 \[ \int \frac {x \left (-\sqrt {-4+x^2}+x^2 \sqrt {-4+x^2}-4 \sqrt {-1+x^2}+x^2 \sqrt {-1+x^2}\right )}{\left (4-5 x^2+x^4\right ) \left (1+\sqrt {-4+x^2}+\sqrt {-1+x^2}\right )} \, dx=2 \text {arctanh}\left (1-\frac {2}{3} \sqrt {-4+x^2}+\frac {2}{3} \sqrt {-1+x^2}\right ) \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. \(249\) vs. \(2(17)=34\).
Time = 0.40 (sec) , antiderivative size = 250, normalized size of antiderivative = 11.90
method | result | size |
elliptic | \(\frac {\sqrt {\left (x^{2}-4\right ) \left (x^{2}-1\right )}\, \left (\frac {\ln \left (x^{2}-5\right )}{4}+\frac {\ln \left (-\frac {5}{2}+x^{2}+\sqrt {x^{4}-5 x^{2}+4}\right )}{4}+\frac {\operatorname {arctanh}\left (\frac {5 x^{2}-17}{4 \sqrt {\left (x^{2}-5\right )^{2}+5 x^{2}-21}}\right )}{4}+\frac {\operatorname {arctanh}\left (\frac {8+2 \sqrt {5}\, \left (x -\sqrt {5}\right )}{4 \sqrt {\left (x -\sqrt {5}\right )^{2}+2 \sqrt {5}\, \left (x -\sqrt {5}\right )+4}}\right )}{4}+\frac {\operatorname {arctanh}\left (\frac {8-2 \sqrt {5}\, \left (x +\sqrt {5}\right )}{4 \sqrt {\left (x +\sqrt {5}\right )^{2}-2 \sqrt {5}\, \left (x +\sqrt {5}\right )+4}}\right )}{4}-\frac {\operatorname {arctanh}\left (\frac {2+2 \sqrt {5}\, \left (x -\sqrt {5}\right )}{2 \sqrt {\left (x -\sqrt {5}\right )^{2}+2 \sqrt {5}\, \left (x -\sqrt {5}\right )+1}}\right )}{4}-\frac {\operatorname {arctanh}\left (\frac {2-2 \sqrt {5}\, \left (x +\sqrt {5}\right )}{2 \sqrt {\left (x +\sqrt {5}\right )^{2}-2 \sqrt {5}\, \left (x +\sqrt {5}\right )+1}}\right )}{4}\right )}{\sqrt {x^{2}-4}\, \sqrt {x^{2}-1}}\) | \(250\) |
default | \(\frac {\ln \left (x^{2}-5\right )}{4}-\frac {\sqrt {\left (-2+x \right )^{2}+4 x -8}+2 \ln \left (x +\sqrt {\left (-2+x \right )^{2}+4 x -8}\right )}{4 \left (2+\sqrt {5}\right ) \left (-2+\sqrt {5}\right )}-\frac {\sqrt {\left (2+x \right )^{2}-4 x -8}-2 \ln \left (x +\sqrt {\left (2+x \right )^{2}-4 x -8}\right )}{4 \left (2+\sqrt {5}\right ) \left (-2+\sqrt {5}\right )}+\frac {\sqrt {\left (x -\sqrt {5}\right )^{2}+2 \sqrt {5}\, \left (x -\sqrt {5}\right )+1}+\sqrt {5}\, \ln \left (x +\sqrt {\left (x -\sqrt {5}\right )^{2}+2 \sqrt {5}\, \left (x -\sqrt {5}\right )+1}\right )-\operatorname {arctanh}\left (\frac {2+2 \sqrt {5}\, \left (x -\sqrt {5}\right )}{2 \sqrt {\left (x -\sqrt {5}\right )^{2}+2 \sqrt {5}\, \left (x -\sqrt {5}\right )+1}}\right )}{\left (\sqrt {5}+1\right ) \left (2+\sqrt {5}\right ) \left (-2+\sqrt {5}\right ) \left (\sqrt {5}-1\right )}+\frac {\sqrt {\left (x +\sqrt {5}\right )^{2}-2 \sqrt {5}\, \left (x +\sqrt {5}\right )+1}-\sqrt {5}\, \ln \left (x +\sqrt {\left (x +\sqrt {5}\right )^{2}-2 \sqrt {5}\, \left (x +\sqrt {5}\right )+1}\right )-\operatorname {arctanh}\left (\frac {2-2 \sqrt {5}\, \left (x +\sqrt {5}\right )}{2 \sqrt {\left (x +\sqrt {5}\right )^{2}-2 \sqrt {5}\, \left (x +\sqrt {5}\right )+1}}\right )}{\left (\sqrt {5}+1\right ) \left (2+\sqrt {5}\right ) \left (-2+\sqrt {5}\right ) \left (\sqrt {5}-1\right )}+\frac {\sqrt {\left (1+x \right )^{2}-2 x -2}-\ln \left (x +\sqrt {\left (1+x \right )^{2}-2 x -2}\right )}{2 \left (\sqrt {5}+1\right ) \left (\sqrt {5}-1\right )}+\frac {\sqrt {\left (-1+x \right )^{2}+2 x -2}+\ln \left (x +\sqrt {\left (-1+x \right )^{2}+2 x -2}\right )}{2 \left (\sqrt {5}+1\right ) \left (\sqrt {5}-1\right )}-\frac {\sqrt {\left (x -\sqrt {5}\right )^{2}+2 \sqrt {5}\, \left (x -\sqrt {5}\right )+4}+\sqrt {5}\, \ln \left (x +\sqrt {\left (x -\sqrt {5}\right )^{2}+2 \sqrt {5}\, \left (x -\sqrt {5}\right )+4}\right )-2 \,\operatorname {arctanh}\left (\frac {8+2 \sqrt {5}\, \left (x -\sqrt {5}\right )}{4 \sqrt {\left (x -\sqrt {5}\right )^{2}+2 \sqrt {5}\, \left (x -\sqrt {5}\right )+4}}\right )}{2 \left (\sqrt {5}+1\right ) \left (2+\sqrt {5}\right ) \left (-2+\sqrt {5}\right ) \left (\sqrt {5}-1\right )}-\frac {\sqrt {\left (x +\sqrt {5}\right )^{2}-2 \sqrt {5}\, \left (x +\sqrt {5}\right )+4}-\sqrt {5}\, \ln \left (x +\sqrt {\left (x +\sqrt {5}\right )^{2}-2 \sqrt {5}\, \left (x +\sqrt {5}\right )+4}\right )-2 \,\operatorname {arctanh}\left (\frac {8-2 \sqrt {5}\, \left (x +\sqrt {5}\right )}{4 \sqrt {\left (x +\sqrt {5}\right )^{2}-2 \sqrt {5}\, \left (x +\sqrt {5}\right )+4}}\right )}{2 \left (\sqrt {5}+1\right ) \left (2+\sqrt {5}\right ) \left (-2+\sqrt {5}\right ) \left (\sqrt {5}-1\right )}+\frac {7 \sqrt {x^{2}-4}\, \sqrt {x^{2}-1}\, \operatorname {arctanh}\left (\frac {5 x^{2}-17}{4 \sqrt {x^{4}-5 x^{2}+4}}\right )}{8 \sqrt {x^{4}-5 x^{2}+4}}+\frac {\sqrt {x^{2}-4}\, \sqrt {x^{2}-1}\, \left (2 \ln \left (-\frac {5}{2}+x^{2}+\sqrt {x^{4}-5 x^{2}+4}\right )-5 \,\operatorname {arctanh}\left (\frac {5 x^{2}-17}{4 \sqrt {x^{4}-5 x^{2}+4}}\right )\right )}{8 \sqrt {x^{4}-5 x^{2}+4}}\) | \(815\) |
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 162 vs. \(2 (17) = 34\).
Time = 0.26 (sec) , antiderivative size = 162, normalized size of antiderivative = 7.71 \[ \int \frac {x \left (-\sqrt {-4+x^2}+x^2 \sqrt {-4+x^2}-4 \sqrt {-1+x^2}+x^2 \sqrt {-1+x^2}\right )}{\left (4-5 x^2+x^4\right ) \left (1+\sqrt {-4+x^2}+\sqrt {-1+x^2}\right )} \, dx=-\frac {1}{4} \, \log \left (4 \, x^{4} - {\left (4 \, x^{2} - 11\right )} \sqrt {x^{2} - 1} \sqrt {x^{2} - 4} - 21 \, x^{2} + 23\right ) - \frac {1}{4} \, \log \left (x^{2} - \sqrt {x^{2} - 1} {\left (x + 2\right )} + 2 \, x - 1\right ) + \frac {1}{4} \, \log \left (x^{2} - \sqrt {x^{2} - 4} {\left (x + 1\right )} + x - 4\right ) - \frac {1}{4} \, \log \left (x^{2} - \sqrt {x^{2} - 4} {\left (x - 1\right )} - x - 4\right ) + \frac {1}{4} \, \log \left (x^{2} - \sqrt {x^{2} - 1} {\left (x - 2\right )} - 2 \, x - 1\right ) + \frac {1}{4} \, \log \left (x^{2} - 5\right ) + \frac {1}{4} \, \log \left (-x^{2} + \sqrt {x^{2} - 1} \sqrt {x^{2} - 4} + 7\right ) \]
[In]
[Out]
Timed out. \[ \int \frac {x \left (-\sqrt {-4+x^2}+x^2 \sqrt {-4+x^2}-4 \sqrt {-1+x^2}+x^2 \sqrt {-1+x^2}\right )}{\left (4-5 x^2+x^4\right ) \left (1+\sqrt {-4+x^2}+\sqrt {-1+x^2}\right )} \, dx=\text {Timed out} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 171 vs. \(2 (17) = 34\).
Time = 0.25 (sec) , antiderivative size = 171, normalized size of antiderivative = 8.14 \[ \int \frac {x \left (-\sqrt {-4+x^2}+x^2 \sqrt {-4+x^2}-4 \sqrt {-1+x^2}+x^2 \sqrt {-1+x^2}\right )}{\left (4-5 x^2+x^4\right ) \left (1+\sqrt {-4+x^2}+\sqrt {-1+x^2}\right )} \, dx=\frac {1}{4} \, \log \left (x + 1\right ) + \frac {3}{8} \, \log \left (x - 1\right ) + \frac {1}{8} \, \log \left (x - 2\right ) + \frac {1}{4} \, \log \left (\frac {2 \, x^{4} + 4 \, {\left (x^{2} - 3\right )} \sqrt {x + 1} \sqrt {x - 1} - 7 \, x^{2} + 2 \, {\left ({\left (x^{2} - 1\right )} \sqrt {x + 1} \sqrt {x - 1} \sqrt {x - 2} + {\left (2 \, x^{2} - 3\right )} \sqrt {x - 2}\right )} \sqrt {x + 2} + 3}{2 \, {\left ({\left (x^{2} - 1\right )} \sqrt {x + 1} \sqrt {x - 1} \sqrt {x - 2} + {\left (2 \, x^{2} - 3\right )} \sqrt {x - 2}\right )}}\right ) + \frac {1}{4} \, \log \left (\frac {{\left (x^{2} - 1\right )} \sqrt {x + 1} \sqrt {x - 1} + 2 \, x^{2} - 3}{{\left (x^{2} - 1\right )} \sqrt {x - 1}}\right ) \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 76 vs. \(2 (17) = 34\).
Time = 0.28 (sec) , antiderivative size = 76, normalized size of antiderivative = 3.62 \[ \int \frac {x \left (-\sqrt {-4+x^2}+x^2 \sqrt {-4+x^2}-4 \sqrt {-1+x^2}+x^2 \sqrt {-1+x^2}\right )}{\left (4-5 x^2+x^4\right ) \left (1+\sqrt {-4+x^2}+\sqrt {-1+x^2}\right )} \, dx=-\frac {1}{2} \, \log \left (\sqrt {x^{2} - 1} - \sqrt {x^{2} - 4} + 1\right ) - \frac {1}{2} \, \log \left (\sqrt {x^{2} - 1} - \sqrt {x^{2} - 4}\right ) + \frac {1}{2} \, \log \left (\sqrt {x^{2} - 1} + 2\right ) + \frac {1}{2} \, \log \left ({\left | -\sqrt {x^{2} - 1} + \sqrt {x^{2} - 4} - 3 \right |}\right ) \]
[In]
[Out]
Time = 2.37 (sec) , antiderivative size = 172, normalized size of antiderivative = 8.19 \[ \int \frac {x \left (-\sqrt {-4+x^2}+x^2 \sqrt {-4+x^2}-4 \sqrt {-1+x^2}+x^2 \sqrt {-1+x^2}\right )}{\left (4-5 x^2+x^4\right ) \left (1+\sqrt {-4+x^2}+\sqrt {-1+x^2}\right )} \, dx=\frac {\ln \left (x-\sqrt {5}\right )}{4}-\mathrm {atanh}\left (\frac {\sqrt {3}-\sqrt {x^2-1}}{\sqrt {x^2-4}}\right )+\frac {\mathrm {atanh}\left (\frac {\sqrt {x^2-1}}{2}\right )}{2}+\frac {\ln \left (x+\sqrt {5}\right )}{4}-\frac {7\,\mathrm {atanh}\left (\frac {4\,\left (\sqrt {3}-\sqrt {x^2-1}\right )}{\sqrt {x^2-4}\,\left (\frac {{\left (\sqrt {3}-\sqrt {x^2-1}\right )}^2}{x^2-4}+1\right )}\right )}{4}+\frac {5\,\mathrm {atanh}\left (\frac {12150\,\left (\sqrt {3}-\sqrt {x^2-1}\right )}{\sqrt {x^2-4}\,\left (\frac {6075\,{\left (\sqrt {3}-\sqrt {x^2-1}\right )}^2}{2\,\left (x^2-4\right )}+\frac {6075}{2}\right )}\right )}{4}-\frac {\mathrm {atanh}\left (\sqrt {x^2-4}\right )}{2} \]
[In]
[Out]