Integrand size = 17, antiderivative size = 43 \[ \int \frac {1}{\sqrt {2+x^2} \left (-1+x^4\right )} \, dx=-\frac {1}{2} \arctan \left (\frac {x}{\sqrt {2+x^2}}\right )-\frac {\text {arctanh}\left (\frac {\sqrt {3} x}{\sqrt {2+x^2}}\right )}{2 \sqrt {3}} \]
[Out]
Time = 0.02 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.235, Rules used = {1189, 385, 212, 209} \[ \int \frac {1}{\sqrt {2+x^2} \left (-1+x^4\right )} \, dx=-\frac {1}{2} \arctan \left (\frac {x}{\sqrt {x^2+2}}\right )-\frac {\text {arctanh}\left (\frac {\sqrt {3} x}{\sqrt {x^2+2}}\right )}{2 \sqrt {3}} \]
[In]
[Out]
Rule 209
Rule 212
Rule 385
Rule 1189
Rubi steps \begin{align*} \text {integral}& = -\left (\frac {1}{2} \int \frac {1}{\left (1-x^2\right ) \sqrt {2+x^2}} \, dx\right )-\frac {1}{2} \int \frac {1}{\left (1+x^2\right ) \sqrt {2+x^2}} \, dx \\ & = -\left (\frac {1}{2} \text {Subst}\left (\int \frac {1}{1-3 x^2} \, dx,x,\frac {x}{\sqrt {2+x^2}}\right )\right )-\frac {1}{2} \text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\frac {x}{\sqrt {2+x^2}}\right ) \\ & = -\frac {1}{2} \arctan \left (\frac {x}{\sqrt {2+x^2}}\right )-\frac {\text {arctanh}\left (\frac {\sqrt {3} x}{\sqrt {2+x^2}}\right )}{2 \sqrt {3}} \\ \end{align*}
Time = 0.16 (sec) , antiderivative size = 57, normalized size of antiderivative = 1.33 \[ \int \frac {1}{\sqrt {2+x^2} \left (-1+x^4\right )} \, dx=\frac {1}{6} \left (3 \arctan \left (1+x^2-x \sqrt {2+x^2}\right )-\sqrt {3} \text {arctanh}\left (\frac {1-x^2+x \sqrt {2+x^2}}{\sqrt {3}}\right )\right ) \]
[In]
[Out]
Time = 0.60 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.86
method | result | size |
pseudoelliptic | \(-\frac {\sqrt {3}\, \operatorname {arctanh}\left (\frac {\sqrt {3}\, \sqrt {x^{2}+2}}{3 x}\right )}{6}+\frac {\arctan \left (\frac {\sqrt {x^{2}+2}}{x}\right )}{2}\) | \(37\) |
default | \(-\frac {\sqrt {3}\, \operatorname {arctanh}\left (\frac {\left (2 x +4\right ) \sqrt {3}}{6 \sqrt {\left (-1+x \right )^{2}+1+2 x}}\right )}{12}+\frac {\sqrt {3}\, \operatorname {arctanh}\left (\frac {\left (-2 x +4\right ) \sqrt {3}}{6 \sqrt {\left (1+x \right )^{2}+1-2 x}}\right )}{12}-\frac {\arctan \left (\frac {x}{\sqrt {x^{2}+2}}\right )}{2}\) | \(70\) |
trager | \(-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \ln \left (\frac {\sqrt {x^{2}+2}\, x +\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right )}{x^{2}+1}\right )}{4}+\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right ) \ln \left (-\frac {-2 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right ) x^{2}+3 \sqrt {x^{2}+2}\, x -\operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right )}{\left (-1+x \right ) \left (1+x \right )}\right )}{12}\) | \(86\) |
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 72 vs. \(2 (31) = 62\).
Time = 0.25 (sec) , antiderivative size = 72, normalized size of antiderivative = 1.67 \[ \int \frac {1}{\sqrt {2+x^2} \left (-1+x^4\right )} \, dx=\frac {1}{12} \, \sqrt {3} \log \left (\frac {4 \, x^{2} - \sqrt {3} {\left (2 \, x^{2} + 1\right )} - \sqrt {x^{2} + 2} {\left (2 \, \sqrt {3} x - 3 \, x\right )} + 2}{x^{2} - 1}\right ) - \frac {1}{2} \, \arctan \left (-x^{2} + \sqrt {x^{2} + 2} x - 1\right ) \]
[In]
[Out]
\[ \int \frac {1}{\sqrt {2+x^2} \left (-1+x^4\right )} \, dx=\int \frac {1}{\left (x - 1\right ) \left (x + 1\right ) \left (x^{2} + 1\right ) \sqrt {x^{2} + 2}}\, dx \]
[In]
[Out]
\[ \int \frac {1}{\sqrt {2+x^2} \left (-1+x^4\right )} \, dx=\int { \frac {1}{{\left (x^{4} - 1\right )} \sqrt {x^{2} + 2}} \,d x } \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 74 vs. \(2 (31) = 62\).
Time = 0.31 (sec) , antiderivative size = 74, normalized size of antiderivative = 1.72 \[ \int \frac {1}{\sqrt {2+x^2} \left (-1+x^4\right )} \, dx=-\frac {1}{12} \, \sqrt {3} \log \left (\frac {{\left | 2 \, {\left (x - \sqrt {x^{2} + 2}\right )}^{2} - 4 \, \sqrt {3} - 8 \right |}}{{\left | 2 \, {\left (x - \sqrt {x^{2} + 2}\right )}^{2} + 4 \, \sqrt {3} - 8 \right |}}\right ) + \frac {1}{2} \, \arctan \left (\frac {1}{2} \, {\left (x - \sqrt {x^{2} + 2}\right )}^{2}\right ) \]
[In]
[Out]
Time = 0.11 (sec) , antiderivative size = 107, normalized size of antiderivative = 2.49 \[ \int \frac {1}{\sqrt {2+x^2} \left (-1+x^4\right )} \, dx=\frac {\sqrt {3}\,\left (\ln \left (x-1\right )-\ln \left (x+\sqrt {3}\,\sqrt {x^2+2}+2\right )\right )}{12}-\frac {\sqrt {3}\,\left (\ln \left (x+1\right )-\ln \left (\sqrt {3}\,\sqrt {x^2+2}-x+2\right )\right )}{12}+\frac {\ln \left (\sqrt {x^2+2}+2-x\,1{}\mathrm {i}\right )\,1{}\mathrm {i}}{4}-\frac {\ln \left (\sqrt {x^2+2}+2+x\,1{}\mathrm {i}\right )\,1{}\mathrm {i}}{4}+\frac {\ln \left (x-\mathrm {i}\right )\,1{}\mathrm {i}}{4}-\frac {\ln \left (x+1{}\mathrm {i}\right )\,1{}\mathrm {i}}{4} \]
[In]
[Out]