Integrand size = 22, antiderivative size = 101 \[ \int \frac {x^4}{\left (-1+x^4\right )^2 \sqrt [4]{x^2+x^4}} \, dx=\frac {\left (-5-2 x^2-13 x^4\right ) \left (x^2+x^4\right )^{3/4}}{80 x \left (-1+x^2\right ) \left (1+x^2\right )^2}-\frac {\arctan \left (\frac {\sqrt [4]{2} x}{\sqrt [4]{x^2+x^4}}\right )}{32 \sqrt [4]{2}}-\frac {\text {arctanh}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{x^2+x^4}}\right )}{32 \sqrt [4]{2}} \]
[Out]
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 4.62 (sec) , antiderivative size = 109, normalized size of antiderivative = 1.08, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {2081, 1268, 477, 524} \[ \int \frac {x^4}{\left (-1+x^4\right )^2 \sqrt [4]{x^2+x^4}} \, dx=\frac {128 x^5 \operatorname {Gamma}\left (\frac {13}{4}\right ) \left (17 \left (-4 x^4-9 x^2+13\right ) \operatorname {Hypergeometric2F1}\left (1,2,\frac {17}{4},-\frac {2 x^2}{1-x^2}\right )-64 x^2 \left (x^2+1\right ) \operatorname {Hypergeometric2F1}\left (2,3,\frac {21}{4},-\frac {2 x^2}{1-x^2}\right )\right )}{89505 \left (1-x^2\right )^3 \left (x^2+1\right ) \sqrt [4]{x^4+x^2} \operatorname {Gamma}\left (\frac {1}{4}\right )} \]
[In]
[Out]
Rule 477
Rule 524
Rule 1268
Rule 2081
Rubi steps \begin{align*} \text {integral}& = \frac {\left (\sqrt {x} \sqrt [4]{1+x^2}\right ) \int \frac {x^{7/2}}{\sqrt [4]{1+x^2} \left (-1+x^4\right )^2} \, dx}{\sqrt [4]{x^2+x^4}} \\ & = \frac {\left (\sqrt {x} \sqrt [4]{1+x^2}\right ) \int \frac {x^{7/2}}{\left (-1+x^2\right )^2 \left (1+x^2\right )^{9/4}} \, dx}{\sqrt [4]{x^2+x^4}} \\ & = \frac {\left (2 \sqrt {x} \sqrt [4]{1+x^2}\right ) \text {Subst}\left (\int \frac {x^8}{\left (-1+x^4\right )^2 \left (1+x^4\right )^{9/4}} \, dx,x,\sqrt {x}\right )}{\sqrt [4]{x^2+x^4}} \\ & = \frac {128 x^5 \operatorname {Gamma}\left (\frac {13}{4}\right ) \left (17 \left (13-9 x^2-4 x^4\right ) \operatorname {Hypergeometric2F1}\left (1,2,\frac {17}{4},-\frac {2 x^2}{1-x^2}\right )-64 x^2 \left (1+x^2\right ) \operatorname {Hypergeometric2F1}\left (2,3,\frac {21}{4},-\frac {2 x^2}{1-x^2}\right )\right )}{89505 \left (1-x^2\right )^3 \left (1+x^2\right ) \sqrt [4]{x^2+x^4} \operatorname {Gamma}\left (\frac {1}{4}\right )} \\ \end{align*}
Time = 0.48 (sec) , antiderivative size = 131, normalized size of antiderivative = 1.30 \[ \int \frac {x^4}{\left (-1+x^4\right )^2 \sqrt [4]{x^2+x^4}} \, dx=-\frac {\sqrt {x} \left (4 \sqrt {x} \left (5+2 x^2+13 x^4\right )+5\ 2^{3/4} \sqrt [4]{1+x^2} \left (-1+x^4\right ) \arctan \left (\frac {\sqrt [4]{2} \sqrt {x}}{\sqrt [4]{1+x^2}}\right )+5\ 2^{3/4} \sqrt [4]{1+x^2} \left (-1+x^4\right ) \text {arctanh}\left (\frac {\sqrt [4]{2} \sqrt {x}}{\sqrt [4]{1+x^2}}\right )\right )}{320 \left (-1+x^4\right ) \sqrt [4]{x^2+x^4}} \]
[In]
[Out]
Time = 4.98 (sec) , antiderivative size = 122, normalized size of antiderivative = 1.21
method | result | size |
pseudoelliptic | \(\frac {5 \left (x^{2} \left (x^{2}+1\right )\right )^{\frac {1}{4}} \left (x^{4}-1\right ) \left (2 \arctan \left (\frac {\left (x^{2} \left (x^{2}+1\right )\right )^{\frac {1}{4}} 2^{\frac {3}{4}}}{2 x}\right )-\ln \left (\frac {2^{\frac {1}{4}} x +\left (x^{2} \left (x^{2}+1\right )\right )^{\frac {1}{4}}}{-2^{\frac {1}{4}} x +\left (x^{2} \left (x^{2}+1\right )\right )^{\frac {1}{4}}}\right )\right ) 2^{\frac {3}{4}}-104 x^{5}-16 x^{3}-40 x}{\left (x^{2} \left (x^{2}+1\right )\right )^{\frac {1}{4}} \left (640 x^{4}-640\right )}\) | \(122\) |
trager | \(-\frac {\left (13 x^{4}+2 x^{2}+5\right ) \left (x^{4}+x^{2}\right )^{\frac {3}{4}}}{80 \left (x^{2}-1\right ) \left (x^{2}+1\right )^{2} x}-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right )^{2}\right ) \ln \left (\frac {-\sqrt {x^{4}+x^{2}}\, \operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right )^{2}\right ) \operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right )^{2} x -2 \left (x^{4}+x^{2}\right )^{\frac {1}{4}} \operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right )^{2} x^{2}+3 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right )^{2}\right ) x^{3}+4 \left (x^{4}+x^{2}\right )^{\frac {3}{4}}+\operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right )^{2}\right ) x}{\left (1+x \right ) \left (-1+x \right ) x}\right )}{128}-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right ) \ln \left (\frac {\sqrt {x^{4}+x^{2}}\, \operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right )^{3} x +2 \left (x^{4}+x^{2}\right )^{\frac {1}{4}} \operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right )^{2} x^{2}+3 \operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right ) x^{3}+4 \left (x^{4}+x^{2}\right )^{\frac {3}{4}}+\operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right ) x}{\left (1+x \right ) \left (-1+x \right ) x}\right )}{128}\) | \(267\) |
risch | \(-\frac {x \left (13 x^{4}+2 x^{2}+5\right )}{80 \left (x^{2}-1\right ) \left (x^{2} \left (x^{2}+1\right )\right )^{\frac {1}{4}} \left (x^{2}+1\right )}-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right ) \ln \left (\frac {\sqrt {x^{4}+x^{2}}\, \operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right )^{3} x +2 \left (x^{4}+x^{2}\right )^{\frac {1}{4}} \operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right )^{2} x^{2}+3 \operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right ) x^{3}+4 \left (x^{4}+x^{2}\right )^{\frac {3}{4}}+\operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right ) x}{\left (1+x \right ) \left (-1+x \right ) x}\right )}{128}+\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right )^{2}\right ) \ln \left (\frac {\sqrt {x^{4}+x^{2}}\, \operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right )^{2}\right ) \operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right )^{2} x -2 \left (x^{4}+x^{2}\right )^{\frac {1}{4}} \operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right )^{2} x^{2}-3 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right )^{2}\right ) x^{3}+4 \left (x^{4}+x^{2}\right )^{\frac {3}{4}}-\operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right )^{2}\right ) x}{\left (1+x \right ) \left (-1+x \right ) x}\right )}{128}\) | \(267\) |
[In]
[Out]
Result contains complex when optimal does not.
Time = 1.01 (sec) , antiderivative size = 404, normalized size of antiderivative = 4.00 \[ \int \frac {x^4}{\left (-1+x^4\right )^2 \sqrt [4]{x^2+x^4}} \, dx=-\frac {5 \cdot 2^{\frac {3}{4}} {\left (x^{7} + x^{5} - x^{3} - x\right )} \log \left (\frac {4 \, \sqrt {2} {\left (x^{4} + x^{2}\right )}^{\frac {1}{4}} x^{2} + 2^{\frac {3}{4}} {\left (3 \, x^{3} + x\right )} + 4 \cdot 2^{\frac {1}{4}} \sqrt {x^{4} + x^{2}} x + 4 \, {\left (x^{4} + x^{2}\right )}^{\frac {3}{4}}}{x^{3} - x}\right ) - 5 \cdot 2^{\frac {3}{4}} {\left (x^{7} + x^{5} - x^{3} - x\right )} \log \left (\frac {4 \, \sqrt {2} {\left (x^{4} + x^{2}\right )}^{\frac {1}{4}} x^{2} - 2^{\frac {3}{4}} {\left (3 \, x^{3} + x\right )} - 4 \cdot 2^{\frac {1}{4}} \sqrt {x^{4} + x^{2}} x + 4 \, {\left (x^{4} + x^{2}\right )}^{\frac {3}{4}}}{x^{3} - x}\right ) + 5 \cdot 2^{\frac {3}{4}} {\left (i \, x^{7} + i \, x^{5} - i \, x^{3} - i \, x\right )} \log \left (-\frac {4 \, \sqrt {2} {\left (x^{4} + x^{2}\right )}^{\frac {1}{4}} x^{2} - 2^{\frac {3}{4}} {\left (3 i \, x^{3} + i \, x\right )} + 4 i \cdot 2^{\frac {1}{4}} \sqrt {x^{4} + x^{2}} x - 4 \, {\left (x^{4} + x^{2}\right )}^{\frac {3}{4}}}{x^{3} - x}\right ) + 5 \cdot 2^{\frac {3}{4}} {\left (-i \, x^{7} - i \, x^{5} + i \, x^{3} + i \, x\right )} \log \left (-\frac {4 \, \sqrt {2} {\left (x^{4} + x^{2}\right )}^{\frac {1}{4}} x^{2} - 2^{\frac {3}{4}} {\left (-3 i \, x^{3} - i \, x\right )} - 4 i \cdot 2^{\frac {1}{4}} \sqrt {x^{4} + x^{2}} x - 4 \, {\left (x^{4} + x^{2}\right )}^{\frac {3}{4}}}{x^{3} - x}\right ) + 16 \, {\left (13 \, x^{4} + 2 \, x^{2} + 5\right )} {\left (x^{4} + x^{2}\right )}^{\frac {3}{4}}}{1280 \, {\left (x^{7} + x^{5} - x^{3} - x\right )}} \]
[In]
[Out]
\[ \int \frac {x^4}{\left (-1+x^4\right )^2 \sqrt [4]{x^2+x^4}} \, dx=\int \frac {x^{4}}{\sqrt [4]{x^{2} \left (x^{2} + 1\right )} \left (x - 1\right )^{2} \left (x + 1\right )^{2} \left (x^{2} + 1\right )^{2}}\, dx \]
[In]
[Out]
\[ \int \frac {x^4}{\left (-1+x^4\right )^2 \sqrt [4]{x^2+x^4}} \, dx=\int { \frac {x^{4}}{{\left (x^{4} + x^{2}\right )}^{\frac {1}{4}} {\left (x^{4} - 1\right )}^{2}} \,d x } \]
[In]
[Out]
none
Time = 0.30 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.80 \[ \int \frac {x^4}{\left (-1+x^4\right )^2 \sqrt [4]{x^2+x^4}} \, dx=\frac {1}{64} \cdot 2^{\frac {3}{4}} \arctan \left (\frac {1}{2} \cdot 2^{\frac {3}{4}} {\left (\frac {1}{x^{2}} + 1\right )}^{\frac {1}{4}}\right ) - \frac {1}{128} \cdot 2^{\frac {3}{4}} \log \left (2^{\frac {1}{4}} + {\left (\frac {1}{x^{2}} + 1\right )}^{\frac {1}{4}}\right ) + \frac {1}{128} \cdot 2^{\frac {3}{4}} \log \left ({\left | -2^{\frac {1}{4}} + {\left (\frac {1}{x^{2}} + 1\right )}^{\frac {1}{4}} \right |}\right ) + \frac {{\left (\frac {1}{x^{2}} + 1\right )}^{\frac {3}{4}}}{16 \, {\left (\frac {1}{x^{2}} - 1\right )}} - \frac {1}{10 \, {\left (\frac {1}{x^{2}} + 1\right )}^{\frac {5}{4}}} \]
[In]
[Out]
Timed out. \[ \int \frac {x^4}{\left (-1+x^4\right )^2 \sqrt [4]{x^2+x^4}} \, dx=\int \frac {x^4}{{\left (x^4+x^2\right )}^{1/4}\,{\left (x^4-1\right )}^2} \,d x \]
[In]
[Out]