Integrand size = 29, antiderivative size = 111 \[ \int \frac {1}{\sqrt [4]{-1-3 x^4-2 x^8+2 x^{12}+3 x^{16}+x^{20}}} \, dx=-\frac {\arctan \left (\frac {\sqrt [4]{-1-3 x^4-2 x^8+2 x^{12}+3 x^{16}+x^{20}}}{\sqrt [4]{2} x \left (1+x^4\right )}\right )}{2 \sqrt [4]{2}}+\frac {\text {arctanh}\left (\frac {\sqrt [4]{-1-3 x^4-2 x^8+2 x^{12}+3 x^{16}+x^{20}}}{\sqrt [4]{2} x \left (1+x^4\right )}\right )}{2 \sqrt [4]{2}} \]
[Out]
Time = 0.04 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.09, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.207, Rules used = {6820, 1986, 385, 218, 212, 209} \[ \int \frac {1}{\sqrt [4]{-1-3 x^4-2 x^8+2 x^{12}+3 x^{16}+x^{20}}} \, dx=\frac {\sqrt [4]{x^4-1} \left (x^4+1\right ) \arctan \left (\frac {\sqrt [4]{2} x}{\sqrt [4]{x^4-1}}\right )}{2 \sqrt [4]{2} \sqrt [4]{-\left (\left (1-x^4\right ) \left (x^4+1\right )^4\right )}}+\frac {\sqrt [4]{x^4-1} \left (x^4+1\right ) \text {arctanh}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{x^4-1}}\right )}{2 \sqrt [4]{2} \sqrt [4]{-\left (\left (1-x^4\right ) \left (x^4+1\right )^4\right )}} \]
[In]
[Out]
Rule 209
Rule 212
Rule 218
Rule 385
Rule 1986
Rule 6820
Rubi steps \begin{align*} \text {integral}& = \int \frac {1}{\sqrt [4]{\left (-1+x^4\right ) \left (1+x^4\right )^4}} \, dx \\ & = \frac {\left (\sqrt [4]{-1+x^4} \left (1+x^4\right )\right ) \int \frac {1}{\sqrt [4]{-1+x^4} \left (1+x^4\right )} \, dx}{\sqrt [4]{\left (-1+x^4\right ) \left (1+x^4\right )^4}} \\ & = \frac {\left (\sqrt [4]{-1+x^4} \left (1+x^4\right )\right ) \text {Subst}\left (\int \frac {1}{1-2 x^4} \, dx,x,\frac {x}{\sqrt [4]{-1+x^4}}\right )}{\sqrt [4]{\left (-1+x^4\right ) \left (1+x^4\right )^4}} \\ & = \frac {\left (\sqrt [4]{-1+x^4} \left (1+x^4\right )\right ) \text {Subst}\left (\int \frac {1}{1-\sqrt {2} x^2} \, dx,x,\frac {x}{\sqrt [4]{-1+x^4}}\right )}{2 \sqrt [4]{\left (-1+x^4\right ) \left (1+x^4\right )^4}}+\frac {\left (\sqrt [4]{-1+x^4} \left (1+x^4\right )\right ) \text {Subst}\left (\int \frac {1}{1+\sqrt {2} x^2} \, dx,x,\frac {x}{\sqrt [4]{-1+x^4}}\right )}{2 \sqrt [4]{\left (-1+x^4\right ) \left (1+x^4\right )^4}} \\ & = \frac {\sqrt [4]{-1+x^4} \left (1+x^4\right ) \arctan \left (\frac {\sqrt [4]{2} x}{\sqrt [4]{-1+x^4}}\right )}{2 \sqrt [4]{2} \sqrt [4]{-\left (\left (1-x^4\right ) \left (1+x^4\right )^4\right )}}+\frac {\sqrt [4]{-1+x^4} \left (1+x^4\right ) \text {arctanh}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{-1+x^4}}\right )}{2 \sqrt [4]{2} \sqrt [4]{-\left (\left (1-x^4\right ) \left (1+x^4\right )^4\right )}} \\ \end{align*}
Time = 0.22 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.68 \[ \int \frac {1}{\sqrt [4]{-1-3 x^4-2 x^8+2 x^{12}+3 x^{16}+x^{20}}} \, dx=\frac {\sqrt [4]{-1+x^4} \left (1+x^4\right ) \left (\arctan \left (\frac {\sqrt [4]{2} x}{\sqrt [4]{-1+x^4}}\right )+\text {arctanh}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{-1+x^4}}\right )\right )}{2 \sqrt [4]{2} \sqrt [4]{\left (-1+x^4\right ) \left (1+x^4\right )^4}} \]
[In]
[Out]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 8.05 (sec) , antiderivative size = 639, normalized size of antiderivative = 5.76
method | result | size |
trager | \(-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right ) \ln \left (\frac {-3 x^{16} \operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right )+2 \left (x^{20}+3 x^{16}+2 x^{12}-2 x^{8}-3 x^{4}-1\right )^{\frac {1}{4}} \operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right )^{2} x^{11}-8 \operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right ) x^{12}-\sqrt {x^{20}+3 x^{16}+2 x^{12}-2 x^{8}-3 x^{4}-1}\, \operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right )^{3} x^{6}+4 \left (x^{20}+3 x^{16}+2 x^{12}-2 x^{8}-3 x^{4}-1\right )^{\frac {1}{4}} \operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right )^{2} x^{7}-6 \operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right ) x^{8}-\sqrt {x^{20}+3 x^{16}+2 x^{12}-2 x^{8}-3 x^{4}-1}\, \operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right )^{3} x^{2}+2 \left (x^{20}+3 x^{16}+2 x^{12}-2 x^{8}-3 x^{4}-1\right )^{\frac {1}{4}} \operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right )^{2} x^{3}+4 \left (x^{20}+3 x^{16}+2 x^{12}-2 x^{8}-3 x^{4}-1\right )^{\frac {3}{4}} x +\operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right )}{\left (x^{4}+1\right )^{4}}\right )}{8}+\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right )^{2}\right ) \ln \left (\frac {3 x^{16} \operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right )^{2}\right )-2 \left (x^{20}+3 x^{16}+2 x^{12}-2 x^{8}-3 x^{4}-1\right )^{\frac {1}{4}} \operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right )^{2} x^{11}+8 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right )^{2}\right ) x^{12}-\sqrt {x^{20}+3 x^{16}+2 x^{12}-2 x^{8}-3 x^{4}-1}\, \operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right )^{2} \operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right )^{2}\right ) x^{6}-4 \left (x^{20}+3 x^{16}+2 x^{12}-2 x^{8}-3 x^{4}-1\right )^{\frac {1}{4}} \operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right )^{2} x^{7}+6 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right )^{2}\right ) x^{8}-\sqrt {x^{20}+3 x^{16}+2 x^{12}-2 x^{8}-3 x^{4}-1}\, \operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right )^{2} \operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right )^{2}\right ) x^{2}-2 \left (x^{20}+3 x^{16}+2 x^{12}-2 x^{8}-3 x^{4}-1\right )^{\frac {1}{4}} \operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right )^{2} x^{3}+4 \left (x^{20}+3 x^{16}+2 x^{12}-2 x^{8}-3 x^{4}-1\right )^{\frac {3}{4}} x -\operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right )^{2}\right )}{\left (x^{4}+1\right )^{4}}\right )}{8}\) | \(639\) |
[In]
[Out]
Result contains complex when optimal does not.
Time = 2.89 (sec) , antiderivative size = 666, normalized size of antiderivative = 6.00 \[ \int \frac {1}{\sqrt [4]{-1-3 x^4-2 x^8+2 x^{12}+3 x^{16}+x^{20}}} \, dx=\frac {1}{16} \cdot 2^{\frac {3}{4}} \log \left (\frac {2^{\frac {3}{4}} {\left (3 \, x^{16} + 8 \, x^{12} + 6 \, x^{8} - 1\right )} + 4 \, \sqrt {2} {\left (x^{20} + 3 \, x^{16} + 2 \, x^{12} - 2 \, x^{8} - 3 \, x^{4} - 1\right )}^{\frac {1}{4}} {\left (x^{11} + 2 \, x^{7} + x^{3}\right )} + 4 \cdot 2^{\frac {1}{4}} \sqrt {x^{20} + 3 \, x^{16} + 2 \, x^{12} - 2 \, x^{8} - 3 \, x^{4} - 1} {\left (x^{6} + x^{2}\right )} + 4 \, {\left (x^{20} + 3 \, x^{16} + 2 \, x^{12} - 2 \, x^{8} - 3 \, x^{4} - 1\right )}^{\frac {3}{4}} x}{x^{16} + 4 \, x^{12} + 6 \, x^{8} + 4 \, x^{4} + 1}\right ) - \frac {1}{16} \cdot 2^{\frac {3}{4}} \log \left (-\frac {2^{\frac {3}{4}} {\left (3 \, x^{16} + 8 \, x^{12} + 6 \, x^{8} - 1\right )} - 4 \, \sqrt {2} {\left (x^{20} + 3 \, x^{16} + 2 \, x^{12} - 2 \, x^{8} - 3 \, x^{4} - 1\right )}^{\frac {1}{4}} {\left (x^{11} + 2 \, x^{7} + x^{3}\right )} + 4 \cdot 2^{\frac {1}{4}} \sqrt {x^{20} + 3 \, x^{16} + 2 \, x^{12} - 2 \, x^{8} - 3 \, x^{4} - 1} {\left (x^{6} + x^{2}\right )} - 4 \, {\left (x^{20} + 3 \, x^{16} + 2 \, x^{12} - 2 \, x^{8} - 3 \, x^{4} - 1\right )}^{\frac {3}{4}} x}{x^{16} + 4 \, x^{12} + 6 \, x^{8} + 4 \, x^{4} + 1}\right ) + \frac {1}{16} i \cdot 2^{\frac {3}{4}} \log \left (\frac {2^{\frac {3}{4}} {\left (3 i \, x^{16} + 8 i \, x^{12} + 6 i \, x^{8} - i\right )} - 4 \, \sqrt {2} {\left (x^{20} + 3 \, x^{16} + 2 \, x^{12} - 2 \, x^{8} - 3 \, x^{4} - 1\right )}^{\frac {1}{4}} {\left (x^{11} + 2 \, x^{7} + x^{3}\right )} - 4 \cdot 2^{\frac {1}{4}} \sqrt {x^{20} + 3 \, x^{16} + 2 \, x^{12} - 2 \, x^{8} - 3 \, x^{4} - 1} {\left (i \, x^{6} + i \, x^{2}\right )} + 4 \, {\left (x^{20} + 3 \, x^{16} + 2 \, x^{12} - 2 \, x^{8} - 3 \, x^{4} - 1\right )}^{\frac {3}{4}} x}{x^{16} + 4 \, x^{12} + 6 \, x^{8} + 4 \, x^{4} + 1}\right ) - \frac {1}{16} i \cdot 2^{\frac {3}{4}} \log \left (\frac {2^{\frac {3}{4}} {\left (-3 i \, x^{16} - 8 i \, x^{12} - 6 i \, x^{8} + i\right )} - 4 \, \sqrt {2} {\left (x^{20} + 3 \, x^{16} + 2 \, x^{12} - 2 \, x^{8} - 3 \, x^{4} - 1\right )}^{\frac {1}{4}} {\left (x^{11} + 2 \, x^{7} + x^{3}\right )} - 4 \cdot 2^{\frac {1}{4}} \sqrt {x^{20} + 3 \, x^{16} + 2 \, x^{12} - 2 \, x^{8} - 3 \, x^{4} - 1} {\left (-i \, x^{6} - i \, x^{2}\right )} + 4 \, {\left (x^{20} + 3 \, x^{16} + 2 \, x^{12} - 2 \, x^{8} - 3 \, x^{4} - 1\right )}^{\frac {3}{4}} x}{x^{16} + 4 \, x^{12} + 6 \, x^{8} + 4 \, x^{4} + 1}\right ) \]
[In]
[Out]
\[ \int \frac {1}{\sqrt [4]{-1-3 x^4-2 x^8+2 x^{12}+3 x^{16}+x^{20}}} \, dx=\int \frac {1}{\sqrt [4]{x^{20} + 3 x^{16} + 2 x^{12} - 2 x^{8} - 3 x^{4} - 1}}\, dx \]
[In]
[Out]
\[ \int \frac {1}{\sqrt [4]{-1-3 x^4-2 x^8+2 x^{12}+3 x^{16}+x^{20}}} \, dx=\int { \frac {1}{{\left (x^{20} + 3 \, x^{16} + 2 \, x^{12} - 2 \, x^{8} - 3 \, x^{4} - 1\right )}^{\frac {1}{4}}} \,d x } \]
[In]
[Out]
\[ \int \frac {1}{\sqrt [4]{-1-3 x^4-2 x^8+2 x^{12}+3 x^{16}+x^{20}}} \, dx=\int { \frac {1}{{\left (x^{20} + 3 \, x^{16} + 2 \, x^{12} - 2 \, x^{8} - 3 \, x^{4} - 1\right )}^{\frac {1}{4}}} \,d x } \]
[In]
[Out]
Timed out. \[ \int \frac {1}{\sqrt [4]{-1-3 x^4-2 x^8+2 x^{12}+3 x^{16}+x^{20}}} \, dx=\int \frac {1}{{\left (x^{20}+3\,x^{16}+2\,x^{12}-2\,x^8-3\,x^4-1\right )}^{1/4}} \,d x \]
[In]
[Out]