Integrand size = 33, antiderivative size = 153 \[ \int \frac {1+x^4}{\left (-1+x^4\right ) \sqrt {1-x-x^2+x^3+x^4}} \, dx=\text {arctanh}\left (\frac {x}{-1+x^2-\sqrt {1-x-x^2+x^3+x^4}}\right )+\frac {1}{4} \text {RootSum}\left [5-16 \text {$\#$1}+14 \text {$\#$1}^2+\text {$\#$1}^4\&,\frac {\log (x)-\log \left (1-x^2+\sqrt {1-x-x^2+x^3+x^4}-x \text {$\#$1}\right )-\log (x) \text {$\#$1}^2+\log \left (1-x^2+\sqrt {1-x-x^2+x^3+x^4}-x \text {$\#$1}\right ) \text {$\#$1}^2}{-4+7 \text {$\#$1}+\text {$\#$1}^3}\&\right ] \]
Time = 0.22 (sec) , antiderivative size = 153, normalized size of antiderivative = 1.00 \[ \int \frac {1+x^4}{\left (-1+x^4\right ) \sqrt {1-x-x^2+x^3+x^4}} \, dx=\text {arctanh}\left (\frac {x}{-1+x^2-\sqrt {1-x-x^2+x^3+x^4}}\right )+\frac {1}{4} \text {RootSum}\left [5-16 \text {$\#$1}+14 \text {$\#$1}^2+\text {$\#$1}^4\&,\frac {\log (x)-\log \left (1-x^2+\sqrt {1-x-x^2+x^3+x^4}-x \text {$\#$1}\right )-\log (x) \text {$\#$1}^2+\log \left (1-x^2+\sqrt {1-x-x^2+x^3+x^4}-x \text {$\#$1}\right ) \text {$\#$1}^2}{-4+7 \text {$\#$1}+\text {$\#$1}^3}\&\right ] \]
ArcTanh[x/(-1 + x^2 - Sqrt[1 - x - x^2 + x^3 + x^4])] + RootSum[5 - 16*#1 + 14*#1^2 + #1^4 & , (Log[x] - Log[1 - x^2 + Sqrt[1 - x - x^2 + x^3 + x^4] - x*#1] - Log[x]*#1^2 + Log[1 - x^2 + Sqrt[1 - x - x^2 + x^3 + x^4] - x*# 1]*#1^2)/(-4 + 7*#1 + #1^3) & ]/4
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {x^4+1}{\left (x^4-1\right ) \sqrt {x^4+x^3-x^2-x+1}} \, dx\) |
| \(\Big \downarrow \) 7276 |
| \(\displaystyle \int \left (\frac {2}{\left (x^4-1\right ) \sqrt {x^4+x^3-x^2-x+1}}+\frac {1}{\sqrt {x^4+x^3-x^2-x+1}}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \int \frac {1}{\sqrt {x^4+x^3-x^2-x+1}}dx-\frac {1}{2} i \int \frac {1}{(i-x) \sqrt {x^4+x^3-x^2-x+1}}dx-\frac {1}{2} \int \frac {1}{(1-x) \sqrt {x^4+x^3-x^2-x+1}}dx-\frac {1}{2} i \int \frac {1}{(x+i) \sqrt {x^4+x^3-x^2-x+1}}dx-\frac {1}{2} \int \frac {1}{(x+1) \sqrt {x^4+x^3-x^2-x+1}}dx\) |
3.22.8.3.1 Defintions of rubi rules used
Int[(u_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{v = RationalFunctionE xpand[u/(a + b*x^n), x]}, Int[v, x] /; SumQ[v]] /; FreeQ[{a, b}, x] && IGtQ [n, 0]
Time = 8.18 (sec) , antiderivative size = 386, normalized size of antiderivative = 2.52
| method | result | size |
| default | \(\frac {-28 \sqrt {\frac {x^{4}+x^{3}-x^{2}-x +1}{\left (\sqrt {13}\, x +x^{2}-3 x -1\right )^{2}}}\, \left (\left (x^{2}-\frac {47}{7} x -1\right ) \sqrt {13}-\frac {26 x^{2}}{7}+\frac {169 x}{7}+\frac {26}{7}\right ) \operatorname {arctanh}\left (\frac {4 \sqrt {\frac {x^{4}+x^{3}-x^{2}-x +1}{\left (\sqrt {13}\, x +x^{2}-3 x -1\right )^{2}}}}{\sqrt {6+2 \sqrt {13}}}\right )+\sqrt {78+26 \sqrt {13}}\, \left (\arctan \left (\frac {35 \sqrt {13}\, \sqrt {\frac {x^{4}+x^{3}-x^{2}-x +1}{\left (\sqrt {13}\, x +x^{2}-3 x -1\right )^{2}}}\, \left (\sqrt {13}+\frac {19}{5}\right ) \sqrt {-2+\sqrt {13}}\, \left (-\sqrt {13}\, x +x^{2}-3 x -1\right ) \left (\sqrt {13}-\frac {26}{7}\right ) \left (\sqrt {13}\, x +x^{2}-3 x -1\right )}{468 \left (x^{4}+x^{3}-x^{2}-x +1\right )}\right ) \sqrt {-2+\sqrt {13}}\, \left (\left (x^{2}-6 x -1\right ) \sqrt {13}-3 x^{2}+22 x +3\right ) \sqrt {\frac {x^{4}+x^{3}-x^{2}-x +1}{\left (\sqrt {13}\, x +x^{2}-3 x -1\right )^{2}}}-14 \,\operatorname {arctanh}\left (\frac {x^{2}+2 x -1}{2 \sqrt {x^{4}+x^{3}-x^{2}-x +1}}\right ) \sqrt {\frac {x^{4}+x^{3}-x^{2}-x +1}{x^{2}}}\, x \left (\sqrt {13}-\frac {26}{7}\right )\right )}{\sqrt {\frac {x^{4}+x^{3}-x^{2}-x +1}{x^{2}}}\, \sqrt {78+26 \sqrt {13}}\, \left (28 \sqrt {13}\, x -104 x \right )}\) | \(386\) |
| pseudoelliptic | \(\frac {-28 \sqrt {\frac {x^{4}+x^{3}-x^{2}-x +1}{\left (\sqrt {13}\, x +x^{2}-3 x -1\right )^{2}}}\, \left (\left (x^{2}-\frac {47}{7} x -1\right ) \sqrt {13}-\frac {26 x^{2}}{7}+\frac {169 x}{7}+\frac {26}{7}\right ) \operatorname {arctanh}\left (\frac {4 \sqrt {\frac {x^{4}+x^{3}-x^{2}-x +1}{\left (\sqrt {13}\, x +x^{2}-3 x -1\right )^{2}}}}{\sqrt {6+2 \sqrt {13}}}\right )+\sqrt {78+26 \sqrt {13}}\, \left (\arctan \left (\frac {35 \sqrt {13}\, \sqrt {\frac {x^{4}+x^{3}-x^{2}-x +1}{\left (\sqrt {13}\, x +x^{2}-3 x -1\right )^{2}}}\, \left (\sqrt {13}+\frac {19}{5}\right ) \sqrt {-2+\sqrt {13}}\, \left (-\sqrt {13}\, x +x^{2}-3 x -1\right ) \left (\sqrt {13}-\frac {26}{7}\right ) \left (\sqrt {13}\, x +x^{2}-3 x -1\right )}{468 \left (x^{4}+x^{3}-x^{2}-x +1\right )}\right ) \sqrt {-2+\sqrt {13}}\, \left (\left (x^{2}-6 x -1\right ) \sqrt {13}-3 x^{2}+22 x +3\right ) \sqrt {\frac {x^{4}+x^{3}-x^{2}-x +1}{\left (\sqrt {13}\, x +x^{2}-3 x -1\right )^{2}}}-14 \,\operatorname {arctanh}\left (\frac {x^{2}+2 x -1}{2 \sqrt {x^{4}+x^{3}-x^{2}-x +1}}\right ) \sqrt {\frac {x^{4}+x^{3}-x^{2}-x +1}{x^{2}}}\, x \left (\sqrt {13}-\frac {26}{7}\right )\right )}{\sqrt {\frac {x^{4}+x^{3}-x^{2}-x +1}{x^{2}}}\, \sqrt {78+26 \sqrt {13}}\, \left (28 \sqrt {13}\, x -104 x \right )}\) | \(386\) |
| trager | \(\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+676 \operatorname {RootOf}\left (208 \textit {\_Z}^{4}+24 \textit {\_Z}^{2}+1\right )^{2}+78\right ) \ln \left (\frac {-9984 \operatorname {RootOf}\left (\textit {\_Z}^{2}+676 \operatorname {RootOf}\left (208 \textit {\_Z}^{4}+24 \textit {\_Z}^{2}+1\right )^{2}+78\right ) \operatorname {RootOf}\left (208 \textit {\_Z}^{4}+24 \textit {\_Z}^{2}+1\right )^{4} x +1088 \operatorname {RootOf}\left (\textit {\_Z}^{2}+676 \operatorname {RootOf}\left (208 \textit {\_Z}^{4}+24 \textit {\_Z}^{2}+1\right )^{2}+78\right ) \operatorname {RootOf}\left (208 \textit {\_Z}^{4}+24 \textit {\_Z}^{2}+1\right )^{2} x^{2}-1028 \operatorname {RootOf}\left (\textit {\_Z}^{2}+676 \operatorname {RootOf}\left (208 \textit {\_Z}^{4}+24 \textit {\_Z}^{2}+1\right )^{2}+78\right ) \operatorname {RootOf}\left (208 \textit {\_Z}^{4}+24 \textit {\_Z}^{2}+1\right )^{2} x +7280 \sqrt {x^{4}+x^{3}-x^{2}-x +1}\, \operatorname {RootOf}\left (208 \textit {\_Z}^{4}+24 \textit {\_Z}^{2}+1\right )^{2}-1088 \operatorname {RootOf}\left (208 \textit {\_Z}^{4}+24 \textit {\_Z}^{2}+1\right )^{2} \operatorname {RootOf}\left (\textit {\_Z}^{2}+676 \operatorname {RootOf}\left (208 \textit {\_Z}^{4}+24 \textit {\_Z}^{2}+1\right )^{2}+78\right )+119 \operatorname {RootOf}\left (\textit {\_Z}^{2}+676 \operatorname {RootOf}\left (208 \textit {\_Z}^{4}+24 \textit {\_Z}^{2}+1\right )^{2}+78\right ) x^{2}+7 \operatorname {RootOf}\left (\textit {\_Z}^{2}+676 \operatorname {RootOf}\left (208 \textit {\_Z}^{4}+24 \textit {\_Z}^{2}+1\right )^{2}+78\right ) x +78 \sqrt {x^{4}+x^{3}-x^{2}-x +1}-119 \operatorname {RootOf}\left (\textit {\_Z}^{2}+676 \operatorname {RootOf}\left (208 \textit {\_Z}^{4}+24 \textit {\_Z}^{2}+1\right )^{2}+78\right )}{{\left (52 x \operatorname {RootOf}\left (208 \textit {\_Z}^{4}+24 \textit {\_Z}^{2}+1\right )^{2}+3 x +2\right )}^{2}}\right )}{52}-\frac {\operatorname {RootOf}\left (208 \textit {\_Z}^{4}+24 \textit {\_Z}^{2}+1\right ) \ln \left (-\frac {-129792 \operatorname {RootOf}\left (208 \textit {\_Z}^{4}+24 \textit {\_Z}^{2}+1\right )^{5} x -14144 \operatorname {RootOf}\left (208 \textit {\_Z}^{4}+24 \textit {\_Z}^{2}+1\right )^{3} x^{2}-16588 \operatorname {RootOf}\left (208 \textit {\_Z}^{4}+24 \textit {\_Z}^{2}+1\right )^{3} x +3640 \sqrt {x^{4}+x^{3}-x^{2}-x +1}\, \operatorname {RootOf}\left (208 \textit {\_Z}^{4}+24 \textit {\_Z}^{2}+1\right )^{2}+14144 \operatorname {RootOf}\left (208 \textit {\_Z}^{4}+24 \textit {\_Z}^{2}+1\right )^{3}-85 \operatorname {RootOf}\left (208 \textit {\_Z}^{4}+24 \textit {\_Z}^{2}+1\right ) x^{2}-95 \operatorname {RootOf}\left (208 \textit {\_Z}^{4}+24 \textit {\_Z}^{2}+1\right ) x +381 \sqrt {x^{4}+x^{3}-x^{2}-x +1}+85 \operatorname {RootOf}\left (208 \textit {\_Z}^{4}+24 \textit {\_Z}^{2}+1\right )}{{\left (52 x \operatorname {RootOf}\left (208 \textit {\_Z}^{4}+24 \textit {\_Z}^{2}+1\right )^{2}+3 x -2\right )}^{2}}\right )}{2}+\frac {\ln \left (-\frac {-x^{2}+2 \sqrt {x^{4}+x^{3}-x^{2}-x +1}-2 x +1}{\left (1+x \right ) \left (-1+x \right )}\right )}{2}\) | \(620\) |
| elliptic | \(\text {Expression too large to display}\) | \(1418170\) |
1/((x^4+x^3-x^2-x+1)/x^2)^(1/2)/(78+26*13^(1/2))^(1/2)*(-28*((x^4+x^3-x^2- x+1)/(13^(1/2)*x+x^2-3*x-1)^2)^(1/2)*((x^2-47/7*x-1)*13^(1/2)-26/7*x^2+169 /7*x+26/7)*arctanh(4*((x^4+x^3-x^2-x+1)/(13^(1/2)*x+x^2-3*x-1)^2)^(1/2)/(6 +2*13^(1/2))^(1/2))+(78+26*13^(1/2))^(1/2)*(arctan(35/468*13^(1/2)*((x^4+x ^3-x^2-x+1)/(13^(1/2)*x+x^2-3*x-1)^2)^(1/2)*(13^(1/2)+19/5)*(-2+13^(1/2))^ (1/2)*(-13^(1/2)*x+x^2-3*x-1)*(13^(1/2)-26/7)*(13^(1/2)*x+x^2-3*x-1)/(x^4+ x^3-x^2-x+1))*(-2+13^(1/2))^(1/2)*((x^2-6*x-1)*13^(1/2)-3*x^2+22*x+3)*((x^ 4+x^3-x^2-x+1)/(13^(1/2)*x+x^2-3*x-1)^2)^(1/2)-14*arctanh(1/2*(x^2+2*x-1)/ (x^4+x^3-x^2-x+1)^(1/2))*((x^4+x^3-x^2-x+1)/x^2)^(1/2)*x*(13^(1/2)-26/7))) /(28*13^(1/2)*x-104*x)
Result contains higher order function than in optimal. Order 3 vs. order 1.
Time = 0.39 (sec) , antiderivative size = 361, normalized size of antiderivative = 2.36 \[ \int \frac {1+x^4}{\left (-1+x^4\right ) \sqrt {1-x-x^2+x^3+x^4}} \, dx=\frac {1}{104} \, \sqrt {13} \sqrt {-2 i - 3} \log \left (\frac {\sqrt {13} \sqrt {-2 i - 3} {\left (\left (544 i + 731\right ) \, x^{4} + \left (1524 i - 466\right ) \, x^{3} + \left (156 i - 1586\right ) \, x^{2} - \left (1524 i - 466\right ) \, x + 544 i + 731\right )} - 26 \, \sqrt {x^{4} + x^{3} - x^{2} - x + 1} {\left (-\left (140 i - 171\right ) \, x^{2} + \left (342 i + 280\right ) \, x + 140 i - 171\right )}}{x^{4} + 2 \, x^{2} + 1}\right ) + \frac {1}{104} \, \sqrt {13} \sqrt {2 i - 3} \log \left (\frac {\sqrt {13} \sqrt {2 i - 3} {\left (-\left (544 i - 731\right ) \, x^{4} - \left (1524 i + 466\right ) \, x^{3} - \left (156 i + 1586\right ) \, x^{2} + \left (1524 i + 466\right ) \, x - 544 i + 731\right )} - 26 \, \sqrt {x^{4} + x^{3} - x^{2} - x + 1} {\left (\left (140 i + 171\right ) \, x^{2} - \left (342 i - 280\right ) \, x - 140 i - 171\right )}}{x^{4} + 2 \, x^{2} + 1}\right ) - \frac {1}{104} \, \sqrt {13} \sqrt {2 i - 3} \log \left (\frac {\sqrt {13} \sqrt {2 i - 3} {\left (\left (544 i - 731\right ) \, x^{4} + \left (1524 i + 466\right ) \, x^{3} + \left (156 i + 1586\right ) \, x^{2} - \left (1524 i + 466\right ) \, x + 544 i - 731\right )} - 26 \, \sqrt {x^{4} + x^{3} - x^{2} - x + 1} {\left (\left (140 i + 171\right ) \, x^{2} - \left (342 i - 280\right ) \, x - 140 i - 171\right )}}{x^{4} + 2 \, x^{2} + 1}\right ) - \frac {1}{104} \, \sqrt {13} \sqrt {-2 i - 3} \log \left (\frac {\sqrt {13} \sqrt {-2 i - 3} {\left (-\left (544 i + 731\right ) \, x^{4} - \left (1524 i - 466\right ) \, x^{3} - \left (156 i - 1586\right ) \, x^{2} + \left (1524 i - 466\right ) \, x - 544 i - 731\right )} - 26 \, \sqrt {x^{4} + x^{3} - x^{2} - x + 1} {\left (-\left (140 i - 171\right ) \, x^{2} + \left (342 i + 280\right ) \, x + 140 i - 171\right )}}{x^{4} + 2 \, x^{2} + 1}\right ) + \frac {1}{2} \, \log \left (-\frac {x^{2} + 2 \, x - 2 \, \sqrt {x^{4} + x^{3} - x^{2} - x + 1} - 1}{x^{2} - 1}\right ) \]
1/104*sqrt(13)*sqrt(-2*I - 3)*log((sqrt(13)*sqrt(-2*I - 3)*((544*I + 731)* x^4 + (1524*I - 466)*x^3 + (156*I - 1586)*x^2 - (1524*I - 466)*x + 544*I + 731) - 26*sqrt(x^4 + x^3 - x^2 - x + 1)*(-(140*I - 171)*x^2 + (342*I + 28 0)*x + 140*I - 171))/(x^4 + 2*x^2 + 1)) + 1/104*sqrt(13)*sqrt(2*I - 3)*log ((sqrt(13)*sqrt(2*I - 3)*(-(544*I - 731)*x^4 - (1524*I + 466)*x^3 - (156*I + 1586)*x^2 + (1524*I + 466)*x - 544*I + 731) - 26*sqrt(x^4 + x^3 - x^2 - x + 1)*((140*I + 171)*x^2 - (342*I - 280)*x - 140*I - 171))/(x^4 + 2*x^2 + 1)) - 1/104*sqrt(13)*sqrt(2*I - 3)*log((sqrt(13)*sqrt(2*I - 3)*((544*I - 731)*x^4 + (1524*I + 466)*x^3 + (156*I + 1586)*x^2 - (1524*I + 466)*x + 5 44*I - 731) - 26*sqrt(x^4 + x^3 - x^2 - x + 1)*((140*I + 171)*x^2 - (342*I - 280)*x - 140*I - 171))/(x^4 + 2*x^2 + 1)) - 1/104*sqrt(13)*sqrt(-2*I - 3)*log((sqrt(13)*sqrt(-2*I - 3)*(-(544*I + 731)*x^4 - (1524*I - 466)*x^3 - (156*I - 1586)*x^2 + (1524*I - 466)*x - 544*I - 731) - 26*sqrt(x^4 + x^3 - x^2 - x + 1)*(-(140*I - 171)*x^2 + (342*I + 280)*x + 140*I - 171))/(x^4 + 2*x^2 + 1)) + 1/2*log(-(x^2 + 2*x - 2*sqrt(x^4 + x^3 - x^2 - x + 1) - 1) /(x^2 - 1))
Not integrable
Time = 2.46 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.22 \[ \int \frac {1+x^4}{\left (-1+x^4\right ) \sqrt {1-x-x^2+x^3+x^4}} \, dx=\int \frac {x^{4} + 1}{\left (x - 1\right ) \left (x + 1\right ) \left (x^{2} + 1\right ) \sqrt {x^{4} + x^{3} - x^{2} - x + 1}}\, dx \]
Not integrable
Time = 0.32 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.22 \[ \int \frac {1+x^4}{\left (-1+x^4\right ) \sqrt {1-x-x^2+x^3+x^4}} \, dx=\int { \frac {x^{4} + 1}{\sqrt {x^{4} + x^{3} - x^{2} - x + 1} {\left (x^{4} - 1\right )}} \,d x } \]
Not integrable
Time = 0.31 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.22 \[ \int \frac {1+x^4}{\left (-1+x^4\right ) \sqrt {1-x-x^2+x^3+x^4}} \, dx=\int { \frac {x^{4} + 1}{\sqrt {x^{4} + x^{3} - x^{2} - x + 1} {\left (x^{4} - 1\right )}} \,d x } \]
Not integrable
Time = 5.76 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.22 \[ \int \frac {1+x^4}{\left (-1+x^4\right ) \sqrt {1-x-x^2+x^3+x^4}} \, dx=\int \frac {x^4+1}{\left (x^4-1\right )\,\sqrt {x^4+x^3-x^2-x+1}} \,d x \]