Integrand size = 33, antiderivative size = 120 \[ \int \frac {-1+x^4}{\left (1+x^4\right ) \sqrt {1-x-x^2+x^3+x^4}} \, dx=\frac {1}{2} \text {RootSum}\left [3-8 \text {$\#$1}+6 \text {$\#$1}^2+\text {$\#$1}^4\&,\frac {\log (x)-\log \left (1-x^2+\sqrt {1-x-x^2+x^3+x^4}-x \text {$\#$1}\right )-\log (x) \text {$\#$1}^2+\log \left (1-x^2+\sqrt {1-x-x^2+x^3+x^4}-x \text {$\#$1}\right ) \text {$\#$1}^2}{-2+3 \text {$\#$1}+\text {$\#$1}^3}\&\right ] \]
Time = 0.00 (sec) , antiderivative size = 120, normalized size of antiderivative = 1.00 \[ \int \frac {-1+x^4}{\left (1+x^4\right ) \sqrt {1-x-x^2+x^3+x^4}} \, dx=\frac {1}{2} \text {RootSum}\left [3-8 \text {$\#$1}+6 \text {$\#$1}^2+\text {$\#$1}^4\&,\frac {\log (x)-\log \left (1-x^2+\sqrt {1-x-x^2+x^3+x^4}-x \text {$\#$1}\right )-\log (x) \text {$\#$1}^2+\log \left (1-x^2+\sqrt {1-x-x^2+x^3+x^4}-x \text {$\#$1}\right ) \text {$\#$1}^2}{-2+3 \text {$\#$1}+\text {$\#$1}^3}\&\right ] \]
RootSum[3 - 8*#1 + 6*#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)/(-2 + 3*#1 + #1^3) & ]/2
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 |
| \(\displaystyle \int \left (\frac {1}{\sqrt {x^4+x^3-x^2-x+1}}-\frac {2}{\left (x^4+1\right ) \sqrt {x^4+x^3-x^2-x+1}}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \int \frac {1}{\sqrt {x^4+x^3-x^2-x+1}}dx-\frac {1}{2} \sqrt [4]{-1} \int \frac {1}{\left (\sqrt [4]{-1}-x\right ) \sqrt {x^4+x^3-x^2-x+1}}dx+\frac {1}{2} (-1)^{3/4} \int \frac {1}{\left (-x-(-1)^{3/4}\right ) \sqrt {x^4+x^3-x^2-x+1}}dx-\frac {1}{2} \sqrt [4]{-1} \int \frac {1}{\left (x+\sqrt [4]{-1}\right ) \sqrt {x^4+x^3-x^2-x+1}}dx+\frac {1}{2} (-1)^{3/4} \int \frac {1}{\left (x-(-1)^{3/4}\right ) \sqrt {x^4+x^3-x^2-x+1}}dx\) |
3.18.83.3.1 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 = 2.33 (sec) , antiderivative size = 256, normalized size of antiderivative = 2.13
| method | result | size |
| default | \(-\frac {\sqrt {\frac {x^{4}+x^{3}-x^{2}-x +1}{\left (x \sqrt {3}+x^{2}-x -1\right )^{2}}}\, \left (\left (3-2 \sqrt {3}\right ) \sqrt {1+\sqrt {3}}\, \operatorname {arctanh}\left (\frac {2 \sqrt {\frac {x^{4}+x^{3}-x^{2}-x +1}{\left (x \sqrt {3}+x^{2}-x -1\right )^{2}}}}{\sqrt {1+\sqrt {3}}}\right )+\sqrt {-3+3 \sqrt {3}}\, \arctan \left (\frac {\sqrt {3}\, \sqrt {-3+3 \sqrt {3}}\, \sqrt {\frac {x^{4}+x^{3}-x^{2}-x +1}{\left (x \sqrt {3}+x^{2}-x -1\right )^{2}}}\, \left (-x \sqrt {3}+x^{2}-x -1\right ) \left (x \sqrt {3}+x^{2}-x -1\right )}{6 x^{4}+6 x^{3}-6 x^{2}-6 x +6}\right )\right ) \left (x \sqrt {3}+x^{2}-x -1\right )}{3 \sqrt {\frac {x^{4}+x^{3}-x^{2}-x +1}{x^{2}}}\, \left (1+\sqrt {3}\right ) \left (-2+\sqrt {3}\right ) x}\) | \(256\) |
| pseudoelliptic | \(-\frac {\sqrt {\frac {x^{4}+x^{3}-x^{2}-x +1}{\left (x \sqrt {3}+x^{2}-x -1\right )^{2}}}\, \left (\left (3-2 \sqrt {3}\right ) \sqrt {1+\sqrt {3}}\, \operatorname {arctanh}\left (\frac {2 \sqrt {\frac {x^{4}+x^{3}-x^{2}-x +1}{\left (x \sqrt {3}+x^{2}-x -1\right )^{2}}}}{\sqrt {1+\sqrt {3}}}\right )+\sqrt {-3+3 \sqrt {3}}\, \arctan \left (\frac {\sqrt {3}\, \sqrt {-3+3 \sqrt {3}}\, \sqrt {\frac {x^{4}+x^{3}-x^{2}-x +1}{\left (x \sqrt {3}+x^{2}-x -1\right )^{2}}}\, \left (-x \sqrt {3}+x^{2}-x -1\right ) \left (x \sqrt {3}+x^{2}-x -1\right )}{6 x^{4}+6 x^{3}-6 x^{2}-6 x +6}\right )\right ) \left (x \sqrt {3}+x^{2}-x -1\right )}{3 \sqrt {\frac {x^{4}+x^{3}-x^{2}-x +1}{x^{2}}}\, \left (1+\sqrt {3}\right ) \left (-2+\sqrt {3}\right ) x}\) | \(256\) |
| trager | \(-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+36 \operatorname {RootOf}\left (48 \textit {\_Z}^{4}+8 \textit {\_Z}^{2}+1\right )^{2}+6\right ) \ln \left (\frac {12 \operatorname {RootOf}\left (48 \textit {\_Z}^{4}+8 \textit {\_Z}^{2}+1\right )^{2} \operatorname {RootOf}\left (\textit {\_Z}^{2}+36 \operatorname {RootOf}\left (48 \textit {\_Z}^{4}+8 \textit {\_Z}^{2}+1\right )^{2}+6\right ) x +48 \sqrt {x^{4}+x^{3}-x^{2}-x +1}\, \operatorname {RootOf}\left (48 \textit {\_Z}^{4}+8 \textit {\_Z}^{2}+1\right )^{2}-3 \operatorname {RootOf}\left (\textit {\_Z}^{2}+36 \operatorname {RootOf}\left (48 \textit {\_Z}^{4}+8 \textit {\_Z}^{2}+1\right )^{2}+6\right ) x^{2}-\operatorname {RootOf}\left (\textit {\_Z}^{2}+36 \operatorname {RootOf}\left (48 \textit {\_Z}^{4}+8 \textit {\_Z}^{2}+1\right )^{2}+6\right ) x +2 \sqrt {x^{4}+x^{3}-x^{2}-x +1}+3 \operatorname {RootOf}\left (\textit {\_Z}^{2}+36 \operatorname {RootOf}\left (48 \textit {\_Z}^{4}+8 \textit {\_Z}^{2}+1\right )^{2}+6\right )}{12 x \operatorname {RootOf}\left (48 \textit {\_Z}^{4}+8 \textit {\_Z}^{2}+1\right )^{2}-x^{2}+x +1}\right )}{6}-\operatorname {RootOf}\left (48 \textit {\_Z}^{4}+8 \textit {\_Z}^{2}+1\right ) \ln \left (-\frac {12 \operatorname {RootOf}\left (48 \textit {\_Z}^{4}+8 \textit {\_Z}^{2}+1\right )^{3} x +8 \sqrt {x^{4}+x^{3}-x^{2}-x +1}\, \operatorname {RootOf}\left (48 \textit {\_Z}^{4}+8 \textit {\_Z}^{2}+1\right )^{2}+3 \operatorname {RootOf}\left (48 \textit {\_Z}^{4}+8 \textit {\_Z}^{2}+1\right ) x^{2}+3 \operatorname {RootOf}\left (48 \textit {\_Z}^{4}+8 \textit {\_Z}^{2}+1\right ) x -3 \operatorname {RootOf}\left (48 \textit {\_Z}^{4}+8 \textit {\_Z}^{2}+1\right )+\sqrt {x^{4}+x^{3}-x^{2}-x +1}}{12 x \operatorname {RootOf}\left (48 \textit {\_Z}^{4}+8 \textit {\_Z}^{2}+1\right )^{2}+x^{2}+x -1}\right )\) | \(398\) |
| elliptic | \(\text {Expression too large to display}\) | \(3623\) |
-1/3*((x^4+x^3-x^2-x+1)/(x*3^(1/2)+x^2-x-1)^2)^(1/2)*((3-2*3^(1/2))*(1+3^( 1/2))^(1/2)*arctanh(2*((x^4+x^3-x^2-x+1)/(x*3^(1/2)+x^2-x-1)^2)^(1/2)/(1+3 ^(1/2))^(1/2))+(-3+3*3^(1/2))^(1/2)*arctan(3^(1/2)*(-3+3*3^(1/2))^(1/2)*(( x^4+x^3-x^2-x+1)/(x*3^(1/2)+x^2-x-1)^2)^(1/2)*(-x*3^(1/2)+x^2-x-1)*(x*3^(1 /2)+x^2-x-1)/(6*x^4+6*x^3-6*x^2-6*x+6)))/((x^4+x^3-x^2-x+1)/x^2)^(1/2)*(x* 3^(1/2)+x^2-x-1)/(1+3^(1/2))/(-2+3^(1/2))/x
Result contains higher order function than in optimal. Order 3 vs. order 1.
Time = 0.35 (sec) , antiderivative size = 475, normalized size of antiderivative = 3.96 \[ \int \frac {-1+x^4}{\left (1+x^4\right ) \sqrt {1-x-x^2+x^3+x^4}} \, dx=\frac {1}{12} \, \sqrt {3} \sqrt {i \, \sqrt {2} - 1} \log \left (\frac {\sqrt {3} {\left (3 \, x^{4} + 2 \, x^{3} - 4 \, x^{2} - 2 \, \sqrt {2} {\left (i \, x^{3} + i \, x^{2} - i \, x\right )} - 2 \, x + 3\right )} \sqrt {i \, \sqrt {2} - 1} - 2 \, \sqrt {x^{4} + x^{3} - x^{2} - x + 1} {\left (x^{2} + \sqrt {2} {\left (2 i \, x^{2} - i \, x - 2 i\right )} + 4 \, x - 1\right )}}{x^{4} + 1}\right ) - \frac {1}{12} \, \sqrt {3} \sqrt {i \, \sqrt {2} - 1} \log \left (-\frac {\sqrt {3} {\left (3 \, x^{4} + 2 \, x^{3} - 4 \, x^{2} + 2 \, \sqrt {2} {\left (-i \, x^{3} - i \, x^{2} + i \, x\right )} - 2 \, x + 3\right )} \sqrt {i \, \sqrt {2} - 1} + 2 \, \sqrt {x^{4} + x^{3} - x^{2} - x + 1} {\left (x^{2} + \sqrt {2} {\left (2 i \, x^{2} - i \, x - 2 i\right )} + 4 \, x - 1\right )}}{x^{4} + 1}\right ) - \frac {1}{12} \, \sqrt {3} \sqrt {-i \, \sqrt {2} - 1} \log \left (-\frac {\sqrt {3} {\left (3 \, x^{4} + 2 \, x^{3} - 4 \, x^{2} + 2 \, \sqrt {2} {\left (i \, x^{3} + i \, x^{2} - i \, x\right )} - 2 \, x + 3\right )} \sqrt {-i \, \sqrt {2} - 1} + 2 \, \sqrt {x^{4} + x^{3} - x^{2} - x + 1} {\left (x^{2} + \sqrt {2} {\left (-2 i \, x^{2} + i \, x + 2 i\right )} + 4 \, x - 1\right )}}{x^{4} + 1}\right ) + \frac {1}{12} \, \sqrt {3} \sqrt {-i \, \sqrt {2} - 1} \log \left (\frac {\sqrt {3} {\left (3 \, x^{4} + 2 \, x^{3} - 4 \, x^{2} - 2 \, \sqrt {2} {\left (-i \, x^{3} - i \, x^{2} + i \, x\right )} - 2 \, x + 3\right )} \sqrt {-i \, \sqrt {2} - 1} - 2 \, \sqrt {x^{4} + x^{3} - x^{2} - x + 1} {\left (x^{2} + \sqrt {2} {\left (-2 i \, x^{2} + i \, x + 2 i\right )} + 4 \, x - 1\right )}}{x^{4} + 1}\right ) \]
1/12*sqrt(3)*sqrt(I*sqrt(2) - 1)*log((sqrt(3)*(3*x^4 + 2*x^3 - 4*x^2 - 2*s qrt(2)*(I*x^3 + I*x^2 - I*x) - 2*x + 3)*sqrt(I*sqrt(2) - 1) - 2*sqrt(x^4 + x^3 - x^2 - x + 1)*(x^2 + sqrt(2)*(2*I*x^2 - I*x - 2*I) + 4*x - 1))/(x^4 + 1)) - 1/12*sqrt(3)*sqrt(I*sqrt(2) - 1)*log(-(sqrt(3)*(3*x^4 + 2*x^3 - 4* x^2 + 2*sqrt(2)*(-I*x^3 - I*x^2 + I*x) - 2*x + 3)*sqrt(I*sqrt(2) - 1) + 2* sqrt(x^4 + x^3 - x^2 - x + 1)*(x^2 + sqrt(2)*(2*I*x^2 - I*x - 2*I) + 4*x - 1))/(x^4 + 1)) - 1/12*sqrt(3)*sqrt(-I*sqrt(2) - 1)*log(-(sqrt(3)*(3*x^4 + 2*x^3 - 4*x^2 + 2*sqrt(2)*(I*x^3 + I*x^2 - I*x) - 2*x + 3)*sqrt(-I*sqrt(2 ) - 1) + 2*sqrt(x^4 + x^3 - x^2 - x + 1)*(x^2 + sqrt(2)*(-2*I*x^2 + I*x + 2*I) + 4*x - 1))/(x^4 + 1)) + 1/12*sqrt(3)*sqrt(-I*sqrt(2) - 1)*log((sqrt( 3)*(3*x^4 + 2*x^3 - 4*x^2 - 2*sqrt(2)*(-I*x^3 - I*x^2 + I*x) - 2*x + 3)*sq rt(-I*sqrt(2) - 1) - 2*sqrt(x^4 + x^3 - x^2 - x + 1)*(x^2 + sqrt(2)*(-2*I* x^2 + I*x + 2*I) + 4*x - 1))/(x^4 + 1))
Not integrable
Time = 2.24 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.28 \[ \int \frac {-1+x^4}{\left (1+x^4\right ) \sqrt {1-x-x^2+x^3+x^4}} \, dx=\int \frac {\left (x - 1\right ) \left (x + 1\right ) \left (x^{2} + 1\right )}{\left (x^{4} + 1\right ) \sqrt {x^{4} + x^{3} - x^{2} - x + 1}}\, dx \]
Not integrable
Time = 0.32 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.28 \[ \int \frac {-1+x^4}{\left (1+x^4\right ) \sqrt {1-x-x^2+x^3+x^4}} \, dx=\int { \frac {x^{4} - 1}{\sqrt {x^{4} + x^{3} - x^{2} - x + 1} {\left (x^{4} + 1\right )}} \,d x } \]
Not integrable
Time = 0.31 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.28 \[ \int \frac {-1+x^4}{\left (1+x^4\right ) \sqrt {1-x-x^2+x^3+x^4}} \, dx=\int { \frac {x^{4} - 1}{\sqrt {x^{4} + x^{3} - x^{2} - x + 1} {\left (x^{4} + 1\right )}} \,d x } \]
Not integrable
Time = 0.00 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.28 \[ \int \frac {-1+x^4}{\left (1+x^4\right ) \sqrt {1-x-x^2+x^3+x^4}} \, dx=\int \frac {x^4-1}{\left (x^4+1\right )\,\sqrt {x^4+x^3-x^2-x+1}} \,d x \]