Integrand size = 29, antiderivative size = 137 \[ \int \frac {-2+x^4}{\sqrt [4]{1+x^4} \left (-1-x^4+2 x^8\right )} \, dx=\frac {\arctan \left (\frac {\sqrt [4]{2} x}{\sqrt [4]{1+x^4}}\right )}{6 \sqrt [4]{2}}+\frac {5 \arctan \left (\frac {\sqrt {2} x \sqrt [4]{1+x^4}}{-x^2+\sqrt {1+x^4}}\right )}{6 \sqrt {2}}+\frac {\text {arctanh}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{1+x^4}}\right )}{6 \sqrt [4]{2}}+\frac {5 \text {arctanh}\left (\frac {\sqrt {2} x \sqrt [4]{1+x^4}}{x^2+\sqrt {1+x^4}}\right )}{6 \sqrt {2}} \]
1/12*arctan(2^(1/4)*x/(x^4+1)^(1/4))*2^(3/4)+5/12*arctan(2^(1/2)*x*(x^4+1) ^(1/4)/(-x^2+(x^4+1)^(1/2)))*2^(1/2)+1/12*arctanh(2^(1/4)*x/(x^4+1)^(1/4)) *2^(3/4)+5/12*arctanh(2^(1/2)*x*(x^4+1)^(1/4)/(x^2+(x^4+1)^(1/2)))*2^(1/2)
Time = 0.38 (sec) , antiderivative size = 126, normalized size of antiderivative = 0.92 \[ \int \frac {-2+x^4}{\sqrt [4]{1+x^4} \left (-1-x^4+2 x^8\right )} \, dx=\frac {\sqrt [4]{2} \arctan \left (\frac {\sqrt [4]{2} x}{\sqrt [4]{1+x^4}}\right )+5 \arctan \left (\frac {\sqrt {2} x \sqrt [4]{1+x^4}}{-x^2+\sqrt {1+x^4}}\right )+\sqrt [4]{2} \text {arctanh}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{1+x^4}}\right )+5 \text {arctanh}\left (\frac {\sqrt {2} x \sqrt [4]{1+x^4}}{x^2+\sqrt {1+x^4}}\right )}{6 \sqrt {2}} \]
(2^(1/4)*ArcTan[(2^(1/4)*x)/(1 + x^4)^(1/4)] + 5*ArcTan[(Sqrt[2]*x*(1 + x^ 4)^(1/4))/(-x^2 + Sqrt[1 + x^4])] + 2^(1/4)*ArcTanh[(2^(1/4)*x)/(1 + x^4)^ (1/4)] + 5*ArcTanh[(Sqrt[2]*x*(1 + x^4)^(1/4))/(x^2 + Sqrt[1 + x^4])])/(6* Sqrt[2])
Time = 0.51 (sec) , antiderivative size = 193, normalized size of antiderivative = 1.41, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.069, Rules used = {7279, 2009}
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.
| \(\displaystyle \int \frac {x^4-2}{\sqrt [4]{x^4+1} \left (2 x^8-x^4-1\right )} \, dx\) |
| \(\Big \downarrow \) 7279 |
| \(\displaystyle \int \left (\frac {10}{3 \sqrt [4]{x^4+1} \left (4 x^4+2\right )}-\frac {4}{3 \sqrt [4]{x^4+1} \left (4 x^4-4\right )}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\arctan \left (\frac {\sqrt [4]{2} x}{\sqrt [4]{x^4+1}}\right )}{6 \sqrt [4]{2}}-\frac {5 \arctan \left (1-\frac {\sqrt {2} x}{\sqrt [4]{x^4+1}}\right )}{6 \sqrt {2}}+\frac {5 \arctan \left (\frac {\sqrt {2} x}{\sqrt [4]{x^4+1}}+1\right )}{6 \sqrt {2}}+\frac {\text {arctanh}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{x^4+1}}\right )}{6 \sqrt [4]{2}}-\frac {5 \log \left (-\frac {\sqrt {2} x}{\sqrt [4]{x^4+1}}+\frac {x^2}{\sqrt {x^4+1}}+1\right )}{12 \sqrt {2}}+\frac {5 \log \left (\frac {\sqrt {2} x}{\sqrt [4]{x^4+1}}+\frac {x^2}{\sqrt {x^4+1}}+1\right )}{12 \sqrt {2}}\) |
ArcTan[(2^(1/4)*x)/(1 + x^4)^(1/4)]/(6*2^(1/4)) - (5*ArcTan[1 - (Sqrt[2]*x )/(1 + x^4)^(1/4)])/(6*Sqrt[2]) + (5*ArcTan[1 + (Sqrt[2]*x)/(1 + x^4)^(1/4 )])/(6*Sqrt[2]) + ArcTanh[(2^(1/4)*x)/(1 + x^4)^(1/4)]/(6*2^(1/4)) - (5*Lo g[1 + x^2/Sqrt[1 + x^4] - (Sqrt[2]*x)/(1 + x^4)^(1/4)])/(12*Sqrt[2]) + (5* Log[1 + x^2/Sqrt[1 + x^4] + (Sqrt[2]*x)/(1 + x^4)^(1/4)])/(12*Sqrt[2])
3.20.52.3.1 Defintions of rubi rules used
Int[(u_)/((a_.) + (b_.)*(x_)^(n_.) + (c_.)*(x_)^(n2_.)), x_Symbol] :> With[ {v = RationalFunctionExpand[u/(a + b*x^n + c*x^(2*n)), x]}, Int[v, x] /; Su mQ[v]] /; FreeQ[{a, b, c}, x] && EqQ[n2, 2*n] && IGtQ[n, 0]
Time = 6.16 (sec) , antiderivative size = 167, normalized size of antiderivative = 1.22
| method | result | size |
| pseudoelliptic | \(-\frac {\arctan \left (\frac {2^{\frac {3}{4}} \left (x^{4}+1\right )^{\frac {1}{4}}}{2 x}\right ) 2^{\frac {3}{4}}}{12}+\frac {\ln \left (\frac {-2^{\frac {1}{4}} x -\left (x^{4}+1\right )^{\frac {1}{4}}}{2^{\frac {1}{4}} x -\left (x^{4}+1\right )^{\frac {1}{4}}}\right ) 2^{\frac {3}{4}}}{24}-\frac {5 \ln \left (\frac {-\left (x^{4}+1\right )^{\frac {1}{4}} x \sqrt {2}+x^{2}+\sqrt {x^{4}+1}}{\left (x^{4}+1\right )^{\frac {1}{4}} x \sqrt {2}+x^{2}+\sqrt {x^{4}+1}}\right ) \sqrt {2}}{24}-\frac {5 \arctan \left (\frac {\left (x^{4}+1\right )^{\frac {1}{4}} \sqrt {2}+x}{x}\right ) \sqrt {2}}{12}-\frac {5 \arctan \left (\frac {\left (x^{4}+1\right )^{\frac {1}{4}} \sqrt {2}-x}{x}\right ) \sqrt {2}}{12}\) | \(167\) |
| trager | \(\text {Expression too large to display}\) | \(642\) |
-1/12*arctan(1/2*2^(3/4)/x*(x^4+1)^(1/4))*2^(3/4)+1/24*ln((-2^(1/4)*x-(x^4 +1)^(1/4))/(2^(1/4)*x-(x^4+1)^(1/4)))*2^(3/4)-5/24*ln((-(x^4+1)^(1/4)*x*2^ (1/2)+x^2+(x^4+1)^(1/2))/((x^4+1)^(1/4)*x*2^(1/2)+x^2+(x^4+1)^(1/2)))*2^(1 /2)-5/12*arctan(((x^4+1)^(1/4)*2^(1/2)+x)/x)*2^(1/2)-5/12*arctan(((x^4+1)^ (1/4)*2^(1/2)-x)/x)*2^(1/2)
Result contains complex when optimal does not.
Time = 6.18 (sec) , antiderivative size = 502, normalized size of antiderivative = 3.66 \[ \int \frac {-2+x^4}{\sqrt [4]{1+x^4} \left (-1-x^4+2 x^8\right )} \, dx=\frac {1}{48} \cdot 2^{\frac {3}{4}} \log \left (\frac {4 \, \sqrt {2} {\left (x^{4} + 1\right )}^{\frac {1}{4}} x^{3} + 4 \cdot 2^{\frac {1}{4}} \sqrt {x^{4} + 1} x^{2} + 2^{\frac {3}{4}} {\left (3 \, x^{4} + 1\right )} + 4 \, {\left (x^{4} + 1\right )}^{\frac {3}{4}} x}{x^{4} - 1}\right ) + \frac {1}{48} i \cdot 2^{\frac {3}{4}} \log \left (-\frac {4 \, \sqrt {2} {\left (x^{4} + 1\right )}^{\frac {1}{4}} x^{3} + 4 i \cdot 2^{\frac {1}{4}} \sqrt {x^{4} + 1} x^{2} - 2^{\frac {3}{4}} {\left (3 i \, x^{4} + i\right )} - 4 \, {\left (x^{4} + 1\right )}^{\frac {3}{4}} x}{x^{4} - 1}\right ) - \frac {1}{48} i \cdot 2^{\frac {3}{4}} \log \left (-\frac {4 \, \sqrt {2} {\left (x^{4} + 1\right )}^{\frac {1}{4}} x^{3} - 4 i \cdot 2^{\frac {1}{4}} \sqrt {x^{4} + 1} x^{2} - 2^{\frac {3}{4}} {\left (-3 i \, x^{4} - i\right )} - 4 \, {\left (x^{4} + 1\right )}^{\frac {3}{4}} x}{x^{4} - 1}\right ) - \frac {1}{48} \cdot 2^{\frac {3}{4}} \log \left (\frac {4 \, \sqrt {2} {\left (x^{4} + 1\right )}^{\frac {1}{4}} x^{3} - 4 \cdot 2^{\frac {1}{4}} \sqrt {x^{4} + 1} x^{2} - 2^{\frac {3}{4}} {\left (3 \, x^{4} + 1\right )} + 4 \, {\left (x^{4} + 1\right )}^{\frac {3}{4}} x}{x^{4} - 1}\right ) - \left (\frac {5}{48} i + \frac {5}{48}\right ) \, \sqrt {2} \log \left (\frac {\left (i + 1\right ) \, \sqrt {2} {\left (x^{4} + 1\right )}^{\frac {1}{4}} x^{3} - 2 i \, \sqrt {x^{4} + 1} x^{2} + \left (i - 1\right ) \, \sqrt {2} {\left (x^{4} + 1\right )}^{\frac {3}{4}} x + 1}{2 \, x^{4} + 1}\right ) + \left (\frac {5}{48} i - \frac {5}{48}\right ) \, \sqrt {2} \log \left (\frac {-\left (i - 1\right ) \, \sqrt {2} {\left (x^{4} + 1\right )}^{\frac {1}{4}} x^{3} + 2 i \, \sqrt {x^{4} + 1} x^{2} - \left (i + 1\right ) \, \sqrt {2} {\left (x^{4} + 1\right )}^{\frac {3}{4}} x + 1}{2 \, x^{4} + 1}\right ) - \left (\frac {5}{48} i - \frac {5}{48}\right ) \, \sqrt {2} \log \left (\frac {\left (i - 1\right ) \, \sqrt {2} {\left (x^{4} + 1\right )}^{\frac {1}{4}} x^{3} + 2 i \, \sqrt {x^{4} + 1} x^{2} + \left (i + 1\right ) \, \sqrt {2} {\left (x^{4} + 1\right )}^{\frac {3}{4}} x + 1}{2 \, x^{4} + 1}\right ) + \left (\frac {5}{48} i + \frac {5}{48}\right ) \, \sqrt {2} \log \left (\frac {-\left (i + 1\right ) \, \sqrt {2} {\left (x^{4} + 1\right )}^{\frac {1}{4}} x^{3} - 2 i \, \sqrt {x^{4} + 1} x^{2} - \left (i - 1\right ) \, \sqrt {2} {\left (x^{4} + 1\right )}^{\frac {3}{4}} x + 1}{2 \, x^{4} + 1}\right ) \]
1/48*2^(3/4)*log((4*sqrt(2)*(x^4 + 1)^(1/4)*x^3 + 4*2^(1/4)*sqrt(x^4 + 1)* x^2 + 2^(3/4)*(3*x^4 + 1) + 4*(x^4 + 1)^(3/4)*x)/(x^4 - 1)) + 1/48*I*2^(3/ 4)*log(-(4*sqrt(2)*(x^4 + 1)^(1/4)*x^3 + 4*I*2^(1/4)*sqrt(x^4 + 1)*x^2 - 2 ^(3/4)*(3*I*x^4 + I) - 4*(x^4 + 1)^(3/4)*x)/(x^4 - 1)) - 1/48*I*2^(3/4)*lo g(-(4*sqrt(2)*(x^4 + 1)^(1/4)*x^3 - 4*I*2^(1/4)*sqrt(x^4 + 1)*x^2 - 2^(3/4 )*(-3*I*x^4 - I) - 4*(x^4 + 1)^(3/4)*x)/(x^4 - 1)) - 1/48*2^(3/4)*log((4*s qrt(2)*(x^4 + 1)^(1/4)*x^3 - 4*2^(1/4)*sqrt(x^4 + 1)*x^2 - 2^(3/4)*(3*x^4 + 1) + 4*(x^4 + 1)^(3/4)*x)/(x^4 - 1)) - (5/48*I + 5/48)*sqrt(2)*log(((I + 1)*sqrt(2)*(x^4 + 1)^(1/4)*x^3 - 2*I*sqrt(x^4 + 1)*x^2 + (I - 1)*sqrt(2)* (x^4 + 1)^(3/4)*x + 1)/(2*x^4 + 1)) + (5/48*I - 5/48)*sqrt(2)*log((-(I - 1 )*sqrt(2)*(x^4 + 1)^(1/4)*x^3 + 2*I*sqrt(x^4 + 1)*x^2 - (I + 1)*sqrt(2)*(x ^4 + 1)^(3/4)*x + 1)/(2*x^4 + 1)) - (5/48*I - 5/48)*sqrt(2)*log(((I - 1)*s qrt(2)*(x^4 + 1)^(1/4)*x^3 + 2*I*sqrt(x^4 + 1)*x^2 + (I + 1)*sqrt(2)*(x^4 + 1)^(3/4)*x + 1)/(2*x^4 + 1)) + (5/48*I + 5/48)*sqrt(2)*log((-(I + 1)*sqr t(2)*(x^4 + 1)^(1/4)*x^3 - 2*I*sqrt(x^4 + 1)*x^2 - (I - 1)*sqrt(2)*(x^4 + 1)^(3/4)*x + 1)/(2*x^4 + 1))
\[ \int \frac {-2+x^4}{\sqrt [4]{1+x^4} \left (-1-x^4+2 x^8\right )} \, dx=\int \frac {x^{4} - 2}{\left (x - 1\right ) \left (x + 1\right ) \left (x^{2} + 1\right ) \sqrt [4]{x^{4} + 1} \cdot \left (2 x^{4} + 1\right )}\, dx \]
\[ \int \frac {-2+x^4}{\sqrt [4]{1+x^4} \left (-1-x^4+2 x^8\right )} \, dx=\int { \frac {x^{4} - 2}{{\left (2 \, x^{8} - x^{4} - 1\right )} {\left (x^{4} + 1\right )}^{\frac {1}{4}}} \,d x } \]
\[ \int \frac {-2+x^4}{\sqrt [4]{1+x^4} \left (-1-x^4+2 x^8\right )} \, dx=\int { \frac {x^{4} - 2}{{\left (2 \, x^{8} - x^{4} - 1\right )} {\left (x^{4} + 1\right )}^{\frac {1}{4}}} \,d x } \]
Timed out. \[ \int \frac {-2+x^4}{\sqrt [4]{1+x^4} \left (-1-x^4+2 x^8\right )} \, dx=-\int \frac {x^4-2}{{\left (x^4+1\right )}^{1/4}\,\left (-2\,x^8+x^4+1\right )} \,d x \]