Integrand size = 21, antiderivative size = 114 \[ \int \frac {-1+x}{\left (1+x+x^2\right ) \sqrt [4]{1+x^4}} \, dx=\frac {\arctan \left (\frac {-\frac {1}{\sqrt {2}}-\sqrt {2} x-\frac {x^2}{\sqrt {2}}+\frac {\sqrt {1+x^4}}{\sqrt {2}}}{(1+x) \sqrt [4]{1+x^4}}\right )}{\sqrt {2}}-\frac {\text {arctanh}\left (\frac {\left (\sqrt {2}+\sqrt {2} x\right ) \sqrt [4]{1+x^4}}{1+2 x+x^2+\sqrt {1+x^4}}\right )}{\sqrt {2}} \]
1/2*arctan((-1/2*2^(1/2)-x*2^(1/2)-1/2*2^(1/2)*x^2+1/2*(x^4+1)^(1/2)*2^(1/ 2))/(1+x)/(x^4+1)^(1/4))*2^(1/2)-1/2*arctanh((2^(1/2)+x*2^(1/2))*(x^4+1)^( 1/4)/(1+2*x+x^2+(x^4+1)^(1/2)))*2^(1/2)
Time = 1.65 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.76 \[ \int \frac {-1+x}{\left (1+x+x^2\right ) \sqrt [4]{1+x^4}} \, dx=\frac {\arctan \left (\frac {-1-2 x-x^2+\sqrt {1+x^4}}{\sqrt {2} (1+x) \sqrt [4]{1+x^4}}\right )-\text {arctanh}\left (\frac {\sqrt {2} (1+x) \sqrt [4]{1+x^4}}{1+2 x+x^2+\sqrt {1+x^4}}\right )}{\sqrt {2}} \]
(ArcTan[(-1 - 2*x - x^2 + Sqrt[1 + x^4])/(Sqrt[2]*(1 + x)*(1 + x^4)^(1/4)) ] - ArcTanh[(Sqrt[2]*(1 + x)*(1 + x^4)^(1/4))/(1 + 2*x + x^2 + Sqrt[1 + x^ 4])])/Sqrt[2]
Result contains higher order function than in optimal. Order 6 vs. order 3 in optimal.
Time = 1.71 (sec) , antiderivative size = 797, normalized size of antiderivative = 6.99, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, 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.
| \(\displaystyle \int \frac {x-1}{\left (x^2+x+1\right ) \sqrt [4]{x^4+1}} \, dx\) |
| \(\Big \downarrow \) 7279 |
| \(\displaystyle \int \left (\frac {1+i \sqrt {3}}{\left (2 x-i \sqrt {3}+1\right ) \sqrt [4]{x^4+1}}+\frac {1-i \sqrt {3}}{\left (2 x+i \sqrt {3}+1\right ) \sqrt [4]{x^4+1}}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {1}{6} \left (1+i \sqrt {3}\right ) \operatorname {AppellF1}\left (\frac {3}{4},\frac {1}{4},1,\frac {7}{4},-x^4,-\frac {2 x^4}{1-i \sqrt {3}}\right ) x^3-\frac {1}{6} \left (1-i \sqrt {3}\right ) \operatorname {AppellF1}\left (\frac {3}{4},\frac {1}{4},1,\frac {7}{4},-x^4,-\frac {2 x^4}{1+i \sqrt {3}}\right ) x^3-\frac {\arctan \left (\frac {x}{\sqrt [4]{-\frac {i-\sqrt {3}}{i+\sqrt {3}}} \sqrt [4]{x^4+1}}\right )}{2 \left (-\frac {i-\sqrt {3}}{i+\sqrt {3}}\right )^{3/4}}-\frac {1}{2} \left (-\frac {i-\sqrt {3}}{i+\sqrt {3}}\right )^{3/4} \arctan \left (\frac {\sqrt [4]{-\frac {i-\sqrt {3}}{i+\sqrt {3}}} x}{\sqrt [4]{x^4+1}}\right )+\frac {1}{2} \left (\frac {1}{2} \left (1-i \sqrt {3}\right )\right )^{3/4} \arctan \left (\frac {\sqrt [4]{2} \sqrt [4]{x^4+1}}{\sqrt [4]{1-i \sqrt {3}}}\right )+\frac {1}{2} \left (\frac {1}{2} \left (1+i \sqrt {3}\right )\right )^{3/4} \arctan \left (\frac {\sqrt [4]{2} \sqrt [4]{x^4+1}}{\sqrt [4]{1+i \sqrt {3}}}\right )-\frac {\text {arctanh}\left (\frac {x}{\sqrt [4]{-\frac {i-\sqrt {3}}{i+\sqrt {3}}} \sqrt [4]{x^4+1}}\right )}{2 \left (-\frac {i-\sqrt {3}}{i+\sqrt {3}}\right )^{3/4}}-\frac {1}{2} \left (-\frac {i-\sqrt {3}}{i+\sqrt {3}}\right )^{3/4} \text {arctanh}\left (\frac {\sqrt [4]{-\frac {i-\sqrt {3}}{i+\sqrt {3}}} x}{\sqrt [4]{x^4+1}}\right )-\frac {1}{2} \left (\frac {1}{2} \left (1-i \sqrt {3}\right )\right )^{3/4} \text {arctanh}\left (\frac {\sqrt [4]{2} \sqrt [4]{x^4+1}}{\sqrt [4]{1-i \sqrt {3}}}\right )-\frac {1}{2} \left (\frac {1}{2} \left (1+i \sqrt {3}\right )\right )^{3/4} \text {arctanh}\left (\frac {\sqrt [4]{2} \sqrt [4]{x^4+1}}{\sqrt [4]{1+i \sqrt {3}}}\right )+\frac {i \sqrt {-x^4} \operatorname {EllipticPi}\left (\frac {1}{2} \left (-i-\sqrt {3}\right ),\arcsin \left (\sqrt [4]{x^4+1}\right ),-1\right )}{2 x^2}-\frac {i \sqrt {-x^4} \operatorname {EllipticPi}\left (\frac {1}{2} \left (i-\sqrt {3}\right ),\arcsin \left (\sqrt [4]{x^4+1}\right ),-1\right )}{2 x^2}-\frac {i \sqrt {-x^4} \operatorname {EllipticPi}\left (\frac {1}{\sqrt {\frac {1}{2} \left (1-i \sqrt {3}\right )}},\arcsin \left (\sqrt [4]{x^4+1}\right ),-1\right )}{2 x^2}+\frac {i \sqrt {-x^4} \operatorname {EllipticPi}\left (\frac {1}{\sqrt {\frac {1}{2} \left (1+i \sqrt {3}\right )}},\arcsin \left (\sqrt [4]{x^4+1}\right ),-1\right )}{2 x^2}\) |
-1/6*((1 + I*Sqrt[3])*x^3*AppellF1[3/4, 1/4, 1, 7/4, -x^4, (-2*x^4)/(1 - I *Sqrt[3])]) - ((1 - I*Sqrt[3])*x^3*AppellF1[3/4, 1/4, 1, 7/4, -x^4, (-2*x^ 4)/(1 + I*Sqrt[3])])/6 - ArcTan[x/((-((I - Sqrt[3])/(I + Sqrt[3])))^(1/4)* (1 + x^4)^(1/4))]/(2*(-((I - Sqrt[3])/(I + Sqrt[3])))^(3/4)) - ((-((I - Sq rt[3])/(I + Sqrt[3])))^(3/4)*ArcTan[((-((I - Sqrt[3])/(I + Sqrt[3])))^(1/4 )*x)/(1 + x^4)^(1/4)])/2 + (((1 - I*Sqrt[3])/2)^(3/4)*ArcTan[(2^(1/4)*(1 + x^4)^(1/4))/(1 - I*Sqrt[3])^(1/4)])/2 + (((1 + I*Sqrt[3])/2)^(3/4)*ArcTan [(2^(1/4)*(1 + x^4)^(1/4))/(1 + I*Sqrt[3])^(1/4)])/2 - ArcTanh[x/((-((I - Sqrt[3])/(I + Sqrt[3])))^(1/4)*(1 + x^4)^(1/4))]/(2*(-((I - Sqrt[3])/(I + Sqrt[3])))^(3/4)) - ((-((I - Sqrt[3])/(I + Sqrt[3])))^(3/4)*ArcTanh[((-((I - Sqrt[3])/(I + Sqrt[3])))^(1/4)*x)/(1 + x^4)^(1/4)])/2 - (((1 - I*Sqrt[3 ])/2)^(3/4)*ArcTanh[(2^(1/4)*(1 + x^4)^(1/4))/(1 - I*Sqrt[3])^(1/4)])/2 - (((1 + I*Sqrt[3])/2)^(3/4)*ArcTanh[(2^(1/4)*(1 + x^4)^(1/4))/(1 + I*Sqrt[3 ])^(1/4)])/2 + ((I/2)*Sqrt[-x^4]*EllipticPi[(-I - Sqrt[3])/2, ArcSin[(1 + x^4)^(1/4)], -1])/x^2 - ((I/2)*Sqrt[-x^4]*EllipticPi[(I - Sqrt[3])/2, ArcS in[(1 + x^4)^(1/4)], -1])/x^2 - ((I/2)*Sqrt[-x^4]*EllipticPi[1/Sqrt[(1 - I *Sqrt[3])/2], ArcSin[(1 + x^4)^(1/4)], -1])/x^2 + ((I/2)*Sqrt[-x^4]*Ellipt icPi[1/Sqrt[(1 + I*Sqrt[3])/2], ArcSin[(1 + x^4)^(1/4)], -1])/x^2
3.17.100.3.1 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]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 3.28 (sec) , antiderivative size = 390, normalized size of antiderivative = 3.42
| method | result | size |
| trager | \(\frac {\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right ) \ln \left (\frac {\sqrt {x^{4}+1}\, \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{3} x^{2}+2 \sqrt {x^{4}+1}\, \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{3} x -\left (x^{4}+1\right )^{\frac {1}{4}} \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{2} x^{3}+\sqrt {x^{4}+1}\, \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{3}-3 \left (x^{4}+1\right )^{\frac {1}{4}} \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{2} x^{2}-3 \left (x^{4}+1\right )^{\frac {1}{4}} \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{2} x +2 \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right ) x^{3}+\left (x^{4}+1\right )^{\frac {3}{4}} x -\left (x^{4}+1\right )^{\frac {1}{4}} \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{2}+3 \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right ) x^{2}+\left (x^{4}+1\right )^{\frac {3}{4}}+2 \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right ) x}{\left (x^{2}+x +1\right )^{2}}\right )}{2}+\frac {\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{3} \ln \left (\frac {2 \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{3} x^{3}+\left (x^{4}+1\right )^{\frac {1}{4}} \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{2} x^{3}+3 \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{3} x^{2}+3 \left (x^{4}+1\right )^{\frac {1}{4}} \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{2} x^{2}+\sqrt {x^{4}+1}\, \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right ) x^{2}+2 \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{3} x +3 \left (x^{4}+1\right )^{\frac {1}{4}} \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{2} x +2 \sqrt {x^{4}+1}\, \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right ) x +\left (x^{4}+1\right )^{\frac {3}{4}} x +\left (x^{4}+1\right )^{\frac {1}{4}} \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{2}+\sqrt {x^{4}+1}\, \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )+\left (x^{4}+1\right )^{\frac {3}{4}}}{\left (x^{2}+x +1\right )^{2}}\right )}{2}\) | \(390\) |
1/2*RootOf(_Z^4+1)*ln(((x^4+1)^(1/2)*RootOf(_Z^4+1)^3*x^2+2*(x^4+1)^(1/2)* RootOf(_Z^4+1)^3*x-(x^4+1)^(1/4)*RootOf(_Z^4+1)^2*x^3+(x^4+1)^(1/2)*RootOf (_Z^4+1)^3-3*(x^4+1)^(1/4)*RootOf(_Z^4+1)^2*x^2-3*(x^4+1)^(1/4)*RootOf(_Z^ 4+1)^2*x+2*RootOf(_Z^4+1)*x^3+(x^4+1)^(3/4)*x-(x^4+1)^(1/4)*RootOf(_Z^4+1) ^2+3*RootOf(_Z^4+1)*x^2+(x^4+1)^(3/4)+2*RootOf(_Z^4+1)*x)/(x^2+x+1)^2)+1/2 *RootOf(_Z^4+1)^3*ln((2*RootOf(_Z^4+1)^3*x^3+(x^4+1)^(1/4)*RootOf(_Z^4+1)^ 2*x^3+3*RootOf(_Z^4+1)^3*x^2+3*(x^4+1)^(1/4)*RootOf(_Z^4+1)^2*x^2+(x^4+1)^ (1/2)*RootOf(_Z^4+1)*x^2+2*RootOf(_Z^4+1)^3*x+3*(x^4+1)^(1/4)*RootOf(_Z^4+ 1)^2*x+2*(x^4+1)^(1/2)*RootOf(_Z^4+1)*x+(x^4+1)^(3/4)*x+(x^4+1)^(1/4)*Root Of(_Z^4+1)^2+(x^4+1)^(1/2)*RootOf(_Z^4+1)+(x^4+1)^(3/4))/(x^2+x+1)^2)
Result contains complex when optimal does not.
Time = 4.55 (sec) , antiderivative size = 413, normalized size of antiderivative = 3.62 \[ \int \frac {-1+x}{\left (1+x+x^2\right ) \sqrt [4]{1+x^4}} \, dx=\left (\frac {1}{8} i - \frac {1}{8}\right ) \, \sqrt {2} \log \left (\frac {\sqrt {2} \sqrt {x^{4} + 1} {\left (\left (i + 1\right ) \, x^{2} + \left (2 i + 2\right ) \, x + i + 1\right )} + 2 \, {\left (x^{4} + 1\right )}^{\frac {3}{4}} {\left (x + 1\right )} + \sqrt {2} {\left (\left (2 i - 2\right ) \, x^{3} + \left (3 i - 3\right ) \, x^{2} + \left (2 i - 2\right ) \, x\right )} - 2 \, {\left (x^{4} + 1\right )}^{\frac {1}{4}} {\left (-i \, x^{3} - 3 i \, x^{2} - 3 i \, x - i\right )}}{x^{4} + 2 \, x^{3} + 3 \, x^{2} + 2 \, x + 1}\right ) - \left (\frac {1}{8} i + \frac {1}{8}\right ) \, \sqrt {2} \log \left (\frac {\sqrt {2} \sqrt {x^{4} + 1} {\left (-\left (i - 1\right ) \, x^{2} - \left (2 i - 2\right ) \, x - i + 1\right )} + 2 \, {\left (x^{4} + 1\right )}^{\frac {3}{4}} {\left (x + 1\right )} + \sqrt {2} {\left (-\left (2 i + 2\right ) \, x^{3} - \left (3 i + 3\right ) \, x^{2} - \left (2 i + 2\right ) \, x\right )} - 2 \, {\left (x^{4} + 1\right )}^{\frac {1}{4}} {\left (i \, x^{3} + 3 i \, x^{2} + 3 i \, x + i\right )}}{x^{4} + 2 \, x^{3} + 3 \, x^{2} + 2 \, x + 1}\right ) + \left (\frac {1}{8} i + \frac {1}{8}\right ) \, \sqrt {2} \log \left (\frac {\sqrt {2} \sqrt {x^{4} + 1} {\left (\left (i - 1\right ) \, x^{2} + \left (2 i - 2\right ) \, x + i - 1\right )} + 2 \, {\left (x^{4} + 1\right )}^{\frac {3}{4}} {\left (x + 1\right )} + \sqrt {2} {\left (\left (2 i + 2\right ) \, x^{3} + \left (3 i + 3\right ) \, x^{2} + \left (2 i + 2\right ) \, x\right )} - 2 \, {\left (x^{4} + 1\right )}^{\frac {1}{4}} {\left (i \, x^{3} + 3 i \, x^{2} + 3 i \, x + i\right )}}{x^{4} + 2 \, x^{3} + 3 \, x^{2} + 2 \, x + 1}\right ) - \left (\frac {1}{8} i - \frac {1}{8}\right ) \, \sqrt {2} \log \left (\frac {\sqrt {2} \sqrt {x^{4} + 1} {\left (-\left (i + 1\right ) \, x^{2} - \left (2 i + 2\right ) \, x - i - 1\right )} + 2 \, {\left (x^{4} + 1\right )}^{\frac {3}{4}} {\left (x + 1\right )} + \sqrt {2} {\left (-\left (2 i - 2\right ) \, x^{3} - \left (3 i - 3\right ) \, x^{2} - \left (2 i - 2\right ) \, x\right )} - 2 \, {\left (x^{4} + 1\right )}^{\frac {1}{4}} {\left (-i \, x^{3} - 3 i \, x^{2} - 3 i \, x - i\right )}}{x^{4} + 2 \, x^{3} + 3 \, x^{2} + 2 \, x + 1}\right ) \]
(1/8*I - 1/8)*sqrt(2)*log((sqrt(2)*sqrt(x^4 + 1)*((I + 1)*x^2 + (2*I + 2)* x + I + 1) + 2*(x^4 + 1)^(3/4)*(x + 1) + sqrt(2)*((2*I - 2)*x^3 + (3*I - 3 )*x^2 + (2*I - 2)*x) - 2*(x^4 + 1)^(1/4)*(-I*x^3 - 3*I*x^2 - 3*I*x - I))/( x^4 + 2*x^3 + 3*x^2 + 2*x + 1)) - (1/8*I + 1/8)*sqrt(2)*log((sqrt(2)*sqrt( x^4 + 1)*(-(I - 1)*x^2 - (2*I - 2)*x - I + 1) + 2*(x^4 + 1)^(3/4)*(x + 1) + sqrt(2)*(-(2*I + 2)*x^3 - (3*I + 3)*x^2 - (2*I + 2)*x) - 2*(x^4 + 1)^(1/ 4)*(I*x^3 + 3*I*x^2 + 3*I*x + I))/(x^4 + 2*x^3 + 3*x^2 + 2*x + 1)) + (1/8* I + 1/8)*sqrt(2)*log((sqrt(2)*sqrt(x^4 + 1)*((I - 1)*x^2 + (2*I - 2)*x + I - 1) + 2*(x^4 + 1)^(3/4)*(x + 1) + sqrt(2)*((2*I + 2)*x^3 + (3*I + 3)*x^2 + (2*I + 2)*x) - 2*(x^4 + 1)^(1/4)*(I*x^3 + 3*I*x^2 + 3*I*x + I))/(x^4 + 2*x^3 + 3*x^2 + 2*x + 1)) - (1/8*I - 1/8)*sqrt(2)*log((sqrt(2)*sqrt(x^4 + 1)*(-(I + 1)*x^2 - (2*I + 2)*x - I - 1) + 2*(x^4 + 1)^(3/4)*(x + 1) + sqrt (2)*(-(2*I - 2)*x^3 - (3*I - 3)*x^2 - (2*I - 2)*x) - 2*(x^4 + 1)^(1/4)*(-I *x^3 - 3*I*x^2 - 3*I*x - I))/(x^4 + 2*x^3 + 3*x^2 + 2*x + 1))
\[ \int \frac {-1+x}{\left (1+x+x^2\right ) \sqrt [4]{1+x^4}} \, dx=\int \frac {x - 1}{\sqrt [4]{x^{4} + 1} \left (x^{2} + x + 1\right )}\, dx \]
\[ \int \frac {-1+x}{\left (1+x+x^2\right ) \sqrt [4]{1+x^4}} \, dx=\int { \frac {x - 1}{{\left (x^{4} + 1\right )}^{\frac {1}{4}} {\left (x^{2} + x + 1\right )}} \,d x } \]
\[ \int \frac {-1+x}{\left (1+x+x^2\right ) \sqrt [4]{1+x^4}} \, dx=\int { \frac {x - 1}{{\left (x^{4} + 1\right )}^{\frac {1}{4}} {\left (x^{2} + x + 1\right )}} \,d x } \]
Timed out. \[ \int \frac {-1+x}{\left (1+x+x^2\right ) \sqrt [4]{1+x^4}} \, dx=\int \frac {x-1}{{\left (x^4+1\right )}^{1/4}\,\left (x^2+x+1\right )} \,d x \]