\(\int \frac {-1+x}{(1+x+x^2) \sqrt [4]{1+x^4}} \, dx\) [1700]

Optimal result

Integrand size = 21, antiderivative size = 114 \[ \int \frac {-1+x}{\left (1+x+x^2\right ) \sqrt [4]{1+x^4}} \, dx=\frac {\arctan \left (\frac {-\frac {1}{\sqrt {2}}-\sqrt {2} x-\frac {x^2}{\sqrt {2}}+\frac {\sqrt {1+x^4}}{\sqrt {2}}}{(1+x) \sqrt [4]{1+x^4}}\right )}{\sqrt {2}}-\frac {\text {arctanh}\left (\frac {\left (\sqrt {2}+\sqrt {2} x\right ) \sqrt [4]{1+x^4}}{1+2 x+x^2+\sqrt {1+x^4}}\right )}{\sqrt {2}} \] Output:

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)
                                                                                    
                                                                                    
 

Mathematica [A] (verified)

Time = 1.64 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.76 \[ \int \frac {-1+x}{\left (1+x+x^2\right ) \sqrt [4]{1+x^4}} \, dx=\frac {\arctan \left (\frac {-1-2 x-x^2+\sqrt {1+x^4}}{\sqrt {2} (1+x) \sqrt [4]{1+x^4}}\right )-\text {arctanh}\left (\frac {\sqrt {2} (1+x) \sqrt [4]{1+x^4}}{1+2 x+x^2+\sqrt {1+x^4}}\right )}{\sqrt {2}} \] Input:

Integrate[(-1 + x)/((1 + x + x^2)*(1 + x^4)^(1/4)),x]
 

Output:

(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]
 

Rubi [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 6 vs. order 3 in optimal.

Time = 1.62 (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}\)

Input:

Int[(-1 + x)/((1 + x + x^2)*(1 + x^4)^(1/4)),x]
 

Output:

-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
 

Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 7279

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]
 

Maple [C] (verified)

Result contains higher order function than in optimal. Order 9 vs. order 3.

Time = 4.00 (sec) , antiderivative size = 392, normalized size of antiderivative = 3.44

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}\)	\(392\)

Input:

int((-1+x)/(x^2+x+1)/(x^4+1)^(1/4),x,method=_RETURNVERBOSE)
 

Output:

-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)*Root 
Of(_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)*Ro 
otOf(_Z^4+1)^2+(x^4+1)^(1/2)*RootOf(_Z^4+1)+(x^4+1)^(3/4))/(x^2+x+1)^2)
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 582 vs. \(2 (90) = 180\).

Time = 4.19 (sec) , antiderivative size = 582, normalized size of antiderivative = 5.11 \[ \int \frac {-1+x}{\left (1+x+x^2\right ) \sqrt [4]{1+x^4}} \, dx=-\frac {1}{4} \, \sqrt {2} \arctan \left (\frac {x^{8} + 4 \, x^{7} + 10 \, x^{6} + 16 \, x^{5} + 19 \, x^{4} + 16 \, x^{3} + \sqrt {2} {\left (x^{5} + 7 \, x^{4} + 15 \, x^{3} + 15 \, x^{2} + 7 \, x + 1\right )} {\left (x^{4} + 1\right )}^{\frac {3}{4}} + 10 \, x^{2} - \sqrt {2} {\left (x^{7} + x^{6} - 6 \, x^{5} - 16 \, x^{4} - 16 \, x^{3} - 6 \, x^{2} + x + 1\right )} {\left (x^{4} + 1\right )}^{\frac {1}{4}} + 2 \, {\left (x^{6} + 4 \, x^{5} + 8 \, x^{4} + 10 \, x^{3} + 8 \, x^{2} + 4 \, x + 1\right )} \sqrt {x^{4} + 1} + 4 \, x + 1}{3 \, x^{8} + 12 \, x^{7} + 14 \, x^{6} - 11 \, x^{4} + 14 \, x^{2} + 12 \, x + 3}\right ) - \frac {1}{4} \, \sqrt {2} \arctan \left (-\frac {x^{8} + 4 \, x^{7} + 10 \, x^{6} + 16 \, x^{5} + 19 \, x^{4} + 16 \, x^{3} - \sqrt {2} {\left (x^{5} + 7 \, x^{4} + 15 \, x^{3} + 15 \, x^{2} + 7 \, x + 1\right )} {\left (x^{4} + 1\right )}^{\frac {3}{4}} + 10 \, x^{2} + \sqrt {2} {\left (x^{7} + x^{6} - 6 \, x^{5} - 16 \, x^{4} - 16 \, x^{3} - 6 \, x^{2} + x + 1\right )} {\left (x^{4} + 1\right )}^{\frac {1}{4}} + 2 \, {\left (x^{6} + 4 \, x^{5} + 8 \, x^{4} + 10 \, x^{3} + 8 \, x^{2} + 4 \, x + 1\right )} \sqrt {x^{4} + 1} + 4 \, x + 1}{3 \, x^{8} + 12 \, x^{7} + 14 \, x^{6} - 11 \, x^{4} + 14 \, x^{2} + 12 \, x + 3}\right ) - \frac {1}{8} \, \sqrt {2} \log \left (\frac {x^{4} + 2 \, x^{3} + \sqrt {2} {\left (x^{4} + 1\right )}^{\frac {3}{4}} {\left (x + 1\right )} + 3 \, x^{2} + \sqrt {2} {\left (x^{4} + 1\right )}^{\frac {1}{4}} {\left (x^{3} + 3 \, x^{2} + 3 \, x + 1\right )} + 2 \, \sqrt {x^{4} + 1} {\left (x^{2} + 2 \, x + 1\right )} + 2 \, x + 1}{x^{4} + 2 \, x^{3} + 3 \, x^{2} + 2 \, x + 1}\right ) + \frac {1}{8} \, \sqrt {2} \log \left (\frac {x^{4} + 2 \, x^{3} - \sqrt {2} {\left (x^{4} + 1\right )}^{\frac {3}{4}} {\left (x + 1\right )} + 3 \, x^{2} - \sqrt {2} {\left (x^{4} + 1\right )}^{\frac {1}{4}} {\left (x^{3} + 3 \, x^{2} + 3 \, x + 1\right )} + 2 \, \sqrt {x^{4} + 1} {\left (x^{2} + 2 \, x + 1\right )} + 2 \, x + 1}{x^{4} + 2 \, x^{3} + 3 \, x^{2} + 2 \, x + 1}\right ) \] Input:

integrate((-1+x)/(x^2+x+1)/(x^4+1)^(1/4),x, algorithm="fricas")
 

Output:

-1/4*sqrt(2)*arctan((x^8 + 4*x^7 + 10*x^6 + 16*x^5 + 19*x^4 + 16*x^3 + sqr 
t(2)*(x^5 + 7*x^4 + 15*x^3 + 15*x^2 + 7*x + 1)*(x^4 + 1)^(3/4) + 10*x^2 - 
sqrt(2)*(x^7 + x^6 - 6*x^5 - 16*x^4 - 16*x^3 - 6*x^2 + x + 1)*(x^4 + 1)^(1 
/4) + 2*(x^6 + 4*x^5 + 8*x^4 + 10*x^3 + 8*x^2 + 4*x + 1)*sqrt(x^4 + 1) + 4 
*x + 1)/(3*x^8 + 12*x^7 + 14*x^6 - 11*x^4 + 14*x^2 + 12*x + 3)) - 1/4*sqrt 
(2)*arctan(-(x^8 + 4*x^7 + 10*x^6 + 16*x^5 + 19*x^4 + 16*x^3 - sqrt(2)*(x^ 
5 + 7*x^4 + 15*x^3 + 15*x^2 + 7*x + 1)*(x^4 + 1)^(3/4) + 10*x^2 + sqrt(2)* 
(x^7 + x^6 - 6*x^5 - 16*x^4 - 16*x^3 - 6*x^2 + x + 1)*(x^4 + 1)^(1/4) + 2* 
(x^6 + 4*x^5 + 8*x^4 + 10*x^3 + 8*x^2 + 4*x + 1)*sqrt(x^4 + 1) + 4*x + 1)/ 
(3*x^8 + 12*x^7 + 14*x^6 - 11*x^4 + 14*x^2 + 12*x + 3)) - 1/8*sqrt(2)*log( 
(x^4 + 2*x^3 + sqrt(2)*(x^4 + 1)^(3/4)*(x + 1) + 3*x^2 + sqrt(2)*(x^4 + 1) 
^(1/4)*(x^3 + 3*x^2 + 3*x + 1) + 2*sqrt(x^4 + 1)*(x^2 + 2*x + 1) + 2*x + 1 
)/(x^4 + 2*x^3 + 3*x^2 + 2*x + 1)) + 1/8*sqrt(2)*log((x^4 + 2*x^3 - sqrt(2 
)*(x^4 + 1)^(3/4)*(x + 1) + 3*x^2 - sqrt(2)*(x^4 + 1)^(1/4)*(x^3 + 3*x^2 + 
 3*x + 1) + 2*sqrt(x^4 + 1)*(x^2 + 2*x + 1) + 2*x + 1)/(x^4 + 2*x^3 + 3*x^ 
2 + 2*x + 1))
 

Sympy [F]

\[ \int \frac {-1+x}{\left (1+x+x^2\right ) \sqrt [4]{1+x^4}} \, dx=\int \frac {x - 1}{\sqrt [4]{x^{4} + 1} \left (x^{2} + x + 1\right )}\, dx \] Input:

integrate((-1+x)/(x**2+x+1)/(x**4+1)**(1/4),x)
                                                                                    
                                                                                    
 

Output:

Integral((x - 1)/((x**4 + 1)**(1/4)*(x**2 + x + 1)), x)
 

Maxima [F]

\[ \int \frac {-1+x}{\left (1+x+x^2\right ) \sqrt [4]{1+x^4}} \, dx=\int { \frac {x - 1}{{\left (x^{4} + 1\right )}^{\frac {1}{4}} {\left (x^{2} + x + 1\right )}} \,d x } \] Input:

integrate((-1+x)/(x^2+x+1)/(x^4+1)^(1/4),x, algorithm="maxima")
 

Output:

integrate((x - 1)/((x^4 + 1)^(1/4)*(x^2 + x + 1)), x)
 

Giac [F]

\[ \int \frac {-1+x}{\left (1+x+x^2\right ) \sqrt [4]{1+x^4}} \, dx=\int { \frac {x - 1}{{\left (x^{4} + 1\right )}^{\frac {1}{4}} {\left (x^{2} + x + 1\right )}} \,d x } \] Input:

integrate((-1+x)/(x^2+x+1)/(x^4+1)^(1/4),x, algorithm="giac")
 

Output:

integrate((x - 1)/((x^4 + 1)^(1/4)*(x^2 + x + 1)), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {-1+x}{\left (1+x+x^2\right ) \sqrt [4]{1+x^4}} \, dx=\int \frac {x-1}{{\left (x^4+1\right )}^{1/4}\,\left (x^2+x+1\right )} \,d x \] Input:

int((x - 1)/((x^4 + 1)^(1/4)*(x + x^2 + 1)),x)
 

Output:

int((x - 1)/((x^4 + 1)^(1/4)*(x + x^2 + 1)), x)
 

Reduce [F]

\[ \int \frac {-1+x}{\left (1+x+x^2\right ) \sqrt [4]{1+x^4}} \, dx=\int \frac {x}{\left (x^{4}+1\right )^{\frac {1}{4}} x^{2}+\left (x^{4}+1\right )^{\frac {1}{4}} x +\left (x^{4}+1\right )^{\frac {1}{4}}}d x -\left (\int \frac {1}{\left (x^{4}+1\right )^{\frac {1}{4}} x^{2}+\left (x^{4}+1\right )^{\frac {1}{4}} x +\left (x^{4}+1\right )^{\frac {1}{4}}}d x \right ) \] Input:

int((-1+x)/(x^2+x+1)/(x^4+1)^(1/4),x)
 

Output:

int(x/((x**4 + 1)**(1/4)*x**2 + (x**4 + 1)**(1/4)*x + (x**4 + 1)**(1/4)),x 
) - int(1/((x**4 + 1)**(1/4)*x**2 + (x**4 + 1)**(1/4)*x + (x**4 + 1)**(1/4 
)),x)