\(\int \frac {1-2 x+2 x^4}{x \sqrt [4]{-1+x^4}} \, dx\) [1840]

Optimal result

Integrand size = 23, antiderivative size = 125 \[ \int \frac {1-2 x+2 x^4}{x \sqrt [4]{-1+x^4}} \, dx=\frac {2}{3} \left (-1+x^4\right )^{3/4}+\arctan \left (\frac {\sqrt [4]{-1+x^4}}{x}\right )+\frac {\arctan \left (\frac {-\frac {1}{\sqrt {2}}+\frac {\sqrt {-1+x^4}}{\sqrt {2}}}{\sqrt [4]{-1+x^4}}\right )}{2 \sqrt {2}}-\text {arctanh}\left (\frac {\sqrt [4]{-1+x^4}}{x}\right )-\frac {\text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{-1+x^4}}{1+\sqrt {-1+x^4}}\right )}{2 \sqrt {2}} \] Output:

2/3*(x^4-1)^(3/4)+arctan((x^4-1)^(1/4)/x)+1/4*arctan((-1/2*2^(1/2)+1/2*(x^ 
4-1)^(1/2)*2^(1/2))/(x^4-1)^(1/4))*2^(1/2)-arctanh((x^4-1)^(1/4)/x)-1/4*ar 
ctanh(2^(1/2)*(x^4-1)^(1/4)/(1+(x^4-1)^(1/2)))*2^(1/2)
                                                                                    
                                                                                    
 

Mathematica [A] (verified)

Time = 5.53 (sec) , antiderivative size = 120, normalized size of antiderivative = 0.96 \[ \int \frac {1-2 x+2 x^4}{x \sqrt [4]{-1+x^4}} \, dx=\frac {2}{3} \left (-1+x^4\right )^{3/4}+\arctan \left (\frac {\sqrt [4]{-1+x^4}}{x}\right )-\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{-1+x^4}}{-1+\sqrt {-1+x^4}}\right )}{2 \sqrt {2}}-\text {arctanh}\left (\frac {\sqrt [4]{-1+x^4}}{x}\right )-\frac {\text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{-1+x^4}}{1+\sqrt {-1+x^4}}\right )}{2 \sqrt {2}} \] Input:

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

Output:

(2*(-1 + x^4)^(3/4))/3 + ArcTan[(-1 + x^4)^(1/4)/x] - ArcTan[(Sqrt[2]*(-1 
+ x^4)^(1/4))/(-1 + Sqrt[-1 + x^4])]/(2*Sqrt[2]) - ArcTanh[(-1 + x^4)^(1/4 
)/x] - ArcTanh[(Sqrt[2]*(-1 + x^4)^(1/4))/(1 + Sqrt[-1 + x^4])]/(2*Sqrt[2] 
)
 

Rubi [A] (verified)

Time = 0.38 (sec) , antiderivative size = 170, normalized size of antiderivative = 1.36, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {2372, 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.

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

\(\Big \downarrow \) 2372

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

\(\Big \downarrow \) 2009

\(\displaystyle -\arctan \left (\frac {x}{\sqrt [4]{x^4-1}}\right )-\frac {\arctan \left (1-\sqrt {2} \sqrt [4]{x^4-1}\right )}{2 \sqrt {2}}+\frac {\arctan \left (\sqrt {2} \sqrt [4]{x^4-1}+1\right )}{2 \sqrt {2}}-\text {arctanh}\left (\frac {x}{\sqrt [4]{x^4-1}}\right )+\frac {2}{3} \left (x^4-1\right )^{3/4}+\frac {\log \left (\sqrt {x^4-1}-\sqrt {2} \sqrt [4]{x^4-1}+1\right )}{4 \sqrt {2}}-\frac {\log \left (\sqrt {x^4-1}+\sqrt {2} \sqrt [4]{x^4-1}+1\right )}{4 \sqrt {2}}\)

Input:

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

Output:

(2*(-1 + x^4)^(3/4))/3 - ArcTan[x/(-1 + x^4)^(1/4)] - ArcTan[1 - Sqrt[2]*( 
-1 + x^4)^(1/4)]/(2*Sqrt[2]) + ArcTan[1 + Sqrt[2]*(-1 + x^4)^(1/4)]/(2*Sqr 
t[2]) - ArcTanh[x/(-1 + x^4)^(1/4)] + Log[1 - Sqrt[2]*(-1 + x^4)^(1/4) + S 
qrt[-1 + x^4]]/(4*Sqrt[2]) - Log[1 + Sqrt[2]*(-1 + x^4)^(1/4) + Sqrt[-1 + 
x^4]]/(4*Sqrt[2])
 

Defintions of rubi rules used

rule 2009

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

rule 2372

Int[(Pq_)*((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Mo 
dule[{q = Expon[Pq, x], j, k}, Int[Sum[((c*x)^(m + j)/c^j)*Sum[Coeff[Pq, x, 
 j + k*(n/2)]*x^(k*(n/2)), {k, 0, 2*((q - j)/n) + 1}]*(a + b*x^n)^p, {j, 0, 
 n/2 - 1}], x]] /; FreeQ[{a, b, c, m, p}, x] && PolyQ[Pq, x] && IGtQ[n/2, 0 
] &&  !PolyQ[Pq, x^(n/2)]
 

Maple [C] (warning: unable to verify)

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

Time = 10.84 (sec) , antiderivative size = 142, normalized size of antiderivative = 1.14

method	result	size
meijerg	\(\frac {\sqrt {2}\, \Gamma \left (\frac {3}{4}\right ) {\left (-\operatorname {signum}\left (x^{4}-1\right )\right )}^{\frac {1}{4}} \left (\frac {\pi \sqrt {2}\, x^{4} \operatorname {hypergeom}\left (\left [1, 1, \frac {5}{4}\right ], \left [2, 2\right ], x^{4}\right )}{4 \Gamma \left (\frac {3}{4}\right )}+\frac {\left (-3 \ln \left (2\right )-\frac {\pi }{2}+4 \ln \left (x \right )+i \pi \right ) \pi \sqrt {2}}{\Gamma \left (\frac {3}{4}\right )}\right )}{8 \pi \operatorname {signum}\left (x^{4}-1\right )^{\frac {1}{4}}}+\frac {{\left (-\operatorname {signum}\left (x^{4}-1\right )\right )}^{\frac {1}{4}} x^{4} \operatorname {hypergeom}\left (\left [\frac {1}{4}, 1\right ], \left [2\right ], x^{4}\right )}{2 \operatorname {signum}\left (x^{4}-1\right )^{\frac {1}{4}}}-\frac {2 {\left (-\operatorname {signum}\left (x^{4}-1\right )\right )}^{\frac {1}{4}} x \operatorname {hypergeom}\left (\left [\frac {1}{4}, \frac {1}{4}\right ], \left [\frac {5}{4}\right ], x^{4}\right )}{\operatorname {signum}\left (x^{4}-1\right )^{\frac {1}{4}}}\)	\(142\)
trager	\(\frac {2 \left (x^{4}-1\right )^{\frac {3}{4}}}{3}-\frac {\ln \left (2 \left (x^{4}-1\right )^{\frac {3}{4}} x +2 x^{2} \sqrt {x^{4}-1}+2 x^{3} \left (x^{4}-1\right )^{\frac {1}{4}}+2 x^{4}-1\right )}{2}+\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \ln \left (2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \sqrt {x^{4}-1}\, x^{2}-2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x^{4}+2 \left (x^{4}-1\right )^{\frac {3}{4}} x -2 x^{3} \left (x^{4}-1\right )^{\frac {1}{4}}+\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right )\right )}{2}-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}-\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right )\right ) \ln \left (-\frac {-\operatorname {RootOf}\left (\textit {\_Z}^{2}-\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right )\right ) x^{4}+2 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right )\right ) \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \sqrt {x^{4}-1}-2 \left (x^{4}-1\right )^{\frac {3}{4}}+2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \left (x^{4}-1\right )^{\frac {1}{4}}+2 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right )\right )}{x^{4}}\right )}{4}+\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \operatorname {RootOf}\left (\textit {\_Z}^{2}-\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right )\right ) \ln \left (\frac {-\operatorname {RootOf}\left (\textit {\_Z}^{2}-\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right )\right ) \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x^{4}+2 \left (x^{4}-1\right )^{\frac {3}{4}}+2 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right )\right ) \sqrt {x^{4}-1}+2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \left (x^{4}-1\right )^{\frac {1}{4}}+2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \operatorname {RootOf}\left (\textit {\_Z}^{2}-\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right )\right )}{x^{4}}\right )}{4}\)	\(347\)

Input:

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

Output:

1/8/Pi*2^(1/2)*GAMMA(3/4)/signum(x^4-1)^(1/4)*(-signum(x^4-1))^(1/4)*(1/4* 
Pi*2^(1/2)/GAMMA(3/4)*x^4*hypergeom([1,1,5/4],[2,2],x^4)+(-3*ln(2)-1/2*Pi+ 
4*ln(x)+I*Pi)*Pi*2^(1/2)/GAMMA(3/4))+1/2/signum(x^4-1)^(1/4)*(-signum(x^4- 
1))^(1/4)*x^4*hypergeom([1/4,1],[2],x^4)-2/signum(x^4-1)^(1/4)*(-signum(x^ 
4-1))^(1/4)*x*hypergeom([1/4,1/4],[5/4],x^4)
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 317 vs. \(2 (96) = 192\).

Time = 11.48 (sec) , antiderivative size = 317, normalized size of antiderivative = 2.54 \[ \int \frac {1-2 x+2 x^4}{x \sqrt [4]{-1+x^4}} \, dx=-\frac {1}{8} \, \sqrt {2} \arctan \left (\frac {x^{8} + 4 \, \sqrt {x^{4} - 1} x^{4} + 2 \, \sqrt {2} {\left (x^{4} - 1\right )}^{\frac {3}{4}} {\left (x^{4} - 4\right )} + 2 \, \sqrt {2} {\left (3 \, x^{4} - 4\right )} {\left (x^{4} - 1\right )}^{\frac {1}{4}}}{x^{8} - 16 \, x^{4} + 16}\right ) - \frac {1}{8} \, \sqrt {2} \arctan \left (-\frac {x^{8} + 4 \, \sqrt {x^{4} - 1} x^{4} - 2 \, \sqrt {2} {\left (x^{4} - 1\right )}^{\frac {3}{4}} {\left (x^{4} - 4\right )} - 2 \, \sqrt {2} {\left (3 \, x^{4} - 4\right )} {\left (x^{4} - 1\right )}^{\frac {1}{4}}}{x^{8} - 16 \, x^{4} + 16}\right ) - \frac {1}{16} \, \sqrt {2} \log \left (\frac {x^{4} + 2 \, \sqrt {2} {\left (x^{4} - 1\right )}^{\frac {3}{4}} + 2 \, \sqrt {2} {\left (x^{4} - 1\right )}^{\frac {1}{4}} + 4 \, \sqrt {x^{4} - 1}}{x^{4}}\right ) + \frac {1}{16} \, \sqrt {2} \log \left (\frac {x^{4} - 2 \, \sqrt {2} {\left (x^{4} - 1\right )}^{\frac {3}{4}} - 2 \, \sqrt {2} {\left (x^{4} - 1\right )}^{\frac {1}{4}} + 4 \, \sqrt {x^{4} - 1}}{x^{4}}\right ) + \frac {2}{3} \, {\left (x^{4} - 1\right )}^{\frac {3}{4}} + \frac {1}{2} \, \arctan \left (2 \, {\left (x^{4} - 1\right )}^{\frac {1}{4}} x^{3} + 2 \, {\left (x^{4} - 1\right )}^{\frac {3}{4}} x\right ) + \frac {1}{2} \, \log \left (-2 \, x^{4} + 2 \, {\left (x^{4} - 1\right )}^{\frac {1}{4}} x^{3} - 2 \, \sqrt {x^{4} - 1} x^{2} + 2 \, {\left (x^{4} - 1\right )}^{\frac {3}{4}} x + 1\right ) \] Input:

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

Output:

-1/8*sqrt(2)*arctan((x^8 + 4*sqrt(x^4 - 1)*x^4 + 2*sqrt(2)*(x^4 - 1)^(3/4) 
*(x^4 - 4) + 2*sqrt(2)*(3*x^4 - 4)*(x^4 - 1)^(1/4))/(x^8 - 16*x^4 + 16)) - 
 1/8*sqrt(2)*arctan(-(x^8 + 4*sqrt(x^4 - 1)*x^4 - 2*sqrt(2)*(x^4 - 1)^(3/4 
)*(x^4 - 4) - 2*sqrt(2)*(3*x^4 - 4)*(x^4 - 1)^(1/4))/(x^8 - 16*x^4 + 16)) 
- 1/16*sqrt(2)*log((x^4 + 2*sqrt(2)*(x^4 - 1)^(3/4) + 2*sqrt(2)*(x^4 - 1)^ 
(1/4) + 4*sqrt(x^4 - 1))/x^4) + 1/16*sqrt(2)*log((x^4 - 2*sqrt(2)*(x^4 - 1 
)^(3/4) - 2*sqrt(2)*(x^4 - 1)^(1/4) + 4*sqrt(x^4 - 1))/x^4) + 2/3*(x^4 - 1 
)^(3/4) + 1/2*arctan(2*(x^4 - 1)^(1/4)*x^3 + 2*(x^4 - 1)^(3/4)*x) + 1/2*lo 
g(-2*x^4 + 2*(x^4 - 1)^(1/4)*x^3 - 2*sqrt(x^4 - 1)*x^2 + 2*(x^4 - 1)^(3/4) 
*x + 1)
 

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 1.62 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.57 \[ \int \frac {1-2 x+2 x^4}{x \sqrt [4]{-1+x^4}} \, dx=- \frac {x e^{- \frac {i \pi }{4}} \Gamma \left (\frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{4}, \frac {1}{4} \\ \frac {5}{4} \end {matrix}\middle | {x^{4}} \right )}}{2 \Gamma \left (\frac {5}{4}\right )} + \frac {2 \left (x^{4} - 1\right )^{\frac {3}{4}}}{3} - \frac {\Gamma \left (\frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{4}, \frac {1}{4} \\ \frac {5}{4} \end {matrix}\middle | {\frac {e^{2 i \pi }}{x^{4}}} \right )}}{4 x \Gamma \left (\frac {5}{4}\right )} \] Input:

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

Output:

-x*exp(-I*pi/4)*gamma(1/4)*hyper((1/4, 1/4), (5/4,), x**4)/(2*gamma(5/4)) 
+ 2*(x**4 - 1)**(3/4)/3 - gamma(1/4)*hyper((1/4, 1/4), (5/4,), exp_polar(2 
*I*pi)/x**4)/(4*x*gamma(5/4))
 

Maxima [A] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 155, normalized size of antiderivative = 1.24 \[ \int \frac {1-2 x+2 x^4}{x \sqrt [4]{-1+x^4}} \, dx=\frac {1}{4} \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + 2 \, {\left (x^{4} - 1\right )}^{\frac {1}{4}}\right )}\right ) + \frac {1}{4} \, \sqrt {2} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - 2 \, {\left (x^{4} - 1\right )}^{\frac {1}{4}}\right )}\right ) - \frac {1}{8} \, \sqrt {2} \log \left (\sqrt {2} {\left (x^{4} - 1\right )}^{\frac {1}{4}} + \sqrt {x^{4} - 1} + 1\right ) + \frac {1}{8} \, \sqrt {2} \log \left (-\sqrt {2} {\left (x^{4} - 1\right )}^{\frac {1}{4}} + \sqrt {x^{4} - 1} + 1\right ) + \frac {2}{3} \, {\left (x^{4} - 1\right )}^{\frac {3}{4}} + \arctan \left (\frac {{\left (x^{4} - 1\right )}^{\frac {1}{4}}}{x}\right ) - \frac {1}{2} \, \log \left (\frac {{\left (x^{4} - 1\right )}^{\frac {1}{4}}}{x} + 1\right ) + \frac {1}{2} \, \log \left (\frac {{\left (x^{4} - 1\right )}^{\frac {1}{4}}}{x} - 1\right ) \] Input:

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

Output:

1/4*sqrt(2)*arctan(1/2*sqrt(2)*(sqrt(2) + 2*(x^4 - 1)^(1/4))) + 1/4*sqrt(2 
)*arctan(-1/2*sqrt(2)*(sqrt(2) - 2*(x^4 - 1)^(1/4))) - 1/8*sqrt(2)*log(sqr 
t(2)*(x^4 - 1)^(1/4) + sqrt(x^4 - 1) + 1) + 1/8*sqrt(2)*log(-sqrt(2)*(x^4 
- 1)^(1/4) + sqrt(x^4 - 1) + 1) + 2/3*(x^4 - 1)^(3/4) + arctan((x^4 - 1)^( 
1/4)/x) - 1/2*log((x^4 - 1)^(1/4)/x + 1) + 1/2*log((x^4 - 1)^(1/4)/x - 1)
 

Giac [F]

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

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

Output:

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

Mupad [B] (verification not implemented)

Time = 8.30 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.65 \[ \int \frac {1-2 x+2 x^4}{x \sqrt [4]{-1+x^4}} \, dx=\frac {2\,{\left (x^4-1\right )}^{3/4}}{3}-\frac {2\,x\,{\left (1-x^4\right )}^{1/4}\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{4},\frac {1}{4};\ \frac {5}{4};\ x^4\right )}{{\left (x^4-1\right )}^{1/4}}+\sqrt {2}\,\mathrm {atan}\left (\sqrt {2}\,{\left (x^4-1\right )}^{1/4}\,\left (\frac {1}{2}-\frac {1}{2}{}\mathrm {i}\right )\right )\,\left (\frac {1}{4}-\frac {1}{4}{}\mathrm {i}\right )+\sqrt {2}\,\mathrm {atan}\left (\sqrt {2}\,{\left (x^4-1\right )}^{1/4}\,\left (\frac {1}{2}+\frac {1}{2}{}\mathrm {i}\right )\right )\,\left (\frac {1}{4}+\frac {1}{4}{}\mathrm {i}\right ) \] Input:

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

Output:

2^(1/2)*atan(2^(1/2)*(x^4 - 1)^(1/4)*(1/2 - 1i/2))*(1/4 - 1i/4) + 2^(1/2)* 
atan(2^(1/2)*(x^4 - 1)^(1/4)*(1/2 + 1i/2))*(1/4 + 1i/4) + (2*(x^4 - 1)^(3/ 
4))/3 - (2*x*(1 - x^4)^(1/4)*hypergeom([1/4, 1/4], 5/4, x^4))/(x^4 - 1)^(1 
/4)
 

Reduce [F]

\[ \int \frac {1-2 x+2 x^4}{x \sqrt [4]{-1+x^4}} \, dx=2 \left (\int \frac {x^{3}}{\left (x^{4}-1\right )^{\frac {1}{4}}}d x \right )-2 \left (\int \frac {1}{\left (x^{4}-1\right )^{\frac {1}{4}}}d x \right )+\int \frac {1}{\left (x^{4}-1\right )^{\frac {1}{4}} x}d x \] Input:

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

Output:

2*int(x**3/(x**4 - 1)**(1/4),x) - 2*int(1/(x**4 - 1)**(1/4),x) + int(1/((x 
**4 - 1)**(1/4)*x),x)