\(\int \frac {-3+2 x}{x \sqrt [4]{-1+x^4}} \, dx\) [1653]

Optimal result

Integrand size = 18, antiderivative size = 112 \[ \int \frac {-3+2 x}{x \sqrt [4]{-1+x^4}} \, dx=-\arctan \left (\frac {\sqrt [4]{-1+x^4}}{x}\right )-\frac {3 \arctan \left (\frac {-\frac {1}{\sqrt {2}}+\frac {\sqrt {-1+x^4}}{\sqrt {2}}}{\sqrt [4]{-1+x^4}}\right )}{2 \sqrt {2}}+\text {arctanh}\left (\frac {\sqrt [4]{-1+x^4}}{x}\right )+\frac {3 \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{-1+x^4}}{1+\sqrt {-1+x^4}}\right )}{2 \sqrt {2}} \] Output:

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

Mathematica [A] (verified)

Time = 5.63 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.96 \[ \int \frac {-3+2 x}{x \sqrt [4]{-1+x^4}} \, dx=-\arctan \left (\frac {\sqrt [4]{-1+x^4}}{x}\right )+\frac {3 \arctan \left (\frac {\sqrt {2} \sqrt [4]{-1+x^4}}{-1+\sqrt {-1+x^4}}\right )}{2 \sqrt {2}}+\text {arctanh}\left (\frac {\sqrt [4]{-1+x^4}}{x}\right )+\frac {3 \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{-1+x^4}}{1+\sqrt {-1+x^4}}\right )}{2 \sqrt {2}} \] Input:

Integrate[(-3 + 2*x)/(x*(-1 + x^4)^(1/4)),x]
 

Output:

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

Rubi [A] (verified)

Time = 0.37 (sec) , antiderivative size = 153, normalized size of antiderivative = 1.37, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, 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.

\(\displaystyle \int \frac {2 x-3}{x \sqrt [4]{x^4-1}} \, dx\)

\(\Big \downarrow \) 2372

\(\displaystyle \int \left (\frac {2}{\sqrt [4]{x^4-1}}-\frac {3}{x \sqrt [4]{x^4-1}}\right )dx\)

\(\Big \downarrow \) 2009

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

Input:

Int[(-3 + 2*x)/(x*(-1 + x^4)^(1/4)),x]
 

Output:

ArcTan[x/(-1 + x^4)^(1/4)] + (3*ArcTan[1 - Sqrt[2]*(-1 + x^4)^(1/4)])/(2*S 
qrt[2]) - (3*ArcTan[1 + Sqrt[2]*(-1 + x^4)^(1/4)])/(2*Sqrt[2]) + ArcTanh[x 
/(-1 + x^4)^(1/4)] - (3*Log[1 - Sqrt[2]*(-1 + x^4)^(1/4) + Sqrt[-1 + x^4]] 
)/(4*Sqrt[2]) + (3*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 = 7.72 (sec) , antiderivative size = 110, normalized size of antiderivative = 0.98

method	result	size
meijerg	\(-\frac {3 \sqrt {2}\, \Gamma \left (\frac {3}{4}\right ) {\left (-\operatorname {signum}\left (x^{4}-1\right )\right )}^{\frac {1}{4}} \left (\frac {\pi \sqrt {2}\, x^{4} \operatorname {hypergeom}\left (\left [1, 1, \frac {5}{4}\right ], \left [2, 2\right ], x^{4}\right )}{4 \Gamma \left (\frac {3}{4}\right )}+\frac {\left (-3 \ln \left (2\right )-\frac {\pi }{2}+4 \ln \left (x \right )+i \pi \right ) \pi \sqrt {2}}{\Gamma \left (\frac {3}{4}\right )}\right )}{8 \pi \operatorname {signum}\left (x^{4}-1\right )^{\frac {1}{4}}}+\frac {2 {\left (-\operatorname {signum}\left (x^{4}-1\right )\right )}^{\frac {1}{4}} x \operatorname {hypergeom}\left (\left [\frac {1}{4}, \frac {1}{4}\right ], \left [\frac {5}{4}\right ], x^{4}\right )}{\operatorname {signum}\left (x^{4}-1\right )^{\frac {1}{4}}}\)	\(110\)
trager	\(-\frac {\ln \left (2 \left (x^{4}-1\right )^{\frac {3}{4}} x -2 x^{2} \sqrt {x^{4}-1}+2 x^{3} \left (x^{4}-1\right )^{\frac {1}{4}}-2 x^{4}+1\right )}{2}+\frac {3 \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right ) \ln \left (\frac {-2 \sqrt {x^{4}-1}\, \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{3}+\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right ) x^{4}+2 \left (x^{4}-1\right )^{\frac {3}{4}}-2 \left (x^{4}-1\right )^{\frac {1}{4}} \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{2}-2 \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )}{x^{4}}\right )}{4}+\frac {3 \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{3} \ln \left (\frac {\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{3} x^{4}+2 \left (x^{4}-1\right )^{\frac {3}{4}}-2 \sqrt {x^{4}-1}\, \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )+2 \left (x^{4}-1\right )^{\frac {1}{4}} \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{2}-2 \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{3}}{x^{4}}\right )}{4}+\frac {\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{2} \ln \left (-2 \sqrt {x^{4}-1}\, \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{2} x^{2}+2 \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{2} x^{4}+2 \left (x^{4}-1\right )^{\frac {3}{4}} x -2 x^{3} \left (x^{4}-1\right )^{\frac {1}{4}}-\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{2}\right )}{2}\)	\(277\)

Input:

int((-3+2*x)/x/(x^4-1)^(1/4),x,method=_RETURNVERBOSE)
 

Output:

-3/8/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))+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. 308 vs. \(2 (87) = 174\).

Time = 11.11 (sec) , antiderivative size = 308, normalized size of antiderivative = 2.75 \[ \int \frac {-3+2 x}{x \sqrt [4]{-1+x^4}} \, dx=\frac {3}{8} \, \sqrt {2} \arctan \left (\frac {x^{8} + 4 \, \sqrt {x^{4} - 1} x^{4} + 2 \, \sqrt {2} {\left (x^{4} - 1\right )}^{\frac {3}{4}} {\left (x^{4} - 4\right )} + 2 \, \sqrt {2} {\left (3 \, x^{4} - 4\right )} {\left (x^{4} - 1\right )}^{\frac {1}{4}}}{x^{8} - 16 \, x^{4} + 16}\right ) + \frac {3}{8} \, \sqrt {2} \arctan \left (-\frac {x^{8} + 4 \, \sqrt {x^{4} - 1} x^{4} - 2 \, \sqrt {2} {\left (x^{4} - 1\right )}^{\frac {3}{4}} {\left (x^{4} - 4\right )} - 2 \, \sqrt {2} {\left (3 \, x^{4} - 4\right )} {\left (x^{4} - 1\right )}^{\frac {1}{4}}}{x^{8} - 16 \, x^{4} + 16}\right ) + \frac {3}{16} \, \sqrt {2} \log \left (\frac {x^{4} + 2 \, \sqrt {2} {\left (x^{4} - 1\right )}^{\frac {3}{4}} + 2 \, \sqrt {2} {\left (x^{4} - 1\right )}^{\frac {1}{4}} + 4 \, \sqrt {x^{4} - 1}}{x^{4}}\right ) - \frac {3}{16} \, \sqrt {2} \log \left (\frac {x^{4} - 2 \, \sqrt {2} {\left (x^{4} - 1\right )}^{\frac {3}{4}} - 2 \, \sqrt {2} {\left (x^{4} - 1\right )}^{\frac {1}{4}} + 4 \, \sqrt {x^{4} - 1}}{x^{4}}\right ) - \frac {1}{2} \, \arctan \left (2 \, {\left (x^{4} - 1\right )}^{\frac {1}{4}} x^{3} + 2 \, {\left (x^{4} - 1\right )}^{\frac {3}{4}} x\right ) + \frac {1}{2} \, \log \left (2 \, x^{4} + 2 \, {\left (x^{4} - 1\right )}^{\frac {1}{4}} x^{3} + 2 \, \sqrt {x^{4} - 1} x^{2} + 2 \, {\left (x^{4} - 1\right )}^{\frac {3}{4}} x - 1\right ) \] Input:

integrate((-3+2*x)/x/(x^4-1)^(1/4),x, algorithm="fricas")
 

Output:

3/8*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)) + 
3/8*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)) + 
 3/16*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) - 3/16*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) - 1/2*arctan(2* 
(x^4 - 1)^(1/4)*x^3 + 2*(x^4 - 1)^(3/4)*x) + 1/2*log(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 = 1.46 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.54 \[ \int \frac {-3+2 x}{x \sqrt [4]{-1+x^4}} \, dx=\frac {x e^{- \frac {i \pi }{4}} \Gamma \left (\frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{4}, \frac {1}{4} \\ \frac {5}{4} \end {matrix}\middle | {x^{4}} \right )}}{2 \Gamma \left (\frac {5}{4}\right )} + \frac {3 \Gamma \left (\frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{4}, \frac {1}{4} \\ \frac {5}{4} \end {matrix}\middle | {\frac {e^{2 i \pi }}{x^{4}}} \right )}}{4 x \Gamma \left (\frac {5}{4}\right )} \] Input:

integrate((-3+2*x)/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)) + 
 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 = 0.11 (sec) , antiderivative size = 148, normalized size of antiderivative = 1.32 \[ \int \frac {-3+2 x}{x \sqrt [4]{-1+x^4}} \, dx=-\frac {3}{4} \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + 2 \, {\left (x^{4} - 1\right )}^{\frac {1}{4}}\right )}\right ) - \frac {3}{4} \, \sqrt {2} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - 2 \, {\left (x^{4} - 1\right )}^{\frac {1}{4}}\right )}\right ) + \frac {3}{8} \, \sqrt {2} \log \left (\sqrt {2} {\left (x^{4} - 1\right )}^{\frac {1}{4}} + \sqrt {x^{4} - 1} + 1\right ) - \frac {3}{8} \, \sqrt {2} \log \left (-\sqrt {2} {\left (x^{4} - 1\right )}^{\frac {1}{4}} + \sqrt {x^{4} - 1} + 1\right ) - \arctan \left (\frac {{\left (x^{4} - 1\right )}^{\frac {1}{4}}}{x}\right ) + \frac {1}{2} \, \log \left (\frac {{\left (x^{4} - 1\right )}^{\frac {1}{4}}}{x} + 1\right ) - \frac {1}{2} \, \log \left (\frac {{\left (x^{4} - 1\right )}^{\frac {1}{4}}}{x} - 1\right ) \] Input:

integrate((-3+2*x)/x/(x^4-1)^(1/4),x, algorithm="maxima")
 

Output:

-3/4*sqrt(2)*arctan(1/2*sqrt(2)*(sqrt(2) + 2*(x^4 - 1)^(1/4))) - 3/4*sqrt( 
2)*arctan(-1/2*sqrt(2)*(sqrt(2) - 2*(x^4 - 1)^(1/4))) + 3/8*sqrt(2)*log(sq 
rt(2)*(x^4 - 1)^(1/4) + sqrt(x^4 - 1) + 1) - 3/8*sqrt(2)*log(-sqrt(2)*(x^4 
 - 1)^(1/4) + sqrt(x^4 - 1) + 1) - 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]

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

integrate((-3+2*x)/x/(x^4-1)^(1/4),x, algorithm="giac")
 

Output:

integrate((2*x - 3)/((x^4 - 1)^(1/4)*x), x)
 

Mupad [B] (verification not implemented)

Time = 8.95 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.64 \[ \int \frac {-3+2 x}{x \sqrt [4]{-1+x^4}} \, dx=\frac {2\,x\,{\left (1-x^4\right )}^{1/4}\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{4},\frac {1}{4};\ \frac {5}{4};\ x^4\right )}{{\left (x^4-1\right )}^{1/4}}+\sqrt {2}\,\mathrm {atan}\left (\sqrt {2}\,{\left (x^4-1\right )}^{1/4}\,\left (\frac {1}{2}-\frac {1}{2}{}\mathrm {i}\right )\right )\,\left (-\frac {3}{4}+\frac {3}{4}{}\mathrm {i}\right )+\sqrt {2}\,\mathrm {atan}\left (\sqrt {2}\,{\left (x^4-1\right )}^{1/4}\,\left (\frac {1}{2}+\frac {1}{2}{}\mathrm {i}\right )\right )\,\left (-\frac {3}{4}-\frac {3}{4}{}\mathrm {i}\right ) \] Input:

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

Output:

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

Reduce [F]

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

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

Output:

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