\(\int \frac {\sqrt [3]{-x+x^3} (8-10 x^2+x^4)}{x^4 (4-2 x^2+x^4)} \, dx\) [1779]

Optimal result

Integrand size = 37, antiderivative size = 120 \[ \int \frac {\sqrt [3]{-x+x^3} \left (8-10 x^2+x^4\right )}{x^4 \left (4-2 x^2+x^4\right )} \, dx=\frac {3 \left (-1+4 x^2\right ) \sqrt [3]{-x+x^3}}{4 x^3}-\frac {1}{8} \text {RootSum}\left [3-6 \text {$\#$1}^3+4 \text {$\#$1}^6\&,\frac {-9 \log (x)+9 \log \left (\sqrt [3]{-x+x^3}-x \text {$\#$1}\right )+4 \log (x) \text {$\#$1}^3-4 \log \left (\sqrt [3]{-x+x^3}-x \text {$\#$1}\right ) \text {$\#$1}^3}{-3 \text {$\#$1}^2+4 \text {$\#$1}^5}\&\right ] \] Output:

Unintegrable
 

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 146, normalized size of antiderivative = 1.22 \[ \int \frac {\sqrt [3]{-x+x^3} \left (8-10 x^2+x^4\right )}{x^4 \left (4-2 x^2+x^4\right )} \, dx=-\frac {\sqrt [3]{x \left (-1+x^2\right )} \left (18 \left (1-4 x^2\right ) \sqrt [3]{-1+x^2}+x^{8/3} \text {RootSum}\left [3-6 \text {$\#$1}^3+4 \text {$\#$1}^6\&,\frac {-18 \log (x)+27 \log \left (\sqrt [3]{-1+x^2}-x^{2/3} \text {$\#$1}\right )+8 \log (x) \text {$\#$1}^3-12 \log \left (\sqrt [3]{-1+x^2}-x^{2/3} \text {$\#$1}\right ) \text {$\#$1}^3}{-3 \text {$\#$1}^2+4 \text {$\#$1}^5}\&\right ]\right )}{24 x^3 \sqrt [3]{-1+x^2}} \] Input:

Integrate[((-x + x^3)^(1/3)*(8 - 10*x^2 + x^4))/(x^4*(4 - 2*x^2 + x^4)),x]
 

Output:

-1/24*((x*(-1 + x^2))^(1/3)*(18*(1 - 4*x^2)*(-1 + x^2)^(1/3) + x^(8/3)*Roo 
tSum[3 - 6*#1^3 + 4*#1^6 & , (-18*Log[x] + 27*Log[(-1 + x^2)^(1/3) - x^(2/ 
3)*#1] + 8*Log[x]*#1^3 - 12*Log[(-1 + x^2)^(1/3) - x^(2/3)*#1]*#1^3)/(-3*# 
1^2 + 4*#1^5) & ]))/(x^3*(-1 + x^2)^(1/3))
 

Rubi [C] (verified)

Result contains complex when optimal does not.

Time = 2.42 (sec) , antiderivative size = 1050, normalized size of antiderivative = 8.75, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.108, Rules used = {2467, 2035, 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.

\(\displaystyle \int \frac {\sqrt [3]{x^3-x} \left (x^4-10 x^2+8\right )}{x^4 \left (x^4-2 x^2+4\right )} \, dx\)

\(\Big \downarrow \) 2467

\(\displaystyle \frac {\sqrt [3]{x^3-x} \int \frac {\sqrt [3]{x^2-1} \left (x^4-10 x^2+8\right )}{x^{11/3} \left (x^4-2 x^2+4\right )}dx}{\sqrt [3]{x} \sqrt [3]{x^2-1}}\)

\(\Big \downarrow \) 2035

\(\displaystyle \frac {3 \sqrt [3]{x^3-x} \int \frac {\sqrt [3]{x^2-1} \left (x^4-10 x^2+8\right )}{x^3 \left (x^4-2 x^2+4\right )}d\sqrt [3]{x}}{\sqrt [3]{x} \sqrt [3]{x^2-1}}\)

\(\Big \downarrow \) 7279

\(\displaystyle \frac {3 \sqrt [3]{x^3-x} \int \left (\frac {x \sqrt [3]{x^2-1} \left (3 x^2-8\right )}{2 \left (x^4-2 x^2+4\right )}-\frac {3 \sqrt [3]{x^2-1}}{2 x}+\frac {2 \sqrt [3]{x^2-1}}{x^3}\right )d\sqrt [3]{x}}{\sqrt [3]{x} \sqrt [3]{x^2-1}}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {3 \sqrt [3]{x^3-x} \left (\frac {\left (x^2-1\right )^{4/3}}{4 x^{8/3}}+\frac {3 \sqrt [3]{x^2-1}}{4 x^{2/3}}-\frac {\left (1+i \sqrt {3}\right ) \arctan \left (\frac {\frac {2 \sqrt [6]{3} x^{2/3}}{\sqrt [3]{-i+\sqrt {3}} \sqrt [3]{x^2-1}}+1}{\sqrt {3}}\right )}{8 \sqrt [3]{3 \left (-i+\sqrt {3}\right )}}+\frac {\arctan \left (\frac {\frac {2 \sqrt [6]{3} x^{2/3}}{\sqrt [3]{-i+\sqrt {3}} \sqrt [3]{x^2-1}}+1}{\sqrt {3}}\right )}{2 \sqrt [3]{3 \left (-i+\sqrt {3}\right )}}+\frac {i \arctan \left (\frac {\frac {2 \sqrt [6]{3} x^{2/3}}{\sqrt [3]{-i+\sqrt {3}} \sqrt [3]{x^2-1}}+1}{\sqrt {3}}\right )}{3^{5/6} \sqrt [3]{-i+\sqrt {3}}}+\frac {\arctan \left (\frac {\frac {2 \sqrt [6]{3} x^{2/3}}{\sqrt [3]{i+\sqrt {3}} \sqrt [3]{x^2-1}}+1}{\sqrt {3}}\right )}{2 \sqrt [3]{3 \left (i+\sqrt {3}\right )}}+\frac {i \left (i+\sqrt {3}\right )^{2/3} \arctan \left (\frac {\frac {2 \sqrt [6]{3} x^{2/3}}{\sqrt [3]{i+\sqrt {3}} \sqrt [3]{x^2-1}}+1}{\sqrt {3}}\right )}{8 \sqrt [3]{3}}-\frac {i \arctan \left (\frac {\frac {2 \sqrt [6]{3} x^{2/3}}{\sqrt [3]{i+\sqrt {3}} \sqrt [3]{x^2-1}}+1}{\sqrt {3}}\right )}{3^{5/6} \sqrt [3]{i+\sqrt {3}}}+\frac {i \log \left (-x^2-i \sqrt {3}+1\right )}{6 \sqrt [3]{3 \left (i+\sqrt {3}\right )}}-\frac {i \left (i+\sqrt {3}\right )^{2/3} \log \left (-x^2-i \sqrt {3}+1\right )}{16\ 3^{5/6}}-\frac {\log \left (-x^2-i \sqrt {3}+1\right )}{4\ 3^{5/6} \sqrt [3]{i+\sqrt {3}}}-\frac {i \log \left (-x^2+i \sqrt {3}+1\right )}{6 \sqrt [3]{3 \left (-i+\sqrt {3}\right )}}+\frac {i \left (-i+\sqrt {3}\right )^{2/3} \log \left (-x^2+i \sqrt {3}+1\right )}{16\ 3^{5/6}}-\frac {\log \left (-x^2+i \sqrt {3}+1\right )}{4\ 3^{5/6} \sqrt [3]{-i+\sqrt {3}}}+\frac {i \log \left (\sqrt [6]{3} x^{2/3}-\sqrt [3]{-i+\sqrt {3}} \sqrt [3]{x^2-1}\right )}{2 \sqrt [3]{3 \left (-i+\sqrt {3}\right )}}-\frac {1}{16} i \sqrt [6]{3} \left (-i+\sqrt {3}\right )^{2/3} \log \left (\sqrt [6]{3} x^{2/3}-\sqrt [3]{-i+\sqrt {3}} \sqrt [3]{x^2-1}\right )+\frac {\sqrt [6]{3} \log \left (\sqrt [6]{3} x^{2/3}-\sqrt [3]{-i+\sqrt {3}} \sqrt [3]{x^2-1}\right )}{4 \sqrt [3]{-i+\sqrt {3}}}-\frac {i \log \left (\sqrt [6]{3} x^{2/3}-\sqrt [3]{i+\sqrt {3}} \sqrt [3]{x^2-1}\right )}{2 \sqrt [3]{3 \left (i+\sqrt {3}\right )}}+\frac {1}{16} i \sqrt [6]{3} \left (i+\sqrt {3}\right )^{2/3} \log \left (\sqrt [6]{3} x^{2/3}-\sqrt [3]{i+\sqrt {3}} \sqrt [3]{x^2-1}\right )+\frac {\sqrt [6]{3} \log \left (\sqrt [6]{3} x^{2/3}-\sqrt [3]{i+\sqrt {3}} \sqrt [3]{x^2-1}\right )}{4 \sqrt [3]{i+\sqrt {3}}}\right )}{\sqrt [3]{x} \sqrt [3]{x^2-1}}\)

Input:

Int[((-x + x^3)^(1/3)*(8 - 10*x^2 + x^4))/(x^4*(4 - 2*x^2 + x^4)),x]
 

Output:

(3*(-x + x^3)^(1/3)*((3*(-1 + x^2)^(1/3))/(4*x^(2/3)) + (-1 + x^2)^(4/3)/( 
4*x^(8/3)) + (I*ArcTan[(1 + (2*3^(1/6)*x^(2/3))/((-I + Sqrt[3])^(1/3)*(-1 
+ x^2)^(1/3)))/Sqrt[3]])/(3^(5/6)*(-I + Sqrt[3])^(1/3)) + ArcTan[(1 + (2*3 
^(1/6)*x^(2/3))/((-I + Sqrt[3])^(1/3)*(-1 + x^2)^(1/3)))/Sqrt[3]]/(2*(3*(- 
I + Sqrt[3]))^(1/3)) - ((1 + I*Sqrt[3])*ArcTan[(1 + (2*3^(1/6)*x^(2/3))/(( 
-I + Sqrt[3])^(1/3)*(-1 + x^2)^(1/3)))/Sqrt[3]])/(8*(3*(-I + Sqrt[3]))^(1/ 
3)) - (I*ArcTan[(1 + (2*3^(1/6)*x^(2/3))/((I + Sqrt[3])^(1/3)*(-1 + x^2)^( 
1/3)))/Sqrt[3]])/(3^(5/6)*(I + Sqrt[3])^(1/3)) + ((I/8)*(I + Sqrt[3])^(2/3 
)*ArcTan[(1 + (2*3^(1/6)*x^(2/3))/((I + Sqrt[3])^(1/3)*(-1 + x^2)^(1/3)))/ 
Sqrt[3]])/3^(1/3) + ArcTan[(1 + (2*3^(1/6)*x^(2/3))/((I + Sqrt[3])^(1/3)*( 
-1 + x^2)^(1/3)))/Sqrt[3]]/(2*(3*(I + Sqrt[3]))^(1/3)) - Log[1 - I*Sqrt[3] 
 - x^2]/(4*3^(5/6)*(I + Sqrt[3])^(1/3)) - ((I/16)*(I + Sqrt[3])^(2/3)*Log[ 
1 - I*Sqrt[3] - x^2])/3^(5/6) + ((I/6)*Log[1 - I*Sqrt[3] - x^2])/(3*(I + S 
qrt[3]))^(1/3) - Log[1 + I*Sqrt[3] - x^2]/(4*3^(5/6)*(-I + Sqrt[3])^(1/3)) 
 + ((I/16)*(-I + Sqrt[3])^(2/3)*Log[1 + I*Sqrt[3] - x^2])/3^(5/6) - ((I/6) 
*Log[1 + I*Sqrt[3] - x^2])/(3*(-I + Sqrt[3]))^(1/3) + (3^(1/6)*Log[3^(1/6) 
*x^(2/3) - (-I + Sqrt[3])^(1/3)*(-1 + x^2)^(1/3)])/(4*(-I + Sqrt[3])^(1/3) 
) - (I/16)*3^(1/6)*(-I + Sqrt[3])^(2/3)*Log[3^(1/6)*x^(2/3) - (-I + Sqrt[3 
])^(1/3)*(-1 + x^2)^(1/3)] + ((I/2)*Log[3^(1/6)*x^(2/3) - (-I + Sqrt[3])^( 
1/3)*(-1 + x^2)^(1/3)])/(3*(-I + Sqrt[3]))^(1/3) + (3^(1/6)*Log[3^(1/6)...
 

Defintions of rubi rules used

rule 2009

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

rule 2035

Int[(Fx_)*(x_)^(m_), x_Symbol] :> With[{k = Denominator[m]}, Simp[k   Subst 
[Int[x^(k*(m + 1) - 1)*SubstPower[Fx, x, k], x], x, x^(1/k)], x]] /; Fracti 
onQ[m] && AlgebraicFunctionQ[Fx, x]
 

rule 2467

Int[(Fx_.)*(Px_)^(p_), x_Symbol] :> With[{r = Expon[Px, x, Min]}, Simp[Px^F 
racPart[p]/(x^(r*FracPart[p])*ExpandToSum[Px/x^r, x]^FracPart[p])   Int[x^( 
p*r)*ExpandToSum[Px/x^r, x]^p*Fx, x], x] /; IGtQ[r, 0]] /; FreeQ[p, x] && P 
olyQ[Px, x] &&  !IntegerQ[p] &&  !MonomialQ[Px, x] &&  !PolyQ[Fx, x]
 

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] (warning: unable to verify)

Result contains higher order function than in optimal. Order 9 vs. order 1.

Time = 33.46 (sec) , antiderivative size = 3329, normalized size of antiderivative = 27.74

\[\text {output too large to display}\]

Input:

int((x^3-x)^(1/3)*(x^4-10*x^2+8)/x^4/(x^4-2*x^2+4),x)
 

Output:

3/4*(4*x^4-5*x^2+1)/x^3*(x*(x^2-1))^(1/3)/(x^2-1)+(1/1836*ln(-(728*RootOf( 
4*_Z^6+1602*_Z^3+177957)^6*RootOf(_Z^3+8*RootOf(4*_Z^6+1602*_Z^3+177957)^3 
+3204)*x^4-3640*RootOf(_Z^3+8*RootOf(4*_Z^6+1602*_Z^3+177957)^3+3204)*Root 
Of(4*_Z^6+1602*_Z^3+177957)^6*x^2-279582*RootOf(4*_Z^6+1602*_Z^3+177957)^3 
*(x^6-2*x^4+x^2)^(1/3)*RootOf(_Z^3+8*RootOf(4*_Z^6+1602*_Z^3+177957)^3+320 
4)^2*x^2-1094106*RootOf(_Z^3+8*RootOf(4*_Z^6+1602*_Z^3+177957)^3+3204)*Roo 
tOf(4*_Z^6+1602*_Z^3+177957)^3*x^4+2912*RootOf(_Z^3+8*RootOf(4*_Z^6+1602*_ 
Z^3+177957)^3+3204)*RootOf(4*_Z^6+1602*_Z^3+177957)^6+279582*(x^6-2*x^4+x^ 
2)^(1/3)*RootOf(4*_Z^6+1602*_Z^3+177957)^3*RootOf(_Z^3+8*RootOf(4*_Z^6+160 
2*_Z^3+177957)^3+3204)^2+1126554*RootOf(_Z^3+8*RootOf(4*_Z^6+1602*_Z^3+177 
957)^3+3204)*RootOf(4*_Z^6+1602*_Z^3+177957)^3*x^2+24512436*RootOf(4*_Z^6+ 
1602*_Z^3+177957)^3*(x^6-2*x^4+x^2)^(2/3)-53446419*(x^6-2*x^4+x^2)^(1/3)*R 
ootOf(_Z^3+8*RootOf(4*_Z^6+1602*_Z^3+177957)^3+3204)^2*x^2-20049822*RootOf 
(_Z^3+8*RootOf(4*_Z^6+1602*_Z^3+177957)^3+3204)*x^4-32448*RootOf(4*_Z^6+16 
02*_Z^3+177957)^3*RootOf(_Z^3+8*RootOf(4*_Z^6+1602*_Z^3+177957)^3+3204)+53 
446419*(x^6-2*x^4+x^2)^(1/3)*RootOf(_Z^3+8*RootOf(4*_Z^6+1602*_Z^3+177957) 
^3+3204)^2+21592116*RootOf(_Z^3+8*RootOf(4*_Z^6+1602*_Z^3+177957)^3+3204)* 
x^2+6843158478*(x^6-2*x^4+x^2)^(2/3)-1542294*RootOf(_Z^3+8*RootOf(4*_Z^6+1 
602*_Z^3+177957)^3+3204))/(2*RootOf(4*_Z^6+1602*_Z^3+177957)^3*x^2-8*RootO 
f(4*_Z^6+1602*_Z^3+177957)^3+171*x^2-1602)/(-1+x)/(1+x))*RootOf(4*_Z^6+...
 

Fricas [F(-2)]

Exception generated. \[ \int \frac {\sqrt [3]{-x+x^3} \left (8-10 x^2+x^4\right )}{x^4 \left (4-2 x^2+x^4\right )} \, dx=\text {Exception raised: TypeError} \] Input:

integrate((x^3-x)^(1/3)*(x^4-10*x^2+8)/x^4/(x^4-2*x^2+4),x, algorithm="fri 
cas")
 

Output:

Exception raised: TypeError >>  Error detected within library code:   inte 
grate: implementation incomplete (trace 0)
 

Sympy [N/A]

Not integrable

Time = 3.10 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.30 \[ \int \frac {\sqrt [3]{-x+x^3} \left (8-10 x^2+x^4\right )}{x^4 \left (4-2 x^2+x^4\right )} \, dx=\int \frac {\sqrt [3]{x \left (x - 1\right ) \left (x + 1\right )} \left (x^{4} - 10 x^{2} + 8\right )}{x^{4} \left (x^{4} - 2 x^{2} + 4\right )}\, dx \] Input:

integrate((x**3-x)**(1/3)*(x**4-10*x**2+8)/x**4/(x**4-2*x**2+4),x)
 

Output:

Integral((x*(x - 1)*(x + 1))**(1/3)*(x**4 - 10*x**2 + 8)/(x**4*(x**4 - 2*x 
**2 + 4)), x)
 

Maxima [N/A]

Not integrable

Time = 0.24 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.31 \[ \int \frac {\sqrt [3]{-x+x^3} \left (8-10 x^2+x^4\right )}{x^4 \left (4-2 x^2+x^4\right )} \, dx=\int { \frac {{\left (x^{4} - 10 \, x^{2} + 8\right )} {\left (x^{3} - x\right )}^{\frac {1}{3}}}{{\left (x^{4} - 2 \, x^{2} + 4\right )} x^{4}} \,d x } \] Input:

integrate((x^3-x)^(1/3)*(x^4-10*x^2+8)/x^4/(x^4-2*x^2+4),x, algorithm="max 
ima")
 

Output:

integrate((x^4 - 10*x^2 + 8)*(x^3 - x)^(1/3)/((x^4 - 2*x^2 + 4)*x^4), x)
 

Giac [F(-2)]

Exception generated. \[ \int \frac {\sqrt [3]{-x+x^3} \left (8-10 x^2+x^4\right )}{x^4 \left (4-2 x^2+x^4\right )} \, dx=\text {Exception raised: RuntimeError} \] Input:

integrate((x^3-x)^(1/3)*(x^4-10*x^2+8)/x^4/(x^4-2*x^2+4),x, algorithm="gia 
c")
 

Output:

Exception raised: RuntimeError >> an error occurred running a Giac command 
:INPUT:sage2OUTPUT:Invalid _EXT in replace_ext Error: Bad Argument Value(3 
072*(-(1/sageVARx)^2+1)^(1/3)*(-(1/sageVARx)^2+1)+9216*(-(1/sageVARx)^2+1) 
^(1/3))/4096
 

Mupad [N/A]

Not integrable

Time = 0.00 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.31 \[ \int \frac {\sqrt [3]{-x+x^3} \left (8-10 x^2+x^4\right )}{x^4 \left (4-2 x^2+x^4\right )} \, dx=\int \frac {{\left (x^3-x\right )}^{1/3}\,\left (x^4-10\,x^2+8\right )}{x^4\,\left (x^4-2\,x^2+4\right )} \,d x \] Input:

int(((x^3 - x)^(1/3)*(x^4 - 10*x^2 + 8))/(x^4*(x^4 - 2*x^2 + 4)),x)
 

Output:

int(((x^3 - x)^(1/3)*(x^4 - 10*x^2 + 8))/(x^4*(x^4 - 2*x^2 + 4)), x)
 

Reduce [N/A]

Not integrable

Time = 0.35 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.82 \[ \int \frac {\sqrt [3]{-x+x^3} \left (8-10 x^2+x^4\right )}{x^4 \left (4-2 x^2+x^4\right )} \, dx=\frac {12 \left (x^{2}-1\right )^{\frac {1}{3}} x^{2}-3 \left (x^{2}-1\right )^{\frac {1}{3}}-8 x^{\frac {8}{3}} \left (\int \frac {\left (x^{2}-1\right )^{\frac {1}{3}} x}{x^{\frac {20}{3}}-3 x^{\frac {14}{3}}+6 x^{\frac {8}{3}}-4 x^{\frac {2}{3}}}d x \right )-10 x^{\frac {8}{3}} \left (\int \frac {x^{\frac {7}{3}} \left (x^{2}-1\right )^{\frac {1}{3}}}{x^{6}-3 x^{4}+6 x^{2}-4}d x \right )}{4 x^{\frac {8}{3}}} \] Input:

int((x^3-x)^(1/3)*(x^4-10*x^2+8)/x^4/(x^4-2*x^2+4),x)
 

Output:

(12*(x**2 - 1)**(1/3)*x**2 - 3*(x**2 - 1)**(1/3) - 8*x**(2/3)*int(((x**2 - 
 1)**(1/3)*x)/(x**(2/3)*x**6 - 3*x**(2/3)*x**4 + 6*x**(2/3)*x**2 - 4*x**(2 
/3)),x)*x**2 - 10*x**(2/3)*int((x**(1/3)*(x**2 - 1)**(1/3)*x**2)/(x**6 - 3 
*x**4 + 6*x**2 - 4),x)*x**2)/(4*x**(2/3)*x**2)