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
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))
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)...
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]
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]
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 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+...
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)
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)
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)
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
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)
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)