Integrand size = 37, antiderivative size = 120 \[ \int \frac {\sqrt [3]{-x+x^3} \left (8-10 x^2+x^4\right )}{x^4 \left (4-2 x^2+x^4\right )} \, dx=\frac {3 \left (-1+4 x^2\right ) \sqrt [3]{-x+x^3}}{4 x^3}-\frac {1}{8} \text {RootSum}\left [3-6 \text {$\#$1}^3+4 \text {$\#$1}^6\&,\frac {-9 \log (x)+9 \log \left (\sqrt [3]{-x+x^3}-x \text {$\#$1}\right )+4 \log (x) \text {$\#$1}^3-4 \log \left (\sqrt [3]{-x+x^3}-x \text {$\#$1}\right ) \text {$\#$1}^3}{-3 \text {$\#$1}^2+4 \text {$\#$1}^5}\&\right ] \]
Time = 0.00 (sec) , antiderivative size = 146, normalized size of antiderivative = 1.22 \[ \int \frac {\sqrt [3]{-x+x^3} \left (8-10 x^2+x^4\right )}{x^4 \left (4-2 x^2+x^4\right )} \, dx=-\frac {\sqrt [3]{x \left (-1+x^2\right )} \left (18 \left (1-4 x^2\right ) \sqrt [3]{-1+x^2}+x^{8/3} \text {RootSum}\left [3-6 \text {$\#$1}^3+4 \text {$\#$1}^6\&,\frac {-18 \log (x)+27 \log \left (\sqrt [3]{-1+x^2}-x^{2/3} \text {$\#$1}\right )+8 \log (x) \text {$\#$1}^3-12 \log \left (\sqrt [3]{-1+x^2}-x^{2/3} \text {$\#$1}\right ) \text {$\#$1}^3}{-3 \text {$\#$1}^2+4 \text {$\#$1}^5}\&\right ]\right )}{24 x^3 \sqrt [3]{-1+x^2}} \]
-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 = 2.43 (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}}\) |
(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)...
3.18.79.3.1 Defintions of rubi rules used
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]
Time = 42.73 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.69
| method | result | size |
| pseudoelliptic | \(\frac {\left (24 x^{2}-6\right ) \left (x^{3}-x \right )^{\frac {1}{3}}+\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (4 \textit {\_Z}^{6}-6 \textit {\_Z}^{3}+3\right )}{\sum }\frac {\left (4 \textit {\_R}^{3}-9\right ) \ln \left (\frac {-\textit {\_R} x +\left (x^{3}-x \right )^{\frac {1}{3}}}{x}\right )}{\textit {\_R}^{2} \left (4 \textit {\_R}^{3}-3\right )}\right ) x^{3}}{8 x^{3}}\) | \(83\) |
| trager | \(\text {Expression too large to display}\) | \(2117\) |
| risch | \(\text {Expression too large to display}\) | \(3327\) |
1/8*((24*x^2-6)*(x^3-x)^(1/3)+sum((4*_R^3-9)*ln((-_R*x+(x^3-x)^(1/3))/x)/_ R^2/(4*_R^3-3),_R=RootOf(4*_Z^6-6*_Z^3+3))*x^3)/x^3
Exception generated. \[ \int \frac {\sqrt [3]{-x+x^3} \left (8-10 x^2+x^4\right )}{x^4 \left (4-2 x^2+x^4\right )} \, dx=\text {Exception raised: TypeError} \]
Exception raised: TypeError >> Error detected within library code: inte grate: implementation incomplete (trace 0)
Not integrable
Time = 2.89 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.30 \[ \int \frac {\sqrt [3]{-x+x^3} \left (8-10 x^2+x^4\right )}{x^4 \left (4-2 x^2+x^4\right )} \, dx=\int \frac {\sqrt [3]{x \left (x - 1\right ) \left (x + 1\right )} \left (x^{4} - 10 x^{2} + 8\right )}{x^{4} \left (x^{4} - 2 x^{2} + 4\right )}\, dx \]
Not integrable
Time = 0.33 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.31 \[ \int \frac {\sqrt [3]{-x+x^3} \left (8-10 x^2+x^4\right )}{x^4 \left (4-2 x^2+x^4\right )} \, dx=\int { \frac {{\left (x^{4} - 10 \, x^{2} + 8\right )} {\left (x^{3} - x\right )}^{\frac {1}{3}}}{{\left (x^{4} - 2 \, x^{2} + 4\right )} x^{4}} \,d x } \]
Exception generated. \[ \int \frac {\sqrt [3]{-x+x^3} \left (8-10 x^2+x^4\right )}{x^4 \left (4-2 x^2+x^4\right )} \, dx=\text {Exception raised: RuntimeError} \]
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 \[ \int \frac {\sqrt [3]{-x+x^3} \left (8-10 x^2+x^4\right )}{x^4 \left (4-2 x^2+x^4\right )} \, dx=\int \frac {{\left (x^3-x\right )}^{1/3}\,\left (x^4-10\,x^2+8\right )}{x^4\,\left (x^4-2\,x^2+4\right )} \,d x \]