Integrand size = 32, antiderivative size = 116 \[ \int \frac {\left (4+x^2\right ) \sqrt [3]{-2 x+x^3}}{x^4 \left (-4-4 x^2+x^4\right )} \, dx=-\frac {3 \left (-2+7 x^2\right ) \sqrt [3]{-2 x+x^3}}{16 x^3}+\frac {1}{16} \text {RootSum}\left [2-4 \text {$\#$1}^3+\text {$\#$1}^6\&,\frac {-6 \log (x)+6 \log \left (\sqrt [3]{-2 x+x^3}-x \text {$\#$1}\right )+11 \log (x) \text {$\#$1}^3-11 \log \left (\sqrt [3]{-2 x+x^3}-x \text {$\#$1}\right ) \text {$\#$1}^3}{-2 \text {$\#$1}^2+\text {$\#$1}^5}\&\right ] \]
Time = 3.10 (sec) , antiderivative size = 142, normalized size of antiderivative = 1.22 \[ \int \frac {\left (4+x^2\right ) \sqrt [3]{-2 x+x^3}}{x^4 \left (-4-4 x^2+x^4\right )} \, dx=\frac {\sqrt [3]{x \left (-2+x^2\right )} \left (9 \left (2-7 x^2\right ) \sqrt [3]{-2+x^2}+x^{8/3} \text {RootSum}\left [2-4 \text {$\#$1}^3+\text {$\#$1}^6\&,\frac {-12 \log (x)+18 \log \left (\sqrt [3]{-2+x^2}-x^{2/3} \text {$\#$1}\right )+22 \log (x) \text {$\#$1}^3-33 \log \left (\sqrt [3]{-2+x^2}-x^{2/3} \text {$\#$1}\right ) \text {$\#$1}^3}{-2 \text {$\#$1}^2+\text {$\#$1}^5}\&\right ]\right )}{48 x^3 \sqrt [3]{-2+x^2}} \]
((x*(-2 + x^2))^(1/3)*(9*(2 - 7*x^2)*(-2 + x^2)^(1/3) + x^(8/3)*RootSum[2 - 4*#1^3 + #1^6 & , (-12*Log[x] + 18*Log[(-2 + x^2)^(1/3) - x^(2/3)*#1] + 22*Log[x]*#1^3 - 33*Log[(-2 + x^2)^(1/3) - x^(2/3)*#1]*#1^3)/(-2*#1^2 + #1 ^5) & ]))/(48*x^3*(-2 + x^2)^(1/3))
Leaf count is larger than twice the leaf count of optimal. \(964\) vs. \(2(116)=232\).
Time = 2.66 (sec) , antiderivative size = 964, normalized size of antiderivative = 8.31, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.156, Rules used = {2467, 25, 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.
\(\displaystyle \int \frac {\left (x^2+4\right ) \sqrt [3]{x^3-2 x}}{x^4 \left (x^4-4 x^2-4\right )} \, dx\) |
\(\Big \downarrow \) 2467 |
\(\displaystyle \frac {\sqrt [3]{x^3-2 x} \int -\frac {\sqrt [3]{x^2-2} \left (x^2+4\right )}{x^{11/3} \left (-x^4+4 x^2+4\right )}dx}{\sqrt [3]{x} \sqrt [3]{x^2-2}}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\frac {\sqrt [3]{x^3-2 x} \int \frac {\sqrt [3]{x^2-2} \left (x^2+4\right )}{x^{11/3} \left (-x^4+4 x^2+4\right )}dx}{\sqrt [3]{x} \sqrt [3]{x^2-2}}\) |
\(\Big \downarrow \) 2035 |
\(\displaystyle -\frac {3 \sqrt [3]{x^3-2 x} \int \frac {\sqrt [3]{x^2-2} \left (x^2+4\right )}{x^3 \left (-x^4+4 x^2+4\right )}d\sqrt [3]{x}}{\sqrt [3]{x} \sqrt [3]{x^2-2}}\) |
\(\Big \downarrow \) 7279 |
\(\displaystyle -\frac {3 \sqrt [3]{x^3-2 x} \int \left (\frac {x \sqrt [3]{x^2-2} \left (3 x^2-16\right )}{4 \left (x^4-4 x^2-4\right )}-\frac {3 \sqrt [3]{x^2-2}}{4 x}+\frac {\sqrt [3]{x^2-2}}{x^3}\right )d\sqrt [3]{x}}{\sqrt [3]{x} \sqrt [3]{x^2-2}}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {3 \sqrt [3]{x^3-2 x} \left (\frac {\left (x^2-2\right )^{4/3}}{16 x^{8/3}}+\frac {3 \sqrt [3]{x^2-2}}{8 x^{2/3}}+\frac {\sqrt {3} \sqrt [3]{3+2 \sqrt {2}} \arctan \left (\frac {\frac {2 \sqrt [3]{2-\sqrt {2}} x^{2/3}}{\sqrt [3]{x^2-2}}+1}{\sqrt {3}}\right )}{16\ 2^{5/6}}+\frac {\sqrt {3} \sqrt [3]{-1+\sqrt {2}} \arctan \left (\frac {\frac {2 \sqrt [3]{2-\sqrt {2}} x^{2/3}}{\sqrt [3]{x^2-2}}+1}{\sqrt {3}}\right )}{16\ 2^{5/6}}-\frac {\sqrt [3]{\frac {1}{2} \left (-1+\sqrt {2}\right )} \arctan \left (\frac {\frac {2 \sqrt [3]{2-\sqrt {2}} x^{2/3}}{\sqrt [3]{x^2-2}}+1}{\sqrt {3}}\right )}{2 \sqrt {3}}+\frac {\sqrt {3} \sqrt [3]{1+\sqrt {2}} \arctan \left (\frac {\frac {2 \sqrt [3]{2+\sqrt {2}} x^{2/3}}{\sqrt [3]{x^2-2}}+1}{\sqrt {3}}\right )}{16\ 2^{5/6}}+\frac {\sqrt [3]{\frac {1}{2} \left (1+\sqrt {2}\right )} \arctan \left (\frac {\frac {2 \sqrt [3]{2+\sqrt {2}} x^{2/3}}{\sqrt [3]{x^2-2}}+1}{\sqrt {3}}\right )}{2 \sqrt {3}}-\frac {\sqrt {3} \sqrt [3]{3-2 \sqrt {2}} \arctan \left (\frac {\frac {2 \sqrt [3]{2+\sqrt {2}} x^{2/3}}{\sqrt [3]{x^2-2}}+1}{\sqrt {3}}\right )}{16\ 2^{5/6}}-\frac {\sqrt [3]{3+2 \sqrt {2}} \log \left (2 \left (1+\sqrt {2}\right )-x^2\right )}{32\ 2^{5/6}}-\frac {\sqrt [3]{-1+\sqrt {2}} \log \left (2 \left (1+\sqrt {2}\right )-x^2\right )}{32\ 2^{5/6}}+\frac {1}{12} \sqrt [3]{\frac {1}{2} \left (-1+\sqrt {2}\right )} \log \left (2 \left (1+\sqrt {2}\right )-x^2\right )-\frac {\sqrt [3]{1+\sqrt {2}} \log \left (x^2-2 \left (1-\sqrt {2}\right )\right )}{32\ 2^{5/6}}-\frac {1}{12} \sqrt [3]{\frac {1}{2} \left (1+\sqrt {2}\right )} \log \left (x^2-2 \left (1-\sqrt {2}\right )\right )+\frac {\sqrt [3]{3-2 \sqrt {2}} \log \left (x^2-2 \left (1-\sqrt {2}\right )\right )}{32\ 2^{5/6}}+\frac {3 \sqrt [3]{3+2 \sqrt {2}} \log \left (\sqrt [3]{2-\sqrt {2}} x^{2/3}-\sqrt [3]{x^2-2}\right )}{32\ 2^{5/6}}+\frac {3 \sqrt [3]{-1+\sqrt {2}} \log \left (\sqrt [3]{2-\sqrt {2}} x^{2/3}-\sqrt [3]{x^2-2}\right )}{32\ 2^{5/6}}-\frac {1}{4} \sqrt [3]{\frac {1}{2} \left (-1+\sqrt {2}\right )} \log \left (\sqrt [3]{2-\sqrt {2}} x^{2/3}-\sqrt [3]{x^2-2}\right )+\frac {3 \sqrt [3]{1+\sqrt {2}} \log \left (\sqrt [3]{2+\sqrt {2}} x^{2/3}-\sqrt [3]{x^2-2}\right )}{32\ 2^{5/6}}+\frac {1}{4} \sqrt [3]{\frac {1}{2} \left (1+\sqrt {2}\right )} \log \left (\sqrt [3]{2+\sqrt {2}} x^{2/3}-\sqrt [3]{x^2-2}\right )-\frac {3 \sqrt [3]{3-2 \sqrt {2}} \log \left (\sqrt [3]{2+\sqrt {2}} x^{2/3}-\sqrt [3]{x^2-2}\right )}{32\ 2^{5/6}}\right )}{\sqrt [3]{x} \sqrt [3]{x^2-2}}\) |
(-3*(-2*x + x^3)^(1/3)*((3*(-2 + x^2)^(1/3))/(8*x^(2/3)) + (-2 + x^2)^(4/3 )/(16*x^(8/3)) - (((-1 + Sqrt[2])/2)^(1/3)*ArcTan[(1 + (2*(2 - Sqrt[2])^(1 /3)*x^(2/3))/(-2 + x^2)^(1/3))/Sqrt[3]])/(2*Sqrt[3]) + (Sqrt[3]*(-1 + Sqrt [2])^(1/3)*ArcTan[(1 + (2*(2 - Sqrt[2])^(1/3)*x^(2/3))/(-2 + x^2)^(1/3))/S qrt[3]])/(16*2^(5/6)) + (Sqrt[3]*(3 + 2*Sqrt[2])^(1/3)*ArcTan[(1 + (2*(2 - Sqrt[2])^(1/3)*x^(2/3))/(-2 + x^2)^(1/3))/Sqrt[3]])/(16*2^(5/6)) - (Sqrt[ 3]*(3 - 2*Sqrt[2])^(1/3)*ArcTan[(1 + (2*(2 + Sqrt[2])^(1/3)*x^(2/3))/(-2 + x^2)^(1/3))/Sqrt[3]])/(16*2^(5/6)) + (((1 + Sqrt[2])/2)^(1/3)*ArcTan[(1 + (2*(2 + Sqrt[2])^(1/3)*x^(2/3))/(-2 + x^2)^(1/3))/Sqrt[3]])/(2*Sqrt[3]) + (Sqrt[3]*(1 + Sqrt[2])^(1/3)*ArcTan[(1 + (2*(2 + Sqrt[2])^(1/3)*x^(2/3))/ (-2 + x^2)^(1/3))/Sqrt[3]])/(16*2^(5/6)) + (((-1 + Sqrt[2])/2)^(1/3)*Log[2 *(1 + Sqrt[2]) - x^2])/12 - ((-1 + Sqrt[2])^(1/3)*Log[2*(1 + Sqrt[2]) - x^ 2])/(32*2^(5/6)) - ((3 + 2*Sqrt[2])^(1/3)*Log[2*(1 + Sqrt[2]) - x^2])/(32* 2^(5/6)) + ((3 - 2*Sqrt[2])^(1/3)*Log[-2*(1 - Sqrt[2]) + x^2])/(32*2^(5/6) ) - (((1 + Sqrt[2])/2)^(1/3)*Log[-2*(1 - Sqrt[2]) + x^2])/12 - ((1 + Sqrt[ 2])^(1/3)*Log[-2*(1 - Sqrt[2]) + x^2])/(32*2^(5/6)) - (((-1 + Sqrt[2])/2)^ (1/3)*Log[(2 - Sqrt[2])^(1/3)*x^(2/3) - (-2 + x^2)^(1/3)])/4 + (3*(-1 + Sq rt[2])^(1/3)*Log[(2 - Sqrt[2])^(1/3)*x^(2/3) - (-2 + x^2)^(1/3)])/(32*2^(5 /6)) + (3*(3 + 2*Sqrt[2])^(1/3)*Log[(2 - Sqrt[2])^(1/3)*x^(2/3) - (-2 + x^ 2)^(1/3)])/(32*2^(5/6)) - (3*(3 - 2*Sqrt[2])^(1/3)*Log[(2 + Sqrt[2])^(1...
3.18.17.3.1 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 = 202.40 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.76
method | result | size |
pseudoelliptic | \(\frac {-\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{6}-4 \textit {\_Z}^{3}+2\right )}{\sum }\frac {\left (11 \textit {\_R}^{3}-6\right ) \ln \left (\frac {-\textit {\_R} x +{\left (x \left (x^{2}-2\right )\right )}^{\frac {1}{3}}}{x}\right )}{\textit {\_R}^{2} \left (\textit {\_R}^{3}-2\right )}\right ) x^{3}-21 {\left (x \left (x^{2}-2\right )\right )}^{\frac {1}{3}} x^{2}+6 {\left (x \left (x^{2}-2\right )\right )}^{\frac {1}{3}}}{16 x^{3}}\) | \(88\) |
trager | \(\text {Expression too large to display}\) | \(7671\) |
risch | \(\text {Expression too large to display}\) | \(14280\) |
1/16*(-sum((11*_R^3-6)*ln((-_R*x+(x*(x^2-2))^(1/3))/x)/_R^2/(_R^3-2),_R=Ro otOf(_Z^6-4*_Z^3+2))*x^3-21*(x*(x^2-2))^(1/3)*x^2+6*(x*(x^2-2))^(1/3))/x^3
Exception generated. \[ \int \frac {\left (4+x^2\right ) \sqrt [3]{-2 x+x^3}}{x^4 \left (-4-4 x^2+x^4\right )} \, dx=\text {Exception raised: TypeError} \]
Exception raised: TypeError >> Error detected within library code: inte grate: implementation incomplete (trace 0)
Not integrable
Time = 50.62 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.25 \[ \int \frac {\left (4+x^2\right ) \sqrt [3]{-2 x+x^3}}{x^4 \left (-4-4 x^2+x^4\right )} \, dx=\int \frac {\sqrt [3]{x \left (x^{2} - 2\right )} \left (x^{2} + 4\right )}{x^{4} \left (x^{4} - 4 x^{2} - 4\right )}\, dx \]
Not integrable
Time = 0.32 (sec) , antiderivative size = 139, normalized size of antiderivative = 1.20 \[ \int \frac {\left (4+x^2\right ) \sqrt [3]{-2 x+x^3}}{x^4 \left (-4-4 x^2+x^4\right )} \, dx=\int { \frac {{\left (x^{3} - 2 \, x\right )}^{\frac {1}{3}} {\left (x^{2} + 4\right )}}{{\left (x^{4} - 4 \, x^{2} - 4\right )} x^{4}} \,d x } \]
3/1120*(36*x^7 - 36*x^5 + 5*(3*x^5 + 2*x^3 - 16*x)*x^2 - 8*x^3 - 128*x)*(x ^2 - 2)^(1/3)/(x^(23/3) - 4*x^(17/3) - 4*x^(11/3)) + integrate(3/70*(36*x^ 6 - 6*x^4 + (18*x^6 + 27*x^4 + 26*x^2 - 304)*x^2 + 12*x^2 - 288)*(x^2 - 2) ^(1/3)/(x^(35/3) - 8*x^(29/3) + 8*x^(23/3) + 32*x^(17/3) + 16*x^(11/3)), x )
Not integrable
Time = 0.39 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.28 \[ \int \frac {\left (4+x^2\right ) \sqrt [3]{-2 x+x^3}}{x^4 \left (-4-4 x^2+x^4\right )} \, dx=\int { \frac {{\left (x^{3} - 2 \, x\right )}^{\frac {1}{3}} {\left (x^{2} + 4\right )}}{{\left (x^{4} - 4 \, x^{2} - 4\right )} x^{4}} \,d x } \]
Not integrable
Time = 6.00 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.30 \[ \int \frac {\left (4+x^2\right ) \sqrt [3]{-2 x+x^3}}{x^4 \left (-4-4 x^2+x^4\right )} \, dx=\int -\frac {\left (x^2+4\right )\,{\left (x^3-2\,x\right )}^{1/3}}{x^4\,\left (-x^4+4\,x^2+4\right )} \,d x \]