Integrand size = 11, antiderivative size = 104 \[ \int \sqrt [3]{-x+x^3} \, dx=\frac {1}{2} x \sqrt [3]{-x+x^3}+\frac {\arctan \left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{-x+x^3}}\right )}{2 \sqrt {3}}+\frac {1}{6} \log \left (-x+\sqrt [3]{-x+x^3}\right )-\frac {1}{12} \log \left (x^2+x \sqrt [3]{-x+x^3}+\left (-x+x^3\right )^{2/3}\right ) \]
1/2*x*(x^3-x)^(1/3)+1/6*arctan(3^(1/2)*x/(x+2*(x^3-x)^(1/3)))*3^(1/2)+1/6* ln(-x+(x^3-x)^(1/3))-1/12*ln(x^2+x*(x^3-x)^(1/3)+(x^3-x)^(2/3))
Time = 0.52 (sec) , antiderivative size = 137, normalized size of antiderivative = 1.32 \[ \int \sqrt [3]{-x+x^3} \, dx=\frac {x^{2/3} \left (-1+x^2\right )^{2/3} \left (6 x^{4/3} \sqrt [3]{-1+x^2}+2 \sqrt {3} \arctan \left (\frac {\sqrt {3} x^{2/3}}{x^{2/3}+2 \sqrt [3]{-1+x^2}}\right )+2 \log \left (-x^{2/3}+\sqrt [3]{-1+x^2}\right )-\log \left (x^{4/3}+x^{2/3} \sqrt [3]{-1+x^2}+\left (-1+x^2\right )^{2/3}\right )\right )}{12 \left (x \left (-1+x^2\right )\right )^{2/3}} \]
(x^(2/3)*(-1 + x^2)^(2/3)*(6*x^(4/3)*(-1 + x^2)^(1/3) + 2*Sqrt[3]*ArcTan[( Sqrt[3]*x^(2/3))/(x^(2/3) + 2*(-1 + x^2)^(1/3))] + 2*Log[-x^(2/3) + (-1 + x^2)^(1/3)] - Log[x^(4/3) + x^(2/3)*(-1 + x^2)^(1/3) + (-1 + x^2)^(2/3)])) /(12*(x*(-1 + x^2))^(2/3))
Time = 0.23 (sec) , antiderivative size = 97, normalized size of antiderivative = 0.93, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.455, Rules used = {1910, 1938, 266, 807, 853}
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.
\(\displaystyle \int \sqrt [3]{x^3-x} \, dx\) |
\(\Big \downarrow \) 1910 |
\(\displaystyle \frac {1}{2} x \sqrt [3]{x^3-x}-\frac {1}{3} \int \frac {x}{\left (x^3-x\right )^{2/3}}dx\) |
\(\Big \downarrow \) 1938 |
\(\displaystyle \frac {1}{2} x \sqrt [3]{x^3-x}-\frac {x^{2/3} \left (x^2-1\right )^{2/3} \int \frac {\sqrt [3]{x}}{\left (x^2-1\right )^{2/3}}dx}{3 \left (x^3-x\right )^{2/3}}\) |
\(\Big \downarrow \) 266 |
\(\displaystyle \frac {1}{2} x \sqrt [3]{x^3-x}-\frac {x^{2/3} \left (x^2-1\right )^{2/3} \int \frac {x}{\left (x^2-1\right )^{2/3}}d\sqrt [3]{x}}{\left (x^3-x\right )^{2/3}}\) |
\(\Big \downarrow \) 807 |
\(\displaystyle \frac {1}{2} x \sqrt [3]{x^3-x}-\frac {x^{2/3} \left (x^2-1\right )^{2/3} \int \frac {x^{2/3}}{(x-1)^{2/3}}dx^{2/3}}{2 \left (x^3-x\right )^{2/3}}\) |
\(\Big \downarrow \) 853 |
\(\displaystyle \frac {1}{2} x \sqrt [3]{x^3-x}-\frac {x^{2/3} \left (x^2-1\right )^{2/3} \left (-\frac {\arctan \left (\frac {\frac {2 x^{2/3}}{\sqrt [3]{x-1}}+1}{\sqrt {3}}\right )}{\sqrt {3}}-\frac {1}{2} \log \left (x^{2/3}-\sqrt [3]{x-1}\right )\right )}{2 \left (x^3-x\right )^{2/3}}\) |
(x*(-x + x^3)^(1/3))/2 - (x^(2/3)*(-1 + x^2)^(2/3)*(-(ArcTan[(1 + (2*x^(2/ 3))/(-1 + x)^(1/3))/Sqrt[3]]/Sqrt[3]) - Log[-(-1 + x)^(1/3) + x^(2/3)]/2)) /(2*(-x + x^3)^(2/3))
3.15.72.3.1 Defintions of rubi rules used
Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{k = De nominator[m]}, Simp[k/c Subst[Int[x^(k*(m + 1) - 1)*(a + b*(x^(2*k)/c^2)) ^p, x], x, (c*x)^(1/k)], x]] /; FreeQ[{a, b, c, p}, x] && FractionQ[m] && I ntBinomialQ[a, b, c, 2, m, p, x]
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = GCD[m + 1, n]}, Simp[1/k Subst[Int[x^((m + 1)/k - 1)*(a + b*x^(n/k))^p, x], x, x^k], x] /; k != 1] /; FreeQ[{a, b, p}, x] && IGtQ[n, 0] && IntegerQ[m]
Int[(x_)/((a_) + (b_.)*(x_)^3)^(2/3), x_Symbol] :> With[{q = Rt[b, 3]}, Sim p[-ArcTan[(1 + 2*q*(x/(a + b*x^3)^(1/3)))/Sqrt[3]]/(Sqrt[3]*q^2), x] - Simp [Log[q*x - (a + b*x^3)^(1/3)]/(2*q^2), x]] /; FreeQ[{a, b}, x]
Int[((a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_.))^(p_), x_Symbol] :> Simp[x*((a*x^j + b*x^n)^p/(n*p + 1)), x] + Simp[a*(n - j)*(p/(n*p + 1)) Int[x^j*(a*x^j + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b}, x] && !IntegerQ[p] && LtQ[0, j, n] && GtQ[p, 0] && NeQ[n*p + 1, 0]
Int[((c_.)*(x_))^(m_.)*((a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_.))^(p_), x_Symbol ] :> Simp[c^IntPart[m]*(c*x)^FracPart[m]*((a*x^j + b*x^n)^FracPart[p]/(x^(F racPart[m] + j*FracPart[p])*(a + b*x^(n - j))^FracPart[p])) Int[x^(m + j* p)*(a + b*x^(n - j))^p, x], x] /; FreeQ[{a, b, c, j, m, n, p}, x] && !Inte gerQ[p] && NeQ[n, j] && PosQ[n - j]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 2.22 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.32
method | result | size |
meijerg | \(\frac {3 \operatorname {signum}\left (x^{2}-1\right )^{\frac {1}{3}} x^{\frac {4}{3}} \operatorname {hypergeom}\left (\left [-\frac {1}{3}, \frac {2}{3}\right ], \left [\frac {5}{3}\right ], x^{2}\right )}{4 {\left (-\operatorname {signum}\left (x^{2}-1\right )\right )}^{\frac {1}{3}}}\) | \(33\) |
pseudoelliptic | \(\frac {x \left (-6 x \left (x^{3}-x \right )^{\frac {1}{3}}+2 \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (x +2 \left (x^{3}-x \right )^{\frac {1}{3}}\right )}{3 x}\right )-2 \ln \left (\frac {-x +\left (x^{3}-x \right )^{\frac {1}{3}}}{x}\right )+\ln \left (\frac {x^{2}+x \left (x^{3}-x \right )^{\frac {1}{3}}+\left (x^{3}-x \right )^{\frac {2}{3}}}{x^{2}}\right )\right )}{12 \left (x^{2}+x \left (x^{3}-x \right )^{\frac {1}{3}}+\left (x^{3}-x \right )^{\frac {2}{3}}\right ) \left (-x +\left (x^{3}-x \right )^{\frac {1}{3}}\right )}\) | \(134\) |
trager | \(\frac {x \left (x^{3}-x \right )^{\frac {1}{3}}}{2}+\frac {\operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \ln \left (1395 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2} x^{2}+6768 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (x^{3}-x \right )^{\frac {2}{3}}+6768 \left (x^{3}-x \right )^{\frac {1}{3}} \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x +7233 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{2}-5580 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2}+7611 \left (x^{3}-x \right )^{\frac {2}{3}}+7611 x \left (x^{3}-x \right )^{\frac {1}{3}}+7766 x^{2}-11727 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )-4942\right )}{2}-\frac {\ln \left (1395 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2} x^{2}-6768 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (x^{3}-x \right )^{\frac {2}{3}}-6768 \left (x^{3}-x \right )^{\frac {1}{3}} \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x -6303 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{2}-5580 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2}+5355 \left (x^{3}-x \right )^{\frac {2}{3}}+5355 x \left (x^{3}-x \right )^{\frac {1}{3}}+5510 x^{2}+8007 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )-1653\right )}{6}-\frac {\ln \left (1395 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2} x^{2}-6768 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (x^{3}-x \right )^{\frac {2}{3}}-6768 \left (x^{3}-x \right )^{\frac {1}{3}} \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x -6303 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{2}-5580 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2}+5355 \left (x^{3}-x \right )^{\frac {2}{3}}+5355 x \left (x^{3}-x \right )^{\frac {1}{3}}+5510 x^{2}+8007 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )-1653\right ) \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )}{2}\) | \(456\) |
risch | \(\frac {x {\left (x \left (x^{2}-1\right )\right )}^{\frac {1}{3}}}{2}+\frac {\left (\frac {\ln \left (-\frac {-35 \operatorname {RootOf}\left (\textit {\_Z}^{2}+6 \textit {\_Z} +36\right )^{2} x^{4}-1956 \operatorname {RootOf}\left (\textit {\_Z}^{2}+6 \textit {\_Z} +36\right ) x^{4}-4104 \left (x^{6}-2 x^{4}+x^{2}\right )^{\frac {1}{3}} \operatorname {RootOf}\left (\textit {\_Z}^{2}+6 \textit {\_Z} +36\right ) x^{2}+175 \operatorname {RootOf}\left (\textit {\_Z}^{2}+6 \textit {\_Z} +36\right )^{2} x^{2}+23364 x^{4}+5850 \operatorname {RootOf}\left (\textit {\_Z}^{2}+6 \textit {\_Z} +36\right ) \left (x^{6}-2 x^{4}+x^{2}\right )^{\frac {2}{3}}-35100 \left (x^{6}-2 x^{4}+x^{2}\right )^{\frac {1}{3}} x^{2}+2010 \operatorname {RootOf}\left (\textit {\_Z}^{2}+6 \textit {\_Z} +36\right ) x^{2}+10476 \left (x^{6}-2 x^{4}+x^{2}\right )^{\frac {2}{3}}+4104 \operatorname {RootOf}\left (\textit {\_Z}^{2}+6 \textit {\_Z} +36\right ) \left (x^{6}-2 x^{4}+x^{2}\right )^{\frac {1}{3}}-140 \operatorname {RootOf}\left (\textit {\_Z}^{2}+6 \textit {\_Z} +36\right )^{2}-38232 x^{2}+35100 \left (x^{6}-2 x^{4}+x^{2}\right )^{\frac {1}{3}}-54 \operatorname {RootOf}\left (\textit {\_Z}^{2}+6 \textit {\_Z} +36\right )+14868}{\left (-1+x \right ) \left (1+x \right )}\right )}{6}+\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+6 \textit {\_Z} +36\right ) \ln \left (\frac {59 \operatorname {RootOf}\left (\textit {\_Z}^{2}+6 \textit {\_Z} +36\right )^{2} x^{4}-3750 \operatorname {RootOf}\left (\textit {\_Z}^{2}+6 \textit {\_Z} +36\right ) x^{4}-1746 \left (x^{6}-2 x^{4}+x^{2}\right )^{\frac {1}{3}} \operatorname {RootOf}\left (\textit {\_Z}^{2}+6 \textit {\_Z} +36\right ) x^{2}-295 \operatorname {RootOf}\left (\textit {\_Z}^{2}+6 \textit {\_Z} +36\right )^{2} x^{2}+12600 x^{4}+5850 \operatorname {RootOf}\left (\textit {\_Z}^{2}+6 \textit {\_Z} +36\right ) \left (x^{6}-2 x^{4}+x^{2}\right )^{\frac {2}{3}}-35100 \left (x^{6}-2 x^{4}+x^{2}\right )^{\frac {1}{3}} x^{2}+5652 \operatorname {RootOf}\left (\textit {\_Z}^{2}+6 \textit {\_Z} +36\right ) x^{2}+24624 \left (x^{6}-2 x^{4}+x^{2}\right )^{\frac {2}{3}}+1746 \operatorname {RootOf}\left (\textit {\_Z}^{2}+6 \textit {\_Z} +36\right ) \left (x^{6}-2 x^{4}+x^{2}\right )^{\frac {1}{3}}+236 \operatorname {RootOf}\left (\textit {\_Z}^{2}+6 \textit {\_Z} +36\right )^{2}-16380 x^{2}+35100 \left (x^{6}-2 x^{4}+x^{2}\right )^{\frac {1}{3}}-1902 \operatorname {RootOf}\left (\textit {\_Z}^{2}+6 \textit {\_Z} +36\right )+3780}{\left (-1+x \right ) \left (1+x \right )}\right )}{36}\right ) {\left (x \left (x^{2}-1\right )\right )}^{\frac {1}{3}} \left (x^{2} \left (x^{2}-1\right )^{2}\right )^{\frac {1}{3}}}{x \left (x^{2}-1\right )}\) | \(536\) |
Time = 0.47 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.95 \[ \int \sqrt [3]{-x+x^3} \, dx=\frac {1}{6} \, \sqrt {3} \arctan \left (-\frac {44032959556 \, \sqrt {3} {\left (x^{3} - x\right )}^{\frac {1}{3}} x + \sqrt {3} {\left (16754327161 \, x^{2} - 2707204793\right )} - 10524305234 \, \sqrt {3} {\left (x^{3} - x\right )}^{\frac {2}{3}}}{81835897185 \, x^{2} - 1102302937}\right ) + \frac {1}{2} \, {\left (x^{3} - x\right )}^{\frac {1}{3}} x + \frac {1}{12} \, \log \left (-3 \, {\left (x^{3} - x\right )}^{\frac {1}{3}} x + 3 \, {\left (x^{3} - x\right )}^{\frac {2}{3}} + 1\right ) \]
1/6*sqrt(3)*arctan(-(44032959556*sqrt(3)*(x^3 - x)^(1/3)*x + sqrt(3)*(1675 4327161*x^2 - 2707204793) - 10524305234*sqrt(3)*(x^3 - x)^(2/3))/(81835897 185*x^2 - 1102302937)) + 1/2*(x^3 - x)^(1/3)*x + 1/12*log(-3*(x^3 - x)^(1/ 3)*x + 3*(x^3 - x)^(2/3) + 1)
\[ \int \sqrt [3]{-x+x^3} \, dx=\int \sqrt [3]{x^{3} - x}\, dx \]
\[ \int \sqrt [3]{-x+x^3} \, dx=\int { {\left (x^{3} - x\right )}^{\frac {1}{3}} \,d x } \]
Time = 0.28 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.74 \[ \int \sqrt [3]{-x+x^3} \, dx=\frac {1}{2} \, x^{2} {\left (-\frac {1}{x^{2}} + 1\right )}^{\frac {1}{3}} - \frac {1}{6} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, {\left (-\frac {1}{x^{2}} + 1\right )}^{\frac {1}{3}} + 1\right )}\right ) - \frac {1}{12} \, \log \left ({\left (-\frac {1}{x^{2}} + 1\right )}^{\frac {2}{3}} + {\left (-\frac {1}{x^{2}} + 1\right )}^{\frac {1}{3}} + 1\right ) + \frac {1}{6} \, \log \left ({\left | {\left (-\frac {1}{x^{2}} + 1\right )}^{\frac {1}{3}} - 1 \right |}\right ) \]
1/2*x^2*(-1/x^2 + 1)^(1/3) - 1/6*sqrt(3)*arctan(1/3*sqrt(3)*(2*(-1/x^2 + 1 )^(1/3) + 1)) - 1/12*log((-1/x^2 + 1)^(2/3) + (-1/x^2 + 1)^(1/3) + 1) + 1/ 6*log(abs((-1/x^2 + 1)^(1/3) - 1))
Time = 6.23 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.28 \[ \int \sqrt [3]{-x+x^3} \, dx=\frac {3\,x\,{\left (x^3-x\right )}^{1/3}\,{{}}_2{\mathrm {F}}_1\left (-\frac {1}{3},\frac {2}{3};\ \frac {5}{3};\ x^2\right )}{4\,{\left (1-x^2\right )}^{1/3}} \]