\(\int \sqrt [3]{-x+x^3} \, dx\) [1472]

Optimal result

Integrand size = 11, antiderivative size = 104 \[ \int \sqrt [3]{-x+x^3} \, dx=\frac {1}{2} x \sqrt [3]{-x+x^3}+\frac {\arctan \left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{-x+x^3}}\right )}{2 \sqrt {3}}+\frac {1}{6} \log \left (-x+\sqrt [3]{-x+x^3}\right )-\frac {1}{12} \log \left (x^2+x \sqrt [3]{-x+x^3}+\left (-x+x^3\right )^{2/3}\right ) \]

[Out]

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))

Rubi [A] (verified)

Time = 0.04 (sec) , antiderivative size = 123, normalized size of antiderivative = 1.18, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.455, Rules used = {2029, 2057, 335, 281, 337} \[ \int \sqrt [3]{-x+x^3} \, dx=\frac {\left (x^2-1\right )^{2/3} x^{2/3} \arctan \left (\frac {\frac {2 x^{2/3}}{\sqrt [3]{x^2-1}}+1}{\sqrt {3}}\right )}{2 \sqrt {3} \left (x^3-x\right )^{2/3}}+\frac {1}{2} \sqrt [3]{x^3-x} x+\frac {\left (x^2-1\right )^{2/3} x^{2/3} \log \left (x^{2/3}-\sqrt [3]{x^2-1}\right )}{4 \left (x^3-x\right )^{2/3}} \]

[In]

Int[(-x + x^3)^(1/3),x]

[Out]

(x*(-x + x^3)^(1/3))/2 + (x^(2/3)*(-1 + x^2)^(2/3)*ArcTan[(1 + (2*x^(2/3))/(-1 + x^2)^(1/3))/Sqrt[3]])/(2*Sqrt
[3]*(-x + x^3)^(2/3)) + (x^(2/3)*(-1 + x^2)^(2/3)*Log[x^(2/3) - (-1 + x^2)^(1/3)])/(4*(-x + x^3)^(2/3))

Rule 281

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = GCD[m + 1, n]}, Dist[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]

Rule 335

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = Denominator[m]}, Dist[k/c, Subst[I
nt[x^(k*(m + 1) - 1)*(a + b*(x^(k*n)/c^n))^p, x], x, (c*x)^(1/k)], x]] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0]
&& FractionQ[m] && IntBinomialQ[a, b, c, n, m, p, x]

Rule 337

Int[(x_)/((a_) + (b_.)*(x_)^3)^(2/3), x_Symbol] :> With[{q = Rt[b, 3]}, Simp[-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]

Rule 2029

Int[((a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_.))^(p_), x_Symbol] :> Simp[x*((a*x^j + b*x^n)^p/(n*p + 1)), x] + Dist[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]

Rule 2057

Int[((c_.)*(x_))^(m_.)*((a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_.))^(p_), x_Symbol] :> Dist[c^IntPart[m]*(c*x)^FracPa
rt[m]*((a*x^j + b*x^n)^FracPart[p]/(x^(FracPart[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] &&  !IntegerQ[p] && NeQ[n, j] && PosQ[n
- j]

Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} x \sqrt [3]{-x+x^3}-\frac {1}{3} \int \frac {x}{\left (-x+x^3\right )^{2/3}} \, dx \\ & = \frac {1}{2} x \sqrt [3]{-x+x^3}-\frac {\left (x^{2/3} \left (-1+x^2\right )^{2/3}\right ) \int \frac {\sqrt [3]{x}}{\left (-1+x^2\right )^{2/3}} \, dx}{3 \left (-x+x^3\right )^{2/3}} \\ & = \frac {1}{2} x \sqrt [3]{-x+x^3}-\frac {\left (x^{2/3} \left (-1+x^2\right )^{2/3}\right ) \text {Subst}\left (\int \frac {x^3}{\left (-1+x^6\right )^{2/3}} \, dx,x,\sqrt [3]{x}\right )}{\left (-x+x^3\right )^{2/3}} \\ & = \frac {1}{2} x \sqrt [3]{-x+x^3}-\frac {\left (x^{2/3} \left (-1+x^2\right )^{2/3}\right ) \text {Subst}\left (\int \frac {x}{\left (-1+x^3\right )^{2/3}} \, dx,x,x^{2/3}\right )}{2 \left (-x+x^3\right )^{2/3}} \\ & = \frac {1}{2} x \sqrt [3]{-x+x^3}+\frac {x^{2/3} \left (-1+x^2\right )^{2/3} \arctan \left (\frac {1+\frac {2 x^{2/3}}{\sqrt [3]{-1+x^2}}}{\sqrt {3}}\right )}{2 \sqrt {3} \left (-x+x^3\right )^{2/3}}+\frac {x^{2/3} \left (-1+x^2\right )^{2/3} \log \left (x^{2/3}-\sqrt [3]{-1+x^2}\right )}{4 \left (-x+x^3\right )^{2/3}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.52 (sec) , antiderivative size = 137, normalized size of antiderivative = 1.32 \[ \int \sqrt [3]{-x+x^3} \, dx=\frac {x^{2/3} \left (-1+x^2\right )^{2/3} \left (6 x^{4/3} \sqrt [3]{-1+x^2}+2 \sqrt {3} \arctan \left (\frac {\sqrt {3} x^{2/3}}{x^{2/3}+2 \sqrt [3]{-1+x^2}}\right )+2 \log \left (-x^{2/3}+\sqrt [3]{-1+x^2}\right )-\log \left (x^{4/3}+x^{2/3} \sqrt [3]{-1+x^2}+\left (-1+x^2\right )^{2/3}\right )\right )}{12 \left (x \left (-1+x^2\right )\right )^{2/3}} \]

[In]

Integrate[(-x + x^3)^(1/3),x]

[Out]

(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))

Maple [C] (warning: unable to verify)

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 )}\)	\(536\)

[In]

int((x^3-x)^(1/3),x,method=_RETURNVERBOSE)

[Out]

3/4*signum(x^2-1)^(1/3)/(-signum(x^2-1))^(1/3)*x^(4/3)*hypergeom([-1/3,2/3],[5/3],x^2)

Fricas [A] (verification not implemented)

none

Time = 0.47 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.95 \[ \int \sqrt [3]{-x+x^3} \, dx=\frac {1}{6} \, \sqrt {3} \arctan \left (-\frac {44032959556 \, \sqrt {3} {\left (x^{3} - x\right )}^{\frac {1}{3}} x + \sqrt {3} {\left (16754327161 \, x^{2} - 2707204793\right )} - 10524305234 \, \sqrt {3} {\left (x^{3} - x\right )}^{\frac {2}{3}}}{81835897185 \, x^{2} - 1102302937}\right ) + \frac {1}{2} \, {\left (x^{3} - x\right )}^{\frac {1}{3}} x + \frac {1}{12} \, \log \left (-3 \, {\left (x^{3} - x\right )}^{\frac {1}{3}} x + 3 \, {\left (x^{3} - x\right )}^{\frac {2}{3}} + 1\right ) \]

[In]

integrate((x^3-x)^(1/3),x, algorithm="fricas")

[Out]

1/6*sqrt(3)*arctan(-(44032959556*sqrt(3)*(x^3 - x)^(1/3)*x + sqrt(3)*(16754327161*x^2 - 2707204793) - 10524305
234*sqrt(3)*(x^3 - x)^(2/3))/(81835897185*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)

Sympy [F]

\[ \int \sqrt [3]{-x+x^3} \, dx=\int \sqrt [3]{x^{3} - x}\, dx \]

[In]

integrate((x**3-x)**(1/3),x)

[Out]

Integral((x**3 - x)**(1/3), x)

Maxima [F]

\[ \int \sqrt [3]{-x+x^3} \, dx=\int { {\left (x^{3} - x\right )}^{\frac {1}{3}} \,d x } \]

[In]

integrate((x^3-x)^(1/3),x, algorithm="maxima")

[Out]

integrate((x^3 - x)^(1/3), x)

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.74 \[ \int \sqrt [3]{-x+x^3} \, dx=\frac {1}{2} \, x^{2} {\left (-\frac {1}{x^{2}} + 1\right )}^{\frac {1}{3}} - \frac {1}{6} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, {\left (-\frac {1}{x^{2}} + 1\right )}^{\frac {1}{3}} + 1\right )}\right ) - \frac {1}{12} \, \log \left ({\left (-\frac {1}{x^{2}} + 1\right )}^{\frac {2}{3}} + {\left (-\frac {1}{x^{2}} + 1\right )}^{\frac {1}{3}} + 1\right ) + \frac {1}{6} \, \log \left ({\left | {\left (-\frac {1}{x^{2}} + 1\right )}^{\frac {1}{3}} - 1 \right |}\right ) \]

[In]

integrate((x^3-x)^(1/3),x, algorithm="giac")

[Out]

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))

Mupad [B] (verification not implemented)

Time = 6.23 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.28 \[ \int \sqrt [3]{-x+x^3} \, dx=\frac {3\,x\,{\left (x^3-x\right )}^{1/3}\,{{}}_2{\mathrm {F}}_1\left (-\frac {1}{3},\frac {2}{3};\ \frac {5}{3};\ x^2\right )}{4\,{\left (1-x^2\right )}^{1/3}} \]

[In]

int((x^3 - x)^(1/3),x)

[Out]

(3*x*(x^3 - x)^(1/3)*hypergeom([-1/3, 2/3], 5/3, x^2))/(4*(1 - x^2)^(1/3))