3.4.6 \(\int \sqrt [3]{x (1-x^2)} \, dx\) [306]

Optimal. Leaf size=93 \[ \frac {1}{2} x \sqrt [3]{x \left (1-x^2\right )}+\frac {\tan ^{-1}\left (\frac {2 x-\sqrt [3]{x \left (1-x^2\right )}}{\sqrt {3} \sqrt [3]{x \left (1-x^2\right )}}\right )}{2 \sqrt {3}}+\frac {\log (x)}{12}-\frac {1}{4} \log \left (x+\sqrt [3]{x \left (1-x^2\right )}\right ) \]

[Out]

1/2*x*(x*(-x^2+1))^(1/3)+1/12*ln(x)-1/4*ln(x+(x*(-x^2+1))^(1/3))+1/6*arctan(1/3*(2*x-(x*(-x^2+1))^(1/3))/(x*(-
x^2+1))^(1/3)*3^(1/2))*3^(1/2)

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Rubi [A]
time = 0.05, antiderivative size = 129, normalized size of antiderivative = 1.39, number of steps used = 6, number of rules used = 6, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.462, Rules used = {2004, 2029, 2057, 335, 281, 337} \begin {gather*} -\frac {\left (1-x^2\right )^{2/3} x^{2/3} \text {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 {1}{2} \sqrt [3]{x-x^3} x-\frac {\left (1-x^2\right )^{2/3} x^{2/3} \log \left (x^{2/3}+\sqrt [3]{1-x^2}\right )}{4 \left (x-x^3\right )^{2/3}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(x*(1 - x^2))^(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 2004

Int[(u_)^(p_), x_Symbol] :> Int[ExpandToSum[u, x]^p, x] /; FreeQ[p, x] && GeneralizedBinomialQ[u, x] &&  !Gene
ralizedBinomialMatchQ[u, 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*} \int \sqrt [3]{x \left (1-x^2\right )} \, dx &=\int \sqrt [3]{x-x^3} \, dx\\ &=\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 {\left (x^{2/3} \left (1-x^2\right )^{2/3}\right ) \text {Subst}\left (\int \frac {x}{1+x^3} \, dx,x,\frac {x^{2/3}}{\sqrt [3]{1-x^2}}\right )}{2 \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 {1}{1+x} \, dx,x,\frac {x^{2/3}}{\sqrt [3]{1-x^2}}\right )}{6 \left (x-x^3\right )^{2/3}}+\frac {\left (x^{2/3} \left (1-x^2\right )^{2/3}\right ) \text {Subst}\left (\int \frac {1+x}{1-x+x^2} \, dx,x,\frac {x^{2/3}}{\sqrt [3]{1-x^2}}\right )}{6 \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} \log \left (1+\frac {x^{2/3}}{\sqrt [3]{1-x^2}}\right )}{6 \left (x-x^3\right )^{2/3}}+\frac {\left (x^{2/3} \left (1-x^2\right )^{2/3}\right ) \text {Subst}\left (\int \frac {-1+2 x}{1-x+x^2} \, dx,x,\frac {x^{2/3}}{\sqrt [3]{1-x^2}}\right )}{12 \left (x-x^3\right )^{2/3}}+\frac {\left (x^{2/3} \left (1-x^2\right )^{2/3}\right ) \text {Subst}\left (\int \frac {1}{1-x+x^2} \, dx,x,\frac {x^{2/3}}{\sqrt [3]{1-x^2}}\right )}{4 \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} \log \left (1+\frac {x^{4/3}}{\left (1-x^2\right )^{2/3}}-\frac {x^{2/3}}{\sqrt [3]{1-x^2}}\right )}{12 \left (x-x^3\right )^{2/3}}-\frac {x^{2/3} \left (1-x^2\right )^{2/3} \log \left (1+\frac {x^{2/3}}{\sqrt [3]{1-x^2}}\right )}{6 \left (x-x^3\right )^{2/3}}-\frac {\left (x^{2/3} \left (1-x^2\right )^{2/3}\right ) \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,-1+\frac {2 x^{2/3}}{\sqrt [3]{1-x^2}}\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} \tan ^{-1}\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 (1+\frac {x^{4/3}}{\left (1-x^2\right )^{2/3}}-\frac {x^{2/3}}{\sqrt [3]{1-x^2}}\right )}{12 \left (x-x^3\right )^{2/3}}-\frac {x^{2/3} \left (1-x^2\right )^{2/3} \log \left (1+\frac {x^{2/3}}{\sqrt [3]{1-x^2}}\right )}{6 \left (x-x^3\right )^{2/3}}\\ \end {align*}

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Mathematica [A]
time = 0.48, size = 137, normalized size = 1.47 \begin {gather*} \frac {\sqrt [3]{x-x^3} \left (6 x^{4/3} \sqrt [3]{-1+x^2}+2 \sqrt {3} \tan ^{-1}\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 \sqrt [3]{x} \sqrt [3]{-1+x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(x*(1 - x^2))^(1/3),x]

[Out]

((x - x^3)^(1/3)*(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^
(1/3)*(-1 + x^2)^(1/3))

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Maple [C] Result contains higher order function than in optimal. Order 5 vs. order 3.
time = 2.05, size = 15, normalized size = 0.16

method	result	size
meijerg	\(\frac {3 x^{\frac {4}{3}} \hypergeom \left (\left [-\frac {1}{3}, \frac {2}{3}\right ], \left [\frac {5}{3}\right ], x^{2}\right )}{4}\)	\(15\)
trager	\(\frac {x \left (-x^{3}+x \right )^{\frac {1}{3}}}{2}+\frac {\RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) \ln \left (1395 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )^{2} x^{2}-6768 \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}} \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) x -7233 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) x^{2}+7611 \left (-x^{3}+x \right )^{\frac {2}{3}}-7611 x \left (-x^{3}+x \right )^{\frac {1}{3}}-5580 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )^{2}+7766 x^{2}+11727 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )-4942\right )}{2}+\frac {\ln \left (1395 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )^{2} x^{2}+6768 \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}} \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) x +6303 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) x^{2}+5355 \left (-x^{3}+x \right )^{\frac {2}{3}}-5355 x \left (-x^{3}+x \right )^{\frac {1}{3}}-5580 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )^{2}+5510 x^{2}-8007 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )-1653\right )}{6}-\frac {\ln \left (1395 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )^{2} x^{2}+6768 \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}} \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) x +6303 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) x^{2}+5355 \left (-x^{3}+x \right )^{\frac {2}{3}}-5355 x \left (-x^{3}+x \right )^{\frac {1}{3}}-5580 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )^{2}+5510 x^{2}-8007 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )-1653\right ) \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 x^{4} \RootOf \left (\textit {\_Z}^{2}+6 \textit {\_Z} +36\right )^{2}+1956 \RootOf \left (\textit {\_Z}^{2}+6 \textit {\_Z} +36\right ) x^{4}-175 x^{2} \RootOf \left (\textit {\_Z}^{2}+6 \textit {\_Z} +36\right )^{2}+4104 \RootOf \left (\textit {\_Z}^{2}+6 \textit {\_Z} +36\right ) \left (x^{6}-2 x^{4}+x^{2}\right )^{\frac {1}{3}} x^{2}-23364 x^{4}-5850 \RootOf \left (\textit {\_Z}^{2}+6 \textit {\_Z} +36\right ) \left (x^{6}-2 x^{4}+x^{2}\right )^{\frac {2}{3}}-2010 \RootOf \left (\textit {\_Z}^{2}+6 \textit {\_Z} +36\right ) x^{2}+35100 \left (x^{6}-2 x^{4}+x^{2}\right )^{\frac {1}{3}} x^{2}+140 \RootOf \left (\textit {\_Z}^{2}+6 \textit {\_Z} +36\right )^{2}-4104 \RootOf \left (\textit {\_Z}^{2}+6 \textit {\_Z} +36\right ) \left (x^{6}-2 x^{4}+x^{2}\right )^{\frac {1}{3}}-10476 \left (x^{6}-2 x^{4}+x^{2}\right )^{\frac {2}{3}}+38232 x^{2}+54 \RootOf \left (\textit {\_Z}^{2}+6 \textit {\_Z} +36\right )-35100 \left (x^{6}-2 x^{4}+x^{2}\right )^{\frac {1}{3}}-14868}{\left (1+x \right ) \left (-1+x \right )}\right )}{6}+\frac {\RootOf \left (\textit {\_Z}^{2}+6 \textit {\_Z} +36\right ) \ln \left (\frac {59 x^{4} \RootOf \left (\textit {\_Z}^{2}+6 \textit {\_Z} +36\right )^{2}-3750 \RootOf \left (\textit {\_Z}^{2}+6 \textit {\_Z} +36\right ) x^{4}-295 x^{2} \RootOf \left (\textit {\_Z}^{2}+6 \textit {\_Z} +36\right )^{2}-1746 \RootOf \left (\textit {\_Z}^{2}+6 \textit {\_Z} +36\right ) \left (x^{6}-2 x^{4}+x^{2}\right )^{\frac {1}{3}} x^{2}+12600 x^{4}+5850 \RootOf \left (\textit {\_Z}^{2}+6 \textit {\_Z} +36\right ) \left (x^{6}-2 x^{4}+x^{2}\right )^{\frac {2}{3}}+5652 \RootOf \left (\textit {\_Z}^{2}+6 \textit {\_Z} +36\right ) x^{2}-35100 \left (x^{6}-2 x^{4}+x^{2}\right )^{\frac {1}{3}} x^{2}+236 \RootOf \left (\textit {\_Z}^{2}+6 \textit {\_Z} +36\right )^{2}+1746 \RootOf \left (\textit {\_Z}^{2}+6 \textit {\_Z} +36\right ) \left (x^{6}-2 x^{4}+x^{2}\right )^{\frac {1}{3}}+24624 \left (x^{6}-2 x^{4}+x^{2}\right )^{\frac {2}{3}}-16380 x^{2}-1902 \RootOf \left (\textit {\_Z}^{2}+6 \textit {\_Z} +36\right )+35100 \left (x^{6}-2 x^{4}+x^{2}\right )^{\frac {1}{3}}+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 )}\)	\(537\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x*(-x^2+1))^(1/3),x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x*(-x^2+1))^(1/3),x, algorithm="maxima")

[Out]

integrate((-(x^2 - 1)*x)^(1/3), x)

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Fricas [A]
time = 1.34, size = 99, normalized size = 1.06 \begin {gather*} -\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 ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x*(-x^2+1))^(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) + 1052430
5234*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)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \sqrt [3]{x \left (1 - x^{2}\right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x*(-x**2+1))**(1/3),x)

[Out]

Integral((x*(1 - x**2))**(1/3), x)

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Giac [A]
time = 0.76, size = 69, normalized size = 0.74 \begin {gather*} \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 ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x*(-x^2+1))^(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))

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Mupad [B]
time = 0.37, size = 29, normalized size = 0.31 \begin {gather*} \frac {3\,x\,{\left (x-x^3\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}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-x*(x^2 - 1))^(1/3),x)

[Out]

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

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