\(\int \frac {(-1+x^3) \sqrt [3]{1+x^3}}{x} \, dx\) [1214]

Optimal result

Integrand size = 18, antiderivative size = 89 \[ \int \frac {\left (-1+x^3\right ) \sqrt [3]{1+x^3}}{x} \, dx=\frac {1}{4} \left (-3+x^3\right ) \sqrt [3]{1+x^3}+\frac {\arctan \left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{1+x^3}}{\sqrt {3}}\right )}{\sqrt {3}}-\frac {1}{3} \log \left (-1+\sqrt [3]{1+x^3}\right )+\frac {1}{6} \log \left (1+\sqrt [3]{1+x^3}+\left (1+x^3\right )^{2/3}\right ) \]

[Out]

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

Rubi [A] (verified)

Time = 0.07 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.84, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.389, Rules used = {457, 81, 52, 59, 632, 210, 31} \[ \int \frac {\left (-1+x^3\right ) \sqrt [3]{1+x^3}}{x} \, dx=\frac {\arctan \left (\frac {2 \sqrt [3]{x^3+1}+1}{\sqrt {3}}\right )}{\sqrt {3}}+\frac {1}{4} \left (x^3+1\right )^{4/3}-\sqrt [3]{x^3+1}-\frac {1}{2} \log \left (1-\sqrt [3]{x^3+1}\right )+\frac {\log (x)}{2} \]

[In]

Int[((-1 + x^3)*(1 + x^3)^(1/3))/x,x]

[Out]

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

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rule 52

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(a + b*x)^(m + 1)*((c + d*x)^n/(b*(
m + n + 1))), x] + Dist[n*((b*c - a*d)/(b*(m + n + 1))), Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a
, b, c, d}, x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !IntegerQ[n] || (G
tQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && IntLinearQ[a, b, c, d, m, n, x]

Rule 59

Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(2/3)), x_Symbol] :> With[{q = Rt[(b*c - a*d)/b, 3]}, Simp[-L
og[RemoveContent[a + b*x, x]]/(2*b*q^2), x] + (-Dist[3/(2*b*q), Subst[Int[1/(q^2 + q*x + x^2), x], x, (c + d*x
)^(1/3)], x] - Dist[3/(2*b*q^2), Subst[Int[1/(q - x), x], x, (c + d*x)^(1/3)], x])] /; FreeQ[{a, b, c, d}, x]
&& PosQ[(b*c - a*d)/b]

Rule 81

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[b*(c + d*x)^
(n + 1)*((e + f*x)^(p + 1)/(d*f*(n + p + 2))), x] + Dist[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)))/(
d*f*(n + p + 2)), Int[(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 2,
0]

Rule 210

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^(-1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])
], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rule 457

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> Dist[1/n, Subst[Int
[x^(Simplify[(m + 1)/n] - 1)*(a + b*x)^p*(c + d*x)^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p, q}, x] &&
NeQ[b*c - a*d, 0] && IntegerQ[Simplify[(m + 1)/n]]

Rule 632

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Dist[-2, Subst[Int[1/Simp[b^2 - 4*a*c - x^2, x], x]
, x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

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

Mathematica [A] (verified)

Time = 0.06 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.96 \[ \int \frac {\left (-1+x^3\right ) \sqrt [3]{1+x^3}}{x} \, dx=\frac {1}{12} \left (3 \left (-3+x^3\right ) \sqrt [3]{1+x^3}+4 \sqrt {3} \arctan \left (\frac {1+2 \sqrt [3]{1+x^3}}{\sqrt {3}}\right )-4 \log \left (-1+\sqrt [3]{1+x^3}\right )+2 \log \left (1+\sqrt [3]{1+x^3}+\left (1+x^3\right )^{2/3}\right )\right ) \]

[In]

Integrate[((-1 + x^3)*(1 + x^3)^(1/3))/x,x]

[Out]

(3*(-3 + x^3)*(1 + x^3)^(1/3) + 4*Sqrt[3]*ArcTan[(1 + 2*(1 + x^3)^(1/3))/Sqrt[3]] - 4*Log[-1 + (1 + x^3)^(1/3)
] + 2*Log[1 + (1 + x^3)^(1/3) + (1 + x^3)^(2/3)])/12

Maple [C] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 3.

Time = 2.29 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.73

method	result	size
meijerg	\(\frac {-\Gamma \left (\frac {2}{3}\right ) x^{3} \operatorname {hypergeom}\left (\left [\frac {2}{3}, 1, 1\right ], \left [2, 2\right ], -x^{3}\right )-3 \left (3+\frac {\pi \sqrt {3}}{6}-\frac {3 \ln \left (3\right )}{2}+3 \ln \left (x \right )\right ) \Gamma \left (\frac {2}{3}\right )}{9 \Gamma \left (\frac {2}{3}\right )}+\frac {x^{3} \operatorname {hypergeom}\left (\left [-\frac {1}{3}, 1\right ], \left [2\right ], -x^{3}\right )}{3}\)	\(65\)
pseudoelliptic	\(\frac {x^{3} \left (x^{3}+1\right )^{\frac {1}{3}}}{4}-\frac {3 \left (x^{3}+1\right )^{\frac {1}{3}}}{4}-\frac {\ln \left (-1+\left (x^{3}+1\right )^{\frac {1}{3}}\right )}{3}+\frac {\ln \left (1+\left (x^{3}+1\right )^{\frac {1}{3}}+\left (x^{3}+1\right )^{\frac {2}{3}}\right )}{6}+\frac {\sqrt {3}\, \arctan \left (\frac {\left (2 \left (x^{3}+1\right )^{\frac {1}{3}}+1\right ) \sqrt {3}}{3}\right )}{3}\)	\(76\)
trager	\(\left (\frac {x^{3}}{4}-\frac {3}{4}\right ) \left (x^{3}+1\right )^{\frac {1}{3}}+\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \ln \left (-\frac {4 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2} x^{3}-17 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{3}-15 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{3}+1\right )^{\frac {2}{3}}+15 x^{3}-4 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2}-15 \left (x^{3}+1\right )^{\frac {1}{3}} \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )+24 \left (x^{3}+1\right )^{\frac {2}{3}}-11 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )+24 \left (x^{3}+1\right )^{\frac {1}{3}}+20}{x^{3}}\right )}{3}-\frac {\ln \left (-\frac {4 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2} x^{3}+9 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{3}+15 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{3}+1\right )^{\frac {2}{3}}+2 x^{3}-4 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2}+15 \left (x^{3}+1\right )^{\frac {1}{3}} \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )+9 \left (x^{3}+1\right )^{\frac {2}{3}}+19 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )+9 \left (x^{3}+1\right )^{\frac {1}{3}}+5}{x^{3}}\right ) \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )}{3}+\frac {\ln \left (-\frac {4 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2} x^{3}+9 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{3}+15 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{3}+1\right )^{\frac {2}{3}}+2 x^{3}-4 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2}+15 \left (x^{3}+1\right )^{\frac {1}{3}} \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )+9 \left (x^{3}+1\right )^{\frac {2}{3}}+19 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )+9 \left (x^{3}+1\right )^{\frac {1}{3}}+5}{x^{3}}\right )}{3}\)	\(404\)

[In]

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

[Out]

1/9/GAMMA(2/3)*(-GAMMA(2/3)*x^3*hypergeom([2/3,1,1],[2,2],-x^3)-3*(3+1/6*Pi*3^(1/2)-3/2*ln(3)+3*ln(x))*GAMMA(2
/3))+1/3*x^3*hypergeom([-1/3,1],[2],-x^3)

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.79 \[ \int \frac {\left (-1+x^3\right ) \sqrt [3]{1+x^3}}{x} \, dx=\frac {1}{3} \, \sqrt {3} \arctan \left (\frac {2}{3} \, \sqrt {3} {\left (x^{3} + 1\right )}^{\frac {1}{3}} + \frac {1}{3} \, \sqrt {3}\right ) + \frac {1}{4} \, {\left (x^{3} + 1\right )}^{\frac {1}{3}} {\left (x^{3} - 3\right )} + \frac {1}{6} \, \log \left ({\left (x^{3} + 1\right )}^{\frac {2}{3}} + {\left (x^{3} + 1\right )}^{\frac {1}{3}} + 1\right ) - \frac {1}{3} \, \log \left ({\left (x^{3} + 1\right )}^{\frac {1}{3}} - 1\right ) \]

[In]

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

[Out]

1/3*sqrt(3)*arctan(2/3*sqrt(3)*(x^3 + 1)^(1/3) + 1/3*sqrt(3)) + 1/4*(x^3 + 1)^(1/3)*(x^3 - 3) + 1/6*log((x^3 +
1)^(2/3) + (x^3 + 1)^(1/3) + 1) - 1/3*log((x^3 + 1)^(1/3) - 1)

Sympy [A] (verification not implemented)

Time = 7.25 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.92 \[ \int \frac {\left (-1+x^3\right ) \sqrt [3]{1+x^3}}{x} \, dx=\frac {\left (x^{3} + 1\right )^{\frac {4}{3}}}{4} - \sqrt [3]{x^{3} + 1} - \frac {\log {\left (\sqrt [3]{x^{3} + 1} - 1 \right )}}{3} + \frac {\log {\left (\left (x^{3} + 1\right )^{\frac {2}{3}} + \sqrt [3]{x^{3} + 1} + 1 \right )}}{6} + \frac {\sqrt {3} \operatorname {atan}{\left (\frac {2 \sqrt {3} \left (\sqrt [3]{x^{3} + 1} + \frac {1}{2}\right )}{3} \right )}}{3} \]

[In]

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

[Out]

(x**3 + 1)**(4/3)/4 - (x**3 + 1)**(1/3) - log((x**3 + 1)**(1/3) - 1)/3 + log((x**3 + 1)**(2/3) + (x**3 + 1)**(
1/3) + 1)/6 + sqrt(3)*atan(2*sqrt(3)*((x**3 + 1)**(1/3) + 1/2)/3)/3

Maxima [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.81 \[ \int \frac {\left (-1+x^3\right ) \sqrt [3]{1+x^3}}{x} \, dx=\frac {1}{3} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, {\left (x^{3} + 1\right )}^{\frac {1}{3}} + 1\right )}\right ) + \frac {1}{4} \, {\left (x^{3} + 1\right )}^{\frac {4}{3}} - {\left (x^{3} + 1\right )}^{\frac {1}{3}} + \frac {1}{6} \, \log \left ({\left (x^{3} + 1\right )}^{\frac {2}{3}} + {\left (x^{3} + 1\right )}^{\frac {1}{3}} + 1\right ) - \frac {1}{3} \, \log \left ({\left (x^{3} + 1\right )}^{\frac {1}{3}} - 1\right ) \]

[In]

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

[Out]

1/3*sqrt(3)*arctan(1/3*sqrt(3)*(2*(x^3 + 1)^(1/3) + 1)) + 1/4*(x^3 + 1)^(4/3) - (x^3 + 1)^(1/3) + 1/6*log((x^3
+ 1)^(2/3) + (x^3 + 1)^(1/3) + 1) - 1/3*log((x^3 + 1)^(1/3) - 1)

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.82 \[ \int \frac {\left (-1+x^3\right ) \sqrt [3]{1+x^3}}{x} \, dx=\frac {1}{3} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, {\left (x^{3} + 1\right )}^{\frac {1}{3}} + 1\right )}\right ) + \frac {1}{4} \, {\left (x^{3} + 1\right )}^{\frac {4}{3}} - {\left (x^{3} + 1\right )}^{\frac {1}{3}} + \frac {1}{6} \, \log \left ({\left (x^{3} + 1\right )}^{\frac {2}{3}} + {\left (x^{3} + 1\right )}^{\frac {1}{3}} + 1\right ) - \frac {1}{3} \, \log \left ({\left | {\left (x^{3} + 1\right )}^{\frac {1}{3}} - 1 \right |}\right ) \]

[In]

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

[Out]

1/3*sqrt(3)*arctan(1/3*sqrt(3)*(2*(x^3 + 1)^(1/3) + 1)) + 1/4*(x^3 + 1)^(4/3) - (x^3 + 1)^(1/3) + 1/6*log((x^3
+ 1)^(2/3) + (x^3 + 1)^(1/3) + 1) - 1/3*log(abs((x^3 + 1)^(1/3) - 1))

Mupad [B] (verification not implemented)

Time = 5.99 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.97 \[ \int \frac {\left (-1+x^3\right ) \sqrt [3]{1+x^3}}{x} \, dx=\frac {{\left (x^3+1\right )}^{4/3}}{4}-{\left (x^3+1\right )}^{1/3}-\frac {\ln \left ({\left (x^3+1\right )}^{1/3}-1\right )}{3}-\ln \left (3\,{\left (x^3+1\right )}^{1/3}+\frac {3}{2}-\frac {\sqrt {3}\,3{}\mathrm {i}}{2}\right )\,\left (-\frac {1}{6}+\frac {\sqrt {3}\,1{}\mathrm {i}}{6}\right )+\ln \left (3\,{\left (x^3+1\right )}^{1/3}+\frac {3}{2}+\frac {\sqrt {3}\,3{}\mathrm {i}}{2}\right )\,\left (\frac {1}{6}+\frac {\sqrt {3}\,1{}\mathrm {i}}{6}\right ) \]

[In]

int(((x^3 - 1)*(x^3 + 1)^(1/3))/x,x)

[Out]

(x^3 + 1)^(4/3)/4 - (x^3 + 1)^(1/3) - log((x^3 + 1)^(1/3) - 1)/3 - log(3*(x^3 + 1)^(1/3) - (3^(1/2)*3i)/2 + 3/
2)*((3^(1/2)*1i)/6 - 1/6) + log((3^(1/2)*3i)/2 + 3*(x^3 + 1)^(1/3) + 3/2)*((3^(1/2)*1i)/6 + 1/6)