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

Optimal result

Integrand size = 21, antiderivative size = 107 \[ \int \frac {(1+x) \sqrt [3]{-1+x^3}}{(-1+x)^2 x} \, dx=-\frac {3 \sqrt [3]{-1+x^3}}{-1+x}+\sqrt {3} \arctan \left (\frac {\sqrt {3} \sqrt [3]{-1+x^3}}{-2+2 x+\sqrt [3]{-1+x^3}}\right )-\log \left (1-x+\sqrt [3]{-1+x^3}\right )+\frac {1}{2} \log \left (1-2 x+x^2+(-1+x) \sqrt [3]{-1+x^3}+\left (-1+x^3\right )^{2/3}\right ) \] Output:

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

Mathematica [A] (verified)

Time = 2.01 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.00 \[ \int \frac {(1+x) \sqrt [3]{-1+x^3}}{(-1+x)^2 x} \, dx=-\frac {3 \sqrt [3]{-1+x^3}}{-1+x}+\sqrt {3} \arctan \left (\frac {\sqrt {3} \sqrt [3]{-1+x^3}}{-2+2 x+\sqrt [3]{-1+x^3}}\right )-\log \left (1-x+\sqrt [3]{-1+x^3}\right )+\frac {1}{2} \log \left (1-2 x+x^2+(-1+x) \sqrt [3]{-1+x^3}+\left (-1+x^3\right )^{2/3}\right ) \] Input:

Integrate[((1 + x)*(-1 + x^3)^(1/3))/((-1 + x)^2*x),x]
 

Output:

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

Rubi [C] (verified)

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

Time = 0.41 (sec) , antiderivative size = 196, normalized size of antiderivative = 1.83, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {2580, 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.

\(\displaystyle \int \frac {(x+1) \sqrt [3]{x^3-1}}{(x-1)^2 x} \, dx\)

\(\Big \downarrow \) 2580

\(\displaystyle \int \left (\frac {3 x^3}{\left (x^3-1\right )^{5/3}}+\frac {5 x}{\left (x^3-1\right )^{5/3}}+\frac {3}{\left (x^3-1\right )^{5/3}}+\frac {1}{\left (x^3-1\right )^{5/3} x}+\frac {x^4}{\left (x^3-1\right )^{5/3}}+\frac {5 x^2}{\left (x^3-1\right )^{5/3}}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {\arctan \left (\frac {\frac {2 x}{\sqrt [3]{x^3-1}}+1}{\sqrt {3}}\right )}{\sqrt {3}}+\frac {\arctan \left (\frac {1-2 \sqrt [3]{x^3-1}}{\sqrt {3}}\right )}{\sqrt {3}}-\frac {3 \left (1-x^3\right )^{2/3} x \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {5}{3},\frac {4}{3},x^3\right )}{\left (x^3-1\right )^{2/3}}-\frac {3 \left (1-x^3\right )^{2/3} x^4 \operatorname {Hypergeometric2F1}\left (\frac {4}{3},\frac {5}{3},\frac {7}{3},x^3\right )}{4 \left (x^3-1\right )^{2/3}}-\frac {3}{\left (x^3-1\right )^{2/3}}-\frac {1}{2} \log \left (x-\sqrt [3]{x^3-1}\right )-\frac {1}{2} \log \left (\sqrt [3]{x^3-1}+1\right )-\frac {3 x^2}{\left (x^3-1\right )^{2/3}}+\frac {\log (x)}{2}\)

Input:

Int[((1 + x)*(-1 + x^3)^(1/3))/((-1 + x)^2*x),x]
 

Output:

-3/(-1 + x^3)^(2/3) - (3*x^2)/(-1 + x^3)^(2/3) - ArcTan[(1 + (2*x)/(-1 + x 
^3)^(1/3))/Sqrt[3]]/Sqrt[3] + ArcTan[(1 - 2*(-1 + x^3)^(1/3))/Sqrt[3]]/Sqr 
t[3] - (3*x*(1 - x^3)^(2/3)*Hypergeometric2F1[1/3, 5/3, 4/3, x^3])/(-1 + x 
^3)^(2/3) - (3*x^4*(1 - x^3)^(2/3)*Hypergeometric2F1[4/3, 5/3, 7/3, x^3])/ 
(4*(-1 + x^3)^(2/3)) + Log[x]/2 - Log[x - (-1 + x^3)^(1/3)]/2 - Log[1 + (- 
1 + x^3)^(1/3)]/2
 

Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 2580

Int[(Px_)*(x_)^(m_.)*((c_) + (d_.)*(x_))^(q_)*((a_) + (b_.)*(x_)^3)^(p_.), 
x_Symbol] :> Int[ExpandIntegrand[(c^3 + d^3*x^3)^q*(a + b*x^3)^p, x^m*(Px/( 
c^2 - c*d*x + d^2*x^2)^q), x], x] /; FreeQ[{a, b, c, d, m, p}, x] && PolyQ[ 
Px, x] && ILtQ[q, 0] && IntegerQ[m] && RationalQ[p] && EqQ[Denominator[p], 
3]
 

Maple [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 9 vs. order 3.

Time = 0.86 (sec) , antiderivative size = 164, normalized size of antiderivative = 1.53

method	result	size
risch	\(-\frac {3 \left (x^{2}+x +1\right )}{\left (x^{3}-1\right )^{\frac {2}{3}}}+\frac {\left (\frac {\left (x^{3}-1\right )^{\frac {2}{3}} {\left (-\operatorname {signum}\left (x^{3}-1\right )\right )}^{\frac {2}{3}} x^{2} \operatorname {hypergeom}\left (\left [\frac {2}{3}, \frac {2}{3}\right ], \left [\frac {5}{3}\right ], x^{3}\right )}{2 {\left (\left (x^{3}-1\right )^{2}\right )}^{\frac {1}{3}} \operatorname {signum}\left (x^{3}-1\right )^{\frac {2}{3}}}-\frac {\left (x^{3}-1\right )^{\frac {2}{3}} {\left (-\operatorname {signum}\left (x^{3}-1\right )\right )}^{\frac {2}{3}} \left (\left (\frac {\pi \sqrt {3}}{6}-\frac {3 \ln \left (3\right )}{2}+3 \ln \left (x \right )+i \pi \right ) \Gamma \left (\frac {2}{3}\right )+\frac {2 \Gamma \left (\frac {2}{3}\right ) x^{3} \operatorname {hypergeom}\left (\left [1, 1, \frac {5}{3}\right ], \left [2, 2\right ], x^{3}\right )}{3}\right )}{3 {\left (\left (x^{3}-1\right )^{2}\right )}^{\frac {1}{3}} \Gamma \left (\frac {2}{3}\right ) \operatorname {signum}\left (x^{3}-1\right )^{\frac {2}{3}}}\right ) {\left (\left (x^{3}-1\right )^{2}\right )}^{\frac {1}{3}}}{\left (x^{3}-1\right )^{\frac {2}{3}}}\)	\(164\)
trager	\(-\frac {3 \left (x^{3}-1\right )^{\frac {1}{3}}}{-1+x}-\ln \left (\frac {12 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2} x^{2}-30 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2} x -99 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{3}-1\right )^{\frac {2}{3}}+127 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{3}-1\right )^{\frac {1}{3}} x -40 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{2}+12 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2}-127 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{3}-1\right )^{\frac {1}{3}}+157 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x +127 \left (x^{3}-1\right )^{\frac {2}{3}}-28 \left (x^{3}-1\right )^{\frac {1}{3}} x -87 x^{2}-40 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )+28 \left (x^{3}-1\right )^{\frac {1}{3}}-58 x -87}{x}\right )+\operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \ln \left (\frac {46 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2} x^{2}-115 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2} x +99 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{3}-1\right )^{\frac {2}{3}}+28 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{3}-1\right )^{\frac {1}{3}} x -173 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{2}+46 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2}-28 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{3}-1\right )^{\frac {1}{3}}+214 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x +28 \left (x^{3}-1\right )^{\frac {2}{3}}-127 \left (x^{3}-1\right )^{\frac {1}{3}} x +145 x^{2}-173 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )+127 \left (x^{3}-1\right )^{\frac {1}{3}}-87 x +145}{x}\right )\)	\(384\)

Input:

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

Output:

-3*(x^2+x+1)/(x^3-1)^(2/3)+(1/2/((x^3-1)^2)^(1/3)*(x^3-1)^(2/3)/signum(x^3 
-1)^(2/3)*(-signum(x^3-1))^(2/3)*x^2*hypergeom([2/3,2/3],[5/3],x^3)-1/3/(( 
x^3-1)^2)^(1/3)*(x^3-1)^(2/3)/GAMMA(2/3)/signum(x^3-1)^(2/3)*(-signum(x^3- 
1))^(2/3)*((1/6*Pi*3^(1/2)-3/2*ln(3)+3*ln(x)+I*Pi)*GAMMA(2/3)+2/3*GAMMA(2/ 
3)*x^3*hypergeom([1,1,5/3],[2,2],x^3)))/(x^3-1)^(2/3)*((x^3-1)^2)^(1/3)
 

Fricas [A] (verification not implemented)

Time = 0.31 (sec) , antiderivative size = 109, normalized size of antiderivative = 1.02 \[ \int \frac {(1+x) \sqrt [3]{-1+x^3}}{(-1+x)^2 x} \, dx=-\frac {2 \, \sqrt {3} {\left (x - 1\right )} \arctan \left (-\frac {4 \, \sqrt {3} {\left (x^{3} - 1\right )}^{\frac {1}{3}} {\left (x - 1\right )} + \sqrt {3} {\left (x^{2} + x + 1\right )} - 2 \, \sqrt {3} {\left (x^{3} - 1\right )}^{\frac {2}{3}}}{3 \, {\left (3 \, x^{2} - 5 \, x + 3\right )}}\right ) + {\left (x - 1\right )} \log \left (\frac {{\left (x^{3} - 1\right )}^{\frac {1}{3}} {\left (x - 1\right )} + x - {\left (x^{3} - 1\right )}^{\frac {2}{3}}}{x}\right ) + 6 \, {\left (x^{3} - 1\right )}^{\frac {1}{3}}}{2 \, {\left (x - 1\right )}} \] Input:

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

Output:

-1/2*(2*sqrt(3)*(x - 1)*arctan(-1/3*(4*sqrt(3)*(x^3 - 1)^(1/3)*(x - 1) + s 
qrt(3)*(x^2 + x + 1) - 2*sqrt(3)*(x^3 - 1)^(2/3))/(3*x^2 - 5*x + 3)) + (x 
- 1)*log(((x^3 - 1)^(1/3)*(x - 1) + x - (x^3 - 1)^(2/3))/x) + 6*(x^3 - 1)^ 
(1/3))/(x - 1)
 

Sympy [F]

\[ \int \frac {(1+x) \sqrt [3]{-1+x^3}}{(-1+x)^2 x} \, dx=\int \frac {\sqrt [3]{\left (x - 1\right ) \left (x^{2} + x + 1\right )} \left (x + 1\right )}{x \left (x - 1\right )^{2}}\, dx \] Input:

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

Output:

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

Maxima [F]

\[ \int \frac {(1+x) \sqrt [3]{-1+x^3}}{(-1+x)^2 x} \, dx=\int { \frac {{\left (x^{3} - 1\right )}^{\frac {1}{3}} {\left (x + 1\right )}}{{\left (x - 1\right )}^{2} x} \,d x } \] Input:

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

Output:

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

Giac [F]

\[ \int \frac {(1+x) \sqrt [3]{-1+x^3}}{(-1+x)^2 x} \, dx=\int { \frac {{\left (x^{3} - 1\right )}^{\frac {1}{3}} {\left (x + 1\right )}}{{\left (x - 1\right )}^{2} x} \,d x } \] Input:

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

Output:

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

Mupad [F(-1)]

Timed out. \[ \int \frac {(1+x) \sqrt [3]{-1+x^3}}{(-1+x)^2 x} \, dx=\int \frac {{\left (x^3-1\right )}^{1/3}\,\left (x+1\right )}{x\,{\left (x-1\right )}^2} \,d x \] Input:

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

Output:

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

Reduce [F]

\[ \int \frac {(1+x) \sqrt [3]{-1+x^3}}{(-1+x)^2 x} \, dx=\frac {-2 \left (x^{3}-1\right )^{\frac {1}{3}}+\left (\int \frac {\left (x^{3}-1\right )^{\frac {1}{3}}}{x^{5}-x^{4}-x^{2}+x}d x \right ) x -\left (\int \frac {\left (x^{3}-1\right )^{\frac {1}{3}}}{x^{5}-x^{4}-x^{2}+x}d x \right )+\left (\int \frac {\left (x^{3}-1\right )^{\frac {1}{3}} x^{2}}{x^{4}-x^{3}-x +1}d x \right ) x -\left (\int \frac {\left (x^{3}-1\right )^{\frac {1}{3}} x^{2}}{x^{4}-x^{3}-x +1}d x \right )}{x -1} \] Input:

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

Output:

( - 2*(x**3 - 1)**(1/3) + int((x**3 - 1)**(1/3)/(x**5 - x**4 - x**2 + x),x 
)*x - int((x**3 - 1)**(1/3)/(x**5 - x**4 - x**2 + x),x) + int(((x**3 - 1)* 
*(1/3)*x**2)/(x**4 - x**3 - x + 1),x)*x - int(((x**3 - 1)**(1/3)*x**2)/(x* 
*4 - x**3 - x + 1),x))/(x - 1)