Integrand size = 21, antiderivative size = 147 \[ \int \frac {(1+x) \left (-1+x^3\right )^{2/3}}{(-1+x)^3 x} \, dx=\frac {-3 \left (-1+x^3\right )^{2/3}-2 \sqrt {3} \left (1-2 x+x^2\right ) \arctan \left (\frac {\sqrt {3} \left (1+x+x^2\right )+4 \sqrt {3} (-1+x) \sqrt [3]{-1+x^3}-2 \sqrt {3} \left (-1+x^3\right )^{2/3}}{3 \left (3-5 x+3 x^2\right )}\right )-\left (1-2 x+x^2\right ) \log \left (\frac {x+(-1+x) \sqrt [3]{-1+x^3}-\left (-1+x^3\right )^{2/3}}{x}\right )}{2 \left (1-2 x+x^2\right )} \] Output:
1/2*(-2*3^(1/2)*(x^2-2*x+1)*arctan(1/3*(4*3^(1/2)*(x^3-1)^(1/3)*(-1+x)+3^( 1/2)*(x^2+x+1)-2*3^(1/2)*(x^3-1)^(2/3))/(3*x^2-5*x+3))-(x^2-2*x+1)*ln(((-1 +x)*(x^3-1)^(1/3)+x-(x^3-1)^(2/3))/x)-3*(x^3-1)^(2/3))/(x^2-2*x+1)
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 10.13 (sec) , antiderivative size = 123, normalized size of antiderivative = 0.84 \[ \int \frac {(1+x) \left (-1+x^3\right )^{2/3}}{(-1+x)^3 x} \, dx=-\frac {3 \left (-1+x^3\right )^{2/3}}{2 (-1+x)^2}+\frac {x \sqrt [3]{1-x^3} \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {1}{3},\frac {4}{3},x^3\right )}{\sqrt [3]{-1+x^3}}+\frac {1}{6} \left (-2 \sqrt {3} \arctan \left (\frac {1-2 \sqrt [3]{-1+x^3}}{\sqrt {3}}\right )-2 \log \left (1+\sqrt [3]{-1+x^3}\right )+\log \left (1-\sqrt [3]{-1+x^3}+\left (-1+x^3\right )^{2/3}\right )\right ) \] Input:
Integrate[((1 + x)*(-1 + x^3)^(2/3))/((-1 + x)^3*x),x]
Output:
(-3*(-1 + x^3)^(2/3))/(2*(-1 + x)^2) + (x*(1 - x^3)^(1/3)*Hypergeometric2F 1[1/3, 1/3, 4/3, x^3])/(-1 + x^3)^(1/3) + (-2*Sqrt[3]*ArcTan[(1 - 2*(-1 + x^3)^(1/3))/Sqrt[3]] - 2*Log[1 + (-1 + x^3)^(1/3)] + Log[1 - (-1 + x^3)^(1 /3) + (-1 + x^3)^(2/3)])/6
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 0.51 (sec) , antiderivative size = 239, normalized size of antiderivative = 1.63, 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.
| \(\displaystyle \int \frac {(x+1) \left (x^3-1\right )^{2/3}}{(x-1)^3 x} \, dx\) |
| \(\Big \downarrow \) 2580 |
| \(\displaystyle \int \left (\frac {13 x^3}{\left (x^3-1\right )^{7/3}}+\frac {9 x}{\left (x^3-1\right )^{7/3}}+\frac {4}{\left (x^3-1\right )^{7/3}}+\frac {1}{\left (x^3-1\right )^{7/3} x}+\frac {x^6}{\left (x^3-1\right )^{7/3}}+\frac {4 x^5}{\left (x^3-1\right )^{7/3}}+\frac {9 x^4}{\left (x^3-1\right )^{7/3}}+\frac {13 x^2}{\left (x^3-1\right )^{7/3}}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\arctan \left (\frac {\frac {2 x}{\sqrt [3]{x^3-1}}+1}{\sqrt {3}}\right )}{\sqrt {3}}-\frac {\arctan \left (\frac {1-2 \sqrt [3]{x^3-1}}{\sqrt {3}}\right )}{\sqrt {3}}+\frac {9 \sqrt [3]{1-x^3} x^5 \operatorname {Hypergeometric2F1}\left (\frac {5}{3},\frac {7}{3},\frac {8}{3},x^3\right )}{5 \sqrt [3]{x^3-1}}+\frac {9 \sqrt [3]{1-x^3} x^2 \operatorname {Hypergeometric2F1}\left (\frac {2}{3},\frac {7}{3},\frac {5}{3},x^3\right )}{2 \sqrt [3]{x^3-1}}+\frac {2 x}{\sqrt [3]{x^3-1}}-\frac {x}{\left (x^3-1\right )^{4/3}}-\frac {3}{\sqrt [3]{x^3-1}}-\frac {9}{2 \left (x^3-1\right )^{4/3}}-\frac {1}{2} \log \left (\sqrt [3]{x^3-1}+1\right )-\frac {1}{2} \log \left (\sqrt [3]{x^3-1}-x\right )-\frac {7 x^4}{2 \left (x^3-1\right )^{4/3}}+\frac {\log (x)}{2}\) |
Input:
Int[((1 + x)*(-1 + x^3)^(2/3))/((-1 + x)^3*x),x]
Output:
-9/(2*(-1 + x^3)^(4/3)) - x/(-1 + x^3)^(4/3) - (7*x^4)/(2*(-1 + x^3)^(4/3) ) - 3/(-1 + x^3)^(1/3) + (2*x)/(-1 + x^3)^(1/3) + ArcTan[(1 + (2*x)/(-1 + x^3)^(1/3))/Sqrt[3]]/Sqrt[3] - ArcTan[(1 - 2*(-1 + x^3)^(1/3))/Sqrt[3]]/Sq rt[3] + (9*x^2*(1 - x^3)^(1/3)*Hypergeometric2F1[2/3, 7/3, 5/3, x^3])/(2*( -1 + x^3)^(1/3)) + (9*x^5*(1 - x^3)^(1/3)*Hypergeometric2F1[5/3, 7/3, 8/3, x^3])/(5*(-1 + x^3)^(1/3)) + Log[x]/2 - Log[1 + (-1 + x^3)^(1/3)]/2 - Log [-x + (-1 + x^3)^(1/3)]/2
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]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.84 (sec) , antiderivative size = 133, normalized size of antiderivative = 0.90
| method | result | size |
| risch | \(-\frac {3 \left (x^{2}+x +1\right )}{2 \left (-1+x \right ) \left (x^{3}-1\right )^{\frac {1}{3}}}+\frac {\sqrt {3}\, \Gamma \left (\frac {2}{3}\right ) {\left (-\operatorname {signum}\left (x^{3}-1\right )\right )}^{\frac {1}{3}} \left (\frac {2 \left (-\frac {\pi \sqrt {3}}{6}-\frac {3 \ln \left (3\right )}{2}+3 \ln \left (x \right )+i \pi \right ) \pi \sqrt {3}}{3 \Gamma \left (\frac {2}{3}\right )}+\frac {2 \pi \sqrt {3}\, x^{3} \operatorname {hypergeom}\left (\left [1, 1, \frac {4}{3}\right ], \left [2, 2\right ], x^{3}\right )}{9 \Gamma \left (\frac {2}{3}\right )}\right )}{6 \pi \operatorname {signum}\left (x^{3}-1\right )^{\frac {1}{3}}}+\frac {{\left (-\operatorname {signum}\left (x^{3}-1\right )\right )}^{\frac {1}{3}} x \operatorname {hypergeom}\left (\left [\frac {1}{3}, \frac {1}{3}\right ], \left [\frac {4}{3}\right ], x^{3}\right )}{\operatorname {signum}\left (x^{3}-1\right )^{\frac {1}{3}}}\) | \(133\) |
| trager | \(-\frac {3 \left (x^{3}-1\right )^{\frac {2}{3}}}{2 \left (-1+x \right )^{2}}+\operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \ln \left (\frac {-58 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2} x^{2}+99 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{3}-1\right )^{\frac {2}{3}}+99 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{3}-1\right )^{\frac {1}{3}} x +145 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2} x +157 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{2}+28 \left (x^{3}-1\right )^{\frac {2}{3}}-99 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{3}-1\right )^{\frac {1}{3}}+28 \left (x^{3}-1\right )^{\frac {1}{3}} x -58 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2}-117 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x -30 x^{2}-28 \left (x^{3}-1\right )^{\frac {1}{3}}+157 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )+18 x -30}{x}\right )-\ln \left (-\frac {58 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2} x^{2}+99 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{3}-1\right )^{\frac {2}{3}}+99 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{3}-1\right )^{\frac {1}{3}} x -145 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2} x +41 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{2}-127 \left (x^{3}-1\right )^{\frac {2}{3}}-99 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{3}-1\right )^{\frac {1}{3}}-127 \left (x^{3}-1\right )^{\frac {1}{3}} x +58 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2}+173 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x -69 x^{2}+127 \left (x^{3}-1\right )^{\frac {1}{3}}+41 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )-46 x -69}{x}\right ) \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )+\ln \left (-\frac {58 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2} x^{2}+99 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{3}-1\right )^{\frac {2}{3}}+99 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{3}-1\right )^{\frac {1}{3}} x -145 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2} x +41 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{2}-127 \left (x^{3}-1\right )^{\frac {2}{3}}-99 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{3}-1\right )^{\frac {1}{3}}-127 \left (x^{3}-1\right )^{\frac {1}{3}} x +58 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2}+173 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x -69 x^{2}+127 \left (x^{3}-1\right )^{\frac {1}{3}}+41 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )-46 x -69}{x}\right )\) | \(573\) |
Input:
int((1+x)*(x^3-1)^(2/3)/(-1+x)^3/x,x,method=_RETURNVERBOSE)
Output:
-3/2*(x^2+x+1)/(-1+x)/(x^3-1)^(1/3)+1/6/Pi*3^(1/2)*GAMMA(2/3)/signum(x^3-1 )^(1/3)*(-signum(x^3-1))^(1/3)*(2/3*(-1/6*Pi*3^(1/2)-3/2*ln(3)+3*ln(x)+I*P i)*Pi*3^(1/2)/GAMMA(2/3)+2/9*Pi*3^(1/2)/GAMMA(2/3)*x^3*hypergeom([1,1,4/3] ,[2,2],x^3))+1/signum(x^3-1)^(1/3)*(-signum(x^3-1))^(1/3)*x*hypergeom([1/3 ,1/3],[4/3],x^3)
Time = 0.25 (sec) , antiderivative size = 125, normalized size of antiderivative = 0.85 \[ \int \frac {(1+x) \left (-1+x^3\right )^{2/3}}{(-1+x)^3 x} \, dx=\frac {2 \, \sqrt {3} {\left (x^{2} - 2 \, x + 1\right )} \arctan \left (-\frac {4 \, \sqrt {3} {\left (x^{3} - 1\right )}^{\frac {1}{3}} {\left (x - 1\right )} + \sqrt {3} {\left (x^{2} + x + 1\right )} - 2 \, \sqrt {3} {\left (x^{3} - 1\right )}^{\frac {2}{3}}}{3 \, {\left (3 \, x^{2} - 5 \, x + 3\right )}}\right ) - {\left (x^{2} - 2 \, x + 1\right )} \log \left (\frac {{\left (x^{3} - 1\right )}^{\frac {1}{3}} {\left (x - 1\right )} + x - {\left (x^{3} - 1\right )}^{\frac {2}{3}}}{x}\right ) - 3 \, {\left (x^{3} - 1\right )}^{\frac {2}{3}}}{2 \, {\left (x^{2} - 2 \, x + 1\right )}} \] Input:
integrate((1+x)*(x^3-1)^(2/3)/(-1+x)^3/x,x, algorithm="fricas")
Output:
1/2*(2*sqrt(3)*(x^2 - 2*x + 1)*arctan(-1/3*(4*sqrt(3)*(x^3 - 1)^(1/3)*(x - 1) + sqrt(3)*(x^2 + x + 1) - 2*sqrt(3)*(x^3 - 1)^(2/3))/(3*x^2 - 5*x + 3) ) - (x^2 - 2*x + 1)*log(((x^3 - 1)^(1/3)*(x - 1) + x - (x^3 - 1)^(2/3))/x) - 3*(x^3 - 1)^(2/3))/(x^2 - 2*x + 1)
\[ \int \frac {(1+x) \left (-1+x^3\right )^{2/3}}{(-1+x)^3 x} \, dx=\int \frac {\left (\left (x - 1\right ) \left (x^{2} + x + 1\right )\right )^{\frac {2}{3}} \left (x + 1\right )}{x \left (x - 1\right )^{3}}\, dx \] Input:
integrate((1+x)*(x**3-1)**(2/3)/(-1+x)**3/x,x)
Output:
Integral(((x - 1)*(x**2 + x + 1))**(2/3)*(x + 1)/(x*(x - 1)**3), x)
\[ \int \frac {(1+x) \left (-1+x^3\right )^{2/3}}{(-1+x)^3 x} \, dx=\int { \frac {{\left (x^{3} - 1\right )}^{\frac {2}{3}} {\left (x + 1\right )}}{{\left (x - 1\right )}^{3} x} \,d x } \] Input:
integrate((1+x)*(x^3-1)^(2/3)/(-1+x)^3/x,x, algorithm="maxima")
Output:
integrate((x^3 - 1)^(2/3)*(x + 1)/((x - 1)^3*x), x)
\[ \int \frac {(1+x) \left (-1+x^3\right )^{2/3}}{(-1+x)^3 x} \, dx=\int { \frac {{\left (x^{3} - 1\right )}^{\frac {2}{3}} {\left (x + 1\right )}}{{\left (x - 1\right )}^{3} x} \,d x } \] Input:
integrate((1+x)*(x^3-1)^(2/3)/(-1+x)^3/x,x, algorithm="giac")
Output:
integrate((x^3 - 1)^(2/3)*(x + 1)/((x - 1)^3*x), x)
Timed out. \[ \int \frac {(1+x) \left (-1+x^3\right )^{2/3}}{(-1+x)^3 x} \, dx=\int \frac {{\left (x^3-1\right )}^{2/3}\,\left (x+1\right )}{x\,{\left (x-1\right )}^3} \,d x \] Input:
int(((x^3 - 1)^(2/3)*(x + 1))/(x*(x - 1)^3),x)
Output:
int(((x^3 - 1)^(2/3)*(x + 1))/(x*(x - 1)^3), x)
\[ \int \frac {(1+x) \left (-1+x^3\right )^{2/3}}{(-1+x)^3 x} \, dx=\frac {-\left (x^{3}-1\right )^{\frac {2}{3}}+\left (\int \frac {\left (x^{3}-1\right )^{\frac {2}{3}}}{x^{6}-2 x^{5}+x^{4}-x^{3}+2 x^{2}-x}d x \right ) x^{2}-2 \left (\int \frac {\left (x^{3}-1\right )^{\frac {2}{3}}}{x^{6}-2 x^{5}+x^{4}-x^{3}+2 x^{2}-x}d x \right ) x +\int \frac {\left (x^{3}-1\right )^{\frac {2}{3}}}{x^{6}-2 x^{5}+x^{4}-x^{3}+2 x^{2}-x}d x +\left (\int \frac {\left (x^{3}-1\right )^{\frac {2}{3}} x^{2}}{x^{5}-2 x^{4}+x^{3}-x^{2}+2 x -1}d x \right ) x^{2}-2 \left (\int \frac {\left (x^{3}-1\right )^{\frac {2}{3}} x^{2}}{x^{5}-2 x^{4}+x^{3}-x^{2}+2 x -1}d x \right ) x +\int \frac {\left (x^{3}-1\right )^{\frac {2}{3}} x^{2}}{x^{5}-2 x^{4}+x^{3}-x^{2}+2 x -1}d x}{x^{2}-2 x +1} \] Input:
int((1+x)*(x^3-1)^(2/3)/(-1+x)^3/x,x)
Output:
( - (x**3 - 1)**(2/3) + int((x**3 - 1)**(2/3)/(x**6 - 2*x**5 + x**4 - x**3 + 2*x**2 - x),x)*x**2 - 2*int((x**3 - 1)**(2/3)/(x**6 - 2*x**5 + x**4 - x **3 + 2*x**2 - x),x)*x + int((x**3 - 1)**(2/3)/(x**6 - 2*x**5 + x**4 - x** 3 + 2*x**2 - x),x) + int(((x**3 - 1)**(2/3)*x**2)/(x**5 - 2*x**4 + x**3 - x**2 + 2*x - 1),x)*x**2 - 2*int(((x**3 - 1)**(2/3)*x**2)/(x**5 - 2*x**4 + x**3 - x**2 + 2*x - 1),x)*x + int(((x**3 - 1)**(2/3)*x**2)/(x**5 - 2*x**4 + x**3 - x**2 + 2*x - 1),x))/(x**2 - 2*x + 1)