Integrand size = 18, antiderivative size = 101 \[ \int \frac {\sqrt [3]{-1+x^3} \left (1+x^3\right )}{x^5} \, dx=\frac {\left (-1-3 x^3\right ) \sqrt [3]{-1+x^3}}{4 x^4}-\frac {\arctan \left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{-1+x^3}}\right )}{\sqrt {3}}-\frac {1}{3} \log \left (-x+\sqrt [3]{-1+x^3}\right )+\frac {1}{6} \log \left (x^2+x \sqrt [3]{-1+x^3}+\left (-1+x^3\right )^{2/3}\right ) \] Output:
1/4*(-3*x^3-1)*(x^3-1)^(1/3)/x^4-1/3*arctan(3^(1/2)*x/(x+2*(x^3-1)^(1/3))) *3^(1/2)-1/3*ln(-x+(x^3-1)^(1/3))+1/6*ln(x^2+x*(x^3-1)^(1/3)+(x^3-1)^(2/3) )
Time = 0.17 (sec) , antiderivative size = 101, normalized size of antiderivative = 1.00 \[ \int \frac {\sqrt [3]{-1+x^3} \left (1+x^3\right )}{x^5} \, dx=\frac {\left (-1-3 x^3\right ) \sqrt [3]{-1+x^3}}{4 x^4}-\frac {\arctan \left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{-1+x^3}}\right )}{\sqrt {3}}-\frac {1}{3} \log \left (-x+\sqrt [3]{-1+x^3}\right )+\frac {1}{6} \log \left (x^2+x \sqrt [3]{-1+x^3}+\left (-1+x^3\right )^{2/3}\right ) \] Input:
Integrate[((-1 + x^3)^(1/3)*(1 + x^3))/x^5,x]
Output:
((-1 - 3*x^3)*(-1 + x^3)^(1/3))/(4*x^4) - ArcTan[(Sqrt[3]*x)/(x + 2*(-1 + x^3)^(1/3))]/Sqrt[3] - Log[-x + (-1 + x^3)^(1/3)]/3 + Log[x^2 + x*(-1 + x^ 3)^(1/3) + (-1 + x^3)^(2/3)]/6
Time = 0.19 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.76, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {953, 809, 853}
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.
\(\displaystyle \int \frac {\sqrt [3]{x^3-1} \left (x^3+1\right )}{x^5} \, dx\) |
\(\Big \downarrow \) 953 |
\(\displaystyle \int \frac {\sqrt [3]{x^3-1}}{x^2}dx+\frac {\left (x^3-1\right )^{4/3}}{4 x^4}\) |
\(\Big \downarrow \) 809 |
\(\displaystyle \int \frac {x}{\left (x^3-1\right )^{2/3}}dx-\frac {\sqrt [3]{x^3-1}}{x}+\frac {\left (x^3-1\right )^{4/3}}{4 x^4}\) |
\(\Big \downarrow \) 853 |
\(\displaystyle -\frac {\arctan \left (\frac {\frac {2 x}{\sqrt [3]{x^3-1}}+1}{\sqrt {3}}\right )}{\sqrt {3}}-\frac {\sqrt [3]{x^3-1}}{x}-\frac {1}{2} \log \left (x-\sqrt [3]{x^3-1}\right )+\frac {\left (x^3-1\right )^{4/3}}{4 x^4}\) |
Input:
Int[((-1 + x^3)^(1/3)*(1 + x^3))/x^5,x]
Output:
-((-1 + x^3)^(1/3)/x) + (-1 + x^3)^(4/3)/(4*x^4) - ArcTan[(1 + (2*x)/(-1 + x^3)^(1/3))/Sqrt[3]]/Sqrt[3] - Log[x - (-1 + x^3)^(1/3)]/2
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c* x)^(m + 1)*((a + b*x^n)^p/(c*(m + 1))), x] - Simp[b*n*(p/(c^n*(m + 1))) I nt[(c*x)^(m + n)*(a + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b, c}, x] && IGtQ [n, 0] && GtQ[p, 0] && LtQ[m, -1] && !ILtQ[(m + n*p + n + 1)/n, 0] && IntB inomialQ[a, b, c, n, m, p, x]
Int[(x_)/((a_) + (b_.)*(x_)^3)^(2/3), x_Symbol] :> With[{q = Rt[b, 3]}, Sim p[-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]
Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n _)), x_Symbol] :> Simp[c*(e*x)^(m + 1)*((a + b*x^n)^(p + 1)/(a*e*(m + 1))), x] + Simp[d/e^n Int[(e*x)^(m + n)*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, c, d, e, m, n, p}, x] && NeQ[b*c - a*d, 0] && EqQ[m + n*(p + 1) + 1, 0] && (IntegerQ[n] || GtQ[e, 0]) && ((GtQ[n, 0] && LtQ[m, -1]) || (LtQ[n, 0] && G tQ[m + n, -1]))
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 1.87 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.57
method | result | size |
risch | \(-\frac {3 x^{6}-2 x^{3}-1}{4 x^{4} \left (x^{3}-1\right )^{\frac {2}{3}}}+\frac {{\left (-\operatorname {signum}\left (x^{3}-1\right )\right )}^{\frac {2}{3}} x^{2} \operatorname {hypergeom}\left (\left [\frac {2}{3}, \frac {2}{3}\right ], \left [\frac {5}{3}\right ], x^{3}\right )}{2 \operatorname {signum}\left (x^{3}-1\right )^{\frac {2}{3}}}\) | \(58\) |
meijerg | \(-\frac {\operatorname {signum}\left (x^{3}-1\right )^{\frac {1}{3}} \operatorname {hypergeom}\left (\left [-\frac {1}{3}, -\frac {1}{3}\right ], \left [\frac {2}{3}\right ], x^{3}\right )}{{\left (-\operatorname {signum}\left (x^{3}-1\right )\right )}^{\frac {1}{3}} x}-\frac {\operatorname {signum}\left (x^{3}-1\right )^{\frac {1}{3}} \left (-x^{3}+1\right )^{\frac {4}{3}}}{4 {\left (-\operatorname {signum}\left (x^{3}-1\right )\right )}^{\frac {1}{3}} x^{4}}\) | \(66\) |
pseudoelliptic | \(\frac {4 \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (x +2 \left (x^{3}-1\right )^{\frac {1}{3}}\right )}{3 x}\right ) x^{4}+2 \ln \left (\frac {x^{2}+x \left (x^{3}-1\right )^{\frac {1}{3}}+\left (x^{3}-1\right )^{\frac {2}{3}}}{x^{2}}\right ) x^{4}-4 \ln \left (\frac {-x +\left (x^{3}-1\right )^{\frac {1}{3}}}{x}\right ) x^{4}-9 x^{3} \left (x^{3}-1\right )^{\frac {1}{3}}-3 \left (x^{3}-1\right )^{\frac {1}{3}}}{12 x^{4}}\) | \(107\) |
trager | \(-\frac {\left (3 x^{3}+1\right ) \left (x^{3}-1\right )^{\frac {1}{3}}}{4 x^{4}}-\frac {\ln \left (-17421502720 \operatorname {RootOf}\left (256 \textit {\_Z}^{2}-16 \textit {\_Z} +1\right )^{2} x^{3}+3103405824 \operatorname {RootOf}\left (256 \textit {\_Z}^{2}-16 \textit {\_Z} +1\right ) \left (x^{3}-1\right )^{\frac {2}{3}} x +21033432144 \left (x^{3}-1\right )^{\frac {1}{3}} \operatorname {RootOf}\left (256 \textit {\_Z}^{2}-16 \textit {\_Z} +1\right ) x^{2}-23047994048 \operatorname {RootOf}\left (256 \textit {\_Z}^{2}-16 \textit {\_Z} +1\right ) x^{3}-1508552373 x \left (x^{3}-1\right )^{\frac {2}{3}}+193962864 x^{2} \left (x^{3}-1\right )^{\frac {1}{3}}+1246536764 x^{3}+139372021760 \operatorname {RootOf}\left (256 \textit {\_Z}^{2}-16 \textit {\_Z} +1\right )^{2}+369329904 \operatorname {RootOf}\left (256 \textit {\_Z}^{2}-16 \textit {\_Z} +1\right )-396625334\right )}{3}+\frac {\ln \left (14505155072 \operatorname {RootOf}\left (256 \textit {\_Z}^{2}-16 \textit {\_Z} +1\right )^{2} x^{3}-3103405824 \operatorname {RootOf}\left (256 \textit {\_Z}^{2}-16 \textit {\_Z} +1\right ) \left (x^{3}-1\right )^{\frac {2}{3}} x +24136837968 \left (x^{3}-1\right )^{\frac {1}{3}} \operatorname {RootOf}\left (256 \textit {\_Z}^{2}-16 \textit {\_Z} +1\right ) x^{2}-21940004336 \operatorname {RootOf}\left (256 \textit {\_Z}^{2}-16 \textit {\_Z} +1\right ) x^{3}-1314589509 x \left (x^{3}-1\right )^{\frac {2}{3}}-193962864 x^{2} \left (x^{3}-1\right )^{\frac {1}{3}}+1565213135 x^{3}-116041240576 \operatorname {RootOf}\left (256 \textit {\_Z}^{2}-16 \textit {\_Z} +1\right )^{2}+22309334240 \operatorname {RootOf}\left (256 \textit {\_Z}^{2}-16 \textit {\_Z} +1\right )-1020791175\right )}{3}-\frac {16 \ln \left (14505155072 \operatorname {RootOf}\left (256 \textit {\_Z}^{2}-16 \textit {\_Z} +1\right )^{2} x^{3}-3103405824 \operatorname {RootOf}\left (256 \textit {\_Z}^{2}-16 \textit {\_Z} +1\right ) \left (x^{3}-1\right )^{\frac {2}{3}} x +24136837968 \left (x^{3}-1\right )^{\frac {1}{3}} \operatorname {RootOf}\left (256 \textit {\_Z}^{2}-16 \textit {\_Z} +1\right ) x^{2}-21940004336 \operatorname {RootOf}\left (256 \textit {\_Z}^{2}-16 \textit {\_Z} +1\right ) x^{3}-1314589509 x \left (x^{3}-1\right )^{\frac {2}{3}}-193962864 x^{2} \left (x^{3}-1\right )^{\frac {1}{3}}+1565213135 x^{3}-116041240576 \operatorname {RootOf}\left (256 \textit {\_Z}^{2}-16 \textit {\_Z} +1\right )^{2}+22309334240 \operatorname {RootOf}\left (256 \textit {\_Z}^{2}-16 \textit {\_Z} +1\right )-1020791175\right ) \operatorname {RootOf}\left (256 \textit {\_Z}^{2}-16 \textit {\_Z} +1\right )}{3}\) | \(446\) |
Input:
int((x^3-1)^(1/3)*(x^3+1)/x^5,x,method=_RETURNVERBOSE)
Output:
-1/4*(3*x^6-2*x^3-1)/x^4/(x^3-1)^(2/3)+1/2/signum(x^3-1)^(2/3)*(-signum(x^ 3-1))^(2/3)*x^2*hypergeom([2/3,2/3],[5/3],x^3)
Time = 0.25 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.11 \[ \int \frac {\sqrt [3]{-1+x^3} \left (1+x^3\right )}{x^5} \, dx=-\frac {4 \, \sqrt {3} x^{4} \arctan \left (-\frac {25382 \, \sqrt {3} {\left (x^{3} - 1\right )}^{\frac {1}{3}} x^{2} - 13720 \, \sqrt {3} {\left (x^{3} - 1\right )}^{\frac {2}{3}} x + \sqrt {3} {\left (5831 \, x^{3} - 7200\right )}}{58653 \, x^{3} - 8000}\right ) + 2 \, x^{4} \log \left (-3 \, {\left (x^{3} - 1\right )}^{\frac {1}{3}} x^{2} + 3 \, {\left (x^{3} - 1\right )}^{\frac {2}{3}} x + 1\right ) + 3 \, {\left (3 \, x^{3} + 1\right )} {\left (x^{3} - 1\right )}^{\frac {1}{3}}}{12 \, x^{4}} \] Input:
integrate((x^3-1)^(1/3)*(x^3+1)/x^5,x, algorithm="fricas")
Output:
-1/12*(4*sqrt(3)*x^4*arctan(-(25382*sqrt(3)*(x^3 - 1)^(1/3)*x^2 - 13720*sq rt(3)*(x^3 - 1)^(2/3)*x + sqrt(3)*(5831*x^3 - 7200))/(58653*x^3 - 8000)) + 2*x^4*log(-3*(x^3 - 1)^(1/3)*x^2 + 3*(x^3 - 1)^(2/3)*x + 1) + 3*(3*x^3 + 1)*(x^3 - 1)^(1/3))/x^4
Result contains complex when optimal does not.
Time = 1.30 (sec) , antiderivative size = 165, normalized size of antiderivative = 1.63 \[ \int \frac {\sqrt [3]{-1+x^3} \left (1+x^3\right )}{x^5} \, dx=\begin {cases} \frac {\sqrt [3]{-1 + \frac {1}{x^{3}}} e^{- \frac {2 i \pi }{3}} \Gamma \left (- \frac {4}{3}\right )}{3 \Gamma \left (- \frac {1}{3}\right )} - \frac {\sqrt [3]{-1 + \frac {1}{x^{3}}} e^{- \frac {2 i \pi }{3}} \Gamma \left (- \frac {4}{3}\right )}{3 x^{3} \Gamma \left (- \frac {1}{3}\right )} & \text {for}\: \frac {1}{\left |{x^{3}}\right |} > 1 \\- \frac {\sqrt [3]{1 - \frac {1}{x^{3}}} \Gamma \left (- \frac {4}{3}\right )}{3 \Gamma \left (- \frac {1}{3}\right )} + \frac {\sqrt [3]{1 - \frac {1}{x^{3}}} \Gamma \left (- \frac {4}{3}\right )}{3 x^{3} \Gamma \left (- \frac {1}{3}\right )} & \text {otherwise} \end {cases} + \frac {e^{\frac {i \pi }{3}} \Gamma \left (- \frac {1}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{3}, - \frac {1}{3} \\ \frac {2}{3} \end {matrix}\middle | {x^{3}} \right )}}{3 x \Gamma \left (\frac {2}{3}\right )} \] Input:
integrate((x**3-1)**(1/3)*(x**3+1)/x**5,x)
Output:
Piecewise(((-1 + x**(-3))**(1/3)*exp(-2*I*pi/3)*gamma(-4/3)/(3*gamma(-1/3) ) - (-1 + x**(-3))**(1/3)*exp(-2*I*pi/3)*gamma(-4/3)/(3*x**3*gamma(-1/3)), 1/Abs(x**3) > 1), (-(1 - 1/x**3)**(1/3)*gamma(-4/3)/(3*gamma(-1/3)) + (1 - 1/x**3)**(1/3)*gamma(-4/3)/(3*x**3*gamma(-1/3)), True)) + exp(I*pi/3)*ga mma(-1/3)*hyper((-1/3, -1/3), (2/3,), x**3)/(3*x*gamma(2/3))
Time = 0.12 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.92 \[ \int \frac {\sqrt [3]{-1+x^3} \left (1+x^3\right )}{x^5} \, dx=\frac {1}{3} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (\frac {2 \, {\left (x^{3} - 1\right )}^{\frac {1}{3}}}{x} + 1\right )}\right ) - \frac {{\left (x^{3} - 1\right )}^{\frac {1}{3}}}{x} + \frac {{\left (x^{3} - 1\right )}^{\frac {4}{3}}}{4 \, x^{4}} + \frac {1}{6} \, \log \left (\frac {{\left (x^{3} - 1\right )}^{\frac {1}{3}}}{x} + \frac {{\left (x^{3} - 1\right )}^{\frac {2}{3}}}{x^{2}} + 1\right ) - \frac {1}{3} \, \log \left (\frac {{\left (x^{3} - 1\right )}^{\frac {1}{3}}}{x} - 1\right ) \] Input:
integrate((x^3-1)^(1/3)*(x^3+1)/x^5,x, algorithm="maxima")
Output:
1/3*sqrt(3)*arctan(1/3*sqrt(3)*(2*(x^3 - 1)^(1/3)/x + 1)) - (x^3 - 1)^(1/3 )/x + 1/4*(x^3 - 1)^(4/3)/x^4 + 1/6*log((x^3 - 1)^(1/3)/x + (x^3 - 1)^(2/3 )/x^2 + 1) - 1/3*log((x^3 - 1)^(1/3)/x - 1)
\[ \int \frac {\sqrt [3]{-1+x^3} \left (1+x^3\right )}{x^5} \, dx=\int { \frac {{\left (x^{3} + 1\right )} {\left (x^{3} - 1\right )}^{\frac {1}{3}}}{x^{5}} \,d x } \] Input:
integrate((x^3-1)^(1/3)*(x^3+1)/x^5,x, algorithm="giac")
Output:
integrate((x^3 + 1)*(x^3 - 1)^(1/3)/x^5, x)
Time = 8.17 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.54 \[ \int \frac {\sqrt [3]{-1+x^3} \left (1+x^3\right )}{x^5} \, dx=-\frac {{\left (x^3-1\right )}^{1/3}-x^3\,{\left (x^3-1\right )}^{1/3}}{4\,x^4}-\frac {{\left (x^3-1\right )}^{1/3}\,{{}}_2{\mathrm {F}}_1\left (-\frac {1}{3},-\frac {1}{3};\ \frac {2}{3};\ x^3\right )}{x\,{\left (1-x^3\right )}^{1/3}} \] Input:
int(((x^3 - 1)^(1/3)*(x^3 + 1))/x^5,x)
Output:
- ((x^3 - 1)^(1/3) - x^3*(x^3 - 1)^(1/3))/(4*x^4) - ((x^3 - 1)^(1/3)*hyper geom([-1/3, -1/3], 2/3, x^3))/(x*(1 - x^3)^(1/3))
\[ \int \frac {\sqrt [3]{-1+x^3} \left (1+x^3\right )}{x^5} \, dx=\frac {\left (x^{3}-1\right )^{\frac {1}{3}} x^{3}-\left (x^{3}-1\right )^{\frac {1}{3}}+4 \left (\int \frac {\left (x^{3}-1\right )^{\frac {1}{3}}}{x^{2}}d x \right ) x^{4}}{4 x^{4}} \] Input:
int((x^3-1)^(1/3)*(x^3+1)/x^5,x)
Output:
((x**3 - 1)**(1/3)*x**3 - (x**3 - 1)**(1/3) + 4*int((x**3 - 1)**(1/3)/x**2 ,x)*x**4)/(4*x**4)