Integrand size = 18, antiderivative size = 101 \[ \int \frac {\left (-1+x^3\right ) \sqrt [3]{1+x^3}}{x^2} \, dx=\frac {\sqrt [3]{1+x^3} \left (3+x^3\right )}{3 x}+\frac {2 \arctan \left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{1+x^3}}\right )}{3 \sqrt {3}}+\frac {2}{9} \log \left (-x+\sqrt [3]{1+x^3}\right )-\frac {1}{9} \log \left (x^2+x \sqrt [3]{1+x^3}+\left (1+x^3\right )^{2/3}\right ) \]
1/3*(x^3+1)^(1/3)*(x^3+3)/x+2/9*arctan(3^(1/2)*x/(x+2*(x^3+1)^(1/3)))*3^(1 /2)+2/9*ln(-x+(x^3+1)^(1/3))-1/9*ln(x^2+x*(x^3+1)^(1/3)+(x^3+1)^(2/3))
Time = 0.17 (sec) , antiderivative size = 97, normalized size of antiderivative = 0.96 \[ \int \frac {\left (-1+x^3\right ) \sqrt [3]{1+x^3}}{x^2} \, dx=\frac {1}{9} \left (\frac {3 \sqrt [3]{1+x^3} \left (3+x^3\right )}{x}+2 \sqrt {3} \arctan \left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{1+x^3}}\right )+2 \log \left (-x+\sqrt [3]{1+x^3}\right )-\log \left (x^2+x \sqrt [3]{1+x^3}+\left (1+x^3\right )^{2/3}\right )\right ) \]
((3*(1 + x^3)^(1/3)*(3 + x^3))/x + 2*Sqrt[3]*ArcTan[(Sqrt[3]*x)/(x + 2*(1 + x^3)^(1/3))] + 2*Log[-x + (1 + x^3)^(1/3)] - Log[x^2 + x*(1 + x^3)^(1/3) + (1 + x^3)^(2/3)])/9
Time = 0.19 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.83, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {955, 811, 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.
\(\displaystyle \int \frac {\left (x^3-1\right ) \sqrt [3]{x^3+1}}{x^2} \, dx\) |
\(\Big \downarrow \) 955 |
\(\displaystyle \frac {\left (x^3+1\right )^{4/3}}{x}-2 \int x \sqrt [3]{x^3+1}dx\) |
\(\Big \downarrow \) 811 |
\(\displaystyle \frac {\left (x^3+1\right )^{4/3}}{x}-2 \left (\frac {1}{3} \int \frac {x}{\left (x^3+1\right )^{2/3}}dx+\frac {1}{3} \sqrt [3]{x^3+1} x^2\right )\) |
\(\Big \downarrow \) 853 |
\(\displaystyle \frac {\left (x^3+1\right )^{4/3}}{x}-2 \left (\frac {1}{3} \left (-\frac {\arctan \left (\frac {\frac {2 x}{\sqrt [3]{x^3+1}}+1}{\sqrt {3}}\right )}{\sqrt {3}}-\frac {1}{2} \log \left (x-\sqrt [3]{x^3+1}\right )\right )+\frac {1}{3} \sqrt [3]{x^3+1} x^2\right )\) |
(1 + x^3)^(4/3)/x - 2*((x^2*(1 + x^3)^(1/3))/3 + (-(ArcTan[(1 + (2*x)/(1 + x^3)^(1/3))/Sqrt[3]]/Sqrt[3]) - Log[x - (1 + x^3)^(1/3)]/2)/3)
3.15.8.3.1 Defintions of rubi rules used
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c* x)^(m + 1)*((a + b*x^n)^p/(c*(m + n*p + 1))), x] + Simp[a*n*(p/(m + n*p + 1 )) Int[(c*x)^m*(a + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b, c, m}, x] && I GtQ[n, 0] && GtQ[p, 0] && NeQ[m + n*p + 1, 0] && IntBinomialQ[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[(a*d*(m + 1) - b*c*(m + n*(p + 1) + 1))/(a*e^n*(m + 1)) Int[(e *x)^(m + n)*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, c, d, e, p}, x] && NeQ[b* c - a*d, 0] && (IntegerQ[n] || GtQ[e, 0]) && ((GtQ[n, 0] && LtQ[m, -1]) || (LtQ[n, 0] && GtQ[m + n, -1])) && !ILtQ[p, -1]
Result contains higher order function than in optimal. Order 5 vs. order 3.
Time = 1.27 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.33
method | result | size |
meijerg | \(\frac {\operatorname {hypergeom}\left (\left [-\frac {1}{3}, -\frac {1}{3}\right ], \left [\frac {2}{3}\right ], -x^{3}\right )}{x}+\frac {x^{2} \operatorname {hypergeom}\left (\left [-\frac {1}{3}, \frac {2}{3}\right ], \left [\frac {5}{3}\right ], -x^{3}\right )}{2}\) | \(33\) |
risch | \(\frac {x^{6}+4 x^{3}+3}{3 x \left (x^{3}+1\right )^{\frac {2}{3}}}-\frac {x^{2} \operatorname {hypergeom}\left (\left [\frac {2}{3}, \frac {2}{3}\right ], \left [\frac {5}{3}\right ], -x^{3}\right )}{3}\) | \(40\) |
pseudoelliptic | \(\frac {-3 \left (x^{3}+1\right )^{\frac {1}{3}} x^{3}+2 \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (x +2 \left (x^{3}+1\right )^{\frac {1}{3}}\right )}{3 x}\right ) x +\ln \left (\frac {x^{2}+x \left (x^{3}+1\right )^{\frac {1}{3}}+\left (x^{3}+1\right )^{\frac {2}{3}}}{x^{2}}\right ) x -2 \ln \left (\frac {-x +\left (x^{3}+1\right )^{\frac {1}{3}}}{x}\right ) x -9 \left (x^{3}+1\right )^{\frac {1}{3}}}{9 \left (\left (x^{3}+1\right )^{\frac {2}{3}}+x \left (x +\left (x^{3}+1\right )^{\frac {1}{3}}\right )\right ) x \left (x -\left (x^{3}+1\right )^{\frac {1}{3}}\right )}\) | \(134\) |
trager | \(\frac {\left (x^{3}+1\right )^{\frac {1}{3}} \left (x^{3}+3\right )}{3 x}+\frac {2 \ln \left (317 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x^{3}-555 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{3}+1\right )^{\frac {2}{3}} x +2358 \left (x^{3}+1\right )^{\frac {1}{3}} \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{2}-1486 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{3}-2358 x \left (x^{3}+1\right )^{\frac {2}{3}}+1803 x^{2} \left (x^{3}+1\right )^{\frac {1}{3}}+872 x^{3}-317 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2}-733 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )+654\right )}{9}+\frac {2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \ln \left (-535 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x^{3}+555 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{3}+1\right )^{\frac {2}{3}} x +1803 \left (x^{3}+1\right )^{\frac {1}{3}} \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{2}-2893 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{3}-1803 x \left (x^{3}+1\right )^{\frac {2}{3}}+2358 x^{2} \left (x^{3}+1\right )^{\frac {1}{3}}-1090 x^{3}+535 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2}-852 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )-436\right )}{9}\) | \(254\) |
Time = 0.42 (sec) , antiderivative size = 105, normalized size of antiderivative = 1.04 \[ \int \frac {\left (-1+x^3\right ) \sqrt [3]{1+x^3}}{x^2} \, dx=\frac {2 \, \sqrt {3} x \arctan \left (-\frac {25382 \, \sqrt {3} {\left (x^{3} + 1\right )}^{\frac {1}{3}} x^{2} - 13720 \, \sqrt {3} {\left (x^{3} + 1\right )}^{\frac {2}{3}} x + \sqrt {3} {\left (5831 \, x^{3} + 7200\right )}}{58653 \, x^{3} + 8000}\right ) + x \log \left (3 \, {\left (x^{3} + 1\right )}^{\frac {1}{3}} x^{2} - 3 \, {\left (x^{3} + 1\right )}^{\frac {2}{3}} x + 1\right ) + 3 \, {\left (x^{3} + 3\right )} {\left (x^{3} + 1\right )}^{\frac {1}{3}}}{9 \, x} \]
1/9*(2*sqrt(3)*x*arctan(-(25382*sqrt(3)*(x^3 + 1)^(1/3)*x^2 - 13720*sqrt(3 )*(x^3 + 1)^(2/3)*x + sqrt(3)*(5831*x^3 + 7200))/(58653*x^3 + 8000)) + x*l og(3*(x^3 + 1)^(1/3)*x^2 - 3*(x^3 + 1)^(2/3)*x + 1) + 3*(x^3 + 3)*(x^3 + 1 )^(1/3))/x
Result contains complex when optimal does not.
Time = 1.44 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.64 \[ \int \frac {\left (-1+x^3\right ) \sqrt [3]{1+x^3}}{x^2} \, dx=\frac {x^{2} \Gamma \left (\frac {2}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{3}, \frac {2}{3} \\ \frac {5}{3} \end {matrix}\middle | {x^{3} e^{i \pi }} \right )}}{3 \Gamma \left (\frac {5}{3}\right )} - \frac {\Gamma \left (- \frac {1}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{3}, - \frac {1}{3} \\ \frac {2}{3} \end {matrix}\middle | {x^{3} e^{i \pi }} \right )}}{3 x \Gamma \left (\frac {2}{3}\right )} \]
x**2*gamma(2/3)*hyper((-1/3, 2/3), (5/3,), x**3*exp_polar(I*pi))/(3*gamma( 5/3)) - gamma(-1/3)*hyper((-1/3, -1/3), (2/3,), x**3*exp_polar(I*pi))/(3*x *gamma(2/3))
Time = 0.28 (sec) , antiderivative size = 105, normalized size of antiderivative = 1.04 \[ \int \frac {\left (-1+x^3\right ) \sqrt [3]{1+x^3}}{x^2} \, dx=-\frac {2}{9} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (\frac {2 \, {\left (x^{3} + 1\right )}^{\frac {1}{3}}}{x} + 1\right )}\right ) + \frac {{\left (x^{3} + 1\right )}^{\frac {1}{3}}}{x} + \frac {{\left (x^{3} + 1\right )}^{\frac {1}{3}}}{3 \, x {\left (\frac {x^{3} + 1}{x^{3}} - 1\right )}} - \frac {1}{9} \, \log \left (\frac {{\left (x^{3} + 1\right )}^{\frac {1}{3}}}{x} + \frac {{\left (x^{3} + 1\right )}^{\frac {2}{3}}}{x^{2}} + 1\right ) + \frac {2}{9} \, \log \left (\frac {{\left (x^{3} + 1\right )}^{\frac {1}{3}}}{x} - 1\right ) \]
-2/9*sqrt(3)*arctan(1/3*sqrt(3)*(2*(x^3 + 1)^(1/3)/x + 1)) + (x^3 + 1)^(1/ 3)/x + 1/3*(x^3 + 1)^(1/3)/(x*((x^3 + 1)/x^3 - 1)) - 1/9*log((x^3 + 1)^(1/ 3)/x + (x^3 + 1)^(2/3)/x^2 + 1) + 2/9*log((x^3 + 1)^(1/3)/x - 1)
\[ \int \frac {\left (-1+x^3\right ) \sqrt [3]{1+x^3}}{x^2} \, dx=\int { \frac {{\left (x^{3} + 1\right )}^{\frac {1}{3}} {\left (x^{3} - 1\right )}}{x^{2}} \,d x } \]
Timed out. \[ \int \frac {\left (-1+x^3\right ) \sqrt [3]{1+x^3}}{x^2} \, dx=\int \frac {\left (x^3-1\right )\,{\left (x^3+1\right )}^{1/3}}{x^2} \,d x \]