Integrand size = 15, antiderivative size = 67 \[ \int \frac {\sqrt [3]{1-x^3}}{x} \, dx=\sqrt [3]{1-x^3}-\frac {\arctan \left (\frac {1+2 \sqrt [3]{1-x^3}}{\sqrt {3}}\right )}{\sqrt {3}}-\frac {\log (x)}{2}+\frac {1}{2} \log \left (1-\sqrt [3]{1-x^3}\right ) \]
(-x^3+1)^(1/3)-1/2*ln(x)+1/2*ln(1-(-x^3+1)^(1/3))-1/3*arctan(1/3*(1+2*(-x^ 3+1)^(1/3))*3^(1/2))*3^(1/2)
Time = 0.07 (sec) , antiderivative size = 88, normalized size of antiderivative = 1.31 \[ \int \frac {\sqrt [3]{1-x^3}}{x} \, dx=\sqrt [3]{1-x^3}-\frac {\arctan \left (\frac {1+2 \sqrt [3]{1-x^3}}{\sqrt {3}}\right )}{\sqrt {3}}+\frac {1}{3} \log \left (-1+\sqrt [3]{1-x^3}\right )-\frac {1}{6} \log \left (1+\sqrt [3]{1-x^3}+\left (1-x^3\right )^{2/3}\right ) \]
(1 - x^3)^(1/3) - ArcTan[(1 + 2*(1 - x^3)^(1/3))/Sqrt[3]]/Sqrt[3] + Log[-1 + (1 - x^3)^(1/3)]/3 - Log[1 + (1 - x^3)^(1/3) + (1 - x^3)^(2/3)]/6
Time = 0.20 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.12, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {798, 60, 69, 16, 1083, 217}
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.
\(\displaystyle \int \frac {\sqrt [3]{1-x^3}}{x} \, dx\) |
\(\Big \downarrow \) 798 |
\(\displaystyle \frac {1}{3} \int \frac {\sqrt [3]{1-x^3}}{x^3}dx^3\) |
\(\Big \downarrow \) 60 |
\(\displaystyle \frac {1}{3} \left (\int \frac {1}{x^3 \left (1-x^3\right )^{2/3}}dx^3+3 \sqrt [3]{1-x^3}\right )\) |
\(\Big \downarrow \) 69 |
\(\displaystyle \frac {1}{3} \left (-\frac {3}{2} \int \frac {1}{1-\sqrt [3]{1-x^3}}d\sqrt [3]{1-x^3}-\frac {3}{2} \int \frac {1}{x^6+\sqrt [3]{1-x^3}+1}d\sqrt [3]{1-x^3}+3 \sqrt [3]{1-x^3}-\frac {1}{2} \log \left (x^3\right )\right )\) |
\(\Big \downarrow \) 16 |
\(\displaystyle \frac {1}{3} \left (-\frac {3}{2} \int \frac {1}{x^6+\sqrt [3]{1-x^3}+1}d\sqrt [3]{1-x^3}+3 \sqrt [3]{1-x^3}-\frac {1}{2} \log \left (x^3\right )+\frac {3}{2} \log \left (1-\sqrt [3]{1-x^3}\right )\right )\) |
\(\Big \downarrow \) 1083 |
\(\displaystyle \frac {1}{3} \left (3 \int \frac {1}{-x^6-3}d\left (2 \sqrt [3]{1-x^3}+1\right )+3 \sqrt [3]{1-x^3}-\frac {1}{2} \log \left (x^3\right )+\frac {3}{2} \log \left (1-\sqrt [3]{1-x^3}\right )\right )\) |
\(\Big \downarrow \) 217 |
\(\displaystyle \frac {1}{3} \left (-\sqrt {3} \arctan \left (\frac {2 \sqrt [3]{1-x^3}+1}{\sqrt {3}}\right )+3 \sqrt [3]{1-x^3}-\frac {\log \left (x^3\right )}{2}+\frac {3}{2} \log \left (1-\sqrt [3]{1-x^3}\right )\right )\) |
(3*(1 - x^3)^(1/3) - Sqrt[3]*ArcTan[(1 + 2*(1 - x^3)^(1/3))/Sqrt[3]] - Log [x^3]/2 + (3*Log[1 - (1 - x^3)^(1/3)])/2)/3
3.1.57.3.1 Defintions of rubi rules used
Int[(c_.)/((a_.) + (b_.)*(x_)), x_Symbol] :> Simp[c*(Log[RemoveContent[a + b*x, x]]/b), x] /; FreeQ[{a, b, c}, x]
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] + Simp[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] && GtQ[n, 0] && NeQ[m + n + 1, 0] && !(IGtQ[m, 0] && ( !Integer Q[n] || (GtQ[m, 0] && LtQ[m - n, 0]))) && !ILtQ[m + n + 2, 0] && IntLinear Q[a, b, c, d, m, n, x]
Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(2/3)), x_Symbol] :> With[ {q = Rt[(b*c - a*d)/b, 3]}, Simp[-Log[RemoveContent[a + b*x, x]]/(2*b*q^2), x] + (-Simp[3/(2*b*q) Subst[Int[1/(q^2 + q*x + x^2), x], x, (c + d*x)^(1 /3)], x] - Simp[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]
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])
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[1/n Subst [Int[x^(Simplify[(m + 1)/n] - 1)*(a + b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]
Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Simp[-2 Subst[I nt[1/Simp[b^2 - 4*a*c - x^2, x], x], x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x]
Result contains higher order function than in optimal. Order 5 vs. order 3.
Time = 1.55 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.73
method | result | size |
meijerg | \(-\frac {-3 \left (3+\frac {\pi \sqrt {3}}{6}-\frac {3 \ln \left (3\right )}{2}+3 \ln \left (x \right )+i \pi \right ) \Gamma \left (\frac {2}{3}\right )+\Gamma \left (\frac {2}{3}\right ) x^{3} {}_{3}^{}{\moversetsp {}{\mundersetsp {}{F_{2}^{}}}}\left (\frac {2}{3},1,1;2,2;x^{3}\right )}{9 \Gamma \left (\frac {2}{3}\right )}\) | \(49\) |
pseudoelliptic | \(\left (-x^{3}+1\right )^{\frac {1}{3}}-\frac {\ln \left (\left (-x^{3}+1\right )^{\frac {2}{3}}+\left (-x^{3}+1\right )^{\frac {1}{3}}+1\right )}{6}-\frac {\arctan \left (\frac {\left (1+2 \left (-x^{3}+1\right )^{\frac {1}{3}}\right ) \sqrt {3}}{3}\right ) \sqrt {3}}{3}+\frac {\ln \left (\left (-x^{3}+1\right )^{\frac {1}{3}}-1\right )}{3}\) | \(72\) |
trager | \(\left (-x^{3}+1\right )^{\frac {1}{3}}+\frac {\ln \left (-\frac {1438 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x^{3}-6979 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{3}+5502 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (-x^{3}+1\right )^{\frac {2}{3}}-9894 x^{3}+19749 \left (-x^{3}+1\right )^{\frac {2}{3}}+5502 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (-x^{3}+1\right )^{\frac {1}{3}}-11504 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2}+19749 \left (-x^{3}+1\right )^{\frac {1}{3}}-6002 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )+8245}{x^{3}}\right ) \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )}{3}-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \ln \left (\frac {-1438 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x^{3}-9855 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{3}+5502 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (-x^{3}+1\right )^{\frac {2}{3}}+1477 x^{3}-14247 \left (-x^{3}+1\right )^{\frac {2}{3}}+5502 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (-x^{3}+1\right )^{\frac {1}{3}}+11504 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2}-14247 \left (-x^{3}+1\right )^{\frac {1}{3}}+17006 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )-2743}{x^{3}}\right )}{3}-\frac {\ln \left (\frac {-1438 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x^{3}-9855 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{3}+5502 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (-x^{3}+1\right )^{\frac {2}{3}}+1477 x^{3}-14247 \left (-x^{3}+1\right )^{\frac {2}{3}}+5502 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (-x^{3}+1\right )^{\frac {1}{3}}+11504 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2}-14247 \left (-x^{3}+1\right )^{\frac {1}{3}}+17006 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )-2743}{x^{3}}\right )}{3}\) | \(380\) |
-1/9/GAMMA(2/3)*(-3*(3+1/6*Pi*3^(1/2)-3/2*ln(3)+3*ln(x)+I*Pi)*GAMMA(2/3)+G AMMA(2/3)*x^3*hypergeom([2/3,1,1],[2,2],x^3))
Time = 0.24 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.09 \[ \int \frac {\sqrt [3]{1-x^3}}{x} \, dx=-\frac {1}{3} \, \sqrt {3} \arctan \left (\frac {2}{3} \, \sqrt {3} {\left (-x^{3} + 1\right )}^{\frac {1}{3}} + \frac {1}{3} \, \sqrt {3}\right ) + {\left (-x^{3} + 1\right )}^{\frac {1}{3}} - \frac {1}{6} \, \log \left ({\left (-x^{3} + 1\right )}^{\frac {2}{3}} + {\left (-x^{3} + 1\right )}^{\frac {1}{3}} + 1\right ) + \frac {1}{3} \, \log \left ({\left (-x^{3} + 1\right )}^{\frac {1}{3}} - 1\right ) \]
-1/3*sqrt(3)*arctan(2/3*sqrt(3)*(-x^3 + 1)^(1/3) + 1/3*sqrt(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)
Result contains complex when optimal does not.
Time = 0.56 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.55 \[ \int \frac {\sqrt [3]{1-x^3}}{x} \, dx=- \frac {x e^{\frac {i \pi }{3}} \Gamma \left (- \frac {1}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{3}, - \frac {1}{3} \\ \frac {2}{3} \end {matrix}\middle | {\frac {1}{x^{3}}} \right )}}{3 \Gamma \left (\frac {2}{3}\right )} \]
Time = 0.28 (sec) , antiderivative size = 71, normalized size of antiderivative = 1.06 \[ \int \frac {\sqrt [3]{1-x^3}}{x} \, dx=-\frac {1}{3} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, {\left (-x^{3} + 1\right )}^{\frac {1}{3}} + 1\right )}\right ) + {\left (-x^{3} + 1\right )}^{\frac {1}{3}} - \frac {1}{6} \, \log \left ({\left (-x^{3} + 1\right )}^{\frac {2}{3}} + {\left (-x^{3} + 1\right )}^{\frac {1}{3}} + 1\right ) + \frac {1}{3} \, \log \left ({\left (-x^{3} + 1\right )}^{\frac {1}{3}} - 1\right ) \]
-1/3*sqrt(3)*arctan(1/3*sqrt(3)*(2*(-x^3 + 1)^(1/3) + 1)) + (-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)
Time = 0.30 (sec) , antiderivative size = 72, normalized size of antiderivative = 1.07 \[ \int \frac {\sqrt [3]{1-x^3}}{x} \, dx=-\frac {1}{3} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, {\left (-x^{3} + 1\right )}^{\frac {1}{3}} + 1\right )}\right ) + {\left (-x^{3} + 1\right )}^{\frac {1}{3}} - \frac {1}{6} \, \log \left ({\left (-x^{3} + 1\right )}^{\frac {2}{3}} + {\left (-x^{3} + 1\right )}^{\frac {1}{3}} + 1\right ) + \frac {1}{3} \, \log \left ({\left | {\left (-x^{3} + 1\right )}^{\frac {1}{3}} - 1 \right |}\right ) \]
-1/3*sqrt(3)*arctan(1/3*sqrt(3)*(2*(-x^3 + 1)^(1/3) + 1)) + (-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))
Time = 0.41 (sec) , antiderivative size = 83, normalized size of antiderivative = 1.24 \[ \int \frac {\sqrt [3]{1-x^3}}{x} \, dx=\frac {\ln \left ({\left (1-x^3\right )}^{1/3}-1\right )}{3}+\ln \left (3\,{\left (1-x^3\right )}^{1/3}+\frac {3}{2}-\frac {\sqrt {3}\,3{}\mathrm {i}}{2}\right )\,\left (-\frac {1}{6}+\frac {\sqrt {3}\,1{}\mathrm {i}}{6}\right )-\ln \left (3\,{\left (1-x^3\right )}^{1/3}+\frac {3}{2}+\frac {\sqrt {3}\,3{}\mathrm {i}}{2}\right )\,\left (\frac {1}{6}+\frac {\sqrt {3}\,1{}\mathrm {i}}{6}\right )+{\left (1-x^3\right )}^{1/3} \]
log((1 - x^3)^(1/3) - 1)/3 + log(3*(1 - x^3)^(1/3) - (3^(1/2)*3i)/2 + 3/2) *((3^(1/2)*1i)/6 - 1/6) - log((3^(1/2)*3i)/2 + 3*(1 - x^3)^(1/3) + 3/2)*(( 3^(1/2)*1i)/6 + 1/6) + (1 - x^3)^(1/3)
\[ \int \frac {\sqrt [3]{1-x^3}}{x} \, dx=\left (-x^{3}+1\right )^{\frac {1}{3}}-\left (\int \frac {\left (-x^{3}+1\right )^{\frac {1}{3}}}{x^{4}-x}d x \right ) \]