Optimal. Leaf size=96 \[ \frac {1}{3} x^2 \sqrt [3]{1+x^3}-\frac {\text {ArcTan}\left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{1+x^3}}\right )}{3 \sqrt {3}}-\frac {1}{9} \log \left (-x+\sqrt [3]{1+x^3}\right )+\frac {1}{18} \log \left (x^2+x \sqrt [3]{1+x^3}+\left (1+x^3\right )^{2/3}\right ) \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.01, antiderivative size = 65, normalized size of antiderivative = 0.68, number of steps
used = 2, number of rules used = 2, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {285, 337}
\begin {gather*} -\frac {\text {ArcTan}\left (\frac {\frac {2 x}{\sqrt [3]{x^3+1}}+1}{\sqrt {3}}\right )}{3 \sqrt {3}}-\frac {1}{6} \log \left (x-\sqrt [3]{x^3+1}\right )+\frac {1}{3} \sqrt [3]{x^3+1} x^2 \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 285
Rule 337
Rubi steps
\begin {align*} \int x \sqrt [3]{1+x^3} \, dx &=\frac {1}{3} x^2 \sqrt [3]{1+x^3}+\frac {1}{3} \int \frac {x}{\left (1+x^3\right )^{2/3}} \, dx\\ &=\frac {1}{3} x^2 \sqrt [3]{1+x^3}+\frac {1}{3} \text {Subst}\left (\int \frac {x}{1-x^3} \, dx,x,\frac {x}{\sqrt [3]{1+x^3}}\right )\\ &=\frac {1}{3} x^2 \sqrt [3]{1+x^3}+\frac {1}{9} \text {Subst}\left (\int \frac {1}{1-x} \, dx,x,\frac {x}{\sqrt [3]{1+x^3}}\right )-\frac {1}{9} \text {Subst}\left (\int \frac {1-x}{1+x+x^2} \, dx,x,\frac {x}{\sqrt [3]{1+x^3}}\right )\\ &=\frac {1}{3} x^2 \sqrt [3]{1+x^3}-\frac {1}{9} \log \left (1-\frac {x}{\sqrt [3]{1+x^3}}\right )+\frac {1}{18} \text {Subst}\left (\int \frac {1+2 x}{1+x+x^2} \, dx,x,\frac {x}{\sqrt [3]{1+x^3}}\right )-\frac {1}{6} \text {Subst}\left (\int \frac {1}{1+x+x^2} \, dx,x,\frac {x}{\sqrt [3]{1+x^3}}\right )\\ &=\frac {1}{3} x^2 \sqrt [3]{1+x^3}-\frac {1}{9} \log \left (1-\frac {x}{\sqrt [3]{1+x^3}}\right )+\frac {1}{18} \log \left (1+\frac {x^2}{\left (1+x^3\right )^{2/3}}+\frac {x}{\sqrt [3]{1+x^3}}\right )+\frac {1}{3} \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+\frac {2 x}{\sqrt [3]{1+x^3}}\right )\\ &=\frac {1}{3} x^2 \sqrt [3]{1+x^3}-\frac {\tan ^{-1}\left (\frac {1+\frac {2 x}{\sqrt [3]{1+x^3}}}{\sqrt {3}}\right )}{3 \sqrt {3}}-\frac {1}{9} \log \left (1-\frac {x}{\sqrt [3]{1+x^3}}\right )+\frac {1}{18} \log \left (1+\frac {x^2}{\left (1+x^3\right )^{2/3}}+\frac {x}{\sqrt [3]{1+x^3}}\right )\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.12, size = 90, normalized size = 0.94 \begin {gather*} \frac {1}{18} \left (6 x^2 \sqrt [3]{1+x^3}-2 \sqrt {3} \text {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 ) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [C] Result contains higher order function than in optimal. Order 5 vs. order
3.
time = 0.94, size = 17, normalized size = 0.18
method | result | size |
meijerg | \(\frac {x^{2} \hypergeom \left (\left [-\frac {1}{3}, \frac {2}{3}\right ], \left [\frac {5}{3}\right ], -x^{3}\right )}{2}\) | \(17\) |
risch | \(\frac {x^{2} \left (x^{3}+1\right )^{\frac {1}{3}}}{3}+\frac {x^{2} \hypergeom \left (\left [\frac {2}{3}, \frac {2}{3}\right ], \left [\frac {5}{3}\right ], -x^{3}\right )}{6}\) | \(30\) |
trager | \(\frac {x^{2} \left (x^{3}+1\right )^{\frac {1}{3}}}{3}+\frac {\RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \ln \left (\RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2} x^{3}+3 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{3}+1\right )^{\frac {2}{3}} x +3 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{3}+1\right )^{\frac {1}{3}} x^{2}+2 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{3}+x^{3}+\RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )+1\right )}{9}-\frac {\ln \left (\RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2} x^{3}-3 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{3}+1\right )^{\frac {2}{3}} x -3 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{3}+1\right )^{\frac {1}{3}} x^{2}-4 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{3}+3 x \left (x^{3}+1\right )^{\frac {2}{3}}+3 x^{2} \left (x^{3}+1\right )^{\frac {1}{3}}+4 x^{3}-\RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )+2\right ) \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )}{9}+\frac {\ln \left (\RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2} x^{3}-3 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{3}+1\right )^{\frac {2}{3}} x -3 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{3}+1\right )^{\frac {1}{3}} x^{2}-4 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{3}+3 x \left (x^{3}+1\right )^{\frac {2}{3}}+3 x^{2} \left (x^{3}+1\right )^{\frac {1}{3}}+4 x^{3}-\RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )+2\right )}{9}\) | \(342\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.46, size = 94, normalized size = 0.98 \begin {gather*} \frac {1}{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}}}{3 \, x {\left (\frac {x^{3} + 1}{x^{3}} - 1\right )}} + \frac {1}{18} \, \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}{9} \, \log \left (\frac {{\left (x^{3} + 1\right )}^{\frac {1}{3}}}{x} - 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 0.36, size = 88, normalized size = 0.92 \begin {gather*} \frac {1}{3} \, {\left (x^{3} + 1\right )}^{\frac {1}{3}} x^{2} + \frac {1}{9} \, \sqrt {3} \arctan \left (\frac {\sqrt {3} x + 2 \, \sqrt {3} {\left (x^{3} + 1\right )}^{\frac {1}{3}}}{3 \, x}\right ) - \frac {1}{9} \, \log \left (-\frac {x - {\left (x^{3} + 1\right )}^{\frac {1}{3}}}{x}\right ) + \frac {1}{18} \, \log \left (\frac {x^{2} + {\left (x^{3} + 1\right )}^{\frac {1}{3}} x + {\left (x^{3} + 1\right )}^{\frac {2}{3}}}{x^{2}}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [C] Result contains complex when optimal does not.
time = 0.55, size = 31, normalized size = 0.32 \begin {gather*} \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 )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int x\,{\left (x^3+1\right )}^{1/3} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________