Optimal. Leaf size=141 \[ \frac {\sqrt [3]{x-1} (x+1)^{2/3} \left ((x-1)^{2/3} \sqrt [3]{x+1}+\frac {1}{3} \log \left (\sqrt [3]{x-1}-\sqrt [3]{x+1}\right )-\frac {1}{6} \log \left ((x-1)^{2/3}+\sqrt [3]{x+1} \sqrt [3]{x-1}+(x+1)^{2/3}\right )-\frac {\tan ^{-1}\left (\frac {\sqrt {3} \sqrt [3]{x+1}}{2 \sqrt [3]{x-1}+\sqrt [3]{x+1}}\right )}{\sqrt {3}}\right )}{\sqrt [3]{(x-1) (x+1)^2}} \]
________________________________________________________________________________________
Rubi [A] time = 0.20, antiderivative size = 191, normalized size of antiderivative = 1.35, number of steps used = 4, number of rules used = 4, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.235, Rules used = {2081, 2077, 80, 60} \begin {gather*} -\frac {(1-x) (x+1)}{\sqrt [3]{x^3+x^2-x-1}}+\frac {(-x-1)^{2/3} \sqrt [3]{x-1} \log \left (\frac {\sqrt [3]{x-1}}{\sqrt [3]{-x-1}}+1\right )}{2 \sqrt [3]{x^3+x^2-x-1}}+\frac {(-x-1)^{2/3} \sqrt [3]{x-1} \log \left (-\frac {8}{3} (x+1)\right )}{6 \sqrt [3]{x^3+x^2-x-1}}+\frac {(-x-1)^{2/3} \sqrt [3]{x-1} \tan ^{-1}\left (\frac {1}{\sqrt {3}}-\frac {2 \sqrt [3]{x-1}}{\sqrt {3} \sqrt [3]{-x-1}}\right )}{\sqrt {3} \sqrt [3]{x^3+x^2-x-1}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 60
Rule 80
Rule 2077
Rule 2081
Rubi steps
\begin {align*} \int \frac {x}{\sqrt [3]{-1-x+x^2+x^3}} \, dx &=\operatorname {Subst}\left (\int \frac {-\frac {1}{3}+x}{\sqrt [3]{-\frac {16}{27}-\frac {4 x}{3}+x^3}} \, dx,x,\frac {1}{3}+x\right )\\ &=\frac {\left (4\ 2^{2/3} (-1-x)^{2/3} \sqrt [3]{-1+x}\right ) \operatorname {Subst}\left (\int \frac {-\frac {1}{3}+x}{\left (-\frac {16}{9}-\frac {8 x}{3}\right )^{2/3} \sqrt [3]{-\frac {16}{9}+\frac {4 x}{3}}} \, dx,x,\frac {1}{3}+x\right )}{3 \sqrt [3]{-1-x+x^2+x^3}}\\ &=-\frac {(1-x) (1+x)}{\sqrt [3]{-1-x+x^2+x^3}}-\frac {\left (4\ 2^{2/3} (-1-x)^{2/3} \sqrt [3]{-1+x}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (-\frac {16}{9}-\frac {8 x}{3}\right )^{2/3} \sqrt [3]{-\frac {16}{9}+\frac {4 x}{3}}} \, dx,x,\frac {1}{3}+x\right )}{9 \sqrt [3]{-1-x+x^2+x^3}}\\ &=-\frac {(1-x) (1+x)}{\sqrt [3]{-1-x+x^2+x^3}}+\frac {(-1-x)^{2/3} \sqrt [3]{-1+x} \tan ^{-1}\left (\frac {1}{\sqrt {3}}-\frac {2 \sqrt [3]{-1+x}}{\sqrt {3} \sqrt [3]{-1-x}}\right )}{\sqrt {3} \sqrt [3]{-1-x+x^2+x^3}}+\frac {(-1-x)^{2/3} \sqrt [3]{-1+x} \log \left (1+\frac {\sqrt [3]{-1+x}}{\sqrt [3]{-1-x}}\right )}{2 \sqrt [3]{-1-x+x^2+x^3}}+\frac {(-1-x)^{2/3} \sqrt [3]{-1+x} \log (1+x)}{6 \sqrt [3]{-1-x+x^2+x^3}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] time = 0.02, size = 58, normalized size = 0.41 \begin {gather*} \frac {(x-1) \left (-\sqrt [3]{2} (x+1)^{2/3} \, _2F_1\left (\frac {2}{3},\frac {2}{3};\frac {5}{3};\frac {1-x}{2}\right )+4 x+4\right )}{4 \sqrt [3]{(x-1) (x+1)^2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [A] time = 5.07, size = 152, normalized size = 1.08 \begin {gather*} \frac {\sqrt [3]{-1+x} (1+x)^{2/3} \left (\frac {2 \sqrt [3]{1+x}}{\sqrt [3]{-1+x} \left (-1+\frac {1+x}{-1+x}\right )}-\frac {\tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{1+x}}{\sqrt {3} \sqrt [3]{-1+x}}\right )}{\sqrt {3}}+\frac {1}{3} \log \left (-1+\frac {\sqrt [3]{1+x}}{\sqrt [3]{-1+x}}\right )-\frac {1}{6} \log \left (1+\frac {\sqrt [3]{1+x}}{\sqrt [3]{-1+x}}+\frac {(1+x)^{2/3}}{(-1+x)^{2/3}}\right )\right )}{\sqrt [3]{(-1+x) (1+x)^2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.47, size = 151, normalized size = 1.07 \begin {gather*} \frac {2 \, \sqrt {3} {\left (x + 1\right )} \arctan \left (\frac {\sqrt {3} {\left (x + 1\right )} + 2 \, \sqrt {3} {\left (x^{3} + x^{2} - x - 1\right )}^{\frac {1}{3}}}{3 \, {\left (x + 1\right )}}\right ) - {\left (x + 1\right )} \log \left (\frac {x^{2} + {\left (x^{3} + x^{2} - x - 1\right )}^{\frac {1}{3}} {\left (x + 1\right )} + 2 \, x + {\left (x^{3} + x^{2} - x - 1\right )}^{\frac {2}{3}} + 1}{x^{2} + 2 \, x + 1}\right ) + 2 \, {\left (x + 1\right )} \log \left (-\frac {x - {\left (x^{3} + x^{2} - x - 1\right )}^{\frac {1}{3}} + 1}{x + 1}\right ) + 6 \, {\left (x^{3} + x^{2} - x - 1\right )}^{\frac {2}{3}}}{6 \, {\left (x + 1\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*} \int \frac {x}{{\left (x^{3} + x^{2} - x - 1\right )}^{\frac {1}{3}}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [C] time = 0.81, size = 403, normalized size = 2.86
method | result | size |
trager | \(\frac {\left (x^{3}+x^{2}-x -1\right )^{\frac {2}{3}}}{1+x}+\frac {\ln \left (\frac {-36 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2} x^{2}+9 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (x^{3}+x^{2}-x -1\right )^{\frac {2}{3}}-9 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (x^{3}+x^{2}-x -1\right )^{\frac {1}{3}} x -36 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2} x -12 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{2}-9 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (x^{3}+x^{2}-x -1\right )^{\frac {1}{3}}-3 x \left (x^{3}+x^{2}-x -1\right )^{\frac {1}{3}}-6 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x -x^{2}-3 \left (x^{3}+x^{2}-x -1\right )^{\frac {1}{3}}+6 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )+1}{1+x}\right )}{3}+\RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \ln \left (-\frac {-18 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2} x^{2}+9 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (x^{3}+x^{2}-x -1\right )^{\frac {2}{3}}-18 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2} x -15 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{2}+3 \left (x^{3}+x^{2}-x -1\right )^{\frac {2}{3}}-3 x \left (x^{3}+x^{2}-x -1\right )^{\frac {1}{3}}-18 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x -2 x^{2}-3 \left (x^{3}+x^{2}-x -1\right )^{\frac {1}{3}}-3 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )-4 x -2}{1+x}\right )\) | \(403\) |
risch | \(\frac {\left (-1+x \right ) \left (1+x \right )}{\left (\left (-1+x \right ) \left (1+x \right )^{2}\right )^{\frac {1}{3}}}+\frac {\RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \ln \left (\frac {2 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x^{2}+2 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x +3 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{3}+x^{2}-x -1\right )^{\frac {2}{3}}+3 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{3}+x^{2}-x -1\right )^{\frac {1}{3}} x +5 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{2}+3 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{3}+x^{2}-x -1\right )^{\frac {1}{3}}+4 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x +2 x^{2}-\RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )-2}{1+x}\right )}{3}-\frac {\ln \left (\frac {2 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x^{2}+2 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x -3 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{3}+x^{2}-x -1\right )^{\frac {2}{3}}-3 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{3}+x^{2}-x -1\right )^{\frac {1}{3}} x -\RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{2}-3 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{3}+x^{2}-x -1\right )^{\frac {1}{3}}-3 \left (x^{3}+x^{2}-x -1\right )^{\frac {2}{3}}-3 x \left (x^{3}+x^{2}-x -1\right )^{\frac {1}{3}}-x^{2}+\RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )-3 \left (x^{3}+x^{2}-x -1\right )^{\frac {1}{3}}-2 x -1}{1+x}\right ) \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )}{3}-\frac {\ln \left (\frac {2 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x^{2}+2 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x -3 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{3}+x^{2}-x -1\right )^{\frac {2}{3}}-3 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{3}+x^{2}-x -1\right )^{\frac {1}{3}} x -\RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{2}-3 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{3}+x^{2}-x -1\right )^{\frac {1}{3}}-3 \left (x^{3}+x^{2}-x -1\right )^{\frac {2}{3}}-3 x \left (x^{3}+x^{2}-x -1\right )^{\frac {1}{3}}-x^{2}+\RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )-3 \left (x^{3}+x^{2}-x -1\right )^{\frac {1}{3}}-2 x -1}{1+x}\right )}{3}\) | \(528\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x}{{\left (x^{3} + x^{2} - x - 1\right )}^{\frac {1}{3}}}\,{d x} \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 \frac {x}{{\left (x^3+x^2-x-1\right )}^{1/3}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x}{\sqrt [3]{\left (x - 1\right ) \left (x + 1\right )^{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________