Optimal. Leaf size=114 \[ -\frac {6 \left (x^3-x^2\right )^{2/3}}{(x-1) x}-\log \left (\sqrt [3]{x^3-x^2}-x\right )+\frac {1}{2} \log \left (x^2+\sqrt [3]{x^3-x^2} x+\left (x^3-x^2\right )^{2/3}\right )+\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} x}{2 \sqrt [3]{x^3-x^2}+x}\right ) \]
________________________________________________________________________________________
Rubi [A] time = 0.09, antiderivative size = 151, normalized size of antiderivative = 1.32, number of steps used = 3, number of rules used = 3, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {2056, 78, 59} \begin {gather*} -\frac {6 x}{\sqrt [3]{x^3-x^2}}-\frac {3 \sqrt [3]{x-1} x^{2/3} \log \left (\frac {\sqrt [3]{x-1}}{\sqrt [3]{x}}-1\right )}{2 \sqrt [3]{x^3-x^2}}-\frac {\sqrt [3]{x-1} x^{2/3} \log (x)}{2 \sqrt [3]{x^3-x^2}}-\frac {\sqrt {3} \sqrt [3]{x-1} x^{2/3} \tan ^{-1}\left (\frac {2 \sqrt [3]{x-1}}{\sqrt {3} \sqrt [3]{x}}+\frac {1}{\sqrt {3}}\right )}{\sqrt [3]{x^3-x^2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 59
Rule 78
Rule 2056
Rubi steps
\begin {align*} \int \frac {1+x}{(-1+x) \sqrt [3]{-x^2+x^3}} \, dx &=\frac {\left (\sqrt [3]{-1+x} x^{2/3}\right ) \int \frac {1+x}{(-1+x)^{4/3} x^{2/3}} \, dx}{\sqrt [3]{-x^2+x^3}}\\ &=-\frac {6 x}{\sqrt [3]{-x^2+x^3}}+\frac {\left (\sqrt [3]{-1+x} x^{2/3}\right ) \int \frac {1}{\sqrt [3]{-1+x} x^{2/3}} \, dx}{\sqrt [3]{-x^2+x^3}}\\ &=-\frac {6 x}{\sqrt [3]{-x^2+x^3}}-\frac {\sqrt {3} \sqrt [3]{-1+x} x^{2/3} \tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{-1+x}}{\sqrt {3} \sqrt [3]{x}}\right )}{\sqrt [3]{-x^2+x^3}}-\frac {3 \sqrt [3]{-1+x} x^{2/3} \log \left (-1+\frac {\sqrt [3]{-1+x}}{\sqrt [3]{x}}\right )}{2 \sqrt [3]{-x^2+x^3}}-\frac {\sqrt [3]{-1+x} x^{2/3} \log (x)}{2 \sqrt [3]{-x^2+x^3}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] time = 0.01, size = 36, normalized size = 0.32 \begin {gather*} -\frac {3 \left (x^{2/3} \, _2F_1\left (-\frac {1}{3},-\frac {1}{3};\frac {2}{3};1-x\right )+x\right )}{\sqrt [3]{(x-1) x^2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [A] time = 0.24, size = 114, normalized size = 1.00 \begin {gather*} -\frac {6 \left (-x^2+x^3\right )^{2/3}}{(-1+x) x}+\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{-x^2+x^3}}\right )-\log \left (-x+\sqrt [3]{-x^2+x^3}\right )+\frac {1}{2} \log \left (x^2+x \sqrt [3]{-x^2+x^3}+\left (-x^2+x^3\right )^{2/3}\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 1.76, size = 137, normalized size = 1.20 \begin {gather*} -\frac {2 \, \sqrt {3} {\left (x^{2} - x\right )} \arctan \left (\frac {\sqrt {3} x + 2 \, \sqrt {3} {\left (x^{3} - x^{2}\right )}^{\frac {1}{3}}}{3 \, x}\right ) + 2 \, {\left (x^{2} - x\right )} \log \left (-\frac {x - {\left (x^{3} - x^{2}\right )}^{\frac {1}{3}}}{x}\right ) - {\left (x^{2} - x\right )} \log \left (\frac {x^{2} + {\left (x^{3} - x^{2}\right )}^{\frac {1}{3}} x + {\left (x^{3} - x^{2}\right )}^{\frac {2}{3}}}{x^{2}}\right ) + 12 \, {\left (x^{3} - x^{2}\right )}^{\frac {2}{3}}}{2 \, {\left (x^{2} - x\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.16, size = 74, normalized size = 0.65 \begin {gather*} -\sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{3}} + 1\right )}\right ) - \frac {6}{{\left (-\frac {1}{x} + 1\right )}^{\frac {1}{3}}} + \frac {1}{2} \, \log \left ({\left (-\frac {1}{x} + 1\right )}^{\frac {2}{3}} + {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{3}} + 1\right ) - \log \left ({\left | {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{3}} - 1 \right |}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [C] time = 0.56, size = 40, normalized size = 0.35
method | result | size |
risch | \(-\frac {6 x}{\left (\left (-1+x \right ) x^{2}\right )^{\frac {1}{3}}}+\frac {3 \left (-\mathrm {signum}\left (-1+x \right )\right )^{\frac {1}{3}} x^{\frac {1}{3}} \hypergeom \left (\left [\frac {1}{3}, \frac {1}{3}\right ], \left [\frac {4}{3}\right ], x\right )}{\mathrm {signum}\left (-1+x \right )^{\frac {1}{3}}}\) | \(40\) |
meijerg | \(-\frac {3 \left (-\mathrm {signum}\left (-1+x \right )\right )^{\frac {1}{3}} x^{\frac {1}{3}}}{\mathrm {signum}\left (-1+x \right )^{\frac {1}{3}} \left (1-x \right )^{\frac {1}{3}}}-\frac {3 \left (-\mathrm {signum}\left (-1+x \right )\right )^{\frac {1}{3}} x^{\frac {4}{3}} \hypergeom \left (\left [\frac {4}{3}, \frac {4}{3}\right ], \left [\frac {7}{3}\right ], x\right )}{4 \mathrm {signum}\left (-1+x \right )^{\frac {1}{3}}}\) | \(54\) |
trager | \(-\frac {6 \left (x^{3}-x^{2}\right )^{\frac {2}{3}}}{\left (-1+x \right ) x}+3 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) \ln \left (\frac {45 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )^{2} x^{2}+72 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) \left (x^{3}-x^{2}\right )^{\frac {2}{3}}+72 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) \left (x^{3}-x^{2}\right )^{\frac {1}{3}} x -90 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )^{2} x +57 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) x^{2}-9 \left (x^{3}-x^{2}\right )^{\frac {2}{3}}-9 x \left (x^{3}-x^{2}\right )^{\frac {1}{3}}-9 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) x -4 x^{2}+x}{x}\right )-3 \ln \left (-\frac {-45 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )^{2} x^{2}+72 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) \left (x^{3}-x^{2}\right )^{\frac {2}{3}}+72 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) \left (x^{3}-x^{2}\right )^{\frac {1}{3}} x +90 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )^{2} x +87 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) x^{2}-15 \left (x^{3}-x^{2}\right )^{\frac {2}{3}}-15 x \left (x^{3}-x^{2}\right )^{\frac {1}{3}}-69 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) x -20 x^{2}+12 x}{x}\right ) \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )+\ln \left (-\frac {-45 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )^{2} x^{2}+72 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) \left (x^{3}-x^{2}\right )^{\frac {2}{3}}+72 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) \left (x^{3}-x^{2}\right )^{\frac {1}{3}} x +90 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )^{2} x +87 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) x^{2}-15 \left (x^{3}-x^{2}\right )^{\frac {2}{3}}-15 x \left (x^{3}-x^{2}\right )^{\frac {1}{3}}-69 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) x -20 x^{2}+12 x}{x}\right )\) | \(511\) |
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 + 1}{{\left (x^{3} - x^{2}\right )}^{\frac {1}{3}} {\left (x - 1\right )}}\,{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+1}{{\left (x^3-x^2\right )}^{1/3}\,\left (x-1\right )} \,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 + 1}{\sqrt [3]{x^{2} \left (x - 1\right )} \left (x - 1\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________