Optimal. Leaf size=108 \[ \frac {\left (-1+x^3\right )^{2/3} \left (12-27 x^3+10 x^6\right )}{30 x^5}+\frac {\text {ArcTan}\left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{-1+x^3}}\right )}{3 \sqrt {3}}-\frac {1}{9} \log \left (-x+\sqrt [3]{-1+x^3}\right )+\frac {1}{18} \log \left (x^2+x \sqrt [3]{-1+x^3}+\left (-1+x^3\right )^{2/3}\right ) \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.02, antiderivative size = 95, normalized size of antiderivative = 0.88, number of steps
used = 5, number of rules used = 5, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {1492, 464, 283,
201, 245} \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} x \left (x^3-1\right )^{2/3}-\frac {1}{6} \log \left (\sqrt [3]{x^3-1}-x\right )+\frac {2 \left (x^3-1\right )^{8/3}}{5 x^5}+\frac {\left (x^3-1\right )^{5/3}}{10 x^2} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 201
Rule 245
Rule 283
Rule 464
Rule 1492
Rubi steps
\begin {align*} \int \frac {\left (-1+x^3\right )^{2/3} \left (-2+x^3+x^6\right )}{x^6} \, dx &=\int \frac {\left (-1+x^3\right )^{5/3} \left (2+x^3\right )}{x^6} \, dx\\ &=\frac {2 \left (-1+x^3\right )^{8/3}}{5 x^5}-\frac {1}{5} \int \frac {\left (-1+x^3\right )^{5/3}}{x^3} \, dx\\ &=\frac {\left (-1+x^3\right )^{5/3}}{10 x^2}+\frac {2 \left (-1+x^3\right )^{8/3}}{5 x^5}-\frac {1}{2} \int \left (-1+x^3\right )^{2/3} \, dx\\ &=-\frac {1}{6} x \left (-1+x^3\right )^{2/3}+\frac {\left (-1+x^3\right )^{5/3}}{10 x^2}+\frac {2 \left (-1+x^3\right )^{8/3}}{5 x^5}+\frac {1}{3} \int \frac {1}{\sqrt [3]{-1+x^3}} \, dx\\ &=-\frac {1}{6} x \left (-1+x^3\right )^{2/3}+\frac {\left (-1+x^3\right )^{5/3}}{10 x^2}+\frac {2 \left (-1+x^3\right )^{8/3}}{5 x^5}+\frac {\tan ^{-1}\left (\frac {1+\frac {2 x}{\sqrt [3]{-1+x^3}}}{\sqrt {3}}\right )}{3 \sqrt {3}}-\frac {1}{6} \log \left (-x+\sqrt [3]{-1+x^3}\right )\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.18, size = 104, normalized size = 0.96 \begin {gather*} \frac {1}{90} \left (\frac {3 \left (-1+x^3\right )^{2/3} \left (12-27 x^3+10 x^6\right )}{x^5}+10 \sqrt {3} \text {ArcTan}\left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{-1+x^3}}\right )-10 \log \left (-x+\sqrt [3]{-1+x^3}\right )+5 \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 9 vs. order
3.
time = 2.73, size = 61, normalized size = 0.56
method | result | size |
risch | \(\frac {10 x^{9}-37 x^{6}+39 x^{3}-12}{30 x^{5} \left (x^{3}-1\right )^{\frac {1}{3}}}+\frac {\left (-\mathrm {signum}\left (x^{3}-1\right )\right )^{\frac {1}{3}} x \hypergeom \left (\left [\frac {1}{3}, \frac {1}{3}\right ], \left [\frac {4}{3}\right ], x^{3}\right )}{3 \mathrm {signum}\left (x^{3}-1\right )^{\frac {1}{3}}}\) | \(61\) |
meijerg | \(\frac {\mathrm {signum}\left (x^{3}-1\right )^{\frac {2}{3}} x \hypergeom \left (\left [-\frac {2}{3}, \frac {1}{3}\right ], \left [\frac {4}{3}\right ], x^{3}\right )}{\left (-\mathrm {signum}\left (x^{3}-1\right )\right )^{\frac {2}{3}}}-\frac {\mathrm {signum}\left (x^{3}-1\right )^{\frac {2}{3}} \hypergeom \left (\left [-\frac {2}{3}, -\frac {2}{3}\right ], \left [\frac {1}{3}\right ], x^{3}\right )}{2 \left (-\mathrm {signum}\left (x^{3}-1\right )\right )^{\frac {2}{3}} x^{2}}+\frac {2 \mathrm {signum}\left (x^{3}-1\right )^{\frac {2}{3}} \left (-x^{3}+1\right )^{\frac {5}{3}}}{5 \left (-\mathrm {signum}\left (x^{3}-1\right )\right )^{\frac {2}{3}} x^{5}}\) | \(95\) |
trager | \(\frac {\left (x^{3}-1\right )^{\frac {2}{3}} \left (10 x^{6}-27 x^{3}+12\right )}{30 x^{5}}-\frac {\ln \left (-69686010880 \RootOf \left (1024 \textit {\_Z}^{2}-32 \textit {\_Z} +1\right )^{2} x^{3}-6206811648 \RootOf \left (1024 \textit {\_Z}^{2}-32 \textit {\_Z} +1\right ) \left (x^{3}-1\right )^{\frac {2}{3}} x -42066864288 \left (x^{3}-1\right )^{\frac {1}{3}} \RootOf \left (1024 \textit {\_Z}^{2}-32 \textit {\_Z} +1\right ) x^{2}+50451363776 \RootOf \left (1024 \textit {\_Z}^{2}-32 \textit {\_Z} +1\right ) x^{3}-1314589509 x \left (x^{3}-1\right )^{\frac {2}{3}}+1508552373 x^{2} \left (x^{3}-1\right )^{\frac {1}{3}}-262015609 x^{3}+557488087040 \RootOf \left (1024 \textit {\_Z}^{2}-32 \textit {\_Z} +1\right )^{2}-35581665248 \RootOf \left (1024 \textit {\_Z}^{2}-32 \textit {\_Z} +1\right )+170879745\right )}{9}+\frac {\ln \left (58020620288 \RootOf \left (1024 \textit {\_Z}^{2}-32 \textit {\_Z} +1\right )^{2} x^{3}+6206811648 \RootOf \left (1024 \textit {\_Z}^{2}-32 \textit {\_Z} +1\right ) \left (x^{3}-1\right )^{\frac {2}{3}} x -48273675936 \left (x^{3}-1\right )^{\frac {1}{3}} \RootOf \left (1024 \textit {\_Z}^{2}-32 \textit {\_Z} +1\right ) x^{2}+40253719904 \RootOf \left (1024 \textit {\_Z}^{2}-32 \textit {\_Z} +1\right ) x^{3}-1508552373 x \left (x^{3}-1\right )^{\frac {2}{3}}+1314589509 x^{2} \left (x^{3}-1\right )^{\frac {1}{3}}+250623626 x^{3}-464164962304 \RootOf \left (1024 \textit {\_Z}^{2}-32 \textit {\_Z} +1\right )^{2}-15608358336 \RootOf \left (1024 \textit {\_Z}^{2}-32 \textit {\_Z} +1\right )-79743881\right )}{9}-\frac {32 \ln \left (58020620288 \RootOf \left (1024 \textit {\_Z}^{2}-32 \textit {\_Z} +1\right )^{2} x^{3}+6206811648 \RootOf \left (1024 \textit {\_Z}^{2}-32 \textit {\_Z} +1\right ) \left (x^{3}-1\right )^{\frac {2}{3}} x -48273675936 \left (x^{3}-1\right )^{\frac {1}{3}} \RootOf \left (1024 \textit {\_Z}^{2}-32 \textit {\_Z} +1\right ) x^{2}+40253719904 \RootOf \left (1024 \textit {\_Z}^{2}-32 \textit {\_Z} +1\right ) x^{3}-1508552373 x \left (x^{3}-1\right )^{\frac {2}{3}}+1314589509 x^{2} \left (x^{3}-1\right )^{\frac {1}{3}}+250623626 x^{3}-464164962304 \RootOf \left (1024 \textit {\_Z}^{2}-32 \textit {\_Z} +1\right )^{2}-15608358336 \RootOf \left (1024 \textit {\_Z}^{2}-32 \textit {\_Z} +1\right )-79743881\right ) \RootOf \left (1024 \textit {\_Z}^{2}-32 \textit {\_Z} +1\right )}{9}\) | \(451\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.46, size = 118, normalized size = 1.09 \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 {2}{3}}}{2 \, x^{2}} - \frac {{\left (x^{3} - 1\right )}^{\frac {2}{3}}}{3 \, x^{2} {\left (\frac {x^{3} - 1}{x^{3}} - 1\right )}} - \frac {2 \, {\left (x^{3} - 1\right )}^{\frac {5}{3}}}{5 \, x^{5}} + \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 = 0.72, size = 117, normalized size = 1.08 \begin {gather*} \frac {10 \, \sqrt {3} x^{5} \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 ) - 5 \, x^{5} \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 (10 \, x^{6} - 27 \, x^{3} + 12\right )} {\left (x^{3} - 1\right )}^{\frac {2}{3}}}{90 \, x^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [C] Result contains complex when optimal does not.
time = 2.08, size = 199, normalized size = 1.84 \begin {gather*} - \frac {x e^{- \frac {i \pi }{3}} \Gamma \left (\frac {1}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {2}{3}, \frac {1}{3} \\ \frac {4}{3} \end {matrix}\middle | {x^{3}} \right )}}{3 \Gamma \left (\frac {4}{3}\right )} - 2 \left (\begin {cases} \frac {\left (-1 + \frac {1}{x^{3}}\right )^{\frac {2}{3}} e^{- \frac {i \pi }{3}} \Gamma \left (- \frac {5}{3}\right )}{3 \Gamma \left (- \frac {2}{3}\right )} - \frac {\left (-1 + \frac {1}{x^{3}}\right )^{\frac {2}{3}} e^{- \frac {i \pi }{3}} \Gamma \left (- \frac {5}{3}\right )}{3 x^{3} \Gamma \left (- \frac {2}{3}\right )} & \text {for}\: \frac {1}{\left |{x^{3}}\right |} > 1 \\- \frac {\left (1 - \frac {1}{x^{3}}\right )^{\frac {2}{3}} \Gamma \left (- \frac {5}{3}\right )}{3 \Gamma \left (- \frac {2}{3}\right )} + \frac {\left (1 - \frac {1}{x^{3}}\right )^{\frac {2}{3}} \Gamma \left (- \frac {5}{3}\right )}{3 x^{3} \Gamma \left (- \frac {2}{3}\right )} & \text {otherwise} \end {cases}\right ) + \frac {e^{\frac {2 i \pi }{3}} \Gamma \left (- \frac {2}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {2}{3}, - \frac {2}{3} \\ \frac {1}{3} \end {matrix}\middle | {x^{3}} \right )}}{3 x^{2} \Gamma \left (\frac {1}{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 \frac {{\left (x^3-1\right )}^{2/3}\,\left (x^6+x^3-2\right )}{x^6} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________