Optimal. Leaf size=151 \[ -\frac {7}{3} 2^{2/3} \log \left (2^{2/3} \sqrt [3]{x^3-1}-2 x\right )+\frac {7\ 2^{2/3} \tan ^{-1}\left (\frac {\sqrt {3} x}{2^{2/3} \sqrt [3]{x^3-1}+x}\right )}{\sqrt {3}}+\frac {7 \log \left (2^{2/3} \sqrt [3]{x^3-1} x+\sqrt [3]{2} \left (x^3-1\right )^{2/3}+2 x^2\right )}{3 \sqrt [3]{2}}+\frac {\left (x^3-1\right )^{2/3} \left (-22 x^6-15 x^3+2\right )}{5 x^5 \left (x^3+1\right )} \]
________________________________________________________________________________________
Rubi [F] time = 1.88, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {\left (-1+x^3\right )^{2/3} \left (-2+2 x^3+x^6\right )}{x^6 \left (1+x^3\right )^2} \, dx \end {gather*}
Verification is not applicable to the result.
[In]
[Out]
Rubi steps
\begin {align*} \int \frac {\left (-1+x^3\right )^{2/3} \left (-2+2 x^3+x^6\right )}{x^6 \left (1+x^3\right )^2} \, dx &=\int \left (-\frac {2 \left (-1+x^3\right )^{2/3}}{x^6}+\frac {6 \left (-1+x^3\right )^{2/3}}{x^3}-\frac {\left (-1+x^3\right )^{2/3}}{3 (1+x)^2}-\frac {8 \left (-1+x^3\right )^{2/3}}{3 (1+x)}+\frac {(-1+x) \left (-1+x^3\right )^{2/3}}{\left (1-x+x^2\right )^2}+\frac {(-15+8 x) \left (-1+x^3\right )^{2/3}}{3 \left (1-x+x^2\right )}\right ) \, dx\\ &=-\left (\frac {1}{3} \int \frac {\left (-1+x^3\right )^{2/3}}{(1+x)^2} \, dx\right )+\frac {1}{3} \int \frac {(-15+8 x) \left (-1+x^3\right )^{2/3}}{1-x+x^2} \, dx-2 \int \frac {\left (-1+x^3\right )^{2/3}}{x^6} \, dx-\frac {8}{3} \int \frac {\left (-1+x^3\right )^{2/3}}{1+x} \, dx+6 \int \frac {\left (-1+x^3\right )^{2/3}}{x^3} \, dx+\int \frac {(-1+x) \left (-1+x^3\right )^{2/3}}{\left (1-x+x^2\right )^2} \, dx\\ &=-\frac {3 \left (-1+x^3\right )^{2/3}}{x^2}-\frac {2 \left (-1+x^3\right )^{5/3}}{5 x^5}-\frac {1}{3} \int \frac {\left (-1+x^3\right )^{2/3}}{(1+x)^2} \, dx+\frac {1}{3} \int \left (\frac {\left (8+\frac {22 i}{\sqrt {3}}\right ) \left (-1+x^3\right )^{2/3}}{-1-i \sqrt {3}+2 x}+\frac {\left (8-\frac {22 i}{\sqrt {3}}\right ) \left (-1+x^3\right )^{2/3}}{-1+i \sqrt {3}+2 x}\right ) \, dx-\frac {8}{3} \int \frac {\left (-1+x^3\right )^{2/3}}{1+x} \, dx+6 \int \frac {1}{\sqrt [3]{-1+x^3}} \, dx+\int \left (-\frac {\left (-1+x^3\right )^{2/3}}{\left (1-x+x^2\right )^2}+\frac {x \left (-1+x^3\right )^{2/3}}{\left (1-x+x^2\right )^2}\right ) \, dx\\ &=-\frac {3 \left (-1+x^3\right )^{2/3}}{x^2}-\frac {2 \left (-1+x^3\right )^{5/3}}{5 x^5}+2 \sqrt {3} \tan ^{-1}\left (\frac {1+\frac {2 x}{\sqrt [3]{-1+x^3}}}{\sqrt {3}}\right )-3 \log \left (-x+\sqrt [3]{-1+x^3}\right )-\frac {1}{3} \int \frac {\left (-1+x^3\right )^{2/3}}{(1+x)^2} \, dx-\frac {8}{3} \int \frac {\left (-1+x^3\right )^{2/3}}{1+x} \, dx+\frac {1}{9} \left (2 \left (12-11 i \sqrt {3}\right )\right ) \int \frac {\left (-1+x^3\right )^{2/3}}{-1+i \sqrt {3}+2 x} \, dx+\frac {1}{9} \left (2 \left (12+11 i \sqrt {3}\right )\right ) \int \frac {\left (-1+x^3\right )^{2/3}}{-1-i \sqrt {3}+2 x} \, dx-\int \frac {\left (-1+x^3\right )^{2/3}}{\left (1-x+x^2\right )^2} \, dx+\int \frac {x \left (-1+x^3\right )^{2/3}}{\left (1-x+x^2\right )^2} \, dx\\ &=-\frac {3 \left (-1+x^3\right )^{2/3}}{x^2}-\frac {2 \left (-1+x^3\right )^{5/3}}{5 x^5}+2 \sqrt {3} \tan ^{-1}\left (\frac {1+\frac {2 x}{\sqrt [3]{-1+x^3}}}{\sqrt {3}}\right )-3 \log \left (-x+\sqrt [3]{-1+x^3}\right )-\frac {1}{3} \int \frac {\left (-1+x^3\right )^{2/3}}{(1+x)^2} \, dx-\frac {8}{3} \int \frac {\left (-1+x^3\right )^{2/3}}{1+x} \, dx+\frac {1}{9} \left (2 \left (12-11 i \sqrt {3}\right )\right ) \int \frac {\left (-1+x^3\right )^{2/3}}{-1+i \sqrt {3}+2 x} \, dx+\frac {1}{9} \left (2 \left (12+11 i \sqrt {3}\right )\right ) \int \frac {\left (-1+x^3\right )^{2/3}}{-1-i \sqrt {3}+2 x} \, dx+\int \left (-\frac {2 \left (1+i \sqrt {3}\right ) \left (-1+x^3\right )^{2/3}}{3 \left (1+i \sqrt {3}-2 x\right )^2}+\frac {2 i \left (-1+x^3\right )^{2/3}}{3 \sqrt {3} \left (1+i \sqrt {3}-2 x\right )}-\frac {2 \left (1-i \sqrt {3}\right ) \left (-1+x^3\right )^{2/3}}{3 \left (-1+i \sqrt {3}+2 x\right )^2}+\frac {2 i \left (-1+x^3\right )^{2/3}}{3 \sqrt {3} \left (-1+i \sqrt {3}+2 x\right )}\right ) \, dx-\int \left (-\frac {4 \left (-1+x^3\right )^{2/3}}{3 \left (1+i \sqrt {3}-2 x\right )^2}+\frac {4 i \left (-1+x^3\right )^{2/3}}{3 \sqrt {3} \left (1+i \sqrt {3}-2 x\right )}-\frac {4 \left (-1+x^3\right )^{2/3}}{3 \left (-1+i \sqrt {3}+2 x\right )^2}+\frac {4 i \left (-1+x^3\right )^{2/3}}{3 \sqrt {3} \left (-1+i \sqrt {3}+2 x\right )}\right ) \, dx\\ &=-\frac {3 \left (-1+x^3\right )^{2/3}}{x^2}-\frac {2 \left (-1+x^3\right )^{5/3}}{5 x^5}+2 \sqrt {3} \tan ^{-1}\left (\frac {1+\frac {2 x}{\sqrt [3]{-1+x^3}}}{\sqrt {3}}\right )-3 \log \left (-x+\sqrt [3]{-1+x^3}\right )-\frac {1}{3} \int \frac {\left (-1+x^3\right )^{2/3}}{(1+x)^2} \, dx+\frac {4}{3} \int \frac {\left (-1+x^3\right )^{2/3}}{\left (1+i \sqrt {3}-2 x\right )^2} \, dx+\frac {4}{3} \int \frac {\left (-1+x^3\right )^{2/3}}{\left (-1+i \sqrt {3}+2 x\right )^2} \, dx-\frac {8}{3} \int \frac {\left (-1+x^3\right )^{2/3}}{1+x} \, dx+\frac {(2 i) \int \frac {\left (-1+x^3\right )^{2/3}}{1+i \sqrt {3}-2 x} \, dx}{3 \sqrt {3}}+\frac {(2 i) \int \frac {\left (-1+x^3\right )^{2/3}}{-1+i \sqrt {3}+2 x} \, dx}{3 \sqrt {3}}-\frac {(4 i) \int \frac {\left (-1+x^3\right )^{2/3}}{1+i \sqrt {3}-2 x} \, dx}{3 \sqrt {3}}-\frac {(4 i) \int \frac {\left (-1+x^3\right )^{2/3}}{-1+i \sqrt {3}+2 x} \, dx}{3 \sqrt {3}}-\frac {1}{3} \left (2 \left (1-i \sqrt {3}\right )\right ) \int \frac {\left (-1+x^3\right )^{2/3}}{\left (-1+i \sqrt {3}+2 x\right )^2} \, dx-\frac {1}{3} \left (2 \left (1+i \sqrt {3}\right )\right ) \int \frac {\left (-1+x^3\right )^{2/3}}{\left (1+i \sqrt {3}-2 x\right )^2} \, dx+\frac {1}{9} \left (2 \left (12-11 i \sqrt {3}\right )\right ) \int \frac {\left (-1+x^3\right )^{2/3}}{-1+i \sqrt {3}+2 x} \, dx+\frac {1}{9} \left (2 \left (12+11 i \sqrt {3}\right )\right ) \int \frac {\left (-1+x^3\right )^{2/3}}{-1-i \sqrt {3}+2 x} \, dx\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.59, size = 146, normalized size = 0.97 \begin {gather*} \frac {7 \left (-2 \log \left (1-\frac {\sqrt [3]{2} x}{\sqrt [3]{1-x^3}}\right )+2 \sqrt {3} \tan ^{-1}\left (\frac {\frac {2 \sqrt [3]{2} x}{\sqrt [3]{1-x^3}}+1}{\sqrt {3}}\right )+\log \left (\frac {\sqrt [3]{2} x}{\sqrt [3]{1-x^3}}+\frac {2^{2/3} x^2}{\left (1-x^3\right )^{2/3}}+1\right )\right )}{3 \sqrt [3]{2}}+\left (x^3-1\right )^{2/3} \left (\frac {2}{5 x^5}-\frac {x}{x^3+1}-\frac {17}{5 x^2}\right ) \end {gather*}
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [A] time = 0.48, size = 151, normalized size = 1.00 \begin {gather*} \frac {\left (-1+x^3\right )^{2/3} \left (2-15 x^3-22 x^6\right )}{5 x^5 \left (1+x^3\right )}+\frac {7\ 2^{2/3} \tan ^{-1}\left (\frac {\sqrt {3} x}{x+2^{2/3} \sqrt [3]{-1+x^3}}\right )}{\sqrt {3}}-\frac {7}{3} 2^{2/3} \log \left (-2 x+2^{2/3} \sqrt [3]{-1+x^3}\right )+\frac {7 \log \left (2 x^2+2^{2/3} x \sqrt [3]{-1+x^3}+\sqrt [3]{2} \left (-1+x^3\right )^{2/3}\right )}{3 \sqrt [3]{2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [B] time = 3.08, size = 294, normalized size = 1.95 \begin {gather*} -\frac {70 \, \sqrt {3} \left (-4\right )^{\frac {1}{3}} {\left (x^{8} + x^{5}\right )} \arctan \left (\frac {3 \, \sqrt {3} \left (-4\right )^{\frac {2}{3}} {\left (5 \, x^{7} + 4 \, x^{4} - x\right )} {\left (x^{3} - 1\right )}^{\frac {2}{3}} + 6 \, \sqrt {3} \left (-4\right )^{\frac {1}{3}} {\left (19 \, x^{8} - 16 \, x^{5} + x^{2}\right )} {\left (x^{3} - 1\right )}^{\frac {1}{3}} - \sqrt {3} {\left (71 \, x^{9} - 111 \, x^{6} + 33 \, x^{3} - 1\right )}}{3 \, {\left (109 \, x^{9} - 105 \, x^{6} + 3 \, x^{3} + 1\right )}}\right ) - 70 \, \left (-4\right )^{\frac {1}{3}} {\left (x^{8} + x^{5}\right )} \log \left (-\frac {3 \, \left (-4\right )^{\frac {2}{3}} {\left (x^{3} - 1\right )}^{\frac {1}{3}} x^{2} - 6 \, {\left (x^{3} - 1\right )}^{\frac {2}{3}} x + \left (-4\right )^{\frac {1}{3}} {\left (x^{3} + 1\right )}}{x^{3} + 1}\right ) + 35 \, \left (-4\right )^{\frac {1}{3}} {\left (x^{8} + x^{5}\right )} \log \left (-\frac {6 \, \left (-4\right )^{\frac {1}{3}} {\left (5 \, x^{4} - x\right )} {\left (x^{3} - 1\right )}^{\frac {2}{3}} - \left (-4\right )^{\frac {2}{3}} {\left (19 \, x^{6} - 16 \, x^{3} + 1\right )} - 24 \, {\left (2 \, x^{5} - x^{2}\right )} {\left (x^{3} - 1\right )}^{\frac {1}{3}}}{x^{6} + 2 \, x^{3} + 1}\right ) + 18 \, {\left (22 \, x^{6} + 15 \, x^{3} - 2\right )} {\left (x^{3} - 1\right )}^{\frac {2}{3}}}{90 \, {\left (x^{8} + x^{5}\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 {{\left (x^{6} + 2 \, x^{3} - 2\right )} {\left (x^{3} - 1\right )}^{\frac {2}{3}}}{{\left (x^{3} + 1\right )}^{2} x^{6}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [C] time = 16.76, size = 635, normalized size = 4.21
method | result | size |
risch | \(-\frac {22 x^{9}-7 x^{6}-17 x^{3}+2}{5 x^{5} \left (x^{3}-1\right )^{\frac {1}{3}} \left (x^{3}+1\right )}+14 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+4\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+4\right )+36 \textit {\_Z}^{2}\right ) \ln \left (\frac {9 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+4\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+4\right )+36 \textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}+4\right )^{3} x^{3}+36 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+4\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+4\right )+36 \textit {\_Z}^{2}\right )^{2} \RootOf \left (\textit {\_Z}^{3}+4\right )^{2} x^{3}+12 \left (x^{3}-1\right )^{\frac {2}{3}} \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+4\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+4\right )+36 \textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}+4\right )^{2} x +4 \left (x^{3}-1\right )^{\frac {1}{3}} \RootOf \left (\textit {\_Z}^{3}+4\right )^{2} x^{2}+30 \left (x^{3}-1\right )^{\frac {1}{3}} \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+4\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+4\right )+36 \textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}+4\right ) x^{2}+3 \RootOf \left (\textit {\_Z}^{3}+4\right ) x^{3}+12 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+4\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+4\right )+36 \textit {\_Z}^{2}\right ) x^{3}+2 x \left (x^{3}-1\right )^{\frac {2}{3}}-3 \RootOf \left (\textit {\_Z}^{3}+4\right )-12 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+4\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+4\right )+36 \textit {\_Z}^{2}\right )}{\left (1+x \right ) \left (x^{2}-x +1\right )}\right )+\frac {7 \RootOf \left (\textit {\_Z}^{3}+4\right ) \ln \left (-\frac {3 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+4\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+4\right )+36 \textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}+4\right )^{3} x^{3}+27 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+4\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+4\right )+36 \textit {\_Z}^{2}\right )^{2} \RootOf \left (\textit {\_Z}^{3}+4\right )^{2} x^{3}+6 \left (x^{3}-1\right )^{\frac {2}{3}} \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+4\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+4\right )+36 \textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}+4\right )^{2} x +2 \left (x^{3}-1\right )^{\frac {1}{3}} \RootOf \left (\textit {\_Z}^{3}+4\right )^{2} x^{2}-3 \left (x^{3}-1\right )^{\frac {1}{3}} \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+4\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+4\right )+36 \textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}+4\right ) x^{2}-3 \RootOf \left (\textit {\_Z}^{3}+4\right ) x^{3}-27 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+4\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+4\right )+36 \textit {\_Z}^{2}\right ) x^{3}-5 x \left (x^{3}-1\right )^{\frac {2}{3}}+\RootOf \left (\textit {\_Z}^{3}+4\right )+9 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+4\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+4\right )+36 \textit {\_Z}^{2}\right )}{\left (1+x \right ) \left (x^{2}-x +1\right )}\right )}{3}\) | \(635\) |
trager | \(\text {Expression too large to display}\) | \(1132\) |
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 {{\left (x^{6} + 2 \, x^{3} - 2\right )} {\left (x^{3} - 1\right )}^{\frac {2}{3}}}{{\left (x^{3} + 1\right )}^{2} x^{6}}\,{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 {{\left (x^3-1\right )}^{2/3}\,\left (x^6+2\,x^3-2\right )}{x^6\,{\left (x^3+1\right )}^2} \,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 {\left (\left (x - 1\right ) \left (x^{2} + x + 1\right )\right )^{\frac {2}{3}} \left (x^{6} + 2 x^{3} - 2\right )}{x^{6} \left (x + 1\right )^{2} \left (x^{2} - x + 1\right )^{2}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________