Optimal. Leaf size=90 \[ \log \left (\sqrt [3]{x^4-1}+x\right )+\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} x}{2 \sqrt [3]{x^4-1}-x}\right )+\frac {3 \left (x^4-1\right )^{2/3}}{2 x^2}-\frac {1}{2} \log \left (-\sqrt [3]{x^4-1} x+\left (x^4-1\right )^{2/3}+x^2\right ) \]
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Rubi [F] time = 1.01, 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^4\right )^{2/3} \left (3+x^4\right )}{x^3 \left (-1+x^3+x^4\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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\begin {align*} \int \frac {\left (-1+x^4\right )^{2/3} \left (3+x^4\right )}{x^3 \left (-1+x^3+x^4\right )} \, dx &=\int \left (-\frac {3 \left (-1+x^4\right )^{2/3}}{x^3}+\frac {(3+4 x) \left (-1+x^4\right )^{2/3}}{-1+x^3+x^4}\right ) \, dx\\ &=-\left (3 \int \frac {\left (-1+x^4\right )^{2/3}}{x^3} \, dx\right )+\int \frac {(3+4 x) \left (-1+x^4\right )^{2/3}}{-1+x^3+x^4} \, dx\\ &=-\left (\frac {3}{2} \operatorname {Subst}\left (\int \frac {\left (-1+x^2\right )^{2/3}}{x^2} \, dx,x,x^2\right )\right )+\int \left (\frac {3 \left (-1+x^4\right )^{2/3}}{-1+x^3+x^4}+\frac {4 x \left (-1+x^4\right )^{2/3}}{-1+x^3+x^4}\right ) \, dx\\ &=\frac {3 \left (-1+x^4\right )^{2/3}}{2 x^2}-2 \operatorname {Subst}\left (\int \frac {1}{\sqrt [3]{-1+x^2}} \, dx,x,x^2\right )+3 \int \frac {\left (-1+x^4\right )^{2/3}}{-1+x^3+x^4} \, dx+4 \int \frac {x \left (-1+x^4\right )^{2/3}}{-1+x^3+x^4} \, dx\\ &=\frac {3 \left (-1+x^4\right )^{2/3}}{2 x^2}+3 \int \frac {\left (-1+x^4\right )^{2/3}}{-1+x^3+x^4} \, dx+4 \int \frac {x \left (-1+x^4\right )^{2/3}}{-1+x^3+x^4} \, dx-\frac {\left (3 \sqrt {x^4}\right ) \operatorname {Subst}\left (\int \frac {x}{\sqrt {1+x^3}} \, dx,x,\sqrt [3]{-1+x^4}\right )}{x^2}\\ &=\frac {3 \left (-1+x^4\right )^{2/3}}{2 x^2}+3 \int \frac {\left (-1+x^4\right )^{2/3}}{-1+x^3+x^4} \, dx+4 \int \frac {x \left (-1+x^4\right )^{2/3}}{-1+x^3+x^4} \, dx-\frac {\left (3 \sqrt {x^4}\right ) \operatorname {Subst}\left (\int \frac {1-\sqrt {3}+x}{\sqrt {1+x^3}} \, dx,x,\sqrt [3]{-1+x^4}\right )}{x^2}-\frac {\left (3 \sqrt {2 \left (2-\sqrt {3}\right )} \sqrt {x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1+x^3}} \, dx,x,\sqrt [3]{-1+x^4}\right )}{x^2}\\ &=\frac {3 \left (-1+x^4\right )^{2/3}}{2 x^2}-\frac {6 x^2}{1+\sqrt {3}+\sqrt [3]{-1+x^4}}+\frac {3 \sqrt [4]{3} \sqrt {2-\sqrt {3}} \left (1+\sqrt [3]{-1+x^4}\right ) \sqrt {\frac {1-\sqrt [3]{-1+x^4}+\left (-1+x^4\right )^{2/3}}{\left (1+\sqrt {3}+\sqrt [3]{-1+x^4}\right )^2}} E\left (\sin ^{-1}\left (\frac {1-\sqrt {3}+\sqrt [3]{-1+x^4}}{1+\sqrt {3}+\sqrt [3]{-1+x^4}}\right )|-7-4 \sqrt {3}\right )}{x^2 \sqrt {\frac {1+\sqrt [3]{-1+x^4}}{\left (1+\sqrt {3}+\sqrt [3]{-1+x^4}\right )^2}}}-\frac {2 \sqrt {2} 3^{3/4} \left (1+\sqrt [3]{-1+x^4}\right ) \sqrt {\frac {1-\sqrt [3]{-1+x^4}+\left (-1+x^4\right )^{2/3}}{\left (1+\sqrt {3}+\sqrt [3]{-1+x^4}\right )^2}} F\left (\sin ^{-1}\left (\frac {1-\sqrt {3}+\sqrt [3]{-1+x^4}}{1+\sqrt {3}+\sqrt [3]{-1+x^4}}\right )|-7-4 \sqrt {3}\right )}{x^2 \sqrt {\frac {1+\sqrt [3]{-1+x^4}}{\left (1+\sqrt {3}+\sqrt [3]{-1+x^4}\right )^2}}}+3 \int \frac {\left (-1+x^4\right )^{2/3}}{-1+x^3+x^4} \, dx+4 \int \frac {x \left (-1+x^4\right )^{2/3}}{-1+x^3+x^4} \, dx\\ \end {align*}
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Mathematica [F] time = 0.24, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (-1+x^4\right )^{2/3} \left (3+x^4\right )}{x^3 \left (-1+x^3+x^4\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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IntegrateAlgebraic [A] time = 1.02, size = 90, normalized size = 1.00 \begin {gather*} \frac {3 \left (-1+x^4\right )^{2/3}}{2 x^2}+\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} x}{-x+2 \sqrt [3]{-1+x^4}}\right )+\log \left (x+\sqrt [3]{-1+x^4}\right )-\frac {1}{2} \log \left (x^2-x \sqrt [3]{-1+x^4}+\left (-1+x^4\right )^{2/3}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 5.35, size = 131, normalized size = 1.46 \begin {gather*} \frac {2 \, \sqrt {3} x^{2} \arctan \left (-\frac {33798185694614068 \, \sqrt {3} {\left (x^{4} - 1\right )}^{\frac {1}{3}} x^{2} - 35774000716806898 \, \sqrt {3} {\left (x^{4} - 1\right )}^{\frac {2}{3}} x + \sqrt {3} {\left (18215948833549379 \, x^{4} - 16570144372161104 \, x^{3} - 18215948833549379\right )}}{18912305915671589 \, x^{4} + 15948583382382344 \, x^{3} - 18912305915671589}\right ) + x^{2} \log \left (\frac {x^{4} + x^{3} + 3 \, {\left (x^{4} - 1\right )}^{\frac {1}{3}} x^{2} + 3 \, {\left (x^{4} - 1\right )}^{\frac {2}{3}} x - 1}{x^{4} + x^{3} - 1}\right ) + 3 \, {\left (x^{4} - 1\right )}^{\frac {2}{3}}}{2 \, x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (x^{4} + 3\right )} {\left (x^{4} - 1\right )}^{\frac {2}{3}}}{{\left (x^{4} + x^{3} - 1\right )} x^{3}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 8.83, size = 267, normalized size = 2.97
method | result | size |
risch | \(\frac {3 \left (x^{4}-1\right )^{\frac {2}{3}}}{2 x^{2}}+\ln \left (\frac {-\RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x^{3}+\RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{4}+\RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{4}-1\right )^{\frac {2}{3}} x -\left (x^{4}-1\right )^{\frac {1}{3}} \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{2}-2 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{3}+x^{4}+2 \left (x^{4}-1\right )^{\frac {2}{3}} x +\left (x^{4}-1\right )^{\frac {1}{3}} x^{2}-x^{3}-\RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )-1}{x^{4}+x^{3}-1}\right )+\RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \ln \left (-\frac {-\RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x^{3}-\RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{4}+\RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{4}-1\right )^{\frac {2}{3}} x +2 \left (x^{4}-1\right )^{\frac {1}{3}} \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{2}-\RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{3}-x^{4}-\left (x^{4}-1\right )^{\frac {2}{3}} x +\left (x^{4}-1\right )^{\frac {1}{3}} x^{2}+\RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )+1}{x^{4}+x^{3}-1}\right )\) | \(267\) |
trager | \(\frac {3 \left (x^{4}-1\right )^{\frac {2}{3}}}{2 x^{2}}+12 \RootOf \left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right ) \ln \left (-\frac {-38450304 \RootOf \left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right )^{2} x^{4}+72094320 \RootOf \left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right )^{2} x^{3}-9421908 \RootOf \left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right ) x^{4}+9414864 \RootOf \left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right ) \left (x^{4}-1\right )^{\frac {2}{3}} x -9414864 \left (x^{4}-1\right )^{\frac {1}{3}} \RootOf \left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right ) x^{2}+9205008 \RootOf \left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right ) x^{3}-533445 x^{4}+14715 \left (x^{4}-1\right )^{\frac {2}{3}} x -14715 \left (x^{4}-1\right )^{\frac {1}{3}} x^{2}+248941 x^{3}+38450304 \RootOf \left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right )^{2}+9421908 \RootOf \left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right )+533445}{x^{4}+x^{3}-1}\right )-12 \ln \left (\frac {38450304 \RootOf \left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right )^{2} x^{4}-72094320 \RootOf \left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right )^{2} x^{3}-3013524 \RootOf \left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right ) x^{4}+9414864 \RootOf \left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right ) \left (x^{4}-1\right )^{\frac {2}{3}} x -9414864 \left (x^{4}-1\right )^{\frac {1}{3}} \RootOf \left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right ) x^{2}-2810712 \RootOf \left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right ) x^{3}+15302 x^{4}+769857 \left (x^{4}-1\right )^{\frac {2}{3}} x -769857 \left (x^{4}-1\right )^{\frac {1}{3}} x^{2}+17488 x^{3}-38450304 \RootOf \left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right )^{2}+3013524 \RootOf \left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right )-15302}{x^{4}+x^{3}-1}\right ) \RootOf \left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right )-\ln \left (\frac {38450304 \RootOf \left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right )^{2} x^{4}-72094320 \RootOf \left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right )^{2} x^{3}-3013524 \RootOf \left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right ) x^{4}+9414864 \RootOf \left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right ) \left (x^{4}-1\right )^{\frac {2}{3}} x -9414864 \left (x^{4}-1\right )^{\frac {1}{3}} \RootOf \left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right ) x^{2}-2810712 \RootOf \left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right ) x^{3}+15302 x^{4}+769857 \left (x^{4}-1\right )^{\frac {2}{3}} x -769857 \left (x^{4}-1\right )^{\frac {1}{3}} x^{2}+17488 x^{3}-38450304 \RootOf \left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right )^{2}+3013524 \RootOf \left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right )-15302}{x^{4}+x^{3}-1}\right )\) | \(601\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (x^{4} + 3\right )} {\left (x^{4} - 1\right )}^{\frac {2}{3}}}{{\left (x^{4} + x^{3} - 1\right )} x^{3}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\left (x^4-1\right )}^{2/3}\,\left (x^4+3\right )}{x^3\,\left (x^4+x^3-1\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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