Optimal. Leaf size=76 \[ -\frac {1}{4} \log \left (\sqrt [3]{x^4-1}+1\right )+\frac {1}{8} \log \left (\left (x^4-1\right )^{2/3}-\sqrt [3]{x^4-1}+1\right )-\frac {1}{4} \sqrt {3} \tan ^{-1}\left (\frac {1}{\sqrt {3}}-\frac {2 \sqrt [3]{x^4-1}}{\sqrt {3}}\right ) \]
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Rubi [A] time = 0.03, antiderivative size = 52, normalized size of antiderivative = 0.68, number of steps used = 5, number of rules used = 5, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.385, Rules used = {266, 56, 618, 204, 31} \begin {gather*} -\frac {3}{8} \log \left (\sqrt [3]{x^4-1}+1\right )-\frac {1}{4} \sqrt {3} \tan ^{-1}\left (\frac {1-2 \sqrt [3]{x^4-1}}{\sqrt {3}}\right )+\frac {\log (x)}{2} \end {gather*}
Antiderivative was successfully verified.
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Rule 31
Rule 56
Rule 204
Rule 266
Rule 618
Rubi steps
\begin {align*} \int \frac {1}{x \sqrt [3]{-1+x^4}} \, dx &=\frac {1}{4} \operatorname {Subst}\left (\int \frac {1}{\sqrt [3]{-1+x} x} \, dx,x,x^4\right )\\ &=\frac {\log (x)}{2}-\frac {3}{8} \operatorname {Subst}\left (\int \frac {1}{1+x} \, dx,x,\sqrt [3]{-1+x^4}\right )+\frac {3}{8} \operatorname {Subst}\left (\int \frac {1}{1-x+x^2} \, dx,x,\sqrt [3]{-1+x^4}\right )\\ &=\frac {\log (x)}{2}-\frac {3}{8} \log \left (1+\sqrt [3]{-1+x^4}\right )-\frac {3}{4} \operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,-1+2 \sqrt [3]{-1+x^4}\right )\\ &=-\frac {1}{4} \sqrt {3} \tan ^{-1}\left (\frac {1-2 \sqrt [3]{-1+x^4}}{\sqrt {3}}\right )+\frac {\log (x)}{2}-\frac {3}{8} \log \left (1+\sqrt [3]{-1+x^4}\right )\\ \end {align*}
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Mathematica [C] time = 0.01, size = 28, normalized size = 0.37 \begin {gather*} \frac {3}{8} \left (x^4-1\right )^{2/3} \, _2F_1\left (\frac {2}{3},1;\frac {5}{3};1-x^4\right ) \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.08, size = 76, normalized size = 1.00 \begin {gather*} -\frac {1}{4} \sqrt {3} \tan ^{-1}\left (\frac {1}{\sqrt {3}}-\frac {2 \sqrt [3]{-1+x^4}}{\sqrt {3}}\right )-\frac {1}{4} \log \left (1+\sqrt [3]{-1+x^4}\right )+\frac {1}{8} \log \left (1-\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 = 0.45, size = 58, normalized size = 0.76 \begin {gather*} \frac {1}{4} \, \sqrt {3} \arctan \left (\frac {2}{3} \, \sqrt {3} {\left (x^{4} - 1\right )}^{\frac {1}{3}} - \frac {1}{3} \, \sqrt {3}\right ) + \frac {1}{8} \, \log \left ({\left (x^{4} - 1\right )}^{\frac {2}{3}} - {\left (x^{4} - 1\right )}^{\frac {1}{3}} + 1\right ) - \frac {1}{4} \, \log \left ({\left (x^{4} - 1\right )}^{\frac {1}{3}} + 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.45, size = 57, normalized size = 0.75 \begin {gather*} \frac {1}{4} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, {\left (x^{4} - 1\right )}^{\frac {1}{3}} - 1\right )}\right ) + \frac {1}{8} \, \log \left ({\left (x^{4} - 1\right )}^{\frac {2}{3}} - {\left (x^{4} - 1\right )}^{\frac {1}{3}} + 1\right ) - \frac {1}{4} \, \log \left ({\left | {\left (x^{4} - 1\right )}^{\frac {1}{3}} + 1 \right |}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 3.42, size = 83, normalized size = 1.09
method | result | size |
meijerg | \(\frac {\sqrt {3}\, \Gamma \left (\frac {2}{3}\right ) \left (-\mathrm {signum}\left (x^{4}-1\right )\right )^{\frac {1}{3}} \left (\frac {2 \pi \sqrt {3}\, x^{4} \hypergeom \left (\left [1, 1, \frac {4}{3}\right ], \left [2, 2\right ], x^{4}\right )}{9 \Gamma \left (\frac {2}{3}\right )}+\frac {2 \left (-\frac {\pi \sqrt {3}}{6}-\frac {3 \ln \relax (3)}{2}+4 \ln \relax (x )+i \pi \right ) \pi \sqrt {3}}{3 \Gamma \left (\frac {2}{3}\right )}\right )}{8 \pi \mathrm {signum}\left (x^{4}-1\right )^{\frac {1}{3}}}\) | \(83\) |
trager | \(\frac {\RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \ln \left (-\frac {-13019 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2} x^{4}+173029 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{4}+333840 x^{4}+186849 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{4}-1\right )^{\frac {2}{3}}-853728 \left (x^{4}-1\right )^{\frac {2}{3}}-186849 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{4}-1\right )^{\frac {1}{3}}+208304 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2}+853728 \left (x^{4}-1\right )^{\frac {1}{3}}-21455 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )-645424}{x^{4}}\right )}{4}-\frac {\ln \left (-\frac {13085182254449 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2} x^{4}+153030407784045 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{4}-422270012895374 x^{4}+103124015072835 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{4}-1\right )^{\frac {2}{3}}+601470785188317 \left (x^{4}-1\right )^{\frac {2}{3}}-103124015072835 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{4}-1\right )^{\frac {1}{3}}-209362916071184 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2}-601470785188317 \left (x^{4}-1\right )^{\frac {1}{3}}+312486931144019 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )+392107869117133}{x^{4}}\right ) \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )}{4}+\frac {\ln \left (-\frac {13085182254449 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2} x^{4}+153030407784045 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{4}-422270012895374 x^{4}+103124015072835 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{4}-1\right )^{\frac {2}{3}}+601470785188317 \left (x^{4}-1\right )^{\frac {2}{3}}-103124015072835 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{4}-1\right )^{\frac {1}{3}}-209362916071184 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2}-601470785188317 \left (x^{4}-1\right )^{\frac {1}{3}}+312486931144019 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )+392107869117133}{x^{4}}\right )}{4}\) | \(389\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.42, size = 56, normalized size = 0.74 \begin {gather*} \frac {1}{4} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, {\left (x^{4} - 1\right )}^{\frac {1}{3}} - 1\right )}\right ) + \frac {1}{8} \, \log \left ({\left (x^{4} - 1\right )}^{\frac {2}{3}} - {\left (x^{4} - 1\right )}^{\frac {1}{3}} + 1\right ) - \frac {1}{4} \, \log \left ({\left (x^{4} - 1\right )}^{\frac {1}{3}} + 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.98, size = 80, normalized size = 1.05 \begin {gather*} -\frac {\ln \left (\frac {9\,{\left (x^4-1\right )}^{1/3}}{16}+\frac {9}{16}\right )}{4}-\ln \left (9\,{\left (-\frac {1}{8}+\frac {\sqrt {3}\,1{}\mathrm {i}}{8}\right )}^2+\frac {9\,{\left (x^4-1\right )}^{1/3}}{16}\right )\,\left (-\frac {1}{8}+\frac {\sqrt {3}\,1{}\mathrm {i}}{8}\right )+\ln \left (9\,{\left (\frac {1}{8}+\frac {\sqrt {3}\,1{}\mathrm {i}}{8}\right )}^2+\frac {9\,{\left (x^4-1\right )}^{1/3}}{16}\right )\,\left (\frac {1}{8}+\frac {\sqrt {3}\,1{}\mathrm {i}}{8}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 0.81, size = 34, normalized size = 0.45 \begin {gather*} - \frac {\Gamma \left (\frac {1}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{3}, \frac {1}{3} \\ \frac {4}{3} \end {matrix}\middle | {\frac {e^{2 i \pi }}{x^{4}}} \right )}}{4 x^{\frac {4}{3}} \Gamma \left (\frac {4}{3}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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