Optimal. Leaf size=92 \[ -\frac {\left (x^5-1\right )^{2/3}}{5 x^5}-\frac {2}{15} \log \left (\sqrt [3]{x^5-1}+1\right )+\frac {1}{15} \log \left (\left (x^5-1\right )^{2/3}-\sqrt [3]{x^5-1}+1\right )-\frac {2 \tan ^{-1}\left (\frac {1}{\sqrt {3}}-\frac {2 \sqrt [3]{x^5-1}}{\sqrt {3}}\right )}{5 \sqrt {3}} \]
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Rubi [A] time = 0.04, antiderivative size = 68, normalized size of antiderivative = 0.74, number of steps used = 6, number of rules used = 6, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.462, Rules used = {266, 47, 56, 618, 204, 31} \begin {gather*} -\frac {\left (x^5-1\right )^{2/3}}{5 x^5}-\frac {1}{5} \log \left (\sqrt [3]{x^5-1}+1\right )-\frac {2 \tan ^{-1}\left (\frac {1-2 \sqrt [3]{x^5-1}}{\sqrt {3}}\right )}{5 \sqrt {3}}+\frac {\log (x)}{3} \end {gather*}
Antiderivative was successfully verified.
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Rule 31
Rule 47
Rule 56
Rule 204
Rule 266
Rule 618
Rubi steps
\begin {align*} \int \frac {\left (-1+x^5\right )^{2/3}}{x^6} \, dx &=\frac {1}{5} \operatorname {Subst}\left (\int \frac {(-1+x)^{2/3}}{x^2} \, dx,x,x^5\right )\\ &=-\frac {\left (-1+x^5\right )^{2/3}}{5 x^5}+\frac {2}{15} \operatorname {Subst}\left (\int \frac {1}{\sqrt [3]{-1+x} x} \, dx,x,x^5\right )\\ &=-\frac {\left (-1+x^5\right )^{2/3}}{5 x^5}+\frac {\log (x)}{3}-\frac {1}{5} \operatorname {Subst}\left (\int \frac {1}{1+x} \, dx,x,\sqrt [3]{-1+x^5}\right )+\frac {1}{5} \operatorname {Subst}\left (\int \frac {1}{1-x+x^2} \, dx,x,\sqrt [3]{-1+x^5}\right )\\ &=-\frac {\left (-1+x^5\right )^{2/3}}{5 x^5}+\frac {\log (x)}{3}-\frac {1}{5} \log \left (1+\sqrt [3]{-1+x^5}\right )-\frac {2}{5} \operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,-1+2 \sqrt [3]{-1+x^5}\right )\\ &=-\frac {\left (-1+x^5\right )^{2/3}}{5 x^5}-\frac {2 \tan ^{-1}\left (\frac {1-2 \sqrt [3]{-1+x^5}}{\sqrt {3}}\right )}{5 \sqrt {3}}+\frac {\log (x)}{3}-\frac {1}{5} \log \left (1+\sqrt [3]{-1+x^5}\right )\\ \end {align*}
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Mathematica [C] time = 0.01, size = 28, normalized size = 0.30 \begin {gather*} \frac {3}{25} \left (x^5-1\right )^{5/3} \, _2F_1\left (\frac {5}{3},2;\frac {8}{3};1-x^5\right ) \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.06, size = 92, normalized size = 1.00 \begin {gather*} -\frac {\left (-1+x^5\right )^{2/3}}{5 x^5}-\frac {2 \tan ^{-1}\left (\frac {1}{\sqrt {3}}-\frac {2 \sqrt [3]{-1+x^5}}{\sqrt {3}}\right )}{5 \sqrt {3}}-\frac {2}{15} \log \left (1+\sqrt [3]{-1+x^5}\right )+\frac {1}{15} \log \left (1-\sqrt [3]{-1+x^5}+\left (-1+x^5\right )^{2/3}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.46, size = 80, normalized size = 0.87 \begin {gather*} \frac {2 \, \sqrt {3} x^{5} \arctan \left (\frac {2}{3} \, \sqrt {3} {\left (x^{5} - 1\right )}^{\frac {1}{3}} - \frac {1}{3} \, \sqrt {3}\right ) + x^{5} \log \left ({\left (x^{5} - 1\right )}^{\frac {2}{3}} - {\left (x^{5} - 1\right )}^{\frac {1}{3}} + 1\right ) - 2 \, x^{5} \log \left ({\left (x^{5} - 1\right )}^{\frac {1}{3}} + 1\right ) - 3 \, {\left (x^{5} - 1\right )}^{\frac {2}{3}}}{15 \, x^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.30, size = 69, normalized size = 0.75 \begin {gather*} \frac {2}{15} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, {\left (x^{5} - 1\right )}^{\frac {1}{3}} - 1\right )}\right ) - \frac {{\left (x^{5} - 1\right )}^{\frac {2}{3}}}{5 \, x^{5}} + \frac {1}{15} \, \log \left ({\left (x^{5} - 1\right )}^{\frac {2}{3}} - {\left (x^{5} - 1\right )}^{\frac {1}{3}} + 1\right ) - \frac {2}{15} \, \log \left ({\left | {\left (x^{5} - 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 = 6.72, size = 96, normalized size = 1.04
method | result | size |
risch | \(-\frac {\left (x^{5}-1\right )^{\frac {2}{3}}}{5 x^{5}}+\frac {\sqrt {3}\, \Gamma \left (\frac {2}{3}\right ) \left (-\mathrm {signum}\left (x^{5}-1\right )\right )^{\frac {1}{3}} \left (\frac {2 \pi \sqrt {3}\, x^{5} \hypergeom \left (\left [1, 1, \frac {4}{3}\right ], \left [2, 2\right ], x^{5}\right )}{9 \Gamma \left (\frac {2}{3}\right )}+\frac {2 \left (-\frac {\pi \sqrt {3}}{6}-\frac {3 \ln \relax (3)}{2}+5 \ln \relax (x )+i \pi \right ) \pi \sqrt {3}}{3 \Gamma \left (\frac {2}{3}\right )}\right )}{15 \pi \mathrm {signum}\left (x^{5}-1\right )^{\frac {1}{3}}}\) | \(96\) |
meijerg | \(\frac {\sqrt {3}\, \Gamma \left (\frac {2}{3}\right ) \mathrm {signum}\left (x^{5}-1\right )^{\frac {2}{3}} \left (-\frac {\pi \sqrt {3}\, x^{5} \hypergeom \left (\left [1, 1, \frac {4}{3}\right ], \left [2, 3\right ], x^{5}\right )}{9 \Gamma \left (\frac {2}{3}\right )}-\frac {2 \left (-\frac {\pi \sqrt {3}}{6}-\frac {3 \ln \relax (3)}{2}-1+5 \ln \relax (x )+i \pi \right ) \pi \sqrt {3}}{3 \Gamma \left (\frac {2}{3}\right )}-\frac {\pi \sqrt {3}}{\Gamma \left (\frac {2}{3}\right ) x^{5}}\right )}{15 \pi \left (-\mathrm {signum}\left (x^{5}-1\right )\right )^{\frac {2}{3}}}\) | \(97\) |
trager | \(-\frac {\left (x^{5}-1\right )^{\frac {2}{3}}}{5 x^{5}}-\frac {2 \ln \left (\frac {30885108600167424 \RootOf \left (9216 \textit {\_Z}^{2}-96 \textit {\_Z} +1\right )^{2} x^{5}+9590347318616160 \RootOf \left (9216 \textit {\_Z}^{2}-96 \textit {\_Z} +1\right ) x^{5}-19140349667550 x^{5}+12071260649204448 \RootOf \left (9216 \textit {\_Z}^{2}-96 \textit {\_Z} +1\right ) \left (x^{5}-1\right )^{\frac {2}{3}}+58267503137446 \left (x^{5}-1\right )^{\frac {2}{3}}-5593680301194816 \RootOf \left (9216 \textit {\_Z}^{2}-96 \textit {\_Z} +1\right ) \left (x^{5}-1\right )^{\frac {1}{3}}-988323475205357568 \RootOf \left (9216 \textit {\_Z}^{2}-96 \textit {\_Z} +1\right )^{2}+184009801566659 \left (x^{5}-1\right )^{\frac {1}{3}}-7369904750343456 \RootOf \left (9216 \textit {\_Z}^{2}-96 \textit {\_Z} +1\right )+18502338011965}{x^{5}}\right )}{15}+\frac {2 \ln \left (-\frac {25005193182296064 \RootOf \left (9216 \textit {\_Z}^{2}-96 \textit {\_Z} +1\right )^{2} x^{5}-8013344512846944 \RootOf \left (9216 \textit {\_Z}^{2}-96 \textit {\_Z} +1\right ) x^{5}-19778361323135 x^{5}+12071260649204448 \RootOf \left (9216 \textit {\_Z}^{2}-96 \textit {\_Z} +1\right ) \left (x^{5}-1\right )^{\frac {2}{3}}-184009801566659 \left (x^{5}-1\right )^{\frac {2}{3}}+17664940950399264 \RootOf \left (9216 \textit {\_Z}^{2}-96 \textit {\_Z} +1\right ) \left (x^{5}-1\right )^{\frac {1}{3}}-800166181833474048 \RootOf \left (9216 \textit {\_Z}^{2}-96 \textit {\_Z} +1\right )^{2}-58267503137446 \left (x^{5}-1\right )^{\frac {1}{3}}+13928744695293504 \RootOf \left (9216 \textit {\_Z}^{2}-96 \textit {\_Z} +1\right )+38918710990685}{x^{5}}\right )}{15}-\frac {64 \ln \left (-\frac {25005193182296064 \RootOf \left (9216 \textit {\_Z}^{2}-96 \textit {\_Z} +1\right )^{2} x^{5}-8013344512846944 \RootOf \left (9216 \textit {\_Z}^{2}-96 \textit {\_Z} +1\right ) x^{5}-19778361323135 x^{5}+12071260649204448 \RootOf \left (9216 \textit {\_Z}^{2}-96 \textit {\_Z} +1\right ) \left (x^{5}-1\right )^{\frac {2}{3}}-184009801566659 \left (x^{5}-1\right )^{\frac {2}{3}}+17664940950399264 \RootOf \left (9216 \textit {\_Z}^{2}-96 \textit {\_Z} +1\right ) \left (x^{5}-1\right )^{\frac {1}{3}}-800166181833474048 \RootOf \left (9216 \textit {\_Z}^{2}-96 \textit {\_Z} +1\right )^{2}-58267503137446 \left (x^{5}-1\right )^{\frac {1}{3}}+13928744695293504 \RootOf \left (9216 \textit {\_Z}^{2}-96 \textit {\_Z} +1\right )+38918710990685}{x^{5}}\right ) \RootOf \left (9216 \textit {\_Z}^{2}-96 \textit {\_Z} +1\right )}{5}\) | \(429\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.47, size = 68, normalized size = 0.74 \begin {gather*} \frac {2}{15} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, {\left (x^{5} - 1\right )}^{\frac {1}{3}} - 1\right )}\right ) - \frac {{\left (x^{5} - 1\right )}^{\frac {2}{3}}}{5 \, x^{5}} + \frac {1}{15} \, \log \left ({\left (x^{5} - 1\right )}^{\frac {2}{3}} - {\left (x^{5} - 1\right )}^{\frac {1}{3}} + 1\right ) - \frac {2}{15} \, \log \left ({\left (x^{5} - 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 = 0.94, size = 92, normalized size = 1.00 \begin {gather*} -\frac {2\,\ln \left (\frac {4\,{\left (x^5-1\right )}^{1/3}}{25}+\frac {4}{25}\right )}{15}-\ln \left (9\,{\left (-\frac {1}{15}+\frac {\sqrt {3}\,1{}\mathrm {i}}{15}\right )}^2+\frac {4\,{\left (x^5-1\right )}^{1/3}}{25}\right )\,\left (-\frac {1}{15}+\frac {\sqrt {3}\,1{}\mathrm {i}}{15}\right )+\ln \left (9\,{\left (\frac {1}{15}+\frac {\sqrt {3}\,1{}\mathrm {i}}{15}\right )}^2+\frac {4\,{\left (x^5-1\right )}^{1/3}}{25}\right )\,\left (\frac {1}{15}+\frac {\sqrt {3}\,1{}\mathrm {i}}{15}\right )-\frac {{\left (x^5-1\right )}^{2/3}}{5\,x^5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 1.09, size = 36, normalized size = 0.39 \begin {gather*} - \frac {\Gamma \left (\frac {1}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {2}{3}, \frac {1}{3} \\ \frac {4}{3} \end {matrix}\middle | {\frac {e^{2 i \pi }}{x^{5}}} \right )}}{5 x^{\frac {5}{3}} \Gamma \left (\frac {4}{3}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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