Optimal. Leaf size=92 \[ \frac {\left (x^4-1\right )^{2/3}}{4 x^4}-\frac {1}{12} \log \left (\sqrt [3]{x^4-1}+1\right )+\frac {1}{24} \log \left (\left (x^4-1\right )^{2/3}-\sqrt [3]{x^4-1}+1\right )-\frac {\tan ^{-1}\left (\frac {1}{\sqrt {3}}-\frac {2 \sqrt [3]{x^4-1}}{\sqrt {3}}\right )}{4 \sqrt {3}} \]
________________________________________________________________________________________
Rubi [A] time = 0.05, 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, 51, 56, 618, 204, 31} \begin {gather*} \frac {\left (x^4-1\right )^{2/3}}{4 x^4}-\frac {1}{8} \log \left (\sqrt [3]{x^4-1}+1\right )-\frac {\tan ^{-1}\left (\frac {1-2 \sqrt [3]{x^4-1}}{\sqrt {3}}\right )}{4 \sqrt {3}}+\frac {\log (x)}{6} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 31
Rule 51
Rule 56
Rule 204
Rule 266
Rule 618
Rubi steps
\begin {align*} \int \frac {1}{x^5 \sqrt [3]{-1+x^4}} \, dx &=\frac {1}{4} \operatorname {Subst}\left (\int \frac {1}{\sqrt [3]{-1+x} x^2} \, dx,x,x^4\right )\\ &=\frac {\left (-1+x^4\right )^{2/3}}{4 x^4}+\frac {1}{12} \operatorname {Subst}\left (\int \frac {1}{\sqrt [3]{-1+x} x} \, dx,x,x^4\right )\\ &=\frac {\left (-1+x^4\right )^{2/3}}{4 x^4}+\frac {\log (x)}{6}-\frac {1}{8} \operatorname {Subst}\left (\int \frac {1}{1+x} \, dx,x,\sqrt [3]{-1+x^4}\right )+\frac {1}{8} \operatorname {Subst}\left (\int \frac {1}{1-x+x^2} \, dx,x,\sqrt [3]{-1+x^4}\right )\\ &=\frac {\left (-1+x^4\right )^{2/3}}{4 x^4}+\frac {\log (x)}{6}-\frac {1}{8} \log \left (1+\sqrt [3]{-1+x^4}\right )-\frac {1}{4} \operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,-1+2 \sqrt [3]{-1+x^4}\right )\\ &=\frac {\left (-1+x^4\right )^{2/3}}{4 x^4}-\frac {\tan ^{-1}\left (\frac {1-2 \sqrt [3]{-1+x^4}}{\sqrt {3}}\right )}{4 \sqrt {3}}+\frac {\log (x)}{6}-\frac {1}{8} \log \left (1+\sqrt [3]{-1+x^4}\right )\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] time = 0.00, size = 28, normalized size = 0.30 \begin {gather*} \frac {3}{8} \left (x^4-1\right )^{2/3} \, _2F_1\left (\frac {2}{3},2;\frac {5}{3};1-x^4\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [A] time = 0.07, size = 92, normalized size = 1.00 \begin {gather*} \frac {\left (-1+x^4\right )^{2/3}}{4 x^4}-\frac {\tan ^{-1}\left (\frac {1}{\sqrt {3}}-\frac {2 \sqrt [3]{-1+x^4}}{\sqrt {3}}\right )}{4 \sqrt {3}}-\frac {1}{12} \log \left (1+\sqrt [3]{-1+x^4}\right )+\frac {1}{24} \log \left (1-\sqrt [3]{-1+x^4}+\left (-1+x^4\right )^{2/3}\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.47, size = 80, normalized size = 0.87 \begin {gather*} \frac {2 \, \sqrt {3} x^{4} \arctan \left (\frac {2}{3} \, \sqrt {3} {\left (x^{4} - 1\right )}^{\frac {1}{3}} - \frac {1}{3} \, \sqrt {3}\right ) + x^{4} \log \left ({\left (x^{4} - 1\right )}^{\frac {2}{3}} - {\left (x^{4} - 1\right )}^{\frac {1}{3}} + 1\right ) - 2 \, x^{4} \log \left ({\left (x^{4} - 1\right )}^{\frac {1}{3}} + 1\right ) + 6 \, {\left (x^{4} - 1\right )}^{\frac {2}{3}}}{24 \, x^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.43, size = 69, normalized size = 0.75 \begin {gather*} \frac {1}{12} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, {\left (x^{4} - 1\right )}^{\frac {1}{3}} - 1\right )}\right ) + \frac {{\left (x^{4} - 1\right )}^{\frac {2}{3}}}{4 \, x^{4}} + \frac {1}{24} \, \log \left ({\left (x^{4} - 1\right )}^{\frac {2}{3}} - {\left (x^{4} - 1\right )}^{\frac {1}{3}} + 1\right ) - \frac {1}{12} \, \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.
[In]
[Out]
________________________________________________________________________________________
maple [C] time = 4.36, size = 96, normalized size = 1.04
method | result | size |
risch | \(\frac {\left (x^{4}-1\right )^{\frac {2}{3}}}{4 x^{4}}+\frac {\sqrt {3}\, \Gamma \left (\frac {2}{3}\right ) \left (-\mathrm {signum}\left (x^{4}-1\right )\right )^{\frac {1}{3}} \left (\frac {2 \pi \sqrt {3}\, x^{4} \hypergeom \left (\left [1, 1, \frac {4}{3}\right ], \left [2, 2\right ], x^{4}\right )}{9 \Gamma \left (\frac {2}{3}\right )}+\frac {2 \left (-\frac {\pi \sqrt {3}}{6}-\frac {3 \ln \relax (3)}{2}+4 \ln \relax (x )+i \pi \right ) \pi \sqrt {3}}{3 \Gamma \left (\frac {2}{3}\right )}\right )}{24 \pi \mathrm {signum}\left (x^{4}-1\right )^{\frac {1}{3}}}\) | \(96\) |
meijerg | \(-\frac {\sqrt {3}\, \Gamma \left (\frac {2}{3}\right ) \left (-\mathrm {signum}\left (x^{4}-1\right )\right )^{\frac {1}{3}} \left (-\frac {4 \pi \sqrt {3}\, x^{4} \hypergeom \left (\left [1, 1, \frac {7}{3}\right ], \left [2, 3\right ], x^{4}\right )}{27 \Gamma \left (\frac {2}{3}\right )}-\frac {2 \left (2-\frac {\pi \sqrt {3}}{6}-\frac {3 \ln \relax (3)}{2}+4 \ln \relax (x )+i \pi \right ) \pi \sqrt {3}}{9 \Gamma \left (\frac {2}{3}\right )}+\frac {2 \pi \sqrt {3}}{3 \Gamma \left (\frac {2}{3}\right ) x^{4}}\right )}{8 \pi \mathrm {signum}\left (x^{4}-1\right )^{\frac {1}{3}}}\) | \(97\) |
trager | \(\frac {\left (x^{4}-1\right )^{\frac {2}{3}}}{4 x^{4}}+\frac {\ln \left (\frac {6749049600 \RootOf \left (518400 \textit {\_Z}^{2}-720 \textit {\_Z} +1\right )^{2} x^{4}+105833520 \RootOf \left (518400 \textit {\_Z}^{2}-720 \textit {\_Z} +1\right ) x^{4}-493850 x^{4}+134531280 \RootOf \left (518400 \textit {\_Z}^{2}-720 \textit {\_Z} +1\right ) \left (x^{4}-1\right )^{\frac {2}{3}}+666879 \left (x^{4}-1\right )^{\frac {2}{3}}-134531280 \RootOf \left (518400 \textit {\_Z}^{2}-720 \textit {\_Z} +1\right ) \left (x^{4}-1\right )^{\frac {1}{3}}-107984793600 \RootOf \left (518400 \textit {\_Z}^{2}-720 \textit {\_Z} +1\right )^{2}-666879 \left (x^{4}-1\right )^{\frac {1}{3}}+284510160 \RootOf \left (518400 \textit {\_Z}^{2}-720 \textit {\_Z} +1\right )+458575}{x^{4}}\right )}{12}-60 \ln \left (\frac {6749049600 \RootOf \left (518400 \textit {\_Z}^{2}-720 \textit {\_Z} +1\right )^{2} x^{4}+105833520 \RootOf \left (518400 \textit {\_Z}^{2}-720 \textit {\_Z} +1\right ) x^{4}-493850 x^{4}+134531280 \RootOf \left (518400 \textit {\_Z}^{2}-720 \textit {\_Z} +1\right ) \left (x^{4}-1\right )^{\frac {2}{3}}+666879 \left (x^{4}-1\right )^{\frac {2}{3}}-134531280 \RootOf \left (518400 \textit {\_Z}^{2}-720 \textit {\_Z} +1\right ) \left (x^{4}-1\right )^{\frac {1}{3}}-107984793600 \RootOf \left (518400 \textit {\_Z}^{2}-720 \textit {\_Z} +1\right )^{2}-666879 \left (x^{4}-1\right )^{\frac {1}{3}}+284510160 \RootOf \left (518400 \textit {\_Z}^{2}-720 \textit {\_Z} +1\right )+458575}{x^{4}}\right ) \RootOf \left (518400 \textit {\_Z}^{2}-720 \textit {\_Z} +1\right )+60 \RootOf \left (518400 \textit {\_Z}^{2}-720 \textit {\_Z} +1\right ) \ln \left (-\frac {-421815600 \RootOf \left (518400 \textit {\_Z}^{2}-720 \textit {\_Z} +1\right )^{2} x^{4}+7786305 \RootOf \left (518400 \textit {\_Z}^{2}-720 \textit {\_Z} +1\right ) x^{4}+20865 x^{4}+8408205 \RootOf \left (518400 \textit {\_Z}^{2}-720 \textit {\_Z} +1\right ) \left (x^{4}-1\right )^{\frac {2}{3}}-53358 \left (x^{4}-1\right )^{\frac {2}{3}}-8408205 \RootOf \left (518400 \textit {\_Z}^{2}-720 \textit {\_Z} +1\right ) \left (x^{4}-1\right )^{\frac {1}{3}}+6749049600 \RootOf \left (518400 \textit {\_Z}^{2}-720 \textit {\_Z} +1\right )^{2}+53358 \left (x^{4}-1\right )^{\frac {1}{3}}-965475 \RootOf \left (518400 \textit {\_Z}^{2}-720 \textit {\_Z} +1\right )-40339}{x^{4}}\right )\) | \(439\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.50, size = 68, normalized size = 0.74 \begin {gather*} \frac {1}{12} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, {\left (x^{4} - 1\right )}^{\frac {1}{3}} - 1\right )}\right ) + \frac {{\left (x^{4} - 1\right )}^{\frac {2}{3}}}{4 \, x^{4}} + \frac {1}{24} \, \log \left ({\left (x^{4} - 1\right )}^{\frac {2}{3}} - {\left (x^{4} - 1\right )}^{\frac {1}{3}} + 1\right ) - \frac {1}{12} \, \log \left ({\left (x^{4} - 1\right )}^{\frac {1}{3}} + 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.94, size = 92, normalized size = 1.00 \begin {gather*} \frac {{\left (x^4-1\right )}^{2/3}}{4\,x^4}-\ln \left (9\,{\left (-\frac {1}{24}+\frac {\sqrt {3}\,1{}\mathrm {i}}{24}\right )}^2+\frac {{\left (x^4-1\right )}^{1/3}}{16}\right )\,\left (-\frac {1}{24}+\frac {\sqrt {3}\,1{}\mathrm {i}}{24}\right )+\ln \left (9\,{\left (\frac {1}{24}+\frac {\sqrt {3}\,1{}\mathrm {i}}{24}\right )}^2+\frac {{\left (x^4-1\right )}^{1/3}}{16}\right )\,\left (\frac {1}{24}+\frac {\sqrt {3}\,1{}\mathrm {i}}{24}\right )-\frac {\ln \left (\frac {{\left (x^4-1\right )}^{1/3}}{16}+\frac {1}{16}\right )}{12} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [C] time = 0.99, size = 34, normalized size = 0.37 \begin {gather*} - \frac {\Gamma \left (\frac {4}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{3}, \frac {4}{3} \\ \frac {7}{3} \end {matrix}\middle | {\frac {e^{2 i \pi }}{x^{4}}} \right )}}{4 x^{\frac {16}{3}} \Gamma \left (\frac {7}{3}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________