Optimal. Leaf size=90 \[ \frac {3 \sqrt [3]{x^4+1}}{x}-\log \left (\sqrt [3]{x^4+1}+x\right )+\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} x}{2 \sqrt [3]{x^4+1}-x}\right )+\frac {1}{2} \log \left (-\sqrt [3]{x^4+1} x+\left (x^4+1\right )^{2/3}+x^2\right ) \]
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Rubi [F] time = 0.74, 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 (-3+x^4\right ) \sqrt [3]{1+x^4}}{x^2 \left (1+x^3+x^4\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {\left (-3+x^4\right ) \sqrt [3]{1+x^4}}{x^2 \left (1+x^3+x^4\right )} \, dx &=\int \left (-\frac {3 \sqrt [3]{1+x^4}}{x^2}+\frac {x (3+4 x) \sqrt [3]{1+x^4}}{1+x^3+x^4}\right ) \, dx\\ &=-\left (3 \int \frac {\sqrt [3]{1+x^4}}{x^2} \, dx\right )+\int \frac {x (3+4 x) \sqrt [3]{1+x^4}}{1+x^3+x^4} \, dx\\ &=\frac {3 \, _2F_1\left (-\frac {1}{3},-\frac {1}{4};\frac {3}{4};-x^4\right )}{x}+\int \left (\frac {3 x \sqrt [3]{1+x^4}}{1+x^3+x^4}+\frac {4 x^2 \sqrt [3]{1+x^4}}{1+x^3+x^4}\right ) \, dx\\ &=\frac {3 \, _2F_1\left (-\frac {1}{3},-\frac {1}{4};\frac {3}{4};-x^4\right )}{x}+3 \int \frac {x \sqrt [3]{1+x^4}}{1+x^3+x^4} \, dx+4 \int \frac {x^2 \sqrt [3]{1+x^4}}{1+x^3+x^4} \, dx\\ \end {align*}
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Mathematica [F] time = 0.26, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (-3+x^4\right ) \sqrt [3]{1+x^4}}{x^2 \left (1+x^3+x^4\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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IntegrateAlgebraic [A] time = 0.87, size = 90, normalized size = 1.00 \begin {gather*} \frac {3 \sqrt [3]{1+x^4}}{x}+\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 = 2.86, size = 122, normalized size = 1.36 \begin {gather*} \frac {2 \, \sqrt {3} x \arctan \left (\frac {2 \, \sqrt {3} {\left (x^{4} + 1\right )}^{\frac {1}{3}} x^{2} + 2 \, \sqrt {3} {\left (x^{4} + 1\right )}^{\frac {2}{3}} x + \sqrt {3} {\left (x^{4} + x^{3} + 1\right )}}{3 \, {\left (x^{4} - x^{3} + 1\right )}}\right ) - x \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 ) + 6 \, {\left (x^{4} + 1\right )}^{\frac {1}{3}}}{2 \, x} \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} + 1\right )}^{\frac {1}{3}} {\left (x^{4} - 3\right )}}{{\left (x^{4} + x^{3} + 1\right )} x^{2}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 8.08, size = 477, normalized size = 5.30
method | result | size |
risch | \(\frac {3 \left (x^{4}+1\right )^{\frac {1}{3}}}{x}+\frac {\left (-\ln \left (-\frac {\RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{7}-x^{8}+\RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{8}+2 x^{4}+1\right )^{\frac {1}{3}} x^{5}-2 \left (x^{8}+2 x^{4}+1\right )^{\frac {1}{3}} x^{5}+2 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{8}+2 x^{4}+1\right )^{\frac {2}{3}} x^{2}+\RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{3}-\left (x^{8}+2 x^{4}+1\right )^{\frac {2}{3}} x^{2}-2 x^{4}+\RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{8}+2 x^{4}+1\right )^{\frac {1}{3}} x -2 \left (x^{8}+2 x^{4}+1\right )^{\frac {1}{3}} x -1}{\left (x^{4}+x^{3}+1\right ) \left (x^{4}+1\right )}\right )+\RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \ln \left (\frac {\RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2} x^{7}+\RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{8}-2 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{7}-x^{8}+\RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{8}+2 x^{4}+1\right )^{\frac {1}{3}} x^{5}+x^{7}+\left (x^{8}+2 x^{4}+1\right )^{\frac {1}{3}} x^{5}+\RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2} x^{3}-\RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{8}+2 x^{4}+1\right )^{\frac {2}{3}} x^{2}+2 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{4}-2 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{3}+2 \left (x^{8}+2 x^{4}+1\right )^{\frac {2}{3}} x^{2}-2 x^{4}+\RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{8}+2 x^{4}+1\right )^{\frac {1}{3}} x +x^{3}+\left (x^{8}+2 x^{4}+1\right )^{\frac {1}{3}} x +\RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )-1}{\left (x^{4}+x^{3}+1\right ) \left (x^{4}+1\right )}\right )\right ) \left (\left (x^{4}+1\right )^{2}\right )^{\frac {1}{3}}}{\left (x^{4}+1\right )^{\frac {2}{3}}}\) | \(477\) |
trager | \(\frac {3 \left (x^{4}+1\right )^{\frac {1}{3}}}{x}-3 \ln \left (\frac {81 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )^{2} x^{4}-162 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )^{2} x^{3}-39 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) x^{4}-45 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) \left (x^{4}+1\right )^{\frac {2}{3}} x +45 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) \left (x^{4}+1\right )^{\frac {1}{3}} x^{2}-3 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) x^{3}-10 x^{4}+42 \left (x^{4}+1\right )^{\frac {2}{3}} x -42 x^{2} \left (x^{4}+1\right )^{\frac {1}{3}}+5 x^{3}+81 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )^{2}-39 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )-10}{x^{4}+x^{3}+1}\right ) \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )+3 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) \ln \left (\frac {81 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )^{2} x^{4}-162 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )^{2} x^{3}-15 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) x^{4}+45 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) \left (x^{4}+1\right )^{\frac {2}{3}} x -45 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) \left (x^{4}+1\right )^{\frac {1}{3}} x^{2}+111 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) x^{3}-14 x^{4}+27 \left (x^{4}+1\right )^{\frac {2}{3}} x -27 x^{2} \left (x^{4}+1\right )^{\frac {1}{3}}-14 x^{3}+81 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )^{2}-15 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )-14}{x^{4}+x^{3}+1}\right )+\ln \left (\frac {81 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )^{2} x^{4}-162 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )^{2} x^{3}-39 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) x^{4}-45 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) \left (x^{4}+1\right )^{\frac {2}{3}} x +45 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) \left (x^{4}+1\right )^{\frac {1}{3}} x^{2}-3 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) x^{3}-10 x^{4}+42 \left (x^{4}+1\right )^{\frac {2}{3}} x -42 x^{2} \left (x^{4}+1\right )^{\frac {1}{3}}+5 x^{3}+81 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )^{2}-39 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )-10}{x^{4}+x^{3}+1}\right )\) | \(598\) |
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} + 1\right )}^{\frac {1}{3}} {\left (x^{4} - 3\right )}}{{\left (x^{4} + x^{3} + 1\right )} x^{2}}\,{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 )}^{1/3}\,\left (x^4-3\right )}{x^2\,\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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