Optimal. Leaf size=81 \[ \frac {1}{2} \log \left (\sqrt [3]{x^4-1}-2 x\right )-\frac {1}{2} \sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} x}{\sqrt [3]{x^4-1}+x}\right )-\frac {1}{4} \log \left (2 \sqrt [3]{x^4-1} x+\left (x^4-1\right )^{2/3}+4 x^2\right ) \]
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Rubi [F] time = 0.64, 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 {3+x^4}{\sqrt [3]{-1+x^4} \left (-1-8 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 {3+x^4}{\sqrt [3]{-1+x^4} \left (-1-8 x^3+x^4\right )} \, dx &=\int \left (\frac {1}{\sqrt [3]{-1+x^4}}+\frac {4 \left (1+2 x^3\right )}{\sqrt [3]{-1+x^4} \left (-1-8 x^3+x^4\right )}\right ) \, dx\\ &=4 \int \frac {1+2 x^3}{\sqrt [3]{-1+x^4} \left (-1-8 x^3+x^4\right )} \, dx+\int \frac {1}{\sqrt [3]{-1+x^4}} \, dx\\ &=4 \int \left (\frac {1}{\sqrt [3]{-1+x^4} \left (-1-8 x^3+x^4\right )}+\frac {2 x^3}{\sqrt [3]{-1+x^4} \left (-1-8 x^3+x^4\right )}\right ) \, dx+\frac {\sqrt [3]{1-x^4} \int \frac {1}{\sqrt [3]{1-x^4}} \, dx}{\sqrt [3]{-1+x^4}}\\ &=\frac {x \sqrt [3]{1-x^4} \, _2F_1\left (\frac {1}{4},\frac {1}{3};\frac {5}{4};x^4\right )}{\sqrt [3]{-1+x^4}}+4 \int \frac {1}{\sqrt [3]{-1+x^4} \left (-1-8 x^3+x^4\right )} \, dx+8 \int \frac {x^3}{\sqrt [3]{-1+x^4} \left (-1-8 x^3+x^4\right )} \, dx\\ \end {align*}
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Mathematica [F] time = 0.15, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {3+x^4}{\sqrt [3]{-1+x^4} \left (-1-8 x^3+x^4\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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IntegrateAlgebraic [A] time = 1.17, size = 81, normalized size = 1.00 \begin {gather*} \frac {1}{2} \log \left (\sqrt [3]{x^4-1}-2 x\right )-\frac {1}{2} \sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} x}{\sqrt [3]{x^4-1}+x}\right )-\frac {1}{4} \log \left (2 \sqrt [3]{x^4-1} x+\left (x^4-1\right )^{2/3}+4 x^2\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 2.30, size = 112, normalized size = 1.38 \begin {gather*} -\frac {1}{2} \, \sqrt {3} \arctan \left (-\frac {8 \, \sqrt {3} {\left (x^{4} - 1\right )}^{\frac {1}{3}} x^{2} - 4 \, \sqrt {3} {\left (x^{4} - 1\right )}^{\frac {2}{3}} x + \sqrt {3} {\left (x^{4} - 8 \, x^{3} - 1\right )}}{3 \, {\left (x^{4} + 8 \, x^{3} - 1\right )}}\right ) + \frac {1}{4} \, \log \left (\frac {x^{4} - 8 \, x^{3} + 12 \, {\left (x^{4} - 1\right )}^{\frac {1}{3}} x^{2} - 6 \, {\left (x^{4} - 1\right )}^{\frac {2}{3}} x - 1}{x^{4} - 8 \, x^{3} - 1}\right ) \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 {x^{4} + 3}{{\left (x^{4} - 8 \, x^{3} - 1\right )} {\left (x^{4} - 1\right )}^{\frac {1}{3}}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 3.11, size = 320, normalized size = 3.95 \begin {gather*} \frac {\ln \left (-\frac {32 \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right )^{2} x^{3}+2 \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right ) x^{4}-4 \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right ) \left (x^{4}-1\right )^{\frac {2}{3}} x -8 \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right ) \left (x^{4}-1\right )^{\frac {1}{3}} x^{2}+32 x^{3} \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right )+x^{4}-4 \left (x^{4}-1\right )^{\frac {2}{3}} x +4 \left (x^{4}-1\right )^{\frac {1}{3}} x^{2}+8 x^{3}-2 \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right )-1}{x^{4}-8 x^{3}-1}\right )}{2}+\RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right ) \ln \left (\frac {32 \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right )^{2} x^{3}-2 \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right ) x^{4}-4 \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right ) \left (x^{4}-1\right )^{\frac {2}{3}} x +16 \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right ) \left (x^{4}-1\right )^{\frac {1}{3}} x^{2}+16 x^{3} \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right )-x^{4}+2 \left (x^{4}-1\right )^{\frac {2}{3}} x +4 \left (x^{4}-1\right )^{\frac {1}{3}} x^{2}+2 \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right )+1}{x^{4}-8 x^{3}-1}\right ) \end {gather*}
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 {x^{4} + 3}{{\left (x^{4} - 8 \, x^{3} - 1\right )} {\left (x^{4} - 1\right )}^{\frac {1}{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 {x^4+3}{{\left (x^4-1\right )}^{1/3}\,\left (-x^4+8\,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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