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 = 0.78, 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 {\sqrt [3]{-1+x^4} \left (3+x^4\right )}{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 {\sqrt [3]{-1+x^4} \left (3+x^4\right )}{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 {\left (3 \sqrt [3]{-1+x^4}\right ) \int \frac {\sqrt [3]{1-x^4}}{x^2} \, dx}{\sqrt [3]{1-x^4}}+\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 \sqrt [3]{-1+x^4} \, _2F_1\left (-\frac {1}{3},-\frac {1}{4};\frac {3}{4};x^4\right )}{x \sqrt [3]{1-x^4}}+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 = 0.30, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt [3]{-1+x^4} \left (3+x^4\right )}{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 = 0.85, size = 90, normalized size = 1.00 \begin {gather*} \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 ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 4.35, size = 128, normalized size = 1.42 \begin {gather*} \frac {2 \, \sqrt {3} x \arctan \left (-\frac {33798185694614068 \, \sqrt {3} {\left (x^{4} - 1\right )}^{\frac {1}{3}} x^{2} - 35774000716806898 \, \sqrt {3} {\left (x^{4} - 1\right )}^{\frac {2}{3}} x + \sqrt {3} {\left (18215948833549379 \, x^{4} - 16570144372161104 \, x^{3} - 18215948833549379\right )}}{18912305915671589 \, x^{4} + 15948583382382344 \, x^{3} - 18912305915671589}\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} + 3\right )} {\left (x^{4} - 1\right )}^{\frac {1}{3}}}{{\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 = 3.63, size = 806, normalized size = 8.96
result too large to display
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} + 3\right )} {\left (x^{4} - 1\right )}^{\frac {1}{3}}}{{\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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