Optimal. Leaf size=206 \[ \frac {\log \left (-2 \sqrt [3]{x^6-1}+2^{2/3} x^2+2^{2/3} x-2^{2/3}\right )}{3\ 2^{2/3}}+\frac {\tan ^{-1}\left (\frac {\sqrt {3} \sqrt [3]{x^6-1}}{\sqrt [3]{x^6-1}+2^{2/3} x^2+2^{2/3} x-2^{2/3}}\right )}{2^{2/3} \sqrt {3}}-\frac {\log \left (2 \left (x^6-1\right )^{2/3}+\sqrt [3]{2} x^4+2 \sqrt [3]{2} x^3-\sqrt [3]{2} x^2+\left (2^{2/3} x^2+2^{2/3} x-2^{2/3}\right ) \sqrt [3]{x^6-1}-2 \sqrt [3]{2} x+\sqrt [3]{2}\right )}{6\ 2^{2/3}} \]
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Rubi [F] time = 0.43, 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 {1+x^2}{\left (-1-x+x^2\right ) \sqrt [3]{-1+x^6}} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {1+x^2}{\left (-1-x+x^2\right ) \sqrt [3]{-1+x^6}} \, dx &=\int \left (\frac {1}{\sqrt [3]{-1+x^6}}+\frac {2+x}{\left (-1-x+x^2\right ) \sqrt [3]{-1+x^6}}\right ) \, dx\\ &=\int \frac {1}{\sqrt [3]{-1+x^6}} \, dx+\int \frac {2+x}{\left (-1-x+x^2\right ) \sqrt [3]{-1+x^6}} \, dx\\ &=\frac {\sqrt [3]{1-x^6} \int \frac {1}{\sqrt [3]{1-x^6}} \, dx}{\sqrt [3]{-1+x^6}}+\int \left (\frac {1+\sqrt {5}}{\left (-1-\sqrt {5}+2 x\right ) \sqrt [3]{-1+x^6}}+\frac {1-\sqrt {5}}{\left (-1+\sqrt {5}+2 x\right ) \sqrt [3]{-1+x^6}}\right ) \, dx\\ &=\frac {x \sqrt [3]{1-x^6} \, _2F_1\left (\frac {1}{6},\frac {1}{3};\frac {7}{6};x^6\right )}{\sqrt [3]{-1+x^6}}+\left (1-\sqrt {5}\right ) \int \frac {1}{\left (-1+\sqrt {5}+2 x\right ) \sqrt [3]{-1+x^6}} \, dx+\left (1+\sqrt {5}\right ) \int \frac {1}{\left (-1-\sqrt {5}+2 x\right ) \sqrt [3]{-1+x^6}} \, dx\\ \end {align*}
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Mathematica [F] time = 0.20, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1+x^2}{\left (-1-x+x^2\right ) \sqrt [3]{-1+x^6}} \, dx \end {gather*}
Verification is not applicable to the result.
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IntegrateAlgebraic [A] time = 6.83, size = 206, normalized size = 1.00 \begin {gather*} \frac {\log \left (-2 \sqrt [3]{x^6-1}+2^{2/3} x^2+2^{2/3} x-2^{2/3}\right )}{3\ 2^{2/3}}+\frac {\tan ^{-1}\left (\frac {\sqrt {3} \sqrt [3]{x^6-1}}{\sqrt [3]{x^6-1}+2^{2/3} x^2+2^{2/3} x-2^{2/3}}\right )}{2^{2/3} \sqrt {3}}-\frac {\log \left (2 \left (x^6-1\right )^{2/3}+\sqrt [3]{2} x^4+2 \sqrt [3]{2} x^3-\sqrt [3]{2} x^2+\left (2^{2/3} x^2+2^{2/3} x-2^{2/3}\right ) \sqrt [3]{x^6-1}-2 \sqrt [3]{2} x+\sqrt [3]{2}\right )}{6\ 2^{2/3}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 33.83, size = 223, normalized size = 1.08 \begin {gather*} -\frac {1}{6} \cdot 4^{\frac {1}{6}} \sqrt {3} \arctan \left (\frac {4^{\frac {1}{6}} \sqrt {3} {\left (2 \cdot 4^{\frac {2}{3}} {\left (x^{6} - 1\right )}^{\frac {2}{3}} {\left (x^{2} + x - 1\right )} - 4^{\frac {1}{3}} {\left (x^{6} - 3 \, x^{5} + 5 \, x^{3} - 3 \, x - 1\right )} - 4 \, {\left (x^{6} - 1\right )}^{\frac {1}{3}} {\left (x^{4} + 2 \, x^{3} - x^{2} - 2 \, x + 1\right )}\right )}}{6 \, {\left (3 \, x^{6} + 3 \, x^{5} - 5 \, x^{3} + 3 \, x - 3\right )}}\right ) - \frac {1}{24} \cdot 4^{\frac {2}{3}} \log \left (\frac {4^{\frac {2}{3}} {\left (x^{6} - 1\right )}^{\frac {2}{3}} + 4^{\frac {1}{3}} {\left (x^{4} + 2 \, x^{3} - x^{2} - 2 \, x + 1\right )} + 2 \, {\left (x^{6} - 1\right )}^{\frac {1}{3}} {\left (x^{2} + x - 1\right )}}{x^{4} - 2 \, x^{3} - x^{2} + 2 \, x + 1}\right ) + \frac {1}{12} \cdot 4^{\frac {2}{3}} \log \left (-\frac {4^{\frac {1}{3}} {\left (x^{2} + x - 1\right )} - 2 \, {\left (x^{6} - 1\right )}^{\frac {1}{3}}}{x^{2} - x - 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^{2} + 1}{{\left (x^{6} - 1\right )}^{\frac {1}{3}} {\left (x^{2} - x - 1\right )}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 30.58, size = 1807, normalized size = 8.77 \begin {gather*} \text {Expression too large to display} \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^{2} + 1}{{\left (x^{6} - 1\right )}^{\frac {1}{3}} {\left (x^{2} - x - 1\right )}}\,{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 = -0.00 \begin {gather*} \int -\frac {x^2+1}{{\left (x^6-1\right )}^{1/3}\,\left (-x^2+x+1\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{2} + 1}{\sqrt [3]{\left (x - 1\right ) \left (x + 1\right ) \left (x^{2} - x + 1\right ) \left (x^{2} + x + 1\right )} \left (x^{2} - x - 1\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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