Optimal. Leaf size=210 \[ -\frac {\text {ArcTan}\left (\frac {\sqrt {3} \sqrt [3]{-1+x^6}}{-2^{2/3}-2^{2/3} x+2^{2/3} x^2+\sqrt [3]{-1+x^6}}\right )}{2^{2/3} \sqrt {3}}-\frac {\log \left (-2^{2/3}-2^{2/3} x+2^{2/3} x^2-2 \sqrt [3]{-1+x^6}\right )}{3\ 2^{2/3}}+\frac {\log \left (\sqrt [3]{2}+2 \sqrt [3]{2} x-\sqrt [3]{2} x^2-2 \sqrt [3]{2} x^3+\sqrt [3]{2} x^4+\left (-2^{2/3}-2^{2/3} x+2^{2/3} x^2\right ) \sqrt [3]{-1+x^6}+2 \left (-1+x^6\right )^{2/3}\right )}{6\ 2^{2/3}} \]
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Rubi [F]
time = 0.21, 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 [A]
time = 0.00, size = 186, normalized size = 0.89 \begin {gather*} \frac {2 \sqrt {3} \text {ArcTan}\left (\frac {\sqrt {3} \sqrt [3]{-1+x^6}}{2^{2/3}+2^{2/3} x-2^{2/3} x^2-\sqrt [3]{-1+x^6}}\right )-2 \log \left (-2^{2/3}-2^{2/3} x+2^{2/3} x^2-2 \sqrt [3]{-1+x^6}\right )+\log \left (\sqrt [3]{2}+2 \sqrt [3]{2} x-\sqrt [3]{2} x^2-2 \sqrt [3]{2} x^3+\sqrt [3]{2} x^4+2^{2/3} \left (-1-x+x^2\right ) \sqrt [3]{-1+x^6}+2 \left (-1+x^6\right )^{2/3}\right )}{6\ 2^{2/3}} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
3.
time = 32.07, size = 1803, normalized size = 8.59
method | result | size |
trager | \(\text {Expression too large to display}\) | \(1803\) |
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*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 12.64, size = 251, normalized size = 1.20 \begin {gather*} -\frac {1}{6} \cdot 4^{\frac {1}{6}} \sqrt {3} \left (-1\right )^{\frac {1}{3}} \arctan \left (\frac {4^{\frac {1}{6}} \sqrt {3} {\left (2 \cdot 4^{\frac {2}{3}} \left (-1\right )^{\frac {2}{3}} {\left (x^{6} - 1\right )}^{\frac {2}{3}} {\left (x^{2} - x - 1\right )} + 4 \, \left (-1\right )^{\frac {1}{3}} {\left (x^{6} - 1\right )}^{\frac {1}{3}} {\left (x^{4} - 2 \, x^{3} - x^{2} + 2 \, x + 1\right )} - 4^{\frac {1}{3}} {\left (x^{6} + 3 \, x^{5} - 5 \, x^{3} + 3 \, 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}} \left (-1\right )^{\frac {1}{3}} \log \left (\frac {4^{\frac {1}{3}} \left (-1\right )^{\frac {2}{3}} {\left (x^{4} - 2 \, x^{3} - x^{2} + 2 \, x + 1\right )} - 4^{\frac {2}{3}} \left (-1\right )^{\frac {1}{3}} {\left (x^{6} - 1\right )}^{\frac {2}{3}} + 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}} \left (-1\right )^{\frac {1}{3}} \log \left (\frac {4^{\frac {1}{3}} {\left (x^{2} - x - 1\right )} + 2 \, \left (-1\right )^{\frac {1}{3}} {\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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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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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \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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