Optimal. Leaf size=261 \[ -\frac {1}{2} \sqrt {\frac {1}{2} \left (2-\sqrt {2}\right )} \text {ArcTan}\left (\frac {\sqrt {2+\sqrt {2}} x \sqrt [4]{-1+x^6}}{-x^2+\sqrt {-1+x^6}}\right )+\frac {1}{2} \sqrt {\frac {1}{2} \left (2+\sqrt {2}\right )} \text {ArcTan}\left (\frac {\left (\sqrt {\frac {2}{2-\sqrt {2}}}-\frac {2}{\sqrt {2-\sqrt {2}}}\right ) x \sqrt [4]{-1+x^6}}{-x^2+\sqrt {-1+x^6}}\right )-\frac {1}{2} \sqrt {\frac {1}{2} \left (2+\sqrt {2}\right )} \tanh ^{-1}\left (\frac {\sqrt {2-\sqrt {2}} x \sqrt [4]{-1+x^6}}{x^2+\sqrt {-1+x^6}}\right )-\frac {1}{2} \sqrt {\frac {1}{2} \left (2-\sqrt {2}\right )} \tanh ^{-1}\left (\frac {\sqrt {2+\sqrt {2}} x \sqrt [4]{-1+x^6}}{x^2+\sqrt {-1+x^6}}\right ) \]
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Rubi [F]
time = 0.97, 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 (2+x^6\right ) \left (-1+x^4+x^6\right )}{\sqrt [4]{-1+x^6} \left (1-2 x^6+x^8+x^{12}\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {\left (2+x^6\right ) \left (-1+x^4+x^6\right )}{\sqrt [4]{-1+x^6} \left (1-2 x^6+x^8+x^{12}\right )} \, dx &=\int \left (\frac {1}{\sqrt [4]{-1+x^6}}-\frac {3-2 x^4-3 x^6+x^8-x^{10}}{\sqrt [4]{-1+x^6} \left (1-2 x^6+x^8+x^{12}\right )}\right ) \, dx\\ &=\int \frac {1}{\sqrt [4]{-1+x^6}} \, dx-\int \frac {3-2 x^4-3 x^6+x^8-x^{10}}{\sqrt [4]{-1+x^6} \left (1-2 x^6+x^8+x^{12}\right )} \, dx\\ &=\frac {\sqrt [4]{1-x^6} \int \frac {1}{\sqrt [4]{1-x^6}} \, dx}{\sqrt [4]{-1+x^6}}-\int \left (\frac {3}{\sqrt [4]{-1+x^6} \left (1-2 x^6+x^8+x^{12}\right )}-\frac {2 x^4}{\sqrt [4]{-1+x^6} \left (1-2 x^6+x^8+x^{12}\right )}-\frac {3 x^6}{\sqrt [4]{-1+x^6} \left (1-2 x^6+x^8+x^{12}\right )}+\frac {x^8}{\sqrt [4]{-1+x^6} \left (1-2 x^6+x^8+x^{12}\right )}-\frac {x^{10}}{\sqrt [4]{-1+x^6} \left (1-2 x^6+x^8+x^{12}\right )}\right ) \, dx\\ &=\frac {x \sqrt [4]{1-x^6} \, _2F_1\left (\frac {1}{6},\frac {1}{4};\frac {7}{6};x^6\right )}{\sqrt [4]{-1+x^6}}+2 \int \frac {x^4}{\sqrt [4]{-1+x^6} \left (1-2 x^6+x^8+x^{12}\right )} \, dx-3 \int \frac {1}{\sqrt [4]{-1+x^6} \left (1-2 x^6+x^8+x^{12}\right )} \, dx+3 \int \frac {x^6}{\sqrt [4]{-1+x^6} \left (1-2 x^6+x^8+x^{12}\right )} \, dx-\int \frac {x^8}{\sqrt [4]{-1+x^6} \left (1-2 x^6+x^8+x^{12}\right )} \, dx+\int \frac {x^{10}}{\sqrt [4]{-1+x^6} \left (1-2 x^6+x^8+x^{12}\right )} \, dx\\ \end {align*}
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Mathematica [A]
time = 8.33, size = 222, normalized size = 0.85 \begin {gather*} -\frac {\sqrt {2+\sqrt {2}} \text {ArcTan}\left (\frac {\sqrt {2-\sqrt {2}} x \sqrt [4]{-1+x^6}}{-x^2+\sqrt {-1+x^6}}\right )+\sqrt {2-\sqrt {2}} \text {ArcTan}\left (\frac {\sqrt {2+\sqrt {2}} x \sqrt [4]{-1+x^6}}{-x^2+\sqrt {-1+x^6}}\right )+\sqrt {2+\sqrt {2}} \tanh ^{-1}\left (\frac {\sqrt {2-\sqrt {2}} x \sqrt [4]{-1+x^6}}{x^2+\sqrt {-1+x^6}}\right )+\sqrt {2-\sqrt {2}} \tanh ^{-1}\left (\frac {\sqrt {2+\sqrt {2}} x \sqrt [4]{-1+x^6}}{x^2+\sqrt {-1+x^6}}\right )}{2 \sqrt {2}} \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 = 27.76, size = 681, normalized size = 2.61
method | result | size |
trager | \(\text {Expression too large to display}\) | \(681\) |
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 [F(-1)] Timed out
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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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (x^{6} + 2\right ) \left (x^{6} + x^{4} - 1\right )}{\sqrt [4]{\left (x - 1\right ) \left (x + 1\right ) \left (x^{2} - x + 1\right ) \left (x^{2} + x + 1\right )} \left (x^{12} + x^{8} - 2 x^{6} + 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 {\left (x^6+2\right )\,\left (x^6+x^4-1\right )}{{\left (x^6-1\right )}^{1/4}\,\left (x^{12}+x^8-2\,x^6+1\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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