Optimal. Leaf size=101 \[ -\frac {\tan ^{-1}\left (\frac {2^{3/4} x \sqrt {2 x^6+1}}{2 \sqrt {2} x^6-x^2+\sqrt {2}}\right )}{4 \sqrt [4]{2}}-\frac {\tanh ^{-1}\left (\frac {2^{3/4} x \sqrt {2 x^6+1}}{2 \sqrt {2} x^6+x^2+\sqrt {2}}\right )}{4 \sqrt [4]{2}} \]
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Rubi [F] time = 0.57, 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 {1+2 x^6} \left (-1+4 x^6\right )}{2+x^4+8 x^6+8 x^{12}} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {\sqrt {1+2 x^6} \left (-1+4 x^6\right )}{2+x^4+8 x^6+8 x^{12}} \, dx &=\int \left (\frac {\sqrt {1+2 x^6}}{-2-x^4-8 x^6-8 x^{12}}+\frac {4 x^6 \sqrt {1+2 x^6}}{2+x^4+8 x^6+8 x^{12}}\right ) \, dx\\ &=4 \int \frac {x^6 \sqrt {1+2 x^6}}{2+x^4+8 x^6+8 x^{12}} \, dx+\int \frac {\sqrt {1+2 x^6}}{-2-x^4-8 x^6-8 x^{12}} \, dx\\ \end {align*}
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Mathematica [F] time = 0.15, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {1+2 x^6} \left (-1+4 x^6\right )}{2+x^4+8 x^6+8 x^{12}} \, dx \end {gather*}
Verification is not applicable to the result.
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IntegrateAlgebraic [A] time = 5.03, size = 101, normalized size = 1.00 \begin {gather*} -\frac {\tan ^{-1}\left (\frac {2^{3/4} x \sqrt {1+2 x^6}}{\sqrt {2}-x^2+2 \sqrt {2} x^6}\right )}{4 \sqrt [4]{2}}-\frac {\tanh ^{-1}\left (\frac {2^{3/4} x \sqrt {1+2 x^6}}{\sqrt {2}+x^2+2 \sqrt {2} x^6}\right )}{4 \sqrt [4]{2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 1.14, size = 1041, normalized size = 10.31
result too large to display
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 (4 \, x^{6} - 1\right )} \sqrt {2 \, x^{6} + 1}}{8 \, x^{12} + 8 \, x^{6} + x^{4} + 2}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 2.42, size = 187, normalized size = 1.85
method | result | size |
trager | \(\frac {\RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}+2\right )^{2}\right ) \ln \left (-\frac {4 \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}+2\right )^{2}\right ) x^{6}+\RootOf \left (\textit {\_Z}^{4}+2\right )^{2} \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}+2\right )^{2}\right ) x^{2}-4 \sqrt {2 x^{6}+1}\, x +2 \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}+2\right )^{2}\right )}{-4 x^{6}+\RootOf \left (\textit {\_Z}^{4}+2\right )^{2} x^{2}-2}\right )}{8}+\frac {\RootOf \left (\textit {\_Z}^{4}+2\right ) \ln \left (\frac {-4 \RootOf \left (\textit {\_Z}^{4}+2\right ) x^{6}+\RootOf \left (\textit {\_Z}^{4}+2\right )^{3} x^{2}+4 \sqrt {2 x^{6}+1}\, x -2 \RootOf \left (\textit {\_Z}^{4}+2\right )}{4 x^{6}+\RootOf \left (\textit {\_Z}^{4}+2\right )^{2} x^{2}+2}\right )}{8}\) | \(187\) |
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 (4 \, x^{6} - 1\right )} \sqrt {2 \, x^{6} + 1}}{8 \, x^{12} + 8 \, x^{6} + x^{4} + 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 {\sqrt {2\,x^6+1}\,\left (4\,x^6-1\right )}{8\,x^{12}+8\,x^6+x^4+2} \,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 {\left (2 x^{3} - 1\right ) \left (2 x^{3} + 1\right ) \sqrt {2 x^{6} + 1}}{8 x^{12} + 8 x^{6} + x^{4} + 2}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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