Optimal. Leaf size=288 \[ \frac {1}{2} \sqrt {3} \text {RootSum}\left [\text {$\#$1}^4-2 \text {$\#$1}^3+4 \text {$\#$1}+4\& ,\frac {\text {$\#$1}^2 \log \left (-\text {$\#$1}+\sqrt {3 x^4-2}+\sqrt {3} x^2\right )-\text {$\#$1} \log \left (-\text {$\#$1}+\sqrt {3 x^4-2}+\sqrt {3} x^2\right )-2 \log \left (-\text {$\#$1}+\sqrt {3 x^4-2}+\sqrt {3} x^2\right )}{2 \text {$\#$1}^3-3 \text {$\#$1}^2+2}\& \right ]-\frac {1}{2} \sqrt {3} \text {RootSum}\left [\text {$\#$1}^4+2 \text {$\#$1}^3-4 \text {$\#$1}+4\& ,\frac {\text {$\#$1}^2 \log \left (-\text {$\#$1}+\sqrt {3 x^4-2}+\sqrt {3} x^2\right )+\text {$\#$1} \log \left (-\text {$\#$1}+\sqrt {3 x^4-2}+\sqrt {3} x^2\right )-2 \log \left (-\text {$\#$1}+\sqrt {3 x^4-2}+\sqrt {3} x^2\right )}{2 \text {$\#$1}^3+3 \text {$\#$1}^2-2}\& \right ]+\frac {\sqrt {3 x^4-2}}{2 x^2} \]
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Rubi [C] time = 0.89, antiderivative size = 197, normalized size of antiderivative = 0.68, number of steps used = 16, number of rules used = 11, integrand size = 36, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.306, Rules used = {6728, 275, 277, 217, 206, 1490, 1293, 1692, 377, 205, 208} \begin {gather*} \frac {\sqrt {3 x^4-2}}{2 x^2}-\frac {\left (-\sqrt {3}+i\right ) \sqrt {\frac {3 \left (-\sqrt {3}+3 i\right )}{\sqrt {3}+i}} \tan ^{-1}\left (\frac {x^2}{\sqrt {\frac {-\sqrt {3}+3 i}{3 \left (\sqrt {3}+i\right )}} \sqrt {3 x^4-2}}\right )}{2 \left (-\sqrt {3}+3 i\right )}-\frac {\left (\sqrt {3}+i\right ) \tanh ^{-1}\left (\frac {\sqrt {-\frac {3 \left (-\sqrt {3}+i\right )}{\sqrt {3}+3 i}} x^2}{\sqrt {3 x^4-2}}\right )}{2 \sqrt {\frac {2}{3} \left (3+i \sqrt {3}\right )}} \end {gather*}
Warning: Unable to verify antiderivative.
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Rule 205
Rule 206
Rule 208
Rule 217
Rule 275
Rule 277
Rule 377
Rule 1293
Rule 1490
Rule 1692
Rule 6728
Rubi steps
\begin {align*} \int \frac {\sqrt {-2+3 x^4} \left (-1+3 x^4\right )}{x^3 \left (1-3 x^4+3 x^8\right )} \, dx &=\int \left (-\frac {\sqrt {-2+3 x^4}}{x^3}+\frac {3 x^5 \sqrt {-2+3 x^4}}{1-3 x^4+3 x^8}\right ) \, dx\\ &=3 \int \frac {x^5 \sqrt {-2+3 x^4}}{1-3 x^4+3 x^8} \, dx-\int \frac {\sqrt {-2+3 x^4}}{x^3} \, dx\\ &=-\left (\frac {1}{2} \operatorname {Subst}\left (\int \frac {\sqrt {-2+3 x^2}}{x^2} \, dx,x,x^2\right )\right )+\frac {3}{2} \operatorname {Subst}\left (\int \frac {x^2 \sqrt {-2+3 x^2}}{1-3 x^2+3 x^4} \, dx,x,x^2\right )\\ &=\frac {\sqrt {-2+3 x^4}}{2 x^2}-\frac {1}{2} \operatorname {Subst}\left (\int \frac {3-3 x^2}{\sqrt {-2+3 x^2} \left (1-3 x^2+3 x^4\right )} \, dx,x,x^2\right )\\ &=\frac {\sqrt {-2+3 x^4}}{2 x^2}-\frac {1}{2} \operatorname {Subst}\left (\int \left (\frac {-3-3 i \sqrt {3}}{\sqrt {-2+3 x^2} \left (-3-i \sqrt {3}+6 x^2\right )}+\frac {-3+3 i \sqrt {3}}{\sqrt {-2+3 x^2} \left (-3+i \sqrt {3}+6 x^2\right )}\right ) \, dx,x,x^2\right )\\ &=\frac {\sqrt {-2+3 x^4}}{2 x^2}+\frac {1}{2} \left (3 \left (1-i \sqrt {3}\right )\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {-2+3 x^2} \left (-3+i \sqrt {3}+6 x^2\right )} \, dx,x,x^2\right )+\frac {1}{2} \left (3 \left (1+i \sqrt {3}\right )\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {-2+3 x^2} \left (-3-i \sqrt {3}+6 x^2\right )} \, dx,x,x^2\right )\\ &=\frac {\sqrt {-2+3 x^4}}{2 x^2}+\frac {1}{2} \left (3 \left (1-i \sqrt {3}\right )\right ) \operatorname {Subst}\left (\int \frac {1}{-3+i \sqrt {3}-\left (12+3 \left (-3+i \sqrt {3}\right )\right ) x^2} \, dx,x,\frac {x^2}{\sqrt {-2+3 x^4}}\right )+\frac {1}{2} \left (3 \left (1+i \sqrt {3}\right )\right ) \operatorname {Subst}\left (\int \frac {1}{-3-i \sqrt {3}-\left (12+3 \left (-3-i \sqrt {3}\right )\right ) x^2} \, dx,x,\frac {x^2}{\sqrt {-2+3 x^4}}\right )\\ &=\frac {\sqrt {-2+3 x^4}}{2 x^2}-\frac {\left (i-\sqrt {3}\right ) \sqrt {\frac {3 \left (3 i-\sqrt {3}\right )}{i+\sqrt {3}}} \tan ^{-1}\left (\frac {x^2}{\sqrt {\frac {3 i-\sqrt {3}}{3 \left (i+\sqrt {3}\right )}} \sqrt {-2+3 x^4}}\right )}{2 \left (3 i-\sqrt {3}\right )}-\frac {\left (i+\sqrt {3}\right ) \tanh ^{-1}\left (\frac {\sqrt {-\frac {3 \left (i-\sqrt {3}\right )}{3 i+\sqrt {3}}} x^2}{\sqrt {-2+3 x^4}}\right )}{2 \sqrt {\frac {2}{3} \left (3+i \sqrt {3}\right )}}\\ \end {align*}
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Mathematica [C] time = 5.31, size = 182, normalized size = 0.63 \begin {gather*} \frac {\sqrt {3 x^4-2}}{2 x^2}+\frac {3 i \left (\tan ^{-1}\left (\frac {\sqrt {-\frac {3 \left (\sqrt {3}-i\right )}{\sqrt {3}+3 i}} x^2}{\sqrt {3 x^4-2}}\right )+\sqrt {-\sqrt {3}+i} \sqrt {-\frac {1}{\sqrt {3}+i}} \tan ^{-1}\left (\frac {x^2}{\sqrt {\frac {-\sqrt {3}+3 i}{3 \sqrt {3}+3 i}} \sqrt {3 x^4-2}}\right )\right )}{\left (\sqrt {3}-3 i\right ) \sqrt {\frac {-3 \sqrt {3}+3 i}{\sqrt {3}+3 i}}} \end {gather*}
Warning: Unable to verify antiderivative.
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IntegrateAlgebraic [A] time = 0.35, size = 288, normalized size = 1.00 \begin {gather*} \frac {\sqrt {-2+3 x^4}}{2 x^2}+\frac {1}{2} \sqrt {3} \text {RootSum}\left [4+4 \text {$\#$1}-2 \text {$\#$1}^3+\text {$\#$1}^4\&,\frac {-2 \log \left (\sqrt {3} x^2+\sqrt {-2+3 x^4}-\text {$\#$1}\right )-\log \left (\sqrt {3} x^2+\sqrt {-2+3 x^4}-\text {$\#$1}\right ) \text {$\#$1}+\log \left (\sqrt {3} x^2+\sqrt {-2+3 x^4}-\text {$\#$1}\right ) \text {$\#$1}^2}{2-3 \text {$\#$1}^2+2 \text {$\#$1}^3}\&\right ]-\frac {1}{2} \sqrt {3} \text {RootSum}\left [4-4 \text {$\#$1}+2 \text {$\#$1}^3+\text {$\#$1}^4\&,\frac {-2 \log \left (\sqrt {3} x^2+\sqrt {-2+3 x^4}-\text {$\#$1}\right )+\log \left (\sqrt {3} x^2+\sqrt {-2+3 x^4}-\text {$\#$1}\right ) \text {$\#$1}+\log \left (\sqrt {3} x^2+\sqrt {-2+3 x^4}-\text {$\#$1}\right ) \text {$\#$1}^2}{-2+3 \text {$\#$1}^2+2 \text {$\#$1}^3}\&\right ] \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.67, size = 1041, normalized size = 3.61
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 1.66, size = 178, normalized size = 0.62 \begin {gather*} \frac {1}{16} \, \sqrt {3} {\left ({\left (\sqrt {3} x^{2} - \sqrt {3 \, x^{4} - 2}\right )}^{4} - 8 \, {\left (\sqrt {3} x^{2} - \sqrt {3 \, x^{4} - 2}\right )}^{2} + 4\right )} \log \left ({\left | -{\left (\sqrt {3} x^{2} - \sqrt {3 \, x^{4} - 2}\right )}^{8} + 4 \, {\left (\sqrt {3} x^{2} - \sqrt {3 \, x^{4} - 2}\right )}^{6} - 24 \, {\left (\sqrt {3} x^{2} - \sqrt {3 \, x^{4} - 2}\right )}^{4} + 16 \, {\left (\sqrt {3} x^{2} - \sqrt {3 \, x^{4} - 2}\right )}^{2} - 16 \right |}\right ) + \frac {2 \, \sqrt {3}}{{\left (\sqrt {3} x^{2} - \sqrt {3 \, x^{4} - 2}\right )}^{2} + 2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 8.00, size = 92, normalized size = 0.32
method | result | size |
risch | \(\frac {\sqrt {3 x^{4}-2}}{2 x^{2}}-\frac {\sqrt {3}\, \left (\munderset {\textit {\_R} =\RootOf \left (\textit {\_Z}^{4}-4 \textit {\_Z}^{3}+24 \textit {\_Z}^{2}-16 \textit {\_Z} +16\right )}{\sum }\frac {\left (\textit {\_R}^{2}-8 \textit {\_R} +4\right ) \ln \left (\left (\sqrt {3 x^{4}-2}-\sqrt {3}\, x^{2}\right )^{2}-\textit {\_R} \right )}{\textit {\_R}^{3}-3 \textit {\_R}^{2}+12 \textit {\_R} -4}\right )}{4}\) | \(92\) |
elliptic | \(\frac {\sqrt {3 x^{4}-2}}{2 x^{2}}-\frac {\sqrt {3}\, \left (\munderset {\textit {\_R} =\RootOf \left (\textit {\_Z}^{4}-4 \textit {\_Z}^{3}+24 \textit {\_Z}^{2}-16 \textit {\_Z} +16\right )}{\sum }\frac {\left (\textit {\_R}^{2}-8 \textit {\_R} +4\right ) \ln \left (\left (\sqrt {3 x^{4}-2}-\sqrt {3}\, x^{2}\right )^{2}-\textit {\_R} \right )}{\textit {\_R}^{3}-3 \textit {\_R}^{2}+12 \textit {\_R} -4}\right )}{4}\) | \(92\) |
default | \(-\frac {\sqrt {3}\, \left (\munderset {\textit {\_R} =\RootOf \left (\textit {\_Z}^{4}+2 \textit {\_Z}^{3}-4 \textit {\_Z} +4\right )}{\sum }\frac {\left (-\textit {\_R}^{2}-\textit {\_R} +2\right ) \ln \left (\sqrt {3 x^{4}-2}-\sqrt {3}\, x^{2}-\textit {\_R} \right )}{2 \textit {\_R}^{3}+3 \textit {\_R}^{2}-2}\right )}{2}-\frac {\sqrt {3}\, \ln \left (\sqrt {3 x^{4}-2}-\sqrt {3}\, x^{2}\right )}{2}-\frac {\sqrt {3}\, \left (\munderset {\textit {\_R} =\RootOf \left (\textit {\_Z}^{4}-2 \textit {\_Z}^{3}+4 \textit {\_Z} +4\right )}{\sum }\frac {\left (\textit {\_R}^{2}-\textit {\_R} -2\right ) \ln \left (\sqrt {3 x^{4}-2}-\sqrt {3}\, x^{2}-\textit {\_R} \right )}{2 \textit {\_R}^{3}-3 \textit {\_R}^{2}+2}\right )}{2}-\frac {\left (3 x^{4}-2\right )^{\frac {3}{2}}}{4 x^{2}}+\frac {3 x^{2} \sqrt {3 x^{4}-2}}{4}-\frac {\ln \left (\sqrt {3}\, x^{2}+\sqrt {3 x^{4}-2}\right ) \sqrt {3}}{2}\) | \(213\) |
trager | \(\frac {\sqrt {3 x^{4}-2}}{2 x^{2}}+\frac {\RootOf \left (\textit {\_Z}^{2}+144 \RootOf \left (6912 \textit {\_Z}^{4}-144 \textit {\_Z}^{2}+1\right )^{2}-3\right ) \ln \left (-\frac {-55296 \RootOf \left (6912 \textit {\_Z}^{4}-144 \textit {\_Z}^{2}+1\right )^{4} x^{4} \RootOf \left (\textit {\_Z}^{2}+144 \RootOf \left (6912 \textit {\_Z}^{4}-144 \textit {\_Z}^{2}+1\right )^{2}-3\right )-720 \RootOf \left (6912 \textit {\_Z}^{4}-144 \textit {\_Z}^{2}+1\right )^{2} \RootOf \left (\textit {\_Z}^{2}+144 \RootOf \left (6912 \textit {\_Z}^{4}-144 \textit {\_Z}^{2}+1\right )^{2}-3\right ) x^{4}+1872 \sqrt {3 x^{4}-2}\, \RootOf \left (6912 \textit {\_Z}^{4}-144 \textit {\_Z}^{2}+1\right )^{2} x^{2}+55296 \RootOf \left (\textit {\_Z}^{2}+144 \RootOf \left (6912 \textit {\_Z}^{4}-144 \textit {\_Z}^{2}+1\right )^{2}-3\right ) \RootOf \left (6912 \textit {\_Z}^{4}-144 \textit {\_Z}^{2}+1\right )^{4}+39 \RootOf \left (\textit {\_Z}^{2}+144 \RootOf \left (6912 \textit {\_Z}^{4}-144 \textit {\_Z}^{2}+1\right )^{2}-3\right ) x^{4}-47 x^{2} \sqrt {3 x^{4}-2}+1104 \RootOf \left (6912 \textit {\_Z}^{4}-144 \textit {\_Z}^{2}+1\right )^{2} \RootOf \left (\textit {\_Z}^{2}+144 \RootOf \left (6912 \textit {\_Z}^{4}-144 \textit {\_Z}^{2}+1\right )^{2}-3\right )-26 \RootOf \left (\textit {\_Z}^{2}+144 \RootOf \left (6912 \textit {\_Z}^{4}-144 \textit {\_Z}^{2}+1\right )^{2}-3\right )}{144 \RootOf \left (6912 \textit {\_Z}^{4}-144 \textit {\_Z}^{2}+1\right )^{2} x^{4}-144 \RootOf \left (6912 \textit {\_Z}^{4}-144 \textit {\_Z}^{2}+1\right )^{2}+1}\right )}{4}-3 \RootOf \left (6912 \textit {\_Z}^{4}-144 \textit {\_Z}^{2}+1\right ) \ln \left (\frac {-165888 \RootOf \left (6912 \textit {\_Z}^{4}-144 \textit {\_Z}^{2}+1\right )^{5} x^{4}+9072 \RootOf \left (6912 \textit {\_Z}^{4}-144 \textit {\_Z}^{2}+1\right )^{3} x^{4}+468 \sqrt {3 x^{4}-2}\, \RootOf \left (6912 \textit {\_Z}^{4}-144 \textit {\_Z}^{2}+1\right )^{2} x^{2}+165888 \RootOf \left (6912 \textit {\_Z}^{4}-144 \textit {\_Z}^{2}+1\right )^{5}+2 x^{2} \sqrt {3 x^{4}-2}-10224 \RootOf \left (6912 \textit {\_Z}^{4}-144 \textit {\_Z}^{2}+1\right )^{3}+63 \RootOf \left (6912 \textit {\_Z}^{4}-144 \textit {\_Z}^{2}+1\right )}{\left (1728 \RootOf \left (6912 \textit {\_Z}^{4}-144 \textit {\_Z}^{2}+1\right )^{3} x^{2}+144 x^{2} \RootOf \left (6912 \textit {\_Z}^{4}-144 \textit {\_Z}^{2}+1\right )^{2}-1728 \RootOf \left (6912 \textit {\_Z}^{4}-144 \textit {\_Z}^{2}+1\right )^{3}-24 \RootOf \left (6912 \textit {\_Z}^{4}-144 \textit {\_Z}^{2}+1\right ) x^{2}-144 \RootOf \left (6912 \textit {\_Z}^{4}-144 \textit {\_Z}^{2}+1\right )^{2}-3 x^{2}+24 \RootOf \left (6912 \textit {\_Z}^{4}-144 \textit {\_Z}^{2}+1\right )+2\right ) \left (1728 \RootOf \left (6912 \textit {\_Z}^{4}-144 \textit {\_Z}^{2}+1\right )^{3} x^{2}-144 x^{2} \RootOf \left (6912 \textit {\_Z}^{4}-144 \textit {\_Z}^{2}+1\right )^{2}-1728 \RootOf \left (6912 \textit {\_Z}^{4}-144 \textit {\_Z}^{2}+1\right )^{3}-24 \RootOf \left (6912 \textit {\_Z}^{4}-144 \textit {\_Z}^{2}+1\right ) x^{2}+144 \RootOf \left (6912 \textit {\_Z}^{4}-144 \textit {\_Z}^{2}+1\right )^{2}+3 x^{2}+24 \RootOf \left (6912 \textit {\_Z}^{4}-144 \textit {\_Z}^{2}+1\right )-2\right )}\right )\) | \(730\) |
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 (3 \, x^{4} - 1\right )} \sqrt {3 \, x^{4} - 2}}{{\left (3 \, x^{8} - 3 \, x^{4} + 1\right )} x^{3}}\,{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 {\left (3\,x^4-1\right )\,\sqrt {3\,x^4-2}}{x^3\,\left (3\,x^8-3\,x^4+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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