Optimal. Leaf size=163 \[ -\sqrt {3} \text {ArcTan}\left (\frac {3 \sqrt {3} x \sqrt [3]{-x+x^4}-3 x^2 \sqrt [3]{-x+x^4}}{-6+2 \sqrt {3} x-3 x \sqrt [3]{-x+x^4}+\sqrt {3} x^2 \sqrt [3]{-x+x^4}}\right )+2 \tanh ^{-1}\left (1-2 x \sqrt [3]{-x+x^4}\right )-\tanh ^{-1}\left (\frac {1+x \sqrt [3]{-x+x^4}}{1+x \sqrt [3]{-x+x^4}+2 x^2 \left (-x+x^4\right )^{2/3}}\right ) \]
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Rubi [F]
time = 0.83, 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 {x^2 \left (-4+7 x^3\right )}{\sqrt [3]{-x+x^4} \left (-1-x^4+x^7\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {x^2 \left (-4+7 x^3\right )}{\sqrt [3]{-x+x^4} \left (-1-x^4+x^7\right )} \, dx &=\frac {\left (\sqrt [3]{x} \sqrt [3]{-1+x^3}\right ) \int \frac {x^{5/3} \left (-4+7 x^3\right )}{\sqrt [3]{-1+x^3} \left (-1-x^4+x^7\right )} \, dx}{\sqrt [3]{-x+x^4}}\\ &=\frac {\left (3 \sqrt [3]{x} \sqrt [3]{-1+x^3}\right ) \text {Subst}\left (\int \frac {x^7 \left (-4+7 x^9\right )}{\sqrt [3]{-1+x^9} \left (-1-x^{12}+x^{21}\right )} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{-x+x^4}}\\ &=\frac {\left (3 \sqrt [3]{x} \sqrt [3]{-1+x^3}\right ) \text {Subst}\left (\int \left (-\frac {4 x^7}{\sqrt [3]{-1+x^9} \left (-1-x^{12}+x^{21}\right )}+\frac {7 x^{16}}{\sqrt [3]{-1+x^9} \left (-1-x^{12}+x^{21}\right )}\right ) \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{-x+x^4}}\\ &=-\frac {\left (12 \sqrt [3]{x} \sqrt [3]{-1+x^3}\right ) \text {Subst}\left (\int \frac {x^7}{\sqrt [3]{-1+x^9} \left (-1-x^{12}+x^{21}\right )} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{-x+x^4}}+\frac {\left (21 \sqrt [3]{x} \sqrt [3]{-1+x^3}\right ) \text {Subst}\left (\int \frac {x^{16}}{\sqrt [3]{-1+x^9} \left (-1-x^{12}+x^{21}\right )} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{-x+x^4}}\\ \end {align*}
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Mathematica [F]
time = 10.12, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^2 \left (-4+7 x^3\right )}{\sqrt [3]{-x+x^4} \left (-1-x^4+x^7\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
3.
time = 14.09, size = 339, normalized size = 2.08
method | result | size |
trager | \(\ln \left (-\frac {89530993272594509387852 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x^{7}-9935189275233670145246431 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{7}-10335580785254469493615216 x^{7}-89530993272594509387852 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x^{4}+9624328758485465306265498 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{4}-x \right )^{\frac {2}{3}} x^{2}+9935189275233670145246431 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{4}+20449832047033328657637351 x^{2} \left (x^{4}-x \right )^{\frac {2}{3}}+10335580785254469493615216 x^{4}+10825503288547863351371853 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{4}-x \right )^{\frac {1}{3}} x -10027471246530585051439424 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2}-9624328758485465306265498 x \left (x^{4}-x \right )^{\frac {1}{3}}-20452583025057649054442492 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )-10427862756551384399808209}{x^{7}-x^{4}-1}\right )+\RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \ln \left (\frac {92281971296914906192993 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x^{7}+10517393749823978909196061 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{7}+10117002239803179560827276 x^{7}-92281971296914906192993 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x^{4}+9624328758485465306265498 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{4}-x \right )^{\frac {2}{3}} x^{2}-10517393749823978909196061 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{4}-10825503288547863351371853 x^{2} \left (x^{4}-x \right )^{\frac {2}{3}}-10117002239803179560827276 x^{4}-20449832047033328657637351 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{4}-x \right )^{\frac {1}{3}} x -10335580785254469493615216 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2}-9624328758485465306265498 x \left (x^{4}-x \right )^{\frac {1}{3}}-9935189275233670145246431 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )+89530993272594509387852}{x^{7}-x^{4}-1}\right )\) | \(339\) |
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 = 1.90, size = 119, normalized size = 0.73 \begin {gather*} -\sqrt {3} \arctan \left (\frac {2 \, \sqrt {3} {\left (x^{4} - x\right )}^{\frac {2}{3}} x^{2} - 4 \, \sqrt {3} {\left (x^{4} - x\right )}^{\frac {1}{3}} x - \sqrt {3} {\left (x^{7} - x^{4}\right )}}{x^{7} - x^{4} + 8}\right ) + \frac {1}{2} \, \log \left (\frac {x^{7} - x^{4} - 3 \, {\left (x^{4} - x\right )}^{\frac {2}{3}} x^{2} + 3 \, {\left (x^{4} - x\right )}^{\frac {1}{3}} x - 1}{x^{7} - x^{4} - 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} \cdot \left (7 x^{3} - 4\right )}{\sqrt [3]{x \left (x - 1\right ) \left (x^{2} + x + 1\right )} \left (x^{7} - x^{4} - 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 = -0.01 \begin {gather*} -\int \frac {x^2\,\left (7\,x^3-4\right )}{{\left (x^4-x\right )}^{1/3}\,\left (-x^7+x^4+1\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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