Optimal. Leaf size=206 \[ \frac {1}{3} \log \left (\sqrt [3]{x^7+x^6-x^5+2 x^3-x}-x\right )-\frac {\tan ^{-1}\left (\frac {\sqrt {3} \sqrt [3]{x^7+x^6-x^5+2 x^3-x}}{\sqrt [3]{x^7+x^6-x^5+2 x^3-x}+2 x}\right )}{\sqrt {3}}-\frac {1}{6} \log \left (x^2+\sqrt [3]{x^7+x^6-x^5+2 x^3-x} x+\left (x^7+x^6-x^5+2 x^3-x\right )^{2/3}\right )-\frac {\sqrt [3]{x^7+x^6-x^5+2 x^3-x} x}{x^6+x^5-x^4+x^2-1} \]
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Rubi [F] time = 4.53, 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 (2-2 x^4+3 x^5+4 x^6\right ) \sqrt [3]{-x+2 x^3-x^5+x^6+x^7}}{\left (-1+x^2-x^4+x^5+x^6\right )^2} \, dx \end {gather*}
Verification is not applicable to the result.
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\begin {align*} \int \frac {\left (2-2 x^4+3 x^5+4 x^6\right ) \sqrt [3]{-x+2 x^3-x^5+x^6+x^7}}{\left (-1+x^2-x^4+x^5+x^6\right )^2} \, dx &=\frac {\sqrt [3]{-x+2 x^3-x^5+x^6+x^7} \int \frac {\sqrt [3]{x} \sqrt [3]{-1+2 x^2-x^4+x^5+x^6} \left (2-2 x^4+3 x^5+4 x^6\right )}{\left (-1+x^2-x^4+x^5+x^6\right )^2} \, dx}{\sqrt [3]{x} \sqrt [3]{-1+2 x^2-x^4+x^5+x^6}}\\ &=\frac {\left (3 \sqrt [3]{-x+2 x^3-x^5+x^6+x^7}\right ) \operatorname {Subst}\left (\int \frac {x^3 \sqrt [3]{-1+2 x^6-x^{12}+x^{15}+x^{18}} \left (2-2 x^{12}+3 x^{15}+4 x^{18}\right )}{\left (-1+x^6-x^{12}+x^{15}+x^{18}\right )^2} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-1+2 x^2-x^4+x^5+x^6}}\\ &=\frac {\left (3 \sqrt [3]{-x+2 x^3-x^5+x^6+x^7}\right ) \operatorname {Subst}\left (\int \left (\frac {\left (-1+6 x^3+x^6-4 x^9-x^{12}+3 x^{15}\right ) \sqrt [3]{-1+2 x^6-x^{12}+x^{15}+x^{18}}}{\left (-1+x^6-x^{12}+x^{15}+x^{18}\right )^2}+\frac {\left (-1+4 x^3\right ) \sqrt [3]{-1+2 x^6-x^{12}+x^{15}+x^{18}}}{-1+x^6-x^{12}+x^{15}+x^{18}}\right ) \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-1+2 x^2-x^4+x^5+x^6}}\\ &=\frac {\left (3 \sqrt [3]{-x+2 x^3-x^5+x^6+x^7}\right ) \operatorname {Subst}\left (\int \frac {\left (-1+6 x^3+x^6-4 x^9-x^{12}+3 x^{15}\right ) \sqrt [3]{-1+2 x^6-x^{12}+x^{15}+x^{18}}}{\left (-1+x^6-x^{12}+x^{15}+x^{18}\right )^2} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-1+2 x^2-x^4+x^5+x^6}}+\frac {\left (3 \sqrt [3]{-x+2 x^3-x^5+x^6+x^7}\right ) \operatorname {Subst}\left (\int \frac {\left (-1+4 x^3\right ) \sqrt [3]{-1+2 x^6-x^{12}+x^{15}+x^{18}}}{-1+x^6-x^{12}+x^{15}+x^{18}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-1+2 x^2-x^4+x^5+x^6}}\\ &=\frac {\left (3 \sqrt [3]{-x+2 x^3-x^5+x^6+x^7}\right ) \operatorname {Subst}\left (\int \left (-\frac {\sqrt [3]{-1+2 x^6-x^{12}+x^{15}+x^{18}}}{\left (-1+x^6-x^{12}+x^{15}+x^{18}\right )^2}+\frac {6 x^3 \sqrt [3]{-1+2 x^6-x^{12}+x^{15}+x^{18}}}{\left (-1+x^6-x^{12}+x^{15}+x^{18}\right )^2}+\frac {x^6 \sqrt [3]{-1+2 x^6-x^{12}+x^{15}+x^{18}}}{\left (-1+x^6-x^{12}+x^{15}+x^{18}\right )^2}-\frac {4 x^9 \sqrt [3]{-1+2 x^6-x^{12}+x^{15}+x^{18}}}{\left (-1+x^6-x^{12}+x^{15}+x^{18}\right )^2}-\frac {x^{12} \sqrt [3]{-1+2 x^6-x^{12}+x^{15}+x^{18}}}{\left (-1+x^6-x^{12}+x^{15}+x^{18}\right )^2}+\frac {3 x^{15} \sqrt [3]{-1+2 x^6-x^{12}+x^{15}+x^{18}}}{\left (-1+x^6-x^{12}+x^{15}+x^{18}\right )^2}\right ) \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-1+2 x^2-x^4+x^5+x^6}}+\frac {\left (3 \sqrt [3]{-x+2 x^3-x^5+x^6+x^7}\right ) \operatorname {Subst}\left (\int \left (\frac {\sqrt [3]{-1+2 x^6-x^{12}+x^{15}+x^{18}}}{1-x^6+x^{12}-x^{15}-x^{18}}+\frac {4 x^3 \sqrt [3]{-1+2 x^6-x^{12}+x^{15}+x^{18}}}{-1+x^6-x^{12}+x^{15}+x^{18}}\right ) \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-1+2 x^2-x^4+x^5+x^6}}\\ &=\frac {\left (3 \sqrt [3]{-x+2 x^3-x^5+x^6+x^7}\right ) \operatorname {Subst}\left (\int \frac {\sqrt [3]{-1+2 x^6-x^{12}+x^{15}+x^{18}}}{1-x^6+x^{12}-x^{15}-x^{18}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-1+2 x^2-x^4+x^5+x^6}}-\frac {\left (3 \sqrt [3]{-x+2 x^3-x^5+x^6+x^7}\right ) \operatorname {Subst}\left (\int \frac {\sqrt [3]{-1+2 x^6-x^{12}+x^{15}+x^{18}}}{\left (-1+x^6-x^{12}+x^{15}+x^{18}\right )^2} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-1+2 x^2-x^4+x^5+x^6}}+\frac {\left (3 \sqrt [3]{-x+2 x^3-x^5+x^6+x^7}\right ) \operatorname {Subst}\left (\int \frac {x^6 \sqrt [3]{-1+2 x^6-x^{12}+x^{15}+x^{18}}}{\left (-1+x^6-x^{12}+x^{15}+x^{18}\right )^2} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-1+2 x^2-x^4+x^5+x^6}}-\frac {\left (3 \sqrt [3]{-x+2 x^3-x^5+x^6+x^7}\right ) \operatorname {Subst}\left (\int \frac {x^{12} \sqrt [3]{-1+2 x^6-x^{12}+x^{15}+x^{18}}}{\left (-1+x^6-x^{12}+x^{15}+x^{18}\right )^2} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-1+2 x^2-x^4+x^5+x^6}}+\frac {\left (9 \sqrt [3]{-x+2 x^3-x^5+x^6+x^7}\right ) \operatorname {Subst}\left (\int \frac {x^{15} \sqrt [3]{-1+2 x^6-x^{12}+x^{15}+x^{18}}}{\left (-1+x^6-x^{12}+x^{15}+x^{18}\right )^2} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-1+2 x^2-x^4+x^5+x^6}}-\frac {\left (12 \sqrt [3]{-x+2 x^3-x^5+x^6+x^7}\right ) \operatorname {Subst}\left (\int \frac {x^9 \sqrt [3]{-1+2 x^6-x^{12}+x^{15}+x^{18}}}{\left (-1+x^6-x^{12}+x^{15}+x^{18}\right )^2} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-1+2 x^2-x^4+x^5+x^6}}+\frac {\left (12 \sqrt [3]{-x+2 x^3-x^5+x^6+x^7}\right ) \operatorname {Subst}\left (\int \frac {x^3 \sqrt [3]{-1+2 x^6-x^{12}+x^{15}+x^{18}}}{-1+x^6-x^{12}+x^{15}+x^{18}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-1+2 x^2-x^4+x^5+x^6}}+\frac {\left (18 \sqrt [3]{-x+2 x^3-x^5+x^6+x^7}\right ) \operatorname {Subst}\left (\int \frac {x^3 \sqrt [3]{-1+2 x^6-x^{12}+x^{15}+x^{18}}}{\left (-1+x^6-x^{12}+x^{15}+x^{18}\right )^2} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-1+2 x^2-x^4+x^5+x^6}}\\ \end {align*}
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Mathematica [F] time = 2.16, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (2-2 x^4+3 x^5+4 x^6\right ) \sqrt [3]{-x+2 x^3-x^5+x^6+x^7}}{\left (-1+x^2-x^4+x^5+x^6\right )^2} \, dx \end {gather*}
Verification is not applicable to the result.
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IntegrateAlgebraic [A] time = 5.46, size = 206, normalized size = 1.00 \begin {gather*} -\frac {x \sqrt [3]{-x+2 x^3-x^5+x^6+x^7}}{-1+x^2-x^4+x^5+x^6}-\frac {\tan ^{-1}\left (\frac {\sqrt {3} \sqrt [3]{-x+2 x^3-x^5+x^6+x^7}}{2 x+\sqrt [3]{-x+2 x^3-x^5+x^6+x^7}}\right )}{\sqrt {3}}+\frac {1}{3} \log \left (-x+\sqrt [3]{-x+2 x^3-x^5+x^6+x^7}\right )-\frac {1}{6} \log \left (x^2+x \sqrt [3]{-x+2 x^3-x^5+x^6+x^7}+\left (-x+2 x^3-x^5+x^6+x^7\right )^{2/3}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 4.93, size = 268, normalized size = 1.30 \begin {gather*} \frac {2 \, \sqrt {3} {\left (x^{6} + x^{5} - x^{4} + x^{2} - 1\right )} \arctan \left (-\frac {2 \, \sqrt {3} {\left (x^{7} + x^{6} - x^{5} + 2 \, x^{3} - x\right )}^{\frac {1}{3}} x + \sqrt {3} {\left (x^{6} + x^{5} - x^{4} + x^{2} - 1\right )} - 2 \, \sqrt {3} {\left (x^{7} + x^{6} - x^{5} + 2 \, x^{3} - x\right )}^{\frac {2}{3}}}{3 \, {\left (x^{6} + x^{5} - x^{4} + 3 \, x^{2} - 1\right )}}\right ) + {\left (x^{6} + x^{5} - x^{4} + x^{2} - 1\right )} \log \left (\frac {x^{6} + x^{5} - x^{4} + x^{2} + 3 \, {\left (x^{7} + x^{6} - x^{5} + 2 \, x^{3} - x\right )}^{\frac {1}{3}} x - 3 \, {\left (x^{7} + x^{6} - x^{5} + 2 \, x^{3} - x\right )}^{\frac {2}{3}} - 1}{x^{6} + x^{5} - x^{4} + x^{2} - 1}\right ) - 6 \, {\left (x^{7} + x^{6} - x^{5} + 2 \, x^{3} - x\right )}^{\frac {1}{3}} x}{6 \, {\left (x^{6} + x^{5} - x^{4} + x^{2} - 1\right )}} \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*} \int \frac {{\left (x^{7} + x^{6} - x^{5} + 2 \, x^{3} - x\right )}^{\frac {1}{3}} {\left (4 \, x^{6} + 3 \, x^{5} - 2 \, x^{4} + 2\right )}}{{\left (x^{6} + x^{5} - x^{4} + x^{2} - 1\right )}^{2}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 26.47, size = 676, normalized size = 3.28
method | result | size |
trager | \(-\frac {x \left (x^{7}+x^{6}-x^{5}+2 x^{3}-x \right )^{\frac {1}{3}}}{x^{6}+x^{5}-x^{4}+x^{2}-1}+\frac {\ln \left (-\frac {-726439828539024 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2} x^{6}-726439828539024 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2} x^{5}-948727463035299 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{6}+726439828539024 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2} x^{4}-948727463035299 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{5}-222287634496275 x^{6}+948727463035299 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{4}-222287634496275 x^{5}+222287634496275 x^{4}+1131297147498108 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (x^{7}+x^{6}-x^{5}+2 x^{3}-x \right )^{\frac {2}{3}}-142851734429343 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (x^{7}+x^{6}-x^{5}+2 x^{3}-x \right )^{\frac {1}{3}} x -1695026266591056 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{2}+47617244809781 \left (x^{7}+x^{6}-x^{5}+2 x^{3}-x \right )^{\frac {2}{3}}+329481804356255 x \left (x^{7}+x^{6}-x^{5}+2 x^{3}-x \right )^{\frac {1}{3}}+726439828539024 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2}-518671147157975 x^{2}+948727463035299 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )+222287634496275}{x^{6}+x^{5}-x^{4}+x^{2}-1}\right )}{3}+\RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \ln \left (\frac {-59576925050199 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2} x^{6}-59576925050199 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2} x^{5}+262005584529741 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{6}+59576925050199 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2} x^{4}+262005584529741 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{5}-148191756330850 x^{6}-262005584529741 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{4}-148191756330850 x^{5}+148191756330850 x^{4}+1131297147498108 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (x^{7}+x^{6}-x^{5}+2 x^{3}-x \right )^{\frac {2}{3}}-988445413068765 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (x^{7}+x^{6}-x^{5}+2 x^{3}-x \right )^{\frac {1}{3}} x +139012825117131 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{2}+329481804356255 \left (x^{7}+x^{6}-x^{5}+2 x^{3}-x \right )^{\frac {2}{3}}+47617244809781 x \left (x^{7}+x^{6}-x^{5}+2 x^{3}-x \right )^{\frac {1}{3}}+59576925050199 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2}-518671147157975 x^{2}-262005584529741 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )+148191756330850}{x^{6}+x^{5}-x^{4}+x^{2}-1}\right )\) | \(676\) |
risch | \(\text {Expression too large to display}\) | \(3644\) |
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 (x^{7} + x^{6} - x^{5} + 2 \, x^{3} - x\right )}^{\frac {1}{3}} {\left (4 \, x^{6} + 3 \, x^{5} - 2 \, x^{4} + 2\right )}}{{\left (x^{6} + x^{5} - x^{4} + x^{2} - 1\right )}^{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 = -0.00 \begin {gather*} \int \frac {\left (4\,x^6+3\,x^5-2\,x^4+2\right )\,{\left (x^7+x^6-x^5+2\,x^3-x\right )}^{1/3}}{{\left (x^6+x^5-x^4+x^2-1\right )}^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 {\sqrt [3]{x \left (x + 1\right ) \left (x^{5} - x^{3} + x^{2} + x - 1\right )} \left (4 x^{6} + 3 x^{5} - 2 x^{4} + 2\right )}{\left (x^{6} + x^{5} - x^{4} + x^{2} - 1\right )^{2}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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