Optimal. Leaf size=245 \[ \frac {1}{2} \sqrt {2+\sqrt {2}} \tan ^{-1}\left (\frac {\sqrt {2+\sqrt {2}} x \sqrt [4]{x^3-1}}{\sqrt {x^3-1}-x^2}\right )-\frac {1}{2} \sqrt {2-\sqrt {2}} \tan ^{-1}\left (\frac {\left (\sqrt {\frac {2}{2-\sqrt {2}}}-\frac {2}{\sqrt {2-\sqrt {2}}}\right ) x \sqrt [4]{x^3-1}}{\sqrt {x^3-1}-x^2}\right )-\frac {1}{2} \sqrt {2-\sqrt {2}} \tanh ^{-1}\left (\frac {\sqrt {2-\sqrt {2}} x \sqrt [4]{x^3-1}}{\sqrt {x^3-1}+x^2}\right )-\frac {1}{2} \sqrt {2+\sqrt {2}} \tanh ^{-1}\left (\frac {\sqrt {2+\sqrt {2}} x \sqrt [4]{x^3-1}}{\sqrt {x^3-1}+x^2}\right ) \]
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Rubi [F] time = 1.42, 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^6 \left (-4+x^3\right )}{\left (-1+x^3\right )^{3/4} \left (1-2 x^3+x^6+x^8\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {x^6 \left (-4+x^3\right )}{\left (-1+x^3\right )^{3/4} \left (1-2 x^3+x^6+x^8\right )} \, dx &=\int \left (\frac {x}{\left (-1+x^3\right )^{3/4}}+\frac {x \left (-1+2 x^3-4 x^5-x^6\right )}{\left (-1+x^3\right )^{3/4} \left (1-2 x^3+x^6+x^8\right )}\right ) \, dx\\ &=\int \frac {x}{\left (-1+x^3\right )^{3/4}} \, dx+\int \frac {x \left (-1+2 x^3-4 x^5-x^6\right )}{\left (-1+x^3\right )^{3/4} \left (1-2 x^3+x^6+x^8\right )} \, dx\\ &=\frac {\left (1-x^3\right )^{3/4} \int \frac {x}{\left (1-x^3\right )^{3/4}} \, dx}{\left (-1+x^3\right )^{3/4}}+\int \left (-\frac {x}{\left (-1+x^3\right )^{3/4} \left (1-2 x^3+x^6+x^8\right )}+\frac {2 x^4}{\left (-1+x^3\right )^{3/4} \left (1-2 x^3+x^6+x^8\right )}-\frac {4 x^6}{\left (-1+x^3\right )^{3/4} \left (1-2 x^3+x^6+x^8\right )}-\frac {x^7}{\left (-1+x^3\right )^{3/4} \left (1-2 x^3+x^6+x^8\right )}\right ) \, dx\\ &=\frac {x^2 \left (1-x^3\right )^{3/4} \, _2F_1\left (\frac {2}{3},\frac {3}{4};\frac {5}{3};x^3\right )}{2 \left (-1+x^3\right )^{3/4}}+2 \int \frac {x^4}{\left (-1+x^3\right )^{3/4} \left (1-2 x^3+x^6+x^8\right )} \, dx-4 \int \frac {x^6}{\left (-1+x^3\right )^{3/4} \left (1-2 x^3+x^6+x^8\right )} \, dx-\int \frac {x}{\left (-1+x^3\right )^{3/4} \left (1-2 x^3+x^6+x^8\right )} \, dx-\int \frac {x^7}{\left (-1+x^3\right )^{3/4} \left (1-2 x^3+x^6+x^8\right )} \, dx\\ \end {align*}
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Mathematica [F] time = 0.29, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^6 \left (-4+x^3\right )}{\left (-1+x^3\right )^{3/4} \left (1-2 x^3+x^6+x^8\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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IntegrateAlgebraic [A] time = 4.00, size = 225, normalized size = 0.92 \begin {gather*} \frac {1}{2} \sqrt {2-\sqrt {2}} \tan ^{-1}\left (\frac {\sqrt {2-\sqrt {2}} x \sqrt [4]{-1+x^3}}{-x^2+\sqrt {-1+x^3}}\right )+\frac {1}{2} \sqrt {2+\sqrt {2}} \tan ^{-1}\left (\frac {\sqrt {2+\sqrt {2}} x \sqrt [4]{-1+x^3}}{-x^2+\sqrt {-1+x^3}}\right )-\frac {1}{2} \sqrt {2-\sqrt {2}} \tanh ^{-1}\left (\frac {\sqrt {2-\sqrt {2}} x \sqrt [4]{-1+x^3}}{x^2+\sqrt {-1+x^3}}\right )-\frac {1}{2} \sqrt {2+\sqrt {2}} \tanh ^{-1}\left (\frac {\sqrt {2+\sqrt {2}} x \sqrt [4]{-1+x^3}}{x^2+\sqrt {-1+x^3}}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.69, size = 1425, normalized size = 5.82
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 (x^{3} - 4\right )} x^{6}}{{\left (x^{8} + x^{6} - 2 \, x^{3} + 1\right )} {\left (x^{3} - 1\right )}^{\frac {3}{4}}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 7.40, size = 468, normalized size = 1.91
method | result | size |
trager | \(\frac {\RootOf \left (\textit {\_Z}^{8}+1\right ) \ln \left (-\frac {\RootOf \left (\textit {\_Z}^{8}+1\right )^{11} x^{4}+\RootOf \left (\textit {\_Z}^{8}+1\right )^{7} x^{3}-\RootOf \left (\textit {\_Z}^{8}+1\right )^{7}+2 \left (x^{3}-1\right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{8}+1\right )^{2} x^{3}-2 \sqrt {x^{3}-1}\, \RootOf \left (\textit {\_Z}^{8}+1\right ) x^{2}+2 \left (x^{3}-1\right )^{\frac {3}{4}} x}{\RootOf \left (\textit {\_Z}^{8}+1\right )^{4} x^{4}-x^{3}+1}\right )}{2}+\frac {\RootOf \left (\textit {\_Z}^{8}+1\right )^{3} \ln \left (-\frac {-\RootOf \left (\textit {\_Z}^{8}+1\right )^{9} x^{4}+2 \left (x^{3}-1\right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{8}+1\right )^{6} x^{3}+\RootOf \left (\textit {\_Z}^{8}+1\right )^{5} x^{3}-2 \sqrt {x^{3}-1}\, \RootOf \left (\textit {\_Z}^{8}+1\right )^{3} x^{2}-\RootOf \left (\textit {\_Z}^{8}+1\right )^{5}+2 \left (x^{3}-1\right )^{\frac {3}{4}} x}{\RootOf \left (\textit {\_Z}^{8}+1\right )^{4} x^{4}+x^{3}-1}\right )}{2}+\frac {\RootOf \left (\textit {\_Z}^{8}+1\right )^{7} \ln \left (-\frac {-2 \sqrt {x^{3}-1}\, \RootOf \left (\textit {\_Z}^{8}+1\right )^{7} x^{2}-2 \left (x^{3}-1\right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{8}+1\right )^{6} x^{3}-\RootOf \left (\textit {\_Z}^{8}+1\right )^{5} x^{4}+2 \left (x^{3}-1\right )^{\frac {3}{4}} x +\RootOf \left (\textit {\_Z}^{8}+1\right ) x^{3}-\RootOf \left (\textit {\_Z}^{8}+1\right )}{\RootOf \left (\textit {\_Z}^{8}+1\right )^{4} x^{4}+x^{3}-1}\right )}{2}-\frac {\RootOf \left (\textit {\_Z}^{8}+1\right )^{5} \ln \left (-\frac {-\RootOf \left (\textit {\_Z}^{8}+1\right )^{7} x^{4}+2 \sqrt {x^{3}-1}\, \RootOf \left (\textit {\_Z}^{8}+1\right )^{5} x^{2}-2 \left (x^{3}-1\right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{8}+1\right )^{2} x^{3}-\RootOf \left (\textit {\_Z}^{8}+1\right )^{3} x^{3}+2 \left (x^{3}-1\right )^{\frac {3}{4}} x +\RootOf \left (\textit {\_Z}^{8}+1\right )^{3}}{\RootOf \left (\textit {\_Z}^{8}+1\right )^{4} x^{4}-x^{3}+1}\right )}{2}\) | \(468\) |
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^{3} - 4\right )} x^{6}}{{\left (x^{8} + x^{6} - 2 \, x^{3} + 1\right )} {\left (x^{3} - 1\right )}^{\frac {3}{4}}}\,{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 {x^6\,\left (x^3-4\right )}{{\left (x^3-1\right )}^{3/4}\,\left (x^8+x^6-2\,x^3+1\right )} \,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 {x^{6} \left (x^{3} - 4\right )}{\left (\left (x - 1\right ) \left (x^{2} + x + 1\right )\right )^{\frac {3}{4}} \left (x^{8} + x^{6} - 2 x^{3} + 1\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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