Optimal. Leaf size=97 \[ -\sqrt {3} \text {ArcTan}\left (\frac {\sqrt {3} x}{-x+2 \sqrt [3]{1-x^3+x^8}}\right )-\log \left (x+\sqrt [3]{1-x^3+x^8}\right )+\frac {1}{2} \log \left (x^2-x \sqrt [3]{1-x^3+x^8}+\left (1-x^3+x^8\right )^{2/3}\right ) \]
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Rubi [F]
time = 1.21, 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 {-3+5 x^8}{\left (1+x^8\right ) \sqrt [3]{1-x^3+x^8}} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {-3+5 x^8}{\left (1+x^8\right ) \sqrt [3]{1-x^3+x^8}} \, dx &=\int \left (\frac {5}{\sqrt [3]{1-x^3+x^8}}-\frac {8}{\left (1+x^8\right ) \sqrt [3]{1-x^3+x^8}}\right ) \, dx\\ &=5 \int \frac {1}{\sqrt [3]{1-x^3+x^8}} \, dx-8 \int \frac {1}{\left (1+x^8\right ) \sqrt [3]{1-x^3+x^8}} \, dx\\ &=5 \int \frac {1}{\sqrt [3]{1-x^3+x^8}} \, dx-8 \int \left (\frac {i}{2 \left (i-x^4\right ) \sqrt [3]{1-x^3+x^8}}+\frac {i}{2 \left (i+x^4\right ) \sqrt [3]{1-x^3+x^8}}\right ) \, dx\\ &=-\left (4 i \int \frac {1}{\left (i-x^4\right ) \sqrt [3]{1-x^3+x^8}} \, dx\right )-4 i \int \frac {1}{\left (i+x^4\right ) \sqrt [3]{1-x^3+x^8}} \, dx+5 \int \frac {1}{\sqrt [3]{1-x^3+x^8}} \, dx\\ &=-\left (4 i \int \left (-\frac {(-1)^{3/4}}{2 \left (\sqrt [4]{-1}-x^2\right ) \sqrt [3]{1-x^3+x^8}}-\frac {(-1)^{3/4}}{2 \left (\sqrt [4]{-1}+x^2\right ) \sqrt [3]{1-x^3+x^8}}\right ) \, dx\right )-4 i \int \left (-\frac {\sqrt [4]{-1}}{2 \left (-(-1)^{3/4}-x^2\right ) \sqrt [3]{1-x^3+x^8}}-\frac {\sqrt [4]{-1}}{2 \left (-(-1)^{3/4}+x^2\right ) \sqrt [3]{1-x^3+x^8}}\right ) \, dx+5 \int \frac {1}{\sqrt [3]{1-x^3+x^8}} \, dx\\ &=5 \int \frac {1}{\sqrt [3]{1-x^3+x^8}} \, dx-\left (2 \sqrt [4]{-1}\right ) \int \frac {1}{\left (\sqrt [4]{-1}-x^2\right ) \sqrt [3]{1-x^3+x^8}} \, dx-\left (2 \sqrt [4]{-1}\right ) \int \frac {1}{\left (\sqrt [4]{-1}+x^2\right ) \sqrt [3]{1-x^3+x^8}} \, dx+\left (2 (-1)^{3/4}\right ) \int \frac {1}{\left (-(-1)^{3/4}-x^2\right ) \sqrt [3]{1-x^3+x^8}} \, dx+\left (2 (-1)^{3/4}\right ) \int \frac {1}{\left (-(-1)^{3/4}+x^2\right ) \sqrt [3]{1-x^3+x^8}} \, dx\\ &=5 \int \frac {1}{\sqrt [3]{1-x^3+x^8}} \, dx-\left (2 \sqrt [4]{-1}\right ) \int \left (-\frac {(-1)^{7/8}}{2 \left (\sqrt [8]{-1}-x\right ) \sqrt [3]{1-x^3+x^8}}-\frac {(-1)^{7/8}}{2 \left (\sqrt [8]{-1}+x\right ) \sqrt [3]{1-x^3+x^8}}\right ) \, dx-\left (2 \sqrt [4]{-1}\right ) \int \left (-\frac {(-1)^{3/8}}{2 \left (-(-1)^{5/8}-x\right ) \sqrt [3]{1-x^3+x^8}}-\frac {(-1)^{3/8}}{2 \left (-(-1)^{5/8}+x\right ) \sqrt [3]{1-x^3+x^8}}\right ) \, dx+\left (2 (-1)^{3/4}\right ) \int \left (\frac {(-1)^{5/8}}{2 \left ((-1)^{3/8}-x\right ) \sqrt [3]{1-x^3+x^8}}+\frac {(-1)^{5/8}}{2 \left ((-1)^{3/8}+x\right ) \sqrt [3]{1-x^3+x^8}}\right ) \, dx+\left (2 (-1)^{3/4}\right ) \int \left (\frac {\sqrt [8]{-1}}{2 \left (-(-1)^{7/8}-x\right ) \sqrt [3]{1-x^3+x^8}}+\frac {\sqrt [8]{-1}}{2 \left (-(-1)^{7/8}+x\right ) \sqrt [3]{1-x^3+x^8}}\right ) \, dx\\ &=5 \int \frac {1}{\sqrt [3]{1-x^3+x^8}} \, dx-\sqrt [8]{-1} \int \frac {1}{\left (\sqrt [8]{-1}-x\right ) \sqrt [3]{1-x^3+x^8}} \, dx-\sqrt [8]{-1} \int \frac {1}{\left (\sqrt [8]{-1}+x\right ) \sqrt [3]{1-x^3+x^8}} \, dx-(-1)^{3/8} \int \frac {1}{\left ((-1)^{3/8}-x\right ) \sqrt [3]{1-x^3+x^8}} \, dx-(-1)^{3/8} \int \frac {1}{\left ((-1)^{3/8}+x\right ) \sqrt [3]{1-x^3+x^8}} \, dx+(-1)^{5/8} \int \frac {1}{\left (-(-1)^{5/8}-x\right ) \sqrt [3]{1-x^3+x^8}} \, dx+(-1)^{5/8} \int \frac {1}{\left (-(-1)^{5/8}+x\right ) \sqrt [3]{1-x^3+x^8}} \, dx+(-1)^{7/8} \int \frac {1}{\left (-(-1)^{7/8}-x\right ) \sqrt [3]{1-x^3+x^8}} \, dx+(-1)^{7/8} \int \frac {1}{\left (-(-1)^{7/8}+x\right ) \sqrt [3]{1-x^3+x^8}} \, dx\\ \end {align*}
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Mathematica [A]
time = 1.48, size = 94, normalized size = 0.97 \begin {gather*} \sqrt {3} \text {ArcTan}\left (\frac {\sqrt {3} x}{x-2 \sqrt [3]{1-x^3+x^8}}\right )-\log \left (x+\sqrt [3]{1-x^3+x^8}\right )+\frac {1}{2} \log \left (x^2-x \sqrt [3]{1-x^3+x^8}+\left (1-x^3+x^8\right )^{2/3}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
3.
time = 1.30, size = 307, normalized size = 3.16
method | result | size |
trager | \(-\ln \left (\frac {\RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{8}-x^{8}+\RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2} x^{3}+\left (x^{8}-x^{3}+1\right )^{\frac {2}{3}} \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x -\RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{8}-x^{3}+1\right )^{\frac {1}{3}} x^{2}-3 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{3}-2 \left (x^{8}-x^{3}+1\right )^{\frac {2}{3}} x -\left (x^{8}-x^{3}+1\right )^{\frac {1}{3}} x^{2}+2 x^{3}+\RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )-1}{x^{8}+1}\right )+\RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \ln \left (-\frac {-\RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{8}+x^{8}+\RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2} x^{3}+\left (x^{8}-x^{3}+1\right )^{\frac {2}{3}} \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x +2 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{8}-x^{3}+1\right )^{\frac {1}{3}} x^{2}+\left (x^{8}-x^{3}+1\right )^{\frac {2}{3}} x -\left (x^{8}-x^{3}+1\right )^{\frac {1}{3}} x^{2}-x^{3}-\RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )+1}{x^{8}+1}\right )\) | \(307\) |
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 = 1.25, size = 121, normalized size = 1.25 \begin {gather*} -\sqrt {3} \arctan \left (\frac {4 \, \sqrt {3} {\left (x^{8} - x^{3} + 1\right )}^{\frac {1}{3}} x^{2} + 2 \, \sqrt {3} {\left (x^{8} - x^{3} + 1\right )}^{\frac {2}{3}} x + \sqrt {3} {\left (x^{8} - x^{3} + 1\right )}}{x^{8} - 9 \, x^{3} + 1}\right ) - \frac {1}{2} \, \log \left (\frac {x^{8} + 3 \, {\left (x^{8} - x^{3} + 1\right )}^{\frac {1}{3}} x^{2} + 3 \, {\left (x^{8} - x^{3} + 1\right )}^{\frac {2}{3}} x + 1}{x^{8} + 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 {5 x^{8} - 3}{\left (x^{8} + 1\right ) \sqrt [3]{x^{8} - x^{3} + 1}}\, 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 {5\,x^8-3}{\left (x^8+1\right )\,{\left (x^8-x^3+1\right )}^{1/3}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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