Optimal. Leaf size=113 \[ \frac {1}{2} \sqrt {\frac {1-x^3}{1+x^3}} \left (1+x^3\right )-\frac {1}{6} \sqrt {\frac {1-x^3}{1+x^3}} \left (1+x^3\right )^2-\frac {1}{9} \left (\frac {1-x^3}{1+x^3}\right )^{3/2} \left (1+x^3\right )^3-\frac {1}{3} \tan ^{-1}\left (\sqrt {\frac {1-x^3}{1+x^3}}\right ) \]
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Rubi [A]
time = 0.07, antiderivative size = 113, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 6, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used = {1981, 1980,
474, 466, 393, 210} \begin {gather*} -\frac {1}{3} \text {ArcTan}\left (\sqrt {\frac {1-x^3}{x^3+1}}\right )-\frac {1}{9} \left (\frac {1-x^3}{x^3+1}\right )^{3/2} \left (x^3+1\right )^3-\frac {1}{6} \sqrt {\frac {1-x^3}{x^3+1}} \left (x^3+1\right )^2+\frac {1}{2} \sqrt {\frac {1-x^3}{x^3+1}} \left (x^3+1\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 210
Rule 393
Rule 466
Rule 474
Rule 1980
Rule 1981
Rubi steps
\begin {align*} \int x^8 \sqrt {\frac {1-x^3}{1+x^3}} \, dx &=-\left (\frac {4}{3} \text {Subst}\left (\int \frac {x^2 \left (-1+x^2\right )^2}{\left (-1-x^2\right )^4} \, dx,x,\sqrt {\frac {1-x^3}{1+x^3}}\right )\right )\\ &=-\frac {1}{9} \left (\frac {1-x^3}{1+x^3}\right )^{3/2} \left (1+x^3\right )^3-\frac {2}{9} \text {Subst}\left (\int \frac {x^2 \left (6-6 x^2\right )}{\left (-1-x^2\right )^3} \, dx,x,\sqrt {\frac {1-x^3}{1+x^3}}\right )\\ &=-\frac {1}{6} \sqrt {\frac {1-x^3}{1+x^3}} \left (1+x^3\right )^2-\frac {1}{9} \left (\frac {1-x^3}{1+x^3}\right )^{3/2} \left (1+x^3\right )^3+\frac {1}{18} \text {Subst}\left (\int \frac {12-24 x^2}{\left (-1-x^2\right )^2} \, dx,x,\sqrt {\frac {1-x^3}{1+x^3}}\right )\\ &=\frac {1}{2} \sqrt {\frac {1-x^3}{1+x^3}} \left (1+x^3\right )-\frac {1}{6} \sqrt {\frac {1-x^3}{1+x^3}} \left (1+x^3\right )^2-\frac {1}{9} \left (\frac {1-x^3}{1+x^3}\right )^{3/2} \left (1+x^3\right )^3+\frac {1}{3} \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,\sqrt {\frac {1-x^3}{1+x^3}}\right )\\ &=\frac {1}{2} \sqrt {\frac {1-x^3}{1+x^3}} \left (1+x^3\right )-\frac {1}{6} \sqrt {\frac {1-x^3}{1+x^3}} \left (1+x^3\right )^2-\frac {1}{9} \left (\frac {1-x^3}{1+x^3}\right )^{3/2} \left (1+x^3\right )^3-\frac {1}{3} \tan ^{-1}\left (\sqrt {\frac {1-x^3}{1+x^3}}\right )\\ \end {align*}
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Mathematica [A]
time = 0.39, size = 95, normalized size = 0.84 \begin {gather*} \frac {\sqrt {\frac {1-x^3}{1+x^3}} \left (\sqrt {1-x^3} \left (4+x^3-x^6+2 x^9\right )-6 \sqrt {1+x^3} \tan ^{-1}\left (\frac {\sqrt {1-x^3}}{\sqrt {1+x^3}}\right )\right )}{18 \sqrt {1-x^3}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.20, size = 80, normalized size = 0.71
method | result | size |
risch | \(\frac {\left (x^{3}+1\right ) \left (2 x^{6}-3 x^{3}+4\right ) \sqrt {-\frac {x^{3}-1}{x^{3}+1}}}{18}-\frac {\arcsin \left (x^{3}\right ) \sqrt {-\frac {x^{3}-1}{x^{3}+1}}\, \sqrt {-\left (x^{3}-1\right ) \left (x^{3}+1\right )}}{6 \left (x^{3}-1\right )}\) | \(80\) |
trager | \(\frac {\left (x^{3}+1\right ) \left (2 x^{6}-3 x^{3}+4\right ) \sqrt {-\frac {x^{3}-1}{x^{3}+1}}}{18}+\frac {\RootOf \left (\textit {\_Z}^{2}+1\right ) \ln \left (\RootOf \left (\textit {\_Z}^{2}+1\right ) \sqrt {-\frac {x^{3}-1}{x^{3}+1}}\, x^{3}+x^{3}+\RootOf \left (\textit {\_Z}^{2}+1\right ) \sqrt {-\frac {x^{3}-1}{x^{3}+1}}\right )}{6}\) | \(99\) |
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 = 0.34, size = 65, normalized size = 0.58 \begin {gather*} \frac {1}{18} \, {\left (2 \, x^{9} - x^{6} + x^{3} + 4\right )} \sqrt {-\frac {x^{3} - 1}{x^{3} + 1}} - \frac {1}{3} \, \arctan \left (\frac {{\left (x^{3} + 1\right )} \sqrt {-\frac {x^{3} - 1}{x^{3} + 1}} - 1}{x^{3}}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
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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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 [B]
time = 2.66, size = 101, normalized size = 0.89 \begin {gather*} \frac {2\,\sqrt {-\frac {x^3-1}{x^3+1}}}{9}-\frac {\mathrm {atan}\left (\sqrt {-\frac {x^3-1}{x^3+1}}\right )}{3}+\frac {x^3\,\sqrt {-\frac {x^3-1}{x^3+1}}}{18}-\frac {x^6\,\sqrt {-\frac {x^3-1}{x^3+1}}}{18}+\frac {x^9\,\sqrt {-\frac {x^3-1}{x^3+1}}}{9} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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