Optimal. Leaf size=37 \[ \frac {2\ 2^{2/3} \tan ^{-1}\left (\frac {\sqrt {3} \left (1+\sqrt [3]{2} x\right )}{\sqrt {1+x^3}}\right )}{\sqrt {3}} \]
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Rubi [A]
time = 0.07, antiderivative size = 37, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {2162, 209}
\begin {gather*} \frac {2\ 2^{2/3} \text {ArcTan}\left (\frac {\sqrt {3} \left (\sqrt [3]{2} x+1\right )}{\sqrt {x^3+1}}\right )}{\sqrt {3}} \end {gather*}
Antiderivative was successfully verified.
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Rule 209
Rule 2162
Rubi steps
\begin {align*} \int \frac {2^{2/3}-2 x}{\left (2^{2/3}+x\right ) \sqrt {1+x^3}} \, dx &=\left (2\ 2^{2/3}\right ) \text {Subst}\left (\int \frac {1}{1+3 x^2} \, dx,x,\frac {1+\sqrt [3]{2} x}{\sqrt {1+x^3}}\right )\\ &=\frac {2\ 2^{2/3} \tan ^{-1}\left (\frac {\sqrt {3} \left (1+\sqrt [3]{2} x\right )}{\sqrt {1+x^3}}\right )}{\sqrt {3}}\\ \end {align*}
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Mathematica [A]
time = 1.25, size = 39, normalized size = 1.05 \begin {gather*} -\frac {2\ 2^{2/3} \tan ^{-1}\left (\frac {\sqrt {1+x^3}}{\sqrt {3} \left (1+\sqrt [3]{2} x\right )}\right )}{\sqrt {3}} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 4 vs. order
3.
time = 1.33, size = 258, normalized size = 6.97
method | result | size |
trager | \(-\frac {\RootOf \left (\textit {\_Z}^{2}+6 \,2^{\frac {1}{3}}\right ) \ln \left (\frac {12 x \sqrt {x^{3}+1}+3 \RootOf \left (\textit {\_Z}^{2}+6 \,2^{\frac {1}{3}}\right ) 2^{\frac {2}{3}} x^{2}-\RootOf \left (\textit {\_Z}^{2}+6 \,2^{\frac {1}{3}}\right ) x^{3}+6 \sqrt {x^{3}+1}\, 2^{\frac {2}{3}}+6 \RootOf \left (\textit {\_Z}^{2}+6 \,2^{\frac {1}{3}}\right ) 2^{\frac {1}{3}} x +2 \RootOf \left (\textit {\_Z}^{2}+6 \,2^{\frac {1}{3}}\right )}{\left (2^{\frac {1}{3}} x +2\right )^{3}}\right )}{3}\) | \(108\) |
default | \(-\frac {4 \left (\frac {3}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {1+x}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x -\frac {1}{2}-\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x -\frac {1}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \EllipticF \left (\sqrt {\frac {1+x}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, \sqrt {\frac {-\frac {3}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )}{\sqrt {x^{3}+1}}+\frac {6 \,2^{\frac {2}{3}} \left (\frac {3}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {1+x}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x -\frac {1}{2}-\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x -\frac {1}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \EllipticPi \left (\sqrt {\frac {1+x}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, \frac {-\frac {3}{2}+\frac {i \sqrt {3}}{2}}{2^{\frac {2}{3}}-1}, \sqrt {\frac {-\frac {3}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )}{\sqrt {x^{3}+1}\, \left (2^{\frac {2}{3}}-1\right )}\) | \(258\) |
elliptic | \(-\frac {4 \left (\frac {3}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {1+x}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x -\frac {1}{2}-\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x -\frac {1}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \EllipticF \left (\sqrt {\frac {1+x}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, \sqrt {\frac {-\frac {3}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )}{\sqrt {x^{3}+1}}+\frac {6 \,2^{\frac {2}{3}} \left (\frac {3}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {1+x}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x -\frac {1}{2}-\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x -\frac {1}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \EllipticPi \left (\sqrt {\frac {1+x}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, \frac {-\frac {3}{2}+\frac {i \sqrt {3}}{2}}{2^{\frac {2}{3}}-1}, \sqrt {\frac {-\frac {3}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )}{\sqrt {x^{3}+1}\, \left (2^{\frac {2}{3}}-1\right )}\) | \(258\) |
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 [B] Leaf count of result is larger than twice the leaf count of optimal. 75 vs.
\(2 (27) = 54\).
time = 0.44, size = 75, normalized size = 2.03 \begin {gather*} \frac {1}{3} \, \sqrt {6} 2^{\frac {1}{6}} \arctan \left (-\frac {\sqrt {6} 2^{\frac {1}{6}} {\left (2 \, x^{5} + 2 \, x^{2} - 2^{\frac {2}{3}} {\left (7 \, x^{4} + 4 \, x\right )} - 2^{\frac {1}{3}} {\left (5 \, x^{3} + 2\right )}\right )} \sqrt {x^{3} + 1}}{12 \, {\left (2 \, x^{6} + 3 \, x^{3} + 1\right )}}\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 \left (- \frac {2^{\frac {2}{3}}}{x \sqrt {x^{3} + 1} + 2^{\frac {2}{3}} \sqrt {x^{3} + 1}}\right )\, dx - \int \frac {2 x}{x \sqrt {x^{3} + 1} + 2^{\frac {2}{3}} \sqrt {x^{3} + 1}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 3.59, size = 70, normalized size = 1.89 \begin {gather*} \frac {2^{2/3}\,\sqrt {3}\,\ln \left (\frac {\left (\sqrt {3}\,1{}\mathrm {i}+\sqrt {x^3+1}+2^{1/3}\,\sqrt {3}\,x\,1{}\mathrm {i}\right )\,{\left (\sqrt {3}\,1{}\mathrm {i}-\sqrt {x^3+1}+2^{1/3}\,\sqrt {3}\,x\,1{}\mathrm {i}\right )}^3}{{\left (x+2^{2/3}\right )}^6}\right )\,1{}\mathrm {i}}{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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