Optimal. Leaf size=119 \[ -\frac {\sqrt {3} \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{2} (1+x)}{\sqrt [3]{1+x^3}}}{\sqrt {3}}\right )}{\sqrt [3]{2}}-\frac {\log \left (1+\frac {2^{2/3} (1+x)^2}{\left (1+x^3\right )^{2/3}}-\frac {\sqrt [3]{2} (1+x)}{\sqrt [3]{1+x^3}}\right )}{2 \sqrt [3]{2}}+\frac {\log \left (1+\frac {\sqrt [3]{2} (1+x)}{\sqrt [3]{1+x^3}}\right )}{\sqrt [3]{2}} \]
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Rubi [B] Leaf count is larger than twice the leaf count of optimal. \(357\) vs. \(2(119)=238\).
time = 0.18, antiderivative size = 357, normalized size of antiderivative = 3.00, number of steps
used = 16, number of rules used = 12, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.522, Rules used = {2183, 384,
502, 2174, 206, 31, 648, 631, 210, 642, 455, 57} \begin {gather*} \frac {\log \left (1-x^3\right )}{3 \sqrt [3]{2}}-\frac {\log \left (\frac {2^{2/3} (x+1)^2}{\left (x^3+1\right )^{2/3}}-\frac {\sqrt [3]{2} (x+1)}{\sqrt [3]{x^3+1}}+1\right )}{3 \sqrt [3]{2}}+\frac {1}{3} 2^{2/3} \log \left (\frac {\sqrt [3]{2} (x+1)}{\sqrt [3]{x^3+1}}+1\right )-\frac {\log \left (\sqrt [3]{2}-\sqrt [3]{x^3+1}\right )}{2 \sqrt [3]{2}}-\frac {\log \left (\sqrt [3]{2} x-\sqrt [3]{x^3+1}\right )}{2 \sqrt [3]{2}}+\frac {\log \left (-2^{2/3} \sqrt [3]{x^3+1}+x+1\right )}{2 \sqrt [3]{2}}+\frac {\tan ^{-1}\left (\frac {\frac {2 \sqrt [3]{2} x}{\sqrt [3]{x^3+1}}+1}{\sqrt {3}}\right )}{\sqrt [3]{2} \sqrt {3}}-\frac {2^{2/3} \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{2} (x+1)}{\sqrt [3]{x^3+1}}}{\sqrt {3}}\right )}{\sqrt {3}}-\frac {\tan ^{-1}\left (\frac {\frac {\sqrt [3]{2} (x+1)}{\sqrt [3]{x^3+1}}+1}{\sqrt {3}}\right )}{\sqrt [3]{2} \sqrt {3}}-\frac {\tan ^{-1}\left (\frac {2^{2/3} \sqrt [3]{x^3+1}+1}{\sqrt {3}}\right )}{\sqrt [3]{2} \sqrt {3}}-\frac {\log \left ((1-x)^2 (x+1)\right )}{6 \sqrt [3]{2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 31
Rule 57
Rule 206
Rule 210
Rule 384
Rule 455
Rule 502
Rule 631
Rule 642
Rule 648
Rule 2174
Rule 2183
Rubi steps
\begin {align*} \int \frac {1-x}{\left (1+x+x^2\right ) \sqrt [3]{1+x^3}} \, dx &=\int \left (\frac {-1-i \sqrt {3}}{\left (1-i \sqrt {3}+2 x\right ) \sqrt [3]{1+x^3}}+\frac {-1+i \sqrt {3}}{\left (1+i \sqrt {3}+2 x\right ) \sqrt [3]{1+x^3}}\right ) \, dx\\ &=\left (-1-i \sqrt {3}\right ) \int \frac {1}{\left (1-i \sqrt {3}+2 x\right ) \sqrt [3]{1+x^3}} \, dx+\left (-1+i \sqrt {3}\right ) \int \frac {1}{\left (1+i \sqrt {3}+2 x\right ) \sqrt [3]{1+x^3}} \, dx\\ &=\frac {\left (3-i \sqrt {3}\right ) \tan ^{-1}\left (\frac {2-\frac {\sqrt [3]{2} \left (1-i \sqrt {3}-2 x\right )}{\sqrt [3]{1+x^3}}}{2 \sqrt {3}}\right )}{2 \sqrt [3]{2} \left (i+\sqrt {3}\right )}-\frac {\left (3+i \sqrt {3}\right ) \tan ^{-1}\left (\frac {2-\frac {\sqrt [3]{2} \left (1+i \sqrt {3}-2 x\right )}{\sqrt [3]{1+x^3}}}{2 \sqrt {3}}\right )}{2 \sqrt [3]{2} \left (i-\sqrt {3}\right )}-\frac {\left (i-\sqrt {3}\right ) \log \left (\left (1-i \sqrt {3}-2 x\right ) \left (1-i \sqrt {3}+2 x\right )^2\right )}{4 \sqrt [3]{2} \left (i+\sqrt {3}\right )}-\frac {\left (i+\sqrt {3}\right ) \log \left (\left (1+i \sqrt {3}-2 x\right ) \left (1+i \sqrt {3}+2 x\right )^2\right )}{4 \sqrt [3]{2} \left (i-\sqrt {3}\right )}+\frac {3 \left (i-\sqrt {3}\right ) \log \left (1-i \sqrt {3}-2 x+2\ 2^{2/3} \sqrt [3]{1+x^3}\right )}{4 \sqrt [3]{2} \left (i+\sqrt {3}\right )}+\frac {3 \left (i+\sqrt {3}\right ) \log \left (1+i \sqrt {3}-2 x+2\ 2^{2/3} \sqrt [3]{1+x^3}\right )}{4 \sqrt [3]{2} \left (i-\sqrt {3}\right )}\\ \end {align*}
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Mathematica [A]
time = 0.86, size = 139, normalized size = 1.17 \begin {gather*} \frac {2 \sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} \sqrt [3]{1+x^3}}{-2 \sqrt [3]{2}-2 \sqrt [3]{2} x+\sqrt [3]{1+x^3}}\right )+2 \log \left (\sqrt [3]{2}+\sqrt [3]{2} x+\sqrt [3]{1+x^3}\right )-\log \left (2^{2/3}+2\ 2^{2/3} x+2^{2/3} x^2-\sqrt [3]{2} (1+x) \sqrt [3]{1+x^3}+\left (1+x^3\right )^{2/3}\right )}{2 \sqrt [3]{2}} \end {gather*}
Antiderivative was successfully verified.
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Mathics [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {cought exception: maximum recursion depth exceeded while calling a Python object} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
3.
time = 5.45, size = 714, normalized size = 6.00
method | result | size |
trager | \(\text {Expression too large to display}\) | \(714\) |
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. 268 vs.
\(2 (93) = 186\).
time = 5.49, size = 268, normalized size = 2.25 \begin {gather*} \frac {1}{6} \, \sqrt {3} 2^{\frac {2}{3}} \arctan \left (\frac {\sqrt {3} 2^{\frac {1}{6}} {\left (2^{\frac {5}{6}} {\left (x^{6} + 7 \, x^{5} + 10 \, x^{4} + 7 \, x^{3} + 10 \, x^{2} + 7 \, x + 1\right )} - 4 \, \sqrt {2} {\left (x^{5} + x^{4} - 3 \, x^{3} - 3 \, x^{2} + x + 1\right )} {\left (x^{3} + 1\right )}^{\frac {1}{3}} + 4 \cdot 2^{\frac {1}{6}} {\left (x^{4} + 4 \, x^{3} + 5 \, x^{2} + 4 \, x + 1\right )} {\left (x^{3} + 1\right )}^{\frac {2}{3}}\right )}}{6 \, {\left (3 \, x^{6} + 9 \, x^{5} + 6 \, x^{4} + x^{3} + 6 \, x^{2} + 9 \, x + 3\right )}}\right ) - \frac {1}{12} \cdot 2^{\frac {2}{3}} \log \left (\frac {2^{\frac {2}{3}} {\left (x^{3} + 1\right )}^{\frac {2}{3}} {\left (x^{2} + 3 \, x + 1\right )} - 2^{\frac {1}{3}} {\left (x^{4} - 3 \, x^{2} + 1\right )} - 4 \, {\left (x^{3} + 1\right )}^{\frac {1}{3}} {\left (x^{2} + x\right )}}{x^{4} + 2 \, x^{3} + 3 \, x^{2} + 2 \, x + 1}\right ) + \frac {1}{6} \cdot 2^{\frac {2}{3}} \log \left (\frac {2^{\frac {2}{3}} {\left (x^{2} + x + 1\right )} + 2 \cdot 2^{\frac {1}{3}} {\left (x^{3} + 1\right )}^{\frac {1}{3}} {\left (x + 1\right )} + 2 \, {\left (x^{3} + 1\right )}^{\frac {2}{3}}}{x^{2} + x + 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 {x}{x^{2} \sqrt [3]{x^{3} + 1} + x \sqrt [3]{x^{3} + 1} + \sqrt [3]{x^{3} + 1}}\, dx - \int \left (- \frac {1}{x^{2} \sqrt [3]{x^{3} + 1} + x \sqrt [3]{x^{3} + 1} + \sqrt [3]{x^{3} + 1}}\right )\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] N/A
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 {x-1}{{\left (x^3+1\right )}^{1/3}\,\left (x^2+x+1\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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