Optimal. Leaf size=138 \[ \frac {\left (-1+x^3\right )^{2/3} \left (2+3 x^3\right )}{10 x^5}+\frac {\text {ArcTan}\left (\frac {\sqrt {3} x}{x+2^{2/3} \sqrt [3]{-1+x^3}}\right )}{\sqrt [3]{2} \sqrt {3}}-\frac {\log \left (-2 x+2^{2/3} \sqrt [3]{-1+x^3}\right )}{3 \sqrt [3]{2}}+\frac {\log \left (2 x^2+2^{2/3} x \sqrt [3]{-1+x^3}+\sqrt [3]{2} \left (-1+x^3\right )^{2/3}\right )}{6 \sqrt [3]{2}} \]
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Rubi [B] Leaf count is larger than twice the leaf count of optimal. \(407\) vs. \(2(138)=276\).
time = 0.51, antiderivative size = 407, normalized size of antiderivative = 2.95, number of
steps used = 22, number of rules used = 16, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.571, Rules used
= {1600, 6857, 277, 270, 2174, 2183, 384, 502, 206, 31, 648, 631, 210, 642, 455, 58}
\begin {gather*} \frac {\text {ArcTan}\left (\frac {1-\frac {\sqrt [3]{2} (1-x)}{\sqrt [3]{x^3-1}}}{\sqrt {3}}\right )}{3 \sqrt [3]{2} \sqrt {3}}-\frac {\text {ArcTan}\left (\frac {\frac {2 \sqrt [3]{2} (1-x)}{\sqrt [3]{x^3-1}}+1}{\sqrt {3}}\right )}{3 \sqrt [3]{2} \sqrt {3}}+\frac {2^{2/3} \text {ArcTan}\left (\frac {\frac {2 \sqrt [3]{2} x}{\sqrt [3]{x^3-1}}+1}{\sqrt {3}}\right )}{3 \sqrt {3}}+\frac {\text {ArcTan}\left (\frac {1-2^{2/3} \sqrt [3]{x^3-1}}{\sqrt {3}}\right )}{3 \sqrt [3]{2} \sqrt {3}}+\frac {\log \left (x^3+1\right )}{18 \sqrt [3]{2}}+\frac {\log \left (1-\frac {\sqrt [3]{2} (1-x)}{\sqrt [3]{x^3-1}}\right )}{9 \sqrt [3]{2}}-\frac {\log \left (\frac {2^{2/3} (1-x)^2}{\left (x^3-1\right )^{2/3}}+\frac {\sqrt [3]{2} (1-x)}{\sqrt [3]{x^3-1}}+1\right )}{18 \sqrt [3]{2}}-\frac {\log \left (\sqrt [3]{2} x-\sqrt [3]{x^3-1}\right )}{3 \sqrt [3]{2}}+\frac {\log \left (\sqrt [3]{x^3-1}+\sqrt [3]{2}\right )}{6 \sqrt [3]{2}}-\frac {\log \left (2^{2/3} \sqrt [3]{x^3-1}-x+1\right )}{6 \sqrt [3]{2}}+\frac {\left (x^3-1\right )^{2/3}}{5 x^5}+\frac {3 \left (x^3-1\right )^{2/3}}{10 x^2}+\frac {\log \left ((1-x) (x+1)^2\right )}{18 \sqrt [3]{2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 31
Rule 58
Rule 206
Rule 210
Rule 270
Rule 277
Rule 384
Rule 455
Rule 502
Rule 631
Rule 642
Rule 648
Rule 1600
Rule 2174
Rule 2183
Rule 6857
Rubi steps
\begin {align*} \int \frac {\left (-1+x^3\right )^{2/3} \left (1+x^3+x^6\right )}{x^6 \left (-1+x^6\right )} \, dx &=\int \frac {1+x^3+x^6}{x^6 \sqrt [3]{-1+x^3} \left (1+x^3\right )} \, dx\\ &=\int \left (\frac {1}{x^6 \sqrt [3]{-1+x^3}}+\frac {1}{3 (1+x) \sqrt [3]{-1+x^3}}+\frac {2-x}{3 \left (1-x+x^2\right ) \sqrt [3]{-1+x^3}}\right ) \, dx\\ &=\frac {1}{3} \int \frac {1}{(1+x) \sqrt [3]{-1+x^3}} \, dx+\frac {1}{3} \int \frac {2-x}{\left (1-x+x^2\right ) \sqrt [3]{-1+x^3}} \, dx+\int \frac {1}{x^6 \sqrt [3]{-1+x^3}} \, dx\\ &=\frac {\left (-1+x^3\right )^{2/3}}{5 x^5}+\frac {\tan ^{-1}\left (\frac {1-\frac {\sqrt [3]{2} (1-x)}{\sqrt [3]{-1+x^3}}}{\sqrt {3}}\right )}{2 \sqrt [3]{2} \sqrt {3}}+\frac {\log \left ((1-x) (1+x)^2\right )}{12 \sqrt [3]{2}}-\frac {\log \left (1-x+2^{2/3} \sqrt [3]{-1+x^3}\right )}{4 \sqrt [3]{2}}+\frac {1}{3} \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+\frac {3}{5} \int \frac {1}{x^3 \sqrt [3]{-1+x^3}} \, dx\\ &=\frac {\left (-1+x^3\right )^{2/3}}{5 x^5}+\frac {3 \left (-1+x^3\right )^{2/3}}{10 x^2}+\frac {\tan ^{-1}\left (\frac {1-\frac {\sqrt [3]{2} (1-x)}{\sqrt [3]{-1+x^3}}}{\sqrt {3}}\right )}{2 \sqrt [3]{2} \sqrt {3}}+\frac {\log \left ((1-x) (1+x)^2\right )}{12 \sqrt [3]{2}}-\frac {\log \left (1-x+2^{2/3} \sqrt [3]{-1+x^3}\right )}{4 \sqrt [3]{2}}+\frac {1}{3} \left (-1-i \sqrt {3}\right ) \int \frac {1}{\left (-1-i \sqrt {3}+2 x\right ) \sqrt [3]{-1+x^3}} \, dx+\frac {1}{3} \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 (-1+x^3\right )^{2/3}}{5 x^5}+\frac {3 \left (-1+x^3\right )^{2/3}}{10 x^2}+\frac {\tan ^{-1}\left (\frac {1-\frac {\sqrt [3]{2} (1-x)}{\sqrt [3]{-1+x^3}}}{\sqrt {3}}\right )}{2 \sqrt [3]{2} \sqrt {3}}+\frac {\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} \sqrt {3}}+\frac {\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} \sqrt {3}}+\frac {\log \left ((1-x) (1+x)^2\right )}{12 \sqrt [3]{2}}+\frac {\log \left (-\left (1-i \sqrt {3}-2 x\right )^2 \left (1-i \sqrt {3}+2 x\right )\right )}{12 \sqrt [3]{2}}+\frac {\log \left (-\left (1+i \sqrt {3}-2 x\right )^2 \left (1+i \sqrt {3}+2 x\right )\right )}{12 \sqrt [3]{2}}-\frac {\log \left (1-x+2^{2/3} \sqrt [3]{-1+x^3}\right )}{4 \sqrt [3]{2}}-\frac {\log \left (-1-i \sqrt {3}-2 x+2\ 2^{2/3} \sqrt [3]{-1+x^3}\right )}{4 \sqrt [3]{2}}-\frac {\log \left (-1+i \sqrt {3}-2 x+2\ 2^{2/3} \sqrt [3]{-1+x^3}\right )}{4 \sqrt [3]{2}}\\ \end {align*}
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Mathematica [A]
time = 0.31, size = 138, normalized size = 1.00 \begin {gather*} \frac {\left (-1+x^3\right )^{2/3} \left (2+3 x^3\right )}{10 x^5}+\frac {\text {ArcTan}\left (\frac {\sqrt {3} x}{x+2^{2/3} \sqrt [3]{-1+x^3}}\right )}{\sqrt [3]{2} \sqrt {3}}-\frac {\log \left (-2 x+2^{2/3} \sqrt [3]{-1+x^3}\right )}{3 \sqrt [3]{2}}+\frac {\log \left (2 x^2+2^{2/3} x \sqrt [3]{-1+x^3}+\sqrt [3]{2} \left (-1+x^3\right )^{2/3}\right )}{6 \sqrt [3]{2}} \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 = 9.77, size = 914, normalized size = 6.62
method | result | size |
risch | \(\text {Expression too large to display}\) | \(914\) |
trager | \(\text {Expression too large to display}\) | \(1137\) |
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. 301 vs.
\(2 (105) = 210\).
time = 1.79, size = 301, normalized size = 2.18 \begin {gather*} -\frac {10 \, \sqrt {6} 2^{\frac {1}{6}} \left (-1\right )^{\frac {1}{3}} x^{5} \arctan \left (\frac {2^{\frac {1}{6}} {\left (6 \, \sqrt {6} 2^{\frac {2}{3}} \left (-1\right )^{\frac {2}{3}} {\left (5 \, x^{7} + 4 \, x^{4} - x\right )} {\left (x^{3} - 1\right )}^{\frac {2}{3}} + 12 \, \sqrt {6} \left (-1\right )^{\frac {1}{3}} {\left (19 \, x^{8} - 16 \, x^{5} + x^{2}\right )} {\left (x^{3} - 1\right )}^{\frac {1}{3}} - \sqrt {6} 2^{\frac {1}{3}} {\left (71 \, x^{9} - 111 \, x^{6} + 33 \, x^{3} - 1\right )}\right )}}{6 \, {\left (109 \, x^{9} - 105 \, x^{6} + 3 \, x^{3} + 1\right )}}\right ) - 10 \cdot 2^{\frac {2}{3}} \left (-1\right )^{\frac {1}{3}} x^{5} \log \left (-\frac {6 \cdot 2^{\frac {1}{3}} \left (-1\right )^{\frac {2}{3}} {\left (x^{3} - 1\right )}^{\frac {1}{3}} x^{2} + 2^{\frac {2}{3}} \left (-1\right )^{\frac {1}{3}} {\left (x^{3} + 1\right )} - 6 \, {\left (x^{3} - 1\right )}^{\frac {2}{3}} x}{x^{3} + 1}\right ) + 5 \cdot 2^{\frac {2}{3}} \left (-1\right )^{\frac {1}{3}} x^{5} \log \left (-\frac {3 \cdot 2^{\frac {2}{3}} \left (-1\right )^{\frac {1}{3}} {\left (5 \, x^{4} - x\right )} {\left (x^{3} - 1\right )}^{\frac {2}{3}} - 2^{\frac {1}{3}} \left (-1\right )^{\frac {2}{3}} {\left (19 \, x^{6} - 16 \, x^{3} + 1\right )} - 12 \, {\left (2 \, x^{5} - x^{2}\right )} {\left (x^{3} - 1\right )}^{\frac {1}{3}}}{x^{6} + 2 \, x^{3} + 1}\right ) - 18 \, {\left (3 \, x^{3} + 2\right )} {\left (x^{3} - 1\right )}^{\frac {2}{3}}}{180 \, x^{5}} \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 {\left (\left (x - 1\right ) \left (x^{2} + x + 1\right )\right )^{\frac {2}{3}} \left (x^{6} + x^{3} + 1\right )}{x^{6} \left (x - 1\right ) \left (x + 1\right ) \left (x^{2} - x + 1\right ) \left (x^{2} + x + 1\right )}\, 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 {{\left (x^3-1\right )}^{2/3}\,\left (x^6+x^3+1\right )}{x^6\,\left (x^6-1\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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