Integrand size = 25, antiderivative size = 214 \[ \int \frac {x}{\sqrt {-1+x^3} \left (-10+6 \sqrt {3}+x^3\right )} \, dx=-\frac {\left (2+\sqrt {3}\right ) \arctan \left (\frac {\sqrt [4]{3} \left (1-\sqrt {3}\right ) (1-x)}{\sqrt {2} \sqrt {-1+x^3}}\right )}{2 \sqrt {2} 3^{3/4}}+\frac {\left (2+\sqrt {3}\right ) \arctan \left (\frac {\left (1+\sqrt {3}\right ) \sqrt {-1+x^3}}{\sqrt {2} 3^{3/4}}\right )}{3 \sqrt {2} 3^{3/4}}+\frac {\left (2+\sqrt {3}\right ) \text {arctanh}\left (\frac {\sqrt [4]{3} \left (1+\sqrt {3}\right ) (1-x)}{\sqrt {2} \sqrt {-1+x^3}}\right )}{6 \sqrt {2} \sqrt [4]{3}}+\frac {\left (2+\sqrt {3}\right ) \text {arctanh}\left (\frac {\sqrt [4]{3} \left (1-\sqrt {3}+2 x\right )}{\sqrt {2} \sqrt {-1+x^3}}\right )}{3 \sqrt {2} \sqrt [4]{3}} \]
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Time = 0.03 (sec) , antiderivative size = 214, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.040, Rules used = {501} \[ \int \frac {x}{\sqrt {-1+x^3} \left (-10+6 \sqrt {3}+x^3\right )} \, dx=-\frac {\left (2+\sqrt {3}\right ) \arctan \left (\frac {\sqrt [4]{3} \left (1-\sqrt {3}\right ) (1-x)}{\sqrt {2} \sqrt {x^3-1}}\right )}{2 \sqrt {2} 3^{3/4}}+\frac {\left (2+\sqrt {3}\right ) \arctan \left (\frac {\left (1+\sqrt {3}\right ) \sqrt {x^3-1}}{\sqrt {2} 3^{3/4}}\right )}{3 \sqrt {2} 3^{3/4}}+\frac {\left (2+\sqrt {3}\right ) \text {arctanh}\left (\frac {\sqrt [4]{3} \left (1+\sqrt {3}\right ) (1-x)}{\sqrt {2} \sqrt {x^3-1}}\right )}{6 \sqrt {2} \sqrt [4]{3}}+\frac {\left (2+\sqrt {3}\right ) \text {arctanh}\left (\frac {\sqrt [4]{3} \left (2 x-\sqrt {3}+1\right )}{\sqrt {2} \sqrt {x^3-1}}\right )}{3 \sqrt {2} \sqrt [4]{3}} \]
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Rule 501
Rubi steps \begin{align*} \text {integral}& = -\frac {\left (2+\sqrt {3}\right ) \arctan \left (\frac {\sqrt [4]{3} \left (1-\sqrt {3}\right ) (1-x)}{\sqrt {2} \sqrt {-1+x^3}}\right )}{2 \sqrt {2} 3^{3/4}}+\frac {\left (2+\sqrt {3}\right ) \arctan \left (\frac {\left (1+\sqrt {3}\right ) \sqrt {-1+x^3}}{\sqrt {2} 3^{3/4}}\right )}{3 \sqrt {2} 3^{3/4}}+\frac {\left (2+\sqrt {3}\right ) \text {arctanh}\left (\frac {\sqrt [4]{3} \left (1+\sqrt {3}\right ) (1-x)}{\sqrt {2} \sqrt {-1+x^3}}\right )}{6 \sqrt {2} \sqrt [4]{3}}+\frac {\left (2+\sqrt {3}\right ) \text {arctanh}\left (\frac {\sqrt [4]{3} \left (1-\sqrt {3}+2 x\right )}{\sqrt {2} \sqrt {-1+x^3}}\right )}{3 \sqrt {2} \sqrt [4]{3}} \\ \end{align*}
Result contains higher order function than in optimal. Order 6 vs. order 3 in optimal.
Time = 10.04 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.32 \[ \int \frac {x}{\sqrt {-1+x^3} \left (-10+6 \sqrt {3}+x^3\right )} \, dx=\frac {x^2 \sqrt {1-x^3} \operatorname {AppellF1}\left (\frac {2}{3},\frac {1}{2},1,\frac {5}{3},x^3,-\frac {x^3}{-10+6 \sqrt {3}}\right )}{4 \left (-5+3 \sqrt {3}\right ) \sqrt {-1+x^3}} \]
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Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 41.40 (sec) , antiderivative size = 352, normalized size of antiderivative = 1.64
method | result | size |
default | \(\frac {2 \left (\sqrt {3}-1\right ) \left (-\frac {3}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {-1+x}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x +\frac {1}{2}-\frac {i \sqrt {3}}{2}}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x +\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \sqrt {3}\, \Pi \left (\sqrt {\frac {-1+x}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, \frac {\left (\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}{3}, \sqrt {\frac {\frac {3}{2}+\frac {i \sqrt {3}}{2}}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )}{3 \left (6 \sqrt {3}-12\right ) \sqrt {x^{3}-1}}-\frac {\sqrt {2}\, \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\operatorname {RootOf}\left (\textit {\_Z}^{2}+\left (1-\sqrt {3}\right ) \textit {\_Z} -2 \sqrt {3}+4\right )}{\sum }\frac {\left (-\underline {\hspace {1.25 ex}}\alpha \sqrt {3}-\underline {\hspace {1.25 ex}}\alpha -2\right ) \left (-3-i \sqrt {3}\right ) \sqrt {\frac {-1+x}{-3-i \sqrt {3}}}\, \sqrt {\frac {-i \sqrt {3}+2 x +1}{3-i \sqrt {3}}}\, \sqrt {\frac {i \sqrt {3}+2 x +1}{i \sqrt {3}+3}}\, \left (1+2 \underline {\hspace {1.25 ex}}\alpha +\underline {\hspace {1.25 ex}}\alpha \sqrt {3}\right ) \Pi \left (\sqrt {\frac {-1+x}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, \frac {i \underline {\hspace {1.25 ex}}\alpha \sqrt {3}}{3}+\frac {i \underline {\hspace {1.25 ex}}\alpha }{2}+\frac {\underline {\hspace {1.25 ex}}\alpha \sqrt {3}}{2}+\underline {\hspace {1.25 ex}}\alpha +\frac {i \sqrt {3}}{6}+\frac {1}{2}, \sqrt {\frac {\frac {3}{2}+\frac {i \sqrt {3}}{2}}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )}{\left (-\sqrt {3}+2 \underline {\hspace {1.25 ex}}\alpha +1\right ) \sqrt {x^{3}-1}}\right )}{18}\) | \(352\) |
elliptic | \(\frac {2 \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}}}\, \left (\frac {2 \left (1-\sqrt {3}\right )^{2} \sqrt {3}}{9}+\frac {\left (1-\sqrt {3}\right )^{2}}{3}+\frac {2 \left (1-\sqrt {3}\right ) \sqrt {3}}{9}+\frac {2}{3}-\frac {\sqrt {3}}{9}\right ) \Pi \left (\sqrt {\frac {-1+x}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, \frac {i \left (1-\sqrt {3}\right )^{2} \sqrt {3}}{6}+\frac {i \left (1-\sqrt {3}\right )^{2}}{3}+\frac {\left (1-\sqrt {3}\right )^{2} \sqrt {3}}{3}+\frac {\left (1-\sqrt {3}\right )^{2}}{2}+\frac {i \left (1-\sqrt {3}\right ) \sqrt {3}}{6}+\frac {i \left (1-\sqrt {3}\right )}{3}+\frac {\left (1-\sqrt {3}\right ) \sqrt {3}}{3}+1+\frac {i}{3}-\frac {\sqrt {3}}{6}+\frac {i \sqrt {3}}{6}, \sqrt {\frac {\frac {3}{2}+\frac {i \sqrt {3}}{2}}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )}{3 \left (1-\sqrt {3}\right ) \sqrt {x^{3}-1}}+\frac {\sqrt {2}\, \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\operatorname {RootOf}\left (\textit {\_Z}^{2}+\left (1-\sqrt {3}\right ) \textit {\_Z} -2 \sqrt {3}+4\right )}{\sum }\frac {\left (-3-i \sqrt {3}\right ) \sqrt {\frac {-1+x}{-3-i \sqrt {3}}}\, \sqrt {\frac {-i \sqrt {3}+2 x +1}{3-i \sqrt {3}}}\, \sqrt {\frac {i \sqrt {3}+2 x +1}{i \sqrt {3}+3}}\, \left (3 \underline {\hspace {1.25 ex}}\alpha ^{2}+3 \underline {\hspace {1.25 ex}}\alpha +3+2 \sqrt {3}\, \left (\underline {\hspace {1.25 ex}}\alpha ^{2}+\underline {\hspace {1.25 ex}}\alpha +1\right )\right ) \Pi \left (\sqrt {\frac {-1+x}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, \frac {i \underline {\hspace {1.25 ex}}\alpha ^{2} \sqrt {3}}{6}+\frac {i \underline {\hspace {1.25 ex}}\alpha ^{2}}{3}+\frac {\underline {\hspace {1.25 ex}}\alpha ^{2} \sqrt {3}}{3}+\frac {\underline {\hspace {1.25 ex}}\alpha ^{2}}{2}+\frac {i \underline {\hspace {1.25 ex}}\alpha \sqrt {3}}{6}+\frac {i \underline {\hspace {1.25 ex}}\alpha }{3}+\frac {\underline {\hspace {1.25 ex}}\alpha \sqrt {3}}{3}+\frac {\underline {\hspace {1.25 ex}}\alpha }{2}+\frac {i \sqrt {3}}{6}+\frac {1}{2}+\frac {i}{3}+\frac {\sqrt {3}}{3}, \sqrt {\frac {\frac {3}{2}+\frac {i \sqrt {3}}{2}}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )}{\underline {\hspace {1.25 ex}}\alpha \sqrt {x^{3}-1}}\right )}{27}\) | \(505\) |
trager | \(\text {Expression too large to display}\) | \(4174\) |
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Leaf count of result is larger than twice the leaf count of optimal. 1538 vs. \(2 (148) = 296\).
Time = 0.39 (sec) , antiderivative size = 1538, normalized size of antiderivative = 7.19 \[ \int \frac {x}{\sqrt {-1+x^3} \left (-10+6 \sqrt {3}+x^3\right )} \, dx=\text {Too large to display} \]
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\[ \int \frac {x}{\sqrt {-1+x^3} \left (-10+6 \sqrt {3}+x^3\right )} \, dx=\int \frac {x}{\sqrt {\left (x - 1\right ) \left (x^{2} + x + 1\right )} \left (x^{3} - 10 + 6 \sqrt {3}\right )}\, dx \]
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\[ \int \frac {x}{\sqrt {-1+x^3} \left (-10+6 \sqrt {3}+x^3\right )} \, dx=\int { \frac {x}{{\left (x^{3} + 6 \, \sqrt {3} - 10\right )} \sqrt {x^{3} - 1}} \,d x } \]
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\[ \int \frac {x}{\sqrt {-1+x^3} \left (-10+6 \sqrt {3}+x^3\right )} \, dx=\int { \frac {x}{{\left (x^{3} + 6 \, \sqrt {3} - 10\right )} \sqrt {x^{3} - 1}} \,d x } \]
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Timed out. \[ \int \frac {x}{\sqrt {-1+x^3} \left (-10+6 \sqrt {3}+x^3\right )} \, dx=\int \frac {x}{\sqrt {x^3-1}\,\left (x^3+6\,\sqrt {3}-10\right )} \,d x \]
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