Integrand size = 25, antiderivative size = 218 \[ \int \frac {x}{\sqrt {1+x^3} \left (10+6 \sqrt {3}+x^3\right )} \, dx=-\frac {\left (2-\sqrt {3}\right ) \arctan \left (\frac {\sqrt [4]{3} \left (1+\sqrt {3}\right ) (1+x)}{\sqrt {2} \sqrt {1+x^3}}\right )}{2 \sqrt {2} 3^{3/4}}-\frac {\left (2-\sqrt {3}\right ) \arctan \left (\frac {\left (1-\sqrt {3}\right ) \sqrt {1+x^3}}{\sqrt {2} 3^{3/4}}\right )}{3 \sqrt {2} 3^{3/4}}-\frac {\left (2-\sqrt {3}\right ) \text {arctanh}\left (\frac {\sqrt [4]{3} \left (1+\sqrt {3}-2 x\right )}{\sqrt {2} \sqrt {1+x^3}}\right )}{3 \sqrt {2} \sqrt [4]{3}}-\frac {\left (2-\sqrt {3}\right ) \text {arctanh}\left (\frac {\sqrt [4]{3} \left (1-\sqrt {3}\right ) (1+x)}{\sqrt {2} \sqrt {1+x^3}}\right )}{6 \sqrt {2} \sqrt [4]{3}} \]
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Time = 0.03 (sec) , antiderivative size = 218, 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 = {500} \[ \int \frac {x}{\sqrt {1+x^3} \left (10+6 \sqrt {3}+x^3\right )} \, dx=-\frac {\left (2-\sqrt {3}\right ) \arctan \left (\frac {\sqrt [4]{3} \left (1+\sqrt {3}\right ) (x+1)}{\sqrt {2} \sqrt {x^3+1}}\right )}{2 \sqrt {2} 3^{3/4}}-\frac {\left (2-\sqrt {3}\right ) \arctan \left (\frac {\left (1-\sqrt {3}\right ) \sqrt {x^3+1}}{\sqrt {2} 3^{3/4}}\right )}{3 \sqrt {2} 3^{3/4}}-\frac {\left (2-\sqrt {3}\right ) \text {arctanh}\left (\frac {\sqrt [4]{3} \left (-2 x+\sqrt {3}+1\right )}{\sqrt {2} \sqrt {x^3+1}}\right )}{3 \sqrt {2} \sqrt [4]{3}}-\frac {\left (2-\sqrt {3}\right ) \text {arctanh}\left (\frac {\sqrt [4]{3} \left (1-\sqrt {3}\right ) (x+1)}{\sqrt {2} \sqrt {x^3+1}}\right )}{6 \sqrt {2} \sqrt [4]{3}} \]
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Rule 500
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}-2 x\right )}{\sqrt {2} \sqrt {1+x^3}}\right )}{3 \sqrt {2} \sqrt [4]{3}}-\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}} \\ \end{align*}
Result contains higher order function than in optimal. Order 6 vs. order 3 in optimal.
Time = 10.05 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.22 \[ \int \frac {x}{\sqrt {1+x^3} \left (10+6 \sqrt {3}+x^3\right )} \, dx=\frac {x^2 \operatorname {AppellF1}\left (\frac {2}{3},\frac {1}{2},1,\frac {5}{3},-x^3,-\frac {x^3}{10+6 \sqrt {3}}\right )}{20+12 \sqrt {3}} \]
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Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 48.35 (sec) , antiderivative size = 355, normalized size of antiderivative = 1.63
method | result | size |
default | \(\frac {2 \left (-1-\sqrt {3}\right ) \left (\frac {3}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {1+x}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x -\frac {1}{2}-\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x -\frac {1}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \sqrt {3}\, \Pi \left (\sqrt {\frac {1+x}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, \frac {\left (-\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}{3}, \sqrt {\frac {-\frac {3}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )}{3 \left (12+6 \sqrt {3}\right ) \sqrt {x^{3}+1}}-\frac {\sqrt {2}\, \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\operatorname {RootOf}\left (\textit {\_Z}^{2}+\left (-1-\sqrt {3}\right ) \textit {\_Z} +2 \sqrt {3}+4\right )}{\sum }\frac {\left (-\underline {\hspace {1.25 ex}}\alpha \sqrt {3}+\underline {\hspace {1.25 ex}}\alpha -2\right ) \left (3-i \sqrt {3}\right ) \sqrt {\frac {1+x}{3-i \sqrt {3}}}\, \sqrt {\frac {-i \sqrt {3}+2 x -1}{-3-i \sqrt {3}}}\, \sqrt {\frac {i \sqrt {3}+2 x -1}{i \sqrt {3}-3}}\, \left (-1+2 \underline {\hspace {1.25 ex}}\alpha -\underline {\hspace {1.25 ex}}\alpha \sqrt {3}\right ) \Pi \left (\sqrt {\frac {1+x}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, -\frac {i \underline {\hspace {1.25 ex}}\alpha }{2}+\frac {i \underline {\hspace {1.25 ex}}\alpha \sqrt {3}}{3}+\frac {\underline {\hspace {1.25 ex}}\alpha \sqrt {3}}{2}-\underline {\hspace {1.25 ex}}\alpha -\frac {i \sqrt {3}}{6}+\frac {1}{2}, \sqrt {\frac {-\frac {3}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )}{\left (-1+2 \underline {\hspace {1.25 ex}}\alpha -\sqrt {3}\right ) \sqrt {x^{3}+1}}\right )}{18}\) | \(355\) |
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 {\left (-1-\sqrt {3}\right )^{2}}{3}+\frac {2 \left (-1-\sqrt {3}\right )^{2} \sqrt {3}}{9}-\frac {2}{3}-\frac {\sqrt {3}}{9}-\frac {2 \left (-1-\sqrt {3}\right ) \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}}{3}-\frac {i \left (-1-\sqrt {3}\right )^{2} \sqrt {3}}{6}-\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 )}{3}+\frac {i \left (-1-\sqrt {3}\right ) \sqrt {3}}{6}+\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}}{3}-\frac {i \underline {\hspace {1.25 ex}}\alpha ^{2} \sqrt {3}}{6}-\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 }{3}+\frac {i \underline {\hspace {1.25 ex}}\alpha \sqrt {3}}{6}+\frac {\underline {\hspace {1.25 ex}}\alpha \sqrt {3}}{3}-\frac {\underline {\hspace {1.25 ex}}\alpha }{2}+\frac {1}{2}+\frac {i}{3}-\frac {i \sqrt {3}}{6}-\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}\) | \(507\) |
trager | \(\text {Expression too large to display}\) | \(4181\) |
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Leaf count of result is larger than twice the leaf count of optimal. 1569 vs. \(2 (148) = 296\).
Time = 0.43 (sec) , antiderivative size = 1569, normalized size of antiderivative = 7.20 \[ \int \frac {x}{\sqrt {1+x^3} \left (10+6 \sqrt {3}+x^3\right )} \, dx=\text {Too large to display} \]
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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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