Integrand size = 12, antiderivative size = 157 \[ \int e^{\frac {2}{3} \coth ^{-1}(x)} x^2 \, dx=\frac {14}{27} \sqrt [3]{1+\frac {1}{x}} \left (\frac {-1+x}{x}\right )^{2/3} x+\frac {4}{9} \sqrt [3]{1+\frac {1}{x}} \left (\frac {-1+x}{x}\right )^{2/3} x^2+\frac {1}{3} \sqrt [3]{1+\frac {1}{x}} \left (\frac {-1+x}{x}\right )^{2/3} x^3-\frac {22 \arctan \left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{\frac {-1+x}{x}}}{\sqrt {3} \sqrt [3]{1+\frac {1}{x}}}\right )}{27 \sqrt {3}}-\frac {11}{27} \log \left (\sqrt [3]{1+\frac {1}{x}}-\sqrt [3]{\frac {-1+x}{x}}\right )-\frac {11 \log (x)}{81} \]
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Time = 0.05 (sec) , antiderivative size = 157, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {6306, 101, 156, 12, 93} \[ \int e^{\frac {2}{3} \coth ^{-1}(x)} x^2 \, dx=-\frac {22 \arctan \left (\frac {2 \sqrt [3]{\frac {x-1}{x}}}{\sqrt {3} \sqrt [3]{\frac {1}{x}+1}}+\frac {1}{\sqrt {3}}\right )}{27 \sqrt {3}}+\frac {1}{3} \sqrt [3]{\frac {1}{x}+1} \left (\frac {x-1}{x}\right )^{2/3} x^3+\frac {4}{9} \sqrt [3]{\frac {1}{x}+1} \left (\frac {x-1}{x}\right )^{2/3} x^2+\frac {14}{27} \sqrt [3]{\frac {1}{x}+1} \left (\frac {x-1}{x}\right )^{2/3} x-\frac {11}{27} \log \left (\sqrt [3]{\frac {1}{x}+1}-\sqrt [3]{\frac {x-1}{x}}\right )-\frac {11 \log (x)}{81} \]
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Rule 12
Rule 93
Rule 101
Rule 156
Rule 6306
Rubi steps \begin{align*} \text {integral}& = -\text {Subst}\left (\int \frac {\sqrt [3]{1+x}}{\sqrt [3]{1-x} x^4} \, dx,x,\frac {1}{x}\right ) \\ & = \frac {1}{3} \sqrt [3]{1+\frac {1}{x}} \left (\frac {-1+x}{x}\right )^{2/3} x^3-\frac {1}{3} \text {Subst}\left (\int \frac {\frac {8}{3}+2 x}{\sqrt [3]{1-x} x^3 (1+x)^{2/3}} \, dx,x,\frac {1}{x}\right ) \\ & = \frac {4}{9} \sqrt [3]{1+\frac {1}{x}} \left (-\frac {1-x}{x}\right )^{2/3} x^2+\frac {1}{3} \sqrt [3]{1+\frac {1}{x}} \left (\frac {-1+x}{x}\right )^{2/3} x^3+\frac {1}{6} \text {Subst}\left (\int \frac {-\frac {28}{9}-\frac {8 x}{3}}{\sqrt [3]{1-x} x^2 (1+x)^{2/3}} \, dx,x,\frac {1}{x}\right ) \\ & = \frac {14}{27} \sqrt [3]{1+\frac {1}{x}} \left (-\frac {1-x}{x}\right )^{2/3} x+\frac {4}{9} \sqrt [3]{1+\frac {1}{x}} \left (-\frac {1-x}{x}\right )^{2/3} x^2+\frac {1}{3} \sqrt [3]{1+\frac {1}{x}} \left (\frac {-1+x}{x}\right )^{2/3} x^3-\frac {1}{6} \text {Subst}\left (\int \frac {44}{27 \sqrt [3]{1-x} x (1+x)^{2/3}} \, dx,x,\frac {1}{x}\right ) \\ & = \frac {14}{27} \sqrt [3]{1+\frac {1}{x}} \left (-\frac {1-x}{x}\right )^{2/3} x+\frac {4}{9} \sqrt [3]{1+\frac {1}{x}} \left (-\frac {1-x}{x}\right )^{2/3} x^2+\frac {1}{3} \sqrt [3]{1+\frac {1}{x}} \left (\frac {-1+x}{x}\right )^{2/3} x^3-\frac {22}{81} \text {Subst}\left (\int \frac {1}{\sqrt [3]{1-x} x (1+x)^{2/3}} \, dx,x,\frac {1}{x}\right ) \\ & = \frac {14}{27} \sqrt [3]{1+\frac {1}{x}} \left (-\frac {1-x}{x}\right )^{2/3} x+\frac {4}{9} \sqrt [3]{1+\frac {1}{x}} \left (-\frac {1-x}{x}\right )^{2/3} x^2+\frac {1}{3} \sqrt [3]{1+\frac {1}{x}} \left (\frac {-1+x}{x}\right )^{2/3} x^3-\frac {22 \arctan \left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{-\frac {1-x}{x}}}{\sqrt {3} \sqrt [3]{1+\frac {1}{x}}}\right )}{27 \sqrt {3}}-\frac {11}{27} \log \left (\sqrt [3]{1+\frac {1}{x}}-\sqrt [3]{-\frac {1-x}{x}}\right )-\frac {11 \log (x)}{81} \\ \end{align*}
Time = 5.27 (sec) , antiderivative size = 189, normalized size of antiderivative = 1.20 \[ \int e^{\frac {2}{3} \coth ^{-1}(x)} x^2 \, dx=\frac {1}{81} \left (\frac {216 e^{\frac {2}{3} \coth ^{-1}(x)}}{\left (-1+e^{2 \coth ^{-1}(x)}\right )^3}+\frac {360 e^{\frac {2}{3} \coth ^{-1}(x)}}{\left (-1+e^{2 \coth ^{-1}(x)}\right )^2}+\frac {210 e^{\frac {2}{3} \coth ^{-1}(x)}}{-1+e^{2 \coth ^{-1}(x)}}+22 \sqrt {3} \arctan \left (\frac {-1+2 e^{\frac {1}{3} \coth ^{-1}(x)}}{\sqrt {3}}\right )-22 \sqrt {3} \arctan \left (\frac {1+2 e^{\frac {1}{3} \coth ^{-1}(x)}}{\sqrt {3}}\right )-22 \log \left (1-e^{\frac {1}{3} \coth ^{-1}(x)}\right )-22 \log \left (1+e^{\frac {1}{3} \coth ^{-1}(x)}\right )+11 \log \left (1-e^{\frac {1}{3} \coth ^{-1}(x)}+e^{\frac {2}{3} \coth ^{-1}(x)}\right )+11 \log \left (1+e^{\frac {1}{3} \coth ^{-1}(x)}+e^{\frac {2}{3} \coth ^{-1}(x)}\right )\right ) \]
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Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.76 (sec) , antiderivative size = 408, normalized size of antiderivative = 2.60
method | result | size |
trager | \(\frac {\left (1+x \right ) \left (9 x^{2}+12 x +14\right ) \left (-\frac {1-x}{1+x}\right )^{\frac {2}{3}}}{27}-\frac {22 \ln \left (-9 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) \left (-\frac {1-x}{1+x}\right )^{\frac {2}{3}} x -9 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) \left (-\frac {1-x}{1+x}\right )^{\frac {2}{3}}+9 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) \left (-\frac {1-x}{1+x}\right )^{\frac {1}{3}} x -36 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )^{2} x +9 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) \left (-\frac {1-x}{1+x}\right )^{\frac {1}{3}}-3 \left (-\frac {1-x}{1+x}\right )^{\frac {1}{3}} x +12 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) x -3 \left (-\frac {1-x}{1+x}\right )^{\frac {1}{3}}-6 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )-x +1\right )}{81}+\frac {22 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) \ln \left (9 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) \left (-\frac {1-x}{1+x}\right )^{\frac {2}{3}} x +9 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) \left (-\frac {1-x}{1+x}\right )^{\frac {2}{3}}-3 \left (-\frac {1-x}{1+x}\right )^{\frac {2}{3}} x +18 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )^{2} x -3 \left (-\frac {1-x}{1+x}\right )^{\frac {2}{3}}+3 \left (-\frac {1-x}{1+x}\right )^{\frac {1}{3}} x -15 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) x +3 \left (-\frac {1-x}{1+x}\right )^{\frac {1}{3}}-3 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )+2 x +2\right )}{27}\) | \(408\) |
risch | \(\frac {\left (9 x^{2}+12 x +14\right ) \left (x -1\right )}{27 \left (\frac {x -1}{1+x}\right )^{\frac {1}{3}}}+\frac {\left (\frac {22 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \ln \left (-\frac {-2 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2} x^{2}+3 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{3}+x^{2}-x -1\right )^{\frac {2}{3}}+3 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{3}+x^{2}-x -1\right )^{\frac {1}{3}} x -2 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2} x +5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{2}+3 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{3}+x^{2}-x -1\right )^{\frac {1}{3}}+4 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x -2 x^{2}-\operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )+2}{1+x}\right )}{81}-\frac {22 \ln \left (\frac {2 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2} x^{2}+3 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{3}+x^{2}-x -1\right )^{\frac {2}{3}}+3 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{3}+x^{2}-x -1\right )^{\frac {1}{3}} x +2 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2} x +\operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{2}-3 \left (x^{3}+x^{2}-x -1\right )^{\frac {2}{3}}+3 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{3}+x^{2}-x -1\right )^{\frac {1}{3}}-3 \left (x^{3}+x^{2}-x -1\right )^{\frac {1}{3}} x -x^{2}-3 \left (x^{3}+x^{2}-x -1\right )^{\frac {1}{3}}-\operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )-2 x -1}{1+x}\right ) \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )}{81}+\frac {22 \ln \left (\frac {2 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2} x^{2}+3 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{3}+x^{2}-x -1\right )^{\frac {2}{3}}+3 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{3}+x^{2}-x -1\right )^{\frac {1}{3}} x +2 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2} x +\operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{2}-3 \left (x^{3}+x^{2}-x -1\right )^{\frac {2}{3}}+3 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{3}+x^{2}-x -1\right )^{\frac {1}{3}}-3 \left (x^{3}+x^{2}-x -1\right )^{\frac {1}{3}} x -x^{2}-3 \left (x^{3}+x^{2}-x -1\right )^{\frac {1}{3}}-\operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )-2 x -1}{1+x}\right )}{81}\right ) \left (\left (1+x \right )^{2} \left (x -1\right )\right )^{\frac {1}{3}}}{\left (\frac {x -1}{1+x}\right )^{\frac {1}{3}} \left (1+x \right )}\) | \(616\) |
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Time = 0.25 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.64 \[ \int e^{\frac {2}{3} \coth ^{-1}(x)} x^2 \, dx=\frac {1}{27} \, {\left (9 \, x^{3} + 21 \, x^{2} + 26 \, x + 14\right )} \left (\frac {x - 1}{x + 1}\right )^{\frac {2}{3}} - \frac {22}{81} \, \sqrt {3} \arctan \left (\frac {2}{3} \, \sqrt {3} \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{3}} + \frac {1}{3} \, \sqrt {3}\right ) + \frac {11}{81} \, \log \left (\left (\frac {x - 1}{x + 1}\right )^{\frac {2}{3}} + \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{3}} + 1\right ) - \frac {22}{81} \, \log \left (\left (\frac {x - 1}{x + 1}\right )^{\frac {1}{3}} - 1\right ) \]
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\[ \int e^{\frac {2}{3} \coth ^{-1}(x)} x^2 \, dx=\int \frac {x^{2}}{\sqrt [3]{\frac {x - 1}{x + 1}}}\, dx \]
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Time = 0.28 (sec) , antiderivative size = 149, normalized size of antiderivative = 0.95 \[ \int e^{\frac {2}{3} \coth ^{-1}(x)} x^2 \, dx=-\frac {22}{81} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{3}} + 1\right )}\right ) - \frac {2 \, {\left (11 \, \left (\frac {x - 1}{x + 1}\right )^{\frac {8}{3}} - 10 \, \left (\frac {x - 1}{x + 1}\right )^{\frac {5}{3}} + 35 \, \left (\frac {x - 1}{x + 1}\right )^{\frac {2}{3}}\right )}}{27 \, {\left (\frac {3 \, {\left (x - 1\right )}}{x + 1} - \frac {3 \, {\left (x - 1\right )}^{2}}{{\left (x + 1\right )}^{2}} + \frac {{\left (x - 1\right )}^{3}}{{\left (x + 1\right )}^{3}} - 1\right )}} + \frac {11}{81} \, \log \left (\left (\frac {x - 1}{x + 1}\right )^{\frac {2}{3}} + \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{3}} + 1\right ) - \frac {22}{81} \, \log \left (\left (\frac {x - 1}{x + 1}\right )^{\frac {1}{3}} - 1\right ) \]
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Time = 0.29 (sec) , antiderivative size = 144, normalized size of antiderivative = 0.92 \[ \int e^{\frac {2}{3} \coth ^{-1}(x)} x^2 \, dx=-\frac {22}{81} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{3}} + 1\right )}\right ) + \frac {2 \, {\left (\frac {10 \, {\left (x - 1\right )} \left (\frac {x - 1}{x + 1}\right )^{\frac {2}{3}}}{x + 1} - \frac {11 \, {\left (x - 1\right )}^{2} \left (\frac {x - 1}{x + 1}\right )^{\frac {2}{3}}}{{\left (x + 1\right )}^{2}} - 35 \, \left (\frac {x - 1}{x + 1}\right )^{\frac {2}{3}}\right )}}{27 \, {\left (\frac {x - 1}{x + 1} - 1\right )}^{3}} + \frac {11}{81} \, \log \left (\left (\frac {x - 1}{x + 1}\right )^{\frac {2}{3}} + \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{3}} + 1\right ) - \frac {22}{81} \, \log \left ({\left | \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{3}} - 1 \right |}\right ) \]
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Time = 4.05 (sec) , antiderivative size = 171, normalized size of antiderivative = 1.09 \[ \int e^{\frac {2}{3} \coth ^{-1}(x)} x^2 \, dx=-\frac {22\,\ln \left (\frac {484\,{\left (\frac {x-1}{x+1}\right )}^{1/3}}{729}-\frac {484}{729}\right )}{81}-\frac {\frac {70\,{\left (\frac {x-1}{x+1}\right )}^{2/3}}{27}-\frac {20\,{\left (\frac {x-1}{x+1}\right )}^{5/3}}{27}+\frac {22\,{\left (\frac {x-1}{x+1}\right )}^{8/3}}{27}}{\frac {3\,\left (x-1\right )}{x+1}-\frac {3\,{\left (x-1\right )}^2}{{\left (x+1\right )}^2}+\frac {{\left (x-1\right )}^3}{{\left (x+1\right )}^3}-1}-\ln \left (\frac {484\,{\left (\frac {x-1}{x+1}\right )}^{1/3}}{729}-9\,{\left (-\frac {11}{81}+\frac {\sqrt {3}\,11{}\mathrm {i}}{81}\right )}^2\right )\,\left (-\frac {11}{81}+\frac {\sqrt {3}\,11{}\mathrm {i}}{81}\right )+\ln \left (\frac {484\,{\left (\frac {x-1}{x+1}\right )}^{1/3}}{729}-9\,{\left (\frac {11}{81}+\frac {\sqrt {3}\,11{}\mathrm {i}}{81}\right )}^2\right )\,\left (\frac {11}{81}+\frac {\sqrt {3}\,11{}\mathrm {i}}{81}\right ) \]
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