Integrand size = 10, antiderivative size = 130 \[ \int e^{\frac {2}{3} \coth ^{-1}(x)} x \, dx=\frac {1}{3} \sqrt [3]{1+\frac {1}{x}} \left (\frac {-1+x}{x}\right )^{2/3} x+\frac {1}{2} \left (1+\frac {1}{x}\right )^{4/3} \left (\frac {-1+x}{x}\right )^{2/3} x^2-\frac {2 \arctan \left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{\frac {-1+x}{x}}}{\sqrt {3} \sqrt [3]{1+\frac {1}{x}}}\right )}{3 \sqrt {3}}-\frac {1}{3} \log \left (\sqrt [3]{1+\frac {1}{x}}-\sqrt [3]{\frac {-1+x}{x}}\right )-\frac {\log (x)}{9} \]
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Time = 0.03 (sec) , antiderivative size = 130, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {6306, 98, 96, 93} \[ \int e^{\frac {2}{3} \coth ^{-1}(x)} x \, dx=-\frac {2 \arctan \left (\frac {2 \sqrt [3]{\frac {x-1}{x}}}{\sqrt {3} \sqrt [3]{\frac {1}{x}+1}}+\frac {1}{\sqrt {3}}\right )}{3 \sqrt {3}}+\frac {1}{2} \left (\frac {1}{x}+1\right )^{4/3} \left (\frac {x-1}{x}\right )^{2/3} x^2+\frac {1}{3} \sqrt [3]{\frac {1}{x}+1} \left (\frac {x-1}{x}\right )^{2/3} x-\frac {1}{3} \log \left (\sqrt [3]{\frac {1}{x}+1}-\sqrt [3]{\frac {x-1}{x}}\right )-\frac {\log (x)}{9} \]
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Rule 93
Rule 96
Rule 98
Rule 6306
Rubi steps \begin{align*} \text {integral}& = -\text {Subst}\left (\int \frac {\sqrt [3]{1+x}}{\sqrt [3]{1-x} x^3} \, dx,x,\frac {1}{x}\right ) \\ & = \frac {1}{2} \left (1+\frac {1}{x}\right )^{4/3} \left (\frac {-1+x}{x}\right )^{2/3} x^2-\frac {1}{3} \text {Subst}\left (\int \frac {\sqrt [3]{1+x}}{\sqrt [3]{1-x} x^2} \, dx,x,\frac {1}{x}\right ) \\ & = \frac {1}{3} \sqrt [3]{1+\frac {1}{x}} \left (-\frac {1-x}{x}\right )^{2/3} x+\frac {1}{2} \left (1+\frac {1}{x}\right )^{4/3} \left (\frac {-1+x}{x}\right )^{2/3} x^2-\frac {2}{9} \text {Subst}\left (\int \frac {1}{\sqrt [3]{1-x} x (1+x)^{2/3}} \, dx,x,\frac {1}{x}\right ) \\ & = \frac {1}{3} \sqrt [3]{1+\frac {1}{x}} \left (-\frac {1-x}{x}\right )^{2/3} x+\frac {1}{2} \left (1+\frac {1}{x}\right )^{4/3} \left (\frac {-1+x}{x}\right )^{2/3} x^2-\frac {2 \arctan \left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{-\frac {1-x}{x}}}{\sqrt {3} \sqrt [3]{1+\frac {1}{x}}}\right )}{3 \sqrt {3}}-\frac {1}{3} \log \left (\sqrt [3]{1+\frac {1}{x}}-\sqrt [3]{-\frac {1-x}{x}}\right )-\frac {\log (x)}{9} \\ \end{align*}
Time = 0.60 (sec) , antiderivative size = 165, normalized size of antiderivative = 1.27 \[ \int e^{\frac {2}{3} \coth ^{-1}(x)} x \, dx=\frac {1}{9} \left (\frac {18 e^{\frac {2}{3} \coth ^{-1}(x)}}{\left (-1+e^{2 \coth ^{-1}(x)}\right )^2}+\frac {24 e^{\frac {2}{3} \coth ^{-1}(x)}}{-1+e^{2 \coth ^{-1}(x)}}+2 \sqrt {3} \arctan \left (\frac {-1+2 e^{\frac {1}{3} \coth ^{-1}(x)}}{\sqrt {3}}\right )-2 \sqrt {3} \arctan \left (\frac {1+2 e^{\frac {1}{3} \coth ^{-1}(x)}}{\sqrt {3}}\right )-2 \log \left (1-e^{\frac {1}{3} \coth ^{-1}(x)}\right )-2 \log \left (1+e^{\frac {1}{3} \coth ^{-1}(x)}\right )+\log \left (1-e^{\frac {1}{3} \coth ^{-1}(x)}+e^{\frac {2}{3} \coth ^{-1}(x)}\right )+\log \left (1+e^{\frac {1}{3} \coth ^{-1}(x)}+e^{\frac {2}{3} \coth ^{-1}(x)}\right )\right ) \]
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Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.71 (sec) , antiderivative size = 403, normalized size of antiderivative = 3.10
method | result | size |
risch | \(\frac {\left (5+3 x \right ) \left (x -1\right )}{6 \left (\frac {x -1}{1+x}\right )^{\frac {1}{3}}}+\frac {\left (-\frac {2 \ln \left (-\frac {4 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2} x^{2}+4 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2} x +3 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{3}+x^{2}-x -1\right )^{\frac {2}{3}}-3 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{3}+x^{2}-x -1\right )^{\frac {1}{3}} x -4 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{2}-3 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{3}+x^{2}-x -1\right )^{\frac {1}{3}}-2 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x +3 \left (x^{3}+x^{2}-x -1\right )^{\frac {1}{3}} x +x^{2}+2 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )+3 \left (x^{3}+x^{2}-x -1\right )^{\frac {1}{3}}-1}{1+x}\right )}{9}+\frac {2 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \ln \left (\frac {2 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2} x^{2}+2 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2} x +3 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{3}+x^{2}-x -1\right )^{\frac {2}{3}}-5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{2}-6 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x -3 \left (x^{3}+x^{2}-x -1\right )^{\frac {2}{3}}+3 \left (x^{3}+x^{2}-x -1\right )^{\frac {1}{3}} x +2 x^{2}-\operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )+3 \left (x^{3}+x^{2}-x -1\right )^{\frac {1}{3}}+4 x +2}{1+x}\right )}{9}\right ) \left (\left (1+x \right )^{2} \left (x -1\right )\right )^{\frac {1}{3}}}{\left (\frac {x -1}{1+x}\right )^{\frac {1}{3}} \left (1+x \right )}\) | \(403\) |
trager | \(\frac {\left (1+x \right ) \left (5+3 x \right ) \left (-\frac {1-x}{1+x}\right )^{\frac {2}{3}}}{6}+\frac {2 \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 +18 \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 {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 -3 \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 {1}{3}}+3 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) x -3 \left (-\frac {1-x}{1+x}\right )^{\frac {2}{3}}-3 \left (-\frac {1-x}{1+x}\right )^{\frac {1}{3}} x -3 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )-3 \left (-\frac {1-x}{1+x}\right )^{\frac {1}{3}}-x -1\right )}{9}-\frac {2 \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 +18 \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 {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 -3 \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 {1}{3}}+3 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) x -3 \left (-\frac {1-x}{1+x}\right )^{\frac {2}{3}}-3 \left (-\frac {1-x}{1+x}\right )^{\frac {1}{3}} x -3 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )-3 \left (-\frac {1-x}{1+x}\right )^{\frac {1}{3}}-x -1\right ) \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )}{3}+\frac {2 \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 +18 \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 {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 -9 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) \left (-\frac {1-x}{1+x}\right )^{\frac {1}{3}}-15 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) x +3 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )+2 x -2\right )}{3}\) | \(663\) |
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Time = 0.25 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.73 \[ \int e^{\frac {2}{3} \coth ^{-1}(x)} x \, dx=\frac {1}{6} \, {\left (3 \, x^{2} + 8 \, x + 5\right )} \left (\frac {x - 1}{x + 1}\right )^{\frac {2}{3}} - \frac {2}{9} \, \sqrt {3} \arctan \left (\frac {2}{3} \, \sqrt {3} \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{3}} + \frac {1}{3} \, \sqrt {3}\right ) + \frac {1}{9} \, \log \left (\left (\frac {x - 1}{x + 1}\right )^{\frac {2}{3}} + \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{3}} + 1\right ) - \frac {2}{9} \, \log \left (\left (\frac {x - 1}{x + 1}\right )^{\frac {1}{3}} - 1\right ) \]
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\[ \int e^{\frac {2}{3} \coth ^{-1}(x)} x \, dx=\int \frac {x}{\sqrt [3]{\frac {x - 1}{x + 1}}}\, dx \]
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Time = 0.27 (sec) , antiderivative size = 123, normalized size of antiderivative = 0.95 \[ \int e^{\frac {2}{3} \coth ^{-1}(x)} x \, dx=-\frac {2}{9} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{3}} + 1\right )}\right ) + \frac {2 \, {\left (\left (\frac {x - 1}{x + 1}\right )^{\frac {5}{3}} - 4 \, \left (\frac {x - 1}{x + 1}\right )^{\frac {2}{3}}\right )}}{3 \, {\left (\frac {2 \, {\left (x - 1\right )}}{x + 1} - \frac {{\left (x - 1\right )}^{2}}{{\left (x + 1\right )}^{2}} - 1\right )}} + \frac {1}{9} \, \log \left (\left (\frac {x - 1}{x + 1}\right )^{\frac {2}{3}} + \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{3}} + 1\right ) - \frac {2}{9} \, \log \left (\left (\frac {x - 1}{x + 1}\right )^{\frac {1}{3}} - 1\right ) \]
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Time = 0.29 (sec) , antiderivative size = 120, normalized size of antiderivative = 0.92 \[ \int e^{\frac {2}{3} \coth ^{-1}(x)} x \, dx=-\frac {2}{9} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{3}} + 1\right )}\right ) - \frac {2 \, {\left (\frac {{\left (x - 1\right )} \left (\frac {x - 1}{x + 1}\right )^{\frac {2}{3}}}{x + 1} - 4 \, \left (\frac {x - 1}{x + 1}\right )^{\frac {2}{3}}\right )}}{3 \, {\left (\frac {x - 1}{x + 1} - 1\right )}^{2}} + \frac {1}{9} \, \log \left (\left (\frac {x - 1}{x + 1}\right )^{\frac {2}{3}} + \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{3}} + 1\right ) - \frac {2}{9} \, \log \left ({\left | \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{3}} - 1 \right |}\right ) \]
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Time = 4.04 (sec) , antiderivative size = 145, normalized size of antiderivative = 1.12 \[ \int e^{\frac {2}{3} \coth ^{-1}(x)} x \, dx=\frac {\frac {8\,{\left (\frac {x-1}{x+1}\right )}^{2/3}}{3}-\frac {2\,{\left (\frac {x-1}{x+1}\right )}^{5/3}}{3}}{\frac {{\left (x-1\right )}^2}{{\left (x+1\right )}^2}-\frac {2\,\left (x-1\right )}{x+1}+1}-\frac {2\,\ln \left (\frac {4\,{\left (\frac {x-1}{x+1}\right )}^{1/3}}{9}-\frac {4}{9}\right )}{9}-\ln \left (\frac {4\,{\left (\frac {x-1}{x+1}\right )}^{1/3}}{9}-9\,{\left (-\frac {1}{9}+\frac {\sqrt {3}\,1{}\mathrm {i}}{9}\right )}^2\right )\,\left (-\frac {1}{9}+\frac {\sqrt {3}\,1{}\mathrm {i}}{9}\right )+\ln \left (\frac {4\,{\left (\frac {x-1}{x+1}\right )}^{1/3}}{9}-9\,{\left (\frac {1}{9}+\frac {\sqrt {3}\,1{}\mathrm {i}}{9}\right )}^2\right )\,\left (\frac {1}{9}+\frac {\sqrt {3}\,1{}\mathrm {i}}{9}\right ) \]
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