Integrand size = 14, antiderivative size = 430 \[ \int \frac {e^{\frac {1}{3} i \arctan (x)}}{x} \, dx=\arctan \left (\sqrt {3}-\frac {2 \sqrt [6]{1-i x}}{\sqrt [6]{1+i x}}\right )-\arctan \left (\sqrt {3}+\frac {2 \sqrt [6]{1-i x}}{\sqrt [6]{1+i x}}\right )+\sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [6]{1+i x}}{\sqrt [6]{1-i x}}}{\sqrt {3}}\right )-\sqrt {3} \arctan \left (\frac {1+\frac {2 \sqrt [6]{1+i x}}{\sqrt [6]{1-i x}}}{\sqrt {3}}\right )-2 \arctan \left (\frac {\sqrt [6]{1-i x}}{\sqrt [6]{1+i x}}\right )-2 \text {arctanh}\left (\frac {\sqrt [6]{1+i x}}{\sqrt [6]{1-i x}}\right )-\frac {1}{2} \sqrt {3} \log \left (1+\frac {\sqrt [3]{1-i x}}{\sqrt [3]{1+i x}}-\frac {\sqrt {3} \sqrt [6]{1-i x}}{\sqrt [6]{1+i x}}\right )+\frac {1}{2} \sqrt {3} \log \left (1+\frac {\sqrt [3]{1-i x}}{\sqrt [3]{1+i x}}+\frac {\sqrt {3} \sqrt [6]{1-i x}}{\sqrt [6]{1+i x}}\right )+\frac {1}{2} \log \left (1-\frac {\sqrt [6]{1+i x}}{\sqrt [6]{1-i x}}+\frac {\sqrt [3]{1+i x}}{\sqrt [3]{1-i x}}\right )-\frac {1}{2} \log \left (1+\frac {\sqrt [6]{1+i x}}{\sqrt [6]{1-i x}}+\frac {\sqrt [3]{1+i x}}{\sqrt [3]{1-i x}}\right ) \]
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Time = 0.34 (sec) , antiderivative size = 430, normalized size of antiderivative = 1.00, number of steps used = 25, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.929, Rules used = {5170, 132, 65, 338, 301, 648, 632, 210, 642, 209, 95, 216, 212} \[ \int \frac {e^{\frac {1}{3} i \arctan (x)}}{x} \, dx=\arctan \left (\sqrt {3}-\frac {2 \sqrt [6]{1-i x}}{\sqrt [6]{1+i x}}\right )-\arctan \left (\sqrt {3}+\frac {2 \sqrt [6]{1-i x}}{\sqrt [6]{1+i x}}\right )+\sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [6]{1+i x}}{\sqrt [6]{1-i x}}}{\sqrt {3}}\right )-\sqrt {3} \arctan \left (\frac {1+\frac {2 \sqrt [6]{1+i x}}{\sqrt [6]{1-i x}}}{\sqrt {3}}\right )-2 \arctan \left (\frac {\sqrt [6]{1-i x}}{\sqrt [6]{1+i x}}\right )-2 \text {arctanh}\left (\frac {\sqrt [6]{1+i x}}{\sqrt [6]{1-i x}}\right )-\frac {1}{2} \sqrt {3} \log \left (\frac {\sqrt [3]{1-i x}}{\sqrt [3]{1+i x}}-\frac {\sqrt {3} \sqrt [6]{1-i x}}{\sqrt [6]{1+i x}}+1\right )+\frac {1}{2} \sqrt {3} \log \left (\frac {\sqrt [3]{1-i x}}{\sqrt [3]{1+i x}}+\frac {\sqrt {3} \sqrt [6]{1-i x}}{\sqrt [6]{1+i x}}+1\right )+\frac {1}{2} \log \left (\frac {\sqrt [3]{1+i x}}{\sqrt [3]{1-i x}}-\frac {\sqrt [6]{1+i x}}{\sqrt [6]{1-i x}}+1\right )-\frac {1}{2} \log \left (\frac {\sqrt [3]{1+i x}}{\sqrt [3]{1-i x}}+\frac {\sqrt [6]{1+i x}}{\sqrt [6]{1-i x}}+1\right ) \]
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Rule 65
Rule 95
Rule 132
Rule 209
Rule 210
Rule 212
Rule 216
Rule 301
Rule 338
Rule 632
Rule 642
Rule 648
Rule 5170
Rubi steps \begin{align*} \text {integral}& = \int \frac {\sqrt [6]{1+i x}}{\sqrt [6]{1-i x} x} \, dx \\ & = i \int \frac {1}{\sqrt [6]{1-i x} (1+i x)^{5/6}} \, dx+\int \frac {1}{\sqrt [6]{1-i x} (1+i x)^{5/6} x} \, dx \\ & = -\left (6 \text {Subst}\left (\int \frac {x^4}{\left (2-x^6\right )^{5/6}} \, dx,x,\sqrt [6]{1-i x}\right )\right )+6 \text {Subst}\left (\int \frac {1}{-1+x^6} \, dx,x,\frac {\sqrt [6]{1+i x}}{\sqrt [6]{1-i x}}\right ) \\ & = -\left (2 \text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\frac {\sqrt [6]{1+i x}}{\sqrt [6]{1-i x}}\right )\right )-2 \text {Subst}\left (\int \frac {1-\frac {x}{2}}{1-x+x^2} \, dx,x,\frac {\sqrt [6]{1+i x}}{\sqrt [6]{1-i x}}\right )-2 \text {Subst}\left (\int \frac {1+\frac {x}{2}}{1+x+x^2} \, dx,x,\frac {\sqrt [6]{1+i x}}{\sqrt [6]{1-i x}}\right )-6 \text {Subst}\left (\int \frac {x^4}{1+x^6} \, dx,x,\frac {\sqrt [6]{1-i x}}{\sqrt [6]{1+i x}}\right ) \\ & = -2 \text {arctanh}\left (\frac {\sqrt [6]{1+i x}}{\sqrt [6]{1-i x}}\right )+\frac {1}{2} \text {Subst}\left (\int \frac {-1+2 x}{1-x+x^2} \, dx,x,\frac {\sqrt [6]{1+i x}}{\sqrt [6]{1-i x}}\right )-\frac {1}{2} \text {Subst}\left (\int \frac {1+2 x}{1+x+x^2} \, dx,x,\frac {\sqrt [6]{1+i x}}{\sqrt [6]{1-i x}}\right )-\frac {3}{2} \text {Subst}\left (\int \frac {1}{1-x+x^2} \, dx,x,\frac {\sqrt [6]{1+i x}}{\sqrt [6]{1-i x}}\right )-\frac {3}{2} \text {Subst}\left (\int \frac {1}{1+x+x^2} \, dx,x,\frac {\sqrt [6]{1+i x}}{\sqrt [6]{1-i x}}\right )-2 \text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\frac {\sqrt [6]{1-i x}}{\sqrt [6]{1+i x}}\right )-2 \text {Subst}\left (\int \frac {-\frac {1}{2}+\frac {\sqrt {3} x}{2}}{1-\sqrt {3} x+x^2} \, dx,x,\frac {\sqrt [6]{1-i x}}{\sqrt [6]{1+i x}}\right )-2 \text {Subst}\left (\int \frac {-\frac {1}{2}-\frac {\sqrt {3} x}{2}}{1+\sqrt {3} x+x^2} \, dx,x,\frac {\sqrt [6]{1-i x}}{\sqrt [6]{1+i x}}\right ) \\ & = -2 \arctan \left (\frac {\sqrt [6]{1-i x}}{\sqrt [6]{1+i x}}\right )-2 \text {arctanh}\left (\frac {\sqrt [6]{1+i x}}{\sqrt [6]{1-i x}}\right )+\frac {1}{2} \log \left (1-\frac {\sqrt [6]{1+i x}}{\sqrt [6]{1-i x}}+\frac {\sqrt [3]{1+i x}}{\sqrt [3]{1-i x}}\right )-\frac {1}{2} \log \left (1+\frac {\sqrt [6]{1+i x}}{\sqrt [6]{1-i x}}+\frac {\sqrt [3]{1+i x}}{\sqrt [3]{1-i x}}\right )-\frac {1}{2} \text {Subst}\left (\int \frac {1}{1-\sqrt {3} x+x^2} \, dx,x,\frac {\sqrt [6]{1-i x}}{\sqrt [6]{1+i x}}\right )-\frac {1}{2} \text {Subst}\left (\int \frac {1}{1+\sqrt {3} x+x^2} \, dx,x,\frac {\sqrt [6]{1-i x}}{\sqrt [6]{1+i x}}\right )+3 \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,-1+\frac {2 \sqrt [6]{1+i x}}{\sqrt [6]{1-i x}}\right )+3 \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+\frac {2 \sqrt [6]{1+i x}}{\sqrt [6]{1-i x}}\right )-\frac {1}{2} \sqrt {3} \text {Subst}\left (\int \frac {-\sqrt {3}+2 x}{1-\sqrt {3} x+x^2} \, dx,x,\frac {\sqrt [6]{1-i x}}{\sqrt [6]{1+i x}}\right )+\frac {1}{2} \sqrt {3} \text {Subst}\left (\int \frac {\sqrt {3}+2 x}{1+\sqrt {3} x+x^2} \, dx,x,\frac {\sqrt [6]{1-i x}}{\sqrt [6]{1+i x}}\right ) \\ & = \sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [6]{1+i x}}{\sqrt [6]{1-i x}}}{\sqrt {3}}\right )-\sqrt {3} \arctan \left (\frac {1+\frac {2 \sqrt [6]{1+i x}}{\sqrt [6]{1-i x}}}{\sqrt {3}}\right )-2 \arctan \left (\frac {\sqrt [6]{1-i x}}{\sqrt [6]{1+i x}}\right )-2 \text {arctanh}\left (\frac {\sqrt [6]{1+i x}}{\sqrt [6]{1-i x}}\right )-\frac {1}{2} \sqrt {3} \log \left (1+\frac {\sqrt [3]{1-i x}}{\sqrt [3]{1+i x}}-\frac {\sqrt {3} \sqrt [6]{1-i x}}{\sqrt [6]{1+i x}}\right )+\frac {1}{2} \sqrt {3} \log \left (1+\frac {\sqrt [3]{1-i x}}{\sqrt [3]{1+i x}}+\frac {\sqrt {3} \sqrt [6]{1-i x}}{\sqrt [6]{1+i x}}\right )+\frac {1}{2} \log \left (1-\frac {\sqrt [6]{1+i x}}{\sqrt [6]{1-i x}}+\frac {\sqrt [3]{1+i x}}{\sqrt [3]{1-i x}}\right )-\frac {1}{2} \log \left (1+\frac {\sqrt [6]{1+i x}}{\sqrt [6]{1-i x}}+\frac {\sqrt [3]{1+i x}}{\sqrt [3]{1-i x}}\right )+\text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,-\sqrt {3}+\frac {2 \sqrt [6]{1-i x}}{\sqrt [6]{1+i x}}\right )+\text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,\sqrt {3}+\frac {2 \sqrt [6]{1-i x}}{\sqrt [6]{1+i x}}\right ) \\ & = \arctan \left (\sqrt {3}-\frac {2 \sqrt [6]{1-i x}}{\sqrt [6]{1+i x}}\right )-\arctan \left (\sqrt {3}+\frac {2 \sqrt [6]{1-i x}}{\sqrt [6]{1+i x}}\right )+\sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [6]{1+i x}}{\sqrt [6]{1-i x}}}{\sqrt {3}}\right )-\sqrt {3} \arctan \left (\frac {1+\frac {2 \sqrt [6]{1+i x}}{\sqrt [6]{1-i x}}}{\sqrt {3}}\right )-2 \arctan \left (\frac {\sqrt [6]{1-i x}}{\sqrt [6]{1+i x}}\right )-2 \text {arctanh}\left (\frac {\sqrt [6]{1+i x}}{\sqrt [6]{1-i x}}\right )-\frac {1}{2} \sqrt {3} \log \left (1+\frac {\sqrt [3]{1-i x}}{\sqrt [3]{1+i x}}-\frac {\sqrt {3} \sqrt [6]{1-i x}}{\sqrt [6]{1+i x}}\right )+\frac {1}{2} \sqrt {3} \log \left (1+\frac {\sqrt [3]{1-i x}}{\sqrt [3]{1+i x}}+\frac {\sqrt {3} \sqrt [6]{1-i x}}{\sqrt [6]{1+i x}}\right )+\frac {1}{2} \log \left (1-\frac {\sqrt [6]{1+i x}}{\sqrt [6]{1-i x}}+\frac {\sqrt [3]{1+i x}}{\sqrt [3]{1-i x}}\right )-\frac {1}{2} \log \left (1+\frac {\sqrt [6]{1+i x}}{\sqrt [6]{1-i x}}+\frac {\sqrt [3]{1+i x}}{\sqrt [3]{1-i x}}\right ) \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 0.02 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.21 \[ \int \frac {e^{\frac {1}{3} i \arctan (x)}}{x} \, dx=-\frac {3 (1-i x)^{5/6} \left (\sqrt [6]{2} (1+i x)^{5/6} \operatorname {Hypergeometric2F1}\left (\frac {5}{6},\frac {5}{6},\frac {11}{6},\frac {1}{2}-\frac {i x}{2}\right )+2 \operatorname {Hypergeometric2F1}\left (\frac {5}{6},1,\frac {11}{6},\frac {i+x}{i-x}\right )\right )}{5 (1+i x)^{5/6}} \]
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\[\int \frac {{\left (\frac {i x +1}{\sqrt {x^{2}+1}}\right )}^{\frac {1}{3}}}{x}d x\]
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none
Time = 0.25 (sec) , antiderivative size = 339, normalized size of antiderivative = 0.79 \[ \int \frac {e^{\frac {1}{3} i \arctan (x)}}{x} \, dx=\frac {1}{2} \, {\left (\sqrt {3} + i\right )} \log \left (\frac {1}{2} \, \sqrt {3} + \left (\frac {i \, \sqrt {x^{2} + 1}}{x + i}\right )^{\frac {1}{3}} + \frac {1}{2} i\right ) + \frac {1}{2} \, {\left (\sqrt {3} - i\right )} \log \left (\frac {1}{2} \, \sqrt {3} + \left (\frac {i \, \sqrt {x^{2} + 1}}{x + i}\right )^{\frac {1}{3}} - \frac {1}{2} i\right ) + \frac {1}{2} \, {\left (-i \, \sqrt {3} - 1\right )} \log \left (\frac {1}{2} i \, \sqrt {3} + \left (\frac {i \, \sqrt {x^{2} + 1}}{x + i}\right )^{\frac {1}{3}} + \frac {1}{2}\right ) + \frac {1}{2} \, {\left (-i \, \sqrt {3} + 1\right )} \log \left (\frac {1}{2} i \, \sqrt {3} + \left (\frac {i \, \sqrt {x^{2} + 1}}{x + i}\right )^{\frac {1}{3}} - \frac {1}{2}\right ) + \frac {1}{2} \, {\left (i \, \sqrt {3} - 1\right )} \log \left (-\frac {1}{2} i \, \sqrt {3} + \left (\frac {i \, \sqrt {x^{2} + 1}}{x + i}\right )^{\frac {1}{3}} + \frac {1}{2}\right ) + \frac {1}{2} \, {\left (i \, \sqrt {3} + 1\right )} \log \left (-\frac {1}{2} i \, \sqrt {3} + \left (\frac {i \, \sqrt {x^{2} + 1}}{x + i}\right )^{\frac {1}{3}} - \frac {1}{2}\right ) - \frac {1}{2} \, {\left (\sqrt {3} - i\right )} \log \left (-\frac {1}{2} \, \sqrt {3} + \left (\frac {i \, \sqrt {x^{2} + 1}}{x + i}\right )^{\frac {1}{3}} + \frac {1}{2} i\right ) - \frac {1}{2} \, {\left (\sqrt {3} + i\right )} \log \left (-\frac {1}{2} \, \sqrt {3} + \left (\frac {i \, \sqrt {x^{2} + 1}}{x + i}\right )^{\frac {1}{3}} - \frac {1}{2} i\right ) - \log \left (\left (\frac {i \, \sqrt {x^{2} + 1}}{x + i}\right )^{\frac {1}{3}} + 1\right ) + i \, \log \left (\left (\frac {i \, \sqrt {x^{2} + 1}}{x + i}\right )^{\frac {1}{3}} + i\right ) - i \, \log \left (\left (\frac {i \, \sqrt {x^{2} + 1}}{x + i}\right )^{\frac {1}{3}} - i\right ) + \log \left (\left (\frac {i \, \sqrt {x^{2} + 1}}{x + i}\right )^{\frac {1}{3}} - 1\right ) \]
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\[ \int \frac {e^{\frac {1}{3} i \arctan (x)}}{x} \, dx=\int \frac {\sqrt [3]{\frac {i \left (x - i\right )}{\sqrt {x^{2} + 1}}}}{x}\, dx \]
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\[ \int \frac {e^{\frac {1}{3} i \arctan (x)}}{x} \, dx=\int { \frac {\left (\frac {i \, x + 1}{\sqrt {x^{2} + 1}}\right )^{\frac {1}{3}}}{x} \,d x } \]
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\[ \int \frac {e^{\frac {1}{3} i \arctan (x)}}{x} \, dx=\int { \frac {\left (\frac {i \, x + 1}{\sqrt {x^{2} + 1}}\right )^{\frac {1}{3}}}{x} \,d x } \]
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Timed out. \[ \int \frac {e^{\frac {1}{3} i \arctan (x)}}{x} \, dx=\int \frac {{\left (\frac {1+x\,1{}\mathrm {i}}{\sqrt {x^2+1}}\right )}^{1/3}}{x} \,d x \]
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