Integrand size = 14, antiderivative size = 319 \[ \int e^{\frac {1}{3} i \arctan (x)} x^2 \, dx=-\frac {19}{54} i (1-i x)^{5/6} \sqrt [6]{1+i x}-\frac {1}{18} i (1-i x)^{5/6} (1+i x)^{7/6}+\frac {1}{3} (1-i x)^{5/6} (1+i x)^{7/6} x+\frac {19}{162} i \arctan \left (\sqrt {3}-\frac {2 \sqrt [6]{1-i x}}{\sqrt [6]{1+i x}}\right )-\frac {19}{162} i \arctan \left (\sqrt {3}+\frac {2 \sqrt [6]{1-i x}}{\sqrt [6]{1+i x}}\right )-\frac {19}{81} i \arctan \left (\frac {\sqrt [6]{1-i x}}{\sqrt [6]{1+i x}}\right )-\frac {19 i \log \left (1+\frac {\sqrt [3]{1-i x}}{\sqrt [3]{1+i x}}-\frac {\sqrt {3} \sqrt [6]{1-i x}}{\sqrt [6]{1+i x}}\right )}{108 \sqrt {3}}+\frac {19 i \log \left (1+\frac {\sqrt [3]{1-i x}}{\sqrt [3]{1+i x}}+\frac {\sqrt {3} \sqrt [6]{1-i x}}{\sqrt [6]{1+i x}}\right )}{108 \sqrt {3}} \]
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Time = 0.29 (sec) , antiderivative size = 319, normalized size of antiderivative = 1.00, number of steps used = 16, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.857, Rules used = {5170, 92, 81, 52, 65, 338, 301, 648, 632, 210, 642, 209} \[ \int e^{\frac {1}{3} i \arctan (x)} x^2 \, dx=\frac {19}{162} i \arctan \left (\sqrt {3}-\frac {2 \sqrt [6]{1-i x}}{\sqrt [6]{1+i x}}\right )-\frac {19}{162} i \arctan \left (\sqrt {3}+\frac {2 \sqrt [6]{1-i x}}{\sqrt [6]{1+i x}}\right )-\frac {19}{81} i \arctan \left (\frac {\sqrt [6]{1-i x}}{\sqrt [6]{1+i x}}\right )+\frac {1}{3} (1-i x)^{5/6} x (1+i x)^{7/6}-\frac {1}{18} i (1-i x)^{5/6} (1+i x)^{7/6}-\frac {19}{54} i (1-i x)^{5/6} \sqrt [6]{1+i x}-\frac {19 i \log \left (\frac {\sqrt [3]{1-i x}}{\sqrt [3]{1+i x}}-\frac {\sqrt {3} \sqrt [6]{1-i x}}{\sqrt [6]{1+i x}}+1\right )}{108 \sqrt {3}}+\frac {19 i \log \left (\frac {\sqrt [3]{1-i x}}{\sqrt [3]{1+i x}}+\frac {\sqrt {3} \sqrt [6]{1-i x}}{\sqrt [6]{1+i x}}+1\right )}{108 \sqrt {3}} \]
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Rule 52
Rule 65
Rule 81
Rule 92
Rule 209
Rule 210
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} x^2}{\sqrt [6]{1-i x}} \, dx \\ & = \frac {1}{3} (1-i x)^{5/6} (1+i x)^{7/6} x+\frac {1}{3} \int \frac {\left (-1-\frac {i x}{3}\right ) \sqrt [6]{1+i x}}{\sqrt [6]{1-i x}} \, dx \\ & = -\frac {1}{18} i (1-i x)^{5/6} (1+i x)^{7/6}+\frac {1}{3} (1-i x)^{5/6} (1+i x)^{7/6} x-\frac {19}{54} \int \frac {\sqrt [6]{1+i x}}{\sqrt [6]{1-i x}} \, dx \\ & = -\frac {19}{54} i (1-i x)^{5/6} \sqrt [6]{1+i x}-\frac {1}{18} i (1-i x)^{5/6} (1+i x)^{7/6}+\frac {1}{3} (1-i x)^{5/6} (1+i x)^{7/6} x-\frac {19}{162} \int \frac {1}{\sqrt [6]{1-i x} (1+i x)^{5/6}} \, dx \\ & = -\frac {19}{54} i (1-i x)^{5/6} \sqrt [6]{1+i x}-\frac {1}{18} i (1-i x)^{5/6} (1+i x)^{7/6}+\frac {1}{3} (1-i x)^{5/6} (1+i x)^{7/6} x-\frac {19}{27} i \text {Subst}\left (\int \frac {x^4}{\left (2-x^6\right )^{5/6}} \, dx,x,\sqrt [6]{1-i x}\right ) \\ & = -\frac {19}{54} i (1-i x)^{5/6} \sqrt [6]{1+i x}-\frac {1}{18} i (1-i x)^{5/6} (1+i x)^{7/6}+\frac {1}{3} (1-i x)^{5/6} (1+i x)^{7/6} x-\frac {19}{27} i \text {Subst}\left (\int \frac {x^4}{1+x^6} \, dx,x,\frac {\sqrt [6]{1-i x}}{\sqrt [6]{1+i x}}\right ) \\ & = -\frac {19}{54} i (1-i x)^{5/6} \sqrt [6]{1+i x}-\frac {1}{18} i (1-i x)^{5/6} (1+i x)^{7/6}+\frac {1}{3} (1-i x)^{5/6} (1+i x)^{7/6} x-\frac {19}{81} i \text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\frac {\sqrt [6]{1-i x}}{\sqrt [6]{1+i x}}\right )-\frac {19}{81} i \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 )-\frac {19}{81} i \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 ) \\ & = -\frac {19}{54} i (1-i x)^{5/6} \sqrt [6]{1+i x}-\frac {1}{18} i (1-i x)^{5/6} (1+i x)^{7/6}+\frac {1}{3} (1-i x)^{5/6} (1+i x)^{7/6} x-\frac {19}{81} i \arctan \left (\frac {\sqrt [6]{1-i x}}{\sqrt [6]{1+i x}}\right )-\frac {19}{324} i \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 {19}{324} i \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 {(19 i) \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 )}{108 \sqrt {3}}+\frac {(19 i) \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 )}{108 \sqrt {3}} \\ & = -\frac {19}{54} i (1-i x)^{5/6} \sqrt [6]{1+i x}-\frac {1}{18} i (1-i x)^{5/6} (1+i x)^{7/6}+\frac {1}{3} (1-i x)^{5/6} (1+i x)^{7/6} x-\frac {19}{81} i \arctan \left (\frac {\sqrt [6]{1-i x}}{\sqrt [6]{1+i x}}\right )-\frac {19 i \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 )}{108 \sqrt {3}}+\frac {19 i \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 )}{108 \sqrt {3}}+\frac {19}{162} i \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 )+\frac {19}{162} i \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 ) \\ & = -\frac {19}{54} i (1-i x)^{5/6} \sqrt [6]{1+i x}-\frac {1}{18} i (1-i x)^{5/6} (1+i x)^{7/6}+\frac {1}{3} (1-i x)^{5/6} (1+i x)^{7/6} x+\frac {19}{162} i \arctan \left (\sqrt {3}-\frac {2 \sqrt [6]{1-i x}}{\sqrt [6]{1+i x}}\right )-\frac {19}{162} i \arctan \left (\sqrt {3}+\frac {2 \sqrt [6]{1-i x}}{\sqrt [6]{1+i x}}\right )-\frac {19}{81} i \arctan \left (\frac {\sqrt [6]{1-i x}}{\sqrt [6]{1+i x}}\right )-\frac {19 i \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 )}{108 \sqrt {3}}+\frac {19 i \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 )}{108 \sqrt {3}} \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 0.03 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.23 \[ \int e^{\frac {1}{3} i \arctan (x)} x^2 \, dx=\frac {1}{90} (1-i x)^{5/6} \left (5 \sqrt [6]{1+i x} \left (-i+7 x+6 i x^2\right )-38 i \sqrt [6]{2} \operatorname {Hypergeometric2F1}\left (-\frac {1}{6},\frac {5}{6},\frac {11}{6},\frac {1}{2}-\frac {i x}{2}\right )\right ) \]
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\[\int {\left (\frac {i x +1}{\sqrt {x^{2}+1}}\right )}^{\frac {1}{3}} x^{2}d x\]
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none
Time = 0.26 (sec) , antiderivative size = 208, normalized size of antiderivative = 0.65 \[ \int e^{\frac {1}{3} i \arctan (x)} x^2 \, dx=-\frac {19}{324} \, {\left (-i \, \sqrt {3} + 1\right )} \log \left (\frac {1}{2} \, \sqrt {3} + \left (\frac {i \, \sqrt {x^{2} + 1}}{x + i}\right )^{\frac {1}{3}} + \frac {1}{2} i\right ) - \frac {19}{324} \, {\left (-i \, \sqrt {3} - 1\right )} \log \left (\frac {1}{2} \, \sqrt {3} + \left (\frac {i \, \sqrt {x^{2} + 1}}{x + i}\right )^{\frac {1}{3}} - \frac {1}{2} i\right ) - \frac {19}{324} \, {\left (i \, \sqrt {3} + 1\right )} \log \left (-\frac {1}{2} \, \sqrt {3} + \left (\frac {i \, \sqrt {x^{2} + 1}}{x + i}\right )^{\frac {1}{3}} + \frac {1}{2} i\right ) - \frac {19}{324} \, {\left (i \, \sqrt {3} - 1\right )} \log \left (-\frac {1}{2} \, \sqrt {3} + \left (\frac {i \, \sqrt {x^{2} + 1}}{x + i}\right )^{\frac {1}{3}} - \frac {1}{2} i\right ) + \frac {1}{54} \, {\left (18 \, x^{3} - 3 i \, x^{2} - x - 22 i\right )} \left (\frac {i \, \sqrt {x^{2} + 1}}{x + i}\right )^{\frac {1}{3}} - \frac {19}{162} \, \log \left (\left (\frac {i \, \sqrt {x^{2} + 1}}{x + i}\right )^{\frac {1}{3}} + i\right ) + \frac {19}{162} \, \log \left (\left (\frac {i \, \sqrt {x^{2} + 1}}{x + i}\right )^{\frac {1}{3}} - i\right ) \]
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\[ \int e^{\frac {1}{3} i \arctan (x)} x^2 \, dx=\int x^{2} \sqrt [3]{\frac {i \left (x - i\right )}{\sqrt {x^{2} + 1}}}\, dx \]
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\[ \int e^{\frac {1}{3} i \arctan (x)} x^2 \, dx=\int { x^{2} \left (\frac {i \, x + 1}{\sqrt {x^{2} + 1}}\right )^{\frac {1}{3}} \,d x } \]
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\[ \int e^{\frac {1}{3} i \arctan (x)} x^2 \, dx=\int { x^{2} \left (\frac {i \, x + 1}{\sqrt {x^{2} + 1}}\right )^{\frac {1}{3}} \,d x } \]
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Timed out. \[ \int e^{\frac {1}{3} i \arctan (x)} x^2 \, dx=\int x^2\,{\left (\frac {1+x\,1{}\mathrm {i}}{\sqrt {x^2+1}}\right )}^{1/3} \,d x \]
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