Integrand size = 10, antiderivative size = 262 \[ \int e^{\frac {1}{3} i \arctan (x)} \, dx=i (1-i x)^{5/6} \sqrt [6]{1+i x}-\frac {1}{3} i \arctan \left (\sqrt {3}-\frac {2 \sqrt [6]{1-i x}}{\sqrt [6]{1+i x}}\right )+\frac {1}{3} i \arctan \left (\sqrt {3}+\frac {2 \sqrt [6]{1-i x}}{\sqrt [6]{1+i x}}\right )+\frac {2}{3} i \arctan \left (\frac {\sqrt [6]{1-i x}}{\sqrt [6]{1+i x}}\right )+\frac {i \log \left (1+\frac {\sqrt [3]{1-i x}}{\sqrt [3]{1+i x}}-\frac {\sqrt {3} \sqrt [6]{1-i x}}{\sqrt [6]{1+i x}}\right )}{2 \sqrt {3}}-\frac {i \log \left (1+\frac {\sqrt [3]{1-i x}}{\sqrt [3]{1+i x}}+\frac {\sqrt {3} \sqrt [6]{1-i x}}{\sqrt [6]{1+i x}}\right )}{2 \sqrt {3}} \]
[Out]
Time = 0.23 (sec) , antiderivative size = 262, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.000, Rules used = {5169, 52, 65, 338, 301, 648, 632, 210, 642, 209} \[ \int e^{\frac {1}{3} i \arctan (x)} \, dx=-\frac {1}{3} i \arctan \left (\sqrt {3}-\frac {2 \sqrt [6]{1-i x}}{\sqrt [6]{1+i x}}\right )+\frac {1}{3} i \arctan \left (\sqrt {3}+\frac {2 \sqrt [6]{1-i x}}{\sqrt [6]{1+i x}}\right )+\frac {2}{3} i \arctan \left (\frac {\sqrt [6]{1-i x}}{\sqrt [6]{1+i x}}\right )+i (1-i x)^{5/6} \sqrt [6]{1+i x}+\frac {i \log \left (\frac {\sqrt [3]{1-i x}}{\sqrt [3]{1+i x}}-\frac {\sqrt {3} \sqrt [6]{1-i x}}{\sqrt [6]{1+i x}}+1\right )}{2 \sqrt {3}}-\frac {i \log \left (\frac {\sqrt [3]{1-i x}}{\sqrt [3]{1+i x}}+\frac {\sqrt {3} \sqrt [6]{1-i x}}{\sqrt [6]{1+i x}}+1\right )}{2 \sqrt {3}} \]
[In]
[Out]
Rule 52
Rule 65
Rule 209
Rule 210
Rule 301
Rule 338
Rule 632
Rule 642
Rule 648
Rule 5169
Rubi steps \begin{align*} \text {integral}& = \int \frac {\sqrt [6]{1+i x}}{\sqrt [6]{1-i x}} \, dx \\ & = i (1-i x)^{5/6} \sqrt [6]{1+i x}+\frac {1}{3} \int \frac {1}{\sqrt [6]{1-i x} (1+i x)^{5/6}} \, dx \\ & = i (1-i x)^{5/6} \sqrt [6]{1+i x}+2 i \text {Subst}\left (\int \frac {x^4}{\left (2-x^6\right )^{5/6}} \, dx,x,\sqrt [6]{1-i x}\right ) \\ & = i (1-i x)^{5/6} \sqrt [6]{1+i x}+2 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 ) \\ & = i (1-i x)^{5/6} \sqrt [6]{1+i x}+\frac {2}{3} 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 {2}{3} 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 {2}{3} 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 ) \\ & = i (1-i x)^{5/6} \sqrt [6]{1+i x}+\frac {2}{3} i \arctan \left (\frac {\sqrt [6]{1-i x}}{\sqrt [6]{1+i x}}\right )+\frac {1}{6} 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 {1}{6} 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 {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 )}{2 \sqrt {3}}-\frac {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 )}{2 \sqrt {3}} \\ & = i (1-i x)^{5/6} \sqrt [6]{1+i x}+\frac {2}{3} i \arctan \left (\frac {\sqrt [6]{1-i x}}{\sqrt [6]{1+i x}}\right )+\frac {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 )}{2 \sqrt {3}}-\frac {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 )}{2 \sqrt {3}}-\frac {1}{3} 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 {1}{3} 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 ) \\ & = i (1-i x)^{5/6} \sqrt [6]{1+i x}-\frac {1}{3} i \arctan \left (\sqrt {3}-\frac {2 \sqrt [6]{1-i x}}{\sqrt [6]{1+i x}}\right )+\frac {1}{3} i \arctan \left (\sqrt {3}+\frac {2 \sqrt [6]{1-i x}}{\sqrt [6]{1+i x}}\right )+\frac {2}{3} i \arctan \left (\frac {\sqrt [6]{1-i x}}{\sqrt [6]{1+i x}}\right )+\frac {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 )}{2 \sqrt {3}}-\frac {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 )}{2 \sqrt {3}} \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 0.02 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.13 \[ \int e^{\frac {1}{3} i \arctan (x)} \, dx=-\frac {12}{7} i e^{\frac {7}{3} i \arctan (x)} \operatorname {Hypergeometric2F1}\left (\frac {7}{6},2,\frac {13}{6},-e^{2 i \arctan (x)}\right ) \]
[In]
[Out]
\[\int {\left (\frac {i x +1}{\sqrt {x^{2}+1}}\right )}^{\frac {1}{3}}d x\]
[In]
[Out]
none
Time = 0.25 (sec) , antiderivative size = 195, normalized size of antiderivative = 0.74 \[ \int e^{\frac {1}{3} i \arctan (x)} \, dx=\frac {1}{6} \, {\left (-i \, \sqrt {3} + 1\right )} \log \left (\frac {1}{2} \, \sqrt {3} + \left (\frac {i \, \sqrt {x^{2} + 1}}{x + i}\right )^{\frac {1}{3}} + \frac {1}{2} i\right ) + \frac {1}{6} \, {\left (-i \, \sqrt {3} - 1\right )} \log \left (\frac {1}{2} \, \sqrt {3} + \left (\frac {i \, \sqrt {x^{2} + 1}}{x + i}\right )^{\frac {1}{3}} - \frac {1}{2} i\right ) + \frac {1}{6} \, {\left (i \, \sqrt {3} + 1\right )} \log \left (-\frac {1}{2} \, \sqrt {3} + \left (\frac {i \, \sqrt {x^{2} + 1}}{x + i}\right )^{\frac {1}{3}} + \frac {1}{2} i\right ) + \frac {1}{6} \, {\left (i \, \sqrt {3} - 1\right )} \log \left (-\frac {1}{2} \, \sqrt {3} + \left (\frac {i \, \sqrt {x^{2} + 1}}{x + i}\right )^{\frac {1}{3}} - \frac {1}{2} i\right ) + {\left (x + i\right )} \left (\frac {i \, \sqrt {x^{2} + 1}}{x + i}\right )^{\frac {1}{3}} + \frac {1}{3} \, \log \left (\left (\frac {i \, \sqrt {x^{2} + 1}}{x + i}\right )^{\frac {1}{3}} + i\right ) - \frac {1}{3} \, \log \left (\left (\frac {i \, \sqrt {x^{2} + 1}}{x + i}\right )^{\frac {1}{3}} - i\right ) \]
[In]
[Out]
\[ \int e^{\frac {1}{3} i \arctan (x)} \, dx=\int \sqrt [3]{\frac {i x + 1}{\sqrt {x^{2} + 1}}}\, dx \]
[In]
[Out]
\[ \int e^{\frac {1}{3} i \arctan (x)} \, dx=\int { \left (\frac {i \, x + 1}{\sqrt {x^{2} + 1}}\right )^{\frac {1}{3}} \,d x } \]
[In]
[Out]
\[ \int e^{\frac {1}{3} i \arctan (x)} \, dx=\int { \left (\frac {i \, x + 1}{\sqrt {x^{2} + 1}}\right )^{\frac {1}{3}} \,d x } \]
[In]
[Out]
Timed out. \[ \int e^{\frac {1}{3} i \arctan (x)} \, dx=\int {\left (\frac {1+x\,1{}\mathrm {i}}{\sqrt {x^2+1}}\right )}^{1/3} \,d x \]
[In]
[Out]