\(\int \frac {e^{\frac {1}{3} i \arctan (x)}}{x} \, dx\) [133]

Optimal result

Integrand size = 14, antiderivative size = 311 \[ \int \frac {e^{\frac {1}{3} i \arctan (x)}}{x} \, dx=\arctan \left (\sqrt {3}-\frac {2 \sqrt [6]{1-i x}}{\sqrt [6]{1+i x}}\right )-\arctan \left (\sqrt {3}+\frac {2 \sqrt [6]{1-i x}}{\sqrt [6]{1+i x}}\right )+\sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [6]{1+i x}}{\sqrt [6]{1-i x}}}{\sqrt {3}}\right )-\sqrt {3} \arctan \left (\frac {1+\frac {2 \sqrt [6]{1+i x}}{\sqrt [6]{1-i x}}}{\sqrt {3}}\right )-2 \arctan \left (\frac {\sqrt [6]{1-i x}}{\sqrt [6]{1+i x}}\right )+\sqrt {3} \text {arctanh}\left (\frac {\sqrt {3} \sqrt [6]{1-i x}}{\left (1+\frac {\sqrt [3]{1-i x}}{\sqrt [3]{1+i x}}\right ) \sqrt [6]{1+i x}}\right )-2 \text {arctanh}\left (\frac {\sqrt [6]{1+i x}}{\sqrt [6]{1-i x}}\right )-\text {arctanh}\left (\frac {\sqrt [6]{1+i x}}{\left (1+\frac {\sqrt [3]{1+i x}}{\sqrt [3]{1-i x}}\right ) \sqrt [6]{1-i x}}\right ) \] Output:

-arctan(-3^(1/2)+2*(1-I*x)^(1/6)/(1+I*x)^(1/6))-arctan(3^(1/2)+2*(1-I*x)^( 
1/6)/(1+I*x)^(1/6))+3^(1/2)*arctan(1/3*(1-2*(1+I*x)^(1/6)/(1-I*x)^(1/6))*3 
^(1/2))-3^(1/2)*arctan(1/3*(1+2*(1+I*x)^(1/6)/(1-I*x)^(1/6))*3^(1/2))-2*ar 
ctan((1-I*x)^(1/6)/(1+I*x)^(1/6))+arctanh(3^(1/2)*(1-I*x)^(1/6)/(1+(1-I*x) 
^(1/3)/(1+I*x)^(1/3))/(1+I*x)^(1/6))*3^(1/2)-2*arctanh((1+I*x)^(1/6)/(1-I* 
x)^(1/6))-arctanh((1+I*x)^(1/6)/(1+(1+I*x)^(1/3)/(1-I*x)^(1/3))/(1-I*x)^(1 
/6))
 

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.

Time = 0.04 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.29 \[ \int \frac {e^{\frac {1}{3} i \arctan (x)}}{x} \, dx=-\frac {3 (1-i x)^{5/6} \left (\sqrt [6]{2} (1+i x)^{5/6} \operatorname {Hypergeometric2F1}\left (\frac {5}{6},\frac {5}{6},\frac {11}{6},\frac {1}{2}-\frac {i x}{2}\right )+2 \operatorname {Hypergeometric2F1}\left (\frac {5}{6},1,\frac {11}{6},\frac {i+x}{i-x}\right )\right )}{5 (1+i x)^{5/6}} \] Input:

Integrate[E^((I/3)*ArcTan[x])/x,x]
 

Output:

(-3*(1 - I*x)^(5/6)*(2^(1/6)*(1 + I*x)^(5/6)*Hypergeometric2F1[5/6, 5/6, 1 
1/6, 1/2 - (I/2)*x] + 2*Hypergeometric2F1[5/6, 1, 11/6, (I + x)/(I - x)])) 
/(5*(1 + I*x)^(5/6))
 

Rubi [A] (warning: unable to verify)

Time = 0.84 (sec) , antiderivative size = 437, normalized size of antiderivative = 1.41, number of steps used = 17, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.143, Rules used = {5585, 140, 73, 104, 754, 27, 219, 854, 824, 27, 216, 1142, 25, 1083, 217, 1103}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \frac {e^{\frac {1}{3} i \arctan (x)}}{x} \, dx\)

\(\Big \downarrow \) 5585

\(\displaystyle \int \frac {\sqrt [6]{1+i x}}{\sqrt [6]{1-i x} x}dx\)

\(\Big \downarrow \) 140

\(\displaystyle i \int \frac {1}{\sqrt [6]{1-i x} (i x+1)^{5/6}}dx+\int \frac {1}{\sqrt [6]{1-i x} (i x+1)^{5/6} x}dx\)

\(\Big \downarrow \) 73

\(\displaystyle \int \frac {1}{\sqrt [6]{1-i x} (i x+1)^{5/6} x}dx-6 \int \frac {(1-i x)^{2/3}}{(i x+1)^{5/6}}d\sqrt [6]{1-i x}\)

\(\Big \downarrow \) 104

\(\displaystyle 6 \int \frac {1}{\frac {i x+1}{1-i x}-1}d\frac {\sqrt [6]{i x+1}}{\sqrt [6]{1-i x}}-6 \int \frac {(1-i x)^{2/3}}{(i x+1)^{5/6}}d\sqrt [6]{1-i x}\)

\(\Big \downarrow \) 754

\(\displaystyle 6 \left (-\frac {1}{3} \int \frac {1}{1-\frac {\sqrt [3]{i x+1}}{\sqrt [3]{1-i x}}}d\frac {\sqrt [6]{i x+1}}{\sqrt [6]{1-i x}}-\frac {1}{3} \int \frac {2-\frac {\sqrt [6]{i x+1}}{\sqrt [6]{1-i x}}}{2 \left (\frac {\sqrt [3]{i x+1}}{\sqrt [3]{1-i x}}-\frac {\sqrt [6]{i x+1}}{\sqrt [6]{1-i x}}+1\right )}d\frac {\sqrt [6]{i x+1}}{\sqrt [6]{1-i x}}-\frac {1}{3} \int \frac {\frac {\sqrt [6]{i x+1}}{\sqrt [6]{1-i x}}+2}{2 \left (\frac {\sqrt [3]{i x+1}}{\sqrt [3]{1-i x}}+\frac {\sqrt [6]{i x+1}}{\sqrt [6]{1-i x}}+1\right )}d\frac {\sqrt [6]{i x+1}}{\sqrt [6]{1-i x}}\right )-6 \int \frac {(1-i x)^{2/3}}{(i x+1)^{5/6}}d\sqrt [6]{1-i x}\)

\(\Big \downarrow \) 27

\(\displaystyle 6 \left (-\frac {1}{3} \int \frac {1}{1-\frac {\sqrt [3]{i x+1}}{\sqrt [3]{1-i x}}}d\frac {\sqrt [6]{i x+1}}{\sqrt [6]{1-i x}}-\frac {1}{6} \int \frac {2-\frac {\sqrt [6]{i x+1}}{\sqrt [6]{1-i x}}}{\frac {\sqrt [3]{i x+1}}{\sqrt [3]{1-i x}}-\frac {\sqrt [6]{i x+1}}{\sqrt [6]{1-i x}}+1}d\frac {\sqrt [6]{i x+1}}{\sqrt [6]{1-i x}}-\frac {1}{6} \int \frac {\frac {\sqrt [6]{i x+1}}{\sqrt [6]{1-i x}}+2}{\frac {\sqrt [3]{i x+1}}{\sqrt [3]{1-i x}}+\frac {\sqrt [6]{i x+1}}{\sqrt [6]{1-i x}}+1}d\frac {\sqrt [6]{i x+1}}{\sqrt [6]{1-i x}}\right )-6 \int \frac {(1-i x)^{2/3}}{(i x+1)^{5/6}}d\sqrt [6]{1-i x}\)

\(\Big \downarrow \) 219

\(\displaystyle 6 \left (-\frac {1}{6} \int \frac {2-\frac {\sqrt [6]{i x+1}}{\sqrt [6]{1-i x}}}{\frac {\sqrt [3]{i x+1}}{\sqrt [3]{1-i x}}-\frac {\sqrt [6]{i x+1}}{\sqrt [6]{1-i x}}+1}d\frac {\sqrt [6]{i x+1}}{\sqrt [6]{1-i x}}-\frac {1}{6} \int \frac {\frac {\sqrt [6]{i x+1}}{\sqrt [6]{1-i x}}+2}{\frac {\sqrt [3]{i x+1}}{\sqrt [3]{1-i x}}+\frac {\sqrt [6]{i x+1}}{\sqrt [6]{1-i x}}+1}d\frac {\sqrt [6]{i x+1}}{\sqrt [6]{1-i x}}-\frac {1}{3} \text {arctanh}\left (\frac {\sqrt [6]{1+i x}}{\sqrt [6]{1-i x}}\right )\right )-6 \int \frac {(1-i x)^{2/3}}{(i x+1)^{5/6}}d\sqrt [6]{1-i x}\)

\(\Big \downarrow \) 854

\(\displaystyle 6 \left (-\frac {1}{6} \int \frac {2-\frac {\sqrt [6]{i x+1}}{\sqrt [6]{1-i x}}}{\frac {\sqrt [3]{i x+1}}{\sqrt [3]{1-i x}}-\frac {\sqrt [6]{i x+1}}{\sqrt [6]{1-i x}}+1}d\frac {\sqrt [6]{i x+1}}{\sqrt [6]{1-i x}}-\frac {1}{6} \int \frac {\frac {\sqrt [6]{i x+1}}{\sqrt [6]{1-i x}}+2}{\frac {\sqrt [3]{i x+1}}{\sqrt [3]{1-i x}}+\frac {\sqrt [6]{i x+1}}{\sqrt [6]{1-i x}}+1}d\frac {\sqrt [6]{i x+1}}{\sqrt [6]{1-i x}}-\frac {1}{3} \text {arctanh}\left (\frac {\sqrt [6]{1+i x}}{\sqrt [6]{1-i x}}\right )\right )-6 \int \frac {(1-i x)^{2/3}}{2-i x}d\frac {\sqrt [6]{1-i x}}{\sqrt [6]{i x+1}}\)

\(\Big \downarrow \) 824

\(\displaystyle 6 \left (-\frac {1}{6} \int \frac {2-\frac {\sqrt [6]{i x+1}}{\sqrt [6]{1-i x}}}{\frac {\sqrt [3]{i x+1}}{\sqrt [3]{1-i x}}-\frac {\sqrt [6]{i x+1}}{\sqrt [6]{1-i x}}+1}d\frac {\sqrt [6]{i x+1}}{\sqrt [6]{1-i x}}-\frac {1}{6} \int \frac {\frac {\sqrt [6]{i x+1}}{\sqrt [6]{1-i x}}+2}{\frac {\sqrt [3]{i x+1}}{\sqrt [3]{1-i x}}+\frac {\sqrt [6]{i x+1}}{\sqrt [6]{1-i x}}+1}d\frac {\sqrt [6]{i x+1}}{\sqrt [6]{1-i x}}-\frac {1}{3} \text {arctanh}\left (\frac {\sqrt [6]{1+i x}}{\sqrt [6]{1-i x}}\right )\right )-6 \left (\frac {1}{3} \int \frac {1}{\sqrt [3]{1-i x}+1}d\frac {\sqrt [6]{1-i x}}{\sqrt [6]{i x+1}}+\frac {1}{3} \int -\frac {1-\frac {\sqrt {3} \sqrt [6]{1-i x}}{\sqrt [6]{i x+1}}}{2 \left (\sqrt [3]{1-i x}-\frac {\sqrt {3} \sqrt [6]{1-i x}}{\sqrt [6]{i x+1}}+1\right )}d\frac {\sqrt [6]{1-i x}}{\sqrt [6]{i x+1}}+\frac {1}{3} \int -\frac {\frac {\sqrt {3} \sqrt [6]{1-i x}}{\sqrt [6]{i x+1}}+1}{2 \left (\sqrt [3]{1-i x}+\frac {\sqrt {3} \sqrt [6]{1-i x}}{\sqrt [6]{i x+1}}+1\right )}d\frac {\sqrt [6]{1-i x}}{\sqrt [6]{i x+1}}\right )\)

\(\Big \downarrow \) 27

\(\displaystyle 6 \left (-\frac {1}{6} \int \frac {2-\frac {\sqrt [6]{i x+1}}{\sqrt [6]{1-i x}}}{\frac {\sqrt [3]{i x+1}}{\sqrt [3]{1-i x}}-\frac {\sqrt [6]{i x+1}}{\sqrt [6]{1-i x}}+1}d\frac {\sqrt [6]{i x+1}}{\sqrt [6]{1-i x}}-\frac {1}{6} \int \frac {\frac {\sqrt [6]{i x+1}}{\sqrt [6]{1-i x}}+2}{\frac {\sqrt [3]{i x+1}}{\sqrt [3]{1-i x}}+\frac {\sqrt [6]{i x+1}}{\sqrt [6]{1-i x}}+1}d\frac {\sqrt [6]{i x+1}}{\sqrt [6]{1-i x}}-\frac {1}{3} \text {arctanh}\left (\frac {\sqrt [6]{1+i x}}{\sqrt [6]{1-i x}}\right )\right )-6 \left (\frac {1}{3} \int \frac {1}{\sqrt [3]{1-i x}+1}d\frac {\sqrt [6]{1-i x}}{\sqrt [6]{i x+1}}-\frac {1}{6} \int \frac {1-\frac {\sqrt {3} \sqrt [6]{1-i x}}{\sqrt [6]{i x+1}}}{\sqrt [3]{1-i x}-\frac {\sqrt {3} \sqrt [6]{1-i x}}{\sqrt [6]{i x+1}}+1}d\frac {\sqrt [6]{1-i x}}{\sqrt [6]{i x+1}}-\frac {1}{6} \int \frac {\frac {\sqrt {3} \sqrt [6]{1-i x}}{\sqrt [6]{i x+1}}+1}{\sqrt [3]{1-i x}+\frac {\sqrt {3} \sqrt [6]{1-i x}}{\sqrt [6]{i x+1}}+1}d\frac {\sqrt [6]{1-i x}}{\sqrt [6]{i x+1}}\right )\)

\(\Big \downarrow \) 216

\(\displaystyle 6 \left (-\frac {1}{6} \int \frac {2-\frac {\sqrt [6]{i x+1}}{\sqrt [6]{1-i x}}}{\frac {\sqrt [3]{i x+1}}{\sqrt [3]{1-i x}}-\frac {\sqrt [6]{i x+1}}{\sqrt [6]{1-i x}}+1}d\frac {\sqrt [6]{i x+1}}{\sqrt [6]{1-i x}}-\frac {1}{6} \int \frac {\frac {\sqrt [6]{i x+1}}{\sqrt [6]{1-i x}}+2}{\frac {\sqrt [3]{i x+1}}{\sqrt [3]{1-i x}}+\frac {\sqrt [6]{i x+1}}{\sqrt [6]{1-i x}}+1}d\frac {\sqrt [6]{i x+1}}{\sqrt [6]{1-i x}}-\frac {1}{3} \text {arctanh}\left (\frac {\sqrt [6]{1+i x}}{\sqrt [6]{1-i x}}\right )\right )-6 \left (-\frac {1}{6} \int \frac {1-\frac {\sqrt {3} \sqrt [6]{1-i x}}{\sqrt [6]{i x+1}}}{\sqrt [3]{1-i x}-\frac {\sqrt {3} \sqrt [6]{1-i x}}{\sqrt [6]{i x+1}}+1}d\frac {\sqrt [6]{1-i x}}{\sqrt [6]{i x+1}}-\frac {1}{6} \int \frac {\frac {\sqrt {3} \sqrt [6]{1-i x}}{\sqrt [6]{i x+1}}+1}{\sqrt [3]{1-i x}+\frac {\sqrt {3} \sqrt [6]{1-i x}}{\sqrt [6]{i x+1}}+1}d\frac {\sqrt [6]{1-i x}}{\sqrt [6]{i x+1}}+\frac {1}{3} \arctan \left (\frac {\sqrt [6]{1-i x}}{\sqrt [6]{1+i x}}\right )\right )\)

\(\Big \downarrow \) 1142

\(\displaystyle 6 \left (-\frac {1}{3} \text {arctanh}\left (\frac {\sqrt [6]{i x+1}}{\sqrt [6]{1-i x}}\right )+\frac {1}{6} \left (\frac {1}{2} \int -\frac {1-\frac {2 \sqrt [6]{i x+1}}{\sqrt [6]{1-i x}}}{\frac {\sqrt [3]{i x+1}}{\sqrt [3]{1-i x}}-\frac {\sqrt [6]{i x+1}}{\sqrt [6]{1-i x}}+1}d\frac {\sqrt [6]{i x+1}}{\sqrt [6]{1-i x}}-\frac {3}{2} \int \frac {1}{\frac {\sqrt [3]{i x+1}}{\sqrt [3]{1-i x}}-\frac {\sqrt [6]{i x+1}}{\sqrt [6]{1-i x}}+1}d\frac {\sqrt [6]{i x+1}}{\sqrt [6]{1-i x}}\right )+\frac {1}{6} \left (-\frac {3}{2} \int \frac {1}{\frac {\sqrt [3]{i x+1}}{\sqrt [3]{1-i x}}+\frac {\sqrt [6]{i x+1}}{\sqrt [6]{1-i x}}+1}d\frac {\sqrt [6]{i x+1}}{\sqrt [6]{1-i x}}-\frac {1}{2} \int \frac {\frac {2 \sqrt [6]{i x+1}}{\sqrt [6]{1-i x}}+1}{\frac {\sqrt [3]{i x+1}}{\sqrt [3]{1-i x}}+\frac {\sqrt [6]{i x+1}}{\sqrt [6]{1-i x}}+1}d\frac {\sqrt [6]{i x+1}}{\sqrt [6]{1-i x}}\right )\right )-6 \left (\frac {1}{3} \arctan \left (\frac {\sqrt [6]{1-i x}}{\sqrt [6]{i x+1}}\right )+\frac {1}{6} \left (\frac {1}{2} \int \frac {1}{\sqrt [3]{1-i x}-\frac {\sqrt {3} \sqrt [6]{1-i x}}{\sqrt [6]{i x+1}}+1}d\frac {\sqrt [6]{1-i x}}{\sqrt [6]{i x+1}}+\frac {1}{2} \sqrt {3} \int -\frac {\sqrt {3}-\frac {2 \sqrt [6]{1-i x}}{\sqrt [6]{i x+1}}}{\sqrt [3]{1-i x}-\frac {\sqrt {3} \sqrt [6]{1-i x}}{\sqrt [6]{i x+1}}+1}d\frac {\sqrt [6]{1-i x}}{\sqrt [6]{i x+1}}\right )+\frac {1}{6} \left (\frac {1}{2} \int \frac {1}{\sqrt [3]{1-i x}+\frac {\sqrt {3} \sqrt [6]{1-i x}}{\sqrt [6]{i x+1}}+1}d\frac {\sqrt [6]{1-i x}}{\sqrt [6]{i x+1}}-\frac {1}{2} \sqrt {3} \int \frac {\frac {2 \sqrt [6]{1-i x}}{\sqrt [6]{i x+1}}+\sqrt {3}}{\sqrt [3]{1-i x}+\frac {\sqrt {3} \sqrt [6]{1-i x}}{\sqrt [6]{i x+1}}+1}d\frac {\sqrt [6]{1-i x}}{\sqrt [6]{i x+1}}\right )\right )\)

\(\Big \downarrow \) 25

\(\displaystyle 6 \left (-\frac {1}{3} \text {arctanh}\left (\frac {\sqrt [6]{i x+1}}{\sqrt [6]{1-i x}}\right )+\frac {1}{6} \left (-\frac {3}{2} \int \frac {1}{\frac {\sqrt [3]{i x+1}}{\sqrt [3]{1-i x}}-\frac {\sqrt [6]{i x+1}}{\sqrt [6]{1-i x}}+1}d\frac {\sqrt [6]{i x+1}}{\sqrt [6]{1-i x}}-\frac {1}{2} \int \frac {1-\frac {2 \sqrt [6]{i x+1}}{\sqrt [6]{1-i x}}}{\frac {\sqrt [3]{i x+1}}{\sqrt [3]{1-i x}}-\frac {\sqrt [6]{i x+1}}{\sqrt [6]{1-i x}}+1}d\frac {\sqrt [6]{i x+1}}{\sqrt [6]{1-i x}}\right )+\frac {1}{6} \left (-\frac {3}{2} \int \frac {1}{\frac {\sqrt [3]{i x+1}}{\sqrt [3]{1-i x}}+\frac {\sqrt [6]{i x+1}}{\sqrt [6]{1-i x}}+1}d\frac {\sqrt [6]{i x+1}}{\sqrt [6]{1-i x}}-\frac {1}{2} \int \frac {\frac {2 \sqrt [6]{i x+1}}{\sqrt [6]{1-i x}}+1}{\frac {\sqrt [3]{i x+1}}{\sqrt [3]{1-i x}}+\frac {\sqrt [6]{i x+1}}{\sqrt [6]{1-i x}}+1}d\frac {\sqrt [6]{i x+1}}{\sqrt [6]{1-i x}}\right )\right )-6 \left (\frac {1}{3} \arctan \left (\frac {\sqrt [6]{1-i x}}{\sqrt [6]{i x+1}}\right )+\frac {1}{6} \left (\frac {1}{2} \int \frac {1}{\sqrt [3]{1-i x}-\frac {\sqrt {3} \sqrt [6]{1-i x}}{\sqrt [6]{i x+1}}+1}d\frac {\sqrt [6]{1-i x}}{\sqrt [6]{i x+1}}-\frac {1}{2} \sqrt {3} \int \frac {\sqrt {3}-\frac {2 \sqrt [6]{1-i x}}{\sqrt [6]{i x+1}}}{\sqrt [3]{1-i x}-\frac {\sqrt {3} \sqrt [6]{1-i x}}{\sqrt [6]{i x+1}}+1}d\frac {\sqrt [6]{1-i x}}{\sqrt [6]{i x+1}}\right )+\frac {1}{6} \left (\frac {1}{2} \int \frac {1}{\sqrt [3]{1-i x}+\frac {\sqrt {3} \sqrt [6]{1-i x}}{\sqrt [6]{i x+1}}+1}d\frac {\sqrt [6]{1-i x}}{\sqrt [6]{i x+1}}-\frac {1}{2} \sqrt {3} \int \frac {\frac {2 \sqrt [6]{1-i x}}{\sqrt [6]{i x+1}}+\sqrt {3}}{\sqrt [3]{1-i x}+\frac {\sqrt {3} \sqrt [6]{1-i x}}{\sqrt [6]{i x+1}}+1}d\frac {\sqrt [6]{1-i x}}{\sqrt [6]{i x+1}}\right )\right )\)

\(\Big \downarrow \) 1083

\(\displaystyle 6 \left (\frac {1}{6} \left (3 \int \frac {1}{-\frac {\sqrt [3]{i x+1}}{\sqrt [3]{1-i x}}-3}d\left (\frac {2 \sqrt [6]{i x+1}}{\sqrt [6]{1-i x}}-1\right )-\frac {1}{2} \int \frac {1-\frac {2 \sqrt [6]{i x+1}}{\sqrt [6]{1-i x}}}{\frac {\sqrt [3]{i x+1}}{\sqrt [3]{1-i x}}-\frac {\sqrt [6]{i x+1}}{\sqrt [6]{1-i x}}+1}d\frac {\sqrt [6]{i x+1}}{\sqrt [6]{1-i x}}\right )+\frac {1}{6} \left (3 \int \frac {1}{-\frac {\sqrt [3]{i x+1}}{\sqrt [3]{1-i x}}-3}d\left (\frac {2 \sqrt [6]{i x+1}}{\sqrt [6]{1-i x}}+1\right )-\frac {1}{2} \int \frac {\frac {2 \sqrt [6]{i x+1}}{\sqrt [6]{1-i x}}+1}{\frac {\sqrt [3]{i x+1}}{\sqrt [3]{1-i x}}+\frac {\sqrt [6]{i x+1}}{\sqrt [6]{1-i x}}+1}d\frac {\sqrt [6]{i x+1}}{\sqrt [6]{1-i x}}\right )-\frac {1}{3} \text {arctanh}\left (\frac {\sqrt [6]{1+i x}}{\sqrt [6]{1-i x}}\right )\right )-6 \left (\frac {1}{6} \left (-\int \frac {1}{-\sqrt [3]{1-i x}-1}d\left (\frac {2 \sqrt [6]{1-i x}}{\sqrt [6]{i x+1}}-\sqrt {3}\right )-\frac {1}{2} \sqrt {3} \int \frac {\sqrt {3}-\frac {2 \sqrt [6]{1-i x}}{\sqrt [6]{i x+1}}}{\sqrt [3]{1-i x}-\frac {\sqrt {3} \sqrt [6]{1-i x}}{\sqrt [6]{i x+1}}+1}d\frac {\sqrt [6]{1-i x}}{\sqrt [6]{i x+1}}\right )+\frac {1}{6} \left (-\int \frac {1}{-\sqrt [3]{1-i x}-1}d\left (\frac {2 \sqrt [6]{1-i x}}{\sqrt [6]{i x+1}}+\sqrt {3}\right )-\frac {1}{2} \sqrt {3} \int \frac {\frac {2 \sqrt [6]{1-i x}}{\sqrt [6]{i x+1}}+\sqrt {3}}{\sqrt [3]{1-i x}+\frac {\sqrt {3} \sqrt [6]{1-i x}}{\sqrt [6]{i x+1}}+1}d\frac {\sqrt [6]{1-i x}}{\sqrt [6]{i x+1}}\right )+\frac {1}{3} \arctan \left (\frac {\sqrt [6]{1-i x}}{\sqrt [6]{1+i x}}\right )\right )\)

\(\Big \downarrow \) 217

\(\displaystyle 6 \left (\frac {1}{6} \left (-\frac {1}{2} \int \frac {1-\frac {2 \sqrt [6]{i x+1}}{\sqrt [6]{1-i x}}}{\frac {\sqrt [3]{i x+1}}{\sqrt [3]{1-i x}}-\frac {\sqrt [6]{i x+1}}{\sqrt [6]{1-i x}}+1}d\frac {\sqrt [6]{i x+1}}{\sqrt [6]{1-i x}}-\sqrt {3} \arctan \left (\frac {-1+\frac {2 \sqrt [6]{1+i x}}{\sqrt [6]{1-i x}}}{\sqrt {3}}\right )\right )+\frac {1}{6} \left (-\frac {1}{2} \int \frac {\frac {2 \sqrt [6]{i x+1}}{\sqrt [6]{1-i x}}+1}{\frac {\sqrt [3]{i x+1}}{\sqrt [3]{1-i x}}+\frac {\sqrt [6]{i x+1}}{\sqrt [6]{1-i x}}+1}d\frac {\sqrt [6]{i x+1}}{\sqrt [6]{1-i x}}-\sqrt {3} \arctan \left (\frac {1+\frac {2 \sqrt [6]{1+i x}}{\sqrt [6]{1-i x}}}{\sqrt {3}}\right )\right )-\frac {1}{3} \text {arctanh}\left (\frac {\sqrt [6]{1+i x}}{\sqrt [6]{1-i x}}\right )\right )-6 \left (\frac {1}{6} \left (-\frac {1}{2} \sqrt {3} \int \frac {\sqrt {3}-\frac {2 \sqrt [6]{1-i x}}{\sqrt [6]{i x+1}}}{\sqrt [3]{1-i x}-\frac {\sqrt {3} \sqrt [6]{1-i x}}{\sqrt [6]{i x+1}}+1}d\frac {\sqrt [6]{1-i x}}{\sqrt [6]{i x+1}}-\arctan \left (\sqrt {3}-\frac {2 \sqrt [6]{1-i x}}{\sqrt [6]{1+i x}}\right )\right )+\frac {1}{6} \left (\arctan \left (\sqrt {3}+\frac {2 \sqrt [6]{1-i x}}{\sqrt [6]{1+i x}}\right )-\frac {1}{2} \sqrt {3} \int \frac {\frac {2 \sqrt [6]{1-i x}}{\sqrt [6]{i x+1}}+\sqrt {3}}{\sqrt [3]{1-i x}+\frac {\sqrt {3} \sqrt [6]{1-i x}}{\sqrt [6]{i x+1}}+1}d\frac {\sqrt [6]{1-i x}}{\sqrt [6]{i x+1}}\right )+\frac {1}{3} \arctan \left (\frac {\sqrt [6]{1-i x}}{\sqrt [6]{1+i x}}\right )\right )\)

\(\Big \downarrow \) 1103

\(\displaystyle 6 \left (\frac {1}{6} \left (\frac {1}{2} \log \left (\frac {\sqrt [3]{1+i x}}{\sqrt [3]{1-i x}}-\frac {\sqrt [6]{1+i x}}{\sqrt [6]{1-i x}}+1\right )-\sqrt {3} \arctan \left (\frac {-1+\frac {2 \sqrt [6]{1+i x}}{\sqrt [6]{1-i x}}}{\sqrt {3}}\right )\right )+\frac {1}{6} \left (-\sqrt {3} \arctan \left (\frac {1+\frac {2 \sqrt [6]{1+i x}}{\sqrt [6]{1-i x}}}{\sqrt {3}}\right )-\frac {1}{2} \log \left (\frac {\sqrt [3]{1+i x}}{\sqrt [3]{1-i x}}+\frac {\sqrt [6]{1+i x}}{\sqrt [6]{1-i x}}+1\right )\right )-\frac {1}{3} \text {arctanh}\left (\frac {\sqrt [6]{1+i x}}{\sqrt [6]{1-i x}}\right )\right )-6 \left (\frac {1}{3} \arctan \left (\frac {\sqrt [6]{1-i x}}{\sqrt [6]{1+i x}}\right )+\frac {1}{6} \left (\frac {1}{2} \sqrt {3} \log \left (\sqrt [3]{1-i x}-\frac {\sqrt {3} \sqrt [6]{1-i x}}{\sqrt [6]{1+i x}}+1\right )-\arctan \left (\sqrt {3}-\frac {2 \sqrt [6]{1-i x}}{\sqrt [6]{1+i x}}\right )\right )+\frac {1}{6} \left (\arctan \left (\sqrt {3}+\frac {2 \sqrt [6]{1-i x}}{\sqrt [6]{1+i x}}\right )-\frac {1}{2} \sqrt {3} \log \left (\sqrt [3]{1-i x}+\frac {\sqrt {3} \sqrt [6]{1-i x}}{\sqrt [6]{1+i x}}+1\right )\right )\right )\)

Input:

Int[E^((I/3)*ArcTan[x])/x,x]
 

Output:

-6*(ArcTan[(1 - I*x)^(1/6)/(1 + I*x)^(1/6)]/3 + (-ArcTan[Sqrt[3] - (2*(1 - 
 I*x)^(1/6))/(1 + I*x)^(1/6)] + (Sqrt[3]*Log[1 + (1 - I*x)^(1/3) - (Sqrt[3 
]*(1 - I*x)^(1/6))/(1 + I*x)^(1/6)])/2)/6 + (ArcTan[Sqrt[3] + (2*(1 - I*x) 
^(1/6))/(1 + I*x)^(1/6)] - (Sqrt[3]*Log[1 + (1 - I*x)^(1/3) + (Sqrt[3]*(1 
- I*x)^(1/6))/(1 + I*x)^(1/6)])/2)/6) + 6*(-1/3*ArcTanh[(1 + I*x)^(1/6)/(1 
 - I*x)^(1/6)] + (-(Sqrt[3]*ArcTan[(-1 + (2*(1 + I*x)^(1/6))/(1 - I*x)^(1/ 
6))/Sqrt[3]]) + Log[1 - (1 + I*x)^(1/6)/(1 - I*x)^(1/6) + (1 + I*x)^(1/3)/ 
(1 - I*x)^(1/3)]/2)/6 + (-(Sqrt[3]*ArcTan[(1 + (2*(1 + I*x)^(1/6))/(1 - I* 
x)^(1/6))/Sqrt[3]]) - Log[1 + (1 + I*x)^(1/6)/(1 - I*x)^(1/6) + (1 + I*x)^ 
(1/3)/(1 - I*x)^(1/3)]/2)/6)
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 73

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ 
{p = Denominator[m]}, Simp[p/b   Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + 
 d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt 
Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL 
inearQ[a, b, c, d, m, n, x]
 

rule 104

Int[(((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x 
_)), x_] :> With[{q = Denominator[m]}, Simp[q   Subst[Int[x^(q*(m + 1) - 1) 
/(b*e - a*f - (d*e - c*f)*x^q), x], x, (a + b*x)^(1/q)/(c + d*x)^(1/q)], x] 
] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[m + n + 1, 0] && RationalQ[n] && L 
tQ[-1, m, 0] && SimplerQ[a + b*x, c + d*x]
 

rule 140

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) 
)^(p_), x_] :> Simp[b*d^(m + n)*f^p   Int[(a + b*x)^(m - 1)/(c + d*x)^m, x] 
, x] + Int[(a + b*x)^(m - 1)*((e + f*x)^p/(c + d*x)^m)*ExpandToSum[(a + b*x 
)*(c + d*x)^(-p - 1) - (b*d^(-p - 1)*f^p)/(e + f*x)^p, x], x] /; FreeQ[{a, 
b, c, d, e, f, m, n}, x] && EqQ[m + n + p + 1, 0] && ILtQ[p, 0] && (GtQ[m, 
0] || SumSimplerQ[m, -1] ||  !(GtQ[n, 0] || SumSimplerQ[n, -1]))
 

rule 216

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*A 
rcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a 
, 0] || GtQ[b, 0])
 

rule 217

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^( 
-1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] & 
& (LtQ[a, 0] || LtQ[b, 0])
 

rule 219

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* 
ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt 
Q[a, 0] || LtQ[b, 0])
 

rule 754

Int[((a_) + (b_.)*(x_)^(n_))^(-1), x_Symbol] :> Module[{r = Numerator[Rt[-a 
/b, n]], s = Denominator[Rt[-a/b, n]], k, u}, Simp[u = Int[(r - s*Cos[(2*k* 
Pi)/n]*x)/(r^2 - 2*r*s*Cos[(2*k*Pi)/n]*x + s^2*x^2), x] + Int[(r + s*Cos[(2 
*k*Pi)/n]*x)/(r^2 + 2*r*s*Cos[(2*k*Pi)/n]*x + s^2*x^2), x]; 2*(r^2/(a*n)) 
 Int[1/(r^2 - s^2*x^2), x] + 2*(r/(a*n))   Sum[u, {k, 1, (n - 2)/4}], x]] / 
; FreeQ[{a, b}, x] && IGtQ[(n - 2)/4, 0] && NegQ[a/b]
 

rule 824

Int[(x_)^(m_.)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Module[{r = Numerator 
[Rt[a/b, n]], s = Denominator[Rt[a/b, n]], k, u}, Simp[u = Int[(r*Cos[(2*k 
- 1)*m*(Pi/n)] - s*Cos[(2*k - 1)*(m + 1)*(Pi/n)]*x)/(r^2 - 2*r*s*Cos[(2*k - 
 1)*(Pi/n)]*x + s^2*x^2), x] + Int[(r*Cos[(2*k - 1)*m*(Pi/n)] + s*Cos[(2*k 
- 1)*(m + 1)*(Pi/n)]*x)/(r^2 + 2*r*s*Cos[(2*k - 1)*(Pi/n)]*x + s^2*x^2), x] 
; 2*(-1)^(m/2)*(r^(m + 2)/(a*n*s^m))   Int[1/(r^2 + s^2*x^2), x] + 2*(r^(m 
+ 1)/(a*n*s^m))   Sum[u, {k, 1, (n - 2)/4}], x]] /; FreeQ[{a, b}, x] && IGt 
Q[(n - 2)/4, 0] && IGtQ[m, 0] && LtQ[m, n - 1] && PosQ[a/b]
 

rule 854

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^(p + (m + 
 1)/n)   Subst[Int[x^m/(1 - b*x^n)^(p + (m + 1)/n + 1), x], x, x/(a + b*x^n 
)^(1/n)], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && LtQ[-1, p, 0] && NeQ[p, - 
2^(-1)] && IntegersQ[m, p + (m + 1)/n]
 

rule 1083

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Simp[-2   Subst[I 
nt[1/Simp[b^2 - 4*a*c - x^2, x], x], x, b + 2*c*x], x] /; FreeQ[{a, b, c}, 
x]
 

rule 1103

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S 
imp[d*(Log[RemoveContent[a + b*x + c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, 
e}, x] && EqQ[2*c*d - b*e, 0]
 

rule 1142

Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S 
imp[(2*c*d - b*e)/(2*c)   Int[1/(a + b*x + c*x^2), x], x] + Simp[e/(2*c) 
Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x]
 

rule 5585

Int[E^(ArcTan[(a_.)*(x_)]*(n_.))*(x_)^(m_.), x_Symbol] :> Int[x^m*((1 - I*a 
*x)^(I*(n/2))/(1 + I*a*x)^(I*(n/2))), x] /; FreeQ[{a, m, n}, x] &&  !Intege 
rQ[(I*n - 1)/2]
 

Maple [F]

\[\int \frac {{\left (\frac {i x +1}{\sqrt {x^{2}+1}}\right )}^{\frac {1}{3}}}{x}d x\]

Input:

int(((1+I*x)/(x^2+1)^(1/2))^(1/3)/x,x)
 

Output:

int(((1+I*x)/(x^2+1)^(1/2))^(1/3)/x,x)
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 339, normalized size of antiderivative = 1.09 \[ \int \frac {e^{\frac {1}{3} i \arctan (x)}}{x} \, dx=\frac {1}{2} \, {\left (\sqrt {3} + i\right )} \log \left (\frac {1}{2} \, \sqrt {3} + \left (\frac {i \, \sqrt {x^{2} + 1}}{x + i}\right )^{\frac {1}{3}} + \frac {1}{2} i\right ) + \frac {1}{2} \, {\left (\sqrt {3} - i\right )} \log \left (\frac {1}{2} \, \sqrt {3} + \left (\frac {i \, \sqrt {x^{2} + 1}}{x + i}\right )^{\frac {1}{3}} - \frac {1}{2} i\right ) + \frac {1}{2} \, {\left (-i \, \sqrt {3} - 1\right )} \log \left (\frac {1}{2} i \, \sqrt {3} + \left (\frac {i \, \sqrt {x^{2} + 1}}{x + i}\right )^{\frac {1}{3}} + \frac {1}{2}\right ) + \frac {1}{2} \, {\left (-i \, \sqrt {3} + 1\right )} \log \left (\frac {1}{2} i \, \sqrt {3} + \left (\frac {i \, \sqrt {x^{2} + 1}}{x + i}\right )^{\frac {1}{3}} - \frac {1}{2}\right ) + \frac {1}{2} \, {\left (i \, \sqrt {3} - 1\right )} \log \left (-\frac {1}{2} i \, \sqrt {3} + \left (\frac {i \, \sqrt {x^{2} + 1}}{x + i}\right )^{\frac {1}{3}} + \frac {1}{2}\right ) + \frac {1}{2} \, {\left (i \, \sqrt {3} + 1\right )} \log \left (-\frac {1}{2} i \, \sqrt {3} + \left (\frac {i \, \sqrt {x^{2} + 1}}{x + i}\right )^{\frac {1}{3}} - \frac {1}{2}\right ) - \frac {1}{2} \, {\left (\sqrt {3} - i\right )} \log \left (-\frac {1}{2} \, \sqrt {3} + \left (\frac {i \, \sqrt {x^{2} + 1}}{x + i}\right )^{\frac {1}{3}} + \frac {1}{2} i\right ) - \frac {1}{2} \, {\left (\sqrt {3} + i\right )} \log \left (-\frac {1}{2} \, \sqrt {3} + \left (\frac {i \, \sqrt {x^{2} + 1}}{x + i}\right )^{\frac {1}{3}} - \frac {1}{2} i\right ) - \log \left (\left (\frac {i \, \sqrt {x^{2} + 1}}{x + i}\right )^{\frac {1}{3}} + 1\right ) + i \, \log \left (\left (\frac {i \, \sqrt {x^{2} + 1}}{x + i}\right )^{\frac {1}{3}} + i\right ) - i \, \log \left (\left (\frac {i \, \sqrt {x^{2} + 1}}{x + i}\right )^{\frac {1}{3}} - i\right ) + \log \left (\left (\frac {i \, \sqrt {x^{2} + 1}}{x + i}\right )^{\frac {1}{3}} - 1\right ) \] Input:

integrate(((1+I*x)/(x^2+1)^(1/2))^(1/3)/x,x, algorithm="fricas")
 

Output:

1/2*(sqrt(3) + I)*log(1/2*sqrt(3) + (I*sqrt(x^2 + 1)/(x + I))^(1/3) + 1/2* 
I) + 1/2*(sqrt(3) - I)*log(1/2*sqrt(3) + (I*sqrt(x^2 + 1)/(x + I))^(1/3) - 
 1/2*I) + 1/2*(-I*sqrt(3) - 1)*log(1/2*I*sqrt(3) + (I*sqrt(x^2 + 1)/(x + I 
))^(1/3) + 1/2) + 1/2*(-I*sqrt(3) + 1)*log(1/2*I*sqrt(3) + (I*sqrt(x^2 + 1 
)/(x + I))^(1/3) - 1/2) + 1/2*(I*sqrt(3) - 1)*log(-1/2*I*sqrt(3) + (I*sqrt 
(x^2 + 1)/(x + I))^(1/3) + 1/2) + 1/2*(I*sqrt(3) + 1)*log(-1/2*I*sqrt(3) + 
 (I*sqrt(x^2 + 1)/(x + I))^(1/3) - 1/2) - 1/2*(sqrt(3) - I)*log(-1/2*sqrt( 
3) + (I*sqrt(x^2 + 1)/(x + I))^(1/3) + 1/2*I) - 1/2*(sqrt(3) + I)*log(-1/2 
*sqrt(3) + (I*sqrt(x^2 + 1)/(x + I))^(1/3) - 1/2*I) - log((I*sqrt(x^2 + 1) 
/(x + I))^(1/3) + 1) + I*log((I*sqrt(x^2 + 1)/(x + I))^(1/3) + I) - I*log( 
(I*sqrt(x^2 + 1)/(x + I))^(1/3) - I) + log((I*sqrt(x^2 + 1)/(x + I))^(1/3) 
 - 1)
 

Sympy [F]

\[ \int \frac {e^{\frac {1}{3} i \arctan (x)}}{x} \, dx=\int \frac {\sqrt [3]{\frac {i \left (x - i\right )}{\sqrt {x^{2} + 1}}}}{x}\, dx \] Input:

integrate(((1+I*x)/(x**2+1)**(1/2))**(1/3)/x,x)
 

Output:

Integral((I*(x - I)/sqrt(x**2 + 1))**(1/3)/x, x)
 

Maxima [F]

\[ \int \frac {e^{\frac {1}{3} i \arctan (x)}}{x} \, dx=\int { \frac {\left (\frac {i \, x + 1}{\sqrt {x^{2} + 1}}\right )^{\frac {1}{3}}}{x} \,d x } \] Input:

integrate(((1+I*x)/(x^2+1)^(1/2))^(1/3)/x,x, algorithm="maxima")
 

Output:

integrate(((I*x + 1)/sqrt(x^2 + 1))^(1/3)/x, x)
 

Giac [F]

\[ \int \frac {e^{\frac {1}{3} i \arctan (x)}}{x} \, dx=\int { \frac {\left (\frac {i \, x + 1}{\sqrt {x^{2} + 1}}\right )^{\frac {1}{3}}}{x} \,d x } \] Input:

integrate(((1+I*x)/(x^2+1)^(1/2))^(1/3)/x,x, algorithm="giac")
 

Output:

integrate(((I*x + 1)/sqrt(x^2 + 1))^(1/3)/x, x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {e^{\frac {1}{3} i \arctan (x)}}{x} \, dx=\int \frac {{\left (\frac {1+x\,1{}\mathrm {i}}{\sqrt {x^2+1}}\right )}^{1/3}}{x} \,d x \] Input:

int(((x*1i + 1)/(x^2 + 1)^(1/2))^(1/3)/x,x)
 

Output:

int(((x*1i + 1)/(x^2 + 1)^(1/2))^(1/3)/x, x)
 

Reduce [F]

\[ \int \frac {e^{\frac {1}{3} i \arctan (x)}}{x} \, dx=\int \frac {\left (i x +1\right )^{\frac {1}{3}}}{\left (x^{2}+1\right )^{\frac {1}{6}} x}d x \] Input:

int(((1+I*x)/(x^2+1)^(1/2))^(1/3)/x,x)
                                                                                    
                                                                                    
 

Output:

int((i*x + 1)**(1/3)/((x**2 + 1)**(1/6)*x),x)