\(\int \frac {e^{\frac {1}{3} \coth ^{-1}(x)}}{x} \, dx\) [124]

Optimal result

Integrand size = 12, antiderivative size = 290 \[ \int \frac {e^{\frac {1}{3} \coth ^{-1}(x)}}{x} \, dx=-\arctan \left (\sqrt {3}-\frac {2 \sqrt [6]{1-\frac {1}{x}}}{\sqrt [6]{1+\frac {1}{x}}}\right )+\arctan \left (\sqrt {3}+\frac {2 \sqrt [6]{1-\frac {1}{x}}}{\sqrt [6]{1+\frac {1}{x}}}\right )-\sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{1-\frac {1}{x}}}}{\sqrt {3}}\right )+\sqrt {3} \arctan \left (\frac {1+\frac {2 \sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{1-\frac {1}{x}}}}{\sqrt {3}}\right )+2 \arctan \left (\frac {\sqrt [6]{1-\frac {1}{x}}}{\sqrt [6]{1+\frac {1}{x}}}\right )-\sqrt {3} \text {arctanh}\left (\frac {\sqrt {3} \sqrt [6]{1-\frac {1}{x}}}{\left (1+\frac {\sqrt [3]{1-\frac {1}{x}}}{\sqrt [3]{1+\frac {1}{x}}}\right ) \sqrt [6]{1+\frac {1}{x}}}\right )+2 \text {arctanh}\left (\frac {\sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{1-\frac {1}{x}}}\right )+\text {arctanh}\left (\frac {\sqrt [6]{1+\frac {1}{x}}}{\left (1+\frac {\sqrt [3]{1+\frac {1}{x}}}{\sqrt [3]{1-\frac {1}{x}}}\right ) \sqrt [6]{1-\frac {1}{x}}}\right ) \] Output:

arctan(-3^(1/2)+2*(1-1/x)^(1/6)/(1+1/x)^(1/6))+arctan(3^(1/2)+2*(1-1/x)^(1 
/6)/(1+1/x)^(1/6))-arctan(1/3*(1-2*(1+1/x)^(1/6)/(1-1/x)^(1/6))*3^(1/2))*3 
^(1/2)+arctan(1/3*(1+2*(1+1/x)^(1/6)/(1-1/x)^(1/6))*3^(1/2))*3^(1/2)+2*arc 
tan((1-1/x)^(1/6)/(1+1/x)^(1/6))-3^(1/2)*arctanh(3^(1/2)*(1-1/x)^(1/6)/(1+ 
(1-1/x)^(1/3)/(1+1/x)^(1/3))/(1+1/x)^(1/6))+2*arctanh((1+1/x)^(1/6)/(1-1/x 
)^(1/6))+arctanh((1+1/x)^(1/6)/(1+(1+1/x)^(1/3)/(1-1/x)^(1/3))/(1-1/x)^(1/ 
6))
 

Mathematica [C] (verified)

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

Time = 0.05 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.09 \[ \int \frac {e^{\frac {1}{3} \coth ^{-1}(x)}}{x} \, dx=\frac {12}{7} e^{\frac {7}{3} \coth ^{-1}(x)} \operatorname {Hypergeometric2F1}\left (\frac {7}{12},1,\frac {19}{12},e^{4 \coth ^{-1}(x)}\right ) \] Input:

Integrate[E^(ArcCoth[x]/3)/x,x]
 

Output:

(12*E^((7*ArcCoth[x])/3)*Hypergeometric2F1[7/12, 1, 19/12, E^(4*ArcCoth[x] 
)])/7
 

Rubi [A] (warning: unable to verify)

Time = 0.89 (sec) , antiderivative size = 371, normalized size of antiderivative = 1.28, number of steps used = 17, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.333, Rules used = {6721, 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.

\(\displaystyle \int \frac {e^{\frac {1}{3} \coth ^{-1}(x)}}{x} \, dx\)

\(\Big \downarrow \) 6721

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

\(\Big \downarrow \) 140

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

\(\Big \downarrow \) 73

\(\displaystyle 6 \int \frac {1}{\left (2-\frac {1}{x^6}\right )^{5/6} x^4}d\sqrt [6]{1-\frac {1}{x}}-\int \frac {x}{\sqrt [6]{1-\frac {1}{x}} \left (1+\frac {1}{x}\right )^{5/6}}d\frac {1}{x}\)

\(\Big \downarrow \) 104

\(\displaystyle 6 \int \frac {1}{\left (2-\frac {1}{x^6}\right )^{5/6} x^4}d\sqrt [6]{1-\frac {1}{x}}-6 \int \frac {1}{\frac {1}{x^6}-1}d\frac {\sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{1-\frac {1}{x}}}\)

\(\Big \downarrow \) 754

\(\displaystyle 6 \int \frac {1}{\left (2-\frac {1}{x^6}\right )^{5/6} x^4}d\sqrt [6]{1-\frac {1}{x}}-6 \left (-\frac {1}{3} \int \frac {1}{1-\frac {1}{x^2}}d\frac {\sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{1-\frac {1}{x}}}-\frac {1}{3} \int \frac {2-\frac {\sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{1-\frac {1}{x}}}}{2 \left (-\frac {\sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{1-\frac {1}{x}}}+\frac {1}{x^2}+1\right )}d\frac {\sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{1-\frac {1}{x}}}-\frac {1}{3} \int \frac {\frac {\sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{1-\frac {1}{x}}}+2}{2 \left (\frac {\sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{1-\frac {1}{x}}}+\frac {1}{x^2}+1\right )}d\frac {\sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{1-\frac {1}{x}}}\right )\)

\(\Big \downarrow \) 27

\(\displaystyle 6 \int \frac {1}{\left (2-\frac {1}{x^6}\right )^{5/6} x^4}d\sqrt [6]{1-\frac {1}{x}}-6 \left (-\frac {1}{3} \int \frac {1}{1-\frac {1}{x^2}}d\frac {\sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{1-\frac {1}{x}}}-\frac {1}{6} \int \frac {2-\frac {\sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{1-\frac {1}{x}}}}{-\frac {\sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{1-\frac {1}{x}}}+\frac {1}{x^2}+1}d\frac {\sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{1-\frac {1}{x}}}-\frac {1}{6} \int \frac {\frac {\sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{1-\frac {1}{x}}}+2}{\frac {\sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{1-\frac {1}{x}}}+\frac {1}{x^2}+1}d\frac {\sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{1-\frac {1}{x}}}\right )\)

\(\Big \downarrow \) 219

\(\displaystyle 6 \int \frac {1}{\left (2-\frac {1}{x^6}\right )^{5/6} x^4}d\sqrt [6]{1-\frac {1}{x}}-6 \left (-\frac {1}{6} \int \frac {2-\frac {\sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{1-\frac {1}{x}}}}{-\frac {\sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{1-\frac {1}{x}}}+\frac {1}{x^2}+1}d\frac {\sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{1-\frac {1}{x}}}-\frac {1}{6} \int \frac {\frac {\sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{1-\frac {1}{x}}}+2}{\frac {\sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{1-\frac {1}{x}}}+\frac {1}{x^2}+1}d\frac {\sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{1-\frac {1}{x}}}-\frac {1}{3} \text {arctanh}\left (\frac {\sqrt [6]{\frac {1}{x}+1}}{\sqrt [6]{1-\frac {1}{x}}}\right )\right )\)

\(\Big \downarrow \) 854

\(\displaystyle 6 \int \frac {1}{\left (1+\frac {1}{x^6}\right ) x^4}d\frac {\sqrt [6]{1-\frac {1}{x}}}{\sqrt [6]{2-\frac {1}{x^6}}}-6 \left (-\frac {1}{6} \int \frac {2-\frac {\sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{1-\frac {1}{x}}}}{-\frac {\sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{1-\frac {1}{x}}}+\frac {1}{x^2}+1}d\frac {\sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{1-\frac {1}{x}}}-\frac {1}{6} \int \frac {\frac {\sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{1-\frac {1}{x}}}+2}{\frac {\sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{1-\frac {1}{x}}}+\frac {1}{x^2}+1}d\frac {\sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{1-\frac {1}{x}}}-\frac {1}{3} \text {arctanh}\left (\frac {\sqrt [6]{\frac {1}{x}+1}}{\sqrt [6]{1-\frac {1}{x}}}\right )\right )\)

\(\Big \downarrow \) 824

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 216

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

\(\Big \downarrow \) 1142

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 1083

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

\(\Big \downarrow \) 217

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

\(\Big \downarrow \) 1103

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

Input:

Int[E^(ArcCoth[x]/3)/x,x]
 

Output:

6*(ArcTan[(1 - x^(-1))^(1/6)/(2 - x^(-6))^(1/6)]/3 + (-ArcTan[Sqrt[3] - (2 
*(1 - x^(-1))^(1/6))/(2 - x^(-6))^(1/6)] + (Sqrt[3]*Log[1 - (Sqrt[3]*(1 - 
x^(-1))^(1/6))/(2 - x^(-6))^(1/6) + x^(-2)])/2)/6 + (ArcTan[Sqrt[3] + (2*( 
1 - x^(-1))^(1/6))/(2 - x^(-6))^(1/6)] - (Sqrt[3]*Log[1 + (Sqrt[3]*(1 - x^ 
(-1))^(1/6))/(2 - x^(-6))^(1/6) + x^(-2)])/2)/6) - 6*(-1/3*ArcTanh[(1 + x^ 
(-1))^(1/6)/(1 - x^(-1))^(1/6)] + (-(Sqrt[3]*ArcTan[(-1 + (2*(1 + x^(-1))^ 
(1/6))/(1 - x^(-1))^(1/6))/Sqrt[3]]) + Log[1 - (1 + x^(-1))^(1/6)/(1 - x^( 
-1))^(1/6) + x^(-2)]/2)/6 + (-(Sqrt[3]*ArcTan[(1 + (2*(1 + x^(-1))^(1/6))/ 
(1 - x^(-1))^(1/6))/Sqrt[3]]) - Log[1 + (1 + x^(-1))^(1/6)/(1 - x^(-1))^(1 
/6) + x^(-2)]/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 6721

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

Maple [C] (verified)

Result contains higher order function than in optimal. Order 9 vs. order 3.

Time = 10.05 (sec) , antiderivative size = 2202, normalized size of antiderivative = 7.59

method	result	size
trager	\(\text {Expression too large to display}\)	\(2202\)

Input:

int(1/((x-1)/(1+x))^(1/6)/x,x,method=_RETURNVERBOSE)
 

Output:

27*RootOf(81*_Z^4-9*_Z^2+1)^3*ln(-(27*RootOf(81*_Z^4-9*_Z^2+1)^3*(-(1-x)/( 
1+x))^(2/3)*x+27*RootOf(81*_Z^4-9*_Z^2+1)^3*(-(1-x)/(1+x))^(2/3)+27*RootOf 
(81*_Z^4-9*_Z^2+1)^3*(-(1-x)/(1+x))^(1/3)*x+18*RootOf(81*_Z^4-9*_Z^2+1)^2* 
(-(1-x)/(1+x))^(1/2)*x+3*RootOf(81*_Z^4-9*_Z^2+1)*(-(1-x)/(1+x))^(2/3)*x+( 
-(1-x)/(1+x))^(5/6)*x+27*RootOf(81*_Z^4-9*_Z^2+1)^3*(-(1-x)/(1+x))^(1/3)+1 
8*RootOf(81*_Z^4-9*_Z^2+1)^2*(-(1-x)/(1+x))^(1/2)+3*RootOf(81*_Z^4-9*_Z^2+ 
1)*(-(1-x)/(1+x))^(2/3)+(-(1-x)/(1+x))^(5/6)-18*RootOf(81*_Z^4-9*_Z^2+1)^3 
*x-9*RootOf(81*_Z^4-9*_Z^2+1)^2*(-(1-x)/(1+x))^(1/6)*x-6*RootOf(81*_Z^4-9* 
_Z^2+1)*(-(1-x)/(1+x))^(1/3)*x-2*(-(1-x)/(1+x))^(1/2)*x-9*RootOf(81*_Z^4-9 
*_Z^2+1)^2*(-(1-x)/(1+x))^(1/6)-6*RootOf(81*_Z^4-9*_Z^2+1)*(-(1-x)/(1+x))^ 
(1/3)-2*(-(1-x)/(1+x))^(1/2)+RootOf(81*_Z^4-9*_Z^2+1)*x)/x)-9*RootOf(81*_Z 
^4-9*_Z^2+1)^2*ln(-54*RootOf(81*_Z^4-9*_Z^2+1)^2*(-(1-x)/(1+x))^(2/3)*x-54 
*RootOf(81*_Z^4-9*_Z^2+1)^2*(-(1-x)/(1+x))^(2/3)-3*(-(1-x)/(1+x))^(5/6)*x+ 
54*RootOf(81*_Z^4-9*_Z^2+1)^2*(-(1-x)/(1+x))^(1/3)*x-3*(-(1-x)/(1+x))^(5/6 
)+3*(-(1-x)/(1+x))^(2/3)*x+54*RootOf(81*_Z^4-9*_Z^2+1)^2*(-(1-x)/(1+x))^(1 
/3)+3*(-(1-x)/(1+x))^(2/3)+6*(-(1-x)/(1+x))^(1/2)*x+6*(-(1-x)/(1+x))^(1/2) 
-3*(-(1-x)/(1+x))^(1/3)*x-18*RootOf(81*_Z^4-9*_Z^2+1)^2-3*(-(1-x)/(1+x))^( 
1/3)-3*(-(1-x)/(1+x))^(1/6)*x-3*(-(1-x)/(1+x))^(1/6)+1)-3*RootOf(81*_Z^4-9 
*_Z^2+1)*ln((54*RootOf(81*_Z^4-9*_Z^2+1)^3*(-(1-x)/(1+x))^(2/3)*x+54*RootO 
f(81*_Z^4-9*_Z^2+1)^3*(-(1-x)/(1+x))^(2/3)-27*RootOf(81*_Z^4-9*_Z^2+1)^...
 

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 264, normalized size of antiderivative = 0.91 \[ \int \frac {e^{\frac {1}{3} \coth ^{-1}(x)}}{x} \, dx=-\sqrt {3} \arctan \left (\frac {2}{3} \, \sqrt {3} \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{6}} + \frac {1}{3} \, \sqrt {3}\right ) - \sqrt {3} \arctan \left (\frac {2}{3} \, \sqrt {3} \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{6}} - \frac {1}{3} \, \sqrt {3}\right ) - \frac {1}{2} \, \sqrt {3} \log \left (\sqrt {3} \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{6}} + \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{3}} + 1\right ) + \frac {1}{2} \, \sqrt {3} \log \left (-\sqrt {3} \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{6}} + \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{3}} + 1\right ) + \arctan \left (\sqrt {3} + 2 \, \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{6}}\right ) + \arctan \left (-\sqrt {3} + 2 \, \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{6}}\right ) + 2 \, \arctan \left (\left (\frac {x - 1}{x + 1}\right )^{\frac {1}{6}}\right ) + \frac {1}{2} \, \log \left (\left (\frac {x - 1}{x + 1}\right )^{\frac {1}{3}} + \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{6}} + 1\right ) - \frac {1}{2} \, \log \left (\left (\frac {x - 1}{x + 1}\right )^{\frac {1}{3}} - \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{6}} + 1\right ) + \log \left (\left (\frac {x - 1}{x + 1}\right )^{\frac {1}{6}} + 1\right ) - \log \left (\left (\frac {x - 1}{x + 1}\right )^{\frac {1}{6}} - 1\right ) \] Input:

integrate(1/((x-1)/(1+x))^(1/6)/x,x, algorithm="fricas")
 

Output:

-sqrt(3)*arctan(2/3*sqrt(3)*((x - 1)/(x + 1))^(1/6) + 1/3*sqrt(3)) - sqrt( 
3)*arctan(2/3*sqrt(3)*((x - 1)/(x + 1))^(1/6) - 1/3*sqrt(3)) - 1/2*sqrt(3) 
*log(sqrt(3)*((x - 1)/(x + 1))^(1/6) + ((x - 1)/(x + 1))^(1/3) + 1) + 1/2* 
sqrt(3)*log(-sqrt(3)*((x - 1)/(x + 1))^(1/6) + ((x - 1)/(x + 1))^(1/3) + 1 
) + arctan(sqrt(3) + 2*((x - 1)/(x + 1))^(1/6)) + arctan(-sqrt(3) + 2*((x 
- 1)/(x + 1))^(1/6)) + 2*arctan(((x - 1)/(x + 1))^(1/6)) + 1/2*log(((x - 1 
)/(x + 1))^(1/3) + ((x - 1)/(x + 1))^(1/6) + 1) - 1/2*log(((x - 1)/(x + 1) 
)^(1/3) - ((x - 1)/(x + 1))^(1/6) + 1) + log(((x - 1)/(x + 1))^(1/6) + 1) 
- log(((x - 1)/(x + 1))^(1/6) - 1)
 

Sympy [F]

\[ \int \frac {e^{\frac {1}{3} \coth ^{-1}(x)}}{x} \, dx=\int \frac {1}{x \sqrt [6]{\frac {x - 1}{x + 1}}}\, dx \] Input:

integrate(1/((x-1)/(1+x))**(1/6)/x,x)
 

Output:

Integral(1/(x*((x - 1)/(x + 1))**(1/6)), x)
 

Maxima [F]

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

integrate(1/((x-1)/(1+x))^(1/6)/x,x, algorithm="maxima")
 

Output:

integrate(1/(x*((x - 1)/(x + 1))^(1/6)), x)
 

Giac [A] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 261, normalized size of antiderivative = 0.90 \[ \int \frac {e^{\frac {1}{3} \coth ^{-1}(x)}}{x} \, dx=-\sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{6}} + 1\right )}\right ) - \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{6}} - 1\right )}\right ) - \frac {1}{2} \, \sqrt {3} \log \left (\sqrt {3} \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{6}} + \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{3}} + 1\right ) + \frac {1}{2} \, \sqrt {3} \log \left (-\sqrt {3} \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{6}} + \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{3}} + 1\right ) + \arctan \left (\sqrt {3} + 2 \, \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{6}}\right ) + \arctan \left (-\sqrt {3} + 2 \, \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{6}}\right ) + 2 \, \arctan \left (\left (\frac {x - 1}{x + 1}\right )^{\frac {1}{6}}\right ) + \frac {1}{2} \, \log \left (\left (\frac {x - 1}{x + 1}\right )^{\frac {1}{3}} + \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{6}} + 1\right ) - \frac {1}{2} \, \log \left (\left (\frac {x - 1}{x + 1}\right )^{\frac {1}{3}} - \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{6}} + 1\right ) + \log \left (\left (\frac {x - 1}{x + 1}\right )^{\frac {1}{6}} + 1\right ) - \log \left ({\left | \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{6}} - 1 \right |}\right ) \] Input:

integrate(1/((x-1)/(1+x))^(1/6)/x,x, algorithm="giac")
 

Output:

-sqrt(3)*arctan(1/3*sqrt(3)*(2*((x - 1)/(x + 1))^(1/6) + 1)) - sqrt(3)*arc 
tan(1/3*sqrt(3)*(2*((x - 1)/(x + 1))^(1/6) - 1)) - 1/2*sqrt(3)*log(sqrt(3) 
*((x - 1)/(x + 1))^(1/6) + ((x - 1)/(x + 1))^(1/3) + 1) + 1/2*sqrt(3)*log( 
-sqrt(3)*((x - 1)/(x + 1))^(1/6) + ((x - 1)/(x + 1))^(1/3) + 1) + arctan(s 
qrt(3) + 2*((x - 1)/(x + 1))^(1/6)) + arctan(-sqrt(3) + 2*((x - 1)/(x + 1) 
)^(1/6)) + 2*arctan(((x - 1)/(x + 1))^(1/6)) + 1/2*log(((x - 1)/(x + 1))^( 
1/3) + ((x - 1)/(x + 1))^(1/6) + 1) - 1/2*log(((x - 1)/(x + 1))^(1/3) - (( 
x - 1)/(x + 1))^(1/6) + 1) + log(((x - 1)/(x + 1))^(1/6) + 1) - log(abs((( 
x - 1)/(x + 1))^(1/6) - 1))
 

Mupad [B] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 167, normalized size of antiderivative = 0.58 \[ \int \frac {e^{\frac {1}{3} \coth ^{-1}(x)}}{x} \, dx=2\,\mathrm {atan}\left ({\left (\frac {x-1}{x+1}\right )}^{1/6}\right )-\mathrm {atan}\left ({\left (\frac {x-1}{x+1}\right )}^{1/6}\,1{}\mathrm {i}\right )\,2{}\mathrm {i}-\mathrm {atan}\left (\frac {{\left (\frac {x-1}{x+1}\right )}^{1/6}\,1486016741376{}\mathrm {i}}{-743008370688+\sqrt {3}\,743008370688{}\mathrm {i}}\right )\,\left (\sqrt {3}-\mathrm {i}\right )-\mathrm {atan}\left (\frac {{\left (\frac {x-1}{x+1}\right )}^{1/6}\,1486016741376{}\mathrm {i}}{743008370688+\sqrt {3}\,743008370688{}\mathrm {i}}\right )\,\left (\sqrt {3}+1{}\mathrm {i}\right )-\mathrm {atan}\left (\frac {1486016741376\,{\left (\frac {x-1}{x+1}\right )}^{1/6}}{-743008370688+\sqrt {3}\,743008370688{}\mathrm {i}}\right )\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )-\mathrm {atan}\left (\frac {1486016741376\,{\left (\frac {x-1}{x+1}\right )}^{1/6}}{743008370688+\sqrt {3}\,743008370688{}\mathrm {i}}\right )\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right ) \] Input:

int(1/(x*((x - 1)/(x + 1))^(1/6)),x)
 

Output:

2*atan(((x - 1)/(x + 1))^(1/6)) - atan(((x - 1)/(x + 1))^(1/6)*1i)*2i - at 
an((((x - 1)/(x + 1))^(1/6)*1486016741376i)/(3^(1/2)*743008370688i - 74300 
8370688))*(3^(1/2) - 1i) - atan((((x - 1)/(x + 1))^(1/6)*1486016741376i)/( 
3^(1/2)*743008370688i + 743008370688))*(3^(1/2) + 1i) - atan((148601674137 
6*((x - 1)/(x + 1))^(1/6))/(3^(1/2)*743008370688i - 743008370688))*(3^(1/2 
)*1i + 1) - atan((1486016741376*((x - 1)/(x + 1))^(1/6))/(3^(1/2)*74300837 
0688i + 743008370688))*(3^(1/2)*1i - 1)
 

Reduce [B] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.31 \[ \int \frac {e^{\frac {1}{3} \coth ^{-1}(x)}}{x} \, dx=\frac {\left (x +1\right )^{\frac {7}{6}} \left (4 \mathit {atan} \left (\left (x -1\right )^{\frac {1}{6}}\right )+2 \mathit {atan} \left (2 \left (x -1\right )^{\frac {1}{6}}-\sqrt {3}\right )+2 \mathit {atan} \left (2 \left (x -1\right )^{\frac {1}{6}}+\sqrt {3}\right )+\sqrt {3}\, \mathrm {log}\left (-\left (x -1\right )^{\frac {1}{6}} \sqrt {3}+\left (x -1\right )^{\frac {1}{3}}+1\right )-\sqrt {3}\, \mathrm {log}\left (\left (x -1\right )^{\frac {1}{6}} \sqrt {3}+\left (x -1\right )^{\frac {1}{3}}+1\right )\right )}{2 x +2} \] Input:

int(1/((x-1)/(1+x))^(1/6)/x,x)
 

Output:

((x + 1)**(7/6)*(4*atan((x - 1)**(1/6)) + 2*atan(2*(x - 1)**(1/6) - sqrt(3 
)) + 2*atan(2*(x - 1)**(1/6) + sqrt(3)) + sqrt(3)*log( - (x - 1)**(1/6)*sq 
rt(3) + (x - 1)**(1/3) + 1) - sqrt(3)*log((x - 1)**(1/6)*sqrt(3) + (x - 1) 
**(1/3) + 1)))/(2*(x + 1))