Integrand size = 14, antiderivative size = 676 \[ \int \frac {e^{\frac {1}{4} \coth ^{-1}(a x)}}{x^2} \, dx=a \left (1-\frac {1}{a x}\right )^{7/8} \sqrt [8]{1+\frac {1}{a x}}-\frac {1}{4} \sqrt {2+\sqrt {2}} a \arctan \left (\frac {\sqrt {2-\sqrt {2}}-\frac {2 \sqrt [8]{1-\frac {1}{a x}}}{\sqrt [8]{1+\frac {1}{a x}}}}{\sqrt {2+\sqrt {2}}}\right )-\frac {1}{4} \sqrt {2-\sqrt {2}} a \arctan \left (\frac {\sqrt {2+\sqrt {2}}-\frac {2 \sqrt [8]{1-\frac {1}{a x}}}{\sqrt [8]{1+\frac {1}{a x}}}}{\sqrt {2-\sqrt {2}}}\right )+\frac {1}{4} \sqrt {2+\sqrt {2}} a \arctan \left (\frac {\sqrt {2-\sqrt {2}}+\frac {2 \sqrt [8]{1-\frac {1}{a x}}}{\sqrt [8]{1+\frac {1}{a x}}}}{\sqrt {2+\sqrt {2}}}\right )+\frac {1}{4} \sqrt {2-\sqrt {2}} a \arctan \left (\frac {\sqrt {2+\sqrt {2}}+\frac {2 \sqrt [8]{1-\frac {1}{a x}}}{\sqrt [8]{1+\frac {1}{a x}}}}{\sqrt {2-\sqrt {2}}}\right )+\frac {1}{8} \sqrt {2-\sqrt {2}} a \log \left (1+\frac {\sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{1+\frac {1}{a x}}}-\frac {\sqrt {2-\sqrt {2}} \sqrt [8]{1-\frac {1}{a x}}}{\sqrt [8]{1+\frac {1}{a x}}}\right )-\frac {1}{8} \sqrt {2-\sqrt {2}} a \log \left (1+\frac {\sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{1+\frac {1}{a x}}}+\frac {\sqrt {2-\sqrt {2}} \sqrt [8]{1-\frac {1}{a x}}}{\sqrt [8]{1+\frac {1}{a x}}}\right )+\frac {1}{8} \sqrt {2+\sqrt {2}} a \log \left (1+\frac {\sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{1+\frac {1}{a x}}}-\frac {\sqrt {2+\sqrt {2}} \sqrt [8]{1-\frac {1}{a x}}}{\sqrt [8]{1+\frac {1}{a x}}}\right )-\frac {1}{8} \sqrt {2+\sqrt {2}} a \log \left (1+\frac {\sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{1+\frac {1}{a x}}}+\frac {\sqrt {2+\sqrt {2}} \sqrt [8]{1-\frac {1}{a x}}}{\sqrt [8]{1+\frac {1}{a x}}}\right ) \]
a*(1-1/a/x)^(7/8)*(1+1/a/x)^(1/8)-1/4*a*arctan((-2*(1-1/a/x)^(1/8)/(1+1/a/ x)^(1/8)+(2+2^(1/2))^(1/2))/(2-2^(1/2))^(1/2))*(2-2^(1/2))^(1/2)+1/4*a*arc tan((2*(1-1/a/x)^(1/8)/(1+1/a/x)^(1/8)+(2+2^(1/2))^(1/2))/(2-2^(1/2))^(1/2 ))*(2-2^(1/2))^(1/2)+1/8*a*ln(1+(1-1/a/x)^(1/4)/(1+1/a/x)^(1/4)-(1-1/a/x)^ (1/8)*(2-2^(1/2))^(1/2)/(1+1/a/x)^(1/8))*(2-2^(1/2))^(1/2)-1/8*a*ln(1+(1-1 /a/x)^(1/4)/(1+1/a/x)^(1/4)+(1-1/a/x)^(1/8)*(2-2^(1/2))^(1/2)/(1+1/a/x)^(1 /8))*(2-2^(1/2))^(1/2)-1/4*a*arctan((-2*(1-1/a/x)^(1/8)/(1+1/a/x)^(1/8)+(2 -2^(1/2))^(1/2))/(2+2^(1/2))^(1/2))*(2+2^(1/2))^(1/2)+1/4*a*arctan((2*(1-1 /a/x)^(1/8)/(1+1/a/x)^(1/8)+(2-2^(1/2))^(1/2))/(2+2^(1/2))^(1/2))*(2+2^(1/ 2))^(1/2)+1/8*a*ln(1+(1-1/a/x)^(1/4)/(1+1/a/x)^(1/4)-(1-1/a/x)^(1/8)*(2+2^ (1/2))^(1/2)/(1+1/a/x)^(1/8))*(2+2^(1/2))^(1/2)-1/8*a*ln(1+(1-1/a/x)^(1/4) /(1+1/a/x)^(1/4)+(1-1/a/x)^(1/8)*(2+2^(1/2))^(1/2)/(1+1/a/x)^(1/8))*(2+2^( 1/2))^(1/2)
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 0.07 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.07 \[ \int \frac {e^{\frac {1}{4} \coth ^{-1}(a x)}}{x^2} \, dx=-2 a e^{\frac {1}{4} \coth ^{-1}(a x)} \left (-\frac {1}{1+e^{2 \coth ^{-1}(a x)}}+\operatorname {Hypergeometric2F1}\left (\frac {1}{8},1,\frac {9}{8},-e^{2 \coth ^{-1}(a x)}\right )\right ) \]
-2*a*E^(ArcCoth[a*x]/4)*(-(1 + E^(2*ArcCoth[a*x]))^(-1) + Hypergeometric2F 1[1/8, 1, 9/8, -E^(2*ArcCoth[a*x])])
Time = 0.76 (sec) , antiderivative size = 624, normalized size of antiderivative = 0.92, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.857, Rules used = {6721, 60, 73, 854, 828, 1442, 1483, 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.
| \(\displaystyle \int \frac {e^{\frac {1}{4} \coth ^{-1}(a x)}}{x^2} \, dx\) |
| \(\Big \downarrow \) 6721 |
| \(\displaystyle -\int \frac {\sqrt [8]{1+\frac {1}{a x}}}{\sqrt [8]{1-\frac {1}{a x}}}d\frac {1}{x}\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle a \left (1-\frac {1}{a x}\right )^{7/8} \sqrt [8]{\frac {1}{a x}+1}-\frac {1}{4} \int \frac {1}{\sqrt [8]{1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{7/8}}d\frac {1}{x}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle 2 a \int \frac {1}{\left (2-\frac {1}{x^8}\right )^{7/8} x^6}d\sqrt [8]{1-\frac {1}{a x}}+a \left (1-\frac {1}{a x}\right )^{7/8} \sqrt [8]{\frac {1}{a x}+1}\) |
| \(\Big \downarrow \) 854 |
| \(\displaystyle 2 a \int \frac {1}{\left (1+\frac {1}{x^8}\right ) x^6}d\frac {\sqrt [8]{1-\frac {1}{a x}}}{\sqrt [8]{2-\frac {1}{x^8}}}+a \left (1-\frac {1}{a x}\right )^{7/8} \sqrt [8]{\frac {1}{a x}+1}\) |
| \(\Big \downarrow \) 828 |
| \(\displaystyle 2 a \left (\frac {\int \frac {1}{\left (1-\frac {\sqrt {2}}{x^2}+\frac {1}{x^4}\right ) x^4}d\frac {\sqrt [8]{1-\frac {1}{a x}}}{\sqrt [8]{2-\frac {1}{x^8}}}}{2 \sqrt {2}}-\frac {\int \frac {1}{\left (1+\frac {\sqrt {2}}{x^2}+\frac {1}{x^4}\right ) x^4}d\frac {\sqrt [8]{1-\frac {1}{a x}}}{\sqrt [8]{2-\frac {1}{x^8}}}}{2 \sqrt {2}}\right )+a \left (1-\frac {1}{a x}\right )^{7/8} \sqrt [8]{\frac {1}{a x}+1}\) |
| \(\Big \downarrow \) 1442 |
| \(\displaystyle 2 a \left (\frac {\frac {\sqrt [8]{1-\frac {1}{a x}}}{\sqrt [8]{2-\frac {1}{x^8}}}-\int \frac {1-\frac {\sqrt {2}}{x^2}}{1-\frac {\sqrt {2}}{x^2}+\frac {1}{x^4}}d\frac {\sqrt [8]{1-\frac {1}{a x}}}{\sqrt [8]{2-\frac {1}{x^8}}}}{2 \sqrt {2}}-\frac {\frac {\sqrt [8]{1-\frac {1}{a x}}}{\sqrt [8]{2-\frac {1}{x^8}}}-\int \frac {1+\frac {\sqrt {2}}{x^2}}{1+\frac {\sqrt {2}}{x^2}+\frac {1}{x^4}}d\frac {\sqrt [8]{1-\frac {1}{a x}}}{\sqrt [8]{2-\frac {1}{x^8}}}}{2 \sqrt {2}}\right )+a \left (1-\frac {1}{a x}\right )^{7/8} \sqrt [8]{\frac {1}{a x}+1}\) |
| \(\Big \downarrow \) 1483 |
| \(\displaystyle 2 a \left (\frac {-\frac {\int \frac {\sqrt {2+\sqrt {2}}-\frac {\left (1+\sqrt {2}\right ) \sqrt [8]{1-\frac {1}{a x}}}{\sqrt [8]{2-\frac {1}{x^8}}}}{-\frac {\sqrt {2+\sqrt {2}} \sqrt [8]{1-\frac {1}{a x}}}{\sqrt [8]{2-\frac {1}{x^8}}}+\frac {1}{x^2}+1}d\frac {\sqrt [8]{1-\frac {1}{a x}}}{\sqrt [8]{2-\frac {1}{x^8}}}}{2 \sqrt {2+\sqrt {2}}}-\frac {\int \frac {\frac {\left (1+\sqrt {2}\right ) \sqrt [8]{1-\frac {1}{a x}}}{\sqrt [8]{2-\frac {1}{x^8}}}+\sqrt {2+\sqrt {2}}}{\frac {\sqrt {2+\sqrt {2}} \sqrt [8]{1-\frac {1}{a x}}}{\sqrt [8]{2-\frac {1}{x^8}}}+\frac {1}{x^2}+1}d\frac {\sqrt [8]{1-\frac {1}{a x}}}{\sqrt [8]{2-\frac {1}{x^8}}}}{2 \sqrt {2+\sqrt {2}}}+\frac {\sqrt [8]{1-\frac {1}{a x}}}{\sqrt [8]{2-\frac {1}{x^8}}}}{2 \sqrt {2}}-\frac {-\frac {\int \frac {\sqrt {2-\sqrt {2}}-\frac {\left (1-\sqrt {2}\right ) \sqrt [8]{1-\frac {1}{a x}}}{\sqrt [8]{2-\frac {1}{x^8}}}}{-\frac {\sqrt {2-\sqrt {2}} \sqrt [8]{1-\frac {1}{a x}}}{\sqrt [8]{2-\frac {1}{x^8}}}+\frac {1}{x^2}+1}d\frac {\sqrt [8]{1-\frac {1}{a x}}}{\sqrt [8]{2-\frac {1}{x^8}}}}{2 \sqrt {2-\sqrt {2}}}-\frac {\int \frac {\frac {\left (1-\sqrt {2}\right ) \sqrt [8]{1-\frac {1}{a x}}}{\sqrt [8]{2-\frac {1}{x^8}}}+\sqrt {2-\sqrt {2}}}{\frac {\sqrt {2-\sqrt {2}} \sqrt [8]{1-\frac {1}{a x}}}{\sqrt [8]{2-\frac {1}{x^8}}}+\frac {1}{x^2}+1}d\frac {\sqrt [8]{1-\frac {1}{a x}}}{\sqrt [8]{2-\frac {1}{x^8}}}}{2 \sqrt {2-\sqrt {2}}}+\frac {\sqrt [8]{1-\frac {1}{a x}}}{\sqrt [8]{2-\frac {1}{x^8}}}}{2 \sqrt {2}}\right )+a \left (1-\frac {1}{a x}\right )^{7/8} \sqrt [8]{\frac {1}{a x}+1}\) |
| \(\Big \downarrow \) 1142 |
| \(\displaystyle 2 \left (\frac {-\frac {-\frac {1}{2} \sqrt {2-\sqrt {2}} \int \frac {1}{-\frac {\sqrt {2+\sqrt {2}} \sqrt [8]{1-\frac {1}{a x}}}{\sqrt [8]{2-\frac {1}{x^8}}}+\frac {1}{x^2}+1}d\frac {\sqrt [8]{1-\frac {1}{a x}}}{\sqrt [8]{2-\frac {1}{x^8}}}-\frac {1}{2} \left (1+\sqrt {2}\right ) \int -\frac {\sqrt {2+\sqrt {2}}-\frac {2 \sqrt [8]{1-\frac {1}{a x}}}{\sqrt [8]{2-\frac {1}{x^8}}}}{-\frac {\sqrt {2+\sqrt {2}} \sqrt [8]{1-\frac {1}{a x}}}{\sqrt [8]{2-\frac {1}{x^8}}}+\frac {1}{x^2}+1}d\frac {\sqrt [8]{1-\frac {1}{a x}}}{\sqrt [8]{2-\frac {1}{x^8}}}}{2 \sqrt {2+\sqrt {2}}}-\frac {\frac {1}{2} \left (1+\sqrt {2}\right ) \int \frac {\frac {2 \sqrt [8]{1-\frac {1}{a x}}}{\sqrt [8]{2-\frac {1}{x^8}}}+\sqrt {2+\sqrt {2}}}{\frac {\sqrt {2+\sqrt {2}} \sqrt [8]{1-\frac {1}{a x}}}{\sqrt [8]{2-\frac {1}{x^8}}}+\frac {1}{x^2}+1}d\frac {\sqrt [8]{1-\frac {1}{a x}}}{\sqrt [8]{2-\frac {1}{x^8}}}-\frac {1}{2} \sqrt {2-\sqrt {2}} \int \frac {1}{\frac {\sqrt {2+\sqrt {2}} \sqrt [8]{1-\frac {1}{a x}}}{\sqrt [8]{2-\frac {1}{x^8}}}+\frac {1}{x^2}+1}d\frac {\sqrt [8]{1-\frac {1}{a x}}}{\sqrt [8]{2-\frac {1}{x^8}}}}{2 \sqrt {2+\sqrt {2}}}+\frac {\sqrt [8]{1-\frac {1}{a x}}}{\sqrt [8]{2-\frac {1}{x^8}}}}{2 \sqrt {2}}-\frac {-\frac {\frac {1}{2} \sqrt {2+\sqrt {2}} \int \frac {1}{-\frac {\sqrt {2-\sqrt {2}} \sqrt [8]{1-\frac {1}{a x}}}{\sqrt [8]{2-\frac {1}{x^8}}}+\frac {1}{x^2}+1}d\frac {\sqrt [8]{1-\frac {1}{a x}}}{\sqrt [8]{2-\frac {1}{x^8}}}-\frac {1}{2} \left (1-\sqrt {2}\right ) \int -\frac {\sqrt {2-\sqrt {2}}-\frac {2 \sqrt [8]{1-\frac {1}{a x}}}{\sqrt [8]{2-\frac {1}{x^8}}}}{-\frac {\sqrt {2-\sqrt {2}} \sqrt [8]{1-\frac {1}{a x}}}{\sqrt [8]{2-\frac {1}{x^8}}}+\frac {1}{x^2}+1}d\frac {\sqrt [8]{1-\frac {1}{a x}}}{\sqrt [8]{2-\frac {1}{x^8}}}}{2 \sqrt {2-\sqrt {2}}}-\frac {\frac {1}{2} \sqrt {2+\sqrt {2}} \int \frac {1}{\frac {\sqrt {2-\sqrt {2}} \sqrt [8]{1-\frac {1}{a x}}}{\sqrt [8]{2-\frac {1}{x^8}}}+\frac {1}{x^2}+1}d\frac {\sqrt [8]{1-\frac {1}{a x}}}{\sqrt [8]{2-\frac {1}{x^8}}}+\frac {1}{2} \left (1-\sqrt {2}\right ) \int \frac {\frac {2 \sqrt [8]{1-\frac {1}{a x}}}{\sqrt [8]{2-\frac {1}{x^8}}}+\sqrt {2-\sqrt {2}}}{\frac {\sqrt {2-\sqrt {2}} \sqrt [8]{1-\frac {1}{a x}}}{\sqrt [8]{2-\frac {1}{x^8}}}+\frac {1}{x^2}+1}d\frac {\sqrt [8]{1-\frac {1}{a x}}}{\sqrt [8]{2-\frac {1}{x^8}}}}{2 \sqrt {2-\sqrt {2}}}+\frac {\sqrt [8]{1-\frac {1}{a x}}}{\sqrt [8]{2-\frac {1}{x^8}}}}{2 \sqrt {2}}\right ) a+\left (1-\frac {1}{a x}\right )^{7/8} \sqrt [8]{1+\frac {1}{a x}} a\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle 2 \left (\frac {-\frac {\frac {1}{2} \left (1+\sqrt {2}\right ) \int \frac {\sqrt {2+\sqrt {2}}-\frac {2 \sqrt [8]{1-\frac {1}{a x}}}{\sqrt [8]{2-\frac {1}{x^8}}}}{-\frac {\sqrt {2+\sqrt {2}} \sqrt [8]{1-\frac {1}{a x}}}{\sqrt [8]{2-\frac {1}{x^8}}}+\frac {1}{x^2}+1}d\frac {\sqrt [8]{1-\frac {1}{a x}}}{\sqrt [8]{2-\frac {1}{x^8}}}-\frac {1}{2} \sqrt {2-\sqrt {2}} \int \frac {1}{-\frac {\sqrt {2+\sqrt {2}} \sqrt [8]{1-\frac {1}{a x}}}{\sqrt [8]{2-\frac {1}{x^8}}}+\frac {1}{x^2}+1}d\frac {\sqrt [8]{1-\frac {1}{a x}}}{\sqrt [8]{2-\frac {1}{x^8}}}}{2 \sqrt {2+\sqrt {2}}}-\frac {\frac {1}{2} \left (1+\sqrt {2}\right ) \int \frac {\frac {2 \sqrt [8]{1-\frac {1}{a x}}}{\sqrt [8]{2-\frac {1}{x^8}}}+\sqrt {2+\sqrt {2}}}{\frac {\sqrt {2+\sqrt {2}} \sqrt [8]{1-\frac {1}{a x}}}{\sqrt [8]{2-\frac {1}{x^8}}}+\frac {1}{x^2}+1}d\frac {\sqrt [8]{1-\frac {1}{a x}}}{\sqrt [8]{2-\frac {1}{x^8}}}-\frac {1}{2} \sqrt {2-\sqrt {2}} \int \frac {1}{\frac {\sqrt {2+\sqrt {2}} \sqrt [8]{1-\frac {1}{a x}}}{\sqrt [8]{2-\frac {1}{x^8}}}+\frac {1}{x^2}+1}d\frac {\sqrt [8]{1-\frac {1}{a x}}}{\sqrt [8]{2-\frac {1}{x^8}}}}{2 \sqrt {2+\sqrt {2}}}+\frac {\sqrt [8]{1-\frac {1}{a x}}}{\sqrt [8]{2-\frac {1}{x^8}}}}{2 \sqrt {2}}-\frac {-\frac {\frac {1}{2} \sqrt {2+\sqrt {2}} \int \frac {1}{-\frac {\sqrt {2-\sqrt {2}} \sqrt [8]{1-\frac {1}{a x}}}{\sqrt [8]{2-\frac {1}{x^8}}}+\frac {1}{x^2}+1}d\frac {\sqrt [8]{1-\frac {1}{a x}}}{\sqrt [8]{2-\frac {1}{x^8}}}+\frac {1}{2} \left (1-\sqrt {2}\right ) \int \frac {\sqrt {2-\sqrt {2}}-\frac {2 \sqrt [8]{1-\frac {1}{a x}}}{\sqrt [8]{2-\frac {1}{x^8}}}}{-\frac {\sqrt {2-\sqrt {2}} \sqrt [8]{1-\frac {1}{a x}}}{\sqrt [8]{2-\frac {1}{x^8}}}+\frac {1}{x^2}+1}d\frac {\sqrt [8]{1-\frac {1}{a x}}}{\sqrt [8]{2-\frac {1}{x^8}}}}{2 \sqrt {2-\sqrt {2}}}-\frac {\frac {1}{2} \sqrt {2+\sqrt {2}} \int \frac {1}{\frac {\sqrt {2-\sqrt {2}} \sqrt [8]{1-\frac {1}{a x}}}{\sqrt [8]{2-\frac {1}{x^8}}}+\frac {1}{x^2}+1}d\frac {\sqrt [8]{1-\frac {1}{a x}}}{\sqrt [8]{2-\frac {1}{x^8}}}+\frac {1}{2} \left (1-\sqrt {2}\right ) \int \frac {\frac {2 \sqrt [8]{1-\frac {1}{a x}}}{\sqrt [8]{2-\frac {1}{x^8}}}+\sqrt {2-\sqrt {2}}}{\frac {\sqrt {2-\sqrt {2}} \sqrt [8]{1-\frac {1}{a x}}}{\sqrt [8]{2-\frac {1}{x^8}}}+\frac {1}{x^2}+1}d\frac {\sqrt [8]{1-\frac {1}{a x}}}{\sqrt [8]{2-\frac {1}{x^8}}}}{2 \sqrt {2-\sqrt {2}}}+\frac {\sqrt [8]{1-\frac {1}{a x}}}{\sqrt [8]{2-\frac {1}{x^8}}}}{2 \sqrt {2}}\right ) a+\left (1-\frac {1}{a x}\right )^{7/8} \sqrt [8]{1+\frac {1}{a x}} a\) |
| \(\Big \downarrow \) 1083 |
| \(\displaystyle 2 \left (\frac {-\frac {\sqrt {2-\sqrt {2}} \int \frac {1}{\sqrt {2}-2-\frac {1}{x^2}}d\left (\frac {2 \sqrt [8]{1-\frac {1}{a x}}}{\sqrt [8]{2-\frac {1}{x^8}}}-\sqrt {2+\sqrt {2}}\right )+\frac {1}{2} \left (1+\sqrt {2}\right ) \int \frac {\sqrt {2+\sqrt {2}}-\frac {2 \sqrt [8]{1-\frac {1}{a x}}}{\sqrt [8]{2-\frac {1}{x^8}}}}{-\frac {\sqrt {2+\sqrt {2}} \sqrt [8]{1-\frac {1}{a x}}}{\sqrt [8]{2-\frac {1}{x^8}}}+\frac {1}{x^2}+1}d\frac {\sqrt [8]{1-\frac {1}{a x}}}{\sqrt [8]{2-\frac {1}{x^8}}}}{2 \sqrt {2+\sqrt {2}}}-\frac {\sqrt {2-\sqrt {2}} \int \frac {1}{\sqrt {2}-2-\frac {1}{x^2}}d\left (\frac {2 \sqrt [8]{1-\frac {1}{a x}}}{\sqrt [8]{2-\frac {1}{x^8}}}+\sqrt {2+\sqrt {2}}\right )+\frac {1}{2} \left (1+\sqrt {2}\right ) \int \frac {\frac {2 \sqrt [8]{1-\frac {1}{a x}}}{\sqrt [8]{2-\frac {1}{x^8}}}+\sqrt {2+\sqrt {2}}}{\frac {\sqrt {2+\sqrt {2}} \sqrt [8]{1-\frac {1}{a x}}}{\sqrt [8]{2-\frac {1}{x^8}}}+\frac {1}{x^2}+1}d\frac {\sqrt [8]{1-\frac {1}{a x}}}{\sqrt [8]{2-\frac {1}{x^8}}}}{2 \sqrt {2+\sqrt {2}}}+\frac {\sqrt [8]{1-\frac {1}{a x}}}{\sqrt [8]{2-\frac {1}{x^8}}}}{2 \sqrt {2}}-\frac {-\frac {\frac {1}{2} \left (1-\sqrt {2}\right ) \int \frac {\sqrt {2-\sqrt {2}}-\frac {2 \sqrt [8]{1-\frac {1}{a x}}}{\sqrt [8]{2-\frac {1}{x^8}}}}{-\frac {\sqrt {2-\sqrt {2}} \sqrt [8]{1-\frac {1}{a x}}}{\sqrt [8]{2-\frac {1}{x^8}}}+\frac {1}{x^2}+1}d\frac {\sqrt [8]{1-\frac {1}{a x}}}{\sqrt [8]{2-\frac {1}{x^8}}}-\sqrt {2+\sqrt {2}} \int \frac {1}{-\sqrt {2}-2-\frac {1}{x^2}}d\left (\frac {2 \sqrt [8]{1-\frac {1}{a x}}}{\sqrt [8]{2-\frac {1}{x^8}}}-\sqrt {2-\sqrt {2}}\right )}{2 \sqrt {2-\sqrt {2}}}-\frac {\frac {1}{2} \left (1-\sqrt {2}\right ) \int \frac {\frac {2 \sqrt [8]{1-\frac {1}{a x}}}{\sqrt [8]{2-\frac {1}{x^8}}}+\sqrt {2-\sqrt {2}}}{\frac {\sqrt {2-\sqrt {2}} \sqrt [8]{1-\frac {1}{a x}}}{\sqrt [8]{2-\frac {1}{x^8}}}+\frac {1}{x^2}+1}d\frac {\sqrt [8]{1-\frac {1}{a x}}}{\sqrt [8]{2-\frac {1}{x^8}}}-\sqrt {2+\sqrt {2}} \int \frac {1}{-\sqrt {2}-2-\frac {1}{x^2}}d\left (\frac {2 \sqrt [8]{1-\frac {1}{a x}}}{\sqrt [8]{2-\frac {1}{x^8}}}+\sqrt {2-\sqrt {2}}\right )}{2 \sqrt {2-\sqrt {2}}}+\frac {\sqrt [8]{1-\frac {1}{a x}}}{\sqrt [8]{2-\frac {1}{x^8}}}}{2 \sqrt {2}}\right ) a+\left (1-\frac {1}{a x}\right )^{7/8} \sqrt [8]{1+\frac {1}{a x}} a\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle 2 \left (\frac {-\frac {\frac {1}{2} \left (1+\sqrt {2}\right ) \int \frac {\sqrt {2+\sqrt {2}}-\frac {2 \sqrt [8]{1-\frac {1}{a x}}}{\sqrt [8]{2-\frac {1}{x^8}}}}{-\frac {\sqrt {2+\sqrt {2}} \sqrt [8]{1-\frac {1}{a x}}}{\sqrt [8]{2-\frac {1}{x^8}}}+\frac {1}{x^2}+1}d\frac {\sqrt [8]{1-\frac {1}{a x}}}{\sqrt [8]{2-\frac {1}{x^8}}}-\arctan \left (\frac {\frac {2 \sqrt [8]{1-\frac {1}{a x}}}{\sqrt [8]{2-\frac {1}{x^8}}}-\sqrt {2+\sqrt {2}}}{\sqrt {2-\sqrt {2}}}\right )}{2 \sqrt {2+\sqrt {2}}}-\frac {\frac {1}{2} \left (1+\sqrt {2}\right ) \int \frac {\frac {2 \sqrt [8]{1-\frac {1}{a x}}}{\sqrt [8]{2-\frac {1}{x^8}}}+\sqrt {2+\sqrt {2}}}{\frac {\sqrt {2+\sqrt {2}} \sqrt [8]{1-\frac {1}{a x}}}{\sqrt [8]{2-\frac {1}{x^8}}}+\frac {1}{x^2}+1}d\frac {\sqrt [8]{1-\frac {1}{a x}}}{\sqrt [8]{2-\frac {1}{x^8}}}-\arctan \left (\frac {\frac {2 \sqrt [8]{1-\frac {1}{a x}}}{\sqrt [8]{2-\frac {1}{x^8}}}+\sqrt {2+\sqrt {2}}}{\sqrt {2-\sqrt {2}}}\right )}{2 \sqrt {2+\sqrt {2}}}+\frac {\sqrt [8]{1-\frac {1}{a x}}}{\sqrt [8]{2-\frac {1}{x^8}}}}{2 \sqrt {2}}-\frac {-\frac {\arctan \left (\frac {\frac {2 \sqrt [8]{1-\frac {1}{a x}}}{\sqrt [8]{2-\frac {1}{x^8}}}-\sqrt {2-\sqrt {2}}}{\sqrt {2+\sqrt {2}}}\right )+\frac {1}{2} \left (1-\sqrt {2}\right ) \int \frac {\sqrt {2-\sqrt {2}}-\frac {2 \sqrt [8]{1-\frac {1}{a x}}}{\sqrt [8]{2-\frac {1}{x^8}}}}{-\frac {\sqrt {2-\sqrt {2}} \sqrt [8]{1-\frac {1}{a x}}}{\sqrt [8]{2-\frac {1}{x^8}}}+\frac {1}{x^2}+1}d\frac {\sqrt [8]{1-\frac {1}{a x}}}{\sqrt [8]{2-\frac {1}{x^8}}}}{2 \sqrt {2-\sqrt {2}}}-\frac {\arctan \left (\frac {\frac {2 \sqrt [8]{1-\frac {1}{a x}}}{\sqrt [8]{2-\frac {1}{x^8}}}+\sqrt {2-\sqrt {2}}}{\sqrt {2+\sqrt {2}}}\right )+\frac {1}{2} \left (1-\sqrt {2}\right ) \int \frac {\frac {2 \sqrt [8]{1-\frac {1}{a x}}}{\sqrt [8]{2-\frac {1}{x^8}}}+\sqrt {2-\sqrt {2}}}{\frac {\sqrt {2-\sqrt {2}} \sqrt [8]{1-\frac {1}{a x}}}{\sqrt [8]{2-\frac {1}{x^8}}}+\frac {1}{x^2}+1}d\frac {\sqrt [8]{1-\frac {1}{a x}}}{\sqrt [8]{2-\frac {1}{x^8}}}}{2 \sqrt {2-\sqrt {2}}}+\frac {\sqrt [8]{1-\frac {1}{a x}}}{\sqrt [8]{2-\frac {1}{x^8}}}}{2 \sqrt {2}}\right ) a+\left (1-\frac {1}{a x}\right )^{7/8} \sqrt [8]{1+\frac {1}{a x}} a\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle 2 a \left (\frac {-\frac {-\arctan \left (\frac {\frac {2 \sqrt [8]{1-\frac {1}{a x}}}{\sqrt [8]{2-\frac {1}{x^8}}}-\sqrt {2+\sqrt {2}}}{\sqrt {2-\sqrt {2}}}\right )-\frac {1}{2} \left (1+\sqrt {2}\right ) \log \left (-\frac {\sqrt {2+\sqrt {2}} \sqrt [8]{1-\frac {1}{a x}}}{\sqrt [8]{2-\frac {1}{x^8}}}+\frac {1}{x^2}+1\right )}{2 \sqrt {2+\sqrt {2}}}-\frac {\frac {1}{2} \left (1+\sqrt {2}\right ) \log \left (\frac {\sqrt {2+\sqrt {2}} \sqrt [8]{1-\frac {1}{a x}}}{\sqrt [8]{2-\frac {1}{x^8}}}+\frac {1}{x^2}+1\right )-\arctan \left (\frac {\frac {2 \sqrt [8]{1-\frac {1}{a x}}}{\sqrt [8]{2-\frac {1}{x^8}}}+\sqrt {2+\sqrt {2}}}{\sqrt {2-\sqrt {2}}}\right )}{2 \sqrt {2+\sqrt {2}}}+\frac {\sqrt [8]{1-\frac {1}{a x}}}{\sqrt [8]{2-\frac {1}{x^8}}}}{2 \sqrt {2}}-\frac {-\frac {\arctan \left (\frac {\frac {2 \sqrt [8]{1-\frac {1}{a x}}}{\sqrt [8]{2-\frac {1}{x^8}}}-\sqrt {2-\sqrt {2}}}{\sqrt {2+\sqrt {2}}}\right )-\frac {1}{2} \left (1-\sqrt {2}\right ) \log \left (-\frac {\sqrt {2-\sqrt {2}} \sqrt [8]{1-\frac {1}{a x}}}{\sqrt [8]{2-\frac {1}{x^8}}}+\frac {1}{x^2}+1\right )}{2 \sqrt {2-\sqrt {2}}}-\frac {\arctan \left (\frac {\frac {2 \sqrt [8]{1-\frac {1}{a x}}}{\sqrt [8]{2-\frac {1}{x^8}}}+\sqrt {2-\sqrt {2}}}{\sqrt {2+\sqrt {2}}}\right )+\frac {1}{2} \left (1-\sqrt {2}\right ) \log \left (\frac {\sqrt {2-\sqrt {2}} \sqrt [8]{1-\frac {1}{a x}}}{\sqrt [8]{2-\frac {1}{x^8}}}+\frac {1}{x^2}+1\right )}{2 \sqrt {2-\sqrt {2}}}+\frac {\sqrt [8]{1-\frac {1}{a x}}}{\sqrt [8]{2-\frac {1}{x^8}}}}{2 \sqrt {2}}\right )+a \left (1-\frac {1}{a x}\right )^{7/8} \sqrt [8]{\frac {1}{a x}+1}\) |
a*(1 - 1/(a*x))^(7/8)*(1 + 1/(a*x))^(1/8) + 2*a*(-1/2*((1 - 1/(a*x))^(1/8) /(2 - x^(-8))^(1/8) - (ArcTan[(-Sqrt[2 - Sqrt[2]] + (2*(1 - 1/(a*x))^(1/8) )/(2 - x^(-8))^(1/8))/Sqrt[2 + Sqrt[2]]] - ((1 - Sqrt[2])*Log[1 - (Sqrt[2 - Sqrt[2]]*(1 - 1/(a*x))^(1/8))/(2 - x^(-8))^(1/8) + x^(-2)])/2)/(2*Sqrt[2 - Sqrt[2]]) - (ArcTan[(Sqrt[2 - Sqrt[2]] + (2*(1 - 1/(a*x))^(1/8))/(2 - x ^(-8))^(1/8))/Sqrt[2 + Sqrt[2]]] + ((1 - Sqrt[2])*Log[1 + (Sqrt[2 - Sqrt[2 ]]*(1 - 1/(a*x))^(1/8))/(2 - x^(-8))^(1/8) + x^(-2)])/2)/(2*Sqrt[2 - Sqrt[ 2]]))/Sqrt[2] + ((1 - 1/(a*x))^(1/8)/(2 - x^(-8))^(1/8) - (-ArcTan[(-Sqrt[ 2 + Sqrt[2]] + (2*(1 - 1/(a*x))^(1/8))/(2 - x^(-8))^(1/8))/Sqrt[2 - Sqrt[2 ]]] - ((1 + Sqrt[2])*Log[1 - (Sqrt[2 + Sqrt[2]]*(1 - 1/(a*x))^(1/8))/(2 - x^(-8))^(1/8) + x^(-2)])/2)/(2*Sqrt[2 + Sqrt[2]]) - (-ArcTan[(Sqrt[2 + Sqr t[2]] + (2*(1 - 1/(a*x))^(1/8))/(2 - x^(-8))^(1/8))/Sqrt[2 - Sqrt[2]]] + ( (1 + Sqrt[2])*Log[1 + (Sqrt[2 + Sqrt[2]]*(1 - 1/(a*x))^(1/8))/(2 - x^(-8)) ^(1/8) + x^(-2)])/2)/(2*Sqrt[2 + Sqrt[2]]))/(2*Sqrt[2]))
3.2.30.3.1 Defintions of rubi rules used
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ (a + b*x)^(m + 1)*((c + d*x)^n/(b*(m + n + 1))), x] + Simp[n*((b*c - a*d)/( b*(m + n + 1))) Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d}, x] && GtQ[n, 0] && NeQ[m + n + 1, 0] && !(IGtQ[m, 0] && ( !Integer Q[n] || (GtQ[m, 0] && LtQ[m - n, 0]))) && !ILtQ[m + n + 2, 0] && IntLinear Q[a, b, c, d, m, n, x]
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]
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])
Int[(x_)^(m_.)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{r = Numerator[R t[a/b, 4]], s = Denominator[Rt[a/b, 4]]}, Simp[s^3/(2*Sqrt[2]*b*r) Int[x^ (m - n/4)/(r^2 - Sqrt[2]*r*s*x^(n/4) + s^2*x^(n/2)), x], x] - Simp[s^3/(2*S qrt[2]*b*r) Int[x^(m - n/4)/(r^2 + Sqrt[2]*r*s*x^(n/4) + s^2*x^(n/2)), x] , x]] /; FreeQ[{a, b}, x] && IGtQ[n/4, 0] && IGtQ[m, 0] && LtQ[m, n - 1] && GtQ[a/b, 0]
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]
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]
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]
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]
Int[((d_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> Simp[d^3*(d*x)^(m - 3)*((a + b*x^2 + c*x^4)^(p + 1)/(c*(m + 4*p + 1))), x] - Simp[d^4/(c*(m + 4*p + 1)) Int[(d*x)^(m - 4)*Simp[a*(m - 3) + b*(m + 2*p - 1)*x^2, x]*(a + b*x^2 + c*x^4)^p, x], x] /; FreeQ[{a, b, c, d, p}, x ] && NeQ[b^2 - 4*a*c, 0] && GtQ[m, 3] && NeQ[m + 4*p + 1, 0] && IntegerQ[2* p] && (IntegerQ[p] || IntegerQ[m])
Int[((d_) + (e_.)*(x_)^2)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] : > With[{q = Rt[a/c, 2]}, With[{r = Rt[2*q - b/c, 2]}, Simp[1/(2*c*q*r) In t[(d*r - (d - e*q)*x)/(q - r*x + x^2), x], x] + Simp[1/(2*c*q*r) Int[(d*r + (d - e*q)*x)/(q + r*x + x^2), x], x]]] /; FreeQ[{a, b, c, d, e}, x] && N eQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && NegQ[b^2 - 4*a*c]
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]
\[\int \frac {1}{\left (\frac {a x -1}{a x +1}\right )^{\frac {1}{8}} x^{2}}d x\]
Result contains complex when optimal does not.
Time = 0.25 (sec) , antiderivative size = 374, normalized size of antiderivative = 0.55 \[ \int \frac {e^{\frac {1}{4} \coth ^{-1}(a x)}}{x^2} \, dx=\frac {-\left (i - 1\right ) \, \sqrt {2} \left (-a^{8}\right )^{\frac {1}{8}} x \log \left (2 \, a^{7} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{8}} + \left (i + 1\right ) \, \sqrt {2} \left (-a^{8}\right )^{\frac {7}{8}}\right ) + \left (i + 1\right ) \, \sqrt {2} \left (-a^{8}\right )^{\frac {1}{8}} x \log \left (2 \, a^{7} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{8}} - \left (i - 1\right ) \, \sqrt {2} \left (-a^{8}\right )^{\frac {7}{8}}\right ) - \left (i + 1\right ) \, \sqrt {2} \left (-a^{8}\right )^{\frac {1}{8}} x \log \left (2 \, a^{7} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{8}} + \left (i - 1\right ) \, \sqrt {2} \left (-a^{8}\right )^{\frac {7}{8}}\right ) + \left (i - 1\right ) \, \sqrt {2} \left (-a^{8}\right )^{\frac {1}{8}} x \log \left (2 \, a^{7} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{8}} - \left (i + 1\right ) \, \sqrt {2} \left (-a^{8}\right )^{\frac {7}{8}}\right ) + 2 \, \left (-a^{8}\right )^{\frac {1}{8}} x \log \left (a^{7} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{8}} + \left (-a^{8}\right )^{\frac {7}{8}}\right ) - 2 i \, \left (-a^{8}\right )^{\frac {1}{8}} x \log \left (a^{7} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{8}} + i \, \left (-a^{8}\right )^{\frac {7}{8}}\right ) + 2 i \, \left (-a^{8}\right )^{\frac {1}{8}} x \log \left (a^{7} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{8}} - i \, \left (-a^{8}\right )^{\frac {7}{8}}\right ) - 2 \, \left (-a^{8}\right )^{\frac {1}{8}} x \log \left (a^{7} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{8}} - \left (-a^{8}\right )^{\frac {7}{8}}\right ) + 8 \, {\left (a x + 1\right )} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {7}{8}}}{8 \, x} \]
1/8*(-(I - 1)*sqrt(2)*(-a^8)^(1/8)*x*log(2*a^7*((a*x - 1)/(a*x + 1))^(1/8) + (I + 1)*sqrt(2)*(-a^8)^(7/8)) + (I + 1)*sqrt(2)*(-a^8)^(1/8)*x*log(2*a^ 7*((a*x - 1)/(a*x + 1))^(1/8) - (I - 1)*sqrt(2)*(-a^8)^(7/8)) - (I + 1)*sq rt(2)*(-a^8)^(1/8)*x*log(2*a^7*((a*x - 1)/(a*x + 1))^(1/8) + (I - 1)*sqrt( 2)*(-a^8)^(7/8)) + (I - 1)*sqrt(2)*(-a^8)^(1/8)*x*log(2*a^7*((a*x - 1)/(a* x + 1))^(1/8) - (I + 1)*sqrt(2)*(-a^8)^(7/8)) + 2*(-a^8)^(1/8)*x*log(a^7*( (a*x - 1)/(a*x + 1))^(1/8) + (-a^8)^(7/8)) - 2*I*(-a^8)^(1/8)*x*log(a^7*(( a*x - 1)/(a*x + 1))^(1/8) + I*(-a^8)^(7/8)) + 2*I*(-a^8)^(1/8)*x*log(a^7*( (a*x - 1)/(a*x + 1))^(1/8) - I*(-a^8)^(7/8)) - 2*(-a^8)^(1/8)*x*log(a^7*(( a*x - 1)/(a*x + 1))^(1/8) - (-a^8)^(7/8)) + 8*(a*x + 1)*((a*x - 1)/(a*x + 1))^(7/8))/x
\[ \int \frac {e^{\frac {1}{4} \coth ^{-1}(a x)}}{x^2} \, dx=\int \frac {1}{x^{2} \sqrt [8]{\frac {a x - 1}{a x + 1}}}\, dx \]
\[ \int \frac {e^{\frac {1}{4} \coth ^{-1}(a x)}}{x^2} \, dx=\int { \frac {1}{x^{2} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{8}}} \,d x } \]
Time = 0.46 (sec) , antiderivative size = 432, normalized size of antiderivative = 0.64 \[ \int \frac {e^{\frac {1}{4} \coth ^{-1}(a x)}}{x^2} \, dx=\frac {1}{8} \, {\left (2 \, \sqrt {-\sqrt {2} + 2} \arctan \left (\frac {\sqrt {\sqrt {2} + 2} + 2 \, \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{8}}}{\sqrt {-\sqrt {2} + 2}}\right ) + 2 \, \sqrt {-\sqrt {2} + 2} \arctan \left (-\frac {\sqrt {\sqrt {2} + 2} - 2 \, \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{8}}}{\sqrt {-\sqrt {2} + 2}}\right ) + 2 \, \sqrt {\sqrt {2} + 2} \arctan \left (\frac {\sqrt {-\sqrt {2} + 2} + 2 \, \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{8}}}{\sqrt {\sqrt {2} + 2}}\right ) + 2 \, \sqrt {\sqrt {2} + 2} \arctan \left (-\frac {\sqrt {-\sqrt {2} + 2} - 2 \, \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{8}}}{\sqrt {\sqrt {2} + 2}}\right ) - \sqrt {\sqrt {2} + 2} \log \left (\sqrt {\sqrt {2} + 2} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{8}} + \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}} + 1\right ) + \sqrt {\sqrt {2} + 2} \log \left (-\sqrt {\sqrt {2} + 2} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{8}} + \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}} + 1\right ) - \sqrt {-\sqrt {2} + 2} \log \left (\sqrt {-\sqrt {2} + 2} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{8}} + \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}} + 1\right ) + \sqrt {-\sqrt {2} + 2} \log \left (-\sqrt {-\sqrt {2} + 2} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{8}} + \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}} + 1\right ) + \frac {16 \, \left (\frac {a x - 1}{a x + 1}\right )^{\frac {7}{8}}}{\frac {a x - 1}{a x + 1} + 1}\right )} a \]
1/8*(2*sqrt(-sqrt(2) + 2)*arctan((sqrt(sqrt(2) + 2) + 2*((a*x - 1)/(a*x + 1))^(1/8))/sqrt(-sqrt(2) + 2)) + 2*sqrt(-sqrt(2) + 2)*arctan(-(sqrt(sqrt(2 ) + 2) - 2*((a*x - 1)/(a*x + 1))^(1/8))/sqrt(-sqrt(2) + 2)) + 2*sqrt(sqrt( 2) + 2)*arctan((sqrt(-sqrt(2) + 2) + 2*((a*x - 1)/(a*x + 1))^(1/8))/sqrt(s qrt(2) + 2)) + 2*sqrt(sqrt(2) + 2)*arctan(-(sqrt(-sqrt(2) + 2) - 2*((a*x - 1)/(a*x + 1))^(1/8))/sqrt(sqrt(2) + 2)) - sqrt(sqrt(2) + 2)*log(sqrt(sqrt (2) + 2)*((a*x - 1)/(a*x + 1))^(1/8) + ((a*x - 1)/(a*x + 1))^(1/4) + 1) + sqrt(sqrt(2) + 2)*log(-sqrt(sqrt(2) + 2)*((a*x - 1)/(a*x + 1))^(1/8) + ((a *x - 1)/(a*x + 1))^(1/4) + 1) - sqrt(-sqrt(2) + 2)*log(sqrt(-sqrt(2) + 2)* ((a*x - 1)/(a*x + 1))^(1/8) + ((a*x - 1)/(a*x + 1))^(1/4) + 1) + sqrt(-sqr t(2) + 2)*log(-sqrt(-sqrt(2) + 2)*((a*x - 1)/(a*x + 1))^(1/8) + ((a*x - 1) /(a*x + 1))^(1/4) + 1) + 16*((a*x - 1)/(a*x + 1))^(7/8)/((a*x - 1)/(a*x + 1) + 1))*a
Time = 4.13 (sec) , antiderivative size = 162, normalized size of antiderivative = 0.24 \[ \int \frac {e^{\frac {1}{4} \coth ^{-1}(a x)}}{x^2} \, dx=\frac {{\left (-1\right )}^{1/8}\,a\,\mathrm {atan}\left ({\left (-1\right )}^{1/8}\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{1/8}\right )}{2}+\frac {{\left (-1\right )}^{1/8}\,a\,\mathrm {atan}\left ({\left (-1\right )}^{1/8}\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{1/8}\,1{}\mathrm {i}\right )\,1{}\mathrm {i}}{2}+\frac {2\,a\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{7/8}}{\frac {a\,x-1}{a\,x+1}+1}+{\left (-1\right )}^{1/8}\,\sqrt {2}\,a\,\mathrm {atan}\left ({\left (-1\right )}^{1/8}\,\sqrt {2}\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{1/8}\,\left (\frac {1}{2}-\frac {1}{2}{}\mathrm {i}\right )\right )\,\left (\frac {1}{4}-\frac {1}{4}{}\mathrm {i}\right )+{\left (-1\right )}^{1/8}\,\sqrt {2}\,a\,\mathrm {atan}\left ({\left (-1\right )}^{1/8}\,\sqrt {2}\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{1/8}\,\left (\frac {1}{2}+\frac {1}{2}{}\mathrm {i}\right )\right )\,\left (\frac {1}{4}+\frac {1}{4}{}\mathrm {i}\right ) \]
((-1)^(1/8)*a*atan((-1)^(1/8)*((a*x - 1)/(a*x + 1))^(1/8)))/2 + ((-1)^(1/8 )*a*atan((-1)^(1/8)*((a*x - 1)/(a*x + 1))^(1/8)*1i)*1i)/2 + (2*a*((a*x - 1 )/(a*x + 1))^(7/8))/((a*x - 1)/(a*x + 1) + 1) + (-1)^(1/8)*2^(1/2)*a*atan( (-1)^(1/8)*2^(1/2)*((a*x - 1)/(a*x + 1))^(1/8)*(1/2 - 1i/2))*(1/4 - 1i/4) + (-1)^(1/8)*2^(1/2)*a*atan((-1)^(1/8)*2^(1/2)*((a*x - 1)/(a*x + 1))^(1/8) *(1/2 + 1i/2))*(1/4 + 1i/4)