Integrand size = 14, antiderivative size = 919 \[ \int \frac {e^{\frac {1}{4} \coth ^{-1}(a x)}}{x} \, dx=-\sqrt {2+\sqrt {2}} \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 )-\sqrt {2-\sqrt {2}} \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 )+\sqrt {2+\sqrt {2}} \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 )+\sqrt {2-\sqrt {2}} \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 )-\sqrt {2} \arctan \left (1-\frac {\sqrt {2} \sqrt [8]{1+\frac {1}{a x}}}{\sqrt [8]{1-\frac {1}{a x}}}\right )+\sqrt {2} \arctan \left (1+\frac {\sqrt {2} \sqrt [8]{1+\frac {1}{a x}}}{\sqrt [8]{1-\frac {1}{a x}}}\right )+2 \arctan \left (\frac {\sqrt [8]{1+\frac {1}{a x}}}{\sqrt [8]{1-\frac {1}{a x}}}\right )+2 \text {arctanh}\left (\frac {\sqrt [8]{1+\frac {1}{a x}}}{\sqrt [8]{1-\frac {1}{a x}}}\right )+\frac {1}{2} \sqrt {2-\sqrt {2}} \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}{2} \sqrt {2-\sqrt {2}} \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}{2} \sqrt {2+\sqrt {2}} \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}{2} \sqrt {2+\sqrt {2}} \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 {\log \left (1-\frac {\sqrt {2} \sqrt [8]{1+\frac {1}{a x}}}{\sqrt [8]{1-\frac {1}{a x}}}+\frac {\sqrt [4]{1+\frac {1}{a x}}}{\sqrt [4]{1-\frac {1}{a x}}}\right )}{\sqrt {2}}+\frac {\log \left (1+\frac {\sqrt {2} \sqrt [8]{1+\frac {1}{a x}}}{\sqrt [8]{1-\frac {1}{a x}}}+\frac {\sqrt [4]{1+\frac {1}{a x}}}{\sqrt [4]{1-\frac {1}{a x}}}\right )}{\sqrt {2}} \]
2*arctan((1+1/a/x)^(1/8)/(1-1/a/x)^(1/8))+2*arctanh((1+1/a/x)^(1/8)/(1-1/a /x)^(1/8))-1/2*ln(1+(1+1/a/x)^(1/4)/(1-1/a/x)^(1/4)-(1+1/a/x)^(1/8)*2^(1/2 )/(1-1/a/x)^(1/8))*2^(1/2)+1/2*ln(1+(1+1/a/x)^(1/4)/(1-1/a/x)^(1/4)+(1+1/a /x)^(1/8)*2^(1/2)/(1-1/a/x)^(1/8))*2^(1/2)-arctan(1-(1+1/a/x)^(1/8)*2^(1/2 )/(1-1/a/x)^(1/8))*2^(1/2)+arctan(1+(1+1/a/x)^(1/8)*2^(1/2)/(1-1/a/x)^(1/8 ))*2^(1/2)-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)+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)+1/2*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)-1/2*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)-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)+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)+1/2*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)-1/2*l n(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 = 0.05 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.03 \[ \int \frac {e^{\frac {1}{4} \coth ^{-1}(a x)}}{x} \, dx=\frac {16}{9} e^{\frac {9}{4} \coth ^{-1}(a x)} \operatorname {Hypergeometric2F1}\left (\frac {9}{16},1,\frac {25}{16},e^{4 \coth ^{-1}(a x)}\right ) \]
Time = 1.17 (sec) , antiderivative size = 866, normalized size of antiderivative = 0.94, number of steps used = 26, number of rules used = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.786, Rules used = {6721, 140, 73, 104, 758, 755, 756, 216, 219, 854, 828, 1442, 1476, 1082, 217, 1479, 25, 27, 1103, 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.
| \(\displaystyle \int \frac {e^{\frac {1}{4} \coth ^{-1}(a x)}}{x} \, dx\) |
| \(\Big \downarrow \) 6721 |
| \(\displaystyle -\int \frac {\sqrt [8]{1+\frac {1}{a x}} x}{\sqrt [8]{1-\frac {1}{a x}}}d\frac {1}{x}\) |
| \(\Big \downarrow \) 140 |
| \(\displaystyle -\frac {\int \frac {1}{\sqrt [8]{1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{7/8}}d\frac {1}{x}}{a}-\int \frac {x}{\sqrt [8]{1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{7/8}}d\frac {1}{x}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle 8 \int \frac {1}{\left (2-\frac {1}{x^8}\right )^{7/8} x^6}d\sqrt [8]{1-\frac {1}{a x}}-\int \frac {x}{\sqrt [8]{1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{7/8}}d\frac {1}{x}\) |
| \(\Big \downarrow \) 104 |
| \(\displaystyle 8 \int \frac {1}{\left (2-\frac {1}{x^8}\right )^{7/8} x^6}d\sqrt [8]{1-\frac {1}{a x}}-8 \int \frac {1}{\frac {1}{x^8}-1}d\frac {\sqrt [8]{1+\frac {1}{a x}}}{\sqrt [8]{1-\frac {1}{a x}}}\) |
| \(\Big \downarrow \) 758 |
| \(\displaystyle 8 \int \frac {1}{\left (2-\frac {1}{x^8}\right )^{7/8} x^6}d\sqrt [8]{1-\frac {1}{a x}}-8 \left (-\frac {1}{2} \int \frac {1}{1-\frac {1}{x^4}}d\frac {\sqrt [8]{1+\frac {1}{a x}}}{\sqrt [8]{1-\frac {1}{a x}}}-\frac {1}{2} \int \frac {1}{1+\frac {1}{x^4}}d\frac {\sqrt [8]{1+\frac {1}{a x}}}{\sqrt [8]{1-\frac {1}{a x}}}\right )\) |
| \(\Big \downarrow \) 755 |
| \(\displaystyle 8 \int \frac {1}{\left (2-\frac {1}{x^8}\right )^{7/8} x^6}d\sqrt [8]{1-\frac {1}{a x}}-8 \left (\frac {1}{2} \left (-\frac {1}{2} \int \frac {1-\frac {1}{x^2}}{1+\frac {1}{x^4}}d\frac {\sqrt [8]{1+\frac {1}{a x}}}{\sqrt [8]{1-\frac {1}{a x}}}-\frac {1}{2} \int \frac {1+\frac {1}{x^2}}{1+\frac {1}{x^4}}d\frac {\sqrt [8]{1+\frac {1}{a x}}}{\sqrt [8]{1-\frac {1}{a x}}}\right )-\frac {1}{2} \int \frac {1}{1-\frac {1}{x^4}}d\frac {\sqrt [8]{1+\frac {1}{a x}}}{\sqrt [8]{1-\frac {1}{a x}}}\right )\) |
| \(\Big \downarrow \) 756 |
| \(\displaystyle 8 \int \frac {1}{\left (2-\frac {1}{x^8}\right )^{7/8} x^6}d\sqrt [8]{1-\frac {1}{a x}}-8 \left (\frac {1}{2} \left (-\frac {1}{2} \int \frac {1}{1-\frac {1}{x^2}}d\frac {\sqrt [8]{1+\frac {1}{a x}}}{\sqrt [8]{1-\frac {1}{a x}}}-\frac {1}{2} \int \frac {1}{1+\frac {1}{x^2}}d\frac {\sqrt [8]{1+\frac {1}{a x}}}{\sqrt [8]{1-\frac {1}{a x}}}\right )+\frac {1}{2} \left (-\frac {1}{2} \int \frac {1-\frac {1}{x^2}}{1+\frac {1}{x^4}}d\frac {\sqrt [8]{1+\frac {1}{a x}}}{\sqrt [8]{1-\frac {1}{a x}}}-\frac {1}{2} \int \frac {1+\frac {1}{x^2}}{1+\frac {1}{x^4}}d\frac {\sqrt [8]{1+\frac {1}{a x}}}{\sqrt [8]{1-\frac {1}{a x}}}\right )\right )\) |
| \(\Big \downarrow \) 216 |
| \(\displaystyle 8 \int \frac {1}{\left (2-\frac {1}{x^8}\right )^{7/8} x^6}d\sqrt [8]{1-\frac {1}{a x}}-8 \left (\frac {1}{2} \left (-\frac {1}{2} \int \frac {1}{1-\frac {1}{x^2}}d\frac {\sqrt [8]{1+\frac {1}{a x}}}{\sqrt [8]{1-\frac {1}{a x}}}-\frac {1}{2} \arctan \left (\frac {\sqrt [8]{\frac {1}{a x}+1}}{\sqrt [8]{1-\frac {1}{a x}}}\right )\right )+\frac {1}{2} \left (-\frac {1}{2} \int \frac {1-\frac {1}{x^2}}{1+\frac {1}{x^4}}d\frac {\sqrt [8]{1+\frac {1}{a x}}}{\sqrt [8]{1-\frac {1}{a x}}}-\frac {1}{2} \int \frac {1+\frac {1}{x^2}}{1+\frac {1}{x^4}}d\frac {\sqrt [8]{1+\frac {1}{a x}}}{\sqrt [8]{1-\frac {1}{a x}}}\right )\right )\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle 8 \int \frac {1}{\left (2-\frac {1}{x^8}\right )^{7/8} x^6}d\sqrt [8]{1-\frac {1}{a x}}-8 \left (\frac {1}{2} \left (-\frac {1}{2} \int \frac {1-\frac {1}{x^2}}{1+\frac {1}{x^4}}d\frac {\sqrt [8]{1+\frac {1}{a x}}}{\sqrt [8]{1-\frac {1}{a x}}}-\frac {1}{2} \int \frac {1+\frac {1}{x^2}}{1+\frac {1}{x^4}}d\frac {\sqrt [8]{1+\frac {1}{a x}}}{\sqrt [8]{1-\frac {1}{a x}}}\right )+\frac {1}{2} \left (-\frac {1}{2} \arctan \left (\frac {\sqrt [8]{\frac {1}{a x}+1}}{\sqrt [8]{1-\frac {1}{a x}}}\right )-\frac {1}{2} \text {arctanh}\left (\frac {\sqrt [8]{\frac {1}{a x}+1}}{\sqrt [8]{1-\frac {1}{a x}}}\right )\right )\right )\) |
| \(\Big \downarrow \) 854 |
| \(\displaystyle 8 \int \frac {1}{\left (1+\frac {1}{x^8}\right ) x^6}d\frac {\sqrt [8]{1-\frac {1}{a x}}}{\sqrt [8]{2-\frac {1}{x^8}}}-8 \left (\frac {1}{2} \left (-\frac {1}{2} \int \frac {1-\frac {1}{x^2}}{1+\frac {1}{x^4}}d\frac {\sqrt [8]{1+\frac {1}{a x}}}{\sqrt [8]{1-\frac {1}{a x}}}-\frac {1}{2} \int \frac {1+\frac {1}{x^2}}{1+\frac {1}{x^4}}d\frac {\sqrt [8]{1+\frac {1}{a x}}}{\sqrt [8]{1-\frac {1}{a x}}}\right )+\frac {1}{2} \left (-\frac {1}{2} \arctan \left (\frac {\sqrt [8]{\frac {1}{a x}+1}}{\sqrt [8]{1-\frac {1}{a x}}}\right )-\frac {1}{2} \text {arctanh}\left (\frac {\sqrt [8]{\frac {1}{a x}+1}}{\sqrt [8]{1-\frac {1}{a x}}}\right )\right )\right )\) |
| \(\Big \downarrow \) 828 |
| \(\displaystyle 8 \left (\frac {\int \frac {1}{\left (1-\frac {\sqrt {2}}{x^2}+\frac {1}{x^4}\right ) x^4}d\frac {\sqrt [8]{1-\frac {1}{a x}}}{\sqrt [8]{2-\frac {1}{x^8}}}}{2 \sqrt {2}}-\frac {\int \frac {1}{\left (1+\frac {\sqrt {2}}{x^2}+\frac {1}{x^4}\right ) x^4}d\frac {\sqrt [8]{1-\frac {1}{a x}}}{\sqrt [8]{2-\frac {1}{x^8}}}}{2 \sqrt {2}}\right )-8 \left (\frac {1}{2} \left (-\frac {1}{2} \int \frac {1-\frac {1}{x^2}}{1+\frac {1}{x^4}}d\frac {\sqrt [8]{1+\frac {1}{a x}}}{\sqrt [8]{1-\frac {1}{a x}}}-\frac {1}{2} \int \frac {1+\frac {1}{x^2}}{1+\frac {1}{x^4}}d\frac {\sqrt [8]{1+\frac {1}{a x}}}{\sqrt [8]{1-\frac {1}{a x}}}\right )+\frac {1}{2} \left (-\frac {1}{2} \arctan \left (\frac {\sqrt [8]{\frac {1}{a x}+1}}{\sqrt [8]{1-\frac {1}{a x}}}\right )-\frac {1}{2} \text {arctanh}\left (\frac {\sqrt [8]{\frac {1}{a x}+1}}{\sqrt [8]{1-\frac {1}{a x}}}\right )\right )\right )\) |
| \(\Big \downarrow \) 1442 |
| \(\displaystyle 8 \left (\frac {\frac {\sqrt [8]{1-\frac {1}{a x}}}{\sqrt [8]{2-\frac {1}{x^8}}}-\int \frac {1-\frac {\sqrt {2}}{x^2}}{1-\frac {\sqrt {2}}{x^2}+\frac {1}{x^4}}d\frac {\sqrt [8]{1-\frac {1}{a x}}}{\sqrt [8]{2-\frac {1}{x^8}}}}{2 \sqrt {2}}-\frac {\frac {\sqrt [8]{1-\frac {1}{a x}}}{\sqrt [8]{2-\frac {1}{x^8}}}-\int \frac {1+\frac {\sqrt {2}}{x^2}}{1+\frac {\sqrt {2}}{x^2}+\frac {1}{x^4}}d\frac {\sqrt [8]{1-\frac {1}{a x}}}{\sqrt [8]{2-\frac {1}{x^8}}}}{2 \sqrt {2}}\right )-8 \left (\frac {1}{2} \left (-\frac {1}{2} \int \frac {1-\frac {1}{x^2}}{1+\frac {1}{x^4}}d\frac {\sqrt [8]{1+\frac {1}{a x}}}{\sqrt [8]{1-\frac {1}{a x}}}-\frac {1}{2} \int \frac {1+\frac {1}{x^2}}{1+\frac {1}{x^4}}d\frac {\sqrt [8]{1+\frac {1}{a x}}}{\sqrt [8]{1-\frac {1}{a x}}}\right )+\frac {1}{2} \left (-\frac {1}{2} \arctan \left (\frac {\sqrt [8]{\frac {1}{a x}+1}}{\sqrt [8]{1-\frac {1}{a x}}}\right )-\frac {1}{2} \text {arctanh}\left (\frac {\sqrt [8]{\frac {1}{a x}+1}}{\sqrt [8]{1-\frac {1}{a x}}}\right )\right )\right )\) |
| \(\Big \downarrow \) 1476 |
| \(\displaystyle 8 \left (\frac {\frac {\sqrt [8]{1-\frac {1}{a x}}}{\sqrt [8]{2-\frac {1}{x^8}}}-\int \frac {1-\frac {\sqrt {2}}{x^2}}{1-\frac {\sqrt {2}}{x^2}+\frac {1}{x^4}}d\frac {\sqrt [8]{1-\frac {1}{a x}}}{\sqrt [8]{2-\frac {1}{x^8}}}}{2 \sqrt {2}}-\frac {\frac {\sqrt [8]{1-\frac {1}{a x}}}{\sqrt [8]{2-\frac {1}{x^8}}}-\int \frac {1+\frac {\sqrt {2}}{x^2}}{1+\frac {\sqrt {2}}{x^2}+\frac {1}{x^4}}d\frac {\sqrt [8]{1-\frac {1}{a x}}}{\sqrt [8]{2-\frac {1}{x^8}}}}{2 \sqrt {2}}\right )-8 \left (\frac {1}{2} \left (\frac {1}{2} \left (-\frac {1}{2} \int \frac {1}{-\frac {\sqrt {2} \sqrt [8]{1+\frac {1}{a x}}}{\sqrt [8]{1-\frac {1}{a x}}}+\frac {1}{x^2}+1}d\frac {\sqrt [8]{1+\frac {1}{a x}}}{\sqrt [8]{1-\frac {1}{a x}}}-\frac {1}{2} \int \frac {1}{\frac {\sqrt {2} \sqrt [8]{1+\frac {1}{a x}}}{\sqrt [8]{1-\frac {1}{a x}}}+\frac {1}{x^2}+1}d\frac {\sqrt [8]{1+\frac {1}{a x}}}{\sqrt [8]{1-\frac {1}{a x}}}\right )-\frac {1}{2} \int \frac {1-\frac {1}{x^2}}{1+\frac {1}{x^4}}d\frac {\sqrt [8]{1+\frac {1}{a x}}}{\sqrt [8]{1-\frac {1}{a x}}}\right )+\frac {1}{2} \left (-\frac {1}{2} \arctan \left (\frac {\sqrt [8]{\frac {1}{a x}+1}}{\sqrt [8]{1-\frac {1}{a x}}}\right )-\frac {1}{2} \text {arctanh}\left (\frac {\sqrt [8]{\frac {1}{a x}+1}}{\sqrt [8]{1-\frac {1}{a x}}}\right )\right )\right )\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle 8 \left (\frac {\frac {\sqrt [8]{1-\frac {1}{a x}}}{\sqrt [8]{2-\frac {1}{x^8}}}-\int \frac {1-\frac {\sqrt {2}}{x^2}}{1-\frac {\sqrt {2}}{x^2}+\frac {1}{x^4}}d\frac {\sqrt [8]{1-\frac {1}{a x}}}{\sqrt [8]{2-\frac {1}{x^8}}}}{2 \sqrt {2}}-\frac {\frac {\sqrt [8]{1-\frac {1}{a x}}}{\sqrt [8]{2-\frac {1}{x^8}}}-\int \frac {1+\frac {\sqrt {2}}{x^2}}{1+\frac {\sqrt {2}}{x^2}+\frac {1}{x^4}}d\frac {\sqrt [8]{1-\frac {1}{a x}}}{\sqrt [8]{2-\frac {1}{x^8}}}}{2 \sqrt {2}}\right )-8 \left (\frac {1}{2} \left (\frac {1}{2} \left (\frac {\int \frac {1}{-1-\frac {1}{x^2}}d\left (\frac {\sqrt {2} \sqrt [8]{1+\frac {1}{a x}}}{\sqrt [8]{1-\frac {1}{a x}}}+1\right )}{\sqrt {2}}-\frac {\int \frac {1}{-1-\frac {1}{x^2}}d\left (1-\frac {\sqrt {2} \sqrt [8]{1+\frac {1}{a x}}}{\sqrt [8]{1-\frac {1}{a x}}}\right )}{\sqrt {2}}\right )-\frac {1}{2} \int \frac {1-\frac {1}{x^2}}{1+\frac {1}{x^4}}d\frac {\sqrt [8]{1+\frac {1}{a x}}}{\sqrt [8]{1-\frac {1}{a x}}}\right )+\frac {1}{2} \left (-\frac {1}{2} \arctan \left (\frac {\sqrt [8]{\frac {1}{a x}+1}}{\sqrt [8]{1-\frac {1}{a x}}}\right )-\frac {1}{2} \text {arctanh}\left (\frac {\sqrt [8]{\frac {1}{a x}+1}}{\sqrt [8]{1-\frac {1}{a x}}}\right )\right )\right )\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle 8 \left (\frac {\frac {\sqrt [8]{1-\frac {1}{a x}}}{\sqrt [8]{2-\frac {1}{x^8}}}-\int \frac {1-\frac {\sqrt {2}}{x^2}}{1-\frac {\sqrt {2}}{x^2}+\frac {1}{x^4}}d\frac {\sqrt [8]{1-\frac {1}{a x}}}{\sqrt [8]{2-\frac {1}{x^8}}}}{2 \sqrt {2}}-\frac {\frac {\sqrt [8]{1-\frac {1}{a x}}}{\sqrt [8]{2-\frac {1}{x^8}}}-\int \frac {1+\frac {\sqrt {2}}{x^2}}{1+\frac {\sqrt {2}}{x^2}+\frac {1}{x^4}}d\frac {\sqrt [8]{1-\frac {1}{a x}}}{\sqrt [8]{2-\frac {1}{x^8}}}}{2 \sqrt {2}}\right )-8 \left (\frac {1}{2} \left (\frac {1}{2} \left (\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [8]{\frac {1}{a x}+1}}{\sqrt [8]{1-\frac {1}{a x}}}\right )}{\sqrt {2}}-\frac {\arctan \left (\frac {\sqrt {2} \sqrt [8]{\frac {1}{a x}+1}}{\sqrt [8]{1-\frac {1}{a x}}}+1\right )}{\sqrt {2}}\right )-\frac {1}{2} \int \frac {1-\frac {1}{x^2}}{1+\frac {1}{x^4}}d\frac {\sqrt [8]{1+\frac {1}{a x}}}{\sqrt [8]{1-\frac {1}{a x}}}\right )+\frac {1}{2} \left (-\frac {1}{2} \arctan \left (\frac {\sqrt [8]{\frac {1}{a x}+1}}{\sqrt [8]{1-\frac {1}{a x}}}\right )-\frac {1}{2} \text {arctanh}\left (\frac {\sqrt [8]{\frac {1}{a x}+1}}{\sqrt [8]{1-\frac {1}{a x}}}\right )\right )\right )\) |
| \(\Big \downarrow \) 1479 |
| \(\displaystyle 8 \left (\frac {\frac {\sqrt [8]{1-\frac {1}{a x}}}{\sqrt [8]{2-\frac {1}{x^8}}}-\int \frac {1-\frac {\sqrt {2}}{x^2}}{1-\frac {\sqrt {2}}{x^2}+\frac {1}{x^4}}d\frac {\sqrt [8]{1-\frac {1}{a x}}}{\sqrt [8]{2-\frac {1}{x^8}}}}{2 \sqrt {2}}-\frac {\frac {\sqrt [8]{1-\frac {1}{a x}}}{\sqrt [8]{2-\frac {1}{x^8}}}-\int \frac {1+\frac {\sqrt {2}}{x^2}}{1+\frac {\sqrt {2}}{x^2}+\frac {1}{x^4}}d\frac {\sqrt [8]{1-\frac {1}{a x}}}{\sqrt [8]{2-\frac {1}{x^8}}}}{2 \sqrt {2}}\right )-8 \left (\frac {1}{2} \left (\frac {1}{2} \left (\frac {\int -\frac {\sqrt {2}-\frac {2 \sqrt [8]{1+\frac {1}{a x}}}{\sqrt [8]{1-\frac {1}{a x}}}}{-\frac {\sqrt {2} \sqrt [8]{1+\frac {1}{a x}}}{\sqrt [8]{1-\frac {1}{a x}}}+\frac {1}{x^2}+1}d\frac {\sqrt [8]{1+\frac {1}{a x}}}{\sqrt [8]{1-\frac {1}{a x}}}}{2 \sqrt {2}}+\frac {\int -\frac {\sqrt {2} \left (\frac {\sqrt {2} \sqrt [8]{1+\frac {1}{a x}}}{\sqrt [8]{1-\frac {1}{a x}}}+1\right )}{\frac {\sqrt {2} \sqrt [8]{1+\frac {1}{a x}}}{\sqrt [8]{1-\frac {1}{a x}}}+\frac {1}{x^2}+1}d\frac {\sqrt [8]{1+\frac {1}{a x}}}{\sqrt [8]{1-\frac {1}{a x}}}}{2 \sqrt {2}}\right )+\frac {1}{2} \left (\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [8]{\frac {1}{a x}+1}}{\sqrt [8]{1-\frac {1}{a x}}}\right )}{\sqrt {2}}-\frac {\arctan \left (\frac {\sqrt {2} \sqrt [8]{\frac {1}{a x}+1}}{\sqrt [8]{1-\frac {1}{a x}}}+1\right )}{\sqrt {2}}\right )\right )+\frac {1}{2} \left (-\frac {1}{2} \arctan \left (\frac {\sqrt [8]{\frac {1}{a x}+1}}{\sqrt [8]{1-\frac {1}{a x}}}\right )-\frac {1}{2} \text {arctanh}\left (\frac {\sqrt [8]{\frac {1}{a x}+1}}{\sqrt [8]{1-\frac {1}{a x}}}\right )\right )\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle 8 \left (\frac {\frac {\sqrt [8]{1-\frac {1}{a x}}}{\sqrt [8]{2-\frac {1}{x^8}}}-\int \frac {1-\frac {\sqrt {2}}{x^2}}{1-\frac {\sqrt {2}}{x^2}+\frac {1}{x^4}}d\frac {\sqrt [8]{1-\frac {1}{a x}}}{\sqrt [8]{2-\frac {1}{x^8}}}}{2 \sqrt {2}}-\frac {\frac {\sqrt [8]{1-\frac {1}{a x}}}{\sqrt [8]{2-\frac {1}{x^8}}}-\int \frac {1+\frac {\sqrt {2}}{x^2}}{1+\frac {\sqrt {2}}{x^2}+\frac {1}{x^4}}d\frac {\sqrt [8]{1-\frac {1}{a x}}}{\sqrt [8]{2-\frac {1}{x^8}}}}{2 \sqrt {2}}\right )-8 \left (\frac {1}{2} \left (\frac {1}{2} \left (-\frac {\int \frac {\sqrt {2}-\frac {2 \sqrt [8]{1+\frac {1}{a x}}}{\sqrt [8]{1-\frac {1}{a x}}}}{-\frac {\sqrt {2} \sqrt [8]{1+\frac {1}{a x}}}{\sqrt [8]{1-\frac {1}{a x}}}+\frac {1}{x^2}+1}d\frac {\sqrt [8]{1+\frac {1}{a x}}}{\sqrt [8]{1-\frac {1}{a x}}}}{2 \sqrt {2}}-\frac {\int \frac {\sqrt {2} \left (\frac {\sqrt {2} \sqrt [8]{1+\frac {1}{a x}}}{\sqrt [8]{1-\frac {1}{a x}}}+1\right )}{\frac {\sqrt {2} \sqrt [8]{1+\frac {1}{a x}}}{\sqrt [8]{1-\frac {1}{a x}}}+\frac {1}{x^2}+1}d\frac {\sqrt [8]{1+\frac {1}{a x}}}{\sqrt [8]{1-\frac {1}{a x}}}}{2 \sqrt {2}}\right )+\frac {1}{2} \left (\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [8]{\frac {1}{a x}+1}}{\sqrt [8]{1-\frac {1}{a x}}}\right )}{\sqrt {2}}-\frac {\arctan \left (\frac {\sqrt {2} \sqrt [8]{\frac {1}{a x}+1}}{\sqrt [8]{1-\frac {1}{a x}}}+1\right )}{\sqrt {2}}\right )\right )+\frac {1}{2} \left (-\frac {1}{2} \arctan \left (\frac {\sqrt [8]{\frac {1}{a x}+1}}{\sqrt [8]{1-\frac {1}{a x}}}\right )-\frac {1}{2} \text {arctanh}\left (\frac {\sqrt [8]{\frac {1}{a x}+1}}{\sqrt [8]{1-\frac {1}{a x}}}\right )\right )\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 8 \left (\frac {\frac {\sqrt [8]{1-\frac {1}{a x}}}{\sqrt [8]{2-\frac {1}{x^8}}}-\int \frac {1-\frac {\sqrt {2}}{x^2}}{1-\frac {\sqrt {2}}{x^2}+\frac {1}{x^4}}d\frac {\sqrt [8]{1-\frac {1}{a x}}}{\sqrt [8]{2-\frac {1}{x^8}}}}{2 \sqrt {2}}-\frac {\frac {\sqrt [8]{1-\frac {1}{a x}}}{\sqrt [8]{2-\frac {1}{x^8}}}-\int \frac {1+\frac {\sqrt {2}}{x^2}}{1+\frac {\sqrt {2}}{x^2}+\frac {1}{x^4}}d\frac {\sqrt [8]{1-\frac {1}{a x}}}{\sqrt [8]{2-\frac {1}{x^8}}}}{2 \sqrt {2}}\right )-8 \left (\frac {1}{2} \left (\frac {1}{2} \left (-\frac {\int \frac {\sqrt {2}-\frac {2 \sqrt [8]{1+\frac {1}{a x}}}{\sqrt [8]{1-\frac {1}{a x}}}}{-\frac {\sqrt {2} \sqrt [8]{1+\frac {1}{a x}}}{\sqrt [8]{1-\frac {1}{a x}}}+\frac {1}{x^2}+1}d\frac {\sqrt [8]{1+\frac {1}{a x}}}{\sqrt [8]{1-\frac {1}{a x}}}}{2 \sqrt {2}}-\frac {1}{2} \int \frac {\frac {\sqrt {2} \sqrt [8]{1+\frac {1}{a x}}}{\sqrt [8]{1-\frac {1}{a x}}}+1}{\frac {\sqrt {2} \sqrt [8]{1+\frac {1}{a x}}}{\sqrt [8]{1-\frac {1}{a x}}}+\frac {1}{x^2}+1}d\frac {\sqrt [8]{1+\frac {1}{a x}}}{\sqrt [8]{1-\frac {1}{a x}}}\right )+\frac {1}{2} \left (\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [8]{\frac {1}{a x}+1}}{\sqrt [8]{1-\frac {1}{a x}}}\right )}{\sqrt {2}}-\frac {\arctan \left (\frac {\sqrt {2} \sqrt [8]{\frac {1}{a x}+1}}{\sqrt [8]{1-\frac {1}{a x}}}+1\right )}{\sqrt {2}}\right )\right )+\frac {1}{2} \left (-\frac {1}{2} \arctan \left (\frac {\sqrt [8]{\frac {1}{a x}+1}}{\sqrt [8]{1-\frac {1}{a x}}}\right )-\frac {1}{2} \text {arctanh}\left (\frac {\sqrt [8]{\frac {1}{a x}+1}}{\sqrt [8]{1-\frac {1}{a x}}}\right )\right )\right )\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle 8 \left (\frac {\frac {\sqrt [8]{1-\frac {1}{a x}}}{\sqrt [8]{2-\frac {1}{x^8}}}-\int \frac {1-\frac {\sqrt {2}}{x^2}}{1-\frac {\sqrt {2}}{x^2}+\frac {1}{x^4}}d\frac {\sqrt [8]{1-\frac {1}{a x}}}{\sqrt [8]{2-\frac {1}{x^8}}}}{2 \sqrt {2}}-\frac {\frac {\sqrt [8]{1-\frac {1}{a x}}}{\sqrt [8]{2-\frac {1}{x^8}}}-\int \frac {1+\frac {\sqrt {2}}{x^2}}{1+\frac {\sqrt {2}}{x^2}+\frac {1}{x^4}}d\frac {\sqrt [8]{1-\frac {1}{a x}}}{\sqrt [8]{2-\frac {1}{x^8}}}}{2 \sqrt {2}}\right )-8 \left (\frac {1}{2} \left (-\frac {1}{2} \arctan \left (\frac {\sqrt [8]{\frac {1}{a x}+1}}{\sqrt [8]{1-\frac {1}{a x}}}\right )-\frac {1}{2} \text {arctanh}\left (\frac {\sqrt [8]{\frac {1}{a x}+1}}{\sqrt [8]{1-\frac {1}{a x}}}\right )\right )+\frac {1}{2} \left (\frac {1}{2} \left (\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [8]{\frac {1}{a x}+1}}{\sqrt [8]{1-\frac {1}{a x}}}\right )}{\sqrt {2}}-\frac {\arctan \left (\frac {\sqrt {2} \sqrt [8]{\frac {1}{a x}+1}}{\sqrt [8]{1-\frac {1}{a x}}}+1\right )}{\sqrt {2}}\right )+\frac {1}{2} \left (\frac {\log \left (-\frac {\sqrt {2} \sqrt [8]{\frac {1}{a x}+1}}{\sqrt [8]{1-\frac {1}{a x}}}+\frac {1}{x^2}+1\right )}{2 \sqrt {2}}-\frac {\log \left (\frac {\sqrt {2} \sqrt [8]{\frac {1}{a x}+1}}{\sqrt [8]{1-\frac {1}{a x}}}+\frac {1}{x^2}+1\right )}{2 \sqrt {2}}\right )\right )\right )\) |
| \(\Big \downarrow \) 1483 |
| \(\displaystyle 8 \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 )-8 \left (\frac {1}{2} \left (-\frac {1}{2} \arctan \left (\frac {\sqrt [8]{1+\frac {1}{a x}}}{\sqrt [8]{1-\frac {1}{a x}}}\right )-\frac {1}{2} \text {arctanh}\left (\frac {\sqrt [8]{1+\frac {1}{a x}}}{\sqrt [8]{1-\frac {1}{a x}}}\right )\right )+\frac {1}{2} \left (\frac {1}{2} \left (\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [8]{1+\frac {1}{a x}}}{\sqrt [8]{1-\frac {1}{a x}}}\right )}{\sqrt {2}}-\frac {\arctan \left (\frac {\sqrt {2} \sqrt [8]{1+\frac {1}{a x}}}{\sqrt [8]{1-\frac {1}{a x}}}+1\right )}{\sqrt {2}}\right )+\frac {1}{2} \left (\frac {\log \left (-\frac {\sqrt {2} \sqrt [8]{1+\frac {1}{a x}}}{\sqrt [8]{1-\frac {1}{a x}}}+\frac {1}{x^2}+1\right )}{2 \sqrt {2}}-\frac {\log \left (\frac {\sqrt {2} \sqrt [8]{1+\frac {1}{a x}}}{\sqrt [8]{1-\frac {1}{a x}}}+\frac {1}{x^2}+1\right )}{2 \sqrt {2}}\right )\right )\right )\) |
| \(\Big \downarrow \) 1142 |
| \(\displaystyle 8 \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 )-8 \left (\frac {1}{2} \left (-\frac {1}{2} \arctan \left (\frac {\sqrt [8]{1+\frac {1}{a x}}}{\sqrt [8]{1-\frac {1}{a x}}}\right )-\frac {1}{2} \text {arctanh}\left (\frac {\sqrt [8]{1+\frac {1}{a x}}}{\sqrt [8]{1-\frac {1}{a x}}}\right )\right )+\frac {1}{2} \left (\frac {1}{2} \left (\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [8]{1+\frac {1}{a x}}}{\sqrt [8]{1-\frac {1}{a x}}}\right )}{\sqrt {2}}-\frac {\arctan \left (\frac {\sqrt {2} \sqrt [8]{1+\frac {1}{a x}}}{\sqrt [8]{1-\frac {1}{a x}}}+1\right )}{\sqrt {2}}\right )+\frac {1}{2} \left (\frac {\log \left (-\frac {\sqrt {2} \sqrt [8]{1+\frac {1}{a x}}}{\sqrt [8]{1-\frac {1}{a x}}}+\frac {1}{x^2}+1\right )}{2 \sqrt {2}}-\frac {\log \left (\frac {\sqrt {2} \sqrt [8]{1+\frac {1}{a x}}}{\sqrt [8]{1-\frac {1}{a x}}}+\frac {1}{x^2}+1\right )}{2 \sqrt {2}}\right )\right )\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle 8 \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 )-8 \left (\frac {1}{2} \left (-\frac {1}{2} \arctan \left (\frac {\sqrt [8]{1+\frac {1}{a x}}}{\sqrt [8]{1-\frac {1}{a x}}}\right )-\frac {1}{2} \text {arctanh}\left (\frac {\sqrt [8]{1+\frac {1}{a x}}}{\sqrt [8]{1-\frac {1}{a x}}}\right )\right )+\frac {1}{2} \left (\frac {1}{2} \left (\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [8]{1+\frac {1}{a x}}}{\sqrt [8]{1-\frac {1}{a x}}}\right )}{\sqrt {2}}-\frac {\arctan \left (\frac {\sqrt {2} \sqrt [8]{1+\frac {1}{a x}}}{\sqrt [8]{1-\frac {1}{a x}}}+1\right )}{\sqrt {2}}\right )+\frac {1}{2} \left (\frac {\log \left (-\frac {\sqrt {2} \sqrt [8]{1+\frac {1}{a x}}}{\sqrt [8]{1-\frac {1}{a x}}}+\frac {1}{x^2}+1\right )}{2 \sqrt {2}}-\frac {\log \left (\frac {\sqrt {2} \sqrt [8]{1+\frac {1}{a x}}}{\sqrt [8]{1-\frac {1}{a x}}}+\frac {1}{x^2}+1\right )}{2 \sqrt {2}}\right )\right )\right )\) |
| \(\Big \downarrow \) 1083 |
| \(\displaystyle 8 \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 )-8 \left (\frac {1}{2} \left (-\frac {1}{2} \arctan \left (\frac {\sqrt [8]{1+\frac {1}{a x}}}{\sqrt [8]{1-\frac {1}{a x}}}\right )-\frac {1}{2} \text {arctanh}\left (\frac {\sqrt [8]{1+\frac {1}{a x}}}{\sqrt [8]{1-\frac {1}{a x}}}\right )\right )+\frac {1}{2} \left (\frac {1}{2} \left (\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [8]{1+\frac {1}{a x}}}{\sqrt [8]{1-\frac {1}{a x}}}\right )}{\sqrt {2}}-\frac {\arctan \left (\frac {\sqrt {2} \sqrt [8]{1+\frac {1}{a x}}}{\sqrt [8]{1-\frac {1}{a x}}}+1\right )}{\sqrt {2}}\right )+\frac {1}{2} \left (\frac {\log \left (-\frac {\sqrt {2} \sqrt [8]{1+\frac {1}{a x}}}{\sqrt [8]{1-\frac {1}{a x}}}+\frac {1}{x^2}+1\right )}{2 \sqrt {2}}-\frac {\log \left (\frac {\sqrt {2} \sqrt [8]{1+\frac {1}{a x}}}{\sqrt [8]{1-\frac {1}{a x}}}+\frac {1}{x^2}+1\right )}{2 \sqrt {2}}\right )\right )\right )\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle 8 \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 )-8 \left (\frac {1}{2} \left (-\frac {1}{2} \arctan \left (\frac {\sqrt [8]{1+\frac {1}{a x}}}{\sqrt [8]{1-\frac {1}{a x}}}\right )-\frac {1}{2} \text {arctanh}\left (\frac {\sqrt [8]{1+\frac {1}{a x}}}{\sqrt [8]{1-\frac {1}{a x}}}\right )\right )+\frac {1}{2} \left (\frac {1}{2} \left (\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [8]{1+\frac {1}{a x}}}{\sqrt [8]{1-\frac {1}{a x}}}\right )}{\sqrt {2}}-\frac {\arctan \left (\frac {\sqrt {2} \sqrt [8]{1+\frac {1}{a x}}}{\sqrt [8]{1-\frac {1}{a x}}}+1\right )}{\sqrt {2}}\right )+\frac {1}{2} \left (\frac {\log \left (-\frac {\sqrt {2} \sqrt [8]{1+\frac {1}{a x}}}{\sqrt [8]{1-\frac {1}{a x}}}+\frac {1}{x^2}+1\right )}{2 \sqrt {2}}-\frac {\log \left (\frac {\sqrt {2} \sqrt [8]{1+\frac {1}{a x}}}{\sqrt [8]{1-\frac {1}{a x}}}+\frac {1}{x^2}+1\right )}{2 \sqrt {2}}\right )\right )\right )\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle 8 \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 )-8 \left (\frac {1}{2} \left (-\frac {1}{2} \arctan \left (\frac {\sqrt [8]{1+\frac {1}{a x}}}{\sqrt [8]{1-\frac {1}{a x}}}\right )-\frac {1}{2} \text {arctanh}\left (\frac {\sqrt [8]{1+\frac {1}{a x}}}{\sqrt [8]{1-\frac {1}{a x}}}\right )\right )+\frac {1}{2} \left (\frac {1}{2} \left (\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [8]{1+\frac {1}{a x}}}{\sqrt [8]{1-\frac {1}{a x}}}\right )}{\sqrt {2}}-\frac {\arctan \left (\frac {\sqrt {2} \sqrt [8]{1+\frac {1}{a x}}}{\sqrt [8]{1-\frac {1}{a x}}}+1\right )}{\sqrt {2}}\right )+\frac {1}{2} \left (\frac {\log \left (-\frac {\sqrt {2} \sqrt [8]{1+\frac {1}{a x}}}{\sqrt [8]{1-\frac {1}{a x}}}+\frac {1}{x^2}+1\right )}{2 \sqrt {2}}-\frac {\log \left (\frac {\sqrt {2} \sqrt [8]{1+\frac {1}{a x}}}{\sqrt [8]{1-\frac {1}{a x}}}+\frac {1}{x^2}+1\right )}{2 \sqrt {2}}\right )\right )\right )\) |
8*(-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 + 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]]))/(2 *Sqrt[2])) - 8*((-1/2*ArcTan[(1 + 1/(a*x))^(1/8)/(1 - 1/(a*x))^(1/8)] - Ar cTanh[(1 + 1/(a*x))^(1/8)/(1 - 1/(a*x))^(1/8)]/2)/2 + ((ArcTan[1 - (Sqrt[2 ]*(1 + 1/(a*x))^(1/8))/(1 - 1/(a*x))^(1/8)]/Sqrt[2] - ArcTan[1 + (Sqrt[2]* (1 + 1/(a*x))^(1/8))/(1 - 1/(a*x))^(1/8)]/Sqrt[2])/2 + (Log[1 - (Sqrt[2]*( 1 + 1/(a*x))^(1/8))/(1 - 1/(a*x))^(1/8) + x^(-2)]/(2*Sqrt[2]) - Log[1 + (S qrt[2]*(1 + 1/(a*x))^(1/8))/(1 - 1/(a*x))^(1/8) + x^(-2)]/(2*Sqrt[2]))/2)/ 2)
3.2.29.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, 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_))^(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]
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]))
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])
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[((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])
Int[((a_) + (b_.)*(x_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[a/b, 2] ], s = Denominator[Rt[a/b, 2]]}, Simp[1/(2*r) Int[(r - s*x^2)/(a + b*x^4) , x], x] + Simp[1/(2*r) Int[(r + s*x^2)/(a + b*x^4), x], x]] /; FreeQ[{a, b}, x] && (GtQ[a/b, 0] || (PosQ[a/b] && AtomQ[SplitProduct[SumBaseQ, a]] & & AtomQ[SplitProduct[SumBaseQ, b]]))
Int[((a_) + (b_.)*(x_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[-a/b, 2 ]], s = Denominator[Rt[-a/b, 2]]}, Simp[r/(2*a) Int[1/(r - s*x^2), x], x] + Simp[r/(2*a) Int[1/(r + s*x^2), x], x]] /; FreeQ[{a, b}, x] && !GtQ[a /b, 0]
Int[((a_) + (b_.)*(x_)^(n_))^(-1), x_Symbol] :> With[{r = Numerator[Rt[-a/b , 2]], s = Denominator[Rt[-a/b, 2]]}, Simp[r/(2*a) Int[1/(r - s*x^(n/2)), x], x] + Simp[r/(2*a) Int[1/(r + s*x^(n/2)), x], x]] /; FreeQ[{a, b}, x] && IGtQ[n/4, 1] && !GtQ[a/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] :> With[{q = 1 - 4*S implify[a*(c/b^2)]}, Simp[-2/b Subst[Int[1/(q - x^2), x], x, 1 + 2*c*(x/b )], x] /; RationalQ[q] && (EqQ[q^2, 1] || !RationalQ[b^2 - 4*a*c])] /; Fre eQ[{a, b, c}, x]
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_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[ 2*(d/e), 2]}, Simp[e/(2*c) Int[1/Simp[d/e + q*x + x^2, x], x], x] + Simp[ e/(2*c) Int[1/Simp[d/e - q*x + x^2, x], x], x]] /; FreeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && PosQ[d*e]
Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[ -2*(d/e), 2]}, Simp[e/(2*c*q) Int[(q - 2*x)/Simp[d/e + q*x - x^2, x], x], x] + Simp[e/(2*c*q) Int[(q + 2*x)/Simp[d/e - q*x - x^2, x], x], x]] /; F reeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && NegQ[d*e]
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]
\[\int \frac {1}{\left (\frac {a x -1}{a x +1}\right )^{\frac {1}{8}} x}d x\]
Result contains complex when optimal does not.
Time = 0.25 (sec) , antiderivative size = 418, normalized size of antiderivative = 0.45 \[ \int \frac {e^{\frac {1}{4} \coth ^{-1}(a x)}}{x} \, dx=-\left (\frac {1}{2} i - \frac {1}{2}\right ) \, \sqrt {2} \left (-1\right )^{\frac {1}{8}} \log \left (\left (i + 1\right ) \, \sqrt {2} \left (-1\right )^{\frac {7}{8}} + 2 \, \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{8}}\right ) + \left (\frac {1}{2} i + \frac {1}{2}\right ) \, \sqrt {2} \left (-1\right )^{\frac {1}{8}} \log \left (-\left (i - 1\right ) \, \sqrt {2} \left (-1\right )^{\frac {7}{8}} + 2 \, \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{8}}\right ) - \left (\frac {1}{2} i + \frac {1}{2}\right ) \, \sqrt {2} \left (-1\right )^{\frac {1}{8}} \log \left (\left (i - 1\right ) \, \sqrt {2} \left (-1\right )^{\frac {7}{8}} + 2 \, \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{8}}\right ) + \left (\frac {1}{2} i - \frac {1}{2}\right ) \, \sqrt {2} \left (-1\right )^{\frac {1}{8}} \log \left (-\left (i + 1\right ) \, \sqrt {2} \left (-1\right )^{\frac {7}{8}} + 2 \, \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{8}}\right ) - \left (\frac {1}{2} i - \frac {1}{2}\right ) \, \sqrt {2} \log \left (\left (i + 1\right ) \, \sqrt {2} + 2 \, \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{8}}\right ) + \left (\frac {1}{2} i + \frac {1}{2}\right ) \, \sqrt {2} \log \left (-\left (i - 1\right ) \, \sqrt {2} + 2 \, \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{8}}\right ) - \left (\frac {1}{2} i + \frac {1}{2}\right ) \, \sqrt {2} \log \left (\left (i - 1\right ) \, \sqrt {2} + 2 \, \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{8}}\right ) + \left (\frac {1}{2} i - \frac {1}{2}\right ) \, \sqrt {2} \log \left (-\left (i + 1\right ) \, \sqrt {2} + 2 \, \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{8}}\right ) + \left (-1\right )^{\frac {1}{8}} \log \left (\left (-1\right )^{\frac {7}{8}} + \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{8}}\right ) - i \, \left (-1\right )^{\frac {1}{8}} \log \left (i \, \left (-1\right )^{\frac {7}{8}} + \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{8}}\right ) + i \, \left (-1\right )^{\frac {1}{8}} \log \left (-i \, \left (-1\right )^{\frac {7}{8}} + \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{8}}\right ) - \left (-1\right )^{\frac {1}{8}} \log \left (-\left (-1\right )^{\frac {7}{8}} + \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{8}}\right ) - 2 \, \arctan \left (\left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{8}}\right ) + \log \left (\left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{8}} + 1\right ) - \log \left (\left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{8}} - 1\right ) \]
-(1/2*I - 1/2)*sqrt(2)*(-1)^(1/8)*log((I + 1)*sqrt(2)*(-1)^(7/8) + 2*((a*x - 1)/(a*x + 1))^(1/8)) + (1/2*I + 1/2)*sqrt(2)*(-1)^(1/8)*log(-(I - 1)*sq rt(2)*(-1)^(7/8) + 2*((a*x - 1)/(a*x + 1))^(1/8)) - (1/2*I + 1/2)*sqrt(2)* (-1)^(1/8)*log((I - 1)*sqrt(2)*(-1)^(7/8) + 2*((a*x - 1)/(a*x + 1))^(1/8)) + (1/2*I - 1/2)*sqrt(2)*(-1)^(1/8)*log(-(I + 1)*sqrt(2)*(-1)^(7/8) + 2*(( a*x - 1)/(a*x + 1))^(1/8)) - (1/2*I - 1/2)*sqrt(2)*log((I + 1)*sqrt(2) + 2 *((a*x - 1)/(a*x + 1))^(1/8)) + (1/2*I + 1/2)*sqrt(2)*log(-(I - 1)*sqrt(2) + 2*((a*x - 1)/(a*x + 1))^(1/8)) - (1/2*I + 1/2)*sqrt(2)*log((I - 1)*sqrt (2) + 2*((a*x - 1)/(a*x + 1))^(1/8)) + (1/2*I - 1/2)*sqrt(2)*log(-(I + 1)* sqrt(2) + 2*((a*x - 1)/(a*x + 1))^(1/8)) + (-1)^(1/8)*log((-1)^(7/8) + ((a *x - 1)/(a*x + 1))^(1/8)) - I*(-1)^(1/8)*log(I*(-1)^(7/8) + ((a*x - 1)/(a* x + 1))^(1/8)) + I*(-1)^(1/8)*log(-I*(-1)^(7/8) + ((a*x - 1)/(a*x + 1))^(1 /8)) - (-1)^(1/8)*log(-(-1)^(7/8) + ((a*x - 1)/(a*x + 1))^(1/8)) - 2*arcta n(((a*x - 1)/(a*x + 1))^(1/8)) + log(((a*x - 1)/(a*x + 1))^(1/8) + 1) - lo g(((a*x - 1)/(a*x + 1))^(1/8) - 1)
\[ \int \frac {e^{\frac {1}{4} \coth ^{-1}(a x)}}{x} \, dx=\int \frac {1}{x \sqrt [8]{\frac {a x - 1}{a x + 1}}}\, dx \]
\[ \int \frac {e^{\frac {1}{4} \coth ^{-1}(a x)}}{x} \, dx=\int { \frac {1}{x \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{8}}} \,d x } \]
Time = 1.04 (sec) , antiderivative size = 661, normalized size of antiderivative = 0.72 \[ \int \frac {e^{\frac {1}{4} \coth ^{-1}(a x)}}{x} \, dx=-\frac {1}{2} \, a {\left (\frac {2 \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + 2 \, \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{8}}\right )}\right )}{a} + \frac {2 \, \sqrt {2} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - 2 \, \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{8}}\right )}\right )}{a} - \frac {\sqrt {2} \log \left (\sqrt {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 )}{a} + \frac {\sqrt {2} \log \left (-\sqrt {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 )}{a} + \frac {4 \, \arctan \left (\left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{8}}\right )}{a} - \frac {2 \, \log \left (\left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{8}} + 1\right )}{a} + \frac {2 \, \log \left (-\left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{8}} + 1\right )}{a} - \frac {4 \, \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 )}{a \sqrt {2 \, \sqrt {2} + 4}} - \frac {4 \, \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 )}{a \sqrt {2 \, \sqrt {2} + 4}} - \frac {4 \, \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 )}{a \sqrt {-2 \, \sqrt {2} + 4}} - \frac {4 \, \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 )}{a \sqrt {-2 \, \sqrt {2} + 4}} + \frac {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 )}{a \sqrt {-2 \, \sqrt {2} + 4}} - \frac {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 )}{a \sqrt {-2 \, \sqrt {2} + 4}} + \frac {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 )}{a \sqrt {2 \, \sqrt {2} + 4}} - \frac {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 )}{a \sqrt {2 \, \sqrt {2} + 4}}\right )} \]
-1/2*a*(2*sqrt(2)*arctan(1/2*sqrt(2)*(sqrt(2) + 2*((a*x - 1)/(a*x + 1))^(1 /8)))/a + 2*sqrt(2)*arctan(-1/2*sqrt(2)*(sqrt(2) - 2*((a*x - 1)/(a*x + 1)) ^(1/8)))/a - sqrt(2)*log(sqrt(2)*((a*x - 1)/(a*x + 1))^(1/8) + ((a*x - 1)/ (a*x + 1))^(1/4) + 1)/a + sqrt(2)*log(-sqrt(2)*((a*x - 1)/(a*x + 1))^(1/8) + ((a*x - 1)/(a*x + 1))^(1/4) + 1)/a + 4*arctan(((a*x - 1)/(a*x + 1))^(1/ 8))/a - 2*log(((a*x - 1)/(a*x + 1))^(1/8) + 1)/a + 2*log(-((a*x - 1)/(a*x + 1))^(1/8) + 1)/a - 4*arctan((sqrt(sqrt(2) + 2) + 2*((a*x - 1)/(a*x + 1)) ^(1/8))/sqrt(-sqrt(2) + 2))/(a*sqrt(2*sqrt(2) + 4)) - 4*arctan(-(sqrt(sqrt (2) + 2) - 2*((a*x - 1)/(a*x + 1))^(1/8))/sqrt(-sqrt(2) + 2))/(a*sqrt(2*sq rt(2) + 4)) - 4*arctan((sqrt(-sqrt(2) + 2) + 2*((a*x - 1)/(a*x + 1))^(1/8) )/sqrt(sqrt(2) + 2))/(a*sqrt(-2*sqrt(2) + 4)) - 4*arctan(-(sqrt(-sqrt(2) + 2) - 2*((a*x - 1)/(a*x + 1))^(1/8))/sqrt(sqrt(2) + 2))/(a*sqrt(-2*sqrt(2) + 4)) + 2*log(sqrt(sqrt(2) + 2)*((a*x - 1)/(a*x + 1))^(1/8) + ((a*x - 1)/ (a*x + 1))^(1/4) + 1)/(a*sqrt(-2*sqrt(2) + 4)) - 2*log(-sqrt(sqrt(2) + 2)* ((a*x - 1)/(a*x + 1))^(1/8) + ((a*x - 1)/(a*x + 1))^(1/4) + 1)/(a*sqrt(-2* sqrt(2) + 4)) + 2*log(sqrt(-sqrt(2) + 2)*((a*x - 1)/(a*x + 1))^(1/8) + ((a *x - 1)/(a*x + 1))^(1/4) + 1)/(a*sqrt(2*sqrt(2) + 4)) - 2*log(-sqrt(-sqrt( 2) + 2)*((a*x - 1)/(a*x + 1))^(1/8) + ((a*x - 1)/(a*x + 1))^(1/4) + 1)/(a* sqrt(2*sqrt(2) + 4)))
Time = 4.28 (sec) , antiderivative size = 648, normalized size of antiderivative = 0.71 \[ \int \frac {e^{\frac {1}{4} \coth ^{-1}(a x)}}{x} \, dx=-\mathrm {atan}\left ({\left (\frac {a\,x-1}{a\,x+1}\right )}^{1/8}\,1{}\mathrm {i}\right )\,2{}\mathrm {i}-2\,\mathrm {atan}\left ({\left (\frac {a\,x-1}{a\,x+1}\right )}^{1/8}\right )+\sqrt {2}\,\mathrm {atan}\left (\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 (-1+1{}\mathrm {i}\right )+\sqrt {2}\,\mathrm {atan}\left (\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 (-1-\mathrm {i}\right )+\mathrm {atan}\left (-\frac {{\left (\frac {a\,x-1}{a\,x+1}\right )}^{1/8}\,1{}\mathrm {i}}{\sqrt {\sqrt {2}-2}}+\frac {{\left (\frac {a\,x-1}{a\,x+1}\right )}^{1/8}\,1{}\mathrm {i}}{\sqrt {\sqrt {2}+2}}+\frac {\sqrt {2}\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{1/8}\,1{}\mathrm {i}}{2\,\sqrt {\sqrt {2}-2}}+\frac {\sqrt {2}\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{1/8}\,1{}\mathrm {i}}{2\,\sqrt {\sqrt {2}+2}}\right )\,\left (\sqrt {\sqrt {2}-2}\,1{}\mathrm {i}+\sqrt {\sqrt {2}+2}\,1{}\mathrm {i}\right )-\mathrm {atan}\left (\frac {{\left (\frac {a\,x-1}{a\,x+1}\right )}^{1/8}\,1{}\mathrm {i}}{\sqrt {\sqrt {2}-2}}+\frac {{\left (\frac {a\,x-1}{a\,x+1}\right )}^{1/8}\,1{}\mathrm {i}}{\sqrt {\sqrt {2}+2}}-\frac {\sqrt {2}\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{1/8}\,1{}\mathrm {i}}{2\,\sqrt {\sqrt {2}-2}}+\frac {\sqrt {2}\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{1/8}\,1{}\mathrm {i}}{2\,\sqrt {\sqrt {2}+2}}\right )\,\left (\sqrt {\sqrt {2}-2}\,1{}\mathrm {i}-\sqrt {\sqrt {2}+2}\,1{}\mathrm {i}\right )-\mathrm {atan}\left (\frac {{\left (\frac {a\,x-1}{a\,x+1}\right )}^{1/8}\,1{}\mathrm {i}}{\sqrt {-\sqrt {2}-2}}-\frac {{\left (\frac {a\,x-1}{a\,x+1}\right )}^{1/8}\,1{}\mathrm {i}}{\sqrt {2-\sqrt {2}}}+\frac {\sqrt {2}\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{1/8}\,1{}\mathrm {i}}{2\,\sqrt {-\sqrt {2}-2}}+\frac {\sqrt {2}\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{1/8}\,1{}\mathrm {i}}{2\,\sqrt {2-\sqrt {2}}}\right )\,\left (\sqrt {-\sqrt {2}-2}\,1{}\mathrm {i}+\sqrt {2-\sqrt {2}}\,1{}\mathrm {i}\right )-\mathrm {atan}\left (\frac {{\left (\frac {a\,x-1}{a\,x+1}\right )}^{1/8}\,1{}\mathrm {i}}{\sqrt {-\sqrt {2}-2}}+\frac {{\left (\frac {a\,x-1}{a\,x+1}\right )}^{1/8}\,1{}\mathrm {i}}{\sqrt {2-\sqrt {2}}}+\frac {\sqrt {2}\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{1/8}\,1{}\mathrm {i}}{2\,\sqrt {-\sqrt {2}-2}}-\frac {\sqrt {2}\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{1/8}\,1{}\mathrm {i}}{2\,\sqrt {2-\sqrt {2}}}\right )\,\left (\sqrt {-\sqrt {2}-2}\,1{}\mathrm {i}-\sqrt {2-\sqrt {2}}\,1{}\mathrm {i}\right ) \]
atan((((a*x - 1)/(a*x + 1))^(1/8)*1i)/(2^(1/2) + 2)^(1/2) - (((a*x - 1)/(a *x + 1))^(1/8)*1i)/(2^(1/2) - 2)^(1/2) + (2^(1/2)*((a*x - 1)/(a*x + 1))^(1 /8)*1i)/(2*(2^(1/2) - 2)^(1/2)) + (2^(1/2)*((a*x - 1)/(a*x + 1))^(1/8)*1i) /(2*(2^(1/2) + 2)^(1/2)))*((2^(1/2) - 2)^(1/2)*1i + (2^(1/2) + 2)^(1/2)*1i ) - 2*atan(((a*x - 1)/(a*x + 1))^(1/8)) - 2^(1/2)*atan(2^(1/2)*((a*x - 1)/ (a*x + 1))^(1/8)*(1/2 - 1i/2))*(1 - 1i) - 2^(1/2)*atan(2^(1/2)*((a*x - 1)/ (a*x + 1))^(1/8)*(1/2 + 1i/2))*(1 + 1i) - atan(((a*x - 1)/(a*x + 1))^(1/8) *1i)*2i - atan((((a*x - 1)/(a*x + 1))^(1/8)*1i)/(2^(1/2) - 2)^(1/2) + (((a *x - 1)/(a*x + 1))^(1/8)*1i)/(2^(1/2) + 2)^(1/2) - (2^(1/2)*((a*x - 1)/(a* x + 1))^(1/8)*1i)/(2*(2^(1/2) - 2)^(1/2)) + (2^(1/2)*((a*x - 1)/(a*x + 1)) ^(1/8)*1i)/(2*(2^(1/2) + 2)^(1/2)))*((2^(1/2) - 2)^(1/2)*1i - (2^(1/2) + 2 )^(1/2)*1i) - atan((((a*x - 1)/(a*x + 1))^(1/8)*1i)/(- 2^(1/2) - 2)^(1/2) - (((a*x - 1)/(a*x + 1))^(1/8)*1i)/(2 - 2^(1/2))^(1/2) + (2^(1/2)*((a*x - 1)/(a*x + 1))^(1/8)*1i)/(2*(- 2^(1/2) - 2)^(1/2)) + (2^(1/2)*((a*x - 1)/(a *x + 1))^(1/8)*1i)/(2*(2 - 2^(1/2))^(1/2)))*((- 2^(1/2) - 2)^(1/2)*1i + (2 - 2^(1/2))^(1/2)*1i) - atan((((a*x - 1)/(a*x + 1))^(1/8)*1i)/(- 2^(1/2) - 2)^(1/2) + (((a*x - 1)/(a*x + 1))^(1/8)*1i)/(2 - 2^(1/2))^(1/2) + (2^(1/2 )*((a*x - 1)/(a*x + 1))^(1/8)*1i)/(2*(- 2^(1/2) - 2)^(1/2)) - (2^(1/2)*((a *x - 1)/(a*x + 1))^(1/8)*1i)/(2*(2 - 2^(1/2))^(1/2)))*((- 2^(1/2) - 2)^(1/ 2)*1i - (2 - 2^(1/2))^(1/2)*1i)