Integrand size = 14, antiderivative size = 553 \[ \int \frac {e^{\frac {1}{4} \coth ^{-1}(a x)}}{x^3} \, dx=\frac {1}{8} a^2 \left (1-\frac {1}{a x}\right )^{7/8} \sqrt [8]{1+\frac {1}{a x}}+\frac {1}{2} a^2 \left (1-\frac {1}{a x}\right )^{7/8} \left (1+\frac {1}{a x}\right )^{9/8}-\frac {1}{32} \sqrt {2+\sqrt {2}} a^2 \arctan \left (\frac {\sqrt {2-\sqrt {2}}-\frac {2 \sqrt [8]{1-\frac {1}{a x}}}{\sqrt [8]{1+\frac {1}{a x}}}}{\sqrt {2+\sqrt {2}}}\right )-\frac {1}{32} \sqrt {2-\sqrt {2}} a^2 \arctan \left (\frac {\sqrt {2+\sqrt {2}}-\frac {2 \sqrt [8]{1-\frac {1}{a x}}}{\sqrt [8]{1+\frac {1}{a x}}}}{\sqrt {2-\sqrt {2}}}\right )+\frac {1}{32} \sqrt {2+\sqrt {2}} a^2 \arctan \left (\frac {\sqrt {2-\sqrt {2}}+\frac {2 \sqrt [8]{1-\frac {1}{a x}}}{\sqrt [8]{1+\frac {1}{a x}}}}{\sqrt {2+\sqrt {2}}}\right )+\frac {1}{32} \sqrt {2-\sqrt {2}} a^2 \arctan \left (\frac {\sqrt {2+\sqrt {2}}+\frac {2 \sqrt [8]{1-\frac {1}{a x}}}{\sqrt [8]{1+\frac {1}{a x}}}}{\sqrt {2-\sqrt {2}}}\right )-\frac {1}{32} \sqrt {2-\sqrt {2}} a^2 \text {arctanh}\left (\frac {\sqrt {2-\sqrt {2}} \sqrt [8]{1-\frac {1}{a x}}}{\left (1+\frac {\sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{1+\frac {1}{a x}}}\right ) \sqrt [8]{1+\frac {1}{a x}}}\right )-\frac {1}{32} \sqrt {2+\sqrt {2}} a^2 \text {arctanh}\left (\frac {\sqrt {2+\sqrt {2}} \sqrt [8]{1-\frac {1}{a x}}}{\left (1+\frac {\sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{1+\frac {1}{a x}}}\right ) \sqrt [8]{1+\frac {1}{a x}}}\right ) \] Output:
1/8*a^2*(1-1/a/x)^(7/8)*(1+1/a/x)^(1/8)+1/2*a^2*(1-1/a/x)^(7/8)*(1+1/a/x)^ (9/8)-1/32*(2+2^(1/2))^(1/2)*a^2*arctan(((2-2^(1/2))^(1/2)-2*(1-1/a/x)^(1/ 8)/(1+1/a/x)^(1/8))/(2+2^(1/2))^(1/2))-1/32*(2-2^(1/2))^(1/2)*a^2*arctan(( (2+2^(1/2))^(1/2)-2*(1-1/a/x)^(1/8)/(1+1/a/x)^(1/8))/(2-2^(1/2))^(1/2))+1/ 32*(2+2^(1/2))^(1/2)*a^2*arctan(((2-2^(1/2))^(1/2)+2*(1-1/a/x)^(1/8)/(1+1/ a/x)^(1/8))/(2+2^(1/2))^(1/2))+1/32*(2-2^(1/2))^(1/2)*a^2*arctan(((2+2^(1/ 2))^(1/2)+2*(1-1/a/x)^(1/8)/(1+1/a/x)^(1/8))/(2-2^(1/2))^(1/2))-1/32*(2-2^ (1/2))^(1/2)*a^2*arctanh((2-2^(1/2))^(1/2)*(1-1/a/x)^(1/8)/(1+(1-1/a/x)^(1 /4)/(1+1/a/x)^(1/4))/(1+1/a/x)^(1/8))-1/32*(2+2^(1/2))^(1/2)*a^2*arctanh(( 2+2^(1/2))^(1/2)*(1-1/a/x)^(1/8)/(1+(1-1/a/x)^(1/4)/(1+1/a/x)^(1/4))/(1+1/ a/x)^(1/8))
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 0.10 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.13 \[ \int \frac {e^{\frac {1}{4} \coth ^{-1}(a x)}}{x^3} \, dx=-\frac {a^2 e^{\frac {1}{4} \coth ^{-1}(a x)} \left (-1-9 e^{2 \coth ^{-1}(a x)}+\left (1+e^{2 \coth ^{-1}(a x)}\right )^2 \operatorname {Hypergeometric2F1}\left (\frac {1}{8},1,\frac {9}{8},-e^{2 \coth ^{-1}(a x)}\right )\right )}{4 \left (1+e^{2 \coth ^{-1}(a x)}\right )^2} \] Input:
Integrate[E^(ArcCoth[a*x]/4)/x^3,x]
Output:
-1/4*(a^2*E^(ArcCoth[a*x]/4)*(-1 - 9*E^(2*ArcCoth[a*x]) + (1 + E^(2*ArcCot h[a*x]))^2*Hypergeometric2F1[1/8, 1, 9/8, -E^(2*ArcCoth[a*x])]))/(1 + E^(2 *ArcCoth[a*x]))^2
Time = 1.44 (sec) , antiderivative size = 665, normalized size of antiderivative = 1.20, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.929, Rules used = {6721, 90, 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.
| \(\displaystyle \int \frac {e^{\frac {1}{4} \coth ^{-1}(a x)}}{x^3} \, dx\) |
| \(\Big \downarrow \) 6721 |
| \(\displaystyle -\int \frac {\sqrt [8]{1+\frac {1}{a x}}}{\sqrt [8]{1-\frac {1}{a x}} x}d\frac {1}{x}\) |
| \(\Big \downarrow \) 90 |
| \(\displaystyle \frac {1}{2} a^2 \left (1-\frac {1}{a x}\right )^{7/8} \left (\frac {1}{a x}+1\right )^{9/8}-\frac {1}{8} a \int \frac {\sqrt [8]{1+\frac {1}{a x}}}{\sqrt [8]{1-\frac {1}{a x}}}d\frac {1}{x}\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle \frac {1}{2} a^2 \left (1-\frac {1}{a x}\right )^{7/8} \left (\frac {1}{a x}+1\right )^{9/8}-\frac {1}{8} a \left (\frac {1}{4} \int \frac {1}{\sqrt [8]{1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{7/8}}d\frac {1}{x}-a \left (1-\frac {1}{a x}\right )^{7/8} \sqrt [8]{\frac {1}{a x}+1}\right )\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {1}{2} a^2 \left (1-\frac {1}{a x}\right )^{7/8} \left (\frac {1}{a x}+1\right )^{9/8}-\frac {1}{8} a \left (-2 a \int \frac {1}{\left (2-\frac {1}{x^8}\right )^{7/8} x^6}d\sqrt [8]{1-\frac {1}{a x}}-a \left (1-\frac {1}{a x}\right )^{7/8} \sqrt [8]{\frac {1}{a x}+1}\right )\) |
| \(\Big \downarrow \) 854 |
| \(\displaystyle \frac {1}{2} a^2 \left (1-\frac {1}{a x}\right )^{7/8} \left (\frac {1}{a x}+1\right )^{9/8}-\frac {1}{8} a \left (-2 a \int \frac {1}{\left (1+\frac {1}{x^8}\right ) x^6}d\frac {\sqrt [8]{1-\frac {1}{a x}}}{\sqrt [8]{2-\frac {1}{x^8}}}-a \left (1-\frac {1}{a x}\right )^{7/8} \sqrt [8]{\frac {1}{a x}+1}\right )\) |
| \(\Big \downarrow \) 828 |
| \(\displaystyle \frac {1}{2} a^2 \left (1-\frac {1}{a x}\right )^{7/8} \left (\frac {1}{a x}+1\right )^{9/8}-\frac {1}{8} a \left (-2 a \left (\frac {\int \frac {1}{\left (1-\frac {\sqrt {2}}{x^2}+\frac {1}{x^4}\right ) x^4}d\frac {\sqrt [8]{1-\frac {1}{a x}}}{\sqrt [8]{2-\frac {1}{x^8}}}}{2 \sqrt {2}}-\frac {\int \frac {1}{\left (1+\frac {\sqrt {2}}{x^2}+\frac {1}{x^4}\right ) x^4}d\frac {\sqrt [8]{1-\frac {1}{a x}}}{\sqrt [8]{2-\frac {1}{x^8}}}}{2 \sqrt {2}}\right )-a \left (1-\frac {1}{a x}\right )^{7/8} \sqrt [8]{\frac {1}{a x}+1}\right )\) |
| \(\Big \downarrow \) 1442 |
| \(\displaystyle \frac {1}{2} a^2 \left (1-\frac {1}{a x}\right )^{7/8} \left (\frac {1}{a x}+1\right )^{9/8}-\frac {1}{8} a \left (-2 a \left (\frac {\frac {\sqrt [8]{1-\frac {1}{a x}}}{\sqrt [8]{2-\frac {1}{x^8}}}-\int \frac {1-\frac {\sqrt {2}}{x^2}}{1-\frac {\sqrt {2}}{x^2}+\frac {1}{x^4}}d\frac {\sqrt [8]{1-\frac {1}{a x}}}{\sqrt [8]{2-\frac {1}{x^8}}}}{2 \sqrt {2}}-\frac {\frac {\sqrt [8]{1-\frac {1}{a x}}}{\sqrt [8]{2-\frac {1}{x^8}}}-\int \frac {1+\frac {\sqrt {2}}{x^2}}{1+\frac {\sqrt {2}}{x^2}+\frac {1}{x^4}}d\frac {\sqrt [8]{1-\frac {1}{a x}}}{\sqrt [8]{2-\frac {1}{x^8}}}}{2 \sqrt {2}}\right )-a \left (1-\frac {1}{a x}\right )^{7/8} \sqrt [8]{\frac {1}{a x}+1}\right )\) |
| \(\Big \downarrow \) 1483 |
| \(\displaystyle \frac {1}{2} a^2 \left (1-\frac {1}{a x}\right )^{7/8} \left (\frac {1}{a x}+1\right )^{9/8}-\frac {1}{8} a \left (-2 a \left (\frac {-\frac {\int \frac {\sqrt {2+\sqrt {2}}-\frac {\left (1+\sqrt {2}\right ) \sqrt [8]{1-\frac {1}{a x}}}{\sqrt [8]{2-\frac {1}{x^8}}}}{-\frac {\sqrt {2+\sqrt {2}} \sqrt [8]{1-\frac {1}{a x}}}{\sqrt [8]{2-\frac {1}{x^8}}}+\frac {1}{x^2}+1}d\frac {\sqrt [8]{1-\frac {1}{a x}}}{\sqrt [8]{2-\frac {1}{x^8}}}}{2 \sqrt {2+\sqrt {2}}}-\frac {\int \frac {\frac {\left (1+\sqrt {2}\right ) \sqrt [8]{1-\frac {1}{a x}}}{\sqrt [8]{2-\frac {1}{x^8}}}+\sqrt {2+\sqrt {2}}}{\frac {\sqrt {2+\sqrt {2}} \sqrt [8]{1-\frac {1}{a x}}}{\sqrt [8]{2-\frac {1}{x^8}}}+\frac {1}{x^2}+1}d\frac {\sqrt [8]{1-\frac {1}{a x}}}{\sqrt [8]{2-\frac {1}{x^8}}}}{2 \sqrt {2+\sqrt {2}}}+\frac {\sqrt [8]{1-\frac {1}{a x}}}{\sqrt [8]{2-\frac {1}{x^8}}}}{2 \sqrt {2}}-\frac {-\frac {\int \frac {\sqrt {2-\sqrt {2}}-\frac {\left (1-\sqrt {2}\right ) \sqrt [8]{1-\frac {1}{a x}}}{\sqrt [8]{2-\frac {1}{x^8}}}}{-\frac {\sqrt {2-\sqrt {2}} \sqrt [8]{1-\frac {1}{a x}}}{\sqrt [8]{2-\frac {1}{x^8}}}+\frac {1}{x^2}+1}d\frac {\sqrt [8]{1-\frac {1}{a x}}}{\sqrt [8]{2-\frac {1}{x^8}}}}{2 \sqrt {2-\sqrt {2}}}-\frac {\int \frac {\frac {\left (1-\sqrt {2}\right ) \sqrt [8]{1-\frac {1}{a x}}}{\sqrt [8]{2-\frac {1}{x^8}}}+\sqrt {2-\sqrt {2}}}{\frac {\sqrt {2-\sqrt {2}} \sqrt [8]{1-\frac {1}{a x}}}{\sqrt [8]{2-\frac {1}{x^8}}}+\frac {1}{x^2}+1}d\frac {\sqrt [8]{1-\frac {1}{a x}}}{\sqrt [8]{2-\frac {1}{x^8}}}}{2 \sqrt {2-\sqrt {2}}}+\frac {\sqrt [8]{1-\frac {1}{a x}}}{\sqrt [8]{2-\frac {1}{x^8}}}}{2 \sqrt {2}}\right )-a \left (1-\frac {1}{a x}\right )^{7/8} \sqrt [8]{\frac {1}{a x}+1}\right )\) |
| \(\Big \downarrow \) 1142 |
| \(\displaystyle \frac {1}{2} a^2 \left (1-\frac {1}{a x}\right )^{7/8} \left (1+\frac {1}{a x}\right )^{9/8}-\frac {1}{8} a \left (-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\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {1}{2} a^2 \left (1-\frac {1}{a x}\right )^{7/8} \left (1+\frac {1}{a x}\right )^{9/8}-\frac {1}{8} a \left (-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\right )\) |
| \(\Big \downarrow \) 1083 |
| \(\displaystyle \frac {1}{2} a^2 \left (1-\frac {1}{a x}\right )^{7/8} \left (1+\frac {1}{a x}\right )^{9/8}-\frac {1}{8} a \left (-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\right )\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {1}{2} a^2 \left (1-\frac {1}{a x}\right )^{7/8} \left (1+\frac {1}{a x}\right )^{9/8}-\frac {1}{8} a \left (-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\right )\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {1}{2} a^2 \left (1-\frac {1}{a x}\right )^{7/8} \left (\frac {1}{a x}+1\right )^{9/8}-\frac {1}{8} a \left (-2 a \left (\frac {-\frac {-\arctan \left (\frac {\frac {2 \sqrt [8]{1-\frac {1}{a x}}}{\sqrt [8]{2-\frac {1}{x^8}}}-\sqrt {2+\sqrt {2}}}{\sqrt {2-\sqrt {2}}}\right )-\frac {1}{2} \left (1+\sqrt {2}\right ) \log \left (-\frac {\sqrt {2+\sqrt {2}} \sqrt [8]{1-\frac {1}{a x}}}{\sqrt [8]{2-\frac {1}{x^8}}}+\frac {1}{x^2}+1\right )}{2 \sqrt {2+\sqrt {2}}}-\frac {\frac {1}{2} \left (1+\sqrt {2}\right ) \log \left (\frac {\sqrt {2+\sqrt {2}} \sqrt [8]{1-\frac {1}{a x}}}{\sqrt [8]{2-\frac {1}{x^8}}}+\frac {1}{x^2}+1\right )-\arctan \left (\frac {\frac {2 \sqrt [8]{1-\frac {1}{a x}}}{\sqrt [8]{2-\frac {1}{x^8}}}+\sqrt {2+\sqrt {2}}}{\sqrt {2-\sqrt {2}}}\right )}{2 \sqrt {2+\sqrt {2}}}+\frac {\sqrt [8]{1-\frac {1}{a x}}}{\sqrt [8]{2-\frac {1}{x^8}}}}{2 \sqrt {2}}-\frac {-\frac {\arctan \left (\frac {\frac {2 \sqrt [8]{1-\frac {1}{a x}}}{\sqrt [8]{2-\frac {1}{x^8}}}-\sqrt {2-\sqrt {2}}}{\sqrt {2+\sqrt {2}}}\right )-\frac {1}{2} \left (1-\sqrt {2}\right ) \log \left (-\frac {\sqrt {2-\sqrt {2}} \sqrt [8]{1-\frac {1}{a x}}}{\sqrt [8]{2-\frac {1}{x^8}}}+\frac {1}{x^2}+1\right )}{2 \sqrt {2-\sqrt {2}}}-\frac {\arctan \left (\frac {\frac {2 \sqrt [8]{1-\frac {1}{a x}}}{\sqrt [8]{2-\frac {1}{x^8}}}+\sqrt {2-\sqrt {2}}}{\sqrt {2+\sqrt {2}}}\right )+\frac {1}{2} \left (1-\sqrt {2}\right ) \log \left (\frac {\sqrt {2-\sqrt {2}} \sqrt [8]{1-\frac {1}{a x}}}{\sqrt [8]{2-\frac {1}{x^8}}}+\frac {1}{x^2}+1\right )}{2 \sqrt {2-\sqrt {2}}}+\frac {\sqrt [8]{1-\frac {1}{a x}}}{\sqrt [8]{2-\frac {1}{x^8}}}}{2 \sqrt {2}}\right )-a \left (1-\frac {1}{a x}\right )^{7/8} \sqrt [8]{\frac {1}{a x}+1}\right )\) |
Input:
Int[E^(ArcCoth[a*x]/4)/x^3,x]
Output:
(a^2*(1 - 1/(a*x))^(7/8)*(1 + 1/(a*x))^(9/8))/2 - (a*(-(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]]) - (A rcTan[(Sqrt[2 - Sqrt[2]] + (2*(1 - 1/(a*x))^(1/8))/(2 - x^(-8))^(1/8))/Sqr t[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
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_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p _.), x_] :> Simp[b*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d*f*(n + p + 2))), x] + Simp[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)))/(d*f*(n + p + 2)) Int[(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 2, 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[(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]
\[\int \frac {1}{\left (\frac {a x -1}{a x +1}\right )^{\frac {1}{8}} x^{3}}d x\]
Input:
int(1/((a*x-1)/(a*x+1))^(1/8)/x^3,x)
Output:
int(1/((a*x-1)/(a*x+1))^(1/8)/x^3,x)
Result contains complex when optimal does not.
Time = 0.10 (sec) , antiderivative size = 399, normalized size of antiderivative = 0.72 \[ \int \frac {e^{\frac {1}{4} \coth ^{-1}(a x)}}{x^3} \, dx=\frac {-\left (i - 1\right ) \, \sqrt {2} \left (-a^{16}\right )^{\frac {1}{8}} x^{2} \log \left (2 \, a^{14} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{8}} + \left (i + 1\right ) \, \sqrt {2} \left (-a^{16}\right )^{\frac {7}{8}}\right ) + \left (i + 1\right ) \, \sqrt {2} \left (-a^{16}\right )^{\frac {1}{8}} x^{2} \log \left (2 \, a^{14} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{8}} - \left (i - 1\right ) \, \sqrt {2} \left (-a^{16}\right )^{\frac {7}{8}}\right ) - \left (i + 1\right ) \, \sqrt {2} \left (-a^{16}\right )^{\frac {1}{8}} x^{2} \log \left (2 \, a^{14} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{8}} + \left (i - 1\right ) \, \sqrt {2} \left (-a^{16}\right )^{\frac {7}{8}}\right ) + \left (i - 1\right ) \, \sqrt {2} \left (-a^{16}\right )^{\frac {1}{8}} x^{2} \log \left (2 \, a^{14} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{8}} - \left (i + 1\right ) \, \sqrt {2} \left (-a^{16}\right )^{\frac {7}{8}}\right ) + 2 \, \left (-a^{16}\right )^{\frac {1}{8}} x^{2} \log \left (a^{14} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{8}} + \left (-a^{16}\right )^{\frac {7}{8}}\right ) - 2 i \, \left (-a^{16}\right )^{\frac {1}{8}} x^{2} \log \left (a^{14} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{8}} + i \, \left (-a^{16}\right )^{\frac {7}{8}}\right ) + 2 i \, \left (-a^{16}\right )^{\frac {1}{8}} x^{2} \log \left (a^{14} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{8}} - i \, \left (-a^{16}\right )^{\frac {7}{8}}\right ) - 2 \, \left (-a^{16}\right )^{\frac {1}{8}} x^{2} \log \left (a^{14} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{8}} - \left (-a^{16}\right )^{\frac {7}{8}}\right ) + 8 \, {\left (5 \, a^{2} x^{2} + 9 \, a x + 4\right )} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {7}{8}}}{64 \, x^{2}} \] Input:
integrate(1/((a*x-1)/(a*x+1))^(1/8)/x^3,x, algorithm="fricas")
Output:
1/64*(-(I - 1)*sqrt(2)*(-a^16)^(1/8)*x^2*log(2*a^14*((a*x - 1)/(a*x + 1))^ (1/8) + (I + 1)*sqrt(2)*(-a^16)^(7/8)) + (I + 1)*sqrt(2)*(-a^16)^(1/8)*x^2 *log(2*a^14*((a*x - 1)/(a*x + 1))^(1/8) - (I - 1)*sqrt(2)*(-a^16)^(7/8)) - (I + 1)*sqrt(2)*(-a^16)^(1/8)*x^2*log(2*a^14*((a*x - 1)/(a*x + 1))^(1/8) + (I - 1)*sqrt(2)*(-a^16)^(7/8)) + (I - 1)*sqrt(2)*(-a^16)^(1/8)*x^2*log(2 *a^14*((a*x - 1)/(a*x + 1))^(1/8) - (I + 1)*sqrt(2)*(-a^16)^(7/8)) + 2*(-a ^16)^(1/8)*x^2*log(a^14*((a*x - 1)/(a*x + 1))^(1/8) + (-a^16)^(7/8)) - 2*I *(-a^16)^(1/8)*x^2*log(a^14*((a*x - 1)/(a*x + 1))^(1/8) + I*(-a^16)^(7/8)) + 2*I*(-a^16)^(1/8)*x^2*log(a^14*((a*x - 1)/(a*x + 1))^(1/8) - I*(-a^16)^ (7/8)) - 2*(-a^16)^(1/8)*x^2*log(a^14*((a*x - 1)/(a*x + 1))^(1/8) - (-a^16 )^(7/8)) + 8*(5*a^2*x^2 + 9*a*x + 4)*((a*x - 1)/(a*x + 1))^(7/8))/x^2
\[ \int \frac {e^{\frac {1}{4} \coth ^{-1}(a x)}}{x^3} \, dx=\int \frac {1}{x^{3} \sqrt [8]{\frac {a x - 1}{a x + 1}}}\, dx \] Input:
integrate(1/((a*x-1)/(a*x+1))**(1/8)/x**3,x)
Output:
Integral(1/(x**3*((a*x - 1)/(a*x + 1))**(1/8)), x)
\[ \int \frac {e^{\frac {1}{4} \coth ^{-1}(a x)}}{x^3} \, dx=\int { \frac {1}{x^{3} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{8}}} \,d x } \] Input:
integrate(1/((a*x-1)/(a*x+1))^(1/8)/x^3,x, algorithm="maxima")
Output:
integrate(1/(x^3*((a*x - 1)/(a*x + 1))^(1/8)), x)
Time = 0.26 (sec) , antiderivative size = 461, normalized size of antiderivative = 0.83 \[ \int \frac {e^{\frac {1}{4} \coth ^{-1}(a x)}}{x^3} \, dx=\frac {1}{64} \, {\left (2 \, a \sqrt {-\sqrt {2} + 2} \arctan \left (\frac {\sqrt {\sqrt {2} + 2} + 2 \, \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{8}}}{\sqrt {-\sqrt {2} + 2}}\right ) + 2 \, a \sqrt {-\sqrt {2} + 2} \arctan \left (-\frac {\sqrt {\sqrt {2} + 2} - 2 \, \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{8}}}{\sqrt {-\sqrt {2} + 2}}\right ) + 2 \, a \sqrt {\sqrt {2} + 2} \arctan \left (\frac {\sqrt {-\sqrt {2} + 2} + 2 \, \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{8}}}{\sqrt {\sqrt {2} + 2}}\right ) + 2 \, a \sqrt {\sqrt {2} + 2} \arctan \left (-\frac {\sqrt {-\sqrt {2} + 2} - 2 \, \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{8}}}{\sqrt {\sqrt {2} + 2}}\right ) - a \sqrt {\sqrt {2} + 2} \log \left (\sqrt {\sqrt {2} + 2} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{8}} + \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}} + 1\right ) + a \sqrt {\sqrt {2} + 2} \log \left (-\sqrt {\sqrt {2} + 2} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{8}} + \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}} + 1\right ) - a \sqrt {-\sqrt {2} + 2} \log \left (\sqrt {-\sqrt {2} + 2} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{8}} + \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}} + 1\right ) + a \sqrt {-\sqrt {2} + 2} \log \left (-\sqrt {-\sqrt {2} + 2} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{8}} + \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}} + 1\right ) + \frac {16 \, {\left (a \left (\frac {a x - 1}{a x + 1}\right )^{\frac {15}{8}} + 9 \, a \left (\frac {a x - 1}{a x + 1}\right )^{\frac {7}{8}}\right )}}{{\left (\frac {a x - 1}{a x + 1} + 1\right )}^{2}}\right )} a \] Input:
integrate(1/((a*x-1)/(a*x+1))^(1/8)/x^3,x, algorithm="giac")
Output:
1/64*(2*a*sqrt(-sqrt(2) + 2)*arctan((sqrt(sqrt(2) + 2) + 2*((a*x - 1)/(a*x + 1))^(1/8))/sqrt(-sqrt(2) + 2)) + 2*a*sqrt(-sqrt(2) + 2)*arctan(-(sqrt(s qrt(2) + 2) - 2*((a*x - 1)/(a*x + 1))^(1/8))/sqrt(-sqrt(2) + 2)) + 2*a*sqr t(sqrt(2) + 2)*arctan((sqrt(-sqrt(2) + 2) + 2*((a*x - 1)/(a*x + 1))^(1/8)) /sqrt(sqrt(2) + 2)) + 2*a*sqrt(sqrt(2) + 2)*arctan(-(sqrt(-sqrt(2) + 2) - 2*((a*x - 1)/(a*x + 1))^(1/8))/sqrt(sqrt(2) + 2)) - a*sqrt(sqrt(2) + 2)*lo g(sqrt(sqrt(2) + 2)*((a*x - 1)/(a*x + 1))^(1/8) + ((a*x - 1)/(a*x + 1))^(1 /4) + 1) + a*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) - a*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) + a*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) + 16*(a*((a*x - 1)/(a*x + 1))^(15/ 8) + 9*a*((a*x - 1)/(a*x + 1))^(7/8))/((a*x - 1)/(a*x + 1) + 1)^2)*a
Time = 0.07 (sec) , antiderivative size = 210, normalized size of antiderivative = 0.38 \[ \int \frac {e^{\frac {1}{4} \coth ^{-1}(a x)}}{x^3} \, dx=\frac {\frac {9\,a^2\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{7/8}}{4}+\frac {a^2\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{15/8}}{4}}{\frac {{\left (a\,x-1\right )}^2}{{\left (a\,x+1\right )}^2}+\frac {2\,\left (a\,x-1\right )}{a\,x+1}+1}+\frac {{\left (-1\right )}^{1/8}\,a^2\,\mathrm {atan}\left ({\left (-1\right )}^{1/8}\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{1/8}\right )}{16}+\frac {{\left (-1\right )}^{1/8}\,a^2\,\mathrm {atan}\left ({\left (-1\right )}^{1/8}\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{1/8}\,1{}\mathrm {i}\right )\,1{}\mathrm {i}}{16}+{\left (-1\right )}^{1/8}\,\sqrt {2}\,a^2\,\mathrm {atan}\left ({\left (-1\right )}^{1/8}\,\sqrt {2}\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{1/8}\,\left (\frac {1}{2}-\frac {1}{2}{}\mathrm {i}\right )\right )\,\left (\frac {1}{32}-\frac {1}{32}{}\mathrm {i}\right )+{\left (-1\right )}^{1/8}\,\sqrt {2}\,a^2\,\mathrm {atan}\left ({\left (-1\right )}^{1/8}\,\sqrt {2}\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{1/8}\,\left (\frac {1}{2}+\frac {1}{2}{}\mathrm {i}\right )\right )\,\left (\frac {1}{32}+\frac {1}{32}{}\mathrm {i}\right ) \] Input:
int(1/(x^3*((a*x - 1)/(a*x + 1))^(1/8)),x)
Output:
((9*a^2*((a*x - 1)/(a*x + 1))^(7/8))/4 + (a^2*((a*x - 1)/(a*x + 1))^(15/8) )/4)/((a*x - 1)^2/(a*x + 1)^2 + (2*(a*x - 1))/(a*x + 1) + 1) + ((-1)^(1/8) *a^2*atan((-1)^(1/8)*((a*x - 1)/(a*x + 1))^(1/8)))/16 + ((-1)^(1/8)*a^2*at an((-1)^(1/8)*((a*x - 1)/(a*x + 1))^(1/8)*1i)*1i)/16 + (-1)^(1/8)*2^(1/2)* a^2*atan((-1)^(1/8)*2^(1/2)*((a*x - 1)/(a*x + 1))^(1/8)*(1/2 - 1i/2))*(1/3 2 - 1i/32) + (-1)^(1/8)*2^(1/2)*a^2*atan((-1)^(1/8)*2^(1/2)*((a*x - 1)/(a* x + 1))^(1/8)*(1/2 + 1i/2))*(1/32 + 1i/32)
\[ \int \frac {e^{\frac {1}{4} \coth ^{-1}(a x)}}{x^3} \, dx=\frac {-36 \left (a x +1\right )^{\frac {7}{8}} \left (a x -1\right )^{\frac {5}{8}} a^{4} x^{4}+16 \left (a x +1\right )^{\frac {7}{8}} \left (a x -1\right )^{\frac {5}{8}} a^{2} x^{2}-8 \left (a x +1\right )^{\frac {7}{8}} \left (a x -1\right )^{\frac {5}{8}} a x -8 \left (a x +1\right )^{\frac {7}{8}} \left (a x -1\right )^{\frac {5}{8}}+36 \left (a x +1\right )^{\frac {7}{8}} \left (a x -1\right )^{\frac {5}{8}} a^{3} x^{3}+36 \left (a x +1\right )^{\frac {3}{4}} \left (a x -1\right )^{\frac {3}{4}} \left (\int \frac {\left (a x +1\right )^{\frac {3}{4}} \sqrt {a x -1}\, x^{2}}{\left (a x +1\right )^{\frac {5}{8}} \left (a x -1\right )^{\frac {5}{8}} a x +\left (a x +1\right )^{\frac {5}{8}} \left (a x -1\right )^{\frac {5}{8}}}d x \right ) a^{5} x^{2}-9 \left (a x +1\right )^{\frac {3}{4}} \left (a x -1\right )^{\frac {3}{4}} \left (\int \frac {\left (a x +1\right )^{\frac {3}{4}} \sqrt {a x -1}\, x}{\left (a x +1\right )^{\frac {5}{8}} \left (a x -1\right )^{\frac {5}{8}} a x +\left (a x +1\right )^{\frac {5}{8}} \left (a x -1\right )^{\frac {5}{8}}}d x \right ) a^{4} x^{2}+26 \left (a x +1\right )^{\frac {3}{4}} \left (a x -1\right )^{\frac {3}{4}} \left (\int \frac {\left (a x +1\right )^{\frac {3}{4}} \sqrt {a x -1}}{\left (a x +1\right )^{\frac {5}{8}} \left (a x -1\right )^{\frac {5}{8}} a \,x^{2}+\left (a x +1\right )^{\frac {5}{8}} \left (a x -1\right )^{\frac {5}{8}} x}d x \right ) a^{2} x^{2}}{16 \left (a x +1\right )^{\frac {3}{4}} \left (a x -1\right )^{\frac {3}{4}} x^{2}} \] Input:
int(1/((a*x-1)/(a*x+1))^(1/8)/x^3,x)
Output:
( - 36*(a*x + 1)**(7/8)*(a*x - 1)**(5/8)*a**4*x**4 + 52*(a*x + 1)**(7/8)*( a*x - 1)**(5/8)*a**2*x**2 - 8*(a*x + 1)**(7/8)*(a*x - 1)**(5/8)*a*x - 8*(a *x + 1)**(7/8)*(a*x - 1)**(5/8) + 36*(a*x + 1)**(7/8)*(a*x - 1)**(5/8)*a** 3*x**3 - 36*(a*x + 1)**(7/8)*(a*x - 1)**(5/8)*a**2*x**2 + 36*(a*x + 1)**(3 /4)*(a*x - 1)**(3/4)*int(((a*x + 1)**(3/4)*sqrt(a*x - 1)*x**2)/((a*x + 1)* *(5/8)*(a*x - 1)**(5/8)*a*x + (a*x + 1)**(5/8)*(a*x - 1)**(5/8)),x)*a**5*x **2 - 9*(a*x + 1)**(3/4)*(a*x - 1)**(3/4)*int(((a*x + 1)**(3/4)*sqrt(a*x - 1)*x)/((a*x + 1)**(5/8)*(a*x - 1)**(5/8)*a*x + (a*x + 1)**(5/8)*(a*x - 1) **(5/8)),x)*a**4*x**2 + 26*(a*x + 1)**(3/4)*(a*x - 1)**(3/4)*int(((a*x + 1 )**(3/4)*sqrt(a*x - 1))/((a*x + 1)**(5/8)*(a*x - 1)**(5/8)*a*x**2 + (a*x + 1)**(5/8)*(a*x - 1)**(5/8)*x),x)*a**2*x**2)/(16*(a*x + 1)**(3/4)*(a*x - 1 )**(3/4)*x**2)