Integrand size = 22, antiderivative size = 327 \[ \int \frac {e^{-3 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^4} \, dx=-\frac {4}{3 a c^4 \left (1-\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{9/2}}-\frac {5}{a c^4 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{9/2}}+\frac {28 \sqrt {1-\frac {1}{a x}}}{9 a c^4 \left (1+\frac {1}{a x}\right )^{9/2}}+\frac {139 \sqrt {1-\frac {1}{a x}}}{63 a c^4 \left (1+\frac {1}{a x}\right )^{7/2}}+\frac {202 \sqrt {1-\frac {1}{a x}}}{105 a c^4 \left (1+\frac {1}{a x}\right )^{5/2}}+\frac {719 \sqrt {1-\frac {1}{a x}}}{315 a c^4 \left (1+\frac {1}{a x}\right )^{3/2}}+\frac {1664 \sqrt {1-\frac {1}{a x}}}{315 a c^4 \sqrt {1+\frac {1}{a x}}}+\frac {x}{c^4 \left (1-\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{9/2}}-\frac {3 \text {arctanh}\left (\sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}\right )}{a c^4} \]
-4/3/a/c^4/(1-1/a/x)^(3/2)/(1+1/a/x)^(9/2)+x/c^4/(1-1/a/x)^(3/2)/(1+1/a/x) ^(9/2)-3*arctanh((1-1/a/x)^(1/2)*(1+1/a/x)^(1/2))/a/c^4-5/a/c^4/(1+1/a/x)^ (9/2)/(1-1/a/x)^(1/2)+28/9*(1-1/a/x)^(1/2)/a/c^4/(1+1/a/x)^(9/2)+139/63*(1 -1/a/x)^(1/2)/a/c^4/(1+1/a/x)^(7/2)+202/105*(1-1/a/x)^(1/2)/a/c^4/(1+1/a/x )^(5/2)+719/315*(1-1/a/x)^(1/2)/a/c^4/(1+1/a/x)^(3/2)+1664/315*(1-1/a/x)^( 1/2)/a/c^4/(1+1/a/x)^(1/2)
Time = 0.53 (sec) , antiderivative size = 117, normalized size of antiderivative = 0.36 \[ \int \frac {e^{-3 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^4} \, dx=\frac {\frac {a \sqrt {1-\frac {1}{a^2 x^2}} x \left (1664+4047 a x-339 a^2 x^2-7399 a^3 x^3-4029 a^4 x^4+2967 a^5 x^5+2669 a^6 x^6+315 a^7 x^7\right )}{315 (-1+a x)^2 (1+a x)^5}-3 \log \left (\left (1+\sqrt {1-\frac {1}{a^2 x^2}}\right ) x\right )}{a c^4} \]
((a*Sqrt[1 - 1/(a^2*x^2)]*x*(1664 + 4047*a*x - 339*a^2*x^2 - 7399*a^3*x^3 - 4029*a^4*x^4 + 2967*a^5*x^5 + 2669*a^6*x^6 + 315*a^7*x^7))/(315*(-1 + a* x)^2*(1 + a*x)^5) - 3*Log[(1 + Sqrt[1 - 1/(a^2*x^2)])*x])/(a*c^4)
Time = 0.48 (sec) , antiderivative size = 314, normalized size of antiderivative = 0.96, number of steps used = 21, number of rules used = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.909, Rules used = {6748, 114, 27, 169, 27, 169, 25, 27, 169, 27, 169, 27, 169, 27, 169, 27, 169, 27, 103, 221}
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.
| \(\displaystyle \int \frac {e^{-3 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^4} \, dx\) |
| \(\Big \downarrow \) 6748 |
| \(\displaystyle -\frac {\int \frac {x^2}{\left (1-\frac {1}{a x}\right )^{5/2} \left (1+\frac {1}{a x}\right )^{11/2}}d\frac {1}{x}}{c^4}\) |
| \(\Big \downarrow \) 114 |
| \(\displaystyle -\frac {-\int \frac {\left (3 a-\frac {7}{x}\right ) x}{a^2 \left (1-\frac {1}{a x}\right )^{5/2} \left (1+\frac {1}{a x}\right )^{11/2}}d\frac {1}{x}-\frac {x}{\left (1-\frac {1}{a x}\right )^{3/2} \left (\frac {1}{a x}+1\right )^{9/2}}}{c^4}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {-\frac {\int \frac {\left (3 a-\frac {7}{x}\right ) x}{\left (1-\frac {1}{a x}\right )^{5/2} \left (1+\frac {1}{a x}\right )^{11/2}}d\frac {1}{x}}{a^2}-\frac {x}{\left (1-\frac {1}{a x}\right )^{3/2} \left (\frac {1}{a x}+1\right )^{9/2}}}{c^4}\) |
| \(\Big \downarrow \) 169 |
| \(\displaystyle -\frac {-\frac {-\frac {1}{3} a \int -\frac {3 \left (3 a-\frac {8}{x}\right ) x}{a \left (1-\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{11/2}}d\frac {1}{x}-\frac {4 a}{3 \left (1-\frac {1}{a x}\right )^{3/2} \left (\frac {1}{a x}+1\right )^{9/2}}}{a^2}-\frac {x}{\left (1-\frac {1}{a x}\right )^{3/2} \left (\frac {1}{a x}+1\right )^{9/2}}}{c^4}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {-\frac {\int \frac {\left (3 a-\frac {8}{x}\right ) x}{\left (1-\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{11/2}}d\frac {1}{x}-\frac {4 a}{3 \left (1-\frac {1}{a x}\right )^{3/2} \left (\frac {1}{a x}+1\right )^{9/2}}}{a^2}-\frac {x}{\left (1-\frac {1}{a x}\right )^{3/2} \left (\frac {1}{a x}+1\right )^{9/2}}}{c^4}\) |
| \(\Big \downarrow \) 169 |
| \(\displaystyle -\frac {-\frac {a \left (-\int -\frac {\left (3 a-\frac {25}{x}\right ) x}{a \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{11/2}}d\frac {1}{x}\right )-\frac {5 a}{\sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{9/2}}-\frac {4 a}{3 \left (1-\frac {1}{a x}\right )^{3/2} \left (\frac {1}{a x}+1\right )^{9/2}}}{a^2}-\frac {x}{\left (1-\frac {1}{a x}\right )^{3/2} \left (\frac {1}{a x}+1\right )^{9/2}}}{c^4}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {-\frac {a \int \frac {\left (3 a-\frac {25}{x}\right ) x}{a \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{11/2}}d\frac {1}{x}-\frac {5 a}{\sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{9/2}}-\frac {4 a}{3 \left (1-\frac {1}{a x}\right )^{3/2} \left (\frac {1}{a x}+1\right )^{9/2}}}{a^2}-\frac {x}{\left (1-\frac {1}{a x}\right )^{3/2} \left (\frac {1}{a x}+1\right )^{9/2}}}{c^4}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {-\frac {\int \frac {\left (3 a-\frac {25}{x}\right ) x}{\sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{11/2}}d\frac {1}{x}-\frac {5 a}{\sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{9/2}}-\frac {4 a}{3 \left (1-\frac {1}{a x}\right )^{3/2} \left (\frac {1}{a x}+1\right )^{9/2}}}{a^2}-\frac {x}{\left (1-\frac {1}{a x}\right )^{3/2} \left (\frac {1}{a x}+1\right )^{9/2}}}{c^4}\) |
| \(\Big \downarrow \) 169 |
| \(\displaystyle -\frac {-\frac {\frac {1}{9} a \int \frac {\left (27 a-\frac {112}{x}\right ) x}{a \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{9/2}}d\frac {1}{x}+\frac {28 a \sqrt {1-\frac {1}{a x}}}{9 \left (\frac {1}{a x}+1\right )^{9/2}}-\frac {5 a}{\sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{9/2}}-\frac {4 a}{3 \left (1-\frac {1}{a x}\right )^{3/2} \left (\frac {1}{a x}+1\right )^{9/2}}}{a^2}-\frac {x}{\left (1-\frac {1}{a x}\right )^{3/2} \left (\frac {1}{a x}+1\right )^{9/2}}}{c^4}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {-\frac {\frac {1}{9} \int \frac {\left (27 a-\frac {112}{x}\right ) x}{\sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{9/2}}d\frac {1}{x}+\frac {28 a \sqrt {1-\frac {1}{a x}}}{9 \left (\frac {1}{a x}+1\right )^{9/2}}-\frac {5 a}{\sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{9/2}}-\frac {4 a}{3 \left (1-\frac {1}{a x}\right )^{3/2} \left (\frac {1}{a x}+1\right )^{9/2}}}{a^2}-\frac {x}{\left (1-\frac {1}{a x}\right )^{3/2} \left (\frac {1}{a x}+1\right )^{9/2}}}{c^4}\) |
| \(\Big \downarrow \) 169 |
| \(\displaystyle -\frac {-\frac {\frac {1}{9} \left (\frac {1}{7} a \int \frac {3 \left (63 a-\frac {139}{x}\right ) x}{a \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{7/2}}d\frac {1}{x}+\frac {139 a \sqrt {1-\frac {1}{a x}}}{7 \left (\frac {1}{a x}+1\right )^{7/2}}\right )+\frac {28 a \sqrt {1-\frac {1}{a x}}}{9 \left (\frac {1}{a x}+1\right )^{9/2}}-\frac {5 a}{\sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{9/2}}-\frac {4 a}{3 \left (1-\frac {1}{a x}\right )^{3/2} \left (\frac {1}{a x}+1\right )^{9/2}}}{a^2}-\frac {x}{\left (1-\frac {1}{a x}\right )^{3/2} \left (\frac {1}{a x}+1\right )^{9/2}}}{c^4}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {-\frac {\frac {1}{9} \left (\frac {3}{7} \int \frac {\left (63 a-\frac {139}{x}\right ) x}{\sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{7/2}}d\frac {1}{x}+\frac {139 a \sqrt {1-\frac {1}{a x}}}{7 \left (\frac {1}{a x}+1\right )^{7/2}}\right )+\frac {28 a \sqrt {1-\frac {1}{a x}}}{9 \left (\frac {1}{a x}+1\right )^{9/2}}-\frac {5 a}{\sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{9/2}}-\frac {4 a}{3 \left (1-\frac {1}{a x}\right )^{3/2} \left (\frac {1}{a x}+1\right )^{9/2}}}{a^2}-\frac {x}{\left (1-\frac {1}{a x}\right )^{3/2} \left (\frac {1}{a x}+1\right )^{9/2}}}{c^4}\) |
| \(\Big \downarrow \) 169 |
| \(\displaystyle -\frac {-\frac {\frac {1}{9} \left (\frac {3}{7} \left (\frac {1}{5} a \int \frac {\left (315 a-\frac {404}{x}\right ) x}{a \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{5/2}}d\frac {1}{x}+\frac {202 a \sqrt {1-\frac {1}{a x}}}{5 \left (\frac {1}{a x}+1\right )^{5/2}}\right )+\frac {139 a \sqrt {1-\frac {1}{a x}}}{7 \left (\frac {1}{a x}+1\right )^{7/2}}\right )+\frac {28 a \sqrt {1-\frac {1}{a x}}}{9 \left (\frac {1}{a x}+1\right )^{9/2}}-\frac {5 a}{\sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{9/2}}-\frac {4 a}{3 \left (1-\frac {1}{a x}\right )^{3/2} \left (\frac {1}{a x}+1\right )^{9/2}}}{a^2}-\frac {x}{\left (1-\frac {1}{a x}\right )^{3/2} \left (\frac {1}{a x}+1\right )^{9/2}}}{c^4}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {-\frac {\frac {1}{9} \left (\frac {3}{7} \left (\frac {1}{5} \int \frac {\left (315 a-\frac {404}{x}\right ) x}{\sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{5/2}}d\frac {1}{x}+\frac {202 a \sqrt {1-\frac {1}{a x}}}{5 \left (\frac {1}{a x}+1\right )^{5/2}}\right )+\frac {139 a \sqrt {1-\frac {1}{a x}}}{7 \left (\frac {1}{a x}+1\right )^{7/2}}\right )+\frac {28 a \sqrt {1-\frac {1}{a x}}}{9 \left (\frac {1}{a x}+1\right )^{9/2}}-\frac {5 a}{\sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{9/2}}-\frac {4 a}{3 \left (1-\frac {1}{a x}\right )^{3/2} \left (\frac {1}{a x}+1\right )^{9/2}}}{a^2}-\frac {x}{\left (1-\frac {1}{a x}\right )^{3/2} \left (\frac {1}{a x}+1\right )^{9/2}}}{c^4}\) |
| \(\Big \downarrow \) 169 |
| \(\displaystyle -\frac {-\frac {\frac {1}{9} \left (\frac {3}{7} \left (\frac {1}{5} \left (\frac {1}{3} a \int \frac {\left (945 a-\frac {719}{x}\right ) x}{a \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{3/2}}d\frac {1}{x}+\frac {719 a \sqrt {1-\frac {1}{a x}}}{3 \left (\frac {1}{a x}+1\right )^{3/2}}\right )+\frac {202 a \sqrt {1-\frac {1}{a x}}}{5 \left (\frac {1}{a x}+1\right )^{5/2}}\right )+\frac {139 a \sqrt {1-\frac {1}{a x}}}{7 \left (\frac {1}{a x}+1\right )^{7/2}}\right )+\frac {28 a \sqrt {1-\frac {1}{a x}}}{9 \left (\frac {1}{a x}+1\right )^{9/2}}-\frac {5 a}{\sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{9/2}}-\frac {4 a}{3 \left (1-\frac {1}{a x}\right )^{3/2} \left (\frac {1}{a x}+1\right )^{9/2}}}{a^2}-\frac {x}{\left (1-\frac {1}{a x}\right )^{3/2} \left (\frac {1}{a x}+1\right )^{9/2}}}{c^4}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {-\frac {\frac {1}{9} \left (\frac {3}{7} \left (\frac {1}{5} \left (\frac {1}{3} \int \frac {\left (945 a-\frac {719}{x}\right ) x}{\sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{3/2}}d\frac {1}{x}+\frac {719 a \sqrt {1-\frac {1}{a x}}}{3 \left (\frac {1}{a x}+1\right )^{3/2}}\right )+\frac {202 a \sqrt {1-\frac {1}{a x}}}{5 \left (\frac {1}{a x}+1\right )^{5/2}}\right )+\frac {139 a \sqrt {1-\frac {1}{a x}}}{7 \left (\frac {1}{a x}+1\right )^{7/2}}\right )+\frac {28 a \sqrt {1-\frac {1}{a x}}}{9 \left (\frac {1}{a x}+1\right )^{9/2}}-\frac {5 a}{\sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{9/2}}-\frac {4 a}{3 \left (1-\frac {1}{a x}\right )^{3/2} \left (\frac {1}{a x}+1\right )^{9/2}}}{a^2}-\frac {x}{\left (1-\frac {1}{a x}\right )^{3/2} \left (\frac {1}{a x}+1\right )^{9/2}}}{c^4}\) |
| \(\Big \downarrow \) 169 |
| \(\displaystyle -\frac {-\frac {\frac {1}{9} \left (\frac {3}{7} \left (\frac {1}{5} \left (\frac {1}{3} \left (a \int \frac {945 x}{\sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}}d\frac {1}{x}+\frac {1664 a \sqrt {1-\frac {1}{a x}}}{\sqrt {\frac {1}{a x}+1}}\right )+\frac {719 a \sqrt {1-\frac {1}{a x}}}{3 \left (\frac {1}{a x}+1\right )^{3/2}}\right )+\frac {202 a \sqrt {1-\frac {1}{a x}}}{5 \left (\frac {1}{a x}+1\right )^{5/2}}\right )+\frac {139 a \sqrt {1-\frac {1}{a x}}}{7 \left (\frac {1}{a x}+1\right )^{7/2}}\right )+\frac {28 a \sqrt {1-\frac {1}{a x}}}{9 \left (\frac {1}{a x}+1\right )^{9/2}}-\frac {5 a}{\sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{9/2}}-\frac {4 a}{3 \left (1-\frac {1}{a x}\right )^{3/2} \left (\frac {1}{a x}+1\right )^{9/2}}}{a^2}-\frac {x}{\left (1-\frac {1}{a x}\right )^{3/2} \left (\frac {1}{a x}+1\right )^{9/2}}}{c^4}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {-\frac {\frac {1}{9} \left (\frac {3}{7} \left (\frac {1}{5} \left (\frac {1}{3} \left (945 a \int \frac {x}{\sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}}d\frac {1}{x}+\frac {1664 a \sqrt {1-\frac {1}{a x}}}{\sqrt {\frac {1}{a x}+1}}\right )+\frac {719 a \sqrt {1-\frac {1}{a x}}}{3 \left (\frac {1}{a x}+1\right )^{3/2}}\right )+\frac {202 a \sqrt {1-\frac {1}{a x}}}{5 \left (\frac {1}{a x}+1\right )^{5/2}}\right )+\frac {139 a \sqrt {1-\frac {1}{a x}}}{7 \left (\frac {1}{a x}+1\right )^{7/2}}\right )+\frac {28 a \sqrt {1-\frac {1}{a x}}}{9 \left (\frac {1}{a x}+1\right )^{9/2}}-\frac {5 a}{\sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{9/2}}-\frac {4 a}{3 \left (1-\frac {1}{a x}\right )^{3/2} \left (\frac {1}{a x}+1\right )^{9/2}}}{a^2}-\frac {x}{\left (1-\frac {1}{a x}\right )^{3/2} \left (\frac {1}{a x}+1\right )^{9/2}}}{c^4}\) |
| \(\Big \downarrow \) 103 |
| \(\displaystyle -\frac {-\frac {\frac {1}{9} \left (\frac {3}{7} \left (\frac {1}{5} \left (\frac {1}{3} \left (\frac {1664 a \sqrt {1-\frac {1}{a x}}}{\sqrt {\frac {1}{a x}+1}}-945 \int \frac {1}{\frac {1}{a}-\frac {1}{a x^2}}d\left (\sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}\right )\right )+\frac {719 a \sqrt {1-\frac {1}{a x}}}{3 \left (\frac {1}{a x}+1\right )^{3/2}}\right )+\frac {202 a \sqrt {1-\frac {1}{a x}}}{5 \left (\frac {1}{a x}+1\right )^{5/2}}\right )+\frac {139 a \sqrt {1-\frac {1}{a x}}}{7 \left (\frac {1}{a x}+1\right )^{7/2}}\right )+\frac {28 a \sqrt {1-\frac {1}{a x}}}{9 \left (\frac {1}{a x}+1\right )^{9/2}}-\frac {5 a}{\sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{9/2}}-\frac {4 a}{3 \left (1-\frac {1}{a x}\right )^{3/2} \left (\frac {1}{a x}+1\right )^{9/2}}}{a^2}-\frac {x}{\left (1-\frac {1}{a x}\right )^{3/2} \left (\frac {1}{a x}+1\right )^{9/2}}}{c^4}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle -\frac {-\frac {\frac {1}{9} \left (\frac {3}{7} \left (\frac {1}{5} \left (\frac {1}{3} \left (\frac {1664 a \sqrt {1-\frac {1}{a x}}}{\sqrt {\frac {1}{a x}+1}}-945 a \text {arctanh}\left (\sqrt {1-\frac {1}{a x}} \sqrt {\frac {1}{a x}+1}\right )\right )+\frac {719 a \sqrt {1-\frac {1}{a x}}}{3 \left (\frac {1}{a x}+1\right )^{3/2}}\right )+\frac {202 a \sqrt {1-\frac {1}{a x}}}{5 \left (\frac {1}{a x}+1\right )^{5/2}}\right )+\frac {139 a \sqrt {1-\frac {1}{a x}}}{7 \left (\frac {1}{a x}+1\right )^{7/2}}\right )+\frac {28 a \sqrt {1-\frac {1}{a x}}}{9 \left (\frac {1}{a x}+1\right )^{9/2}}-\frac {5 a}{\sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{9/2}}-\frac {4 a}{3 \left (1-\frac {1}{a x}\right )^{3/2} \left (\frac {1}{a x}+1\right )^{9/2}}}{a^2}-\frac {x}{\left (1-\frac {1}{a x}\right )^{3/2} \left (\frac {1}{a x}+1\right )^{9/2}}}{c^4}\) |
-((-(x/((1 - 1/(a*x))^(3/2)*(1 + 1/(a*x))^(9/2))) - ((-4*a)/(3*(1 - 1/(a*x ))^(3/2)*(1 + 1/(a*x))^(9/2)) - (5*a)/(Sqrt[1 - 1/(a*x)]*(1 + 1/(a*x))^(9/ 2)) + (28*a*Sqrt[1 - 1/(a*x)])/(9*(1 + 1/(a*x))^(9/2)) + ((139*a*Sqrt[1 - 1/(a*x)])/(7*(1 + 1/(a*x))^(7/2)) + (3*((202*a*Sqrt[1 - 1/(a*x)])/(5*(1 + 1/(a*x))^(5/2)) + ((719*a*Sqrt[1 - 1/(a*x)])/(3*(1 + 1/(a*x))^(3/2)) + ((1 664*a*Sqrt[1 - 1/(a*x)])/Sqrt[1 + 1/(a*x)] - 945*a*ArcTanh[Sqrt[1 - 1/(a*x )]*Sqrt[1 + 1/(a*x)]])/3)/5))/7)/9)/a^2)/c^4)
3.9.29.3.1 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[1/(Sqrt[(a_.) + (b_.)*(x_)]*Sqrt[(c_.) + (d_.)*(x_)]*((e_.) + (f_.)*(x_ ))), x_] :> Simp[b*f Subst[Int[1/(d*(b*e - a*f)^2 + b*f^2*x^2), x], x, Sq rt[a + b*x]*Sqrt[c + d*x]], x] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[2*b*d *e - f*(b*c + a*d), 0]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_), x_] :> Simp[b*(a + b*x)^(m + 1)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1 )/((m + 1)*(b*c - a*d)*(b*e - a*f))), x] + Simp[1/((m + 1)*(b*c - a*d)*(b*e - a*f)) Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p*Simp[a*d*f*(m + 1) - b*(d*e*(m + n + 2) + c*f*(m + p + 2)) - b*d*f*(m + n + p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && ILtQ[m, -1] && (IntegerQ[n] || IntegersQ[2*n, 2*p] || ILtQ[m + n + p + 3, 0])
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_)*((g_.) + (h_.)*(x_)), x_] :> Simp[(b*g - a*h)*(a + b*x)^(m + 1)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/((m + 1)*(b*c - a*d)*(b*e - a*f))), x] + S imp[1/((m + 1)*(b*c - a*d)*(b*e - a*f)) Int[(a + b*x)^(m + 1)*(c + d*x)^n *(e + f*x)^p*Simp[(a*d*f*g - b*(d*e + c*f)*g + b*c*e*h)*(m + 1) - (b*g - a* h)*(d*e*(n + 1) + c*f*(p + 1)) - d*f*(b*g - a*h)*(m + n + p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && LtQ[m, -1] && IntegersQ[ 2*m, 2*n, 2*p]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x /Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
Int[E^(ArcCoth[(a_.)*(x_)]*(n_.))*((c_) + (d_.)/(x_)^2)^(p_.), x_Symbol] :> Simp[-c^p Subst[Int[(1 - x/a)^(p - n/2)*((1 + x/a)^(p + n/2)/x^2), x], x , 1/x], x] /; FreeQ[{a, c, d, n, p}, x] && EqQ[c + a^2*d, 0] && !IntegerQ[ n/2] && (IntegerQ[p] || GtQ[c, 0]) && !IntegersQ[2*p, p + n/2]
Time = 0.25 (sec) , antiderivative size = 355, normalized size of antiderivative = 1.09
| method | result | size |
| risch | \(\frac {\left (a x +1\right ) \sqrt {\frac {a x -1}{a x +1}}}{a \,c^{4}}+\frac {\left (-\frac {3 \ln \left (\frac {a^{2} x}{\sqrt {a^{2}}}+\sqrt {a^{2} x^{2}-1}\right )}{a^{8} \sqrt {a^{2}}}-\frac {59 \sqrt {a^{2} \left (x +\frac {1}{a}\right )^{2}-2 a \left (x +\frac {1}{a}\right )}}{252 a^{13} \left (x +\frac {1}{a}\right )^{4}}+\frac {1507 \sqrt {a^{2} \left (x +\frac {1}{a}\right )^{2}-2 a \left (x +\frac {1}{a}\right )}}{1680 a^{12} \left (x +\frac {1}{a}\right )^{3}}-\frac {691 \sqrt {a^{2} \left (x +\frac {1}{a}\right )^{2}-2 a \left (x +\frac {1}{a}\right )}}{315 a^{11} \left (x +\frac {1}{a}\right )^{2}}+\frac {113591 \sqrt {a^{2} \left (x +\frac {1}{a}\right )^{2}-2 a \left (x +\frac {1}{a}\right )}}{20160 a^{10} \left (x +\frac {1}{a}\right )}+\frac {\sqrt {a^{2} \left (x +\frac {1}{a}\right )^{2}-2 a \left (x +\frac {1}{a}\right )}}{36 a^{14} \left (x +\frac {1}{a}\right )^{5}}-\frac {\sqrt {\left (x -\frac {1}{a}\right )^{2} a^{2}+2 \left (x -\frac {1}{a}\right ) a}}{96 a^{11} \left (x -\frac {1}{a}\right )^{2}}-\frac {31 \sqrt {\left (x -\frac {1}{a}\right )^{2} a^{2}+2 \left (x -\frac {1}{a}\right ) a}}{192 a^{10} \left (x -\frac {1}{a}\right )}\right ) a^{8} \sqrt {\frac {a x -1}{a x +1}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}}{c^{4} \left (a x -1\right )}\) | \(355\) |
| default | \(-\frac {\left (-138915 \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}\, a^{9} x^{9}+120960 \ln \left (\frac {a^{2} x +\sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}}{\sqrt {a^{2}}}\right ) a^{10} x^{9}+98595 \left (\left (a x -1\right ) \left (a x +1\right )\right )^{\frac {3}{2}} \sqrt {a^{2}}\, a^{7} x^{7}-416745 \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}\, a^{8} x^{8}+362880 \ln \left (\frac {a^{2} x +\sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}}{\sqrt {a^{2}}}\right ) a^{9} x^{8}+75113 \left (\left (a x -1\right ) \left (a x +1\right )\right )^{\frac {3}{2}} \sqrt {a^{2}}\, a^{6} x^{6}-240861 \left (\left (a x -1\right ) \left (a x +1\right )\right )^{\frac {3}{2}} \sqrt {a^{2}}\, a^{5} x^{5}+1111320 \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}\, a^{6} x^{6}-967680 \ln \left (\frac {a^{2} x +\sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}}{\sqrt {a^{2}}}\right ) a^{7} x^{6}-178863 \left (\left (a x -1\right ) \left (a x +1\right )\right )^{\frac {3}{2}} \sqrt {a^{2}}\, a^{4} x^{4}+833490 \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}\, a^{5} x^{5}-725760 \ln \left (\frac {a^{2} x +\sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}}{\sqrt {a^{2}}}\right ) a^{6} x^{5}+252497 \left (\left (a x -1\right ) \left (a x +1\right )\right )^{\frac {3}{2}} \sqrt {a^{2}}\, a^{3} x^{3}-833490 \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}\, a^{4} x^{4}+725760 \ln \left (\frac {a^{2} x +\sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}}{\sqrt {a^{2}}}\right ) a^{5} x^{4}+182307 \left (\left (a x -1\right ) \left (a x +1\right )\right )^{\frac {3}{2}} \sqrt {a^{2}}\, a^{2} x^{2}-1111320 \sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, a^{3} x^{3}+967680 \ln \left (\frac {a^{2} x +\sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}}{\sqrt {a^{2}}}\right ) a^{4} x^{3}-101271 \sqrt {a^{2}}\, \left (\left (a x -1\right ) \left (a x +1\right )\right )^{\frac {3}{2}} a x -74077 \left (\left (a x -1\right ) \left (a x +1\right )\right )^{\frac {3}{2}} \sqrt {a^{2}}+416745 \sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, a x -362880 \ln \left (\frac {a^{2} x +\sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}}{\sqrt {a^{2}}}\right ) a^{2} x +138915 \sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}-120960 a \ln \left (\frac {a^{2} x +\sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}}{\sqrt {a^{2}}}\right )\right ) \left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}}}{40320 a \sqrt {a^{2}}\, \left (a x +1\right )^{4} c^{4} \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \left (a x -1\right )^{4}}\) | \(766\) |
1/a*(a*x+1)/c^4*((a*x-1)/(a*x+1))^(1/2)+(-3/a^8*ln(a^2*x/(a^2)^(1/2)+(a^2* x^2-1)^(1/2))/(a^2)^(1/2)-59/252/a^13/(x+1/a)^4*(a^2*(x+1/a)^2-2*a*(x+1/a) )^(1/2)+1507/1680/a^12/(x+1/a)^3*(a^2*(x+1/a)^2-2*a*(x+1/a))^(1/2)-691/315 /a^11/(x+1/a)^2*(a^2*(x+1/a)^2-2*a*(x+1/a))^(1/2)+113591/20160/a^10/(x+1/a )*(a^2*(x+1/a)^2-2*a*(x+1/a))^(1/2)+1/36/a^14/(x+1/a)^5*(a^2*(x+1/a)^2-2*a *(x+1/a))^(1/2)-1/96/a^11/(x-1/a)^2*((x-1/a)^2*a^2+2*(x-1/a)*a)^(1/2)-31/1 92/a^10/(x-1/a)*((x-1/a)^2*a^2+2*(x-1/a)*a)^(1/2))*a^8/c^4*((a*x-1)/(a*x+1 ))^(1/2)*((a*x-1)*(a*x+1))^(1/2)/(a*x-1)
Time = 0.25 (sec) , antiderivative size = 275, normalized size of antiderivative = 0.84 \[ \int \frac {e^{-3 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^4} \, dx=-\frac {945 \, {\left (a^{6} x^{6} + 2 \, a^{5} x^{5} - a^{4} x^{4} - 4 \, a^{3} x^{3} - a^{2} x^{2} + 2 \, a x + 1\right )} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right ) - 945 \, {\left (a^{6} x^{6} + 2 \, a^{5} x^{5} - a^{4} x^{4} - 4 \, a^{3} x^{3} - a^{2} x^{2} + 2 \, a x + 1\right )} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right ) - {\left (315 \, a^{7} x^{7} + 2669 \, a^{6} x^{6} + 2967 \, a^{5} x^{5} - 4029 \, a^{4} x^{4} - 7399 \, a^{3} x^{3} - 339 \, a^{2} x^{2} + 4047 \, a x + 1664\right )} \sqrt {\frac {a x - 1}{a x + 1}}}{315 \, {\left (a^{7} c^{4} x^{6} + 2 \, a^{6} c^{4} x^{5} - a^{5} c^{4} x^{4} - 4 \, a^{4} c^{4} x^{3} - a^{3} c^{4} x^{2} + 2 \, a^{2} c^{4} x + a c^{4}\right )}} \]
-1/315*(945*(a^6*x^6 + 2*a^5*x^5 - a^4*x^4 - 4*a^3*x^3 - a^2*x^2 + 2*a*x + 1)*log(sqrt((a*x - 1)/(a*x + 1)) + 1) - 945*(a^6*x^6 + 2*a^5*x^5 - a^4*x^ 4 - 4*a^3*x^3 - a^2*x^2 + 2*a*x + 1)*log(sqrt((a*x - 1)/(a*x + 1)) - 1) - (315*a^7*x^7 + 2669*a^6*x^6 + 2967*a^5*x^5 - 4029*a^4*x^4 - 7399*a^3*x^3 - 339*a^2*x^2 + 4047*a*x + 1664)*sqrt((a*x - 1)/(a*x + 1)))/(a^7*c^4*x^6 + 2*a^6*c^4*x^5 - a^5*c^4*x^4 - 4*a^4*c^4*x^3 - a^3*c^4*x^2 + 2*a^2*c^4*x + a*c^4)
Timed out. \[ \int \frac {e^{-3 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^4} \, dx=\text {Timed out} \]
Time = 0.20 (sec) , antiderivative size = 231, normalized size of antiderivative = 0.71 \[ \int \frac {e^{-3 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^4} \, dx=\frac {1}{20160} \, a {\left (\frac {105 \, {\left (\frac {29 \, {\left (a x - 1\right )}}{a x + 1} - \frac {414 \, {\left (a x - 1\right )}^{2}}{{\left (a x + 1\right )}^{2}} + 1\right )}}{a^{2} c^{4} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {5}{2}} - a^{2} c^{4} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}}} + \frac {35 \, \left (\frac {a x - 1}{a x + 1}\right )^{\frac {9}{2}} + 450 \, \left (\frac {a x - 1}{a x + 1}\right )^{\frac {7}{2}} + 2961 \, \left (\frac {a x - 1}{a x + 1}\right )^{\frac {5}{2}} + 14700 \, \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}} + 95445 \, \sqrt {\frac {a x - 1}{a x + 1}}}{a^{2} c^{4}} - \frac {60480 \, \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right )}{a^{2} c^{4}} + \frac {60480 \, \log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right )}{a^{2} c^{4}}\right )} \]
1/20160*a*(105*(29*(a*x - 1)/(a*x + 1) - 414*(a*x - 1)^2/(a*x + 1)^2 + 1)/ (a^2*c^4*((a*x - 1)/(a*x + 1))^(5/2) - a^2*c^4*((a*x - 1)/(a*x + 1))^(3/2) ) + (35*((a*x - 1)/(a*x + 1))^(9/2) + 450*((a*x - 1)/(a*x + 1))^(7/2) + 29 61*((a*x - 1)/(a*x + 1))^(5/2) + 14700*((a*x - 1)/(a*x + 1))^(3/2) + 95445 *sqrt((a*x - 1)/(a*x + 1)))/(a^2*c^4) - 60480*log(sqrt((a*x - 1)/(a*x + 1) ) + 1)/(a^2*c^4) + 60480*log(sqrt((a*x - 1)/(a*x + 1)) - 1)/(a^2*c^4))
\[ \int \frac {e^{-3 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^4} \, dx=\int { \frac {\left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}}}{{\left (c - \frac {c}{a^{2} x^{2}}\right )}^{4}} \,d x } \]
Time = 0.06 (sec) , antiderivative size = 224, normalized size of antiderivative = 0.69 \[ \int \frac {e^{-3 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^4} \, dx=\frac {303\,\sqrt {\frac {a\,x-1}{a\,x+1}}}{64\,a\,c^4}-\frac {\frac {29\,\left (a\,x-1\right )}{3\,\left (a\,x+1\right )}-\frac {138\,{\left (a\,x-1\right )}^2}{{\left (a\,x+1\right )}^2}+\frac {1}{3}}{64\,a\,c^4\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{3/2}-64\,a\,c^4\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{5/2}}+\frac {35\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{3/2}}{48\,a\,c^4}+\frac {47\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{5/2}}{320\,a\,c^4}+\frac {5\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{7/2}}{224\,a\,c^4}+\frac {{\left (\frac {a\,x-1}{a\,x+1}\right )}^{9/2}}{576\,a\,c^4}+\frac {\mathrm {atan}\left (\sqrt {\frac {a\,x-1}{a\,x+1}}\,1{}\mathrm {i}\right )\,6{}\mathrm {i}}{a\,c^4} \]
(303*((a*x - 1)/(a*x + 1))^(1/2))/(64*a*c^4) - ((29*(a*x - 1))/(3*(a*x + 1 )) - (138*(a*x - 1)^2)/(a*x + 1)^2 + 1/3)/(64*a*c^4*((a*x - 1)/(a*x + 1))^ (3/2) - 64*a*c^4*((a*x - 1)/(a*x + 1))^(5/2)) + (35*((a*x - 1)/(a*x + 1))^ (3/2))/(48*a*c^4) + (47*((a*x - 1)/(a*x + 1))^(5/2))/(320*a*c^4) + (5*((a* x - 1)/(a*x + 1))^(7/2))/(224*a*c^4) + ((a*x - 1)/(a*x + 1))^(9/2)/(576*a* c^4) + (atan(((a*x - 1)/(a*x + 1))^(1/2)*1i)*6i)/(a*c^4)