Integrand size = 22, antiderivative size = 329 \[ \int \frac {e^{3 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^4} \, dx=-\frac {10}{9 a c^4 \left (1-\frac {1}{a x}\right )^{9/2} \left (1+\frac {1}{a x}\right )^{3/2}}-\frac {29}{21 a c^4 \left (1-\frac {1}{a x}\right )^{7/2} \left (1+\frac {1}{a x}\right )^{3/2}}-\frac {208}{105 a c^4 \left (1-\frac {1}{a x}\right )^{5/2} \left (1+\frac {1}{a x}\right )^{3/2}}-\frac {1147}{315 a c^4 \left (1-\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{3/2}}-\frac {1462}{105 a c^4 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{3/2}}+\frac {2609 \sqrt {1-\frac {1}{a x}}}{315 a c^4 \left (1+\frac {1}{a x}\right )^{3/2}}+\frac {1664 \sqrt {1-\frac {1}{a x}}}{315 a c^4 \sqrt {1+\frac {1}{a x}}}+\frac {x}{c^4 \left (1-\frac {1}{a x}\right )^{9/2} \left (1+\frac {1}{a x}\right )^{3/2}}+\frac {3 \text {arctanh}\left (\sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}\right )}{a c^4} \]
-10/9/a/c^4/(1-1/a/x)^(9/2)/(1+1/a/x)^(3/2)-29/21/a/c^4/(1-1/a/x)^(7/2)/(1 +1/a/x)^(3/2)-208/105/a/c^4/(1-1/a/x)^(5/2)/(1+1/a/x)^(3/2)-1147/315/a/c^4 /(1-1/a/x)^(3/2)/(1+1/a/x)^(3/2)+x/c^4/(1-1/a/x)^(9/2)/(1+1/a/x)^(3/2)+3*a rctanh((1-1/a/x)^(1/2)*(1+1/a/x)^(1/2))/a/c^4-1462/105/a/c^4/(1+1/a/x)^(3/ 2)/(1-1/a/x)^(1/2)+2609/315*(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 \[ \int \frac {e^{3 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^4} \, dx=\frac {\frac {a \sqrt {1-\frac {1}{a^2 x^2}} x \left (-1664+4047 a x+339 a^2 x^2-7399 a^3 x^3+4029 a^4 x^4+2967 a^5 x^5-2669 a^6 x^6+315 a^7 x^7\right )}{315 (-1+a x)^5 (1+a x)^2}+3 \log \left (\left (1+\sqrt {1-\frac {1}{a^2 x^2}}\right ) x\right )}{a c^4} \]
((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)^5*(1 + a*x)^2) + 3*Log[(1 + Sqrt[1 - 1/(a^2*x^2)])*x])/(a*c^4)
Time = 0.48 (sec) , antiderivative size = 313, normalized size of antiderivative = 0.95, number of steps used = 24, number of rules used = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.045, Rules used = {6748, 114, 25, 27, 169, 27, 169, 25, 27, 169, 25, 27, 169, 27, 169, 25, 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 |
| \(\displaystyle -\frac {\int \frac {x^2}{\left (1-\frac {1}{a x}\right )^{11/2} \left (1+\frac {1}{a x}\right )^{5/2}}d\frac {1}{x}}{c^4}\) |
| \(\Big \downarrow \) 114 |
| \(\displaystyle -\frac {-\int -\frac {\left (3 a+\frac {7}{x}\right ) x}{a^2 \left (1-\frac {1}{a x}\right )^{11/2} \left (1+\frac {1}{a x}\right )^{5/2}}d\frac {1}{x}-\frac {x}{\left (1-\frac {1}{a x}\right )^{9/2} \left (\frac {1}{a x}+1\right )^{3/2}}}{c^4}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\int \frac {\left (3 a+\frac {7}{x}\right ) x}{a^2 \left (1-\frac {1}{a x}\right )^{11/2} \left (1+\frac {1}{a x}\right )^{5/2}}d\frac {1}{x}-\frac {x}{\left (1-\frac {1}{a x}\right )^{9/2} \left (\frac {1}{a x}+1\right )^{3/2}}}{c^4}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\frac {\int \frac {\left (3 a+\frac {7}{x}\right ) x}{\left (1-\frac {1}{a x}\right )^{11/2} \left (1+\frac {1}{a x}\right )^{5/2}}d\frac {1}{x}}{a^2}-\frac {x}{\left (1-\frac {1}{a x}\right )^{9/2} \left (\frac {1}{a x}+1\right )^{3/2}}}{c^4}\) |
| \(\Big \downarrow \) 169 |
| \(\displaystyle -\frac {\frac {\frac {10 a}{9 \left (1-\frac {1}{a x}\right )^{9/2} \left (\frac {1}{a x}+1\right )^{3/2}}-\frac {1}{9} a \int -\frac {3 \left (9 a+\frac {20}{x}\right ) x}{a \left (1-\frac {1}{a x}\right )^{9/2} \left (1+\frac {1}{a x}\right )^{5/2}}d\frac {1}{x}}{a^2}-\frac {x}{\left (1-\frac {1}{a x}\right )^{9/2} \left (\frac {1}{a x}+1\right )^{3/2}}}{c^4}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\frac {\frac {1}{3} \int \frac {\left (9 a+\frac {20}{x}\right ) x}{\left (1-\frac {1}{a x}\right )^{9/2} \left (1+\frac {1}{a x}\right )^{5/2}}d\frac {1}{x}+\frac {10 a}{9 \left (1-\frac {1}{a x}\right )^{9/2} \left (\frac {1}{a x}+1\right )^{3/2}}}{a^2}-\frac {x}{\left (1-\frac {1}{a x}\right )^{9/2} \left (\frac {1}{a x}+1\right )^{3/2}}}{c^4}\) |
| \(\Big \downarrow \) 169 |
| \(\displaystyle -\frac {\frac {\frac {1}{3} \left (\frac {29 a}{7 \left (1-\frac {1}{a x}\right )^{7/2} \left (\frac {1}{a x}+1\right )^{3/2}}-\frac {1}{7} a \int -\frac {\left (63 a+\frac {145}{x}\right ) x}{a \left (1-\frac {1}{a x}\right )^{7/2} \left (1+\frac {1}{a x}\right )^{5/2}}d\frac {1}{x}\right )+\frac {10 a}{9 \left (1-\frac {1}{a x}\right )^{9/2} \left (\frac {1}{a x}+1\right )^{3/2}}}{a^2}-\frac {x}{\left (1-\frac {1}{a x}\right )^{9/2} \left (\frac {1}{a x}+1\right )^{3/2}}}{c^4}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\frac {\frac {1}{3} \left (\frac {1}{7} a \int \frac {\left (63 a+\frac {145}{x}\right ) x}{a \left (1-\frac {1}{a x}\right )^{7/2} \left (1+\frac {1}{a x}\right )^{5/2}}d\frac {1}{x}+\frac {29 a}{7 \left (1-\frac {1}{a x}\right )^{7/2} \left (\frac {1}{a x}+1\right )^{3/2}}\right )+\frac {10 a}{9 \left (1-\frac {1}{a x}\right )^{9/2} \left (\frac {1}{a x}+1\right )^{3/2}}}{a^2}-\frac {x}{\left (1-\frac {1}{a x}\right )^{9/2} \left (\frac {1}{a x}+1\right )^{3/2}}}{c^4}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\frac {\frac {1}{3} \left (\frac {1}{7} \int \frac {\left (63 a+\frac {145}{x}\right ) x}{\left (1-\frac {1}{a x}\right )^{7/2} \left (1+\frac {1}{a x}\right )^{5/2}}d\frac {1}{x}+\frac {29 a}{7 \left (1-\frac {1}{a x}\right )^{7/2} \left (\frac {1}{a x}+1\right )^{3/2}}\right )+\frac {10 a}{9 \left (1-\frac {1}{a x}\right )^{9/2} \left (\frac {1}{a x}+1\right )^{3/2}}}{a^2}-\frac {x}{\left (1-\frac {1}{a x}\right )^{9/2} \left (\frac {1}{a x}+1\right )^{3/2}}}{c^4}\) |
| \(\Big \downarrow \) 169 |
| \(\displaystyle -\frac {\frac {\frac {1}{3} \left (\frac {1}{7} \left (\frac {208 a}{5 \left (1-\frac {1}{a x}\right )^{5/2} \left (\frac {1}{a x}+1\right )^{3/2}}-\frac {1}{5} a \int -\frac {\left (315 a+\frac {832}{x}\right ) x}{a \left (1-\frac {1}{a x}\right )^{5/2} \left (1+\frac {1}{a x}\right )^{5/2}}d\frac {1}{x}\right )+\frac {29 a}{7 \left (1-\frac {1}{a x}\right )^{7/2} \left (\frac {1}{a x}+1\right )^{3/2}}\right )+\frac {10 a}{9 \left (1-\frac {1}{a x}\right )^{9/2} \left (\frac {1}{a x}+1\right )^{3/2}}}{a^2}-\frac {x}{\left (1-\frac {1}{a x}\right )^{9/2} \left (\frac {1}{a x}+1\right )^{3/2}}}{c^4}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\frac {\frac {1}{3} \left (\frac {1}{7} \left (\frac {1}{5} a \int \frac {\left (315 a+\frac {832}{x}\right ) x}{a \left (1-\frac {1}{a x}\right )^{5/2} \left (1+\frac {1}{a x}\right )^{5/2}}d\frac {1}{x}+\frac {208 a}{5 \left (1-\frac {1}{a x}\right )^{5/2} \left (\frac {1}{a x}+1\right )^{3/2}}\right )+\frac {29 a}{7 \left (1-\frac {1}{a x}\right )^{7/2} \left (\frac {1}{a x}+1\right )^{3/2}}\right )+\frac {10 a}{9 \left (1-\frac {1}{a x}\right )^{9/2} \left (\frac {1}{a x}+1\right )^{3/2}}}{a^2}-\frac {x}{\left (1-\frac {1}{a x}\right )^{9/2} \left (\frac {1}{a x}+1\right )^{3/2}}}{c^4}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\frac {\frac {1}{3} \left (\frac {1}{7} \left (\frac {1}{5} \int \frac {\left (315 a+\frac {832}{x}\right ) x}{\left (1-\frac {1}{a x}\right )^{5/2} \left (1+\frac {1}{a x}\right )^{5/2}}d\frac {1}{x}+\frac {208 a}{5 \left (1-\frac {1}{a x}\right )^{5/2} \left (\frac {1}{a x}+1\right )^{3/2}}\right )+\frac {29 a}{7 \left (1-\frac {1}{a x}\right )^{7/2} \left (\frac {1}{a x}+1\right )^{3/2}}\right )+\frac {10 a}{9 \left (1-\frac {1}{a x}\right )^{9/2} \left (\frac {1}{a x}+1\right )^{3/2}}}{a^2}-\frac {x}{\left (1-\frac {1}{a x}\right )^{9/2} \left (\frac {1}{a x}+1\right )^{3/2}}}{c^4}\) |
| \(\Big \downarrow \) 169 |
| \(\displaystyle -\frac {\frac {\frac {1}{3} \left (\frac {1}{7} \left (\frac {1}{5} \left (\frac {1147 a}{3 \left (1-\frac {1}{a x}\right )^{3/2} \left (\frac {1}{a x}+1\right )^{3/2}}-\frac {1}{3} a \int -\frac {3 \left (315 a+\frac {1147}{x}\right ) x}{a \left (1-\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{5/2}}d\frac {1}{x}\right )+\frac {208 a}{5 \left (1-\frac {1}{a x}\right )^{5/2} \left (\frac {1}{a x}+1\right )^{3/2}}\right )+\frac {29 a}{7 \left (1-\frac {1}{a x}\right )^{7/2} \left (\frac {1}{a x}+1\right )^{3/2}}\right )+\frac {10 a}{9 \left (1-\frac {1}{a x}\right )^{9/2} \left (\frac {1}{a x}+1\right )^{3/2}}}{a^2}-\frac {x}{\left (1-\frac {1}{a x}\right )^{9/2} \left (\frac {1}{a x}+1\right )^{3/2}}}{c^4}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\frac {\frac {1}{3} \left (\frac {1}{7} \left (\frac {1}{5} \left (\int \frac {\left (315 a+\frac {1147}{x}\right ) x}{\left (1-\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{5/2}}d\frac {1}{x}+\frac {1147 a}{3 \left (1-\frac {1}{a x}\right )^{3/2} \left (\frac {1}{a x}+1\right )^{3/2}}\right )+\frac {208 a}{5 \left (1-\frac {1}{a x}\right )^{5/2} \left (\frac {1}{a x}+1\right )^{3/2}}\right )+\frac {29 a}{7 \left (1-\frac {1}{a x}\right )^{7/2} \left (\frac {1}{a x}+1\right )^{3/2}}\right )+\frac {10 a}{9 \left (1-\frac {1}{a x}\right )^{9/2} \left (\frac {1}{a x}+1\right )^{3/2}}}{a^2}-\frac {x}{\left (1-\frac {1}{a x}\right )^{9/2} \left (\frac {1}{a x}+1\right )^{3/2}}}{c^4}\) |
| \(\Big \downarrow \) 169 |
| \(\displaystyle -\frac {\frac {\frac {1}{3} \left (\frac {1}{7} \left (\frac {1}{5} \left (a \left (-\int -\frac {\left (315 a+\frac {2924}{x}\right ) x}{a \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{5/2}}d\frac {1}{x}\right )+\frac {1462 a}{\sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{3/2}}+\frac {1147 a}{3 \left (1-\frac {1}{a x}\right )^{3/2} \left (\frac {1}{a x}+1\right )^{3/2}}\right )+\frac {208 a}{5 \left (1-\frac {1}{a x}\right )^{5/2} \left (\frac {1}{a x}+1\right )^{3/2}}\right )+\frac {29 a}{7 \left (1-\frac {1}{a x}\right )^{7/2} \left (\frac {1}{a x}+1\right )^{3/2}}\right )+\frac {10 a}{9 \left (1-\frac {1}{a x}\right )^{9/2} \left (\frac {1}{a x}+1\right )^{3/2}}}{a^2}-\frac {x}{\left (1-\frac {1}{a x}\right )^{9/2} \left (\frac {1}{a x}+1\right )^{3/2}}}{c^4}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\frac {\frac {1}{3} \left (\frac {1}{7} \left (\frac {1}{5} \left (a \int \frac {\left (315 a+\frac {2924}{x}\right ) x}{a \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{5/2}}d\frac {1}{x}+\frac {1462 a}{\sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{3/2}}+\frac {1147 a}{3 \left (1-\frac {1}{a x}\right )^{3/2} \left (\frac {1}{a x}+1\right )^{3/2}}\right )+\frac {208 a}{5 \left (1-\frac {1}{a x}\right )^{5/2} \left (\frac {1}{a x}+1\right )^{3/2}}\right )+\frac {29 a}{7 \left (1-\frac {1}{a x}\right )^{7/2} \left (\frac {1}{a x}+1\right )^{3/2}}\right )+\frac {10 a}{9 \left (1-\frac {1}{a x}\right )^{9/2} \left (\frac {1}{a x}+1\right )^{3/2}}}{a^2}-\frac {x}{\left (1-\frac {1}{a x}\right )^{9/2} \left (\frac {1}{a x}+1\right )^{3/2}}}{c^4}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\frac {\frac {1}{3} \left (\frac {1}{7} \left (\frac {1}{5} \left (\int \frac {\left (315 a+\frac {2924}{x}\right ) x}{\sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{5/2}}d\frac {1}{x}+\frac {1462 a}{\sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{3/2}}+\frac {1147 a}{3 \left (1-\frac {1}{a x}\right )^{3/2} \left (\frac {1}{a x}+1\right )^{3/2}}\right )+\frac {208 a}{5 \left (1-\frac {1}{a x}\right )^{5/2} \left (\frac {1}{a x}+1\right )^{3/2}}\right )+\frac {29 a}{7 \left (1-\frac {1}{a x}\right )^{7/2} \left (\frac {1}{a x}+1\right )^{3/2}}\right )+\frac {10 a}{9 \left (1-\frac {1}{a x}\right )^{9/2} \left (\frac {1}{a x}+1\right )^{3/2}}}{a^2}-\frac {x}{\left (1-\frac {1}{a x}\right )^{9/2} \left (\frac {1}{a x}+1\right )^{3/2}}}{c^4}\) |
| \(\Big \downarrow \) 169 |
| \(\displaystyle -\frac {\frac {\frac {1}{3} \left (\frac {1}{7} \left (\frac {1}{5} \left (\frac {1}{3} a \int \frac {\left (945 a+\frac {2609}{x}\right ) x}{a \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{3/2}}d\frac {1}{x}-\frac {2609 a \sqrt {1-\frac {1}{a x}}}{3 \left (\frac {1}{a x}+1\right )^{3/2}}+\frac {1462 a}{\sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{3/2}}+\frac {1147 a}{3 \left (1-\frac {1}{a x}\right )^{3/2} \left (\frac {1}{a x}+1\right )^{3/2}}\right )+\frac {208 a}{5 \left (1-\frac {1}{a x}\right )^{5/2} \left (\frac {1}{a x}+1\right )^{3/2}}\right )+\frac {29 a}{7 \left (1-\frac {1}{a x}\right )^{7/2} \left (\frac {1}{a x}+1\right )^{3/2}}\right )+\frac {10 a}{9 \left (1-\frac {1}{a x}\right )^{9/2} \left (\frac {1}{a x}+1\right )^{3/2}}}{a^2}-\frac {x}{\left (1-\frac {1}{a x}\right )^{9/2} \left (\frac {1}{a x}+1\right )^{3/2}}}{c^4}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\frac {\frac {1}{3} \left (\frac {1}{7} \left (\frac {1}{5} \left (\frac {1}{3} \int \frac {\left (945 a+\frac {2609}{x}\right ) x}{\sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{3/2}}d\frac {1}{x}-\frac {2609 a \sqrt {1-\frac {1}{a x}}}{3 \left (\frac {1}{a x}+1\right )^{3/2}}+\frac {1462 a}{\sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{3/2}}+\frac {1147 a}{3 \left (1-\frac {1}{a x}\right )^{3/2} \left (\frac {1}{a x}+1\right )^{3/2}}\right )+\frac {208 a}{5 \left (1-\frac {1}{a x}\right )^{5/2} \left (\frac {1}{a x}+1\right )^{3/2}}\right )+\frac {29 a}{7 \left (1-\frac {1}{a x}\right )^{7/2} \left (\frac {1}{a x}+1\right )^{3/2}}\right )+\frac {10 a}{9 \left (1-\frac {1}{a x}\right )^{9/2} \left (\frac {1}{a x}+1\right )^{3/2}}}{a^2}-\frac {x}{\left (1-\frac {1}{a x}\right )^{9/2} \left (\frac {1}{a x}+1\right )^{3/2}}}{c^4}\) |
| \(\Big \downarrow \) 169 |
| \(\displaystyle -\frac {\frac {\frac {1}{3} \left (\frac {1}{7} \left (\frac {1}{5} \left (\frac {1}{3} \left (a \int \frac {945 x}{\sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}}d\frac {1}{x}-\frac {1664 a \sqrt {1-\frac {1}{a x}}}{\sqrt {\frac {1}{a x}+1}}\right )-\frac {2609 a \sqrt {1-\frac {1}{a x}}}{3 \left (\frac {1}{a x}+1\right )^{3/2}}+\frac {1462 a}{\sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{3/2}}+\frac {1147 a}{3 \left (1-\frac {1}{a x}\right )^{3/2} \left (\frac {1}{a x}+1\right )^{3/2}}\right )+\frac {208 a}{5 \left (1-\frac {1}{a x}\right )^{5/2} \left (\frac {1}{a x}+1\right )^{3/2}}\right )+\frac {29 a}{7 \left (1-\frac {1}{a x}\right )^{7/2} \left (\frac {1}{a x}+1\right )^{3/2}}\right )+\frac {10 a}{9 \left (1-\frac {1}{a x}\right )^{9/2} \left (\frac {1}{a x}+1\right )^{3/2}}}{a^2}-\frac {x}{\left (1-\frac {1}{a x}\right )^{9/2} \left (\frac {1}{a x}+1\right )^{3/2}}}{c^4}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\frac {\frac {1}{3} \left (\frac {1}{7} \left (\frac {1}{5} \left (\frac {1}{3} \left (945 a \int \frac {x}{\sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}}d\frac {1}{x}-\frac {1664 a \sqrt {1-\frac {1}{a x}}}{\sqrt {\frac {1}{a x}+1}}\right )-\frac {2609 a \sqrt {1-\frac {1}{a x}}}{3 \left (\frac {1}{a x}+1\right )^{3/2}}+\frac {1462 a}{\sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{3/2}}+\frac {1147 a}{3 \left (1-\frac {1}{a x}\right )^{3/2} \left (\frac {1}{a x}+1\right )^{3/2}}\right )+\frac {208 a}{5 \left (1-\frac {1}{a x}\right )^{5/2} \left (\frac {1}{a x}+1\right )^{3/2}}\right )+\frac {29 a}{7 \left (1-\frac {1}{a x}\right )^{7/2} \left (\frac {1}{a x}+1\right )^{3/2}}\right )+\frac {10 a}{9 \left (1-\frac {1}{a x}\right )^{9/2} \left (\frac {1}{a x}+1\right )^{3/2}}}{a^2}-\frac {x}{\left (1-\frac {1}{a x}\right )^{9/2} \left (\frac {1}{a x}+1\right )^{3/2}}}{c^4}\) |
| \(\Big \downarrow \) 103 |
| \(\displaystyle -\frac {\frac {\frac {1}{3} \left (\frac {1}{7} \left (\frac {1}{5} \left (\frac {1}{3} \left (-945 \int \frac {1}{\frac {1}{a}-\frac {1}{a x^2}}d\left (\sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}\right )-\frac {1664 a \sqrt {1-\frac {1}{a x}}}{\sqrt {\frac {1}{a x}+1}}\right )-\frac {2609 a \sqrt {1-\frac {1}{a x}}}{3 \left (\frac {1}{a x}+1\right )^{3/2}}+\frac {1462 a}{\sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{3/2}}+\frac {1147 a}{3 \left (1-\frac {1}{a x}\right )^{3/2} \left (\frac {1}{a x}+1\right )^{3/2}}\right )+\frac {208 a}{5 \left (1-\frac {1}{a x}\right )^{5/2} \left (\frac {1}{a x}+1\right )^{3/2}}\right )+\frac {29 a}{7 \left (1-\frac {1}{a x}\right )^{7/2} \left (\frac {1}{a x}+1\right )^{3/2}}\right )+\frac {10 a}{9 \left (1-\frac {1}{a x}\right )^{9/2} \left (\frac {1}{a x}+1\right )^{3/2}}}{a^2}-\frac {x}{\left (1-\frac {1}{a x}\right )^{9/2} \left (\frac {1}{a x}+1\right )^{3/2}}}{c^4}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle -\frac {\frac {\frac {1}{3} \left (\frac {1}{7} \left (\frac {1}{5} \left (\frac {1}{3} \left (-945 a \text {arctanh}\left (\sqrt {1-\frac {1}{a x}} \sqrt {\frac {1}{a x}+1}\right )-\frac {1664 a \sqrt {1-\frac {1}{a x}}}{\sqrt {\frac {1}{a x}+1}}\right )-\frac {2609 a \sqrt {1-\frac {1}{a x}}}{3 \left (\frac {1}{a x}+1\right )^{3/2}}+\frac {1462 a}{\sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{3/2}}+\frac {1147 a}{3 \left (1-\frac {1}{a x}\right )^{3/2} \left (\frac {1}{a x}+1\right )^{3/2}}\right )+\frac {208 a}{5 \left (1-\frac {1}{a x}\right )^{5/2} \left (\frac {1}{a x}+1\right )^{3/2}}\right )+\frac {29 a}{7 \left (1-\frac {1}{a x}\right )^{7/2} \left (\frac {1}{a x}+1\right )^{3/2}}\right )+\frac {10 a}{9 \left (1-\frac {1}{a x}\right )^{9/2} \left (\frac {1}{a x}+1\right )^{3/2}}}{a^2}-\frac {x}{\left (1-\frac {1}{a x}\right )^{9/2} \left (\frac {1}{a x}+1\right )^{3/2}}}{c^4}\) |
-((-(x/((1 - 1/(a*x))^(9/2)*(1 + 1/(a*x))^(3/2))) + ((10*a)/(9*(1 - 1/(a*x ))^(9/2)*(1 + 1/(a*x))^(3/2)) + ((29*a)/(7*(1 - 1/(a*x))^(7/2)*(1 + 1/(a*x ))^(3/2)) + ((208*a)/(5*(1 - 1/(a*x))^(5/2)*(1 + 1/(a*x))^(3/2)) + ((1147* a)/(3*(1 - 1/(a*x))^(3/2)*(1 + 1/(a*x))^(3/2)) + (1462*a)/(Sqrt[1 - 1/(a*x )]*(1 + 1/(a*x))^(3/2)) - (2609*a*Sqrt[1 - 1/(a*x)])/(3*(1 + 1/(a*x))^(3/2 )) + ((-1664*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)/3)/a^2)/c^4)
3.8.96.3.1 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 = 373, normalized size of antiderivative = 1.13
| method | result | size |
| risch | \(\frac {a x -1}{a \,c^{4} \sqrt {\frac {a x -1}{a x +1}}}+\frac {\left (\frac {3 \ln \left (\frac {a^{2} x}{\sqrt {a^{2}}}+\sqrt {a^{2} x^{2}-1}\right )}{a^{8} \sqrt {a^{2}}}-\frac {\sqrt {\left (x -\frac {1}{a}\right )^{2} a^{2}+2 \left (x -\frac {1}{a}\right ) a}}{36 a^{14} \left (x -\frac {1}{a}\right )^{5}}-\frac {59 \sqrt {\left (x -\frac {1}{a}\right )^{2} a^{2}+2 \left (x -\frac {1}{a}\right ) a}}{252 a^{13} \left (x -\frac {1}{a}\right )^{4}}-\frac {1507 \sqrt {\left (x -\frac {1}{a}\right )^{2} a^{2}+2 \left (x -\frac {1}{a}\right ) a}}{1680 a^{12} \left (x -\frac {1}{a}\right )^{3}}-\frac {691 \sqrt {\left (x -\frac {1}{a}\right )^{2} a^{2}+2 \left (x -\frac {1}{a}\right ) a}}{315 a^{11} \left (x -\frac {1}{a}\right )^{2}}-\frac {113591 \sqrt {\left (x -\frac {1}{a}\right )^{2} a^{2}+2 \left (x -\frac {1}{a}\right ) a}}{20160 a^{10} \left (x -\frac {1}{a}\right )}-\frac {\sqrt {a^{2} \left (x +\frac {1}{a}\right )^{2}-2 a \left (x +\frac {1}{a}\right )}}{96 a^{11} \left (x +\frac {1}{a}\right )^{2}}+\frac {31 \sqrt {a^{2} \left (x +\frac {1}{a}\right )^{2}-2 a \left (x +\frac {1}{a}\right )}}{192 a^{10} \left (x +\frac {1}{a}\right )}\right ) a^{8} \sqrt {\left (a x -1\right ) \left (a x +1\right )}}{c^{4} \sqrt {\frac {a x -1}{a x +1}}\, \left (a x +1\right )}\) | \(373\) |
| default | \(-\frac {-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 )}{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} \left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}}}\) | \(766\) |
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)-1/36/a^14/(x-1/a)^5*((x-1/a)^2*a^2+2*(x-1/a)*a)^( 1/2)-59/252/a^13/(x-1/a)^4*((x-1/a)^2*a^2+2*(x-1/a)*a)^(1/2)-1507/1680/a^1 2/(x-1/a)^3*((x-1/a)^2*a^2+2*(x-1/a)*a)^(1/2)-691/315/a^11/(x-1/a)^2*((x-1 /a)^2*a^2+2*(x-1/a)*a)^(1/2)-113591/20160/a^10/(x-1/a)*((x-1/a)^2*a^2+2*(x -1/a)*a)^(1/2)-1/96/a^11/(x+1/a)^2*(a^2*(x+1/a)^2-2*a*(x+1/a))^(1/2)+31/19 2/a^10/(x+1/a)*(a^2*(x+1/a)^2-2*a*(x+1/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 = 0.26 (sec) , antiderivative size = 248, normalized size of antiderivative = 0.75 \[ \int \frac {e^{3 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^4} \, dx=\frac {945 \, {\left (a^{6} x^{6} - 4 \, a^{5} x^{5} + 5 \, a^{4} x^{4} - 5 \, a^{2} x^{2} + 4 \, a x - 1\right )} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right ) - 945 \, {\left (a^{6} x^{6} - 4 \, a^{5} x^{5} + 5 \, a^{4} x^{4} - 5 \, a^{2} x^{2} + 4 \, a x - 1\right )} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right ) + {\left (315 \, a^{7} x^{7} - 2669 \, a^{6} x^{6} + 2967 \, a^{5} x^{5} + 4029 \, a^{4} x^{4} - 7399 \, a^{3} x^{3} + 339 \, a^{2} x^{2} + 4047 \, a x - 1664\right )} \sqrt {\frac {a x - 1}{a x + 1}}}{315 \, {\left (a^{7} c^{4} x^{6} - 4 \, a^{6} c^{4} x^{5} + 5 \, a^{5} c^{4} x^{4} - 5 \, a^{3} c^{4} x^{2} + 4 \, a^{2} c^{4} x - a c^{4}\right )}} \]
1/315*(945*(a^6*x^6 - 4*a^5*x^5 + 5*a^4*x^4 - 5*a^2*x^2 + 4*a*x - 1)*log(s qrt((a*x - 1)/(a*x + 1)) + 1) - 945*(a^6*x^6 - 4*a^5*x^5 + 5*a^4*x^4 - 5*a ^2*x^2 + 4*a*x - 1)*log(sqrt((a*x - 1)/(a*x + 1)) - 1) + (315*a^7*x^7 - 26 69*a^6*x^6 + 2967*a^5*x^5 + 4029*a^4*x^4 - 7399*a^3*x^3 + 339*a^2*x^2 + 40 47*a*x - 1664)*sqrt((a*x - 1)/(a*x + 1)))/(a^7*c^4*x^6 - 4*a^6*c^4*x^5 + 5 *a^5*c^4*x^4 - 5*a^3*c^4*x^2 + 4*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 = 226, normalized size of antiderivative = 0.69 \[ \int \frac {e^{3 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^4} \, dx=\frac {1}{20160} \, a {\left (\frac {\frac {415 \, {\left (a x - 1\right )}}{a x + 1} + \frac {2511 \, {\left (a x - 1\right )}^{2}}{{\left (a x + 1\right )}^{2}} + \frac {11739 \, {\left (a x - 1\right )}^{3}}{{\left (a x + 1\right )}^{3}} + \frac {80745 \, {\left (a x - 1\right )}^{4}}{{\left (a x + 1\right )}^{4}} - \frac {135765 \, {\left (a x - 1\right )}^{5}}{{\left (a x + 1\right )}^{5}} + 35}{a^{2} c^{4} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {11}{2}} - a^{2} c^{4} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {9}{2}}} + \frac {105 \, {\left (\left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}} + 30 \, \sqrt {\frac {a x - 1}{a x + 1}}\right )}}{a^{2} c^{4}} + \frac {60480 \, \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right )}{a^{2} c^{4}} - \frac {60480 \, \log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right )}{a^{2} c^{4}}\right )} \]
1/20160*a*((415*(a*x - 1)/(a*x + 1) + 2511*(a*x - 1)^2/(a*x + 1)^2 + 11739 *(a*x - 1)^3/(a*x + 1)^3 + 80745*(a*x - 1)^4/(a*x + 1)^4 - 135765*(a*x - 1 )^5/(a*x + 1)^5 + 35)/(a^2*c^4*((a*x - 1)/(a*x + 1))^(11/2) - a^2*c^4*((a* x - 1)/(a*x + 1))^(9/2)) + 105*(((a*x - 1)/(a*x + 1))^(3/2) + 30*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))
\[ \int \frac {e^{3 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^4} \, dx=\int { \frac {1}{{\left (c - \frac {c}{a^{2} x^{2}}\right )}^{4} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}}} \,d x } \]
Time = 0.11 (sec) , antiderivative size = 203, normalized size of antiderivative = 0.62 \[ \int \frac {e^{3 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^4} \, dx=\frac {5\,\sqrt {\frac {a\,x-1}{a\,x+1}}}{32\,a\,c^4}-\frac {\frac {279\,{\left (a\,x-1\right )}^2}{35\,{\left (a\,x+1\right )}^2}+\frac {559\,{\left (a\,x-1\right )}^3}{15\,{\left (a\,x+1\right )}^3}+\frac {769\,{\left (a\,x-1\right )}^4}{3\,{\left (a\,x+1\right )}^4}-\frac {431\,{\left (a\,x-1\right )}^5}{{\left (a\,x+1\right )}^5}+\frac {83\,\left (a\,x-1\right )}{63\,\left (a\,x+1\right )}+\frac {1}{9}}{64\,a\,c^4\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{9/2}-64\,a\,c^4\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{11/2}}+\frac {{\left (\frac {a\,x-1}{a\,x+1}\right )}^{3/2}}{192\,a\,c^4}-\frac {\mathrm {atan}\left (\sqrt {\frac {a\,x-1}{a\,x+1}}\,1{}\mathrm {i}\right )\,6{}\mathrm {i}}{a\,c^4} \]
(5*((a*x - 1)/(a*x + 1))^(1/2))/(32*a*c^4) - ((279*(a*x - 1)^2)/(35*(a*x + 1)^2) + (559*(a*x - 1)^3)/(15*(a*x + 1)^3) + (769*(a*x - 1)^4)/(3*(a*x + 1)^4) - (431*(a*x - 1)^5)/(a*x + 1)^5 + (83*(a*x - 1))/(63*(a*x + 1)) + 1/ 9)/(64*a*c^4*((a*x - 1)/(a*x + 1))^(9/2) - 64*a*c^4*((a*x - 1)/(a*x + 1))^ (11/2)) + ((a*x - 1)/(a*x + 1))^(3/2)/(192*a*c^4) - (atan(((a*x - 1)/(a*x + 1))^(1/2)*1i)*6i)/(a*c^4)