Integrand size = 22, antiderivative size = 329 \[ \int \frac {e^{-\coth ^{-1}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^4} \, dx=-\frac {6}{5 a c^4 \left (1-\frac {1}{a x}\right )^{5/2} \left (1+\frac {1}{a x}\right )^{7/2}}-\frac {31}{15 a c^4 \left (1-\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{7/2}}-\frac {28}{3 a c^4 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{7/2}}+\frac {115 \sqrt {1-\frac {1}{a x}}}{21 a c^4 \left (1+\frac {1}{a x}\right )^{7/2}}+\frac {122 \sqrt {1-\frac {1}{a x}}}{35 a c^4 \left (1+\frac {1}{a x}\right )^{5/2}}+\frac {93 \sqrt {1-\frac {1}{a x}}}{35 a c^4 \left (1+\frac {1}{a x}\right )^{3/2}}+\frac {128 \sqrt {1-\frac {1}{a x}}}{35 a c^4 \sqrt {1+\frac {1}{a x}}}+\frac {x}{c^4 \left (1-\frac {1}{a x}\right )^{5/2} \left (1+\frac {1}{a x}\right )^{7/2}}-\frac {\text {arctanh}\left (\sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}\right )}{a c^4} \]
-6/5/a/c^4/(1-1/a/x)^(5/2)/(1+1/a/x)^(7/2)-31/15/a/c^4/(1-1/a/x)^(3/2)/(1+ 1/a/x)^(7/2)+x/c^4/(1-1/a/x)^(5/2)/(1+1/a/x)^(7/2)-arctanh((1-1/a/x)^(1/2) *(1+1/a/x)^(1/2))/a/c^4-28/3/a/c^4/(1+1/a/x)^(7/2)/(1-1/a/x)^(1/2)+115/21* (1-1/a/x)^(1/2)/a/c^4/(1+1/a/x)^(7/2)+122/35*(1-1/a/x)^(1/2)/a/c^4/(1+1/a/ x)^(5/2)+93/35*(1-1/a/x)^(1/2)/a/c^4/(1+1/a/x)^(3/2)+128/35*(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 \[ \int \frac {e^{-\coth ^{-1}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^4} \, dx=\frac {\frac {a \sqrt {1-\frac {1}{a^2 x^2}} x \left (-384-279 a x+1065 a^2 x^2+715 a^3 x^3-965 a^4 x^4-559 a^5 x^5+281 a^6 x^6+105 a^7 x^7\right )}{105 (-1+a x)^3 (1+a x)^4}-\log \left (\left (1+\sqrt {1-\frac {1}{a^2 x^2}}\right ) x\right )}{a c^4} \]
((a*Sqrt[1 - 1/(a^2*x^2)]*x*(-384 - 279*a*x + 1065*a^2*x^2 + 715*a^3*x^3 - 965*a^4*x^4 - 559*a^5*x^5 + 281*a^6*x^6 + 105*a^7*x^7))/(105*(-1 + a*x)^3 *(1 + a*x)^4) - Log[(1 + Sqrt[1 - 1/(a^2*x^2)])*x])/(a*c^4)
Time = 0.46 (sec) , antiderivative size = 312, normalized size of antiderivative = 0.95, number of steps used = 22, number of rules used = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.955, Rules used = {6748, 114, 27, 169, 25, 27, 169, 27, 169, 25, 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.
| \(\displaystyle \int \frac {e^{-\coth ^{-1}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^4} \, dx\) |
| \(\Big \downarrow \) 6748 |
| \(\displaystyle -\frac {\int \frac {x^2}{\left (1-\frac {1}{a x}\right )^{7/2} \left (1+\frac {1}{a x}\right )^{9/2}}d\frac {1}{x}}{c^4}\) |
| \(\Big \downarrow \) 114 |
| \(\displaystyle -\frac {-\int \frac {\left (a-\frac {7}{x}\right ) x}{a^2 \left (1-\frac {1}{a x}\right )^{7/2} \left (1+\frac {1}{a x}\right )^{9/2}}d\frac {1}{x}-\frac {x}{\left (1-\frac {1}{a x}\right )^{5/2} \left (\frac {1}{a x}+1\right )^{7/2}}}{c^4}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {-\frac {\int \frac {\left (a-\frac {7}{x}\right ) x}{\left (1-\frac {1}{a x}\right )^{7/2} \left (1+\frac {1}{a x}\right )^{9/2}}d\frac {1}{x}}{a^2}-\frac {x}{\left (1-\frac {1}{a x}\right )^{5/2} \left (\frac {1}{a x}+1\right )^{7/2}}}{c^4}\) |
| \(\Big \downarrow \) 169 |
| \(\displaystyle -\frac {-\frac {-\frac {1}{5} a \int -\frac {\left (5 a-\frac {36}{x}\right ) x}{a \left (1-\frac {1}{a x}\right )^{5/2} \left (1+\frac {1}{a x}\right )^{9/2}}d\frac {1}{x}-\frac {6 a}{5 \left (1-\frac {1}{a x}\right )^{5/2} \left (\frac {1}{a x}+1\right )^{7/2}}}{a^2}-\frac {x}{\left (1-\frac {1}{a x}\right )^{5/2} \left (\frac {1}{a x}+1\right )^{7/2}}}{c^4}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {-\frac {\frac {1}{5} a \int \frac {\left (5 a-\frac {36}{x}\right ) x}{a \left (1-\frac {1}{a x}\right )^{5/2} \left (1+\frac {1}{a x}\right )^{9/2}}d\frac {1}{x}-\frac {6 a}{5 \left (1-\frac {1}{a x}\right )^{5/2} \left (\frac {1}{a x}+1\right )^{7/2}}}{a^2}-\frac {x}{\left (1-\frac {1}{a x}\right )^{5/2} \left (\frac {1}{a x}+1\right )^{7/2}}}{c^4}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {-\frac {\frac {1}{5} \int \frac {\left (5 a-\frac {36}{x}\right ) x}{\left (1-\frac {1}{a x}\right )^{5/2} \left (1+\frac {1}{a x}\right )^{9/2}}d\frac {1}{x}-\frac {6 a}{5 \left (1-\frac {1}{a x}\right )^{5/2} \left (\frac {1}{a x}+1\right )^{7/2}}}{a^2}-\frac {x}{\left (1-\frac {1}{a x}\right )^{5/2} \left (\frac {1}{a x}+1\right )^{7/2}}}{c^4}\) |
| \(\Big \downarrow \) 169 |
| \(\displaystyle -\frac {-\frac {\frac {1}{5} \left (-\frac {1}{3} a \int -\frac {5 \left (3 a-\frac {31}{x}\right ) x}{a \left (1-\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{9/2}}d\frac {1}{x}-\frac {31 a}{3 \left (1-\frac {1}{a x}\right )^{3/2} \left (\frac {1}{a x}+1\right )^{7/2}}\right )-\frac {6 a}{5 \left (1-\frac {1}{a x}\right )^{5/2} \left (\frac {1}{a x}+1\right )^{7/2}}}{a^2}-\frac {x}{\left (1-\frac {1}{a x}\right )^{5/2} \left (\frac {1}{a x}+1\right )^{7/2}}}{c^4}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {-\frac {\frac {1}{5} \left (\frac {5}{3} \int \frac {\left (3 a-\frac {31}{x}\right ) x}{\left (1-\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{9/2}}d\frac {1}{x}-\frac {31 a}{3 \left (1-\frac {1}{a x}\right )^{3/2} \left (\frac {1}{a x}+1\right )^{7/2}}\right )-\frac {6 a}{5 \left (1-\frac {1}{a x}\right )^{5/2} \left (\frac {1}{a x}+1\right )^{7/2}}}{a^2}-\frac {x}{\left (1-\frac {1}{a x}\right )^{5/2} \left (\frac {1}{a x}+1\right )^{7/2}}}{c^4}\) |
| \(\Big \downarrow \) 169 |
| \(\displaystyle -\frac {-\frac {\frac {1}{5} \left (\frac {5}{3} \left (a \left (-\int -\frac {\left (3 a-\frac {112}{x}\right ) x}{a \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{9/2}}d\frac {1}{x}\right )-\frac {28 a}{\sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{7/2}}\right )-\frac {31 a}{3 \left (1-\frac {1}{a x}\right )^{3/2} \left (\frac {1}{a x}+1\right )^{7/2}}\right )-\frac {6 a}{5 \left (1-\frac {1}{a x}\right )^{5/2} \left (\frac {1}{a x}+1\right )^{7/2}}}{a^2}-\frac {x}{\left (1-\frac {1}{a x}\right )^{5/2} \left (\frac {1}{a x}+1\right )^{7/2}}}{c^4}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {-\frac {\frac {1}{5} \left (\frac {5}{3} \left (a \int \frac {\left (3 a-\frac {112}{x}\right ) x}{a \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{9/2}}d\frac {1}{x}-\frac {28 a}{\sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{7/2}}\right )-\frac {31 a}{3 \left (1-\frac {1}{a x}\right )^{3/2} \left (\frac {1}{a x}+1\right )^{7/2}}\right )-\frac {6 a}{5 \left (1-\frac {1}{a x}\right )^{5/2} \left (\frac {1}{a x}+1\right )^{7/2}}}{a^2}-\frac {x}{\left (1-\frac {1}{a x}\right )^{5/2} \left (\frac {1}{a x}+1\right )^{7/2}}}{c^4}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {-\frac {\frac {1}{5} \left (\frac {5}{3} \left (\int \frac {\left (3 a-\frac {112}{x}\right ) x}{\sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{9/2}}d\frac {1}{x}-\frac {28 a}{\sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{7/2}}\right )-\frac {31 a}{3 \left (1-\frac {1}{a x}\right )^{3/2} \left (\frac {1}{a x}+1\right )^{7/2}}\right )-\frac {6 a}{5 \left (1-\frac {1}{a x}\right )^{5/2} \left (\frac {1}{a x}+1\right )^{7/2}}}{a^2}-\frac {x}{\left (1-\frac {1}{a x}\right )^{5/2} \left (\frac {1}{a x}+1\right )^{7/2}}}{c^4}\) |
| \(\Big \downarrow \) 169 |
| \(\displaystyle -\frac {-\frac {\frac {1}{5} \left (\frac {5}{3} \left (\frac {1}{7} a \int \frac {3 \left (7 a-\frac {115}{x}\right ) x}{a \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{7/2}}d\frac {1}{x}+\frac {115 a \sqrt {1-\frac {1}{a x}}}{7 \left (\frac {1}{a x}+1\right )^{7/2}}-\frac {28 a}{\sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{7/2}}\right )-\frac {31 a}{3 \left (1-\frac {1}{a x}\right )^{3/2} \left (\frac {1}{a x}+1\right )^{7/2}}\right )-\frac {6 a}{5 \left (1-\frac {1}{a x}\right )^{5/2} \left (\frac {1}{a x}+1\right )^{7/2}}}{a^2}-\frac {x}{\left (1-\frac {1}{a x}\right )^{5/2} \left (\frac {1}{a x}+1\right )^{7/2}}}{c^4}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {-\frac {\frac {1}{5} \left (\frac {5}{3} \left (\frac {3}{7} \int \frac {\left (7 a-\frac {115}{x}\right ) x}{\sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{7/2}}d\frac {1}{x}+\frac {115 a \sqrt {1-\frac {1}{a x}}}{7 \left (\frac {1}{a x}+1\right )^{7/2}}-\frac {28 a}{\sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{7/2}}\right )-\frac {31 a}{3 \left (1-\frac {1}{a x}\right )^{3/2} \left (\frac {1}{a x}+1\right )^{7/2}}\right )-\frac {6 a}{5 \left (1-\frac {1}{a x}\right )^{5/2} \left (\frac {1}{a x}+1\right )^{7/2}}}{a^2}-\frac {x}{\left (1-\frac {1}{a x}\right )^{5/2} \left (\frac {1}{a x}+1\right )^{7/2}}}{c^4}\) |
| \(\Big \downarrow \) 169 |
| \(\displaystyle -\frac {-\frac {\frac {1}{5} \left (\frac {5}{3} \left (\frac {3}{7} \left (\frac {1}{5} a \int \frac {\left (35 a-\frac {244}{x}\right ) x}{a \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{5/2}}d\frac {1}{x}+\frac {122 a \sqrt {1-\frac {1}{a x}}}{5 \left (\frac {1}{a x}+1\right )^{5/2}}\right )+\frac {115 a \sqrt {1-\frac {1}{a x}}}{7 \left (\frac {1}{a x}+1\right )^{7/2}}-\frac {28 a}{\sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{7/2}}\right )-\frac {31 a}{3 \left (1-\frac {1}{a x}\right )^{3/2} \left (\frac {1}{a x}+1\right )^{7/2}}\right )-\frac {6 a}{5 \left (1-\frac {1}{a x}\right )^{5/2} \left (\frac {1}{a x}+1\right )^{7/2}}}{a^2}-\frac {x}{\left (1-\frac {1}{a x}\right )^{5/2} \left (\frac {1}{a x}+1\right )^{7/2}}}{c^4}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {-\frac {\frac {1}{5} \left (\frac {5}{3} \left (\frac {3}{7} \left (\frac {1}{5} \int \frac {\left (35 a-\frac {244}{x}\right ) x}{\sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{5/2}}d\frac {1}{x}+\frac {122 a \sqrt {1-\frac {1}{a x}}}{5 \left (\frac {1}{a x}+1\right )^{5/2}}\right )+\frac {115 a \sqrt {1-\frac {1}{a x}}}{7 \left (\frac {1}{a x}+1\right )^{7/2}}-\frac {28 a}{\sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{7/2}}\right )-\frac {31 a}{3 \left (1-\frac {1}{a x}\right )^{3/2} \left (\frac {1}{a x}+1\right )^{7/2}}\right )-\frac {6 a}{5 \left (1-\frac {1}{a x}\right )^{5/2} \left (\frac {1}{a x}+1\right )^{7/2}}}{a^2}-\frac {x}{\left (1-\frac {1}{a x}\right )^{5/2} \left (\frac {1}{a x}+1\right )^{7/2}}}{c^4}\) |
| \(\Big \downarrow \) 169 |
| \(\displaystyle -\frac {-\frac {\frac {1}{5} \left (\frac {5}{3} \left (\frac {3}{7} \left (\frac {1}{5} \left (\frac {1}{3} a \int \frac {3 \left (35 a-\frac {93}{x}\right ) x}{a \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{3/2}}d\frac {1}{x}+\frac {93 a \sqrt {1-\frac {1}{a x}}}{\left (\frac {1}{a x}+1\right )^{3/2}}\right )+\frac {122 a \sqrt {1-\frac {1}{a x}}}{5 \left (\frac {1}{a x}+1\right )^{5/2}}\right )+\frac {115 a \sqrt {1-\frac {1}{a x}}}{7 \left (\frac {1}{a x}+1\right )^{7/2}}-\frac {28 a}{\sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{7/2}}\right )-\frac {31 a}{3 \left (1-\frac {1}{a x}\right )^{3/2} \left (\frac {1}{a x}+1\right )^{7/2}}\right )-\frac {6 a}{5 \left (1-\frac {1}{a x}\right )^{5/2} \left (\frac {1}{a x}+1\right )^{7/2}}}{a^2}-\frac {x}{\left (1-\frac {1}{a x}\right )^{5/2} \left (\frac {1}{a x}+1\right )^{7/2}}}{c^4}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {-\frac {\frac {1}{5} \left (\frac {5}{3} \left (\frac {3}{7} \left (\frac {1}{5} \left (\int \frac {\left (35 a-\frac {93}{x}\right ) x}{\sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{3/2}}d\frac {1}{x}+\frac {93 a \sqrt {1-\frac {1}{a x}}}{\left (\frac {1}{a x}+1\right )^{3/2}}\right )+\frac {122 a \sqrt {1-\frac {1}{a x}}}{5 \left (\frac {1}{a x}+1\right )^{5/2}}\right )+\frac {115 a \sqrt {1-\frac {1}{a x}}}{7 \left (\frac {1}{a x}+1\right )^{7/2}}-\frac {28 a}{\sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{7/2}}\right )-\frac {31 a}{3 \left (1-\frac {1}{a x}\right )^{3/2} \left (\frac {1}{a x}+1\right )^{7/2}}\right )-\frac {6 a}{5 \left (1-\frac {1}{a x}\right )^{5/2} \left (\frac {1}{a x}+1\right )^{7/2}}}{a^2}-\frac {x}{\left (1-\frac {1}{a x}\right )^{5/2} \left (\frac {1}{a x}+1\right )^{7/2}}}{c^4}\) |
| \(\Big \downarrow \) 169 |
| \(\displaystyle -\frac {-\frac {\frac {1}{5} \left (\frac {5}{3} \left (\frac {3}{7} \left (\frac {1}{5} \left (a \int \frac {35 x}{\sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}}d\frac {1}{x}+\frac {128 a \sqrt {1-\frac {1}{a x}}}{\sqrt {\frac {1}{a x}+1}}+\frac {93 a \sqrt {1-\frac {1}{a x}}}{\left (\frac {1}{a x}+1\right )^{3/2}}\right )+\frac {122 a \sqrt {1-\frac {1}{a x}}}{5 \left (\frac {1}{a x}+1\right )^{5/2}}\right )+\frac {115 a \sqrt {1-\frac {1}{a x}}}{7 \left (\frac {1}{a x}+1\right )^{7/2}}-\frac {28 a}{\sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{7/2}}\right )-\frac {31 a}{3 \left (1-\frac {1}{a x}\right )^{3/2} \left (\frac {1}{a x}+1\right )^{7/2}}\right )-\frac {6 a}{5 \left (1-\frac {1}{a x}\right )^{5/2} \left (\frac {1}{a x}+1\right )^{7/2}}}{a^2}-\frac {x}{\left (1-\frac {1}{a x}\right )^{5/2} \left (\frac {1}{a x}+1\right )^{7/2}}}{c^4}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {-\frac {\frac {1}{5} \left (\frac {5}{3} \left (\frac {3}{7} \left (\frac {1}{5} \left (35 a \int \frac {x}{\sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}}d\frac {1}{x}+\frac {128 a \sqrt {1-\frac {1}{a x}}}{\sqrt {\frac {1}{a x}+1}}+\frac {93 a \sqrt {1-\frac {1}{a x}}}{\left (\frac {1}{a x}+1\right )^{3/2}}\right )+\frac {122 a \sqrt {1-\frac {1}{a x}}}{5 \left (\frac {1}{a x}+1\right )^{5/2}}\right )+\frac {115 a \sqrt {1-\frac {1}{a x}}}{7 \left (\frac {1}{a x}+1\right )^{7/2}}-\frac {28 a}{\sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{7/2}}\right )-\frac {31 a}{3 \left (1-\frac {1}{a x}\right )^{3/2} \left (\frac {1}{a x}+1\right )^{7/2}}\right )-\frac {6 a}{5 \left (1-\frac {1}{a x}\right )^{5/2} \left (\frac {1}{a x}+1\right )^{7/2}}}{a^2}-\frac {x}{\left (1-\frac {1}{a x}\right )^{5/2} \left (\frac {1}{a x}+1\right )^{7/2}}}{c^4}\) |
| \(\Big \downarrow \) 103 |
| \(\displaystyle -\frac {-\frac {\frac {1}{5} \left (\frac {5}{3} \left (\frac {3}{7} \left (\frac {1}{5} \left (-35 \int \frac {1}{\frac {1}{a}-\frac {1}{a x^2}}d\left (\sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}\right )+\frac {128 a \sqrt {1-\frac {1}{a x}}}{\sqrt {\frac {1}{a x}+1}}+\frac {93 a \sqrt {1-\frac {1}{a x}}}{\left (\frac {1}{a x}+1\right )^{3/2}}\right )+\frac {122 a \sqrt {1-\frac {1}{a x}}}{5 \left (\frac {1}{a x}+1\right )^{5/2}}\right )+\frac {115 a \sqrt {1-\frac {1}{a x}}}{7 \left (\frac {1}{a x}+1\right )^{7/2}}-\frac {28 a}{\sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{7/2}}\right )-\frac {31 a}{3 \left (1-\frac {1}{a x}\right )^{3/2} \left (\frac {1}{a x}+1\right )^{7/2}}\right )-\frac {6 a}{5 \left (1-\frac {1}{a x}\right )^{5/2} \left (\frac {1}{a x}+1\right )^{7/2}}}{a^2}-\frac {x}{\left (1-\frac {1}{a x}\right )^{5/2} \left (\frac {1}{a x}+1\right )^{7/2}}}{c^4}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle -\frac {-\frac {\frac {1}{5} \left (\frac {5}{3} \left (\frac {3}{7} \left (\frac {1}{5} \left (-35 a \text {arctanh}\left (\sqrt {1-\frac {1}{a x}} \sqrt {\frac {1}{a x}+1}\right )+\frac {128 a \sqrt {1-\frac {1}{a x}}}{\sqrt {\frac {1}{a x}+1}}+\frac {93 a \sqrt {1-\frac {1}{a x}}}{\left (\frac {1}{a x}+1\right )^{3/2}}\right )+\frac {122 a \sqrt {1-\frac {1}{a x}}}{5 \left (\frac {1}{a x}+1\right )^{5/2}}\right )+\frac {115 a \sqrt {1-\frac {1}{a x}}}{7 \left (\frac {1}{a x}+1\right )^{7/2}}-\frac {28 a}{\sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{7/2}}\right )-\frac {31 a}{3 \left (1-\frac {1}{a x}\right )^{3/2} \left (\frac {1}{a x}+1\right )^{7/2}}\right )-\frac {6 a}{5 \left (1-\frac {1}{a x}\right )^{5/2} \left (\frac {1}{a x}+1\right )^{7/2}}}{a^2}-\frac {x}{\left (1-\frac {1}{a x}\right )^{5/2} \left (\frac {1}{a x}+1\right )^{7/2}}}{c^4}\) |
-((-(x/((1 - 1/(a*x))^(5/2)*(1 + 1/(a*x))^(7/2))) - ((-6*a)/(5*(1 - 1/(a*x ))^(5/2)*(1 + 1/(a*x))^(7/2)) + ((-31*a)/(3*(1 - 1/(a*x))^(3/2)*(1 + 1/(a* x))^(7/2)) + (5*((-28*a)/(Sqrt[1 - 1/(a*x)]*(1 + 1/(a*x))^(7/2)) + (115*a* Sqrt[1 - 1/(a*x)])/(7*(1 + 1/(a*x))^(7/2)) + (3*((122*a*Sqrt[1 - 1/(a*x)]) /(5*(1 + 1/(a*x))^(5/2)) + ((93*a*Sqrt[1 - 1/(a*x)])/(1 + 1/(a*x))^(3/2) + (128*a*Sqrt[1 - 1/(a*x)])/Sqrt[1 + 1/(a*x)] - 35*a*ArcTanh[Sqrt[1 - 1/(a* x)]*Sqrt[1 + 1/(a*x)]])/5))/7))/3)/5)/a^2)/c^4)
3.9.13.3.1 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 = 0.24 (sec) , antiderivative size = 361, normalized size of antiderivative = 1.10
| method | result | size |
| risch | \(\frac {\left (a x +1\right ) \sqrt {\frac {a x -1}{a x +1}}}{a \,c^{4}}+\frac {\left (-\frac {\ln \left (\frac {a^{2} x}{\sqrt {a^{2}}}+\sqrt {a^{2} x^{2}-1}\right )}{a^{8} \sqrt {a^{2}}}-\frac {\sqrt {a^{2} \left (x +\frac {1}{a}\right )^{2}-2 a \left (x +\frac {1}{a}\right )}}{56 a^{13} \left (x +\frac {1}{a}\right )^{4}}+\frac {17 \sqrt {a^{2} \left (x +\frac {1}{a}\right )^{2}-2 a \left (x +\frac {1}{a}\right )}}{112 a^{12} \left (x +\frac {1}{a}\right )^{3}}-\frac {211 \sqrt {a^{2} \left (x +\frac {1}{a}\right )^{2}-2 a \left (x +\frac {1}{a}\right )}}{336 a^{11} \left (x +\frac {1}{a}\right )^{2}}+\frac {1657 \sqrt {a^{2} \left (x +\frac {1}{a}\right )^{2}-2 a \left (x +\frac {1}{a}\right )}}{672 a^{10} \left (x +\frac {1}{a}\right )}-\frac {7 \sqrt {\left (x -\frac {1}{a}\right )^{2} a^{2}+2 \left (x -\frac {1}{a}\right ) a}}{60 a^{11} \left (x -\frac {1}{a}\right )^{2}}-\frac {379 \sqrt {\left (x -\frac {1}{a}\right )^{2} a^{2}+2 \left (x -\frac {1}{a}\right ) a}}{480 a^{10} \left (x -\frac {1}{a}\right )}-\frac {\sqrt {\left (x -\frac {1}{a}\right )^{2} a^{2}+2 \left (x -\frac {1}{a}\right ) a}}{80 a^{12} \left (x -\frac {1}{a}\right )^{3}}\right ) a^{8} \sqrt {\frac {a x -1}{a x +1}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}}{c^{4} \left (a x -1\right )}\) | \(361\) |
| default | \(\text {Expression too large to display}\) | \(898\) |
1/a*(a*x+1)/c^4*((a*x-1)/(a*x+1))^(1/2)+(-1/a^8*ln(a^2*x/(a^2)^(1/2)+(a^2* x^2-1)^(1/2))/(a^2)^(1/2)-1/56/a^13/(x+1/a)^4*(a^2*(x+1/a)^2-2*a*(x+1/a))^ (1/2)+17/112/a^12/(x+1/a)^3*(a^2*(x+1/a)^2-2*a*(x+1/a))^(1/2)-211/336/a^11 /(x+1/a)^2*(a^2*(x+1/a)^2-2*a*(x+1/a))^(1/2)+1657/672/a^10/(x+1/a)*(a^2*(x +1/a)^2-2*a*(x+1/a))^(1/2)-7/60/a^11/(x-1/a)^2*((x-1/a)^2*a^2+2*(x-1/a)*a) ^(1/2)-379/480/a^10/(x-1/a)*((x-1/a)^2*a^2+2*(x-1/a)*a)^(1/2)-1/80/a^12/(x -1/a)^3*((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 = 205, normalized size of antiderivative = 0.62 \[ \int \frac {e^{-\coth ^{-1}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^4} \, dx=-\frac {105 \, {\left (a^{6} x^{6} - 3 \, a^{4} x^{4} + 3 \, a^{2} x^{2} - 1\right )} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right ) - 105 \, {\left (a^{6} x^{6} - 3 \, a^{4} x^{4} + 3 \, a^{2} x^{2} - 1\right )} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right ) - {\left (105 \, a^{7} x^{7} + 281 \, a^{6} x^{6} - 559 \, a^{5} x^{5} - 965 \, a^{4} x^{4} + 715 \, a^{3} x^{3} + 1065 \, a^{2} x^{2} - 279 \, a x - 384\right )} \sqrt {\frac {a x - 1}{a x + 1}}}{105 \, {\left (a^{7} c^{4} x^{6} - 3 \, a^{5} c^{4} x^{4} + 3 \, a^{3} c^{4} x^{2} - a c^{4}\right )}} \]
-1/105*(105*(a^6*x^6 - 3*a^4*x^4 + 3*a^2*x^2 - 1)*log(sqrt((a*x - 1)/(a*x + 1)) + 1) - 105*(a^6*x^6 - 3*a^4*x^4 + 3*a^2*x^2 - 1)*log(sqrt((a*x - 1)/ (a*x + 1)) - 1) - (105*a^7*x^7 + 281*a^6*x^6 - 559*a^5*x^5 - 965*a^4*x^4 + 715*a^3*x^3 + 1065*a^2*x^2 - 279*a*x - 384)*sqrt((a*x - 1)/(a*x + 1)))/(a ^7*c^4*x^6 - 3*a^5*c^4*x^4 + 3*a^3*c^4*x^2 - a*c^4)
\[ \int \frac {e^{-\coth ^{-1}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^4} \, dx=\frac {a^{8} \int \frac {x^{8} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a^{8} x^{8} - 4 a^{6} x^{6} + 6 a^{4} x^{4} - 4 a^{2} x^{2} + 1}\, dx}{c^{4}} \]
a**8*Integral(x**8*sqrt(a*x/(a*x + 1) - 1/(a*x + 1))/(a**8*x**8 - 4*a**6*x **6 + 6*a**4*x**4 - 4*a**2*x**2 + 1), x)/c**4
Time = 0.20 (sec) , antiderivative size = 231, normalized size of antiderivative = 0.70 \[ \int \frac {e^{-\coth ^{-1}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^4} \, dx=\frac {1}{6720} \, a {\left (\frac {7 \, {\left (\frac {47 \, {\left (a x - 1\right )}}{a x + 1} + \frac {655 \, {\left (a x - 1\right )}^{2}}{{\left (a x + 1\right )}^{2}} - \frac {2625 \, {\left (a x - 1\right )}^{3}}{{\left (a x + 1\right )}^{3}} + 3\right )}}{a^{2} c^{4} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {7}{2}} - a^{2} c^{4} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {5}{2}}} + \frac {5 \, {\left (3 \, \left (\frac {a x - 1}{a x + 1}\right )^{\frac {7}{2}} + 42 \, \left (\frac {a x - 1}{a x + 1}\right )^{\frac {5}{2}} + 329 \, \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}} + 2940 \, \sqrt {\frac {a x - 1}{a x + 1}}\right )}}{a^{2} c^{4}} - \frac {6720 \, \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right )}{a^{2} c^{4}} + \frac {6720 \, \log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right )}{a^{2} c^{4}}\right )} \]
1/6720*a*(7*(47*(a*x - 1)/(a*x + 1) + 655*(a*x - 1)^2/(a*x + 1)^2 - 2625*( a*x - 1)^3/(a*x + 1)^3 + 3)/(a^2*c^4*((a*x - 1)/(a*x + 1))^(7/2) - a^2*c^4 *((a*x - 1)/(a*x + 1))^(5/2)) + 5*(3*((a*x - 1)/(a*x + 1))^(7/2) + 42*((a* x - 1)/(a*x + 1))^(5/2) + 329*((a*x - 1)/(a*x + 1))^(3/2) + 2940*sqrt((a*x - 1)/(a*x + 1)))/(a^2*c^4) - 6720*log(sqrt((a*x - 1)/(a*x + 1)) + 1)/(a^2 *c^4) + 6720*log(sqrt((a*x - 1)/(a*x + 1)) - 1)/(a^2*c^4))
\[ \int \frac {e^{-\coth ^{-1}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^4} \, dx=\int { \frac {\sqrt {\frac {a x - 1}{a x + 1}}}{{\left (c - \frac {c}{a^{2} x^{2}}\right )}^{4}} \,d x } \]
Time = 4.27 (sec) , antiderivative size = 217, normalized size of antiderivative = 0.66 \[ \int \frac {e^{-\coth ^{-1}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^4} \, dx=\frac {35\,\sqrt {\frac {a\,x-1}{a\,x+1}}}{16\,a\,c^4}-\frac {\frac {131\,{\left (a\,x-1\right )}^2}{3\,{\left (a\,x+1\right )}^2}-\frac {175\,{\left (a\,x-1\right )}^3}{{\left (a\,x+1\right )}^3}+\frac {47\,\left (a\,x-1\right )}{15\,\left (a\,x+1\right )}+\frac {1}{5}}{64\,a\,c^4\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{5/2}-64\,a\,c^4\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{7/2}}+\frac {47\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{3/2}}{192\,a\,c^4}+\frac {{\left (\frac {a\,x-1}{a\,x+1}\right )}^{5/2}}{32\,a\,c^4}+\frac {{\left (\frac {a\,x-1}{a\,x+1}\right )}^{7/2}}{448\,a\,c^4}+\frac {\mathrm {atan}\left (\sqrt {\frac {a\,x-1}{a\,x+1}}\,1{}\mathrm {i}\right )\,2{}\mathrm {i}}{a\,c^4} \]
(35*((a*x - 1)/(a*x + 1))^(1/2))/(16*a*c^4) - ((131*(a*x - 1)^2)/(3*(a*x + 1)^2) - (175*(a*x - 1)^3)/(a*x + 1)^3 + (47*(a*x - 1))/(15*(a*x + 1)) + 1 /5)/(64*a*c^4*((a*x - 1)/(a*x + 1))^(5/2) - 64*a*c^4*((a*x - 1)/(a*x + 1)) ^(7/2)) + (47*((a*x - 1)/(a*x + 1))^(3/2))/(192*a*c^4) + ((a*x - 1)/(a*x + 1))^(5/2)/(32*a*c^4) + ((a*x - 1)/(a*x + 1))^(7/2)/(448*a*c^4) + (atan((( a*x - 1)/(a*x + 1))^(1/2)*1i)*2i)/(a*c^4)