Integrand size = 20, antiderivative size = 170 \[ \int \frac {e^{\coth ^{-1}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^4} \, dx=-\frac {a+\frac {1}{x}}{7 a^2 c^4 \left (1-\frac {1}{a^2 x^2}\right )^{7/2}}-\frac {7 a+\frac {13}{x}}{35 a^2 c^4 \left (1-\frac {1}{a^2 x^2}\right )^{5/2}}-\frac {35 a+\frac {87}{x}}{105 a^2 c^4 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}-\frac {35 a+\frac {93}{x}}{35 a^2 c^4 \sqrt {1-\frac {1}{a^2 x^2}}}+\frac {\sqrt {1-\frac {1}{a^2 x^2}} x}{c^4}+\frac {\text {arctanh}\left (\sqrt {1-\frac {1}{a^2 x^2}}\right )}{a c^4} \] Output:
-1/7*(a+1/x)/a^2/c^4/(1-1/a^2/x^2)^(7/2)-1/35*(7*a+13/x)/a^2/c^4/(1-1/a^2/ x^2)^(5/2)-1/105*(35*a+87/x)/a^2/c^4/(1-1/a^2/x^2)^(3/2)-1/35*(35*a+93/x)/ a^2/c^4/(1-1/a^2/x^2)^(1/2)+(1-1/a^2/x^2)^(1/2)*x/c^4+arctanh((1-1/a^2/x^2 )^(1/2))/a/c^4
Time = 0.72 (sec) , antiderivative size = 115, normalized size of antiderivative = 0.68 \[ \int \frac {e^{\coth ^{-1}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^4} \, dx=\frac {\frac {a \sqrt {1-\frac {1}{a^2 x^2}} x \left (384-279 a x-1065 a^2 x^2+715 a^3 x^3+965 a^4 x^4-559 a^5 x^5-281 a^6 x^6+105 a^7 x^7\right )}{105 (-1+a x)^4 (1+a x)^3}+\log \left (\left (1+\sqrt {1-\frac {1}{a^2 x^2}}\right ) x\right )}{a c^4} \] Input:
Integrate[E^ArcCoth[a*x]/(c - c/(a^2*x^2))^4,x]
Output:
((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)^4* (1 + a*x)^3) + Log[(1 + Sqrt[1 - 1/(a^2*x^2)])*x])/(a*c^4)
Time = 0.85 (sec) , antiderivative size = 307, normalized size of antiderivative = 1.81, number of steps used = 23, number of rules used = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.100, Rules used = {6748, 114, 25, 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 |
| \(\displaystyle -\frac {\int \frac {x^2}{\left (1-\frac {1}{a x}\right )^{9/2} \left (1+\frac {1}{a x}\right )^{7/2}}d\frac {1}{x}}{c^4}\) |
| \(\Big \downarrow \) 114 |
| \(\displaystyle -\frac {-\int -\frac {\left (a+\frac {7}{x}\right ) x}{a^2 \left (1-\frac {1}{a x}\right )^{9/2} \left (1+\frac {1}{a x}\right )^{7/2}}d\frac {1}{x}-\frac {x}{\left (1-\frac {1}{a x}\right )^{7/2} \left (\frac {1}{a x}+1\right )^{5/2}}}{c^4}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\int \frac {\left (a+\frac {7}{x}\right ) x}{a^2 \left (1-\frac {1}{a x}\right )^{9/2} \left (1+\frac {1}{a x}\right )^{7/2}}d\frac {1}{x}-\frac {x}{\left (1-\frac {1}{a x}\right )^{7/2} \left (\frac {1}{a x}+1\right )^{5/2}}}{c^4}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\frac {\int \frac {\left (a+\frac {7}{x}\right ) x}{\left (1-\frac {1}{a x}\right )^{9/2} \left (1+\frac {1}{a x}\right )^{7/2}}d\frac {1}{x}}{a^2}-\frac {x}{\left (1-\frac {1}{a x}\right )^{7/2} \left (\frac {1}{a x}+1\right )^{5/2}}}{c^4}\) |
| \(\Big \downarrow \) 169 |
| \(\displaystyle -\frac {\frac {\frac {8 a}{7 \left (1-\frac {1}{a x}\right )^{7/2} \left (\frac {1}{a x}+1\right )^{5/2}}-\frac {1}{7} a \int -\frac {\left (7 a+\frac {48}{x}\right ) x}{a \left (1-\frac {1}{a x}\right )^{7/2} \left (1+\frac {1}{a x}\right )^{7/2}}d\frac {1}{x}}{a^2}-\frac {x}{\left (1-\frac {1}{a x}\right )^{7/2} \left (\frac {1}{a x}+1\right )^{5/2}}}{c^4}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\frac {\frac {1}{7} a \int \frac {\left (7 a+\frac {48}{x}\right ) x}{a \left (1-\frac {1}{a x}\right )^{7/2} \left (1+\frac {1}{a x}\right )^{7/2}}d\frac {1}{x}+\frac {8 a}{7 \left (1-\frac {1}{a x}\right )^{7/2} \left (\frac {1}{a x}+1\right )^{5/2}}}{a^2}-\frac {x}{\left (1-\frac {1}{a x}\right )^{7/2} \left (\frac {1}{a x}+1\right )^{5/2}}}{c^4}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\frac {\frac {1}{7} \int \frac {\left (7 a+\frac {48}{x}\right ) x}{\left (1-\frac {1}{a x}\right )^{7/2} \left (1+\frac {1}{a x}\right )^{7/2}}d\frac {1}{x}+\frac {8 a}{7 \left (1-\frac {1}{a x}\right )^{7/2} \left (\frac {1}{a x}+1\right )^{5/2}}}{a^2}-\frac {x}{\left (1-\frac {1}{a x}\right )^{7/2} \left (\frac {1}{a x}+1\right )^{5/2}}}{c^4}\) |
| \(\Big \downarrow \) 169 |
| \(\displaystyle -\frac {\frac {\frac {1}{7} \left (\frac {11 a}{\left (1-\frac {1}{a x}\right )^{5/2} \left (\frac {1}{a x}+1\right )^{5/2}}-\frac {1}{5} a \int -\frac {5 \left (7 a+\frac {55}{x}\right ) x}{a \left (1-\frac {1}{a x}\right )^{5/2} \left (1+\frac {1}{a x}\right )^{7/2}}d\frac {1}{x}\right )+\frac {8 a}{7 \left (1-\frac {1}{a x}\right )^{7/2} \left (\frac {1}{a x}+1\right )^{5/2}}}{a^2}-\frac {x}{\left (1-\frac {1}{a x}\right )^{7/2} \left (\frac {1}{a x}+1\right )^{5/2}}}{c^4}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\frac {\frac {1}{7} \left (\int \frac {\left (7 a+\frac {55}{x}\right ) x}{\left (1-\frac {1}{a x}\right )^{5/2} \left (1+\frac {1}{a x}\right )^{7/2}}d\frac {1}{x}+\frac {11 a}{\left (1-\frac {1}{a x}\right )^{5/2} \left (\frac {1}{a x}+1\right )^{5/2}}\right )+\frac {8 a}{7 \left (1-\frac {1}{a x}\right )^{7/2} \left (\frac {1}{a x}+1\right )^{5/2}}}{a^2}-\frac {x}{\left (1-\frac {1}{a x}\right )^{7/2} \left (\frac {1}{a x}+1\right )^{5/2}}}{c^4}\) |
| \(\Big \downarrow \) 169 |
| \(\displaystyle -\frac {\frac {\frac {1}{7} \left (-\frac {1}{3} a \int -\frac {\left (21 a+\frac {248}{x}\right ) x}{a \left (1-\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{7/2}}d\frac {1}{x}+\frac {62 a}{3 \left (1-\frac {1}{a x}\right )^{3/2} \left (\frac {1}{a x}+1\right )^{5/2}}+\frac {11 a}{\left (1-\frac {1}{a x}\right )^{5/2} \left (\frac {1}{a x}+1\right )^{5/2}}\right )+\frac {8 a}{7 \left (1-\frac {1}{a x}\right )^{7/2} \left (\frac {1}{a x}+1\right )^{5/2}}}{a^2}-\frac {x}{\left (1-\frac {1}{a x}\right )^{7/2} \left (\frac {1}{a x}+1\right )^{5/2}}}{c^4}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\frac {\frac {1}{7} \left (\frac {1}{3} a \int \frac {\left (21 a+\frac {248}{x}\right ) x}{a \left (1-\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{7/2}}d\frac {1}{x}+\frac {62 a}{3 \left (1-\frac {1}{a x}\right )^{3/2} \left (\frac {1}{a x}+1\right )^{5/2}}+\frac {11 a}{\left (1-\frac {1}{a x}\right )^{5/2} \left (\frac {1}{a x}+1\right )^{5/2}}\right )+\frac {8 a}{7 \left (1-\frac {1}{a x}\right )^{7/2} \left (\frac {1}{a x}+1\right )^{5/2}}}{a^2}-\frac {x}{\left (1-\frac {1}{a x}\right )^{7/2} \left (\frac {1}{a x}+1\right )^{5/2}}}{c^4}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\frac {\frac {1}{7} \left (\frac {1}{3} \int \frac {\left (21 a+\frac {248}{x}\right ) x}{\left (1-\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{7/2}}d\frac {1}{x}+\frac {62 a}{3 \left (1-\frac {1}{a x}\right )^{3/2} \left (\frac {1}{a x}+1\right )^{5/2}}+\frac {11 a}{\left (1-\frac {1}{a x}\right )^{5/2} \left (\frac {1}{a x}+1\right )^{5/2}}\right )+\frac {8 a}{7 \left (1-\frac {1}{a x}\right )^{7/2} \left (\frac {1}{a x}+1\right )^{5/2}}}{a^2}-\frac {x}{\left (1-\frac {1}{a x}\right )^{7/2} \left (\frac {1}{a x}+1\right )^{5/2}}}{c^4}\) |
| \(\Big \downarrow \) 169 |
| \(\displaystyle -\frac {\frac {\frac {1}{7} \left (\frac {1}{3} \left (\frac {269 a}{\sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{5/2}}-a \int -\frac {3 \left (7 a+\frac {269}{x}\right ) x}{a \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{7/2}}d\frac {1}{x}\right )+\frac {62 a}{3 \left (1-\frac {1}{a x}\right )^{3/2} \left (\frac {1}{a x}+1\right )^{5/2}}+\frac {11 a}{\left (1-\frac {1}{a x}\right )^{5/2} \left (\frac {1}{a x}+1\right )^{5/2}}\right )+\frac {8 a}{7 \left (1-\frac {1}{a x}\right )^{7/2} \left (\frac {1}{a x}+1\right )^{5/2}}}{a^2}-\frac {x}{\left (1-\frac {1}{a x}\right )^{7/2} \left (\frac {1}{a x}+1\right )^{5/2}}}{c^4}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\frac {\frac {1}{7} \left (\frac {1}{3} \left (3 \int \frac {\left (7 a+\frac {269}{x}\right ) x}{\sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{7/2}}d\frac {1}{x}+\frac {269 a}{\sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{5/2}}\right )+\frac {62 a}{3 \left (1-\frac {1}{a x}\right )^{3/2} \left (\frac {1}{a x}+1\right )^{5/2}}+\frac {11 a}{\left (1-\frac {1}{a x}\right )^{5/2} \left (\frac {1}{a x}+1\right )^{5/2}}\right )+\frac {8 a}{7 \left (1-\frac {1}{a x}\right )^{7/2} \left (\frac {1}{a x}+1\right )^{5/2}}}{a^2}-\frac {x}{\left (1-\frac {1}{a x}\right )^{7/2} \left (\frac {1}{a x}+1\right )^{5/2}}}{c^4}\) |
| \(\Big \downarrow \) 169 |
| \(\displaystyle -\frac {\frac {\frac {1}{7} \left (\frac {1}{3} \left (3 \left (\frac {1}{5} a \int \frac {\left (35 a+\frac {524}{x}\right ) x}{a \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{5/2}}d\frac {1}{x}-\frac {262 a \sqrt {1-\frac {1}{a x}}}{5 \left (\frac {1}{a x}+1\right )^{5/2}}\right )+\frac {269 a}{\sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{5/2}}\right )+\frac {62 a}{3 \left (1-\frac {1}{a x}\right )^{3/2} \left (\frac {1}{a x}+1\right )^{5/2}}+\frac {11 a}{\left (1-\frac {1}{a x}\right )^{5/2} \left (\frac {1}{a x}+1\right )^{5/2}}\right )+\frac {8 a}{7 \left (1-\frac {1}{a x}\right )^{7/2} \left (\frac {1}{a x}+1\right )^{5/2}}}{a^2}-\frac {x}{\left (1-\frac {1}{a x}\right )^{7/2} \left (\frac {1}{a x}+1\right )^{5/2}}}{c^4}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\frac {\frac {1}{7} \left (\frac {1}{3} \left (3 \left (\frac {1}{5} \int \frac {\left (35 a+\frac {524}{x}\right ) x}{\sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{5/2}}d\frac {1}{x}-\frac {262 a \sqrt {1-\frac {1}{a x}}}{5 \left (\frac {1}{a x}+1\right )^{5/2}}\right )+\frac {269 a}{\sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{5/2}}\right )+\frac {62 a}{3 \left (1-\frac {1}{a x}\right )^{3/2} \left (\frac {1}{a x}+1\right )^{5/2}}+\frac {11 a}{\left (1-\frac {1}{a x}\right )^{5/2} \left (\frac {1}{a x}+1\right )^{5/2}}\right )+\frac {8 a}{7 \left (1-\frac {1}{a x}\right )^{7/2} \left (\frac {1}{a x}+1\right )^{5/2}}}{a^2}-\frac {x}{\left (1-\frac {1}{a x}\right )^{7/2} \left (\frac {1}{a x}+1\right )^{5/2}}}{c^4}\) |
| \(\Big \downarrow \) 169 |
| \(\displaystyle -\frac {\frac {\frac {1}{7} \left (\frac {1}{3} \left (3 \left (\frac {1}{5} \left (\frac {1}{3} a \int \frac {3 \left (35 a+\frac {163}{x}\right ) x}{a \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{3/2}}d\frac {1}{x}-\frac {163 a \sqrt {1-\frac {1}{a x}}}{\left (\frac {1}{a x}+1\right )^{3/2}}\right )-\frac {262 a \sqrt {1-\frac {1}{a x}}}{5 \left (\frac {1}{a x}+1\right )^{5/2}}\right )+\frac {269 a}{\sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{5/2}}\right )+\frac {62 a}{3 \left (1-\frac {1}{a x}\right )^{3/2} \left (\frac {1}{a x}+1\right )^{5/2}}+\frac {11 a}{\left (1-\frac {1}{a x}\right )^{5/2} \left (\frac {1}{a x}+1\right )^{5/2}}\right )+\frac {8 a}{7 \left (1-\frac {1}{a x}\right )^{7/2} \left (\frac {1}{a x}+1\right )^{5/2}}}{a^2}-\frac {x}{\left (1-\frac {1}{a x}\right )^{7/2} \left (\frac {1}{a x}+1\right )^{5/2}}}{c^4}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\frac {\frac {1}{7} \left (\frac {1}{3} \left (3 \left (\frac {1}{5} \left (\int \frac {\left (35 a+\frac {163}{x}\right ) x}{\sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{3/2}}d\frac {1}{x}-\frac {163 a \sqrt {1-\frac {1}{a x}}}{\left (\frac {1}{a x}+1\right )^{3/2}}\right )-\frac {262 a \sqrt {1-\frac {1}{a x}}}{5 \left (\frac {1}{a x}+1\right )^{5/2}}\right )+\frac {269 a}{\sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{5/2}}\right )+\frac {62 a}{3 \left (1-\frac {1}{a x}\right )^{3/2} \left (\frac {1}{a x}+1\right )^{5/2}}+\frac {11 a}{\left (1-\frac {1}{a x}\right )^{5/2} \left (\frac {1}{a x}+1\right )^{5/2}}\right )+\frac {8 a}{7 \left (1-\frac {1}{a x}\right )^{7/2} \left (\frac {1}{a x}+1\right )^{5/2}}}{a^2}-\frac {x}{\left (1-\frac {1}{a x}\right )^{7/2} \left (\frac {1}{a x}+1\right )^{5/2}}}{c^4}\) |
| \(\Big \downarrow \) 169 |
| \(\displaystyle -\frac {\frac {\frac {1}{7} \left (\frac {1}{3} \left (3 \left (\frac {1}{5} \left (a \int \frac {35 x}{\sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}}d\frac {1}{x}-\frac {128 a \sqrt {1-\frac {1}{a x}}}{\sqrt {\frac {1}{a x}+1}}-\frac {163 a \sqrt {1-\frac {1}{a x}}}{\left (\frac {1}{a x}+1\right )^{3/2}}\right )-\frac {262 a \sqrt {1-\frac {1}{a x}}}{5 \left (\frac {1}{a x}+1\right )^{5/2}}\right )+\frac {269 a}{\sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{5/2}}\right )+\frac {62 a}{3 \left (1-\frac {1}{a x}\right )^{3/2} \left (\frac {1}{a x}+1\right )^{5/2}}+\frac {11 a}{\left (1-\frac {1}{a x}\right )^{5/2} \left (\frac {1}{a x}+1\right )^{5/2}}\right )+\frac {8 a}{7 \left (1-\frac {1}{a x}\right )^{7/2} \left (\frac {1}{a x}+1\right )^{5/2}}}{a^2}-\frac {x}{\left (1-\frac {1}{a x}\right )^{7/2} \left (\frac {1}{a x}+1\right )^{5/2}}}{c^4}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\frac {\frac {1}{7} \left (\frac {1}{3} \left (3 \left (\frac {1}{5} \left (35 a \int \frac {x}{\sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}}d\frac {1}{x}-\frac {128 a \sqrt {1-\frac {1}{a x}}}{\sqrt {\frac {1}{a x}+1}}-\frac {163 a \sqrt {1-\frac {1}{a x}}}{\left (\frac {1}{a x}+1\right )^{3/2}}\right )-\frac {262 a \sqrt {1-\frac {1}{a x}}}{5 \left (\frac {1}{a x}+1\right )^{5/2}}\right )+\frac {269 a}{\sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{5/2}}\right )+\frac {62 a}{3 \left (1-\frac {1}{a x}\right )^{3/2} \left (\frac {1}{a x}+1\right )^{5/2}}+\frac {11 a}{\left (1-\frac {1}{a x}\right )^{5/2} \left (\frac {1}{a x}+1\right )^{5/2}}\right )+\frac {8 a}{7 \left (1-\frac {1}{a x}\right )^{7/2} \left (\frac {1}{a x}+1\right )^{5/2}}}{a^2}-\frac {x}{\left (1-\frac {1}{a x}\right )^{7/2} \left (\frac {1}{a x}+1\right )^{5/2}}}{c^4}\) |
| \(\Big \downarrow \) 103 |
| \(\displaystyle -\frac {\frac {\frac {1}{7} \left (\frac {1}{3} \left (3 \left (\frac {1}{5} \left (-35 \int \frac {1}{\frac {1}{a}-\frac {1}{a x^2}}d\left (\sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}\right )-\frac {128 a \sqrt {1-\frac {1}{a x}}}{\sqrt {\frac {1}{a x}+1}}-\frac {163 a \sqrt {1-\frac {1}{a x}}}{\left (\frac {1}{a x}+1\right )^{3/2}}\right )-\frac {262 a \sqrt {1-\frac {1}{a x}}}{5 \left (\frac {1}{a x}+1\right )^{5/2}}\right )+\frac {269 a}{\sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{5/2}}\right )+\frac {62 a}{3 \left (1-\frac {1}{a x}\right )^{3/2} \left (\frac {1}{a x}+1\right )^{5/2}}+\frac {11 a}{\left (1-\frac {1}{a x}\right )^{5/2} \left (\frac {1}{a x}+1\right )^{5/2}}\right )+\frac {8 a}{7 \left (1-\frac {1}{a x}\right )^{7/2} \left (\frac {1}{a x}+1\right )^{5/2}}}{a^2}-\frac {x}{\left (1-\frac {1}{a x}\right )^{7/2} \left (\frac {1}{a x}+1\right )^{5/2}}}{c^4}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle -\frac {\frac {\frac {1}{7} \left (\frac {1}{3} \left (3 \left (\frac {1}{5} \left (-35 a \text {arctanh}\left (\sqrt {1-\frac {1}{a x}} \sqrt {\frac {1}{a x}+1}\right )-\frac {128 a \sqrt {1-\frac {1}{a x}}}{\sqrt {\frac {1}{a x}+1}}-\frac {163 a \sqrt {1-\frac {1}{a x}}}{\left (\frac {1}{a x}+1\right )^{3/2}}\right )-\frac {262 a \sqrt {1-\frac {1}{a x}}}{5 \left (\frac {1}{a x}+1\right )^{5/2}}\right )+\frac {269 a}{\sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{5/2}}\right )+\frac {62 a}{3 \left (1-\frac {1}{a x}\right )^{3/2} \left (\frac {1}{a x}+1\right )^{5/2}}+\frac {11 a}{\left (1-\frac {1}{a x}\right )^{5/2} \left (\frac {1}{a x}+1\right )^{5/2}}\right )+\frac {8 a}{7 \left (1-\frac {1}{a x}\right )^{7/2} \left (\frac {1}{a x}+1\right )^{5/2}}}{a^2}-\frac {x}{\left (1-\frac {1}{a x}\right )^{7/2} \left (\frac {1}{a x}+1\right )^{5/2}}}{c^4}\) |
Input:
Int[E^ArcCoth[a*x]/(c - c/(a^2*x^2))^4,x]
Output:
-((-(x/((1 - 1/(a*x))^(7/2)*(1 + 1/(a*x))^(5/2))) + ((8*a)/(7*(1 - 1/(a*x) )^(7/2)*(1 + 1/(a*x))^(5/2)) + ((11*a)/((1 - 1/(a*x))^(5/2)*(1 + 1/(a*x))^ (5/2)) + (62*a)/(3*(1 - 1/(a*x))^(3/2)*(1 + 1/(a*x))^(5/2)) + ((269*a)/(Sq rt[1 - 1/(a*x)]*(1 + 1/(a*x))^(5/2)) + 3*((-262*a*Sqrt[1 - 1/(a*x)])/(5*(1 + 1/(a*x))^(5/2)) + ((-163*a*Sqrt[1 - 1/(a*x)])/(1 + 1/(a*x))^(3/2) - (12 8*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))/3)/7)/a^2)/c^4)
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]
Leaf count of result is larger than twice the leaf count of optimal. \(365\) vs. \(2(150)=300\).
Time = 0.20 (sec) , antiderivative size = 366, normalized size of antiderivative = 2.15
| method | result | size |
| risch | \(\frac {a x -1}{a \,c^{4} \sqrt {\frac {a x -1}{a x +1}}}+\frac {\left (\frac {\ln \left (\frac {a^{2} x}{\sqrt {a^{2}}}+\sqrt {a^{2} x^{2}-1}\right )}{a^{8} \sqrt {a^{2}}}-\frac {1657 \sqrt {\left (x -\frac {1}{a}\right )^{2} a^{2}+2 a \left (x -\frac {1}{a}\right )}}{672 a^{10} \left (x -\frac {1}{a}\right )}-\frac {\sqrt {\left (x -\frac {1}{a}\right )^{2} a^{2}+2 a \left (x -\frac {1}{a}\right )}}{56 a^{13} \left (x -\frac {1}{a}\right )^{4}}-\frac {17 \sqrt {\left (x -\frac {1}{a}\right )^{2} a^{2}+2 a \left (x -\frac {1}{a}\right )}}{112 a^{12} \left (x -\frac {1}{a}\right )^{3}}-\frac {211 \sqrt {\left (x -\frac {1}{a}\right )^{2} a^{2}+2 a \left (x -\frac {1}{a}\right )}}{336 a^{11} \left (x -\frac {1}{a}\right )^{2}}+\frac {\sqrt {a^{2} \left (x +\frac {1}{a}\right )^{2}-2 a \left (x +\frac {1}{a}\right )}}{80 a^{12} \left (x +\frac {1}{a}\right )^{3}}-\frac {7 \sqrt {a^{2} \left (x +\frac {1}{a}\right )^{2}-2 a \left (x +\frac {1}{a}\right )}}{60 a^{11} \left (x +\frac {1}{a}\right )^{2}}+\frac {379 \sqrt {a^{2} \left (x +\frac {1}{a}\right )^{2}-2 a \left (x +\frac {1}{a}\right )}}{480 a^{10} \left (x +\frac {1}{a}\right )}\right ) a^{8} \sqrt {\left (a x -1\right ) \left (a x +1\right )}}{c^{4} \sqrt {\frac {a x -1}{a x +1}}\, \left (a x +1\right )}\) | \(366\) |
| default | \(\text {Expression too large to display}\) | \(898\) |
Input:
int(1/((a*x-1)/(a*x+1))^(1/2)/(c-c/a^2/x^2)^4,x,method=_RETURNVERBOSE)
Output:
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)-1657/672/a^10/(x-1/a)*((x-1/a)^2*a^2+2*a*(x-1/a)) ^(1/2)-1/56/a^13/(x-1/a)^4*((x-1/a)^2*a^2+2*a*(x-1/a))^(1/2)-17/112/a^12/( x-1/a)^3*((x-1/a)^2*a^2+2*a*(x-1/a))^(1/2)-211/336/a^11/(x-1/a)^2*((x-1/a) ^2*a^2+2*a*(x-1/a))^(1/2)+1/80/a^12/(x+1/a)^3*(a^2*(x+1/a)^2-2*a*(x+1/a))^ (1/2)-7/60/a^11/(x+1/a)^2*(a^2*(x+1/a)^2-2*a*(x+1/a))^(1/2)+379/480/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 = 0.11 (sec) , antiderivative size = 274, normalized size of antiderivative = 1.61 \[ \int \frac {e^{\coth ^{-1}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^4} \, dx=\frac {105 \, {\left (a^{6} x^{6} - 2 \, a^{5} x^{5} - a^{4} x^{4} + 4 \, a^{3} x^{3} - a^{2} x^{2} - 2 \, a x + 1\right )} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right ) - 105 \, {\left (a^{6} x^{6} - 2 \, a^{5} x^{5} - a^{4} x^{4} + 4 \, a^{3} x^{3} - a^{2} x^{2} - 2 \, a x + 1\right )} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right ) + {\left (105 \, a^{7} x^{7} - 281 \, a^{6} x^{6} - 559 \, a^{5} x^{5} + 965 \, a^{4} x^{4} + 715 \, a^{3} x^{3} - 1065 \, a^{2} x^{2} - 279 \, a x + 384\right )} \sqrt {\frac {a x - 1}{a x + 1}}}{105 \, {\left (a^{7} c^{4} x^{6} - 2 \, a^{6} c^{4} x^{5} - a^{5} c^{4} x^{4} + 4 \, a^{4} c^{4} x^{3} - a^{3} c^{4} x^{2} - 2 \, a^{2} c^{4} x + a c^{4}\right )}} \] Input:
integrate(1/((a*x-1)/(a*x+1))^(1/2)/(c-c/a^2/x^2)^4,x, algorithm="fricas")
Output:
1/105*(105*(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) - 105*(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) + ( 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 - 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)
\[ \int \frac {e^{\coth ^{-1}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^4} \, dx=\frac {a^{8} \int \frac {x^{8}}{a^{8} x^{8} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}} - 4 a^{6} x^{6} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}} + 6 a^{4} x^{4} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}} - 4 a^{2} x^{2} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}} + \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}\, dx}{c^{4}} \] Input:
integrate(1/((a*x-1)/(a*x+1))**(1/2)/(c-c/a**2/x**2)**4,x)
Output:
a**8*Integral(x**8/(a**8*x**8*sqrt(a*x/(a*x + 1) - 1/(a*x + 1)) - 4*a**6*x **6*sqrt(a*x/(a*x + 1) - 1/(a*x + 1)) + 6*a**4*x**4*sqrt(a*x/(a*x + 1) - 1 /(a*x + 1)) - 4*a**2*x**2*sqrt(a*x/(a*x + 1) - 1/(a*x + 1)) + sqrt(a*x/(a* x + 1) - 1/(a*x + 1))), x)/c**4
Time = 0.03 (sec) , antiderivative size = 230, normalized size of antiderivative = 1.35 \[ \int \frac {e^{\coth ^{-1}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^4} \, dx=\frac {1}{6720} \, a {\left (\frac {5 \, {\left (\frac {39 \, {\left (a x - 1\right )}}{a x + 1} + \frac {287 \, {\left (a x - 1\right )}^{2}}{{\left (a x + 1\right )}^{2}} + \frac {2611 \, {\left (a x - 1\right )}^{3}}{{\left (a x + 1\right )}^{3}} - \frac {5628 \, {\left (a x - 1\right )}^{4}}{{\left (a x + 1\right )}^{4}} + 3\right )}}{a^{2} c^{4} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {9}{2}} - a^{2} c^{4} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {7}{2}}} + \frac {7 \, {\left (3 \, \left (\frac {a x - 1}{a x + 1}\right )^{\frac {5}{2}} + 50 \, \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}} + 705 \, \sqrt {\frac {a x - 1}{a x + 1}}\right )}}{a^{2} c^{4}} + \frac {6720 \, \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right )}{a^{2} c^{4}} - \frac {6720 \, \log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right )}{a^{2} c^{4}}\right )} \] Input:
integrate(1/((a*x-1)/(a*x+1))^(1/2)/(c-c/a^2/x^2)^4,x, algorithm="maxima")
Output:
1/6720*a*(5*(39*(a*x - 1)/(a*x + 1) + 287*(a*x - 1)^2/(a*x + 1)^2 + 2611*( a*x - 1)^3/(a*x + 1)^3 - 5628*(a*x - 1)^4/(a*x + 1)^4 + 3)/(a^2*c^4*((a*x - 1)/(a*x + 1))^(9/2) - a^2*c^4*((a*x - 1)/(a*x + 1))^(7/2)) + 7*(3*((a*x - 1)/(a*x + 1))^(5/2) + 50*((a*x - 1)/(a*x + 1))^(3/2) + 705*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))
\[ \int \frac {e^{\coth ^{-1}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^4} \, dx=\int { \frac {1}{{\left (c - \frac {c}{a^{2} x^{2}}\right )}^{4} \sqrt {\frac {a x - 1}{a x + 1}}} \,d x } \] Input:
integrate(1/((a*x-1)/(a*x+1))^(1/2)/(c-c/a^2/x^2)^4,x, algorithm="giac")
Output:
integrate(1/((c - c/(a^2*x^2))^4*sqrt((a*x - 1)/(a*x + 1))), x)
Time = 13.22 (sec) , antiderivative size = 210, normalized size of antiderivative = 1.24 \[ \int \frac {e^{\coth ^{-1}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^4} \, dx=\frac {47\,\sqrt {\frac {a\,x-1}{a\,x+1}}}{64\,a\,c^4}-\frac {\frac {41\,{\left (a\,x-1\right )}^2}{3\,{\left (a\,x+1\right )}^2}+\frac {373\,{\left (a\,x-1\right )}^3}{3\,{\left (a\,x+1\right )}^3}-\frac {268\,{\left (a\,x-1\right )}^4}{{\left (a\,x+1\right )}^4}+\frac {13\,\left (a\,x-1\right )}{7\,\left (a\,x+1\right )}+\frac {1}{7}}{64\,a\,c^4\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{7/2}-64\,a\,c^4\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{9/2}}+\frac {5\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{3/2}}{96\,a\,c^4}+\frac {{\left (\frac {a\,x-1}{a\,x+1}\right )}^{5/2}}{320\,a\,c^4}-\frac {\mathrm {atan}\left (\sqrt {\frac {a\,x-1}{a\,x+1}}\,1{}\mathrm {i}\right )\,2{}\mathrm {i}}{a\,c^4} \] Input:
int(1/((c - c/(a^2*x^2))^4*((a*x - 1)/(a*x + 1))^(1/2)),x)
Output:
(47*((a*x - 1)/(a*x + 1))^(1/2))/(64*a*c^4) - ((41*(a*x - 1)^2)/(3*(a*x + 1)^2) + (373*(a*x - 1)^3)/(3*(a*x + 1)^3) - (268*(a*x - 1)^4)/(a*x + 1)^4 + (13*(a*x - 1))/(7*(a*x + 1)) + 1/7)/(64*a*c^4*((a*x - 1)/(a*x + 1))^(7/2 ) - 64*a*c^4*((a*x - 1)/(a*x + 1))^(9/2)) + (5*((a*x - 1)/(a*x + 1))^(3/2) )/(96*a*c^4) + ((a*x - 1)/(a*x + 1))^(5/2)/(320*a*c^4) - (atan(((a*x - 1)/ (a*x + 1))^(1/2)*1i)*2i)/(a*c^4)
Time = 0.26 (sec) , antiderivative size = 322, normalized size of antiderivative = 1.89 \[ \int \frac {e^{\coth ^{-1}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^4} \, dx=\frac {840 \sqrt {a x -1}\, \mathrm {log}\left (\frac {\sqrt {a x -1}+\sqrt {a x +1}}{\sqrt {2}}\right ) a^{6} x^{6}-2520 \sqrt {a x -1}\, \mathrm {log}\left (\frac {\sqrt {a x -1}+\sqrt {a x +1}}{\sqrt {2}}\right ) a^{4} x^{4}+2520 \sqrt {a x -1}\, \mathrm {log}\left (\frac {\sqrt {a x -1}+\sqrt {a x +1}}{\sqrt {2}}\right ) a^{2} x^{2}-840 \sqrt {a x -1}\, \mathrm {log}\left (\frac {\sqrt {a x -1}+\sqrt {a x +1}}{\sqrt {2}}\right )-1501 \sqrt {a x -1}\, a^{6} x^{6}+4503 \sqrt {a x -1}\, a^{4} x^{4}-4503 \sqrt {a x -1}\, a^{2} x^{2}+1501 \sqrt {a x -1}+420 \sqrt {a x +1}\, a^{7} x^{7}-1124 \sqrt {a x +1}\, a^{6} x^{6}-2236 \sqrt {a x +1}\, a^{5} x^{5}+3860 \sqrt {a x +1}\, a^{4} x^{4}+2860 \sqrt {a x +1}\, a^{3} x^{3}-4260 \sqrt {a x +1}\, a^{2} x^{2}-1116 \sqrt {a x +1}\, a x +1536 \sqrt {a x +1}}{420 \sqrt {a x -1}\, a \,c^{4} \left (a^{6} x^{6}-3 a^{4} x^{4}+3 a^{2} x^{2}-1\right )} \] Input:
int(1/((a*x-1)/(a*x+1))^(1/2)/(c-c/a^2/x^2)^4,x)
Output:
(840*sqrt(a*x - 1)*log((sqrt(a*x - 1) + sqrt(a*x + 1))/sqrt(2))*a**6*x**6 - 2520*sqrt(a*x - 1)*log((sqrt(a*x - 1) + sqrt(a*x + 1))/sqrt(2))*a**4*x** 4 + 2520*sqrt(a*x - 1)*log((sqrt(a*x - 1) + sqrt(a*x + 1))/sqrt(2))*a**2*x **2 - 840*sqrt(a*x - 1)*log((sqrt(a*x - 1) + sqrt(a*x + 1))/sqrt(2)) - 150 1*sqrt(a*x - 1)*a**6*x**6 + 4503*sqrt(a*x - 1)*a**4*x**4 - 4503*sqrt(a*x - 1)*a**2*x**2 + 1501*sqrt(a*x - 1) + 420*sqrt(a*x + 1)*a**7*x**7 - 1124*sq rt(a*x + 1)*a**6*x**6 - 2236*sqrt(a*x + 1)*a**5*x**5 + 3860*sqrt(a*x + 1)* a**4*x**4 + 2860*sqrt(a*x + 1)*a**3*x**3 - 4260*sqrt(a*x + 1)*a**2*x**2 - 1116*sqrt(a*x + 1)*a*x + 1536*sqrt(a*x + 1))/(420*sqrt(a*x - 1)*a*c**4*(a* *6*x**6 - 3*a**4*x**4 + 3*a**2*x**2 - 1))