Integrand size = 22, antiderivative size = 169 \[ \int e^{-\coth ^{-1}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^4 \, dx=\frac {c^4 \left (1-\frac {1}{a^2 x^2}\right )^{5/2} \left (6 a+\frac {35}{x}\right )}{30 a^2}+\frac {c^4 \left (1-\frac {1}{a^2 x^2}\right )^{3/2} \left (8 a+\frac {35}{x}\right )}{24 a^2}+\frac {c^4 \sqrt {1-\frac {1}{a^2 x^2}} \left (16 a+\frac {35}{x}\right )}{16 a^2}+\frac {c^4 \left (1-\frac {1}{a^2 x^2}\right )^{7/2} \left (7 a+\frac {1}{x}\right ) x}{7 a}+\frac {35 c^4 \csc ^{-1}(a x)}{16 a}-\frac {c^4 \text {arctanh}\left (\sqrt {1-\frac {1}{a^2 x^2}}\right )}{a} \] Output:
1/30*c^4*(1-1/a^2/x^2)^(5/2)*(6*a+35/x)/a^2+1/24*c^4*(1-1/a^2/x^2)^(3/2)*( 8*a+35/x)/a^2+1/16*c^4*(1-1/a^2/x^2)^(1/2)*(16*a+35/x)/a^2+1/7*c^4*(1-1/a^ 2/x^2)^(7/2)*(7*a+1/x)*x/a+35/16*c^4*arccsc(a*x)/a-c^4*arctanh((1-1/a^2/x^ 2)^(1/2))/a
Time = 0.42 (sec) , antiderivative size = 120, normalized size of antiderivative = 0.71 \[ \int e^{-\coth ^{-1}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^4 \, dx=\frac {c^4 \left (\frac {\sqrt {1-\frac {1}{a^2 x^2}} \left (-240+280 a x+1056 a^2 x^2-1330 a^3 x^3-1952 a^4 x^4+3045 a^5 x^5+2816 a^6 x^6+1680 a^7 x^7\right )}{x^6}+3675 a^6 \arcsin \left (\frac {1}{a x}\right )-1680 a^6 \log \left (\left (1+\sqrt {1-\frac {1}{a^2 x^2}}\right ) x\right )\right )}{1680 a^7} \] Input:
Integrate[(c - c/(a^2*x^2))^4/E^ArcCoth[a*x],x]
Output:
(c^4*((Sqrt[1 - 1/(a^2*x^2)]*(-240 + 280*a*x + 1056*a^2*x^2 - 1330*a^3*x^3 - 1952*a^4*x^4 + 3045*a^5*x^5 + 2816*a^6*x^6 + 1680*a^7*x^7))/x^6 + 3675* a^6*ArcSin[1/(a*x)] - 1680*a^6*Log[(1 + Sqrt[1 - 1/(a^2*x^2)])*x]))/(1680* a^7)
Time = 0.91 (sec) , antiderivative size = 328, normalized size of antiderivative = 1.94, number of steps used = 26, number of rules used = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.136, Rules used = {6748, 108, 25, 27, 171, 27, 171, 27, 171, 27, 171, 27, 171, 27, 171, 25, 27, 171, 25, 27, 175, 39, 103, 221, 223}
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.
| \(\displaystyle \int \left (c-\frac {c}{a^2 x^2}\right )^4 e^{-\coth ^{-1}(a x)} \, dx\) |
| \(\Big \downarrow \) 6748 |
| \(\displaystyle -c^4 \int \left (1-\frac {1}{a x}\right )^{9/2} \left (1+\frac {1}{a x}\right )^{7/2} x^2d\frac {1}{x}\) |
| \(\Big \downarrow \) 108 |
| \(\displaystyle -c^4 \left (\int -\frac {\left (a+\frac {8}{x}\right ) \left (1-\frac {1}{a x}\right )^{7/2} \left (1+\frac {1}{a x}\right )^{5/2} x}{a^2}d\frac {1}{x}-x \left (1-\frac {1}{a x}\right )^{9/2} \left (\frac {1}{a x}+1\right )^{7/2}\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -c^4 \left (x \left (-\left (\frac {1}{a x}+1\right )^{7/2}\right ) \left (1-\frac {1}{a x}\right )^{9/2}-\int \frac {\left (a+\frac {8}{x}\right ) \left (1-\frac {1}{a x}\right )^{7/2} \left (1+\frac {1}{a x}\right )^{5/2} x}{a^2}d\frac {1}{x}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -c^4 \left (x \left (-\left (\frac {1}{a x}+1\right )^{7/2}\right ) \left (1-\frac {1}{a x}\right )^{9/2}-\frac {\int \left (a+\frac {8}{x}\right ) \left (1-\frac {1}{a x}\right )^{7/2} \left (1+\frac {1}{a x}\right )^{5/2} xd\frac {1}{x}}{a^2}\right )\) |
| \(\Big \downarrow \) 171 |
| \(\displaystyle -c^4 \left (x \left (-\left (\frac {1}{a x}+1\right )^{7/2}\right ) \left (1-\frac {1}{a x}\right )^{9/2}-\frac {\frac {1}{7} a \int \frac {7 \left (a+\frac {7}{x}\right ) \left (1-\frac {1}{a x}\right )^{5/2} \left (1+\frac {1}{a x}\right )^{5/2} x}{a}d\frac {1}{x}+\frac {8}{7} a \left (1-\frac {1}{a x}\right )^{7/2} \left (\frac {1}{a x}+1\right )^{7/2}}{a^2}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -c^4 \left (x \left (-\left (\frac {1}{a x}+1\right )^{7/2}\right ) \left (1-\frac {1}{a x}\right )^{9/2}-\frac {\int \left (a+\frac {7}{x}\right ) \left (1-\frac {1}{a x}\right )^{5/2} \left (1+\frac {1}{a x}\right )^{5/2} xd\frac {1}{x}+\frac {8}{7} a \left (1-\frac {1}{a x}\right )^{7/2} \left (\frac {1}{a x}+1\right )^{7/2}}{a^2}\right )\) |
| \(\Big \downarrow \) 171 |
| \(\displaystyle -c^4 \left (x \left (-\left (\frac {1}{a x}+1\right )^{7/2}\right ) \left (1-\frac {1}{a x}\right )^{9/2}-\frac {\frac {1}{6} a \int \frac {\left (6 a+\frac {29}{x}\right ) \left (1-\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{5/2} x}{a}d\frac {1}{x}+\frac {8}{7} a \left (1-\frac {1}{a x}\right )^{7/2} \left (\frac {1}{a x}+1\right )^{7/2}+\frac {7}{6} a \left (1-\frac {1}{a x}\right )^{5/2} \left (\frac {1}{a x}+1\right )^{7/2}}{a^2}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -c^4 \left (x \left (-\left (\frac {1}{a x}+1\right )^{7/2}\right ) \left (1-\frac {1}{a x}\right )^{9/2}-\frac {\frac {1}{6} \int \left (6 a+\frac {29}{x}\right ) \left (1-\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{5/2} xd\frac {1}{x}+\frac {8}{7} a \left (1-\frac {1}{a x}\right )^{7/2} \left (\frac {1}{a x}+1\right )^{7/2}+\frac {7}{6} a \left (1-\frac {1}{a x}\right )^{5/2} \left (\frac {1}{a x}+1\right )^{7/2}}{a^2}\right )\) |
| \(\Big \downarrow \) 171 |
| \(\displaystyle -c^4 \left (x \left (-\left (\frac {1}{a x}+1\right )^{7/2}\right ) \left (1-\frac {1}{a x}\right )^{9/2}-\frac {\frac {1}{6} \left (\frac {1}{5} a \int \frac {3 \left (10 a+\frac {19}{x}\right ) \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{5/2} x}{a}d\frac {1}{x}+\frac {29}{5} a \left (1-\frac {1}{a x}\right )^{3/2} \left (\frac {1}{a x}+1\right )^{7/2}\right )+\frac {8}{7} a \left (1-\frac {1}{a x}\right )^{7/2} \left (\frac {1}{a x}+1\right )^{7/2}+\frac {7}{6} a \left (1-\frac {1}{a x}\right )^{5/2} \left (\frac {1}{a x}+1\right )^{7/2}}{a^2}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -c^4 \left (x \left (-\left (\frac {1}{a x}+1\right )^{7/2}\right ) \left (1-\frac {1}{a x}\right )^{9/2}-\frac {\frac {1}{6} \left (\frac {3}{5} \int \left (10 a+\frac {19}{x}\right ) \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{5/2} xd\frac {1}{x}+\frac {29}{5} a \left (1-\frac {1}{a x}\right )^{3/2} \left (\frac {1}{a x}+1\right )^{7/2}\right )+\frac {8}{7} a \left (1-\frac {1}{a x}\right )^{7/2} \left (\frac {1}{a x}+1\right )^{7/2}+\frac {7}{6} a \left (1-\frac {1}{a x}\right )^{5/2} \left (\frac {1}{a x}+1\right )^{7/2}}{a^2}\right )\) |
| \(\Big \downarrow \) 171 |
| \(\displaystyle -c^4 \left (x \left (-\left (\frac {1}{a x}+1\right )^{7/2}\right ) \left (1-\frac {1}{a x}\right )^{9/2}-\frac {\frac {1}{6} \left (\frac {3}{5} \left (\frac {1}{4} a \int \frac {\left (40 a-\frac {21}{x}\right ) \left (1+\frac {1}{a x}\right )^{5/2} x}{a \sqrt {1-\frac {1}{a x}}}d\frac {1}{x}+\frac {19}{4} a \sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{7/2}\right )+\frac {29}{5} a \left (1-\frac {1}{a x}\right )^{3/2} \left (\frac {1}{a x}+1\right )^{7/2}\right )+\frac {8}{7} a \left (1-\frac {1}{a x}\right )^{7/2} \left (\frac {1}{a x}+1\right )^{7/2}+\frac {7}{6} a \left (1-\frac {1}{a x}\right )^{5/2} \left (\frac {1}{a x}+1\right )^{7/2}}{a^2}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -c^4 \left (x \left (-\left (\frac {1}{a x}+1\right )^{7/2}\right ) \left (1-\frac {1}{a x}\right )^{9/2}-\frac {\frac {1}{6} \left (\frac {3}{5} \left (\frac {1}{4} \int \frac {\left (40 a-\frac {21}{x}\right ) \left (1+\frac {1}{a x}\right )^{5/2} x}{\sqrt {1-\frac {1}{a x}}}d\frac {1}{x}+\frac {19}{4} a \sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{7/2}\right )+\frac {29}{5} a \left (1-\frac {1}{a x}\right )^{3/2} \left (\frac {1}{a x}+1\right )^{7/2}\right )+\frac {8}{7} a \left (1-\frac {1}{a x}\right )^{7/2} \left (\frac {1}{a x}+1\right )^{7/2}+\frac {7}{6} a \left (1-\frac {1}{a x}\right )^{5/2} \left (\frac {1}{a x}+1\right )^{7/2}}{a^2}\right )\) |
| \(\Big \downarrow \) 171 |
| \(\displaystyle -c^4 \left (x \left (-\left (\frac {1}{a x}+1\right )^{7/2}\right ) \left (1-\frac {1}{a x}\right )^{9/2}-\frac {\frac {1}{6} \left (\frac {3}{5} \left (\frac {1}{4} \left (7 a \sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{5/2}-\frac {1}{3} a \int -\frac {15 \left (8 a+\frac {1}{x}\right ) \left (1+\frac {1}{a x}\right )^{3/2} x}{a \sqrt {1-\frac {1}{a x}}}d\frac {1}{x}\right )+\frac {19}{4} a \sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{7/2}\right )+\frac {29}{5} a \left (1-\frac {1}{a x}\right )^{3/2} \left (\frac {1}{a x}+1\right )^{7/2}\right )+\frac {8}{7} a \left (1-\frac {1}{a x}\right )^{7/2} \left (\frac {1}{a x}+1\right )^{7/2}+\frac {7}{6} a \left (1-\frac {1}{a x}\right )^{5/2} \left (\frac {1}{a x}+1\right )^{7/2}}{a^2}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -c^4 \left (x \left (-\left (\frac {1}{a x}+1\right )^{7/2}\right ) \left (1-\frac {1}{a x}\right )^{9/2}-\frac {\frac {1}{6} \left (\frac {3}{5} \left (\frac {1}{4} \left (5 \int \frac {\left (8 a+\frac {1}{x}\right ) \left (1+\frac {1}{a x}\right )^{3/2} x}{\sqrt {1-\frac {1}{a x}}}d\frac {1}{x}+7 a \sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{5/2}\right )+\frac {19}{4} a \sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{7/2}\right )+\frac {29}{5} a \left (1-\frac {1}{a x}\right )^{3/2} \left (\frac {1}{a x}+1\right )^{7/2}\right )+\frac {8}{7} a \left (1-\frac {1}{a x}\right )^{7/2} \left (\frac {1}{a x}+1\right )^{7/2}+\frac {7}{6} a \left (1-\frac {1}{a x}\right )^{5/2} \left (\frac {1}{a x}+1\right )^{7/2}}{a^2}\right )\) |
| \(\Big \downarrow \) 171 |
| \(\displaystyle -c^4 \left (x \left (-\left (\frac {1}{a x}+1\right )^{7/2}\right ) \left (1-\frac {1}{a x}\right )^{9/2}-\frac {\frac {1}{6} \left (\frac {3}{5} \left (\frac {1}{4} \left (5 \left (-\frac {1}{2} a \int -\frac {\left (16 a+\frac {19}{x}\right ) \sqrt {1+\frac {1}{a x}} x}{a \sqrt {1-\frac {1}{a x}}}d\frac {1}{x}-\frac {1}{2} a \sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{3/2}\right )+7 a \sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{5/2}\right )+\frac {19}{4} a \sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{7/2}\right )+\frac {29}{5} a \left (1-\frac {1}{a x}\right )^{3/2} \left (\frac {1}{a x}+1\right )^{7/2}\right )+\frac {8}{7} a \left (1-\frac {1}{a x}\right )^{7/2} \left (\frac {1}{a x}+1\right )^{7/2}+\frac {7}{6} a \left (1-\frac {1}{a x}\right )^{5/2} \left (\frac {1}{a x}+1\right )^{7/2}}{a^2}\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -c^4 \left (x \left (-\left (\frac {1}{a x}+1\right )^{7/2}\right ) \left (1-\frac {1}{a x}\right )^{9/2}-\frac {\frac {1}{6} \left (\frac {3}{5} \left (\frac {1}{4} \left (5 \left (\frac {1}{2} a \int \frac {\left (16 a+\frac {19}{x}\right ) \sqrt {1+\frac {1}{a x}} x}{a \sqrt {1-\frac {1}{a x}}}d\frac {1}{x}-\frac {1}{2} a \sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{3/2}\right )+7 a \sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{5/2}\right )+\frac {19}{4} a \sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{7/2}\right )+\frac {29}{5} a \left (1-\frac {1}{a x}\right )^{3/2} \left (\frac {1}{a x}+1\right )^{7/2}\right )+\frac {8}{7} a \left (1-\frac {1}{a x}\right )^{7/2} \left (\frac {1}{a x}+1\right )^{7/2}+\frac {7}{6} a \left (1-\frac {1}{a x}\right )^{5/2} \left (\frac {1}{a x}+1\right )^{7/2}}{a^2}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -c^4 \left (x \left (-\left (\frac {1}{a x}+1\right )^{7/2}\right ) \left (1-\frac {1}{a x}\right )^{9/2}-\frac {\frac {1}{6} \left (\frac {3}{5} \left (\frac {1}{4} \left (5 \left (\frac {1}{2} \int \frac {\left (16 a+\frac {19}{x}\right ) \sqrt {1+\frac {1}{a x}} x}{\sqrt {1-\frac {1}{a x}}}d\frac {1}{x}-\frac {1}{2} a \sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{3/2}\right )+7 a \sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{5/2}\right )+\frac {19}{4} a \sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{7/2}\right )+\frac {29}{5} a \left (1-\frac {1}{a x}\right )^{3/2} \left (\frac {1}{a x}+1\right )^{7/2}\right )+\frac {8}{7} a \left (1-\frac {1}{a x}\right )^{7/2} \left (\frac {1}{a x}+1\right )^{7/2}+\frac {7}{6} a \left (1-\frac {1}{a x}\right )^{5/2} \left (\frac {1}{a x}+1\right )^{7/2}}{a^2}\right )\) |
| \(\Big \downarrow \) 171 |
| \(\displaystyle -c^4 \left (x \left (-\left (\frac {1}{a x}+1\right )^{7/2}\right ) \left (1-\frac {1}{a x}\right )^{9/2}-\frac {\frac {1}{6} \left (\frac {3}{5} \left (\frac {1}{4} \left (5 \left (\frac {1}{2} \left (a \left (-\int -\frac {\left (16 a+\frac {35}{x}\right ) x}{a \sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}}d\frac {1}{x}\right )-19 a \sqrt {1-\frac {1}{a x}} \sqrt {\frac {1}{a x}+1}\right )-\frac {1}{2} a \sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{3/2}\right )+7 a \sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{5/2}\right )+\frac {19}{4} a \sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{7/2}\right )+\frac {29}{5} a \left (1-\frac {1}{a x}\right )^{3/2} \left (\frac {1}{a x}+1\right )^{7/2}\right )+\frac {8}{7} a \left (1-\frac {1}{a x}\right )^{7/2} \left (\frac {1}{a x}+1\right )^{7/2}+\frac {7}{6} a \left (1-\frac {1}{a x}\right )^{5/2} \left (\frac {1}{a x}+1\right )^{7/2}}{a^2}\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -c^4 \left (x \left (-\left (\frac {1}{a x}+1\right )^{7/2}\right ) \left (1-\frac {1}{a x}\right )^{9/2}-\frac {\frac {1}{6} \left (\frac {3}{5} \left (\frac {1}{4} \left (5 \left (\frac {1}{2} \left (a \int \frac {\left (16 a+\frac {35}{x}\right ) x}{a \sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}}d\frac {1}{x}-19 a \sqrt {1-\frac {1}{a x}} \sqrt {\frac {1}{a x}+1}\right )-\frac {1}{2} a \sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{3/2}\right )+7 a \sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{5/2}\right )+\frac {19}{4} a \sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{7/2}\right )+\frac {29}{5} a \left (1-\frac {1}{a x}\right )^{3/2} \left (\frac {1}{a x}+1\right )^{7/2}\right )+\frac {8}{7} a \left (1-\frac {1}{a x}\right )^{7/2} \left (\frac {1}{a x}+1\right )^{7/2}+\frac {7}{6} a \left (1-\frac {1}{a x}\right )^{5/2} \left (\frac {1}{a x}+1\right )^{7/2}}{a^2}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -c^4 \left (x \left (-\left (\frac {1}{a x}+1\right )^{7/2}\right ) \left (1-\frac {1}{a x}\right )^{9/2}-\frac {\frac {1}{6} \left (\frac {3}{5} \left (\frac {1}{4} \left (5 \left (\frac {1}{2} \left (\int \frac {\left (16 a+\frac {35}{x}\right ) x}{\sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}}d\frac {1}{x}-19 a \sqrt {1-\frac {1}{a x}} \sqrt {\frac {1}{a x}+1}\right )-\frac {1}{2} a \sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{3/2}\right )+7 a \sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{5/2}\right )+\frac {19}{4} a \sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{7/2}\right )+\frac {29}{5} a \left (1-\frac {1}{a x}\right )^{3/2} \left (\frac {1}{a x}+1\right )^{7/2}\right )+\frac {8}{7} a \left (1-\frac {1}{a x}\right )^{7/2} \left (\frac {1}{a x}+1\right )^{7/2}+\frac {7}{6} a \left (1-\frac {1}{a x}\right )^{5/2} \left (\frac {1}{a x}+1\right )^{7/2}}{a^2}\right )\) |
| \(\Big \downarrow \) 175 |
| \(\displaystyle -c^4 \left (x \left (-\left (\frac {1}{a x}+1\right )^{7/2}\right ) \left (1-\frac {1}{a x}\right )^{9/2}-\frac {\frac {1}{6} \left (\frac {3}{5} \left (\frac {1}{4} \left (5 \left (\frac {1}{2} \left (16 a \int \frac {x}{\sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}}d\frac {1}{x}+35 \int \frac {1}{\sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}}d\frac {1}{x}-19 a \sqrt {1-\frac {1}{a x}} \sqrt {\frac {1}{a x}+1}\right )-\frac {1}{2} a \sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{3/2}\right )+7 a \sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{5/2}\right )+\frac {19}{4} a \sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{7/2}\right )+\frac {29}{5} a \left (1-\frac {1}{a x}\right )^{3/2} \left (\frac {1}{a x}+1\right )^{7/2}\right )+\frac {8}{7} a \left (1-\frac {1}{a x}\right )^{7/2} \left (\frac {1}{a x}+1\right )^{7/2}+\frac {7}{6} a \left (1-\frac {1}{a x}\right )^{5/2} \left (\frac {1}{a x}+1\right )^{7/2}}{a^2}\right )\) |
| \(\Big \downarrow \) 39 |
| \(\displaystyle -c^4 \left (x \left (-\left (\frac {1}{a x}+1\right )^{7/2}\right ) \left (1-\frac {1}{a x}\right )^{9/2}-\frac {\frac {1}{6} \left (\frac {3}{5} \left (\frac {1}{4} \left (5 \left (\frac {1}{2} \left (35 \int \frac {1}{\sqrt {1-\frac {1}{a^2 x^2}}}d\frac {1}{x}+16 a \int \frac {x}{\sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}}d\frac {1}{x}-19 a \sqrt {1-\frac {1}{a x}} \sqrt {\frac {1}{a x}+1}\right )-\frac {1}{2} a \sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{3/2}\right )+7 a \sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{5/2}\right )+\frac {19}{4} a \sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{7/2}\right )+\frac {29}{5} a \left (1-\frac {1}{a x}\right )^{3/2} \left (\frac {1}{a x}+1\right )^{7/2}\right )+\frac {8}{7} a \left (1-\frac {1}{a x}\right )^{7/2} \left (\frac {1}{a x}+1\right )^{7/2}+\frac {7}{6} a \left (1-\frac {1}{a x}\right )^{5/2} \left (\frac {1}{a x}+1\right )^{7/2}}{a^2}\right )\) |
| \(\Big \downarrow \) 103 |
| \(\displaystyle -c^4 \left (x \left (-\left (\frac {1}{a x}+1\right )^{7/2}\right ) \left (1-\frac {1}{a x}\right )^{9/2}-\frac {\frac {1}{6} \left (\frac {3}{5} \left (\frac {1}{4} \left (5 \left (\frac {1}{2} \left (35 \int \frac {1}{\sqrt {1-\frac {1}{a^2 x^2}}}d\frac {1}{x}-16 \int \frac {1}{\frac {1}{a}-\frac {1}{a x^2}}d\left (\sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}\right )-19 a \sqrt {1-\frac {1}{a x}} \sqrt {\frac {1}{a x}+1}\right )-\frac {1}{2} a \sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{3/2}\right )+7 a \sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{5/2}\right )+\frac {19}{4} a \sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{7/2}\right )+\frac {29}{5} a \left (1-\frac {1}{a x}\right )^{3/2} \left (\frac {1}{a x}+1\right )^{7/2}\right )+\frac {8}{7} a \left (1-\frac {1}{a x}\right )^{7/2} \left (\frac {1}{a x}+1\right )^{7/2}+\frac {7}{6} a \left (1-\frac {1}{a x}\right )^{5/2} \left (\frac {1}{a x}+1\right )^{7/2}}{a^2}\right )\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle -c^4 \left (x \left (-\left (\frac {1}{a x}+1\right )^{7/2}\right ) \left (1-\frac {1}{a x}\right )^{9/2}-\frac {\frac {1}{6} \left (\frac {3}{5} \left (\frac {1}{4} \left (5 \left (\frac {1}{2} \left (35 \int \frac {1}{\sqrt {1-\frac {1}{a^2 x^2}}}d\frac {1}{x}-16 a \text {arctanh}\left (\sqrt {1-\frac {1}{a x}} \sqrt {\frac {1}{a x}+1}\right )-19 a \sqrt {1-\frac {1}{a x}} \sqrt {\frac {1}{a x}+1}\right )-\frac {1}{2} a \sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{3/2}\right )+7 a \sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{5/2}\right )+\frac {19}{4} a \sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{7/2}\right )+\frac {29}{5} a \left (1-\frac {1}{a x}\right )^{3/2} \left (\frac {1}{a x}+1\right )^{7/2}\right )+\frac {8}{7} a \left (1-\frac {1}{a x}\right )^{7/2} \left (\frac {1}{a x}+1\right )^{7/2}+\frac {7}{6} a \left (1-\frac {1}{a x}\right )^{5/2} \left (\frac {1}{a x}+1\right )^{7/2}}{a^2}\right )\) |
| \(\Big \downarrow \) 223 |
| \(\displaystyle -c^4 \left (x \left (-\left (\frac {1}{a x}+1\right )^{7/2}\right ) \left (1-\frac {1}{a x}\right )^{9/2}-\frac {\frac {1}{6} \left (\frac {3}{5} \left (\frac {1}{4} \left (5 \left (\frac {1}{2} \left (35 a \arcsin \left (\frac {1}{a x}\right )-16 a \text {arctanh}\left (\sqrt {1-\frac {1}{a x}} \sqrt {\frac {1}{a x}+1}\right )-19 a \sqrt {1-\frac {1}{a x}} \sqrt {\frac {1}{a x}+1}\right )-\frac {1}{2} a \sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{3/2}\right )+7 a \sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{5/2}\right )+\frac {19}{4} a \sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{7/2}\right )+\frac {29}{5} a \left (1-\frac {1}{a x}\right )^{3/2} \left (\frac {1}{a x}+1\right )^{7/2}\right )+\frac {8}{7} a \left (1-\frac {1}{a x}\right )^{7/2} \left (\frac {1}{a x}+1\right )^{7/2}+\frac {7}{6} a \left (1-\frac {1}{a x}\right )^{5/2} \left (\frac {1}{a x}+1\right )^{7/2}}{a^2}\right )\) |
Input:
Int[(c - c/(a^2*x^2))^4/E^ArcCoth[a*x],x]
Output:
-(c^4*(-((1 - 1/(a*x))^(9/2)*(1 + 1/(a*x))^(7/2)*x) - ((7*a*(1 - 1/(a*x))^ (5/2)*(1 + 1/(a*x))^(7/2))/6 + (8*a*(1 - 1/(a*x))^(7/2)*(1 + 1/(a*x))^(7/2 ))/7 + ((29*a*(1 - 1/(a*x))^(3/2)*(1 + 1/(a*x))^(7/2))/5 + (3*((19*a*Sqrt[ 1 - 1/(a*x)]*(1 + 1/(a*x))^(7/2))/4 + (7*a*Sqrt[1 - 1/(a*x)]*(1 + 1/(a*x)) ^(5/2) + 5*(-1/2*(a*Sqrt[1 - 1/(a*x)]*(1 + 1/(a*x))^(3/2)) + (-19*a*Sqrt[1 - 1/(a*x)]*Sqrt[1 + 1/(a*x)] + 35*a*ArcSin[1/(a*x)] - 16*a*ArcTanh[Sqrt[1 - 1/(a*x)]*Sqrt[1 + 1/(a*x)]])/2))/4))/5)/6)/a^2))
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_) + (b_.)*(x_))^(m_.)*((c_) + (d_.)*(x_))^(m_.), x_Symbol] :> Int[( a*c + b*d*x^2)^m, x] /; FreeQ[{a, b, c, d, m}, x] && EqQ[b*c + a*d, 0] && ( IntegerQ[m] || (GtQ[a, 0] && GtQ[c, 0]))
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[(a + b*x)^(m + 1)*(c + d*x)^n*((e + f*x)^p/(b*(m + 1))) , x] - Simp[1/(b*(m + 1)) Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*(e + f* x)^(p - 1)*Simp[d*e*n + c*f*p + d*f*(n + p)*x, x], x], x] /; FreeQ[{a, b, c , d, e, f}, x] && LtQ[m, -1] && GtQ[n, 0] && GtQ[p, 0] && (IntegersQ[2*m, 2 *n, 2*p] || IntegersQ[m, n + p] || IntegersQ[p, m + n])
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_)*((g_.) + (h_.)*(x_)), x_] :> Simp[h*(a + b*x)^m*(c + d*x)^(n + 1)*(( e + f*x)^(p + 1)/(d*f*(m + n + p + 2))), x] + Simp[1/(d*f*(m + n + p + 2)) Int[(a + b*x)^(m - 1)*(c + d*x)^n*(e + f*x)^p*Simp[a*d*f*g*(m + n + p + 2 ) - h*(b*c*e*m + a*(d*e*(n + 1) + c*f*(p + 1))) + (b*d*f*g*(m + n + p + 2) + h*(a*d*f*m - b*(d*e*(m + n + 1) + c*f*(m + p + 1))))*x, x], x], x] /; Fre eQ[{a, b, c, d, e, f, g, h, n, p}, x] && GtQ[m, 0] && NeQ[m + n + p + 2, 0] && IntegersQ[2*m, 2*n, 2*p]
Int[(((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_ )))/((a_.) + (b_.)*(x_)), x_] :> Simp[h/b Int[(c + d*x)^n*(e + f*x)^p, x] , x] + Simp[(b*g - a*h)/b Int[(c + d*x)^n*((e + f*x)^p/(a + b*x)), x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x]
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[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSin[Rt[-b, 2]*(x/Sqrt [a])]/Rt[-b, 2], x] /; FreeQ[{a, b}, x] && GtQ[a, 0] && NegQ[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 = 0.12 (sec) , antiderivative size = 189, normalized size of antiderivative = 1.12
| method | result | size |
| risch | \(\frac {\left (a x +1\right ) \left (2816 x^{6} a^{6}+3045 a^{5} x^{5}-1952 a^{4} x^{4}-1330 a^{3} x^{3}+1056 a^{2} x^{2}+280 a x -240\right ) c^{4} \sqrt {\frac {a x -1}{a x +1}}}{1680 x^{7} a^{8}}+\frac {\left (\frac {35 a^{7} \arctan \left (\frac {1}{\sqrt {a^{2} x^{2}-1}}\right )}{16}-\frac {a^{8} \ln \left (\frac {a^{2} x}{\sqrt {a^{2}}}+\sqrt {a^{2} x^{2}-1}\right )}{\sqrt {a^{2}}}+a^{7} \sqrt {\left (a x -1\right ) \left (a x +1\right )}\right ) c^{4} \sqrt {\frac {a x -1}{a x +1}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}}{a^{8} \left (a x -1\right )}\) | \(189\) |
| default | \(-\frac {\sqrt {\frac {a x -1}{a x +1}}\, \left (a x +1\right ) c^{4} \left (-1680 \sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}\, a^{8} x^{8}+1680 \left (a^{2} x^{2}-1\right )^{\frac {3}{2}} \sqrt {a^{2}}\, a^{6} x^{6}-3675 a^{7} x^{7} \sqrt {a^{2}}\, \sqrt {a^{2} x^{2}-1}-3675 a^{7} x^{7} \sqrt {a^{2}}\, \arctan \left (\frac {1}{\sqrt {a^{2} x^{2}-1}}\right )+1680 \ln \left (\frac {a^{2} x +\sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) a^{8} x^{7}+1995 \left (a^{2} x^{2}-1\right )^{\frac {3}{2}} \sqrt {a^{2}}\, a^{5} x^{5}-1136 \left (a^{2} x^{2}-1\right )^{\frac {3}{2}} \sqrt {a^{2}}\, a^{4} x^{4}-1050 \sqrt {a^{2}}\, \left (a^{2} x^{2}-1\right )^{\frac {3}{2}} a^{3} x^{3}+816 \sqrt {a^{2}}\, \left (a^{2} x^{2}-1\right )^{\frac {3}{2}} a^{2} x^{2}+280 \sqrt {a^{2}}\, \left (a^{2} x^{2}-1\right )^{\frac {3}{2}} a x -240 \left (a^{2} x^{2}-1\right )^{\frac {3}{2}} \sqrt {a^{2}}\right )}{1680 \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, a^{8} x^{7} \sqrt {a^{2}}}\) | \(320\) |
Input:
int((c-c/a^2/x^2)^4*((a*x-1)/(a*x+1))^(1/2),x,method=_RETURNVERBOSE)
Output:
1/1680*(a*x+1)*(2816*a^6*x^6+3045*a^5*x^5-1952*a^4*x^4-1330*a^3*x^3+1056*a ^2*x^2+280*a*x-240)/x^7*c^4/a^8*((a*x-1)/(a*x+1))^(1/2)+(35/16*a^7*arctan( 1/(a^2*x^2-1)^(1/2))-a^8*ln(a^2*x/(a^2)^(1/2)+(a^2*x^2-1)^(1/2))/(a^2)^(1/ 2)+a^7*((a*x-1)*(a*x+1))^(1/2))*c^4/a^8*((a*x-1)/(a*x+1))^(1/2)*((a*x-1)*( a*x+1))^(1/2)/(a*x-1)
Time = 0.10 (sec) , antiderivative size = 201, normalized size of antiderivative = 1.19 \[ \int e^{-\coth ^{-1}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^4 \, dx=-\frac {7350 \, a^{7} c^{4} x^{7} \arctan \left (\sqrt {\frac {a x - 1}{a x + 1}}\right ) + 1680 \, a^{7} c^{4} x^{7} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right ) - 1680 \, a^{7} c^{4} x^{7} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right ) - {\left (1680 \, a^{8} c^{4} x^{8} + 4496 \, a^{7} c^{4} x^{7} + 5861 \, a^{6} c^{4} x^{6} + 1093 \, a^{5} c^{4} x^{5} - 3282 \, a^{4} c^{4} x^{4} - 274 \, a^{3} c^{4} x^{3} + 1336 \, a^{2} c^{4} x^{2} + 40 \, a c^{4} x - 240 \, c^{4}\right )} \sqrt {\frac {a x - 1}{a x + 1}}}{1680 \, a^{8} x^{7}} \] Input:
integrate((c-c/a^2/x^2)^4*((a*x-1)/(a*x+1))^(1/2),x, algorithm="fricas")
Output:
-1/1680*(7350*a^7*c^4*x^7*arctan(sqrt((a*x - 1)/(a*x + 1))) + 1680*a^7*c^4 *x^7*log(sqrt((a*x - 1)/(a*x + 1)) + 1) - 1680*a^7*c^4*x^7*log(sqrt((a*x - 1)/(a*x + 1)) - 1) - (1680*a^8*c^4*x^8 + 4496*a^7*c^4*x^7 + 5861*a^6*c^4* x^6 + 1093*a^5*c^4*x^5 - 3282*a^4*c^4*x^4 - 274*a^3*c^4*x^3 + 1336*a^2*c^4 *x^2 + 40*a*c^4*x - 240*c^4)*sqrt((a*x - 1)/(a*x + 1)))/(a^8*x^7)
\[ \int e^{-\coth ^{-1}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^4 \, dx=\frac {c^{4} \left (\int a^{8} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}\, dx + \int \frac {\sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{x^{8}}\, dx + \int \left (- \frac {4 a^{2} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{x^{6}}\right )\, dx + \int \frac {6 a^{4} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{x^{4}}\, dx + \int \left (- \frac {4 a^{6} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{x^{2}}\right )\, dx\right )}{a^{8}} \] Input:
integrate((c-c/a**2/x**2)**4*((a*x-1)/(a*x+1))**(1/2),x)
Output:
c**4*(Integral(a**8*sqrt(a*x/(a*x + 1) - 1/(a*x + 1)), x) + Integral(sqrt( a*x/(a*x + 1) - 1/(a*x + 1))/x**8, x) + Integral(-4*a**2*sqrt(a*x/(a*x + 1 ) - 1/(a*x + 1))/x**6, x) + Integral(6*a**4*sqrt(a*x/(a*x + 1) - 1/(a*x + 1))/x**4, x) + Integral(-4*a**6*sqrt(a*x/(a*x + 1) - 1/(a*x + 1))/x**2, x) )/a**8
Leaf count of result is larger than twice the leaf count of optimal. 380 vs. \(2 (149) = 298\).
Time = 0.11 (sec) , antiderivative size = 380, normalized size of antiderivative = 2.25 \[ \int e^{-\coth ^{-1}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^4 \, dx=-\frac {1}{840} \, {\left (\frac {3675 \, c^{4} \arctan \left (\sqrt {\frac {a x - 1}{a x + 1}}\right )}{a^{2}} + \frac {840 \, c^{4} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right )}{a^{2}} - \frac {840 \, c^{4} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right )}{a^{2}} - \frac {1995 \, c^{4} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {15}{2}} + 10185 \, c^{4} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {13}{2}} + 17619 \, c^{4} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {11}{2}} + 4569 \, c^{4} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {9}{2}} + 71801 \, c^{4} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {7}{2}} + 72051 \, c^{4} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {5}{2}} + 31465 \, c^{4} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}} + 5355 \, c^{4} \sqrt {\frac {a x - 1}{a x + 1}}}{\frac {6 \, {\left (a x - 1\right )} a^{2}}{a x + 1} + \frac {14 \, {\left (a x - 1\right )}^{2} a^{2}}{{\left (a x + 1\right )}^{2}} + \frac {14 \, {\left (a x - 1\right )}^{3} a^{2}}{{\left (a x + 1\right )}^{3}} - \frac {14 \, {\left (a x - 1\right )}^{5} a^{2}}{{\left (a x + 1\right )}^{5}} - \frac {14 \, {\left (a x - 1\right )}^{6} a^{2}}{{\left (a x + 1\right )}^{6}} - \frac {6 \, {\left (a x - 1\right )}^{7} a^{2}}{{\left (a x + 1\right )}^{7}} - \frac {{\left (a x - 1\right )}^{8} a^{2}}{{\left (a x + 1\right )}^{8}} + a^{2}}\right )} a \] Input:
integrate((c-c/a^2/x^2)^4*((a*x-1)/(a*x+1))^(1/2),x, algorithm="maxima")
Output:
-1/840*(3675*c^4*arctan(sqrt((a*x - 1)/(a*x + 1)))/a^2 + 840*c^4*log(sqrt( (a*x - 1)/(a*x + 1)) + 1)/a^2 - 840*c^4*log(sqrt((a*x - 1)/(a*x + 1)) - 1) /a^2 - (1995*c^4*((a*x - 1)/(a*x + 1))^(15/2) + 10185*c^4*((a*x - 1)/(a*x + 1))^(13/2) + 17619*c^4*((a*x - 1)/(a*x + 1))^(11/2) + 4569*c^4*((a*x - 1 )/(a*x + 1))^(9/2) + 71801*c^4*((a*x - 1)/(a*x + 1))^(7/2) + 72051*c^4*((a *x - 1)/(a*x + 1))^(5/2) + 31465*c^4*((a*x - 1)/(a*x + 1))^(3/2) + 5355*c^ 4*sqrt((a*x - 1)/(a*x + 1)))/(6*(a*x - 1)*a^2/(a*x + 1) + 14*(a*x - 1)^2*a ^2/(a*x + 1)^2 + 14*(a*x - 1)^3*a^2/(a*x + 1)^3 - 14*(a*x - 1)^5*a^2/(a*x + 1)^5 - 14*(a*x - 1)^6*a^2/(a*x + 1)^6 - 6*(a*x - 1)^7*a^2/(a*x + 1)^7 - (a*x - 1)^8*a^2/(a*x + 1)^8 + a^2))*a
Leaf count of result is larger than twice the leaf count of optimal. 524 vs. \(2 (149) = 298\).
Time = 0.16 (sec) , antiderivative size = 524, normalized size of antiderivative = 3.10 \[ \int e^{-\coth ^{-1}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^4 \, dx =\text {Too large to display} \] Input:
integrate((c-c/a^2/x^2)^4*((a*x-1)/(a*x+1))^(1/2),x, algorithm="giac")
Output:
-35/8*c^4*arctan(-x*abs(a) + sqrt(a^2*x^2 - 1))*sgn(a*x + 1)/a + c^4*log(a bs(-x*abs(a) + sqrt(a^2*x^2 - 1)))*sgn(a*x + 1)/abs(a) + sqrt(a^2*x^2 - 1) *c^4*sgn(a*x + 1)/a - 1/840*(3045*(x*abs(a) - sqrt(a^2*x^2 - 1))^13*c^4*ab s(a)*sgn(a*x + 1) - 6720*(x*abs(a) - sqrt(a^2*x^2 - 1))^12*a*c^4*sgn(a*x + 1) + 6860*(x*abs(a) - sqrt(a^2*x^2 - 1))^11*c^4*abs(a)*sgn(a*x + 1) - 201 60*(x*abs(a) - sqrt(a^2*x^2 - 1))^10*a*c^4*sgn(a*x + 1) + 9065*(x*abs(a) - sqrt(a^2*x^2 - 1))^9*c^4*abs(a)*sgn(a*x + 1) - 49280*(x*abs(a) - sqrt(a^2 *x^2 - 1))^8*a*c^4*sgn(a*x + 1) - 49280*(x*abs(a) - sqrt(a^2*x^2 - 1))^6*a *c^4*sgn(a*x + 1) - 9065*(x*abs(a) - sqrt(a^2*x^2 - 1))^5*c^4*abs(a)*sgn(a *x + 1) - 38976*(x*abs(a) - sqrt(a^2*x^2 - 1))^4*a*c^4*sgn(a*x + 1) - 6860 *(x*abs(a) - sqrt(a^2*x^2 - 1))^3*c^4*abs(a)*sgn(a*x + 1) - 12992*(x*abs(a ) - sqrt(a^2*x^2 - 1))^2*a*c^4*sgn(a*x + 1) - 3045*(x*abs(a) - sqrt(a^2*x^ 2 - 1))*c^4*abs(a)*sgn(a*x + 1) - 2816*a*c^4*sgn(a*x + 1))/(((x*abs(a) - s qrt(a^2*x^2 - 1))^2 + 1)^7*a*abs(a))
Time = 0.12 (sec) , antiderivative size = 332, normalized size of antiderivative = 1.96 \[ \int e^{-\coth ^{-1}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^4 \, dx=\frac {\frac {51\,c^4\,\sqrt {\frac {a\,x-1}{a\,x+1}}}{8}+\frac {899\,c^4\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{3/2}}{24}+\frac {3431\,c^4\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{5/2}}{40}+\frac {71801\,c^4\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{7/2}}{840}+\frac {1523\,c^4\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{9/2}}{280}+\frac {839\,c^4\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{11/2}}{40}+\frac {97\,c^4\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{13/2}}{8}+\frac {19\,c^4\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{15/2}}{8}}{a+\frac {6\,a\,\left (a\,x-1\right )}{a\,x+1}+\frac {14\,a\,{\left (a\,x-1\right )}^2}{{\left (a\,x+1\right )}^2}+\frac {14\,a\,{\left (a\,x-1\right )}^3}{{\left (a\,x+1\right )}^3}-\frac {14\,a\,{\left (a\,x-1\right )}^5}{{\left (a\,x+1\right )}^5}-\frac {14\,a\,{\left (a\,x-1\right )}^6}{{\left (a\,x+1\right )}^6}-\frac {6\,a\,{\left (a\,x-1\right )}^7}{{\left (a\,x+1\right )}^7}-\frac {a\,{\left (a\,x-1\right )}^8}{{\left (a\,x+1\right )}^8}}-\frac {35\,c^4\,\mathrm {atan}\left (\sqrt {\frac {a\,x-1}{a\,x+1}}\right )}{8\,a}-\frac {2\,c^4\,\mathrm {atanh}\left (\sqrt {\frac {a\,x-1}{a\,x+1}}\right )}{a} \] Input:
int((c - c/(a^2*x^2))^4*((a*x - 1)/(a*x + 1))^(1/2),x)
Output:
((51*c^4*((a*x - 1)/(a*x + 1))^(1/2))/8 + (899*c^4*((a*x - 1)/(a*x + 1))^( 3/2))/24 + (3431*c^4*((a*x - 1)/(a*x + 1))^(5/2))/40 + (71801*c^4*((a*x - 1)/(a*x + 1))^(7/2))/840 + (1523*c^4*((a*x - 1)/(a*x + 1))^(9/2))/280 + (8 39*c^4*((a*x - 1)/(a*x + 1))^(11/2))/40 + (97*c^4*((a*x - 1)/(a*x + 1))^(1 3/2))/8 + (19*c^4*((a*x - 1)/(a*x + 1))^(15/2))/8)/(a + (6*a*(a*x - 1))/(a *x + 1) + (14*a*(a*x - 1)^2)/(a*x + 1)^2 + (14*a*(a*x - 1)^3)/(a*x + 1)^3 - (14*a*(a*x - 1)^5)/(a*x + 1)^5 - (14*a*(a*x - 1)^6)/(a*x + 1)^6 - (6*a*( a*x - 1)^7)/(a*x + 1)^7 - (a*(a*x - 1)^8)/(a*x + 1)^8) - (35*c^4*atan(((a* x - 1)/(a*x + 1))^(1/2)))/(8*a) - (2*c^4*atanh(((a*x - 1)/(a*x + 1))^(1/2) ))/a
Time = 0.20 (sec) , antiderivative size = 243, normalized size of antiderivative = 1.44 \[ \int e^{-\coth ^{-1}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^4 \, dx=\frac {c^{4} \left (-7350 \mathit {atan} \left (\sqrt {a x -1}+\sqrt {a x +1}-1\right ) a^{7} x^{7}+7350 \mathit {atan} \left (\sqrt {a x -1}+\sqrt {a x +1}+1\right ) a^{7} x^{7}+1680 \sqrt {a x +1}\, \sqrt {a x -1}\, a^{7} x^{7}+2816 \sqrt {a x +1}\, \sqrt {a x -1}\, a^{6} x^{6}+3045 \sqrt {a x +1}\, \sqrt {a x -1}\, a^{5} x^{5}-1952 \sqrt {a x +1}\, \sqrt {a x -1}\, a^{4} x^{4}-1330 \sqrt {a x +1}\, \sqrt {a x -1}\, a^{3} x^{3}+1056 \sqrt {a x +1}\, \sqrt {a x -1}\, a^{2} x^{2}+280 \sqrt {a x +1}\, \sqrt {a x -1}\, a x -240 \sqrt {a x +1}\, \sqrt {a x -1}-3360 \,\mathrm {log}\left (\frac {\sqrt {a x -1}+\sqrt {a x +1}}{\sqrt {2}}\right ) a^{7} x^{7}-896 a^{7} x^{7}\right )}{1680 a^{8} x^{7}} \] Input:
int((c-c/a^2/x^2)^4*((a*x-1)/(a*x+1))^(1/2),x)
Output:
(c**4*( - 7350*atan(sqrt(a*x - 1) + sqrt(a*x + 1) - 1)*a**7*x**7 + 7350*at an(sqrt(a*x - 1) + sqrt(a*x + 1) + 1)*a**7*x**7 + 1680*sqrt(a*x + 1)*sqrt( a*x - 1)*a**7*x**7 + 2816*sqrt(a*x + 1)*sqrt(a*x - 1)*a**6*x**6 + 3045*sqr t(a*x + 1)*sqrt(a*x - 1)*a**5*x**5 - 1952*sqrt(a*x + 1)*sqrt(a*x - 1)*a**4 *x**4 - 1330*sqrt(a*x + 1)*sqrt(a*x - 1)*a**3*x**3 + 1056*sqrt(a*x + 1)*sq rt(a*x - 1)*a**2*x**2 + 280*sqrt(a*x + 1)*sqrt(a*x - 1)*a*x - 240*sqrt(a*x + 1)*sqrt(a*x - 1) - 3360*log((sqrt(a*x - 1) + sqrt(a*x + 1))/sqrt(2))*a* *7*x**7 - 896*a**7*x**7))/(1680*a**8*x**7)