Integrand size = 22, antiderivative size = 180 \[ \int e^{3 \coth ^{-1}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^4 \, dx=\frac {c^4 \left (1-\frac {1}{a^2 x^2}\right )^{7/2}}{7 a}-\frac {c^4 \left (1-\frac {1}{a^2 x^2}\right )^{5/2} \left (6 a-\frac {5}{x}\right )}{10 a^2}-\frac {c^4 \left (1-\frac {1}{a^2 x^2}\right )^{3/2} \left (8 a-\frac {5}{x}\right )}{8 a^2}-\frac {3 c^4 \sqrt {1-\frac {1}{a^2 x^2}} \left (16 a-\frac {5}{x}\right )}{16 a^2}+c^4 \left (1-\frac {1}{a^2 x^2}\right )^{7/2} x+\frac {15 c^4 \csc ^{-1}(a x)}{16 a}+\frac {3 c^4 \text {arctanh}\left (\sqrt {1-\frac {1}{a^2 x^2}}\right )}{a} \] Output:
1/7*c^4*(1-1/a^2/x^2)^(7/2)/a-1/10*c^4*(1-1/a^2/x^2)^(5/2)*(6*a-5/x)/a^2-1 /8*c^4*(1-1/a^2/x^2)^(3/2)*(8*a-5/x)/a^2-3/16*c^4*(1-1/a^2/x^2)^(1/2)*(16* a-5/x)/a^2+c^4*(1-1/a^2/x^2)^(7/2)*x+15/16*c^4*arccsc(a*x)/a+3*c^4*arctanh ((1-1/a^2/x^2)^(1/2))/a
Time = 0.36 (sec) , antiderivative size = 126, normalized size of antiderivative = 0.70 \[ \int e^{3 \coth ^{-1}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^4 \, dx=\frac {c^4 \left (\sqrt {1-\frac {1}{a^2 x^2}} \left (-80-280 a x-96 a^2 x^2+770 a^3 x^3+992 a^4 x^4-525 a^5 x^5-2496 a^6 x^6+560 a^7 x^7\right )+525 a^6 x^6 \arcsin \left (\frac {1}{a x}\right )+1680 a^6 x^6 \log \left (\left (1+\sqrt {1-\frac {1}{a^2 x^2}}\right ) x\right )\right )}{560 a^7 x^6} \] Input:
Integrate[E^(3*ArcCoth[a*x])*(c - c/(a^2*x^2))^4,x]
Output:
(c^4*(Sqrt[1 - 1/(a^2*x^2)]*(-80 - 280*a*x - 96*a^2*x^2 + 770*a^3*x^3 + 99 2*a^4*x^4 - 525*a^5*x^5 - 2496*a^6*x^6 + 560*a^7*x^7) + 525*a^6*x^6*ArcSin [1/(a*x)] + 1680*a^6*x^6*Log[(1 + Sqrt[1 - 1/(a^2*x^2)])*x]))/(560*a^7*x^6 )
Time = 0.88 (sec) , antiderivative size = 336, normalized size of antiderivative = 1.87, number of steps used = 25, number of rules used = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.091, Rules used = {6748, 108, 27, 171, 27, 171, 27, 171, 25, 27, 171, 27, 171, 27, 171, 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.
| \(\displaystyle \int \left (c-\frac {c}{a^2 x^2}\right )^4 e^{3 \coth ^{-1}(a x)} \, dx\) |
| \(\Big \downarrow \) 6748 |
| \(\displaystyle -c^4 \int \left (1-\frac {1}{a x}\right )^{5/2} \left (1+\frac {1}{a x}\right )^{11/2} x^2d\frac {1}{x}\) |
| \(\Big \downarrow \) 108 |
| \(\displaystyle -c^4 \left (\int \frac {\left (3 a-\frac {8}{x}\right ) \left (1-\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{9/2} x}{a^2}d\frac {1}{x}-x \left (1-\frac {1}{a x}\right )^{5/2} \left (\frac {1}{a x}+1\right )^{11/2}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -c^4 \left (\frac {\int \left (3 a-\frac {8}{x}\right ) \left (1-\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{9/2} xd\frac {1}{x}}{a^2}-x \left (1-\frac {1}{a x}\right )^{5/2} \left (\frac {1}{a x}+1\right )^{11/2}\right )\) |
| \(\Big \downarrow \) 171 |
| \(\displaystyle -c^4 \left (\frac {\frac {1}{7} a \int \frac {3 \left (7 a-\frac {15}{x}\right ) \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{9/2} x}{a}d\frac {1}{x}-\frac {8}{7} a \left (1-\frac {1}{a x}\right )^{3/2} \left (\frac {1}{a x}+1\right )^{11/2}}{a^2}-x \left (1-\frac {1}{a x}\right )^{5/2} \left (\frac {1}{a x}+1\right )^{11/2}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -c^4 \left (\frac {\frac {3}{7} \int \left (7 a-\frac {15}{x}\right ) \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{9/2} xd\frac {1}{x}-\frac {8}{7} a \left (1-\frac {1}{a x}\right )^{3/2} \left (\frac {1}{a x}+1\right )^{11/2}}{a^2}-x \left (1-\frac {1}{a x}\right )^{5/2} \left (\frac {1}{a x}+1\right )^{11/2}\right )\) |
| \(\Big \downarrow \) 171 |
| \(\displaystyle -c^4 \left (\frac {\frac {3}{7} \left (\frac {1}{6} a \int \frac {3 \left (14 a-\frac {19}{x}\right ) \left (1+\frac {1}{a x}\right )^{9/2} x}{a \sqrt {1-\frac {1}{a x}}}d\frac {1}{x}-\frac {5}{2} a \sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{11/2}\right )-\frac {8}{7} a \left (1-\frac {1}{a x}\right )^{3/2} \left (\frac {1}{a x}+1\right )^{11/2}}{a^2}-x \left (1-\frac {1}{a x}\right )^{5/2} \left (\frac {1}{a x}+1\right )^{11/2}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -c^4 \left (\frac {\frac {3}{7} \left (\frac {1}{2} \int \frac {\left (14 a-\frac {19}{x}\right ) \left (1+\frac {1}{a x}\right )^{9/2} x}{\sqrt {1-\frac {1}{a x}}}d\frac {1}{x}-\frac {5}{2} a \sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{11/2}\right )-\frac {8}{7} a \left (1-\frac {1}{a x}\right )^{3/2} \left (\frac {1}{a x}+1\right )^{11/2}}{a^2}-x \left (1-\frac {1}{a x}\right )^{5/2} \left (\frac {1}{a x}+1\right )^{11/2}\right )\) |
| \(\Big \downarrow \) 171 |
| \(\displaystyle -c^4 \left (\frac {\frac {3}{7} \left (\frac {1}{2} \left (\frac {19}{5} a \sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{9/2}-\frac {1}{5} a \int -\frac {\left (70 a-\frac {101}{x}\right ) \left (1+\frac {1}{a x}\right )^{7/2} x}{a \sqrt {1-\frac {1}{a x}}}d\frac {1}{x}\right )-\frac {5}{2} a \sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{11/2}\right )-\frac {8}{7} a \left (1-\frac {1}{a x}\right )^{3/2} \left (\frac {1}{a x}+1\right )^{11/2}}{a^2}-x \left (1-\frac {1}{a x}\right )^{5/2} \left (\frac {1}{a x}+1\right )^{11/2}\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -c^4 \left (\frac {\frac {3}{7} \left (\frac {1}{2} \left (\frac {1}{5} a \int \frac {\left (70 a-\frac {101}{x}\right ) \left (1+\frac {1}{a x}\right )^{7/2} x}{a \sqrt {1-\frac {1}{a x}}}d\frac {1}{x}+\frac {19}{5} a \sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{9/2}\right )-\frac {5}{2} a \sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{11/2}\right )-\frac {8}{7} a \left (1-\frac {1}{a x}\right )^{3/2} \left (\frac {1}{a x}+1\right )^{11/2}}{a^2}-x \left (1-\frac {1}{a x}\right )^{5/2} \left (\frac {1}{a x}+1\right )^{11/2}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -c^4 \left (\frac {\frac {3}{7} \left (\frac {1}{2} \left (\frac {1}{5} \int \frac {\left (70 a-\frac {101}{x}\right ) \left (1+\frac {1}{a x}\right )^{7/2} x}{\sqrt {1-\frac {1}{a x}}}d\frac {1}{x}+\frac {19}{5} a \sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{9/2}\right )-\frac {5}{2} a \sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{11/2}\right )-\frac {8}{7} a \left (1-\frac {1}{a x}\right )^{3/2} \left (\frac {1}{a x}+1\right )^{11/2}}{a^2}-x \left (1-\frac {1}{a x}\right )^{5/2} \left (\frac {1}{a x}+1\right )^{11/2}\right )\) |
| \(\Big \downarrow \) 171 |
| \(\displaystyle -c^4 \left (\frac {\frac {3}{7} \left (\frac {1}{2} \left (\frac {1}{5} \left (\frac {101}{4} a \sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{7/2}-\frac {1}{4} a \int -\frac {7 \left (40 a-\frac {61}{x}\right ) \left (1+\frac {1}{a x}\right )^{5/2} x}{a \sqrt {1-\frac {1}{a x}}}d\frac {1}{x}\right )+\frac {19}{5} a \sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{9/2}\right )-\frac {5}{2} a \sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{11/2}\right )-\frac {8}{7} a \left (1-\frac {1}{a x}\right )^{3/2} \left (\frac {1}{a x}+1\right )^{11/2}}{a^2}-x \left (1-\frac {1}{a x}\right )^{5/2} \left (\frac {1}{a x}+1\right )^{11/2}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -c^4 \left (\frac {\frac {3}{7} \left (\frac {1}{2} \left (\frac {1}{5} \left (\frac {7}{4} \int \frac {\left (40 a-\frac {61}{x}\right ) \left (1+\frac {1}{a x}\right )^{5/2} x}{\sqrt {1-\frac {1}{a x}}}d\frac {1}{x}+\frac {101}{4} a \sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{7/2}\right )+\frac {19}{5} a \sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{9/2}\right )-\frac {5}{2} a \sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{11/2}\right )-\frac {8}{7} a \left (1-\frac {1}{a x}\right )^{3/2} \left (\frac {1}{a x}+1\right )^{11/2}}{a^2}-x \left (1-\frac {1}{a x}\right )^{5/2} \left (\frac {1}{a x}+1\right )^{11/2}\right )\) |
| \(\Big \downarrow \) 171 |
| \(\displaystyle -c^4 \left (\frac {\frac {3}{7} \left (\frac {1}{2} \left (\frac {1}{5} \left (\frac {7}{4} \left (\frac {61}{3} a \sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{5/2}-\frac {1}{3} a \int -\frac {5 \left (24 a-\frac {37}{x}\right ) \left (1+\frac {1}{a x}\right )^{3/2} x}{a \sqrt {1-\frac {1}{a x}}}d\frac {1}{x}\right )+\frac {101}{4} a \sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{7/2}\right )+\frac {19}{5} a \sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{9/2}\right )-\frac {5}{2} a \sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{11/2}\right )-\frac {8}{7} a \left (1-\frac {1}{a x}\right )^{3/2} \left (\frac {1}{a x}+1\right )^{11/2}}{a^2}-x \left (1-\frac {1}{a x}\right )^{5/2} \left (\frac {1}{a x}+1\right )^{11/2}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -c^4 \left (\frac {\frac {3}{7} \left (\frac {1}{2} \left (\frac {1}{5} \left (\frac {7}{4} \left (\frac {5}{3} \int \frac {\left (24 a-\frac {37}{x}\right ) \left (1+\frac {1}{a x}\right )^{3/2} x}{\sqrt {1-\frac {1}{a x}}}d\frac {1}{x}+\frac {61}{3} a \sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{5/2}\right )+\frac {101}{4} a \sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{7/2}\right )+\frac {19}{5} a \sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{9/2}\right )-\frac {5}{2} a \sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{11/2}\right )-\frac {8}{7} a \left (1-\frac {1}{a x}\right )^{3/2} \left (\frac {1}{a x}+1\right )^{11/2}}{a^2}-x \left (1-\frac {1}{a x}\right )^{5/2} \left (\frac {1}{a x}+1\right )^{11/2}\right )\) |
| \(\Big \downarrow \) 171 |
| \(\displaystyle -c^4 \left (\frac {\frac {3}{7} \left (\frac {1}{2} \left (\frac {1}{5} \left (\frac {7}{4} \left (\frac {5}{3} \left (\frac {37}{2} a \sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{3/2}-\frac {1}{2} a \int -\frac {3 \left (16 a-\frac {21}{x}\right ) \sqrt {1+\frac {1}{a x}} x}{a \sqrt {1-\frac {1}{a x}}}d\frac {1}{x}\right )+\frac {61}{3} a \sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{5/2}\right )+\frac {101}{4} a \sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{7/2}\right )+\frac {19}{5} a \sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{9/2}\right )-\frac {5}{2} a \sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{11/2}\right )-\frac {8}{7} a \left (1-\frac {1}{a x}\right )^{3/2} \left (\frac {1}{a x}+1\right )^{11/2}}{a^2}-x \left (1-\frac {1}{a x}\right )^{5/2} \left (\frac {1}{a x}+1\right )^{11/2}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -c^4 \left (\frac {\frac {3}{7} \left (\frac {1}{2} \left (\frac {1}{5} \left (\frac {7}{4} \left (\frac {5}{3} \left (\frac {3}{2} \int \frac {\left (16 a-\frac {21}{x}\right ) \sqrt {1+\frac {1}{a x}} x}{\sqrt {1-\frac {1}{a x}}}d\frac {1}{x}+\frac {37}{2} a \sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{3/2}\right )+\frac {61}{3} a \sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{5/2}\right )+\frac {101}{4} a \sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{7/2}\right )+\frac {19}{5} a \sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{9/2}\right )-\frac {5}{2} a \sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{11/2}\right )-\frac {8}{7} a \left (1-\frac {1}{a x}\right )^{3/2} \left (\frac {1}{a x}+1\right )^{11/2}}{a^2}-x \left (1-\frac {1}{a x}\right )^{5/2} \left (\frac {1}{a x}+1\right )^{11/2}\right )\) |
| \(\Big \downarrow \) 171 |
| \(\displaystyle -c^4 \left (\frac {\frac {3}{7} \left (\frac {1}{2} \left (\frac {1}{5} \left (\frac {7}{4} \left (\frac {5}{3} \left (\frac {3}{2} \left (21 a \sqrt {1-\frac {1}{a x}} \sqrt {\frac {1}{a x}+1}-a \int -\frac {\left (16 a-\frac {5}{x}\right ) x}{a \sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}}d\frac {1}{x}\right )+\frac {37}{2} a \sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{3/2}\right )+\frac {61}{3} a \sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{5/2}\right )+\frac {101}{4} a \sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{7/2}\right )+\frac {19}{5} a \sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{9/2}\right )-\frac {5}{2} a \sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{11/2}\right )-\frac {8}{7} a \left (1-\frac {1}{a x}\right )^{3/2} \left (\frac {1}{a x}+1\right )^{11/2}}{a^2}-x \left (1-\frac {1}{a x}\right )^{5/2} \left (\frac {1}{a x}+1\right )^{11/2}\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -c^4 \left (\frac {\frac {3}{7} \left (\frac {1}{2} \left (\frac {1}{5} \left (\frac {7}{4} \left (\frac {5}{3} \left (\frac {3}{2} \left (a \int \frac {\left (16 a-\frac {5}{x}\right ) x}{a \sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}}d\frac {1}{x}+21 a \sqrt {1-\frac {1}{a x}} \sqrt {\frac {1}{a x}+1}\right )+\frac {37}{2} a \sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{3/2}\right )+\frac {61}{3} a \sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{5/2}\right )+\frac {101}{4} a \sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{7/2}\right )+\frac {19}{5} a \sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{9/2}\right )-\frac {5}{2} a \sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{11/2}\right )-\frac {8}{7} a \left (1-\frac {1}{a x}\right )^{3/2} \left (\frac {1}{a x}+1\right )^{11/2}}{a^2}-x \left (1-\frac {1}{a x}\right )^{5/2} \left (\frac {1}{a x}+1\right )^{11/2}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -c^4 \left (\frac {\frac {3}{7} \left (\frac {1}{2} \left (\frac {1}{5} \left (\frac {7}{4} \left (\frac {5}{3} \left (\frac {3}{2} \left (\int \frac {\left (16 a-\frac {5}{x}\right ) x}{\sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}}d\frac {1}{x}+21 a \sqrt {1-\frac {1}{a x}} \sqrt {\frac {1}{a x}+1}\right )+\frac {37}{2} a \sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{3/2}\right )+\frac {61}{3} a \sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{5/2}\right )+\frac {101}{4} a \sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{7/2}\right )+\frac {19}{5} a \sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{9/2}\right )-\frac {5}{2} a \sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{11/2}\right )-\frac {8}{7} a \left (1-\frac {1}{a x}\right )^{3/2} \left (\frac {1}{a x}+1\right )^{11/2}}{a^2}-x \left (1-\frac {1}{a x}\right )^{5/2} \left (\frac {1}{a x}+1\right )^{11/2}\right )\) |
| \(\Big \downarrow \) 175 |
| \(\displaystyle -c^4 \left (\frac {\frac {3}{7} \left (\frac {1}{2} \left (\frac {1}{5} \left (\frac {7}{4} \left (\frac {5}{3} \left (\frac {3}{2} \left (16 a \int \frac {x}{\sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}}d\frac {1}{x}-5 \int \frac {1}{\sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}}d\frac {1}{x}+21 a \sqrt {1-\frac {1}{a x}} \sqrt {\frac {1}{a x}+1}\right )+\frac {37}{2} a \sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{3/2}\right )+\frac {61}{3} a \sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{5/2}\right )+\frac {101}{4} a \sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{7/2}\right )+\frac {19}{5} a \sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{9/2}\right )-\frac {5}{2} a \sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{11/2}\right )-\frac {8}{7} a \left (1-\frac {1}{a x}\right )^{3/2} \left (\frac {1}{a x}+1\right )^{11/2}}{a^2}-x \left (1-\frac {1}{a x}\right )^{5/2} \left (\frac {1}{a x}+1\right )^{11/2}\right )\) |
| \(\Big \downarrow \) 39 |
| \(\displaystyle -c^4 \left (\frac {\frac {3}{7} \left (\frac {1}{2} \left (\frac {1}{5} \left (\frac {7}{4} \left (\frac {5}{3} \left (\frac {3}{2} \left (-5 \int \frac {1}{\sqrt {1-\frac {1}{a^2 x^2}}}d\frac {1}{x}+16 a \int \frac {x}{\sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}}d\frac {1}{x}+21 a \sqrt {1-\frac {1}{a x}} \sqrt {\frac {1}{a x}+1}\right )+\frac {37}{2} a \sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{3/2}\right )+\frac {61}{3} a \sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{5/2}\right )+\frac {101}{4} a \sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{7/2}\right )+\frac {19}{5} a \sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{9/2}\right )-\frac {5}{2} a \sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{11/2}\right )-\frac {8}{7} a \left (1-\frac {1}{a x}\right )^{3/2} \left (\frac {1}{a x}+1\right )^{11/2}}{a^2}-x \left (1-\frac {1}{a x}\right )^{5/2} \left (\frac {1}{a x}+1\right )^{11/2}\right )\) |
| \(\Big \downarrow \) 103 |
| \(\displaystyle -c^4 \left (\frac {\frac {3}{7} \left (\frac {1}{2} \left (\frac {1}{5} \left (\frac {7}{4} \left (\frac {5}{3} \left (\frac {3}{2} \left (-5 \int \frac {1}{\sqrt {1-\frac {1}{a^2 x^2}}}d\frac {1}{x}-16 \int \frac {1}{\frac {1}{a}-\frac {1}{a x^2}}d\left (\sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}\right )+21 a \sqrt {1-\frac {1}{a x}} \sqrt {\frac {1}{a x}+1}\right )+\frac {37}{2} a \sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{3/2}\right )+\frac {61}{3} a \sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{5/2}\right )+\frac {101}{4} a \sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{7/2}\right )+\frac {19}{5} a \sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{9/2}\right )-\frac {5}{2} a \sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{11/2}\right )-\frac {8}{7} a \left (1-\frac {1}{a x}\right )^{3/2} \left (\frac {1}{a x}+1\right )^{11/2}}{a^2}-x \left (1-\frac {1}{a x}\right )^{5/2} \left (\frac {1}{a x}+1\right )^{11/2}\right )\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle -c^4 \left (\frac {\frac {3}{7} \left (\frac {1}{2} \left (\frac {1}{5} \left (\frac {7}{4} \left (\frac {5}{3} \left (\frac {3}{2} \left (-5 \int \frac {1}{\sqrt {1-\frac {1}{a^2 x^2}}}d\frac {1}{x}-16 a \text {arctanh}\left (\sqrt {1-\frac {1}{a x}} \sqrt {\frac {1}{a x}+1}\right )+21 a \sqrt {1-\frac {1}{a x}} \sqrt {\frac {1}{a x}+1}\right )+\frac {37}{2} a \sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{3/2}\right )+\frac {61}{3} a \sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{5/2}\right )+\frac {101}{4} a \sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{7/2}\right )+\frac {19}{5} a \sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{9/2}\right )-\frac {5}{2} a \sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{11/2}\right )-\frac {8}{7} a \left (1-\frac {1}{a x}\right )^{3/2} \left (\frac {1}{a x}+1\right )^{11/2}}{a^2}-x \left (1-\frac {1}{a x}\right )^{5/2} \left (\frac {1}{a x}+1\right )^{11/2}\right )\) |
| \(\Big \downarrow \) 223 |
| \(\displaystyle -c^4 \left (\frac {\frac {3}{7} \left (\frac {1}{2} \left (\frac {1}{5} \left (\frac {7}{4} \left (\frac {5}{3} \left (\frac {3}{2} \left (-5 a \arcsin \left (\frac {1}{a x}\right )-16 a \text {arctanh}\left (\sqrt {1-\frac {1}{a x}} \sqrt {\frac {1}{a x}+1}\right )+21 a \sqrt {1-\frac {1}{a x}} \sqrt {\frac {1}{a x}+1}\right )+\frac {37}{2} a \sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{3/2}\right )+\frac {61}{3} a \sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{5/2}\right )+\frac {101}{4} a \sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{7/2}\right )+\frac {19}{5} a \sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{9/2}\right )-\frac {5}{2} a \sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{11/2}\right )-\frac {8}{7} a \left (1-\frac {1}{a x}\right )^{3/2} \left (\frac {1}{a x}+1\right )^{11/2}}{a^2}-x \left (1-\frac {1}{a x}\right )^{5/2} \left (\frac {1}{a x}+1\right )^{11/2}\right )\) |
Input:
Int[E^(3*ArcCoth[a*x])*(c - c/(a^2*x^2))^4,x]
Output:
-(c^4*(-((1 - 1/(a*x))^(5/2)*(1 + 1/(a*x))^(11/2)*x) + ((-8*a*(1 - 1/(a*x) )^(3/2)*(1 + 1/(a*x))^(11/2))/7 + (3*((-5*a*Sqrt[1 - 1/(a*x)]*(1 + 1/(a*x) )^(11/2))/2 + ((19*a*Sqrt[1 - 1/(a*x)]*(1 + 1/(a*x))^(9/2))/5 + ((101*a*Sq rt[1 - 1/(a*x)]*(1 + 1/(a*x))^(7/2))/4 + (7*((61*a*Sqrt[1 - 1/(a*x)]*(1 + 1/(a*x))^(5/2))/3 + (5*((37*a*Sqrt[1 - 1/(a*x)]*(1 + 1/(a*x))^(3/2))/2 + ( 3*(21*a*Sqrt[1 - 1/(a*x)]*Sqrt[1 + 1/(a*x)] - 5*a*ArcSin[1/(a*x)] - 16*a*A rcTanh[Sqrt[1 - 1/(a*x)]*Sqrt[1 + 1/(a*x)]]))/2))/3))/4)/5)/2))/7)/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 = 180, normalized size of antiderivative = 1.00
| method | result | size |
| risch | \(\frac {\left (a x -1\right ) \left (560 a^{7} x^{7}-2496 x^{6} a^{6}-525 a^{5} x^{5}+992 a^{4} x^{4}+770 a^{3} x^{3}-96 a^{2} x^{2}-280 a x -80\right ) c^{4}}{560 x^{7} a^{8} \sqrt {\frac {a x -1}{a x +1}}}+\frac {\left (\frac {15 a^{7} \arctan \left (\frac {1}{\sqrt {a^{2} x^{2}-1}}\right )}{16}+\frac {3 a^{8} \ln \left (\frac {a^{2} x}{\sqrt {a^{2}}}+\sqrt {a^{2} x^{2}-1}\right )}{\sqrt {a^{2}}}\right ) c^{4} \sqrt {\left (a x -1\right ) \left (a x +1\right )}}{a^{8} \left (a x +1\right ) \sqrt {\frac {a x -1}{a x +1}}}\) | \(180\) |
| default | \(\frac {\left (a x -1\right )^{2} c^{4} \left (-1680 \sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}\, a^{8} x^{8}+1680 \left (a^{2} x^{2}-1\right )^{\frac {3}{2}} \sqrt {a^{2}}\, a^{6} x^{6}+525 a^{7} x^{7} \sqrt {a^{2}}\, \sqrt {a^{2} x^{2}-1}+525 a^{7} x^{7} \sqrt {a^{2}}\, \arctan \left (\frac {1}{\sqrt {a^{2} x^{2}-1}}\right )+1680 \ln \left (\frac {a^{2} x +\sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) a^{8} x^{7}+35 \left (a^{2} x^{2}-1\right )^{\frac {3}{2}} \sqrt {a^{2}}\, a^{5} x^{5}-816 \left (a^{2} x^{2}-1\right )^{\frac {3}{2}} \sqrt {a^{2}}\, a^{4} x^{4}-490 \sqrt {a^{2}}\, \left (a^{2} x^{2}-1\right )^{\frac {3}{2}} a^{3} x^{3}+176 \sqrt {a^{2}}\, \left (a^{2} x^{2}-1\right )^{\frac {3}{2}} a^{2} x^{2}+280 \sqrt {a^{2}}\, \left (a^{2} x^{2}-1\right )^{\frac {3}{2}} a x +80 \left (a^{2} x^{2}-1\right )^{\frac {3}{2}} \sqrt {a^{2}}\right )}{560 \left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}} \left (a x +1\right ) \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, a^{8} x^{7} \sqrt {a^{2}}}\) | \(329\) |
Input:
int(1/((a*x-1)/(a*x+1))^(3/2)*(c-c/a^2/x^2)^4,x,method=_RETURNVERBOSE)
Output:
1/560*(a*x-1)*(560*a^7*x^7-2496*a^6*x^6-525*a^5*x^5+992*a^4*x^4+770*a^3*x^ 3-96*a^2*x^2-280*a*x-80)/x^7*c^4/a^8/((a*x-1)/(a*x+1))^(1/2)+(15/16*a^7*ar ctan(1/(a^2*x^2-1)^(1/2))+3*a^8*ln(a^2*x/(a^2)^(1/2)+(a^2*x^2-1)^(1/2))/(a ^2)^(1/2))*c^4/a^8/(a*x+1)/((a*x-1)/(a*x+1))^(1/2)*((a*x-1)*(a*x+1))^(1/2)
Time = 0.12 (sec) , antiderivative size = 201, normalized size of antiderivative = 1.12 \[ \int e^{3 \coth ^{-1}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^4 \, dx=-\frac {1050 \, a^{7} c^{4} x^{7} \arctan \left (\sqrt {\frac {a x - 1}{a x + 1}}\right ) - 1680 \, a^{7} c^{4} x^{7} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right ) + 1680 \, a^{7} c^{4} x^{7} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right ) - {\left (560 \, a^{8} c^{4} x^{8} - 1936 \, a^{7} c^{4} x^{7} - 3021 \, a^{6} c^{4} x^{6} + 467 \, a^{5} c^{4} x^{5} + 1762 \, a^{4} c^{4} x^{4} + 674 \, a^{3} c^{4} x^{3} - 376 \, a^{2} c^{4} x^{2} - 360 \, a c^{4} x - 80 \, c^{4}\right )} \sqrt {\frac {a x - 1}{a x + 1}}}{560 \, a^{8} x^{7}} \] Input:
integrate(1/((a*x-1)/(a*x+1))^(3/2)*(c-c/a^2/x^2)^4,x, algorithm="fricas")
Output:
-1/560*(1050*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) - (560*a^8*c^4*x^8 - 1936*a^7*c^4*x^7 - 3021*a^6*c^4*x^ 6 + 467*a^5*c^4*x^5 + 1762*a^4*c^4*x^4 + 674*a^3*c^4*x^3 - 376*a^2*c^4*x^2 - 360*a*c^4*x - 80*c^4)*sqrt((a*x - 1)/(a*x + 1)))/(a^8*x^7)
\[ \int e^{3 \coth ^{-1}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^4 \, dx=\frac {c^{4} \left (\int \left (- \frac {4 a^{2}}{\frac {a x^{7} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x + 1} - \frac {x^{6} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x + 1}}\right )\, dx + \int \frac {6 a^{4}}{\frac {a x^{5} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x + 1} - \frac {x^{4} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x + 1}}\, dx + \int \left (- \frac {4 a^{6}}{\frac {a x^{3} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x + 1} - \frac {x^{2} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x + 1}}\right )\, dx + \int \frac {a^{8}}{\frac {a x \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x + 1} - \frac {\sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x + 1}}\, dx + \int \frac {1}{\frac {a x^{9} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x + 1} - \frac {x^{8} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x + 1}}\, dx\right )}{a^{8}} \] Input:
integrate(1/((a*x-1)/(a*x+1))**(3/2)*(c-c/a**2/x**2)**4,x)
Output:
c**4*(Integral(-4*a**2/(a*x**7*sqrt(a*x/(a*x + 1) - 1/(a*x + 1))/(a*x + 1) - x**6*sqrt(a*x/(a*x + 1) - 1/(a*x + 1))/(a*x + 1)), x) + Integral(6*a**4 /(a*x**5*sqrt(a*x/(a*x + 1) - 1/(a*x + 1))/(a*x + 1) - x**4*sqrt(a*x/(a*x + 1) - 1/(a*x + 1))/(a*x + 1)), x) + Integral(-4*a**6/(a*x**3*sqrt(a*x/(a* x + 1) - 1/(a*x + 1))/(a*x + 1) - x**2*sqrt(a*x/(a*x + 1) - 1/(a*x + 1))/( a*x + 1)), x) + Integral(a**8/(a*x*sqrt(a*x/(a*x + 1) - 1/(a*x + 1))/(a*x + 1) - sqrt(a*x/(a*x + 1) - 1/(a*x + 1))/(a*x + 1)), x) + Integral(1/(a*x* *9*sqrt(a*x/(a*x + 1) - 1/(a*x + 1))/(a*x + 1) - x**8*sqrt(a*x/(a*x + 1) - 1/(a*x + 1))/(a*x + 1)), x))/a**8
Leaf count of result is larger than twice the leaf count of optimal. 380 vs. \(2 (158) = 316\).
Time = 0.12 (sec) , antiderivative size = 380, normalized size of antiderivative = 2.11 \[ \int e^{3 \coth ^{-1}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^4 \, dx=-\frac {1}{280} \, {\left (\frac {525 \, c^{4} \arctan \left (\sqrt {\frac {a x - 1}{a x + 1}}\right )}{a^{2}} - \frac {840 \, c^{4} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right )}{a^{2}} + \frac {840 \, c^{4} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right )}{a^{2}} - \frac {2205 \, c^{4} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {15}{2}} + 13615 \, c^{4} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {13}{2}} + 33621 \, c^{4} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {11}{2}} + 39071 \, c^{4} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {9}{2}} + 12799 \, c^{4} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {7}{2}} - 20811 \, c^{4} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {5}{2}} - 7665 \, c^{4} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}} - 1155 \, c^{4} \sqrt {\frac {a x - 1}{a x + 1}}}{\frac {6 \, {\left (a x - 1\right )} a^{2}}{a x + 1} + \frac {14 \, {\left (a x - 1\right )}^{2} a^{2}}{{\left (a x + 1\right )}^{2}} + \frac {14 \, {\left (a x - 1\right )}^{3} a^{2}}{{\left (a x + 1\right )}^{3}} - \frac {14 \, {\left (a x - 1\right )}^{5} a^{2}}{{\left (a x + 1\right )}^{5}} - \frac {14 \, {\left (a x - 1\right )}^{6} a^{2}}{{\left (a x + 1\right )}^{6}} - \frac {6 \, {\left (a x - 1\right )}^{7} a^{2}}{{\left (a x + 1\right )}^{7}} - \frac {{\left (a x - 1\right )}^{8} a^{2}}{{\left (a x + 1\right )}^{8}} + a^{2}}\right )} a \] Input:
integrate(1/((a*x-1)/(a*x+1))^(3/2)*(c-c/a^2/x^2)^4,x, algorithm="maxima")
Output:
-1/280*(525*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 - (2205*c^4*((a*x - 1)/(a*x + 1))^(15/2) + 13615*c^4*((a*x - 1)/(a*x + 1))^(13/2) + 33621*c^4*((a*x - 1)/(a*x + 1))^(11/2) + 39071*c^4*((a*x - 1 )/(a*x + 1))^(9/2) + 12799*c^4*((a*x - 1)/(a*x + 1))^(7/2) - 20811*c^4*((a *x - 1)/(a*x + 1))^(5/2) - 7665*c^4*((a*x - 1)/(a*x + 1))^(3/2) - 1155*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. 461 vs. \(2 (158) = 316\).
Time = 0.16 (sec) , antiderivative size = 461, normalized size of antiderivative = 2.56 \[ \int e^{3 \coth ^{-1}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^4 \, dx=-\frac {15 \, c^{4} \arctan \left (-x {\left | a \right |} + \sqrt {a^{2} x^{2} - 1}\right )}{8 \, a \mathrm {sgn}\left (a x + 1\right )} - \frac {3 \, c^{4} \log \left ({\left | -x {\left | a \right |} + \sqrt {a^{2} x^{2} - 1} \right |}\right )}{{\left | a \right |} \mathrm {sgn}\left (a x + 1\right )} + \frac {\sqrt {a^{2} x^{2} - 1} c^{4}}{a \mathrm {sgn}\left (a x + 1\right )} + \frac {525 \, {\left (x {\left | a \right |} - \sqrt {a^{2} x^{2} - 1}\right )}^{13} c^{4} {\left | a \right |} - 4480 \, {\left (x {\left | a \right |} - \sqrt {a^{2} x^{2} - 1}\right )}^{12} a c^{4} - 980 \, {\left (x {\left | a \right |} - \sqrt {a^{2} x^{2} - 1}\right )}^{11} c^{4} {\left | a \right |} - 20160 \, {\left (x {\left | a \right |} - \sqrt {a^{2} x^{2} - 1}\right )}^{10} a c^{4} + 945 \, {\left (x {\left | a \right |} - \sqrt {a^{2} x^{2} - 1}\right )}^{9} c^{4} {\left | a \right |} - 38080 \, {\left (x {\left | a \right |} - \sqrt {a^{2} x^{2} - 1}\right )}^{8} a c^{4} - 49280 \, {\left (x {\left | a \right |} - \sqrt {a^{2} x^{2} - 1}\right )}^{6} a c^{4} - 945 \, {\left (x {\left | a \right |} - \sqrt {a^{2} x^{2} - 1}\right )}^{5} c^{4} {\left | a \right |} - 32256 \, {\left (x {\left | a \right |} - \sqrt {a^{2} x^{2} - 1}\right )}^{4} a c^{4} + 980 \, {\left (x {\left | a \right |} - \sqrt {a^{2} x^{2} - 1}\right )}^{3} c^{4} {\left | a \right |} - 12992 \, {\left (x {\left | a \right |} - \sqrt {a^{2} x^{2} - 1}\right )}^{2} a c^{4} - 525 \, {\left (x {\left | a \right |} - \sqrt {a^{2} x^{2} - 1}\right )} c^{4} {\left | a \right |} - 2496 \, a c^{4}}{280 \, {\left ({\left (x {\left | a \right |} - \sqrt {a^{2} x^{2} - 1}\right )}^{2} + 1\right )}^{7} a {\left | a \right |} \mathrm {sgn}\left (a x + 1\right )} \] Input:
integrate(1/((a*x-1)/(a*x+1))^(3/2)*(c-c/a^2/x^2)^4,x, algorithm="giac")
Output:
-15/8*c^4*arctan(-x*abs(a) + sqrt(a^2*x^2 - 1))/(a*sgn(a*x + 1)) - 3*c^4*l og(abs(-x*abs(a) + sqrt(a^2*x^2 - 1)))/(abs(a)*sgn(a*x + 1)) + sqrt(a^2*x^ 2 - 1)*c^4/(a*sgn(a*x + 1)) + 1/280*(525*(x*abs(a) - sqrt(a^2*x^2 - 1))^13 *c^4*abs(a) - 4480*(x*abs(a) - sqrt(a^2*x^2 - 1))^12*a*c^4 - 980*(x*abs(a) - sqrt(a^2*x^2 - 1))^11*c^4*abs(a) - 20160*(x*abs(a) - sqrt(a^2*x^2 - 1)) ^10*a*c^4 + 945*(x*abs(a) - sqrt(a^2*x^2 - 1))^9*c^4*abs(a) - 38080*(x*abs (a) - sqrt(a^2*x^2 - 1))^8*a*c^4 - 49280*(x*abs(a) - sqrt(a^2*x^2 - 1))^6* a*c^4 - 945*(x*abs(a) - sqrt(a^2*x^2 - 1))^5*c^4*abs(a) - 32256*(x*abs(a) - sqrt(a^2*x^2 - 1))^4*a*c^4 + 980*(x*abs(a) - sqrt(a^2*x^2 - 1))^3*c^4*ab s(a) - 12992*(x*abs(a) - sqrt(a^2*x^2 - 1))^2*a*c^4 - 525*(x*abs(a) - sqrt (a^2*x^2 - 1))*c^4*abs(a) - 2496*a*c^4)/(((x*abs(a) - sqrt(a^2*x^2 - 1))^2 + 1)^7*a*abs(a)*sgn(a*x + 1))
Time = 0.14 (sec) , antiderivative size = 332, normalized size of antiderivative = 1.84 \[ \int e^{3 \coth ^{-1}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^4 \, dx=\frac {\frac {12799\,c^4\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{7/2}}{280}-\frac {219\,c^4\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{3/2}}{8}-\frac {2973\,c^4\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{5/2}}{40}-\frac {33\,c^4\,\sqrt {\frac {a\,x-1}{a\,x+1}}}{8}+\frac {39071\,c^4\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{9/2}}{280}+\frac {4803\,c^4\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{11/2}}{40}+\frac {389\,c^4\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{13/2}}{8}+\frac {63\,c^4\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{15/2}}{8}}{a+\frac {6\,a\,\left (a\,x-1\right )}{a\,x+1}+\frac {14\,a\,{\left (a\,x-1\right )}^2}{{\left (a\,x+1\right )}^2}+\frac {14\,a\,{\left (a\,x-1\right )}^3}{{\left (a\,x+1\right )}^3}-\frac {14\,a\,{\left (a\,x-1\right )}^5}{{\left (a\,x+1\right )}^5}-\frac {14\,a\,{\left (a\,x-1\right )}^6}{{\left (a\,x+1\right )}^6}-\frac {6\,a\,{\left (a\,x-1\right )}^7}{{\left (a\,x+1\right )}^7}-\frac {a\,{\left (a\,x-1\right )}^8}{{\left (a\,x+1\right )}^8}}-\frac {15\,c^4\,\mathrm {atan}\left (\sqrt {\frac {a\,x-1}{a\,x+1}}\right )}{8\,a}+\frac {6\,c^4\,\mathrm {atanh}\left (\sqrt {\frac {a\,x-1}{a\,x+1}}\right )}{a} \] Input:
int((c - c/(a^2*x^2))^4/((a*x - 1)/(a*x + 1))^(3/2),x)
Output:
((12799*c^4*((a*x - 1)/(a*x + 1))^(7/2))/280 - (219*c^4*((a*x - 1)/(a*x + 1))^(3/2))/8 - (2973*c^4*((a*x - 1)/(a*x + 1))^(5/2))/40 - (33*c^4*((a*x - 1)/(a*x + 1))^(1/2))/8 + (39071*c^4*((a*x - 1)/(a*x + 1))^(9/2))/280 + (4 803*c^4*((a*x - 1)/(a*x + 1))^(11/2))/40 + (389*c^4*((a*x - 1)/(a*x + 1))^ (13/2))/8 + (63*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) - (15*c^4*atan((( a*x - 1)/(a*x + 1))^(1/2)))/(8*a) + (6*c^4*atanh(((a*x - 1)/(a*x + 1))^(1/ 2)))/a
Time = 0.21 (sec) , antiderivative size = 243, normalized size of antiderivative = 1.35 \[ \int e^{3 \coth ^{-1}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^4 \, dx=\frac {c^{4} \left (-1050 \mathit {atan} \left (\sqrt {a x -1}+\sqrt {a x +1}-1\right ) a^{7} x^{7}+1050 \mathit {atan} \left (\sqrt {a x -1}+\sqrt {a x +1}+1\right ) a^{7} x^{7}+560 \sqrt {a x +1}\, \sqrt {a x -1}\, a^{7} x^{7}-2496 \sqrt {a x +1}\, \sqrt {a x -1}\, a^{6} x^{6}-525 \sqrt {a x +1}\, \sqrt {a x -1}\, a^{5} x^{5}+992 \sqrt {a x +1}\, \sqrt {a x -1}\, a^{4} x^{4}+770 \sqrt {a x +1}\, \sqrt {a x -1}\, a^{3} x^{3}-96 \sqrt {a x +1}\, \sqrt {a x -1}\, a^{2} x^{2}-280 \sqrt {a x +1}\, \sqrt {a x -1}\, a x -80 \sqrt {a x +1}\, \sqrt {a x -1}+3360 \,\mathrm {log}\left (\frac {\sqrt {a x -1}+\sqrt {a x +1}}{\sqrt {2}}\right ) a^{7} x^{7}+1216 a^{7} x^{7}\right )}{560 a^{8} x^{7}} \] Input:
int(1/((a*x-1)/(a*x+1))^(3/2)*(c-c/a^2/x^2)^4,x)
Output:
(c**4*( - 1050*atan(sqrt(a*x - 1) + sqrt(a*x + 1) - 1)*a**7*x**7 + 1050*at an(sqrt(a*x - 1) + sqrt(a*x + 1) + 1)*a**7*x**7 + 560*sqrt(a*x + 1)*sqrt(a *x - 1)*a**7*x**7 - 2496*sqrt(a*x + 1)*sqrt(a*x - 1)*a**6*x**6 - 525*sqrt( a*x + 1)*sqrt(a*x - 1)*a**5*x**5 + 992*sqrt(a*x + 1)*sqrt(a*x - 1)*a**4*x* *4 + 770*sqrt(a*x + 1)*sqrt(a*x - 1)*a**3*x**3 - 96*sqrt(a*x + 1)*sqrt(a*x - 1)*a**2*x**2 - 280*sqrt(a*x + 1)*sqrt(a*x - 1)*a*x - 80*sqrt(a*x + 1)*s qrt(a*x - 1) + 3360*log((sqrt(a*x - 1) + sqrt(a*x + 1))/sqrt(2))*a**7*x**7 + 1216*a**7*x**7))/(560*a**8*x**7)