Integrand size = 22, antiderivative size = 269 \[ \int e^{-3 \coth ^{-1}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^3 \, dx=\frac {21 c^3 \sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}}{8 a}+\frac {3 c^3 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{3/2}}{8 a}+\frac {5 c^3 \left (1-\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{3/2}}{4 a}+\frac {27 c^3 \left (1-\frac {1}{a x}\right )^{5/2} \left (1+\frac {1}{a x}\right )^{3/2}}{20 a}+\frac {6 c^3 \left (1-\frac {1}{a x}\right )^{7/2} \left (1+\frac {1}{a x}\right )^{3/2}}{5 a}+c^3 \left (1-\frac {1}{a x}\right )^{9/2} \left (1+\frac {1}{a x}\right )^{3/2} x+\frac {3 c^3 \csc ^{-1}(a x)}{8 a}-\frac {3 c^3 \text {arctanh}\left (\sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}\right )}{a} \]
5/4*c^3*(1-1/a/x)^(3/2)*(1+1/a/x)^(3/2)/a+27/20*c^3*(1-1/a/x)^(5/2)*(1+1/a /x)^(3/2)/a+6/5*c^3*(1-1/a/x)^(7/2)*(1+1/a/x)^(3/2)/a+c^3*(1-1/a/x)^(9/2)* (1+1/a/x)^(3/2)*x+3/8*c^3*arccsc(a*x)/a-3*c^3*arctanh((1-1/a/x)^(1/2)*(1+1 /a/x)^(1/2))/a+3/8*c^3*(1+1/a/x)^(3/2)*(1-1/a/x)^(1/2)/a+21/8*c^3*(1-1/a/x )^(1/2)*(1+1/a/x)^(1/2)/a
Time = 0.25 (sec) , antiderivative size = 110, normalized size of antiderivative = 0.41 \[ \int e^{-3 \coth ^{-1}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^3 \, dx=\frac {c^3 \left (\sqrt {1-\frac {1}{a^2 x^2}} \left (-8+30 a x-24 a^2 x^2-55 a^3 x^3+152 a^4 x^4+40 a^5 x^5\right )+15 a^4 x^4 \arcsin \left (\frac {1}{a x}\right )-120 a^4 x^4 \log \left (\left (1+\sqrt {1-\frac {1}{a^2 x^2}}\right ) x\right )\right )}{40 a^5 x^4} \]
(c^3*(Sqrt[1 - 1/(a^2*x^2)]*(-8 + 30*a*x - 24*a^2*x^2 - 55*a^3*x^3 + 152*a ^4*x^4 + 40*a^5*x^5) + 15*a^4*x^4*ArcSin[1/(a*x)] - 120*a^4*x^4*Log[(1 + S qrt[1 - 1/(a^2*x^2)])*x]))/(40*a^5*x^4)
Time = 0.43 (sec) , antiderivative size = 257, normalized size of antiderivative = 0.96, number of steps used = 20, number of rules used = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.864, Rules used = {6748, 108, 27, 171, 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.
| \(\displaystyle \int \left (c-\frac {c}{a^2 x^2}\right )^3 e^{-3 \coth ^{-1}(a x)} \, dx\) |
| \(\Big \downarrow \) 6748 |
| \(\displaystyle -c^3 \int \left (1-\frac {1}{a x}\right )^{9/2} \left (1+\frac {1}{a x}\right )^{3/2} x^2d\frac {1}{x}\) |
| \(\Big \downarrow \) 108 |
| \(\displaystyle -c^3 \left (\int -\frac {3 \left (a+\frac {2}{x}\right ) \left (1-\frac {1}{a x}\right )^{7/2} \sqrt {1+\frac {1}{a x}} x}{a^2}d\frac {1}{x}-x \left (1-\frac {1}{a x}\right )^{9/2} \left (\frac {1}{a x}+1\right )^{3/2}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -c^3 \left (x \left (-\left (\frac {1}{a x}+1\right )^{3/2}\right ) \left (1-\frac {1}{a x}\right )^{9/2}-\frac {3 \int \left (a+\frac {2}{x}\right ) \left (1-\frac {1}{a x}\right )^{7/2} \sqrt {1+\frac {1}{a x}} xd\frac {1}{x}}{a^2}\right )\) |
| \(\Big \downarrow \) 171 |
| \(\displaystyle -c^3 \left (x \left (-\left (\frac {1}{a x}+1\right )^{3/2}\right ) \left (1-\frac {1}{a x}\right )^{9/2}-\frac {3 \left (\frac {1}{5} a \int \frac {\left (5 a+\frac {9}{x}\right ) \left (1-\frac {1}{a x}\right )^{5/2} \sqrt {1+\frac {1}{a x}} x}{a}d\frac {1}{x}+\frac {2}{5} a \left (\frac {1}{a x}+1\right )^{3/2} \left (1-\frac {1}{a x}\right )^{7/2}\right )}{a^2}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -c^3 \left (x \left (-\left (\frac {1}{a x}+1\right )^{3/2}\right ) \left (1-\frac {1}{a x}\right )^{9/2}-\frac {3 \left (\frac {1}{5} \int \left (5 a+\frac {9}{x}\right ) \left (1-\frac {1}{a x}\right )^{5/2} \sqrt {1+\frac {1}{a x}} xd\frac {1}{x}+\frac {2}{5} a \left (\frac {1}{a x}+1\right )^{3/2} \left (1-\frac {1}{a x}\right )^{7/2}\right )}{a^2}\right )\) |
| \(\Big \downarrow \) 171 |
| \(\displaystyle -c^3 \left (x \left (-\left (\frac {1}{a x}+1\right )^{3/2}\right ) \left (1-\frac {1}{a x}\right )^{9/2}-\frac {3 \left (\frac {1}{5} \left (\frac {1}{4} a \int \frac {5 \left (4 a+\frac {5}{x}\right ) \left (1-\frac {1}{a x}\right )^{3/2} \sqrt {1+\frac {1}{a x}} x}{a}d\frac {1}{x}+\frac {9}{4} a \left (\frac {1}{a x}+1\right )^{3/2} \left (1-\frac {1}{a x}\right )^{5/2}\right )+\frac {2}{5} a \left (\frac {1}{a x}+1\right )^{3/2} \left (1-\frac {1}{a x}\right )^{7/2}\right )}{a^2}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -c^3 \left (x \left (-\left (\frac {1}{a x}+1\right )^{3/2}\right ) \left (1-\frac {1}{a x}\right )^{9/2}-\frac {3 \left (\frac {1}{5} \left (\frac {5}{4} \int \left (4 a+\frac {5}{x}\right ) \left (1-\frac {1}{a x}\right )^{3/2} \sqrt {1+\frac {1}{a x}} xd\frac {1}{x}+\frac {9}{4} a \left (\frac {1}{a x}+1\right )^{3/2} \left (1-\frac {1}{a x}\right )^{5/2}\right )+\frac {2}{5} a \left (\frac {1}{a x}+1\right )^{3/2} \left (1-\frac {1}{a x}\right )^{7/2}\right )}{a^2}\right )\) |
| \(\Big \downarrow \) 171 |
| \(\displaystyle -c^3 \left (x \left (-\left (\frac {1}{a x}+1\right )^{3/2}\right ) \left (1-\frac {1}{a x}\right )^{9/2}-\frac {3 \left (\frac {1}{5} \left (\frac {5}{4} \left (\frac {1}{3} a \int \frac {3 \left (4 a+\frac {1}{x}\right ) \sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}} x}{a}d\frac {1}{x}+\frac {5}{3} a \left (1-\frac {1}{a x}\right )^{3/2} \left (\frac {1}{a x}+1\right )^{3/2}\right )+\frac {9}{4} a \left (\frac {1}{a x}+1\right )^{3/2} \left (1-\frac {1}{a x}\right )^{5/2}\right )+\frac {2}{5} a \left (\frac {1}{a x}+1\right )^{3/2} \left (1-\frac {1}{a x}\right )^{7/2}\right )}{a^2}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -c^3 \left (x \left (-\left (\frac {1}{a x}+1\right )^{3/2}\right ) \left (1-\frac {1}{a x}\right )^{9/2}-\frac {3 \left (\frac {1}{5} \left (\frac {5}{4} \left (\int \left (4 a+\frac {1}{x}\right ) \sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}} xd\frac {1}{x}+\frac {5}{3} a \left (1-\frac {1}{a x}\right )^{3/2} \left (\frac {1}{a x}+1\right )^{3/2}\right )+\frac {9}{4} a \left (\frac {1}{a x}+1\right )^{3/2} \left (1-\frac {1}{a x}\right )^{5/2}\right )+\frac {2}{5} a \left (\frac {1}{a x}+1\right )^{3/2} \left (1-\frac {1}{a x}\right )^{7/2}\right )}{a^2}\right )\) |
| \(\Big \downarrow \) 171 |
| \(\displaystyle -c^3 \left (x \left (-\left (\frac {1}{a x}+1\right )^{3/2}\right ) \left (1-\frac {1}{a x}\right )^{9/2}-\frac {3 \left (\frac {1}{5} \left (\frac {5}{4} \left (\frac {1}{2} a \int \frac {\left (8 a-\frac {7}{x}\right ) \sqrt {1+\frac {1}{a x}} x}{a \sqrt {1-\frac {1}{a x}}}d\frac {1}{x}+\frac {5}{3} a \left (1-\frac {1}{a x}\right )^{3/2} \left (\frac {1}{a x}+1\right )^{3/2}+\frac {1}{2} a \sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{3/2}\right )+\frac {9}{4} a \left (\frac {1}{a x}+1\right )^{3/2} \left (1-\frac {1}{a x}\right )^{5/2}\right )+\frac {2}{5} a \left (\frac {1}{a x}+1\right )^{3/2} \left (1-\frac {1}{a x}\right )^{7/2}\right )}{a^2}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -c^3 \left (x \left (-\left (\frac {1}{a x}+1\right )^{3/2}\right ) \left (1-\frac {1}{a x}\right )^{9/2}-\frac {3 \left (\frac {1}{5} \left (\frac {5}{4} \left (\frac {1}{2} \int \frac {\left (8 a-\frac {7}{x}\right ) \sqrt {1+\frac {1}{a x}} x}{\sqrt {1-\frac {1}{a x}}}d\frac {1}{x}+\frac {5}{3} a \left (1-\frac {1}{a x}\right )^{3/2} \left (\frac {1}{a x}+1\right )^{3/2}+\frac {1}{2} a \sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{3/2}\right )+\frac {9}{4} a \left (\frac {1}{a x}+1\right )^{3/2} \left (1-\frac {1}{a x}\right )^{5/2}\right )+\frac {2}{5} a \left (\frac {1}{a x}+1\right )^{3/2} \left (1-\frac {1}{a x}\right )^{7/2}\right )}{a^2}\right )\) |
| \(\Big \downarrow \) 171 |
| \(\displaystyle -c^3 \left (x \left (-\left (\frac {1}{a x}+1\right )^{3/2}\right ) \left (1-\frac {1}{a x}\right )^{9/2}-\frac {3 \left (\frac {1}{5} \left (\frac {5}{4} \left (\frac {1}{2} \left (7 a \sqrt {1-\frac {1}{a x}} \sqrt {\frac {1}{a x}+1}-a \int -\frac {\left (8 a+\frac {1}{x}\right ) x}{a \sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}}d\frac {1}{x}\right )+\frac {5}{3} a \left (1-\frac {1}{a x}\right )^{3/2} \left (\frac {1}{a x}+1\right )^{3/2}+\frac {1}{2} a \sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{3/2}\right )+\frac {9}{4} a \left (\frac {1}{a x}+1\right )^{3/2} \left (1-\frac {1}{a x}\right )^{5/2}\right )+\frac {2}{5} a \left (\frac {1}{a x}+1\right )^{3/2} \left (1-\frac {1}{a x}\right )^{7/2}\right )}{a^2}\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -c^3 \left (x \left (-\left (\frac {1}{a x}+1\right )^{3/2}\right ) \left (1-\frac {1}{a x}\right )^{9/2}-\frac {3 \left (\frac {1}{5} \left (\frac {5}{4} \left (\frac {1}{2} \left (a \int \frac {\left (8 a+\frac {1}{x}\right ) x}{a \sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}}d\frac {1}{x}+7 a \sqrt {1-\frac {1}{a x}} \sqrt {\frac {1}{a x}+1}\right )+\frac {5}{3} a \left (1-\frac {1}{a x}\right )^{3/2} \left (\frac {1}{a x}+1\right )^{3/2}+\frac {1}{2} a \sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{3/2}\right )+\frac {9}{4} a \left (\frac {1}{a x}+1\right )^{3/2} \left (1-\frac {1}{a x}\right )^{5/2}\right )+\frac {2}{5} a \left (\frac {1}{a x}+1\right )^{3/2} \left (1-\frac {1}{a x}\right )^{7/2}\right )}{a^2}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -c^3 \left (x \left (-\left (\frac {1}{a x}+1\right )^{3/2}\right ) \left (1-\frac {1}{a x}\right )^{9/2}-\frac {3 \left (\frac {1}{5} \left (\frac {5}{4} \left (\frac {1}{2} \left (\int \frac {\left (8 a+\frac {1}{x}\right ) x}{\sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}}d\frac {1}{x}+7 a \sqrt {1-\frac {1}{a x}} \sqrt {\frac {1}{a x}+1}\right )+\frac {5}{3} a \left (1-\frac {1}{a x}\right )^{3/2} \left (\frac {1}{a x}+1\right )^{3/2}+\frac {1}{2} a \sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{3/2}\right )+\frac {9}{4} a \left (\frac {1}{a x}+1\right )^{3/2} \left (1-\frac {1}{a x}\right )^{5/2}\right )+\frac {2}{5} a \left (\frac {1}{a x}+1\right )^{3/2} \left (1-\frac {1}{a x}\right )^{7/2}\right )}{a^2}\right )\) |
| \(\Big \downarrow \) 175 |
| \(\displaystyle -c^3 \left (x \left (-\left (\frac {1}{a x}+1\right )^{3/2}\right ) \left (1-\frac {1}{a x}\right )^{9/2}-\frac {3 \left (\frac {1}{5} \left (\frac {5}{4} \left (\frac {1}{2} \left (8 a \int \frac {x}{\sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}}d\frac {1}{x}+\int \frac {1}{\sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}}d\frac {1}{x}+7 a \sqrt {1-\frac {1}{a x}} \sqrt {\frac {1}{a x}+1}\right )+\frac {5}{3} a \left (1-\frac {1}{a x}\right )^{3/2} \left (\frac {1}{a x}+1\right )^{3/2}+\frac {1}{2} a \sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{3/2}\right )+\frac {9}{4} a \left (\frac {1}{a x}+1\right )^{3/2} \left (1-\frac {1}{a x}\right )^{5/2}\right )+\frac {2}{5} a \left (\frac {1}{a x}+1\right )^{3/2} \left (1-\frac {1}{a x}\right )^{7/2}\right )}{a^2}\right )\) |
| \(\Big \downarrow \) 39 |
| \(\displaystyle -c^3 \left (x \left (-\left (\frac {1}{a x}+1\right )^{3/2}\right ) \left (1-\frac {1}{a x}\right )^{9/2}-\frac {3 \left (\frac {1}{5} \left (\frac {5}{4} \left (\frac {1}{2} \left (\int \frac {1}{\sqrt {1-\frac {1}{a^2 x^2}}}d\frac {1}{x}+8 a \int \frac {x}{\sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}}d\frac {1}{x}+7 a \sqrt {1-\frac {1}{a x}} \sqrt {\frac {1}{a x}+1}\right )+\frac {5}{3} a \left (1-\frac {1}{a x}\right )^{3/2} \left (\frac {1}{a x}+1\right )^{3/2}+\frac {1}{2} a \sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{3/2}\right )+\frac {9}{4} a \left (\frac {1}{a x}+1\right )^{3/2} \left (1-\frac {1}{a x}\right )^{5/2}\right )+\frac {2}{5} a \left (\frac {1}{a x}+1\right )^{3/2} \left (1-\frac {1}{a x}\right )^{7/2}\right )}{a^2}\right )\) |
| \(\Big \downarrow \) 103 |
| \(\displaystyle -c^3 \left (x \left (-\left (\frac {1}{a x}+1\right )^{3/2}\right ) \left (1-\frac {1}{a x}\right )^{9/2}-\frac {3 \left (\frac {1}{5} \left (\frac {5}{4} \left (\frac {1}{2} \left (\int \frac {1}{\sqrt {1-\frac {1}{a^2 x^2}}}d\frac {1}{x}-8 \int \frac {1}{\frac {1}{a}-\frac {1}{a x^2}}d\left (\sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}\right )+7 a \sqrt {1-\frac {1}{a x}} \sqrt {\frac {1}{a x}+1}\right )+\frac {5}{3} a \left (1-\frac {1}{a x}\right )^{3/2} \left (\frac {1}{a x}+1\right )^{3/2}+\frac {1}{2} a \sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{3/2}\right )+\frac {9}{4} a \left (\frac {1}{a x}+1\right )^{3/2} \left (1-\frac {1}{a x}\right )^{5/2}\right )+\frac {2}{5} a \left (\frac {1}{a x}+1\right )^{3/2} \left (1-\frac {1}{a x}\right )^{7/2}\right )}{a^2}\right )\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle -c^3 \left (x \left (-\left (\frac {1}{a x}+1\right )^{3/2}\right ) \left (1-\frac {1}{a x}\right )^{9/2}-\frac {3 \left (\frac {1}{5} \left (\frac {5}{4} \left (\frac {1}{2} \left (\int \frac {1}{\sqrt {1-\frac {1}{a^2 x^2}}}d\frac {1}{x}-8 a \text {arctanh}\left (\sqrt {1-\frac {1}{a x}} \sqrt {\frac {1}{a x}+1}\right )+7 a \sqrt {1-\frac {1}{a x}} \sqrt {\frac {1}{a x}+1}\right )+\frac {5}{3} a \left (1-\frac {1}{a x}\right )^{3/2} \left (\frac {1}{a x}+1\right )^{3/2}+\frac {1}{2} a \sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{3/2}\right )+\frac {9}{4} a \left (\frac {1}{a x}+1\right )^{3/2} \left (1-\frac {1}{a x}\right )^{5/2}\right )+\frac {2}{5} a \left (\frac {1}{a x}+1\right )^{3/2} \left (1-\frac {1}{a x}\right )^{7/2}\right )}{a^2}\right )\) |
| \(\Big \downarrow \) 223 |
| \(\displaystyle -c^3 \left (x \left (-\left (\frac {1}{a x}+1\right )^{3/2}\right ) \left (1-\frac {1}{a x}\right )^{9/2}-\frac {3 \left (\frac {1}{5} \left (\frac {5}{4} \left (\frac {1}{2} \left (a \arcsin \left (\frac {1}{a x}\right )-8 a \text {arctanh}\left (\sqrt {1-\frac {1}{a x}} \sqrt {\frac {1}{a x}+1}\right )+7 a \sqrt {1-\frac {1}{a x}} \sqrt {\frac {1}{a x}+1}\right )+\frac {5}{3} a \left (1-\frac {1}{a x}\right )^{3/2} \left (\frac {1}{a x}+1\right )^{3/2}+\frac {1}{2} a \sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{3/2}\right )+\frac {9}{4} a \left (\frac {1}{a x}+1\right )^{3/2} \left (1-\frac {1}{a x}\right )^{5/2}\right )+\frac {2}{5} a \left (\frac {1}{a x}+1\right )^{3/2} \left (1-\frac {1}{a x}\right )^{7/2}\right )}{a^2}\right )\) |
-(c^3*(-((1 - 1/(a*x))^(9/2)*(1 + 1/(a*x))^(3/2)*x) - (3*((2*a*(1 - 1/(a*x ))^(7/2)*(1 + 1/(a*x))^(3/2))/5 + ((9*a*(1 - 1/(a*x))^(5/2)*(1 + 1/(a*x))^ (3/2))/4 + (5*((a*Sqrt[1 - 1/(a*x)]*(1 + 1/(a*x))^(3/2))/2 + (5*a*(1 - 1/( a*x))^(3/2)*(1 + 1/(a*x))^(3/2))/3 + (7*a*Sqrt[1 - 1/(a*x)]*Sqrt[1 + 1/(a* x)] + a*ArcSin[1/(a*x)] - 8*a*ArcTanh[Sqrt[1 - 1/(a*x)]*Sqrt[1 + 1/(a*x)]] )/2))/4)/5))/a^2))
3.9.23.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((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 = 0.10 (sec) , antiderivative size = 173, normalized size of antiderivative = 0.64
| method | result | size |
| risch | \(\frac {\left (a x +1\right ) \left (152 a^{4} x^{4}-55 a^{3} x^{3}-24 a^{2} x^{2}+30 a x -8\right ) c^{3} \sqrt {\frac {a x -1}{a x +1}}}{40 x^{5} a^{6}}+\frac {\left (-\frac {3 a^{6} \ln \left (\frac {a^{2} x}{\sqrt {a^{2}}}+\sqrt {a^{2} x^{2}-1}\right )}{\sqrt {a^{2}}}+\frac {3 a^{5} \arctan \left (\frac {1}{\sqrt {a^{2} x^{2}-1}}\right )}{8}+a^{5} \sqrt {\left (a x -1\right ) \left (a x +1\right )}\right ) c^{3} \sqrt {\frac {a x -1}{a x +1}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}}{a^{6} \left (a x -1\right )}\) | \(173\) |
| default | \(-\frac {\left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}} \left (a x +1\right )^{2} c^{3} \left (-120 \sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}\, a^{6} x^{6}+120 \left (a^{2} x^{2}-1\right )^{\frac {3}{2}} \sqrt {a^{2}}\, a^{4} x^{4}-15 a^{5} x^{5} \sqrt {a^{2}}\, \sqrt {a^{2} x^{2}-1}-15 a^{5} x^{5} \sqrt {a^{2}}\, \arctan \left (\frac {1}{\sqrt {a^{2} x^{2}-1}}\right )+120 \ln \left (\frac {a^{2} x +\sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) a^{6} x^{5}-25 \left (a^{2} x^{2}-1\right )^{\frac {3}{2}} \sqrt {a^{2}}\, a^{3} x^{3}-32 \left (a^{2} x^{2}-1\right )^{\frac {3}{2}} \sqrt {a^{2}}\, a^{2} x^{2}+30 \left (a^{2} x^{2}-1\right )^{\frac {3}{2}} \sqrt {a^{2}}\, a x -8 \left (a^{2} x^{2}-1\right )^{\frac {3}{2}} \sqrt {a^{2}}\right )}{40 \left (a x -1\right ) \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, a^{6} x^{5} \sqrt {a^{2}}}\) | \(281\) |
1/40*(a*x+1)*(152*a^4*x^4-55*a^3*x^3-24*a^2*x^2+30*a*x-8)/x^5*c^3/a^6*((a* x-1)/(a*x+1))^(1/2)+(-3*a^6*ln(a^2*x/(a^2)^(1/2)+(a^2*x^2-1)^(1/2))/(a^2)^ (1/2)+3/8*a^5*arctan(1/(a^2*x^2-1)^(1/2))+a^5*((a*x-1)*(a*x+1))^(1/2))*c^3 /a^6*((a*x-1)/(a*x+1))^(1/2)/(a*x-1)*((a*x-1)*(a*x+1))^(1/2)
Time = 0.26 (sec) , antiderivative size = 179, normalized size of antiderivative = 0.67 \[ \int e^{-3 \coth ^{-1}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^3 \, dx=-\frac {30 \, a^{5} c^{3} x^{5} \arctan \left (\sqrt {\frac {a x - 1}{a x + 1}}\right ) + 120 \, a^{5} c^{3} x^{5} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right ) - 120 \, a^{5} c^{3} x^{5} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right ) - {\left (40 \, a^{6} c^{3} x^{6} + 192 \, a^{5} c^{3} x^{5} + 97 \, a^{4} c^{3} x^{4} - 79 \, a^{3} c^{3} x^{3} + 6 \, a^{2} c^{3} x^{2} + 22 \, a c^{3} x - 8 \, c^{3}\right )} \sqrt {\frac {a x - 1}{a x + 1}}}{40 \, a^{6} x^{5}} \]
-1/40*(30*a^5*c^3*x^5*arctan(sqrt((a*x - 1)/(a*x + 1))) + 120*a^5*c^3*x^5* log(sqrt((a*x - 1)/(a*x + 1)) + 1) - 120*a^5*c^3*x^5*log(sqrt((a*x - 1)/(a *x + 1)) - 1) - (40*a^6*c^3*x^6 + 192*a^5*c^3*x^5 + 97*a^4*c^3*x^4 - 79*a^ 3*c^3*x^3 + 6*a^2*c^3*x^2 + 22*a*c^3*x - 8*c^3)*sqrt((a*x - 1)/(a*x + 1))) /(a^6*x^5)
\[ \int e^{-3 \coth ^{-1}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^3 \, dx=\frac {c^{3} \left (\int \frac {\sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x^{7} + x^{6}}\, dx + \int \left (- \frac {a \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x^{6} + x^{5}}\right )\, dx + \int \left (- \frac {3 a^{2} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x^{5} + x^{4}}\right )\, dx + \int \frac {3 a^{3} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x^{4} + x^{3}}\, dx + \int \frac {3 a^{4} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x^{3} + x^{2}}\, dx + \int \left (- \frac {3 a^{5} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x^{2} + x}\right )\, dx + \int \left (- \frac {a^{6} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x + 1}\right )\, dx + \int \frac {a^{7} x \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x + 1}\, dx\right )}{a^{6}} \]
c**3*(Integral(sqrt(a*x/(a*x + 1) - 1/(a*x + 1))/(a*x**7 + x**6), x) + Int egral(-a*sqrt(a*x/(a*x + 1) - 1/(a*x + 1))/(a*x**6 + x**5), x) + Integral( -3*a**2*sqrt(a*x/(a*x + 1) - 1/(a*x + 1))/(a*x**5 + x**4), x) + Integral(3 *a**3*sqrt(a*x/(a*x + 1) - 1/(a*x + 1))/(a*x**4 + x**3), x) + Integral(3*a **4*sqrt(a*x/(a*x + 1) - 1/(a*x + 1))/(a*x**3 + x**2), x) + Integral(-3*a* *5*sqrt(a*x/(a*x + 1) - 1/(a*x + 1))/(a*x**2 + x), x) + Integral(-a**6*sqr t(a*x/(a*x + 1) - 1/(a*x + 1))/(a*x + 1), x) + Integral(a**7*x*sqrt(a*x/(a *x + 1) - 1/(a*x + 1))/(a*x + 1), x))/a**6
Time = 0.28 (sec) , antiderivative size = 301, normalized size of antiderivative = 1.12 \[ \int e^{-3 \coth ^{-1}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^3 \, dx=-\frac {1}{20} \, {\left (\frac {15 \, c^{3} \arctan \left (\sqrt {\frac {a x - 1}{a x + 1}}\right )}{a^{2}} + \frac {60 \, c^{3} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right )}{a^{2}} - \frac {60 \, c^{3} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right )}{a^{2}} + \frac {105 \, c^{3} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {11}{2}} + 465 \, c^{3} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {9}{2}} - 298 \, c^{3} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {7}{2}} - 842 \, c^{3} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {5}{2}} - 575 \, c^{3} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}} - 135 \, c^{3} \sqrt {\frac {a x - 1}{a x + 1}}}{\frac {4 \, {\left (a x - 1\right )} a^{2}}{a x + 1} + \frac {5 \, {\left (a x - 1\right )}^{2} a^{2}}{{\left (a x + 1\right )}^{2}} - \frac {5 \, {\left (a x - 1\right )}^{4} a^{2}}{{\left (a x + 1\right )}^{4}} - \frac {4 \, {\left (a x - 1\right )}^{5} a^{2}}{{\left (a x + 1\right )}^{5}} - \frac {{\left (a x - 1\right )}^{6} a^{2}}{{\left (a x + 1\right )}^{6}} + a^{2}}\right )} a \]
-1/20*(15*c^3*arctan(sqrt((a*x - 1)/(a*x + 1)))/a^2 + 60*c^3*log(sqrt((a*x - 1)/(a*x + 1)) + 1)/a^2 - 60*c^3*log(sqrt((a*x - 1)/(a*x + 1)) - 1)/a^2 + (105*c^3*((a*x - 1)/(a*x + 1))^(11/2) + 465*c^3*((a*x - 1)/(a*x + 1))^(9 /2) - 298*c^3*((a*x - 1)/(a*x + 1))^(7/2) - 842*c^3*((a*x - 1)/(a*x + 1))^ (5/2) - 575*c^3*((a*x - 1)/(a*x + 1))^(3/2) - 135*c^3*sqrt((a*x - 1)/(a*x + 1)))/(4*(a*x - 1)*a^2/(a*x + 1) + 5*(a*x - 1)^2*a^2/(a*x + 1)^2 - 5*(a*x - 1)^4*a^2/(a*x + 1)^4 - 4*(a*x - 1)^5*a^2/(a*x + 1)^5 - (a*x - 1)^6*a^2/ (a*x + 1)^6 + a^2))*a
Time = 0.30 (sec) , antiderivative size = 395, normalized size of antiderivative = 1.47 \[ \int e^{-3 \coth ^{-1}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^3 \, dx=-\frac {3 \, c^{3} \arctan \left (-x {\left | a \right |} + \sqrt {a^{2} x^{2} - 1}\right ) \mathrm {sgn}\left (a x + 1\right )}{4 \, a} + \frac {3 \, c^{3} \log \left ({\left | -x {\left | a \right |} + \sqrt {a^{2} x^{2} - 1} \right |}\right ) \mathrm {sgn}\left (a x + 1\right )}{{\left | a \right |}} + \frac {\sqrt {a^{2} x^{2} - 1} c^{3} \mathrm {sgn}\left (a x + 1\right )}{a} + \frac {55 \, {\left (x {\left | a \right |} - \sqrt {a^{2} x^{2} - 1}\right )}^{9} c^{3} {\left | a \right |} \mathrm {sgn}\left (a x + 1\right ) + 200 \, {\left (x {\left | a \right |} - \sqrt {a^{2} x^{2} - 1}\right )}^{8} a c^{3} \mathrm {sgn}\left (a x + 1\right ) - 10 \, {\left (x {\left | a \right |} - \sqrt {a^{2} x^{2} - 1}\right )}^{7} c^{3} {\left | a \right |} \mathrm {sgn}\left (a x + 1\right ) + 720 \, {\left (x {\left | a \right |} - \sqrt {a^{2} x^{2} - 1}\right )}^{6} a c^{3} \mathrm {sgn}\left (a x + 1\right ) + 800 \, {\left (x {\left | a \right |} - \sqrt {a^{2} x^{2} - 1}\right )}^{4} a c^{3} \mathrm {sgn}\left (a x + 1\right ) + 10 \, {\left (x {\left | a \right |} - \sqrt {a^{2} x^{2} - 1}\right )}^{3} c^{3} {\left | a \right |} \mathrm {sgn}\left (a x + 1\right ) + 560 \, {\left (x {\left | a \right |} - \sqrt {a^{2} x^{2} - 1}\right )}^{2} a c^{3} \mathrm {sgn}\left (a x + 1\right ) - 55 \, {\left (x {\left | a \right |} - \sqrt {a^{2} x^{2} - 1}\right )} c^{3} {\left | a \right |} \mathrm {sgn}\left (a x + 1\right ) + 152 \, a c^{3} \mathrm {sgn}\left (a x + 1\right )}{20 \, {\left ({\left (x {\left | a \right |} - \sqrt {a^{2} x^{2} - 1}\right )}^{2} + 1\right )}^{5} a {\left | a \right |}} \]
-3/4*c^3*arctan(-x*abs(a) + sqrt(a^2*x^2 - 1))*sgn(a*x + 1)/a + 3*c^3*log( abs(-x*abs(a) + sqrt(a^2*x^2 - 1)))*sgn(a*x + 1)/abs(a) + sqrt(a^2*x^2 - 1 )*c^3*sgn(a*x + 1)/a + 1/20*(55*(x*abs(a) - sqrt(a^2*x^2 - 1))^9*c^3*abs(a )*sgn(a*x + 1) + 200*(x*abs(a) - sqrt(a^2*x^2 - 1))^8*a*c^3*sgn(a*x + 1) - 10*(x*abs(a) - sqrt(a^2*x^2 - 1))^7*c^3*abs(a)*sgn(a*x + 1) + 720*(x*abs( a) - sqrt(a^2*x^2 - 1))^6*a*c^3*sgn(a*x + 1) + 800*(x*abs(a) - sqrt(a^2*x^ 2 - 1))^4*a*c^3*sgn(a*x + 1) + 10*(x*abs(a) - sqrt(a^2*x^2 - 1))^3*c^3*abs (a)*sgn(a*x + 1) + 560*(x*abs(a) - sqrt(a^2*x^2 - 1))^2*a*c^3*sgn(a*x + 1) - 55*(x*abs(a) - sqrt(a^2*x^2 - 1))*c^3*abs(a)*sgn(a*x + 1) + 152*a*c^3*s gn(a*x + 1))/(((x*abs(a) - sqrt(a^2*x^2 - 1))^2 + 1)^5*a*abs(a))
Time = 0.11 (sec) , antiderivative size = 258, normalized size of antiderivative = 0.96 \[ \int e^{-3 \coth ^{-1}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^3 \, dx=\frac {\frac {27\,c^3\,\sqrt {\frac {a\,x-1}{a\,x+1}}}{4}+\frac {115\,c^3\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{3/2}}{4}+\frac {421\,c^3\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{5/2}}{10}+\frac {149\,c^3\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{7/2}}{10}-\frac {93\,c^3\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{9/2}}{4}-\frac {21\,c^3\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{11/2}}{4}}{a+\frac {4\,a\,\left (a\,x-1\right )}{a\,x+1}+\frac {5\,a\,{\left (a\,x-1\right )}^2}{{\left (a\,x+1\right )}^2}-\frac {5\,a\,{\left (a\,x-1\right )}^4}{{\left (a\,x+1\right )}^4}-\frac {4\,a\,{\left (a\,x-1\right )}^5}{{\left (a\,x+1\right )}^5}-\frac {a\,{\left (a\,x-1\right )}^6}{{\left (a\,x+1\right )}^6}}-\frac {3\,c^3\,\mathrm {atan}\left (\sqrt {\frac {a\,x-1}{a\,x+1}}\right )}{4\,a}-\frac {6\,c^3\,\mathrm {atanh}\left (\sqrt {\frac {a\,x-1}{a\,x+1}}\right )}{a} \]
((27*c^3*((a*x - 1)/(a*x + 1))^(1/2))/4 + (115*c^3*((a*x - 1)/(a*x + 1))^( 3/2))/4 + (421*c^3*((a*x - 1)/(a*x + 1))^(5/2))/10 + (149*c^3*((a*x - 1)/( a*x + 1))^(7/2))/10 - (93*c^3*((a*x - 1)/(a*x + 1))^(9/2))/4 - (21*c^3*((a *x - 1)/(a*x + 1))^(11/2))/4)/(a + (4*a*(a*x - 1))/(a*x + 1) + (5*a*(a*x - 1)^2)/(a*x + 1)^2 - (5*a*(a*x - 1)^4)/(a*x + 1)^4 - (4*a*(a*x - 1)^5)/(a* x + 1)^5 - (a*(a*x - 1)^6)/(a*x + 1)^6) - (3*c^3*atan(((a*x - 1)/(a*x + 1) )^(1/2)))/(4*a) - (6*c^3*atanh(((a*x - 1)/(a*x + 1))^(1/2)))/a