Integrand size = 22, antiderivative size = 269 \[ \int e^{-\coth ^{-1}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^3 \, dx=-\frac {7 c^3 \sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}}{8 a}+\frac {c^3 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{3/2}}{24 a}+\frac {11 c^3 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{5/2}}{12 a}+\frac {5 c^3 \left (1-\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{5/2}}{4 a}+\frac {6 c^3 \left (1-\frac {1}{a x}\right )^{5/2} \left (1+\frac {1}{a x}\right )^{5/2}}{5 a}+c^3 \left (1-\frac {1}{a x}\right )^{7/2} \left (1+\frac {1}{a x}\right )^{5/2} x+\frac {15 c^3 \csc ^{-1}(a x)}{8 a}-\frac {c^3 \text {arctanh}\left (\sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}\right )}{a} \]
5/4*c^3*(1-1/a/x)^(3/2)*(1+1/a/x)^(5/2)/a+6/5*c^3*(1-1/a/x)^(5/2)*(1+1/a/x )^(5/2)/a+c^3*(1-1/a/x)^(7/2)*(1+1/a/x)^(5/2)*x+15/8*c^3*arccsc(a*x)/a-c^3 *arctanh((1-1/a/x)^(1/2)*(1+1/a/x)^(1/2))/a+1/24*c^3*(1+1/a/x)^(3/2)*(1-1/ a/x)^(1/2)/a+11/12*c^3*(1+1/a/x)^(5/2)*(1-1/a/x)^(1/2)/a-7/8*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 \[ \int e^{-\coth ^{-1}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^3 \, dx=\frac {c^3 \left (\sqrt {1-\frac {1}{a^2 x^2}} \left (24-30 a x-88 a^2 x^2+135 a^3 x^3+184 a^4 x^4+120 a^5 x^5\right )+225 a^4 x^4 \arcsin \left (\frac {1}{a x}\right )-120 a^4 x^4 \log \left (\left (1+\sqrt {1-\frac {1}{a^2 x^2}}\right ) x\right )\right )}{120 a^5 x^4} \]
(c^3*(Sqrt[1 - 1/(a^2*x^2)]*(24 - 30*a*x - 88*a^2*x^2 + 135*a^3*x^3 + 184* a^4*x^4 + 120*a^5*x^5) + 225*a^4*x^4*ArcSin[1/(a*x)] - 120*a^4*x^4*Log[(1 + Sqrt[1 - 1/(a^2*x^2)])*x]))/(120*a^5*x^4)
Time = 0.45 (sec) , antiderivative size = 258, normalized size of antiderivative = 0.96, number of steps used = 21, number of rules used = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.909, Rules used = {6748, 108, 25, 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.
| \(\displaystyle \int \left (c-\frac {c}{a^2 x^2}\right )^3 e^{-\coth ^{-1}(a x)} \, dx\) |
| \(\Big \downarrow \) 6748 |
| \(\displaystyle -c^3 \int \left (1-\frac {1}{a x}\right )^{7/2} \left (1+\frac {1}{a x}\right )^{5/2} x^2d\frac {1}{x}\) |
| \(\Big \downarrow \) 108 |
| \(\displaystyle -c^3 \left (\int -\frac {\left (a+\frac {6}{x}\right ) \left (1-\frac {1}{a x}\right )^{5/2} \left (1+\frac {1}{a x}\right )^{3/2} x}{a^2}d\frac {1}{x}-x \left (1-\frac {1}{a x}\right )^{7/2} \left (\frac {1}{a x}+1\right )^{5/2}\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -c^3 \left (x \left (-\left (\frac {1}{a x}+1\right )^{5/2}\right ) \left (1-\frac {1}{a x}\right )^{7/2}-\int \frac {\left (a+\frac {6}{x}\right ) \left (1-\frac {1}{a x}\right )^{5/2} \left (1+\frac {1}{a x}\right )^{3/2} x}{a^2}d\frac {1}{x}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -c^3 \left (x \left (-\left (\frac {1}{a x}+1\right )^{5/2}\right ) \left (1-\frac {1}{a x}\right )^{7/2}-\frac {\int \left (a+\frac {6}{x}\right ) \left (1-\frac {1}{a x}\right )^{5/2} \left (1+\frac {1}{a x}\right )^{3/2} xd\frac {1}{x}}{a^2}\right )\) |
| \(\Big \downarrow \) 171 |
| \(\displaystyle -c^3 \left (x \left (-\left (\frac {1}{a x}+1\right )^{5/2}\right ) \left (1-\frac {1}{a x}\right )^{7/2}-\frac {\frac {1}{5} a \int \frac {5 \left (a+\frac {5}{x}\right ) \left (1-\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{3/2} x}{a}d\frac {1}{x}+\frac {6}{5} a \left (1-\frac {1}{a x}\right )^{5/2} \left (\frac {1}{a x}+1\right )^{5/2}}{a^2}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -c^3 \left (x \left (-\left (\frac {1}{a x}+1\right )^{5/2}\right ) \left (1-\frac {1}{a x}\right )^{7/2}-\frac {\int \left (a+\frac {5}{x}\right ) \left (1-\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{3/2} xd\frac {1}{x}+\frac {6}{5} a \left (1-\frac {1}{a x}\right )^{5/2} \left (\frac {1}{a x}+1\right )^{5/2}}{a^2}\right )\) |
| \(\Big \downarrow \) 171 |
| \(\displaystyle -c^3 \left (x \left (-\left (\frac {1}{a x}+1\right )^{5/2}\right ) \left (1-\frac {1}{a x}\right )^{7/2}-\frac {\frac {1}{4} a \int \frac {\left (4 a+\frac {11}{x}\right ) \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{3/2} x}{a}d\frac {1}{x}+\frac {6}{5} a \left (1-\frac {1}{a x}\right )^{5/2} \left (\frac {1}{a x}+1\right )^{5/2}+\frac {5}{4} a \left (1-\frac {1}{a x}\right )^{3/2} \left (\frac {1}{a x}+1\right )^{5/2}}{a^2}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -c^3 \left (x \left (-\left (\frac {1}{a x}+1\right )^{5/2}\right ) \left (1-\frac {1}{a x}\right )^{7/2}-\frac {\frac {1}{4} \int \left (4 a+\frac {11}{x}\right ) \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{3/2} xd\frac {1}{x}+\frac {6}{5} a \left (1-\frac {1}{a x}\right )^{5/2} \left (\frac {1}{a x}+1\right )^{5/2}+\frac {5}{4} a \left (1-\frac {1}{a x}\right )^{3/2} \left (\frac {1}{a x}+1\right )^{5/2}}{a^2}\right )\) |
| \(\Big \downarrow \) 171 |
| \(\displaystyle -c^3 \left (x \left (-\left (\frac {1}{a x}+1\right )^{5/2}\right ) \left (1-\frac {1}{a x}\right )^{7/2}-\frac {\frac {1}{4} \left (\frac {1}{3} a \int \frac {\left (12 a-\frac {1}{x}\right ) \left (1+\frac {1}{a x}\right )^{3/2} x}{a \sqrt {1-\frac {1}{a x}}}d\frac {1}{x}+\frac {11}{3} a \sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{5/2}\right )+\frac {6}{5} a \left (1-\frac {1}{a x}\right )^{5/2} \left (\frac {1}{a x}+1\right )^{5/2}+\frac {5}{4} a \left (1-\frac {1}{a x}\right )^{3/2} \left (\frac {1}{a x}+1\right )^{5/2}}{a^2}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -c^3 \left (x \left (-\left (\frac {1}{a x}+1\right )^{5/2}\right ) \left (1-\frac {1}{a x}\right )^{7/2}-\frac {\frac {1}{4} \left (\frac {1}{3} \int \frac {\left (12 a-\frac {1}{x}\right ) \left (1+\frac {1}{a x}\right )^{3/2} x}{\sqrt {1-\frac {1}{a x}}}d\frac {1}{x}+\frac {11}{3} a \sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{5/2}\right )+\frac {6}{5} a \left (1-\frac {1}{a x}\right )^{5/2} \left (\frac {1}{a x}+1\right )^{5/2}+\frac {5}{4} a \left (1-\frac {1}{a x}\right )^{3/2} \left (\frac {1}{a x}+1\right )^{5/2}}{a^2}\right )\) |
| \(\Big \downarrow \) 171 |
| \(\displaystyle -c^3 \left (x \left (-\left (\frac {1}{a x}+1\right )^{5/2}\right ) \left (1-\frac {1}{a x}\right )^{7/2}-\frac {\frac {1}{4} \left (\frac {1}{3} \left (\frac {1}{2} a \sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{3/2}-\frac {1}{2} a \int -\frac {3 \left (8 a+\frac {7}{x}\right ) \sqrt {1+\frac {1}{a x}} x}{a \sqrt {1-\frac {1}{a x}}}d\frac {1}{x}\right )+\frac {11}{3} a \sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{5/2}\right )+\frac {6}{5} a \left (1-\frac {1}{a x}\right )^{5/2} \left (\frac {1}{a x}+1\right )^{5/2}+\frac {5}{4} a \left (1-\frac {1}{a x}\right )^{3/2} \left (\frac {1}{a x}+1\right )^{5/2}}{a^2}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -c^3 \left (x \left (-\left (\frac {1}{a x}+1\right )^{5/2}\right ) \left (1-\frac {1}{a x}\right )^{7/2}-\frac {\frac {1}{4} \left (\frac {1}{3} \left (\frac {3}{2} \int \frac {\left (8 a+\frac {7}{x}\right ) \sqrt {1+\frac {1}{a x}} x}{\sqrt {1-\frac {1}{a x}}}d\frac {1}{x}+\frac {1}{2} a \sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{3/2}\right )+\frac {11}{3} a \sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{5/2}\right )+\frac {6}{5} a \left (1-\frac {1}{a x}\right )^{5/2} \left (\frac {1}{a x}+1\right )^{5/2}+\frac {5}{4} a \left (1-\frac {1}{a x}\right )^{3/2} \left (\frac {1}{a x}+1\right )^{5/2}}{a^2}\right )\) |
| \(\Big \downarrow \) 171 |
| \(\displaystyle -c^3 \left (x \left (-\left (\frac {1}{a x}+1\right )^{5/2}\right ) \left (1-\frac {1}{a x}\right )^{7/2}-\frac {\frac {1}{4} \left (\frac {1}{3} \left (\frac {3}{2} \left (a \left (-\int -\frac {\left (8 a+\frac {15}{x}\right ) x}{a \sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}}d\frac {1}{x}\right )-7 a \sqrt {1-\frac {1}{a x}} \sqrt {\frac {1}{a x}+1}\right )+\frac {1}{2} a \sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{3/2}\right )+\frac {11}{3} a \sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{5/2}\right )+\frac {6}{5} a \left (1-\frac {1}{a x}\right )^{5/2} \left (\frac {1}{a x}+1\right )^{5/2}+\frac {5}{4} a \left (1-\frac {1}{a x}\right )^{3/2} \left (\frac {1}{a x}+1\right )^{5/2}}{a^2}\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -c^3 \left (x \left (-\left (\frac {1}{a x}+1\right )^{5/2}\right ) \left (1-\frac {1}{a x}\right )^{7/2}-\frac {\frac {1}{4} \left (\frac {1}{3} \left (\frac {3}{2} \left (a \int \frac {\left (8 a+\frac {15}{x}\right ) x}{a \sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}}d\frac {1}{x}-7 a \sqrt {1-\frac {1}{a x}} \sqrt {\frac {1}{a x}+1}\right )+\frac {1}{2} a \sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{3/2}\right )+\frac {11}{3} a \sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{5/2}\right )+\frac {6}{5} a \left (1-\frac {1}{a x}\right )^{5/2} \left (\frac {1}{a x}+1\right )^{5/2}+\frac {5}{4} a \left (1-\frac {1}{a x}\right )^{3/2} \left (\frac {1}{a x}+1\right )^{5/2}}{a^2}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -c^3 \left (x \left (-\left (\frac {1}{a x}+1\right )^{5/2}\right ) \left (1-\frac {1}{a x}\right )^{7/2}-\frac {\frac {1}{4} \left (\frac {1}{3} \left (\frac {3}{2} \left (\int \frac {\left (8 a+\frac {15}{x}\right ) x}{\sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}}d\frac {1}{x}-7 a \sqrt {1-\frac {1}{a x}} \sqrt {\frac {1}{a x}+1}\right )+\frac {1}{2} a \sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{3/2}\right )+\frac {11}{3} a \sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{5/2}\right )+\frac {6}{5} a \left (1-\frac {1}{a x}\right )^{5/2} \left (\frac {1}{a x}+1\right )^{5/2}+\frac {5}{4} a \left (1-\frac {1}{a x}\right )^{3/2} \left (\frac {1}{a x}+1\right )^{5/2}}{a^2}\right )\) |
| \(\Big \downarrow \) 175 |
| \(\displaystyle -c^3 \left (x \left (-\left (\frac {1}{a x}+1\right )^{5/2}\right ) \left (1-\frac {1}{a x}\right )^{7/2}-\frac {\frac {1}{4} \left (\frac {1}{3} \left (\frac {3}{2} \left (8 a \int \frac {x}{\sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}}d\frac {1}{x}+15 \int \frac {1}{\sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}}d\frac {1}{x}-7 a \sqrt {1-\frac {1}{a x}} \sqrt {\frac {1}{a x}+1}\right )+\frac {1}{2} a \sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{3/2}\right )+\frac {11}{3} a \sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{5/2}\right )+\frac {6}{5} a \left (1-\frac {1}{a x}\right )^{5/2} \left (\frac {1}{a x}+1\right )^{5/2}+\frac {5}{4} a \left (1-\frac {1}{a x}\right )^{3/2} \left (\frac {1}{a x}+1\right )^{5/2}}{a^2}\right )\) |
| \(\Big \downarrow \) 39 |
| \(\displaystyle -c^3 \left (x \left (-\left (\frac {1}{a x}+1\right )^{5/2}\right ) \left (1-\frac {1}{a x}\right )^{7/2}-\frac {\frac {1}{4} \left (\frac {1}{3} \left (\frac {3}{2} \left (15 \int \frac {1}{\sqrt {1-\frac {1}{a^2 x^2}}}d\frac {1}{x}+8 a \int \frac {x}{\sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}}d\frac {1}{x}-7 a \sqrt {1-\frac {1}{a x}} \sqrt {\frac {1}{a x}+1}\right )+\frac {1}{2} a \sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{3/2}\right )+\frac {11}{3} a \sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{5/2}\right )+\frac {6}{5} a \left (1-\frac {1}{a x}\right )^{5/2} \left (\frac {1}{a x}+1\right )^{5/2}+\frac {5}{4} a \left (1-\frac {1}{a x}\right )^{3/2} \left (\frac {1}{a x}+1\right )^{5/2}}{a^2}\right )\) |
| \(\Big \downarrow \) 103 |
| \(\displaystyle -c^3 \left (x \left (-\left (\frac {1}{a x}+1\right )^{5/2}\right ) \left (1-\frac {1}{a x}\right )^{7/2}-\frac {\frac {1}{4} \left (\frac {1}{3} \left (\frac {3}{2} \left (15 \int \frac {1}{\sqrt {1-\frac {1}{a^2 x^2}}}d\frac {1}{x}-8 \int \frac {1}{\frac {1}{a}-\frac {1}{a x^2}}d\left (\sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}\right )-7 a \sqrt {1-\frac {1}{a x}} \sqrt {\frac {1}{a x}+1}\right )+\frac {1}{2} a \sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{3/2}\right )+\frac {11}{3} a \sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{5/2}\right )+\frac {6}{5} a \left (1-\frac {1}{a x}\right )^{5/2} \left (\frac {1}{a x}+1\right )^{5/2}+\frac {5}{4} a \left (1-\frac {1}{a x}\right )^{3/2} \left (\frac {1}{a x}+1\right )^{5/2}}{a^2}\right )\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle -c^3 \left (x \left (-\left (\frac {1}{a x}+1\right )^{5/2}\right ) \left (1-\frac {1}{a x}\right )^{7/2}-\frac {\frac {1}{4} \left (\frac {1}{3} \left (\frac {3}{2} \left (15 \int \frac {1}{\sqrt {1-\frac {1}{a^2 x^2}}}d\frac {1}{x}-8 a \text {arctanh}\left (\sqrt {1-\frac {1}{a x}} \sqrt {\frac {1}{a x}+1}\right )-7 a \sqrt {1-\frac {1}{a x}} \sqrt {\frac {1}{a x}+1}\right )+\frac {1}{2} a \sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{3/2}\right )+\frac {11}{3} a \sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{5/2}\right )+\frac {6}{5} a \left (1-\frac {1}{a x}\right )^{5/2} \left (\frac {1}{a x}+1\right )^{5/2}+\frac {5}{4} a \left (1-\frac {1}{a x}\right )^{3/2} \left (\frac {1}{a x}+1\right )^{5/2}}{a^2}\right )\) |
| \(\Big \downarrow \) 223 |
| \(\displaystyle -c^3 \left (x \left (-\left (\frac {1}{a x}+1\right )^{5/2}\right ) \left (1-\frac {1}{a x}\right )^{7/2}-\frac {\frac {1}{4} \left (\frac {1}{3} \left (\frac {3}{2} \left (15 a \arcsin \left (\frac {1}{a x}\right )-8 a \text {arctanh}\left (\sqrt {1-\frac {1}{a x}} \sqrt {\frac {1}{a x}+1}\right )-7 a \sqrt {1-\frac {1}{a x}} \sqrt {\frac {1}{a x}+1}\right )+\frac {1}{2} a \sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{3/2}\right )+\frac {11}{3} a \sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{5/2}\right )+\frac {6}{5} a \left (1-\frac {1}{a x}\right )^{5/2} \left (\frac {1}{a x}+1\right )^{5/2}+\frac {5}{4} a \left (1-\frac {1}{a x}\right )^{3/2} \left (\frac {1}{a x}+1\right )^{5/2}}{a^2}\right )\) |
-(c^3*(-((1 - 1/(a*x))^(7/2)*(1 + 1/(a*x))^(5/2)*x) - ((5*a*(1 - 1/(a*x))^ (3/2)*(1 + 1/(a*x))^(5/2))/4 + (6*a*(1 - 1/(a*x))^(5/2)*(1 + 1/(a*x))^(5/2 ))/5 + ((11*a*Sqrt[1 - 1/(a*x)]*(1 + 1/(a*x))^(5/2))/3 + ((a*Sqrt[1 - 1/(a *x)]*(1 + 1/(a*x))^(3/2))/2 + (3*(-7*a*Sqrt[1 - 1/(a*x)]*Sqrt[1 + 1/(a*x)] + 15*a*ArcSin[1/(a*x)] - 8*a*ArcTanh[Sqrt[1 - 1/(a*x)]*Sqrt[1 + 1/(a*x)]] ))/2)/3)/4)/a^2))
3.9.7.3.1 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 | \(\frac {\left (a x +1\right ) \left (184 a^{4} x^{4}+135 a^{3} x^{3}-88 a^{2} x^{2}-30 a x +24\right ) c^{3} \sqrt {\frac {a x -1}{a x +1}}}{120 x^{5} a^{6}}+\frac {\left (-\frac {a^{6} \ln \left (\frac {a^{2} x}{\sqrt {a^{2}}}+\sqrt {a^{2} x^{2}-1}\right )}{\sqrt {a^{2}}}+\frac {15 a^{5} \arctan \left (\frac {1}{\sqrt {a^{2} x^{2}-1}}\right )}{8}+a^{5} \sqrt {\left (a x -1\right ) \left (a x +1\right )}\right ) c^{3} \sqrt {\frac {a x -1}{a x +1}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}}{a^{6} \left (a x -1\right )}\) | \(173\) |
| default | \(-\frac {\sqrt {\frac {a x -1}{a x +1}}\, \left (a x +1\right ) c^{3} \left (-120 \sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}\, a^{6} x^{6}+120 \left (a^{2} x^{2}-1\right )^{\frac {3}{2}} \sqrt {a^{2}}\, a^{4} x^{4}-225 a^{5} x^{5} \sqrt {a^{2}}\, \sqrt {a^{2} x^{2}-1}-225 a^{5} x^{5} \sqrt {a^{2}}\, \arctan \left (\frac {1}{\sqrt {a^{2} x^{2}-1}}\right )+120 \ln \left (\frac {a^{2} x +\sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) a^{6} x^{5}+105 \left (a^{2} x^{2}-1\right )^{\frac {3}{2}} \sqrt {a^{2}}\, a^{3} x^{3}-64 \left (a^{2} x^{2}-1\right )^{\frac {3}{2}} \sqrt {a^{2}}\, a^{2} x^{2}-30 \left (a^{2} x^{2}-1\right )^{\frac {3}{2}} \sqrt {a^{2}}\, a x +24 \left (a^{2} x^{2}-1\right )^{\frac {3}{2}} \sqrt {a^{2}}\right )}{120 \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, a^{6} x^{5} \sqrt {a^{2}}}\) | \(272\) |
1/120*(a*x+1)*(184*a^4*x^4+135*a^3*x^3-88*a^2*x^2-30*a*x+24)/x^5*c^3/a^6*( (a*x-1)/(a*x+1))^(1/2)+(-a^6*ln(a^2*x/(a^2)^(1/2)+(a^2*x^2-1)^(1/2))/(a^2) ^(1/2)+15/8*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))^(1/2)/(a*x-1)
Time = 0.26 (sec) , antiderivative size = 179, normalized size of antiderivative = 0.67 \[ \int e^{-\coth ^{-1}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^3 \, dx=-\frac {450 \, a^{5} c^{3} x^{5} \arctan \left (\sqrt {\frac {a x - 1}{a x + 1}}\right ) + 120 \, a^{5} c^{3} x^{5} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right ) - 120 \, a^{5} c^{3} x^{5} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right ) - {\left (120 \, a^{6} c^{3} x^{6} + 304 \, a^{5} c^{3} x^{5} + 319 \, a^{4} c^{3} x^{4} + 47 \, a^{3} c^{3} x^{3} - 118 \, a^{2} c^{3} x^{2} - 6 \, a c^{3} x + 24 \, c^{3}\right )} \sqrt {\frac {a x - 1}{a x + 1}}}{120 \, a^{6} x^{5}} \]
-1/120*(450*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) - (120*a^6*c^3*x^6 + 304*a^5*c^3*x^5 + 319*a^4*c^3*x^4 + 4 7*a^3*c^3*x^3 - 118*a^2*c^3*x^2 - 6*a*c^3*x + 24*c^3)*sqrt((a*x - 1)/(a*x + 1)))/(a^6*x^5)
\[ \int e^{-\coth ^{-1}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^3 \, dx=\frac {c^{3} \left (\int a^{6} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}\, dx + \int \left (- \frac {\sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{x^{6}}\right )\, dx + \int \frac {3 a^{2} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{x^{4}}\, dx + \int \left (- \frac {3 a^{4} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{x^{2}}\right )\, dx\right )}{a^{6}} \]
c**3*(Integral(a**6*sqrt(a*x/(a*x + 1) - 1/(a*x + 1)), x) + Integral(-sqrt (a*x/(a*x + 1) - 1/(a*x + 1))/x**6, x) + Integral(3*a**2*sqrt(a*x/(a*x + 1 ) - 1/(a*x + 1))/x**4, x) + Integral(-3*a**4*sqrt(a*x/(a*x + 1) - 1/(a*x + 1))/x**2, x))/a**6
Time = 0.28 (sec) , antiderivative size = 302, normalized size of antiderivative = 1.12 \[ \int e^{-\coth ^{-1}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^3 \, dx=-\frac {1}{60} \, {\left (\frac {225 \, c^{3} \arctan \left (\sqrt {\frac {a x - 1}{a x + 1}}\right )}{a^{2}} + \frac {60 \, c^{3} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right )}{a^{2}} - \frac {60 \, c^{3} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right )}{a^{2}} - \frac {105 \, c^{3} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {11}{2}} + 305 \, c^{3} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {9}{2}} + 86 \, c^{3} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {7}{2}} + 1654 \, c^{3} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {5}{2}} + 1345 \, c^{3} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}} + 345 \, c^{3} \sqrt {\frac {a x - 1}{a x + 1}}}{\frac {4 \, {\left (a x - 1\right )} a^{2}}{a x + 1} + \frac {5 \, {\left (a x - 1\right )}^{2} a^{2}}{{\left (a x + 1\right )}^{2}} - \frac {5 \, {\left (a x - 1\right )}^{4} a^{2}}{{\left (a x + 1\right )}^{4}} - \frac {4 \, {\left (a x - 1\right )}^{5} a^{2}}{{\left (a x + 1\right )}^{5}} - \frac {{\left (a x - 1\right )}^{6} a^{2}}{{\left (a x + 1\right )}^{6}} + a^{2}}\right )} a \]
-1/60*(225*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) + 305*c^3*((a*x - 1)/(a*x + 1))^( 9/2) + 86*c^3*((a*x - 1)/(a*x + 1))^(7/2) + 1654*c^3*((a*x - 1)/(a*x + 1)) ^(5/2) + 1345*c^3*((a*x - 1)/(a*x + 1))^(3/2) + 345*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 = 0.31 (sec) , antiderivative size = 394, normalized size of antiderivative = 1.46 \[ \int e^{-\coth ^{-1}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^3 \, dx=-\frac {15 \, c^{3} \arctan \left (-x {\left | a \right |} + \sqrt {a^{2} x^{2} - 1}\right ) \mathrm {sgn}\left (a x + 1\right )}{4 \, a} + \frac {c^{3} \log \left ({\left | -x {\left | a \right |} + \sqrt {a^{2} x^{2} - 1} \right |}\right ) \mathrm {sgn}\left (a x + 1\right )}{{\left | a \right |}} + \frac {\sqrt {a^{2} x^{2} - 1} c^{3} \mathrm {sgn}\left (a x + 1\right )}{a} - \frac {135 \, {\left (x {\left | a \right |} - \sqrt {a^{2} x^{2} - 1}\right )}^{9} c^{3} {\left | a \right |} \mathrm {sgn}\left (a x + 1\right ) - 360 \, {\left (x {\left | a \right |} - \sqrt {a^{2} x^{2} - 1}\right )}^{8} a c^{3} \mathrm {sgn}\left (a x + 1\right ) + 150 \, {\left (x {\left | a \right |} - \sqrt {a^{2} x^{2} - 1}\right )}^{7} c^{3} {\left | a \right |} \mathrm {sgn}\left (a x + 1\right ) - 720 \, {\left (x {\left | a \right |} - \sqrt {a^{2} x^{2} - 1}\right )}^{6} a c^{3} \mathrm {sgn}\left (a x + 1\right ) - 1120 \, {\left (x {\left | a \right |} - \sqrt {a^{2} x^{2} - 1}\right )}^{4} a c^{3} \mathrm {sgn}\left (a x + 1\right ) - 150 \, {\left (x {\left | a \right |} - \sqrt {a^{2} x^{2} - 1}\right )}^{3} c^{3} {\left | a \right |} \mathrm {sgn}\left (a x + 1\right ) - 560 \, {\left (x {\left | a \right |} - \sqrt {a^{2} x^{2} - 1}\right )}^{2} a c^{3} \mathrm {sgn}\left (a x + 1\right ) - 135 \, {\left (x {\left | a \right |} - \sqrt {a^{2} x^{2} - 1}\right )} c^{3} {\left | a \right |} \mathrm {sgn}\left (a x + 1\right ) - 184 \, a c^{3} \mathrm {sgn}\left (a x + 1\right )}{60 \, {\left ({\left (x {\left | a \right |} - \sqrt {a^{2} x^{2} - 1}\right )}^{2} + 1\right )}^{5} a {\left | a \right |}} \]
-15/4*c^3*arctan(-x*abs(a) + sqrt(a^2*x^2 - 1))*sgn(a*x + 1)/a + c^3*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^3*sgn(a*x + 1)/a - 1/60*(135*(x*abs(a) - sqrt(a^2*x^2 - 1))^9*c^3*abs(a )*sgn(a*x + 1) - 360*(x*abs(a) - sqrt(a^2*x^2 - 1))^8*a*c^3*sgn(a*x + 1) + 150*(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) - 1120*(x*abs(a) - sqrt(a^2* x^2 - 1))^4*a*c^3*sgn(a*x + 1) - 150*(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) - 135*(x*abs(a) - sqrt(a^2*x^2 - 1))*c^3*abs(a)*sgn(a*x + 1) - 184*a*c ^3*sgn(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 \[ \int e^{-\coth ^{-1}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^3 \, dx=\frac {\frac {23\,c^3\,\sqrt {\frac {a\,x-1}{a\,x+1}}}{4}+\frac {269\,c^3\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{3/2}}{12}+\frac {827\,c^3\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{5/2}}{30}+\frac {43\,c^3\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{7/2}}{30}+\frac {61\,c^3\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{9/2}}{12}+\frac {7\,c^3\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{11/2}}{4}}{a+\frac {4\,a\,\left (a\,x-1\right )}{a\,x+1}+\frac {5\,a\,{\left (a\,x-1\right )}^2}{{\left (a\,x+1\right )}^2}-\frac {5\,a\,{\left (a\,x-1\right )}^4}{{\left (a\,x+1\right )}^4}-\frac {4\,a\,{\left (a\,x-1\right )}^5}{{\left (a\,x+1\right )}^5}-\frac {a\,{\left (a\,x-1\right )}^6}{{\left (a\,x+1\right )}^6}}-\frac {15\,c^3\,\mathrm {atan}\left (\sqrt {\frac {a\,x-1}{a\,x+1}}\right )}{4\,a}-\frac {2\,c^3\,\mathrm {atanh}\left (\sqrt {\frac {a\,x-1}{a\,x+1}}\right )}{a} \]
((23*c^3*((a*x - 1)/(a*x + 1))^(1/2))/4 + (269*c^3*((a*x - 1)/(a*x + 1))^( 3/2))/12 + (827*c^3*((a*x - 1)/(a*x + 1))^(5/2))/30 + (43*c^3*((a*x - 1)/( a*x + 1))^(7/2))/30 + (61*c^3*((a*x - 1)/(a*x + 1))^(9/2))/12 + (7*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) - (15*c^3*atan(((a*x - 1)/(a*x + 1 ))^(1/2)))/(4*a) - (2*c^3*atanh(((a*x - 1)/(a*x + 1))^(1/2)))/a