Integrand size = 21, antiderivative size = 385 \[ \int \frac {\text {arctanh}(a x)^3}{\left (1-a^2 x^2\right )^{9/2}} \, dx=-\frac {6}{2401 a \left (1-a^2 x^2\right )^{7/2}}-\frac {2664}{214375 a \left (1-a^2 x^2\right )^{5/2}}-\frac {30256}{385875 a \left (1-a^2 x^2\right )^{3/2}}-\frac {413312}{128625 a \sqrt {1-a^2 x^2}}+\frac {6 x \text {arctanh}(a x)}{343 \left (1-a^2 x^2\right )^{7/2}}+\frac {2664 x \text {arctanh}(a x)}{42875 \left (1-a^2 x^2\right )^{5/2}}+\frac {30256 x \text {arctanh}(a x)}{128625 \left (1-a^2 x^2\right )^{3/2}}+\frac {413312 x \text {arctanh}(a x)}{128625 \sqrt {1-a^2 x^2}}-\frac {3 \text {arctanh}(a x)^2}{49 a \left (1-a^2 x^2\right )^{7/2}}-\frac {18 \text {arctanh}(a x)^2}{175 a \left (1-a^2 x^2\right )^{5/2}}-\frac {8 \text {arctanh}(a x)^2}{35 a \left (1-a^2 x^2\right )^{3/2}}-\frac {48 \text {arctanh}(a x)^2}{35 a \sqrt {1-a^2 x^2}}+\frac {x \text {arctanh}(a x)^3}{7 \left (1-a^2 x^2\right )^{7/2}}+\frac {6 x \text {arctanh}(a x)^3}{35 \left (1-a^2 x^2\right )^{5/2}}+\frac {8 x \text {arctanh}(a x)^3}{35 \left (1-a^2 x^2\right )^{3/2}}+\frac {16 x \text {arctanh}(a x)^3}{35 \sqrt {1-a^2 x^2}} \] Output:
-6/2401/a/(-a^2*x^2+1)^(7/2)-2664/214375/a/(-a^2*x^2+1)^(5/2)-30256/385875 /a/(-a^2*x^2+1)^(3/2)-413312/128625/a/(-a^2*x^2+1)^(1/2)+6/343*x*arctanh(a *x)/(-a^2*x^2+1)^(7/2)+2664/42875*x*arctanh(a*x)/(-a^2*x^2+1)^(5/2)+30256/ 128625*x*arctanh(a*x)/(-a^2*x^2+1)^(3/2)+413312/128625*x*arctanh(a*x)/(-a^ 2*x^2+1)^(1/2)-3/49*arctanh(a*x)^2/a/(-a^2*x^2+1)^(7/2)-18/175*arctanh(a*x )^2/a/(-a^2*x^2+1)^(5/2)-8/35*arctanh(a*x)^2/a/(-a^2*x^2+1)^(3/2)-48/35*ar ctanh(a*x)^2/a/(-a^2*x^2+1)^(1/2)+1/7*x*arctanh(a*x)^3/(-a^2*x^2+1)^(7/2)+ 6/35*x*arctanh(a*x)^3/(-a^2*x^2+1)^(5/2)+8/35*x*arctanh(a*x)^3/(-a^2*x^2+1 )^(3/2)+16/35*x*arctanh(a*x)^3/(-a^2*x^2+1)^(1/2)
Time = 0.09 (sec) , antiderivative size = 151, normalized size of antiderivative = 0.39 \[ \int \frac {\text {arctanh}(a x)^3}{\left (1-a^2 x^2\right )^{9/2}} \, dx=\frac {-44658302+132479032 a^2 x^2-131252240 a^4 x^4+43397760 a^6 x^6-210 a x \left (-226905+654220 a^2 x^2-635096 a^4 x^4+206656 a^6 x^6\right ) \text {arctanh}(a x)+11025 \left (-2161+5726 a^2 x^2-5320 a^4 x^4+1680 a^6 x^6\right ) \text {arctanh}(a x)^2-385875 a x \left (-35+70 a^2 x^2-56 a^4 x^4+16 a^6 x^6\right ) \text {arctanh}(a x)^3}{13505625 a \left (1-a^2 x^2\right )^{7/2}} \] Input:
Integrate[ArcTanh[a*x]^3/(1 - a^2*x^2)^(9/2),x]
Output:
(-44658302 + 132479032*a^2*x^2 - 131252240*a^4*x^4 + 43397760*a^6*x^6 - 21 0*a*x*(-226905 + 654220*a^2*x^2 - 635096*a^4*x^4 + 206656*a^6*x^6)*ArcTanh [a*x] + 11025*(-2161 + 5726*a^2*x^2 - 5320*a^4*x^4 + 1680*a^6*x^6)*ArcTanh [a*x]^2 - 385875*a*x*(-35 + 70*a^2*x^2 - 56*a^4*x^4 + 16*a^6*x^6)*ArcTanh[ a*x]^3)/(13505625*a*(1 - a^2*x^2)^(7/2))
Time = 3.80 (sec) , antiderivative size = 687, normalized size of antiderivative = 1.78, number of steps used = 14, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.667, Rules used = {6526, 6522, 6522, 6522, 6520, 6526, 6522, 6522, 6520, 6526, 6522, 6520, 6524, 6520}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {\text {arctanh}(a x)^3}{\left (1-a^2 x^2\right )^{9/2}} \, dx\) |
| \(\Big \downarrow \) 6526 |
| \(\displaystyle \frac {6}{49} \int \frac {\text {arctanh}(a x)}{\left (1-a^2 x^2\right )^{9/2}}dx+\frac {6}{7} \int \frac {\text {arctanh}(a x)^3}{\left (1-a^2 x^2\right )^{7/2}}dx+\frac {x \text {arctanh}(a x)^3}{7 \left (1-a^2 x^2\right )^{7/2}}-\frac {3 \text {arctanh}(a x)^2}{49 a \left (1-a^2 x^2\right )^{7/2}}\) |
| \(\Big \downarrow \) 6522 |
| \(\displaystyle \frac {6}{49} \left (\frac {6}{7} \int \frac {\text {arctanh}(a x)}{\left (1-a^2 x^2\right )^{7/2}}dx+\frac {x \text {arctanh}(a x)}{7 \left (1-a^2 x^2\right )^{7/2}}-\frac {1}{49 a \left (1-a^2 x^2\right )^{7/2}}\right )+\frac {6}{7} \int \frac {\text {arctanh}(a x)^3}{\left (1-a^2 x^2\right )^{7/2}}dx+\frac {x \text {arctanh}(a x)^3}{7 \left (1-a^2 x^2\right )^{7/2}}-\frac {3 \text {arctanh}(a x)^2}{49 a \left (1-a^2 x^2\right )^{7/2}}\) |
| \(\Big \downarrow \) 6522 |
| \(\displaystyle \frac {6}{49} \left (\frac {6}{7} \left (\frac {4}{5} \int \frac {\text {arctanh}(a x)}{\left (1-a^2 x^2\right )^{5/2}}dx+\frac {x \text {arctanh}(a x)}{5 \left (1-a^2 x^2\right )^{5/2}}-\frac {1}{25 a \left (1-a^2 x^2\right )^{5/2}}\right )+\frac {x \text {arctanh}(a x)}{7 \left (1-a^2 x^2\right )^{7/2}}-\frac {1}{49 a \left (1-a^2 x^2\right )^{7/2}}\right )+\frac {6}{7} \int \frac {\text {arctanh}(a x)^3}{\left (1-a^2 x^2\right )^{7/2}}dx+\frac {x \text {arctanh}(a x)^3}{7 \left (1-a^2 x^2\right )^{7/2}}-\frac {3 \text {arctanh}(a x)^2}{49 a \left (1-a^2 x^2\right )^{7/2}}\) |
| \(\Big \downarrow \) 6522 |
| \(\displaystyle \frac {6}{49} \left (\frac {6}{7} \left (\frac {4}{5} \left (\frac {2}{3} \int \frac {\text {arctanh}(a x)}{\left (1-a^2 x^2\right )^{3/2}}dx+\frac {x \text {arctanh}(a x)}{3 \left (1-a^2 x^2\right )^{3/2}}-\frac {1}{9 a \left (1-a^2 x^2\right )^{3/2}}\right )+\frac {x \text {arctanh}(a x)}{5 \left (1-a^2 x^2\right )^{5/2}}-\frac {1}{25 a \left (1-a^2 x^2\right )^{5/2}}\right )+\frac {x \text {arctanh}(a x)}{7 \left (1-a^2 x^2\right )^{7/2}}-\frac {1}{49 a \left (1-a^2 x^2\right )^{7/2}}\right )+\frac {6}{7} \int \frac {\text {arctanh}(a x)^3}{\left (1-a^2 x^2\right )^{7/2}}dx+\frac {x \text {arctanh}(a x)^3}{7 \left (1-a^2 x^2\right )^{7/2}}-\frac {3 \text {arctanh}(a x)^2}{49 a \left (1-a^2 x^2\right )^{7/2}}\) |
| \(\Big \downarrow \) 6520 |
| \(\displaystyle \frac {6}{7} \int \frac {\text {arctanh}(a x)^3}{\left (1-a^2 x^2\right )^{7/2}}dx+\frac {x \text {arctanh}(a x)^3}{7 \left (1-a^2 x^2\right )^{7/2}}-\frac {3 \text {arctanh}(a x)^2}{49 a \left (1-a^2 x^2\right )^{7/2}}+\frac {6}{49} \left (\frac {x \text {arctanh}(a x)}{7 \left (1-a^2 x^2\right )^{7/2}}+\frac {6}{7} \left (\frac {x \text {arctanh}(a x)}{5 \left (1-a^2 x^2\right )^{5/2}}+\frac {4}{5} \left (\frac {x \text {arctanh}(a x)}{3 \left (1-a^2 x^2\right )^{3/2}}+\frac {2}{3} \left (\frac {x \text {arctanh}(a x)}{\sqrt {1-a^2 x^2}}-\frac {1}{a \sqrt {1-a^2 x^2}}\right )-\frac {1}{9 a \left (1-a^2 x^2\right )^{3/2}}\right )-\frac {1}{25 a \left (1-a^2 x^2\right )^{5/2}}\right )-\frac {1}{49 a \left (1-a^2 x^2\right )^{7/2}}\right )\) |
| \(\Big \downarrow \) 6526 |
| \(\displaystyle \frac {6}{7} \left (\frac {6}{25} \int \frac {\text {arctanh}(a x)}{\left (1-a^2 x^2\right )^{7/2}}dx+\frac {4}{5} \int \frac {\text {arctanh}(a x)^3}{\left (1-a^2 x^2\right )^{5/2}}dx+\frac {x \text {arctanh}(a x)^3}{5 \left (1-a^2 x^2\right )^{5/2}}-\frac {3 \text {arctanh}(a x)^2}{25 a \left (1-a^2 x^2\right )^{5/2}}\right )+\frac {x \text {arctanh}(a x)^3}{7 \left (1-a^2 x^2\right )^{7/2}}-\frac {3 \text {arctanh}(a x)^2}{49 a \left (1-a^2 x^2\right )^{7/2}}+\frac {6}{49} \left (\frac {x \text {arctanh}(a x)}{7 \left (1-a^2 x^2\right )^{7/2}}+\frac {6}{7} \left (\frac {x \text {arctanh}(a x)}{5 \left (1-a^2 x^2\right )^{5/2}}+\frac {4}{5} \left (\frac {x \text {arctanh}(a x)}{3 \left (1-a^2 x^2\right )^{3/2}}+\frac {2}{3} \left (\frac {x \text {arctanh}(a x)}{\sqrt {1-a^2 x^2}}-\frac {1}{a \sqrt {1-a^2 x^2}}\right )-\frac {1}{9 a \left (1-a^2 x^2\right )^{3/2}}\right )-\frac {1}{25 a \left (1-a^2 x^2\right )^{5/2}}\right )-\frac {1}{49 a \left (1-a^2 x^2\right )^{7/2}}\right )\) |
| \(\Big \downarrow \) 6522 |
| \(\displaystyle \frac {6}{7} \left (\frac {6}{25} \left (\frac {4}{5} \int \frac {\text {arctanh}(a x)}{\left (1-a^2 x^2\right )^{5/2}}dx+\frac {x \text {arctanh}(a x)}{5 \left (1-a^2 x^2\right )^{5/2}}-\frac {1}{25 a \left (1-a^2 x^2\right )^{5/2}}\right )+\frac {4}{5} \int \frac {\text {arctanh}(a x)^3}{\left (1-a^2 x^2\right )^{5/2}}dx+\frac {x \text {arctanh}(a x)^3}{5 \left (1-a^2 x^2\right )^{5/2}}-\frac {3 \text {arctanh}(a x)^2}{25 a \left (1-a^2 x^2\right )^{5/2}}\right )+\frac {x \text {arctanh}(a x)^3}{7 \left (1-a^2 x^2\right )^{7/2}}-\frac {3 \text {arctanh}(a x)^2}{49 a \left (1-a^2 x^2\right )^{7/2}}+\frac {6}{49} \left (\frac {x \text {arctanh}(a x)}{7 \left (1-a^2 x^2\right )^{7/2}}+\frac {6}{7} \left (\frac {x \text {arctanh}(a x)}{5 \left (1-a^2 x^2\right )^{5/2}}+\frac {4}{5} \left (\frac {x \text {arctanh}(a x)}{3 \left (1-a^2 x^2\right )^{3/2}}+\frac {2}{3} \left (\frac {x \text {arctanh}(a x)}{\sqrt {1-a^2 x^2}}-\frac {1}{a \sqrt {1-a^2 x^2}}\right )-\frac {1}{9 a \left (1-a^2 x^2\right )^{3/2}}\right )-\frac {1}{25 a \left (1-a^2 x^2\right )^{5/2}}\right )-\frac {1}{49 a \left (1-a^2 x^2\right )^{7/2}}\right )\) |
| \(\Big \downarrow \) 6522 |
| \(\displaystyle \frac {6}{7} \left (\frac {6}{25} \left (\frac {4}{5} \left (\frac {2}{3} \int \frac {\text {arctanh}(a x)}{\left (1-a^2 x^2\right )^{3/2}}dx+\frac {x \text {arctanh}(a x)}{3 \left (1-a^2 x^2\right )^{3/2}}-\frac {1}{9 a \left (1-a^2 x^2\right )^{3/2}}\right )+\frac {x \text {arctanh}(a x)}{5 \left (1-a^2 x^2\right )^{5/2}}-\frac {1}{25 a \left (1-a^2 x^2\right )^{5/2}}\right )+\frac {4}{5} \int \frac {\text {arctanh}(a x)^3}{\left (1-a^2 x^2\right )^{5/2}}dx+\frac {x \text {arctanh}(a x)^3}{5 \left (1-a^2 x^2\right )^{5/2}}-\frac {3 \text {arctanh}(a x)^2}{25 a \left (1-a^2 x^2\right )^{5/2}}\right )+\frac {x \text {arctanh}(a x)^3}{7 \left (1-a^2 x^2\right )^{7/2}}-\frac {3 \text {arctanh}(a x)^2}{49 a \left (1-a^2 x^2\right )^{7/2}}+\frac {6}{49} \left (\frac {x \text {arctanh}(a x)}{7 \left (1-a^2 x^2\right )^{7/2}}+\frac {6}{7} \left (\frac {x \text {arctanh}(a x)}{5 \left (1-a^2 x^2\right )^{5/2}}+\frac {4}{5} \left (\frac {x \text {arctanh}(a x)}{3 \left (1-a^2 x^2\right )^{3/2}}+\frac {2}{3} \left (\frac {x \text {arctanh}(a x)}{\sqrt {1-a^2 x^2}}-\frac {1}{a \sqrt {1-a^2 x^2}}\right )-\frac {1}{9 a \left (1-a^2 x^2\right )^{3/2}}\right )-\frac {1}{25 a \left (1-a^2 x^2\right )^{5/2}}\right )-\frac {1}{49 a \left (1-a^2 x^2\right )^{7/2}}\right )\) |
| \(\Big \downarrow \) 6520 |
| \(\displaystyle \frac {6}{7} \left (\frac {4}{5} \int \frac {\text {arctanh}(a x)^3}{\left (1-a^2 x^2\right )^{5/2}}dx+\frac {x \text {arctanh}(a x)^3}{5 \left (1-a^2 x^2\right )^{5/2}}-\frac {3 \text {arctanh}(a x)^2}{25 a \left (1-a^2 x^2\right )^{5/2}}+\frac {6}{25} \left (\frac {x \text {arctanh}(a x)}{5 \left (1-a^2 x^2\right )^{5/2}}+\frac {4}{5} \left (\frac {x \text {arctanh}(a x)}{3 \left (1-a^2 x^2\right )^{3/2}}+\frac {2}{3} \left (\frac {x \text {arctanh}(a x)}{\sqrt {1-a^2 x^2}}-\frac {1}{a \sqrt {1-a^2 x^2}}\right )-\frac {1}{9 a \left (1-a^2 x^2\right )^{3/2}}\right )-\frac {1}{25 a \left (1-a^2 x^2\right )^{5/2}}\right )\right )+\frac {x \text {arctanh}(a x)^3}{7 \left (1-a^2 x^2\right )^{7/2}}-\frac {3 \text {arctanh}(a x)^2}{49 a \left (1-a^2 x^2\right )^{7/2}}+\frac {6}{49} \left (\frac {x \text {arctanh}(a x)}{7 \left (1-a^2 x^2\right )^{7/2}}+\frac {6}{7} \left (\frac {x \text {arctanh}(a x)}{5 \left (1-a^2 x^2\right )^{5/2}}+\frac {4}{5} \left (\frac {x \text {arctanh}(a x)}{3 \left (1-a^2 x^2\right )^{3/2}}+\frac {2}{3} \left (\frac {x \text {arctanh}(a x)}{\sqrt {1-a^2 x^2}}-\frac {1}{a \sqrt {1-a^2 x^2}}\right )-\frac {1}{9 a \left (1-a^2 x^2\right )^{3/2}}\right )-\frac {1}{25 a \left (1-a^2 x^2\right )^{5/2}}\right )-\frac {1}{49 a \left (1-a^2 x^2\right )^{7/2}}\right )\) |
| \(\Big \downarrow \) 6526 |
| \(\displaystyle \frac {6}{7} \left (\frac {4}{5} \left (\frac {2}{3} \int \frac {\text {arctanh}(a x)}{\left (1-a^2 x^2\right )^{5/2}}dx+\frac {2}{3} \int \frac {\text {arctanh}(a x)^3}{\left (1-a^2 x^2\right )^{3/2}}dx+\frac {x \text {arctanh}(a x)^3}{3 \left (1-a^2 x^2\right )^{3/2}}-\frac {\text {arctanh}(a x)^2}{3 a \left (1-a^2 x^2\right )^{3/2}}\right )+\frac {x \text {arctanh}(a x)^3}{5 \left (1-a^2 x^2\right )^{5/2}}-\frac {3 \text {arctanh}(a x)^2}{25 a \left (1-a^2 x^2\right )^{5/2}}+\frac {6}{25} \left (\frac {x \text {arctanh}(a x)}{5 \left (1-a^2 x^2\right )^{5/2}}+\frac {4}{5} \left (\frac {x \text {arctanh}(a x)}{3 \left (1-a^2 x^2\right )^{3/2}}+\frac {2}{3} \left (\frac {x \text {arctanh}(a x)}{\sqrt {1-a^2 x^2}}-\frac {1}{a \sqrt {1-a^2 x^2}}\right )-\frac {1}{9 a \left (1-a^2 x^2\right )^{3/2}}\right )-\frac {1}{25 a \left (1-a^2 x^2\right )^{5/2}}\right )\right )+\frac {x \text {arctanh}(a x)^3}{7 \left (1-a^2 x^2\right )^{7/2}}-\frac {3 \text {arctanh}(a x)^2}{49 a \left (1-a^2 x^2\right )^{7/2}}+\frac {6}{49} \left (\frac {x \text {arctanh}(a x)}{7 \left (1-a^2 x^2\right )^{7/2}}+\frac {6}{7} \left (\frac {x \text {arctanh}(a x)}{5 \left (1-a^2 x^2\right )^{5/2}}+\frac {4}{5} \left (\frac {x \text {arctanh}(a x)}{3 \left (1-a^2 x^2\right )^{3/2}}+\frac {2}{3} \left (\frac {x \text {arctanh}(a x)}{\sqrt {1-a^2 x^2}}-\frac {1}{a \sqrt {1-a^2 x^2}}\right )-\frac {1}{9 a \left (1-a^2 x^2\right )^{3/2}}\right )-\frac {1}{25 a \left (1-a^2 x^2\right )^{5/2}}\right )-\frac {1}{49 a \left (1-a^2 x^2\right )^{7/2}}\right )\) |
| \(\Big \downarrow \) 6522 |
| \(\displaystyle \frac {6}{7} \left (\frac {4}{5} \left (\frac {2}{3} \left (\frac {2}{3} \int \frac {\text {arctanh}(a x)}{\left (1-a^2 x^2\right )^{3/2}}dx+\frac {x \text {arctanh}(a x)}{3 \left (1-a^2 x^2\right )^{3/2}}-\frac {1}{9 a \left (1-a^2 x^2\right )^{3/2}}\right )+\frac {2}{3} \int \frac {\text {arctanh}(a x)^3}{\left (1-a^2 x^2\right )^{3/2}}dx+\frac {x \text {arctanh}(a x)^3}{3 \left (1-a^2 x^2\right )^{3/2}}-\frac {\text {arctanh}(a x)^2}{3 a \left (1-a^2 x^2\right )^{3/2}}\right )+\frac {x \text {arctanh}(a x)^3}{5 \left (1-a^2 x^2\right )^{5/2}}-\frac {3 \text {arctanh}(a x)^2}{25 a \left (1-a^2 x^2\right )^{5/2}}+\frac {6}{25} \left (\frac {x \text {arctanh}(a x)}{5 \left (1-a^2 x^2\right )^{5/2}}+\frac {4}{5} \left (\frac {x \text {arctanh}(a x)}{3 \left (1-a^2 x^2\right )^{3/2}}+\frac {2}{3} \left (\frac {x \text {arctanh}(a x)}{\sqrt {1-a^2 x^2}}-\frac {1}{a \sqrt {1-a^2 x^2}}\right )-\frac {1}{9 a \left (1-a^2 x^2\right )^{3/2}}\right )-\frac {1}{25 a \left (1-a^2 x^2\right )^{5/2}}\right )\right )+\frac {x \text {arctanh}(a x)^3}{7 \left (1-a^2 x^2\right )^{7/2}}-\frac {3 \text {arctanh}(a x)^2}{49 a \left (1-a^2 x^2\right )^{7/2}}+\frac {6}{49} \left (\frac {x \text {arctanh}(a x)}{7 \left (1-a^2 x^2\right )^{7/2}}+\frac {6}{7} \left (\frac {x \text {arctanh}(a x)}{5 \left (1-a^2 x^2\right )^{5/2}}+\frac {4}{5} \left (\frac {x \text {arctanh}(a x)}{3 \left (1-a^2 x^2\right )^{3/2}}+\frac {2}{3} \left (\frac {x \text {arctanh}(a x)}{\sqrt {1-a^2 x^2}}-\frac {1}{a \sqrt {1-a^2 x^2}}\right )-\frac {1}{9 a \left (1-a^2 x^2\right )^{3/2}}\right )-\frac {1}{25 a \left (1-a^2 x^2\right )^{5/2}}\right )-\frac {1}{49 a \left (1-a^2 x^2\right )^{7/2}}\right )\) |
| \(\Big \downarrow \) 6520 |
| \(\displaystyle \frac {6}{7} \left (\frac {4}{5} \left (\frac {2}{3} \int \frac {\text {arctanh}(a x)^3}{\left (1-a^2 x^2\right )^{3/2}}dx+\frac {x \text {arctanh}(a x)^3}{3 \left (1-a^2 x^2\right )^{3/2}}-\frac {\text {arctanh}(a x)^2}{3 a \left (1-a^2 x^2\right )^{3/2}}+\frac {2}{3} \left (\frac {x \text {arctanh}(a x)}{3 \left (1-a^2 x^2\right )^{3/2}}+\frac {2}{3} \left (\frac {x \text {arctanh}(a x)}{\sqrt {1-a^2 x^2}}-\frac {1}{a \sqrt {1-a^2 x^2}}\right )-\frac {1}{9 a \left (1-a^2 x^2\right )^{3/2}}\right )\right )+\frac {x \text {arctanh}(a x)^3}{5 \left (1-a^2 x^2\right )^{5/2}}-\frac {3 \text {arctanh}(a x)^2}{25 a \left (1-a^2 x^2\right )^{5/2}}+\frac {6}{25} \left (\frac {x \text {arctanh}(a x)}{5 \left (1-a^2 x^2\right )^{5/2}}+\frac {4}{5} \left (\frac {x \text {arctanh}(a x)}{3 \left (1-a^2 x^2\right )^{3/2}}+\frac {2}{3} \left (\frac {x \text {arctanh}(a x)}{\sqrt {1-a^2 x^2}}-\frac {1}{a \sqrt {1-a^2 x^2}}\right )-\frac {1}{9 a \left (1-a^2 x^2\right )^{3/2}}\right )-\frac {1}{25 a \left (1-a^2 x^2\right )^{5/2}}\right )\right )+\frac {x \text {arctanh}(a x)^3}{7 \left (1-a^2 x^2\right )^{7/2}}-\frac {3 \text {arctanh}(a x)^2}{49 a \left (1-a^2 x^2\right )^{7/2}}+\frac {6}{49} \left (\frac {x \text {arctanh}(a x)}{7 \left (1-a^2 x^2\right )^{7/2}}+\frac {6}{7} \left (\frac {x \text {arctanh}(a x)}{5 \left (1-a^2 x^2\right )^{5/2}}+\frac {4}{5} \left (\frac {x \text {arctanh}(a x)}{3 \left (1-a^2 x^2\right )^{3/2}}+\frac {2}{3} \left (\frac {x \text {arctanh}(a x)}{\sqrt {1-a^2 x^2}}-\frac {1}{a \sqrt {1-a^2 x^2}}\right )-\frac {1}{9 a \left (1-a^2 x^2\right )^{3/2}}\right )-\frac {1}{25 a \left (1-a^2 x^2\right )^{5/2}}\right )-\frac {1}{49 a \left (1-a^2 x^2\right )^{7/2}}\right )\) |
| \(\Big \downarrow \) 6524 |
| \(\displaystyle \frac {6}{7} \left (\frac {4}{5} \left (\frac {2}{3} \left (6 \int \frac {\text {arctanh}(a x)}{\left (1-a^2 x^2\right )^{3/2}}dx+\frac {x \text {arctanh}(a x)^3}{\sqrt {1-a^2 x^2}}-\frac {3 \text {arctanh}(a x)^2}{a \sqrt {1-a^2 x^2}}\right )+\frac {x \text {arctanh}(a x)^3}{3 \left (1-a^2 x^2\right )^{3/2}}-\frac {\text {arctanh}(a x)^2}{3 a \left (1-a^2 x^2\right )^{3/2}}+\frac {2}{3} \left (\frac {x \text {arctanh}(a x)}{3 \left (1-a^2 x^2\right )^{3/2}}+\frac {2}{3} \left (\frac {x \text {arctanh}(a x)}{\sqrt {1-a^2 x^2}}-\frac {1}{a \sqrt {1-a^2 x^2}}\right )-\frac {1}{9 a \left (1-a^2 x^2\right )^{3/2}}\right )\right )+\frac {x \text {arctanh}(a x)^3}{5 \left (1-a^2 x^2\right )^{5/2}}-\frac {3 \text {arctanh}(a x)^2}{25 a \left (1-a^2 x^2\right )^{5/2}}+\frac {6}{25} \left (\frac {x \text {arctanh}(a x)}{5 \left (1-a^2 x^2\right )^{5/2}}+\frac {4}{5} \left (\frac {x \text {arctanh}(a x)}{3 \left (1-a^2 x^2\right )^{3/2}}+\frac {2}{3} \left (\frac {x \text {arctanh}(a x)}{\sqrt {1-a^2 x^2}}-\frac {1}{a \sqrt {1-a^2 x^2}}\right )-\frac {1}{9 a \left (1-a^2 x^2\right )^{3/2}}\right )-\frac {1}{25 a \left (1-a^2 x^2\right )^{5/2}}\right )\right )+\frac {x \text {arctanh}(a x)^3}{7 \left (1-a^2 x^2\right )^{7/2}}-\frac {3 \text {arctanh}(a x)^2}{49 a \left (1-a^2 x^2\right )^{7/2}}+\frac {6}{49} \left (\frac {x \text {arctanh}(a x)}{7 \left (1-a^2 x^2\right )^{7/2}}+\frac {6}{7} \left (\frac {x \text {arctanh}(a x)}{5 \left (1-a^2 x^2\right )^{5/2}}+\frac {4}{5} \left (\frac {x \text {arctanh}(a x)}{3 \left (1-a^2 x^2\right )^{3/2}}+\frac {2}{3} \left (\frac {x \text {arctanh}(a x)}{\sqrt {1-a^2 x^2}}-\frac {1}{a \sqrt {1-a^2 x^2}}\right )-\frac {1}{9 a \left (1-a^2 x^2\right )^{3/2}}\right )-\frac {1}{25 a \left (1-a^2 x^2\right )^{5/2}}\right )-\frac {1}{49 a \left (1-a^2 x^2\right )^{7/2}}\right )\) |
| \(\Big \downarrow \) 6520 |
| \(\displaystyle \frac {x \text {arctanh}(a x)^3}{7 \left (1-a^2 x^2\right )^{7/2}}-\frac {3 \text {arctanh}(a x)^2}{49 a \left (1-a^2 x^2\right )^{7/2}}+\frac {6}{49} \left (\frac {x \text {arctanh}(a x)}{7 \left (1-a^2 x^2\right )^{7/2}}+\frac {6}{7} \left (\frac {x \text {arctanh}(a x)}{5 \left (1-a^2 x^2\right )^{5/2}}+\frac {4}{5} \left (\frac {x \text {arctanh}(a x)}{3 \left (1-a^2 x^2\right )^{3/2}}+\frac {2}{3} \left (\frac {x \text {arctanh}(a x)}{\sqrt {1-a^2 x^2}}-\frac {1}{a \sqrt {1-a^2 x^2}}\right )-\frac {1}{9 a \left (1-a^2 x^2\right )^{3/2}}\right )-\frac {1}{25 a \left (1-a^2 x^2\right )^{5/2}}\right )-\frac {1}{49 a \left (1-a^2 x^2\right )^{7/2}}\right )+\frac {6}{7} \left (\frac {x \text {arctanh}(a x)^3}{5 \left (1-a^2 x^2\right )^{5/2}}-\frac {3 \text {arctanh}(a x)^2}{25 a \left (1-a^2 x^2\right )^{5/2}}+\frac {6}{25} \left (\frac {x \text {arctanh}(a x)}{5 \left (1-a^2 x^2\right )^{5/2}}+\frac {4}{5} \left (\frac {x \text {arctanh}(a x)}{3 \left (1-a^2 x^2\right )^{3/2}}+\frac {2}{3} \left (\frac {x \text {arctanh}(a x)}{\sqrt {1-a^2 x^2}}-\frac {1}{a \sqrt {1-a^2 x^2}}\right )-\frac {1}{9 a \left (1-a^2 x^2\right )^{3/2}}\right )-\frac {1}{25 a \left (1-a^2 x^2\right )^{5/2}}\right )+\frac {4}{5} \left (\frac {x \text {arctanh}(a x)^3}{3 \left (1-a^2 x^2\right )^{3/2}}-\frac {\text {arctanh}(a x)^2}{3 a \left (1-a^2 x^2\right )^{3/2}}+\frac {2}{3} \left (\frac {x \text {arctanh}(a x)}{3 \left (1-a^2 x^2\right )^{3/2}}+\frac {2}{3} \left (\frac {x \text {arctanh}(a x)}{\sqrt {1-a^2 x^2}}-\frac {1}{a \sqrt {1-a^2 x^2}}\right )-\frac {1}{9 a \left (1-a^2 x^2\right )^{3/2}}\right )+\frac {2}{3} \left (\frac {x \text {arctanh}(a x)^3}{\sqrt {1-a^2 x^2}}-\frac {3 \text {arctanh}(a x)^2}{a \sqrt {1-a^2 x^2}}+6 \left (\frac {x \text {arctanh}(a x)}{\sqrt {1-a^2 x^2}}-\frac {1}{a \sqrt {1-a^2 x^2}}\right )\right )\right )\right )\) |
Input:
Int[ArcTanh[a*x]^3/(1 - a^2*x^2)^(9/2),x]
Output:
(-3*ArcTanh[a*x]^2)/(49*a*(1 - a^2*x^2)^(7/2)) + (x*ArcTanh[a*x]^3)/(7*(1 - a^2*x^2)^(7/2)) + (6*(-1/49*1/(a*(1 - a^2*x^2)^(7/2)) + (x*ArcTanh[a*x]) /(7*(1 - a^2*x^2)^(7/2)) + (6*(-1/25*1/(a*(1 - a^2*x^2)^(5/2)) + (x*ArcTan h[a*x])/(5*(1 - a^2*x^2)^(5/2)) + (4*(-1/9*1/(a*(1 - a^2*x^2)^(3/2)) + (x* ArcTanh[a*x])/(3*(1 - a^2*x^2)^(3/2)) + (2*(-(1/(a*Sqrt[1 - a^2*x^2])) + ( x*ArcTanh[a*x])/Sqrt[1 - a^2*x^2]))/3))/5))/7))/49 + (6*((-3*ArcTanh[a*x]^ 2)/(25*a*(1 - a^2*x^2)^(5/2)) + (x*ArcTanh[a*x]^3)/(5*(1 - a^2*x^2)^(5/2)) + (6*(-1/25*1/(a*(1 - a^2*x^2)^(5/2)) + (x*ArcTanh[a*x])/(5*(1 - a^2*x^2) ^(5/2)) + (4*(-1/9*1/(a*(1 - a^2*x^2)^(3/2)) + (x*ArcTanh[a*x])/(3*(1 - a^ 2*x^2)^(3/2)) + (2*(-(1/(a*Sqrt[1 - a^2*x^2])) + (x*ArcTanh[a*x])/Sqrt[1 - a^2*x^2]))/3))/5))/25 + (4*(-1/3*ArcTanh[a*x]^2/(a*(1 - a^2*x^2)^(3/2)) + (x*ArcTanh[a*x]^3)/(3*(1 - a^2*x^2)^(3/2)) + (2*(-1/9*1/(a*(1 - a^2*x^2)^ (3/2)) + (x*ArcTanh[a*x])/(3*(1 - a^2*x^2)^(3/2)) + (2*(-(1/(a*Sqrt[1 - a^ 2*x^2])) + (x*ArcTanh[a*x])/Sqrt[1 - a^2*x^2]))/3))/3 + (2*((-3*ArcTanh[a* x]^2)/(a*Sqrt[1 - a^2*x^2]) + (x*ArcTanh[a*x]^3)/Sqrt[1 - a^2*x^2] + 6*(-( 1/(a*Sqrt[1 - a^2*x^2])) + (x*ArcTanh[a*x])/Sqrt[1 - a^2*x^2])))/3))/5))/7
Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))/((d_) + (e_.)*(x_)^2)^(3/2), x_Symb ol] :> Simp[-b/(c*d*Sqrt[d + e*x^2]), x] + Simp[x*((a + b*ArcTanh[c*x])/(d* Sqrt[d + e*x^2])), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0]
Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))*((d_) + (e_.)*(x_)^2)^(q_), x_Symbo l] :> Simp[(-b)*((d + e*x^2)^(q + 1)/(4*c*d*(q + 1)^2)), x] + (-Simp[x*(d + e*x^2)^(q + 1)*((a + b*ArcTanh[c*x])/(2*d*(q + 1))), x] + Simp[(2*q + 3)/( 2*d*(q + 1)) Int[(d + e*x^2)^(q + 1)*(a + b*ArcTanh[c*x]), x], x]) /; Fre eQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && LtQ[q, -1] && NeQ[q, -3/2]
Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_)/((d_) + (e_.)*(x_)^2)^(3/2), x _Symbol] :> Simp[(-b)*p*((a + b*ArcTanh[c*x])^(p - 1)/(c*d*Sqrt[d + e*x^2]) ), x] + (Simp[x*((a + b*ArcTanh[c*x])^p/(d*Sqrt[d + e*x^2])), x] + Simp[b^2 *p*(p - 1) Int[(a + b*ArcTanh[c*x])^(p - 2)/(d + e*x^2)^(3/2), x], x]) /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && GtQ[p, 1]
Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_)*((d_) + (e_.)*(x_)^2)^(q_), x_ Symbol] :> Simp[(-b)*p*(d + e*x^2)^(q + 1)*((a + b*ArcTanh[c*x])^(p - 1)/(4 *c*d*(q + 1)^2)), x] + (-Simp[x*(d + e*x^2)^(q + 1)*((a + b*ArcTanh[c*x])^p /(2*d*(q + 1))), x] + Simp[(2*q + 3)/(2*d*(q + 1)) Int[(d + e*x^2)^(q + 1 )*(a + b*ArcTanh[c*x])^p, x], x] + Simp[b^2*p*((p - 1)/(4*(q + 1)^2)) Int [(d + e*x^2)^q*(a + b*ArcTanh[c*x])^(p - 2), x], x]) /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && LtQ[q, -1] && GtQ[p, 1] && NeQ[q, -3/2]
Time = 0.55 (sec) , antiderivative size = 201, normalized size of antiderivative = 0.52
| method | result | size |
| default | \(-\frac {\sqrt {-a^{2} x^{2}+1}\, \left (6174000 \operatorname {arctanh}\left (a x \right )^{3} a^{7} x^{7}+43397760 \,\operatorname {arctanh}\left (a x \right ) a^{7} x^{7}-18522000 \operatorname {arctanh}\left (a x \right )^{2} a^{6} x^{6}-21609000 \operatorname {arctanh}\left (a x \right )^{3} a^{5} x^{5}-43397760 a^{6} x^{6}-133370160 \,\operatorname {arctanh}\left (a x \right ) a^{5} x^{5}+58653000 a^{4} x^{4} \operatorname {arctanh}\left (a x \right )^{2}+27011250 \operatorname {arctanh}\left (a x \right )^{3} a^{3} x^{3}+131252240 a^{4} x^{4}+137386200 a^{3} x^{3} \operatorname {arctanh}\left (a x \right )-63129150 a^{2} x^{2} \operatorname {arctanh}\left (a x \right )^{2}-13505625 \operatorname {arctanh}\left (a x \right )^{3} a x -132479032 a^{2} x^{2}-47650050 a x \,\operatorname {arctanh}\left (a x \right )+23825025 \operatorname {arctanh}\left (a x \right )^{2}+44658302\right )}{13505625 a \left (a^{2} x^{2}-1\right )^{4}}\) | \(201\) |
| orering | \(\frac {\left (\frac {4959744}{8575} a^{10} x^{11}-\frac {12708544}{5145} a^{8} x^{9}+\frac {525524992}{128625} a^{6} x^{7}-\frac {4829742448}{1500625} a^{4} x^{5}+\frac {5255006528}{4501875} a^{2} x^{3}-\frac {643043504}{4501875} x \right ) \operatorname {arctanh}\left (a x \right )^{3}}{\left (-a^{2} x^{2}+1\right )^{\frac {9}{2}}}+\frac {2 \left (a x +1\right )^{2} \left (a x -1\right )^{2} \left (3797304000 a^{8} x^{8}-11800533360 a^{6} x^{6}+12538703380 a^{4} x^{4}-4851468572 a^{2} x^{2}+311860177\right ) \left (\frac {3 \operatorname {arctanh}\left (a x \right )^{2} a}{\left (-a^{2} x^{2}+1\right )^{\frac {11}{2}}}+\frac {9 \operatorname {arctanh}\left (a x \right )^{3} a^{2} x}{\left (-a^{2} x^{2}+1\right )^{\frac {11}{2}}}\right )}{40516875 a^{2}}+\frac {8 x \left (a x +1\right )^{3} \left (a x -1\right )^{3} \left (30740080 a^{6} x^{6}-93014460 a^{4} x^{4}+93941547 a^{2} x^{2}-31695292\right ) \left (\frac {6 \,\operatorname {arctanh}\left (a x \right ) a^{2}}{\left (-a^{2} x^{2}+1\right )^{\frac {13}{2}}}+\frac {60 \operatorname {arctanh}\left (a x \right )^{2} a^{3} x}{\left (-a^{2} x^{2}+1\right )^{\frac {13}{2}}}+\frac {99 \operatorname {arctanh}\left (a x \right )^{3} a^{4} x^{2}}{\left (-a^{2} x^{2}+1\right )^{\frac {13}{2}}}+\frac {9 \operatorname {arctanh}\left (a x \right )^{3} a^{2}}{\left (-a^{2} x^{2}+1\right )^{\frac {11}{2}}}\right )}{13505625 a^{2}}+\frac {\left (21698880 a^{6} x^{6}-65626120 a^{4} x^{4}+66239516 a^{2} x^{2}-22329151\right ) \left (a x -1\right )^{4} \left (a x +1\right )^{4} \left (\frac {6 a^{3}}{\left (-a^{2} x^{2}+1\right )^{\frac {15}{2}}}+\frac {198 \,\operatorname {arctanh}\left (a x \right ) a^{4} x}{\left (-a^{2} x^{2}+1\right )^{\frac {15}{2}}}+\frac {1077 \operatorname {arctanh}\left (a x \right )^{2} a^{5} x^{2}}{\left (-a^{2} x^{2}+1\right )^{\frac {15}{2}}}+\frac {87 \operatorname {arctanh}\left (a x \right )^{2} a^{3}}{\left (-a^{2} x^{2}+1\right )^{\frac {13}{2}}}+\frac {1287 \operatorname {arctanh}\left (a x \right )^{3} a^{6} x^{3}}{\left (-a^{2} x^{2}+1\right )^{\frac {15}{2}}}+\frac {297 \operatorname {arctanh}\left (a x \right )^{3} a^{4} x}{\left (-a^{2} x^{2}+1\right )^{\frac {13}{2}}}\right )}{40516875 a^{4}}\) | \(489\) |
Input:
int(arctanh(a*x)^3/(-a^2*x^2+1)^(9/2),x,method=_RETURNVERBOSE)
Output:
-1/13505625/a*(-a^2*x^2+1)^(1/2)*(6174000*arctanh(a*x)^3*a^7*x^7+43397760* arctanh(a*x)*a^7*x^7-18522000*arctanh(a*x)^2*a^6*x^6-21609000*arctanh(a*x) ^3*a^5*x^5-43397760*a^6*x^6-133370160*arctanh(a*x)*a^5*x^5+58653000*a^4*x^ 4*arctanh(a*x)^2+27011250*arctanh(a*x)^3*a^3*x^3+131252240*a^4*x^4+1373862 00*a^3*x^3*arctanh(a*x)-63129150*a^2*x^2*arctanh(a*x)^2-13505625*arctanh(a *x)^3*a*x-132479032*a^2*x^2-47650050*a*x*arctanh(a*x)+23825025*arctanh(a*x )^2+44658302)/(a^2*x^2-1)^4
Time = 0.10 (sec) , antiderivative size = 214, normalized size of antiderivative = 0.56 \[ \int \frac {\text {arctanh}(a x)^3}{\left (1-a^2 x^2\right )^{9/2}} \, dx=\frac {{\left (347182080 \, a^{6} x^{6} - 1050017920 \, a^{4} x^{4} + 1059832256 \, a^{2} x^{2} - 385875 \, {\left (16 \, a^{7} x^{7} - 56 \, a^{5} x^{5} + 70 \, a^{3} x^{3} - 35 \, a x\right )} \log \left (-\frac {a x + 1}{a x - 1}\right )^{3} + 22050 \, {\left (1680 \, a^{6} x^{6} - 5320 \, a^{4} x^{4} + 5726 \, a^{2} x^{2} - 2161\right )} \log \left (-\frac {a x + 1}{a x - 1}\right )^{2} - 840 \, {\left (206656 \, a^{7} x^{7} - 635096 \, a^{5} x^{5} + 654220 \, a^{3} x^{3} - 226905 \, a x\right )} \log \left (-\frac {a x + 1}{a x - 1}\right ) - 357266416\right )} \sqrt {-a^{2} x^{2} + 1}}{108045000 \, {\left (a^{9} x^{8} - 4 \, a^{7} x^{6} + 6 \, a^{5} x^{4} - 4 \, a^{3} x^{2} + a\right )}} \] Input:
integrate(arctanh(a*x)^3/(-a^2*x^2+1)^(9/2),x, algorithm="fricas")
Output:
1/108045000*(347182080*a^6*x^6 - 1050017920*a^4*x^4 + 1059832256*a^2*x^2 - 385875*(16*a^7*x^7 - 56*a^5*x^5 + 70*a^3*x^3 - 35*a*x)*log(-(a*x + 1)/(a* x - 1))^3 + 22050*(1680*a^6*x^6 - 5320*a^4*x^4 + 5726*a^2*x^2 - 2161)*log( -(a*x + 1)/(a*x - 1))^2 - 840*(206656*a^7*x^7 - 635096*a^5*x^5 + 654220*a^ 3*x^3 - 226905*a*x)*log(-(a*x + 1)/(a*x - 1)) - 357266416)*sqrt(-a^2*x^2 + 1)/(a^9*x^8 - 4*a^7*x^6 + 6*a^5*x^4 - 4*a^3*x^2 + a)
\[ \int \frac {\text {arctanh}(a x)^3}{\left (1-a^2 x^2\right )^{9/2}} \, dx=\int \frac {\operatorname {atanh}^{3}{\left (a x \right )}}{\left (- \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac {9}{2}}}\, dx \] Input:
integrate(atanh(a*x)**3/(-a**2*x**2+1)**(9/2),x)
Output:
Integral(atanh(a*x)**3/(-(a*x - 1)*(a*x + 1))**(9/2), x)
\[ \int \frac {\text {arctanh}(a x)^3}{\left (1-a^2 x^2\right )^{9/2}} \, dx=\int { \frac {\operatorname {artanh}\left (a x\right )^{3}}{{\left (-a^{2} x^{2} + 1\right )}^{\frac {9}{2}}} \,d x } \] Input:
integrate(arctanh(a*x)^3/(-a^2*x^2+1)^(9/2),x, algorithm="maxima")
Output:
integrate(arctanh(a*x)^3/(-a^2*x^2 + 1)^(9/2), x)
\[ \int \frac {\text {arctanh}(a x)^3}{\left (1-a^2 x^2\right )^{9/2}} \, dx=\int { \frac {\operatorname {artanh}\left (a x\right )^{3}}{{\left (-a^{2} x^{2} + 1\right )}^{\frac {9}{2}}} \,d x } \] Input:
integrate(arctanh(a*x)^3/(-a^2*x^2+1)^(9/2),x, algorithm="giac")
Output:
integrate(arctanh(a*x)^3/(-a^2*x^2 + 1)^(9/2), x)
Timed out. \[ \int \frac {\text {arctanh}(a x)^3}{\left (1-a^2 x^2\right )^{9/2}} \, dx=\int \frac {{\mathrm {atanh}\left (a\,x\right )}^3}{{\left (1-a^2\,x^2\right )}^{9/2}} \,d x \] Input:
int(atanh(a*x)^3/(1 - a^2*x^2)^(9/2),x)
Output:
int(atanh(a*x)^3/(1 - a^2*x^2)^(9/2), x)
\[ \int \frac {\text {arctanh}(a x)^3}{\left (1-a^2 x^2\right )^{9/2}} \, dx=\int \frac {\mathit {atanh} \left (a x \right )^{3}}{\sqrt {-a^{2} x^{2}+1}\, a^{8} x^{8}-4 \sqrt {-a^{2} x^{2}+1}\, a^{6} x^{6}+6 \sqrt {-a^{2} x^{2}+1}\, a^{4} x^{4}-4 \sqrt {-a^{2} x^{2}+1}\, a^{2} x^{2}+\sqrt {-a^{2} x^{2}+1}}d x \] Input:
int(atanh(a*x)^3/(-a^2*x^2+1)^(9/2),x)
Output:
int(atanh(a*x)**3/(sqrt( - a**2*x**2 + 1)*a**8*x**8 - 4*sqrt( - a**2*x**2 + 1)*a**6*x**6 + 6*sqrt( - a**2*x**2 + 1)*a**4*x**4 - 4*sqrt( - a**2*x**2 + 1)*a**2*x**2 + sqrt( - a**2*x**2 + 1)),x)