Integrand size = 20, antiderivative size = 370 \[ \int \left (c-a^2 c x^2\right )^3 \arccos (a x)^3 \, dx=-\frac {413312 c^3 \sqrt {1-a^2 x^2}}{128625 a}-\frac {30256 c^3 \left (1-a^2 x^2\right )^{3/2}}{385875 a}-\frac {2664 c^3 \left (1-a^2 x^2\right )^{5/2}}{214375 a}-\frac {6 c^3 \left (1-a^2 x^2\right )^{7/2}}{2401 a}-\frac {4322 c^3 x \arccos (a x)}{1225}+\frac {1514 a^2 c^3 x^3 \arccos (a x)}{3675}-\frac {702 a^4 c^3 x^5 \arccos (a x)}{6125}+\frac {6}{343} a^6 c^3 x^7 \arccos (a x)+\frac {48 c^3 \sqrt {1-a^2 x^2} \arccos (a x)^2}{35 a}+\frac {8 c^3 \left (1-a^2 x^2\right )^{3/2} \arccos (a x)^2}{35 a}+\frac {18 c^3 \left (1-a^2 x^2\right )^{5/2} \arccos (a x)^2}{175 a}+\frac {3 c^3 \left (1-a^2 x^2\right )^{7/2} \arccos (a x)^2}{49 a}+\frac {16}{35} c^3 x \arccos (a x)^3+\frac {8}{35} c^3 x \left (1-a^2 x^2\right ) \arccos (a x)^3+\frac {6}{35} c^3 x \left (1-a^2 x^2\right )^2 \arccos (a x)^3+\frac {1}{7} c^3 x \left (1-a^2 x^2\right )^3 \arccos (a x)^3 \] Output:
-413312/128625*c^3*(-a^2*x^2+1)^(1/2)/a-30256/385875*c^3*(-a^2*x^2+1)^(3/2 )/a-2664/214375*c^3*(-a^2*x^2+1)^(5/2)/a-6/2401*c^3*(-a^2*x^2+1)^(7/2)/a-4 322/1225*c^3*x*arccos(a*x)+1514/3675*a^2*c^3*x^3*arccos(a*x)-702/6125*a^4* c^3*x^5*arccos(a*x)+6/343*a^6*c^3*x^7*arccos(a*x)+48/35*c^3*(-a^2*x^2+1)^( 1/2)*arccos(a*x)^2/a+8/35*c^3*(-a^2*x^2+1)^(3/2)*arccos(a*x)^2/a+18/175*c^ 3*(-a^2*x^2+1)^(5/2)*arccos(a*x)^2/a+3/49*c^3*(-a^2*x^2+1)^(7/2)*arccos(a* x)^2/a+16/35*c^3*x*arccos(a*x)^3+8/35*c^3*x*(-a^2*x^2+1)*arccos(a*x)^3+6/3 5*c^3*x*(-a^2*x^2+1)^2*arccos(a*x)^3+1/7*c^3*x*(-a^2*x^2+1)^3*arccos(a*x)^ 3
Time = 0.35 (sec) , antiderivative size = 171, normalized size of antiderivative = 0.46 \[ \int \left (c-a^2 c x^2\right )^3 \arccos (a x)^3 \, dx=\frac {c^3 \left (2 \sqrt {1-a^2 x^2} \left (22329151-747937 a^2 x^2+134541 a^4 x^4-16875 a^6 x^6\right )+210 a x \left (-226905+26495 a^2 x^2-7371 a^4 x^4+1125 a^6 x^6\right ) \arccos (a x)+11025 \sqrt {1-a^2 x^2} \left (-2161+757 a^2 x^2-351 a^4 x^4+75 a^6 x^6\right ) \arccos (a x)^2-385875 a x \left (-35+35 a^2 x^2-21 a^4 x^4+5 a^6 x^6\right ) \arccos (a x)^3\right )}{13505625 a} \] Input:
Integrate[(c - a^2*c*x^2)^3*ArcCos[a*x]^3,x]
Output:
(c^3*(2*Sqrt[1 - a^2*x^2]*(22329151 - 747937*a^2*x^2 + 134541*a^4*x^4 - 16 875*a^6*x^6) + 210*a*x*(-226905 + 26495*a^2*x^2 - 7371*a^4*x^4 + 1125*a^6* x^6)*ArcCos[a*x] + 11025*Sqrt[1 - a^2*x^2]*(-2161 + 757*a^2*x^2 - 351*a^4* x^4 + 75*a^6*x^6)*ArcCos[a*x]^2 - 385875*a*x*(-35 + 35*a^2*x^2 - 21*a^4*x^ 4 + 5*a^6*x^6)*ArcCos[a*x]^3))/(13505625*a)
Time = 2.54 (sec) , antiderivative size = 581, normalized size of antiderivative = 1.57, number of steps used = 19, number of rules used = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.900, Rules used = {5159, 27, 5159, 5159, 5131, 5183, 5131, 241, 5155, 27, 353, 53, 1576, 1140, 2009, 2331, 2389, 2009}
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.
| \(\displaystyle \int \arccos (a x)^3 \left (c-a^2 c x^2\right )^3 \, dx\) |
| \(\Big \downarrow \) 5159 |
| \(\displaystyle \frac {3}{7} a c^3 \int x \left (1-a^2 x^2\right )^{5/2} \arccos (a x)^2dx+\frac {6}{7} c \int c^2 \left (1-a^2 x^2\right )^2 \arccos (a x)^3dx+\frac {1}{7} c^3 x \left (1-a^2 x^2\right )^3 \arccos (a x)^3\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {3}{7} a c^3 \int x \left (1-a^2 x^2\right )^{5/2} \arccos (a x)^2dx+\frac {6}{7} c^3 \int \left (1-a^2 x^2\right )^2 \arccos (a x)^3dx+\frac {1}{7} c^3 x \left (1-a^2 x^2\right )^3 \arccos (a x)^3\) |
| \(\Big \downarrow \) 5159 |
| \(\displaystyle \frac {3}{7} a c^3 \int x \left (1-a^2 x^2\right )^{5/2} \arccos (a x)^2dx+\frac {6}{7} c^3 \left (\frac {3}{5} a \int x \left (1-a^2 x^2\right )^{3/2} \arccos (a x)^2dx+\frac {4}{5} \int \left (1-a^2 x^2\right ) \arccos (a x)^3dx+\frac {1}{5} x \left (1-a^2 x^2\right )^2 \arccos (a x)^3\right )+\frac {1}{7} c^3 x \left (1-a^2 x^2\right )^3 \arccos (a x)^3\) |
| \(\Big \downarrow \) 5159 |
| \(\displaystyle \frac {3}{7} a c^3 \int x \left (1-a^2 x^2\right )^{5/2} \arccos (a x)^2dx+\frac {6}{7} c^3 \left (\frac {3}{5} a \int x \left (1-a^2 x^2\right )^{3/2} \arccos (a x)^2dx+\frac {4}{5} \left (a \int x \sqrt {1-a^2 x^2} \arccos (a x)^2dx+\frac {2}{3} \int \arccos (a x)^3dx+\frac {1}{3} x \left (1-a^2 x^2\right ) \arccos (a x)^3\right )+\frac {1}{5} x \left (1-a^2 x^2\right )^2 \arccos (a x)^3\right )+\frac {1}{7} c^3 x \left (1-a^2 x^2\right )^3 \arccos (a x)^3\) |
| \(\Big \downarrow \) 5131 |
| \(\displaystyle \frac {6}{7} c^3 \left (\frac {4}{5} \left (\frac {2}{3} \left (3 a \int \frac {x \arccos (a x)^2}{\sqrt {1-a^2 x^2}}dx+x \arccos (a x)^3\right )+a \int x \sqrt {1-a^2 x^2} \arccos (a x)^2dx+\frac {1}{3} x \left (1-a^2 x^2\right ) \arccos (a x)^3\right )+\frac {3}{5} a \int x \left (1-a^2 x^2\right )^{3/2} \arccos (a x)^2dx+\frac {1}{5} x \left (1-a^2 x^2\right )^2 \arccos (a x)^3\right )+\frac {3}{7} a c^3 \int x \left (1-a^2 x^2\right )^{5/2} \arccos (a x)^2dx+\frac {1}{7} c^3 x \left (1-a^2 x^2\right )^3 \arccos (a x)^3\) |
| \(\Big \downarrow \) 5183 |
| \(\displaystyle \frac {6}{7} c^3 \left (\frac {4}{5} \left (\frac {2}{3} \left (3 a \left (-\frac {2 \int \arccos (a x)dx}{a}-\frac {\sqrt {1-a^2 x^2} \arccos (a x)^2}{a^2}\right )+x \arccos (a x)^3\right )+a \left (-\frac {2 \int \left (1-a^2 x^2\right ) \arccos (a x)dx}{3 a}-\frac {\left (1-a^2 x^2\right )^{3/2} \arccos (a x)^2}{3 a^2}\right )+\frac {1}{3} x \left (1-a^2 x^2\right ) \arccos (a x)^3\right )+\frac {3}{5} a \left (-\frac {2 \int \left (1-a^2 x^2\right )^2 \arccos (a x)dx}{5 a}-\frac {\left (1-a^2 x^2\right )^{5/2} \arccos (a x)^2}{5 a^2}\right )+\frac {1}{5} x \left (1-a^2 x^2\right )^2 \arccos (a x)^3\right )+\frac {3}{7} a c^3 \left (-\frac {2 \int \left (1-a^2 x^2\right )^3 \arccos (a x)dx}{7 a}-\frac {\left (1-a^2 x^2\right )^{7/2} \arccos (a x)^2}{7 a^2}\right )+\frac {1}{7} c^3 x \left (1-a^2 x^2\right )^3 \arccos (a x)^3\) |
| \(\Big \downarrow \) 5131 |
| \(\displaystyle \frac {6}{7} c^3 \left (\frac {4}{5} \left (\frac {2}{3} \left (3 a \left (-\frac {2 \left (a \int \frac {x}{\sqrt {1-a^2 x^2}}dx+x \arccos (a x)\right )}{a}-\frac {\sqrt {1-a^2 x^2} \arccos (a x)^2}{a^2}\right )+x \arccos (a x)^3\right )+a \left (-\frac {2 \int \left (1-a^2 x^2\right ) \arccos (a x)dx}{3 a}-\frac {\left (1-a^2 x^2\right )^{3/2} \arccos (a x)^2}{3 a^2}\right )+\frac {1}{3} x \left (1-a^2 x^2\right ) \arccos (a x)^3\right )+\frac {3}{5} a \left (-\frac {2 \int \left (1-a^2 x^2\right )^2 \arccos (a x)dx}{5 a}-\frac {\left (1-a^2 x^2\right )^{5/2} \arccos (a x)^2}{5 a^2}\right )+\frac {1}{5} x \left (1-a^2 x^2\right )^2 \arccos (a x)^3\right )+\frac {3}{7} a c^3 \left (-\frac {2 \int \left (1-a^2 x^2\right )^3 \arccos (a x)dx}{7 a}-\frac {\left (1-a^2 x^2\right )^{7/2} \arccos (a x)^2}{7 a^2}\right )+\frac {1}{7} c^3 x \left (1-a^2 x^2\right )^3 \arccos (a x)^3\) |
| \(\Big \downarrow \) 241 |
| \(\displaystyle \frac {6}{7} c^3 \left (\frac {4}{5} \left (a \left (-\frac {2 \int \left (1-a^2 x^2\right ) \arccos (a x)dx}{3 a}-\frac {\left (1-a^2 x^2\right )^{3/2} \arccos (a x)^2}{3 a^2}\right )+\frac {1}{3} x \left (1-a^2 x^2\right ) \arccos (a x)^3+\frac {2}{3} \left (3 a \left (-\frac {\sqrt {1-a^2 x^2} \arccos (a x)^2}{a^2}-\frac {2 \left (x \arccos (a x)-\frac {\sqrt {1-a^2 x^2}}{a}\right )}{a}\right )+x \arccos (a x)^3\right )\right )+\frac {3}{5} a \left (-\frac {2 \int \left (1-a^2 x^2\right )^2 \arccos (a x)dx}{5 a}-\frac {\left (1-a^2 x^2\right )^{5/2} \arccos (a x)^2}{5 a^2}\right )+\frac {1}{5} x \left (1-a^2 x^2\right )^2 \arccos (a x)^3\right )+\frac {3}{7} a c^3 \left (-\frac {2 \int \left (1-a^2 x^2\right )^3 \arccos (a x)dx}{7 a}-\frac {\left (1-a^2 x^2\right )^{7/2} \arccos (a x)^2}{7 a^2}\right )+\frac {1}{7} c^3 x \left (1-a^2 x^2\right )^3 \arccos (a x)^3\) |
| \(\Big \downarrow \) 5155 |
| \(\displaystyle \frac {6}{7} c^3 \left (\frac {4}{5} \left (a \left (-\frac {2 \left (a \int \frac {x \left (3-a^2 x^2\right )}{3 \sqrt {1-a^2 x^2}}dx-\frac {1}{3} a^2 x^3 \arccos (a x)+x \arccos (a x)\right )}{3 a}-\frac {\left (1-a^2 x^2\right )^{3/2} \arccos (a x)^2}{3 a^2}\right )+\frac {1}{3} x \left (1-a^2 x^2\right ) \arccos (a x)^3+\frac {2}{3} \left (3 a \left (-\frac {\sqrt {1-a^2 x^2} \arccos (a x)^2}{a^2}-\frac {2 \left (x \arccos (a x)-\frac {\sqrt {1-a^2 x^2}}{a}\right )}{a}\right )+x \arccos (a x)^3\right )\right )+\frac {3}{5} a \left (-\frac {2 \left (a \int \frac {x \left (3 a^4 x^4-10 a^2 x^2+15\right )}{15 \sqrt {1-a^2 x^2}}dx+\frac {1}{5} a^4 x^5 \arccos (a x)-\frac {2}{3} a^2 x^3 \arccos (a x)+x \arccos (a x)\right )}{5 a}-\frac {\left (1-a^2 x^2\right )^{5/2} \arccos (a x)^2}{5 a^2}\right )+\frac {1}{5} x \left (1-a^2 x^2\right )^2 \arccos (a x)^3\right )+\frac {3}{7} a c^3 \left (-\frac {2 \left (a \int \frac {x \left (-5 a^6 x^6+21 a^4 x^4-35 a^2 x^2+35\right )}{35 \sqrt {1-a^2 x^2}}dx-\frac {1}{7} a^6 x^7 \arccos (a x)+\frac {3}{5} a^4 x^5 \arccos (a x)-a^2 x^3 \arccos (a x)+x \arccos (a x)\right )}{7 a}-\frac {\left (1-a^2 x^2\right )^{7/2} \arccos (a x)^2}{7 a^2}\right )+\frac {1}{7} c^3 x \left (1-a^2 x^2\right )^3 \arccos (a x)^3\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {6}{7} c^3 \left (\frac {4}{5} \left (a \left (-\frac {2 \left (\frac {1}{3} a \int \frac {x \left (3-a^2 x^2\right )}{\sqrt {1-a^2 x^2}}dx-\frac {1}{3} a^2 x^3 \arccos (a x)+x \arccos (a x)\right )}{3 a}-\frac {\left (1-a^2 x^2\right )^{3/2} \arccos (a x)^2}{3 a^2}\right )+\frac {1}{3} x \left (1-a^2 x^2\right ) \arccos (a x)^3+\frac {2}{3} \left (3 a \left (-\frac {\sqrt {1-a^2 x^2} \arccos (a x)^2}{a^2}-\frac {2 \left (x \arccos (a x)-\frac {\sqrt {1-a^2 x^2}}{a}\right )}{a}\right )+x \arccos (a x)^3\right )\right )+\frac {3}{5} a \left (-\frac {2 \left (\frac {1}{15} a \int \frac {x \left (3 a^4 x^4-10 a^2 x^2+15\right )}{\sqrt {1-a^2 x^2}}dx+\frac {1}{5} a^4 x^5 \arccos (a x)-\frac {2}{3} a^2 x^3 \arccos (a x)+x \arccos (a x)\right )}{5 a}-\frac {\left (1-a^2 x^2\right )^{5/2} \arccos (a x)^2}{5 a^2}\right )+\frac {1}{5} x \left (1-a^2 x^2\right )^2 \arccos (a x)^3\right )+\frac {3}{7} a c^3 \left (-\frac {2 \left (\frac {1}{35} a \int \frac {x \left (-5 a^6 x^6+21 a^4 x^4-35 a^2 x^2+35\right )}{\sqrt {1-a^2 x^2}}dx-\frac {1}{7} a^6 x^7 \arccos (a x)+\frac {3}{5} a^4 x^5 \arccos (a x)-a^2 x^3 \arccos (a x)+x \arccos (a x)\right )}{7 a}-\frac {\left (1-a^2 x^2\right )^{7/2} \arccos (a x)^2}{7 a^2}\right )+\frac {1}{7} c^3 x \left (1-a^2 x^2\right )^3 \arccos (a x)^3\) |
| \(\Big \downarrow \) 353 |
| \(\displaystyle \frac {6}{7} c^3 \left (\frac {4}{5} \left (a \left (-\frac {2 \left (\frac {1}{6} a \int \frac {3-a^2 x^2}{\sqrt {1-a^2 x^2}}dx^2-\frac {1}{3} a^2 x^3 \arccos (a x)+x \arccos (a x)\right )}{3 a}-\frac {\left (1-a^2 x^2\right )^{3/2} \arccos (a x)^2}{3 a^2}\right )+\frac {1}{3} x \left (1-a^2 x^2\right ) \arccos (a x)^3+\frac {2}{3} \left (3 a \left (-\frac {\sqrt {1-a^2 x^2} \arccos (a x)^2}{a^2}-\frac {2 \left (x \arccos (a x)-\frac {\sqrt {1-a^2 x^2}}{a}\right )}{a}\right )+x \arccos (a x)^3\right )\right )+\frac {3}{5} a \left (-\frac {2 \left (\frac {1}{15} a \int \frac {x \left (3 a^4 x^4-10 a^2 x^2+15\right )}{\sqrt {1-a^2 x^2}}dx+\frac {1}{5} a^4 x^5 \arccos (a x)-\frac {2}{3} a^2 x^3 \arccos (a x)+x \arccos (a x)\right )}{5 a}-\frac {\left (1-a^2 x^2\right )^{5/2} \arccos (a x)^2}{5 a^2}\right )+\frac {1}{5} x \left (1-a^2 x^2\right )^2 \arccos (a x)^3\right )+\frac {3}{7} a c^3 \left (-\frac {2 \left (\frac {1}{35} a \int \frac {x \left (-5 a^6 x^6+21 a^4 x^4-35 a^2 x^2+35\right )}{\sqrt {1-a^2 x^2}}dx-\frac {1}{7} a^6 x^7 \arccos (a x)+\frac {3}{5} a^4 x^5 \arccos (a x)-a^2 x^3 \arccos (a x)+x \arccos (a x)\right )}{7 a}-\frac {\left (1-a^2 x^2\right )^{7/2} \arccos (a x)^2}{7 a^2}\right )+\frac {1}{7} c^3 x \left (1-a^2 x^2\right )^3 \arccos (a x)^3\) |
| \(\Big \downarrow \) 53 |
| \(\displaystyle \frac {6}{7} c^3 \left (\frac {4}{5} \left (a \left (-\frac {2 \left (\frac {1}{6} a \int \left (\sqrt {1-a^2 x^2}+\frac {2}{\sqrt {1-a^2 x^2}}\right )dx^2-\frac {1}{3} a^2 x^3 \arccos (a x)+x \arccos (a x)\right )}{3 a}-\frac {\left (1-a^2 x^2\right )^{3/2} \arccos (a x)^2}{3 a^2}\right )+\frac {1}{3} x \left (1-a^2 x^2\right ) \arccos (a x)^3+\frac {2}{3} \left (3 a \left (-\frac {\sqrt {1-a^2 x^2} \arccos (a x)^2}{a^2}-\frac {2 \left (x \arccos (a x)-\frac {\sqrt {1-a^2 x^2}}{a}\right )}{a}\right )+x \arccos (a x)^3\right )\right )+\frac {3}{5} a \left (-\frac {2 \left (\frac {1}{15} a \int \frac {x \left (3 a^4 x^4-10 a^2 x^2+15\right )}{\sqrt {1-a^2 x^2}}dx+\frac {1}{5} a^4 x^5 \arccos (a x)-\frac {2}{3} a^2 x^3 \arccos (a x)+x \arccos (a x)\right )}{5 a}-\frac {\left (1-a^2 x^2\right )^{5/2} \arccos (a x)^2}{5 a^2}\right )+\frac {1}{5} x \left (1-a^2 x^2\right )^2 \arccos (a x)^3\right )+\frac {3}{7} a c^3 \left (-\frac {2 \left (\frac {1}{35} a \int \frac {x \left (-5 a^6 x^6+21 a^4 x^4-35 a^2 x^2+35\right )}{\sqrt {1-a^2 x^2}}dx-\frac {1}{7} a^6 x^7 \arccos (a x)+\frac {3}{5} a^4 x^5 \arccos (a x)-a^2 x^3 \arccos (a x)+x \arccos (a x)\right )}{7 a}-\frac {\left (1-a^2 x^2\right )^{7/2} \arccos (a x)^2}{7 a^2}\right )+\frac {1}{7} c^3 x \left (1-a^2 x^2\right )^3 \arccos (a x)^3\) |
| \(\Big \downarrow \) 1576 |
| \(\displaystyle \frac {6}{7} c^3 \left (\frac {4}{5} \left (a \left (-\frac {2 \left (\frac {1}{6} a \int \left (\sqrt {1-a^2 x^2}+\frac {2}{\sqrt {1-a^2 x^2}}\right )dx^2-\frac {1}{3} a^2 x^3 \arccos (a x)+x \arccos (a x)\right )}{3 a}-\frac {\left (1-a^2 x^2\right )^{3/2} \arccos (a x)^2}{3 a^2}\right )+\frac {1}{3} x \left (1-a^2 x^2\right ) \arccos (a x)^3+\frac {2}{3} \left (3 a \left (-\frac {\sqrt {1-a^2 x^2} \arccos (a x)^2}{a^2}-\frac {2 \left (x \arccos (a x)-\frac {\sqrt {1-a^2 x^2}}{a}\right )}{a}\right )+x \arccos (a x)^3\right )\right )+\frac {3}{5} a \left (-\frac {2 \left (\frac {1}{30} a \int \frac {3 a^4 x^4-10 a^2 x^2+15}{\sqrt {1-a^2 x^2}}dx^2+\frac {1}{5} a^4 x^5 \arccos (a x)-\frac {2}{3} a^2 x^3 \arccos (a x)+x \arccos (a x)\right )}{5 a}-\frac {\left (1-a^2 x^2\right )^{5/2} \arccos (a x)^2}{5 a^2}\right )+\frac {1}{5} x \left (1-a^2 x^2\right )^2 \arccos (a x)^3\right )+\frac {3}{7} a c^3 \left (-\frac {2 \left (\frac {1}{35} a \int \frac {x \left (-5 a^6 x^6+21 a^4 x^4-35 a^2 x^2+35\right )}{\sqrt {1-a^2 x^2}}dx-\frac {1}{7} a^6 x^7 \arccos (a x)+\frac {3}{5} a^4 x^5 \arccos (a x)-a^2 x^3 \arccos (a x)+x \arccos (a x)\right )}{7 a}-\frac {\left (1-a^2 x^2\right )^{7/2} \arccos (a x)^2}{7 a^2}\right )+\frac {1}{7} c^3 x \left (1-a^2 x^2\right )^3 \arccos (a x)^3\) |
| \(\Big \downarrow \) 1140 |
| \(\displaystyle \frac {6}{7} c^3 \left (\frac {4}{5} \left (a \left (-\frac {2 \left (\frac {1}{6} a \int \left (\sqrt {1-a^2 x^2}+\frac {2}{\sqrt {1-a^2 x^2}}\right )dx^2-\frac {1}{3} a^2 x^3 \arccos (a x)+x \arccos (a x)\right )}{3 a}-\frac {\left (1-a^2 x^2\right )^{3/2} \arccos (a x)^2}{3 a^2}\right )+\frac {1}{3} x \left (1-a^2 x^2\right ) \arccos (a x)^3+\frac {2}{3} \left (3 a \left (-\frac {\sqrt {1-a^2 x^2} \arccos (a x)^2}{a^2}-\frac {2 \left (x \arccos (a x)-\frac {\sqrt {1-a^2 x^2}}{a}\right )}{a}\right )+x \arccos (a x)^3\right )\right )+\frac {3}{5} a \left (-\frac {2 \left (\frac {1}{30} a \int \left (3 \left (1-a^2 x^2\right )^{3/2}+4 \sqrt {1-a^2 x^2}+\frac {8}{\sqrt {1-a^2 x^2}}\right )dx^2+\frac {1}{5} a^4 x^5 \arccos (a x)-\frac {2}{3} a^2 x^3 \arccos (a x)+x \arccos (a x)\right )}{5 a}-\frac {\left (1-a^2 x^2\right )^{5/2} \arccos (a x)^2}{5 a^2}\right )+\frac {1}{5} x \left (1-a^2 x^2\right )^2 \arccos (a x)^3\right )+\frac {3}{7} a c^3 \left (-\frac {2 \left (\frac {1}{35} a \int \frac {x \left (-5 a^6 x^6+21 a^4 x^4-35 a^2 x^2+35\right )}{\sqrt {1-a^2 x^2}}dx-\frac {1}{7} a^6 x^7 \arccos (a x)+\frac {3}{5} a^4 x^5 \arccos (a x)-a^2 x^3 \arccos (a x)+x \arccos (a x)\right )}{7 a}-\frac {\left (1-a^2 x^2\right )^{7/2} \arccos (a x)^2}{7 a^2}\right )+\frac {1}{7} c^3 x \left (1-a^2 x^2\right )^3 \arccos (a x)^3\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {3}{7} a c^3 \left (-\frac {2 \left (\frac {1}{35} a \int \frac {x \left (-5 a^6 x^6+21 a^4 x^4-35 a^2 x^2+35\right )}{\sqrt {1-a^2 x^2}}dx-\frac {1}{7} a^6 x^7 \arccos (a x)+\frac {3}{5} a^4 x^5 \arccos (a x)-a^2 x^3 \arccos (a x)+x \arccos (a x)\right )}{7 a}-\frac {\left (1-a^2 x^2\right )^{7/2} \arccos (a x)^2}{7 a^2}\right )+\frac {1}{7} c^3 x \left (1-a^2 x^2\right )^3 \arccos (a x)^3+\frac {6}{7} c^3 \left (\frac {1}{5} x \left (1-a^2 x^2\right )^2 \arccos (a x)^3+\frac {4}{5} \left (\frac {1}{3} x \left (1-a^2 x^2\right ) \arccos (a x)^3+\frac {2}{3} \left (3 a \left (-\frac {\sqrt {1-a^2 x^2} \arccos (a x)^2}{a^2}-\frac {2 \left (x \arccos (a x)-\frac {\sqrt {1-a^2 x^2}}{a}\right )}{a}\right )+x \arccos (a x)^3\right )+a \left (-\frac {\left (1-a^2 x^2\right )^{3/2} \arccos (a x)^2}{3 a^2}-\frac {2 \left (-\frac {1}{3} a^2 x^3 \arccos (a x)+\frac {1}{6} a \left (-\frac {2 \left (1-a^2 x^2\right )^{3/2}}{3 a^2}-\frac {4 \sqrt {1-a^2 x^2}}{a^2}\right )+x \arccos (a x)\right )}{3 a}\right )\right )+\frac {3}{5} a \left (-\frac {\left (1-a^2 x^2\right )^{5/2} \arccos (a x)^2}{5 a^2}-\frac {2 \left (\frac {1}{5} a^4 x^5 \arccos (a x)-\frac {2}{3} a^2 x^3 \arccos (a x)+\frac {1}{30} a \left (-\frac {6 \left (1-a^2 x^2\right )^{5/2}}{5 a^2}-\frac {8 \left (1-a^2 x^2\right )^{3/2}}{3 a^2}-\frac {16 \sqrt {1-a^2 x^2}}{a^2}\right )+x \arccos (a x)\right )}{5 a}\right )\right )\) |
| \(\Big \downarrow \) 2331 |
| \(\displaystyle \frac {3}{7} a c^3 \left (-\frac {2 \left (\frac {1}{70} a \int \frac {-5 a^6 x^6+21 a^4 x^4-35 a^2 x^2+35}{\sqrt {1-a^2 x^2}}dx^2-\frac {1}{7} a^6 x^7 \arccos (a x)+\frac {3}{5} a^4 x^5 \arccos (a x)-a^2 x^3 \arccos (a x)+x \arccos (a x)\right )}{7 a}-\frac {\left (1-a^2 x^2\right )^{7/2} \arccos (a x)^2}{7 a^2}\right )+\frac {1}{7} c^3 x \left (1-a^2 x^2\right )^3 \arccos (a x)^3+\frac {6}{7} c^3 \left (\frac {1}{5} x \left (1-a^2 x^2\right )^2 \arccos (a x)^3+\frac {4}{5} \left (\frac {1}{3} x \left (1-a^2 x^2\right ) \arccos (a x)^3+\frac {2}{3} \left (3 a \left (-\frac {\sqrt {1-a^2 x^2} \arccos (a x)^2}{a^2}-\frac {2 \left (x \arccos (a x)-\frac {\sqrt {1-a^2 x^2}}{a}\right )}{a}\right )+x \arccos (a x)^3\right )+a \left (-\frac {\left (1-a^2 x^2\right )^{3/2} \arccos (a x)^2}{3 a^2}-\frac {2 \left (-\frac {1}{3} a^2 x^3 \arccos (a x)+\frac {1}{6} a \left (-\frac {2 \left (1-a^2 x^2\right )^{3/2}}{3 a^2}-\frac {4 \sqrt {1-a^2 x^2}}{a^2}\right )+x \arccos (a x)\right )}{3 a}\right )\right )+\frac {3}{5} a \left (-\frac {\left (1-a^2 x^2\right )^{5/2} \arccos (a x)^2}{5 a^2}-\frac {2 \left (\frac {1}{5} a^4 x^5 \arccos (a x)-\frac {2}{3} a^2 x^3 \arccos (a x)+\frac {1}{30} a \left (-\frac {6 \left (1-a^2 x^2\right )^{5/2}}{5 a^2}-\frac {8 \left (1-a^2 x^2\right )^{3/2}}{3 a^2}-\frac {16 \sqrt {1-a^2 x^2}}{a^2}\right )+x \arccos (a x)\right )}{5 a}\right )\right )\) |
| \(\Big \downarrow \) 2389 |
| \(\displaystyle \frac {3}{7} a c^3 \left (-\frac {2 \left (\frac {1}{70} a \int \left (5 \left (1-a^2 x^2\right )^{5/2}+6 \left (1-a^2 x^2\right )^{3/2}+8 \sqrt {1-a^2 x^2}+\frac {16}{\sqrt {1-a^2 x^2}}\right )dx^2-\frac {1}{7} a^6 x^7 \arccos (a x)+\frac {3}{5} a^4 x^5 \arccos (a x)-a^2 x^3 \arccos (a x)+x \arccos (a x)\right )}{7 a}-\frac {\left (1-a^2 x^2\right )^{7/2} \arccos (a x)^2}{7 a^2}\right )+\frac {1}{7} c^3 x \left (1-a^2 x^2\right )^3 \arccos (a x)^3+\frac {6}{7} c^3 \left (\frac {1}{5} x \left (1-a^2 x^2\right )^2 \arccos (a x)^3+\frac {4}{5} \left (\frac {1}{3} x \left (1-a^2 x^2\right ) \arccos (a x)^3+\frac {2}{3} \left (3 a \left (-\frac {\sqrt {1-a^2 x^2} \arccos (a x)^2}{a^2}-\frac {2 \left (x \arccos (a x)-\frac {\sqrt {1-a^2 x^2}}{a}\right )}{a}\right )+x \arccos (a x)^3\right )+a \left (-\frac {\left (1-a^2 x^2\right )^{3/2} \arccos (a x)^2}{3 a^2}-\frac {2 \left (-\frac {1}{3} a^2 x^3 \arccos (a x)+\frac {1}{6} a \left (-\frac {2 \left (1-a^2 x^2\right )^{3/2}}{3 a^2}-\frac {4 \sqrt {1-a^2 x^2}}{a^2}\right )+x \arccos (a x)\right )}{3 a}\right )\right )+\frac {3}{5} a \left (-\frac {\left (1-a^2 x^2\right )^{5/2} \arccos (a x)^2}{5 a^2}-\frac {2 \left (\frac {1}{5} a^4 x^5 \arccos (a x)-\frac {2}{3} a^2 x^3 \arccos (a x)+\frac {1}{30} a \left (-\frac {6 \left (1-a^2 x^2\right )^{5/2}}{5 a^2}-\frac {8 \left (1-a^2 x^2\right )^{3/2}}{3 a^2}-\frac {16 \sqrt {1-a^2 x^2}}{a^2}\right )+x \arccos (a x)\right )}{5 a}\right )\right )\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{7} c^3 x \left (1-a^2 x^2\right )^3 \arccos (a x)^3+\frac {6}{7} c^3 \left (\frac {1}{5} x \left (1-a^2 x^2\right )^2 \arccos (a x)^3+\frac {4}{5} \left (\frac {1}{3} x \left (1-a^2 x^2\right ) \arccos (a x)^3+\frac {2}{3} \left (3 a \left (-\frac {\sqrt {1-a^2 x^2} \arccos (a x)^2}{a^2}-\frac {2 \left (x \arccos (a x)-\frac {\sqrt {1-a^2 x^2}}{a}\right )}{a}\right )+x \arccos (a x)^3\right )+a \left (-\frac {\left (1-a^2 x^2\right )^{3/2} \arccos (a x)^2}{3 a^2}-\frac {2 \left (-\frac {1}{3} a^2 x^3 \arccos (a x)+\frac {1}{6} a \left (-\frac {2 \left (1-a^2 x^2\right )^{3/2}}{3 a^2}-\frac {4 \sqrt {1-a^2 x^2}}{a^2}\right )+x \arccos (a x)\right )}{3 a}\right )\right )+\frac {3}{5} a \left (-\frac {\left (1-a^2 x^2\right )^{5/2} \arccos (a x)^2}{5 a^2}-\frac {2 \left (\frac {1}{5} a^4 x^5 \arccos (a x)-\frac {2}{3} a^2 x^3 \arccos (a x)+\frac {1}{30} a \left (-\frac {6 \left (1-a^2 x^2\right )^{5/2}}{5 a^2}-\frac {8 \left (1-a^2 x^2\right )^{3/2}}{3 a^2}-\frac {16 \sqrt {1-a^2 x^2}}{a^2}\right )+x \arccos (a x)\right )}{5 a}\right )\right )+\frac {3}{7} a c^3 \left (-\frac {\left (1-a^2 x^2\right )^{7/2} \arccos (a x)^2}{7 a^2}-\frac {2 \left (-\frac {1}{7} a^6 x^7 \arccos (a x)+\frac {3}{5} a^4 x^5 \arccos (a x)-a^2 x^3 \arccos (a x)+\frac {1}{70} a \left (-\frac {10 \left (1-a^2 x^2\right )^{7/2}}{7 a^2}-\frac {12 \left (1-a^2 x^2\right )^{5/2}}{5 a^2}-\frac {16 \left (1-a^2 x^2\right )^{3/2}}{3 a^2}-\frac {32 \sqrt {1-a^2 x^2}}{a^2}\right )+x \arccos (a x)\right )}{7 a}\right )\) |
Input:
Int[(c - a^2*c*x^2)^3*ArcCos[a*x]^3,x]
Output:
(c^3*x*(1 - a^2*x^2)^3*ArcCos[a*x]^3)/7 + (3*a*c^3*(-1/7*((1 - a^2*x^2)^(7 /2)*ArcCos[a*x]^2)/a^2 - (2*((a*((-32*Sqrt[1 - a^2*x^2])/a^2 - (16*(1 - a^ 2*x^2)^(3/2))/(3*a^2) - (12*(1 - a^2*x^2)^(5/2))/(5*a^2) - (10*(1 - a^2*x^ 2)^(7/2))/(7*a^2)))/70 + x*ArcCos[a*x] - a^2*x^3*ArcCos[a*x] + (3*a^4*x^5* ArcCos[a*x])/5 - (a^6*x^7*ArcCos[a*x])/7))/(7*a)))/7 + (6*c^3*((x*(1 - a^2 *x^2)^2*ArcCos[a*x]^3)/5 + (3*a*(-1/5*((1 - a^2*x^2)^(5/2)*ArcCos[a*x]^2)/ a^2 - (2*((a*((-16*Sqrt[1 - a^2*x^2])/a^2 - (8*(1 - a^2*x^2)^(3/2))/(3*a^2 ) - (6*(1 - a^2*x^2)^(5/2))/(5*a^2)))/30 + x*ArcCos[a*x] - (2*a^2*x^3*ArcC os[a*x])/3 + (a^4*x^5*ArcCos[a*x])/5))/(5*a)))/5 + (4*((x*(1 - a^2*x^2)*Ar cCos[a*x]^3)/3 + a*(-1/3*((1 - a^2*x^2)^(3/2)*ArcCos[a*x]^2)/a^2 - (2*((a* ((-4*Sqrt[1 - a^2*x^2])/a^2 - (2*(1 - a^2*x^2)^(3/2))/(3*a^2)))/6 + x*ArcC os[a*x] - (a^2*x^3*ArcCos[a*x])/3))/(3*a)) + (2*(x*ArcCos[a*x]^3 + 3*a*(-( (Sqrt[1 - a^2*x^2]*ArcCos[a*x]^2)/a^2) - (2*(-(Sqrt[1 - a^2*x^2]/a) + x*Ar cCos[a*x]))/a)))/3))/5))/7
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_))^(n_.), x_Symbol] :> Int [ExpandIntegrand[(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0] && LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])
Int[(x_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(a + b*x^2)^(p + 1)/ (2*b*(p + 1)), x] /; FreeQ[{a, b, p}, x] && NeQ[p, -1]
Int[(x_)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(q_.), x_Symbol] :> Simp[1/2 Subst[Int[(a + b*x)^p*(c + d*x)^q, x], x, x^2], x] /; FreeQ[ {a, b, c, d, p, q}, x] && NeQ[b*c - a*d, 0]
Int[((d_.) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x _Symbol] :> Int[ExpandIntegrand[(d + e*x)^m*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && IGtQ[p, 0]
Int[(x_)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^( p_.), x_Symbol] :> Simp[1/2 Subst[Int[(d + e*x)^q*(a + b*x + c*x^2)^p, x] , x, x^2], x] /; FreeQ[{a, b, c, d, e, p, q}, x]
Int[(Pq_)*(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[1/2 S ubst[Int[x^((m - 1)/2)*SubstFor[x^2, Pq, x]*(a + b*x)^p, x], x, x^2], x] /; FreeQ[{a, b, p}, x] && PolyQ[Pq, x^2] && IntegerQ[(m - 1)/2]
Int[(Pq_)*((a_) + (b_.)*(x_)^(n_.))^(p_.), x_Symbol] :> Int[ExpandIntegrand [Pq*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, n}, x] && PolyQ[Pq, x] && (IGtQ[p , 0] || EqQ[n, 1])
Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_.), x_Symbol] :> Simp[x*(a + b*Ar cCos[c*x])^n, x] + Simp[b*c*n Int[x*((a + b*ArcCos[c*x])^(n - 1)/Sqrt[1 - c^2*x^2]), x], x] /; FreeQ[{a, b, c}, x] && GtQ[n, 0]
Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbo l] :> With[{u = IntHide[(d + e*x^2)^p, x]}, Simp[(a + b*ArcCos[c*x]) u, x ] + Simp[b*c Int[SimplifyIntegrand[u/Sqrt[1 - c^2*x^2], x], x], x]] /; Fr eeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && IGtQ[p, 0]
Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_.)*((d_) + (e_.)*(x_)^2)^(p_.), x _Symbol] :> Simp[x*(d + e*x^2)^p*((a + b*ArcCos[c*x])^n/(2*p + 1)), x] + (S imp[2*d*(p/(2*p + 1)) Int[(d + e*x^2)^(p - 1)*(a + b*ArcCos[c*x])^n, x], x] + Simp[b*c*(n/(2*p + 1))*Simp[(d + e*x^2)^p/(1 - c^2*x^2)^p] Int[x*(1 - c^2*x^2)^(p - 1/2)*(a + b*ArcCos[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c , d, e}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && GtQ[p, 0]
Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_.)*(x_)*((d_) + (e_.)*(x_)^2)^(p_ .), x_Symbol] :> Simp[(d + e*x^2)^(p + 1)*((a + b*ArcCos[c*x])^n/(2*e*(p + 1))), x] - Simp[b*(n/(2*c*(p + 1)))*Simp[(d + e*x^2)^p/(1 - c^2*x^2)^p] I nt[(1 - c^2*x^2)^(p + 1/2)*(a + b*ArcCos[c*x])^(n - 1), x], x] /; FreeQ[{a, b, c, d, e, p}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && NeQ[p, -1]
Time = 0.33 (sec) , antiderivative size = 278, normalized size of antiderivative = 0.75
| method | result | size |
| derivativedivides | \(-\frac {c^{3} \left (1929375 \arccos \left (a x \right )^{3} a^{7} x^{7}-826875 \sqrt {-a^{2} x^{2}+1}\, \arccos \left (a x \right )^{2} a^{6} x^{6}-8103375 a^{5} x^{5} \arccos \left (a x \right )^{3}-236250 \arccos \left (a x \right ) a^{7} x^{7}+3869775 \sqrt {-a^{2} x^{2}+1}\, \arccos \left (a x \right )^{2} a^{4} x^{4}+33750 a^{6} x^{6} \sqrt {-a^{2} x^{2}+1}+13505625 a^{3} x^{3} \arccos \left (a x \right )^{3}+1547910 a^{5} x^{5} \arccos \left (a x \right )-8345925 x^{2} \sqrt {-a^{2} x^{2}+1}\, \arccos \left (a x \right )^{2} a^{2}-269082 a^{4} x^{4} \sqrt {-a^{2} x^{2}+1}-13505625 a x \arccos \left (a x \right )^{3}-5563950 a^{3} x^{3} \arccos \left (a x \right )+23825025 \arccos \left (a x \right )^{2} \sqrt {-a^{2} x^{2}+1}+1495874 a^{2} x^{2} \sqrt {-a^{2} x^{2}+1}+47650050 a x \arccos \left (a x \right )-44658302 \sqrt {-a^{2} x^{2}+1}\right )}{13505625 a}\) | \(278\) |
| default | \(-\frac {c^{3} \left (1929375 \arccos \left (a x \right )^{3} a^{7} x^{7}-826875 \sqrt {-a^{2} x^{2}+1}\, \arccos \left (a x \right )^{2} a^{6} x^{6}-8103375 a^{5} x^{5} \arccos \left (a x \right )^{3}-236250 \arccos \left (a x \right ) a^{7} x^{7}+3869775 \sqrt {-a^{2} x^{2}+1}\, \arccos \left (a x \right )^{2} a^{4} x^{4}+33750 a^{6} x^{6} \sqrt {-a^{2} x^{2}+1}+13505625 a^{3} x^{3} \arccos \left (a x \right )^{3}+1547910 a^{5} x^{5} \arccos \left (a x \right )-8345925 x^{2} \sqrt {-a^{2} x^{2}+1}\, \arccos \left (a x \right )^{2} a^{2}-269082 a^{4} x^{4} \sqrt {-a^{2} x^{2}+1}-13505625 a x \arccos \left (a x \right )^{3}-5563950 a^{3} x^{3} \arccos \left (a x \right )+23825025 \arccos \left (a x \right )^{2} \sqrt {-a^{2} x^{2}+1}+1495874 a^{2} x^{2} \sqrt {-a^{2} x^{2}+1}+47650050 a x \arccos \left (a x \right )-44658302 \sqrt {-a^{2} x^{2}+1}\right )}{13505625 a}\) | \(278\) |
| orering | \(\frac {x \left (6215625 a^{8} x^{8}-37489212 a^{6} x^{6}+126346014 a^{4} x^{4}-1949470892 a^{2} x^{2}-879660415\right ) \left (-a^{2} c \,x^{2}+c \right )^{3} \arccos \left (a x \right )^{3}}{13505625 \left (a x -1\right ) \left (a x +1\right ) \left (a^{2} x^{2}-1\right )^{3}}-\frac {\left (3661875 a^{8} x^{8}-25166511 a^{6} x^{6}+108592495 a^{4} x^{4}-2313484037 a^{2} x^{2}-355770542\right ) \left (-6 \left (-a^{2} c \,x^{2}+c \right )^{2} \arccos \left (a x \right )^{3} c x \,a^{2}-\frac {3 \left (-a^{2} c \,x^{2}+c \right )^{3} \arccos \left (a x \right )^{2} a}{\sqrt {-a^{2} x^{2}+1}}\right )}{40516875 a^{2} \left (a x -1\right ) \left (a x +1\right ) \left (a^{2} x^{2}-1\right )^{2}}+\frac {2 x \left (12375 a^{6} x^{6}-93069 a^{4} x^{4}+466701 a^{2} x^{2}-11958743\right ) \left (24 \left (-a^{2} c \,x^{2}+c \right ) \arccos \left (a x \right )^{3} c^{2} x^{2} a^{4}+\frac {36 \left (-a^{2} c \,x^{2}+c \right )^{2} \arccos \left (a x \right )^{2} c x \,a^{3}}{\sqrt {-a^{2} x^{2}+1}}-6 \left (-a^{2} c \,x^{2}+c \right )^{2} \arccos \left (a x \right )^{3} c \,a^{2}+\frac {6 \left (-a^{2} c \,x^{2}+c \right )^{3} \arccos \left (a x \right ) a^{2}}{-a^{2} x^{2}+1}-\frac {3 \left (-a^{2} c \,x^{2}+c \right )^{3} \arccos \left (a x \right )^{2} a^{3} x}{\left (-a^{2} x^{2}+1\right )^{\frac {3}{2}}}\right )}{2701125 a^{2} \left (a x -1\right ) \left (a x +1\right ) \left (a^{2} x^{2}-1\right )}-\frac {\left (16875 a^{6} x^{6}-134541 a^{4} x^{4}+747937 a^{2} x^{2}-22329151\right ) \left (-48 c^{3} x^{3} a^{6} \arccos \left (a x \right )^{3}-\frac {216 \left (-a^{2} c \,x^{2}+c \right ) \arccos \left (a x \right )^{2} c^{2} x^{2} a^{5}}{\sqrt {-a^{2} x^{2}+1}}+72 \left (-a^{2} c \,x^{2}+c \right ) \arccos \left (a x \right )^{3} c^{2} x \,a^{4}-\frac {108 \left (-a^{2} c \,x^{2}+c \right )^{2} \arccos \left (a x \right ) c x \,a^{4}}{-a^{2} x^{2}+1}+\frac {54 \left (-a^{2} c \,x^{2}+c \right )^{2} \arccos \left (a x \right )^{2} c \,a^{3}}{\sqrt {-a^{2} x^{2}+1}}+\frac {54 \left (-a^{2} c \,x^{2}+c \right )^{2} \arccos \left (a x \right )^{2} c \,x^{2} a^{5}}{\left (-a^{2} x^{2}+1\right )^{\frac {3}{2}}}-\frac {6 \left (-a^{2} c \,x^{2}+c \right )^{3} a^{3}}{\left (-a^{2} x^{2}+1\right )^{\frac {3}{2}}}+\frac {18 \left (-a^{2} c \,x^{2}+c \right )^{3} \arccos \left (a x \right ) a^{4} x}{\left (-a^{2} x^{2}+1\right )^{2}}-\frac {9 \left (-a^{2} c \,x^{2}+c \right )^{3} \arccos \left (a x \right )^{2} a^{5} x^{2}}{\left (-a^{2} x^{2}+1\right )^{\frac {5}{2}}}-\frac {3 \left (-a^{2} c \,x^{2}+c \right )^{3} \arccos \left (a x \right )^{2} a^{3}}{\left (-a^{2} x^{2}+1\right )^{\frac {3}{2}}}\right )}{40516875 a^{4} \left (a x -1\right ) \left (a x +1\right )}\) | \(810\) |
Input:
int((-a^2*c*x^2+c)^3*arccos(a*x)^3,x,method=_RETURNVERBOSE)
Output:
-1/13505625/a*c^3*(1929375*arccos(a*x)^3*a^7*x^7-826875*(-a^2*x^2+1)^(1/2) *arccos(a*x)^2*a^6*x^6-8103375*a^5*x^5*arccos(a*x)^3-236250*arccos(a*x)*a^ 7*x^7+3869775*(-a^2*x^2+1)^(1/2)*arccos(a*x)^2*a^4*x^4+33750*a^6*x^6*(-a^2 *x^2+1)^(1/2)+13505625*a^3*x^3*arccos(a*x)^3+1547910*a^5*x^5*arccos(a*x)-8 345925*x^2*(-a^2*x^2+1)^(1/2)*arccos(a*x)^2*a^2-269082*a^4*x^4*(-a^2*x^2+1 )^(1/2)-13505625*a*x*arccos(a*x)^3-5563950*a^3*x^3*arccos(a*x)+23825025*ar ccos(a*x)^2*(-a^2*x^2+1)^(1/2)+1495874*a^2*x^2*(-a^2*x^2+1)^(1/2)+47650050 *a*x*arccos(a*x)-44658302*(-a^2*x^2+1)^(1/2))
Time = 0.10 (sec) , antiderivative size = 201, normalized size of antiderivative = 0.54 \[ \int \left (c-a^2 c x^2\right )^3 \arccos (a x)^3 \, dx=-\frac {385875 \, {\left (5 \, a^{7} c^{3} x^{7} - 21 \, a^{5} c^{3} x^{5} + 35 \, a^{3} c^{3} x^{3} - 35 \, a c^{3} x\right )} \arccos \left (a x\right )^{3} - 210 \, {\left (1125 \, a^{7} c^{3} x^{7} - 7371 \, a^{5} c^{3} x^{5} + 26495 \, a^{3} c^{3} x^{3} - 226905 \, a c^{3} x\right )} \arccos \left (a x\right ) + {\left (33750 \, a^{6} c^{3} x^{6} - 269082 \, a^{4} c^{3} x^{4} + 1495874 \, a^{2} c^{3} x^{2} - 44658302 \, c^{3} - 11025 \, {\left (75 \, a^{6} c^{3} x^{6} - 351 \, a^{4} c^{3} x^{4} + 757 \, a^{2} c^{3} x^{2} - 2161 \, c^{3}\right )} \arccos \left (a x\right )^{2}\right )} \sqrt {-a^{2} x^{2} + 1}}{13505625 \, a} \] Input:
integrate((-a^2*c*x^2+c)^3*arccos(a*x)^3,x, algorithm="fricas")
Output:
-1/13505625*(385875*(5*a^7*c^3*x^7 - 21*a^5*c^3*x^5 + 35*a^3*c^3*x^3 - 35* a*c^3*x)*arccos(a*x)^3 - 210*(1125*a^7*c^3*x^7 - 7371*a^5*c^3*x^5 + 26495* a^3*c^3*x^3 - 226905*a*c^3*x)*arccos(a*x) + (33750*a^6*c^3*x^6 - 269082*a^ 4*c^3*x^4 + 1495874*a^2*c^3*x^2 - 44658302*c^3 - 11025*(75*a^6*c^3*x^6 - 3 51*a^4*c^3*x^4 + 757*a^2*c^3*x^2 - 2161*c^3)*arccos(a*x)^2)*sqrt(-a^2*x^2 + 1))/a
Time = 1.05 (sec) , antiderivative size = 364, normalized size of antiderivative = 0.98 \[ \int \left (c-a^2 c x^2\right )^3 \arccos (a x)^3 \, dx=\begin {cases} - \frac {a^{6} c^{3} x^{7} \operatorname {acos}^{3}{\left (a x \right )}}{7} + \frac {6 a^{6} c^{3} x^{7} \operatorname {acos}{\left (a x \right )}}{343} + \frac {3 a^{5} c^{3} x^{6} \sqrt {- a^{2} x^{2} + 1} \operatorname {acos}^{2}{\left (a x \right )}}{49} - \frac {6 a^{5} c^{3} x^{6} \sqrt {- a^{2} x^{2} + 1}}{2401} + \frac {3 a^{4} c^{3} x^{5} \operatorname {acos}^{3}{\left (a x \right )}}{5} - \frac {702 a^{4} c^{3} x^{5} \operatorname {acos}{\left (a x \right )}}{6125} - \frac {351 a^{3} c^{3} x^{4} \sqrt {- a^{2} x^{2} + 1} \operatorname {acos}^{2}{\left (a x \right )}}{1225} + \frac {29898 a^{3} c^{3} x^{4} \sqrt {- a^{2} x^{2} + 1}}{1500625} - a^{2} c^{3} x^{3} \operatorname {acos}^{3}{\left (a x \right )} + \frac {1514 a^{2} c^{3} x^{3} \operatorname {acos}{\left (a x \right )}}{3675} + \frac {757 a c^{3} x^{2} \sqrt {- a^{2} x^{2} + 1} \operatorname {acos}^{2}{\left (a x \right )}}{1225} - \frac {1495874 a c^{3} x^{2} \sqrt {- a^{2} x^{2} + 1}}{13505625} + c^{3} x \operatorname {acos}^{3}{\left (a x \right )} - \frac {4322 c^{3} x \operatorname {acos}{\left (a x \right )}}{1225} - \frac {2161 c^{3} \sqrt {- a^{2} x^{2} + 1} \operatorname {acos}^{2}{\left (a x \right )}}{1225 a} + \frac {44658302 c^{3} \sqrt {- a^{2} x^{2} + 1}}{13505625 a} & \text {for}\: a \neq 0 \\\frac {\pi ^{3} c^{3} x}{8} & \text {otherwise} \end {cases} \] Input:
integrate((-a**2*c*x**2+c)**3*acos(a*x)**3,x)
Output:
Piecewise((-a**6*c**3*x**7*acos(a*x)**3/7 + 6*a**6*c**3*x**7*acos(a*x)/343 + 3*a**5*c**3*x**6*sqrt(-a**2*x**2 + 1)*acos(a*x)**2/49 - 6*a**5*c**3*x** 6*sqrt(-a**2*x**2 + 1)/2401 + 3*a**4*c**3*x**5*acos(a*x)**3/5 - 702*a**4*c **3*x**5*acos(a*x)/6125 - 351*a**3*c**3*x**4*sqrt(-a**2*x**2 + 1)*acos(a*x )**2/1225 + 29898*a**3*c**3*x**4*sqrt(-a**2*x**2 + 1)/1500625 - a**2*c**3* x**3*acos(a*x)**3 + 1514*a**2*c**3*x**3*acos(a*x)/3675 + 757*a*c**3*x**2*s qrt(-a**2*x**2 + 1)*acos(a*x)**2/1225 - 1495874*a*c**3*x**2*sqrt(-a**2*x** 2 + 1)/13505625 + c**3*x*acos(a*x)**3 - 4322*c**3*x*acos(a*x)/1225 - 2161* c**3*sqrt(-a**2*x**2 + 1)*acos(a*x)**2/(1225*a) + 44658302*c**3*sqrt(-a**2 *x**2 + 1)/(13505625*a), Ne(a, 0)), (pi**3*c**3*x/8, True))
Time = 0.13 (sec) , antiderivative size = 284, normalized size of antiderivative = 0.77 \[ \int \left (c-a^2 c x^2\right )^3 \arccos (a x)^3 \, dx=\frac {1}{1225} \, {\left (75 \, \sqrt {-a^{2} x^{2} + 1} a^{4} c^{3} x^{6} - 351 \, \sqrt {-a^{2} x^{2} + 1} a^{2} c^{3} x^{4} + 757 \, \sqrt {-a^{2} x^{2} + 1} c^{3} x^{2} - \frac {2161 \, \sqrt {-a^{2} x^{2} + 1} c^{3}}{a^{2}}\right )} a \arccos \left (a x\right )^{2} - \frac {1}{35} \, {\left (5 \, a^{6} c^{3} x^{7} - 21 \, a^{4} c^{3} x^{5} + 35 \, a^{2} c^{3} x^{3} - 35 \, c^{3} x\right )} \arccos \left (a x\right )^{3} - \frac {2}{13505625} \, {\left (16875 \, \sqrt {-a^{2} x^{2} + 1} a^{4} c^{3} x^{6} - 134541 \, \sqrt {-a^{2} x^{2} + 1} a^{2} c^{3} x^{4} + 747937 \, \sqrt {-a^{2} x^{2} + 1} c^{3} x^{2} - \frac {22329151 \, \sqrt {-a^{2} x^{2} + 1} c^{3}}{a^{2}} - \frac {105 \, {\left (1125 \, a^{6} c^{3} x^{7} - 7371 \, a^{4} c^{3} x^{5} + 26495 \, a^{2} c^{3} x^{3} - 226905 \, c^{3} x\right )} \arccos \left (a x\right )}{a}\right )} a \] Input:
integrate((-a^2*c*x^2+c)^3*arccos(a*x)^3,x, algorithm="maxima")
Output:
1/1225*(75*sqrt(-a^2*x^2 + 1)*a^4*c^3*x^6 - 351*sqrt(-a^2*x^2 + 1)*a^2*c^3 *x^4 + 757*sqrt(-a^2*x^2 + 1)*c^3*x^2 - 2161*sqrt(-a^2*x^2 + 1)*c^3/a^2)*a *arccos(a*x)^2 - 1/35*(5*a^6*c^3*x^7 - 21*a^4*c^3*x^5 + 35*a^2*c^3*x^3 - 3 5*c^3*x)*arccos(a*x)^3 - 2/13505625*(16875*sqrt(-a^2*x^2 + 1)*a^4*c^3*x^6 - 134541*sqrt(-a^2*x^2 + 1)*a^2*c^3*x^4 + 747937*sqrt(-a^2*x^2 + 1)*c^3*x^ 2 - 22329151*sqrt(-a^2*x^2 + 1)*c^3/a^2 - 105*(1125*a^6*c^3*x^7 - 7371*a^4 *c^3*x^5 + 26495*a^2*c^3*x^3 - 226905*c^3*x)*arccos(a*x)/a)*a
Time = 0.16 (sec) , antiderivative size = 316, normalized size of antiderivative = 0.85 \[ \int \left (c-a^2 c x^2\right )^3 \arccos (a x)^3 \, dx=-\frac {1}{7} \, a^{6} c^{3} x^{7} \arccos \left (a x\right )^{3} + \frac {6}{343} \, a^{6} c^{3} x^{7} \arccos \left (a x\right ) + \frac {3}{49} \, \sqrt {-a^{2} x^{2} + 1} a^{5} c^{3} x^{6} \arccos \left (a x\right )^{2} + \frac {3}{5} \, a^{4} c^{3} x^{5} \arccos \left (a x\right )^{3} - \frac {6}{2401} \, \sqrt {-a^{2} x^{2} + 1} a^{5} c^{3} x^{6} - \frac {702}{6125} \, a^{4} c^{3} x^{5} \arccos \left (a x\right ) - \frac {351}{1225} \, \sqrt {-a^{2} x^{2} + 1} a^{3} c^{3} x^{4} \arccos \left (a x\right )^{2} - a^{2} c^{3} x^{3} \arccos \left (a x\right )^{3} + \frac {29898}{1500625} \, \sqrt {-a^{2} x^{2} + 1} a^{3} c^{3} x^{4} + \frac {1514}{3675} \, a^{2} c^{3} x^{3} \arccos \left (a x\right ) + \frac {757}{1225} \, \sqrt {-a^{2} x^{2} + 1} a c^{3} x^{2} \arccos \left (a x\right )^{2} + c^{3} x \arccos \left (a x\right )^{3} - \frac {1495874}{13505625} \, \sqrt {-a^{2} x^{2} + 1} a c^{3} x^{2} - \frac {4322}{1225} \, c^{3} x \arccos \left (a x\right ) - \frac {2161 \, \sqrt {-a^{2} x^{2} + 1} c^{3} \arccos \left (a x\right )^{2}}{1225 \, a} + \frac {44658302 \, \sqrt {-a^{2} x^{2} + 1} c^{3}}{13505625 \, a} \] Input:
integrate((-a^2*c*x^2+c)^3*arccos(a*x)^3,x, algorithm="giac")
Output:
-1/7*a^6*c^3*x^7*arccos(a*x)^3 + 6/343*a^6*c^3*x^7*arccos(a*x) + 3/49*sqrt (-a^2*x^2 + 1)*a^5*c^3*x^6*arccos(a*x)^2 + 3/5*a^4*c^3*x^5*arccos(a*x)^3 - 6/2401*sqrt(-a^2*x^2 + 1)*a^5*c^3*x^6 - 702/6125*a^4*c^3*x^5*arccos(a*x) - 351/1225*sqrt(-a^2*x^2 + 1)*a^3*c^3*x^4*arccos(a*x)^2 - a^2*c^3*x^3*arcc os(a*x)^3 + 29898/1500625*sqrt(-a^2*x^2 + 1)*a^3*c^3*x^4 + 1514/3675*a^2*c ^3*x^3*arccos(a*x) + 757/1225*sqrt(-a^2*x^2 + 1)*a*c^3*x^2*arccos(a*x)^2 + c^3*x*arccos(a*x)^3 - 1495874/13505625*sqrt(-a^2*x^2 + 1)*a*c^3*x^2 - 432 2/1225*c^3*x*arccos(a*x) - 2161/1225*sqrt(-a^2*x^2 + 1)*c^3*arccos(a*x)^2/ a + 44658302/13505625*sqrt(-a^2*x^2 + 1)*c^3/a
Timed out. \[ \int \left (c-a^2 c x^2\right )^3 \arccos (a x)^3 \, dx=\int {\mathrm {acos}\left (a\,x\right )}^3\,{\left (c-a^2\,c\,x^2\right )}^3 \,d x \] Input:
int(acos(a*x)^3*(c - a^2*c*x^2)^3,x)
Output:
int(acos(a*x)^3*(c - a^2*c*x^2)^3, x)
\[ \int \left (c-a^2 c x^2\right )^3 \arccos (a x)^3 \, dx=\frac {c^{3} \left (\mathit {acos} \left (a x \right )^{3} a x -3 \sqrt {-a^{2} x^{2}+1}\, \mathit {acos} \left (a x \right )^{2}-6 \mathit {acos} \left (a x \right ) a x +6 \sqrt {-a^{2} x^{2}+1}-\left (\int \mathit {acos} \left (a x \right )^{3} x^{6}d x \right ) a^{7}+3 \left (\int \mathit {acos} \left (a x \right )^{3} x^{4}d x \right ) a^{5}-3 \left (\int \mathit {acos} \left (a x \right )^{3} x^{2}d x \right ) a^{3}\right )}{a} \] Input:
int((-a^2*c*x^2+c)^3*acos(a*x)^3,x)
Output:
(c**3*(acos(a*x)**3*a*x - 3*sqrt( - a**2*x**2 + 1)*acos(a*x)**2 - 6*acos(a *x)*a*x + 6*sqrt( - a**2*x**2 + 1) - int(acos(a*x)**3*x**6,x)*a**7 + 3*int (acos(a*x)**3*x**4,x)*a**5 - 3*int(acos(a*x)**3*x**2,x)*a**3))/a