Integrand size = 24, antiderivative size = 446 \[ \int \left (d-c^2 d x^2\right )^3 (a+b \arccos (c x))^3 \, dx=\frac {413312 b^3 d^3 \sqrt {1-c^2 x^2}}{128625 c}+\frac {30256 b^3 d^3 \left (1-c^2 x^2\right )^{3/2}}{385875 c}+\frac {2664 b^3 d^3 \left (1-c^2 x^2\right )^{5/2}}{214375 c}+\frac {6 b^3 d^3 \left (1-c^2 x^2\right )^{7/2}}{2401 c}-\frac {4322 b^2 d^3 x (a+b \arccos (c x))}{1225}+\frac {1514 b^2 c^2 d^3 x^3 (a+b \arccos (c x))}{3675}-\frac {702 b^2 c^4 d^3 x^5 (a+b \arccos (c x))}{6125}+\frac {6}{343} b^2 c^6 d^3 x^7 (a+b \arccos (c x))-\frac {48 b d^3 \sqrt {1-c^2 x^2} (a+b \arccos (c x))^2}{35 c}-\frac {8 b d^3 \left (1-c^2 x^2\right )^{3/2} (a+b \arccos (c x))^2}{35 c}-\frac {18 b d^3 \left (1-c^2 x^2\right )^{5/2} (a+b \arccos (c x))^2}{175 c}-\frac {3 b d^3 \left (1-c^2 x^2\right )^{7/2} (a+b \arccos (c x))^2}{49 c}+\frac {16}{35} d^3 x (a+b \arccos (c x))^3+\frac {8}{35} d^3 x \left (1-c^2 x^2\right ) (a+b \arccos (c x))^3+\frac {6}{35} d^3 x \left (1-c^2 x^2\right )^2 (a+b \arccos (c x))^3+\frac {1}{7} d^3 x \left (1-c^2 x^2\right )^3 (a+b \arccos (c x))^3 \] Output:
413312/128625*b^3*d^3*(-c^2*x^2+1)^(1/2)/c+30256/385875*b^3*d^3*(-c^2*x^2+ 1)^(3/2)/c+2664/214375*b^3*d^3*(-c^2*x^2+1)^(5/2)/c+6/2401*b^3*d^3*(-c^2*x ^2+1)^(7/2)/c-4322/1225*b^2*d^3*x*(a+b*arccos(c*x))+1514/3675*b^2*c^2*d^3* x^3*(a+b*arccos(c*x))-702/6125*b^2*c^4*d^3*x^5*(a+b*arccos(c*x))+6/343*b^2 *c^6*d^3*x^7*(a+b*arccos(c*x))-48/35*b*d^3*(-c^2*x^2+1)^(1/2)*(a+b*arccos( c*x))^2/c-8/35*b*d^3*(-c^2*x^2+1)^(3/2)*(a+b*arccos(c*x))^2/c-18/175*b*d^3 *(-c^2*x^2+1)^(5/2)*(a+b*arccos(c*x))^2/c-3/49*b*d^3*(-c^2*x^2+1)^(7/2)*(a +b*arccos(c*x))^2/c+16/35*d^3*x*(a+b*arccos(c*x))^3+8/35*d^3*x*(-c^2*x^2+1 )*(a+b*arccos(c*x))^3+6/35*d^3*x*(-c^2*x^2+1)^2*(a+b*arccos(c*x))^3+1/7*d^ 3*x*(-c^2*x^2+1)^3*(a+b*arccos(c*x))^3
Time = 1.50 (sec) , antiderivative size = 411, normalized size of antiderivative = 0.92 \[ \int \left (d-c^2 d x^2\right )^3 (a+b \arccos (c x))^3 \, dx=\frac {d^3 \left (2 b^3 \sqrt {1-c^2 x^2} \left (22329151-747937 c^2 x^2+134541 c^4 x^4-16875 c^6 x^6\right )-385875 a^3 c x \left (-35+35 c^2 x^2-21 c^4 x^4+5 c^6 x^6\right )+11025 a^2 b \sqrt {1-c^2 x^2} \left (-2161+757 c^2 x^2-351 c^4 x^4+75 c^6 x^6\right )+210 a b^2 c x \left (-226905+26495 c^2 x^2-7371 c^4 x^4+1125 c^6 x^6\right )+105 b \left (-11025 a^2 c x \left (-35+35 c^2 x^2-21 c^4 x^4+5 c^6 x^6\right )+210 a b \sqrt {1-c^2 x^2} \left (-2161+757 c^2 x^2-351 c^4 x^4+75 c^6 x^6\right )+2 b^2 c x \left (-226905+26495 c^2 x^2-7371 c^4 x^4+1125 c^6 x^6\right )\right ) \arccos (c x)+11025 b^2 \left (-105 a c x \left (-35+35 c^2 x^2-21 c^4 x^4+5 c^6 x^6\right )+b \sqrt {1-c^2 x^2} \left (-2161+757 c^2 x^2-351 c^4 x^4+75 c^6 x^6\right )\right ) \arccos (c x)^2-385875 b^3 c x \left (-35+35 c^2 x^2-21 c^4 x^4+5 c^6 x^6\right ) \arccos (c x)^3\right )}{13505625 c} \] Input:
Integrate[(d - c^2*d*x^2)^3*(a + b*ArcCos[c*x])^3,x]
Output:
(d^3*(2*b^3*Sqrt[1 - c^2*x^2]*(22329151 - 747937*c^2*x^2 + 134541*c^4*x^4 - 16875*c^6*x^6) - 385875*a^3*c*x*(-35 + 35*c^2*x^2 - 21*c^4*x^4 + 5*c^6*x ^6) + 11025*a^2*b*Sqrt[1 - c^2*x^2]*(-2161 + 757*c^2*x^2 - 351*c^4*x^4 + 7 5*c^6*x^6) + 210*a*b^2*c*x*(-226905 + 26495*c^2*x^2 - 7371*c^4*x^4 + 1125* c^6*x^6) + 105*b*(-11025*a^2*c*x*(-35 + 35*c^2*x^2 - 21*c^4*x^4 + 5*c^6*x^ 6) + 210*a*b*Sqrt[1 - c^2*x^2]*(-2161 + 757*c^2*x^2 - 351*c^4*x^4 + 75*c^6 *x^6) + 2*b^2*c*x*(-226905 + 26495*c^2*x^2 - 7371*c^4*x^4 + 1125*c^6*x^6)) *ArcCos[c*x] + 11025*b^2*(-105*a*c*x*(-35 + 35*c^2*x^2 - 21*c^4*x^4 + 5*c^ 6*x^6) + b*Sqrt[1 - c^2*x^2]*(-2161 + 757*c^2*x^2 - 351*c^4*x^4 + 75*c^6*x ^6))*ArcCos[c*x]^2 - 385875*b^3*c*x*(-35 + 35*c^2*x^2 - 21*c^4*x^4 + 5*c^6 *x^6)*ArcCos[c*x]^3))/(13505625*c)
Time = 3.16 (sec) , antiderivative size = 665, normalized size of antiderivative = 1.49, number of steps used = 18, number of rules used = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.708, Rules used = {5159, 27, 5159, 5159, 5131, 5183, 2009, 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.
| \(\displaystyle \int \left (d-c^2 d x^2\right )^3 (a+b \arccos (c x))^3 \, dx\) |
| \(\Big \downarrow \) 5159 |
| \(\displaystyle \frac {3}{7} b c d^3 \int x \left (1-c^2 x^2\right )^{5/2} (a+b \arccos (c x))^2dx+\frac {6}{7} d \int d^2 \left (1-c^2 x^2\right )^2 (a+b \arccos (c x))^3dx+\frac {1}{7} d^3 x \left (1-c^2 x^2\right )^3 (a+b \arccos (c x))^3\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {3}{7} b c d^3 \int x \left (1-c^2 x^2\right )^{5/2} (a+b \arccos (c x))^2dx+\frac {6}{7} d^3 \int \left (1-c^2 x^2\right )^2 (a+b \arccos (c x))^3dx+\frac {1}{7} d^3 x \left (1-c^2 x^2\right )^3 (a+b \arccos (c x))^3\) |
| \(\Big \downarrow \) 5159 |
| \(\displaystyle \frac {3}{7} b c d^3 \int x \left (1-c^2 x^2\right )^{5/2} (a+b \arccos (c x))^2dx+\frac {6}{7} d^3 \left (\frac {3}{5} b c \int x \left (1-c^2 x^2\right )^{3/2} (a+b \arccos (c x))^2dx+\frac {4}{5} \int \left (1-c^2 x^2\right ) (a+b \arccos (c x))^3dx+\frac {1}{5} x \left (1-c^2 x^2\right )^2 (a+b \arccos (c x))^3\right )+\frac {1}{7} d^3 x \left (1-c^2 x^2\right )^3 (a+b \arccos (c x))^3\) |
| \(\Big \downarrow \) 5159 |
| \(\displaystyle \frac {3}{7} b c d^3 \int x \left (1-c^2 x^2\right )^{5/2} (a+b \arccos (c x))^2dx+\frac {6}{7} d^3 \left (\frac {3}{5} b c \int x \left (1-c^2 x^2\right )^{3/2} (a+b \arccos (c x))^2dx+\frac {4}{5} \left (b c \int x \sqrt {1-c^2 x^2} (a+b \arccos (c x))^2dx+\frac {2}{3} \int (a+b \arccos (c x))^3dx+\frac {1}{3} x \left (1-c^2 x^2\right ) (a+b \arccos (c x))^3\right )+\frac {1}{5} x \left (1-c^2 x^2\right )^2 (a+b \arccos (c x))^3\right )+\frac {1}{7} d^3 x \left (1-c^2 x^2\right )^3 (a+b \arccos (c x))^3\) |
| \(\Big \downarrow \) 5131 |
| \(\displaystyle \frac {6}{7} d^3 \left (\frac {4}{5} \left (\frac {2}{3} \left (3 b c \int \frac {x (a+b \arccos (c x))^2}{\sqrt {1-c^2 x^2}}dx+x (a+b \arccos (c x))^3\right )+b c \int x \sqrt {1-c^2 x^2} (a+b \arccos (c x))^2dx+\frac {1}{3} x \left (1-c^2 x^2\right ) (a+b \arccos (c x))^3\right )+\frac {3}{5} b c \int x \left (1-c^2 x^2\right )^{3/2} (a+b \arccos (c x))^2dx+\frac {1}{5} x \left (1-c^2 x^2\right )^2 (a+b \arccos (c x))^3\right )+\frac {3}{7} b c d^3 \int x \left (1-c^2 x^2\right )^{5/2} (a+b \arccos (c x))^2dx+\frac {1}{7} d^3 x \left (1-c^2 x^2\right )^3 (a+b \arccos (c x))^3\) |
| \(\Big \downarrow \) 5183 |
| \(\displaystyle \frac {6}{7} d^3 \left (\frac {4}{5} \left (\frac {2}{3} \left (3 b c \left (-\frac {2 b \int (a+b \arccos (c x))dx}{c}-\frac {\sqrt {1-c^2 x^2} (a+b \arccos (c x))^2}{c^2}\right )+x (a+b \arccos (c x))^3\right )+b c \left (-\frac {2 b \int \left (1-c^2 x^2\right ) (a+b \arccos (c x))dx}{3 c}-\frac {\left (1-c^2 x^2\right )^{3/2} (a+b \arccos (c x))^2}{3 c^2}\right )+\frac {1}{3} x \left (1-c^2 x^2\right ) (a+b \arccos (c x))^3\right )+\frac {3}{5} b c \left (-\frac {2 b \int \left (1-c^2 x^2\right )^2 (a+b \arccos (c x))dx}{5 c}-\frac {\left (1-c^2 x^2\right )^{5/2} (a+b \arccos (c x))^2}{5 c^2}\right )+\frac {1}{5} x \left (1-c^2 x^2\right )^2 (a+b \arccos (c x))^3\right )+\frac {3}{7} b c d^3 \left (-\frac {2 b \int \left (1-c^2 x^2\right )^3 (a+b \arccos (c x))dx}{7 c}-\frac {\left (1-c^2 x^2\right )^{7/2} (a+b \arccos (c x))^2}{7 c^2}\right )+\frac {1}{7} d^3 x \left (1-c^2 x^2\right )^3 (a+b \arccos (c x))^3\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {6}{7} d^3 \left (\frac {4}{5} \left (b c \left (-\frac {2 b \int \left (1-c^2 x^2\right ) (a+b \arccos (c x))dx}{3 c}-\frac {\left (1-c^2 x^2\right )^{3/2} (a+b \arccos (c x))^2}{3 c^2}\right )+\frac {1}{3} x \left (1-c^2 x^2\right ) (a+b \arccos (c x))^3+\frac {2}{3} \left (3 b c \left (-\frac {\sqrt {1-c^2 x^2} (a+b \arccos (c x))^2}{c^2}-\frac {2 b \left (a x+b x \arccos (c x)-\frac {b \sqrt {1-c^2 x^2}}{c}\right )}{c}\right )+x (a+b \arccos (c x))^3\right )\right )+\frac {3}{5} b c \left (-\frac {2 b \int \left (1-c^2 x^2\right )^2 (a+b \arccos (c x))dx}{5 c}-\frac {\left (1-c^2 x^2\right )^{5/2} (a+b \arccos (c x))^2}{5 c^2}\right )+\frac {1}{5} x \left (1-c^2 x^2\right )^2 (a+b \arccos (c x))^3\right )+\frac {3}{7} b c d^3 \left (-\frac {2 b \int \left (1-c^2 x^2\right )^3 (a+b \arccos (c x))dx}{7 c}-\frac {\left (1-c^2 x^2\right )^{7/2} (a+b \arccos (c x))^2}{7 c^2}\right )+\frac {1}{7} d^3 x \left (1-c^2 x^2\right )^3 (a+b \arccos (c x))^3\) |
| \(\Big \downarrow \) 5155 |
| \(\displaystyle \frac {6}{7} d^3 \left (\frac {4}{5} \left (b c \left (-\frac {2 b \left (b c \int \frac {x \left (3-c^2 x^2\right )}{3 \sqrt {1-c^2 x^2}}dx-\frac {1}{3} c^2 x^3 (a+b \arccos (c x))+x (a+b \arccos (c x))\right )}{3 c}-\frac {\left (1-c^2 x^2\right )^{3/2} (a+b \arccos (c x))^2}{3 c^2}\right )+\frac {1}{3} x \left (1-c^2 x^2\right ) (a+b \arccos (c x))^3+\frac {2}{3} \left (3 b c \left (-\frac {\sqrt {1-c^2 x^2} (a+b \arccos (c x))^2}{c^2}-\frac {2 b \left (a x+b x \arccos (c x)-\frac {b \sqrt {1-c^2 x^2}}{c}\right )}{c}\right )+x (a+b \arccos (c x))^3\right )\right )+\frac {3}{5} b c \left (-\frac {2 b \left (b c \int \frac {x \left (3 c^4 x^4-10 c^2 x^2+15\right )}{15 \sqrt {1-c^2 x^2}}dx+\frac {1}{5} c^4 x^5 (a+b \arccos (c x))-\frac {2}{3} c^2 x^3 (a+b \arccos (c x))+x (a+b \arccos (c x))\right )}{5 c}-\frac {\left (1-c^2 x^2\right )^{5/2} (a+b \arccos (c x))^2}{5 c^2}\right )+\frac {1}{5} x \left (1-c^2 x^2\right )^2 (a+b \arccos (c x))^3\right )+\frac {3}{7} b c d^3 \left (-\frac {2 b \left (b c \int \frac {x \left (-5 c^6 x^6+21 c^4 x^4-35 c^2 x^2+35\right )}{35 \sqrt {1-c^2 x^2}}dx-\frac {1}{7} c^6 x^7 (a+b \arccos (c x))+\frac {3}{5} c^4 x^5 (a+b \arccos (c x))-c^2 x^3 (a+b \arccos (c x))+x (a+b \arccos (c x))\right )}{7 c}-\frac {\left (1-c^2 x^2\right )^{7/2} (a+b \arccos (c x))^2}{7 c^2}\right )+\frac {1}{7} d^3 x \left (1-c^2 x^2\right )^3 (a+b \arccos (c x))^3\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {6}{7} d^3 \left (\frac {4}{5} \left (b c \left (-\frac {2 b \left (\frac {1}{3} b c \int \frac {x \left (3-c^2 x^2\right )}{\sqrt {1-c^2 x^2}}dx-\frac {1}{3} c^2 x^3 (a+b \arccos (c x))+x (a+b \arccos (c x))\right )}{3 c}-\frac {\left (1-c^2 x^2\right )^{3/2} (a+b \arccos (c x))^2}{3 c^2}\right )+\frac {1}{3} x \left (1-c^2 x^2\right ) (a+b \arccos (c x))^3+\frac {2}{3} \left (3 b c \left (-\frac {\sqrt {1-c^2 x^2} (a+b \arccos (c x))^2}{c^2}-\frac {2 b \left (a x+b x \arccos (c x)-\frac {b \sqrt {1-c^2 x^2}}{c}\right )}{c}\right )+x (a+b \arccos (c x))^3\right )\right )+\frac {3}{5} b c \left (-\frac {2 b \left (\frac {1}{15} b c \int \frac {x \left (3 c^4 x^4-10 c^2 x^2+15\right )}{\sqrt {1-c^2 x^2}}dx+\frac {1}{5} c^4 x^5 (a+b \arccos (c x))-\frac {2}{3} c^2 x^3 (a+b \arccos (c x))+x (a+b \arccos (c x))\right )}{5 c}-\frac {\left (1-c^2 x^2\right )^{5/2} (a+b \arccos (c x))^2}{5 c^2}\right )+\frac {1}{5} x \left (1-c^2 x^2\right )^2 (a+b \arccos (c x))^3\right )+\frac {3}{7} b c d^3 \left (-\frac {2 b \left (\frac {1}{35} b c \int \frac {x \left (-5 c^6 x^6+21 c^4 x^4-35 c^2 x^2+35\right )}{\sqrt {1-c^2 x^2}}dx-\frac {1}{7} c^6 x^7 (a+b \arccos (c x))+\frac {3}{5} c^4 x^5 (a+b \arccos (c x))-c^2 x^3 (a+b \arccos (c x))+x (a+b \arccos (c x))\right )}{7 c}-\frac {\left (1-c^2 x^2\right )^{7/2} (a+b \arccos (c x))^2}{7 c^2}\right )+\frac {1}{7} d^3 x \left (1-c^2 x^2\right )^3 (a+b \arccos (c x))^3\) |
| \(\Big \downarrow \) 353 |
| \(\displaystyle \frac {6}{7} d^3 \left (\frac {4}{5} \left (b c \left (-\frac {2 b \left (\frac {1}{6} b c \int \frac {3-c^2 x^2}{\sqrt {1-c^2 x^2}}dx^2-\frac {1}{3} c^2 x^3 (a+b \arccos (c x))+x (a+b \arccos (c x))\right )}{3 c}-\frac {\left (1-c^2 x^2\right )^{3/2} (a+b \arccos (c x))^2}{3 c^2}\right )+\frac {1}{3} x \left (1-c^2 x^2\right ) (a+b \arccos (c x))^3+\frac {2}{3} \left (3 b c \left (-\frac {\sqrt {1-c^2 x^2} (a+b \arccos (c x))^2}{c^2}-\frac {2 b \left (a x+b x \arccos (c x)-\frac {b \sqrt {1-c^2 x^2}}{c}\right )}{c}\right )+x (a+b \arccos (c x))^3\right )\right )+\frac {3}{5} b c \left (-\frac {2 b \left (\frac {1}{15} b c \int \frac {x \left (3 c^4 x^4-10 c^2 x^2+15\right )}{\sqrt {1-c^2 x^2}}dx+\frac {1}{5} c^4 x^5 (a+b \arccos (c x))-\frac {2}{3} c^2 x^3 (a+b \arccos (c x))+x (a+b \arccos (c x))\right )}{5 c}-\frac {\left (1-c^2 x^2\right )^{5/2} (a+b \arccos (c x))^2}{5 c^2}\right )+\frac {1}{5} x \left (1-c^2 x^2\right )^2 (a+b \arccos (c x))^3\right )+\frac {3}{7} b c d^3 \left (-\frac {2 b \left (\frac {1}{35} b c \int \frac {x \left (-5 c^6 x^6+21 c^4 x^4-35 c^2 x^2+35\right )}{\sqrt {1-c^2 x^2}}dx-\frac {1}{7} c^6 x^7 (a+b \arccos (c x))+\frac {3}{5} c^4 x^5 (a+b \arccos (c x))-c^2 x^3 (a+b \arccos (c x))+x (a+b \arccos (c x))\right )}{7 c}-\frac {\left (1-c^2 x^2\right )^{7/2} (a+b \arccos (c x))^2}{7 c^2}\right )+\frac {1}{7} d^3 x \left (1-c^2 x^2\right )^3 (a+b \arccos (c x))^3\) |
| \(\Big \downarrow \) 53 |
| \(\displaystyle \frac {6}{7} d^3 \left (\frac {4}{5} \left (b c \left (-\frac {2 b \left (\frac {1}{6} b c \int \left (\sqrt {1-c^2 x^2}+\frac {2}{\sqrt {1-c^2 x^2}}\right )dx^2-\frac {1}{3} c^2 x^3 (a+b \arccos (c x))+x (a+b \arccos (c x))\right )}{3 c}-\frac {\left (1-c^2 x^2\right )^{3/2} (a+b \arccos (c x))^2}{3 c^2}\right )+\frac {1}{3} x \left (1-c^2 x^2\right ) (a+b \arccos (c x))^3+\frac {2}{3} \left (3 b c \left (-\frac {\sqrt {1-c^2 x^2} (a+b \arccos (c x))^2}{c^2}-\frac {2 b \left (a x+b x \arccos (c x)-\frac {b \sqrt {1-c^2 x^2}}{c}\right )}{c}\right )+x (a+b \arccos (c x))^3\right )\right )+\frac {3}{5} b c \left (-\frac {2 b \left (\frac {1}{15} b c \int \frac {x \left (3 c^4 x^4-10 c^2 x^2+15\right )}{\sqrt {1-c^2 x^2}}dx+\frac {1}{5} c^4 x^5 (a+b \arccos (c x))-\frac {2}{3} c^2 x^3 (a+b \arccos (c x))+x (a+b \arccos (c x))\right )}{5 c}-\frac {\left (1-c^2 x^2\right )^{5/2} (a+b \arccos (c x))^2}{5 c^2}\right )+\frac {1}{5} x \left (1-c^2 x^2\right )^2 (a+b \arccos (c x))^3\right )+\frac {3}{7} b c d^3 \left (-\frac {2 b \left (\frac {1}{35} b c \int \frac {x \left (-5 c^6 x^6+21 c^4 x^4-35 c^2 x^2+35\right )}{\sqrt {1-c^2 x^2}}dx-\frac {1}{7} c^6 x^7 (a+b \arccos (c x))+\frac {3}{5} c^4 x^5 (a+b \arccos (c x))-c^2 x^3 (a+b \arccos (c x))+x (a+b \arccos (c x))\right )}{7 c}-\frac {\left (1-c^2 x^2\right )^{7/2} (a+b \arccos (c x))^2}{7 c^2}\right )+\frac {1}{7} d^3 x \left (1-c^2 x^2\right )^3 (a+b \arccos (c x))^3\) |
| \(\Big \downarrow \) 1576 |
| \(\displaystyle \frac {6}{7} d^3 \left (\frac {4}{5} \left (b c \left (-\frac {2 b \left (\frac {1}{6} b c \int \left (\sqrt {1-c^2 x^2}+\frac {2}{\sqrt {1-c^2 x^2}}\right )dx^2-\frac {1}{3} c^2 x^3 (a+b \arccos (c x))+x (a+b \arccos (c x))\right )}{3 c}-\frac {\left (1-c^2 x^2\right )^{3/2} (a+b \arccos (c x))^2}{3 c^2}\right )+\frac {1}{3} x \left (1-c^2 x^2\right ) (a+b \arccos (c x))^3+\frac {2}{3} \left (3 b c \left (-\frac {\sqrt {1-c^2 x^2} (a+b \arccos (c x))^2}{c^2}-\frac {2 b \left (a x+b x \arccos (c x)-\frac {b \sqrt {1-c^2 x^2}}{c}\right )}{c}\right )+x (a+b \arccos (c x))^3\right )\right )+\frac {3}{5} b c \left (-\frac {2 b \left (\frac {1}{30} b c \int \frac {3 c^4 x^4-10 c^2 x^2+15}{\sqrt {1-c^2 x^2}}dx^2+\frac {1}{5} c^4 x^5 (a+b \arccos (c x))-\frac {2}{3} c^2 x^3 (a+b \arccos (c x))+x (a+b \arccos (c x))\right )}{5 c}-\frac {\left (1-c^2 x^2\right )^{5/2} (a+b \arccos (c x))^2}{5 c^2}\right )+\frac {1}{5} x \left (1-c^2 x^2\right )^2 (a+b \arccos (c x))^3\right )+\frac {3}{7} b c d^3 \left (-\frac {2 b \left (\frac {1}{35} b c \int \frac {x \left (-5 c^6 x^6+21 c^4 x^4-35 c^2 x^2+35\right )}{\sqrt {1-c^2 x^2}}dx-\frac {1}{7} c^6 x^7 (a+b \arccos (c x))+\frac {3}{5} c^4 x^5 (a+b \arccos (c x))-c^2 x^3 (a+b \arccos (c x))+x (a+b \arccos (c x))\right )}{7 c}-\frac {\left (1-c^2 x^2\right )^{7/2} (a+b \arccos (c x))^2}{7 c^2}\right )+\frac {1}{7} d^3 x \left (1-c^2 x^2\right )^3 (a+b \arccos (c x))^3\) |
| \(\Big \downarrow \) 1140 |
| \(\displaystyle \frac {6}{7} d^3 \left (\frac {4}{5} \left (b c \left (-\frac {2 b \left (\frac {1}{6} b c \int \left (\sqrt {1-c^2 x^2}+\frac {2}{\sqrt {1-c^2 x^2}}\right )dx^2-\frac {1}{3} c^2 x^3 (a+b \arccos (c x))+x (a+b \arccos (c x))\right )}{3 c}-\frac {\left (1-c^2 x^2\right )^{3/2} (a+b \arccos (c x))^2}{3 c^2}\right )+\frac {1}{3} x \left (1-c^2 x^2\right ) (a+b \arccos (c x))^3+\frac {2}{3} \left (3 b c \left (-\frac {\sqrt {1-c^2 x^2} (a+b \arccos (c x))^2}{c^2}-\frac {2 b \left (a x+b x \arccos (c x)-\frac {b \sqrt {1-c^2 x^2}}{c}\right )}{c}\right )+x (a+b \arccos (c x))^3\right )\right )+\frac {3}{5} b c \left (-\frac {2 b \left (\frac {1}{30} b c \int \left (3 \left (1-c^2 x^2\right )^{3/2}+4 \sqrt {1-c^2 x^2}+\frac {8}{\sqrt {1-c^2 x^2}}\right )dx^2+\frac {1}{5} c^4 x^5 (a+b \arccos (c x))-\frac {2}{3} c^2 x^3 (a+b \arccos (c x))+x (a+b \arccos (c x))\right )}{5 c}-\frac {\left (1-c^2 x^2\right )^{5/2} (a+b \arccos (c x))^2}{5 c^2}\right )+\frac {1}{5} x \left (1-c^2 x^2\right )^2 (a+b \arccos (c x))^3\right )+\frac {3}{7} b c d^3 \left (-\frac {2 b \left (\frac {1}{35} b c \int \frac {x \left (-5 c^6 x^6+21 c^4 x^4-35 c^2 x^2+35\right )}{\sqrt {1-c^2 x^2}}dx-\frac {1}{7} c^6 x^7 (a+b \arccos (c x))+\frac {3}{5} c^4 x^5 (a+b \arccos (c x))-c^2 x^3 (a+b \arccos (c x))+x (a+b \arccos (c x))\right )}{7 c}-\frac {\left (1-c^2 x^2\right )^{7/2} (a+b \arccos (c x))^2}{7 c^2}\right )+\frac {1}{7} d^3 x \left (1-c^2 x^2\right )^3 (a+b \arccos (c x))^3\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {3}{7} b c d^3 \left (-\frac {2 b \left (\frac {1}{35} b c \int \frac {x \left (-5 c^6 x^6+21 c^4 x^4-35 c^2 x^2+35\right )}{\sqrt {1-c^2 x^2}}dx-\frac {1}{7} c^6 x^7 (a+b \arccos (c x))+\frac {3}{5} c^4 x^5 (a+b \arccos (c x))-c^2 x^3 (a+b \arccos (c x))+x (a+b \arccos (c x))\right )}{7 c}-\frac {\left (1-c^2 x^2\right )^{7/2} (a+b \arccos (c x))^2}{7 c^2}\right )+\frac {1}{7} d^3 x \left (1-c^2 x^2\right )^3 (a+b \arccos (c x))^3+\frac {6}{7} d^3 \left (\frac {1}{5} x \left (1-c^2 x^2\right )^2 (a+b \arccos (c x))^3+\frac {4}{5} \left (\frac {1}{3} x \left (1-c^2 x^2\right ) (a+b \arccos (c x))^3+\frac {2}{3} \left (3 b c \left (-\frac {\sqrt {1-c^2 x^2} (a+b \arccos (c x))^2}{c^2}-\frac {2 b \left (a x+b x \arccos (c x)-\frac {b \sqrt {1-c^2 x^2}}{c}\right )}{c}\right )+x (a+b \arccos (c x))^3\right )+b c \left (-\frac {\left (1-c^2 x^2\right )^{3/2} (a+b \arccos (c x))^2}{3 c^2}-\frac {2 b \left (-\frac {1}{3} c^2 x^3 (a+b \arccos (c x))+x (a+b \arccos (c x))+\frac {1}{6} b c \left (-\frac {2 \left (1-c^2 x^2\right )^{3/2}}{3 c^2}-\frac {4 \sqrt {1-c^2 x^2}}{c^2}\right )\right )}{3 c}\right )\right )+\frac {3}{5} b c \left (-\frac {\left (1-c^2 x^2\right )^{5/2} (a+b \arccos (c x))^2}{5 c^2}-\frac {2 b \left (\frac {1}{5} c^4 x^5 (a+b \arccos (c x))-\frac {2}{3} c^2 x^3 (a+b \arccos (c x))+x (a+b \arccos (c x))+\frac {1}{30} b c \left (-\frac {6 \left (1-c^2 x^2\right )^{5/2}}{5 c^2}-\frac {8 \left (1-c^2 x^2\right )^{3/2}}{3 c^2}-\frac {16 \sqrt {1-c^2 x^2}}{c^2}\right )\right )}{5 c}\right )\right )\) |
| \(\Big \downarrow \) 2331 |
| \(\displaystyle \frac {3}{7} b c d^3 \left (-\frac {2 b \left (\frac {1}{70} b c \int \frac {-5 c^6 x^6+21 c^4 x^4-35 c^2 x^2+35}{\sqrt {1-c^2 x^2}}dx^2-\frac {1}{7} c^6 x^7 (a+b \arccos (c x))+\frac {3}{5} c^4 x^5 (a+b \arccos (c x))-c^2 x^3 (a+b \arccos (c x))+x (a+b \arccos (c x))\right )}{7 c}-\frac {\left (1-c^2 x^2\right )^{7/2} (a+b \arccos (c x))^2}{7 c^2}\right )+\frac {1}{7} d^3 x \left (1-c^2 x^2\right )^3 (a+b \arccos (c x))^3+\frac {6}{7} d^3 \left (\frac {1}{5} x \left (1-c^2 x^2\right )^2 (a+b \arccos (c x))^3+\frac {4}{5} \left (\frac {1}{3} x \left (1-c^2 x^2\right ) (a+b \arccos (c x))^3+\frac {2}{3} \left (3 b c \left (-\frac {\sqrt {1-c^2 x^2} (a+b \arccos (c x))^2}{c^2}-\frac {2 b \left (a x+b x \arccos (c x)-\frac {b \sqrt {1-c^2 x^2}}{c}\right )}{c}\right )+x (a+b \arccos (c x))^3\right )+b c \left (-\frac {\left (1-c^2 x^2\right )^{3/2} (a+b \arccos (c x))^2}{3 c^2}-\frac {2 b \left (-\frac {1}{3} c^2 x^3 (a+b \arccos (c x))+x (a+b \arccos (c x))+\frac {1}{6} b c \left (-\frac {2 \left (1-c^2 x^2\right )^{3/2}}{3 c^2}-\frac {4 \sqrt {1-c^2 x^2}}{c^2}\right )\right )}{3 c}\right )\right )+\frac {3}{5} b c \left (-\frac {\left (1-c^2 x^2\right )^{5/2} (a+b \arccos (c x))^2}{5 c^2}-\frac {2 b \left (\frac {1}{5} c^4 x^5 (a+b \arccos (c x))-\frac {2}{3} c^2 x^3 (a+b \arccos (c x))+x (a+b \arccos (c x))+\frac {1}{30} b c \left (-\frac {6 \left (1-c^2 x^2\right )^{5/2}}{5 c^2}-\frac {8 \left (1-c^2 x^2\right )^{3/2}}{3 c^2}-\frac {16 \sqrt {1-c^2 x^2}}{c^2}\right )\right )}{5 c}\right )\right )\) |
| \(\Big \downarrow \) 2389 |
| \(\displaystyle \frac {3}{7} b c d^3 \left (-\frac {2 b \left (\frac {1}{70} b c \int \left (5 \left (1-c^2 x^2\right )^{5/2}+6 \left (1-c^2 x^2\right )^{3/2}+8 \sqrt {1-c^2 x^2}+\frac {16}{\sqrt {1-c^2 x^2}}\right )dx^2-\frac {1}{7} c^6 x^7 (a+b \arccos (c x))+\frac {3}{5} c^4 x^5 (a+b \arccos (c x))-c^2 x^3 (a+b \arccos (c x))+x (a+b \arccos (c x))\right )}{7 c}-\frac {\left (1-c^2 x^2\right )^{7/2} (a+b \arccos (c x))^2}{7 c^2}\right )+\frac {1}{7} d^3 x \left (1-c^2 x^2\right )^3 (a+b \arccos (c x))^3+\frac {6}{7} d^3 \left (\frac {1}{5} x \left (1-c^2 x^2\right )^2 (a+b \arccos (c x))^3+\frac {4}{5} \left (\frac {1}{3} x \left (1-c^2 x^2\right ) (a+b \arccos (c x))^3+\frac {2}{3} \left (3 b c \left (-\frac {\sqrt {1-c^2 x^2} (a+b \arccos (c x))^2}{c^2}-\frac {2 b \left (a x+b x \arccos (c x)-\frac {b \sqrt {1-c^2 x^2}}{c}\right )}{c}\right )+x (a+b \arccos (c x))^3\right )+b c \left (-\frac {\left (1-c^2 x^2\right )^{3/2} (a+b \arccos (c x))^2}{3 c^2}-\frac {2 b \left (-\frac {1}{3} c^2 x^3 (a+b \arccos (c x))+x (a+b \arccos (c x))+\frac {1}{6} b c \left (-\frac {2 \left (1-c^2 x^2\right )^{3/2}}{3 c^2}-\frac {4 \sqrt {1-c^2 x^2}}{c^2}\right )\right )}{3 c}\right )\right )+\frac {3}{5} b c \left (-\frac {\left (1-c^2 x^2\right )^{5/2} (a+b \arccos (c x))^2}{5 c^2}-\frac {2 b \left (\frac {1}{5} c^4 x^5 (a+b \arccos (c x))-\frac {2}{3} c^2 x^3 (a+b \arccos (c x))+x (a+b \arccos (c x))+\frac {1}{30} b c \left (-\frac {6 \left (1-c^2 x^2\right )^{5/2}}{5 c^2}-\frac {8 \left (1-c^2 x^2\right )^{3/2}}{3 c^2}-\frac {16 \sqrt {1-c^2 x^2}}{c^2}\right )\right )}{5 c}\right )\right )\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{7} d^3 x \left (1-c^2 x^2\right )^3 (a+b \arccos (c x))^3+\frac {6}{7} d^3 \left (\frac {1}{5} x \left (1-c^2 x^2\right )^2 (a+b \arccos (c x))^3+\frac {4}{5} \left (\frac {1}{3} x \left (1-c^2 x^2\right ) (a+b \arccos (c x))^3+\frac {2}{3} \left (3 b c \left (-\frac {\sqrt {1-c^2 x^2} (a+b \arccos (c x))^2}{c^2}-\frac {2 b \left (a x+b x \arccos (c x)-\frac {b \sqrt {1-c^2 x^2}}{c}\right )}{c}\right )+x (a+b \arccos (c x))^3\right )+b c \left (-\frac {\left (1-c^2 x^2\right )^{3/2} (a+b \arccos (c x))^2}{3 c^2}-\frac {2 b \left (-\frac {1}{3} c^2 x^3 (a+b \arccos (c x))+x (a+b \arccos (c x))+\frac {1}{6} b c \left (-\frac {2 \left (1-c^2 x^2\right )^{3/2}}{3 c^2}-\frac {4 \sqrt {1-c^2 x^2}}{c^2}\right )\right )}{3 c}\right )\right )+\frac {3}{5} b c \left (-\frac {\left (1-c^2 x^2\right )^{5/2} (a+b \arccos (c x))^2}{5 c^2}-\frac {2 b \left (\frac {1}{5} c^4 x^5 (a+b \arccos (c x))-\frac {2}{3} c^2 x^3 (a+b \arccos (c x))+x (a+b \arccos (c x))+\frac {1}{30} b c \left (-\frac {6 \left (1-c^2 x^2\right )^{5/2}}{5 c^2}-\frac {8 \left (1-c^2 x^2\right )^{3/2}}{3 c^2}-\frac {16 \sqrt {1-c^2 x^2}}{c^2}\right )\right )}{5 c}\right )\right )+\frac {3}{7} b c d^3 \left (-\frac {\left (1-c^2 x^2\right )^{7/2} (a+b \arccos (c x))^2}{7 c^2}-\frac {2 b \left (-\frac {1}{7} c^6 x^7 (a+b \arccos (c x))+\frac {3}{5} c^4 x^5 (a+b \arccos (c x))-c^2 x^3 (a+b \arccos (c x))+x (a+b \arccos (c x))+\frac {1}{70} b c \left (-\frac {10 \left (1-c^2 x^2\right )^{7/2}}{7 c^2}-\frac {12 \left (1-c^2 x^2\right )^{5/2}}{5 c^2}-\frac {16 \left (1-c^2 x^2\right )^{3/2}}{3 c^2}-\frac {32 \sqrt {1-c^2 x^2}}{c^2}\right )\right )}{7 c}\right )\) |
Input:
Int[(d - c^2*d*x^2)^3*(a + b*ArcCos[c*x])^3,x]
Output:
(d^3*x*(1 - c^2*x^2)^3*(a + b*ArcCos[c*x])^3)/7 + (3*b*c*d^3*(-1/7*((1 - c ^2*x^2)^(7/2)*(a + b*ArcCos[c*x])^2)/c^2 - (2*b*((b*c*((-32*Sqrt[1 - c^2*x ^2])/c^2 - (16*(1 - c^2*x^2)^(3/2))/(3*c^2) - (12*(1 - c^2*x^2)^(5/2))/(5* c^2) - (10*(1 - c^2*x^2)^(7/2))/(7*c^2)))/70 + x*(a + b*ArcCos[c*x]) - c^2 *x^3*(a + b*ArcCos[c*x]) + (3*c^4*x^5*(a + b*ArcCos[c*x]))/5 - (c^6*x^7*(a + b*ArcCos[c*x]))/7))/(7*c)))/7 + (6*d^3*((x*(1 - c^2*x^2)^2*(a + b*ArcCo s[c*x])^3)/5 + (3*b*c*(-1/5*((1 - c^2*x^2)^(5/2)*(a + b*ArcCos[c*x])^2)/c^ 2 - (2*b*((b*c*((-16*Sqrt[1 - c^2*x^2])/c^2 - (8*(1 - c^2*x^2)^(3/2))/(3*c ^2) - (6*(1 - c^2*x^2)^(5/2))/(5*c^2)))/30 + x*(a + b*ArcCos[c*x]) - (2*c^ 2*x^3*(a + b*ArcCos[c*x]))/3 + (c^4*x^5*(a + b*ArcCos[c*x]))/5))/(5*c)))/5 + (4*((x*(1 - c^2*x^2)*(a + b*ArcCos[c*x])^3)/3 + b*c*(-1/3*((1 - c^2*x^2 )^(3/2)*(a + b*ArcCos[c*x])^2)/c^2 - (2*b*((b*c*((-4*Sqrt[1 - c^2*x^2])/c^ 2 - (2*(1 - c^2*x^2)^(3/2))/(3*c^2)))/6 + x*(a + b*ArcCos[c*x]) - (c^2*x^3 *(a + b*ArcCos[c*x]))/3))/(3*c)) + (2*(x*(a + b*ArcCos[c*x])^3 + 3*b*c*(-( (Sqrt[1 - c^2*x^2]*(a + b*ArcCos[c*x])^2)/c^2) - (2*b*(a*x - (b*Sqrt[1 - c ^2*x^2])/c + b*x*ArcCos[c*x]))/c)))/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_.)*((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 = 0.54 (sec) , antiderivative size = 715, normalized size of antiderivative = 1.60
| method | result | size |
| derivativedivides | \(\frac {-d^{3} a^{3} \left (\frac {1}{7} c^{7} x^{7}-\frac {3}{5} c^{5} x^{5}+c^{3} x^{3}-c x \right )-d^{3} b^{3} \left (\frac {\arccos \left (c x \right )^{3} \left (5 c^{6} x^{6}-21 c^{4} x^{4}+35 c^{2} x^{2}-35\right ) c x}{35}-\frac {3 \arccos \left (c x \right )^{2} \left (c^{2} x^{2}-1\right )^{3} \sqrt {-c^{2} x^{2}+1}}{49}-\frac {6 \arccos \left (c x \right ) \left (5 c^{6} x^{6}-21 c^{4} x^{4}+35 c^{2} x^{2}-35\right ) c x}{1715}+\frac {6 \left (c^{2} x^{2}-1\right )^{3} \sqrt {-c^{2} x^{2}+1}}{2401}-\frac {2664 \left (c^{2} x^{2}-1\right )^{2} \sqrt {-c^{2} x^{2}+1}}{214375}+\frac {30256 \left (c^{2} x^{2}-1\right ) \sqrt {-c^{2} x^{2}+1}}{385875}-\frac {413312 \sqrt {-c^{2} x^{2}+1}}{128625}+\frac {18 \arccos \left (c x \right )^{2} \left (c^{2} x^{2}-1\right )^{2} \sqrt {-c^{2} x^{2}+1}}{175}+\frac {12 \arccos \left (c x \right ) \left (3 c^{4} x^{4}-10 c^{2} x^{2}+15\right ) c x}{875}-\frac {8 \arccos \left (c x \right )^{2} \left (c^{2} x^{2}-1\right ) \sqrt {-c^{2} x^{2}+1}}{35}-\frac {16 \arccos \left (c x \right ) \left (c^{2} x^{2}-3\right ) c x}{105}+\frac {48 \arccos \left (c x \right )^{2} \sqrt {-c^{2} x^{2}+1}}{35}+\frac {96 c x \arccos \left (c x \right )}{35}\right )-3 d^{3} a \,b^{2} \left (\frac {\arccos \left (c x \right )^{2} \left (5 c^{6} x^{6}-21 c^{4} x^{4}+35 c^{2} x^{2}-35\right ) c x}{35}-\frac {2 \arccos \left (c x \right ) \left (c^{2} x^{2}-1\right )^{3} \sqrt {-c^{2} x^{2}+1}}{49}-\frac {2 \left (5 c^{6} x^{6}-21 c^{4} x^{4}+35 c^{2} x^{2}-35\right ) c x}{1715}+\frac {12 \arccos \left (c x \right ) \left (c^{2} x^{2}-1\right )^{2} \sqrt {-c^{2} x^{2}+1}}{175}+\frac {4 \left (3 c^{4} x^{4}-10 c^{2} x^{2}+15\right ) c x}{875}-\frac {16 \arccos \left (c x \right ) \left (c^{2} x^{2}-1\right ) \sqrt {-c^{2} x^{2}+1}}{105}-\frac {16 \left (c^{2} x^{2}-3\right ) c x}{315}+\frac {32 c x}{35}+\frac {32 \arccos \left (c x \right ) \sqrt {-c^{2} x^{2}+1}}{35}\right )-3 d^{3} a^{2} b \left (\frac {\arccos \left (c x \right ) c^{7} x^{7}}{7}-\frac {3 \arccos \left (c x \right ) c^{5} x^{5}}{5}+c^{3} x^{3} \arccos \left (c x \right )-c x \arccos \left (c x \right )+\frac {2161 \sqrt {-c^{2} x^{2}+1}}{3675}-\frac {757 c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}}{3675}+\frac {117 c^{4} x^{4} \sqrt {-c^{2} x^{2}+1}}{1225}-\frac {c^{6} x^{6} \sqrt {-c^{2} x^{2}+1}}{49}\right )}{c}\) | \(715\) |
| default | \(\frac {-d^{3} a^{3} \left (\frac {1}{7} c^{7} x^{7}-\frac {3}{5} c^{5} x^{5}+c^{3} x^{3}-c x \right )-d^{3} b^{3} \left (\frac {\arccos \left (c x \right )^{3} \left (5 c^{6} x^{6}-21 c^{4} x^{4}+35 c^{2} x^{2}-35\right ) c x}{35}-\frac {3 \arccos \left (c x \right )^{2} \left (c^{2} x^{2}-1\right )^{3} \sqrt {-c^{2} x^{2}+1}}{49}-\frac {6 \arccos \left (c x \right ) \left (5 c^{6} x^{6}-21 c^{4} x^{4}+35 c^{2} x^{2}-35\right ) c x}{1715}+\frac {6 \left (c^{2} x^{2}-1\right )^{3} \sqrt {-c^{2} x^{2}+1}}{2401}-\frac {2664 \left (c^{2} x^{2}-1\right )^{2} \sqrt {-c^{2} x^{2}+1}}{214375}+\frac {30256 \left (c^{2} x^{2}-1\right ) \sqrt {-c^{2} x^{2}+1}}{385875}-\frac {413312 \sqrt {-c^{2} x^{2}+1}}{128625}+\frac {18 \arccos \left (c x \right )^{2} \left (c^{2} x^{2}-1\right )^{2} \sqrt {-c^{2} x^{2}+1}}{175}+\frac {12 \arccos \left (c x \right ) \left (3 c^{4} x^{4}-10 c^{2} x^{2}+15\right ) c x}{875}-\frac {8 \arccos \left (c x \right )^{2} \left (c^{2} x^{2}-1\right ) \sqrt {-c^{2} x^{2}+1}}{35}-\frac {16 \arccos \left (c x \right ) \left (c^{2} x^{2}-3\right ) c x}{105}+\frac {48 \arccos \left (c x \right )^{2} \sqrt {-c^{2} x^{2}+1}}{35}+\frac {96 c x \arccos \left (c x \right )}{35}\right )-3 d^{3} a \,b^{2} \left (\frac {\arccos \left (c x \right )^{2} \left (5 c^{6} x^{6}-21 c^{4} x^{4}+35 c^{2} x^{2}-35\right ) c x}{35}-\frac {2 \arccos \left (c x \right ) \left (c^{2} x^{2}-1\right )^{3} \sqrt {-c^{2} x^{2}+1}}{49}-\frac {2 \left (5 c^{6} x^{6}-21 c^{4} x^{4}+35 c^{2} x^{2}-35\right ) c x}{1715}+\frac {12 \arccos \left (c x \right ) \left (c^{2} x^{2}-1\right )^{2} \sqrt {-c^{2} x^{2}+1}}{175}+\frac {4 \left (3 c^{4} x^{4}-10 c^{2} x^{2}+15\right ) c x}{875}-\frac {16 \arccos \left (c x \right ) \left (c^{2} x^{2}-1\right ) \sqrt {-c^{2} x^{2}+1}}{105}-\frac {16 \left (c^{2} x^{2}-3\right ) c x}{315}+\frac {32 c x}{35}+\frac {32 \arccos \left (c x \right ) \sqrt {-c^{2} x^{2}+1}}{35}\right )-3 d^{3} a^{2} b \left (\frac {\arccos \left (c x \right ) c^{7} x^{7}}{7}-\frac {3 \arccos \left (c x \right ) c^{5} x^{5}}{5}+c^{3} x^{3} \arccos \left (c x \right )-c x \arccos \left (c x \right )+\frac {2161 \sqrt {-c^{2} x^{2}+1}}{3675}-\frac {757 c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}}{3675}+\frac {117 c^{4} x^{4} \sqrt {-c^{2} x^{2}+1}}{1225}-\frac {c^{6} x^{6} \sqrt {-c^{2} x^{2}+1}}{49}\right )}{c}\) | \(715\) |
| parts | \(-d^{3} a^{3} \left (\frac {1}{7} c^{6} x^{7}-\frac {3}{5} c^{4} x^{5}+c^{2} x^{3}-x \right )-\frac {d^{3} b^{3} \left (\frac {\arccos \left (c x \right )^{3} \left (5 c^{6} x^{6}-21 c^{4} x^{4}+35 c^{2} x^{2}-35\right ) c x}{35}-\frac {3 \arccos \left (c x \right )^{2} \left (c^{2} x^{2}-1\right )^{3} \sqrt {-c^{2} x^{2}+1}}{49}-\frac {6 \arccos \left (c x \right ) \left (5 c^{6} x^{6}-21 c^{4} x^{4}+35 c^{2} x^{2}-35\right ) c x}{1715}+\frac {6 \left (c^{2} x^{2}-1\right )^{3} \sqrt {-c^{2} x^{2}+1}}{2401}-\frac {2664 \left (c^{2} x^{2}-1\right )^{2} \sqrt {-c^{2} x^{2}+1}}{214375}+\frac {30256 \left (c^{2} x^{2}-1\right ) \sqrt {-c^{2} x^{2}+1}}{385875}-\frac {413312 \sqrt {-c^{2} x^{2}+1}}{128625}+\frac {18 \arccos \left (c x \right )^{2} \left (c^{2} x^{2}-1\right )^{2} \sqrt {-c^{2} x^{2}+1}}{175}+\frac {12 \arccos \left (c x \right ) \left (3 c^{4} x^{4}-10 c^{2} x^{2}+15\right ) c x}{875}-\frac {8 \arccos \left (c x \right )^{2} \left (c^{2} x^{2}-1\right ) \sqrt {-c^{2} x^{2}+1}}{35}-\frac {16 \arccos \left (c x \right ) \left (c^{2} x^{2}-3\right ) c x}{105}+\frac {48 \arccos \left (c x \right )^{2} \sqrt {-c^{2} x^{2}+1}}{35}+\frac {96 c x \arccos \left (c x \right )}{35}\right )}{c}-\frac {3 d^{3} a \,b^{2} \left (\frac {\arccos \left (c x \right )^{2} \left (5 c^{6} x^{6}-21 c^{4} x^{4}+35 c^{2} x^{2}-35\right ) c x}{35}-\frac {2 \arccos \left (c x \right ) \left (c^{2} x^{2}-1\right )^{3} \sqrt {-c^{2} x^{2}+1}}{49}-\frac {2 \left (5 c^{6} x^{6}-21 c^{4} x^{4}+35 c^{2} x^{2}-35\right ) c x}{1715}+\frac {12 \arccos \left (c x \right ) \left (c^{2} x^{2}-1\right )^{2} \sqrt {-c^{2} x^{2}+1}}{175}+\frac {4 \left (3 c^{4} x^{4}-10 c^{2} x^{2}+15\right ) c x}{875}-\frac {16 \arccos \left (c x \right ) \left (c^{2} x^{2}-1\right ) \sqrt {-c^{2} x^{2}+1}}{105}-\frac {16 \left (c^{2} x^{2}-3\right ) c x}{315}+\frac {32 c x}{35}+\frac {32 \arccos \left (c x \right ) \sqrt {-c^{2} x^{2}+1}}{35}\right )}{c}-\frac {3 d^{3} a^{2} b \left (\frac {\arccos \left (c x \right ) c^{7} x^{7}}{7}-\frac {3 \arccos \left (c x \right ) c^{5} x^{5}}{5}+c^{3} x^{3} \arccos \left (c x \right )-c x \arccos \left (c x \right )+\frac {2161 \sqrt {-c^{2} x^{2}+1}}{3675}-\frac {757 c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}}{3675}+\frac {117 c^{4} x^{4} \sqrt {-c^{2} x^{2}+1}}{1225}-\frac {c^{6} x^{6} \sqrt {-c^{2} x^{2}+1}}{49}\right )}{c}\) | \(719\) |
| orering | \(\frac {x \left (6215625 c^{8} x^{8}-37489212 c^{6} x^{6}+126346014 c^{4} x^{4}-1949470892 c^{2} x^{2}-879660415\right ) \left (-c^{2} d \,x^{2}+d \right )^{3} \left (a +b \arccos \left (c x \right )\right )^{3}}{13505625 \left (c x -1\right ) \left (c x +1\right ) \left (c^{2} x^{2}-1\right )^{3}}-\frac {\left (3661875 c^{8} x^{8}-25166511 c^{6} x^{6}+108592495 c^{4} x^{4}-2313484037 c^{2} x^{2}-355770542\right ) \left (-6 \left (-c^{2} d \,x^{2}+d \right )^{2} \left (a +b \arccos \left (c x \right )\right )^{3} d \,c^{2} x -\frac {3 \left (-c^{2} d \,x^{2}+d \right )^{3} \left (a +b \arccos \left (c x \right )\right )^{2} b c}{\sqrt {-c^{2} x^{2}+1}}\right )}{40516875 c^{2} \left (c x -1\right ) \left (c x +1\right ) \left (c^{2} x^{2}-1\right )^{2}}+\frac {2 x \left (12375 c^{6} x^{6}-93069 c^{4} x^{4}+466701 c^{2} x^{2}-11958743\right ) \left (24 \left (-c^{2} d \,x^{2}+d \right ) \left (a +b \arccos \left (c x \right )\right )^{3} d^{2} c^{4} x^{2}+\frac {36 \left (-c^{2} d \,x^{2}+d \right )^{2} \left (a +b \arccos \left (c x \right )\right )^{2} d \,c^{3} x b}{\sqrt {-c^{2} x^{2}+1}}-6 \left (-c^{2} d \,x^{2}+d \right )^{2} \left (a +b \arccos \left (c x \right )\right )^{3} d \,c^{2}+\frac {6 \left (-c^{2} d \,x^{2}+d \right )^{3} \left (a +b \arccos \left (c x \right )\right ) b^{2} c^{2}}{-c^{2} x^{2}+1}-\frac {3 \left (-c^{2} d \,x^{2}+d \right )^{3} \left (a +b \arccos \left (c x \right )\right )^{2} b \,c^{3} x}{\left (-c^{2} x^{2}+1\right )^{\frac {3}{2}}}\right )}{2701125 c^{2} \left (c x -1\right ) \left (c x +1\right ) \left (c^{2} x^{2}-1\right )}-\frac {\left (16875 c^{6} x^{6}-134541 c^{4} x^{4}+747937 c^{2} x^{2}-22329151\right ) \left (-48 d^{3} c^{6} x^{3} \left (a +b \arccos \left (c x \right )\right )^{3}-\frac {216 \left (-c^{2} d \,x^{2}+d \right ) \left (a +b \arccos \left (c x \right )\right )^{2} d^{2} c^{5} x^{2} b}{\sqrt {-c^{2} x^{2}+1}}+72 \left (-c^{2} d \,x^{2}+d \right ) \left (a +b \arccos \left (c x \right )\right )^{3} d^{2} c^{4} x -\frac {108 \left (-c^{2} d \,x^{2}+d \right )^{2} \left (a +b \arccos \left (c x \right )\right ) d \,c^{4} x \,b^{2}}{-c^{2} x^{2}+1}+\frac {54 \left (-c^{2} d \,x^{2}+d \right )^{2} \left (a +b \arccos \left (c x \right )\right )^{2} d \,c^{3} b}{\sqrt {-c^{2} x^{2}+1}}+\frac {54 \left (-c^{2} d \,x^{2}+d \right )^{2} \left (a +b \arccos \left (c x \right )\right )^{2} d \,c^{5} x^{2} b}{\left (-c^{2} x^{2}+1\right )^{\frac {3}{2}}}-\frac {6 \left (-c^{2} d \,x^{2}+d \right )^{3} b^{3} c^{3}}{\left (-c^{2} x^{2}+1\right )^{\frac {3}{2}}}+\frac {18 \left (-c^{2} d \,x^{2}+d \right )^{3} \left (a +b \arccos \left (c x \right )\right ) b^{2} c^{4} x}{\left (-c^{2} x^{2}+1\right )^{2}}-\frac {9 \left (-c^{2} d \,x^{2}+d \right )^{3} \left (a +b \arccos \left (c x \right )\right )^{2} b \,c^{5} x^{2}}{\left (-c^{2} x^{2}+1\right )^{\frac {5}{2}}}-\frac {3 \left (-c^{2} d \,x^{2}+d \right )^{3} \left (a +b \arccos \left (c x \right )\right )^{2} b \,c^{3}}{\left (-c^{2} x^{2}+1\right )^{\frac {3}{2}}}\right )}{40516875 c^{4} \left (c x -1\right ) \left (c x +1\right )}\) | \(898\) |
Input:
int((-c^2*d*x^2+d)^3*(a+b*arccos(c*x))^3,x,method=_RETURNVERBOSE)
Output:
1/c*(-d^3*a^3*(1/7*c^7*x^7-3/5*c^5*x^5+c^3*x^3-c*x)-d^3*b^3*(1/35*arccos(c *x)^3*(5*c^6*x^6-21*c^4*x^4+35*c^2*x^2-35)*c*x-3/49*arccos(c*x)^2*(c^2*x^2 -1)^3*(-c^2*x^2+1)^(1/2)-6/1715*arccos(c*x)*(5*c^6*x^6-21*c^4*x^4+35*c^2*x ^2-35)*c*x+6/2401*(c^2*x^2-1)^3*(-c^2*x^2+1)^(1/2)-2664/214375*(c^2*x^2-1) ^2*(-c^2*x^2+1)^(1/2)+30256/385875*(c^2*x^2-1)*(-c^2*x^2+1)^(1/2)-413312/1 28625*(-c^2*x^2+1)^(1/2)+18/175*arccos(c*x)^2*(c^2*x^2-1)^2*(-c^2*x^2+1)^( 1/2)+12/875*arccos(c*x)*(3*c^4*x^4-10*c^2*x^2+15)*c*x-8/35*arccos(c*x)^2*( c^2*x^2-1)*(-c^2*x^2+1)^(1/2)-16/105*arccos(c*x)*(c^2*x^2-3)*c*x+48/35*arc cos(c*x)^2*(-c^2*x^2+1)^(1/2)+96/35*c*x*arccos(c*x))-3*d^3*a*b^2*(1/35*arc cos(c*x)^2*(5*c^6*x^6-21*c^4*x^4+35*c^2*x^2-35)*c*x-2/49*arccos(c*x)*(c^2* x^2-1)^3*(-c^2*x^2+1)^(1/2)-2/1715*(5*c^6*x^6-21*c^4*x^4+35*c^2*x^2-35)*c* x+12/175*arccos(c*x)*(c^2*x^2-1)^2*(-c^2*x^2+1)^(1/2)+4/875*(3*c^4*x^4-10* c^2*x^2+15)*c*x-16/105*arccos(c*x)*(c^2*x^2-1)*(-c^2*x^2+1)^(1/2)-16/315*( c^2*x^2-3)*c*x+32/35*c*x+32/35*arccos(c*x)*(-c^2*x^2+1)^(1/2))-3*d^3*a^2*b *(1/7*arccos(c*x)*c^7*x^7-3/5*arccos(c*x)*c^5*x^5+c^3*x^3*arccos(c*x)-c*x* arccos(c*x)+2161/3675*(-c^2*x^2+1)^(1/2)-757/3675*c^2*x^2*(-c^2*x^2+1)^(1/ 2)+117/1225*c^4*x^4*(-c^2*x^2+1)^(1/2)-1/49*c^6*x^6*(-c^2*x^2+1)^(1/2)))
Time = 0.15 (sec) , antiderivative size = 535, normalized size of antiderivative = 1.20 \[ \int \left (d-c^2 d x^2\right )^3 (a+b \arccos (c x))^3 \, dx=-\frac {39375 \, {\left (49 \, a^{3} - 6 \, a b^{2}\right )} c^{7} d^{3} x^{7} - 6615 \, {\left (1225 \, a^{3} - 234 \, a b^{2}\right )} c^{5} d^{3} x^{5} + 3675 \, {\left (3675 \, a^{3} - 1514 \, a b^{2}\right )} c^{3} d^{3} x^{3} - 11025 \, {\left (1225 \, a^{3} - 4322 \, a b^{2}\right )} c d^{3} x + 385875 \, {\left (5 \, b^{3} c^{7} d^{3} x^{7} - 21 \, b^{3} c^{5} d^{3} x^{5} + 35 \, b^{3} c^{3} d^{3} x^{3} - 35 \, b^{3} c d^{3} x\right )} \arccos \left (c x\right )^{3} + 1157625 \, {\left (5 \, a b^{2} c^{7} d^{3} x^{7} - 21 \, a b^{2} c^{5} d^{3} x^{5} + 35 \, a b^{2} c^{3} d^{3} x^{3} - 35 \, a b^{2} c d^{3} x\right )} \arccos \left (c x\right )^{2} + 105 \, {\left (1125 \, {\left (49 \, a^{2} b - 2 \, b^{3}\right )} c^{7} d^{3} x^{7} - 189 \, {\left (1225 \, a^{2} b - 78 \, b^{3}\right )} c^{5} d^{3} x^{5} + 35 \, {\left (11025 \, a^{2} b - 1514 \, b^{3}\right )} c^{3} d^{3} x^{3} - 105 \, {\left (3675 \, a^{2} b - 4322 \, b^{3}\right )} c d^{3} x\right )} \arccos \left (c x\right ) - {\left (16875 \, {\left (49 \, a^{2} b - 2 \, b^{3}\right )} c^{6} d^{3} x^{6} - 81 \, {\left (47775 \, a^{2} b - 3322 \, b^{3}\right )} c^{4} d^{3} x^{4} + {\left (8345925 \, a^{2} b - 1495874 \, b^{3}\right )} c^{2} d^{3} x^{2} - {\left (23825025 \, a^{2} b - 44658302 \, b^{3}\right )} d^{3} + 11025 \, {\left (75 \, b^{3} c^{6} d^{3} x^{6} - 351 \, b^{3} c^{4} d^{3} x^{4} + 757 \, b^{3} c^{2} d^{3} x^{2} - 2161 \, b^{3} d^{3}\right )} \arccos \left (c x\right )^{2} + 22050 \, {\left (75 \, a b^{2} c^{6} d^{3} x^{6} - 351 \, a b^{2} c^{4} d^{3} x^{4} + 757 \, a b^{2} c^{2} d^{3} x^{2} - 2161 \, a b^{2} d^{3}\right )} \arccos \left (c x\right )\right )} \sqrt {-c^{2} x^{2} + 1}}{13505625 \, c} \] Input:
integrate((-c^2*d*x^2+d)^3*(a+b*arccos(c*x))^3,x, algorithm="fricas")
Output:
-1/13505625*(39375*(49*a^3 - 6*a*b^2)*c^7*d^3*x^7 - 6615*(1225*a^3 - 234*a *b^2)*c^5*d^3*x^5 + 3675*(3675*a^3 - 1514*a*b^2)*c^3*d^3*x^3 - 11025*(1225 *a^3 - 4322*a*b^2)*c*d^3*x + 385875*(5*b^3*c^7*d^3*x^7 - 21*b^3*c^5*d^3*x^ 5 + 35*b^3*c^3*d^3*x^3 - 35*b^3*c*d^3*x)*arccos(c*x)^3 + 1157625*(5*a*b^2* c^7*d^3*x^7 - 21*a*b^2*c^5*d^3*x^5 + 35*a*b^2*c^3*d^3*x^3 - 35*a*b^2*c*d^3 *x)*arccos(c*x)^2 + 105*(1125*(49*a^2*b - 2*b^3)*c^7*d^3*x^7 - 189*(1225*a ^2*b - 78*b^3)*c^5*d^3*x^5 + 35*(11025*a^2*b - 1514*b^3)*c^3*d^3*x^3 - 105 *(3675*a^2*b - 4322*b^3)*c*d^3*x)*arccos(c*x) - (16875*(49*a^2*b - 2*b^3)* c^6*d^3*x^6 - 81*(47775*a^2*b - 3322*b^3)*c^4*d^3*x^4 + (8345925*a^2*b - 1 495874*b^3)*c^2*d^3*x^2 - (23825025*a^2*b - 44658302*b^3)*d^3 + 11025*(75* b^3*c^6*d^3*x^6 - 351*b^3*c^4*d^3*x^4 + 757*b^3*c^2*d^3*x^2 - 2161*b^3*d^3 )*arccos(c*x)^2 + 22050*(75*a*b^2*c^6*d^3*x^6 - 351*a*b^2*c^4*d^3*x^4 + 75 7*a*b^2*c^2*d^3*x^2 - 2161*a*b^2*d^3)*arccos(c*x))*sqrt(-c^2*x^2 + 1))/c
Leaf count of result is larger than twice the leaf count of optimal. 978 vs. \(2 (422) = 844\).
Time = 1.22 (sec) , antiderivative size = 978, normalized size of antiderivative = 2.19 \[ \int \left (d-c^2 d x^2\right )^3 (a+b \arccos (c x))^3 \, dx =\text {Too large to display} \] Input:
integrate((-c**2*d*x**2+d)**3*(a+b*acos(c*x))**3,x)
Output:
Piecewise((-a**3*c**6*d**3*x**7/7 + 3*a**3*c**4*d**3*x**5/5 - a**3*c**2*d* *3*x**3 + a**3*d**3*x - 3*a**2*b*c**6*d**3*x**7*acos(c*x)/7 + 3*a**2*b*c** 5*d**3*x**6*sqrt(-c**2*x**2 + 1)/49 + 9*a**2*b*c**4*d**3*x**5*acos(c*x)/5 - 351*a**2*b*c**3*d**3*x**4*sqrt(-c**2*x**2 + 1)/1225 - 3*a**2*b*c**2*d**3 *x**3*acos(c*x) + 757*a**2*b*c*d**3*x**2*sqrt(-c**2*x**2 + 1)/1225 + 3*a** 2*b*d**3*x*acos(c*x) - 2161*a**2*b*d**3*sqrt(-c**2*x**2 + 1)/(1225*c) - 3* a*b**2*c**6*d**3*x**7*acos(c*x)**2/7 + 6*a*b**2*c**6*d**3*x**7/343 + 6*a*b **2*c**5*d**3*x**6*sqrt(-c**2*x**2 + 1)*acos(c*x)/49 + 9*a*b**2*c**4*d**3* x**5*acos(c*x)**2/5 - 702*a*b**2*c**4*d**3*x**5/6125 - 702*a*b**2*c**3*d** 3*x**4*sqrt(-c**2*x**2 + 1)*acos(c*x)/1225 - 3*a*b**2*c**2*d**3*x**3*acos( c*x)**2 + 1514*a*b**2*c**2*d**3*x**3/3675 + 1514*a*b**2*c*d**3*x**2*sqrt(- c**2*x**2 + 1)*acos(c*x)/1225 + 3*a*b**2*d**3*x*acos(c*x)**2 - 4322*a*b**2 *d**3*x/1225 - 4322*a*b**2*d**3*sqrt(-c**2*x**2 + 1)*acos(c*x)/(1225*c) - b**3*c**6*d**3*x**7*acos(c*x)**3/7 + 6*b**3*c**6*d**3*x**7*acos(c*x)/343 + 3*b**3*c**5*d**3*x**6*sqrt(-c**2*x**2 + 1)*acos(c*x)**2/49 - 6*b**3*c**5* d**3*x**6*sqrt(-c**2*x**2 + 1)/2401 + 3*b**3*c**4*d**3*x**5*acos(c*x)**3/5 - 702*b**3*c**4*d**3*x**5*acos(c*x)/6125 - 351*b**3*c**3*d**3*x**4*sqrt(- c**2*x**2 + 1)*acos(c*x)**2/1225 + 29898*b**3*c**3*d**3*x**4*sqrt(-c**2*x* *2 + 1)/1500625 - b**3*c**2*d**3*x**3*acos(c*x)**3 + 1514*b**3*c**2*d**3*x **3*acos(c*x)/3675 + 757*b**3*c*d**3*x**2*sqrt(-c**2*x**2 + 1)*acos(c*x...
Leaf count of result is larger than twice the leaf count of optimal. 1389 vs. \(2 (395) = 790\).
Time = 0.22 (sec) , antiderivative size = 1389, normalized size of antiderivative = 3.11 \[ \int \left (d-c^2 d x^2\right )^3 (a+b \arccos (c x))^3 \, dx=\text {Too large to display} \] Input:
integrate((-c^2*d*x^2+d)^3*(a+b*arccos(c*x))^3,x, algorithm="maxima")
Output:
-1/7*b^3*c^6*d^3*x^7*arccos(c*x)^3 - 3/7*a*b^2*c^6*d^3*x^7*arccos(c*x)^2 - 1/7*a^3*c^6*d^3*x^7 + 3/5*b^3*c^4*d^3*x^5*arccos(c*x)^3 + 9/5*a*b^2*c^4*d ^3*x^5*arccos(c*x)^2 + 3/5*a^3*c^4*d^3*x^5 - b^3*c^2*d^3*x^3*arccos(c*x)^3 - 3/245*(35*x^7*arccos(c*x) - (5*sqrt(-c^2*x^2 + 1)*x^6/c^2 + 6*sqrt(-c^2 *x^2 + 1)*x^4/c^4 + 8*sqrt(-c^2*x^2 + 1)*x^2/c^6 + 16*sqrt(-c^2*x^2 + 1)/c ^8)*c)*a^2*b*c^6*d^3 + 2/8575*(105*(5*sqrt(-c^2*x^2 + 1)*x^6/c^2 + 6*sqrt( -c^2*x^2 + 1)*x^4/c^4 + 8*sqrt(-c^2*x^2 + 1)*x^2/c^6 + 16*sqrt(-c^2*x^2 + 1)/c^8)*c*arccos(c*x) + (75*c^6*x^7 + 126*c^4*x^5 + 280*c^2*x^3 + 1680*x)/ c^6)*a*b^2*c^6*d^3 + 1/900375*(11025*(5*sqrt(-c^2*x^2 + 1)*x^6/c^2 + 6*sqr t(-c^2*x^2 + 1)*x^4/c^4 + 8*sqrt(-c^2*x^2 + 1)*x^2/c^6 + 16*sqrt(-c^2*x^2 + 1)/c^8)*c*arccos(c*x)^2 - 2*c*((1125*sqrt(-c^2*x^2 + 1)*c^4*x^6 + 3996*s qrt(-c^2*x^2 + 1)*c^2*x^4 + 15128*sqrt(-c^2*x^2 + 1)*x^2 + 206656*sqrt(-c^ 2*x^2 + 1)/c^2)/c^6 - 105*(75*c^6*x^7 + 126*c^4*x^5 + 280*c^2*x^3 + 1680*x )*arccos(c*x)/c^7))*b^3*c^6*d^3 - 3*a*b^2*c^2*d^3*x^3*arccos(c*x)^2 + 3/25 *(15*x^5*arccos(c*x) - (3*sqrt(-c^2*x^2 + 1)*x^4/c^2 + 4*sqrt(-c^2*x^2 + 1 )*x^2/c^4 + 8*sqrt(-c^2*x^2 + 1)/c^6)*c)*a^2*b*c^4*d^3 - 2/125*(15*(3*sqrt (-c^2*x^2 + 1)*x^4/c^2 + 4*sqrt(-c^2*x^2 + 1)*x^2/c^4 + 8*sqrt(-c^2*x^2 + 1)/c^6)*c*arccos(c*x) + (9*c^4*x^5 + 20*c^2*x^3 + 120*x)/c^4)*a*b^2*c^4*d^ 3 - 1/1875*(225*(3*sqrt(-c^2*x^2 + 1)*x^4/c^2 + 4*sqrt(-c^2*x^2 + 1)*x^2/c ^4 + 8*sqrt(-c^2*x^2 + 1)/c^6)*c*arccos(c*x)^2 - 2*c*((27*sqrt(-c^2*x^2...
Leaf count of result is larger than twice the leaf count of optimal. 841 vs. \(2 (395) = 790\).
Time = 0.21 (sec) , antiderivative size = 841, normalized size of antiderivative = 1.89 \[ \int \left (d-c^2 d x^2\right )^3 (a+b \arccos (c x))^3 \, dx =\text {Too large to display} \] Input:
integrate((-c^2*d*x^2+d)^3*(a+b*arccos(c*x))^3,x, algorithm="giac")
Output:
-1/7*b^3*c^6*d^3*x^7*arccos(c*x)^3 - 3/7*a*b^2*c^6*d^3*x^7*arccos(c*x)^2 - 3/7*a^2*b*c^6*d^3*x^7*arccos(c*x) + 6/343*b^3*c^6*d^3*x^7*arccos(c*x) + 3 /49*sqrt(-c^2*x^2 + 1)*b^3*c^5*d^3*x^6*arccos(c*x)^2 - 1/7*a^3*c^6*d^3*x^7 + 6/343*a*b^2*c^6*d^3*x^7 + 6/49*sqrt(-c^2*x^2 + 1)*a*b^2*c^5*d^3*x^6*arc cos(c*x) + 3/5*b^3*c^4*d^3*x^5*arccos(c*x)^3 + 3/49*sqrt(-c^2*x^2 + 1)*a^2 *b*c^5*d^3*x^6 - 6/2401*sqrt(-c^2*x^2 + 1)*b^3*c^5*d^3*x^6 + 9/5*a*b^2*c^4 *d^3*x^5*arccos(c*x)^2 + 9/5*a^2*b*c^4*d^3*x^5*arccos(c*x) - 702/6125*b^3* c^4*d^3*x^5*arccos(c*x) - 351/1225*sqrt(-c^2*x^2 + 1)*b^3*c^3*d^3*x^4*arcc os(c*x)^2 + 3/5*a^3*c^4*d^3*x^5 - 702/6125*a*b^2*c^4*d^3*x^5 - 702/1225*sq rt(-c^2*x^2 + 1)*a*b^2*c^3*d^3*x^4*arccos(c*x) - b^3*c^2*d^3*x^3*arccos(c* x)^3 - 351/1225*sqrt(-c^2*x^2 + 1)*a^2*b*c^3*d^3*x^4 + 29898/1500625*sqrt( -c^2*x^2 + 1)*b^3*c^3*d^3*x^4 - 3*a*b^2*c^2*d^3*x^3*arccos(c*x)^2 - 3*a^2* b*c^2*d^3*x^3*arccos(c*x) + 1514/3675*b^3*c^2*d^3*x^3*arccos(c*x) + 757/12 25*sqrt(-c^2*x^2 + 1)*b^3*c*d^3*x^2*arccos(c*x)^2 - a^3*c^2*d^3*x^3 + 1514 /3675*a*b^2*c^2*d^3*x^3 + 1514/1225*sqrt(-c^2*x^2 + 1)*a*b^2*c*d^3*x^2*arc cos(c*x) + b^3*d^3*x*arccos(c*x)^3 + 757/1225*sqrt(-c^2*x^2 + 1)*a^2*b*c*d ^3*x^2 - 1495874/13505625*sqrt(-c^2*x^2 + 1)*b^3*c*d^3*x^2 + 3*a*b^2*d^3*x *arccos(c*x)^2 + 3*a^2*b*d^3*x*arccos(c*x) - 4322/1225*b^3*d^3*x*arccos(c* x) - 2161/1225*sqrt(-c^2*x^2 + 1)*b^3*d^3*arccos(c*x)^2/c + a^3*d^3*x - 43 22/1225*a*b^2*d^3*x - 4322/1225*sqrt(-c^2*x^2 + 1)*a*b^2*d^3*arccos(c*x...
Timed out. \[ \int \left (d-c^2 d x^2\right )^3 (a+b \arccos (c x))^3 \, dx=\int {\left (a+b\,\mathrm {acos}\left (c\,x\right )\right )}^3\,{\left (d-c^2\,d\,x^2\right )}^3 \,d x \] Input:
int((a + b*acos(c*x))^3*(d - c^2*d*x^2)^3,x)
Output:
int((a + b*acos(c*x))^3*(d - c^2*d*x^2)^3, x)
\[ \int \left (d-c^2 d x^2\right )^3 (a+b \arccos (c x))^3 \, dx=\frac {d^{3} \left (-2161 \sqrt {-c^{2} x^{2}+1}\, a^{2} b -175 a^{3} c^{7} x^{7}+735 a^{3} c^{5} x^{5}-1225 a^{3} c^{3} x^{3}-11025 \left (\int \mathit {acos} \left (c x \right )^{2} x^{2}d x \right ) a \,b^{2} c^{3}-3675 \left (\int \mathit {acos} \left (c x \right )^{3} x^{2}d x \right ) b^{3} c^{3}+7350 \sqrt {-c^{2} x^{2}+1}\, b^{3}-525 \mathit {acos} \left (c x \right ) a^{2} b \,c^{7} x^{7}+2205 \mathit {acos} \left (c x \right ) a^{2} b \,c^{5} x^{5}+75 \sqrt {-c^{2} x^{2}+1}\, a^{2} b \,c^{6} x^{6}-351 \sqrt {-c^{2} x^{2}+1}\, a^{2} b \,c^{4} x^{4}+1225 a^{3} c x +1225 \mathit {acos} \left (c x \right )^{3} b^{3} c x -7350 \sqrt {-c^{2} x^{2}+1}\, \mathit {acos} \left (c x \right ) a \,b^{2}-7350 \mathit {acos} \left (c x \right ) b^{3} c x -7350 a \,b^{2} c x -3675 \left (\int \mathit {acos} \left (c x \right )^{2} x^{6}d x \right ) a \,b^{2} c^{7}+11025 \left (\int \mathit {acos} \left (c x \right )^{2} x^{4}d x \right ) a \,b^{2} c^{5}-3675 \mathit {acos} \left (c x \right ) a^{2} b \,c^{3} x^{3}+3675 \mathit {acos} \left (c x \right ) a^{2} b c x -3675 \sqrt {-c^{2} x^{2}+1}\, \mathit {acos} \left (c x \right )^{2} b^{3}+3675 \mathit {acos} \left (c x \right )^{2} a \,b^{2} c x +757 \sqrt {-c^{2} x^{2}+1}\, a^{2} b \,c^{2} x^{2}+3675 \left (\int \mathit {acos} \left (c x \right )^{3} x^{4}d x \right ) b^{3} c^{5}-1225 \left (\int \mathit {acos} \left (c x \right )^{3} x^{6}d x \right ) b^{3} c^{7}\right )}{1225 c} \] Input:
int((-c^2*d*x^2+d)^3*(a+b*acos(c*x))^3,x)
Output:
(d**3*(1225*acos(c*x)**3*b**3*c*x - 3675*sqrt( - c**2*x**2 + 1)*acos(c*x)* *2*b**3 + 3675*acos(c*x)**2*a*b**2*c*x - 7350*sqrt( - c**2*x**2 + 1)*acos( c*x)*a*b**2 - 525*acos(c*x)*a**2*b*c**7*x**7 + 2205*acos(c*x)*a**2*b*c**5* x**5 - 3675*acos(c*x)*a**2*b*c**3*x**3 + 3675*acos(c*x)*a**2*b*c*x - 7350* acos(c*x)*b**3*c*x + 75*sqrt( - c**2*x**2 + 1)*a**2*b*c**6*x**6 - 351*sqrt ( - c**2*x**2 + 1)*a**2*b*c**4*x**4 + 757*sqrt( - c**2*x**2 + 1)*a**2*b*c* *2*x**2 - 2161*sqrt( - c**2*x**2 + 1)*a**2*b + 7350*sqrt( - c**2*x**2 + 1) *b**3 - 1225*int(acos(c*x)**3*x**6,x)*b**3*c**7 + 3675*int(acos(c*x)**3*x* *4,x)*b**3*c**5 - 3675*int(acos(c*x)**3*x**2,x)*b**3*c**3 - 3675*int(acos( c*x)**2*x**6,x)*a*b**2*c**7 + 11025*int(acos(c*x)**2*x**4,x)*a*b**2*c**5 - 11025*int(acos(c*x)**2*x**2,x)*a*b**2*c**3 - 175*a**3*c**7*x**7 + 735*a** 3*c**5*x**5 - 1225*a**3*c**3*x**3 + 1225*a**3*c*x - 7350*a*b**2*c*x))/(122 5*c)