Integrand size = 27, antiderivative size = 302 \[ \int x^3 \left (d-c^2 d x^2\right )^2 (a+b \arccos (c x))^2 \, dx=-\frac {73 b^2 d^2 x^2}{3072 c^2}-\frac {73 b^2 d^2 x^4}{9216}+\frac {43 b^2 c^2 d^2 x^6}{3456}-\frac {1}{256} b^2 c^4 d^2 x^8+\frac {73 b d^2 x \sqrt {1-c^2 x^2} (a+b \arccos (c x))}{1536 c^3}+\frac {73 b d^2 x^3 \sqrt {1-c^2 x^2} (a+b \arccos (c x))}{2304 c}-\frac {25}{576} b c d^2 x^5 \sqrt {1-c^2 x^2} (a+b \arccos (c x))-\frac {1}{32} b c d^2 x^5 \left (1-c^2 x^2\right )^{3/2} (a+b \arccos (c x))-\frac {73 d^2 (a+b \arccos (c x))^2}{3072 c^4}+\frac {1}{24} d^2 x^4 (a+b \arccos (c x))^2+\frac {1}{12} d^2 x^4 \left (1-c^2 x^2\right ) (a+b \arccos (c x))^2+\frac {1}{8} d^2 x^4 \left (1-c^2 x^2\right )^2 (a+b \arccos (c x))^2 \] Output:
-73/3072*b^2*d^2*x^2/c^2-73/9216*b^2*d^2*x^4+43/3456*b^2*c^2*d^2*x^6-1/256 *b^2*c^4*d^2*x^8+73/1536*b*d^2*x*(-c^2*x^2+1)^(1/2)*(a+b*arccos(c*x))/c^3+ 73/2304*b*d^2*x^3*(-c^2*x^2+1)^(1/2)*(a+b*arccos(c*x))/c-25/576*b*c*d^2*x^ 5*(-c^2*x^2+1)^(1/2)*(a+b*arccos(c*x))-1/32*b*c*d^2*x^5*(-c^2*x^2+1)^(3/2) *(a+b*arccos(c*x))-73/3072*d^2*(a+b*arccos(c*x))^2/c^4+1/24*d^2*x^4*(a+b*a rccos(c*x))^2+1/12*d^2*x^4*(-c^2*x^2+1)*(a+b*arccos(c*x))^2+1/8*d^2*x^4*(- c^2*x^2+1)^2*(a+b*arccos(c*x))^2
Time = 0.25 (sec) , antiderivative size = 246, normalized size of antiderivative = 0.81 \[ \int x^3 \left (d-c^2 d x^2\right )^2 (a+b \arccos (c x))^2 \, dx=\frac {d^2 \left (-c x \left (-1152 a^2 c^3 x^3 \left (6-8 c^2 x^2+3 c^4 x^4\right )+b^2 c x \left (657+219 c^2 x^2-344 c^4 x^4+108 c^6 x^6\right )+6 a b \sqrt {1-c^2 x^2} \left (219+146 c^2 x^2-344 c^4 x^4+144 c^6 x^6\right )\right )+6 b c x \left (384 a c^3 x^3 \left (6-8 c^2 x^2+3 c^4 x^4\right )-b \sqrt {1-c^2 x^2} \left (219+146 c^2 x^2-344 c^4 x^4+144 c^6 x^6\right )\right ) \arccos (c x)+9 b^2 \left (-73+768 c^4 x^4-1024 c^6 x^6+384 c^8 x^8\right ) \arccos (c x)^2+1314 a b \arcsin (c x)\right )}{27648 c^4} \] Input:
Integrate[x^3*(d - c^2*d*x^2)^2*(a + b*ArcCos[c*x])^2,x]
Output:
(d^2*(-(c*x*(-1152*a^2*c^3*x^3*(6 - 8*c^2*x^2 + 3*c^4*x^4) + b^2*c*x*(657 + 219*c^2*x^2 - 344*c^4*x^4 + 108*c^6*x^6) + 6*a*b*Sqrt[1 - c^2*x^2]*(219 + 146*c^2*x^2 - 344*c^4*x^4 + 144*c^6*x^6))) + 6*b*c*x*(384*a*c^3*x^3*(6 - 8*c^2*x^2 + 3*c^4*x^4) - b*Sqrt[1 - c^2*x^2]*(219 + 146*c^2*x^2 - 344*c^4 *x^4 + 144*c^6*x^6))*ArcCos[c*x] + 9*b^2*(-73 + 768*c^4*x^4 - 1024*c^6*x^6 + 384*c^8*x^8)*ArcCos[c*x]^2 + 1314*a*b*ArcSin[c*x]))/(27648*c^4)
Time = 2.42 (sec) , antiderivative size = 594, normalized size of antiderivative = 1.97, number of steps used = 13, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.481, Rules used = {5203, 27, 5203, 244, 2009, 5139, 5199, 15, 5211, 15, 5211, 15, 5153}
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.
| \(\displaystyle \int x^3 \left (d-c^2 d x^2\right )^2 (a+b \arccos (c x))^2 \, dx\) |
| \(\Big \downarrow \) 5203 |
| \(\displaystyle \frac {1}{4} b c d^2 \int x^4 \left (1-c^2 x^2\right )^{3/2} (a+b \arccos (c x))dx+\frac {1}{2} d \int d x^3 \left (1-c^2 x^2\right ) (a+b \arccos (c x))^2dx+\frac {1}{8} d^2 x^4 \left (1-c^2 x^2\right )^2 (a+b \arccos (c x))^2\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{4} b c d^2 \int x^4 \left (1-c^2 x^2\right )^{3/2} (a+b \arccos (c x))dx+\frac {1}{2} d^2 \int x^3 \left (1-c^2 x^2\right ) (a+b \arccos (c x))^2dx+\frac {1}{8} d^2 x^4 \left (1-c^2 x^2\right )^2 (a+b \arccos (c x))^2\) |
| \(\Big \downarrow \) 5203 |
| \(\displaystyle \frac {1}{4} b c d^2 \left (\frac {3}{8} \int x^4 \sqrt {1-c^2 x^2} (a+b \arccos (c x))dx+\frac {1}{8} b c \int x^5 \left (1-c^2 x^2\right )dx+\frac {1}{8} x^5 \left (1-c^2 x^2\right )^{3/2} (a+b \arccos (c x))\right )+\frac {1}{2} d^2 \left (\frac {1}{3} b c \int x^4 \sqrt {1-c^2 x^2} (a+b \arccos (c x))dx+\frac {1}{3} \int x^3 (a+b \arccos (c x))^2dx+\frac {1}{6} x^4 \left (1-c^2 x^2\right ) (a+b \arccos (c x))^2\right )+\frac {1}{8} d^2 x^4 \left (1-c^2 x^2\right )^2 (a+b \arccos (c x))^2\) |
| \(\Big \downarrow \) 244 |
| \(\displaystyle \frac {1}{2} d^2 \left (\frac {1}{3} b c \int x^4 \sqrt {1-c^2 x^2} (a+b \arccos (c x))dx+\frac {1}{3} \int x^3 (a+b \arccos (c x))^2dx+\frac {1}{6} x^4 \left (1-c^2 x^2\right ) (a+b \arccos (c x))^2\right )+\frac {1}{4} b c d^2 \left (\frac {3}{8} \int x^4 \sqrt {1-c^2 x^2} (a+b \arccos (c x))dx+\frac {1}{8} b c \int \left (x^5-c^2 x^7\right )dx+\frac {1}{8} x^5 \left (1-c^2 x^2\right )^{3/2} (a+b \arccos (c x))\right )+\frac {1}{8} d^2 x^4 \left (1-c^2 x^2\right )^2 (a+b \arccos (c x))^2\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{2} d^2 \left (\frac {1}{3} b c \int x^4 \sqrt {1-c^2 x^2} (a+b \arccos (c x))dx+\frac {1}{3} \int x^3 (a+b \arccos (c x))^2dx+\frac {1}{6} x^4 \left (1-c^2 x^2\right ) (a+b \arccos (c x))^2\right )+\frac {1}{4} b c d^2 \left (\frac {3}{8} \int x^4 \sqrt {1-c^2 x^2} (a+b \arccos (c x))dx+\frac {1}{8} x^5 \left (1-c^2 x^2\right )^{3/2} (a+b \arccos (c x))+\frac {1}{8} b c \left (\frac {x^6}{6}-\frac {c^2 x^8}{8}\right )\right )+\frac {1}{8} d^2 x^4 \left (1-c^2 x^2\right )^2 (a+b \arccos (c x))^2\) |
| \(\Big \downarrow \) 5139 |
| \(\displaystyle \frac {1}{2} d^2 \left (\frac {1}{3} \left (\frac {1}{2} b c \int \frac {x^4 (a+b \arccos (c x))}{\sqrt {1-c^2 x^2}}dx+\frac {1}{4} x^4 (a+b \arccos (c x))^2\right )+\frac {1}{3} b c \int x^4 \sqrt {1-c^2 x^2} (a+b \arccos (c x))dx+\frac {1}{6} x^4 \left (1-c^2 x^2\right ) (a+b \arccos (c x))^2\right )+\frac {1}{4} b c d^2 \left (\frac {3}{8} \int x^4 \sqrt {1-c^2 x^2} (a+b \arccos (c x))dx+\frac {1}{8} x^5 \left (1-c^2 x^2\right )^{3/2} (a+b \arccos (c x))+\frac {1}{8} b c \left (\frac {x^6}{6}-\frac {c^2 x^8}{8}\right )\right )+\frac {1}{8} d^2 x^4 \left (1-c^2 x^2\right )^2 (a+b \arccos (c x))^2\) |
| \(\Big \downarrow \) 5199 |
| \(\displaystyle \frac {1}{2} d^2 \left (\frac {1}{3} \left (\frac {1}{2} b c \int \frac {x^4 (a+b \arccos (c x))}{\sqrt {1-c^2 x^2}}dx+\frac {1}{4} x^4 (a+b \arccos (c x))^2\right )+\frac {1}{3} b c \left (\frac {1}{6} \int \frac {x^4 (a+b \arccos (c x))}{\sqrt {1-c^2 x^2}}dx+\frac {1}{6} b c \int x^5dx+\frac {1}{6} x^5 \sqrt {1-c^2 x^2} (a+b \arccos (c x))\right )+\frac {1}{6} x^4 \left (1-c^2 x^2\right ) (a+b \arccos (c x))^2\right )+\frac {1}{4} b c d^2 \left (\frac {3}{8} \left (\frac {1}{6} \int \frac {x^4 (a+b \arccos (c x))}{\sqrt {1-c^2 x^2}}dx+\frac {1}{6} b c \int x^5dx+\frac {1}{6} x^5 \sqrt {1-c^2 x^2} (a+b \arccos (c x))\right )+\frac {1}{8} x^5 \left (1-c^2 x^2\right )^{3/2} (a+b \arccos (c x))+\frac {1}{8} b c \left (\frac {x^6}{6}-\frac {c^2 x^8}{8}\right )\right )+\frac {1}{8} d^2 x^4 \left (1-c^2 x^2\right )^2 (a+b \arccos (c x))^2\) |
| \(\Big \downarrow \) 15 |
| \(\displaystyle \frac {1}{2} d^2 \left (\frac {1}{3} \left (\frac {1}{2} b c \int \frac {x^4 (a+b \arccos (c x))}{\sqrt {1-c^2 x^2}}dx+\frac {1}{4} x^4 (a+b \arccos (c x))^2\right )+\frac {1}{3} b c \left (\frac {1}{6} \int \frac {x^4 (a+b \arccos (c x))}{\sqrt {1-c^2 x^2}}dx+\frac {1}{6} x^5 \sqrt {1-c^2 x^2} (a+b \arccos (c x))+\frac {1}{36} b c x^6\right )+\frac {1}{6} x^4 \left (1-c^2 x^2\right ) (a+b \arccos (c x))^2\right )+\frac {1}{4} b c d^2 \left (\frac {3}{8} \left (\frac {1}{6} \int \frac {x^4 (a+b \arccos (c x))}{\sqrt {1-c^2 x^2}}dx+\frac {1}{6} x^5 \sqrt {1-c^2 x^2} (a+b \arccos (c x))+\frac {1}{36} b c x^6\right )+\frac {1}{8} x^5 \left (1-c^2 x^2\right )^{3/2} (a+b \arccos (c x))+\frac {1}{8} b c \left (\frac {x^6}{6}-\frac {c^2 x^8}{8}\right )\right )+\frac {1}{8} d^2 x^4 \left (1-c^2 x^2\right )^2 (a+b \arccos (c x))^2\) |
| \(\Big \downarrow \) 5211 |
| \(\displaystyle \frac {1}{4} b c d^2 \left (\frac {3}{8} \left (\frac {1}{6} \left (\frac {3 \int \frac {x^2 (a+b \arccos (c x))}{\sqrt {1-c^2 x^2}}dx}{4 c^2}-\frac {b \int x^3dx}{4 c}-\frac {x^3 \sqrt {1-c^2 x^2} (a+b \arccos (c x))}{4 c^2}\right )+\frac {1}{6} x^5 \sqrt {1-c^2 x^2} (a+b \arccos (c x))+\frac {1}{36} b c x^6\right )+\frac {1}{8} x^5 \left (1-c^2 x^2\right )^{3/2} (a+b \arccos (c x))+\frac {1}{8} b c \left (\frac {x^6}{6}-\frac {c^2 x^8}{8}\right )\right )+\frac {1}{2} d^2 \left (\frac {1}{3} \left (\frac {1}{2} b c \left (\frac {3 \int \frac {x^2 (a+b \arccos (c x))}{\sqrt {1-c^2 x^2}}dx}{4 c^2}-\frac {b \int x^3dx}{4 c}-\frac {x^3 \sqrt {1-c^2 x^2} (a+b \arccos (c x))}{4 c^2}\right )+\frac {1}{4} x^4 (a+b \arccos (c x))^2\right )+\frac {1}{3} b c \left (\frac {1}{6} \left (\frac {3 \int \frac {x^2 (a+b \arccos (c x))}{\sqrt {1-c^2 x^2}}dx}{4 c^2}-\frac {b \int x^3dx}{4 c}-\frac {x^3 \sqrt {1-c^2 x^2} (a+b \arccos (c x))}{4 c^2}\right )+\frac {1}{6} x^5 \sqrt {1-c^2 x^2} (a+b \arccos (c x))+\frac {1}{36} b c x^6\right )+\frac {1}{6} x^4 \left (1-c^2 x^2\right ) (a+b \arccos (c x))^2\right )+\frac {1}{8} d^2 x^4 \left (1-c^2 x^2\right )^2 (a+b \arccos (c x))^2\) |
| \(\Big \downarrow \) 15 |
| \(\displaystyle \frac {1}{2} d^2 \left (\frac {1}{3} \left (\frac {1}{2} b c \left (\frac {3 \int \frac {x^2 (a+b \arccos (c x))}{\sqrt {1-c^2 x^2}}dx}{4 c^2}-\frac {x^3 \sqrt {1-c^2 x^2} (a+b \arccos (c x))}{4 c^2}-\frac {b x^4}{16 c}\right )+\frac {1}{4} x^4 (a+b \arccos (c x))^2\right )+\frac {1}{3} b c \left (\frac {1}{6} \left (\frac {3 \int \frac {x^2 (a+b \arccos (c x))}{\sqrt {1-c^2 x^2}}dx}{4 c^2}-\frac {x^3 \sqrt {1-c^2 x^2} (a+b \arccos (c x))}{4 c^2}-\frac {b x^4}{16 c}\right )+\frac {1}{6} x^5 \sqrt {1-c^2 x^2} (a+b \arccos (c x))+\frac {1}{36} b c x^6\right )+\frac {1}{6} x^4 \left (1-c^2 x^2\right ) (a+b \arccos (c x))^2\right )+\frac {1}{4} b c d^2 \left (\frac {3}{8} \left (\frac {1}{6} \left (\frac {3 \int \frac {x^2 (a+b \arccos (c x))}{\sqrt {1-c^2 x^2}}dx}{4 c^2}-\frac {x^3 \sqrt {1-c^2 x^2} (a+b \arccos (c x))}{4 c^2}-\frac {b x^4}{16 c}\right )+\frac {1}{6} x^5 \sqrt {1-c^2 x^2} (a+b \arccos (c x))+\frac {1}{36} b c x^6\right )+\frac {1}{8} x^5 \left (1-c^2 x^2\right )^{3/2} (a+b \arccos (c x))+\frac {1}{8} b c \left (\frac {x^6}{6}-\frac {c^2 x^8}{8}\right )\right )+\frac {1}{8} d^2 x^4 \left (1-c^2 x^2\right )^2 (a+b \arccos (c x))^2\) |
| \(\Big \downarrow \) 5211 |
| \(\displaystyle \frac {1}{2} d^2 \left (\frac {1}{3} \left (\frac {1}{2} b c \left (\frac {3 \left (\frac {\int \frac {a+b \arccos (c x)}{\sqrt {1-c^2 x^2}}dx}{2 c^2}-\frac {b \int xdx}{2 c}-\frac {x \sqrt {1-c^2 x^2} (a+b \arccos (c x))}{2 c^2}\right )}{4 c^2}-\frac {x^3 \sqrt {1-c^2 x^2} (a+b \arccos (c x))}{4 c^2}-\frac {b x^4}{16 c}\right )+\frac {1}{4} x^4 (a+b \arccos (c x))^2\right )+\frac {1}{3} b c \left (\frac {1}{6} \left (\frac {3 \left (\frac {\int \frac {a+b \arccos (c x)}{\sqrt {1-c^2 x^2}}dx}{2 c^2}-\frac {b \int xdx}{2 c}-\frac {x \sqrt {1-c^2 x^2} (a+b \arccos (c x))}{2 c^2}\right )}{4 c^2}-\frac {x^3 \sqrt {1-c^2 x^2} (a+b \arccos (c x))}{4 c^2}-\frac {b x^4}{16 c}\right )+\frac {1}{6} x^5 \sqrt {1-c^2 x^2} (a+b \arccos (c x))+\frac {1}{36} b c x^6\right )+\frac {1}{6} x^4 \left (1-c^2 x^2\right ) (a+b \arccos (c x))^2\right )+\frac {1}{4} b c d^2 \left (\frac {3}{8} \left (\frac {1}{6} \left (\frac {3 \left (\frac {\int \frac {a+b \arccos (c x)}{\sqrt {1-c^2 x^2}}dx}{2 c^2}-\frac {b \int xdx}{2 c}-\frac {x \sqrt {1-c^2 x^2} (a+b \arccos (c x))}{2 c^2}\right )}{4 c^2}-\frac {x^3 \sqrt {1-c^2 x^2} (a+b \arccos (c x))}{4 c^2}-\frac {b x^4}{16 c}\right )+\frac {1}{6} x^5 \sqrt {1-c^2 x^2} (a+b \arccos (c x))+\frac {1}{36} b c x^6\right )+\frac {1}{8} x^5 \left (1-c^2 x^2\right )^{3/2} (a+b \arccos (c x))+\frac {1}{8} b c \left (\frac {x^6}{6}-\frac {c^2 x^8}{8}\right )\right )+\frac {1}{8} d^2 x^4 \left (1-c^2 x^2\right )^2 (a+b \arccos (c x))^2\) |
| \(\Big \downarrow \) 15 |
| \(\displaystyle \frac {1}{2} d^2 \left (\frac {1}{3} \left (\frac {1}{2} b c \left (\frac {3 \left (\frac {\int \frac {a+b \arccos (c x)}{\sqrt {1-c^2 x^2}}dx}{2 c^2}-\frac {x \sqrt {1-c^2 x^2} (a+b \arccos (c x))}{2 c^2}-\frac {b x^2}{4 c}\right )}{4 c^2}-\frac {x^3 \sqrt {1-c^2 x^2} (a+b \arccos (c x))}{4 c^2}-\frac {b x^4}{16 c}\right )+\frac {1}{4} x^4 (a+b \arccos (c x))^2\right )+\frac {1}{3} b c \left (\frac {1}{6} \left (\frac {3 \left (\frac {\int \frac {a+b \arccos (c x)}{\sqrt {1-c^2 x^2}}dx}{2 c^2}-\frac {x \sqrt {1-c^2 x^2} (a+b \arccos (c x))}{2 c^2}-\frac {b x^2}{4 c}\right )}{4 c^2}-\frac {x^3 \sqrt {1-c^2 x^2} (a+b \arccos (c x))}{4 c^2}-\frac {b x^4}{16 c}\right )+\frac {1}{6} x^5 \sqrt {1-c^2 x^2} (a+b \arccos (c x))+\frac {1}{36} b c x^6\right )+\frac {1}{6} x^4 \left (1-c^2 x^2\right ) (a+b \arccos (c x))^2\right )+\frac {1}{4} b c d^2 \left (\frac {3}{8} \left (\frac {1}{6} \left (\frac {3 \left (\frac {\int \frac {a+b \arccos (c x)}{\sqrt {1-c^2 x^2}}dx}{2 c^2}-\frac {x \sqrt {1-c^2 x^2} (a+b \arccos (c x))}{2 c^2}-\frac {b x^2}{4 c}\right )}{4 c^2}-\frac {x^3 \sqrt {1-c^2 x^2} (a+b \arccos (c x))}{4 c^2}-\frac {b x^4}{16 c}\right )+\frac {1}{6} x^5 \sqrt {1-c^2 x^2} (a+b \arccos (c x))+\frac {1}{36} b c x^6\right )+\frac {1}{8} x^5 \left (1-c^2 x^2\right )^{3/2} (a+b \arccos (c x))+\frac {1}{8} b c \left (\frac {x^6}{6}-\frac {c^2 x^8}{8}\right )\right )+\frac {1}{8} d^2 x^4 \left (1-c^2 x^2\right )^2 (a+b \arccos (c x))^2\) |
| \(\Big \downarrow \) 5153 |
| \(\displaystyle \frac {1}{8} d^2 x^4 \left (1-c^2 x^2\right )^2 (a+b \arccos (c x))^2+\frac {1}{2} d^2 \left (\frac {1}{6} x^4 \left (1-c^2 x^2\right ) (a+b \arccos (c x))^2+\frac {1}{3} \left (\frac {1}{2} b c \left (-\frac {x^3 \sqrt {1-c^2 x^2} (a+b \arccos (c x))}{4 c^2}+\frac {3 \left (-\frac {(a+b \arccos (c x))^2}{4 b c^3}-\frac {x \sqrt {1-c^2 x^2} (a+b \arccos (c x))}{2 c^2}-\frac {b x^2}{4 c}\right )}{4 c^2}-\frac {b x^4}{16 c}\right )+\frac {1}{4} x^4 (a+b \arccos (c x))^2\right )+\frac {1}{3} b c \left (\frac {1}{6} x^5 \sqrt {1-c^2 x^2} (a+b \arccos (c x))+\frac {1}{6} \left (-\frac {x^3 \sqrt {1-c^2 x^2} (a+b \arccos (c x))}{4 c^2}+\frac {3 \left (-\frac {(a+b \arccos (c x))^2}{4 b c^3}-\frac {x \sqrt {1-c^2 x^2} (a+b \arccos (c x))}{2 c^2}-\frac {b x^2}{4 c}\right )}{4 c^2}-\frac {b x^4}{16 c}\right )+\frac {1}{36} b c x^6\right )\right )+\frac {1}{4} b c d^2 \left (\frac {1}{8} x^5 \left (1-c^2 x^2\right )^{3/2} (a+b \arccos (c x))+\frac {3}{8} \left (\frac {1}{6} x^5 \sqrt {1-c^2 x^2} (a+b \arccos (c x))+\frac {1}{6} \left (-\frac {x^3 \sqrt {1-c^2 x^2} (a+b \arccos (c x))}{4 c^2}+\frac {3 \left (-\frac {(a+b \arccos (c x))^2}{4 b c^3}-\frac {x \sqrt {1-c^2 x^2} (a+b \arccos (c x))}{2 c^2}-\frac {b x^2}{4 c}\right )}{4 c^2}-\frac {b x^4}{16 c}\right )+\frac {1}{36} b c x^6\right )+\frac {1}{8} b c \left (\frac {x^6}{6}-\frac {c^2 x^8}{8}\right )\right )\) |
Input:
Int[x^3*(d - c^2*d*x^2)^2*(a + b*ArcCos[c*x])^2,x]
Output:
(d^2*x^4*(1 - c^2*x^2)^2*(a + b*ArcCos[c*x])^2)/8 + (b*c*d^2*((b*c*(x^6/6 - (c^2*x^8)/8))/8 + (x^5*(1 - c^2*x^2)^(3/2)*(a + b*ArcCos[c*x]))/8 + (3*( (b*c*x^6)/36 + (x^5*Sqrt[1 - c^2*x^2]*(a + b*ArcCos[c*x]))/6 + (-1/16*(b*x ^4)/c - (x^3*Sqrt[1 - c^2*x^2]*(a + b*ArcCos[c*x]))/(4*c^2) + (3*(-1/4*(b* x^2)/c - (x*Sqrt[1 - c^2*x^2]*(a + b*ArcCos[c*x]))/(2*c^2) - (a + b*ArcCos [c*x])^2/(4*b*c^3)))/(4*c^2))/6))/8))/4 + (d^2*((x^4*(1 - c^2*x^2)*(a + b* ArcCos[c*x])^2)/6 + (b*c*((b*c*x^6)/36 + (x^5*Sqrt[1 - c^2*x^2]*(a + b*Arc Cos[c*x]))/6 + (-1/16*(b*x^4)/c - (x^3*Sqrt[1 - c^2*x^2]*(a + b*ArcCos[c*x ]))/(4*c^2) + (3*(-1/4*(b*x^2)/c - (x*Sqrt[1 - c^2*x^2]*(a + b*ArcCos[c*x] ))/(2*c^2) - (a + b*ArcCos[c*x])^2/(4*b*c^3)))/(4*c^2))/6))/3 + ((x^4*(a + b*ArcCos[c*x])^2)/4 + (b*c*(-1/16*(b*x^4)/c - (x^3*Sqrt[1 - c^2*x^2]*(a + b*ArcCos[c*x]))/(4*c^2) + (3*(-1/4*(b*x^2)/c - (x*Sqrt[1 - c^2*x^2]*(a + b*ArcCos[c*x]))/(2*c^2) - (a + b*ArcCos[c*x])^2/(4*b*c^3)))/(4*c^2)))/2)/3 ))/2
Int[(a_.)*(x_)^(m_.), x_Symbol] :> Simp[a*(x^(m + 1)/(m + 1)), x] /; FreeQ[ {a, m}, x] && NeQ[m, -1]
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Int[Expand Integrand[(c*x)^m*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, m}, x] && IGtQ[p , 0]
Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_.)*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[(d*x)^(m + 1)*((a + b*ArcCos[c*x])^n/(d*(m + 1))), x] + Simp[b*c*(n /(d*(m + 1))) Int[(d*x)^(m + 1)*((a + b*ArcCos[c*x])^(n - 1)/Sqrt[1 - c^2 *x^2]), x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] && NeQ[m, -1]
Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_.)/Sqrt[(d_) + (e_.)*(x_)^2], x_S ymbol] :> Simp[(-(b*c*(n + 1))^(-1))*Simp[Sqrt[1 - c^2*x^2]/Sqrt[d + e*x^2] ]*(a + b*ArcCos[c*x])^(n + 1), x] /; FreeQ[{a, b, c, d, e, n}, x] && EqQ[c^ 2*d + e, 0] && NeQ[n, -1]
Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Simp[(f*x)^(m + 1)*Sqrt[d + e*x^2]*((a + b*ArcC os[c*x])^n/(f*(m + 2))), x] + (Simp[(1/(m + 2))*Simp[Sqrt[d + e*x^2]/Sqrt[1 - c^2*x^2]] Int[(f*x)^m*((a + b*ArcCos[c*x])^n/Sqrt[1 - c^2*x^2]), x], x ] + Simp[b*c*(n/(f*(m + 2)))*Simp[Sqrt[d + e*x^2]/Sqrt[1 - c^2*x^2]] Int[ (f*x)^(m + 1)*(a + b*ArcCos[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && (IGtQ[m, -2] || EqQ[n, 1])
Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*((d_) + (e_. )*(x_)^2)^(p_.), x_Symbol] :> Simp[(f*x)^(m + 1)*(d + e*x^2)^p*((a + b*ArcC os[c*x])^n/(f*(m + 2*p + 1))), x] + (Simp[2*d*(p/(m + 2*p + 1)) Int[(f*x) ^m*(d + e*x^2)^(p - 1)*(a + b*ArcCos[c*x])^n, x], x] + Simp[b*c*(n/(f*(m + 2*p + 1)))*Simp[(d + e*x^2)^p/(1 - c^2*x^2)^p] Int[(f*x)^(m + 1)*(1 - c^2 *x^2)^(p - 1/2)*(a + b*ArcCos[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && GtQ[p, 0] && !LtQ[m, -1]
Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*((d_) + (e_. )*(x_)^2)^(p_), x_Symbol] :> Simp[f*(f*x)^(m - 1)*(d + e*x^2)^(p + 1)*((a + b*ArcCos[c*x])^n/(e*(m + 2*p + 1))), x] + (Simp[f^2*((m - 1)/(c^2*(m + 2*p + 1))) Int[(f*x)^(m - 2)*(d + e*x^2)^p*(a + b*ArcCos[c*x])^n, x], x] - S imp[b*f*(n/(c*(m + 2*p + 1)))*Simp[(d + e*x^2)^p/(1 - c^2*x^2)^p] Int[(f* x)^(m - 1)*(1 - c^2*x^2)^(p + 1/2)*(a + b*ArcCos[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d, e, f, p}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && IGtQ[m , 1] && NeQ[m + 2*p + 1, 0]
Time = 0.42 (sec) , antiderivative size = 439, normalized size of antiderivative = 1.45
| method | result | size |
| parts | \(d^{2} a^{2} \left (\frac {1}{8} c^{4} x^{8}-\frac {1}{3} c^{2} x^{6}+\frac {1}{4} x^{4}\right )+\frac {d^{2} b^{2} \left (\frac {\arccos \left (c x \right )^{2} \left (c^{2} x^{2}-1\right )^{3}}{6}+\frac {\arccos \left (c x \right ) \left (-8 c^{5} x^{5} \sqrt {-c^{2} x^{2}+1}+26 c^{3} x^{3} \sqrt {-c^{2} x^{2}+1}-33 c x \sqrt {-c^{2} x^{2}+1}+15 \arccos \left (c x \right )\right )}{144}-\frac {55 \arccos \left (c x \right )^{2}}{3072}-\frac {c^{6} x^{6}}{108}+\frac {13 c^{4} x^{4}}{288}-\frac {247 c^{2} x^{2}}{3072}+\frac {\arccos \left (c x \right )^{2} \left (c^{2} x^{2}-1\right )^{4}}{8}-\frac {\arccos \left (c x \right ) \left (48 c^{7} x^{7} \sqrt {-c^{2} x^{2}+1}-200 c^{5} x^{5} \sqrt {-c^{2} x^{2}+1}+326 c^{3} x^{3} \sqrt {-c^{2} x^{2}+1}-279 c x \sqrt {-c^{2} x^{2}+1}+105 \arccos \left (c x \right )\right )}{1536}-\frac {\left (c^{2} x^{2}-1\right )^{4}}{256}+\frac {7 \left (c^{2} x^{2}-1\right )^{3}}{1152}-\frac {35 \left (c^{2} x^{2}-1\right )^{2}}{3072}-\frac {35}{1024}\right )}{c^{4}}+\frac {2 d^{2} a b \left (\frac {\arccos \left (c x \right ) c^{8} x^{8}}{8}-\frac {\arccos \left (c x \right ) c^{6} x^{6}}{3}+\frac {c^{4} x^{4} \arccos \left (c x \right )}{4}-\frac {73 c^{3} x^{3} \sqrt {-c^{2} x^{2}+1}}{4608}-\frac {73 c x \sqrt {-c^{2} x^{2}+1}}{3072}+\frac {73 \arcsin \left (c x \right )}{3072}+\frac {43 c^{5} x^{5} \sqrt {-c^{2} x^{2}+1}}{1152}-\frac {c^{7} x^{7} \sqrt {-c^{2} x^{2}+1}}{64}\right )}{c^{4}}\) | \(439\) |
| derivativedivides | \(\frac {d^{2} a^{2} \left (\frac {1}{8} c^{8} x^{8}-\frac {1}{3} c^{6} x^{6}+\frac {1}{4} c^{4} x^{4}\right )+d^{2} b^{2} \left (\frac {\arccos \left (c x \right )^{2} \left (c^{2} x^{2}-1\right )^{3}}{6}+\frac {\arccos \left (c x \right ) \left (-8 c^{5} x^{5} \sqrt {-c^{2} x^{2}+1}+26 c^{3} x^{3} \sqrt {-c^{2} x^{2}+1}-33 c x \sqrt {-c^{2} x^{2}+1}+15 \arccos \left (c x \right )\right )}{144}-\frac {55 \arccos \left (c x \right )^{2}}{3072}-\frac {c^{6} x^{6}}{108}+\frac {13 c^{4} x^{4}}{288}-\frac {247 c^{2} x^{2}}{3072}+\frac {\arccos \left (c x \right )^{2} \left (c^{2} x^{2}-1\right )^{4}}{8}-\frac {\arccos \left (c x \right ) \left (48 c^{7} x^{7} \sqrt {-c^{2} x^{2}+1}-200 c^{5} x^{5} \sqrt {-c^{2} x^{2}+1}+326 c^{3} x^{3} \sqrt {-c^{2} x^{2}+1}-279 c x \sqrt {-c^{2} x^{2}+1}+105 \arccos \left (c x \right )\right )}{1536}-\frac {\left (c^{2} x^{2}-1\right )^{4}}{256}+\frac {7 \left (c^{2} x^{2}-1\right )^{3}}{1152}-\frac {35 \left (c^{2} x^{2}-1\right )^{2}}{3072}-\frac {35}{1024}\right )+2 d^{2} a b \left (\frac {\arccos \left (c x \right ) c^{8} x^{8}}{8}-\frac {\arccos \left (c x \right ) c^{6} x^{6}}{3}+\frac {c^{4} x^{4} \arccos \left (c x \right )}{4}-\frac {73 c^{3} x^{3} \sqrt {-c^{2} x^{2}+1}}{4608}-\frac {73 c x \sqrt {-c^{2} x^{2}+1}}{3072}+\frac {73 \arcsin \left (c x \right )}{3072}+\frac {43 c^{5} x^{5} \sqrt {-c^{2} x^{2}+1}}{1152}-\frac {c^{7} x^{7} \sqrt {-c^{2} x^{2}+1}}{64}\right )}{c^{4}}\) | \(440\) |
| default | \(\frac {d^{2} a^{2} \left (\frac {1}{8} c^{8} x^{8}-\frac {1}{3} c^{6} x^{6}+\frac {1}{4} c^{4} x^{4}\right )+d^{2} b^{2} \left (\frac {\arccos \left (c x \right )^{2} \left (c^{2} x^{2}-1\right )^{3}}{6}+\frac {\arccos \left (c x \right ) \left (-8 c^{5} x^{5} \sqrt {-c^{2} x^{2}+1}+26 c^{3} x^{3} \sqrt {-c^{2} x^{2}+1}-33 c x \sqrt {-c^{2} x^{2}+1}+15 \arccos \left (c x \right )\right )}{144}-\frac {55 \arccos \left (c x \right )^{2}}{3072}-\frac {c^{6} x^{6}}{108}+\frac {13 c^{4} x^{4}}{288}-\frac {247 c^{2} x^{2}}{3072}+\frac {\arccos \left (c x \right )^{2} \left (c^{2} x^{2}-1\right )^{4}}{8}-\frac {\arccos \left (c x \right ) \left (48 c^{7} x^{7} \sqrt {-c^{2} x^{2}+1}-200 c^{5} x^{5} \sqrt {-c^{2} x^{2}+1}+326 c^{3} x^{3} \sqrt {-c^{2} x^{2}+1}-279 c x \sqrt {-c^{2} x^{2}+1}+105 \arccos \left (c x \right )\right )}{1536}-\frac {\left (c^{2} x^{2}-1\right )^{4}}{256}+\frac {7 \left (c^{2} x^{2}-1\right )^{3}}{1152}-\frac {35 \left (c^{2} x^{2}-1\right )^{2}}{3072}-\frac {35}{1024}\right )+2 d^{2} a b \left (\frac {\arccos \left (c x \right ) c^{8} x^{8}}{8}-\frac {\arccos \left (c x \right ) c^{6} x^{6}}{3}+\frac {c^{4} x^{4} \arccos \left (c x \right )}{4}-\frac {73 c^{3} x^{3} \sqrt {-c^{2} x^{2}+1}}{4608}-\frac {73 c x \sqrt {-c^{2} x^{2}+1}}{3072}+\frac {73 \arcsin \left (c x \right )}{3072}+\frac {43 c^{5} x^{5} \sqrt {-c^{2} x^{2}+1}}{1152}-\frac {c^{7} x^{7} \sqrt {-c^{2} x^{2}+1}}{64}\right )}{c^{4}}\) | \(440\) |
| orering | \(\frac {\left (18252 c^{10} x^{10}-69716 c^{8} x^{8}+87751 c^{6} x^{6}-492 c^{4} x^{4}-36135 c^{2} x^{2}+13140\right ) \left (-c^{2} d \,x^{2}+d \right )^{2} \left (a +b \arccos \left (c x \right )\right )^{2}}{55296 c^{4} \left (c x -1\right ) \left (c x +1\right ) \left (c^{2} x^{2}-1\right )^{2}}-\frac {\left (2268 c^{8} x^{8}-8048 c^{6} x^{6}+7851 c^{4} x^{4}+7665 c^{2} x^{2}-5256\right ) \left (3 x^{2} \left (-c^{2} d \,x^{2}+d \right )^{2} \left (a +b \arccos \left (c x \right )\right )^{2}-4 x^{4} \left (-c^{2} d \,x^{2}+d \right ) \left (a +b \arccos \left (c x \right )\right )^{2} d \,c^{2}-\frac {2 x^{3} \left (-c^{2} d \,x^{2}+d \right )^{2} \left (a +b \arccos \left (c x \right )\right ) b c}{\sqrt {-c^{2} x^{2}+1}}\right )}{55296 x^{2} c^{4} \left (c x -1\right ) \left (c x +1\right ) \left (c^{2} x^{2}-1\right )}+\frac {\left (108 c^{6} x^{6}-344 c^{4} x^{4}+219 c^{2} x^{2}+657\right ) \left (6 x \left (-c^{2} d \,x^{2}+d \right )^{2} \left (a +b \arccos \left (c x \right )\right )^{2}-28 x^{3} \left (-c^{2} d \,x^{2}+d \right ) \left (a +b \arccos \left (c x \right )\right )^{2} d \,c^{2}-\frac {12 x^{2} \left (-c^{2} d \,x^{2}+d \right )^{2} \left (a +b \arccos \left (c x \right )\right ) b c}{\sqrt {-c^{2} x^{2}+1}}+8 x^{5} d^{2} c^{4} \left (a +b \arccos \left (c x \right )\right )^{2}+\frac {16 x^{4} \left (-c^{2} d \,x^{2}+d \right ) \left (a +b \arccos \left (c x \right )\right ) d \,c^{3} b}{\sqrt {-c^{2} x^{2}+1}}+\frac {2 x^{3} \left (-c^{2} d \,x^{2}+d \right )^{2} b^{2} c^{2}}{-c^{2} x^{2}+1}-\frac {2 x^{4} \left (-c^{2} d \,x^{2}+d \right )^{2} \left (a +b \arccos \left (c x \right )\right ) b \,c^{3}}{\left (-c^{2} x^{2}+1\right )^{\frac {3}{2}}}\right )}{55296 x \,c^{4} \left (c x -1\right ) \left (c x +1\right )}\) | \(548\) |
Input:
int(x^3*(-c^2*d*x^2+d)^2*(a+b*arccos(c*x))^2,x,method=_RETURNVERBOSE)
Output:
d^2*a^2*(1/8*c^4*x^8-1/3*c^2*x^6+1/4*x^4)+d^2*b^2/c^4*(1/6*arccos(c*x)^2*( c^2*x^2-1)^3+1/144*arccos(c*x)*(-8*c^5*x^5*(-c^2*x^2+1)^(1/2)+26*c^3*x^3*( -c^2*x^2+1)^(1/2)-33*c*x*(-c^2*x^2+1)^(1/2)+15*arccos(c*x))-55/3072*arccos (c*x)^2-1/108*c^6*x^6+13/288*c^4*x^4-247/3072*c^2*x^2+1/8*arccos(c*x)^2*(c ^2*x^2-1)^4-1/1536*arccos(c*x)*(48*c^7*x^7*(-c^2*x^2+1)^(1/2)-200*c^5*x^5* (-c^2*x^2+1)^(1/2)+326*c^3*x^3*(-c^2*x^2+1)^(1/2)-279*c*x*(-c^2*x^2+1)^(1/ 2)+105*arccos(c*x))-1/256*(c^2*x^2-1)^4+7/1152*(c^2*x^2-1)^3-35/3072*(c^2* x^2-1)^2-35/1024)+2*d^2*a*b/c^4*(1/8*arccos(c*x)*c^8*x^8-1/3*arccos(c*x)*c ^6*x^6+1/4*c^4*x^4*arccos(c*x)-73/4608*c^3*x^3*(-c^2*x^2+1)^(1/2)-73/3072* c*x*(-c^2*x^2+1)^(1/2)+73/3072*arcsin(c*x)+43/1152*c^5*x^5*(-c^2*x^2+1)^(1 /2)-1/64*c^7*x^7*(-c^2*x^2+1)^(1/2))
Time = 0.11 (sec) , antiderivative size = 319, normalized size of antiderivative = 1.06 \[ \int x^3 \left (d-c^2 d x^2\right )^2 (a+b \arccos (c x))^2 \, dx=\frac {108 \, {\left (32 \, a^{2} - b^{2}\right )} c^{8} d^{2} x^{8} - 8 \, {\left (1152 \, a^{2} - 43 \, b^{2}\right )} c^{6} d^{2} x^{6} + 3 \, {\left (2304 \, a^{2} - 73 \, b^{2}\right )} c^{4} d^{2} x^{4} - 657 \, b^{2} c^{2} d^{2} x^{2} + 9 \, {\left (384 \, b^{2} c^{8} d^{2} x^{8} - 1024 \, b^{2} c^{6} d^{2} x^{6} + 768 \, b^{2} c^{4} d^{2} x^{4} - 73 \, b^{2} d^{2}\right )} \arccos \left (c x\right )^{2} + 18 \, {\left (384 \, a b c^{8} d^{2} x^{8} - 1024 \, a b c^{6} d^{2} x^{6} + 768 \, a b c^{4} d^{2} x^{4} - 73 \, a b d^{2}\right )} \arccos \left (c x\right ) - 6 \, {\left (144 \, a b c^{7} d^{2} x^{7} - 344 \, a b c^{5} d^{2} x^{5} + 146 \, a b c^{3} d^{2} x^{3} + 219 \, a b c d^{2} x + {\left (144 \, b^{2} c^{7} d^{2} x^{7} - 344 \, b^{2} c^{5} d^{2} x^{5} + 146 \, b^{2} c^{3} d^{2} x^{3} + 219 \, b^{2} c d^{2} x\right )} \arccos \left (c x\right )\right )} \sqrt {-c^{2} x^{2} + 1}}{27648 \, c^{4}} \] Input:
integrate(x^3*(-c^2*d*x^2+d)^2*(a+b*arccos(c*x))^2,x, algorithm="fricas")
Output:
1/27648*(108*(32*a^2 - b^2)*c^8*d^2*x^8 - 8*(1152*a^2 - 43*b^2)*c^6*d^2*x^ 6 + 3*(2304*a^2 - 73*b^2)*c^4*d^2*x^4 - 657*b^2*c^2*d^2*x^2 + 9*(384*b^2*c ^8*d^2*x^8 - 1024*b^2*c^6*d^2*x^6 + 768*b^2*c^4*d^2*x^4 - 73*b^2*d^2)*arcc os(c*x)^2 + 18*(384*a*b*c^8*d^2*x^8 - 1024*a*b*c^6*d^2*x^6 + 768*a*b*c^4*d ^2*x^4 - 73*a*b*d^2)*arccos(c*x) - 6*(144*a*b*c^7*d^2*x^7 - 344*a*b*c^5*d^ 2*x^5 + 146*a*b*c^3*d^2*x^3 + 219*a*b*c*d^2*x + (144*b^2*c^7*d^2*x^7 - 344 *b^2*c^5*d^2*x^5 + 146*b^2*c^3*d^2*x^3 + 219*b^2*c*d^2*x)*arccos(c*x))*sqr t(-c^2*x^2 + 1))/c^4
Time = 1.29 (sec) , antiderivative size = 520, normalized size of antiderivative = 1.72 \[ \int x^3 \left (d-c^2 d x^2\right )^2 (a+b \arccos (c x))^2 \, dx=\begin {cases} \frac {a^{2} c^{4} d^{2} x^{8}}{8} - \frac {a^{2} c^{2} d^{2} x^{6}}{3} + \frac {a^{2} d^{2} x^{4}}{4} + \frac {a b c^{4} d^{2} x^{8} \operatorname {acos}{\left (c x \right )}}{4} - \frac {a b c^{3} d^{2} x^{7} \sqrt {- c^{2} x^{2} + 1}}{32} - \frac {2 a b c^{2} d^{2} x^{6} \operatorname {acos}{\left (c x \right )}}{3} + \frac {43 a b c d^{2} x^{5} \sqrt {- c^{2} x^{2} + 1}}{576} + \frac {a b d^{2} x^{4} \operatorname {acos}{\left (c x \right )}}{2} - \frac {73 a b d^{2} x^{3} \sqrt {- c^{2} x^{2} + 1}}{2304 c} - \frac {73 a b d^{2} x \sqrt {- c^{2} x^{2} + 1}}{1536 c^{3}} - \frac {73 a b d^{2} \operatorname {acos}{\left (c x \right )}}{1536 c^{4}} + \frac {b^{2} c^{4} d^{2} x^{8} \operatorname {acos}^{2}{\left (c x \right )}}{8} - \frac {b^{2} c^{4} d^{2} x^{8}}{256} - \frac {b^{2} c^{3} d^{2} x^{7} \sqrt {- c^{2} x^{2} + 1} \operatorname {acos}{\left (c x \right )}}{32} - \frac {b^{2} c^{2} d^{2} x^{6} \operatorname {acos}^{2}{\left (c x \right )}}{3} + \frac {43 b^{2} c^{2} d^{2} x^{6}}{3456} + \frac {43 b^{2} c d^{2} x^{5} \sqrt {- c^{2} x^{2} + 1} \operatorname {acos}{\left (c x \right )}}{576} + \frac {b^{2} d^{2} x^{4} \operatorname {acos}^{2}{\left (c x \right )}}{4} - \frac {73 b^{2} d^{2} x^{4}}{9216} - \frac {73 b^{2} d^{2} x^{3} \sqrt {- c^{2} x^{2} + 1} \operatorname {acos}{\left (c x \right )}}{2304 c} - \frac {73 b^{2} d^{2} x^{2}}{3072 c^{2}} - \frac {73 b^{2} d^{2} x \sqrt {- c^{2} x^{2} + 1} \operatorname {acos}{\left (c x \right )}}{1536 c^{3}} - \frac {73 b^{2} d^{2} \operatorname {acos}^{2}{\left (c x \right )}}{3072 c^{4}} & \text {for}\: c \neq 0 \\\frac {d^{2} x^{4} \left (a + \frac {\pi b}{2}\right )^{2}}{4} & \text {otherwise} \end {cases} \] Input:
integrate(x**3*(-c**2*d*x**2+d)**2*(a+b*acos(c*x))**2,x)
Output:
Piecewise((a**2*c**4*d**2*x**8/8 - a**2*c**2*d**2*x**6/3 + a**2*d**2*x**4/ 4 + a*b*c**4*d**2*x**8*acos(c*x)/4 - a*b*c**3*d**2*x**7*sqrt(-c**2*x**2 + 1)/32 - 2*a*b*c**2*d**2*x**6*acos(c*x)/3 + 43*a*b*c*d**2*x**5*sqrt(-c**2*x **2 + 1)/576 + a*b*d**2*x**4*acos(c*x)/2 - 73*a*b*d**2*x**3*sqrt(-c**2*x** 2 + 1)/(2304*c) - 73*a*b*d**2*x*sqrt(-c**2*x**2 + 1)/(1536*c**3) - 73*a*b* d**2*acos(c*x)/(1536*c**4) + b**2*c**4*d**2*x**8*acos(c*x)**2/8 - b**2*c** 4*d**2*x**8/256 - b**2*c**3*d**2*x**7*sqrt(-c**2*x**2 + 1)*acos(c*x)/32 - b**2*c**2*d**2*x**6*acos(c*x)**2/3 + 43*b**2*c**2*d**2*x**6/3456 + 43*b**2 *c*d**2*x**5*sqrt(-c**2*x**2 + 1)*acos(c*x)/576 + b**2*d**2*x**4*acos(c*x) **2/4 - 73*b**2*d**2*x**4/9216 - 73*b**2*d**2*x**3*sqrt(-c**2*x**2 + 1)*ac os(c*x)/(2304*c) - 73*b**2*d**2*x**2/(3072*c**2) - 73*b**2*d**2*x*sqrt(-c* *2*x**2 + 1)*acos(c*x)/(1536*c**3) - 73*b**2*d**2*acos(c*x)**2/(3072*c**4) , Ne(c, 0)), (d**2*x**4*(a + pi*b/2)**2/4, True))
\[ \int x^3 \left (d-c^2 d x^2\right )^2 (a+b \arccos (c x))^2 \, dx=\int { {\left (c^{2} d x^{2} - d\right )}^{2} {\left (b \arccos \left (c x\right ) + a\right )}^{2} x^{3} \,d x } \] Input:
integrate(x^3*(-c^2*d*x^2+d)^2*(a+b*arccos(c*x))^2,x, algorithm="maxima")
Output:
1/8*a^2*c^4*d^2*x^8 - 1/3*a^2*c^2*d^2*x^6 + 1/1536*(384*x^8*arccos(c*x) - (48*sqrt(-c^2*x^2 + 1)*x^7/c^2 + 56*sqrt(-c^2*x^2 + 1)*x^5/c^4 + 70*sqrt(- c^2*x^2 + 1)*x^3/c^6 + 105*sqrt(-c^2*x^2 + 1)*x/c^8 - 105*arcsin(c*x)/c^9) *c)*a*b*c^4*d^2 + 1/4*a^2*d^2*x^4 - 1/72*(48*x^6*arccos(c*x) - (8*sqrt(-c^ 2*x^2 + 1)*x^5/c^2 + 10*sqrt(-c^2*x^2 + 1)*x^3/c^4 + 15*sqrt(-c^2*x^2 + 1) *x/c^6 - 15*arcsin(c*x)/c^7)*c)*a*b*c^2*d^2 + 1/16*(8*x^4*arccos(c*x) - (2 *sqrt(-c^2*x^2 + 1)*x^3/c^2 + 3*sqrt(-c^2*x^2 + 1)*x/c^4 - 3*arcsin(c*x)/c ^5)*c)*a*b*d^2 + 1/24*(3*b^2*c^4*d^2*x^8 - 8*b^2*c^2*d^2*x^6 + 6*b^2*d^2*x ^4)*arctan2(sqrt(c*x + 1)*sqrt(-c*x + 1), c*x)^2 - integrate(1/12*(3*b^2*c ^5*d^2*x^8 - 8*b^2*c^3*d^2*x^6 + 6*b^2*c*d^2*x^4)*sqrt(c*x + 1)*sqrt(-c*x + 1)*arctan2(sqrt(c*x + 1)*sqrt(-c*x + 1), c*x)/(c^2*x^2 - 1), x)
Time = 0.17 (sec) , antiderivative size = 452, normalized size of antiderivative = 1.50 \[ \int x^3 \left (d-c^2 d x^2\right )^2 (a+b \arccos (c x))^2 \, dx=\frac {1}{8} \, b^{2} c^{4} d^{2} x^{8} \arccos \left (c x\right )^{2} + \frac {1}{4} \, a b c^{4} d^{2} x^{8} \arccos \left (c x\right ) + \frac {1}{8} \, a^{2} c^{4} d^{2} x^{8} - \frac {1}{256} \, b^{2} c^{4} d^{2} x^{8} - \frac {1}{32} \, \sqrt {-c^{2} x^{2} + 1} b^{2} c^{3} d^{2} x^{7} \arccos \left (c x\right ) - \frac {1}{32} \, \sqrt {-c^{2} x^{2} + 1} a b c^{3} d^{2} x^{7} - \frac {1}{3} \, b^{2} c^{2} d^{2} x^{6} \arccos \left (c x\right )^{2} - \frac {2}{3} \, a b c^{2} d^{2} x^{6} \arccos \left (c x\right ) - \frac {1}{3} \, a^{2} c^{2} d^{2} x^{6} + \frac {43}{3456} \, b^{2} c^{2} d^{2} x^{6} + \frac {43}{576} \, \sqrt {-c^{2} x^{2} + 1} b^{2} c d^{2} x^{5} \arccos \left (c x\right ) + \frac {43}{576} \, \sqrt {-c^{2} x^{2} + 1} a b c d^{2} x^{5} + \frac {1}{4} \, b^{2} d^{2} x^{4} \arccos \left (c x\right )^{2} + \frac {1}{2} \, a b d^{2} x^{4} \arccos \left (c x\right ) + \frac {1}{4} \, a^{2} d^{2} x^{4} - \frac {73}{9216} \, b^{2} d^{2} x^{4} - \frac {73 \, \sqrt {-c^{2} x^{2} + 1} b^{2} d^{2} x^{3} \arccos \left (c x\right )}{2304 \, c} - \frac {73 \, \sqrt {-c^{2} x^{2} + 1} a b d^{2} x^{3}}{2304 \, c} - \frac {73 \, b^{2} d^{2} x^{2}}{3072 \, c^{2}} - \frac {73 \, \sqrt {-c^{2} x^{2} + 1} b^{2} d^{2} x \arccos \left (c x\right )}{1536 \, c^{3}} - \frac {73 \, \sqrt {-c^{2} x^{2} + 1} a b d^{2} x}{1536 \, c^{3}} - \frac {73 \, b^{2} d^{2} \arccos \left (c x\right )^{2}}{3072 \, c^{4}} - \frac {73 \, a b d^{2} \arccos \left (c x\right )}{1536 \, c^{4}} + \frac {10645 \, b^{2} d^{2}}{884736 \, c^{4}} \] Input:
integrate(x^3*(-c^2*d*x^2+d)^2*(a+b*arccos(c*x))^2,x, algorithm="giac")
Output:
1/8*b^2*c^4*d^2*x^8*arccos(c*x)^2 + 1/4*a*b*c^4*d^2*x^8*arccos(c*x) + 1/8* a^2*c^4*d^2*x^8 - 1/256*b^2*c^4*d^2*x^8 - 1/32*sqrt(-c^2*x^2 + 1)*b^2*c^3* d^2*x^7*arccos(c*x) - 1/32*sqrt(-c^2*x^2 + 1)*a*b*c^3*d^2*x^7 - 1/3*b^2*c^ 2*d^2*x^6*arccos(c*x)^2 - 2/3*a*b*c^2*d^2*x^6*arccos(c*x) - 1/3*a^2*c^2*d^ 2*x^6 + 43/3456*b^2*c^2*d^2*x^6 + 43/576*sqrt(-c^2*x^2 + 1)*b^2*c*d^2*x^5* arccos(c*x) + 43/576*sqrt(-c^2*x^2 + 1)*a*b*c*d^2*x^5 + 1/4*b^2*d^2*x^4*ar ccos(c*x)^2 + 1/2*a*b*d^2*x^4*arccos(c*x) + 1/4*a^2*d^2*x^4 - 73/9216*b^2* d^2*x^4 - 73/2304*sqrt(-c^2*x^2 + 1)*b^2*d^2*x^3*arccos(c*x)/c - 73/2304*s qrt(-c^2*x^2 + 1)*a*b*d^2*x^3/c - 73/3072*b^2*d^2*x^2/c^2 - 73/1536*sqrt(- c^2*x^2 + 1)*b^2*d^2*x*arccos(c*x)/c^3 - 73/1536*sqrt(-c^2*x^2 + 1)*a*b*d^ 2*x/c^3 - 73/3072*b^2*d^2*arccos(c*x)^2/c^4 - 73/1536*a*b*d^2*arccos(c*x)/ c^4 + 10645/884736*b^2*d^2/c^4
Timed out. \[ \int x^3 \left (d-c^2 d x^2\right )^2 (a+b \arccos (c x))^2 \, dx=\int x^3\,{\left (a+b\,\mathrm {acos}\left (c\,x\right )\right )}^2\,{\left (d-c^2\,d\,x^2\right )}^2 \,d x \] Input:
int(x^3*(a + b*acos(c*x))^2*(d - c^2*d*x^2)^2,x)
Output:
int(x^3*(a + b*acos(c*x))^2*(d - c^2*d*x^2)^2, x)
\[ \int x^3 \left (d-c^2 d x^2\right )^2 (a+b \arccos (c x))^2 \, dx=\frac {d^{2} \left (1152 \mathit {acos} \left (c x \right ) a b \,c^{8} x^{8}-3072 \mathit {acos} \left (c x \right ) a b \,c^{6} x^{6}+2304 \mathit {acos} \left (c x \right ) a b \,c^{4} x^{4}+219 \mathit {asin} \left (c x \right ) a b -144 \sqrt {-c^{2} x^{2}+1}\, a b \,c^{7} x^{7}+344 \sqrt {-c^{2} x^{2}+1}\, a b \,c^{5} x^{5}-146 \sqrt {-c^{2} x^{2}+1}\, a b \,c^{3} x^{3}-219 \sqrt {-c^{2} x^{2}+1}\, a b c x +4608 \left (\int \mathit {acos} \left (c x \right )^{2} x^{7}d x \right ) b^{2} c^{8}-9216 \left (\int \mathit {acos} \left (c x \right )^{2} x^{5}d x \right ) b^{2} c^{6}+4608 \left (\int \mathit {acos} \left (c x \right )^{2} x^{3}d x \right ) b^{2} c^{4}+576 a^{2} c^{8} x^{8}-1536 a^{2} c^{6} x^{6}+1152 a^{2} c^{4} x^{4}\right )}{4608 c^{4}} \] Input:
int(x^3*(-c^2*d*x^2+d)^2*(a+b*acos(c*x))^2,x)
Output:
(d**2*(1152*acos(c*x)*a*b*c**8*x**8 - 3072*acos(c*x)*a*b*c**6*x**6 + 2304* acos(c*x)*a*b*c**4*x**4 + 219*asin(c*x)*a*b - 144*sqrt( - c**2*x**2 + 1)*a *b*c**7*x**7 + 344*sqrt( - c**2*x**2 + 1)*a*b*c**5*x**5 - 146*sqrt( - c**2 *x**2 + 1)*a*b*c**3*x**3 - 219*sqrt( - c**2*x**2 + 1)*a*b*c*x + 4608*int(a cos(c*x)**2*x**7,x)*b**2*c**8 - 9216*int(acos(c*x)**2*x**5,x)*b**2*c**6 + 4608*int(acos(c*x)**2*x**3,x)*b**2*c**4 + 576*a**2*c**8*x**8 - 1536*a**2*c **6*x**6 + 1152*a**2*c**4*x**4))/(4608*c**4)