Integrand size = 27, antiderivative size = 371 \[ \int \frac {\left (d-c^2 d x^2\right )^3 (a+b \arcsin (c x))^2}{x^3} \, dx=-\frac {21}{32} b^2 c^4 d^3 x^2+\frac {1}{32} b^2 c^6 d^3 x^4+\frac {3}{16} b c^3 d^3 x \sqrt {1-c^2 x^2} (a+b \arcsin (c x))-\frac {7}{8} b c^3 d^3 x \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))-\frac {b c d^3 \left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))}{x}+\frac {3}{32} c^2 d^3 (a+b \arcsin (c x))^2-\frac {3}{2} c^2 d^3 \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2-\frac {3}{4} c^2 d^3 \left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))^2-\frac {d^3 \left (1-c^2 x^2\right )^3 (a+b \arcsin (c x))^2}{2 x^2}+\frac {i c^2 d^3 (a+b \arcsin (c x))^3}{b}-3 c^2 d^3 (a+b \arcsin (c x))^2 \log \left (1-e^{2 i \arcsin (c x)}\right )+b^2 c^2 d^3 \log (x)+3 i b c^2 d^3 (a+b \arcsin (c x)) \operatorname {PolyLog}\left (2,e^{2 i \arcsin (c x)}\right )-\frac {3}{2} b^2 c^2 d^3 \operatorname {PolyLog}\left (3,e^{2 i \arcsin (c x)}\right ) \]
-21/32*b^2*c^4*d^3*x^2+1/32*b^2*c^6*d^3*x^4-7/8*b*c^3*d^3*x*(-c^2*x^2+1)^( 3/2)*(a+b*arcsin(c*x))-b*c*d^3*(-c^2*x^2+1)^(5/2)*(a+b*arcsin(c*x))/x+3/32 *c^2*d^3*(a+b*arcsin(c*x))^2-3/2*c^2*d^3*(-c^2*x^2+1)*(a+b*arcsin(c*x))^2- 3/4*c^2*d^3*(-c^2*x^2+1)^2*(a+b*arcsin(c*x))^2-1/2*d^3*(-c^2*x^2+1)^3*(a+b *arcsin(c*x))^2/x^2+I*c^2*d^3*(a+b*arcsin(c*x))^3/b-3*c^2*d^3*(a+b*arcsin( c*x))^2*ln(1-(I*c*x+(-c^2*x^2+1)^(1/2))^2)+b^2*c^2*d^3*ln(x)+3*I*b*c^2*d^3 *(a+b*arcsin(c*x))*polylog(2,(I*c*x+(-c^2*x^2+1)^(1/2))^2)-3/2*b^2*c^2*d^3 *polylog(3,(I*c*x+(-c^2*x^2+1)^(1/2))^2)+3/16*b*c^3*d^3*x*(a+b*arcsin(c*x) )*(-c^2*x^2+1)^(1/2)
Time = 0.56 (sec) , antiderivative size = 556, normalized size of antiderivative = 1.50 \[ \int \frac {\left (d-c^2 d x^2\right )^3 (a+b \arcsin (c x))^2}{x^3} \, dx=-\frac {d^3 \left (128 a^2-32 i b^2 c^2 \pi ^3 x^2-384 a^2 c^4 x^4+64 a^2 c^6 x^6+256 a b c x \sqrt {1-c^2 x^2}-336 a b c^3 x^3 \sqrt {1-c^2 x^2}+32 a b c^5 x^5 \sqrt {1-c^2 x^2}+256 a b \arcsin (c x)-768 a b c^4 x^4 \arcsin (c x)+128 a b c^6 x^6 \arcsin (c x)+256 b^2 c x \sqrt {1-c^2 x^2} \arcsin (c x)+128 b^2 \arcsin (c x)^2-768 i a b c^2 x^2 \arcsin (c x)^2+256 i b^2 c^2 x^2 \arcsin (c x)^3+672 a b c^2 x^2 \arctan \left (\frac {c x}{-1+\sqrt {1-c^2 x^2}}\right )-80 b^2 c^2 x^2 \cos (2 \arcsin (c x))+160 b^2 c^2 x^2 \arcsin (c x)^2 \cos (2 \arcsin (c x))-b^2 c^2 x^2 \cos (4 \arcsin (c x))+8 b^2 c^2 x^2 \arcsin (c x)^2 \cos (4 \arcsin (c x))+768 b^2 c^2 x^2 \arcsin (c x)^2 \log \left (1-e^{-2 i \arcsin (c x)}\right )+1536 a b c^2 x^2 \arcsin (c x) \log \left (1-e^{2 i \arcsin (c x)}\right )+768 a^2 c^2 x^2 \log (x)-256 b^2 c^2 x^2 \log (c x)+768 i b^2 c^2 x^2 \arcsin (c x) \operatorname {PolyLog}\left (2,e^{-2 i \arcsin (c x)}\right )-768 i a b c^2 x^2 \operatorname {PolyLog}\left (2,e^{2 i \arcsin (c x)}\right )+384 b^2 c^2 x^2 \operatorname {PolyLog}\left (3,e^{-2 i \arcsin (c x)}\right )-160 b^2 c^2 x^2 \arcsin (c x) \sin (2 \arcsin (c x))-4 b^2 c^2 x^2 \arcsin (c x) \sin (4 \arcsin (c x))\right )}{256 x^2} \]
-1/256*(d^3*(128*a^2 - (32*I)*b^2*c^2*Pi^3*x^2 - 384*a^2*c^4*x^4 + 64*a^2* c^6*x^6 + 256*a*b*c*x*Sqrt[1 - c^2*x^2] - 336*a*b*c^3*x^3*Sqrt[1 - c^2*x^2 ] + 32*a*b*c^5*x^5*Sqrt[1 - c^2*x^2] + 256*a*b*ArcSin[c*x] - 768*a*b*c^4*x ^4*ArcSin[c*x] + 128*a*b*c^6*x^6*ArcSin[c*x] + 256*b^2*c*x*Sqrt[1 - c^2*x^ 2]*ArcSin[c*x] + 128*b^2*ArcSin[c*x]^2 - (768*I)*a*b*c^2*x^2*ArcSin[c*x]^2 + (256*I)*b^2*c^2*x^2*ArcSin[c*x]^3 + 672*a*b*c^2*x^2*ArcTan[(c*x)/(-1 + Sqrt[1 - c^2*x^2])] - 80*b^2*c^2*x^2*Cos[2*ArcSin[c*x]] + 160*b^2*c^2*x^2* ArcSin[c*x]^2*Cos[2*ArcSin[c*x]] - b^2*c^2*x^2*Cos[4*ArcSin[c*x]] + 8*b^2* c^2*x^2*ArcSin[c*x]^2*Cos[4*ArcSin[c*x]] + 768*b^2*c^2*x^2*ArcSin[c*x]^2*L og[1 - E^((-2*I)*ArcSin[c*x])] + 1536*a*b*c^2*x^2*ArcSin[c*x]*Log[1 - E^(( 2*I)*ArcSin[c*x])] + 768*a^2*c^2*x^2*Log[x] - 256*b^2*c^2*x^2*Log[c*x] + ( 768*I)*b^2*c^2*x^2*ArcSin[c*x]*PolyLog[2, E^((-2*I)*ArcSin[c*x])] - (768*I )*a*b*c^2*x^2*PolyLog[2, E^((2*I)*ArcSin[c*x])] + 384*b^2*c^2*x^2*PolyLog[ 3, E^((-2*I)*ArcSin[c*x])] - 160*b^2*c^2*x^2*ArcSin[c*x]*Sin[2*ArcSin[c*x] ] - 4*b^2*c^2*x^2*ArcSin[c*x]*Sin[4*ArcSin[c*x]]))/x^2
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {\left (d-c^2 d x^2\right )^3 (a+b \arcsin (c x))^2}{x^3} \, dx\) |
| \(\Big \downarrow \) 5200 |
| \(\displaystyle b c d^3 \int \frac {\left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))}{x^2}dx-3 c^2 d \int \frac {d^2 \left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))^2}{x}dx-\frac {d^3 \left (1-c^2 x^2\right )^3 (a+b \arcsin (c x))^2}{2 x^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle b c d^3 \int \frac {\left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))}{x^2}dx-3 c^2 d^3 \int \frac {\left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))^2}{x}dx-\frac {d^3 \left (1-c^2 x^2\right )^3 (a+b \arcsin (c x))^2}{2 x^2}\) |
| \(\Big \downarrow \) 5200 |
| \(\displaystyle b c d^3 \left (-5 c^2 \int \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))dx+b c \int \frac {\left (1-c^2 x^2\right )^2}{x}dx-\frac {\left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))}{x}\right )-3 c^2 d^3 \int \frac {\left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))^2}{x}dx-\frac {d^3 \left (1-c^2 x^2\right )^3 (a+b \arcsin (c x))^2}{2 x^2}\) |
| \(\Big \downarrow \) 243 |
| \(\displaystyle b c d^3 \left (-5 c^2 \int \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))dx+\frac {1}{2} b c \int \frac {\left (1-c^2 x^2\right )^2}{x^2}dx^2-\frac {\left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))}{x}\right )-3 c^2 d^3 \int \frac {\left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))^2}{x}dx-\frac {d^3 \left (1-c^2 x^2\right )^3 (a+b \arcsin (c x))^2}{2 x^2}\) |
| \(\Big \downarrow \) 49 |
| \(\displaystyle -3 c^2 d^3 \int \frac {\left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))^2}{x}dx+b c d^3 \left (-5 c^2 \int \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))dx+\frac {1}{2} b c \int \left (x^2 c^4-2 c^2+\frac {1}{x^2}\right )dx^2-\frac {\left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))}{x}\right )-\frac {d^3 \left (1-c^2 x^2\right )^3 (a+b \arcsin (c x))^2}{2 x^2}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -3 c^2 d^3 \int \frac {\left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))^2}{x}dx+b c d^3 \left (-5 c^2 \int \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))dx-\frac {\left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))}{x}+\frac {1}{2} b c \left (\frac {c^4 x^4}{2}-2 c^2 x^2+\log \left (x^2\right )\right )\right )-\frac {d^3 \left (1-c^2 x^2\right )^3 (a+b \arcsin (c x))^2}{2 x^2}\) |
| \(\Big \downarrow \) 5158 |
| \(\displaystyle -3 c^2 d^3 \int \frac {\left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))^2}{x}dx+b c d^3 \left (-5 c^2 \left (\frac {3}{4} \int \sqrt {1-c^2 x^2} (a+b \arcsin (c x))dx-\frac {1}{4} b c \int x \left (1-c^2 x^2\right )dx+\frac {1}{4} x \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))\right )-\frac {\left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))}{x}+\frac {1}{2} b c \left (\frac {c^4 x^4}{2}-2 c^2 x^2+\log \left (x^2\right )\right )\right )-\frac {d^3 \left (1-c^2 x^2\right )^3 (a+b \arcsin (c x))^2}{2 x^2}\) |
| \(\Big \downarrow \) 244 |
| \(\displaystyle -3 c^2 d^3 \int \frac {\left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))^2}{x}dx+b c d^3 \left (-5 c^2 \left (\frac {3}{4} \int \sqrt {1-c^2 x^2} (a+b \arcsin (c x))dx-\frac {1}{4} b c \int \left (x-c^2 x^3\right )dx+\frac {1}{4} x \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))\right )-\frac {\left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))}{x}+\frac {1}{2} b c \left (\frac {c^4 x^4}{2}-2 c^2 x^2+\log \left (x^2\right )\right )\right )-\frac {d^3 \left (1-c^2 x^2\right )^3 (a+b \arcsin (c x))^2}{2 x^2}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -3 c^2 d^3 \int \frac {\left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))^2}{x}dx+b c d^3 \left (-5 c^2 \left (\frac {3}{4} \int \sqrt {1-c^2 x^2} (a+b \arcsin (c x))dx+\frac {1}{4} x \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))-\frac {1}{4} b c \left (\frac {x^2}{2}-\frac {c^2 x^4}{4}\right )\right )-\frac {\left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))}{x}+\frac {1}{2} b c \left (\frac {c^4 x^4}{2}-2 c^2 x^2+\log \left (x^2\right )\right )\right )-\frac {d^3 \left (1-c^2 x^2\right )^3 (a+b \arcsin (c x))^2}{2 x^2}\) |
| \(\Big \downarrow \) 5156 |
| \(\displaystyle -3 c^2 d^3 \int \frac {\left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))^2}{x}dx+b c d^3 \left (-5 c^2 \left (\frac {3}{4} \left (\frac {1}{2} \int \frac {a+b \arcsin (c x)}{\sqrt {1-c^2 x^2}}dx-\frac {1}{2} b c \int xdx+\frac {1}{2} x \sqrt {1-c^2 x^2} (a+b \arcsin (c x))\right )+\frac {1}{4} x \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))-\frac {1}{4} b c \left (\frac {x^2}{2}-\frac {c^2 x^4}{4}\right )\right )-\frac {\left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))}{x}+\frac {1}{2} b c \left (\frac {c^4 x^4}{2}-2 c^2 x^2+\log \left (x^2\right )\right )\right )-\frac {d^3 \left (1-c^2 x^2\right )^3 (a+b \arcsin (c x))^2}{2 x^2}\) |
| \(\Big \downarrow \) 15 |
| \(\displaystyle -3 c^2 d^3 \int \frac {\left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))^2}{x}dx+b c d^3 \left (-5 c^2 \left (\frac {3}{4} \left (\frac {1}{2} \int \frac {a+b \arcsin (c x)}{\sqrt {1-c^2 x^2}}dx+\frac {1}{2} x \sqrt {1-c^2 x^2} (a+b \arcsin (c x))-\frac {1}{4} b c x^2\right )+\frac {1}{4} x \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))-\frac {1}{4} b c \left (\frac {x^2}{2}-\frac {c^2 x^4}{4}\right )\right )-\frac {\left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))}{x}+\frac {1}{2} b c \left (\frac {c^4 x^4}{2}-2 c^2 x^2+\log \left (x^2\right )\right )\right )-\frac {d^3 \left (1-c^2 x^2\right )^3 (a+b \arcsin (c x))^2}{2 x^2}\) |
| \(\Big \downarrow \) 5152 |
| \(\displaystyle -3 c^2 d^3 \int \frac {\left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))^2}{x}dx-\frac {d^3 \left (1-c^2 x^2\right )^3 (a+b \arcsin (c x))^2}{2 x^2}+b c d^3 \left (-\frac {\left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))}{x}-5 c^2 \left (\frac {1}{4} x \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))+\frac {3}{4} \left (\frac {1}{2} x \sqrt {1-c^2 x^2} (a+b \arcsin (c x))+\frac {(a+b \arcsin (c x))^2}{4 b c}-\frac {1}{4} b c x^2\right )-\frac {1}{4} b c \left (\frac {x^2}{2}-\frac {c^2 x^4}{4}\right )\right )+\frac {1}{2} b c \left (\frac {c^4 x^4}{2}-2 c^2 x^2+\log \left (x^2\right )\right )\right )\) |
| \(\Big \downarrow \) 5202 |
| \(\displaystyle -3 c^2 d^3 \left (-\frac {1}{2} b c \int \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))dx+\int \frac {\left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2}{x}dx+\frac {1}{4} \left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))^2\right )-\frac {d^3 \left (1-c^2 x^2\right )^3 (a+b \arcsin (c x))^2}{2 x^2}+b c d^3 \left (-\frac {\left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))}{x}-5 c^2 \left (\frac {1}{4} x \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))+\frac {3}{4} \left (\frac {1}{2} x \sqrt {1-c^2 x^2} (a+b \arcsin (c x))+\frac {(a+b \arcsin (c x))^2}{4 b c}-\frac {1}{4} b c x^2\right )-\frac {1}{4} b c \left (\frac {x^2}{2}-\frac {c^2 x^4}{4}\right )\right )+\frac {1}{2} b c \left (\frac {c^4 x^4}{2}-2 c^2 x^2+\log \left (x^2\right )\right )\right )\) |
| \(\Big \downarrow \) 5158 |
| \(\displaystyle -3 c^2 d^3 \left (-\frac {1}{2} b c \left (\frac {3}{4} \int \sqrt {1-c^2 x^2} (a+b \arcsin (c x))dx-\frac {1}{4} b c \int x \left (1-c^2 x^2\right )dx+\frac {1}{4} x \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))\right )+\int \frac {\left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2}{x}dx+\frac {1}{4} \left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))^2\right )-\frac {d^3 \left (1-c^2 x^2\right )^3 (a+b \arcsin (c x))^2}{2 x^2}+b c d^3 \left (-\frac {\left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))}{x}-5 c^2 \left (\frac {1}{4} x \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))+\frac {3}{4} \left (\frac {1}{2} x \sqrt {1-c^2 x^2} (a+b \arcsin (c x))+\frac {(a+b \arcsin (c x))^2}{4 b c}-\frac {1}{4} b c x^2\right )-\frac {1}{4} b c \left (\frac {x^2}{2}-\frac {c^2 x^4}{4}\right )\right )+\frac {1}{2} b c \left (\frac {c^4 x^4}{2}-2 c^2 x^2+\log \left (x^2\right )\right )\right )\) |
| \(\Big \downarrow \) 244 |
| \(\displaystyle -3 c^2 d^3 \left (\int \frac {\left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2}{x}dx-\frac {1}{2} b c \left (\frac {3}{4} \int \sqrt {1-c^2 x^2} (a+b \arcsin (c x))dx-\frac {1}{4} b c \int \left (x-c^2 x^3\right )dx+\frac {1}{4} x \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))\right )+\frac {1}{4} \left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))^2\right )-\frac {d^3 \left (1-c^2 x^2\right )^3 (a+b \arcsin (c x))^2}{2 x^2}+b c d^3 \left (-\frac {\left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))}{x}-5 c^2 \left (\frac {1}{4} x \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))+\frac {3}{4} \left (\frac {1}{2} x \sqrt {1-c^2 x^2} (a+b \arcsin (c x))+\frac {(a+b \arcsin (c x))^2}{4 b c}-\frac {1}{4} b c x^2\right )-\frac {1}{4} b c \left (\frac {x^2}{2}-\frac {c^2 x^4}{4}\right )\right )+\frac {1}{2} b c \left (\frac {c^4 x^4}{2}-2 c^2 x^2+\log \left (x^2\right )\right )\right )\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -3 c^2 d^3 \left (\int \frac {\left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2}{x}dx-\frac {1}{2} b c \left (\frac {3}{4} \int \sqrt {1-c^2 x^2} (a+b \arcsin (c x))dx+\frac {1}{4} x \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))-\frac {1}{4} b c \left (\frac {x^2}{2}-\frac {c^2 x^4}{4}\right )\right )+\frac {1}{4} \left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))^2\right )-\frac {d^3 \left (1-c^2 x^2\right )^3 (a+b \arcsin (c x))^2}{2 x^2}+b c d^3 \left (-\frac {\left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))}{x}-5 c^2 \left (\frac {1}{4} x \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))+\frac {3}{4} \left (\frac {1}{2} x \sqrt {1-c^2 x^2} (a+b \arcsin (c x))+\frac {(a+b \arcsin (c x))^2}{4 b c}-\frac {1}{4} b c x^2\right )-\frac {1}{4} b c \left (\frac {x^2}{2}-\frac {c^2 x^4}{4}\right )\right )+\frac {1}{2} b c \left (\frac {c^4 x^4}{2}-2 c^2 x^2+\log \left (x^2\right )\right )\right )\) |
| \(\Big \downarrow \) 5156 |
| \(\displaystyle -3 c^2 d^3 \left (\int \frac {\left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2}{x}dx-\frac {1}{2} b c \left (\frac {3}{4} \left (\frac {1}{2} \int \frac {a+b \arcsin (c x)}{\sqrt {1-c^2 x^2}}dx-\frac {1}{2} b c \int xdx+\frac {1}{2} x \sqrt {1-c^2 x^2} (a+b \arcsin (c x))\right )+\frac {1}{4} x \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))-\frac {1}{4} b c \left (\frac {x^2}{2}-\frac {c^2 x^4}{4}\right )\right )+\frac {1}{4} \left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))^2\right )-\frac {d^3 \left (1-c^2 x^2\right )^3 (a+b \arcsin (c x))^2}{2 x^2}+b c d^3 \left (-\frac {\left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))}{x}-5 c^2 \left (\frac {1}{4} x \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))+\frac {3}{4} \left (\frac {1}{2} x \sqrt {1-c^2 x^2} (a+b \arcsin (c x))+\frac {(a+b \arcsin (c x))^2}{4 b c}-\frac {1}{4} b c x^2\right )-\frac {1}{4} b c \left (\frac {x^2}{2}-\frac {c^2 x^4}{4}\right )\right )+\frac {1}{2} b c \left (\frac {c^4 x^4}{2}-2 c^2 x^2+\log \left (x^2\right )\right )\right )\) |
| \(\Big \downarrow \) 15 |
| \(\displaystyle -3 c^2 d^3 \left (\int \frac {\left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2}{x}dx-\frac {1}{2} b c \left (\frac {3}{4} \left (\frac {1}{2} \int \frac {a+b \arcsin (c x)}{\sqrt {1-c^2 x^2}}dx+\frac {1}{2} x \sqrt {1-c^2 x^2} (a+b \arcsin (c x))-\frac {1}{4} b c x^2\right )+\frac {1}{4} x \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))-\frac {1}{4} b c \left (\frac {x^2}{2}-\frac {c^2 x^4}{4}\right )\right )+\frac {1}{4} \left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))^2\right )-\frac {d^3 \left (1-c^2 x^2\right )^3 (a+b \arcsin (c x))^2}{2 x^2}+b c d^3 \left (-\frac {\left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))}{x}-5 c^2 \left (\frac {1}{4} x \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))+\frac {3}{4} \left (\frac {1}{2} x \sqrt {1-c^2 x^2} (a+b \arcsin (c x))+\frac {(a+b \arcsin (c x))^2}{4 b c}-\frac {1}{4} b c x^2\right )-\frac {1}{4} b c \left (\frac {x^2}{2}-\frac {c^2 x^4}{4}\right )\right )+\frac {1}{2} b c \left (\frac {c^4 x^4}{2}-2 c^2 x^2+\log \left (x^2\right )\right )\right )\) |
| \(\Big \downarrow \) 5152 |
| \(\displaystyle -3 c^2 d^3 \left (\int \frac {\left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2}{x}dx+\frac {1}{4} \left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))^2-\frac {1}{2} b c \left (\frac {1}{4} x \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))+\frac {3}{4} \left (\frac {1}{2} x \sqrt {1-c^2 x^2} (a+b \arcsin (c x))+\frac {(a+b \arcsin (c x))^2}{4 b c}-\frac {1}{4} b c x^2\right )-\frac {1}{4} b c \left (\frac {x^2}{2}-\frac {c^2 x^4}{4}\right )\right )\right )-\frac {d^3 \left (1-c^2 x^2\right )^3 (a+b \arcsin (c x))^2}{2 x^2}+b c d^3 \left (-\frac {\left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))}{x}-5 c^2 \left (\frac {1}{4} x \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))+\frac {3}{4} \left (\frac {1}{2} x \sqrt {1-c^2 x^2} (a+b \arcsin (c x))+\frac {(a+b \arcsin (c x))^2}{4 b c}-\frac {1}{4} b c x^2\right )-\frac {1}{4} b c \left (\frac {x^2}{2}-\frac {c^2 x^4}{4}\right )\right )+\frac {1}{2} b c \left (\frac {c^4 x^4}{2}-2 c^2 x^2+\log \left (x^2\right )\right )\right )\) |
| \(\Big \downarrow \) 5202 |
| \(\displaystyle -3 c^2 d^3 \left (-b c \int \sqrt {1-c^2 x^2} (a+b \arcsin (c x))dx+\int \frac {(a+b \arcsin (c x))^2}{x}dx+\frac {1}{4} \left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))^2+\frac {1}{2} \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2-\frac {1}{2} b c \left (\frac {1}{4} x \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))+\frac {3}{4} \left (\frac {1}{2} x \sqrt {1-c^2 x^2} (a+b \arcsin (c x))+\frac {(a+b \arcsin (c x))^2}{4 b c}-\frac {1}{4} b c x^2\right )-\frac {1}{4} b c \left (\frac {x^2}{2}-\frac {c^2 x^4}{4}\right )\right )\right )-\frac {d^3 \left (1-c^2 x^2\right )^3 (a+b \arcsin (c x))^2}{2 x^2}+b c d^3 \left (-\frac {\left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))}{x}-5 c^2 \left (\frac {1}{4} x \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))+\frac {3}{4} \left (\frac {1}{2} x \sqrt {1-c^2 x^2} (a+b \arcsin (c x))+\frac {(a+b \arcsin (c x))^2}{4 b c}-\frac {1}{4} b c x^2\right )-\frac {1}{4} b c \left (\frac {x^2}{2}-\frac {c^2 x^4}{4}\right )\right )+\frac {1}{2} b c \left (\frac {c^4 x^4}{2}-2 c^2 x^2+\log \left (x^2\right )\right )\right )\) |
| \(\Big \downarrow \) 5136 |
| \(\displaystyle -3 c^2 d^3 \left (-b c \int \sqrt {1-c^2 x^2} (a+b \arcsin (c x))dx+\int \frac {\sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2}{c x}d\arcsin (c x)+\frac {1}{4} \left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))^2+\frac {1}{2} \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2-\frac {1}{2} b c \left (\frac {1}{4} x \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))+\frac {3}{4} \left (\frac {1}{2} x \sqrt {1-c^2 x^2} (a+b \arcsin (c x))+\frac {(a+b \arcsin (c x))^2}{4 b c}-\frac {1}{4} b c x^2\right )-\frac {1}{4} b c \left (\frac {x^2}{2}-\frac {c^2 x^4}{4}\right )\right )\right )-\frac {d^3 \left (1-c^2 x^2\right )^3 (a+b \arcsin (c x))^2}{2 x^2}+b c d^3 \left (-\frac {\left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))}{x}-5 c^2 \left (\frac {1}{4} x \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))+\frac {3}{4} \left (\frac {1}{2} x \sqrt {1-c^2 x^2} (a+b \arcsin (c x))+\frac {(a+b \arcsin (c x))^2}{4 b c}-\frac {1}{4} b c x^2\right )-\frac {1}{4} b c \left (\frac {x^2}{2}-\frac {c^2 x^4}{4}\right )\right )+\frac {1}{2} b c \left (\frac {c^4 x^4}{2}-2 c^2 x^2+\log \left (x^2\right )\right )\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -3 c^2 d^3 \left (-b c \int \sqrt {1-c^2 x^2} (a+b \arcsin (c x))dx+\int -(a+b \arcsin (c x))^2 \tan \left (\arcsin (c x)+\frac {\pi }{2}\right )d\arcsin (c x)+\frac {1}{4} \left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))^2+\frac {1}{2} \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2-\frac {1}{2} b c \left (\frac {1}{4} x \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))+\frac {3}{4} \left (\frac {1}{2} x \sqrt {1-c^2 x^2} (a+b \arcsin (c x))+\frac {(a+b \arcsin (c x))^2}{4 b c}-\frac {1}{4} b c x^2\right )-\frac {1}{4} b c \left (\frac {x^2}{2}-\frac {c^2 x^4}{4}\right )\right )\right )-\frac {d^3 \left (1-c^2 x^2\right )^3 (a+b \arcsin (c x))^2}{2 x^2}+b c d^3 \left (-\frac {\left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))}{x}-5 c^2 \left (\frac {1}{4} x \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))+\frac {3}{4} \left (\frac {1}{2} x \sqrt {1-c^2 x^2} (a+b \arcsin (c x))+\frac {(a+b \arcsin (c x))^2}{4 b c}-\frac {1}{4} b c x^2\right )-\frac {1}{4} b c \left (\frac {x^2}{2}-\frac {c^2 x^4}{4}\right )\right )+\frac {1}{2} b c \left (\frac {c^4 x^4}{2}-2 c^2 x^2+\log \left (x^2\right )\right )\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -3 c^2 d^3 \left (-b c \int \sqrt {1-c^2 x^2} (a+b \arcsin (c x))dx-\int (a+b \arcsin (c x))^2 \tan \left (\arcsin (c x)+\frac {\pi }{2}\right )d\arcsin (c x)+\frac {1}{4} \left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))^2+\frac {1}{2} \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2-\frac {1}{2} b c \left (\frac {1}{4} x \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))+\frac {3}{4} \left (\frac {1}{2} x \sqrt {1-c^2 x^2} (a+b \arcsin (c x))+\frac {(a+b \arcsin (c x))^2}{4 b c}-\frac {1}{4} b c x^2\right )-\frac {1}{4} b c \left (\frac {x^2}{2}-\frac {c^2 x^4}{4}\right )\right )\right )-\frac {d^3 \left (1-c^2 x^2\right )^3 (a+b \arcsin (c x))^2}{2 x^2}+b c d^3 \left (-\frac {\left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))}{x}-5 c^2 \left (\frac {1}{4} x \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))+\frac {3}{4} \left (\frac {1}{2} x \sqrt {1-c^2 x^2} (a+b \arcsin (c x))+\frac {(a+b \arcsin (c x))^2}{4 b c}-\frac {1}{4} b c x^2\right )-\frac {1}{4} b c \left (\frac {x^2}{2}-\frac {c^2 x^4}{4}\right )\right )+\frac {1}{2} b c \left (\frac {c^4 x^4}{2}-2 c^2 x^2+\log \left (x^2\right )\right )\right )\) |
| \(\Big \downarrow \) 4200 |
| \(\displaystyle -3 c^2 d^3 \left (-b c \int \sqrt {1-c^2 x^2} (a+b \arcsin (c x))dx+2 i \int -\frac {e^{2 i \arcsin (c x)} (a+b \arcsin (c x))^2}{1-e^{2 i \arcsin (c x)}}d\arcsin (c x)+\frac {1}{4} \left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))^2+\frac {1}{2} \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2-\frac {1}{2} b c \left (\frac {1}{4} x \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))+\frac {3}{4} \left (\frac {1}{2} x \sqrt {1-c^2 x^2} (a+b \arcsin (c x))+\frac {(a+b \arcsin (c x))^2}{4 b c}-\frac {1}{4} b c x^2\right )-\frac {1}{4} b c \left (\frac {x^2}{2}-\frac {c^2 x^4}{4}\right )\right )-\frac {i (a+b \arcsin (c x))^3}{3 b}\right )-\frac {d^3 \left (1-c^2 x^2\right )^3 (a+b \arcsin (c x))^2}{2 x^2}+b c d^3 \left (-\frac {\left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))}{x}-5 c^2 \left (\frac {1}{4} x \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))+\frac {3}{4} \left (\frac {1}{2} x \sqrt {1-c^2 x^2} (a+b \arcsin (c x))+\frac {(a+b \arcsin (c x))^2}{4 b c}-\frac {1}{4} b c x^2\right )-\frac {1}{4} b c \left (\frac {x^2}{2}-\frac {c^2 x^4}{4}\right )\right )+\frac {1}{2} b c \left (\frac {c^4 x^4}{2}-2 c^2 x^2+\log \left (x^2\right )\right )\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -3 c^2 d^3 \left (-b c \int \sqrt {1-c^2 x^2} (a+b \arcsin (c x))dx-2 i \int \frac {e^{2 i \arcsin (c x)} (a+b \arcsin (c x))^2}{1-e^{2 i \arcsin (c x)}}d\arcsin (c x)+\frac {1}{4} \left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))^2+\frac {1}{2} \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2-\frac {1}{2} b c \left (\frac {1}{4} x \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))+\frac {3}{4} \left (\frac {1}{2} x \sqrt {1-c^2 x^2} (a+b \arcsin (c x))+\frac {(a+b \arcsin (c x))^2}{4 b c}-\frac {1}{4} b c x^2\right )-\frac {1}{4} b c \left (\frac {x^2}{2}-\frac {c^2 x^4}{4}\right )\right )-\frac {i (a+b \arcsin (c x))^3}{3 b}\right )-\frac {d^3 \left (1-c^2 x^2\right )^3 (a+b \arcsin (c x))^2}{2 x^2}+b c d^3 \left (-\frac {\left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))}{x}-5 c^2 \left (\frac {1}{4} x \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))+\frac {3}{4} \left (\frac {1}{2} x \sqrt {1-c^2 x^2} (a+b \arcsin (c x))+\frac {(a+b \arcsin (c x))^2}{4 b c}-\frac {1}{4} b c x^2\right )-\frac {1}{4} b c \left (\frac {x^2}{2}-\frac {c^2 x^4}{4}\right )\right )+\frac {1}{2} b c \left (\frac {c^4 x^4}{2}-2 c^2 x^2+\log \left (x^2\right )\right )\right )\) |
| \(\Big \downarrow \) 2620 |
| \(\displaystyle -3 c^2 d^3 \left (-b c \int \sqrt {1-c^2 x^2} (a+b \arcsin (c x))dx-2 i \left (\frac {1}{2} i \log \left (1-e^{2 i \arcsin (c x)}\right ) (a+b \arcsin (c x))^2-i b \int (a+b \arcsin (c x)) \log \left (1-e^{2 i \arcsin (c x)}\right )d\arcsin (c x)\right )+\frac {1}{4} \left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))^2+\frac {1}{2} \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2-\frac {1}{2} b c \left (\frac {1}{4} x \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))+\frac {3}{4} \left (\frac {1}{2} x \sqrt {1-c^2 x^2} (a+b \arcsin (c x))+\frac {(a+b \arcsin (c x))^2}{4 b c}-\frac {1}{4} b c x^2\right )-\frac {1}{4} b c \left (\frac {x^2}{2}-\frac {c^2 x^4}{4}\right )\right )-\frac {i (a+b \arcsin (c x))^3}{3 b}\right )-\frac {d^3 \left (1-c^2 x^2\right )^3 (a+b \arcsin (c x))^2}{2 x^2}+b c d^3 \left (-\frac {\left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))}{x}-5 c^2 \left (\frac {1}{4} x \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))+\frac {3}{4} \left (\frac {1}{2} x \sqrt {1-c^2 x^2} (a+b \arcsin (c x))+\frac {(a+b \arcsin (c x))^2}{4 b c}-\frac {1}{4} b c x^2\right )-\frac {1}{4} b c \left (\frac {x^2}{2}-\frac {c^2 x^4}{4}\right )\right )+\frac {1}{2} b c \left (\frac {c^4 x^4}{2}-2 c^2 x^2+\log \left (x^2\right )\right )\right )\) |
| \(\Big \downarrow \) 3011 |
| \(\displaystyle -3 c^2 d^3 \left (-b c \int \sqrt {1-c^2 x^2} (a+b \arcsin (c x))dx-2 i \left (\frac {1}{2} i \log \left (1-e^{2 i \arcsin (c x)}\right ) (a+b \arcsin (c x))^2-i b \left (\frac {1}{2} i \operatorname {PolyLog}\left (2,e^{2 i \arcsin (c x)}\right ) (a+b \arcsin (c x))-\frac {1}{2} i b \int \operatorname {PolyLog}\left (2,e^{2 i \arcsin (c x)}\right )d\arcsin (c x)\right )\right )+\frac {1}{4} \left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))^2+\frac {1}{2} \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2-\frac {1}{2} b c \left (\frac {1}{4} x \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))+\frac {3}{4} \left (\frac {1}{2} x \sqrt {1-c^2 x^2} (a+b \arcsin (c x))+\frac {(a+b \arcsin (c x))^2}{4 b c}-\frac {1}{4} b c x^2\right )-\frac {1}{4} b c \left (\frac {x^2}{2}-\frac {c^2 x^4}{4}\right )\right )-\frac {i (a+b \arcsin (c x))^3}{3 b}\right )-\frac {d^3 \left (1-c^2 x^2\right )^3 (a+b \arcsin (c x))^2}{2 x^2}+b c d^3 \left (-\frac {\left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))}{x}-5 c^2 \left (\frac {1}{4} x \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))+\frac {3}{4} \left (\frac {1}{2} x \sqrt {1-c^2 x^2} (a+b \arcsin (c x))+\frac {(a+b \arcsin (c x))^2}{4 b c}-\frac {1}{4} b c x^2\right )-\frac {1}{4} b c \left (\frac {x^2}{2}-\frac {c^2 x^4}{4}\right )\right )+\frac {1}{2} b c \left (\frac {c^4 x^4}{2}-2 c^2 x^2+\log \left (x^2\right )\right )\right )\) |
| \(\Big \downarrow \) 2720 |
| \(\displaystyle -3 c^2 d^3 \left (-b c \int \sqrt {1-c^2 x^2} (a+b \arcsin (c x))dx-2 i \left (\frac {1}{2} i \log \left (1-e^{2 i \arcsin (c x)}\right ) (a+b \arcsin (c x))^2-i b \left (\frac {1}{2} i \operatorname {PolyLog}\left (2,e^{2 i \arcsin (c x)}\right ) (a+b \arcsin (c x))-\frac {1}{4} b \int e^{-2 i \arcsin (c x)} \operatorname {PolyLog}\left (2,e^{2 i \arcsin (c x)}\right )de^{2 i \arcsin (c x)}\right )\right )+\frac {1}{4} \left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))^2+\frac {1}{2} \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2-\frac {1}{2} b c \left (\frac {1}{4} x \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))+\frac {3}{4} \left (\frac {1}{2} x \sqrt {1-c^2 x^2} (a+b \arcsin (c x))+\frac {(a+b \arcsin (c x))^2}{4 b c}-\frac {1}{4} b c x^2\right )-\frac {1}{4} b c \left (\frac {x^2}{2}-\frac {c^2 x^4}{4}\right )\right )-\frac {i (a+b \arcsin (c x))^3}{3 b}\right )-\frac {d^3 \left (1-c^2 x^2\right )^3 (a+b \arcsin (c x))^2}{2 x^2}+b c d^3 \left (-\frac {\left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))}{x}-5 c^2 \left (\frac {1}{4} x \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))+\frac {3}{4} \left (\frac {1}{2} x \sqrt {1-c^2 x^2} (a+b \arcsin (c x))+\frac {(a+b \arcsin (c x))^2}{4 b c}-\frac {1}{4} b c x^2\right )-\frac {1}{4} b c \left (\frac {x^2}{2}-\frac {c^2 x^4}{4}\right )\right )+\frac {1}{2} b c \left (\frac {c^4 x^4}{2}-2 c^2 x^2+\log \left (x^2\right )\right )\right )\) |
| \(\Big \downarrow \) 5156 |
| \(\displaystyle -3 c^2 d^3 \left (-b c \left (\frac {1}{2} \int \frac {a+b \arcsin (c x)}{\sqrt {1-c^2 x^2}}dx-\frac {1}{2} b c \int xdx+\frac {1}{2} x \sqrt {1-c^2 x^2} (a+b \arcsin (c x))\right )-2 i \left (\frac {1}{2} i \log \left (1-e^{2 i \arcsin (c x)}\right ) (a+b \arcsin (c x))^2-i b \left (\frac {1}{2} i \operatorname {PolyLog}\left (2,e^{2 i \arcsin (c x)}\right ) (a+b \arcsin (c x))-\frac {1}{4} b \int e^{-2 i \arcsin (c x)} \operatorname {PolyLog}\left (2,e^{2 i \arcsin (c x)}\right )de^{2 i \arcsin (c x)}\right )\right )+\frac {1}{4} \left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))^2+\frac {1}{2} \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2-\frac {1}{2} b c \left (\frac {1}{4} x \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))+\frac {3}{4} \left (\frac {1}{2} x \sqrt {1-c^2 x^2} (a+b \arcsin (c x))+\frac {(a+b \arcsin (c x))^2}{4 b c}-\frac {1}{4} b c x^2\right )-\frac {1}{4} b c \left (\frac {x^2}{2}-\frac {c^2 x^4}{4}\right )\right )-\frac {i (a+b \arcsin (c x))^3}{3 b}\right )-\frac {d^3 \left (1-c^2 x^2\right )^3 (a+b \arcsin (c x))^2}{2 x^2}+b c d^3 \left (-\frac {\left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))}{x}-5 c^2 \left (\frac {1}{4} x \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))+\frac {3}{4} \left (\frac {1}{2} x \sqrt {1-c^2 x^2} (a+b \arcsin (c x))+\frac {(a+b \arcsin (c x))^2}{4 b c}-\frac {1}{4} b c x^2\right )-\frac {1}{4} b c \left (\frac {x^2}{2}-\frac {c^2 x^4}{4}\right )\right )+\frac {1}{2} b c \left (\frac {c^4 x^4}{2}-2 c^2 x^2+\log \left (x^2\right )\right )\right )\) |
| \(\Big \downarrow \) 15 |
| \(\displaystyle -3 c^2 d^3 \left (-b c \left (\frac {1}{2} \int \frac {a+b \arcsin (c x)}{\sqrt {1-c^2 x^2}}dx+\frac {1}{2} x \sqrt {1-c^2 x^2} (a+b \arcsin (c x))-\frac {1}{4} b c x^2\right )-2 i \left (\frac {1}{2} i \log \left (1-e^{2 i \arcsin (c x)}\right ) (a+b \arcsin (c x))^2-i b \left (\frac {1}{2} i \operatorname {PolyLog}\left (2,e^{2 i \arcsin (c x)}\right ) (a+b \arcsin (c x))-\frac {1}{4} b \int e^{-2 i \arcsin (c x)} \operatorname {PolyLog}\left (2,e^{2 i \arcsin (c x)}\right )de^{2 i \arcsin (c x)}\right )\right )+\frac {1}{4} \left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))^2+\frac {1}{2} \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2-\frac {1}{2} b c \left (\frac {1}{4} x \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))+\frac {3}{4} \left (\frac {1}{2} x \sqrt {1-c^2 x^2} (a+b \arcsin (c x))+\frac {(a+b \arcsin (c x))^2}{4 b c}-\frac {1}{4} b c x^2\right )-\frac {1}{4} b c \left (\frac {x^2}{2}-\frac {c^2 x^4}{4}\right )\right )-\frac {i (a+b \arcsin (c x))^3}{3 b}\right )-\frac {d^3 \left (1-c^2 x^2\right )^3 (a+b \arcsin (c x))^2}{2 x^2}+b c d^3 \left (-\frac {\left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))}{x}-5 c^2 \left (\frac {1}{4} x \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))+\frac {3}{4} \left (\frac {1}{2} x \sqrt {1-c^2 x^2} (a+b \arcsin (c x))+\frac {(a+b \arcsin (c x))^2}{4 b c}-\frac {1}{4} b c x^2\right )-\frac {1}{4} b c \left (\frac {x^2}{2}-\frac {c^2 x^4}{4}\right )\right )+\frac {1}{2} b c \left (\frac {c^4 x^4}{2}-2 c^2 x^2+\log \left (x^2\right )\right )\right )\) |
3.2.81.3.1 Defintions of rubi rules used
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[((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}, x] && IGtQ[m, 0] && IGtQ[m + n + 2, 0]
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[1/2 Subst[In t[x^((m - 1)/2)*(a + b*x)^p, x], x, x^2], x] /; FreeQ[{a, b, m, p}, x] && I ntegerQ[(m - 1)/2]
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[(((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.))/ ((a_) + (b_.)*((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)), x_Symbol] :> Simp [((c + d*x)^m/(b*f*g*n*Log[F]))*Log[1 + b*((F^(g*(e + f*x)))^n/a)], x] - Si mp[d*(m/(b*f*g*n*Log[F])) Int[(c + d*x)^(m - 1)*Log[1 + b*((F^(g*(e + f*x )))^n/a)], x], x] /; FreeQ[{F, a, b, c, d, e, f, g, n}, x] && IGtQ[m, 0]
Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Simp[v/D[v, x] Subst[Int[FunctionOfExponentialFunction[u, x]/x, x], x, v], x]] /; Funct ionOfExponentialQ[u, x] && !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; FreeQ [{a, m, n}, x] && IntegerQ[m*n]] && !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x)) *(F_)[v_] /; FreeQ[{a, b, c}, x] && InverseFunctionQ[F[x]]]
Int[Log[1 + (e_.)*((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.)]*((f_.) + (g_.) *(x_))^(m_.), x_Symbol] :> Simp[(-(f + g*x)^m)*(PolyLog[2, (-e)*(F^(c*(a + b*x)))^n]/(b*c*n*Log[F])), x] + Simp[g*(m/(b*c*n*Log[F])) Int[(f + g*x)^( m - 1)*PolyLog[2, (-e)*(F^(c*(a + b*x)))^n], x], x] /; FreeQ[{F, a, b, c, e , f, g, n}, x] && GtQ[m, 0]
Int[((c_.) + (d_.)*(x_))^(m_.)*tan[(e_.) + Pi*(k_.) + (f_.)*(x_)], x_Symbol ] :> Simp[I*((c + d*x)^(m + 1)/(d*(m + 1))), x] - Simp[2*I Int[(c + d*x)^ m*E^(2*I*k*Pi)*(E^(2*I*(e + f*x))/(1 + E^(2*I*k*Pi)*E^(2*I*(e + f*x)))), x] , x] /; FreeQ[{c, d, e, f}, x] && IntegerQ[4*k] && IGtQ[m, 0]
Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)/(x_), x_Symbol] :> Subst[Int[( a + b*x)^n*Cot[x], x], x, ArcSin[c*x]] /; FreeQ[{a, b, c}, x] && IGtQ[n, 0]
Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)/Sqrt[(d_) + (e_.)*(x_)^2], x_S ymbol] :> Simp[(1/(b*c*(n + 1)))*Simp[Sqrt[1 - c^2*x^2]/Sqrt[d + e*x^2]]*(a + b*ArcSin[c*x])^(n + 1), x] /; FreeQ[{a, b, c, d, e, n}, x] && EqQ[c^2*d + e, 0] && NeQ[n, -1]
Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*Sqrt[(d_) + (e_.)*(x_)^2], x_S ymbol] :> Simp[x*Sqrt[d + e*x^2]*((a + b*ArcSin[c*x])^n/2), x] + (Simp[(1/2 )*Simp[Sqrt[d + e*x^2]/Sqrt[1 - c^2*x^2]] Int[(a + b*ArcSin[c*x])^n/Sqrt[ 1 - c^2*x^2], x], x] - Simp[b*c*(n/2)*Simp[Sqrt[d + e*x^2]/Sqrt[1 - c^2*x^2 ]] Int[x*(a + b*ArcSin[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d, e}, x ] && EqQ[c^2*d + e, 0] && GtQ[n, 0]
Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((d_) + (e_.)*(x_)^2)^(p_.), x _Symbol] :> Simp[x*(d + e*x^2)^p*((a + b*ArcSin[c*x])^n/(2*p + 1)), x] + (S imp[2*d*(p/(2*p + 1)) Int[(d + e*x^2)^(p - 1)*(a + b*ArcSin[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*ArcSin[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_.) + ArcSin[(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*ArcS in[c*x])^n/(f*(m + 1))), x] + (-Simp[2*e*(p/(f^2*(m + 1))) Int[(f*x)^(m + 2)*(d + e*x^2)^(p - 1)*(a + b*ArcSin[c*x])^n, x], x] - Simp[b*c*(n/(f*(m + 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*ArcSin[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d, e, f} , x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && GtQ[p, 0] && LtQ[m, -1]
Int[((a_.) + ArcSin[(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*ArcS in[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*ArcSin[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*ArcSin[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]
Time = 0.33 (sec) , antiderivative size = 708, normalized size of antiderivative = 1.91
| method | result | size |
| derivativedivides | \(c^{2} \left (-d^{3} a^{2} \left (\frac {c^{4} x^{4}}{4}-\frac {3 c^{2} x^{2}}{2}+\frac {1}{2 c^{2} x^{2}}+3 \ln \left (c x \right )\right )-d^{3} b^{2} \left (-6 i \arcsin \left (c x \right ) \operatorname {polylog}\left (2, -i c x -\sqrt {-c^{2} x^{2}+1}\right )-\frac {5 \left (2 i \arcsin \left (c x \right )+2 \arcsin \left (c x \right )^{2}-1\right ) \left (2 c^{2} x^{2}-2 i c x \sqrt {-c^{2} x^{2}+1}-1\right )}{32}-\frac {5 \left (2 i c x \sqrt {-c^{2} x^{2}+1}+2 c^{2} x^{2}-1\right ) \left (-2 i \arcsin \left (c x \right )+2 \arcsin \left (c x \right )^{2}-1\right )}{32}+\frac {\arcsin \left (c x \right ) \left (-2 i c^{2} x^{2}+2 c x \sqrt {-c^{2} x^{2}+1}+\arcsin \left (c x \right )\right )}{2 c^{2} x^{2}}-\ln \left (i c x +\sqrt {-c^{2} x^{2}+1}-1\right )+2 \ln \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )-\ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )-i \arcsin \left (c x \right )^{3}+3 \arcsin \left (c x \right )^{2} \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )-6 i \arcsin \left (c x \right ) \operatorname {polylog}\left (2, i c x +\sqrt {-c^{2} x^{2}+1}\right )+6 \operatorname {polylog}\left (3, -i c x -\sqrt {-c^{2} x^{2}+1}\right )+3 \arcsin \left (c x \right )^{2} \ln \left (1-i c x -\sqrt {-c^{2} x^{2}+1}\right )+6 \operatorname {polylog}\left (3, i c x +\sqrt {-c^{2} x^{2}+1}\right )+\frac {\left (8 \arcsin \left (c x \right )^{2}-1\right ) \cos \left (4 \arcsin \left (c x \right )\right )}{256}-\frac {\arcsin \left (c x \right ) \sin \left (4 \arcsin \left (c x \right )\right )}{64}\right )-2 d^{3} a b \left (-\frac {3 i \arcsin \left (c x \right )^{2}}{2}-\frac {5 \left (i+2 \arcsin \left (c x \right )\right ) \left (2 c^{2} x^{2}-2 i c x \sqrt {-c^{2} x^{2}+1}-1\right )}{32}-\frac {5 \left (2 i c x \sqrt {-c^{2} x^{2}+1}+2 c^{2} x^{2}-1\right ) \left (-i+2 \arcsin \left (c x \right )\right )}{32}+\frac {-i c^{2} x^{2}+c x \sqrt {-c^{2} x^{2}+1}+\arcsin \left (c x \right )}{2 c^{2} x^{2}}+3 \arcsin \left (c x \right ) \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )+3 \arcsin \left (c x \right ) \ln \left (1-i c x -\sqrt {-c^{2} x^{2}+1}\right )-3 i \operatorname {polylog}\left (2, -i c x -\sqrt {-c^{2} x^{2}+1}\right )-3 i \operatorname {polylog}\left (2, i c x +\sqrt {-c^{2} x^{2}+1}\right )+\frac {\arcsin \left (c x \right ) \cos \left (4 \arcsin \left (c x \right )\right )}{32}-\frac {\sin \left (4 \arcsin \left (c x \right )\right )}{128}\right )\right )\) | \(708\) |
| default | \(c^{2} \left (-d^{3} a^{2} \left (\frac {c^{4} x^{4}}{4}-\frac {3 c^{2} x^{2}}{2}+\frac {1}{2 c^{2} x^{2}}+3 \ln \left (c x \right )\right )-d^{3} b^{2} \left (-6 i \arcsin \left (c x \right ) \operatorname {polylog}\left (2, -i c x -\sqrt {-c^{2} x^{2}+1}\right )-\frac {5 \left (2 i \arcsin \left (c x \right )+2 \arcsin \left (c x \right )^{2}-1\right ) \left (2 c^{2} x^{2}-2 i c x \sqrt {-c^{2} x^{2}+1}-1\right )}{32}-\frac {5 \left (2 i c x \sqrt {-c^{2} x^{2}+1}+2 c^{2} x^{2}-1\right ) \left (-2 i \arcsin \left (c x \right )+2 \arcsin \left (c x \right )^{2}-1\right )}{32}+\frac {\arcsin \left (c x \right ) \left (-2 i c^{2} x^{2}+2 c x \sqrt {-c^{2} x^{2}+1}+\arcsin \left (c x \right )\right )}{2 c^{2} x^{2}}-\ln \left (i c x +\sqrt {-c^{2} x^{2}+1}-1\right )+2 \ln \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )-\ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )-i \arcsin \left (c x \right )^{3}+3 \arcsin \left (c x \right )^{2} \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )-6 i \arcsin \left (c x \right ) \operatorname {polylog}\left (2, i c x +\sqrt {-c^{2} x^{2}+1}\right )+6 \operatorname {polylog}\left (3, -i c x -\sqrt {-c^{2} x^{2}+1}\right )+3 \arcsin \left (c x \right )^{2} \ln \left (1-i c x -\sqrt {-c^{2} x^{2}+1}\right )+6 \operatorname {polylog}\left (3, i c x +\sqrt {-c^{2} x^{2}+1}\right )+\frac {\left (8 \arcsin \left (c x \right )^{2}-1\right ) \cos \left (4 \arcsin \left (c x \right )\right )}{256}-\frac {\arcsin \left (c x \right ) \sin \left (4 \arcsin \left (c x \right )\right )}{64}\right )-2 d^{3} a b \left (-\frac {3 i \arcsin \left (c x \right )^{2}}{2}-\frac {5 \left (i+2 \arcsin \left (c x \right )\right ) \left (2 c^{2} x^{2}-2 i c x \sqrt {-c^{2} x^{2}+1}-1\right )}{32}-\frac {5 \left (2 i c x \sqrt {-c^{2} x^{2}+1}+2 c^{2} x^{2}-1\right ) \left (-i+2 \arcsin \left (c x \right )\right )}{32}+\frac {-i c^{2} x^{2}+c x \sqrt {-c^{2} x^{2}+1}+\arcsin \left (c x \right )}{2 c^{2} x^{2}}+3 \arcsin \left (c x \right ) \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )+3 \arcsin \left (c x \right ) \ln \left (1-i c x -\sqrt {-c^{2} x^{2}+1}\right )-3 i \operatorname {polylog}\left (2, -i c x -\sqrt {-c^{2} x^{2}+1}\right )-3 i \operatorname {polylog}\left (2, i c x +\sqrt {-c^{2} x^{2}+1}\right )+\frac {\arcsin \left (c x \right ) \cos \left (4 \arcsin \left (c x \right )\right )}{32}-\frac {\sin \left (4 \arcsin \left (c x \right )\right )}{128}\right )\right )\) | \(708\) |
| parts | \(-d^{3} a^{2} \left (\frac {c^{6} x^{4}}{4}-\frac {3 c^{4} x^{2}}{2}+\frac {1}{2 x^{2}}+3 c^{2} \ln \left (x \right )\right )-d^{3} b^{2} c^{2} \left (-6 i \arcsin \left (c x \right ) \operatorname {polylog}\left (2, -i c x -\sqrt {-c^{2} x^{2}+1}\right )-\frac {5 \left (2 i \arcsin \left (c x \right )+2 \arcsin \left (c x \right )^{2}-1\right ) \left (2 c^{2} x^{2}-2 i c x \sqrt {-c^{2} x^{2}+1}-1\right )}{32}-\frac {5 \left (2 i c x \sqrt {-c^{2} x^{2}+1}+2 c^{2} x^{2}-1\right ) \left (-2 i \arcsin \left (c x \right )+2 \arcsin \left (c x \right )^{2}-1\right )}{32}+\frac {\arcsin \left (c x \right ) \left (-2 i c^{2} x^{2}+2 c x \sqrt {-c^{2} x^{2}+1}+\arcsin \left (c x \right )\right )}{2 c^{2} x^{2}}-\ln \left (i c x +\sqrt {-c^{2} x^{2}+1}-1\right )+2 \ln \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )-\ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )-i \arcsin \left (c x \right )^{3}+3 \arcsin \left (c x \right )^{2} \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )-6 i \arcsin \left (c x \right ) \operatorname {polylog}\left (2, i c x +\sqrt {-c^{2} x^{2}+1}\right )+6 \operatorname {polylog}\left (3, -i c x -\sqrt {-c^{2} x^{2}+1}\right )+3 \arcsin \left (c x \right )^{2} \ln \left (1-i c x -\sqrt {-c^{2} x^{2}+1}\right )+6 \operatorname {polylog}\left (3, i c x +\sqrt {-c^{2} x^{2}+1}\right )+\frac {\left (8 \arcsin \left (c x \right )^{2}-1\right ) \cos \left (4 \arcsin \left (c x \right )\right )}{256}-\frac {\arcsin \left (c x \right ) \sin \left (4 \arcsin \left (c x \right )\right )}{64}\right )-2 d^{3} a b \,c^{2} \left (-\frac {3 i \arcsin \left (c x \right )^{2}}{2}-\frac {5 \left (i+2 \arcsin \left (c x \right )\right ) \left (2 c^{2} x^{2}-2 i c x \sqrt {-c^{2} x^{2}+1}-1\right )}{32}-\frac {5 \left (2 i c x \sqrt {-c^{2} x^{2}+1}+2 c^{2} x^{2}-1\right ) \left (-i+2 \arcsin \left (c x \right )\right )}{32}+\frac {-i c^{2} x^{2}+c x \sqrt {-c^{2} x^{2}+1}+\arcsin \left (c x \right )}{2 c^{2} x^{2}}+3 \arcsin \left (c x \right ) \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )+3 \arcsin \left (c x \right ) \ln \left (1-i c x -\sqrt {-c^{2} x^{2}+1}\right )-3 i \operatorname {polylog}\left (2, -i c x -\sqrt {-c^{2} x^{2}+1}\right )-3 i \operatorname {polylog}\left (2, i c x +\sqrt {-c^{2} x^{2}+1}\right )+\frac {\arcsin \left (c x \right ) \cos \left (4 \arcsin \left (c x \right )\right )}{32}-\frac {\sin \left (4 \arcsin \left (c x \right )\right )}{128}\right )\) | \(708\) |
c^2*(-d^3*a^2*(1/4*c^4*x^4-3/2*c^2*x^2+1/2/c^2/x^2+3*ln(c*x))-d^3*b^2*(-6* I*arcsin(c*x)*polylog(2,-I*c*x-(-c^2*x^2+1)^(1/2))-5/32*(2*I*arcsin(c*x)+2 *arcsin(c*x)^2-1)*(2*c^2*x^2-2*I*c*x*(-c^2*x^2+1)^(1/2)-1)-5/32*(2*I*c*x*( -c^2*x^2+1)^(1/2)+2*c^2*x^2-1)*(-2*I*arcsin(c*x)+2*arcsin(c*x)^2-1)+1/2*ar csin(c*x)*(-2*I*c^2*x^2+2*c*x*(-c^2*x^2+1)^(1/2)+arcsin(c*x))/c^2/x^2-ln(I *c*x+(-c^2*x^2+1)^(1/2)-1)+2*ln(I*c*x+(-c^2*x^2+1)^(1/2))-ln(1+I*c*x+(-c^2 *x^2+1)^(1/2))-I*arcsin(c*x)^3+3*arcsin(c*x)^2*ln(1+I*c*x+(-c^2*x^2+1)^(1/ 2))-6*I*arcsin(c*x)*polylog(2,I*c*x+(-c^2*x^2+1)^(1/2))+6*polylog(3,-I*c*x -(-c^2*x^2+1)^(1/2))+3*arcsin(c*x)^2*ln(1-I*c*x-(-c^2*x^2+1)^(1/2))+6*poly log(3,I*c*x+(-c^2*x^2+1)^(1/2))+1/256*(8*arcsin(c*x)^2-1)*cos(4*arcsin(c*x ))-1/64*arcsin(c*x)*sin(4*arcsin(c*x)))-2*d^3*a*b*(-3/2*I*arcsin(c*x)^2-5/ 32*(I+2*arcsin(c*x))*(2*c^2*x^2-2*I*c*x*(-c^2*x^2+1)^(1/2)-1)-5/32*(2*I*c* x*(-c^2*x^2+1)^(1/2)+2*c^2*x^2-1)*(-I+2*arcsin(c*x))+1/2*(-I*c^2*x^2+c*x*( -c^2*x^2+1)^(1/2)+arcsin(c*x))/c^2/x^2+3*arcsin(c*x)*ln(1+I*c*x+(-c^2*x^2+ 1)^(1/2))+3*arcsin(c*x)*ln(1-I*c*x-(-c^2*x^2+1)^(1/2))-3*I*polylog(2,-I*c* x-(-c^2*x^2+1)^(1/2))-3*I*polylog(2,I*c*x+(-c^2*x^2+1)^(1/2))+1/32*arcsin( c*x)*cos(4*arcsin(c*x))-1/128*sin(4*arcsin(c*x))))
\[ \int \frac {\left (d-c^2 d x^2\right )^3 (a+b \arcsin (c x))^2}{x^3} \, dx=\int { -\frac {{\left (c^{2} d x^{2} - d\right )}^{3} {\left (b \arcsin \left (c x\right ) + a\right )}^{2}}{x^{3}} \,d x } \]
integral(-(a^2*c^6*d^3*x^6 - 3*a^2*c^4*d^3*x^4 + 3*a^2*c^2*d^3*x^2 - a^2*d ^3 + (b^2*c^6*d^3*x^6 - 3*b^2*c^4*d^3*x^4 + 3*b^2*c^2*d^3*x^2 - b^2*d^3)*a rcsin(c*x)^2 + 2*(a*b*c^6*d^3*x^6 - 3*a*b*c^4*d^3*x^4 + 3*a*b*c^2*d^3*x^2 - a*b*d^3)*arcsin(c*x))/x^3, x)
\[ \int \frac {\left (d-c^2 d x^2\right )^3 (a+b \arcsin (c x))^2}{x^3} \, dx=- d^{3} \left (\int \left (- \frac {a^{2}}{x^{3}}\right )\, dx + \int \frac {3 a^{2} c^{2}}{x}\, dx + \int \left (- 3 a^{2} c^{4} x\right )\, dx + \int a^{2} c^{6} x^{3}\, dx + \int \left (- \frac {b^{2} \operatorname {asin}^{2}{\left (c x \right )}}{x^{3}}\right )\, dx + \int \left (- \frac {2 a b \operatorname {asin}{\left (c x \right )}}{x^{3}}\right )\, dx + \int \frac {3 b^{2} c^{2} \operatorname {asin}^{2}{\left (c x \right )}}{x}\, dx + \int \left (- 3 b^{2} c^{4} x \operatorname {asin}^{2}{\left (c x \right )}\right )\, dx + \int b^{2} c^{6} x^{3} \operatorname {asin}^{2}{\left (c x \right )}\, dx + \int \frac {6 a b c^{2} \operatorname {asin}{\left (c x \right )}}{x}\, dx + \int \left (- 6 a b c^{4} x \operatorname {asin}{\left (c x \right )}\right )\, dx + \int 2 a b c^{6} x^{3} \operatorname {asin}{\left (c x \right )}\, dx\right ) \]
-d**3*(Integral(-a**2/x**3, x) + Integral(3*a**2*c**2/x, x) + Integral(-3* a**2*c**4*x, x) + Integral(a**2*c**6*x**3, x) + Integral(-b**2*asin(c*x)** 2/x**3, x) + Integral(-2*a*b*asin(c*x)/x**3, x) + Integral(3*b**2*c**2*asi n(c*x)**2/x, x) + Integral(-3*b**2*c**4*x*asin(c*x)**2, x) + Integral(b**2 *c**6*x**3*asin(c*x)**2, x) + Integral(6*a*b*c**2*asin(c*x)/x, x) + Integr al(-6*a*b*c**4*x*asin(c*x), x) + Integral(2*a*b*c**6*x**3*asin(c*x), x))
\[ \int \frac {\left (d-c^2 d x^2\right )^3 (a+b \arcsin (c x))^2}{x^3} \, dx=\int { -\frac {{\left (c^{2} d x^{2} - d\right )}^{3} {\left (b \arcsin \left (c x\right ) + a\right )}^{2}}{x^{3}} \,d x } \]
-1/4*a^2*c^6*d^3*x^4 + 3/2*a^2*c^4*d^3*x^2 - 3*a^2*c^2*d^3*log(x) - a*b*d^ 3*(sqrt(-c^2*x^2 + 1)*c/x + arcsin(c*x)/x^2) - 1/2*a^2*d^3/x^2 - integrate (((b^2*c^6*d^3*x^6 - 3*b^2*c^4*d^3*x^4 + 3*b^2*c^2*d^3*x^2 - b^2*d^3)*arct an2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1))^2 + 2*(a*b*c^6*d^3*x^6 - 3*a*b*c^4* d^3*x^4 + 3*a*b*c^2*d^3*x^2)*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1)))/x ^3, x)
\[ \int \frac {\left (d-c^2 d x^2\right )^3 (a+b \arcsin (c x))^2}{x^3} \, dx=\int { -\frac {{\left (c^{2} d x^{2} - d\right )}^{3} {\left (b \arcsin \left (c x\right ) + a\right )}^{2}}{x^{3}} \,d x } \]
Timed out. \[ \int \frac {\left (d-c^2 d x^2\right )^3 (a+b \arcsin (c x))^2}{x^3} \, dx=\int \frac {{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^2\,{\left (d-c^2\,d\,x^2\right )}^3}{x^3} \,d x \]