Integrand size = 27, antiderivative size = 287 \[ \int \frac {\left (d-c^2 d x^2\right )^2 (a+b \arcsin (c x))^2}{x^3} \, dx=-\frac {1}{4} b^2 c^4 d^2 x^2-\frac {1}{2} b c^3 d^2 x \sqrt {1-c^2 x^2} (a+b \arcsin (c x))-\frac {b c d^2 \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))}{x}-\frac {1}{4} c^2 d^2 (a+b \arcsin (c x))^2-c^2 d^2 \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2-\frac {d^2 \left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))^2}{2 x^2}+\frac {2 i c^2 d^2 (a+b \arcsin (c x))^3}{3 b}-2 c^2 d^2 (a+b \arcsin (c x))^2 \log \left (1-e^{2 i \arcsin (c x)}\right )+b^2 c^2 d^2 \log (x)+2 i b c^2 d^2 (a+b \arcsin (c x)) \operatorname {PolyLog}\left (2,e^{2 i \arcsin (c x)}\right )-b^2 c^2 d^2 \operatorname {PolyLog}\left (3,e^{2 i \arcsin (c x)}\right ) \]
-1/4*b^2*c^4*d^2*x^2-b*c*d^2*(-c^2*x^2+1)^(3/2)*(a+b*arcsin(c*x))/x-1/4*c^ 2*d^2*(a+b*arcsin(c*x))^2-c^2*d^2*(-c^2*x^2+1)*(a+b*arcsin(c*x))^2-1/2*d^2 *(-c^2*x^2+1)^2*(a+b*arcsin(c*x))^2/x^2+2/3*I*c^2*d^2*(a+b*arcsin(c*x))^3/ b-2*c^2*d^2*(a+b*arcsin(c*x))^2*ln(1-(I*c*x+(-c^2*x^2+1)^(1/2))^2)+b^2*c^2 *d^2*ln(x)+2*I*b*c^2*d^2*(a+b*arcsin(c*x))*polylog(2,(I*c*x+(-c^2*x^2+1)^( 1/2))^2)-b^2*c^2*d^2*polylog(3,(I*c*x+(-c^2*x^2+1)^(1/2))^2)-1/2*b*c^3*d^2 *x*(a+b*arcsin(c*x))*(-c^2*x^2+1)^(1/2)
Time = 0.74 (sec) , antiderivative size = 361, normalized size of antiderivative = 1.26 \[ \int \frac {\left (d-c^2 d x^2\right )^2 (a+b \arcsin (c x))^2}{x^3} \, dx=\frac {1}{2} d^2 \left (-\frac {a^2}{x^2}+a^2 c^4 x^2+2 a b c^4 x^2 \arcsin (c x)-\frac {2 a b \left (c x \sqrt {1-c^2 x^2}+\arcsin (c x)\right )}{x^2}+a b c^2 \left (c x \sqrt {1-c^2 x^2}-2 \arctan \left (\frac {c x}{-1+\sqrt {1-c^2 x^2}}\right )\right )-\frac {1}{4} b^2 c^2 \left (-1+2 \arcsin (c x)^2\right ) \cos (2 \arcsin (c x))-8 a b c^2 \arcsin (c x) \log \left (1-e^{2 i \arcsin (c x)}\right )-4 a^2 c^2 \log (x)-\frac {b^2 \left (2 c x \sqrt {1-c^2 x^2} \arcsin (c x)+\arcsin (c x)^2-2 c^2 x^2 \log (c x)\right )}{x^2}+4 i a b c^2 \left (\arcsin (c x)^2+\operatorname {PolyLog}\left (2,e^{2 i \arcsin (c x)}\right )\right )+\frac {1}{6} i b^2 c^2 \left (\pi ^3-8 \arcsin (c x)^3+24 i \arcsin (c x)^2 \log \left (1-e^{-2 i \arcsin (c x)}\right )-24 \arcsin (c x) \operatorname {PolyLog}\left (2,e^{-2 i \arcsin (c x)}\right )+12 i \operatorname {PolyLog}\left (3,e^{-2 i \arcsin (c x)}\right )\right )+\frac {1}{2} b^2 c^2 \arcsin (c x) \sin (2 \arcsin (c x))\right ) \]
(d^2*(-(a^2/x^2) + a^2*c^4*x^2 + 2*a*b*c^4*x^2*ArcSin[c*x] - (2*a*b*(c*x*S qrt[1 - c^2*x^2] + ArcSin[c*x]))/x^2 + a*b*c^2*(c*x*Sqrt[1 - c^2*x^2] - 2* ArcTan[(c*x)/(-1 + Sqrt[1 - c^2*x^2])]) - (b^2*c^2*(-1 + 2*ArcSin[c*x]^2)* Cos[2*ArcSin[c*x]])/4 - 8*a*b*c^2*ArcSin[c*x]*Log[1 - E^((2*I)*ArcSin[c*x] )] - 4*a^2*c^2*Log[x] - (b^2*(2*c*x*Sqrt[1 - c^2*x^2]*ArcSin[c*x] + ArcSin [c*x]^2 - 2*c^2*x^2*Log[c*x]))/x^2 + (4*I)*a*b*c^2*(ArcSin[c*x]^2 + PolyLo g[2, E^((2*I)*ArcSin[c*x])]) + (I/6)*b^2*c^2*(Pi^3 - 8*ArcSin[c*x]^3 + (24 *I)*ArcSin[c*x]^2*Log[1 - E^((-2*I)*ArcSin[c*x])] - 24*ArcSin[c*x]*PolyLog [2, E^((-2*I)*ArcSin[c*x])] + (12*I)*PolyLog[3, E^((-2*I)*ArcSin[c*x])]) + (b^2*c^2*ArcSin[c*x]*Sin[2*ArcSin[c*x]])/2))/2
Time = 2.05 (sec) , antiderivative size = 343, normalized size of antiderivative = 1.20, number of steps used = 22, number of rules used = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.778, Rules used = {5200, 27, 5200, 244, 2009, 5156, 15, 5152, 5202, 5136, 3042, 25, 4200, 25, 2620, 3011, 2720, 5156, 15, 5152, 7143}
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.
| \(\displaystyle \int \frac {\left (d-c^2 d x^2\right )^2 (a+b \arcsin (c x))^2}{x^3} \, dx\) |
| \(\Big \downarrow \) 5200 |
| \(\displaystyle b c d^2 \int \frac {\left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))}{x^2}dx-2 c^2 d \int \frac {d \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2}{x}dx-\frac {d^2 \left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))^2}{2 x^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle b c d^2 \int \frac {\left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))}{x^2}dx-2 c^2 d^2 \int \frac {\left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2}{x}dx-\frac {d^2 \left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))^2}{2 x^2}\) |
| \(\Big \downarrow \) 5200 |
| \(\displaystyle b c d^2 \left (-3 c^2 \int \sqrt {1-c^2 x^2} (a+b \arcsin (c x))dx+b c \int \frac {1-c^2 x^2}{x}dx-\frac {\left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))}{x}\right )-2 c^2 d^2 \int \frac {\left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2}{x}dx-\frac {d^2 \left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))^2}{2 x^2}\) |
| \(\Big \downarrow \) 244 |
| \(\displaystyle b c d^2 \left (-3 c^2 \int \sqrt {1-c^2 x^2} (a+b \arcsin (c x))dx+b c \int \left (\frac {1}{x}-c^2 x\right )dx-\frac {\left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))}{x}\right )-2 c^2 d^2 \int \frac {\left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2}{x}dx-\frac {d^2 \left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))^2}{2 x^2}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -2 c^2 d^2 \int \frac {\left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2}{x}dx+b c d^2 \left (-3 c^2 \int \sqrt {1-c^2 x^2} (a+b \arcsin (c x))dx-\frac {\left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))}{x}+b c \left (\log (x)-\frac {c^2 x^2}{2}\right )\right )-\frac {d^2 \left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))^2}{2 x^2}\) |
| \(\Big \downarrow \) 5156 |
| \(\displaystyle -2 c^2 d^2 \int \frac {\left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2}{x}dx+b c d^2 \left (-3 c^2 \left (\frac {1}{2} \int \frac {a+b \arcsin (c x)}{\sqrt {1-c^2 x^2}}dx-\frac {1}{2} b c \int xdx+\frac {1}{2} x \sqrt {1-c^2 x^2} (a+b \arcsin (c x))\right )-\frac {\left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))}{x}+b c \left (\log (x)-\frac {c^2 x^2}{2}\right )\right )-\frac {d^2 \left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))^2}{2 x^2}\) |
| \(\Big \downarrow \) 15 |
| \(\displaystyle -2 c^2 d^2 \int \frac {\left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2}{x}dx+b c d^2 \left (-3 c^2 \left (\frac {1}{2} \int \frac {a+b \arcsin (c x)}{\sqrt {1-c^2 x^2}}dx+\frac {1}{2} x \sqrt {1-c^2 x^2} (a+b \arcsin (c x))-\frac {1}{4} b c x^2\right )-\frac {\left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))}{x}+b c \left (\log (x)-\frac {c^2 x^2}{2}\right )\right )-\frac {d^2 \left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))^2}{2 x^2}\) |
| \(\Big \downarrow \) 5152 |
| \(\displaystyle -2 c^2 d^2 \int \frac {\left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2}{x}dx-\frac {d^2 \left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))^2}{2 x^2}+b c d^2 \left (-3 c^2 \left (\frac {1}{2} x \sqrt {1-c^2 x^2} (a+b \arcsin (c x))+\frac {(a+b \arcsin (c x))^2}{4 b c}-\frac {1}{4} b c x^2\right )-\frac {\left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))}{x}+b c \left (\log (x)-\frac {c^2 x^2}{2}\right )\right )\) |
| \(\Big \downarrow \) 5202 |
| \(\displaystyle -2 c^2 d^2 \left (-b c \int \sqrt {1-c^2 x^2} (a+b \arcsin (c x))dx+\int \frac {(a+b \arcsin (c x))^2}{x}dx+\frac {1}{2} \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2\right )-\frac {d^2 \left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))^2}{2 x^2}+b c d^2 \left (-3 c^2 \left (\frac {1}{2} x \sqrt {1-c^2 x^2} (a+b \arcsin (c x))+\frac {(a+b \arcsin (c x))^2}{4 b c}-\frac {1}{4} b c x^2\right )-\frac {\left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))}{x}+b c \left (\log (x)-\frac {c^2 x^2}{2}\right )\right )\) |
| \(\Big \downarrow \) 5136 |
| \(\displaystyle -2 c^2 d^2 \left (-b c \int \sqrt {1-c^2 x^2} (a+b \arcsin (c x))dx+\int \frac {\sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2}{c x}d\arcsin (c x)+\frac {1}{2} \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2\right )-\frac {d^2 \left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))^2}{2 x^2}+b c d^2 \left (-3 c^2 \left (\frac {1}{2} x \sqrt {1-c^2 x^2} (a+b \arcsin (c x))+\frac {(a+b \arcsin (c x))^2}{4 b c}-\frac {1}{4} b c x^2\right )-\frac {\left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))}{x}+b c \left (\log (x)-\frac {c^2 x^2}{2}\right )\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -2 c^2 d^2 \left (-b c \int \sqrt {1-c^2 x^2} (a+b \arcsin (c x))dx+\int -(a+b \arcsin (c x))^2 \tan \left (\arcsin (c x)+\frac {\pi }{2}\right )d\arcsin (c x)+\frac {1}{2} \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2\right )-\frac {d^2 \left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))^2}{2 x^2}+b c d^2 \left (-3 c^2 \left (\frac {1}{2} x \sqrt {1-c^2 x^2} (a+b \arcsin (c x))+\frac {(a+b \arcsin (c x))^2}{4 b c}-\frac {1}{4} b c x^2\right )-\frac {\left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))}{x}+b c \left (\log (x)-\frac {c^2 x^2}{2}\right )\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -2 c^2 d^2 \left (-b c \int \sqrt {1-c^2 x^2} (a+b \arcsin (c x))dx-\int (a+b \arcsin (c x))^2 \tan \left (\arcsin (c x)+\frac {\pi }{2}\right )d\arcsin (c x)+\frac {1}{2} \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2\right )-\frac {d^2 \left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))^2}{2 x^2}+b c d^2 \left (-3 c^2 \left (\frac {1}{2} x \sqrt {1-c^2 x^2} (a+b \arcsin (c x))+\frac {(a+b \arcsin (c x))^2}{4 b c}-\frac {1}{4} b c x^2\right )-\frac {\left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))}{x}+b c \left (\log (x)-\frac {c^2 x^2}{2}\right )\right )\) |
| \(\Big \downarrow \) 4200 |
| \(\displaystyle -2 c^2 d^2 \left (-b c \int \sqrt {1-c^2 x^2} (a+b \arcsin (c x))dx+2 i \int -\frac {e^{2 i \arcsin (c x)} (a+b \arcsin (c x))^2}{1-e^{2 i \arcsin (c x)}}d\arcsin (c x)+\frac {1}{2} \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2-\frac {i (a+b \arcsin (c x))^3}{3 b}\right )-\frac {d^2 \left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))^2}{2 x^2}+b c d^2 \left (-3 c^2 \left (\frac {1}{2} x \sqrt {1-c^2 x^2} (a+b \arcsin (c x))+\frac {(a+b \arcsin (c x))^2}{4 b c}-\frac {1}{4} b c x^2\right )-\frac {\left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))}{x}+b c \left (\log (x)-\frac {c^2 x^2}{2}\right )\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -2 c^2 d^2 \left (-b c \int \sqrt {1-c^2 x^2} (a+b \arcsin (c x))dx-2 i \int \frac {e^{2 i \arcsin (c x)} (a+b \arcsin (c x))^2}{1-e^{2 i \arcsin (c x)}}d\arcsin (c x)+\frac {1}{2} \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2-\frac {i (a+b \arcsin (c x))^3}{3 b}\right )-\frac {d^2 \left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))^2}{2 x^2}+b c d^2 \left (-3 c^2 \left (\frac {1}{2} x \sqrt {1-c^2 x^2} (a+b \arcsin (c x))+\frac {(a+b \arcsin (c x))^2}{4 b c}-\frac {1}{4} b c x^2\right )-\frac {\left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))}{x}+b c \left (\log (x)-\frac {c^2 x^2}{2}\right )\right )\) |
| \(\Big \downarrow \) 2620 |
| \(\displaystyle -2 c^2 d^2 \left (-b c \int \sqrt {1-c^2 x^2} (a+b \arcsin (c x))dx-2 i \left (\frac {1}{2} i \log \left (1-e^{2 i \arcsin (c x)}\right ) (a+b \arcsin (c x))^2-i b \int (a+b \arcsin (c x)) \log \left (1-e^{2 i \arcsin (c x)}\right )d\arcsin (c x)\right )+\frac {1}{2} \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2-\frac {i (a+b \arcsin (c x))^3}{3 b}\right )-\frac {d^2 \left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))^2}{2 x^2}+b c d^2 \left (-3 c^2 \left (\frac {1}{2} x \sqrt {1-c^2 x^2} (a+b \arcsin (c x))+\frac {(a+b \arcsin (c x))^2}{4 b c}-\frac {1}{4} b c x^2\right )-\frac {\left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))}{x}+b c \left (\log (x)-\frac {c^2 x^2}{2}\right )\right )\) |
| \(\Big \downarrow \) 3011 |
| \(\displaystyle -2 c^2 d^2 \left (-b c \int \sqrt {1-c^2 x^2} (a+b \arcsin (c x))dx-2 i \left (\frac {1}{2} i \log \left (1-e^{2 i \arcsin (c x)}\right ) (a+b \arcsin (c x))^2-i b \left (\frac {1}{2} i \operatorname {PolyLog}\left (2,e^{2 i \arcsin (c x)}\right ) (a+b \arcsin (c x))-\frac {1}{2} i b \int \operatorname {PolyLog}\left (2,e^{2 i \arcsin (c x)}\right )d\arcsin (c x)\right )\right )+\frac {1}{2} \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2-\frac {i (a+b \arcsin (c x))^3}{3 b}\right )-\frac {d^2 \left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))^2}{2 x^2}+b c d^2 \left (-3 c^2 \left (\frac {1}{2} x \sqrt {1-c^2 x^2} (a+b \arcsin (c x))+\frac {(a+b \arcsin (c x))^2}{4 b c}-\frac {1}{4} b c x^2\right )-\frac {\left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))}{x}+b c \left (\log (x)-\frac {c^2 x^2}{2}\right )\right )\) |
| \(\Big \downarrow \) 2720 |
| \(\displaystyle -2 c^2 d^2 \left (-b c \int \sqrt {1-c^2 x^2} (a+b \arcsin (c x))dx-2 i \left (\frac {1}{2} i \log \left (1-e^{2 i \arcsin (c x)}\right ) (a+b \arcsin (c x))^2-i b \left (\frac {1}{2} i \operatorname {PolyLog}\left (2,e^{2 i \arcsin (c x)}\right ) (a+b \arcsin (c x))-\frac {1}{4} b \int e^{-2 i \arcsin (c x)} \operatorname {PolyLog}\left (2,e^{2 i \arcsin (c x)}\right )de^{2 i \arcsin (c x)}\right )\right )+\frac {1}{2} \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2-\frac {i (a+b \arcsin (c x))^3}{3 b}\right )-\frac {d^2 \left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))^2}{2 x^2}+b c d^2 \left (-3 c^2 \left (\frac {1}{2} x \sqrt {1-c^2 x^2} (a+b \arcsin (c x))+\frac {(a+b \arcsin (c x))^2}{4 b c}-\frac {1}{4} b c x^2\right )-\frac {\left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))}{x}+b c \left (\log (x)-\frac {c^2 x^2}{2}\right )\right )\) |
| \(\Big \downarrow \) 5156 |
| \(\displaystyle -2 c^2 d^2 \left (-b c \left (\frac {1}{2} \int \frac {a+b \arcsin (c x)}{\sqrt {1-c^2 x^2}}dx-\frac {1}{2} b c \int xdx+\frac {1}{2} x \sqrt {1-c^2 x^2} (a+b \arcsin (c x))\right )-2 i \left (\frac {1}{2} i \log \left (1-e^{2 i \arcsin (c x)}\right ) (a+b \arcsin (c x))^2-i b \left (\frac {1}{2} i \operatorname {PolyLog}\left (2,e^{2 i \arcsin (c x)}\right ) (a+b \arcsin (c x))-\frac {1}{4} b \int e^{-2 i \arcsin (c x)} \operatorname {PolyLog}\left (2,e^{2 i \arcsin (c x)}\right )de^{2 i \arcsin (c x)}\right )\right )+\frac {1}{2} \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2-\frac {i (a+b \arcsin (c x))^3}{3 b}\right )-\frac {d^2 \left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))^2}{2 x^2}+b c d^2 \left (-3 c^2 \left (\frac {1}{2} x \sqrt {1-c^2 x^2} (a+b \arcsin (c x))+\frac {(a+b \arcsin (c x))^2}{4 b c}-\frac {1}{4} b c x^2\right )-\frac {\left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))}{x}+b c \left (\log (x)-\frac {c^2 x^2}{2}\right )\right )\) |
| \(\Big \downarrow \) 15 |
| \(\displaystyle -2 c^2 d^2 \left (-b c \left (\frac {1}{2} \int \frac {a+b \arcsin (c x)}{\sqrt {1-c^2 x^2}}dx+\frac {1}{2} x \sqrt {1-c^2 x^2} (a+b \arcsin (c x))-\frac {1}{4} b c x^2\right )-2 i \left (\frac {1}{2} i \log \left (1-e^{2 i \arcsin (c x)}\right ) (a+b \arcsin (c x))^2-i b \left (\frac {1}{2} i \operatorname {PolyLog}\left (2,e^{2 i \arcsin (c x)}\right ) (a+b \arcsin (c x))-\frac {1}{4} b \int e^{-2 i \arcsin (c x)} \operatorname {PolyLog}\left (2,e^{2 i \arcsin (c x)}\right )de^{2 i \arcsin (c x)}\right )\right )+\frac {1}{2} \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2-\frac {i (a+b \arcsin (c x))^3}{3 b}\right )-\frac {d^2 \left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))^2}{2 x^2}+b c d^2 \left (-3 c^2 \left (\frac {1}{2} x \sqrt {1-c^2 x^2} (a+b \arcsin (c x))+\frac {(a+b \arcsin (c x))^2}{4 b c}-\frac {1}{4} b c x^2\right )-\frac {\left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))}{x}+b c \left (\log (x)-\frac {c^2 x^2}{2}\right )\right )\) |
| \(\Big \downarrow \) 5152 |
| \(\displaystyle -2 c^2 d^2 \left (-2 i \left (\frac {1}{2} i \log \left (1-e^{2 i \arcsin (c x)}\right ) (a+b \arcsin (c x))^2-i b \left (\frac {1}{2} i \operatorname {PolyLog}\left (2,e^{2 i \arcsin (c x)}\right ) (a+b \arcsin (c x))-\frac {1}{4} b \int e^{-2 i \arcsin (c x)} \operatorname {PolyLog}\left (2,e^{2 i \arcsin (c x)}\right )de^{2 i \arcsin (c x)}\right )\right )+\frac {1}{2} \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2-b c \left (\frac {1}{2} x \sqrt {1-c^2 x^2} (a+b \arcsin (c x))+\frac {(a+b \arcsin (c x))^2}{4 b c}-\frac {1}{4} b c x^2\right )-\frac {i (a+b \arcsin (c x))^3}{3 b}\right )-\frac {d^2 \left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))^2}{2 x^2}+b c d^2 \left (-3 c^2 \left (\frac {1}{2} x \sqrt {1-c^2 x^2} (a+b \arcsin (c x))+\frac {(a+b \arcsin (c x))^2}{4 b c}-\frac {1}{4} b c x^2\right )-\frac {\left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))}{x}+b c \left (\log (x)-\frac {c^2 x^2}{2}\right )\right )\) |
| \(\Big \downarrow \) 7143 |
| \(\displaystyle -2 c^2 d^2 \left (\frac {1}{2} \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2-b c \left (\frac {1}{2} x \sqrt {1-c^2 x^2} (a+b \arcsin (c x))+\frac {(a+b \arcsin (c x))^2}{4 b c}-\frac {1}{4} b c x^2\right )-2 i \left (\frac {1}{2} i \log \left (1-e^{2 i \arcsin (c x)}\right ) (a+b \arcsin (c x))^2-i b \left (\frac {1}{2} i \operatorname {PolyLog}\left (2,e^{2 i \arcsin (c x)}\right ) (a+b \arcsin (c x))-\frac {1}{4} b \operatorname {PolyLog}\left (3,e^{2 i \arcsin (c x)}\right )\right )\right )-\frac {i (a+b \arcsin (c x))^3}{3 b}\right )-\frac {d^2 \left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))^2}{2 x^2}+b c d^2 \left (-3 c^2 \left (\frac {1}{2} x \sqrt {1-c^2 x^2} (a+b \arcsin (c x))+\frac {(a+b \arcsin (c x))^2}{4 b c}-\frac {1}{4} b c x^2\right )-\frac {\left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))}{x}+b c \left (\log (x)-\frac {c^2 x^2}{2}\right )\right )\) |
-1/2*(d^2*(1 - c^2*x^2)^2*(a + b*ArcSin[c*x])^2)/x^2 + b*c*d^2*(-(((1 - c^ 2*x^2)^(3/2)*(a + b*ArcSin[c*x]))/x) - 3*c^2*(-1/4*(b*c*x^2) + (x*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/2 + (a + b*ArcSin[c*x])^2/(4*b*c)) + b*c*(- 1/2*(c^2*x^2) + Log[x])) - 2*c^2*d^2*(((1 - c^2*x^2)*(a + b*ArcSin[c*x])^2 )/2 - ((I/3)*(a + b*ArcSin[c*x])^3)/b - b*c*(-1/4*(b*c*x^2) + (x*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/2 + (a + b*ArcSin[c*x])^2/(4*b*c)) - (2*I)*( (I/2)*(a + b*ArcSin[c*x])^2*Log[1 - E^((2*I)*ArcSin[c*x])] - I*b*((I/2)*(a + b*ArcSin[c*x])*PolyLog[2, E^((2*I)*ArcSin[c*x])] - (b*PolyLog[3, E^((2* I)*ArcSin[c*x])])/4)))
3.2.72.3.1 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[((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_.)*((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]
Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_S ymbol] :> Simp[PolyLog[n + 1, c*(a + b*x)^p]/(e*p), x] /; FreeQ[{a, b, c, d , e, n, p}, x] && EqQ[b*d, a*e]
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 639 vs. \(2 (301 ) = 602\).
Time = 0.31 (sec) , antiderivative size = 640, normalized size of antiderivative = 2.23
| method | result | size |
| parts | \(d^{2} a^{2} \left (\frac {c^{4} x^{2}}{2}-\frac {1}{2 x^{2}}-2 c^{2} \ln \left (x \right )\right )+d^{2} b^{2} c^{2} \left (\frac {2 i \arcsin \left (c x \right )^{3}}{3}+\frac {\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 )}{16}+\frac {\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 )}{16}-\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 )-2 \arcsin \left (c x \right )^{2} \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )+4 i \arcsin \left (c x \right ) \operatorname {polylog}\left (2, -i c x -\sqrt {-c^{2} x^{2}+1}\right )-4 \operatorname {polylog}\left (3, -i c x -\sqrt {-c^{2} x^{2}+1}\right )-2 \arcsin \left (c x \right )^{2} \ln \left (1-i c x -\sqrt {-c^{2} x^{2}+1}\right )+4 i \arcsin \left (c x \right ) \operatorname {polylog}\left (2, i c x +\sqrt {-c^{2} x^{2}+1}\right )-4 \operatorname {polylog}\left (3, i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )+2 d^{2} a b \,c^{2} \left (i \arcsin \left (c x \right )^{2}+\frac {\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 )}{16}+\frac {\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 )}{16}-\frac {-i c^{2} x^{2}+c x \sqrt {-c^{2} x^{2}+1}+\arcsin \left (c x \right )}{2 c^{2} x^{2}}-2 \arcsin \left (c x \right ) \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )-2 \arcsin \left (c x \right ) \ln \left (1-i c x -\sqrt {-c^{2} x^{2}+1}\right )+2 i \operatorname {polylog}\left (2, -i c x -\sqrt {-c^{2} x^{2}+1}\right )+2 i \operatorname {polylog}\left (2, i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )\) | \(640\) |
| derivativedivides | \(c^{2} \left (d^{2} a^{2} \left (\frac {c^{2} x^{2}}{2}-\frac {1}{2 c^{2} x^{2}}-2 \ln \left (c x \right )\right )+d^{2} b^{2} \left (\frac {2 i \arcsin \left (c x \right )^{3}}{3}+\frac {\left (2 i \arcsin \left (c x \right )+2 \arcsin \left (c x \right )^{2}-1\right ) \left (2 c^{2} x^{2}-2 i c x \sqrt {-c^{2} x^{2}+1}-1\right )}{16}+\frac {\left (2 i c x \sqrt {-c^{2} x^{2}+1}+2 c^{2} x^{2}-1\right ) \left (-2 i \arcsin \left (c x \right )+2 \arcsin \left (c x \right )^{2}-1\right )}{16}-\frac {\arcsin \left (c x \right ) \left (-2 i c^{2} x^{2}+2 c x \sqrt {-c^{2} x^{2}+1}+\arcsin \left (c x \right )\right )}{2 c^{2} x^{2}}+\ln \left (i c x +\sqrt {-c^{2} x^{2}+1}-1\right )-2 \ln \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )+\ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )-2 \arcsin \left (c x \right )^{2} \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )+4 i \arcsin \left (c x \right ) \operatorname {polylog}\left (2, -i c x -\sqrt {-c^{2} x^{2}+1}\right )-4 \operatorname {polylog}\left (3, -i c x -\sqrt {-c^{2} x^{2}+1}\right )-2 \arcsin \left (c x \right )^{2} \ln \left (1-i c x -\sqrt {-c^{2} x^{2}+1}\right )+4 i \arcsin \left (c x \right ) \operatorname {polylog}\left (2, i c x +\sqrt {-c^{2} x^{2}+1}\right )-4 \operatorname {polylog}\left (3, i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )+2 i d^{2} a b \arcsin \left (c x \right )^{2}+\frac {d^{2} a b \sqrt {-c^{2} x^{2}+1}\, c x}{2}+d^{2} a b \arcsin \left (c x \right ) c^{2} x^{2}-\frac {d^{2} a b \arcsin \left (c x \right )}{2}+i d^{2} a b -\frac {d^{2} a b \sqrt {-c^{2} x^{2}+1}}{c x}-\frac {d^{2} a b \arcsin \left (c x \right )}{c^{2} x^{2}}-4 d^{2} a b \arcsin \left (c x \right ) \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )-4 d^{2} a b \arcsin \left (c x \right ) \ln \left (1-i c x -\sqrt {-c^{2} x^{2}+1}\right )+4 i d^{2} a b \operatorname {polylog}\left (2, -i c x -\sqrt {-c^{2} x^{2}+1}\right )+4 i d^{2} a b \operatorname {polylog}\left (2, i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )\) | \(642\) |
| default | \(c^{2} \left (d^{2} a^{2} \left (\frac {c^{2} x^{2}}{2}-\frac {1}{2 c^{2} x^{2}}-2 \ln \left (c x \right )\right )+d^{2} b^{2} \left (\frac {2 i \arcsin \left (c x \right )^{3}}{3}+\frac {\left (2 i \arcsin \left (c x \right )+2 \arcsin \left (c x \right )^{2}-1\right ) \left (2 c^{2} x^{2}-2 i c x \sqrt {-c^{2} x^{2}+1}-1\right )}{16}+\frac {\left (2 i c x \sqrt {-c^{2} x^{2}+1}+2 c^{2} x^{2}-1\right ) \left (-2 i \arcsin \left (c x \right )+2 \arcsin \left (c x \right )^{2}-1\right )}{16}-\frac {\arcsin \left (c x \right ) \left (-2 i c^{2} x^{2}+2 c x \sqrt {-c^{2} x^{2}+1}+\arcsin \left (c x \right )\right )}{2 c^{2} x^{2}}+\ln \left (i c x +\sqrt {-c^{2} x^{2}+1}-1\right )-2 \ln \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )+\ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )-2 \arcsin \left (c x \right )^{2} \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )+4 i \arcsin \left (c x \right ) \operatorname {polylog}\left (2, -i c x -\sqrt {-c^{2} x^{2}+1}\right )-4 \operatorname {polylog}\left (3, -i c x -\sqrt {-c^{2} x^{2}+1}\right )-2 \arcsin \left (c x \right )^{2} \ln \left (1-i c x -\sqrt {-c^{2} x^{2}+1}\right )+4 i \arcsin \left (c x \right ) \operatorname {polylog}\left (2, i c x +\sqrt {-c^{2} x^{2}+1}\right )-4 \operatorname {polylog}\left (3, i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )+2 i d^{2} a b \arcsin \left (c x \right )^{2}+\frac {d^{2} a b \sqrt {-c^{2} x^{2}+1}\, c x}{2}+d^{2} a b \arcsin \left (c x \right ) c^{2} x^{2}-\frac {d^{2} a b \arcsin \left (c x \right )}{2}+i d^{2} a b -\frac {d^{2} a b \sqrt {-c^{2} x^{2}+1}}{c x}-\frac {d^{2} a b \arcsin \left (c x \right )}{c^{2} x^{2}}-4 d^{2} a b \arcsin \left (c x \right ) \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )-4 d^{2} a b \arcsin \left (c x \right ) \ln \left (1-i c x -\sqrt {-c^{2} x^{2}+1}\right )+4 i d^{2} a b \operatorname {polylog}\left (2, -i c x -\sqrt {-c^{2} x^{2}+1}\right )+4 i d^{2} a b \operatorname {polylog}\left (2, i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )\) | \(642\) |
d^2*a^2*(1/2*c^4*x^2-1/2/x^2-2*c^2*ln(x))+d^2*b^2*c^2*(2/3*I*arcsin(c*x)^3 +1/16*(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)+1/16*(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*arcsin(c*x)*(-2*I*c^2*x^2+2*c*x*(-c^2*x^2+1)^(1/2)+a rcsin(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))-2*arcsin(c*x)^2*ln(1+I*c*x+(-c^2*x^ 2+1)^(1/2))+4*I*arcsin(c*x)*polylog(2,-I*c*x-(-c^2*x^2+1)^(1/2))-4*polylog (3,-I*c*x-(-c^2*x^2+1)^(1/2))-2*arcsin(c*x)^2*ln(1-I*c*x-(-c^2*x^2+1)^(1/2 ))+4*I*arcsin(c*x)*polylog(2,I*c*x+(-c^2*x^2+1)^(1/2))-4*polylog(3,I*c*x+( -c^2*x^2+1)^(1/2)))+2*d^2*a*b*c^2*(I*arcsin(c*x)^2+1/16*(I+2*arcsin(c*x))* (2*c^2*x^2-2*I*c*x*(-c^2*x^2+1)^(1/2)-1)+1/16*(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)+arc sin(c*x))/c^2/x^2-2*arcsin(c*x)*ln(1+I*c*x+(-c^2*x^2+1)^(1/2))-2*arcsin(c* x)*ln(1-I*c*x-(-c^2*x^2+1)^(1/2))+2*I*polylog(2,-I*c*x-(-c^2*x^2+1)^(1/2)) +2*I*polylog(2,I*c*x+(-c^2*x^2+1)^(1/2)))
\[ \int \frac {\left (d-c^2 d x^2\right )^2 (a+b \arcsin (c x))^2}{x^3} \, dx=\int { \frac {{\left (c^{2} d x^{2} - d\right )}^{2} {\left (b \arcsin \left (c x\right ) + a\right )}^{2}}{x^{3}} \,d x } \]
integral((a^2*c^4*d^2*x^4 - 2*a^2*c^2*d^2*x^2 + a^2*d^2 + (b^2*c^4*d^2*x^4 - 2*b^2*c^2*d^2*x^2 + b^2*d^2)*arcsin(c*x)^2 + 2*(a*b*c^4*d^2*x^4 - 2*a*b *c^2*d^2*x^2 + a*b*d^2)*arcsin(c*x))/x^3, x)
\[ \int \frac {\left (d-c^2 d x^2\right )^2 (a+b \arcsin (c x))^2}{x^3} \, dx=d^{2} \left (\int \frac {a^{2}}{x^{3}}\, dx + \int \left (- \frac {2 a^{2} c^{2}}{x}\right )\, dx + \int a^{2} c^{4} x\, dx + \int \frac {b^{2} \operatorname {asin}^{2}{\left (c x \right )}}{x^{3}}\, dx + \int \frac {2 a b \operatorname {asin}{\left (c x \right )}}{x^{3}}\, dx + \int \left (- \frac {2 b^{2} c^{2} \operatorname {asin}^{2}{\left (c x \right )}}{x}\right )\, dx + \int b^{2} c^{4} x \operatorname {asin}^{2}{\left (c x \right )}\, dx + \int \left (- \frac {4 a b c^{2} \operatorname {asin}{\left (c x \right )}}{x}\right )\, dx + \int 2 a b c^{4} x \operatorname {asin}{\left (c x \right )}\, dx\right ) \]
d**2*(Integral(a**2/x**3, x) + Integral(-2*a**2*c**2/x, x) + Integral(a**2 *c**4*x, x) + Integral(b**2*asin(c*x)**2/x**3, x) + Integral(2*a*b*asin(c* x)/x**3, x) + Integral(-2*b**2*c**2*asin(c*x)**2/x, x) + Integral(b**2*c** 4*x*asin(c*x)**2, x) + Integral(-4*a*b*c**2*asin(c*x)/x, x) + Integral(2*a *b*c**4*x*asin(c*x), x))
\[ \int \frac {\left (d-c^2 d x^2\right )^2 (a+b \arcsin (c x))^2}{x^3} \, dx=\int { \frac {{\left (c^{2} d x^{2} - d\right )}^{2} {\left (b \arcsin \left (c x\right ) + a\right )}^{2}}{x^{3}} \,d x } \]
1/2*a^2*c^4*d^2*x^2 - 2*a^2*c^2*d^2*log(x) - a*b*d^2*(sqrt(-c^2*x^2 + 1)*c /x + arcsin(c*x)/x^2) - 1/2*a^2*d^2/x^2 + integrate(((b^2*c^4*d^2*x^4 - 2* b^2*c^2*d^2*x^2 + b^2*d^2)*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1))^2 + 2*(a*b*c^4*d^2*x^4 - 2*a*b*c^2*d^2*x^2)*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c *x + 1)))/x^3, x)
\[ \int \frac {\left (d-c^2 d x^2\right )^2 (a+b \arcsin (c x))^2}{x^3} \, dx=\int { \frac {{\left (c^{2} d x^{2} - d\right )}^{2} {\left (b \arcsin \left (c x\right ) + a\right )}^{2}}{x^{3}} \,d x } \]
Timed out. \[ \int \frac {\left (d-c^2 d x^2\right )^2 (a+b \arcsin (c x))^2}{x^3} \, dx=\int \frac {{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^2\,{\left (d-c^2\,d\,x^2\right )}^2}{x^3} \,d x \]