Integrand size = 27, antiderivative size = 271 \[ \int \frac {\left (d-c^2 d x^2\right )^2 (a+b \arcsin (c x))^2}{x} \, dx=\frac {13}{32} b^2 c^2 d^2 x^2-\frac {1}{32} b^2 c^4 d^2 x^4-\frac {11}{16} b c d^2 x \sqrt {1-c^2 x^2} (a+b \arcsin (c x))-\frac {1}{8} b c d^2 x \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))-\frac {11}{32} d^2 (a+b \arcsin (c x))^2+\frac {1}{2} d^2 \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2+\frac {1}{4} d^2 \left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))^2-\frac {i d^2 (a+b \arcsin (c x))^3}{3 b}+d^2 (a+b \arcsin (c x))^2 \log \left (1-e^{2 i \arcsin (c x)}\right )-i b d^2 (a+b \arcsin (c x)) \operatorname {PolyLog}\left (2,e^{2 i \arcsin (c x)}\right )+\frac {1}{2} b^2 d^2 \operatorname {PolyLog}\left (3,e^{2 i \arcsin (c x)}\right ) \]
13/32*b^2*c^2*d^2*x^2-1/32*b^2*c^4*d^2*x^4-1/8*b*c*d^2*x*(-c^2*x^2+1)^(3/2 )*(a+b*arcsin(c*x))-11/32*d^2*(a+b*arcsin(c*x))^2+1/2*d^2*(-c^2*x^2+1)*(a+ b*arcsin(c*x))^2+1/4*d^2*(-c^2*x^2+1)^2*(a+b*arcsin(c*x))^2-1/3*I*d^2*(a+b *arcsin(c*x))^3/b+d^2*(a+b*arcsin(c*x))^2*ln(1-(I*c*x+(-c^2*x^2+1)^(1/2))^ 2)-I*b*d^2*(a+b*arcsin(c*x))*polylog(2,(I*c*x+(-c^2*x^2+1)^(1/2))^2)+1/2*b ^2*d^2*polylog(3,(I*c*x+(-c^2*x^2+1)^(1/2))^2)-11/16*b*c*d^2*x*(a+b*arcsin (c*x))*(-c^2*x^2+1)^(1/2)
Time = 0.45 (sec) , antiderivative size = 371, normalized size of antiderivative = 1.37 \[ \int \frac {\left (d-c^2 d x^2\right )^2 (a+b \arcsin (c x))^2}{x} \, dx=\frac {1}{768} d^2 \left (-32 i b^2 \pi ^3-768 a^2 c^2 x^2+192 a^2 c^4 x^4-624 a b c x \sqrt {1-c^2 x^2}+96 a b c^3 x^3 \sqrt {1-c^2 x^2}-1536 a b c^2 x^2 \arcsin (c x)+384 a b c^4 x^4 \arcsin (c x)-768 i a b \arcsin (c x)^2+256 i b^2 \arcsin (c x)^3+1248 a b \arctan \left (\frac {c x}{-1+\sqrt {1-c^2 x^2}}\right )-144 b^2 \cos (2 \arcsin (c x))+288 b^2 \arcsin (c x)^2 \cos (2 \arcsin (c x))-3 b^2 \cos (4 \arcsin (c x))+24 b^2 \arcsin (c x)^2 \cos (4 \arcsin (c x))+768 b^2 \arcsin (c x)^2 \log \left (1-e^{-2 i \arcsin (c x)}\right )+1536 a b \arcsin (c x) \log \left (1-e^{2 i \arcsin (c x)}\right )+768 a^2 \log (c x)+768 i b^2 \arcsin (c x) \operatorname {PolyLog}\left (2,e^{-2 i \arcsin (c x)}\right )-768 i a b \operatorname {PolyLog}\left (2,e^{2 i \arcsin (c x)}\right )+384 b^2 \operatorname {PolyLog}\left (3,e^{-2 i \arcsin (c x)}\right )-288 b^2 \arcsin (c x) \sin (2 \arcsin (c x))-12 b^2 \arcsin (c x) \sin (4 \arcsin (c x))\right ) \]
(d^2*((-32*I)*b^2*Pi^3 - 768*a^2*c^2*x^2 + 192*a^2*c^4*x^4 - 624*a*b*c*x*S qrt[1 - c^2*x^2] + 96*a*b*c^3*x^3*Sqrt[1 - c^2*x^2] - 1536*a*b*c^2*x^2*Arc Sin[c*x] + 384*a*b*c^4*x^4*ArcSin[c*x] - (768*I)*a*b*ArcSin[c*x]^2 + (256* I)*b^2*ArcSin[c*x]^3 + 1248*a*b*ArcTan[(c*x)/(-1 + Sqrt[1 - c^2*x^2])] - 1 44*b^2*Cos[2*ArcSin[c*x]] + 288*b^2*ArcSin[c*x]^2*Cos[2*ArcSin[c*x]] - 3*b ^2*Cos[4*ArcSin[c*x]] + 24*b^2*ArcSin[c*x]^2*Cos[4*ArcSin[c*x]] + 768*b^2* ArcSin[c*x]^2*Log[1 - E^((-2*I)*ArcSin[c*x])] + 1536*a*b*ArcSin[c*x]*Log[1 - E^((2*I)*ArcSin[c*x])] + 768*a^2*Log[c*x] + (768*I)*b^2*ArcSin[c*x]*Pol yLog[2, E^((-2*I)*ArcSin[c*x])] - (768*I)*a*b*PolyLog[2, E^((2*I)*ArcSin[c *x])] + 384*b^2*PolyLog[3, E^((-2*I)*ArcSin[c*x])] - 288*b^2*ArcSin[c*x]*S in[2*ArcSin[c*x]] - 12*b^2*ArcSin[c*x]*Sin[4*ArcSin[c*x]]))/768
Time = 2.01 (sec) , antiderivative size = 346, normalized size of antiderivative = 1.28, number of steps used = 22, number of rules used = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.778, Rules used = {5202, 27, 5158, 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.
| \(\displaystyle \int \frac {\left (d-c^2 d x^2\right )^2 (a+b \arcsin (c x))^2}{x} \, dx\) |
| \(\Big \downarrow \) 5202 |
| \(\displaystyle -\frac {1}{2} b c d^2 \int \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))dx+d \int \frac {d \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2}{x}dx+\frac {1}{4} d^2 \left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))^2\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {1}{2} b c d^2 \int \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))dx+d^2 \int \frac {\left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2}{x}dx+\frac {1}{4} d^2 \left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))^2\) |
| \(\Big \downarrow \) 5158 |
| \(\displaystyle -\frac {1}{2} b c d^2 \left (\frac {3}{4} \int \sqrt {1-c^2 x^2} (a+b \arcsin (c x))dx-\frac {1}{4} b c \int x \left (1-c^2 x^2\right )dx+\frac {1}{4} x \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))\right )+d^2 \int \frac {\left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2}{x}dx+\frac {1}{4} d^2 \left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))^2\) |
| \(\Big \downarrow \) 244 |
| \(\displaystyle d^2 \int \frac {\left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2}{x}dx-\frac {1}{2} b c d^2 \left (\frac {3}{4} \int \sqrt {1-c^2 x^2} (a+b \arcsin (c x))dx-\frac {1}{4} b c \int \left (x-c^2 x^3\right )dx+\frac {1}{4} x \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))\right )+\frac {1}{4} d^2 \left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))^2\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle d^2 \int \frac {\left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2}{x}dx-\frac {1}{2} b c d^2 \left (\frac {3}{4} \int \sqrt {1-c^2 x^2} (a+b \arcsin (c x))dx+\frac {1}{4} x \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))-\frac {1}{4} b c \left (\frac {x^2}{2}-\frac {c^2 x^4}{4}\right )\right )+\frac {1}{4} d^2 \left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))^2\) |
| \(\Big \downarrow \) 5156 |
| \(\displaystyle d^2 \int \frac {\left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2}{x}dx-\frac {1}{2} b c d^2 \left (\frac {3}{4} \left (\frac {1}{2} \int \frac {a+b \arcsin (c x)}{\sqrt {1-c^2 x^2}}dx-\frac {1}{2} b c \int xdx+\frac {1}{2} x \sqrt {1-c^2 x^2} (a+b \arcsin (c x))\right )+\frac {1}{4} x \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))-\frac {1}{4} b c \left (\frac {x^2}{2}-\frac {c^2 x^4}{4}\right )\right )+\frac {1}{4} d^2 \left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))^2\) |
| \(\Big \downarrow \) 15 |
| \(\displaystyle d^2 \int \frac {\left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2}{x}dx-\frac {1}{2} b c d^2 \left (\frac {3}{4} \left (\frac {1}{2} \int \frac {a+b \arcsin (c x)}{\sqrt {1-c^2 x^2}}dx+\frac {1}{2} x \sqrt {1-c^2 x^2} (a+b \arcsin (c x))-\frac {1}{4} b c x^2\right )+\frac {1}{4} x \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))-\frac {1}{4} b c \left (\frac {x^2}{2}-\frac {c^2 x^4}{4}\right )\right )+\frac {1}{4} d^2 \left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))^2\) |
| \(\Big \downarrow \) 5152 |
| \(\displaystyle d^2 \int \frac {\left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2}{x}dx+\frac {1}{4} d^2 \left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))^2-\frac {1}{2} b c d^2 \left (\frac {1}{4} x \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))+\frac {3}{4} \left (\frac {1}{2} x \sqrt {1-c^2 x^2} (a+b \arcsin (c x))+\frac {(a+b \arcsin (c x))^2}{4 b c}-\frac {1}{4} b c x^2\right )-\frac {1}{4} b c \left (\frac {x^2}{2}-\frac {c^2 x^4}{4}\right )\right )\) |
| \(\Big \downarrow \) 5202 |
| \(\displaystyle d^2 \left (-b c \int \sqrt {1-c^2 x^2} (a+b \arcsin (c x))dx+\int \frac {(a+b \arcsin (c x))^2}{x}dx+\frac {1}{2} \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2\right )+\frac {1}{4} d^2 \left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))^2-\frac {1}{2} b c d^2 \left (\frac {1}{4} x \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))+\frac {3}{4} \left (\frac {1}{2} x \sqrt {1-c^2 x^2} (a+b \arcsin (c x))+\frac {(a+b \arcsin (c x))^2}{4 b c}-\frac {1}{4} b c x^2\right )-\frac {1}{4} b c \left (\frac {x^2}{2}-\frac {c^2 x^4}{4}\right )\right )\) |
| \(\Big \downarrow \) 5136 |
| \(\displaystyle d^2 \left (-b c \int \sqrt {1-c^2 x^2} (a+b \arcsin (c x))dx+\int \frac {\sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2}{c x}d\arcsin (c x)+\frac {1}{2} \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2\right )+\frac {1}{4} d^2 \left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))^2-\frac {1}{2} b c d^2 \left (\frac {1}{4} x \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))+\frac {3}{4} \left (\frac {1}{2} x \sqrt {1-c^2 x^2} (a+b \arcsin (c x))+\frac {(a+b \arcsin (c x))^2}{4 b c}-\frac {1}{4} b c x^2\right )-\frac {1}{4} b c \left (\frac {x^2}{2}-\frac {c^2 x^4}{4}\right )\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle d^2 \left (-b c \int \sqrt {1-c^2 x^2} (a+b \arcsin (c x))dx+\int -(a+b \arcsin (c x))^2 \tan \left (\arcsin (c x)+\frac {\pi }{2}\right )d\arcsin (c x)+\frac {1}{2} \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2\right )+\frac {1}{4} d^2 \left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))^2-\frac {1}{2} b c d^2 \left (\frac {1}{4} x \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))+\frac {3}{4} \left (\frac {1}{2} x \sqrt {1-c^2 x^2} (a+b \arcsin (c x))+\frac {(a+b \arcsin (c x))^2}{4 b c}-\frac {1}{4} b c x^2\right )-\frac {1}{4} b c \left (\frac {x^2}{2}-\frac {c^2 x^4}{4}\right )\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle d^2 \left (-b c \int \sqrt {1-c^2 x^2} (a+b \arcsin (c x))dx-\int (a+b \arcsin (c x))^2 \tan \left (\arcsin (c x)+\frac {\pi }{2}\right )d\arcsin (c x)+\frac {1}{2} \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2\right )+\frac {1}{4} d^2 \left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))^2-\frac {1}{2} b c d^2 \left (\frac {1}{4} x \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))+\frac {3}{4} \left (\frac {1}{2} x \sqrt {1-c^2 x^2} (a+b \arcsin (c x))+\frac {(a+b \arcsin (c x))^2}{4 b c}-\frac {1}{4} b c x^2\right )-\frac {1}{4} b c \left (\frac {x^2}{2}-\frac {c^2 x^4}{4}\right )\right )\) |
| \(\Big \downarrow \) 4200 |
| \(\displaystyle d^2 \left (-b c \int \sqrt {1-c^2 x^2} (a+b \arcsin (c x))dx+2 i \int -\frac {e^{2 i \arcsin (c x)} (a+b \arcsin (c x))^2}{1-e^{2 i \arcsin (c x)}}d\arcsin (c x)+\frac {1}{2} \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2-\frac {i (a+b \arcsin (c x))^3}{3 b}\right )+\frac {1}{4} d^2 \left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))^2-\frac {1}{2} b c d^2 \left (\frac {1}{4} x \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))+\frac {3}{4} \left (\frac {1}{2} x \sqrt {1-c^2 x^2} (a+b \arcsin (c x))+\frac {(a+b \arcsin (c x))^2}{4 b c}-\frac {1}{4} b c x^2\right )-\frac {1}{4} b c \left (\frac {x^2}{2}-\frac {c^2 x^4}{4}\right )\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle d^2 \left (-b c \int \sqrt {1-c^2 x^2} (a+b \arcsin (c x))dx-2 i \int \frac {e^{2 i \arcsin (c x)} (a+b \arcsin (c x))^2}{1-e^{2 i \arcsin (c x)}}d\arcsin (c x)+\frac {1}{2} \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2-\frac {i (a+b \arcsin (c x))^3}{3 b}\right )+\frac {1}{4} d^2 \left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))^2-\frac {1}{2} b c d^2 \left (\frac {1}{4} x \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))+\frac {3}{4} \left (\frac {1}{2} x \sqrt {1-c^2 x^2} (a+b \arcsin (c x))+\frac {(a+b \arcsin (c x))^2}{4 b c}-\frac {1}{4} b c x^2\right )-\frac {1}{4} b c \left (\frac {x^2}{2}-\frac {c^2 x^4}{4}\right )\right )\) |
| \(\Big \downarrow \) 2620 |
| \(\displaystyle d^2 \left (-b c \int \sqrt {1-c^2 x^2} (a+b \arcsin (c x))dx-2 i \left (\frac {1}{2} i \log \left (1-e^{2 i \arcsin (c x)}\right ) (a+b \arcsin (c x))^2-i b \int (a+b \arcsin (c x)) \log \left (1-e^{2 i \arcsin (c x)}\right )d\arcsin (c x)\right )+\frac {1}{2} \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2-\frac {i (a+b \arcsin (c x))^3}{3 b}\right )+\frac {1}{4} d^2 \left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))^2-\frac {1}{2} b c d^2 \left (\frac {1}{4} x \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))+\frac {3}{4} \left (\frac {1}{2} x \sqrt {1-c^2 x^2} (a+b \arcsin (c x))+\frac {(a+b \arcsin (c x))^2}{4 b c}-\frac {1}{4} b c x^2\right )-\frac {1}{4} b c \left (\frac {x^2}{2}-\frac {c^2 x^4}{4}\right )\right )\) |
| \(\Big \downarrow \) 3011 |
| \(\displaystyle d^2 \left (-b c \int \sqrt {1-c^2 x^2} (a+b \arcsin (c x))dx-2 i \left (\frac {1}{2} i \log \left (1-e^{2 i \arcsin (c x)}\right ) (a+b \arcsin (c x))^2-i b \left (\frac {1}{2} i \operatorname {PolyLog}\left (2,e^{2 i \arcsin (c x)}\right ) (a+b \arcsin (c x))-\frac {1}{2} i b \int \operatorname {PolyLog}\left (2,e^{2 i \arcsin (c x)}\right )d\arcsin (c x)\right )\right )+\frac {1}{2} \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2-\frac {i (a+b \arcsin (c x))^3}{3 b}\right )+\frac {1}{4} d^2 \left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))^2-\frac {1}{2} b c d^2 \left (\frac {1}{4} x \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))+\frac {3}{4} \left (\frac {1}{2} x \sqrt {1-c^2 x^2} (a+b \arcsin (c x))+\frac {(a+b \arcsin (c x))^2}{4 b c}-\frac {1}{4} b c x^2\right )-\frac {1}{4} b c \left (\frac {x^2}{2}-\frac {c^2 x^4}{4}\right )\right )\) |
| \(\Big \downarrow \) 2720 |
| \(\displaystyle d^2 \left (-b c \int \sqrt {1-c^2 x^2} (a+b \arcsin (c x))dx-2 i \left (\frac {1}{2} i \log \left (1-e^{2 i \arcsin (c x)}\right ) (a+b \arcsin (c x))^2-i b \left (\frac {1}{2} i \operatorname {PolyLog}\left (2,e^{2 i \arcsin (c x)}\right ) (a+b \arcsin (c x))-\frac {1}{4} b \int e^{-2 i \arcsin (c x)} \operatorname {PolyLog}\left (2,e^{2 i \arcsin (c x)}\right )de^{2 i \arcsin (c x)}\right )\right )+\frac {1}{2} \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2-\frac {i (a+b \arcsin (c x))^3}{3 b}\right )+\frac {1}{4} d^2 \left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))^2-\frac {1}{2} b c d^2 \left (\frac {1}{4} x \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))+\frac {3}{4} \left (\frac {1}{2} x \sqrt {1-c^2 x^2} (a+b \arcsin (c x))+\frac {(a+b \arcsin (c x))^2}{4 b c}-\frac {1}{4} b c x^2\right )-\frac {1}{4} b c \left (\frac {x^2}{2}-\frac {c^2 x^4}{4}\right )\right )\) |
| \(\Big \downarrow \) 5156 |
| \(\displaystyle d^2 \left (-b c \left (\frac {1}{2} \int \frac {a+b \arcsin (c x)}{\sqrt {1-c^2 x^2}}dx-\frac {1}{2} b c \int xdx+\frac {1}{2} x \sqrt {1-c^2 x^2} (a+b \arcsin (c x))\right )-2 i \left (\frac {1}{2} i \log \left (1-e^{2 i \arcsin (c x)}\right ) (a+b \arcsin (c x))^2-i b \left (\frac {1}{2} i \operatorname {PolyLog}\left (2,e^{2 i \arcsin (c x)}\right ) (a+b \arcsin (c x))-\frac {1}{4} b \int e^{-2 i \arcsin (c x)} \operatorname {PolyLog}\left (2,e^{2 i \arcsin (c x)}\right )de^{2 i \arcsin (c x)}\right )\right )+\frac {1}{2} \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2-\frac {i (a+b \arcsin (c x))^3}{3 b}\right )+\frac {1}{4} d^2 \left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))^2-\frac {1}{2} b c d^2 \left (\frac {1}{4} x \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))+\frac {3}{4} \left (\frac {1}{2} x \sqrt {1-c^2 x^2} (a+b \arcsin (c x))+\frac {(a+b \arcsin (c x))^2}{4 b c}-\frac {1}{4} b c x^2\right )-\frac {1}{4} b c \left (\frac {x^2}{2}-\frac {c^2 x^4}{4}\right )\right )\) |
| \(\Big \downarrow \) 15 |
| \(\displaystyle d^2 \left (-b c \left (\frac {1}{2} \int \frac {a+b \arcsin (c x)}{\sqrt {1-c^2 x^2}}dx+\frac {1}{2} x \sqrt {1-c^2 x^2} (a+b \arcsin (c x))-\frac {1}{4} b c x^2\right )-2 i \left (\frac {1}{2} i \log \left (1-e^{2 i \arcsin (c x)}\right ) (a+b \arcsin (c x))^2-i b \left (\frac {1}{2} i \operatorname {PolyLog}\left (2,e^{2 i \arcsin (c x)}\right ) (a+b \arcsin (c x))-\frac {1}{4} b \int e^{-2 i \arcsin (c x)} \operatorname {PolyLog}\left (2,e^{2 i \arcsin (c x)}\right )de^{2 i \arcsin (c x)}\right )\right )+\frac {1}{2} \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2-\frac {i (a+b \arcsin (c x))^3}{3 b}\right )+\frac {1}{4} d^2 \left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))^2-\frac {1}{2} b c d^2 \left (\frac {1}{4} x \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))+\frac {3}{4} \left (\frac {1}{2} x \sqrt {1-c^2 x^2} (a+b \arcsin (c x))+\frac {(a+b \arcsin (c x))^2}{4 b c}-\frac {1}{4} b c x^2\right )-\frac {1}{4} b c \left (\frac {x^2}{2}-\frac {c^2 x^4}{4}\right )\right )\) |
| \(\Big \downarrow \) 5152 |
| \(\displaystyle d^2 \left (-2 i \left (\frac {1}{2} i \log \left (1-e^{2 i \arcsin (c x)}\right ) (a+b \arcsin (c x))^2-i b \left (\frac {1}{2} i \operatorname {PolyLog}\left (2,e^{2 i \arcsin (c x)}\right ) (a+b \arcsin (c x))-\frac {1}{4} b \int e^{-2 i \arcsin (c x)} \operatorname {PolyLog}\left (2,e^{2 i \arcsin (c x)}\right )de^{2 i \arcsin (c x)}\right )\right )+\frac {1}{2} \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2-b c \left (\frac {1}{2} x \sqrt {1-c^2 x^2} (a+b \arcsin (c x))+\frac {(a+b \arcsin (c x))^2}{4 b c}-\frac {1}{4} b c x^2\right )-\frac {i (a+b \arcsin (c x))^3}{3 b}\right )+\frac {1}{4} d^2 \left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))^2-\frac {1}{2} b c d^2 \left (\frac {1}{4} x \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))+\frac {3}{4} \left (\frac {1}{2} x \sqrt {1-c^2 x^2} (a+b \arcsin (c x))+\frac {(a+b \arcsin (c x))^2}{4 b c}-\frac {1}{4} b c x^2\right )-\frac {1}{4} b c \left (\frac {x^2}{2}-\frac {c^2 x^4}{4}\right )\right )\) |
| \(\Big \downarrow \) 7143 |
| \(\displaystyle d^2 \left (\frac {1}{2} \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2-b c \left (\frac {1}{2} x \sqrt {1-c^2 x^2} (a+b \arcsin (c x))+\frac {(a+b \arcsin (c x))^2}{4 b c}-\frac {1}{4} b c x^2\right )-2 i \left (\frac {1}{2} i \log \left (1-e^{2 i \arcsin (c x)}\right ) (a+b \arcsin (c x))^2-i b \left (\frac {1}{2} i \operatorname {PolyLog}\left (2,e^{2 i \arcsin (c x)}\right ) (a+b \arcsin (c x))-\frac {1}{4} b \operatorname {PolyLog}\left (3,e^{2 i \arcsin (c x)}\right )\right )\right )-\frac {i (a+b \arcsin (c x))^3}{3 b}\right )+\frac {1}{4} d^2 \left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))^2-\frac {1}{2} b c d^2 \left (\frac {1}{4} x \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))+\frac {3}{4} \left (\frac {1}{2} x \sqrt {1-c^2 x^2} (a+b \arcsin (c x))+\frac {(a+b \arcsin (c x))^2}{4 b c}-\frac {1}{4} b c x^2\right )-\frac {1}{4} b c \left (\frac {x^2}{2}-\frac {c^2 x^4}{4}\right )\right )\) |
(d^2*(1 - c^2*x^2)^2*(a + b*ArcSin[c*x])^2)/4 - (b*c*d^2*(-1/4*(b*c*(x^2/2 - (c^2*x^4)/4)) + (x*(1 - c^2*x^2)^(3/2)*(a + b*ArcSin[c*x]))/4 + (3*(-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)))/4))/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*Ar cSin[c*x])*PolyLog[2, E^((2*I)*ArcSin[c*x])] - (b*PolyLog[3, E^((2*I)*ArcS in[c*x])])/4)))
3.2.70.3.1 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_.)*((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 + 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]
Time = 0.18 (sec) , antiderivative size = 428, normalized size of antiderivative = 1.58
| method | result | size |
| parts | \(d^{2} a^{2} \left (\frac {c^{4} x^{4}}{4}-c^{2} x^{2}+\ln \left (x \right )\right )+d^{2} b^{2} \left (-\frac {i \arcsin \left (c x \right )^{3}}{3}+\arcsin \left (c x \right )^{2} \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )-2 i \arcsin \left (c x \right ) \operatorname {polylog}\left (2, -i c x -\sqrt {-c^{2} x^{2}+1}\right )+2 \operatorname {polylog}\left (3, -i c x -\sqrt {-c^{2} x^{2}+1}\right )+\arcsin \left (c x \right )^{2} \ln \left (1-i c x -\sqrt {-c^{2} x^{2}+1}\right )-2 i \arcsin \left (c x \right ) \operatorname {polylog}\left (2, i c x +\sqrt {-c^{2} x^{2}+1}\right )+2 \operatorname {polylog}\left (3, i c x +\sqrt {-c^{2} x^{2}+1}\right )+\frac {\left (8 \arcsin \left (c x \right )^{2}-1\right ) \cos \left (4 \arcsin \left (c x \right )\right )}{256}-\frac {\arcsin \left (c x \right ) \sin \left (4 \arcsin \left (c x \right )\right )}{64}+\frac {3 \left (2 \arcsin \left (c x \right )^{2}-1\right ) \cos \left (2 \arcsin \left (c x \right )\right )}{16}-\frac {3 \arcsin \left (c x \right ) \sin \left (2 \arcsin \left (c x \right )\right )}{8}\right )+2 d^{2} a b \left (-\frac {i \arcsin \left (c x \right )^{2}}{2}+\arcsin \left (c x \right ) \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )-i \operatorname {polylog}\left (2, -i c x -\sqrt {-c^{2} x^{2}+1}\right )+\arcsin \left (c x \right ) \ln \left (1-i c x -\sqrt {-c^{2} x^{2}+1}\right )-i \operatorname {polylog}\left (2, i c x +\sqrt {-c^{2} x^{2}+1}\right )+\frac {\arcsin \left (c x \right ) \cos \left (4 \arcsin \left (c x \right )\right )}{32}-\frac {\sin \left (4 \arcsin \left (c x \right )\right )}{128}+\frac {3 \arcsin \left (c x \right ) \cos \left (2 \arcsin \left (c x \right )\right )}{8}-\frac {3 \sin \left (2 \arcsin \left (c x \right )\right )}{16}\right )\) | \(428\) |
| derivativedivides | \(d^{2} a^{2} \left (\frac {c^{4} x^{4}}{4}-c^{2} x^{2}+\ln \left (c x \right )\right )+d^{2} b^{2} \left (-\frac {i \arcsin \left (c x \right )^{3}}{3}+\arcsin \left (c x \right )^{2} \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )-2 i \arcsin \left (c x \right ) \operatorname {polylog}\left (2, -i c x -\sqrt {-c^{2} x^{2}+1}\right )+2 \operatorname {polylog}\left (3, -i c x -\sqrt {-c^{2} x^{2}+1}\right )+\arcsin \left (c x \right )^{2} \ln \left (1-i c x -\sqrt {-c^{2} x^{2}+1}\right )-2 i \arcsin \left (c x \right ) \operatorname {polylog}\left (2, i c x +\sqrt {-c^{2} x^{2}+1}\right )+2 \operatorname {polylog}\left (3, i c x +\sqrt {-c^{2} x^{2}+1}\right )+\frac {\left (8 \arcsin \left (c x \right )^{2}-1\right ) \cos \left (4 \arcsin \left (c x \right )\right )}{256}-\frac {\arcsin \left (c x \right ) \sin \left (4 \arcsin \left (c x \right )\right )}{64}+\frac {3 \left (2 \arcsin \left (c x \right )^{2}-1\right ) \cos \left (2 \arcsin \left (c x \right )\right )}{16}-\frac {3 \arcsin \left (c x \right ) \sin \left (2 \arcsin \left (c x \right )\right )}{8}\right )+2 d^{2} a b \left (-\frac {i \arcsin \left (c x \right )^{2}}{2}+\arcsin \left (c x \right ) \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )-i \operatorname {polylog}\left (2, -i c x -\sqrt {-c^{2} x^{2}+1}\right )+\arcsin \left (c x \right ) \ln \left (1-i c x -\sqrt {-c^{2} x^{2}+1}\right )-i \operatorname {polylog}\left (2, i c x +\sqrt {-c^{2} x^{2}+1}\right )+\frac {\arcsin \left (c x \right ) \cos \left (4 \arcsin \left (c x \right )\right )}{32}-\frac {\sin \left (4 \arcsin \left (c x \right )\right )}{128}+\frac {3 \arcsin \left (c x \right ) \cos \left (2 \arcsin \left (c x \right )\right )}{8}-\frac {3 \sin \left (2 \arcsin \left (c x \right )\right )}{16}\right )\) | \(430\) |
| default | \(d^{2} a^{2} \left (\frac {c^{4} x^{4}}{4}-c^{2} x^{2}+\ln \left (c x \right )\right )+d^{2} b^{2} \left (-\frac {i \arcsin \left (c x \right )^{3}}{3}+\arcsin \left (c x \right )^{2} \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )-2 i \arcsin \left (c x \right ) \operatorname {polylog}\left (2, -i c x -\sqrt {-c^{2} x^{2}+1}\right )+2 \operatorname {polylog}\left (3, -i c x -\sqrt {-c^{2} x^{2}+1}\right )+\arcsin \left (c x \right )^{2} \ln \left (1-i c x -\sqrt {-c^{2} x^{2}+1}\right )-2 i \arcsin \left (c x \right ) \operatorname {polylog}\left (2, i c x +\sqrt {-c^{2} x^{2}+1}\right )+2 \operatorname {polylog}\left (3, i c x +\sqrt {-c^{2} x^{2}+1}\right )+\frac {\left (8 \arcsin \left (c x \right )^{2}-1\right ) \cos \left (4 \arcsin \left (c x \right )\right )}{256}-\frac {\arcsin \left (c x \right ) \sin \left (4 \arcsin \left (c x \right )\right )}{64}+\frac {3 \left (2 \arcsin \left (c x \right )^{2}-1\right ) \cos \left (2 \arcsin \left (c x \right )\right )}{16}-\frac {3 \arcsin \left (c x \right ) \sin \left (2 \arcsin \left (c x \right )\right )}{8}\right )+2 d^{2} a b \left (-\frac {i \arcsin \left (c x \right )^{2}}{2}+\arcsin \left (c x \right ) \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )-i \operatorname {polylog}\left (2, -i c x -\sqrt {-c^{2} x^{2}+1}\right )+\arcsin \left (c x \right ) \ln \left (1-i c x -\sqrt {-c^{2} x^{2}+1}\right )-i \operatorname {polylog}\left (2, i c x +\sqrt {-c^{2} x^{2}+1}\right )+\frac {\arcsin \left (c x \right ) \cos \left (4 \arcsin \left (c x \right )\right )}{32}-\frac {\sin \left (4 \arcsin \left (c x \right )\right )}{128}+\frac {3 \arcsin \left (c x \right ) \cos \left (2 \arcsin \left (c x \right )\right )}{8}-\frac {3 \sin \left (2 \arcsin \left (c x \right )\right )}{16}\right )\) | \(430\) |
d^2*a^2*(1/4*c^4*x^4-c^2*x^2+ln(x))+d^2*b^2*(-1/3*I*arcsin(c*x)^3+arcsin(c *x)^2*ln(1+I*c*x+(-c^2*x^2+1)^(1/2))-2*I*arcsin(c*x)*polylog(2,-I*c*x-(-c^ 2*x^2+1)^(1/2))+2*polylog(3,-I*c*x-(-c^2*x^2+1)^(1/2))+arcsin(c*x)^2*ln(1- I*c*x-(-c^2*x^2+1)^(1/2))-2*I*arcsin(c*x)*polylog(2,I*c*x+(-c^2*x^2+1)^(1/ 2))+2*polylog(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))+3/16*(2*arcsin(c*x)^2-1)* cos(2*arcsin(c*x))-3/8*arcsin(c*x)*sin(2*arcsin(c*x)))+2*d^2*a*b*(-1/2*I*a rcsin(c*x)^2+arcsin(c*x)*ln(1+I*c*x+(-c^2*x^2+1)^(1/2))-I*polylog(2,-I*c*x -(-c^2*x^2+1)^(1/2))+arcsin(c*x)*ln(1-I*c*x-(-c^2*x^2+1)^(1/2))-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))+3/8*arcsin(c*x)*cos(2*arcsin(c*x))-3/16*sin(2*arcsin(c*x)))
\[ \int \frac {\left (d-c^2 d x^2\right )^2 (a+b \arcsin (c x))^2}{x} \, dx=\int { \frac {{\left (c^{2} d x^{2} - d\right )}^{2} {\left (b \arcsin \left (c x\right ) + a\right )}^{2}}{x} \,d x } \]
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, x)
\[ \int \frac {\left (d-c^2 d x^2\right )^2 (a+b \arcsin (c x))^2}{x} \, dx=d^{2} \left (\int \frac {a^{2}}{x}\, dx + \int \left (- 2 a^{2} c^{2} x\right )\, dx + \int a^{2} c^{4} x^{3}\, dx + \int \frac {b^{2} \operatorname {asin}^{2}{\left (c x \right )}}{x}\, dx + \int \frac {2 a b \operatorname {asin}{\left (c x \right )}}{x}\, dx + \int \left (- 2 b^{2} c^{2} x \operatorname {asin}^{2}{\left (c x \right )}\right )\, dx + \int b^{2} c^{4} x^{3} \operatorname {asin}^{2}{\left (c x \right )}\, dx + \int \left (- 4 a b c^{2} x \operatorname {asin}{\left (c x \right )}\right )\, dx + \int 2 a b c^{4} x^{3} \operatorname {asin}{\left (c x \right )}\, dx\right ) \]
d**2*(Integral(a**2/x, x) + Integral(-2*a**2*c**2*x, x) + Integral(a**2*c* *4*x**3, x) + Integral(b**2*asin(c*x)**2/x, x) + Integral(2*a*b*asin(c*x)/ x, x) + Integral(-2*b**2*c**2*x*asin(c*x)**2, x) + Integral(b**2*c**4*x**3 *asin(c*x)**2, x) + Integral(-4*a*b*c**2*x*asin(c*x), x) + Integral(2*a*b* c**4*x**3*asin(c*x), x))
\[ \int \frac {\left (d-c^2 d x^2\right )^2 (a+b \arcsin (c x))^2}{x} \, dx=\int { \frac {{\left (c^{2} d x^{2} - d\right )}^{2} {\left (b \arcsin \left (c x\right ) + a\right )}^{2}}{x} \,d x } \]
1/4*a^2*c^4*d^2*x^4 - a^2*c^2*d^2*x^2 + a^2*d^2*log(x) + 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 + a*b*d^2)*arctan2(c *x, sqrt(c*x + 1)*sqrt(-c*x + 1)))/x, x)
\[ \int \frac {\left (d-c^2 d x^2\right )^2 (a+b \arcsin (c x))^2}{x} \, dx=\int { \frac {{\left (c^{2} d x^{2} - d\right )}^{2} {\left (b \arcsin \left (c x\right ) + a\right )}^{2}}{x} \,d x } \]
Timed out. \[ \int \frac {\left (d-c^2 d x^2\right )^2 (a+b \arcsin (c x))^2}{x} \, dx=\int \frac {{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^2\,{\left (d-c^2\,d\,x^2\right )}^2}{x} \,d x \]