Integrand size = 27, antiderivative size = 296 \[ \int \frac {(a+b \arcsin (c x))^2}{x \left (d-c^2 d x^2\right )^3} \, dx=\frac {b^2}{12 d^3 \left (1-c^2 x^2\right )}-\frac {b c x (a+b \arcsin (c x))}{6 d^3 \left (1-c^2 x^2\right )^{3/2}}-\frac {4 b c x (a+b \arcsin (c x))}{3 d^3 \sqrt {1-c^2 x^2}}+\frac {(a+b \arcsin (c x))^2}{4 d^3 \left (1-c^2 x^2\right )^2}+\frac {(a+b \arcsin (c x))^2}{2 d^3 \left (1-c^2 x^2\right )}-\frac {2 (a+b \arcsin (c x))^2 \text {arctanh}\left (e^{2 i \arcsin (c x)}\right )}{d^3}-\frac {2 b^2 \log \left (1-c^2 x^2\right )}{3 d^3}+\frac {i b (a+b \arcsin (c x)) \operatorname {PolyLog}\left (2,-e^{2 i \arcsin (c x)}\right )}{d^3}-\frac {i b (a+b \arcsin (c x)) \operatorname {PolyLog}\left (2,e^{2 i \arcsin (c x)}\right )}{d^3}-\frac {b^2 \operatorname {PolyLog}\left (3,-e^{2 i \arcsin (c x)}\right )}{2 d^3}+\frac {b^2 \operatorname {PolyLog}\left (3,e^{2 i \arcsin (c x)}\right )}{2 d^3} \]
1/12*b^2/d^3/(-c^2*x^2+1)-1/6*b*c*x*(a+b*arcsin(c*x))/d^3/(-c^2*x^2+1)^(3/ 2)+1/4*(a+b*arcsin(c*x))^2/d^3/(-c^2*x^2+1)^2+1/2*(a+b*arcsin(c*x))^2/d^3/ (-c^2*x^2+1)-2*(a+b*arcsin(c*x))^2*arctanh((I*c*x+(-c^2*x^2+1)^(1/2))^2)/d ^3-2/3*b^2*ln(-c^2*x^2+1)/d^3+I*b*(a+b*arcsin(c*x))*polylog(2,-(I*c*x+(-c^ 2*x^2+1)^(1/2))^2)/d^3-I*b*(a+b*arcsin(c*x))*polylog(2,(I*c*x+(-c^2*x^2+1) ^(1/2))^2)/d^3-1/2*b^2*polylog(3,-(I*c*x+(-c^2*x^2+1)^(1/2))^2)/d^3+1/2*b^ 2*polylog(3,(I*c*x+(-c^2*x^2+1)^(1/2))^2)/d^3-4/3*b*c*x*(a+b*arcsin(c*x))/ d^3/(-c^2*x^2+1)^(1/2)
Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(756\) vs. \(2(296)=592\).
Time = 4.56 (sec) , antiderivative size = 756, normalized size of antiderivative = 2.55 \[ \int \frac {(a+b \arcsin (c x))^2}{x \left (d-c^2 d x^2\right )^3} \, dx=\frac {\frac {6 a^2}{\left (-1+c^2 x^2\right )^2}-\frac {12 a^2}{-1+c^2 x^2}+\frac {15 a b \left (\sqrt {1-c^2 x^2}-\arcsin (c x)\right )}{-1+c x}+\frac {15 a b \left (\sqrt {1-c^2 x^2}+\arcsin (c x)\right )}{1+c x}+\frac {a b \left ((-2+c x) \sqrt {1-c^2 x^2}+3 \arcsin (c x)\right )}{(-1+c x)^2}+\frac {a b \left ((2+c x) \sqrt {1-c^2 x^2}+3 \arcsin (c x)\right )}{(1+c x)^2}+48 a b \arcsin (c x) \log \left (1-e^{2 i \arcsin (c x)}\right )+24 a^2 \log (c x)-12 a^2 \log \left (1-c^2 x^2\right )+12 a b \left (i \arcsin (c x)^2+\arcsin (c x) \left (-3 i \pi -4 \log \left (1+i e^{i \arcsin (c x)}\right )\right )+2 \pi \left (-2 \log \left (1+e^{-i \arcsin (c x)}\right )+\log \left (1+i e^{i \arcsin (c x)}\right )+2 \log \left (\cos \left (\frac {1}{2} \arcsin (c x)\right )\right )-\log \left (-\cos \left (\frac {1}{4} (\pi +2 \arcsin (c x))\right )\right )\right )+4 i \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right )\right )+12 a b \left (i \arcsin (c x)^2+\arcsin (c x) \left (-i \pi -4 \log \left (1-i e^{i \arcsin (c x)}\right )\right )+2 \pi \left (-2 \log \left (1+e^{-i \arcsin (c x)}\right )-\log \left (1-i e^{i \arcsin (c x)}\right )+2 \log \left (\cos \left (\frac {1}{2} \arcsin (c x)\right )\right )+\log \left (\sin \left (\frac {1}{4} (\pi +2 \arcsin (c x))\right )\right )\right )+4 i \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right )\right )-24 i a b \left (\arcsin (c x)^2+\operatorname {PolyLog}\left (2,e^{2 i \arcsin (c x)}\right )\right )-b^2 \left (i \pi ^3+\frac {2}{-1+c^2 x^2}+\frac {4 c x \arcsin (c x)}{\left (1-c^2 x^2\right )^{3/2}}+\frac {32 c x \arcsin (c x)}{\sqrt {1-c^2 x^2}}-\frac {6 \arcsin (c x)^2}{\left (-1+c^2 x^2\right )^2}+\frac {12 \arcsin (c x)^2}{-1+c^2 x^2}-16 i \arcsin (c x)^3-24 \arcsin (c x)^2 \log \left (1-e^{-2 i \arcsin (c x)}\right )+24 \arcsin (c x)^2 \log \left (1+e^{2 i \arcsin (c x)}\right )+16 \log \left (1-c^2 x^2\right )-24 i \arcsin (c x) \operatorname {PolyLog}\left (2,e^{-2 i \arcsin (c x)}\right )-24 i \arcsin (c x) \operatorname {PolyLog}\left (2,-e^{2 i \arcsin (c x)}\right )-12 \operatorname {PolyLog}\left (3,e^{-2 i \arcsin (c x)}\right )+12 \operatorname {PolyLog}\left (3,-e^{2 i \arcsin (c x)}\right )\right )}{24 d^3} \]
((6*a^2)/(-1 + c^2*x^2)^2 - (12*a^2)/(-1 + c^2*x^2) + (15*a*b*(Sqrt[1 - c^ 2*x^2] - ArcSin[c*x]))/(-1 + c*x) + (15*a*b*(Sqrt[1 - c^2*x^2] + ArcSin[c* x]))/(1 + c*x) + (a*b*((-2 + c*x)*Sqrt[1 - c^2*x^2] + 3*ArcSin[c*x]))/(-1 + c*x)^2 + (a*b*((2 + c*x)*Sqrt[1 - c^2*x^2] + 3*ArcSin[c*x]))/(1 + c*x)^2 + 48*a*b*ArcSin[c*x]*Log[1 - E^((2*I)*ArcSin[c*x])] + 24*a^2*Log[c*x] - 1 2*a^2*Log[1 - c^2*x^2] + 12*a*b*(I*ArcSin[c*x]^2 + ArcSin[c*x]*((-3*I)*Pi - 4*Log[1 + I*E^(I*ArcSin[c*x])]) + 2*Pi*(-2*Log[1 + E^((-I)*ArcSin[c*x])] + Log[1 + I*E^(I*ArcSin[c*x])] + 2*Log[Cos[ArcSin[c*x]/2]] - Log[-Cos[(Pi + 2*ArcSin[c*x])/4]]) + (4*I)*PolyLog[2, (-I)*E^(I*ArcSin[c*x])]) + 12*a* b*(I*ArcSin[c*x]^2 + ArcSin[c*x]*((-I)*Pi - 4*Log[1 - I*E^(I*ArcSin[c*x])] ) + 2*Pi*(-2*Log[1 + E^((-I)*ArcSin[c*x])] - Log[1 - I*E^(I*ArcSin[c*x])] + 2*Log[Cos[ArcSin[c*x]/2]] + Log[Sin[(Pi + 2*ArcSin[c*x])/4]]) + (4*I)*Po lyLog[2, I*E^(I*ArcSin[c*x])]) - (24*I)*a*b*(ArcSin[c*x]^2 + PolyLog[2, E^ ((2*I)*ArcSin[c*x])]) - b^2*(I*Pi^3 + 2/(-1 + c^2*x^2) + (4*c*x*ArcSin[c*x ])/(1 - c^2*x^2)^(3/2) + (32*c*x*ArcSin[c*x])/Sqrt[1 - c^2*x^2] - (6*ArcSi n[c*x]^2)/(-1 + c^2*x^2)^2 + (12*ArcSin[c*x]^2)/(-1 + c^2*x^2) - (16*I)*Ar cSin[c*x]^3 - 24*ArcSin[c*x]^2*Log[1 - E^((-2*I)*ArcSin[c*x])] + 24*ArcSin [c*x]^2*Log[1 + E^((2*I)*ArcSin[c*x])] + 16*Log[1 - c^2*x^2] - (24*I)*ArcS in[c*x]*PolyLog[2, E^((-2*I)*ArcSin[c*x])] - (24*I)*ArcSin[c*x]*PolyLog[2, -E^((2*I)*ArcSin[c*x])] - 12*PolyLog[3, E^((-2*I)*ArcSin[c*x])] + 12*P...
Time = 1.84 (sec) , antiderivative size = 337, normalized size of antiderivative = 1.14, number of steps used = 17, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.593, Rules used = {5208, 27, 5162, 241, 5160, 240, 5208, 5160, 240, 5184, 4919, 3042, 4671, 3011, 2720, 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.
| \(\displaystyle \int \frac {(a+b \arcsin (c x))^2}{x \left (d-c^2 d x^2\right )^3} \, dx\) |
| \(\Big \downarrow \) 5208 |
| \(\displaystyle -\frac {b c \int \frac {a+b \arcsin (c x)}{\left (1-c^2 x^2\right )^{5/2}}dx}{2 d^3}+\frac {\int \frac {(a+b \arcsin (c x))^2}{d^2 x \left (1-c^2 x^2\right )^2}dx}{d}+\frac {(a+b \arcsin (c x))^2}{4 d^3 \left (1-c^2 x^2\right )^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {b c \int \frac {a+b \arcsin (c x)}{\left (1-c^2 x^2\right )^{5/2}}dx}{2 d^3}+\frac {\int \frac {(a+b \arcsin (c x))^2}{x \left (1-c^2 x^2\right )^2}dx}{d^3}+\frac {(a+b \arcsin (c x))^2}{4 d^3 \left (1-c^2 x^2\right )^2}\) |
| \(\Big \downarrow \) 5162 |
| \(\displaystyle -\frac {b c \left (\frac {2}{3} \int \frac {a+b \arcsin (c x)}{\left (1-c^2 x^2\right )^{3/2}}dx-\frac {1}{3} b c \int \frac {x}{\left (1-c^2 x^2\right )^2}dx+\frac {x (a+b \arcsin (c x))}{3 \left (1-c^2 x^2\right )^{3/2}}\right )}{2 d^3}+\frac {\int \frac {(a+b \arcsin (c x))^2}{x \left (1-c^2 x^2\right )^2}dx}{d^3}+\frac {(a+b \arcsin (c x))^2}{4 d^3 \left (1-c^2 x^2\right )^2}\) |
| \(\Big \downarrow \) 241 |
| \(\displaystyle -\frac {b c \left (\frac {2}{3} \int \frac {a+b \arcsin (c x)}{\left (1-c^2 x^2\right )^{3/2}}dx+\frac {x (a+b \arcsin (c x))}{3 \left (1-c^2 x^2\right )^{3/2}}-\frac {b}{6 c \left (1-c^2 x^2\right )}\right )}{2 d^3}+\frac {\int \frac {(a+b \arcsin (c x))^2}{x \left (1-c^2 x^2\right )^2}dx}{d^3}+\frac {(a+b \arcsin (c x))^2}{4 d^3 \left (1-c^2 x^2\right )^2}\) |
| \(\Big \downarrow \) 5160 |
| \(\displaystyle -\frac {b c \left (\frac {2}{3} \left (\frac {x (a+b \arcsin (c x))}{\sqrt {1-c^2 x^2}}-b c \int \frac {x}{1-c^2 x^2}dx\right )+\frac {x (a+b \arcsin (c x))}{3 \left (1-c^2 x^2\right )^{3/2}}-\frac {b}{6 c \left (1-c^2 x^2\right )}\right )}{2 d^3}+\frac {\int \frac {(a+b \arcsin (c x))^2}{x \left (1-c^2 x^2\right )^2}dx}{d^3}+\frac {(a+b \arcsin (c x))^2}{4 d^3 \left (1-c^2 x^2\right )^2}\) |
| \(\Big \downarrow \) 240 |
| \(\displaystyle \frac {\int \frac {(a+b \arcsin (c x))^2}{x \left (1-c^2 x^2\right )^2}dx}{d^3}+\frac {(a+b \arcsin (c x))^2}{4 d^3 \left (1-c^2 x^2\right )^2}-\frac {b c \left (\frac {x (a+b \arcsin (c x))}{3 \left (1-c^2 x^2\right )^{3/2}}+\frac {2}{3} \left (\frac {x (a+b \arcsin (c x))}{\sqrt {1-c^2 x^2}}+\frac {b \log \left (1-c^2 x^2\right )}{2 c}\right )-\frac {b}{6 c \left (1-c^2 x^2\right )}\right )}{2 d^3}\) |
| \(\Big \downarrow \) 5208 |
| \(\displaystyle \frac {-b c \int \frac {a+b \arcsin (c x)}{\left (1-c^2 x^2\right )^{3/2}}dx+\int \frac {(a+b \arcsin (c x))^2}{x \left (1-c^2 x^2\right )}dx+\frac {(a+b \arcsin (c x))^2}{2 \left (1-c^2 x^2\right )}}{d^3}+\frac {(a+b \arcsin (c x))^2}{4 d^3 \left (1-c^2 x^2\right )^2}-\frac {b c \left (\frac {x (a+b \arcsin (c x))}{3 \left (1-c^2 x^2\right )^{3/2}}+\frac {2}{3} \left (\frac {x (a+b \arcsin (c x))}{\sqrt {1-c^2 x^2}}+\frac {b \log \left (1-c^2 x^2\right )}{2 c}\right )-\frac {b}{6 c \left (1-c^2 x^2\right )}\right )}{2 d^3}\) |
| \(\Big \downarrow \) 5160 |
| \(\displaystyle \frac {-b c \left (\frac {x (a+b \arcsin (c x))}{\sqrt {1-c^2 x^2}}-b c \int \frac {x}{1-c^2 x^2}dx\right )+\int \frac {(a+b \arcsin (c x))^2}{x \left (1-c^2 x^2\right )}dx+\frac {(a+b \arcsin (c x))^2}{2 \left (1-c^2 x^2\right )}}{d^3}+\frac {(a+b \arcsin (c x))^2}{4 d^3 \left (1-c^2 x^2\right )^2}-\frac {b c \left (\frac {x (a+b \arcsin (c x))}{3 \left (1-c^2 x^2\right )^{3/2}}+\frac {2}{3} \left (\frac {x (a+b \arcsin (c x))}{\sqrt {1-c^2 x^2}}+\frac {b \log \left (1-c^2 x^2\right )}{2 c}\right )-\frac {b}{6 c \left (1-c^2 x^2\right )}\right )}{2 d^3}\) |
| \(\Big \downarrow \) 240 |
| \(\displaystyle \frac {\int \frac {(a+b \arcsin (c x))^2}{x \left (1-c^2 x^2\right )}dx+\frac {(a+b \arcsin (c x))^2}{2 \left (1-c^2 x^2\right )}-b c \left (\frac {x (a+b \arcsin (c x))}{\sqrt {1-c^2 x^2}}+\frac {b \log \left (1-c^2 x^2\right )}{2 c}\right )}{d^3}+\frac {(a+b \arcsin (c x))^2}{4 d^3 \left (1-c^2 x^2\right )^2}-\frac {b c \left (\frac {x (a+b \arcsin (c x))}{3 \left (1-c^2 x^2\right )^{3/2}}+\frac {2}{3} \left (\frac {x (a+b \arcsin (c x))}{\sqrt {1-c^2 x^2}}+\frac {b \log \left (1-c^2 x^2\right )}{2 c}\right )-\frac {b}{6 c \left (1-c^2 x^2\right )}\right )}{2 d^3}\) |
| \(\Big \downarrow \) 5184 |
| \(\displaystyle \frac {\int \frac {(a+b \arcsin (c x))^2}{c x \sqrt {1-c^2 x^2}}d\arcsin (c x)+\frac {(a+b \arcsin (c x))^2}{2 \left (1-c^2 x^2\right )}-b c \left (\frac {x (a+b \arcsin (c x))}{\sqrt {1-c^2 x^2}}+\frac {b \log \left (1-c^2 x^2\right )}{2 c}\right )}{d^3}+\frac {(a+b \arcsin (c x))^2}{4 d^3 \left (1-c^2 x^2\right )^2}-\frac {b c \left (\frac {x (a+b \arcsin (c x))}{3 \left (1-c^2 x^2\right )^{3/2}}+\frac {2}{3} \left (\frac {x (a+b \arcsin (c x))}{\sqrt {1-c^2 x^2}}+\frac {b \log \left (1-c^2 x^2\right )}{2 c}\right )-\frac {b}{6 c \left (1-c^2 x^2\right )}\right )}{2 d^3}\) |
| \(\Big \downarrow \) 4919 |
| \(\displaystyle \frac {2 \int (a+b \arcsin (c x))^2 \csc (2 \arcsin (c x))d\arcsin (c x)+\frac {(a+b \arcsin (c x))^2}{2 \left (1-c^2 x^2\right )}-b c \left (\frac {x (a+b \arcsin (c x))}{\sqrt {1-c^2 x^2}}+\frac {b \log \left (1-c^2 x^2\right )}{2 c}\right )}{d^3}+\frac {(a+b \arcsin (c x))^2}{4 d^3 \left (1-c^2 x^2\right )^2}-\frac {b c \left (\frac {x (a+b \arcsin (c x))}{3 \left (1-c^2 x^2\right )^{3/2}}+\frac {2}{3} \left (\frac {x (a+b \arcsin (c x))}{\sqrt {1-c^2 x^2}}+\frac {b \log \left (1-c^2 x^2\right )}{2 c}\right )-\frac {b}{6 c \left (1-c^2 x^2\right )}\right )}{2 d^3}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2 \int (a+b \arcsin (c x))^2 \csc (2 \arcsin (c x))d\arcsin (c x)+\frac {(a+b \arcsin (c x))^2}{2 \left (1-c^2 x^2\right )}-b c \left (\frac {x (a+b \arcsin (c x))}{\sqrt {1-c^2 x^2}}+\frac {b \log \left (1-c^2 x^2\right )}{2 c}\right )}{d^3}+\frac {(a+b \arcsin (c x))^2}{4 d^3 \left (1-c^2 x^2\right )^2}-\frac {b c \left (\frac {x (a+b \arcsin (c x))}{3 \left (1-c^2 x^2\right )^{3/2}}+\frac {2}{3} \left (\frac {x (a+b \arcsin (c x))}{\sqrt {1-c^2 x^2}}+\frac {b \log \left (1-c^2 x^2\right )}{2 c}\right )-\frac {b}{6 c \left (1-c^2 x^2\right )}\right )}{2 d^3}\) |
| \(\Big \downarrow \) 4671 |
| \(\displaystyle \frac {2 \left (-b \int (a+b \arcsin (c x)) \log \left (1-e^{2 i \arcsin (c x)}\right )d\arcsin (c x)+b \int (a+b \arcsin (c x)) \log \left (1+e^{2 i \arcsin (c x)}\right )d\arcsin (c x)-\text {arctanh}\left (e^{2 i \arcsin (c x)}\right ) (a+b \arcsin (c x))^2\right )+\frac {(a+b \arcsin (c x))^2}{2 \left (1-c^2 x^2\right )}-b c \left (\frac {x (a+b \arcsin (c x))}{\sqrt {1-c^2 x^2}}+\frac {b \log \left (1-c^2 x^2\right )}{2 c}\right )}{d^3}+\frac {(a+b \arcsin (c x))^2}{4 d^3 \left (1-c^2 x^2\right )^2}-\frac {b c \left (\frac {x (a+b \arcsin (c x))}{3 \left (1-c^2 x^2\right )^{3/2}}+\frac {2}{3} \left (\frac {x (a+b \arcsin (c x))}{\sqrt {1-c^2 x^2}}+\frac {b \log \left (1-c^2 x^2\right )}{2 c}\right )-\frac {b}{6 c \left (1-c^2 x^2\right )}\right )}{2 d^3}\) |
| \(\Big \downarrow \) 3011 |
| \(\displaystyle \frac {2 \left (b \left (\frac {1}{2} i \operatorname {PolyLog}\left (2,-e^{2 i \arcsin (c x)}\right ) (a+b \arcsin (c x))-\frac {1}{2} i b \int \operatorname {PolyLog}\left (2,-e^{2 i \arcsin (c x)}\right )d\arcsin (c x)\right )-b \left (\frac {1}{2} i \operatorname {PolyLog}\left (2,e^{2 i \arcsin (c x)}\right ) (a+b \arcsin (c x))-\frac {1}{2} i b \int \operatorname {PolyLog}\left (2,e^{2 i \arcsin (c x)}\right )d\arcsin (c x)\right )-\text {arctanh}\left (e^{2 i \arcsin (c x)}\right ) (a+b \arcsin (c x))^2\right )+\frac {(a+b \arcsin (c x))^2}{2 \left (1-c^2 x^2\right )}-b c \left (\frac {x (a+b \arcsin (c x))}{\sqrt {1-c^2 x^2}}+\frac {b \log \left (1-c^2 x^2\right )}{2 c}\right )}{d^3}+\frac {(a+b \arcsin (c x))^2}{4 d^3 \left (1-c^2 x^2\right )^2}-\frac {b c \left (\frac {x (a+b \arcsin (c x))}{3 \left (1-c^2 x^2\right )^{3/2}}+\frac {2}{3} \left (\frac {x (a+b \arcsin (c x))}{\sqrt {1-c^2 x^2}}+\frac {b \log \left (1-c^2 x^2\right )}{2 c}\right )-\frac {b}{6 c \left (1-c^2 x^2\right )}\right )}{2 d^3}\) |
| \(\Big \downarrow \) 2720 |
| \(\displaystyle \frac {2 \left (b \left (\frac {1}{2} i \operatorname {PolyLog}\left (2,-e^{2 i \arcsin (c x)}\right ) (a+b \arcsin (c x))-\frac {1}{4} b \int e^{-2 i \arcsin (c x)} \operatorname {PolyLog}\left (2,-e^{2 i \arcsin (c x)}\right )de^{2 i \arcsin (c x)}\right )-b \left (\frac {1}{2} i \operatorname {PolyLog}\left (2,e^{2 i \arcsin (c x)}\right ) (a+b \arcsin (c x))-\frac {1}{4} b \int e^{-2 i \arcsin (c x)} \operatorname {PolyLog}\left (2,e^{2 i \arcsin (c x)}\right )de^{2 i \arcsin (c x)}\right )-\text {arctanh}\left (e^{2 i \arcsin (c x)}\right ) (a+b \arcsin (c x))^2\right )+\frac {(a+b \arcsin (c x))^2}{2 \left (1-c^2 x^2\right )}-b c \left (\frac {x (a+b \arcsin (c x))}{\sqrt {1-c^2 x^2}}+\frac {b \log \left (1-c^2 x^2\right )}{2 c}\right )}{d^3}+\frac {(a+b \arcsin (c x))^2}{4 d^3 \left (1-c^2 x^2\right )^2}-\frac {b c \left (\frac {x (a+b \arcsin (c x))}{3 \left (1-c^2 x^2\right )^{3/2}}+\frac {2}{3} \left (\frac {x (a+b \arcsin (c x))}{\sqrt {1-c^2 x^2}}+\frac {b \log \left (1-c^2 x^2\right )}{2 c}\right )-\frac {b}{6 c \left (1-c^2 x^2\right )}\right )}{2 d^3}\) |
| \(\Big \downarrow \) 7143 |
| \(\displaystyle \frac {2 \left (-\text {arctanh}\left (e^{2 i \arcsin (c x)}\right ) (a+b \arcsin (c x))^2+b \left (\frac {1}{2} i \operatorname {PolyLog}\left (2,-e^{2 i \arcsin (c x)}\right ) (a+b \arcsin (c x))-\frac {1}{4} b \operatorname {PolyLog}\left (3,-e^{2 i \arcsin (c x)}\right )\right )-b \left (\frac {1}{2} i \operatorname {PolyLog}\left (2,e^{2 i \arcsin (c x)}\right ) (a+b \arcsin (c x))-\frac {1}{4} b \operatorname {PolyLog}\left (3,e^{2 i \arcsin (c x)}\right )\right )\right )+\frac {(a+b \arcsin (c x))^2}{2 \left (1-c^2 x^2\right )}-b c \left (\frac {x (a+b \arcsin (c x))}{\sqrt {1-c^2 x^2}}+\frac {b \log \left (1-c^2 x^2\right )}{2 c}\right )}{d^3}+\frac {(a+b \arcsin (c x))^2}{4 d^3 \left (1-c^2 x^2\right )^2}-\frac {b c \left (\frac {x (a+b \arcsin (c x))}{3 \left (1-c^2 x^2\right )^{3/2}}+\frac {2}{3} \left (\frac {x (a+b \arcsin (c x))}{\sqrt {1-c^2 x^2}}+\frac {b \log \left (1-c^2 x^2\right )}{2 c}\right )-\frac {b}{6 c \left (1-c^2 x^2\right )}\right )}{2 d^3}\) |
(a + b*ArcSin[c*x])^2/(4*d^3*(1 - c^2*x^2)^2) - (b*c*(-1/6*b/(c*(1 - c^2*x ^2)) + (x*(a + b*ArcSin[c*x]))/(3*(1 - c^2*x^2)^(3/2)) + (2*((x*(a + b*Arc Sin[c*x]))/Sqrt[1 - c^2*x^2] + (b*Log[1 - c^2*x^2])/(2*c)))/3))/(2*d^3) + ((a + b*ArcSin[c*x])^2/(2*(1 - c^2*x^2)) - b*c*((x*(a + b*ArcSin[c*x]))/Sq rt[1 - c^2*x^2] + (b*Log[1 - c^2*x^2])/(2*c)) + 2*(-((a + b*ArcSin[c*x])^2 *ArcTanh[E^((2*I)*ArcSin[c*x])]) + 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) - 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)))/d^3
3.3.6.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[(x_)/((a_) + (b_.)*(x_)^2), x_Symbol] :> Simp[Log[RemoveContent[a + b*x ^2, x]]/(2*b), x] /; FreeQ[{a, b}, x]
Int[(x_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(a + b*x^2)^(p + 1)/ (2*b*(p + 1)), x] /; FreeQ[{a, b, p}, x] && NeQ[p, -1]
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[csc[(e_.) + (f_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[- 2*(c + d*x)^m*(ArcTanh[E^(I*(e + f*x))]/f), x] + (-Simp[d*(m/f) Int[(c + d*x)^(m - 1)*Log[1 - E^(I*(e + f*x))], x], x] + Simp[d*(m/f) Int[(c + d*x )^(m - 1)*Log[1 + E^(I*(e + f*x))], x], x]) /; FreeQ[{c, d, e, f}, x] && IG tQ[m, 0]
Int[Csc[(a_.) + (b_.)*(x_)]^(n_.)*((c_.) + (d_.)*(x_))^(m_.)*Sec[(a_.) + (b _.)*(x_)]^(n_.), x_Symbol] :> Simp[2^n Int[(c + d*x)^m*Csc[2*a + 2*b*x]^n , x], x] /; FreeQ[{a, b, c, d, m}, x] && IntegerQ[n] && RationalQ[m]
Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)/((d_) + (e_.)*(x_)^2)^(3/2), x _Symbol] :> Simp[x*((a + b*ArcSin[c*x])^n/(d*Sqrt[d + e*x^2])), x] - Simp[b *c*(n/d)*Simp[Sqrt[1 - c^2*x^2]/Sqrt[d + e*x^2]] Int[x*((a + b*ArcSin[c*x ])^(n - 1)/(1 - c^2*x^2)), 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 + 1)*((a + b*ArcSin[c*x])^n/(2*d*(p + 1 ))), x] + (Simp[(2*p + 3)/(2*d*(p + 1)) Int[(d + e*x^2)^(p + 1)*(a + b*Ar cSin[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] && LtQ[p, -1] && NeQ[p, -3/2]
Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)/((x_)*((d_) + (e_.)*(x_)^2)), x_Symbol] :> Simp[1/d Subst[Int[(a + b*x)^n/(Cos[x]*Sin[x]), x], x, ArcSi n[c*x]], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && IGtQ[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 + 1)*((a + b*ArcSin[c*x])^n/(2*d*f*(p + 1))), x] + (Simp[(m + 2*p + 3)/(2*d*(p + 1)) Int[(f*x)^m*(d + e*x^2)^(p + 1)*(a + b*ArcSin[c*x])^n, x], x] + Simp[b*c *(n/(2*f*(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] && LtQ[p, -1] && !G tQ[m, 1] && (IntegerQ[m] || IntegerQ[p] || EqQ[n, 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. 758 vs. \(2 (324 ) = 648\).
Time = 0.44 (sec) , antiderivative size = 759, normalized size of antiderivative = 2.56
| method | result | size |
| parts | \(-\frac {a^{2} \left (-\ln \left (x \right )-\frac {1}{16 \left (c x -1\right )^{2}}+\frac {5}{16 \left (c x -1\right )}+\frac {\ln \left (c x -1\right )}{2}-\frac {1}{16 \left (c x +1\right )^{2}}-\frac {5}{16 \left (c x +1\right )}+\frac {\ln \left (c x +1\right )}{2}\right )}{d^{3}}-\frac {b^{2} \left (\frac {16 i \arcsin \left (c x \right ) x^{4} c^{4}-16 \sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right ) c^{3} x^{3}+6 \arcsin \left (c x \right )^{2} x^{2} c^{2}-32 i \arcsin \left (c x \right ) x^{2} c^{2}+18 \sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right ) x c -9 \arcsin \left (c x \right )^{2}+16 i \arcsin \left (c x \right )+c^{2} x^{2}-1}{12 c^{4} x^{4}-24 c^{2} x^{2}+12}+\frac {4 \ln \left (1+\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{2}\right )}{3}-\frac {8 \ln \left (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 )+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+\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{2}\right )-i \arcsin \left (c x \right ) \operatorname {polylog}\left (2, -\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{2}\right )+\frac {\operatorname {polylog}\left (3, -\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{2}\right )}{2}-\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 )\right )}{d^{3}}-\frac {2 a b \left (\frac {8 i c^{4} x^{4}-8 c^{3} x^{3} \sqrt {-c^{2} x^{2}+1}+6 c^{2} x^{2} \arcsin \left (c x \right )-16 i c^{2} x^{2}+9 c x \sqrt {-c^{2} x^{2}+1}-9 \arcsin \left (c x \right )+8 i}{12 c^{4} x^{4}-24 c^{2} x^{2}+12}-\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+\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{2}\right )-\frac {i \operatorname {polylog}\left (2, -\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{2}\right )}{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 )\right )}{d^{3}}\) | \(759\) |
| derivativedivides | \(-\frac {a^{2} \left (-\ln \left (c x \right )-\frac {1}{16 \left (c x -1\right )^{2}}+\frac {5}{16 \left (c x -1\right )}+\frac {\ln \left (c x -1\right )}{2}-\frac {1}{16 \left (c x +1\right )^{2}}-\frac {5}{16 \left (c x +1\right )}+\frac {\ln \left (c x +1\right )}{2}\right )}{d^{3}}-\frac {b^{2} \left (\frac {16 i \arcsin \left (c x \right ) x^{4} c^{4}-16 \sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right ) c^{3} x^{3}+6 \arcsin \left (c x \right )^{2} x^{2} c^{2}-32 i \arcsin \left (c x \right ) x^{2} c^{2}+18 \sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right ) x c -9 \arcsin \left (c x \right )^{2}+16 i \arcsin \left (c x \right )+c^{2} x^{2}-1}{12 c^{4} x^{4}-24 c^{2} x^{2}+12}+\frac {4 \ln \left (1+\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{2}\right )}{3}-\frac {8 \ln \left (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 )+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+\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{2}\right )-i \arcsin \left (c x \right ) \operatorname {polylog}\left (2, -\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{2}\right )+\frac {\operatorname {polylog}\left (3, -\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{2}\right )}{2}-\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 )\right )}{d^{3}}-\frac {2 a b \left (\frac {8 i c^{4} x^{4}-8 c^{3} x^{3} \sqrt {-c^{2} x^{2}+1}+6 c^{2} x^{2} \arcsin \left (c x \right )-16 i c^{2} x^{2}+9 c x \sqrt {-c^{2} x^{2}+1}-9 \arcsin \left (c x \right )+8 i}{12 c^{4} x^{4}-24 c^{2} x^{2}+12}-\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+\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{2}\right )-\frac {i \operatorname {polylog}\left (2, -\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{2}\right )}{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 )\right )}{d^{3}}\) | \(761\) |
| default | \(-\frac {a^{2} \left (-\ln \left (c x \right )-\frac {1}{16 \left (c x -1\right )^{2}}+\frac {5}{16 \left (c x -1\right )}+\frac {\ln \left (c x -1\right )}{2}-\frac {1}{16 \left (c x +1\right )^{2}}-\frac {5}{16 \left (c x +1\right )}+\frac {\ln \left (c x +1\right )}{2}\right )}{d^{3}}-\frac {b^{2} \left (\frac {16 i \arcsin \left (c x \right ) x^{4} c^{4}-16 \sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right ) c^{3} x^{3}+6 \arcsin \left (c x \right )^{2} x^{2} c^{2}-32 i \arcsin \left (c x \right ) x^{2} c^{2}+18 \sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right ) x c -9 \arcsin \left (c x \right )^{2}+16 i \arcsin \left (c x \right )+c^{2} x^{2}-1}{12 c^{4} x^{4}-24 c^{2} x^{2}+12}+\frac {4 \ln \left (1+\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{2}\right )}{3}-\frac {8 \ln \left (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 )+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+\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{2}\right )-i \arcsin \left (c x \right ) \operatorname {polylog}\left (2, -\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{2}\right )+\frac {\operatorname {polylog}\left (3, -\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{2}\right )}{2}-\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 )\right )}{d^{3}}-\frac {2 a b \left (\frac {8 i c^{4} x^{4}-8 c^{3} x^{3} \sqrt {-c^{2} x^{2}+1}+6 c^{2} x^{2} \arcsin \left (c x \right )-16 i c^{2} x^{2}+9 c x \sqrt {-c^{2} x^{2}+1}-9 \arcsin \left (c x \right )+8 i}{12 c^{4} x^{4}-24 c^{2} x^{2}+12}-\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+\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{2}\right )-\frac {i \operatorname {polylog}\left (2, -\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{2}\right )}{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 )\right )}{d^{3}}\) | \(761\) |
-a^2/d^3*(-ln(x)-1/16/(c*x-1)^2+5/16/(c*x-1)+1/2*ln(c*x-1)-1/16/(c*x+1)^2- 5/16/(c*x+1)+1/2*ln(c*x+1))-b^2/d^3*(1/12*(16*I*arcsin(c*x)*c^4*x^4-16*(-c ^2*x^2+1)^(1/2)*arcsin(c*x)*c^3*x^3+6*arcsin(c*x)^2*x^2*c^2-32*I*arcsin(c* x)*c^2*x^2+18*(-c^2*x^2+1)^(1/2)*arcsin(c*x)*x*c-9*arcsin(c*x)^2+16*I*arcs in(c*x)+c^2*x^2-1)/(c^4*x^4-2*c^2*x^2+1)+4/3*ln(1+(I*c*x+(-c^2*x^2+1)^(1/2 ))^2)-8/3*ln(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))+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)+1/2*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))+ 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)))-2*a*b/d^3*(1/12*(8*I*c^4*x^4-8*c^3*x^3*(-c^2*x^2+1)^(1/2) +6*c^2*x^2*arcsin(c*x)-16*I*c^2*x^2+9*c*x*(-c^2*x^2+1)^(1/2)-9*arcsin(c*x) +8*I)/(c^4*x^4-2*c^2*x^2+1)-arcsin(c*x)*ln(1+I*c*x+(-c^2*x^2+1)^(1/2))+I*p olylog(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))^2)-1/2*I*polylog(2,-(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))+I*polylog(2,I*c*x+(-c^2*x^2+1)^(1/2)))
\[ \int \frac {(a+b \arcsin (c x))^2}{x \left (d-c^2 d x^2\right )^3} \, dx=\int { -\frac {{\left (b \arcsin \left (c x\right ) + a\right )}^{2}}{{\left (c^{2} d x^{2} - d\right )}^{3} x} \,d x } \]
integral(-(b^2*arcsin(c*x)^2 + 2*a*b*arcsin(c*x) + a^2)/(c^6*d^3*x^7 - 3*c ^4*d^3*x^5 + 3*c^2*d^3*x^3 - d^3*x), x)
\[ \int \frac {(a+b \arcsin (c x))^2}{x \left (d-c^2 d x^2\right )^3} \, dx=- \frac {\int \frac {a^{2}}{c^{6} x^{7} - 3 c^{4} x^{5} + 3 c^{2} x^{3} - x}\, dx + \int \frac {b^{2} \operatorname {asin}^{2}{\left (c x \right )}}{c^{6} x^{7} - 3 c^{4} x^{5} + 3 c^{2} x^{3} - x}\, dx + \int \frac {2 a b \operatorname {asin}{\left (c x \right )}}{c^{6} x^{7} - 3 c^{4} x^{5} + 3 c^{2} x^{3} - x}\, dx}{d^{3}} \]
-(Integral(a**2/(c**6*x**7 - 3*c**4*x**5 + 3*c**2*x**3 - x), x) + Integral (b**2*asin(c*x)**2/(c**6*x**7 - 3*c**4*x**5 + 3*c**2*x**3 - x), x) + Integ ral(2*a*b*asin(c*x)/(c**6*x**7 - 3*c**4*x**5 + 3*c**2*x**3 - x), x))/d**3
\[ \int \frac {(a+b \arcsin (c x))^2}{x \left (d-c^2 d x^2\right )^3} \, dx=\int { -\frac {{\left (b \arcsin \left (c x\right ) + a\right )}^{2}}{{\left (c^{2} d x^{2} - d\right )}^{3} x} \,d x } \]
-1/4*a^2*((2*c^2*x^2 - 3)/(c^4*d^3*x^4 - 2*c^2*d^3*x^2 + d^3) + 2*log(c*x + 1)/d^3 + 2*log(c*x - 1)/d^3 - 4*log(x)/d^3) - integrate((b^2*arctan2(c*x , sqrt(c*x + 1)*sqrt(-c*x + 1))^2 + 2*a*b*arctan2(c*x, sqrt(c*x + 1)*sqrt( -c*x + 1)))/(c^6*d^3*x^7 - 3*c^4*d^3*x^5 + 3*c^2*d^3*x^3 - d^3*x), x)
\[ \int \frac {(a+b \arcsin (c x))^2}{x \left (d-c^2 d x^2\right )^3} \, dx=\int { -\frac {{\left (b \arcsin \left (c x\right ) + a\right )}^{2}}{{\left (c^{2} d x^{2} - d\right )}^{3} x} \,d x } \]
Timed out. \[ \int \frac {(a+b \arcsin (c x))^2}{x \left (d-c^2 d x^2\right )^3} \, dx=\int \frac {{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^2}{x\,{\left (d-c^2\,d\,x^2\right )}^3} \,d x \]