3.3.1 \(\int \frac {x^4 (a+b \arcsin (c x))^2}{(d-c^2 d x^2)^3} \, dx\) [201]

3.3.1.1 Optimal result

Integrand size = 27, antiderivative size = 343 \[ \int \frac {x^4 (a+b \arcsin (c x))^2}{\left (d-c^2 d x^2\right )^3} \, dx=\frac {b^2 x}{12 c^4 d^3 \left (1-c^2 x^2\right )}-\frac {b (a+b \arcsin (c x))}{6 c^5 d^3 \left (1-c^2 x^2\right )^{3/2}}+\frac {5 b (a+b \arcsin (c x))}{4 c^5 d^3 \sqrt {1-c^2 x^2}}+\frac {x^3 (a+b \arcsin (c x))^2}{4 c^2 d^3 \left (1-c^2 x^2\right )^2}-\frac {3 x (a+b \arcsin (c x))^2}{8 c^4 d^3 \left (1-c^2 x^2\right )}-\frac {3 i (a+b \arcsin (c x))^2 \arctan \left (e^{i \arcsin (c x)}\right )}{4 c^5 d^3}-\frac {7 b^2 \text {arctanh}(c x)}{6 c^5 d^3}+\frac {3 i b (a+b \arcsin (c x)) \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right )}{4 c^5 d^3}-\frac {3 i b (a+b \arcsin (c x)) \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right )}{4 c^5 d^3}-\frac {3 b^2 \operatorname {PolyLog}\left (3,-i e^{i \arcsin (c x)}\right )}{4 c^5 d^3}+\frac {3 b^2 \operatorname {PolyLog}\left (3,i e^{i \arcsin (c x)}\right )}{4 c^5 d^3} \]

output

1/12*b^2*x/c^4/d^3/(-c^2*x^2+1)-1/6*b*(a+b*arcsin(c*x))/c^5/d^3/(-c^2*x^2+ 
1)^(3/2)+1/4*x^3*(a+b*arcsin(c*x))^2/c^2/d^3/(-c^2*x^2+1)^2-3/8*x*(a+b*arc 
sin(c*x))^2/c^4/d^3/(-c^2*x^2+1)-3/4*I*(a+b*arcsin(c*x))^2*arctan(I*c*x+(- 
c^2*x^2+1)^(1/2))/c^5/d^3-7/6*b^2*arctanh(c*x)/c^5/d^3+3/4*I*b*(a+b*arcsin 
(c*x))*polylog(2,-I*(I*c*x+(-c^2*x^2+1)^(1/2)))/c^5/d^3-3/4*I*b*(a+b*arcsi 
n(c*x))*polylog(2,I*(I*c*x+(-c^2*x^2+1)^(1/2)))/c^5/d^3-3/4*b^2*polylog(3, 
-I*(I*c*x+(-c^2*x^2+1)^(1/2)))/c^5/d^3+3/4*b^2*polylog(3,I*(I*c*x+(-c^2*x^ 
2+1)^(1/2)))/c^5/d^3+5/4*b*(a+b*arcsin(c*x))/c^5/d^3/(-c^2*x^2+1)^(1/2)
 

3.3.1.2 Mathematica [A] (verified)

Time = 5.77 (sec) , antiderivative size = 667, normalized size of antiderivative = 1.94 \[ \int \frac {x^4 (a+b \arcsin (c x))^2}{\left (d-c^2 d x^2\right )^3} \, dx=\frac {\frac {24 a^2 c x}{\left (-1+c^2 x^2\right )^2}+\frac {60 a^2 c x}{-1+c^2 x^2}-\frac {60 a b \left (\sqrt {1-c^2 x^2}-\arcsin (c x)\right )}{-1+c x}+\frac {60 a b \left (\sqrt {1-c^2 x^2}+\arcsin (c x)\right )}{1+c x}+\frac {4 a b \left ((-2+c x) \sqrt {1-c^2 x^2}+3 \arcsin (c x)\right )}{(-1+c x)^2}-\frac {4 a b \left ((2+c x) \sqrt {1-c^2 x^2}+3 \arcsin (c x)\right )}{(1+c x)^2}-18 a^2 \log (1-c x)+18 a^2 \log (1+c x)+18 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 )+18 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 )+8 b^2 \left (-9 i \arcsin (c x)^2 \arctan \left (e^{i \arcsin (c x)}\right )-14 \text {arctanh}(c x)+9 i \arcsin (c x) \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right )-9 i \arcsin (c x) \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right )-9 \operatorname {PolyLog}\left (3,-i e^{i \arcsin (c x)}\right )+9 \operatorname {PolyLog}\left (3,i e^{i \arcsin (c x)}\right )\right )+\frac {b^2 \left (\arcsin (c x) \left (74 \sqrt {1-c^2 x^2}+30 \cos (3 \arcsin (c x))\right )+3 \arcsin (c x)^2 (3 c x-5 \sin (3 \arcsin (c x)))+2 (c x+\sin (3 \arcsin (c x)))\right )}{\left (-1+c^2 x^2\right )^2}}{96 c^5 d^3} \]

input

Integrate[(x^4*(a + b*ArcSin[c*x])^2)/(d - c^2*d*x^2)^3,x]
 

output

((24*a^2*c*x)/(-1 + c^2*x^2)^2 + (60*a^2*c*x)/(-1 + c^2*x^2) - (60*a*b*(Sq 
rt[1 - c^2*x^2] - ArcSin[c*x]))/(-1 + c*x) + (60*a*b*(Sqrt[1 - c^2*x^2] + 
ArcSin[c*x]))/(1 + c*x) + (4*a*b*((-2 + c*x)*Sqrt[1 - c^2*x^2] + 3*ArcSin[ 
c*x]))/(-1 + c*x)^2 - (4*a*b*((2 + c*x)*Sqrt[1 - c^2*x^2] + 3*ArcSin[c*x]) 
)/(1 + c*x)^2 - 18*a^2*Log[1 - c*x] + 18*a^2*Log[1 + c*x] + 18*a*b*(I*ArcS 
in[c*x]^2 + ArcSin[c*x]*((-3*I)*Pi - 4*Log[1 + I*E^(I*ArcSin[c*x])]) + 2*P 
i*(-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])]) + 18*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[(P 
i + 2*ArcSin[c*x])/4]]) - (4*I)*PolyLog[2, I*E^(I*ArcSin[c*x])]) + 8*b^2*( 
(-9*I)*ArcSin[c*x]^2*ArcTan[E^(I*ArcSin[c*x])] - 14*ArcTanh[c*x] + (9*I)*A 
rcSin[c*x]*PolyLog[2, (-I)*E^(I*ArcSin[c*x])] - (9*I)*ArcSin[c*x]*PolyLog[ 
2, I*E^(I*ArcSin[c*x])] - 9*PolyLog[3, (-I)*E^(I*ArcSin[c*x])] + 9*PolyLog 
[3, I*E^(I*ArcSin[c*x])]) + (b^2*(ArcSin[c*x]*(74*Sqrt[1 - c^2*x^2] + 30*C 
os[3*ArcSin[c*x]]) + 3*ArcSin[c*x]^2*(3*c*x - 5*Sin[3*ArcSin[c*x]]) + 2*(c 
*x + Sin[3*ArcSin[c*x]])))/(-1 + c^2*x^2)^2)/(96*c^5*d^3)
 

3.3.1.3 Rubi [A] (verified)

Time = 1.91 (sec) , antiderivative size = 360, normalized size of antiderivative = 1.05, number of steps used = 16, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.556, Rules used = {5206, 27, 5194, 27, 298, 219, 5206, 5164, 3042, 4669, 3011, 2720, 5182, 219, 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.

\(\displaystyle \int \frac {x^4 (a+b \arcsin (c x))^2}{\left (d-c^2 d x^2\right )^3} \, dx\)

\(\Big \downarrow \) 5206

\(\displaystyle -\frac {b \int \frac {x^3 (a+b \arcsin (c x))}{\left (1-c^2 x^2\right )^{5/2}}dx}{2 c d^3}-\frac {3 \int \frac {x^2 (a+b \arcsin (c x))^2}{d^2 \left (1-c^2 x^2\right )^2}dx}{4 c^2 d}+\frac {x^3 (a+b \arcsin (c x))^2}{4 c^2 d^3 \left (1-c^2 x^2\right )^2}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {3 \int \frac {x^2 (a+b \arcsin (c x))^2}{\left (1-c^2 x^2\right )^2}dx}{4 c^2 d^3}-\frac {b \int \frac {x^3 (a+b \arcsin (c x))}{\left (1-c^2 x^2\right )^{5/2}}dx}{2 c d^3}+\frac {x^3 (a+b \arcsin (c x))^2}{4 c^2 d^3 \left (1-c^2 x^2\right )^2}\)

\(\Big \downarrow \) 5194

\(\displaystyle -\frac {3 \int \frac {x^2 (a+b \arcsin (c x))^2}{\left (1-c^2 x^2\right )^2}dx}{4 c^2 d^3}-\frac {b \left (-b c \int -\frac {2-3 c^2 x^2}{3 c^4 \left (1-c^2 x^2\right )^2}dx-\frac {a+b \arcsin (c x)}{c^4 \sqrt {1-c^2 x^2}}+\frac {a+b \arcsin (c x)}{3 c^4 \left (1-c^2 x^2\right )^{3/2}}\right )}{2 c d^3}+\frac {x^3 (a+b \arcsin (c x))^2}{4 c^2 d^3 \left (1-c^2 x^2\right )^2}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {3 \int \frac {x^2 (a+b \arcsin (c x))^2}{\left (1-c^2 x^2\right )^2}dx}{4 c^2 d^3}-\frac {b \left (\frac {b \int \frac {2-3 c^2 x^2}{\left (1-c^2 x^2\right )^2}dx}{3 c^3}-\frac {a+b \arcsin (c x)}{c^4 \sqrt {1-c^2 x^2}}+\frac {a+b \arcsin (c x)}{3 c^4 \left (1-c^2 x^2\right )^{3/2}}\right )}{2 c d^3}+\frac {x^3 (a+b \arcsin (c x))^2}{4 c^2 d^3 \left (1-c^2 x^2\right )^2}\)

\(\Big \downarrow \) 298

\(\displaystyle -\frac {3 \int \frac {x^2 (a+b \arcsin (c x))^2}{\left (1-c^2 x^2\right )^2}dx}{4 c^2 d^3}-\frac {b \left (\frac {b \left (\frac {5}{2} \int \frac {1}{1-c^2 x^2}dx-\frac {x}{2 \left (1-c^2 x^2\right )}\right )}{3 c^3}-\frac {a+b \arcsin (c x)}{c^4 \sqrt {1-c^2 x^2}}+\frac {a+b \arcsin (c x)}{3 c^4 \left (1-c^2 x^2\right )^{3/2}}\right )}{2 c d^3}+\frac {x^3 (a+b \arcsin (c x))^2}{4 c^2 d^3 \left (1-c^2 x^2\right )^2}\)

\(\Big \downarrow \) 219

\(\displaystyle -\frac {3 \int \frac {x^2 (a+b \arcsin (c x))^2}{\left (1-c^2 x^2\right )^2}dx}{4 c^2 d^3}-\frac {b \left (-\frac {a+b \arcsin (c x)}{c^4 \sqrt {1-c^2 x^2}}+\frac {a+b \arcsin (c x)}{3 c^4 \left (1-c^2 x^2\right )^{3/2}}+\frac {b \left (\frac {5 \text {arctanh}(c x)}{2 c}-\frac {x}{2 \left (1-c^2 x^2\right )}\right )}{3 c^3}\right )}{2 c d^3}+\frac {x^3 (a+b \arcsin (c x))^2}{4 c^2 d^3 \left (1-c^2 x^2\right )^2}\)

\(\Big \downarrow \) 5206

\(\displaystyle -\frac {3 \left (-\frac {b \int \frac {x (a+b \arcsin (c x))}{\left (1-c^2 x^2\right )^{3/2}}dx}{c}-\frac {\int \frac {(a+b \arcsin (c x))^2}{1-c^2 x^2}dx}{2 c^2}+\frac {x (a+b \arcsin (c x))^2}{2 c^2 \left (1-c^2 x^2\right )}\right )}{4 c^2 d^3}-\frac {b \left (-\frac {a+b \arcsin (c x)}{c^4 \sqrt {1-c^2 x^2}}+\frac {a+b \arcsin (c x)}{3 c^4 \left (1-c^2 x^2\right )^{3/2}}+\frac {b \left (\frac {5 \text {arctanh}(c x)}{2 c}-\frac {x}{2 \left (1-c^2 x^2\right )}\right )}{3 c^3}\right )}{2 c d^3}+\frac {x^3 (a+b \arcsin (c x))^2}{4 c^2 d^3 \left (1-c^2 x^2\right )^2}\)

\(\Big \downarrow \) 5164

\(\displaystyle -\frac {3 \left (-\frac {b \int \frac {x (a+b \arcsin (c x))}{\left (1-c^2 x^2\right )^{3/2}}dx}{c}-\frac {\int \frac {(a+b \arcsin (c x))^2}{\sqrt {1-c^2 x^2}}d\arcsin (c x)}{2 c^3}+\frac {x (a+b \arcsin (c x))^2}{2 c^2 \left (1-c^2 x^2\right )}\right )}{4 c^2 d^3}-\frac {b \left (-\frac {a+b \arcsin (c x)}{c^4 \sqrt {1-c^2 x^2}}+\frac {a+b \arcsin (c x)}{3 c^4 \left (1-c^2 x^2\right )^{3/2}}+\frac {b \left (\frac {5 \text {arctanh}(c x)}{2 c}-\frac {x}{2 \left (1-c^2 x^2\right )}\right )}{3 c^3}\right )}{2 c d^3}+\frac {x^3 (a+b \arcsin (c x))^2}{4 c^2 d^3 \left (1-c^2 x^2\right )^2}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {3 \left (-\frac {\int (a+b \arcsin (c x))^2 \csc \left (\arcsin (c x)+\frac {\pi }{2}\right )d\arcsin (c x)}{2 c^3}-\frac {b \int \frac {x (a+b \arcsin (c x))}{\left (1-c^2 x^2\right )^{3/2}}dx}{c}+\frac {x (a+b \arcsin (c x))^2}{2 c^2 \left (1-c^2 x^2\right )}\right )}{4 c^2 d^3}-\frac {b \left (-\frac {a+b \arcsin (c x)}{c^4 \sqrt {1-c^2 x^2}}+\frac {a+b \arcsin (c x)}{3 c^4 \left (1-c^2 x^2\right )^{3/2}}+\frac {b \left (\frac {5 \text {arctanh}(c x)}{2 c}-\frac {x}{2 \left (1-c^2 x^2\right )}\right )}{3 c^3}\right )}{2 c d^3}+\frac {x^3 (a+b \arcsin (c x))^2}{4 c^2 d^3 \left (1-c^2 x^2\right )^2}\)

\(\Big \downarrow \) 4669

\(\displaystyle -\frac {3 \left (-\frac {-2 b \int (a+b \arcsin (c x)) \log \left (1-i e^{i \arcsin (c x)}\right )d\arcsin (c x)+2 b \int (a+b \arcsin (c x)) \log \left (1+i e^{i \arcsin (c x)}\right )d\arcsin (c x)-2 i \arctan \left (e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))^2}{2 c^3}-\frac {b \int \frac {x (a+b \arcsin (c x))}{\left (1-c^2 x^2\right )^{3/2}}dx}{c}+\frac {x (a+b \arcsin (c x))^2}{2 c^2 \left (1-c^2 x^2\right )}\right )}{4 c^2 d^3}-\frac {b \left (-\frac {a+b \arcsin (c x)}{c^4 \sqrt {1-c^2 x^2}}+\frac {a+b \arcsin (c x)}{3 c^4 \left (1-c^2 x^2\right )^{3/2}}+\frac {b \left (\frac {5 \text {arctanh}(c x)}{2 c}-\frac {x}{2 \left (1-c^2 x^2\right )}\right )}{3 c^3}\right )}{2 c d^3}+\frac {x^3 (a+b \arcsin (c x))^2}{4 c^2 d^3 \left (1-c^2 x^2\right )^2}\)

\(\Big \downarrow \) 3011

\(\displaystyle -\frac {3 \left (-\frac {2 b \left (i \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))-i b \int \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right )d\arcsin (c x)\right )-2 b \left (i \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))-i b \int \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right )d\arcsin (c x)\right )-2 i \arctan \left (e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))^2}{2 c^3}-\frac {b \int \frac {x (a+b \arcsin (c x))}{\left (1-c^2 x^2\right )^{3/2}}dx}{c}+\frac {x (a+b \arcsin (c x))^2}{2 c^2 \left (1-c^2 x^2\right )}\right )}{4 c^2 d^3}-\frac {b \left (-\frac {a+b \arcsin (c x)}{c^4 \sqrt {1-c^2 x^2}}+\frac {a+b \arcsin (c x)}{3 c^4 \left (1-c^2 x^2\right )^{3/2}}+\frac {b \left (\frac {5 \text {arctanh}(c x)}{2 c}-\frac {x}{2 \left (1-c^2 x^2\right )}\right )}{3 c^3}\right )}{2 c d^3}+\frac {x^3 (a+b \arcsin (c x))^2}{4 c^2 d^3 \left (1-c^2 x^2\right )^2}\)

\(\Big \downarrow \) 2720

\(\displaystyle -\frac {3 \left (-\frac {2 b \left (i \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))-b \int e^{-i \arcsin (c x)} \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right )de^{i \arcsin (c x)}\right )-2 b \left (i \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))-b \int e^{-i \arcsin (c x)} \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right )de^{i \arcsin (c x)}\right )-2 i \arctan \left (e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))^2}{2 c^3}-\frac {b \int \frac {x (a+b \arcsin (c x))}{\left (1-c^2 x^2\right )^{3/2}}dx}{c}+\frac {x (a+b \arcsin (c x))^2}{2 c^2 \left (1-c^2 x^2\right )}\right )}{4 c^2 d^3}-\frac {b \left (-\frac {a+b \arcsin (c x)}{c^4 \sqrt {1-c^2 x^2}}+\frac {a+b \arcsin (c x)}{3 c^4 \left (1-c^2 x^2\right )^{3/2}}+\frac {b \left (\frac {5 \text {arctanh}(c x)}{2 c}-\frac {x}{2 \left (1-c^2 x^2\right )}\right )}{3 c^3}\right )}{2 c d^3}+\frac {x^3 (a+b \arcsin (c x))^2}{4 c^2 d^3 \left (1-c^2 x^2\right )^2}\)

\(\Big \downarrow \) 5182

\(\displaystyle -\frac {3 \left (-\frac {2 b \left (i \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))-b \int e^{-i \arcsin (c x)} \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right )de^{i \arcsin (c x)}\right )-2 b \left (i \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))-b \int e^{-i \arcsin (c x)} \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right )de^{i \arcsin (c x)}\right )-2 i \arctan \left (e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))^2}{2 c^3}-\frac {b \left (\frac {a+b \arcsin (c x)}{c^2 \sqrt {1-c^2 x^2}}-\frac {b \int \frac {1}{1-c^2 x^2}dx}{c}\right )}{c}+\frac {x (a+b \arcsin (c x))^2}{2 c^2 \left (1-c^2 x^2\right )}\right )}{4 c^2 d^3}-\frac {b \left (-\frac {a+b \arcsin (c x)}{c^4 \sqrt {1-c^2 x^2}}+\frac {a+b \arcsin (c x)}{3 c^4 \left (1-c^2 x^2\right )^{3/2}}+\frac {b \left (\frac {5 \text {arctanh}(c x)}{2 c}-\frac {x}{2 \left (1-c^2 x^2\right )}\right )}{3 c^3}\right )}{2 c d^3}+\frac {x^3 (a+b \arcsin (c x))^2}{4 c^2 d^3 \left (1-c^2 x^2\right )^2}\)

\(\Big \downarrow \) 219

\(\displaystyle -\frac {3 \left (-\frac {2 b \left (i \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))-b \int e^{-i \arcsin (c x)} \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right )de^{i \arcsin (c x)}\right )-2 b \left (i \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))-b \int e^{-i \arcsin (c x)} \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right )de^{i \arcsin (c x)}\right )-2 i \arctan \left (e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))^2}{2 c^3}-\frac {b \left (\frac {a+b \arcsin (c x)}{c^2 \sqrt {1-c^2 x^2}}-\frac {b \text {arctanh}(c x)}{c^2}\right )}{c}+\frac {x (a+b \arcsin (c x))^2}{2 c^2 \left (1-c^2 x^2\right )}\right )}{4 c^2 d^3}-\frac {b \left (-\frac {a+b \arcsin (c x)}{c^4 \sqrt {1-c^2 x^2}}+\frac {a+b \arcsin (c x)}{3 c^4 \left (1-c^2 x^2\right )^{3/2}}+\frac {b \left (\frac {5 \text {arctanh}(c x)}{2 c}-\frac {x}{2 \left (1-c^2 x^2\right )}\right )}{3 c^3}\right )}{2 c d^3}+\frac {x^3 (a+b \arcsin (c x))^2}{4 c^2 d^3 \left (1-c^2 x^2\right )^2}\)

\(\Big \downarrow \) 7143

\(\displaystyle -\frac {3 \left (-\frac {-2 i \arctan \left (e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))^2+2 b \left (i \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))-b \operatorname {PolyLog}\left (3,-i e^{i \arcsin (c x)}\right )\right )-2 b \left (i \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))-b \operatorname {PolyLog}\left (3,i e^{i \arcsin (c x)}\right )\right )}{2 c^3}-\frac {b \left (\frac {a+b \arcsin (c x)}{c^2 \sqrt {1-c^2 x^2}}-\frac {b \text {arctanh}(c x)}{c^2}\right )}{c}+\frac {x (a+b \arcsin (c x))^2}{2 c^2 \left (1-c^2 x^2\right )}\right )}{4 c^2 d^3}-\frac {b \left (-\frac {a+b \arcsin (c x)}{c^4 \sqrt {1-c^2 x^2}}+\frac {a+b \arcsin (c x)}{3 c^4 \left (1-c^2 x^2\right )^{3/2}}+\frac {b \left (\frac {5 \text {arctanh}(c x)}{2 c}-\frac {x}{2 \left (1-c^2 x^2\right )}\right )}{3 c^3}\right )}{2 c d^3}+\frac {x^3 (a+b \arcsin (c x))^2}{4 c^2 d^3 \left (1-c^2 x^2\right )^2}\)

input

Int[(x^4*(a + b*ArcSin[c*x])^2)/(d - c^2*d*x^2)^3,x]
 

output

(x^3*(a + b*ArcSin[c*x])^2)/(4*c^2*d^3*(1 - c^2*x^2)^2) - (b*((a + b*ArcSi 
n[c*x])/(3*c^4*(1 - c^2*x^2)^(3/2)) - (a + b*ArcSin[c*x])/(c^4*Sqrt[1 - c^ 
2*x^2]) + (b*(-1/2*x/(1 - c^2*x^2) + (5*ArcTanh[c*x])/(2*c)))/(3*c^3)))/(2 
*c*d^3) - (3*((x*(a + b*ArcSin[c*x])^2)/(2*c^2*(1 - c^2*x^2)) - (b*((a + b 
*ArcSin[c*x])/(c^2*Sqrt[1 - c^2*x^2]) - (b*ArcTanh[c*x])/c^2))/c - ((-2*I) 
*(a + b*ArcSin[c*x])^2*ArcTan[E^(I*ArcSin[c*x])] + 2*b*(I*(a + b*ArcSin[c* 
x])*PolyLog[2, (-I)*E^(I*ArcSin[c*x])] - b*PolyLog[3, (-I)*E^(I*ArcSin[c*x 
])]) - 2*b*(I*(a + b*ArcSin[c*x])*PolyLog[2, I*E^(I*ArcSin[c*x])] - b*Poly 
Log[3, I*E^(I*ArcSin[c*x])]))/(2*c^3)))/(4*c^2*d^3)
 

3.3.1.3.1 Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 219

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* 
ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt 
Q[a, 0] || LtQ[b, 0])
 

rule 298

Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2), x_Symbol] :> Simp[(-( 
b*c - a*d))*x*((a + b*x^2)^(p + 1)/(2*a*b*(p + 1))), x] - Simp[(a*d - b*c*( 
2*p + 3))/(2*a*b*(p + 1))   Int[(a + b*x^2)^(p + 1), x], x] /; FreeQ[{a, b, 
 c, d, p}, x] && NeQ[b*c - a*d, 0] && (LtQ[p, -1] || ILtQ[1/2 + p, 0])
 

rule 2720

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]]]
 

rule 3011

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]
 

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 4669

Int[csc[(e_.) + Pi*(k_.) + (f_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.), x_Symbol 
] :> Simp[-2*(c + d*x)^m*(ArcTanh[E^(I*k*Pi)*E^(I*(e + f*x))]/f), x] + (-Si 
mp[d*(m/f)   Int[(c + d*x)^(m - 1)*Log[1 - E^(I*k*Pi)*E^(I*(e + f*x))], x], 
 x] + Simp[d*(m/f)   Int[(c + d*x)^(m - 1)*Log[1 + E^(I*k*Pi)*E^(I*(e + f*x 
))], x], x]) /; FreeQ[{c, d, e, f}, x] && IntegerQ[2*k] && IGtQ[m, 0]
 

rule 5164

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)/((d_) + (e_.)*(x_)^2), x_Symbo 
l] :> Simp[1/(c*d)   Subst[Int[(a + b*x)^n*Sec[x], x], x, ArcSin[c*x]], x] 
/; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && IGtQ[n, 0]
 

rule 5182

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*(x_)*((d_) + (e_.)*(x_)^2)^(p_ 
.), x_Symbol] :> Simp[(d + e*x^2)^(p + 1)*((a + b*ArcSin[c*x])^n/(2*e*(p + 
1))), x] + Simp[b*(n/(2*c*(p + 1)))*Simp[(d + e*x^2)^p/(1 - c^2*x^2)^p]   I 
nt[(1 - c^2*x^2)^(p + 1/2)*(a + b*ArcSin[c*x])^(n - 1), x], x] /; FreeQ[{a, 
 b, c, d, e, p}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && NeQ[p, -1]
 

rule 5194

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))*(x_)^(m_)*((d_) + (e_.)*(x_)^2)^(p_) 
, x_Symbol] :> With[{u = IntHide[x^m*(d + e*x^2)^p, x]}, Simp[(a + b*ArcSin 
[c*x])   u, x] - Simp[b*c*Simp[Sqrt[d + e*x^2]/Sqrt[1 - c^2*x^2]]   Int[Sim 
plifyIntegrand[u/Sqrt[d + e*x^2], x], x], x]] /; FreeQ[{a, b, c, d, e}, x] 
&& EqQ[c^2*d + e, 0] && IntegerQ[p - 1/2] && NeQ[p, -2^(-1)] && (IGtQ[(m + 
1)/2, 0] || ILtQ[(m + 2*p + 3)/2, 0])
 

rule 5206

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*((d_) + (e_. 
)*(x_)^2)^(p_), x_Symbol] :> Simp[f*(f*x)^(m - 1)*(d + e*x^2)^(p + 1)*((a + 
 b*ArcSin[c*x])^n/(2*e*(p + 1))), x] + (-Simp[f^2*((m - 1)/(2*e*(p + 1))) 
 Int[(f*x)^(m - 2)*(d + e*x^2)^(p + 1)*(a + b*ArcSin[c*x])^n, x], x] + Simp 
[b*f*(n/(2*c*(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}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && LtQ[p, -1] && IG 
tQ[m, 1]
 

rule 7143

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]
 

3.3.1.4 Maple [A] (verified)

Time = 0.44 (sec) , antiderivative size = 570, normalized size of antiderivative = 1.66

method	result	size
derivativedivides	\(\frac {-\frac {a^{2} \left (-\frac {1}{16 \left (c x -1\right )^{2}}-\frac {5}{16 \left (c x -1\right )}+\frac {3 \ln \left (c x -1\right )}{16}+\frac {1}{16 \left (c x +1\right )^{2}}-\frac {5}{16 \left (c x +1\right )}-\frac {3 \ln \left (c x +1\right )}{16}\right )}{d^{3}}-\frac {b^{2} \left (-\frac {15 c^{3} x^{3} \arcsin \left (c x \right )^{2}-30 \sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right ) x^{2} c^{2}-9 c x \arcsin \left (c x \right )^{2}-2 c^{3} x^{3}+26 \arcsin \left (c x \right ) \sqrt {-c^{2} x^{2}+1}+2 c x}{24 \left (c^{4} x^{4}-2 c^{2} x^{2}+1\right )}+\frac {3 \arcsin \left (c x \right )^{2} \ln \left (1+i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{8}-\frac {3 i \arcsin \left (c x \right ) \operatorname {polylog}\left (2, -i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{4}+\frac {3 \operatorname {polylog}\left (3, -i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{4}-\frac {3 \arcsin \left (c x \right )^{2} \ln \left (1-i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{8}+\frac {3 i \arcsin \left (c x \right ) \operatorname {polylog}\left (2, i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{4}-\frac {3 \operatorname {polylog}\left (3, i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{4}-\frac {7 i \arctan \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )}{3}\right )}{d^{3}}-\frac {2 a b \left (-\frac {15 c^{3} x^{3} \arcsin \left (c x \right )-15 c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}-9 c x \arcsin \left (c x \right )+13 \sqrt {-c^{2} x^{2}+1}}{24 \left (c^{4} x^{4}-2 c^{2} x^{2}+1\right )}+\frac {3 \arcsin \left (c x \right ) \ln \left (1+i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{8}-\frac {3 \arcsin \left (c x \right ) \ln \left (1-i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{8}-\frac {3 i \operatorname {dilog}\left (1+i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{8}+\frac {3 i \operatorname {dilog}\left (1-i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{8}\right )}{d^{3}}}{c^{5}}\)	\(570\)
default	\(\frac {-\frac {a^{2} \left (-\frac {1}{16 \left (c x -1\right )^{2}}-\frac {5}{16 \left (c x -1\right )}+\frac {3 \ln \left (c x -1\right )}{16}+\frac {1}{16 \left (c x +1\right )^{2}}-\frac {5}{16 \left (c x +1\right )}-\frac {3 \ln \left (c x +1\right )}{16}\right )}{d^{3}}-\frac {b^{2} \left (-\frac {15 c^{3} x^{3} \arcsin \left (c x \right )^{2}-30 \sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right ) x^{2} c^{2}-9 c x \arcsin \left (c x \right )^{2}-2 c^{3} x^{3}+26 \arcsin \left (c x \right ) \sqrt {-c^{2} x^{2}+1}+2 c x}{24 \left (c^{4} x^{4}-2 c^{2} x^{2}+1\right )}+\frac {3 \arcsin \left (c x \right )^{2} \ln \left (1+i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{8}-\frac {3 i \arcsin \left (c x \right ) \operatorname {polylog}\left (2, -i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{4}+\frac {3 \operatorname {polylog}\left (3, -i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{4}-\frac {3 \arcsin \left (c x \right )^{2} \ln \left (1-i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{8}+\frac {3 i \arcsin \left (c x \right ) \operatorname {polylog}\left (2, i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{4}-\frac {3 \operatorname {polylog}\left (3, i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{4}-\frac {7 i \arctan \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )}{3}\right )}{d^{3}}-\frac {2 a b \left (-\frac {15 c^{3} x^{3} \arcsin \left (c x \right )-15 c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}-9 c x \arcsin \left (c x \right )+13 \sqrt {-c^{2} x^{2}+1}}{24 \left (c^{4} x^{4}-2 c^{2} x^{2}+1\right )}+\frac {3 \arcsin \left (c x \right ) \ln \left (1+i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{8}-\frac {3 \arcsin \left (c x \right ) \ln \left (1-i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{8}-\frac {3 i \operatorname {dilog}\left (1+i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{8}+\frac {3 i \operatorname {dilog}\left (1-i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{8}\right )}{d^{3}}}{c^{5}}\)	\(570\)
parts	\(-\frac {a^{2} \left (-\frac {1}{16 c^{5} \left (c x -1\right )^{2}}-\frac {5}{16 c^{5} \left (c x -1\right )}+\frac {3 \ln \left (c x -1\right )}{16 c^{5}}+\frac {1}{16 c^{5} \left (c x +1\right )^{2}}-\frac {5}{16 c^{5} \left (c x +1\right )}-\frac {3 \ln \left (c x +1\right )}{16 c^{5}}\right )}{d^{3}}-\frac {b^{2} \left (-\frac {15 c^{3} x^{3} \arcsin \left (c x \right )^{2}-30 \sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right ) x^{2} c^{2}-9 c x \arcsin \left (c x \right )^{2}-2 c^{3} x^{3}+26 \arcsin \left (c x \right ) \sqrt {-c^{2} x^{2}+1}+2 c x}{24 \left (c^{4} x^{4}-2 c^{2} x^{2}+1\right )}+\frac {3 \arcsin \left (c x \right )^{2} \ln \left (1+i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{8}-\frac {3 i \arcsin \left (c x \right ) \operatorname {polylog}\left (2, -i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{4}+\frac {3 \operatorname {polylog}\left (3, -i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{4}-\frac {3 \arcsin \left (c x \right )^{2} \ln \left (1-i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{8}+\frac {3 i \arcsin \left (c x \right ) \operatorname {polylog}\left (2, i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{4}-\frac {3 \operatorname {polylog}\left (3, i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{4}-\frac {7 i \arctan \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )}{3}\right )}{d^{3} c^{5}}-\frac {2 a b \left (-\frac {15 c^{3} x^{3} \arcsin \left (c x \right )-15 c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}-9 c x \arcsin \left (c x \right )+13 \sqrt {-c^{2} x^{2}+1}}{24 \left (c^{4} x^{4}-2 c^{2} x^{2}+1\right )}+\frac {3 \arcsin \left (c x \right ) \ln \left (1+i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{8}-\frac {3 \arcsin \left (c x \right ) \ln \left (1-i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{8}-\frac {3 i \operatorname {dilog}\left (1+i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{8}+\frac {3 i \operatorname {dilog}\left (1-i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{8}\right )}{d^{3} c^{5}}\)	\(590\)

input

int(x^4*(a+b*arcsin(c*x))^2/(-c^2*d*x^2+d)^3,x,method=_RETURNVERBOSE)
 

output

1/c^5*(-a^2/d^3*(-1/16/(c*x-1)^2-5/16/(c*x-1)+3/16*ln(c*x-1)+1/16/(c*x+1)^ 
2-5/16/(c*x+1)-3/16*ln(c*x+1))-b^2/d^3*(-1/24*(15*c^3*x^3*arcsin(c*x)^2-30 
*(-c^2*x^2+1)^(1/2)*arcsin(c*x)*x^2*c^2-9*c*x*arcsin(c*x)^2-2*c^3*x^3+26*a 
rcsin(c*x)*(-c^2*x^2+1)^(1/2)+2*c*x)/(c^4*x^4-2*c^2*x^2+1)+3/8*arcsin(c*x) 
^2*ln(1+I*(I*c*x+(-c^2*x^2+1)^(1/2)))-3/4*I*arcsin(c*x)*polylog(2,-I*(I*c* 
x+(-c^2*x^2+1)^(1/2)))+3/4*polylog(3,-I*(I*c*x+(-c^2*x^2+1)^(1/2)))-3/8*ar 
csin(c*x)^2*ln(1-I*(I*c*x+(-c^2*x^2+1)^(1/2)))+3/4*I*arcsin(c*x)*polylog(2 
,I*(I*c*x+(-c^2*x^2+1)^(1/2)))-3/4*polylog(3,I*(I*c*x+(-c^2*x^2+1)^(1/2))) 
-7/3*I*arctan(I*c*x+(-c^2*x^2+1)^(1/2)))-2*a*b/d^3*(-1/24*(15*c^3*x^3*arcs 
in(c*x)-15*c^2*x^2*(-c^2*x^2+1)^(1/2)-9*c*x*arcsin(c*x)+13*(-c^2*x^2+1)^(1 
/2))/(c^4*x^4-2*c^2*x^2+1)+3/8*arcsin(c*x)*ln(1+I*(I*c*x+(-c^2*x^2+1)^(1/2 
)))-3/8*arcsin(c*x)*ln(1-I*(I*c*x+(-c^2*x^2+1)^(1/2)))-3/8*I*dilog(1+I*(I* 
c*x+(-c^2*x^2+1)^(1/2)))+3/8*I*dilog(1-I*(I*c*x+(-c^2*x^2+1)^(1/2)))))
 

3.3.1.5 Fricas [F]

\[ \int \frac {x^4 (a+b \arcsin (c x))^2}{\left (d-c^2 d x^2\right )^3} \, dx=\int { -\frac {{\left (b \arcsin \left (c x\right ) + a\right )}^{2} x^{4}}{{\left (c^{2} d x^{2} - d\right )}^{3}} \,d x } \]

input

integrate(x^4*(a+b*arcsin(c*x))^2/(-c^2*d*x^2+d)^3,x, algorithm="fricas")
 

output

integral(-(b^2*x^4*arcsin(c*x)^2 + 2*a*b*x^4*arcsin(c*x) + a^2*x^4)/(c^6*d 
^3*x^6 - 3*c^4*d^3*x^4 + 3*c^2*d^3*x^2 - d^3), x)
 

3.3.1.6 Sympy [F]

\[ \int \frac {x^4 (a+b \arcsin (c x))^2}{\left (d-c^2 d x^2\right )^3} \, dx=- \frac {\int \frac {a^{2} x^{4}}{c^{6} x^{6} - 3 c^{4} x^{4} + 3 c^{2} x^{2} - 1}\, dx + \int \frac {b^{2} x^{4} \operatorname {asin}^{2}{\left (c x \right )}}{c^{6} x^{6} - 3 c^{4} x^{4} + 3 c^{2} x^{2} - 1}\, dx + \int \frac {2 a b x^{4} \operatorname {asin}{\left (c x \right )}}{c^{6} x^{6} - 3 c^{4} x^{4} + 3 c^{2} x^{2} - 1}\, dx}{d^{3}} \]

input

integrate(x**4*(a+b*asin(c*x))**2/(-c**2*d*x**2+d)**3,x)
 

output

-(Integral(a**2*x**4/(c**6*x**6 - 3*c**4*x**4 + 3*c**2*x**2 - 1), x) + Int 
egral(b**2*x**4*asin(c*x)**2/(c**6*x**6 - 3*c**4*x**4 + 3*c**2*x**2 - 1), 
x) + Integral(2*a*b*x**4*asin(c*x)/(c**6*x**6 - 3*c**4*x**4 + 3*c**2*x**2 
- 1), x))/d**3
 

3.3.1.7 Maxima [F]

\[ \int \frac {x^4 (a+b \arcsin (c x))^2}{\left (d-c^2 d x^2\right )^3} \, dx=\int { -\frac {{\left (b \arcsin \left (c x\right ) + a\right )}^{2} x^{4}}{{\left (c^{2} d x^{2} - d\right )}^{3}} \,d x } \]

input

integrate(x^4*(a+b*arcsin(c*x))^2/(-c^2*d*x^2+d)^3,x, algorithm="maxima")
 

output

1/16*a^2*(2*(5*c^2*x^3 - 3*x)/(c^8*d^3*x^4 - 2*c^6*d^3*x^2 + c^4*d^3) + 3* 
log(c*x + 1)/(c^5*d^3) - 3*log(c*x - 1)/(c^5*d^3)) + 1/16*(3*(b^2*c^4*x^4 
- 2*b^2*c^2*x^2 + b^2)*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1))^2*log(c* 
x + 1) - 3*(b^2*c^4*x^4 - 2*b^2*c^2*x^2 + b^2)*arctan2(c*x, sqrt(c*x + 1)* 
sqrt(-c*x + 1))^2*log(-c*x + 1) + 2*(5*b^2*c^3*x^3 - 3*b^2*c*x)*arctan2(c* 
x, sqrt(c*x + 1)*sqrt(-c*x + 1))^2 + 16*(c^9*d^3*x^4 - 2*c^7*d^3*x^2 + c^5 
*d^3)*integrate(-1/8*(16*a*b*c^4*x^4*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x 
+ 1)) - (3*(b^2*c^4*x^4 - 2*b^2*c^2*x^2 + b^2)*arctan2(c*x, sqrt(c*x + 1)* 
sqrt(-c*x + 1))*log(c*x + 1) - 3*(b^2*c^4*x^4 - 2*b^2*c^2*x^2 + b^2)*arcta 
n2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1))*log(-c*x + 1) + 2*(5*b^2*c^3*x^3 - 3 
*b^2*c*x)*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1)))*sqrt(c*x + 1)*sqrt(- 
c*x + 1))/(c^10*d^3*x^6 - 3*c^8*d^3*x^4 + 3*c^6*d^3*x^2 - c^4*d^3), x))/(c 
^9*d^3*x^4 - 2*c^7*d^3*x^2 + c^5*d^3)
 

3.3.1.8 Giac [F]

\[ \int \frac {x^4 (a+b \arcsin (c x))^2}{\left (d-c^2 d x^2\right )^3} \, dx=\int { -\frac {{\left (b \arcsin \left (c x\right ) + a\right )}^{2} x^{4}}{{\left (c^{2} d x^{2} - d\right )}^{3}} \,d x } \]

input

integrate(x^4*(a+b*arcsin(c*x))^2/(-c^2*d*x^2+d)^3,x, algorithm="giac")
 

output

integrate(-(b*arcsin(c*x) + a)^2*x^4/(c^2*d*x^2 - d)^3, x)
 

3.3.1.9 Mupad [F(-1)]

Timed out. \[ \int \frac {x^4 (a+b \arcsin (c x))^2}{\left (d-c^2 d x^2\right )^3} \, dx=\int \frac {x^4\,{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^2}{{\left (d-c^2\,d\,x^2\right )}^3} \,d x \]

input

int((x^4*(a + b*asin(c*x))^2)/(d - c^2*d*x^2)^3,x)
 

output

int((x^4*(a + b*asin(c*x))^2)/(d - c^2*d*x^2)^3, x)