\(\int \frac {x^4 (a+b \arcsin (c x))^2}{(d-c^2 d x^2)^2} \, dx\) [189]

Optimal result

Integrand size = 27, antiderivative size = 300 \[ \int \frac {x^4 (a+b \arcsin (c x))^2}{\left (d-c^2 d x^2\right )^2} \, dx=-\frac {2 b^2 x}{c^4 d^2}-\frac {b (a+b \arcsin (c x))}{c^5 d^2 \sqrt {1-c^2 x^2}}+\frac {2 b \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{c^5 d^2}+\frac {3 x (a+b \arcsin (c x))^2}{2 c^4 d^2}+\frac {x^3 (a+b \arcsin (c x))^2}{2 c^2 d^2 \left (1-c^2 x^2\right )}+\frac {3 i (a+b \arcsin (c x))^2 \arctan \left (e^{i \arcsin (c x)}\right )}{c^5 d^2}+\frac {b^2 \text {arctanh}(c x)}{c^5 d^2}-\frac {3 i b (a+b \arcsin (c x)) \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right )}{c^5 d^2}+\frac {3 i b (a+b \arcsin (c x)) \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right )}{c^5 d^2}+\frac {3 b^2 \operatorname {PolyLog}\left (3,-i e^{i \arcsin (c x)}\right )}{c^5 d^2}-\frac {3 b^2 \operatorname {PolyLog}\left (3,i e^{i \arcsin (c x)}\right )}{c^5 d^2} \] Output:

-2*b^2*x/c^4/d^2-b*(a+b*arcsin(c*x))/c^5/d^2/(-c^2*x^2+1)^(1/2)+2*b*(-c^2* 
x^2+1)^(1/2)*(a+b*arcsin(c*x))/c^5/d^2+3/2*x*(a+b*arcsin(c*x))^2/c^4/d^2+1 
/2*x^3*(a+b*arcsin(c*x))^2/c^2/d^2/(-c^2*x^2+1)+3*I*(a+b*arcsin(c*x))^2*ar 
ctan(I*c*x+(-c^2*x^2+1)^(1/2))/c^5/d^2+b^2*arctanh(c*x)/c^5/d^2-3*I*b*(a+b 
*arcsin(c*x))*polylog(2,-I*(I*c*x+(-c^2*x^2+1)^(1/2)))/c^5/d^2+3*I*b*(a+b* 
arcsin(c*x))*polylog(2,I*(I*c*x+(-c^2*x^2+1)^(1/2)))/c^5/d^2+3*b^2*polylog 
(3,-I*(I*c*x+(-c^2*x^2+1)^(1/2)))/c^5/d^2-3*b^2*polylog(3,I*(I*c*x+(-c^2*x 
^2+1)^(1/2)))/c^5/d^2
 

Mathematica [B] (verified)

Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(614\) vs. \(2(300)=600\).

Time = 2.35 (sec) , antiderivative size = 614, normalized size of antiderivative = 2.05 \[ \int \frac {x^4 (a+b \arcsin (c x))^2}{\left (d-c^2 d x^2\right )^2} \, dx=\frac {4 a^2 c x+\frac {8 b^2 c^3 x^3}{1-c^2 x^2}+8 a b \sqrt {1-c^2 x^2}+\frac {2 a b \sqrt {1-c^2 x^2}}{-1+c x}-\frac {2 a b \sqrt {1-c^2 x^2}}{1+c x}-\frac {2 a^2 c x}{-1+c^2 x^2}+\frac {8 b^2 c x}{-1+c^2 x^2}+4 b^2 \coth ^{-1}(c x)+6 i a b \pi \arcsin (c x)+8 a b c x \arcsin (c x)-\frac {2 a b \arcsin (c x)}{-1+c x}-\frac {2 a b \arcsin (c x)}{1+c x}+\frac {2 b^2 \arcsin (c x)}{\sqrt {1-c^2 x^2}}-\frac {6 b^2 c^2 x^2 \arcsin (c x)}{\sqrt {1-c^2 x^2}}+2 b^2 \sqrt {1-c^2 x^2} \arcsin (c x)+\frac {6 b^2 c x \arcsin (c x)^2}{1-c^2 x^2}+\frac {4 b^2 c^3 x^3 \arcsin (c x)^2}{-1+c^2 x^2}+12 i b^2 \arcsin (c x)^2 \arctan \left (e^{i \arcsin (c x)}\right )-6 a b \pi \log \left (1-i e^{i \arcsin (c x)}\right )-12 a b \arcsin (c x) \log \left (1-i e^{i \arcsin (c x)}\right )-6 a b \pi \log \left (1+i e^{i \arcsin (c x)}\right )+12 a b \arcsin (c x) \log \left (1+i e^{i \arcsin (c x)}\right )+3 a^2 \log (1-c x)-3 a^2 \log (1+c x)+6 a b \pi \log \left (-\cos \left (\frac {1}{4} (\pi +2 \arcsin (c x))\right )\right )+6 a b \pi \log \left (\sin \left (\frac {1}{4} (\pi +2 \arcsin (c x))\right )\right )-12 i b (a+b \arcsin (c x)) \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right )+12 i b (a+b \arcsin (c x)) \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right )+12 b^2 \operatorname {PolyLog}\left (3,-i e^{i \arcsin (c x)}\right )-12 b^2 \operatorname {PolyLog}\left (3,i e^{i \arcsin (c x)}\right )}{4 c^5 d^2} \] Input:

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

Output:

(4*a^2*c*x + (8*b^2*c^3*x^3)/(1 - c^2*x^2) + 8*a*b*Sqrt[1 - c^2*x^2] + (2* 
a*b*Sqrt[1 - c^2*x^2])/(-1 + c*x) - (2*a*b*Sqrt[1 - c^2*x^2])/(1 + c*x) - 
(2*a^2*c*x)/(-1 + c^2*x^2) + (8*b^2*c*x)/(-1 + c^2*x^2) + 4*b^2*ArcCoth[c* 
x] + (6*I)*a*b*Pi*ArcSin[c*x] + 8*a*b*c*x*ArcSin[c*x] - (2*a*b*ArcSin[c*x] 
)/(-1 + c*x) - (2*a*b*ArcSin[c*x])/(1 + c*x) + (2*b^2*ArcSin[c*x])/Sqrt[1 
- c^2*x^2] - (6*b^2*c^2*x^2*ArcSin[c*x])/Sqrt[1 - c^2*x^2] + 2*b^2*Sqrt[1 
- c^2*x^2]*ArcSin[c*x] + (6*b^2*c*x*ArcSin[c*x]^2)/(1 - c^2*x^2) + (4*b^2* 
c^3*x^3*ArcSin[c*x]^2)/(-1 + c^2*x^2) + (12*I)*b^2*ArcSin[c*x]^2*ArcTan[E^ 
(I*ArcSin[c*x])] - 6*a*b*Pi*Log[1 - I*E^(I*ArcSin[c*x])] - 12*a*b*ArcSin[c 
*x]*Log[1 - I*E^(I*ArcSin[c*x])] - 6*a*b*Pi*Log[1 + I*E^(I*ArcSin[c*x])] + 
 12*a*b*ArcSin[c*x]*Log[1 + I*E^(I*ArcSin[c*x])] + 3*a^2*Log[1 - c*x] - 3* 
a^2*Log[1 + c*x] + 6*a*b*Pi*Log[-Cos[(Pi + 2*ArcSin[c*x])/4]] + 6*a*b*Pi*L 
og[Sin[(Pi + 2*ArcSin[c*x])/4]] - (12*I)*b*(a + b*ArcSin[c*x])*PolyLog[2, 
(-I)*E^(I*ArcSin[c*x])] + (12*I)*b*(a + b*ArcSin[c*x])*PolyLog[2, I*E^(I*A 
rcSin[c*x])] + 12*b^2*PolyLog[3, (-I)*E^(I*ArcSin[c*x])] - 12*b^2*PolyLog[ 
3, I*E^(I*ArcSin[c*x])])/(4*c^5*d^2)
 

Rubi [A] (verified)

Time = 2.05 (sec) , antiderivative size = 313, normalized size of antiderivative = 1.04, 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, 299, 219, 5210, 5164, 3042, 4669, 3011, 2720, 5182, 24, 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.

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

\(\Big \downarrow \) 5206

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 5194

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 299

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

\(\Big \downarrow \) 219

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

\(\Big \downarrow \) 5210

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

\(\Big \downarrow \) 5164

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4669

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

\(\Big \downarrow \) 3011

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

\(\Big \downarrow \) 2720

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

\(\Big \downarrow \) 5182

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

\(\Big \downarrow \) 24

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

\(\Big \downarrow \) 7143

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

Input:

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

Output:

(x^3*(a + b*ArcSin[c*x])^2)/(2*c^2*d^2*(1 - c^2*x^2)) - (b*((a + b*ArcSin[ 
c*x])/(c^4*Sqrt[1 - c^2*x^2]) + (Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/c^ 
4 - (b*(x + ArcTanh[c*x]/c))/c^3))/(c*d^2) - (3*(-((x*(a + b*ArcSin[c*x])^ 
2)/c^2) + (2*b*((b*x)/c - (Sqrt[1 - c^2*x^2]*(a + b*ArcSin[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*A 
rcSin[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])]))/c^3))/(2*c^2*d^2)
 

Defintions of rubi rules used

rule 24

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]
 

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 299

Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2), x_Symbol] :> Simp[d*x 
*((a + b*x^2)^(p + 1)/(b*(2*p + 3))), x] - Simp[(a*d - b*c*(2*p + 3))/(b*(2 
*p + 3))   Int[(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - 
 a*d, 0] && NeQ[2*p + 3, 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 5210

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/(e*(m + 2*p + 1))), x] + (Simp[f^2*((m - 1)/(c^2*(m + 2*p 
 + 1)))   Int[(f*x)^(m - 2)*(d + e*x^2)^p*(a + b*ArcSin[c*x])^n, x], x] + S 
imp[b*f*(n/(c*(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, p}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && IGtQ[m 
, 1] && NeQ[m + 2*p + 1, 0]
 

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]
 

Maple [A] (verified)

Time = 0.68 (sec) , antiderivative size = 618, normalized size of antiderivative = 2.06

method	result	size
derivativedivides	\(\frac {\frac {a^{2} \left (c x -\frac {1}{4 \left (c x +1\right )}-\frac {3 \ln \left (c x +1\right )}{4}-\frac {1}{4 \left (c x -1\right )}+\frac {3 \ln \left (c x -1\right )}{4}\right )}{d^{2}}+\frac {2 b^{2} \arcsin \left (c x \right ) \sqrt {-c^{2} x^{2}+1}}{d^{2}}+\frac {b^{2} \arcsin \left (c x \right )^{2} c x}{d^{2}}-\frac {2 b^{2} c x}{d^{2}}-\frac {b^{2} \arcsin \left (c x \right )^{2} c x}{2 d^{2} \left (c^{2} x^{2}-1\right )}+\frac {b^{2} \arcsin \left (c x \right ) \sqrt {-c^{2} x^{2}+1}}{d^{2} \left (c^{2} x^{2}-1\right )}-\frac {3 b^{2} \arcsin \left (c x \right )^{2} \ln \left (1-i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{2 d^{2}}-\frac {2 i b^{2} \arctan \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )}{d^{2}}-\frac {3 b^{2} \operatorname {polylog}\left (3, i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{d^{2}}+\frac {3 b^{2} \arcsin \left (c x \right )^{2} \ln \left (1+i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{2 d^{2}}+\frac {3 i b^{2} \arcsin \left (c x \right ) \operatorname {polylog}\left (2, i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{d^{2}}+\frac {3 b^{2} \operatorname {polylog}\left (3, -i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{d^{2}}-\frac {3 i b^{2} \arcsin \left (c x \right ) \operatorname {polylog}\left (2, -i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{d^{2}}+\frac {2 a b \sqrt {-c^{2} x^{2}+1}}{d^{2}}+\frac {2 a b \arcsin \left (c x \right ) c x}{d^{2}}-\frac {a b \arcsin \left (c x \right ) c x}{d^{2} \left (c^{2} x^{2}-1\right )}+\frac {a b \sqrt {-c^{2} x^{2}+1}}{d^{2} \left (c^{2} x^{2}-1\right )}+\frac {3 a b \arcsin \left (c x \right ) \ln \left (1+i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{d^{2}}-\frac {3 a b \arcsin \left (c x \right ) \ln \left (1-i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{d^{2}}-\frac {3 i a b \operatorname {dilog}\left (1+i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{d^{2}}+\frac {3 i a b \operatorname {dilog}\left (1-i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{d^{2}}}{c^{5}}\)	\(618\)
default	\(\frac {\frac {a^{2} \left (c x -\frac {1}{4 \left (c x +1\right )}-\frac {3 \ln \left (c x +1\right )}{4}-\frac {1}{4 \left (c x -1\right )}+\frac {3 \ln \left (c x -1\right )}{4}\right )}{d^{2}}+\frac {2 b^{2} \arcsin \left (c x \right ) \sqrt {-c^{2} x^{2}+1}}{d^{2}}+\frac {b^{2} \arcsin \left (c x \right )^{2} c x}{d^{2}}-\frac {2 b^{2} c x}{d^{2}}-\frac {b^{2} \arcsin \left (c x \right )^{2} c x}{2 d^{2} \left (c^{2} x^{2}-1\right )}+\frac {b^{2} \arcsin \left (c x \right ) \sqrt {-c^{2} x^{2}+1}}{d^{2} \left (c^{2} x^{2}-1\right )}-\frac {3 b^{2} \arcsin \left (c x \right )^{2} \ln \left (1-i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{2 d^{2}}-\frac {2 i b^{2} \arctan \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )}{d^{2}}-\frac {3 b^{2} \operatorname {polylog}\left (3, i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{d^{2}}+\frac {3 b^{2} \arcsin \left (c x \right )^{2} \ln \left (1+i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{2 d^{2}}+\frac {3 i b^{2} \arcsin \left (c x \right ) \operatorname {polylog}\left (2, i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{d^{2}}+\frac {3 b^{2} \operatorname {polylog}\left (3, -i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{d^{2}}-\frac {3 i b^{2} \arcsin \left (c x \right ) \operatorname {polylog}\left (2, -i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{d^{2}}+\frac {2 a b \sqrt {-c^{2} x^{2}+1}}{d^{2}}+\frac {2 a b \arcsin \left (c x \right ) c x}{d^{2}}-\frac {a b \arcsin \left (c x \right ) c x}{d^{2} \left (c^{2} x^{2}-1\right )}+\frac {a b \sqrt {-c^{2} x^{2}+1}}{d^{2} \left (c^{2} x^{2}-1\right )}+\frac {3 a b \arcsin \left (c x \right ) \ln \left (1+i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{d^{2}}-\frac {3 a b \arcsin \left (c x \right ) \ln \left (1-i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{d^{2}}-\frac {3 i a b \operatorname {dilog}\left (1+i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{d^{2}}+\frac {3 i a b \operatorname {dilog}\left (1-i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{d^{2}}}{c^{5}}\)	\(618\)
parts	\(\frac {a^{2} \left (\frac {x}{c^{4}}-\frac {1}{4 c^{5} \left (c x -1\right )}+\frac {3 \ln \left (c x -1\right )}{4 c^{5}}-\frac {1}{4 c^{5} \left (c x +1\right )}-\frac {3 \ln \left (c x +1\right )}{4 c^{5}}\right )}{d^{2}}+\frac {2 b^{2} \sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right )}{d^{2} c^{5}}+\frac {b^{2} \arcsin \left (c x \right )^{2} x}{d^{2} c^{4}}-\frac {2 b^{2} x}{c^{4} d^{2}}-\frac {b^{2} \arcsin \left (c x \right )^{2} x}{2 d^{2} c^{4} \left (c^{2} x^{2}-1\right )}+\frac {b^{2} \arcsin \left (c x \right ) \sqrt {-c^{2} x^{2}+1}}{d^{2} c^{5} \left (c^{2} x^{2}-1\right )}-\frac {3 b^{2} \arcsin \left (c x \right )^{2} \ln \left (1-i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{2 d^{2} c^{5}}-\frac {3 i a b \operatorname {dilog}\left (1+i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{d^{2} c^{5}}-\frac {3 b^{2} \operatorname {polylog}\left (3, i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{c^{5} d^{2}}+\frac {3 b^{2} \arcsin \left (c x \right )^{2} \ln \left (1+i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{2 d^{2} c^{5}}-\frac {3 i b^{2} \arcsin \left (c x \right ) \operatorname {polylog}\left (2, -i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{d^{2} c^{5}}+\frac {3 b^{2} \operatorname {polylog}\left (3, -i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{c^{5} d^{2}}+\frac {3 i b^{2} \arcsin \left (c x \right ) \operatorname {polylog}\left (2, i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{d^{2} c^{5}}+\frac {2 a b \sqrt {-c^{2} x^{2}+1}}{d^{2} c^{5}}+\frac {2 a b \arcsin \left (c x \right ) x}{d^{2} c^{4}}-\frac {a b \arcsin \left (c x \right ) x}{d^{2} c^{4} \left (c^{2} x^{2}-1\right )}+\frac {a b \sqrt {-c^{2} x^{2}+1}}{d^{2} c^{5} \left (c^{2} x^{2}-1\right )}+\frac {3 a b \arcsin \left (c x \right ) \ln \left (1+i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{d^{2} c^{5}}-\frac {3 a b \arcsin \left (c x \right ) \ln \left (1-i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{d^{2} c^{5}}+\frac {3 i a b \operatorname {dilog}\left (1-i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{d^{2} c^{5}}-\frac {2 i b^{2} \arctan \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )}{d^{2} c^{5}}\)	\(683\)

Input:

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

Output:

1/c^5*(a^2/d^2*(c*x-1/4/(c*x+1)-3/4*ln(c*x+1)-1/4/(c*x-1)+3/4*ln(c*x-1))+2 
*b^2/d^2*arcsin(c*x)*(-c^2*x^2+1)^(1/2)+b^2/d^2*arcsin(c*x)^2*c*x-2*b^2/d^ 
2*c*x-1/2*b^2/d^2/(c^2*x^2-1)*arcsin(c*x)^2*c*x+b^2/d^2/(c^2*x^2-1)*arcsin 
(c*x)*(-c^2*x^2+1)^(1/2)-3/2*b^2/d^2*arcsin(c*x)^2*ln(1-I*(I*c*x+(-c^2*x^2 
+1)^(1/2)))-2*I*b^2/d^2*arctan(I*c*x+(-c^2*x^2+1)^(1/2))-3*b^2/d^2*polylog 
(3,I*(I*c*x+(-c^2*x^2+1)^(1/2)))+3/2*b^2/d^2*arcsin(c*x)^2*ln(1+I*(I*c*x+( 
-c^2*x^2+1)^(1/2)))+3*I*b^2/d^2*arcsin(c*x)*polylog(2,I*(I*c*x+(-c^2*x^2+1 
)^(1/2)))+3*b^2/d^2*polylog(3,-I*(I*c*x+(-c^2*x^2+1)^(1/2)))-3*I*b^2/d^2*a 
rcsin(c*x)*polylog(2,-I*(I*c*x+(-c^2*x^2+1)^(1/2)))+2*a*b/d^2*(-c^2*x^2+1) 
^(1/2)+2*a*b/d^2*arcsin(c*x)*c*x-a*b/d^2/(c^2*x^2-1)*arcsin(c*x)*c*x+a*b/d 
^2/(c^2*x^2-1)*(-c^2*x^2+1)^(1/2)+3*a*b/d^2*arcsin(c*x)*ln(1+I*(I*c*x+(-c^ 
2*x^2+1)^(1/2)))-3*a*b/d^2*arcsin(c*x)*ln(1-I*(I*c*x+(-c^2*x^2+1)^(1/2)))- 
3*I*a*b/d^2*dilog(1+I*(I*c*x+(-c^2*x^2+1)^(1/2)))+3*I*a*b/d^2*dilog(1-I*(I 
*c*x+(-c^2*x^2+1)^(1/2))))
 

Fricas [F]

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

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

-1/4*a^2*(2*x/(c^6*d^2*x^2 - c^4*d^2) - 4*x/(c^4*d^2) + 3*log(c*x + 1)/(c^ 
5*d^2) - 3*log(c*x - 1)/(c^5*d^2)) - 1/4*(3*(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^2*x^2 - b^2)*ar 
ctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1))^2*log(-c*x + 1) - 2*(2*b^2*c^3*x^ 
3 - 3*b^2*c*x)*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1))^2 + 4*(c^7*d^2*x 
^2 - c^5*d^2)*integrate(-1/2*(4*a*b*c^4*x^4*arctan2(c*x, sqrt(c*x + 1)*sqr 
t(-c*x + 1)) - (3*(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^2*x^2 - b^2)*arctan2(c*x, sqrt(c*x + 1)*sqr 
t(-c*x + 1))*log(-c*x + 1) - 2*(2*b^2*c^3*x^3 - 3*b^2*c*x)*arctan2(c*x, sq 
rt(c*x + 1)*sqrt(-c*x + 1)))*sqrt(c*x + 1)*sqrt(-c*x + 1))/(c^8*d^2*x^4 - 
2*c^6*d^2*x^2 + c^4*d^2), x))/(c^7*d^2*x^2 - c^5*d^2)
 

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {x^4 (a+b \arcsin (c x))^2}{\left (d-c^2 d x^2\right )^2} \, dx=\frac {8 \left (\int \frac {\mathit {asin} \left (c x \right ) x^{4}}{c^{4} x^{4}-2 c^{2} x^{2}+1}d x \right ) a b \,c^{7} x^{2}-8 \left (\int \frac {\mathit {asin} \left (c x \right ) x^{4}}{c^{4} x^{4}-2 c^{2} x^{2}+1}d x \right ) a b \,c^{5}+4 \left (\int \frac {\mathit {asin} \left (c x \right )^{2} x^{4}}{c^{4} x^{4}-2 c^{2} x^{2}+1}d x \right ) b^{2} c^{7} x^{2}-4 \left (\int \frac {\mathit {asin} \left (c x \right )^{2} x^{4}}{c^{4} x^{4}-2 c^{2} x^{2}+1}d x \right ) b^{2} c^{5}+3 \,\mathrm {log}\left (c^{2} x -c \right ) a^{2} c^{2} x^{2}-3 \,\mathrm {log}\left (c^{2} x -c \right ) a^{2}-3 \,\mathrm {log}\left (c^{2} x +c \right ) a^{2} c^{2} x^{2}+3 \,\mathrm {log}\left (c^{2} x +c \right ) a^{2}+4 a^{2} c^{3} x^{3}-6 a^{2} c x}{4 c^{5} d^{2} \left (c^{2} x^{2}-1\right )} \] Input:

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

Output:

(8*int((asin(c*x)*x**4)/(c**4*x**4 - 2*c**2*x**2 + 1),x)*a*b*c**7*x**2 - 8 
*int((asin(c*x)*x**4)/(c**4*x**4 - 2*c**2*x**2 + 1),x)*a*b*c**5 + 4*int((a 
sin(c*x)**2*x**4)/(c**4*x**4 - 2*c**2*x**2 + 1),x)*b**2*c**7*x**2 - 4*int( 
(asin(c*x)**2*x**4)/(c**4*x**4 - 2*c**2*x**2 + 1),x)*b**2*c**5 + 3*log(c** 
2*x - c)*a**2*c**2*x**2 - 3*log(c**2*x - c)*a**2 - 3*log(c**2*x + c)*a**2* 
c**2*x**2 + 3*log(c**2*x + c)*a**2 + 4*a**2*c**3*x**3 - 6*a**2*c*x)/(4*c** 
5*d**2*(c**2*x**2 - 1))