3.3.94 \(\int \frac {\arcsin (a x)^3}{(c-a^2 c x^2)^3} \, dx\) [294]

3.3.94.1 Optimal result

Integrand size = 20, antiderivative size = 455 \[ \int \frac {\arcsin (a x)^3}{\left (c-a^2 c x^2\right )^3} \, dx=-\frac {1}{4 a c^3 \sqrt {1-a^2 x^2}}+\frac {x \arcsin (a x)}{4 c^3 \left (1-a^2 x^2\right )}-\frac {\arcsin (a x)^2}{4 a c^3 \left (1-a^2 x^2\right )^{3/2}}-\frac {9 \arcsin (a x)^2}{8 a c^3 \sqrt {1-a^2 x^2}}+\frac {x \arcsin (a x)^3}{4 c^3 \left (1-a^2 x^2\right )^2}+\frac {3 x \arcsin (a x)^3}{8 c^3 \left (1-a^2 x^2\right )}-\frac {5 i \arcsin (a x) \arctan \left (e^{i \arcsin (a x)}\right )}{a c^3}-\frac {3 i \arcsin (a x)^3 \arctan \left (e^{i \arcsin (a x)}\right )}{4 a c^3}+\frac {5 i \operatorname {PolyLog}\left (2,-i e^{i \arcsin (a x)}\right )}{2 a c^3}+\frac {9 i \arcsin (a x)^2 \operatorname {PolyLog}\left (2,-i e^{i \arcsin (a x)}\right )}{8 a c^3}-\frac {5 i \operatorname {PolyLog}\left (2,i e^{i \arcsin (a x)}\right )}{2 a c^3}-\frac {9 i \arcsin (a x)^2 \operatorname {PolyLog}\left (2,i e^{i \arcsin (a x)}\right )}{8 a c^3}-\frac {9 \arcsin (a x) \operatorname {PolyLog}\left (3,-i e^{i \arcsin (a x)}\right )}{4 a c^3}+\frac {9 \arcsin (a x) \operatorname {PolyLog}\left (3,i e^{i \arcsin (a x)}\right )}{4 a c^3}-\frac {9 i \operatorname {PolyLog}\left (4,-i e^{i \arcsin (a x)}\right )}{4 a c^3}+\frac {9 i \operatorname {PolyLog}\left (4,i e^{i \arcsin (a x)}\right )}{4 a c^3} \]

output

1/4*x*arcsin(a*x)/c^3/(-a^2*x^2+1)-1/4*arcsin(a*x)^2/a/c^3/(-a^2*x^2+1)^(3 
/2)+1/4*x*arcsin(a*x)^3/c^3/(-a^2*x^2+1)^2+3/8*x*arcsin(a*x)^3/c^3/(-a^2*x 
^2+1)-3/4*I*arcsin(a*x)^3*arctan(I*a*x+(-a^2*x^2+1)^(1/2))/a/c^3+9/4*I*pol 
ylog(4,I*(I*a*x+(-a^2*x^2+1)^(1/2)))/a/c^3-5*I*arcsin(a*x)*arctan(I*a*x+(- 
a^2*x^2+1)^(1/2))/a/c^3+9/8*I*arcsin(a*x)^2*polylog(2,-I*(I*a*x+(-a^2*x^2+ 
1)^(1/2)))/a/c^3-9/8*I*arcsin(a*x)^2*polylog(2,I*(I*a*x+(-a^2*x^2+1)^(1/2) 
))/a/c^3+5/2*I*polylog(2,-I*(I*a*x+(-a^2*x^2+1)^(1/2)))/a/c^3-9/4*arcsin(a 
*x)*polylog(3,-I*(I*a*x+(-a^2*x^2+1)^(1/2)))/a/c^3+9/4*arcsin(a*x)*polylog 
(3,I*(I*a*x+(-a^2*x^2+1)^(1/2)))/a/c^3-5/2*I*polylog(2,I*(I*a*x+(-a^2*x^2+ 
1)^(1/2)))/a/c^3-9/4*I*polylog(4,-I*(I*a*x+(-a^2*x^2+1)^(1/2)))/a/c^3-1/4/ 
a/c^3/(-a^2*x^2+1)^(1/2)-9/8*arcsin(a*x)^2/a/c^3/(-a^2*x^2+1)^(1/2)
 

3.3.94.2 Mathematica [B] (verified)

Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(1544\) vs. \(2(455)=910\).

Time = 12.76 (sec) , antiderivative size = 1544, normalized size of antiderivative = 3.39 \[ \int \frac {\arcsin (a x)^3}{\left (c-a^2 c x^2\right )^3} \, dx =\text {Too large to display} \]

input

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

output

-(((1 + 5*ArcSin[a*x]^2)/4 - (5*(ArcSin[a*x]*(Log[1 - I*E^(I*ArcSin[a*x])] 
 - Log[1 + I*E^(I*ArcSin[a*x])]) + I*(PolyLog[2, (-I)*E^(I*ArcSin[a*x])] - 
 PolyLog[2, I*E^(I*ArcSin[a*x])])))/2 - (3*((Pi^3*Log[Cot[(Pi/2 - ArcSin[a 
*x])/2]])/8 + (3*Pi^2*((Pi/2 - ArcSin[a*x])*(Log[1 - E^(I*(Pi/2 - ArcSin[a 
*x]))] - Log[1 + E^(I*(Pi/2 - ArcSin[a*x]))]) + I*(PolyLog[2, -E^(I*(Pi/2 
- ArcSin[a*x]))] - PolyLog[2, E^(I*(Pi/2 - ArcSin[a*x]))])))/4 - (3*Pi*((P 
i/2 - ArcSin[a*x])^2*(Log[1 - E^(I*(Pi/2 - ArcSin[a*x]))] - Log[1 + E^(I*( 
Pi/2 - ArcSin[a*x]))]) + (2*I)*(Pi/2 - ArcSin[a*x])*(PolyLog[2, -E^(I*(Pi/ 
2 - ArcSin[a*x]))] - PolyLog[2, E^(I*(Pi/2 - ArcSin[a*x]))]) + 2*(-PolyLog 
[3, -E^(I*(Pi/2 - ArcSin[a*x]))] + PolyLog[3, E^(I*(Pi/2 - ArcSin[a*x]))]) 
))/2 + 8*((I/64)*(Pi/2 - ArcSin[a*x])^4 + (I/4)*(Pi/2 + (-1/2*Pi + ArcSin[ 
a*x])/2)^4 - ((Pi/2 - ArcSin[a*x])^3*Log[1 + E^(I*(Pi/2 - ArcSin[a*x]))])/ 
8 - (Pi^3*(I*(Pi/2 + (-1/2*Pi + ArcSin[a*x])/2) - Log[1 + E^((2*I)*(Pi/2 + 
 (-1/2*Pi + ArcSin[a*x])/2))]))/8 - (Pi/2 + (-1/2*Pi + ArcSin[a*x])/2)^3*L 
og[1 + E^((2*I)*(Pi/2 + (-1/2*Pi + ArcSin[a*x])/2))] + ((3*I)/8)*(Pi/2 - A 
rcSin[a*x])^2*PolyLog[2, -E^(I*(Pi/2 - ArcSin[a*x]))] + (3*Pi^2*((I/2)*(Pi 
/2 + (-1/2*Pi + ArcSin[a*x])/2)^2 - (Pi/2 + (-1/2*Pi + ArcSin[a*x])/2)*Log 
[1 + E^((2*I)*(Pi/2 + (-1/2*Pi + ArcSin[a*x])/2))] + (I/2)*PolyLog[2, -E^( 
(2*I)*(Pi/2 + (-1/2*Pi + ArcSin[a*x])/2))]))/4 + ((3*I)/2)*(Pi/2 + (-1/2*P 
i + ArcSin[a*x])/2)^2*PolyLog[2, -E^((2*I)*(Pi/2 + (-1/2*Pi + ArcSin[a*...
 

3.3.94.3 Rubi [A] (verified)

Time = 2.68 (sec) , antiderivative size = 483, normalized size of antiderivative = 1.06, number of steps used = 19, number of rules used = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.900, Rules used = {5162, 27, 5162, 5164, 3042, 4669, 3011, 5182, 5162, 241, 5164, 3042, 4669, 2715, 2838, 7163, 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.

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

\(\Big \downarrow \) 5162

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 5162

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

\(\Big \downarrow \) 5164

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4669

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

\(\Big \downarrow \) 3011

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

\(\Big \downarrow \) 5182

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

\(\Big \downarrow \) 5162

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

\(\Big \downarrow \) 241

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

\(\Big \downarrow \) 5164

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4669

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

\(\Big \downarrow \) 2715

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

\(\Big \downarrow \) 2838

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

\(\Big \downarrow \) 7163

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

\(\Big \downarrow \) 2720

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

\(\Big \downarrow \) 7143

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

input

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

output

(x*ArcSin[a*x]^3)/(4*c^3*(1 - a^2*x^2)^2) - (3*a*(ArcSin[a*x]^2/(3*a^2*(1 
- a^2*x^2)^(3/2)) - (2*(-1/2*1/(a*Sqrt[1 - a^2*x^2]) + (x*ArcSin[a*x])/(2* 
(1 - a^2*x^2)) + ((-2*I)*ArcSin[a*x]*ArcTan[E^(I*ArcSin[a*x])] + I*PolyLog 
[2, (-I)*E^(I*ArcSin[a*x])] - I*PolyLog[2, I*E^(I*ArcSin[a*x])])/(2*a)))/( 
3*a)))/(4*c^3) + (3*((x*ArcSin[a*x]^3)/(2*(1 - a^2*x^2)) - (3*a*(ArcSin[a* 
x]^2/(a^2*Sqrt[1 - a^2*x^2]) - (2*((-2*I)*ArcSin[a*x]*ArcTan[E^(I*ArcSin[a 
*x])] + I*PolyLog[2, (-I)*E^(I*ArcSin[a*x])] - I*PolyLog[2, I*E^(I*ArcSin[ 
a*x])]))/a^2))/2 + ((-2*I)*ArcSin[a*x]^3*ArcTan[E^(I*ArcSin[a*x])] + 3*(I* 
ArcSin[a*x]^2*PolyLog[2, (-I)*E^(I*ArcSin[a*x])] - (2*I)*((-I)*ArcSin[a*x] 
*PolyLog[3, (-I)*E^(I*ArcSin[a*x])] + PolyLog[4, (-I)*E^(I*ArcSin[a*x])])) 
 - 3*(I*ArcSin[a*x]^2*PolyLog[2, I*E^(I*ArcSin[a*x])] - (2*I)*((-I)*ArcSin 
[a*x]*PolyLog[3, I*E^(I*ArcSin[a*x])] + PolyLog[4, I*E^(I*ArcSin[a*x])]))) 
/(2*a)))/(4*c^3)
 

3.3.94.3.1 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 241

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]
 

rule 2715

Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] 
:> Simp[1/(d*e*n*Log[F])   Subst[Int[Log[a + b*x]/x, x], x, (F^(e*(c + d*x) 
))^n], x] /; FreeQ[{F, a, b, c, d, e, n}, x] && GtQ[a, 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 2838

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> Simp[-PolyLog[2 
, (-c)*e*x^n]/n, x] /; FreeQ[{c, d, e, n}, x] && EqQ[c*d, 1]
 

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 5162

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]
 

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 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]
 

rule 7163

Int[((e_.) + (f_.)*(x_))^(m_.)*PolyLog[n_, (d_.)*((F_)^((c_.)*((a_.) + (b_. 
)*(x_))))^(p_.)], x_Symbol] :> Simp[(e + f*x)^m*(PolyLog[n + 1, d*(F^(c*(a 
+ b*x)))^p]/(b*c*p*Log[F])), x] - Simp[f*(m/(b*c*p*Log[F]))   Int[(e + f*x) 
^(m - 1)*PolyLog[n + 1, d*(F^(c*(a + b*x)))^p], x], x] /; FreeQ[{F, a, b, c 
, d, e, f, n, p}, x] && GtQ[m, 0]
 

3.3.94.4 Maple [A] (verified)

Time = 0.25 (sec) , antiderivative size = 543, normalized size of antiderivative = 1.19

method	result	size
derivativedivides	\(\frac {-\frac {3 a^{3} x^{3} \arcsin \left (a x \right )^{3}-9 \arcsin \left (a x \right )^{2} \sqrt {-a^{2} x^{2}+1}\, a^{2} x^{2}-5 a x \arcsin \left (a x \right )^{3}+2 a^{3} x^{3} \arcsin \left (a x \right )+11 \arcsin \left (a x \right )^{2} \sqrt {-a^{2} x^{2}+1}-2 a^{2} x^{2} \sqrt {-a^{2} x^{2}+1}-2 a x \arcsin \left (a x \right )+2 \sqrt {-a^{2} x^{2}+1}}{8 \left (a^{4} x^{4}-2 a^{2} x^{2}+1\right ) c^{3}}-\frac {3 \arcsin \left (a x \right )^{3} \ln \left (1+i \left (i a x +\sqrt {-a^{2} x^{2}+1}\right )\right )}{8 c^{3}}+\frac {9 i \arcsin \left (a x \right )^{2} \operatorname {polylog}\left (2, -i \left (i a x +\sqrt {-a^{2} x^{2}+1}\right )\right )}{8 c^{3}}-\frac {9 \arcsin \left (a x \right ) \operatorname {polylog}\left (3, -i \left (i a x +\sqrt {-a^{2} x^{2}+1}\right )\right )}{4 c^{3}}-\frac {9 i \operatorname {polylog}\left (4, -i \left (i a x +\sqrt {-a^{2} x^{2}+1}\right )\right )}{4 c^{3}}+\frac {3 \arcsin \left (a x \right )^{3} \ln \left (1-i \left (i a x +\sqrt {-a^{2} x^{2}+1}\right )\right )}{8 c^{3}}-\frac {9 i \arcsin \left (a x \right )^{2} \operatorname {polylog}\left (2, i \left (i a x +\sqrt {-a^{2} x^{2}+1}\right )\right )}{8 c^{3}}+\frac {9 \arcsin \left (a x \right ) \operatorname {polylog}\left (3, i \left (i a x +\sqrt {-a^{2} x^{2}+1}\right )\right )}{4 c^{3}}+\frac {9 i \operatorname {polylog}\left (4, i \left (i a x +\sqrt {-a^{2} x^{2}+1}\right )\right )}{4 c^{3}}-\frac {5 \arcsin \left (a x \right ) \ln \left (1+i \left (i a x +\sqrt {-a^{2} x^{2}+1}\right )\right )}{2 c^{3}}+\frac {5 \arcsin \left (a x \right ) \ln \left (1-i \left (i a x +\sqrt {-a^{2} x^{2}+1}\right )\right )}{2 c^{3}}+\frac {5 i \operatorname {dilog}\left (1+i \left (i a x +\sqrt {-a^{2} x^{2}+1}\right )\right )}{2 c^{3}}-\frac {5 i \operatorname {dilog}\left (1-i \left (i a x +\sqrt {-a^{2} x^{2}+1}\right )\right )}{2 c^{3}}}{a}\)	\(543\)
default	\(\frac {-\frac {3 a^{3} x^{3} \arcsin \left (a x \right )^{3}-9 \arcsin \left (a x \right )^{2} \sqrt {-a^{2} x^{2}+1}\, a^{2} x^{2}-5 a x \arcsin \left (a x \right )^{3}+2 a^{3} x^{3} \arcsin \left (a x \right )+11 \arcsin \left (a x \right )^{2} \sqrt {-a^{2} x^{2}+1}-2 a^{2} x^{2} \sqrt {-a^{2} x^{2}+1}-2 a x \arcsin \left (a x \right )+2 \sqrt {-a^{2} x^{2}+1}}{8 \left (a^{4} x^{4}-2 a^{2} x^{2}+1\right ) c^{3}}-\frac {3 \arcsin \left (a x \right )^{3} \ln \left (1+i \left (i a x +\sqrt {-a^{2} x^{2}+1}\right )\right )}{8 c^{3}}+\frac {9 i \arcsin \left (a x \right )^{2} \operatorname {polylog}\left (2, -i \left (i a x +\sqrt {-a^{2} x^{2}+1}\right )\right )}{8 c^{3}}-\frac {9 \arcsin \left (a x \right ) \operatorname {polylog}\left (3, -i \left (i a x +\sqrt {-a^{2} x^{2}+1}\right )\right )}{4 c^{3}}-\frac {9 i \operatorname {polylog}\left (4, -i \left (i a x +\sqrt {-a^{2} x^{2}+1}\right )\right )}{4 c^{3}}+\frac {3 \arcsin \left (a x \right )^{3} \ln \left (1-i \left (i a x +\sqrt {-a^{2} x^{2}+1}\right )\right )}{8 c^{3}}-\frac {9 i \arcsin \left (a x \right )^{2} \operatorname {polylog}\left (2, i \left (i a x +\sqrt {-a^{2} x^{2}+1}\right )\right )}{8 c^{3}}+\frac {9 \arcsin \left (a x \right ) \operatorname {polylog}\left (3, i \left (i a x +\sqrt {-a^{2} x^{2}+1}\right )\right )}{4 c^{3}}+\frac {9 i \operatorname {polylog}\left (4, i \left (i a x +\sqrt {-a^{2} x^{2}+1}\right )\right )}{4 c^{3}}-\frac {5 \arcsin \left (a x \right ) \ln \left (1+i \left (i a x +\sqrt {-a^{2} x^{2}+1}\right )\right )}{2 c^{3}}+\frac {5 \arcsin \left (a x \right ) \ln \left (1-i \left (i a x +\sqrt {-a^{2} x^{2}+1}\right )\right )}{2 c^{3}}+\frac {5 i \operatorname {dilog}\left (1+i \left (i a x +\sqrt {-a^{2} x^{2}+1}\right )\right )}{2 c^{3}}-\frac {5 i \operatorname {dilog}\left (1-i \left (i a x +\sqrt {-a^{2} x^{2}+1}\right )\right )}{2 c^{3}}}{a}\)	\(543\)

input

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

output

1/a*(-1/8*(3*a^3*x^3*arcsin(a*x)^3-9*arcsin(a*x)^2*(-a^2*x^2+1)^(1/2)*a^2* 
x^2-5*a*x*arcsin(a*x)^3+2*a^3*x^3*arcsin(a*x)+11*arcsin(a*x)^2*(-a^2*x^2+1 
)^(1/2)-2*a^2*x^2*(-a^2*x^2+1)^(1/2)-2*a*x*arcsin(a*x)+2*(-a^2*x^2+1)^(1/2 
))/(a^4*x^4-2*a^2*x^2+1)/c^3-3/8/c^3*arcsin(a*x)^3*ln(1+I*(I*a*x+(-a^2*x^2 
+1)^(1/2)))+9/8*I/c^3*arcsin(a*x)^2*polylog(2,-I*(I*a*x+(-a^2*x^2+1)^(1/2) 
))-9/4/c^3*arcsin(a*x)*polylog(3,-I*(I*a*x+(-a^2*x^2+1)^(1/2)))-9/4*I/c^3* 
polylog(4,-I*(I*a*x+(-a^2*x^2+1)^(1/2)))+3/8/c^3*arcsin(a*x)^3*ln(1-I*(I*a 
*x+(-a^2*x^2+1)^(1/2)))-9/8*I/c^3*arcsin(a*x)^2*polylog(2,I*(I*a*x+(-a^2*x 
^2+1)^(1/2)))+9/4/c^3*arcsin(a*x)*polylog(3,I*(I*a*x+(-a^2*x^2+1)^(1/2)))+ 
9/4*I/c^3*polylog(4,I*(I*a*x+(-a^2*x^2+1)^(1/2)))-5/2/c^3*arcsin(a*x)*ln(1 
+I*(I*a*x+(-a^2*x^2+1)^(1/2)))+5/2/c^3*arcsin(a*x)*ln(1-I*(I*a*x+(-a^2*x^2 
+1)^(1/2)))+5/2*I/c^3*dilog(1+I*(I*a*x+(-a^2*x^2+1)^(1/2)))-5/2*I/c^3*dilo 
g(1-I*(I*a*x+(-a^2*x^2+1)^(1/2))))
 

3.3.94.5 Fricas [F]

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

input

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

output

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

3.3.94.6 Sympy [F]

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

input

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

output

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

3.3.94.7 Maxima [A] (verification not implemented)

Time = 0.66 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.17 \[ \int \frac {\arcsin (a x)^3}{\left (c-a^2 c x^2\right )^3} \, dx=-\frac {1}{16} \, {\left (\frac {2 \, {\left (3 \, a^{2} x^{3} - 5 \, x\right )}}{a^{4} c^{3} x^{4} - 2 \, a^{2} c^{3} x^{2} + c^{3}} - \frac {3 \, \log \left (a x + 1\right )}{a c^{3}} + \frac {3 \, \log \left (a x - 1\right )}{a c^{3}}\right )} \arcsin \left (a x\right )^{3} \]

input

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

output

-1/16*(2*(3*a^2*x^3 - 5*x)/(a^4*c^3*x^4 - 2*a^2*c^3*x^2 + c^3) - 3*log(a*x 
 + 1)/(a*c^3) + 3*log(a*x - 1)/(a*c^3))*arcsin(a*x)^3
 

3.3.94.8 Giac [F]

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

input

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

output

integrate(-arcsin(a*x)^3/(a^2*c*x^2 - c)^3, x)
 

3.3.94.9 Mupad [F(-1)]

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

input

int(asin(a*x)^3/(c - a^2*c*x^2)^3,x)
 

output

int(asin(a*x)^3/(c - a^2*c*x^2)^3, x)