3.3.93 \(\int \frac {\arcsin (a x)^3}{(c-a^2 c x^2)^2} \, dx\) [293]

3.3.93.1 Optimal result

Integrand size = 20, antiderivative size = 337 \[ \int \frac {\arcsin (a x)^3}{\left (c-a^2 c x^2\right )^2} \, dx=-\frac {3 \arcsin (a x)^2}{2 a c^2 \sqrt {1-a^2 x^2}}+\frac {x \arcsin (a x)^3}{2 c^2 \left (1-a^2 x^2\right )}-\frac {6 i \arcsin (a x) \arctan \left (e^{i \arcsin (a x)}\right )}{a c^2}-\frac {i \arcsin (a x)^3 \arctan \left (e^{i \arcsin (a x)}\right )}{a c^2}+\frac {3 i \operatorname {PolyLog}\left (2,-i e^{i \arcsin (a x)}\right )}{a c^2}+\frac {3 i \arcsin (a x)^2 \operatorname {PolyLog}\left (2,-i e^{i \arcsin (a x)}\right )}{2 a c^2}-\frac {3 i \operatorname {PolyLog}\left (2,i e^{i \arcsin (a x)}\right )}{a c^2}-\frac {3 i \arcsin (a x)^2 \operatorname {PolyLog}\left (2,i e^{i \arcsin (a x)}\right )}{2 a c^2}-\frac {3 \arcsin (a x) \operatorname {PolyLog}\left (3,-i e^{i \arcsin (a x)}\right )}{a c^2}+\frac {3 \arcsin (a x) \operatorname {PolyLog}\left (3,i e^{i \arcsin (a x)}\right )}{a c^2}-\frac {3 i \operatorname {PolyLog}\left (4,-i e^{i \arcsin (a x)}\right )}{a c^2}+\frac {3 i \operatorname {PolyLog}\left (4,i e^{i \arcsin (a x)}\right )}{a c^2} \]

output

1/2*x*arcsin(a*x)^3/c^2/(-a^2*x^2+1)-6*I*arcsin(a*x)*arctan(I*a*x+(-a^2*x^ 
2+1)^(1/2))/a/c^2-I*arcsin(a*x)^3*arctan(I*a*x+(-a^2*x^2+1)^(1/2))/a/c^2+3 
*I*polylog(2,-I*(I*a*x+(-a^2*x^2+1)^(1/2)))/a/c^2+3/2*I*arcsin(a*x)^2*poly 
log(2,-I*(I*a*x+(-a^2*x^2+1)^(1/2)))/a/c^2-3*I*polylog(2,I*(I*a*x+(-a^2*x^ 
2+1)^(1/2)))/a/c^2-3/2*I*arcsin(a*x)^2*polylog(2,I*(I*a*x+(-a^2*x^2+1)^(1/ 
2)))/a/c^2-3*arcsin(a*x)*polylog(3,-I*(I*a*x+(-a^2*x^2+1)^(1/2)))/a/c^2+3* 
arcsin(a*x)*polylog(3,I*(I*a*x+(-a^2*x^2+1)^(1/2)))/a/c^2-3*I*polylog(4,-I 
*(I*a*x+(-a^2*x^2+1)^(1/2)))/a/c^2+3*I*polylog(4,I*(I*a*x+(-a^2*x^2+1)^(1/ 
2)))/a/c^2-3/2*arcsin(a*x)^2/a/c^2/(-a^2*x^2+1)^(1/2)
 

3.3.93.2 Mathematica [A] (verified)

Time = 0.40 (sec) , antiderivative size = 234, normalized size of antiderivative = 0.69 \[ \int \frac {\arcsin (a x)^3}{\left (c-a^2 c x^2\right )^2} \, dx=\frac {-\frac {3 \arcsin (a x)^2}{\sqrt {1-a^2 x^2}}+\frac {a x \arcsin (a x)^3}{1-a^2 x^2}-12 i \arcsin (a x) \arctan \left (e^{i \arcsin (a x)}\right )-2 i \arcsin (a x)^3 \arctan \left (e^{i \arcsin (a x)}\right )+3 i \left (2+\arcsin (a x)^2\right ) \operatorname {PolyLog}\left (2,-i e^{i \arcsin (a x)}\right )-3 i \left (2+\arcsin (a x)^2\right ) \operatorname {PolyLog}\left (2,i e^{i \arcsin (a x)}\right )-6 \arcsin (a x) \operatorname {PolyLog}\left (3,-i e^{i \arcsin (a x)}\right )+6 \arcsin (a x) \operatorname {PolyLog}\left (3,i e^{i \arcsin (a x)}\right )-6 i \operatorname {PolyLog}\left (4,-i e^{i \arcsin (a x)}\right )+6 i \operatorname {PolyLog}\left (4,i e^{i \arcsin (a x)}\right )}{2 a c^2} \]

input

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

output

((-3*ArcSin[a*x]^2)/Sqrt[1 - a^2*x^2] + (a*x*ArcSin[a*x]^3)/(1 - a^2*x^2) 
- (12*I)*ArcSin[a*x]*ArcTan[E^(I*ArcSin[a*x])] - (2*I)*ArcSin[a*x]^3*ArcTa 
n[E^(I*ArcSin[a*x])] + (3*I)*(2 + ArcSin[a*x]^2)*PolyLog[2, (-I)*E^(I*ArcS 
in[a*x])] - (3*I)*(2 + ArcSin[a*x]^2)*PolyLog[2, I*E^(I*ArcSin[a*x])] - 6* 
ArcSin[a*x]*PolyLog[3, (-I)*E^(I*ArcSin[a*x])] + 6*ArcSin[a*x]*PolyLog[3, 
I*E^(I*ArcSin[a*x])] - (6*I)*PolyLog[4, (-I)*E^(I*ArcSin[a*x])] + (6*I)*Po 
lyLog[4, I*E^(I*ArcSin[a*x])])/(2*a*c^2)
 

3.3.93.3 Rubi [A] (verified)

Time = 1.63 (sec) , antiderivative size = 305, normalized size of antiderivative = 0.91, number of steps used = 16, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.750, Rules used = {5162, 27, 5164, 3042, 4669, 3011, 5182, 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.

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

\(\Big \downarrow \) 5162

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 5164

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4669

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

\(\Big \downarrow \) 3011

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

\(\Big \downarrow \) 5182

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

\(\Big \downarrow \) 5164

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4669

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

\(\Big \downarrow \) 2715

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

\(\Big \downarrow \) 2838

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

\(\Big \downarrow \) 7163

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

\(\Big \downarrow \) 2720

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

\(\Big \downarrow \) 7143

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

input

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

output

(x*ArcSin[a*x]^3)/(2*c^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 
*c^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*Arc 
Sin[a*x]^2*PolyLog[2, I*E^(I*ArcSin[a*x])] - (2*I)*((-I)*ArcSin[a*x]*PolyL 
og[3, I*E^(I*ArcSin[a*x])] + PolyLog[4, I*E^(I*ArcSin[a*x])])))/(2*a*c^2)
 

3.3.93.3.1 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 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]
 

3.3.93.4 Maple [A] (verified)

Time = 0.23 (sec) , antiderivative size = 438, normalized size of antiderivative = 1.30

method	result	size
derivativedivides	\(\frac {-\frac {\arcsin \left (a x \right )^{2} \left (a x \arcsin \left (a x \right )-3 \sqrt {-a^{2} x^{2}+1}\right )}{2 \left (a^{2} x^{2}-1\right ) c^{2}}-\frac {\arcsin \left (a x \right )^{3} \ln \left (1+i \left (i a x +\sqrt {-a^{2} x^{2}+1}\right )\right )}{2 c^{2}}+\frac {3 i \arcsin \left (a x \right )^{2} \operatorname {polylog}\left (2, -i \left (i a x +\sqrt {-a^{2} x^{2}+1}\right )\right )}{2 c^{2}}-\frac {3 \arcsin \left (a x \right ) \operatorname {polylog}\left (3, -i \left (i a x +\sqrt {-a^{2} x^{2}+1}\right )\right )}{c^{2}}-\frac {3 i \operatorname {polylog}\left (4, -i \left (i a x +\sqrt {-a^{2} x^{2}+1}\right )\right )}{c^{2}}+\frac {\arcsin \left (a x \right )^{3} \ln \left (1-i \left (i a x +\sqrt {-a^{2} x^{2}+1}\right )\right )}{2 c^{2}}-\frac {3 i \arcsin \left (a x \right )^{2} \operatorname {polylog}\left (2, i \left (i a x +\sqrt {-a^{2} x^{2}+1}\right )\right )}{2 c^{2}}+\frac {3 \arcsin \left (a x \right ) \operatorname {polylog}\left (3, i \left (i a x +\sqrt {-a^{2} x^{2}+1}\right )\right )}{c^{2}}+\frac {3 i \operatorname {polylog}\left (4, i \left (i a x +\sqrt {-a^{2} x^{2}+1}\right )\right )}{c^{2}}-\frac {3 \arcsin \left (a x \right ) \ln \left (1+i \left (i a x +\sqrt {-a^{2} x^{2}+1}\right )\right )}{c^{2}}+\frac {3 \arcsin \left (a x \right ) \ln \left (1-i \left (i a x +\sqrt {-a^{2} x^{2}+1}\right )\right )}{c^{2}}+\frac {3 i \operatorname {dilog}\left (1+i \left (i a x +\sqrt {-a^{2} x^{2}+1}\right )\right )}{c^{2}}-\frac {3 i \operatorname {dilog}\left (1-i \left (i a x +\sqrt {-a^{2} x^{2}+1}\right )\right )}{c^{2}}}{a}\)	\(438\)
default	\(\frac {-\frac {\arcsin \left (a x \right )^{2} \left (a x \arcsin \left (a x \right )-3 \sqrt {-a^{2} x^{2}+1}\right )}{2 \left (a^{2} x^{2}-1\right ) c^{2}}-\frac {\arcsin \left (a x \right )^{3} \ln \left (1+i \left (i a x +\sqrt {-a^{2} x^{2}+1}\right )\right )}{2 c^{2}}+\frac {3 i \arcsin \left (a x \right )^{2} \operatorname {polylog}\left (2, -i \left (i a x +\sqrt {-a^{2} x^{2}+1}\right )\right )}{2 c^{2}}-\frac {3 \arcsin \left (a x \right ) \operatorname {polylog}\left (3, -i \left (i a x +\sqrt {-a^{2} x^{2}+1}\right )\right )}{c^{2}}-\frac {3 i \operatorname {polylog}\left (4, -i \left (i a x +\sqrt {-a^{2} x^{2}+1}\right )\right )}{c^{2}}+\frac {\arcsin \left (a x \right )^{3} \ln \left (1-i \left (i a x +\sqrt {-a^{2} x^{2}+1}\right )\right )}{2 c^{2}}-\frac {3 i \arcsin \left (a x \right )^{2} \operatorname {polylog}\left (2, i \left (i a x +\sqrt {-a^{2} x^{2}+1}\right )\right )}{2 c^{2}}+\frac {3 \arcsin \left (a x \right ) \operatorname {polylog}\left (3, i \left (i a x +\sqrt {-a^{2} x^{2}+1}\right )\right )}{c^{2}}+\frac {3 i \operatorname {polylog}\left (4, i \left (i a x +\sqrt {-a^{2} x^{2}+1}\right )\right )}{c^{2}}-\frac {3 \arcsin \left (a x \right ) \ln \left (1+i \left (i a x +\sqrt {-a^{2} x^{2}+1}\right )\right )}{c^{2}}+\frac {3 \arcsin \left (a x \right ) \ln \left (1-i \left (i a x +\sqrt {-a^{2} x^{2}+1}\right )\right )}{c^{2}}+\frac {3 i \operatorname {dilog}\left (1+i \left (i a x +\sqrt {-a^{2} x^{2}+1}\right )\right )}{c^{2}}-\frac {3 i \operatorname {dilog}\left (1-i \left (i a x +\sqrt {-a^{2} x^{2}+1}\right )\right )}{c^{2}}}{a}\)	\(438\)

input

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

output

1/a*(-1/2/(a^2*x^2-1)*arcsin(a*x)^2*(a*x*arcsin(a*x)-3*(-a^2*x^2+1)^(1/2)) 
/c^2-1/2/c^2*arcsin(a*x)^3*ln(1+I*(I*a*x+(-a^2*x^2+1)^(1/2)))+3/2*I/c^2*ar 
csin(a*x)^2*polylog(2,-I*(I*a*x+(-a^2*x^2+1)^(1/2)))-3/c^2*arcsin(a*x)*pol 
ylog(3,-I*(I*a*x+(-a^2*x^2+1)^(1/2)))-3*I/c^2*polylog(4,-I*(I*a*x+(-a^2*x^ 
2+1)^(1/2)))+1/2/c^2*arcsin(a*x)^3*ln(1-I*(I*a*x+(-a^2*x^2+1)^(1/2)))-3/2* 
I/c^2*arcsin(a*x)^2*polylog(2,I*(I*a*x+(-a^2*x^2+1)^(1/2)))+3/c^2*arcsin(a 
*x)*polylog(3,I*(I*a*x+(-a^2*x^2+1)^(1/2)))+3*I/c^2*polylog(4,I*(I*a*x+(-a 
^2*x^2+1)^(1/2)))-3/c^2*arcsin(a*x)*ln(1+I*(I*a*x+(-a^2*x^2+1)^(1/2)))+3/c 
^2*arcsin(a*x)*ln(1-I*(I*a*x+(-a^2*x^2+1)^(1/2)))+3*I/c^2*dilog(1+I*(I*a*x 
+(-a^2*x^2+1)^(1/2)))-3*I/c^2*dilog(1-I*(I*a*x+(-a^2*x^2+1)^(1/2))))
 

3.3.93.5 Fricas [F]

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

input

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

output

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

3.3.93.6 Sympy [F]

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

input

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

output

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

3.3.93.7 Maxima [A] (verification not implemented)

Time = 0.55 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.17 \[ \int \frac {\arcsin (a x)^3}{\left (c-a^2 c x^2\right )^2} \, dx=-\frac {1}{4} \, {\left (\frac {2 \, x}{a^{2} c^{2} x^{2} - c^{2}} - \frac {\log \left (a x + 1\right )}{a c^{2}} + \frac {\log \left (a x - 1\right )}{a c^{2}}\right )} \arcsin \left (a x\right )^{3} \]

input

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

output

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

3.3.93.8 Giac [F]

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

input

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

output

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

3.3.93.9 Mupad [F(-1)]

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

input

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

output

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