3.1.41 \(\int \frac {\arcsin (a x)^4}{x^4} \, dx\) [41]

3.1.41.1 Optimal result

Integrand size = 10, antiderivative size = 276 \[ \int \frac {\arcsin (a x)^4}{x^4} \, dx=-\frac {2 a^2 \arcsin (a x)^2}{x}-\frac {2 a \sqrt {1-a^2 x^2} \arcsin (a x)^3}{3 x^2}-\frac {\arcsin (a x)^4}{3 x^3}-8 a^3 \arcsin (a x) \text {arctanh}\left (e^{i \arcsin (a x)}\right )-\frac {4}{3} a^3 \arcsin (a x)^3 \text {arctanh}\left (e^{i \arcsin (a x)}\right )+4 i a^3 \operatorname {PolyLog}\left (2,-e^{i \arcsin (a x)}\right )+2 i a^3 \arcsin (a x)^2 \operatorname {PolyLog}\left (2,-e^{i \arcsin (a x)}\right )-4 i a^3 \operatorname {PolyLog}\left (2,e^{i \arcsin (a x)}\right )-2 i a^3 \arcsin (a x)^2 \operatorname {PolyLog}\left (2,e^{i \arcsin (a x)}\right )-4 a^3 \arcsin (a x) \operatorname {PolyLog}\left (3,-e^{i \arcsin (a x)}\right )+4 a^3 \arcsin (a x) \operatorname {PolyLog}\left (3,e^{i \arcsin (a x)}\right )-4 i a^3 \operatorname {PolyLog}\left (4,-e^{i \arcsin (a x)}\right )+4 i a^3 \operatorname {PolyLog}\left (4,e^{i \arcsin (a x)}\right ) \]

output

-2*a^2*arcsin(a*x)^2/x-1/3*arcsin(a*x)^4/x^3-8*a^3*arcsin(a*x)*arctanh(I*a 
*x+(-a^2*x^2+1)^(1/2))-4/3*a^3*arcsin(a*x)^3*arctanh(I*a*x+(-a^2*x^2+1)^(1 
/2))+4*I*a^3*polylog(2,-I*a*x-(-a^2*x^2+1)^(1/2))+2*I*a^3*arcsin(a*x)^2*po 
lylog(2,-I*a*x-(-a^2*x^2+1)^(1/2))-4*I*a^3*polylog(2,I*a*x+(-a^2*x^2+1)^(1 
/2))-2*I*a^3*arcsin(a*x)^2*polylog(2,I*a*x+(-a^2*x^2+1)^(1/2))-4*a^3*arcsi 
n(a*x)*polylog(3,-I*a*x-(-a^2*x^2+1)^(1/2))+4*a^3*arcsin(a*x)*polylog(3,I* 
a*x+(-a^2*x^2+1)^(1/2))-4*I*a^3*polylog(4,-I*a*x-(-a^2*x^2+1)^(1/2))+4*I*a 
^3*polylog(4,I*a*x+(-a^2*x^2+1)^(1/2))-2/3*a*arcsin(a*x)^3*(-a^2*x^2+1)^(1 
/2)/x^2
 

3.1.41.2 Mathematica [A] (verified)

Time = 3.42 (sec) , antiderivative size = 399, normalized size of antiderivative = 1.45 \[ \int \frac {\arcsin (a x)^4}{x^4} \, dx=\frac {1}{24} a^3 \left (-2 i \pi ^4+4 i \arcsin (a x)^4-24 \arcsin (a x)^2 \cot \left (\frac {1}{2} \arcsin (a x)\right )-2 \arcsin (a x)^4 \cot \left (\frac {1}{2} \arcsin (a x)\right )-4 \arcsin (a x)^3 \csc ^2\left (\frac {1}{2} \arcsin (a x)\right )-\frac {1}{2} a x \arcsin (a x)^4 \csc ^4\left (\frac {1}{2} \arcsin (a x)\right )+16 \arcsin (a x)^3 \log \left (1-e^{-i \arcsin (a x)}\right )+96 \arcsin (a x) \log \left (1-e^{i \arcsin (a x)}\right )-96 \arcsin (a x) \log \left (1+e^{i \arcsin (a x)}\right )-16 \arcsin (a x)^3 \log \left (1+e^{i \arcsin (a x)}\right )+48 i \arcsin (a x)^2 \operatorname {PolyLog}\left (2,e^{-i \arcsin (a x)}\right )+48 i \left (2+\arcsin (a x)^2\right ) \operatorname {PolyLog}\left (2,-e^{i \arcsin (a x)}\right )-96 i \operatorname {PolyLog}\left (2,e^{i \arcsin (a x)}\right )+96 \arcsin (a x) \operatorname {PolyLog}\left (3,e^{-i \arcsin (a x)}\right )-96 \arcsin (a x) \operatorname {PolyLog}\left (3,-e^{i \arcsin (a x)}\right )-96 i \operatorname {PolyLog}\left (4,e^{-i \arcsin (a x)}\right )-96 i \operatorname {PolyLog}\left (4,-e^{i \arcsin (a x)}\right )+4 \arcsin (a x)^3 \sec ^2\left (\frac {1}{2} \arcsin (a x)\right )-\frac {8 \arcsin (a x)^4 \sin ^4\left (\frac {1}{2} \arcsin (a x)\right )}{a^3 x^3}-24 \arcsin (a x)^2 \tan \left (\frac {1}{2} \arcsin (a x)\right )-2 \arcsin (a x)^4 \tan \left (\frac {1}{2} \arcsin (a x)\right )\right ) \]

input

Integrate[ArcSin[a*x]^4/x^4,x]
 

output

(a^3*((-2*I)*Pi^4 + (4*I)*ArcSin[a*x]^4 - 24*ArcSin[a*x]^2*Cot[ArcSin[a*x] 
/2] - 2*ArcSin[a*x]^4*Cot[ArcSin[a*x]/2] - 4*ArcSin[a*x]^3*Csc[ArcSin[a*x] 
/2]^2 - (a*x*ArcSin[a*x]^4*Csc[ArcSin[a*x]/2]^4)/2 + 16*ArcSin[a*x]^3*Log[ 
1 - E^((-I)*ArcSin[a*x])] + 96*ArcSin[a*x]*Log[1 - E^(I*ArcSin[a*x])] - 96 
*ArcSin[a*x]*Log[1 + E^(I*ArcSin[a*x])] - 16*ArcSin[a*x]^3*Log[1 + E^(I*Ar 
cSin[a*x])] + (48*I)*ArcSin[a*x]^2*PolyLog[2, E^((-I)*ArcSin[a*x])] + (48* 
I)*(2 + ArcSin[a*x]^2)*PolyLog[2, -E^(I*ArcSin[a*x])] - (96*I)*PolyLog[2, 
E^(I*ArcSin[a*x])] + 96*ArcSin[a*x]*PolyLog[3, E^((-I)*ArcSin[a*x])] - 96* 
ArcSin[a*x]*PolyLog[3, -E^(I*ArcSin[a*x])] - (96*I)*PolyLog[4, E^((-I)*Arc 
Sin[a*x])] - (96*I)*PolyLog[4, -E^(I*ArcSin[a*x])] + 4*ArcSin[a*x]^3*Sec[A 
rcSin[a*x]/2]^2 - (8*ArcSin[a*x]^4*Sin[ArcSin[a*x]/2]^4)/(a^3*x^3) - 24*Ar 
cSin[a*x]^2*Tan[ArcSin[a*x]/2] - 2*ArcSin[a*x]^4*Tan[ArcSin[a*x]/2]))/24
 

3.1.41.3 Rubi [A] (verified)

Time = 1.32 (sec) , antiderivative size = 276, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.200, Rules used = {5138, 5204, 5138, 5218, 3042, 4671, 2715, 2838, 3011, 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.

\(\displaystyle \int \frac {\arcsin (a x)^4}{x^4} \, dx\)

\(\Big \downarrow \) 5138

\(\displaystyle \frac {4}{3} a \int \frac {\arcsin (a x)^3}{x^3 \sqrt {1-a^2 x^2}}dx-\frac {\arcsin (a x)^4}{3 x^3}\)

\(\Big \downarrow \) 5204

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

\(\Big \downarrow \) 5138

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

\(\Big \downarrow \) 5218

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4671

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

\(\Big \downarrow \) 2715

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

\(\Big \downarrow \) 2838

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

\(\Big \downarrow \) 3011

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

\(\Big \downarrow \) 7163

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

\(\Big \downarrow \) 2720

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

\(\Big \downarrow \) 7143

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

input

Int[ArcSin[a*x]^4/x^4,x]
 

output

-1/3*ArcSin[a*x]^4/x^3 + (4*a*(-1/2*(Sqrt[1 - a^2*x^2]*ArcSin[a*x]^3)/x^2 
+ (3*a*(-(ArcSin[a*x]^2/x) + 2*a*(-2*ArcSin[a*x]*ArcTanh[E^(I*ArcSin[a*x]) 
] + I*PolyLog[2, -E^(I*ArcSin[a*x])] - I*PolyLog[2, E^(I*ArcSin[a*x])])))/ 
2 + (a^2*(-2*ArcSin[a*x]^3*ArcTanh[E^(I*ArcSin[a*x])] + 3*(I*ArcSin[a*x]^2 
*PolyLog[2, -E^(I*ArcSin[a*x])] - (2*I)*((-I)*ArcSin[a*x]*PolyLog[3, -E^(I 
*ArcSin[a*x])] + PolyLog[4, -E^(I*ArcSin[a*x])])) - 3*(I*ArcSin[a*x]^2*Pol 
yLog[2, E^(I*ArcSin[a*x])] - (2*I)*((-I)*ArcSin[a*x]*PolyLog[3, E^(I*ArcSi 
n[a*x])] + PolyLog[4, E^(I*ArcSin[a*x])]))))/2))/3
 

3.1.41.3.1 Defintions of rubi rules used

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 4671

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

rule 5138

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((d_.)*(x_))^(m_.), x_Symbol] 
:> Simp[(d*x)^(m + 1)*((a + b*ArcSin[c*x])^n/(d*(m + 1))), x] - Simp[b*c*(n 
/(d*(m + 1)))   Int[(d*x)^(m + 1)*((a + b*ArcSin[c*x])^(n - 1)/Sqrt[1 - c^2 
*x^2]), x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] && NeQ[m, -1]
 

rule 5204

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*((d_) + (e_. 
)*(x_)^2)^(p_), x_Symbol] :> Simp[(f*x)^(m + 1)*(d + e*x^2)^(p + 1)*((a + b 
*ArcSin[c*x])^n/(d*f*(m + 1))), x] + (Simp[c^2*((m + 2*p + 3)/(f^2*(m + 1)) 
)   Int[(f*x)^(m + 2)*(d + e*x^2)^p*(a + b*ArcSin[c*x])^n, x], x] - Simp[b* 
c*(n/(f*(m + 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] && ILtQ[m, -1]
 

rule 5218

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

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]
 

3.1.41.4 Maple [A] (verified)

Time = 0.10 (sec) , antiderivative size = 377, normalized size of antiderivative = 1.37

method	result	size
derivativedivides	\(a^{3} \left (-\frac {\arcsin \left (a x \right )^{2} \left (2 \arcsin \left (a x \right ) \sqrt {-a^{2} x^{2}+1}\, a x +\arcsin \left (a x \right )^{2}+6 a^{2} x^{2}\right )}{3 a^{3} x^{3}}+\frac {2 \arcsin \left (a x \right )^{3} \ln \left (1-i a x -\sqrt {-a^{2} x^{2}+1}\right )}{3}-2 i \arcsin \left (a x \right )^{2} \operatorname {polylog}\left (2, i a x +\sqrt {-a^{2} x^{2}+1}\right )+4 \arcsin \left (a x \right ) \operatorname {polylog}\left (3, i a x +\sqrt {-a^{2} x^{2}+1}\right )+4 i \operatorname {polylog}\left (4, i a x +\sqrt {-a^{2} x^{2}+1}\right )-\frac {2 \arcsin \left (a x \right )^{3} \ln \left (1+i a x +\sqrt {-a^{2} x^{2}+1}\right )}{3}+2 i \arcsin \left (a x \right )^{2} \operatorname {polylog}\left (2, -i a x -\sqrt {-a^{2} x^{2}+1}\right )-4 \arcsin \left (a x \right ) \operatorname {polylog}\left (3, -i a x -\sqrt {-a^{2} x^{2}+1}\right )-4 i \operatorname {polylog}\left (4, -i a x -\sqrt {-a^{2} x^{2}+1}\right )+4 \arcsin \left (a x \right ) \ln \left (1-i a x -\sqrt {-a^{2} x^{2}+1}\right )-4 i \operatorname {polylog}\left (2, i a x +\sqrt {-a^{2} x^{2}+1}\right )-4 \arcsin \left (a x \right ) \ln \left (1+i a x +\sqrt {-a^{2} x^{2}+1}\right )+4 i \operatorname {polylog}\left (2, -i a x -\sqrt {-a^{2} x^{2}+1}\right )\right )\)	\(377\)
default	\(a^{3} \left (-\frac {\arcsin \left (a x \right )^{2} \left (2 \arcsin \left (a x \right ) \sqrt {-a^{2} x^{2}+1}\, a x +\arcsin \left (a x \right )^{2}+6 a^{2} x^{2}\right )}{3 a^{3} x^{3}}+\frac {2 \arcsin \left (a x \right )^{3} \ln \left (1-i a x -\sqrt {-a^{2} x^{2}+1}\right )}{3}-2 i \arcsin \left (a x \right )^{2} \operatorname {polylog}\left (2, i a x +\sqrt {-a^{2} x^{2}+1}\right )+4 \arcsin \left (a x \right ) \operatorname {polylog}\left (3, i a x +\sqrt {-a^{2} x^{2}+1}\right )+4 i \operatorname {polylog}\left (4, i a x +\sqrt {-a^{2} x^{2}+1}\right )-\frac {2 \arcsin \left (a x \right )^{3} \ln \left (1+i a x +\sqrt {-a^{2} x^{2}+1}\right )}{3}+2 i \arcsin \left (a x \right )^{2} \operatorname {polylog}\left (2, -i a x -\sqrt {-a^{2} x^{2}+1}\right )-4 \arcsin \left (a x \right ) \operatorname {polylog}\left (3, -i a x -\sqrt {-a^{2} x^{2}+1}\right )-4 i \operatorname {polylog}\left (4, -i a x -\sqrt {-a^{2} x^{2}+1}\right )+4 \arcsin \left (a x \right ) \ln \left (1-i a x -\sqrt {-a^{2} x^{2}+1}\right )-4 i \operatorname {polylog}\left (2, i a x +\sqrt {-a^{2} x^{2}+1}\right )-4 \arcsin \left (a x \right ) \ln \left (1+i a x +\sqrt {-a^{2} x^{2}+1}\right )+4 i \operatorname {polylog}\left (2, -i a x -\sqrt {-a^{2} x^{2}+1}\right )\right )\)	\(377\)

input

int(arcsin(a*x)^4/x^4,x,method=_RETURNVERBOSE)
 

output

a^3*(-1/3/a^3/x^3*arcsin(a*x)^2*(2*arcsin(a*x)*(-a^2*x^2+1)^(1/2)*a*x+arcs 
in(a*x)^2+6*a^2*x^2)+2/3*arcsin(a*x)^3*ln(1-I*a*x-(-a^2*x^2+1)^(1/2))-2*I* 
arcsin(a*x)^2*polylog(2,I*a*x+(-a^2*x^2+1)^(1/2))+4*arcsin(a*x)*polylog(3, 
I*a*x+(-a^2*x^2+1)^(1/2))+4*I*polylog(4,I*a*x+(-a^2*x^2+1)^(1/2))-2/3*arcs 
in(a*x)^3*ln(1+I*a*x+(-a^2*x^2+1)^(1/2))+2*I*arcsin(a*x)^2*polylog(2,-I*a* 
x-(-a^2*x^2+1)^(1/2))-4*arcsin(a*x)*polylog(3,-I*a*x-(-a^2*x^2+1)^(1/2))-4 
*I*polylog(4,-I*a*x-(-a^2*x^2+1)^(1/2))+4*arcsin(a*x)*ln(1-I*a*x-(-a^2*x^2 
+1)^(1/2))-4*I*polylog(2,I*a*x+(-a^2*x^2+1)^(1/2))-4*arcsin(a*x)*ln(1+I*a* 
x+(-a^2*x^2+1)^(1/2))+4*I*polylog(2,-I*a*x-(-a^2*x^2+1)^(1/2)))
 

3.1.41.5 Fricas [F]

\[ \int \frac {\arcsin (a x)^4}{x^4} \, dx=\int { \frac {\arcsin \left (a x\right )^{4}}{x^{4}} \,d x } \]

input

integrate(arcsin(a*x)^4/x^4,x, algorithm="fricas")
 

output

integral(arcsin(a*x)^4/x^4, x)
 

3.1.41.6 Sympy [F]

\[ \int \frac {\arcsin (a x)^4}{x^4} \, dx=\int \frac {\operatorname {asin}^{4}{\left (a x \right )}}{x^{4}}\, dx \]

input

integrate(asin(a*x)**4/x**4,x)
 

output

Integral(asin(a*x)**4/x**4, x)
 

3.1.41.7 Maxima [F]

\[ \int \frac {\arcsin (a x)^4}{x^4} \, dx=\int { \frac {\arcsin \left (a x\right )^{4}}{x^{4}} \,d x } \]

input

integrate(arcsin(a*x)^4/x^4,x, algorithm="maxima")
 

output

-1/3*(12*a*x^3*integrate(1/3*sqrt(a*x + 1)*sqrt(-a*x + 1)*arctan2(a*x, sqr 
t(a*x + 1)*sqrt(-a*x + 1))^3/(a^2*x^5 - x^3), x) + arctan2(a*x, sqrt(a*x + 
 1)*sqrt(-a*x + 1))^4)/x^3
 

3.1.41.8 Giac [F]

\[ \int \frac {\arcsin (a x)^4}{x^4} \, dx=\int { \frac {\arcsin \left (a x\right )^{4}}{x^{4}} \,d x } \]

input

integrate(arcsin(a*x)^4/x^4,x, algorithm="giac")
 

output

integrate(arcsin(a*x)^4/x^4, x)
 

3.1.41.9 Mupad [F(-1)]

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

input

int(asin(a*x)^4/x^4,x)
 

output

int(asin(a*x)^4/x^4, x)