3.4.8 \(\int \frac {\arcsin (a x)^3}{x \sqrt {1-a^2 x^2}} \, dx\) [308]

3.4.8.1 Optimal result

Integrand size = 24, antiderivative size = 138 \[ \int \frac {\arcsin (a x)^3}{x \sqrt {1-a^2 x^2}} \, dx=-2 \arcsin (a x)^3 \text {arctanh}\left (e^{i \arcsin (a x)}\right )+3 i \arcsin (a x)^2 \operatorname {PolyLog}\left (2,-e^{i \arcsin (a x)}\right )-3 i \arcsin (a x)^2 \operatorname {PolyLog}\left (2,e^{i \arcsin (a x)}\right )-6 \arcsin (a x) \operatorname {PolyLog}\left (3,-e^{i \arcsin (a x)}\right )+6 \arcsin (a x) \operatorname {PolyLog}\left (3,e^{i \arcsin (a x)}\right )-6 i \operatorname {PolyLog}\left (4,-e^{i \arcsin (a x)}\right )+6 i \operatorname {PolyLog}\left (4,e^{i \arcsin (a x)}\right ) \]

output

-2*arcsin(a*x)^3*arctanh(I*a*x+(-a^2*x^2+1)^(1/2))+3*I*arcsin(a*x)^2*polyl 
og(2,-I*a*x-(-a^2*x^2+1)^(1/2))-3*I*arcsin(a*x)^2*polylog(2,I*a*x+(-a^2*x^ 
2+1)^(1/2))-6*arcsin(a*x)*polylog(3,-I*a*x-(-a^2*x^2+1)^(1/2))+6*arcsin(a* 
x)*polylog(3,I*a*x+(-a^2*x^2+1)^(1/2))-6*I*polylog(4,-I*a*x-(-a^2*x^2+1)^( 
1/2))+6*I*polylog(4,I*a*x+(-a^2*x^2+1)^(1/2))
 

3.4.8.2 Mathematica [A] (verified)

Time = 0.16 (sec) , antiderivative size = 180, normalized size of antiderivative = 1.30 \[ \int \frac {\arcsin (a x)^3}{x \sqrt {1-a^2 x^2}} \, dx=-\frac {1}{8} i \left (\pi ^4-2 \arcsin (a x)^4+8 i \arcsin (a x)^3 \log \left (1-e^{-i \arcsin (a x)}\right )-8 i \arcsin (a x)^3 \log \left (1+e^{i \arcsin (a x)}\right )-24 \arcsin (a x)^2 \operatorname {PolyLog}\left (2,e^{-i \arcsin (a x)}\right )-24 \arcsin (a x)^2 \operatorname {PolyLog}\left (2,-e^{i \arcsin (a x)}\right )+48 i \arcsin (a x) \operatorname {PolyLog}\left (3,e^{-i \arcsin (a x)}\right )-48 i \arcsin (a x) \operatorname {PolyLog}\left (3,-e^{i \arcsin (a x)}\right )+48 \operatorname {PolyLog}\left (4,e^{-i \arcsin (a x)}\right )+48 \operatorname {PolyLog}\left (4,-e^{i \arcsin (a x)}\right )\right ) \]

input

Integrate[ArcSin[a*x]^3/(x*Sqrt[1 - a^2*x^2]),x]
 

output

(-1/8*I)*(Pi^4 - 2*ArcSin[a*x]^4 + (8*I)*ArcSin[a*x]^3*Log[1 - E^((-I)*Arc 
Sin[a*x])] - (8*I)*ArcSin[a*x]^3*Log[1 + E^(I*ArcSin[a*x])] - 24*ArcSin[a* 
x]^2*PolyLog[2, E^((-I)*ArcSin[a*x])] - 24*ArcSin[a*x]^2*PolyLog[2, -E^(I* 
ArcSin[a*x])] + (48*I)*ArcSin[a*x]*PolyLog[3, E^((-I)*ArcSin[a*x])] - (48* 
I)*ArcSin[a*x]*PolyLog[3, -E^(I*ArcSin[a*x])] + 48*PolyLog[4, E^((-I)*ArcS 
in[a*x])] + 48*PolyLog[4, -E^(I*ArcSin[a*x])])
 

3.4.8.3 Rubi [A] (verified)

Time = 0.59 (sec) , antiderivative size = 150, normalized size of antiderivative = 1.09, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.292, Rules used = {5218, 3042, 4671, 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.

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

\(\Big \downarrow \) 5218

\(\displaystyle \int \frac {\arcsin (a x)^3}{a x}d\arcsin (a x)\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \arcsin (a x)^3 \csc (\arcsin (a x))d\arcsin (a x)\)

\(\Big \downarrow \) 4671

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

\(\Big \downarrow \) 3011

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

\(\Big \downarrow \) 7163

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

\(\Big \downarrow \) 2720

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

\(\Big \downarrow \) 7143

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

input

Int[ArcSin[a*x]^3/(x*Sqrt[1 - a^2*x^2]),x]
 

output

-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*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])]))
 

3.4.8.3.1 Defintions of rubi rules used

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 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 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]
 

3.4.8.4 Maple [A] (verified)

Time = 0.18 (sec) , antiderivative size = 225, normalized size of antiderivative = 1.63

method	result	size
default	\(i \left (i \arcsin \left (a x \right )^{3} \ln \left (1+i a x +\sqrt {-a^{2} x^{2}+1}\right )+3 \arcsin \left (a x \right )^{2} \operatorname {polylog}\left (2, -i a x -\sqrt {-a^{2} x^{2}+1}\right )+6 i \arcsin \left (a x \right ) \operatorname {polylog}\left (3, -i a x -\sqrt {-a^{2} x^{2}+1}\right )-6 \operatorname {polylog}\left (4, -i a x -\sqrt {-a^{2} x^{2}+1}\right )-i \arcsin \left (a x \right )^{3} \ln \left (1-i a x -\sqrt {-a^{2} x^{2}+1}\right )-3 \arcsin \left (a x \right )^{2} \operatorname {polylog}\left (2, i a x +\sqrt {-a^{2} x^{2}+1}\right )-6 i \arcsin \left (a x \right ) \operatorname {polylog}\left (3, i a x +\sqrt {-a^{2} x^{2}+1}\right )+6 \operatorname {polylog}\left (4, i a x +\sqrt {-a^{2} x^{2}+1}\right )\right )\)	\(225\)

input

int(arcsin(a*x)^3/x/(-a^2*x^2+1)^(1/2),x,method=_RETURNVERBOSE)
 

output

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

3.4.8.5 Fricas [F]

\[ \int \frac {\arcsin (a x)^3}{x \sqrt {1-a^2 x^2}} \, dx=\int { \frac {\arcsin \left (a x\right )^{3}}{\sqrt {-a^{2} x^{2} + 1} x} \,d x } \]

input

integrate(arcsin(a*x)^3/x/(-a^2*x^2+1)^(1/2),x, algorithm="fricas")
 

output

integral(-sqrt(-a^2*x^2 + 1)*arcsin(a*x)^3/(a^2*x^3 - x), x)
 

3.4.8.6 Sympy [F]

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

input

integrate(asin(a*x)**3/x/(-a**2*x**2+1)**(1/2),x)
 

output

Integral(asin(a*x)**3/(x*sqrt(-(a*x - 1)*(a*x + 1))), x)
 

3.4.8.7 Maxima [F]

\[ \int \frac {\arcsin (a x)^3}{x \sqrt {1-a^2 x^2}} \, dx=\int { \frac {\arcsin \left (a x\right )^{3}}{\sqrt {-a^{2} x^{2} + 1} x} \,d x } \]

input

integrate(arcsin(a*x)^3/x/(-a^2*x^2+1)^(1/2),x, algorithm="maxima")
 

output

integrate(arcsin(a*x)^3/(sqrt(-a^2*x^2 + 1)*x), x)
 

3.4.8.8 Giac [F]

\[ \int \frac {\arcsin (a x)^3}{x \sqrt {1-a^2 x^2}} \, dx=\int { \frac {\arcsin \left (a x\right )^{3}}{\sqrt {-a^{2} x^{2} + 1} x} \,d x } \]

input

integrate(arcsin(a*x)^3/x/(-a^2*x^2+1)^(1/2),x, algorithm="giac")
 

output

integrate(arcsin(a*x)^3/(sqrt(-a^2*x^2 + 1)*x), x)
 

3.4.8.9 Mupad [F(-1)]

Timed out. \[ \int \frac {\arcsin (a x)^3}{x \sqrt {1-a^2 x^2}} \, dx=\int \frac {{\mathrm {asin}\left (a\,x\right )}^3}{x\,\sqrt {1-a^2\,x^2}} \,d x \]

input

int(asin(a*x)^3/(x*(1 - a^2*x^2)^(1/2)),x)
 

output

int(asin(a*x)^3/(x*(1 - a^2*x^2)^(1/2)), x)