\(\int \frac {\arccos (a x)^3}{x^3 \sqrt {1-a^2 x^2}} \, dx\) [312]

Optimal result

Integrand size = 24, antiderivative size = 264 \[ \int \frac {\arccos (a x)^3}{x^3 \sqrt {1-a^2 x^2}} \, dx=-\frac {3 a \arccos (a x)^2}{2 x}-\frac {\sqrt {1-a^2 x^2} \arccos (a x)^3}{2 x^2}-6 a^2 \arccos (a x) \text {arctanh}\left (e^{i \arccos (a x)}\right )-a^2 \arccos (a x)^3 \text {arctanh}\left (e^{i \arccos (a x)}\right )+3 i a^2 \operatorname {PolyLog}\left (2,-e^{i \arccos (a x)}\right )+\frac {3}{2} i a^2 \arccos (a x)^2 \operatorname {PolyLog}\left (2,-e^{i \arccos (a x)}\right )-3 i a^2 \operatorname {PolyLog}\left (2,e^{i \arccos (a x)}\right )-\frac {3}{2} i a^2 \arccos (a x)^2 \operatorname {PolyLog}\left (2,e^{i \arccos (a x)}\right )-3 a^2 \arccos (a x) \operatorname {PolyLog}\left (3,-e^{i \arccos (a x)}\right )+3 a^2 \arccos (a x) \operatorname {PolyLog}\left (3,e^{i \arccos (a x)}\right )-3 i a^2 \operatorname {PolyLog}\left (4,-e^{i \arccos (a x)}\right )+3 i a^2 \operatorname {PolyLog}\left (4,e^{i \arccos (a x)}\right ) \] Output:

-3/2*a*arccos(a*x)^2/x-1/2*(-a^2*x^2+1)^(1/2)*arccos(a*x)^3/x^2-6*a^2*arcc 
os(a*x)*arctanh(a*x+I*(-a^2*x^2+1)^(1/2))-a^2*arccos(a*x)^3*arctanh(a*x+I* 
(-a^2*x^2+1)^(1/2))+3*I*a^2*polylog(2,-a*x-I*(-a^2*x^2+1)^(1/2))+3/2*I*a^2 
*arccos(a*x)^2*polylog(2,-a*x-I*(-a^2*x^2+1)^(1/2))-3*I*a^2*polylog(2,a*x+ 
I*(-a^2*x^2+1)^(1/2))-3/2*I*a^2*arccos(a*x)^2*polylog(2,a*x+I*(-a^2*x^2+1) 
^(1/2))-3*a^2*arccos(a*x)*polylog(3,-a*x-I*(-a^2*x^2+1)^(1/2))+3*a^2*arcco 
s(a*x)*polylog(3,a*x+I*(-a^2*x^2+1)^(1/2))-3*I*a^2*polylog(4,-a*x-I*(-a^2* 
x^2+1)^(1/2))+3*I*a^2*polylog(4,a*x+I*(-a^2*x^2+1)^(1/2))
 

Mathematica [A] (verified)

Time = 0.55 (sec) , antiderivative size = 246, normalized size of antiderivative = 0.93 \[ \int \frac {\arccos (a x)^3}{x^3 \sqrt {1-a^2 x^2}} \, dx=\frac {1}{2} \left (\frac {3 a \arccos (a x)^2}{x}-\frac {\sqrt {1-a^2 x^2} \arccos (a x)^3}{x^2}+12 i a^2 \arccos (a x) \arctan \left (e^{i \arccos (a x)}\right )+2 i a^2 \arccos (a x)^3 \arctan \left (e^{i \arccos (a x)}\right )-3 i a^2 \left (2+\arccos (a x)^2\right ) \operatorname {PolyLog}\left (2,-i e^{i \arccos (a x)}\right )+3 i a^2 \left (2+\arccos (a x)^2\right ) \operatorname {PolyLog}\left (2,i e^{i \arccos (a x)}\right )+6 a^2 \arccos (a x) \operatorname {PolyLog}\left (3,-i e^{i \arccos (a x)}\right )-6 a^2 \arccos (a x) \operatorname {PolyLog}\left (3,i e^{i \arccos (a x)}\right )+6 i a^2 \operatorname {PolyLog}\left (4,-i e^{i \arccos (a x)}\right )-6 i a^2 \operatorname {PolyLog}\left (4,i e^{i \arccos (a x)}\right )\right ) \] Input:

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

Output:

((3*a*ArcCos[a*x]^2)/x - (Sqrt[1 - a^2*x^2]*ArcCos[a*x]^3)/x^2 + (12*I)*a^ 
2*ArcCos[a*x]*ArcTan[E^(I*ArcCos[a*x])] + (2*I)*a^2*ArcCos[a*x]^3*ArcTan[E 
^(I*ArcCos[a*x])] - (3*I)*a^2*(2 + ArcCos[a*x]^2)*PolyLog[2, (-I)*E^(I*Arc 
Cos[a*x])] + (3*I)*a^2*(2 + ArcCos[a*x]^2)*PolyLog[2, I*E^(I*ArcCos[a*x])] 
 + 6*a^2*ArcCos[a*x]*PolyLog[3, (-I)*E^(I*ArcCos[a*x])] - 6*a^2*ArcCos[a*x 
]*PolyLog[3, I*E^(I*ArcCos[a*x])] + (6*I)*a^2*PolyLog[4, (-I)*E^(I*ArcCos[ 
a*x])] - (6*I)*a^2*PolyLog[4, I*E^(I*ArcCos[a*x])])/2
 

Rubi [A] (verified)

Time = 1.41 (sec) , antiderivative size = 285, normalized size of antiderivative = 1.08, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.458, Rules used = {5205, 5139, 5219, 3042, 4669, 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.

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

\(\Big \downarrow \) 5205

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

\(\Big \downarrow \) 5139

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

\(\Big \downarrow \) 5219

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4669

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

\(\Big \downarrow \) 2715

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

\(\Big \downarrow \) 2838

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

\(\Big \downarrow \) 3011

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

\(\Big \downarrow \) 7163

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

\(\Big \downarrow \) 2720

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

\(\Big \downarrow \) 7143

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

Input:

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

Output:

-1/2*(Sqrt[1 - a^2*x^2]*ArcCos[a*x]^3)/x^2 - (3*a*(-(ArcCos[a*x]^2/x) + 2* 
a*((-2*I)*ArcCos[a*x]*ArcTan[E^(I*ArcCos[a*x])] + I*PolyLog[2, (-I)*E^(I*A 
rcCos[a*x])] - I*PolyLog[2, I*E^(I*ArcCos[a*x])])))/2 - (a^2*((-2*I)*ArcCo 
s[a*x]^3*ArcTan[E^(I*ArcCos[a*x])] + 3*(I*ArcCos[a*x]^2*PolyLog[2, (-I)*E^ 
(I*ArcCos[a*x])] - (2*I)*((-I)*ArcCos[a*x]*PolyLog[3, (-I)*E^(I*ArcCos[a*x 
])] + PolyLog[4, (-I)*E^(I*ArcCos[a*x])])) - 3*(I*ArcCos[a*x]^2*PolyLog[2, 
 I*E^(I*ArcCos[a*x])] - (2*I)*((-I)*ArcCos[a*x]*PolyLog[3, I*E^(I*ArcCos[a 
*x])] + PolyLog[4, I*E^(I*ArcCos[a*x])]))))/2
 

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 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 5139

Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_.)*((d_.)*(x_))^(m_.), x_Symbol] 
:> Simp[(d*x)^(m + 1)*((a + b*ArcCos[c*x])^n/(d*(m + 1))), x] + Simp[b*c*(n 
/(d*(m + 1)))   Int[(d*x)^(m + 1)*((a + b*ArcCos[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 5205

Int[((a_.) + ArcCos[(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 
*ArcCos[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*ArcCos[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*ArcCos[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 5219

Int[(((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_.)*(x_)^(m_))/Sqrt[(d_) + (e_.)* 
(x_)^2], x_Symbol] :> Simp[(-(c^(m + 1))^(-1))*Simp[Sqrt[1 - c^2*x^2]/Sqrt[ 
d + e*x^2]]   Subst[Int[(a + b*x)^n*Cos[x]^m, x], x, ArcCos[c*x]], x] /; Fr 
eeQ[{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]
 

Maple [A] (verified)

Time = 0.50 (sec) , antiderivative size = 439, normalized size of antiderivative = 1.66

method	result	size
default	\(-\frac {\sqrt {-a^{2} x^{2}+1}\, \arccos \left (a x \right )^{2} \left (a^{2} x^{2} \arccos \left (a x \right )+3 \sqrt {-a^{2} x^{2}+1}\, a x -\arccos \left (a x \right )\right )}{2 x^{2} \left (a^{2} x^{2}-1\right )}-\frac {a^{2} \left (\arccos \left (a x \right )^{3} \ln \left (1-i \left (a x +i \sqrt {-a^{2} x^{2}+1}\right )\right )-\arccos \left (a x \right )^{3} \ln \left (1+i \left (a x +i \sqrt {-a^{2} x^{2}+1}\right )\right )-3 i \arccos \left (a x \right )^{2} \operatorname {polylog}\left (2, i \left (a x +i \sqrt {-a^{2} x^{2}+1}\right )\right )+3 i \arccos \left (a x \right )^{2} \operatorname {polylog}\left (2, -i \left (a x +i \sqrt {-a^{2} x^{2}+1}\right )\right )+6 \arccos \left (a x \right ) \ln \left (1-i \left (a x +i \sqrt {-a^{2} x^{2}+1}\right )\right )+6 \arccos \left (a x \right ) \operatorname {polylog}\left (3, i \left (a x +i \sqrt {-a^{2} x^{2}+1}\right )\right )-6 \arccos \left (a x \right ) \ln \left (1+i \left (a x +i \sqrt {-a^{2} x^{2}+1}\right )\right )-6 \arccos \left (a x \right ) \operatorname {polylog}\left (3, -i \left (a x +i \sqrt {-a^{2} x^{2}+1}\right )\right )+6 i \operatorname {polylog}\left (4, i \left (a x +i \sqrt {-a^{2} x^{2}+1}\right )\right )+6 i \operatorname {dilog}\left (1+i \left (a x +i \sqrt {-a^{2} x^{2}+1}\right )\right )-6 i \operatorname {dilog}\left (1-i \left (a x +i \sqrt {-a^{2} x^{2}+1}\right )\right )-6 i \operatorname {polylog}\left (4, -i \left (a x +i \sqrt {-a^{2} x^{2}+1}\right )\right )\right )}{2}\)	\(439\)

Input:

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

Output:

-1/2*(-a^2*x^2+1)^(1/2)/x^2/(a^2*x^2-1)*arccos(a*x)^2*(a^2*x^2*arccos(a*x) 
+3*(-a^2*x^2+1)^(1/2)*a*x-arccos(a*x))-1/2*a^2*(arccos(a*x)^3*ln(1-I*(a*x+ 
I*(-a^2*x^2+1)^(1/2)))-arccos(a*x)^3*ln(1+I*(a*x+I*(-a^2*x^2+1)^(1/2)))-3* 
I*arccos(a*x)^2*polylog(2,I*(a*x+I*(-a^2*x^2+1)^(1/2)))+3*I*arccos(a*x)^2* 
polylog(2,-I*(a*x+I*(-a^2*x^2+1)^(1/2)))+6*arccos(a*x)*ln(1-I*(a*x+I*(-a^2 
*x^2+1)^(1/2)))+6*arccos(a*x)*polylog(3,I*(a*x+I*(-a^2*x^2+1)^(1/2)))-6*ar 
ccos(a*x)*ln(1+I*(a*x+I*(-a^2*x^2+1)^(1/2)))-6*arccos(a*x)*polylog(3,-I*(a 
*x+I*(-a^2*x^2+1)^(1/2)))+6*I*polylog(4,I*(a*x+I*(-a^2*x^2+1)^(1/2)))+6*I* 
dilog(1+I*(a*x+I*(-a^2*x^2+1)^(1/2)))-6*I*dilog(1-I*(a*x+I*(-a^2*x^2+1)^(1 
/2)))-6*I*polylog(4,-I*(a*x+I*(-a^2*x^2+1)^(1/2))))
 

Fricas [F]

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

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

Output:

integral(-sqrt(-a^2*x^2 + 1)*arccos(a*x)^3/(a^2*x^5 - x^3), x)
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

int(acos(a*x)**3/(sqrt( - a**2*x**2 + 1)*x**3),x)