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

3.1.41.1 Optimal result

Integrand size = 10, antiderivative size = 304 \[ \int \frac {\arccos (a x)^4}{x^4} \, dx=-\frac {2 a^2 \arccos (a x)^2}{x}+\frac {2 a \sqrt {1-a^2 x^2} \arccos (a x)^3}{3 x^2}-\frac {\arccos (a x)^4}{3 x^3}-8 i a^3 \arccos (a x) \arctan \left (e^{i \arccos (a x)}\right )-\frac {4}{3} i a^3 \arccos (a x)^3 \arctan \left (e^{i \arccos (a x)}\right )+4 i a^3 \operatorname {PolyLog}\left (2,-i e^{i \arccos (a x)}\right )+2 i a^3 \arccos (a x)^2 \operatorname {PolyLog}\left (2,-i e^{i \arccos (a x)}\right )-4 i a^3 \operatorname {PolyLog}\left (2,i e^{i \arccos (a x)}\right )-2 i a^3 \arccos (a x)^2 \operatorname {PolyLog}\left (2,i e^{i \arccos (a x)}\right )-4 a^3 \arccos (a x) \operatorname {PolyLog}\left (3,-i e^{i \arccos (a x)}\right )+4 a^3 \arccos (a x) \operatorname {PolyLog}\left (3,i e^{i \arccos (a x)}\right )-4 i a^3 \operatorname {PolyLog}\left (4,-i e^{i \arccos (a x)}\right )+4 i a^3 \operatorname {PolyLog}\left (4,i e^{i \arccos (a x)}\right ) \]

output

-2*a^2*arccos(a*x)^2/x-1/3*arccos(a*x)^4/x^3-8*I*a^3*arccos(a*x)*arctan(a* 
x+I*(-a^2*x^2+1)^(1/2))-4/3*I*a^3*arccos(a*x)^3*arctan(a*x+I*(-a^2*x^2+1)^ 
(1/2))+4*I*a^3*polylog(2,-I*(a*x+I*(-a^2*x^2+1)^(1/2)))+2*I*a^3*arccos(a*x 
)^2*polylog(2,-I*(a*x+I*(-a^2*x^2+1)^(1/2)))-4*I*a^3*polylog(2,I*(a*x+I*(- 
a^2*x^2+1)^(1/2)))-2*I*a^3*arccos(a*x)^2*polylog(2,I*(a*x+I*(-a^2*x^2+1)^( 
1/2)))-4*a^3*arccos(a*x)*polylog(3,-I*(a*x+I*(-a^2*x^2+1)^(1/2)))+4*a^3*ar 
ccos(a*x)*polylog(3,I*(a*x+I*(-a^2*x^2+1)^(1/2)))-4*I*a^3*polylog(4,-I*(a* 
x+I*(-a^2*x^2+1)^(1/2)))+4*I*a^3*polylog(4,I*(a*x+I*(-a^2*x^2+1)^(1/2)))+2 
/3*a*arccos(a*x)^3*(-a^2*x^2+1)^(1/2)/x^2
 

3.1.41.2 Mathematica [B] (verified)

Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(1475\) vs. \(2(304)=608\).

Time = 12.06 (sec) , antiderivative size = 1475, normalized size of antiderivative = 4.85 \[ \int \frac {\arccos (a x)^4}{x^4} \, dx =\text {Too large to display} \]

input

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

output

a^3*(-1/6*(ArcCos[a*x]^2*(12 + ArcCos[a*x]^2)) + 4*(ArcCos[a*x]*(Log[1 - I 
*E^(I*ArcCos[a*x])] - Log[1 + I*E^(I*ArcCos[a*x])]) + I*(PolyLog[2, (-I)*E 
^(I*ArcCos[a*x])] - PolyLog[2, I*E^(I*ArcCos[a*x])])) + (2*((Pi^3*Log[Cot[ 
(Pi/2 - ArcCos[a*x])/2]])/8 + (3*Pi^2*((Pi/2 - ArcCos[a*x])*(Log[1 - E^(I* 
(Pi/2 - ArcCos[a*x]))] - Log[1 + E^(I*(Pi/2 - ArcCos[a*x]))]) + I*(PolyLog 
[2, -E^(I*(Pi/2 - ArcCos[a*x]))] - PolyLog[2, E^(I*(Pi/2 - ArcCos[a*x]))]) 
))/4 - (3*Pi*((Pi/2 - ArcCos[a*x])^2*(Log[1 - E^(I*(Pi/2 - ArcCos[a*x]))] 
- Log[1 + E^(I*(Pi/2 - ArcCos[a*x]))]) + (2*I)*(Pi/2 - ArcCos[a*x])*(PolyL 
og[2, -E^(I*(Pi/2 - ArcCos[a*x]))] - PolyLog[2, E^(I*(Pi/2 - ArcCos[a*x])) 
]) + 2*(-PolyLog[3, -E^(I*(Pi/2 - ArcCos[a*x]))] + PolyLog[3, E^(I*(Pi/2 - 
 ArcCos[a*x]))])))/2 + 8*((I/64)*(Pi/2 - ArcCos[a*x])^4 + (I/4)*(Pi/2 + (- 
1/2*Pi + ArcCos[a*x])/2)^4 - ((Pi/2 - ArcCos[a*x])^3*Log[1 + E^(I*(Pi/2 - 
ArcCos[a*x]))])/8 - (Pi^3*(I*(Pi/2 + (-1/2*Pi + ArcCos[a*x])/2) - Log[1 + 
E^((2*I)*(Pi/2 + (-1/2*Pi + ArcCos[a*x])/2))]))/8 - (Pi/2 + (-1/2*Pi + Arc 
Cos[a*x])/2)^3*Log[1 + E^((2*I)*(Pi/2 + (-1/2*Pi + ArcCos[a*x])/2))] + ((3 
*I)/8)*(Pi/2 - ArcCos[a*x])^2*PolyLog[2, -E^(I*(Pi/2 - ArcCos[a*x]))] + (3 
*Pi^2*((I/2)*(Pi/2 + (-1/2*Pi + ArcCos[a*x])/2)^2 - (Pi/2 + (-1/2*Pi + Arc 
Cos[a*x])/2)*Log[1 + E^((2*I)*(Pi/2 + (-1/2*Pi + ArcCos[a*x])/2))] + (I/2) 
*PolyLog[2, -E^((2*I)*(Pi/2 + (-1/2*Pi + ArcCos[a*x])/2))]))/4 + ((3*I)/2) 
*(Pi/2 + (-1/2*Pi + ArcCos[a*x])/2)^2*PolyLog[2, -E^((2*I)*(Pi/2 + (-1/...
 

3.1.41.3 Rubi [A] (verified)

Time = 1.37 (sec) , antiderivative size = 304, 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 = {5139, 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.

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

\(\Big \downarrow \) 5139

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

\(\Big \downarrow \) 5205

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

\(\Big \downarrow \) 5139

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

\(\Big \downarrow \) 5219

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4669

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

\(\Big \downarrow \) 2715

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

\(\Big \downarrow \) 2838

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

\(\Big \downarrow \) 3011

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

\(\Big \downarrow \) 7163

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

\(\Big \downarrow \) 2720

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

\(\Big \downarrow \) 7143

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

input

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

output

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

3.1.41.4 Maple [A] (verified)

Time = 1.29 (sec) , antiderivative size = 419, normalized size of antiderivative = 1.38

method	result	size
derivativedivides	\(a^{3} \left (-\frac {\arccos \left (a x \right )^{2} \left (-2 \sqrt {-a^{2} x^{2}+1}\, \arccos \left (a x \right ) a x +\arccos \left (a x \right )^{2}+6 a^{2} x^{2}\right )}{3 a^{3} x^{3}}-\frac {2 \arccos \left (a x \right )^{3} \ln \left (1+i \left (i \sqrt {-a^{2} x^{2}+1}+a x \right )\right )}{3}+2 i \operatorname {polylog}\left (2, -i \left (i \sqrt {-a^{2} x^{2}+1}+a x \right )\right ) \arccos \left (a x \right )^{2}-4 \arccos \left (a x \right ) \operatorname {polylog}\left (3, -i \left (i \sqrt {-a^{2} x^{2}+1}+a x \right )\right )-4 i \operatorname {polylog}\left (4, -i \left (i \sqrt {-a^{2} x^{2}+1}+a x \right )\right )+\frac {2 \arccos \left (a x \right )^{3} \ln \left (1-i \left (i \sqrt {-a^{2} x^{2}+1}+a x \right )\right )}{3}-2 i \operatorname {polylog}\left (2, i \left (i \sqrt {-a^{2} x^{2}+1}+a x \right )\right ) \arccos \left (a x \right )^{2}+4 \arccos \left (a x \right ) \operatorname {polylog}\left (3, i \left (i \sqrt {-a^{2} x^{2}+1}+a x \right )\right )+4 i \operatorname {polylog}\left (4, i \left (i \sqrt {-a^{2} x^{2}+1}+a x \right )\right )-4 \arccos \left (a x \right ) \ln \left (1+i \left (i \sqrt {-a^{2} x^{2}+1}+a x \right )\right )+4 \arccos \left (a x \right ) \ln \left (1-i \left (i \sqrt {-a^{2} x^{2}+1}+a x \right )\right )+4 i \operatorname {dilog}\left (1+i \left (i \sqrt {-a^{2} x^{2}+1}+a x \right )\right )-4 i \operatorname {dilog}\left (1-i \left (i \sqrt {-a^{2} x^{2}+1}+a x \right )\right )\right )\)	\(419\)
default	\(a^{3} \left (-\frac {\arccos \left (a x \right )^{2} \left (-2 \sqrt {-a^{2} x^{2}+1}\, \arccos \left (a x \right ) a x +\arccos \left (a x \right )^{2}+6 a^{2} x^{2}\right )}{3 a^{3} x^{3}}-\frac {2 \arccos \left (a x \right )^{3} \ln \left (1+i \left (i \sqrt {-a^{2} x^{2}+1}+a x \right )\right )}{3}+2 i \operatorname {polylog}\left (2, -i \left (i \sqrt {-a^{2} x^{2}+1}+a x \right )\right ) \arccos \left (a x \right )^{2}-4 \arccos \left (a x \right ) \operatorname {polylog}\left (3, -i \left (i \sqrt {-a^{2} x^{2}+1}+a x \right )\right )-4 i \operatorname {polylog}\left (4, -i \left (i \sqrt {-a^{2} x^{2}+1}+a x \right )\right )+\frac {2 \arccos \left (a x \right )^{3} \ln \left (1-i \left (i \sqrt {-a^{2} x^{2}+1}+a x \right )\right )}{3}-2 i \operatorname {polylog}\left (2, i \left (i \sqrt {-a^{2} x^{2}+1}+a x \right )\right ) \arccos \left (a x \right )^{2}+4 \arccos \left (a x \right ) \operatorname {polylog}\left (3, i \left (i \sqrt {-a^{2} x^{2}+1}+a x \right )\right )+4 i \operatorname {polylog}\left (4, i \left (i \sqrt {-a^{2} x^{2}+1}+a x \right )\right )-4 \arccos \left (a x \right ) \ln \left (1+i \left (i \sqrt {-a^{2} x^{2}+1}+a x \right )\right )+4 \arccos \left (a x \right ) \ln \left (1-i \left (i \sqrt {-a^{2} x^{2}+1}+a x \right )\right )+4 i \operatorname {dilog}\left (1+i \left (i \sqrt {-a^{2} x^{2}+1}+a x \right )\right )-4 i \operatorname {dilog}\left (1-i \left (i \sqrt {-a^{2} x^{2}+1}+a x \right )\right )\right )\)	\(419\)

input

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

output

a^3*(-1/3/a^3/x^3*arccos(a*x)^2*(-2*(-a^2*x^2+1)^(1/2)*arccos(a*x)*a*x+arc 
cos(a*x)^2+6*a^2*x^2)-2/3*arccos(a*x)^3*ln(1+I*(I*(-a^2*x^2+1)^(1/2)+a*x)) 
+2*I*polylog(2,-I*(I*(-a^2*x^2+1)^(1/2)+a*x))*arccos(a*x)^2-4*arccos(a*x)* 
polylog(3,-I*(I*(-a^2*x^2+1)^(1/2)+a*x))-4*I*polylog(4,-I*(I*(-a^2*x^2+1)^ 
(1/2)+a*x))+2/3*arccos(a*x)^3*ln(1-I*(I*(-a^2*x^2+1)^(1/2)+a*x))-2*I*polyl 
og(2,I*(I*(-a^2*x^2+1)^(1/2)+a*x))*arccos(a*x)^2+4*arccos(a*x)*polylog(3,I 
*(I*(-a^2*x^2+1)^(1/2)+a*x))+4*I*polylog(4,I*(I*(-a^2*x^2+1)^(1/2)+a*x))-4 
*arccos(a*x)*ln(1+I*(I*(-a^2*x^2+1)^(1/2)+a*x))+4*arccos(a*x)*ln(1-I*(I*(- 
a^2*x^2+1)^(1/2)+a*x))+4*I*dilog(1+I*(I*(-a^2*x^2+1)^(1/2)+a*x))-4*I*dilog 
(1-I*(I*(-a^2*x^2+1)^(1/2)+a*x)))
 

3.1.41.5 Fricas [F]

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

input

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

output

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

3.1.41.6 Sympy [F]

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

input

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

output

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

3.1.41.7 Maxima [F]

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

input

integrate(arccos(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(sqrt(a*x 
+ 1)*sqrt(-a*x + 1), a*x)^3/(a^2*x^5 - x^3), x) - arctan2(sqrt(a*x + 1)*sq 
rt(-a*x + 1), a*x)^4)/x^3
 

3.1.41.8 Giac [F]

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

input

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

output

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

3.1.41.9 Mupad [F(-1)]

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

input

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

output

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