3.1.39 \(\int \frac {\arccos (a x)^4}{x^2} \, dx\) [39]

3.1.39.1 Optimal result

Integrand size = 10, antiderivative size = 176 \[ \int \frac {\arccos (a x)^4}{x^2} \, dx=-\frac {\arccos (a x)^4}{x}-8 i a \arccos (a x)^3 \arctan \left (e^{i \arccos (a x)}\right )+12 i a \arccos (a x)^2 \operatorname {PolyLog}\left (2,-i e^{i \arccos (a x)}\right )-12 i a \arccos (a x)^2 \operatorname {PolyLog}\left (2,i e^{i \arccos (a x)}\right )-24 a \arccos (a x) \operatorname {PolyLog}\left (3,-i e^{i \arccos (a x)}\right )+24 a \arccos (a x) \operatorname {PolyLog}\left (3,i e^{i \arccos (a x)}\right )-24 i a \operatorname {PolyLog}\left (4,-i e^{i \arccos (a x)}\right )+24 i a \operatorname {PolyLog}\left (4,i e^{i \arccos (a x)}\right ) \]

output

-arccos(a*x)^4/x-8*I*a*arccos(a*x)^3*arctan(a*x+I*(-a^2*x^2+1)^(1/2))+12*I 
*a*arccos(a*x)^2*polylog(2,-I*(a*x+I*(-a^2*x^2+1)^(1/2)))-12*I*a*arccos(a* 
x)^2*polylog(2,I*(a*x+I*(-a^2*x^2+1)^(1/2)))-24*a*arccos(a*x)*polylog(3,-I 
*(a*x+I*(-a^2*x^2+1)^(1/2)))+24*a*arccos(a*x)*polylog(3,I*(a*x+I*(-a^2*x^2 
+1)^(1/2)))-24*I*a*polylog(4,-I*(a*x+I*(-a^2*x^2+1)^(1/2)))+24*I*a*polylog 
(4,I*(a*x+I*(-a^2*x^2+1)^(1/2)))
 

3.1.39.2 Mathematica [B] (verified)

Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(549\) vs. \(2(176)=352\).

Time = 0.74 (sec) , antiderivative size = 549, normalized size of antiderivative = 3.12 \[ \int \frac {\arccos (a x)^4}{x^2} \, dx=a \left (-\frac {7 i \pi ^4}{16}-\frac {1}{2} i \pi ^3 \arccos (a x)+\frac {3}{2} i \pi ^2 \arccos (a x)^2-2 i \pi \arccos (a x)^3+i \arccos (a x)^4-\frac {\arccos (a x)^4}{a x}+3 \pi ^2 \arccos (a x) \log \left (1-i e^{-i \arccos (a x)}\right )-6 \pi \arccos (a x)^2 \log \left (1-i e^{-i \arccos (a x)}\right )-\frac {1}{2} \pi ^3 \log \left (1+i e^{-i \arccos (a x)}\right )+4 \arccos (a x)^3 \log \left (1+i e^{-i \arccos (a x)}\right )+\frac {1}{2} \pi ^3 \log \left (1+i e^{i \arccos (a x)}\right )-3 \pi ^2 \arccos (a x) \log \left (1+i e^{i \arccos (a x)}\right )+6 \pi \arccos (a x)^2 \log \left (1+i e^{i \arccos (a x)}\right )-4 \arccos (a x)^3 \log \left (1+i e^{i \arccos (a x)}\right )+\frac {1}{2} \pi ^3 \log \left (\tan \left (\frac {1}{4} (\pi +2 \arccos (a x))\right )\right )+12 i \arccos (a x)^2 \operatorname {PolyLog}\left (2,-i e^{-i \arccos (a x)}\right )+3 i \pi (\pi -4 \arccos (a x)) \operatorname {PolyLog}\left (2,i e^{-i \arccos (a x)}\right )+3 i \pi ^2 \operatorname {PolyLog}\left (2,-i e^{i \arccos (a x)}\right )-12 i \pi \arccos (a x) \operatorname {PolyLog}\left (2,-i e^{i \arccos (a x)}\right )+12 i \arccos (a x)^2 \operatorname {PolyLog}\left (2,-i e^{i \arccos (a x)}\right )+24 \arccos (a x) \operatorname {PolyLog}\left (3,-i e^{-i \arccos (a x)}\right )-12 \pi \operatorname {PolyLog}\left (3,i e^{-i \arccos (a x)}\right )+12 \pi \operatorname {PolyLog}\left (3,-i e^{i \arccos (a x)}\right )-24 \arccos (a x) \operatorname {PolyLog}\left (3,-i e^{i \arccos (a x)}\right )-24 i \operatorname {PolyLog}\left (4,-i e^{-i \arccos (a x)}\right )-24 i \operatorname {PolyLog}\left (4,-i e^{i \arccos (a x)}\right )\right ) \]

input

Integrate[ArcCos[a*x]^4/x^2,x]
 

output

a*(((-7*I)/16)*Pi^4 - (I/2)*Pi^3*ArcCos[a*x] + ((3*I)/2)*Pi^2*ArcCos[a*x]^ 
2 - (2*I)*Pi*ArcCos[a*x]^3 + I*ArcCos[a*x]^4 - ArcCos[a*x]^4/(a*x) + 3*Pi^ 
2*ArcCos[a*x]*Log[1 - I/E^(I*ArcCos[a*x])] - 6*Pi*ArcCos[a*x]^2*Log[1 - I/ 
E^(I*ArcCos[a*x])] - (Pi^3*Log[1 + I/E^(I*ArcCos[a*x])])/2 + 4*ArcCos[a*x] 
^3*Log[1 + I/E^(I*ArcCos[a*x])] + (Pi^3*Log[1 + I*E^(I*ArcCos[a*x])])/2 - 
3*Pi^2*ArcCos[a*x]*Log[1 + I*E^(I*ArcCos[a*x])] + 6*Pi*ArcCos[a*x]^2*Log[1 
 + I*E^(I*ArcCos[a*x])] - 4*ArcCos[a*x]^3*Log[1 + I*E^(I*ArcCos[a*x])] + ( 
Pi^3*Log[Tan[(Pi + 2*ArcCos[a*x])/4]])/2 + (12*I)*ArcCos[a*x]^2*PolyLog[2, 
 (-I)/E^(I*ArcCos[a*x])] + (3*I)*Pi*(Pi - 4*ArcCos[a*x])*PolyLog[2, I/E^(I 
*ArcCos[a*x])] + (3*I)*Pi^2*PolyLog[2, (-I)*E^(I*ArcCos[a*x])] - (12*I)*Pi 
*ArcCos[a*x]*PolyLog[2, (-I)*E^(I*ArcCos[a*x])] + (12*I)*ArcCos[a*x]^2*Pol 
yLog[2, (-I)*E^(I*ArcCos[a*x])] + 24*ArcCos[a*x]*PolyLog[3, (-I)/E^(I*ArcC 
os[a*x])] - 12*Pi*PolyLog[3, I/E^(I*ArcCos[a*x])] + 12*Pi*PolyLog[3, (-I)* 
E^(I*ArcCos[a*x])] - 24*ArcCos[a*x]*PolyLog[3, (-I)*E^(I*ArcCos[a*x])] - ( 
24*I)*PolyLog[4, (-I)/E^(I*ArcCos[a*x])] - (24*I)*PolyLog[4, (-I)*E^(I*Arc 
Cos[a*x])])
 

3.1.39.3 Rubi [A] (verified)

Time = 0.69 (sec) , antiderivative size = 185, normalized size of antiderivative = 1.05, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.800, Rules used = {5139, 5219, 3042, 4669, 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.

\(\displaystyle \int \frac {\arccos (a x)^4}{x^2} \, dx\)

\(\Big \downarrow \) 5139

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

\(\Big \downarrow \) 5219

\(\displaystyle 4 a \int \frac {\arccos (a x)^3}{a x}d\arccos (a x)-\frac {\arccos (a x)^4}{x}\)

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4669

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

\(\Big \downarrow \) 3011

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

\(\Big \downarrow \) 7163

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

\(\Big \downarrow \) 2720

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

\(\Big \downarrow \) 7143

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

input

Int[ArcCos[a*x]^4/x^2,x]
 

output

-(ArcCos[a*x]^4/x) + 4*a*((-2*I)*ArcCos[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)*ArcCo 
s[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 
])])))
 

3.1.39.3.1 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 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 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]
 

3.1.39.4 Maple [F]

\[\int \frac {\arccos \left (a x \right )^{4}}{x^{2}}d x\]

input

int(arccos(a*x)^4/x^2,x)
 

output

int(arccos(a*x)^4/x^2,x)
 

3.1.39.5 Fricas [F]

\[ \int \frac {\arccos (a x)^4}{x^2} \, dx=\int { \frac {\arccos \left (a x\right )^{4}}{x^{2}} \,d x } \]

input

integrate(arccos(a*x)^4/x^2,x, algorithm="fricas")
 

output

integral(arccos(a*x)^4/x^2, x)
 

3.1.39.6 Sympy [F]

\[ \int \frac {\arccos (a x)^4}{x^2} \, dx=\int \frac {\operatorname {acos}^{4}{\left (a x \right )}}{x^{2}}\, dx \]

input

integrate(acos(a*x)**4/x**2,x)
 

output

Integral(acos(a*x)**4/x**2, x)
 

3.1.39.7 Maxima [F]

\[ \int \frac {\arccos (a x)^4}{x^2} \, dx=\int { \frac {\arccos \left (a x\right )^{4}}{x^{2}} \,d x } \]

input

integrate(arccos(a*x)^4/x^2,x, algorithm="maxima")
 

output

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

3.1.39.8 Giac [F]

\[ \int \frac {\arccos (a x)^4}{x^2} \, dx=\int { \frac {\arccos \left (a x\right )^{4}}{x^{2}} \,d x } \]

input

integrate(arccos(a*x)^4/x^2,x, algorithm="giac")
 

output

integrate(arccos(a*x)^4/x^2, x)
 

3.1.39.9 Mupad [F(-1)]

Timed out. \[ \int \frac {\arccos (a x)^4}{x^2} \, dx=\int \frac {{\mathrm {acos}\left (a\,x\right )}^4}{x^2} \,d x \]

input

int(acos(a*x)^4/x^2,x)
 

output

int(acos(a*x)^4/x^2, x)