3.1.28 \(\int \frac {\arccos (a x)^3}{x^2} \, dx\) [28]

3.1.28.1 Optimal result

Integrand size = 10, antiderivative size = 122 \[ \int \frac {\arccos (a x)^3}{x^2} \, dx=-\frac {\arccos (a x)^3}{x}-6 i a \arccos (a x)^2 \arctan \left (e^{i \arccos (a x)}\right )+6 i a \arccos (a x) \operatorname {PolyLog}\left (2,-i e^{i \arccos (a x)}\right )-6 i a \arccos (a x) \operatorname {PolyLog}\left (2,i e^{i \arccos (a x)}\right )-6 a \operatorname {PolyLog}\left (3,-i e^{i \arccos (a x)}\right )+6 a \operatorname {PolyLog}\left (3,i e^{i \arccos (a x)}\right ) \]

output

-arccos(a*x)^3/x-6*I*a*arccos(a*x)^2*arctan(a*x+I*(-a^2*x^2+1)^(1/2))+6*I* 
a*arccos(a*x)*polylog(2,-I*(a*x+I*(-a^2*x^2+1)^(1/2)))-6*I*a*arccos(a*x)*p 
olylog(2,I*(a*x+I*(-a^2*x^2+1)^(1/2)))-6*a*polylog(3,-I*(a*x+I*(-a^2*x^2+1 
)^(1/2)))+6*a*polylog(3,I*(a*x+I*(-a^2*x^2+1)^(1/2)))
 

3.1.28.2 Mathematica [A] (verified)

Time = 0.08 (sec) , antiderivative size = 139, normalized size of antiderivative = 1.14 \[ \int \frac {\arccos (a x)^3}{x^2} \, dx=-\frac {\arccos (a x)^3}{x}+3 a \left (\arccos (a x)^2 \left (\log \left (1-i e^{i \arccos (a x)}\right )-\log \left (1+i e^{i \arccos (a x)}\right )\right )+2 i \arccos (a x) \left (\operatorname {PolyLog}\left (2,-i e^{i \arccos (a x)}\right )-\operatorname {PolyLog}\left (2,i e^{i \arccos (a x)}\right )\right )-2 \operatorname {PolyLog}\left (3,-i e^{i \arccos (a x)}\right )+2 \operatorname {PolyLog}\left (3,i e^{i \arccos (a x)}\right )\right ) \]

input

Integrate[ArcCos[a*x]^3/x^2,x]
 

output

-(ArcCos[a*x]^3/x) + 3*a*(ArcCos[a*x]^2*(Log[1 - I*E^(I*ArcCos[a*x])] - Lo 
g[1 + I*E^(I*ArcCos[a*x])]) + (2*I)*ArcCos[a*x]*(PolyLog[2, (-I)*E^(I*ArcC 
os[a*x])] - PolyLog[2, I*E^(I*ArcCos[a*x])]) - 2*PolyLog[3, (-I)*E^(I*ArcC 
os[a*x])] + 2*PolyLog[3, I*E^(I*ArcCos[a*x])])
 

3.1.28.3 Rubi [A] (verified)

Time = 0.55 (sec) , antiderivative size = 127, normalized size of antiderivative = 1.04, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.700, Rules used = {5139, 5219, 3042, 4669, 3011, 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.

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

\(\Big \downarrow \) 5139

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

\(\Big \downarrow \) 5219

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4669

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

\(\Big \downarrow \) 3011

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

\(\Big \downarrow \) 2720

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

\(\Big \downarrow \) 7143

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

input

Int[ArcCos[a*x]^3/x^2,x]
 

output

-(ArcCos[a*x]^3/x) + 3*a*((-2*I)*ArcCos[a*x]^2*ArcTan[E^(I*ArcCos[a*x])] + 
 2*(I*ArcCos[a*x]*PolyLog[2, (-I)*E^(I*ArcCos[a*x])] - PolyLog[3, (-I)*E^( 
I*ArcCos[a*x])]) - 2*(I*ArcCos[a*x]*PolyLog[2, I*E^(I*ArcCos[a*x])] - Poly 
Log[3, I*E^(I*ArcCos[a*x])]))
 

3.1.28.3.1 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]
 

3.1.28.4 Maple [F]

\[\int \frac {\arccos \left (a x \right )^{3}}{x^{2}}d x\]

input

int(arccos(a*x)^3/x^2,x)
 

output

int(arccos(a*x)^3/x^2,x)
 

3.1.28.5 Fricas [F]

\[ \int \frac {\arccos (a x)^3}{x^2} \, dx=\int { \frac {\arccos \left (a x\right )^{3}}{x^{2}} \,d x } \]

input

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

output

integral(arccos(a*x)^3/x^2, x)
 

3.1.28.6 Sympy [F]

\[ \int \frac {\arccos (a x)^3}{x^2} \, dx=\int \frac {\operatorname {acos}^{3}{\left (a x \right )}}{x^{2}}\, dx \]

input

integrate(acos(a*x)**3/x**2,x)
 

output

Integral(acos(a*x)**3/x**2, x)
 

3.1.28.7 Maxima [F]

\[ \int \frac {\arccos (a x)^3}{x^2} \, dx=\int { \frac {\arccos \left (a x\right )^{3}}{x^{2}} \,d x } \]

input

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

output

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

3.1.28.8 Giac [F]

\[ \int \frac {\arccos (a x)^3}{x^2} \, dx=\int { \frac {\arccos \left (a x\right )^{3}}{x^{2}} \,d x } \]

input

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

output

integrate(arccos(a*x)^3/x^2, x)
 

3.1.28.9 Mupad [F(-1)]

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

input

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

output

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