\(\int \frac {\arccos (a x)^2}{x} \, dx\) [17]

Optimal result

Integrand size = 10, antiderivative size = 73 \[ \int \frac {\arccos (a x)^2}{x} \, dx=-\frac {1}{3} i \arccos (a x)^3+\arccos (a x)^2 \log \left (1+e^{2 i \arccos (a x)}\right )-i \arccos (a x) \operatorname {PolyLog}\left (2,-e^{2 i \arccos (a x)}\right )+\frac {1}{2} \operatorname {PolyLog}\left (3,-e^{2 i \arccos (a x)}\right ) \] Output:

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

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.00 \[ \int \frac {\arccos (a x)^2}{x} \, dx=-\frac {1}{3} i \arccos (a x)^3+\arccos (a x)^2 \log \left (1+e^{2 i \arccos (a x)}\right )-i \arccos (a x) \operatorname {PolyLog}\left (2,-e^{2 i \arccos (a x)}\right )+\frac {1}{2} \operatorname {PolyLog}\left (3,-e^{2 i \arccos (a x)}\right ) \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.46 (sec) , antiderivative size = 90, normalized size of antiderivative = 1.23, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.700, Rules used = {5137, 3042, 4202, 2620, 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.

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

\(\Big \downarrow \) 5137

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

\(\Big \downarrow \) 3042

\(\displaystyle -\int \arccos (a x)^2 \tan (\arccos (a x))d\arccos (a x)\)

\(\Big \downarrow \) 4202

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

\(\Big \downarrow \) 2620

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

\(\Big \downarrow \) 3011

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

\(\Big \downarrow \) 2720

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

\(\Big \downarrow \) 7143

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

Input:

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

Output:

(-1/3*I)*ArcCos[a*x]^3 + (2*I)*((-1/2*I)*ArcCos[a*x]^2*Log[1 + E^((2*I)*Ar 
cCos[a*x])] + I*((I/2)*ArcCos[a*x]*PolyLog[2, -E^((2*I)*ArcCos[a*x])] - Po 
lyLog[3, -E^((2*I)*ArcCos[a*x])]/4))
 

Defintions of rubi rules used

rule 2620

Int[(((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.))/ 
((a_) + (b_.)*((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)), x_Symbol] :> Simp 
[((c + d*x)^m/(b*f*g*n*Log[F]))*Log[1 + b*((F^(g*(e + f*x)))^n/a)], x] - Si 
mp[d*(m/(b*f*g*n*Log[F]))   Int[(c + d*x)^(m - 1)*Log[1 + b*((F^(g*(e + f*x 
)))^n/a)], x], x] /; FreeQ[{F, a, b, c, d, e, f, g, n}, x] && IGtQ[m, 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 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 4202

Int[((c_.) + (d_.)*(x_))^(m_.)*tan[(e_.) + (f_.)*(x_)], x_Symbol] :> Simp[I 
*((c + d*x)^(m + 1)/(d*(m + 1))), x] - Simp[2*I   Int[(c + d*x)^m*(E^(2*I*( 
e + f*x))/(1 + E^(2*I*(e + f*x)))), x], x] /; FreeQ[{c, d, e, f}, x] && IGt 
Q[m, 0]
 

rule 5137

Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_.)/(x_), x_Symbol] :> -Subst[Int[ 
(a + b*x)^n*Tan[x], x], x, ArcCos[c*x]] /; FreeQ[{a, b, c}, x] && IGtQ[n, 0 
]
 

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]
 

Maple [A] (verified)

Time = 0.19 (sec) , antiderivative size = 101, normalized size of antiderivative = 1.38

method	result	size
derivativedivides	\(-\frac {i \arccos \left (a x \right )^{3}}{3}+\arccos \left (a x \right )^{2} \ln \left (1+\left (a x +i \sqrt {-a^{2} x^{2}+1}\right )^{2}\right )-i \arccos \left (a x \right ) \operatorname {polylog}\left (2, -\left (a x +i \sqrt {-a^{2} x^{2}+1}\right )^{2}\right )+\frac {\operatorname {polylog}\left (3, -\left (a x +i \sqrt {-a^{2} x^{2}+1}\right )^{2}\right )}{2}\)	\(101\)
default	\(-\frac {i \arccos \left (a x \right )^{3}}{3}+\arccos \left (a x \right )^{2} \ln \left (1+\left (a x +i \sqrt {-a^{2} x^{2}+1}\right )^{2}\right )-i \arccos \left (a x \right ) \operatorname {polylog}\left (2, -\left (a x +i \sqrt {-a^{2} x^{2}+1}\right )^{2}\right )+\frac {\operatorname {polylog}\left (3, -\left (a x +i \sqrt {-a^{2} x^{2}+1}\right )^{2}\right )}{2}\)	\(101\)

Input:

int(arccos(a*x)^2/x,x,method=_RETURNVERBOSE)
 

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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