\(\int \frac {\arccos (a x)^2}{x^4} \, dx\) [20]

Optimal result

Integrand size = 10, antiderivative size = 124 \[ \int \frac {\arccos (a x)^2}{x^4} \, dx=-\frac {a^2}{3 x}+\frac {a \sqrt {1-a^2 x^2} \arccos (a x)}{3 x^2}-\frac {\arccos (a x)^2}{3 x^3}-\frac {2}{3} i a^3 \arccos (a x) \arctan \left (e^{i \arccos (a x)}\right )+\frac {1}{3} i a^3 \operatorname {PolyLog}\left (2,-i e^{i \arccos (a x)}\right )-\frac {1}{3} i a^3 \operatorname {PolyLog}\left (2,i e^{i \arccos (a x)}\right ) \] Output:

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

Mathematica [A] (verified)

Time = 0.43 (sec) , antiderivative size = 152, normalized size of antiderivative = 1.23 \[ \int \frac {\arccos (a x)^2}{x^4} \, dx=-\frac {a^2 x^2-a x \sqrt {1-a^2 x^2} \arccos (a x)+\arccos (a x)^2-a^3 x^3 \arccos (a x) \log \left (1-i e^{i \arccos (a x)}\right )+a^3 x^3 \arccos (a x) \log \left (1+i e^{i \arccos (a x)}\right )-i a^3 x^3 \operatorname {PolyLog}\left (2,-i e^{i \arccos (a x)}\right )+i a^3 x^3 \operatorname {PolyLog}\left (2,i e^{i \arccos (a x)}\right )}{3 x^3} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.59 (sec) , antiderivative size = 120, normalized size of antiderivative = 0.97, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.800, Rules used = {5139, 5205, 15, 5219, 3042, 4669, 2715, 2838}

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.

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

\(\Big \downarrow \) 5139

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

\(\Big \downarrow \) 5205

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

\(\Big \downarrow \) 15

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

\(\Big \downarrow \) 5219

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4669

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

\(\Big \downarrow \) 2715

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

\(\Big \downarrow \) 2838

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

Input:

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

Output:

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

Defintions of rubi rules used

rule 15

Int[(a_.)*(x_)^(m_.), x_Symbol] :> Simp[a*(x^(m + 1)/(m + 1)), x] /; FreeQ[ 
{a, m}, x] && NeQ[m, -1]
 

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 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 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]
 

Maple [A] (verified)

Time = 0.31 (sec) , antiderivative size = 166, normalized size of antiderivative = 1.34

method	result	size
derivativedivides	\(a^{3} \left (-\frac {-\arccos \left (a x \right ) \sqrt {-a^{2} x^{2}+1}\, a x +\arccos \left (a x \right )^{2}+a^{2} x^{2}}{3 a^{3} x^{3}}-\frac {\arccos \left (a x \right ) \ln \left (1+i \left (a x +i \sqrt {-a^{2} x^{2}+1}\right )\right )}{3}+\frac {\arccos \left (a x \right ) \ln \left (1-i \left (a x +i \sqrt {-a^{2} x^{2}+1}\right )\right )}{3}+\frac {i \operatorname {dilog}\left (1+i \left (a x +i \sqrt {-a^{2} x^{2}+1}\right )\right )}{3}-\frac {i \operatorname {dilog}\left (1-i \left (a x +i \sqrt {-a^{2} x^{2}+1}\right )\right )}{3}\right )\)	\(166\)
default	\(a^{3} \left (-\frac {-\arccos \left (a x \right ) \sqrt {-a^{2} x^{2}+1}\, a x +\arccos \left (a x \right )^{2}+a^{2} x^{2}}{3 a^{3} x^{3}}-\frac {\arccos \left (a x \right ) \ln \left (1+i \left (a x +i \sqrt {-a^{2} x^{2}+1}\right )\right )}{3}+\frac {\arccos \left (a x \right ) \ln \left (1-i \left (a x +i \sqrt {-a^{2} x^{2}+1}\right )\right )}{3}+\frac {i \operatorname {dilog}\left (1+i \left (a x +i \sqrt {-a^{2} x^{2}+1}\right )\right )}{3}-\frac {i \operatorname {dilog}\left (1-i \left (a x +i \sqrt {-a^{2} x^{2}+1}\right )\right )}{3}\right )\)	\(166\)

Input:

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

Output:

a^3*(-1/3*(-arccos(a*x)*(-a^2*x^2+1)^(1/2)*a*x+arccos(a*x)^2+a^2*x^2)/a^3/ 
x^3-1/3*arccos(a*x)*ln(1+I*(a*x+I*(-a^2*x^2+1)^(1/2)))+1/3*arccos(a*x)*ln( 
1-I*(a*x+I*(-a^2*x^2+1)^(1/2)))+1/3*I*dilog(1+I*(a*x+I*(-a^2*x^2+1)^(1/2)) 
)-1/3*I*dilog(1-I*(a*x+I*(-a^2*x^2+1)^(1/2))))
 

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

1/3*(6*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)/(a^2*x^5 - x^3), x) - arctan2(sqrt(a*x + 1)*sqrt( 
-a*x + 1), a*x)^2)/x^3
 

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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