\(\int \frac {\arccos (a x)}{x^3 \sqrt {1-a^2 x^2}} \, dx\) [110]

Optimal result

Integrand size = 22, antiderivative size = 98 \[ \int \frac {\arccos (a x)}{x^3 \sqrt {1-a^2 x^2}} \, dx=-\frac {a}{2 x}-\frac {\sqrt {1-a^2 x^2} \arccos (a x)}{2 x^2}-a^2 \arccos (a x) \text {arctanh}\left (e^{i \arccos (a x)}\right )+\frac {1}{2} i a^2 \operatorname {PolyLog}\left (2,-e^{i \arccos (a x)}\right )-\frac {1}{2} i a^2 \operatorname {PolyLog}\left (2,e^{i \arccos (a x)}\right ) \] Output:

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

Mathematica [A] (verified)

Time = 0.45 (sec) , antiderivative size = 126, normalized size of antiderivative = 1.29 \[ \int \frac {\arccos (a x)}{x^3 \sqrt {1-a^2 x^2}} \, dx=\frac {1}{2} a^2 \left (-\frac {-1+\frac {\sqrt {1-a^2 x^2} \arccos (a x)}{a x}}{a x}-\arccos (a x) \left (\log \left (1-i e^{i \arccos (a x)}\right )-\log \left (1+i e^{i \arccos (a x)}\right )\right )-i \operatorname {PolyLog}\left (2,-i e^{i \arccos (a x)}\right )+i \operatorname {PolyLog}\left (2,i e^{i \arccos (a x)}\right )\right ) \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.58 (sec) , antiderivative size = 101, normalized size of antiderivative = 1.03, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.318, Rules used = {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.

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

\(\Big \downarrow \) 5205

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

\(\Big \downarrow \) 15

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

\(\Big \downarrow \) 5219

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4669

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

\(\Big \downarrow \) 2715

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

\(\Big \downarrow \) 2838

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

Input:

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

Output:

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*Po 
lyLog[2, I*E^(I*ArcCos[a*x])]))/2
 

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 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 = 1.12 (sec) , antiderivative size = 186, normalized size of antiderivative = 1.90

method	result	size
default	\(-\frac {\sqrt {-a^{2} x^{2}+1}\, \left (a^{2} x^{2} \arccos \left (a x \right )+\sqrt {-a^{2} x^{2}+1}\, a x -\arccos \left (a x \right )\right )}{2 \left (a^{2} x^{2}-1\right ) x^{2}}+\frac {a^{2} \left (\arccos \left (a x \right ) \ln \left (1+i \left (a x +i \sqrt {-a^{2} x^{2}+1}\right )\right )-\arccos \left (a x \right ) \ln \left (1-i \left (a x +i \sqrt {-a^{2} x^{2}+1}\right )\right )-i \operatorname {dilog}\left (1+i \left (a x +i \sqrt {-a^{2} x^{2}+1}\right )\right )+i \operatorname {dilog}\left (1-i \left (a x +i \sqrt {-a^{2} x^{2}+1}\right )\right )\right )}{2}\)	\(186\)

Input:

int(arccos(a*x)/x^3/(-a^2*x^2+1)^(1/2),x,method=_RETURNVERBOSE)
 

Output:

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

Fricas [F]

\[ \int \frac {\arccos (a x)}{x^3 \sqrt {1-a^2 x^2}} \, dx=\int { \frac {\arccos \left (a x\right )}{\sqrt {-a^{2} x^{2} + 1} x^{3}} \,d x } \] Input:

integrate(arccos(a*x)/x^3/(-a^2*x^2+1)^(1/2),x, algorithm="fricas")
 

Output:

integral(-sqrt(-a^2*x^2 + 1)*arccos(a*x)/(a^2*x^5 - x^3), x)
 

Sympy [F]

\[ \int \frac {\arccos (a x)}{x^3 \sqrt {1-a^2 x^2}} \, dx=\int \frac {\operatorname {acos}{\left (a x \right )}}{x^{3} \sqrt {- \left (a x - 1\right ) \left (a x + 1\right )}}\, dx \] Input:

integrate(acos(a*x)/x**3/(-a**2*x**2+1)**(1/2),x)
 

Output:

Integral(acos(a*x)/(x**3*sqrt(-(a*x - 1)*(a*x + 1))), x)
 

Maxima [F]

\[ \int \frac {\arccos (a x)}{x^3 \sqrt {1-a^2 x^2}} \, dx=\int { \frac {\arccos \left (a x\right )}{\sqrt {-a^{2} x^{2} + 1} x^{3}} \,d x } \] Input:

integrate(arccos(a*x)/x^3/(-a^2*x^2+1)^(1/2),x, algorithm="maxima")
                                                                                    
                                                                                    
 

Output:

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

Giac [F]

\[ \int \frac {\arccos (a x)}{x^3 \sqrt {1-a^2 x^2}} \, dx=\int { \frac {\arccos \left (a x\right )}{\sqrt {-a^{2} x^{2} + 1} x^{3}} \,d x } \] Input:

integrate(arccos(a*x)/x^3/(-a^2*x^2+1)^(1/2),x, algorithm="giac")
 

Output:

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

Mupad [F(-1)]

Timed out. \[ \int \frac {\arccos (a x)}{x^3 \sqrt {1-a^2 x^2}} \, dx=\int \frac {\mathrm {acos}\left (a\,x\right )}{x^3\,\sqrt {1-a^2\,x^2}} \,d x \] Input:

int(acos(a*x)/(x^3*(1 - a^2*x^2)^(1/2)),x)
 

Output:

int(acos(a*x)/(x^3*(1 - a^2*x^2)^(1/2)), x)
 

Reduce [F]

\[ \int \frac {\arccos (a x)}{x^3 \sqrt {1-a^2 x^2}} \, dx=\int \frac {\mathit {acos} \left (a x \right )}{\sqrt {-a^{2} x^{2}+1}\, x^{3}}d x \] Input:

int(acos(a*x)/x^3/(-a^2*x^2+1)^(1/2),x)
 

Output:

int(acos(a*x)/(sqrt( - a**2*x**2 + 1)*x**3),x)