\(\int \frac {(a+b \arccos (c x))^2}{x^2} \, dx\) [152]

Optimal result

Integrand size = 14, antiderivative size = 89 \[ \int \frac {(a+b \arccos (c x))^2}{x^2} \, dx=-\frac {(a+b \arccos (c x))^2}{x}-4 i b c (a+b \arccos (c x)) \arctan \left (e^{i \arccos (c x)}\right )+2 i b^2 c \operatorname {PolyLog}\left (2,-i e^{i \arccos (c x)}\right )-2 i b^2 c \operatorname {PolyLog}\left (2,i e^{i \arccos (c x)}\right ) \] Output:

-(a+b*arccos(c*x))^2/x-4*I*b*c*(a+b*arccos(c*x))*arctan(c*x+I*(-c^2*x^2+1) 
^(1/2))+2*I*b^2*c*polylog(2,-I*(c*x+I*(-c^2*x^2+1)^(1/2)))-2*I*b^2*c*polyl 
og(2,I*(c*x+I*(-c^2*x^2+1)^(1/2)))
 

Mathematica [A] (verified)

Time = 0.14 (sec) , antiderivative size = 134, normalized size of antiderivative = 1.51 \[ \int \frac {(a+b \arccos (c x))^2}{x^2} \, dx=-\frac {a^2+2 a b \left (\arccos (c x)-c x \text {arctanh}\left (\sqrt {1-c^2 x^2}\right )\right )+b^2 \left (\arccos (c x)^2-2 c x \left (\arccos (c x) \left (\log \left (1-i e^{i \arccos (c x)}\right )-\log \left (1+i e^{i \arccos (c x)}\right )\right )+i \left (\operatorname {PolyLog}\left (2,-i e^{i \arccos (c x)}\right )-\operatorname {PolyLog}\left (2,i e^{i \arccos (c x)}\right )\right )\right )\right )}{x} \] Input:

Integrate[(a + b*ArcCos[c*x])^2/x^2,x]
 

Output:

-((a^2 + 2*a*b*(ArcCos[c*x] - c*x*ArcTanh[Sqrt[1 - c^2*x^2]]) + b^2*(ArcCo 
s[c*x]^2 - 2*c*x*(ArcCos[c*x]*(Log[1 - I*E^(I*ArcCos[c*x])] - Log[1 + I*E^ 
(I*ArcCos[c*x])]) + I*(PolyLog[2, (-I)*E^(I*ArcCos[c*x])] - PolyLog[2, I*E 
^(I*ArcCos[c*x])]))))/x)
 

Rubi [A] (verified)

Time = 0.47 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.97, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {5139, 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.

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

\(\Big \downarrow \) 5139

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

\(\Big \downarrow \) 5219

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4669

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

\(\Big \downarrow \) 2715

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

\(\Big \downarrow \) 2838

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

Input:

Int[(a + b*ArcCos[c*x])^2/x^2,x]
 

Output:

-((a + b*ArcCos[c*x])^2/x) + 2*b*c*((-2*I)*(a + b*ArcCos[c*x])*ArcTan[E^(I 
*ArcCos[c*x])] + I*b*PolyLog[2, (-I)*E^(I*ArcCos[c*x])] - I*b*PolyLog[2, I 
*E^(I*ArcCos[c*x])])
 

Defintions of rubi rules used

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 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 = 0.22 (sec) , antiderivative size = 179, normalized size of antiderivative = 2.01

method	result	size
parts	\(-\frac {a^{2}}{x}+b^{2} c \left (-\frac {\arccos \left (c x \right )^{2}}{c x}-2 \arccos \left (c x \right ) \ln \left (1+i \left (c x +i \sqrt {-c^{2} x^{2}+1}\right )\right )+2 \arccos \left (c x \right ) \ln \left (1-i \left (c x +i \sqrt {-c^{2} x^{2}+1}\right )\right )+2 i \operatorname {dilog}\left (1+i \left (c x +i \sqrt {-c^{2} x^{2}+1}\right )\right )-2 i \operatorname {dilog}\left (1-i \left (c x +i \sqrt {-c^{2} x^{2}+1}\right )\right )\right )+2 a b c \left (-\frac {\arccos \left (c x \right )}{c x}+\operatorname {arctanh}\left (\frac {1}{\sqrt {-c^{2} x^{2}+1}}\right )\right )\)	\(179\)
derivativedivides	\(c \left (-\frac {a^{2}}{c x}+b^{2} \left (-\frac {\arccos \left (c x \right )^{2}}{c x}-2 \arccos \left (c x \right ) \ln \left (1+i \left (c x +i \sqrt {-c^{2} x^{2}+1}\right )\right )+2 \arccos \left (c x \right ) \ln \left (1-i \left (c x +i \sqrt {-c^{2} x^{2}+1}\right )\right )+2 i \operatorname {dilog}\left (1+i \left (c x +i \sqrt {-c^{2} x^{2}+1}\right )\right )-2 i \operatorname {dilog}\left (1-i \left (c x +i \sqrt {-c^{2} x^{2}+1}\right )\right )\right )+2 a b \left (-\frac {\arccos \left (c x \right )}{c x}+\operatorname {arctanh}\left (\frac {1}{\sqrt {-c^{2} x^{2}+1}}\right )\right )\right )\)	\(182\)
default	\(c \left (-\frac {a^{2}}{c x}+b^{2} \left (-\frac {\arccos \left (c x \right )^{2}}{c x}-2 \arccos \left (c x \right ) \ln \left (1+i \left (c x +i \sqrt {-c^{2} x^{2}+1}\right )\right )+2 \arccos \left (c x \right ) \ln \left (1-i \left (c x +i \sqrt {-c^{2} x^{2}+1}\right )\right )+2 i \operatorname {dilog}\left (1+i \left (c x +i \sqrt {-c^{2} x^{2}+1}\right )\right )-2 i \operatorname {dilog}\left (1-i \left (c x +i \sqrt {-c^{2} x^{2}+1}\right )\right )\right )+2 a b \left (-\frac {\arccos \left (c x \right )}{c x}+\operatorname {arctanh}\left (\frac {1}{\sqrt {-c^{2} x^{2}+1}}\right )\right )\right )\)	\(182\)

Input:

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

Output:

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

Fricas [F]

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

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

Output:

integral((b^2*arccos(c*x)^2 + 2*a*b*arccos(c*x) + a^2)/x^2, x)
 

Sympy [F]

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

integrate((a+b*acos(c*x))**2/x**2,x)
 

Output:

Integral((a + b*acos(c*x))**2/x**2, x)
 

Maxima [F]

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

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

Output:

2*(c*log(2*sqrt(-c^2*x^2 + 1)/abs(x) + 2/abs(x)) - arccos(c*x)/x)*a*b + (2 
*c*x*integrate(sqrt(c*x + 1)*sqrt(-c*x + 1)*arctan2(sqrt(c*x + 1)*sqrt(-c* 
x + 1), c*x)/(c^2*x^3 - x), x) - arctan2(sqrt(c*x + 1)*sqrt(-c*x + 1), c*x 
)^2)*b^2/x - a^2/x
 

Giac [F(-2)]

Exception generated. \[ \int \frac {(a+b \arccos (c x))^2}{x^2} \, dx=\text {Exception raised: RuntimeError} \] Input:

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

Output:

Exception raised: RuntimeError >> an error occurred running a Giac command 
:INPUT:sage2OUTPUT:sym2poly/r2sym(const gen & e,const index_m & i,const ve 
cteur & l) Error: Bad Argument Value
 

Mupad [F(-1)]

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

int((a + b*acos(c*x))^2/x^2,x)
 

Output:

int((a + b*acos(c*x))^2/x^2, x)
 

Reduce [F]

\[ \int \frac {(a+b \arccos (c x))^2}{x^2} \, dx=\frac {-2 \mathit {acos} \left (c x \right ) a b +\left (\int \frac {\mathit {acos} \left (c x \right )^{2}}{x^{2}}d x \right ) b^{2} x -2 \,\mathrm {log}\left (\tan \left (\frac {\mathit {asin} \left (c x \right )}{2}\right )\right ) a b c x -a^{2}}{x} \] Input:

int((a+b*acos(c*x))^2/x^2,x)
 

Output:

( - 2*acos(c*x)*a*b + int(acos(c*x)**2/x**2,x)*b**2*x - 2*log(tan(asin(c*x 
)/2))*a*b*c*x - a**2)/x