\(\int \frac {\arccos (a x)^2}{(c-a^2 c x^2)^{3/2}} \, dx\) [275]

Optimal result

Integrand size = 22, antiderivative size = 179 \[ \int \frac {\arccos (a x)^2}{\left (c-a^2 c x^2\right )^{3/2}} \, dx=\frac {x \arccos (a x)^2}{c \sqrt {c-a^2 c x^2}}-\frac {i \sqrt {1-a^2 x^2} \arccos (a x)^2}{a c \sqrt {c-a^2 c x^2}}+\frac {2 \sqrt {1-a^2 x^2} \arccos (a x) \log \left (1+e^{2 i \arccos (a x)}\right )}{a c \sqrt {c-a^2 c x^2}}-\frac {i \sqrt {1-a^2 x^2} \operatorname {PolyLog}\left (2,-e^{2 i \arccos (a x)}\right )}{a c \sqrt {c-a^2 c x^2}} \] Output:

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

Mathematica [A] (verified)

Time = 0.24 (sec) , antiderivative size = 111, normalized size of antiderivative = 0.62 \[ \int \frac {\arccos (a x)^2}{\left (c-a^2 c x^2\right )^{3/2}} \, dx=-\frac {-\arccos (a x) \left (a x \arccos (a x)+i \sqrt {1-a^2 x^2} \left (\arccos (a x)+2 i \log \left (1-e^{2 i \arccos (a x)}\right )\right )\right )-i \sqrt {1-a^2 x^2} \operatorname {PolyLog}\left (2,e^{2 i \arccos (a x)}\right )}{a c \sqrt {c \left (1-a^2 x^2\right )}} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.59 (sec) , antiderivative size = 123, normalized size of antiderivative = 0.69, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.409, Rules used = {5161, 5181, 3042, 25, 4200, 25, 2620, 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.

\(\displaystyle \int \frac {\arccos (a x)^2}{\left (c-a^2 c x^2\right )^{3/2}} \, dx\)

\(\Big \downarrow \) 5161

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

\(\Big \downarrow \) 5181

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 4200

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 2620

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

\(\Big \downarrow \) 2715

\(\displaystyle \frac {x \arccos (a x)^2}{c \sqrt {c-a^2 c x^2}}-\frac {2 \sqrt {1-a^2 x^2} \left (-2 i \left (\frac {1}{2} i \arccos (a x) \log \left (1-e^{2 i \arccos (a x)}\right )-\frac {1}{4} \int e^{-2 i \arccos (a x)} \log \left (1-e^{2 i \arccos (a x)}\right )de^{2 i \arccos (a x)}\right )-\frac {1}{2} i \arccos (a x)^2\right )}{a c \sqrt {c-a^2 c x^2}}\)

\(\Big \downarrow \) 2838

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

Input:

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

Output:

(x*ArcCos[a*x]^2)/(c*Sqrt[c - a^2*c*x^2]) - (2*Sqrt[1 - a^2*x^2]*((-1/2*I) 
*ArcCos[a*x]^2 - (2*I)*((I/2)*ArcCos[a*x]*Log[1 - E^((2*I)*ArcCos[a*x])] + 
 PolyLog[2, E^((2*I)*ArcCos[a*x])]/4)))/(a*c*Sqrt[c - a^2*c*x^2])
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

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 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 4200

Int[((c_.) + (d_.)*(x_))^(m_.)*tan[(e_.) + Pi*(k_.) + (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*k*Pi)*(E^(2*I*(e + f*x))/(1 + E^(2*I*k*Pi)*E^(2*I*(e + f*x)))), x] 
, x] /; FreeQ[{c, d, e, f}, x] && IntegerQ[4*k] && IGtQ[m, 0]
 

rule 5161

Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_.)/((d_) + (e_.)*(x_)^2)^(3/2), x 
_Symbol] :> Simp[x*((a + b*ArcCos[c*x])^n/(d*Sqrt[d + e*x^2])), x] + Simp[b 
*c*(n/d)*Simp[Sqrt[1 - c^2*x^2]/Sqrt[d + e*x^2]]   Int[x*((a + b*ArcCos[c*x 
])^(n - 1)/(1 - c^2*x^2)), x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d 
 + e, 0] && GtQ[n, 0]
 

rule 5181

Int[(((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_.)*(x_))/((d_) + (e_.)*(x_)^2), 
x_Symbol] :> Simp[1/e   Subst[Int[(a + b*x)^n*Cot[x], x], x, ArcCos[c*x]], 
x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && IGtQ[n, 0]
 

Maple [A] (verified)

Time = 0.46 (sec) , antiderivative size = 213, normalized size of antiderivative = 1.19

method	result	size
default	\(-\frac {\sqrt {-c \left (a^{2} x^{2}-1\right )}\, \left (-i \sqrt {-a^{2} x^{2}+1}+a x \right ) \arccos \left (a x \right )^{2}}{\left (a^{2} x^{2}-1\right ) a \,c^{2}}-\frac {2 i \sqrt {-a^{2} x^{2}+1}\, \sqrt {-c \left (a^{2} x^{2}-1\right )}\, \left (i \arccos \left (a x \right ) \ln \left (1-a x -i \sqrt {-a^{2} x^{2}+1}\right )+i \arccos \left (a x \right ) \ln \left (1+a x +i \sqrt {-a^{2} x^{2}+1}\right )+\arccos \left (a x \right )^{2}+\operatorname {polylog}\left (2, a x +i \sqrt {-a^{2} x^{2}+1}\right )+\operatorname {polylog}\left (2, -a x -i \sqrt {-a^{2} x^{2}+1}\right )\right )}{\left (a^{2} x^{2}-1\right ) a \,c^{2}}\)	\(213\)

Input:

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

Output:

-(-c*(a^2*x^2-1))^(1/2)*(-I*(-a^2*x^2+1)^(1/2)+a*x)*arccos(a*x)^2/(a^2*x^2 
-1)/a/c^2-2*I*(-a^2*x^2+1)^(1/2)*(-c*(a^2*x^2-1))^(1/2)*(I*arccos(a*x)*ln( 
1-a*x-I*(-a^2*x^2+1)^(1/2))+I*arccos(a*x)*ln(1+a*x+I*(-a^2*x^2+1)^(1/2))+a 
rccos(a*x)^2+polylog(2,a*x+I*(-a^2*x^2+1)^(1/2))+polylog(2,-a*x-I*(-a^2*x^ 
2+1)^(1/2)))/(a^2*x^2-1)/a/c^2
 

Fricas [F]

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

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

Output:

integral(sqrt(-a^2*c*x^2 + c)*arccos(a*x)^2/(a^4*c^2*x^4 - 2*a^2*c^2*x^2 + 
 c^2), x)
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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