\(\int \frac {\arccos (a x)^3}{(c-a^2 c x^2)^{3/2}} \, dx\) [301]

Optimal result

Integrand size = 22, antiderivative size = 238 \[ \int \frac {\arccos (a x)^3}{\left (c-a^2 c x^2\right )^{3/2}} \, dx=\frac {x \arccos (a x)^3}{c \sqrt {c-a^2 c x^2}}-\frac {i \sqrt {1-a^2 x^2} \arccos (a x)^3}{a c \sqrt {c-a^2 c x^2}}+\frac {3 \sqrt {1-a^2 x^2} \arccos (a x)^2 \log \left (1+e^{2 i \arccos (a x)}\right )}{a c \sqrt {c-a^2 c x^2}}-\frac {3 i \sqrt {1-a^2 x^2} \arccos (a x) \operatorname {PolyLog}\left (2,-e^{2 i \arccos (a x)}\right )}{a c \sqrt {c-a^2 c x^2}}+\frac {3 \sqrt {1-a^2 x^2} \operatorname {PolyLog}\left (3,-e^{2 i \arccos (a x)}\right )}{2 a c \sqrt {c-a^2 c x^2}} \] Output:

x*arccos(a*x)^3/c/(-a^2*c*x^2+c)^(1/2)-I*(-a^2*x^2+1)^(1/2)*arccos(a*x)^3/ 
a/c/(-a^2*c*x^2+c)^(1/2)+3*(-a^2*x^2+1)^(1/2)*arccos(a*x)^2*ln(1+(a*x+I*(- 
a^2*x^2+1)^(1/2))^2)/a/c/(-a^2*c*x^2+c)^(1/2)-3*I*(-a^2*x^2+1)^(1/2)*arcco 
s(a*x)*polylog(2,-(a*x+I*(-a^2*x^2+1)^(1/2))^2)/a/c/(-a^2*c*x^2+c)^(1/2)+3 
/2*(-a^2*x^2+1)^(1/2)*polylog(3,-(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 = 0.18 (sec) , antiderivative size = 181, normalized size of antiderivative = 0.76 \[ \int \frac {\arccos (a x)^3}{\left (c-a^2 c x^2\right )^{3/2}} \, dx=-\frac {-i \pi ^3 \sqrt {1-a^2 x^2}-8 a x \arccos (a x)^3+8 i \sqrt {1-a^2 x^2} \arccos (a x)^3+24 \sqrt {1-a^2 x^2} \arccos (a x)^2 \log \left (1-e^{-2 i \arccos (a x)}\right )+24 i \sqrt {1-a^2 x^2} \arccos (a x) \operatorname {PolyLog}\left (2,e^{-2 i \arccos (a x)}\right )+12 \sqrt {1-a^2 x^2} \operatorname {PolyLog}\left (3,e^{-2 i \arccos (a x)}\right )}{8 a c \sqrt {c \left (1-a^2 x^2\right )}} \] Input:

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

Output:

-1/8*((-I)*Pi^3*Sqrt[1 - a^2*x^2] - 8*a*x*ArcCos[a*x]^3 + (8*I)*Sqrt[1 - a 
^2*x^2]*ArcCos[a*x]^3 + 24*Sqrt[1 - a^2*x^2]*ArcCos[a*x]^2*Log[1 - E^((-2* 
I)*ArcCos[a*x])] + (24*I)*Sqrt[1 - a^2*x^2]*ArcCos[a*x]*PolyLog[2, E^((-2* 
I)*ArcCos[a*x])] + 12*Sqrt[1 - a^2*x^2]*PolyLog[3, E^((-2*I)*ArcCos[a*x])] 
)/(a*c*Sqrt[c*(1 - a^2*x^2)])
 

Rubi [A] (verified)

Time = 0.74 (sec) , antiderivative size = 152, normalized size of antiderivative = 0.64, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.455, Rules used = {5161, 5181, 3042, 25, 4200, 25, 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.

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

\(\Big \downarrow \) 5161

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

\(\Big \downarrow \) 5181

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 4200

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 2620

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

\(\Big \downarrow \) 3011

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

\(\Big \downarrow \) 2720

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

\(\Big \downarrow \) 7143

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

Input:

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

Output:

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

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 = 0.00 (sec) , antiderivative size = 278, normalized size of antiderivative = 1.17

method	result	size
default	\(-\frac {\sqrt {-c \left (a^{2} x^{2}-1\right )}\, \left (-i \sqrt {-a^{2} x^{2}+1}+a x \right ) \arccos \left (a x \right )^{3}}{c^{2} a \left (a^{2} x^{2}-1\right )}-\frac {\sqrt {-a^{2} x^{2}+1}\, \sqrt {-c \left (a^{2} x^{2}-1\right )}\, \left (2 i \arccos \left (a x \right )^{3}-3 \arccos \left (a x \right )^{2} \ln \left (1+a x +i \sqrt {-a^{2} x^{2}+1}\right )-3 \arccos \left (a x \right )^{2} \ln \left (1-a x -i \sqrt {-a^{2} x^{2}+1}\right )+6 i \arccos \left (a x \right ) \operatorname {polylog}\left (2, -a x -i \sqrt {-a^{2} x^{2}+1}\right )+6 i \arccos \left (a x \right ) \operatorname {polylog}\left (2, a x +i \sqrt {-a^{2} x^{2}+1}\right )-6 \operatorname {polylog}\left (3, -a x -i \sqrt {-a^{2} x^{2}+1}\right )-6 \operatorname {polylog}\left (3, a x +i \sqrt {-a^{2} x^{2}+1}\right )\right )}{c^{2} a \left (a^{2} x^{2}-1\right )}\)	\(278\)

Input:

int(arccos(a*x)^3/(-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)^3/c^2/a/(a 
^2*x^2-1)-(-a^2*x^2+1)^(1/2)*(-c*(a^2*x^2-1))^(1/2)*(2*I*arccos(a*x)^3-3*a 
rccos(a*x)^2*ln(1+a*x+I*(-a^2*x^2+1)^(1/2))-3*arccos(a*x)^2*ln(1-a*x-I*(-a 
^2*x^2+1)^(1/2))+6*I*arccos(a*x)*polylog(2,-a*x-I*(-a^2*x^2+1)^(1/2))+6*I* 
arccos(a*x)*polylog(2,a*x+I*(-a^2*x^2+1)^(1/2))-6*polylog(3,-a*x-I*(-a^2*x 
^2+1)^(1/2))-6*polylog(3,a*x+I*(-a^2*x^2+1)^(1/2)))/c^2/a/(a^2*x^2-1)
                                                                                    
                                                                                    
 

Fricas [F]

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

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

Output:

integral(sqrt(-a^2*c*x^2 + c)*arccos(a*x)^3/(a^4*c^2*x^4 - 2*a^2*c^2*x^2 + 
 c^2), x)
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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