\(\int \frac {\arccos (a x)^3}{(c-a^2 c x^2)^{5/2}} \, dx\) [67]

Optimal result

Integrand size = 22, antiderivative size = 387 \[ \int \frac {\arccos (a x)^3}{\left (c-a^2 c x^2\right )^{5/2}} \, dx=\frac {x \arccos (a x)}{c^2 \sqrt {c-a^2 c x^2}}+\frac {\arccos (a x)^2}{2 a c^2 \sqrt {1-a^2 x^2} \sqrt {c-a^2 c x^2}}+\frac {x \arccos (a x)^3}{3 c \left (c-a^2 c x^2\right )^{3/2}}+\frac {2 x \arccos (a x)^3}{3 c^2 \sqrt {c-a^2 c x^2}}+\frac {2 i \sqrt {1-a^2 x^2} \arccos (a x)^3}{3 a c^2 \sqrt {c-a^2 c x^2}}-\frac {2 \sqrt {1-a^2 x^2} \arccos (a x)^2 \log \left (1-e^{2 i \arccos (a x)}\right )}{a c^2 \sqrt {c-a^2 c x^2}}-\frac {\sqrt {1-a^2 x^2} \log \left (1-a^2 x^2\right )}{2 a c^2 \sqrt {c-a^2 c x^2}}+\frac {2 i \sqrt {1-a^2 x^2} \arccos (a x) \operatorname {PolyLog}\left (2,e^{2 i \arccos (a x)}\right )}{a c^2 \sqrt {c-a^2 c x^2}}-\frac {\sqrt {1-a^2 x^2} \operatorname {PolyLog}\left (3,e^{2 i \arccos (a x)}\right )}{a c^2 \sqrt {c-a^2 c x^2}} \] Output:

x*arccos(a*x)/c^2/(-a^2*c*x^2+c)^(1/2)+1/2*arccos(a*x)^2/a/c^2/(-a^2*x^2+1 
)^(1/2)/(-a^2*c*x^2+c)^(1/2)+1/3*x*arccos(a*x)^3/c/(-a^2*c*x^2+c)^(3/2)+2/ 
3*x*arccos(a*x)^3/c^2/(-a^2*c*x^2+c)^(1/2)+2/3*I*(-a^2*x^2+1)^(1/2)*arccos 
(a*x)^3/a/c^2/(-a^2*c*x^2+c)^(1/2)-2*(-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^2/(-a^2*c*x^2+c)^(1/2)-1/2*(-a^2*x^2+1) 
^(1/2)*ln(-a^2*x^2+1)/a/c^2/(-a^2*c*x^2+c)^(1/2)+2*I*(-a^2*x^2+1)^(1/2)*ar 
ccos(a*x)*polylog(2,(a*x+I*(-a^2*x^2+1)^(1/2))^2)/a/c^2/(-a^2*c*x^2+c)^(1/ 
2)-(-a^2*x^2+1)^(1/2)*polylog(3,(a*x+I*(-a^2*x^2+1)^(1/2))^2)/a/c^2/(-a^2* 
c*x^2+c)^(1/2)
 

Mathematica [A] (verified)

Time = 0.62 (sec) , antiderivative size = 216, normalized size of antiderivative = 0.56 \[ \int \frac {\arccos (a x)^3}{\left (c-a^2 c x^2\right )^{5/2}} \, dx=-\frac {\left (1-a^2 x^2\right )^{3/2} \left (-i \pi ^3-\frac {12 a x \arccos (a x)}{\sqrt {1-a^2 x^2}}+\frac {6 \arccos (a x)^2}{-1+a^2 x^2}+8 i \arccos (a x)^3-\frac {4 a x \arccos (a x)^3}{\left (1-a^2 x^2\right )^{3/2}}-\frac {8 a x \arccos (a x)^3}{\sqrt {1-a^2 x^2}}+24 \arccos (a x)^2 \log \left (1-e^{-2 i \arccos (a x)}\right )+6 \log \left (1-a^2 x^2\right )+24 i \arccos (a x) \operatorname {PolyLog}\left (2,e^{-2 i \arccos (a x)}\right )+12 \operatorname {PolyLog}\left (3,e^{-2 i \arccos (a x)}\right )\right )}{12 a c \left (c-a^2 c x^2\right )^{3/2}} \] Input:

Integrate[ArcCos[a*x]^3/(c - a^2*c*x^2)^(5/2),x]
 

Output:

-1/12*((1 - a^2*x^2)^(3/2)*((-I)*Pi^3 - (12*a*x*ArcCos[a*x])/Sqrt[1 - a^2* 
x^2] + (6*ArcCos[a*x]^2)/(-1 + a^2*x^2) + (8*I)*ArcCos[a*x]^3 - (4*a*x*Arc 
Cos[a*x]^3)/(1 - a^2*x^2)^(3/2) - (8*a*x*ArcCos[a*x]^3)/Sqrt[1 - a^2*x^2] 
+ 24*ArcCos[a*x]^2*Log[1 - E^((-2*I)*ArcCos[a*x])] + 6*Log[1 - a^2*x^2] + 
(24*I)*ArcCos[a*x]*PolyLog[2, E^((-2*I)*ArcCos[a*x])] + 12*PolyLog[3, E^(( 
-2*I)*ArcCos[a*x])]))/(a*c*(c - a^2*c*x^2)^(3/2))
 

Rubi [A] (verified)

Time = 1.67 (sec) , antiderivative size = 292, normalized size of antiderivative = 0.75, number of steps used = 15, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.636, Rules used = {5163, 5161, 5181, 3042, 25, 4200, 25, 2620, 3011, 2720, 5183, 5161, 240, 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.

\(\displaystyle \int \frac {\arccos (a x)^3}{\left (c-a^2 c x^2\right )^{5/2}} \, dx\)

\(\Big \downarrow \) 5163

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

\(\Big \downarrow \) 5161

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

\(\Big \downarrow \) 5181

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 4200

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 2620

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

\(\Big \downarrow \) 3011

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

\(\Big \downarrow \) 2720

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

\(\Big \downarrow \) 5183

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

\(\Big \downarrow \) 5161

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

\(\Big \downarrow \) 240

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

\(\Big \downarrow \) 7143

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

Input:

Int[ArcCos[a*x]^3/(c - a^2*c*x^2)^(5/2),x]
 

Output:

(x*ArcCos[a*x]^3)/(3*c*(c - a^2*c*x^2)^(3/2)) + (a*Sqrt[1 - a^2*x^2]*(ArcC 
os[a*x]^2/(2*a^2*(1 - a^2*x^2)) + ((x*ArcCos[a*x])/Sqrt[1 - a^2*x^2] - Log 
[1 - a^2*x^2]/(2*a))/a))/(c^2*Sqrt[c - a^2*c*x^2]) + (2*((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)*ArcCo 
s[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])))/(3*c)
 

Defintions of rubi rules used

rule 25

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

rule 240

Int[(x_)/((a_) + (b_.)*(x_)^2), x_Symbol] :> Simp[Log[RemoveContent[a + b*x 
^2, x]]/(2*b), x] /; FreeQ[{a, b}, 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 5163

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

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 5183

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

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 = 0.82 (sec) , antiderivative size = 673, normalized size of antiderivative = 1.74

method	result	size
default	\(-\frac {\sqrt {-c \left (a^{2} x^{2}-1\right )}\, \left (-2 i \sqrt {-a^{2} x^{2}+1}\, a^{2} x^{2}+2 a^{3} x^{3}+2 i \sqrt {-a^{2} x^{2}+1}-3 a x \right ) \arccos \left (a x \right ) \left (-6 i \arccos \left (a x \right ) a^{4} x^{4}+6 \sqrt {-a^{2} x^{2}+1}\, \arccos \left (a x \right ) a^{3} x^{3}-6 i \sqrt {-a^{2} x^{2}+1}\, a^{3} x^{3}-6 a^{4} x^{4}+6 a^{2} x^{2} \arccos \left (a x \right )^{2}+12 i \arccos \left (a x \right ) a^{2} x^{2}-9 \arccos \left (a x \right ) \sqrt {-a^{2} x^{2}+1}\, a x +6 i \sqrt {-a^{2} x^{2}+1}\, a x +18 a^{2} x^{2}-8 \arccos \left (a x \right )^{2}-6 i \arccos \left (a x \right )-12\right )}{6 c^{3} \left (3 a^{6} x^{6}-10 a^{4} x^{4}+11 a^{2} x^{2}-4\right ) a}+\frac {\sqrt {-c \left (a^{2} x^{2}-1\right )}\, \sqrt {-a^{2} x^{2}+1}\, \ln \left (1+a x +i \sqrt {-a^{2} x^{2}+1}\right )}{a \,c^{3} \left (a^{2} x^{2}-1\right )}-\frac {2 \sqrt {-c \left (a^{2} x^{2}-1\right )}\, \sqrt {-a^{2} x^{2}+1}\, \ln \left (a x +i \sqrt {-a^{2} x^{2}+1}\right )}{a \,c^{3} \left (a^{2} x^{2}-1\right )}+\frac {\sqrt {-c \left (a^{2} x^{2}-1\right )}\, \sqrt {-a^{2} x^{2}+1}\, \ln \left (i \sqrt {-a^{2} x^{2}+1}+a x -1\right )}{a \,c^{3} \left (a^{2} x^{2}-1\right )}-\frac {2 \sqrt {-c \left (a^{2} x^{2}-1\right )}\, \sqrt {-a^{2} x^{2}+1}\, \left (2 i \arccos \left (a x \right )^{3}-3 \arccos \left (a x \right )^{2} \ln \left (1+a x +i \sqrt {-a^{2} x^{2}+1}\right )-3 \arccos \left (a x \right )^{2} \ln \left (1-a x -i \sqrt {-a^{2} x^{2}+1}\right )+6 i \arccos \left (a x \right ) \operatorname {polylog}\left (2, -a x -i \sqrt {-a^{2} x^{2}+1}\right )+6 i \arccos \left (a x \right ) \operatorname {polylog}\left (2, a x +i \sqrt {-a^{2} x^{2}+1}\right )-6 \operatorname {polylog}\left (3, -a x -i \sqrt {-a^{2} x^{2}+1}\right )-6 \operatorname {polylog}\left (3, a x +i \sqrt {-a^{2} x^{2}+1}\right )\right )}{3 a \,c^{3} \left (a^{2} x^{2}-1\right )}\)	\(673\)

Input:

int(arccos(a*x)^3/(-a^2*c*x^2+c)^(5/2),x,method=_RETURNVERBOSE)
 

Output:

-1/6*(-c*(a^2*x^2-1))^(1/2)*(-2*I*(-a^2*x^2+1)^(1/2)*a^2*x^2+2*a^3*x^3+2*I 
*(-a^2*x^2+1)^(1/2)-3*a*x)*arccos(a*x)*(-6*I*arccos(a*x)*a^4*x^4+6*(-a^2*x 
^2+1)^(1/2)*arccos(a*x)*a^3*x^3-6*I*(-a^2*x^2+1)^(1/2)*a^3*x^3-6*a^4*x^4+6 
*a^2*x^2*arccos(a*x)^2+12*I*arccos(a*x)*a^2*x^2-9*arccos(a*x)*(-a^2*x^2+1) 
^(1/2)*a*x+6*I*(-a^2*x^2+1)^(1/2)*a*x+18*a^2*x^2-8*arccos(a*x)^2-6*I*arcco 
s(a*x)-12)/c^3/(3*a^6*x^6-10*a^4*x^4+11*a^2*x^2-4)/a+(-c*(a^2*x^2-1))^(1/2 
)*(-a^2*x^2+1)^(1/2)/a/c^3/(a^2*x^2-1)*ln(1+a*x+I*(-a^2*x^2+1)^(1/2))-2*(- 
c*(a^2*x^2-1))^(1/2)*(-a^2*x^2+1)^(1/2)/a/c^3/(a^2*x^2-1)*ln(a*x+I*(-a^2*x 
^2+1)^(1/2))+(-c*(a^2*x^2-1))^(1/2)*(-a^2*x^2+1)^(1/2)/a/c^3/(a^2*x^2-1)*l 
n(I*(-a^2*x^2+1)^(1/2)+a*x-1)-2/3*(-c*(a^2*x^2-1))^(1/2)*(-a^2*x^2+1)^(1/2 
)*(2*I*arccos(a*x)^3-3*arccos(a*x)^2*ln(1+a*x+I*(-a^2*x^2+1)^(1/2))-3*arcc 
os(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)) 
)/a/c^3/(a^2*x^2-1)
 

Fricas [F]

\[ \int \frac {\arccos (a x)^3}{\left (c-a^2 c x^2\right )^{5/2}} \, dx=\int { \frac {\arccos \left (a x\right )^{3}}{{\left (-a^{2} c x^{2} + c\right )}^{\frac {5}{2}}} \,d x } \] Input:

integrate(arccos(a*x)^3/(-a^2*c*x^2+c)^(5/2),x, algorithm="fricas")
 

Output:

integral(-sqrt(-a^2*c*x^2 + c)*arccos(a*x)^3/(a^6*c^3*x^6 - 3*a^4*c^3*x^4 
+ 3*a^2*c^3*x^2 - c^3), x)
 

Sympy [F]

\[ \int \frac {\arccos (a x)^3}{\left (c-a^2 c x^2\right )^{5/2}} \, dx=\int \frac {\operatorname {acos}^{3}{\left (a x \right )}}{\left (- c \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac {5}{2}}}\, dx \] Input:

integrate(acos(a*x)**3/(-a**2*c*x**2+c)**(5/2),x)
 

Output:

Integral(acos(a*x)**3/(-c*(a*x - 1)*(a*x + 1))**(5/2), x)
 

Maxima [F]

\[ \int \frac {\arccos (a x)^3}{\left (c-a^2 c x^2\right )^{5/2}} \, dx=\int { \frac {\arccos \left (a x\right )^{3}}{{\left (-a^{2} c x^{2} + c\right )}^{\frac {5}{2}}} \,d x } \] Input:

integrate(arccos(a*x)^3/(-a^2*c*x^2+c)^(5/2),x, algorithm="maxima")
 

Output:

integrate(arccos(a*x)^3/(-a^2*c*x^2 + c)^(5/2), x)
 

Giac [F(-2)]

Exception generated. \[ \int \frac {\arccos (a x)^3}{\left (c-a^2 c x^2\right )^{5/2}} \, dx=\text {Exception raised: RuntimeError} \] Input:

integrate(arccos(a*x)^3/(-a^2*c*x^2+c)^(5/2),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. \[ \int \frac {\arccos (a x)^3}{\left (c-a^2 c x^2\right )^{5/2}} \, dx=\int \frac {{\mathrm {acos}\left (a\,x\right )}^3}{{\left (c-a^2\,c\,x^2\right )}^{5/2}} \,d x \] Input:

int(acos(a*x)^3/(c - a^2*c*x^2)^(5/2),x)
 

Output:

int(acos(a*x)^3/(c - a^2*c*x^2)^(5/2), x)
 

Reduce [F]

\[ \int \frac {\arccos (a x)^3}{\left (c-a^2 c x^2\right )^{5/2}} \, dx=\frac {\int \frac {\mathit {acos} \left (a x \right )^{3}}{\sqrt {-a^{2} x^{2}+1}\, a^{4} x^{4}-2 \sqrt {-a^{2} x^{2}+1}\, a^{2} x^{2}+\sqrt {-a^{2} x^{2}+1}}d x}{\sqrt {c}\, c^{2}} \] Input:

int(acos(a*x)^3/(-a^2*c*x^2+c)^(5/2),x)
 

Output:

int(acos(a*x)**3/(sqrt( - a**2*x**2 + 1)*a**4*x**4 - 2*sqrt( - a**2*x**2 + 
 1)*a**2*x**2 + sqrt( - a**2*x**2 + 1)),x)/(sqrt(c)*c**2)