\(\int \frac {x^4 (a+b \arccos (c x))^2}{d-c^2 d x^2} \, dx\) [185]

Optimal result

Integrand size = 27, antiderivative size = 297 \[ \int \frac {x^4 (a+b \arccos (c x))^2}{d-c^2 d x^2} \, dx=\frac {22 b^2 x}{9 c^4 d}+\frac {2 b^2 x^3}{27 c^2 d}-\frac {22 b \sqrt {1-c^2 x^2} (a+b \arccos (c x))}{9 c^5 d}-\frac {2 b x^2 \sqrt {1-c^2 x^2} (a+b \arccos (c x))}{9 c^3 d}-\frac {x (a+b \arccos (c x))^2}{c^4 d}-\frac {x^3 (a+b \arccos (c x))^2}{3 c^2 d}-\frac {2 i (a+b \arccos (c x))^2 \arctan \left (e^{i \arccos (c x)}\right )}{c^5 d}+\frac {2 i b (a+b \arccos (c x)) \operatorname {PolyLog}\left (2,-i e^{i \arccos (c x)}\right )}{c^5 d}-\frac {2 i b (a+b \arccos (c x)) \operatorname {PolyLog}\left (2,i e^{i \arccos (c x)}\right )}{c^5 d}-\frac {2 b^2 \operatorname {PolyLog}\left (3,-i e^{i \arccos (c x)}\right )}{c^5 d}+\frac {2 b^2 \operatorname {PolyLog}\left (3,i e^{i \arccos (c x)}\right )}{c^5 d} \] Output:

22/9*b^2*x/c^4/d+2/27*b^2*x^3/c^2/d-22/9*b*(-c^2*x^2+1)^(1/2)*(a+b*arccos( 
c*x))/c^5/d-2/9*b*x^2*(-c^2*x^2+1)^(1/2)*(a+b*arccos(c*x))/c^3/d-x*(a+b*ar 
ccos(c*x))^2/c^4/d-1/3*x^3*(a+b*arccos(c*x))^2/c^2/d-2*I*(a+b*arccos(c*x)) 
^2*arctan(c*x+I*(-c^2*x^2+1)^(1/2))/c^5/d+2*I*b*(a+b*arccos(c*x))*polylog( 
2,-I*(c*x+I*(-c^2*x^2+1)^(1/2)))/c^5/d-2*I*b*(a+b*arccos(c*x))*polylog(2,I 
*(c*x+I*(-c^2*x^2+1)^(1/2)))/c^5/d-2*b^2*polylog(3,-I*(c*x+I*(-c^2*x^2+1)^ 
(1/2)))/c^5/d+2*b^2*polylog(3,I*(c*x+I*(-c^2*x^2+1)^(1/2)))/c^5/d
 

Mathematica [A] (verified)

Time = 0.49 (sec) , antiderivative size = 389, normalized size of antiderivative = 1.31 \[ \int \frac {x^4 (a+b \arccos (c x))^2}{d-c^2 d x^2} \, dx=\frac {-108 a^2 c x+270 b^2 c x-36 a^2 c^3 x^3+264 a b \sqrt {1-c^2 x^2}+24 a b c^2 x^2 \sqrt {1-c^2 x^2}-216 a b c x \arccos (c x)-72 a b c^3 x^3 \arccos (c x)+270 b^2 \sqrt {1-c^2 x^2} \arccos (c x)-135 b^2 c x \arccos (c x)^2+2 b^2 \cos (3 \arccos (c x))-9 b^2 \arccos (c x)^2 \cos (3 \arccos (c x))-216 a b \arccos (c x) \log \left (1-e^{i \arccos (c x)}\right )-108 b^2 \arccos (c x)^2 \log \left (1-e^{i \arccos (c x)}\right )+216 a b \arccos (c x) \log \left (1+e^{i \arccos (c x)}\right )+108 b^2 \arccos (c x)^2 \log \left (1+e^{i \arccos (c x)}\right )-54 a^2 \log (1-c x)+54 a^2 \log (1+c x)-216 i b (a+b \arccos (c x)) \operatorname {PolyLog}\left (2,-e^{i \arccos (c x)}\right )+216 i b (a+b \arccos (c x)) \operatorname {PolyLog}\left (2,e^{i \arccos (c x)}\right )+216 b^2 \operatorname {PolyLog}\left (3,-e^{i \arccos (c x)}\right )-216 b^2 \operatorname {PolyLog}\left (3,e^{i \arccos (c x)}\right )+6 b^2 \arccos (c x) \sin (3 \arccos (c x))}{108 c^5 d} \] Input:

Integrate[(x^4*(a + b*ArcCos[c*x])^2)/(d - c^2*d*x^2),x]
 

Output:

(-108*a^2*c*x + 270*b^2*c*x - 36*a^2*c^3*x^3 + 264*a*b*Sqrt[1 - c^2*x^2] + 
 24*a*b*c^2*x^2*Sqrt[1 - c^2*x^2] - 216*a*b*c*x*ArcCos[c*x] - 72*a*b*c^3*x 
^3*ArcCos[c*x] + 270*b^2*Sqrt[1 - c^2*x^2]*ArcCos[c*x] - 135*b^2*c*x*ArcCo 
s[c*x]^2 + 2*b^2*Cos[3*ArcCos[c*x]] - 9*b^2*ArcCos[c*x]^2*Cos[3*ArcCos[c*x 
]] - 216*a*b*ArcCos[c*x]*Log[1 - E^(I*ArcCos[c*x])] - 108*b^2*ArcCos[c*x]^ 
2*Log[1 - E^(I*ArcCos[c*x])] + 216*a*b*ArcCos[c*x]*Log[1 + E^(I*ArcCos[c*x 
])] + 108*b^2*ArcCos[c*x]^2*Log[1 + E^(I*ArcCos[c*x])] - 54*a^2*Log[1 - c* 
x] + 54*a^2*Log[1 + c*x] - (216*I)*b*(a + b*ArcCos[c*x])*PolyLog[2, -E^(I* 
ArcCos[c*x])] + (216*I)*b*(a + b*ArcCos[c*x])*PolyLog[2, E^(I*ArcCos[c*x]) 
] + 216*b^2*PolyLog[3, -E^(I*ArcCos[c*x])] - 216*b^2*PolyLog[3, E^(I*ArcCo 
s[c*x])] + 6*b^2*ArcCos[c*x]*Sin[3*ArcCos[c*x]])/(108*c^5*d)
 

Rubi [A] (verified)

Time = 1.91 (sec) , antiderivative size = 305, normalized size of antiderivative = 1.03, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.444, Rules used = {5211, 27, 5211, 15, 5165, 3042, 4671, 3011, 2720, 5183, 24, 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.

\(\displaystyle \int \frac {x^4 (a+b \arccos (c x))^2}{d-c^2 d x^2} \, dx\)

\(\Big \downarrow \) 5211

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 5211

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

\(\Big \downarrow \) 15

\(\displaystyle \frac {-\frac {2 b \int \frac {x (a+b \arccos (c x))}{\sqrt {1-c^2 x^2}}dx}{c}+\frac {\int \frac {(a+b \arccos (c x))^2}{1-c^2 x^2}dx}{c^2}-\frac {x (a+b \arccos (c x))^2}{c^2}}{c^2 d}-\frac {2 b \left (\frac {2 \int \frac {x (a+b \arccos (c x))}{\sqrt {1-c^2 x^2}}dx}{3 c^2}-\frac {x^2 \sqrt {1-c^2 x^2} (a+b \arccos (c x))}{3 c^2}-\frac {b x^3}{9 c}\right )}{3 c d}-\frac {x^3 (a+b \arccos (c x))^2}{3 c^2 d}\)

\(\Big \downarrow \) 5165

\(\displaystyle -\frac {2 b \left (\frac {2 \int \frac {x (a+b \arccos (c x))}{\sqrt {1-c^2 x^2}}dx}{3 c^2}-\frac {x^2 \sqrt {1-c^2 x^2} (a+b \arccos (c x))}{3 c^2}-\frac {b x^3}{9 c}\right )}{3 c d}+\frac {-\frac {2 b \int \frac {x (a+b \arccos (c x))}{\sqrt {1-c^2 x^2}}dx}{c}-\frac {\int \frac {(a+b \arccos (c x))^2}{\sqrt {1-c^2 x^2}}d\arccos (c x)}{c^3}-\frac {x (a+b \arccos (c x))^2}{c^2}}{c^2 d}-\frac {x^3 (a+b \arccos (c x))^2}{3 c^2 d}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {2 b \left (\frac {2 \int \frac {x (a+b \arccos (c x))}{\sqrt {1-c^2 x^2}}dx}{3 c^2}-\frac {x^2 \sqrt {1-c^2 x^2} (a+b \arccos (c x))}{3 c^2}-\frac {b x^3}{9 c}\right )}{3 c d}+\frac {-\frac {\int (a+b \arccos (c x))^2 \csc (\arccos (c x))d\arccos (c x)}{c^3}-\frac {2 b \int \frac {x (a+b \arccos (c x))}{\sqrt {1-c^2 x^2}}dx}{c}-\frac {x (a+b \arccos (c x))^2}{c^2}}{c^2 d}-\frac {x^3 (a+b \arccos (c x))^2}{3 c^2 d}\)

\(\Big \downarrow \) 4671

\(\displaystyle \frac {-\frac {-2 b \int (a+b \arccos (c x)) \log \left (1-e^{i \arccos (c x)}\right )d\arccos (c x)+2 b \int (a+b \arccos (c x)) \log \left (1+e^{i \arccos (c x)}\right )d\arccos (c x)-2 \text {arctanh}\left (e^{i \arccos (c x)}\right ) (a+b \arccos (c x))^2}{c^3}-\frac {2 b \int \frac {x (a+b \arccos (c x))}{\sqrt {1-c^2 x^2}}dx}{c}-\frac {x (a+b \arccos (c x))^2}{c^2}}{c^2 d}-\frac {2 b \left (\frac {2 \int \frac {x (a+b \arccos (c x))}{\sqrt {1-c^2 x^2}}dx}{3 c^2}-\frac {x^2 \sqrt {1-c^2 x^2} (a+b \arccos (c x))}{3 c^2}-\frac {b x^3}{9 c}\right )}{3 c d}-\frac {x^3 (a+b \arccos (c x))^2}{3 c^2 d}\)

\(\Big \downarrow \) 3011

\(\displaystyle \frac {-\frac {2 b \left (i \operatorname {PolyLog}\left (2,-e^{i \arccos (c x)}\right ) (a+b \arccos (c x))-i b \int \operatorname {PolyLog}\left (2,-e^{i \arccos (c x)}\right )d\arccos (c x)\right )-2 b \left (i \operatorname {PolyLog}\left (2,e^{i \arccos (c x)}\right ) (a+b \arccos (c x))-i b \int \operatorname {PolyLog}\left (2,e^{i \arccos (c x)}\right )d\arccos (c x)\right )-2 \text {arctanh}\left (e^{i \arccos (c x)}\right ) (a+b \arccos (c x))^2}{c^3}-\frac {2 b \int \frac {x (a+b \arccos (c x))}{\sqrt {1-c^2 x^2}}dx}{c}-\frac {x (a+b \arccos (c x))^2}{c^2}}{c^2 d}-\frac {2 b \left (\frac {2 \int \frac {x (a+b \arccos (c x))}{\sqrt {1-c^2 x^2}}dx}{3 c^2}-\frac {x^2 \sqrt {1-c^2 x^2} (a+b \arccos (c x))}{3 c^2}-\frac {b x^3}{9 c}\right )}{3 c d}-\frac {x^3 (a+b \arccos (c x))^2}{3 c^2 d}\)

\(\Big \downarrow \) 2720

\(\displaystyle \frac {-\frac {2 b \left (i \operatorname {PolyLog}\left (2,-e^{i \arccos (c x)}\right ) (a+b \arccos (c x))-b \int e^{-i \arccos (c x)} \operatorname {PolyLog}\left (2,-e^{i \arccos (c x)}\right )de^{i \arccos (c x)}\right )-2 b \left (i \operatorname {PolyLog}\left (2,e^{i \arccos (c x)}\right ) (a+b \arccos (c x))-b \int e^{-i \arccos (c x)} \operatorname {PolyLog}\left (2,e^{i \arccos (c x)}\right )de^{i \arccos (c x)}\right )-2 \text {arctanh}\left (e^{i \arccos (c x)}\right ) (a+b \arccos (c x))^2}{c^3}-\frac {2 b \int \frac {x (a+b \arccos (c x))}{\sqrt {1-c^2 x^2}}dx}{c}-\frac {x (a+b \arccos (c x))^2}{c^2}}{c^2 d}-\frac {2 b \left (\frac {2 \int \frac {x (a+b \arccos (c x))}{\sqrt {1-c^2 x^2}}dx}{3 c^2}-\frac {x^2 \sqrt {1-c^2 x^2} (a+b \arccos (c x))}{3 c^2}-\frac {b x^3}{9 c}\right )}{3 c d}-\frac {x^3 (a+b \arccos (c x))^2}{3 c^2 d}\)

\(\Big \downarrow \) 5183

\(\displaystyle \frac {-\frac {2 b \left (i \operatorname {PolyLog}\left (2,-e^{i \arccos (c x)}\right ) (a+b \arccos (c x))-b \int e^{-i \arccos (c x)} \operatorname {PolyLog}\left (2,-e^{i \arccos (c x)}\right )de^{i \arccos (c x)}\right )-2 b \left (i \operatorname {PolyLog}\left (2,e^{i \arccos (c x)}\right ) (a+b \arccos (c x))-b \int e^{-i \arccos (c x)} \operatorname {PolyLog}\left (2,e^{i \arccos (c x)}\right )de^{i \arccos (c x)}\right )-2 \text {arctanh}\left (e^{i \arccos (c x)}\right ) (a+b \arccos (c x))^2}{c^3}-\frac {2 b \left (-\frac {b \int 1dx}{c}-\frac {\sqrt {1-c^2 x^2} (a+b \arccos (c x))}{c^2}\right )}{c}-\frac {x (a+b \arccos (c x))^2}{c^2}}{c^2 d}-\frac {2 b \left (\frac {2 \left (-\frac {b \int 1dx}{c}-\frac {\sqrt {1-c^2 x^2} (a+b \arccos (c x))}{c^2}\right )}{3 c^2}-\frac {x^2 \sqrt {1-c^2 x^2} (a+b \arccos (c x))}{3 c^2}-\frac {b x^3}{9 c}\right )}{3 c d}-\frac {x^3 (a+b \arccos (c x))^2}{3 c^2 d}\)

\(\Big \downarrow \) 24

\(\displaystyle \frac {-\frac {2 b \left (i \operatorname {PolyLog}\left (2,-e^{i \arccos (c x)}\right ) (a+b \arccos (c x))-b \int e^{-i \arccos (c x)} \operatorname {PolyLog}\left (2,-e^{i \arccos (c x)}\right )de^{i \arccos (c x)}\right )-2 b \left (i \operatorname {PolyLog}\left (2,e^{i \arccos (c x)}\right ) (a+b \arccos (c x))-b \int e^{-i \arccos (c x)} \operatorname {PolyLog}\left (2,e^{i \arccos (c x)}\right )de^{i \arccos (c x)}\right )-2 \text {arctanh}\left (e^{i \arccos (c x)}\right ) (a+b \arccos (c x))^2}{c^3}-\frac {2 b \left (-\frac {\sqrt {1-c^2 x^2} (a+b \arccos (c x))}{c^2}-\frac {b x}{c}\right )}{c}-\frac {x (a+b \arccos (c x))^2}{c^2}}{c^2 d}-\frac {x^3 (a+b \arccos (c x))^2}{3 c^2 d}-\frac {2 b \left (-\frac {x^2 \sqrt {1-c^2 x^2} (a+b \arccos (c x))}{3 c^2}+\frac {2 \left (-\frac {\sqrt {1-c^2 x^2} (a+b \arccos (c x))}{c^2}-\frac {b x}{c}\right )}{3 c^2}-\frac {b x^3}{9 c}\right )}{3 c d}\)

\(\Big \downarrow \) 7143

\(\displaystyle \frac {-\frac {-2 \text {arctanh}\left (e^{i \arccos (c x)}\right ) (a+b \arccos (c x))^2+2 b \left (i \operatorname {PolyLog}\left (2,-e^{i \arccos (c x)}\right ) (a+b \arccos (c x))-b \operatorname {PolyLog}\left (3,-e^{i \arccos (c x)}\right )\right )-2 b \left (i \operatorname {PolyLog}\left (2,e^{i \arccos (c x)}\right ) (a+b \arccos (c x))-b \operatorname {PolyLog}\left (3,e^{i \arccos (c x)}\right )\right )}{c^3}-\frac {2 b \left (-\frac {\sqrt {1-c^2 x^2} (a+b \arccos (c x))}{c^2}-\frac {b x}{c}\right )}{c}-\frac {x (a+b \arccos (c x))^2}{c^2}}{c^2 d}-\frac {x^3 (a+b \arccos (c x))^2}{3 c^2 d}-\frac {2 b \left (-\frac {x^2 \sqrt {1-c^2 x^2} (a+b \arccos (c x))}{3 c^2}+\frac {2 \left (-\frac {\sqrt {1-c^2 x^2} (a+b \arccos (c x))}{c^2}-\frac {b x}{c}\right )}{3 c^2}-\frac {b x^3}{9 c}\right )}{3 c d}\)

Input:

Int[(x^4*(a + b*ArcCos[c*x])^2)/(d - c^2*d*x^2),x]
 

Output:

-1/3*(x^3*(a + b*ArcCos[c*x])^2)/(c^2*d) - (2*b*(-1/9*(b*x^3)/c - (x^2*Sqr 
t[1 - c^2*x^2]*(a + b*ArcCos[c*x]))/(3*c^2) + (2*(-((b*x)/c) - (Sqrt[1 - c 
^2*x^2]*(a + b*ArcCos[c*x]))/c^2))/(3*c^2)))/(3*c*d) + (-((x*(a + b*ArcCos 
[c*x])^2)/c^2) - (2*b*(-((b*x)/c) - (Sqrt[1 - c^2*x^2]*(a + b*ArcCos[c*x]) 
)/c^2))/c - (-2*(a + b*ArcCos[c*x])^2*ArcTanh[E^(I*ArcCos[c*x])] + 2*b*(I* 
(a + b*ArcCos[c*x])*PolyLog[2, -E^(I*ArcCos[c*x])] - b*PolyLog[3, -E^(I*Ar 
cCos[c*x])]) - 2*b*(I*(a + b*ArcCos[c*x])*PolyLog[2, E^(I*ArcCos[c*x])] - 
b*PolyLog[3, E^(I*ArcCos[c*x])]))/c^3)/(c^2*d)
 

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 24

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]
 

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

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 4671

Int[csc[(e_.) + (f_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[- 
2*(c + d*x)^m*(ArcTanh[E^(I*(e + f*x))]/f), x] + (-Simp[d*(m/f)   Int[(c + 
d*x)^(m - 1)*Log[1 - E^(I*(e + f*x))], x], x] + Simp[d*(m/f)   Int[(c + d*x 
)^(m - 1)*Log[1 + E^(I*(e + f*x))], x], x]) /; FreeQ[{c, d, e, f}, x] && IG 
tQ[m, 0]
 

rule 5165

Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_.)/((d_) + (e_.)*(x_)^2), x_Symbo 
l] :> Simp[-(c*d)^(-1)   Subst[Int[(a + b*x)^n*Csc[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 5211

Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*((d_) + (e_. 
)*(x_)^2)^(p_), x_Symbol] :> Simp[f*(f*x)^(m - 1)*(d + e*x^2)^(p + 1)*((a + 
 b*ArcCos[c*x])^n/(e*(m + 2*p + 1))), x] + (Simp[f^2*((m - 1)/(c^2*(m + 2*p 
 + 1)))   Int[(f*x)^(m - 2)*(d + e*x^2)^p*(a + b*ArcCos[c*x])^n, x], x] - S 
imp[b*f*(n/(c*(m + 2*p + 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] && IGtQ[m 
, 1] && NeQ[m + 2*p + 1, 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 = 0.57 (sec) , antiderivative size = 533, normalized size of antiderivative = 1.79

method	result	size
derivativedivides	\(\frac {-\frac {a^{2} \left (\frac {c^{3} x^{3}}{3}+c x +\frac {\ln \left (c x -1\right )}{2}-\frac {\ln \left (c x +1\right )}{2}\right )}{d}+\frac {5 b^{2} \arccos \left (c x \right ) \sqrt {-c^{2} x^{2}+1}}{2 d}-\frac {5 b^{2} \arccos \left (c x \right )^{2} c x}{4 d}+\frac {5 b^{2} c x}{2 d}-\frac {b^{2} \arccos \left (c x \right )^{2} \ln \left (1-c x -i \sqrt {-c^{2} x^{2}+1}\right )}{d}-\frac {2 i a b \operatorname {polylog}\left (2, -c x -i \sqrt {-c^{2} x^{2}+1}\right )}{d}-\frac {2 b^{2} \operatorname {polylog}\left (3, c x +i \sqrt {-c^{2} x^{2}+1}\right )}{d}+\frac {b^{2} \arccos \left (c x \right )^{2} \ln \left (1+c x +i \sqrt {-c^{2} x^{2}+1}\right )}{d}-\frac {2 i b^{2} \arccos \left (c x \right ) \operatorname {polylog}\left (2, -c x -i \sqrt {-c^{2} x^{2}+1}\right )}{d}+\frac {2 b^{2} \operatorname {polylog}\left (3, -c x -i \sqrt {-c^{2} x^{2}+1}\right )}{d}-\frac {b^{2} \arccos \left (c x \right )^{2} \cos \left (3 \arccos \left (c x \right )\right )}{12 d}+\frac {b^{2} \cos \left (3 \arccos \left (c x \right )\right )}{54 d}+\frac {b^{2} \arccos \left (c x \right ) \sin \left (3 \arccos \left (c x \right )\right )}{18 d}+\frac {5 a b \sqrt {-c^{2} x^{2}+1}}{2 d}-\frac {5 a b \arccos \left (c x \right ) c x}{2 d}-\frac {2 a b \arccos \left (c x \right ) \ln \left (1-c x -i \sqrt {-c^{2} x^{2}+1}\right )}{d}+\frac {2 a b \arccos \left (c x \right ) \ln \left (1+c x +i \sqrt {-c^{2} x^{2}+1}\right )}{d}+\frac {2 i b^{2} \arccos \left (c x \right ) \operatorname {polylog}\left (2, c x +i \sqrt {-c^{2} x^{2}+1}\right )}{d}+\frac {2 i a b \operatorname {polylog}\left (2, c x +i \sqrt {-c^{2} x^{2}+1}\right )}{d}-\frac {a b \arccos \left (c x \right ) \cos \left (3 \arccos \left (c x \right )\right )}{6 d}+\frac {a b \sin \left (3 \arccos \left (c x \right )\right )}{18 d}}{c^{5}}\)	\(533\)
default	\(\frac {-\frac {a^{2} \left (\frac {c^{3} x^{3}}{3}+c x +\frac {\ln \left (c x -1\right )}{2}-\frac {\ln \left (c x +1\right )}{2}\right )}{d}+\frac {5 b^{2} \arccos \left (c x \right ) \sqrt {-c^{2} x^{2}+1}}{2 d}-\frac {5 b^{2} \arccos \left (c x \right )^{2} c x}{4 d}+\frac {5 b^{2} c x}{2 d}-\frac {b^{2} \arccos \left (c x \right )^{2} \ln \left (1-c x -i \sqrt {-c^{2} x^{2}+1}\right )}{d}-\frac {2 i a b \operatorname {polylog}\left (2, -c x -i \sqrt {-c^{2} x^{2}+1}\right )}{d}-\frac {2 b^{2} \operatorname {polylog}\left (3, c x +i \sqrt {-c^{2} x^{2}+1}\right )}{d}+\frac {b^{2} \arccos \left (c x \right )^{2} \ln \left (1+c x +i \sqrt {-c^{2} x^{2}+1}\right )}{d}-\frac {2 i b^{2} \arccos \left (c x \right ) \operatorname {polylog}\left (2, -c x -i \sqrt {-c^{2} x^{2}+1}\right )}{d}+\frac {2 b^{2} \operatorname {polylog}\left (3, -c x -i \sqrt {-c^{2} x^{2}+1}\right )}{d}-\frac {b^{2} \arccos \left (c x \right )^{2} \cos \left (3 \arccos \left (c x \right )\right )}{12 d}+\frac {b^{2} \cos \left (3 \arccos \left (c x \right )\right )}{54 d}+\frac {b^{2} \arccos \left (c x \right ) \sin \left (3 \arccos \left (c x \right )\right )}{18 d}+\frac {5 a b \sqrt {-c^{2} x^{2}+1}}{2 d}-\frac {5 a b \arccos \left (c x \right ) c x}{2 d}-\frac {2 a b \arccos \left (c x \right ) \ln \left (1-c x -i \sqrt {-c^{2} x^{2}+1}\right )}{d}+\frac {2 a b \arccos \left (c x \right ) \ln \left (1+c x +i \sqrt {-c^{2} x^{2}+1}\right )}{d}+\frac {2 i b^{2} \arccos \left (c x \right ) \operatorname {polylog}\left (2, c x +i \sqrt {-c^{2} x^{2}+1}\right )}{d}+\frac {2 i a b \operatorname {polylog}\left (2, c x +i \sqrt {-c^{2} x^{2}+1}\right )}{d}-\frac {a b \arccos \left (c x \right ) \cos \left (3 \arccos \left (c x \right )\right )}{6 d}+\frac {a b \sin \left (3 \arccos \left (c x \right )\right )}{18 d}}{c^{5}}\)	\(533\)
parts	\(-\frac {a^{2} \left (\frac {\frac {1}{3} c^{2} x^{3}+x}{c^{4}}+\frac {\ln \left (c x -1\right )}{2 c^{5}}-\frac {\ln \left (c x +1\right )}{2 c^{5}}\right )}{d}-\frac {b^{2} \left (\frac {5 \left (\arccos \left (c x \right )^{2}-2+2 i \arccos \left (c x \right )\right ) \left (c x +i \sqrt {-c^{2} x^{2}+1}\right )}{8}+\frac {5 \left (-i \sqrt {-c^{2} x^{2}+1}+c x \right ) \left (\arccos \left (c x \right )^{2}-2-2 i \arccos \left (c x \right )\right )}{8}+\arccos \left (c x \right )^{2} \ln \left (1-c x -i \sqrt {-c^{2} x^{2}+1}\right )-2 i \arccos \left (c x \right ) \operatorname {polylog}\left (2, c x +i \sqrt {-c^{2} x^{2}+1}\right )+2 \operatorname {polylog}\left (3, c x +i \sqrt {-c^{2} x^{2}+1}\right )-\arccos \left (c x \right )^{2} \ln \left (1+c x +i \sqrt {-c^{2} x^{2}+1}\right )+2 i \arccos \left (c x \right ) \operatorname {polylog}\left (2, -c x -i \sqrt {-c^{2} x^{2}+1}\right )-2 \operatorname {polylog}\left (3, -c x -i \sqrt {-c^{2} x^{2}+1}\right )+\frac {\left (9 \arccos \left (c x \right )^{2}-2\right ) \cos \left (3 \arccos \left (c x \right )\right )}{108}-\frac {\arccos \left (c x \right ) \sin \left (3 \arccos \left (c x \right )\right )}{18}\right )}{d \,c^{5}}+\frac {5 a b \sqrt {-c^{2} x^{2}+1}}{2 d \,c^{5}}-\frac {5 a b \arccos \left (c x \right ) x}{2 d \,c^{4}}-\frac {2 a b \arccos \left (c x \right ) \ln \left (1-c x -i \sqrt {-c^{2} x^{2}+1}\right )}{d \,c^{5}}+\frac {2 a b \arccos \left (c x \right ) \ln \left (1+c x +i \sqrt {-c^{2} x^{2}+1}\right )}{d \,c^{5}}+\frac {2 i a b \operatorname {polylog}\left (2, c x +i \sqrt {-c^{2} x^{2}+1}\right )}{d \,c^{5}}-\frac {2 i a b \operatorname {polylog}\left (2, -c x -i \sqrt {-c^{2} x^{2}+1}\right )}{d \,c^{5}}-\frac {a b \arccos \left (c x \right ) \cos \left (3 \arccos \left (c x \right )\right )}{6 d \,c^{5}}+\frac {a b \sin \left (3 \arccos \left (c x \right )\right )}{18 d \,c^{5}}\)	\(536\)

Input:

int(x^4*(a+b*arccos(c*x))^2/(-c^2*d*x^2+d),x,method=_RETURNVERBOSE)
 

Output:

1/c^5*(-a^2/d*(1/3*c^3*x^3+c*x+1/2*ln(c*x-1)-1/2*ln(c*x+1))+5/2*b^2/d*arcc 
os(c*x)*(-c^2*x^2+1)^(1/2)-5/4*b^2/d*arccos(c*x)^2*c*x+5/2*b^2/d*c*x-b^2/d 
*arccos(c*x)^2*ln(1-c*x-I*(-c^2*x^2+1)^(1/2))-2*I*a*b/d*polylog(2,-c*x-I*( 
-c^2*x^2+1)^(1/2))-2*b^2/d*polylog(3,c*x+I*(-c^2*x^2+1)^(1/2))+b^2/d*arcco 
s(c*x)^2*ln(1+c*x+I*(-c^2*x^2+1)^(1/2))-2*I*b^2/d*arccos(c*x)*polylog(2,-c 
*x-I*(-c^2*x^2+1)^(1/2))+2*b^2/d*polylog(3,-c*x-I*(-c^2*x^2+1)^(1/2))-1/12 
*b^2/d*arccos(c*x)^2*cos(3*arccos(c*x))+1/54*b^2/d*cos(3*arccos(c*x))+1/18 
*b^2/d*arccos(c*x)*sin(3*arccos(c*x))+5/2*a*b/d*(-c^2*x^2+1)^(1/2)-5/2*a*b 
/d*arccos(c*x)*c*x-2*a*b/d*arccos(c*x)*ln(1-c*x-I*(-c^2*x^2+1)^(1/2))+2*a* 
b/d*arccos(c*x)*ln(1+c*x+I*(-c^2*x^2+1)^(1/2))+2*I*b^2/d*arccos(c*x)*polyl 
og(2,c*x+I*(-c^2*x^2+1)^(1/2))+2*I*a*b/d*polylog(2,c*x+I*(-c^2*x^2+1)^(1/2 
))-1/6*a*b/d*arccos(c*x)*cos(3*arccos(c*x))+1/18*a*b/d*sin(3*arccos(c*x)))
 

Fricas [F]

\[ \int \frac {x^4 (a+b \arccos (c x))^2}{d-c^2 d x^2} \, dx=\int { -\frac {{\left (b \arccos \left (c x\right ) + a\right )}^{2} x^{4}}{c^{2} d x^{2} - d} \,d x } \] Input:

integrate(x^4*(a+b*arccos(c*x))^2/(-c^2*d*x^2+d),x, algorithm="fricas")
 

Output:

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

Sympy [F]

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

integrate(x**4*(a+b*acos(c*x))**2/(-c**2*d*x**2+d),x)
                                                                                    
                                                                                    
 

Output:

-(Integral(a**2*x**4/(c**2*x**2 - 1), x) + Integral(b**2*x**4*acos(c*x)**2 
/(c**2*x**2 - 1), x) + Integral(2*a*b*x**4*acos(c*x)/(c**2*x**2 - 1), x))/ 
d
 

Maxima [F]

\[ \int \frac {x^4 (a+b \arccos (c x))^2}{d-c^2 d x^2} \, dx=\int { -\frac {{\left (b \arccos \left (c x\right ) + a\right )}^{2} x^{4}}{c^{2} d x^{2} - d} \,d x } \] Input:

integrate(x^4*(a+b*arccos(c*x))^2/(-c^2*d*x^2+d),x, algorithm="maxima")
 

Output:

-1/6*a^2*(2*(c^2*x^3 + 3*x)/(c^4*d) - 3*log(c*x + 1)/(c^5*d) + 3*log(c*x - 
 1)/(c^5*d)) - 1/6*(6*c^5*d*integrate(1/3*(6*a*b*c^4*x^4*arctan2(sqrt(c*x 
+ 1)*sqrt(-c*x + 1), c*x) - (2*b^2*c^3*x^3 + 6*b^2*c*x - 3*b^2*log(c*x + 1 
) + 3*b^2*log(-c*x + 1))*sqrt(c*x + 1)*sqrt(-c*x + 1)*arctan2(sqrt(c*x + 1 
)*sqrt(-c*x + 1), c*x))/(c^6*d*x^2 - c^4*d), x) + (2*b^2*c^3*x^3 + 6*b^2*c 
*x - 3*b^2*log(c*x + 1) + 3*b^2*log(-c*x + 1))*arctan2(sqrt(c*x + 1)*sqrt( 
-c*x + 1), c*x)^2)/(c^5*d)
 

Giac [F]

\[ \int \frac {x^4 (a+b \arccos (c x))^2}{d-c^2 d x^2} \, dx=\int { -\frac {{\left (b \arccos \left (c x\right ) + a\right )}^{2} x^{4}}{c^{2} d x^{2} - d} \,d x } \] Input:

integrate(x^4*(a+b*arccos(c*x))^2/(-c^2*d*x^2+d),x, algorithm="giac")
 

Output:

integrate(-(b*arccos(c*x) + a)^2*x^4/(c^2*d*x^2 - d), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {x^4 (a+b \arccos (c x))^2}{d-c^2 d x^2} \, dx=\int \frac {x^4\,{\left (a+b\,\mathrm {acos}\left (c\,x\right )\right )}^2}{d-c^2\,d\,x^2} \,d x \] Input:

int((x^4*(a + b*acos(c*x))^2)/(d - c^2*d*x^2),x)
 

Output:

int((x^4*(a + b*acos(c*x))^2)/(d - c^2*d*x^2), x)
 

Reduce [F]

\[ \int \frac {x^4 (a+b \arccos (c x))^2}{d-c^2 d x^2} \, dx=\frac {-12 \left (\int \frac {\mathit {acos} \left (c x \right ) x^{4}}{c^{2} x^{2}-1}d x \right ) a b \,c^{5}-6 \left (\int \frac {\mathit {acos} \left (c x \right )^{2} x^{4}}{c^{2} x^{2}-1}d x \right ) b^{2} c^{5}-3 \,\mathrm {log}\left (c^{2} x -c \right ) a^{2}+3 \,\mathrm {log}\left (c^{2} x +c \right ) a^{2}-2 a^{2} c^{3} x^{3}-6 a^{2} c x}{6 c^{5} d} \] Input:

int(x^4*(a+b*acos(c*x))^2/(-c^2*d*x^2+d),x)
 

Output:

( - 12*int((acos(c*x)*x**4)/(c**2*x**2 - 1),x)*a*b*c**5 - 6*int((acos(c*x) 
**2*x**4)/(c**2*x**2 - 1),x)*b**2*c**5 - 3*log(c**2*x - c)*a**2 + 3*log(c* 
*2*x + c)*a**2 - 2*a**2*c**3*x**3 - 6*a**2*c*x)/(6*c**5*d)