\(\int \frac {x^3 (a+b \arccos (c x))^2}{(d-c^2 d x^2)^2} \, dx\) [195]

Optimal result

Integrand size = 27, antiderivative size = 227 \[ \int \frac {x^3 (a+b \arccos (c x))^2}{\left (d-c^2 d x^2\right )^2} \, dx=-\frac {b x (a+b \arccos (c x))}{c^3 d^2 \sqrt {1-c^2 x^2}}+\frac {(a+b \arccos (c x))^2}{2 c^4 d^2}+\frac {x^2 (a+b \arccos (c x))^2}{2 c^2 d^2 \left (1-c^2 x^2\right )}-\frac {i (a+b \arccos (c x))^3}{3 b c^4 d^2}+\frac {(a+b \arccos (c x))^2 \log \left (1+e^{2 i \arccos (c x)}\right )}{c^4 d^2}-\frac {b^2 \log \left (1-c^2 x^2\right )}{2 c^4 d^2}-\frac {i b (a+b \arccos (c x)) \operatorname {PolyLog}\left (2,-e^{2 i \arccos (c x)}\right )}{c^4 d^2}+\frac {b^2 \operatorname {PolyLog}\left (3,-e^{2 i \arccos (c x)}\right )}{2 c^4 d^2} \] Output:

-b*x*(a+b*arccos(c*x))/c^3/d^2/(-c^2*x^2+1)^(1/2)+1/2*(a+b*arccos(c*x))^2/ 
c^4/d^2+1/2*x^2*(a+b*arccos(c*x))^2/c^2/d^2/(-c^2*x^2+1)-1/3*I*(a+b*arccos 
(c*x))^3/b/c^4/d^2+(a+b*arccos(c*x))^2*ln(1+(c*x+I*(-c^2*x^2+1)^(1/2))^2)/ 
c^4/d^2-1/2*b^2*ln(-c^2*x^2+1)/c^4/d^2-I*b*(a+b*arccos(c*x))*polylog(2,-(c 
*x+I*(-c^2*x^2+1)^(1/2))^2)/c^4/d^2+1/2*b^2*polylog(3,-(c*x+I*(-c^2*x^2+1) 
^(1/2))^2)/c^4/d^2
 

Mathematica [A] (verified)

Time = 0.92 (sec) , antiderivative size = 344, normalized size of antiderivative = 1.52 \[ \int \frac {x^3 (a+b \arccos (c x))^2}{\left (d-c^2 d x^2\right )^2} \, dx=\frac {\frac {a^2}{1-c^2 x^2}-\frac {a b \left (\sqrt {1-c^2 x^2}-\arccos (c x)\right )}{1+c x}-\frac {a b \left (\sqrt {1-c^2 x^2}+\arccos (c x)\right )}{-1+c x}+a^2 \log \left (1-c^2 x^2\right )-i a b \left (\arccos (c x) \left (\arccos (c x)+4 i \log \left (1+e^{i \arccos (c x)}\right )\right )+4 \operatorname {PolyLog}\left (2,-e^{i \arccos (c x)}\right )\right )-i a b \left (\arccos (c x) \left (\arccos (c x)+4 i \log \left (1-e^{i \arccos (c x)}\right )\right )+4 \operatorname {PolyLog}\left (2,e^{i \arccos (c x)}\right )\right )+2 b^2 \left (-\frac {i \pi ^3}{24}+\frac {c x \arccos (c x)}{\sqrt {1-c^2 x^2}}+\frac {\arccos (c x)^2}{2-2 c^2 x^2}+\frac {1}{3} i \arccos (c x)^3+\arccos (c x)^2 \log \left (1-e^{-2 i \arccos (c x)}\right )-\frac {1}{2} \log \left (1-c^2 x^2\right )+i \arccos (c x) \operatorname {PolyLog}\left (2,e^{-2 i \arccos (c x)}\right )+\frac {1}{2} \operatorname {PolyLog}\left (3,e^{-2 i \arccos (c x)}\right )\right )}{2 c^4 d^2} \] Input:

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

Output:

(a^2/(1 - c^2*x^2) - (a*b*(Sqrt[1 - c^2*x^2] - ArcCos[c*x]))/(1 + c*x) - ( 
a*b*(Sqrt[1 - c^2*x^2] + ArcCos[c*x]))/(-1 + c*x) + a^2*Log[1 - c^2*x^2] - 
 I*a*b*(ArcCos[c*x]*(ArcCos[c*x] + (4*I)*Log[1 + E^(I*ArcCos[c*x])]) + 4*P 
olyLog[2, -E^(I*ArcCos[c*x])]) - I*a*b*(ArcCos[c*x]*(ArcCos[c*x] + (4*I)*L 
og[1 - E^(I*ArcCos[c*x])]) + 4*PolyLog[2, E^(I*ArcCos[c*x])]) + 2*b^2*((-1 
/24*I)*Pi^3 + (c*x*ArcCos[c*x])/Sqrt[1 - c^2*x^2] + ArcCos[c*x]^2/(2 - 2*c 
^2*x^2) + (I/3)*ArcCos[c*x]^3 + ArcCos[c*x]^2*Log[1 - E^((-2*I)*ArcCos[c*x 
])] - Log[1 - c^2*x^2]/2 + I*ArcCos[c*x]*PolyLog[2, E^((-2*I)*ArcCos[c*x]) 
] + PolyLog[3, E^((-2*I)*ArcCos[c*x])]/2))/(2*c^4*d^2)
 

Rubi [A] (verified)

Time = 1.65 (sec) , antiderivative size = 223, normalized size of antiderivative = 0.98, number of steps used = 15, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.519, Rules used = {5207, 27, 5181, 3042, 25, 4200, 25, 2620, 3011, 2720, 5207, 240, 5153, 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.

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

\(\Big \downarrow \) 5207

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 5181

\(\displaystyle \frac {b \int \frac {x^2 (a+b \arccos (c x))}{\left (1-c^2 x^2\right )^{3/2}}dx}{c d^2}+\frac {\int \frac {c x (a+b \arccos (c x))^2}{\sqrt {1-c^2 x^2}}d\arccos (c x)}{c^4 d^2}+\frac {x^2 (a+b \arccos (c x))^2}{2 c^2 d^2 \left (1-c^2 x^2\right )}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\int -(a+b \arccos (c x))^2 \tan \left (\arccos (c x)+\frac {\pi }{2}\right )d\arccos (c x)}{c^4 d^2}+\frac {b \int \frac {x^2 (a+b \arccos (c x))}{\left (1-c^2 x^2\right )^{3/2}}dx}{c d^2}+\frac {x^2 (a+b \arccos (c x))^2}{2 c^2 d^2 \left (1-c^2 x^2\right )}\)

\(\Big \downarrow \) 25

\(\displaystyle -\frac {\int (a+b \arccos (c x))^2 \tan \left (\arccos (c x)+\frac {\pi }{2}\right )d\arccos (c x)}{c^4 d^2}+\frac {b \int \frac {x^2 (a+b \arccos (c x))}{\left (1-c^2 x^2\right )^{3/2}}dx}{c d^2}+\frac {x^2 (a+b \arccos (c x))^2}{2 c^2 d^2 \left (1-c^2 x^2\right )}\)

\(\Big \downarrow \) 4200

\(\displaystyle \frac {2 i \int -\frac {e^{2 i \arccos (c x)} (a+b \arccos (c x))^2}{1-e^{2 i \arccos (c x)}}d\arccos (c x)-\frac {i (a+b \arccos (c x))^3}{3 b}}{c^4 d^2}+\frac {b \int \frac {x^2 (a+b \arccos (c x))}{\left (1-c^2 x^2\right )^{3/2}}dx}{c d^2}+\frac {x^2 (a+b \arccos (c x))^2}{2 c^2 d^2 \left (1-c^2 x^2\right )}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {-2 i \int \frac {e^{2 i \arccos (c x)} (a+b \arccos (c x))^2}{1-e^{2 i \arccos (c x)}}d\arccos (c x)-\frac {i (a+b \arccos (c x))^3}{3 b}}{c^4 d^2}+\frac {b \int \frac {x^2 (a+b \arccos (c x))}{\left (1-c^2 x^2\right )^{3/2}}dx}{c d^2}+\frac {x^2 (a+b \arccos (c x))^2}{2 c^2 d^2 \left (1-c^2 x^2\right )}\)

\(\Big \downarrow \) 2620

\(\displaystyle \frac {-2 i \left (\frac {1}{2} i \log \left (1-e^{2 i \arccos (c x)}\right ) (a+b \arccos (c x))^2-i b \int (a+b \arccos (c x)) \log \left (1-e^{2 i \arccos (c x)}\right )d\arccos (c x)\right )-\frac {i (a+b \arccos (c x))^3}{3 b}}{c^4 d^2}+\frac {b \int \frac {x^2 (a+b \arccos (c x))}{\left (1-c^2 x^2\right )^{3/2}}dx}{c d^2}+\frac {x^2 (a+b \arccos (c x))^2}{2 c^2 d^2 \left (1-c^2 x^2\right )}\)

\(\Big \downarrow \) 3011

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

\(\Big \downarrow \) 2720

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

\(\Big \downarrow \) 5207

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

\(\Big \downarrow \) 240

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

\(\Big \downarrow \) 5153

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

\(\Big \downarrow \) 7143

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

Input:

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

Output:

(x^2*(a + b*ArcCos[c*x])^2)/(2*c^2*d^2*(1 - c^2*x^2)) + (b*((x*(a + b*ArcC 
os[c*x]))/(c^2*Sqrt[1 - c^2*x^2]) + (a + b*ArcCos[c*x])^2/(2*b*c^3) - (b*L 
og[1 - c^2*x^2])/(2*c^3)))/(c*d^2) + (((-1/3*I)*(a + b*ArcCos[c*x])^3)/b - 
 (2*I)*((I/2)*(a + b*ArcCos[c*x])^2*Log[1 - E^((2*I)*ArcCos[c*x])] - I*b*( 
(I/2)*(a + b*ArcCos[c*x])*PolyLog[2, E^((2*I)*ArcCos[c*x])] - (b*PolyLog[3 
, E^((2*I)*ArcCos[c*x])])/4)))/(c^4*d^2)
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], 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 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 5153

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

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 5207

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/(2*e*(p + 1))), x] + (-Simp[f^2*((m - 1)/(2*e*(p + 1))) 
 Int[(f*x)^(m - 2)*(d + e*x^2)^(p + 1)*(a + b*ArcCos[c*x])^n, x], x] - Simp 
[b*f*(n/(2*c*(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}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && LtQ[p, -1] && IG 
tQ[m, 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 [B] (verified)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 505 vs. \(2 (246 ) = 492\).

Time = 0.46 (sec) , antiderivative size = 506, normalized size of antiderivative = 2.23

method	result	size
derivativedivides	\(\frac {\frac {a^{2} \left (-\frac {1}{4 \left (c x -1\right )}+\frac {\ln \left (c x -1\right )}{2}+\frac {1}{4 c x +4}+\frac {\ln \left (c x +1\right )}{2}\right )}{d^{2}}+\frac {b^{2} \left (-\frac {i \arccos \left (c x \right )^{3}}{3}-\frac {\left (2 i c^{2} x^{2}+2 c x \sqrt {-c^{2} x^{2}+1}-2 i+\arccos \left (c x \right )\right ) \arccos \left (c x \right )}{2 \left (c^{2} x^{2}-1\right )}+\arccos \left (c x \right )^{2} \ln \left (1+c x +i \sqrt {-c^{2} x^{2}+1}\right )-2 i \arccos \left (c x \right ) \operatorname {polylog}\left (2, -c x -i \sqrt {-c^{2} x^{2}+1}\right )+2 \operatorname {polylog}\left (3, -c x -i \sqrt {-c^{2} x^{2}+1}\right )+\arccos \left (c x \right )^{2} \ln \left (1-c x -i \sqrt {-c^{2} x^{2}+1}\right )-2 i \arccos \left (c x \right ) \operatorname {polylog}\left (2, c x +i \sqrt {-c^{2} x^{2}+1}\right )+2 \operatorname {polylog}\left (3, c x +i \sqrt {-c^{2} x^{2}+1}\right )+2 \ln \left (c x +i \sqrt {-c^{2} x^{2}+1}\right )-\ln \left (1+c x +i \sqrt {-c^{2} x^{2}+1}\right )-\ln \left (i \sqrt {-c^{2} x^{2}+1}+c x -1\right )\right )}{d^{2}}+\frac {2 a b \left (-\frac {i \arccos \left (c x \right )^{2}}{2}-\frac {i c^{2} x^{2}+c x \sqrt {-c^{2} x^{2}+1}+\arccos \left (c x \right )-i}{2 \left (c^{2} x^{2}-1\right )}+\arccos \left (c x \right ) \ln \left (1+c x +i \sqrt {-c^{2} x^{2}+1}\right )+\arccos \left (c x \right ) \ln \left (1-c x -i \sqrt {-c^{2} x^{2}+1}\right )-i \operatorname {polylog}\left (2, -c x -i \sqrt {-c^{2} x^{2}+1}\right )-i \operatorname {polylog}\left (2, c x +i \sqrt {-c^{2} x^{2}+1}\right )\right )}{d^{2}}}{c^{4}}\)	\(506\)
default	\(\frac {\frac {a^{2} \left (-\frac {1}{4 \left (c x -1\right )}+\frac {\ln \left (c x -1\right )}{2}+\frac {1}{4 c x +4}+\frac {\ln \left (c x +1\right )}{2}\right )}{d^{2}}+\frac {b^{2} \left (-\frac {i \arccos \left (c x \right )^{3}}{3}-\frac {\left (2 i c^{2} x^{2}+2 c x \sqrt {-c^{2} x^{2}+1}-2 i+\arccos \left (c x \right )\right ) \arccos \left (c x \right )}{2 \left (c^{2} x^{2}-1\right )}+\arccos \left (c x \right )^{2} \ln \left (1+c x +i \sqrt {-c^{2} x^{2}+1}\right )-2 i \arccos \left (c x \right ) \operatorname {polylog}\left (2, -c x -i \sqrt {-c^{2} x^{2}+1}\right )+2 \operatorname {polylog}\left (3, -c x -i \sqrt {-c^{2} x^{2}+1}\right )+\arccos \left (c x \right )^{2} \ln \left (1-c x -i \sqrt {-c^{2} x^{2}+1}\right )-2 i \arccos \left (c x \right ) \operatorname {polylog}\left (2, c x +i \sqrt {-c^{2} x^{2}+1}\right )+2 \operatorname {polylog}\left (3, c x +i \sqrt {-c^{2} x^{2}+1}\right )+2 \ln \left (c x +i \sqrt {-c^{2} x^{2}+1}\right )-\ln \left (1+c x +i \sqrt {-c^{2} x^{2}+1}\right )-\ln \left (i \sqrt {-c^{2} x^{2}+1}+c x -1\right )\right )}{d^{2}}+\frac {2 a b \left (-\frac {i \arccos \left (c x \right )^{2}}{2}-\frac {i c^{2} x^{2}+c x \sqrt {-c^{2} x^{2}+1}+\arccos \left (c x \right )-i}{2 \left (c^{2} x^{2}-1\right )}+\arccos \left (c x \right ) \ln \left (1+c x +i \sqrt {-c^{2} x^{2}+1}\right )+\arccos \left (c x \right ) \ln \left (1-c x -i \sqrt {-c^{2} x^{2}+1}\right )-i \operatorname {polylog}\left (2, -c x -i \sqrt {-c^{2} x^{2}+1}\right )-i \operatorname {polylog}\left (2, c x +i \sqrt {-c^{2} x^{2}+1}\right )\right )}{d^{2}}}{c^{4}}\)	\(506\)
parts	\(\frac {a^{2} \left (-\frac {1}{4 c^{4} \left (c x -1\right )}+\frac {\ln \left (c x -1\right )}{2 c^{4}}+\frac {1}{4 c^{4} \left (c x +1\right )}+\frac {\ln \left (c x +1\right )}{2 c^{4}}\right )}{d^{2}}+\frac {b^{2} \left (-\frac {i \arccos \left (c x \right )^{3}}{3}-\frac {\left (2 i c^{2} x^{2}+2 c x \sqrt {-c^{2} x^{2}+1}-2 i+\arccos \left (c x \right )\right ) \arccos \left (c x \right )}{2 \left (c^{2} x^{2}-1\right )}+\arccos \left (c x \right )^{2} \ln \left (1+c x +i \sqrt {-c^{2} x^{2}+1}\right )-2 i \arccos \left (c x \right ) \operatorname {polylog}\left (2, -c x -i \sqrt {-c^{2} x^{2}+1}\right )+2 \operatorname {polylog}\left (3, -c x -i \sqrt {-c^{2} x^{2}+1}\right )+\arccos \left (c x \right )^{2} \ln \left (1-c x -i \sqrt {-c^{2} x^{2}+1}\right )-2 i \arccos \left (c x \right ) \operatorname {polylog}\left (2, c x +i \sqrt {-c^{2} x^{2}+1}\right )+2 \operatorname {polylog}\left (3, c x +i \sqrt {-c^{2} x^{2}+1}\right )+2 \ln \left (c x +i \sqrt {-c^{2} x^{2}+1}\right )-\ln \left (1+c x +i \sqrt {-c^{2} x^{2}+1}\right )-\ln \left (i \sqrt {-c^{2} x^{2}+1}+c x -1\right )\right )}{d^{2} c^{4}}+\frac {2 a b \left (-\frac {i \arccos \left (c x \right )^{2}}{2}-\frac {i c^{2} x^{2}+c x \sqrt {-c^{2} x^{2}+1}+\arccos \left (c x \right )-i}{2 \left (c^{2} x^{2}-1\right )}+\arccos \left (c x \right ) \ln \left (1+c x +i \sqrt {-c^{2} x^{2}+1}\right )+\arccos \left (c x \right ) \ln \left (1-c x -i \sqrt {-c^{2} x^{2}+1}\right )-i \operatorname {polylog}\left (2, -c x -i \sqrt {-c^{2} x^{2}+1}\right )-i \operatorname {polylog}\left (2, c x +i \sqrt {-c^{2} x^{2}+1}\right )\right )}{d^{2} c^{4}}\)	\(520\)

Input:

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

Output:

1/c^4*(a^2/d^2*(-1/4/(c*x-1)+1/2*ln(c*x-1)+1/4/(c*x+1)+1/2*ln(c*x+1))+b^2/ 
d^2*(-1/3*I*arccos(c*x)^3-1/2*(2*I*c^2*x^2+2*c*x*(-c^2*x^2+1)^(1/2)-2*I+ar 
ccos(c*x))*arccos(c*x)/(c^2*x^2-1)+arccos(c*x)^2*ln(1+c*x+I*(-c^2*x^2+1)^( 
1/2))-2*I*arccos(c*x)*polylog(2,-c*x-I*(-c^2*x^2+1)^(1/2))+2*polylog(3,-c* 
x-I*(-c^2*x^2+1)^(1/2))+arccos(c*x)^2*ln(1-c*x-I*(-c^2*x^2+1)^(1/2))-2*I*a 
rccos(c*x)*polylog(2,c*x+I*(-c^2*x^2+1)^(1/2))+2*polylog(3,c*x+I*(-c^2*x^2 
+1)^(1/2))+2*ln(c*x+I*(-c^2*x^2+1)^(1/2))-ln(1+c*x+I*(-c^2*x^2+1)^(1/2))-l 
n(I*(-c^2*x^2+1)^(1/2)+c*x-1))+2*a*b/d^2*(-1/2*I*arccos(c*x)^2-1/2*(I*c^2* 
x^2+c*x*(-c^2*x^2+1)^(1/2)+arccos(c*x)-I)/(c^2*x^2-1)+arccos(c*x)*ln(1+c*x 
+I*(-c^2*x^2+1)^(1/2))+arccos(c*x)*ln(1-c*x-I*(-c^2*x^2+1)^(1/2))-I*polylo 
g(2,-c*x-I*(-c^2*x^2+1)^(1/2))-I*polylog(2,c*x+I*(-c^2*x^2+1)^(1/2))))
 

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

(Integral(a**2*x**3/(c**4*x**4 - 2*c**2*x**2 + 1), x) + Integral(b**2*x**3 
*acos(c*x)**2/(c**4*x**4 - 2*c**2*x**2 + 1), x) + Integral(2*a*b*x**3*acos 
(c*x)/(c**4*x**4 - 2*c**2*x**2 + 1), x))/d**2
 

Maxima [F]

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

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

Output:

-1/2*a^2*(1/(c^6*d^2*x^2 - c^4*d^2) - log(c^2*x^2 - 1)/(c^4*d^2)) - 1/2*(( 
b^2 - (b^2*c^2*x^2 - b^2)*log(c*x + 1) - (b^2*c^2*x^2 - b^2)*log(-c*x + 1) 
)*arctan2(sqrt(c*x + 1)*sqrt(-c*x + 1), c*x)^2 + 2*(c^6*d^2*x^2 - c^4*d^2) 
*integrate(-(2*a*b*c^3*x^3*arctan2(sqrt(c*x + 1)*sqrt(-c*x + 1), c*x) + (b 
^2 - (b^2*c^2*x^2 - b^2)*log(c*x + 1) - (b^2*c^2*x^2 - 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^7*d^2*x^4 - 2*c^5*d^2*x^2 + c^3*d^2), x))/(c^6*d^2*x^2 - c^4*d^2)
 

Giac [F(-2)]

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

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

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

Output:

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

Reduce [F]

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

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

Output:

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