\(\int \frac {(a+b \arccos (c x))^2}{x^2 (d-c^2 d x^2)} \, dx\) [191]

Optimal result

Integrand size = 27, antiderivative size = 238 \[ \int \frac {(a+b \arccos (c x))^2}{x^2 \left (d-c^2 d x^2\right )} \, dx=-\frac {(a+b \arccos (c x))^2}{d x}-\frac {2 i c (a+b \arccos (c x))^2 \arctan \left (e^{i \arccos (c x)}\right )}{d}-\frac {4 b c (a+b \arccos (c x)) \text {arctanh}\left (e^{i \arccos (c x)}\right )}{d}+\frac {2 i b^2 c \operatorname {PolyLog}\left (2,-e^{i \arccos (c x)}\right )}{d}+\frac {2 i b c (a+b \arccos (c x)) \operatorname {PolyLog}\left (2,-i e^{i \arccos (c x)}\right )}{d}-\frac {2 i b c (a+b \arccos (c x)) \operatorname {PolyLog}\left (2,i e^{i \arccos (c x)}\right )}{d}-\frac {2 i b^2 c \operatorname {PolyLog}\left (2,e^{i \arccos (c x)}\right )}{d}-\frac {2 b^2 c \operatorname {PolyLog}\left (3,-i e^{i \arccos (c x)}\right )}{d}+\frac {2 b^2 c \operatorname {PolyLog}\left (3,i e^{i \arccos (c x)}\right )}{d} \] Output:

-(a+b*arccos(c*x))^2/d/x-2*I*c*(a+b*arccos(c*x))^2*arctan(c*x+I*(-c^2*x^2+ 
1)^(1/2))/d-4*b*c*(a+b*arccos(c*x))*arctanh(c*x+I*(-c^2*x^2+1)^(1/2))/d+2* 
I*b^2*c*polylog(2,-c*x-I*(-c^2*x^2+1)^(1/2))/d+2*I*b*c*(a+b*arccos(c*x))*p 
olylog(2,-I*(c*x+I*(-c^2*x^2+1)^(1/2)))/d-2*I*b*c*(a+b*arccos(c*x))*polylo 
g(2,I*(c*x+I*(-c^2*x^2+1)^(1/2)))/d-2*I*b^2*c*polylog(2,c*x+I*(-c^2*x^2+1) 
^(1/2))/d-2*b^2*c*polylog(3,-I*(c*x+I*(-c^2*x^2+1)^(1/2)))/d+2*b^2*c*polyl 
og(3,I*(c*x+I*(-c^2*x^2+1)^(1/2)))/d
 

Mathematica [A] (verified)

Time = 0.79 (sec) , antiderivative size = 410, normalized size of antiderivative = 1.72 \[ \int \frac {(a+b \arccos (c x))^2}{x^2 \left (d-c^2 d x^2\right )} \, dx=-\frac {a^2}{d x}-\frac {a^2 c \log (1-c x)}{2 d}+\frac {a^2 c \log (1+c x)}{2 d}-\frac {2 a b c \left (\frac {\arccos (c x)}{c x}+\log (c x)-\log \left (1+\sqrt {1-c^2 x^2}\right )+\frac {1}{2} \left (\frac {1}{2} i \arccos (c x)^2-2 \arccos (c x) \log \left (1+e^{i \arccos (c x)}\right )+2 i \operatorname {PolyLog}\left (2,-e^{i \arccos (c x)}\right )\right )+\frac {1}{2} \left (2 \arccos (c x) \log \left (1-e^{i \arccos (c x)}\right )-2 i \left (\frac {1}{4} \arccos (c x)^2+\operatorname {PolyLog}\left (2,e^{i \arccos (c x)}\right )\right )\right )\right )}{d}-\frac {b^2 c \left (\frac {\arccos (c x)^2}{c x}+\arccos (c x)^2 \left (\log \left (1-e^{i \arccos (c x)}\right )-\log \left (1+e^{i \arccos (c x)}\right )\right )-2 \left (\arccos (c x) \left (\log \left (1-i e^{i \arccos (c x)}\right )-\log \left (1+i e^{i \arccos (c x)}\right )\right )+i \left (\operatorname {PolyLog}\left (2,-i e^{i \arccos (c x)}\right )-\operatorname {PolyLog}\left (2,i e^{i \arccos (c x)}\right )\right )\right )+2 i \arccos (c x) \left (\operatorname {PolyLog}\left (2,-e^{i \arccos (c x)}\right )-\operatorname {PolyLog}\left (2,e^{i \arccos (c x)}\right )\right )+2 \left (-\operatorname {PolyLog}\left (3,-e^{i \arccos (c x)}\right )+\operatorname {PolyLog}\left (3,e^{i \arccos (c x)}\right )\right )\right )}{d} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 1.63 (sec) , antiderivative size = 212, normalized size of antiderivative = 0.89, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.481, Rules used = {5205, 27, 5165, 3042, 4671, 3011, 2720, 5219, 3042, 4669, 2715, 2838, 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.

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

\(\Big \downarrow \) 5205

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 5165

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4671

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

\(\Big \downarrow \) 3011

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

\(\Big \downarrow \) 2720

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

\(\Big \downarrow \) 5219

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4669

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

\(\Big \downarrow \) 2715

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

\(\Big \downarrow \) 2838

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

\(\Big \downarrow \) 7143

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

Input:

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

Output:

-((a + b*ArcCos[c*x])^2/(d*x)) + (2*b*c*((-2*I)*(a + b*ArcCos[c*x])*ArcTan 
[E^(I*ArcCos[c*x])] + I*b*PolyLog[2, (-I)*E^(I*ArcCos[c*x])] - I*b*PolyLog 
[2, I*E^(I*ArcCos[c*x])]))/d - (c*(-2*(a + b*ArcCos[c*x])^2*ArcTanh[E^(I*A 
rcCos[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])]) - 2*b*(I*(a + b*ArcCos[c*x])*PolyLog[2, 
 E^(I*ArcCos[c*x])] - b*PolyLog[3, E^(I*ArcCos[c*x])])))/d
 

Defintions of rubi rules used

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 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 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 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 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 4669

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

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 5205

Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*((d_) + (e_. 
)*(x_)^2)^(p_), x_Symbol] :> Simp[(f*x)^(m + 1)*(d + e*x^2)^(p + 1)*((a + b 
*ArcCos[c*x])^n/(d*f*(m + 1))), x] + (Simp[c^2*((m + 2*p + 3)/(f^2*(m + 1)) 
)   Int[(f*x)^(m + 2)*(d + e*x^2)^p*(a + b*ArcCos[c*x])^n, x], x] + Simp[b* 
c*(n/(f*(m + 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] && ILtQ[m, -1]
 

rule 5219

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

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 = 0.62 (sec) , antiderivative size = 452, normalized size of antiderivative = 1.90

method	result	size
parts	\(-\frac {a^{2} \left (\frac {c \ln \left (c x -1\right )}{2}+\frac {1}{x}-\frac {c \ln \left (c x +1\right )}{2}\right )}{d}-\frac {b^{2} c \left (\frac {\arccos \left (c x \right )^{2}}{c x}+\arccos \left (c x \right )^{2} \ln \left (1-c x -i \sqrt {-c^{2} x^{2}+1}\right )-\arccos \left (c x \right )^{2} \ln \left (1+c x +i \sqrt {-c^{2} x^{2}+1}\right )+2 \arccos \left (c x \right ) \ln \left (1+i \left (c x +i \sqrt {-c^{2} x^{2}+1}\right )\right )-2 \arccos \left (c x \right ) \ln \left (1-i \left (c x +i \sqrt {-c^{2} x^{2}+1}\right )\right )+2 \operatorname {polylog}\left (3, c x +i \sqrt {-c^{2} x^{2}+1}\right )-2 \operatorname {polylog}\left (3, -c x -i \sqrt {-c^{2} x^{2}+1}\right )-2 i \arccos \left (c x \right ) \operatorname {polylog}\left (2, c x +i \sqrt {-c^{2} x^{2}+1}\right )+2 i \arccos \left (c x \right ) \operatorname {polylog}\left (2, -c x -i \sqrt {-c^{2} x^{2}+1}\right )-2 i \operatorname {dilog}\left (1+i \left (c x +i \sqrt {-c^{2} x^{2}+1}\right )\right )+2 i \operatorname {dilog}\left (1-i \left (c x +i \sqrt {-c^{2} x^{2}+1}\right )\right )\right )}{d}-\frac {2 a b c \left (\frac {\arccos \left (c x \right )}{c x}+i \operatorname {dilog}\left (1+c x +i \sqrt {-c^{2} x^{2}+1}\right )+2 i \arctan \left (c x +i \sqrt {-c^{2} x^{2}+1}\right )+i \operatorname {dilog}\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )-\arccos \left (c x \right ) \ln \left (1+c x +i \sqrt {-c^{2} x^{2}+1}\right )\right )}{d}\)	\(452\)
derivativedivides	\(c \left (-\frac {a^{2} \left (\frac {\ln \left (c x -1\right )}{2}+\frac {1}{c x}-\frac {\ln \left (c x +1\right )}{2}\right )}{d}-\frac {b^{2} \left (\frac {\arccos \left (c x \right )^{2}}{c x}+\arccos \left (c x \right )^{2} \ln \left (1-c x -i \sqrt {-c^{2} x^{2}+1}\right )-\arccos \left (c x \right )^{2} \ln \left (1+c x +i \sqrt {-c^{2} x^{2}+1}\right )+2 \arccos \left (c x \right ) \ln \left (1+i \left (c x +i \sqrt {-c^{2} x^{2}+1}\right )\right )-2 \arccos \left (c x \right ) \ln \left (1-i \left (c x +i \sqrt {-c^{2} x^{2}+1}\right )\right )+2 \operatorname {polylog}\left (3, c x +i \sqrt {-c^{2} x^{2}+1}\right )-2 \operatorname {polylog}\left (3, -c x -i \sqrt {-c^{2} x^{2}+1}\right )-2 i \arccos \left (c x \right ) \operatorname {polylog}\left (2, c x +i \sqrt {-c^{2} x^{2}+1}\right )+2 i \arccos \left (c x \right ) \operatorname {polylog}\left (2, -c x -i \sqrt {-c^{2} x^{2}+1}\right )-2 i \operatorname {dilog}\left (1+i \left (c x +i \sqrt {-c^{2} x^{2}+1}\right )\right )+2 i \operatorname {dilog}\left (1-i \left (c x +i \sqrt {-c^{2} x^{2}+1}\right )\right )\right )}{d}-\frac {2 a b \left (\frac {\arccos \left (c x \right )}{c x}+i \operatorname {dilog}\left (1+c x +i \sqrt {-c^{2} x^{2}+1}\right )+2 i \arctan \left (c x +i \sqrt {-c^{2} x^{2}+1}\right )+i \operatorname {dilog}\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )-\arccos \left (c x \right ) \ln \left (1+c x +i \sqrt {-c^{2} x^{2}+1}\right )\right )}{d}\right )\)	\(454\)
default	\(c \left (-\frac {a^{2} \left (\frac {\ln \left (c x -1\right )}{2}+\frac {1}{c x}-\frac {\ln \left (c x +1\right )}{2}\right )}{d}-\frac {b^{2} \left (\frac {\arccos \left (c x \right )^{2}}{c x}+\arccos \left (c x \right )^{2} \ln \left (1-c x -i \sqrt {-c^{2} x^{2}+1}\right )-\arccos \left (c x \right )^{2} \ln \left (1+c x +i \sqrt {-c^{2} x^{2}+1}\right )+2 \arccos \left (c x \right ) \ln \left (1+i \left (c x +i \sqrt {-c^{2} x^{2}+1}\right )\right )-2 \arccos \left (c x \right ) \ln \left (1-i \left (c x +i \sqrt {-c^{2} x^{2}+1}\right )\right )+2 \operatorname {polylog}\left (3, c x +i \sqrt {-c^{2} x^{2}+1}\right )-2 \operatorname {polylog}\left (3, -c x -i \sqrt {-c^{2} x^{2}+1}\right )-2 i \arccos \left (c x \right ) \operatorname {polylog}\left (2, c x +i \sqrt {-c^{2} x^{2}+1}\right )+2 i \arccos \left (c x \right ) \operatorname {polylog}\left (2, -c x -i \sqrt {-c^{2} x^{2}+1}\right )-2 i \operatorname {dilog}\left (1+i \left (c x +i \sqrt {-c^{2} x^{2}+1}\right )\right )+2 i \operatorname {dilog}\left (1-i \left (c x +i \sqrt {-c^{2} x^{2}+1}\right )\right )\right )}{d}-\frac {2 a b \left (\frac {\arccos \left (c x \right )}{c x}+i \operatorname {dilog}\left (1+c x +i \sqrt {-c^{2} x^{2}+1}\right )+2 i \arctan \left (c x +i \sqrt {-c^{2} x^{2}+1}\right )+i \operatorname {dilog}\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )-\arccos \left (c x \right ) \ln \left (1+c x +i \sqrt {-c^{2} x^{2}+1}\right )\right )}{d}\right )\)	\(454\)

Input:

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

Output:

-a^2/d*(1/2*c*ln(c*x-1)+1/x-1/2*c*ln(c*x+1))-b^2/d*c*(arccos(c*x)^2/c/x+ar 
ccos(c*x)^2*ln(1-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*arccos(c*x)*ln(1+I*(c*x+I*(-c^2*x^2+1)^(1/2)))-2*arccos(c* 
x)*ln(1-I*(c*x+I*(-c^2*x^2+1)^(1/2)))+2*polylog(3,c*x+I*(-c^2*x^2+1)^(1/2) 
)-2*polylog(3,-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*I*arccos(c*x)*polylog(2,-c*x-I*(-c^2*x^2+1)^(1/2))-2* 
I*dilog(1+I*(c*x+I*(-c^2*x^2+1)^(1/2)))+2*I*dilog(1-I*(c*x+I*(-c^2*x^2+1)^ 
(1/2))))-2*a*b/d*c*(arccos(c*x)/c/x+I*dilog(1+c*x+I*(-c^2*x^2+1)^(1/2))+2* 
I*arctan(c*x+I*(-c^2*x^2+1)^(1/2))+I*dilog(c*x+I*(-c^2*x^2+1)^(1/2))-arcco 
s(c*x)*ln(1+c*x+I*(-c^2*x^2+1)^(1/2)))
 

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F(-2)]

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

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

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

Output:

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

Reduce [F]

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

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

Output:

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