\(\int \frac {(a+b \arccos (c x))^2}{(d-c^2 d x^2)^2} \, dx\) [198]

Optimal result

Integrand size = 24, antiderivative size = 230 \[ \int \frac {(a+b \arccos (c x))^2}{\left (d-c^2 d x^2\right )^2} \, dx=-\frac {b (a+b \arccos (c x))}{c d^2 \sqrt {1-c^2 x^2}}+\frac {x (a+b \arccos (c x))^2}{2 d^2 \left (1-c^2 x^2\right )}-\frac {i (a+b \arccos (c x))^2 \arctan \left (e^{i \arccos (c x)}\right )}{c d^2}+\frac {b^2 \text {arctanh}(c x)}{c d^2}+\frac {i b (a+b \arccos (c x)) \operatorname {PolyLog}\left (2,-i e^{i \arccos (c x)}\right )}{c d^2}-\frac {i b (a+b \arccos (c x)) \operatorname {PolyLog}\left (2,i e^{i \arccos (c x)}\right )}{c d^2}-\frac {b^2 \operatorname {PolyLog}\left (3,-i e^{i \arccos (c x)}\right )}{c d^2}+\frac {b^2 \operatorname {PolyLog}\left (3,i e^{i \arccos (c x)}\right )}{c d^2} \] Output:

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

Mathematica [A] (verified)

Time = 2.61 (sec) , antiderivative size = 401, normalized size of antiderivative = 1.74 \[ \int \frac {(a+b \arccos (c x))^2}{\left (d-c^2 d x^2\right )^2} \, dx=\frac {-\frac {4 a^2 x}{-1+c^2 x^2}-\frac {2 a^2 \log (1-c x)}{c}+\frac {2 a^2 \log (1+c x)}{c}+\frac {4 a b \left (\frac {\sqrt {1-c^2 x^2}}{1-c x}+\frac {\sqrt {1-c^2 x^2}}{1+c x}+\frac {\arccos (c x)}{1-c x}-\frac {\arccos (c x)}{1+c x}-2 \arccos (c x) \log \left (1-e^{i \arccos (c x)}\right )+2 \arccos (c x) \log \left (1+e^{i \arccos (c x)}\right )-2 i \operatorname {PolyLog}\left (2,-e^{i \arccos (c x)}\right )+2 i \operatorname {PolyLog}\left (2,e^{i \arccos (c x)}\right )\right )}{c}+\frac {b^2 \left (4 \arccos (c x) \cot \left (\frac {1}{2} \arccos (c x)\right )+\arccos (c x)^2 \csc ^2\left (\frac {1}{2} \arccos (c x)\right )-4 \arccos (c x)^2 \left (\log \left (1-e^{i \arccos (c x)}\right )-\log \left (1+e^{i \arccos (c x)}\right )\right )-8 \log \left (\tan \left (\frac {1}{2} \arccos (c x)\right )\right )-8 i \arccos (c x) \left (\operatorname {PolyLog}\left (2,-e^{i \arccos (c x)}\right )-\operatorname {PolyLog}\left (2,e^{i \arccos (c x)}\right )\right )+8 \left (\operatorname {PolyLog}\left (3,-e^{i \arccos (c x)}\right )-\operatorname {PolyLog}\left (3,e^{i \arccos (c x)}\right )\right )-\arccos (c x)^2 \sec ^2\left (\frac {1}{2} \arccos (c x)\right )+4 \arccos (c x) \tan \left (\frac {1}{2} \arccos (c x)\right )\right )}{c}}{8 d^2} \] Input:

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

Output:

((-4*a^2*x)/(-1 + c^2*x^2) - (2*a^2*Log[1 - c*x])/c + (2*a^2*Log[1 + c*x]) 
/c + (4*a*b*(Sqrt[1 - c^2*x^2]/(1 - c*x) + Sqrt[1 - c^2*x^2]/(1 + c*x) + A 
rcCos[c*x]/(1 - c*x) - ArcCos[c*x]/(1 + c*x) - 2*ArcCos[c*x]*Log[1 - E^(I* 
ArcCos[c*x])] + 2*ArcCos[c*x]*Log[1 + E^(I*ArcCos[c*x])] - (2*I)*PolyLog[2 
, -E^(I*ArcCos[c*x])] + (2*I)*PolyLog[2, E^(I*ArcCos[c*x])]))/c + (b^2*(4* 
ArcCos[c*x]*Cot[ArcCos[c*x]/2] + ArcCos[c*x]^2*Csc[ArcCos[c*x]/2]^2 - 4*Ar 
cCos[c*x]^2*(Log[1 - E^(I*ArcCos[c*x])] - Log[1 + E^(I*ArcCos[c*x])]) - 8* 
Log[Tan[ArcCos[c*x]/2]] - (8*I)*ArcCos[c*x]*(PolyLog[2, -E^(I*ArcCos[c*x]) 
] - PolyLog[2, E^(I*ArcCos[c*x])]) + 8*(PolyLog[3, -E^(I*ArcCos[c*x])] - P 
olyLog[3, E^(I*ArcCos[c*x])]) - ArcCos[c*x]^2*Sec[ArcCos[c*x]/2]^2 + 4*Arc 
Cos[c*x]*Tan[ArcCos[c*x]/2]))/c)/(8*d^2)
 

Rubi [A] (verified)

Time = 1.16 (sec) , antiderivative size = 197, normalized size of antiderivative = 0.86, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {5163, 27, 5165, 3042, 4671, 3011, 2720, 5183, 219, 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.

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

\(\Big \downarrow \) 5163

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 5165

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4671

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

\(\Big \downarrow \) 3011

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

\(\Big \downarrow \) 2720

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

\(\Big \downarrow \) 5183

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

\(\Big \downarrow \) 219

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

\(\Big \downarrow \) 7143

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

Input:

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

Output:

(x*(a + b*ArcCos[c*x])^2)/(2*d^2*(1 - c^2*x^2)) + (b*c*((a + b*ArcCos[c*x] 
)/(c^2*Sqrt[1 - c^2*x^2]) + (b*ArcTanh[c*x])/c^2))/d^2 - (-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*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] 
)]))/(2*c*d^2)
 

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 219

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* 
ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt 
Q[a, 0] || LtQ[b, 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 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 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 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 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 = 0.29 (sec) , antiderivative size = 423, normalized size of antiderivative = 1.84

method	result	size
derivativedivides	\(\frac {\frac {a^{2} \left (-\frac {1}{4 \left (c x -1\right )}-\frac {\ln \left (c x -1\right )}{4}-\frac {1}{4 \left (c x +1\right )}+\frac {\ln \left (c x +1\right )}{4}\right )}{d^{2}}+\frac {b^{2} \left (-\frac {\arccos \left (c x \right ) \left (c x \arccos \left (c x \right )+2 \sqrt {-c^{2} x^{2}+1}\right )}{2 \left (c^{2} x^{2}-1\right )}-\frac {\arccos \left (c x \right )^{2} \ln \left (1-c x -i \sqrt {-c^{2} x^{2}+1}\right )}{2}+i \arccos \left (c x \right ) \operatorname {polylog}\left (2, c x +i \sqrt {-c^{2} x^{2}+1}\right )-\operatorname {polylog}\left (3, c x +i \sqrt {-c^{2} x^{2}+1}\right )+\frac {\arccos \left (c x \right )^{2} \ln \left (1+c x +i \sqrt {-c^{2} x^{2}+1}\right )}{2}-i \arccos \left (c x \right ) \operatorname {polylog}\left (2, -c x -i \sqrt {-c^{2} x^{2}+1}\right )+\operatorname {polylog}\left (3, -c x -i \sqrt {-c^{2} x^{2}+1}\right )+2 \,\operatorname {arctanh}\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )\right )}{d^{2}}+\frac {2 a b \left (-\frac {c x \arccos \left (c x \right )+\sqrt {-c^{2} x^{2}+1}}{2 \left (c^{2} x^{2}-1\right )}-\frac {\arccos \left (c x \right ) \ln \left (1-c x -i \sqrt {-c^{2} x^{2}+1}\right )}{2}+\frac {i \operatorname {polylog}\left (2, c x +i \sqrt {-c^{2} x^{2}+1}\right )}{2}+\frac {\arccos \left (c x \right ) \ln \left (1+c x +i \sqrt {-c^{2} x^{2}+1}\right )}{2}-\frac {i \operatorname {polylog}\left (2, -c x -i \sqrt {-c^{2} x^{2}+1}\right )}{2}\right )}{d^{2}}}{c}\)	\(423\)
default	\(\frac {\frac {a^{2} \left (-\frac {1}{4 \left (c x -1\right )}-\frac {\ln \left (c x -1\right )}{4}-\frac {1}{4 \left (c x +1\right )}+\frac {\ln \left (c x +1\right )}{4}\right )}{d^{2}}+\frac {b^{2} \left (-\frac {\arccos \left (c x \right ) \left (c x \arccos \left (c x \right )+2 \sqrt {-c^{2} x^{2}+1}\right )}{2 \left (c^{2} x^{2}-1\right )}-\frac {\arccos \left (c x \right )^{2} \ln \left (1-c x -i \sqrt {-c^{2} x^{2}+1}\right )}{2}+i \arccos \left (c x \right ) \operatorname {polylog}\left (2, c x +i \sqrt {-c^{2} x^{2}+1}\right )-\operatorname {polylog}\left (3, c x +i \sqrt {-c^{2} x^{2}+1}\right )+\frac {\arccos \left (c x \right )^{2} \ln \left (1+c x +i \sqrt {-c^{2} x^{2}+1}\right )}{2}-i \arccos \left (c x \right ) \operatorname {polylog}\left (2, -c x -i \sqrt {-c^{2} x^{2}+1}\right )+\operatorname {polylog}\left (3, -c x -i \sqrt {-c^{2} x^{2}+1}\right )+2 \,\operatorname {arctanh}\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )\right )}{d^{2}}+\frac {2 a b \left (-\frac {c x \arccos \left (c x \right )+\sqrt {-c^{2} x^{2}+1}}{2 \left (c^{2} x^{2}-1\right )}-\frac {\arccos \left (c x \right ) \ln \left (1-c x -i \sqrt {-c^{2} x^{2}+1}\right )}{2}+\frac {i \operatorname {polylog}\left (2, c x +i \sqrt {-c^{2} x^{2}+1}\right )}{2}+\frac {\arccos \left (c x \right ) \ln \left (1+c x +i \sqrt {-c^{2} x^{2}+1}\right )}{2}-\frac {i \operatorname {polylog}\left (2, -c x -i \sqrt {-c^{2} x^{2}+1}\right )}{2}\right )}{d^{2}}}{c}\)	\(423\)
parts	\(\frac {a^{2} \left (-\frac {1}{4 c \left (c x -1\right )}-\frac {\ln \left (c x -1\right )}{4 c}-\frac {1}{4 c \left (c x +1\right )}+\frac {\ln \left (c x +1\right )}{4 c}\right )}{d^{2}}+\frac {b^{2} \left (-\frac {\arccos \left (c x \right ) \left (c x \arccos \left (c x \right )+2 \sqrt {-c^{2} x^{2}+1}\right )}{2 \left (c^{2} x^{2}-1\right )}-\frac {\arccos \left (c x \right )^{2} \ln \left (1-c x -i \sqrt {-c^{2} x^{2}+1}\right )}{2}+i \arccos \left (c x \right ) \operatorname {polylog}\left (2, c x +i \sqrt {-c^{2} x^{2}+1}\right )-\operatorname {polylog}\left (3, c x +i \sqrt {-c^{2} x^{2}+1}\right )+\frac {\arccos \left (c x \right )^{2} \ln \left (1+c x +i \sqrt {-c^{2} x^{2}+1}\right )}{2}-i \arccos \left (c x \right ) \operatorname {polylog}\left (2, -c x -i \sqrt {-c^{2} x^{2}+1}\right )+\operatorname {polylog}\left (3, -c x -i \sqrt {-c^{2} x^{2}+1}\right )+2 \,\operatorname {arctanh}\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )\right )}{d^{2} c}+\frac {2 a b \left (-\frac {c x \arccos \left (c x \right )+\sqrt {-c^{2} x^{2}+1}}{2 \left (c^{2} x^{2}-1\right )}-\frac {\arccos \left (c x \right ) \ln \left (1-c x -i \sqrt {-c^{2} x^{2}+1}\right )}{2}+\frac {i \operatorname {polylog}\left (2, c x +i \sqrt {-c^{2} x^{2}+1}\right )}{2}+\frac {\arccos \left (c x \right ) \ln \left (1+c x +i \sqrt {-c^{2} x^{2}+1}\right )}{2}-\frac {i \operatorname {polylog}\left (2, -c x -i \sqrt {-c^{2} x^{2}+1}\right )}{2}\right )}{d^{2} c}\)	\(437\)

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F(-2)]

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

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

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

Output:

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

Reduce [F]

\[ \int \frac {(a+b \arccos (c x))^2}{\left (d-c^2 d x^2\right )^2} \, dx=\frac {8 \left (\int \frac {\mathit {acos} \left (c x \right )}{c^{4} x^{4}-2 c^{2} x^{2}+1}d x \right ) a b \,c^{3} x^{2}-8 \left (\int \frac {\mathit {acos} \left (c x \right )}{c^{4} x^{4}-2 c^{2} x^{2}+1}d x \right ) a b c +4 \left (\int \frac {\mathit {acos} \left (c x \right )^{2}}{c^{4} x^{4}-2 c^{2} x^{2}+1}d x \right ) b^{2} c^{3} x^{2}-4 \left (\int \frac {\mathit {acos} \left (c x \right )^{2}}{c^{4} x^{4}-2 c^{2} x^{2}+1}d x \right ) b^{2} c -\mathrm {log}\left (c^{2} x -c \right ) a^{2} c^{2} x^{2}+\mathrm {log}\left (c^{2} x -c \right ) a^{2}+\mathrm {log}\left (c^{2} x +c \right ) a^{2} c^{2} x^{2}-\mathrm {log}\left (c^{2} x +c \right ) a^{2}-2 a^{2} c x}{4 c \,d^{2} \left (c^{2} x^{2}-1\right )} \] Input:

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

Output:

(8*int(acos(c*x)/(c**4*x**4 - 2*c**2*x**2 + 1),x)*a*b*c**3*x**2 - 8*int(ac 
os(c*x)/(c**4*x**4 - 2*c**2*x**2 + 1),x)*a*b*c + 4*int(acos(c*x)**2/(c**4* 
x**4 - 2*c**2*x**2 + 1),x)*b**2*c**3*x**2 - 4*int(acos(c*x)**2/(c**4*x**4 
- 2*c**2*x**2 + 1),x)*b**2*c - 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 - 2*a** 
2*c*x)/(4*c*d**2*(c**2*x**2 - 1))