\(\int \frac {x^2 (a+b \arccos (c x))^2}{d-c^2 d x^2} \, dx\) [187]

Optimal result

Integrand size = 27, antiderivative size = 218 \[ \int \frac {x^2 (a+b \arccos (c x))^2}{d-c^2 d x^2} \, dx=\frac {2 b^2 x}{c^2 d}-\frac {2 b \sqrt {1-c^2 x^2} (a+b \arccos (c x))}{c^3 d}-\frac {x (a+b \arccos (c x))^2}{c^2 d}-\frac {2 i (a+b \arccos (c x))^2 \arctan \left (e^{i \arccos (c x)}\right )}{c^3 d}+\frac {2 i b (a+b \arccos (c x)) \operatorname {PolyLog}\left (2,-i e^{i \arccos (c x)}\right )}{c^3 d}-\frac {2 i b (a+b \arccos (c x)) \operatorname {PolyLog}\left (2,i e^{i \arccos (c x)}\right )}{c^3 d}-\frac {2 b^2 \operatorname {PolyLog}\left (3,-i e^{i \arccos (c x)}\right )}{c^3 d}+\frac {2 b^2 \operatorname {PolyLog}\left (3,i e^{i \arccos (c x)}\right )}{c^3 d} \] Output:

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

Mathematica [A] (verified)

Time = 0.72 (sec) , antiderivative size = 293, normalized size of antiderivative = 1.34 \[ \int \frac {x^2 (a+b \arccos (c x))^2}{d-c^2 d x^2} \, dx=\frac {-2 a^2 c x+4 b^2 c x+4 a b \sqrt {1-c^2 x^2}-4 a b c x \arccos (c x)+4 b^2 \sqrt {1-c^2 x^2} \arccos (c x)-2 b^2 c x \arccos (c x)^2-4 a b \arccos (c x) \log \left (1-e^{i \arccos (c x)}\right )-2 b^2 \arccos (c x)^2 \log \left (1-e^{i \arccos (c x)}\right )+4 a b \arccos (c x) \log \left (1+e^{i \arccos (c x)}\right )+2 b^2 \arccos (c x)^2 \log \left (1+e^{i \arccos (c x)}\right )-a^2 \log (1-c x)+a^2 \log (1+c x)-4 i b (a+b \arccos (c x)) \operatorname {PolyLog}\left (2,-e^{i \arccos (c x)}\right )+4 i b (a+b \arccos (c x)) \operatorname {PolyLog}\left (2,e^{i \arccos (c x)}\right )+4 b^2 \operatorname {PolyLog}\left (3,-e^{i \arccos (c x)}\right )-4 b^2 \operatorname {PolyLog}\left (3,e^{i \arccos (c x)}\right )}{2 c^3 d} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 1.18 (sec) , antiderivative size = 186, normalized size of antiderivative = 0.85, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.370, Rules used = {5211, 27, 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.

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

\(\Big \downarrow \) 5211

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 5165

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4671

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

\(\Big \downarrow \) 3011

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

\(\Big \downarrow \) 2720

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

\(\Big \downarrow \) 5183

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

\(\Big \downarrow \) 24

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

\(\Big \downarrow \) 7143

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

Input:

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

Output:

-((x*(a + b*ArcCos[c*x])^2)/(c^2*d)) - (2*b*(-((b*x)/c) - (Sqrt[1 - c^2*x^ 
2]*(a + b*ArcCos[c*x]))/c^2))/(c*d) - (-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])*PolyLo 
g[2, E^(I*ArcCos[c*x])] - b*PolyLog[3, E^(I*ArcCos[c*x])]))/(c^3*d)
 

Defintions of rubi rules used

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 = 0.48 (sec) , antiderivative size = 438, normalized size of antiderivative = 2.01

method	result	size
derivativedivides	\(\frac {-\frac {a^{2} \left (c x +\frac {\ln \left (c x -1\right )}{2}-\frac {\ln \left (c x +1\right )}{2}\right )}{d}+\frac {2 b^{2} \arccos \left (c x \right ) \sqrt {-c^{2} x^{2}+1}}{d}+\frac {b^{2} \arccos \left (c x \right )^{2} \ln \left (1+c x +i \sqrt {-c^{2} x^{2}+1}\right )}{d}-\frac {b^{2} \arccos \left (c x \right )^{2} \ln \left (1-c x -i \sqrt {-c^{2} x^{2}+1}\right )}{d}-\frac {b^{2} \arccos \left (c x \right )^{2} c x}{d}-\frac {2 i b^{2} \arccos \left (c x \right ) \operatorname {polylog}\left (2, -c x -i \sqrt {-c^{2} x^{2}+1}\right )}{d}-\frac {2 i a b \operatorname {polylog}\left (2, -c x -i \sqrt {-c^{2} x^{2}+1}\right )}{d}+\frac {2 b^{2} \operatorname {polylog}\left (3, -c x -i \sqrt {-c^{2} x^{2}+1}\right )}{d}-\frac {2 b^{2} \operatorname {polylog}\left (3, c x +i \sqrt {-c^{2} x^{2}+1}\right )}{d}+\frac {2 b^{2} c x}{d}+\frac {2 a b \sqrt {-c^{2} x^{2}+1}}{d}+\frac {2 a b \arccos \left (c x \right ) \ln \left (1+c x +i \sqrt {-c^{2} x^{2}+1}\right )}{d}-\frac {2 a b \arccos \left (c x \right ) \ln \left (1-c x -i \sqrt {-c^{2} x^{2}+1}\right )}{d}-\frac {2 a b \arccos \left (c x \right ) c x}{d}+\frac {2 i b^{2} \arccos \left (c x \right ) \operatorname {polylog}\left (2, c x +i \sqrt {-c^{2} x^{2}+1}\right )}{d}+\frac {2 i a b \operatorname {polylog}\left (2, c x +i \sqrt {-c^{2} x^{2}+1}\right )}{d}}{c^{3}}\)	\(438\)
default	\(\frac {-\frac {a^{2} \left (c x +\frac {\ln \left (c x -1\right )}{2}-\frac {\ln \left (c x +1\right )}{2}\right )}{d}+\frac {2 b^{2} \arccos \left (c x \right ) \sqrt {-c^{2} x^{2}+1}}{d}+\frac {b^{2} \arccos \left (c x \right )^{2} \ln \left (1+c x +i \sqrt {-c^{2} x^{2}+1}\right )}{d}-\frac {b^{2} \arccos \left (c x \right )^{2} \ln \left (1-c x -i \sqrt {-c^{2} x^{2}+1}\right )}{d}-\frac {b^{2} \arccos \left (c x \right )^{2} c x}{d}-\frac {2 i b^{2} \arccos \left (c x \right ) \operatorname {polylog}\left (2, -c x -i \sqrt {-c^{2} x^{2}+1}\right )}{d}-\frac {2 i a b \operatorname {polylog}\left (2, -c x -i \sqrt {-c^{2} x^{2}+1}\right )}{d}+\frac {2 b^{2} \operatorname {polylog}\left (3, -c x -i \sqrt {-c^{2} x^{2}+1}\right )}{d}-\frac {2 b^{2} \operatorname {polylog}\left (3, c x +i \sqrt {-c^{2} x^{2}+1}\right )}{d}+\frac {2 b^{2} c x}{d}+\frac {2 a b \sqrt {-c^{2} x^{2}+1}}{d}+\frac {2 a b \arccos \left (c x \right ) \ln \left (1+c x +i \sqrt {-c^{2} x^{2}+1}\right )}{d}-\frac {2 a b \arccos \left (c x \right ) \ln \left (1-c x -i \sqrt {-c^{2} x^{2}+1}\right )}{d}-\frac {2 a b \arccos \left (c x \right ) c x}{d}+\frac {2 i b^{2} \arccos \left (c x \right ) \operatorname {polylog}\left (2, c x +i \sqrt {-c^{2} x^{2}+1}\right )}{d}+\frac {2 i a b \operatorname {polylog}\left (2, c x +i \sqrt {-c^{2} x^{2}+1}\right )}{d}}{c^{3}}\)	\(438\)
parts	\(-\frac {a^{2} \left (\frac {x}{c^{2}}+\frac {\ln \left (c x -1\right )}{2 c^{3}}-\frac {\ln \left (c x +1\right )}{2 c^{3}}\right )}{d}+\frac {2 b^{2} \sqrt {-c^{2} x^{2}+1}\, \arccos \left (c x \right )}{d \,c^{3}}-\frac {b^{2} \arccos \left (c x \right )^{2} x}{d \,c^{2}}-\frac {b^{2} \arccos \left (c x \right )^{2} \ln \left (1-c x -i \sqrt {-c^{2} x^{2}+1}\right )}{d \,c^{3}}+\frac {2 i b^{2} \arccos \left (c x \right ) \operatorname {polylog}\left (2, c x +i \sqrt {-c^{2} x^{2}+1}\right )}{d \,c^{3}}+\frac {b^{2} \arccos \left (c x \right )^{2} \ln \left (1+c x +i \sqrt {-c^{2} x^{2}+1}\right )}{d \,c^{3}}-\frac {2 i a b \operatorname {polylog}\left (2, -c x -i \sqrt {-c^{2} x^{2}+1}\right )}{d \,c^{3}}+\frac {2 b^{2} x}{c^{2} d}-\frac {2 b^{2} \operatorname {polylog}\left (3, c x +i \sqrt {-c^{2} x^{2}+1}\right )}{d \,c^{3}}+\frac {2 b^{2} \operatorname {polylog}\left (3, -c x -i \sqrt {-c^{2} x^{2}+1}\right )}{d \,c^{3}}+\frac {2 a b \sqrt {-c^{2} x^{2}+1}}{d \,c^{3}}-\frac {2 a b \arccos \left (c x \right ) x}{d \,c^{2}}-\frac {2 a b \arccos \left (c x \right ) \ln \left (1-c x -i \sqrt {-c^{2} x^{2}+1}\right )}{d \,c^{3}}+\frac {2 i a b \operatorname {polylog}\left (2, c x +i \sqrt {-c^{2} x^{2}+1}\right )}{d \,c^{3}}+\frac {2 a b \arccos \left (c x \right ) \ln \left (1+c x +i \sqrt {-c^{2} x^{2}+1}\right )}{d \,c^{3}}-\frac {2 i b^{2} \arccos \left (c x \right ) \operatorname {polylog}\left (2, -c x -i \sqrt {-c^{2} x^{2}+1}\right )}{d \,c^{3}}\)	\(484\)

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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