\(\int \frac {x^2 (a+b \arccos (c x))}{(d-c^2 d x^2)^3} \, dx\) [50]

Optimal result

Integrand size = 25, antiderivative size = 202 \[ \int \frac {x^2 (a+b \arccos (c x))}{\left (d-c^2 d x^2\right )^3} \, dx=-\frac {b}{12 c^3 d^3 \left (1-c^2 x^2\right )^{3/2}}+\frac {b}{8 c^3 d^3 \sqrt {1-c^2 x^2}}+\frac {x (a+b \arccos (c x))}{4 c^2 d^3 \left (1-c^2 x^2\right )^2}-\frac {x (a+b \arccos (c x))}{8 c^2 d^3 \left (1-c^2 x^2\right )}+\frac {i (a+b \arccos (c x)) \arctan \left (e^{i \arccos (c x)}\right )}{4 c^3 d^3}-\frac {i b \operatorname {PolyLog}\left (2,-i e^{i \arccos (c x)}\right )}{8 c^3 d^3}+\frac {i b \operatorname {PolyLog}\left (2,i e^{i \arccos (c x)}\right )}{8 c^3 d^3} \] Output:

-1/12*b/c^3/d^3/(-c^2*x^2+1)^(3/2)+1/8*b/c^3/d^3/(-c^2*x^2+1)^(1/2)+1/4*x* 
(a+b*arccos(c*x))/c^2/d^3/(-c^2*x^2+1)^2-1/8*x*(a+b*arccos(c*x))/c^2/d^3/( 
-c^2*x^2+1)+1/4*I*(a+b*arccos(c*x))*arctan(c*x+I*(-c^2*x^2+1)^(1/2))/c^3/d 
^3-1/8*I*b*polylog(2,-I*(c*x+I*(-c^2*x^2+1)^(1/2)))/c^3/d^3+1/8*I*b*polylo 
g(2,I*(c*x+I*(-c^2*x^2+1)^(1/2)))/c^3/d^3
 

Mathematica [A] (verified)

Time = 0.27 (sec) , antiderivative size = 363, normalized size of antiderivative = 1.80 \[ \int \frac {x^2 (a+b \arccos (c x))}{\left (d-c^2 d x^2\right )^3} \, dx=\frac {a x}{4 c^2 d^3 \left (-1+c^2 x^2\right )^2}+\frac {a x}{8 c^2 d^3 \left (-1+c^2 x^2\right )}+\frac {a \log (1-c x)}{16 c^3 d^3}-\frac {a \log (1+c x)}{16 c^3 d^3}-\frac {b \left (\frac {(-2+c x) \sqrt {1-c^2 x^2}-3 \arccos (c x)}{48 c^3 (-1+c x)^2}-\frac {(2+c x) \sqrt {1-c^2 x^2}-3 \arccos (c x)}{48 c^3 (1+c x)^2}+\frac {\sqrt {1-c^2 x^2}-\arccos (c x)}{16 c^2 \left (c+c^2 x\right )}+\frac {\sqrt {1-c^2 x^2}+\arccos (c x)}{16 c^2 \left (c-c^2 x\right )}+\frac {-\frac {i \arccos (c x)^2}{2 c}+\frac {2 \arccos (c x) \log \left (1+e^{i \arccos (c x)}\right )}{c}-\frac {2 i \operatorname {PolyLog}\left (2,-e^{i \arccos (c x)}\right )}{c}}{16 c^2}+\frac {i \left (\arccos (c x) \left (\arccos (c x)+4 i \log \left (1-e^{i \arccos (c x)}\right )\right )+4 \operatorname {PolyLog}\left (2,e^{i \arccos (c x)}\right )\right )}{32 c^3}\right )}{d^3} \] Input:

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

Output:

(a*x)/(4*c^2*d^3*(-1 + c^2*x^2)^2) + (a*x)/(8*c^2*d^3*(-1 + c^2*x^2)) + (a 
*Log[1 - c*x])/(16*c^3*d^3) - (a*Log[1 + c*x])/(16*c^3*d^3) - (b*(((-2 + c 
*x)*Sqrt[1 - c^2*x^2] - 3*ArcCos[c*x])/(48*c^3*(-1 + c*x)^2) - ((2 + c*x)* 
Sqrt[1 - c^2*x^2] - 3*ArcCos[c*x])/(48*c^3*(1 + c*x)^2) + (Sqrt[1 - c^2*x^ 
2] - ArcCos[c*x])/(16*c^2*(c + c^2*x)) + (Sqrt[1 - c^2*x^2] + ArcCos[c*x]) 
/(16*c^2*(c - c^2*x)) + (((-1/2*I)*ArcCos[c*x]^2)/c + (2*ArcCos[c*x]*Log[1 
 + E^(I*ArcCos[c*x])])/c - ((2*I)*PolyLog[2, -E^(I*ArcCos[c*x])])/c)/(16*c 
^2) + ((I/32)*(ArcCos[c*x]*(ArcCos[c*x] + (4*I)*Log[1 - E^(I*ArcCos[c*x])] 
) + 4*PolyLog[2, E^(I*ArcCos[c*x])]))/c^3))/d^3
 

Rubi [A] (verified)

Time = 0.74 (sec) , antiderivative size = 180, normalized size of antiderivative = 0.89, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {5207, 27, 241, 5163, 241, 5165, 3042, 4671, 2715, 2838}

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.

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

\(\Big \downarrow \) 5207

\(\displaystyle -\frac {\int \frac {a+b \arccos (c x)}{d^2 \left (1-c^2 x^2\right )^2}dx}{4 c^2 d}+\frac {b \int \frac {x}{\left (1-c^2 x^2\right )^{5/2}}dx}{4 c d^3}+\frac {x (a+b \arccos (c x))}{4 c^2 d^3 \left (1-c^2 x^2\right )^2}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {\int \frac {a+b \arccos (c x)}{\left (1-c^2 x^2\right )^2}dx}{4 c^2 d^3}+\frac {b \int \frac {x}{\left (1-c^2 x^2\right )^{5/2}}dx}{4 c d^3}+\frac {x (a+b \arccos (c x))}{4 c^2 d^3 \left (1-c^2 x^2\right )^2}\)

\(\Big \downarrow \) 241

\(\displaystyle -\frac {\int \frac {a+b \arccos (c x)}{\left (1-c^2 x^2\right )^2}dx}{4 c^2 d^3}+\frac {x (a+b \arccos (c x))}{4 c^2 d^3 \left (1-c^2 x^2\right )^2}+\frac {b}{12 c^3 d^3 \left (1-c^2 x^2\right )^{3/2}}\)

\(\Big \downarrow \) 5163

\(\displaystyle -\frac {\frac {1}{2} \int \frac {a+b \arccos (c x)}{1-c^2 x^2}dx+\frac {1}{2} b c \int \frac {x}{\left (1-c^2 x^2\right )^{3/2}}dx+\frac {x (a+b \arccos (c x))}{2 \left (1-c^2 x^2\right )}}{4 c^2 d^3}+\frac {x (a+b \arccos (c x))}{4 c^2 d^3 \left (1-c^2 x^2\right )^2}+\frac {b}{12 c^3 d^3 \left (1-c^2 x^2\right )^{3/2}}\)

\(\Big \downarrow \) 241

\(\displaystyle -\frac {\frac {1}{2} \int \frac {a+b \arccos (c x)}{1-c^2 x^2}dx+\frac {x (a+b \arccos (c x))}{2 \left (1-c^2 x^2\right )}+\frac {b}{2 c \sqrt {1-c^2 x^2}}}{4 c^2 d^3}+\frac {x (a+b \arccos (c x))}{4 c^2 d^3 \left (1-c^2 x^2\right )^2}+\frac {b}{12 c^3 d^3 \left (1-c^2 x^2\right )^{3/2}}\)

\(\Big \downarrow \) 5165

\(\displaystyle -\frac {-\frac {\int \frac {a+b \arccos (c x)}{\sqrt {1-c^2 x^2}}d\arccos (c x)}{2 c}+\frac {x (a+b \arccos (c x))}{2 \left (1-c^2 x^2\right )}+\frac {b}{2 c \sqrt {1-c^2 x^2}}}{4 c^2 d^3}+\frac {x (a+b \arccos (c x))}{4 c^2 d^3 \left (1-c^2 x^2\right )^2}+\frac {b}{12 c^3 d^3 \left (1-c^2 x^2\right )^{3/2}}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {-\frac {\int (a+b \arccos (c x)) \csc (\arccos (c x))d\arccos (c x)}{2 c}+\frac {x (a+b \arccos (c x))}{2 \left (1-c^2 x^2\right )}+\frac {b}{2 c \sqrt {1-c^2 x^2}}}{4 c^2 d^3}+\frac {x (a+b \arccos (c x))}{4 c^2 d^3 \left (1-c^2 x^2\right )^2}+\frac {b}{12 c^3 d^3 \left (1-c^2 x^2\right )^{3/2}}\)

\(\Big \downarrow \) 4671

\(\displaystyle -\frac {-\frac {-b \int \log \left (1-e^{i \arccos (c x)}\right )d\arccos (c x)+b \int \log \left (1+e^{i \arccos (c x)}\right )d\arccos (c x)-2 \text {arctanh}\left (e^{i \arccos (c x)}\right ) (a+b \arccos (c x))}{2 c}+\frac {x (a+b \arccos (c x))}{2 \left (1-c^2 x^2\right )}+\frac {b}{2 c \sqrt {1-c^2 x^2}}}{4 c^2 d^3}+\frac {x (a+b \arccos (c x))}{4 c^2 d^3 \left (1-c^2 x^2\right )^2}+\frac {b}{12 c^3 d^3 \left (1-c^2 x^2\right )^{3/2}}\)

\(\Big \downarrow \) 2715

\(\displaystyle -\frac {-\frac {i b \int e^{-i \arccos (c x)} \log \left (1-e^{i \arccos (c x)}\right )de^{i \arccos (c x)}-i b \int e^{-i \arccos (c x)} \log \left (1+e^{i \arccos (c x)}\right )de^{i \arccos (c x)}-2 \text {arctanh}\left (e^{i \arccos (c x)}\right ) (a+b \arccos (c x))}{2 c}+\frac {x (a+b \arccos (c x))}{2 \left (1-c^2 x^2\right )}+\frac {b}{2 c \sqrt {1-c^2 x^2}}}{4 c^2 d^3}+\frac {x (a+b \arccos (c x))}{4 c^2 d^3 \left (1-c^2 x^2\right )^2}+\frac {b}{12 c^3 d^3 \left (1-c^2 x^2\right )^{3/2}}\)

\(\Big \downarrow \) 2838

\(\displaystyle -\frac {-\frac {-2 \text {arctanh}\left (e^{i \arccos (c x)}\right ) (a+b \arccos (c x))+i b \operatorname {PolyLog}\left (2,-e^{i \arccos (c x)}\right )-i b \operatorname {PolyLog}\left (2,e^{i \arccos (c x)}\right )}{2 c}+\frac {x (a+b \arccos (c x))}{2 \left (1-c^2 x^2\right )}+\frac {b}{2 c \sqrt {1-c^2 x^2}}}{4 c^2 d^3}+\frac {x (a+b \arccos (c x))}{4 c^2 d^3 \left (1-c^2 x^2\right )^2}+\frac {b}{12 c^3 d^3 \left (1-c^2 x^2\right )^{3/2}}\)

Input:

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

Output:

b/(12*c^3*d^3*(1 - c^2*x^2)^(3/2)) + (x*(a + b*ArcCos[c*x]))/(4*c^2*d^3*(1 
 - c^2*x^2)^2) - (b/(2*c*Sqrt[1 - c^2*x^2]) + (x*(a + b*ArcCos[c*x]))/(2*( 
1 - c^2*x^2)) - (-2*(a + b*ArcCos[c*x])*ArcTanh[E^(I*ArcCos[c*x])] + I*b*P 
olyLog[2, -E^(I*ArcCos[c*x])] - I*b*PolyLog[2, E^(I*ArcCos[c*x])])/(2*c))/ 
(4*c^2*d^3)
 

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 241

Int[(x_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(a + b*x^2)^(p + 1)/ 
(2*b*(p + 1)), x] /; FreeQ[{a, b, p}, x] && NeQ[p, -1]
 

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 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 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 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]
 

Maple [A] (verified)

Time = 0.73 (sec) , antiderivative size = 252, normalized size of antiderivative = 1.25

method	result	size
derivativedivides	\(\frac {-\frac {a \left (-\frac {1}{16 \left (c x -1\right )^{2}}-\frac {1}{16 \left (c x -1\right )}-\frac {\ln \left (c x -1\right )}{16}+\frac {1}{16 \left (c x +1\right )^{2}}-\frac {1}{16 \left (c x +1\right )}+\frac {\ln \left (c x +1\right )}{16}\right )}{d^{3}}-\frac {b \left (-\frac {3 c^{3} x^{3} \arccos \left (c x \right )+3 c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}+3 c x \arccos \left (c x \right )-\sqrt {-c^{2} x^{2}+1}}{24 \left (c^{4} x^{4}-2 c^{2} x^{2}+1\right )}-\frac {\arccos \left (c x \right ) \ln \left (1-c x -i \sqrt {-c^{2} x^{2}+1}\right )}{8}+\frac {i \operatorname {polylog}\left (2, c x +i \sqrt {-c^{2} x^{2}+1}\right )}{8}+\frac {\arccos \left (c x \right ) \ln \left (1+c x +i \sqrt {-c^{2} x^{2}+1}\right )}{8}-\frac {i \operatorname {polylog}\left (2, -c x -i \sqrt {-c^{2} x^{2}+1}\right )}{8}\right )}{d^{3}}}{c^{3}}\)	\(252\)
default	\(\frac {-\frac {a \left (-\frac {1}{16 \left (c x -1\right )^{2}}-\frac {1}{16 \left (c x -1\right )}-\frac {\ln \left (c x -1\right )}{16}+\frac {1}{16 \left (c x +1\right )^{2}}-\frac {1}{16 \left (c x +1\right )}+\frac {\ln \left (c x +1\right )}{16}\right )}{d^{3}}-\frac {b \left (-\frac {3 c^{3} x^{3} \arccos \left (c x \right )+3 c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}+3 c x \arccos \left (c x \right )-\sqrt {-c^{2} x^{2}+1}}{24 \left (c^{4} x^{4}-2 c^{2} x^{2}+1\right )}-\frac {\arccos \left (c x \right ) \ln \left (1-c x -i \sqrt {-c^{2} x^{2}+1}\right )}{8}+\frac {i \operatorname {polylog}\left (2, c x +i \sqrt {-c^{2} x^{2}+1}\right )}{8}+\frac {\arccos \left (c x \right ) \ln \left (1+c x +i \sqrt {-c^{2} x^{2}+1}\right )}{8}-\frac {i \operatorname {polylog}\left (2, -c x -i \sqrt {-c^{2} x^{2}+1}\right )}{8}\right )}{d^{3}}}{c^{3}}\)	\(252\)
parts	\(-\frac {a \left (-\frac {1}{16 c^{3} \left (c x -1\right )^{2}}-\frac {1}{16 c^{3} \left (c x -1\right )}-\frac {\ln \left (c x -1\right )}{16 c^{3}}+\frac {1}{16 c^{3} \left (c x +1\right )^{2}}-\frac {1}{16 c^{3} \left (c x +1\right )}+\frac {\ln \left (c x +1\right )}{16 c^{3}}\right )}{d^{3}}-\frac {b \left (-\frac {3 c^{3} x^{3} \arccos \left (c x \right )+3 c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}+3 c x \arccos \left (c x \right )-\sqrt {-c^{2} x^{2}+1}}{24 \left (c^{4} x^{4}-2 c^{2} x^{2}+1\right )}-\frac {\arccos \left (c x \right ) \ln \left (1-c x -i \sqrt {-c^{2} x^{2}+1}\right )}{8}+\frac {i \operatorname {polylog}\left (2, c x +i \sqrt {-c^{2} x^{2}+1}\right )}{8}+\frac {\arccos \left (c x \right ) \ln \left (1+c x +i \sqrt {-c^{2} x^{2}+1}\right )}{8}-\frac {i \operatorname {polylog}\left (2, -c x -i \sqrt {-c^{2} x^{2}+1}\right )}{8}\right )}{d^{3} c^{3}}\)	\(269\)

Input:

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

Output:

1/c^3*(-a/d^3*(-1/16/(c*x-1)^2-1/16/(c*x-1)-1/16*ln(c*x-1)+1/16/(c*x+1)^2- 
1/16/(c*x+1)+1/16*ln(c*x+1))-b/d^3*(-1/24*(3*c^3*x^3*arccos(c*x)+3*c^2*x^2 
*(-c^2*x^2+1)^(1/2)+3*c*x*arccos(c*x)-(-c^2*x^2+1)^(1/2))/(c^4*x^4-2*c^2*x 
^2+1)-1/8*arccos(c*x)*ln(1-c*x-I*(-c^2*x^2+1)^(1/2))+1/8*I*polylog(2,c*x+I 
*(-c^2*x^2+1)^(1/2))+1/8*arccos(c*x)*ln(1+c*x+I*(-c^2*x^2+1)^(1/2))-1/8*I* 
polylog(2,-c*x-I*(-c^2*x^2+1)^(1/2))))
 

Fricas [F]

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

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

Output:

integral(-(b*x^2*arccos(c*x) + a*x^2)/(c^6*d^3*x^6 - 3*c^4*d^3*x^4 + 3*c^2 
*d^3*x^2 - d^3), x)
 

Sympy [F]

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

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

Output:

-(Integral(a*x**2/(c**6*x**6 - 3*c**4*x**4 + 3*c**2*x**2 - 1), x) + Integr 
al(b*x**2*acos(c*x)/(c**6*x**6 - 3*c**4*x**4 + 3*c**2*x**2 - 1), x))/d**3
 

Maxima [F]

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

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

Output:

1/16*a*(2*(c^2*x^3 + x)/(c^6*d^3*x^4 - 2*c^4*d^3*x^2 + c^2*d^3) - log(c*x 
+ 1)/(c^3*d^3) + log(c*x - 1)/(c^3*d^3)) + 1/16*((2*c^3*x^3 + 2*c*x - (c^4 
*x^4 - 2*c^2*x^2 + 1)*log(c*x + 1) + (c^4*x^4 - 2*c^2*x^2 + 1)*log(-c*x + 
1))*arctan2(sqrt(c*x + 1)*sqrt(-c*x + 1), c*x) + 16*(c^7*d^3*x^4 - 2*c^5*d 
^3*x^2 + c^3*d^3)*integrate(-1/16*(2*c^3*x^3 + 2*c*x - (c^4*x^4 - 2*c^2*x^ 
2 + 1)*log(c*x + 1) + (c^4*x^4 - 2*c^2*x^2 + 1)*log(-c*x + 1))*sqrt(c*x + 
1)*sqrt(-c*x + 1)/(c^8*d^3*x^6 - 3*c^6*d^3*x^4 + 3*c^4*d^3*x^2 - c^2*d^3), 
 x))*b/(c^7*d^3*x^4 - 2*c^5*d^3*x^2 + c^3*d^3)
 

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {x^2 (a+b \arccos (c x))}{\left (d-c^2 d x^2\right )^3} \, dx=\frac {-16 \left (\int \frac {\mathit {acos} \left (c x \right ) x^{2}}{c^{6} x^{6}-3 c^{4} x^{4}+3 c^{2} x^{2}-1}d x \right ) b \,c^{7} x^{4}+32 \left (\int \frac {\mathit {acos} \left (c x \right ) x^{2}}{c^{6} x^{6}-3 c^{4} x^{4}+3 c^{2} x^{2}-1}d x \right ) b \,c^{5} x^{2}-16 \left (\int \frac {\mathit {acos} \left (c x \right ) x^{2}}{c^{6} x^{6}-3 c^{4} x^{4}+3 c^{2} x^{2}-1}d x \right ) b \,c^{3}+\mathrm {log}\left (c^{2} x -c \right ) a \,c^{4} x^{4}-2 \,\mathrm {log}\left (c^{2} x -c \right ) a \,c^{2} x^{2}+\mathrm {log}\left (c^{2} x -c \right ) a -\mathrm {log}\left (c^{2} x +c \right ) a \,c^{4} x^{4}+2 \,\mathrm {log}\left (c^{2} x +c \right ) a \,c^{2} x^{2}-\mathrm {log}\left (c^{2} x +c \right ) a +2 a \,c^{3} x^{3}+2 a c x}{16 c^{3} d^{3} \left (c^{4} x^{4}-2 c^{2} x^{2}+1\right )} \] Input:

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

Output:

( - 16*int((acos(c*x)*x**2)/(c**6*x**6 - 3*c**4*x**4 + 3*c**2*x**2 - 1),x) 
*b*c**7*x**4 + 32*int((acos(c*x)*x**2)/(c**6*x**6 - 3*c**4*x**4 + 3*c**2*x 
**2 - 1),x)*b*c**5*x**2 - 16*int((acos(c*x)*x**2)/(c**6*x**6 - 3*c**4*x**4 
 + 3*c**2*x**2 - 1),x)*b*c**3 + log(c**2*x - c)*a*c**4*x**4 - 2*log(c**2*x 
 - c)*a*c**2*x**2 + log(c**2*x - c)*a - log(c**2*x + c)*a*c**4*x**4 + 2*lo 
g(c**2*x + c)*a*c**2*x**2 - log(c**2*x + c)*a + 2*a*c**3*x**3 + 2*a*c*x)/( 
16*c**3*d**3*(c**4*x**4 - 2*c**2*x**2 + 1))