\(\int \frac {x (a+b \arccos (c x))^2}{(d-c^2 d x^2)^3} \, dx\) [206]

Optimal result

Integrand size = 25, antiderivative size = 150 \[ \int \frac {x (a+b \arccos (c x))^2}{\left (d-c^2 d x^2\right )^3} \, dx=\frac {b^2}{12 c^2 d^3 \left (1-c^2 x^2\right )}-\frac {b x (a+b \arccos (c x))}{6 c d^3 \left (1-c^2 x^2\right )^{3/2}}-\frac {b x (a+b \arccos (c x))}{3 c d^3 \sqrt {1-c^2 x^2}}+\frac {(a+b \arccos (c x))^2}{4 c^2 d^3 \left (1-c^2 x^2\right )^2}-\frac {b^2 \log \left (1-c^2 x^2\right )}{6 c^2 d^3} \] Output:

1/12*b^2/c^2/d^3/(-c^2*x^2+1)-1/6*b*x*(a+b*arccos(c*x))/c/d^3/(-c^2*x^2+1) 
^(3/2)-1/3*b*x*(a+b*arccos(c*x))/c/d^3/(-c^2*x^2+1)^(1/2)+1/4*(a+b*arccos( 
c*x))^2/c^2/d^3/(-c^2*x^2+1)^2-1/6*b^2*ln(-c^2*x^2+1)/c^2/d^3
 

Mathematica [A] (verified)

Time = 0.69 (sec) , antiderivative size = 162, normalized size of antiderivative = 1.08 \[ \int \frac {x (a+b \arccos (c x))^2}{\left (d-c^2 d x^2\right )^3} \, dx=\frac {3 a^2+b^2-b^2 c^2 x^2+6 a b c x \sqrt {1-c^2 x^2}-4 a b c^3 x^3 \sqrt {1-c^2 x^2}+2 b \left (3 a+b c x \left (3-2 c^2 x^2\right ) \sqrt {1-c^2 x^2}\right ) \arccos (c x)+3 b^2 \arccos (c x)^2-2 b^2 \left (-1+c^2 x^2\right )^2 \log \left (1-c^2 x^2\right )}{12 c^2 d^3 \left (-1+c^2 x^2\right )^2} \] Input:

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

Output:

(3*a^2 + b^2 - b^2*c^2*x^2 + 6*a*b*c*x*Sqrt[1 - c^2*x^2] - 4*a*b*c^3*x^3*S 
qrt[1 - c^2*x^2] + 2*b*(3*a + b*c*x*(3 - 2*c^2*x^2)*Sqrt[1 - c^2*x^2])*Arc 
Cos[c*x] + 3*b^2*ArcCos[c*x]^2 - 2*b^2*(-1 + c^2*x^2)^2*Log[1 - c^2*x^2])/ 
(12*c^2*d^3*(-1 + c^2*x^2)^2)
 

Rubi [A] (verified)

Time = 0.47 (sec) , antiderivative size = 140, normalized size of antiderivative = 0.93, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {5183, 5163, 241, 5161, 240}

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.

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

\(\Big \downarrow \) 5183

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

\(\Big \downarrow \) 5163

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

\(\Big \downarrow \) 241

\(\displaystyle \frac {b \left (\frac {2}{3} \int \frac {a+b \arccos (c x)}{\left (1-c^2 x^2\right )^{3/2}}dx+\frac {x (a+b \arccos (c x))}{3 \left (1-c^2 x^2\right )^{3/2}}+\frac {b}{6 c \left (1-c^2 x^2\right )}\right )}{2 c d^3}+\frac {(a+b \arccos (c x))^2}{4 c^2 d^3 \left (1-c^2 x^2\right )^2}\)

\(\Big \downarrow \) 5161

\(\displaystyle \frac {b \left (\frac {2}{3} \left (b c \int \frac {x}{1-c^2 x^2}dx+\frac {x (a+b \arccos (c x))}{\sqrt {1-c^2 x^2}}\right )+\frac {x (a+b \arccos (c x))}{3 \left (1-c^2 x^2\right )^{3/2}}+\frac {b}{6 c \left (1-c^2 x^2\right )}\right )}{2 c d^3}+\frac {(a+b \arccos (c x))^2}{4 c^2 d^3 \left (1-c^2 x^2\right )^2}\)

\(\Big \downarrow \) 240

\(\displaystyle \frac {(a+b \arccos (c x))^2}{4 c^2 d^3 \left (1-c^2 x^2\right )^2}+\frac {b \left (\frac {x (a+b \arccos (c x))}{3 \left (1-c^2 x^2\right )^{3/2}}+\frac {2}{3} \left (\frac {x (a+b \arccos (c x))}{\sqrt {1-c^2 x^2}}-\frac {b \log \left (1-c^2 x^2\right )}{2 c}\right )+\frac {b}{6 c \left (1-c^2 x^2\right )}\right )}{2 c d^3}\)

Input:

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

Output:

(a + b*ArcCos[c*x])^2/(4*c^2*d^3*(1 - c^2*x^2)^2) + (b*(b/(6*c*(1 - c^2*x^ 
2)) + (x*(a + b*ArcCos[c*x]))/(3*(1 - c^2*x^2)^(3/2)) + (2*((x*(a + b*ArcC 
os[c*x]))/Sqrt[1 - c^2*x^2] - (b*Log[1 - c^2*x^2])/(2*c)))/3))/(2*c*d^3)
 

Defintions of rubi rules used

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 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 5161

Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_.)/((d_) + (e_.)*(x_)^2)^(3/2), x 
_Symbol] :> Simp[x*((a + b*ArcCos[c*x])^n/(d*Sqrt[d + e*x^2])), x] + Simp[b 
*c*(n/d)*Simp[Sqrt[1 - c^2*x^2]/Sqrt[d + e*x^2]]   Int[x*((a + b*ArcCos[c*x 
])^(n - 1)/(1 - c^2*x^2)), x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d 
 + e, 0] && GtQ[n, 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 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]
 

Maple [A] (verified)

Time = 0.29 (sec) , antiderivative size = 270, normalized size of antiderivative = 1.80

method	result	size
derivativedivides	\(\frac {\frac {a^{2}}{4 d^{3} \left (c^{2} x^{2}-1\right )^{2}}-\frac {b^{2} \left (-\frac {\arccos \left (c x \right )^{2}}{4 \left (c^{2} x^{2}-1\right )^{2}}-\frac {c x \arccos \left (c x \right ) \sqrt {-c^{2} x^{2}+1}}{6 \left (c^{2} x^{2}-1\right )^{2}}+\frac {1}{12 c^{2} x^{2}-12}+\frac {\sqrt {-c^{2} x^{2}+1}\, \arccos \left (c x \right ) x c}{3 c^{2} x^{2}-3}+\frac {\ln \left (-c^{2} x^{2}+1\right )}{6}\right )}{d^{3}}-\frac {2 a b \left (-\frac {\arccos \left (c x \right )}{4 \left (c^{2} x^{2}-1\right )^{2}}-\frac {\sqrt {-\left (c x -1\right )^{2}-2 c x +2}}{48 \left (c x -1\right )^{2}}+\frac {\sqrt {-\left (c x -1\right )^{2}-2 c x +2}}{12 c x -12}+\frac {\sqrt {-\left (c x +1\right )^{2}+2 c x +2}}{48 \left (c x +1\right )^{2}}+\frac {\sqrt {-\left (c x +1\right )^{2}+2 c x +2}}{12 c x +12}\right )}{d^{3}}}{c^{2}}\)	\(270\)
default	\(\frac {\frac {a^{2}}{4 d^{3} \left (c^{2} x^{2}-1\right )^{2}}-\frac {b^{2} \left (-\frac {\arccos \left (c x \right )^{2}}{4 \left (c^{2} x^{2}-1\right )^{2}}-\frac {c x \arccos \left (c x \right ) \sqrt {-c^{2} x^{2}+1}}{6 \left (c^{2} x^{2}-1\right )^{2}}+\frac {1}{12 c^{2} x^{2}-12}+\frac {\sqrt {-c^{2} x^{2}+1}\, \arccos \left (c x \right ) x c}{3 c^{2} x^{2}-3}+\frac {\ln \left (-c^{2} x^{2}+1\right )}{6}\right )}{d^{3}}-\frac {2 a b \left (-\frac {\arccos \left (c x \right )}{4 \left (c^{2} x^{2}-1\right )^{2}}-\frac {\sqrt {-\left (c x -1\right )^{2}-2 c x +2}}{48 \left (c x -1\right )^{2}}+\frac {\sqrt {-\left (c x -1\right )^{2}-2 c x +2}}{12 c x -12}+\frac {\sqrt {-\left (c x +1\right )^{2}+2 c x +2}}{48 \left (c x +1\right )^{2}}+\frac {\sqrt {-\left (c x +1\right )^{2}+2 c x +2}}{12 c x +12}\right )}{d^{3}}}{c^{2}}\)	\(270\)
parts	\(\frac {a^{2}}{4 d^{3} c^{2} \left (c^{2} x^{2}-1\right )^{2}}-\frac {b^{2} \left (-\frac {\arccos \left (c x \right )^{2}}{4 \left (c^{2} x^{2}-1\right )^{2}}-\frac {c x \arccos \left (c x \right ) \sqrt {-c^{2} x^{2}+1}}{6 \left (c^{2} x^{2}-1\right )^{2}}+\frac {1}{12 c^{2} x^{2}-12}+\frac {\sqrt {-c^{2} x^{2}+1}\, \arccos \left (c x \right ) x c}{3 c^{2} x^{2}-3}+\frac {\ln \left (-c^{2} x^{2}+1\right )}{6}\right )}{d^{3} c^{2}}-\frac {2 a b \left (-\frac {\arccos \left (c x \right )}{4 \left (c^{2} x^{2}-1\right )^{2}}-\frac {\sqrt {-\left (c x -1\right )^{2}-2 c x +2}}{48 \left (c x -1\right )^{2}}+\frac {\sqrt {-\left (c x -1\right )^{2}-2 c x +2}}{12 c x -12}+\frac {\sqrt {-\left (c x +1\right )^{2}+2 c x +2}}{48 \left (c x +1\right )^{2}}+\frac {\sqrt {-\left (c x +1\right )^{2}+2 c x +2}}{12 c x +12}\right )}{d^{3} c^{2}}\)	\(275\)

Input:

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

Output:

1/c^2*(1/4*a^2/d^3/(c^2*x^2-1)^2-b^2/d^3*(-1/4/(c^2*x^2-1)^2*arccos(c*x)^2 
-1/6*c*x/(c^2*x^2-1)^2*arccos(c*x)*(-c^2*x^2+1)^(1/2)+1/12/(c^2*x^2-1)+1/3 
*c*x/(c^2*x^2-1)*arccos(c*x)*(-c^2*x^2+1)^(1/2)+1/6*ln(-c^2*x^2+1))-2*a*b/ 
d^3*(-1/4/(c^2*x^2-1)^2*arccos(c*x)-1/48/(c*x-1)^2*(-(c*x-1)^2-2*c*x+2)^(1 
/2)+1/12/(c*x-1)*(-(c*x-1)^2-2*c*x+2)^(1/2)+1/48/(c*x+1)^2*(-(c*x+1)^2+2*c 
*x+2)^(1/2)+1/12/(c*x+1)*(-(c*x+1)^2+2*c*x+2)^(1/2)))
 

Fricas [A] (verification not implemented)

Time = 0.18 (sec) , antiderivative size = 235, normalized size of antiderivative = 1.57 \[ \int \frac {x (a+b \arccos (c x))^2}{\left (d-c^2 d x^2\right )^3} \, dx=-\frac {b^{2} c^{2} x^{2} - 3 \, b^{2} \arccos \left (c x\right )^{2} - 3 \, a^{2} - b^{2} + 6 \, {\left (a b c^{4} x^{4} - 2 \, a b c^{2} x^{2}\right )} \arccos \left (c x\right ) - 6 \, {\left (a b c^{4} x^{4} - 2 \, a b c^{2} x^{2} + a b\right )} \arctan \left (\frac {\sqrt {-c^{2} x^{2} + 1} c x}{c^{2} x^{2} - 1}\right ) + 2 \, {\left (b^{2} c^{4} x^{4} - 2 \, b^{2} c^{2} x^{2} + b^{2}\right )} \log \left (c^{2} x^{2} - 1\right ) + 2 \, {\left (2 \, a b c^{3} x^{3} - 3 \, a b c x + {\left (2 \, b^{2} c^{3} x^{3} - 3 \, b^{2} c x\right )} \arccos \left (c x\right )\right )} \sqrt {-c^{2} x^{2} + 1}}{12 \, {\left (c^{6} d^{3} x^{4} - 2 \, c^{4} d^{3} x^{2} + c^{2} d^{3}\right )}} \] Input:

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

Output:

-1/12*(b^2*c^2*x^2 - 3*b^2*arccos(c*x)^2 - 3*a^2 - b^2 + 6*(a*b*c^4*x^4 - 
2*a*b*c^2*x^2)*arccos(c*x) - 6*(a*b*c^4*x^4 - 2*a*b*c^2*x^2 + a*b)*arctan( 
sqrt(-c^2*x^2 + 1)*c*x/(c^2*x^2 - 1)) + 2*(b^2*c^4*x^4 - 2*b^2*c^2*x^2 + b 
^2)*log(c^2*x^2 - 1) + 2*(2*a*b*c^3*x^3 - 3*a*b*c*x + (2*b^2*c^3*x^3 - 3*b 
^2*c*x)*arccos(c*x))*sqrt(-c^2*x^2 + 1))/(c^6*d^3*x^4 - 2*c^4*d^3*x^2 + c^ 
2*d^3)
                                                                                    
                                                                                    
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 395 vs. \(2 (134) = 268\).

Time = 0.21 (sec) , antiderivative size = 395, normalized size of antiderivative = 2.63 \[ \int \frac {x (a+b \arccos (c x))^2}{\left (d-c^2 d x^2\right )^3} \, dx=\frac {b^{2} c^{2} x^{4} \arccos \left (c x\right )^{2}}{4 \, {\left (c^{2} x^{2} - 1\right )}^{2} d^{3}} + \frac {a b c^{2} x^{4} \arccos \left (c x\right )}{2 \, {\left (c^{2} x^{2} - 1\right )}^{2} d^{3}} + \frac {a^{2} c^{2} x^{4}}{4 \, {\left (c^{2} x^{2} - 1\right )}^{2} d^{3}} - \frac {b^{2} c x^{3} \arccos \left (c x\right )}{6 \, {\left (c^{2} x^{2} - 1\right )} \sqrt {-c^{2} x^{2} + 1} d^{3}} - \frac {b^{2} x^{2} \arccos \left (c x\right )^{2}}{2 \, {\left (c^{2} x^{2} - 1\right )} d^{3}} - \frac {a b c x^{3}}{6 \, {\left (c^{2} x^{2} - 1\right )} \sqrt {-c^{2} x^{2} + 1} d^{3}} - \frac {a b x^{2} \arccos \left (c x\right )}{{\left (c^{2} x^{2} - 1\right )} d^{3}} - \frac {a^{2} x^{2}}{2 \, {\left (c^{2} x^{2} - 1\right )} d^{3}} - \frac {b^{2} x^{2}}{12 \, {\left (c^{2} x^{2} - 1\right )} d^{3}} + \frac {b^{2} x \arccos \left (c x\right )}{2 \, \sqrt {-c^{2} x^{2} + 1} c d^{3}} + \frac {b^{2} \arccos \left (c x\right )^{2}}{4 \, c^{2} d^{3}} + \frac {a b x}{2 \, \sqrt {-c^{2} x^{2} + 1} c d^{3}} + \frac {a b \arccos \left (c x\right )}{2 \, c^{2} d^{3}} - \frac {b^{2} \log \left (2\right )}{3 \, c^{2} d^{3}} - \frac {b^{2} \log \left ({\left | -c^{2} x^{2} + 1 \right |}\right )}{6 \, c^{2} d^{3}} + \frac {a^{2}}{4 \, c^{2} d^{3}} + \frac {b^{2}}{12 \, c^{2} d^{3}} \] Input:

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

Output:

1/4*b^2*c^2*x^4*arccos(c*x)^2/((c^2*x^2 - 1)^2*d^3) + 1/2*a*b*c^2*x^4*arcc 
os(c*x)/((c^2*x^2 - 1)^2*d^3) + 1/4*a^2*c^2*x^4/((c^2*x^2 - 1)^2*d^3) - 1/ 
6*b^2*c*x^3*arccos(c*x)/((c^2*x^2 - 1)*sqrt(-c^2*x^2 + 1)*d^3) - 1/2*b^2*x 
^2*arccos(c*x)^2/((c^2*x^2 - 1)*d^3) - 1/6*a*b*c*x^3/((c^2*x^2 - 1)*sqrt(- 
c^2*x^2 + 1)*d^3) - a*b*x^2*arccos(c*x)/((c^2*x^2 - 1)*d^3) - 1/2*a^2*x^2/ 
((c^2*x^2 - 1)*d^3) - 1/12*b^2*x^2/((c^2*x^2 - 1)*d^3) + 1/2*b^2*x*arccos( 
c*x)/(sqrt(-c^2*x^2 + 1)*c*d^3) + 1/4*b^2*arccos(c*x)^2/(c^2*d^3) + 1/2*a* 
b*x/(sqrt(-c^2*x^2 + 1)*c*d^3) + 1/2*a*b*arccos(c*x)/(c^2*d^3) - 1/3*b^2*l 
og(2)/(c^2*d^3) - 1/6*b^2*log(abs(-c^2*x^2 + 1))/(c^2*d^3) + 1/4*a^2/(c^2* 
d^3) + 1/12*b^2/(c^2*d^3)
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {x (a+b \arccos (c x))^2}{\left (d-c^2 d x^2\right )^3} \, dx=\frac {-8 \left (\int \frac {\mathit {acos} \left (c x \right ) x}{c^{6} x^{6}-3 c^{4} x^{4}+3 c^{2} x^{2}-1}d x \right ) a b \,c^{6} x^{4}+16 \left (\int \frac {\mathit {acos} \left (c x \right ) x}{c^{6} x^{6}-3 c^{4} x^{4}+3 c^{2} x^{2}-1}d x \right ) a b \,c^{4} x^{2}-8 \left (\int \frac {\mathit {acos} \left (c x \right ) x}{c^{6} x^{6}-3 c^{4} x^{4}+3 c^{2} x^{2}-1}d x \right ) a b \,c^{2}-4 \left (\int \frac {\mathit {acos} \left (c x \right )^{2} x}{c^{6} x^{6}-3 c^{4} x^{4}+3 c^{2} x^{2}-1}d x \right ) b^{2} c^{6} x^{4}+8 \left (\int \frac {\mathit {acos} \left (c x \right )^{2} x}{c^{6} x^{6}-3 c^{4} x^{4}+3 c^{2} x^{2}-1}d x \right ) b^{2} c^{4} x^{2}-4 \left (\int \frac {\mathit {acos} \left (c x \right )^{2} x}{c^{6} x^{6}-3 c^{4} x^{4}+3 c^{2} x^{2}-1}d x \right ) b^{2} c^{2}+a^{2}}{4 c^{2} d^{3} \left (c^{4} x^{4}-2 c^{2} x^{2}+1\right )} \] Input:

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

Output:

( - 8*int((acos(c*x)*x)/(c**6*x**6 - 3*c**4*x**4 + 3*c**2*x**2 - 1),x)*a*b 
*c**6*x**4 + 16*int((acos(c*x)*x)/(c**6*x**6 - 3*c**4*x**4 + 3*c**2*x**2 - 
 1),x)*a*b*c**4*x**2 - 8*int((acos(c*x)*x)/(c**6*x**6 - 3*c**4*x**4 + 3*c* 
*2*x**2 - 1),x)*a*b*c**2 - 4*int((acos(c*x)**2*x)/(c**6*x**6 - 3*c**4*x**4 
 + 3*c**2*x**2 - 1),x)*b**2*c**6*x**4 + 8*int((acos(c*x)**2*x)/(c**6*x**6 
- 3*c**4*x**4 + 3*c**2*x**2 - 1),x)*b**2*c**4*x**2 - 4*int((acos(c*x)**2*x 
)/(c**6*x**6 - 3*c**4*x**4 + 3*c**2*x**2 - 1),x)*b**2*c**2 + a**2)/(4*c**2 
*d**3*(c**4*x**4 - 2*c**2*x**2 + 1))