\(\int (d-c^2 d x^2) (a+b \arccos (c x))^2 \, dx\) [9]

Optimal result

Integrand size = 22, antiderivative size = 128 \[ \int \left (d-c^2 d x^2\right ) (a+b \arccos (c x))^2 \, dx=-\frac {14}{9} b^2 d x+\frac {2}{27} b^2 c^2 d x^3-\frac {4 b d \sqrt {1-c^2 x^2} (a+b \arccos (c x))}{3 c}-\frac {2 b d \left (1-c^2 x^2\right )^{3/2} (a+b \arccos (c x))}{9 c}+\frac {2}{3} d x (a+b \arccos (c x))^2+\frac {1}{3} d x \left (1-c^2 x^2\right ) (a+b \arccos (c x))^2 \] Output:

-14/9*b^2*d*x+2/27*b^2*c^2*d*x^3-4/3*b*d*(-c^2*x^2+1)^(1/2)*(a+b*arccos(c* 
x))/c-2/9*b*d*(-c^2*x^2+1)^(3/2)*(a+b*arccos(c*x))/c+2/3*d*x*(a+b*arccos(c 
*x))^2+1/3*d*x*(-c^2*x^2+1)*(a+b*arccos(c*x))^2
 

Mathematica [A] (verified)

Time = 0.16 (sec) , antiderivative size = 138, normalized size of antiderivative = 1.08 \[ \int \left (d-c^2 d x^2\right ) (a+b \arccos (c x))^2 \, dx=\frac {d \left (2 b^2 c x \left (-21+c^2 x^2\right )+6 a b \sqrt {1-c^2 x^2} \left (-7+c^2 x^2\right )-9 a^2 c x \left (-3+c^2 x^2\right )+6 b \left (b \sqrt {1-c^2 x^2} \left (-7+c^2 x^2\right )+a \left (9 c x-3 c^3 x^3\right )\right ) \arccos (c x)-9 b^2 c x \left (-3+c^2 x^2\right ) \arccos (c x)^2\right )}{27 c} \] Input:

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

Output:

(d*(2*b^2*c*x*(-21 + c^2*x^2) + 6*a*b*Sqrt[1 - c^2*x^2]*(-7 + c^2*x^2) - 9 
*a^2*c*x*(-3 + c^2*x^2) + 6*b*(b*Sqrt[1 - c^2*x^2]*(-7 + c^2*x^2) + a*(9*c 
*x - 3*c^3*x^3))*ArcCos[c*x] - 9*b^2*c*x*(-3 + c^2*x^2)*ArcCos[c*x]^2))/(2 
7*c)
 

Rubi [A] (verified)

Time = 0.55 (sec) , antiderivative size = 141, normalized size of antiderivative = 1.10, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {5159, 5131, 5183, 24, 2009}

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.

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

\(\Big \downarrow \) 5159

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

\(\Big \downarrow \) 5131

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

\(\Big \downarrow \) 5183

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

\(\Big \downarrow \) 24

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

(d*x*(1 - c^2*x^2)*(a + b*ArcCos[c*x])^2)/3 + (2*b*c*d*(-1/3*(b*(x - (c^2* 
x^3)/3))/c - ((1 - c^2*x^2)^(3/2)*(a + b*ArcCos[c*x]))/(3*c^2)))/3 + (2*d* 
(x*(a + b*ArcCos[c*x])^2 + 2*b*c*(-((b*x)/c) - (Sqrt[1 - c^2*x^2]*(a + b*A 
rcCos[c*x]))/c^2)))/3
 

Defintions of rubi rules used

rule 24

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 5131

Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_.), x_Symbol] :> Simp[x*(a + b*Ar 
cCos[c*x])^n, x] + Simp[b*c*n   Int[x*((a + b*ArcCos[c*x])^(n - 1)/Sqrt[1 - 
 c^2*x^2]), x], x] /; FreeQ[{a, b, c}, x] && GtQ[n, 0]
 

rule 5159

Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_.)*((d_) + (e_.)*(x_)^2)^(p_.), x 
_Symbol] :> Simp[x*(d + e*x^2)^p*((a + b*ArcCos[c*x])^n/(2*p + 1)), x] + (S 
imp[2*d*(p/(2*p + 1))   Int[(d + e*x^2)^(p - 1)*(a + b*ArcCos[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] && GtQ[p, 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]
 

Maple [A] (verified)

Time = 0.28 (sec) , antiderivative size = 173, normalized size of antiderivative = 1.35

method	result	size
derivativedivides	\(\frac {-d \,a^{2} \left (\frac {1}{3} c^{3} x^{3}-c x \right )-d \,b^{2} \left (\frac {\arccos \left (c x \right )^{2} \left (c^{2} x^{2}-3\right ) c x}{3}+\frac {4 c x}{3}+\frac {4 \arccos \left (c x \right ) \sqrt {-c^{2} x^{2}+1}}{3}-\frac {2 \arccos \left (c x \right ) \left (c^{2} x^{2}-1\right ) \sqrt {-c^{2} x^{2}+1}}{9}-\frac {2 \left (c^{2} x^{2}-3\right ) c x}{27}\right )-2 d a b \left (\frac {c^{3} x^{3} \arccos \left (c x \right )}{3}-c x \arccos \left (c x \right )-\frac {c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}}{9}+\frac {7 \sqrt {-c^{2} x^{2}+1}}{9}\right )}{c}\)	\(173\)
default	\(\frac {-d \,a^{2} \left (\frac {1}{3} c^{3} x^{3}-c x \right )-d \,b^{2} \left (\frac {\arccos \left (c x \right )^{2} \left (c^{2} x^{2}-3\right ) c x}{3}+\frac {4 c x}{3}+\frac {4 \arccos \left (c x \right ) \sqrt {-c^{2} x^{2}+1}}{3}-\frac {2 \arccos \left (c x \right ) \left (c^{2} x^{2}-1\right ) \sqrt {-c^{2} x^{2}+1}}{9}-\frac {2 \left (c^{2} x^{2}-3\right ) c x}{27}\right )-2 d a b \left (\frac {c^{3} x^{3} \arccos \left (c x \right )}{3}-c x \arccos \left (c x \right )-\frac {c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}}{9}+\frac {7 \sqrt {-c^{2} x^{2}+1}}{9}\right )}{c}\)	\(173\)
parts	\(-d \,a^{2} \left (\frac {1}{3} c^{2} x^{3}-x \right )-\frac {d \,b^{2} \left (\frac {\arccos \left (c x \right )^{2} \left (c^{2} x^{2}-3\right ) c x}{3}+\frac {4 c x}{3}+\frac {4 \arccos \left (c x \right ) \sqrt {-c^{2} x^{2}+1}}{3}-\frac {2 \arccos \left (c x \right ) \left (c^{2} x^{2}-1\right ) \sqrt {-c^{2} x^{2}+1}}{9}-\frac {2 \left (c^{2} x^{2}-3\right ) c x}{27}\right )}{c}-\frac {2 d a b \left (\frac {c^{3} x^{3} \arccos \left (c x \right )}{3}-c x \arccos \left (c x \right )-\frac {c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}}{9}+\frac {7 \sqrt {-c^{2} x^{2}+1}}{9}\right )}{c}\)	\(174\)
orering	\(\frac {x \left (19 c^{4} x^{4}-166 c^{2} x^{2}+27\right ) \left (-c^{2} d \,x^{2}+d \right ) \left (a +b \arccos \left (c x \right )\right )^{2}}{27 \left (c^{2} x^{2}-1\right )^{2}}-\frac {\left (2 c^{4} x^{4}-29 c^{2} x^{2}+7\right ) \left (-2 d \,c^{2} x \left (a +b \arccos \left (c x \right )\right )^{2}-\frac {2 \left (-c^{2} d \,x^{2}+d \right ) \left (a +b \arccos \left (c x \right )\right ) b c}{\sqrt {-c^{2} x^{2}+1}}\right )}{9 c^{2} \left (c^{2} x^{2}-1\right )}+\frac {x \left (c^{2} x^{2}-21\right ) \left (-2 d \,c^{2} \left (a +b \arccos \left (c x \right )\right )^{2}+\frac {8 d \,c^{3} x \left (a +b \arccos \left (c x \right )\right ) b}{\sqrt {-c^{2} x^{2}+1}}+\frac {2 \left (-c^{2} d \,x^{2}+d \right ) b^{2} c^{2}}{-c^{2} x^{2}+1}-\frac {2 \left (-c^{2} d \,x^{2}+d \right ) \left (a +b \arccos \left (c x \right )\right ) b \,c^{3} x}{\left (-c^{2} x^{2}+1\right )^{\frac {3}{2}}}\right )}{27 c^{2}}\)	\(271\)

Input:

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

Output:

1/c*(-d*a^2*(1/3*c^3*x^3-c*x)-d*b^2*(1/3*arccos(c*x)^2*(c^2*x^2-3)*c*x+4/3 
*c*x+4/3*arccos(c*x)*(-c^2*x^2+1)^(1/2)-2/9*arccos(c*x)*(c^2*x^2-1)*(-c^2* 
x^2+1)^(1/2)-2/27*(c^2*x^2-3)*c*x)-2*d*a*b*(1/3*c^3*x^3*arccos(c*x)-c*x*ar 
ccos(c*x)-1/9*c^2*x^2*(-c^2*x^2+1)^(1/2)+7/9*(-c^2*x^2+1)^(1/2)))
 

Fricas [A] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 146, normalized size of antiderivative = 1.14 \[ \int \left (d-c^2 d x^2\right ) (a+b \arccos (c x))^2 \, dx=-\frac {{\left (9 \, a^{2} - 2 \, b^{2}\right )} c^{3} d x^{3} - 3 \, {\left (9 \, a^{2} - 14 \, b^{2}\right )} c d x + 9 \, {\left (b^{2} c^{3} d x^{3} - 3 \, b^{2} c d x\right )} \arccos \left (c x\right )^{2} + 18 \, {\left (a b c^{3} d x^{3} - 3 \, a b c d x\right )} \arccos \left (c x\right ) - 6 \, {\left (a b c^{2} d x^{2} - 7 \, a b d + {\left (b^{2} c^{2} d x^{2} - 7 \, b^{2} d\right )} \arccos \left (c x\right )\right )} \sqrt {-c^{2} x^{2} + 1}}{27 \, c} \] Input:

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

Output:

-1/27*((9*a^2 - 2*b^2)*c^3*d*x^3 - 3*(9*a^2 - 14*b^2)*c*d*x + 9*(b^2*c^3*d 
*x^3 - 3*b^2*c*d*x)*arccos(c*x)^2 + 18*(a*b*c^3*d*x^3 - 3*a*b*c*d*x)*arcco 
s(c*x) - 6*(a*b*c^2*d*x^2 - 7*a*b*d + (b^2*c^2*d*x^2 - 7*b^2*d)*arccos(c*x 
))*sqrt(-c^2*x^2 + 1))/c
                                                                                    
                                                                                    
 

Sympy [A] (verification not implemented)

Time = 0.20 (sec) , antiderivative size = 230, normalized size of antiderivative = 1.80 \[ \int \left (d-c^2 d x^2\right ) (a+b \arccos (c x))^2 \, dx=\begin {cases} - \frac {a^{2} c^{2} d x^{3}}{3} + a^{2} d x - \frac {2 a b c^{2} d x^{3} \operatorname {acos}{\left (c x \right )}}{3} + \frac {2 a b c d x^{2} \sqrt {- c^{2} x^{2} + 1}}{9} + 2 a b d x \operatorname {acos}{\left (c x \right )} - \frac {14 a b d \sqrt {- c^{2} x^{2} + 1}}{9 c} - \frac {b^{2} c^{2} d x^{3} \operatorname {acos}^{2}{\left (c x \right )}}{3} + \frac {2 b^{2} c^{2} d x^{3}}{27} + \frac {2 b^{2} c d x^{2} \sqrt {- c^{2} x^{2} + 1} \operatorname {acos}{\left (c x \right )}}{9} + b^{2} d x \operatorname {acos}^{2}{\left (c x \right )} - \frac {14 b^{2} d x}{9} - \frac {14 b^{2} d \sqrt {- c^{2} x^{2} + 1} \operatorname {acos}{\left (c x \right )}}{9 c} & \text {for}\: c \neq 0 \\d x \left (a + \frac {\pi b}{2}\right )^{2} & \text {otherwise} \end {cases} \] Input:

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

Output:

Piecewise((-a**2*c**2*d*x**3/3 + a**2*d*x - 2*a*b*c**2*d*x**3*acos(c*x)/3 
+ 2*a*b*c*d*x**2*sqrt(-c**2*x**2 + 1)/9 + 2*a*b*d*x*acos(c*x) - 14*a*b*d*s 
qrt(-c**2*x**2 + 1)/(9*c) - b**2*c**2*d*x**3*acos(c*x)**2/3 + 2*b**2*c**2* 
d*x**3/27 + 2*b**2*c*d*x**2*sqrt(-c**2*x**2 + 1)*acos(c*x)/9 + b**2*d*x*ac 
os(c*x)**2 - 14*b**2*d*x/9 - 14*b**2*d*sqrt(-c**2*x**2 + 1)*acos(c*x)/(9*c 
), Ne(c, 0)), (d*x*(a + pi*b/2)**2, True))
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 234 vs. \(2 (111) = 222\).

Time = 0.13 (sec) , antiderivative size = 234, normalized size of antiderivative = 1.83 \[ \int \left (d-c^2 d x^2\right ) (a+b \arccos (c x))^2 \, dx=-\frac {1}{3} \, b^{2} c^{2} d x^{3} \arccos \left (c x\right )^{2} - \frac {1}{3} \, a^{2} c^{2} d x^{3} - \frac {2}{9} \, {\left (3 \, x^{3} \arccos \left (c x\right ) - c {\left (\frac {\sqrt {-c^{2} x^{2} + 1} x^{2}}{c^{2}} + \frac {2 \, \sqrt {-c^{2} x^{2} + 1}}{c^{4}}\right )}\right )} a b c^{2} d + \frac {2}{27} \, {\left (3 \, c {\left (\frac {\sqrt {-c^{2} x^{2} + 1} x^{2}}{c^{2}} + \frac {2 \, \sqrt {-c^{2} x^{2} + 1}}{c^{4}}\right )} \arccos \left (c x\right ) + \frac {c^{2} x^{3} + 6 \, x}{c^{2}}\right )} b^{2} c^{2} d + b^{2} d x \arccos \left (c x\right )^{2} - 2 \, b^{2} d {\left (x + \frac {\sqrt {-c^{2} x^{2} + 1} \arccos \left (c x\right )}{c}\right )} + a^{2} d x + \frac {2 \, {\left (c x \arccos \left (c x\right ) - \sqrt {-c^{2} x^{2} + 1}\right )} a b d}{c} \] Input:

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

Output:

-1/3*b^2*c^2*d*x^3*arccos(c*x)^2 - 1/3*a^2*c^2*d*x^3 - 2/9*(3*x^3*arccos(c 
*x) - c*(sqrt(-c^2*x^2 + 1)*x^2/c^2 + 2*sqrt(-c^2*x^2 + 1)/c^4))*a*b*c^2*d 
 + 2/27*(3*c*(sqrt(-c^2*x^2 + 1)*x^2/c^2 + 2*sqrt(-c^2*x^2 + 1)/c^4)*arcco 
s(c*x) + (c^2*x^3 + 6*x)/c^2)*b^2*c^2*d + b^2*d*x*arccos(c*x)^2 - 2*b^2*d* 
(x + sqrt(-c^2*x^2 + 1)*arccos(c*x)/c) + a^2*d*x + 2*(c*x*arccos(c*x) - sq 
rt(-c^2*x^2 + 1))*a*b*d/c
 

Giac [A] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 185, normalized size of antiderivative = 1.45 \[ \int \left (d-c^2 d x^2\right ) (a+b \arccos (c x))^2 \, dx=-\frac {1}{3} \, b^{2} c^{2} d x^{3} \arccos \left (c x\right )^{2} - \frac {2}{3} \, a b c^{2} d x^{3} \arccos \left (c x\right ) - \frac {1}{3} \, a^{2} c^{2} d x^{3} + \frac {2}{27} \, b^{2} c^{2} d x^{3} + \frac {2}{9} \, \sqrt {-c^{2} x^{2} + 1} b^{2} c d x^{2} \arccos \left (c x\right ) + \frac {2}{9} \, \sqrt {-c^{2} x^{2} + 1} a b c d x^{2} + b^{2} d x \arccos \left (c x\right )^{2} + 2 \, a b d x \arccos \left (c x\right ) + a^{2} d x - \frac {14}{9} \, b^{2} d x - \frac {14 \, \sqrt {-c^{2} x^{2} + 1} b^{2} d \arccos \left (c x\right )}{9 \, c} - \frac {14 \, \sqrt {-c^{2} x^{2} + 1} a b d}{9 \, c} \] Input:

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

Output:

-1/3*b^2*c^2*d*x^3*arccos(c*x)^2 - 2/3*a*b*c^2*d*x^3*arccos(c*x) - 1/3*a^2 
*c^2*d*x^3 + 2/27*b^2*c^2*d*x^3 + 2/9*sqrt(-c^2*x^2 + 1)*b^2*c*d*x^2*arcco 
s(c*x) + 2/9*sqrt(-c^2*x^2 + 1)*a*b*c*d*x^2 + b^2*d*x*arccos(c*x)^2 + 2*a* 
b*d*x*arccos(c*x) + a^2*d*x - 14/9*b^2*d*x - 14/9*sqrt(-c^2*x^2 + 1)*b^2*d 
*arccos(c*x)/c - 14/9*sqrt(-c^2*x^2 + 1)*a*b*d/c
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \left (d-c^2 d x^2\right ) (a+b \arccos (c x))^2 \, dx=\frac {d \left (9 \mathit {acos} \left (c x \right )^{2} b^{2} c x -18 \sqrt {-c^{2} x^{2}+1}\, \mathit {acos} \left (c x \right ) b^{2}-6 \mathit {acos} \left (c x \right ) a b \,c^{3} x^{3}+18 \mathit {acos} \left (c x \right ) a b c x +2 \sqrt {-c^{2} x^{2}+1}\, a b \,c^{2} x^{2}-14 \sqrt {-c^{2} x^{2}+1}\, a b -9 \left (\int \mathit {acos} \left (c x \right )^{2} x^{2}d x \right ) b^{2} c^{3}-3 a^{2} c^{3} x^{3}+9 a^{2} c x -18 b^{2} c x \right )}{9 c} \] Input:

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

Output:

(d*(9*acos(c*x)**2*b**2*c*x - 18*sqrt( - c**2*x**2 + 1)*acos(c*x)*b**2 - 6 
*acos(c*x)*a*b*c**3*x**3 + 18*acos(c*x)*a*b*c*x + 2*sqrt( - c**2*x**2 + 1) 
*a*b*c**2*x**2 - 14*sqrt( - c**2*x**2 + 1)*a*b - 9*int(acos(c*x)**2*x**2,x 
)*b**2*c**3 - 3*a**2*c**3*x**3 + 9*a**2*c*x - 18*b**2*c*x))/(9*c)