\(\int (d-c^2 d x^2)^2 (a+b \arccos (c x))^2 \, dx\) [8]

Optimal result

Integrand size = 24, antiderivative size = 219 \[ \int \left (d-c^2 d x^2\right )^2 (a+b \arccos (c x))^2 \, dx=-\frac {298}{225} b^2 d^2 x+\frac {76}{675} b^2 c^2 d^2 x^3-\frac {2}{125} b^2 c^4 d^2 x^5-\frac {16 b d^2 \sqrt {1-c^2 x^2} (a+b \arccos (c x))}{15 c}-\frac {8 b d^2 \left (1-c^2 x^2\right )^{3/2} (a+b \arccos (c x))}{45 c}-\frac {2 b d^2 \left (1-c^2 x^2\right )^{5/2} (a+b \arccos (c x))}{25 c}+\frac {8}{15} d^2 x (a+b \arccos (c x))^2+\frac {4}{15} d^2 x \left (1-c^2 x^2\right ) (a+b \arccos (c x))^2+\frac {1}{5} d^2 x \left (1-c^2 x^2\right )^2 (a+b \arccos (c x))^2 \] Output:

-298/225*b^2*d^2*x+76/675*b^2*c^2*d^2*x^3-2/125*b^2*c^4*d^2*x^5-16/15*b*d^ 
2*(-c^2*x^2+1)^(1/2)*(a+b*arccos(c*x))/c-8/45*b*d^2*(-c^2*x^2+1)^(3/2)*(a+ 
b*arccos(c*x))/c-2/25*b*d^2*(-c^2*x^2+1)^(5/2)*(a+b*arccos(c*x))/c+8/15*d^ 
2*x*(a+b*arccos(c*x))^2+4/15*d^2*x*(-c^2*x^2+1)*(a+b*arccos(c*x))^2+1/5*d^ 
2*x*(-c^2*x^2+1)^2*(a+b*arccos(c*x))^2
 

Mathematica [A] (verified)

Time = 1.21 (sec) , antiderivative size = 193, normalized size of antiderivative = 0.88 \[ \int \left (d-c^2 d x^2\right )^2 (a+b \arccos (c x))^2 \, dx=\frac {d^2 \left (225 a^2 c x \left (15-10 c^2 x^2+3 c^4 x^4\right )-30 a b \sqrt {1-c^2 x^2} \left (149-38 c^2 x^2+9 c^4 x^4\right )-2 b^2 c x \left (2235-190 c^2 x^2+27 c^4 x^4\right )-30 b \left (-15 a c x \left (15-10 c^2 x^2+3 c^4 x^4\right )+b \sqrt {1-c^2 x^2} \left (149-38 c^2 x^2+9 c^4 x^4\right )\right ) \arccos (c x)+225 b^2 c x \left (15-10 c^2 x^2+3 c^4 x^4\right ) \arccos (c x)^2\right )}{3375 c} \] Input:

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

Output:

(d^2*(225*a^2*c*x*(15 - 10*c^2*x^2 + 3*c^4*x^4) - 30*a*b*Sqrt[1 - c^2*x^2] 
*(149 - 38*c^2*x^2 + 9*c^4*x^4) - 2*b^2*c*x*(2235 - 190*c^2*x^2 + 27*c^4*x 
^4) - 30*b*(-15*a*c*x*(15 - 10*c^2*x^2 + 3*c^4*x^4) + b*Sqrt[1 - c^2*x^2]* 
(149 - 38*c^2*x^2 + 9*c^4*x^4))*ArcCos[c*x] + 225*b^2*c*x*(15 - 10*c^2*x^2 
 + 3*c^4*x^4)*ArcCos[c*x]^2))/(3375*c)
 

Rubi [A] (verified)

Time = 0.95 (sec) , antiderivative size = 245, normalized size of antiderivative = 1.12, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {5159, 27, 5159, 5131, 5183, 24, 210, 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.

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

\(\Big \downarrow \) 5159

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 5159

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

\(\Big \downarrow \) 5131

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

\(\Big \downarrow \) 5183

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

\(\Big \downarrow \) 24

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

\(\Big \downarrow \) 210

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

(d^2*x*(1 - c^2*x^2)^2*(a + b*ArcCos[c*x])^2)/5 + (2*b*c*d^2*(-1/5*(b*(x - 
 (2*c^2*x^3)/3 + (c^4*x^5)/5))/c - ((1 - c^2*x^2)^(5/2)*(a + b*ArcCos[c*x] 
))/(5*c^2)))/5 + (4*d^2*((x*(1 - c^2*x^2)*(a + b*ArcCos[c*x])^2)/3 + (2*b* 
c*(-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*(x*(a + b*ArcCos[c*x])^2 + 2*b*c*(-((b*x)/c) - (Sqrt[1 
- c^2*x^2]*(a + b*ArcCos[c*x]))/c^2)))/3))/5
 

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 210

Int[((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Int[ExpandIntegrand[(a + b*x^2 
)^p, x], x] /; FreeQ[{a, b}, x] && IGtQ[p, 0]
 

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 = 0.32 (sec) , antiderivative size = 275, normalized size of antiderivative = 1.26

method	result	size
derivativedivides	\(\frac {d^{2} a^{2} \left (\frac {1}{5} c^{5} x^{5}-\frac {2}{3} c^{3} x^{3}+c x \right )+d^{2} b^{2} \left (\frac {\arccos \left (c x \right )^{2} \left (3 c^{4} x^{4}-10 c^{2} x^{2}+15\right ) c x}{15}-\frac {2 \arccos \left (c x \right ) \left (c^{2} x^{2}-1\right )^{2} \sqrt {-c^{2} x^{2}+1}}{25}-\frac {2 \left (3 c^{4} x^{4}-10 c^{2} x^{2}+15\right ) c x}{375}+\frac {8 \arccos \left (c x \right ) \left (c^{2} x^{2}-1\right ) \sqrt {-c^{2} x^{2}+1}}{45}+\frac {8 \left (c^{2} x^{2}-3\right ) c x}{135}-\frac {16 c x}{15}-\frac {16 \arccos \left (c x \right ) \sqrt {-c^{2} x^{2}+1}}{15}\right )+2 d^{2} a b \left (\frac {\arccos \left (c x \right ) c^{5} x^{5}}{5}-\frac {2 c^{3} x^{3} \arccos \left (c x \right )}{3}+c x \arccos \left (c x \right )-\frac {149 \sqrt {-c^{2} x^{2}+1}}{225}+\frac {38 c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}}{225}-\frac {c^{4} x^{4} \sqrt {-c^{2} x^{2}+1}}{25}\right )}{c}\)	\(275\)
default	\(\frac {d^{2} a^{2} \left (\frac {1}{5} c^{5} x^{5}-\frac {2}{3} c^{3} x^{3}+c x \right )+d^{2} b^{2} \left (\frac {\arccos \left (c x \right )^{2} \left (3 c^{4} x^{4}-10 c^{2} x^{2}+15\right ) c x}{15}-\frac {2 \arccos \left (c x \right ) \left (c^{2} x^{2}-1\right )^{2} \sqrt {-c^{2} x^{2}+1}}{25}-\frac {2 \left (3 c^{4} x^{4}-10 c^{2} x^{2}+15\right ) c x}{375}+\frac {8 \arccos \left (c x \right ) \left (c^{2} x^{2}-1\right ) \sqrt {-c^{2} x^{2}+1}}{45}+\frac {8 \left (c^{2} x^{2}-3\right ) c x}{135}-\frac {16 c x}{15}-\frac {16 \arccos \left (c x \right ) \sqrt {-c^{2} x^{2}+1}}{15}\right )+2 d^{2} a b \left (\frac {\arccos \left (c x \right ) c^{5} x^{5}}{5}-\frac {2 c^{3} x^{3} \arccos \left (c x \right )}{3}+c x \arccos \left (c x \right )-\frac {149 \sqrt {-c^{2} x^{2}+1}}{225}+\frac {38 c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}}{225}-\frac {c^{4} x^{4} \sqrt {-c^{2} x^{2}+1}}{25}\right )}{c}\)	\(275\)
parts	\(d^{2} a^{2} \left (\frac {1}{5} c^{4} x^{5}-\frac {2}{3} c^{2} x^{3}+x \right )+\frac {d^{2} b^{2} \left (\frac {\arccos \left (c x \right )^{2} \left (3 c^{4} x^{4}-10 c^{2} x^{2}+15\right ) c x}{15}-\frac {2 \arccos \left (c x \right ) \left (c^{2} x^{2}-1\right )^{2} \sqrt {-c^{2} x^{2}+1}}{25}-\frac {2 \left (3 c^{4} x^{4}-10 c^{2} x^{2}+15\right ) c x}{375}+\frac {8 \arccos \left (c x \right ) \left (c^{2} x^{2}-1\right ) \sqrt {-c^{2} x^{2}+1}}{45}+\frac {8 \left (c^{2} x^{2}-3\right ) c x}{135}-\frac {16 c x}{15}-\frac {16 \arccos \left (c x \right ) \sqrt {-c^{2} x^{2}+1}}{15}\right )}{c}+\frac {2 d^{2} a b \left (\frac {\arccos \left (c x \right ) c^{5} x^{5}}{5}-\frac {2 c^{3} x^{3} \arccos \left (c x \right )}{3}+c x \arccos \left (c x \right )-\frac {149 \sqrt {-c^{2} x^{2}+1}}{225}+\frac {38 c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}}{225}-\frac {c^{4} x^{4} \sqrt {-c^{2} x^{2}+1}}{25}\right )}{c}\)	\(275\)
orering	\(\frac {x \left (1647 c^{6} x^{6}-8677 c^{4} x^{4}+51845 c^{2} x^{2}-3375\right ) \left (-c^{2} d \,x^{2}+d \right )^{2} \left (a +b \arccos \left (c x \right )\right )^{2}}{3375 \left (c x -1\right ) \left (c x +1\right ) \left (c^{2} x^{2}-1\right )^{2}}-\frac {\left (324 c^{6} x^{6}-2035 c^{4} x^{4}+18450 c^{2} x^{2}-2235\right ) \left (-4 \left (-c^{2} d \,x^{2}+d \right ) \left (a +b \arccos \left (c x \right )\right )^{2} d \,c^{2} x -\frac {2 \left (-c^{2} d \,x^{2}+d \right )^{2} \left (a +b \arccos \left (c x \right )\right ) b c}{\sqrt {-c^{2} x^{2}+1}}\right )}{3375 c^{2} \left (c x -1\right ) \left (c x +1\right ) \left (c^{2} x^{2}-1\right )}+\frac {x \left (27 c^{4} x^{4}-190 c^{2} x^{2}+2235\right ) \left (8 d^{2} c^{4} x^{2} \left (a +b \arccos \left (c x \right )\right )^{2}+\frac {16 \left (-c^{2} d \,x^{2}+d \right ) \left (a +b \arccos \left (c x \right )\right ) d \,c^{3} x b}{\sqrt {-c^{2} x^{2}+1}}-4 \left (-c^{2} d \,x^{2}+d \right ) \left (a +b \arccos \left (c x \right )\right )^{2} d \,c^{2}+\frac {2 \left (-c^{2} d \,x^{2}+d \right )^{2} b^{2} c^{2}}{-c^{2} x^{2}+1}-\frac {2 \left (-c^{2} d \,x^{2}+d \right )^{2} \left (a +b \arccos \left (c x \right )\right ) b \,c^{3} x}{\left (-c^{2} x^{2}+1\right )^{\frac {3}{2}}}\right )}{3375 c^{2} \left (c x -1\right ) \left (c x +1\right )}\)	\(400\)

Input:

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

Output:

1/c*(d^2*a^2*(1/5*c^5*x^5-2/3*c^3*x^3+c*x)+d^2*b^2*(1/15*arccos(c*x)^2*(3* 
c^4*x^4-10*c^2*x^2+15)*c*x-2/25*arccos(c*x)*(c^2*x^2-1)^2*(-c^2*x^2+1)^(1/ 
2)-2/375*(3*c^4*x^4-10*c^2*x^2+15)*c*x+8/45*arccos(c*x)*(c^2*x^2-1)*(-c^2* 
x^2+1)^(1/2)+8/135*(c^2*x^2-3)*c*x-16/15*c*x-16/15*arccos(c*x)*(-c^2*x^2+1 
)^(1/2))+2*d^2*a*b*(1/5*arccos(c*x)*c^5*x^5-2/3*c^3*x^3*arccos(c*x)+c*x*ar 
ccos(c*x)-149/225*(-c^2*x^2+1)^(1/2)+38/225*c^2*x^2*(-c^2*x^2+1)^(1/2)-1/2 
5*c^4*x^4*(-c^2*x^2+1)^(1/2)))
 

Fricas [A] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 247, normalized size of antiderivative = 1.13 \[ \int \left (d-c^2 d x^2\right )^2 (a+b \arccos (c x))^2 \, dx=\frac {27 \, {\left (25 \, a^{2} - 2 \, b^{2}\right )} c^{5} d^{2} x^{5} - 10 \, {\left (225 \, a^{2} - 38 \, b^{2}\right )} c^{3} d^{2} x^{3} + 15 \, {\left (225 \, a^{2} - 298 \, b^{2}\right )} c d^{2} x + 225 \, {\left (3 \, b^{2} c^{5} d^{2} x^{5} - 10 \, b^{2} c^{3} d^{2} x^{3} + 15 \, b^{2} c d^{2} x\right )} \arccos \left (c x\right )^{2} + 450 \, {\left (3 \, a b c^{5} d^{2} x^{5} - 10 \, a b c^{3} d^{2} x^{3} + 15 \, a b c d^{2} x\right )} \arccos \left (c x\right ) - 30 \, {\left (9 \, a b c^{4} d^{2} x^{4} - 38 \, a b c^{2} d^{2} x^{2} + 149 \, a b d^{2} + {\left (9 \, b^{2} c^{4} d^{2} x^{4} - 38 \, b^{2} c^{2} d^{2} x^{2} + 149 \, b^{2} d^{2}\right )} \arccos \left (c x\right )\right )} \sqrt {-c^{2} x^{2} + 1}}{3375 \, c} \] Input:

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

Output:

1/3375*(27*(25*a^2 - 2*b^2)*c^5*d^2*x^5 - 10*(225*a^2 - 38*b^2)*c^3*d^2*x^ 
3 + 15*(225*a^2 - 298*b^2)*c*d^2*x + 225*(3*b^2*c^5*d^2*x^5 - 10*b^2*c^3*d 
^2*x^3 + 15*b^2*c*d^2*x)*arccos(c*x)^2 + 450*(3*a*b*c^5*d^2*x^5 - 10*a*b*c 
^3*d^2*x^3 + 15*a*b*c*d^2*x)*arccos(c*x) - 30*(9*a*b*c^4*d^2*x^4 - 38*a*b* 
c^2*d^2*x^2 + 149*a*b*d^2 + (9*b^2*c^4*d^2*x^4 - 38*b^2*c^2*d^2*x^2 + 149* 
b^2*d^2)*arccos(c*x))*sqrt(-c^2*x^2 + 1))/c
 

Sympy [A] (verification not implemented)

Time = 0.44 (sec) , antiderivative size = 394, normalized size of antiderivative = 1.80 \[ \int \left (d-c^2 d x^2\right )^2 (a+b \arccos (c x))^2 \, dx=\begin {cases} \frac {a^{2} c^{4} d^{2} x^{5}}{5} - \frac {2 a^{2} c^{2} d^{2} x^{3}}{3} + a^{2} d^{2} x + \frac {2 a b c^{4} d^{2} x^{5} \operatorname {acos}{\left (c x \right )}}{5} - \frac {2 a b c^{3} d^{2} x^{4} \sqrt {- c^{2} x^{2} + 1}}{25} - \frac {4 a b c^{2} d^{2} x^{3} \operatorname {acos}{\left (c x \right )}}{3} + \frac {76 a b c d^{2} x^{2} \sqrt {- c^{2} x^{2} + 1}}{225} + 2 a b d^{2} x \operatorname {acos}{\left (c x \right )} - \frac {298 a b d^{2} \sqrt {- c^{2} x^{2} + 1}}{225 c} + \frac {b^{2} c^{4} d^{2} x^{5} \operatorname {acos}^{2}{\left (c x \right )}}{5} - \frac {2 b^{2} c^{4} d^{2} x^{5}}{125} - \frac {2 b^{2} c^{3} d^{2} x^{4} \sqrt {- c^{2} x^{2} + 1} \operatorname {acos}{\left (c x \right )}}{25} - \frac {2 b^{2} c^{2} d^{2} x^{3} \operatorname {acos}^{2}{\left (c x \right )}}{3} + \frac {76 b^{2} c^{2} d^{2} x^{3}}{675} + \frac {76 b^{2} c d^{2} x^{2} \sqrt {- c^{2} x^{2} + 1} \operatorname {acos}{\left (c x \right )}}{225} + b^{2} d^{2} x \operatorname {acos}^{2}{\left (c x \right )} - \frac {298 b^{2} d^{2} x}{225} - \frac {298 b^{2} d^{2} \sqrt {- c^{2} x^{2} + 1} \operatorname {acos}{\left (c x \right )}}{225 c} & \text {for}\: c \neq 0 \\d^{2} x \left (a + \frac {\pi b}{2}\right )^{2} & \text {otherwise} \end {cases} \] Input:

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

Output:

Piecewise((a**2*c**4*d**2*x**5/5 - 2*a**2*c**2*d**2*x**3/3 + a**2*d**2*x + 
 2*a*b*c**4*d**2*x**5*acos(c*x)/5 - 2*a*b*c**3*d**2*x**4*sqrt(-c**2*x**2 + 
 1)/25 - 4*a*b*c**2*d**2*x**3*acos(c*x)/3 + 76*a*b*c*d**2*x**2*sqrt(-c**2* 
x**2 + 1)/225 + 2*a*b*d**2*x*acos(c*x) - 298*a*b*d**2*sqrt(-c**2*x**2 + 1) 
/(225*c) + b**2*c**4*d**2*x**5*acos(c*x)**2/5 - 2*b**2*c**4*d**2*x**5/125 
- 2*b**2*c**3*d**2*x**4*sqrt(-c**2*x**2 + 1)*acos(c*x)/25 - 2*b**2*c**2*d* 
*2*x**3*acos(c*x)**2/3 + 76*b**2*c**2*d**2*x**3/675 + 76*b**2*c*d**2*x**2* 
sqrt(-c**2*x**2 + 1)*acos(c*x)/225 + b**2*d**2*x*acos(c*x)**2 - 298*b**2*d 
**2*x/225 - 298*b**2*d**2*sqrt(-c**2*x**2 + 1)*acos(c*x)/(225*c), Ne(c, 0) 
), (d**2*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. 466 vs. \(2 (193) = 386\).

Time = 0.16 (sec) , antiderivative size = 466, normalized size of antiderivative = 2.13 \[ \int \left (d-c^2 d x^2\right )^2 (a+b \arccos (c x))^2 \, dx=\frac {1}{5} \, b^{2} c^{4} d^{2} x^{5} \arccos \left (c x\right )^{2} + \frac {1}{5} \, a^{2} c^{4} d^{2} x^{5} - \frac {2}{3} \, b^{2} c^{2} d^{2} x^{3} \arccos \left (c x\right )^{2} + \frac {2}{75} \, {\left (15 \, x^{5} \arccos \left (c x\right ) - {\left (\frac {3 \, \sqrt {-c^{2} x^{2} + 1} x^{4}}{c^{2}} + \frac {4 \, \sqrt {-c^{2} x^{2} + 1} x^{2}}{c^{4}} + \frac {8 \, \sqrt {-c^{2} x^{2} + 1}}{c^{6}}\right )} c\right )} a b c^{4} d^{2} - \frac {2}{1125} \, {\left (15 \, {\left (\frac {3 \, \sqrt {-c^{2} x^{2} + 1} x^{4}}{c^{2}} + \frac {4 \, \sqrt {-c^{2} x^{2} + 1} x^{2}}{c^{4}} + \frac {8 \, \sqrt {-c^{2} x^{2} + 1}}{c^{6}}\right )} c \arccos \left (c x\right ) + \frac {9 \, c^{4} x^{5} + 20 \, c^{2} x^{3} + 120 \, x}{c^{4}}\right )} b^{2} c^{4} d^{2} - \frac {2}{3} \, a^{2} c^{2} d^{2} x^{3} - \frac {4}{9} \, {\left (3 \, x^{3} \arccos \left (c x\right ) - c {\left (\frac {\sqrt {-c^{2} x^{2} + 1} x^{2}}{c^{2}} + \frac {2 \, \sqrt {-c^{2} x^{2} + 1}}{c^{4}}\right )}\right )} a b c^{2} d^{2} + \frac {4}{27} \, {\left (3 \, c {\left (\frac {\sqrt {-c^{2} x^{2} + 1} x^{2}}{c^{2}} + \frac {2 \, \sqrt {-c^{2} x^{2} + 1}}{c^{4}}\right )} \arccos \left (c x\right ) + \frac {c^{2} x^{3} + 6 \, x}{c^{2}}\right )} b^{2} c^{2} d^{2} + b^{2} d^{2} x \arccos \left (c x\right )^{2} - 2 \, b^{2} d^{2} {\left (x + \frac {\sqrt {-c^{2} x^{2} + 1} \arccos \left (c x\right )}{c}\right )} + a^{2} d^{2} x + \frac {2 \, {\left (c x \arccos \left (c x\right ) - \sqrt {-c^{2} x^{2} + 1}\right )} a b d^{2}}{c} \] Input:

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

Output:

1/5*b^2*c^4*d^2*x^5*arccos(c*x)^2 + 1/5*a^2*c^4*d^2*x^5 - 2/3*b^2*c^2*d^2* 
x^3*arccos(c*x)^2 + 2/75*(15*x^5*arccos(c*x) - (3*sqrt(-c^2*x^2 + 1)*x^4/c 
^2 + 4*sqrt(-c^2*x^2 + 1)*x^2/c^4 + 8*sqrt(-c^2*x^2 + 1)/c^6)*c)*a*b*c^4*d 
^2 - 2/1125*(15*(3*sqrt(-c^2*x^2 + 1)*x^4/c^2 + 4*sqrt(-c^2*x^2 + 1)*x^2/c 
^4 + 8*sqrt(-c^2*x^2 + 1)/c^6)*c*arccos(c*x) + (9*c^4*x^5 + 20*c^2*x^3 + 1 
20*x)/c^4)*b^2*c^4*d^2 - 2/3*a^2*c^2*d^2*x^3 - 4/9*(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 + 4/2 
7*(3*c*(sqrt(-c^2*x^2 + 1)*x^2/c^2 + 2*sqrt(-c^2*x^2 + 1)/c^4)*arccos(c*x) 
 + (c^2*x^3 + 6*x)/c^2)*b^2*c^2*d^2 + b^2*d^2*x*arccos(c*x)^2 - 2*b^2*d^2* 
(x + sqrt(-c^2*x^2 + 1)*arccos(c*x)/c) + a^2*d^2*x + 2*(c*x*arccos(c*x) - 
sqrt(-c^2*x^2 + 1))*a*b*d^2/c
 

Giac [A] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 329, normalized size of antiderivative = 1.50 \[ \int \left (d-c^2 d x^2\right )^2 (a+b \arccos (c x))^2 \, dx=\frac {1}{5} \, b^{2} c^{4} d^{2} x^{5} \arccos \left (c x\right )^{2} + \frac {2}{5} \, a b c^{4} d^{2} x^{5} \arccos \left (c x\right ) + \frac {1}{5} \, a^{2} c^{4} d^{2} x^{5} - \frac {2}{125} \, b^{2} c^{4} d^{2} x^{5} - \frac {2}{25} \, \sqrt {-c^{2} x^{2} + 1} b^{2} c^{3} d^{2} x^{4} \arccos \left (c x\right ) - \frac {2}{25} \, \sqrt {-c^{2} x^{2} + 1} a b c^{3} d^{2} x^{4} - \frac {2}{3} \, b^{2} c^{2} d^{2} x^{3} \arccos \left (c x\right )^{2} - \frac {4}{3} \, a b c^{2} d^{2} x^{3} \arccos \left (c x\right ) - \frac {2}{3} \, a^{2} c^{2} d^{2} x^{3} + \frac {76}{675} \, b^{2} c^{2} d^{2} x^{3} + \frac {76}{225} \, \sqrt {-c^{2} x^{2} + 1} b^{2} c d^{2} x^{2} \arccos \left (c x\right ) + \frac {76}{225} \, \sqrt {-c^{2} x^{2} + 1} a b c d^{2} x^{2} + b^{2} d^{2} x \arccos \left (c x\right )^{2} + 2 \, a b d^{2} x \arccos \left (c x\right ) + a^{2} d^{2} x - \frac {298}{225} \, b^{2} d^{2} x - \frac {298 \, \sqrt {-c^{2} x^{2} + 1} b^{2} d^{2} \arccos \left (c x\right )}{225 \, c} - \frac {298 \, \sqrt {-c^{2} x^{2} + 1} a b d^{2}}{225 \, c} \] Input:

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

Output:

1/5*b^2*c^4*d^2*x^5*arccos(c*x)^2 + 2/5*a*b*c^4*d^2*x^5*arccos(c*x) + 1/5* 
a^2*c^4*d^2*x^5 - 2/125*b^2*c^4*d^2*x^5 - 2/25*sqrt(-c^2*x^2 + 1)*b^2*c^3* 
d^2*x^4*arccos(c*x) - 2/25*sqrt(-c^2*x^2 + 1)*a*b*c^3*d^2*x^4 - 2/3*b^2*c^ 
2*d^2*x^3*arccos(c*x)^2 - 4/3*a*b*c^2*d^2*x^3*arccos(c*x) - 2/3*a^2*c^2*d^ 
2*x^3 + 76/675*b^2*c^2*d^2*x^3 + 76/225*sqrt(-c^2*x^2 + 1)*b^2*c*d^2*x^2*a 
rccos(c*x) + 76/225*sqrt(-c^2*x^2 + 1)*a*b*c*d^2*x^2 + b^2*d^2*x*arccos(c* 
x)^2 + 2*a*b*d^2*x*arccos(c*x) + a^2*d^2*x - 298/225*b^2*d^2*x - 298/225*s 
qrt(-c^2*x^2 + 1)*b^2*d^2*arccos(c*x)/c - 298/225*sqrt(-c^2*x^2 + 1)*a*b*d 
^2/c
 

Mupad [F(-1)]

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

\[ \int \left (d-c^2 d x^2\right )^2 (a+b \arccos (c x))^2 \, dx=\frac {d^{2} \left (225 \mathit {acos} \left (c x \right )^{2} b^{2} c x -450 \sqrt {-c^{2} x^{2}+1}\, \mathit {acos} \left (c x \right ) b^{2}+90 \mathit {acos} \left (c x \right ) a b \,c^{5} x^{5}-300 \mathit {acos} \left (c x \right ) a b \,c^{3} x^{3}+450 \mathit {acos} \left (c x \right ) a b c x -18 \sqrt {-c^{2} x^{2}+1}\, a b \,c^{4} x^{4}+76 \sqrt {-c^{2} x^{2}+1}\, a b \,c^{2} x^{2}-298 \sqrt {-c^{2} x^{2}+1}\, a b +225 \left (\int \mathit {acos} \left (c x \right )^{2} x^{4}d x \right ) b^{2} c^{5}-450 \left (\int \mathit {acos} \left (c x \right )^{2} x^{2}d x \right ) b^{2} c^{3}+45 a^{2} c^{5} x^{5}-150 a^{2} c^{3} x^{3}+225 a^{2} c x -450 b^{2} c x \right )}{225 c} \] Input:

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

Output:

(d**2*(225*acos(c*x)**2*b**2*c*x - 450*sqrt( - c**2*x**2 + 1)*acos(c*x)*b* 
*2 + 90*acos(c*x)*a*b*c**5*x**5 - 300*acos(c*x)*a*b*c**3*x**3 + 450*acos(c 
*x)*a*b*c*x - 18*sqrt( - c**2*x**2 + 1)*a*b*c**4*x**4 + 76*sqrt( - c**2*x* 
*2 + 1)*a*b*c**2*x**2 - 298*sqrt( - c**2*x**2 + 1)*a*b + 225*int(acos(c*x) 
**2*x**4,x)*b**2*c**5 - 450*int(acos(c*x)**2*x**2,x)*b**2*c**3 + 45*a**2*c 
**5*x**5 - 150*a**2*c**3*x**3 + 225*a**2*c*x - 450*b**2*c*x))/(225*c)