\(\int (d-c^2 d x^2)^3 (a+b \arccos (c x))^2 \, dx\) [7]

Optimal result

Integrand size = 24, antiderivative size = 298 \[ \int \left (d-c^2 d x^2\right )^3 (a+b \arccos (c x))^2 \, dx=-\frac {4322 b^2 d^3 x}{3675}+\frac {1514 b^2 c^2 d^3 x^3}{11025}-\frac {234 b^2 c^4 d^3 x^5}{6125}+\frac {2}{343} b^2 c^6 d^3 x^7-\frac {32 b d^3 \sqrt {1-c^2 x^2} (a+b \arccos (c x))}{35 c}-\frac {16 b d^3 \left (1-c^2 x^2\right )^{3/2} (a+b \arccos (c x))}{105 c}-\frac {12 b d^3 \left (1-c^2 x^2\right )^{5/2} (a+b \arccos (c x))}{175 c}-\frac {2 b d^3 \left (1-c^2 x^2\right )^{7/2} (a+b \arccos (c x))}{49 c}+\frac {16}{35} d^3 x (a+b \arccos (c x))^2+\frac {8}{35} d^3 x \left (1-c^2 x^2\right ) (a+b \arccos (c x))^2+\frac {6}{35} d^3 x \left (1-c^2 x^2\right )^2 (a+b \arccos (c x))^2+\frac {1}{7} d^3 x \left (1-c^2 x^2\right )^3 (a+b \arccos (c x))^2 \] Output:

-4322/3675*b^2*d^3*x+1514/11025*b^2*c^2*d^3*x^3-234/6125*b^2*c^4*d^3*x^5+2 
/343*b^2*c^6*d^3*x^7-32/35*b*d^3*(-c^2*x^2+1)^(1/2)*(a+b*arccos(c*x))/c-16 
/105*b*d^3*(-c^2*x^2+1)^(3/2)*(a+b*arccos(c*x))/c-12/175*b*d^3*(-c^2*x^2+1 
)^(5/2)*(a+b*arccos(c*x))/c-2/49*b*d^3*(-c^2*x^2+1)^(7/2)*(a+b*arccos(c*x) 
)/c+16/35*d^3*x*(a+b*arccos(c*x))^2+8/35*d^3*x*(-c^2*x^2+1)*(a+b*arccos(c* 
x))^2+6/35*d^3*x*(-c^2*x^2+1)^2*(a+b*arccos(c*x))^2+1/7*d^3*x*(-c^2*x^2+1) 
^3*(a+b*arccos(c*x))^2
 

Mathematica [A] (verified)

Time = 1.34 (sec) , antiderivative size = 241, normalized size of antiderivative = 0.81 \[ \int \left (d-c^2 d x^2\right )^3 (a+b \arccos (c x))^2 \, dx=\frac {d^3 \left (-11025 a^2 c x \left (-35+35 c^2 x^2-21 c^4 x^4+5 c^6 x^6\right )+210 a b \sqrt {1-c^2 x^2} \left (-2161+757 c^2 x^2-351 c^4 x^4+75 c^6 x^6\right )+2 b^2 c x \left (-226905+26495 c^2 x^2-7371 c^4 x^4+1125 c^6 x^6\right )+210 b \left (-105 a c x \left (-35+35 c^2 x^2-21 c^4 x^4+5 c^6 x^6\right )+b \sqrt {1-c^2 x^2} \left (-2161+757 c^2 x^2-351 c^4 x^4+75 c^6 x^6\right )\right ) \arccos (c x)-11025 b^2 c x \left (-35+35 c^2 x^2-21 c^4 x^4+5 c^6 x^6\right ) \arccos (c x)^2\right )}{385875 c} \] Input:

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

Output:

(d^3*(-11025*a^2*c*x*(-35 + 35*c^2*x^2 - 21*c^4*x^4 + 5*c^6*x^6) + 210*a*b 
*Sqrt[1 - c^2*x^2]*(-2161 + 757*c^2*x^2 - 351*c^4*x^4 + 75*c^6*x^6) + 2*b^ 
2*c*x*(-226905 + 26495*c^2*x^2 - 7371*c^4*x^4 + 1125*c^6*x^6) + 210*b*(-10 
5*a*c*x*(-35 + 35*c^2*x^2 - 21*c^4*x^4 + 5*c^6*x^6) + b*Sqrt[1 - c^2*x^2]* 
(-2161 + 757*c^2*x^2 - 351*c^4*x^4 + 75*c^6*x^6))*ArcCos[c*x] - 11025*b^2* 
c*x*(-35 + 35*c^2*x^2 - 21*c^4*x^4 + 5*c^6*x^6)*ArcCos[c*x]^2))/(385875*c)
 

Rubi [A] (verified)

Time = 1.32 (sec) , antiderivative size = 351, normalized size of antiderivative = 1.18, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {5159, 27, 5159, 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.

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

\(\Big \downarrow \) 5159

\(\displaystyle \frac {2}{7} b c d^3 \int x \left (1-c^2 x^2\right )^{5/2} (a+b \arccos (c x))dx+\frac {6}{7} d \int d^2 \left (1-c^2 x^2\right )^2 (a+b \arccos (c x))^2dx+\frac {1}{7} d^3 x \left (1-c^2 x^2\right )^3 (a+b \arccos (c x))^2\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {2}{7} b c d^3 \int x \left (1-c^2 x^2\right )^{5/2} (a+b \arccos (c x))dx+\frac {6}{7} d^3 \int \left (1-c^2 x^2\right )^2 (a+b \arccos (c x))^2dx+\frac {1}{7} d^3 x \left (1-c^2 x^2\right )^3 (a+b \arccos (c x))^2\)

\(\Big \downarrow \) 5159

\(\displaystyle \frac {2}{7} b c d^3 \int x \left (1-c^2 x^2\right )^{5/2} (a+b \arccos (c x))dx+\frac {6}{7} d^3 \left (\frac {2}{5} b c \int x \left (1-c^2 x^2\right )^{3/2} (a+b \arccos (c x))dx+\frac {4}{5} \int \left (1-c^2 x^2\right ) (a+b \arccos (c x))^2dx+\frac {1}{5} x \left (1-c^2 x^2\right )^2 (a+b \arccos (c x))^2\right )+\frac {1}{7} d^3 x \left (1-c^2 x^2\right )^3 (a+b \arccos (c x))^2\)

\(\Big \downarrow \) 5159

\(\displaystyle \frac {2}{7} b c d^3 \int x \left (1-c^2 x^2\right )^{5/2} (a+b \arccos (c x))dx+\frac {6}{7} d^3 \left (\frac {2}{5} b c \int x \left (1-c^2 x^2\right )^{3/2} (a+b \arccos (c x))dx+\frac {4}{5} \left (\frac {2}{3} b c \int x \sqrt {1-c^2 x^2} (a+b \arccos (c x))dx+\frac {2}{3} \int (a+b \arccos (c x))^2dx+\frac {1}{3} x \left (1-c^2 x^2\right ) (a+b \arccos (c x))^2\right )+\frac {1}{5} x \left (1-c^2 x^2\right )^2 (a+b \arccos (c x))^2\right )+\frac {1}{7} d^3 x \left (1-c^2 x^2\right )^3 (a+b \arccos (c x))^2\)

\(\Big \downarrow \) 5131

\(\displaystyle \frac {6}{7} d^3 \left (\frac {4}{5} \left (\frac {2}{3} \left (2 b c \int \frac {x (a+b \arccos (c x))}{\sqrt {1-c^2 x^2}}dx+x (a+b \arccos (c x))^2\right )+\frac {2}{3} b c \int x \sqrt {1-c^2 x^2} (a+b \arccos (c x))dx+\frac {1}{3} x \left (1-c^2 x^2\right ) (a+b \arccos (c x))^2\right )+\frac {2}{5} b c \int x \left (1-c^2 x^2\right )^{3/2} (a+b \arccos (c x))dx+\frac {1}{5} x \left (1-c^2 x^2\right )^2 (a+b \arccos (c x))^2\right )+\frac {2}{7} b c d^3 \int x \left (1-c^2 x^2\right )^{5/2} (a+b \arccos (c x))dx+\frac {1}{7} d^3 x \left (1-c^2 x^2\right )^3 (a+b \arccos (c x))^2\)

\(\Big \downarrow \) 5183

\(\displaystyle \frac {6}{7} d^3 \left (\frac {4}{5} \left (\frac {2}{3} \left (2 b c \left (-\frac {b \int 1dx}{c}-\frac {\sqrt {1-c^2 x^2} (a+b \arccos (c x))}{c^2}\right )+x (a+b \arccos (c x))^2\right )+\frac {2}{3} b c \left (-\frac {b \int \left (1-c^2 x^2\right )dx}{3 c}-\frac {\left (1-c^2 x^2\right )^{3/2} (a+b \arccos (c x))}{3 c^2}\right )+\frac {1}{3} x \left (1-c^2 x^2\right ) (a+b \arccos (c x))^2\right )+\frac {2}{5} b c \left (-\frac {b \int \left (1-c^2 x^2\right )^2dx}{5 c}-\frac {\left (1-c^2 x^2\right )^{5/2} (a+b \arccos (c x))}{5 c^2}\right )+\frac {1}{5} x \left (1-c^2 x^2\right )^2 (a+b \arccos (c x))^2\right )+\frac {2}{7} b c d^3 \left (-\frac {b \int \left (1-c^2 x^2\right )^3dx}{7 c}-\frac {\left (1-c^2 x^2\right )^{7/2} (a+b \arccos (c x))}{7 c^2}\right )+\frac {1}{7} d^3 x \left (1-c^2 x^2\right )^3 (a+b \arccos (c x))^2\)

\(\Big \downarrow \) 24

\(\displaystyle \frac {6}{7} d^3 \left (\frac {4}{5} \left (\frac {2}{3} b c \left (-\frac {b \int \left (1-c^2 x^2\right )dx}{3 c}-\frac {\left (1-c^2 x^2\right )^{3/2} (a+b \arccos (c x))}{3 c^2}\right )+\frac {1}{3} x \left (1-c^2 x^2\right ) (a+b \arccos (c x))^2+\frac {2}{3} \left (2 b c \left (-\frac {\sqrt {1-c^2 x^2} (a+b \arccos (c x))}{c^2}-\frac {b x}{c}\right )+x (a+b \arccos (c x))^2\right )\right )+\frac {2}{5} b c \left (-\frac {b \int \left (1-c^2 x^2\right )^2dx}{5 c}-\frac {\left (1-c^2 x^2\right )^{5/2} (a+b \arccos (c x))}{5 c^2}\right )+\frac {1}{5} x \left (1-c^2 x^2\right )^2 (a+b \arccos (c x))^2\right )+\frac {2}{7} b c d^3 \left (-\frac {b \int \left (1-c^2 x^2\right )^3dx}{7 c}-\frac {\left (1-c^2 x^2\right )^{7/2} (a+b \arccos (c x))}{7 c^2}\right )+\frac {1}{7} d^3 x \left (1-c^2 x^2\right )^3 (a+b \arccos (c x))^2\)

\(\Big \downarrow \) 210

\(\displaystyle \frac {6}{7} d^3 \left (\frac {4}{5} \left (\frac {2}{3} b c \left (-\frac {b \int \left (1-c^2 x^2\right )dx}{3 c}-\frac {\left (1-c^2 x^2\right )^{3/2} (a+b \arccos (c x))}{3 c^2}\right )+\frac {1}{3} x \left (1-c^2 x^2\right ) (a+b \arccos (c x))^2+\frac {2}{3} \left (2 b c \left (-\frac {\sqrt {1-c^2 x^2} (a+b \arccos (c x))}{c^2}-\frac {b x}{c}\right )+x (a+b \arccos (c x))^2\right )\right )+\frac {2}{5} b c \left (-\frac {b \int \left (c^4 x^4-2 c^2 x^2+1\right )dx}{5 c}-\frac {\left (1-c^2 x^2\right )^{5/2} (a+b \arccos (c x))}{5 c^2}\right )+\frac {1}{5} x \left (1-c^2 x^2\right )^2 (a+b \arccos (c x))^2\right )+\frac {2}{7} b c d^3 \left (-\frac {b \int \left (-c^6 x^6+3 c^4 x^4-3 c^2 x^2+1\right )dx}{7 c}-\frac {\left (1-c^2 x^2\right )^{7/2} (a+b \arccos (c x))}{7 c^2}\right )+\frac {1}{7} d^3 x \left (1-c^2 x^2\right )^3 (a+b \arccos (c x))^2\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {1}{7} d^3 x \left (1-c^2 x^2\right )^3 (a+b \arccos (c x))^2+\frac {6}{7} d^3 \left (\frac {1}{5} x \left (1-c^2 x^2\right )^2 (a+b \arccos (c x))^2+\frac {4}{5} \left (\frac {1}{3} x \left (1-c^2 x^2\right ) (a+b \arccos (c x))^2+\frac {2}{3} \left (2 b c \left (-\frac {\sqrt {1-c^2 x^2} (a+b \arccos (c x))}{c^2}-\frac {b x}{c}\right )+x (a+b \arccos (c x))^2\right )+\frac {2}{3} b c \left (-\frac {\left (1-c^2 x^2\right )^{3/2} (a+b \arccos (c x))}{3 c^2}-\frac {b \left (x-\frac {c^2 x^3}{3}\right )}{3 c}\right )\right )+\frac {2}{5} b c \left (-\frac {\left (1-c^2 x^2\right )^{5/2} (a+b \arccos (c x))}{5 c^2}-\frac {b \left (\frac {c^4 x^5}{5}-\frac {2 c^2 x^3}{3}+x\right )}{5 c}\right )\right )+\frac {2}{7} b c d^3 \left (-\frac {\left (1-c^2 x^2\right )^{7/2} (a+b \arccos (c x))}{7 c^2}-\frac {b \left (-\frac {1}{7} c^6 x^7+\frac {3 c^4 x^5}{5}-c^2 x^3+x\right )}{7 c}\right )\)

Input:

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

Output:

(d^3*x*(1 - c^2*x^2)^3*(a + b*ArcCos[c*x])^2)/7 + (2*b*c*d^3*(-1/7*(b*(x - 
 c^2*x^3 + (3*c^4*x^5)/5 - (c^6*x^7)/7))/c - ((1 - c^2*x^2)^(7/2)*(a + b*A 
rcCos[c*x]))/(7*c^2)))/7 + (6*d^3*((x*(1 - c^2*x^2)^2*(a + b*ArcCos[c*x])^ 
2)/5 + (2*b*c*(-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*((x*(1 - c^2*x^2)*(a + b*Ar 
cCos[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))/7
 

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 = 0.71 (sec) , antiderivative size = 384, normalized size of antiderivative = 1.29

method	result	size
derivativedivides	\(\frac {-d^{3} a^{2} \left (\frac {1}{7} c^{7} x^{7}-\frac {3}{5} c^{5} x^{5}+c^{3} x^{3}-c x \right )-d^{3} b^{2} \left (\frac {\arccos \left (c x \right )^{2} \left (5 c^{6} x^{6}-21 c^{4} x^{4}+35 c^{2} x^{2}-35\right ) c x}{35}-\frac {2 \arccos \left (c x \right ) \left (c^{2} x^{2}-1\right )^{3} \sqrt {-c^{2} x^{2}+1}}{49}-\frac {2 \left (5 c^{6} x^{6}-21 c^{4} x^{4}+35 c^{2} x^{2}-35\right ) c x}{1715}+\frac {12 \arccos \left (c x \right ) \left (c^{2} x^{2}-1\right )^{2} \sqrt {-c^{2} x^{2}+1}}{175}+\frac {4 \left (3 c^{4} x^{4}-10 c^{2} x^{2}+15\right ) c x}{875}-\frac {16 \arccos \left (c x \right ) \left (c^{2} x^{2}-1\right ) \sqrt {-c^{2} x^{2}+1}}{105}-\frac {16 \left (c^{2} x^{2}-3\right ) c x}{315}+\frac {32 c x}{35}+\frac {32 \arccos \left (c x \right ) \sqrt {-c^{2} x^{2}+1}}{35}\right )-2 d^{3} a b \left (\frac {\arccos \left (c x \right ) c^{7} x^{7}}{7}-\frac {3 \arccos \left (c x \right ) c^{5} x^{5}}{5}+c^{3} x^{3} \arccos \left (c x \right )-c x \arccos \left (c x \right )+\frac {2161 \sqrt {-c^{2} x^{2}+1}}{3675}-\frac {757 c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}}{3675}+\frac {117 c^{4} x^{4} \sqrt {-c^{2} x^{2}+1}}{1225}-\frac {c^{6} x^{6} \sqrt {-c^{2} x^{2}+1}}{49}\right )}{c}\)	\(384\)
default	\(\frac {-d^{3} a^{2} \left (\frac {1}{7} c^{7} x^{7}-\frac {3}{5} c^{5} x^{5}+c^{3} x^{3}-c x \right )-d^{3} b^{2} \left (\frac {\arccos \left (c x \right )^{2} \left (5 c^{6} x^{6}-21 c^{4} x^{4}+35 c^{2} x^{2}-35\right ) c x}{35}-\frac {2 \arccos \left (c x \right ) \left (c^{2} x^{2}-1\right )^{3} \sqrt {-c^{2} x^{2}+1}}{49}-\frac {2 \left (5 c^{6} x^{6}-21 c^{4} x^{4}+35 c^{2} x^{2}-35\right ) c x}{1715}+\frac {12 \arccos \left (c x \right ) \left (c^{2} x^{2}-1\right )^{2} \sqrt {-c^{2} x^{2}+1}}{175}+\frac {4 \left (3 c^{4} x^{4}-10 c^{2} x^{2}+15\right ) c x}{875}-\frac {16 \arccos \left (c x \right ) \left (c^{2} x^{2}-1\right ) \sqrt {-c^{2} x^{2}+1}}{105}-\frac {16 \left (c^{2} x^{2}-3\right ) c x}{315}+\frac {32 c x}{35}+\frac {32 \arccos \left (c x \right ) \sqrt {-c^{2} x^{2}+1}}{35}\right )-2 d^{3} a b \left (\frac {\arccos \left (c x \right ) c^{7} x^{7}}{7}-\frac {3 \arccos \left (c x \right ) c^{5} x^{5}}{5}+c^{3} x^{3} \arccos \left (c x \right )-c x \arccos \left (c x \right )+\frac {2161 \sqrt {-c^{2} x^{2}+1}}{3675}-\frac {757 c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}}{3675}+\frac {117 c^{4} x^{4} \sqrt {-c^{2} x^{2}+1}}{1225}-\frac {c^{6} x^{6} \sqrt {-c^{2} x^{2}+1}}{49}\right )}{c}\)	\(384\)
parts	\(-d^{3} a^{2} \left (\frac {1}{7} c^{6} x^{7}-\frac {3}{5} c^{4} x^{5}+c^{2} x^{3}-x \right )-\frac {d^{3} b^{2} \left (\frac {\arccos \left (c x \right )^{2} \left (5 c^{6} x^{6}-21 c^{4} x^{4}+35 c^{2} x^{2}-35\right ) c x}{35}-\frac {2 \arccos \left (c x \right ) \left (c^{2} x^{2}-1\right )^{3} \sqrt {-c^{2} x^{2}+1}}{49}-\frac {2 \left (5 c^{6} x^{6}-21 c^{4} x^{4}+35 c^{2} x^{2}-35\right ) c x}{1715}+\frac {12 \arccos \left (c x \right ) \left (c^{2} x^{2}-1\right )^{2} \sqrt {-c^{2} x^{2}+1}}{175}+\frac {4 \left (3 c^{4} x^{4}-10 c^{2} x^{2}+15\right ) c x}{875}-\frac {16 \arccos \left (c x \right ) \left (c^{2} x^{2}-1\right ) \sqrt {-c^{2} x^{2}+1}}{105}-\frac {16 \left (c^{2} x^{2}-3\right ) c x}{315}+\frac {32 c x}{35}+\frac {32 \arccos \left (c x \right ) \sqrt {-c^{2} x^{2}+1}}{35}\right )}{c}-\frac {2 d^{3} a b \left (\frac {\arccos \left (c x \right ) c^{7} x^{7}}{7}-\frac {3 \arccos \left (c x \right ) c^{5} x^{5}}{5}+c^{3} x^{3} \arccos \left (c x \right )-c x \arccos \left (c x \right )+\frac {2161 \sqrt {-c^{2} x^{2}+1}}{3675}-\frac {757 c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}}{3675}+\frac {117 c^{4} x^{4} \sqrt {-c^{2} x^{2}+1}}{1225}-\frac {c^{6} x^{6} \sqrt {-c^{2} x^{2}+1}}{49}\right )}{c}\)	\(385\)
orering	\(\frac {x \left (47625 c^{8} x^{8}-271212 c^{6} x^{6}+741678 c^{4} x^{4}-3539900 c^{2} x^{2}+128625\right ) \left (-c^{2} d \,x^{2}+d \right )^{3} \left (a +b \arccos \left (c x \right )\right )^{2}}{128625 \left (c x -1\right )^{2} \left (c x +1\right )^{2} \left (c^{2} x^{2}-1\right )^{2}}-\frac {\left (20250 c^{8} x^{8}-125811 c^{6} x^{6}+407785 c^{4} x^{4}-2802345 c^{2} x^{2}+226905\right ) \left (-6 \left (-c^{2} d \,x^{2}+d \right )^{2} \left (a +b \arccos \left (c x \right )\right )^{2} d \,c^{2} x -\frac {2 \left (-c^{2} d \,x^{2}+d \right )^{3} \left (a +b \arccos \left (c x \right )\right ) b c}{\sqrt {-c^{2} x^{2}+1}}\right )}{385875 c^{2} \left (c x -1\right )^{2} \left (c x +1\right )^{2} \left (c^{2} x^{2}-1\right )}+\frac {x \left (1125 c^{6} x^{6}-7371 c^{4} x^{4}+26495 c^{2} x^{2}-226905\right ) \left (24 \left (-c^{2} d \,x^{2}+d \right ) \left (a +b \arccos \left (c x \right )\right )^{2} d^{2} c^{4} x^{2}+\frac {24 \left (-c^{2} d \,x^{2}+d \right )^{2} \left (a +b \arccos \left (c x \right )\right ) d \,c^{3} x b}{\sqrt {-c^{2} x^{2}+1}}-6 \left (-c^{2} d \,x^{2}+d \right )^{2} \left (a +b \arccos \left (c x \right )\right )^{2} d \,c^{2}+\frac {2 \left (-c^{2} d \,x^{2}+d \right )^{3} b^{2} c^{2}}{-c^{2} x^{2}+1}-\frac {2 \left (-c^{2} d \,x^{2}+d \right )^{3} \left (a +b \arccos \left (c x \right )\right ) b \,c^{3} x}{\left (-c^{2} x^{2}+1\right )^{\frac {3}{2}}}\right )}{385875 c^{2} \left (c x -1\right )^{2} \left (c x +1\right )^{2}}\)	\(441\)

Input:

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

Output:

1/c*(-d^3*a^2*(1/7*c^7*x^7-3/5*c^5*x^5+c^3*x^3-c*x)-d^3*b^2*(1/35*arccos(c 
*x)^2*(5*c^6*x^6-21*c^4*x^4+35*c^2*x^2-35)*c*x-2/49*arccos(c*x)*(c^2*x^2-1 
)^3*(-c^2*x^2+1)^(1/2)-2/1715*(5*c^6*x^6-21*c^4*x^4+35*c^2*x^2-35)*c*x+12/ 
175*arccos(c*x)*(c^2*x^2-1)^2*(-c^2*x^2+1)^(1/2)+4/875*(3*c^4*x^4-10*c^2*x 
^2+15)*c*x-16/105*arccos(c*x)*(c^2*x^2-1)*(-c^2*x^2+1)^(1/2)-16/315*(c^2*x 
^2-3)*c*x+32/35*c*x+32/35*arccos(c*x)*(-c^2*x^2+1)^(1/2))-2*d^3*a*b*(1/7*a 
rccos(c*x)*c^7*x^7-3/5*arccos(c*x)*c^5*x^5+c^3*x^3*arccos(c*x)-c*x*arccos( 
c*x)+2161/3675*(-c^2*x^2+1)^(1/2)-757/3675*c^2*x^2*(-c^2*x^2+1)^(1/2)+117/ 
1225*c^4*x^4*(-c^2*x^2+1)^(1/2)-1/49*c^6*x^6*(-c^2*x^2+1)^(1/2)))
 

Fricas [A] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 323, normalized size of antiderivative = 1.08 \[ \int \left (d-c^2 d x^2\right )^3 (a+b \arccos (c x))^2 \, dx=-\frac {1125 \, {\left (49 \, a^{2} - 2 \, b^{2}\right )} c^{7} d^{3} x^{7} - 189 \, {\left (1225 \, a^{2} - 78 \, b^{2}\right )} c^{5} d^{3} x^{5} + 35 \, {\left (11025 \, a^{2} - 1514 \, b^{2}\right )} c^{3} d^{3} x^{3} - 105 \, {\left (3675 \, a^{2} - 4322 \, b^{2}\right )} c d^{3} x + 11025 \, {\left (5 \, b^{2} c^{7} d^{3} x^{7} - 21 \, b^{2} c^{5} d^{3} x^{5} + 35 \, b^{2} c^{3} d^{3} x^{3} - 35 \, b^{2} c d^{3} x\right )} \arccos \left (c x\right )^{2} + 22050 \, {\left (5 \, a b c^{7} d^{3} x^{7} - 21 \, a b c^{5} d^{3} x^{5} + 35 \, a b c^{3} d^{3} x^{3} - 35 \, a b c d^{3} x\right )} \arccos \left (c x\right ) - 210 \, {\left (75 \, a b c^{6} d^{3} x^{6} - 351 \, a b c^{4} d^{3} x^{4} + 757 \, a b c^{2} d^{3} x^{2} - 2161 \, a b d^{3} + {\left (75 \, b^{2} c^{6} d^{3} x^{6} - 351 \, b^{2} c^{4} d^{3} x^{4} + 757 \, b^{2} c^{2} d^{3} x^{2} - 2161 \, b^{2} d^{3}\right )} \arccos \left (c x\right )\right )} \sqrt {-c^{2} x^{2} + 1}}{385875 \, c} \] Input:

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

Output:

-1/385875*(1125*(49*a^2 - 2*b^2)*c^7*d^3*x^7 - 189*(1225*a^2 - 78*b^2)*c^5 
*d^3*x^5 + 35*(11025*a^2 - 1514*b^2)*c^3*d^3*x^3 - 105*(3675*a^2 - 4322*b^ 
2)*c*d^3*x + 11025*(5*b^2*c^7*d^3*x^7 - 21*b^2*c^5*d^3*x^5 + 35*b^2*c^3*d^ 
3*x^3 - 35*b^2*c*d^3*x)*arccos(c*x)^2 + 22050*(5*a*b*c^7*d^3*x^7 - 21*a*b* 
c^5*d^3*x^5 + 35*a*b*c^3*d^3*x^3 - 35*a*b*c*d^3*x)*arccos(c*x) - 210*(75*a 
*b*c^6*d^3*x^6 - 351*a*b*c^4*d^3*x^4 + 757*a*b*c^2*d^3*x^2 - 2161*a*b*d^3 
+ (75*b^2*c^6*d^3*x^6 - 351*b^2*c^4*d^3*x^4 + 757*b^2*c^2*d^3*x^2 - 2161*b 
^2*d^3)*arccos(c*x))*sqrt(-c^2*x^2 + 1))/c
 

Sympy [A] (verification not implemented)

Time = 0.80 (sec) , antiderivative size = 529, normalized size of antiderivative = 1.78 \[ \int \left (d-c^2 d x^2\right )^3 (a+b \arccos (c x))^2 \, dx=\begin {cases} - \frac {a^{2} c^{6} d^{3} x^{7}}{7} + \frac {3 a^{2} c^{4} d^{3} x^{5}}{5} - a^{2} c^{2} d^{3} x^{3} + a^{2} d^{3} x - \frac {2 a b c^{6} d^{3} x^{7} \operatorname {acos}{\left (c x \right )}}{7} + \frac {2 a b c^{5} d^{3} x^{6} \sqrt {- c^{2} x^{2} + 1}}{49} + \frac {6 a b c^{4} d^{3} x^{5} \operatorname {acos}{\left (c x \right )}}{5} - \frac {234 a b c^{3} d^{3} x^{4} \sqrt {- c^{2} x^{2} + 1}}{1225} - 2 a b c^{2} d^{3} x^{3} \operatorname {acos}{\left (c x \right )} + \frac {1514 a b c d^{3} x^{2} \sqrt {- c^{2} x^{2} + 1}}{3675} + 2 a b d^{3} x \operatorname {acos}{\left (c x \right )} - \frac {4322 a b d^{3} \sqrt {- c^{2} x^{2} + 1}}{3675 c} - \frac {b^{2} c^{6} d^{3} x^{7} \operatorname {acos}^{2}{\left (c x \right )}}{7} + \frac {2 b^{2} c^{6} d^{3} x^{7}}{343} + \frac {2 b^{2} c^{5} d^{3} x^{6} \sqrt {- c^{2} x^{2} + 1} \operatorname {acos}{\left (c x \right )}}{49} + \frac {3 b^{2} c^{4} d^{3} x^{5} \operatorname {acos}^{2}{\left (c x \right )}}{5} - \frac {234 b^{2} c^{4} d^{3} x^{5}}{6125} - \frac {234 b^{2} c^{3} d^{3} x^{4} \sqrt {- c^{2} x^{2} + 1} \operatorname {acos}{\left (c x \right )}}{1225} - b^{2} c^{2} d^{3} x^{3} \operatorname {acos}^{2}{\left (c x \right )} + \frac {1514 b^{2} c^{2} d^{3} x^{3}}{11025} + \frac {1514 b^{2} c d^{3} x^{2} \sqrt {- c^{2} x^{2} + 1} \operatorname {acos}{\left (c x \right )}}{3675} + b^{2} d^{3} x \operatorname {acos}^{2}{\left (c x \right )} - \frac {4322 b^{2} d^{3} x}{3675} - \frac {4322 b^{2} d^{3} \sqrt {- c^{2} x^{2} + 1} \operatorname {acos}{\left (c x \right )}}{3675 c} & \text {for}\: c \neq 0 \\d^{3} x \left (a + \frac {\pi b}{2}\right )^{2} & \text {otherwise} \end {cases} \] Input:

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

Output:

Piecewise((-a**2*c**6*d**3*x**7/7 + 3*a**2*c**4*d**3*x**5/5 - a**2*c**2*d* 
*3*x**3 + a**2*d**3*x - 2*a*b*c**6*d**3*x**7*acos(c*x)/7 + 2*a*b*c**5*d**3 
*x**6*sqrt(-c**2*x**2 + 1)/49 + 6*a*b*c**4*d**3*x**5*acos(c*x)/5 - 234*a*b 
*c**3*d**3*x**4*sqrt(-c**2*x**2 + 1)/1225 - 2*a*b*c**2*d**3*x**3*acos(c*x) 
 + 1514*a*b*c*d**3*x**2*sqrt(-c**2*x**2 + 1)/3675 + 2*a*b*d**3*x*acos(c*x) 
 - 4322*a*b*d**3*sqrt(-c**2*x**2 + 1)/(3675*c) - b**2*c**6*d**3*x**7*acos( 
c*x)**2/7 + 2*b**2*c**6*d**3*x**7/343 + 2*b**2*c**5*d**3*x**6*sqrt(-c**2*x 
**2 + 1)*acos(c*x)/49 + 3*b**2*c**4*d**3*x**5*acos(c*x)**2/5 - 234*b**2*c* 
*4*d**3*x**5/6125 - 234*b**2*c**3*d**3*x**4*sqrt(-c**2*x**2 + 1)*acos(c*x) 
/1225 - b**2*c**2*d**3*x**3*acos(c*x)**2 + 1514*b**2*c**2*d**3*x**3/11025 
+ 1514*b**2*c*d**3*x**2*sqrt(-c**2*x**2 + 1)*acos(c*x)/3675 + b**2*d**3*x* 
acos(c*x)**2 - 4322*b**2*d**3*x/3675 - 4322*b**2*d**3*sqrt(-c**2*x**2 + 1) 
*acos(c*x)/(3675*c), Ne(c, 0)), (d**3*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. 730 vs. \(2 (263) = 526\).

Time = 0.16 (sec) , antiderivative size = 730, normalized size of antiderivative = 2.45 \[ \int \left (d-c^2 d x^2\right )^3 (a+b \arccos (c x))^2 \, dx =\text {Too large to display} \] Input:

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

Output:

-1/7*b^2*c^6*d^3*x^7*arccos(c*x)^2 - 1/7*a^2*c^6*d^3*x^7 + 3/5*b^2*c^4*d^3 
*x^5*arccos(c*x)^2 + 3/5*a^2*c^4*d^3*x^5 - 2/245*(35*x^7*arccos(c*x) - (5* 
sqrt(-c^2*x^2 + 1)*x^6/c^2 + 6*sqrt(-c^2*x^2 + 1)*x^4/c^4 + 8*sqrt(-c^2*x^ 
2 + 1)*x^2/c^6 + 16*sqrt(-c^2*x^2 + 1)/c^8)*c)*a*b*c^6*d^3 + 2/25725*(105* 
(5*sqrt(-c^2*x^2 + 1)*x^6/c^2 + 6*sqrt(-c^2*x^2 + 1)*x^4/c^4 + 8*sqrt(-c^2 
*x^2 + 1)*x^2/c^6 + 16*sqrt(-c^2*x^2 + 1)/c^8)*c*arccos(c*x) + (75*c^6*x^7 
 + 126*c^4*x^5 + 280*c^2*x^3 + 1680*x)/c^6)*b^2*c^6*d^3 - b^2*c^2*d^3*x^3* 
arccos(c*x)^2 + 2/25*(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^3 - 
 2/375*(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 + 120*x) 
/c^4)*b^2*c^4*d^3 - a^2*c^2*d^3*x^3 - 2/3*(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^3 + 2/9*(3*c*(sq 
rt(-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^3 + b^2*d^3*x*arccos(c*x)^2 - 2*b^2*d^3*(x + sqrt( 
-c^2*x^2 + 1)*arccos(c*x)/c) + a^2*d^3*x + 2*(c*x*arccos(c*x) - sqrt(-c^2* 
x^2 + 1))*a*b*d^3/c
 

Giac [A] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 449, normalized size of antiderivative = 1.51 \[ \int \left (d-c^2 d x^2\right )^3 (a+b \arccos (c x))^2 \, dx=-\frac {1}{7} \, b^{2} c^{6} d^{3} x^{7} \arccos \left (c x\right )^{2} - \frac {2}{7} \, a b c^{6} d^{3} x^{7} \arccos \left (c x\right ) - \frac {1}{7} \, a^{2} c^{6} d^{3} x^{7} + \frac {2}{343} \, b^{2} c^{6} d^{3} x^{7} + \frac {2}{49} \, \sqrt {-c^{2} x^{2} + 1} b^{2} c^{5} d^{3} x^{6} \arccos \left (c x\right ) + \frac {2}{49} \, \sqrt {-c^{2} x^{2} + 1} a b c^{5} d^{3} x^{6} + \frac {3}{5} \, b^{2} c^{4} d^{3} x^{5} \arccos \left (c x\right )^{2} + \frac {6}{5} \, a b c^{4} d^{3} x^{5} \arccos \left (c x\right ) + \frac {3}{5} \, a^{2} c^{4} d^{3} x^{5} - \frac {234}{6125} \, b^{2} c^{4} d^{3} x^{5} - \frac {234}{1225} \, \sqrt {-c^{2} x^{2} + 1} b^{2} c^{3} d^{3} x^{4} \arccos \left (c x\right ) - \frac {234}{1225} \, \sqrt {-c^{2} x^{2} + 1} a b c^{3} d^{3} x^{4} - b^{2} c^{2} d^{3} x^{3} \arccos \left (c x\right )^{2} - 2 \, a b c^{2} d^{3} x^{3} \arccos \left (c x\right ) - a^{2} c^{2} d^{3} x^{3} + \frac {1514}{11025} \, b^{2} c^{2} d^{3} x^{3} + \frac {1514}{3675} \, \sqrt {-c^{2} x^{2} + 1} b^{2} c d^{3} x^{2} \arccos \left (c x\right ) + \frac {1514}{3675} \, \sqrt {-c^{2} x^{2} + 1} a b c d^{3} x^{2} + b^{2} d^{3} x \arccos \left (c x\right )^{2} + 2 \, a b d^{3} x \arccos \left (c x\right ) + a^{2} d^{3} x - \frac {4322}{3675} \, b^{2} d^{3} x - \frac {4322 \, \sqrt {-c^{2} x^{2} + 1} b^{2} d^{3} \arccos \left (c x\right )}{3675 \, c} - \frac {4322 \, \sqrt {-c^{2} x^{2} + 1} a b d^{3}}{3675 \, c} \] Input:

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

Output:

-1/7*b^2*c^6*d^3*x^7*arccos(c*x)^2 - 2/7*a*b*c^6*d^3*x^7*arccos(c*x) - 1/7 
*a^2*c^6*d^3*x^7 + 2/343*b^2*c^6*d^3*x^7 + 2/49*sqrt(-c^2*x^2 + 1)*b^2*c^5 
*d^3*x^6*arccos(c*x) + 2/49*sqrt(-c^2*x^2 + 1)*a*b*c^5*d^3*x^6 + 3/5*b^2*c 
^4*d^3*x^5*arccos(c*x)^2 + 6/5*a*b*c^4*d^3*x^5*arccos(c*x) + 3/5*a^2*c^4*d 
^3*x^5 - 234/6125*b^2*c^4*d^3*x^5 - 234/1225*sqrt(-c^2*x^2 + 1)*b^2*c^3*d^ 
3*x^4*arccos(c*x) - 234/1225*sqrt(-c^2*x^2 + 1)*a*b*c^3*d^3*x^4 - b^2*c^2* 
d^3*x^3*arccos(c*x)^2 - 2*a*b*c^2*d^3*x^3*arccos(c*x) - a^2*c^2*d^3*x^3 + 
1514/11025*b^2*c^2*d^3*x^3 + 1514/3675*sqrt(-c^2*x^2 + 1)*b^2*c*d^3*x^2*ar 
ccos(c*x) + 1514/3675*sqrt(-c^2*x^2 + 1)*a*b*c*d^3*x^2 + b^2*d^3*x*arccos( 
c*x)^2 + 2*a*b*d^3*x*arccos(c*x) + a^2*d^3*x - 4322/3675*b^2*d^3*x - 4322/ 
3675*sqrt(-c^2*x^2 + 1)*b^2*d^3*arccos(c*x)/c - 4322/3675*sqrt(-c^2*x^2 + 
1)*a*b*d^3/c
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \left (d-c^2 d x^2\right )^3 (a+b \arccos (c x))^2 \, dx=\frac {d^{3} \left (3675 \mathit {acos} \left (c x \right )^{2} b^{2} c x -7350 \sqrt {-c^{2} x^{2}+1}\, \mathit {acos} \left (c x \right ) b^{2}-1050 \mathit {acos} \left (c x \right ) a b \,c^{7} x^{7}+4410 \mathit {acos} \left (c x \right ) a b \,c^{5} x^{5}-7350 \mathit {acos} \left (c x \right ) a b \,c^{3} x^{3}+7350 \mathit {acos} \left (c x \right ) a b c x +150 \sqrt {-c^{2} x^{2}+1}\, a b \,c^{6} x^{6}-702 \sqrt {-c^{2} x^{2}+1}\, a b \,c^{4} x^{4}+1514 \sqrt {-c^{2} x^{2}+1}\, a b \,c^{2} x^{2}-4322 \sqrt {-c^{2} x^{2}+1}\, a b -3675 \left (\int \mathit {acos} \left (c x \right )^{2} x^{6}d x \right ) b^{2} c^{7}+11025 \left (\int \mathit {acos} \left (c x \right )^{2} x^{4}d x \right ) b^{2} c^{5}-11025 \left (\int \mathit {acos} \left (c x \right )^{2} x^{2}d x \right ) b^{2} c^{3}-525 a^{2} c^{7} x^{7}+2205 a^{2} c^{5} x^{5}-3675 a^{2} c^{3} x^{3}+3675 a^{2} c x -7350 b^{2} c x \right )}{3675 c} \] Input:

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

Output:

(d**3*(3675*acos(c*x)**2*b**2*c*x - 7350*sqrt( - c**2*x**2 + 1)*acos(c*x)* 
b**2 - 1050*acos(c*x)*a*b*c**7*x**7 + 4410*acos(c*x)*a*b*c**5*x**5 - 7350* 
acos(c*x)*a*b*c**3*x**3 + 7350*acos(c*x)*a*b*c*x + 150*sqrt( - c**2*x**2 + 
 1)*a*b*c**6*x**6 - 702*sqrt( - c**2*x**2 + 1)*a*b*c**4*x**4 + 1514*sqrt( 
- c**2*x**2 + 1)*a*b*c**2*x**2 - 4322*sqrt( - c**2*x**2 + 1)*a*b - 3675*in 
t(acos(c*x)**2*x**6,x)*b**2*c**7 + 11025*int(acos(c*x)**2*x**4,x)*b**2*c** 
5 - 11025*int(acos(c*x)**2*x**2,x)*b**2*c**3 - 525*a**2*c**7*x**7 + 2205*a 
**2*c**5*x**5 - 3675*a**2*c**3*x**3 + 3675*a**2*c*x - 7350*b**2*c*x))/(367 
5*c)