\(\int x^3 (d-c^2 d x^2) (a+b \arccos (c x))^2 \, dx\) [159]

Optimal result

Integrand size = 25, antiderivative size = 202 \[ \int x^3 \left (d-c^2 d x^2\right ) (a+b \arccos (c x))^2 \, dx=-\frac {b^2 d x^2}{24 c^2}-\frac {1}{72} b^2 d x^4+\frac {1}{108} b^2 c^2 d x^6+\frac {b d x \sqrt {1-c^2 x^2} (a+b \arccos (c x))}{12 c^3}+\frac {b d x^3 \sqrt {1-c^2 x^2} (a+b \arccos (c x))}{18 c}-\frac {1}{18} b c d x^5 \sqrt {1-c^2 x^2} (a+b \arccos (c x))-\frac {d (a+b \arccos (c x))^2}{24 c^4}+\frac {1}{12} d x^4 (a+b \arccos (c x))^2+\frac {1}{6} d x^4 \left (1-c^2 x^2\right ) (a+b \arccos (c x))^2 \] Output:

-1/24*b^2*d*x^2/c^2-1/72*b^2*d*x^4+1/108*b^2*c^2*d*x^6+1/12*b*d*x*(-c^2*x^ 
2+1)^(1/2)*(a+b*arccos(c*x))/c^3+1/18*b*d*x^3*(-c^2*x^2+1)^(1/2)*(a+b*arcc 
os(c*x))/c-1/18*b*c*d*x^5*(-c^2*x^2+1)^(1/2)*(a+b*arccos(c*x))-1/24*d*(a+b 
*arccos(c*x))^2/c^4+1/12*d*x^4*(a+b*arccos(c*x))^2+1/6*d*x^4*(-c^2*x^2+1)* 
(a+b*arccos(c*x))^2
 

Mathematica [A] (verified)

Time = 0.22 (sec) , antiderivative size = 195, normalized size of antiderivative = 0.97 \[ \int x^3 \left (d-c^2 d x^2\right ) (a+b \arccos (c x))^2 \, dx=-\frac {d \left (c x \left (18 a^2 c^3 x^3 \left (-3+2 c^2 x^2\right )-6 a b \sqrt {1-c^2 x^2} \left (-3-2 c^2 x^2+2 c^4 x^4\right )+b^2 \left (9 c x+3 c^3 x^3-2 c^5 x^5\right )\right )+6 b c x \left (6 a c^3 x^3 \left (-3+2 c^2 x^2\right )+b \sqrt {1-c^2 x^2} \left (3+2 c^2 x^2-2 c^4 x^4\right )\right ) \arccos (c x)+9 b^2 \left (1-6 c^4 x^4+4 c^6 x^6\right ) \arccos (c x)^2-18 a b \arcsin (c x)\right )}{216 c^4} \] Input:

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

Output:

-1/216*(d*(c*x*(18*a^2*c^3*x^3*(-3 + 2*c^2*x^2) - 6*a*b*Sqrt[1 - c^2*x^2]* 
(-3 - 2*c^2*x^2 + 2*c^4*x^4) + b^2*(9*c*x + 3*c^3*x^3 - 2*c^5*x^5)) + 6*b* 
c*x*(6*a*c^3*x^3*(-3 + 2*c^2*x^2) + b*Sqrt[1 - c^2*x^2]*(3 + 2*c^2*x^2 - 2 
*c^4*x^4))*ArcCos[c*x] + 9*b^2*(1 - 6*c^4*x^4 + 4*c^6*x^6)*ArcCos[c*x]^2 - 
 18*a*b*ArcSin[c*x]))/c^4
 

Rubi [A] (verified)

Time = 1.39 (sec) , antiderivative size = 334, normalized size of antiderivative = 1.65, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.360, Rules used = {5203, 5139, 5199, 15, 5211, 15, 5211, 15, 5153}

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.

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

\(\Big \downarrow \) 5203

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

\(\Big \downarrow \) 5139

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

\(\Big \downarrow \) 5199

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

\(\Big \downarrow \) 15

\(\displaystyle \frac {1}{3} d \left (\frac {1}{2} b c \int \frac {x^4 (a+b \arccos (c x))}{\sqrt {1-c^2 x^2}}dx+\frac {1}{4} x^4 (a+b \arccos (c x))^2\right )+\frac {1}{3} b c d \left (\frac {1}{6} \int \frac {x^4 (a+b \arccos (c x))}{\sqrt {1-c^2 x^2}}dx+\frac {1}{6} x^5 \sqrt {1-c^2 x^2} (a+b \arccos (c x))+\frac {1}{36} b c x^6\right )+\frac {1}{6} d x^4 \left (1-c^2 x^2\right ) (a+b \arccos (c x))^2\)

\(\Big \downarrow \) 5211

\(\displaystyle \frac {1}{3} d \left (\frac {1}{2} b c \left (\frac {3 \int \frac {x^2 (a+b \arccos (c x))}{\sqrt {1-c^2 x^2}}dx}{4 c^2}-\frac {b \int x^3dx}{4 c}-\frac {x^3 \sqrt {1-c^2 x^2} (a+b \arccos (c x))}{4 c^2}\right )+\frac {1}{4} x^4 (a+b \arccos (c x))^2\right )+\frac {1}{3} b c d \left (\frac {1}{6} \left (\frac {3 \int \frac {x^2 (a+b \arccos (c x))}{\sqrt {1-c^2 x^2}}dx}{4 c^2}-\frac {b \int x^3dx}{4 c}-\frac {x^3 \sqrt {1-c^2 x^2} (a+b \arccos (c x))}{4 c^2}\right )+\frac {1}{6} x^5 \sqrt {1-c^2 x^2} (a+b \arccos (c x))+\frac {1}{36} b c x^6\right )+\frac {1}{6} d x^4 \left (1-c^2 x^2\right ) (a+b \arccos (c x))^2\)

\(\Big \downarrow \) 15

\(\displaystyle \frac {1}{3} d \left (\frac {1}{2} b c \left (\frac {3 \int \frac {x^2 (a+b \arccos (c x))}{\sqrt {1-c^2 x^2}}dx}{4 c^2}-\frac {x^3 \sqrt {1-c^2 x^2} (a+b \arccos (c x))}{4 c^2}-\frac {b x^4}{16 c}\right )+\frac {1}{4} x^4 (a+b \arccos (c x))^2\right )+\frac {1}{3} b c d \left (\frac {1}{6} \left (\frac {3 \int \frac {x^2 (a+b \arccos (c x))}{\sqrt {1-c^2 x^2}}dx}{4 c^2}-\frac {x^3 \sqrt {1-c^2 x^2} (a+b \arccos (c x))}{4 c^2}-\frac {b x^4}{16 c}\right )+\frac {1}{6} x^5 \sqrt {1-c^2 x^2} (a+b \arccos (c x))+\frac {1}{36} b c x^6\right )+\frac {1}{6} d x^4 \left (1-c^2 x^2\right ) (a+b \arccos (c x))^2\)

\(\Big \downarrow \) 5211

\(\displaystyle \frac {1}{3} d \left (\frac {1}{2} b c \left (\frac {3 \left (\frac {\int \frac {a+b \arccos (c x)}{\sqrt {1-c^2 x^2}}dx}{2 c^2}-\frac {b \int xdx}{2 c}-\frac {x \sqrt {1-c^2 x^2} (a+b \arccos (c x))}{2 c^2}\right )}{4 c^2}-\frac {x^3 \sqrt {1-c^2 x^2} (a+b \arccos (c x))}{4 c^2}-\frac {b x^4}{16 c}\right )+\frac {1}{4} x^4 (a+b \arccos (c x))^2\right )+\frac {1}{3} b c d \left (\frac {1}{6} \left (\frac {3 \left (\frac {\int \frac {a+b \arccos (c x)}{\sqrt {1-c^2 x^2}}dx}{2 c^2}-\frac {b \int xdx}{2 c}-\frac {x \sqrt {1-c^2 x^2} (a+b \arccos (c x))}{2 c^2}\right )}{4 c^2}-\frac {x^3 \sqrt {1-c^2 x^2} (a+b \arccos (c x))}{4 c^2}-\frac {b x^4}{16 c}\right )+\frac {1}{6} x^5 \sqrt {1-c^2 x^2} (a+b \arccos (c x))+\frac {1}{36} b c x^6\right )+\frac {1}{6} d x^4 \left (1-c^2 x^2\right ) (a+b \arccos (c x))^2\)

\(\Big \downarrow \) 15

\(\displaystyle \frac {1}{3} d \left (\frac {1}{2} b c \left (\frac {3 \left (\frac {\int \frac {a+b \arccos (c x)}{\sqrt {1-c^2 x^2}}dx}{2 c^2}-\frac {x \sqrt {1-c^2 x^2} (a+b \arccos (c x))}{2 c^2}-\frac {b x^2}{4 c}\right )}{4 c^2}-\frac {x^3 \sqrt {1-c^2 x^2} (a+b \arccos (c x))}{4 c^2}-\frac {b x^4}{16 c}\right )+\frac {1}{4} x^4 (a+b \arccos (c x))^2\right )+\frac {1}{3} b c d \left (\frac {1}{6} \left (\frac {3 \left (\frac {\int \frac {a+b \arccos (c x)}{\sqrt {1-c^2 x^2}}dx}{2 c^2}-\frac {x \sqrt {1-c^2 x^2} (a+b \arccos (c x))}{2 c^2}-\frac {b x^2}{4 c}\right )}{4 c^2}-\frac {x^3 \sqrt {1-c^2 x^2} (a+b \arccos (c x))}{4 c^2}-\frac {b x^4}{16 c}\right )+\frac {1}{6} x^5 \sqrt {1-c^2 x^2} (a+b \arccos (c x))+\frac {1}{36} b c x^6\right )+\frac {1}{6} d x^4 \left (1-c^2 x^2\right ) (a+b \arccos (c x))^2\)

\(\Big \downarrow \) 5153

\(\displaystyle \frac {1}{6} d x^4 \left (1-c^2 x^2\right ) (a+b \arccos (c x))^2+\frac {1}{3} d \left (\frac {1}{2} b c \left (-\frac {x^3 \sqrt {1-c^2 x^2} (a+b \arccos (c x))}{4 c^2}+\frac {3 \left (-\frac {(a+b \arccos (c x))^2}{4 b c^3}-\frac {x \sqrt {1-c^2 x^2} (a+b \arccos (c x))}{2 c^2}-\frac {b x^2}{4 c}\right )}{4 c^2}-\frac {b x^4}{16 c}\right )+\frac {1}{4} x^4 (a+b \arccos (c x))^2\right )+\frac {1}{3} b c d \left (\frac {1}{6} x^5 \sqrt {1-c^2 x^2} (a+b \arccos (c x))+\frac {1}{6} \left (-\frac {x^3 \sqrt {1-c^2 x^2} (a+b \arccos (c x))}{4 c^2}+\frac {3 \left (-\frac {(a+b \arccos (c x))^2}{4 b c^3}-\frac {x \sqrt {1-c^2 x^2} (a+b \arccos (c x))}{2 c^2}-\frac {b x^2}{4 c}\right )}{4 c^2}-\frac {b x^4}{16 c}\right )+\frac {1}{36} b c x^6\right )\)

Input:

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

Output:

(d*x^4*(1 - c^2*x^2)*(a + b*ArcCos[c*x])^2)/6 + (b*c*d*((b*c*x^6)/36 + (x^ 
5*Sqrt[1 - c^2*x^2]*(a + b*ArcCos[c*x]))/6 + (-1/16*(b*x^4)/c - (x^3*Sqrt[ 
1 - c^2*x^2]*(a + b*ArcCos[c*x]))/(4*c^2) + (3*(-1/4*(b*x^2)/c - (x*Sqrt[1 
 - c^2*x^2]*(a + b*ArcCos[c*x]))/(2*c^2) - (a + b*ArcCos[c*x])^2/(4*b*c^3) 
))/(4*c^2))/6))/3 + (d*((x^4*(a + b*ArcCos[c*x])^2)/4 + (b*c*(-1/16*(b*x^4 
)/c - (x^3*Sqrt[1 - c^2*x^2]*(a + b*ArcCos[c*x]))/(4*c^2) + (3*(-1/4*(b*x^ 
2)/c - (x*Sqrt[1 - c^2*x^2]*(a + b*ArcCos[c*x]))/(2*c^2) - (a + b*ArcCos[c 
*x])^2/(4*b*c^3)))/(4*c^2)))/2))/3
 

Defintions of rubi rules used

rule 15

Int[(a_.)*(x_)^(m_.), x_Symbol] :> Simp[a*(x^(m + 1)/(m + 1)), x] /; FreeQ[ 
{a, m}, x] && NeQ[m, -1]
 

rule 5139

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

rule 5153

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

rule 5199

Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*Sqrt[(d_) + 
(e_.)*(x_)^2], x_Symbol] :> Simp[(f*x)^(m + 1)*Sqrt[d + e*x^2]*((a + b*ArcC 
os[c*x])^n/(f*(m + 2))), x] + (Simp[(1/(m + 2))*Simp[Sqrt[d + e*x^2]/Sqrt[1 
 - c^2*x^2]]   Int[(f*x)^m*((a + b*ArcCos[c*x])^n/Sqrt[1 - c^2*x^2]), x], x 
] + Simp[b*c*(n/(f*(m + 2)))*Simp[Sqrt[d + e*x^2]/Sqrt[1 - c^2*x^2]]   Int[ 
(f*x)^(m + 1)*(a + b*ArcCos[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d, e, 
 f, m}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && (IGtQ[m, -2] || EqQ[n, 1])
 

rule 5203

Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*((d_) + (e_. 
)*(x_)^2)^(p_.), x_Symbol] :> Simp[(f*x)^(m + 1)*(d + e*x^2)^p*((a + b*ArcC 
os[c*x])^n/(f*(m + 2*p + 1))), x] + (Simp[2*d*(p/(m + 2*p + 1))   Int[(f*x) 
^m*(d + e*x^2)^(p - 1)*(a + b*ArcCos[c*x])^n, x], x] + Simp[b*c*(n/(f*(m + 
2*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, m}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && GtQ[p, 0] &&  !LtQ[m, -1]
 

rule 5211

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/(e*(m + 2*p + 1))), x] + (Simp[f^2*((m - 1)/(c^2*(m + 2*p 
 + 1)))   Int[(f*x)^(m - 2)*(d + e*x^2)^p*(a + b*ArcCos[c*x])^n, x], x] - S 
imp[b*f*(n/(c*(m + 2*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, p}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && IGtQ[m 
, 1] && NeQ[m + 2*p + 1, 0]
 

Maple [A] (verified)

Time = 0.27 (sec) , antiderivative size = 329, normalized size of antiderivative = 1.63

method	result	size
parts	\(-d \,a^{2} \left (\frac {1}{6} c^{2} x^{6}-\frac {1}{4} x^{4}\right )-\frac {d \,b^{2} \left (\frac {\arccos \left (c x \right )^{2} \left (c^{2} x^{2}-1\right )^{2}}{4}-\frac {\arccos \left (c x \right ) \left (2 c^{3} x^{3} \sqrt {-c^{2} x^{2}+1}-5 c x \sqrt {-c^{2} x^{2}+1}+3 \arccos \left (c x \right )\right )}{16}+\frac {\arccos \left (c x \right )^{2}}{24}-\frac {\left (2 c^{2} x^{2}-5\right )^{2}}{128}+\frac {\arccos \left (c x \right )^{2} \left (c^{2} x^{2}-1\right )^{3}}{6}+\frac {\arccos \left (c x \right ) \left (-8 c^{5} x^{5} \sqrt {-c^{2} x^{2}+1}+26 c^{3} x^{3} \sqrt {-c^{2} x^{2}+1}-33 c x \sqrt {-c^{2} x^{2}+1}+15 \arccos \left (c x \right )\right )}{144}-\frac {c^{6} x^{6}}{108}+\frac {13 c^{4} x^{4}}{288}-\frac {11 c^{2} x^{2}}{96}\right )}{c^{4}}-\frac {2 d a b \left (\frac {\arccos \left (c x \right ) c^{6} x^{6}}{6}-\frac {c^{4} x^{4} \arccos \left (c x \right )}{4}+\frac {c^{3} x^{3} \sqrt {-c^{2} x^{2}+1}}{36}+\frac {c x \sqrt {-c^{2} x^{2}+1}}{24}-\frac {\arcsin \left (c x \right )}{24}-\frac {c^{5} x^{5} \sqrt {-c^{2} x^{2}+1}}{36}\right )}{c^{4}}\)	\(329\)
derivativedivides	\(\frac {-d \,a^{2} \left (\frac {1}{6} c^{6} x^{6}-\frac {1}{4} c^{4} x^{4}\right )-d \,b^{2} \left (\frac {\arccos \left (c x \right )^{2} \left (c^{2} x^{2}-1\right )^{2}}{4}-\frac {\arccos \left (c x \right ) \left (2 c^{3} x^{3} \sqrt {-c^{2} x^{2}+1}-5 c x \sqrt {-c^{2} x^{2}+1}+3 \arccos \left (c x \right )\right )}{16}+\frac {\arccos \left (c x \right )^{2}}{24}-\frac {\left (2 c^{2} x^{2}-5\right )^{2}}{128}+\frac {\arccos \left (c x \right )^{2} \left (c^{2} x^{2}-1\right )^{3}}{6}+\frac {\arccos \left (c x \right ) \left (-8 c^{5} x^{5} \sqrt {-c^{2} x^{2}+1}+26 c^{3} x^{3} \sqrt {-c^{2} x^{2}+1}-33 c x \sqrt {-c^{2} x^{2}+1}+15 \arccos \left (c x \right )\right )}{144}-\frac {c^{6} x^{6}}{108}+\frac {13 c^{4} x^{4}}{288}-\frac {11 c^{2} x^{2}}{96}\right )-2 d a b \left (\frac {\arccos \left (c x \right ) c^{6} x^{6}}{6}-\frac {c^{4} x^{4} \arccos \left (c x \right )}{4}+\frac {c^{3} x^{3} \sqrt {-c^{2} x^{2}+1}}{36}+\frac {c x \sqrt {-c^{2} x^{2}+1}}{24}-\frac {\arcsin \left (c x \right )}{24}-\frac {c^{5} x^{5} \sqrt {-c^{2} x^{2}+1}}{36}\right )}{c^{4}}\)	\(330\)
default	\(\frac {-d \,a^{2} \left (\frac {1}{6} c^{6} x^{6}-\frac {1}{4} c^{4} x^{4}\right )-d \,b^{2} \left (\frac {\arccos \left (c x \right )^{2} \left (c^{2} x^{2}-1\right )^{2}}{4}-\frac {\arccos \left (c x \right ) \left (2 c^{3} x^{3} \sqrt {-c^{2} x^{2}+1}-5 c x \sqrt {-c^{2} x^{2}+1}+3 \arccos \left (c x \right )\right )}{16}+\frac {\arccos \left (c x \right )^{2}}{24}-\frac {\left (2 c^{2} x^{2}-5\right )^{2}}{128}+\frac {\arccos \left (c x \right )^{2} \left (c^{2} x^{2}-1\right )^{3}}{6}+\frac {\arccos \left (c x \right ) \left (-8 c^{5} x^{5} \sqrt {-c^{2} x^{2}+1}+26 c^{3} x^{3} \sqrt {-c^{2} x^{2}+1}-33 c x \sqrt {-c^{2} x^{2}+1}+15 \arccos \left (c x \right )\right )}{144}-\frac {c^{6} x^{6}}{108}+\frac {13 c^{4} x^{4}}{288}-\frac {11 c^{2} x^{2}}{96}\right )-2 d a b \left (\frac {\arccos \left (c x \right ) c^{6} x^{6}}{6}-\frac {c^{4} x^{4} \arccos \left (c x \right )}{4}+\frac {c^{3} x^{3} \sqrt {-c^{2} x^{2}+1}}{36}+\frac {c x \sqrt {-c^{2} x^{2}+1}}{24}-\frac {\arcsin \left (c x \right )}{24}-\frac {c^{5} x^{5} \sqrt {-c^{2} x^{2}+1}}{36}\right )}{c^{4}}\)	\(330\)
orering	\(\frac {\left (182 c^{8} x^{8}-473 c^{6} x^{6}+42 c^{4} x^{4}+369 c^{2} x^{2}-180\right ) \left (-c^{2} d \,x^{2}+d \right ) \left (a +b \arccos \left (c x \right )\right )^{2}}{432 c^{4} \left (c^{2} x^{2}-1\right )^{2}}-\frac {\left (10 c^{6} x^{6}-21 c^{4} x^{4}-23 c^{2} x^{2}+24\right ) \left (3 x^{2} \left (-c^{2} d \,x^{2}+d \right ) \left (a +b \arccos \left (c x \right )\right )^{2}-2 x^{4} d \,c^{2} \left (a +b \arccos \left (c x \right )\right )^{2}-\frac {2 x^{3} \left (-c^{2} d \,x^{2}+d \right ) \left (a +b \arccos \left (c x \right )\right ) b c}{\sqrt {-c^{2} x^{2}+1}}\right )}{144 x^{2} c^{4} \left (c^{2} x^{2}-1\right )}+\frac {\left (2 c^{4} x^{4}-3 c^{2} x^{2}-9\right ) \left (6 x \left (-c^{2} d \,x^{2}+d \right ) \left (a +b \arccos \left (c x \right )\right )^{2}-14 x^{3} d \,c^{2} \left (a +b \arccos \left (c x \right )\right )^{2}-\frac {12 x^{2} \left (-c^{2} d \,x^{2}+d \right ) \left (a +b \arccos \left (c x \right )\right ) b c}{\sqrt {-c^{2} x^{2}+1}}+\frac {8 x^{4} d \,c^{3} \left (a +b \arccos \left (c x \right )\right ) b}{\sqrt {-c^{2} x^{2}+1}}+\frac {2 x^{3} \left (-c^{2} d \,x^{2}+d \right ) b^{2} c^{2}}{-c^{2} x^{2}+1}-\frac {2 x^{4} \left (-c^{2} d \,x^{2}+d \right ) \left (a +b \arccos \left (c x \right )\right ) b \,c^{3}}{\left (-c^{2} x^{2}+1\right )^{\frac {3}{2}}}\right )}{432 x \,c^{4}}\)	\(414\)

Input:

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

Output:

-d*a^2*(1/6*c^2*x^6-1/4*x^4)-d*b^2/c^4*(1/4*arccos(c*x)^2*(c^2*x^2-1)^2-1/ 
16*arccos(c*x)*(2*c^3*x^3*(-c^2*x^2+1)^(1/2)-5*c*x*(-c^2*x^2+1)^(1/2)+3*ar 
ccos(c*x))+1/24*arccos(c*x)^2-1/128*(2*c^2*x^2-5)^2+1/6*arccos(c*x)^2*(c^2 
*x^2-1)^3+1/144*arccos(c*x)*(-8*c^5*x^5*(-c^2*x^2+1)^(1/2)+26*c^3*x^3*(-c^ 
2*x^2+1)^(1/2)-33*c*x*(-c^2*x^2+1)^(1/2)+15*arccos(c*x))-1/108*c^6*x^6+13/ 
288*c^4*x^4-11/96*c^2*x^2)-2*d*a*b/c^4*(1/6*arccos(c*x)*c^6*x^6-1/4*c^4*x^ 
4*arccos(c*x)+1/36*c^3*x^3*(-c^2*x^2+1)^(1/2)+1/24*c*x*(-c^2*x^2+1)^(1/2)- 
1/24*arcsin(c*x)-1/36*c^5*x^5*(-c^2*x^2+1)^(1/2))
 

Fricas [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 211, normalized size of antiderivative = 1.04 \[ \int x^3 \left (d-c^2 d x^2\right ) (a+b \arccos (c x))^2 \, dx=-\frac {2 \, {\left (18 \, a^{2} - b^{2}\right )} c^{6} d x^{6} - 3 \, {\left (18 \, a^{2} - b^{2}\right )} c^{4} d x^{4} + 9 \, b^{2} c^{2} d x^{2} + 9 \, {\left (4 \, b^{2} c^{6} d x^{6} - 6 \, b^{2} c^{4} d x^{4} + b^{2} d\right )} \arccos \left (c x\right )^{2} + 18 \, {\left (4 \, a b c^{6} d x^{6} - 6 \, a b c^{4} d x^{4} + a b d\right )} \arccos \left (c x\right ) - 6 \, {\left (2 \, a b c^{5} d x^{5} - 2 \, a b c^{3} d x^{3} - 3 \, a b c d x + {\left (2 \, b^{2} c^{5} d x^{5} - 2 \, b^{2} c^{3} d x^{3} - 3 \, b^{2} c d x\right )} \arccos \left (c x\right )\right )} \sqrt {-c^{2} x^{2} + 1}}{216 \, c^{4}} \] Input:

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

Output:

-1/216*(2*(18*a^2 - b^2)*c^6*d*x^6 - 3*(18*a^2 - b^2)*c^4*d*x^4 + 9*b^2*c^ 
2*d*x^2 + 9*(4*b^2*c^6*d*x^6 - 6*b^2*c^4*d*x^4 + b^2*d)*arccos(c*x)^2 + 18 
*(4*a*b*c^6*d*x^6 - 6*a*b*c^4*d*x^4 + a*b*d)*arccos(c*x) - 6*(2*a*b*c^5*d* 
x^5 - 2*a*b*c^3*d*x^3 - 3*a*b*c*d*x + (2*b^2*c^5*d*x^5 - 2*b^2*c^3*d*x^3 - 
 3*b^2*c*d*x)*arccos(c*x))*sqrt(-c^2*x^2 + 1))/c^4
 

Sympy [A] (verification not implemented)

Time = 0.66 (sec) , antiderivative size = 337, normalized size of antiderivative = 1.67 \[ \int x^3 \left (d-c^2 d x^2\right ) (a+b \arccos (c x))^2 \, dx=\begin {cases} - \frac {a^{2} c^{2} d x^{6}}{6} + \frac {a^{2} d x^{4}}{4} - \frac {a b c^{2} d x^{6} \operatorname {acos}{\left (c x \right )}}{3} + \frac {a b c d x^{5} \sqrt {- c^{2} x^{2} + 1}}{18} + \frac {a b d x^{4} \operatorname {acos}{\left (c x \right )}}{2} - \frac {a b d x^{3} \sqrt {- c^{2} x^{2} + 1}}{18 c} - \frac {a b d x \sqrt {- c^{2} x^{2} + 1}}{12 c^{3}} - \frac {a b d \operatorname {acos}{\left (c x \right )}}{12 c^{4}} - \frac {b^{2} c^{2} d x^{6} \operatorname {acos}^{2}{\left (c x \right )}}{6} + \frac {b^{2} c^{2} d x^{6}}{108} + \frac {b^{2} c d x^{5} \sqrt {- c^{2} x^{2} + 1} \operatorname {acos}{\left (c x \right )}}{18} + \frac {b^{2} d x^{4} \operatorname {acos}^{2}{\left (c x \right )}}{4} - \frac {b^{2} d x^{4}}{72} - \frac {b^{2} d x^{3} \sqrt {- c^{2} x^{2} + 1} \operatorname {acos}{\left (c x \right )}}{18 c} - \frac {b^{2} d x^{2}}{24 c^{2}} - \frac {b^{2} d x \sqrt {- c^{2} x^{2} + 1} \operatorname {acos}{\left (c x \right )}}{12 c^{3}} - \frac {b^{2} d \operatorname {acos}^{2}{\left (c x \right )}}{24 c^{4}} & \text {for}\: c \neq 0 \\\frac {d x^{4} \left (a + \frac {\pi b}{2}\right )^{2}}{4} & \text {otherwise} \end {cases} \] Input:

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

Output:

Piecewise((-a**2*c**2*d*x**6/6 + a**2*d*x**4/4 - a*b*c**2*d*x**6*acos(c*x) 
/3 + a*b*c*d*x**5*sqrt(-c**2*x**2 + 1)/18 + a*b*d*x**4*acos(c*x)/2 - a*b*d 
*x**3*sqrt(-c**2*x**2 + 1)/(18*c) - a*b*d*x*sqrt(-c**2*x**2 + 1)/(12*c**3) 
 - a*b*d*acos(c*x)/(12*c**4) - b**2*c**2*d*x**6*acos(c*x)**2/6 + b**2*c**2 
*d*x**6/108 + b**2*c*d*x**5*sqrt(-c**2*x**2 + 1)*acos(c*x)/18 + b**2*d*x** 
4*acos(c*x)**2/4 - b**2*d*x**4/72 - b**2*d*x**3*sqrt(-c**2*x**2 + 1)*acos( 
c*x)/(18*c) - b**2*d*x**2/(24*c**2) - b**2*d*x*sqrt(-c**2*x**2 + 1)*acos(c 
*x)/(12*c**3) - b**2*d*acos(c*x)**2/(24*c**4), Ne(c, 0)), (d*x**4*(a + pi* 
b/2)**2/4, True))
 

Maxima [F]

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

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

Output:

-1/6*a^2*c^2*d*x^6 + 1/4*a^2*d*x^4 - 1/144*(48*x^6*arccos(c*x) - (8*sqrt(- 
c^2*x^2 + 1)*x^5/c^2 + 10*sqrt(-c^2*x^2 + 1)*x^3/c^4 + 15*sqrt(-c^2*x^2 + 
1)*x/c^6 - 15*arcsin(c*x)/c^7)*c)*a*b*c^2*d + 1/16*(8*x^4*arccos(c*x) - (2 
*sqrt(-c^2*x^2 + 1)*x^3/c^2 + 3*sqrt(-c^2*x^2 + 1)*x/c^4 - 3*arcsin(c*x)/c 
^5)*c)*a*b*d - 1/12*(2*b^2*c^2*d*x^6 - 3*b^2*d*x^4)*arctan2(sqrt(c*x + 1)* 
sqrt(-c*x + 1), c*x)^2 + integrate(1/6*(2*b^2*c^3*d*x^6 - 3*b^2*c*d*x^4)*s 
qrt(c*x + 1)*sqrt(-c*x + 1)*arctan2(sqrt(c*x + 1)*sqrt(-c*x + 1), c*x)/(c^ 
2*x^2 - 1), x)
 

Giac [A] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 296, normalized size of antiderivative = 1.47 \[ \int x^3 \left (d-c^2 d x^2\right ) (a+b \arccos (c x))^2 \, dx=-\frac {1}{6} \, b^{2} c^{2} d x^{6} \arccos \left (c x\right )^{2} - \frac {1}{3} \, a b c^{2} d x^{6} \arccos \left (c x\right ) - \frac {1}{6} \, a^{2} c^{2} d x^{6} + \frac {1}{108} \, b^{2} c^{2} d x^{6} + \frac {1}{18} \, \sqrt {-c^{2} x^{2} + 1} b^{2} c d x^{5} \arccos \left (c x\right ) + \frac {1}{18} \, \sqrt {-c^{2} x^{2} + 1} a b c d x^{5} + \frac {1}{4} \, b^{2} d x^{4} \arccos \left (c x\right )^{2} + \frac {1}{2} \, a b d x^{4} \arccos \left (c x\right ) + \frac {1}{4} \, a^{2} d x^{4} - \frac {1}{72} \, b^{2} d x^{4} - \frac {\sqrt {-c^{2} x^{2} + 1} b^{2} d x^{3} \arccos \left (c x\right )}{18 \, c} - \frac {\sqrt {-c^{2} x^{2} + 1} a b d x^{3}}{18 \, c} - \frac {b^{2} d x^{2}}{24 \, c^{2}} - \frac {\sqrt {-c^{2} x^{2} + 1} b^{2} d x \arccos \left (c x\right )}{12 \, c^{3}} - \frac {\sqrt {-c^{2} x^{2} + 1} a b d x}{12 \, c^{3}} - \frac {b^{2} d \arccos \left (c x\right )^{2}}{24 \, c^{4}} - \frac {a b d \arccos \left (c x\right )}{12 \, c^{4}} + \frac {5 \, b^{2} d}{216 \, c^{4}} \] Input:

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

Output:

-1/6*b^2*c^2*d*x^6*arccos(c*x)^2 - 1/3*a*b*c^2*d*x^6*arccos(c*x) - 1/6*a^2 
*c^2*d*x^6 + 1/108*b^2*c^2*d*x^6 + 1/18*sqrt(-c^2*x^2 + 1)*b^2*c*d*x^5*arc 
cos(c*x) + 1/18*sqrt(-c^2*x^2 + 1)*a*b*c*d*x^5 + 1/4*b^2*d*x^4*arccos(c*x) 
^2 + 1/2*a*b*d*x^4*arccos(c*x) + 1/4*a^2*d*x^4 - 1/72*b^2*d*x^4 - 1/18*sqr 
t(-c^2*x^2 + 1)*b^2*d*x^3*arccos(c*x)/c - 1/18*sqrt(-c^2*x^2 + 1)*a*b*d*x^ 
3/c - 1/24*b^2*d*x^2/c^2 - 1/12*sqrt(-c^2*x^2 + 1)*b^2*d*x*arccos(c*x)/c^3 
 - 1/12*sqrt(-c^2*x^2 + 1)*a*b*d*x/c^3 - 1/24*b^2*d*arccos(c*x)^2/c^4 - 1/ 
12*a*b*d*arccos(c*x)/c^4 + 5/216*b^2*d/c^4
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int x^3 \left (d-c^2 d x^2\right ) (a+b \arccos (c x))^2 \, dx=\frac {d \left (-12 \mathit {acos} \left (c x \right ) a b \,c^{6} x^{6}+18 \mathit {acos} \left (c x \right ) a b \,c^{4} x^{4}+3 \mathit {asin} \left (c x \right ) a b +2 \sqrt {-c^{2} x^{2}+1}\, a b \,c^{5} x^{5}-2 \sqrt {-c^{2} x^{2}+1}\, a b \,c^{3} x^{3}-3 \sqrt {-c^{2} x^{2}+1}\, a b c x -36 \left (\int \mathit {acos} \left (c x \right )^{2} x^{5}d x \right ) b^{2} c^{6}+36 \left (\int \mathit {acos} \left (c x \right )^{2} x^{3}d x \right ) b^{2} c^{4}-6 a^{2} c^{6} x^{6}+9 a^{2} c^{4} x^{4}\right )}{36 c^{4}} \] Input:

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

Output:

(d*( - 12*acos(c*x)*a*b*c**6*x**6 + 18*acos(c*x)*a*b*c**4*x**4 + 3*asin(c* 
x)*a*b + 2*sqrt( - c**2*x**2 + 1)*a*b*c**5*x**5 - 2*sqrt( - c**2*x**2 + 1) 
*a*b*c**3*x**3 - 3*sqrt( - c**2*x**2 + 1)*a*b*c*x - 36*int(acos(c*x)**2*x* 
*5,x)*b**2*c**6 + 36*int(acos(c*x)**2*x**3,x)*b**2*c**4 - 6*a**2*c**6*x**6 
 + 9*a**2*c**4*x**4))/(36*c**4)