\(\int (c-a^2 c x^2) \arccos (a x)^3 \, dx\) [293]

Optimal result

Integrand size = 18, antiderivative size = 158 \[ \int \left (c-a^2 c x^2\right ) \arccos (a x)^3 \, dx=-\frac {40 c \sqrt {1-a^2 x^2}}{9 a}-\frac {2 c \left (1-a^2 x^2\right )^{3/2}}{27 a}-\frac {14}{3} c x \arccos (a x)+\frac {2}{9} a^2 c x^3 \arccos (a x)+\frac {2 c \sqrt {1-a^2 x^2} \arccos (a x)^2}{a}+\frac {c \left (1-a^2 x^2\right )^{3/2} \arccos (a x)^2}{3 a}+\frac {2}{3} c x \arccos (a x)^3+\frac {1}{3} c x \left (1-a^2 x^2\right ) \arccos (a x)^3 \] Output:

-40/9*c*(-a^2*x^2+1)^(1/2)/a-2/27*c*(-a^2*x^2+1)^(3/2)/a-14/3*c*x*arccos(a 
*x)+2/9*a^2*c*x^3*arccos(a*x)+2*c*(-a^2*x^2+1)^(1/2)*arccos(a*x)^2/a+1/3*c 
*(-a^2*x^2+1)^(3/2)*arccos(a*x)^2/a+2/3*c*x*arccos(a*x)^3+1/3*c*x*(-a^2*x^ 
2+1)*arccos(a*x)^3
 

Mathematica [A] (verified)

Time = 0.08 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.64 \[ \int \left (c-a^2 c x^2\right ) \arccos (a x)^3 \, dx=\frac {c \left (-2 \sqrt {1-a^2 x^2} \left (-61+a^2 x^2\right )+6 a x \left (-21+a^2 x^2\right ) \arccos (a x)+9 \sqrt {1-a^2 x^2} \left (-7+a^2 x^2\right ) \arccos (a x)^2-9 a x \left (-3+a^2 x^2\right ) \arccos (a x)^3\right )}{27 a} \] Input:

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

Output:

(c*(-2*Sqrt[1 - a^2*x^2]*(-61 + a^2*x^2) + 6*a*x*(-21 + a^2*x^2)*ArcCos[a* 
x] + 9*Sqrt[1 - a^2*x^2]*(-7 + a^2*x^2)*ArcCos[a*x]^2 - 9*a*x*(-3 + a^2*x^ 
2)*ArcCos[a*x]^3))/(27*a)
 

Rubi [A] (verified)

Time = 0.81 (sec) , antiderivative size = 202, normalized size of antiderivative = 1.28, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.556, Rules used = {5159, 5131, 5183, 5131, 241, 5155, 27, 353, 53, 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.

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

\(\Big \downarrow \) 5159

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

\(\Big \downarrow \) 5131

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

\(\Big \downarrow \) 5183

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

\(\Big \downarrow \) 5131

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

\(\Big \downarrow \) 241

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

\(\Big \downarrow \) 5155

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 353

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

\(\Big \downarrow \) 53

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

(c*x*(1 - a^2*x^2)*ArcCos[a*x]^3)/3 + a*c*(-1/3*((1 - a^2*x^2)^(3/2)*ArcCo 
s[a*x]^2)/a^2 - (2*((a*((-4*Sqrt[1 - a^2*x^2])/a^2 - (2*(1 - a^2*x^2)^(3/2 
))/(3*a^2)))/6 + x*ArcCos[a*x] - (a^2*x^3*ArcCos[a*x])/3))/(3*a)) + (2*c*( 
x*ArcCos[a*x]^3 + 3*a*(-((Sqrt[1 - a^2*x^2]*ArcCos[a*x]^2)/a^2) - (2*(-(Sq 
rt[1 - a^2*x^2]/a) + x*ArcCos[a*x]))/a)))/3
 

Defintions of rubi rules used

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 53

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int 
[ExpandIntegrand[(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, 
x] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0] && LeQ[7*m + 4*n + 4, 0]) 
|| LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])
 

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 353

Int[(x_)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(q_.), x_Symbol] 
 :> Simp[1/2   Subst[Int[(a + b*x)^p*(c + d*x)^q, x], x, x^2], x] /; FreeQ[ 
{a, b, c, d, p, q}, x] && NeQ[b*c - a*d, 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 5155

Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbo 
l] :> With[{u = IntHide[(d + e*x^2)^p, x]}, Simp[(a + b*ArcCos[c*x])   u, x 
] + Simp[b*c   Int[SimplifyIntegrand[u/Sqrt[1 - c^2*x^2], x], x], x]] /; Fr 
eeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && IGtQ[p, 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 = 0.25 (sec) , antiderivative size = 132, normalized size of antiderivative = 0.84

method	result	size
derivativedivides	\(-\frac {c \left (9 a^{3} x^{3} \arccos \left (a x \right )^{3}-9 x^{2} \sqrt {-a^{2} x^{2}+1}\, \arccos \left (a x \right )^{2} a^{2}-27 a x \arccos \left (a x \right )^{3}-6 a^{3} x^{3} \arccos \left (a x \right )+63 \arccos \left (a x \right )^{2} \sqrt {-a^{2} x^{2}+1}+2 a^{2} x^{2} \sqrt {-a^{2} x^{2}+1}+126 a x \arccos \left (a x \right )-122 \sqrt {-a^{2} x^{2}+1}\right )}{27 a}\)	\(132\)
default	\(-\frac {c \left (9 a^{3} x^{3} \arccos \left (a x \right )^{3}-9 x^{2} \sqrt {-a^{2} x^{2}+1}\, \arccos \left (a x \right )^{2} a^{2}-27 a x \arccos \left (a x \right )^{3}-6 a^{3} x^{3} \arccos \left (a x \right )+63 \arccos \left (a x \right )^{2} \sqrt {-a^{2} x^{2}+1}+2 a^{2} x^{2} \sqrt {-a^{2} x^{2}+1}+126 a x \arccos \left (a x \right )-122 \sqrt {-a^{2} x^{2}+1}\right )}{27 a}\)	\(132\)
orering	\(\frac {5 x \left (13 a^{4} x^{4}-194 a^{2} x^{2}-179\right ) \left (-a^{2} c \,x^{2}+c \right ) \arccos \left (a x \right )^{3}}{81 \left (a^{2} x^{2}-1\right )^{2}}-\frac {\left (25 a^{4} x^{4}-683 a^{2} x^{2}-242\right ) \left (-2 c x \,a^{2} \arccos \left (a x \right )^{3}-\frac {3 \left (-a^{2} c \,x^{2}+c \right ) \arccos \left (a x \right )^{2} a}{\sqrt {-a^{2} x^{2}+1}}\right )}{81 a^{2} \left (a^{2} x^{2}-1\right )}+\frac {2 x \left (a^{2} x^{2}-41\right ) \left (-2 c \,a^{2} \arccos \left (a x \right )^{3}+\frac {12 c x \,a^{3} \arccos \left (a x \right )^{2}}{\sqrt {-a^{2} x^{2}+1}}+\frac {6 \left (-a^{2} c \,x^{2}+c \right ) \arccos \left (a x \right ) a^{2}}{-a^{2} x^{2}+1}-\frac {3 \left (-a^{2} c \,x^{2}+c \right ) \arccos \left (a x \right )^{2} a^{3} x}{\left (-a^{2} x^{2}+1\right )^{\frac {3}{2}}}\right )}{27 a^{2}}-\frac {\left (a^{2} x^{2}-61\right ) \left (a x -1\right ) \left (a x +1\right ) \left (\frac {18 c \,a^{3} \arccos \left (a x \right )^{2}}{\sqrt {-a^{2} x^{2}+1}}-\frac {36 c x \,a^{4} \arccos \left (a x \right )}{-a^{2} x^{2}+1}+\frac {18 c \,x^{2} a^{5} \arccos \left (a x \right )^{2}}{\left (-a^{2} x^{2}+1\right )^{\frac {3}{2}}}-\frac {6 \left (-a^{2} c \,x^{2}+c \right ) a^{3}}{\left (-a^{2} x^{2}+1\right )^{\frac {3}{2}}}+\frac {18 \left (-a^{2} c \,x^{2}+c \right ) \arccos \left (a x \right ) a^{4} x}{\left (-a^{2} x^{2}+1\right )^{2}}-\frac {9 \left (-a^{2} c \,x^{2}+c \right ) \arccos \left (a x \right )^{2} a^{5} x^{2}}{\left (-a^{2} x^{2}+1\right )^{\frac {5}{2}}}-\frac {3 \left (-a^{2} c \,x^{2}+c \right ) \arccos \left (a x \right )^{2} a^{3}}{\left (-a^{2} x^{2}+1\right )^{\frac {3}{2}}}\right )}{81 a^{4}}\)	\(482\)

Input:

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

Output:

-1/27/a*c*(9*a^3*x^3*arccos(a*x)^3-9*x^2*(-a^2*x^2+1)^(1/2)*arccos(a*x)^2* 
a^2-27*a*x*arccos(a*x)^3-6*a^3*x^3*arccos(a*x)+63*arccos(a*x)^2*(-a^2*x^2+ 
1)^(1/2)+2*a^2*x^2*(-a^2*x^2+1)^(1/2)+126*a*x*arccos(a*x)-122*(-a^2*x^2+1) 
^(1/2))
 

Fricas [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.59 \[ \int \left (c-a^2 c x^2\right ) \arccos (a x)^3 \, dx=-\frac {9 \, {\left (a^{3} c x^{3} - 3 \, a c x\right )} \arccos \left (a x\right )^{3} - 6 \, {\left (a^{3} c x^{3} - 21 \, a c x\right )} \arccos \left (a x\right ) + {\left (2 \, a^{2} c x^{2} - 9 \, {\left (a^{2} c x^{2} - 7 \, c\right )} \arccos \left (a x\right )^{2} - 122 \, c\right )} \sqrt {-a^{2} x^{2} + 1}}{27 \, a} \] Input:

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

Output:

-1/27*(9*(a^3*c*x^3 - 3*a*c*x)*arccos(a*x)^3 - 6*(a^3*c*x^3 - 21*a*c*x)*ar 
ccos(a*x) + (2*a^2*c*x^2 - 9*(a^2*c*x^2 - 7*c)*arccos(a*x)^2 - 122*c)*sqrt 
(-a^2*x^2 + 1))/a
 

Sympy [A] (verification not implemented)

Time = 0.27 (sec) , antiderivative size = 156, normalized size of antiderivative = 0.99 \[ \int \left (c-a^2 c x^2\right ) \arccos (a x)^3 \, dx=\begin {cases} - \frac {a^{2} c x^{3} \operatorname {acos}^{3}{\left (a x \right )}}{3} + \frac {2 a^{2} c x^{3} \operatorname {acos}{\left (a x \right )}}{9} + \frac {a c x^{2} \sqrt {- a^{2} x^{2} + 1} \operatorname {acos}^{2}{\left (a x \right )}}{3} - \frac {2 a c x^{2} \sqrt {- a^{2} x^{2} + 1}}{27} + c x \operatorname {acos}^{3}{\left (a x \right )} - \frac {14 c x \operatorname {acos}{\left (a x \right )}}{3} - \frac {7 c \sqrt {- a^{2} x^{2} + 1} \operatorname {acos}^{2}{\left (a x \right )}}{3 a} + \frac {122 c \sqrt {- a^{2} x^{2} + 1}}{27 a} & \text {for}\: a \neq 0 \\\frac {\pi ^{3} c x}{8} & \text {otherwise} \end {cases} \] Input:

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

Output:

Piecewise((-a**2*c*x**3*acos(a*x)**3/3 + 2*a**2*c*x**3*acos(a*x)/9 + a*c*x 
**2*sqrt(-a**2*x**2 + 1)*acos(a*x)**2/3 - 2*a*c*x**2*sqrt(-a**2*x**2 + 1)/ 
27 + c*x*acos(a*x)**3 - 14*c*x*acos(a*x)/3 - 7*c*sqrt(-a**2*x**2 + 1)*acos 
(a*x)**2/(3*a) + 122*c*sqrt(-a**2*x**2 + 1)/(27*a), Ne(a, 0)), (pi**3*c*x/ 
8, True))
 

Maxima [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 128, normalized size of antiderivative = 0.81 \[ \int \left (c-a^2 c x^2\right ) \arccos (a x)^3 \, dx=\frac {1}{3} \, {\left (\sqrt {-a^{2} x^{2} + 1} c x^{2} - \frac {7 \, \sqrt {-a^{2} x^{2} + 1} c}{a^{2}}\right )} a \arccos \left (a x\right )^{2} - \frac {1}{3} \, {\left (a^{2} c x^{3} - 3 \, c x\right )} \arccos \left (a x\right )^{3} - \frac {2}{27} \, {\left (\sqrt {-a^{2} x^{2} + 1} c x^{2} - \frac {3 \, {\left (a^{2} c x^{3} - 21 \, c x\right )} \arccos \left (a x\right )}{a} - \frac {61 \, \sqrt {-a^{2} x^{2} + 1} c}{a^{2}}\right )} a \] Input:

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

Output:

1/3*(sqrt(-a^2*x^2 + 1)*c*x^2 - 7*sqrt(-a^2*x^2 + 1)*c/a^2)*a*arccos(a*x)^ 
2 - 1/3*(a^2*c*x^3 - 3*c*x)*arccos(a*x)^3 - 2/27*(sqrt(-a^2*x^2 + 1)*c*x^2 
 - 3*(a^2*c*x^3 - 21*c*x)*arccos(a*x)/a - 61*sqrt(-a^2*x^2 + 1)*c/a^2)*a
 

Giac [A] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 132, normalized size of antiderivative = 0.84 \[ \int \left (c-a^2 c x^2\right ) \arccos (a x)^3 \, dx=-\frac {1}{3} \, a^{2} c x^{3} \arccos \left (a x\right )^{3} + \frac {2}{9} \, a^{2} c x^{3} \arccos \left (a x\right ) + \frac {1}{3} \, \sqrt {-a^{2} x^{2} + 1} a c x^{2} \arccos \left (a x\right )^{2} + c x \arccos \left (a x\right )^{3} - \frac {2}{27} \, \sqrt {-a^{2} x^{2} + 1} a c x^{2} - \frac {14}{3} \, c x \arccos \left (a x\right ) - \frac {7 \, \sqrt {-a^{2} x^{2} + 1} c \arccos \left (a x\right )^{2}}{3 \, a} + \frac {122 \, \sqrt {-a^{2} x^{2} + 1} c}{27 \, a} \] Input:

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

Output:

-1/3*a^2*c*x^3*arccos(a*x)^3 + 2/9*a^2*c*x^3*arccos(a*x) + 1/3*sqrt(-a^2*x 
^2 + 1)*a*c*x^2*arccos(a*x)^2 + c*x*arccos(a*x)^3 - 2/27*sqrt(-a^2*x^2 + 1 
)*a*c*x^2 - 14/3*c*x*arccos(a*x) - 7/3*sqrt(-a^2*x^2 + 1)*c*arccos(a*x)^2/ 
a + 122/27*sqrt(-a^2*x^2 + 1)*c/a
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

(c*(acos(a*x)**3*a*x - 3*sqrt( - a**2*x**2 + 1)*acos(a*x)**2 - 6*acos(a*x) 
*a*x + 6*sqrt( - a**2*x**2 + 1) - int(acos(a*x)**3*x**2,x)*a**3))/a