\(\int \sqrt {c-a^2 c x^2} \arccos (a x)^3 \, dx\) [64]

Optimal result

Integrand size = 22, antiderivative size = 215 \[ \int \sqrt {c-a^2 c x^2} \arccos (a x)^3 \, dx=-\frac {3 a x^2 \sqrt {c-a^2 c x^2}}{8 \sqrt {1-a^2 x^2}}-\frac {3}{4} x \sqrt {c-a^2 c x^2} \arccos (a x)-\frac {3 \sqrt {c-a^2 c x^2} \arccos (a x)^2}{8 a \sqrt {1-a^2 x^2}}+\frac {3 a x^2 \sqrt {c-a^2 c x^2} \arccos (a x)^2}{4 \sqrt {1-a^2 x^2}}+\frac {1}{2} x \sqrt {c-a^2 c x^2} \arccos (a x)^3-\frac {\sqrt {c-a^2 c x^2} \arccos (a x)^4}{8 a \sqrt {1-a^2 x^2}} \] Output:

-3/8*a*x^2*(-a^2*c*x^2+c)^(1/2)/(-a^2*x^2+1)^(1/2)-3/4*x*(-a^2*c*x^2+c)^(1 
/2)*arccos(a*x)-3/8*(-a^2*c*x^2+c)^(1/2)*arccos(a*x)^2/a/(-a^2*x^2+1)^(1/2 
)+3/4*a*x^2*(-a^2*c*x^2+c)^(1/2)*arccos(a*x)^2/(-a^2*x^2+1)^(1/2)+1/2*x*(- 
a^2*c*x^2+c)^(1/2)*arccos(a*x)^3-1/8*(-a^2*c*x^2+c)^(1/2)*arccos(a*x)^4/a/ 
(-a^2*x^2+1)^(1/2)
 

Mathematica [A] (verified)

Time = 0.17 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.40 \[ \int \sqrt {c-a^2 c x^2} \arccos (a x)^3 \, dx=-\frac {\sqrt {c \left (1-a^2 x^2\right )} \left (\left (3-6 \arccos (a x)^2\right ) \cos (2 \arccos (a x))+2 \arccos (a x) \left (\arccos (a x)^3+\left (3-2 \arccos (a x)^2\right ) \sin (2 \arccos (a x))\right )\right )}{16 a \sqrt {1-a^2 x^2}} \] Input:

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

Output:

-1/16*(Sqrt[c*(1 - a^2*x^2)]*((3 - 6*ArcCos[a*x]^2)*Cos[2*ArcCos[a*x]] + 2 
*ArcCos[a*x]*(ArcCos[a*x]^3 + (3 - 2*ArcCos[a*x]^2)*Sin[2*ArcCos[a*x]])))/ 
(a*Sqrt[1 - a^2*x^2])
 

Rubi [A] (verified)

Time = 0.89 (sec) , antiderivative size = 169, normalized size of antiderivative = 0.79, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {5157, 5139, 5153, 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.

\(\displaystyle \int \arccos (a x)^3 \sqrt {c-a^2 c x^2} \, dx\)

\(\Big \downarrow \) 5157

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

\(\Big \downarrow \) 5139

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

\(\Big \downarrow \) 5153

\(\displaystyle \frac {3 a \sqrt {c-a^2 c x^2} \left (a \int \frac {x^2 \arccos (a x)}{\sqrt {1-a^2 x^2}}dx+\frac {1}{2} x^2 \arccos (a x)^2\right )}{2 \sqrt {1-a^2 x^2}}-\frac {\arccos (a x)^4 \sqrt {c-a^2 c x^2}}{8 a \sqrt {1-a^2 x^2}}+\frac {1}{2} x \arccos (a x)^3 \sqrt {c-a^2 c x^2}\)

\(\Big \downarrow \) 5211

\(\displaystyle \frac {3 a \sqrt {c-a^2 c x^2} \left (a \left (\frac {\int \frac {\arccos (a x)}{\sqrt {1-a^2 x^2}}dx}{2 a^2}-\frac {\int xdx}{2 a}-\frac {x \sqrt {1-a^2 x^2} \arccos (a x)}{2 a^2}\right )+\frac {1}{2} x^2 \arccos (a x)^2\right )}{2 \sqrt {1-a^2 x^2}}-\frac {\arccos (a x)^4 \sqrt {c-a^2 c x^2}}{8 a \sqrt {1-a^2 x^2}}+\frac {1}{2} x \arccos (a x)^3 \sqrt {c-a^2 c x^2}\)

\(\Big \downarrow \) 15

\(\displaystyle \frac {3 a \sqrt {c-a^2 c x^2} \left (a \left (\frac {\int \frac {\arccos (a x)}{\sqrt {1-a^2 x^2}}dx}{2 a^2}-\frac {x \sqrt {1-a^2 x^2} \arccos (a x)}{2 a^2}-\frac {x^2}{4 a}\right )+\frac {1}{2} x^2 \arccos (a x)^2\right )}{2 \sqrt {1-a^2 x^2}}-\frac {\arccos (a x)^4 \sqrt {c-a^2 c x^2}}{8 a \sqrt {1-a^2 x^2}}+\frac {1}{2} x \arccos (a x)^3 \sqrt {c-a^2 c x^2}\)

\(\Big \downarrow \) 5153

\(\displaystyle -\frac {\arccos (a x)^4 \sqrt {c-a^2 c x^2}}{8 a \sqrt {1-a^2 x^2}}+\frac {1}{2} x \arccos (a x)^3 \sqrt {c-a^2 c x^2}+\frac {3 a \left (a \left (-\frac {\arccos (a x)^2}{4 a^3}-\frac {x \sqrt {1-a^2 x^2} \arccos (a x)}{2 a^2}-\frac {x^2}{4 a}\right )+\frac {1}{2} x^2 \arccos (a x)^2\right ) \sqrt {c-a^2 c x^2}}{2 \sqrt {1-a^2 x^2}}\)

Input:

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

Output:

(x*Sqrt[c - a^2*c*x^2]*ArcCos[a*x]^3)/2 - (Sqrt[c - a^2*c*x^2]*ArcCos[a*x] 
^4)/(8*a*Sqrt[1 - a^2*x^2]) + (3*a*Sqrt[c - a^2*c*x^2]*((x^2*ArcCos[a*x]^2 
)/2 + a*(-1/4*x^2/a - (x*Sqrt[1 - a^2*x^2]*ArcCos[a*x])/(2*a^2) - ArcCos[a 
*x]^2/(4*a^3))))/(2*Sqrt[1 - a^2*x^2])
 

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 5157

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

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 [C] (verified)

Result contains complex when optimal does not.

Time = 0.47 (sec) , antiderivative size = 260, normalized size of antiderivative = 1.21

method	result	size
default	\(\frac {\sqrt {-c \left (a^{2} x^{2}-1\right )}\, \sqrt {-a^{2} x^{2}+1}\, \arccos \left (a x \right )^{4}}{8 a \left (a^{2} x^{2}-1\right )}+\frac {\sqrt {-c \left (a^{2} x^{2}-1\right )}\, \left (2 a^{3} x^{3}-2 a x +2 i \sqrt {-a^{2} x^{2}+1}\, a^{2} x^{2}-i \sqrt {-a^{2} x^{2}+1}\right ) \left (6 i \arccos \left (a x \right )^{2}+4 \arccos \left (a x \right )^{3}-3 i-6 \arccos \left (a x \right )\right )}{32 a \left (a^{2} x^{2}-1\right )}+\frac {\sqrt {-c \left (a^{2} x^{2}-1\right )}\, \left (-2 i \sqrt {-a^{2} x^{2}+1}\, a^{2} x^{2}+2 a^{3} x^{3}+i \sqrt {-a^{2} x^{2}+1}-2 a x \right ) \left (-6 i \arccos \left (a x \right )^{2}+4 \arccos \left (a x \right )^{3}+3 i-6 \arccos \left (a x \right )\right )}{32 a \left (a^{2} x^{2}-1\right )}\)	\(260\)

Input:

int((-a^2*c*x^2+c)^(1/2)*arccos(a*x)^3,x,method=_RETURNVERBOSE)
 

Output:

1/8*(-c*(a^2*x^2-1))^(1/2)*(-a^2*x^2+1)^(1/2)/a/(a^2*x^2-1)*arccos(a*x)^4+ 
1/32*(-c*(a^2*x^2-1))^(1/2)*(2*a^3*x^3-2*a*x+2*I*(-a^2*x^2+1)^(1/2)*a^2*x^ 
2-I*(-a^2*x^2+1)^(1/2))*(6*I*arccos(a*x)^2+4*arccos(a*x)^3-3*I-6*arccos(a* 
x))/a/(a^2*x^2-1)+1/32*(-c*(a^2*x^2-1))^(1/2)*(-2*I*(-a^2*x^2+1)^(1/2)*a^2 
*x^2+2*a^3*x^3+I*(-a^2*x^2+1)^(1/2)-2*a*x)*(-6*I*arccos(a*x)^2+4*arccos(a* 
x)^3+3*I-6*arccos(a*x))/a/(a^2*x^2-1)
 

Fricas [F]

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

integrate((-a^2*c*x^2+c)^(1/2)*arccos(a*x)^3,x, algorithm="fricas")
 

Output:

integral(sqrt(-a^2*c*x^2 + c)*arccos(a*x)^3, x)
 

Sympy [F]

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

integrate((-a**2*c*x**2+c)**(1/2)*acos(a*x)**3,x)
 

Output:

Integral(sqrt(-c*(a*x - 1)*(a*x + 1))*acos(a*x)**3, x)
 

Maxima [F]

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

integrate((-a^2*c*x^2+c)^(1/2)*arccos(a*x)^3,x, algorithm="maxima")
 

Output:

integrate(sqrt(-a^2*c*x^2 + c)*arccos(a*x)^3, x)
 

Giac [F(-2)]

Exception generated. \[ \int \sqrt {c-a^2 c x^2} \arccos (a x)^3 \, dx=\text {Exception raised: TypeError} \] Input:

integrate((-a^2*c*x^2+c)^(1/2)*arccos(a*x)^3,x, algorithm="giac")
 

Output:

Exception raised: TypeError >> an error occurred running a Giac command:IN 
PUT:sage2:=int(sage0,sageVARx):;OUTPUT:sym2poly/r2sym(const gen & e,const 
index_m & i,const vecteur & l) Error: Bad Argument Value
 

Mupad [F(-1)]

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

int(acos(a*x)^3*(c - a^2*c*x^2)^(1/2),x)
 

Output:

int(acos(a*x)^3*(c - a^2*c*x^2)^(1/2), x)
 

Reduce [F]

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

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

Output:

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