\(\int \frac {x^4 \arccos (a x)^2}{\sqrt {1-a^2 x^2}} \, dx\) [266]

Optimal result

Integrand size = 24, antiderivative size = 157 \[ \int \frac {x^4 \arccos (a x)^2}{\sqrt {1-a^2 x^2}} \, dx=\frac {15 x \sqrt {1-a^2 x^2}}{64 a^4}+\frac {x^3 \sqrt {1-a^2 x^2}}{32 a^2}-\frac {15 \arccos (a x)}{64 a^5}+\frac {3 x^2 \arccos (a x)}{8 a^3}+\frac {x^4 \arccos (a x)}{8 a}-\frac {3 x \sqrt {1-a^2 x^2} \arccos (a x)^2}{8 a^4}-\frac {x^3 \sqrt {1-a^2 x^2} \arccos (a x)^2}{4 a^2}+\frac {\arccos (a x)^3}{8 a^5} \] Output:

15/64*x*(-a^2*x^2+1)^(1/2)/a^4+1/32*x^3*(-a^2*x^2+1)^(1/2)/a^2-15/64*arcco 
s(a*x)/a^5+3/8*x^2*arccos(a*x)/a^3+1/8*x^4*arccos(a*x)/a-3/8*x*(-a^2*x^2+1 
)^(1/2)*arccos(a*x)^2/a^4-1/4*x^3*(-a^2*x^2+1)^(1/2)*arccos(a*x)^2/a^2+1/8 
*arccos(a*x)^3/a^5
 

Mathematica [A] (verified)

Time = 0.08 (sec) , antiderivative size = 105, normalized size of antiderivative = 0.67 \[ \int \frac {x^4 \arccos (a x)^2}{\sqrt {1-a^2 x^2}} \, dx=-\frac {-a x \sqrt {1-a^2 x^2} \left (15+2 a^2 x^2\right )+8 a^2 x^2 \left (3+a^2 x^2\right ) \arccos (a x)+8 a x \sqrt {1-a^2 x^2} \left (3+2 a^2 x^2\right ) \arccos (a x)^2+8 \arccos (a x)^3+15 \arcsin (a x)}{64 a^5} \] Input:

Integrate[(x^4*ArcCos[a*x]^2)/Sqrt[1 - a^2*x^2],x]
 

Output:

-1/64*(-(a*x*Sqrt[1 - a^2*x^2]*(15 + 2*a^2*x^2)) + 8*a^2*x^2*(3 + a^2*x^2) 
*ArcCos[a*x] + 8*a*x*Sqrt[1 - a^2*x^2]*(3 + 2*a^2*x^2)*ArcCos[a*x]^2 + 8*A 
rcCos[a*x]^3 + 15*ArcSin[a*x])/a^5
 

Rubi [A] (verified)

Time = 0.96 (sec) , antiderivative size = 226, normalized size of antiderivative = 1.44, number of steps used = 10, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {5211, 5139, 262, 262, 223, 5211, 5139, 262, 223, 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.

\(\displaystyle \int \frac {x^4 \arccos (a x)^2}{\sqrt {1-a^2 x^2}} \, dx\)

\(\Big \downarrow \) 5211

\(\displaystyle \frac {3 \int \frac {x^2 \arccos (a x)^2}{\sqrt {1-a^2 x^2}}dx}{4 a^2}-\frac {\int x^3 \arccos (a x)dx}{2 a}-\frac {x^3 \sqrt {1-a^2 x^2} \arccos (a x)^2}{4 a^2}\)

\(\Big \downarrow \) 5139

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

\(\Big \downarrow \) 262

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

\(\Big \downarrow \) 262

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

\(\Big \downarrow \) 223

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

\(\Big \downarrow \) 5211

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

\(\Big \downarrow \) 5139

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

\(\Big \downarrow \) 262

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

\(\Big \downarrow \) 223

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

\(\Big \downarrow \) 5153

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

Input:

Int[(x^4*ArcCos[a*x]^2)/Sqrt[1 - a^2*x^2],x]
 

Output:

-1/4*(x^3*Sqrt[1 - a^2*x^2]*ArcCos[a*x]^2)/a^2 - ((x^4*ArcCos[a*x])/4 + (a 
*(-1/4*(x^3*Sqrt[1 - a^2*x^2])/a^2 + (3*(-1/2*(x*Sqrt[1 - a^2*x^2])/a^2 + 
ArcSin[a*x]/(2*a^3)))/(4*a^2)))/4)/(2*a) + (3*(-1/2*(x*Sqrt[1 - a^2*x^2]*A 
rcCos[a*x]^2)/a^2 - ArcCos[a*x]^3/(6*a^3) - ((x^2*ArcCos[a*x])/2 + (a*(-1/ 
2*(x*Sqrt[1 - a^2*x^2])/a^2 + ArcSin[a*x]/(2*a^3)))/2)/a))/(4*a^2)
 

Defintions of rubi rules used

rule 223

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSin[Rt[-b, 2]*(x/Sqrt 
[a])]/Rt[-b, 2], x] /; FreeQ[{a, b}, x] && GtQ[a, 0] && NegQ[b]
 

rule 262

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[c*(c*x) 
^(m - 1)*((a + b*x^2)^(p + 1)/(b*(m + 2*p + 1))), x] - Simp[a*c^2*((m - 1)/ 
(b*(m + 2*p + 1)))   Int[(c*x)^(m - 2)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b 
, c, p}, x] && GtQ[m, 2 - 1] && NeQ[m + 2*p + 1, 0] && IntBinomialQ[a, b, c 
, 2, m, p, x]
 

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 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 = 0.33 (sec) , antiderivative size = 129, normalized size of antiderivative = 0.82

method	result	size
default	\(-\frac {16 \arccos \left (a x \right )^{2} \sqrt {-a^{2} x^{2}+1}\, a^{3} x^{3}+8 a^{4} x^{4} \arccos \left (a x \right )-2 a^{3} x^{3} \sqrt {-a^{2} x^{2}+1}+24 \arccos \left (a x \right )^{2} \sqrt {-a^{2} x^{2}+1}\, a x +24 a^{2} x^{2} \arccos \left (a x \right )+8 \arccos \left (a x \right )^{3}-15 \sqrt {-a^{2} x^{2}+1}\, a x -15 \arccos \left (a x \right )}{64 a^{5}}\)	\(129\)

Input:

int(x^4*arccos(a*x)^2/(-a^2*x^2+1)^(1/2),x,method=_RETURNVERBOSE)
 

Output:

-1/64*(16*arccos(a*x)^2*(-a^2*x^2+1)^(1/2)*a^3*x^3+8*a^4*x^4*arccos(a*x)-2 
*a^3*x^3*(-a^2*x^2+1)^(1/2)+24*arccos(a*x)^2*(-a^2*x^2+1)^(1/2)*a*x+24*a^2 
*x^2*arccos(a*x)+8*arccos(a*x)^3-15*(-a^2*x^2+1)^(1/2)*a*x-15*arccos(a*x)) 
/a^5
 

Fricas [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.54 \[ \int \frac {x^4 \arccos (a x)^2}{\sqrt {1-a^2 x^2}} \, dx=-\frac {8 \, \arccos \left (a x\right )^{3} + {\left (8 \, a^{4} x^{4} + 24 \, a^{2} x^{2} - 15\right )} \arccos \left (a x\right ) - {\left (2 \, a^{3} x^{3} - 8 \, {\left (2 \, a^{3} x^{3} + 3 \, a x\right )} \arccos \left (a x\right )^{2} + 15 \, a x\right )} \sqrt {-a^{2} x^{2} + 1}}{64 \, a^{5}} \] Input:

integrate(x^4*arccos(a*x)^2/(-a^2*x^2+1)^(1/2),x, algorithm="fricas")
 

Output:

-1/64*(8*arccos(a*x)^3 + (8*a^4*x^4 + 24*a^2*x^2 - 15)*arccos(a*x) - (2*a^ 
3*x^3 - 8*(2*a^3*x^3 + 3*a*x)*arccos(a*x)^2 + 15*a*x)*sqrt(-a^2*x^2 + 1))/ 
a^5
 

Sympy [A] (verification not implemented)

Time = 0.52 (sec) , antiderivative size = 153, normalized size of antiderivative = 0.97 \[ \int \frac {x^4 \arccos (a x)^2}{\sqrt {1-a^2 x^2}} \, dx=\begin {cases} - \frac {x^{4} \operatorname {acos}{\left (a x \right )}}{8 a} - \frac {x^{3} \sqrt {- a^{2} x^{2} + 1} \operatorname {acos}^{2}{\left (a x \right )}}{4 a^{2}} + \frac {x^{3} \sqrt {- a^{2} x^{2} + 1}}{32 a^{2}} - \frac {3 x^{2} \operatorname {acos}{\left (a x \right )}}{8 a^{3}} - \frac {3 x \sqrt {- a^{2} x^{2} + 1} \operatorname {acos}^{2}{\left (a x \right )}}{8 a^{4}} + \frac {15 x \sqrt {- a^{2} x^{2} + 1}}{64 a^{4}} - \frac {\operatorname {acos}^{3}{\left (a x \right )}}{8 a^{5}} + \frac {15 \operatorname {acos}{\left (a x \right )}}{64 a^{5}} & \text {for}\: a \neq 0 \\\frac {\pi ^{2} x^{5}}{20} & \text {otherwise} \end {cases} \] Input:

integrate(x**4*acos(a*x)**2/(-a**2*x**2+1)**(1/2),x)
 

Output:

Piecewise((-x**4*acos(a*x)/(8*a) - x**3*sqrt(-a**2*x**2 + 1)*acos(a*x)**2/ 
(4*a**2) + x**3*sqrt(-a**2*x**2 + 1)/(32*a**2) - 3*x**2*acos(a*x)/(8*a**3) 
 - 3*x*sqrt(-a**2*x**2 + 1)*acos(a*x)**2/(8*a**4) + 15*x*sqrt(-a**2*x**2 + 
 1)/(64*a**4) - acos(a*x)**3/(8*a**5) + 15*acos(a*x)/(64*a**5), Ne(a, 0)), 
 (pi**2*x**5/20, True))
 

Maxima [F]

\[ \int \frac {x^4 \arccos (a x)^2}{\sqrt {1-a^2 x^2}} \, dx=\int { \frac {x^{4} \arccos \left (a x\right )^{2}}{\sqrt {-a^{2} x^{2} + 1}} \,d x } \] Input:

integrate(x^4*arccos(a*x)^2/(-a^2*x^2+1)^(1/2),x, algorithm="maxima")
 

Output:

integrate(x^4*arccos(a*x)^2/sqrt(-a^2*x^2 + 1), x)
 

Giac [A] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 133, normalized size of antiderivative = 0.85 \[ \int \frac {x^4 \arccos (a x)^2}{\sqrt {1-a^2 x^2}} \, dx=-\frac {x^{4} \arccos \left (a x\right )}{8 \, a} - \frac {\sqrt {-a^{2} x^{2} + 1} x^{3} \arccos \left (a x\right )^{2}}{4 \, a^{2}} + \frac {\sqrt {-a^{2} x^{2} + 1} x^{3}}{32 \, a^{2}} - \frac {3 \, x^{2} \arccos \left (a x\right )}{8 \, a^{3}} - \frac {3 \, \sqrt {-a^{2} x^{2} + 1} x \arccos \left (a x\right )^{2}}{8 \, a^{4}} - \frac {\arccos \left (a x\right )^{3}}{8 \, a^{5}} + \frac {15 \, \sqrt {-a^{2} x^{2} + 1} x}{64 \, a^{4}} + \frac {15 \, \arccos \left (a x\right )}{64 \, a^{5}} \] Input:

integrate(x^4*arccos(a*x)^2/(-a^2*x^2+1)^(1/2),x, algorithm="giac")
 

Output:

-1/8*x^4*arccos(a*x)/a - 1/4*sqrt(-a^2*x^2 + 1)*x^3*arccos(a*x)^2/a^2 + 1/ 
32*sqrt(-a^2*x^2 + 1)*x^3/a^2 - 3/8*x^2*arccos(a*x)/a^3 - 3/8*sqrt(-a^2*x^ 
2 + 1)*x*arccos(a*x)^2/a^4 - 1/8*arccos(a*x)^3/a^5 + 15/64*sqrt(-a^2*x^2 + 
 1)*x/a^4 + 15/64*arccos(a*x)/a^5
 

Mupad [F(-1)]

Timed out. \[ \int \frac {x^4 \arccos (a x)^2}{\sqrt {1-a^2 x^2}} \, dx=\int \frac {x^4\,{\mathrm {acos}\left (a\,x\right )}^2}{\sqrt {1-a^2\,x^2}} \,d x \] Input:

int((x^4*acos(a*x)^2)/(1 - a^2*x^2)^(1/2),x)
 

Output:

int((x^4*acos(a*x)^2)/(1 - a^2*x^2)^(1/2), x)
 

Reduce [F]

\[ \int \frac {x^4 \arccos (a x)^2}{\sqrt {1-a^2 x^2}} \, dx=\int \frac {\mathit {acos} \left (a x \right )^{2} x^{4}}{\sqrt {-a^{2} x^{2}+1}}d x \] Input:

int(x^4*acos(a*x)^2/(-a^2*x^2+1)^(1/2),x)
 

Output:

int((acos(a*x)**2*x**4)/sqrt( - a**2*x**2 + 1),x)