\(\int x^4 \arccos (a x)^4 \, dx\) [33]

Optimal result

Integrand size = 10, antiderivative size = 250 \[ \int x^4 \arccos (a x)^4 \, dx=\frac {16576 x}{5625 a^4}+\frac {1088 x^3}{16875 a^2}+\frac {24 x^5}{3125}+\frac {16576 \sqrt {1-a^2 x^2} \arccos (a x)}{5625 a^5}+\frac {1088 x^2 \sqrt {1-a^2 x^2} \arccos (a x)}{5625 a^3}+\frac {24 x^4 \sqrt {1-a^2 x^2} \arccos (a x)}{625 a}-\frac {32 x \arccos (a x)^2}{25 a^4}-\frac {16 x^3 \arccos (a x)^2}{75 a^2}-\frac {12}{125} x^5 \arccos (a x)^2-\frac {32 \sqrt {1-a^2 x^2} \arccos (a x)^3}{75 a^5}-\frac {16 x^2 \sqrt {1-a^2 x^2} \arccos (a x)^3}{75 a^3}-\frac {4 x^4 \sqrt {1-a^2 x^2} \arccos (a x)^3}{25 a}+\frac {1}{5} x^5 \arccos (a x)^4 \] Output:

16576/5625*x/a^4+1088/16875*x^3/a^2+24/3125*x^5+16576/5625*(-a^2*x^2+1)^(1 
/2)*arccos(a*x)/a^5+1088/5625*x^2*(-a^2*x^2+1)^(1/2)*arccos(a*x)/a^3+24/62 
5*x^4*(-a^2*x^2+1)^(1/2)*arccos(a*x)/a-32/25*x*arccos(a*x)^2/a^4-16/75*x^3 
*arccos(a*x)^2/a^2-12/125*x^5*arccos(a*x)^2-32/75*(-a^2*x^2+1)^(1/2)*arcco 
s(a*x)^3/a^5-16/75*x^2*(-a^2*x^2+1)^(1/2)*arccos(a*x)^3/a^3-4/25*x^4*(-a^2 
*x^2+1)^(1/2)*arccos(a*x)^3/a+1/5*x^5*arccos(a*x)^4
 

Mathematica [A] (verified)

Time = 0.05 (sec) , antiderivative size = 150, normalized size of antiderivative = 0.60 \[ \int x^4 \arccos (a x)^4 \, dx=\frac {8 a x \left (31080+680 a^2 x^2+81 a^4 x^4\right )+120 \sqrt {1-a^2 x^2} \left (2072+136 a^2 x^2+27 a^4 x^4\right ) \arccos (a x)-900 a x \left (120+20 a^2 x^2+9 a^4 x^4\right ) \arccos (a x)^2-4500 \sqrt {1-a^2 x^2} \left (8+4 a^2 x^2+3 a^4 x^4\right ) \arccos (a x)^3+16875 a^5 x^5 \arccos (a x)^4}{84375 a^5} \] Input:

Integrate[x^4*ArcCos[a*x]^4,x]
 

Output:

(8*a*x*(31080 + 680*a^2*x^2 + 81*a^4*x^4) + 120*Sqrt[1 - a^2*x^2]*(2072 + 
136*a^2*x^2 + 27*a^4*x^4)*ArcCos[a*x] - 900*a*x*(120 + 20*a^2*x^2 + 9*a^4* 
x^4)*ArcCos[a*x]^2 - 4500*Sqrt[1 - a^2*x^2]*(8 + 4*a^2*x^2 + 3*a^4*x^4)*Ar 
cCos[a*x]^3 + 16875*a^5*x^5*ArcCos[a*x]^4)/(84375*a^5)
 

Rubi [A] (verified)

Time = 2.34 (sec) , antiderivative size = 416, normalized size of antiderivative = 1.66, number of steps used = 14, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.400, Rules used = {5139, 5211, 5139, 5211, 15, 5139, 5183, 5131, 5183, 24, 5211, 15, 5183, 24}

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.

\(\displaystyle \int x^4 \arccos (a x)^4 \, dx\)

\(\Big \downarrow \) 5139

\(\displaystyle \frac {4}{5} a \int \frac {x^5 \arccos (a x)^3}{\sqrt {1-a^2 x^2}}dx+\frac {1}{5} x^5 \arccos (a x)^4\)

\(\Big \downarrow \) 5211

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

\(\Big \downarrow \) 5139

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

\(\Big \downarrow \) 5211

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

\(\Big \downarrow \) 15

\(\displaystyle \frac {4}{5} a \left (\frac {4 \left (\frac {2 \int \frac {x \arccos (a x)^3}{\sqrt {1-a^2 x^2}}dx}{3 a^2}-\frac {\int x^2 \arccos (a x)^2dx}{a}-\frac {x^2 \sqrt {1-a^2 x^2} \arccos (a x)^3}{3 a^2}\right )}{5 a^2}-\frac {3 \left (\frac {2}{5} a \left (\frac {4 \int \frac {x^3 \arccos (a x)}{\sqrt {1-a^2 x^2}}dx}{5 a^2}-\frac {x^4 \sqrt {1-a^2 x^2} \arccos (a x)}{5 a^2}-\frac {x^5}{25 a}\right )+\frac {1}{5} x^5 \arccos (a x)^2\right )}{5 a}-\frac {x^4 \sqrt {1-a^2 x^2} \arccos (a x)^3}{5 a^2}\right )+\frac {1}{5} x^5 \arccos (a x)^4\)

\(\Big \downarrow \) 5139

\(\displaystyle \frac {4}{5} a \left (\frac {4 \left (\frac {2 \int \frac {x \arccos (a x)^3}{\sqrt {1-a^2 x^2}}dx}{3 a^2}-\frac {\frac {2}{3} a \int \frac {x^3 \arccos (a x)}{\sqrt {1-a^2 x^2}}dx+\frac {1}{3} x^3 \arccos (a x)^2}{a}-\frac {x^2 \sqrt {1-a^2 x^2} \arccos (a x)^3}{3 a^2}\right )}{5 a^2}-\frac {3 \left (\frac {2}{5} a \left (\frac {4 \int \frac {x^3 \arccos (a x)}{\sqrt {1-a^2 x^2}}dx}{5 a^2}-\frac {x^4 \sqrt {1-a^2 x^2} \arccos (a x)}{5 a^2}-\frac {x^5}{25 a}\right )+\frac {1}{5} x^5 \arccos (a x)^2\right )}{5 a}-\frac {x^4 \sqrt {1-a^2 x^2} \arccos (a x)^3}{5 a^2}\right )+\frac {1}{5} x^5 \arccos (a x)^4\)

\(\Big \downarrow \) 5183

\(\displaystyle \frac {4}{5} a \left (\frac {4 \left (\frac {2 \left (-\frac {3 \int \arccos (a x)^2dx}{a}-\frac {\sqrt {1-a^2 x^2} \arccos (a x)^3}{a^2}\right )}{3 a^2}-\frac {\frac {2}{3} a \int \frac {x^3 \arccos (a x)}{\sqrt {1-a^2 x^2}}dx+\frac {1}{3} x^3 \arccos (a x)^2}{a}-\frac {x^2 \sqrt {1-a^2 x^2} \arccos (a x)^3}{3 a^2}\right )}{5 a^2}-\frac {3 \left (\frac {2}{5} a \left (\frac {4 \int \frac {x^3 \arccos (a x)}{\sqrt {1-a^2 x^2}}dx}{5 a^2}-\frac {x^4 \sqrt {1-a^2 x^2} \arccos (a x)}{5 a^2}-\frac {x^5}{25 a}\right )+\frac {1}{5} x^5 \arccos (a x)^2\right )}{5 a}-\frac {x^4 \sqrt {1-a^2 x^2} \arccos (a x)^3}{5 a^2}\right )+\frac {1}{5} x^5 \arccos (a x)^4\)

\(\Big \downarrow \) 5131

\(\displaystyle \frac {4}{5} a \left (\frac {4 \left (\frac {2 \left (-\frac {3 \left (2 a \int \frac {x \arccos (a x)}{\sqrt {1-a^2 x^2}}dx+x \arccos (a x)^2\right )}{a}-\frac {\sqrt {1-a^2 x^2} \arccos (a x)^3}{a^2}\right )}{3 a^2}-\frac {\frac {2}{3} a \int \frac {x^3 \arccos (a x)}{\sqrt {1-a^2 x^2}}dx+\frac {1}{3} x^3 \arccos (a x)^2}{a}-\frac {x^2 \sqrt {1-a^2 x^2} \arccos (a x)^3}{3 a^2}\right )}{5 a^2}-\frac {3 \left (\frac {2}{5} a \left (\frac {4 \int \frac {x^3 \arccos (a x)}{\sqrt {1-a^2 x^2}}dx}{5 a^2}-\frac {x^4 \sqrt {1-a^2 x^2} \arccos (a x)}{5 a^2}-\frac {x^5}{25 a}\right )+\frac {1}{5} x^5 \arccos (a x)^2\right )}{5 a}-\frac {x^4 \sqrt {1-a^2 x^2} \arccos (a x)^3}{5 a^2}\right )+\frac {1}{5} x^5 \arccos (a x)^4\)

\(\Big \downarrow \) 5183

\(\displaystyle \frac {4}{5} a \left (\frac {4 \left (\frac {2 \left (-\frac {3 \left (2 a \left (-\frac {\int 1dx}{a}-\frac {\sqrt {1-a^2 x^2} \arccos (a x)}{a^2}\right )+x \arccos (a x)^2\right )}{a}-\frac {\sqrt {1-a^2 x^2} \arccos (a x)^3}{a^2}\right )}{3 a^2}-\frac {\frac {2}{3} a \int \frac {x^3 \arccos (a x)}{\sqrt {1-a^2 x^2}}dx+\frac {1}{3} x^3 \arccos (a x)^2}{a}-\frac {x^2 \sqrt {1-a^2 x^2} \arccos (a x)^3}{3 a^2}\right )}{5 a^2}-\frac {3 \left (\frac {2}{5} a \left (\frac {4 \int \frac {x^3 \arccos (a x)}{\sqrt {1-a^2 x^2}}dx}{5 a^2}-\frac {x^4 \sqrt {1-a^2 x^2} \arccos (a x)}{5 a^2}-\frac {x^5}{25 a}\right )+\frac {1}{5} x^5 \arccos (a x)^2\right )}{5 a}-\frac {x^4 \sqrt {1-a^2 x^2} \arccos (a x)^3}{5 a^2}\right )+\frac {1}{5} x^5 \arccos (a x)^4\)

\(\Big \downarrow \) 24

\(\displaystyle \frac {4}{5} a \left (\frac {4 \left (-\frac {\frac {2}{3} a \int \frac {x^3 \arccos (a x)}{\sqrt {1-a^2 x^2}}dx+\frac {1}{3} x^3 \arccos (a x)^2}{a}-\frac {x^2 \sqrt {1-a^2 x^2} \arccos (a x)^3}{3 a^2}+\frac {2 \left (-\frac {\sqrt {1-a^2 x^2} \arccos (a x)^3}{a^2}-\frac {3 \left (2 a \left (-\frac {\sqrt {1-a^2 x^2} \arccos (a x)}{a^2}-\frac {x}{a}\right )+x \arccos (a x)^2\right )}{a}\right )}{3 a^2}\right )}{5 a^2}-\frac {3 \left (\frac {2}{5} a \left (\frac {4 \int \frac {x^3 \arccos (a x)}{\sqrt {1-a^2 x^2}}dx}{5 a^2}-\frac {x^4 \sqrt {1-a^2 x^2} \arccos (a x)}{5 a^2}-\frac {x^5}{25 a}\right )+\frac {1}{5} x^5 \arccos (a x)^2\right )}{5 a}-\frac {x^4 \sqrt {1-a^2 x^2} \arccos (a x)^3}{5 a^2}\right )+\frac {1}{5} x^5 \arccos (a x)^4\)

\(\Big \downarrow \) 5211

\(\displaystyle \frac {4}{5} a \left (\frac {4 \left (-\frac {\frac {2}{3} a \left (\frac {2 \int \frac {x \arccos (a x)}{\sqrt {1-a^2 x^2}}dx}{3 a^2}-\frac {\int x^2dx}{3 a}-\frac {x^2 \sqrt {1-a^2 x^2} \arccos (a x)}{3 a^2}\right )+\frac {1}{3} x^3 \arccos (a x)^2}{a}-\frac {x^2 \sqrt {1-a^2 x^2} \arccos (a x)^3}{3 a^2}+\frac {2 \left (-\frac {\sqrt {1-a^2 x^2} \arccos (a x)^3}{a^2}-\frac {3 \left (2 a \left (-\frac {\sqrt {1-a^2 x^2} \arccos (a x)}{a^2}-\frac {x}{a}\right )+x \arccos (a x)^2\right )}{a}\right )}{3 a^2}\right )}{5 a^2}-\frac {3 \left (\frac {2}{5} a \left (\frac {4 \left (\frac {2 \int \frac {x \arccos (a x)}{\sqrt {1-a^2 x^2}}dx}{3 a^2}-\frac {\int x^2dx}{3 a}-\frac {x^2 \sqrt {1-a^2 x^2} \arccos (a x)}{3 a^2}\right )}{5 a^2}-\frac {x^4 \sqrt {1-a^2 x^2} \arccos (a x)}{5 a^2}-\frac {x^5}{25 a}\right )+\frac {1}{5} x^5 \arccos (a x)^2\right )}{5 a}-\frac {x^4 \sqrt {1-a^2 x^2} \arccos (a x)^3}{5 a^2}\right )+\frac {1}{5} x^5 \arccos (a x)^4\)

\(\Big \downarrow \) 15

\(\displaystyle \frac {4}{5} a \left (\frac {4 \left (-\frac {\frac {2}{3} a \left (\frac {2 \int \frac {x \arccos (a x)}{\sqrt {1-a^2 x^2}}dx}{3 a^2}-\frac {x^2 \sqrt {1-a^2 x^2} \arccos (a x)}{3 a^2}-\frac {x^3}{9 a}\right )+\frac {1}{3} x^3 \arccos (a x)^2}{a}-\frac {x^2 \sqrt {1-a^2 x^2} \arccos (a x)^3}{3 a^2}+\frac {2 \left (-\frac {\sqrt {1-a^2 x^2} \arccos (a x)^3}{a^2}-\frac {3 \left (2 a \left (-\frac {\sqrt {1-a^2 x^2} \arccos (a x)}{a^2}-\frac {x}{a}\right )+x \arccos (a x)^2\right )}{a}\right )}{3 a^2}\right )}{5 a^2}-\frac {3 \left (\frac {2}{5} a \left (\frac {4 \left (\frac {2 \int \frac {x \arccos (a x)}{\sqrt {1-a^2 x^2}}dx}{3 a^2}-\frac {x^2 \sqrt {1-a^2 x^2} \arccos (a x)}{3 a^2}-\frac {x^3}{9 a}\right )}{5 a^2}-\frac {x^4 \sqrt {1-a^2 x^2} \arccos (a x)}{5 a^2}-\frac {x^5}{25 a}\right )+\frac {1}{5} x^5 \arccos (a x)^2\right )}{5 a}-\frac {x^4 \sqrt {1-a^2 x^2} \arccos (a x)^3}{5 a^2}\right )+\frac {1}{5} x^5 \arccos (a x)^4\)

\(\Big \downarrow \) 5183

\(\displaystyle \frac {4}{5} a \left (\frac {4 \left (-\frac {\frac {2}{3} a \left (\frac {2 \left (-\frac {\int 1dx}{a}-\frac {\sqrt {1-a^2 x^2} \arccos (a x)}{a^2}\right )}{3 a^2}-\frac {x^2 \sqrt {1-a^2 x^2} \arccos (a x)}{3 a^2}-\frac {x^3}{9 a}\right )+\frac {1}{3} x^3 \arccos (a x)^2}{a}-\frac {x^2 \sqrt {1-a^2 x^2} \arccos (a x)^3}{3 a^2}+\frac {2 \left (-\frac {\sqrt {1-a^2 x^2} \arccos (a x)^3}{a^2}-\frac {3 \left (2 a \left (-\frac {\sqrt {1-a^2 x^2} \arccos (a x)}{a^2}-\frac {x}{a}\right )+x \arccos (a x)^2\right )}{a}\right )}{3 a^2}\right )}{5 a^2}-\frac {3 \left (\frac {2}{5} a \left (\frac {4 \left (\frac {2 \left (-\frac {\int 1dx}{a}-\frac {\sqrt {1-a^2 x^2} \arccos (a x)}{a^2}\right )}{3 a^2}-\frac {x^2 \sqrt {1-a^2 x^2} \arccos (a x)}{3 a^2}-\frac {x^3}{9 a}\right )}{5 a^2}-\frac {x^4 \sqrt {1-a^2 x^2} \arccos (a x)}{5 a^2}-\frac {x^5}{25 a}\right )+\frac {1}{5} x^5 \arccos (a x)^2\right )}{5 a}-\frac {x^4 \sqrt {1-a^2 x^2} \arccos (a x)^3}{5 a^2}\right )+\frac {1}{5} x^5 \arccos (a x)^4\)

\(\Big \downarrow \) 24

\(\displaystyle \frac {4}{5} a \left (-\frac {x^4 \sqrt {1-a^2 x^2} \arccos (a x)^3}{5 a^2}+\frac {4 \left (-\frac {x^2 \sqrt {1-a^2 x^2} \arccos (a x)^3}{3 a^2}+\frac {2 \left (-\frac {\sqrt {1-a^2 x^2} \arccos (a x)^3}{a^2}-\frac {3 \left (2 a \left (-\frac {\sqrt {1-a^2 x^2} \arccos (a x)}{a^2}-\frac {x}{a}\right )+x \arccos (a x)^2\right )}{a}\right )}{3 a^2}-\frac {\frac {2}{3} a \left (-\frac {x^2 \sqrt {1-a^2 x^2} \arccos (a x)}{3 a^2}+\frac {2 \left (-\frac {\sqrt {1-a^2 x^2} \arccos (a x)}{a^2}-\frac {x}{a}\right )}{3 a^2}-\frac {x^3}{9 a}\right )+\frac {1}{3} x^3 \arccos (a x)^2}{a}\right )}{5 a^2}-\frac {3 \left (\frac {2}{5} a \left (-\frac {x^4 \sqrt {1-a^2 x^2} \arccos (a x)}{5 a^2}+\frac {4 \left (-\frac {x^2 \sqrt {1-a^2 x^2} \arccos (a x)}{3 a^2}+\frac {2 \left (-\frac {\sqrt {1-a^2 x^2} \arccos (a x)}{a^2}-\frac {x}{a}\right )}{3 a^2}-\frac {x^3}{9 a}\right )}{5 a^2}-\frac {x^5}{25 a}\right )+\frac {1}{5} x^5 \arccos (a x)^2\right )}{5 a}\right )+\frac {1}{5} x^5 \arccos (a x)^4\)

Input:

Int[x^4*ArcCos[a*x]^4,x]
 

Output:

(x^5*ArcCos[a*x]^4)/5 + (4*a*(-1/5*(x^4*Sqrt[1 - a^2*x^2]*ArcCos[a*x]^3)/a 
^2 - (3*((x^5*ArcCos[a*x]^2)/5 + (2*a*(-1/25*x^5/a - (x^4*Sqrt[1 - a^2*x^2 
]*ArcCos[a*x])/(5*a^2) + (4*(-1/9*x^3/a - (x^2*Sqrt[1 - a^2*x^2]*ArcCos[a* 
x])/(3*a^2) + (2*(-(x/a) - (Sqrt[1 - a^2*x^2]*ArcCos[a*x])/a^2))/(3*a^2))) 
/(5*a^2)))/5))/(5*a) + (4*(-1/3*(x^2*Sqrt[1 - a^2*x^2]*ArcCos[a*x]^3)/a^2 
- ((x^3*ArcCos[a*x]^2)/3 + (2*a*(-1/9*x^3/a - (x^2*Sqrt[1 - a^2*x^2]*ArcCo 
s[a*x])/(3*a^2) + (2*(-(x/a) - (Sqrt[1 - a^2*x^2]*ArcCos[a*x])/a^2))/(3*a^ 
2)))/3)/a + (2*(-((Sqrt[1 - a^2*x^2]*ArcCos[a*x]^3)/a^2) - (3*(x*ArcCos[a* 
x]^2 + 2*a*(-(x/a) - (Sqrt[1 - a^2*x^2]*ArcCos[a*x])/a^2)))/a))/(3*a^2)))/ 
(5*a^2)))/5
 

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 24

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]
 

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 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 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]
 

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 = 0.50 (sec) , antiderivative size = 197, normalized size of antiderivative = 0.79

method	result	size
derivativedivides	\(\frac {\frac {a^{5} x^{5} \arccos \left (a x \right )^{4}}{5}-\frac {4 \arccos \left (a x \right )^{3} \left (3 a^{4} x^{4}+4 a^{2} x^{2}+8\right ) \sqrt {-a^{2} x^{2}+1}}{75}-\frac {12 a^{5} x^{5} \arccos \left (a x \right )^{2}}{125}+\frac {8 \arccos \left (a x \right ) \left (3 a^{4} x^{4}+4 a^{2} x^{2}+8\right ) \sqrt {-a^{2} x^{2}+1}}{625}+\frac {24 a^{5} x^{5}}{3125}+\frac {1088 a^{3} x^{3}}{16875}+\frac {16576 a x}{5625}-\frac {16 a^{3} x^{3} \arccos \left (a x \right )^{2}}{75}+\frac {32 \arccos \left (a x \right ) \left (a^{2} x^{2}+2\right ) \sqrt {-a^{2} x^{2}+1}}{225}-\frac {32 a x \arccos \left (a x \right )^{2}}{25}+\frac {64 \arccos \left (a x \right ) \sqrt {-a^{2} x^{2}+1}}{25}}{a^{5}}\)	\(197\)
default	\(\frac {\frac {a^{5} x^{5} \arccos \left (a x \right )^{4}}{5}-\frac {4 \arccos \left (a x \right )^{3} \left (3 a^{4} x^{4}+4 a^{2} x^{2}+8\right ) \sqrt {-a^{2} x^{2}+1}}{75}-\frac {12 a^{5} x^{5} \arccos \left (a x \right )^{2}}{125}+\frac {8 \arccos \left (a x \right ) \left (3 a^{4} x^{4}+4 a^{2} x^{2}+8\right ) \sqrt {-a^{2} x^{2}+1}}{625}+\frac {24 a^{5} x^{5}}{3125}+\frac {1088 a^{3} x^{3}}{16875}+\frac {16576 a x}{5625}-\frac {16 a^{3} x^{3} \arccos \left (a x \right )^{2}}{75}+\frac {32 \arccos \left (a x \right ) \left (a^{2} x^{2}+2\right ) \sqrt {-a^{2} x^{2}+1}}{225}-\frac {32 a x \arccos \left (a x \right )^{2}}{25}+\frac {64 \arccos \left (a x \right ) \sqrt {-a^{2} x^{2}+1}}{25}}{a^{5}}\)	\(197\)
orering	\(\frac {\left (170181 a^{8} x^{8}+190880 a^{6} x^{6}+9375680 a^{4} x^{4}-37873920 a^{2} x^{2}[phone]\right ) \arccos \left (a x \right )^{4}}{253125 a^{8} x^{3}}-\frac {2 \left (26730 a^{8} x^{8}+61339 a^{6} x^{6}+3095500 a^{4} x^{4}-11225760 a^{2} x^{2}+8391600\right ) \left (4 x^{3} \arccos \left (a x \right )^{4}-\frac {4 x^{4} \arccos \left (a x \right )^{3} a}{\sqrt {-a^{2} x^{2}+1}}\right )}{253125 a^{8} x^{6}}+\frac {\left (10125 a^{8} x^{8}+35798 a^{6} x^{6}+1866400 a^{4} x^{4}-5919360 a^{2} x^{2}+4102560\right ) \left (12 x^{2} \arccos \left (a x \right )^{4}-\frac {32 x^{3} \arccos \left (a x \right )^{3} a}{\sqrt {-a^{2} x^{2}+1}}+\frac {12 x^{4} \arccos \left (a x \right )^{2} a^{2}}{-a^{2} x^{2}+1}-\frac {4 x^{5} \arccos \left (a x \right )^{3} a^{3}}{\left (-a^{2} x^{2}+1\right )^{\frac {3}{2}}}\right )}{253125 a^{8} x^{5}}-\frac {\left (a x -1\right ) \left (a x +1\right ) \left (1215 a^{6} x^{6}+7139 a^{4} x^{4}+328960 a^{2} x^{2}-528360\right ) \left (24 x \arccos \left (a x \right )^{4}-\frac {144 x^{2} \arccos \left (a x \right )^{3} a}{\sqrt {-a^{2} x^{2}+1}}+\frac {144 x^{3} \arccos \left (a x \right )^{2} a^{2}}{-a^{2} x^{2}+1}-\frac {52 x^{4} \arccos \left (a x \right )^{3} a^{3}}{\left (-a^{2} x^{2}+1\right )^{\frac {3}{2}}}-\frac {24 x^{4} \arccos \left (a x \right ) a^{3}}{\left (-a^{2} x^{2}+1\right )^{\frac {3}{2}}}+\frac {36 x^{5} \arccos \left (a x \right )^{2} a^{4}}{\left (-a^{2} x^{2}+1\right )^{2}}-\frac {12 x^{6} \arccos \left (a x \right )^{3} a^{5}}{\left (-a^{2} x^{2}+1\right )^{\frac {5}{2}}}\right )}{253125 x^{4} a^{8}}+\frac {\left (81 a^{4} x^{4}+680 a^{2} x^{2}+31080\right ) \left (a x -1\right )^{2} \left (a x +1\right )^{2} \left (24 \arccos \left (a x \right )^{4}-\frac {384 x \arccos \left (a x \right )^{3} a}{\sqrt {-a^{2} x^{2}+1}}+\frac {864 x^{2} \arccos \left (a x \right )^{2} a^{2}}{-a^{2} x^{2}+1}-\frac {352 x^{3} \arccos \left (a x \right )^{3} a^{3}}{\left (-a^{2} x^{2}+1\right )^{\frac {3}{2}}}-\frac {384 x^{3} \arccos \left (a x \right ) a^{3}}{\left (-a^{2} x^{2}+1\right )^{\frac {3}{2}}}+\frac {624 x^{4} \arccos \left (a x \right )^{2} a^{4}}{\left (-a^{2} x^{2}+1\right )^{2}}-\frac {228 x^{5} \arccos \left (a x \right )^{3} a^{5}}{\left (-a^{2} x^{2}+1\right )^{\frac {5}{2}}}+\frac {24 x^{4} a^{4}}{\left (-a^{2} x^{2}+1\right )^{2}}-\frac {144 x^{5} \arccos \left (a x \right ) a^{5}}{\left (-a^{2} x^{2}+1\right )^{\frac {5}{2}}}+\frac {180 x^{6} \arccos \left (a x \right )^{2} a^{6}}{\left (-a^{2} x^{2}+1\right )^{3}}-\frac {60 x^{7} \arccos \left (a x \right )^{3} a^{7}}{\left (-a^{2} x^{2}+1\right )^{\frac {7}{2}}}\right )}{253125 x^{3} a^{8}}\)	\(759\)

Input:

int(x^4*arccos(a*x)^4,x,method=_RETURNVERBOSE)
 

Output:

1/a^5*(1/5*a^5*x^5*arccos(a*x)^4-4/75*arccos(a*x)^3*(3*a^4*x^4+4*a^2*x^2+8 
)*(-a^2*x^2+1)^(1/2)-12/125*a^5*x^5*arccos(a*x)^2+8/625*arccos(a*x)*(3*a^4 
*x^4+4*a^2*x^2+8)*(-a^2*x^2+1)^(1/2)+24/3125*a^5*x^5+1088/16875*a^3*x^3+16 
576/5625*a*x-16/75*a^3*x^3*arccos(a*x)^2+32/225*arccos(a*x)*(a^2*x^2+2)*(- 
a^2*x^2+1)^(1/2)-32/25*a*x*arccos(a*x)^2+64/25*arccos(a*x)*(-a^2*x^2+1)^(1 
/2))
 

Fricas [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 134, normalized size of antiderivative = 0.54 \[ \int x^4 \arccos (a x)^4 \, dx=\frac {16875 \, a^{5} x^{5} \arccos \left (a x\right )^{4} + 648 \, a^{5} x^{5} + 5440 \, a^{3} x^{3} - 900 \, {\left (9 \, a^{5} x^{5} + 20 \, a^{3} x^{3} + 120 \, a x\right )} \arccos \left (a x\right )^{2} + 248640 \, a x - 60 \, \sqrt {-a^{2} x^{2} + 1} {\left (75 \, {\left (3 \, a^{4} x^{4} + 4 \, a^{2} x^{2} + 8\right )} \arccos \left (a x\right )^{3} - 2 \, {\left (27 \, a^{4} x^{4} + 136 \, a^{2} x^{2} + 2072\right )} \arccos \left (a x\right )\right )}}{84375 \, a^{5}} \] Input:

integrate(x^4*arccos(a*x)^4,x, algorithm="fricas")
 

Output:

1/84375*(16875*a^5*x^5*arccos(a*x)^4 + 648*a^5*x^5 + 5440*a^3*x^3 - 900*(9 
*a^5*x^5 + 20*a^3*x^3 + 120*a*x)*arccos(a*x)^2 + 248640*a*x - 60*sqrt(-a^2 
*x^2 + 1)*(75*(3*a^4*x^4 + 4*a^2*x^2 + 8)*arccos(a*x)^3 - 2*(27*a^4*x^4 + 
136*a^2*x^2 + 2072)*arccos(a*x)))/a^5
 

Sympy [A] (verification not implemented)

Time = 0.79 (sec) , antiderivative size = 248, normalized size of antiderivative = 0.99 \[ \int x^4 \arccos (a x)^4 \, dx=\begin {cases} \frac {x^{5} \operatorname {acos}^{4}{\left (a x \right )}}{5} - \frac {12 x^{5} \operatorname {acos}^{2}{\left (a x \right )}}{125} + \frac {24 x^{5}}{3125} - \frac {4 x^{4} \sqrt {- a^{2} x^{2} + 1} \operatorname {acos}^{3}{\left (a x \right )}}{25 a} + \frac {24 x^{4} \sqrt {- a^{2} x^{2} + 1} \operatorname {acos}{\left (a x \right )}}{625 a} - \frac {16 x^{3} \operatorname {acos}^{2}{\left (a x \right )}}{75 a^{2}} + \frac {1088 x^{3}}{16875 a^{2}} - \frac {16 x^{2} \sqrt {- a^{2} x^{2} + 1} \operatorname {acos}^{3}{\left (a x \right )}}{75 a^{3}} + \frac {1088 x^{2} \sqrt {- a^{2} x^{2} + 1} \operatorname {acos}{\left (a x \right )}}{5625 a^{3}} - \frac {32 x \operatorname {acos}^{2}{\left (a x \right )}}{25 a^{4}} + \frac {16576 x}{5625 a^{4}} - \frac {32 \sqrt {- a^{2} x^{2} + 1} \operatorname {acos}^{3}{\left (a x \right )}}{75 a^{5}} + \frac {16576 \sqrt {- a^{2} x^{2} + 1} \operatorname {acos}{\left (a x \right )}}{5625 a^{5}} & \text {for}\: a \neq 0 \\\frac {\pi ^{4} x^{5}}{80} & \text {otherwise} \end {cases} \] Input:

integrate(x**4*acos(a*x)**4,x)
 

Output:

Piecewise((x**5*acos(a*x)**4/5 - 12*x**5*acos(a*x)**2/125 + 24*x**5/3125 - 
 4*x**4*sqrt(-a**2*x**2 + 1)*acos(a*x)**3/(25*a) + 24*x**4*sqrt(-a**2*x**2 
 + 1)*acos(a*x)/(625*a) - 16*x**3*acos(a*x)**2/(75*a**2) + 1088*x**3/(1687 
5*a**2) - 16*x**2*sqrt(-a**2*x**2 + 1)*acos(a*x)**3/(75*a**3) + 1088*x**2* 
sqrt(-a**2*x**2 + 1)*acos(a*x)/(5625*a**3) - 32*x*acos(a*x)**2/(25*a**4) + 
 16576*x/(5625*a**4) - 32*sqrt(-a**2*x**2 + 1)*acos(a*x)**3/(75*a**5) + 16 
576*sqrt(-a**2*x**2 + 1)*acos(a*x)/(5625*a**5), Ne(a, 0)), (pi**4*x**5/80, 
 True))
 

Maxima [A] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 206, normalized size of antiderivative = 0.82 \[ \int x^4 \arccos (a x)^4 \, dx=\frac {1}{5} \, x^{5} \arccos \left (a x\right )^{4} - \frac {4}{75} \, {\left (\frac {3 \, \sqrt {-a^{2} x^{2} + 1} x^{4}}{a^{2}} + \frac {4 \, \sqrt {-a^{2} x^{2} + 1} x^{2}}{a^{4}} + \frac {8 \, \sqrt {-a^{2} x^{2} + 1}}{a^{6}}\right )} a \arccos \left (a x\right )^{3} + \frac {4}{84375} \, {\left (2 \, a {\left (\frac {15 \, {\left (27 \, \sqrt {-a^{2} x^{2} + 1} a^{2} x^{4} + 136 \, \sqrt {-a^{2} x^{2} + 1} x^{2} + \frac {2072 \, \sqrt {-a^{2} x^{2} + 1}}{a^{2}}\right )} \arccos \left (a x\right )}{a^{5}} + \frac {81 \, a^{4} x^{5} + 680 \, a^{2} x^{3} + 31080 \, x}{a^{6}}\right )} - \frac {225 \, {\left (9 \, a^{4} x^{5} + 20 \, a^{2} x^{3} + 120 \, x\right )} \arccos \left (a x\right )^{2}}{a^{5}}\right )} a \] Input:

integrate(x^4*arccos(a*x)^4,x, algorithm="maxima")
 

Output:

1/5*x^5*arccos(a*x)^4 - 4/75*(3*sqrt(-a^2*x^2 + 1)*x^4/a^2 + 4*sqrt(-a^2*x 
^2 + 1)*x^2/a^4 + 8*sqrt(-a^2*x^2 + 1)/a^6)*a*arccos(a*x)^3 + 4/84375*(2*a 
*(15*(27*sqrt(-a^2*x^2 + 1)*a^2*x^4 + 136*sqrt(-a^2*x^2 + 1)*x^2 + 2072*sq 
rt(-a^2*x^2 + 1)/a^2)*arccos(a*x)/a^5 + (81*a^4*x^5 + 680*a^2*x^3 + 31080* 
x)/a^6) - 225*(9*a^4*x^5 + 20*a^2*x^3 + 120*x)*arccos(a*x)^2/a^5)*a
 

Giac [A] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 212, normalized size of antiderivative = 0.85 \[ \int x^4 \arccos (a x)^4 \, dx=\frac {1}{5} \, x^{5} \arccos \left (a x\right )^{4} - \frac {12}{125} \, x^{5} \arccos \left (a x\right )^{2} - \frac {4 \, \sqrt {-a^{2} x^{2} + 1} x^{4} \arccos \left (a x\right )^{3}}{25 \, a} + \frac {24}{3125} \, x^{5} + \frac {24 \, \sqrt {-a^{2} x^{2} + 1} x^{4} \arccos \left (a x\right )}{625 \, a} - \frac {16 \, x^{3} \arccos \left (a x\right )^{2}}{75 \, a^{2}} - \frac {16 \, \sqrt {-a^{2} x^{2} + 1} x^{2} \arccos \left (a x\right )^{3}}{75 \, a^{3}} + \frac {1088 \, x^{3}}{16875 \, a^{2}} + \frac {1088 \, \sqrt {-a^{2} x^{2} + 1} x^{2} \arccos \left (a x\right )}{5625 \, a^{3}} - \frac {32 \, x \arccos \left (a x\right )^{2}}{25 \, a^{4}} - \frac {32 \, \sqrt {-a^{2} x^{2} + 1} \arccos \left (a x\right )^{3}}{75 \, a^{5}} + \frac {16576 \, x}{5625 \, a^{4}} + \frac {16576 \, \sqrt {-a^{2} x^{2} + 1} \arccos \left (a x\right )}{5625 \, a^{5}} \] Input:

integrate(x^4*arccos(a*x)^4,x, algorithm="giac")
 

Output:

1/5*x^5*arccos(a*x)^4 - 12/125*x^5*arccos(a*x)^2 - 4/25*sqrt(-a^2*x^2 + 1) 
*x^4*arccos(a*x)^3/a + 24/3125*x^5 + 24/625*sqrt(-a^2*x^2 + 1)*x^4*arccos( 
a*x)/a - 16/75*x^3*arccos(a*x)^2/a^2 - 16/75*sqrt(-a^2*x^2 + 1)*x^2*arccos 
(a*x)^3/a^3 + 1088/16875*x^3/a^2 + 1088/5625*sqrt(-a^2*x^2 + 1)*x^2*arccos 
(a*x)/a^3 - 32/25*x*arccos(a*x)^2/a^4 - 32/75*sqrt(-a^2*x^2 + 1)*arccos(a* 
x)^3/a^5 + 16576/5625*x/a^4 + 16576/5625*sqrt(-a^2*x^2 + 1)*arccos(a*x)/a^ 
5
 

Mupad [F(-1)]

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

int(x^4*acos(a*x)^4,x)
 

Output:

int(x^4*acos(a*x)^4, x)
 

Reduce [F]

\[ \int x^4 \arccos (a x)^4 \, dx=\int \mathit {acos} \left (a x \right )^{4} x^{4}d x \] Input:

int(x^4*acos(a*x)^4,x)
 

Output:

int(acos(a*x)**4*x**4,x)