\(\int x^5 \arccos (a x)^4 \, dx\) [32]

Optimal result

Integrand size = 10, antiderivative size = 282 \[ \int x^5 \arccos (a x)^4 \, dx=\frac {245 x^2}{1152 a^4}+\frac {65 x^4}{3456 a^2}+\frac {x^6}{324}+\frac {245 x \sqrt {1-a^2 x^2} \arccos (a x)}{576 a^5}+\frac {65 x^3 \sqrt {1-a^2 x^2} \arccos (a x)}{864 a^3}+\frac {x^5 \sqrt {1-a^2 x^2} \arccos (a x)}{54 a}+\frac {245 \arccos (a x)^2}{1152 a^6}-\frac {5 x^2 \arccos (a x)^2}{16 a^4}-\frac {5 x^4 \arccos (a x)^2}{48 a^2}-\frac {1}{18} x^6 \arccos (a x)^2-\frac {5 x \sqrt {1-a^2 x^2} \arccos (a x)^3}{24 a^5}-\frac {5 x^3 \sqrt {1-a^2 x^2} \arccos (a x)^3}{36 a^3}-\frac {x^5 \sqrt {1-a^2 x^2} \arccos (a x)^3}{9 a}-\frac {5 \arccos (a x)^4}{96 a^6}+\frac {1}{6} x^6 \arccos (a x)^4 \] Output:

245/1152*x^2/a^4+65/3456*x^4/a^2+1/324*x^6+245/576*x*(-a^2*x^2+1)^(1/2)*ar 
ccos(a*x)/a^5+65/864*x^3*(-a^2*x^2+1)^(1/2)*arccos(a*x)/a^3+1/54*x^5*(-a^2 
*x^2+1)^(1/2)*arccos(a*x)/a+245/1152*arccos(a*x)^2/a^6-5/16*x^2*arccos(a*x 
)^2/a^4-5/48*x^4*arccos(a*x)^2/a^2-1/18*x^6*arccos(a*x)^2-5/24*x*(-a^2*x^2 
+1)^(1/2)*arccos(a*x)^3/a^5-5/36*x^3*(-a^2*x^2+1)^(1/2)*arccos(a*x)^3/a^3- 
1/9*x^5*(-a^2*x^2+1)^(1/2)*arccos(a*x)^3/a-5/96*arccos(a*x)^4/a^6+1/6*x^6* 
arccos(a*x)^4
 

Mathematica [A] (verified)

Time = 0.06 (sec) , antiderivative size = 167, normalized size of antiderivative = 0.59 \[ \int x^5 \arccos (a x)^4 \, dx=\frac {a^2 x^2 \left (2205+195 a^2 x^2+32 a^4 x^4\right )+6 a x \sqrt {1-a^2 x^2} \left (735+130 a^2 x^2+32 a^4 x^4\right ) \arccos (a x)-9 \left (-245+360 a^2 x^2+120 a^4 x^4+64 a^6 x^6\right ) \arccos (a x)^2-144 a x \sqrt {1-a^2 x^2} \left (15+10 a^2 x^2+8 a^4 x^4\right ) \arccos (a x)^3+108 \left (-5+16 a^6 x^6\right ) \arccos (a x)^4}{10368 a^6} \] Input:

Integrate[x^5*ArcCos[a*x]^4,x]
 

Output:

(a^2*x^2*(2205 + 195*a^2*x^2 + 32*a^4*x^4) + 6*a*x*Sqrt[1 - a^2*x^2]*(735 
+ 130*a^2*x^2 + 32*a^4*x^4)*ArcCos[a*x] - 9*(-245 + 360*a^2*x^2 + 120*a^4* 
x^4 + 64*a^6*x^6)*ArcCos[a*x]^2 - 144*a*x*Sqrt[1 - a^2*x^2]*(15 + 10*a^2*x 
^2 + 8*a^4*x^4)*ArcCos[a*x]^3 + 108*(-5 + 16*a^6*x^6)*ArcCos[a*x]^4)/(1036 
8*a^6)
 

Rubi [A] (verified)

Time = 2.62 (sec) , antiderivative size = 500, normalized size of antiderivative = 1.77, number of steps used = 13, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.300, Rules used = {5139, 5211, 5139, 5211, 15, 5139, 5211, 15, 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.

\(\displaystyle \int x^5 \arccos (a x)^4 \, dx\)

\(\Big \downarrow \) 5139

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

\(\Big \downarrow \) 5211

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

\(\Big \downarrow \) 5139

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

\(\Big \downarrow \) 5211

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

\(\Big \downarrow \) 15

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

\(\Big \downarrow \) 5139

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

\(\Big \downarrow \) 5211

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

\(\Big \downarrow \) 15

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

\(\Big \downarrow \) 5139

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

\(\Big \downarrow \) 5153

\(\displaystyle \frac {2}{3} a \left (-\frac {\frac {1}{3} a \left (\frac {5 \left (\frac {3 \int \frac {x^2 \arccos (a x)}{\sqrt {1-a^2 x^2}}dx}{4 a^2}-\frac {x^3 \sqrt {1-a^2 x^2} \arccos (a x)}{4 a^2}-\frac {x^4}{16 a}\right )}{6 a^2}-\frac {x^5 \sqrt {1-a^2 x^2} \arccos (a x)}{6 a^2}-\frac {x^6}{36 a}\right )+\frac {1}{6} x^6 \arccos (a x)^2}{2 a}+\frac {5 \left (-\frac {3 \left (\frac {1}{2} a \left (\frac {3 \int \frac {x^2 \arccos (a x)}{\sqrt {1-a^2 x^2}}dx}{4 a^2}-\frac {x^3 \sqrt {1-a^2 x^2} \arccos (a x)}{4 a^2}-\frac {x^4}{16 a}\right )+\frac {1}{4} x^4 \arccos (a x)^2\right )}{4 a}+\frac {3 \left (-\frac {3 \left (a \int \frac {x^2 \arccos (a x)}{\sqrt {1-a^2 x^2}}dx+\frac {1}{2} x^2 \arccos (a x)^2\right )}{2 a}-\frac {\arccos (a x)^4}{8 a^3}-\frac {x \sqrt {1-a^2 x^2} \arccos (a x)^3}{2 a^2}\right )}{4 a^2}-\frac {x^3 \sqrt {1-a^2 x^2} \arccos (a x)^3}{4 a^2}\right )}{6 a^2}-\frac {x^5 \sqrt {1-a^2 x^2} \arccos (a x)^3}{6 a^2}\right )+\frac {1}{6} x^6 \arccos (a x)^4\)

\(\Big \downarrow \) 5211

\(\displaystyle \frac {2}{3} a \left (-\frac {\frac {1}{3} a \left (\frac {5 \left (\frac {3 \left (\frac {\int \frac {\arccos (a x)}{\sqrt {1-a^2 x^2}}dx}{2 a^2}-\frac {\int xdx}{2 a}-\frac {x \sqrt {1-a^2 x^2} \arccos (a x)}{2 a^2}\right )}{4 a^2}-\frac {x^3 \sqrt {1-a^2 x^2} \arccos (a x)}{4 a^2}-\frac {x^4}{16 a}\right )}{6 a^2}-\frac {x^5 \sqrt {1-a^2 x^2} \arccos (a x)}{6 a^2}-\frac {x^6}{36 a}\right )+\frac {1}{6} x^6 \arccos (a x)^2}{2 a}+\frac {5 \left (-\frac {3 \left (\frac {1}{2} a \left (\frac {3 \left (\frac {\int \frac {\arccos (a x)}{\sqrt {1-a^2 x^2}}dx}{2 a^2}-\frac {\int xdx}{2 a}-\frac {x \sqrt {1-a^2 x^2} \arccos (a x)}{2 a^2}\right )}{4 a^2}-\frac {x^3 \sqrt {1-a^2 x^2} \arccos (a x)}{4 a^2}-\frac {x^4}{16 a}\right )+\frac {1}{4} x^4 \arccos (a x)^2\right )}{4 a}+\frac {3 \left (-\frac {3 \left (a \left (\frac {\int \frac {\arccos (a x)}{\sqrt {1-a^2 x^2}}dx}{2 a^2}-\frac {\int xdx}{2 a}-\frac {x \sqrt {1-a^2 x^2} \arccos (a x)}{2 a^2}\right )+\frac {1}{2} x^2 \arccos (a x)^2\right )}{2 a}-\frac {\arccos (a x)^4}{8 a^3}-\frac {x \sqrt {1-a^2 x^2} \arccos (a x)^3}{2 a^2}\right )}{4 a^2}-\frac {x^3 \sqrt {1-a^2 x^2} \arccos (a x)^3}{4 a^2}\right )}{6 a^2}-\frac {x^5 \sqrt {1-a^2 x^2} \arccos (a x)^3}{6 a^2}\right )+\frac {1}{6} x^6 \arccos (a x)^4\)

\(\Big \downarrow \) 15

\(\displaystyle \frac {2}{3} a \left (-\frac {\frac {1}{3} a \left (\frac {5 \left (\frac {3 \left (\frac {\int \frac {\arccos (a x)}{\sqrt {1-a^2 x^2}}dx}{2 a^2}-\frac {x \sqrt {1-a^2 x^2} \arccos (a x)}{2 a^2}-\frac {x^2}{4 a}\right )}{4 a^2}-\frac {x^3 \sqrt {1-a^2 x^2} \arccos (a x)}{4 a^2}-\frac {x^4}{16 a}\right )}{6 a^2}-\frac {x^5 \sqrt {1-a^2 x^2} \arccos (a x)}{6 a^2}-\frac {x^6}{36 a}\right )+\frac {1}{6} x^6 \arccos (a x)^2}{2 a}+\frac {5 \left (-\frac {3 \left (\frac {1}{2} a \left (\frac {3 \left (\frac {\int \frac {\arccos (a x)}{\sqrt {1-a^2 x^2}}dx}{2 a^2}-\frac {x \sqrt {1-a^2 x^2} \arccos (a x)}{2 a^2}-\frac {x^2}{4 a}\right )}{4 a^2}-\frac {x^3 \sqrt {1-a^2 x^2} \arccos (a x)}{4 a^2}-\frac {x^4}{16 a}\right )+\frac {1}{4} x^4 \arccos (a x)^2\right )}{4 a}+\frac {3 \left (-\frac {3 \left (a \left (\frac {\int \frac {\arccos (a x)}{\sqrt {1-a^2 x^2}}dx}{2 a^2}-\frac {x \sqrt {1-a^2 x^2} \arccos (a x)}{2 a^2}-\frac {x^2}{4 a}\right )+\frac {1}{2} x^2 \arccos (a x)^2\right )}{2 a}-\frac {\arccos (a x)^4}{8 a^3}-\frac {x \sqrt {1-a^2 x^2} \arccos (a x)^3}{2 a^2}\right )}{4 a^2}-\frac {x^3 \sqrt {1-a^2 x^2} \arccos (a x)^3}{4 a^2}\right )}{6 a^2}-\frac {x^5 \sqrt {1-a^2 x^2} \arccos (a x)^3}{6 a^2}\right )+\frac {1}{6} x^6 \arccos (a x)^4\)

\(\Big \downarrow \) 5153

\(\displaystyle \frac {2}{3} a \left (-\frac {x^5 \sqrt {1-a^2 x^2} \arccos (a x)^3}{6 a^2}+\frac {5 \left (-\frac {x^3 \sqrt {1-a^2 x^2} \arccos (a x)^3}{4 a^2}+\frac {3 \left (-\frac {\arccos (a x)^4}{8 a^3}-\frac {x \sqrt {1-a^2 x^2} \arccos (a x)^3}{2 a^2}-\frac {3 \left (a \left (-\frac {\arccos (a x)^2}{4 a^3}-\frac {x \sqrt {1-a^2 x^2} \arccos (a x)}{2 a^2}-\frac {x^2}{4 a}\right )+\frac {1}{2} x^2 \arccos (a x)^2\right )}{2 a}\right )}{4 a^2}-\frac {3 \left (\frac {1}{2} a \left (-\frac {x^3 \sqrt {1-a^2 x^2} \arccos (a x)}{4 a^2}+\frac {3 \left (-\frac {\arccos (a x)^2}{4 a^3}-\frac {x \sqrt {1-a^2 x^2} \arccos (a x)}{2 a^2}-\frac {x^2}{4 a}\right )}{4 a^2}-\frac {x^4}{16 a}\right )+\frac {1}{4} x^4 \arccos (a x)^2\right )}{4 a}\right )}{6 a^2}-\frac {\frac {1}{3} a \left (-\frac {x^5 \sqrt {1-a^2 x^2} \arccos (a x)}{6 a^2}+\frac {5 \left (-\frac {x^3 \sqrt {1-a^2 x^2} \arccos (a x)}{4 a^2}+\frac {3 \left (-\frac {\arccos (a x)^2}{4 a^3}-\frac {x \sqrt {1-a^2 x^2} \arccos (a x)}{2 a^2}-\frac {x^2}{4 a}\right )}{4 a^2}-\frac {x^4}{16 a}\right )}{6 a^2}-\frac {x^6}{36 a}\right )+\frac {1}{6} x^6 \arccos (a x)^2}{2 a}\right )+\frac {1}{6} x^6 \arccos (a x)^4\)

Input:

Int[x^5*ArcCos[a*x]^4,x]
 

Output:

(x^6*ArcCos[a*x]^4)/6 + (2*a*(-1/6*(x^5*Sqrt[1 - a^2*x^2]*ArcCos[a*x]^3)/a 
^2 - ((x^6*ArcCos[a*x]^2)/6 + (a*(-1/36*x^6/a - (x^5*Sqrt[1 - a^2*x^2]*Arc 
Cos[a*x])/(6*a^2) + (5*(-1/16*x^4/a - (x^3*Sqrt[1 - a^2*x^2]*ArcCos[a*x])/ 
(4*a^2) + (3*(-1/4*x^2/a - (x*Sqrt[1 - a^2*x^2]*ArcCos[a*x])/(2*a^2) - Arc 
Cos[a*x]^2/(4*a^3)))/(4*a^2)))/(6*a^2)))/3)/(2*a) + (5*(-1/4*(x^3*Sqrt[1 - 
 a^2*x^2]*ArcCos[a*x]^3)/a^2 - (3*((x^4*ArcCos[a*x]^2)/4 + (a*(-1/16*x^4/a 
 - (x^3*Sqrt[1 - a^2*x^2]*ArcCos[a*x])/(4*a^2) + (3*(-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)))/(4*a^2)))/2))/ 
(4*a) + (3*(-1/2*(x*Sqrt[1 - a^2*x^2]*ArcCos[a*x]^3)/a^2 - ArcCos[a*x]^4/( 
8*a^3) - (3*((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*a)))/(4*a^2)))/(6*a^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 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 = 0.63 (sec) , antiderivative size = 332, normalized size of antiderivative = 1.18

method	result	size
derivativedivides	\(\frac {\frac {a^{6} x^{6} \arccos \left (a x \right )^{4}}{6}-\frac {\arccos \left (a x \right )^{3} \left (8 \sqrt {-a^{2} x^{2}+1}\, a^{5} x^{5}+10 a^{3} x^{3} \sqrt {-a^{2} x^{2}+1}+15 \sqrt {-a^{2} x^{2}+1}\, a x +15 \arccos \left (a x \right )\right )}{72}-\frac {\arccos \left (a x \right )^{2} a^{6} x^{6}}{18}+\frac {\arccos \left (a x \right ) \left (8 \sqrt {-a^{2} x^{2}+1}\, a^{5} x^{5}+10 a^{3} x^{3} \sqrt {-a^{2} x^{2}+1}+15 \sqrt {-a^{2} x^{2}+1}\, a x +15 \arccos \left (a x \right )\right )}{432}-\frac {245 \arccos \left (a x \right )^{2}}{1152}+\frac {a^{6} x^{6}}{324}+\frac {5 a^{4} x^{4}}{864}+\frac {25 a^{2} x^{2}}{144}-\frac {5 a^{4} x^{4} \arccos \left (a x \right )^{2}}{48}+\frac {5 \arccos \left (a x \right ) \left (2 a^{3} x^{3} \sqrt {-a^{2} x^{2}+1}+3 \sqrt {-a^{2} x^{2}+1}\, a x +3 \arccos \left (a x \right )\right )}{192}+\frac {5 \left (2 a^{2} x^{2}+3\right )^{2}}{1536}-\frac {5 a^{2} x^{2} \arccos \left (a x \right )^{2}}{16}+\frac {5 \arccos \left (a x \right ) \left (\sqrt {-a^{2} x^{2}+1}\, a x +\arccos \left (a x \right )\right )}{16}-\frac {5}{32}+\frac {5 \arccos \left (a x \right )^{4}}{32}}{a^{6}}\)	\(332\)
default	\(\frac {\frac {a^{6} x^{6} \arccos \left (a x \right )^{4}}{6}-\frac {\arccos \left (a x \right )^{3} \left (8 \sqrt {-a^{2} x^{2}+1}\, a^{5} x^{5}+10 a^{3} x^{3} \sqrt {-a^{2} x^{2}+1}+15 \sqrt {-a^{2} x^{2}+1}\, a x +15 \arccos \left (a x \right )\right )}{72}-\frac {\arccos \left (a x \right )^{2} a^{6} x^{6}}{18}+\frac {\arccos \left (a x \right ) \left (8 \sqrt {-a^{2} x^{2}+1}\, a^{5} x^{5}+10 a^{3} x^{3} \sqrt {-a^{2} x^{2}+1}+15 \sqrt {-a^{2} x^{2}+1}\, a x +15 \arccos \left (a x \right )\right )}{432}-\frac {245 \arccos \left (a x \right )^{2}}{1152}+\frac {a^{6} x^{6}}{324}+\frac {5 a^{4} x^{4}}{864}+\frac {25 a^{2} x^{2}}{144}-\frac {5 a^{4} x^{4} \arccos \left (a x \right )^{2}}{48}+\frac {5 \arccos \left (a x \right ) \left (2 a^{3} x^{3} \sqrt {-a^{2} x^{2}+1}+3 \sqrt {-a^{2} x^{2}+1}\, a x +3 \arccos \left (a x \right )\right )}{192}+\frac {5 \left (2 a^{2} x^{2}+3\right )^{2}}{1536}-\frac {5 a^{2} x^{2} \arccos \left (a x \right )^{2}}{16}+\frac {5 \arccos \left (a x \right ) \left (\sqrt {-a^{2} x^{2}+1}\, a x +\arccos \left (a x \right )\right )}{16}-\frac {5}{32}+\frac {5 \arccos \left (a x \right )^{4}}{32}}{a^{6}}\)	\(332\)
orering	\(\frac {\left (148832 a^{8} x^{8}+112095 a^{6} x^{6}+1448055 a^{4} x^{4}-6263460 a^{2} x^{2}+4762800\right ) \arccos \left (a x \right )^{4}}{248832 a^{8} x^{2}}-\frac {\left (40800 a^{8} x^{8}+62431 a^{6} x^{6}+819975 a^{4} x^{4}-3151260 a^{2} x^{2}+2293200\right ) \left (5 x^{4} \arccos \left (a x \right )^{4}-\frac {4 x^{5} \arccos \left (a x \right )^{3} a}{\sqrt {-a^{2} x^{2}+1}}\right )}{248832 a^{8} x^{6}}+\frac {\left (6560 a^{8} x^{8}+15331 a^{6} x^{6}+205275 a^{4} x^{4}-687330 a^{2} x^{2}+467460\right ) \left (20 x^{3} \arccos \left (a x \right )^{4}-\frac {40 x^{4} \arccos \left (a x \right )^{3} a}{\sqrt {-a^{2} x^{2}+1}}+\frac {12 x^{5} \arccos \left (a x \right )^{2} a^{2}}{-a^{2} x^{2}+1}-\frac {4 x^{6} \arccos \left (a x \right )^{3} a^{3}}{\left (-a^{2} x^{2}+1\right )^{\frac {3}{2}}}\right )}{248832 a^{8} x^{5}}-\frac {\left (a x -1\right ) \left (a x +1\right ) \left (320 a^{6} x^{6}+1339 a^{4} x^{4}+15300 a^{2} x^{2}-24255\right ) \left (60 x^{2} \arccos \left (a x \right )^{4}-\frac {240 x^{3} \arccos \left (a x \right )^{3} a}{\sqrt {-a^{2} x^{2}+1}}+\frac {180 x^{4} \arccos \left (a x \right )^{2} a^{2}}{-a^{2} x^{2}+1}-\frac {64 x^{5} \arccos \left (a x \right )^{3} a^{3}}{\left (-a^{2} x^{2}+1\right )^{\frac {3}{2}}}-\frac {24 x^{5} \arccos \left (a x \right ) a^{3}}{\left (-a^{2} x^{2}+1\right )^{\frac {3}{2}}}+\frac {36 x^{6} \arccos \left (a x \right )^{2} a^{4}}{\left (-a^{2} x^{2}+1\right )^{2}}-\frac {12 x^{7} \arccos \left (a x \right )^{3} a^{5}}{\left (-a^{2} x^{2}+1\right )^{\frac {5}{2}}}\right )}{124416 x^{4} a^{8}}+\frac {\left (32 a^{4} x^{4}+195 a^{2} x^{2}+2205\right ) \left (a x -1\right )^{2} \left (a x +1\right )^{2} \left (120 x \arccos \left (a x \right )^{4}-\frac {960 x^{2} \arccos \left (a x \right )^{3} a}{\sqrt {-a^{2} x^{2}+1}}+\frac {1440 x^{3} \arccos \left (a x \right )^{2} a^{2}}{-a^{2} x^{2}+1}-\frac {560 x^{4} \arccos \left (a x \right )^{3} a^{3}}{\left (-a^{2} x^{2}+1\right )^{\frac {3}{2}}}-\frac {480 x^{4} \arccos \left (a x \right ) a^{3}}{\left (-a^{2} x^{2}+1\right )^{\frac {3}{2}}}+\frac {768 x^{5} \arccos \left (a x \right )^{2} a^{4}}{\left (-a^{2} x^{2}+1\right )^{2}}-\frac {276 x^{6} \arccos \left (a x \right )^{3} a^{5}}{\left (-a^{2} x^{2}+1\right )^{\frac {5}{2}}}+\frac {24 x^{5} a^{4}}{\left (-a^{2} x^{2}+1\right )^{2}}-\frac {144 x^{6} \arccos \left (a x \right ) a^{5}}{\left (-a^{2} x^{2}+1\right )^{\frac {5}{2}}}+\frac {180 x^{7} \arccos \left (a x \right )^{2} a^{6}}{\left (-a^{2} x^{2}+1\right )^{3}}-\frac {60 x^{8} \arccos \left (a x \right )^{3} a^{7}}{\left (-a^{2} x^{2}+1\right )^{\frac {7}{2}}}\right )}{248832 x^{3} a^{8}}\)	\(764\)

Input:

int(x^5*arccos(a*x)^4,x,method=_RETURNVERBOSE)
 

Output:

1/a^6*(1/6*a^6*x^6*arccos(a*x)^4-1/72*arccos(a*x)^3*(8*(-a^2*x^2+1)^(1/2)* 
a^5*x^5+10*a^3*x^3*(-a^2*x^2+1)^(1/2)+15*(-a^2*x^2+1)^(1/2)*a*x+15*arccos( 
a*x))-1/18*arccos(a*x)^2*a^6*x^6+1/432*arccos(a*x)*(8*(-a^2*x^2+1)^(1/2)*a 
^5*x^5+10*a^3*x^3*(-a^2*x^2+1)^(1/2)+15*(-a^2*x^2+1)^(1/2)*a*x+15*arccos(a 
*x))-245/1152*arccos(a*x)^2+1/324*a^6*x^6+5/864*a^4*x^4+25/144*a^2*x^2-5/4 
8*a^4*x^4*arccos(a*x)^2+5/192*arccos(a*x)*(2*a^3*x^3*(-a^2*x^2+1)^(1/2)+3* 
(-a^2*x^2+1)^(1/2)*a*x+3*arccos(a*x))+5/1536*(2*a^2*x^2+3)^2-5/16*a^2*x^2* 
arccos(a*x)^2+5/16*arccos(a*x)*((-a^2*x^2+1)^(1/2)*a*x+arccos(a*x))-5/32+5 
/32*arccos(a*x)^4)
 

Fricas [A] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 153, normalized size of antiderivative = 0.54 \[ \int x^5 \arccos (a x)^4 \, dx=\frac {32 \, a^{6} x^{6} + 195 \, a^{4} x^{4} + 108 \, {\left (16 \, a^{6} x^{6} - 5\right )} \arccos \left (a x\right )^{4} + 2205 \, a^{2} x^{2} - 9 \, {\left (64 \, a^{6} x^{6} + 120 \, a^{4} x^{4} + 360 \, a^{2} x^{2} - 245\right )} \arccos \left (a x\right )^{2} - 6 \, \sqrt {-a^{2} x^{2} + 1} {\left (24 \, {\left (8 \, a^{5} x^{5} + 10 \, a^{3} x^{3} + 15 \, a x\right )} \arccos \left (a x\right )^{3} - {\left (32 \, a^{5} x^{5} + 130 \, a^{3} x^{3} + 735 \, a x\right )} \arccos \left (a x\right )\right )}}{10368 \, a^{6}} \] Input:

integrate(x^5*arccos(a*x)^4,x, algorithm="fricas")
 

Output:

1/10368*(32*a^6*x^6 + 195*a^4*x^4 + 108*(16*a^6*x^6 - 5)*arccos(a*x)^4 + 2 
205*a^2*x^2 - 9*(64*a^6*x^6 + 120*a^4*x^4 + 360*a^2*x^2 - 245)*arccos(a*x) 
^2 - 6*sqrt(-a^2*x^2 + 1)*(24*(8*a^5*x^5 + 10*a^3*x^3 + 15*a*x)*arccos(a*x 
)^3 - (32*a^5*x^5 + 130*a^3*x^3 + 735*a*x)*arccos(a*x)))/a^6
 

Sympy [A] (verification not implemented)

Time = 1.08 (sec) , antiderivative size = 275, normalized size of antiderivative = 0.98 \[ \int x^5 \arccos (a x)^4 \, dx=\begin {cases} \frac {x^{6} \operatorname {acos}^{4}{\left (a x \right )}}{6} - \frac {x^{6} \operatorname {acos}^{2}{\left (a x \right )}}{18} + \frac {x^{6}}{324} - \frac {x^{5} \sqrt {- a^{2} x^{2} + 1} \operatorname {acos}^{3}{\left (a x \right )}}{9 a} + \frac {x^{5} \sqrt {- a^{2} x^{2} + 1} \operatorname {acos}{\left (a x \right )}}{54 a} - \frac {5 x^{4} \operatorname {acos}^{2}{\left (a x \right )}}{48 a^{2}} + \frac {65 x^{4}}{3456 a^{2}} - \frac {5 x^{3} \sqrt {- a^{2} x^{2} + 1} \operatorname {acos}^{3}{\left (a x \right )}}{36 a^{3}} + \frac {65 x^{3} \sqrt {- a^{2} x^{2} + 1} \operatorname {acos}{\left (a x \right )}}{864 a^{3}} - \frac {5 x^{2} \operatorname {acos}^{2}{\left (a x \right )}}{16 a^{4}} + \frac {245 x^{2}}{1152 a^{4}} - \frac {5 x \sqrt {- a^{2} x^{2} + 1} \operatorname {acos}^{3}{\left (a x \right )}}{24 a^{5}} + \frac {245 x \sqrt {- a^{2} x^{2} + 1} \operatorname {acos}{\left (a x \right )}}{576 a^{5}} - \frac {5 \operatorname {acos}^{4}{\left (a x \right )}}{96 a^{6}} + \frac {245 \operatorname {acos}^{2}{\left (a x \right )}}{1152 a^{6}} & \text {for}\: a \neq 0 \\\frac {\pi ^{4} x^{6}}{96} & \text {otherwise} \end {cases} \] Input:

integrate(x**5*acos(a*x)**4,x)
 

Output:

Piecewise((x**6*acos(a*x)**4/6 - x**6*acos(a*x)**2/18 + x**6/324 - x**5*sq 
rt(-a**2*x**2 + 1)*acos(a*x)**3/(9*a) + x**5*sqrt(-a**2*x**2 + 1)*acos(a*x 
)/(54*a) - 5*x**4*acos(a*x)**2/(48*a**2) + 65*x**4/(3456*a**2) - 5*x**3*sq 
rt(-a**2*x**2 + 1)*acos(a*x)**3/(36*a**3) + 65*x**3*sqrt(-a**2*x**2 + 1)*a 
cos(a*x)/(864*a**3) - 5*x**2*acos(a*x)**2/(16*a**4) + 245*x**2/(1152*a**4) 
 - 5*x*sqrt(-a**2*x**2 + 1)*acos(a*x)**3/(24*a**5) + 245*x*sqrt(-a**2*x**2 
 + 1)*acos(a*x)/(576*a**5) - 5*acos(a*x)**4/(96*a**6) + 245*acos(a*x)**2/( 
1152*a**6), Ne(a, 0)), (pi**4*x**6/96, True))
 

Maxima [F]

\[ \int x^5 \arccos (a x)^4 \, dx=\int { x^{5} \arccos \left (a x\right )^{4} \,d x } \] Input:

integrate(x^5*arccos(a*x)^4,x, algorithm="maxima")
 

Output:

1/6*x^6*arctan2(sqrt(a*x + 1)*sqrt(-a*x + 1), a*x)^4 - 2*a*integrate(1/3*s 
qrt(a*x + 1)*sqrt(-a*x + 1)*x^6*arctan2(sqrt(a*x + 1)*sqrt(-a*x + 1), a*x) 
^3/(a^2*x^2 - 1), x)
 

Giac [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 245, normalized size of antiderivative = 0.87 \[ \int x^5 \arccos (a x)^4 \, dx=\frac {1}{6} \, x^{6} \arccos \left (a x\right )^{4} - \frac {1}{18} \, x^{6} \arccos \left (a x\right )^{2} - \frac {\sqrt {-a^{2} x^{2} + 1} x^{5} \arccos \left (a x\right )^{3}}{9 \, a} + \frac {1}{324} \, x^{6} + \frac {\sqrt {-a^{2} x^{2} + 1} x^{5} \arccos \left (a x\right )}{54 \, a} - \frac {5 \, x^{4} \arccos \left (a x\right )^{2}}{48 \, a^{2}} - \frac {5 \, \sqrt {-a^{2} x^{2} + 1} x^{3} \arccos \left (a x\right )^{3}}{36 \, a^{3}} + \frac {65 \, x^{4}}{3456 \, a^{2}} + \frac {65 \, \sqrt {-a^{2} x^{2} + 1} x^{3} \arccos \left (a x\right )}{864 \, a^{3}} - \frac {5 \, x^{2} \arccos \left (a x\right )^{2}}{16 \, a^{4}} - \frac {5 \, \sqrt {-a^{2} x^{2} + 1} x \arccos \left (a x\right )^{3}}{24 \, a^{5}} + \frac {245 \, x^{2}}{1152 \, a^{4}} - \frac {5 \, \arccos \left (a x\right )^{4}}{96 \, a^{6}} + \frac {245 \, \sqrt {-a^{2} x^{2} + 1} x \arccos \left (a x\right )}{576 \, a^{5}} + \frac {245 \, \arccos \left (a x\right )^{2}}{1152 \, a^{6}} - \frac {9485}{82944 \, a^{6}} \] Input:

integrate(x^5*arccos(a*x)^4,x, algorithm="giac")
 

Output:

1/6*x^6*arccos(a*x)^4 - 1/18*x^6*arccos(a*x)^2 - 1/9*sqrt(-a^2*x^2 + 1)*x^ 
5*arccos(a*x)^3/a + 1/324*x^6 + 1/54*sqrt(-a^2*x^2 + 1)*x^5*arccos(a*x)/a 
- 5/48*x^4*arccos(a*x)^2/a^2 - 5/36*sqrt(-a^2*x^2 + 1)*x^3*arccos(a*x)^3/a 
^3 + 65/3456*x^4/a^2 + 65/864*sqrt(-a^2*x^2 + 1)*x^3*arccos(a*x)/a^3 - 5/1 
6*x^2*arccos(a*x)^2/a^4 - 5/24*sqrt(-a^2*x^2 + 1)*x*arccos(a*x)^3/a^5 + 24 
5/1152*x^2/a^4 - 5/96*arccos(a*x)^4/a^6 + 245/576*sqrt(-a^2*x^2 + 1)*x*arc 
cos(a*x)/a^5 + 245/1152*arccos(a*x)^2/a^6 - 9485/82944/a^6
 

Mupad [F(-1)]

Timed out. \[ \int x^5 \arccos (a x)^4 \, dx=\int x^5\,{\mathrm {acos}\left (a\,x\right )}^4 \,d x \] Input:

int(x^5*acos(a*x)^4,x)
 

Output:

int(x^5*acos(a*x)^4, x)
 

Reduce [F]

\[ \int x^5 \arccos (a x)^4 \, dx=\int \mathit {acos} \left (a x \right )^{4} x^{5}d x \] Input:

int(x^5*acos(a*x)^4,x)
 

Output:

int(acos(a*x)**4*x**5,x)