\(\int (c-a^2 c x^2)^{3/2} \arccos (a x)^3 \, dx\) [63]

Optimal result

Integrand size = 22, antiderivative size = 351 \[ \int \left (c-a^2 c x^2\right )^{3/2} \arccos (a x)^3 \, dx=-\frac {45 a c x^2 \sqrt {c-a^2 c x^2}}{128 \sqrt {1-a^2 x^2}}+\frac {3 c \left (1-a^2 x^2\right )^{3/2} \sqrt {c-a^2 c x^2}}{128 a}-\frac {45}{64} c x \sqrt {c-a^2 c x^2} \arccos (a x)-\frac {3}{32} x \left (c-a^2 c x^2\right )^{3/2} \arccos (a x)-\frac {27 c \sqrt {c-a^2 c x^2} \arccos (a x)^2}{128 a \sqrt {1-a^2 x^2}}+\frac {9 a c x^2 \sqrt {c-a^2 c x^2} \arccos (a x)^2}{16 \sqrt {1-a^2 x^2}}-\frac {3 c \left (1-a^2 x^2\right )^{3/2} \sqrt {c-a^2 c x^2} \arccos (a x)^2}{16 a}+\frac {3}{8} c x \sqrt {c-a^2 c x^2} \arccos (a x)^3+\frac {1}{4} x \left (c-a^2 c x^2\right )^{3/2} \arccos (a x)^3-\frac {3 c \sqrt {c-a^2 c x^2} \arccos (a x)^4}{32 a \sqrt {1-a^2 x^2}} \] Output:

-45/128*a*c*x^2*(-a^2*c*x^2+c)^(1/2)/(-a^2*x^2+1)^(1/2)+3/128*c*(-a^2*x^2+ 
1)^(3/2)*(-a^2*c*x^2+c)^(1/2)/a-45/64*c*x*(-a^2*c*x^2+c)^(1/2)*arccos(a*x) 
-3/32*x*(-a^2*c*x^2+c)^(3/2)*arccos(a*x)-27/128*c*(-a^2*c*x^2+c)^(1/2)*arc 
cos(a*x)^2/a/(-a^2*x^2+1)^(1/2)+9/16*a*c*x^2*(-a^2*c*x^2+c)^(1/2)*arccos(a 
*x)^2/(-a^2*x^2+1)^(1/2)-3/16*c*(-a^2*x^2+1)^(3/2)*(-a^2*c*x^2+c)^(1/2)*ar 
ccos(a*x)^2/a+3/8*c*x*(-a^2*c*x^2+c)^(1/2)*arccos(a*x)^3+1/4*x*(-a^2*c*x^2 
+c)^(3/2)*arccos(a*x)^3-3/32*c*(-a^2*c*x^2+c)^(1/2)*arccos(a*x)^4/a/(-a^2* 
x^2+1)^(1/2)
 

Mathematica [A] (verified)

Time = 0.30 (sec) , antiderivative size = 138, normalized size of antiderivative = 0.39 \[ \int \left (c-a^2 c x^2\right )^{3/2} \arccos (a x)^3 \, dx=-\frac {c \sqrt {c-a^2 c x^2} \left (96 \arccos (a x)^4-3 (-64 \cos (2 \arccos (a x))+\cos (4 \arccos (a x)))+24 \arccos (a x)^2 (-16 \cos (2 \arccos (a x))+\cos (4 \arccos (a x)))-12 \arccos (a x) (-32 \sin (2 \arccos (a x))+\sin (4 \arccos (a x)))+32 \arccos (a x)^3 (-8 \sin (2 \arccos (a x))+\sin (4 \arccos (a x)))\right )}{1024 a \sqrt {1-a^2 x^2}} \] Input:

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

Output:

-1/1024*(c*Sqrt[c - a^2*c*x^2]*(96*ArcCos[a*x]^4 - 3*(-64*Cos[2*ArcCos[a*x 
]] + Cos[4*ArcCos[a*x]]) + 24*ArcCos[a*x]^2*(-16*Cos[2*ArcCos[a*x]] + Cos[ 
4*ArcCos[a*x]]) - 12*ArcCos[a*x]*(-32*Sin[2*ArcCos[a*x]] + Sin[4*ArcCos[a* 
x]]) + 32*ArcCos[a*x]^3*(-8*Sin[2*ArcCos[a*x]] + Sin[4*ArcCos[a*x]])))/(a* 
Sqrt[1 - a^2*x^2])
 

Rubi [A] (verified)

Time = 2.46 (sec) , antiderivative size = 365, normalized size of antiderivative = 1.04, number of steps used = 14, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.636, Rules used = {5159, 5157, 5139, 5153, 5183, 5159, 244, 2009, 5157, 15, 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.

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

\(\Big \downarrow \) 5159

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

\(\Big \downarrow \) 5157

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

\(\Big \downarrow \) 5139

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

\(\Big \downarrow \) 5153

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

\(\Big \downarrow \) 5183

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

\(\Big \downarrow \) 5159

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

\(\Big \downarrow \) 244

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

\(\Big \downarrow \) 2009

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

\(\Big \downarrow \) 5157

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

\(\Big \downarrow \) 15

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

\(\Big \downarrow \) 5153

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

\(\Big \downarrow \) 5211

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

\(\Big \downarrow \) 15

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

\(\Big \downarrow \) 5153

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

Input:

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

Output:

(x*(c - a^2*c*x^2)^(3/2)*ArcCos[a*x]^3)/4 + (3*c*((x*Sqrt[c - a^2*c*x^2]*A 
rcCos[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])))/4 + (3*a*c*Sqrt[c - a^2*c*x^2]*(-1/4*((1 - a^2*x^2)^2*Arc 
Cos[a*x]^2)/a^2 - ((a*(x^2/2 - (a^2*x^4)/4))/4 + (x*(1 - a^2*x^2)^(3/2)*Ar 
cCos[a*x])/4 + (3*((a*x^2)/4 + (x*Sqrt[1 - a^2*x^2]*ArcCos[a*x])/2 - ArcCo 
s[a*x]^2/(4*a)))/4)/(2*a)))/(4*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 244

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Int[Expand 
Integrand[(c*x)^m*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, m}, x] && IGtQ[p 
, 0]
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

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 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]
 

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 = 0.80 (sec) , antiderivative size = 533, normalized size of antiderivative = 1.52

method	result	size
default	\(\frac {3 \sqrt {-c \left (a^{2} x^{2}-1\right )}\, \sqrt {-a^{2} x^{2}+1}\, \arccos \left (a x \right )^{4} c}{32 a \left (a^{2} x^{2}-1\right )}-\frac {\sqrt {-c \left (a^{2} x^{2}-1\right )}\, \left (8 a^{5} x^{5}-12 a^{3} x^{3}+8 i \sqrt {-a^{2} x^{2}+1}\, a^{4} x^{4}+4 a x -8 i \sqrt {-a^{2} x^{2}+1}\, a^{2} x^{2}+i \sqrt {-a^{2} x^{2}+1}\right ) \left (24 i \arccos \left (a x \right )^{2}+32 \arccos \left (a x \right )^{3}-3 i-12 \arccos \left (a x \right )\right ) c}{2048 a \left (a^{2} x^{2}-1\right )}+\frac {\sqrt {-c \left (a^{2} x^{2}-1\right )}\, \left (2 a^{3} x^{3}-2 a x +2 i \sqrt {-a^{2} x^{2}+1}\, a^{2} x^{2}-i \sqrt {-a^{2} x^{2}+1}\right ) \left (6 i \arccos \left (a x \right )^{2}+4 \arccos \left (a x \right )^{3}-3 i-6 \arccos \left (a x \right )\right ) c}{32 a \left (a^{2} x^{2}-1\right )}+\frac {\sqrt {-c \left (a^{2} x^{2}-1\right )}\, \left (-2 i \sqrt {-a^{2} x^{2}+1}\, a^{2} x^{2}+2 a^{3} x^{3}+i \sqrt {-a^{2} x^{2}+1}-2 a x \right ) \left (-6 i \arccos \left (a x \right )^{2}+4 \arccos \left (a x \right )^{3}+3 i-6 \arccos \left (a x \right )\right ) c}{32 a \left (a^{2} x^{2}-1\right )}-\frac {\sqrt {-c \left (a^{2} x^{2}-1\right )}\, \left (-8 i \sqrt {-a^{2} x^{2}+1}\, a^{4} x^{4}+8 a^{5} x^{5}+8 i \sqrt {-a^{2} x^{2}+1}\, a^{2} x^{2}-12 a^{3} x^{3}-i \sqrt {-a^{2} x^{2}+1}+4 a x \right ) \left (-24 i \arccos \left (a x \right )^{2}+32 \arccos \left (a x \right )^{3}+3 i-12 \arccos \left (a x \right )\right ) c}{2048 a \left (a^{2} x^{2}-1\right )}\)	\(533\)

Input:

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

Output:

3/32*(-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 
*c-1/2048*(-c*(a^2*x^2-1))^(1/2)*(8*a^5*x^5-12*a^3*x^3+8*I*(-a^2*x^2+1)^(1 
/2)*a^4*x^4+4*a*x-8*I*(-a^2*x^2+1)^(1/2)*a^2*x^2+I*(-a^2*x^2+1)^(1/2))*(24 
*I*arccos(a*x)^2+32*arccos(a*x)^3-3*I-12*arccos(a*x))*c/a/(a^2*x^2-1)+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))* 
c/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))*c/a/(a^2*x^2-1)-1/2048*(-c*(a^2*x^2-1))^(1/2)*(-8*I* 
(-a^2*x^2+1)^(1/2)*a^4*x^4+8*a^5*x^5+8*I*(-a^2*x^2+1)^(1/2)*a^2*x^2-12*a^3 
*x^3-I*(-a^2*x^2+1)^(1/2)+4*a*x)*(-24*I*arccos(a*x)^2+32*arccos(a*x)^3+3*I 
-12*arccos(a*x))*c/a/(a^2*x^2-1)
 

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F(-2)]

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

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

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

Output:

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

Reduce [F]

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

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

Output:

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