\(\int (a^2-x^2)^{3/2} \arccos (\frac {x}{a})^{3/2} \, dx\) [462]

Optimal result

Integrand size = 24, antiderivative size = 359 \[ \int \left (a^2-x^2\right )^{3/2} \arccos \left (\frac {x}{a}\right )^{3/2} \, dx=\frac {27 a^3 \sqrt {a^2-x^2} \sqrt {\arccos \left (\frac {x}{a}\right )}}{256 \sqrt {1-\frac {x^2}{a^2}}}-\frac {9 a x^2 \sqrt {a^2-x^2} \sqrt {\arccos \left (\frac {x}{a}\right )}}{32 \sqrt {1-\frac {x^2}{a^2}}}+\frac {3 \left (a^2-x^2\right )^{5/2} \sqrt {\arccos \left (\frac {x}{a}\right )}}{32 a \sqrt {1-\frac {x^2}{a^2}}}+\frac {3}{8} a^2 x \sqrt {a^2-x^2} \arccos \left (\frac {x}{a}\right )^{3/2}+\frac {1}{4} x \left (a^2-x^2\right )^{3/2} \arccos \left (\frac {x}{a}\right )^{3/2}+\frac {3 a^3 \sqrt {a^2-x^2} \arccos \left (\frac {x}{a}\right )^{5/2}}{20 \sqrt {1-\frac {x^2}{a^2}}}-\frac {3 a^3 \sqrt {\frac {\pi }{2}} \sqrt {a^2-x^2} \operatorname {FresnelC}\left (2 \sqrt {\frac {2}{\pi }} \sqrt {\arccos \left (\frac {x}{a}\right )}\right )}{512 \sqrt {1-\frac {x^2}{a^2}}}-\frac {3 a^3 \sqrt {\pi } \sqrt {a^2-x^2} \operatorname {FresnelC}\left (\frac {2 \sqrt {\arccos \left (\frac {x}{a}\right )}}{\sqrt {\pi }}\right )}{32 \sqrt {1-\frac {x^2}{a^2}}} \] Output:

27/256*a^3*(a^2-x^2)^(1/2)*arccos(x/a)^(1/2)/(1-x^2/a^2)^(1/2)-9/32*a*x^2* 
(a^2-x^2)^(1/2)*arccos(x/a)^(1/2)/(1-x^2/a^2)^(1/2)+3/32*(a^2-x^2)^(5/2)*a 
rccos(x/a)^(1/2)/a/(1-x^2/a^2)^(1/2)+3/8*a^2*x*(a^2-x^2)^(1/2)*arccos(x/a) 
^(3/2)+1/4*x*(a^2-x^2)^(3/2)*arccos(x/a)^(3/2)+3/20*a^3*(a^2-x^2)^(1/2)*ar 
ccos(x/a)^(5/2)/(1-x^2/a^2)^(1/2)-3/1024*a^3*2^(1/2)*Pi^(1/2)*(a^2-x^2)^(1 
/2)*FresnelC(2*2^(1/2)/Pi^(1/2)*arccos(x/a)^(1/2))/(1-x^2/a^2)^(1/2)-3/32* 
a^3*Pi^(1/2)*(a^2-x^2)^(1/2)*FresnelC(2*arccos(x/a)^(1/2)/Pi^(1/2))/(1-x^2 
/a^2)^(1/2)
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 0.33 (sec) , antiderivative size = 209, normalized size of antiderivative = 0.58 \[ \int \left (a^2-x^2\right )^{3/2} \arccos \left (\frac {x}{a}\right )^{3/2} \, dx=-\frac {a^3 \sqrt {a^2-x^2} \left (240 \sqrt {\pi } \sqrt {\arccos \left (\frac {x}{a}\right )^2} \operatorname {FresnelC}\left (\frac {2 \sqrt {\arccos \left (\frac {x}{a}\right )}}{\sqrt {\pi }}\right )+\sqrt {\arccos \left (\frac {x}{a}\right )} \left (5 \sqrt {i \arccos \left (\frac {x}{a}\right )} \Gamma \left (\frac {5}{2},-4 i \arccos \left (\frac {x}{a}\right )\right )+5 \sqrt {-i \arccos \left (\frac {x}{a}\right )} \Gamma \left (\frac {5}{2},4 i \arccos \left (\frac {x}{a}\right )\right )+32 \sqrt {\arccos \left (\frac {x}{a}\right )^2} \left (12 \arccos \left (\frac {x}{a}\right )^2-15 \cos \left (2 \arccos \left (\frac {x}{a}\right )\right )-20 \arccos \left (\frac {x}{a}\right ) \sin \left (2 \arccos \left (\frac {x}{a}\right )\right )\right )\right )\right )}{2560 \sqrt {1-\frac {x^2}{a^2}} \sqrt {\arccos \left (\frac {x}{a}\right )^2}} \] Input:

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

Output:

-1/2560*(a^3*Sqrt[a^2 - x^2]*(240*Sqrt[Pi]*Sqrt[ArcCos[x/a]^2]*FresnelC[(2 
*Sqrt[ArcCos[x/a]])/Sqrt[Pi]] + Sqrt[ArcCos[x/a]]*(5*Sqrt[I*ArcCos[x/a]]*G 
amma[5/2, (-4*I)*ArcCos[x/a]] + 5*Sqrt[(-I)*ArcCos[x/a]]*Gamma[5/2, (4*I)* 
ArcCos[x/a]] + 32*Sqrt[ArcCos[x/a]^2]*(12*ArcCos[x/a]^2 - 15*Cos[2*ArcCos[ 
x/a]] - 20*ArcCos[x/a]*Sin[2*ArcCos[x/a]]))))/(Sqrt[1 - x^2/a^2]*Sqrt[ArcC 
os[x/a]^2])
 

Rubi [A] (verified)

Time = 2.44 (sec) , antiderivative size = 348, normalized size of antiderivative = 0.97, number of steps used = 15, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.583, Rules used = {5159, 27, 5157, 5141, 5153, 5183, 5169, 3042, 3793, 2009, 5225, 3042, 3793, 2009}

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.

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

\(\Big \downarrow \) 5159

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 5157

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

\(\Big \downarrow \) 5141

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

\(\Big \downarrow \) 5153

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

\(\Big \downarrow \) 5183

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

\(\Big \downarrow \) 5169

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3793

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

\(\Big \downarrow \) 2009

\(\displaystyle \frac {3}{4} a^2 \left (\frac {3 \sqrt {a^2-x^2} \left (\frac {\int \frac {x^2}{\sqrt {1-\frac {x^2}{a^2}} \sqrt {\arccos \left (\frac {x}{a}\right )}}dx}{4 a}+\frac {1}{2} x^2 \sqrt {\arccos \left (\frac {x}{a}\right )}\right )}{4 a \sqrt {1-\frac {x^2}{a^2}}}-\frac {a \sqrt {a^2-x^2} \arccos \left (\frac {x}{a}\right )^{5/2}}{5 \sqrt {1-\frac {x^2}{a^2}}}+\frac {1}{2} x \sqrt {a^2-x^2} \arccos \left (\frac {x}{a}\right )^{3/2}\right )+\frac {1}{4} x \left (a^2-x^2\right )^{3/2} \arccos \left (\frac {x}{a}\right )^{3/2}+\frac {3 \sqrt {a^2-x^2} \left (\frac {1}{8} a^4 \left (\frac {1}{8} \sqrt {\frac {\pi }{2}} \operatorname {FresnelC}\left (2 \sqrt {\frac {2}{\pi }} \sqrt {\arccos \left (\frac {x}{a}\right )}\right )-\frac {1}{2} \sqrt {\pi } \operatorname {FresnelC}\left (\frac {2 \sqrt {\arccos \left (\frac {x}{a}\right )}}{\sqrt {\pi }}\right )+\frac {3}{4} \sqrt {\arccos \left (\frac {x}{a}\right )}\right )-\frac {1}{4} \left (a^2-x^2\right )^2 \sqrt {\arccos \left (\frac {x}{a}\right )}\right )}{8 a \sqrt {1-\frac {x^2}{a^2}}}\)

\(\Big \downarrow \) 5225

\(\displaystyle \frac {3}{4} a^2 \left (\frac {3 \sqrt {a^2-x^2} \left (\frac {1}{2} x^2 \sqrt {\arccos \left (\frac {x}{a}\right )}-\frac {1}{4} a^2 \int \frac {x^2}{a^2 \sqrt {\arccos \left (\frac {x}{a}\right )}}d\arccos \left (\frac {x}{a}\right )\right )}{4 a \sqrt {1-\frac {x^2}{a^2}}}-\frac {a \sqrt {a^2-x^2} \arccos \left (\frac {x}{a}\right )^{5/2}}{5 \sqrt {1-\frac {x^2}{a^2}}}+\frac {1}{2} x \sqrt {a^2-x^2} \arccos \left (\frac {x}{a}\right )^{3/2}\right )+\frac {1}{4} x \left (a^2-x^2\right )^{3/2} \arccos \left (\frac {x}{a}\right )^{3/2}+\frac {3 \sqrt {a^2-x^2} \left (\frac {1}{8} a^4 \left (\frac {1}{8} \sqrt {\frac {\pi }{2}} \operatorname {FresnelC}\left (2 \sqrt {\frac {2}{\pi }} \sqrt {\arccos \left (\frac {x}{a}\right )}\right )-\frac {1}{2} \sqrt {\pi } \operatorname {FresnelC}\left (\frac {2 \sqrt {\arccos \left (\frac {x}{a}\right )}}{\sqrt {\pi }}\right )+\frac {3}{4} \sqrt {\arccos \left (\frac {x}{a}\right )}\right )-\frac {1}{4} \left (a^2-x^2\right )^2 \sqrt {\arccos \left (\frac {x}{a}\right )}\right )}{8 a \sqrt {1-\frac {x^2}{a^2}}}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {3}{4} a^2 \left (\frac {3 \sqrt {a^2-x^2} \left (\frac {1}{2} x^2 \sqrt {\arccos \left (\frac {x}{a}\right )}-\frac {1}{4} a^2 \int \frac {\sin \left (\arccos \left (\frac {x}{a}\right )+\frac {\pi }{2}\right )^2}{\sqrt {\arccos \left (\frac {x}{a}\right )}}d\arccos \left (\frac {x}{a}\right )\right )}{4 a \sqrt {1-\frac {x^2}{a^2}}}-\frac {a \sqrt {a^2-x^2} \arccos \left (\frac {x}{a}\right )^{5/2}}{5 \sqrt {1-\frac {x^2}{a^2}}}+\frac {1}{2} x \sqrt {a^2-x^2} \arccos \left (\frac {x}{a}\right )^{3/2}\right )+\frac {1}{4} x \left (a^2-x^2\right )^{3/2} \arccos \left (\frac {x}{a}\right )^{3/2}+\frac {3 \sqrt {a^2-x^2} \left (\frac {1}{8} a^4 \left (\frac {1}{8} \sqrt {\frac {\pi }{2}} \operatorname {FresnelC}\left (2 \sqrt {\frac {2}{\pi }} \sqrt {\arccos \left (\frac {x}{a}\right )}\right )-\frac {1}{2} \sqrt {\pi } \operatorname {FresnelC}\left (\frac {2 \sqrt {\arccos \left (\frac {x}{a}\right )}}{\sqrt {\pi }}\right )+\frac {3}{4} \sqrt {\arccos \left (\frac {x}{a}\right )}\right )-\frac {1}{4} \left (a^2-x^2\right )^2 \sqrt {\arccos \left (\frac {x}{a}\right )}\right )}{8 a \sqrt {1-\frac {x^2}{a^2}}}\)

\(\Big \downarrow \) 3793

\(\displaystyle \frac {3}{4} a^2 \left (\frac {3 \sqrt {a^2-x^2} \left (\frac {1}{2} x^2 \sqrt {\arccos \left (\frac {x}{a}\right )}-\frac {1}{4} a^2 \int \left (\frac {\cos \left (2 \arccos \left (\frac {x}{a}\right )\right )}{2 \sqrt {\arccos \left (\frac {x}{a}\right )}}+\frac {1}{2 \sqrt {\arccos \left (\frac {x}{a}\right )}}\right )d\arccos \left (\frac {x}{a}\right )\right )}{4 a \sqrt {1-\frac {x^2}{a^2}}}-\frac {a \sqrt {a^2-x^2} \arccos \left (\frac {x}{a}\right )^{5/2}}{5 \sqrt {1-\frac {x^2}{a^2}}}+\frac {1}{2} x \sqrt {a^2-x^2} \arccos \left (\frac {x}{a}\right )^{3/2}\right )+\frac {1}{4} x \left (a^2-x^2\right )^{3/2} \arccos \left (\frac {x}{a}\right )^{3/2}+\frac {3 \sqrt {a^2-x^2} \left (\frac {1}{8} a^4 \left (\frac {1}{8} \sqrt {\frac {\pi }{2}} \operatorname {FresnelC}\left (2 \sqrt {\frac {2}{\pi }} \sqrt {\arccos \left (\frac {x}{a}\right )}\right )-\frac {1}{2} \sqrt {\pi } \operatorname {FresnelC}\left (\frac {2 \sqrt {\arccos \left (\frac {x}{a}\right )}}{\sqrt {\pi }}\right )+\frac {3}{4} \sqrt {\arccos \left (\frac {x}{a}\right )}\right )-\frac {1}{4} \left (a^2-x^2\right )^2 \sqrt {\arccos \left (\frac {x}{a}\right )}\right )}{8 a \sqrt {1-\frac {x^2}{a^2}}}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {3}{4} a^2 \left (\frac {3 \sqrt {a^2-x^2} \left (\frac {1}{2} x^2 \sqrt {\arccos \left (\frac {x}{a}\right )}-\frac {1}{4} a^2 \left (\frac {1}{2} \sqrt {\pi } \operatorname {FresnelC}\left (\frac {2 \sqrt {\arccos \left (\frac {x}{a}\right )}}{\sqrt {\pi }}\right )+\sqrt {\arccos \left (\frac {x}{a}\right )}\right )\right )}{4 a \sqrt {1-\frac {x^2}{a^2}}}-\frac {a \sqrt {a^2-x^2} \arccos \left (\frac {x}{a}\right )^{5/2}}{5 \sqrt {1-\frac {x^2}{a^2}}}+\frac {1}{2} x \sqrt {a^2-x^2} \arccos \left (\frac {x}{a}\right )^{3/2}\right )+\frac {1}{4} x \left (a^2-x^2\right )^{3/2} \arccos \left (\frac {x}{a}\right )^{3/2}+\frac {3 \sqrt {a^2-x^2} \left (\frac {1}{8} a^4 \left (\frac {1}{8} \sqrt {\frac {\pi }{2}} \operatorname {FresnelC}\left (2 \sqrt {\frac {2}{\pi }} \sqrt {\arccos \left (\frac {x}{a}\right )}\right )-\frac {1}{2} \sqrt {\pi } \operatorname {FresnelC}\left (\frac {2 \sqrt {\arccos \left (\frac {x}{a}\right )}}{\sqrt {\pi }}\right )+\frac {3}{4} \sqrt {\arccos \left (\frac {x}{a}\right )}\right )-\frac {1}{4} \left (a^2-x^2\right )^2 \sqrt {\arccos \left (\frac {x}{a}\right )}\right )}{8 a \sqrt {1-\frac {x^2}{a^2}}}\)

Input:

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

Output:

(x*(a^2 - x^2)^(3/2)*ArcCos[x/a]^(3/2))/4 + (3*Sqrt[a^2 - x^2]*(-1/4*((a^2 
 - x^2)^2*Sqrt[ArcCos[x/a]]) + (a^4*((3*Sqrt[ArcCos[x/a]])/4 + (Sqrt[Pi/2] 
*FresnelC[2*Sqrt[2/Pi]*Sqrt[ArcCos[x/a]]])/8 - (Sqrt[Pi]*FresnelC[(2*Sqrt[ 
ArcCos[x/a]])/Sqrt[Pi]])/2))/8))/(8*a*Sqrt[1 - x^2/a^2]) + (3*a^2*((x*Sqrt 
[a^2 - x^2]*ArcCos[x/a]^(3/2))/2 - (a*Sqrt[a^2 - x^2]*ArcCos[x/a]^(5/2))/( 
5*Sqrt[1 - x^2/a^2]) + (3*Sqrt[a^2 - x^2]*((x^2*Sqrt[ArcCos[x/a]])/2 - (a^ 
2*(Sqrt[ArcCos[x/a]] + (Sqrt[Pi]*FresnelC[(2*Sqrt[ArcCos[x/a]])/Sqrt[Pi]]) 
/2))/4))/(4*a*Sqrt[1 - x^2/a^2])))/4
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 2009

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

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 3793

Int[((c_.) + (d_.)*(x_))^(m_)*sin[(e_.) + (f_.)*(x_)]^(n_), x_Symbol] :> In 
t[ExpandTrigReduce[(c + d*x)^m, Sin[e + f*x]^n, x], x] /; FreeQ[{c, d, e, f 
, m}, x] && IGtQ[n, 1] && ( !RationalQ[m] || (GeQ[m, -1] && LtQ[m, 1]))
 

rule 5141

Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_)*(x_)^(m_.), x_Symbol] :> Simp[x 
^(m + 1)*((a + b*ArcCos[c*x])^n/(m + 1)), x] + Simp[b*c*(n/(m + 1))   Int[x 
^(m + 1)*((a + b*ArcCos[c*x])^(n - 1)/Sqrt[1 - c^2*x^2]), x], x] /; FreeQ[{ 
a, b, c}, x] && IGtQ[m, 0] && GtQ[n, 0]
 

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 5169

Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_.)*((d_) + (e_.)*(x_)^2)^(p_.), x 
_Symbol] :> Simp[(-(b*c)^(-1))*Simp[(d + e*x^2)^p/(1 - c^2*x^2)^p]   Subst[ 
Int[x^n*Sin[-a/b + x/b]^(2*p + 1), x], x, a + b*ArcCos[c*x]], x] /; FreeQ[{ 
a, b, c, d, e, n}, x] && EqQ[c^2*d + e, 0] && IGtQ[2*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 5225

Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_.)*(x_)^(m_.)*((d_) + (e_.)*(x_)^ 
2)^(p_.), x_Symbol] :> Simp[(-(b*c^(m + 1))^(-1))*Simp[(d + e*x^2)^p/(1 - c 
^2*x^2)^p]   Subst[Int[x^n*Cos[-a/b + x/b]^m*Sin[-a/b + x/b]^(2*p + 1), x], 
 x, a + b*ArcCos[c*x]], x] /; FreeQ[{a, b, c, d, e, n}, x] && EqQ[c^2*d + e 
, 0] && IGtQ[2*p + 2, 0] && IGtQ[m, 0]
 

Maple [F]

\[\int \left (a^{2}-x^{2}\right )^{\frac {3}{2}} \arccos \left (\frac {x}{a}\right )^{\frac {3}{2}}d x\]

Input:

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

Output:

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

Fricas [F(-2)]

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

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

Output:

Exception raised: TypeError >>  Error detected within library code:   inte 
grate: implementation incomplete (constant residues)
 

Sympy [F(-1)]

Timed out. \[ \int \left (a^2-x^2\right )^{3/2} \arccos \left (\frac {x}{a}\right )^{3/2} \, dx=\text {Timed out} \] Input:

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

Output:

Timed out
 

Maxima [F(-2)]

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

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

Output:

Exception raised: RuntimeError >> ECL says: expt: undefined: 0 to a negati 
ve exponent.
 

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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