\(\int \frac {\arccos (a x)^2}{(c-a^2 c x^2)^{7/2}} \, dx\) [277]

Optimal result

Integrand size = 22, antiderivative size = 390 \[ \int \frac {\arccos (a x)^2}{\left (c-a^2 c x^2\right )^{7/2}} \, dx=\frac {x}{3 c^3 \sqrt {c-a^2 c x^2}}+\frac {x}{30 c^3 \left (1-a^2 x^2\right ) \sqrt {c-a^2 c x^2}}-\frac {\arccos (a x)}{10 a c^3 \left (1-a^2 x^2\right )^{3/2} \sqrt {c-a^2 c x^2}}-\frac {4 \arccos (a x)}{15 a c^3 \sqrt {1-a^2 x^2} \sqrt {c-a^2 c x^2}}+\frac {x \arccos (a x)^2}{5 c \left (c-a^2 c x^2\right )^{5/2}}+\frac {4 x \arccos (a x)^2}{15 c^2 \left (c-a^2 c x^2\right )^{3/2}}+\frac {8 x \arccos (a x)^2}{15 c^3 \sqrt {c-a^2 c x^2}}-\frac {8 i \sqrt {1-a^2 x^2} \arccos (a x)^2}{15 a c^3 \sqrt {c-a^2 c x^2}}+\frac {16 \sqrt {1-a^2 x^2} \arccos (a x) \log \left (1+e^{2 i \arccos (a x)}\right )}{15 a c^3 \sqrt {c-a^2 c x^2}}-\frac {8 i \sqrt {1-a^2 x^2} \operatorname {PolyLog}\left (2,-e^{2 i \arccos (a x)}\right )}{15 a c^3 \sqrt {c-a^2 c x^2}} \] Output:

1/3*x/c^3/(-a^2*c*x^2+c)^(1/2)+1/30*x/c^3/(-a^2*x^2+1)/(-a^2*c*x^2+c)^(1/2 
)-1/10*arccos(a*x)/a/c^3/(-a^2*x^2+1)^(3/2)/(-a^2*c*x^2+c)^(1/2)-4/15*arcc 
os(a*x)/a/c^3/(-a^2*x^2+1)^(1/2)/(-a^2*c*x^2+c)^(1/2)+1/5*x*arccos(a*x)^2/ 
c/(-a^2*c*x^2+c)^(5/2)+4/15*x*arccos(a*x)^2/c^2/(-a^2*c*x^2+c)^(3/2)+8/15* 
x*arccos(a*x)^2/c^3/(-a^2*c*x^2+c)^(1/2)-8/15*I*(-a^2*x^2+1)^(1/2)*arccos( 
a*x)^2/a/c^3/(-a^2*c*x^2+c)^(1/2)+16/15*(-a^2*x^2+1)^(1/2)*arccos(a*x)*ln( 
1+(a*x+I*(-a^2*x^2+1)^(1/2))^2)/a/c^3/(-a^2*c*x^2+c)^(1/2)-8/15*I*(-a^2*x^ 
2+1)^(1/2)*polylog(2,-(a*x+I*(-a^2*x^2+1)^(1/2))^2)/a/c^3/(-a^2*c*x^2+c)^( 
1/2)
 

Mathematica [A] (verified)

Time = 0.69 (sec) , antiderivative size = 170, normalized size of antiderivative = 0.44 \[ \int \frac {\arccos (a x)^2}{\left (c-a^2 c x^2\right )^{7/2}} \, dx=\frac {\sqrt {1-a^2 x^2} \left (\frac {a x \left (11-21 a^2 x^2+10 a^4 x^4+2 \left (15-20 a^2 x^2+8 a^4 x^4\right ) \arccos (a x)^2\right )}{\left (1-a^2 x^2\right )^{5/2}}+\arccos (a x) \left (\frac {11-8 a^2 x^2}{\left (-1+a^2 x^2\right )^2}+16 i \arccos (a x)-32 \log \left (1-e^{2 i \arccos (a x)}\right )\right )+16 i \operatorname {PolyLog}\left (2,e^{2 i \arccos (a x)}\right )\right )}{30 a c^3 \sqrt {c-a^2 c x^2}} \] Input:

Integrate[ArcCos[a*x]^2/(c - a^2*c*x^2)^(7/2),x]
 

Output:

(Sqrt[1 - a^2*x^2]*((a*x*(11 - 21*a^2*x^2 + 10*a^4*x^4 + 2*(15 - 20*a^2*x^ 
2 + 8*a^4*x^4)*ArcCos[a*x]^2))/(1 - a^2*x^2)^(5/2) + ArcCos[a*x]*((11 - 8* 
a^2*x^2)/(-1 + a^2*x^2)^2 + (16*I)*ArcCos[a*x] - 32*Log[1 - E^((2*I)*ArcCo 
s[a*x])]) + (16*I)*PolyLog[2, E^((2*I)*ArcCos[a*x])]))/(30*a*c^3*Sqrt[c - 
a^2*c*x^2])
 

Rubi [A] (verified)

Time = 1.85 (sec) , antiderivative size = 387, normalized size of antiderivative = 0.99, number of steps used = 16, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.682, Rules used = {5163, 5163, 5161, 5181, 3042, 25, 4200, 25, 2620, 2715, 2838, 5183, 208, 209, 208}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \frac {\arccos (a x)^2}{\left (c-a^2 c x^2\right )^{7/2}} \, dx\)

\(\Big \downarrow \) 5163

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

\(\Big \downarrow \) 5163

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

\(\Big \downarrow \) 5161

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

\(\Big \downarrow \) 5181

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 4200

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 2620

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

\(\Big \downarrow \) 2715

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

\(\Big \downarrow \) 2838

\(\displaystyle \frac {2 a \sqrt {1-a^2 x^2} \int \frac {x \arccos (a x)}{\left (1-a^2 x^2\right )^3}dx}{5 c^3 \sqrt {c-a^2 c x^2}}+\frac {4 \left (\frac {2 a \sqrt {1-a^2 x^2} \int \frac {x \arccos (a x)}{\left (1-a^2 x^2\right )^2}dx}{3 c^2 \sqrt {c-a^2 c x^2}}+\frac {2 \left (\frac {x \arccos (a x)^2}{c \sqrt {c-a^2 c x^2}}-\frac {2 \sqrt {1-a^2 x^2} \left (-2 i \left (\frac {1}{4} \operatorname {PolyLog}\left (2,e^{2 i \arccos (a x)}\right )+\frac {1}{2} i \arccos (a x) \log \left (1-e^{2 i \arccos (a x)}\right )\right )-\frac {1}{2} i \arccos (a x)^2\right )}{a c \sqrt {c-a^2 c x^2}}\right )}{3 c}+\frac {x \arccos (a x)^2}{3 c \left (c-a^2 c x^2\right )^{3/2}}\right )}{5 c}+\frac {x \arccos (a x)^2}{5 c \left (c-a^2 c x^2\right )^{5/2}}\)

\(\Big \downarrow \) 5183

\(\displaystyle \frac {2 a \sqrt {1-a^2 x^2} \left (\frac {\int \frac {1}{\left (1-a^2 x^2\right )^{5/2}}dx}{4 a}+\frac {\arccos (a x)}{4 a^2 \left (1-a^2 x^2\right )^2}\right )}{5 c^3 \sqrt {c-a^2 c x^2}}+\frac {4 \left (\frac {2 a \sqrt {1-a^2 x^2} \left (\frac {\int \frac {1}{\left (1-a^2 x^2\right )^{3/2}}dx}{2 a}+\frac {\arccos (a x)}{2 a^2 \left (1-a^2 x^2\right )}\right )}{3 c^2 \sqrt {c-a^2 c x^2}}+\frac {2 \left (\frac {x \arccos (a x)^2}{c \sqrt {c-a^2 c x^2}}-\frac {2 \sqrt {1-a^2 x^2} \left (-2 i \left (\frac {1}{4} \operatorname {PolyLog}\left (2,e^{2 i \arccos (a x)}\right )+\frac {1}{2} i \arccos (a x) \log \left (1-e^{2 i \arccos (a x)}\right )\right )-\frac {1}{2} i \arccos (a x)^2\right )}{a c \sqrt {c-a^2 c x^2}}\right )}{3 c}+\frac {x \arccos (a x)^2}{3 c \left (c-a^2 c x^2\right )^{3/2}}\right )}{5 c}+\frac {x \arccos (a x)^2}{5 c \left (c-a^2 c x^2\right )^{5/2}}\)

\(\Big \downarrow \) 208

\(\displaystyle \frac {2 a \sqrt {1-a^2 x^2} \left (\frac {\int \frac {1}{\left (1-a^2 x^2\right )^{5/2}}dx}{4 a}+\frac {\arccos (a x)}{4 a^2 \left (1-a^2 x^2\right )^2}\right )}{5 c^3 \sqrt {c-a^2 c x^2}}+\frac {4 \left (\frac {2 a \sqrt {1-a^2 x^2} \left (\frac {\arccos (a x)}{2 a^2 \left (1-a^2 x^2\right )}+\frac {x}{2 a \sqrt {1-a^2 x^2}}\right )}{3 c^2 \sqrt {c-a^2 c x^2}}+\frac {2 \left (\frac {x \arccos (a x)^2}{c \sqrt {c-a^2 c x^2}}-\frac {2 \sqrt {1-a^2 x^2} \left (-2 i \left (\frac {1}{4} \operatorname {PolyLog}\left (2,e^{2 i \arccos (a x)}\right )+\frac {1}{2} i \arccos (a x) \log \left (1-e^{2 i \arccos (a x)}\right )\right )-\frac {1}{2} i \arccos (a x)^2\right )}{a c \sqrt {c-a^2 c x^2}}\right )}{3 c}+\frac {x \arccos (a x)^2}{3 c \left (c-a^2 c x^2\right )^{3/2}}\right )}{5 c}+\frac {x \arccos (a x)^2}{5 c \left (c-a^2 c x^2\right )^{5/2}}\)

\(\Big \downarrow \) 209

\(\displaystyle \frac {2 a \sqrt {1-a^2 x^2} \left (\frac {\frac {2}{3} \int \frac {1}{\left (1-a^2 x^2\right )^{3/2}}dx+\frac {x}{3 \left (1-a^2 x^2\right )^{3/2}}}{4 a}+\frac {\arccos (a x)}{4 a^2 \left (1-a^2 x^2\right )^2}\right )}{5 c^3 \sqrt {c-a^2 c x^2}}+\frac {4 \left (\frac {2 a \sqrt {1-a^2 x^2} \left (\frac {\arccos (a x)}{2 a^2 \left (1-a^2 x^2\right )}+\frac {x}{2 a \sqrt {1-a^2 x^2}}\right )}{3 c^2 \sqrt {c-a^2 c x^2}}+\frac {2 \left (\frac {x \arccos (a x)^2}{c \sqrt {c-a^2 c x^2}}-\frac {2 \sqrt {1-a^2 x^2} \left (-2 i \left (\frac {1}{4} \operatorname {PolyLog}\left (2,e^{2 i \arccos (a x)}\right )+\frac {1}{2} i \arccos (a x) \log \left (1-e^{2 i \arccos (a x)}\right )\right )-\frac {1}{2} i \arccos (a x)^2\right )}{a c \sqrt {c-a^2 c x^2}}\right )}{3 c}+\frac {x \arccos (a x)^2}{3 c \left (c-a^2 c x^2\right )^{3/2}}\right )}{5 c}+\frac {x \arccos (a x)^2}{5 c \left (c-a^2 c x^2\right )^{5/2}}\)

\(\Big \downarrow \) 208

\(\displaystyle \frac {2 a \sqrt {1-a^2 x^2} \left (\frac {\arccos (a x)}{4 a^2 \left (1-a^2 x^2\right )^2}+\frac {\frac {2 x}{3 \sqrt {1-a^2 x^2}}+\frac {x}{3 \left (1-a^2 x^2\right )^{3/2}}}{4 a}\right )}{5 c^3 \sqrt {c-a^2 c x^2}}+\frac {4 \left (\frac {2 a \sqrt {1-a^2 x^2} \left (\frac {\arccos (a x)}{2 a^2 \left (1-a^2 x^2\right )}+\frac {x}{2 a \sqrt {1-a^2 x^2}}\right )}{3 c^2 \sqrt {c-a^2 c x^2}}+\frac {2 \left (\frac {x \arccos (a x)^2}{c \sqrt {c-a^2 c x^2}}-\frac {2 \sqrt {1-a^2 x^2} \left (-2 i \left (\frac {1}{4} \operatorname {PolyLog}\left (2,e^{2 i \arccos (a x)}\right )+\frac {1}{2} i \arccos (a x) \log \left (1-e^{2 i \arccos (a x)}\right )\right )-\frac {1}{2} i \arccos (a x)^2\right )}{a c \sqrt {c-a^2 c x^2}}\right )}{3 c}+\frac {x \arccos (a x)^2}{3 c \left (c-a^2 c x^2\right )^{3/2}}\right )}{5 c}+\frac {x \arccos (a x)^2}{5 c \left (c-a^2 c x^2\right )^{5/2}}\)

Input:

Int[ArcCos[a*x]^2/(c - a^2*c*x^2)^(7/2),x]
 

Output:

(x*ArcCos[a*x]^2)/(5*c*(c - a^2*c*x^2)^(5/2)) + (2*a*Sqrt[1 - a^2*x^2]*((x 
/(3*(1 - a^2*x^2)^(3/2)) + (2*x)/(3*Sqrt[1 - a^2*x^2]))/(4*a) + ArcCos[a*x 
]/(4*a^2*(1 - a^2*x^2)^2)))/(5*c^3*Sqrt[c - a^2*c*x^2]) + (4*((x*ArcCos[a* 
x]^2)/(3*c*(c - a^2*c*x^2)^(3/2)) + (2*a*Sqrt[1 - a^2*x^2]*(x/(2*a*Sqrt[1 
- a^2*x^2]) + ArcCos[a*x]/(2*a^2*(1 - a^2*x^2))))/(3*c^2*Sqrt[c - a^2*c*x^ 
2]) + (2*((x*ArcCos[a*x]^2)/(c*Sqrt[c - a^2*c*x^2]) - (2*Sqrt[1 - a^2*x^2] 
*((-1/2*I)*ArcCos[a*x]^2 - (2*I)*((I/2)*ArcCos[a*x]*Log[1 - E^((2*I)*ArcCo 
s[a*x])] + PolyLog[2, E^((2*I)*ArcCos[a*x])]/4)))/(a*c*Sqrt[c - a^2*c*x^2] 
)))/(3*c)))/(5*c)
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 208

Int[((a_) + (b_.)*(x_)^2)^(-3/2), x_Symbol] :> Simp[x/(a*Sqrt[a + b*x^2]), 
x] /; FreeQ[{a, b}, x]
 

rule 209

Int[((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(-x)*((a + b*x^2)^(p + 1) 
/(2*a*(p + 1))), x] + Simp[(2*p + 3)/(2*a*(p + 1))   Int[(a + b*x^2)^(p + 1 
), x], x] /; FreeQ[{a, b}, x] && ILtQ[p + 3/2, 0]
 

rule 2620

Int[(((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.))/ 
((a_) + (b_.)*((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)), x_Symbol] :> Simp 
[((c + d*x)^m/(b*f*g*n*Log[F]))*Log[1 + b*((F^(g*(e + f*x)))^n/a)], x] - Si 
mp[d*(m/(b*f*g*n*Log[F]))   Int[(c + d*x)^(m - 1)*Log[1 + b*((F^(g*(e + f*x 
)))^n/a)], x], x] /; FreeQ[{F, a, b, c, d, e, f, g, n}, x] && IGtQ[m, 0]
 

rule 2715

Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] 
:> Simp[1/(d*e*n*Log[F])   Subst[Int[Log[a + b*x]/x, x], x, (F^(e*(c + d*x) 
))^n], x] /; FreeQ[{F, a, b, c, d, e, n}, x] && GtQ[a, 0]
 

rule 2838

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> Simp[-PolyLog[2 
, (-c)*e*x^n]/n, x] /; FreeQ[{c, d, e, n}, x] && EqQ[c*d, 1]
 

rule 3042

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

rule 4200

Int[((c_.) + (d_.)*(x_))^(m_.)*tan[(e_.) + Pi*(k_.) + (f_.)*(x_)], x_Symbol 
] :> Simp[I*((c + d*x)^(m + 1)/(d*(m + 1))), x] - Simp[2*I   Int[(c + d*x)^ 
m*E^(2*I*k*Pi)*(E^(2*I*(e + f*x))/(1 + E^(2*I*k*Pi)*E^(2*I*(e + f*x)))), x] 
, x] /; FreeQ[{c, d, e, f}, x] && IntegerQ[4*k] && IGtQ[m, 0]
 

rule 5161

Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_.)/((d_) + (e_.)*(x_)^2)^(3/2), x 
_Symbol] :> Simp[x*((a + b*ArcCos[c*x])^n/(d*Sqrt[d + e*x^2])), x] + Simp[b 
*c*(n/d)*Simp[Sqrt[1 - c^2*x^2]/Sqrt[d + e*x^2]]   Int[x*((a + b*ArcCos[c*x 
])^(n - 1)/(1 - c^2*x^2)), x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d 
 + e, 0] && GtQ[n, 0]
 

rule 5163

Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_.)*((d_) + (e_.)*(x_)^2)^(p_), x_ 
Symbol] :> Simp[(-x)*(d + e*x^2)^(p + 1)*((a + b*ArcCos[c*x])^n/(2*d*(p + 1 
))), x] + (Simp[(2*p + 3)/(2*d*(p + 1))   Int[(d + e*x^2)^(p + 1)*(a + b*Ar 
cCos[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] && LtQ[p, 
 -1] && NeQ[p, -3/2]
 

rule 5181

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

Maple [A] (verified)

Time = 0.63 (sec) , antiderivative size = 600, normalized size of antiderivative = 1.54

method	result	size
default	\(-\frac {\sqrt {-c \left (a^{2} x^{2}-1\right )}\, \left (-8 i \sqrt {-a^{2} x^{2}+1}\, a^{4} x^{4}+8 a^{5} x^{5}+16 i \sqrt {-a^{2} x^{2}+1}\, a^{2} x^{2}-20 a^{3} x^{3}-8 i \sqrt {-a^{2} x^{2}+1}+15 a x \right ) \left (456 i \arccos \left (a x \right ) a^{4} x^{4}-64 \arccos \left (a x \right ) \sqrt {-a^{2} x^{2}+1}\, a^{7} x^{7}-280 i \arccos \left (a x \right ) a^{6} x^{6}+32 a^{8} x^{8}+32 i \sqrt {-a^{2} x^{2}+1}\, a^{7} x^{7}+248 \arccos \left (a x \right ) \sqrt {-a^{2} x^{2}+1}\, a^{5} x^{5}+64 i \arccos \left (a x \right ) a^{8} x^{8}-142 a^{6} x^{6}+80 a^{4} x^{4} \arccos \left (a x \right )^{2}-126 i \sqrt {-a^{2} x^{2}+1}\, a^{5} x^{5}-340 \sqrt {-a^{2} x^{2}+1}\, \arccos \left (a x \right ) a^{3} x^{3}+156 i \sqrt {-a^{2} x^{2}+1}\, a^{3} x^{3}+265 a^{4} x^{4}-190 a^{2} x^{2} \arccos \left (a x \right )^{2}+88 i \arccos \left (a x \right )+165 \arccos \left (a x \right ) \sqrt {-a^{2} x^{2}+1}\, a x -62 i \sqrt {-a^{2} x^{2}+1}\, a x -235 a^{2} x^{2}+128 \arccos \left (a x \right )^{2}-328 i \arccos \left (a x \right ) a^{2} x^{2}+80\right )}{30 c^{4} \left (40 a^{10} x^{10}-215 a^{8} x^{8}+469 a^{6} x^{6}-517 a^{4} x^{4}+287 a^{2} x^{2}-64\right ) a}-\frac {16 i \sqrt {-a^{2} x^{2}+1}\, \sqrt {-c \left (a^{2} x^{2}-1\right )}\, \left (i \arccos \left (a x \right ) \ln \left (1-a x -i \sqrt {-a^{2} x^{2}+1}\right )+i \arccos \left (a x \right ) \ln \left (1+a x +i \sqrt {-a^{2} x^{2}+1}\right )+\arccos \left (a x \right )^{2}+\operatorname {polylog}\left (2, a x +i \sqrt {-a^{2} x^{2}+1}\right )+\operatorname {polylog}\left (2, -a x -i \sqrt {-a^{2} x^{2}+1}\right )\right )}{15 c^{4} \left (a^{2} x^{2}-1\right ) a}\)	\(600\)

Input:

int(arccos(a*x)^2/(-a^2*c*x^2+c)^(7/2),x,method=_RETURNVERBOSE)
 

Output:

-1/30*(-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+16 
*I*(-a^2*x^2+1)^(1/2)*a^2*x^2-20*a^3*x^3-8*I*(-a^2*x^2+1)^(1/2)+15*a*x)*(4 
56*I*arccos(a*x)*a^4*x^4-64*arccos(a*x)*(-a^2*x^2+1)^(1/2)*a^7*x^7-280*I*a 
rccos(a*x)*a^6*x^6+32*a^8*x^8+32*I*(-a^2*x^2+1)^(1/2)*a^7*x^7+248*arccos(a 
*x)*(-a^2*x^2+1)^(1/2)*a^5*x^5+64*I*arccos(a*x)*a^8*x^8-142*a^6*x^6+80*a^4 
*x^4*arccos(a*x)^2-126*I*(-a^2*x^2+1)^(1/2)*a^5*x^5-340*(-a^2*x^2+1)^(1/2) 
*arccos(a*x)*a^3*x^3+156*I*(-a^2*x^2+1)^(1/2)*a^3*x^3+265*a^4*x^4-190*a^2* 
x^2*arccos(a*x)^2+88*I*arccos(a*x)+165*arccos(a*x)*(-a^2*x^2+1)^(1/2)*a*x- 
62*I*(-a^2*x^2+1)^(1/2)*a*x-235*a^2*x^2+128*arccos(a*x)^2-328*I*arccos(a*x 
)*a^2*x^2+80)/c^4/(40*a^10*x^10-215*a^8*x^8+469*a^6*x^6-517*a^4*x^4+287*a^ 
2*x^2-64)/a-16/15*I*(-a^2*x^2+1)^(1/2)*(-c*(a^2*x^2-1))^(1/2)*(I*arccos(a* 
x)*ln(1-a*x-I*(-a^2*x^2+1)^(1/2))+I*arccos(a*x)*ln(1+a*x+I*(-a^2*x^2+1)^(1 
/2))+arccos(a*x)^2+polylog(2,a*x+I*(-a^2*x^2+1)^(1/2))+polylog(2,-a*x-I*(- 
a^2*x^2+1)^(1/2)))/c^4/(a^2*x^2-1)/a
 

Fricas [F]

\[ \int \frac {\arccos (a x)^2}{\left (c-a^2 c x^2\right )^{7/2}} \, dx=\int { \frac {\arccos \left (a x\right )^{2}}{{\left (-a^{2} c x^{2} + c\right )}^{\frac {7}{2}}} \,d x } \] Input:

integrate(arccos(a*x)^2/(-a^2*c*x^2+c)^(7/2),x, algorithm="fricas")
 

Output:

integral(sqrt(-a^2*c*x^2 + c)*arccos(a*x)^2/(a^8*c^4*x^8 - 4*a^6*c^4*x^6 + 
 6*a^4*c^4*x^4 - 4*a^2*c^4*x^2 + c^4), x)
 

Sympy [F]

\[ \int \frac {\arccos (a x)^2}{\left (c-a^2 c x^2\right )^{7/2}} \, dx=\int \frac {\operatorname {acos}^{2}{\left (a x \right )}}{\left (- c \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac {7}{2}}}\, dx \] Input:

integrate(acos(a*x)**2/(-a**2*c*x**2+c)**(7/2),x)
 

Output:

Integral(acos(a*x)**2/(-c*(a*x - 1)*(a*x + 1))**(7/2), x)
 

Maxima [F]

\[ \int \frac {\arccos (a x)^2}{\left (c-a^2 c x^2\right )^{7/2}} \, dx=\int { \frac {\arccos \left (a x\right )^{2}}{{\left (-a^{2} c x^{2} + c\right )}^{\frac {7}{2}}} \,d x } \] Input:

integrate(arccos(a*x)^2/(-a^2*c*x^2+c)^(7/2),x, algorithm="maxima")
 

Output:

integrate(arccos(a*x)^2/(-a^2*c*x^2 + c)^(7/2), x)
 

Giac [F(-2)]

Exception generated. \[ \int \frac {\arccos (a x)^2}{\left (c-a^2 c x^2\right )^{7/2}} \, dx=\text {Exception raised: RuntimeError} \] Input:

integrate(arccos(a*x)^2/(-a^2*c*x^2+c)^(7/2),x, algorithm="giac")
 

Output:

Exception raised: RuntimeError >> an error occurred running a Giac command 
:INPUT:sage2OUTPUT:sym2poly/r2sym(const gen & e,const index_m & i,const ve 
cteur & l) Error: Bad Argument Value
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\arccos (a x)^2}{\left (c-a^2 c x^2\right )^{7/2}} \, dx=\int \frac {{\mathrm {acos}\left (a\,x\right )}^2}{{\left (c-a^2\,c\,x^2\right )}^{7/2}} \,d x \] Input:

int(acos(a*x)^2/(c - a^2*c*x^2)^(7/2),x)
 

Output:

int(acos(a*x)^2/(c - a^2*c*x^2)^(7/2), x)
 

Reduce [F]

\[ \int \frac {\arccos (a x)^2}{\left (c-a^2 c x^2\right )^{7/2}} \, dx=-\frac {\int \frac {\mathit {acos} \left (a x \right )^{2}}{\sqrt {-a^{2} x^{2}+1}\, a^{6} x^{6}-3 \sqrt {-a^{2} x^{2}+1}\, a^{4} x^{4}+3 \sqrt {-a^{2} x^{2}+1}\, a^{2} x^{2}-\sqrt {-a^{2} x^{2}+1}}d x}{\sqrt {c}\, c^{3}} \] Input:

int(acos(a*x)^2/(-a^2*c*x^2+c)^(7/2),x)
 

Output:

( - int(acos(a*x)**2/(sqrt( - a**2*x**2 + 1)*a**6*x**6 - 3*sqrt( - a**2*x* 
*2 + 1)*a**4*x**4 + 3*sqrt( - a**2*x**2 + 1)*a**2*x**2 - sqrt( - a**2*x**2 
 + 1)),x))/(sqrt(c)*c**3)