\(\int \frac {\arccos (a x)^3}{(c-a^2 c x^2)^2} \, dx\) [295]

Optimal result

Integrand size = 20, antiderivative size = 337 \[ \int \frac {\arccos (a x)^3}{\left (c-a^2 c x^2\right )^2} \, dx=-\frac {3 \arccos (a x)^2}{2 a c^2 \sqrt {1-a^2 x^2}}+\frac {x \arccos (a x)^3}{2 c^2 \left (1-a^2 x^2\right )}-\frac {6 i \arccos (a x) \arctan \left (e^{i \arccos (a x)}\right )}{a c^2}-\frac {i \arccos (a x)^3 \arctan \left (e^{i \arccos (a x)}\right )}{a c^2}+\frac {3 i \operatorname {PolyLog}\left (2,-i e^{i \arccos (a x)}\right )}{a c^2}+\frac {3 i \arccos (a x)^2 \operatorname {PolyLog}\left (2,-i e^{i \arccos (a x)}\right )}{2 a c^2}-\frac {3 i \operatorname {PolyLog}\left (2,i e^{i \arccos (a x)}\right )}{a c^2}-\frac {3 i \arccos (a x)^2 \operatorname {PolyLog}\left (2,i e^{i \arccos (a x)}\right )}{2 a c^2}-\frac {3 \arccos (a x) \operatorname {PolyLog}\left (3,-i e^{i \arccos (a x)}\right )}{a c^2}+\frac {3 \arccos (a x) \operatorname {PolyLog}\left (3,i e^{i \arccos (a x)}\right )}{a c^2}-\frac {3 i \operatorname {PolyLog}\left (4,-i e^{i \arccos (a x)}\right )}{a c^2}+\frac {3 i \operatorname {PolyLog}\left (4,i e^{i \arccos (a x)}\right )}{a c^2} \] Output:

-3/2*arccos(a*x)^2/a/c^2/(-a^2*x^2+1)^(1/2)+1/2*x*arccos(a*x)^3/c^2/(-a^2* 
x^2+1)-6*I*arccos(a*x)*arctan(a*x+I*(-a^2*x^2+1)^(1/2))/a/c^2-I*arccos(a*x 
)^3*arctan(a*x+I*(-a^2*x^2+1)^(1/2))/a/c^2+3*I*polylog(2,-I*(a*x+I*(-a^2*x 
^2+1)^(1/2)))/a/c^2+3/2*I*arccos(a*x)^2*polylog(2,-I*(a*x+I*(-a^2*x^2+1)^( 
1/2)))/a/c^2-3*I*polylog(2,I*(a*x+I*(-a^2*x^2+1)^(1/2)))/a/c^2-3/2*I*arcco 
s(a*x)^2*polylog(2,I*(a*x+I*(-a^2*x^2+1)^(1/2)))/a/c^2-3*arccos(a*x)*polyl 
og(3,-I*(a*x+I*(-a^2*x^2+1)^(1/2)))/a/c^2+3*arccos(a*x)*polylog(3,I*(a*x+I 
*(-a^2*x^2+1)^(1/2)))/a/c^2-3*I*polylog(4,-I*(a*x+I*(-a^2*x^2+1)^(1/2)))/a 
/c^2+3*I*polylog(4,I*(a*x+I*(-a^2*x^2+1)^(1/2)))/a/c^2
 

Mathematica [A] (verified)

Time = 1.51 (sec) , antiderivative size = 320, normalized size of antiderivative = 0.95 \[ \int \frac {\arccos (a x)^3}{\left (c-a^2 c x^2\right )^2} \, dx=\frac {i \pi ^4-2 i \arccos (a x)^4+12 \arccos (a x)^2 \cot \left (\frac {1}{2} \arccos (a x)\right )+2 \arccos (a x)^3 \csc ^2\left (\frac {1}{2} \arccos (a x)\right )-8 \arccos (a x)^3 \log \left (1-e^{-i \arccos (a x)}\right )-48 \arccos (a x) \log \left (1-e^{i \arccos (a x)}\right )+48 \arccos (a x) \log \left (1+e^{i \arccos (a x)}\right )+8 \arccos (a x)^3 \log \left (1+e^{i \arccos (a x)}\right )-24 i \arccos (a x)^2 \operatorname {PolyLog}\left (2,e^{-i \arccos (a x)}\right )-24 i \left (2+\arccos (a x)^2\right ) \operatorname {PolyLog}\left (2,-e^{i \arccos (a x)}\right )+48 i \operatorname {PolyLog}\left (2,e^{i \arccos (a x)}\right )-48 \arccos (a x) \operatorname {PolyLog}\left (3,e^{-i \arccos (a x)}\right )+48 \arccos (a x) \operatorname {PolyLog}\left (3,-e^{i \arccos (a x)}\right )+48 i \operatorname {PolyLog}\left (4,e^{-i \arccos (a x)}\right )+48 i \operatorname {PolyLog}\left (4,-e^{i \arccos (a x)}\right )-2 \arccos (a x)^3 \sec ^2\left (\frac {1}{2} \arccos (a x)\right )+12 \arccos (a x)^2 \tan \left (\frac {1}{2} \arccos (a x)\right )}{16 a c^2} \] Input:

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

Output:

(I*Pi^4 - (2*I)*ArcCos[a*x]^4 + 12*ArcCos[a*x]^2*Cot[ArcCos[a*x]/2] + 2*Ar 
cCos[a*x]^3*Csc[ArcCos[a*x]/2]^2 - 8*ArcCos[a*x]^3*Log[1 - E^((-I)*ArcCos[ 
a*x])] - 48*ArcCos[a*x]*Log[1 - E^(I*ArcCos[a*x])] + 48*ArcCos[a*x]*Log[1 
+ E^(I*ArcCos[a*x])] + 8*ArcCos[a*x]^3*Log[1 + E^(I*ArcCos[a*x])] - (24*I) 
*ArcCos[a*x]^2*PolyLog[2, E^((-I)*ArcCos[a*x])] - (24*I)*(2 + ArcCos[a*x]^ 
2)*PolyLog[2, -E^(I*ArcCos[a*x])] + (48*I)*PolyLog[2, E^(I*ArcCos[a*x])] - 
 48*ArcCos[a*x]*PolyLog[3, E^((-I)*ArcCos[a*x])] + 48*ArcCos[a*x]*PolyLog[ 
3, -E^(I*ArcCos[a*x])] + (48*I)*PolyLog[4, E^((-I)*ArcCos[a*x])] + (48*I)* 
PolyLog[4, -E^(I*ArcCos[a*x])] - 2*ArcCos[a*x]^3*Sec[ArcCos[a*x]/2]^2 + 12 
*ArcCos[a*x]^2*Tan[ArcCos[a*x]/2])/(16*a*c^2)
 

Rubi [A] (verified)

Time = 1.75 (sec) , antiderivative size = 277, normalized size of antiderivative = 0.82, number of steps used = 16, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.750, Rules used = {5163, 27, 5165, 3042, 4671, 3011, 5183, 5165, 3042, 4671, 2715, 2838, 7163, 2720, 7143}

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.

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

\(\Big \downarrow \) 5163

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 5165

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4671

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

\(\Big \downarrow \) 3011

\(\displaystyle \frac {3 a \int \frac {x \arccos (a x)^2}{\left (1-a^2 x^2\right )^{3/2}}dx}{2 c^2}-\frac {3 \left (i \arccos (a x)^2 \operatorname {PolyLog}\left (2,-e^{i \arccos (a x)}\right )-2 i \int \arccos (a x) \operatorname {PolyLog}\left (2,-e^{i \arccos (a x)}\right )d\arccos (a x)\right )-3 \left (i \arccos (a x)^2 \operatorname {PolyLog}\left (2,e^{i \arccos (a x)}\right )-2 i \int \arccos (a x) \operatorname {PolyLog}\left (2,e^{i \arccos (a x)}\right )d\arccos (a x)\right )-2 \arccos (a x)^3 \text {arctanh}\left (e^{i \arccos (a x)}\right )}{2 a c^2}+\frac {x \arccos (a x)^3}{2 c^2 \left (1-a^2 x^2\right )}\)

\(\Big \downarrow \) 5183

\(\displaystyle \frac {3 a \left (\frac {2 \int \frac {\arccos (a x)}{1-a^2 x^2}dx}{a}+\frac {\arccos (a x)^2}{a^2 \sqrt {1-a^2 x^2}}\right )}{2 c^2}-\frac {3 \left (i \arccos (a x)^2 \operatorname {PolyLog}\left (2,-e^{i \arccos (a x)}\right )-2 i \int \arccos (a x) \operatorname {PolyLog}\left (2,-e^{i \arccos (a x)}\right )d\arccos (a x)\right )-3 \left (i \arccos (a x)^2 \operatorname {PolyLog}\left (2,e^{i \arccos (a x)}\right )-2 i \int \arccos (a x) \operatorname {PolyLog}\left (2,e^{i \arccos (a x)}\right )d\arccos (a x)\right )-2 \arccos (a x)^3 \text {arctanh}\left (e^{i \arccos (a x)}\right )}{2 a c^2}+\frac {x \arccos (a x)^3}{2 c^2 \left (1-a^2 x^2\right )}\)

\(\Big \downarrow \) 5165

\(\displaystyle \frac {3 a \left (\frac {\arccos (a x)^2}{a^2 \sqrt {1-a^2 x^2}}-\frac {2 \int \frac {\arccos (a x)}{\sqrt {1-a^2 x^2}}d\arccos (a x)}{a^2}\right )}{2 c^2}-\frac {3 \left (i \arccos (a x)^2 \operatorname {PolyLog}\left (2,-e^{i \arccos (a x)}\right )-2 i \int \arccos (a x) \operatorname {PolyLog}\left (2,-e^{i \arccos (a x)}\right )d\arccos (a x)\right )-3 \left (i \arccos (a x)^2 \operatorname {PolyLog}\left (2,e^{i \arccos (a x)}\right )-2 i \int \arccos (a x) \operatorname {PolyLog}\left (2,e^{i \arccos (a x)}\right )d\arccos (a x)\right )-2 \arccos (a x)^3 \text {arctanh}\left (e^{i \arccos (a x)}\right )}{2 a c^2}+\frac {x \arccos (a x)^3}{2 c^2 \left (1-a^2 x^2\right )}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {3 a \left (\frac {\arccos (a x)^2}{a^2 \sqrt {1-a^2 x^2}}-\frac {2 \int \arccos (a x) \csc (\arccos (a x))d\arccos (a x)}{a^2}\right )}{2 c^2}-\frac {3 \left (i \arccos (a x)^2 \operatorname {PolyLog}\left (2,-e^{i \arccos (a x)}\right )-2 i \int \arccos (a x) \operatorname {PolyLog}\left (2,-e^{i \arccos (a x)}\right )d\arccos (a x)\right )-3 \left (i \arccos (a x)^2 \operatorname {PolyLog}\left (2,e^{i \arccos (a x)}\right )-2 i \int \arccos (a x) \operatorname {PolyLog}\left (2,e^{i \arccos (a x)}\right )d\arccos (a x)\right )-2 \arccos (a x)^3 \text {arctanh}\left (e^{i \arccos (a x)}\right )}{2 a c^2}+\frac {x \arccos (a x)^3}{2 c^2 \left (1-a^2 x^2\right )}\)

\(\Big \downarrow \) 4671

\(\displaystyle \frac {3 a \left (\frac {\arccos (a x)^2}{a^2 \sqrt {1-a^2 x^2}}-\frac {2 \left (-\int \log \left (1-e^{i \arccos (a x)}\right )d\arccos (a x)+\int \log \left (1+e^{i \arccos (a x)}\right )d\arccos (a x)-2 \arccos (a x) \text {arctanh}\left (e^{i \arccos (a x)}\right )\right )}{a^2}\right )}{2 c^2}-\frac {3 \left (i \arccos (a x)^2 \operatorname {PolyLog}\left (2,-e^{i \arccos (a x)}\right )-2 i \int \arccos (a x) \operatorname {PolyLog}\left (2,-e^{i \arccos (a x)}\right )d\arccos (a x)\right )-3 \left (i \arccos (a x)^2 \operatorname {PolyLog}\left (2,e^{i \arccos (a x)}\right )-2 i \int \arccos (a x) \operatorname {PolyLog}\left (2,e^{i \arccos (a x)}\right )d\arccos (a x)\right )-2 \arccos (a x)^3 \text {arctanh}\left (e^{i \arccos (a x)}\right )}{2 a c^2}+\frac {x \arccos (a x)^3}{2 c^2 \left (1-a^2 x^2\right )}\)

\(\Big \downarrow \) 2715

\(\displaystyle \frac {3 a \left (\frac {\arccos (a x)^2}{a^2 \sqrt {1-a^2 x^2}}-\frac {2 \left (i \int e^{-i \arccos (a x)} \log \left (1-e^{i \arccos (a x)}\right )de^{i \arccos (a x)}-i \int e^{-i \arccos (a x)} \log \left (1+e^{i \arccos (a x)}\right )de^{i \arccos (a x)}-2 \arccos (a x) \text {arctanh}\left (e^{i \arccos (a x)}\right )\right )}{a^2}\right )}{2 c^2}-\frac {3 \left (i \arccos (a x)^2 \operatorname {PolyLog}\left (2,-e^{i \arccos (a x)}\right )-2 i \int \arccos (a x) \operatorname {PolyLog}\left (2,-e^{i \arccos (a x)}\right )d\arccos (a x)\right )-3 \left (i \arccos (a x)^2 \operatorname {PolyLog}\left (2,e^{i \arccos (a x)}\right )-2 i \int \arccos (a x) \operatorname {PolyLog}\left (2,e^{i \arccos (a x)}\right )d\arccos (a x)\right )-2 \arccos (a x)^3 \text {arctanh}\left (e^{i \arccos (a x)}\right )}{2 a c^2}+\frac {x \arccos (a x)^3}{2 c^2 \left (1-a^2 x^2\right )}\)

\(\Big \downarrow \) 2838

\(\displaystyle -\frac {3 \left (i \arccos (a x)^2 \operatorname {PolyLog}\left (2,-e^{i \arccos (a x)}\right )-2 i \int \arccos (a x) \operatorname {PolyLog}\left (2,-e^{i \arccos (a x)}\right )d\arccos (a x)\right )-3 \left (i \arccos (a x)^2 \operatorname {PolyLog}\left (2,e^{i \arccos (a x)}\right )-2 i \int \arccos (a x) \operatorname {PolyLog}\left (2,e^{i \arccos (a x)}\right )d\arccos (a x)\right )-2 \arccos (a x)^3 \text {arctanh}\left (e^{i \arccos (a x)}\right )}{2 a c^2}+\frac {3 a \left (\frac {\arccos (a x)^2}{a^2 \sqrt {1-a^2 x^2}}-\frac {2 \left (-2 \arccos (a x) \text {arctanh}\left (e^{i \arccos (a x)}\right )+i \operatorname {PolyLog}\left (2,-e^{i \arccos (a x)}\right )-i \operatorname {PolyLog}\left (2,e^{i \arccos (a x)}\right )\right )}{a^2}\right )}{2 c^2}+\frac {x \arccos (a x)^3}{2 c^2 \left (1-a^2 x^2\right )}\)

\(\Big \downarrow \) 7163

\(\displaystyle -\frac {3 \left (i \arccos (a x)^2 \operatorname {PolyLog}\left (2,-e^{i \arccos (a x)}\right )-2 i \left (i \int \operatorname {PolyLog}\left (3,-e^{i \arccos (a x)}\right )d\arccos (a x)-i \arccos (a x) \operatorname {PolyLog}\left (3,-e^{i \arccos (a x)}\right )\right )\right )-3 \left (i \arccos (a x)^2 \operatorname {PolyLog}\left (2,e^{i \arccos (a x)}\right )-2 i \left (i \int \operatorname {PolyLog}\left (3,e^{i \arccos (a x)}\right )d\arccos (a x)-i \arccos (a x) \operatorname {PolyLog}\left (3,e^{i \arccos (a x)}\right )\right )\right )-2 \arccos (a x)^3 \text {arctanh}\left (e^{i \arccos (a x)}\right )}{2 a c^2}+\frac {3 a \left (\frac {\arccos (a x)^2}{a^2 \sqrt {1-a^2 x^2}}-\frac {2 \left (-2 \arccos (a x) \text {arctanh}\left (e^{i \arccos (a x)}\right )+i \operatorname {PolyLog}\left (2,-e^{i \arccos (a x)}\right )-i \operatorname {PolyLog}\left (2,e^{i \arccos (a x)}\right )\right )}{a^2}\right )}{2 c^2}+\frac {x \arccos (a x)^3}{2 c^2 \left (1-a^2 x^2\right )}\)

\(\Big \downarrow \) 2720

\(\displaystyle -\frac {3 \left (i \arccos (a x)^2 \operatorname {PolyLog}\left (2,-e^{i \arccos (a x)}\right )-2 i \left (\int e^{-i \arccos (a x)} \operatorname {PolyLog}\left (3,-e^{i \arccos (a x)}\right )de^{i \arccos (a x)}-i \arccos (a x) \operatorname {PolyLog}\left (3,-e^{i \arccos (a x)}\right )\right )\right )-3 \left (i \arccos (a x)^2 \operatorname {PolyLog}\left (2,e^{i \arccos (a x)}\right )-2 i \left (\int e^{-i \arccos (a x)} \operatorname {PolyLog}\left (3,e^{i \arccos (a x)}\right )de^{i \arccos (a x)}-i \arccos (a x) \operatorname {PolyLog}\left (3,e^{i \arccos (a x)}\right )\right )\right )-2 \arccos (a x)^3 \text {arctanh}\left (e^{i \arccos (a x)}\right )}{2 a c^2}+\frac {3 a \left (\frac {\arccos (a x)^2}{a^2 \sqrt {1-a^2 x^2}}-\frac {2 \left (-2 \arccos (a x) \text {arctanh}\left (e^{i \arccos (a x)}\right )+i \operatorname {PolyLog}\left (2,-e^{i \arccos (a x)}\right )-i \operatorname {PolyLog}\left (2,e^{i \arccos (a x)}\right )\right )}{a^2}\right )}{2 c^2}+\frac {x \arccos (a x)^3}{2 c^2 \left (1-a^2 x^2\right )}\)

\(\Big \downarrow \) 7143

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

Input:

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

Output:

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

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 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 2720

Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Simp[v/D[v, x] 
   Subst[Int[FunctionOfExponentialFunction[u, x]/x, x], x, v], x]] /; Funct 
ionOfExponentialQ[u, x] &&  !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; FreeQ 
[{a, m, n}, x] && IntegerQ[m*n]] &&  !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x)) 
*(F_)[v_] /; FreeQ[{a, b, c}, x] && InverseFunctionQ[F[x]]]
 

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 3011

Int[Log[1 + (e_.)*((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.)]*((f_.) + (g_.) 
*(x_))^(m_.), x_Symbol] :> Simp[(-(f + g*x)^m)*(PolyLog[2, (-e)*(F^(c*(a + 
b*x)))^n]/(b*c*n*Log[F])), x] + Simp[g*(m/(b*c*n*Log[F]))   Int[(f + g*x)^( 
m - 1)*PolyLog[2, (-e)*(F^(c*(a + b*x)))^n], x], x] /; FreeQ[{F, a, b, c, e 
, f, g, n}, x] && GtQ[m, 0]
 

rule 3042

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

rule 4671

Int[csc[(e_.) + (f_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[- 
2*(c + d*x)^m*(ArcTanh[E^(I*(e + f*x))]/f), x] + (-Simp[d*(m/f)   Int[(c + 
d*x)^(m - 1)*Log[1 - E^(I*(e + f*x))], x], x] + Simp[d*(m/f)   Int[(c + d*x 
)^(m - 1)*Log[1 + E^(I*(e + f*x))], x], x]) /; FreeQ[{c, d, e, f}, x] && IG 
tQ[m, 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 5165

Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_.)/((d_) + (e_.)*(x_)^2), x_Symbo 
l] :> Simp[-(c*d)^(-1)   Subst[Int[(a + b*x)^n*Csc[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]
 

rule 7143

Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_S 
ymbol] :> Simp[PolyLog[n + 1, c*(a + b*x)^p]/(e*p), x] /; FreeQ[{a, b, c, d 
, e, n, p}, x] && EqQ[b*d, a*e]
 

rule 7163

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

Maple [A] (verified)

Time = 0.40 (sec) , antiderivative size = 414, normalized size of antiderivative = 1.23

method	result	size
derivativedivides	\(\frac {-\frac {\arccos \left (a x \right )^{2} \left (a x \arccos \left (a x \right )+3 \sqrt {-a^{2} x^{2}+1}\right )}{2 \left (a^{2} x^{2}-1\right ) c^{2}}-\frac {\arccos \left (a x \right )^{3} \ln \left (1-a x -i \sqrt {-a^{2} x^{2}+1}\right )}{2 c^{2}}+\frac {3 i \arccos \left (a x \right )^{2} \operatorname {polylog}\left (2, a x +i \sqrt {-a^{2} x^{2}+1}\right )}{2 c^{2}}-\frac {3 \arccos \left (a x \right ) \operatorname {polylog}\left (3, a x +i \sqrt {-a^{2} x^{2}+1}\right )}{c^{2}}-\frac {3 i \operatorname {polylog}\left (4, a x +i \sqrt {-a^{2} x^{2}+1}\right )}{c^{2}}+\frac {\arccos \left (a x \right )^{3} \ln \left (1+a x +i \sqrt {-a^{2} x^{2}+1}\right )}{2 c^{2}}-\frac {3 i \arccos \left (a x \right )^{2} \operatorname {polylog}\left (2, -a x -i \sqrt {-a^{2} x^{2}+1}\right )}{2 c^{2}}+\frac {3 \arccos \left (a x \right ) \operatorname {polylog}\left (3, -a x -i \sqrt {-a^{2} x^{2}+1}\right )}{c^{2}}+\frac {3 i \operatorname {polylog}\left (4, -a x -i \sqrt {-a^{2} x^{2}+1}\right )}{c^{2}}-\frac {3 \arccos \left (a x \right ) \ln \left (1-a x -i \sqrt {-a^{2} x^{2}+1}\right )}{c^{2}}+\frac {3 i \operatorname {polylog}\left (2, a x +i \sqrt {-a^{2} x^{2}+1}\right )}{c^{2}}+\frac {3 \arccos \left (a x \right ) \ln \left (1+a x +i \sqrt {-a^{2} x^{2}+1}\right )}{c^{2}}-\frac {3 i \operatorname {polylog}\left (2, -a x -i \sqrt {-a^{2} x^{2}+1}\right )}{c^{2}}}{a}\)	\(414\)
default	\(\frac {-\frac {\arccos \left (a x \right )^{2} \left (a x \arccos \left (a x \right )+3 \sqrt {-a^{2} x^{2}+1}\right )}{2 \left (a^{2} x^{2}-1\right ) c^{2}}-\frac {\arccos \left (a x \right )^{3} \ln \left (1-a x -i \sqrt {-a^{2} x^{2}+1}\right )}{2 c^{2}}+\frac {3 i \arccos \left (a x \right )^{2} \operatorname {polylog}\left (2, a x +i \sqrt {-a^{2} x^{2}+1}\right )}{2 c^{2}}-\frac {3 \arccos \left (a x \right ) \operatorname {polylog}\left (3, a x +i \sqrt {-a^{2} x^{2}+1}\right )}{c^{2}}-\frac {3 i \operatorname {polylog}\left (4, a x +i \sqrt {-a^{2} x^{2}+1}\right )}{c^{2}}+\frac {\arccos \left (a x \right )^{3} \ln \left (1+a x +i \sqrt {-a^{2} x^{2}+1}\right )}{2 c^{2}}-\frac {3 i \arccos \left (a x \right )^{2} \operatorname {polylog}\left (2, -a x -i \sqrt {-a^{2} x^{2}+1}\right )}{2 c^{2}}+\frac {3 \arccos \left (a x \right ) \operatorname {polylog}\left (3, -a x -i \sqrt {-a^{2} x^{2}+1}\right )}{c^{2}}+\frac {3 i \operatorname {polylog}\left (4, -a x -i \sqrt {-a^{2} x^{2}+1}\right )}{c^{2}}-\frac {3 \arccos \left (a x \right ) \ln \left (1-a x -i \sqrt {-a^{2} x^{2}+1}\right )}{c^{2}}+\frac {3 i \operatorname {polylog}\left (2, a x +i \sqrt {-a^{2} x^{2}+1}\right )}{c^{2}}+\frac {3 \arccos \left (a x \right ) \ln \left (1+a x +i \sqrt {-a^{2} x^{2}+1}\right )}{c^{2}}-\frac {3 i \operatorname {polylog}\left (2, -a x -i \sqrt {-a^{2} x^{2}+1}\right )}{c^{2}}}{a}\)	\(414\)

Input:

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

Output:

1/a*(-1/2/(a^2*x^2-1)*arccos(a*x)^2*(a*x*arccos(a*x)+3*(-a^2*x^2+1)^(1/2)) 
/c^2-1/2/c^2*arccos(a*x)^3*ln(1-a*x-I*(-a^2*x^2+1)^(1/2))+3/2*I/c^2*arccos 
(a*x)^2*polylog(2,a*x+I*(-a^2*x^2+1)^(1/2))-3/c^2*arccos(a*x)*polylog(3,a* 
x+I*(-a^2*x^2+1)^(1/2))-3*I/c^2*polylog(4,a*x+I*(-a^2*x^2+1)^(1/2))+1/2/c^ 
2*arccos(a*x)^3*ln(1+a*x+I*(-a^2*x^2+1)^(1/2))-3/2*I/c^2*arccos(a*x)^2*pol 
ylog(2,-a*x-I*(-a^2*x^2+1)^(1/2))+3/c^2*arccos(a*x)*polylog(3,-a*x-I*(-a^2 
*x^2+1)^(1/2))+3*I/c^2*polylog(4,-a*x-I*(-a^2*x^2+1)^(1/2))-3/c^2*arccos(a 
*x)*ln(1-a*x-I*(-a^2*x^2+1)^(1/2))+3*I/c^2*polylog(2,a*x+I*(-a^2*x^2+1)^(1 
/2))+3/c^2*arccos(a*x)*ln(1+a*x+I*(-a^2*x^2+1)^(1/2))-3*I/c^2*polylog(2,-a 
*x-I*(-a^2*x^2+1)^(1/2)))
 

Fricas [F]

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

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

Output:

integral(arccos(a*x)^3/(a^4*c^2*x^4 - 2*a^2*c^2*x^2 + c^2), x)
 

Sympy [F]

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

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

Output:

Integral(acos(a*x)**3/(a**4*x**4 - 2*a**2*x**2 + 1), x)/c**2
 

Maxima [F]

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

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

Output:

-1/4*((2*a*x - (a^2*x^2 - 1)*log(a*x + 1) + (a^2*x^2 - 1)*log(-a*x + 1))*a 
rctan2(sqrt(a*x + 1)*sqrt(-a*x + 1), a*x)^3 + 4*(a^3*c^2*x^2 - a*c^2)*inte 
grate(-3/4*(2*a*x - (a^2*x^2 - 1)*log(a*x + 1) + (a^2*x^2 - 1)*log(-a*x + 
1))*sqrt(a*x + 1)*sqrt(-a*x + 1)*arctan2(sqrt(a*x + 1)*sqrt(-a*x + 1), a*x 
)^2/(a^4*c^2*x^4 - 2*a^2*c^2*x^2 + c^2), x))/(a^3*c^2*x^2 - a*c^2)
 

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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