\(\int \frac {x^4 (a+b \arccos (c x))^2}{(d-c^2 d x^2)^2} \, dx\) [194]

Optimal result

Integrand size = 27, antiderivative size = 300 \[ \int \frac {x^4 (a+b \arccos (c x))^2}{\left (d-c^2 d x^2\right )^2} \, dx=-\frac {2 b^2 x}{c^4 d^2}-\frac {b (a+b \arccos (c x))}{c^5 d^2 \sqrt {1-c^2 x^2}}+\frac {2 b \sqrt {1-c^2 x^2} (a+b \arccos (c x))}{c^5 d^2}+\frac {3 x (a+b \arccos (c x))^2}{2 c^4 d^2}+\frac {x^3 (a+b \arccos (c x))^2}{2 c^2 d^2 \left (1-c^2 x^2\right )}+\frac {3 i (a+b \arccos (c x))^2 \arctan \left (e^{i \arccos (c x)}\right )}{c^5 d^2}+\frac {b^2 \text {arctanh}(c x)}{c^5 d^2}-\frac {3 i b (a+b \arccos (c x)) \operatorname {PolyLog}\left (2,-i e^{i \arccos (c x)}\right )}{c^5 d^2}+\frac {3 i b (a+b \arccos (c x)) \operatorname {PolyLog}\left (2,i e^{i \arccos (c x)}\right )}{c^5 d^2}+\frac {3 b^2 \operatorname {PolyLog}\left (3,-i e^{i \arccos (c x)}\right )}{c^5 d^2}-\frac {3 b^2 \operatorname {PolyLog}\left (3,i e^{i \arccos (c x)}\right )}{c^5 d^2} \] Output:

-2*b^2*x/c^4/d^2-b*(a+b*arccos(c*x))/c^5/d^2/(-c^2*x^2+1)^(1/2)+2*b*(-c^2* 
x^2+1)^(1/2)*(a+b*arccos(c*x))/c^5/d^2+3/2*x*(a+b*arccos(c*x))^2/c^4/d^2+1 
/2*x^3*(a+b*arccos(c*x))^2/c^2/d^2/(-c^2*x^2+1)+3*I*(a+b*arccos(c*x))^2*ar 
ctan(c*x+I*(-c^2*x^2+1)^(1/2))/c^5/d^2+b^2*arctanh(c*x)/c^5/d^2-3*I*b*(a+b 
*arccos(c*x))*polylog(2,-I*(c*x+I*(-c^2*x^2+1)^(1/2)))/c^5/d^2+3*I*b*(a+b* 
arccos(c*x))*polylog(2,I*(c*x+I*(-c^2*x^2+1)^(1/2)))/c^5/d^2+3*b^2*polylog 
(3,-I*(c*x+I*(-c^2*x^2+1)^(1/2)))/c^5/d^2-3*b^2*polylog(3,I*(c*x+I*(-c^2*x 
^2+1)^(1/2)))/c^5/d^2
 

Mathematica [A] (verified)

Time = 4.67 (sec) , antiderivative size = 513, normalized size of antiderivative = 1.71 \[ \int \frac {x^4 (a+b \arccos (c x))^2}{\left (d-c^2 d x^2\right )^2} \, dx=\frac {\frac {8 a^2 x}{c^4}+\frac {4 a^2 x}{c^4-c^6 x^2}+\frac {6 a^2 \log (1-c x)}{c^5}-\frac {6 a^2 \log (1+c x)}{c^5}+\frac {8 a b \left (\sqrt {1-c^2 x^2}-2 c^2 x^2 \sqrt {1-c^2 x^2}-3 c x \arccos (c x)+2 c^3 x^3 \arccos (c x)-3 \arccos (c x) \log \left (1-e^{i \arccos (c x)}\right )+3 c^2 x^2 \arccos (c x) \log \left (1-e^{i \arccos (c x)}\right )+3 \arccos (c x) \log \left (1+e^{i \arccos (c x)}\right )-3 c^2 x^2 \arccos (c x) \log \left (1+e^{i \arccos (c x)}\right )+3 i \left (-1+c^2 x^2\right ) \operatorname {PolyLog}\left (2,-e^{i \arccos (c x)}\right )-3 i \left (-1+c^2 x^2\right ) \operatorname {PolyLog}\left (2,e^{i \arccos (c x)}\right )\right )}{c^5 \left (-1+c^2 x^2\right )}+\frac {b^2 \left (-16 \sqrt {1-c^2 x^2} \arccos (c x)+8 c x \left (-2+\arccos (c x)^2\right )+4 \arccos (c x) \cot \left (\frac {1}{2} \arccos (c x)\right )+\arccos (c x)^2 \csc ^2\left (\frac {1}{2} \arccos (c x)\right )-8 \log \left (\tan \left (\frac {1}{2} \arccos (c x)\right )\right )+12 \left (\arccos (c x)^2 \left (\log \left (1-e^{i \arccos (c x)}\right )-\log \left (1+e^{i \arccos (c x)}\right )\right )+2 i \arccos (c x) \left (\operatorname {PolyLog}\left (2,-e^{i \arccos (c x)}\right )-\operatorname {PolyLog}\left (2,e^{i \arccos (c x)}\right )\right )+2 \left (-\operatorname {PolyLog}\left (3,-e^{i \arccos (c x)}\right )+\operatorname {PolyLog}\left (3,e^{i \arccos (c x)}\right )\right )\right )-\arccos (c x)^2 \sec ^2\left (\frac {1}{2} \arccos (c x)\right )+4 \arccos (c x) \tan \left (\frac {1}{2} \arccos (c x)\right )\right )}{c^5}}{8 d^2} \] Input:

Integrate[(x^4*(a + b*ArcCos[c*x])^2)/(d - c^2*d*x^2)^2,x]
 

Output:

((8*a^2*x)/c^4 + (4*a^2*x)/(c^4 - c^6*x^2) + (6*a^2*Log[1 - c*x])/c^5 - (6 
*a^2*Log[1 + c*x])/c^5 + (8*a*b*(Sqrt[1 - c^2*x^2] - 2*c^2*x^2*Sqrt[1 - c^ 
2*x^2] - 3*c*x*ArcCos[c*x] + 2*c^3*x^3*ArcCos[c*x] - 3*ArcCos[c*x]*Log[1 - 
 E^(I*ArcCos[c*x])] + 3*c^2*x^2*ArcCos[c*x]*Log[1 - E^(I*ArcCos[c*x])] + 3 
*ArcCos[c*x]*Log[1 + E^(I*ArcCos[c*x])] - 3*c^2*x^2*ArcCos[c*x]*Log[1 + E^ 
(I*ArcCos[c*x])] + (3*I)*(-1 + c^2*x^2)*PolyLog[2, -E^(I*ArcCos[c*x])] - ( 
3*I)*(-1 + c^2*x^2)*PolyLog[2, E^(I*ArcCos[c*x])]))/(c^5*(-1 + c^2*x^2)) + 
 (b^2*(-16*Sqrt[1 - c^2*x^2]*ArcCos[c*x] + 8*c*x*(-2 + ArcCos[c*x]^2) + 4* 
ArcCos[c*x]*Cot[ArcCos[c*x]/2] + ArcCos[c*x]^2*Csc[ArcCos[c*x]/2]^2 - 8*Lo 
g[Tan[ArcCos[c*x]/2]] + 12*(ArcCos[c*x]^2*(Log[1 - E^(I*ArcCos[c*x])] - Lo 
g[1 + E^(I*ArcCos[c*x])]) + (2*I)*ArcCos[c*x]*(PolyLog[2, -E^(I*ArcCos[c*x 
])] - PolyLog[2, E^(I*ArcCos[c*x])]) + 2*(-PolyLog[3, -E^(I*ArcCos[c*x])] 
+ PolyLog[3, E^(I*ArcCos[c*x])])) - ArcCos[c*x]^2*Sec[ArcCos[c*x]/2]^2 + 4 
*ArcCos[c*x]*Tan[ArcCos[c*x]/2]))/c^5)/(8*d^2)
 

Rubi [A] (verified)

Time = 1.91 (sec) , antiderivative size = 299, normalized size of antiderivative = 1.00, number of steps used = 16, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.556, Rules used = {5207, 27, 5195, 27, 299, 219, 5211, 5165, 3042, 4671, 3011, 2720, 5183, 24, 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.

\(\displaystyle \int \frac {x^4 (a+b \arccos (c x))^2}{\left (d-c^2 d x^2\right )^2} \, dx\)

\(\Big \downarrow \) 5207

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 5195

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 299

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

\(\Big \downarrow \) 219

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

\(\Big \downarrow \) 5211

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

\(\Big \downarrow \) 5165

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4671

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

\(\Big \downarrow \) 3011

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

\(\Big \downarrow \) 2720

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

\(\Big \downarrow \) 5183

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

\(\Big \downarrow \) 24

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

\(\Big \downarrow \) 7143

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

Input:

Int[(x^4*(a + b*ArcCos[c*x])^2)/(d - c^2*d*x^2)^2,x]
 

Output:

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

Defintions of rubi rules used

rule 24

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]
 

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 219

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* 
ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt 
Q[a, 0] || LtQ[b, 0])
 

rule 299

Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2), x_Symbol] :> Simp[d*x 
*((a + b*x^2)^(p + 1)/(b*(2*p + 3))), x] - Simp[(a*d - b*c*(2*p + 3))/(b*(2 
*p + 3))   Int[(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - 
 a*d, 0] && NeQ[2*p + 3, 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 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 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 5195

Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))*(x_)^(m_)*((d_) + (e_.)*(x_)^2)^(p_) 
, x_Symbol] :> With[{u = IntHide[x^m*(d + e*x^2)^p, x]}, Simp[(a + b*ArcCos 
[c*x])   u, x] + Simp[b*c*Simp[Sqrt[d + e*x^2]/Sqrt[1 - c^2*x^2]]   Int[Sim 
plifyIntegrand[u/Sqrt[d + e*x^2], x], x], x]] /; FreeQ[{a, b, c, d, e}, x] 
&& EqQ[c^2*d + e, 0] && IntegerQ[p - 1/2] && NeQ[p, -2^(-1)] && (IGtQ[(m + 
1)/2, 0] || ILtQ[(m + 2*p + 3)/2, 0])
 

rule 5207

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/(2*e*(p + 1))), x] + (-Simp[f^2*((m - 1)/(2*e*(p + 1))) 
 Int[(f*x)^(m - 2)*(d + e*x^2)^(p + 1)*(a + b*ArcCos[c*x])^n, x], x] - Simp 
[b*f*(n/(2*c*(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}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && LtQ[p, -1] && IG 
tQ[m, 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]
 

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]
 

Maple [A] (verified)

Time = 0.64 (sec) , antiderivative size = 599, normalized size of antiderivative = 2.00

method	result	size
derivativedivides	\(\frac {\frac {a^{2} \left (c x -\frac {1}{4 \left (c x -1\right )}+\frac {3 \ln \left (c x -1\right )}{4}-\frac {1}{4 \left (c x +1\right )}-\frac {3 \ln \left (c x +1\right )}{4}\right )}{d^{2}}-\frac {2 b^{2} \arccos \left (c x \right ) \sqrt {-c^{2} x^{2}+1}}{d^{2}}+\frac {b^{2} \arccos \left (c x \right )^{2} c x}{d^{2}}-\frac {2 b^{2} c x}{d^{2}}-\frac {b^{2} \arccos \left (c x \right )^{2} c x}{2 d^{2} \left (c^{2} x^{2}-1\right )}-\frac {b^{2} \arccos \left (c x \right ) \sqrt {-c^{2} x^{2}+1}}{d^{2} \left (c^{2} x^{2}-1\right )}+\frac {2 b^{2} \operatorname {arctanh}\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )}{d^{2}}-\frac {3 b^{2} \arccos \left (c x \right )^{2} \ln \left (1+c x +i \sqrt {-c^{2} x^{2}+1}\right )}{2 d^{2}}-\frac {3 i b^{2} \arccos \left (c x \right ) \operatorname {polylog}\left (2, c x +i \sqrt {-c^{2} x^{2}+1}\right )}{d^{2}}-\frac {3 b^{2} \operatorname {polylog}\left (3, -c x -i \sqrt {-c^{2} x^{2}+1}\right )}{d^{2}}+\frac {3 b^{2} \arccos \left (c x \right )^{2} \ln \left (1-c x -i \sqrt {-c^{2} x^{2}+1}\right )}{2 d^{2}}+\frac {3 i a b \operatorname {polylog}\left (2, -c x -i \sqrt {-c^{2} x^{2}+1}\right )}{d^{2}}+\frac {3 b^{2} \operatorname {polylog}\left (3, c x +i \sqrt {-c^{2} x^{2}+1}\right )}{d^{2}}-\frac {2 a b \sqrt {-c^{2} x^{2}+1}}{d^{2}}+\frac {2 a b \arccos \left (c x \right ) c x}{d^{2}}-\frac {a b \arccos \left (c x \right ) c x}{d^{2} \left (c^{2} x^{2}-1\right )}-\frac {a b \sqrt {-c^{2} x^{2}+1}}{d^{2} \left (c^{2} x^{2}-1\right )}+\frac {3 a b \arccos \left (c x \right ) \ln \left (1-c x -i \sqrt {-c^{2} x^{2}+1}\right )}{d^{2}}-\frac {3 i a b \operatorname {polylog}\left (2, c x +i \sqrt {-c^{2} x^{2}+1}\right )}{d^{2}}-\frac {3 a b \arccos \left (c x \right ) \ln \left (1+c x +i \sqrt {-c^{2} x^{2}+1}\right )}{d^{2}}+\frac {3 i b^{2} \arccos \left (c x \right ) \operatorname {polylog}\left (2, -c x -i \sqrt {-c^{2} x^{2}+1}\right )}{d^{2}}}{c^{5}}\)	\(599\)
default	\(\frac {\frac {a^{2} \left (c x -\frac {1}{4 \left (c x -1\right )}+\frac {3 \ln \left (c x -1\right )}{4}-\frac {1}{4 \left (c x +1\right )}-\frac {3 \ln \left (c x +1\right )}{4}\right )}{d^{2}}-\frac {2 b^{2} \arccos \left (c x \right ) \sqrt {-c^{2} x^{2}+1}}{d^{2}}+\frac {b^{2} \arccos \left (c x \right )^{2} c x}{d^{2}}-\frac {2 b^{2} c x}{d^{2}}-\frac {b^{2} \arccos \left (c x \right )^{2} c x}{2 d^{2} \left (c^{2} x^{2}-1\right )}-\frac {b^{2} \arccos \left (c x \right ) \sqrt {-c^{2} x^{2}+1}}{d^{2} \left (c^{2} x^{2}-1\right )}+\frac {2 b^{2} \operatorname {arctanh}\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )}{d^{2}}-\frac {3 b^{2} \arccos \left (c x \right )^{2} \ln \left (1+c x +i \sqrt {-c^{2} x^{2}+1}\right )}{2 d^{2}}-\frac {3 i b^{2} \arccos \left (c x \right ) \operatorname {polylog}\left (2, c x +i \sqrt {-c^{2} x^{2}+1}\right )}{d^{2}}-\frac {3 b^{2} \operatorname {polylog}\left (3, -c x -i \sqrt {-c^{2} x^{2}+1}\right )}{d^{2}}+\frac {3 b^{2} \arccos \left (c x \right )^{2} \ln \left (1-c x -i \sqrt {-c^{2} x^{2}+1}\right )}{2 d^{2}}+\frac {3 i a b \operatorname {polylog}\left (2, -c x -i \sqrt {-c^{2} x^{2}+1}\right )}{d^{2}}+\frac {3 b^{2} \operatorname {polylog}\left (3, c x +i \sqrt {-c^{2} x^{2}+1}\right )}{d^{2}}-\frac {2 a b \sqrt {-c^{2} x^{2}+1}}{d^{2}}+\frac {2 a b \arccos \left (c x \right ) c x}{d^{2}}-\frac {a b \arccos \left (c x \right ) c x}{d^{2} \left (c^{2} x^{2}-1\right )}-\frac {a b \sqrt {-c^{2} x^{2}+1}}{d^{2} \left (c^{2} x^{2}-1\right )}+\frac {3 a b \arccos \left (c x \right ) \ln \left (1-c x -i \sqrt {-c^{2} x^{2}+1}\right )}{d^{2}}-\frac {3 i a b \operatorname {polylog}\left (2, c x +i \sqrt {-c^{2} x^{2}+1}\right )}{d^{2}}-\frac {3 a b \arccos \left (c x \right ) \ln \left (1+c x +i \sqrt {-c^{2} x^{2}+1}\right )}{d^{2}}+\frac {3 i b^{2} \arccos \left (c x \right ) \operatorname {polylog}\left (2, -c x -i \sqrt {-c^{2} x^{2}+1}\right )}{d^{2}}}{c^{5}}\)	\(599\)
parts	\(\frac {a^{2} \left (\frac {x}{c^{4}}-\frac {1}{4 c^{5} \left (c x -1\right )}+\frac {3 \ln \left (c x -1\right )}{4 c^{5}}-\frac {1}{4 c^{5} \left (c x +1\right )}-\frac {3 \ln \left (c x +1\right )}{4 c^{5}}\right )}{d^{2}}+\frac {b^{2} \arccos \left (c x \right )^{2} x}{d^{2} c^{4}}-\frac {2 b^{2} \arccos \left (c x \right ) \sqrt {-c^{2} x^{2}+1}}{d^{2} c^{5}}-\frac {2 b^{2} x}{c^{4} d^{2}}-\frac {b^{2} \arccos \left (c x \right )^{2} x}{2 d^{2} c^{4} \left (c^{2} x^{2}-1\right )}-\frac {b^{2} \arccos \left (c x \right ) \sqrt {-c^{2} x^{2}+1}}{d^{2} c^{5} \left (c^{2} x^{2}-1\right )}+\frac {2 b^{2} \operatorname {arctanh}\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )}{d^{2} c^{5}}-\frac {3 b^{2} \arccos \left (c x \right )^{2} \ln \left (1+c x +i \sqrt {-c^{2} x^{2}+1}\right )}{2 d^{2} c^{5}}-\frac {3 i a b \operatorname {polylog}\left (2, c x +i \sqrt {-c^{2} x^{2}+1}\right )}{d^{2} c^{5}}-\frac {3 b^{2} \operatorname {polylog}\left (3, -c x -i \sqrt {-c^{2} x^{2}+1}\right )}{d^{2} c^{5}}+\frac {3 b^{2} \arccos \left (c x \right )^{2} \ln \left (1-c x -i \sqrt {-c^{2} x^{2}+1}\right )}{2 d^{2} c^{5}}+\frac {3 i b^{2} \arccos \left (c x \right ) \operatorname {polylog}\left (2, -c x -i \sqrt {-c^{2} x^{2}+1}\right )}{d^{2} c^{5}}+\frac {3 b^{2} \operatorname {polylog}\left (3, c x +i \sqrt {-c^{2} x^{2}+1}\right )}{d^{2} c^{5}}+\frac {2 a b \arccos \left (c x \right ) x}{d^{2} c^{4}}-\frac {2 a b \sqrt {-c^{2} x^{2}+1}}{d^{2} c^{5}}-\frac {a b \arccos \left (c x \right ) x}{d^{2} c^{4} \left (c^{2} x^{2}-1\right )}-\frac {a b \sqrt {-c^{2} x^{2}+1}}{d^{2} c^{5} \left (c^{2} x^{2}-1\right )}+\frac {3 a b \arccos \left (c x \right ) \ln \left (1-c x -i \sqrt {-c^{2} x^{2}+1}\right )}{d^{2} c^{5}}-\frac {3 i b^{2} \arccos \left (c x \right ) \operatorname {polylog}\left (2, c x +i \sqrt {-c^{2} x^{2}+1}\right )}{d^{2} c^{5}}-\frac {3 a b \arccos \left (c x \right ) \ln \left (1+c x +i \sqrt {-c^{2} x^{2}+1}\right )}{d^{2} c^{5}}+\frac {3 i a b \operatorname {polylog}\left (2, -c x -i \sqrt {-c^{2} x^{2}+1}\right )}{d^{2} c^{5}}\)	\(664\)

Input:

int(x^4*(a+b*arccos(c*x))^2/(-c^2*d*x^2+d)^2,x,method=_RETURNVERBOSE)
 

Output:

1/c^5*(a^2/d^2*(c*x-1/4/(c*x-1)+3/4*ln(c*x-1)-1/4/(c*x+1)-3/4*ln(c*x+1))-2 
*b^2/d^2*arccos(c*x)*(-c^2*x^2+1)^(1/2)+b^2/d^2*arccos(c*x)^2*c*x-2*b^2/d^ 
2*c*x-1/2*b^2/d^2/(c^2*x^2-1)*arccos(c*x)^2*c*x-b^2/d^2/(c^2*x^2-1)*arccos 
(c*x)*(-c^2*x^2+1)^(1/2)+2*b^2/d^2*arctanh(c*x+I*(-c^2*x^2+1)^(1/2))-3/2*b 
^2/d^2*arccos(c*x)^2*ln(1+c*x+I*(-c^2*x^2+1)^(1/2))-3*I*b^2/d^2*arccos(c*x 
)*polylog(2,c*x+I*(-c^2*x^2+1)^(1/2))-3*b^2/d^2*polylog(3,-c*x-I*(-c^2*x^2 
+1)^(1/2))+3/2*b^2/d^2*arccos(c*x)^2*ln(1-c*x-I*(-c^2*x^2+1)^(1/2))+3*I*a* 
b/d^2*polylog(2,-c*x-I*(-c^2*x^2+1)^(1/2))+3*b^2/d^2*polylog(3,c*x+I*(-c^2 
*x^2+1)^(1/2))-2*a*b/d^2*(-c^2*x^2+1)^(1/2)+2*a*b/d^2*arccos(c*x)*c*x-a*b/ 
d^2/(c^2*x^2-1)*arccos(c*x)*c*x-a*b/d^2/(c^2*x^2-1)*(-c^2*x^2+1)^(1/2)+3*a 
*b/d^2*arccos(c*x)*ln(1-c*x-I*(-c^2*x^2+1)^(1/2))-3*I*a*b/d^2*polylog(2,c* 
x+I*(-c^2*x^2+1)^(1/2))-3*a*b/d^2*arccos(c*x)*ln(1+c*x+I*(-c^2*x^2+1)^(1/2 
))+3*I*b^2/d^2*arccos(c*x)*polylog(2,-c*x-I*(-c^2*x^2+1)^(1/2)))
 

Fricas [F]

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

integrate(x^4*(a+b*arccos(c*x))^2/(-c^2*d*x^2+d)^2,x, algorithm="fricas")
 

Output:

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

Sympy [F]

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

integrate(x**4*(a+b*acos(c*x))**2/(-c**2*d*x**2+d)**2,x)
 

Output:

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

Maxima [F]

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

integrate(x^4*(a+b*arccos(c*x))^2/(-c^2*d*x^2+d)^2,x, algorithm="maxima")
 

Output:

-1/4*a^2*(2*x/(c^6*d^2*x^2 - c^4*d^2) - 4*x/(c^4*d^2) + 3*log(c*x + 1)/(c^ 
5*d^2) - 3*log(c*x - 1)/(c^5*d^2)) + 1/4*((4*b^2*c^3*x^3 - 6*b^2*c*x - 3*( 
b^2*c^2*x^2 - b^2)*log(c*x + 1) + 3*(b^2*c^2*x^2 - b^2)*log(-c*x + 1))*arc 
tan2(sqrt(c*x + 1)*sqrt(-c*x + 1), c*x)^2 + 4*(c^7*d^2*x^2 - c^5*d^2)*inte 
grate(1/2*(4*a*b*c^4*x^4*arctan2(sqrt(c*x + 1)*sqrt(-c*x + 1), c*x) - (4*b 
^2*c^3*x^3 - 6*b^2*c*x - 3*(b^2*c^2*x^2 - b^2)*log(c*x + 1) + 3*(b^2*c^2*x 
^2 - b^2)*log(-c*x + 1))*sqrt(c*x + 1)*sqrt(-c*x + 1)*arctan2(sqrt(c*x + 1 
)*sqrt(-c*x + 1), c*x))/(c^8*d^2*x^4 - 2*c^6*d^2*x^2 + c^4*d^2), x))/(c^7* 
d^2*x^2 - c^5*d^2)
 

Giac [F]

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

integrate(x^4*(a+b*arccos(c*x))^2/(-c^2*d*x^2+d)^2,x, algorithm="giac")
 

Output:

integrate((b*arccos(c*x) + a)^2*x^4/(c^2*d*x^2 - d)^2, x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {x^4 (a+b \arccos (c x))^2}{\left (d-c^2 d x^2\right )^2} \, dx=\int \frac {x^4\,{\left (a+b\,\mathrm {acos}\left (c\,x\right )\right )}^2}{{\left (d-c^2\,d\,x^2\right )}^2} \,d x \] Input:

int((x^4*(a + b*acos(c*x))^2)/(d - c^2*d*x^2)^2,x)
 

Output:

int((x^4*(a + b*acos(c*x))^2)/(d - c^2*d*x^2)^2, x)
 

Reduce [F]

\[ \int \frac {x^4 (a+b \arccos (c x))^2}{\left (d-c^2 d x^2\right )^2} \, dx=\frac {8 \left (\int \frac {\mathit {acos} \left (c x \right ) x^{4}}{c^{4} x^{4}-2 c^{2} x^{2}+1}d x \right ) a b \,c^{7} x^{2}-8 \left (\int \frac {\mathit {acos} \left (c x \right ) x^{4}}{c^{4} x^{4}-2 c^{2} x^{2}+1}d x \right ) a b \,c^{5}+4 \left (\int \frac {\mathit {acos} \left (c x \right )^{2} x^{4}}{c^{4} x^{4}-2 c^{2} x^{2}+1}d x \right ) b^{2} c^{7} x^{2}-4 \left (\int \frac {\mathit {acos} \left (c x \right )^{2} x^{4}}{c^{4} x^{4}-2 c^{2} x^{2}+1}d x \right ) b^{2} c^{5}+3 \,\mathrm {log}\left (c^{2} x -c \right ) a^{2} c^{2} x^{2}-3 \,\mathrm {log}\left (c^{2} x -c \right ) a^{2}-3 \,\mathrm {log}\left (c^{2} x +c \right ) a^{2} c^{2} x^{2}+3 \,\mathrm {log}\left (c^{2} x +c \right ) a^{2}+4 a^{2} c^{3} x^{3}-6 a^{2} c x}{4 c^{5} d^{2} \left (c^{2} x^{2}-1\right )} \] Input:

int(x^4*(a+b*acos(c*x))^2/(-c^2*d*x^2+d)^2,x)
 

Output:

(8*int((acos(c*x)*x**4)/(c**4*x**4 - 2*c**2*x**2 + 1),x)*a*b*c**7*x**2 - 8 
*int((acos(c*x)*x**4)/(c**4*x**4 - 2*c**2*x**2 + 1),x)*a*b*c**5 + 4*int((a 
cos(c*x)**2*x**4)/(c**4*x**4 - 2*c**2*x**2 + 1),x)*b**2*c**7*x**2 - 4*int( 
(acos(c*x)**2*x**4)/(c**4*x**4 - 2*c**2*x**2 + 1),x)*b**2*c**5 + 3*log(c** 
2*x - c)*a**2*c**2*x**2 - 3*log(c**2*x - c)*a**2 - 3*log(c**2*x + c)*a**2* 
c**2*x**2 + 3*log(c**2*x + c)*a**2 + 4*a**2*c**3*x**3 - 6*a**2*c*x)/(4*c** 
5*d**2*(c**2*x**2 - 1))