\(\int \frac {(d-c^2 d x^2)^2 (a+b \arccos (c x))^2}{x} \, dx\) [172]

Optimal result

Integrand size = 27, antiderivative size = 277 \[ \int \frac {\left (d-c^2 d x^2\right )^2 (a+b \arccos (c x))^2}{x} \, dx=\frac {11}{32} b^2 c^2 d^2 x^2-\frac {1}{32} b^2 d^2 \left (1-c^2 x^2\right )^2-\frac {11}{16} b c d^2 x \sqrt {1-c^2 x^2} (a+b \arccos (c x))-\frac {1}{8} b c d^2 x \left (1-c^2 x^2\right )^{3/2} (a+b \arccos (c x))-\frac {11}{32} d^2 (a+b \arccos (c x))^2+\frac {1}{2} d^2 \left (1-c^2 x^2\right ) (a+b \arccos (c x))^2+\frac {1}{4} d^2 \left (1-c^2 x^2\right )^2 (a+b \arccos (c x))^2-\frac {i d^2 (a+b \arccos (c x))^3}{3 b}+d^2 (a+b \arccos (c x))^2 \log \left (1-e^{2 i \arccos (c x)}\right )-i b d^2 (a+b \arccos (c x)) \operatorname {PolyLog}\left (2,e^{2 i \arccos (c x)}\right )+\frac {1}{2} b^2 d^2 \operatorname {PolyLog}\left (3,e^{2 i \arccos (c x)}\right ) \] Output:

11/32*b^2*c^2*d^2*x^2-1/32*b^2*d^2*(-c^2*x^2+1)^2-11/16*b*c*d^2*x*(-c^2*x^ 
2+1)^(1/2)*(a+b*arccos(c*x))-1/8*b*c*d^2*x*(-c^2*x^2+1)^(3/2)*(a+b*arccos( 
c*x))-11/32*d^2*(a+b*arccos(c*x))^2+1/2*d^2*(-c^2*x^2+1)*(a+b*arccos(c*x)) 
^2+1/4*d^2*(-c^2*x^2+1)^2*(a+b*arccos(c*x))^2-1/3*I*d^2*(a+b*arccos(c*x))^ 
3/b+d^2*(a+b*arccos(c*x))^2*ln(1-(c*x+I*(-c^2*x^2+1)^(1/2))^2)-I*b*d^2*(a+ 
b*arccos(c*x))*polylog(2,(c*x+I*(-c^2*x^2+1)^(1/2))^2)+1/2*b^2*d^2*polylog 
(3,(c*x+I*(-c^2*x^2+1)^(1/2))^2)
 

Mathematica [A] (verified)

Time = 0.48 (sec) , antiderivative size = 345, normalized size of antiderivative = 1.25 \[ \int \frac {\left (d-c^2 d x^2\right )^2 (a+b \arccos (c x))^2}{x} \, dx=\frac {1}{768} d^2 \left (-768 a^2 c^2 x^2+192 a^2 c^4 x^4+624 a b c x \sqrt {1-c^2 x^2}-96 a b c^3 x^3 \sqrt {1-c^2 x^2}-1536 a b c^2 x^2 \arccos (c x)+384 a b c^4 x^4 \arccos (c x)-768 i a b \arccos (c x)^2-256 i b^2 \arccos (c x)^3-1248 a b \arctan \left (\frac {c x}{-1+\sqrt {1-c^2 x^2}}\right )+144 b^2 \cos (2 \arccos (c x))-288 b^2 \arccos (c x)^2 \cos (2 \arccos (c x))-3 b^2 \cos (4 \arccos (c x))+24 b^2 \arccos (c x)^2 \cos (4 \arccos (c x))+1536 a b \arccos (c x) \log \left (1+e^{2 i \arccos (c x)}\right )+768 b^2 \arccos (c x)^2 \log \left (1+e^{2 i \arccos (c x)}\right )+768 a^2 \log (c x)-768 i b (a+b \arccos (c x)) \operatorname {PolyLog}\left (2,-e^{2 i \arccos (c x)}\right )+384 b^2 \operatorname {PolyLog}\left (3,-e^{2 i \arccos (c x)}\right )+288 b^2 \arccos (c x) \sin (2 \arccos (c x))-12 b^2 \arccos (c x) \sin (4 \arccos (c x))\right ) \] Input:

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

Output:

(d^2*(-768*a^2*c^2*x^2 + 192*a^2*c^4*x^4 + 624*a*b*c*x*Sqrt[1 - c^2*x^2] - 
 96*a*b*c^3*x^3*Sqrt[1 - c^2*x^2] - 1536*a*b*c^2*x^2*ArcCos[c*x] + 384*a*b 
*c^4*x^4*ArcCos[c*x] - (768*I)*a*b*ArcCos[c*x]^2 - (256*I)*b^2*ArcCos[c*x] 
^3 - 1248*a*b*ArcTan[(c*x)/(-1 + Sqrt[1 - c^2*x^2])] + 144*b^2*Cos[2*ArcCo 
s[c*x]] - 288*b^2*ArcCos[c*x]^2*Cos[2*ArcCos[c*x]] - 3*b^2*Cos[4*ArcCos[c* 
x]] + 24*b^2*ArcCos[c*x]^2*Cos[4*ArcCos[c*x]] + 1536*a*b*ArcCos[c*x]*Log[1 
 + E^((2*I)*ArcCos[c*x])] + 768*b^2*ArcCos[c*x]^2*Log[1 + E^((2*I)*ArcCos[ 
c*x])] + 768*a^2*Log[c*x] - (768*I)*b*(a + b*ArcCos[c*x])*PolyLog[2, -E^(( 
2*I)*ArcCos[c*x])] + 384*b^2*PolyLog[3, -E^((2*I)*ArcCos[c*x])] + 288*b^2* 
ArcCos[c*x]*Sin[2*ArcCos[c*x]] - 12*b^2*ArcCos[c*x]*Sin[4*ArcCos[c*x]]))/7 
68
 

Rubi [A] (verified)

Time = 2.23 (sec) , antiderivative size = 347, normalized size of antiderivative = 1.25, number of steps used = 20, number of rules used = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.704, Rules used = {5203, 27, 5159, 244, 2009, 5157, 15, 5153, 5203, 5137, 3042, 4202, 2620, 3011, 2720, 5157, 15, 5153, 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.

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

\(\Big \downarrow \) 5203

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 5159

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

\(\Big \downarrow \) 244

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

\(\Big \downarrow \) 2009

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

\(\Big \downarrow \) 5157

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

\(\Big \downarrow \) 15

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

\(\Big \downarrow \) 5153

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

\(\Big \downarrow \) 5203

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

\(\Big \downarrow \) 5137

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4202

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

\(\Big \downarrow \) 2620

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

\(\Big \downarrow \) 3011

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

\(\Big \downarrow \) 2720

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

\(\Big \downarrow \) 5157

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

\(\Big \downarrow \) 15

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

\(\Big \downarrow \) 5153

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

\(\Big \downarrow \) 7143

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

Input:

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

Output:

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

Defintions of rubi rules used

rule 15

Int[(a_.)*(x_)^(m_.), x_Symbol] :> Simp[a*(x^(m + 1)/(m + 1)), x] /; FreeQ[ 
{a, m}, x] && NeQ[m, -1]
 

rule 27

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

rule 244

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

rule 2009

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

rule 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 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 4202

Int[((c_.) + (d_.)*(x_))^(m_.)*tan[(e_.) + (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*( 
e + f*x))/(1 + E^(2*I*(e + f*x)))), x], x] /; FreeQ[{c, d, e, f}, x] && IGt 
Q[m, 0]
 

rule 5137

Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_.)/(x_), x_Symbol] :> -Subst[Int[ 
(a + b*x)^n*Tan[x], x], x, ArcCos[c*x]] /; FreeQ[{a, b, c}, x] && IGtQ[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 5203

Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*((d_) + (e_. 
)*(x_)^2)^(p_.), x_Symbol] :> Simp[(f*x)^(m + 1)*(d + e*x^2)^p*((a + b*ArcC 
os[c*x])^n/(f*(m + 2*p + 1))), x] + (Simp[2*d*(p/(m + 2*p + 1))   Int[(f*x) 
^m*(d + e*x^2)^(p - 1)*(a + b*ArcCos[c*x])^n, x], x] + Simp[b*c*(n/(f*(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, m}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && GtQ[p, 0] &&  !LtQ[m, -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]
 

Maple [A] (verified)

Time = 0.51 (sec) , antiderivative size = 424, normalized size of antiderivative = 1.53

method	result	size
derivativedivides	\(d^{2} a^{2} \left (\frac {c^{4} x^{4}}{4}-c^{2} x^{2}+\ln \left (c x \right )\right )+d^{2} b^{2} \left (-\frac {i \arccos \left (c x \right )^{3}}{3}-\frac {3 \left (2 \arccos \left (c x \right )^{2}-1+2 i \arccos \left (c x \right )\right ) \left (2 c^{2} x^{2}-1+2 i \sqrt {-c^{2} x^{2}+1}\, x c \right )}{32}-\frac {3 \left (-2 i \sqrt {-c^{2} x^{2}+1}\, x c +2 c^{2} x^{2}-1\right ) \left (2 \arccos \left (c x \right )^{2}-1-2 i \arccos \left (c x \right )\right )}{32}+\arccos \left (c x \right )^{2} \ln \left (1+\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )^{2}\right )-i \arccos \left (c x \right ) \operatorname {polylog}\left (2, -\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )^{2}\right )+\frac {\operatorname {polylog}\left (3, -\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )^{2}\right )}{2}+\frac {\left (8 \arccos \left (c x \right )^{2}-1\right ) \cos \left (4 \arccos \left (c x \right )\right )}{256}-\frac {\arccos \left (c x \right ) \sin \left (4 \arccos \left (c x \right )\right )}{64}\right )-i d^{2} a b \arccos \left (c x \right )^{2}+\frac {3 d^{2} a b \sqrt {-c^{2} x^{2}+1}\, c x}{4}-\frac {3 d^{2} a b \arccos \left (c x \right ) c^{2} x^{2}}{2}+\frac {3 d^{2} a b \arccos \left (c x \right )}{4}+2 d^{2} a b \arccos \left (c x \right ) \ln \left (1+\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )^{2}\right )-i d^{2} a b \operatorname {polylog}\left (2, -\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )^{2}\right )+\frac {d^{2} a b \arccos \left (c x \right ) \cos \left (4 \arccos \left (c x \right )\right )}{16}-\frac {d^{2} a b \sin \left (4 \arccos \left (c x \right )\right )}{64}\)	\(424\)
default	\(d^{2} a^{2} \left (\frac {c^{4} x^{4}}{4}-c^{2} x^{2}+\ln \left (c x \right )\right )+d^{2} b^{2} \left (-\frac {i \arccos \left (c x \right )^{3}}{3}-\frac {3 \left (2 \arccos \left (c x \right )^{2}-1+2 i \arccos \left (c x \right )\right ) \left (2 c^{2} x^{2}-1+2 i \sqrt {-c^{2} x^{2}+1}\, x c \right )}{32}-\frac {3 \left (-2 i \sqrt {-c^{2} x^{2}+1}\, x c +2 c^{2} x^{2}-1\right ) \left (2 \arccos \left (c x \right )^{2}-1-2 i \arccos \left (c x \right )\right )}{32}+\arccos \left (c x \right )^{2} \ln \left (1+\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )^{2}\right )-i \arccos \left (c x \right ) \operatorname {polylog}\left (2, -\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )^{2}\right )+\frac {\operatorname {polylog}\left (3, -\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )^{2}\right )}{2}+\frac {\left (8 \arccos \left (c x \right )^{2}-1\right ) \cos \left (4 \arccos \left (c x \right )\right )}{256}-\frac {\arccos \left (c x \right ) \sin \left (4 \arccos \left (c x \right )\right )}{64}\right )-i d^{2} a b \arccos \left (c x \right )^{2}+\frac {3 d^{2} a b \sqrt {-c^{2} x^{2}+1}\, c x}{4}-\frac {3 d^{2} a b \arccos \left (c x \right ) c^{2} x^{2}}{2}+\frac {3 d^{2} a b \arccos \left (c x \right )}{4}+2 d^{2} a b \arccos \left (c x \right ) \ln \left (1+\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )^{2}\right )-i d^{2} a b \operatorname {polylog}\left (2, -\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )^{2}\right )+\frac {d^{2} a b \arccos \left (c x \right ) \cos \left (4 \arccos \left (c x \right )\right )}{16}-\frac {d^{2} a b \sin \left (4 \arccos \left (c x \right )\right )}{64}\)	\(424\)
parts	\(d^{2} a^{2} \left (\frac {c^{4} x^{4}}{4}-c^{2} x^{2}+\ln \left (x \right )\right )-i d^{2} a b \arccos \left (c x \right )^{2}+\frac {3 d^{2} b^{2} \sqrt {-c^{2} x^{2}+1}\, \arccos \left (c x \right ) x c}{4}-\frac {3 d^{2} b^{2} \arccos \left (c x \right )^{2} x^{2} c^{2}}{4}+\frac {3 b^{2} c^{2} d^{2} x^{2}}{8}+\frac {3 d^{2} b^{2} \arccos \left (c x \right )^{2}}{8}-\frac {3 d^{2} b^{2}}{16}+d^{2} b^{2} \arccos \left (c x \right )^{2} \ln \left (1+\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )^{2}\right )-\frac {i d^{2} b^{2} \arccos \left (c x \right )^{3}}{3}+\frac {d^{2} b^{2} \operatorname {polylog}\left (3, -\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )^{2}\right )}{2}+\frac {d^{2} b^{2} \cos \left (4 \arccos \left (c x \right )\right ) \arccos \left (c x \right )^{2}}{32}-\frac {d^{2} b^{2} \cos \left (4 \arccos \left (c x \right )\right )}{256}-\frac {d^{2} b^{2} \arccos \left (c x \right ) \sin \left (4 \arccos \left (c x \right )\right )}{64}-i d^{2} a b \operatorname {polylog}\left (2, -\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )^{2}\right )+\frac {3 d^{2} a b \sqrt {-c^{2} x^{2}+1}\, c x}{4}-\frac {3 d^{2} a b \arccos \left (c x \right ) c^{2} x^{2}}{2}+\frac {3 d^{2} a b \arccos \left (c x \right )}{4}+2 d^{2} a b \arccos \left (c x \right ) \ln \left (1+\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )^{2}\right )-i d^{2} b^{2} \arccos \left (c x \right ) \operatorname {polylog}\left (2, -\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )^{2}\right )+\frac {d^{2} a b \arccos \left (c x \right ) \cos \left (4 \arccos \left (c x \right )\right )}{16}-\frac {d^{2} a b \sin \left (4 \arccos \left (c x \right )\right )}{64}\)	\(451\)

Input:

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

Output:

d^2*a^2*(1/4*c^4*x^4-c^2*x^2+ln(c*x))+d^2*b^2*(-1/3*I*arccos(c*x)^3-3/32*( 
2*arccos(c*x)^2-1+2*I*arccos(c*x))*(2*c^2*x^2-1+2*I*(-c^2*x^2+1)^(1/2)*c*x 
)-3/32*(-2*I*(-c^2*x^2+1)^(1/2)*c*x+2*c^2*x^2-1)*(2*arccos(c*x)^2-1-2*I*ar 
ccos(c*x))+arccos(c*x)^2*ln(1+(c*x+I*(-c^2*x^2+1)^(1/2))^2)-I*arccos(c*x)* 
polylog(2,-(c*x+I*(-c^2*x^2+1)^(1/2))^2)+1/2*polylog(3,-(c*x+I*(-c^2*x^2+1 
)^(1/2))^2)+1/256*(8*arccos(c*x)^2-1)*cos(4*arccos(c*x))-1/64*arccos(c*x)* 
sin(4*arccos(c*x)))-I*d^2*a*b*arccos(c*x)^2+3/4*d^2*a*b*(-c^2*x^2+1)^(1/2) 
*c*x-3/2*d^2*a*b*arccos(c*x)*c^2*x^2+3/4*d^2*a*b*arccos(c*x)+2*d^2*a*b*arc 
cos(c*x)*ln(1+(c*x+I*(-c^2*x^2+1)^(1/2))^2)-I*d^2*a*b*polylog(2,-(c*x+I*(- 
c^2*x^2+1)^(1/2))^2)+1/16*d^2*a*b*arccos(c*x)*cos(4*arccos(c*x))-1/64*d^2* 
a*b*sin(4*arccos(c*x))
 

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

1/4*a^2*c^4*d^2*x^4 - a^2*c^2*d^2*x^2 + a^2*d^2*log(x) + integrate(((b^2*c 
^4*d^2*x^4 - 2*b^2*c^2*d^2*x^2 + b^2*d^2)*arctan2(sqrt(c*x + 1)*sqrt(-c*x 
+ 1), c*x)^2 + 2*(a*b*c^4*d^2*x^4 - 2*a*b*c^2*d^2*x^2 + a*b*d^2)*arctan2(s 
qrt(c*x + 1)*sqrt(-c*x + 1), c*x))/x, x)
 

Giac [F(-2)]

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

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

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

Output:

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

Reduce [F]

\[ \int \frac {\left (d-c^2 d x^2\right )^2 (a+b \arccos (c x))^2}{x} \, dx=\frac {d^{2} \left (-16 \mathit {acos} \left (c x \right )^{2} b^{2} c^{2} x^{2}+8 \mathit {acos} \left (c x \right )^{2} b^{2}+16 \sqrt {-c^{2} x^{2}+1}\, \mathit {acos} \left (c x \right ) b^{2} c x +8 \mathit {acos} \left (c x \right ) a b \,c^{4} x^{4}-32 \mathit {acos} \left (c x \right ) a b \,c^{2} x^{2}-13 \mathit {asin} \left (c x \right ) a b -2 \sqrt {-c^{2} x^{2}+1}\, a b \,c^{3} x^{3}+13 \sqrt {-c^{2} x^{2}+1}\, a b c x +32 \left (\int \frac {\mathit {acos} \left (c x \right )}{x}d x \right ) a b +16 \left (\int \frac {\mathit {acos} \left (c x \right )^{2}}{x}d x \right ) b^{2}+16 \left (\int \mathit {acos} \left (c x \right )^{2} x^{3}d x \right ) b^{2} c^{4}+16 \,\mathrm {log}\left (x \right ) a^{2}+4 a^{2} c^{4} x^{4}-16 a^{2} c^{2} x^{2}+8 b^{2} c^{2} x^{2}\right )}{16} \] Input:

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

Output:

(d**2*( - 16*acos(c*x)**2*b**2*c**2*x**2 + 8*acos(c*x)**2*b**2 + 16*sqrt( 
- c**2*x**2 + 1)*acos(c*x)*b**2*c*x + 8*acos(c*x)*a*b*c**4*x**4 - 32*acos( 
c*x)*a*b*c**2*x**2 - 13*asin(c*x)*a*b - 2*sqrt( - c**2*x**2 + 1)*a*b*c**3* 
x**3 + 13*sqrt( - c**2*x**2 + 1)*a*b*c*x + 32*int(acos(c*x)/x,x)*a*b + 16* 
int(acos(c*x)**2/x,x)*b**2 + 16*int(acos(c*x)**2*x**3,x)*b**2*c**4 + 16*lo 
g(x)*a**2 + 4*a**2*c**4*x**4 - 16*a**2*c**2*x**2 + 8*b**2*c**2*x**2))/16