\(\int \frac {(d-c^2 d x^2)^3 (a+b \arccos (c x))^2}{x^4} \, dx\) [184]

Optimal result

Integrand size = 27, antiderivative size = 348 \[ \int \frac {\left (d-c^2 d x^2\right )^3 (a+b \arccos (c x))^2}{x^4} \, dx=-\frac {b^2 c^2 d^3}{3 x}-\frac {50}{9} b^2 c^4 d^3 x+\frac {2}{27} b^2 c^6 d^3 x^3+5 b c^3 d^3 \sqrt {1-c^2 x^2} (a+b \arccos (c x))-\frac {1}{9} b c^3 d^3 \left (1-c^2 x^2\right )^{3/2} (a+b \arccos (c x))-\frac {b c d^3 \left (1-c^2 x^2\right )^{5/2} (a+b \arccos (c x))}{3 x^2}+\frac {16}{3} c^4 d^3 x (a+b \arccos (c x))^2+\frac {8}{3} c^4 d^3 x \left (1-c^2 x^2\right ) (a+b \arccos (c x))^2+\frac {2 c^2 d^3 \left (1-c^2 x^2\right )^2 (a+b \arccos (c x))^2}{x}-\frac {d^3 \left (1-c^2 x^2\right )^3 (a+b \arccos (c x))^2}{3 x^3}+\frac {34}{3} b c^3 d^3 (a+b \arccos (c x)) \text {arctanh}\left (e^{i \arccos (c x)}\right )-\frac {17}{3} i b^2 c^3 d^3 \operatorname {PolyLog}\left (2,-e^{i \arccos (c x)}\right )+\frac {17}{3} i b^2 c^3 d^3 \operatorname {PolyLog}\left (2,e^{i \arccos (c x)}\right ) \] Output:

-1/3*b^2*c^2*d^3/x-50/9*b^2*c^4*d^3*x+2/27*b^2*c^6*d^3*x^3+5*b*c^3*d^3*(-c 
^2*x^2+1)^(1/2)*(a+b*arccos(c*x))-1/9*b*c^3*d^3*(-c^2*x^2+1)^(3/2)*(a+b*ar 
ccos(c*x))-1/3*b*c*d^3*(-c^2*x^2+1)^(5/2)*(a+b*arccos(c*x))/x^2+16/3*c^4*d 
^3*x*(a+b*arccos(c*x))^2+8/3*c^4*d^3*x*(-c^2*x^2+1)*(a+b*arccos(c*x))^2+2* 
c^2*d^3*(-c^2*x^2+1)^2*(a+b*arccos(c*x))^2/x-1/3*d^3*(-c^2*x^2+1)^3*(a+b*a 
rccos(c*x))^2/x^3+34/3*b*c^3*d^3*(a+b*arccos(c*x))*arctanh(c*x+I*(-c^2*x^2 
+1)^(1/2))-17/3*I*b^2*c^3*d^3*polylog(2,-c*x-I*(-c^2*x^2+1)^(1/2))+17/3*I* 
b^2*c^3*d^3*polylog(2,c*x+I*(-c^2*x^2+1)^(1/2))
 

Mathematica [A] (verified)

Time = 0.86 (sec) , antiderivative size = 492, normalized size of antiderivative = 1.41 \[ \int \frac {\left (d-c^2 d x^2\right )^3 (a+b \arccos (c x))^2}{x^4} \, dx=-\frac {d^3 \left (9 a^2-81 a^2 c^2 x^2+9 b^2 c^2 x^2-81 a^2 c^4 x^4+150 b^2 c^4 x^4+9 a^2 c^6 x^6-2 b^2 c^6 x^6-9 a b c x \sqrt {1-c^2 x^2}+150 a b c^3 x^3 \sqrt {1-c^2 x^2}-6 a b c^5 x^5 \sqrt {1-c^2 x^2}+18 a b \arccos (c x)-162 a b c^2 x^2 \arccos (c x)-162 a b c^4 x^4 \arccos (c x)+18 a b c^6 x^6 \arccos (c x)-9 b^2 c x \sqrt {1-c^2 x^2} \arccos (c x)+150 b^2 c^3 x^3 \sqrt {1-c^2 x^2} \arccos (c x)-6 b^2 c^5 x^5 \sqrt {1-c^2 x^2} \arccos (c x)+9 b^2 \arccos (c x)^2-81 b^2 c^2 x^2 \arccos (c x)^2-81 b^2 c^4 x^4 \arccos (c x)^2+9 b^2 c^6 x^6 \arccos (c x)^2+153 a b c^3 x^3 \text {arctanh}\left (\sqrt {1-c^2 x^2}\right )+153 b^2 c^3 x^3 \arccos (c x) \log \left (1-i e^{i \arccos (c x)}\right )-153 b^2 c^3 x^3 \arccos (c x) \log \left (1+i e^{i \arccos (c x)}\right )+153 i b^2 c^3 x^3 \operatorname {PolyLog}\left (2,-i e^{i \arccos (c x)}\right )-153 i b^2 c^3 x^3 \operatorname {PolyLog}\left (2,i e^{i \arccos (c x)}\right )\right )}{27 x^3} \] Input:

Integrate[((d - c^2*d*x^2)^3*(a + b*ArcCos[c*x])^2)/x^4,x]
 

Output:

-1/27*(d^3*(9*a^2 - 81*a^2*c^2*x^2 + 9*b^2*c^2*x^2 - 81*a^2*c^4*x^4 + 150* 
b^2*c^4*x^4 + 9*a^2*c^6*x^6 - 2*b^2*c^6*x^6 - 9*a*b*c*x*Sqrt[1 - c^2*x^2] 
+ 150*a*b*c^3*x^3*Sqrt[1 - c^2*x^2] - 6*a*b*c^5*x^5*Sqrt[1 - c^2*x^2] + 18 
*a*b*ArcCos[c*x] - 162*a*b*c^2*x^2*ArcCos[c*x] - 162*a*b*c^4*x^4*ArcCos[c* 
x] + 18*a*b*c^6*x^6*ArcCos[c*x] - 9*b^2*c*x*Sqrt[1 - c^2*x^2]*ArcCos[c*x] 
+ 150*b^2*c^3*x^3*Sqrt[1 - c^2*x^2]*ArcCos[c*x] - 6*b^2*c^5*x^5*Sqrt[1 - c 
^2*x^2]*ArcCos[c*x] + 9*b^2*ArcCos[c*x]^2 - 81*b^2*c^2*x^2*ArcCos[c*x]^2 - 
 81*b^2*c^4*x^4*ArcCos[c*x]^2 + 9*b^2*c^6*x^6*ArcCos[c*x]^2 + 153*a*b*c^3* 
x^3*ArcTanh[Sqrt[1 - c^2*x^2]] + 153*b^2*c^3*x^3*ArcCos[c*x]*Log[1 - I*E^( 
I*ArcCos[c*x])] - 153*b^2*c^3*x^3*ArcCos[c*x]*Log[1 + I*E^(I*ArcCos[c*x])] 
 + (153*I)*b^2*c^3*x^3*PolyLog[2, (-I)*E^(I*ArcCos[c*x])] - (153*I)*b^2*c^ 
3*x^3*PolyLog[2, I*E^(I*ArcCos[c*x])]))/x^3
 

Rubi [A] (verified)

Time = 3.40 (sec) , antiderivative size = 564, normalized size of antiderivative = 1.62, number of steps used = 20, number of rules used = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.704, Rules used = {5201, 27, 5201, 244, 2009, 5159, 5131, 5183, 24, 2009, 5203, 2009, 5199, 24, 5219, 3042, 4669, 2715, 2838}

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.

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

\(\Big \downarrow \) 5201

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 5201

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

\(\Big \downarrow \) 244

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

\(\Big \downarrow \) 2009

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

\(\Big \downarrow \) 5159

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

\(\Big \downarrow \) 5131

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

\(\Big \downarrow \) 5183

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

\(\Big \downarrow \) 24

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

\(\Big \downarrow \) 2009

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

\(\Big \downarrow \) 5203

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

\(\Big \downarrow \) 2009

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

\(\Big \downarrow \) 5199

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

\(\Big \downarrow \) 24

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

\(\Big \downarrow \) 5219

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4669

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

\(\Big \downarrow \) 2715

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

\(\Big \downarrow \) 2838

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

Input:

Int[((d - c^2*d*x^2)^3*(a + b*ArcCos[c*x])^2)/x^4,x]
 

Output:

-1/3*(d^3*(1 - c^2*x^2)^3*(a + b*ArcCos[c*x])^2)/x^3 - 2*c^2*d^3*(-(((1 - 
c^2*x^2)^2*(a + b*ArcCos[c*x])^2)/x) - 4*c^2*((x*(1 - c^2*x^2)*(a + b*ArcC 
os[c*x])^2)/3 + (2*b*c*(-1/3*(b*(x - (c^2*x^3)/3))/c - ((1 - c^2*x^2)^(3/2 
)*(a + b*ArcCos[c*x]))/(3*c^2)))/3 + (2*(x*(a + b*ArcCos[c*x])^2 + 2*b*c*( 
-((b*x)/c) - (Sqrt[1 - c^2*x^2]*(a + b*ArcCos[c*x]))/c^2)))/3) - 2*b*c*(b* 
c*x + (b*c*(x - (c^2*x^3)/3))/3 + Sqrt[1 - c^2*x^2]*(a + b*ArcCos[c*x]) + 
((1 - c^2*x^2)^(3/2)*(a + b*ArcCos[c*x]))/3 + (2*I)*(a + b*ArcCos[c*x])*Ar 
cTan[E^(I*ArcCos[c*x])] - I*b*PolyLog[2, (-I)*E^(I*ArcCos[c*x])] + I*b*Pol 
yLog[2, I*E^(I*ArcCos[c*x])])) - (2*b*c*d^3*(-1/2*(b*c*(-x^(-1) - 2*c^2*x 
+ (c^4*x^3)/3)) - ((1 - c^2*x^2)^(5/2)*(a + b*ArcCos[c*x]))/(2*x^2) - (5*c 
^2*(b*c*x + (b*c*(x - (c^2*x^3)/3))/3 + Sqrt[1 - c^2*x^2]*(a + b*ArcCos[c* 
x]) + ((1 - c^2*x^2)^(3/2)*(a + b*ArcCos[c*x]))/3 + (2*I)*(a + b*ArcCos[c* 
x])*ArcTan[E^(I*ArcCos[c*x])] - I*b*PolyLog[2, (-I)*E^(I*ArcCos[c*x])] + I 
*b*PolyLog[2, I*E^(I*ArcCos[c*x])]))/2))/3
 

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 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 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 4669

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

rule 5131

Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_.), x_Symbol] :> Simp[x*(a + b*Ar 
cCos[c*x])^n, x] + Simp[b*c*n   Int[x*((a + b*ArcCos[c*x])^(n - 1)/Sqrt[1 - 
 c^2*x^2]), x], x] /; FreeQ[{a, b, c}, x] && 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 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 5199

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

rule 5201

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

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 5219

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

Maple [A] (verified)

Time = 0.75 (sec) , antiderivative size = 504, normalized size of antiderivative = 1.45

method	result	size
derivativedivides	\(c^{3} \left (-d^{3} a^{2} \left (\frac {c^{3} x^{3}}{3}-3 c x +\frac {1}{3 c^{3} x^{3}}-\frac {3}{c x}\right )-\frac {d^{3} b^{2}}{3 c x}-\frac {50 d^{3} b^{2} c x}{9}+\frac {2 d^{3} b^{2} c^{3} x^{3}}{27}+\frac {2 d^{3} b^{2} \arccos \left (c x \right ) \sqrt {-c^{2} x^{2}+1}\, c^{2} x^{2}}{9}+\frac {17 i d^{3} b^{2} \operatorname {dilog}\left (1-i \left (c x +i \sqrt {-c^{2} x^{2}+1}\right )\right )}{3}-\frac {17 d^{3} b^{2} \arccos \left (c x \right ) \ln \left (1-i \left (c x +i \sqrt {-c^{2} x^{2}+1}\right )\right )}{3}+\frac {3 d^{3} b^{2} \arccos \left (c x \right )^{2}}{c x}-\frac {d^{3} b^{2} \arccos \left (c x \right )^{2}}{3 c^{3} x^{3}}-\frac {d^{3} b^{2} \arccos \left (c x \right )^{2} c^{3} x^{3}}{3}+3 d^{3} b^{2} \arccos \left (c x \right )^{2} c x +\frac {17 d^{3} b^{2} \arccos \left (c x \right ) \ln \left (1+i \left (c x +i \sqrt {-c^{2} x^{2}+1}\right )\right )}{3}-\frac {17 i d^{3} b^{2} \operatorname {dilog}\left (1+i \left (c x +i \sqrt {-c^{2} x^{2}+1}\right )\right )}{3}+\frac {d^{3} b^{2} \arccos \left (c x \right ) \sqrt {-c^{2} x^{2}+1}}{3 c^{2} x^{2}}-\frac {50 d^{3} b^{2} \arccos \left (c x \right ) \sqrt {-c^{2} x^{2}+1}}{9}-2 d^{3} a b \left (\frac {c^{3} x^{3} \arccos \left (c x \right )}{3}-3 c x \arccos \left (c x \right )+\frac {\arccos \left (c x \right )}{3 c^{3} x^{3}}-\frac {3 \arccos \left (c x \right )}{c x}-\frac {\sqrt {-c^{2} x^{2}+1}}{6 c^{2} x^{2}}+\frac {17 \,\operatorname {arctanh}\left (\frac {1}{\sqrt {-c^{2} x^{2}+1}}\right )}{6}-\frac {c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}}{9}+\frac {25 \sqrt {-c^{2} x^{2}+1}}{9}\right )\right )\)	\(504\)
default	\(c^{3} \left (-d^{3} a^{2} \left (\frac {c^{3} x^{3}}{3}-3 c x +\frac {1}{3 c^{3} x^{3}}-\frac {3}{c x}\right )-\frac {d^{3} b^{2}}{3 c x}-\frac {50 d^{3} b^{2} c x}{9}+\frac {2 d^{3} b^{2} c^{3} x^{3}}{27}+\frac {2 d^{3} b^{2} \arccos \left (c x \right ) \sqrt {-c^{2} x^{2}+1}\, c^{2} x^{2}}{9}+\frac {17 i d^{3} b^{2} \operatorname {dilog}\left (1-i \left (c x +i \sqrt {-c^{2} x^{2}+1}\right )\right )}{3}-\frac {17 d^{3} b^{2} \arccos \left (c x \right ) \ln \left (1-i \left (c x +i \sqrt {-c^{2} x^{2}+1}\right )\right )}{3}+\frac {3 d^{3} b^{2} \arccos \left (c x \right )^{2}}{c x}-\frac {d^{3} b^{2} \arccos \left (c x \right )^{2}}{3 c^{3} x^{3}}-\frac {d^{3} b^{2} \arccos \left (c x \right )^{2} c^{3} x^{3}}{3}+3 d^{3} b^{2} \arccos \left (c x \right )^{2} c x +\frac {17 d^{3} b^{2} \arccos \left (c x \right ) \ln \left (1+i \left (c x +i \sqrt {-c^{2} x^{2}+1}\right )\right )}{3}-\frac {17 i d^{3} b^{2} \operatorname {dilog}\left (1+i \left (c x +i \sqrt {-c^{2} x^{2}+1}\right )\right )}{3}+\frac {d^{3} b^{2} \arccos \left (c x \right ) \sqrt {-c^{2} x^{2}+1}}{3 c^{2} x^{2}}-\frac {50 d^{3} b^{2} \arccos \left (c x \right ) \sqrt {-c^{2} x^{2}+1}}{9}-2 d^{3} a b \left (\frac {c^{3} x^{3} \arccos \left (c x \right )}{3}-3 c x \arccos \left (c x \right )+\frac {\arccos \left (c x \right )}{3 c^{3} x^{3}}-\frac {3 \arccos \left (c x \right )}{c x}-\frac {\sqrt {-c^{2} x^{2}+1}}{6 c^{2} x^{2}}+\frac {17 \,\operatorname {arctanh}\left (\frac {1}{\sqrt {-c^{2} x^{2}+1}}\right )}{6}-\frac {c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}}{9}+\frac {25 \sqrt {-c^{2} x^{2}+1}}{9}\right )\right )\)	\(504\)
parts	\(-d^{3} a^{2} \left (\frac {c^{6} x^{3}}{3}-3 c^{4} x -\frac {3 c^{2}}{x}+\frac {1}{3 x^{3}}\right )+\frac {d^{3} b^{2} c \sqrt {-c^{2} x^{2}+1}\, \arccos \left (c x \right )}{3 x^{2}}+\frac {2 b^{2} c^{6} d^{3} x^{3}}{27}-\frac {50 b^{2} c^{4} d^{3} x}{9}-\frac {b^{2} c^{2} d^{3}}{3 x}+\frac {2 d^{3} b^{2} c^{5} \arccos \left (c x \right ) \sqrt {-c^{2} x^{2}+1}\, x^{2}}{9}+\frac {3 d^{3} b^{2} c^{2} \arccos \left (c x \right )^{2}}{x}-\frac {d^{3} b^{2} \arccos \left (c x \right )^{2}}{3 x^{3}}-\frac {d^{3} b^{2} c^{6} \arccos \left (c x \right )^{2} x^{3}}{3}+3 d^{3} b^{2} c^{4} \arccos \left (c x \right )^{2} x -\frac {50 d^{3} b^{2} c^{3} \arccos \left (c x \right ) \sqrt {-c^{2} x^{2}+1}}{9}+\frac {17 i d^{3} b^{2} c^{3} \operatorname {dilog}\left (1-i \left (c x +i \sqrt {-c^{2} x^{2}+1}\right )\right )}{3}-\frac {17 i d^{3} b^{2} c^{3} \operatorname {dilog}\left (1+i \left (c x +i \sqrt {-c^{2} x^{2}+1}\right )\right )}{3}+\frac {17 d^{3} b^{2} c^{3} \arccos \left (c x \right ) \ln \left (1+i \left (c x +i \sqrt {-c^{2} x^{2}+1}\right )\right )}{3}-\frac {17 d^{3} b^{2} c^{3} \arccos \left (c x \right ) \ln \left (1-i \left (c x +i \sqrt {-c^{2} x^{2}+1}\right )\right )}{3}-2 d^{3} a b \,c^{3} \left (\frac {c^{3} x^{3} \arccos \left (c x \right )}{3}-3 c x \arccos \left (c x \right )+\frac {\arccos \left (c x \right )}{3 c^{3} x^{3}}-\frac {3 \arccos \left (c x \right )}{c x}-\frac {\sqrt {-c^{2} x^{2}+1}}{6 c^{2} x^{2}}+\frac {17 \,\operatorname {arctanh}\left (\frac {1}{\sqrt {-c^{2} x^{2}+1}}\right )}{6}-\frac {c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}}{9}+\frac {25 \sqrt {-c^{2} x^{2}+1}}{9}\right )\)	\(516\)

Input:

int((-c^2*d*x^2+d)^3*(a+b*arccos(c*x))^2/x^4,x,method=_RETURNVERBOSE)
 

Output:

c^3*(-d^3*a^2*(1/3*c^3*x^3-3*c*x+1/3/c^3/x^3-3/c/x)-1/3*d^3*b^2/c/x-50/9*d 
^3*b^2*c*x+2/27*d^3*b^2*c^3*x^3+2/9*d^3*b^2*arccos(c*x)*(-c^2*x^2+1)^(1/2) 
*c^2*x^2+17/3*I*d^3*b^2*dilog(1-I*(c*x+I*(-c^2*x^2+1)^(1/2)))-17/3*d^3*b^2 
*arccos(c*x)*ln(1-I*(c*x+I*(-c^2*x^2+1)^(1/2)))+3*d^3*b^2/c/x*arccos(c*x)^ 
2-1/3*d^3*b^2/c^3/x^3*arccos(c*x)^2-1/3*d^3*b^2*arccos(c*x)^2*c^3*x^3+3*d^ 
3*b^2*arccos(c*x)^2*c*x+17/3*d^3*b^2*arccos(c*x)*ln(1+I*(c*x+I*(-c^2*x^2+1 
)^(1/2)))-17/3*I*d^3*b^2*dilog(1+I*(c*x+I*(-c^2*x^2+1)^(1/2)))+1/3*d^3*b^2 
/c^2/x^2*arccos(c*x)*(-c^2*x^2+1)^(1/2)-50/9*d^3*b^2*arccos(c*x)*(-c^2*x^2 
+1)^(1/2)-2*d^3*a*b*(1/3*c^3*x^3*arccos(c*x)-3*c*x*arccos(c*x)+1/3*arccos( 
c*x)/c^3/x^3-3*arccos(c*x)/c/x-1/6/c^2/x^2*(-c^2*x^2+1)^(1/2)+17/6*arctanh 
(1/(-c^2*x^2+1)^(1/2))-1/9*c^2*x^2*(-c^2*x^2+1)^(1/2)+25/9*(-c^2*x^2+1)^(1 
/2)))
 

Fricas [F]

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

integrate((-c^2*d*x^2+d)^3*(a+b*arccos(c*x))^2/x^4,x, algorithm="fricas")
                                                                                    
                                                                                    
 

Output:

integral(-(a^2*c^6*d^3*x^6 - 3*a^2*c^4*d^3*x^4 + 3*a^2*c^2*d^3*x^2 - a^2*d 
^3 + (b^2*c^6*d^3*x^6 - 3*b^2*c^4*d^3*x^4 + 3*b^2*c^2*d^3*x^2 - b^2*d^3)*a 
rccos(c*x)^2 + 2*(a*b*c^6*d^3*x^6 - 3*a*b*c^4*d^3*x^4 + 3*a*b*c^2*d^3*x^2 
- a*b*d^3)*arccos(c*x))/x^4, x)
 

Sympy [F]

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

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

Output:

-d**3*(Integral(-3*a**2*c**4, x) + Integral(-a**2/x**4, x) + Integral(3*a* 
*2*c**2/x**2, x) + Integral(a**2*c**6*x**2, x) + Integral(-3*b**2*c**4*aco 
s(c*x)**2, x) + Integral(-b**2*acos(c*x)**2/x**4, x) + Integral(-6*a*b*c** 
4*acos(c*x), x) + Integral(-2*a*b*acos(c*x)/x**4, x) + Integral(3*b**2*c** 
2*acos(c*x)**2/x**2, x) + Integral(b**2*c**6*x**2*acos(c*x)**2, x) + Integ 
ral(6*a*b*c**2*acos(c*x)/x**2, x) + Integral(2*a*b*c**6*x**2*acos(c*x), x) 
)
 

Maxima [F]

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

integrate((-c^2*d*x^2+d)^3*(a+b*arccos(c*x))^2/x^4,x, algorithm="maxima")
 

Output:

-1/3*a^2*c^6*d^3*x^3 - 2/9*(3*x^3*arccos(c*x) - c*(sqrt(-c^2*x^2 + 1)*x^2/ 
c^2 + 2*sqrt(-c^2*x^2 + 1)/c^4))*a*b*c^6*d^3 + 3*b^2*c^4*d^3*x*arccos(c*x) 
^2 - 6*b^2*c^4*d^3*(x + sqrt(-c^2*x^2 + 1)*arccos(c*x)/c) + 3*a^2*c^4*d^3* 
x + 6*(c*x*arccos(c*x) - sqrt(-c^2*x^2 + 1))*a*b*c^3*d^3 - 6*(c*log(2*sqrt 
(-c^2*x^2 + 1)/abs(x) + 2/abs(x)) - arccos(c*x)/x)*a*b*c^2*d^3 + 1/3*((c^2 
*log(2*sqrt(-c^2*x^2 + 1)/abs(x) + 2/abs(x)) + sqrt(-c^2*x^2 + 1)/x^2)*c - 
 2*arccos(c*x)/x^3)*a*b*d^3 + 3*a^2*c^2*d^3/x - 1/3*a^2*d^3/x^3 + 1/3*(3*x 
^3*integrate(2/3*(b^2*c^7*d^3*x^6 - 9*b^2*c^3*d^3*x^2 + b^2*c*d^3)*sqrt(c* 
x + 1)*sqrt(-c*x + 1)*arctan2(sqrt(c*x + 1)*sqrt(-c*x + 1), c*x)/(c^2*x^5 
- x^3), x) - (b^2*c^6*d^3*x^6 - 9*b^2*c^2*d^3*x^2 + b^2*d^3)*arctan2(sqrt( 
c*x + 1)*sqrt(-c*x + 1), c*x)^2)/x^3
 

Giac [F(-1)]

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

integrate((-c^2*d*x^2+d)^3*(a+b*arccos(c*x))^2/x^4,x, algorithm="giac")
 

Output:

Timed out
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {\left (d-c^2 d x^2\right )^3 (a+b \arccos (c x))^2}{x^4} \, dx=\frac {d^{3} \left (27 \mathit {acos} \left (c x \right )^{2} b^{2} c^{4} x^{4}-54 \sqrt {-c^{2} x^{2}+1}\, \mathit {acos} \left (c x \right ) b^{2} c^{3} x^{3}-6 \mathit {acos} \left (c x \right ) a b \,c^{6} x^{6}+54 \mathit {acos} \left (c x \right ) a b \,c^{4} x^{4}+54 \mathit {acos} \left (c x \right ) a b \,c^{2} x^{2}-6 \mathit {acos} \left (c x \right ) a b +2 \sqrt {-c^{2} x^{2}+1}\, a b \,c^{5} x^{5}-50 \sqrt {-c^{2} x^{2}+1}\, a b \,c^{3} x^{3}+3 \sqrt {-c^{2} x^{2}+1}\, a b c x +9 \left (\int \frac {\mathit {acos} \left (c x \right )^{2}}{x^{4}}d x \right ) b^{2} x^{3}-27 \left (\int \frac {\mathit {acos} \left (c x \right )^{2}}{x^{2}}d x \right ) b^{2} c^{2} x^{3}-9 \left (\int \mathit {acos} \left (c x \right )^{2} x^{2}d x \right ) b^{2} c^{6} x^{3}+51 \,\mathrm {log}\left (\tan \left (\frac {\mathit {asin} \left (c x \right )}{2}\right )\right ) a b \,c^{3} x^{3}-3 a^{2} c^{6} x^{6}+27 a^{2} c^{4} x^{4}+27 a^{2} c^{2} x^{2}-3 a^{2}-54 b^{2} c^{4} x^{4}\right )}{9 x^{3}} \] Input:

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

Output:

(d**3*(27*acos(c*x)**2*b**2*c**4*x**4 - 54*sqrt( - c**2*x**2 + 1)*acos(c*x 
)*b**2*c**3*x**3 - 6*acos(c*x)*a*b*c**6*x**6 + 54*acos(c*x)*a*b*c**4*x**4 
+ 54*acos(c*x)*a*b*c**2*x**2 - 6*acos(c*x)*a*b + 2*sqrt( - c**2*x**2 + 1)* 
a*b*c**5*x**5 - 50*sqrt( - c**2*x**2 + 1)*a*b*c**3*x**3 + 3*sqrt( - c**2*x 
**2 + 1)*a*b*c*x + 9*int(acos(c*x)**2/x**4,x)*b**2*x**3 - 27*int(acos(c*x) 
**2/x**2,x)*b**2*c**2*x**3 - 9*int(acos(c*x)**2*x**2,x)*b**2*c**6*x**3 + 5 
1*log(tan(asin(c*x)/2))*a*b*c**3*x**3 - 3*a**2*c**6*x**6 + 27*a**2*c**4*x* 
*4 + 27*a**2*c**2*x**2 - 3*a**2 - 54*b**2*c**4*x**4))/(9*x**3)