\(\int \frac {(d-c^2 d x^2)^2 (a+b \arccos (c x))^2}{x^4} \, dx\) [175]

Optimal result

Integrand size = 27, antiderivative size = 268 \[ \int \frac {\left (d-c^2 d x^2\right )^2 (a+b \arccos (c x))^2}{x^4} \, dx=-\frac {b^2 c^2 d^2}{3 x}-2 b^2 c^4 d^2 x+\frac {5}{3} b c^3 d^2 \sqrt {1-c^2 x^2} (a+b \arccos (c x))-\frac {b c d^2 \left (1-c^2 x^2\right )^{3/2} (a+b \arccos (c x))}{3 x^2}+\frac {8}{3} c^4 d^2 x (a+b \arccos (c x))^2+\frac {4 c^2 d^2 \left (1-c^2 x^2\right ) (a+b \arccos (c x))^2}{3 x}-\frac {d^2 \left (1-c^2 x^2\right )^2 (a+b \arccos (c x))^2}{3 x^3}+\frac {22}{3} b c^3 d^2 (a+b \arccos (c x)) \text {arctanh}\left (e^{i \arccos (c x)}\right )-\frac {11}{3} i b^2 c^3 d^2 \operatorname {PolyLog}\left (2,-e^{i \arccos (c x)}\right )+\frac {11}{3} i b^2 c^3 d^2 \operatorname {PolyLog}\left (2,e^{i \arccos (c x)}\right ) \] Output:

-1/3*b^2*c^2*d^2/x-2*b^2*c^4*d^2*x+5/3*b*c^3*d^2*(-c^2*x^2+1)^(1/2)*(a+b*a 
rccos(c*x))-1/3*b*c*d^2*(-c^2*x^2+1)^(3/2)*(a+b*arccos(c*x))/x^2+8/3*c^4*d 
^2*x*(a+b*arccos(c*x))^2+4/3*c^2*d^2*(-c^2*x^2+1)*(a+b*arccos(c*x))^2/x-1/ 
3*d^2*(-c^2*x^2+1)^2*(a+b*arccos(c*x))^2/x^3+22/3*b*c^3*d^2*(a+b*arccos(c* 
x))*arctanh(c*x+I*(-c^2*x^2+1)^(1/2))-11/3*I*b^2*c^3*d^2*polylog(2,-c*x-I* 
(-c^2*x^2+1)^(1/2))+11/3*I*b^2*c^3*d^2*polylog(2,c*x+I*(-c^2*x^2+1)^(1/2))
 

Mathematica [A] (verified)

Time = 0.64 (sec) , antiderivative size = 384, normalized size of antiderivative = 1.43 \[ \int \frac {\left (d-c^2 d x^2\right )^2 (a+b \arccos (c x))^2}{x^4} \, dx=\frac {d^2 \left (-a^2+6 a^2 c^2 x^2-b^2 c^2 x^2+3 a^2 c^4 x^4-6 b^2 c^4 x^4+a b c x \sqrt {1-c^2 x^2}-6 a b c^3 x^3 \sqrt {1-c^2 x^2}-2 a b \arccos (c x)+12 a b c^2 x^2 \arccos (c x)+6 a b c^4 x^4 \arccos (c x)+b^2 c x \sqrt {1-c^2 x^2} \arccos (c x)-6 b^2 c^3 x^3 \sqrt {1-c^2 x^2} \arccos (c x)-b^2 \arccos (c x)^2+6 b^2 c^2 x^2 \arccos (c x)^2+3 b^2 c^4 x^4 \arccos (c x)^2-11 a b c^3 x^3 \text {arctanh}\left (\sqrt {1-c^2 x^2}\right )-11 b^2 c^3 x^3 \arccos (c x) \log \left (1-i e^{i \arccos (c x)}\right )+11 b^2 c^3 x^3 \arccos (c x) \log \left (1+i e^{i \arccos (c x)}\right )-11 i b^2 c^3 x^3 \operatorname {PolyLog}\left (2,-i e^{i \arccos (c x)}\right )+11 i b^2 c^3 x^3 \operatorname {PolyLog}\left (2,i e^{i \arccos (c x)}\right )\right )}{3 x^3} \] Input:

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

Output:

(d^2*(-a^2 + 6*a^2*c^2*x^2 - b^2*c^2*x^2 + 3*a^2*c^4*x^4 - 6*b^2*c^4*x^4 + 
 a*b*c*x*Sqrt[1 - c^2*x^2] - 6*a*b*c^3*x^3*Sqrt[1 - c^2*x^2] - 2*a*b*ArcCo 
s[c*x] + 12*a*b*c^2*x^2*ArcCos[c*x] + 6*a*b*c^4*x^4*ArcCos[c*x] + b^2*c*x* 
Sqrt[1 - c^2*x^2]*ArcCos[c*x] - 6*b^2*c^3*x^3*Sqrt[1 - c^2*x^2]*ArcCos[c*x 
] - b^2*ArcCos[c*x]^2 + 6*b^2*c^2*x^2*ArcCos[c*x]^2 + 3*b^2*c^4*x^4*ArcCos 
[c*x]^2 - 11*a*b*c^3*x^3*ArcTanh[Sqrt[1 - c^2*x^2]] - 11*b^2*c^3*x^3*ArcCo 
s[c*x]*Log[1 - I*E^(I*ArcCos[c*x])] + 11*b^2*c^3*x^3*ArcCos[c*x]*Log[1 + I 
*E^(I*ArcCos[c*x])] - (11*I)*b^2*c^3*x^3*PolyLog[2, (-I)*E^(I*ArcCos[c*x]) 
] + (11*I)*b^2*c^3*x^3*PolyLog[2, I*E^(I*ArcCos[c*x])]))/(3*x^3)
 

Rubi [A] (verified)

Time = 2.21 (sec) , antiderivative size = 380, normalized size of antiderivative = 1.42, number of steps used = 16, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.556, Rules used = {5201, 27, 5201, 244, 2009, 5131, 5183, 24, 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.

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

\(\Big \downarrow \) 5201

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 5201

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

\(\Big \downarrow \) 244

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

\(\Big \downarrow \) 2009

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

\(\Big \downarrow \) 5131

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

\(\Big \downarrow \) 5183

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

\(\Big \downarrow \) 24

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

\(\Big \downarrow \) 5199

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

\(\Big \downarrow \) 24

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

\(\Big \downarrow \) 5219

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4669

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

\(\Big \downarrow \) 2715

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

\(\Big \downarrow \) 2838

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

Input:

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

Output:

-1/3*(d^2*(1 - c^2*x^2)^2*(a + b*ArcCos[c*x])^2)/x^3 - (4*c^2*d^2*(-(((1 - 
 c^2*x^2)*(a + b*ArcCos[c*x])^2)/x) - 2*c^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)) - 2*b*c*(b* 
c*x + Sqrt[1 - c^2*x^2]*(a + b*ArcCos[c*x]) + (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])])))/3 - (2*b*c*d^2*(-1/2*(b*c*(-x^(-1) - c^2*x 
)) - ((1 - c^2*x^2)^(3/2)*(a + b*ArcCos[c*x]))/(2*x^2) - (3*c^2*(b*c*x + S 
qrt[1 - c^2*x^2]*(a + b*ArcCos[c*x]) + (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 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 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 = 0.52 (sec) , antiderivative size = 396, normalized size of antiderivative = 1.48

method	result	size
derivativedivides	\(c^{3} \left (d^{2} a^{2} \left (c x -\frac {1}{3 c^{3} x^{3}}+\frac {2}{c x}\right )-2 d^{2} b^{2} \arccos \left (c x \right ) \sqrt {-c^{2} x^{2}+1}+d^{2} b^{2} \arccos \left (c x \right )^{2} c x -2 d^{2} b^{2} c x +\frac {2 d^{2} b^{2} \arccos \left (c x \right )^{2}}{c x}+\frac {d^{2} b^{2} \arccos \left (c x \right ) \sqrt {-c^{2} x^{2}+1}}{3 c^{2} x^{2}}-\frac {d^{2} b^{2} \arccos \left (c x \right )^{2}}{3 c^{3} x^{3}}-\frac {d^{2} b^{2}}{3 c x}+\frac {11 d^{2} b^{2} \arccos \left (c x \right ) \ln \left (1+i \left (c x +i \sqrt {-c^{2} x^{2}+1}\right )\right )}{3}-\frac {11 d^{2} b^{2} \arccos \left (c x \right ) \ln \left (1-i \left (c x +i \sqrt {-c^{2} x^{2}+1}\right )\right )}{3}-\frac {11 i d^{2} b^{2} \operatorname {dilog}\left (1+i \left (c x +i \sqrt {-c^{2} x^{2}+1}\right )\right )}{3}+\frac {11 i d^{2} b^{2} \operatorname {dilog}\left (1-i \left (c x +i \sqrt {-c^{2} x^{2}+1}\right )\right )}{3}+2 d^{2} a b \left (c x \arccos \left (c x \right )-\frac {\arccos \left (c x \right )}{3 c^{3} x^{3}}+\frac {2 \arccos \left (c x \right )}{c x}+\frac {\sqrt {-c^{2} x^{2}+1}}{6 c^{2} x^{2}}-\frac {11 \,\operatorname {arctanh}\left (\frac {1}{\sqrt {-c^{2} x^{2}+1}}\right )}{6}-\sqrt {-c^{2} x^{2}+1}\right )\right )\)	\(396\)
default	\(c^{3} \left (d^{2} a^{2} \left (c x -\frac {1}{3 c^{3} x^{3}}+\frac {2}{c x}\right )-2 d^{2} b^{2} \arccos \left (c x \right ) \sqrt {-c^{2} x^{2}+1}+d^{2} b^{2} \arccos \left (c x \right )^{2} c x -2 d^{2} b^{2} c x +\frac {2 d^{2} b^{2} \arccos \left (c x \right )^{2}}{c x}+\frac {d^{2} b^{2} \arccos \left (c x \right ) \sqrt {-c^{2} x^{2}+1}}{3 c^{2} x^{2}}-\frac {d^{2} b^{2} \arccos \left (c x \right )^{2}}{3 c^{3} x^{3}}-\frac {d^{2} b^{2}}{3 c x}+\frac {11 d^{2} b^{2} \arccos \left (c x \right ) \ln \left (1+i \left (c x +i \sqrt {-c^{2} x^{2}+1}\right )\right )}{3}-\frac {11 d^{2} b^{2} \arccos \left (c x \right ) \ln \left (1-i \left (c x +i \sqrt {-c^{2} x^{2}+1}\right )\right )}{3}-\frac {11 i d^{2} b^{2} \operatorname {dilog}\left (1+i \left (c x +i \sqrt {-c^{2} x^{2}+1}\right )\right )}{3}+\frac {11 i d^{2} b^{2} \operatorname {dilog}\left (1-i \left (c x +i \sqrt {-c^{2} x^{2}+1}\right )\right )}{3}+2 d^{2} a b \left (c x \arccos \left (c x \right )-\frac {\arccos \left (c x \right )}{3 c^{3} x^{3}}+\frac {2 \arccos \left (c x \right )}{c x}+\frac {\sqrt {-c^{2} x^{2}+1}}{6 c^{2} x^{2}}-\frac {11 \,\operatorname {arctanh}\left (\frac {1}{\sqrt {-c^{2} x^{2}+1}}\right )}{6}-\sqrt {-c^{2} x^{2}+1}\right )\right )\)	\(396\)
parts	\(d^{2} a^{2} \left (c^{4} x +\frac {2 c^{2}}{x}-\frac {1}{3 x^{3}}\right )-2 d^{2} b^{2} c^{3} \sqrt {-c^{2} x^{2}+1}\, \arccos \left (c x \right )+d^{2} b^{2} c^{4} \arccos \left (c x \right )^{2} x -2 b^{2} c^{4} d^{2} x +\frac {2 d^{2} b^{2} c^{2} \arccos \left (c x \right )^{2}}{x}+\frac {d^{2} b^{2} c \sqrt {-c^{2} x^{2}+1}\, \arccos \left (c x \right )}{3 x^{2}}-\frac {b^{2} c^{2} d^{2}}{3 x}-\frac {d^{2} b^{2} \arccos \left (c x \right )^{2}}{3 x^{3}}+\frac {11 d^{2} b^{2} c^{3} \arccos \left (c x \right ) \ln \left (1+i \left (c x +i \sqrt {-c^{2} x^{2}+1}\right )\right )}{3}-\frac {11 d^{2} b^{2} c^{3} \arccos \left (c x \right ) \ln \left (1-i \left (c x +i \sqrt {-c^{2} x^{2}+1}\right )\right )}{3}-\frac {11 i d^{2} b^{2} c^{3} \operatorname {dilog}\left (1+i \left (c x +i \sqrt {-c^{2} x^{2}+1}\right )\right )}{3}+\frac {11 i d^{2} b^{2} c^{3} \operatorname {dilog}\left (1-i \left (c x +i \sqrt {-c^{2} x^{2}+1}\right )\right )}{3}+2 d^{2} a b \,c^{3} \left (c x \arccos \left (c x \right )-\frac {\arccos \left (c x \right )}{3 c^{3} x^{3}}+\frac {2 \arccos \left (c x \right )}{c x}+\frac {\sqrt {-c^{2} x^{2}+1}}{6 c^{2} x^{2}}-\frac {11 \,\operatorname {arctanh}\left (\frac {1}{\sqrt {-c^{2} x^{2}+1}}\right )}{6}-\sqrt {-c^{2} x^{2}+1}\right )\)	\(408\)

Input:

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

Output:

c^3*(d^2*a^2*(c*x-1/3/c^3/x^3+2/c/x)-2*d^2*b^2*arccos(c*x)*(-c^2*x^2+1)^(1 
/2)+d^2*b^2*arccos(c*x)^2*c*x-2*d^2*b^2*c*x+2*d^2*b^2*arccos(c*x)^2/c/x+1/ 
3*d^2*b^2/c^2/x^2*arccos(c*x)*(-c^2*x^2+1)^(1/2)-1/3*d^2*b^2/c^3/x^3*arcco 
s(c*x)^2-1/3*d^2*b^2/c/x+11/3*d^2*b^2*arccos(c*x)*ln(1+I*(c*x+I*(-c^2*x^2+ 
1)^(1/2)))-11/3*d^2*b^2*arccos(c*x)*ln(1-I*(c*x+I*(-c^2*x^2+1)^(1/2)))-11/ 
3*I*d^2*b^2*dilog(1+I*(c*x+I*(-c^2*x^2+1)^(1/2)))+11/3*I*d^2*b^2*dilog(1-I 
*(c*x+I*(-c^2*x^2+1)^(1/2)))+2*d^2*a*b*(c*x*arccos(c*x)-1/3*arccos(c*x)/c^ 
3/x^3+2*arccos(c*x)/c/x+1/6/c^2/x^2*(-c^2*x^2+1)^(1/2)-11/6*arctanh(1/(-c^ 
2*x^2+1)^(1/2))-(-c^2*x^2+1)^(1/2)))
 

Fricas [F]

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

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F(-1)]

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

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

Output:

Timed out
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {\left (d-c^2 d x^2\right )^2 (a+b \arccos (c x))^2}{x^4} \, dx=\frac {d^{2} \left (3 \mathit {acos} \left (c x \right )^{2} b^{2} c^{4} x^{4}-6 \sqrt {-c^{2} x^{2}+1}\, \mathit {acos} \left (c x \right ) b^{2} c^{3} x^{3}+6 \mathit {acos} \left (c x \right ) a b \,c^{4} x^{4}+12 \mathit {acos} \left (c x \right ) a b \,c^{2} x^{2}-2 \mathit {acos} \left (c x \right ) a b -6 \sqrt {-c^{2} x^{2}+1}\, a b \,c^{3} x^{3}+\sqrt {-c^{2} x^{2}+1}\, a b c x +3 \left (\int \frac {\mathit {acos} \left (c x \right )^{2}}{x^{4}}d x \right ) b^{2} x^{3}-6 \left (\int \frac {\mathit {acos} \left (c x \right )^{2}}{x^{2}}d x \right ) b^{2} c^{2} x^{3}+11 \,\mathrm {log}\left (\tan \left (\frac {\mathit {asin} \left (c x \right )}{2}\right )\right ) a b \,c^{3} x^{3}+3 a^{2} c^{4} x^{4}+6 a^{2} c^{2} x^{2}-a^{2}-6 b^{2} c^{4} x^{4}\right )}{3 x^{3}} \] Input:

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

Output:

(d**2*(3*acos(c*x)**2*b**2*c**4*x**4 - 6*sqrt( - c**2*x**2 + 1)*acos(c*x)* 
b**2*c**3*x**3 + 6*acos(c*x)*a*b*c**4*x**4 + 12*acos(c*x)*a*b*c**2*x**2 - 
2*acos(c*x)*a*b - 6*sqrt( - c**2*x**2 + 1)*a*b*c**3*x**3 + sqrt( - c**2*x* 
*2 + 1)*a*b*c*x + 3*int(acos(c*x)**2/x**4,x)*b**2*x**3 - 6*int(acos(c*x)** 
2/x**2,x)*b**2*c**2*x**3 + 11*log(tan(asin(c*x)/2))*a*b*c**3*x**3 + 3*a**2 
*c**4*x**4 + 6*a**2*c**2*x**2 - a**2 - 6*b**2*c**4*x**4))/(3*x**3)