\(\int \frac {a+b \arccos (c x)}{x^4 (d+e x^2)} \, dx\) [634]

Optimal result

Integrand size = 21, antiderivative size = 649 \[ \int \frac {a+b \arccos (c x)}{x^4 \left (d+e x^2\right )} \, dx=-\frac {b c \sqrt {1-c^2 x^2}}{6 d x^2}-\frac {a+b \arccos (c x)}{3 d x^3}+\frac {e (a+b \arccos (c x))}{d^2 x}-\frac {b c^3 \text {arctanh}\left (\sqrt {1-c^2 x^2}\right )}{6 d}+\frac {b c e \text {arctanh}\left (\sqrt {1-c^2 x^2}\right )}{d^2}+\frac {e^{3/2} (a+b \arccos (c x)) \log \left (1-\frac {\sqrt {e} e^{i \arccos (c x)}}{i c \sqrt {-d}-\sqrt {c^2 d+e}}\right )}{2 (-d)^{5/2}}-\frac {e^{3/2} (a+b \arccos (c x)) \log \left (1+\frac {\sqrt {e} e^{i \arccos (c x)}}{i c \sqrt {-d}-\sqrt {c^2 d+e}}\right )}{2 (-d)^{5/2}}+\frac {e^{3/2} (a+b \arccos (c x)) \log \left (1-\frac {\sqrt {e} e^{i \arccos (c x)}}{i c \sqrt {-d}+\sqrt {c^2 d+e}}\right )}{2 (-d)^{5/2}}-\frac {e^{3/2} (a+b \arccos (c x)) \log \left (1+\frac {\sqrt {e} e^{i \arccos (c x)}}{i c \sqrt {-d}+\sqrt {c^2 d+e}}\right )}{2 (-d)^{5/2}}+\frac {i b e^{3/2} \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} e^{i \arccos (c x)}}{i c \sqrt {-d}-\sqrt {c^2 d+e}}\right )}{2 (-d)^{5/2}}-\frac {i b e^{3/2} \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{i \arccos (c x)}}{i c \sqrt {-d}-\sqrt {c^2 d+e}}\right )}{2 (-d)^{5/2}}+\frac {i b e^{3/2} \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} e^{i \arccos (c x)}}{i c \sqrt {-d}+\sqrt {c^2 d+e}}\right )}{2 (-d)^{5/2}}-\frac {i b e^{3/2} \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{i \arccos (c x)}}{i c \sqrt {-d}+\sqrt {c^2 d+e}}\right )}{2 (-d)^{5/2}} \] Output:

-1/6*b*c*(-c^2*x^2+1)^(1/2)/d/x^2-1/3*(a+b*arccos(c*x))/d/x^3+e*(a+b*arcco 
s(c*x))/d^2/x-1/6*b*c^3*arctanh((-c^2*x^2+1)^(1/2))/d+b*c*e*arctanh((-c^2* 
x^2+1)^(1/2))/d^2+1/2*e^(3/2)*(a+b*arccos(c*x))*ln(1-e^(1/2)*(c*x+I*(-c^2* 
x^2+1)^(1/2))/(I*c*(-d)^(1/2)-(c^2*d+e)^(1/2)))/(-d)^(5/2)-1/2*e^(3/2)*(a+ 
b*arccos(c*x))*ln(1+e^(1/2)*(c*x+I*(-c^2*x^2+1)^(1/2))/(I*c*(-d)^(1/2)-(c^ 
2*d+e)^(1/2)))/(-d)^(5/2)+1/2*e^(3/2)*(a+b*arccos(c*x))*ln(1-e^(1/2)*(c*x+ 
I*(-c^2*x^2+1)^(1/2))/(I*c*(-d)^(1/2)+(c^2*d+e)^(1/2)))/(-d)^(5/2)-1/2*e^( 
3/2)*(a+b*arccos(c*x))*ln(1+e^(1/2)*(c*x+I*(-c^2*x^2+1)^(1/2))/(I*c*(-d)^( 
1/2)+(c^2*d+e)^(1/2)))/(-d)^(5/2)+1/2*I*b*e^(3/2)*polylog(2,-e^(1/2)*(c*x+ 
I*(-c^2*x^2+1)^(1/2))/(I*c*(-d)^(1/2)-(c^2*d+e)^(1/2)))/(-d)^(5/2)-1/2*I*b 
*e^(3/2)*polylog(2,e^(1/2)*(c*x+I*(-c^2*x^2+1)^(1/2))/(I*c*(-d)^(1/2)-(c^2 
*d+e)^(1/2)))/(-d)^(5/2)+1/2*I*b*e^(3/2)*polylog(2,-e^(1/2)*(c*x+I*(-c^2*x 
^2+1)^(1/2))/(I*c*(-d)^(1/2)+(c^2*d+e)^(1/2)))/(-d)^(5/2)-1/2*I*b*e^(3/2)* 
polylog(2,e^(1/2)*(c*x+I*(-c^2*x^2+1)^(1/2))/(I*c*(-d)^(1/2)+(c^2*d+e)^(1/ 
2)))/(-d)^(5/2)
 

Mathematica [A] (verified)

Time = 0.52 (sec) , antiderivative size = 1002, normalized size of antiderivative = 1.54 \[ \int \frac {a+b \arccos (c x)}{x^4 \left (d+e x^2\right )} \, dx =\text {Too large to display} \] Input:

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

Output:

-1/3*a/(d*x^3) + (a*e)/(d^2*x) + (a*e^(3/2)*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/d 
^(5/2) + b*(-((e*(-(ArcCos[c*x]/x) - c*Log[x] + c*Log[1 + Sqrt[1 - c^2*x^2 
]]))/d^2) + ((c*Sqrt[1 - c^2*x^2])/(6*x^2) - ArcCos[c*x]/(3*x^3) - (c^3*Lo 
g[x])/6 + (c^3*Log[1 + Sqrt[1 - c^2*x^2]])/6)/d + (e^(3/2)*(ArcCos[c*x]^2 
- 8*ArcSin[Sqrt[1 + (I*c*Sqrt[d])/Sqrt[e]]/Sqrt[2]]*ArcTan[((c*Sqrt[d] + I 
*Sqrt[e])*Tan[ArcCos[c*x]/2])/Sqrt[c^2*d + e]] + (2*I)*ArcCos[c*x]*Log[1 - 
 (I*(-(c*Sqrt[d]) + Sqrt[c^2*d + e])*E^(I*ArcCos[c*x]))/Sqrt[e]] + (4*I)*A 
rcSin[Sqrt[1 + (I*c*Sqrt[d])/Sqrt[e]]/Sqrt[2]]*Log[1 - (I*(-(c*Sqrt[d]) + 
Sqrt[c^2*d + e])*E^(I*ArcCos[c*x]))/Sqrt[e]] + (2*I)*ArcCos[c*x]*Log[1 + ( 
I*(c*Sqrt[d] + Sqrt[c^2*d + e])*E^(I*ArcCos[c*x]))/Sqrt[e]] - (4*I)*ArcSin 
[Sqrt[1 + (I*c*Sqrt[d])/Sqrt[e]]/Sqrt[2]]*Log[1 + (I*(c*Sqrt[d] + Sqrt[c^2 
*d + e])*E^(I*ArcCos[c*x]))/Sqrt[e]] + 2*PolyLog[2, (I*(-(c*Sqrt[d]) + Sqr 
t[c^2*d + e])*E^(I*ArcCos[c*x]))/Sqrt[e]] + 2*PolyLog[2, ((-I)*(c*Sqrt[d] 
+ Sqrt[c^2*d + e])*E^(I*ArcCos[c*x]))/Sqrt[e]]))/(4*d^(5/2)) - (e^(3/2)*(A 
rcCos[c*x]^2 - 8*ArcSin[Sqrt[1 - (I*c*Sqrt[d])/Sqrt[e]]/Sqrt[2]]*ArcTan[(( 
c*Sqrt[d] - I*Sqrt[e])*Tan[ArcCos[c*x]/2])/Sqrt[c^2*d + e]] + (2*I)*ArcCos 
[c*x]*Log[1 + (I*(-(c*Sqrt[d]) + Sqrt[c^2*d + e])*E^(I*ArcCos[c*x]))/Sqrt[ 
e]] + (4*I)*ArcSin[Sqrt[1 - (I*c*Sqrt[d])/Sqrt[e]]/Sqrt[2]]*Log[1 + (I*(-( 
c*Sqrt[d]) + Sqrt[c^2*d + e])*E^(I*ArcCos[c*x]))/Sqrt[e]] + (2*I)*ArcCos[c 
*x]*Log[1 - (I*(c*Sqrt[d] + Sqrt[c^2*d + e])*E^(I*ArcCos[c*x]))/Sqrt[e]...
 

Rubi [A] (verified)

Time = 1.44 (sec) , antiderivative size = 650, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {5233, 2009}

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.

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

\(\Big \downarrow \) 5233

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

\(\Big \downarrow \) 2009

\(\displaystyle \frac {e^{3/2} (a+b \arccos (c x)) \log \left (1-\frac {\sqrt {e} e^{i \arccos (c x)}}{c \sqrt {-d}-i \sqrt {c^2 d+e}}\right )}{2 (-d)^{5/2}}-\frac {e^{3/2} (a+b \arccos (c x)) \log \left (1+\frac {\sqrt {e} e^{i \arccos (c x)}}{c \sqrt {-d}-i \sqrt {c^2 d+e}}\right )}{2 (-d)^{5/2}}+\frac {e^{3/2} (a+b \arccos (c x)) \log \left (1-\frac {\sqrt {e} e^{i \arccos (c x)}}{c \sqrt {-d}+i \sqrt {c^2 d+e}}\right )}{2 (-d)^{5/2}}-\frac {e^{3/2} (a+b \arccos (c x)) \log \left (1+\frac {\sqrt {e} e^{i \arccos (c x)}}{c \sqrt {-d}+i \sqrt {c^2 d+e}}\right )}{2 (-d)^{5/2}}+\frac {e (a+b \arccos (c x))}{d^2 x}-\frac {a+b \arccos (c x)}{3 d x^3}+\frac {i b e^{3/2} \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} e^{i \arccos (c x)}}{c \sqrt {-d}-i \sqrt {d c^2+e}}\right )}{2 (-d)^{5/2}}-\frac {i b e^{3/2} \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{i \arccos (c x)}}{c \sqrt {-d}-i \sqrt {d c^2+e}}\right )}{2 (-d)^{5/2}}+\frac {i b e^{3/2} \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} e^{i \arccos (c x)}}{\sqrt {-d} c+i \sqrt {d c^2+e}}\right )}{2 (-d)^{5/2}}-\frac {i b e^{3/2} \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{i \arccos (c x)}}{\sqrt {-d} c+i \sqrt {d c^2+e}}\right )}{2 (-d)^{5/2}}-\frac {b c e \text {arctanh}\left (\sqrt {1-c^2 x^2}\right )}{d^2}+\frac {b c^3 \text {arctanh}\left (\sqrt {1-c^2 x^2}\right )}{6 d}+\frac {b c \sqrt {1-c^2 x^2}}{6 d x^2}\)

Input:

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

Output:

(b*c*Sqrt[1 - c^2*x^2])/(6*d*x^2) - (a + b*ArcCos[c*x])/(3*d*x^3) + (e*(a 
+ b*ArcCos[c*x]))/(d^2*x) + (b*c^3*ArcTanh[Sqrt[1 - c^2*x^2]])/(6*d) - (b* 
c*e*ArcTanh[Sqrt[1 - c^2*x^2]])/d^2 + (e^(3/2)*(a + b*ArcCos[c*x])*Log[1 - 
 (Sqrt[e]*E^(I*ArcCos[c*x]))/(c*Sqrt[-d] - I*Sqrt[c^2*d + e])])/(2*(-d)^(5 
/2)) - (e^(3/2)*(a + b*ArcCos[c*x])*Log[1 + (Sqrt[e]*E^(I*ArcCos[c*x]))/(c 
*Sqrt[-d] - I*Sqrt[c^2*d + e])])/(2*(-d)^(5/2)) + (e^(3/2)*(a + b*ArcCos[c 
*x])*Log[1 - (Sqrt[e]*E^(I*ArcCos[c*x]))/(c*Sqrt[-d] + I*Sqrt[c^2*d + e])] 
)/(2*(-d)^(5/2)) - (e^(3/2)*(a + b*ArcCos[c*x])*Log[1 + (Sqrt[e]*E^(I*ArcC 
os[c*x]))/(c*Sqrt[-d] + I*Sqrt[c^2*d + e])])/(2*(-d)^(5/2)) + ((I/2)*b*e^( 
3/2)*PolyLog[2, -((Sqrt[e]*E^(I*ArcCos[c*x]))/(c*Sqrt[-d] - I*Sqrt[c^2*d + 
 e]))])/(-d)^(5/2) - ((I/2)*b*e^(3/2)*PolyLog[2, (Sqrt[e]*E^(I*ArcCos[c*x] 
))/(c*Sqrt[-d] - I*Sqrt[c^2*d + e])])/(-d)^(5/2) + ((I/2)*b*e^(3/2)*PolyLo 
g[2, -((Sqrt[e]*E^(I*ArcCos[c*x]))/(c*Sqrt[-d] + I*Sqrt[c^2*d + e]))])/(-d 
)^(5/2) - ((I/2)*b*e^(3/2)*PolyLog[2, (Sqrt[e]*E^(I*ArcCos[c*x]))/(c*Sqrt[ 
-d] + I*Sqrt[c^2*d + e])])/(-d)^(5/2)
 

Defintions of rubi rules used

rule 2009

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

rule 5233

Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_.)*((d_) + (e_ 
.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*ArcCos[c*x])^n, ( 
f*x)^m*(d + e*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[c^2*d + 
 e, 0] && IGtQ[n, 0] && IntegerQ[p] && IntegerQ[m]
 

Maple [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 219.78 (sec) , antiderivative size = 426, normalized size of antiderivative = 0.66

method	result	size
parts	\(a \left (-\frac {1}{3 d \,x^{3}}+\frac {e}{d^{2} x}+\frac {e^{2} \arctan \left (\frac {e x}{\sqrt {d e}}\right )}{d^{2} \sqrt {d e}}\right )-\frac {i b \left (8 \arctan \left (c x +i \sqrt {-c^{2} x^{2}+1}\right ) c^{7} d^{2} x^{3}+4 i \sqrt {-c^{2} x^{2}+1}\, c^{5} d^{2} x -8 i \arccos \left (c x \right ) c^{4} d^{2}+24 i \arccos \left (c x \right ) c^{4} d e \,x^{2}-48 \arctan \left (c x +i \sqrt {-c^{2} x^{2}+1}\right ) c^{5} d e \,x^{3}+3 \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (e \,\textit {\_Z}^{4}+\left (4 c^{2} d +2 e \right ) \textit {\_Z}^{2}+e \right )}{\sum }\frac {\left (4 \textit {\_R1}^{2} c^{2} d +\textit {\_R1}^{2} e +e \right ) \left (i \arccos \left (c x \right ) \ln \left (\frac {\textit {\_R1} -c x -i \sqrt {-c^{2} x^{2}+1}}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {\textit {\_R1} -c x -i \sqrt {-c^{2} x^{2}+1}}{\textit {\_R1}}\right )\right )}{\textit {\_R1} \left (\textit {\_R1}^{2} e +2 c^{2} d +e \right )}\right ) e^{2} c^{3} x^{3}-3 \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (e \,\textit {\_Z}^{4}+\left (4 c^{2} d +2 e \right ) \textit {\_Z}^{2}+e \right )}{\sum }\frac {\left (\textit {\_R1}^{2} e +4 c^{2} d +e \right ) \left (i \arccos \left (c x \right ) \ln \left (\frac {\textit {\_R1} -c x -i \sqrt {-c^{2} x^{2}+1}}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {\textit {\_R1} -c x -i \sqrt {-c^{2} x^{2}+1}}{\textit {\_R1}}\right )\right )}{\textit {\_R1} \left (\textit {\_R1}^{2} e +2 c^{2} d +e \right )}\right ) e^{2} c^{3} x^{3}\right )}{24 c^{4} x^{3} d^{3}}\)	\(426\)
derivativedivides	\(c^{3} \left (-\frac {a}{3 d \,c^{3} x^{3}}+\frac {a e}{c^{3} d^{2} x}+\frac {a \,e^{2} \arctan \left (\frac {e x}{\sqrt {d e}}\right )}{c^{3} d^{2} \sqrt {d e}}-\frac {i b \left (8 \arctan \left (c x +i \sqrt {-c^{2} x^{2}+1}\right ) c^{7} d^{2} x^{3}+4 i \sqrt {-c^{2} x^{2}+1}\, c^{5} d^{2} x -8 i \arccos \left (c x \right ) c^{4} d^{2}+24 i \arccos \left (c x \right ) c^{4} d e \,x^{2}-48 \arctan \left (c x +i \sqrt {-c^{2} x^{2}+1}\right ) c^{5} d e \,x^{3}+3 \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (e \,\textit {\_Z}^{4}+\left (4 c^{2} d +2 e \right ) \textit {\_Z}^{2}+e \right )}{\sum }\frac {\left (4 \textit {\_R1}^{2} c^{2} d +\textit {\_R1}^{2} e +e \right ) \left (i \arccos \left (c x \right ) \ln \left (\frac {\textit {\_R1} -c x -i \sqrt {-c^{2} x^{2}+1}}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {\textit {\_R1} -c x -i \sqrt {-c^{2} x^{2}+1}}{\textit {\_R1}}\right )\right )}{\textit {\_R1} \left (\textit {\_R1}^{2} e +2 c^{2} d +e \right )}\right ) e^{2} c^{3} x^{3}-3 \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (e \,\textit {\_Z}^{4}+\left (4 c^{2} d +2 e \right ) \textit {\_Z}^{2}+e \right )}{\sum }\frac {\left (\textit {\_R1}^{2} e +4 c^{2} d +e \right ) \left (i \arccos \left (c x \right ) \ln \left (\frac {\textit {\_R1} -c x -i \sqrt {-c^{2} x^{2}+1}}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {\textit {\_R1} -c x -i \sqrt {-c^{2} x^{2}+1}}{\textit {\_R1}}\right )\right )}{\textit {\_R1} \left (\textit {\_R1}^{2} e +2 c^{2} d +e \right )}\right ) e^{2} c^{3} x^{3}\right )}{24 c^{7} x^{3} d^{3}}\right )\)	\(439\)
default	\(c^{3} \left (-\frac {a}{3 d \,c^{3} x^{3}}+\frac {a e}{c^{3} d^{2} x}+\frac {a \,e^{2} \arctan \left (\frac {e x}{\sqrt {d e}}\right )}{c^{3} d^{2} \sqrt {d e}}-\frac {i b \left (8 \arctan \left (c x +i \sqrt {-c^{2} x^{2}+1}\right ) c^{7} d^{2} x^{3}+4 i \sqrt {-c^{2} x^{2}+1}\, c^{5} d^{2} x -8 i \arccos \left (c x \right ) c^{4} d^{2}+24 i \arccos \left (c x \right ) c^{4} d e \,x^{2}-48 \arctan \left (c x +i \sqrt {-c^{2} x^{2}+1}\right ) c^{5} d e \,x^{3}+3 \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (e \,\textit {\_Z}^{4}+\left (4 c^{2} d +2 e \right ) \textit {\_Z}^{2}+e \right )}{\sum }\frac {\left (4 \textit {\_R1}^{2} c^{2} d +\textit {\_R1}^{2} e +e \right ) \left (i \arccos \left (c x \right ) \ln \left (\frac {\textit {\_R1} -c x -i \sqrt {-c^{2} x^{2}+1}}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {\textit {\_R1} -c x -i \sqrt {-c^{2} x^{2}+1}}{\textit {\_R1}}\right )\right )}{\textit {\_R1} \left (\textit {\_R1}^{2} e +2 c^{2} d +e \right )}\right ) e^{2} c^{3} x^{3}-3 \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (e \,\textit {\_Z}^{4}+\left (4 c^{2} d +2 e \right ) \textit {\_Z}^{2}+e \right )}{\sum }\frac {\left (\textit {\_R1}^{2} e +4 c^{2} d +e \right ) \left (i \arccos \left (c x \right ) \ln \left (\frac {\textit {\_R1} -c x -i \sqrt {-c^{2} x^{2}+1}}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {\textit {\_R1} -c x -i \sqrt {-c^{2} x^{2}+1}}{\textit {\_R1}}\right )\right )}{\textit {\_R1} \left (\textit {\_R1}^{2} e +2 c^{2} d +e \right )}\right ) e^{2} c^{3} x^{3}\right )}{24 c^{7} x^{3} d^{3}}\right )\)	\(439\)

Input:

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

Output:

a*(-1/3/d/x^3+e/d^2/x+e^2/d^2/(d*e)^(1/2)*arctan(e*x/(d*e)^(1/2)))-1/24*I* 
b/c^4*(8*arctan(c*x+I*(-c^2*x^2+1)^(1/2))*c^7*d^2*x^3+4*I*(-c^2*x^2+1)^(1/ 
2)*c^5*d^2*x-8*I*arccos(c*x)*c^4*d^2+24*I*arccos(c*x)*c^4*d*e*x^2-48*arcta 
n(c*x+I*(-c^2*x^2+1)^(1/2))*c^5*d*e*x^3+3*sum((4*_R1^2*c^2*d+_R1^2*e+e)/_R 
1/(_R1^2*e+2*c^2*d+e)*(I*arccos(c*x)*ln((_R1-c*x-I*(-c^2*x^2+1)^(1/2))/_R1 
)+dilog((_R1-c*x-I*(-c^2*x^2+1)^(1/2))/_R1)),_R1=RootOf(e*_Z^4+(4*c^2*d+2* 
e)*_Z^2+e))*e^2*c^3*x^3-3*sum((_R1^2*e+4*c^2*d+e)/_R1/(_R1^2*e+2*c^2*d+e)* 
(I*arccos(c*x)*ln((_R1-c*x-I*(-c^2*x^2+1)^(1/2))/_R1)+dilog((_R1-c*x-I*(-c 
^2*x^2+1)^(1/2))/_R1)),_R1=RootOf(e*_Z^4+(4*c^2*d+2*e)*_Z^2+e))*e^2*c^3*x^ 
3)/x^3/d^3
 

Fricas [F]

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

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

Output:

integral((b*arccos(c*x) + a)/(e*x^6 + d*x^4), x)
 

Sympy [F]

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

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

Output:

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

Maxima [F(-2)]

Exception generated. \[ \int \frac {a+b \arccos (c x)}{x^4 \left (d+e x^2\right )} \, dx=\text {Exception raised: ValueError} \] Input:

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

Output:

Exception raised: ValueError >> Computation failed since Maxima requested 
additional constraints; using the 'assume' command before evaluation *may* 
 help (example of legal syntax is 'assume(e>0)', see `assume?` for more de 
tails)Is e
 

Giac [F(-2)]

Exception generated. \[ \int \frac {a+b \arccos (c x)}{x^4 \left (d+e x^2\right )} \, dx=\text {Exception raised: RuntimeError} \] Input:

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

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

Output:

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

Reduce [F]

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

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

Output:

(3*sqrt(e)*sqrt(d)*atan((e*x)/(sqrt(e)*sqrt(d)))*a*e*x**3 + 3*int(acos(c*x 
)/(d*x**4 + e*x**6),x)*b*d**3*x**3 - a*d**2 + 3*a*d*e*x**2)/(3*d**3*x**3)