\(\int \frac {a+b \arccos (c x)}{x^3 (d+e x^2)} \, dx\) [633]

Optimal result

Integrand size = 21, antiderivative size = 573 \[ \int \frac {a+b \arccos (c x)}{x^3 \left (d+e x^2\right )} \, dx=-\frac {b c \sqrt {1-c^2 x^2}}{2 d x}-\frac {a+b \arccos (c x)}{2 d x^2}+\frac {e (a+b \arccos (c x)) \log \left (1-\frac {\sqrt {e} e^{i \arccos (c x)}}{i c \sqrt {-d}-\sqrt {c^2 d+e}}\right )}{2 d^2}+\frac {e (a+b \arccos (c x)) \log \left (1+\frac {\sqrt {e} e^{i \arccos (c x)}}{i c \sqrt {-d}-\sqrt {c^2 d+e}}\right )}{2 d^2}+\frac {e (a+b \arccos (c x)) \log \left (1-\frac {\sqrt {e} e^{i \arccos (c x)}}{i c \sqrt {-d}+\sqrt {c^2 d+e}}\right )}{2 d^2}+\frac {e (a+b \arccos (c x)) \log \left (1+\frac {\sqrt {e} e^{i \arccos (c x)}}{i c \sqrt {-d}+\sqrt {c^2 d+e}}\right )}{2 d^2}-\frac {e (a+b \arccos (c x)) \log \left (1-e^{2 i \arccos (c x)}\right )}{d^2}-\frac {i b e \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} e^{i \arccos (c x)}}{i c \sqrt {-d}-\sqrt {c^2 d+e}}\right )}{2 d^2}-\frac {i b e \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{i \arccos (c x)}}{i c \sqrt {-d}-\sqrt {c^2 d+e}}\right )}{2 d^2}-\frac {i b e \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} e^{i \arccos (c x)}}{i c \sqrt {-d}+\sqrt {c^2 d+e}}\right )}{2 d^2}-\frac {i b e \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{i \arccos (c x)}}{i c \sqrt {-d}+\sqrt {c^2 d+e}}\right )}{2 d^2}+\frac {i b e \operatorname {PolyLog}\left (2,e^{2 i \arccos (c x)}\right )}{2 d^2} \] Output:

-1/2*b*c*(-c^2*x^2+1)^(1/2)/d/x-1/2*(a+b*arccos(c*x))/d/x^2+1/2*e*(a+b*arc 
cos(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^2+1/2*e*(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^2+1/2*e*(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^2+1/2* 
e*(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^2-e*(a+b*arccos(c*x))*ln(1-(c*x+I*(-c^2*x^2+1)^(1/2) 
)^2)/d^2-1/2*I*b*e*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^2-1/2*I*b*e*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^2-1/2*I*b*e*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^2-1/2*I* 
b*e*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^2+1/2*I*b*e*polylog(2,(c*x+I*(-c^2*x^2+1)^(1/2))^2)/d^2
 

Mathematica [A] (verified)

Time = 0.16 (sec) , antiderivative size = 966, normalized size of antiderivative = 1.69 \[ \int \frac {a+b \arccos (c x)}{x^3 \left (d+e x^2\right )} \, dx =\text {Too large to display} \] Input:

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

Output:

-1/2*a/(d*x^2) - (a*e*Log[x])/d^2 + (a*e*Log[d + e*x^2])/(2*d^2) + b*((c*x 
*Sqrt[1 - c^2*x^2] - ArcCos[c*x])/(2*d*x^2) - ((I/4)*e*(ArcCos[c*x]^2 - 8* 
ArcSin[Sqrt[1 + (I*c*Sqrt[d])/Sqrt[e]]/Sqrt[2]]*ArcTan[((c*Sqrt[d] + I*Sqr 
t[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)*ArcSi 
n[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[Sqr 
t[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]) + Sqrt[c^ 
2*d + e])*E^(I*ArcCos[c*x]))/Sqrt[e]] + 2*PolyLog[2, ((-I)*(c*Sqrt[d] + Sq 
rt[c^2*d + e])*E^(I*ArcCos[c*x]))/Sqrt[e]]))/d^2 - ((I/4)*e*(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)* 
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]] - (4*I)*ArcSi 
n[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]...
 

Rubi [A] (verified)

Time = 1.46 (sec) , antiderivative size = 573, 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.

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

\(\Big \downarrow \) 5233

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

(b*c*Sqrt[1 - c^2*x^2])/(2*d*x) - (a + b*ArcCos[c*x])/(2*d*x^2) + (e*(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^2) + (e*(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^2) + (e*(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^2) + (e*(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^2) - (e*(a + b*ArcCos[c*x])*Log[1 + E^((2 
*I)*ArcCos[c*x])])/d^2 - ((I/2)*b*e*PolyLog[2, -((Sqrt[e]*E^(I*ArcCos[c*x] 
))/(c*Sqrt[-d] - I*Sqrt[c^2*d + e]))])/d^2 - ((I/2)*b*e*PolyLog[2, (Sqrt[e 
]*E^(I*ArcCos[c*x]))/(c*Sqrt[-d] - I*Sqrt[c^2*d + e])])/d^2 - ((I/2)*b*e*P 
olyLog[2, -((Sqrt[e]*E^(I*ArcCos[c*x]))/(c*Sqrt[-d] + I*Sqrt[c^2*d + e]))] 
)/d^2 - ((I/2)*b*e*PolyLog[2, (Sqrt[e]*E^(I*ArcCos[c*x]))/(c*Sqrt[-d] + I* 
Sqrt[c^2*d + e])])/d^2 + ((I/2)*b*e*PolyLog[2, -E^((2*I)*ArcCos[c*x])])/d^ 
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 = 1.95 (sec) , antiderivative size = 470, normalized size of antiderivative = 0.82

method	result	size
parts	\(a \left (-\frac {1}{2 d \,x^{2}}-\frac {e \ln \left (x \right )}{d^{2}}+\frac {e \ln \left (e \,x^{2}+d \right )}{2 d^{2}}\right )+b \,c^{2} \left (-\frac {-i c^{2} x^{2}-c x \sqrt {-c^{2} x^{2}+1}+\arccos \left (c x \right )}{2 c^{2} x^{2} d}-\frac {e \arccos \left (c x \right ) \ln \left (1+i \left (c x +i \sqrt {-c^{2} x^{2}+1}\right )\right )}{d^{2} c^{2}}-\frac {e \arccos \left (c x \right ) \ln \left (1-i \left (c x +i \sqrt {-c^{2} x^{2}+1}\right )\right )}{d^{2} c^{2}}+\frac {i e \operatorname {dilog}\left (1+i \left (c x +i \sqrt {-c^{2} x^{2}+1}\right )\right )}{d^{2} c^{2}}+\frac {i e \operatorname {dilog}\left (1-i \left (c x +i \sqrt {-c^{2} x^{2}+1}\right )\right )}{d^{2} c^{2}}-\frac {i e^{2} \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (e \,\textit {\_Z}^{4}+\left (4 c^{2} d +2 e \right ) \textit {\_Z}^{2}+e \right )}{\sum }\frac {\left (\textit {\_R1}^{2}+1\right ) \left (i \arccos \left (c x \right ) \ln \left (\frac {\textit {\_R1} -c x -i \sqrt {-c^{2} x^{2}+1}}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {\textit {\_R1} -c x -i \sqrt {-c^{2} x^{2}+1}}{\textit {\_R1}}\right )\right )}{\textit {\_R1}^{2} e +2 c^{2} d +e}\right )}{4 d^{2} c^{2}}-\frac {i e \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (e \,\textit {\_Z}^{4}+\left (4 c^{2} d +2 e \right ) \textit {\_Z}^{2}+e \right )}{\sum }\frac {\left (\textit {\_R1}^{2} e +4 c^{2} d +e \right ) \left (i \arccos \left (c x \right ) \ln \left (\frac {\textit {\_R1} -c x -i \sqrt {-c^{2} x^{2}+1}}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {\textit {\_R1} -c x -i \sqrt {-c^{2} x^{2}+1}}{\textit {\_R1}}\right )\right )}{\textit {\_R1}^{2} e +2 c^{2} d +e}\right )}{4 d^{2} c^{2}}\right )\)	\(470\)
derivativedivides	\(c^{2} \left (\frac {a e \ln \left (c^{2} e \,x^{2}+c^{2} d \right )}{2 c^{2} d^{2}}-\frac {a}{2 d \,c^{2} x^{2}}-\frac {a e \ln \left (c x \right )}{c^{2} d^{2}}+b \,c^{2} \left (-\frac {-i c^{2} x^{2}-c x \sqrt {-c^{2} x^{2}+1}+\arccos \left (c x \right )}{2 c^{4} x^{2} d}-\frac {i e^{2} \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (e \,\textit {\_Z}^{4}+\left (4 c^{2} d +2 e \right ) \textit {\_Z}^{2}+e \right )}{\sum }\frac {\left (\textit {\_R1}^{2}+1\right ) \left (i \arccos \left (c x \right ) \ln \left (\frac {\textit {\_R1} -c x -i \sqrt {-c^{2} x^{2}+1}}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {\textit {\_R1} -c x -i \sqrt {-c^{2} x^{2}+1}}{\textit {\_R1}}\right )\right )}{\textit {\_R1}^{2} e +2 c^{2} d +e}\right )}{4 d^{2} c^{4}}-\frac {e \arccos \left (c x \right ) \ln \left (1+i \left (c x +i \sqrt {-c^{2} x^{2}+1}\right )\right )}{d^{2} c^{4}}-\frac {e \arccos \left (c x \right ) \ln \left (1-i \left (c x +i \sqrt {-c^{2} x^{2}+1}\right )\right )}{d^{2} c^{4}}+\frac {i e \operatorname {dilog}\left (1+i \left (c x +i \sqrt {-c^{2} x^{2}+1}\right )\right )}{d^{2} c^{4}}+\frac {i e \operatorname {dilog}\left (1-i \left (c x +i \sqrt {-c^{2} x^{2}+1}\right )\right )}{d^{2} c^{4}}-\frac {i e \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (e \,\textit {\_Z}^{4}+\left (4 c^{2} d +2 e \right ) \textit {\_Z}^{2}+e \right )}{\sum }\frac {\left (\textit {\_R1}^{2} e +4 c^{2} d +e \right ) \left (i \arccos \left (c x \right ) \ln \left (\frac {\textit {\_R1} -c x -i \sqrt {-c^{2} x^{2}+1}}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {\textit {\_R1} -c x -i \sqrt {-c^{2} x^{2}+1}}{\textit {\_R1}}\right )\right )}{\textit {\_R1}^{2} e +2 c^{2} d +e}\right )}{4 d^{2} c^{4}}\right )\right )\)	\(492\)
default	\(c^{2} \left (\frac {a e \ln \left (c^{2} e \,x^{2}+c^{2} d \right )}{2 c^{2} d^{2}}-\frac {a}{2 d \,c^{2} x^{2}}-\frac {a e \ln \left (c x \right )}{c^{2} d^{2}}+b \,c^{2} \left (-\frac {-i c^{2} x^{2}-c x \sqrt {-c^{2} x^{2}+1}+\arccos \left (c x \right )}{2 c^{4} x^{2} d}-\frac {i e^{2} \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (e \,\textit {\_Z}^{4}+\left (4 c^{2} d +2 e \right ) \textit {\_Z}^{2}+e \right )}{\sum }\frac {\left (\textit {\_R1}^{2}+1\right ) \left (i \arccos \left (c x \right ) \ln \left (\frac {\textit {\_R1} -c x -i \sqrt {-c^{2} x^{2}+1}}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {\textit {\_R1} -c x -i \sqrt {-c^{2} x^{2}+1}}{\textit {\_R1}}\right )\right )}{\textit {\_R1}^{2} e +2 c^{2} d +e}\right )}{4 d^{2} c^{4}}-\frac {e \arccos \left (c x \right ) \ln \left (1+i \left (c x +i \sqrt {-c^{2} x^{2}+1}\right )\right )}{d^{2} c^{4}}-\frac {e \arccos \left (c x \right ) \ln \left (1-i \left (c x +i \sqrt {-c^{2} x^{2}+1}\right )\right )}{d^{2} c^{4}}+\frac {i e \operatorname {dilog}\left (1+i \left (c x +i \sqrt {-c^{2} x^{2}+1}\right )\right )}{d^{2} c^{4}}+\frac {i e \operatorname {dilog}\left (1-i \left (c x +i \sqrt {-c^{2} x^{2}+1}\right )\right )}{d^{2} c^{4}}-\frac {i e \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (e \,\textit {\_Z}^{4}+\left (4 c^{2} d +2 e \right ) \textit {\_Z}^{2}+e \right )}{\sum }\frac {\left (\textit {\_R1}^{2} e +4 c^{2} d +e \right ) \left (i \arccos \left (c x \right ) \ln \left (\frac {\textit {\_R1} -c x -i \sqrt {-c^{2} x^{2}+1}}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {\textit {\_R1} -c x -i \sqrt {-c^{2} x^{2}+1}}{\textit {\_R1}}\right )\right )}{\textit {\_R1}^{2} e +2 c^{2} d +e}\right )}{4 d^{2} c^{4}}\right )\right )\)	\(492\)

Input:

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

Output:

a*(-1/2/d/x^2-e/d^2*ln(x)+1/2*e/d^2*ln(e*x^2+d))+b*c^2*(-1/2*(-I*c^2*x^2-c 
*x*(-c^2*x^2+1)^(1/2)+arccos(c*x))/c^2/x^2/d-e/d^2/c^2*arccos(c*x)*ln(1+I* 
(c*x+I*(-c^2*x^2+1)^(1/2)))-e/d^2/c^2*arccos(c*x)*ln(1-I*(c*x+I*(-c^2*x^2+ 
1)^(1/2)))+I*e/d^2/c^2*dilog(1+I*(c*x+I*(-c^2*x^2+1)^(1/2)))+I*e/d^2/c^2*d 
ilog(1-I*(c*x+I*(-c^2*x^2+1)^(1/2)))-1/4*I*e^2/d^2*sum((_R1^2+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((_R 
1-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)) 
/c^2-1/4*I*e/d^2*sum((_R1^2*e+4*c^2*d+e)/(_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))/c^2)
 

Fricas [F]

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

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

Output:

integral((b*arccos(c*x) + a)/(e*x^5 + d*x^3), x)
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

1/2*a*(e*log(e*x^2 + d)/d^2 - 2*e*log(x)/d^2 - 1/(d*x^2)) + b*integrate(ar 
ctan2(sqrt(c*x + 1)*sqrt(-c*x + 1), c*x)/(e*x^5 + d*x^3), x)
 

Giac [F(-2)]

Exception generated. \[ \int \frac {a+b \arccos (c x)}{x^3 \left (d+e x^2\right )} \, dx=\text {Exception raised: RuntimeError} \] Input:

integrate((a+b*arccos(c*x))/x^3/(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. \[ \int \frac {a+b \arccos (c x)}{x^3 \left (d+e x^2\right )} \, dx=\int \frac {a+b\,\mathrm {acos}\left (c\,x\right )}{x^3\,\left (e\,x^2+d\right )} \,d x \] Input:

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

Output:

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

Reduce [F]

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

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

Output:

(2*int(acos(c*x)/(d*x**3 + e*x**5),x)*b*d**2*x**2 + log(d + e*x**2)*a*e*x* 
*2 - 2*log(x)*a*e*x**2 - a*d)/(2*d**2*x**2)