\(\int \frac {a+b \arcsin (c x)}{x^4 (d+e x^2)} \, dx\) [458]

Optimal result

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

-1/6*b*c*(-c^2*x^2+1)^(1/2)/d/x^2-1/3*(a+b*arcsin(c*x))/d/x^3+e*(a+b*arcsi 
n(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*arcsin(c*x))*ln(1-e^(1/2)*(I*c*x+(-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*arcsin(c*x))*ln(1+e^(1/2)*(I*c*x+(-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*arcsin(c*x))*ln(1-e^(1/2)*(I*c* 
x+(-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*arcsin(c*x))*ln(1+e^(1/2)*(I*c*x+(-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)*(I*c* 
x+(-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)*(I*c*x+(-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)*(I*c*x+(-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)*(I*c*x+(-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 = 0.28 (sec) , antiderivative size = 531, normalized size of antiderivative = 0.82 \[ \int \frac {a+b \arcsin (c x)}{x^4 \left (d+e x^2\right )} \, dx=-\frac {a}{3 d x^3}+\frac {a e}{d^2 x}+\frac {a e^{3/2} \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{d^{5/2}}+b \left (-\frac {e \left (-\frac {\arcsin (c x)}{x}-c \text {arctanh}\left (\sqrt {1-c^2 x^2}\right )\right )}{d^2}-\frac {c x \sqrt {1-c^2 x^2}+2 \arcsin (c x)+c^3 x^3 \text {arctanh}\left (\sqrt {1-c^2 x^2}\right )}{6 d x^3}-\frac {e^{3/2} \left (\arcsin (c x) \left (\arcsin (c x)+2 i \left (\log \left (1+\frac {\sqrt {e} e^{i \arcsin (c x)}}{c \sqrt {d}-\sqrt {c^2 d+e}}\right )+\log \left (1+\frac {\sqrt {e} e^{i \arcsin (c x)}}{c \sqrt {d}+\sqrt {c^2 d+e}}\right )\right )\right )+2 \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{i \arcsin (c x)}}{-c \sqrt {d}+\sqrt {c^2 d+e}}\right )+2 \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} e^{i \arcsin (c x)}}{c \sqrt {d}+\sqrt {c^2 d+e}}\right )\right )}{4 d^{5/2}}+\frac {e^{3/2} \left (\arcsin (c x) \left (\arcsin (c x)+2 i \left (\log \left (1+\frac {\sqrt {e} e^{i \arcsin (c x)}}{-c \sqrt {d}+\sqrt {c^2 d+e}}\right )+\log \left (1-\frac {\sqrt {e} e^{i \arcsin (c x)}}{c \sqrt {d}+\sqrt {c^2 d+e}}\right )\right )\right )+2 \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{i \arcsin (c x)}}{c \sqrt {d}-\sqrt {c^2 d+e}}\right )+2 \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{i \arcsin (c x)}}{c \sqrt {d}+\sqrt {c^2 d+e}}\right )\right )}{4 d^{5/2}}\right ) \] Input:

Integrate[(a + b*ArcSin[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*(-(ArcSin[c*x]/x) - c*ArcTanh[Sqrt[1 - c^2*x^2]]))/d^2) - 
 (c*x*Sqrt[1 - c^2*x^2] + 2*ArcSin[c*x] + c^3*x^3*ArcTanh[Sqrt[1 - c^2*x^2 
]])/(6*d*x^3) - (e^(3/2)*(ArcSin[c*x]*(ArcSin[c*x] + (2*I)*(Log[1 + (Sqrt[ 
e]*E^(I*ArcSin[c*x]))/(c*Sqrt[d] - Sqrt[c^2*d + e])] + Log[1 + (Sqrt[e]*E^ 
(I*ArcSin[c*x]))/(c*Sqrt[d] + Sqrt[c^2*d + e])])) + 2*PolyLog[2, (Sqrt[e]* 
E^(I*ArcSin[c*x]))/(-(c*Sqrt[d]) + Sqrt[c^2*d + e])] + 2*PolyLog[2, -((Sqr 
t[e]*E^(I*ArcSin[c*x]))/(c*Sqrt[d] + Sqrt[c^2*d + e]))]))/(4*d^(5/2)) + (e 
^(3/2)*(ArcSin[c*x]*(ArcSin[c*x] + (2*I)*(Log[1 + (Sqrt[e]*E^(I*ArcSin[c*x 
]))/(-(c*Sqrt[d]) + Sqrt[c^2*d + e])] + Log[1 - (Sqrt[e]*E^(I*ArcSin[c*x]) 
)/(c*Sqrt[d] + Sqrt[c^2*d + e])])) + 2*PolyLog[2, (Sqrt[e]*E^(I*ArcSin[c*x 
]))/(c*Sqrt[d] - Sqrt[c^2*d + e])] + 2*PolyLog[2, (Sqrt[e]*E^(I*ArcSin[c*x 
]))/(c*Sqrt[d] + Sqrt[c^2*d + e])]))/(4*d^(5/2)))
 

Rubi [A] (verified)

Time = 1.45 (sec) , antiderivative size = 649, 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 = {5232, 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.

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

\(\Big \downarrow \) 5232

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

-1/6*(b*c*Sqrt[1 - c^2*x^2])/(d*x^2) - (a + b*ArcSin[c*x])/(3*d*x^3) + (e* 
(a + b*ArcSin[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*ArcSin[c*x])*Log[ 
1 - (Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] - Sqrt[c^2*d + e])])/(2*(-d) 
^(5/2)) - (e^(3/2)*(a + b*ArcSin[c*x])*Log[1 + (Sqrt[e]*E^(I*ArcSin[c*x])) 
/(I*c*Sqrt[-d] - Sqrt[c^2*d + e])])/(2*(-d)^(5/2)) + (e^(3/2)*(a + b*ArcSi 
n[c*x])*Log[1 - (Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] + Sqrt[c^2*d + e 
])])/(2*(-d)^(5/2)) - (e^(3/2)*(a + b*ArcSin[c*x])*Log[1 + (Sqrt[e]*E^(I*A 
rcSin[c*x]))/(I*c*Sqrt[-d] + Sqrt[c^2*d + e])])/(2*(-d)^(5/2)) + ((I/2)*b* 
e^(3/2)*PolyLog[2, -((Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] - Sqrt[c^2* 
d + e]))])/(-d)^(5/2) - ((I/2)*b*e^(3/2)*PolyLog[2, (Sqrt[e]*E^(I*ArcSin[c 
*x]))/(I*c*Sqrt[-d] - Sqrt[c^2*d + e])])/(-d)^(5/2) + ((I/2)*b*e^(3/2)*Pol 
yLog[2, -((Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] + Sqrt[c^2*d + e]))])/ 
(-d)^(5/2) - ((I/2)*b*e^(3/2)*PolyLog[2, (Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c* 
Sqrt[-d] + 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 5232

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_.)*((d_) + (e_ 
.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*ArcSin[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 = 356.98 (sec) , antiderivative size = 491, normalized size of antiderivative = 0.76

method	result	size
parts	\(a \left (-\frac {1}{3 d \,x^{3}}+\frac {e}{d^{2} x}+\frac {e^{2} \arctan \left (\frac {e x}{\sqrt {d e}}\right )}{d^{2} \sqrt {d e}}\right )-\frac {b \left (4 \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right ) c^{7} d^{2} x^{3}-4 \ln \left (i c x +\sqrt {-c^{2} x^{2}+1}-1\right ) c^{7} d^{2} x^{3}+4 \sqrt {-c^{2} x^{2}+1}\, c^{5} d^{2} x -24 \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right ) c^{5} d e \,x^{3}+24 \ln \left (i c x +\sqrt {-c^{2} x^{2}+1}-1\right ) c^{5} d e \,x^{3}-24 \arcsin \left (c x \right ) c^{4} d e \,x^{2}+8 c^{4} d^{2} \arcsin \left (c x \right )+3 \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (e \,\textit {\_Z}^{4}+\left (-4 c^{2} d -2 e \right ) \textit {\_Z}^{2}+e \right )}{\sum }\frac {\left (\textit {\_R1}^{2} e -4 c^{2} d -e \right ) \left (i \arcsin \left (c x \right ) \ln \left (\frac {\textit {\_R1} -i c x -\sqrt {-c^{2} x^{2}+1}}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {\textit {\_R1} -i c x -\sqrt {-c^{2} x^{2}+1}}{\textit {\_R1}}\right )\right )}{\textit {\_R1} \left (\textit {\_R1}^{2} e -2 c^{2} d -e \right )}\right ) c^{3} x^{3} e^{2}-3 \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (e \,\textit {\_Z}^{4}+\left (-4 c^{2} d -2 e \right ) \textit {\_Z}^{2}+e \right )}{\sum }\frac {\left (4 \textit {\_R1}^{2} c^{2} d +\textit {\_R1}^{2} e -e \right ) \left (i \arcsin \left (c x \right ) \ln \left (\frac {\textit {\_R1} -i c x -\sqrt {-c^{2} x^{2}+1}}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {\textit {\_R1} -i c x -\sqrt {-c^{2} x^{2}+1}}{\textit {\_R1}}\right )\right )}{\textit {\_R1} \left (\textit {\_R1}^{2} e -2 c^{2} d -e \right )}\right ) c^{3} x^{3} e^{2}\right )}{24 c^{4} x^{3} d^{3}}\)	\(491\)
derivativedivides	\(c^{3} \left (\frac {a \,e^{2} \arctan \left (\frac {e x}{\sqrt {d e}}\right )}{c^{3} d^{2} \sqrt {d e}}-\frac {a}{3 d \,c^{3} x^{3}}+\frac {a e}{c^{3} d^{2} x}-\frac {b \left (-4 \ln \left (i c x +\sqrt {-c^{2} x^{2}+1}-1\right ) c^{7} d^{2} x^{3}+4 \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right ) c^{7} d^{2} x^{3}+4 \sqrt {-c^{2} x^{2}+1}\, c^{5} d^{2} x +24 \ln \left (i c x +\sqrt {-c^{2} x^{2}+1}-1\right ) c^{5} d e \,x^{3}-24 \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right ) c^{5} d e \,x^{3}+8 c^{4} d^{2} \arcsin \left (c x \right )-24 \arcsin \left (c x \right ) c^{4} d e \,x^{2}+3 \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (e \,\textit {\_Z}^{4}+\left (-4 c^{2} d -2 e \right ) \textit {\_Z}^{2}+e \right )}{\sum }\frac {\left (-\textit {\_R1}^{2} e +4 c^{2} d +e \right ) \left (i \arcsin \left (c x \right ) \ln \left (\frac {\textit {\_R1} -i c x -\sqrt {-c^{2} x^{2}+1}}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {\textit {\_R1} -i c x -\sqrt {-c^{2} x^{2}+1}}{\textit {\_R1}}\right )\right )}{\textit {\_R1} \left (-\textit {\_R1}^{2} e +2 c^{2} d +e \right )}\right ) e^{2} c^{3} x^{3}+3 \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (e \,\textit {\_Z}^{4}+\left (-4 c^{2} d -2 e \right ) \textit {\_Z}^{2}+e \right )}{\sum }\frac {\left (4 \textit {\_R1}^{2} c^{2} d +\textit {\_R1}^{2} e -e \right ) \left (i \arcsin \left (c x \right ) \ln \left (\frac {\textit {\_R1} -i c x -\sqrt {-c^{2} x^{2}+1}}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {\textit {\_R1} -i c x -\sqrt {-c^{2} x^{2}+1}}{\textit {\_R1}}\right )\right )}{\textit {\_R1} \left (-\textit {\_R1}^{2} e +2 c^{2} d +e \right )}\right ) e^{2} c^{3} x^{3}\right )}{24 c^{7} x^{3} d^{3}}\right )\)	\(501\)
default	\(c^{3} \left (\frac {a \,e^{2} \arctan \left (\frac {e x}{\sqrt {d e}}\right )}{c^{3} d^{2} \sqrt {d e}}-\frac {a}{3 d \,c^{3} x^{3}}+\frac {a e}{c^{3} d^{2} x}-\frac {b \left (-4 \ln \left (i c x +\sqrt {-c^{2} x^{2}+1}-1\right ) c^{7} d^{2} x^{3}+4 \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right ) c^{7} d^{2} x^{3}+4 \sqrt {-c^{2} x^{2}+1}\, c^{5} d^{2} x +24 \ln \left (i c x +\sqrt {-c^{2} x^{2}+1}-1\right ) c^{5} d e \,x^{3}-24 \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right ) c^{5} d e \,x^{3}+8 c^{4} d^{2} \arcsin \left (c x \right )-24 \arcsin \left (c x \right ) c^{4} d e \,x^{2}+3 \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (e \,\textit {\_Z}^{4}+\left (-4 c^{2} d -2 e \right ) \textit {\_Z}^{2}+e \right )}{\sum }\frac {\left (-\textit {\_R1}^{2} e +4 c^{2} d +e \right ) \left (i \arcsin \left (c x \right ) \ln \left (\frac {\textit {\_R1} -i c x -\sqrt {-c^{2} x^{2}+1}}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {\textit {\_R1} -i c x -\sqrt {-c^{2} x^{2}+1}}{\textit {\_R1}}\right )\right )}{\textit {\_R1} \left (-\textit {\_R1}^{2} e +2 c^{2} d +e \right )}\right ) e^{2} c^{3} x^{3}+3 \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (e \,\textit {\_Z}^{4}+\left (-4 c^{2} d -2 e \right ) \textit {\_Z}^{2}+e \right )}{\sum }\frac {\left (4 \textit {\_R1}^{2} c^{2} d +\textit {\_R1}^{2} e -e \right ) \left (i \arcsin \left (c x \right ) \ln \left (\frac {\textit {\_R1} -i c x -\sqrt {-c^{2} x^{2}+1}}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {\textit {\_R1} -i c x -\sqrt {-c^{2} x^{2}+1}}{\textit {\_R1}}\right )\right )}{\textit {\_R1} \left (-\textit {\_R1}^{2} e +2 c^{2} d +e \right )}\right ) e^{2} c^{3} x^{3}\right )}{24 c^{7} x^{3} d^{3}}\right )\)	\(501\)

Input:

int((a+b*arcsin(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*b/ 
c^4*(4*ln(1+I*c*x+(-c^2*x^2+1)^(1/2))*c^7*d^2*x^3-4*ln(I*c*x+(-c^2*x^2+1)^ 
(1/2)-1)*c^7*d^2*x^3+4*(-c^2*x^2+1)^(1/2)*c^5*d^2*x-24*ln(1+I*c*x+(-c^2*x^ 
2+1)^(1/2))*c^5*d*e*x^3+24*ln(I*c*x+(-c^2*x^2+1)^(1/2)-1)*c^5*d*e*x^3-24*a 
rcsin(c*x)*c^4*d*e*x^2+8*c^4*d^2*arcsin(c*x)+3*sum((_R1^2*e-4*c^2*d-e)/_R1 
/(_R1^2*e-2*c^2*d-e)*(I*arcsin(c*x)*ln((_R1-I*c*x-(-c^2*x^2+1)^(1/2))/_R1) 
+dilog((_R1-I*c*x-(-c^2*x^2+1)^(1/2))/_R1)),_R1=RootOf(e*_Z^4+(-4*c^2*d-2* 
e)*_Z^2+e))*c^3*x^3*e^2-3*sum((4*_R1^2*c^2*d+_R1^2*e-e)/_R1/(_R1^2*e-2*c^2 
*d-e)*(I*arcsin(c*x)*ln((_R1-I*c*x-(-c^2*x^2+1)^(1/2))/_R1)+dilog((_R1-I*c 
*x-(-c^2*x^2+1)^(1/2))/_R1)),_R1=RootOf(e*_Z^4+(-4*c^2*d-2*e)*_Z^2+e))*c^3 
*x^3*e^2)/x^3/d^3
 

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F(-2)]

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

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

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

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

Output:

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

Reduce [F]

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

int((a+b*asin(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(asin(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)