\(\int \frac {a+b \arcsin (c x)}{x^3 (d+e x^2)^2} \, dx\) [462]

Optimal result

Integrand size = 21, antiderivative size = 632 \[ \int \frac {a+b \arcsin (c x)}{x^3 \left (d+e x^2\right )^2} \, dx=-\frac {b c \sqrt {1-c^2 x^2}}{2 d^2 x}-\frac {a+b \arcsin (c x)}{2 d^2 x^2}-\frac {e (a+b \arcsin (c x))}{2 d^2 \left (d+e x^2\right )}+\frac {b c e \arctan \left (\frac {\sqrt {c^2 d+e} x}{\sqrt {d} \sqrt {1-c^2 x^2}}\right )}{2 d^{5/2} \sqrt {c^2 d+e}}+\frac {e (a+b \arcsin (c x)) \log \left (1-\frac {\sqrt {e} e^{i \arcsin (c x)}}{i c \sqrt {-d}-\sqrt {c^2 d+e}}\right )}{d^3}+\frac {e (a+b \arcsin (c x)) \log \left (1+\frac {\sqrt {e} e^{i \arcsin (c x)}}{i c \sqrt {-d}-\sqrt {c^2 d+e}}\right )}{d^3}+\frac {e (a+b \arcsin (c x)) \log \left (1-\frac {\sqrt {e} e^{i \arcsin (c x)}}{i c \sqrt {-d}+\sqrt {c^2 d+e}}\right )}{d^3}+\frac {e (a+b \arcsin (c x)) \log \left (1+\frac {\sqrt {e} e^{i \arcsin (c x)}}{i c \sqrt {-d}+\sqrt {c^2 d+e}}\right )}{d^3}-\frac {2 e (a+b \arcsin (c x)) \log \left (1-e^{2 i \arcsin (c x)}\right )}{d^3}-\frac {i b e \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} e^{i \arcsin (c x)}}{i c \sqrt {-d}-\sqrt {c^2 d+e}}\right )}{d^3}-\frac {i b e \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{i \arcsin (c x)}}{i c \sqrt {-d}-\sqrt {c^2 d+e}}\right )}{d^3}-\frac {i b e \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} e^{i \arcsin (c x)}}{i c \sqrt {-d}+\sqrt {c^2 d+e}}\right )}{d^3}-\frac {i b e \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{i \arcsin (c x)}}{i c \sqrt {-d}+\sqrt {c^2 d+e}}\right )}{d^3}+\frac {i b e \operatorname {PolyLog}\left (2,e^{2 i \arcsin (c x)}\right )}{d^3} \] Output:

-1/2*b*c*(-c^2*x^2+1)^(1/2)/d^2/x-1/2*(a+b*arcsin(c*x))/d^2/x^2-1/2*e*(a+b 
*arcsin(c*x))/d^2/(e*x^2+d)+1/2*b*c*e*arctan((c^2*d+e)^(1/2)*x/d^(1/2)/(-c 
^2*x^2+1)^(1/2))/d^(5/2)/(c^2*d+e)^(1/2)+e*(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^3+e*(a+b*ar 
csin(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^3+e*(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^3+e*(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^3-2*e*(a+b*ar 
csin(c*x))*ln(1-(I*c*x+(-c^2*x^2+1)^(1/2))^2)/d^3-I*b*e*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^3-I*b*e*po 
lylog(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^3-I*b*e*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^3-I*b*e*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^3+I*b*e*polylog(2,(I*c*x+(-c^2*x^2+1)^ 
(1/2))^2)/d^3
 

Mathematica [A] (verified)

Time = 1.59 (sec) , antiderivative size = 687, normalized size of antiderivative = 1.09 \[ \int \frac {a+b \arcsin (c x)}{x^3 \left (d+e x^2\right )^2} \, dx=-\frac {\frac {2 a d}{x^2}+\frac {2 a d e}{d+e x^2}+8 a e \log (x)-4 a e \log \left (d+e x^2\right )+b \left (\frac {2 d \left (c x \sqrt {1-c^2 x^2}+\arcsin (c x)\right )}{x^2}+\sqrt {d} e \left (\frac {\arcsin (c x)}{\sqrt {d}+i \sqrt {e} x}-\frac {c \arctan \left (\frac {i \sqrt {e}+c^2 \sqrt {d} x}{\sqrt {c^2 d+e} \sqrt {1-c^2 x^2}}\right )}{\sqrt {c^2 d+e}}\right )-i \sqrt {d} e \left (-\frac {\arcsin (c x)}{i \sqrt {d}+\sqrt {e} x}-\frac {c \text {arctanh}\left (\frac {\sqrt {e}+i c^2 \sqrt {d} x}{\sqrt {c^2 d+e} \sqrt {1-c^2 x^2}}\right )}{\sqrt {c^2 d+e}}\right )+2 i e \left (\arcsin (c x) \left (\arcsin (c x)+2 i \left (\log \left (1+\frac {\sqrt {e} e^{i \arcsin (c x)}}{c \sqrt {d}-\sqrt {c^2 d+e}}\right )+\log \left (1+\frac {\sqrt {e} e^{i \arcsin (c x)}}{c \sqrt {d}+\sqrt {c^2 d+e}}\right )\right )\right )+2 \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{i \arcsin (c x)}}{-c \sqrt {d}+\sqrt {c^2 d+e}}\right )+2 \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} e^{i \arcsin (c x)}}{c \sqrt {d}+\sqrt {c^2 d+e}}\right )\right )+2 i e \left (\arcsin (c x) \left (\arcsin (c x)+2 i \left (\log \left (1+\frac {\sqrt {e} e^{i \arcsin (c x)}}{-c \sqrt {d}+\sqrt {c^2 d+e}}\right )+\log \left (1-\frac {\sqrt {e} e^{i \arcsin (c x)}}{c \sqrt {d}+\sqrt {c^2 d+e}}\right )\right )\right )+2 \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{i \arcsin (c x)}}{c \sqrt {d}-\sqrt {c^2 d+e}}\right )+2 \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{i \arcsin (c x)}}{c \sqrt {d}+\sqrt {c^2 d+e}}\right )\right )-4 i e \left (\arcsin (c x) \left (\arcsin (c x)+2 i \log \left (1-e^{2 i \arcsin (c x)}\right )\right )+\operatorname {PolyLog}\left (2,e^{2 i \arcsin (c x)}\right )\right )\right )}{4 d^3} \] Input:

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

Output:

-1/4*((2*a*d)/x^2 + (2*a*d*e)/(d + e*x^2) + 8*a*e*Log[x] - 4*a*e*Log[d + e 
*x^2] + b*((2*d*(c*x*Sqrt[1 - c^2*x^2] + ArcSin[c*x]))/x^2 + Sqrt[d]*e*(Ar 
cSin[c*x]/(Sqrt[d] + I*Sqrt[e]*x) - (c*ArcTan[(I*Sqrt[e] + c^2*Sqrt[d]*x)/ 
(Sqrt[c^2*d + e]*Sqrt[1 - c^2*x^2])])/Sqrt[c^2*d + e]) - I*Sqrt[d]*e*(-(Ar 
cSin[c*x]/(I*Sqrt[d] + Sqrt[e]*x)) - (c*ArcTanh[(Sqrt[e] + I*c^2*Sqrt[d]*x 
)/(Sqrt[c^2*d + e]*Sqrt[1 - c^2*x^2])])/Sqrt[c^2*d + e]) + (2*I)*e*(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] + Sq 
rt[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]))]) + (2*I)*e*(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*I)*e* 
(ArcSin[c*x]*(ArcSin[c*x] + (2*I)*Log[1 - E^((2*I)*ArcSin[c*x])]) + PolyLo 
g[2, E^((2*I)*ArcSin[c*x])])))/d^3
 

Rubi [A] (verified)

Time = 1.65 (sec) , antiderivative size = 632, 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.

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

\(\Big \downarrow \) 5232

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

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

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 = 2.93 (sec) , antiderivative size = 589, normalized size of antiderivative = 0.93

method	result	size
parts	\(-\frac {a}{2 d^{2} x^{2}}-\frac {2 a e \ln \left (x \right )}{d^{3}}+\frac {a e \ln \left (e \,x^{2}+d \right )}{d^{3}}-\frac {a e}{2 d^{2} \left (e \,x^{2}+d \right )}+b \,c^{2} \left (-\frac {-i c^{4} d \,x^{2}-i e \,c^{4} x^{4}+\sqrt {-c^{2} x^{2}+1}\, c^{3} d x +\sqrt {-c^{2} x^{2}+1}\, e \,c^{3} x^{3}+\arcsin \left (c x \right ) c^{2} d +2 \arcsin \left (c x \right ) c^{2} e \,x^{2}}{2 c^{2} x^{2} d^{2} \left (c^{2} e \,x^{2}+c^{2} d \right )}-\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 \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}^{2} e -2 c^{2} d -e}\right )}{2 d^{3} 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 \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}^{2} e -2 c^{2} d -e}\right )}{2 d^{3} c^{2}}-\frac {i \sqrt {c^{2} d \left (c^{2} d +e \right )}\, \operatorname {arctanh}\left (\frac {2 e \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{2}-4 c^{2} d -2 e}{4 \sqrt {c^{4} d^{2}+c^{2} d e}}\right ) e}{2 d^{3} c^{2} \left (c^{2} d +e \right )}+\frac {2 i e \operatorname {dilog}\left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )}{d^{3} c^{2}}-\frac {2 i e \operatorname {dilog}\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )}{d^{3} c^{2}}-\frac {2 e \arcsin \left (c x \right ) \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )}{d^{3} c^{2}}\right )\)	\(589\)
derivativedivides	\(c^{2} \left (\frac {a e \ln \left (c^{2} e \,x^{2}+c^{2} d \right )}{c^{2} d^{3}}-\frac {a e}{2 d^{2} \left (c^{2} e \,x^{2}+c^{2} d \right )}-\frac {a}{2 d^{2} c^{2} x^{2}}-\frac {2 a e \ln \left (c x \right )}{c^{2} d^{3}}+b \,c^{4} \left (-\frac {-i c^{4} d \,x^{2}-i e \,c^{4} x^{4}+\sqrt {-c^{2} x^{2}+1}\, c^{3} d x +\sqrt {-c^{2} x^{2}+1}\, e \,c^{3} x^{3}+\arcsin \left (c x \right ) c^{2} d +2 \arcsin \left (c x \right ) c^{2} e \,x^{2}}{2 c^{6} x^{2} d^{2} \left (c^{2} e \,x^{2}+c^{2} d \right )}+\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 \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}^{2} e +2 c^{2} d +e}\right )}{2 c^{6} d^{3}}-\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 \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}^{2} e +2 c^{2} d +e}\right )}{2 c^{6} d^{3}}+\frac {i \sqrt {c^{2} d \left (c^{2} d +e \right )}\, \operatorname {arctanh}\left (\frac {4 c^{2} d -2 e \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{2}+2 e}{4 \sqrt {c^{4} d^{2}+c^{2} d e}}\right ) e}{2 c^{6} d^{3} \left (c^{2} d +e \right )}+\frac {2 i e \operatorname {dilog}\left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )}{c^{6} d^{3}}-\frac {2 i e \operatorname {dilog}\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )}{c^{6} d^{3}}-\frac {2 e \arcsin \left (c x \right ) \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )}{c^{6} d^{3}}\right )\right )\)	\(615\)
default	\(c^{2} \left (\frac {a e \ln \left (c^{2} e \,x^{2}+c^{2} d \right )}{c^{2} d^{3}}-\frac {a e}{2 d^{2} \left (c^{2} e \,x^{2}+c^{2} d \right )}-\frac {a}{2 d^{2} c^{2} x^{2}}-\frac {2 a e \ln \left (c x \right )}{c^{2} d^{3}}+b \,c^{4} \left (-\frac {-i c^{4} d \,x^{2}-i e \,c^{4} x^{4}+\sqrt {-c^{2} x^{2}+1}\, c^{3} d x +\sqrt {-c^{2} x^{2}+1}\, e \,c^{3} x^{3}+\arcsin \left (c x \right ) c^{2} d +2 \arcsin \left (c x \right ) c^{2} e \,x^{2}}{2 c^{6} x^{2} d^{2} \left (c^{2} e \,x^{2}+c^{2} d \right )}+\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 \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}^{2} e +2 c^{2} d +e}\right )}{2 c^{6} d^{3}}-\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 \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}^{2} e +2 c^{2} d +e}\right )}{2 c^{6} d^{3}}+\frac {i \sqrt {c^{2} d \left (c^{2} d +e \right )}\, \operatorname {arctanh}\left (\frac {4 c^{2} d -2 e \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{2}+2 e}{4 \sqrt {c^{4} d^{2}+c^{2} d e}}\right ) e}{2 c^{6} d^{3} \left (c^{2} d +e \right )}+\frac {2 i e \operatorname {dilog}\left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )}{c^{6} d^{3}}-\frac {2 i e \operatorname {dilog}\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )}{c^{6} d^{3}}-\frac {2 e \arcsin \left (c x \right ) \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )}{c^{6} d^{3}}\right )\right )\)	\(615\)

Input:

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

Output:

-1/2*a/d^2/x^2-2*a/d^3*e*ln(x)+a/d^3*e*ln(e*x^2+d)-1/2*a/d^2*e/(e*x^2+d)+b 
*c^2*(-1/2*(-I*c^4*d*x^2-I*e*c^4*x^4+(-c^2*x^2+1)^(1/2)*c^3*d*x+(-c^2*x^2+ 
1)^(1/2)*e*c^3*x^3+arcsin(c*x)*c^2*d+2*arcsin(c*x)*c^2*e*x^2)/c^2/x^2/d^2/ 
(c^2*e*x^2+c^2*d)-1/2*I*e/d^3*sum((_R1^2*e-4*c^2*d-e)/(_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^2-1/2*I 
*e^2/d^3*sum((_R1^2-1)/(_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=Root 
Of(e*_Z^4+(-4*c^2*d-2*e)*_Z^2+e))/c^2-1/2*I*(c^2*d*(c^2*d+e))^(1/2)/d^3/c^ 
2/(c^2*d+e)*arctanh(1/4*(2*e*(I*c*x+(-c^2*x^2+1)^(1/2))^2-4*c^2*d-2*e)/(c^ 
4*d^2+c^2*d*e)^(1/2))*e+2*I*e/d^3/c^2*dilog(1+I*c*x+(-c^2*x^2+1)^(1/2))-2* 
I*e/d^3*dilog(I*c*x+(-c^2*x^2+1)^(1/2))/c^2-2*e/d^3/c^2*arcsin(c*x)*ln(1+I 
*c*x+(-c^2*x^2+1)^(1/2)))
 

Fricas [F]

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

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

Output:

integral((b*arcsin(c*x) + a)/(e^2*x^7 + 2*d*e*x^5 + d^2*x^3), x)
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F(-1)]

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

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

Output:

Timed out
                                                                                    
                                                                                    
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {a+b \arcsin (c x)}{x^3 \left (d+e x^2\right )^2} \, dx=\frac {2 \left (\int \frac {\mathit {asin} \left (c x \right )}{e^{2} x^{7}+2 d e \,x^{5}+d^{2} x^{3}}d x \right ) b \,d^{4} x^{2}+2 \left (\int \frac {\mathit {asin} \left (c x \right )}{e^{2} x^{7}+2 d e \,x^{5}+d^{2} x^{3}}d x \right ) b \,d^{3} e \,x^{4}+2 \,\mathrm {log}\left (e \,x^{2}+d \right ) a d e \,x^{2}+2 \,\mathrm {log}\left (e \,x^{2}+d \right ) a \,e^{2} x^{4}-4 \,\mathrm {log}\left (x \right ) a d e \,x^{2}-4 \,\mathrm {log}\left (x \right ) a \,e^{2} x^{4}-a \,d^{2}+2 a \,e^{2} x^{4}}{2 d^{3} x^{2} \left (e \,x^{2}+d \right )} \] Input:

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

Output:

(2*int(asin(c*x)/(d**2*x**3 + 2*d*e*x**5 + e**2*x**7),x)*b*d**4*x**2 + 2*i 
nt(asin(c*x)/(d**2*x**3 + 2*d*e*x**5 + e**2*x**7),x)*b*d**3*e*x**4 + 2*log 
(d + e*x**2)*a*d*e*x**2 + 2*log(d + e*x**2)*a*e**2*x**4 - 4*log(x)*a*d*e*x 
**2 - 4*log(x)*a*e**2*x**4 - a*d**2 + 2*a*e**2*x**4)/(2*d**3*x**2*(d + e*x 
**2))