\(\int \frac {a+b \arcsin (c x)}{x (d+e x^2)^2} \, dx\) [461]

Optimal result

Integrand size = 21, antiderivative size = 597 \[ \int \frac {a+b \arcsin (c x)}{x \left (d+e x^2\right )^2} \, dx=\frac {a+b \arcsin (c x)}{2 d \left (d+e x^2\right )}-\frac {b c \arctan \left (\frac {\sqrt {c^2 d+e} x}{\sqrt {d} \sqrt {1-c^2 x^2}}\right )}{2 d^{3/2} \sqrt {c^2 d+e}}-\frac {(a+b \arcsin (c x)) \log \left (1-\frac {\sqrt {e} e^{i \arcsin (c x)}}{i c \sqrt {-d}-\sqrt {c^2 d+e}}\right )}{2 d^2}-\frac {(a+b \arcsin (c x)) \log \left (1+\frac {\sqrt {e} e^{i \arcsin (c x)}}{i c \sqrt {-d}-\sqrt {c^2 d+e}}\right )}{2 d^2}-\frac {(a+b \arcsin (c x)) \log \left (1-\frac {\sqrt {e} e^{i \arcsin (c x)}}{i c \sqrt {-d}+\sqrt {c^2 d+e}}\right )}{2 d^2}-\frac {(a+b \arcsin (c x)) \log \left (1+\frac {\sqrt {e} e^{i \arcsin (c x)}}{i c \sqrt {-d}+\sqrt {c^2 d+e}}\right )}{2 d^2}+\frac {(a+b \arcsin (c x)) \log \left (1-e^{2 i \arcsin (c x)}\right )}{d^2}+\frac {i b \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} e^{i \arcsin (c x)}}{i c \sqrt {-d}-\sqrt {c^2 d+e}}\right )}{2 d^2}+\frac {i b \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{i \arcsin (c x)}}{i c \sqrt {-d}-\sqrt {c^2 d+e}}\right )}{2 d^2}+\frac {i b \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} e^{i \arcsin (c x)}}{i c \sqrt {-d}+\sqrt {c^2 d+e}}\right )}{2 d^2}+\frac {i b \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{i \arcsin (c x)}}{i c \sqrt {-d}+\sqrt {c^2 d+e}}\right )}{2 d^2}-\frac {i b \operatorname {PolyLog}\left (2,e^{2 i \arcsin (c x)}\right )}{2 d^2} \] Output:

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

Mathematica [A] (verified)

Time = 0.78 (sec) , antiderivative size = 640, normalized size of antiderivative = 1.07 \[ \int \frac {a+b \arcsin (c x)}{x \left (d+e x^2\right )^2} \, dx=\frac {a}{2 d^2+2 d e x^2}+\frac {a \log (x)}{d^2}-\frac {a \log \left (d+e x^2\right )}{2 d^2}+\frac {b \left (\frac {\sqrt {d} \arcsin (c x)}{\sqrt {d}-i \sqrt {e} x}+\frac {\sqrt {d} \arcsin (c x)}{\sqrt {d}+i \sqrt {e} x}-\frac {c \sqrt {d} \arctan \left (\frac {i \sqrt {e}+c^2 \sqrt {d} x}{\sqrt {c^2 d+e} \sqrt {1-c^2 x^2}}\right )}{\sqrt {c^2 d+e}}+\frac {i c \sqrt {d} \text {arctanh}\left (\frac {\sqrt {e}+i c^2 \sqrt {d} x}{\sqrt {c^2 d+e} \sqrt {1-c^2 x^2}}\right )}{\sqrt {c^2 d+e}}-2 \arcsin (c x) \log \left (1+\frac {\sqrt {e} e^{i \arcsin (c x)}}{c \sqrt {d}-\sqrt {c^2 d+e}}\right )-2 \arcsin (c x) \log \left (1+\frac {\sqrt {e} e^{i \arcsin (c x)}}{-c \sqrt {d}+\sqrt {c^2 d+e}}\right )-2 \arcsin (c x) \log \left (1-\frac {\sqrt {e} e^{i \arcsin (c x)}}{c \sqrt {d}+\sqrt {c^2 d+e}}\right )-2 \arcsin (c x) \log \left (1+\frac {\sqrt {e} e^{i \arcsin (c x)}}{c \sqrt {d}+\sqrt {c^2 d+e}}\right )+4 \arcsin (c x) \log \left (1-e^{2 i \arcsin (c x)}\right )+2 i \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{i \arcsin (c x)}}{c \sqrt {d}-\sqrt {c^2 d+e}}\right )+2 i \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{i \arcsin (c x)}}{-c \sqrt {d}+\sqrt {c^2 d+e}}\right )+2 i \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} e^{i \arcsin (c x)}}{c \sqrt {d}+\sqrt {c^2 d+e}}\right )+2 i \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{i \arcsin (c x)}}{c \sqrt {d}+\sqrt {c^2 d+e}}\right )-2 i \operatorname {PolyLog}\left (2,e^{2 i \arcsin (c x)}\right )\right )}{4 d^2} \] Input:

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

Output:

a/(2*d^2 + 2*d*e*x^2) + (a*Log[x])/d^2 - (a*Log[d + e*x^2])/(2*d^2) + (b*( 
(Sqrt[d]*ArcSin[c*x])/(Sqrt[d] - I*Sqrt[e]*x) + (Sqrt[d]*ArcSin[c*x])/(Sqr 
t[d] + I*Sqrt[e]*x) - (c*Sqrt[d]*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*c*Sqrt[d]*ArcTanh[(Sq 
rt[e] + I*c^2*Sqrt[d]*x)/(Sqrt[c^2*d + e]*Sqrt[1 - c^2*x^2])])/Sqrt[c^2*d 
+ e] - 2*ArcSin[c*x]*Log[1 + (Sqrt[e]*E^(I*ArcSin[c*x]))/(c*Sqrt[d] - Sqrt 
[c^2*d + e])] - 2*ArcSin[c*x]*Log[1 + (Sqrt[e]*E^(I*ArcSin[c*x]))/(-(c*Sqr 
t[d]) + Sqrt[c^2*d + e])] - 2*ArcSin[c*x]*Log[1 - (Sqrt[e]*E^(I*ArcSin[c*x 
]))/(c*Sqrt[d] + Sqrt[c^2*d + e])] - 2*ArcSin[c*x]*Log[1 + (Sqrt[e]*E^(I*A 
rcSin[c*x]))/(c*Sqrt[d] + Sqrt[c^2*d + e])] + 4*ArcSin[c*x]*Log[1 - E^((2* 
I)*ArcSin[c*x])] + (2*I)*PolyLog[2, (Sqrt[e]*E^(I*ArcSin[c*x]))/(c*Sqrt[d] 
 - Sqrt[c^2*d + e])] + (2*I)*PolyLog[2, (Sqrt[e]*E^(I*ArcSin[c*x]))/(-(c*S 
qrt[d]) + Sqrt[c^2*d + e])] + (2*I)*PolyLog[2, -((Sqrt[e]*E^(I*ArcSin[c*x] 
))/(c*Sqrt[d] + Sqrt[c^2*d + e]))] + (2*I)*PolyLog[2, (Sqrt[e]*E^(I*ArcSin 
[c*x]))/(c*Sqrt[d] + Sqrt[c^2*d + e])] - (2*I)*PolyLog[2, E^((2*I)*ArcSin[ 
c*x])]))/(4*d^2)
 

Rubi [A] (verified)

Time = 1.52 (sec) , antiderivative size = 597, 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.

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

\(\Big \downarrow \) 5232

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

(a + b*ArcSin[c*x])/(2*d*(d + e*x^2)) - (b*c*ArcTan[(Sqrt[c^2*d + e]*x)/(S 
qrt[d]*Sqrt[1 - c^2*x^2])])/(2*d^(3/2)*Sqrt[c^2*d + e]) - ((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^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^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^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^2) + ((a + b*ArcSin[c*x])*Log[1 - E^((2*I)*ArcSin[c*x])]) 
/d^2 + ((I/2)*b*PolyLog[2, -((Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] - S 
qrt[c^2*d + e]))])/d^2 + ((I/2)*b*PolyLog[2, (Sqrt[e]*E^(I*ArcSin[c*x]))/( 
I*c*Sqrt[-d] - Sqrt[c^2*d + e])])/d^2 + ((I/2)*b*PolyLog[2, -((Sqrt[e]*E^( 
I*ArcSin[c*x]))/(I*c*Sqrt[-d] + Sqrt[c^2*d + e]))])/d^2 + ((I/2)*b*PolyLog 
[2, (Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] + Sqrt[c^2*d + e])])/d^2 - ( 
(I/2)*b*PolyLog[2, E^((2*I)*ArcSin[c*x])])/d^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 = 11.19 (sec) , antiderivative size = 468, normalized size of antiderivative = 0.78

method	result	size
parts	\(\frac {a \ln \left (x \right )}{d^{2}}-\frac {a \ln \left (e \,x^{2}+d \right )}{2 d^{2}}+\frac {a}{2 d \left (e \,x^{2}+d \right )}+b \left (\frac {c^{2} \arcsin \left (c x \right )}{2 d \left (c^{2} e \,x^{2}+c^{2} d \right )}+\frac {i \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (e \,\textit {\_Z}^{4}+\left (-4 c^{2} d -2 e \right ) \textit {\_Z}^{2}+e \right )}{\sum }\frac {\left (\textit {\_R1}^{2} e -4 c^{2} d -e \right ) \left (i \arcsin \left (c x \right ) \ln \left (\frac {\textit {\_R1} -i c x -\sqrt {-c^{2} x^{2}+1}}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {\textit {\_R1} -i c x -\sqrt {-c^{2} x^{2}+1}}{\textit {\_R1}}\right )\right )}{\textit {\_R1}^{2} e -2 c^{2} d -e}\right )}{4 d^{2}}+\frac {i \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (e \,\textit {\_Z}^{4}+\left (-4 c^{2} d -2 e \right ) \textit {\_Z}^{2}+e \right )}{\sum }\frac {\left (\textit {\_R1}^{2}-1\right ) \left (i \arcsin \left (c x \right ) \ln \left (\frac {\textit {\_R1} -i c x -\sqrt {-c^{2} x^{2}+1}}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {\textit {\_R1} -i c x -\sqrt {-c^{2} x^{2}+1}}{\textit {\_R1}}\right )\right )}{\textit {\_R1}^{2} e -2 c^{2} d -e}\right ) e}{4 d^{2}}+\frac {i \sqrt {c^{2} d \left (c^{2} d +e \right )}\, \operatorname {arctanh}\left (\frac {2 e \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{2}-4 c^{2} d -2 e}{4 \sqrt {c^{4} d^{2}+c^{2} d e}}\right )}{2 d^{2} \left (c^{2} d +e \right )}+\frac {i \operatorname {dilog}\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )}{d^{2}}-\frac {i \operatorname {dilog}\left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )}{d^{2}}+\frac {\arcsin \left (c x \right ) \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )}{d^{2}}\right )\)	\(468\)
derivativedivides	\(\frac {a \,c^{2}}{2 d \left (c^{2} e \,x^{2}+c^{2} d \right )}-\frac {a \ln \left (c^{2} e \,x^{2}+c^{2} d \right )}{2 d^{2}}+\frac {a \ln \left (c x \right )}{d^{2}}+\frac {b \,c^{2} \arcsin \left (c x \right )}{2 d \left (c^{2} e \,x^{2}+c^{2} d \right )}-\frac {i b \operatorname {dilog}\left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )}{d^{2}}+\frac {i b \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (e \,\textit {\_Z}^{4}+\left (-4 c^{2} d -2 e \right ) \textit {\_Z}^{2}+e \right )}{\sum }\frac {\left (-\textit {\_R1}^{2} e +4 c^{2} d +e \right ) \left (i \arcsin \left (c x \right ) \ln \left (\frac {\textit {\_R1} -i c x -\sqrt {-c^{2} x^{2}+1}}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {\textit {\_R1} -i c x -\sqrt {-c^{2} x^{2}+1}}{\textit {\_R1}}\right )\right )}{-\textit {\_R1}^{2} e +2 c^{2} d +e}\right )}{4 d^{2}}-\frac {i b \sqrt {c^{2} d \left (c^{2} d +e \right )}\, \operatorname {arctanh}\left (\frac {4 c^{2} d -2 e \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{2}+2 e}{4 \sqrt {c^{4} d^{2}+c^{2} d e}}\right )}{2 d^{2} \left (c^{2} d +e \right )}+\frac {i b \operatorname {dilog}\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )}{d^{2}}-\frac {i b \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (e \,\textit {\_Z}^{4}+\left (-4 c^{2} d -2 e \right ) \textit {\_Z}^{2}+e \right )}{\sum }\frac {\left (\textit {\_R1}^{2}-1\right ) \left (i \arcsin \left (c x \right ) \ln \left (\frac {\textit {\_R1} -i c x -\sqrt {-c^{2} x^{2}+1}}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {\textit {\_R1} -i c x -\sqrt {-c^{2} x^{2}+1}}{\textit {\_R1}}\right )\right )}{-\textit {\_R1}^{2} e +2 c^{2} d +e}\right ) e}{4 d^{2}}+\frac {b \arcsin \left (c x \right ) \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )}{d^{2}}\)	\(488\)
default	\(\frac {a \,c^{2}}{2 d \left (c^{2} e \,x^{2}+c^{2} d \right )}-\frac {a \ln \left (c^{2} e \,x^{2}+c^{2} d \right )}{2 d^{2}}+\frac {a \ln \left (c x \right )}{d^{2}}+\frac {b \,c^{2} \arcsin \left (c x \right )}{2 d \left (c^{2} e \,x^{2}+c^{2} d \right )}-\frac {i b \operatorname {dilog}\left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )}{d^{2}}+\frac {i b \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (e \,\textit {\_Z}^{4}+\left (-4 c^{2} d -2 e \right ) \textit {\_Z}^{2}+e \right )}{\sum }\frac {\left (-\textit {\_R1}^{2} e +4 c^{2} d +e \right ) \left (i \arcsin \left (c x \right ) \ln \left (\frac {\textit {\_R1} -i c x -\sqrt {-c^{2} x^{2}+1}}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {\textit {\_R1} -i c x -\sqrt {-c^{2} x^{2}+1}}{\textit {\_R1}}\right )\right )}{-\textit {\_R1}^{2} e +2 c^{2} d +e}\right )}{4 d^{2}}-\frac {i b \sqrt {c^{2} d \left (c^{2} d +e \right )}\, \operatorname {arctanh}\left (\frac {4 c^{2} d -2 e \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{2}+2 e}{4 \sqrt {c^{4} d^{2}+c^{2} d e}}\right )}{2 d^{2} \left (c^{2} d +e \right )}+\frac {i b \operatorname {dilog}\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )}{d^{2}}-\frac {i b \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (e \,\textit {\_Z}^{4}+\left (-4 c^{2} d -2 e \right ) \textit {\_Z}^{2}+e \right )}{\sum }\frac {\left (\textit {\_R1}^{2}-1\right ) \left (i \arcsin \left (c x \right ) \ln \left (\frac {\textit {\_R1} -i c x -\sqrt {-c^{2} x^{2}+1}}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {\textit {\_R1} -i c x -\sqrt {-c^{2} x^{2}+1}}{\textit {\_R1}}\right )\right )}{-\textit {\_R1}^{2} e +2 c^{2} d +e}\right ) e}{4 d^{2}}+\frac {b \arcsin \left (c x \right ) \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )}{d^{2}}\)	\(488\)

Input:

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

Output:

a/d^2*ln(x)-1/2*a/d^2*ln(e*x^2+d)+1/2*a/d/(e*x^2+d)+b*(1/2*c^2*arcsin(c*x) 
/d/(c^2*e*x^2+c^2*d)+1/4*I/d^2*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))+1/4*I/d^ 
2*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=RootOf(e*_Z 
^4+(-4*c^2*d-2*e)*_Z^2+e))*e+1/2*I*(c^2*d*(c^2*d+e))^(1/2)/d^2/(c^2*d+e)*a 
rctanh(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))+I/d^2*dilog(I*c*x+(-c^2*x^2+1)^(1/2))-I/d^2*dilog(1+I*c*x+(-c^2*x 
^2+1)^(1/2))+1/d^2*arcsin(c*x)*ln(1+I*c*x+(-c^2*x^2+1)^(1/2)))
 

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F(-1)]

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

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

Output:

Timed out
                                                                                    
                                                                                    
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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