\(\int \frac {a+b \arcsin (c x)}{(d+e x^2)^2} \, dx\) [465]

Optimal result

Integrand size = 18, antiderivative size = 757 \[ \int \frac {a+b \arcsin (c x)}{\left (d+e x^2\right )^2} \, dx=-\frac {a+b \arcsin (c x)}{4 d \sqrt {e} \left (\sqrt {-d}-\sqrt {e} x\right )}+\frac {a+b \arcsin (c x)}{4 d \sqrt {e} \left (\sqrt {-d}+\sqrt {e} x\right )}+\frac {b c \text {arctanh}\left (\frac {\sqrt {e}-c^2 \sqrt {-d} x}{\sqrt {c^2 d+e} \sqrt {1-c^2 x^2}}\right )}{4 d \sqrt {e} \sqrt {c^2 d+e}}+\frac {b c \text {arctanh}\left (\frac {\sqrt {e}+c^2 \sqrt {-d} x}{\sqrt {c^2 d+e} \sqrt {1-c^2 x^2}}\right )}{4 d \sqrt {e} \sqrt {c^2 d+e}}-\frac {(a+b \arcsin (c x)) \log \left (1-\frac {\sqrt {e} e^{i \arcsin (c x)}}{i c \sqrt {-d}-\sqrt {c^2 d+e}}\right )}{4 (-d)^{3/2} \sqrt {e}}+\frac {(a+b \arcsin (c x)) \log \left (1+\frac {\sqrt {e} e^{i \arcsin (c x)}}{i c \sqrt {-d}-\sqrt {c^2 d+e}}\right )}{4 (-d)^{3/2} \sqrt {e}}-\frac {(a+b \arcsin (c x)) \log \left (1-\frac {\sqrt {e} e^{i \arcsin (c x)}}{i c \sqrt {-d}+\sqrt {c^2 d+e}}\right )}{4 (-d)^{3/2} \sqrt {e}}+\frac {(a+b \arcsin (c x)) \log \left (1+\frac {\sqrt {e} e^{i \arcsin (c x)}}{i c \sqrt {-d}+\sqrt {c^2 d+e}}\right )}{4 (-d)^{3/2} \sqrt {e}}-\frac {i b \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} e^{i \arcsin (c x)}}{i c \sqrt {-d}-\sqrt {c^2 d+e}}\right )}{4 (-d)^{3/2} \sqrt {e}}+\frac {i b \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{i \arcsin (c x)}}{i c \sqrt {-d}-\sqrt {c^2 d+e}}\right )}{4 (-d)^{3/2} \sqrt {e}}-\frac {i b \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} e^{i \arcsin (c x)}}{i c \sqrt {-d}+\sqrt {c^2 d+e}}\right )}{4 (-d)^{3/2} \sqrt {e}}+\frac {i b \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{i \arcsin (c x)}}{i c \sqrt {-d}+\sqrt {c^2 d+e}}\right )}{4 (-d)^{3/2} \sqrt {e}} \] Output:

-1/4*(a+b*arcsin(c*x))/d/e^(1/2)/((-d)^(1/2)-e^(1/2)*x)+1/4*(a+b*arcsin(c* 
x))/d/e^(1/2)/((-d)^(1/2)+e^(1/2)*x)+1/4*b*c*arctanh((e^(1/2)-c^2*(-d)^(1/ 
2)*x)/(c^2*d+e)^(1/2)/(-c^2*x^2+1)^(1/2))/d/e^(1/2)/(c^2*d+e)^(1/2)+1/4*b* 
c*arctanh((e^(1/2)+c^2*(-d)^(1/2)*x)/(c^2*d+e)^(1/2)/(-c^2*x^2+1)^(1/2))/d 
/e^(1/2)/(c^2*d+e)^(1/2)-1/4*(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^(1/2)+1/4*(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^(1/2)-1/4*(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^(1/2)+ 
1/4*(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^(1/2)-1/4*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)^(3/2)/e^(1/2)+ 
1/4*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)^(3/2)/e^(1/2)-1/4*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)^(3/2)/e^(1/2)+1/4*I*b*p 
olylog(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)^(3/2)/e^(1/2)
 

Mathematica [A] (verified)

Time = 1.11 (sec) , antiderivative size = 591, normalized size of antiderivative = 0.78 \[ \int \frac {a+b \arcsin (c x)}{\left (d+e x^2\right )^2} \, dx=\frac {1}{2} \left (\frac {a x}{d^2+d e x^2}+\frac {a \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{d^{3/2} \sqrt {e}}+\frac {b \left (i \sqrt {d} \left (\frac {\arcsin (c x)}{\sqrt {d}+i \sqrt {e} x}-\frac {c \arctan \left (\frac {i \sqrt {e}+c^2 \sqrt {d} x}{\sqrt {c^2 d+e} \sqrt {1-c^2 x^2}}\right )}{\sqrt {c^2 d+e}}\right )+\sqrt {d} \left (\frac {\arcsin (c x)}{i \sqrt {d}+\sqrt {e} x}+\frac {c \text {arctanh}\left (\frac {\sqrt {e}+i c^2 \sqrt {d} x}{\sqrt {c^2 d+e} \sqrt {1-c^2 x^2}}\right )}{\sqrt {c^2 d+e}}\right )+i \arcsin (c x) \left (\log \left (1+\frac {\sqrt {e} e^{i \arcsin (c x)}}{-c \sqrt {d}+\sqrt {c^2 d+e}}\right )+\log \left (1-\frac {\sqrt {e} e^{i \arcsin (c x)}}{c \sqrt {d}+\sqrt {c^2 d+e}}\right )\right )-i \arcsin (c x) \left (\log \left (1+\frac {\sqrt {e} e^{i \arcsin (c x)}}{c \sqrt {d}-\sqrt {c^2 d+e}}\right )+\log \left (1+\frac {\sqrt {e} e^{i \arcsin (c x)}}{c \sqrt {d}+\sqrt {c^2 d+e}}\right )\right )+\operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{i \arcsin (c x)}}{c \sqrt {d}-\sqrt {c^2 d+e}}\right )-\operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{i \arcsin (c x)}}{-c \sqrt {d}+\sqrt {c^2 d+e}}\right )-\operatorname {PolyLog}\left (2,-\frac {\sqrt {e} e^{i \arcsin (c x)}}{c \sqrt {d}+\sqrt {c^2 d+e}}\right )+\operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{i \arcsin (c x)}}{c \sqrt {d}+\sqrt {c^2 d+e}}\right )\right )}{2 d^{3/2} \sqrt {e}}\right ) \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 1.62 (sec) , antiderivative size = 757, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {5172, 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.

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

\(\Big \downarrow \) 5172

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

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {(a+b \arcsin (c x)) \log \left (1-\frac {\sqrt {e} e^{i \arcsin (c x)}}{-\sqrt {c^2 d+e}+i c \sqrt {-d}}\right )}{4 (-d)^{3/2} \sqrt {e}}+\frac {(a+b \arcsin (c x)) \log \left (1+\frac {\sqrt {e} e^{i \arcsin (c x)}}{-\sqrt {c^2 d+e}+i c \sqrt {-d}}\right )}{4 (-d)^{3/2} \sqrt {e}}-\frac {(a+b \arcsin (c x)) \log \left (1-\frac {\sqrt {e} e^{i \arcsin (c x)}}{\sqrt {c^2 d+e}+i c \sqrt {-d}}\right )}{4 (-d)^{3/2} \sqrt {e}}+\frac {(a+b \arcsin (c x)) \log \left (1+\frac {\sqrt {e} e^{i \arcsin (c x)}}{\sqrt {c^2 d+e}+i c \sqrt {-d}}\right )}{4 (-d)^{3/2} \sqrt {e}}-\frac {a+b \arcsin (c x)}{4 d \sqrt {e} \left (\sqrt {-d}-\sqrt {e} x\right )}+\frac {a+b \arcsin (c x)}{4 d \sqrt {e} \left (\sqrt {-d}+\sqrt {e} x\right )}-\frac {i b \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} e^{i \arcsin (c x)}}{i c \sqrt {-d}-\sqrt {d c^2+e}}\right )}{4 (-d)^{3/2} \sqrt {e}}+\frac {i b \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{i \arcsin (c x)}}{i c \sqrt {-d}-\sqrt {d c^2+e}}\right )}{4 (-d)^{3/2} \sqrt {e}}-\frac {i b \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} e^{i \arcsin (c x)}}{i \sqrt {-d} c+\sqrt {d c^2+e}}\right )}{4 (-d)^{3/2} \sqrt {e}}+\frac {i b \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{i \arcsin (c x)}}{i \sqrt {-d} c+\sqrt {d c^2+e}}\right )}{4 (-d)^{3/2} \sqrt {e}}+\frac {b c \text {arctanh}\left (\frac {\sqrt {e}-c^2 \sqrt {-d} x}{\sqrt {1-c^2 x^2} \sqrt {c^2 d+e}}\right )}{4 d \sqrt {e} \sqrt {c^2 d+e}}+\frac {b c \text {arctanh}\left (\frac {c^2 \sqrt {-d} x+\sqrt {e}}{\sqrt {1-c^2 x^2} \sqrt {c^2 d+e}}\right )}{4 d \sqrt {e} \sqrt {c^2 d+e}}\)

Input:

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

Output:

-1/4*(a + b*ArcSin[c*x])/(d*Sqrt[e]*(Sqrt[-d] - Sqrt[e]*x)) + (a + b*ArcSi 
n[c*x])/(4*d*Sqrt[e]*(Sqrt[-d] + Sqrt[e]*x)) + (b*c*ArcTanh[(Sqrt[e] - c^2 
*Sqrt[-d]*x)/(Sqrt[c^2*d + e]*Sqrt[1 - c^2*x^2])])/(4*d*Sqrt[e]*Sqrt[c^2*d 
 + e]) + (b*c*ArcTanh[(Sqrt[e] + c^2*Sqrt[-d]*x)/(Sqrt[c^2*d + e]*Sqrt[1 - 
 c^2*x^2])])/(4*d*Sqrt[e]*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])])/(4*(-d)^(3/ 
2)*Sqrt[e]) + ((a + b*ArcSin[c*x])*Log[1 + (Sqrt[e]*E^(I*ArcSin[c*x]))/(I* 
c*Sqrt[-d] - Sqrt[c^2*d + e])])/(4*(-d)^(3/2)*Sqrt[e]) - ((a + b*ArcSin[c* 
x])*Log[1 - (Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] + Sqrt[c^2*d + e])]) 
/(4*(-d)^(3/2)*Sqrt[e]) + ((a + b*ArcSin[c*x])*Log[1 + (Sqrt[e]*E^(I*ArcSi 
n[c*x]))/(I*c*Sqrt[-d] + Sqrt[c^2*d + e])])/(4*(-d)^(3/2)*Sqrt[e]) - ((I/4 
)*b*PolyLog[2, -((Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] - Sqrt[c^2*d + 
e]))])/((-d)^(3/2)*Sqrt[e]) + ((I/4)*b*PolyLog[2, (Sqrt[e]*E^(I*ArcSin[c*x 
]))/(I*c*Sqrt[-d] - Sqrt[c^2*d + e])])/((-d)^(3/2)*Sqrt[e]) - ((I/4)*b*Pol 
yLog[2, -((Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] + Sqrt[c^2*d + e]))])/ 
((-d)^(3/2)*Sqrt[e]) + ((I/4)*b*PolyLog[2, (Sqrt[e]*E^(I*ArcSin[c*x]))/(I* 
c*Sqrt[-d] + Sqrt[c^2*d + e])])/((-d)^(3/2)*Sqrt[e])
 

Defintions of rubi rules used

rule 2009

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

rule 5172

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((d_) + (e_.)*(x_)^2)^(p_.), x 
_Symbol] :> Int[ExpandIntegrand[(a + b*ArcSin[c*x])^n, (d + e*x^2)^p, x], x 
] /; FreeQ[{a, b, c, d, e, n}, x] && NeQ[c^2*d + e, 0] && IntegerQ[p] && (G 
tQ[p, 0] || IGtQ[n, 0])
 

Maple [C] (warning: unable to verify)

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

Time = 3.81 (sec) , antiderivative size = 843, normalized size of antiderivative = 1.11

\[\frac {\frac {a \,c^{3} x}{2 d \left (c^{2} e \,x^{2}+c^{2} d \right )}+\frac {a c \arctan \left (\frac {e x}{\sqrt {d e}}\right )}{2 d \sqrt {d e}}+b \,c^{4} \left (\frac {\arcsin \left (c x \right ) x}{2 c d \left (c^{2} e \,x^{2}+c^{2} d \right )}-\frac {\sqrt {\left (2 c^{2} d +2 \sqrt {c^{2} d \left (c^{2} d +e \right )}+e \right ) e}\, \left (-2 \sqrt {c^{2} d \left (c^{2} d +e \right )}\, c^{2} d +2 c^{4} d^{2}+2 c^{2} d e -\sqrt {c^{2} d \left (c^{2} d +e \right )}\, e \right ) \operatorname {arctanh}\left (\frac {e \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )}{\sqrt {\left (2 c^{2} d +2 \sqrt {c^{2} d \left (c^{2} d +e \right )}+e \right ) e}}\right )}{2 c^{2} d \left (c^{2} d +e \right ) e^{3}}+\frac {\sqrt {\left (2 c^{2} d +2 \sqrt {c^{2} d \left (c^{2} d +e \right )}+e \right ) e}\, \left (2 c^{2} d -2 \sqrt {c^{2} d \left (c^{2} d +e \right )}+e \right ) \operatorname {arctanh}\left (\frac {e \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )}{\sqrt {\left (2 c^{2} d +2 \sqrt {c^{2} d \left (c^{2} d +e \right )}+e \right ) e}}\right )}{2 c^{2} d \,e^{3}}-\frac {\sqrt {-e \left (2 c^{2} d -2 \sqrt {c^{2} d \left (c^{2} d +e \right )}+e \right )}\, \left (2 c^{4} d^{2}+2 \sqrt {c^{2} d \left (c^{2} d +e \right )}\, c^{2} d +2 c^{2} d e +\sqrt {c^{2} d \left (c^{2} d +e \right )}\, e \right ) \arctan \left (\frac {e \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )}{\sqrt {\left (-2 c^{2} d +2 \sqrt {c^{2} d \left (c^{2} d +e \right )}-e \right ) e}}\right )}{2 c^{2} d \left (c^{2} d +e \right ) e^{3}}+\frac {\sqrt {-e \left (2 c^{2} d -2 \sqrt {c^{2} d \left (c^{2} d +e \right )}+e \right )}\, \left (2 c^{2} d +2 \sqrt {c^{2} d \left (c^{2} d +e \right )}+e \right ) \arctan \left (\frac {e \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )}{\sqrt {\left (-2 c^{2} d +2 \sqrt {c^{2} d \left (c^{2} d +e \right )}-e \right ) e}}\right )}{2 c^{2} d \,e^{3}}+\frac {\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 {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 )}{\textit {\_R1} \left (\textit {\_R1}^{2} e -2 c^{2} d -e \right )}}{4 c^{2} d}+\frac {\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 {\textit {\_R1} \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}}{4 c^{2} d}\right )}{c}\]

Input:

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

Output:

1/c*(1/2*a*c^3*x/d/(c^2*e*x^2+c^2*d)+1/2*a*c/d/(d*e)^(1/2)*arctan(e*x/(d*e 
)^(1/2))+b*c^4*(1/2*arcsin(c*x)/c*x/d/(c^2*e*x^2+c^2*d)-1/2*((2*c^2*d+2*(c 
^2*d*(c^2*d+e))^(1/2)+e)*e)^(1/2)*(-2*(c^2*d*(c^2*d+e))^(1/2)*c^2*d+2*c^4* 
d^2+2*c^2*d*e-(c^2*d*(c^2*d+e))^(1/2)*e)*arctanh(e*(I*c*x+(-c^2*x^2+1)^(1/ 
2))/((2*c^2*d+2*(c^2*d*(c^2*d+e))^(1/2)+e)*e)^(1/2))/c^2/d/(c^2*d+e)/e^3+1 
/2*((2*c^2*d+2*(c^2*d*(c^2*d+e))^(1/2)+e)*e)^(1/2)*(2*c^2*d-2*(c^2*d*(c^2* 
d+e))^(1/2)+e)*arctanh(e*(I*c*x+(-c^2*x^2+1)^(1/2))/((2*c^2*d+2*(c^2*d*(c^ 
2*d+e))^(1/2)+e)*e)^(1/2))/c^2/d/e^3-1/2*(-e*(2*c^2*d-2*(c^2*d*(c^2*d+e))^ 
(1/2)+e))^(1/2)*(2*c^4*d^2+2*(c^2*d*(c^2*d+e))^(1/2)*c^2*d+2*c^2*d*e+(c^2* 
d*(c^2*d+e))^(1/2)*e)*arctan(e*(I*c*x+(-c^2*x^2+1)^(1/2))/((-2*c^2*d+2*(c^ 
2*d*(c^2*d+e))^(1/2)-e)*e)^(1/2))/c^2/d/(c^2*d+e)/e^3+1/2*(-e*(2*c^2*d-2*( 
c^2*d*(c^2*d+e))^(1/2)+e))^(1/2)*(2*c^2*d+2*(c^2*d*(c^2*d+e))^(1/2)+e)*arc 
tan(e*(I*c*x+(-c^2*x^2+1)^(1/2))/((-2*c^2*d+2*(c^2*d*(c^2*d+e))^(1/2)-e)*e 
)^(1/2))/c^2/d/e^3+1/4/c^2/d*sum(1/_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))+1/4/c^2/d*sum(_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))))
 

Fricas [F]

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

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

Output:

integral((b*arcsin(c*x) + a)/(e^2*x^4 + 2*d*e*x^2 + d^2), x)
 

Sympy [F]

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

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

Output:

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

Maxima [F(-2)]

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

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

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

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

Output:

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

Reduce [F]

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

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

Output:

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