\(\int \frac {a+b \arcsin (c x)}{x^2 (d+e x^2)^2} \, dx\) [466]

Optimal result

Integrand size = 21, antiderivative size = 795 \[ \int \frac {a+b \arcsin (c x)}{x^2 \left (d+e x^2\right )^2} \, dx=-\frac {a+b \arcsin (c x)}{d^2 x}+\frac {\sqrt {e} (a+b \arcsin (c x))}{4 d^2 \left (\sqrt {-d}-\sqrt {e} x\right )}-\frac {\sqrt {e} (a+b \arcsin (c x))}{4 d^2 \left (\sqrt {-d}+\sqrt {e} x\right )}-\frac {b c \sqrt {e} \text {arctanh}\left (\frac {\sqrt {e}-c^2 \sqrt {-d} x}{\sqrt {c^2 d+e} \sqrt {1-c^2 x^2}}\right )}{4 d^2 \sqrt {c^2 d+e}}-\frac {b c \sqrt {e} \text {arctanh}\left (\frac {\sqrt {e}+c^2 \sqrt {-d} x}{\sqrt {c^2 d+e} \sqrt {1-c^2 x^2}}\right )}{4 d^2 \sqrt {c^2 d+e}}-\frac {b c \text {arctanh}\left (\sqrt {1-c^2 x^2}\right )}{d^2}-\frac {3 \sqrt {e} (a+b \arcsin (c x)) \log \left (1-\frac {\sqrt {e} e^{i \arcsin (c x)}}{i c \sqrt {-d}-\sqrt {c^2 d+e}}\right )}{4 (-d)^{5/2}}+\frac {3 \sqrt {e} (a+b \arcsin (c x)) \log \left (1+\frac {\sqrt {e} e^{i \arcsin (c x)}}{i c \sqrt {-d}-\sqrt {c^2 d+e}}\right )}{4 (-d)^{5/2}}-\frac {3 \sqrt {e} (a+b \arcsin (c x)) \log \left (1-\frac {\sqrt {e} e^{i \arcsin (c x)}}{i c \sqrt {-d}+\sqrt {c^2 d+e}}\right )}{4 (-d)^{5/2}}+\frac {3 \sqrt {e} (a+b \arcsin (c x)) \log \left (1+\frac {\sqrt {e} e^{i \arcsin (c x)}}{i c \sqrt {-d}+\sqrt {c^2 d+e}}\right )}{4 (-d)^{5/2}}-\frac {3 i b \sqrt {e} \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} e^{i \arcsin (c x)}}{i c \sqrt {-d}-\sqrt {c^2 d+e}}\right )}{4 (-d)^{5/2}}+\frac {3 i b \sqrt {e} \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{i \arcsin (c x)}}{i c \sqrt {-d}-\sqrt {c^2 d+e}}\right )}{4 (-d)^{5/2}}-\frac {3 i b \sqrt {e} \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} e^{i \arcsin (c x)}}{i c \sqrt {-d}+\sqrt {c^2 d+e}}\right )}{4 (-d)^{5/2}}+\frac {3 i b \sqrt {e} \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{i \arcsin (c x)}}{i c \sqrt {-d}+\sqrt {c^2 d+e}}\right )}{4 (-d)^{5/2}} \] Output:

-(a+b*arcsin(c*x))/d^2/x+1/4*e^(1/2)*(a+b*arcsin(c*x))/d^2/((-d)^(1/2)-e^( 
1/2)*x)-1/4*e^(1/2)*(a+b*arcsin(c*x))/d^2/((-d)^(1/2)+e^(1/2)*x)-1/4*b*c*e 
^(1/2)*arctanh((e^(1/2)-c^2*(-d)^(1/2)*x)/(c^2*d+e)^(1/2)/(-c^2*x^2+1)^(1/ 
2))/d^2/(c^2*d+e)^(1/2)-1/4*b*c*e^(1/2)*arctanh((e^(1/2)+c^2*(-d)^(1/2)*x) 
/(c^2*d+e)^(1/2)/(-c^2*x^2+1)^(1/2))/d^2/(c^2*d+e)^(1/2)-b*c*arctanh((-c^2 
*x^2+1)^(1/2))/d^2-3/4*e^(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)^(5/2)+3/4*e^(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)^(5/2)-3/4*e^(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)^(5/2)+3/4*e^ 
(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)^(5/2)-3/4*I*b*e^(1/2)*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)+3/4*I* 
b*e^(1/2)*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)-3/4*I*b*e^(1/2)*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)+3/4*I*b*e^(1/2) 
*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 = 1.02 (sec) , antiderivative size = 672, normalized size of antiderivative = 0.85 \[ \int \frac {a+b \arcsin (c x)}{x^2 \left (d+e x^2\right )^2} \, dx=\frac {-\frac {8 a \sqrt {d}}{x}-\frac {4 a \sqrt {d} e x}{d+e x^2}-12 a \sqrt {e} \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )+b \left (-2 i \sqrt {d} \sqrt {e} \left (\frac {\arcsin (c x)}{\sqrt {d}+i \sqrt {e} x}-\frac {c \arctan \left (\frac {i \sqrt {e}+c^2 \sqrt {d} x}{\sqrt {c^2 d+e} \sqrt {1-c^2 x^2}}\right )}{\sqrt {c^2 d+e}}\right )+2 \sqrt {d} \sqrt {e} \left (-\frac {\arcsin (c x)}{i \sqrt {d}+\sqrt {e} x}-\frac {c \text {arctanh}\left (\frac {\sqrt {e}+i c^2 \sqrt {d} x}{\sqrt {c^2 d+e} \sqrt {1-c^2 x^2}}\right )}{\sqrt {c^2 d+e}}\right )-\frac {8 \sqrt {d} \left (\arcsin (c x)+c x \text {arctanh}\left (\sqrt {1-c^2 x^2}\right )\right )}{x}+3 \sqrt {e} \left (\arcsin (c x) \left (\arcsin (c x)+2 i \left (\log \left (1+\frac {\sqrt {e} e^{i \arcsin (c x)}}{c \sqrt {d}-\sqrt {c^2 d+e}}\right )+\log \left (1+\frac {\sqrt {e} e^{i \arcsin (c x)}}{c \sqrt {d}+\sqrt {c^2 d+e}}\right )\right )\right )+2 \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{i \arcsin (c x)}}{-c \sqrt {d}+\sqrt {c^2 d+e}}\right )+2 \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} e^{i \arcsin (c x)}}{c \sqrt {d}+\sqrt {c^2 d+e}}\right )\right )-3 \sqrt {e} \left (\arcsin (c x) \left (\arcsin (c x)+2 i \left (\log \left (1+\frac {\sqrt {e} e^{i \arcsin (c x)}}{-c \sqrt {d}+\sqrt {c^2 d+e}}\right )+\log \left (1-\frac {\sqrt {e} e^{i \arcsin (c x)}}{c \sqrt {d}+\sqrt {c^2 d+e}}\right )\right )\right )+2 \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{i \arcsin (c x)}}{c \sqrt {d}-\sqrt {c^2 d+e}}\right )+2 \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{i \arcsin (c x)}}{c \sqrt {d}+\sqrt {c^2 d+e}}\right )\right )\right )}{8 d^{5/2}} \] Input:

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

Output:

((-8*a*Sqrt[d])/x - (4*a*Sqrt[d]*e*x)/(d + e*x^2) - 12*a*Sqrt[e]*ArcTan[(S 
qrt[e]*x)/Sqrt[d]] + b*((-2*I)*Sqrt[d]*Sqrt[e]*(ArcSin[c*x]/(Sqrt[d] + I*S 
qrt[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]) + 2*Sqrt[d]*Sqrt[e]*(-(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]) - (8*Sqrt[d]*(ArcSin[c*x] + c*x*Arc 
Tanh[Sqrt[1 - c^2*x^2]]))/x + 3*Sqrt[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*PolyL 
og[2, (Sqrt[e]*E^(I*ArcSin[c*x]))/(-(c*Sqrt[d]) + Sqrt[c^2*d + e])] + 2*Po 
lyLog[2, -((Sqrt[e]*E^(I*ArcSin[c*x]))/(c*Sqrt[d] + Sqrt[c^2*d + e]))]) - 
3*Sqrt[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])])))/(8*d^(5/2))
 

Rubi [A] (verified)

Time = 2.59 (sec) , antiderivative size = 795, 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.

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

\(\Big \downarrow \) 5232

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

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {3 \sqrt {e} \log \left (1-\frac {\sqrt {e} e^{i \arcsin (c x)}}{i c \sqrt {-d}-\sqrt {d c^2+e}}\right ) (a+b \arcsin (c x))}{4 (-d)^{5/2}}+\frac {3 \sqrt {e} \log \left (\frac {e^{i \arcsin (c x)} \sqrt {e}}{i c \sqrt {-d}-\sqrt {d c^2+e}}+1\right ) (a+b \arcsin (c x))}{4 (-d)^{5/2}}-\frac {3 \sqrt {e} \log \left (1-\frac {\sqrt {e} e^{i \arcsin (c x)}}{i \sqrt {-d} c+\sqrt {d c^2+e}}\right ) (a+b \arcsin (c x))}{4 (-d)^{5/2}}+\frac {3 \sqrt {e} \log \left (\frac {e^{i \arcsin (c x)} \sqrt {e}}{i \sqrt {-d} c+\sqrt {d c^2+e}}+1\right ) (a+b \arcsin (c x))}{4 (-d)^{5/2}}-\frac {a+b \arcsin (c x)}{d^2 x}+\frac {\sqrt {e} (a+b \arcsin (c x))}{4 d^2 \left (\sqrt {-d}-\sqrt {e} x\right )}-\frac {\sqrt {e} (a+b \arcsin (c x))}{4 d^2 \left (\sqrt {e} x+\sqrt {-d}\right )}-\frac {b c \sqrt {e} \text {arctanh}\left (\frac {\sqrt {e}-c^2 \sqrt {-d} x}{\sqrt {d c^2+e} \sqrt {1-c^2 x^2}}\right )}{4 d^2 \sqrt {d c^2+e}}-\frac {b c \sqrt {e} \text {arctanh}\left (\frac {\sqrt {-d} x c^2+\sqrt {e}}{\sqrt {d c^2+e} \sqrt {1-c^2 x^2}}\right )}{4 d^2 \sqrt {d c^2+e}}-\frac {b c \text {arctanh}\left (\sqrt {1-c^2 x^2}\right )}{d^2}-\frac {3 i b \sqrt {e} \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} e^{i \arcsin (c x)}}{i c \sqrt {-d}-\sqrt {d c^2+e}}\right )}{4 (-d)^{5/2}}+\frac {3 i b \sqrt {e} \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{i \arcsin (c x)}}{i c \sqrt {-d}-\sqrt {d c^2+e}}\right )}{4 (-d)^{5/2}}-\frac {3 i b \sqrt {e} \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} e^{i \arcsin (c x)}}{i \sqrt {-d} c+\sqrt {d c^2+e}}\right )}{4 (-d)^{5/2}}+\frac {3 i b \sqrt {e} \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{i \arcsin (c x)}}{i \sqrt {-d} c+\sqrt {d c^2+e}}\right )}{4 (-d)^{5/2}}\)

Input:

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

Output:

-((a + b*ArcSin[c*x])/(d^2*x)) + (Sqrt[e]*(a + b*ArcSin[c*x]))/(4*d^2*(Sqr 
t[-d] - Sqrt[e]*x)) - (Sqrt[e]*(a + b*ArcSin[c*x]))/(4*d^2*(Sqrt[-d] + Sqr 
t[e]*x)) - (b*c*Sqrt[e]*ArcTanh[(Sqrt[e] - c^2*Sqrt[-d]*x)/(Sqrt[c^2*d + e 
]*Sqrt[1 - c^2*x^2])])/(4*d^2*Sqrt[c^2*d + e]) - (b*c*Sqrt[e]*ArcTanh[(Sqr 
t[e] + c^2*Sqrt[-d]*x)/(Sqrt[c^2*d + e]*Sqrt[1 - c^2*x^2])])/(4*d^2*Sqrt[c 
^2*d + e]) - (b*c*ArcTanh[Sqrt[1 - c^2*x^2]])/d^2 - (3*Sqrt[e]*(a + b*ArcS 
in[c*x])*Log[1 - (Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] - Sqrt[c^2*d + 
e])])/(4*(-d)^(5/2)) + (3*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)^(5/2)) - (3*Sqr 
t[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)^(5/2)) + (3*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) 
^(5/2)) - (((3*I)/4)*b*Sqrt[e]*PolyLog[2, -((Sqrt[e]*E^(I*ArcSin[c*x]))/(I 
*c*Sqrt[-d] - Sqrt[c^2*d + e]))])/(-d)^(5/2) + (((3*I)/4)*b*Sqrt[e]*PolyLo 
g[2, (Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] - Sqrt[c^2*d + e])])/(-d)^( 
5/2) - (((3*I)/4)*b*Sqrt[e]*PolyLog[2, -((Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c* 
Sqrt[-d] + Sqrt[c^2*d + e]))])/(-d)^(5/2) + (((3*I)/4)*b*Sqrt[e]*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 = 6.72 (sec) , antiderivative size = 965, normalized size of antiderivative = 1.21

\[\text {Expression too large to display}\]

Input:

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

Output:

c*(-1/2*a/d^2*e*c*x/(c^2*e*x^2+c^2*d)-3/2*a/c/d^2*e/(d*e)^(1/2)*arctan(e*x 
/(d*e)^(1/2))-a/d^2/c/x+b*c^4*(-1/2/c^5/x*arcsin(c*x)*(3*c^2*e*x^2+2*c^2*d 
)/d^2/(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-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^4/d^2/e^2-1/2*(-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)*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^4/d^2/e^2-3/16/d^3/c^6*e*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))+3/16/d^3/c^6*e*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))-1/d 
^2/c^4*ln(1+I*c*x+(-c^2*x^2+1)^(1/2))+1/d^2/c^4*ln(I*c*x+(-c^2*x^2+1)^(1/2 
)-1)+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) 
)/d^2/c^4/(c^2*d+e)/e^2+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...
 

Fricas [F]

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

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

Output:

integral((b*arcsin(c*x) + a)/(e^2*x^6 + 2*d*e*x^4 + d^2*x^2), x)
 

Sympy [F]

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

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

Output:

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

Maxima [F(-2)]

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

integrate((a+b*arcsin(c*x))/x^2/(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(-1)]

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

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

Output:

Timed out
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

( - 3*sqrt(e)*sqrt(d)*atan((e*x)/(sqrt(e)*sqrt(d)))*a*d*x - 3*sqrt(e)*sqrt 
(d)*atan((e*x)/(sqrt(e)*sqrt(d)))*a*e*x**3 + 2*int(asin(c*x)/(d**2*x**2 + 
2*d*e*x**4 + e**2*x**6),x)*b*d**4*x + 2*int(asin(c*x)/(d**2*x**2 + 2*d*e*x 
**4 + e**2*x**6),x)*b*d**3*e*x**3 - 2*a*d**2 - 3*a*d*e*x**2)/(2*d**3*x*(d 
+ e*x**2))