\(\int \frac {a+b \arcsin (c x)}{x (d+e x^2)^3} \, dx\) [470]

Optimal result

Integrand size = 21, antiderivative size = 727 \[ \int \frac {a+b \arcsin (c x)}{x \left (d+e x^2\right )^3} \, dx=-\frac {b c e x \sqrt {1-c^2 x^2}}{8 d^2 \left (c^2 d+e\right ) \left (d+e x^2\right )}+\frac {a+b \arcsin (c x)}{4 d \left (d+e x^2\right )^2}+\frac {a+b \arcsin (c x)}{2 d^2 \left (d+e x^2\right )}-\frac {b c \arctan \left (\frac {\sqrt {c^2 d+e} x}{\sqrt {d} \sqrt {1-c^2 x^2}}\right )}{2 d^{5/2} \sqrt {c^2 d+e}}-\frac {b c \left (2 c^2 d+e\right ) \arctan \left (\frac {\sqrt {c^2 d+e} x}{\sqrt {d} \sqrt {1-c^2 x^2}}\right )}{8 d^{5/2} \left (c^2 d+e\right )^{3/2}}-\frac {(a+b \arcsin (c x)) \log \left (1-\frac {\sqrt {e} e^{i \arcsin (c x)}}{i c \sqrt {-d}-\sqrt {c^2 d+e}}\right )}{2 d^3}-\frac {(a+b \arcsin (c x)) \log \left (1+\frac {\sqrt {e} e^{i \arcsin (c x)}}{i c \sqrt {-d}-\sqrt {c^2 d+e}}\right )}{2 d^3}-\frac {(a+b \arcsin (c x)) \log \left (1-\frac {\sqrt {e} e^{i \arcsin (c x)}}{i c \sqrt {-d}+\sqrt {c^2 d+e}}\right )}{2 d^3}-\frac {(a+b \arcsin (c x)) \log \left (1+\frac {\sqrt {e} e^{i \arcsin (c x)}}{i c \sqrt {-d}+\sqrt {c^2 d+e}}\right )}{2 d^3}+\frac {(a+b \arcsin (c x)) \log \left (1-e^{2 i \arcsin (c x)}\right )}{d^3}+\frac {i b \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} e^{i \arcsin (c x)}}{i c \sqrt {-d}-\sqrt {c^2 d+e}}\right )}{2 d^3}+\frac {i b \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{i \arcsin (c x)}}{i c \sqrt {-d}-\sqrt {c^2 d+e}}\right )}{2 d^3}+\frac {i b \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} e^{i \arcsin (c x)}}{i c \sqrt {-d}+\sqrt {c^2 d+e}}\right )}{2 d^3}+\frac {i b \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{i \arcsin (c x)}}{i c \sqrt {-d}+\sqrt {c^2 d+e}}\right )}{2 d^3}-\frac {i b \operatorname {PolyLog}\left (2,e^{2 i \arcsin (c x)}\right )}{2 d^3} \] Output:

-1/8*b*c*e*x*(-c^2*x^2+1)^(1/2)/d^2/(c^2*d+e)/(e*x^2+d)+1/4*(a+b*arcsin(c* 
x))/d/(e*x^2+d)^2+1/2*(a+b*arcsin(c*x))/d^2/(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^(5/2)/(c^2*d+e)^(1/2)-1/8*b*c*( 
2*c^2*d+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)^(3/2)-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^3-1/2*(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^3-1/2*(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^3-1/2*(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^3+(a+b*arcsin(c*x))*ln(1-(I*c*x+ 
(-c^2*x^2+1)^(1/2))^2)/d^3+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^3+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^3+1/2*I*b*poly 
log(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+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^3-1/2*I*b*polylog(2,(I*c*x+(-c^2*x^2+1)^(1/2))^2)/d^3
 

Mathematica [A] (warning: unable to verify)

Time = 3.70 (sec) , antiderivative size = 1022, normalized size of antiderivative = 1.41 \[ \int \frac {a+b \arcsin (c x)}{x \left (d+e x^2\right )^3} \, dx =\text {Too large to display} \] Input:

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

Output:

(-((b*c*d*Sqrt[e]*Sqrt[1 - c^2*x^2])/((c^2*d + e)*((-I)*Sqrt[d] + Sqrt[e]* 
x))) - (b*c*d*Sqrt[e]*Sqrt[1 - c^2*x^2])/((c^2*d + e)*(I*Sqrt[d] + Sqrt[e] 
*x)) + (4*a*d^2)/(d + e*x^2)^2 + (8*a*d)/(d + e*x^2) + (b*d*ArcSin[c*x])/( 
Sqrt[d] - I*Sqrt[e]*x)^2 + (5*b*Sqrt[d]*ArcSin[c*x])/(Sqrt[d] - I*Sqrt[e]* 
x) + (b*d*ArcSin[c*x])/(Sqrt[d] + I*Sqrt[e]*x)^2 + (5*b*Sqrt[d]*ArcSin[c*x 
])/(Sqrt[d] + I*Sqrt[e]*x) - (5*b*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] + ((5*I)*b*c*Sq 
rt[d]*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*b*ArcSin[c*x]*Log[1 + (Sqrt[e]*E^(I*ArcSin[c*x]) 
)/(c*Sqrt[d] - Sqrt[c^2*d + e])] - 8*b*ArcSin[c*x]*Log[1 + (Sqrt[e]*E^(I*A 
rcSin[c*x]))/(-(c*Sqrt[d]) + Sqrt[c^2*d + e])] - 8*b*ArcSin[c*x]*Log[1 - ( 
Sqrt[e]*E^(I*ArcSin[c*x]))/(c*Sqrt[d] + Sqrt[c^2*d + e])] - 8*b*ArcSin[c*x 
]*Log[1 + (Sqrt[e]*E^(I*ArcSin[c*x]))/(c*Sqrt[d] + Sqrt[c^2*d + e])] + 16* 
b*ArcSin[c*x]*Log[1 - E^((2*I)*ArcSin[c*x])] + 16*a*Log[x] - 8*a*Log[d + e 
*x^2] - (I*b*c^3*d^(3/2)*Log[(e*Sqrt[c^2*d + e]*(Sqrt[e] - I*c^2*Sqrt[d]*x 
 + Sqrt[c^2*d + e]*Sqrt[1 - c^2*x^2]))/(c^3*(d + I*Sqrt[d]*Sqrt[e]*x))])/( 
c^2*d + e)^(3/2) + (I*b*c^3*d^(3/2)*Log[(e*Sqrt[c^2*d + e]*(Sqrt[e] + I*c^ 
2*Sqrt[d]*x + Sqrt[c^2*d + e]*Sqrt[1 - c^2*x^2]))/(c^3*(d - I*Sqrt[d]*Sqrt 
[e]*x))])/(c^2*d + e)^(3/2) + (8*I)*b*PolyLog[2, (Sqrt[e]*E^(I*ArcSin[c*x] 
))/(c*Sqrt[d] - Sqrt[c^2*d + e])] + (8*I)*b*PolyLog[2, (Sqrt[e]*E^(I*Ar...
 

Rubi [A] (verified)

Time = 1.66 (sec) , antiderivative size = 727, 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.

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

\(\Big \downarrow \) 5232

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

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {(a+b \arcsin (c x)) \log \left (1-\frac {\sqrt {e} e^{i \arcsin (c x)}}{-\sqrt {c^2 d+e}+i c \sqrt {-d}}\right )}{2 d^3}-\frac {(a+b \arcsin (c x)) \log \left (1+\frac {\sqrt {e} e^{i \arcsin (c x)}}{-\sqrt {c^2 d+e}+i c \sqrt {-d}}\right )}{2 d^3}-\frac {(a+b \arcsin (c x)) \log \left (1-\frac {\sqrt {e} e^{i \arcsin (c x)}}{\sqrt {c^2 d+e}+i c \sqrt {-d}}\right )}{2 d^3}-\frac {(a+b \arcsin (c x)) \log \left (1+\frac {\sqrt {e} e^{i \arcsin (c x)}}{\sqrt {c^2 d+e}+i c \sqrt {-d}}\right )}{2 d^3}+\frac {\log \left (1-e^{2 i \arcsin (c x)}\right ) (a+b \arcsin (c x))}{d^3}+\frac {a+b \arcsin (c x)}{2 d^2 \left (d+e x^2\right )}+\frac {a+b \arcsin (c x)}{4 d \left (d+e x^2\right )^2}+\frac {i b \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} e^{i \arcsin (c x)}}{i c \sqrt {-d}-\sqrt {d c^2+e}}\right )}{2 d^3}+\frac {i b \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{i \arcsin (c x)}}{i c \sqrt {-d}-\sqrt {d c^2+e}}\right )}{2 d^3}+\frac {i b \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} e^{i \arcsin (c x)}}{i \sqrt {-d} c+\sqrt {d c^2+e}}\right )}{2 d^3}+\frac {i b \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{i \arcsin (c x)}}{i \sqrt {-d} c+\sqrt {d c^2+e}}\right )}{2 d^3}-\frac {i b \operatorname {PolyLog}\left (2,e^{2 i \arcsin (c x)}\right )}{2 d^3}-\frac {b c \left (2 c^2 d+e\right ) \arctan \left (\frac {x \sqrt {c^2 d+e}}{\sqrt {d} \sqrt {1-c^2 x^2}}\right )}{8 d^{5/2} \left (c^2 d+e\right )^{3/2}}-\frac {b c \arctan \left (\frac {x \sqrt {c^2 d+e}}{\sqrt {d} \sqrt {1-c^2 x^2}}\right )}{2 d^{5/2} \sqrt {c^2 d+e}}-\frac {b c e x \sqrt {1-c^2 x^2}}{8 d^2 \left (c^2 d+e\right ) \left (d+e x^2\right )}\)

Input:

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

Output:

-1/8*(b*c*e*x*Sqrt[1 - c^2*x^2])/(d^2*(c^2*d + e)*(d + e*x^2)) + (a + b*Ar 
cSin[c*x])/(4*d*(d + e*x^2)^2) + (a + b*ArcSin[c*x])/(2*d^2*(d + e*x^2)) - 
 (b*c*ArcTan[(Sqrt[c^2*d + e]*x)/(Sqrt[d]*Sqrt[1 - c^2*x^2])])/(2*d^(5/2)* 
Sqrt[c^2*d + e]) - (b*c*(2*c^2*d + e)*ArcTan[(Sqrt[c^2*d + e]*x)/(Sqrt[d]* 
Sqrt[1 - c^2*x^2])])/(8*d^(5/2)*(c^2*d + 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^3) - ((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^3) - ((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^3) - ((a + b*Arc 
Sin[c*x])*Log[1 + (Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] + Sqrt[c^2*d + 
 e])])/(2*d^3) + ((a + b*ArcSin[c*x])*Log[1 - E^((2*I)*ArcSin[c*x])])/d^3 
+ ((I/2)*b*PolyLog[2, -((Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] - Sqrt[c 
^2*d + e]))])/d^3 + ((I/2)*b*PolyLog[2, (Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*S 
qrt[-d] - Sqrt[c^2*d + e])])/d^3 + ((I/2)*b*PolyLog[2, -((Sqrt[e]*E^(I*Arc 
Sin[c*x]))/(I*c*Sqrt[-d] + Sqrt[c^2*d + e]))])/d^3 + ((I/2)*b*PolyLog[2, ( 
Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] + Sqrt[c^2*d + e])])/d^3 - ((I/2) 
*b*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 = 4.48 (sec) , antiderivative size = 1130, normalized size of antiderivative = 1.55

method	result	size
parts	\(\text {Expression too large to display}\)	\(1130\)
derivativedivides	\(\text {Expression too large to display}\)	\(1174\)
default	\(\text {Expression too large to display}\)	\(1174\)

Input:

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

Output:

a/d^3*ln(x)+1/4*a/d/(e*x^2+d)^2-1/2*a/d^3*ln(e*x^2+d)+1/2*a/d^2/(e*x^2+d)+ 
b*(1/8*c^2*(6*c^4*d^2*arcsin(c*x)+4*arcsin(c*x)*c^4*d*e*x^2+I*c^4*d^2+2*I* 
c^4*d*e*x^2+I*e^2*c^4*x^4-(-c^2*x^2+1)^(1/2)*c^3*d*e*x-(-c^2*x^2+1)^(1/2)* 
e^2*c^3*x^3+6*c^2*d*e*arcsin(c*x)+4*arcsin(c*x)*e^2*c^2*x^2)/d^2/(c^2*e*x^ 
2+c^2*d)^2/(c^2*d+e)+5/8*I*(c^2*d*(c^2*d+e))^(1/2)/(c^2*d+e)^2/d^3*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-I/(c^2*d+e)*c^2/d^2*dilog(1+I*c*x+(-c^2*x^2+1)^(1/2))+I/(c^2*d+e)*c^2 
/d^2*dilog(I*c*x+(-c^2*x^2+1)^(1/2))+1/(c^2*d+e)*c^2/d^2*arcsin(c*x)*ln(1+ 
I*c*x+(-c^2*x^2+1)^(1/2))+1/4*I/(c^2*d+e)*c^2/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/(c^2*d+e)/d^3*e*arcsin(c*x)*ln(1+I*c*x+(-c^2*x^2+1)^(1/2))+1/4*I/(c^2 
*d+e)/d^3*e*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/(c^2*d+e)/d^3*e^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))+3/4*I*(c^2*d*(c^2*d+e))^(1/2)/(c^2*d+e)^2/d^2*c^2*ar 
ctanh(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/(c^2*d+e)/d^3*e*dilog(1+I*c*x+(-c^2*x^2+1)^(1/2))+I/(c^2*d+e)...
 

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F(-1)]

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

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

Output:

Timed out
                                                                                    
                                                                                    
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {a+b \arcsin (c x)}{x \left (d+e x^2\right )^3} \, dx=\frac {4 \left (\int \frac {\mathit {asin} \left (c x \right )}{e^{3} x^{7}+3 d \,e^{2} x^{5}+3 d^{2} e \,x^{3}+d^{3} x}d x \right ) b \,d^{5}+8 \left (\int \frac {\mathit {asin} \left (c x \right )}{e^{3} x^{7}+3 d \,e^{2} x^{5}+3 d^{2} e \,x^{3}+d^{3} x}d x \right ) b \,d^{4} e \,x^{2}+4 \left (\int \frac {\mathit {asin} \left (c x \right )}{e^{3} x^{7}+3 d \,e^{2} x^{5}+3 d^{2} e \,x^{3}+d^{3} x}d x \right ) b \,d^{3} e^{2} x^{4}-2 \,\mathrm {log}\left (e \,x^{2}+d \right ) a \,d^{2}-4 \,\mathrm {log}\left (e \,x^{2}+d \right ) a d e \,x^{2}-2 \,\mathrm {log}\left (e \,x^{2}+d \right ) a \,e^{2} x^{4}+4 \,\mathrm {log}\left (x \right ) a \,d^{2}+8 \,\mathrm {log}\left (x \right ) a d e \,x^{2}+4 \,\mathrm {log}\left (x \right ) a \,e^{2} x^{4}+2 a \,d^{2}-a \,e^{2} x^{4}}{4 d^{3} \left (e^{2} x^{4}+2 d e \,x^{2}+d^{2}\right )} \] Input:

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

Output:

(4*int(asin(c*x)/(d**3*x + 3*d**2*e*x**3 + 3*d*e**2*x**5 + e**3*x**7),x)*b 
*d**5 + 8*int(asin(c*x)/(d**3*x + 3*d**2*e*x**3 + 3*d*e**2*x**5 + e**3*x** 
7),x)*b*d**4*e*x**2 + 4*int(asin(c*x)/(d**3*x + 3*d**2*e*x**3 + 3*d*e**2*x 
**5 + e**3*x**7),x)*b*d**3*e**2*x**4 - 2*log(d + e*x**2)*a*d**2 - 4*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**2 + 8 
*log(x)*a*d*e*x**2 + 4*log(x)*a*e**2*x**4 + 2*a*d**2 - a*e**2*x**4)/(4*d** 
3*(d**2 + 2*d*e*x**2 + e**2*x**4))