\(\int \frac {(d+e x^2)^2}{(a+b \arcsin (c x))^2} \, dx\) [160]

Optimal result

Integrand size = 20, antiderivative size = 498 \[ \int \frac {\left (d+e x^2\right )^2}{(a+b \arcsin (c x))^2} \, dx=-\frac {d^2 \sqrt {1-c^2 x^2}}{b c (a+b \arcsin (c x))}-\frac {2 d e x^2 \sqrt {1-c^2 x^2}}{b c (a+b \arcsin (c x))}-\frac {e^2 x^4 \sqrt {1-c^2 x^2}}{b c (a+b \arcsin (c x))}+\frac {d^2 \operatorname {CosIntegral}\left (\frac {a+b \arcsin (c x)}{b}\right ) \sin \left (\frac {a}{b}\right )}{b^2 c}+\frac {d e \operatorname {CosIntegral}\left (\frac {a+b \arcsin (c x)}{b}\right ) \sin \left (\frac {a}{b}\right )}{2 b^2 c^3}+\frac {e^2 \operatorname {CosIntegral}\left (\frac {a+b \arcsin (c x)}{b}\right ) \sin \left (\frac {a}{b}\right )}{8 b^2 c^5}-\frac {3 d e \operatorname {CosIntegral}\left (\frac {3 (a+b \arcsin (c x))}{b}\right ) \sin \left (\frac {3 a}{b}\right )}{2 b^2 c^3}-\frac {9 e^2 \operatorname {CosIntegral}\left (\frac {3 (a+b \arcsin (c x))}{b}\right ) \sin \left (\frac {3 a}{b}\right )}{16 b^2 c^5}+\frac {5 e^2 \operatorname {CosIntegral}\left (\frac {5 (a+b \arcsin (c x))}{b}\right ) \sin \left (\frac {5 a}{b}\right )}{16 b^2 c^5}-\frac {d^2 \cos \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a+b \arcsin (c x)}{b}\right )}{b^2 c}-\frac {d e \cos \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a+b \arcsin (c x)}{b}\right )}{2 b^2 c^3}-\frac {e^2 \cos \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a+b \arcsin (c x)}{b}\right )}{8 b^2 c^5}+\frac {3 d e \cos \left (\frac {3 a}{b}\right ) \text {Si}\left (\frac {3 (a+b \arcsin (c x))}{b}\right )}{2 b^2 c^3}+\frac {9 e^2 \cos \left (\frac {3 a}{b}\right ) \text {Si}\left (\frac {3 (a+b \arcsin (c x))}{b}\right )}{16 b^2 c^5}-\frac {5 e^2 \cos \left (\frac {5 a}{b}\right ) \text {Si}\left (\frac {5 (a+b \arcsin (c x))}{b}\right )}{16 b^2 c^5} \] Output:

-d^2*(-c^2*x^2+1)^(1/2)/b/c/(a+b*arcsin(c*x))-2*d*e*x^2*(-c^2*x^2+1)^(1/2) 
/b/c/(a+b*arcsin(c*x))-e^2*x^4*(-c^2*x^2+1)^(1/2)/b/c/(a+b*arcsin(c*x))+d^ 
2*Ci((a+b*arcsin(c*x))/b)*sin(a/b)/b^2/c+1/2*d*e*Ci((a+b*arcsin(c*x))/b)*s 
in(a/b)/b^2/c^3+1/8*e^2*Ci((a+b*arcsin(c*x))/b)*sin(a/b)/b^2/c^5-3/2*d*e*C 
i(3*(a+b*arcsin(c*x))/b)*sin(3*a/b)/b^2/c^3-9/16*e^2*Ci(3*(a+b*arcsin(c*x) 
)/b)*sin(3*a/b)/b^2/c^5+5/16*e^2*Ci(5*(a+b*arcsin(c*x))/b)*sin(5*a/b)/b^2/ 
c^5-d^2*cos(a/b)*Si((a+b*arcsin(c*x))/b)/b^2/c-1/2*d*e*cos(a/b)*Si((a+b*ar 
csin(c*x))/b)/b^2/c^3-1/8*e^2*cos(a/b)*Si((a+b*arcsin(c*x))/b)/b^2/c^5+3/2 
*d*e*cos(3*a/b)*Si(3*(a+b*arcsin(c*x))/b)/b^2/c^3+9/16*e^2*cos(3*a/b)*Si(3 
*(a+b*arcsin(c*x))/b)/b^2/c^5-5/16*e^2*cos(5*a/b)*Si(5*(a+b*arcsin(c*x))/b 
)/b^2/c^5
 

Mathematica [A] (verified)

Time = 1.67 (sec) , antiderivative size = 359, normalized size of antiderivative = 0.72 \[ \int \frac {\left (d+e x^2\right )^2}{(a+b \arcsin (c x))^2} \, dx=-\frac {\frac {16 b c^4 d^2 \sqrt {1-c^2 x^2}}{a+b \arcsin (c x)}+\frac {32 b c^4 d e x^2 \sqrt {1-c^2 x^2}}{a+b \arcsin (c x)}+\frac {16 b c^4 e^2 x^4 \sqrt {1-c^2 x^2}}{a+b \arcsin (c x)}-2 \left (8 c^4 d^2+4 c^2 d e+e^2\right ) \operatorname {CosIntegral}\left (\frac {a}{b}+\arcsin (c x)\right ) \sin \left (\frac {a}{b}\right )+3 e \left (8 c^2 d+3 e\right ) \operatorname {CosIntegral}\left (3 \left (\frac {a}{b}+\arcsin (c x)\right )\right ) \sin \left (\frac {3 a}{b}\right )-5 e^2 \operatorname {CosIntegral}\left (5 \left (\frac {a}{b}+\arcsin (c x)\right )\right ) \sin \left (\frac {5 a}{b}\right )+16 c^4 d^2 \cos \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a}{b}+\arcsin (c x)\right )+8 c^2 d e \cos \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a}{b}+\arcsin (c x)\right )+2 e^2 \cos \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a}{b}+\arcsin (c x)\right )-24 c^2 d e \cos \left (\frac {3 a}{b}\right ) \text {Si}\left (3 \left (\frac {a}{b}+\arcsin (c x)\right )\right )-9 e^2 \cos \left (\frac {3 a}{b}\right ) \text {Si}\left (3 \left (\frac {a}{b}+\arcsin (c x)\right )\right )+5 e^2 \cos \left (\frac {5 a}{b}\right ) \text {Si}\left (5 \left (\frac {a}{b}+\arcsin (c x)\right )\right )}{16 b^2 c^5} \] Input:

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

Output:

-1/16*((16*b*c^4*d^2*Sqrt[1 - c^2*x^2])/(a + b*ArcSin[c*x]) + (32*b*c^4*d* 
e*x^2*Sqrt[1 - c^2*x^2])/(a + b*ArcSin[c*x]) + (16*b*c^4*e^2*x^4*Sqrt[1 - 
c^2*x^2])/(a + b*ArcSin[c*x]) - 2*(8*c^4*d^2 + 4*c^2*d*e + e^2)*CosIntegra 
l[a/b + ArcSin[c*x]]*Sin[a/b] + 3*e*(8*c^2*d + 3*e)*CosIntegral[3*(a/b + A 
rcSin[c*x])]*Sin[(3*a)/b] - 5*e^2*CosIntegral[5*(a/b + ArcSin[c*x])]*Sin[( 
5*a)/b] + 16*c^4*d^2*Cos[a/b]*SinIntegral[a/b + ArcSin[c*x]] + 8*c^2*d*e*C 
os[a/b]*SinIntegral[a/b + ArcSin[c*x]] + 2*e^2*Cos[a/b]*SinIntegral[a/b + 
ArcSin[c*x]] - 24*c^2*d*e*Cos[(3*a)/b]*SinIntegral[3*(a/b + ArcSin[c*x])] 
- 9*e^2*Cos[(3*a)/b]*SinIntegral[3*(a/b + ArcSin[c*x])] + 5*e^2*Cos[(5*a)/ 
b]*SinIntegral[5*(a/b + ArcSin[c*x])])/(b^2*c^5)
 

Rubi [A] (verified)

Time = 1.17 (sec) , antiderivative size = 498, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, 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.

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

\(\Big \downarrow \) 5172

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

\(\Big \downarrow \) 2009

\(\displaystyle \frac {e^2 \sin \left (\frac {a}{b}\right ) \operatorname {CosIntegral}\left (\frac {a+b \arcsin (c x)}{b}\right )}{8 b^2 c^5}-\frac {9 e^2 \sin \left (\frac {3 a}{b}\right ) \operatorname {CosIntegral}\left (\frac {3 (a+b \arcsin (c x))}{b}\right )}{16 b^2 c^5}+\frac {5 e^2 \sin \left (\frac {5 a}{b}\right ) \operatorname {CosIntegral}\left (\frac {5 (a+b \arcsin (c x))}{b}\right )}{16 b^2 c^5}-\frac {e^2 \cos \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a+b \arcsin (c x)}{b}\right )}{8 b^2 c^5}+\frac {9 e^2 \cos \left (\frac {3 a}{b}\right ) \text {Si}\left (\frac {3 (a+b \arcsin (c x))}{b}\right )}{16 b^2 c^5}-\frac {5 e^2 \cos \left (\frac {5 a}{b}\right ) \text {Si}\left (\frac {5 (a+b \arcsin (c x))}{b}\right )}{16 b^2 c^5}+\frac {d e \sin \left (\frac {a}{b}\right ) \operatorname {CosIntegral}\left (\frac {a+b \arcsin (c x)}{b}\right )}{2 b^2 c^3}-\frac {3 d e \sin \left (\frac {3 a}{b}\right ) \operatorname {CosIntegral}\left (\frac {3 (a+b \arcsin (c x))}{b}\right )}{2 b^2 c^3}-\frac {d e \cos \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a+b \arcsin (c x)}{b}\right )}{2 b^2 c^3}+\frac {3 d e \cos \left (\frac {3 a}{b}\right ) \text {Si}\left (\frac {3 (a+b \arcsin (c x))}{b}\right )}{2 b^2 c^3}+\frac {d^2 \sin \left (\frac {a}{b}\right ) \operatorname {CosIntegral}\left (\frac {a+b \arcsin (c x)}{b}\right )}{b^2 c}-\frac {d^2 \cos \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a+b \arcsin (c x)}{b}\right )}{b^2 c}-\frac {d^2 \sqrt {1-c^2 x^2}}{b c (a+b \arcsin (c x))}-\frac {2 d e x^2 \sqrt {1-c^2 x^2}}{b c (a+b \arcsin (c x))}-\frac {e^2 x^4 \sqrt {1-c^2 x^2}}{b c (a+b \arcsin (c x))}\)

Input:

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

Output:

-((d^2*Sqrt[1 - c^2*x^2])/(b*c*(a + b*ArcSin[c*x]))) - (2*d*e*x^2*Sqrt[1 - 
 c^2*x^2])/(b*c*(a + b*ArcSin[c*x])) - (e^2*x^4*Sqrt[1 - c^2*x^2])/(b*c*(a 
 + b*ArcSin[c*x])) + (d^2*CosIntegral[(a + b*ArcSin[c*x])/b]*Sin[a/b])/(b^ 
2*c) + (d*e*CosIntegral[(a + b*ArcSin[c*x])/b]*Sin[a/b])/(2*b^2*c^3) + (e^ 
2*CosIntegral[(a + b*ArcSin[c*x])/b]*Sin[a/b])/(8*b^2*c^5) - (3*d*e*CosInt 
egral[(3*(a + b*ArcSin[c*x]))/b]*Sin[(3*a)/b])/(2*b^2*c^3) - (9*e^2*CosInt 
egral[(3*(a + b*ArcSin[c*x]))/b]*Sin[(3*a)/b])/(16*b^2*c^5) + (5*e^2*CosIn 
tegral[(5*(a + b*ArcSin[c*x]))/b]*Sin[(5*a)/b])/(16*b^2*c^5) - (d^2*Cos[a/ 
b]*SinIntegral[(a + b*ArcSin[c*x])/b])/(b^2*c) - (d*e*Cos[a/b]*SinIntegral 
[(a + b*ArcSin[c*x])/b])/(2*b^2*c^3) - (e^2*Cos[a/b]*SinIntegral[(a + b*Ar 
cSin[c*x])/b])/(8*b^2*c^5) + (3*d*e*Cos[(3*a)/b]*SinIntegral[(3*(a + b*Arc 
Sin[c*x]))/b])/(2*b^2*c^3) + (9*e^2*Cos[(3*a)/b]*SinIntegral[(3*(a + b*Arc 
Sin[c*x]))/b])/(16*b^2*c^5) - (5*e^2*Cos[(5*a)/b]*SinIntegral[(5*(a + b*Ar 
cSin[c*x]))/b])/(16*b^2*c^5)
 

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 [A] (verified)

Time = 0.27 (sec) , antiderivative size = 795, normalized size of antiderivative = 1.60

method	result	size
derivativedivides	\(\text {Expression too large to display}\)	\(795\)
default	\(\text {Expression too large to display}\)	\(795\)

Input:

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

Output:

-1/16/c^5*(2*(-c^2*x^2+1)^(1/2)*b*e^2+16*arcsin(c*x)*Si(arcsin(c*x)+a/b)*c 
os(a/b)*b*c^4*d^2-16*arcsin(c*x)*Ci(arcsin(c*x)+a/b)*sin(a/b)*b*c^4*d^2+8* 
Si(arcsin(c*x)+a/b)*cos(a/b)*a*c^2*d*e+16*Si(arcsin(c*x)+a/b)*cos(a/b)*a*c 
^4*d^2-16*Ci(arcsin(c*x)+a/b)*sin(a/b)*a*c^4*d^2-8*cos(3*arcsin(c*x))*b*c^ 
2*d*e-9*arcsin(c*x)*Si(3*arcsin(c*x)+3*a/b)*cos(3*a/b)*b*e^2+9*arcsin(c*x) 
*Ci(3*arcsin(c*x)+3*a/b)*sin(3*a/b)*b*e^2+5*arcsin(c*x)*Si(5*arcsin(c*x)+5 
*a/b)*cos(5*a/b)*b*e^2-5*arcsin(c*x)*Ci(5*arcsin(c*x)+5*a/b)*sin(5*a/b)*b* 
e^2+2*arcsin(c*x)*Si(arcsin(c*x)+a/b)*cos(a/b)*b*e^2-2*arcsin(c*x)*Ci(arcs 
in(c*x)+a/b)*sin(a/b)*b*e^2+8*(-c^2*x^2+1)^(1/2)*b*c^2*d*e-3*cos(3*arcsin( 
c*x))*b*e^2+cos(5*arcsin(c*x))*b*e^2+8*arcsin(c*x)*Si(arcsin(c*x)+a/b)*cos 
(a/b)*b*c^2*d*e-8*arcsin(c*x)*Ci(arcsin(c*x)+a/b)*sin(a/b)*b*c^2*d*e-24*ar 
csin(c*x)*Si(3*arcsin(c*x)+3*a/b)*cos(3*a/b)*b*c^2*d*e+24*arcsin(c*x)*Ci(3 
*arcsin(c*x)+3*a/b)*sin(3*a/b)*b*c^2*d*e-9*Si(3*arcsin(c*x)+3*a/b)*cos(3*a 
/b)*a*e^2+9*Ci(3*arcsin(c*x)+3*a/b)*sin(3*a/b)*a*e^2+5*Si(5*arcsin(c*x)+5* 
a/b)*cos(5*a/b)*a*e^2-5*Ci(5*arcsin(c*x)+5*a/b)*sin(5*a/b)*a*e^2+2*Si(arcs 
in(c*x)+a/b)*cos(a/b)*a*e^2-2*Ci(arcsin(c*x)+a/b)*sin(a/b)*a*e^2+16*(-c^2* 
x^2+1)^(1/2)*b*c^4*d^2-8*Ci(arcsin(c*x)+a/b)*sin(a/b)*a*c^2*d*e-24*Si(3*ar 
csin(c*x)+3*a/b)*cos(3*a/b)*a*c^2*d*e+24*Ci(3*arcsin(c*x)+3*a/b)*sin(3*a/b 
)*a*c^2*d*e)/(a+b*arcsin(c*x))/b^2
 

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2337 vs. \(2 (472) = 944\).

Time = 0.21 (sec) , antiderivative size = 2337, normalized size of antiderivative = 4.69 \[ \int \frac {\left (d+e x^2\right )^2}{(a+b \arcsin (c x))^2} \, dx=\text {Too large to display} \] Input:

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

Output:

5*b*e^2*arcsin(c*x)*cos(a/b)^4*cos_integral(5*a/b + 5*arcsin(c*x))*sin(a/b 
)/(b^3*c^5*arcsin(c*x) + a*b^2*c^5) - 6*b*c^2*d*e*arcsin(c*x)*cos(a/b)^2*c 
os_integral(3*a/b + 3*arcsin(c*x))*sin(a/b)/(b^3*c^5*arcsin(c*x) + a*b^2*c 
^5) + b*c^4*d^2*arcsin(c*x)*cos_integral(a/b + arcsin(c*x))*sin(a/b)/(b^3* 
c^5*arcsin(c*x) + a*b^2*c^5) - 5*b*e^2*arcsin(c*x)*cos(a/b)^5*sin_integral 
(5*a/b + 5*arcsin(c*x))/(b^3*c^5*arcsin(c*x) + a*b^2*c^5) + 6*b*c^2*d*e*ar 
csin(c*x)*cos(a/b)^3*sin_integral(3*a/b + 3*arcsin(c*x))/(b^3*c^5*arcsin(c 
*x) + a*b^2*c^5) - b*c^4*d^2*arcsin(c*x)*cos(a/b)*sin_integral(a/b + arcsi 
n(c*x))/(b^3*c^5*arcsin(c*x) + a*b^2*c^5) + 5*a*e^2*cos(a/b)^4*cos_integra 
l(5*a/b + 5*arcsin(c*x))*sin(a/b)/(b^3*c^5*arcsin(c*x) + a*b^2*c^5) - 6*a* 
c^2*d*e*cos(a/b)^2*cos_integral(3*a/b + 3*arcsin(c*x))*sin(a/b)/(b^3*c^5*a 
rcsin(c*x) + a*b^2*c^5) + a*c^4*d^2*cos_integral(a/b + arcsin(c*x))*sin(a/ 
b)/(b^3*c^5*arcsin(c*x) + a*b^2*c^5) - 5*a*e^2*cos(a/b)^5*sin_integral(5*a 
/b + 5*arcsin(c*x))/(b^3*c^5*arcsin(c*x) + a*b^2*c^5) + 6*a*c^2*d*e*cos(a/ 
b)^3*sin_integral(3*a/b + 3*arcsin(c*x))/(b^3*c^5*arcsin(c*x) + a*b^2*c^5) 
 - a*c^4*d^2*cos(a/b)*sin_integral(a/b + arcsin(c*x))/(b^3*c^5*arcsin(c*x) 
 + a*b^2*c^5) - 15/4*b*e^2*arcsin(c*x)*cos(a/b)^2*cos_integral(5*a/b + 5*a 
rcsin(c*x))*sin(a/b)/(b^3*c^5*arcsin(c*x) + a*b^2*c^5) + 3/2*b*c^2*d*e*arc 
sin(c*x)*cos_integral(3*a/b + 3*arcsin(c*x))*sin(a/b)/(b^3*c^5*arcsin(c*x) 
 + a*b^2*c^5) - 9/4*b*e^2*arcsin(c*x)*cos(a/b)^2*cos_integral(3*a/b + 3...
                                                                                    
                                                                                    
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {\left (d+e x^2\right )^2}{(a+b \arcsin (c x))^2} \, dx=\left (\int \frac {x^{4}}{\mathit {asin} \left (c x \right )^{2} b^{2}+2 \mathit {asin} \left (c x \right ) a b +a^{2}}d x \right ) e^{2}+2 \left (\int \frac {x^{2}}{\mathit {asin} \left (c x \right )^{2} b^{2}+2 \mathit {asin} \left (c x \right ) a b +a^{2}}d x \right ) d e +\left (\int \frac {1}{\mathit {asin} \left (c x \right )^{2} b^{2}+2 \mathit {asin} \left (c x \right ) a b +a^{2}}d x \right ) d^{2} \] Input:

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

Output:

int(x**4/(asin(c*x)**2*b**2 + 2*asin(c*x)*a*b + a**2),x)*e**2 + 2*int(x**2 
/(asin(c*x)**2*b**2 + 2*asin(c*x)*a*b + a**2),x)*d*e + int(1/(asin(c*x)**2 
*b**2 + 2*asin(c*x)*a*b + a**2),x)*d**2