\(\int \frac {(d-c^2 d x^2)^2}{(a+b \arcsin (c x))^2} \, dx\) [33]

Optimal result

Integrand size = 24, antiderivative size = 236 \[ \int \frac {\left (d-c^2 d x^2\right )^2}{(a+b \arcsin (c x))^2} \, dx=-\frac {d^2 \left (1-c^2 x^2\right )^{5/2}}{b c (a+b \arcsin (c x))}+\frac {5 d^2 \operatorname {CosIntegral}\left (\frac {a+b \arcsin (c x)}{b}\right ) \sin \left (\frac {a}{b}\right )}{8 b^2 c}+\frac {15 d^2 \operatorname {CosIntegral}\left (\frac {3 (a+b \arcsin (c x))}{b}\right ) \sin \left (\frac {3 a}{b}\right )}{16 b^2 c}+\frac {5 d^2 \operatorname {CosIntegral}\left (\frac {5 (a+b \arcsin (c x))}{b}\right ) \sin \left (\frac {5 a}{b}\right )}{16 b^2 c}-\frac {5 d^2 \cos \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a+b \arcsin (c x)}{b}\right )}{8 b^2 c}-\frac {15 d^2 \cos \left (\frac {3 a}{b}\right ) \text {Si}\left (\frac {3 (a+b \arcsin (c x))}{b}\right )}{16 b^2 c}-\frac {5 d^2 \cos \left (\frac {5 a}{b}\right ) \text {Si}\left (\frac {5 (a+b \arcsin (c x))}{b}\right )}{16 b^2 c} \] Output:

-d^2*(-c^2*x^2+1)^(5/2)/b/c/(a+b*arcsin(c*x))+5/8*d^2*Ci((a+b*arcsin(c*x)) 
/b)*sin(a/b)/b^2/c+15/16*d^2*Ci(3*(a+b*arcsin(c*x))/b)*sin(3*a/b)/b^2/c+5/ 
16*d^2*Ci(5*(a+b*arcsin(c*x))/b)*sin(5*a/b)/b^2/c-5/8*d^2*cos(a/b)*Si((a+b 
*arcsin(c*x))/b)/b^2/c-15/16*d^2*cos(3*a/b)*Si(3*(a+b*arcsin(c*x))/b)/b^2/ 
c-5/16*d^2*cos(5*a/b)*Si(5*(a+b*arcsin(c*x))/b)/b^2/c
 

Mathematica [A] (verified)

Time = 0.86 (sec) , antiderivative size = 233, normalized size of antiderivative = 0.99 \[ \int \frac {\left (d-c^2 d x^2\right )^2}{(a+b \arcsin (c x))^2} \, dx=\frac {d^2 \left (-\frac {16 b \sqrt {1-c^2 x^2}}{a+b \arcsin (c x)}+\frac {32 b c^2 x^2 \sqrt {1-c^2 x^2}}{a+b \arcsin (c x)}-\frac {16 b c^4 x^4 \sqrt {1-c^2 x^2}}{a+b \arcsin (c x)}+10 \operatorname {CosIntegral}\left (\frac {a}{b}+\arcsin (c x)\right ) \sin \left (\frac {a}{b}\right )+15 \operatorname {CosIntegral}\left (3 \left (\frac {a}{b}+\arcsin (c x)\right )\right ) \sin \left (\frac {3 a}{b}\right )+5 \operatorname {CosIntegral}\left (5 \left (\frac {a}{b}+\arcsin (c x)\right )\right ) \sin \left (\frac {5 a}{b}\right )-10 \cos \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a}{b}+\arcsin (c x)\right )-15 \cos \left (\frac {3 a}{b}\right ) \text {Si}\left (3 \left (\frac {a}{b}+\arcsin (c x)\right )\right )-5 \cos \left (\frac {5 a}{b}\right ) \text {Si}\left (5 \left (\frac {a}{b}+\arcsin (c x)\right )\right )\right )}{16 b^2 c} \] Input:

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

Output:

(d^2*((-16*b*Sqrt[1 - c^2*x^2])/(a + b*ArcSin[c*x]) + (32*b*c^2*x^2*Sqrt[1 
 - c^2*x^2])/(a + b*ArcSin[c*x]) - (16*b*c^4*x^4*Sqrt[1 - c^2*x^2])/(a + b 
*ArcSin[c*x]) + 10*CosIntegral[a/b + ArcSin[c*x]]*Sin[a/b] + 15*CosIntegra 
l[3*(a/b + ArcSin[c*x])]*Sin[(3*a)/b] + 5*CosIntegral[5*(a/b + ArcSin[c*x] 
)]*Sin[(5*a)/b] - 10*Cos[a/b]*SinIntegral[a/b + ArcSin[c*x]] - 15*Cos[(3*a 
)/b]*SinIntegral[3*(a/b + ArcSin[c*x])] - 5*Cos[(5*a)/b]*SinIntegral[5*(a/ 
b + ArcSin[c*x])]))/(16*b^2*c)
 

Rubi [A] (verified)

Time = 0.65 (sec) , antiderivative size = 194, normalized size of antiderivative = 0.82, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {5166, 5224, 25, 4906, 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.

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

\(\Big \downarrow \) 5166

\(\displaystyle -\frac {5 c d^2 \int \frac {x \left (1-c^2 x^2\right )^{3/2}}{a+b \arcsin (c x)}dx}{b}-\frac {d^2 \left (1-c^2 x^2\right )^{5/2}}{b c (a+b \arcsin (c x))}\)

\(\Big \downarrow \) 5224

\(\displaystyle -\frac {5 d^2 \int -\frac {\cos ^4\left (\frac {a}{b}-\frac {a+b \arcsin (c x)}{b}\right ) \sin \left (\frac {a}{b}-\frac {a+b \arcsin (c x)}{b}\right )}{a+b \arcsin (c x)}d(a+b \arcsin (c x))}{b^2 c}-\frac {d^2 \left (1-c^2 x^2\right )^{5/2}}{b c (a+b \arcsin (c x))}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {5 d^2 \int \frac {\cos ^4\left (\frac {a}{b}-\frac {a+b \arcsin (c x)}{b}\right ) \sin \left (\frac {a}{b}-\frac {a+b \arcsin (c x)}{b}\right )}{a+b \arcsin (c x)}d(a+b \arcsin (c x))}{b^2 c}-\frac {d^2 \left (1-c^2 x^2\right )^{5/2}}{b c (a+b \arcsin (c x))}\)

\(\Big \downarrow \) 4906

\(\displaystyle \frac {5 d^2 \int \left (\frac {\sin \left (\frac {5 a}{b}-\frac {5 (a+b \arcsin (c x))}{b}\right )}{16 (a+b \arcsin (c x))}+\frac {3 \sin \left (\frac {3 a}{b}-\frac {3 (a+b \arcsin (c x))}{b}\right )}{16 (a+b \arcsin (c x))}+\frac {\sin \left (\frac {a}{b}-\frac {a+b \arcsin (c x)}{b}\right )}{8 (a+b \arcsin (c x))}\right )d(a+b \arcsin (c x))}{b^2 c}-\frac {d^2 \left (1-c^2 x^2\right )^{5/2}}{b c (a+b \arcsin (c x))}\)

\(\Big \downarrow \) 2009

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

Input:

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

Output:

-((d^2*(1 - c^2*x^2)^(5/2))/(b*c*(a + b*ArcSin[c*x]))) - (5*d^2*(-1/8*(Cos 
Integral[(a + b*ArcSin[c*x])/b]*Sin[a/b]) - (3*CosIntegral[(3*(a + b*ArcSi 
n[c*x]))/b]*Sin[(3*a)/b])/16 - (CosIntegral[(5*(a + b*ArcSin[c*x]))/b]*Sin 
[(5*a)/b])/16 + (Cos[a/b]*SinIntegral[(a + b*ArcSin[c*x])/b])/8 + (3*Cos[( 
3*a)/b]*SinIntegral[(3*(a + b*ArcSin[c*x]))/b])/16 + (Cos[(5*a)/b]*SinInte 
gral[(5*(a + b*ArcSin[c*x]))/b])/16))/(b^2*c)
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 2009

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

rule 4906

Int[Cos[(a_.) + (b_.)*(x_)]^(p_.)*((c_.) + (d_.)*(x_))^(m_.)*Sin[(a_.) + (b 
_.)*(x_)]^(n_.), x_Symbol] :> Int[ExpandTrigReduce[(c + d*x)^m, Sin[a + b*x 
]^n*Cos[a + b*x]^p, x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] && IG 
tQ[p, 0]
 

rule 5166

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_)*((d_) + (e_.)*(x_)^2)^(p_.), x_ 
Symbol] :> Simp[Sqrt[1 - c^2*x^2]*(d + e*x^2)^p*((a + b*ArcSin[c*x])^(n + 1 
)/(b*c*(n + 1))), x] + Simp[c*((2*p + 1)/(b*(n + 1)))*Simp[(d + e*x^2)^p/(1 
 - c^2*x^2)^p]   Int[x*(1 - c^2*x^2)^(p - 1/2)*(a + b*ArcSin[c*x])^(n + 1), 
 x], x] /; FreeQ[{a, b, c, d, e, p}, x] && EqQ[c^2*d + e, 0] && LtQ[n, -1]
 

rule 5224

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*(x_)^(m_.)*((d_) + (e_.)*(x_)^ 
2)^(p_.), x_Symbol] :> Simp[(1/(b*c^(m + 1)))*Simp[(d + e*x^2)^p/(1 - c^2*x 
^2)^p]   Subst[Int[x^n*Sin[-a/b + x/b]^m*Cos[-a/b + x/b]^(2*p + 1), x], x, 
a + b*ArcSin[c*x]], x] /; FreeQ[{a, b, c, d, e, n}, x] && EqQ[c^2*d + e, 0] 
 && IGtQ[2*p + 2, 0] && IGtQ[m, 0]
 

Maple [A] (verified)

Time = 0.09 (sec) , antiderivative size = 353, normalized size of antiderivative = 1.50

method	result	size
derivativedivides	\(-\frac {d^{2} \left (5 \arcsin \left (c x \right ) \operatorname {Si}\left (5 \arcsin \left (c x \right )+\frac {5 a}{b}\right ) \cos \left (\frac {5 a}{b}\right ) b -5 \arcsin \left (c x \right ) \operatorname {Ci}\left (5 \arcsin \left (c x \right )+\frac {5 a}{b}\right ) \sin \left (\frac {5 a}{b}\right ) b +15 \arcsin \left (c x \right ) \operatorname {Si}\left (3 \arcsin \left (c x \right )+\frac {3 a}{b}\right ) \cos \left (\frac {3 a}{b}\right ) b -15 \arcsin \left (c x \right ) \operatorname {Ci}\left (3 \arcsin \left (c x \right )+\frac {3 a}{b}\right ) \sin \left (\frac {3 a}{b}\right ) b +10 \arcsin \left (c x \right ) \operatorname {Si}\left (\arcsin \left (c x \right )+\frac {a}{b}\right ) \cos \left (\frac {a}{b}\right ) b -10 \arcsin \left (c x \right ) \operatorname {Ci}\left (\arcsin \left (c x \right )+\frac {a}{b}\right ) \sin \left (\frac {a}{b}\right ) b +5 \,\operatorname {Si}\left (5 \arcsin \left (c x \right )+\frac {5 a}{b}\right ) \cos \left (\frac {5 a}{b}\right ) a -5 \,\operatorname {Ci}\left (5 \arcsin \left (c x \right )+\frac {5 a}{b}\right ) \sin \left (\frac {5 a}{b}\right ) a +15 \,\operatorname {Si}\left (3 \arcsin \left (c x \right )+\frac {3 a}{b}\right ) \cos \left (\frac {3 a}{b}\right ) a -15 \,\operatorname {Ci}\left (3 \arcsin \left (c x \right )+\frac {3 a}{b}\right ) \sin \left (\frac {3 a}{b}\right ) a +10 \,\operatorname {Si}\left (\arcsin \left (c x \right )+\frac {a}{b}\right ) \cos \left (\frac {a}{b}\right ) a -10 \,\operatorname {Ci}\left (\arcsin \left (c x \right )+\frac {a}{b}\right ) \sin \left (\frac {a}{b}\right ) a +10 \sqrt {-c^{2} x^{2}+1}\, b +\cos \left (5 \arcsin \left (c x \right )\right ) b +5 \cos \left (3 \arcsin \left (c x \right )\right ) b \right )}{16 c \left (a +b \arcsin \left (c x \right )\right ) b^{2}}\)	\(353\)
default	\(-\frac {d^{2} \left (5 \arcsin \left (c x \right ) \operatorname {Si}\left (5 \arcsin \left (c x \right )+\frac {5 a}{b}\right ) \cos \left (\frac {5 a}{b}\right ) b -5 \arcsin \left (c x \right ) \operatorname {Ci}\left (5 \arcsin \left (c x \right )+\frac {5 a}{b}\right ) \sin \left (\frac {5 a}{b}\right ) b +15 \arcsin \left (c x \right ) \operatorname {Si}\left (3 \arcsin \left (c x \right )+\frac {3 a}{b}\right ) \cos \left (\frac {3 a}{b}\right ) b -15 \arcsin \left (c x \right ) \operatorname {Ci}\left (3 \arcsin \left (c x \right )+\frac {3 a}{b}\right ) \sin \left (\frac {3 a}{b}\right ) b +10 \arcsin \left (c x \right ) \operatorname {Si}\left (\arcsin \left (c x \right )+\frac {a}{b}\right ) \cos \left (\frac {a}{b}\right ) b -10 \arcsin \left (c x \right ) \operatorname {Ci}\left (\arcsin \left (c x \right )+\frac {a}{b}\right ) \sin \left (\frac {a}{b}\right ) b +5 \,\operatorname {Si}\left (5 \arcsin \left (c x \right )+\frac {5 a}{b}\right ) \cos \left (\frac {5 a}{b}\right ) a -5 \,\operatorname {Ci}\left (5 \arcsin \left (c x \right )+\frac {5 a}{b}\right ) \sin \left (\frac {5 a}{b}\right ) a +15 \,\operatorname {Si}\left (3 \arcsin \left (c x \right )+\frac {3 a}{b}\right ) \cos \left (\frac {3 a}{b}\right ) a -15 \,\operatorname {Ci}\left (3 \arcsin \left (c x \right )+\frac {3 a}{b}\right ) \sin \left (\frac {3 a}{b}\right ) a +10 \,\operatorname {Si}\left (\arcsin \left (c x \right )+\frac {a}{b}\right ) \cos \left (\frac {a}{b}\right ) a -10 \,\operatorname {Ci}\left (\arcsin \left (c x \right )+\frac {a}{b}\right ) \sin \left (\frac {a}{b}\right ) a +10 \sqrt {-c^{2} x^{2}+1}\, b +\cos \left (5 \arcsin \left (c x \right )\right ) b +5 \cos \left (3 \arcsin \left (c x \right )\right ) b \right )}{16 c \left (a +b \arcsin \left (c x \right )\right ) b^{2}}\)	\(353\)

Input:

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

Output:

-1/16/c*d^2*(5*arcsin(c*x)*Si(5*arcsin(c*x)+5*a/b)*cos(5*a/b)*b-5*arcsin(c 
*x)*Ci(5*arcsin(c*x)+5*a/b)*sin(5*a/b)*b+15*arcsin(c*x)*Si(3*arcsin(c*x)+3 
*a/b)*cos(3*a/b)*b-15*arcsin(c*x)*Ci(3*arcsin(c*x)+3*a/b)*sin(3*a/b)*b+10* 
arcsin(c*x)*Si(arcsin(c*x)+a/b)*cos(a/b)*b-10*arcsin(c*x)*Ci(arcsin(c*x)+a 
/b)*sin(a/b)*b+5*Si(5*arcsin(c*x)+5*a/b)*cos(5*a/b)*a-5*Ci(5*arcsin(c*x)+5 
*a/b)*sin(5*a/b)*a+15*Si(3*arcsin(c*x)+3*a/b)*cos(3*a/b)*a-15*Ci(3*arcsin( 
c*x)+3*a/b)*sin(3*a/b)*a+10*Si(arcsin(c*x)+a/b)*cos(a/b)*a-10*Ci(arcsin(c* 
x)+a/b)*sin(a/b)*a+10*(-c^2*x^2+1)^(1/2)*b+cos(5*arcsin(c*x))*b+5*cos(3*ar 
csin(c*x))*b)/(a+b*arcsin(c*x))/b^2
 

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

d**2*(Integral(-2*c**2*x**2/(a**2 + 2*a*b*asin(c*x) + b**2*asin(c*x)**2), 
x) + Integral(c**4*x**4/(a**2 + 2*a*b*asin(c*x) + b**2*asin(c*x)**2), x) + 
 Integral(1/(a**2 + 2*a*b*asin(c*x) + b**2*asin(c*x)**2), x))
 

Maxima [F]

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

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

Output:

-((c^4*d^2*x^4 - 2*c^2*d^2*x^2 + d^2)*sqrt(c*x + 1)*sqrt(-c*x + 1) - (b^2* 
c*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1)) + a*b*c)*integrate(5*(c^3*d^2 
*x^3 - c*d^2*x)*sqrt(c*x + 1)*sqrt(-c*x + 1)/(b^2*arctan2(c*x, sqrt(c*x + 
1)*sqrt(-c*x + 1)) + a*b), 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. 1200 vs. \(2 (222) = 444\).

Time = 0.23 (sec) , antiderivative size = 1200, normalized size of antiderivative = 5.08 \[ \int \frac {\left (d-c^2 d x^2\right )^2}{(a+b \arcsin (c x))^2} \, dx=\text {Too large to display} \] Input:

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

Output:

5*b*d^2*arcsin(c*x)*cos(a/b)^4*cos_integral(5*a/b + 5*arcsin(c*x))*sin(a/b 
)/(b^3*c*arcsin(c*x) + a*b^2*c) - 5*b*d^2*arcsin(c*x)*cos(a/b)^5*sin_integ 
ral(5*a/b + 5*arcsin(c*x))/(b^3*c*arcsin(c*x) + a*b^2*c) + 5*a*d^2*cos(a/b 
)^4*cos_integral(5*a/b + 5*arcsin(c*x))*sin(a/b)/(b^3*c*arcsin(c*x) + a*b^ 
2*c) - 5*a*d^2*cos(a/b)^5*sin_integral(5*a/b + 5*arcsin(c*x))/(b^3*c*arcsi 
n(c*x) + a*b^2*c) - 15/4*b*d^2*arcsin(c*x)*cos(a/b)^2*cos_integral(5*a/b + 
 5*arcsin(c*x))*sin(a/b)/(b^3*c*arcsin(c*x) + a*b^2*c) + 15/4*b*d^2*arcsin 
(c*x)*cos(a/b)^2*cos_integral(3*a/b + 3*arcsin(c*x))*sin(a/b)/(b^3*c*arcsi 
n(c*x) + a*b^2*c) + 25/4*b*d^2*arcsin(c*x)*cos(a/b)^3*sin_integral(5*a/b + 
 5*arcsin(c*x))/(b^3*c*arcsin(c*x) + a*b^2*c) - 15/4*b*d^2*arcsin(c*x)*cos 
(a/b)^3*sin_integral(3*a/b + 3*arcsin(c*x))/(b^3*c*arcsin(c*x) + a*b^2*c) 
- 15/4*a*d^2*cos(a/b)^2*cos_integral(5*a/b + 5*arcsin(c*x))*sin(a/b)/(b^3* 
c*arcsin(c*x) + a*b^2*c) + 15/4*a*d^2*cos(a/b)^2*cos_integral(3*a/b + 3*ar 
csin(c*x))*sin(a/b)/(b^3*c*arcsin(c*x) + a*b^2*c) + 25/4*a*d^2*cos(a/b)^3* 
sin_integral(5*a/b + 5*arcsin(c*x))/(b^3*c*arcsin(c*x) + a*b^2*c) - 15/4*a 
*d^2*cos(a/b)^3*sin_integral(3*a/b + 3*arcsin(c*x))/(b^3*c*arcsin(c*x) + a 
*b^2*c) + 5/16*b*d^2*arcsin(c*x)*cos_integral(5*a/b + 5*arcsin(c*x))*sin(a 
/b)/(b^3*c*arcsin(c*x) + a*b^2*c) - 15/16*b*d^2*arcsin(c*x)*cos_integral(3 
*a/b + 3*arcsin(c*x))*sin(a/b)/(b^3*c*arcsin(c*x) + a*b^2*c) + 5/8*b*d^2*a 
rcsin(c*x)*cos_integral(a/b + arcsin(c*x))*sin(a/b)/(b^3*c*arcsin(c*x) ...
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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