\(\int \frac {(d-c^2 d x^2)^3}{(a+b \arcsin (c x))^2} \, dx\) [32]

Optimal result

Integrand size = 24, antiderivative size = 304 \[ \int \frac {\left (d-c^2 d x^2\right )^3}{(a+b \arcsin (c x))^2} \, dx=-\frac {d^3 \left (1-c^2 x^2\right )^{7/2}}{b c (a+b \arcsin (c x))}+\frac {35 d^3 \operatorname {CosIntegral}\left (\frac {a+b \arcsin (c x)}{b}\right ) \sin \left (\frac {a}{b}\right )}{64 b^2 c}+\frac {63 d^3 \operatorname {CosIntegral}\left (\frac {3 (a+b \arcsin (c x))}{b}\right ) \sin \left (\frac {3 a}{b}\right )}{64 b^2 c}+\frac {35 d^3 \operatorname {CosIntegral}\left (\frac {5 (a+b \arcsin (c x))}{b}\right ) \sin \left (\frac {5 a}{b}\right )}{64 b^2 c}+\frac {7 d^3 \operatorname {CosIntegral}\left (\frac {7 (a+b \arcsin (c x))}{b}\right ) \sin \left (\frac {7 a}{b}\right )}{64 b^2 c}-\frac {35 d^3 \cos \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a+b \arcsin (c x)}{b}\right )}{64 b^2 c}-\frac {63 d^3 \cos \left (\frac {3 a}{b}\right ) \text {Si}\left (\frac {3 (a+b \arcsin (c x))}{b}\right )}{64 b^2 c}-\frac {35 d^3 \cos \left (\frac {5 a}{b}\right ) \text {Si}\left (\frac {5 (a+b \arcsin (c x))}{b}\right )}{64 b^2 c}-\frac {7 d^3 \cos \left (\frac {7 a}{b}\right ) \text {Si}\left (\frac {7 (a+b \arcsin (c x))}{b}\right )}{64 b^2 c} \] Output:

-d^3*(-c^2*x^2+1)^(7/2)/b/c/(a+b*arcsin(c*x))+35/64*d^3*Ci((a+b*arcsin(c*x 
))/b)*sin(a/b)/b^2/c+63/64*d^3*Ci(3*(a+b*arcsin(c*x))/b)*sin(3*a/b)/b^2/c+ 
35/64*d^3*Ci(5*(a+b*arcsin(c*x))/b)*sin(5*a/b)/b^2/c+7/64*d^3*Ci(7*(a+b*ar 
csin(c*x))/b)*sin(7*a/b)/b^2/c-35/64*d^3*cos(a/b)*Si((a+b*arcsin(c*x))/b)/ 
b^2/c-63/64*d^3*cos(3*a/b)*Si(3*(a+b*arcsin(c*x))/b)/b^2/c-35/64*d^3*cos(5 
*a/b)*Si(5*(a+b*arcsin(c*x))/b)/b^2/c-7/64*d^3*cos(7*a/b)*Si(7*(a+b*arcsin 
(c*x))/b)/b^2/c
 

Mathematica [A] (verified)

Time = 1.84 (sec) , antiderivative size = 310, normalized size of antiderivative = 1.02 \[ \int \frac {\left (d-c^2 d x^2\right )^3}{(a+b \arcsin (c x))^2} \, dx=\frac {d^3 \left (-\frac {64 b \sqrt {1-c^2 x^2}}{a+b \arcsin (c x)}+\frac {192 b c^2 x^2 \sqrt {1-c^2 x^2}}{a+b \arcsin (c x)}-\frac {192 b c^4 x^4 \sqrt {1-c^2 x^2}}{a+b \arcsin (c x)}+\frac {64 b c^6 x^6 \sqrt {1-c^2 x^2}}{a+b \arcsin (c x)}+35 \operatorname {CosIntegral}\left (\frac {a}{b}+\arcsin (c x)\right ) \sin \left (\frac {a}{b}\right )+63 \operatorname {CosIntegral}\left (3 \left (\frac {a}{b}+\arcsin (c x)\right )\right ) \sin \left (\frac {3 a}{b}\right )+35 \operatorname {CosIntegral}\left (5 \left (\frac {a}{b}+\arcsin (c x)\right )\right ) \sin \left (\frac {5 a}{b}\right )+7 \operatorname {CosIntegral}\left (7 \left (\frac {a}{b}+\arcsin (c x)\right )\right ) \sin \left (\frac {7 a}{b}\right )-35 \cos \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a}{b}+\arcsin (c x)\right )-63 \cos \left (\frac {3 a}{b}\right ) \text {Si}\left (3 \left (\frac {a}{b}+\arcsin (c x)\right )\right )-35 \cos \left (\frac {5 a}{b}\right ) \text {Si}\left (5 \left (\frac {a}{b}+\arcsin (c x)\right )\right )-7 \cos \left (\frac {7 a}{b}\right ) \text {Si}\left (7 \left (\frac {a}{b}+\arcsin (c x)\right )\right )\right )}{64 b^2 c} \] Input:

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

Output:

(d^3*((-64*b*Sqrt[1 - c^2*x^2])/(a + b*ArcSin[c*x]) + (192*b*c^2*x^2*Sqrt[ 
1 - c^2*x^2])/(a + b*ArcSin[c*x]) - (192*b*c^4*x^4*Sqrt[1 - c^2*x^2])/(a + 
 b*ArcSin[c*x]) + (64*b*c^6*x^6*Sqrt[1 - c^2*x^2])/(a + b*ArcSin[c*x]) + 3 
5*CosIntegral[a/b + ArcSin[c*x]]*Sin[a/b] + 63*CosIntegral[3*(a/b + ArcSin 
[c*x])]*Sin[(3*a)/b] + 35*CosIntegral[5*(a/b + ArcSin[c*x])]*Sin[(5*a)/b] 
+ 7*CosIntegral[7*(a/b + ArcSin[c*x])]*Sin[(7*a)/b] - 35*Cos[a/b]*SinInteg 
ral[a/b + ArcSin[c*x]] - 63*Cos[(3*a)/b]*SinIntegral[3*(a/b + ArcSin[c*x]) 
] - 35*Cos[(5*a)/b]*SinIntegral[5*(a/b + ArcSin[c*x])] - 7*Cos[(7*a)/b]*Si 
nIntegral[7*(a/b + ArcSin[c*x])]))/(64*b^2*c)
 

Rubi [A] (verified)

Time = 0.72 (sec) , antiderivative size = 244, normalized size of antiderivative = 0.80, 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.

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

\(\Big \downarrow \) 5166

\(\displaystyle -\frac {7 c d^3 \int \frac {x \left (1-c^2 x^2\right )^{5/2}}{a+b \arcsin (c x)}dx}{b}-\frac {d^3 \left (1-c^2 x^2\right )^{7/2}}{b c (a+b \arcsin (c x))}\)

\(\Big \downarrow \) 5224

\(\displaystyle -\frac {7 d^3 \int -\frac {\cos ^6\left (\frac {a}{b}-\frac {a+b \arcsin (c x)}{b}\right ) \sin \left (\frac {a}{b}-\frac {a+b \arcsin (c x)}{b}\right )}{a+b \arcsin (c x)}d(a+b \arcsin (c x))}{b^2 c}-\frac {d^3 \left (1-c^2 x^2\right )^{7/2}}{b c (a+b \arcsin (c x))}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {7 d^3 \int \frac {\cos ^6\left (\frac {a}{b}-\frac {a+b \arcsin (c x)}{b}\right ) \sin \left (\frac {a}{b}-\frac {a+b \arcsin (c x)}{b}\right )}{a+b \arcsin (c x)}d(a+b \arcsin (c x))}{b^2 c}-\frac {d^3 \left (1-c^2 x^2\right )^{7/2}}{b c (a+b \arcsin (c x))}\)

\(\Big \downarrow \) 4906

\(\displaystyle \frac {7 d^3 \int \left (\frac {\sin \left (\frac {7 a}{b}-\frac {7 (a+b \arcsin (c x))}{b}\right )}{64 (a+b \arcsin (c x))}+\frac {5 \sin \left (\frac {5 a}{b}-\frac {5 (a+b \arcsin (c x))}{b}\right )}{64 (a+b \arcsin (c x))}+\frac {9 \sin \left (\frac {3 a}{b}-\frac {3 (a+b \arcsin (c x))}{b}\right )}{64 (a+b \arcsin (c x))}+\frac {5 \sin \left (\frac {a}{b}-\frac {a+b \arcsin (c x)}{b}\right )}{64 (a+b \arcsin (c x))}\right )d(a+b \arcsin (c x))}{b^2 c}-\frac {d^3 \left (1-c^2 x^2\right )^{7/2}}{b c (a+b \arcsin (c x))}\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {7 d^3 \left (-\frac {5}{64} \sin \left (\frac {a}{b}\right ) \operatorname {CosIntegral}\left (\frac {a+b \arcsin (c x)}{b}\right )-\frac {9}{64} \sin \left (\frac {3 a}{b}\right ) \operatorname {CosIntegral}\left (\frac {3 (a+b \arcsin (c x))}{b}\right )-\frac {5}{64} \sin \left (\frac {5 a}{b}\right ) \operatorname {CosIntegral}\left (\frac {5 (a+b \arcsin (c x))}{b}\right )-\frac {1}{64} \sin \left (\frac {7 a}{b}\right ) \operatorname {CosIntegral}\left (\frac {7 (a+b \arcsin (c x))}{b}\right )+\frac {5}{64} \cos \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a+b \arcsin (c x)}{b}\right )+\frac {9}{64} \cos \left (\frac {3 a}{b}\right ) \text {Si}\left (\frac {3 (a+b \arcsin (c x))}{b}\right )+\frac {5}{64} \cos \left (\frac {5 a}{b}\right ) \text {Si}\left (\frac {5 (a+b \arcsin (c x))}{b}\right )+\frac {1}{64} \cos \left (\frac {7 a}{b}\right ) \text {Si}\left (\frac {7 (a+b \arcsin (c x))}{b}\right )\right )}{b^2 c}-\frac {d^3 \left (1-c^2 x^2\right )^{7/2}}{b c (a+b \arcsin (c x))}\)

Input:

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

Output:

-((d^3*(1 - c^2*x^2)^(7/2))/(b*c*(a + b*ArcSin[c*x]))) - (7*d^3*((-5*CosIn 
tegral[(a + b*ArcSin[c*x])/b]*Sin[a/b])/64 - (9*CosIntegral[(3*(a + b*ArcS 
in[c*x]))/b]*Sin[(3*a)/b])/64 - (5*CosIntegral[(5*(a + b*ArcSin[c*x]))/b]* 
Sin[(5*a)/b])/64 - (CosIntegral[(7*(a + b*ArcSin[c*x]))/b]*Sin[(7*a)/b])/6 
4 + (5*Cos[a/b]*SinIntegral[(a + b*ArcSin[c*x])/b])/64 + (9*Cos[(3*a)/b]*S 
inIntegral[(3*(a + b*ArcSin[c*x]))/b])/64 + (5*Cos[(5*a)/b]*SinIntegral[(5 
*(a + b*ArcSin[c*x]))/b])/64 + (Cos[(7*a)/b]*SinIntegral[(7*(a + b*ArcSin[ 
c*x]))/b])/64))/(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 = 0.08 (sec) , antiderivative size = 467, normalized size of antiderivative = 1.54

method	result	size
derivativedivides	\(-\frac {d^{3} \left (7 \arcsin \left (c x \right ) \operatorname {Si}\left (7 \arcsin \left (c x \right )+\frac {7 a}{b}\right ) \cos \left (\frac {7 a}{b}\right ) b -7 \arcsin \left (c x \right ) \operatorname {Ci}\left (7 \arcsin \left (c x \right )+\frac {7 a}{b}\right ) \sin \left (\frac {7 a}{b}\right ) b +35 \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 -35 \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 +63 \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 -63 \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 +35 \arcsin \left (c x \right ) \operatorname {Si}\left (\arcsin \left (c x \right )+\frac {a}{b}\right ) \cos \left (\frac {a}{b}\right ) b -35 \arcsin \left (c x \right ) \operatorname {Ci}\left (\arcsin \left (c x \right )+\frac {a}{b}\right ) \sin \left (\frac {a}{b}\right ) b +7 \,\operatorname {Si}\left (7 \arcsin \left (c x \right )+\frac {7 a}{b}\right ) \cos \left (\frac {7 a}{b}\right ) a -7 \,\operatorname {Ci}\left (7 \arcsin \left (c x \right )+\frac {7 a}{b}\right ) \sin \left (\frac {7 a}{b}\right ) a +35 \,\operatorname {Si}\left (5 \arcsin \left (c x \right )+\frac {5 a}{b}\right ) \cos \left (\frac {5 a}{b}\right ) a -35 \,\operatorname {Ci}\left (5 \arcsin \left (c x \right )+\frac {5 a}{b}\right ) \sin \left (\frac {5 a}{b}\right ) a +63 \,\operatorname {Si}\left (3 \arcsin \left (c x \right )+\frac {3 a}{b}\right ) \cos \left (\frac {3 a}{b}\right ) a -63 \,\operatorname {Ci}\left (3 \arcsin \left (c x \right )+\frac {3 a}{b}\right ) \sin \left (\frac {3 a}{b}\right ) a +35 \,\operatorname {Si}\left (\arcsin \left (c x \right )+\frac {a}{b}\right ) \cos \left (\frac {a}{b}\right ) a -35 \,\operatorname {Ci}\left (\arcsin \left (c x \right )+\frac {a}{b}\right ) \sin \left (\frac {a}{b}\right ) a +35 \sqrt {-c^{2} x^{2}+1}\, b +\cos \left (7 \arcsin \left (c x \right )\right ) b +7 \cos \left (5 \arcsin \left (c x \right )\right ) b +21 \cos \left (3 \arcsin \left (c x \right )\right ) b \right )}{64 c \left (a +b \arcsin \left (c x \right )\right ) b^{2}}\)	\(467\)
default	\(-\frac {d^{3} \left (7 \arcsin \left (c x \right ) \operatorname {Si}\left (7 \arcsin \left (c x \right )+\frac {7 a}{b}\right ) \cos \left (\frac {7 a}{b}\right ) b -7 \arcsin \left (c x \right ) \operatorname {Ci}\left (7 \arcsin \left (c x \right )+\frac {7 a}{b}\right ) \sin \left (\frac {7 a}{b}\right ) b +35 \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 -35 \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 +63 \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 -63 \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 +35 \arcsin \left (c x \right ) \operatorname {Si}\left (\arcsin \left (c x \right )+\frac {a}{b}\right ) \cos \left (\frac {a}{b}\right ) b -35 \arcsin \left (c x \right ) \operatorname {Ci}\left (\arcsin \left (c x \right )+\frac {a}{b}\right ) \sin \left (\frac {a}{b}\right ) b +7 \,\operatorname {Si}\left (7 \arcsin \left (c x \right )+\frac {7 a}{b}\right ) \cos \left (\frac {7 a}{b}\right ) a -7 \,\operatorname {Ci}\left (7 \arcsin \left (c x \right )+\frac {7 a}{b}\right ) \sin \left (\frac {7 a}{b}\right ) a +35 \,\operatorname {Si}\left (5 \arcsin \left (c x \right )+\frac {5 a}{b}\right ) \cos \left (\frac {5 a}{b}\right ) a -35 \,\operatorname {Ci}\left (5 \arcsin \left (c x \right )+\frac {5 a}{b}\right ) \sin \left (\frac {5 a}{b}\right ) a +63 \,\operatorname {Si}\left (3 \arcsin \left (c x \right )+\frac {3 a}{b}\right ) \cos \left (\frac {3 a}{b}\right ) a -63 \,\operatorname {Ci}\left (3 \arcsin \left (c x \right )+\frac {3 a}{b}\right ) \sin \left (\frac {3 a}{b}\right ) a +35 \,\operatorname {Si}\left (\arcsin \left (c x \right )+\frac {a}{b}\right ) \cos \left (\frac {a}{b}\right ) a -35 \,\operatorname {Ci}\left (\arcsin \left (c x \right )+\frac {a}{b}\right ) \sin \left (\frac {a}{b}\right ) a +35 \sqrt {-c^{2} x^{2}+1}\, b +\cos \left (7 \arcsin \left (c x \right )\right ) b +7 \cos \left (5 \arcsin \left (c x \right )\right ) b +21 \cos \left (3 \arcsin \left (c x \right )\right ) b \right )}{64 c \left (a +b \arcsin \left (c x \right )\right ) b^{2}}\)	\(467\)

Input:

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

Output:

-1/64/c*d^3*(7*arcsin(c*x)*Si(7*arcsin(c*x)+7*a/b)*cos(7*a/b)*b-7*arcsin(c 
*x)*Ci(7*arcsin(c*x)+7*a/b)*sin(7*a/b)*b+35*arcsin(c*x)*Si(5*arcsin(c*x)+5 
*a/b)*cos(5*a/b)*b-35*arcsin(c*x)*Ci(5*arcsin(c*x)+5*a/b)*sin(5*a/b)*b+63* 
arcsin(c*x)*Si(3*arcsin(c*x)+3*a/b)*cos(3*a/b)*b-63*arcsin(c*x)*Ci(3*arcsi 
n(c*x)+3*a/b)*sin(3*a/b)*b+35*arcsin(c*x)*Si(arcsin(c*x)+a/b)*cos(a/b)*b-3 
5*arcsin(c*x)*Ci(arcsin(c*x)+a/b)*sin(a/b)*b+7*Si(7*arcsin(c*x)+7*a/b)*cos 
(7*a/b)*a-7*Ci(7*arcsin(c*x)+7*a/b)*sin(7*a/b)*a+35*Si(5*arcsin(c*x)+5*a/b 
)*cos(5*a/b)*a-35*Ci(5*arcsin(c*x)+5*a/b)*sin(5*a/b)*a+63*Si(3*arcsin(c*x) 
+3*a/b)*cos(3*a/b)*a-63*Ci(3*arcsin(c*x)+3*a/b)*sin(3*a/b)*a+35*Si(arcsin( 
c*x)+a/b)*cos(a/b)*a-35*Ci(arcsin(c*x)+a/b)*sin(a/b)*a+35*(-c^2*x^2+1)^(1/ 
2)*b+cos(7*arcsin(c*x))*b+7*cos(5*arcsin(c*x))*b+21*cos(3*arcsin(c*x))*b)/ 
(a+b*arcsin(c*x))/b^2
 

Fricas [F]

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

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

Output:

integral(-(c^6*d^3*x^6 - 3*c^4*d^3*x^4 + 3*c^2*d^3*x^2 - d^3)/(b^2*arcsin( 
c*x)^2 + 2*a*b*arcsin(c*x) + a^2), x)
 

Sympy [F]

\[ \int \frac {\left (d-c^2 d x^2\right )^3}{(a+b \arcsin (c x))^2} \, dx=- d^{3} \left (\int \frac {3 c^{2} x^{2}}{a^{2} + 2 a b \operatorname {asin}{\left (c x \right )} + b^{2} \operatorname {asin}^{2}{\left (c x \right )}}\, dx + \int \left (- \frac {3 c^{4} x^{4}}{a^{2} + 2 a b \operatorname {asin}{\left (c x \right )} + b^{2} \operatorname {asin}^{2}{\left (c x \right )}}\right )\, dx + \int \frac {c^{6} x^{6}}{a^{2} + 2 a b \operatorname {asin}{\left (c x \right )} + b^{2} \operatorname {asin}^{2}{\left (c x \right )}}\, dx + \int \left (- \frac {1}{a^{2} + 2 a b \operatorname {asin}{\left (c x \right )} + b^{2} \operatorname {asin}^{2}{\left (c x \right )}}\right )\, dx\right ) \] Input:

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

Output:

-d**3*(Integral(3*c**2*x**2/(a**2 + 2*a*b*asin(c*x) + b**2*asin(c*x)**2), 
x) + Integral(-3*c**4*x**4/(a**2 + 2*a*b*asin(c*x) + b**2*asin(c*x)**2), x 
) + Integral(c**6*x**6/(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]

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

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

Output:

((c^6*d^3*x^6 - 3*c^4*d^3*x^4 + 3*c^2*d^3*x^2 - d^3)*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)*integ 
rate(7*(c^5*d^3*x^5 - 2*c^3*d^3*x^3 + c*d^3*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*arcta 
n2(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. 1995 vs. \(2 (286) = 572\).

Time = 0.25 (sec) , antiderivative size = 1995, normalized size of antiderivative = 6.56 \[ \int \frac {\left (d-c^2 d x^2\right )^3}{(a+b \arcsin (c x))^2} \, dx=\text {Too large to display} \] Input:

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

Output:

7*b*d^3*arcsin(c*x)*cos(a/b)^6*cos_integral(7*a/b + 7*arcsin(c*x))*sin(a/b 
)/(b^3*c*arcsin(c*x) + a*b^2*c) - 7*b*d^3*arcsin(c*x)*cos(a/b)^7*sin_integ 
ral(7*a/b + 7*arcsin(c*x))/(b^3*c*arcsin(c*x) + a*b^2*c) + 7*a*d^3*cos(a/b 
)^6*cos_integral(7*a/b + 7*arcsin(c*x))*sin(a/b)/(b^3*c*arcsin(c*x) + a*b^ 
2*c) - 7*a*d^3*cos(a/b)^7*sin_integral(7*a/b + 7*arcsin(c*x))/(b^3*c*arcsi 
n(c*x) + a*b^2*c) - 35/4*b*d^3*arcsin(c*x)*cos(a/b)^4*cos_integral(7*a/b + 
 7*arcsin(c*x))*sin(a/b)/(b^3*c*arcsin(c*x) + a*b^2*c) + 35/4*b*d^3*arcsin 
(c*x)*cos(a/b)^4*cos_integral(5*a/b + 5*arcsin(c*x))*sin(a/b)/(b^3*c*arcsi 
n(c*x) + a*b^2*c) + 49/4*b*d^3*arcsin(c*x)*cos(a/b)^5*sin_integral(7*a/b + 
 7*arcsin(c*x))/(b^3*c*arcsin(c*x) + a*b^2*c) - 35/4*b*d^3*arcsin(c*x)*cos 
(a/b)^5*sin_integral(5*a/b + 5*arcsin(c*x))/(b^3*c*arcsin(c*x) + a*b^2*c) 
- 35/4*a*d^3*cos(a/b)^4*cos_integral(7*a/b + 7*arcsin(c*x))*sin(a/b)/(b^3* 
c*arcsin(c*x) + a*b^2*c) + 35/4*a*d^3*cos(a/b)^4*cos_integral(5*a/b + 5*ar 
csin(c*x))*sin(a/b)/(b^3*c*arcsin(c*x) + a*b^2*c) + 49/4*a*d^3*cos(a/b)^5* 
sin_integral(7*a/b + 7*arcsin(c*x))/(b^3*c*arcsin(c*x) + a*b^2*c) - 35/4*a 
*d^3*cos(a/b)^5*sin_integral(5*a/b + 5*arcsin(c*x))/(b^3*c*arcsin(c*x) + a 
*b^2*c) + 21/8*b*d^3*arcsin(c*x)*cos(a/b)^2*cos_integral(7*a/b + 7*arcsin( 
c*x))*sin(a/b)/(b^3*c*arcsin(c*x) + a*b^2*c) - 105/16*b*d^3*arcsin(c*x)*co 
s(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) + 63/16*b*d^3*arcsin(c*x)*cos(a/b)^2*cos_integral(3*a/b + 3*a...
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {\left (d-c^2 d x^2\right )^3}{(a+b \arcsin (c x))^2} \, dx=d^{3} \left (-\left (\int \frac {x^{6}}{\mathit {asin} \left (c x \right )^{2} b^{2}+2 \mathit {asin} \left (c x \right ) a b +a^{2}}d x \right ) c^{6}+3 \left (\int \frac {x^{4}}{\mathit {asin} \left (c x \right )^{2} b^{2}+2 \mathit {asin} \left (c x \right ) a b +a^{2}}d x \right ) c^{4}-3 \left (\int \frac {x^{2}}{\mathit {asin} \left (c x \right )^{2} b^{2}+2 \mathit {asin} \left (c x \right ) a b +a^{2}}d x \right ) c^{2}+\int \frac {1}{\mathit {asin} \left (c x \right )^{2} b^{2}+2 \mathit {asin} \left (c x \right ) a b +a^{2}}d x \right ) \] Input:

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

Output:

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