\(\int \frac {(c-a^2 c x^2)^3}{\arcsin (a x)^2} \, dx\) [22]

Optimal result

Integrand size = 20, antiderivative size = 95 \[ \int \frac {\left (c-a^2 c x^2\right )^3}{\arcsin (a x)^2} \, dx=-\frac {c^3 \left (1-a^2 x^2\right )^{7/2}}{a \arcsin (a x)}-\frac {35 c^3 \text {Si}(\arcsin (a x))}{64 a}-\frac {63 c^3 \text {Si}(3 \arcsin (a x))}{64 a}-\frac {35 c^3 \text {Si}(5 \arcsin (a x))}{64 a}-\frac {7 c^3 \text {Si}(7 \arcsin (a x))}{64 a} \] Output:

-c^3*(-a^2*x^2+1)^(7/2)/a/arcsin(a*x)-35/64*c^3*Si(arcsin(a*x))/a-63/64*c^ 
3*Si(3*arcsin(a*x))/a-35/64*c^3*Si(5*arcsin(a*x))/a-7/64*c^3*Si(7*arcsin(a 
*x))/a
 

Mathematica [A] (verified)

Time = 0.40 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.87 \[ \int \frac {\left (c-a^2 c x^2\right )^3}{\arcsin (a x)^2} \, dx=-\frac {c^3 \left (64 \left (1-a^2 x^2\right )^{7/2}+35 \arcsin (a x) \text {Si}(\arcsin (a x))+63 \arcsin (a x) \text {Si}(3 \arcsin (a x))+35 \arcsin (a x) \text {Si}(5 \arcsin (a x))+7 \arcsin (a x) \text {Si}(7 \arcsin (a x))\right )}{64 a \arcsin (a x)} \] Input:

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

Output:

-1/64*(c^3*(64*(1 - a^2*x^2)^(7/2) + 35*ArcSin[a*x]*SinIntegral[ArcSin[a*x 
]] + 63*ArcSin[a*x]*SinIntegral[3*ArcSin[a*x]] + 35*ArcSin[a*x]*SinIntegra 
l[5*ArcSin[a*x]] + 7*ArcSin[a*x]*SinIntegral[7*ArcSin[a*x]]))/(a*ArcSin[a* 
x])
 

Rubi [A] (verified)

Time = 0.48 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.84, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {5166, 5224, 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.

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

\(\Big \downarrow \) 5166

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

\(\Big \downarrow \) 5224

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

\(\Big \downarrow \) 4906

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

-((c^3*(1 - a^2*x^2)^(7/2))/(a*ArcSin[a*x])) - (7*c^3*((5*SinIntegral[ArcS 
in[a*x]])/64 + (9*SinIntegral[3*ArcSin[a*x]])/64 + (5*SinIntegral[5*ArcSin 
[a*x]])/64 + SinIntegral[7*ArcSin[a*x]]/64))/a
 

Defintions of rubi rules used

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 = 105, normalized size of antiderivative = 1.11

method	result	size
derivativedivides	\(-\frac {c^{3} \left (35 \,\operatorname {Si}\left (\arcsin \left (a x \right )\right ) \arcsin \left (a x \right )+63 \,\operatorname {Si}\left (3 \arcsin \left (a x \right )\right ) \arcsin \left (a x \right )+35 \,\operatorname {Si}\left (5 \arcsin \left (a x \right )\right ) \arcsin \left (a x \right )+7 \,\operatorname {Si}\left (7 \arcsin \left (a x \right )\right ) \arcsin \left (a x \right )+35 \sqrt {-a^{2} x^{2}+1}+21 \cos \left (3 \arcsin \left (a x \right )\right )+7 \cos \left (5 \arcsin \left (a x \right )\right )+\cos \left (7 \arcsin \left (a x \right )\right )\right )}{64 a \arcsin \left (a x \right )}\)	\(105\)
default	\(-\frac {c^{3} \left (35 \,\operatorname {Si}\left (\arcsin \left (a x \right )\right ) \arcsin \left (a x \right )+63 \,\operatorname {Si}\left (3 \arcsin \left (a x \right )\right ) \arcsin \left (a x \right )+35 \,\operatorname {Si}\left (5 \arcsin \left (a x \right )\right ) \arcsin \left (a x \right )+7 \,\operatorname {Si}\left (7 \arcsin \left (a x \right )\right ) \arcsin \left (a x \right )+35 \sqrt {-a^{2} x^{2}+1}+21 \cos \left (3 \arcsin \left (a x \right )\right )+7 \cos \left (5 \arcsin \left (a x \right )\right )+\cos \left (7 \arcsin \left (a x \right )\right )\right )}{64 a \arcsin \left (a x \right )}\)	\(105\)

Input:

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

Output:

-1/64/a*c^3*(35*Si(arcsin(a*x))*arcsin(a*x)+63*Si(3*arcsin(a*x))*arcsin(a* 
x)+35*Si(5*arcsin(a*x))*arcsin(a*x)+7*Si(7*arcsin(a*x))*arcsin(a*x)+35*(-a 
^2*x^2+1)^(1/2)+21*cos(3*arcsin(a*x))+7*cos(5*arcsin(a*x))+cos(7*arcsin(a* 
x)))/arcsin(a*x)
 

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

-c**3*(Integral(3*a**2*x**2/asin(a*x)**2, x) + Integral(-3*a**4*x**4/asin( 
a*x)**2, x) + Integral(a**6*x**6/asin(a*x)**2, x) + Integral(-1/asin(a*x)* 
*2, x))
 

Maxima [F]

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

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

Output:

-(a*arctan2(a*x, sqrt(a*x + 1)*sqrt(-a*x + 1))*integrate(7*(a^5*c^3*x^5 - 
2*a^3*c^3*x^3 + a*c^3*x)*sqrt(a*x + 1)*sqrt(-a*x + 1)/arctan2(a*x, sqrt(a* 
x + 1)*sqrt(-a*x + 1)), x) - (a^6*c^3*x^6 - 3*a^4*c^3*x^4 + 3*a^2*c^3*x^2 
- c^3)*sqrt(a*x + 1)*sqrt(-a*x + 1))/(a*arctan2(a*x, sqrt(a*x + 1)*sqrt(-a 
*x + 1)))
 

Giac [A] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 95, normalized size of antiderivative = 1.00 \[ \int \frac {\left (c-a^2 c x^2\right )^3}{\arcsin (a x)^2} \, dx=\frac {{\left (a^{2} x^{2} - 1\right )}^{3} \sqrt {-a^{2} x^{2} + 1} c^{3}}{a \arcsin \left (a x\right )} - \frac {7 \, c^{3} \operatorname {Si}\left (7 \, \arcsin \left (a x\right )\right )}{64 \, a} - \frac {35 \, c^{3} \operatorname {Si}\left (5 \, \arcsin \left (a x\right )\right )}{64 \, a} - \frac {63 \, c^{3} \operatorname {Si}\left (3 \, \arcsin \left (a x\right )\right )}{64 \, a} - \frac {35 \, c^{3} \operatorname {Si}\left (\arcsin \left (a x\right )\right )}{64 \, a} \] Input:

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

Output:

(a^2*x^2 - 1)^3*sqrt(-a^2*x^2 + 1)*c^3/(a*arcsin(a*x)) - 7/64*c^3*sin_inte 
gral(7*arcsin(a*x))/a - 35/64*c^3*sin_integral(5*arcsin(a*x))/a - 63/64*c^ 
3*sin_integral(3*arcsin(a*x))/a - 35/64*c^3*sin_integral(arcsin(a*x))/a
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

c**3*( - int(x**6/asin(a*x)**2,x)*a**6 + 3*int(x**4/asin(a*x)**2,x)*a**4 - 
 3*int(x**2/asin(a*x)**2,x)*a**2 + int(1/asin(a*x)**2,x))