\(\int \frac {(d-c^2 d x^2)^2}{a+b \arcsin (c x)} \, dx\) [28]

Optimal result

Integrand size = 24, antiderivative size = 201 \[ \int \frac {\left (d-c^2 d x^2\right )^2}{a+b \arcsin (c x)} \, dx=\frac {5 d^2 \cos \left (\frac {a}{b}\right ) \operatorname {CosIntegral}\left (\frac {a+b \arcsin (c x)}{b}\right )}{8 b c}+\frac {5 d^2 \cos \left (\frac {3 a}{b}\right ) \operatorname {CosIntegral}\left (\frac {3 (a+b \arcsin (c x))}{b}\right )}{16 b c}+\frac {d^2 \cos \left (\frac {5 a}{b}\right ) \operatorname {CosIntegral}\left (\frac {5 (a+b \arcsin (c x))}{b}\right )}{16 b c}+\frac {5 d^2 \sin \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a+b \arcsin (c x)}{b}\right )}{8 b c}+\frac {5 d^2 \sin \left (\frac {3 a}{b}\right ) \text {Si}\left (\frac {3 (a+b \arcsin (c x))}{b}\right )}{16 b c}+\frac {d^2 \sin \left (\frac {5 a}{b}\right ) \text {Si}\left (\frac {5 (a+b \arcsin (c x))}{b}\right )}{16 b c} \] Output:

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

Mathematica [A] (verified)

Time = 0.48 (sec) , antiderivative size = 138, normalized size of antiderivative = 0.69 \[ \int \frac {\left (d-c^2 d x^2\right )^2}{a+b \arcsin (c x)} \, dx=\frac {d^2 \left (10 \cos \left (\frac {a}{b}\right ) \operatorname {CosIntegral}\left (\frac {a}{b}+\arcsin (c x)\right )+5 \cos \left (\frac {3 a}{b}\right ) \operatorname {CosIntegral}\left (3 \left (\frac {a}{b}+\arcsin (c x)\right )\right )+\cos \left (\frac {5 a}{b}\right ) \operatorname {CosIntegral}\left (5 \left (\frac {a}{b}+\arcsin (c x)\right )\right )+10 \sin \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a}{b}+\arcsin (c x)\right )+5 \sin \left (\frac {3 a}{b}\right ) \text {Si}\left (3 \left (\frac {a}{b}+\arcsin (c x)\right )\right )+\sin \left (\frac {5 a}{b}\right ) \text {Si}\left (5 \left (\frac {a}{b}+\arcsin (c x)\right )\right )\right )}{16 b c} \] Input:

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

Output:

(d^2*(10*Cos[a/b]*CosIntegral[a/b + ArcSin[c*x]] + 5*Cos[(3*a)/b]*CosInteg 
ral[3*(a/b + ArcSin[c*x])] + Cos[(5*a)/b]*CosIntegral[5*(a/b + ArcSin[c*x] 
)] + 10*Sin[a/b]*SinIntegral[a/b + ArcSin[c*x]] + 5*Sin[(3*a)/b]*SinIntegr 
al[3*(a/b + ArcSin[c*x])] + Sin[(5*a)/b]*SinIntegral[5*(a/b + ArcSin[c*x]) 
]))/(16*b*c)
 

Rubi [A] (verified)

Time = 0.52 (sec) , antiderivative size = 157, normalized size of antiderivative = 0.78, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {5168, 3042, 3793, 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.

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

\(\Big \downarrow \) 5168

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3793

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

(d^2*((5*Cos[a/b]*CosIntegral[(a + b*ArcSin[c*x])/b])/8 + (5*Cos[(3*a)/b]* 
CosIntegral[(3*(a + b*ArcSin[c*x]))/b])/16 + (Cos[(5*a)/b]*CosIntegral[(5* 
(a + b*ArcSin[c*x]))/b])/16 + (5*Sin[a/b]*SinIntegral[(a + b*ArcSin[c*x])/ 
b])/8 + (5*Sin[(3*a)/b]*SinIntegral[(3*(a + b*ArcSin[c*x]))/b])/16 + (Sin[ 
(5*a)/b]*SinIntegral[(5*(a + b*ArcSin[c*x]))/b])/16))/(b*c)
 

Defintions of rubi rules used

rule 2009

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

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 3793

Int[((c_.) + (d_.)*(x_))^(m_)*sin[(e_.) + (f_.)*(x_)]^(n_), x_Symbol] :> In 
t[ExpandTrigReduce[(c + d*x)^m, Sin[e + f*x]^n, x], x] /; FreeQ[{c, d, e, f 
, m}, x] && IGtQ[n, 1] && ( !RationalQ[m] || (GeQ[m, -1] && LtQ[m, 1]))
 

rule 5168

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((d_) + (e_.)*(x_)^2)^(p_.), x 
_Symbol] :> Simp[(1/(b*c))*Simp[(d + e*x^2)^p/(1 - c^2*x^2)^p]   Subst[Int[ 
x^n*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, 0]
 

Maple [A] (verified)

Time = 0.08 (sec) , antiderivative size = 141, normalized size of antiderivative = 0.70

method	result	size
derivativedivides	\(\frac {d^{2} \left (\operatorname {Si}\left (5 \arcsin \left (c x \right )+\frac {5 a}{b}\right ) \sin \left (\frac {5 a}{b}\right )+\operatorname {Ci}\left (5 \arcsin \left (c x \right )+\frac {5 a}{b}\right ) \cos \left (\frac {5 a}{b}\right )+5 \,\operatorname {Si}\left (3 \arcsin \left (c x \right )+\frac {3 a}{b}\right ) \sin \left (\frac {3 a}{b}\right )+5 \,\operatorname {Ci}\left (3 \arcsin \left (c x \right )+\frac {3 a}{b}\right ) \cos \left (\frac {3 a}{b}\right )+10 \,\operatorname {Si}\left (\arcsin \left (c x \right )+\frac {a}{b}\right ) \sin \left (\frac {a}{b}\right )+10 \,\operatorname {Ci}\left (\arcsin \left (c x \right )+\frac {a}{b}\right ) \cos \left (\frac {a}{b}\right )\right )}{16 c b}\)	\(141\)
default	\(\frac {d^{2} \left (\operatorname {Si}\left (5 \arcsin \left (c x \right )+\frac {5 a}{b}\right ) \sin \left (\frac {5 a}{b}\right )+\operatorname {Ci}\left (5 \arcsin \left (c x \right )+\frac {5 a}{b}\right ) \cos \left (\frac {5 a}{b}\right )+5 \,\operatorname {Si}\left (3 \arcsin \left (c x \right )+\frac {3 a}{b}\right ) \sin \left (\frac {3 a}{b}\right )+5 \,\operatorname {Ci}\left (3 \arcsin \left (c x \right )+\frac {3 a}{b}\right ) \cos \left (\frac {3 a}{b}\right )+10 \,\operatorname {Si}\left (\arcsin \left (c x \right )+\frac {a}{b}\right ) \sin \left (\frac {a}{b}\right )+10 \,\operatorname {Ci}\left (\arcsin \left (c x \right )+\frac {a}{b}\right ) \cos \left (\frac {a}{b}\right )\right )}{16 c b}\)	\(141\)

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 395 vs. \(2 (189) = 378\).

Time = 0.15 (sec) , antiderivative size = 395, normalized size of antiderivative = 1.97 \[ \int \frac {\left (d-c^2 d x^2\right )^2}{a+b \arcsin (c x)} \, dx=\frac {d^{2} \cos \left (\frac {a}{b}\right )^{5} \operatorname {Ci}\left (\frac {5 \, a}{b} + 5 \, \arcsin \left (c x\right )\right )}{b c} + \frac {d^{2} \cos \left (\frac {a}{b}\right )^{4} \sin \left (\frac {a}{b}\right ) \operatorname {Si}\left (\frac {5 \, a}{b} + 5 \, \arcsin \left (c x\right )\right )}{b c} - \frac {5 \, d^{2} \cos \left (\frac {a}{b}\right )^{3} \operatorname {Ci}\left (\frac {5 \, a}{b} + 5 \, \arcsin \left (c x\right )\right )}{4 \, b c} + \frac {5 \, d^{2} \cos \left (\frac {a}{b}\right )^{3} \operatorname {Ci}\left (\frac {3 \, a}{b} + 3 \, \arcsin \left (c x\right )\right )}{4 \, b c} - \frac {3 \, d^{2} \cos \left (\frac {a}{b}\right )^{2} \sin \left (\frac {a}{b}\right ) \operatorname {Si}\left (\frac {5 \, a}{b} + 5 \, \arcsin \left (c x\right )\right )}{4 \, b c} + \frac {5 \, d^{2} \cos \left (\frac {a}{b}\right )^{2} \sin \left (\frac {a}{b}\right ) \operatorname {Si}\left (\frac {3 \, a}{b} + 3 \, \arcsin \left (c x\right )\right )}{4 \, b c} + \frac {5 \, d^{2} \cos \left (\frac {a}{b}\right ) \operatorname {Ci}\left (\frac {5 \, a}{b} + 5 \, \arcsin \left (c x\right )\right )}{16 \, b c} - \frac {15 \, d^{2} \cos \left (\frac {a}{b}\right ) \operatorname {Ci}\left (\frac {3 \, a}{b} + 3 \, \arcsin \left (c x\right )\right )}{16 \, b c} + \frac {5 \, d^{2} \cos \left (\frac {a}{b}\right ) \operatorname {Ci}\left (\frac {a}{b} + \arcsin \left (c x\right )\right )}{8 \, b c} + \frac {d^{2} \sin \left (\frac {a}{b}\right ) \operatorname {Si}\left (\frac {5 \, a}{b} + 5 \, \arcsin \left (c x\right )\right )}{16 \, b c} - \frac {5 \, d^{2} \sin \left (\frac {a}{b}\right ) \operatorname {Si}\left (\frac {3 \, a}{b} + 3 \, \arcsin \left (c x\right )\right )}{16 \, b c} + \frac {5 \, d^{2} \sin \left (\frac {a}{b}\right ) \operatorname {Si}\left (\frac {a}{b} + \arcsin \left (c x\right )\right )}{8 \, b c} \] Input:

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

Output:

d^2*cos(a/b)^5*cos_integral(5*a/b + 5*arcsin(c*x))/(b*c) + d^2*cos(a/b)^4* 
sin(a/b)*sin_integral(5*a/b + 5*arcsin(c*x))/(b*c) - 5/4*d^2*cos(a/b)^3*co 
s_integral(5*a/b + 5*arcsin(c*x))/(b*c) + 5/4*d^2*cos(a/b)^3*cos_integral( 
3*a/b + 3*arcsin(c*x))/(b*c) - 3/4*d^2*cos(a/b)^2*sin(a/b)*sin_integral(5* 
a/b + 5*arcsin(c*x))/(b*c) + 5/4*d^2*cos(a/b)^2*sin(a/b)*sin_integral(3*a/ 
b + 3*arcsin(c*x))/(b*c) + 5/16*d^2*cos(a/b)*cos_integral(5*a/b + 5*arcsin 
(c*x))/(b*c) - 15/16*d^2*cos(a/b)*cos_integral(3*a/b + 3*arcsin(c*x))/(b*c 
) + 5/8*d^2*cos(a/b)*cos_integral(a/b + arcsin(c*x))/(b*c) + 1/16*d^2*sin( 
a/b)*sin_integral(5*a/b + 5*arcsin(c*x))/(b*c) - 5/16*d^2*sin(a/b)*sin_int 
egral(3*a/b + 3*arcsin(c*x))/(b*c) + 5/8*d^2*sin(a/b)*sin_integral(a/b + a 
rcsin(c*x))/(b*c)
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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