\(\int \frac {x}{\sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2} \, dx\) [376]

Optimal result

Integrand size = 26, antiderivative size = 72 \[ \int \frac {x}{\sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2} \, dx=-\frac {x}{b c (a+b \arcsin (c x))}+\frac {\cos \left (\frac {a}{b}\right ) \operatorname {CosIntegral}\left (\frac {a+b \arcsin (c x)}{b}\right )}{b^2 c^2}+\frac {\sin \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a+b \arcsin (c x)}{b}\right )}{b^2 c^2} \] Output:

-x/b/c/(a+b*arcsin(c*x))+cos(a/b)*Ci((a+b*arcsin(c*x))/b)/b^2/c^2+sin(a/b) 
*Si((a+b*arcsin(c*x))/b)/b^2/c^2
 

Mathematica [A] (verified)

Time = 0.10 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.82 \[ \int \frac {x}{\sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2} \, dx=\frac {-\frac {b c x}{a+b \arcsin (c x)}+\cos \left (\frac {a}{b}\right ) \operatorname {CosIntegral}\left (\frac {a}{b}+\arcsin (c x)\right )+\sin \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a}{b}+\arcsin (c x)\right )}{b^2 c^2} \] Input:

Integrate[x/(Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^2),x]
 

Output:

(-((b*c*x)/(a + b*ArcSin[c*x])) + Cos[a/b]*CosIntegral[a/b + ArcSin[c*x]] 
+ Sin[a/b]*SinIntegral[a/b + ArcSin[c*x]])/(b^2*c^2)
 

Rubi [A] (verified)

Time = 0.57 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.94, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {5222, 5134, 3042, 3784, 25, 3042, 3780, 3783}

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.

\(\displaystyle \int \frac {x}{\sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2} \, dx\)

\(\Big \downarrow \) 5222

\(\displaystyle \frac {\int \frac {1}{a+b \arcsin (c x)}dx}{b c}-\frac {x}{b c (a+b \arcsin (c x))}\)

\(\Big \downarrow \) 5134

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3784

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3780

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

\(\Big \downarrow \) 3783

\(\displaystyle \frac {\cos \left (\frac {a}{b}\right ) \operatorname {CosIntegral}\left (\frac {a+b \arcsin (c x)}{b}\right )+\sin \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a+b \arcsin (c x)}{b}\right )}{b^2 c^2}-\frac {x}{b c (a+b \arcsin (c x))}\)

Input:

Int[x/(Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^2),x]
 

Output:

-(x/(b*c*(a + b*ArcSin[c*x]))) + (Cos[a/b]*CosIntegral[(a + b*ArcSin[c*x]) 
/b] + Sin[a/b]*SinIntegral[(a + b*ArcSin[c*x])/b])/(b^2*c^2)
 

Defintions of rubi rules used

rule 25

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

rule 3042

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

rule 3780

Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[SinInte 
gral[e + f*x]/d, x] /; FreeQ[{c, d, e, f}, x] && EqQ[d*e - c*f, 0]
 

rule 3783

Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[CosInte 
gral[e - Pi/2 + f*x]/d, x] /; FreeQ[{c, d, e, f}, x] && EqQ[d*(e - Pi/2) - 
c*f, 0]
 

rule 3784

Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[Cos[(d* 
e - c*f)/d]   Int[Sin[c*(f/d) + f*x]/(c + d*x), x], x] + Simp[Sin[(d*e - c* 
f)/d]   Int[Cos[c*(f/d) + f*x]/(c + d*x), x], x] /; FreeQ[{c, d, e, f}, x] 
&& NeQ[d*e - c*f, 0]
 

rule 5134

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Simp[1/(b*c)   Su 
bst[Int[x^n*Cos[-a/b + x/b], x], x, a + b*ArcSin[c*x]], x] /; FreeQ[{a, b, 
c, n}, x]
 

rule 5222

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

Maple [A] (verified)

Time = 0.44 (sec) , antiderivative size = 108, normalized size of antiderivative = 1.50

method	result	size
default	\(\frac {\arcsin \left (c x \right ) \operatorname {Si}\left (\arcsin \left (c x \right )+\frac {a}{b}\right ) \sin \left (\frac {a}{b}\right ) b +\arcsin \left (c x \right ) \operatorname {Ci}\left (\arcsin \left (c x \right )+\frac {a}{b}\right ) \cos \left (\frac {a}{b}\right ) b +\operatorname {Si}\left (\arcsin \left (c x \right )+\frac {a}{b}\right ) \sin \left (\frac {a}{b}\right ) a +\operatorname {Ci}\left (\arcsin \left (c x \right )+\frac {a}{b}\right ) \cos \left (\frac {a}{b}\right ) a -x b c}{c^{2} \left (a +b \arcsin \left (c x \right )\right ) b^{2}}\)	\(108\)

Input:

int(x/(-c^2*x^2+1)^(1/2)/(a+b*arcsin(c*x))^2,x,method=_RETURNVERBOSE)
 

Output:

1/c^2*(arcsin(c*x)*Si(arcsin(c*x)+a/b)*sin(a/b)*b+arcsin(c*x)*Ci(arcsin(c* 
x)+a/b)*cos(a/b)*b+Si(arcsin(c*x)+a/b)*sin(a/b)*a+Ci(arcsin(c*x)+a/b)*cos( 
a/b)*a-x*b*c)/(a+b*arcsin(c*x))/b^2
 

Fricas [F]

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

integrate(x/(-c^2*x^2+1)^(1/2)/(a+b*arcsin(c*x))^2,x, algorithm="fricas")
 

Output:

integral(-sqrt(-c^2*x^2 + 1)*x/(a^2*c^2*x^2 + (b^2*c^2*x^2 - b^2)*arcsin(c 
*x)^2 - a^2 + 2*(a*b*c^2*x^2 - a*b)*arcsin(c*x)), x)
 

Sympy [F]

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

integrate(x/(-c**2*x**2+1)**(1/2)/(a+b*asin(c*x))**2,x)
 

Output:

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

Maxima [F]

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

integrate(x/(-c^2*x^2+1)^(1/2)/(a+b*arcsin(c*x))^2,x, algorithm="maxima")
                                                                                    
                                                                                    
 

Output:

((b^2*c*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1)) + a*b*c)*integrate(1/(b 
^2*c*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1)) + a*b*c), x) - x)/(b^2*c*a 
rctan2(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. 200 vs. \(2 (72) = 144\).

Time = 0.22 (sec) , antiderivative size = 200, normalized size of antiderivative = 2.78 \[ \int \frac {x}{\sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2} \, dx=\frac {b \arcsin \left (c x\right ) \cos \left (\frac {a}{b}\right ) \operatorname {Ci}\left (\frac {a}{b} + \arcsin \left (c x\right )\right )}{b^{3} c^{2} \arcsin \left (c x\right ) + a b^{2} c^{2}} + \frac {b \arcsin \left (c x\right ) \sin \left (\frac {a}{b}\right ) \operatorname {Si}\left (\frac {a}{b} + \arcsin \left (c x\right )\right )}{b^{3} c^{2} \arcsin \left (c x\right ) + a b^{2} c^{2}} - \frac {b c x}{b^{3} c^{2} \arcsin \left (c x\right ) + a b^{2} c^{2}} + \frac {a \cos \left (\frac {a}{b}\right ) \operatorname {Ci}\left (\frac {a}{b} + \arcsin \left (c x\right )\right )}{b^{3} c^{2} \arcsin \left (c x\right ) + a b^{2} c^{2}} + \frac {a \sin \left (\frac {a}{b}\right ) \operatorname {Si}\left (\frac {a}{b} + \arcsin \left (c x\right )\right )}{b^{3} c^{2} \arcsin \left (c x\right ) + a b^{2} c^{2}} \] Input:

integrate(x/(-c^2*x^2+1)^(1/2)/(a+b*arcsin(c*x))^2,x, algorithm="giac")
 

Output:

b*arcsin(c*x)*cos(a/b)*cos_integral(a/b + arcsin(c*x))/(b^3*c^2*arcsin(c*x 
) + a*b^2*c^2) + b*arcsin(c*x)*sin(a/b)*sin_integral(a/b + arcsin(c*x))/(b 
^3*c^2*arcsin(c*x) + a*b^2*c^2) - b*c*x/(b^3*c^2*arcsin(c*x) + a*b^2*c^2) 
+ a*cos(a/b)*cos_integral(a/b + arcsin(c*x))/(b^3*c^2*arcsin(c*x) + a*b^2* 
c^2) + a*sin(a/b)*sin_integral(a/b + arcsin(c*x))/(b^3*c^2*arcsin(c*x) + a 
*b^2*c^2)
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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