\(\int \frac {x}{(a+b \arcsin (c x))^2} \, dx\) [49]

Optimal result

Integrand size = 12, antiderivative size = 90 \[ \int \frac {x}{(a+b \arcsin (c x))^2} \, dx=-\frac {x \sqrt {1-c^2 x^2}}{b c (a+b \arcsin (c x))}+\frac {\cos \left (\frac {2 a}{b}\right ) \operatorname {CosIntegral}\left (\frac {2 (a+b \arcsin (c x))}{b}\right )}{b^2 c^2}+\frac {\sin \left (\frac {2 a}{b}\right ) \text {Si}\left (\frac {2 (a+b \arcsin (c x))}{b}\right )}{b^2 c^2} \] Output:

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

Mathematica [A] (verified)

Time = 0.27 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.88 \[ \int \frac {x}{(a+b \arcsin (c x))^2} \, dx=\frac {-\frac {b c x \sqrt {1-c^2 x^2}}{a+b \arcsin (c x)}+\cos \left (\frac {2 a}{b}\right ) \operatorname {CosIntegral}\left (2 \left (\frac {a}{b}+\arcsin (c x)\right )\right )+\sin \left (\frac {2 a}{b}\right ) \text {Si}\left (2 \left (\frac {a}{b}+\arcsin (c x)\right )\right )}{b^2 c^2} \] Input:

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

Output:

(-((b*c*x*Sqrt[1 - c^2*x^2])/(a + b*ArcSin[c*x])) + Cos[(2*a)/b]*CosIntegr 
al[2*(a/b + ArcSin[c*x])] + Sin[(2*a)/b]*SinIntegral[2*(a/b + ArcSin[c*x]) 
])/(b^2*c^2)
 

Rubi [A] (verified)

Time = 0.49 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.96, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.583, Rules used = {5142, 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.

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

\(\Big \downarrow \) 5142

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

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\int \frac {\sin \left (\frac {2 a}{b}-\frac {2 (a+b \arcsin (c x))}{b}+\frac {\pi }{2}\right )}{a+b \arcsin (c x)}d(a+b \arcsin (c x))}{b^2 c^2}-\frac {x \sqrt {1-c^2 x^2}}{b c (a+b \arcsin (c x))}\)

\(\Big \downarrow \) 3784

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3780

\(\displaystyle \frac {\cos \left (\frac {2 a}{b}\right ) \int \frac {\sin \left (\frac {2 (a+b \arcsin (c x))}{b}+\frac {\pi }{2}\right )}{a+b \arcsin (c x)}d(a+b \arcsin (c x))+\sin \left (\frac {2 a}{b}\right ) \text {Si}\left (\frac {2 (a+b \arcsin (c x))}{b}\right )}{b^2 c^2}-\frac {x \sqrt {1-c^2 x^2}}{b c (a+b \arcsin (c x))}\)

\(\Big \downarrow \) 3783

\(\displaystyle \frac {\cos \left (\frac {2 a}{b}\right ) \operatorname {CosIntegral}\left (\frac {2 (a+b \arcsin (c x))}{b}\right )+\sin \left (\frac {2 a}{b}\right ) \text {Si}\left (\frac {2 (a+b \arcsin (c x))}{b}\right )}{b^2 c^2}-\frac {x \sqrt {1-c^2 x^2}}{b c (a+b \arcsin (c x))}\)

Input:

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

Output:

-((x*Sqrt[1 - c^2*x^2])/(b*c*(a + b*ArcSin[c*x]))) + (Cos[(2*a)/b]*CosInte 
gral[(2*(a + b*ArcSin[c*x]))/b] + Sin[(2*a)/b]*SinIntegral[(2*(a + b*ArcSi 
n[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 5142

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_)*(x_)^(m_.), x_Symbol] :> Simp[x 
^m*Sqrt[1 - c^2*x^2]*((a + b*ArcSin[c*x])^(n + 1)/(b*c*(n + 1))), x] - Simp 
[1/(b^2*c^(m + 1)*(n + 1))   Subst[Int[ExpandTrigReduce[x^(n + 1), Sin[-a/b 
 + x/b]^(m - 1)*(m - (m + 1)*Sin[-a/b + x/b]^2), x], x], x, a + b*ArcSin[c* 
x]], x] /; FreeQ[{a, b, c}, x] && IGtQ[m, 0] && GeQ[n, -2] && LtQ[n, -1]
 

Maple [A] (verified)

Time = 0.06 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.86

method	result	size
derivativedivides	\(\frac {-\frac {\sin \left (2 \arcsin \left (c x \right )\right )}{2 \left (a +b \arcsin \left (c x \right )\right ) b}+\frac {\operatorname {Si}\left (2 \arcsin \left (c x \right )+\frac {2 a}{b}\right ) \sin \left (\frac {2 a}{b}\right )+\operatorname {Ci}\left (2 \arcsin \left (c x \right )+\frac {2 a}{b}\right ) \cos \left (\frac {2 a}{b}\right )}{b^{2}}}{c^{2}}\)	\(77\)
default	\(\frac {-\frac {\sin \left (2 \arcsin \left (c x \right )\right )}{2 \left (a +b \arcsin \left (c x \right )\right ) b}+\frac {\operatorname {Si}\left (2 \arcsin \left (c x \right )+\frac {2 a}{b}\right ) \sin \left (\frac {2 a}{b}\right )+\operatorname {Ci}\left (2 \arcsin \left (c x \right )+\frac {2 a}{b}\right ) \cos \left (\frac {2 a}{b}\right )}{b^{2}}}{c^{2}}\)	\(77\)

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

-(sqrt(c*x + 1)*sqrt(-c*x + 1)*x - (b^2*c*arctan2(c*x, sqrt(c*x + 1)*sqrt( 
-c*x + 1)) + a*b*c)*integrate((2*c^2*x^2 - 1)*sqrt(c*x + 1)*sqrt(-c*x + 1) 
/(a*b*c^3*x^2 - a*b*c + (b^2*c^3*x^2 - b^2*c)*arctan2(c*x, sqrt(c*x + 1)*s 
qrt(-c*x + 1))), 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. 326 vs. \(2 (88) = 176\).

Time = 0.14 (sec) , antiderivative size = 326, normalized size of antiderivative = 3.62 \[ \int \frac {x}{(a+b \arcsin (c x))^2} \, dx=\frac {2 \, b \arcsin \left (c x\right ) \cos \left (\frac {a}{b}\right )^{2} \operatorname {Ci}\left (\frac {2 \, a}{b} + 2 \, \arcsin \left (c x\right )\right )}{b^{3} c^{2} \arcsin \left (c x\right ) + a b^{2} c^{2}} + \frac {2 \, b \arcsin \left (c x\right ) \cos \left (\frac {a}{b}\right ) \sin \left (\frac {a}{b}\right ) \operatorname {Si}\left (\frac {2 \, a}{b} + 2 \, \arcsin \left (c x\right )\right )}{b^{3} c^{2} \arcsin \left (c x\right ) + a b^{2} c^{2}} + \frac {2 \, a \cos \left (\frac {a}{b}\right )^{2} \operatorname {Ci}\left (\frac {2 \, a}{b} + 2 \, \arcsin \left (c x\right )\right )}{b^{3} c^{2} \arcsin \left (c x\right ) + a b^{2} c^{2}} + \frac {2 \, a \cos \left (\frac {a}{b}\right ) \sin \left (\frac {a}{b}\right ) \operatorname {Si}\left (\frac {2 \, a}{b} + 2 \, \arcsin \left (c x\right )\right )}{b^{3} c^{2} \arcsin \left (c x\right ) + a b^{2} c^{2}} - \frac {\sqrt {-c^{2} x^{2} + 1} b c x}{b^{3} c^{2} \arcsin \left (c x\right ) + a b^{2} c^{2}} - \frac {b \arcsin \left (c x\right ) \operatorname {Ci}\left (\frac {2 \, a}{b} + 2 \, \arcsin \left (c x\right )\right )}{b^{3} c^{2} \arcsin \left (c x\right ) + a b^{2} c^{2}} - \frac {a \operatorname {Ci}\left (\frac {2 \, a}{b} + 2 \, \arcsin \left (c x\right )\right )}{b^{3} c^{2} \arcsin \left (c x\right ) + a b^{2} c^{2}} \] Input:

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

Output:

2*b*arcsin(c*x)*cos(a/b)^2*cos_integral(2*a/b + 2*arcsin(c*x))/(b^3*c^2*ar 
csin(c*x) + a*b^2*c^2) + 2*b*arcsin(c*x)*cos(a/b)*sin(a/b)*sin_integral(2* 
a/b + 2*arcsin(c*x))/(b^3*c^2*arcsin(c*x) + a*b^2*c^2) + 2*a*cos(a/b)^2*co 
s_integral(2*a/b + 2*arcsin(c*x))/(b^3*c^2*arcsin(c*x) + a*b^2*c^2) + 2*a* 
cos(a/b)*sin(a/b)*sin_integral(2*a/b + 2*arcsin(c*x))/(b^3*c^2*arcsin(c*x) 
 + a*b^2*c^2) - sqrt(-c^2*x^2 + 1)*b*c*x/(b^3*c^2*arcsin(c*x) + a*b^2*c^2) 
 - b*arcsin(c*x)*cos_integral(2*a/b + 2*arcsin(c*x))/(b^3*c^2*arcsin(c*x) 
+ a*b^2*c^2) - a*cos_integral(2*a/b + 2*arcsin(c*x))/(b^3*c^2*arcsin(c*x) 
+ a*b^2*c^2)
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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