\(\int \frac {1}{(a+b \arcsin (c x))^2} \, dx\) [24]

Optimal result

Integrand size = 10, antiderivative size = 86 \[ \int \frac {1}{(a+b \arcsin (c x))^2} \, dx=-\frac {\sqrt {1-c^2 x^2}}{b c (a+b \arcsin (c x))}+\frac {\operatorname {CosIntegral}\left (\frac {a+b \arcsin (c x)}{b}\right ) \sin \left (\frac {a}{b}\right )}{b^2 c}-\frac {\cos \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a+b \arcsin (c x)}{b}\right )}{b^2 c} \] Output:

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

Mathematica [A] (verified)

Time = 0.05 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.84 \[ \int \frac {1}{(a+b \arcsin (c x))^2} \, dx=\frac {-\frac {b \sqrt {1-c^2 x^2}}{a+b \arcsin (c x)}+\operatorname {CosIntegral}\left (\frac {a}{b}+\arcsin (c x)\right ) \sin \left (\frac {a}{b}\right )-\cos \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a}{b}+\arcsin (c x)\right )}{b^2 c} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.59 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.97, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.900, Rules used = {5132, 5224, 25, 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.

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

\(\Big \downarrow \) 5132

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

\(\Big \downarrow \) 5224

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3784

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3780

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

\(\Big \downarrow \) 3783

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

Input:

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

Output:

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

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 5132

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Simp[Sqrt[1 - c^2 
*x^2]*((a + b*ArcSin[c*x])^(n + 1)/(b*c*(n + 1))), x] + Simp[c/(b*(n + 1)) 
  Int[x*((a + b*ArcSin[c*x])^(n + 1)/Sqrt[1 - c^2*x^2]), x], x] /; FreeQ[{a 
, b, c}, x] && 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 = 0.10 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.87

method	result	size
derivativedivides	\(\frac {-\frac {\sqrt {-c^{2} x^{2}+1}}{\left (a +b \arcsin \left (c x \right )\right ) b}+\frac {\operatorname {Ci}\left (\arcsin \left (c x \right )+\frac {a}{b}\right ) \sin \left (\frac {a}{b}\right )-\operatorname {Si}\left (\arcsin \left (c x \right )+\frac {a}{b}\right ) \cos \left (\frac {a}{b}\right )}{b^{2}}}{c}\)	\(75\)
default	\(\frac {-\frac {\sqrt {-c^{2} x^{2}+1}}{\left (a +b \arcsin \left (c x \right )\right ) b}+\frac {\operatorname {Ci}\left (\arcsin \left (c x \right )+\frac {a}{b}\right ) \sin \left (\frac {a}{b}\right )-\operatorname {Si}\left (\arcsin \left (c x \right )+\frac {a}{b}\right ) \cos \left (\frac {a}{b}\right )}{b^{2}}}{c}\)	\(75\)

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

((b^2*c^2*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1)) + a*b*c^2)*integrate( 
sqrt(c*x + 1)*sqrt(-c*x + 1)*x/(a*b*c^2*x^2 - a*b + (b^2*c^2*x^2 - b^2)*ar 
ctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1))), x) - 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)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 192 vs. \(2 (84) = 168\).

Time = 0.12 (sec) , antiderivative size = 192, normalized size of antiderivative = 2.23 \[ \int \frac {1}{(a+b \arcsin (c x))^2} \, dx=\frac {b \arcsin \left (c x\right ) \operatorname {Ci}\left (\frac {a}{b} + \arcsin \left (c x\right )\right ) \sin \left (\frac {a}{b}\right )}{b^{3} c \arcsin \left (c x\right ) + a b^{2} c} - \frac {b \arcsin \left (c x\right ) \cos \left (\frac {a}{b}\right ) \operatorname {Si}\left (\frac {a}{b} + \arcsin \left (c x\right )\right )}{b^{3} c \arcsin \left (c x\right ) + a b^{2} c} + \frac {a \operatorname {Ci}\left (\frac {a}{b} + \arcsin \left (c x\right )\right ) \sin \left (\frac {a}{b}\right )}{b^{3} c \arcsin \left (c x\right ) + a b^{2} c} - \frac {a \cos \left (\frac {a}{b}\right ) \operatorname {Si}\left (\frac {a}{b} + \arcsin \left (c x\right )\right )}{b^{3} c \arcsin \left (c x\right ) + a b^{2} c} - \frac {\sqrt {-c^{2} x^{2} + 1} b}{b^{3} c \arcsin \left (c x\right ) + a b^{2} c} \] Input:

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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