3.2.70 \(\int \frac {1}{(a+b \arcsin (c x))^3} \, dx\) [170]

3.2.70.1 Optimal result

Integrand size = 10, antiderivative size = 111 \[ \int \frac {1}{(a+b \arcsin (c x))^3} \, dx=-\frac {\sqrt {1-c^2 x^2}}{2 b c (a+b \arcsin (c x))^2}+\frac {x}{2 b^2 (a+b \arcsin (c x))}-\frac {\cos \left (\frac {a}{b}\right ) \operatorname {CosIntegral}\left (\frac {a+b \arcsin (c x)}{b}\right )}{2 b^3 c}-\frac {\sin \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a+b \arcsin (c x)}{b}\right )}{2 b^3 c} \]

output

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

3.2.70.2 Mathematica [A] (verified)

Time = 0.32 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.84 \[ \int \frac {1}{(a+b \arcsin (c x))^3} \, dx=-\frac {\frac {b \left (\frac {b \sqrt {1-c^2 x^2}}{c}-x (a+b \arcsin (c x))\right )}{(a+b \arcsin (c x))^2}+\frac {\cos \left (\frac {a}{b}\right ) \operatorname {CosIntegral}\left (\frac {a}{b}+\arcsin (c x)\right )}{c}+\frac {\sin \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a}{b}+\arcsin (c x)\right )}{c}}{2 b^3} \]

input

Integrate[(a + b*ArcSin[c*x])^(-3),x]
 

output

-1/2*((b*((b*Sqrt[1 - c^2*x^2])/c - x*(a + b*ArcSin[c*x])))/(a + b*ArcSin[ 
c*x])^2 + (Cos[a/b]*CosIntegral[a/b + ArcSin[c*x]])/c + (Sin[a/b]*SinInteg 
ral[a/b + ArcSin[c*x]])/c)/b^3
 

3.2.70.3 Rubi [A] (verified)

Time = 0.65 (sec) , antiderivative size = 111, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.900, Rules used = {5132, 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.

\(\displaystyle \int \frac {1}{(a+b \arcsin (c x))^3} \, dx\)

\(\Big \downarrow \) 5132

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

\(\Big \downarrow \) 5222

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

\(\Big \downarrow \) 5134

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3784

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3780

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

\(\Big \downarrow \) 3783

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

input

Int[(a + b*ArcSin[c*x])^(-3),x]
 

output

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

3.2.70.3.1 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 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]
 

3.2.70.4 Maple [A] (verified)

Time = 0.03 (sec) , antiderivative size = 138, normalized size of antiderivative = 1.24

method	result	size
derivativedivides	\(\frac {-\frac {\sqrt {-c^{2} x^{2}+1}}{2 \left (a +b \arcsin \left (c x \right )\right )^{2} b}-\frac {\arcsin \left (c x \right ) \operatorname {Si}\left (\arcsin \left (c x \right )+\frac {a}{b}\right ) \sin \left (\frac {a}{b}\right ) b +\arcsin \left (c x \right ) \operatorname {Ci}\left (\arcsin \left (c x \right )+\frac {a}{b}\right ) \cos \left (\frac {a}{b}\right ) b +\operatorname {Si}\left (\arcsin \left (c x \right )+\frac {a}{b}\right ) \sin \left (\frac {a}{b}\right ) a +\operatorname {Ci}\left (\arcsin \left (c x \right )+\frac {a}{b}\right ) \cos \left (\frac {a}{b}\right ) a -x b c}{2 \left (a +b \arcsin \left (c x \right )\right ) b^{3}}}{c}\)	\(138\)
default	\(\frac {-\frac {\sqrt {-c^{2} x^{2}+1}}{2 \left (a +b \arcsin \left (c x \right )\right )^{2} b}-\frac {\arcsin \left (c x \right ) \operatorname {Si}\left (\arcsin \left (c x \right )+\frac {a}{b}\right ) \sin \left (\frac {a}{b}\right ) b +\arcsin \left (c x \right ) \operatorname {Ci}\left (\arcsin \left (c x \right )+\frac {a}{b}\right ) \cos \left (\frac {a}{b}\right ) b +\operatorname {Si}\left (\arcsin \left (c x \right )+\frac {a}{b}\right ) \sin \left (\frac {a}{b}\right ) a +\operatorname {Ci}\left (\arcsin \left (c x \right )+\frac {a}{b}\right ) \cos \left (\frac {a}{b}\right ) a -x b c}{2 \left (a +b \arcsin \left (c x \right )\right ) b^{3}}}{c}\)	\(138\)

input

int(1/(a+b*arcsin(c*x))^3,x,method=_RETURNVERBOSE)
 

output

1/c*(-1/2*(-c^2*x^2+1)^(1/2)/(a+b*arcsin(c*x))^2/b-1/2*(arcsin(c*x)*Si(arc 
sin(c*x)+a/b)*sin(a/b)*b+arcsin(c*x)*Ci(arcsin(c*x)+a/b)*cos(a/b)*b+Si(arc 
sin(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^3)
 

3.2.70.5 Fricas [F]

\[ \int \frac {1}{(a+b \arcsin (c x))^3} \, dx=\int { \frac {1}{{\left (b \arcsin \left (c x\right ) + a\right )}^{3}} \,d x } \]

input

integrate(1/(a+b*arcsin(c*x))^3,x, algorithm="fricas")
 

output

integral(1/(b^3*arcsin(c*x)^3 + 3*a*b^2*arcsin(c*x)^2 + 3*a^2*b*arcsin(c*x 
) + a^3), x)
 

3.2.70.6 Sympy [F]

\[ \int \frac {1}{(a+b \arcsin (c x))^3} \, dx=\int \frac {1}{\left (a + b \operatorname {asin}{\left (c x \right )}\right )^{3}}\, dx \]

input

integrate(1/(a+b*asin(c*x))**3,x)
 

output

Integral((a + b*asin(c*x))**(-3), x)
 

3.2.70.7 Maxima [F]

\[ \int \frac {1}{(a+b \arcsin (c x))^3} \, dx=\int { \frac {1}{{\left (b \arcsin \left (c x\right ) + a\right )}^{3}} \,d x } \]

input

integrate(1/(a+b*arcsin(c*x))^3,x, algorithm="maxima")
 

output

1/2*(b*c*x*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1)) + a*c*x - sqrt(c*x + 
 1)*sqrt(-c*x + 1)*b - 2*(b^4*c*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1)) 
^2 + 2*a*b^3*c*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1)) + a^2*b^2*c)*int 
egrate(1/2/(b^3*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1)) + a*b^2), x))/( 
b^4*c*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1))^2 + 2*a*b^3*c*arctan2(c*x 
, sqrt(c*x + 1)*sqrt(-c*x + 1)) + a^2*b^2*c)
 

3.2.70.8 Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 482 vs. \(2 (101) = 202\).

Time = 0.33 (sec) , antiderivative size = 482, normalized size of antiderivative = 4.34 \[ \int \frac {1}{(a+b \arcsin (c x))^3} \, dx=-\frac {b^{2} \arcsin \left (c x\right )^{2} \cos \left (\frac {a}{b}\right ) \operatorname {Ci}\left (\frac {a}{b} + \arcsin \left (c x\right )\right )}{2 \, {\left (b^{5} c \arcsin \left (c x\right )^{2} + 2 \, a b^{4} c \arcsin \left (c x\right ) + a^{2} b^{3} c\right )}} - \frac {b^{2} \arcsin \left (c x\right )^{2} \sin \left (\frac {a}{b}\right ) \operatorname {Si}\left (\frac {a}{b} + \arcsin \left (c x\right )\right )}{2 \, {\left (b^{5} c \arcsin \left (c x\right )^{2} + 2 \, a b^{4} c \arcsin \left (c x\right ) + a^{2} b^{3} c\right )}} + \frac {b^{2} c x \arcsin \left (c x\right )}{2 \, {\left (b^{5} c \arcsin \left (c x\right )^{2} + 2 \, a b^{4} c \arcsin \left (c x\right ) + a^{2} b^{3} c\right )}} - \frac {a 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^{5} c \arcsin \left (c x\right )^{2} + 2 \, a b^{4} c \arcsin \left (c x\right ) + a^{2} b^{3} c} - \frac {a 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^{5} c \arcsin \left (c x\right )^{2} + 2 \, a b^{4} c \arcsin \left (c x\right ) + a^{2} b^{3} c} + \frac {a b c x}{2 \, {\left (b^{5} c \arcsin \left (c x\right )^{2} + 2 \, a b^{4} c \arcsin \left (c x\right ) + a^{2} b^{3} c\right )}} - \frac {a^{2} \cos \left (\frac {a}{b}\right ) \operatorname {Ci}\left (\frac {a}{b} + \arcsin \left (c x\right )\right )}{2 \, {\left (b^{5} c \arcsin \left (c x\right )^{2} + 2 \, a b^{4} c \arcsin \left (c x\right ) + a^{2} b^{3} c\right )}} - \frac {a^{2} \sin \left (\frac {a}{b}\right ) \operatorname {Si}\left (\frac {a}{b} + \arcsin \left (c x\right )\right )}{2 \, {\left (b^{5} c \arcsin \left (c x\right )^{2} + 2 \, a b^{4} c \arcsin \left (c x\right ) + a^{2} b^{3} c\right )}} - \frac {\sqrt {-c^{2} x^{2} + 1} b^{2}}{2 \, {\left (b^{5} c \arcsin \left (c x\right )^{2} + 2 \, a b^{4} c \arcsin \left (c x\right ) + a^{2} b^{3} c\right )}} \]

input

integrate(1/(a+b*arcsin(c*x))^3,x, algorithm="giac")
 

output

-1/2*b^2*arcsin(c*x)^2*cos(a/b)*cos_integral(a/b + arcsin(c*x))/(b^5*c*arc 
sin(c*x)^2 + 2*a*b^4*c*arcsin(c*x) + a^2*b^3*c) - 1/2*b^2*arcsin(c*x)^2*si 
n(a/b)*sin_integral(a/b + arcsin(c*x))/(b^5*c*arcsin(c*x)^2 + 2*a*b^4*c*ar 
csin(c*x) + a^2*b^3*c) + 1/2*b^2*c*x*arcsin(c*x)/(b^5*c*arcsin(c*x)^2 + 2* 
a*b^4*c*arcsin(c*x) + a^2*b^3*c) - a*b*arcsin(c*x)*cos(a/b)*cos_integral(a 
/b + arcsin(c*x))/(b^5*c*arcsin(c*x)^2 + 2*a*b^4*c*arcsin(c*x) + a^2*b^3*c 
) - a*b*arcsin(c*x)*sin(a/b)*sin_integral(a/b + arcsin(c*x))/(b^5*c*arcsin 
(c*x)^2 + 2*a*b^4*c*arcsin(c*x) + a^2*b^3*c) + 1/2*a*b*c*x/(b^5*c*arcsin(c 
*x)^2 + 2*a*b^4*c*arcsin(c*x) + a^2*b^3*c) - 1/2*a^2*cos(a/b)*cos_integral 
(a/b + arcsin(c*x))/(b^5*c*arcsin(c*x)^2 + 2*a*b^4*c*arcsin(c*x) + a^2*b^3 
*c) - 1/2*a^2*sin(a/b)*sin_integral(a/b + arcsin(c*x))/(b^5*c*arcsin(c*x)^ 
2 + 2*a*b^4*c*arcsin(c*x) + a^2*b^3*c) - 1/2*sqrt(-c^2*x^2 + 1)*b^2/(b^5*c 
*arcsin(c*x)^2 + 2*a*b^4*c*arcsin(c*x) + a^2*b^3*c)
 

3.2.70.9 Mupad [F(-1)]

Timed out. \[ \int \frac {1}{(a+b \arcsin (c x))^3} \, dx=\int \frac {1}{{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^3} \,d x \]

input

int(1/(a + b*asin(c*x))^3,x)
 

output

int(1/(a + b*asin(c*x))^3, x)