3.2.37 \(\int \frac {\arcsin (a+b x)^2}{x^3} \, dx\) [137]

3.2.37.1 Optimal result

Integrand size = 12, antiderivative size = 272 \[ \int \frac {\arcsin (a+b x)^2}{x^3} \, dx=-\frac {b \sqrt {1-(a+b x)^2} \arcsin (a+b x)}{\left (1-a^2\right ) x}-\frac {\arcsin (a+b x)^2}{2 x^2}-\frac {i a b^2 \arcsin (a+b x) \log \left (1+\frac {i e^{i \arcsin (a+b x)}}{a-\sqrt {-1+a^2}}\right )}{\left (-1+a^2\right )^{3/2}}+\frac {i a b^2 \arcsin (a+b x) \log \left (1+\frac {i e^{i \arcsin (a+b x)}}{a+\sqrt {-1+a^2}}\right )}{\left (-1+a^2\right )^{3/2}}+\frac {b^2 \log (x)}{1-a^2}-\frac {a b^2 \operatorname {PolyLog}\left (2,-\frac {i e^{i \arcsin (a+b x)}}{a-\sqrt {-1+a^2}}\right )}{\left (-1+a^2\right )^{3/2}}+\frac {a b^2 \operatorname {PolyLog}\left (2,-\frac {i e^{i \arcsin (a+b x)}}{a+\sqrt {-1+a^2}}\right )}{\left (-1+a^2\right )^{3/2}} \]

output

-1/2*arcsin(b*x+a)^2/x^2+b^2*ln(x)/(-a^2+1)-I*a*b^2*arcsin(b*x+a)*ln(1+I*( 
I*(b*x+a)+(1-(b*x+a)^2)^(1/2))/(a-(a^2-1)^(1/2)))/(a^2-1)^(3/2)+I*a*b^2*ar 
csin(b*x+a)*ln(1+I*(I*(b*x+a)+(1-(b*x+a)^2)^(1/2))/(a+(a^2-1)^(1/2)))/(a^2 
-1)^(3/2)-a*b^2*polylog(2,-I*(I*(b*x+a)+(1-(b*x+a)^2)^(1/2))/(a-(a^2-1)^(1 
/2)))/(a^2-1)^(3/2)+a*b^2*polylog(2,-I*(I*(b*x+a)+(1-(b*x+a)^2)^(1/2))/(a+ 
(a^2-1)^(1/2)))/(a^2-1)^(3/2)-b*arcsin(b*x+a)*(1-(b*x+a)^2)^(1/2)/(-a^2+1) 
/x
 

3.2.37.2 Mathematica [A] (verified)

Time = 0.15 (sec) , antiderivative size = 314, normalized size of antiderivative = 1.15 \[ \int \frac {\arcsin (a+b x)^2}{x^3} \, dx=\frac {2 \sqrt {-1+a^2} b x \sqrt {1-(a+b x)^2} \arcsin (a+b x)+\sqrt {-1+a^2} \arcsin (a+b x)^2-a^2 \sqrt {-1+a^2} \arcsin (a+b x)^2-2 i a b^2 x^2 \arcsin (a+b x) \log \left (\frac {a-\sqrt {-1+a^2}+i e^{i \arcsin (a+b x)}}{a-\sqrt {-1+a^2}}\right )+2 i a b^2 x^2 \arcsin (a+b x) \log \left (\frac {a+\sqrt {-1+a^2}+i e^{i \arcsin (a+b x)}}{a+\sqrt {-1+a^2}}\right )-2 \sqrt {-1+a^2} b^2 x^2 \log (x)-2 a b^2 x^2 \operatorname {PolyLog}\left (2,\frac {i e^{i \arcsin (a+b x)}}{-a+\sqrt {-1+a^2}}\right )+2 a b^2 x^2 \operatorname {PolyLog}\left (2,-\frac {i e^{i \arcsin (a+b x)}}{a+\sqrt {-1+a^2}}\right )}{2 \left (-1+a^2\right )^{3/2} x^2} \]

input

Integrate[ArcSin[a + b*x]^2/x^3,x]
 

output

(2*Sqrt[-1 + a^2]*b*x*Sqrt[1 - (a + b*x)^2]*ArcSin[a + b*x] + Sqrt[-1 + a^ 
2]*ArcSin[a + b*x]^2 - a^2*Sqrt[-1 + a^2]*ArcSin[a + b*x]^2 - (2*I)*a*b^2* 
x^2*ArcSin[a + b*x]*Log[(a - Sqrt[-1 + a^2] + I*E^(I*ArcSin[a + b*x]))/(a 
- Sqrt[-1 + a^2])] + (2*I)*a*b^2*x^2*ArcSin[a + b*x]*Log[(a + Sqrt[-1 + a^ 
2] + I*E^(I*ArcSin[a + b*x]))/(a + Sqrt[-1 + a^2])] - 2*Sqrt[-1 + a^2]*b^2 
*x^2*Log[x] - 2*a*b^2*x^2*PolyLog[2, (I*E^(I*ArcSin[a + b*x]))/(-a + Sqrt[ 
-1 + a^2])] + 2*a*b^2*x^2*PolyLog[2, ((-I)*E^(I*ArcSin[a + b*x]))/(a + Sqr 
t[-1 + a^2])])/(2*(-1 + a^2)^(3/2)*x^2)
 

3.2.37.3 Rubi [A] (warning: unable to verify)

Time = 1.04 (sec) , antiderivative size = 279, normalized size of antiderivative = 1.03, number of steps used = 18, number of rules used = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.417, Rules used = {5304, 25, 27, 5242, 5272, 3042, 3805, 3042, 3147, 16, 3804, 25, 2694, 27, 2620, 2715, 2838}

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.

\(\displaystyle \int \frac {\arcsin (a+b x)^2}{x^3} \, dx\)

\(\Big \downarrow \) 5304

\(\displaystyle \frac {\int \frac {\arcsin (a+b x)^2}{x^3}d(a+b x)}{b}\)

\(\Big \downarrow \) 25

\(\displaystyle -\frac {\int -\frac {\arcsin (a+b x)^2}{x^3}d(a+b x)}{b}\)

\(\Big \downarrow \) 27

\(\displaystyle -b^2 \int -\frac {\arcsin (a+b x)^2}{b^3 x^3}d(a+b x)\)

\(\Big \downarrow \) 5242

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

\(\Big \downarrow \) 5272

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3805

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3147

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

\(\Big \downarrow \) 16

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

\(\Big \downarrow \) 3804

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 2694

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 2620

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

\(\Big \downarrow \) 2715

\(\displaystyle -b^2 \left (-\frac {2 a \left (\frac {i \left (-i \int e^{-i \arcsin (a+b x)} \log \left (1+\frac {i e^{i \arcsin (a+b x)}}{a+\sqrt {a^2-1}}\right )de^{i \arcsin (a+b x)}-\arcsin (a+b x) \log \left (1+\frac {i e^{i \arcsin (a+b x)}}{\sqrt {a^2-1}+a}\right )\right )}{2 \sqrt {a^2-1}}-\frac {i \left (-i \int e^{-i \arcsin (a+b x)} \log \left (1+\frac {i e^{i \arcsin (a+b x)}}{a-\sqrt {a^2-1}}\right )de^{i \arcsin (a+b x)}-\arcsin (a+b x) \log \left (1+\frac {i e^{i \arcsin (a+b x)}}{a-\sqrt {a^2-1}}\right )\right )}{2 \sqrt {a^2-1}}\right )}{1-a^2}+\frac {\sqrt {1-(a+b x)^2} \arcsin (a+b x)}{\left (1-a^2\right ) b x}-\frac {\log (2 a+b x)}{1-a^2}+\frac {\arcsin (a+b x)^2}{2 b^2 x^2}\right )\)

\(\Big \downarrow \) 2838

\(\displaystyle -b^2 \left (-\frac {2 a \left (\frac {i \left (i \operatorname {PolyLog}\left (2,-\frac {i e^{i \arcsin (a+b x)}}{a+\sqrt {a^2-1}}\right )-\arcsin (a+b x) \log \left (1+\frac {i e^{i \arcsin (a+b x)}}{\sqrt {a^2-1}+a}\right )\right )}{2 \sqrt {a^2-1}}-\frac {i \left (i \operatorname {PolyLog}\left (2,-\frac {i e^{i \arcsin (a+b x)}}{a-\sqrt {a^2-1}}\right )-\arcsin (a+b x) \log \left (1+\frac {i e^{i \arcsin (a+b x)}}{a-\sqrt {a^2-1}}\right )\right )}{2 \sqrt {a^2-1}}\right )}{1-a^2}+\frac {\sqrt {1-(a+b x)^2} \arcsin (a+b x)}{\left (1-a^2\right ) b x}-\frac {\log (2 a+b x)}{1-a^2}+\frac {\arcsin (a+b x)^2}{2 b^2 x^2}\right )\)

input

Int[ArcSin[a + b*x]^2/x^3,x]
 

output

-(b^2*((Sqrt[1 - (a + b*x)^2]*ArcSin[a + b*x])/((1 - a^2)*b*x) + ArcSin[a 
+ b*x]^2/(2*b^2*x^2) - Log[2*a + b*x]/(1 - a^2) - (2*a*(((-1/2*I)*(-(ArcSi 
n[a + b*x]*Log[1 + (I*E^(I*ArcSin[a + b*x]))/(a - Sqrt[-1 + a^2])]) + I*Po 
lyLog[2, ((-I)*E^(I*ArcSin[a + b*x]))/(a - Sqrt[-1 + a^2])]))/Sqrt[-1 + a^ 
2] + ((I/2)*(-(ArcSin[a + b*x]*Log[1 + (I*E^(I*ArcSin[a + b*x]))/(a + Sqrt 
[-1 + a^2])]) + I*PolyLog[2, ((-I)*E^(I*ArcSin[a + b*x]))/(a + Sqrt[-1 + a 
^2])]))/Sqrt[-1 + a^2]))/(1 - a^2)))
 

3.2.37.3.1 Defintions of rubi rules used

rule 16

Int[(c_.)/((a_.) + (b_.)*(x_)), x_Symbol] :> Simp[c*(Log[RemoveContent[a + 
b*x, x]]/b), x] /; FreeQ[{a, b, c}, x]
 

rule 25

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

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 2620

Int[(((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.))/ 
((a_) + (b_.)*((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)), x_Symbol] :> Simp 
[((c + d*x)^m/(b*f*g*n*Log[F]))*Log[1 + b*((F^(g*(e + f*x)))^n/a)], x] - Si 
mp[d*(m/(b*f*g*n*Log[F]))   Int[(c + d*x)^(m - 1)*Log[1 + b*((F^(g*(e + f*x 
)))^n/a)], x], x] /; FreeQ[{F, a, b, c, d, e, f, g, n}, x] && IGtQ[m, 0]
 

rule 2694

Int[((F_)^(u_)*((f_.) + (g_.)*(x_))^(m_.))/((a_.) + (b_.)*(F_)^(u_) + (c_.) 
*(F_)^(v_)), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Simp[2*(c/q)   Int 
[(f + g*x)^m*(F^u/(b - q + 2*c*F^u)), x], x] - Simp[2*(c/q)   Int[(f + g*x) 
^m*(F^u/(b + q + 2*c*F^u)), x], x]] /; FreeQ[{F, a, b, c, f, g}, x] && EqQ[ 
v, 2*u] && LinearQ[u, x] && NeQ[b^2 - 4*a*c, 0] && IGtQ[m, 0]
 

rule 2715

Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] 
:> Simp[1/(d*e*n*Log[F])   Subst[Int[Log[a + b*x]/x, x], x, (F^(e*(c + d*x) 
))^n], x] /; FreeQ[{F, a, b, c, d, e, n}, x] && GtQ[a, 0]
 

rule 2838

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> Simp[-PolyLog[2 
, (-c)*e*x^n]/n, x] /; FreeQ[{c, d, e, n}, x] && EqQ[c*d, 1]
 

rule 3042

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

rule 3147

Int[cos[(e_.) + (f_.)*(x_)]^(p_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m 
_.), x_Symbol] :> Simp[1/(b^p*f)   Subst[Int[(a + x)^m*(b^2 - x^2)^((p - 1) 
/2), x], x, b*Sin[e + f*x]], x] /; FreeQ[{a, b, e, f, m}, x] && IntegerQ[(p 
 - 1)/2] && NeQ[a^2 - b^2, 0]
 

rule 3804

Int[((c_.) + (d_.)*(x_))^(m_.)/((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]), x_Sy 
mbol] :> Simp[2   Int[(c + d*x)^m*(E^(I*(e + f*x))/(I*b + 2*a*E^(I*(e + f*x 
)) - I*b*E^(2*I*(e + f*x)))), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ 
[a^2 - b^2, 0] && IGtQ[m, 0]
 

rule 3805

Int[((c_.) + (d_.)*(x_))^(m_.)/((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^2, x_ 
Symbol] :> Simp[b*(c + d*x)^m*(Cos[e + f*x]/(f*(a^2 - b^2)*(a + b*Sin[e + f 
*x]))), x] + (Simp[a/(a^2 - b^2)   Int[(c + d*x)^m/(a + b*Sin[e + f*x]), x] 
, x] - Simp[b*d*(m/(f*(a^2 - b^2)))   Int[(c + d*x)^(m - 1)*(Cos[e + f*x]/( 
a + b*Sin[e + f*x])), x], x]) /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[a^2 - 
b^2, 0] && IGtQ[m, 0]
 

rule 5242

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

rule 5272

Int[(((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((f_) + (g_.)*(x_))^(m_.))/Sq 
rt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Simp[1/(c^(m + 1)*Sqrt[d])   Subst[In 
t[(a + b*x)^n*(c*f + g*Sin[x])^m, x], x, ArcSin[c*x]], x] /; FreeQ[{a, b, c 
, d, e, f, g, n}, x] && EqQ[c^2*d + e, 0] && IntegerQ[m] && GtQ[d, 0] && (G 
tQ[m, 0] || IGtQ[n, 0])
 

rule 5304

Int[((a_.) + ArcSin[(c_) + (d_.)*(x_)]*(b_.))^(n_.)*((e_.) + (f_.)*(x_))^(m 
_.), x_Symbol] :> Simp[1/d   Subst[Int[((d*e - c*f)/d + f*(x/d))^m*(a + b*A 
rcSin[x])^n, x], x, c + d*x], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x]
 

3.2.37.4 Maple [A] (verified)

Time = 1.29 (sec) , antiderivative size = 521, normalized size of antiderivative = 1.92

method	result	size
derivativedivides	\(b^{2} \left (-\frac {\arcsin \left (b x +a \right ) \left (-\arcsin \left (b x +a \right )+2 i a^{2}-4 i a \left (b x +a \right )+a^{2} \arcsin \left (b x +a \right )+2 a \sqrt {1-\left (b x +a \right )^{2}}+2 i \left (b x +a \right )^{2}-2 \left (b x +a \right ) \sqrt {1-\left (b x +a \right )^{2}}\right )}{2 \left (a^{2}-1\right ) b^{2} x^{2}}-\frac {\ln \left (i \left (i \left (b x +a \right )+\sqrt {1-\left (b x +a \right )^{2}}\right )^{2}+2 a \left (i \left (b x +a \right )+\sqrt {1-\left (b x +a \right )^{2}}\right )-i\right )}{a^{2}-1}+\frac {2 \ln \left (i \left (b x +a \right )+\sqrt {1-\left (b x +a \right )^{2}}\right )}{a^{2}-1}+\frac {\sqrt {-a^{2}+1}\, a \arcsin \left (b x +a \right ) \ln \left (\frac {i a +\sqrt {-a^{2}+1}-i \left (b x +a \right )-\sqrt {1-\left (b x +a \right )^{2}}}{i a +\sqrt {-a^{2}+1}}\right )}{\left (a^{2}-1\right )^{2}}-\frac {\sqrt {-a^{2}+1}\, a \arcsin \left (b x +a \right ) \ln \left (\frac {i a -\sqrt {-a^{2}+1}-i \left (b x +a \right )-\sqrt {1-\left (b x +a \right )^{2}}}{i a -\sqrt {-a^{2}+1}}\right )}{\left (a^{2}-1\right )^{2}}-\frac {i \sqrt {-a^{2}+1}\, \operatorname {dilog}\left (\frac {i a +\sqrt {-a^{2}+1}-i \left (b x +a \right )-\sqrt {1-\left (b x +a \right )^{2}}}{i a +\sqrt {-a^{2}+1}}\right ) a}{\left (a^{2}-1\right )^{2}}+\frac {i \sqrt {-a^{2}+1}\, \operatorname {dilog}\left (\frac {i a -\sqrt {-a^{2}+1}-i \left (b x +a \right )-\sqrt {1-\left (b x +a \right )^{2}}}{i a -\sqrt {-a^{2}+1}}\right ) a}{\left (a^{2}-1\right )^{2}}\right )\)	\(521\)
default	\(b^{2} \left (-\frac {\arcsin \left (b x +a \right ) \left (-\arcsin \left (b x +a \right )+2 i a^{2}-4 i a \left (b x +a \right )+a^{2} \arcsin \left (b x +a \right )+2 a \sqrt {1-\left (b x +a \right )^{2}}+2 i \left (b x +a \right )^{2}-2 \left (b x +a \right ) \sqrt {1-\left (b x +a \right )^{2}}\right )}{2 \left (a^{2}-1\right ) b^{2} x^{2}}-\frac {\ln \left (i \left (i \left (b x +a \right )+\sqrt {1-\left (b x +a \right )^{2}}\right )^{2}+2 a \left (i \left (b x +a \right )+\sqrt {1-\left (b x +a \right )^{2}}\right )-i\right )}{a^{2}-1}+\frac {2 \ln \left (i \left (b x +a \right )+\sqrt {1-\left (b x +a \right )^{2}}\right )}{a^{2}-1}+\frac {\sqrt {-a^{2}+1}\, a \arcsin \left (b x +a \right ) \ln \left (\frac {i a +\sqrt {-a^{2}+1}-i \left (b x +a \right )-\sqrt {1-\left (b x +a \right )^{2}}}{i a +\sqrt {-a^{2}+1}}\right )}{\left (a^{2}-1\right )^{2}}-\frac {\sqrt {-a^{2}+1}\, a \arcsin \left (b x +a \right ) \ln \left (\frac {i a -\sqrt {-a^{2}+1}-i \left (b x +a \right )-\sqrt {1-\left (b x +a \right )^{2}}}{i a -\sqrt {-a^{2}+1}}\right )}{\left (a^{2}-1\right )^{2}}-\frac {i \sqrt {-a^{2}+1}\, \operatorname {dilog}\left (\frac {i a +\sqrt {-a^{2}+1}-i \left (b x +a \right )-\sqrt {1-\left (b x +a \right )^{2}}}{i a +\sqrt {-a^{2}+1}}\right ) a}{\left (a^{2}-1\right )^{2}}+\frac {i \sqrt {-a^{2}+1}\, \operatorname {dilog}\left (\frac {i a -\sqrt {-a^{2}+1}-i \left (b x +a \right )-\sqrt {1-\left (b x +a \right )^{2}}}{i a -\sqrt {-a^{2}+1}}\right ) a}{\left (a^{2}-1\right )^{2}}\right )\)	\(521\)

input

int(arcsin(b*x+a)^2/x^3,x,method=_RETURNVERBOSE)
 

output

b^2*(-1/2*arcsin(b*x+a)*(-arcsin(b*x+a)+2*I*a^2-4*I*a*(b*x+a)+a^2*arcsin(b 
*x+a)+2*a*(1-(b*x+a)^2)^(1/2)+2*I*(b*x+a)^2-2*(b*x+a)*(1-(b*x+a)^2)^(1/2)) 
/(a^2-1)/b^2/x^2-1/(a^2-1)*ln(I*(I*(b*x+a)+(1-(b*x+a)^2)^(1/2))^2+2*a*(I*( 
b*x+a)+(1-(b*x+a)^2)^(1/2))-I)+2/(a^2-1)*ln(I*(b*x+a)+(1-(b*x+a)^2)^(1/2)) 
+(-a^2+1)^(1/2)/(a^2-1)^2*a*arcsin(b*x+a)*ln((I*a+(-a^2+1)^(1/2)-I*(b*x+a) 
-(1-(b*x+a)^2)^(1/2))/(I*a+(-a^2+1)^(1/2)))-(-a^2+1)^(1/2)/(a^2-1)^2*a*arc 
sin(b*x+a)*ln((I*a-(-a^2+1)^(1/2)-I*(b*x+a)-(1-(b*x+a)^2)^(1/2))/(I*a-(-a^ 
2+1)^(1/2)))-I*(-a^2+1)^(1/2)/(a^2-1)^2*dilog((I*a+(-a^2+1)^(1/2)-I*(b*x+a 
)-(1-(b*x+a)^2)^(1/2))/(I*a+(-a^2+1)^(1/2)))*a+I*(-a^2+1)^(1/2)/(a^2-1)^2* 
dilog((I*a-(-a^2+1)^(1/2)-I*(b*x+a)-(1-(b*x+a)^2)^(1/2))/(I*a-(-a^2+1)^(1/ 
2)))*a)
 

3.2.37.5 Fricas [F]

\[ \int \frac {\arcsin (a+b x)^2}{x^3} \, dx=\int { \frac {\arcsin \left (b x + a\right )^{2}}{x^{3}} \,d x } \]

input

integrate(arcsin(b*x+a)^2/x^3,x, algorithm="fricas")
 

output

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

3.2.37.6 Sympy [F]

\[ \int \frac {\arcsin (a+b x)^2}{x^3} \, dx=\int \frac {\operatorname {asin}^{2}{\left (a + b x \right )}}{x^{3}}\, dx \]

input

integrate(asin(b*x+a)**2/x**3,x)
 

output

Integral(asin(a + b*x)**2/x**3, x)
 

3.2.37.7 Maxima [F(-2)]

Exception generated. \[ \int \frac {\arcsin (a+b x)^2}{x^3} \, dx=\text {Exception raised: ValueError} \]

input

integrate(arcsin(b*x+a)^2/x^3,x, algorithm="maxima")
 

output

Exception raised: ValueError >> Computation failed since Maxima requested 
additional constraints; using the 'assume' command before evaluation *may* 
 help (example of legal syntax is 'assume(a-1>0)', see `assume?` for more 
details)Is
 

3.2.37.8 Giac [F]

\[ \int \frac {\arcsin (a+b x)^2}{x^3} \, dx=\int { \frac {\arcsin \left (b x + a\right )^{2}}{x^{3}} \,d x } \]

input

integrate(arcsin(b*x+a)^2/x^3,x, algorithm="giac")
 

output

integrate(arcsin(b*x + a)^2/x^3, x)
 

3.2.37.9 Mupad [F(-1)]

Timed out. \[ \int \frac {\arcsin (a+b x)^2}{x^3} \, dx=\int \frac {{\mathrm {asin}\left (a+b\,x\right )}^2}{x^3} \,d x \]

input

int(asin(a + b*x)^2/x^3,x)
 

output

int(asin(a + b*x)^2/x^3, x)