3.2.42 \(\int \frac {\arcsin (a+b x)^3}{x^2} \, dx\) [142]

3.2.42.1 Optimal result

Integrand size = 12, antiderivative size = 316 \[ \int \frac {\arcsin (a+b x)^3}{x^2} \, dx=-\frac {\arcsin (a+b x)^3}{x}+\frac {3 i b \arcsin (a+b x)^2 \log \left (1+\frac {i e^{i \arcsin (a+b x)}}{a-\sqrt {-1+a^2}}\right )}{\sqrt {-1+a^2}}-\frac {3 i b \arcsin (a+b x)^2 \log \left (1+\frac {i e^{i \arcsin (a+b x)}}{a+\sqrt {-1+a^2}}\right )}{\sqrt {-1+a^2}}+\frac {6 b \arcsin (a+b x) \operatorname {PolyLog}\left (2,-\frac {i e^{i \arcsin (a+b x)}}{a-\sqrt {-1+a^2}}\right )}{\sqrt {-1+a^2}}-\frac {6 b \arcsin (a+b x) \operatorname {PolyLog}\left (2,-\frac {i e^{i \arcsin (a+b x)}}{a+\sqrt {-1+a^2}}\right )}{\sqrt {-1+a^2}}+\frac {6 i b \operatorname {PolyLog}\left (3,-\frac {i e^{i \arcsin (a+b x)}}{a-\sqrt {-1+a^2}}\right )}{\sqrt {-1+a^2}}-\frac {6 i b \operatorname {PolyLog}\left (3,-\frac {i e^{i \arcsin (a+b x)}}{a+\sqrt {-1+a^2}}\right )}{\sqrt {-1+a^2}} \]

output

-arcsin(b*x+a)^3/x+3*I*b*arcsin(b*x+a)^2*ln(1+I*(I*(b*x+a)+(1-(b*x+a)^2)^( 
1/2))/(a-(a^2-1)^(1/2)))/(a^2-1)^(1/2)-3*I*b*arcsin(b*x+a)^2*ln(1+I*(I*(b* 
x+a)+(1-(b*x+a)^2)^(1/2))/(a+(a^2-1)^(1/2)))/(a^2-1)^(1/2)+6*b*arcsin(b*x+ 
a)*polylog(2,-I*(I*(b*x+a)+(1-(b*x+a)^2)^(1/2))/(a-(a^2-1)^(1/2)))/(a^2-1) 
^(1/2)-6*b*arcsin(b*x+a)*polylog(2,-I*(I*(b*x+a)+(1-(b*x+a)^2)^(1/2))/(a+( 
a^2-1)^(1/2)))/(a^2-1)^(1/2)+6*I*b*polylog(3,-I*(I*(b*x+a)+(1-(b*x+a)^2)^( 
1/2))/(a-(a^2-1)^(1/2)))/(a^2-1)^(1/2)-6*I*b*polylog(3,-I*(I*(b*x+a)+(1-(b 
*x+a)^2)^(1/2))/(a+(a^2-1)^(1/2)))/(a^2-1)^(1/2)
 

3.2.42.2 Mathematica [A] (verified)

Time = 0.15 (sec) , antiderivative size = 309, normalized size of antiderivative = 0.98 \[ \int \frac {\arcsin (a+b x)^3}{x^2} \, dx=-\frac {\sqrt {-1+a^2} \arcsin (a+b x)^3-3 i b x \arcsin (a+b x)^2 \log \left (\frac {a-\sqrt {-1+a^2}+i e^{i \arcsin (a+b x)}}{a-\sqrt {-1+a^2}}\right )+3 i b x \arcsin (a+b x)^2 \log \left (\frac {a+\sqrt {-1+a^2}+i e^{i \arcsin (a+b x)}}{a+\sqrt {-1+a^2}}\right )-6 b x \arcsin (a+b x) \operatorname {PolyLog}\left (2,\frac {i e^{i \arcsin (a+b x)}}{-a+\sqrt {-1+a^2}}\right )+6 b x \arcsin (a+b x) \operatorname {PolyLog}\left (2,-\frac {i e^{i \arcsin (a+b x)}}{a+\sqrt {-1+a^2}}\right )-6 i b x \operatorname {PolyLog}\left (3,\frac {i e^{i \arcsin (a+b x)}}{-a+\sqrt {-1+a^2}}\right )+6 i b x \operatorname {PolyLog}\left (3,-\frac {i e^{i \arcsin (a+b x)}}{a+\sqrt {-1+a^2}}\right )}{\sqrt {-1+a^2} x} \]

input

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

output

-((Sqrt[-1 + a^2]*ArcSin[a + b*x]^3 - (3*I)*b*x*ArcSin[a + b*x]^2*Log[(a - 
 Sqrt[-1 + a^2] + I*E^(I*ArcSin[a + b*x]))/(a - Sqrt[-1 + a^2])] + (3*I)*b 
*x*ArcSin[a + b*x]^2*Log[(a + Sqrt[-1 + a^2] + I*E^(I*ArcSin[a + b*x]))/(a 
 + Sqrt[-1 + a^2])] - 6*b*x*ArcSin[a + b*x]*PolyLog[2, (I*E^(I*ArcSin[a + 
b*x]))/(-a + Sqrt[-1 + a^2])] + 6*b*x*ArcSin[a + b*x]*PolyLog[2, ((-I)*E^( 
I*ArcSin[a + b*x]))/(a + Sqrt[-1 + a^2])] - (6*I)*b*x*PolyLog[3, (I*E^(I*A 
rcSin[a + b*x]))/(-a + Sqrt[-1 + a^2])] + (6*I)*b*x*PolyLog[3, ((-I)*E^(I* 
ArcSin[a + b*x]))/(a + Sqrt[-1 + a^2])])/(Sqrt[-1 + a^2]*x))
 

3.2.42.3 Rubi [A] (verified)

Time = 1.13 (sec) , antiderivative size = 298, normalized size of antiderivative = 0.94, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.083, Rules used = {5304, 27, 5242, 5272, 3042, 3804, 25, 2694, 27, 2620, 3011, 2720, 7143}

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.

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

\(\Big \downarrow \) 5304

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 5242

\(\displaystyle b \left (-3 \int -\frac {\arcsin (a+b x)^2}{b x \sqrt {1-(a+b x)^2}}d(a+b x)-\frac {\arcsin (a+b x)^3}{b x}\right )\)

\(\Big \downarrow \) 5272

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3804

\(\displaystyle b \left (-\frac {\arcsin (a+b x)^3}{b x}-6 \int -\frac {e^{i \arcsin (a+b x)} \arcsin (a+b x)^2}{-2 e^{i \arcsin (a+b x)} a-i e^{2 i \arcsin (a+b x)}+i}d\arcsin (a+b x)\right )\)

\(\Big \downarrow \) 25

\(\displaystyle b \left (-\frac {\arcsin (a+b x)^3}{b x}+6 \int \frac {e^{i \arcsin (a+b x)} \arcsin (a+b x)^2}{-2 e^{i \arcsin (a+b x)} a-i e^{2 i \arcsin (a+b x)}+i}d\arcsin (a+b x)\right )\)

\(\Big \downarrow \) 2694

\(\displaystyle b \left (-\frac {\arcsin (a+b x)^3}{b x}+6 \left (\frac {i \int -\frac {e^{i \arcsin (a+b x)} \arcsin (a+b x)^2}{2 \left (a+i e^{i \arcsin (a+b x)}-\sqrt {a^2-1}\right )}d\arcsin (a+b x)}{\sqrt {a^2-1}}-\frac {i \int -\frac {e^{i \arcsin (a+b x)} \arcsin (a+b x)^2}{2 \left (a+i e^{i \arcsin (a+b x)}+\sqrt {a^2-1}\right )}d\arcsin (a+b x)}{\sqrt {a^2-1}}\right )\right )\)

\(\Big \downarrow \) 27

\(\displaystyle b \left (-\frac {\arcsin (a+b x)^3}{b x}+6 \left (\frac {i \int \frac {e^{i \arcsin (a+b x)} \arcsin (a+b x)^2}{a+i e^{i \arcsin (a+b x)}+\sqrt {a^2-1}}d\arcsin (a+b x)}{2 \sqrt {a^2-1}}-\frac {i \int \frac {e^{i \arcsin (a+b x)} \arcsin (a+b x)^2}{a+i e^{i \arcsin (a+b x)}-\sqrt {a^2-1}}d\arcsin (a+b x)}{2 \sqrt {a^2-1}}\right )\right )\)

\(\Big \downarrow \) 2620

\(\displaystyle b \left (-\frac {\arcsin (a+b x)^3}{b x}+6 \left (\frac {i \left (2 \int \arcsin (a+b x) \log \left (1+\frac {i e^{i \arcsin (a+b x)}}{a+\sqrt {a^2-1}}\right )d\arcsin (a+b x)-\arcsin (a+b x)^2 \log \left (1+\frac {i e^{i \arcsin (a+b x)}}{\sqrt {a^2-1}+a}\right )\right )}{2 \sqrt {a^2-1}}-\frac {i \left (2 \int \arcsin (a+b x) \log \left (1+\frac {i e^{i \arcsin (a+b x)}}{a-\sqrt {a^2-1}}\right )d\arcsin (a+b x)-\arcsin (a+b x)^2 \log \left (1+\frac {i e^{i \arcsin (a+b x)}}{a-\sqrt {a^2-1}}\right )\right )}{2 \sqrt {a^2-1}}\right )\right )\)

\(\Big \downarrow \) 3011

\(\displaystyle b \left (-\frac {\arcsin (a+b x)^3}{b x}+6 \left (\frac {i \left (2 \left (i \arcsin (a+b x) \operatorname {PolyLog}\left (2,-\frac {i e^{i \arcsin (a+b x)}}{a+\sqrt {a^2-1}}\right )-i \int \operatorname {PolyLog}\left (2,-\frac {i e^{i \arcsin (a+b x)}}{a+\sqrt {a^2-1}}\right )d\arcsin (a+b x)\right )-\arcsin (a+b x)^2 \log \left (1+\frac {i e^{i \arcsin (a+b x)}}{\sqrt {a^2-1}+a}\right )\right )}{2 \sqrt {a^2-1}}-\frac {i \left (2 \left (i \arcsin (a+b x) \operatorname {PolyLog}\left (2,-\frac {i e^{i \arcsin (a+b x)}}{a-\sqrt {a^2-1}}\right )-i \int \operatorname {PolyLog}\left (2,-\frac {i e^{i \arcsin (a+b x)}}{a-\sqrt {a^2-1}}\right )d\arcsin (a+b x)\right )-\arcsin (a+b x)^2 \log \left (1+\frac {i e^{i \arcsin (a+b x)}}{a-\sqrt {a^2-1}}\right )\right )}{2 \sqrt {a^2-1}}\right )\right )\)

\(\Big \downarrow \) 2720

\(\displaystyle b \left (-\frac {\arcsin (a+b x)^3}{b x}+6 \left (\frac {i \left (2 \left (i \arcsin (a+b x) \operatorname {PolyLog}\left (2,-\frac {i e^{i \arcsin (a+b x)}}{a+\sqrt {a^2-1}}\right )-\int e^{-i \arcsin (a+b x)} \operatorname {PolyLog}\left (2,-\frac {i e^{i \arcsin (a+b x)}}{a+\sqrt {a^2-1}}\right )de^{i \arcsin (a+b x)}\right )-\arcsin (a+b x)^2 \log \left (1+\frac {i e^{i \arcsin (a+b x)}}{\sqrt {a^2-1}+a}\right )\right )}{2 \sqrt {a^2-1}}-\frac {i \left (2 \left (i \arcsin (a+b x) \operatorname {PolyLog}\left (2,-\frac {i e^{i \arcsin (a+b x)}}{a-\sqrt {a^2-1}}\right )-\int e^{-i \arcsin (a+b x)} \operatorname {PolyLog}\left (2,-\frac {i e^{i \arcsin (a+b x)}}{a-\sqrt {a^2-1}}\right )de^{i \arcsin (a+b x)}\right )-\arcsin (a+b x)^2 \log \left (1+\frac {i e^{i \arcsin (a+b x)}}{a-\sqrt {a^2-1}}\right )\right )}{2 \sqrt {a^2-1}}\right )\right )\)

\(\Big \downarrow \) 7143

\(\displaystyle b \left (-\frac {\arcsin (a+b x)^3}{b x}+6 \left (\frac {i \left (2 \left (i \arcsin (a+b x) \operatorname {PolyLog}\left (2,-\frac {i e^{i \arcsin (a+b x)}}{a+\sqrt {a^2-1}}\right )-\operatorname {PolyLog}\left (3,-\frac {i e^{i \arcsin (a+b x)}}{a+\sqrt {a^2-1}}\right )\right )-\arcsin (a+b x)^2 \log \left (1+\frac {i e^{i \arcsin (a+b x)}}{\sqrt {a^2-1}+a}\right )\right )}{2 \sqrt {a^2-1}}-\frac {i \left (2 \left (i \arcsin (a+b x) \operatorname {PolyLog}\left (2,-\frac {i e^{i \arcsin (a+b x)}}{a-\sqrt {a^2-1}}\right )-\operatorname {PolyLog}\left (3,-\frac {i e^{i \arcsin (a+b x)}}{a-\sqrt {a^2-1}}\right )\right )-\arcsin (a+b x)^2 \log \left (1+\frac {i e^{i \arcsin (a+b x)}}{a-\sqrt {a^2-1}}\right )\right )}{2 \sqrt {a^2-1}}\right )\right )\)

input

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

output

b*(-(ArcSin[a + b*x]^3/(b*x)) + 6*(((-1/2*I)*(-(ArcSin[a + b*x]^2*Log[1 + 
(I*E^(I*ArcSin[a + b*x]))/(a - Sqrt[-1 + a^2])]) + 2*(I*ArcSin[a + b*x]*Po 
lyLog[2, ((-I)*E^(I*ArcSin[a + b*x]))/(a - Sqrt[-1 + a^2])] - PolyLog[3, ( 
(-I)*E^(I*ArcSin[a + b*x]))/(a - Sqrt[-1 + a^2])])))/Sqrt[-1 + a^2] + ((I/ 
2)*(-(ArcSin[a + b*x]^2*Log[1 + (I*E^(I*ArcSin[a + b*x]))/(a + Sqrt[-1 + a 
^2])]) + 2*(I*ArcSin[a + b*x]*PolyLog[2, ((-I)*E^(I*ArcSin[a + b*x]))/(a + 
 Sqrt[-1 + a^2])] - PolyLog[3, ((-I)*E^(I*ArcSin[a + b*x]))/(a + Sqrt[-1 + 
 a^2])])))/Sqrt[-1 + a^2]))
 

3.2.42.3.1 Defintions of rubi rules used

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 2720

Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Simp[v/D[v, x] 
   Subst[Int[FunctionOfExponentialFunction[u, x]/x, x], x, v], x]] /; Funct 
ionOfExponentialQ[u, x] &&  !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; FreeQ 
[{a, m, n}, x] && IntegerQ[m*n]] &&  !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x)) 
*(F_)[v_] /; FreeQ[{a, b, c}, x] && InverseFunctionQ[F[x]]]
 

rule 3011

Int[Log[1 + (e_.)*((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.)]*((f_.) + (g_.) 
*(x_))^(m_.), x_Symbol] :> Simp[(-(f + g*x)^m)*(PolyLog[2, (-e)*(F^(c*(a + 
b*x)))^n]/(b*c*n*Log[F])), x] + Simp[g*(m/(b*c*n*Log[F]))   Int[(f + g*x)^( 
m - 1)*PolyLog[2, (-e)*(F^(c*(a + b*x)))^n], x], x] /; FreeQ[{F, a, b, c, e 
, f, g, n}, x] && GtQ[m, 0]
 

rule 3042

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

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 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]
 

rule 7143

Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_S 
ymbol] :> Simp[PolyLog[n + 1, c*(a + b*x)^p]/(e*p), x] /; FreeQ[{a, b, c, d 
, e, n, p}, x] && EqQ[b*d, a*e]
 

3.2.42.4 Maple [F]

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

input

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

output

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

3.2.42.5 Fricas [F]

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

input

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

output

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

3.2.42.6 Sympy [F]

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

input

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

output

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

3.2.42.7 Maxima [F(-2)]

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

input

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

3.2.42.8 Giac [F]

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

input

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

output

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

3.2.42.9 Mupad [F(-1)]

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

input

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

output

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