\(\int \frac {\arcsin (a x)}{x^3 \sqrt {1-a^2 x^2}} \, dx\) [106]

Optimal result

Integrand size = 22, antiderivative size = 98 \[ \int \frac {\arcsin (a x)}{x^3 \sqrt {1-a^2 x^2}} \, dx=-\frac {a}{2 x}-\frac {\sqrt {1-a^2 x^2} \arcsin (a x)}{2 x^2}-a^2 \arcsin (a x) \text {arctanh}\left (e^{i \arcsin (a x)}\right )+\frac {1}{2} i a^2 \operatorname {PolyLog}\left (2,-e^{i \arcsin (a x)}\right )-\frac {1}{2} i a^2 \operatorname {PolyLog}\left (2,e^{i \arcsin (a x)}\right ) \] Output:

-1/2*a/x-1/2*(-a^2*x^2+1)^(1/2)*arcsin(a*x)/x^2-a^2*arcsin(a*x)*arctanh(I* 
a*x+(-a^2*x^2+1)^(1/2))+1/2*I*a^2*polylog(2,-I*a*x-(-a^2*x^2+1)^(1/2))-1/2 
*I*a^2*polylog(2,I*a*x+(-a^2*x^2+1)^(1/2))
 

Mathematica [A] (verified)

Time = 0.61 (sec) , antiderivative size = 137, normalized size of antiderivative = 1.40 \[ \int \frac {\arcsin (a x)}{x^3 \sqrt {1-a^2 x^2}} \, dx=\frac {1}{8} a^2 \left (-2 \cot \left (\frac {1}{2} \arcsin (a x)\right )-\arcsin (a x) \csc ^2\left (\frac {1}{2} \arcsin (a x)\right )+4 \arcsin (a x) \log \left (1-e^{i \arcsin (a x)}\right )-4 \arcsin (a x) \log \left (1+e^{i \arcsin (a x)}\right )+4 i \operatorname {PolyLog}\left (2,-e^{i \arcsin (a x)}\right )-4 i \operatorname {PolyLog}\left (2,e^{i \arcsin (a x)}\right )+\arcsin (a x) \sec ^2\left (\frac {1}{2} \arcsin (a x)\right )-2 \tan \left (\frac {1}{2} \arcsin (a x)\right )\right ) \] Input:

Integrate[ArcSin[a*x]/(x^3*Sqrt[1 - a^2*x^2]),x]
 

Output:

(a^2*(-2*Cot[ArcSin[a*x]/2] - ArcSin[a*x]*Csc[ArcSin[a*x]/2]^2 + 4*ArcSin[ 
a*x]*Log[1 - E^(I*ArcSin[a*x])] - 4*ArcSin[a*x]*Log[1 + E^(I*ArcSin[a*x])] 
 + (4*I)*PolyLog[2, -E^(I*ArcSin[a*x])] - (4*I)*PolyLog[2, E^(I*ArcSin[a*x 
])] + ArcSin[a*x]*Sec[ArcSin[a*x]/2]^2 - 2*Tan[ArcSin[a*x]/2]))/8
 

Rubi [A] (verified)

Time = 0.52 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.95, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.318, Rules used = {5204, 15, 5218, 3042, 4671, 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.

\(\displaystyle \int \frac {\arcsin (a x)}{x^3 \sqrt {1-a^2 x^2}} \, dx\)

\(\Big \downarrow \) 5204

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

\(\Big \downarrow \) 15

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

\(\Big \downarrow \) 5218

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

\(\Big \downarrow \) 3042

\(\displaystyle \frac {1}{2} a^2 \int \arcsin (a x) \csc (\arcsin (a x))d\arcsin (a x)-\frac {\sqrt {1-a^2 x^2} \arcsin (a x)}{2 x^2}-\frac {a}{2 x}\)

\(\Big \downarrow \) 4671

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

\(\Big \downarrow \) 2715

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

\(\Big \downarrow \) 2838

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

Input:

Int[ArcSin[a*x]/(x^3*Sqrt[1 - a^2*x^2]),x]
 

Output:

-1/2*a/x - (Sqrt[1 - a^2*x^2]*ArcSin[a*x])/(2*x^2) + (a^2*(-2*ArcSin[a*x]* 
ArcTanh[E^(I*ArcSin[a*x])] + I*PolyLog[2, -E^(I*ArcSin[a*x])] - I*PolyLog[ 
2, E^(I*ArcSin[a*x])]))/2
 

Defintions of rubi rules used

rule 15

Int[(a_.)*(x_)^(m_.), x_Symbol] :> Simp[a*(x^(m + 1)/(m + 1)), x] /; FreeQ[ 
{a, m}, x] && NeQ[m, -1]
 

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 4671

Int[csc[(e_.) + (f_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[- 
2*(c + d*x)^m*(ArcTanh[E^(I*(e + f*x))]/f), x] + (-Simp[d*(m/f)   Int[(c + 
d*x)^(m - 1)*Log[1 - E^(I*(e + f*x))], x], x] + Simp[d*(m/f)   Int[(c + d*x 
)^(m - 1)*Log[1 + E^(I*(e + f*x))], x], x]) /; FreeQ[{c, d, e, f}, x] && IG 
tQ[m, 0]
 

rule 5204

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*((d_) + (e_. 
)*(x_)^2)^(p_), x_Symbol] :> Simp[(f*x)^(m + 1)*(d + e*x^2)^(p + 1)*((a + b 
*ArcSin[c*x])^n/(d*f*(m + 1))), x] + (Simp[c^2*((m + 2*p + 3)/(f^2*(m + 1)) 
)   Int[(f*x)^(m + 2)*(d + e*x^2)^p*(a + b*ArcSin[c*x])^n, x], x] - Simp[b* 
c*(n/(f*(m + 1)))*Simp[(d + e*x^2)^p/(1 - c^2*x^2)^p]   Int[(f*x)^(m + 1)*( 
1 - c^2*x^2)^(p + 1/2)*(a + b*ArcSin[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, 
 c, d, e, f, p}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && ILtQ[m, -1]
 

rule 5218

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

Maple [A] (verified)

Time = 0.74 (sec) , antiderivative size = 171, normalized size of antiderivative = 1.74

method	result	size
default	\(-\frac {\sqrt {-a^{2} x^{2}+1}\, \left (\arcsin \left (a x \right ) a^{2} x^{2}-\sqrt {-a^{2} x^{2}+1}\, x a -\arcsin \left (a x \right )\right )}{2 \left (a^{2} x^{2}-1\right ) x^{2}}+\frac {i a^{2} \left (i \arcsin \left (a x \right ) \ln \left (1+i a x +\sqrt {-a^{2} x^{2}+1}\right )-i \arcsin \left (a x \right ) \ln \left (1-i a x -\sqrt {-a^{2} x^{2}+1}\right )+\operatorname {polylog}\left (2, -i a x -\sqrt {-a^{2} x^{2}+1}\right )-\operatorname {polylog}\left (2, i a x +\sqrt {-a^{2} x^{2}+1}\right )\right )}{2}\)	\(171\)

Input:

int(arcsin(a*x)/x^3/(-a^2*x^2+1)^(1/2),x,method=_RETURNVERBOSE)
 

Output:

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

Fricas [F]

\[ \int \frac {\arcsin (a x)}{x^3 \sqrt {1-a^2 x^2}} \, dx=\int { \frac {\arcsin \left (a x\right )}{\sqrt {-a^{2} x^{2} + 1} x^{3}} \,d x } \] Input:

integrate(arcsin(a*x)/x^3/(-a^2*x^2+1)^(1/2),x, algorithm="fricas")
 

Output:

integral(-sqrt(-a^2*x^2 + 1)*arcsin(a*x)/(a^2*x^5 - x^3), x)
 

Sympy [F]

\[ \int \frac {\arcsin (a x)}{x^3 \sqrt {1-a^2 x^2}} \, dx=\int \frac {\operatorname {asin}{\left (a x \right )}}{x^{3} \sqrt {- \left (a x - 1\right ) \left (a x + 1\right )}}\, dx \] Input:

integrate(asin(a*x)/x**3/(-a**2*x**2+1)**(1/2),x)
 

Output:

Integral(asin(a*x)/(x**3*sqrt(-(a*x - 1)*(a*x + 1))), x)
 

Maxima [F]

\[ \int \frac {\arcsin (a x)}{x^3 \sqrt {1-a^2 x^2}} \, dx=\int { \frac {\arcsin \left (a x\right )}{\sqrt {-a^{2} x^{2} + 1} x^{3}} \,d x } \] Input:

integrate(arcsin(a*x)/x^3/(-a^2*x^2+1)^(1/2),x, algorithm="maxima")
                                                                                    
                                                                                    
 

Output:

integrate(arcsin(a*x)/(sqrt(-a^2*x^2 + 1)*x^3), x)
 

Giac [F]

\[ \int \frac {\arcsin (a x)}{x^3 \sqrt {1-a^2 x^2}} \, dx=\int { \frac {\arcsin \left (a x\right )}{\sqrt {-a^{2} x^{2} + 1} x^{3}} \,d x } \] Input:

integrate(arcsin(a*x)/x^3/(-a^2*x^2+1)^(1/2),x, algorithm="giac")
 

Output:

integrate(arcsin(a*x)/(sqrt(-a^2*x^2 + 1)*x^3), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\arcsin (a x)}{x^3 \sqrt {1-a^2 x^2}} \, dx=\int \frac {\mathrm {asin}\left (a\,x\right )}{x^3\,\sqrt {1-a^2\,x^2}} \,d x \] Input:

int(asin(a*x)/(x^3*(1 - a^2*x^2)^(1/2)),x)
 

Output:

int(asin(a*x)/(x^3*(1 - a^2*x^2)^(1/2)), x)
 

Reduce [F]

\[ \int \frac {\arcsin (a x)}{x^3 \sqrt {1-a^2 x^2}} \, dx=\int \frac {\mathit {asin} \left (a x \right )}{\sqrt {-a^{2} x^{2}+1}\, x^{3}}d x \] Input:

int(asin(a*x)/x^3/(-a^2*x^2+1)^(1/2),x)
 

Output:

int(asin(a*x)/(sqrt( - a**2*x**2 + 1)*x**3),x)