\(\int \frac {\arcsin (a x)}{x \sqrt {1-a^2 x^2}} \, dx\) [104]

Optimal result

Integrand size = 22, antiderivative size = 52 \[ \int \frac {\arcsin (a x)}{x \sqrt {1-a^2 x^2}} \, dx=-2 \arcsin (a x) \text {arctanh}\left (e^{i \arcsin (a x)}\right )+i \operatorname {PolyLog}\left (2,-e^{i \arcsin (a x)}\right )-i \operatorname {PolyLog}\left (2,e^{i \arcsin (a x)}\right ) \] Output:

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

Mathematica [A] (verified)

Time = 0.09 (sec) , antiderivative size = 71, normalized size of antiderivative = 1.37 \[ \int \frac {\arcsin (a x)}{x \sqrt {1-a^2 x^2}} \, dx=\arcsin (a x) \left (\log \left (1-e^{i \arcsin (a x)}\right )-\log \left (1+e^{i \arcsin (a x)}\right )\right )+i \operatorname {PolyLog}\left (2,-e^{i \arcsin (a x)}\right )-i \operatorname {PolyLog}\left (2,e^{i \arcsin (a x)}\right ) \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.37 (sec) , antiderivative size = 52, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {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.

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

\(\Big \downarrow \) 5218

\(\displaystyle \int \frac {\arcsin (a x)}{a x}d\arcsin (a x)\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \arcsin (a x) \csc (\arcsin (a x))d\arcsin (a x)\)

\(\Big \downarrow \) 4671

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

\(\Big \downarrow \) 2715

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

\(\Big \downarrow \) 2838

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

Input:

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

Output:

-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])]
 

Defintions of rubi rules used

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 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 = 0.24 (sec) , antiderivative size = 103, normalized size of antiderivative = 1.98

method	result	size
default	\(\arcsin \left (a x \right ) \ln \left (1-i a x -\sqrt {-a^{2} x^{2}+1}\right )-\arcsin \left (a x \right ) \ln \left (1+i a x +\sqrt {-a^{2} x^{2}+1}\right )+i \operatorname {dilog}\left (1+i a x +\sqrt {-a^{2} x^{2}+1}\right )-i \operatorname {dilog}\left (1-i a x -\sqrt {-a^{2} x^{2}+1}\right )\)	\(103\)

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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