3.7.65 \(\int \frac {\arcsin (x)}{x (1-x^2)^{3/2}} \, dx\) [665]

3.7.65.1 Optimal result

Integrand size = 17, antiderivative size = 62 \[ \int \frac {\arcsin (x)}{x \left (1-x^2\right )^{3/2}} \, dx=\frac {\arcsin (x)}{\sqrt {1-x^2}}-2 \arcsin (x) \text {arctanh}\left (e^{i \arcsin (x)}\right )-\text {arctanh}(x)+i \operatorname {PolyLog}\left (2,-e^{i \arcsin (x)}\right )-i \operatorname {PolyLog}\left (2,e^{i \arcsin (x)}\right ) \]

output

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

3.7.65.2 Mathematica [A] (verified)

Time = 0.16 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.81 \[ \int \frac {\arcsin (x)}{x \left (1-x^2\right )^{3/2}} \, dx=\frac {\arcsin (x)}{\sqrt {1-x^2}}+\arcsin (x) \log \left (1-e^{i \arcsin (x)}\right )-\arcsin (x) \log \left (1+e^{i \arcsin (x)}\right )+\log \left (\cos \left (\frac {\arcsin (x)}{2}\right )-\sin \left (\frac {\arcsin (x)}{2}\right )\right )-\log \left (\cos \left (\frac {\arcsin (x)}{2}\right )+\sin \left (\frac {\arcsin (x)}{2}\right )\right )+i \operatorname {PolyLog}\left (2,-e^{i \arcsin (x)}\right )-i \operatorname {PolyLog}\left (2,e^{i \arcsin (x)}\right ) \]

input

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

output

ArcSin[x]/Sqrt[1 - x^2] + ArcSin[x]*Log[1 - E^(I*ArcSin[x])] - ArcSin[x]*L 
og[1 + E^(I*ArcSin[x])] + Log[Cos[ArcSin[x]/2] - Sin[ArcSin[x]/2]] - Log[C 
os[ArcSin[x]/2] + Sin[ArcSin[x]/2]] + I*PolyLog[2, -E^(I*ArcSin[x])] - I*P 
olyLog[2, E^(I*ArcSin[x])]
 

3.7.65.3 Rubi [A] (verified)

Time = 0.40 (sec) , antiderivative size = 62, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.412, Rules used = {5208, 219, 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.

\(\displaystyle \int \frac {\arcsin (x)}{x \left (1-x^2\right )^{3/2}} \, dx\)

\(\Big \downarrow \) 5208

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

\(\Big \downarrow \) 219

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

\(\Big \downarrow \) 5218

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4671

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

\(\Big \downarrow \) 2715

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

\(\Big \downarrow \) 2838

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

input

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

output

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

3.7.65.3.1 Defintions of rubi rules used

rule 219

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* 
ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt 
Q[a, 0] || LtQ[b, 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 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 5208

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/(2*d*f*(p + 1))), x] + (Simp[(m + 2*p + 3)/(2*d*(p + 1)) 
   Int[(f*x)^m*(d + e*x^2)^(p + 1)*(a + b*ArcSin[c*x])^n, x], x] + Simp[b*c 
*(n/(2*f*(p + 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, m}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && LtQ[p, -1] &&  !G 
tQ[m, 1] && (IntegerQ[m] || IntegerQ[p] || EqQ[n, 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]
 

3.7.65.4 Maple [B] (verified)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 429 vs. \(2 (76 ) = 152\).

Time = 0.69 (sec) , antiderivative size = 430, normalized size of antiderivative = 6.94

method	result	size
default	\(-\frac {\sqrt {-x^{2}+1}\, \arcsin \left (x \right )}{x^{2}-1}-\frac {i \left (\ln \left (i x +\sqrt {-x^{2}+1}+1\right )-\ln \left (i x +\sqrt {-x^{2}+1}-1\right )-2 \arctan \left (i x +\sqrt {-x^{2}+1}\right )\right )}{2}+\frac {i \left (i \arcsin \left (x \right ) \ln \left (i x +\sqrt {-x^{2}+1}+1\right )+\arcsin \left (x \right ) \ln \left (1+i \left (i x +\sqrt {-x^{2}+1}\right )\right )-\arcsin \left (x \right ) \ln \left (1-i \left (i x +\sqrt {-x^{2}+1}\right )\right )-i \operatorname {dilog}\left (1+i \left (i x +\sqrt {-x^{2}+1}\right )\right )+i \operatorname {dilog}\left (1-i \left (i x +\sqrt {-x^{2}+1}\right )\right )+\operatorname {dilog}\left (i x +\sqrt {-x^{2}+1}+1\right )+\operatorname {dilog}\left (i x +\sqrt {-x^{2}+1}\right )\right )}{2}+\frac {i \left (i \arcsin \left (x \right ) \ln \left (i x +\sqrt {-x^{2}+1}+1\right )-\arcsin \left (x \right ) \ln \left (1+i \left (i x +\sqrt {-x^{2}+1}\right )\right )+\arcsin \left (x \right ) \ln \left (1-i \left (i x +\sqrt {-x^{2}+1}\right )\right )+i \operatorname {dilog}\left (1+i \left (i x +\sqrt {-x^{2}+1}\right )\right )-i \operatorname {dilog}\left (1-i \left (i x +\sqrt {-x^{2}+1}\right )\right )+\operatorname {dilog}\left (i x +\sqrt {-x^{2}+1}+1\right )+\operatorname {dilog}\left (i x +\sqrt {-x^{2}+1}\right )\right )}{2}+\frac {i \left (\ln \left (i x +\sqrt {-x^{2}+1}+1\right )-\ln \left (i x +\sqrt {-x^{2}+1}-1\right )+2 \arctan \left (i x +\sqrt {-x^{2}+1}\right )\right )}{2}\)	\(430\)

input

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

output

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

3.7.65.5 Fricas [F]

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

input

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

output

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

3.7.65.6 Sympy [F]

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

input

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

output

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

3.7.65.7 Maxima [F]

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

input

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

output

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

3.7.65.8 Giac [F]

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

input

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

output

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

3.7.65.9 Mupad [F(-1)]

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

input

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

output

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

3.7.65.10 Reduce [F]

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

input

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

output

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