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

Optimal result

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

[Out]

-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)

Rubi [A] (verified)

Time = 0.10 (sec) , antiderivative size = 62, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.353, Rules used = {4793, 4803, 4268, 2317, 2438, 212} \[ \int \frac {\arcsin (x)}{x \left (1-x^2\right )^{3/2}} \, dx=-2 \arcsin (x) \text {arctanh}\left (e^{i \arcsin (x)}\right )+i \operatorname {PolyLog}\left (2,-e^{i \arcsin (x)}\right )-i \operatorname {PolyLog}\left (2,e^{i \arcsin (x)}\right )+\frac {\arcsin (x)}{\sqrt {1-x^2}}-\text {arctanh}(x) \]

[In]

Int[ArcSin[x]/(x*(1 - x^2)^(3/2)),x]

[Out]

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])]

Rule 212

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] && (GtQ[a, 0] || LtQ[b, 0])

Rule 2317

Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] :> Dist[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 2438

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 4268

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] + (-Dist[d*(m/f), Int[(c + d*x)^(m - 1)*Log[1 - E^(I*(e + f*x))], x], x] + Dist[d*(m/f), Int[(c +
d*x)^(m - 1)*Log[1 + E^(I*(e + f*x))], x], x]) /; FreeQ[{c, d, e, f}, x] && IGtQ[m, 0]

Rule 4793

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] + (Dist[(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] + Dist[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]) /; Fre
eQ[{a, b, c, d, e, f, m}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && LtQ[p, -1] &&  !GtQ[m, 1] && (IntegerQ[m] ||
IntegerQ[p] || EqQ[n, 1])

Rule 4803

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

Rubi steps \begin{align*} \text {integral}& = \frac {\arcsin (x)}{\sqrt {1-x^2}}-\int \frac {1}{1-x^2} \, dx+\int \frac {\arcsin (x)}{x \sqrt {1-x^2}} \, dx \\ & = \frac {\arcsin (x)}{\sqrt {1-x^2}}-\text {arctanh}(x)+\text {Subst}(\int x \csc (x) \, dx,x,\arcsin (x)) \\ & = \frac {\arcsin (x)}{\sqrt {1-x^2}}-2 \arcsin (x) \text {arctanh}\left (e^{i \arcsin (x)}\right )-\text {arctanh}(x)-\text {Subst}\left (\int \log \left (1-e^{i x}\right ) \, dx,x,\arcsin (x)\right )+\text {Subst}\left (\int \log \left (1+e^{i x}\right ) \, dx,x,\arcsin (x)\right ) \\ & = \frac {\arcsin (x)}{\sqrt {1-x^2}}-2 \arcsin (x) \text {arctanh}\left (e^{i \arcsin (x)}\right )-\text {arctanh}(x)+i \text {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{i \arcsin (x)}\right )-i \text {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{i \arcsin (x)}\right ) \\ & = \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 ) \\ \end{align*}

Mathematica [A] (verified)

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

[In]

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

[Out]

ArcSin[x]/Sqrt[1 - x^2] + ArcSin[x]*Log[1 - E^(I*ArcSin[x])] - ArcSin[x]*Log[1 + E^(I*ArcSin[x])] + Log[Cos[Ar
cSin[x]/2] - Sin[ArcSin[x]/2]] - Log[Cos[ArcSin[x]/2] + Sin[ArcSin[x]/2]] + I*PolyLog[2, -E^(I*ArcSin[x])] - I
*PolyLog[2, E^(I*ArcSin[x])]

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

[In]

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

[Out]

-(-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)))

Fricas [F]

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

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

int(asin(x)/(x*(1 - x^2)^(3/2)),x)

[Out]

int(asin(x)/(x*(1 - x^2)^(3/2)), x)