\(\int (\frac {x}{1-x^2}+\frac {1}{\sqrt {1-x^2} \arcsin (x)}) \, dx\) [473]

Optimal result

Integrand size = 28, antiderivative size = 16 \[ \int \left (\frac {x}{1-x^2}+\frac {1}{\sqrt {1-x^2} \arcsin (x)}\right ) \, dx=-\frac {1}{2} \log \left (1-x^2\right )+\log (\arcsin (x)) \]

[Out]

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

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {266, 4735} \[ \int \left (\frac {x}{1-x^2}+\frac {1}{\sqrt {1-x^2} \arcsin (x)}\right ) \, dx=\log (\arcsin (x))-\frac {1}{2} \log \left (1-x^2\right ) \]

[In]

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

[Out]

-1/2*Log[1 - x^2] + Log[ArcSin[x]]

Rule 266

Int[(x_)^(m_.)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Simp[Log[RemoveContent[a + b*x^n, x]]/(b*n), x] /; FreeQ
[{a, b, m, n}, x] && EqQ[m, n - 1]

Rule 4735

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

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

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00 \[ \int \left (\frac {x}{1-x^2}+\frac {1}{\sqrt {1-x^2} \arcsin (x)}\right ) \, dx=-\frac {1}{2} \log \left (1-x^2\right )+\log (\arcsin (x)) \]

[In]

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

[Out]

-1/2*Log[1 - x^2] + Log[ArcSin[x]]

Maple [A] (verified)

Time = 0.49 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.06

method	result	size
default	\(-\frac {\ln \left (x -1\right )}{2}-\frac {\ln \left (x +1\right )}{2}+\ln \left (\arcsin \left (x \right )\right )\)	\(17\)
parts	\(-\frac {\ln \left (x -1\right )}{2}-\frac {\ln \left (x +1\right )}{2}+\ln \left (\arcsin \left (x \right )\right )\)	\(17\)

[In]

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

[Out]

-1/2*ln(x-1)-1/2*ln(x+1)+ln(arcsin(x))

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.88 \[ \int \left (\frac {x}{1-x^2}+\frac {1}{\sqrt {1-x^2} \arcsin (x)}\right ) \, dx=-\frac {1}{2} \, \log \left (x^{2} - 1\right ) + \log \left (-\arcsin \left (x\right )\right ) \]

[In]

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

[Out]

-1/2*log(x^2 - 1) + log(-arcsin(x))

Sympy [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.75 \[ \int \left (\frac {x}{1-x^2}+\frac {1}{\sqrt {1-x^2} \arcsin (x)}\right ) \, dx=- \frac {\log {\left (x^{2} - 1 \right )}}{2} + \log {\left (\operatorname {asin}{\left (x \right )} \right )} \]

[In]

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

[Out]

-log(x**2 - 1)/2 + log(asin(x))

Maxima [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.75 \[ \int \left (\frac {x}{1-x^2}+\frac {1}{\sqrt {1-x^2} \arcsin (x)}\right ) \, dx=-\frac {1}{2} \, \log \left (x^{2} - 1\right ) + \log \left (\arcsin \left (x\right )\right ) \]

[In]

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

[Out]

-1/2*log(x^2 - 1) + log(arcsin(x))

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.88 \[ \int \left (\frac {x}{1-x^2}+\frac {1}{\sqrt {1-x^2} \arcsin (x)}\right ) \, dx=-\frac {1}{2} \, \log \left ({\left | x^{2} - 1 \right |}\right ) + \log \left ({\left | \arcsin \left (x\right ) \right |}\right ) \]

[In]

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

[Out]

-1/2*log(abs(x^2 - 1)) + log(abs(arcsin(x)))

Mupad [B] (verification not implemented)

Time = 0.34 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.75 \[ \int \left (\frac {x}{1-x^2}+\frac {1}{\sqrt {1-x^2} \arcsin (x)}\right ) \, dx=\ln \left (\mathrm {asin}\left (x\right )\right )-\frac {\ln \left (x^2-1\right )}{2} \]

[In]

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

[Out]

log(asin(x)) - log(x^2 - 1)/2