3.2.5 \(\int \frac {1}{x (1-x^2)} \, dx\) [105]

Optimal. Leaf size=14 \[ \log \left (\frac {x}{\sqrt {1-x^2}}\right ) \]

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 0.00, antiderivative size = 15, normalized size of antiderivative = 1.07, number of steps used = 4, number of rules used = 4, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {272, 36, 31, 29} \begin {gather*} \log (x)-\frac {1}{2} \log \left (1-x^2\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/(x*(1 - x^2)),x]

[Out]

Log[x] - Log[1 - x^2]/2

Rule 29

Int[(x_)^(-1), x_Symbol] :> Simp[Log[x], x]

Rule 31

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

Rule 36

Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :> Dist[b/(b*c - a*d), Int[1/(a + b*x), x], x] -
Dist[d/(b*c - a*d), Int[1/(c + d*x), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0]

Rule 272

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rubi steps

\begin {gather*} \begin {aligned} \text {Integral} &=\frac {1}{2} \text {Subst}\left (\int \frac {1}{(1-x) x} \, dx,x,x^2\right )\\ &=\frac {1}{2} \text {Subst}\left (\int \frac {1}{1-x} \, dx,x,x^2\right )+\frac {1}{2} \text {Subst}\left (\int \frac {1}{x} \, dx,x,x^2\right )\\ &=\log (x)-\frac {1}{2} \log \left (1-x^2\right )\\ \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [A]
time = 0.00, size = 15, normalized size = 1.07 \begin {gather*} \log (x)-\frac {1}{2} \log \left (1-x^2\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/(x*(1 - x^2)),x]

[Out]

Log[x] - Log[1 - x^2]/2

________________________________________________________________________________________

Maple [A]
time = 0.02, size = 16, normalized size = 1.14

method	result	size
risch	\(\ln \left (x \right )-\frac {\ln \left (x^{2}-1\right )}{2}\)	\(12\)
default	\(-\frac {\ln \left (1+x \right )}{2}-\frac {\ln \left (x -1\right )}{2}+\ln \left (x \right )\)	\(16\)
norman	\(-\frac {\ln \left (1+x \right )}{2}-\frac {\ln \left (x -1\right )}{2}+\ln \left (x \right )\)	\(16\)
parallelrisch	\(-\frac {\ln \left (1+x \right )}{2}-\frac {\ln \left (x -1\right )}{2}+\ln \left (x \right )\)	\(16\)
meijerg	\(\ln \left (x \right )+\frac {i \pi }{2}-\frac {\ln \left (-x^{2}+1\right )}{2}\)	\(18\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [A]
time = 0.34, size = 15, normalized size = 1.07 \begin {gather*} -\frac {1}{2} \, \log \left (x^{2} - 1\right ) + \frac {1}{2} \, \log \left (x^{2}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*log(x^2 - 1) + 1/2*log(x^2)

________________________________________________________________________________________

Fricas [A]
time = 0.55, size = 11, normalized size = 0.79 \begin {gather*} -\frac {1}{2} \, \log \left (x^{2} - 1\right ) + \log \left (x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*log(x^2 - 1) + log(x)

________________________________________________________________________________________

Sympy [A]
time = 0.03, size = 10, normalized size = 0.71 \begin {gather*} \log {\left (x \right )} - \frac {\log {\left (x^{2} - 1 \right )}}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(x) - log(x**2 - 1)/2

________________________________________________________________________________________

Giac [A]
time = 0.44, size = 16, normalized size = 1.14 \begin {gather*} \frac {1}{2} \, \log \left (x^{2}\right ) - \frac {1}{2} \, \log \left ({\left | x^{2} - 1 \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*log(x^2) - 1/2*log(abs(x^2 - 1))

________________________________________________________________________________________

Mupad [B]
time = 0.10, size = 11, normalized size = 0.79 \begin {gather*} \ln \left (x\right )-\frac {\ln \left (x^2-1\right )}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-1/(x*(x^2 - 1)),x)

[Out]

log(x) - log(x^2 - 1)/2

________________________________________________________________________________________