3.1.92 \(\int \frac {1}{1-x^2} \, dx\) [92]

Optimal. Leaf size=2 \[ \text {arctanh}(x) \]

[Out]

arctanh(x)

________________________________________________________________________________________

Rubi [A]
time = 0.00, antiderivative size = 2, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {212} \begin {gather*} \text {arctanh}(x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(1 - x^2)^(-1),x]

[Out]

ArcTanh[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])

Rubi steps

\begin {gather*} \begin {aligned} \text {Integral} &=\text {arctanh}(x)\\ \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(19\) vs. \(2(2)=4\).
time = 0.00, size = 19, normalized size = 9.50 \begin {gather*} -\frac {1}{2} \log (1-x)+\frac {1}{2} \log (1+x) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(1 - x^2)^(-1),x]

[Out]

-1/2*Log[1 - x] + Log[1 + x]/2

________________________________________________________________________________________

Maple [A]
time = 0.00, size = 3, normalized size = 1.50

method	result	size
default	\(\arctanh \left (x \right )\)	\(3\)
meijerg	\(\arctanh \left (x \right )\)	\(3\)
norman	\(-\frac {\ln \left (x -1\right )}{2}+\frac {\ln \left (1+x \right )}{2}\)	\(14\)
risch	\(-\frac {\ln \left (x -1\right )}{2}+\frac {\ln \left (1+x \right )}{2}\)	\(14\)
parallelrisch	\(-\frac {\ln \left (x -1\right )}{2}+\frac {\ln \left (1+x \right )}{2}\)	\(14\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

arctanh(x)

________________________________________________________________________________________

Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 13 vs. \(2 (2) = 4\).
time = 0.35, size = 13, normalized size = 6.50 \begin {gather*} \frac {1}{2} \, \log \left (x + 1\right ) - \frac {1}{2} \, \log \left (x - 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*log(x + 1) - 1/2*log(x - 1)

________________________________________________________________________________________

Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 13 vs. \(2 (2) = 4\).
time = 0.59, size = 13, normalized size = 6.50 \begin {gather*} \frac {1}{2} \, \log \left (x + 1\right ) - \frac {1}{2} \, \log \left (x - 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*log(x + 1) - 1/2*log(x - 1)

________________________________________________________________________________________

Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 12 vs. \(2 (2) = 4\).
time = 0.03, size = 12, normalized size = 6.00 \begin {gather*} - \frac {\log {\left (x - 1 \right )}}{2} + \frac {\log {\left (x + 1 \right )}}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-log(x - 1)/2 + log(x + 1)/2

________________________________________________________________________________________

Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 15 vs. \(2 (2) = 4\).
time = 0.44, size = 15, normalized size = 7.50 \begin {gather*} \frac {1}{2} \, \log \left ({\left | x + 1 \right |}\right ) - \frac {1}{2} \, \log \left ({\left | x - 1 \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Mupad [B]
time = 0.00, size = 2, normalized size = 1.00 \begin {gather*} \mathrm {atanh}\left (x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-1/(x^2 - 1),x)

[Out]

atanh(x)

________________________________________________________________________________________