3.1.93 \(\int \frac {1}{-1+x^2} \, dx\) [93]

Optimal. Leaf size=4 \[ -\coth ^{-1}(x) \]

[Out]

-arccoth(x)

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Rubi [A]
time = 0.00, antiderivative size = 4, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {213} \begin {gather*} -\text {arctanh}(x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-1 + x^2)^(-1),x]

[Out]

-ArcTanh[x]

Rule 213

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

Rubi steps

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

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(19\) vs. \(2(4)=8\).
time = 0.00, size = 19, normalized size = 4.75 \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]

Log[1 - x]/2 - Log[1 + x]/2

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Maple [A]
time = 0.02, size = 5, normalized size = 1.25

method	result	size
default	\(-\arctanh \left (x \right )\)	\(5\)
meijerg	\(-\arctanh \left (x \right )\)	\(5\)
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)

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 13 vs. \(2 (4) = 8\).
time = 0.34, size = 13, normalized size = 3.25 \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)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 13 vs. \(2 (4) = 8\).
time = 0.57, size = 13, normalized size = 3.25 \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)

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 12 vs. \(2 (3) = 6\).
time = 0.02, size = 12, normalized size = 3.00 \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

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 15 vs. \(2 (4) = 8\).
time = 0.46, size = 15, normalized size = 3.75 \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))

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Mupad [B]
time = 0.10, size = 4, 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)

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