3.136 \(\int \frac {1}{-1+x^4} \, dx\)

Optimal. Leaf size=13 \[ -\frac {1}{2} \tan ^{-1}(x)-\frac {1}{2} \tanh ^{-1}(x) \]

[Out]

-1/2*arctan(x)-1/2*arctanh(x)

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Rubi [A]  time = 0.00, antiderivative size = 13, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {212, 206, 203} \[ -\frac {1}{2} \tan ^{-1}(x)-\frac {1}{2} \tanh ^{-1}(x) \]

Antiderivative was successfully verified.

[In]

Int[(-1 + x^4)^(-1),x]

[Out]

-ArcTan[x]/2 - ArcTanh[x]/2

Rule 203

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

Rule 206

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

Rule 212

Int[((a_) + (b_.)*(x_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[-(a/b), 2]], s = Denominator[Rt[-(a/b), 2]
]}, Dist[r/(2*a), Int[1/(r - s*x^2), x], x] + Dist[r/(2*a), Int[1/(r + s*x^2), x], x]] /; FreeQ[{a, b}, x] &&
!GtQ[a/b, 0]

Rubi steps

\begin {align*} \int \frac {1}{-1+x^4} \, dx &=-\left (\frac {1}{2} \int \frac {1}{1-x^2} \, dx\right )-\frac {1}{2} \int \frac {1}{1+x^2} \, dx\\ &=-\frac {1}{2} \tan ^{-1}(x)-\frac {1}{2} \tanh ^{-1}(x)\\ \end {align*}

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Mathematica [A]  time = 0.00, size = 25, normalized size = 1.92 \[ \frac {1}{4} \log (1-x)-\frac {1}{4} \log (x+1)-\frac {1}{2} \tan ^{-1}(x) \]

Antiderivative was successfully verified.

[In]

Integrate[(-1 + x^4)^(-1),x]

[Out]

-1/2*ArcTan[x] + Log[1 - x]/4 - Log[1 + x]/4

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fricas [A]  time = 0.40, size = 17, normalized size = 1.31 \[ -\frac {1}{2} \, \arctan \relax (x) - \frac {1}{4} \, \log \left (x + 1\right ) + \frac {1}{4} \, \log \left (x - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(x^4-1),x, algorithm="fricas")

[Out]

-1/2*arctan(x) - 1/4*log(x + 1) + 1/4*log(x - 1)

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giac [B]  time = 0.01, size = 19, normalized size = 1.46 \[ -\frac {1}{2} \, \arctan \relax (x) - \frac {1}{4} \, \log \left ({\left | x + 1 \right |}\right ) + \frac {1}{4} \, \log \left ({\left | x - 1 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(x^4-1),x, algorithm="giac")

[Out]

-1/2*arctan(x) - 1/4*log(abs(x + 1)) + 1/4*log(abs(x - 1))

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maple [A]  time = 0.00, size = 10, normalized size = 0.77 \[ -\frac {\arctanh \relax (x )}{2}-\frac {\arctan \relax (x )}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^4-1),x)

[Out]

-1/2*arctan(x)-1/2*arctanh(x)

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maxima [A]  time = 1.37, size = 17, normalized size = 1.31 \[ -\frac {1}{2} \, \arctan \relax (x) - \frac {1}{4} \, \log \left (x + 1\right ) + \frac {1}{4} \, \log \left (x - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(x^4-1),x, algorithm="maxima")

[Out]

-1/2*arctan(x) - 1/4*log(x + 1) + 1/4*log(x - 1)

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mupad [B]  time = 0.03, size = 9, normalized size = 0.69 \[ -\frac {\mathrm {atan}\relax (x)}{2}-\frac {\mathrm {atanh}\relax (x)}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^4 - 1),x)

[Out]

- atan(x)/2 - atanh(x)/2

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sympy [A]  time = 0.13, size = 17, normalized size = 1.31 \[ \frac {\log {\left (x - 1 \right )}}{4} - \frac {\log {\left (x + 1 \right )}}{4} - \frac {\operatorname {atan}{\relax (x )}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(x**4-1),x)

[Out]

log(x - 1)/4 - log(x + 1)/4 - atan(x)/2

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