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

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

[Out]

-ArcTan[x]/2 - ArcTanh[x]/2

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Rubi [A]  time = 0.0031654, antiderivative size = 13, normalized size of antiderivative = 1., 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 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]

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 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])

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*}

Mathematica [A]  time = 0.0035192, 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]

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

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Maple [A]  time = 0., size = 10, normalized size = 0.8 \begin{align*} -{\frac{\arctan \left ( x \right ) }{2}}-{\frac{{\it Artanh} \left ( x \right ) }{2}} \end{align*}

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 = 1.41753, size = 23, normalized size = 1.77 \begin{align*} -\frac{1}{2} \, \arctan \left (x\right ) - \frac{1}{4} \, \log \left (x + 1\right ) + \frac{1}{4} \, \log \left (x - 1\right ) \end{align*}

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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Fricas [A]  time = 1.88316, size = 68, normalized size = 5.23 \begin{align*} -\frac{1}{2} \, \arctan \left (x\right ) - \frac{1}{4} \, \log \left (x + 1\right ) + \frac{1}{4} \, \log \left (x - 1\right ) \end{align*}

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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Sympy [A]  time = 0.114985, size = 17, normalized size = 1.31 \begin{align*} \frac{\log{\left (x - 1 \right )}}{4} - \frac{\log{\left (x + 1 \right )}}{4} - \frac{\operatorname{atan}{\left (x \right )}}{2} \end{align*}

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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Giac [B]  time = 1.08715, size = 26, normalized size = 2. \begin{align*} -\frac{1}{2} \, \arctan \left (x\right ) - \frac{1}{4} \, \log \left ({\left | x + 1 \right |}\right ) + \frac{1}{4} \, \log \left ({\left | x - 1 \right |}\right ) \end{align*}

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))