3.36 \(\int \frac{1}{-2+x^4} \, dx\)

Optimal. Leaf size=35 \[ -\frac{\tan ^{-1}\left (\frac{x}{\sqrt [4]{2}}\right )}{2\ 2^{3/4}}-\frac{\tanh ^{-1}\left (\frac{x}{\sqrt [4]{2}}\right )}{2\ 2^{3/4}} \]

[Out]

-ArcTan[x/2^(1/4)]/(2*2^(3/4)) - ArcTanh[x/2^(1/4)]/(2*2^(3/4))

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Rubi [A]  time = 0.0107502, antiderivative size = 35, 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} \[ -\frac{\tan ^{-1}\left (\frac{x}{\sqrt [4]{2}}\right )}{2\ 2^{3/4}}-\frac{\tanh ^{-1}\left (\frac{x}{\sqrt [4]{2}}\right )}{2\ 2^{3/4}} \]

Antiderivative was successfully verified.

[In]

Int[(-2 + x^4)^(-1),x]

[Out]

-ArcTan[x/2^(1/4)]/(2*2^(3/4)) - ArcTanh[x/2^(1/4)]/(2*2^(3/4))

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}{-2+x^4} \, dx &=-\frac{\int \frac{1}{\sqrt{2}-x^2} \, dx}{2 \sqrt{2}}-\frac{\int \frac{1}{\sqrt{2}+x^2} \, dx}{2 \sqrt{2}}\\ &=-\frac{\tan ^{-1}\left (\frac{x}{\sqrt [4]{2}}\right )}{2\ 2^{3/4}}-\frac{\tanh ^{-1}\left (\frac{x}{\sqrt [4]{2}}\right )}{2\ 2^{3/4}}\\ \end{align*}

Mathematica [A]  time = 0.0177323, size = 43, normalized size = 1.23 \[ -\frac{-\log \left (2-2^{3/4} x\right )+\log \left (2^{3/4} x+2\right )+2 \tan ^{-1}\left (\frac{x}{\sqrt [4]{2}}\right )}{4\ 2^{3/4}} \]

Antiderivative was successfully verified.

[In]

Integrate[(-2 + x^4)^(-1),x]

[Out]

-(2*ArcTan[x/2^(1/4)] - Log[2 - 2^(3/4)*x] + Log[2 + 2^(3/4)*x])/(4*2^(3/4))

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Maple [A]  time = 0.002, size = 35, normalized size = 1. \begin{align*} -{\frac{\sqrt [4]{2}}{4}\arctan \left ({\frac{x{2}^{{\frac{3}{4}}}}{2}} \right ) }-{\frac{\sqrt [4]{2}}{8}\ln \left ({\frac{x+\sqrt [4]{2}}{x-\sqrt [4]{2}}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^4-2),x)

[Out]

-1/4*arctan(1/2*x*2^(3/4))*2^(1/4)-1/8*2^(1/4)*ln((x+2^(1/4))/(x-2^(1/4)))

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Maxima [A]  time = 1.44758, size = 46, normalized size = 1.31 \begin{align*} -\frac{1}{4} \cdot 2^{\frac{1}{4}} \arctan \left (\frac{1}{2} \cdot 2^{\frac{3}{4}} x\right ) + \frac{1}{8} \cdot 2^{\frac{1}{4}} \log \left (\frac{x - 2^{\frac{1}{4}}}{x + 2^{\frac{1}{4}}}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(x^4-2),x, algorithm="maxima")

[Out]

-1/4*2^(1/4)*arctan(1/2*2^(3/4)*x) + 1/8*2^(1/4)*log((x - 2^(1/4))/(x + 2^(1/4)))

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Fricas [B]  time = 2.07077, size = 201, normalized size = 5.74 \begin{align*} \frac{1}{8} \cdot 8^{\frac{3}{4}} \arctan \left (\frac{1}{4} \cdot 8^{\frac{1}{4}} \sqrt{2} \sqrt{2 \, x^{2} + 2 \, \sqrt{2}} - \frac{1}{2} \cdot 8^{\frac{1}{4}} x\right ) - \frac{1}{32} \cdot 8^{\frac{3}{4}} \log \left (4 \, x + 8^{\frac{3}{4}}\right ) + \frac{1}{32} \cdot 8^{\frac{3}{4}} \log \left (4 \, x - 8^{\frac{3}{4}}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(x^4-2),x, algorithm="fricas")

[Out]

1/8*8^(3/4)*arctan(1/4*8^(1/4)*sqrt(2)*sqrt(2*x^2 + 2*sqrt(2)) - 1/2*8^(1/4)*x) - 1/32*8^(3/4)*log(4*x + 8^(3/
4)) + 1/32*8^(3/4)*log(4*x - 8^(3/4))

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Sympy [A]  time = 0.317221, size = 46, normalized size = 1.31 \begin{align*} \frac{\sqrt [4]{2} \log{\left (x - \sqrt [4]{2} \right )}}{8} - \frac{\sqrt [4]{2} \log{\left (x + \sqrt [4]{2} \right )}}{8} - \frac{\sqrt [4]{2} \operatorname{atan}{\left (\frac{2^{\frac{3}{4}} x}{2} \right )}}{4} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(x**4-2),x)

[Out]

2**(1/4)*log(x - 2**(1/4))/8 - 2**(1/4)*log(x + 2**(1/4))/8 - 2**(1/4)*atan(2**(3/4)*x/2)/4

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Giac [A]  time = 1.11874, size = 53, normalized size = 1.51 \begin{align*} -\frac{1}{4} \cdot 2^{\frac{1}{4}} \arctan \left (\frac{1}{2} \cdot 2^{\frac{3}{4}} x\right ) - \frac{1}{8} \cdot 2^{\frac{1}{4}} \log \left ({\left | x + 2^{\frac{1}{4}} \right |}\right ) + \frac{1}{8} \cdot 2^{\frac{1}{4}} \log \left ({\left | x - 2^{\frac{1}{4}} \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(x^4-2),x, algorithm="giac")

[Out]

-1/4*2^(1/4)*arctan(1/2*2^(3/4)*x) - 1/8*2^(1/4)*log(abs(x + 2^(1/4))) + 1/8*2^(1/4)*log(abs(x - 2^(1/4)))