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]

-1/4*arctan(1/2*x*2^(3/4))*2^(1/4)-1/4*arctanh(1/2*x*2^(3/4))*2^(1/4)

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Rubi [A]  time = 0.01, antiderivative size = 35, 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} \[ -\frac {\tan ^{-1}\left (\frac {x}{\sqrt [4]{2}}\right )}{2\ 2^{3/4}}-\frac {\tanh ^{-1}\left (\frac {x}{\sqrt [4]{2}}\right )}{2\ 2^{3/4}} \]

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 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}{-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*}

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Mathematica [A]  time = 0.02, 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]

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

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fricas [B]  time = 0.44, size = 63, normalized size = 1.80 \[ \frac {1}{8} \cdot 8^{\frac {3}{4}} \arctan \left (\frac {1}{4} \cdot 8^{\frac {1}{4}} \sqrt {2} \sqrt {2 \, x^{2} + 2 \, \sqrt {2}} - \frac {1}{2} \cdot 8^{\frac {1}{4}} x\right ) - \frac {1}{32} \cdot 8^{\frac {3}{4}} \log \left (4 \, x + 8^{\frac {3}{4}}\right ) + \frac {1}{32} \cdot 8^{\frac {3}{4}} \log \left (4 \, x - 8^{\frac {3}{4}}\right ) \]

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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giac [A]  time = 1.29, size = 39, normalized size = 1.11 \[ -\frac {1}{4} \cdot 2^{\frac {1}{4}} \arctan \left (\frac {1}{2} \cdot 2^{\frac {3}{4}} x\right ) - \frac {1}{8} \cdot 2^{\frac {1}{4}} \log \left ({\left | x + 2^{\frac {1}{4}} \right |}\right ) + \frac {1}{8} \cdot 2^{\frac {1}{4}} \log \left ({\left | x - 2^{\frac {1}{4}} \right |}\right ) \]

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)))

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maple [A]  time = 0.00, size = 35, normalized size = 1.00 \[ -\frac {2^{\frac {1}{4}} \arctan \left (\frac {2^{\frac {3}{4}} x}{2}\right )}{4}-\frac {2^{\frac {1}{4}} \ln \left (\frac {x +2^{\frac {1}{4}}}{x -2^{\frac {1}{4}}}\right )}{8} \]

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 = 0.98, size = 34, normalized size = 0.97 \[ -\frac {1}{4} \cdot 2^{\frac {1}{4}} \arctan \left (\frac {1}{2} \cdot 2^{\frac {3}{4}} x\right ) + \frac {1}{8} \cdot 2^{\frac {1}{4}} \log \left (\frac {x - 2^{\frac {1}{4}}}{x + 2^{\frac {1}{4}}}\right ) \]

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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mupad [B]  time = 0.16, size = 20, normalized size = 0.57 \[ -\frac {2^{1/4}\,\left (\mathrm {atan}\left (\frac {2^{3/4}\,x}{2}\right )+\mathrm {atanh}\left (\frac {2^{3/4}\,x}{2}\right )\right )}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^4 - 2),x)

[Out]

-(2^(1/4)*(atan((2^(3/4)*x)/2) + atanh((2^(3/4)*x)/2)))/4

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sympy [A]  time = 0.31, size = 46, normalized size = 1.31 \[ \frac {\sqrt [4]{2} \log {\left (x - \sqrt [4]{2} \right )}}{8} - \frac {\sqrt [4]{2} \log {\left (x + \sqrt [4]{2} \right )}}{8} - \frac {\sqrt [4]{2} \operatorname {atan}{\left (\frac {2^{\frac {3}{4}} x}{2} \right )}}{4} \]

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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