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

Optimal. Leaf size=46 \[ \frac{\log \left (x^2+\sqrt{2} x+1\right )}{2 \sqrt{2}}-\frac{\log \left (x^2-\sqrt{2} x+1\right )}{2 \sqrt{2}} \]

[Out]

-Log[1 - Sqrt[2]*x + x^2]/(2*Sqrt[2]) + Log[1 + Sqrt[2]*x + x^2]/(2*Sqrt[2])

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Rubi [A]  time = 0.0194266, antiderivative size = 46, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {1165, 628} \[ \frac{\log \left (x^2+\sqrt{2} x+1\right )}{2 \sqrt{2}}-\frac{\log \left (x^2-\sqrt{2} x+1\right )}{2 \sqrt{2}} \]

Antiderivative was successfully verified.

[In]

Int[(1 - x^2)/(1 + x^4),x]

[Out]

-Log[1 - Sqrt[2]*x + x^2]/(2*Sqrt[2]) + Log[1 + Sqrt[2]*x + x^2]/(2*Sqrt[2])

Rule 1165

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[(-2*d)/e, 2]}, Dist[e/(2*c*q), Int[
(q - 2*x)/Simp[d/e + q*x - x^2, x], x], x] + Dist[e/(2*c*q), Int[(q + 2*x)/Simp[d/e - q*x - x^2, x], x], x]] /
; FreeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && NegQ[d*e]

Rule 628

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(d*Log[RemoveContent[a + b*x +
c*x^2, x]])/b, x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rubi steps

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

Mathematica [A]  time = 0.0109317, size = 40, normalized size = 0.87 \[ \frac{\log \left (x^2+\sqrt{2} x+1\right )-\log \left (-x^2+\sqrt{2} x-1\right )}{2 \sqrt{2}} \]

Antiderivative was successfully verified.

[In]

Integrate[(1 - x^2)/(1 + x^4),x]

[Out]

(-Log[-1 + Sqrt[2]*x - x^2] + Log[1 + Sqrt[2]*x + x^2])/(2*Sqrt[2])

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Maple [A]  time = 0.042, size = 62, normalized size = 1.4 \begin{align*}{\frac{\sqrt{2}}{8}\ln \left ({\frac{1+{x}^{2}+x\sqrt{2}}{1+{x}^{2}-x\sqrt{2}}} \right ) }-{\frac{\sqrt{2}}{8}\ln \left ({\frac{1+{x}^{2}-x\sqrt{2}}{1+{x}^{2}+x\sqrt{2}}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.45449, size = 46, normalized size = 1. \begin{align*} \frac{1}{4} \, \sqrt{2} \log \left (x^{2} + \sqrt{2} x + 1\right ) - \frac{1}{4} \, \sqrt{2} \log \left (x^{2} - \sqrt{2} x + 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*sqrt(2)*log(x^2 + sqrt(2)*x + 1) - 1/4*sqrt(2)*log(x^2 - sqrt(2)*x + 1)

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Fricas [A]  time = 1.36964, size = 92, normalized size = 2. \begin{align*} \frac{1}{4} \, \sqrt{2} \log \left (\frac{x^{4} + 4 \, x^{2} + 2 \, \sqrt{2}{\left (x^{3} + x\right )} + 1}{x^{4} + 1}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*sqrt(2)*log((x^4 + 4*x^2 + 2*sqrt(2)*(x^3 + x) + 1)/(x^4 + 1))

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Sympy [A]  time = 0.097355, size = 39, normalized size = 0.85 \begin{align*} - \frac{\sqrt{2} \log{\left (x^{2} - \sqrt{2} x + 1 \right )}}{4} + \frac{\sqrt{2} \log{\left (x^{2} + \sqrt{2} x + 1 \right )}}{4} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.10006, size = 46, normalized size = 1. \begin{align*} \frac{1}{4} \, \sqrt{2} \log \left (x^{2} + \sqrt{2} x + 1\right ) - \frac{1}{4} \, \sqrt{2} \log \left (x^{2} - \sqrt{2} x + 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*sqrt(2)*log(x^2 + sqrt(2)*x + 1) - 1/4*sqrt(2)*log(x^2 - sqrt(2)*x + 1)