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

Optimal. Leaf size=49 \[ \frac{1}{2} \tan ^{-1}\left (\frac{x \left (x^2+1\right )}{\sqrt{1-x^4}}\right )+\frac{1}{2} \tanh ^{-1}\left (\frac{x \left (1-x^2\right )}{\sqrt{1-x^4}}\right ) \]

[Out]

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

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Rubi [A]  time = 0.0072942, antiderivative size = 49, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.053, Rules used = {405} \[ \frac{1}{2} \tan ^{-1}\left (\frac{x \left (x^2+1\right )}{\sqrt{1-x^4}}\right )+\frac{1}{2} \tanh ^{-1}\left (\frac{x \left (1-x^2\right )}{\sqrt{1-x^4}}\right ) \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[1 - x^4]/(1 + x^4),x]

[Out]

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

Rule 405

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

Rubi steps

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

Mathematica [C]  time = 0.0968655, size = 110, normalized size = 2.24 \[ -\frac{5 x \sqrt{1-x^4} F_1\left (\frac{1}{4};-\frac{1}{2},1;\frac{5}{4};x^4,-x^4\right )}{\left (x^4+1\right ) \left (2 x^4 \left (2 F_1\left (\frac{5}{4};-\frac{1}{2},2;\frac{9}{4};x^4,-x^4\right )+F_1\left (\frac{5}{4};\frac{1}{2},1;\frac{9}{4};x^4,-x^4\right )\right )-5 F_1\left (\frac{1}{4};-\frac{1}{2},1;\frac{5}{4};x^4,-x^4\right )\right )} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[Sqrt[1 - x^4]/(1 + x^4),x]

[Out]

(-5*x*Sqrt[1 - x^4]*AppellF1[1/4, -1/2, 1, 5/4, x^4, -x^4])/((1 + x^4)*(-5*AppellF1[1/4, -1/2, 1, 5/4, x^4, -x
^4] + 2*x^4*(2*AppellF1[5/4, -1/2, 2, 9/4, x^4, -x^4] + AppellF1[5/4, 1/2, 1, 9/4, x^4, -x^4])))

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Maple [B]  time = 0.013, size = 100, normalized size = 2. \begin{align*}{\frac{1}{4}\arctan \left ( -{\frac{1}{x}\sqrt{-{x}^{4}+1}}+1 \right ) }-{\frac{1}{8}\ln \left ({ \left ({\frac{-{x}^{4}+1}{2\,{x}^{2}}}-{\frac{1}{x}\sqrt{-{x}^{4}+1}}+1 \right ) \left ({\frac{-{x}^{4}+1}{2\,{x}^{2}}}+{\frac{1}{x}\sqrt{-{x}^{4}+1}}+1 \right ) ^{-1}} \right ) }-{\frac{1}{4}\arctan \left ({\frac{1}{x}\sqrt{-{x}^{4}+1}}+1 \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{-x^{4} + 1}}{x^{4} + 1}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(sqrt(-x^4 + 1)/(x^4 + 1), x)

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Fricas [A]  time = 2.51574, size = 138, normalized size = 2.82 \begin{align*} -\frac{1}{2} \, \arctan \left (\frac{\sqrt{-x^{4} + 1} x}{x^{2} - 1}\right ) + \frac{1}{4} \, \log \left (-\frac{x^{4} - 2 \, x^{2} - 2 \, \sqrt{-x^{4} + 1} x - 1}{x^{4} + 1}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{- \left (x - 1\right ) \left (x + 1\right ) \left (x^{2} + 1\right )}}{x^{4} + 1}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(sqrt(-(x - 1)*(x + 1)*(x**2 + 1))/(x**4 + 1), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{-x^{4} + 1}}{x^{4} + 1}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(sqrt(-x^4 + 1)/(x^4 + 1), x)