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

Optimal. Leaf size=22 \[ \tan ^{-1}\left (\frac{x}{\sqrt{\sqrt{x^4+1}-x^2}}\right ) \]

[Out]

ArcTan[x/Sqrt[-x^2 + Sqrt[1 + x^4]]]

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Rubi [A]  time = 0.064396, antiderivative size = 22, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.074, Rules used = {2128, 203} \[ \tan ^{-1}\left (\frac{x}{\sqrt{\sqrt{x^4+1}-x^2}}\right ) \]

Antiderivative was successfully verified.

[In]

Int[1/((1 + x^4)*Sqrt[-x^2 + Sqrt[1 + x^4]]),x]

[Out]

ArcTan[x/Sqrt[-x^2 + Sqrt[1 + x^4]]]

Rule 2128

Int[1/(((a_) + (b_.)*(x_)^(n_.))*Sqrt[(c_.)*(x_)^2 + (d_.)*((a_) + (b_.)*(x_)^(n_.))^(p_.)]), x_Symbol] :> Dis
t[1/a, Subst[Int[1/(1 - c*x^2), x], x, x/Sqrt[c*x^2 + d*(a + b*x^n)^(2/n)]], x] /; FreeQ[{a, b, c, d, n}, x] &
& EqQ[p, 2/n]

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

Mathematica [A]  time = 1.00667, size = 24, normalized size = 1.09 \[ \cot ^{-1}\left (\frac{\sqrt{\sqrt{x^4+1}-x^2}}{x}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[1/((1 + x^4)*Sqrt[-x^2 + Sqrt[1 + x^4]]),x]

[Out]

ArcCot[Sqrt[-x^2 + Sqrt[1 + x^4]]/x]

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Maple [F]  time = 0.024, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{4}+1}{\frac{1}{\sqrt{-{x}^{2}+\sqrt{{x}^{4}+1}}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B]  time = 6.07889, size = 149, normalized size = 6.77 \begin{align*} -\frac{1}{4} \, \arctan \left (\frac{4 \,{\left (10 \, x^{7} - 6 \, x^{3} +{\left (7 \, x^{5} - x\right )} \sqrt{x^{4} + 1}\right )} \sqrt{-x^{2} + \sqrt{x^{4} + 1}}}{17 \, x^{8} - 46 \, x^{4} + 1}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*arctan(4*(10*x^7 - 6*x^3 + (7*x^5 - x)*sqrt(x^4 + 1))*sqrt(-x^2 + sqrt(x^4 + 1))/(17*x^8 - 46*x^4 + 1))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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