3.4.28 \(\int \frac {1}{(1+x^4) \sqrt {-x^2+\sqrt {1+x^4}}} \, dx\) [328]

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

[Out]

arctan(x/(-x^2+(x^4+1)^(1/2))^(1/2))

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Rubi [A]
time = 0.04, antiderivative size = 22, normalized size of antiderivative = 1.00, 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 = {2153, 209} \begin {gather*} \text {ArcTan}\left (\frac {x}{\sqrt {\sqrt {x^4+1}-x^2}}\right ) \end {gather*}

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 209

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*ArcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])

Rule 2153

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]

Rubi steps

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

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Mathematica [C] Result contains complex when optimal does not.
time = 0.22, size = 96, normalized size = 4.36 \begin {gather*} i \tanh ^{-1}\left (\sqrt {2}+\sqrt {2} x^4-i x^3 \sqrt {-x^2+\sqrt {1+x^4}}+\frac {\sqrt {1+x^4} \left (-2 x^2+i \sqrt {2} x \sqrt {-x^2+\sqrt {1+x^4}}\right )}{\sqrt {2}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

I*ArcTanh[Sqrt[2] + Sqrt[2]*x^4 - I*x^3*Sqrt[-x^2 + Sqrt[1 + x^4]] + (Sqrt[1 + x^4]*(-2*x^2 + I*Sqrt[2]*x*Sqrt
[-x^2 + Sqrt[1 + x^4]]))/Sqrt[2]]

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Maple [F]
time = 0.02, size = 0, normalized size = 0.00 \[\int \frac {1}{\left (x^{4}+1\right ) \sqrt {-x^{2}+\sqrt {x^{4}+1}}}\, dx\]

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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] Leaf count of result is larger than twice the leaf count of optimal. 62 vs. \(2 (18) = 36\).
time = 2.62, size = 62, normalized size = 2.82 \begin {gather*} -\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 {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\sqrt {- x^{2} + \sqrt {x^{4} + 1}} \left (x^{4} + 1\right )}\, dx \end {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.05 \begin {gather*} \int \frac {1}{\sqrt {\sqrt {x^4+1}-x^2}\,\left (x^4+1\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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