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

Optimal. Leaf size=33 \[ \frac {\tan ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {\sqrt {x^4+1}-x^2}}\right )}{\sqrt {2}} \]

[Out]

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

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Rubi [A]  time = 0.06, antiderivative size = 33, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.069, Rules used = {2132, 203} \[ \frac {\tan ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {\sqrt {x^4+1}-x^2}}\right )}{\sqrt {2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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 2132

Int[Sqrt[(c_.)*(x_)^2 + (d_.)*Sqrt[(a_) + (b_.)*(x_)^4]]/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> Dist[d, Subst
[Int[1/(1 - 2*c*x^2), x], x, x/Sqrt[c*x^2 + d*Sqrt[a + b*x^4]]], x] /; FreeQ[{a, b, c, d}, x] && EqQ[c^2 - b*d
^2, 0]

Rubi steps

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

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Mathematica [A]  time = 0.01, size = 33, normalized size = 1.00 \[ \frac {\tan ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {\sqrt {x^4+1}-x^2}}\right )}{\sqrt {2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 1.62, size = 29, normalized size = 0.88 \[ -\frac {1}{2} \, \sqrt {2} \arctan \left (\frac {\sqrt {2} \sqrt {-x^{2} + \sqrt {x^{4} + 1}}}{2 \, x}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {-x^{2} + \sqrt {x^{4} + 1}}}{\sqrt {x^{4} + 1}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [C]  time = 0.10, size = 22, normalized size = 0.67 \[ -\frac {\sqrt {2}\, \hypergeom \left (\left [\frac {1}{2}, \frac {3}{4}, \frac {5}{4}\right ], \left [\frac {3}{2}, \frac {3}{2}\right ], -\frac {1}{x^{4}}\right )}{4 x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*2^(1/2)/x^2*hypergeom([1/2,3/4,5/4],[3/2,3/2],-1/x^4)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {-x^{2} + \sqrt {x^{4} + 1}}}{\sqrt {x^{4} + 1}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.03 \[ \int \frac {\sqrt {\sqrt {x^4+1}-x^2}}{\sqrt {x^4+1}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 0.86, size = 15, normalized size = 0.45 \[ \frac {{G_{3, 3}^{2, 2}\left (\begin {matrix} \frac {1}{2}, 1 & 1 \\\frac {1}{4}, \frac {3}{4} & 0 \end {matrix} \middle | {x^{4}} \right )}}{4 \sqrt {\pi }} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

meijerg(((1/2, 1), (1,)), ((1/4, 3/4), (0,)), x**4)/(4*sqrt(pi))

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