3.1.15 \(\int \frac {\sqrt {-x^2+\sqrt {1+x^4}}}{\sqrt {1+x^4}} \, dx\) [15]

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

[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 = 0.03, 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 = {2157, 209} \begin {gather*} \frac {\tan ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {\sqrt {x^4+1}-x^2}}\right )}{\sqrt {2}} \end {gather*}

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 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 2157

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 &=\text {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 = 0.14, size = 33, normalized size = 1.00 \begin {gather*} \frac {\tan ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {-x^2+\sqrt {1+x^4}}}\right )}{\sqrt {2}} \end {gather*}

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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Mathics [C] Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
time = 1.99, size = 21, normalized size = 0.64 \begin {gather*} \frac {\text {meijerg}\left [\left \{\left \{\frac {1}{2},1\right \},\left \{1\right \}\right \},\left \{\left \{\frac {1}{4},\frac {3}{4}\right \},\left \{0\right \}\right \},x^4\right ]}{4 \sqrt {\text {Pi}}} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

meijerg[{{1 / 2, 1}, {1}}, {{1 / 4, 3 / 4}, {0}}, x ^ 4] / (4 Sqrt[Pi])

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Maple [C] Result contains higher order function than in optimal. Order 5 vs. order 3.
time = 0.10, size = 22, normalized size = 0.67

method	result	size
meijerg	\(-\frac {\sqrt {2}\, \hypergeom \left (\left [\frac {1}{2}, \frac {3}{4}, \frac {5}{4}\right ], \left [\frac {3}{2}, \frac {3}{2}\right ], -\frac {1}{x^{4}}\right )}{4 x^{2}}\)	\(22\)

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,method=_RETURNVERBOSE)

[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 \begin {gather*} \text {Failed to integrate} \end {gather*}

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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Fricas [A]
time = 0.44, size = 29, normalized size = 0.88 \begin {gather*} -\frac {1}{2} \, \sqrt {2} \arctan \left (\frac {\sqrt {2} \sqrt {-x^{2} + \sqrt {x^{4} + 1}}}{2 \, x}\right ) \end {gather*}

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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Sympy [A]
time = 0.44, size = 15, normalized size = 0.45 \begin {gather*} \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 }} \end {gather*}

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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Giac [F] N/A
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((-x^2+(x^4+1)^(1/2))^(1/2)/(x^4+1)^(1/2),x)

[Out]

Could not integrate

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

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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