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

Optimal. Leaf size=79 \[ \frac {1}{4} \sqrt {\sqrt {x^4+1}+x^2} x-\frac {x}{8 \left (\sqrt {x^4+1}+x^2\right )^{3/2}}-\frac {\tan ^{-1}\left (\sqrt {2} x \sqrt {\sqrt {x^4+1}+x^2}\right )}{8 \sqrt {2}} \]

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Rubi [F]  time = 0.10, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {x^2}{\sqrt {x^2+\sqrt {1+x^4}}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[x^2/Sqrt[x^2 + Sqrt[1 + x^4]],x]

[Out]

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

Rubi steps

\begin {align*} \int \frac {x^2}{\sqrt {x^2+\sqrt {1+x^4}}} \, dx &=\int \frac {x^2}{\sqrt {x^2+\sqrt {1+x^4}}} \, dx\\ \end {align*}

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Mathematica [B]  time = 3.30, size = 194, normalized size = 2.46 \begin {gather*} \frac {\sqrt {2} x^2 \left (\sqrt {x^4+1}+x^2\right ) \left (8 x^8+9 x^4+8 \sqrt {x^4+1} x^6+5 \sqrt {x^4+1} x^2+1\right )-\sqrt {x^2 \left (\sqrt {x^4+1}+x^2\right )} \left (4 x^8+5 x^4+4 \sqrt {x^4+1} x^6+3 \sqrt {x^4+1} x^2+1\right ) \tan ^{-1}\left (\sqrt {\left (\sqrt {x^4+1}+x^2\right )^2-1}\right )}{8 \sqrt {2} \left (\sqrt {x^4+1}+x^2\right )^{5/2} \left (x^5+\sqrt {x^4+1} x^3+x\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[2]*x^2*(x^2 + Sqrt[1 + x^4])*(1 + 9*x^4 + 8*x^8 + 5*x^2*Sqrt[1 + x^4] + 8*x^6*Sqrt[1 + x^4]) - Sqrt[x^2*
(x^2 + Sqrt[1 + x^4])]*(1 + 5*x^4 + 4*x^8 + 3*x^2*Sqrt[1 + x^4] + 4*x^6*Sqrt[1 + x^4])*ArcTan[Sqrt[-1 + (x^2 +
Sqrt[1 + x^4])^2]])/(8*Sqrt[2]*(x^2 + Sqrt[1 + x^4])^(5/2)*(x + x^5 + x^3*Sqrt[1 + x^4]))

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IntegrateAlgebraic [A]  time = 0.15, size = 79, normalized size = 1.00 \begin {gather*} -\frac {x}{8 \left (x^2+\sqrt {1+x^4}\right )^{3/2}}+\frac {1}{4} x \sqrt {x^2+\sqrt {1+x^4}}-\frac {\tan ^{-1}\left (\sqrt {2} x \sqrt {x^2+\sqrt {1+x^4}}\right )}{8 \sqrt {2}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[x^2/Sqrt[x^2 + Sqrt[1 + x^4]],x]

[Out]

-1/8*x/(x^2 + Sqrt[1 + x^4])^(3/2) + (x*Sqrt[x^2 + Sqrt[1 + x^4]])/4 - ArcTan[Sqrt[2]*x*Sqrt[x^2 + Sqrt[1 + x^
4]]]/(8*Sqrt[2])

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fricas [A]  time = 0.83, size = 81, normalized size = 1.03 \begin {gather*} -\frac {1}{8} \, {\left (2 \, x^{5} - 2 \, \sqrt {x^{4} + 1} x^{3} - x\right )} \sqrt {x^{2} + \sqrt {x^{4} + 1}} + \frac {1}{16} \, \sqrt {2} \arctan \left (-\frac {{\left (\sqrt {2} x^{2} - \sqrt {2} \sqrt {x^{4} + 1}\right )} \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^2+(x^4+1)^(1/2))^(1/2),x, algorithm="fricas")

[Out]

-1/8*(2*x^5 - 2*sqrt(x^4 + 1)*x^3 - x)*sqrt(x^2 + sqrt(x^4 + 1)) + 1/16*sqrt(2)*arctan(-1/2*(sqrt(2)*x^2 - sqr
t(2)*sqrt(x^4 + 1))*sqrt(x^2 + sqrt(x^4 + 1))/x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [C]  time = 0.03, size = 22, normalized size = 0.28

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2/(x^2+(x^4+1)^(1/2))^(1/2),x,method=_RETURNVERBOSE)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {x^2}{\sqrt {\sqrt {x^4+1}+x^2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 0.73, size = 15, normalized size = 0.19 \begin {gather*} \frac {{G_{3, 3}^{2, 2}\left (\begin {matrix} \frac {3}{2}, 1 & 2 \\\frac {3}{4}, \frac {5}{4} & 0 \end {matrix} \middle | {x^{4}} \right )}}{16 \sqrt {\pi }} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

meijerg(((3/2, 1), (2,)), ((3/4, 5/4), (0,)), x**4)/(16*sqrt(pi))

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