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

Optimal. Leaf size=101 \[ \frac {2 \sqrt {x^4+1} x^4+\left (2 x^4-1\right ) x^2}{8 x \sqrt {\sqrt {x^4+1}+x^2}}+\frac {\tanh ^{-1}\left (\frac {\sqrt {2} x \sqrt {\sqrt {x^4+1}+x^2}}{\sqrt {x^4+1}+x^2+1}\right )}{4 \sqrt {2}} \]

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Rubi [F]  time = 0.11, 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 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 x^2 \sqrt {x^2+\sqrt {1+x^4}} \, dx &=\int x^2 \sqrt {x^2+\sqrt {1+x^4}} \, dx\\ \end {align*}

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Mathematica [B]  time = 0.73, size = 250, normalized size = 2.48 \begin {gather*} \frac {x \sqrt {x^4+1} \sqrt {\sqrt {x^4+1}+x^2} \sqrt {x^2 \left (\sqrt {x^4+1}+x^2\right )} \left (2 x^4+2 \sqrt {x^4+1} x^2+1\right )^2 \left (\sqrt {2} \sqrt {-x^2 \left (\sqrt {x^4+1}+x^2\right )} \left (2 x^4+2 \sqrt {x^4+1} x^2-1\right )-\left (\sqrt {x^4+1}+x^2\right ) \sin ^{-1}\left (\sqrt {x^4+1}+x^2\right )\right )}{8 \sqrt {2} \sqrt {-x^4 \left (2 x^4+2 \sqrt {x^4+1} x^2+1\right )} \left (16 x^{12}+28 x^8+13 x^4+16 \sqrt {x^4+1} x^{10}+20 \sqrt {x^4+1} x^6+5 \sqrt {x^4+1} x^2+1\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(x*Sqrt[1 + x^4]*Sqrt[x^2 + Sqrt[1 + x^4]]*Sqrt[x^2*(x^2 + Sqrt[1 + x^4])]*(1 + 2*x^4 + 2*x^2*Sqrt[1 + x^4])^2
*(Sqrt[2]*Sqrt[-(x^2*(x^2 + Sqrt[1 + x^4]))]*(-1 + 2*x^4 + 2*x^2*Sqrt[1 + x^4]) - (x^2 + Sqrt[1 + x^4])*ArcSin
[x^2 + Sqrt[1 + x^4]]))/(8*Sqrt[2]*Sqrt[-(x^4*(1 + 2*x^4 + 2*x^2*Sqrt[1 + x^4]))]*(1 + 13*x^4 + 28*x^8 + 16*x^
12 + 5*x^2*Sqrt[1 + x^4] + 20*x^6*Sqrt[1 + x^4] + 16*x^10*Sqrt[1 + x^4]))

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IntegrateAlgebraic [A]  time = 0.36, size = 101, normalized size = 1.00 \begin {gather*} \frac {2 x^4 \sqrt {1+x^4}+x^2 \left (-1+2 x^4\right )}{8 x \sqrt {x^2+\sqrt {1+x^4}}}+\frac {\tanh ^{-1}\left (\frac {\sqrt {2} x \sqrt {x^2+\sqrt {1+x^4}}}{1+x^2+\sqrt {1+x^4}}\right )}{4 \sqrt {2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 0.90, size = 92, normalized size = 0.91 \begin {gather*} \frac {1}{8} \, {\left (3 \, x^{3} - \sqrt {x^{4} + 1} x\right )} \sqrt {x^{2} + \sqrt {x^{4} + 1}} + \frac {1}{32} \, \sqrt {2} \log \left (4 \, x^{4} + 4 \, \sqrt {x^{4} + 1} x^{2} + 2 \, {\left (\sqrt {2} x^{3} + \sqrt {2} \sqrt {x^{4} + 1} x\right )} \sqrt {x^{2} + \sqrt {x^{4} + 1}} + 1\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*(3*x^3 - sqrt(x^4 + 1)*x)*sqrt(x^2 + sqrt(x^4 + 1)) + 1/32*sqrt(2)*log(4*x^4 + 4*sqrt(x^4 + 1)*x^2 + 2*(sq
rt(2)*x^3 + sqrt(2)*sqrt(x^4 + 1)*x)*sqrt(x^2 + sqrt(x^4 + 1)) + 1)

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

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maple [C]  time = 0.08, size = 62, normalized size = 0.61

method	result	size
meijerg	\(\frac {\frac {5 \sqrt {\pi }\, \sqrt {2}\, \hypergeom \left (\left [1, 1, \frac {7}{4}, \frac {9}{4}\right ], \left [2, \frac {5}{2}, 3\right ], -\frac {1}{x^{4}}\right )}{32 x^{4}}-\frac {\left (1-4 \ln \relax (2)-4 \ln \relax (x )\right ) \sqrt {\pi }\, \sqrt {2}}{2}+4 \sqrt {\pi }\, \sqrt {2}\, x^{4}}{16 \sqrt {\pi }}\)	\(62\)

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]

1/16/Pi^(1/2)*(5/32*Pi^(1/2)*2^(1/2)/x^4*hypergeom([1,1,7/4,9/4],[2,5/2,3],-1/x^4)-1/2*(1-4*ln(2)-4*ln(x))*Pi^
(1/2)*2^(1/2)+4*Pi^(1/2)*2^(1/2)*x^4)

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int 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 = 1.25, size = 17, normalized size = 0.17 \begin {gather*} - \frac {{G_{3, 3}^{2, 2}\left (\begin {matrix} 2, 1 & \frac {3}{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(((2, 1), (3/2,)), ((3/4, 5/4), (0,)), x**4)/(16*sqrt(pi))

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