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

Optimal. Leaf size=133 \[ \frac {x \left (16 x^8+36 x^4+9\right ) \sqrt {\sqrt {x^4+1}+x^2}+x \sqrt {x^4+1} \sqrt {\sqrt {x^4+1}+x^2} \left (16 x^6+28 x^2\right )}{48 \left (2 x^4+1\right )+96 \sqrt {x^4+1} x^2}-\frac {3 \tan ^{-1}\left (\sqrt {2} x \sqrt {\sqrt {x^4+1}+x^2}\right )}{16 \sqrt {2}} \]

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Rubi [F]  time = 0.34, 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 {1+x^4} \sqrt {x^2+\sqrt {1+x^4}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [A]  time = 0.23, size = 189, normalized size = 1.42 \begin {gather*} \frac {\sqrt {x^4+1} \sqrt {x^2 \left (\sqrt {x^4+1}+x^2\right )} \left (\sqrt {2} \sqrt {x^2 \left (\sqrt {x^4+1}+x^2\right )} \left (16 x^8+36 x^4+16 \sqrt {x^4+1} x^6+28 \sqrt {x^4+1} x^2+9\right )-9 \left (2 x^4+2 \sqrt {x^4+1} x^2+1\right ) \tan ^{-1}\left (\sqrt {\left (\sqrt {x^4+1}+x^2\right )^2-1}\right )\right )}{48 \sqrt {2} \left (\sqrt {x^4+1}+x^2\right )^{3/2} \left (x^5+\sqrt {x^4+1} x^3+x\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[1 + x^4]*Sqrt[x^2*(x^2 + Sqrt[1 + x^4])]*(Sqrt[2]*Sqrt[x^2*(x^2 + Sqrt[1 + x^4])]*(9 + 36*x^4 + 16*x^8 +
28*x^2*Sqrt[1 + x^4] + 16*x^6*Sqrt[1 + x^4]) - 9*(1 + 2*x^4 + 2*x^2*Sqrt[1 + x^4])*ArcTan[Sqrt[-1 + (x^2 + Sq
rt[1 + x^4])^2]]))/(48*Sqrt[2]*(x^2 + Sqrt[1 + x^4])^(3/2)*(x + x^5 + x^3*Sqrt[1 + x^4]))

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IntegrateAlgebraic [A]  time = 0.16, size = 133, normalized size = 1.00 \begin {gather*} \frac {x \sqrt {1+x^4} \left (28 x^2+16 x^6\right ) \sqrt {x^2+\sqrt {1+x^4}}+x \left (9+36 x^4+16 x^8\right ) \sqrt {x^2+\sqrt {1+x^4}}}{96 x^2 \sqrt {1+x^4}+48 \left (1+2 x^4\right )}-\frac {3 \tan ^{-1}\left (\sqrt {2} x \sqrt {x^2+\sqrt {1+x^4}}\right )}{16 \sqrt {2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(x*Sqrt[1 + x^4]*(28*x^2 + 16*x^6)*Sqrt[x^2 + Sqrt[1 + x^4]] + x*(9 + 36*x^4 + 16*x^8)*Sqrt[x^2 + Sqrt[1 + x^4
]])/(96*x^2*Sqrt[1 + x^4] + 48*(1 + 2*x^4)) - (3*ArcTan[Sqrt[2]*x*Sqrt[x^2 + Sqrt[1 + x^4]]])/(16*Sqrt[2])

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fricas [A]  time = 1.05, size = 81, normalized size = 0.61 \begin {gather*} -\frac {1}{48} \, {\left (2 \, x^{5} - 10 \, \sqrt {x^{4} + 1} x^{3} - 9 \, x\right )} \sqrt {x^{2} + \sqrt {x^{4} + 1}} + \frac {3}{32} \, \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^4+1)^(1/2)*(x^2+(x^4+1)^(1/2))^(1/2),x, algorithm="fricas")

[Out]

-1/48*(2*x^5 - 10*sqrt(x^4 + 1)*x^3 - 9*x)*sqrt(x^2 + sqrt(x^4 + 1)) + 3/32*sqrt(2)*arctan(-1/2*(sqrt(2)*x^2 -
sqrt(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 \sqrt {x^{4} + 1} \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^4+1)^(1/2)*(x^2+(x^4+1)^(1/2))^(1/2),x, algorithm="giac")

[Out]

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

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maple [F]  time = 0.02, size = 0, normalized size = 0.00 \[\int x^{2} \sqrt {x^{4}+1}\, \sqrt {x^{2}+\sqrt {x^{4}+1}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int(x^2*(x^4+1)^(1/2)*(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*} \int \sqrt {x^{4} + 1} \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^4+1)^(1/2)*(x^2+(x^4+1)^(1/2))^(1/2),x, algorithm="maxima")

[Out]

integrate(sqrt(x^4 + 1)*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 {x^4+1}\,\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^4 + 1)^(1/2) + x^2)^(1/2),x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(x**2*sqrt(x**2 + sqrt(x**4 + 1))*sqrt(x**4 + 1), x)

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