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

Optimal. Leaf size=79 \[ \frac {1}{4} \sqrt {\sqrt {x^4+1}+x^2} x+\frac {x}{8 \left (\sqrt {x^4+1}+x^2\right )^{3/2}}+\frac {5 \tan ^{-1}\left (\sqrt {2} x \sqrt {\sqrt {x^4+1}+x^2}\right )}{8 \sqrt {2}} \]

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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 {\sqrt {1+x^4}}{\sqrt {x^2+\sqrt {1+x^4}}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [B]  time = 0.96, size = 200, normalized size = 2.53 \begin {gather*} \frac {\sqrt {2} \left (\sqrt {x^4+1}+x^2\right ) \left (16 x^8+18 x^4+16 \sqrt {x^4+1} x^6+10 \sqrt {x^4+1} x^2+3\right ) x^2+5 \sqrt {x^2 \left (\sqrt {x^4+1}+x^2\right )} \left (8 x^8+8 x^4+8 \sqrt {x^4+1} x^6+4 \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} x \sqrt {\sqrt {x^4+1}+x^2} \left (2 x^4+2 \sqrt {x^4+1} x^2+1\right )^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[2]*x^2*(x^2 + Sqrt[1 + x^4])*(3 + 18*x^4 + 16*x^8 + 10*x^2*Sqrt[1 + x^4] + 16*x^6*Sqrt[1 + x^4]) + 5*Sqr
t[x^2*(x^2 + Sqrt[1 + x^4])]*(1 + 8*x^4 + 8*x^8 + 4*x^2*Sqrt[1 + x^4] + 8*x^6*Sqrt[1 + x^4])*ArcTan[Sqrt[-1 +
(x^2 + Sqrt[1 + x^4])^2]])/(8*Sqrt[2]*x*Sqrt[x^2 + Sqrt[1 + x^4]]*(1 + 2*x^4 + 2*x^2*Sqrt[1 + x^4])^2)

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IntegrateAlgebraic [A]  time = 0.16, 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 {5 \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[Sqrt[1 + x^4]/Sqrt[x^2 + Sqrt[1 + x^4]],x]

[Out]

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

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fricas [A]  time = 0.94, size = 81, normalized size = 1.03 \begin {gather*} \frac {1}{8} \, {\left (2 \, x^{5} - 2 \, \sqrt {x^{4} + 1} x^{3} + 3 \, x\right )} \sqrt {x^{2} + \sqrt {x^{4} + 1}} - \frac {5}{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^4+1)^(1/2)/(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 + 3*x)*sqrt(x^2 + sqrt(x^4 + 1)) - 5/16*sqrt(2)*arctan(-1/2*(sqrt(2)*x^2 - sq
rt(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 {\sqrt {x^{4} + 1}}{\sqrt {x^{2} + \sqrt {x^{4} + 1}}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((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)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int((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 \frac {\sqrt {x^{4} + 1}}{\sqrt {x^{2} + \sqrt {x^{4} + 1}}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((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)

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

[Out]

int((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 \frac {\sqrt {x^{4} + 1}}{\sqrt {x^{2} + \sqrt {x^{4} + 1}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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