3.23.67 \(\int \frac {\sqrt {x^2+\sqrt {1+x^4}}}{1+x^2} \, dx\) [2267]

Optimal. Leaf size=171 \[ \sqrt {-1+\sqrt {2}} \text {ArcTan}\left (\frac {\sqrt {2 \left (1+\sqrt {2}\right )} x \sqrt {x^2+\sqrt {1+x^4}}}{1+x^2+\sqrt {1+x^4}}\right )+\sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {2} x \sqrt {x^2+\sqrt {1+x^4}}}{1+x^2+\sqrt {1+x^4}}\right )-\sqrt {1+\sqrt {2}} \tanh ^{-1}\left (\frac {\sqrt {2 \left (-1+\sqrt {2}\right )} x \sqrt {x^2+\sqrt {1+x^4}}}{1+x^2+\sqrt {1+x^4}}\right ) \]

[Out]

(2^(1/2)-1)^(1/2)*arctan((2+2*2^(1/2))^(1/2)*x*(x^2+(x^4+1)^(1/2))^(1/2)/(1+x^2+(x^4+1)^(1/2)))+arctanh(2^(1/2
)*x*(x^2+(x^4+1)^(1/2))^(1/2)/(1+x^2+(x^4+1)^(1/2)))*2^(1/2)-(1+2^(1/2))^(1/2)*arctanh((-2+2*2^(1/2))^(1/2)*x*
(x^2+(x^4+1)^(1/2))^(1/2)/(1+x^2+(x^4+1)^(1/2)))

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Rubi [F]
time = 0.20, 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 {x^2+\sqrt {1+x^4}}}{1+x^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

(I/2)*Defer[Int][Sqrt[x^2 + Sqrt[1 + x^4]]/(I - x), x] + (I/2)*Defer[Int][Sqrt[x^2 + Sqrt[1 + x^4]]/(I + x), x
]

Rubi steps

\begin {align*} \int \frac {\sqrt {x^2+\sqrt {1+x^4}}}{1+x^2} \, dx &=\int \left (\frac {i \sqrt {x^2+\sqrt {1+x^4}}}{2 (i-x)}+\frac {i \sqrt {x^2+\sqrt {1+x^4}}}{2 (i+x)}\right ) \, dx\\ &=\frac {1}{2} i \int \frac {\sqrt {x^2+\sqrt {1+x^4}}}{i-x} \, dx+\frac {1}{2} i \int \frac {\sqrt {x^2+\sqrt {1+x^4}}}{i+x} \, dx\\ \end {align*}

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Mathematica [A]
time = 0.59, size = 171, normalized size = 1.00 \begin {gather*} \sqrt {-1+\sqrt {2}} \text {ArcTan}\left (\frac {\sqrt {\frac {1}{2}+\frac {1}{\sqrt {2}}} \left (-1+x^2+\sqrt {1+x^4}\right )}{x \sqrt {x^2+\sqrt {1+x^4}}}\right )+\sqrt {2} \tanh ^{-1}\left (\frac {-1+x^2+\sqrt {1+x^4}}{\sqrt {2} x \sqrt {x^2+\sqrt {1+x^4}}}\right )-\sqrt {1+\sqrt {2}} \tanh ^{-1}\left (\frac {-1+x^2+\sqrt {1+x^4}}{\sqrt {2 \left (1+\sqrt {2}\right )} x \sqrt {x^2+\sqrt {1+x^4}}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]
time = 0.01, size = 0, normalized size = 0.00 \[\int \frac {\sqrt {x^{2}+\sqrt {x^{4}+1}}}{x^{2}+1}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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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^2+1),x, algorithm="maxima")

[Out]

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 333 vs. \(2 (133) = 266\).
time = 1.16, size = 333, normalized size = 1.95 \begin {gather*} \sqrt {\sqrt {2} - 1} \arctan \left (\frac {{\left (\sqrt {2} x^{2} + x^{2} + \sqrt {x^{4} + 1} {\left ({\left (\sqrt {2} + 1\right )} \sqrt {-2 \, \sqrt {2} + 3} - \sqrt {2} - 1\right )} - {\left (x^{2} + \sqrt {2} {\left (x^{2} + 2\right )} + 3\right )} \sqrt {-2 \, \sqrt {2} + 3} + 1\right )} \sqrt {x^{2} + \sqrt {x^{4} + 1}} \sqrt {\sqrt {2} - 1}}{2 \, x}\right ) + \frac {1}{4} \, \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 ) - \frac {1}{4} \, \sqrt {\sqrt {2} + 1} \log \left (\frac {\sqrt {2} x^{2} + 2 \, x^{2} + {\left (x^{3} + \sqrt {2} x - \sqrt {x^{4} + 1} x + x\right )} \sqrt {x^{2} + \sqrt {x^{4} + 1}} \sqrt {\sqrt {2} + 1} + \sqrt {x^{4} + 1} {\left (\sqrt {2} + 1\right )}}{x^{2} + 1}\right ) + \frac {1}{4} \, \sqrt {\sqrt {2} + 1} \log \left (\frac {\sqrt {2} x^{2} + 2 \, x^{2} - {\left (x^{3} + \sqrt {2} x - \sqrt {x^{4} + 1} x + x\right )} \sqrt {x^{2} + \sqrt {x^{4} + 1}} \sqrt {\sqrt {2} + 1} + \sqrt {x^{4} + 1} {\left (\sqrt {2} + 1\right )}}{x^{2} + 1}\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^2+1),x, algorithm="fricas")

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F]
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^2+1),x, algorithm="giac")

[Out]

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

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

[Out]

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

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