3.28.1 \(\int \frac {\sqrt {1+x^4} \sqrt {x^2+\sqrt {1+x^4}}}{1+x^2} \, dx\) [2701]

Optimal. Leaf size=245 \[ \frac {1}{2} x \sqrt {x^2+\sqrt {1+x^4}}-\frac {\text {ArcTan}\left (\frac {\sqrt {2} x \sqrt {x^2+\sqrt {1+x^4}}}{1+x^2+\sqrt {1+x^4}}\right )}{\sqrt {2}}+\sqrt {2 \left (-1+\sqrt {2}\right )} \text {ArcTan}\left (\frac {\sqrt {2 \left (1+\sqrt {2}\right )} x \sqrt {x^2+\sqrt {1+x^4}}}{1+x^2+\sqrt {1+x^4}}\right )-\sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {2} x \sqrt {x^2+\sqrt {1+x^4}}}{1+x^2+\sqrt {1+x^4}}\right )+\sqrt {2 \left (1+\sqrt {2}\right )} \tanh ^{-1}\left (\frac {\sqrt {2 \left (-1+\sqrt {2}\right )} x \sqrt {x^2+\sqrt {1+x^4}}}{1+x^2+\sqrt {1+x^4}}\right ) \]

[Out]

1/2*x*(x^2+(x^4+1)^(1/2))^(1/2)-1/2*arctan(2^(1/2)*x*(x^2+(x^4+1)^(1/2))^(1/2)/(1+x^2+(x^4+1)^(1/2)))*2^(1/2)+
(-2+2*2^(1/2))^(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)+(2+2*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 = 0.61, 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}}}{1+x^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [A]
time = 0.91, size = 245, normalized size = 1.00 \begin {gather*} \frac {1}{2} x \sqrt {x^2+\sqrt {1+x^4}}-\frac {\text {ArcTan}\left (\frac {-1+x^2+\sqrt {1+x^4}}{\sqrt {2} x \sqrt {x^2+\sqrt {1+x^4}}}\right )}{\sqrt {2}}+\sqrt {2 \left (-1+\sqrt {2}\right )} \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 {2 \left (1+\sqrt {2}\right )} \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[1 + x^4]*Sqrt[x^2 + Sqrt[1 + x^4]])/(1 + x^2),x]

[Out]

(x*Sqrt[x^2 + Sqrt[1 + x^4]])/2 - ArcTan[(-1 + x^2 + Sqrt[1 + x^4])/(Sqrt[2]*x*Sqrt[x^2 + Sqrt[1 + x^4]])]/Sqr
t[2] + Sqrt[2*(-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[2*(1 + Sqr
t[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 = 0.04, size = 0, normalized size = 0.00 \[\int \frac {\sqrt {x^{4}+1}\, \sqrt {x^{2}+\sqrt {x^{4}+1}}}{x^{2}+1}\, dx\]

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^2+1),x)

[Out]

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

[Out]

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

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Fricas [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \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^2+1),x, algorithm="fricas")

[Out]

Timed out

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

[Out]

Integral(sqrt(x**2 + sqrt(x**4 + 1))*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^4+1)^(1/2)*(x^2+(x^4+1)^(1/2))^(1/2)/(x^2+1),x, algorithm="giac")

[Out]

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

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

[Out]

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

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