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

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

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Rubi [F]  time = 0.83, 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 [F]  time = 0.18, size = 0, normalized size = 0.00 \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]

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

[Out]

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

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IntegrateAlgebraic [A]  time = 1.31, size = 329, normalized size = 1.34 \begin {gather*} \frac {1}{2} x \sqrt {x^2+\sqrt {1+x^4}}-\frac {\tan ^{-1}\left (\frac {-\frac {1}{\sqrt {2}}+\frac {x^2}{\sqrt {2}}+\frac {\sqrt {1+x^4}}{\sqrt {2}}}{x \sqrt {x^2+\sqrt {1+x^4}}}\right )}{\sqrt {2}}+\sqrt {2 \left (-1+\sqrt {2}\right )} \tan ^{-1}\left (\frac {-\sqrt {\frac {1}{2}+\frac {1}{\sqrt {2}}}+\sqrt {\frac {1}{2}+\frac {1}{\sqrt {2}}} x^2+\sqrt {\frac {1}{2}+\frac {1}{\sqrt {2}}} \sqrt {1+x^4}}{x \sqrt {x^2+\sqrt {1+x^4}}}\right )-\sqrt {2} \tanh ^{-1}\left (\frac {-\frac {1}{\sqrt {2}}+\frac {x^2}{\sqrt {2}}+\frac {\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 {-\sqrt {-\frac {1}{2}+\frac {1}{\sqrt {2}}}+\sqrt {-\frac {1}{2}+\frac {1}{\sqrt {2}}} x^2+\sqrt {-\frac {1}{2}+\frac {1}{\sqrt {2}}} \sqrt {1+x^4}}{x \sqrt {x^2+\sqrt {1+x^4}}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(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/Sqrt[2]) + x^2/Sqrt[2] + Sqrt[1 + x^4]/Sqrt[2])/(x*Sqrt[x^2 + Sq
rt[1 + x^4]])]/Sqrt[2] + Sqrt[2*(-1 + Sqrt[2])]*ArcTan[(-Sqrt[1/2 + 1/Sqrt[2]] + Sqrt[1/2 + 1/Sqrt[2]]*x^2 + S
qrt[1/2 + 1/Sqrt[2]]*Sqrt[1 + x^4])/(x*Sqrt[x^2 + Sqrt[1 + x^4]])] - Sqrt[2]*ArcTanh[(-(1/Sqrt[2]) + x^2/Sqrt[
2] + Sqrt[1 + x^4]/Sqrt[2])/(x*Sqrt[x^2 + Sqrt[1 + x^4]])] + Sqrt[2*(1 + Sqrt[2])]*ArcTanh[(-Sqrt[-1/2 + 1/Sqr
t[2]] + Sqrt[-1/2 + 1/Sqrt[2]]*x^2 + Sqrt[-1/2 + 1/Sqrt[2]]*Sqrt[1 + x^4])/(x*Sqrt[x^2 + Sqrt[1 + x^4]])]

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fricas [F(-1)]  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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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}}}{x^{2} + 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^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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maple [F]  time = 0.05, 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*} \int \frac {\sqrt {x^{4} + 1} \sqrt {x^{2} + \sqrt {x^{4} + 1}}}{x^{2} + 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^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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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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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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