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

Optimal. Leaf size=362 \[ \frac {x \left (x^4-3\right )}{2 \left (x^4-1\right ) \sqrt {\sqrt {x^4+1}+x^2}}+\frac {1}{2} \sqrt {\frac {1}{2} \left (7+5 \sqrt {2}\right )} \tan ^{-1}\left (\frac {\sqrt {2 \left (\sqrt {2}-1\right )} x \sqrt {\sqrt {x^4+1}+x^2}}{\sqrt {x^4+1}+x^2+1}\right )-\frac {1}{2} \sqrt {\frac {1}{2} \left (7+5 \sqrt {2}\right )} \tan ^{-1}\left (\frac {\sqrt {2 \left (1+\sqrt {2}\right )} x \sqrt {\sqrt {x^4+1}+x^2}}{\sqrt {x^4+1}+x^2+1}\right )+\frac {\tanh ^{-1}\left (\frac {\sqrt {2} x \sqrt {\sqrt {x^4+1}+x^2}}{\sqrt {x^4+1}+x^2+1}\right )}{\sqrt {2}}-\frac {1}{2} \sqrt {\frac {1}{2} \left (5 \sqrt {2}-7\right )} \tanh ^{-1}\left (\frac {\sqrt {2 \left (\sqrt {2}-1\right )} x \sqrt {\sqrt {x^4+1}+x^2}}{\sqrt {x^4+1}+x^2+1}\right )-\frac {1}{2} \sqrt {\frac {1}{2} \left (5 \sqrt {2}-7\right )} \tanh ^{-1}\left (\frac {\sqrt {2 \left (1+\sqrt {2}\right )} x \sqrt {\sqrt {x^4+1}+x^2}}{\sqrt {x^4+1}+x^2+1}\right ) \]

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Rubi [F]  time = 1.70, 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 {\left (1+x^4\right )^2}{\left (-1+x^4\right )^2 \sqrt {x^2+\sqrt {1+x^4}}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [F]  time = 0.53, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (1+x^4\right )^2}{\left (-1+x^4\right )^2 \sqrt {x^2+\sqrt {1+x^4}}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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IntegrateAlgebraic [A]  time = 6.85, size = 483, normalized size = 1.33 \begin {gather*} \frac {x \left (-3+x^4\right )}{2 \left (-1+x^4\right ) \sqrt {x^2+\sqrt {1+x^4}}}+\frac {1}{2} \sqrt {\frac {7}{2}+\frac {5}{\sqrt {2}}} \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 )-\frac {1}{2} \sqrt {\frac {7}{2}+\frac {5}{\sqrt {2}}} \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 )+\frac {\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}}-\frac {1}{2} \sqrt {-\frac {7}{2}+\frac {5}{\sqrt {2}}} \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 )-\frac {1}{2} \sqrt {-\frac {7}{2}+\frac {5}{\sqrt {2}}} \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[(1 + x^4)^2/((-1 + x^4)^2*Sqrt[x^2 + Sqrt[1 + x^4]]),x]

[Out]

(x*(-3 + x^4))/(2*(-1 + x^4)*Sqrt[x^2 + Sqrt[1 + x^4]]) + (Sqrt[7/2 + 5/Sqrt[2]]*ArcTan[(-Sqrt[-1/2 + 1/Sqrt[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]])])/2 - (S
qrt[7/2 + 5/Sqrt[2]]*ArcTan[(-Sqrt[1/2 + 1/Sqrt[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]])])/2 + ArcTanh[(-(1/Sqrt[2]) + x^2/Sqrt[2] + Sqrt[1 + x^4]/Sqrt[2])/(x*S
qrt[x^2 + Sqrt[1 + x^4]])]/Sqrt[2] - (Sqrt[-7/2 + 5/Sqrt[2]]*ArcTanh[(-Sqrt[-1/2 + 1/Sqrt[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]])])/2 - (Sqrt[-7/2 + 5/Sqrt[2
]]*ArcTanh[(-Sqrt[1/2 + 1/Sqrt[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]])])/2

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fricas [B]  time = 6.35, size = 553, normalized size = 1.53 \begin {gather*} -\frac {4 \, \sqrt {2} {\left (x^{4} - 1\right )} \sqrt {5 \, \sqrt {2} + 7} \arctan \left (\frac {2 \, {\left (6 \, x^{7} + 10 \, x^{3} - \sqrt {2} {\left (5 \, x^{7} + 7 \, x^{3}\right )} - {\left (x^{5} - 2 \, \sqrt {2} {\left (x^{5} + x\right )} + 3 \, x\right )} \sqrt {x^{4} + 1}\right )} \sqrt {x^{2} + \sqrt {x^{4} + 1}} \sqrt {5 \, \sqrt {2} + 7} - {\left (5 \, x^{8} + 10 \, x^{4} - \sqrt {2} {\left (3 \, x^{8} + 4 \, x^{4} + 1\right )} - 2 \, {\left (x^{6} + 3 \, x^{2} - 2 \, \sqrt {2} {\left (x^{6} + x^{2}\right )}\right )} \sqrt {x^{4} + 1} + 1\right )} \sqrt {5 \, \sqrt {2} + 7} \sqrt {\sqrt {2} - 1}}{7 \, x^{8} + 10 \, x^{4} - 1}\right ) + \sqrt {2} {\left (x^{4} - 1\right )} \sqrt {5 \, \sqrt {2} - 7} \log \left (\frac {2 \, {\left (\sqrt {2} x^{3} + 2 \, x^{3} + \sqrt {x^{4} + 1} {\left (\sqrt {2} x + x\right )}\right )} \sqrt {x^{2} + \sqrt {x^{4} + 1}} + {\left (17 \, x^{4} + 2 \, \sqrt {2} {\left (6 \, x^{4} + 1\right )} + 2 \, \sqrt {x^{4} + 1} {\left (5 \, \sqrt {2} x^{2} + 7 \, x^{2}\right )} + 3\right )} \sqrt {5 \, \sqrt {2} - 7}}{x^{4} - 1}\right ) - \sqrt {2} {\left (x^{4} - 1\right )} \sqrt {5 \, \sqrt {2} - 7} \log \left (\frac {2 \, {\left (\sqrt {2} x^{3} + 2 \, x^{3} + \sqrt {x^{4} + 1} {\left (\sqrt {2} x + x\right )}\right )} \sqrt {x^{2} + \sqrt {x^{4} + 1}} - {\left (17 \, x^{4} + 2 \, \sqrt {2} {\left (6 \, x^{4} + 1\right )} + 2 \, \sqrt {x^{4} + 1} {\left (5 \, \sqrt {2} x^{2} + 7 \, x^{2}\right )} + 3\right )} \sqrt {5 \, \sqrt {2} - 7}}{x^{4} - 1}\right ) - 2 \, \sqrt {2} {\left (x^{4} - 1\right )} \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 ) + 8 \, {\left (x^{7} - 3 \, x^{3} - {\left (x^{5} - 3 \, x\right )} \sqrt {x^{4} + 1}\right )} \sqrt {x^{2} + \sqrt {x^{4} + 1}}}{16 \, {\left (x^{4} - 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^4+1)^2/(x^4-1)^2/(x^2+(x^4+1)^(1/2))^(1/2),x, algorithm="fricas")

[Out]

-1/16*(4*sqrt(2)*(x^4 - 1)*sqrt(5*sqrt(2) + 7)*arctan((2*(6*x^7 + 10*x^3 - sqrt(2)*(5*x^7 + 7*x^3) - (x^5 - 2*
sqrt(2)*(x^5 + x) + 3*x)*sqrt(x^4 + 1))*sqrt(x^2 + sqrt(x^4 + 1))*sqrt(5*sqrt(2) + 7) - (5*x^8 + 10*x^4 - sqrt
(2)*(3*x^8 + 4*x^4 + 1) - 2*(x^6 + 3*x^2 - 2*sqrt(2)*(x^6 + x^2))*sqrt(x^4 + 1) + 1)*sqrt(5*sqrt(2) + 7)*sqrt(
sqrt(2) - 1))/(7*x^8 + 10*x^4 - 1)) + sqrt(2)*(x^4 - 1)*sqrt(5*sqrt(2) - 7)*log((2*(sqrt(2)*x^3 + 2*x^3 + sqrt
(x^4 + 1)*(sqrt(2)*x + x))*sqrt(x^2 + sqrt(x^4 + 1)) + (17*x^4 + 2*sqrt(2)*(6*x^4 + 1) + 2*sqrt(x^4 + 1)*(5*sq
rt(2)*x^2 + 7*x^2) + 3)*sqrt(5*sqrt(2) - 7))/(x^4 - 1)) - sqrt(2)*(x^4 - 1)*sqrt(5*sqrt(2) - 7)*log((2*(sqrt(2
)*x^3 + 2*x^3 + sqrt(x^4 + 1)*(sqrt(2)*x + x))*sqrt(x^2 + sqrt(x^4 + 1)) - (17*x^4 + 2*sqrt(2)*(6*x^4 + 1) + 2
*sqrt(x^4 + 1)*(5*sqrt(2)*x^2 + 7*x^2) + 3)*sqrt(5*sqrt(2) - 7))/(x^4 - 1)) - 2*sqrt(2)*(x^4 - 1)*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) + 8*(x^7 - 3*x^
3 - (x^5 - 3*x)*sqrt(x^4 + 1))*sqrt(x^2 + sqrt(x^4 + 1)))/(x^4 - 1)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (x^{4} + 1\right )}^{2}}{{\left (x^{4} - 1\right )}^{2} \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)^2/(x^4-1)^2/(x^2+(x^4+1)^(1/2))^(1/2),x, algorithm="giac")

[Out]

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

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maple [F]  time = 0.12, size = 0, normalized size = 0.00 \[\int \frac {\left (x^{4}+1\right )^{2}}{\left (x^{4}-1\right )^{2} \sqrt {x^{2}+\sqrt {x^{4}+1}}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int((x^4+1)^2/(x^4-1)^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 {{\left (x^{4} + 1\right )}^{2}}{{\left (x^{4} - 1\right )}^{2} \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)^2/(x^4-1)^2/(x^2+(x^4+1)^(1/2))^(1/2),x, algorithm="maxima")

[Out]

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

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

[Out]

int((x^4 + 1)^2/((x^4 - 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 {\left (x^{4} + 1\right )^{2}}{\left (x - 1\right )^{2} \left (x + 1\right )^{2} \left (x^{2} + 1\right )^{2} \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)**2/(x**4-1)**2/(x**2+(x**4+1)**(1/2))**(1/2),x)

[Out]

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

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