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

Optimal. Leaf size=311 \[ \sqrt {\frac {1}{2} \left (\sqrt {2}-1\right )} \tan ^{-1}\left (\frac {\sqrt {2 \left (\sqrt {2}-1\right )} x \sqrt {\sqrt {x^4+1}+x^2}}{\sqrt {x^4+1}+x^2+1}\right )-\sqrt {\frac {1}{2} \left (\sqrt {2}-1\right )} \tan ^{-1}\left (\frac {\sqrt {2 \left (1+\sqrt {2}\right )} x \sqrt {\sqrt {x^4+1}+x^2}}{\sqrt {x^4+1}+x^2+1}\right )+\sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {2} x \sqrt {\sqrt {x^4+1}+x^2}}{\sqrt {x^4+1}+x^2+1}\right )-\sqrt {\frac {1}{2} \left (1+\sqrt {2}\right )} \tanh ^{-1}\left (\frac {\sqrt {2 \left (\sqrt {2}-1\right )} x \sqrt {\sqrt {x^4+1}+x^2}}{\sqrt {x^4+1}+x^2+1}\right )-\sqrt {\frac {1}{2} \left (1+\sqrt {2}\right )} \tanh ^{-1}\left (\frac {\sqrt {2 \left (1+\sqrt {2}\right )} x \sqrt {\sqrt {x^4+1}+x^2}}{\sqrt {x^4+1}+x^2+1}\right ) \]

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Rubi [F]  time = 1.79, 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^4} \, 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^4),x]

[Out]

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

Rubi steps

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

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Mathematica [F]  time = 0.27, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {1+x^4} \sqrt {x^2+\sqrt {1+x^4}}}{-1+x^4} \, 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^4),x]

[Out]

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

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IntegrateAlgebraic [A]  time = 1.85, size = 440, normalized size = 1.41 \begin {gather*} \sqrt {\frac {1}{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 {\frac {1}{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 {\frac {1}{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 )-\sqrt {\frac {1}{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^4),x]

[Out]

Sqrt[(-1 + Sqrt[2])/2]*ArcTan[(-Sqrt[-1/2 + 1/Sqrt[2]] + Sqrt[-1/2 + 1/Sqrt[2]]*x^2 + Sqrt[-1/2 + 1/Sqrt[2]]*S
qrt[1 + x^4])/(x*Sqrt[x^2 + Sqrt[1 + x^4]])] - Sqrt[(-1 + Sqrt[2])/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]])] + 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[(1 + Sqrt[2])/2]*ArcTa
nh[(-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]])] - Sqrt[(1 + Sqrt[2])/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]])]

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fricas [A]  time = 4.06, size = 433, normalized size = 1.39 \begin {gather*} -\frac {1}{2} \, \sqrt {2} \sqrt {\sqrt {2} - 1} \arctan \left (\frac {x^{8} - 2 \, x^{4} - 2 \, {\left (2 \, x^{7} - 2 \, x^{3} + \sqrt {2} {\left (3 \, x^{7} + x^{3}\right )} - {\left (4 \, \sqrt {2} x^{5} + 5 \, x^{5} - x\right )} \sqrt {x^{4} + 1}\right )} \sqrt {x^{2} + \sqrt {x^{4} + 1}} \sqrt {\sqrt {2} - 1} - 2 \, \sqrt {2} {\left (x^{8} + 3 \, x^{4}\right )} - 2 \, {\left (3 \, x^{6} + x^{2} + \sqrt {2} {\left (x^{6} - x^{2}\right )}\right )} \sqrt {x^{4} + 1} + 1}{7 \, x^{8} + 10 \, x^{4} - 1}\right ) - \frac {1}{8} \, \sqrt {2} \sqrt {\sqrt {2} + 1} \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 (2 \, \sqrt {2} x^{4} + 3 \, x^{4} + 2 \, \sqrt {x^{4} + 1} {\left (\sqrt {2} x^{2} + x^{2}\right )} + 1\right )} \sqrt {\sqrt {2} + 1}}{x^{4} - 1}\right ) + \frac {1}{8} \, \sqrt {2} \sqrt {\sqrt {2} + 1} \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 (2 \, \sqrt {2} x^{4} + 3 \, x^{4} + 2 \, \sqrt {x^{4} + 1} {\left (\sqrt {2} x^{2} + x^{2}\right )} + 1\right )} \sqrt {\sqrt {2} + 1}}{x^{4} - 1}\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 ) \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^4-1),x, algorithm="fricas")

[Out]

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

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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^{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^4-1),x, algorithm="giac")

[Out]

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

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maple [F]  time = 0.06, size = 0, normalized size = 0.00 \[\int \frac {\sqrt {x^{4}+1}\, \sqrt {x^{2}+\sqrt {x^{4}+1}}}{x^{4}-1}\, dx\]

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^4-1),x)

[Out]

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

[Out]

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

[Out]

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

[Out]

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

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