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

Optimal. Leaf size=261 \[ \frac {x \left (x^2-5\right )}{2 \left (x^2-1\right ) \sqrt {\sqrt {x^4+1}+x^2}}-4 \sqrt {2} \tan ^{-1}\left (\frac {\sqrt {2} x \sqrt {\sqrt {x^4+1}+x^2}}{\sqrt {x^4+1}+x^2+1}\right )+\sqrt {2 \left (7+5 \sqrt {2}\right )} \tan ^{-1}\left (\frac {\sqrt {2 \left (\sqrt {2}-1\right )} x \sqrt {\sqrt {x^4+1}+x^2}}{\sqrt {x^4+1}+x^2+1}\right )+\frac {\tanh ^{-1}\left (\frac {\sqrt {2} x \sqrt {\sqrt {x^4+1}+x^2}}{\sqrt {x^4+1}+x^2+1}\right )}{\sqrt {2}}-\sqrt {2 \left (5 \sqrt {2}-7\right )} \tanh ^{-1}\left (\frac {\sqrt {2 \left (1+\sqrt {2}\right )} x \sqrt {\sqrt {x^4+1}+x^2}}{\sqrt {x^4+1}+x^2+1}\right ) \]

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Rubi [F]  time = 0.72, 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^2\right )^2}{\left (-1+x^2\right )^2 \sqrt {x^2+\sqrt {1+x^4}}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [F]  time = 0.30, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (1+x^2\right )^2}{\left (-1+x^2\right )^2 \sqrt {x^2+\sqrt {1+x^4}}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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IntegrateAlgebraic [A]  time = 2.22, size = 341, normalized size = 1.31 \begin {gather*} \frac {x \left (-5+x^2\right )}{2 \left (-1+x^2\right ) \sqrt {x^2+\sqrt {1+x^4}}}-4 \sqrt {2} \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 {14+10 \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}}-\sqrt {-14+10 \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^2)^2/((-1 + x^2)^2*Sqrt[x^2 + Sqrt[1 + x^4]]),x]

[Out]

(x*(-5 + x^2))/(2*(-1 + x^2)*Sqrt[x^2 + Sqrt[1 + x^4]]) - 4*Sqrt[2]*ArcTan[(-(1/Sqrt[2]) + x^2/Sqrt[2] + Sqrt[
1 + x^4]/Sqrt[2])/(x*Sqrt[x^2 + Sqrt[1 + x^4]])] + Sqrt[14 + 10*Sqrt[2]]*ArcTan[(-Sqrt[-1/2 + 1/Sqrt[2]] + Sqr
t[-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]])] + ArcTanh[(-(1/
Sqrt[2]) + x^2/Sqrt[2] + Sqrt[1 + x^4]/Sqrt[2])/(x*Sqrt[x^2 + Sqrt[1 + x^4]])]/Sqrt[2] - Sqrt[-14 + 10*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]])]

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fricas [B]  time = 5.84, size = 517, normalized size = 1.98 \begin {gather*} \frac {8 \, {\left (x^{2} - 1\right )} \sqrt {10 \, \sqrt {2} + 14} \arctan \left (\frac {{\left (4 \, x^{2} - 2 \, \sqrt {2} {\left (x^{2} + 3\right )} + \sqrt {x^{4} + 1} {\left ({\left (\sqrt {2} - 2\right )} \sqrt {-8 \, \sqrt {2} + 12} + 2 \, \sqrt {2} - 4\right )} + {\left (2 \, x^{2} - \sqrt {2} {\left (x^{2} + 1\right )}\right )} \sqrt {-8 \, \sqrt {2} + 12} + 8\right )} \sqrt {x^{2} + \sqrt {x^{4} + 1}} \sqrt {10 \, \sqrt {2} + 14}}{8 \, x}\right ) + 16 \, \sqrt {2} {\left (x^{2} - 1\right )} \arctan \left (-\frac {{\left (\sqrt {2} x^{2} - \sqrt {2} \sqrt {x^{4} + 1}\right )} \sqrt {x^{2} + \sqrt {x^{4} + 1}}}{2 \, x}\right ) + \sqrt {2} {\left (x^{2} - 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 ) + 2 \, {\left (x^{2} - 1\right )} \sqrt {10 \, \sqrt {2} - 14} \log \left (-\frac {2 \, \sqrt {2} x^{2} + 4 \, x^{2} + {\left (4 \, x^{3} + \sqrt {2} {\left (3 \, x^{3} - 7 \, x\right )} - \sqrt {x^{4} + 1} {\left (3 \, \sqrt {2} x + 4 \, x\right )} - 10 \, x\right )} \sqrt {x^{2} + \sqrt {x^{4} + 1}} \sqrt {10 \, \sqrt {2} - 14} + 2 \, \sqrt {x^{4} + 1} {\left (\sqrt {2} + 1\right )}}{x^{2} - 1}\right ) - 2 \, {\left (x^{2} - 1\right )} \sqrt {10 \, \sqrt {2} - 14} \log \left (-\frac {2 \, \sqrt {2} x^{2} + 4 \, x^{2} - {\left (4 \, x^{3} + \sqrt {2} {\left (3 \, x^{3} - 7 \, x\right )} - \sqrt {x^{4} + 1} {\left (3 \, \sqrt {2} x + 4 \, x\right )} - 10 \, x\right )} \sqrt {x^{2} + \sqrt {x^{4} + 1}} \sqrt {10 \, \sqrt {2} - 14} + 2 \, \sqrt {x^{4} + 1} {\left (\sqrt {2} + 1\right )}}{x^{2} - 1}\right ) - 4 \, {\left (x^{5} - 5 \, x^{3} - \sqrt {x^{4} + 1} {\left (x^{3} - 5 \, x\right )}\right )} \sqrt {x^{2} + \sqrt {x^{4} + 1}}}{8 \, {\left (x^{2} - 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*(8*(x^2 - 1)*sqrt(10*sqrt(2) + 14)*arctan(1/8*(4*x^2 - 2*sqrt(2)*(x^2 + 3) + sqrt(x^4 + 1)*((sqrt(2) - 2)*
sqrt(-8*sqrt(2) + 12) + 2*sqrt(2) - 4) + (2*x^2 - sqrt(2)*(x^2 + 1))*sqrt(-8*sqrt(2) + 12) + 8)*sqrt(x^2 + sqr
t(x^4 + 1))*sqrt(10*sqrt(2) + 14)/x) + 16*sqrt(2)*(x^2 - 1)*arctan(-1/2*(sqrt(2)*x^2 - sqrt(2)*sqrt(x^4 + 1))*
sqrt(x^2 + sqrt(x^4 + 1))/x) + sqrt(2)*(x^2 - 1)*log(4*x^4 + 4*sqrt(x^4 + 1)*x^2 + 2*(sqrt(2)*x^3 + sqrt(2)*sq
rt(x^4 + 1)*x)*sqrt(x^2 + sqrt(x^4 + 1)) + 1) + 2*(x^2 - 1)*sqrt(10*sqrt(2) - 14)*log(-(2*sqrt(2)*x^2 + 4*x^2
+ (4*x^3 + sqrt(2)*(3*x^3 - 7*x) - sqrt(x^4 + 1)*(3*sqrt(2)*x + 4*x) - 10*x)*sqrt(x^2 + sqrt(x^4 + 1))*sqrt(10
*sqrt(2) - 14) + 2*sqrt(x^4 + 1)*(sqrt(2) + 1))/(x^2 - 1)) - 2*(x^2 - 1)*sqrt(10*sqrt(2) - 14)*log(-(2*sqrt(2)
*x^2 + 4*x^2 - (4*x^3 + sqrt(2)*(3*x^3 - 7*x) - sqrt(x^4 + 1)*(3*sqrt(2)*x + 4*x) - 10*x)*sqrt(x^2 + sqrt(x^4
+ 1))*sqrt(10*sqrt(2) - 14) + 2*sqrt(x^4 + 1)*(sqrt(2) + 1))/(x^2 - 1)) - 4*(x^5 - 5*x^3 - sqrt(x^4 + 1)*(x^3
- 5*x))*sqrt(x^2 + sqrt(x^4 + 1)))/(x^2 - 1)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [F]  time = 0.11, size = 0, normalized size = 0.00 \[\int \frac {\left (x^{2}+1\right )^{2}}{\left (x^{2}-1\right )^{2} \sqrt {x^{2}+\sqrt {x^{4}+1}}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

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

[Out]

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

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