3.28.68 \(\int \frac {(-1+x^2)^2}{(1+x^2)^2 \sqrt {x^2+\sqrt {1+x^4}}} \, dx\) [2768]

Optimal. Leaf size=262 \[ \frac {x \left (5+x^2\right )}{2 \left (1+x^2\right ) \sqrt {x^2+\sqrt {1+x^4}}}+4 \sqrt {2} \text {ArcTan}\left (\frac {\sqrt {2} x \sqrt {x^2+\sqrt {1+x^4}}}{1+x^2+\sqrt {1+x^4}}\right )-\sqrt {2 \left (7+5 \sqrt {2}\right )} \text {ArcTan}\left (\frac {\sqrt {2 \left (1+\sqrt {2}\right )} x \sqrt {x^2+\sqrt {1+x^4}}}{1+x^2+\sqrt {1+x^4}}\right )+\frac {\tanh ^{-1}\left (\frac {\sqrt {2} x \sqrt {x^2+\sqrt {1+x^4}}}{1+x^2+\sqrt {1+x^4}}\right )}{\sqrt {2}}-\sqrt {2 \left (-7+5 \sqrt {2}\right )} \tanh ^{-1}\left (\frac {\sqrt {2 \left (-1+\sqrt {2}\right )} x \sqrt {x^2+\sqrt {1+x^4}}}{1+x^2+\sqrt {1+x^4}}\right ) \]

[Out]

1/2*x*(x^2+5)/(x^2+1)/(x^2+(x^4+1)^(1/2))^(1/2)+4*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)-(14+10*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
)))+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)-(-14+10*2^(1/2))^(1/2)*arct
anh((-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 = 0.83, 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 {\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/((I - x)^2*Sqrt[x^2 + Sqrt[1 + x^4]]), x] - I*Defer[
Int][1/((I - x)*Sqrt[x^2 + Sqrt[1 + x^4]]), x] - Defer[Int][1/((I + x)^2*Sqrt[x^2 + Sqrt[1 + x^4]]), x] - I*De
fer[Int][1/((I + x)*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 {4}{\left (1+x^2\right )^2 \sqrt {x^2+\sqrt {1+x^4}}}-\frac {4}{\left (1+x^2\right ) \sqrt {x^2+\sqrt {1+x^4}}}\right ) \, dx\\ &=4 \int \frac {1}{\left (1+x^2\right )^2 \sqrt {x^2+\sqrt {1+x^4}}} \, dx-4 \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 (4 \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\right )+4 \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+\int \frac {1}{\sqrt {x^2+\sqrt {1+x^4}}} \, dx\\ &=-\left (2 i \int \frac {1}{(i-x) \sqrt {x^2+\sqrt {1+x^4}}} \, dx\right )-2 i \int \frac {1}{(i+x) \sqrt {x^2+\sqrt {1+x^4}}} \, dx-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}{(i-x)^2 \sqrt {x^2+\sqrt {1+x^4}}} \, dx-\int \frac {1}{(i+x)^2 \sqrt {x^2+\sqrt {1+x^4}}} \, dx\\ &=-\left (2 i \int \frac {1}{(i-x) \sqrt {x^2+\sqrt {1+x^4}}} \, dx\right )-2 i \int \frac {1}{(i+x) \sqrt {x^2+\sqrt {1+x^4}}} \, dx-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-\int \frac {1}{(i-x)^2 \sqrt {x^2+\sqrt {1+x^4}}} \, dx-\int \frac {1}{(i+x)^2 \sqrt {x^2+\sqrt {1+x^4}}} \, dx\\ &=i \int \frac {1}{(i-x) \sqrt {x^2+\sqrt {1+x^4}}} \, dx+i \int \frac {1}{(i+x) \sqrt {x^2+\sqrt {1+x^4}}} \, dx-2 i \int \frac {1}{(i-x) \sqrt {x^2+\sqrt {1+x^4}}} \, dx-2 i \int \frac {1}{(i+x) \sqrt {x^2+\sqrt {1+x^4}}} \, dx+\int \frac {1}{\sqrt {x^2+\sqrt {1+x^4}}} \, dx-\int \frac {1}{(i-x)^2 \sqrt {x^2+\sqrt {1+x^4}}} \, dx-\int \frac {1}{(i+x)^2 \sqrt {x^2+\sqrt {1+x^4}}} \, dx\\ \end {align*}

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Mathematica [A]
time = 1.53, size = 262, normalized size = 1.00 \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} \text {ArcTan}\left (\frac {-1+x^2+\sqrt {1+x^4}}{\sqrt {2} x \sqrt {x^2+\sqrt {1+x^4}}}\right )-\sqrt {2 \left (7+5 \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 )+\frac {\tanh ^{-1}\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 (-7+5 \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[(-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 + x^2 + Sqrt[1 + x^4])/(Sqrt[2]*x
*Sqrt[x^2 + Sqrt[1 + x^4]])] - Sqrt[2*(7 + 5*Sqrt[2])]*ArcTan[(Sqrt[1/2 + 1/Sqrt[2]]*(-1 + x^2 + Sqrt[1 + x^4]
))/(x*Sqrt[x^2 + Sqrt[1 + x^4]])] + ArcTanh[(-1 + x^2 + Sqrt[1 + x^4])/(Sqrt[2]*x*Sqrt[x^2 + Sqrt[1 + x^4]])]/
Sqrt[2] - Sqrt[2*(-7 + 5*Sqrt[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 = 0.08, 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*} \text {Failed to integrate} \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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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 517 vs. \(2 (205) = 410\).
time = 7.78, size = 517, normalized size = 1.97 \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 + sq
rt(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)*s
qrt(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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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\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**2-1)**2/(x**2+1)**2/(x**2+(x**4+1)**(1/2))**(1/2),x)

[Out]

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