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

Optimal. Leaf size=242 \[ \frac {x}{2 \sqrt {\sqrt {x^4+1}+x^2}}+2 \sqrt {2} \tan ^{-1}\left (\frac {\sqrt {2} x \sqrt {\sqrt {x^4+1}+x^2}}{\sqrt {x^4+1}+x^2+1}\right )-2 \sqrt {1+\sqrt {2}} \tan ^{-1}\left (\frac {\sqrt {2 \left (1+\sqrt {2}\right )} x \sqrt {\sqrt {x^4+1}+x^2}}{\sqrt {x^4+1}+x^2+1}\right )+\frac {\tanh ^{-1}\left (\frac {\sqrt {2} x \sqrt {\sqrt {x^4+1}+x^2}}{\sqrt {x^4+1}+x^2+1}\right )}{\sqrt {2}}-2 \sqrt {\sqrt {2}-1} \tanh ^{-1}\left (\frac {\sqrt {2 \left (\sqrt {2}-1\right )} x \sqrt {\sqrt {x^4+1}+x^2}}{\sqrt {x^4+1}+x^2+1}\right ) \]

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Rubi [F]  time = 0.54, 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 {-1+x^2}{\left (1+x^2\right ) \sqrt {x^2+\sqrt {1+x^4}}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [F]  time = 0.25, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {-1+x^2}{\left (1+x^2\right ) \sqrt {x^2+\sqrt {1+x^4}}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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IntegrateAlgebraic [A]  time = 1.05, size = 326, normalized size = 1.35 \begin {gather*} \frac {x}{2 \sqrt {x^2+\sqrt {1+x^4}}}+2 \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 )-2 \sqrt {1+\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}}-2 \sqrt {-1+\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)/((1 + x^2)*Sqrt[x^2 + Sqrt[1 + x^4]]),x]

[Out]

x/(2*Sqrt[x^2 + Sqrt[1 + x^4]]) + 2*Sqrt[2]*ArcTan[(-(1/Sqrt[2]) + x^2/Sqrt[2] + Sqrt[1 + x^4]/Sqrt[2])/(x*Sqr
t[x^2 + Sqrt[1 + x^4]])] - 2*Sqrt[1 + Sqrt[2]]*ArcTan[(-Sqrt[1/2 + 1/Sqrt[2]] + Sqrt[1/2 + 1/Sqrt[2]]*x^2 + Sq
rt[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] - 2*Sqrt[-1 + 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 = 4.03, size = 420, normalized size = 1.74 \begin {gather*} -\frac {1}{2} \, {\left (x^{3} - \sqrt {x^{4} + 1} x\right )} \sqrt {x^{2} + \sqrt {x^{4} + 1}} - 2 \, \sqrt {\sqrt {2} + 1} \arctan \left (\frac {{\left (2 \, x^{2} - {\left (x^{2} + \sqrt {2} + 1\right )} \sqrt {-8 \, \sqrt {2} + 12} + \sqrt {x^{4} + 1} {\left (\sqrt {-8 \, \sqrt {2} + 12} - 2\right )} + 2 \, \sqrt {2} - 2\right )} \sqrt {x^{2} + \sqrt {x^{4} + 1}} \sqrt {\sqrt {2} + 1}}{4 \, x}\right ) - \sqrt {2} \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 ) + \frac {1}{8} \, \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 ) - \frac {1}{2} \, \sqrt {\sqrt {2} - 1} \log \left (\frac {2 \, {\left (\sqrt {2} x^{2} + 2 \, x^{2} + {\left (x^{3} + \sqrt {2} {\left (x^{3} + 2 \, x\right )} - \sqrt {x^{4} + 1} {\left (\sqrt {2} x + x\right )} + 3 \, x\right )} \sqrt {x^{2} + \sqrt {x^{4} + 1}} \sqrt {\sqrt {2} - 1} + \sqrt {x^{4} + 1} {\left (\sqrt {2} + 1\right )}\right )}}{x^{2} + 1}\right ) + \frac {1}{2} \, \sqrt {\sqrt {2} - 1} \log \left (\frac {2 \, {\left (\sqrt {2} x^{2} + 2 \, x^{2} - {\left (x^{3} + \sqrt {2} {\left (x^{3} + 2 \, x\right )} - \sqrt {x^{4} + 1} {\left (\sqrt {2} x + x\right )} + 3 \, x\right )} \sqrt {x^{2} + \sqrt {x^{4} + 1}} \sqrt {\sqrt {2} - 1} + \sqrt {x^{4} + 1} {\left (\sqrt {2} + 1\right )}\right )}}{x^{2} + 1}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*(x^3 - sqrt(x^4 + 1)*x)*sqrt(x^2 + sqrt(x^4 + 1)) - 2*sqrt(sqrt(2) + 1)*arctan(1/4*(2*x^2 - (x^2 + sqrt(2
) + 1)*sqrt(-8*sqrt(2) + 12) + sqrt(x^4 + 1)*(sqrt(-8*sqrt(2) + 12) - 2) + 2*sqrt(2) - 2)*sqrt(x^2 + sqrt(x^4
+ 1))*sqrt(sqrt(2) + 1)/x) - sqrt(2)*arctan(-1/2*(sqrt(2)*x^2 - sqrt(2)*sqrt(x^4 + 1))*sqrt(x^2 + sqrt(x^4 + 1
))/x) + 1/8*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 + sqr
t(x^4 + 1)) + 1) - 1/2*sqrt(sqrt(2) - 1)*log(2*(sqrt(2)*x^2 + 2*x^2 + (x^3 + sqrt(2)*(x^3 + 2*x) - sqrt(x^4 +
1)*(sqrt(2)*x + x) + 3*x)*sqrt(x^2 + sqrt(x^4 + 1))*sqrt(sqrt(2) - 1) + sqrt(x^4 + 1)*(sqrt(2) + 1))/(x^2 + 1)
) + 1/2*sqrt(sqrt(2) - 1)*log(2*(sqrt(2)*x^2 + 2*x^2 - (x^3 + sqrt(2)*(x^3 + 2*x) - sqrt(x^4 + 1)*(sqrt(2)*x +
x) + 3*x)*sqrt(x^2 + sqrt(x^4 + 1))*sqrt(sqrt(2) - 1) + sqrt(x^4 + 1)*(sqrt(2) + 1))/(x^2 + 1))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [F]  time = 0.04, size = 0, normalized size = 0.00 \[\int \frac {x^{2}-1}{\left (x^{2}+1\right ) \sqrt {x^{2}+\sqrt {x^{4}+1}}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

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

[Out]

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

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