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

Optimal. Leaf size=223 \[ \frac {\left (x^2+1\right ) x^2+\sqrt {x^4+1} x^2}{x \left (x^2+1\right ) \sqrt {\sqrt {x^4+1}+x^2}}+\sqrt {2} \tan ^{-1}\left (\frac {\sqrt {2} x \sqrt {\sqrt {x^4+1}+x^2}}{\sqrt {x^4+1}+x^2+1}\right )-\sqrt {1+\sqrt {2}} \tan ^{-1}\left (\frac {\sqrt {2 \left (1+\sqrt {2}\right )} x \sqrt {\sqrt {x^4+1}+x^2}}{\sqrt {x^4+1}+x^2+1}\right )-\sqrt {\sqrt {2}-1} \tanh ^{-1}\left (\frac {\sqrt {2 \left (\sqrt {2}-1\right )} x \sqrt {\sqrt {x^4+1}+x^2}}{\sqrt {x^4+1}+x^2+1}\right ) \]

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Rubi [F]  time = 3.36, 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 {1+x^4} \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[1 + x^4]*Sqrt[x^2 + Sqrt[1 + x^4]]),x]

[Out]

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

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Mathematica [F]  time = 0.55, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (-1+x^2\right )^2}{\left (1+x^2\right )^2 \sqrt {1+x^4} \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[1 + x^4]*Sqrt[x^2 + Sqrt[1 + x^4]]),x]

[Out]

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

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IntegrateAlgebraic [A]  time = 0.92, size = 294, normalized size = 1.32 \begin {gather*} \frac {x^2 \left (1+x^2\right )+x^2 \sqrt {1+x^4}}{x \left (1+x^2\right ) \sqrt {x^2+\sqrt {1+x^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 {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 )-\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)^2/((1 + x^2)^2*Sqrt[1 + x^4]*Sqrt[x^2 + Sqrt[1 + x^4]]),x]

[Out]

(x^2*(1 + x^2) + x^2*Sqrt[1 + x^4])/(x*(1 + x^2)*Sqrt[x^2 + Sqrt[1 + x^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[1 + Sqrt[2]]*ArcTan[(-Sqrt[1/2 + 1/S
qrt[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]])] - Sq
rt[-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 = 2.14, size = 385, normalized size = 1.73 \begin {gather*} -\frac {4 \, {\left (x^{2} + 1\right )} \sqrt {\sqrt {2} + 1} \arctan \left (\frac {{\left (x^{2} - {\left (x^{2} + \sqrt {2} + 1\right )} \sqrt {-2 \, \sqrt {2} + 3} + \sqrt {x^{4} + 1} {\left (\sqrt {-2 \, \sqrt {2} + 3} - 1\right )} + \sqrt {2} - 1\right )} \sqrt {x^{2} + \sqrt {x^{4} + 1}} \sqrt {\sqrt {2} + 1}}{2 \, x}\right ) + 2 \, \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 ) + {\left (x^{2} + 1\right )} \sqrt {\sqrt {2} - 1} \log \left (\frac {\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 )}}{x^{2} + 1}\right ) - {\left (x^{2} + 1\right )} \sqrt {\sqrt {2} - 1} \log \left (\frac {\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 )}}{x^{2} + 1}\right ) + 4 \, {\left (x^{3} - \sqrt {x^{4} + 1} x - x\right )} \sqrt {x^{2} + \sqrt {x^{4} + 1}}}{4 \, {\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^4+1)^(1/2)/(x^2+(x^4+1)^(1/2))^(1/2),x, algorithm="fricas")

[Out]

-1/4*(4*(x^2 + 1)*sqrt(sqrt(2) + 1)*arctan(1/2*(x^2 - (x^2 + sqrt(2) + 1)*sqrt(-2*sqrt(2) + 3) + sqrt(x^4 + 1)
*(sqrt(-2*sqrt(2) + 3) - 1) + sqrt(2) - 1)*sqrt(x^2 + sqrt(x^4 + 1))*sqrt(sqrt(2) + 1)/x) + 2*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) + (x^2 + 1)*sqrt(sqrt(2) - 1)
*log((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)) - (x^2 + 1)*sqrt(sqrt(2) - 1)*log((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))*s
qrt(sqrt(2) - 1) + sqrt(x^4 + 1)*(sqrt(2) + 1))/(x^2 + 1)) + 4*(x^3 - sqrt(x^4 + 1)*x - 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^{4} + 1} \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^4+1)^(1/2)/(x^2+(x^4+1)^(1/2))^(1/2),x, algorithm="giac")

[Out]

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

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maple [F]  time = 0.09, size = 0, normalized size = 0.00 \[\int \frac {\left (x^{2}-1\right )^{2}}{\left (x^{2}+1\right )^{2} \sqrt {x^{4}+1}\, \sqrt {x^{2}+\sqrt {x^{4}+1}}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

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

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