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

Optimal. Leaf size=152 \[ \frac {2 x}{\sqrt {\sqrt {x^2+1}+1}}-2 \tan ^{-1}\left (\frac {x}{\sqrt {\sqrt {x^2+1}+1}}\right )+\sqrt {2} \tan ^{-1}\left (\frac {x}{\sqrt {2} \sqrt {\sqrt {x^2+1}+1}}\right )-\sqrt {1+\sqrt {2}} \tan ^{-1}\left (\frac {x}{\sqrt {1+\sqrt {2}} \sqrt {\sqrt {x^2+1}+1}}\right )-\sqrt {\sqrt {2}-1} \tanh ^{-1}\left (\frac {x}{\sqrt {\sqrt {2}-1} \sqrt {\sqrt {x^2+1}+1}}\right ) \]

________________________________________________________________________________________

Rubi [F]  time = 0.88, 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^4}{\left (-1+x^4\right ) \sqrt {1+\sqrt {1+x^2}}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

Defer[Int][1/Sqrt[1 + Sqrt[1 + x^2]], x] - (I/2)*Defer[Int][1/((I - x)*Sqrt[1 + Sqrt[1 + x^2]]), x] - Defer[In
t][1/((1 - x)*Sqrt[1 + Sqrt[1 + x^2]]), x]/2 - (I/2)*Defer[Int][1/((I + x)*Sqrt[1 + Sqrt[1 + x^2]]), x] - Defe
r[Int][1/((1 + x)*Sqrt[1 + Sqrt[1 + x^2]]), x]/2

Rubi steps

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

________________________________________________________________________________________

Mathematica [F]  time = 0.26, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1+x^4}{\left (-1+x^4\right ) \sqrt {1+\sqrt {1+x^2}}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

________________________________________________________________________________________

IntegrateAlgebraic [A]  time = 0.49, size = 152, normalized size = 1.00 \begin {gather*} \frac {2 x}{\sqrt {1+\sqrt {1+x^2}}}-2 \tan ^{-1}\left (\frac {x}{\sqrt {1+\sqrt {1+x^2}}}\right )+\sqrt {2} \tan ^{-1}\left (\frac {x}{\sqrt {2} \sqrt {1+\sqrt {1+x^2}}}\right )-\sqrt {1+\sqrt {2}} \tan ^{-1}\left (\frac {\sqrt {-1+\sqrt {2}} x}{\sqrt {1+\sqrt {1+x^2}}}\right )-\sqrt {-1+\sqrt {2}} \tanh ^{-1}\left (\frac {\sqrt {1+\sqrt {2}} x}{\sqrt {1+\sqrt {1+x^2}}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*x)/Sqrt[1 + Sqrt[1 + x^2]] - 2*ArcTan[x/Sqrt[1 + Sqrt[1 + x^2]]] + Sqrt[2]*ArcTan[x/(Sqrt[2]*Sqrt[1 + Sqrt[
1 + x^2]])] - Sqrt[1 + Sqrt[2]]*ArcTan[(Sqrt[-1 + Sqrt[2]]*x)/Sqrt[1 + Sqrt[1 + x^2]]] - Sqrt[-1 + Sqrt[2]]*Ar
cTanh[(Sqrt[1 + Sqrt[2]]*x)/Sqrt[1 + Sqrt[1 + x^2]]]

________________________________________________________________________________________

fricas [B]  time = 21.35, size = 520, normalized size = 3.42 \begin {gather*} \frac {4 \, x \sqrt {\sqrt {2} + 1} \arctan \left (-\frac {{\left (71 \, x^{5} - 874 \, x^{3} - \sqrt {2} {\left (61 \, x^{5} - 548 \, x^{3} - 81 \, x\right )} - 2 \, {\left (51 \, x^{3} - 2 \, \sqrt {2} {\left (5 \, x^{3} - 127 \, x\right )} - 335 \, x\right )} \sqrt {x^{2} + 1} - 173 \, x\right )} \sqrt {3821 \, \sqrt {2} + 4841} \sqrt {\sqrt {2} + 1} - 4802 \, {\left (x^{4} - 6 \, x^{2} + \sqrt {2} {\left (3 \, x^{2} + 1\right )} + {\left (x^{2} + \sqrt {2} {\left (x^{2} - 1\right )} + 3\right )} \sqrt {x^{2} + 1} - 3\right )} \sqrt {\sqrt {2} + 1} \sqrt {\sqrt {x^{2} + 1} + 1}}{2401 \, {\left (x^{5} - 10 \, x^{3} - 7 \, x\right )}}\right ) - 4 \, \sqrt {2} x \arctan \left (\frac {\sqrt {2} \sqrt {\sqrt {x^{2} + 1} + 1}}{x}\right ) + x \sqrt {\sqrt {2} - 1} \log \left (-\frac {{\left (51 \, x^{3} - 2 \, \sqrt {2} {\left (5 \, x^{3} + 66 \, x\right )} + 2 \, \sqrt {x^{2} + 1} {\left (61 \, \sqrt {2} x - 71 \, x\right )} + 193 \, x\right )} \sqrt {\sqrt {2} - 1} + 2 \, {\left (71 \, x^{2} - \sqrt {2} {\left (61 \, x^{2} + 132\right )} + \sqrt {x^{2} + 1} {\left (132 \, \sqrt {2} - 193\right )} + 193\right )} \sqrt {\sqrt {x^{2} + 1} + 1}}{x^{3} - x}\right ) - x \sqrt {\sqrt {2} - 1} \log \left (\frac {{\left (51 \, x^{3} - 2 \, \sqrt {2} {\left (5 \, x^{3} + 66 \, x\right )} + 2 \, \sqrt {x^{2} + 1} {\left (61 \, \sqrt {2} x - 71 \, x\right )} + 193 \, x\right )} \sqrt {\sqrt {2} - 1} - 2 \, {\left (71 \, x^{2} - \sqrt {2} {\left (61 \, x^{2} + 132\right )} + \sqrt {x^{2} + 1} {\left (132 \, \sqrt {2} - 193\right )} + 193\right )} \sqrt {\sqrt {x^{2} + 1} + 1}}{x^{3} - x}\right ) + 2 \, x \arctan \left (\frac {4 \, {\left (x^{4} - 12 \, x^{2} + {\left (5 \, x^{2} - 3\right )} \sqrt {x^{2} + 1} + 3\right )} \sqrt {\sqrt {x^{2} + 1} + 1}}{x^{5} - 46 \, x^{3} + 17 \, x}\right ) + 8 \, \sqrt {\sqrt {x^{2} + 1} + 1} {\left (\sqrt {x^{2} + 1} - 1\right )}}{4 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*(4*x*sqrt(sqrt(2) + 1)*arctan(-1/2401*((71*x^5 - 874*x^3 - sqrt(2)*(61*x^5 - 548*x^3 - 81*x) - 2*(51*x^3 -
2*sqrt(2)*(5*x^3 - 127*x) - 335*x)*sqrt(x^2 + 1) - 173*x)*sqrt(3821*sqrt(2) + 4841)*sqrt(sqrt(2) + 1) - 4802*
(x^4 - 6*x^2 + sqrt(2)*(3*x^2 + 1) + (x^2 + sqrt(2)*(x^2 - 1) + 3)*sqrt(x^2 + 1) - 3)*sqrt(sqrt(2) + 1)*sqrt(s
qrt(x^2 + 1) + 1))/(x^5 - 10*x^3 - 7*x)) - 4*sqrt(2)*x*arctan(sqrt(2)*sqrt(sqrt(x^2 + 1) + 1)/x) + x*sqrt(sqrt
(2) - 1)*log(-((51*x^3 - 2*sqrt(2)*(5*x^3 + 66*x) + 2*sqrt(x^2 + 1)*(61*sqrt(2)*x - 71*x) + 193*x)*sqrt(sqrt(2
) - 1) + 2*(71*x^2 - sqrt(2)*(61*x^2 + 132) + sqrt(x^2 + 1)*(132*sqrt(2) - 193) + 193)*sqrt(sqrt(x^2 + 1) + 1)
)/(x^3 - x)) - x*sqrt(sqrt(2) - 1)*log(((51*x^3 - 2*sqrt(2)*(5*x^3 + 66*x) + 2*sqrt(x^2 + 1)*(61*sqrt(2)*x - 7
1*x) + 193*x)*sqrt(sqrt(2) - 1) - 2*(71*x^2 - sqrt(2)*(61*x^2 + 132) + sqrt(x^2 + 1)*(132*sqrt(2) - 193) + 193
)*sqrt(sqrt(x^2 + 1) + 1))/(x^3 - x)) + 2*x*arctan(4*(x^4 - 12*x^2 + (5*x^2 - 3)*sqrt(x^2 + 1) + 3)*sqrt(sqrt(
x^2 + 1) + 1)/(x^5 - 46*x^3 + 17*x)) + 8*sqrt(sqrt(x^2 + 1) + 1)*(sqrt(x^2 + 1) - 1))/x

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maple [F]  time = 0.05, size = 0, normalized size = 0.00 \[\int \frac {x^{4}+1}{\left (x^{4}-1\right ) \sqrt {1+\sqrt {x^{2}+1}}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {x^4+1}{\left (x^4-1\right )\,\sqrt {\sqrt {x^2+1}+1}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________