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

Optimal. Leaf size=274 \[ \sqrt {\sqrt {x^4+1}+x^2}-\sqrt {\sqrt {2}-1} \tan ^{-1}\left (\frac {\sqrt {\sqrt {x^4+1}+x^2}}{\sqrt {\sqrt {2}-1}}\right )+\sqrt {\sqrt {2}-1} \tan ^{-1}\left (\frac {\sqrt {2 \left (\sqrt {2}-1\right )} x \sqrt {\sqrt {x^4+1}+x^2}}{\sqrt {x^4+1}+x^2+1}\right )-\sqrt {1+\sqrt {2}} \tanh ^{-1}\left (\frac {\sqrt {\sqrt {x^4+1}+x^2}}{\sqrt {1+\sqrt {2}}}\right )-\sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {2} x \sqrt {\sqrt {x^4+1}+x^2}}{\sqrt {x^4+1}+x^2+1}\right )+\sqrt {1+\sqrt {2}} \tanh ^{-1}\left (\frac {\sqrt {2 \left (1+\sqrt {2}\right )} x \sqrt {\sqrt {x^4+1}+x^2}}{\sqrt {x^4+1}+x^2+1}\right ) \]

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Rubi [F]  time = 0.11, 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 {\sqrt {x^2+\sqrt {1+x^4}}}{1+x} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [F]  time = 0.18, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {x^2+\sqrt {1+x^4}}}{1+x} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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IntegrateAlgebraic [A]  time = 1.37, size = 274, normalized size = 1.00 \begin {gather*} \sqrt {x^2+\sqrt {1+x^4}}-\sqrt {-1+\sqrt {2}} \tan ^{-1}\left (\sqrt {1+\sqrt {2}} \sqrt {x^2+\sqrt {1+x^4}}\right )+\sqrt {-1+\sqrt {2}} \tan ^{-1}\left (\frac {\sqrt {-2+2 \sqrt {2}} x \sqrt {x^2+\sqrt {1+x^4}}}{1+x^2+\sqrt {1+x^4}}\right )-\sqrt {1+\sqrt {2}} \tanh ^{-1}\left (\sqrt {-1+\sqrt {2}} \sqrt {x^2+\sqrt {1+x^4}}\right )-\sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {2} x \sqrt {x^2+\sqrt {1+x^4}}}{1+x^2+\sqrt {1+x^4}}\right )+\sqrt {1+\sqrt {2}} \tanh ^{-1}\left (\frac {\sqrt {2+2 \sqrt {2}} x \sqrt {x^2+\sqrt {1+x^4}}}{1+x^2+\sqrt {1+x^4}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

Sqrt[x^2 + Sqrt[1 + x^4]] - Sqrt[-1 + Sqrt[2]]*ArcTan[Sqrt[1 + Sqrt[2]]*Sqrt[x^2 + Sqrt[1 + x^4]]] + Sqrt[-1 +
Sqrt[2]]*ArcTan[(Sqrt[-2 + 2*Sqrt[2]]*x*Sqrt[x^2 + Sqrt[1 + x^4]])/(1 + x^2 + Sqrt[1 + x^4])] - Sqrt[1 + Sqrt
[2]]*ArcTanh[Sqrt[-1 + Sqrt[2]]*Sqrt[x^2 + Sqrt[1 + x^4]]] - Sqrt[2]*ArcTanh[(Sqrt[2]*x*Sqrt[x^2 + Sqrt[1 + x^
4]])/(1 + x^2 + Sqrt[1 + x^4])] + Sqrt[1 + Sqrt[2]]*ArcTanh[(Sqrt[2 + 2*Sqrt[2]]*x*Sqrt[x^2 + Sqrt[1 + x^4]])/
(1 + x^2 + Sqrt[1 + x^4])]

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fricas [B]  time = 1.89, size = 436, normalized size = 1.59 \begin {gather*} \sqrt {\sqrt {2} - 1} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} {\left (x^{2} + 1\right )} + 2 \, \sqrt {x^{4} + 1}\right )} \sqrt {\sqrt {2} + 1} \sqrt {\sqrt {2} - 1} - 2 \, {\left (x^{3} - x^{2} - \sqrt {2} {\left (x^{2} - x\right )} - \sqrt {x^{4} + 1} {\left (x - \sqrt {2} - 1\right )} + x + 1\right )} \sqrt {x^{2} + \sqrt {x^{4} + 1}} \sqrt {\sqrt {2} - 1}}{2 \, {\left (x^{2} - 2 \, x + 1\right )}}\right ) + \frac {1}{4} \, \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}{4} \, \sqrt {\sqrt {2} + 1} \log \left (-\frac {2 \, {\left ({\left (2 \, x^{3} - \sqrt {2} {\left (x^{3} - x^{2} - x - 1\right )} + \sqrt {x^{4} + 1} {\left (\sqrt {2} {\left (x - 1\right )} - 2 \, x\right )} - 2\right )} \sqrt {x^{2} + \sqrt {x^{4} + 1}} + {\left (2 \, x^{2} - \sqrt {2} {\left (x^{2} + 1\right )} + 2 \, \sqrt {x^{4} + 1} {\left (\sqrt {2} - 1\right )} + 2\right )} \sqrt {\sqrt {2} + 1}\right )}}{x^{2} + 2 \, x + 1}\right ) - \frac {1}{4} \, \sqrt {\sqrt {2} + 1} \log \left (-\frac {2 \, {\left ({\left (2 \, x^{3} - \sqrt {2} {\left (x^{3} - x^{2} - x - 1\right )} + \sqrt {x^{4} + 1} {\left (\sqrt {2} {\left (x - 1\right )} - 2 \, x\right )} - 2\right )} \sqrt {x^{2} + \sqrt {x^{4} + 1}} - {\left (2 \, x^{2} - \sqrt {2} {\left (x^{2} + 1\right )} + 2 \, \sqrt {x^{4} + 1} {\left (\sqrt {2} - 1\right )} + 2\right )} \sqrt {\sqrt {2} + 1}\right )}}{x^{2} + 2 \, x + 1}\right ) + \sqrt {x^{2} + \sqrt {x^{4} + 1}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [F]  time = 0.03, size = 0, normalized size = 0.00 \[\int \frac {\sqrt {x^{2}+\sqrt {x^{4}+1}}}{1+x}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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