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

Optimal. Leaf size=87 \[ 2 \text {RootSum}\left [\text {$\#$1}^4+2 \text {$\#$1}^3-1\& ,\frac {\text {$\#$1} \log \left (\sqrt {\sqrt {x^2+1}+x}-\text {$\#$1}\right )+\log \left (\sqrt {\sqrt {x^2+1}+x}-\text {$\#$1}\right )}{2 \text {$\#$1}+3}\& \right ]-\log \left (\sqrt {x^2+1}+x\right ) \]

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Rubi [F]  time = 1.01, 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+\sqrt {x+\sqrt {1+x^2}}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

Log[-1 - 2*x^3 + x^4]/4 + Defer[Int][x^2/(-1 - 2*x^3 + x^4), x]/2 + Defer[Int][(x*Sqrt[1 + x^2])/(-1 - 2*x^3 +
x^4), x] + Defer[Int][(x*Sqrt[x + Sqrt[1 + x^2]])/(-1 - 2*x^3 + x^4), x] - Defer[Int][(x^2*Sqrt[x + Sqrt[1 +
x^2]])/(-1 - 2*x^3 + x^4), x] - Defer[Int][(Sqrt[1 + x^2]*Sqrt[x + Sqrt[1 + x^2]])/(-1 - 2*x^3 + x^4), x]

Rubi steps

\begin {align*} \int \frac {1}{x+\sqrt {x+\sqrt {1+x^2}}} \, dx &=\int \left (\frac {(-1+x) x^2}{-1-2 x^3+x^4}+\frac {x \sqrt {1+x^2}}{-1-2 x^3+x^4}+\frac {x \sqrt {x+\sqrt {1+x^2}}}{-1-2 x^3+x^4}-\frac {x^2 \sqrt {x+\sqrt {1+x^2}}}{-1-2 x^3+x^4}-\frac {\sqrt {1+x^2} \sqrt {x+\sqrt {1+x^2}}}{-1-2 x^3+x^4}\right ) \, dx\\ &=\int \frac {(-1+x) x^2}{-1-2 x^3+x^4} \, dx+\int \frac {x \sqrt {1+x^2}}{-1-2 x^3+x^4} \, dx+\int \frac {x \sqrt {x+\sqrt {1+x^2}}}{-1-2 x^3+x^4} \, dx-\int \frac {x^2 \sqrt {x+\sqrt {1+x^2}}}{-1-2 x^3+x^4} \, dx-\int \frac {\sqrt {1+x^2} \sqrt {x+\sqrt {1+x^2}}}{-1-2 x^3+x^4} \, dx\\ &=\frac {1}{4} \log \left (-1-2 x^3+x^4\right )+\frac {1}{4} \int \frac {2 x^2}{-1-2 x^3+x^4} \, dx+\int \frac {x \sqrt {1+x^2}}{-1-2 x^3+x^4} \, dx+\int \frac {x \sqrt {x+\sqrt {1+x^2}}}{-1-2 x^3+x^4} \, dx-\int \frac {x^2 \sqrt {x+\sqrt {1+x^2}}}{-1-2 x^3+x^4} \, dx-\int \frac {\sqrt {1+x^2} \sqrt {x+\sqrt {1+x^2}}}{-1-2 x^3+x^4} \, dx\\ &=\frac {1}{4} \log \left (-1-2 x^3+x^4\right )+\frac {1}{2} \int \frac {x^2}{-1-2 x^3+x^4} \, dx+\int \frac {x \sqrt {1+x^2}}{-1-2 x^3+x^4} \, dx+\int \frac {x \sqrt {x+\sqrt {1+x^2}}}{-1-2 x^3+x^4} \, dx-\int \frac {x^2 \sqrt {x+\sqrt {1+x^2}}}{-1-2 x^3+x^4} \, dx-\int \frac {\sqrt {1+x^2} \sqrt {x+\sqrt {1+x^2}}}{-1-2 x^3+x^4} \, dx\\ \end {align*}

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Mathematica [F]  time = 0.23, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{x+\sqrt {x+\sqrt {1+x^2}}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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IntegrateAlgebraic [A]  time = 0.61, size = 87, normalized size = 1.00 \begin {gather*} -\log \left (x+\sqrt {1+x^2}\right )+2 \text {RootSum}\left [-1+2 \text {$\#$1}^3+\text {$\#$1}^4\&,\frac {\log \left (\sqrt {x+\sqrt {1+x^2}}-\text {$\#$1}\right )+\log \left (\sqrt {x+\sqrt {1+x^2}}-\text {$\#$1}\right ) \text {$\#$1}}{3+2 \text {$\#$1}}\&\right ] \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-Log[x + Sqrt[1 + x^2]] + 2*RootSum[-1 + 2*#1^3 + #1^4 & , (Log[Sqrt[x + Sqrt[1 + x^2]] - #1] + Log[Sqrt[x + S
qrt[1 + x^2]] - #1]*#1)/(3 + 2*#1) & ]

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fricas [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int(1/(x+(x+(x^2+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 {1}{x + \sqrt {x + \sqrt {x^{2} + 1}}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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