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