Optimal. Leaf size=236 \[ \frac {\left (-24993-2680 x-21570 x^2-4096 x^3+30720 x^4\right ) \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}+\left (1712+1814 x+4096 x^2+3072 x^3\right ) \sqrt {x+\sqrt {1+x^2}} \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}+\sqrt {1+x^2} \left (\left (-632-36930 x-4096 x^2+30720 x^3\right ) \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}+\left (278+4096 x+3072 x^2\right ) \sqrt {x+\sqrt {1+x^2}} \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}\right )}{26880 \left (x+\sqrt {1+x^2}\right )^{5/2}}-\frac {263}{256} \tanh ^{-1}\left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}\right ) \]
[Out]
________________________________________________________________________________________
Rubi [F]
time = 0.34, 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 {1+x^2} \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{\sqrt {x+\sqrt {1+x^2}}} \, dx \end {gather*}
Verification is not applicable to the result.
[In]
[Out]
Rubi steps
\begin {align*} \int \frac {\sqrt {1+x^2} \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{\sqrt {x+\sqrt {1+x^2}}} \, dx &=\int \frac {\sqrt {1+x^2} \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{\sqrt {x+\sqrt {1+x^2}}} \, dx\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.40, size = 169, normalized size = 0.72 \begin {gather*} \frac {\frac {\sqrt {1+\sqrt {x+\sqrt {1+x^2}}} \left (-24993-2680 x-21570 x^2-4096 x^3+30720 x^4+2 \left (856+907 x+2048 x^2+1536 x^3\right ) \sqrt {x+\sqrt {1+x^2}}+\sqrt {1+x^2} \left (-632-36930 x-4096 x^2+30720 x^3+\left (278+4096 x+3072 x^2\right ) \sqrt {x+\sqrt {1+x^2}}\right )\right )}{\left (x+\sqrt {1+x^2}\right )^{5/2}}-27615 \tanh ^{-1}\left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}\right )}{26880} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F]
time = 0.03, size = 0, normalized size = 0.00 \[\int \frac {\sqrt {x^{2}+1}\, \sqrt {1+\sqrt {x +\sqrt {x^{2}+1}}}}{\sqrt {x +\sqrt {x^{2}+1}}}\, dx\]
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*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 0.37, size = 129, normalized size = 0.55 \begin {gather*} -\frac {1}{26880} \, {\left (672 \, x^{2} - 2 \, \sqrt {x^{2} + 1} {\left (336 \, x + 139\right )} - {\left (10752 \, x^{3} + 784 \, x^{2} - {\left (10752 \, x^{2} + 784 \, x + 24993\right )} \sqrt {x^{2} + 1} + 38049 \, x - 632\right )} \sqrt {x + \sqrt {x^{2} + 1}} - 1258 \, x - 1712\right )} \sqrt {\sqrt {x + \sqrt {x^{2} + 1}} + 1} - \frac {263}{512} \, \log \left (\sqrt {\sqrt {x + \sqrt {x^{2} + 1}} + 1} + 1\right ) + \frac {263}{512} \, \log \left (\sqrt {\sqrt {x + \sqrt {x^{2} + 1}} + 1} - 1\right ) \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 {\sqrt {x^{2} + 1} \sqrt {\sqrt {x + \sqrt {x^{2} + 1}} + 1}}{\sqrt {x + \sqrt {x^{2} + 1}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F(-1)] Timed out
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]
________________________________________________________________________________________
Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {\sqrt {\sqrt {x+\sqrt {x^2+1}}+1}\,\sqrt {x^2+1}}{\sqrt {x+\sqrt {x^2+1}}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________