Optimal. Leaf size=50 \[ \frac {\tan ^{-1}\left (\frac {x}{\sqrt {2} \sqrt {\sqrt {x^2+1}+1}}\right )}{\sqrt {2}}-\frac {\sqrt {\sqrt {x^2+1}+1}}{x} \]
________________________________________________________________________________________
Rubi [F] time = 0.08, 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+\sqrt {1+x^2}}}{x^2} \, dx \end {gather*}
Verification is not applicable to the result.
[In]
[Out]
Rubi steps
\begin {align*} \int \frac {\sqrt {1+\sqrt {1+x^2}}}{x^2} \, dx &=\int \frac {\sqrt {1+\sqrt {1+x^2}}}{x^2} \, dx\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.12, size = 78, normalized size = 1.56 \begin {gather*} \frac {\left (\sqrt {x^2+1}-1\right ) \left (\sqrt {x^2+1}+1\right )^{3/2} \left (\sqrt {2} \sqrt {\sqrt {x^2+1}-1} \tan ^{-1}\left (\frac {\sqrt {\sqrt {x^2+1}-1}}{\sqrt {2}}\right )-2\right )}{2 x^3} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [A] time = 0.11, size = 50, normalized size = 1.00 \begin {gather*} -\frac {\sqrt {1+\sqrt {1+x^2}}}{x}+\frac {\tan ^{-1}\left (\frac {x}{\sqrt {2} \sqrt {1+\sqrt {1+x^2}}}\right )}{\sqrt {2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 2.60, size = 43, normalized size = 0.86 \begin {gather*} -\frac {\sqrt {2} x \arctan \left (\frac {\sqrt {2} \sqrt {\sqrt {x^{2} + 1} + 1}}{x}\right ) + 2 \, \sqrt {\sqrt {x^{2} + 1} + 1}}{2 \, x} \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 {\sqrt {\sqrt {x^{2} + 1} + 1}}{x^{2}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [C] time = 0.03, size = 22, normalized size = 0.44
method | result | size |
meijerg | \(-\frac {\sqrt {2}\, \hypergeom \left (\left [-\frac {1}{2}, -\frac {1}{4}, \frac {1}{4}\right ], \left [\frac {1}{2}, \frac {1}{2}\right ], -x^{2}\right )}{x}\) | \(22\) |
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 {\sqrt {\sqrt {x^{2} + 1} + 1}}{x^{2}}\,{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 = -0.02 \begin {gather*} \int \frac {\sqrt {\sqrt {x^2+1}+1}}{x^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [C] time = 0.88, size = 37, normalized size = 0.74 \begin {gather*} \frac {\Gamma \left (- \frac {1}{4}\right ) \Gamma \left (\frac {1}{4}\right ) {{}_{3}F_{2}\left (\begin {matrix} - \frac {1}{2}, - \frac {1}{4}, \frac {1}{4} \\ \frac {1}{2}, \frac {1}{2} \end {matrix}\middle | {x^{2} e^{i \pi }} \right )}}{4 \pi x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________