Optimal. Leaf size=70 \[ -\frac {\sqrt {x^2+\sqrt {1+x^4}}}{x}+\sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {2} x \sqrt {x^2+\sqrt {1+x^4}}}{1+x^2+\sqrt {1+x^4}}\right ) \]
[Out]
________________________________________________________________________________________
Rubi [F]
time = 0.06, 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}}}{x^2} \, dx \end {gather*}
Verification is not applicable to the result.
[In]
[Out]
Rubi steps
\begin {align*} \int \frac {\sqrt {x^2+\sqrt {1+x^4}}}{x^2} \, dx &=\int \frac {\sqrt {x^2+\sqrt {1+x^4}}}{x^2} \, dx\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.22, size = 70, normalized size = 1.00 \begin {gather*} -\frac {\sqrt {x^2+\sqrt {1+x^4}}}{x}+\sqrt {2} \tanh ^{-1}\left (\frac {\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]
[Out]
________________________________________________________________________________________
Maple [C] Result contains higher order function than in optimal. Order 5 vs. order
3.
time = 0.09, size = 51, normalized size = 0.73
method | result | size |
meijerg | \(\frac {-\frac {\sqrt {\pi }\, \sqrt {2}\, \hypergeom \left (\left [\frac {3}{4}, 1, 1, \frac {5}{4}\right ], \left [\frac {3}{2}, 2, 2\right ], -\frac {1}{x^{4}}\right )}{2 x^{4}}-4 \left (-4 \ln \left (2\right )+4-4 \ln \left (x \right )\right ) \sqrt {\pi }\, \sqrt {2}}{16 \sqrt {\pi }}\) | \(51\) |
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 = 0.63, size = 81, normalized size = 1.16 \begin {gather*} \frac {\sqrt {2} x \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 ) - 4 \, \sqrt {x^{2} + \sqrt {x^{4} + 1}}}{4 \, x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [C] Result contains complex when optimal does not.
time = 1.93, size = 53, normalized size = 0.76 \begin {gather*} - \frac {\log {\left (\frac {1}{x^{4}} \right )} \Gamma \left (\frac {1}{4}\right ) \Gamma \left (\frac {3}{4}\right )}{4 \pi } - \frac {\Gamma \left (\frac {3}{4}\right ) \Gamma \left (\frac {5}{4}\right ) {{}_{4}F_{3}\left (\begin {matrix} \frac {3}{4}, 1, 1, \frac {5}{4} \\ \frac {3}{2}, 2, 2 \end {matrix}\middle | {\frac {e^{i \pi }}{x^{4}}} \right )}}{8 \pi x^{4}} \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*} \text {could not integrate} \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 {\sqrt {\sqrt {x^4+1}+x^2}}{x^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________