Optimal. Leaf size=47 \[ \sqrt {\frac {2}{3}} \tanh ^{-1}\left (\frac {\sqrt {6} x+\sqrt {6}}{-\sqrt {x^4+4 x+3}+x^2+2 x+1}\right ) \]
________________________________________________________________________________________
Rubi [F] time = 0.01, 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}{\sqrt {3+4 x+x^4}} \, dx \end {gather*}
Verification is not applicable to the result.
[In]
[Out]
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {3+4 x+x^4}} \, dx &=\int \frac {1}{\sqrt {3+4 x+x^4}} \, dx\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.02, size = 57, normalized size = 1.21 \begin {gather*} \frac {(x+1) \sqrt {x^2-2 x+3} \tanh ^{-1}\left (\frac {\sqrt {\frac {2}{3}} (x-2)}{\sqrt {x^2-2 x+3}}\right )}{\sqrt {6} \sqrt {x^4+4 x+3}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [A] time = 0.28, size = 47, normalized size = 1.00 \begin {gather*} \sqrt {\frac {2}{3}} \tanh ^{-1}\left (\frac {\sqrt {6} x+\sqrt {6}}{-\sqrt {x^4+4 x+3}+x^2+2 x+1}\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.38, size = 66, normalized size = 1.40 \begin {gather*} \frac {1}{6} \, \sqrt {3} \sqrt {2} \log \left (-\frac {\sqrt {3} \sqrt {2} {\left (x^{2} - x - 2\right )} + 2 \, x^{2} + \sqrt {x^{4} + 4 \, x + 3} {\left (\sqrt {3} \sqrt {2} + 3\right )} - 2 \, x - 4}{x^{2} + 2 \, x + 1}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.48, size = 61, normalized size = 1.30 \begin {gather*} \frac {\sqrt {6} \log \left (\frac {{\left | -2 \, x - 2 \, \sqrt {6} + 2 \, \sqrt {x^{2} - 2 \, x + 3} - 2 \right |}}{{\left | -2 \, x + 2 \, \sqrt {6} + 2 \, \sqrt {x^{2} - 2 \, x + 3} - 2 \right |}}\right )}{6 \, \mathrm {sgn}\left (x + 1\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.01, size = 48, normalized size = 1.02 \begin {gather*} \frac {\left (1+x \right ) \sqrt {x^{2}-2 x +3}\, \sqrt {6}\, \arctanh \left (\frac {\left (-2+x \right ) \sqrt {6}}{3 \sqrt {x^{2}-2 x +3}}\right )}{6 \sqrt {x^{4}+4 x +3}} \end {gather*}
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}{\sqrt {x^{4} + 4 \, x + 3}}\,{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 {1}{\sqrt {x^4+4\,x+3}} \,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}{\sqrt {x^{4} + 4 x + 3}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________