\(\int \frac {1}{\sqrt {3+4 x+x^4}} \, dx\) [596]

Optimal result

Integrand size = 12, antiderivative size = 47 \[ \int \frac {1}{\sqrt {3+4 x+x^4}} \, dx=\sqrt {\frac {2}{3}} \text {arctanh}\left (\frac {\sqrt {6}+\sqrt {6} x}{1+2 x+x^2-\sqrt {3+4 x+x^4}}\right ) \]

[Out]

1/3*6^(1/2)*arctanh((6^(1/2)+x*6^(1/2))/(1+2*x+x^2-(x^4+4*x+3)^(1/2)))

Rubi [F]

\[ \int \frac {1}{\sqrt {3+4 x+x^4}} \, dx=\int \frac {1}{\sqrt {3+4 x+x^4}} \, dx \]

[In]

Int[1/Sqrt[3 + 4*x + x^4],x]

[Out]

Defer[Int][1/Sqrt[3 + 4*x + x^4], x]

Rubi steps \begin{align*} \text {integral}& = \int \frac {1}{\sqrt {3+4 x+x^4}} \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 0.12 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.26 \[ \int \frac {1}{\sqrt {3+4 x+x^4}} \, dx=\frac {\sqrt {\frac {2}{3}} (1+x) \sqrt {3-2 x+x^2} \text {arctanh}\left (\frac {1+x-\sqrt {3-2 x+x^2}}{\sqrt {6}}\right )}{\sqrt {3+4 x+x^4}} \]

[In]

Integrate[1/Sqrt[3 + 4*x + x^4],x]

[Out]

(Sqrt[2/3]*(1 + x)*Sqrt[3 - 2*x + x^2]*ArcTanh[(1 + x - Sqrt[3 - 2*x + x^2])/Sqrt[6]])/Sqrt[3 + 4*x + x^4]

Maple [A] (verified)

Time = 0.53 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.02

method	result	size
default	\(\frac {\left (1+x \right ) \sqrt {x^{2}-2 x +3}\, \sqrt {6}\, \operatorname {arctanh}\left (\frac {\left (x -2\right ) \sqrt {6}}{3 \sqrt {x^{2}-2 x +3}}\right )}{6 \sqrt {x^{4}+4 x +3}}\)	\(48\)
trager	\(\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}-6\right ) \ln \left (\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}-6\right ) x^{2}-\operatorname {RootOf}\left (\textit {\_Z}^{2}-6\right ) x +3 \sqrt {x^{4}+4 x +3}-2 \operatorname {RootOf}\left (\textit {\_Z}^{2}-6\right )}{\left (1+x \right )^{2}}\right )}{6}\)	\(56\)

[In]

int(1/(x^4+4*x+3)^(1/2),x,method=_RETURNVERBOSE)

[Out]

1/6/(x^4+4*x+3)^(1/2)*(1+x)*(x^2-2*x+3)^(1/2)*6^(1/2)*arctanh(1/3*(x-2)*6^(1/2)/(x^2-2*x+3)^(1/2))

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 66, normalized size of antiderivative = 1.40 \[ \int \frac {1}{\sqrt {3+4 x+x^4}} \, dx=\frac {1}{6} \, \sqrt {3} \sqrt {2} \log \left (-\frac {\sqrt {3} \sqrt {2} {\left (x^{2} - x - 2\right )} + 2 \, x^{2} + \sqrt {x^{4} + 4 \, x + 3} {\left (\sqrt {3} \sqrt {2} + 3\right )} - 2 \, x - 4}{x^{2} + 2 \, x + 1}\right ) \]

[In]

integrate(1/(x^4+4*x+3)^(1/2),x, algorithm="fricas")

[Out]

1/6*sqrt(3)*sqrt(2)*log(-(sqrt(3)*sqrt(2)*(x^2 - x - 2) + 2*x^2 + sqrt(x^4 + 4*x + 3)*(sqrt(3)*sqrt(2) + 3) -
2*x - 4)/(x^2 + 2*x + 1))

Sympy [F]

\[ \int \frac {1}{\sqrt {3+4 x+x^4}} \, dx=\int \frac {1}{\sqrt {x^{4} + 4 x + 3}}\, dx \]

[In]

integrate(1/(x**4+4*x+3)**(1/2),x)

[Out]

Integral(1/sqrt(x**4 + 4*x + 3), x)

Maxima [F]

\[ \int \frac {1}{\sqrt {3+4 x+x^4}} \, dx=\int { \frac {1}{\sqrt {x^{4} + 4 \, x + 3}} \,d x } \]

[In]

integrate(1/(x^4+4*x+3)^(1/2),x, algorithm="maxima")

[Out]

integrate(1/sqrt(x^4 + 4*x + 3), x)

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 61, normalized size of antiderivative = 1.30 \[ \int \frac {1}{\sqrt {3+4 x+x^4}} \, dx=\frac {\sqrt {6} \log \left (\frac {{\left | -2 \, x - 2 \, \sqrt {6} + 2 \, \sqrt {x^{2} - 2 \, x + 3} - 2 \right |}}{{\left | -2 \, x + 2 \, \sqrt {6} + 2 \, \sqrt {x^{2} - 2 \, x + 3} - 2 \right |}}\right )}{6 \, \mathrm {sgn}\left (x + 1\right )} \]

[In]

integrate(1/(x^4+4*x+3)^(1/2),x, algorithm="giac")

[Out]

1/6*sqrt(6)*log(abs(-2*x - 2*sqrt(6) + 2*sqrt(x^2 - 2*x + 3) - 2)/abs(-2*x + 2*sqrt(6) + 2*sqrt(x^2 - 2*x + 3)
- 2))/sgn(x + 1)

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{\sqrt {3+4 x+x^4}} \, dx=\int \frac {1}{\sqrt {x^4+4\,x+3}} \,d x \]

[In]

int(1/(4*x + x^4 + 3)^(1/2),x)

[Out]

int(1/(4*x + x^4 + 3)^(1/2), x)