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

Optimal. Leaf size=47 \[ \sqrt {\frac {2}{3}} \tanh ^{-1}\left (\frac {\sqrt {6} x+\sqrt {6}}{-\sqrt {x^4+4 x+3}+x^2+2 x+1}\right ) \]

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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]

Int[1/Sqrt[3 + 4*x + x^4],x]

[Out]

Defer[Int][1/Sqrt[3 + 4*x + x^4], x]

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*}

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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]

Integrate[1/Sqrt[3 + 4*x + x^4],x]

[Out]

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

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IntegrateAlgebraic [A]  time = 0.23, size = 47, normalized size = 1.00 \begin {gather*} \sqrt {\frac {2}{3}} \tanh ^{-1}\left (\frac {\sqrt {6}+\sqrt {6} x}{1+2 x+x^2-\sqrt {3+4 x+x^4}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[1/Sqrt[3 + 4*x + x^4],x]

[Out]

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

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fricas [A]  time = 0.43, 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]

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))

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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]

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)

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maple [A]  time = 0.16, size = 48, normalized size = 1.02

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

Verification of antiderivative is not currently implemented for this CAS.

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

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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]

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

[Out]

integrate(1/sqrt(x^4 + 4*x + 3), x)

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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]

int(1/(4*x + x^4 + 3)^(1/2),x)

[Out]

int(1/(4*x + x^4 + 3)^(1/2), x)

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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]

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

[Out]

Integral(1/sqrt(x**4 + 4*x + 3), x)

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