3.903 \(\int \frac{-1+2 x}{1+8 x+4 x^2} \, dx\)

Optimal. Leaf size=49 \[ \frac{1}{4} \left (1-\sqrt{3}\right ) \log \left (2 x-\sqrt{3}+2\right )+\frac{1}{4} \left (1+\sqrt{3}\right ) \log \left (2 x+\sqrt{3}+2\right ) \]

[Out]

((1 - Sqrt[3])*Log[2 - Sqrt[3] + 2*x])/4 + ((1 + Sqrt[3])*Log[2 + Sqrt[3] + 2*x]
)/4

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Rubi [A]  time = 0.0534583, antiderivative size = 49, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111 \[ \frac{1}{4} \left (1-\sqrt{3}\right ) \log \left (2 x-\sqrt{3}+2\right )+\frac{1}{4} \left (1+\sqrt{3}\right ) \log \left (2 x+\sqrt{3}+2\right ) \]

Antiderivative was successfully verified.

[In]  Int[(-1 + 2*x)/(1 + 8*x + 4*x^2),x]

[Out]

((1 - Sqrt[3])*Log[2 - Sqrt[3] + 2*x])/4 + ((1 + Sqrt[3])*Log[2 + Sqrt[3] + 2*x]
)/4

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Rubi in Sympy [A]  time = 5.20711, size = 53, normalized size = 1.08 \[ - \frac{\sqrt{3} \left (- 4 \sqrt{3} + 12\right ) \log{\left (2 x - \sqrt{3} + 2 \right )}}{48} + \frac{\sqrt{3} \left (4 \sqrt{3} + 12\right ) \log{\left (2 x + \sqrt{3} + 2 \right )}}{48} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((-1+2*x)/(4*x**2+8*x+1),x)

[Out]

-sqrt(3)*(-4*sqrt(3) + 12)*log(2*x - sqrt(3) + 2)/48 + sqrt(3)*(4*sqrt(3) + 12)*
log(2*x + sqrt(3) + 2)/48

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Mathematica [A]  time = 0.0386482, size = 44, normalized size = 0.9 \[ \frac{1}{4} \left (\left (1+\sqrt{3}\right ) \log \left (2 x+\sqrt{3}+2\right )-\left (\sqrt{3}-1\right ) \log \left (-2 x+\sqrt{3}-2\right )\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[(-1 + 2*x)/(1 + 8*x + 4*x^2),x]

[Out]

(-((-1 + Sqrt[3])*Log[-2 + Sqrt[3] - 2*x]) + (1 + Sqrt[3])*Log[2 + Sqrt[3] + 2*x
])/4

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Maple [A]  time = 0.005, size = 31, normalized size = 0.6 \[{\frac{\ln \left ( 4\,{x}^{2}+8\,x+1 \right ) }{4}}+{\frac{\sqrt{3}}{2}{\it Artanh} \left ({\frac{ \left ( 8\,x+8 \right ) \sqrt{3}}{12}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((2*x-1)/(4*x^2+8*x+1),x)

[Out]

1/4*ln(4*x^2+8*x+1)+1/2*3^(1/2)*arctanh(1/12*(8*x+8)*3^(1/2))

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Maxima [A]  time = 0.767363, size = 55, normalized size = 1.12 \[ -\frac{1}{4} \, \sqrt{3} \log \left (\frac{2 \, x - \sqrt{3} + 2}{2 \, x + \sqrt{3} + 2}\right ) + \frac{1}{4} \, \log \left (4 \, x^{2} + 8 \, x + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((2*x - 1)/(4*x^2 + 8*x + 1),x, algorithm="maxima")

[Out]

-1/4*sqrt(3)*log((2*x - sqrt(3) + 2)/(2*x + sqrt(3) + 2)) + 1/4*log(4*x^2 + 8*x
+ 1)

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Fricas [A]  time = 0.283647, size = 69, normalized size = 1.41 \[ \frac{1}{4} \, \sqrt{3} \log \left (\frac{4 \, x^{2} + 4 \, \sqrt{3}{\left (x + 1\right )} + 8 \, x + 7}{4 \, x^{2} + 8 \, x + 1}\right ) + \frac{1}{4} \, \log \left (4 \, x^{2} + 8 \, x + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((2*x - 1)/(4*x^2 + 8*x + 1),x, algorithm="fricas")

[Out]

1/4*sqrt(3)*log((4*x^2 + 4*sqrt(3)*(x + 1) + 8*x + 7)/(4*x^2 + 8*x + 1)) + 1/4*l
og(4*x^2 + 8*x + 1)

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Sympy [A]  time = 0.20839, size = 42, normalized size = 0.86 \[ \left (- \frac{\sqrt{3}}{4} + \frac{1}{4}\right ) \log{\left (x - \frac{\sqrt{3}}{2} + 1 \right )} + \left (\frac{1}{4} + \frac{\sqrt{3}}{4}\right ) \log{\left (x + \frac{\sqrt{3}}{2} + 1 \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-1+2*x)/(4*x**2+8*x+1),x)

[Out]

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

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GIAC/XCAS [A]  time = 0.271463, size = 62, normalized size = 1.27 \[ -\frac{1}{4} \, \sqrt{3}{\rm ln}\left (\frac{{\left | 8 \, x - 4 \, \sqrt{3} + 8 \right |}}{{\left | 8 \, x + 4 \, \sqrt{3} + 8 \right |}}\right ) + \frac{1}{4} \,{\rm ln}\left ({\left | 4 \, x^{2} + 8 \, x + 1 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((2*x - 1)/(4*x^2 + 8*x + 1),x, algorithm="giac")

[Out]

-1/4*sqrt(3)*ln(abs(8*x - 4*sqrt(3) + 8)/abs(8*x + 4*sqrt(3) + 8)) + 1/4*ln(abs(
4*x^2 + 8*x + 1))