3.980 \(\int \frac{1-x}{x \sqrt{1+3 x+x^2}} \, dx\)

Optimal. Leaf size=19 \[ -2 \tanh ^{-1}\left (\frac{x+1}{\sqrt{x^2+3 x+1}}\right ) \]

[Out]

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

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Rubi [A]  time = 0.048265, antiderivative size = 19, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095 \[ -2 \tanh ^{-1}\left (\frac{x+1}{\sqrt{x^2+3 x+1}}\right ) \]

Antiderivative was successfully verified.

[In]  Int[(1 - x)/(x*Sqrt[1 + 3*x + x^2]),x]

[Out]

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

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Rubi in Sympy [A]  time = 6.718, size = 22, normalized size = 1.16 \[ - 2 \operatorname{atanh}{\left (\frac{2 x + 2}{2 \sqrt{x^{2} + 3 x + 1}} \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((1-x)/x/(x**2+3*x+1)**(1/2),x)

[Out]

-2*atanh((2*x + 2)/(2*sqrt(x**2 + 3*x + 1)))

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Mathematica [B]  time = 0.0197564, size = 47, normalized size = 2.47 \[ -\log \left (2 \sqrt{x^2+3 x+1}+2 x+3\right )-\log \left (2 \sqrt{x^2+3 x+1}+3 x+2\right )+\log (x) \]

Antiderivative was successfully verified.

[In]  Integrate[(1 - x)/(x*Sqrt[1 + 3*x + x^2]),x]

[Out]

Log[x] - Log[3 + 2*x + 2*Sqrt[1 + 3*x + x^2]] - Log[2 + 3*x + 2*Sqrt[1 + 3*x + x
^2]]

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Maple [B]  time = 0.011, size = 38, normalized size = 2. \[ -\ln \left ( x+{\frac{3}{2}}+\sqrt{{x}^{2}+3\,x+1} \right ) -{\it Artanh} \left ({\frac{2+3\,x}{2}{\frac{1}{\sqrt{{x}^{2}+3\,x+1}}}} \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((1-x)/x/(x^2+3*x+1)^(1/2),x)

[Out]

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

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Maxima [A]  time = 0.711901, size = 65, normalized size = 3.42 \[ -\log \left (2 \, x + 2 \, \sqrt{x^{2} + 3 \, x + 1} + 3\right ) - \log \left (\frac{2 \, \sqrt{x^{2} + 3 \, x + 1}}{{\left | x \right |}} + \frac{2}{{\left | x \right |}} + 3\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(x - 1)/(sqrt(x^2 + 3*x + 1)*x),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.271761, size = 63, normalized size = 3.32 \[ \log \left (4 \, x^{2} - \sqrt{x^{2} + 3 \, x + 1}{\left (4 \, x + 5\right )} + 11 \, x + 5\right ) - \log \left (-x + \sqrt{x^{2} + 3 \, x + 1} + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(x - 1)/(sqrt(x^2 + 3*x + 1)*x),x, algorithm="fricas")

[Out]

log(4*x^2 - sqrt(x^2 + 3*x + 1)*(4*x + 5) + 11*x + 5) - log(-x + sqrt(x^2 + 3*x
+ 1) + 1)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \int \left (- \frac{1}{x \sqrt{x^{2} + 3 x + 1}}\right )\, dx - \int \frac{1}{\sqrt{x^{2} + 3 x + 1}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((1-x)/x/(x**2+3*x+1)**(1/2),x)

[Out]

-Integral(-1/(x*sqrt(x**2 + 3*x + 1)), x) - Integral(1/sqrt(x**2 + 3*x + 1), x)

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GIAC/XCAS [A]  time = 0.280176, size = 76, normalized size = 4. \[ -{\rm ln}\left ({\left | -x + \sqrt{x^{2} + 3 \, x + 1} + 1 \right |}\right ) +{\rm ln}\left ({\left | -x + \sqrt{x^{2} + 3 \, x + 1} - 1 \right |}\right ) +{\rm ln}\left ({\left | -2 \, x + 2 \, \sqrt{x^{2} + 3 \, x + 1} - 3 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(x - 1)/(sqrt(x^2 + 3*x + 1)*x),x, algorithm="giac")

[Out]

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